# Microscopic Interactions and Macroscopic Properties Final Report (1st

Microscopic Interactions and Macroscopic Properties Final Report (1st

Microscopic Interactions and Macroscopic Properties Final Report (1st

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Deutsche<br />

Forschungsgemeinschaft<br />

Macromolecular<br />

Systems:<br />

<strong>Microscopic</strong><br />

<strong>Interactions</strong> <strong>and</strong><br />

<strong>Macroscopic</strong> <strong>Properties</strong><br />

Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

Deutsche<br />

Forschungsgemeinschaft<br />

Macromolecular Systems:<br />

<strong>Microscopic</strong> <strong>Interactions</strong><br />

<strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

<strong>Final</strong> report of the<br />

collaborative research centre 213,<br />

``Topospezifische Chemie und Toposelektive<br />

Spektroskopie von MakromolekÏlsystemen:<br />

Mikroskopische Wechselwirkung und<br />

Makroskopische Funktion'', 1984^1995<br />

Edited by<br />

HeinzHoffmann,MarkusSchwoerer<strong>and</strong><br />

Thomas Vogtmann<br />

Collaborative Research Centres

Deutsche Forschungsgemeinschaft<br />

Kennedyallee 40, D-53175 Bonn, Federal Republic of Germany<br />

Postal address: D-53175 Bonn<br />

Phone: ++49/228/885-1<br />

Telefax: ++49/228/885-2777<br />

E-Mail: (X.400): S = postmaster; P = dfg; A = d400; C = de<br />

E-Mail: (Internet RFC 822): postmaster@dfg.de<br />

Internet: http://www.dfg.de<br />

This book was carefully produced. Nevertheless, editors, authors <strong>and</strong> publisher do not warrant the information<br />

contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations,<br />

procedural details or other items may inadvertently be inaccurate.<br />

Library of Congress Card No.: applied for<br />

A catalogue record for this book is available from the British Library.<br />

Die Deutsche Bibliothek – CIP Cataloguing-in-Publication Data<br />

A catalogue record for this publication is available from Die Deutsche Bibliothek<br />

ISBN 3-527-27726-9<br />

© WILEY-VCH Verlag GmbH, D-69469 Weinheim (Federal Republic of Germany), 2000<br />

Printed on acid-free <strong>and</strong> chlorine-free paper<br />

All rights reserved (including those of translation into other languages). No part of this book may be reproduced<br />

in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine<br />

language without written permission from the publishers. Registered names, trademarks, etc. used in this book,<br />

even when not specifically marked as such, are not to be considered unprotected by law.<br />

Cover Design <strong>and</strong> Typography: Dieter Hüsken<br />

Composition: ProSatz Unger, D-69469 Weinheim<br />

Printing: betz-druck gmbh, D-64291 Darmstadt<br />

Bookbindung: J. Schäffer GmbH & Co. KG, D-67269 Grünstadt<br />

Printed in the Federal Republic of Germany

Contents<br />

Preface ...................................................<br />

List of Contributors ..........................................<br />

XV<br />

XVII<br />

I<br />

Mainly Solids<br />

1 Model Systems for Photoconductive Material ..................... 3<br />

Harald Meyer <strong>and</strong> Dietrich Haarer<br />

1.1 Introduction . . ............................................. 3<br />

1.2 Experimental techniques . . . ................................... 4<br />

1.3 Transport models ............................................ 5<br />

1.4 Results ................................................... 7<br />

1.4.1 Moleculary doped polymers . ................................... 7<br />

1.4.2 Side-chain polymers ......................................... 9<br />

1.4.3 Conjugated systems .......................................... 10<br />

1.5 Outlook .................................................. 12<br />

References . . . ............................................. 13<br />

2 Novel Photoconductive Polymers ............................... 15<br />

Jörg Bettenhausen <strong>and</strong> Peter Strohriegl<br />

2.1 Introduction . . ............................................. 15<br />

2.2 Liquid crystalline oxadiazoles <strong>and</strong> thiadiazoles ...................... 16<br />

2.2.1 The basic idea . ............................................. 16<br />

2.2.2 Monomer synthesis .......................................... 17<br />

2.2.3 Oligo- <strong>and</strong> polysiloxanes with pendant oxadiazole groups .............. 19<br />

2.2.4 Photoconductivity measurements ................................ 21<br />

2.3 Starburst oxadiazole compounds ................................. 22<br />

2.3.1 Motivation . . . ............................................. 22<br />

2.3.2 Synthesis of starburst oxadiazole compounds ....................... 24<br />

2.3.3 Thermal properties .......................................... 27<br />

References . . . ............................................. 30<br />

V

Contents<br />

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems ...... 31<br />

Alex<strong>and</strong>er Blumen<br />

3.1 General aspects ............................................. 31<br />

3.2 Photoconductivity ........................................... 32<br />

3.3 The Matheron-de-Marsily model ................................ 36<br />

3.4 Polymer chains in MdM flow fields .............................. 38<br />

3.5 Conclusions . . ............................................. 42<br />

Acknowledgements .......................................... 42<br />

References . . . ............................................. 43<br />

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation,<br />

Specific Heat <strong>and</strong> Thermal Conductivity in Polymers ............... 44<br />

Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz,<br />

Junyun Li, <strong>and</strong> Frank Pobell<br />

4.1 Introduction . . ............................................. 44<br />

4.2 Phenomenological theory for heat release . . ........................ 46<br />

4.2.1 Generalities . . . ............................................. 46<br />

4.2.2 The st<strong>and</strong>ard tunneling model with infinite cooling rate . .............. 47<br />

4.2.3 Influence of higher-order tunneling processes <strong>and</strong> a finite cooling rate . . . . 49<br />

4.2.4 The influence of a constant <strong>and</strong> thermally activated relaxation rate ....... 52<br />

4.3 Experimental details ......................................... 54<br />

4.4 Experimental results <strong>and</strong> discussion .............................. 55<br />

4.4.1 Specific heat <strong>and</strong> thermal conductivity ............................ 55<br />

4.4.2 Internal friction <strong>and</strong> sound velocity .............................. 58<br />

4.4.3 Heat release . . ............................................. 62<br />

4.5 Conclusions . . ............................................. 65<br />

Acknowledgements .......................................... 66<br />

References . . . ............................................. 66<br />

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures 68<br />

Hans Maier, Karl-Peter Müller, Siegbert Jahn, <strong>and</strong> Dietrich Haarer<br />

5.1 Introduction . . ............................................. 68<br />

5.2 The optical cryostat .......................................... 69<br />

5.3 Theoretical considerations . . ................................... 71<br />

5.4 Temperature dependence . . . ................................... 73<br />

5.5 Time dependence ............................................ 74<br />

References . . . ............................................. 76<br />

6 Optically Induced Spectral Diffusion in Polymers Containing Water<br />

Molecules: A TLS Model System ............................... 78<br />

Klaus Barth, Dietrich Haarer, <strong>and</strong> Wolfgang Richter<br />

6.1 Introduction . . ............................................. 78<br />

6.2 Experimental setup for burning <strong>and</strong> detecting spectral holes ............ 79<br />

6.3 Reversible line broadening phenomena ............................ 80<br />

6.4 Induced spectral diffusion . . ................................... 83<br />

References . . . ............................................. 87<br />

VI

Contents<br />

7 Slave-Boson Approach to Strongly Correlated Electron Systems ....... 88<br />

Holger Fehske, Martin Deeg, <strong>and</strong> Helmut Büttner<br />

7.1 Introduction . . ............................................. 88<br />

7.2 Slave-boson theory for the t-t'-J model ............................ 90<br />

7.2.1 SU(2)-invariant slave-particle representation ........................ 90<br />

7.2.2 Functional integral formulation ................................. 93<br />

7.2.3 Saddle-point approximation . ................................... 95<br />

7.2.4 Magnetic phase diagram of the t-t'-J model ........................ 97<br />

7.3 Comparison with experiments .................................. 102<br />

7.3.1 Normal-state transport properties ................................ 102<br />

7.3.2 Magnetic correlations <strong>and</strong> spin dynamics . . ........................ 105<br />

7.3.3 Inelastic neutron scattering measurements . . ........................ 106<br />

7.4 Summary ................................................. 109<br />

References . . . ............................................. 110<br />

8 Non-Linear Excitations <strong>and</strong> the Electronic Structure of Conjugated<br />

Polymers ................................................. 113<br />

Klaus Fesser<br />

8.1 Introduction . . ............................................. 113<br />

8.2 Models ................................................... 114<br />

8.3 Disorder .................................................. 116<br />

8.4 Non-linear excitations ........................................ 117<br />

8.5 Perspective . . . ............................................. 119<br />

Acknowledgements .......................................... 120<br />

References . . . ............................................. 120<br />

9 Diacetylene Single Crystals ................................... 122<br />

Markus Schwoerer, Elmar Dormann, Thomas Vogtmann,<br />

<strong>and</strong> Andreas Feldner<br />

9.1 Introduction . . ............................................. 122<br />

9.2 Photopolymerization ......................................... 129<br />

9.2.1 Carbenes .................................................. 129<br />

9.2.2 Intermediate photoproducts . ................................... 131<br />

9.2.3 Electronic structure of dicarbenes ................................ 131<br />

9.2.3.1 Electron spin resonance of quintet states ( 5 DC n ) .................... 131<br />

9.2.3.2 ENDOR of quintet states . . . ................................... 136<br />

9.2.3.3 ESR <strong>and</strong> ENDOR of triplet dicarbenes 3 DC n ....................... 139<br />

9.2.4 Flash photolysis <strong>and</strong> reaction dynamics of diradicals .................. 141<br />

9.3 Holography . . . ............................................. 144<br />

9.3.1 Theory ................................................... 145<br />

9.3.2 Experimental setup .......................................... 147<br />

9.3.3 General characterization . . . ................................... 149<br />

9.3.4 Angular selectivity .......................................... 150<br />

9.3.5 Prepolymerized samples ....................................... 152<br />

9.3.6 Chain length, polymer profile, <strong>and</strong> grating profiles ................... 152<br />

9.3.7 Multrecording . ............................................. 154<br />

VII

Contents<br />

9.3.8 Holography . . . ............................................. 154<br />

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity .................................. 155<br />

9.4.1 Dielectric properties of diacetylenes .............................. 156<br />

9.4.1.1 Correlation of polymer content <strong>and</strong> electric permittivity . .............. 156<br />

9.4.1.2 Application to topospecifically modified diacetylenes . . . .............. 158<br />

9.4.1.3 Additional applications ....................................... 159<br />

9.4.2 Pyroelectric diacetylenes . . . ................................... 159<br />

9.4.2.1 IPUDO ................................................... 159<br />

9.4.2.2 NP/4-MPU . . . ............................................. 160<br />

9.4.2.3 DNP/MNP . . . ............................................. 161<br />

9.4.2.4 Spurious piezo <strong>and</strong> pyroelectricity of diacetylenes .................... 161<br />

9.4.3 The ferroelectric diacetylene DNP ............................... 162<br />

9.4.4 Summary ................................................. 166<br />

9.5 Non-linear optical properties ................................... 167<br />

9.5.1 Aims of investigation ........................................ 167<br />

9.5.2 Experimental setup .......................................... 167<br />

9.5.3 Theoretical approaches ....................................... 170<br />

9.5.4 Sample preparation .......................................... 170<br />

9.5.5 Value <strong>and</strong> phase of the third order susceptibility w (3) .................. 170<br />

9.5.6 Relaxation of the singlet exciton ................................ 171<br />

9.5.7 The w (3) tensor components . ................................... 172<br />

9.5.8 Signal saturation ............................................ 173<br />

9.5.9 Spectral dispersion, phase, <strong>and</strong> relaxation of w (5) ..................... 174<br />

9.5.10 Conclusion . . . ............................................. 176<br />

References . . . ............................................. 177<br />

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids .......... 181<br />

Thomas Giering, Peter Geißinger, Wolfgang Richter, <strong>and</strong> Dietrich Haarer<br />

10.1 Introduction . . ............................................. 181<br />

10.2 Stochastic theory ............................................ 183<br />

10.3 Rare gases ................................................. 186<br />

10.4 Experimental . . ............................................. 187<br />

10.5 Inhomogeneous absorption lines ................................. 188<br />

10.6 Pressure effects ............................................. 190<br />

10.7 Rare gas mixtures ........................................... 191<br />

10.8 Summary ................................................. 194<br />

Acknowledgements .......................................... 195<br />

References . . . ............................................. 195<br />

II<br />

Mainly Micelles, Polymers, <strong>and</strong> Liquid Crystals<br />

11 The Micellar Structures <strong>and</strong> the <strong>Macroscopic</strong> <strong>Properties</strong> of Surfactant<br />

Solutions ................................................. 199<br />

Heinz Hoffmann<br />

11.1 General behaviour of surfactants ................................ 199<br />

VIII

Contents<br />

11.2 From globular micelles towards bilayers . . . ........................ 200<br />

11.3 Viscoelastic solutions with entangled rods . ........................ 202<br />

11.3.1 General behaviour ........................................... 202<br />

11.3.2 Viscoelastic systems ......................................... 205<br />

11.3.3 Mechanisms for the different scaling behaviour ..................... 209<br />

11.4 Viscoelastic solutions with multilamellar vesicles .................... 211<br />

11.4.1 The conditions for the existence of vesicles ........................ 211<br />

11.4.2 Freeze fracture electron microscopy .............................. 212<br />

11.4.3 Rheological properties ........................................ 213<br />

11.4.4 Model for the shear modulus ................................... 217<br />

11.5 Ringing gels . . ............................................. 220<br />

11.5.1 Introduction . . ............................................. 220<br />

11.5.2 The aminoxide system ........................................ 221<br />

11.5.3 The bis-(2-ethylhexyl)sulfosuccinate system ........................ 224<br />

11.5.4 PEO-PPO-PEO block copolymers ................................ 226<br />

11.6 Lyotropic mesophases ........................................ 227<br />

11.6.1 Introduction . . ............................................. 227<br />

11.6.2 Nematic phases <strong>and</strong> their properties .............................. 228<br />

11.6.3 Cholesteric phases <strong>and</strong> their properties ............................ 232<br />

11.6.4 Vesicle phases <strong>and</strong> L 3 phases ................................... 233<br />

11.7 Shear induced phenomena . . ................................... 236<br />

11.7.1 General ................................................... 236<br />

11.7.2 Under what conditions do we find drag-reducing surfactants? ........... 236<br />

11.8 SANS measurements on micellar systems . . ........................ 239<br />

11.9 A new rheometer ............................................ 243<br />

References . . . ............................................. 247<br />

12 Photophysics of J Aggregates .................................. 251<br />

Hermann Pschierer, Hauke Wendt, <strong>and</strong> Josef Friedrich<br />

12.1 Introduction . . ............................................. 251<br />

12.2 Basic aspects of pressure <strong>and</strong> electric field phenomena in hole burning<br />

spectroscopy of J aggregates ................................... 252<br />

12.3 Experimental . . ............................................. 253<br />

12.4 Results ................................................... 254<br />

12.5 Discussion . . . ............................................. 256<br />

12.5.1 Pressure phenomena ......................................... 256<br />

12.5.2 Electric field-induced phenomena ............................... 258<br />

Acknowledgements .......................................... 258<br />

References . . . ............................................. 259<br />

13 Convection Instabilities in Nematic Liquid Crystals ................ 260<br />

Lorenz Kramer <strong>and</strong> Werner Pesch<br />

13.1 Introduction . . ............................................. 260<br />

13.2 Basic equations <strong>and</strong> instability mechanisms ........................ 264<br />

13.2.1 The director equation ......................................... 264<br />

13.2.2 The velocity field ........................................... 266<br />

IX

Contents<br />

13.2.3 Electroconvection ........................................... 267<br />

13.2.3.1 The st<strong>and</strong>ard model .......................................... 267<br />

13.2.3.2 The weak electrolyte model . ................................... 268<br />

13.2.4 Rayleigh-Bénard convection . ................................... 269<br />

13.3 Theoretical analysis .......................................... 269<br />

13.4 Rayleigh-Bénard convection . ................................... 275<br />

13.5 Electrohydrodynamic convection ................................ 278<br />

13.5.1 Linear theory <strong>and</strong> type of bifurcation ............................. 278<br />

13.5.2 Results of Ginzburg-L<strong>and</strong>au equation ............................. 279<br />

13.5.3 Beyond the Ginzburg-L<strong>and</strong>au equation ............................ 281<br />

13.5.3.1 Experimental results ......................................... 281<br />

13.5.3.2 Theoretical results <strong>and</strong> discussion ................................ 282<br />

13.6 Concluding remarks .......................................... 286<br />

Acknowledgements .......................................... 288<br />

Note added . . . ............................................. 289<br />

References . . . ............................................. 290<br />

14 Preparation <strong>and</strong> <strong>Properties</strong> of Ionic <strong>and</strong> Surface Modified<br />

Micronetworks ............................................. 295<br />

Michael Mirke, Ralf Grottenmüller, <strong>and</strong> Manfred Schmidt<br />

14.1 Introduction . . ............................................. 295<br />

14.2 Polymerization in normal microemulsion . . ........................ 295<br />

14.2.1 Mechanism <strong>and</strong> size control . ................................... 295<br />

14.2.2 Surface functionalization of microgels ............................ 298<br />

14.3 Polymerization in inverse microemulsion . . ........................ 299<br />

14.3.1 Preparation of ionic microgels .................................. 299<br />

14.3.2 <strong>Properties</strong> of ionic microgels <strong>and</strong> interparticle interaction .............. 299<br />

14.4 Conclusion <strong>and</strong> relevance to future work . . ........................ 303<br />

References . . . ............................................. 304<br />

15 Ferrocene-Containing Polymers ................................ 305<br />

Oskar Nuyken,Volker Burkhardt, Thomas Pöhlmann, Max Herberhold,<br />

Fred Jochen Litterst, <strong>and</strong> Christian Hübsch<br />

15.1 Introduction . . ............................................. 305<br />

15.2 Addition polymers ........................................... 306<br />

15.2.1 Radical polymerization ....................................... 306<br />

15.2.2 Radical copolymerization . . . ................................... 308<br />

15.2.3 Anionic polymerization of VFc ................................. 308<br />

15.2.3.1 Living polymerization ........................................ 309<br />

15.2.3.2 Block copolymers ........................................... 312<br />

15.2.4 Polymeranalogeous reactions ................................... 314<br />

15.3 Polymers with ferrocene units in the main chain ..................... 315<br />

15.3.1 Polycondensation ............................................ 315<br />

15.3.2 Polymers by addition of dithiols to diolefins ........................ 316<br />

15.3.2.1 Radical reaction ............................................ 316<br />

15.3.2.2 Base catalyzed reactions . . . ................................... 316<br />

X

Contents<br />

15.3.2.3 Acid catalyzed reactions . . . ................................... 318<br />

15.3.3 1,1'-dimercapto-ferrocene as initiator ............................. 319<br />

15.3.4 Reductive coupling .......................................... 320<br />

15.4 Mößbauer studies of polymers containing ferrocene .................. 320<br />

References . . . ............................................. 323<br />

16 Transfer of Vibrational Energy in Dye-Doped Polymers ............. 325<br />

Johannes Baier, Thomas Dahinten, <strong>and</strong> Alois Seilmeier<br />

16.1 Introduction . . ............................................. 325<br />

16.2 Experimental . . ............................................. 326<br />

16.3 Results <strong>and</strong> discussion ........................................ 327<br />

16.4 Summary ................................................. 332<br />

References . . . ............................................. 332<br />

17 Picosecond Laser Induced Photophysical Processes of Thiophene<br />

Oligomers ................................................ 333<br />

Dieter Grebner, Matthias Helbig, <strong>and</strong> Sabine Rentsch<br />

17.1 Introduction . . ............................................. 333<br />

17.2 Experimental . . ............................................. 334<br />

17.3 Spectroscopic properties of oligothiophenes ........................ 337<br />

17.4 Results ................................................... 337<br />

17.4.1 Picosecond-transient spectra of oligothiophenes in solution ............. 337<br />

17.4.2 Time behaviour of transient spectra .............................. 339<br />

17.4.3 Size dependence of spectroscopic properties of oligothiophenes ......... 341<br />

17.4.4 Size dependence of the kinetic behaviour of oligothiophenes ............ 342<br />

17.5 Discussion . . . ............................................. 343<br />

References . . . ............................................. 343<br />

18 Topospecific Chemistry at Surfaces ............................. 344<br />

Hans Ludwig Krauss<br />

18.1 Introduction . . ............................................. 344<br />

18.1.1 The problem . . ............................................. 344<br />

18.1.2 Preparative <strong>and</strong> analytical methods ............................... 344<br />

18.1.3 Industrial applications ........................................ 345<br />

18.1.4 The st<strong>and</strong>ard procedures of the Phillips process ..................... 345<br />

18.1.5 Earlier work . . ............................................. 346<br />

18.2 The support . . . ............................................. 346<br />

18.2.1 Unmodified silica ........................................... 346<br />

18.2.2 Modified silica ............................................. 347<br />

18.2.3 Others .................................................... 348<br />

18.3 Transition metal surface compounds .............................. 348<br />

18.3.1 The metals . . . ............................................. 348<br />

18.3.2 Impregnation <strong>and</strong> activation . ................................... 349<br />

18.4 Coordinatively unsaturated sites ................................. 350<br />

18.4.1 Reduction of saturated surface compounds . ........................ 350<br />

18.4.2 Elimination of lig<strong>and</strong>s ........................................ 352<br />

XI

Contents<br />

18.5 Physical properties of the coordinatively unsaturated sites .............. 353<br />

18.5.1 Topologically different sites . ................................... 353<br />

18.5.2 Optical <strong>and</strong> magnetic properties ................................. 353<br />

18.6 Chemical properties of the coordinatively unsaturated sites ............. 355<br />

18.6.1 Survey of catalytic reactions ................................... 355<br />

18.6.2 Olefin polymerization ........................................ 356<br />

18.6.3 Other catalytic reactions . . . ................................... 360<br />

18.7 Deactivation . . ............................................. 361<br />

18.8 Summary <strong>and</strong> outlook ........................................ 362<br />

References . . . ............................................. 363<br />

III<br />

Biopolymers<br />

19 Site-Directed Spectroscopy <strong>and</strong> Site-Directed Chemistry of Biopolymers 369<br />

Stefan Limmer, Günther Ott, <strong>and</strong> Mathias Sprinzl<br />

19.1 Introduction . . ............................................. 369<br />

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes . . 370<br />

19.2.1 Stability of tRNA-derived acceptor stem duplexes .................... 371<br />

19.2.2 Manganese ion binding sites at RNA duplexes ...................... 374<br />

19.2.3 Structural determination of short RNA duplexes by 2D NMR spectroscopy . 377<br />

19.2.4 NMR derived model of the tRNA Ala acceptor arm ................... 379<br />

19.2.5 Chemical shifts <strong>and</strong> scalar coupling as an indicator of RNA structure in the<br />

vicinity of a G-U pair ........................................ 380<br />

19.2.6 Structure of aminoacyl-tRNA <strong>and</strong> transacylation of the aminoacyl residue . . 382<br />

19.3 Structure of elongation factor Tu ................................ 383<br />

19.3.1 Sequence of Thermus thermophilus EF-Tu.......................... 384<br />

19.3.2 Crystallization, X-ray analysis, <strong>and</strong> the tertiary structure . .............. 386<br />

19.3.3 Nucleotide binding <strong>and</strong> GTPase reaction . . ........................ 387<br />

19.3.4 Mechanism of GTP induced conformational change of EF-Tu ........... 388<br />

19.3.5 Aminoacyl-tRNA in complex with EF-Tu 7 GTP ..................... 389<br />

19.3.6<br />

1 H NMR of yeast Phe-tRNA Phe EF-Tu7GTP complex . . .............. 391<br />

19.3.7<br />

13 C NMR studies of the Val-tRNA Val EF-Tu7GTP ternary complex ....... 393<br />

19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA ..................... 395<br />

19.3.9 EF-Tu interaction with EF-Ts ................................... 395<br />

19. 3.10 Site-directed mutagenesis of EF-Tu ............................... 396<br />

19.4 Summary <strong>and</strong> conclusions . . ................................... 397<br />

Acknowledgement ........................................... 397<br />

References . . . ............................................. 398<br />

20 Spectroscopic Probes of Surfactant Systems <strong>and</strong> Biopolymers ......... 401<br />

Alex<strong>and</strong>er Wokaun<br />

20.1 Introduction . . ............................................. 401<br />

20.2 Diffusion in surfactant systems .................................. 402<br />

20.2.1 Structural characteristics of micellar solutions, cubic phases, <strong>and</strong> multilamellar<br />

vesicles from NMR self-diffusion measurements .............. 402<br />

XII

Contents<br />

20.2.2 Probing of mobilities in multilamellar vesicles by forced Rayleigh scattering 405<br />

20.2.3 Dimensionality of diffusion in lyotropic mesophases from fluorescence<br />

quenching ................................................. 409<br />

20.2.4 Summary of results .......................................... 412<br />

20.3 Vibrational spectroscopy <strong>and</strong> conformational analysis of oligonucleotides . . 413<br />

20.3.1 Spectroscopic characterization of right <strong>and</strong> left-helical forms of a hexadecanucleotide<br />

duplex ........................................ 413<br />

20.3.2 SERS spectra of deoxyribonucleotides ............................ 415<br />

20.3.3 Studies of chromophore-DNA interaction by vibrational spectroscopy . . . . . 417<br />

20.3.4 Summary of results .......................................... 418<br />

20.4 Related projects carried out within the framework of the Collaborative<br />

Research Centre ............................................ 419<br />

20.5 Remarks <strong>and</strong> acknowledgements ................................ 421<br />

References . . . ............................................. 422<br />

21 Energy Transport by Lattice Solitons in a-Helical Proteins ........... 424<br />

Franz-Georg Mertens, Dieter Hochstrasser, <strong>and</strong> Helmut Büttner<br />

21.1 Introduction . . ............................................. 424<br />

21.2 The model ................................................. 426<br />

21.3 Quasicontinuum approximation ................................. 428<br />

21.4 Velocity range for the quasicontinuum approach ..................... 431<br />

21.5 Solitary waves for realistic parameter values ........................ 432<br />

21.6 Iterative method <strong>and</strong> stability ................................... 434<br />

21.7 Conclusion . . . ............................................. 437<br />

References . . . ............................................. 438<br />

IV<br />

Appendix<br />

22 Documentation of the Collaborative Research Centre 213 ............ 443<br />

Markus Schwoerer <strong>and</strong> Heinz Hoffmann<br />

22.1 List of Members ............................................ 443<br />

22.2 Heads of Projects (Teilprojektleiter) .............................. 444<br />

22.2.1 Projektbereich A: Gemeinsame Einrichtungen ...................... 444<br />

22.2.2 Projektbereich B: Festkörper ................................... 444<br />

22.2.3 Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen<br />

und Polymere . ............................................. 445<br />

22.2.4 Projektbereich D: Biopolymere ................................. 446<br />

22.3 Guests .................................................... 447<br />

22.4 Co-workers . . . ............................................. 450<br />

22.5 International Cooperation . . . ................................... 457<br />

22.6 Funding .................................................. 459<br />

XIII

Preface<br />

At the end of 1983, about eight years after the inauguration of the University of Bayreuth<br />

the Deutsche Forschungsgemeinschaft (DFG) agreed to establish the Collaborative Research<br />

Centre 213, TOPOMAC, for the promotion of basic research in chemistry <strong>and</strong> physics of<br />

macromolecular systems. The title of TOPOMAC in its full length, “Topospezifische Chemie<br />

und Toposelektive Spektroskopie von Makromolekülsystemen: Mikroskopische Wechselwirkung<br />

und Makroskopische Funktion” expessed the intention of the original applicants:<br />

a productive cooperation between physicists, chemists <strong>and</strong> biochemists across the mutual<br />

borders of their original research fields. Until the end of 1995 TOPOMAC was supported by<br />

the Deutsche Forschungsgemeinschaft with 28 MDM.<br />

The present book documents the achievements of TOPOMAC. It is not a minute addition<br />

of all the results which have been published in periodical journals but rather a survey of<br />

important research fields of members of TOPOMAC. The articles have been written towards<br />

or after the end of the support period as both, review <strong>and</strong> original publications.<br />

The book covers the fields of:<br />

. Electronic, photoelectric, thermal, dielectric, optical <strong>and</strong> magnetic properties of macromolecular<br />

solids (polymers <strong>and</strong> polymer crystals),<br />

. Micellar structures, J-aggregates, liquid crystals, µ-gels, ferrocene-containing polymers<br />

<strong>and</strong> topospecific chemistry at surfaces <strong>and</strong> in single crystals, <strong>and</strong><br />

. Biopolymers <strong>and</strong> surfactant systems as studied by site directed spectroscopy <strong>and</strong> site directed<br />

chemistry <strong>and</strong> also by the theory of energy transport.<br />

During the period of its support TOPOMAC had 30 members (Teilprojektleiter). Only<br />

ten of them have been members for the entire period, mainly because twenty times a call<br />

from other universities or research institutions reached one of the members of the Collaborative<br />

Research Centre 213. Fourteen of them followed this call <strong>and</strong> left the Collaborative Research<br />

Centre 213. Less than two years after the end of the period of support some of the remaining<br />

former members of Collaborative Research Centre 213 together with young <strong>and</strong><br />

new faculty members began to continue the formal cooperation between chemists <strong>and</strong> physicists<br />

in the field of macromolecular research. Their actual cooperation in the meantime<br />

never had been terminated.<br />

The editors would like to express the sincere thanks of the members of TOPOMAC to<br />

the foreign guests of TOPOMAC, to the Deutsche Forschungsgemeinschaft, to the University<br />

XV

Preface<br />

of Bayreuth <strong>and</strong> also to the Freistaat Bayern. Our foreign guests stayed for long or short periods<br />

between one year <strong>and</strong> one day. They have contributed in an essential <strong>and</strong> special manner<br />

to the success on our research fields. They also strongly intensified the national <strong>and</strong> international<br />

scientific relations of both, the members <strong>and</strong> the research students of TOPO-<br />

MAC. The cooperation of the speakers of TOPOMAC with Dr. Funk from the Deutsche Forschungsgemeinschaft<br />

office throughout the entire period of support was excellent. The help<br />

of the former president of the university, Dr. K. D. Wolff <strong>and</strong> his chancellor, W. P. Hentschel<br />

as well as the continuous support by the late Ministerialrat G. Grote <strong>and</strong> by J. Großkreutz<br />

from the Bayerisches Staatsministerium für Unterricht, Kultus, Wissenschaft und Kunst has<br />

been an essential stimulus for the scientific members of TOPOMAC. Last but not least we<br />

thank Doris Buntkowski for her faithful <strong>and</strong> reliable work as our secretary.<br />

The Editors<br />

XVI

List of Contributors<br />

Alex<strong>and</strong>er Blumen<br />

Theoretische Polymerphysik<br />

Universität Freiburg<br />

Herrmann-Herder-Straße 3<br />

79104 Freiburg<br />

Elmar Dormann<br />

Physikalisches Institut<br />

Universität Karlsruhe<br />

Engesserstr. 7<br />

67131 Karlsruhe<br />

Pablo Esquinazi<br />

Institut für Experimentelle Physik II<br />

Universität Leipzig<br />

Linnestraße 5<br />

04103 Leipzig<br />

Klaus Fesser<br />

Fachbereich Physik<br />

Universität Greifswald<br />

Domstraße 10a<br />

17489 Greifswald<br />

Josef Friedrich<br />

Lehrstuhl für Physik<br />

Technische Universität München<br />

85350 Freising-Weihenstephan<br />

Holger Fehske<br />

Theoretische Physik I<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Dietrich Haarer<br />

Experimentalphysik IV<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Heinz Hoffmann<br />

Physikalische Chemie I<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Lorenz Kramer<br />

Theoretische Physik II<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Hans Ludwig Krauss<br />

Heunischstraße 5 b<br />

96049 Bamberg<br />

Franz G. Mertens<br />

Theoretische Physik<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Oskar Nuyken<br />

Lehrstuhl für Makromolekulare Stoffe<br />

Technische Universität München<br />

Lichtenbergstraße 4<br />

85747 München<br />

XVII

List of Contributors<br />

Sabine Rentsch<br />

Institut für Optik und Quantenelektronik<br />

Friedrich Schiller Universität Jena<br />

Max-Wien-Platz 1<br />

07743 Jena<br />

Wolfgang Richter<br />

Experimentalphysik IV<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Paul Rösch<br />

Biopolymere<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Manfred Schmidt<br />

Institut für Physikalische Chemie<br />

Universität Mainz<br />

Welder-Weg 11<br />

55099 Mainz<br />

Markus Schwoerer<br />

Experimentalphysik II<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Alois Seilmeier<br />

Physikalisches Institut<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Mathias Sprinzl<br />

Biochemie<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Peter Strohriegel<br />

Makromolekulare Chemie I<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Thomas Vogtmann<br />

Experimentalphysik II<br />

Universität Bayreuth<br />

Universitätsstraße 30<br />

95447 Bayreuth<br />

Alex<strong>and</strong>er Wokaun<br />

Bereich F5<br />

Paul Scherrer Institut<br />

CH-5232 Villingen<br />

XVIII

I<br />

Mainly Solids<br />

Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

1 Model Systems for Photoconductive Materials<br />

Harald Meyer <strong>and</strong> Dietrich Haarer<br />

1.1 Introduction<br />

The effect of photoconductivity, i. e. the increase of the electrical conductivity of a material<br />

upon illumination with light of suitable photon energy, was first discovered in selenium by<br />

W. Smith in 1873.<br />

A typical application for these materials is the xerographic process [2, 3], which was<br />

developed by C. F. Carlson in 1942 [1]. Here, a photoconducting film on top of a grounded<br />

electrode is homogeneously charged by a corona discharge. Typical field strengths are up to<br />

10 6 V/m. In a second step the image of the original document is projected onto the film. At<br />

the areas where the film is illuminated charge carriers are generated in the photoconducting<br />

film. One species traverses the film <strong>and</strong> recombines at the grounded electrode whereas the<br />

oppositely charged species neutralizes the surface charges. With this step the original image<br />

is transferred into an electrostatic image on the photoconductor film. Subsequently, small toner<br />

particles are deposited on the photoconductor. They stick to the charged regions <strong>and</strong><br />

thus generate a real image. In the next step, this image is transferred onto paper <strong>and</strong> fixed<br />

by thermally fusing the toner particles on the paper.<br />

The process for a laser printer is similar except for the fact that the image is written<br />

onto the photoconductor directly by a laser or a diode array. If the toner particles are fused<br />

directly onto the photoconductor instead of transferring them onto paper the photoconductor<br />

can be used as an offset printing master [4].<br />

Potential systems for commercial use have to meet several requirements:<br />

a) sensitivity in the visible region of the spectrum;<br />

b) good charge transport properties, i. e. charge carrier mobility in excess of 10 –7 cm 2 /Vs<br />

[3] <strong>and</strong> minor trapping effects;<br />

c) good mechanical <strong>and</strong> dielectrical properties;<br />

d) excellent film forming properties <strong>and</strong> possibility of manufacturing defect free large area<br />

films;<br />

e) mechanical flexibility for the use in small sized desktop devices.<br />

The last two requirements cannot be met neither by organic nor inorganic crystalline<br />

materials. Therefore both, organic <strong>and</strong> inorganic amorphous photoconductors, have been de-<br />

Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1<br />

3

1 Model Systems for Photoconductive Materials<br />

veloped. Inorganic materials like e. g. a-Se, As 2 Se 3 or alloys of Se <strong>and</strong> Te show good transport<br />

properties with mobilities in the range of 0.1 cm 2 /Vs at room temperature [7], <strong>and</strong> high<br />

sensitivity for visible light together with moderate mechanical <strong>and</strong> dielectric properties. One<br />

of their most important disadvantages compared to organic systems is that most of these materials<br />

are highly toxic.<br />

In the past 30 years, since H. Hoegl discovered photoconductivity in the organic polymer<br />

poly(N-vinylcarbazole) (PVK) (Fig. 1.2) [5, 6], organic materials have almost completely<br />

replaced their inorganic counterparts, although the effective mobility in these systems is<br />

typically 5 orders of magnitude lower as compared to a-Se.<br />

Organic photoconductors can be regarded as large or medium b<strong>and</strong>gap semiconductors.<br />

A typical value for commercially applied systems is 3.5 eV for carbazole derivatives<br />

(Section 1.4). This inherent lack of sensitivity in the visible region has been overcome by<br />

using charge transfer systems [5] or multilayer systems with additional charge generation<br />

layers. For a review see e. g. Ref. [3]. On the other h<strong>and</strong>, the large b<strong>and</strong>gap of organic materials<br />

virtually eliminates the influence of thermally generated charge carriers <strong>and</strong> thus improves<br />

the dielectric properties as compared to low b<strong>and</strong>gap materials.<br />

Therefore the main target of the works, which will be described in the following, was<br />

to identify the key factors which limit the charge carrier mobilities in organic systems <strong>and</strong><br />

to develop new high mobility materials.<br />

1.2 Experimental techniques<br />

Besides the xerographic discharge method [3, 10], the time-of-flight (TOF) technique is generally<br />

used to study the charge carrier transport of thin organic films. Here, the photoconducting<br />

film with a typical thickness of 10 mm is s<strong>and</strong>wiched between two electrodes (Fig. 1.1). Electron-hole<br />

pairs are generated by the energy hn of a strongly absorbed laser pulse, which is irradiated<br />

through one of the semitransparent electrodes. For most organic materials the charge<br />

carrier generation process can be described by an Onsager model, either in its one or three dimensional<br />

form [11–13].<br />

The wavelength of the laser is chosen to ensure that the penetration depth is considerably<br />

less than the sample thickness. Thus the charge carriers are generated close to the illuminated<br />

surface. Under the influence of an externally applied electrical field the electronhole<br />

pairs are separated <strong>and</strong> one species, depending on the polarity of the external field, immediately<br />

recombines at the illuminated electrode. The opposite charged species drifts<br />

through the sample, thus giving rise to a time dependent photocurrent I p (t).<br />

For a quantitative analysis the knowledge about the electrical field inside the sample<br />

is necessary. This can be achieved by performing the measurements in the small signal limit,<br />

i. e. disturbations due to space charge effects are negligible <strong>and</strong> the electric field in the bulk<br />

is determined by the externally applied field. It turns out, that this condition is fulfilled as<br />

long as the generated photocharge Q tot is less than 10 % of the charge Q = CU, which is<br />

4

1.3 Transport models<br />

Figure 1.1: Principle setup for TOF experiments.<br />

stored on the electrodes. With typical experimental parameters (sample thickness d =10mm,<br />

sample area A =4cm 2 ,10 21 monomer units per cm 3 , voltage U = 300 V, sample capacitance<br />

C = 1 nF) the above criterion leads to an upper limit for the photocharges of Q tot

1 Model Systems for Photoconductive Materials<br />

neglected, or in other words, the detrapping always occurs to states at e d . The traps can be<br />

caused by static disorder as well as by self trapping due to polaronic effects or both.<br />

Mathematically equivalent with the MT model is the Continuous Time R<strong>and</strong>om Walk<br />

(CTRW) model [47–49], where the exponential trap distribution density g (e) in the MT<br />

model,<br />

<br />

"<br />

g…"† /exp ; …2†<br />

k B T 0<br />

corresponds to the well-known algebraic distribution of hopping times in the CTRW model<br />

[45],<br />

…t† /t 1 ; …3†<br />

where the disorder parameter a in Eq. 3 corresponds to the dimensionless temperature<br />

(a = T/T 0 ).<br />

A variety of experimental data [16, 25, 26, 33, 34] can be described by the empirical<br />

formula,<br />

<br />

ef f ˆ 0 exp<br />

p<br />

" 0 E<br />

k B T ef f<br />

<br />

; …4†<br />

first proposed by Gill [33], where e 0 is the zero field activation energy, E the electrical field<br />

<strong>and</strong> b the Poole-Frenkel factor. This equation describes the thermal activated release from<br />

localized traps where the activation energy is lowered by an external field according to the<br />

Poole-Frenkel effect [37].<br />

The effective temperature T eff is related to the physical temperature T by<br />

1<br />

ˆ 1<br />

T ef f T<br />

1<br />

T 0<br />

:<br />

…5†<br />

Initially, the characteristic temperature T 0 simply was an empirical parameter. In<br />

Section 1.4, however, we shall see that in certain cases this parameter can be interpreted microscopically.<br />

An alternative approach [28, 50–54] is based on the assumption that the density of<br />

states can be modelled by a Gaussian distribution. Charge carrier transport occurs via direct<br />

hopping between the localized sites. In general, the differences between the two models are<br />

too small to be detected experimentally. Since our data can be quantitatively explained<br />

within the framework of multiple trapping we will restrict the discussion to this model.<br />

6

1.4 Results<br />

1.4 Results<br />

As stated in Section 1.1, organic photoconductors can be regarded as large or medium b<strong>and</strong>gap<br />

semiconductors. In contrast to inorganic crystalline semiconductors the electronic coupling between<br />

adjacent sites is weak. Therefore the electronic states in these materials are primarily of<br />

molecular character <strong>and</strong> the charge carrier transport has to be regarded as an incoherent hopping<br />

process between two neighbouring sites. For the following discussion organic photoconductors<br />

will be divided into three main groups namely molecularly doped polymers (Section<br />

1.4.1), side-chain polymers (Section 1.4.2.), <strong>and</strong> conjugated systems (Section 1.4.3).<br />

1.4.1 Molecularly doped polymers<br />

In molecularly doped polymers (MDPs) the transport molecules are molecularly dispersed in<br />

an inert matrix. Due to crystallization the maximum concentration of chromophores, which<br />

can be achieved, is typically in the range of 50 mol%.<br />

Typical chemical compounds include oxadiazole derivatives [14], pyrazolines [15–<br />

18], hydrazones [19–22], carbazole derivatives [23–26], triphenylmethane (TPM) derivatives<br />

[27, 28], triphenylamine (TPA) derivatives [29, 30], <strong>and</strong> TAPC [31], which can be regarded<br />

as a dimer of TPA. The charge carrier mobilities at room temperature are typically<br />

in the range from 10 –6 cm 2 /Vs for N-isopropylcarbazole [25] to 10 –4 cm 2 /Vs for p-diethylaminobenzaldehyde<br />

diphenyl hydrazone (DEH) [22].<br />

In order to study the effect of chemical substitutions of the respective transport molecule<br />

N-isopropylcarbazole (NIPC) <strong>and</strong> derivatives thereof, with electron donor as well as acceptor<br />

substituents (Fig. 1.2), have been investigated in a polycarbonate host [26].<br />

As compared by 3,6-dibromo-N-isopropylcarbazole (DBr-NIPC) the effective hole<br />

mobility of 3,6-dimethoxy-N-isopropylcarbazole (DMO-NIPC) is slightly decreased. The<br />

reason for this behaviour is the fact that the cation, which is relevant for the transport process,<br />

is stabilized by electron donating substituents like the methoxy group, as can be seen<br />

from the shift of the ionisation potentials [26]. This leads to stronger localization of the<br />

charge within the aromatic rings as compared to derivatives with moderate electron acceptors<br />

like bromine. Therefore the spatial overlap of the electronic states of adjacent molecules,<br />

which are involved in the transport process, is reduced.<br />

With strong acceptors like the nitro groups in 3,6-dinitro-N-isopropylcarbazole (DN-<br />

NIPC) the ionisation potential is already larger than for the host matrix polycarbonate. In<br />

this case even the matrix acts as a trap for holes, thus preventing efficient charge carrier<br />

transport [26].<br />

Since the effective mobility is determined by the spatial overlap of the transport states,<br />

the concentration dependence of the effective mobility can be described by<br />

<br />

ef f / r 2 exp<br />

<br />

2r<br />

; …6†<br />

r 0<br />

7

1 Model Systems for Photoconductive Materials<br />

N<br />

H 3 CO<br />

N<br />

N<br />

N<br />

N<br />

C 2 H 5<br />

R<br />

R<br />

BTA<br />

R: H: NIPC<br />

Br: DBr-NIPC<br />

OCH 3 : DMO-NIPC<br />

NO 2 : DN-NIPC<br />

Si<br />

Si<br />

O<br />

CH 3<br />

PMPS<br />

n<br />

Ca n<br />

CH CH 2<br />

PVK<br />

(CH 2 ) n<br />

Ca<br />

n<br />

R<br />

Polysiloxan<br />

R<br />

n<br />

Ca:<br />

N<br />

R:<br />

O<br />

DPOP-PPV<br />

Figure 1.2: Chemical structure of photoconducting materials. Abbreviations see text.<br />

H<br />

PPV<br />

where r is the mean distance between adjacent transport molecules <strong>and</strong> r 0 the wave function<br />

decay parameter. For carbazole derivatives r 0 is typically in the range 1.3–1.6 Å, depending<br />

on the substituted groups [26].<br />

In recent years, also percolation models have been successfully applied to experimental<br />

data [55] as an alternative approach to model the transport properties of doped disordered<br />

systems. For polycarbonate doped with a derivative of benztriazole (BTA) (Fig. 1.2) it has<br />

been shown that the concentration dependence of the effective mobility can be described at<br />

low concentrations by [55]<br />

ef f ˆ 0 …p p c † t …7†<br />

for p>p c , where p is the concentration of transport molecules, p c the percolation threshold,<br />

<strong>and</strong> t the critical exponent. The experimental values (p c = 0.095 <strong>and</strong> t = 2.46) are consistent<br />

with 3D-continuum percolation calculations [56].<br />

Since this system shows good transport properties over a wide range of concentrations,<br />

the influence of extrinsic traps could be studied [57]. Here, a sample containing 25 wt% of<br />

BTA in a polycarbonate host was doped with small amounts of the organic dye astrazone<br />

orange (AO). It turned out that the effective mobility is reduced by a factor of 2 when the<br />

8

1.4 Results<br />

concentration of AO is increased to 0.5 wt%. This finding becomes also important when<br />

thinking of new applications like organic electroluminescent devices. One way to increase<br />

the efficiency in these devices is to molecularly disperse the luminescent moiety into a<br />

charge transport material. But there is a trade-off between efficient trapping of the charge<br />

carriers – which is improved by higher concentrations of the luminescent groups – <strong>and</strong> concentration<br />

quenching of the fluorescence, which can be avoided by keeping the concentration<br />

as low as possible. The above described measurements [57] show that organic dyes can be<br />

very efficient traps for charge carriers. Therefore the concentration of the luminescent moiety<br />

can be as low as 2% or less [61], thus avoiding concentration quenching without loosing<br />

high trapping efficiency.<br />

1.4.2 Side-chain polymers<br />

In side-chain polymers, where the chromophore is covalently bonded to a polymer main<br />

chain, the concentration of transport molecules can be increased as compared to MDPs without<br />

causing crystallization. Because the underlying physical mechanism of charge carrier<br />

transport (nearest neighbour hopping between weakly coupled, localized states) is the same<br />

for both MDPs <strong>and</strong> side-chain polymers, the transport properties are qualitatively similar.<br />

This finding offers the opportunity to tailor the thermal <strong>and</strong> mechanical properties of<br />

the photoconductor without seriously affecting the transport properties. This can be achieved<br />

either by variation of the spacer length between the chromophore <strong>and</strong> the backbone or by variation<br />

of the backbone itself. A typical example for this group is PVK [12, 32, 34, 35]. Here,<br />

the transition from dispersive to non-dispersive transport could be observed [34]. Due to the<br />

fact, that the TOF curves have been measured over 10 decades in time, it was possible by<br />

means of a numerical inverse Laplace transform [34] to calculate from the measured photocurrent<br />

the trapping rate distribution, which is a measure for the density of localized states.<br />

In polysiloxane derivatives with pendant carbazolyl groups [58, 59] (Fig. 1.2) the effect<br />

of a variation of the spacer length was investigated. Starting from a spacer length of 3 or 5<br />

methylene units up to 6 or 11 groups, the glass transition temperature drops from its initial value<br />

of 51 8Cor78Cdownto–58Cor–458C for the C 11 spacer. It turns out, that the zero field<br />

activation energy e act for the hole transport remains unchanged <strong>and</strong> is the same for NIPC or<br />

PVK, e act = 0.51 eV [58]. This indicates, that the activation energy in side-chain polymers<br />

with pendant carbazolyl groups is dominated by intrinsic properties of the carbazolyl moiety<br />

<strong>and</strong> is not much affected by other factors like the morphology of the polymer [58].<br />

However, the absolute values for m eff differ by roughly one order of magnitude, where the<br />

compound with the shortest spacer shows an effective mobility in the range of 10 –6 cm 2 /Vs<br />

(T = 300 K, E =3710 5 V/cm). For the materials with a spacer length of 5 <strong>and</strong> 6 methylene<br />

units the mobility is lower by one order of magnitude <strong>and</strong> therefore comparable to the mobility<br />

in NIPC or PVK. A striking feature, when comparing NIPC with PVK, is the fact that<br />

polycarbonate doped with 20 wt% of NIPC shows the same hole mobility at room temperature<br />

as compared to PVK where the carbazole concentration amounts to 86 wt%. Obviously,<br />

the covalent attachment of the chromophore to a stiff backbone, like PVK with its glass<br />

transition temperature of 227 8C, induces a mutual orientation of adjacent pendant groups,<br />

9

1 Model Systems for Photoconductive Materials<br />

which reduces the electronic coupling between the two sites. Therefore it is desirable to<br />

induce a well-defined amount of flexibility in both, the spacer <strong>and</strong> the backbone, to allow<br />

for mutual reorientation of the chromophores. For a polysiloxane backbone with carbazolyl<br />

transport groups the optimum spacer length n is equal to 3.<br />

1.4.3 Conjugated systems<br />

In contrast to the two main groups described above the charge carrier transport in conjugated<br />

systems occurs via the polymer backbone. Various substituents are used to tailor the<br />

mechanical properties <strong>and</strong> the processibility. For instance, the chain can be a skeleton with<br />

quasi-s-conjugation like in polysilanes [36, 38–42] or polygermylenes [39].<br />

The second group of main chain polymers contains a p-conjugated backbone like<br />

polyacetylene, polythiophene, polypyrrole, polyphenylene, poly(phenylenevinylene) (PPV),<br />

<strong>and</strong> their derivatives. For a review see [43, 44].<br />

Both types of conjugated systems have been investigated. In the case of poly(methylphenyl<br />

silane) (PMPS), a quasi-s-conjugated polymer (Fig. 1.2), non-dispersive transport<br />

has been found down to a temperature of 250 K [60]. For lower temperatures, the transport<br />

is dispersive, which indicates that the transport in these materials is controlled by traps.<br />

Compared to the materials described above, the depths of the relevant traps are much lower.<br />

The zero field activation energy turns out to be as low as 0.37 eV as compared to 0.5 eV for<br />

materials containing carbazole. This gives a hole mobility at room temperature of roughly<br />

10 –3 cm 2 /Vs, which is 3 orders of magnitude higher than for the materials described above.<br />

The high mobility <strong>and</strong> the transparency in the visible region makes PMPS also useful for<br />

applications in multilayer electroluminescent devices [61–63].<br />

Besides the s-bonded PMPS, we investigated oligomeric conjugated compounds based<br />

on carbazole [64]. First experiments with a trimer of a carbazole derivative showed that this<br />

material is soluble in most of the common organic solvents. It forms a low molecular weight<br />

glass <strong>and</strong> shows no tendency to crystallize even after several months. In this material the<br />

p-system has been extended over three monomer units, which results in remarkably high<br />

hole mobilities as compared to the monomer model compound NIPC. First TOF experiments<br />

show an increase in mobility by roughly two orders of magnitude as compared to unconjugated<br />

carbazole systems, which reaches 2 7 10 –4 cm 2 /Vs [64].<br />

The second group of p-conjugated polymers contains one p z -orbital per carbon atom<br />

which st<strong>and</strong>s perpendicular to the plane of the s-bonded skeleton of the main chain. The interaction<br />

between these orbitals leads to electronic states, which are delocalized – at least<br />

partly – along the conjugated chain. As compared to the intrachain coupling, the intrachain<br />

coupling is lower by typically one or two orders of magnitude [66] thus leading to a large<br />

anisotropy of the electronic <strong>and</strong> optical properties. These materials can be regarded as basically<br />

one-dimensional organic semiconductors with b<strong>and</strong>gaps in the visible region of the<br />

spectrum, e. g. 1.5 eV (trans-(CH) x ) [43], 2.0 eV (polythiophene), 2.5 eV (polyphenylene vinylene),<br />

<strong>and</strong> 3.0 eV [poly(para-phenylene)] [67].<br />

We investigated the charge transport properties of the p-conjugated polymer poly<br />

(para-phenylenevinylene) <strong>and</strong> its soluble substituted derivative DPOP-PPV (Fig. 1.2). In<br />

10

1.4 Results<br />

contrast to e. g. (CH) x , PPV can be prepared in undoped form. If oxygen is carefully heated<br />

out, PPV exhibits excellent film forming properties as well as thermal stability under ambient<br />

conditions. In recent years, PPV has drawn additional attention due to its use in organic<br />

electroluminescent devices [45].<br />

In the following we will show that the charge carrier transport in these materials can<br />

be described by the conventional models mentioned before, which have been developed for<br />

the characterization of disordered molecular systems.<br />

Based on the MT model a method has been developed to calculate the distribution of<br />

capture rates from the measured photocurrent [68, 69]. In contrast to Ref. [34], the method<br />

is based on a Fourier transform technique with better numerical stability as compared to the<br />

inverse Laplace transform used in Ref. [34].<br />

With this approach the capture rate density o c is given by<br />

! c …" ! †k B T ˆ 2<br />

p<br />

Q tot sin <br />

t mic<br />

I…!†<br />

<br />

!<br />

! ; …8†<br />

where Q tot is the total photogenerated charge, t mic the microscopic transit time, <strong>and</strong> I (o) the<br />

Fourier transform of the measured photocurrent, <strong>and</strong> f the phase shift (Eq. 9). For DPOP-<br />

PPV, t mic = 7.5610 –10 s for the given experimental conditions.<br />

<br />

ˆ tan 1 Im I…!†<br />

Re I…!†<br />

…9†<br />

The trap depth e o is related to the frequency o by the expression<br />

<br />

! ˆ r 0 exp<br />

<br />

" !<br />

; …10†<br />

k B T<br />

where r 0 is the attempt-to-escape frequency (r 0 =10 10 s –1 for DPOP-PPV).<br />

In DPOP-PPV the effective mobility is field dependent <strong>and</strong> thermally activated according<br />

to Gill’s formula (Eq. 4) [33] with a characteristic temperature T 0 = 465 K (Fig. 1.3).<br />

The calculated trap density g (e) is exponential down to a pronounced cutoff energy e d ,<br />

<br />

g…"† /exp<br />

<br />

"<br />

: …11†<br />

k B T 0<br />

This cutoff energy corresponds exactly to the measured activation energy of the effective<br />

mobility. Furthermore, the initially simply empirical parameter T 0 in Gill’s formula<br />

(Eq. 4) could be correlated with the decay constant k B T 0 of the trap distribution.<br />

By calculating the total number of trapping events it can be seen that the typical distance<br />

which a charge carrier travels between two trapping events is of the order of 4 Å. This<br />

value is comparable to the intrachain distance <strong>and</strong> indicates that the transport in these materials<br />

can be best described by conventional hopping between closely neighbouring sites. No<br />

evidence for b<strong>and</strong>like transport has been found.<br />

11

µ eff<br />

[cm 2 /Vs]<br />

1 Model Systems for Photoconductive Materials<br />

10 -2<br />

10 -3<br />

10 -4<br />

350 K 300 K 270 K<br />

230 K<br />

500 kV/cm<br />

350 kV/cm<br />

200 kV/cm<br />

75 kV/cm<br />

0kV/cm<br />

10 -5<br />

10 -6<br />

10 -7<br />

10 2<br />

10 0<br />

T 0<br />

= 465K<br />

10 -2<br />

10 -4<br />

10 -6<br />

10 -8<br />

10 -10<br />

0 2 4 6 8<br />

10 -8<br />

2.5 3.0 3.5 4.0 4.5<br />

1/T [1000/K]<br />

Figure 1.3: large frame: effective mobility m eff of DPOP-PPV as a function of temperature for some<br />

electrical fields, data points for zero field were extrapolated from field dependent measurements; inset:<br />

Arrhenius fits for different electrical fields. For details see text. Data from [68, 69].<br />

For the case of PPV the dispersion of the photocurrent is too large for transit time determination.<br />

It can be concluded that even at room temperature the release times from the<br />

deepest traps, which control the transport properties, are in the range of 1 s or less. This corresponds<br />

to an effective mobility of less than 10 –8 cm 2 /Vs [69]. Based on TSC measurements<br />

from different groups [70], we conclude that this behaviour is primarily caused by the<br />

existence of grain boundaries.<br />

It is reasonable to assume that in conjugated polymers with a rigid backbone the preferred<br />

orientation of the main chain will be parallel to the film surface. Therefore, the<br />

charge carrier transport through the film has to occur perpendicular to the backbone, i. e.<br />

perpendicular to the direction with high intrinsic mobility, which would mean a principle<br />

limit for this class of materials for this kind of applications.<br />

1.5 Outlook<br />

As described above, disordered organic materials have been developed with effective hole<br />

mobilities of up to 10 –3 cm 2 /Vs together with good mechanical <strong>and</strong> dielectric properties.<br />

However it seems that for disordered systems a considerable improvement of this value will<br />

12

References<br />

be difficult to achieve, because for all investigated material groups the charge carrier transport<br />

is limited by the localized nature of the electronic states <strong>and</strong> by the hopping mechanism<br />

of the transport process.<br />

Based on recent other works [8, 9], we think that for developing even higher mobility<br />

materials it is essential to improve the structural order of the material. This may open new<br />

application fields, e. g. the use of active organic materials in semiconductor devices. One<br />

promising way is the use of highly ordered liquid crystals. With these materials, however,<br />

charge carrier mobilities up to 0.1 cm 2 /Vs or even more are realistic, which would make<br />

them comparable to single crystalline systems [65].<br />

References<br />

1. C. F. Carlson: US Pat. 2297691 (1942)<br />

2. D. Haarer: Angew. Makromol. Chem., 183, 197 (1990)<br />

3. D.M. Pai, B.E. Springett: Rev. Mod. Phys., 65(1), 163 (1993)<br />

4. K.-W. Klüpfel, M. Tomanck, F. Endermann: German Patent No. DE-B-11 45184 (1963)<br />

5. H. Hoegl, O. Süs, W. Neugebauer, Kalle AG: German Patent No. DBP 1068115. Chem. Abstr., 55,<br />

20742 a (1961)<br />

6. H. Hoegl: J. Phys. Chem., 69(3), 755 (1965)<br />

7. G. Pfister: Phys. Rev. Lett., 36(5), 271 (1976)<br />

8. D. Adam, F. Closs, T. Frey, D. Funhoff, D. Haarer, H. Ringsdorf, P. Schuhmacher, K. Siemensmeyer:<br />

Phys. Rev. Lett., 70(4), 457 (1993)<br />

9. D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf,<br />

D. Haarer: Nature, 371, 141 (1994)<br />

10. H. Tung Li, P.J. Regensburger: J. Appl. Phys., 34, 1730 (1963)<br />

11. L. Onsager: Phys. Rev. B, 54, 554 (1938)<br />

12. H. Kaul, D. Haarer: Ber. Bunsenges. Phys. Chem., 91, 845 (1987)<br />

13. M. Gailberger, H. Bässler: Phys. Rev. B, 44(16), 8643 (1991)<br />

14. Y. Kanemitsu, Y. Sugimoto: Phys. Rev. B, 16(21), 14 182 (1992)<br />

15. Y. Sano, K. Kato, M. Yokoyama, Y. Shirota: H. Mikawa, Mol. Cryst. Liq. Cryst. 36(1–2), 137<br />

(1976)<br />

16. A. Peled, L.B. Schein, D. Glatz: Phys. Rev. B, 41(15), 10835 (1990)<br />

17. Y. Kanemitsu, H. Funada, Y. Masumoto: J. Appl. Phys., 71(1), 300 (1992)<br />

18. H. Bässler, P.M. Borsenberger: Chem. Phys., 177(3), 763 (1993)<br />

19. P.N. S<strong>and</strong>a, T. Takamori, D.B. Dove: J. Appl. Phys., 64(3), 1229 (1988)<br />

20. J.X. Mack, L.B. Schein, A. Peled: Phys. Rev. B, 39, 7500 (1989)<br />

21. Y. Kanemitsu, H. Funada, S. Imamura: J. Appl. Phys., 67(9), 4152 (1990)<br />

22. P.M. Borsenberger, L.T. Pautmeier, H. Bässler: Phys. Rev. B, 46(19), 12145 (1992)<br />

23. J. Mort, G. Pfister, S. Grammatica: Solid State Commun., 18, 693 (1976)<br />

24. P.M. Borsenberger, L.E. Contois, A.I. Ateya: J. Appl. Phys., 50(2), 914 (1979)<br />

25. S.J. Santos Lemus, J. Hirsch: Philos. Mag. B, 53(1), 25 (1986)<br />

26. D. Haarer, H. Meyer, P. Strohriegl, D. Naegele: Makromol. Chem., 192(3), 617 (1991)<br />

27. D.M. Pai, J.F. Yanus, M. Stolka, D. Renfer, W.W. Limburg: Philos. Mag., 48(6), 505 (1983)<br />

28. H. Bässler: Philos. Mag., 50(3), 347 (1984)<br />

29. G. Pfister: Phys. Rev. B, 16(8), 3676 (1977)<br />

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1 Model Systems for Photoconductive Materials<br />

30. P.M. Borsenberger: J. Appl. Phys., 68(12), 6263 (1990)<br />

31. P.M. Borsenberger L. Pautmeier, R. Richert, H. Bässler: J. Chem. Phys., 94(12), 8276 (1991)<br />

32. J.M. Pearson, M. Stolka: Polymer Monographs, Vol. 6: Poly(N-vinylcarbazole), Gordon <strong>and</strong> Breach<br />

Sci. Publ., New York (1981)<br />

33. W.D. Gill: J. Appl. Phys., 43, 5033 (1972)<br />

34. E. Müller-Horsche, D. Haarer, H. Scher: Phys. Rev. B, 35, 1273 (1987)<br />

35. F.C. Bos, D.M. Burl<strong>and</strong>: Phys. Rev. Lett., 58, 152 (1987)<br />

36. R.G. Kepler, J.M. Zeigler, L.A. Harrah, S.R. Kurtz: Phys. Rev. B, 35(6), 2818 (1987)<br />

37. J. Frenekl: Phys. Rev., 54, 647 (1938)<br />

38. K. Shimakawa, T. Okada, O. Imagawa: J. Non-Cryst. Solids, 114(1), 345 (1989)<br />

39. M. Abkowitz, M. Stolka: Solid State Commun., 78(4), 269 (1991)<br />

40. H. Kaul: PhD thesis, Universität Bayreuth (1991)<br />

41. E. Brynda, S. Nespurek, W. Schnabel: Chem. Phys., 175(2–3), 459 (1993)<br />

42. V. Cimrova, S. Nespurek, R. Kuzel, W. Schnabel: Synth. Met., 67(1–3), 103 (1994)<br />

43. A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.-P. Su: Rev. Mod. Phys., 60(3), 781 (1988)<br />

44. J. L. Brédas, R. Silbey (eds.): Conjugated Polymers, Kluwer Academic Publishers, Dordrecht<br />

(1991)<br />

45. J.H. Burroughes, D.D.C. Bradley, A.R. Brown, R.N. Marks, K. Mackay, R.H. Friend, P.L. Burns,<br />

A.B. Holmes: Nature, 347(6293), 539 (1990)<br />

46. F.W. Schmidlin: Phys. Rev. B, 16(6), 2362 (1977)<br />

47. H. Scher, E.W. Montroll: Phys. Rev. B, 12(6), 2455 (1975)<br />

48. J. Nool<strong>and</strong>i: Phys. Rev. B, 16(10), 4474 (1977)<br />

49. G. Pfister, H. Scher, Adv. in Phys, 27(5), 747 (1978)<br />

50. P.M. Borsenberger, L. Pautmeier, H. Bässler: J. Chem. Phys., 94(8), 5447 (1991)<br />

51. G. Schönherr, H. Bässler, M. Silver: Philos. Mag. B, 44(3), 369 (1981)<br />

52. H. Bässler, G. Schönherr, M. Abkowitz, D.M. Pai: Phys. Rev. B, 26(6), 3105 (1982)<br />

53. L. Pautmeier, R. Richert, H. Bässler: Philos. Mag. Lett., 59(6), 325 (1989)<br />

54. L. Pautmeier, R. Richert, H. Bässler: Synth. Met., 37, 271 (1990)<br />

55. H. Domes, R. Leyrer, D. Haarer, A. Blumen: Phys. Rev. B, 36(8), 4522 (1987)<br />

56. D. Stauffer: Introduction to Percolation Theory, Taylor <strong>and</strong> Francis, London (1985)<br />

57. H. Domes: PhD thesis, Universität Bayreuth (1988)<br />

58. H. Domes, R. Fischer, D. Haarer, R. Strohriegl: Makromol. Chem., 190, 165 (1989)<br />

59. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58(2), 101 (1988)<br />

60. H. Kaul: PhD thesis, Universität Bayreuth (1991)<br />

61. H. Suzuki, H. Meyer, J. Simmerer, J. Yang, D. Haarer: Adv. Mater., 5, 743 (1993)<br />

62. H. Suzuki, H. Meyer, S. Hoshino, D. Haarer: J. Appl. Phys. in press (1995)<br />

63. J. Kido, K. Nagai, Y. Okamoto, T. Skotheim: Appl. Phys. Lett., 59, 2760 (1991)<br />

64. C. Beginn, J.V. Grazulevicius, P. Strohriegl,J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195,<br />

2353 (1994)<br />

65. N. Karl: in: K. Sumino (ed.): Defect Control in Semiconductors, Elsevier Science Publishers,<br />

North Holl<strong>and</strong> (1990)<br />

66. P. Gomes da Costa, R.G. D<strong>and</strong>rea, E.M. Conwell: Phys. Rev. B, 47(4), 1800 (1993)<br />

67. G. Leising, K. Pichler, F. Stelzer: in: H. Kuzmany, M. Mehring, S. Roth, (eds.): Springer Series in<br />

Solid-State Sciences, Vol. 91: Electronic <strong>Properties</strong> of Conjugated Polymers III, Springer, Berlin<br />

(1989)<br />

68. H. Meyer: PhD thesis, Universität Bayreuth (1994)<br />

69. H. Meyer, D. Haarer, H. Naarmann, H.H. Hörhold: Phys. Rev. B, in press (1995)<br />

70. M. Onoda, D.H. Park, K. Yoshino: J. Phys. (London), Condens. Matter, 1(1), 113 (1989)<br />

14

2 Novel Photoconductive Polymers<br />

Jörg Bettenhausen <strong>and</strong> Peter Strohriegl<br />

2.1 Introduction<br />

Electrophotography is the only area in which the conductivity of sophisticated organic materials<br />

<strong>and</strong> polymers is exploited in a large scale industrial process today. Photoconductors are<br />

characterized by an increase of electrical conductivity upon irradiation. According to this definition<br />

photoconductive materials are insulators in the dark <strong>and</strong> become semiconductors if<br />

illuminated. In contrast to electrically conductive compounds photoconductors do not contain<br />

free carriers of charge. In photoconductors these carriers are generated by the action of<br />

light.<br />

The discovery of photoconductivity dates back to 1873 when W. Smith found the effect<br />

in selenium. Based on this discovery C. F. Carlson developed the principles of the xerographic<br />

process already in 1938.<br />

Photoconductivity in organic polymers was first discovered in 1957 by H. Hoegl, who<br />

found that poly(N-vinylcarbazole) (PVK) <strong>and</strong> charge transfer complexes of PVK with electron<br />

acceptors like 2,4,7-trinitrofluorenone act as photoconductors [1].<br />

Besides the application of photoconductive polymers in photocopiers these materials<br />

are also widely used in laser printers in the last years. The third area in which photoconductors<br />

are applied is the manufacturing of electrophotographic printing plates.<br />

The organic photoconductors used in practice are based on two types of systems. The<br />

first one are polymers in which the photoconductive moiety is part of the polymer, for example<br />

a pendant or in-chain group. The second group involves low molecular weight compounds<br />

imbedded in a polymer matrix. These so-called moleculary doped polymers are<br />

widely used today.<br />

One interesting class of photoconductive materials are oxadiazoles. It is known for a<br />

long time that derivatives of 1,3,4-oxadiazole are good photoconductors. Compound 1 for<br />

example is described in patents <strong>and</strong> was frequently applied in photocopiers [2].<br />

1 2<br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1<br />

15

2 Novel Photoconductive Polymers<br />

The oxadiazole 1 is a hole transport material since the electron withdrawing effect of<br />

the oxadiazole group with three electronegative heteroatoms is overcompensated by two<br />

electron donating amino groups.<br />

Within the last years, oxadiazoles like 2-(biphenyl)-5-(4-tert.butylphenyl)-1,3,4-<br />

oxadiazole (PBD) 2 have been frequently applied in organic light emitting diodes [3]. Here<br />

the electron withdrawing oxadiazole unit dominates the electronic properties <strong>and</strong> the oxadiazole<br />

compounds act as electron injection <strong>and</strong> transport layers. Furthermore, 2,5-diphenyloxadiazoles<br />

have been used as building blocks in thermostable polymers <strong>and</strong> they are highly<br />

fluorescent as well.<br />

In this paper the synthesis <strong>and</strong> characterization of a number of novel low molecular<br />

weight oxadiazole derivatives <strong>and</strong> polymers is described. The compounds show different molecular<br />

shapes, e. g. rod-like <strong>and</strong> star-shaped structures have been realized.<br />

In addition to the compounds described here, a series of different main chain <strong>and</strong> side<br />

group polymers with oxadiazole moieties are presently synthesized in our research group [4].<br />

2.2 Liquid crystalline oxadiazoles <strong>and</strong> thiadiazoles<br />

2.2.1 The basic idea<br />

The aim of this work is the preparation of photoconductive compounds with high carrier mobilities.<br />

There are a number of indications that the carrier mobility, i. e. the speed of the<br />

charge particles within the sample, depends on the geometric arrangement of the photoconductive<br />

moieties.<br />

At room temperature for instance, the mobility in single crystals of aromatic compounds<br />

like anthracene or perylene is very high, i. e. in the range of 10 –1 cm 2 /Vs [5]. In<br />

amorphous polymers the carrier mobilities are orders of magnitude smaller <strong>and</strong> typical values<br />

are in the range of 10 –6 <strong>and</strong> 10 –8 cm 2 /Vs [6].<br />

So we were interested to investigate if the higher order in the liquid crystalline state<br />

leads to higher mobilities in comparison to less ordered amorphous solids. In liquid crystalline<br />

polymers a macroscopic orientation of the photoconductive groups in the mesophase<br />

can be achieved by means of electric or magnetic fields. The orientation can be frozen in by<br />

lowering the temperature below the glass transition. By this, materials with a degree of order<br />

between a perfect single crystal <strong>and</strong> totally disordered amorphous polymers are obtained.<br />

With this in mind we started the synthesis of a series of liquid crystalline model compounds<br />

<strong>and</strong> polymers in which the side groups possess both mesogenic <strong>and</strong> photoconductive<br />

properties.<br />

16

2.2 Liquid crystalline oxadiazoles <strong>and</strong> thiadiazoles<br />

2.2.2 Monomer synthesis<br />

In the past years we have synthesized a variety of compounds with oxadiazole <strong>and</strong> thiadiazole<br />

moieties [7]:<br />

3 X=S 4 X=O<br />

The rod-like thiadiazole derivative 3 is liquid crystalline <strong>and</strong> shows a smectic as well<br />

as a nematic LC phase with the following phase behaviour: c 103 s a 180 n 203 i. An acrylate<br />

monomer was prepared by hydroboration of the terminal double bond <strong>and</strong> subsequent<br />

esterification with acryloyl chloride <strong>and</strong> then polymerized. The resulting polymer 5 exhibits<br />

a mesophase which has not yet been identified.<br />

5<br />

Unfortunately the clearing point of the polymer is at 246 8C, where the material starts<br />

to decompose. Therefore orientation in an electric field was not possible.<br />

In contrast to the thiadiazole the oxadiazole 4 is neither liquid crystalline as monomer<br />

nor as polymer. The reason for the lack of mesogenic properties is that the substitution of<br />

the sulphur by oxygen introduces a bend into the molecule which prevents the formation of<br />

a LC phase.<br />

Recently liquid crystalline oxadiazoles (Scheme 2.1) have been described [8]. Here<br />

the oxadiazole is coupled to at least two benzene or cyclohexane rings. Thereby an increase<br />

of the mesophase stability is achieved.<br />

Scheme 2.1: Liquid crystalline oxadiazole compounds [8].<br />

17

2 Novel Photoconductive Polymers<br />

6 7<br />

8<br />

9<br />

10-13<br />

10 R=Ph-OC 6<br />

H 13<br />

12 R = Ph-N(CH 3<br />

) 2<br />

11 R=C 7<br />

H 15<br />

13 R = Ph-N(CH 3<br />

)C 6<br />

H 13<br />

Scheme 2.2: Monomer synthesis.<br />

Our aim was to synthesize a series of different oxadiazole monomers which are functionalized<br />

for the preparation of polymers. Scheme 2.2 shows the synthesis of the monomeric<br />

1,3,4-oxadiazoles. The oxadiazole moiety is formed by a cyclisation reaction of the<br />

diacylhydrazine derivatives 8 with phosphorous oxychloride. The unsymmetrical bishydrazides<br />

8 are prepared by treatment of benzoyl chlorides 6 with the appropriate monohydrazides<br />

7. The last reaction step is an esterification with 4-(5-hexenyloxy)benzoyl chloride.<br />

The esterification reaction is essential in this case because of two reasons. First the rod-like<br />

18

2.2 Liquid crystalline oxadiazoles <strong>and</strong> thiadiazoles<br />

mesogen is formed, <strong>and</strong> second a functional group for the preparation of LC polymers is introduced.<br />

Additionally the oxadiazole monomer 14 was synthesized, in which the ester group is<br />

replaced by a biphenyl unit. By this it is possible to investigate the influence of the ester<br />

function on the photoconductive properties of the liquid crystalline compounds.<br />

14<br />

The liquid crystalline behaviour <strong>and</strong> the phase transitions of the monomeric oxadiazole<br />

derivatives 10–14 were determined by DSC <strong>and</strong> polarizing microscopy. In Tab. 2.1 the<br />

phase transition temperatures are summarized:<br />

Table 2.1: Phase behaviour of the monomeric oxadiazoles 10–14<br />

Compound<br />

Transition temperature in 8C<br />

10 k 141 n 167 i<br />

11 k 90 s a 116 i<br />

12 k 162 n 179 i<br />

13 k 127 (s a 111) i<br />

14 k 72 s a 77 i<br />

Except compound 13 all monomers are enantiotropic liquid crystalline <strong>and</strong> show nematic<br />

or smectic mesophases. Only derivative 13 shows a monotropic s a phase.<br />

2.2.3 Oligo <strong>and</strong> polysiloxanes with pendant oxadiazole groups<br />

The next step was the synthesis of liquid crystalline polysiloxanes with oxadiazole groups in<br />

the mesogenic unit. The polymers were prepared by a polymeranalogous reaction of the<br />

monomers 10 <strong>and</strong> 11 with poly(hydrogenmethylsiloxane) 15.<br />

Both polymers are liquid crystalline. The transition into the isotropic phase takes place<br />

at about 200 8C <strong>and</strong> therefore at a much higher temperature than in the monomers. The high<br />

clearing temperature makes the orientation of the polymers difficult which is preferably carried<br />

out near the clearing point. Up to now it was not possible to prepare well-defined polymers<br />

with the monomers 12–14. A possible reason is the inactivation of the Pt catalyst by<br />

the amino groups.<br />

If the cyclic tetrasiloxane 18, which is commercially available in high purity, is used<br />

instead of 15, well-defined monodisperse model compounds are obtained [9]. These com-<br />

19

2 Novel Photoconductive Polymers<br />

10, 11 15<br />

16 R=Ph-OC 6<br />

H 13<br />

17 R=C 7<br />

H 15<br />

Scheme 2.3: Synthesis of polysiloxanes with pendant oxadiazole groups.<br />

pounds can be highly purified, for example by preparative gel permeation chromatography.<br />

This is very important for photoconductors, because it is well-known that even traces of impurities<br />

may reduce the carrier mobility.<br />

18<br />

So the monomeric compounds 10 <strong>and</strong> 12–14 have been reacted with 18 to yield the<br />

following tetrameric derivatives:<br />

19 R=Ph-OC 6<br />

H 13<br />

,n=1 21 R = Ph-N(CH 3<br />

)C 6<br />

H 13<br />

,n=<br />

20 R = Ph-N(CH 3<br />

) 2<br />

,n=1 22 R = Ph-N(CH 3<br />

)C 6<br />

H 13<br />

,n=<br />

Scheme 2.4: Tetrasiloxanes with pendant oxadiazole groups.<br />

20

2.2 Liquid crystalline oxadiazoles <strong>and</strong> thiadiazoles<br />

All cyclosiloxanes are liquid crystalline <strong>and</strong> show several advantages compared to the<br />

polymers. So the cyclic siloxanes form stable glasses but their clearing points are much<br />

lower. For example compounds 21 <strong>and</strong> 22 possess clearing temperatures at only 102 8C.<br />

The phase behaviour of the tetramers are listed in Tab. 2.2.<br />

Table 2.2: Phase behaviour of the tetrameric siloxanes 19–22.<br />

Comp.<br />

Transition temperature in 8C<br />

19 g 54<br />

a)<br />

m 1 108<br />

a)<br />

m 2 117 s c 155 n 185 i<br />

20 g 67 s a 138 n 181 i<br />

21 g 34 k 66 n 102 i<br />

22 g 38 s a 82 n 102 i<br />

a) Mesophase not identified<br />

2.2.4 Photoconductivity measurements<br />

For the characterization of the novel photoconductive compounds two different experimental<br />

techniques have been used. Some measurements were made by BASF AG using a steady-state<br />

method. These experiments allow a quick characterization, but in contrast to the time-of-flight<br />

(TOF) method no statements about transient photocurrents <strong>and</strong> carrier mobilities are possible.<br />

The TOF measurements were carried out in the group of D. Haarer, Universität Bayreuth.<br />

The tetrameric cyclosiloxanes are very suitable compounds for physical investigations.<br />

Because of the low viscosity the preparation of samples is much easier than in the case of<br />

the polymers. Additionally the dark currents of the tetramers are very low.<br />

All the compounds are photoconductive, as shown by the steady-state method. For example,<br />

the measurements of the thiadiazole 3 showed a distinct rise of the photocurrent at<br />

the transition from the crystalline to the mesophase, as shown in Fig. 2.1. The decrease of<br />

Figure 2.1: Temperature dependence of the photocurrent of the thiadiazole 3.<br />

21

2 Novel Photoconductive Polymers<br />

the current at the transition from the smectic to the nematic phase is directly correlated with<br />

the loss of order. These results show that the basic idea of this work is correct.<br />

Nevertheless, we were not able to carry out TOF measurements with calamitic monomers,<br />

because the dark currents in the liquid crystalline phases were too high. As mentioned<br />

before, such measurements are possible with the cyclic tetramers. The transient photocurrents<br />

of the tetrasiloxanes 19, 21, 22 illustrate that the carrier transport is totally dispersive,<br />

i. e. dominated by deep traps in which the charge carriers are captured. This is a typical behaviour<br />

of amorphous polymers. No transit time could be detected <strong>and</strong> no statements about<br />

the carrier mobility can be made for the rod-like mesogens.<br />

In contrast it was shown in the past years that discotic liquid crystals like hexapentyloxytriphenylene<br />

(HPT) 23 have carrier mobilities up to 10 –3 cm 2 /Vs in the liquid crystalline<br />

D ho phase [10]. Here, the disc-shaped molecules are ideally stacked above each other<br />

<strong>and</strong> therewith allow a fast carrier transport. The discotic hexathioether 24 with a highly ordered<br />

helical columnar (H) phase exhibits mobilities up to 10 –1 cm 2 /Vs, which are almost as<br />

high as in organic single crystals [11].<br />

6<br />

6<br />

23 24<br />

One important observation during the photoconductivity measurements of the tetramer<br />

19 was, that it showed almost the same photocurrent for holes <strong>and</strong> electrons. This was interesting<br />

because of the lack of electron transporting materials <strong>and</strong> led us synthesize a variety<br />

of starburst oxadiazole compounds, which are described in the next Chapter.<br />

2.3 Starburst oxadiazole compounds<br />

2.3.1 Motivation<br />

The synthesis of novel materials with high carrier mobilities is one of the major goals in the<br />

field of photoconductive polymers. Within the last years different approaches have been pursued<br />

to reach this goal. First, the photoconductive properties of conjugated polymers like<br />

poly(phenylenevinylene) <strong>and</strong> poly(methyl phenylsilane) have been investigated [12]. Another<br />

approach are liquid crystals which are the topic of the first Chapter. The third way to realize<br />

22

2.3 Starburst oxadiazole compounds<br />

the goal are glasses of large extended aromatic amines. The best investigated representatives<br />

of this class of compounds are N,N'-diphenyl-N,N'-bis(3-methylphenyl)-[1,1'-biphenyl]-4,4'-<br />

diamine (TPD) 25 from Xerox [13] <strong>and</strong> 1,1-bis(di-4-tolyl-aminophenyl)cyclohexane (TAPC)<br />

26 from Kodak [14].<br />

Both compounds are derivatives of triphenylamine, a well-known photoconductor. If<br />

thin films of TPD <strong>and</strong> TAPC are prepared by vacuum evaporation both compounds form<br />

metastable glasses. In such glasses carrier mobilities up to 10 –3 cm 2 /Vs for TPD [13] <strong>and</strong><br />

10 –2 cm 2 /Vs for TAPC [14] have been reported. But both TAPC <strong>and</strong> TPD glasses are metastable<br />

<strong>and</strong> have a strong tendency to crystallize. If the molecules are imbedded in a polymer<br />

matrix, e. g. polycarbonate or polystyrene, morphologically stable materials are formed, but<br />

the mobilities decrease drastically [13].<br />

25 26<br />

Several attempts have been made to overcome the problems with metastable TPD <strong>and</strong><br />

TAPC glasses. Recently we described the synthesis of 3,6-bis[(9-hexyl-3-carbazolyl)ethynyl]-9-hexylcarbazole<br />

27, a trimeric model compound of poly(carbazolylene ethynylene)<br />

[15]. This material shows a glass transition temperature of 41 8C <strong>and</strong> forms glasses which<br />

are stable for more than a year. Mobilities up to 10 –4 cm 2 /Vs (E =67 10 5 V/cm, T =308C)<br />

have been measured by the time-of-flight technique.<br />

27<br />

Another interesting approach are starburst compounds with high glass transition temperatures.<br />

So 4,4',4@-tris(N-carbazolyl)triphenylamine, which has been published recently<br />

[16], shows a glass transition at 151 8C <strong>and</strong> forms a stable glass.<br />

Beginning with the work of Thomalia [17] in 1986, starburst molecules (dendrimers)<br />

have achieved enormous interest within the last years. Dendrimers are highly branched regular<br />

molecules, which are usually prepared by stepwise reactions. In many cases the behaviour<br />

of dendritic macromolecules are different from that of linear polymers, e. g. the former<br />

23

2 Novel Photoconductive Polymers<br />

usually show enhanced solubility. The reasons for these differences are the unique three-dimensional<br />

structure <strong>and</strong> the large number of chain ends in dendrimers.<br />

Two different methods have been developed for the stepwise synthesis of starburst<br />

dendrimers:<br />

a) the divergent approach, where the synthesis starts from a core molecule with two or<br />

more reactive groups;<br />

b) the convergent approach, in which the synthesis starts at the outer sphere of the dendrimer.<br />

In the divergent approach, the reaction of the core molecule with two or more reagents<br />

containing at least two protected branching sites is followed by removal of the protecting<br />

groups <strong>and</strong> subsequent reaction of the liberated reactive groups which leads to the starburst<br />

molecule of the <strong>1st</strong> generation. The process is repeated until the desired size is reached. In<br />

the convergent approach the synthesis starts at what will become the outer surface of the<br />

dendrimer. Step by step large dendrimer arms are prepared <strong>and</strong> finally the completed arms<br />

are coupled to the core. Both methods produce well-defined large dendritic molecules whose<br />

structures are manifolds of the building blocks. They allow structural as well as functional<br />

group variation.<br />

On the other h<strong>and</strong>, hyperbranched polymers can be synthesized in a one-step reaction<br />

using highly functionalized monomers of the type A x B, where x is 2 or larger. This method<br />

does not yield polymers of such a well-defined structure, but has the advantage to provide<br />

rapidly large quantities of material. The control of the degree of branching is difficult <strong>and</strong><br />

mainly depends on statistics, steric effects, <strong>and</strong> the reactivity of the functional groups.<br />

The aim of our work is the synthesis of starburst compounds with oxadiazole moieties.<br />

In contrast to the triphenylamine <strong>and</strong> carbazole derivatives oxadiazoles are strong electron<br />

acceptors. Therefore their transport characteristics can be switched from hole transport materials<br />

like 2,5-(4-diethylaminophenyl)-oxadiazole 1, in which the electron withdrawing effect<br />

of the oxadiazole ring is overcompensated by the two electron donating amino groups,<br />

to electron transport materials like 2-(biphenyl)-5-(4-tert.butylphenyl)-oxadiazole (PBD) 2.<br />

Recently, Saito demonstrated that oxadiazoles in a polycarbonate matrix show electron transport<br />

with mobilities up to 10 –5 cm 2 /Vs [18]. Such electron transport materials are attractive<br />

for the use in copiers <strong>and</strong> in organic light-emitting diodes.<br />

2.3.2 Synthesis of starburst oxadiazole compounds<br />

Starburst oxadiazole compounds have been mentioned for the first time in a thermodynamic<br />

study of the structure-property-relationship in low molecular weight organic glasses [19,<br />

20]. Upon rapid cooling these compounds form glasses with T g s between 77 <strong>and</strong> 169 8C.<br />

We have used three different approaches for the preparation of starburst oxadiazole<br />

compounds, which are schematically shown in Fig. 2.2.<br />

The methods A <strong>and</strong> B can be compared to the divergent synthesis of dendrimers.<br />

Method A starts from a central core <strong>and</strong> involves the reaction of the acid chloride groups<br />

<strong>and</strong> a subsequent cyclisation with phosphorous oxychloride to the oxadiazole ring (Chap-<br />

24

O<br />

O<br />

Cl<br />

C<br />

C<br />

Cl<br />

O<br />

C<br />

Cl<br />

Method A<br />

Method B<br />

O<br />

H 2N NH C<br />

O<br />

O<br />

O<br />

O<br />

C<br />

NH<br />

NH<br />

C<br />

C<br />

NH NH C<br />

O<br />

O<br />

C<br />

NH<br />

NH<br />

C<br />

C<br />

N N<br />

N NH<br />

N<br />

N<br />

O<br />

N N O<br />

O<br />

N<br />

N<br />

=core<br />

= shell<br />

Method C<br />

X<br />

Z<br />

+<br />

Y<br />

X<br />

X<br />

Z<br />

Z<br />

X = OH, C CH Y = Br, F Z = O , C C<br />

=core<br />

=shell<br />

Figure 2.2: The different approaches to starburst oxadiazoles.<br />

25

2 Novel Photoconductive Polymers<br />

ter 1). The main difficulty in this case is that three functional groups must react in one step.<br />

Partially incomplete cyclisation causes problems, because the products contain one or two<br />

bishydrazides <strong>and</strong> are difficult to separate from the target compounds, which have three oxadiazole<br />

groups. Therefore we recently developed a totally different route (method B). The<br />

compounds 31 a–f were prepared by the reaction of benzene tricarbonyl chloride 30 with<br />

the tetrazoles 29. The latter were synthesized from benzonitriles 28 with potassium azide<br />

(Scheme 2.5). With the loss of nitrogen the tetrazole ring is transferred to the oxadiazoles<br />

[21]. This gives an easy access to unsymmetrically substituted oxadiazoles which we used<br />

for the first time in the synthesis of starburst oxadiazole compounds.<br />

In the third approach (method C) which is similar to the convergent synthesis a preformed<br />

oxadiazole precursor is prepared by stepwise synthesis <strong>and</strong> finally coupled to the<br />

core molecule.<br />

R 1<br />

R 2 CN + NaN 3<br />

NH 4 Cl<br />

R 1<br />

R 2<br />

R 3<br />

R 3<br />

28 29<br />

C<br />

N<br />

N<br />

NH<br />

N<br />

Cl<br />

O<br />

C<br />

O<br />

C<br />

Cl<br />

O<br />

C<br />

Cl<br />

+ 3<br />

R 2<br />

R 1<br />

R 3<br />

C<br />

N<br />

N<br />

NH<br />

N<br />

31a-f<br />

pyridine<br />

2h, reflux<br />

31a-f<br />

30 29<br />

Scheme 2.5: Synthesis of starburst oxadiazoles via tetrazole intermediates.<br />

For the preparation of the novel starburst molecules, different core molecules have<br />

been used: 1,3,5-benzene tricarboxylic acid chloride, 1,3,5-tris(4-benzoyl)benzene, 1,3,5-<br />

triethynylbenzene, <strong>and</strong> 4,4',4@trihydroxytriphenylamine.<br />

Except benzenetricarboxylic acid which is commercially available, all core molecules<br />

were synthesized following well-known literature procedures [22, 23].<br />

The structures of the starburst oxadiazole compounds are summarized in Tab. 2.3.<br />

31 c <strong>and</strong> 31g have been described before [19, 20], but no synthetisis procedure was given.<br />

The oxadiazoles 31a–h have been prepared according to method B (Fig. 2.3), starting<br />

from benzene tricarboxylic acid chloride.<br />

Compound 31g with tert.-butyl substituents was synthesized by method A, too. So a<br />

comparison of methods A <strong>and</strong> B is possible. The tetrazole route (method B) exhibits several<br />

advantages compared to the ring closure with dehydrating agents (method A). One advantage<br />

is the short reaction time. The reaction is finished within 2 hours whereas heating for<br />

1–3 days is necessary if the ring closure is carried out with phosphorous oxychloride. Even<br />

more important is the facile work up procedure. In the case of the dehydration with POCl 3<br />

column chromatography <strong>and</strong> subsequent recrystallization is necessary to purify the product<br />

which is finally obtained in 33% yield. In contrast only one filtration on a short silica gel<br />

26

2.3 Starburst oxadiazole compounds<br />

Table 2.3: Structures of the starburst oxadiazole compounds 31–34.<br />

Comp. Core a) Shell a)<br />

31<br />

N<br />

O<br />

N<br />

R 1<br />

R 3<br />

R 2<br />

31a R 1 =R 3 =H,R 2 =CH 3<br />

31b R 2 =R 3 =CH 3 ,R 1 =H<br />

31c R 1 =R 3 =CF 3 ,R 2 =H<br />

31d R 1 =R 3 =H,R 2 =C 2 H 5<br />

31e R 1 =R 3 =H,R 2 =OC 2 H 5<br />

31f R 1 =R 3 =H,R 2 = CH(CH 3 ) 2<br />

31g R 1 =R 3 =H,R 2 = C(CH 3 ) 3<br />

31h R 1 =R 3 =H,R 2 = N(C 2 H 5 ) 2<br />

N N<br />

32 R<br />

R = C(CH 3 ) 3<br />

O<br />

C<br />

C<br />

C C<br />

N N<br />

33 R<br />

R = C(CH 3 ) 3<br />

C<br />

C<br />

O<br />

N N<br />

34 N<br />

O<br />

R R = C(CH 3 ) 3<br />

O<br />

a) cf. Figure 2.2.<br />

column is sufficient for the purification of 31 g, prepared by the tetrazole route. In this case<br />

the yield of the pure product is 69%. The starburst oxadiazole 32 has been prepared by<br />

method B too, whereas for 33 <strong>and</strong> 34 method C (Fig. 2.2) was used. The identity of all compounds<br />

was confirmed by NMR <strong>and</strong> FTIR spectroscopy.<br />

2.3.3 Thermal properties<br />

The thermal behaviour of the starburst molecules has been investigated by differential scanning<br />

calorimetry <strong>and</strong> thermogravimetric measurements.<br />

27

2 Novel Photoconductive Polymers<br />

Table 2.4: Thermal properties of the starburst compounds 31–34.<br />

a)<br />

a)<br />

Compound Molecular mass T g T m<br />

[g/mol] [8C] [8C] [8C]<br />

31a 553 – 334 348<br />

31b 595 – 333 358<br />

31c 919 – 250 336<br />

31d 595 – 276 360<br />

31e 643 108 259 337<br />

31f 637 97 225 349<br />

31g 679 142 257 366<br />

31h 724 128 299 329<br />

32 907 165 297 409<br />

33 979 – 320 356<br />

34 1122 137 – 391<br />

a) 3rd heating, determined by DSC with 20 K/min<br />

b) onset of decomposition in nitrogen, thermogravimetric measurement with 10 K/min<br />

T dec<br />

b)<br />

Among the compounds 31 a–h with a small benzene core both crystalline <strong>and</strong> glass<br />

forming materials exist. So 31 a–d with small methyl, ethyl, or trifluoromethyl substituents<br />

show only melting points in the DSC experiment with cooling rates of 20 K/min. Naito <strong>and</strong><br />

Miura reported that it is possible to obtain glasses even with small methyl or trifluoromethyl<br />

substituents if the compounds are rapidly cooled with liquid nitrogen [19]. In contrast, the<br />

compounds 31 e with ethoxy, 31f with iso-propyl, 31g with tert-butyl substituents, <strong>and</strong> 31h<br />

with diethylamino groups form glasses upon cooling with 20 K/min in the DSC equipment.<br />

When the amorphous samples are heated again the glass transition appears first but on<br />

further heating the samples start to recrystallize <strong>and</strong> consequently show a melting point at<br />

higher temperatures. Compound 32 with the triphenylbenzene core behaves similar like<br />

31e–h. The most stable glasses are formed by 34 with the triphenylamine core. The DSC<br />

diagram of the novel glass forming oxadiazole compound is shown below. Upon both, heating<br />

<strong>and</strong> cooling, only a glass transition is observed at 137 8C (Fig. 2.3). In our DSC experi-<br />

Figure 2.3: DSC scan of the starburst oxadiazole compound 34, 3rd heating <strong>and</strong> cooling, with 20 K/min.<br />

28

2.3 Starburst oxadiazole compounds<br />

ments we never found any evidence for crystallization of 34. Even in the first heating no<br />

melting point is observed up to 350 8C. Consequently, transparent amorphous films are obtained<br />

from the starburst oxadiazole compound 34 by solvent casting.<br />

The thermal behavior of compound 33 is somewhat different. No reproducible DSC<br />

scans are obtained in subsequent heating-cooling cycles. We attribute this to thermal crosslinking<br />

of the triple bonds [24].<br />

The thermal stability has been monitored by thermogravimetric measurements. In<br />

most cases the onset of decomposition is in the range from 330–370 8C. The oxadiazole<br />

compound 32 with a triphenyl benzene core shows a somewhat higher thermal stability up<br />

to 410 8C.<br />

(CH 3 ) 3 C<br />

N<br />

O<br />

N<br />

O<br />

C(CH 3 ) 3<br />

N<br />

N O<br />

O<br />

N<br />

O<br />

N<br />

O<br />

N<br />

C(CH 3 ) 3<br />

34<br />

The starburst oxadiazole compounds are now being tested as electron injection <strong>and</strong><br />

transport layer in organic LEDs <strong>and</strong> as photoconductors. First tests of two layer LEDs with<br />

PPV show that the novel materials possess properties comparable to 2 but have the great advantage<br />

to show no recrystallization if thin films were made by spin-coating. We will report<br />

on these measurements in the near future.<br />

29

2 Novel Photoconductive Polymers<br />

References<br />

1. H. Hoegl, O. Süs, W. Neugebauer, Kalle AG: DBP 1068115, Chem. Abstr., 55, 20742 a (1961),<br />

H. Hoegl: J. Phys. Chem., 69, 755 (1965)<br />

2. W. Wiedemann: Chem.-Ztg., 106, 275 (1982) <strong>and</strong> references therein<br />

3. A.R. Brown, D.D.C. Bradley, J.H. Burroughes, R.H. Friend, N.C. Green-ham, P.L. Burn, A.B.<br />

Holmes, A. Kraft: Appl. Phys. Lett., 61, 2793 (1992)<br />

4. E. Buchwald, M. Meier, S. Karg, W. Rieß, M. Schwoerer, P. Pösch, H.-W. Schmidt, P. Strohriegl:<br />

Adv. Mater., 7, 839 (1995), M. Greczmiel, P. Pösch, H.-W. Schmidt, P. Strohriegl, E. Buchwald,<br />

M. Meier, W. Rieß, M. Schwoerer: Makromol. Symposia, 102, 371 (1996)<br />

5. W. Warta, R. Stehle, N. Karl: Appl. Phys., A36, 163 (1985)<br />

6. P.M. Borsenberger, D.S. Weiss: in: Organic Photoreceptors For Imaging Systems, M. Dekker, New<br />

York, (1993)<br />

7. R.G. Müller: PhD thesis, Bayreuth, (1992)<br />

8. D. Girdziunaite, C. Tschierske, E. Novotna, H. Kresse, A. Hetzheim: Liq. Cryst., 10, 397 (1991)<br />

9. F.H. Kreuzer, D. Andrejewski, W. Haas, N. Häberle, G. Riepl, P. Spes: Mol. Cryst. Liq. Cryst.,<br />

199, 345 (1991)<br />

10. D. Adam, F. Closs, T. Frey, D. Funhoff, D. Haarer, H. Ringsdorf, P. Schuhmacher, K. Siemensmeyer:<br />

Phys. Rev. Lett., 70, 457 (1993)<br />

11. D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf,<br />

D. Haarer: Nature, 371, 141 (1994)<br />

12. M. Gailberger, H. Bässler: Phys. Rev. B, 44, 8643 (1991), M.A. Abkowitz, M.J. Rice, M. Stolka:<br />

Phil. Mag. B, 61, 25 (1990)<br />

13. M. Stolka, J.F. Yanus, D.M. Pai: J. Phys. Chem., 88, 4707 (1984)<br />

14. P.M. Borsenberger, L. Pautmeier, R. Richert, H. Bässler: J. Phys. Chem., 94, 8276 (1991)<br />

15. C. Beginn, J.V. Grazulevicius, P. Strohriegl, J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195,<br />

2353 (1994)<br />

16. Y. Kuwabara, H. Ogawa, H. Inoda, N. Noma, Y. Shirota: Adv. Mater., 6, 677 (1994)<br />

17. D.A. Tomalia, H. Baker, J. Dewald, M. Hall, G. Kallos, S. Martin, J. Roeck, J. Ryder, P. Smith:<br />

Macromolecules, 19, 2466 (1986)<br />

18. H. Tokuhisa, M. Era, T. Tsutsui, S. Saito: Appl. Phys. Lett., 66, 3433 (1995)<br />

19. K. Naito, A. Miura: J. Phys. Chem., 97, 6240,(1993)<br />

20. S. Egusa, Y. Watanabe: EP 553950 A2 to Toshiba Corp.<br />

21. R. Huisgen, J. Sauer, H.J. Sturm, J.H. Markgraf: Chem. Ber., 93, 2106 (1960)<br />

22. E. Weber, M. Hecker, E. Koepp, W. Orlia, M. Czugler, I. Csöregh: J. Chem. Soc., Perkin Trans. II,<br />

1251 (1988)<br />

23. S.J.G. Linkletter, G.A. Pearson, R.I. Walter: J. Am. Chem. Soc., 99, 5269 (1977)<br />

24. S.-C. Lin, E.M. Pearce: in: High-Performance Thermosets – Chemistry, <strong>Properties</strong>, Applications,<br />

Carl Hanser Verlag, Munich, 137 (1994)<br />

30

3 Theoretical Aspects of Anomalous Diffusion<br />

in Complex Systems<br />

Alex<strong>and</strong>er Blumen<br />

3.1 General aspects<br />

Many relaxation phenomena in the solid phase depend on diffusion, which is usually treated<br />

in the framework of statistical methods. Regular diffusion, known as Brownian motion, is<br />

characterized by a linear increase of the mean-squared displacement with time. On the other<br />

h<strong>and</strong>, for a whole series of phenomena this simple relation does not hold; their temporal<br />

evolution of the mean-squared displacement is non-linear <strong>and</strong> thus obeys at long times:<br />

<br />

r 2 <br />

…t† t<br />

<br />

…1†<br />

with g 0 1. Relation 1 is referred to as anomalous diffusion. In the case that g < 1 one denotes<br />

the behaviour as subdiffusive. A subdiffusive pattern of motion often results from disorder<br />

[1–4]. One has to note, however, that the asymptotic law, Eq. 1, emerges only when<br />

the disorder influences the motion on all scales. In the case that g > 1 the motion is termed<br />

superdiffusive. A classical example for superdiffusive behaviour is furnished by the motion<br />

of particles in a turbulent flow. In this paper we focus on several models for anomalous diffusion<br />

which involve polymeric systems.<br />

Now, the mean-squared displacement is a basic characteristic feature for the motion<br />

but, as an averaged quantity, it can provide only restricted information about the basic microscopic<br />

mechanisms involved. In several works we have also studied the propagator P(r,t),<br />

the probability to be at r at time t having started at the origin at t = 0. Space limitations prevent<br />

us from going into details. Here we focus on Ar 2 (t)S <strong>and</strong> refer to Klafter et al. <strong>and</strong> Zumofen<br />

et al. [4, 5] for in-depth analyses of the propagator.<br />

From the beginning, the role of time-dependent aspects in disordered media has to be<br />

emphasized. In usual r<strong>and</strong>om walk problems it is often assumed that the disorder is<br />

quenched, so that the dynamics evolves over a static substrate, i. e. the geometrical or the energetical<br />

disorder is frozen in. However, it is also possible that the dynamical processes are<br />

directly under the influence of temporal (possibly medium induced) fluctuations.<br />

First, in Section 3.2 we concentrate on photoconductivity, whose canonical description<br />

involves the continuous time r<strong>and</strong>om walk (CTRW) approach [1, 4, 6–8]. Basic ingredients<br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1<br />

31

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

in CTRWs are waiting time distributions with long time-tails. For photoconductivity one<br />

usually has g 1 shows up.<br />

An interesting dynamical feature is observed when the object to move has itself an internal<br />

structure; that is the case, for instance, for a polymer. In this case, because of the connections<br />

which exist between different segments of the polymer, the motion gets a very rich<br />

structure. The details of this situation will be discussed in Section 3.4.<br />

<strong>Final</strong>ly, we close in Section 3.5 with some conclusions.<br />

3.2 Photoconductivity<br />

A major field of application of scaling ideas in the time domain is photoconductivity [1–3,<br />

7, 12]; here the use of polymeric photoconductors in printing <strong>and</strong> copying represents one of<br />

the most sophisticated applications for organic materials. Such materials are nowadays<br />

superior to their inorganic counterparts. Now, the issue at stake in amorphous photoconductors<br />

is the appearance of dispersive transport, as contrasted to the familiar diffusive Gaussian<br />

behaviour. One observes experimentally that the motion of the charge carriers becomes<br />

slower <strong>and</strong> slower with the passage of time, a situation mirrored by Eq. 1 with g t T one finds<br />

I t 1 : …4†<br />

For instance measurements on polysiloxanes with pendant carbazole groups [13] follow<br />

Eqs. 3 <strong>and</strong> 4 with g = 0.58 very well over four decades in time, see below. In fact this is<br />

32

3.2 Photoconductivity<br />

typical for g significantly lower than one. However, one should note that the behaviour of<br />

I (t) is complex, since for g > 1 in the long-time limit the transport behaviour is non-dispersive.<br />

This is related to the existence of a finite mean waiting-time t = g/(g – 1). It follows<br />

that around g = 1 there is a crossover from dispersive to non-dispersive behaviour [13], as is<br />

also confirmed experimentally.<br />

From the preceding discussion it is obvious that the transport of charge carriers is<br />

much influenced by the waiting-time distributions (WTD) between hops. Physically the<br />

WTD arise from the local disorder in the sample. For large disorder the WTD are not exponential<br />

but decay much more slowly. In line with the preceding arguments one takes for the<br />

WTD expressions c (t) which behave at long times algebraically [7, 12]<br />

…t† t 1 ; …5†<br />

where 0 < g < 1. This choice for c (t) reproduces Eqs. 2–4, see below. Equation 5 is not valid<br />

near the time-origin. Therefore in calculations one prefers to work with functions welldefined<br />

for t 6 0. A suitable choice is for instance the series<br />

…t† ˆ1 a<br />

a<br />

X 1<br />

nˆ1<br />

a n b n exp … b n t† …6†<br />

with a < 1 <strong>and</strong> b < a –1 [8]. This function is everywhere continuous <strong>and</strong> finite, <strong>and</strong> for purely<br />

imaginary t it turns into the Weierstrass function. One verifies readily that for large t Eq. 6<br />

obeys Eq. 5 <strong>and</strong> that the corresponding g in Eq. 5 is g =lna/ln b. Equation 6 is very useful,<br />

since it allows to vary g freely by a judicious choice of a <strong>and</strong> b. Another choice for c (t)<br />

which is continuous for t 6 0is<br />

…t† ˆ…1 ‡ t† :<br />

…7†<br />

Here again the long-time behaviour follows Eq. 5.<br />

We now recall the basic ingredients of r<strong>and</strong>om walks in continuous time, the so-called<br />

CTRW [6–8]. Let c (r, t) be the probability distribution of making a step of length r in the<br />

time interval t to t +dt. The total transition probability in this time interval is<br />

…t† ˆX<br />

r<br />

…r;t†:<br />

…8†<br />

Furthermore the survival probability at the initial site is<br />

…t† ˆ1<br />

R t<br />

0<br />

…† d;<br />

…9†<br />

so that, switching to the Laplace space (t ? u), one has<br />

…u† ˆ‰1 …u†Š=u: …10†<br />

33

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

The probability density Z(r,t) of just arriving at r in the time interval t to t +dt obeys<br />

the iterative relation<br />

…r;t†ˆP<br />

r 0<br />

R t<br />

0<br />

…r 0 ;† …r r 0 ;t † d ‡ …t† r;0 ; …11†<br />

in which the initial condition of starting at t = 0 from r = 0 is incorporated. One then has<br />

for the probability P (r,t), that the particle is at r at time t,<br />

P …r;t†ˆRt<br />

…r;t † …† d: …12†<br />

0<br />

Now P (r, t) also obeys an iterative relation:<br />

P …r;t†ˆX<br />

r 0<br />

R t<br />

0<br />

P …r 0 ;† …r r 0 ;t † d ‡ …t† r;0 …13†<br />

as may be seen either by inspection or, more formally, by using Eqs. 11 <strong>and</strong> 12. Clearly, a<br />

description of such convolutions, Eqs. 11–13, is more compact in Fourier-Laplace space.<br />

For P (k, u) one has from Eq. 13:<br />

P …k;u†ˆP …k;u† …k;u†‡ …u†<br />

…14†<br />

with the immediate solution<br />

P …k;u†ˆ …u†=‰ 1 …k;u† Š ˆ 1 …u†<br />

u<br />

1<br />

1 …k;u† : …15†<br />

The last expression generalizes the usual diffusion relation to r<strong>and</strong>om media, in which<br />

spatial <strong>and</strong> temporal aspects are coupled through c (k,u). The analysis is much simplified if<br />

such aspects decouple, which is the case for instance when the disorder is mainly energetic<br />

<strong>and</strong> the r<strong>and</strong>om walker moves over a rather regular lattice. In the decoupled case:<br />

…r;t†ˆ …r† …t† :<br />

…16†<br />

From this it follows immediately that also c (k,u) =l(k) c (u) is decoupled. In the decoupled<br />

scheme P (k, u) takes the form<br />

P …k;u†ˆ1 …u†<br />

u<br />

1<br />

1 …k† …u† ; …17†<br />

where l (k) = P qp(q)e –ik7q is the structure function of the infinite lattice <strong>and</strong> p (q) denotes<br />

the probability that a step extends over the distance q.<br />

34

3.2 Photoconductivity<br />

Coupling is very important for superlinear behaviour. Thus for the c (r,t) given by<br />

Eq. 16 the mean-squared displacement is either divergent or increases sublinearly or at most<br />

linearly in time. In order to obtain finite Ar 2 (t)S with a superlinear temporal behaviour,<br />

coupled c (r,t) forms have to be used [4]. An example is the WTD<br />

…r;t†ˆAr …r t †; …18†<br />

in which the d-function couples r <strong>and</strong> t. This WTD leads to Lévy walks.<br />

Let us now focus on the mean-squared displacement. Evidently, one has<br />

<br />

r 2 R<br />

…t† ˆ r 2 P …r;t† dr ˆ rk 2 P …k;t† j kˆ0<br />

…19†<br />

from which, in the decoupled scheme, using Eqs. 5 <strong>and</strong> 17 it follows that in the absence of<br />

any bias Eq. 2 is fulfilled. The procedure is more readily examplified by calculating the current<br />

from the mean displacement of the carrier Ar(t)S in a biasing field [8]. Setting L –1 for<br />

the inverse Laplace transform it follows:<br />

P<br />

r…t† ˆ rP …r;t†ˆir k P …k;t† kˆ0 ˆ L 1 … ir k P …k;u† j kˆ0 †<br />

r<br />

<br />

ˆ L 1<br />

@P…k;u†<br />

@<br />

<br />

ir k …k† j kˆ0<br />

ˆ1<br />

<br />

ˆ L 1 …u†<br />

u ‰ 1 …u† Š<br />

<br />

<br />

q ;<br />

where in the last line we work in the decoupled scheme <strong>and</strong> AqS = P q qp (q) is the mean displacement<br />

per hop. In the presence of a bias AqS 0 0. Thus, setting |AqS| = 1, we have for the<br />

current I (t) in an infinite lattice of any dimension,<br />

I …t† ˆ d <br />

r…t† <br />

<br />

dt ˆ L 1 …u†<br />

: …21†<br />

1 …u†<br />

For a finite chain of N sites Eq. 21 takes in the Laplace space the form [3, 13]:<br />

I…u† ˆXN<br />

nˆ1<br />

‰ …u† Š n ˆ<br />

…u†<br />

1 …u†<br />

…20†<br />

<br />

1 ‰ …u† Š N : …22†<br />

It is precisely this function which was used in Refs. [13, 14] to analyse, together with<br />

the WTD Eq. 6, the time-of-flight currents of photoconductive carriers in polysiloxanes with<br />

pendant carbazole groups. In this work one achieved with g = 0.58 a good agreement with<br />

the experimental findings. The short time behaviour indeed obeys Eq. 3, whereas at long<br />

times the form of Eq. 4 is reproduced fairly well.<br />

35

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

3.3 The Matheron-de-Marsily model<br />

As mentioned in the introduction, superdiffusive (enhanced) behaviour is often found in turbulent<br />

flows. In this Section we adopt a picture different from CTRW. We focus on the influence<br />

of r<strong>and</strong>omly distributed biasing external fields <strong>and</strong> follow the description of Oshanin<br />

<strong>and</strong> Blumen [11]. We take a three-dimensional (3D) solvent for which the flow is parallel to<br />

the Y-axis. The direction <strong>and</strong> magnitude of the flow depend only on the X coordinate of the<br />

position vector [9] so that V Y , the non-vanishing component of the flow field, obeys:<br />

V Y …X;Y;Z†ˆVX ‰ Š: …23†<br />

Here furthermore V[X] is a r<strong>and</strong>om function of X. Geometrically the system consists<br />

of parallel layers perpendicular to the X-axis. In each layer the value of V Y is constant but<br />

varies from layer to layer [9].<br />

We assume the r<strong>and</strong>om function V[X] of Eq. 23 to be Gaussian with zero mean,<br />

AV[X]S = 0, <strong>and</strong> with the covariance<br />

<br />

VX ‰ 1 ŠVX ‰ 2 Š ˆ … j X1 X 2 j†: …24†<br />

Here the brackets denote configurational averages, which are conveniently expressed<br />

through Fourier integrals,<br />

… jX 1 X 2 j† ˆ R1 1<br />

dwQ…w† exp ‰ iw…X 1 X 2 † Š: …25†<br />

Now many possibilities for Q(w) can be envisaged. For simplicity we take here only a<br />

flat spectrum, Q(w) =W/2p, as in the original Matheron-de-Marsily (MdM) model [9] in<br />

which the flows are delta-correlated,<br />

… jX 1 X 2 j† ˆ W …X 1 X 2 †: …26†<br />

We start from the Langevin dynamics of a single spherical bead subject to the MdM flow.<br />

This allows us to display enhanced (superlinear) diffusion in a simple situation [9, 10, 22]. The<br />

study of the dynamics of Rouse polymers in such flows is deferred to the next Section.<br />

Let R(t) be the position of the center of mass of the bead at time t, <strong>and</strong> we assume<br />

that R(0) = 0. The components X(t), Y(t) <strong>and</strong> Z(t) ofR(t) obey the following Langevin equations<br />

m d2 X<br />

dt 2 ˆ dX<br />

dt ‡ f X…t† ;<br />

m d2 Z<br />

dt 2 ˆ dZ<br />

dt ‡ f Z…t† ;<br />

…27†<br />

…28†<br />

36

3.3 The Matheron-de-Marsily model<br />

<br />

m d2 Y<br />

dt 2 ˆ dY<br />

dt<br />

<br />

VX ‰ Š<br />

‡ f Y …t† : …29†<br />

Here m denotes the mass of the bead <strong>and</strong> z the friction constant. The terms f X (t), f Y (t),<br />

<strong>and</strong> f Z (t) give the r<strong>and</strong>om (thermal-noise) forces exerted on the bead by the solvent molecules.<br />

These forces are Gaussian, with the moments<br />

f i …t† ˆ0<br />

…30†<br />

<strong>and</strong><br />

f i …t† f j …t 0 †ˆ2T i;j …t t 0 † ; …31†<br />

where i,j B {X,Y,Z}. The dash st<strong>and</strong>s for thermal averaging, d i,j is the Kronecker-delta <strong>and</strong><br />

the temperature T is measured in units of the Boltzmann constant k B .<br />

Conventionally the acceleration terms in Eqs. 27–29 are neglected, since they are<br />

small relative to the other terms [15]. This leads to<br />

dX<br />

dt ˆ f X…t† ;<br />

dZ<br />

dt ˆ f Z…t† ;<br />

dY<br />

dt ˆ V‰X…t†Š ‡ f Y…t† :<br />

…32†<br />

…33†<br />

…34†<br />

Note that in Eq. 34 the X <strong>and</strong> Y coordinates are coupled. Equations 32 <strong>and</strong> 33 are<br />

readily solved,<br />

X…t† ˆ 1 Rt<br />

df X …† ;<br />

Z…t† ˆ 1 Rt<br />

df Z …† :<br />

0<br />

0<br />

…35†<br />

…36†<br />

Thus the bead undergoes a conventional diffusive motion between the layers (along<br />

the X-axis) <strong>and</strong> in the Z direction. One sees it readily by evaluating, say:<br />

X 2 …t† ˆ 2Rt R<br />

d t<br />

1 d 2 f X … 1 † f X … 2 †ˆ2 1 T Rt R<br />

d t<br />

1 d 2 … 1 2 †ˆ2…T=† t; …37†<br />

0 0<br />

0 0<br />

so that X 2 …t† ˆ2D 1 t, where D 1 = T/z is the diffusion coefficient of a single bead.<br />

37

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

A similar procedure can be performed with respect to the solution of Eq. 34,<br />

Y…t† ˆRt<br />

0<br />

dV‰ X…† Š‡ 1 Rt<br />

df Y …† :<br />

0<br />

…38†<br />

The averaging involves now both, the thermal noise <strong>and</strong> the configurational disorder:<br />

Y 2 …t† ˆ 2 Rt R t<br />

d 1 d 2 f Y … 1 † f Y … 2 †‡ Rt R t<br />

d 1<br />

0<br />

0<br />

0<br />

0<br />

<br />

<br />

d 2 V‰X… 1 †Š V‰X… 2 †Š<br />

ˆ 2D 1 t ‡…W=2† Rt R t R 1<br />

d 1 d 2 dw exp ‰iw…X… 1 † X… 2 ††Š; …39†<br />

1<br />

0<br />

0<br />

where in the last line use was made of the representation Eqs. 25 <strong>and</strong> 26 with the flat spectrum<br />

of the delta-function. The remaining average on the rhs. of Eq. 39 is readily evaluated<br />

by remembering that it is the characteristic functional of the Brownian trajectory<br />

… 1 ; 2 ; w† exp‰iw…X… 1 † X… 2 ††Š ˆ exp w 2 D 1 j 1 2 j<br />

: …40†<br />

Inserting Eq. 40 into Eq. 39 one recovers the following result for the average squared<br />

displacement (ASD) in the direction of the flow field,<br />

<br />

Y 2 …t† ˆ 2D1 t ‡ 4W 3<br />

<br />

t 3 1=2<br />

: …41†<br />

p D 1<br />

One should remark that at times greater than t c =9pD 1 3 /(4W 2 ) the superlinear growth<br />

hY 2 …t†i t 3=2 in Eq. 41 dominates <strong>and</strong> the first, diffusive term can be neglected. One has<br />

then a superdiffusive behaviour with an exponent of g = 3/2 in Eq. 1 [9].<br />

We now turn to the analysis of the behaviour of polymers in MdM flow fields.<br />

3.4 Polymer chains in MdM flow fields<br />

The conformational properties <strong>and</strong> the dynamics of polymers in solutions under various<br />

types of flows have been a subject of considerable interest within the last decades. Much<br />

progress has been gained in the explanation of experimental data for systems in which the<br />

flow velocities are given functions in space <strong>and</strong> time, see Refs. [16–19]. On the other h<strong>and</strong>,<br />

the behaviour of polymers in r<strong>and</strong>om flows is less understood. In recent works [11] we<br />

(Oshanin <strong>and</strong> Blumen) succeeded in establishing analytically the behaviour of Rouse polymers<br />

[20] in MdM flow fields. The presentation here follows closely Ref. [11].<br />

38

3.4 Polymer chains in MdM flow fields<br />

In the Rouse model N monomers (beads) are coupled to each other via harmonic<br />

springs [16, 17, 20]. As is well-known, the forces are of entropic origin. It is customary to<br />

revert to a continuous picture in which n, the bead’s running number, takes real values. For a<br />

detailed discussion see Doi <strong>and</strong> Edwards [17]. The Langevin equations of motion for such a<br />

polymer in the MdM flow field are<br />

@X n…t†<br />

@t<br />

@Z n…t†<br />

@t<br />

@Y n…t†<br />

@t<br />

ˆ K @2 X n …t†<br />

@n 2 ‡ f x …n; t† ; …42†<br />

ˆ K @2 Z n …t†<br />

@n 2 ‡ f Z …n; t† ; …43†<br />

ˆ K @2 Y n …t†<br />

@n 2 ‡ V‰X N …t†Š ‡ f Y …n; t† ; …44†<br />

see Ref. [11]. Equations 42–44 are the generalization of Eqs. 27–29 to polymers. They are to<br />

be solved subject to the Rouse boundary conditions at the chain’s ends, n = 0 <strong>and</strong> n = N [17]:<br />

@X n …t†<br />

@n<br />

ˆ @Y n…t†<br />

@n<br />

ˆ @Z n…t†<br />

@n<br />

ˆ 0 :<br />

…45†<br />

As before, the fluctuating forces on the rhs. of Eqs. 42–44 are Gaussian <strong>and</strong> also<br />

delta-correlated with respect to the running index [11, 17]. For the X <strong>and</strong> Z components,<br />

which are not subject to the flow, the procedure is st<strong>and</strong>ard [17]. Say, for the averaged X<br />

component of the end-to-end vector one has<br />

P 2 X …t† ˆ…X 0…t† X N …t†† 2 ˆ b2 N<br />

; …46†<br />

3<br />

where b is the so-called persistence length.<br />

Furthermore, the X component of the radius of gyration is<br />

X 2 g ˆ 1<br />

2N 2 Z N<br />

0<br />

Z N<br />

0<br />

dndm …X n …t†<br />

X m …t†† 2 ˆ b2 N<br />

18 : …47†<br />

For isotropic situations (in the absence of flow fields) the end-to-end vector P R <strong>and</strong><br />

the radius of gyration R g of the polymer are related to P X <strong>and</strong> X g through P R 2 =3P X 2 <strong>and</strong><br />

R g 2 =3X g 2 . Because of the anisotropy one has to consider in the MdM model the different<br />

components separately.<br />

The dynamics of a flexible polymer chain is richer than that of a single bead. In the<br />

Rouse model the dynamics of X n (t) depends essentially on the time of observation t <strong>and</strong> on<br />

the Rouse time t R ,<br />

R ˆ b2 N 2<br />

3p 2 T : …48†<br />

39

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

Now t R is the largest internal relaxation time of the chain [17, 20]. Exemplarily for<br />

t P t R one finds for the mean-squared displacement of a bead of the chain, say the zeroth one,<br />

<br />

…X 0 …t† X 0 …0†† 2 ˆ 2b D 1=2<br />

1t<br />

: …49†<br />

3p<br />

Equation 49 is subdiffusive with g = 1/2 in Eq. 1. This is due to the fact that the trajectory<br />

of a bead in a chain is spatially confined by its neighbours. In the limit t p t R the<br />

chain diffuses as one entity <strong>and</strong> the bead’s trajectory follows mainly the motion of the<br />

chain’s center of mass. The chain’s center of mass obeys<br />

X 2 …t† ˆ2D R …N† t<br />

…50†<br />

with D R (N) =D 1 /N { T/(zN).<br />

Let us now turn to the Y component of the center of mass for which Eq. 44 leads readily<br />

to<br />

Y…t† 1 N<br />

so that we obtain<br />

Z N<br />

0<br />

dnY n …t† ˆ 1 Z t<br />

d<br />

N<br />

0<br />

Z N<br />

0<br />

<br />

dn VX ‰ n …† Š‡ 1 f Y …n; † ; …51†<br />

<br />

Y 2 …t† ˆ 2DR …N† t ‡ W Z t Z t Z 1<br />

d 1 d 2<br />

2N<br />

0 0 1<br />

dwg…w; 1 ; 2 †;<br />

…52†<br />

where g(w;t 1 ,t 2 ) denotes the dynamic structure factor of the chain,<br />

g…w; 1 ; 2 †ˆ 1<br />

N<br />

Z N Z N<br />

0<br />

0<br />

dn dm exp‰iw…X n … 1 † X m … 2 ††Š : …53†<br />

It turns out [11], that all these integrations can be performed analytically. In the longtime<br />

limit t p t R one finds<br />

<br />

Y 2 …t† ˆ 2DR …N† t ‡ 4W 3<br />

whereas for t P t R the short-time behaviour is given by<br />

40<br />

<br />

N 1=2<br />

t 3=2 <br />

1 O…t 1=2 <br />

† ; …54†<br />

pT<br />

p<br />

<br />

Y 2 …t† 2DR …N† t ‡ 3 W <br />

bN 1=2 t2 1 O…t 1=4 <br />

† : …55†

3.4 Polymer chains in MdM flow fields<br />

The interpretation of Eq. 54 is that at long times the t 3/2 dynamics dominates the picture;<br />

the Rouse chain behaves like a compact bead. At short times the term t 2 may become<br />

important. This g = 2 case in Eq. 1 is called ballistic; at very short times the center of mass<br />

of the chain hardly moves <strong>and</strong> it practically does not change the flow pattern to which it is<br />

subjected.<br />

We remark that at short times the motion of the center of mass of the chain <strong>and</strong><br />

the motion of a tagged bead are characterized by different dependences on time. The<br />

mean-squared displacement along the Y-axis of a tagged bead, say the zeroth one, grows<br />

in time as<br />

<br />

Y0 2…t†<br />

<br />

e2DR …N† t ‡ W 1=4<br />

D<br />

b1=2 1<br />

t 7=4 ; …56†<br />

see Ref. [11]. This g = 7/4 dependence is, of course, related to the fact that the segmental<br />

motion at short times is confined, the number of distinct flow layers visited by the bead<br />

growing as t 1/4 . Results of such fractal-type behaviour may be formulated exactly [21, 22].<br />

We now turn to the question of the elongation of the Rouse chain in the MdM flow<br />

<strong>and</strong> sketch the evaluation of the end-to-end distance along the Y-axis [11].<br />

The solution of Eq. 44 under the boundary conditions Eq. 45 has the form of a Fourier<br />

series,<br />

Y n …t† ˆY…0;t†‡2 X1<br />

pˆ1<br />

<br />

cos ppn <br />

Y…p; t† ;<br />

N<br />

where the Y(p,t), p = 0, 1, …, denote the normal coordinates [17]. At t = 0 the chain is assumed<br />

to be in thermal equilibrium, i. e. to have a Gaussian conformation. This can be accounted for<br />

automatically by stipulating it to be subject to the thermal fluctuations since t =–?. Furthermore,<br />

the MdM flow fields are switched on at t = 0. This leads to<br />

…57†<br />

Y…p; t† ˆ 1 Rt 1<br />

d exp… p 2 …t †= R † ~ f Y …p; †<br />

‡ 1 N<br />

Z t<br />

0<br />

d<br />

Z N<br />

0<br />

<br />

dn cos ppn <br />

V‰X n …t †Š exp… p 2 = R † ; …58†<br />

N<br />

where the functions ~ f Y denote the Fourier components of the thermal fluctuations [17]. The<br />

Y component of the end-to-end vector follows now from:<br />

P Y …t† ˆY 0 …t†<br />

Y N …t† ˆ2 P1<br />

… 1 … 1† p †Y…p; t† : …59†<br />

pˆ1<br />

After performing the averaging over both, the thermal fluctuations <strong>and</strong> the realizations<br />

of the flow fields, one obtains from Eqs. 58 <strong>and</strong> 59 in the long-time limit the equilibrium<br />

value of the end-to-end vector for delta-correlated flows [11],<br />

41

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems<br />

<br />

P 2 Y …1† b 2 <br />

N ˆ 1 ‡ CW2 bN 5=2 <br />

3<br />

T 2 ; …60†<br />

where C is a constant. Therefore, a Rouse polymer stretches along the Y-axis under MdM<br />

flows, taking the form of a prolate ellipsoid. The correction term to the Gaussian behaviour<br />

grows as N 7/2 .<br />

It is now interesting to confront this finding with the situation in simple shear flows,<br />

for which the result is [18, 19]:<br />

P 2 Y …t† ˆb2 N<br />

3<br />

<br />

1 ‡ 3<br />

6480<br />

_ 2 2 b 4 N 4<br />

T 2<br />

<br />

: …61†<br />

In Eq. 61 _ st<strong>and</strong>s for the (constant) shear rate. Because all flow lines point now in<br />

the same direction, the correction term shows a stronger N-dependence <strong>and</strong> obeys a N 5 law.<br />

3.5 Conclusions<br />

In this review several situations were presented, which lead naturally to the appearance of<br />

anomalous diffusion. Besides the already well-discussed subdiffusive behaviour, which is often<br />

seen in disordered media, we also considered superdiffusive dynamics, such as encountered<br />

in layered r<strong>and</strong>om flows. Anomalous diffusion was analysed through several models<br />

<strong>and</strong> we focussed on the behaviour of polymeric materials under such conditions. The basic<br />

aspect underlying these phenomena is dynamical scaling, which is often encountered experimentally<br />

<strong>and</strong> theoretically.<br />

On the other h<strong>and</strong> not all systems scale with time, <strong>and</strong> care is required in applying the<br />

models presented here; one has to be aware of intrinsic limitations of the scaling range (e. g.<br />

size limitations, other temporal scales involved). Because of this, close cooperations between<br />

experimentalists <strong>and</strong> theoreticians are much needed for the analysis of systems similar to the<br />

ones described here. Furthermore, despite the success of the now-closing Collaborative Research<br />

Center 213, much work still remains to be done.<br />

Acknowledgements<br />

The research collaboration with Prof. D. Haarer, Prof. J. Klafter, Dr. G. Oshanin, Dr. H.<br />

Schnörer, <strong>and</strong> Dr. G. Zumofen in our joint work reviewed here was always very helpful <strong>and</strong><br />

pleasant. The support of the Deutsche Forschungsgemeinschaft through the Sonder-<br />

42

References<br />

forschungsbereich 213 <strong>and</strong> Sonderforschungsbereich 60 was fundamental for the whole project.<br />

Additional help was provided by the Fonds der Chemischen Industrie <strong>and</strong> – in the later<br />

stages – by the PROCOPE-Program of the DAAD.<br />

References<br />

1. A. Blumen, J. Klafter, G. Zumofen: in: I. Zschokke (ed.): Optical Spectroscopy of Glasses, Reidel,<br />

Dordrecht, p. 199 (1986)<br />

2. D. Haarer, A. Blumen: Angew. Chem. Int. Ed. Engl., 27, 1210 (1988)<br />

3. A. Blumen, H. Schnörer: Angew. Chem. Int. Ed. Engl., 29, 113 (1990)<br />

4. G. Zumofen, J. Klafter, A. Blumen: in: R. Richert, A. Blumen. (eds.): Disorder Effects on Relaxational<br />

Processes: Glasses, Polymers, Proteins, Springer, Berlin, p. 251 (1994)<br />

5. J. Klafter, G. Zumofen, A. Blumen: J. Phys., A24, 4835 (1991)<br />

6. E.W. Montroll, G.H. Weiss: J. Math. Phys., 6, 167 (1965)<br />

7. H. Scher, M. Lax: Phys. Rev. B, 7, 4491; 4502 (1973)<br />

8. M.F. Shlesinger: J.Stat. Phys., 36, 639 (1984)<br />

9. G. Matheron, G. de Marsily: Water Resour. Res., 16, 901 (1980)<br />

10. G. Zumofen, J. Klafter, A. Blumen: Phys. Rev. A, 42, 4601 (1990)<br />

11. G. Oshanin, A. Blumen: Macromol. Theory Simul. 4, 87 (1995); G. Oshanin, A. Blumen: Phys.<br />

Rev. E, 49, 4185 (1994)<br />

12. H. Scher, E.W. Montroll: Phys. Rev. B, 12, 2455 (1975)<br />

13. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58, 101 (1988)<br />

14. D. Haarer, H. Schnörer, A. Blumen: Dynamical Processes in Condensed Molecular Systems, in: J.<br />

Klafter, J. Jortner, A. Blumen (eds.): World Scientific, Singapore, p. 107 (1989)<br />

15. M. Fixman: J. Chem. Phys., 42, 3831 (1965)<br />

16. P.G. de Gennes: in: Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, N.Y., (1979)<br />

17. M. Doi, S.F. Edwards: in: The Theory of Polymer Dynamics, Oxford Univ. Press, Oxford, (1986)<br />

18. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager: in: Dynamics of Polymeric Liquids, Vol.2,<br />

2nd Ed. Wiley, New York, (1987)<br />

19. W. Carl, W. Bruns: Macromol. Theory Simul., 3, 295 (1994)<br />

20. P.E. Rouse: J. Chem. Phys., 21, 1273 (1953)<br />

21. J.-U. Sommer, A. Blumen: Croat. Chem. Acta, 69, 793 (1996)<br />

22. G. Zumofen, J. Klafter, A. Blumen: J.Stat. Phys., 65, 991 (1991)<br />

43

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong><br />

Attenuation, Specific Heat <strong>and</strong> Thermal Conductivity<br />

in Polymers<br />

Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz, Junyun Li,<br />

<strong>and</strong> Frank Pobell<br />

We have measured the long time (t =5 h to 200 h) heat release of polymethylmethacrylate<br />

(PMMA) <strong>and</strong> polystyrene (PS) at 0.070 K ^ T ^ 0.300 K. After cooling from a temperature<br />

(the charging temperature) of 80 K the heat release in PMMA shows a t –1 -dependence<br />

in the measured time <strong>and</strong> temperature ranges in agreement with the tunneling model. In contrast,<br />

for PS we observe strong deviations from a t –1 -dependence <strong>and</strong> a heat release smaller<br />

than in PMMA in by a factor of ten, in apparent contradiction to specific heat <strong>and</strong> thermal<br />

conductivity data for PS.<br />

To compare the heat release with other low-temperature properties <strong>and</strong> to verify the consistency<br />

of the tunneling model we have measured also the acoustical properties (sound velocity<br />

<strong>and</strong> attenuation), the specific heat <strong>and</strong> the thermal conductivity of PMMA <strong>and</strong> PS in the<br />

temperature ranges 0.070 K ^ T ^ 100 K, 0.070 K ^ T ^ 0.200 K <strong>and</strong> 0.3 K ^ T ^ 4K,<br />

respectively. We show that the anomalous time dependence of the heat release of PS is due to<br />

the thermally activated relaxation of energy states with excitation energies above 15 K.<br />

4.1 Introduction<br />

A disordered material releases heat after cooling it from an equilibrium or charging temperature<br />

T 1 to a measuring temperature T 0 [1]. This heat release _Q (T 1 ,T 0 , t) depends on the<br />

charging temperature T 1 as well as on the temperature T 0 , at which the measurement is performed,<br />

<strong>and</strong> the elapsed time during <strong>and</strong> after cooling [2, 3]. The time-dependent heat release,<br />

observed in several disordered systems, is a consequence of the long-time relaxation<br />

of the low-energy excitations, identified as two-level tunneling systems (TS) [4–6]. As a<br />

consequence of the finite relaxation time of the TS through their interaction with thermal<br />

phonons [7] the specific heat depends also on the time scale of the experiment [5], as first<br />

measured by Zimmermann <strong>and</strong> Weber [1].<br />

In recent publications [5, 6] it was shown that within the tunneling model we can<br />

quantitatively underst<strong>and</strong> the observed temperature <strong>and</strong> time dependence of the specific heat<br />

44 Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

4.1 Introduction<br />

<strong>and</strong> heat release of vitreous silica (SiO 2 ) over ten orders of magnitude in time. It has also<br />

been pointed out that the measurements of the heat release at workable long times is probably<br />

the only feasible method to obtain at least approximately the low-energy limit of the<br />

tunnel splitting D 0 of the distribution function of TS. This low-energy limit is considered in<br />

the literature by the cut-off parameter u min =(D 0 /E) min of the distribution function (E is the<br />

energy splitting of the TS) <strong>and</strong> is introduced to keep the number of TS finite, avoiding divergences<br />

in the calculated properties.<br />

The interpretation of the heat release data in terms of the tunneling model is a difficult<br />

task due to the not well-known:<br />

a) temperature <strong>and</strong> time dependence of the specific heat due to TS at T >3K;<br />

b) influence of relaxation processes of the TS at T > 3 K other than one-phonon tunneling<br />

relaxation, i. e. high-order phonon tunneling <strong>and</strong> thermally activated relaxation;<br />

c) influence of the cooling procedure.<br />

In a recently published paper Parshin <strong>and</strong> Sahling [8] showed the complexity of the interpretation<br />

of the heat release data when thermally activated relaxation is taken into account<br />

within the framework of the soft potential model. Further theoretical work on the residual<br />

properties of two-level systems <strong>and</strong> its dependence on the cooling procedure has been published<br />

by Brey <strong>and</strong> Prados [9].<br />

In this paper we present a further example of the complexity in the interpretation of<br />

the heat release data <strong>and</strong> an experimental proof of the influence of the relaxation, probably<br />

by thermally activated processes, of excited states that contribute to the heat release of the<br />

TS at low temperatures. We have studied the long-time heat release of two amorphous polymers<br />

with similar low-temperature specific heats <strong>and</strong> thermal conductivities, cooled under<br />

similar conditions. In spite of those similarities we have found a large difference in the absolute<br />

value <strong>and</strong> in the time dependence of the heat release between the two polymers when<br />

cooled from temperatures above 15 K. The similarities <strong>and</strong> differences in the low-temperature<br />

properties, their interpretation within the tunneling model, <strong>and</strong> the influence of thermally<br />

activated relaxation are the main scope of this work. Preliminary results were published<br />

in Ref. [10].<br />

The paper is organized in five sections. In Section 4.2 we describe the phenomenological<br />

theory for the heat release, based on the tunneling model <strong>and</strong> the influence of different<br />

relaxation rates, <strong>and</strong> the cooling process. In Section 4.3 we briefly describe the experimental<br />

procedures <strong>and</strong> samples. In Section 4.4 we show <strong>and</strong> discuss the experimental results. Conclusions<br />

are drawn in Section 4.5.<br />

45

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

4.2 Phenomenological theory for the heat release<br />

4.2.1 Generalities<br />

New theoretical work for the calculation of the heat release within the soft potential model<br />

has been published recently [8]. In order to simplify the calculations, to minimize the number<br />

of free parameters, <strong>and</strong> to assure a more transparent interpretation of the results we<br />

decided, however, to interpret our results in terms of the st<strong>and</strong>ard tunneling model. According<br />

to Ref. [8] <strong>and</strong> to our numerical calculations the differences between the soft potential<br />

<strong>and</strong> st<strong>and</strong>ard tunneling model are not significant at the low temperatures of our measurements.<br />

The st<strong>and</strong>ard tunneling model, including a thermally activated relaxation of the twolevel<br />

systems, was used successfully to interpret the acoustical properties of vitreous silica<br />

in eight orders of magnitude in phonon frequency from approximately 0.1 K up to room<br />

temperature [11].<br />

For a system of N two-level systems with energy difference E, the difference in the population<br />

of the two levels at a given temperature T <strong>and</strong> at thermal equilibrium is given by<br />

n 0 ˆ N tanh…E=2k B T† :<br />

…1†<br />

If the thermodynamic equilibrium of the system is slightly perturbed, e. g. the system<br />

is rapidly cooled to a temperature T, the dynamical behaviour of the population difference<br />

can be calculated according to the relaxation time approximation formula:<br />

d…n…t†<br />

n 0 …T††<br />

dt<br />

ˆ n…t† n 0 …T†<br />

…E;T†<br />

; …2†<br />

where t(E,T) is the relaxation time of a tunneling system with energy E at a temperature T.<br />

The heat released by the N two-level systems after the temperature change is given by<br />

_Q ˆ _nE=2 :<br />

…3†<br />

The problem of calculating _Q simplifies to calculating n from Eq. 2. However, the<br />

cooling process has to be taken into account. In this case Eq. 2 must be rewritten as<br />

_n ˆ @n 0<br />

@T<br />

dT<br />

dt<br />

n n 0<br />

<br />

: …4†<br />

Equation 4 describes the response of two-level systems during a temperature change<br />

given by the function T(t). In the general case Eq. 4 has to be solved numerically since n 0<br />

<strong>and</strong> t are temperature-dependent variables.<br />

46

4.2 Phenomenological theory for the heat release<br />

4.2.2 The st<strong>and</strong>ard tunneling model with infinite cooling rate<br />

If N two-level systems are in thermal equilibrium at a temperature T 1 <strong>and</strong> they are cooled to T 0<br />

with infinite cooling rate, i. e. the time to cool the sample from T 1 to T 0 is zero, from Eq. 2 we<br />

obtain<br />

n…t† ˆ…n 0 …T 1 † n 0 …T 0 †† exp… t= …E;T 0 †† ‡ n 0 …T 0 † : …5†<br />

The st<strong>and</strong>ard tunneling model assumes that at low temperatures (T < 2 K) the<br />

one-phonon process is the dominating mechanism. The one-phonon relaxation rate is given<br />

by<br />

1<br />

p ˆ 1<br />

pm u2<br />

…6†<br />

with<br />

1<br />

pm ˆ AE3 coth…E=2K B T† ;<br />

…7†<br />

where A =(g 2 l /n 5 l +2g 2 t n 5 t )/2 pr –4 . The indices l, t refer to the longitudinal <strong>and</strong> transversal<br />

phonon branches, r is the mass density, g l <strong>and</strong> g t are the coupling constants between phonons<br />

<strong>and</strong> TS, <strong>and</strong> u=D 0 /E (D 0 is the tunneling splitting). For symmetrical TS, i. e. D 0 = E (u = 1),<br />

t p reaches its minimum value t pm .<br />

Furthermore, the st<strong>and</strong>ard tunneling model assumes that the distribution function of<br />

TS is constant in terms of two independent variables, namely the asymmetry D <strong>and</strong> the tunneling<br />

parameter l, i.e.<br />

P…;†dd ˆ Pdd:<br />

…8†<br />

According Ref. [7] the distribution function can be written in terms of the variables E<br />

<strong>and</strong> u as<br />

P…E;u† ˆ P=u…1 u 2 † 1=2 : …9†<br />

It is also convenient to have the distribution function in terms of the asymmetry <strong>and</strong><br />

the barrier height V between the potential wells. Following the work of Tielbürger et al. [11]<br />

<strong>and</strong> assuming two well-defined harmonic potentials, it can be shown that in a first approximation<br />

l = V/E 0 where E 0 represents the zero-point energy. In this case the distribution function<br />

is<br />

Pdd ˆ<br />

P<br />

E 0<br />

ddV :<br />

…10†<br />

Replacing the total number N of TS with the integrals in E <strong>and</strong> u, the heat release is<br />

given by<br />

47

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

_Q ˆ PV Z 1 Z<br />

E<br />

E<br />

1<br />

dEE tanh tanh<br />

E 0 2k B T 0 2k B T 1<br />

0<br />

u min<br />

du<br />

p<br />

u 1 u 2<br />

1 …T 0 † exp… t=…T 0 †† ;<br />

(11)<br />

where V is the sample volume <strong>and</strong> u min is the cut-off in the distribution function at u ?0.<br />

Figure 4.1 shows the heat release at T 0 = 0.1 K as a function of time for different<br />

charging temperatures T 1 following Eq. 11 <strong>and</strong> taking into account one-phonon relaxation<br />

rate, Eqs. 6 <strong>and</strong> 7. The calculation was performed with parameters appropriated for vitreous<br />

silica, Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 <strong>and</strong> u min =5610 –8 [5]. We observe that<br />

at short times <strong>and</strong> small T 1 the heat release _Q ! t –1 T 2 1. This dependence follows from the<br />

logarithmic time dependence of the specific heat <strong>and</strong> holds for tPt m /u 2 min. In this limit the<br />

heat release follows the often used approximation from Eq. 11 [1]:<br />

_Q<br />

p2<br />

24 k2 B PV…T 2 1 T 2 0 † 1 t ; …12†<br />

where _Q <strong>and</strong> P are measured in W <strong>and</strong> (Jg) –1 . We recognize in Fig. 4.1, however, that at<br />

longer times the theory deviates from the t –1 -dependence. This deviation comes from the<br />

exponential term in Eq. 11 that decreases strongly at long times <strong>and</strong> is determined by the<br />

product Au 2 min. The smaller this product the larger is the time-range where the approximation<br />

given by Eq. 12 holds. The influence of the cut-off u min can be recognized comparing the<br />

calculated heat release in Fig. 4.1 with that in Fig. 4.2 (dashed lines) calculated with a<br />

smaller u min .<br />

In Fig. 4.1 we note also that at large times the heat release becomes independent of<br />

T 1 . This feature was recognized experimentally in Refs. [2, 3] <strong>and</strong> was discussed in Ref. [6].<br />

This T 1 -independence within the assumptions described above is a direct consequence of the<br />

finite number of TS given by the cut-off u min . We should note, however, that in SiO 2 a saturation<br />

of the heat release for large charging temperatures (T 1 > 20 K) is observed <strong>and</strong> still<br />

a t –1 -dependence was measured [2]. This result cannot be explained within the st<strong>and</strong>ard tun-<br />

Figure 4.1: Heat release as a function of time t (in s) according to the st<strong>and</strong>ard tunneling model <strong>and</strong> infinite<br />

cooling rate with Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 <strong>and</strong> u min =5610 –8 at a measuring<br />

temperature of 0.1 K <strong>and</strong> at different charging temperatures T 1 , bottom: 5 K, top: 80 K.<br />

48

4.2 Phenomenological theory for the heat release<br />

Figure 4.2: Heat release as a function of time t (in s) within the st<strong>and</strong>ard tunneling model <strong>and</strong> infinite<br />

cooling rate. The parameters are the same as in the previous figure but with u min =5610 –9 (dashed<br />

lines) <strong>and</strong> an additional constant (energy-independent) relaxation time t TA =10 3 s (continuous lines).<br />

neling model. The reason for this discrepancy is the relaxation rate, which is different from<br />

the one-phonon process at higher temperatures <strong>and</strong> the influence of a finite cooling rate (see<br />

below).<br />

4.2.3 Influence of higher-order tunneling processes <strong>and</strong> a finite cooling rate<br />

At temperatures above 3 K tunneling processes of higher-order are, in principle, possible. One<br />

particular high-order process was studied theoretically in Ref. [12] <strong>and</strong> is similar to the optical<br />

Raman process; the relaxation time of the TS is given by an interaction that involves two phonons.<br />

Following Ref. [12] we added this Raman relaxation rate to the one-phonon relaxation<br />

rate (Eq. 6). Following Ref. [12] a new free parameter, i. e. the coupling constant R for the Raman<br />

process, is assumed. Due to its strong temperature dependence (T 7 ), the Raman process<br />

can influence the population difference of the energy states of the TS mainly at high temperatures<br />

(T > 5 K) during the cooling process. We should note that for infinite cooling rate <strong>and</strong> for<br />

the temperature range of our measurements (T 0^1 K) the contribution of the Raman process<br />

to the time <strong>and</strong> temperature dependence of the heat release is negligible. This follows from Eq. 5<br />

where only the relaxation of the TS at the measurement temperature T 0 enters.<br />

In order to take into account the cooling process we used the algorithm explained below.<br />

The zero-time point is chosen as the time when the cooling process is started. The function<br />

n (E,u,t) represents the difference in population of the TS energy states at any time t<br />

<strong>and</strong> n 0 (E,T) means this difference in equilibrium (t ? ?) at a temperature T. Analog to<br />

Eq. 11 the heat release can be written as:<br />

_Q ˆ 1<br />

2 PV<br />

Z 1<br />

0<br />

dEE<br />

Z 1<br />

u min<br />

du<br />

p 1 …T 0 † e 1 …T 0 † t …n 0 …E;T 0 † n…E; u; t ˆ 0†† …13†<br />

u 1 u 2<br />

49

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

Note that the population difference n depends on E <strong>and</strong> u because the relaxation rates,<br />

that influence the relaxation of the TS, depend on these variables. As already described<br />

above, to obtain the function n we need to solve the differential equation given by Eq. 4. In<br />

general this is only possible by numerical methods.<br />

The time dependence of the temperature during the cooling process performed in this<br />

work can be well approximated by a linear function given by<br />

T…t† ˆT 1 ‡ T 0 T 1<br />

t A<br />

: …14†<br />

We assume that the sample is at t =0(t A )atT 1 (T 0 ); t A is the time needed to cool the<br />

sample from T 1 to T 0 . The calculations have been done splitting the function T(t) inN steps<br />

of time Dt (Fig. 4.3). During the time Dt the temperature remains constant <strong>and</strong> n (E,u,t)<br />

shows an exponential behaviour given by Eq. 5; this is qualitatively shown in Fig. 4.3. To<br />

obtain the population difference at the time Dt cooling from T 1 at t = 0 for a given E <strong>and</strong> u<br />

we calculate iteratively the value<br />

n…t ˆ 0 ‡ t† ˆn 0 …0 ‡ t† …n 0 …0 ‡ t† n…0† e 1 …T…0‡t††t : …15†<br />

The calculations given in this work have been made taking into account 10 to 200<br />

time-steps or iterations depending on the convergence of the numerical results.<br />

Figure 4.4 shows the heat release as a function of time taking into account one-phonon<br />

(dashed lines) <strong>and</strong> the Raman process (Rk 7<br />

B = 100 s –1 K –7 ) with a cooling time<br />

t A = 3000 s from T 1 to T 0 = 0.1 K. Taking into account our experimental time scale the<br />

calculations were performed for the time interval 10 4 s ^ t ^ 10 6 s only. For one-phonon<br />

Figure 4.3: Time dependence of the sample temperature during a cooling process <strong>and</strong> its splitting in N<br />

steps (dashed lines) assumed for numerical calculations. Bottom: qualitative behaviour of the population<br />

difference of the tunneling systems n 0 (T) as a function of time.<br />

50

4.2 Phenomenological theory for the heat release<br />

Figure 4.4: Heat release as a function of time for a finite cooling time T A = 3000 s <strong>and</strong> with (continuous<br />

lines) or without (dashed lines) Raman processes.<br />

process only, our calculations indicate that a finite cooling rate with t A = 3000 s to 5000 s<br />

has a negligible influence (< 3%) on the heat release in our experimental time scale (compare<br />

the dashed lines in Fig. 4.4 with the dashed lines in Fig. 4.2 obtained with an infinite<br />

cooling rate).<br />

The influence of the Raman process at T 0 = 0.1 K can be well observed if we take<br />

into account a finite cooling rate. In comparison with the results using the one-phonon process<br />

only <strong>and</strong> for T 1 >20K, _Q is smaller <strong>and</strong> shows a slightly different time dependence.<br />

Note that the results for T 1 6 20 K resemble those obtained taking into account the one-phonon<br />

process only but with a smaller charging temperature T 1 . We note also that at T 1^ 20 K<br />

the heat release still shows a T 1 2 -dependence but reaches its saturation at a smaller T 1 in<br />

comparison with the results with the one-phonon process only (Fig. 4.4).<br />

We conclude that taking into account Raman processes, that influence the relaxation<br />

rate of TS at temperatures larger than 2 K, it is possible to underst<strong>and</strong> qualitatively the results<br />

of, for example, SiO 2 where _Q reached a saturation at T & 10 K but still shows a t –1 -<br />

dependence at t ^ 3610 5 s [2]. This result is only valid if the product Au 2 min in Eq. 7 remains<br />

small enough.<br />

Because of the uncertainty in the coupling constant R between TS <strong>and</strong> phonons a<br />

quantitative comparison, however, of the heat release results with the theory taking into account<br />

the contribution of higher-order processes is not useful. Phenomenologically <strong>and</strong> in order<br />

to simplify the calculations, one can take into account the influence of higher-order processes<br />

by choosing an appropriate smaller charging temperature than the real one. In that<br />

sense charging temperatures T 1 6 10 K can be considered for comparison with theory as<br />

free parameters. From our calculation we learn that it is practically impossible to obtain reliable<br />

information on the density of states of TS from heat release experiments for charging<br />

temperatures in the region of the T 1 -saturation of _Q.<br />

51

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

4.2.4 The influence of a constant <strong>and</strong> thermally activated relaxation rate<br />

There are different attempts to link the high-temperature (T > 10 K) with the low-temperature<br />

properties of disordered solids [13, 14]. In particular, it has been proposed that the maximum<br />

in the attenuation of phonons observed in amorphous materials at T > 10 K can be interpreted<br />

assuming a thermally activated relaxation of the two-level systems. In this Section<br />

we discuss the influence of thermally activated relaxation rate on the heat release for a finite<br />

cooling rate.<br />

For a single activation barrier V 0 between the two potential wells <strong>and</strong> at a measuring<br />

temperature T 0 the thermally activated relaxation rate is given by<br />

1<br />

TA ˆ 1<br />

0 e V 0=k B T 0<br />

: …16†<br />

Taking into account this relaxation rate in addition to quantum tunneling (Eq. 6) <strong>and</strong><br />

assuming that both processes are independent, the total relaxation rate can be written as<br />

1 ˆ 1<br />

P ‡ 1<br />

TA :<br />

…17†<br />

Obviously, at a given temperature T 0 <strong>and</strong> for a single activation barrier we add a constant<br />

relaxation rate to the tunneling process. Figure 4.2 shows the time dependence of the<br />

heat release at different temperatures T 1 <strong>and</strong> using Eq. 17 with t TA =10 3 s. As expected, at<br />

t P t TA the introduction of an additional constant relaxation rate does not influence the heat<br />

release. At t & t TA the heat release is larger than taking only into account the one-phonon<br />

process (Fig. 4.2). At longer times _Q decreases exponentially with time. This decrease can<br />

be easily understood: the energy levels with long (tunneling) relaxation time have been depopulated<br />

at t & t TA at a larger rate. Since the total amount of heat released by the TS<br />

should be finite a decrease below the st<strong>and</strong>ard result (dashed lines in Fig. 4.2) is expected.<br />

To take into account the distribution of potential barriers V <strong>and</strong> a thermally activated<br />

rate we will follow the approach used by Tielbürger et al. [11] <strong>and</strong> transform Eq. 11 in a double<br />

integral in the variables D <strong>and</strong> V using the approximation l = V/E 0 <strong>and</strong> D 0 & 2E 0 e –l /p:<br />

_Q ˆ PV<br />

2<br />

Z 1<br />

0<br />

d<br />

ZV max<br />

0<br />

dV E <br />

E<br />

tanh<br />

E 0 2k B T 0<br />

<br />

E<br />

tanh 1 …T 0 † exp… t=…T 0 †† : …18†<br />

2k B T 1<br />

For the following discussion it is not relevant to introduce in Eq. 18 a specific distribution<br />

of potential barriers V [11]. This is done below for comparison with experimental results<br />

<strong>and</strong> for computing the acoustic properties. After finding the range of potential barriers<br />

relevant for the heat release we divided the calculations in two regions: T 1 ^ 1 K <strong>and</strong><br />

T 1 > 1 K as described below.<br />

In order to find the range of potential barriers relevant for the heat release in our measuring<br />

time <strong>and</strong> temperature ranges we have calculated it following Eq. 18, splitting the V-<br />

limits of the inner integral from V min = V max –10KtoV max , using only the one-phonon process,<br />

<strong>and</strong> measuring temperature T 0 = 0.200 K. We have recognized that for V < 130 K <strong>and</strong><br />

52

4.2 Phenomenological theory for the heat release<br />

at t&10 3 s the TS are mostly relaxed <strong>and</strong> do not contribute appreciably to the heat release<br />

in our measured time range (10 4 s ^ t ^ 10 6 s) any more. The TS that relax through the<br />

one-phonon process <strong>and</strong> are relevant to the heat release are found to be those with potential<br />

barriers 130 K < V < 200 K. The contribution of tunneling systems with V > 200 K to the<br />

heat release is negligible.<br />

At T 1 < 1 K the contribution of a thermally activated rate for TS with potential barriers<br />

130 K < V < 200 K is irrelevant, i. e. using t&10 –13 s (from acoustic measurements,<br />

see below) we obtain a t TA (V = 130 K k B ) of several years. The potential barriers which are<br />

relevant in our time <strong>and</strong> temperature range through a thermally activated rate are V/k B 1 K. Figure 4.5 shows the calculated heat release as a function of T 1 at a<br />

given time t = 1.5610 5 s taking into account the cooling process <strong>and</strong> one-phonon process<br />

as well as thermally activated relaxation rate. The numerical results (Fig. 4.5) show clearly a<br />

saturation of _Q for T 1 > 5 K, i. e. at lower temperatures in comparison with the results without<br />

thermally activated relaxation, in qualitative agreement with the results assuming Raman<br />

processes <strong>and</strong> experimental results [2]. It is also interesting to note that a thermally activated<br />

relaxation rate decreases slightly the exponent in the time dependence of the heat release,<br />

i. e. _Q ! t –a with a < 1. This result is shown in Section 4.4.<br />

Figure 4.5: Heat release at t = 1.5610 5 s as a function of charging temperature T 1 calculated with <strong>and</strong><br />

without thermally activated processes <strong>and</strong> a finite cooling rate.<br />

53

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

4.3 Experimental details<br />

The experimental setup for measuring the heat release <strong>and</strong> the specific heat consists of a calorimeter<br />

on which the sample is mounted. The samples were cooled from 80 K to 0.050 K<br />

in about (5–8)610 3 s. The calorimeter is attached to a holder through a thermal resistance<br />

or a superconducting heat switch made of an Al strip. The holder is screwed to the mixing<br />

chamber of a top loading dilution refrigerator.<br />

Two sample holders were used, one from plastic <strong>and</strong> a second from Teflon. The first<br />

one showed a heat release approximately proportional to t –1 whereas for the second one no<br />

heat release was measured in agreement with previous measurements [2].<br />

We used two different methods to measure the heat release. Firstly, after cooling to<br />

low temperatures (T&0.060 K) the heat switch was opened <strong>and</strong> the time dependence of the<br />

sample temperature (warmup rate) was measured till about 0.120 K. Then the sample was<br />

cooled again to about 0.050 K before starting another warmup run. The heat release is obtained<br />

from the slope of the temperature in the warmup run dT (t)/dt with the knowledge of<br />

the specific heat of the sample plus calorimeter C(T), i. e. _Q = C(T) dT (t)/dt. The measurements<br />

were completely automated with a personal computer. The background heat leak was<br />

about (0.03–0.50) nW at 0.090 K <strong>and</strong> at t & 10 5 s depending on the sample holder <strong>and</strong> the<br />

vibration of the connecting cables. To test the measuring setup <strong>and</strong> procedure we have produced<br />

well-known heat leaks with a electrical heater, fixed to the sample.<br />

The second method for measuring the heat release is based as before on the measurement<br />

of the time dependence of the sample temperature at constant bath (mixing chamber)<br />

temperature. The difference lies on the selection of a fixed thermal resistance R th between<br />

sample <strong>and</strong> holder that enables the semiadiabatic measurement of the heat release in a time<br />

scale larger than the intrinsic thermal relaxation time of the arrangement. Approximately<br />

30 minutes after reaching a constant bath temperature T b the decrease of the sample temperature<br />

T s with time provides directly the heat release, i. e. _Q =(T s – T b )/R th (T s ,T b ). The<br />

measurement of the thermal resistance together with the background contribution to the heat<br />

release is made in situ after reaching the time-independent minimum temperature. Within<br />

experimental error (&10 %) both methods show the same results.<br />

The specific heat of both polymers was measured with the heat pulse technique in semiadiabatic<br />

fashion. The thermal conductivity was measured with a top loading 3 He refrigerator<br />

using the st<strong>and</strong>ard procedure. The acoustic properties, sound velocity <strong>and</strong> attenuation, were<br />

measured with the vibrating reed technique [16] in the frequency range (0.2–3) kHz.<br />

Both polymers were prepared following st<strong>and</strong>ard procedures. The PMMA sample had<br />

additionally 10 –2 mol% of tetra-4-tert.butyl-phthalocyamin (dye molecule) because it was<br />

used in an early optical hole burning experiment [15]. For the specific heat <strong>and</strong> heat release<br />

measurements the mass <strong>and</strong> density of the PMMA sample were determined as 11.97 g <strong>and</strong><br />

1.15 g/cm 3 , respectively. We have measured two PS samples prepared from different<br />

batches. The densities of these samples were 1.05 g/cm 3 , <strong>and</strong> the masses 11.4 g (sample<br />

PS1) <strong>and</strong> 38.0 g (sample PS2).<br />

For the thermal conductivity measurements two slices were cut from the bulk samples<br />

with length l = 1 cm, width w = 2 mm, <strong>and</strong> thickness d = 300 mm. For the acoustic measurements<br />

the reeds had the geometry l = (0.7–1) cm, w = (0.1–0.3) cm <strong>and</strong> d = (100–300) mm.<br />

54

4.4 Experimental results <strong>and</strong> discussion<br />

4.4 Experimental results <strong>and</strong> discussion<br />

4.4.1 Specific heat <strong>and</strong> thermal conductivity<br />

Figure 4.6 shows for both polymers the specific heat devided by the temperature as a square<br />

function of temperature. These measurements were performed only when the heat release is<br />

negligible (t & 10 6 s). For t ?? the specific heat can be written in terms of two contributions:<br />

c (T) =c 1 T + c 3 T 3 with coefficients for the tunneling system c 1 <strong>and</strong> phonon contribution<br />

c 3 . From Fig. 4.6 we obtain the values c 1 = (3.0+0.3) mJ/gK 2 <strong>and</strong> c 3 = (93 + 18) mJ/gK 4<br />

for PMMA, <strong>and</strong> c 1 = (4.6+0.5) mJ/gK 2 <strong>and</strong> c 3 = (77 +23) mJ/gK 4 for PS. The values of the<br />

linear term are in fair agreement with those published by Stephens [17, 18] c 1 = 4.6 mJ/gK 2<br />

for PMMA <strong>and</strong> c 1 = 5.1 mJ/gK 2 for [PS], but we obtained larger phonon contribution<br />

(Stephens: c 3 =29mJ/gK 4 for PMMA <strong>and</strong> c 3 =45mJ/gK 4 for PS). In Figs. 4.7 <strong>and</strong> 4.8 we<br />

compare our specific heat results with data from Ref. [18]. Within the scatter of the data it<br />

is difficult to conclude whether the c values are really different.<br />

Figure 4.6: Specific heat divided by the temperature as function of the squared temperature for PMMA<br />

(a) <strong>and</strong> PS (b).<br />

Figure 4.7: Comparison of our specific heat for PMMA with data from Ref. [18].<br />

55

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

Figure 4.8: Comparison of our specific heat for PS with data from Ref. [18].<br />

According to the tunneling model <strong>and</strong> for t ?? the specific heat due to the TS is given<br />

by [5]:<br />

c…T† ˆp2<br />

6 k2 2<br />

BPV ln… †T ˆ c 1 T:<br />

u min<br />

…19†<br />

If we assume (from heat release data, see below) u min =10 –10 for PMMA <strong>and</strong><br />

u min =6610 –9 for PS <strong>and</strong> from the measured c 1 we obtain a density of states P = 4.0610 38<br />

(7.5610 38 ) 1/Jg for PMMA (PS) in reasonable agreement with the values obtained in earlier<br />

specific heat measurements [17] <strong>and</strong> also from acoustic measurements for PMMA [24].<br />

Recent optical hole burning experiments on PMMA <strong>and</strong> PS [15] indicate that the density of<br />

states of TS for PS is about two times larger than for PMMA in reasonable agreement with<br />

our specific heat measurements.<br />

Figure 4.9: Thermal conductivity as a function of temperature for PMMA <strong>and</strong> PS.<br />

56

4.4 Experimental results <strong>and</strong> discussion<br />

Figure 4.9 shows the thermal conductivity for PMMA <strong>and</strong> PS as a function of temperature.<br />

For PMMA <strong>and</strong> at T < 0.7 K we obtain k =28T 1.84 610 –3 W/mK 2.84 <strong>and</strong> for PS<br />

k =19T 1.93 610 –3 W/mK 2.93 , in agreement with earlier measurements [17, 19–22], (Figs. 4.10<br />

<strong>and</strong> 4.11), <strong>and</strong> with measurements in epoxies [23]. According to the tunneling model <strong>and</strong> for<br />

T < 1 K the thermal conductivity is given by:<br />

…T† ˆk3 B v<br />

6p 2 … P 2 † 1 T 2 ; …20†<br />

where v is the sound velocity <strong>and</strong> r the mass density. From Eq. 20 <strong>and</strong> with the sound velocity<br />

from Ref. [17] we obtain Pg 2 = 9.2610 5 J/m 3 (11.5610 5 J/m 3 ) for PMMA (PS). If we<br />

replace the density of states P from the specific heat results we obtain similar values for the<br />

coupling constant between TS <strong>and</strong> phonons in both materials g = 0.28 eV (0.27 eV) for<br />

PMMA (PS). The results for the thermal conductivity indicate that the density of states of<br />

TS for the PS sample is about a factor of two larger than for the PMMA sample.<br />

Figure 4.10: Comparison of our thermal conductivity data for PMMA with earlier publications. The<br />

dashed line has been calculated within the tunneling model with parameters taken from our internal friction<br />

data.<br />

Figure 4.11: Comparison of our thermal conductivity data for PS with earlier publications. The dashed<br />

line has been calculated within the tunneling model with parameters taken from our internal friction<br />

data.<br />

57

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

4.4.2 Internal friction <strong>and</strong> sound velocity<br />

Figures 4.12 <strong>and</strong> 4.13 show the internal friction of PMMA <strong>and</strong> PS. Below 1 K the internal<br />

friction of PMMA <strong>and</strong> PS is approximately temperature-independent. According to the tunneling<br />

model the sound absorption should be temperature-independent in the region ot p P1<br />

(fulfilled in our temperature range) where o =2pv is the phonon frequency <strong>and</strong> t p the relaxation<br />

time of the TS (Eq. 6). In this temperature range a simple relationship is found for<br />

the internal friction [11, 16]<br />

Q 1 ˆ p<br />

2 C …21†<br />

Figure 4.12: Squares: PMMA internal friction as a function of temperature at a frequency of 535 Hz.<br />

Full circles: indicate the attenuation data taken from the ultrasonic measurements at 15 MHz from<br />

Ref. [24]. Solid <strong>and</strong> dotted lines: the calculated values for v=535 Hz <strong>and</strong> v=15 MHz following the<br />

modified tunneling model considering a thermally activated relaxation rate. For more details see text.<br />

Figure 4.13: Squares <strong>and</strong> Circles: PS internal friction as a function of temperature at two different frequencies.<br />

Solid line: the calculated values for v=0.24 kHz following the modified tunneling model described<br />

in the text.<br />

58

4.4 Experimental results <strong>and</strong> discussion<br />

with the constant C = Pg 2 /ru 2 <strong>and</strong> the sound velocity u. Equation 21 should hold even if<br />

higher-order processes dominate the relaxation of the TS (if the relaxation rate is proportional<br />

to u 2 !D 0 2 , Eq. 6). From the temperature-independent value of the internal friction<br />

<strong>and</strong> assuming no background contribution (e. g. due to the clampling) we obtain for PMMA<br />

(PS) C % 2.6610 –4 (8.3610 –4 ). Taking into account the mass density, sound velocity, <strong>and</strong><br />

the coupling constant obtained from the thermal conductivity <strong>and</strong> specific heat results described<br />

above, the values for the parameters C from the internal friction indicate that the<br />

density of states of TS for PS is larger than for PMMA by a factor &2.5. This is slightly<br />

larger than that obtained from the specific heat (about 1.5). This might be attributed to the<br />

unknown clamping contribution to the measured attenuation (which is always present) or to<br />

different u min values (Eq. 19). In Figs. 4.10 <strong>and</strong> 4.11 we show the calculated thermal conductivity<br />

using Eq. 20 with Pg 2 values from our internal friction at the plateau. Excellent agreement<br />

is obtained for PMMA (Fig. 4.10) but for PS the thermal conductivity is by a factor<br />

of two smaller. This difference might be attributed partially to the background contribution<br />

which is not subtracted from the measurement before computing the parameter C.<br />

The behaviour of the internal friction above about 3 K is qualitatively different for<br />

both polymers. PMMA shows a decrease in the internal friction reaching a minimum at<br />

about 30 K <strong>and</strong> increasing monotonously to the highest temperature of our experiment,<br />

120 K (not shown in Fig. 4.12) in very good agreement with previous work at similar frequencies<br />

[25]. On the contrary, the internal friction of PS shows the typical temperature dependence<br />

measured for other amorphous materials, for example SiO 2 [16]. It increases,<br />

reaching a frequency-dependent maximum at T&37 K (;40 K), (Fig. 4.13), <strong>and</strong> increases<br />

again at T > 60 K (70 K) at the frequency 0.24 kHz (3.2 kHz). This internal friction increase<br />

at T > 60 K for PS together with the increase at T > 30 K for PMMA will not be discussed<br />

here. Instead, we will discuss the behaviour of the internal friction at lower temperatures in<br />

terms of an extension of the tunneling model following the procedure described in Ref. [11].<br />

We assume the simplified relation for a thermally activated relaxation rate given by<br />

Eq. 16 <strong>and</strong> add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials<br />

<strong>and</strong> within the approximations given by Eq. 10 <strong>and</strong> with l = V/E 0 , it can be shown that the<br />

internal friction increases linear with temperature just above the temperature-independent region<br />

(plateau) [11]:<br />

Q 1 ˆ pCk BT<br />

E 0<br />

: …22†<br />

This increase is observed for PS, (Fig. 4.13); applying Eq. 22 to the results below<br />

20 K we obtain a zero point energy E 0 =13+ 2 K. In the same temperature region <strong>and</strong> due<br />

to the influence of the thermally activated relaxation the relative change of the sound velocity<br />

can be written as [11]:<br />

uu<br />

u ˆ Ck BT<br />

E 0<br />

ln…! 0 † : …23†<br />

A nearly linear temperature dependence of uu/u is observed in both polymers at temperatures<br />

about below 50 K (Figs. 4.14 <strong>and</strong> 4.15). At T ~ 1 K a crossover, due to tunneling<br />

<strong>and</strong> due to thermally activated relaxation, from the linear T-dependence to the logarithmic<br />

59

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

Figure 4.14: Relative change of sound velocity at v=535 Hz for PMMA. The solid line represents the<br />

linear temperature dependence of the sound velocity at temperatures below 20 K used for the numerical<br />

calculations. Inset: the same data but below 10 K in a semilogarithmic scale.<br />

Figure 4.15: Relative change of sound velocity at two different frequencies for PS. The solid line was<br />

calculated according to the modified tunneling model described in the text. Inset: the same data but below<br />

10 K in a semilogarithmic scale.<br />

T-dependence can be observed for both polymers (insets in Figs. 4.14 <strong>and</strong> 4.15). Although<br />

the tunneling model with the assumption of thermally activated relaxation of the TS provides<br />

a reasonable fit for the linear temperature dependence, its origin is still controversial. We<br />

note that a linear temperature dependence of the sound velocity above a few Kelvin is a<br />

rather general behaviour observed in several amorphous [26, 27], disordered [28], <strong>and</strong> polycrystalline<br />

metals [29]. Nava [28] argued recently against an interpretation in terms of thermally<br />

activated relaxation of the TS for the linear T-dependence of the sound velocity. However,<br />

new acoustical results in polycrystalline materials indicate a linear T-dependence of the<br />

sound velocity comparable with those found in amorphous materials [29].<br />

Applying Eq. 23 to the measurements at the two phonon frequencies for PS <strong>and</strong> with<br />

the value of E 0 from the internal friction we obtain t 0 % (1 +0.5)610 –17 s. As discussed<br />

in Ref. [11] the meaning of the prefactor t 0 is still unknown. The very small value of t 0<br />

found for PS in comparison with the one for SiO 2 (t 0 %10 –13 s) should be taken as an ef-<br />

60

4.4 Experimental results <strong>and</strong> discussion<br />

fective value until a systematic comparison of the applied model to other experimental data<br />

is available.<br />

To obtain a maximum in the internal friction an upper limit of the potential height<br />

must be assumed. Instead of a cut-off in the distribution P (D, V) <strong>and</strong> following Ref. [11], a<br />

Gaussian distribution with a width s 0 will be assumed:<br />

P…;V†ˆ<br />

P<br />

E 0<br />

e … V2 =2 2 0 † : …24†<br />

Figures 4.13 <strong>and</strong> 4.15 show the result of numerical calculations with only the width of<br />

the distribution s as free parameter. The fits were obtained assuming s 0 = 1200 K <strong>and</strong> with<br />

the values C = 8.3610 –4 , v = 0.24 kHz, t 0 =10 –17 s <strong>and</strong> E 0 = 13 K using the equations<br />

given in Ref. [11]. A reasonable agreement with the experimental data is achieved for both<br />

acoustic properties in the expected temperature region. The minimum obtained numerically<br />

at T ~ 3 K (Fig. 4.13) has been discussed in Ref. [11] <strong>and</strong> is attributed to the difference in<br />

the number of TS contributing to the relaxation process in the tunneling or thermally activated<br />

regime.<br />

As pointed out above, the internal friction of PMMA behaves differently from PS<br />

above the plateau. It decreases at T > 3 K <strong>and</strong> reaches a minimum at T ~ 30 K. The predicted<br />

linear temperature dependence of the internal friction (Eq. 22), at the measured frequency<br />

(v = 535 Hz) is not observed for PMMA. It is tempting to interpret the decrease of<br />

the internal friction results above 3 K assuming an upper bound of the TS density of states<br />

around 15 K. This assumption has been indeed used to interpret the irreversible line broadening<br />

of the optical hole burning experiments [30]. However, ultrasonic attenuation measurements<br />

at 15 MHz for PMMA [24] show clearly a thermally activated maximum at 12 K.<br />

Therefore we have decided to search for a set of parameters that might explain the low <strong>and</strong><br />

high frequency results under the assumptions described above <strong>and</strong> as done for PS.<br />

From the sound velocity data below 20 K (Fig. 4.14) <strong>and</strong> assuming the validity of<br />

Eq. 23 (ot P1) we obtain (E 0 /k B )/ln (ot 0 ) %C/(u ln (u)/uT) ~ 0.9 +0.2. If we assume<br />

t 0 ~10 –17 s like PS, we obtain E 0 /k B ~ 30 K. With this value we are not able to obtain a reasonable<br />

set of parameters that explain the data. If we take the value for SiO 2 from Ref. [11],<br />

t 0 % 10 –13 s, we obtain E 0 /k B % (20 + 5) K, a value comparable to that for SiO 2 . After several<br />

trials, eventually we have found a set of parameters that fits reasonably the low <strong>and</strong><br />

high frequency measurements without the assumption of an energy cut-off.<br />

In Fig. 4.12 we compare the experimental results of the internal friction with the numerical<br />

calculations for frequencies at 535 Hz <strong>and</strong> at 15 MHz using E 0 /k B =10K,t 0 =10 –13 s,<br />

<strong>and</strong> a rather small distribution width s 0 = 150 K. The model reproduces fairly well the position<br />

of the maximum in the attenuation at 15 MHz as well as its relative value in comparison<br />

with the plateau [24]. It is worth pointing out, that for PMMA, since the changes produced by<br />

the free parameters E 0 , t 0 <strong>and</strong> s 0 compete with each other, it is not possible to find another set<br />

of values which can explain the results as well as we can. It is important to note that the internal<br />

friction data for both polymers indicate a much smaller thermally activated contribution to<br />

the phonon attenuation for PMMA than for PS, in spite of the fact that in the quantum tunneling<br />

regime both samples show similar results. This difference may influence the low-temperature<br />

heat release because the number of excited states that are available after the cooling process<br />

will be different.<br />

61

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

4.4.3 Heat release<br />

Figure 4.16 shows the heat release as a function of time for the 12 g PMMA sample at<br />

T = 0.090 K <strong>and</strong> after cooling from 80 K. The heat release follows very well the predicted<br />

t –1 law from the tunneling model (Eq. 12). The same result was obtained in different runs<br />

with slightly different cooling rates <strong>and</strong> measuring temperatures 0.070 K^T 0 ^0.110 K.<br />

For finite cooling rates <strong>and</strong> for charging temperatures T 1 p 1 K this independence cooling<br />

rate is to be expected (Section 4.2.3).<br />

As discussed in Section 4.2 the unknown higher-order relaxation processes are taken<br />

into account through an effective charging temperature T 1 . With the tunneling systems density<br />

of states from the specific heat <strong>and</strong> using Eq. 12 we obtain T 1 % 26 K. As stated in<br />

Section 4.2 for charging temperatures T > 10 K it is not possible to obtain reliable values of<br />

the density of states for TS from heat release measurements.<br />

From the observed t –1 -dependence <strong>and</strong> using the value for the coupling constant between<br />

TS <strong>and</strong> phonons from Ref. [24], Ak B 3 = 1.6610 9 K –3 s –1 , we obtain an upper limit for<br />

the parameter u min

4.4 Experimental results <strong>and</strong> discussion<br />

<strong>and</strong> (5). Even worse, the computed curve (5) with thermal activation shows much smaller<br />

values for the heat release by almost two orders of magnitude. This result indicates that too<br />

many energy states were depopulated during the cooling process through thermal activation.<br />

This disagreement might be ascribed to the assumed potential distribution function (Eq. 24)<br />

or the approximation l = V/E 0 . Curve (2), which is calculated without thermally activated relaxation,<br />

lies only a factor of two higher than the measured one (Fig. 4.16) <strong>and</strong> is in qualitative<br />

agreement with the acoustical data where no influence from thermal activation were taken<br />

into account.<br />

For PS, according to the specific heat <strong>and</strong> acoustical data (at T < 3 K) <strong>and</strong> due to a larger<br />

density of states of tunneling systems, we expected a larger heat release than for PMMA.<br />

Surprisingly, the opposite is observed. After cooling from 80 K with the same cooling rate,<br />

the heat release for PS measured at the same temperature as before was at least of a factor of<br />

ten smaller than for PMMA (Figure 4.16), <strong>and</strong> it does not follow the t –1 -dependence.<br />

In order to underst<strong>and</strong> the observed deviations we have measured the heat release of<br />

PS at lower charging temperatures. Figure 4.17 shows the heat release as a function of time<br />

at charging temperatures 0.5 K^T 1 ^1 K. The heat release follows very well the t –1 -dependence<br />

as well as an increase with T 1 2 (Eq. 12, Fig. 4.18). It is interesting to note that a<br />

fair quantitative agreement with the prediction of the tunneling model is obtained. If we calculate<br />

the density of states of tunneling systems P from these data with Eq. 12 (straight line<br />

in Fig. 4.18) we obtain P = 9.0610 –38 J –1 g –1 which is similar to the one obtained from the<br />

specific heat assuming u min =6610 –9 .<br />

Figure 4.17: Heat release as a function of time (in hours) for Polystyrene at a measuring temperature<br />

T 0 = 0.200 K cooling from different T 1 . The straight lines have a t –1 -dependence.<br />

At higher temperatures T 1 we observe the expected saturation of the heat release<br />

(Fig. 4.18), however, no deviation from the t –1 -dependence has been measured within experimental<br />

error. Deviations are observed if we charge the sample at temperatures T 1 >15K<br />

(Fig. 4.16). In this case we have cooled the sample from 80 K <strong>and</strong> measured the heat release<br />

at 0.090 K (sample PS1) <strong>and</strong> 0.300 K (sample PS2). According to theoretical estimates the<br />

observed difference in the heat release between the two PS samples is not attributed to the<br />

difference in measuring temperatures. The rather abrupt saturation of the heat release at<br />

T 1 6 7 K is attributed to the depopulation of the excited states through thermally activated<br />

relaxation in the cooling process (Figs. 4.5 <strong>and</strong> 4.18), as it will become clear below.<br />

63

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

Figure 4.18: Heat release multiplied by the time t as a function of the difference between the squares of<br />

charging temperature T 1 <strong>and</strong> measuring temperature T 0 for PS. The solid line follows the theoretical<br />

prediction with a density of states of tunneling systems 1.45 larger than the one obtained from the specific<br />

heat results.<br />

The following experiment provides further verification that the non-simple time dependence<br />

of the heat release for PS is due to energy states excited above 15 K <strong>and</strong> depopulation<br />

during the cooling process, very likely through thermally activated relaxation. We have<br />

charged the sample at 80 K for several hours <strong>and</strong> cooled it then to 1 K. We have left the sample<br />

for 22 h at 1 K <strong>and</strong> later continued to 0.3 K, the temperature at which the heat release<br />

was measured (Curve (1) in Fig. 4.19). We observe now a clear deviation from the t –1 law.<br />

That means that even after 22 h at 1 K the energy states excited above 15 K have not relaxed<br />

completely. After 60 h at 0.3 K we warmed the sample to 1 K for 17 h, cooled it to 0.3 K<br />

<strong>and</strong> measured the heat release (Curve (2) in Fig. 4.19). The heat release follows the theoretical<br />

t –1 -dependence very well.<br />

We have calculated the heat release taking one-phonon,thermally activated processes,<br />

<strong>and</strong> the experimental cooling process into account. We used the same parameters obtained<br />

from the fit to the acoustical data (Figs. 4.13 <strong>and</strong> 4.15). With only one free parameter, u min ,<br />

Figure 4.19: Heat release as a function of time for PS. (1): cooled from 80 K to 1 K <strong>and</strong> leaving the<br />

sample 17 h at this temperature, T 0 = 0.300 K. (2): the sample was warmed to a charging temperature<br />

T 1 = 1 K for 22 h <strong>and</strong> cooled again to T 0 = 0.300 K. The solid line in (1) is only a guide <strong>and</strong> the solid<br />

line (2) has a t –1 -dependence.<br />

64

4.5 Conclusions<br />

we can reproduce the measured heat release of PS reasonably well (Fig. 4.16). In this figure<br />

we present the two curves (3) <strong>and</strong> (4) using the parameter u min =10 –10 <strong>and</strong> 6610 –9 . This<br />

fit indicates that we can observe the influence of a finite u min through the steeper decrease<br />

of the heat release at long times. Curve (1) in Fig. 4.16 was calculated with parameters for<br />

PS but without thermally activated relaxation.<br />

4.5 Conclusions<br />

We conclude that thermal conductivity, specific heat <strong>and</strong> acoustical properties at low temperatures<br />

(T < 1 K) can be quantitatively interpreted within the tunneling model for PMMA<br />

<strong>and</strong> PS. At higher temperatures we have measured very different temperature dependence of<br />

the internal friction for the two polymers. At the used frequencies <strong>and</strong> at 0.1 K^T^120 K<br />

PMMA shows no thermally activated dissipation peak. PMMA shows a minimum in the internal<br />

friction at T ~ 30 K where PS shows a maximum. Nevertheless, within the tunneling<br />

model <strong>and</strong> introducing a thermally activated relaxation rate as well as a Gaussian distributon<br />

density of states of tunneling systems, it is possible to underst<strong>and</strong> the acoustical properties.<br />

From the fits we may conclude that the density of states of tunneling systems for PMMA is<br />

restricted to smaller energies, i. e. a smaller width s 0 , than for PS.<br />

Although the thermal conductivity, specific heat, <strong>and</strong> sound velocity temperature dependence<br />

for both samples are similar, the heat release shows different time dependences as<br />

well as different values in apparent contradiction with the other measured low-temperature<br />

properties. Taking into account the internal friction results, it is tempting to correlate the differences<br />

in the heat release between both samples to the difference in the contribution of<br />

thermally activated processes in the relaxation of the tunneling states during the cooling process.<br />

We have demonstrated that the deviations of the time dependence of the heat release<br />

from the t –1 -dependence predicted by the tunneling model is due to the relaxation of energy<br />

states excited above 15 K. The smaller values of the measured heat release of PS in comparison<br />

with the st<strong>and</strong>ard tunneling model can be explained by the depopulation of energy<br />

states relaxing mainly by thermally activated processes during the cooling process.<br />

Without detailed knowledge of the short relaxation times at T 1 > 1 K, a quantitative<br />

fit or even a qualitative underst<strong>and</strong>ing of the heat release _Q(T 1 ,T 0 ,t) data for high charging<br />

temperatures seems to be impossible. Therefore, the heat release with high charging temperatures<br />

can be hardly used to obtain information on the density of states of tunneling systems.<br />

Heat release experiments at high measuring temperatures T 0 > 1 K are necessary to<br />

obtain information on the dynamics of the tunneling systems with a relaxation rate not given<br />

by the tunneling one-phonon process. These experiments may clarify the influence of the<br />

two different relaxation mechanisms discussed nowadays in the literature to interpret the<br />

low-temperature properties of amorphous solids above 1 K, namely incoherent tunneling<br />

65

4 Low-Temperature Heat Release, Sound Velocity <strong>and</strong> Attenuation, …<br />

[31] <strong>and</strong> thermal activation within the soft potential model [8]. Both approaches have been<br />

used with impressive success in the last years. It is, however, not yet clear if incoherent tunneling,<br />

a mixture of the two processes (incoherent <strong>and</strong> thermal activation), or only thermal<br />

activation is the main relaxation mechanism of the TS above 1 K.<br />

Acknowledgements<br />

We wish to acknowledge W. Joy for the sample preparation, <strong>and</strong> S. Hunklinger, D. Haarer<br />

<strong>and</strong> J. Friedrich for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft<br />

through the Sonderforschungsbereich 213. A. Nittke was supported by<br />

the “Graduiertenkolleg Po 88/13” of the Deutsche Forschungsgemeinschaft.<br />

References<br />

1. J. Zimmerman, G. Weber: Phys. Rev. Lett., 46, 661 (1981)<br />

2. M. Schwark, F. Pobell, M. Kubota, R.M. Mueller: J. Low Temp. Phys., 58, 171 (1985)<br />

3. S. Sahling, A. Sahling, M. Kolac: Solid State Commun., 65, 1031 (1988)<br />

4. P.W. Anderson, B.I. Halperin, C.M. Varma: Phil. Mag., 25, 1 (1972); W.A. Phillips, J. Low Temp.<br />

Physics 7, 351 (1972)<br />

5. M. Deye, P. Esquinazi: Z. Phys. B – Condensed Matter 76, 283 (1989)<br />

6. M. Deye, P. Esquinazi: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Phonon 89, World Scientific,<br />

Singapore, p. 468, (1990)<br />

7. J. Jäckle: Z. Phys., 257, 212 (1972)<br />

8. D. Parshin, S. Sahling: Phys. Rev. B, 47, 5677 (1993)<br />

9. J.J. Brey, A. Prados: Phys. Rev. B, 43, 8350 (1991)<br />

10. P. Esquinazi, M. Scherl, Li Junyun, F. Pobell: in: M. Meissner, R. Pohl (eds): 4th International<br />

Conference on Phonon Scattering in Condensed Matter, Springer Series in Solid State Sciences,<br />

Vol. 112, p. 287 (1993)<br />

11. D. Tielbürger, R. Merz, R. Ehrenfels, S. Hunklinger: Phys. Rev. B, 45, 2750 (1992)<br />

12. P. Doussineau, C. Frenois, R. G. Leisure, A. Levelut, J. Y. Prieur: J. Phys. (Paris), 41, 1193 (1980)<br />

13. S. Hunklinger, W. Arnold: in: W. P. Mason, R. N. Thurston (eds.): Physical Acoustics, Vol XII,<br />

Academic Press, New York, 1976<br />

14. W.A. Phillips: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Phonon 89, World Scientific, Singapore,<br />

p. 367 (1990)<br />

15. K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991)<br />

16. A.K. Raychaudhuri, S. Hunklinger: Z. Phys. B – Condensed Matter, 57, 113 (1984)<br />

17. R.B. Stephens: Phys. Rev. B, 8, 2896 (1973)<br />

18. R.B. Stephens, G.S. Cieloszyk, G.L. Salinger: Physics Letters, 28A, 215 (1972)<br />

66

References<br />

19. D. Cahill, R. Pohel: Phys. Rev. B, 35, 4067 (1987)<br />

20. C. Choy, G. Salinger, Y. Chiang: J. of Applied Physics, 41, 597 (1970)<br />

21. J. Freeman, A. C. Anderson: Phys. Rev. B, 34, 5684 (1986)<br />

22. J. Mack, J. Freeman, A.C. Anderson: J. of Non Crystalline Solids, 91, 391 (1987)<br />

23. G. Hartwig: Progr. Colloid <strong>and</strong> Polymer Sci., 64, 56 (1978)<br />

24. G. Federle, S. Hunklinger: J. de Physique, C9(12), Tome 43, p. C9–505 (1982); G. Federle: PhD<br />

Thesis, M. Planck Stuttgart, 1983, unpublished; A.K. Raychaudhuri: in: T.V. Ramakrishnan, M.<br />

Raj Lakshmi (eds.): Non-Debye Relaxation in Condensed Matter, World Scientific, p. 193 (1987)<br />

25. J. Crissman, J. Sauer, E. Woodward: J. of Polymer Science, A2, 5075 (1964)<br />

26. G. Bellesa: Phys. Rev. Lett., 40, 1456 (1978)<br />

27. J.-Y. Duquesne, G. Bellesa: J. de Physique Lettres, 40, L-193 (1979)<br />

28. R. Nava, R. Oentrich: J. of Alloys <strong>and</strong> Compounds, in press; R. Nava, Phys. Rev. B, 49, 4295<br />

(1994)<br />

29. E. Gagemidze, P. Esquinazi, R. König: Europhys. Letters, 31, 13 (1995)<br />

30. W. Köhler, J. Zollfrank, J. Friedrich: Phys. Rev. B, 39, 5414 (1989)<br />

31. A. Würger: in: From coherent tunneling to relaxation, Springer Tracts in Modern Physics,Vol. 135,<br />

(1997)<br />

67

5 Spectral Diffusion due to Tunneling Processes<br />

at very low Temperatures<br />

Hans Maier, Karl-Peter Müller, Siegbert Jahn, <strong>and</strong> Dietrich Haarer<br />

5.1 Introduction<br />

Pioneered by the work of Zeller <strong>and</strong> Pohl [1] it was discovered in the early 1970s that amorphous<br />

solids show low-temperature thermal <strong>and</strong> acoustic properties which are very different<br />

from those observed in crystals. For reviews see for example Refs. [2, 3]. The most wellknown<br />

of these anomalous properties is the specific heat, which is in general considerably<br />

larger than would be expected from the Debye model <strong>and</strong> varies linear with temperature in<br />

contrast to the Debye T 3 -dependence. Other anomalies are the temperature dependence of<br />

the thermal conductivity, the properties of phonon echoes, <strong>and</strong> ultrasonic absorption. These<br />

features seem to be quite universal for all kinds of amorphous solids, irrespective of their<br />

chemical composition <strong>and</strong> structure, i. e. inorganic as well as organic or polymeric.<br />

Very soon after the first experimental evidence for these anomalous low-temperature<br />

excitations Anderson et al. [4] <strong>and</strong> independently Phillips [5] developed a theoretical description,<br />

called the tunneling model [4, 5]. It is based on the assumption that the low energy<br />

degrees of freedom in amorphous solids arise from tunneling motions of atoms or groups of<br />

atoms between local energetic minima which should exist in any disordered solid. If only<br />

the lowest energy level of each minimum is considered, this leads to two eigenstates. For<br />

this reason, these degrees of freedom are often referred to as two-level system (TLS). An<br />

important consequence of this model is the formal analogy to a system of particles with<br />

spin 1/2, which for example immediately explains the existence of phonon echoes. Another<br />

very striking feature of the tunneling model was the prediction of a time-dependent specific<br />

heat [4]. This was confirmed experimentally several years later [6]. Furthermore, the model<br />

can explain the heat release [7] of amorphous solids, which is a time-dependent non-equilibrium<br />

phenomenon. From these kinds of experiments, it could be concluded that TLS dynamics<br />

occur on time scales extending over many orders of magnitude.<br />

The sensitivity of optical experiments on amorphous solids was hindered, in many instances,<br />

by the large inhomogeneous broadening which arises from the distribution of local<br />

environments of the involved optical transitions. Therefore a very important step was the<br />

discovery of persistent spectral hole burning in 1974 [8, 9]. This method of high-resolution<br />

laser spectroscopy eliminates the inhomogeneous effects induced by the static disorder in<br />

68 Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

5.2 The optical cryostat<br />

amorphous hosts. For this purpose dye molecules are embedded in the solid which change<br />

their absorption spectra when irradiated with resonant narrow b<strong>and</strong> laser light. This produces<br />

a dip in the inhomogeneous absorption b<strong>and</strong>, called a spectral hole. A narrow spectral hole<br />

can be regarded as a highly sensitive spectral probe for any kind of distortion of the matrix,<br />

which induces small spectral changes [10]. In this way variations of strain fields caused by<br />

pressure changes of almost some 10 hPa can be detected [11]. Spectral holes are also sensitive<br />

to small changes of other external parameters including electric fields [12, 13].<br />

After observing quite a few anomalous properties of optical transitions in glasses <strong>and</strong><br />

attributing them to the dynamics of TLS [14], the tunneling model was adopted by Reinecke<br />

[15] to explain the low-temperature line widths of optical transitions in amorphous solids<br />

using the concept of spectral diffusion. This concept had originally been developed for the<br />

description of spin resonance experiments [16] <strong>and</strong> had already been applied to the theoretical<br />

treatment of the above mentioned ultrasonic properties of glasses [17]. Soon after this<br />

step, the possibility of a connection between thermal <strong>and</strong> optical properties of amorphous<br />

solids was supported by the observation of time dependence of spectral hole widths [18].<br />

The application of the tunneling model to the description of spectroscopic properties<br />

proposed an important link suggesting a correlation between the specific heat <strong>and</strong> the optical<br />

line width. This implies that, besides the mentioned calorimetric <strong>and</strong> acoustic methods,<br />

optical spectroscopy yields information about the dynamics of the same low energy excitations<br />

of amorphous solids, which dominates the calorimetric experiments. A crucial point in<br />

establishing this connection is the temperature dependence of these physical solid state parameters.<br />

The tunneling model in its original form predicts linear temperature dependence for<br />

the specific heat as well as for the optical line width. This prediction, however, is only valid<br />

at temperatures where the contribution of phonons is of minor importance. Earlier hole burning<br />

measurements on the temperature dependence of optical line widths [19] indicated the<br />

necessity of exp<strong>and</strong>ing the temperature range accessible to optical absorption spectroscopy<br />

to values far below 1 Kelvin in order to exclude any other mechanisms except TLS dynamics.<br />

For this reason we have constructed a 3 He/ 4 He dilution refrigerator with optical<br />

windows allowing transmission spectroscopy at temperatures down to 0.025 K, a temperature<br />

regime in which only very few data from optical spectroscopy existed before [20, 21].<br />

In addition, cooling with a 3 He/ 4 He mixture can be performed continuously for very long<br />

times (in principle unlimited) if the necessary care in cryostat design <strong>and</strong> operation is taken.<br />

In our experiments we have reached operation times up to three months.<br />

5.2 The optical cryostat<br />

In the design of a cryostat for optical spectroscopy two main problems have to be overcome.<br />

In an absorption experiment the sample has to dissipate the absorbed energy in order to<br />

avoid overheat during the measurement. Therefore the best possible thermal contact to the<br />

cold stage is of extreme importance. The second problem is how to guide the laser light to<br />

the sample without creating an intolerably large additional heat leak, for example by infrared<br />

69

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures<br />

irradiation from the windows. Since various construction methods for optical cryostats in the<br />

temperature range below 0.500 K did not yield the desired results, special care had to be taken<br />

in the design of our dilution refrigerator. For example it had been reported [22] that for<br />

samples located outside the mixing chamber temperatures below 0.300 K could not be<br />

reached, even if the best thermal contact is realized by pressing the samples to the wall of<br />

the mixing chamber with indium metal as contact material. In another kind of experiment a<br />

glass fibre winding around the cold finger of a dilution refrigerator shows a significant temperature<br />

gradient with respect to the mixing chamber below 0.100 K [20]. For these reasons,<br />

we decided to place the sample directly into the dilute phase of the mixing chamber. This<br />

way optimal thermal contact <strong>and</strong> a constant temperature of the sample was achieved via the<br />

superfluid 4 He. For performing optical experiments the mixing chamber was designed as a<br />

glass cylinder.<br />

The optical path, used in our cryostat, is shown in Figure 5.1 [23]. Three sets of windows<br />

are mounted on cold copper shields to eliminate most of the room temperature radiation<br />

in three steps: at liquid nitrogen temperature (77 K), liquid helium temperature (4.2 K),<br />

<strong>and</strong> at approximately 1 K, the latter being achieved by cooling with the pumped 3 He of the<br />

distillation chamber. In contrast to other constructions, the vessel containing the dilution refrigerator<br />

part of the cryostat is in our design not surrounded by the helium tank which has<br />

the advantage that the laser beam is not scattered by boiling liquid helium. We estimate that<br />

the total heat leak caused by room temperature radiation is below 50 nW. The minimum<br />

temperature reached is 0.024 K. The cooling power is about 6 mW at 0.100 K. Although this<br />

is a very low value, it is sufficient since the power of the absorbed laser light was always far<br />

below 1 mW. The light powers used for hole burning in our experiments were on the order of<br />

several nanowatts to several tens of nanowatts depending on the temperature. For hole detection<br />

the power is reduced to the picowatt range. The cooling power of our cryostat <strong>and</strong> the<br />

thermal conductivity of the involved polymers are high enough to guarantee a uniform temperature<br />

distribution with no significant sample heating during the hole burning process.<br />

4<br />

He - shield mixing - chamber<br />

LN 2 - shield 1-K - shield<br />

sample<br />

Figure 5.1: Alignment of optical windows <strong>and</strong> sample in the cryostat. The sample is placed inside the<br />

mixing chamber of the dilution refrigerator. The shields are connected to the distillation chamber, the<br />

helium tank, <strong>and</strong> the nitrogen tank, respectively. The liquid N 2 windows possess an infrared reflective<br />

coating.<br />

70

5.3 Theoretical considerations<br />

The temperature is measured with a RuO 2 thick film resistor (TFR) supplied by Phillips<br />

(type RC-01). The heat capacity <strong>and</strong> thermal relaxation time of the TFR are equivalent<br />

to those of a carbon resistor but its thermal reproducibility during temperature cycles is better<br />

[24]. The resistor is calibrated against several primary <strong>and</strong> secondary thermometers including<br />

a NBS fixed point device, CMN <strong>and</strong> Ge, thermometers. Its accuracy is better than<br />

2% in the temperature range between 4.2 K <strong>and</strong> 0.025 K. The thermometer is placed into<br />

the dilute phase of the mixing chamber in direct thermal contact to the sample. The resistance<br />

is measured using a four wire ac resistance bridge (AVS-46 RV Elektroniikka, Finl<strong>and</strong>).<br />

At the minimum temperature, the energy dissipated in the resistor is less than 0.5 pW.<br />

5.3 Theoretical considerations<br />

In the tunneling model [4] the TLS was described by two parameters (Fig. 5.2), the asymmetry<br />

parameter D <strong>and</strong> the overlap parameter l, which contains the barrier height, the distance<br />

of the two minima, <strong>and</strong> the mass of the tunneling particle. They are related with the total energy<br />

splitting E of the two levels <strong>and</strong> the tunneling matrix element D 0 by the relations<br />

q<br />

E ˆ 2 ‡ 2 0<br />

<strong>and</strong> 0 ˆ k exp… † ; …1†<br />

where kO is a typical zero point energy. One of the most important ingredients of the TLS<br />

model is the distribution function characterizing the two parameters. The originally used assumption<br />

for this function, which has also been applied to explain spectral diffusion [15,<br />

17], is a flat distribution in both parameters, namely<br />

P…;†dd ˆ Pdd:<br />

…2†<br />

Figure 5.2: Double minimum potential as a model for a TLS. All relevant parameters are shown in the<br />

figure.<br />

71

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures<br />

It is convenient, however, to use experimentally accessible quantities as variables, i. e.<br />

the energy E <strong>and</strong> a dimensionless relaxation rate R, which can be defined as: R = r/r max =<br />

D 2 0 /E 2 . The transformation to the new variables yields for the distribution function [25]:<br />

1<br />

P…E;R† ˆ P p : …3†<br />

2R 1 R<br />

A very important feature of this distribution is the fact that P (E, R) is independent of<br />

E. In order to keep the total number of TLS finite, some cut-off value for R ? 0 has to be<br />

introduced [3], which is usually denoted by R min .<br />

It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results<br />

in spectral diffusion, which shows up in the experiment as a time <strong>and</strong> temperature-dependent<br />

Lorentzian line broadening. The width of this Lorentzian line must be calculated by<br />

averaging over the distribution of energies <strong>and</strong> relaxation rates P(E, R). It can be written as:<br />

…t; T† ˆ2p2 <br />

3k C ij<br />

<br />

<br />

Z Emax<br />

E min<br />

Z 1<br />

dEn…T†<br />

R min<br />

dR E P…E;R†…1 exp‰ tr max RŠ† : …4†<br />

Here AC ij S represents an average coupling constant between the TLS <strong>and</strong> the optical<br />

transition, n (T) is a thermal occupation factor of the TLS states. For the system which is investigated<br />

in this work, the parameter r max can be estimated from experimental data on ultrasonic<br />

attenuation [26] to be on the order of 10 7 s –1 at a temperature of 0.100 K <strong>and</strong> even larger<br />

at higher temperatures. Therefore, the condition t 7r max p 1 is well fulfilled on all time<br />

scales exceeding several microseconds. In this limit the rate integration in Eq. 4 can be performed<br />

analytically [27], which leads to the well-known logarithmic time dependence of<br />

spectral diffusion:<br />

!…t; T† ˆ p2 <br />

3k C <br />

ij P k B T ln…r max t† : …5†<br />

Equation 5 represents the theoretical prediction of the tunneling model for the time<br />

<strong>and</strong> temperature-dependent broadening of spectral holes. This result, however, is the result<br />

of a particular form of the density of tunneling states P (E, R), which is based on the a<br />

priori assumption of Eq. 2. A uniform density of states in D is a physically reasonable<br />

choice; the independence of l, however, is difficult to justify, since l consists of several<br />

parameters [28]. The temperature dependence of spectral diffusion is dominated by D the<br />

time evolution stems mainly from l. Therefore, these two predictions from Eq. 5 do not<br />

have the same validity. In our experiments we have investigated time <strong>and</strong> temperature dependence<br />

separately.<br />

72

5.4 Temperature dependence<br />

5.4 Temperature dependence<br />

In these experiments, we investigated polystyrene (PS) doped with phthalocyanine <strong>and</strong> polymethylmethacrylate<br />

(PMMA) doped with tetra-4-tert-butyl phthalocyanine. Both sample<br />

materials have also been investigated by heat release <strong>and</strong> specific heat measurements [29].<br />

The samples had optical densities of about 0.4 at a typical thickness of 3 mm. The samples<br />

were prepared by bulk polymerisation of the solution of the dye in the monomer.<br />

The observed hole shapes are Lorentzian with no detectable deviations in all cases. To<br />

eliminate saturation effects at least 10 holes with different energies were burned at each temperature<br />

<strong>and</strong> the homogeneous line width was determined by extrapolation to zero burning<br />

fluence [30]. Due to their low relative depths the extrapolation was done assuming a linear<br />

dependence of the hole area on energy [10]. A weak variation of the line width with light<br />

power was observed at our minimum temperature of 0.025 K for PMMA. We attribute this<br />

variation to sample heating.<br />

The temperature dependence of the hole width is shown in Fig. 5.3 [23]. The results are:<br />

a) both systems display a linear temperature dependence over the whole temperature range<br />

from 0.500 K down to 0.025 K <strong>and</strong> no crossover to a constant line width is seen. The linearity<br />

of the data plot shows that the density of states P (E, R) is indeed independent of the<br />

TLS energy;<br />

b) the relative magnitude of the line width in the two systems correlates with the respective<br />

specific heat [31];<br />

c) the extrapolated line width for T ? 0 is nearly the same for both systems, confirming<br />

that in an amorphous system the line width reaches the lifetime limited value G h = 1/2 pT 1<br />

(Heisenberg limit) of the electronic transition only when T ?0. This limit is reached in our<br />

experiment within 10%!<br />

Figure 5.3: Temperature dependence of the hole burning line width for PS <strong>and</strong> PMMA between<br />

0.025 K <strong>and</strong> 0.500 K.<br />

73

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures<br />

In the case of PS, where unsubstituted phthalocyanine was used as dye, the result for<br />

the extrapolated line width G (T ? 0) is in very good agreement with the fluorescence lifetime<br />

measurements of this molecule [32]. We are not aware of measurements of the fluorescence<br />

lifetime of substituted phthalocyanine, but our results show that there is no major difference<br />

between both dyes as far as their excited state lifetimes are concerned.<br />

These results imply that in amorphous solids there are dynamical processes of twolevel<br />

systems with an energy spectrum extending down to values as low as k B 70.025 K.<br />

These low-energy excitations dominate the line widths of optical transitions even at low<br />

temperatures <strong>and</strong> the lifetime limited value can only be reached by extrapolating to zero<br />

temperature.<br />

5.5 Time dependence<br />

After the system has been cooled down spectral holes are burned immediately <strong>and</strong> after<br />

some delay times, while the sample was at constant temperature, we observe a hole broadening<br />

which depends on this delay time. This behaviour is demonstrated in Fig. 5.4. Dataset A<br />

is the broadening of a hole which is burned immediately after the sample (phthalocyanine in<br />

PMMA) has been cooled from liquid nitrogen temperature to 0.300 K. The curves show the<br />

evolution of holes burned about one day, three days, <strong>and</strong> one week after cooling down. The<br />

amount of spectral diffusion in a given time interval decreases continuously until a small residual<br />

effect becomes observable, which is independent of the time delay after reaching the<br />

final temperature. This is represented by dataset B in Fig. 5.4.<br />

A<br />

B<br />

Figure 5.4: Time evolution of spectral holes burned at different times after sample cooling. Datasets<br />

corresponding to A <strong>and</strong> B are investigated in the following.<br />

74

5.5 Time dependence<br />

Investigating the temperature dependence of the time evolution of the hole width [33],<br />

we performed experiments on a PMMA sample doped with phthalocyanine at different temperatures<br />

T, varying from 0.100 K to 1 K. For each of the temperatures T = 0.100, 0.300,<br />

0.500, <strong>and</strong> 0.700 K, the respective experiments were carried out in a separate run. It took<br />

about one hour to cool the system from 77 K to 4 K. The time for cooling down from 4 K<br />

to the final temperature T depends slightly on T, but is in the range of about (6+1) hours.<br />

Immediately after reaching T, holes are burned <strong>and</strong> their subsequent broadening is observed<br />

for about one week. These experiments yield data corresponding to set A in Fig. 5.4.<br />

The results of these measurements are shown in Fig. 5.5 [33]. In this plot the time origin for<br />

each run is identical, namely the time at which the cooling down procedure from 77 K was<br />

started. For better visibility, however, the data corresponding to T = 0.500, 0.300, <strong>and</strong><br />

0.100 K are shifted by 0.5, 1.0, <strong>and</strong> 1.5 orders of magnitude along the logarithmic time axis,<br />

respectively. Except for the data taken at 0.500 K the total observation time was 6–10 days.<br />

Figure 5.5: Broadening of holes burned immediately after reaching T. Data shifted for better visibility<br />

(see text).<br />

Keeping the samples at constant temperature new holes are burned after 1–2 weeks<br />

<strong>and</strong> their broadening is observed for several hours, corresponding to set B in Fig. 5.4. These<br />

data are shown in Fig. 5.6 [33]. Here the observation time is 2–16 hours. Note that the range<br />

of the DG-axis is about 10 times less as compared with Fig. 5.5.<br />

Comparing our data of Figs. 5.5 <strong>and</strong> 5.6, it is obvious that the measurements which are<br />

started immediately after cooling down show no variation of the observed broadening with the<br />

phonon bath temperature, while the residual behaviour observed many days later exhibits a<br />

rather pronounced dependence on the experimental temperature. It has to be emphasized that<br />

the data shown in Fig. 5.5 <strong>and</strong> the corresponding data in Fig. 5.6 for the same values of Twere<br />

measured without any change of temperature in between the two subsequent experiments.<br />

In spite of the variation of the final temperature over almost one order of magnitude,<br />

all four datasets in Fig. 5.5 show the same hole broadening behaviour.<br />

We attribute this to the existence of a non-equilibrium state of the TLS ensemble due<br />

to the fast cooling procedure: The large amount of spectral diffusion observed after fast<br />

75

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures<br />

Figure 5.6: Broadening of holes burnt after 1–2 weeks at low temperature.<br />

cooling is due to the relaxation of TLS to their thermal equilibrium [34]. This relaxation<br />

process is known to produce the thermal phenomenon of heat release [7]. Thus, it can be<br />

concluded that we have established an important connection between optical <strong>and</strong> calorimetric<br />

phenomena.<br />

The data of Fig. 5.6 exhibit a pronounced dependence on temperature, as can be expected<br />

from theoretical considerations outlined in Section 5.3 <strong>and</strong> from the non-time resolved<br />

experiments of Section 5.4. The time evolution, however, shows a distinct non-logarithmic<br />

behaviour in contradiction to the prediction of the tunneling model. For short times<br />

the Fig. 5.6 shows clearly that the hole broadening starts fairly logarithmic. We have performed<br />

transient hole burning experiments on the millisecond time scale, which also yielded<br />

a logarithmic behaviour [35]. At longer times, however, the diffusional broadening increases<br />

faster than logarithmic. This increase occurs later at lower temperatures, which is to be expected<br />

from the temperature dependence of the relaxation rates. These new features are in<br />

qualitative agreement with theoretical work that invokes strong coupling of TLS with phonons<br />

[28, 36].<br />

We believe that our experimental work shows that the method of spectral hole burning<br />

spectroscopy is suitable for studying the low-temperature properties of amorphous solids.<br />

Especially for the investigation of the long-time behaviour of TLS equilibrium dynamics,<br />

where other methods cease to function, it is a very powerful experimental technique.<br />

References<br />

1. R.C. Zeller, R.O. Pohl: Phys. Rev. B, 4, 2029 (1971)<br />

2. Amorphous Solids. Low Temperature <strong>Properties</strong>, in:W.A. Phillips (ed.): Topics in current Physics,<br />

Vol. 24, Springer, Berlin, (1981)<br />

76

References<br />

3. S. Hunklinger, A. K. Raychaudhuri: in: D. F. Brewer (ed.), Progress in Low Temp. Physics,Vol. IX,<br />

Elsevier Science, Amsterdam, p. 265 (1986)<br />

4. P.W. Anderson, B.I. Halperin, C.M. Varma: Philos. Mag., 25, 1 (1972)<br />

5. W.A. Phillips: J. Low Temp. Phys., 7, 351 (1972)<br />

6. M.T. Loponen, R.C. Dynes,V. Narayanamurti, J.P. Garno: Phys. Rev. Lett., 45, 457 (1980)<br />

7. J. Zimmermann, G. Weber: Phys. Rev. Lett., 46, 661 (1981)<br />

8. B.M. Kharlamov, R.I. Personov, L.A. Bykovskaya: Opt. Commun., 12, 191 (1974)<br />

9. A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane: JETP Lett., 20, 216 (1974)<br />

10. J. Friedrich, D. Haarer: Angew. Chem., 96, 96 (1984); Angew. Chem. Int. Ed. Engl., 23, 113<br />

(1984)<br />

11. Th. Sesselmann, W. Richter, D. Haarer, H. Morawitz: Phys. Rev. B, 36(14), 7601 (1987)<br />

12. A.P. Marchetti, M. Scozzafara, R.H. Young: Chem. Phys. Lett., 51(3), 424 (1977)<br />

13. V.D. Samoilenko, N.V. Rasumova, R.I. Personov: Opt. Spectr., 52(4), 580 (in Russian) (1982)<br />

14. P.M. Selzer, D.L. Huber, D.S. Hamilton, W.M. Yen, M.J. Weber: Phys. Rev. Lett., 36, 813 (1976)<br />

15. T.L. Reinecke: Solid State Commun., 32, 1103 (1979)<br />

16. J.R. Klauder, P.W. Anderson: Phys. Rev., 125(3), 912 (1962)<br />

17. J.L. Black, B. I. Halperin: Phys. Rev. B, 16, 2879 (1977)<br />

18. W. Breinl, J. Friedrich, D. Haarer: J. Chem. Phys., 81(9), 3915 (1984)<br />

19. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88(1), 679 (1988)<br />

20. M.M. Broer, B. Golding, W.H. Haemmerle, J.R. Simpson, D.L. Huber: Phys. Rev. B, 33, 4160<br />

(1986)<br />

21. A. Gorokhovskii,V. Korrovits, V. Palm, M. Trummal: Chem. Phys. Lett., 125, 355 (1986)<br />

22. Korrovits, M. Trummal: Proceedings of the Academy of Sciences of the Estonian SSR, 35(2), 198<br />

(1986)<br />

23. K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991)<br />

24. W.A. Bosch, F. Mathu, H.C. Meijer, R.W. Willekers: Cryogenics, 26, 3 (1985)<br />

25. J. Jäckle: Z. Phys., 257, 212 (1972)<br />

26. G. Federle: PhD thesis, Max-Planck-Institut für Festkörperforschung Stuttgart (1983)<br />

27. S. Hunklinger, M. Schmidt: Z. Phys. B, 54, 93 (1984)<br />

28. K. Kassner: Z. Phys. B, 81, 245 (1990)<br />

29. A. Nittke, M. Scherl, P. Esquinazi, W. Lorenz, Junyun Li, F. Pobell: J. Low Temp. Phys., 98(5/6),<br />

517 (1995)<br />

30. L. Kador, G. Schulte, D. Haarer: J. Phys. Chem., 90, 1264 (1986)<br />

31. K.P. Müller: PhD thesis, Universität Bayreuth (1991)<br />

32. W.H. Chen, K.E. Rieckhoff, E.M. Voigt, L.W. Thewalt: Mol. Phys., 67(6), 1439 (1989)<br />

33. H. Maier, D. Haarer: J. Lum., 64, 87 (1995)<br />

34. S. Jahn, K.-P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)<br />

35. S. Jahn, D. Haarer, B.M. Kharlamov: Chem. Phys. Lett., 181, 31 (1991)<br />

36. K. Kassner, R. Silbey: J. Phys. Condens. Matter, 1, 4599 (1989)<br />

77

6 Optically Induced Spectral Diffusion in Polymers<br />

Containing Water Molecules: A TLS Model System<br />

Klaus Barth, Dietrich Haarer, <strong>and</strong> Wolfgang Richter<br />

6.1 Introduction<br />

In the past decade much experimental <strong>and</strong> theoretical work was done on spectral diffusion in<br />

amorphous solids. It turned out that at low temperatures, optical dephasing phenomena [1]<br />

as well as the numerous caloric data [2] can be well described by assuming the presence of<br />

low-energy excitations, the so-called two-level systems (TLS). Photochemical hole burning<br />

[3] <strong>and</strong> optical echo experiments [4] have provided experimental evidence for different dynamics<br />

of optical transitions in glassy systems as compared to crystalline matrices. For<br />

glassy organic solids the TLS concept was first proposed by Small <strong>and</strong> co-workers [5, 6]<br />

<strong>and</strong> was later used by Reinecke [7] to explain low-temperature optical line widths. Especially<br />

the photochemical hole burning data over long observation periods (observation times<br />

longer than three months) at temperatures down to 0.050 K allowed a critical test of the theoretical<br />

approach within the tunneling model [8]. Theoretical <strong>and</strong> experimental results to<br />

this topic are also given in this book by H. Maier et al.<br />

At temperatures above 1 K tunneling transitions <strong>and</strong> localized vibrations influence the<br />

physical properties [9]. Both types of mechanisms have recently been incorporated in the<br />

soft potential model [10] which contains the well-known tunneling model as a special case.<br />

A microscopic interpretation of optical line broadening phenomena in terms of e. g.<br />

local excitations of the matrix or switching of optically addressed chemical groups suffered<br />

from the lack of experimental data. In the past, several experimental techniques have been<br />

developed to shed light on the microscopic mechanisms of the phenomena. These techniques<br />

are all based on the generation of phonons in the matrix material, especially in the neighbourhood<br />

of the spectral probe. Heat pulses from external heaters [11] produce a broad distribution<br />

of phonon frequencies inside the chromophore host system <strong>and</strong> permit the investigation<br />

of the dynamics of barrier crossings <strong>and</strong> the coupling between the TLS <strong>and</strong> the dye<br />

molecules [12]. Light-induced phonon generation in hole burning systems can in principle<br />

be achieved either by exciting non-radiative decay processes in appropriate chromophores<br />

[13] or by direct absorption of IR light in the matrix material [13, 14]. The non-radiative decay<br />

method, which has been used to study transient <strong>and</strong> irreversible spectral diffusion processes,<br />

yields also a broad distribution of phonon frequencies <strong>and</strong> gives rise to similar experimental<br />

processes as e. g. the heat pulse technique. On the other h<strong>and</strong>, the IR absorption<br />

78 Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

6.2 Experimental setup for burning <strong>and</strong> detecting spectral holes<br />

method where the sample is illuminated with narrow b<strong>and</strong> IR light allows us to generate<br />

phonons of high density <strong>and</strong> of a comparatively small energy distribution in the direct neighbourhood<br />

of the optical probe. The contribution of the broad b<strong>and</strong> black-body background<br />

radiation can be minimized by using an appropriate intensity of a narrow b<strong>and</strong> IR source.<br />

For the optical investigation of amorphous systems, photochemical hole burning (PHB)<br />

is a high-resolution technique <strong>and</strong> a powerful tool to examine local sites in polymers at low<br />

temperatures. Dye molecules embedded in polymers at low concentration exhibit strongly inhomogeneously<br />

broadened absorption b<strong>and</strong>s in a conventional spectroscopic experiment. Selecting<br />

an ensemble of these dye molecules with narrow b<strong>and</strong> laser light, which gives rise to a<br />

photoreaction, leads to a sharp spectral dip at the laser frequency, the so-called spectral hole.<br />

Due to the high resolution, this method is appropriate for detecting small perturbations in the<br />

matrix such as a transient change of the phonon distribution or permanent matrix rearrangements<br />

in the vicinity of the dye molecules. Such rearrangements can be due to a configurational<br />

change of parts of the polymer main chain or due to a reorientation of small molecules<br />

(water), which may be embedded in the matrix in addition to the dye molecules.<br />

In the following, two different experimental conditions are discussed which are based<br />

on the absorption of IR light. Both experimental situations lead to a change of the local interaction<br />

of the optical probe with its local matrix environment. In the first part the phonons<br />

are treated as a time-dependent temperature bath. In the second part experiments are discussed<br />

where local groups of the matrix are selectively addressed by IR radiation of very<br />

low intensity. The observed enhanced spectral diffusion shows a characteristic dependence<br />

on the IR frequency <strong>and</strong> coincides with only a few of the numerous IR absorption b<strong>and</strong>s. A<br />

quantitative description of this new process together with the identification of the resonant<br />

vibrations is given within a simple kinetic model.<br />

6.2 Experimental setup for burning <strong>and</strong> detecting spectral holes<br />

The experimental setup for burning <strong>and</strong> detecting spectral holes with a narrow b<strong>and</strong> laser is<br />

described elsewhere [16]. The experiments were performed at 1.8 K in a bath cryostat in<br />

which the sample was immersed in superfluid helium. As samples we used different polymeric<br />

matrices such as polymethylmethacrylate (PMMA), polyamide (PA), polyethylene, polystyrene,<br />

etc. doped with free base phthalocyanine (H 2 Pc) at low concentration (10 –2 mol%). Their<br />

preparation is described in Ref. [16]. The water content of the samples has been controlled by<br />

heating them under vacuum conditions. Because of its strong hydrophilic nature PMMA has<br />

a high capacity of water absorption at atmospheric conditions. In order to investigate the influence<br />

of the water molecules the experiments, described in Section 6.4, were carried out either<br />

with a nearly water-free PMMA sample or with a PMMA sample containing water molecules<br />

under equilibrium conditions. The optical windows of the cryostat consisted of crystalline<br />

BaF 2 which allowed the study of hole burning spectra in the lowest vibrational b<strong>and</strong> of the<br />

electronic S 0 ? S 1 transition of H 2 Pc at about 0.69 mm as well as under IR illumination between<br />

2 mm <strong>and</strong> 11 mm. In this setup, the broad b<strong>and</strong> background radiation with an emission<br />

79

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules<br />

maximum at 10 mm has an integrated intensity of about 300 mW/cm 2 at the location of the<br />

sample. As IR radiation sources we used a Globar with a maximum emission intensity near<br />

3 mm <strong>and</strong> different CO 2 laser lines around 10 mm.<br />

An acousto-optic modulator was used for reducing the intensity of the CO 2 laser down<br />

to 1 W/cm 2 <strong>and</strong> for switching with times some microseconds. The radiation of the Globar<br />

was dispersed in a monochromator <strong>and</strong> with different dielectric filters to a b<strong>and</strong>width between<br />

0.05 mm <strong>and</strong> 0.2 mm with typical intensities around 100 mW/cm 2 .<br />

6.3 Reversible line broadening phenomena<br />

In this Section the mechanisms of reversible line broadening phenomena under continuous<br />

or pulsed irradiation conditions are discussed. Phonons are generated by illuminating the<br />

polymeric sample with the light of a single CO 2 laser line. The wavelength of the CO 2 laser<br />

was selected to generate high frequency phonons only within a small penetration depth of<br />

less than 10 mm. The dye molecules, which are homogeneously distributed throughout the<br />

sample thickness of some 100 mm, are a local probe for the increase in the phonon density<br />

during the IR irradiation. Due to the thermalisation processes, phonons show a broad energy<br />

distribution. This change of the phonon distribution gives rise to an increased effective temperature<br />

under cw irradiation conditions. Because of the high cooling rate of the surrounding<br />

superfluid helium, typical irradiation intensities of about 1 W/cm 2 were used to obtain a<br />

significant increase in the phonon density. The influence on the homogeneous line width of<br />

the dye molecules can be investigated in a zero burning fluence experiment, i. e. the limit of<br />

the hole width is given by extrapolating to zero burning fluence. The lower curve in Fig. 6.1,<br />

which was measured without any IR irradiation, yields an extrapolated value for the line<br />

Figure 6.1: Zero burning fluence experiment under different phonon density conditions in polyamide<br />

doped with H 2 Pc (see text).<br />

80

6.3 Reversible line broadening phenomena<br />

width of about 375 MHz. The two upper curves show increased hole widths due to an enhanced<br />

phonon generation either by IR irradiation during the burning process only(triangles)<br />

or during the detection process only(circles).<br />

The profile of a spectral hole can be written as<br />

Z 1 1<br />

A…! ! L †/<br />

nz…! 0 ! L †z…! ! 0 †d! 0 ; …1†<br />

where nz(o' – o L ) is the site distribution function of the molecular line contributing to the<br />

hole spectrum <strong>and</strong> z(o – o') is the absorption profile of a single molecule.<br />

On the time scale of this experiment (up to some minutes) the spectral hole profiles<br />

are mainly reversible (see also Fig. 6.2 <strong>and</strong> Fig. 6.3). Therefore, the homogeneous line profile<br />

function z(o) is much more affected by the altered phonon density than the site distribution<br />

function nz(o). Thus, it is obvious that the two upper curves in Fig. 6.1 are similar <strong>and</strong><br />

give for the hole width nearly the same zero burning fluence value of 440 MHz. Both curves<br />

are well separated from the lower curve which was obtained without any additional phonon<br />

generation. Applying an effective temperature model the difference between these two zero<br />

burning fluence values corresponds to a temperature change of about 0.3 K within the sample.<br />

Because of the strongly inhomogeneous experimental conditions during phonon generation<br />

<strong>and</strong> phonon diffusion through the sample, however, each different subensemble of dye<br />

molecules experiences a different distribution in phonon energies <strong>and</strong>, as a consequence, a<br />

bath description in terms of one single temperature is not appropriate. By using different op-<br />

Figure 6.2: Lower part: Spectral profile of a photochemical hole with <strong>and</strong> without phonon generation<br />

during the detection process. Upper part: Directly measured difference between these two profiles as<br />

obtained by modulating the IR light at a frequency of 150 Hz.<br />

81

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules<br />

Figure 6.3: Decay of the transmission maximum during a laser pulse.<br />

tical energies for the phonon generation, it is in principle possible to study the contribution<br />

of different matrix vibrations on the dephasing of a single absorber function z(o).<br />

The time scale of the hole burning experiments discussed above is several minutes. It<br />

is about 3 orders of magnitude larger than the typical thermal relaxation time of the sample<br />

as given by<br />

ˆ d2 c<br />

<br />

; …2†<br />

where d is the dimension (thickness) of the sample, c the specific heat, r the mass density,<br />

<strong>and</strong> k the thermal conductivity. For the temperature of our experiment (1.8 K) <strong>and</strong> a typical<br />

sample dimension of 1 mm the thermal relaxation time t is calculated to 10 ms.<br />

A consequence of this short relaxation time is shown in Fig. 6.2 <strong>and</strong> Fig. 6.3. The two<br />

spectral holes in the lower part of Fig. 6.2 were obtained with <strong>and</strong> without phonon generation<br />

during the hole detection process. A modulation of the IR light during the detection<br />

process results in a continuous switching between these two hole profiles. In the upper part<br />

of Fig. 6.2, a modulation frequency of 150 Hz was used during the scan of the optical frequency.<br />

The switching time between the two hole profiles is of the same order of magnitude<br />

as the thermal relaxation time. The continuous switching between the two hole profiles is<br />

detected with lock-in technique. The result shows that the phonon generation <strong>and</strong> relaxation<br />

process is sufficiently fast to follow the 150 Hz modulation of the IR light.<br />

A time-resolved experiment is shown in Fig. 6.3. The optical transmission at the peak<br />

of a hole spectrum was monitored while the sample was being illuminated with a single<br />

CO 2 laser pulse of 600 ms duration. The decay of the transmission signal cannot be described<br />

with a single exponential function. It is well represented by the superposition of two<br />

exponentials with a fast contribution t 1 & 10 ms <strong>and</strong> a slow contribution t 2 & 150 ms. The<br />

fast contribution is close to the thermal relaxation time at the temperature of the helium<br />

bath. The slow contribution contains all the inhomogeneous experimental conditions as mentioned<br />

above, in particular a temperature gradient inside the sample that gives rise to a variation<br />

of the thermal relaxation time. Since the amplitude of the fast contribution is larger by<br />

a factor of 4, the switching time between the two curves in Fig. 6.2 is mainly determined by<br />

the fast contribution t 1 .<br />

82

6.4 Induced spectral diffusion<br />

The experimental results of this Section show that optically generated phonons can be<br />

used to study the transient broadening of the optical line shape of a single absorber. In principle,<br />

the dependence on the phonon frequency can be studied in such an experiment. In the<br />

same experiment the time-resolved dynamics of phonon diffusion <strong>and</strong> phonon decay in polymeric<br />

systems can be investigated via the dephasing mechanism of the optical probe.<br />

6.4 Induced spectral diffusion<br />

In this Section we will demonstrate that the resonant absorption of IR light in a polymeric<br />

system can also lead to irreversible line broadening phenomena. Even at a very low irradiation<br />

intensity (a typical value is 100 mW/cm 2 ) the investigation of specific types of relaxation<br />

processes is possible. Such low irradiation intensities do not increase the temperature of<br />

the matrix by more than 0.01 K <strong>and</strong> hence yield no measurable dephasing effect via a<br />

change of the phonon distribution. Because of the irreversible nature of the optical line<br />

broadening, such phenomena are usually ascribed to spectral diffusion processes. The IR induced<br />

spectral diffusion is caused by addressing specific moieties within the matrix via the<br />

excitation of local vibrations. In the following weakly bound water molecules are studied.<br />

When a change in the spatial configuration takes place during the relaxation process of a<br />

water molecule the optical transition of a neighbouring dye molecule will be affected. Optical<br />

absorbers in glasses are therefore very sensitive probes for local rearrangements <strong>and</strong> are<br />

suited for a microscopic study of this topic.<br />

Due to the hydrophilic nature of the matrix the investigated guest host system, PMMA<br />

doped with H 2 Pc, contains a comparatively large number of water molecules (up to one volume<br />

percent). The experimental procedure is the following: A spectral hole is burned <strong>and</strong> its<br />

spectrum is repeatedly recorded while the sample is being exposed to IR radiation of a certain<br />

wavelength at a constant irradiation intensity. Figure 6.4 shows the typical increase of the hole<br />

Figure 6.4: Time evolution of the hole broadening induced by different IR wavelengths. From top to<br />

bottom: 2.80 mm, 2.86 mm, 2.90 mm, 2.96 mm, <strong>and</strong> 3.04 mm.<br />

83

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules<br />

Figure 6.5: Transmission spectra of PMMA without (curve A) <strong>and</strong> with (curve B) natural water content.<br />

The quotient C = B/A shows the H 2 O absorption lines. Dependence of the hole broadening on the<br />

IR wavelength after t 0 = 35 min (curve D) due to induced spectral diffusion.<br />

width <strong>and</strong> the strong dependence of this behaviour on the IR wavelength. To illustrate the pronounced<br />

spectral selectivity, one value of each hole broadening curve at t 0 = 35 min is plotted<br />

in Fig. 6.5 (curve D) versus the corresponding IR wavelength. The data points do not simply<br />

reflect the absorption behaviour of the sample. Although the PMMA matrix has many absorption<br />

b<strong>and</strong>s between 2 mm <strong>and</strong> 11 mm (curve B in Fig. 6.5), there are only two distinct wavelengths<br />

which give rise to induced spectral diffusion processes.<br />

The small concentration of water molecules in the matrix gives rise to non-saturated<br />

absorption b<strong>and</strong>s near 2.8 mm <strong>and</strong> 6.1 mm. They correspond to the fundamental stretching<br />

<strong>and</strong> bending vibrations of H 2 O, respectively. The difference spectrum (curve C in Fig. 6.5)<br />

of water free <strong>and</strong> water saturated PMMA yields the exact position of the H 2 O absorption<br />

b<strong>and</strong>s, which are in good agreement with the resonances found in our experiment. On an exp<strong>and</strong>ed<br />

scale in Fig. 6.5, even the asymmetric shape of the inhomogeneous H 2 O b<strong>and</strong> is reflected<br />

by the experimental data. Replacing protonated with deuterated water yields a red<br />

shift of the IR resonances which also agrees very well with the corresponding absorption<br />

spectra of heavy water.<br />

In analogy to the tunneling model, which is based on the assumption of two potential<br />

minima, we assume two stable sites for each water molecule (Fig. 6.6). Since the experimental<br />

results indicate that a permanent change in the matrix occurs transitions between the two<br />

sites are only allowed via the first excited vibrational states. Using such a simple four-level<br />

model for the underlying kinetics, we are able to explain the temporal behaviour of the induced<br />

spectral diffusion in a quantitative fashion.<br />

84

6.4 Induced spectral diffusion<br />

Figure 6.6: Energy level scheme of a H 2 O molecule in a two-site model.<br />

The two ground states are labelled 1 <strong>and</strong> 2, the first vibrationally excited states 3 <strong>and</strong><br />

4, respectively. The IR induced <strong>and</strong> the spontaneous transition rate for each site are denoted<br />

by the Einstein coefficients B <strong>and</strong> A, respectively. In our notation, B is proportional to the irradiated<br />

IR intensity. Transitions between the two sites are denoted by the conversion rate k.<br />

The transitions shown in Fig. 6.6 lead to a system of 4 coupled linear rate equations.<br />

Since we used in our experiments very low intensities of the IR light, we consider only the<br />

limit B P A where the population of the excited levels 3 <strong>and</strong> 4 is negligible at all times. The<br />

result is a time-dependent number of flips between the two ground states n f = n 1?2 + n 2?1 .<br />

Only those flips are taken into account which contribute to a change in the total configuration<br />

of all water molecules with respect to the time t = 0 of the hole burning process.<br />

According to Reinecke [7] the width of the spectral diffusion kernel Do is proportional<br />

to the number of flips n f . Though using the approximation that each flip yields the<br />

same contribution to Do, when taking into account the decrease of the IR intensity across<br />

the sample thickness as well as a r<strong>and</strong>om orientation of the H 2 O molecules, the result for<br />

Do can only be calculated numerically. Using a Taylor expansion, however, a very good analytical<br />

approximation of the width of the induced spectral diffusion is given by<br />

!…t† ˆG N <br />

0<br />

4pk 1 1 ‡ 2Bp !<br />

f t b<br />

: …3†<br />

b<br />

The difference between this formula <strong>and</strong> the numerical result is less than 5%. p f is the<br />

flip probability of a single H 2 O molecule as defined by p f = k /(A + k), b is a dimensionless<br />

geometry parameter of about 2, N 0 is the density of the H 2 O molecules <strong>and</strong> G is the coupling<br />

constant between dye <strong>and</strong> water molecules. Equation 3 describes very well the dependence<br />

of the broadening of a spectral hole on the number of absorbed IR photons (Fig. 6.4).<br />

It is possible to perform a test of the model. Equation 3 contains only two fitting parameters,<br />

the flip probability p f , <strong>and</strong> the coupling constant G. These two parameters can be obtained<br />

from a single experiment.<br />

An example is given in Fig. 6.7. One set of the experimental data (filled circles) is obtained<br />

at an IR intensity of 80 µW/cm 2 . The solid line is a fit according to Eq. 3 <strong>and</strong> yields<br />

a flip probability of 18% <strong>and</strong> a coupling constant of G = 3.6 7 10 –44 Jcm 3 .(Note: the slight<br />

difference with respect to the data given in reference [17] is due to the extension of the<br />

model by including the r<strong>and</strong>om orientation of the water molecules <strong>and</strong> the decrease of the<br />

IR intensity according to Beer’s law.) With these two values it is possible to calculate the<br />

85

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules<br />

Time (s)<br />

Figure 6.7: Test of the model. Hole broadening versus IR irradiation time for three different IR intensities.<br />

The solid line is a fit, the dashed lines are calculated according to Equation 3.<br />

time dependence of the induced spectral diffusion for any other irradiation intensity. The<br />

dotted lines in Fig. 6.7 are calculated with Eq. 3 for the obtained values <strong>and</strong> the intensities<br />

230 µW/cm 2 <strong>and</strong> 20 µW/cm 2 . The experimental results under these conditions, represented<br />

by the squares <strong>and</strong> triangles, are in very good agreement with the theoretical predictions.<br />

The high quantum yield of 18% for the flip of a single water molecule becomes obvious<br />

by a comparison of the number of absorbed IR photons (typically 10 14 s –1 ) with the<br />

total number of water molecules (about 10 18 ) in the sample volume. Since a strong induced<br />

spectral diffusion is observed within several minutes, a local reorientation process with a<br />

high quantum yield must be involved.<br />

The coupling strength between water <strong>and</strong> dye molecules can be estimated by taking<br />

into account the static dipole moment of the H 2 O molecule. A reorientation of the H 2 O molecules<br />

affects the dye molecules by the concomitant change in the local electric field at the<br />

location of the chromophores. Assuming a mean distance of 15 Å <strong>and</strong> inserting data from<br />

Stark effect experiments [18], one obtains a value of approximately 2710 –44 Jcm 3 for the<br />

coupling constant which is very close to our fitted value. Together with the resonant phenomena<br />

discussed above, this yields a detailed microscopic underst<strong>and</strong>ing of this special<br />

kind of interaction.<br />

We can conclude that spectral diffusion induced by the absorption of IR photons in<br />

the frequency range 2–11 µm is a resonant process. Its microscopic origin is a spatial rearrangement<br />

of weakly bound water molecules after vibrational excitation. Assuming a configurational<br />

model with only two possible sites for each water molecule <strong>and</strong> applying the theoretical<br />

model of spectral diffusion by Reinecke [7], an analytical description of the resonant<br />

hole broadening behaviour can be obtained. A variation of the IR irradiation intensity shows<br />

that the model description yields excellent agreement with the experimental data.<br />

86

References<br />

References<br />

1. S. Jahn, K. P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)<br />

2. M. Deye, P. Esquinazi: Zeitschr. für Phys. B, 39, 283 (1989)<br />

3. J. Friedrich, D. Haarer: Angew. Chem., Int. Ed. Engl,. 23, 113 (1984)<br />

4. C. A. Walsh, M. Berg, L. R. Narashiman, M. D. Fayer: Chem. Phys. Lett., 130, 6 (1986)<br />

5. J. M. Hayes G. J. Small: Chem. Phys., 27, 151 (1978)<br />

6. G. J. Small: Persistent nonphotochemical hole burning <strong>and</strong> the dephasing of impurity electronic<br />

transitions in organic glasses, in: V. M. Agranovich R. M. Hochstrasser (eds.): Spectroscopy <strong>and</strong><br />

Excitation Dynamics of Condensed Molecular Systems, North-Holl<strong>and</strong>, Amsterdam, p. 515 (1983)<br />

7. T. L. Reinecke: Sol. Stat. Com., 32, 1103 (1979)<br />

8. H. Maier, D. Haarer: J. Lumin., 64, 87 (1995)<br />

9. R. Greenfield, Y. S. Bai, M. D. Fayer: Chem. Phys. Lett., 170, 133 (1990)<br />

10. D. A. Parshin: Phys. Sol. State, 36, 991 (1994)<br />

11. U. Bogner: Phys. Rev. Lett., 37, 909 (1976)<br />

12. T. Attenberger, K. Beck, U. Bogner: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Proc. of the<br />

3rd Int. Conf. on Phonon Physics, p. 555<br />

13. A. A. Gorokhovskii, G. S. Zavt, V. V. Palm: JETP Lett., 48, 369 (1988)<br />

14. W. Richter, Th. Sesselmann, D. Haarer: Chem. Phys. Lett., 159, 235 (1989)<br />

15. W. Richter, M. Lieberth, D. Haarer: JOSA B, 9, 715 (1992)<br />

16. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88, 679 (1988)<br />

17. K. Barth W. Richter: J. of Lumin., 64, 63 (1995)<br />

18. R. B. Altmann, I. Renge, L. Kador, D. Haarer: J. Chem. Phys., 97, 5316 (1992)<br />

87

7 Slave-Boson Approach to Strongly Correlated Electron<br />

Systems<br />

Holger Fehske, Martin Deeg, <strong>and</strong> Helmut Büttner<br />

7.1 Introduction<br />

The problem of underst<strong>and</strong>ing high-temperature superconductivity has been a challenge to<br />

theoreticians from a wide variety of fields. Many theoretical investigations have been carried<br />

out in order to identify the mechanism of this fascinating phenomenon as well as to establish<br />

the canonical model itself [1]. Although excellent progress is being made in deducing<br />

a consistent description of the high-temperature superconductivity systems from a<br />

first-principles theory, at present no microscopic theory can account for their unconventional<br />

normal-phase data in its entirety. The interplay of charge <strong>and</strong> spin dynamics in the<br />

normal state seems to hold the key to the underst<strong>and</strong>ing of the physical mechanism behind<br />

high-temperature superconductivity in the cuprates. Both macroscopic measurements of<br />

transport <strong>and</strong> magnetic properties as well as microscopic measurements probing the charge<br />

<strong>and</strong> spin excitation spectra are fundamental in establishing the anomalous normal-state<br />

properties of these materials [1].<br />

The nature of spin excitations of the high-temperature superconductivity cuprates has<br />

been experimentally studied by means of nuclear magnetic/quadrupole resonance (NMR/<br />

NQR) <strong>and</strong> inelastic neutron scattering (INS) techniques clarifying the persistence of strong<br />

antiferromagnetic (AFM) correlations in the normal <strong>and</strong> superconducting states [2]. Detailed<br />

investigations of the wave-vector dependence of the low-frequency spin fluctuation spectrum<br />

have revealed remarkable differences between the YBa 2 Cu 3 O 6+x <strong>and</strong> La 2–x Sr x CuO 4 families<br />

at low doping level x; the dynamic structure factor S (~q, o) keeps its maximum at (p,p) in<br />

the YBCO system, while in LSCO, the peaks are displaced from the commensurate position<br />

to the four incommensurate wave-vectors p(1 +q 0 , 1), p (1,1+q 0 ), where q 0 F2x. On the<br />

other h<strong>and</strong>, the analysis of the NMR data shows that the relaxation rates on the planar Cu<br />

sites are similar in all materials.<br />

Two striking features are associated with 63 T 1 –1 :<br />

a) as a result of strong local spin fluctuations on 63 Cu sites it is enhanced by one order of<br />

magnitude over the oxygen rate 17 T 1 –1 ; <strong>and</strong><br />

b) in sharp contrast to the Korringa-like behaviour at the planar 17 O sites, its temperature<br />

dependence does not follow the Korringa law [2].<br />

88 Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

7.1 Introduction<br />

The physical origin of the contrasting ~q-dependence of the spin fluctuation spectrum<br />

is still under discussion. Millis <strong>and</strong> Monien [3] have argued that the spin dynamics <strong>and</strong>, in<br />

particular, the temperature dependence of the spin susceptibility w s (T) in LSCO are caused<br />

by a spin density wave instability, whereas in the YBCO family they are due to in-plane<br />

AFM fluctuations <strong>and</strong> a novel non-Fermi liquid spin singlet pairing of electrons in adjacent<br />

planes. On the other h<strong>and</strong>, the magnetic properties are intimately related to the energy b<strong>and</strong><br />

dispersion of the non-interacting system within a Fermi liquid based framework, i. e., in this<br />

way the observed spin dynamics can be attributed to different Fermi surface (FS) geometry<br />

of LSCO-type <strong>and</strong> YBCO-type, respectively. Along this line, details of the spin fluctuation<br />

spectrum are studied using a nearly antiferromagnetic Fermi liquid approach by Monthoux<br />

<strong>and</strong> Pines [4]. From a more microscopic point of view, the important effects of FS shape on<br />

the magnetic properties were confirmed by Si et al. [5] within a large Coulomb-U auxiliary<br />

boson scheme <strong>and</strong> by Fukuyama <strong>and</strong> co-workers [6] on the basis of a resonating valence<br />

bond (RVB) slave-boson mean-field approach to the extended t-J model. Furthermore, Ito et<br />

al. [7] have recently reported that the charge transport in the CuO 2 plane is determined by<br />

dominant spin scattering, i. e., the spin dynamics are manifest in the extraordinary transport<br />

properties of the high-temperature superconductivity cuprates as well.<br />

Encouraged by these findings it is the aim of this report to study magnetic <strong>and</strong> transport<br />

phenomena of high-temperature superconductors probably in terms of the most simple<br />

effective one-b<strong>and</strong> model describing both correlation <strong>and</strong> b<strong>and</strong> structure effects, the socalled<br />

t-t'-J model:<br />

H t t 0 J ˆ t X hi;ji;<br />

~c y i ~c j t 0 X<br />

h<br />

hi;jii;<br />

~c y i ~c j ‡ J X ij h i<br />

<br />

~n i ~n j<br />

~S i<br />

~S j<br />

4<br />

: …1†<br />

H t–t'–J acts in a projected Hilbert space without double occupancy, where ~c …y†<br />

c …y†<br />

i …1 ~n i † is the electron annihilation (creation) operator, ~S i ˆ 1<br />

P<br />

0 ~cy i ~ 0 ~c i<br />

i =<br />

0;<br />

2<br />

<strong>and</strong><br />

~n i ˆ P ~cy i ~c i. J measures the AFM exchange interaction, t <strong>and</strong> t' denotes hopping processes<br />

between nearest-neighbour (NN; Ai,jS) <strong>and</strong> next nearest-neighbour (NNN; AAi,jSS) sites<br />

on a square lattice. Compared to the original t-J model the t'-term incorporates several important<br />

effects near half-filling. Starting from a rather complex three-b<strong>and</strong> Hubbard or Emery<br />

model [8] for the CuO 2 planes, quantum cluster calculations [9] have revealed that the<br />

relative large direct transfer between NN oxygen sites (t pp < t pd /2) leads to a sizeable NNN<br />

hopping t' in the context of an effective one-b<strong>and</strong> description. More recently the t'-term has<br />

been introduced to reproduce the FS geometry observed in ARPES experiments [6, 10–12].<br />

Fitting the quasi-particle dispersion relation<br />

" ~k ˆ 2t…cos k x ‡ cos k y † 4t 0 cos k x cos k y ; …2†<br />

involved in Eq. 1 to experimental <strong>and</strong> b<strong>and</strong> theory results yields t in the order of 0.3 eV <strong>and</strong>,<br />

e. g., for the case of YBCO, t'&–0.4 t [4, 5, 13]. Moreover, Tohyama <strong>and</strong> Maekawa [12]<br />

have emphasized that a t-t'-J model with t' > 0 can be used to describe the electron-doped<br />

systems, e. g. Nd 2–x Ce x CuO 4 (NCCO). In this case one has to shift the momentum ~ k ? ~ k +<br />

(p,p) [12], i. e., within a b<strong>and</strong>-filling scenario one obtains a hole pocket-like FS centred at<br />

(p,p)-point which shrinks with increasing doping [14]. <strong>Final</strong>ly, as pointed out by Lee [15],<br />

89

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

in a locally AFM environment doped holes can propagate coherently only on the same sublattice<br />

without disturbing spins. Therefore, the t'-term coupling the same sublattice becomes<br />

crucial for the low-lying magnetic excitations. This clearly is a correlation effect related to<br />

the NNN hopping processes.<br />

7.2 Slave-boson theory for the t-t'-J model<br />

Apart from numerical techniques, the slave-particle methods have been employed extensively<br />

in studying the effects of electron-electron correlation in the (extended) t-J model [6, 16–21].<br />

The two commonly used approaches, the NZA slave-boson [16, 17] <strong>and</strong> slave-fermion [18]<br />

schemes, however, yield quite different results concerning the (mean-field) ground-state phase<br />

diagram <strong>and</strong> spin/charge excitations for this model [22]. As yet, the relationship between both<br />

types of slave-field theories is not well understood. Within the NZA formulation, for example,<br />

the Hamiltonian can be solved by a mean-field approximation with the (uniform) RVB order<br />

parameter. A serious difficulty of this approach is the absence of AFM correlations [18, 22],<br />

e. g., in the half-filled case, the lowest energy state, the energy of which is considerable higher<br />

than numerical estimates indicate, does not satisfy the Marshall sign rule <strong>and</strong> fails to show the<br />

expected long-range Néel order [23]. On the other h<strong>and</strong>, the mean-field slave-fermion schemes<br />

[18] are known to give reasonable results for the spin susceptibility as well as for the spin correlation<br />

length [21, 22], but also suffer from neglecting important correlation effects, especially<br />

the fermion charge degrees of freedom are not described sufficiently well. In contrast, the fourfield<br />

slave-boson (SB) technique, introduced by Kotliar <strong>and</strong> Ruckenstein (KR) [24] in the context<br />

of the Hubbard model, has the advantage of treating spin <strong>and</strong> charge degrees of freedom on<br />

an equal footing. Starting from the scalar KR SB representation of the Hubbard model, one may<br />

generate an intersite exchange interaction via a loop expansion in the coherent state functional<br />

integral [19]. However, the effective t-J Lagrangian, derived in this way, contains only an Ising<br />

interaction term, i. e., important spin-flip exchange processes are neglected. As we shall see below,<br />

to bosonize the complete exchange interaction term one has to use the spin-rotation-invariant<br />

(SRI) extension of the KR SB theory from the beginning [25, 26].<br />

7.2.1 SU(2)-invariant slave-particle representation<br />

For the sake of definiteness, we return to the extended t-J Hamiltonian (Eq. 1), that may be<br />

cast into the form<br />

H t t 0 J ˆ X<br />

t ij<br />

~ y i<br />

i;j<br />

~ j<br />

J<br />

4<br />

X <br />

hiji<br />

~ y i <br />

<br />

~ i ~ y<br />

j ~ <br />

j : …3†<br />

90

7.2 Slave-boson theory for the t-t'J model<br />

In order to bring out the SU(2) symmetry of the system, we have used in Eq. 3 a spinor<br />

representation, where the one-row [one-column] matrices ~ y i ˆ…~ y i ; ~ i †‰ ~ i Š are built up<br />

by the projected fermion creation [annihilation] operators ~ y i ~cy i ‰ ~ i ~c i Š: ‰ Š denotes<br />

the contravariant [covariant] four-component vector (m =0,x, y, z) of Pauli’s matrices.<br />

To preserve SRI, we apply to H t-J the manifest [SU(2) 6 U(1)] invariant SB scheme [26,<br />

27] based on the SRI SB approach developed for the Hubbard model by Li et al. [28]. Accordingly,<br />

we define scalar boson fields e ({) i <strong>and</strong> bosonic matrix operators p ({) i (representing<br />

empty <strong>and</strong> singly occupied sites, respectively) <strong>and</strong> pseudofermion spinor fields C ({) i in the<br />

following way:<br />

j0 i i ˆ e y i jvaci;<br />

j i i ˆ X y<br />

<br />

i py i vac<br />

j i: …4†<br />

The unphysical states in the extended Fock space of pseudofermionic <strong>and</strong> bosonic<br />

states are eliminated by imposing two sets of local constraints:<br />

C …1†<br />

i<br />

ˆ e y i e i ‡ 2Trp y i p i<br />

1 ˆ 0 ; …5†<br />

expressing the completeness of the bosonic projectors, <strong>and</strong><br />

C …2†<br />

i<br />

ˆ i <br />

y<br />

i ‡ 2p y i p i<br />

o ˆ 0 ;<br />

…6†<br />

relating the pseudofermion number to the number of p-type slave-bosons (hereafter underbars<br />

denote a 262 matrix in the spin variables). Obviously, double occupancy has been projected<br />

out. In the transformation of the fermionic spinor fields analogous to [26, 28],<br />

~ i ! z i i ; …7†<br />

the non-linear bosonic hopping operators<br />

z i ˆ‰ o 2p y i p i Š 1=2 e y i ‰1 ‡ ey i e i ‡ 2Trpy i p i Š1=2 p i<br />

‰…1 e y i e i † o 2 ~p y i ~p i Š 1=2 …8†<br />

yield a correlation-induced b<strong>and</strong> renormalization where ~p ({)<br />

irr' = rr'p ({)<br />

i,–r',r. Exploiting the<br />

SU(2),O(3) homomorphism, the matrix operators p i<br />

may be decomposed into scalar (singlet)<br />

p o ({) <strong>and</strong> vector (triplet) ~p i =(p ix , p iy , p iz ) components as<br />

p …y†<br />

i<br />

ˆ 1<br />

2<br />

X<br />

<br />

p …y†<br />

i ;<br />

…9†<br />

where the p im obey the usual Bose commutation rules ‰p i; p y j 0Šˆ ij 0. Consequently, the pseudofermions<br />

i ˆ… i; i † T satisfy anticommutation relations of the form f i ; y<br />

j 0gˆ ij 0.<br />

91

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

This way the interaction term is converted due to<br />

~ y i ~ i ! 2Trp y i p i<br />

; …10†<br />

where, more explicitly, the locally defined particle number <strong>and</strong> spin operators are just<br />

~n i ! ~n i …p i †ˆ2Trp y p ˆ P p y p ;<br />

i i i i<br />

~S i ! ~S i …p i †ˆTrp y i<br />

<br />

~p i ˆ 1<br />

2 …py io ~p i ‡ ~p y i p io i~p y i ~p i † : …11†<br />

Actually, the components of the bosonized SB spin operator act as generators of rotations<br />

in spin space, i. e., ~S i …p i † satisfies the spin algebra. Since C (1) i <strong>and</strong> C …2†<br />

i commute with<br />

the SB t-J Hamiltonian, the constraints in Eq. 5 <strong>and</strong> Eq. 6 can be ensured by introducing the<br />

time-independent Lagrange multipliers …1†<br />

i <strong>and</strong> …2†<br />

i<br />

ˆ P<br />

…2†<br />

i .1 As a consequence, the<br />

Hamiltonian of the t-J model (Eq. 3) in terms of the slave-boson <strong>and</strong> pseudofermion operators<br />

has to be replaced by<br />

H SB<br />

t t 0 J ˆ P<br />

t ij<br />

i;j<br />

‡ P i<br />

y<br />

i zy i z j j<br />

… …1†<br />

i C …1†<br />

i<br />

J P <br />

<br />

Trp y i p i<br />

Trp y j p j<br />

hiji<br />

‡ Tr …2†<br />

i C …2†<br />

i † : …12†<br />

In the physical subspace, H SB<br />

t–J possesses the same matrix elements for the basis states<br />

(Eq. 4) as the original t-J Hamiltonian (Eq. 3) for the purely fermionic states. To verify this<br />

directly, special attention has to be paid to the bosonization of AFM exchange interaction<br />

term<br />

<br />

~ y <br />

i ~ i<br />

~ y j ~ <br />

j ˆ 2 P …~c y i ~cy j ~c j ~c i <br />

~c y i ~cy j ~c j ~c i † ; …13†<br />

<br />

<br />

which includes besides the Ising exchange contributions # i " j $ <br />

#i " j the spin-flip processes<br />

# i " j $ "i # j . Let us emphasize that the matrix elements of the spin-flip terms<br />

are not reproduced in the scalar KR SB approach [19]. By contrast, within the SRI SB approach<br />

it is a straightforward exercise to show that these contributions can be expressed in<br />

terms of the p ({) irr' :<br />

X<br />

p y i py j 0 p j p <br />

1<br />

0 i i j ˆ<br />

4 <br />

<br />

i j :<br />

0<br />

…14†<br />

Thus our SRI SB scheme (Eqs. 4–12) provides a consistent bosonization of the extended<br />

t-J model.<br />

1 Note that additional constraints do not exist. Especially, if the constraint ~p i ~p i ˆ 0 is added [25] the<br />

spin algebra is not satisfied.<br />

92

7.2 Slave-boson theory for the t-t'J model<br />

7.2.2 Functional integral formulation<br />

To proceed further, it is convenient to represent the gr<strong>and</strong> canonical partition function for<br />

the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29]<br />

over Grassmann fermionic <strong>and</strong> complex bosonic fields as<br />

Z<br />

R <br />

Z ˆ D‰ ; ŠD‰e ;eŠD‰p ; p Š d‰ …1† Š d‰ …2†<br />

dL…†<br />

Š e 0 ; …15†<br />

<br />

L…† ˆP<br />

e i …@ ‡ …1†<br />

i †e i ‡ 2Trp T ‰…@ ‡ …1†<br />

i † o …2†T<br />

i<br />

‡ P <br />

<br />

Trp ~p i i Trp~p Trp p j j i i Trpp j j<br />

hi;ji<br />

‡ P <br />

<br />

i …@ † o ‡ …2†<br />

i i ‡ P t ij<br />

i z i z j j :<br />

i<br />

i;j<br />

i<br />

i<br />

Šp T i<br />

…1†<br />

i<br />

<br />

…16†<br />

Apart from the above symmetry considerations the SB functional-integral formalism<br />

reveals additional global <strong>and</strong> local gauge invariances [25, 30–32].<br />

The action S ˆ R <br />

0<br />

dL…† is invariant under the following site <strong>and</strong> time-dependent<br />

phase transformations (y i (t)w i<br />

(t)), i. e., under the local symmetry group SU(2) 6U(1),<br />

i ! e ii e i i i ;<br />

e i ! e i e i i<br />

; …17†<br />

p i<br />

! p i<br />

e i i ;<br />

provided that the (five) Lagrange parameters act as time-dependent gauge fields, l (1) i (t) <strong>and</strong><br />

…t†, absorbing the time derivatives of the phase factors:<br />

l …2†<br />

i<br />

…1†<br />

i<br />

! …1†<br />

i ‡ i _ i ;<br />

…2†<br />

i<br />

! e i i …2†<br />

i e i i i _ i<br />

‡ i _ i o : …18†<br />

Next, in the continuum limit, the radial gauge is introduced by representing the Bose<br />

fields by modulus <strong>and</strong> phase,<br />

e i ˆ je i je i'…ei† ;<br />

p i ˆ 1<br />

2<br />

X<br />

<br />

p i<br />

e<br />

i …p † i ;<br />

<br />

…19†<br />

93

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

(for a time-discretized version of this gauge fixing, see [31]). Exploiting the gauge freedom<br />

of the action, we can now fix the five real-valued coefficients y i (t), w im (t) to remove five<br />

phases (j i (t), f im (t)) of the Bose fields e i , p im in the radial gauge. As a consequence, all<br />

the Bose fields {e i , p im } become real, in contrast to the Hubbard model, where one SB field<br />

remains complex [25, 32]. At this point one should notice that the particle number <strong>and</strong> spin<br />

operators are changed into ~n i …p i †ˆP p2 i ; ~S i …p i †ˆp io ~p i (the previous notations e i , p io<br />

<strong>and</strong> ~p i now denote the radial parts of the corresponding Bose fields). Then, using the familiar<br />

identity for Gaussian integrals over Grassmann fields [29],<br />

R<br />

D‰ ; Š e ‰ G 1 Š 0 <br />

ˆ e Tr ln‰ G 1Š ; …20†<br />

the fermionic degrees of freedom can be integrated out <strong>and</strong> we obtain the following exact representation<br />

of the gr<strong>and</strong> canonical partition function<br />

Z ˆ R D‰Š e S ef f<br />

…21†<br />

in terms of the real-valued bosonic fields f ia ,<br />

i …† ˆ… i …†† ˆ …e i ;p io ; …2†<br />

io ;…1† i<br />

; p ix ; …2†<br />

ix ; p iy; …2†<br />

iy ; p iz; …2†<br />

iz † :<br />

…22†<br />

In Eq. 21 the effective bosonic action S eff takes the form<br />

(<br />

S ef f ˆ R d P<br />

0 i<br />

<br />

…1†<br />

i<br />

e 2 i ‡ P <br />

<br />

…1†<br />

i<br />

…2†<br />

io<br />

<br />

p 2 i<br />

2p io ~p i ~ …2†<br />

i<br />

…1†<br />

i<br />

<br />

‡J P )<br />

1<br />

p io ~p i ~p j p jo<br />

hiji<br />

4 …p2 io ‡ ~p2 i †…p2 jo ‡ ~p2 j †<br />

h<br />

i<br />

Tr ij; 0 ; 0 ln G ij; 1 † ; …23†<br />

where the (inverse) SB Green propagator is given by<br />

h<br />

<br />

Gij; 1 0…; 0 †ˆ @ ‡ …2†<br />

io<br />

0<br />

i<br />

~ …2†<br />

i ~ 0 ij … 0 †<br />

t ij …z y i z j† 0 ; 0…1 ij† : …24†<br />

It may be remarked that since the bosons are taken to be real, their kinetic terms,<br />

being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary<br />

conditions imposed on Bose fields (f ia (b) =f ia (0)). Strictly speaking it follows from this<br />

property that all the Bose fields do no longer have dynamics of their own [25].<br />

94

7.2 Slave-boson theory for the t-t'J model<br />

7.2.3 Saddle-point approximation<br />

The evaluation of Eq. 23 proceeds via the saddle-point expansion 2 , where at the first level<br />

of approximation we look for an extremum S … i † of the bosonized action S eff with respect<br />

to the Bose <strong>and</strong> Lagrange multiplier fields f ia :<br />

@S ef f<br />

@ i<br />

ˆ!<br />

0 ) S ˆ S ef f<br />

<br />

iˆ i<br />

: …25†<br />

The physically relevant saddle-point F i is determined to give the lowest free energy<br />

(per site)<br />

f t t 0 J ˆ =N ‡ n ; …26†<br />

where at given mean electron density n =1–d, the chemical potential m is fixed by the requirement<br />

n ˆ 1 @<br />

N @<br />

…27†<br />

( O ˆ S= denotes the gr<strong>and</strong> canonical potential). Obviously, an unrestricted minimization of<br />

the free energy functional is impossible for an infinite system. To keep the problem tractable,<br />

we use the ansatz<br />

~m i ˆ m~u i ; ~u i ˆ…cos ~Q ~R i ; sin ~Q ~R i ; 0† …28†<br />

for the local magnetization ~m i ˆ 2 ~S i ˆ 2p io ~p i . Following earlier analyses of spiral states<br />

for the Hubbard model [34] the unit vector ~u i is chosen as a local spin quantization axis<br />

pointing in opposite directions on different sublattices. Thereby, the order parameter wavevector<br />

~Q is introduced as a new variational parameter to describe several magnetic ordered<br />

states: PM, FM ( ~Q = 0), AFM ( ~Q =(p,p)), <strong>and</strong> incommensurate (1,1)-spiral ( ~Q =(Q, Q)),<br />

(1,p)-spiral ( ~Q =(Q,p)) <strong>and</strong> (0,1)-spiral ( ~Q = (0,Q)) states. Note that since fluctuations of<br />

the charge density are not incorporated the scalar Bose fields are homogeneous: e i = e, p io =<br />

p o , l (1) i = l (1) , <strong>and</strong> l (2) io = l (2) o . The vector fields exhibit the same spatial variation as the magnetization:<br />

~p i ˆ p~u i <strong>and</strong> l ~ …2†<br />

i ˆ l …2† ~u i . Then transforming Eq. 23 in the ( k; ~ o)-representation<br />

the trace in S eff can be easily performed to give the free energy functional<br />

2 Mean-field approximations to the functional integral are achieved by replacing the bosonic fields by<br />

their time averaged values. For the Hubbard model, a comparison of the paramagnetic <strong>and</strong> antiferromagnetic<br />

SB solutions with quantum Monte Carlo results shows that such a mean-field like approach<br />

yields an excellent quantitative agreement for local observables [33] <strong>and</strong> therefore can give a qualitative<br />

correct picture with relatively little effort.<br />

95

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

where<br />

f t t 0 J ˆ …1† …e 2 ‡ p 2 o ‡ p2 1† o …2† …p2 o ‡ p2 † 2 …2† p o p ‡ n<br />

<br />

<br />

‡J p 2 1<br />

o p2 …cos Q x ‡ cos Q y †<br />

2 …p2 o ‡ p2 † 2<br />

‡ 1<br />

N<br />

X<br />

ln‰1 n ~k Š ; …29†<br />

~k<br />

n ~k ˆ‰expf…E ~k<br />

†g ‡ 1Š<br />

1<br />

…30†<br />

holds. The renormalized single-particle energies E ~k … ˆ†are obtained by diagonalizing<br />

the kinetic part of Eq. 23 [26]. Requiring that f t-J be stationary with respect to the variation of<br />

the magnetic order vector, the wave-vector ~Q can be obtained from the extremal condition<br />

sin Q x;y ˆ 1<br />

Jp 2 o p2 1<br />

N<br />

X<br />

~k<br />

n ~k<br />

@E ~k<br />

@Q x;y<br />

:<br />

…31†<br />

If one substitutes Eq. 31 together with f a (obtained from the solution of the coupled<br />

self-consistency equations (Eq. 25)) into Eq. 29, the free energy of the t-t'-J model is obtained<br />

as<br />

f t t 0 J ˆ J<br />

4 ‰m2 …cos Q x ‡ cos Q y † 2n 2 Š‡ …2† m ‡… o …2† †n<br />

1 X <br />

ln 1 ‡ e …E k ~ ; …32†<br />

N<br />

~k<br />

where the quasi-particle energy takes on the form<br />

E ~k ˆ …1 ‡ †…" ~k ‡ " ~k<br />

†=y ‡ …2†<br />

~Q o<br />

h i 1=2<br />

2 …" ~k " ~k ~Q †2 =y ‡‰m…" ~k ‡ " ~k ~Q †=y ‡ …2† Š 2<br />

…33†<br />

with m=2 p o p <strong>and</strong> y=(1+d) 2 – m 2 .<br />

In particular, at the spatially uniform paramagnetic saddle-point, F …PM† =(e, p o , l o (2) ,<br />

l (1) ; 0, 0; 0,0; 0,0) the remaining bosonic fields<br />

96<br />

e 2 ˆ ;<br />

p 2 o ˆ 1 ;<br />

…1† ˆ<br />

…2†<br />

o<br />

ˆ<br />

2 ‡ 3 2<br />

1 2 ~"…0† ;<br />

2<br />

~"…0† …1 †J; …34†<br />

2<br />

…1 ‡ †

7.2 Slave-boson theory for the t-t'J model<br />

are explicitly given in terms of J, d <strong>and</strong> a single energy parameter ~" (0) defined by<br />

~"…~q† ˆ 2<br />

N<br />

X<br />

" ~k ~q<br />

… E ~k † …35†<br />

~k<br />

(at T=0). Here, the quasi-particle energy E ~k is<br />

E ~k ˆ 2" ~k =…1 ‡ †‡ …2†<br />

0 ; …36†<br />

<strong>and</strong> m can be determined from d =1– 2 N<br />

free energy becomes simply<br />

P<br />

~k … E ~ k<br />

†. <strong>Final</strong>ly, in this approximation, the<br />

f …PM†<br />

t t 0 J ˆ z2 ~"…0†<br />

1<br />

2 J …1 †2 ‡ …1 † : …37†<br />

7.2.4 Magnetic phase diagram of the t-t'-J model<br />

In the numerical evaluation of the self-consistency loop we proceed as follows: at given<br />

model parameters J <strong>and</strong> d, we solve the remaining saddle-point equations for m, l (1) , l o<br />

(2)<br />

<strong>and</strong> l (2) together with the integral equation for m using an iteration technique. Then, in an<br />

outer loop, the order parameter wave-vector ~Q is obtained from the extremal Eq. 31 by<br />

means of a secant method. Convergence is achieved if all quantities are determined with relative<br />

error less than 10 –6 . Note that our numerical procedure allows for the investigation of<br />

different metastable symmetry-broken states corresponding to local minima of the variational<br />

free energy functional (Eq. 32).<br />

At first, let us consider the case t' = 0, i. e., the pure t-J model. The resulting groundstate<br />

phase diagram in the J-d plane is shown in Fig. 7.1. For the case d = 0 (Heisenberg<br />

model), we obtain an AFM ground state. At J=0, the FM is lowest in energy up to a hole concentration<br />

of 0.327, where a first-order transition to the (1, p)-spiral takes place; above<br />

d = 0.39, we find a degenerate ground state with wave-vector (0,p). At d = 0.63 the PM becomes<br />

the lowest in energy state (second-order transition). This coincides with the U ?? SB<br />

results of the Hubbard model (J=4t 2 /U) [35]. The PM-FM instability occurs at d PM-FM = 0.33<br />

[32]. In contrast, the slave-fermion phase diagram [20] exhibits a much larger FM region.<br />

This can be taken as an indication that correlation effects are treated less accurate within the<br />

slave-fermion mean-field scheme [20, 36]. For finite exchange interaction J the (1,1)-spiral<br />

is the ground state at small doping. With increasing d a transition to the FM takes place,<br />

which becomes unstable against the (1,p)-spiral at larger doping concentrations. For<br />

J/t > 0.08, we find a transition from (1,1)-spirals to (1,p)-spirals at about d & 0.2. In<br />

Figure 7.1, the dotted line separates the (1, p)-spiral state from the region, where the (1, p)<br />

<strong>and</strong> (0, p) states are degenerate. If we admit only homogeneous phases, the phase boundaries<br />

(1,1)-spiral ⇔ (1,p)-spiral at d & 0.2 <strong>and</strong> (1,p)/(0, p)-phase ⇔ PM at d &0.63 remain<br />

nearly unchanged at larger values of J. However, analyzing the thermodynamic stability of<br />

97

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

Figure 7.1: SB ground-state phase diagram of the t-J model.<br />

the saddle-point solutions (i. e. the curvature of f t J …d), one observes a tendency towards<br />

phase separation into hole-rich <strong>and</strong> AFM regions above the ‘diagonal’ solid curve in the left<br />

upper part of the J-d phase diagram (see below).<br />

Figure 7.2 displays the variation of the extremal spiral wave-vector ~Q as function of<br />

doping. At d =0, ~Q =(p, p) indicates the AFM order. At low doping the (1,1)-spiral order vector<br />

decreases approximately linear. With decreasing J the AFM exchange is weakened, consequently<br />

the deviation of the order vector from (p, p) increases. The discontinuities reflect the<br />

first-order transition from (1,1)-spirals to (1, p)-spirals. For J/t =0.05 the transition to the FM<br />

state takes place, with Q x jumping down to zero. The (1,p)-spiral wave-vector shows a monotonous<br />

decrease of Q x until at d = 0.63 the transition to the PM (Q x = 0) occurs. Comparing<br />

the magnitude of the theoretical order vector of the spiral solutions to results from inelastic<br />

neutron scattering experiments on La 2–x Sr x CuO 4 [37], we find good agreement for an exchange<br />

interaction strength of J/t =0.4 (which seems to be a reasonable value with respect to<br />

the strong electron correlations observed in the high-temperature superconductors).<br />

Figure 7.2: The x-component of the spiral wave-vector ~Q as a function of doping d (compared with experiments<br />

[_] on LSCO [37]).<br />

98

7.2 Slave-boson theory for the t-t'J model<br />

Next, we investigate the ground-state properties of the t-t'-J model, where we fix<br />

J/t =0.4. Comparing the free energies of several (homogeneous) symmetry-broken states,<br />

we obtain the SB phase diagram shown in the t'/t-d plane in Fig. 7.3. Obviously, we can distinguish<br />

two regions. In the parameter region |t'/t| ^ 0.2, Figure 7.3 resembles the groundstate<br />

phase diagram of the pure t-J model (Fig. 7.1), i. e., we found large regions with incommensurate<br />

spiral order. However, compared to the pure t-J model, the t'-term stabilizes Néel<br />

order in a finite d region near half-filling. The increasing stability of AFM configurations<br />

can be intuitively understood because the t'-term moves electrons without disturbing the<br />

Néel-like background [12]. For larger ratios |t'/t| we have a completely different situation. In<br />

this parameter regime only commensurate states (AFM, FM, PM) occur, where for<br />

t' < t' c = –1.4 t we obtain the AFM state for all d. Note the rather large differences to the value<br />

of t' c obtained within a semiclassical (1/N)-expansion [38]. By varying the exchange coupling<br />

J, the phase boundaries in the t'/t-d plane are not much affected, e. g. for J/t =1 <strong>and</strong><br />

t'/t =–0.4, the transitions AFM ⇔ (1,1)-spiral <strong>and</strong> (1,1)-spiral ⇔ FM take place at d = 0.17<br />

<strong>and</strong> d = 0.6, respectively. We would like to point out that the main qualitative features of our<br />

SB phase diagram do confirm recent studies of magnetic long-range order in the t-t'-J<br />

model [38, 39].<br />

Figure 7.3: Restricted SB ground-state phase diagram of the 2D t-t'-J model at J/t =0.4.<br />

In Fig. 7.4 we plot the order parameter wave-vector as a function of doping at t'/t<br />

= +0.16 <strong>and</strong> t'/t =–0.4. The behaviour of Q x reflects a series of transitions AFM ⇔ (1,1)-<br />

spiral ⇔ (1, p)-spiral ⇔ PM. The corresponding (sublattice) magnetization abruptly changes<br />

at the (1,1)-spiral ⇔ (1, p)-spiral first-order transition. From Fig. 7.4 the asymmetry between<br />

hole (t' < 0) <strong>and</strong> electron doping (t' > 0) becomes evident. In contrast to recent Hartree-<br />

Fock results for the Hubbard model [40] we found the AFM phase near half-filling for both<br />

electron-doped <strong>and</strong> hole-doped cases (provided t'( 0). In the absence of t ' hopping, for arbitrarily<br />

small doping the AFM is found to be unstable against the (1,1)-spiral phase (cf.<br />

Fig. 7.3). Obviously, the AFM correlations are strongly enhanced by a positive t'-term,<br />

which is also in qualitative agreement with exact diagonalization studies of the t-t'-J model<br />

[12] <strong>and</strong> confirms the experimental findings for the electron-doped system NCCO [41, 42].<br />

Note that the stability region of the AFM phase agrees surprisingly well with the combined<br />

99

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

Figure 7.4: The x-component of the SB spiral wave-vector ~Q away from half-filling for the negative values<br />

t'/t =–0.16 (solid) <strong>and</strong> t'/t =–0.4 (long-dashed), i. e., hole doping (d > 0), <strong>and</strong> for the positive one<br />

t'/t =0.16 (dashed), i. e., electron doping.<br />

phase diagram for La 2–x Sr x CuO 4 <strong>and</strong> Nd 2–x Sr x CuO 4 obtained from neutron scattering [43]<br />

<strong>and</strong> muon spin relaxation measurements [44], respectively. For the YBCO parameter t'/t =<br />

–0.4, the AFM disappears around d = 0.1 whereas in the phase diagram of YBCO, determined<br />

by neutron diffraction [45], this transition takes place at about x = 0.4 oxygen content.<br />

However, there exits strong evidence that at least up to x = 0.2 no holes are transferred<br />

from CuO chains to CO 2 planes.<br />

As we have already noted, the phase diagrams in Fig. 7.1 <strong>and</strong> Fig. 7.3 result from the<br />

relative stability of various homogeneous states. On the other h<strong>and</strong>, there are arguments for<br />

the existence of inhomogeneous, e. g., phase-separated states in the t-J <strong>and</strong> related models<br />

[46]. Using very different methods, it was realized by several groups [47–51], that at large<br />

J/t the ground state of the t-J model separates into hole-poor (AFM) <strong>and</strong> hole-rich regions.<br />

Unfortunately, in the physically interesting regime of small exchange coupling (J/t ~ 0.2–<br />

0.4) <strong>and</strong> low doping level this point is still controversial. To gain more insight into the phenomenon<br />

of phase separation in t-J-type models of strongly correlated electrons it seems to<br />

be important to investigate the effect of an additional NNN hopping term t' as well. Therefore<br />

we study the free energy as a function of hole density d, where a (concave) convex curvature<br />

indicates local thermodynamic (in)stability implying a (negative) positive inverse isothermal<br />

compressibility k –1 = n 2 @2 f<br />

If k –1 < 0, the domain of the two-phase regime is determined<br />

performing a Maxwell construction for the anomalous increase of the chemical poten-<br />

@n 2 :<br />

tial m by doping.<br />

The results of our analysis of thermodynamic stability are depicted in Figs. 7.5a <strong>and</strong><br />

7.5 b for the t-J <strong>and</strong> t-t'-J model, respectively. The boundary of phase separation for the t-J<br />

model is given by the solid curve in the J-d plane shown in Fig. 7.5 a. As can be seen from<br />

Fig. 7.1, the different phase separated domains are built up by the (AFM) states at half-filling<br />

(d = 0) <strong>and</strong> the corresponding hole-rich state on the right boundary of the respective region.<br />

At J=0, where we recover the U ?? result of the Hubbard model [32], the free energy<br />

is a convex function Vd, i. e., our SRI SB theory does not support recent arguments<br />

[47] for phase separation in this limit. In the opposite limit of large J, complete charge se-<br />

100

7.2 Slave-boson theory for the t-t'J model<br />

Figure 7.5: Phase diagram of the t-(t')-J model including phase separated states. The phase separation<br />

boundary (solid curve) for the t-J model is shown in the J-d plane (a). We include the transition lines of<br />

Refs. [49] (dashed), [50] (dotted), <strong>and</strong> [51] (chain dashed). The triangles are the Lanczos results of<br />

Ref. [47]. The phase diagram of the t-t'-J is calculated in the t'/t-d plane at J/t =0.4 (b). Here the twophase<br />

region consists of AFM <strong>and</strong> (1,1)-spiral states. For further explanation see text.<br />

paration takes place for J/t > J PS /t=4.0, which seems to be an essentially classical result.<br />

From exact diagonalization (ED) we obtain J PS /t=4.1 +0.1 [52] compared with 3.8 derived<br />

by means of a high-temperature expansion [51]. We note that the homogeneous magnetic<br />

phases are always unstable close to half-filling (provided J/t > 0). This is in qualitative<br />

agreement with results obtained from ED studies [47] as well as from semiclassical [49] or<br />

renormalization-group calculations [50]. Also plotted in Fig. 7.5 a are the results of the hightemperature<br />

expansion method [51], where phase separation may occur only above a critical<br />

exchange J/t =1.2 as d?0, contrary to all the other approaches. The line separating the<br />

two-phase region from the stable states was determined by Marder et al. [49] within a semiclassical<br />

theory to vary as J/t =4 d 2 whereas our theory yields an approximately linear dependence<br />

at small d. We believe that, due to an improved treatment of spin correlations in<br />

our approach, the region of incomplete phase separation is reduced. The instability towards<br />

phase separation at small J can be taken as an indication that charge correlations may play<br />

an important role as well.<br />

The dotted lines of zero inverse compressibility in Fig. 7.5b show that also for |t'/t| >0<br />

there is a finite range of d over which the (1,1)-spiral is locally unstable. Similar results were<br />

recently obtained by Psaltakis <strong>and</strong> Papanicolaou [38]. But it is important to stress that in our<br />

theory the AFM state is locally stable for both signs of t'. In addition, based on the Maxwell<br />

construction, we can show that near half-filling the AFM state remains also globally stable<br />

against phase separation for the t-t'-J model (cf. Fig. 7.5 b, where the two-phase region is<br />

bounded by the solid lines). At larger values of |t'/t| 6 0.5, the phase-separated region is due<br />

to the first-order nature of the transition AFM ⇔ (1,1)-spiral (dashed curve) [38].<br />

<strong>Final</strong>ly to demonstrate the quality of our SRI SB approach, for the t-J model the expectation<br />

value of the kinetic energy is compared with the results from ED for a finite 16-<br />

site [48] (36-site [52]) lattice in Fig. 7.6. We find an excellent agreement between SB results<br />

<strong>and</strong> ED data. Obviously, this result does not depend on the interaction strength J, i. e. the SB<br />

101

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

Figure 7.6: Expectation value of the kinetic energy as a function of doping at several interaction<br />

strengths J in comparison to ED results for the 16 (36) site lattice [48, 52].<br />

theory well describes important correlation effects. Note that AH t S t–J /t is directly related to<br />

the effective transfer amplitude of the renormalized quasi-particle b<strong>and</strong>, which is taken as input<br />

for the calculation of transport coefficients in the following section.<br />

7.3 Comparison with experiments<br />

7.3.1 Normal-state transport properties<br />

As one of the main normal-state puzzles of the CuO 2 based high-temperature superconductors,<br />

the anomalous transport properties, in particular the temperature <strong>and</strong> doping dependence<br />

of the Hall resistivity R H (T,d) [53–56], has been under extensive experimental <strong>and</strong><br />

theoretical study. Quasi-particle transport measurements suggest a small density of charge<br />

carriers <strong>and</strong> hence a small pocket-like Fermi surface (FS) [57]. On the other h<strong>and</strong>, direct angle-resolved<br />

photoemission (ARPES) probes of the FS [14, 58] yield a large FS which satisfies<br />

Luttinger’s theorem <strong>and</strong> might be well described by (LDA) b<strong>and</strong> structure calculations<br />

[13, 59]. In principle, this contrasting behaviour is found for hole-doped (La 2–x Sr x CuO 4 ,<br />

YBa 2 Cu 3 O 6+x ) <strong>and</strong> electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides as well.<br />

Adopting the hypothesis of Trugman [60], most of the normal-state properties of the<br />

cuprates may be explained by the dressing of quasi-particles due to magnetic interactions<br />

<strong>and</strong> the subsequent modification of their dispersion relation. Then, once the quasi-particle<br />

b<strong>and</strong> E ~k has been obtained, the Hall resistivity R H = s xyz /s xx s yy can be calculated in the relaxation<br />

time approximation, using st<strong>and</strong>ard formulas for the transport coefficients:<br />

102

7.3 Comparison with experiments<br />

ˆ<br />

ˆ<br />

e 2 X @n ~k<br />

k 2 u<br />

V k ~ u k ~ ;<br />

@E ~k<br />

~k<br />

e 3 2 X<br />

k 4 cV<br />

~k<br />

u ~ k<br />

" u ~ k<br />

@u ~ k<br />

@k <br />

@n ~k<br />

@E ~k<br />

:<br />

(38)<br />

Here n ~k is given by Eq. 30,V denotes the volume of the unit cell, e lkg is the completely<br />

antisymmetric tensor, <strong>and</strong> u ~ k ˆ @E ~k =@k . Note that R H does not depend on the relaxation<br />

time t.<br />

To make the discussion more quantitative, let us now consider the doping dependence of<br />

R H (d,T) in terms of the t-t'-J model using the saddle-point <strong>and</strong> relaxation time approximations,<br />

where FS <strong>and</strong> correlation effects are involved via the renormalized SB b<strong>and</strong> E ~k (Eq. 33). As we<br />

have pointed out above, in our approach the SB quasi-particle b<strong>and</strong> dispersion E ~k has to be determined<br />

in a self-consistent way at each doping level d. This should be in contrast to the NZA<br />

SB mean-field approach to the t-t'-J model of Chi <strong>and</strong> Nagi [61] where, in the J ? 0 limit, the<br />

calculation of transport properties is based on the simple replacement " ~k ! ~" ~k =<br />

–2td [(cos k x + cos k y )+2(t'/t)cos k x cos k y ] of the non-interacting b<strong>and</strong> dispersion (Eq. 2).<br />

Figure 7.7 shows the theoretical Hall resistivity as a function of carrier density in<br />

comparison to experiments on LSCO [53], YBCO [54] <strong>and</strong> NCCO [55, 56]. In the LSCO<br />

<strong>and</strong> NCCO systems, the concentration of chemically doped charge carriers in the CuO 2<br />

planes (d) definitely agrees with the composition (x) of the substitutes Sr <strong>and</strong> Ce. This simple<br />

relation, however, no longer holds for YBCO, i. e., the number of holes transferred into<br />

the planes does not increase linearly with the oxygen content. Indeed, the magnetic properties<br />

indicate d & 0uptox=0.2 [45]. In order to compare our theoretical model with the<br />

R H (x) data found on oxygen-doped YBCO, we use the relation d =(x – 0.2)/2 [62].<br />

Figure 7.7: Doping dependence of the Hall resistivity for hole-doped (left panel) <strong>and</strong> electron-doped<br />

(right panel) systems. The slave-boson results for J/t =0.4 <strong>and</strong> different ratios t'/t = 0 (solid), –0.16<br />

(dashed), –0.4 (chain dotted) <strong>and</strong> t'/t =0.16 (dotted) are compared with experiments on LSCO (_) [53]<br />

(at 80 K), YBCO (O) [54] (at 100 K) <strong>and</strong> NCCO (Z) [55, 56] (at 80 K), respectively. The inset shows<br />

the temperature dependence of R H for t'/t =0atd = 0.1 (short dashed) <strong>and</strong> d = 0.15 (dotted).<br />

103

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

Figure 7.7 clearly demonstrates the importance of the NNN transfer term t' for a consistent<br />

theoretical description of the experimental Hall data. For J/t =0.4, an excellent agreement<br />

with experiments on LSCO <strong>and</strong> NCCO, including the sign change of R H (d) at a very similar<br />

value, can be achieved using the parameter values t'/t =0 <strong>and</strong> t'/t = 0.16, respectively. It<br />

should be noted that in the case of LSCO we obtain t'/t =0 from R H (d) while LDA calculations<br />

yield a ratio t'/t =–0.16 [13, 59]. For YBa 2 Cu 3 O 6+x , where the experiments [54] give<br />

R H > 0 up to x=1, a negative t'-term suffices to give the correct tendency to R H (d). Using<br />

t'/t =–0.4, our theory yields a sign change of R H at d & 0.7. The strong increase (decrease)<br />

of the positive (negative) Hall coefficient as d?0 can be attributed to the formation of<br />

small hole (electron) pockets in the FS, which is a correlation effect. A recent analysis of resistivity<br />

saturation in LSCO [57], based on Boltzmann transport, has been taken as an indication<br />

of a small FS as well, however, the existence of a pocket-like FS is still a subtle <strong>and</strong><br />

unresolved issue. We found that the temperature dependence of R H (d,T) is at least in qualitative<br />

agreement with experiments on LSCO (see inset Fig. 7.7 (left panel)). Note that the<br />

quasi-particle dispersion E ~k exhibits extremely flat minima implying the presence of a new<br />

small energy scale D. Therefore, when the temperature becomes comparable to D the hole<br />

pockets are washed out <strong>and</strong> a sign change of R H occurs (cf. Fig. 7.7 at fixed d (inset)). 3<br />

The FS of the interacting system are shown in Fig. 7.8 for typical ratios t'/t at J/t =0.4<br />

<strong>and</strong> d = +0.1, where the diagonal (1,1)-spiral phase is lowest in energy. As observed for t-J<br />

<strong>and</strong> Hubbard models as well [64], we obtain small hole (or electron) pockets with a volume<br />

! |d|. The calculated FS are very anisotropic. As |d| increases, the pockets grow, until the<br />

FS topology changes completely at a critical doping value d c (R H (d c ) = 0 (cf. Fig 7.7), reflecting<br />

the transition from hole to electron carriers for t'/t < 0 <strong>and</strong> vice versa for t'/t >0.<br />

Figure 7.8: Quasi-particle Fermi surface in the (1,1)-spiral phase at J/t =0.4 for d = 0.1 (hole-doped<br />

system) <strong>and</strong> d = –0.1 (electron-doped system).<br />

3 We recently learned of a related exact diagonalization study of Dagotto et al. [63], where, similar in<br />

conclusion, the doping <strong>and</strong> temperature dependence of R H was calculated using a strongly renormalized<br />

flat quasi-particle dispersion.<br />

104

7.3 Comparison with experiments<br />

We want to point out that the renormalization of the quasi-particle b<strong>and</strong> E ~k strongly depends<br />

on both interaction strength J <strong>and</strong> doping level d [62] which, in fact, calls into question<br />

the frequently used rigid b<strong>and</strong> approximation. Due to the strong coupling of spin <strong>and</strong><br />

charge dynamics the characteristic energy scale for the coherent motion of the charge carriers<br />

is J <strong>and</strong> not t (provided t > J).<br />

7.3.2 Magnetic correlations <strong>and</strong> spin dynamics<br />

In this Section we try to underst<strong>and</strong> the INS <strong>and</strong> NMR experiments on the basis of the SRI<br />

SB mean-field theory. Using a generalized RPA expression for the spin susceptibility <strong>and</strong> assuming<br />

that the AFM correlations are spatially filtered by various ~q-dependent hyperfine<br />

form factors, we focus, in particular, on the spin dynamical properties in the paraphase of<br />

the t-t'-J model. Our starting point is an RPA-like form for the exchange-enhanced spin susceptibility<br />

[5, 6, 65]<br />

o …~q; !†<br />

s …~q; !† ˆ<br />

1 ‡ J 2 …cos q x ‡ cos q y † o …~q; !†<br />

: …39†<br />

The irreducible part 4<br />

o …~q; i! m †ˆ<br />

2 X<br />

N<br />

~k;n<br />

G… ~ k; i! n † G… ~ k ‡ ~q; i! n‡m †<br />

…40†<br />

contains the (dressed) SB Green propagators G… k; ~ i! n †ˆ‰i! n E ~k ‡ ~Š 1 describing noninteracting<br />

electrons with the renormalized b<strong>and</strong> structure E ~k …d† (Eq. 26).<br />

Once the dynamic spin susceptibility has been obtained both, INS measurements <strong>and</strong><br />

NMR experiments, can be explored. Probed by INS from the fluctuation-dissipation theorem<br />

the q-dependent <strong>and</strong> o-dependent spin structure factor is related to the dynamical susceptibility<br />

by<br />

S …~q; !† ˆ 1 1<br />

p 1 exp… k!† Im s…~q; !† …41†<br />

On the other h<strong>and</strong>, the nuclear spin-lattice relaxation rate a T –1 1a (a =k,k), e. g. for a<br />

field H a applied parallel to the c-axis, given by [66]<br />

a T 1k T 1 k B<br />

1ˆ<br />

2 2 B k lim 1<br />

!!0 N<br />

X<br />

~q<br />

a F 2 ? …~q† Im s…~q; !†<br />

k!<br />

…42†<br />

4 o n = 2np/b [o m = (2m +1)p/b] denote the fermionic [bosonic] Matsubara frequencies.<br />

105

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

<strong>and</strong> the transverse spin-spin relaxation rate, T –1 2G , for the RKKY coupling of the nuclear Cu<br />

spins, given by [67]<br />

T 2<br />

2G ˆ<br />

2<br />

0<br />

123<br />

c 1 X 2<br />

8k 2 …2 B † 4 Fk 2 N<br />

…~q† 1 X<br />

4<br />

s…~q† @ Fk 2 N<br />

…~q† s…~q† A 5 ; …43†<br />

~q<br />

provide local, atomic site (a = {63, 17} specific information. Here m B denotes the Bohr<br />

magneton <strong>and</strong> the constant c=0.69 is the natural-abundance fraction of the 63 Cu isotope.<br />

The form factors are given for 63 Cu <strong>and</strong> the planar 17 O nucleus as [68]<br />

63 F …~q† ˆA ‡ 2B…cos q x ‡ cos q y † ; …44†<br />

~q<br />

17 F …~q† ˆ2C cos q x<br />

2 : …45†<br />

Together with the anisotropy of the Cu relaxation rates the measurements of the<br />

Knight shift, a K ˆ 2<br />

a B n k lim ~q!0 a F …~q† s …~q; ! ˆ 0†, have been used to determine the hyperfine<br />

coupling constants A a , B <strong>and</strong> C on the basis of the Mila-Rice Hamiltonian [69]. Following<br />

Ref. [66] we take A || & –4B, A k & 0.84B, C & 0.87B <strong>and</strong> B & 3.3610 –7 eV, where<br />

63 gk = 7.5610 –24 erg/G <strong>and</strong> 17 gk = 3.8610 –24 erg/G.<br />

As we know, the RPA susceptibility contains an unphysical instability of the paramagnetic<br />

phase at some particular wave-vector ~q below a critical doping d c as signaled by the<br />

zero in the denominator of Eq. 39 at o = 0 (Stoner condition). Therefore the use of Eq. 39<br />

only makes sense if the system is far from the magnetic instability, i. e., d > d c , where for<br />

J=0.4 we have d c (t'/t =–0.16) H 0.27 <strong>and</strong> d c (–0.4) H d c (0) H 0.17.<br />

7.3.3 Inelastic neutron scattering measurements<br />

We begin with a discussion of the RPA dynamical spin structure factor S (~q; o†. The ~qdependence<br />

of S (~q; o† along the main symmetry axis of the Brillouin zone is shown in<br />

Fig. 7.9 at J=0.4 <strong>and</strong> hole density d = 0.3 for different ratios t'/t. Here the temperature<br />

is T=35 K <strong>and</strong> the frequency ko = 0.010 eV. For LSCO-type parameters (t'/t =0,<br />

–0.16) we found four pronounced incommensurate peaks located at the points<br />

p (1+q o , 1), p (1,1+q o ). The incommensurate modulation wave-vectors move with increasing<br />

doping level d away from the corner of the Brillouin zone along the directions<br />

(1, p) or(p, 1) (square lattice notation). Note, that the incommensurate peak position obtained<br />

from a three-b<strong>and</strong> RPA calculation of S (~q; o† [5] can be parametrized consistent<br />

with the experimental observation that q 0 H 2 x [37], while all the effective one-b<strong>and</strong><br />

RPA approaches [6, 10] yield an incommensurability scaling rather as q 0 H d. A more<br />

detailed investigation of the LSCO-type ~q-scans show that the ~q-variation of S(~q; o† is<br />

mainly governed by that of w o …~q; o† <strong>and</strong>, in accordance with experiments [70], the in-<br />

106

7.3 Comparison with experiments<br />

commensurate peaks considerably broaden when temperature or energy transfer are increased<br />

[71]. By contrast, the same plot for YBCO-type (t'/t =–0.4) parameters shows a<br />

broad <strong>and</strong> nearly T-independent [71] maximum around the (p, p)-point [72] (Fig. 7.9)<br />

which, due to the flat topology of w o , mainly reflects the ~q-dependence<br />

J (~q) =J (cosq x + cosq y ) (cf. Eq. 39). In this way, our calculations confirm recent arguments<br />

[5, 6, 73] for the importance of b<strong>and</strong> structure (Fermi surface geometry) effects<br />

in explaining the difference between observed LSCO <strong>and</strong> YBCO spin dynamics. Nevertheless,<br />

whether the incommensurate signals arise from an intrinsic magnetic structure or<br />

whether they result from the formation of domains (charge superstructures) in the LSCO<br />

system remains unanswered by INS [2, 68].<br />

Figure 7.9: Dynamic magnetic structure factor S(~q; o† is plotted along the (1,1) <strong>and</strong> (1, p)-directions of<br />

the Brillouin zone for different ratios t'/t.<br />

In a next step, we calculate the longitudinal or spin-lattice relaxation rate, T –1 1 , using the<br />

hyperfine form factors (Eq. 45). In Figure 7.10 the temperature dependences of 63 T –1 1k <strong>and</strong><br />

17 T –1 1k (inset) are shown for d = 0.35 <strong>and</strong> t'/t =–0.4 in comparison to experiments [74] on<br />

fully oxygenated YBCO materials (x =1). Although our theory does not succeed in giving<br />

the correct amplitude of a T –1 1k the qualitative features of the NMR data are described surprisingly<br />

well. Obviously, the broad magnetic peak in S (~q; o† at the AFM wave-vector<br />

~Q AFM =(p,p) strongly enhances the relaxation rate on Cu sites while, due to a geometrical<br />

cancellation ( 17 F a (~q ~Q AFM † 0†, the corresponding oxygen rate is rather insensitive to<br />

nearly commensurate AFM fluctuations <strong>and</strong> therefore is governed by the long wavelength<br />

part ~q 0 of the spin susceptibility [2]. For 63 Cu the nominal Korringa ratio S : (1/T 1 TK 2 S )<br />

(K S denotes the spin part of the Knight shift) is at least one order of magnitude larger. As<br />

can be seen from Fig. 7.10, for YBa 2 Cu 3 O 7 -type parameters, a Korringa (1/T 1 ! T) dependence<br />

(dotted line) holds at both Cu <strong>and</strong> O sites below T* ~ 120K, demonstrating the existence<br />

of a characteristic temperature T* as well as in all other near-optimum T c compounds<br />

[2, 75]. T* is in good agreement with the coherence energy scale suggested experimentally<br />

[75]. Above T* the 17 O NMR relaxation remains linear whereas the 63 Cu relaxation time<br />

does not follow the Korringa law.<br />

107

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

Figure 7.10: Spin-lattice relaxation rates 63 T –1 1k <strong>and</strong> 17 T –1 1k (inset) as a function of temperature. SB results<br />

((+); t'/t =–0.4, d = 0.35) are compared with experiments (D) on YBa 2 Cu 3 O 7 [74].<br />

In the oxygen-deficient compound YBa 2 Cu 3 O 6.6 , 1/( 63 T 1k T) shows a broad maximum<br />

at about 150 K (Fig. 7.11), which reflects a strong deviation from the canonical Korringa behaviour.<br />

In the normal-state regime the theoretical results agree even quantitatively with the<br />

experimental 63 Cu NMR data [76]. At this point it is important to stress that the present theory<br />

incorporates considerable b<strong>and</strong> renormalization effects already via w 0 …~q; o†, especially<br />

at low doping level. Thus a rather moderate strength J = 0.4 of the AFM exchange interaction<br />

yields the experimentally observed enhancement of 63 T 1k . In striking contrast to the optimally<br />

doped YBCO system, the Korringa relation is no longer satisfied for the planar 17 O<br />

nucleus sites in the underdoped material (cf. inset Fig. 7.11). Instead, a different behaviour<br />

17 T 1k T 17 K S = const was suggested to hold down to T c [76]. The unconventional T-scaling of<br />

17 T 1k has been taken as a signature for another important feature of the normal-state spin dynamics,<br />

the so-called spin-gap behaviour [2].<br />

Figure 7.11: 63 Cu <strong>and</strong> planar 17 O relaxation data (^) for underdoped YBa 2 Cu 3 O 6.6 [76] are plotted vs<br />

temperature. Theoretical results (6) are given at t'/t =–0.4, <strong>and</strong> d = 0.2.<br />

108

7.4 Summary<br />

Complementary measurements of the transverse spin-spin relaxation rate, T –1 2G ,have<br />

provided further insights into the drastic change in the magnetic properties when passing<br />

from the overdoped to the underdoped regime [67, 77]. As experimentally observed, we<br />

found that T –1 2G increases (decreases) with increasing (decreasing) hole doping (temperature)<br />

[78]. In order to detect the opening of a spin-pseudogap as a function of T, a powerful technique<br />

is to measure the ratio T 2G /T 1 T [79] which is nearly constant above ~ 200 K for deoxygenated<br />

YBa 2 Cu 3 O 6.6 [67]. The calculated temperature dependence of this quantity is<br />

shown in Fig. 7.12 together with recent experimental results [67]. Most notably, the opening<br />

of a spin-pseudo gap at T* ~ 135 K [79], i. e. well above T c , is clearly seen as a decrease of<br />

below T*. Note that for YBa 2 Cu 3 O 7 , as predicted by Fermi liquid theories, the ratio T 2 2G/T 1 T<br />

is approximately constant above 150 K [2].<br />

Figure 7.12: T 2G /( 63 T 1k T) in YBa 2 Cu 3 O 6.6 (^) as measured by Takigawa [67] compared with the SB<br />

data (6) att'/t =–0.4, <strong>and</strong> d = 0.2.<br />

7.4 Summary<br />

In this work we have used a spin-rotation-invariant SB approach to investigate magnetic <strong>and</strong><br />

transport properties of the 2D t-t'-J model. Our main results are the following (see also [81]):<br />

a) We present a detailed magnetic ground-state phase diagram of the 2D t-(t')-Jmodel, including<br />

incommensurate magnetic structures <strong>and</strong> phase separated states. At finite t', a main<br />

feature of the phase diagram, we would like to emphasize, is the existence of an AFM state<br />

away from half-filling, which is locally <strong>and</strong> also globally stable against phase separation.<br />

This result agrees with the experimentally observed AFM long-range order in the weakly<br />

doped LSCO <strong>and</strong> YBCO compounds. In contrast, for the simple t-J model we observe no<br />

AFM long-range order at any finite doping due to phase separation.<br />

109

7 Slave-Boson Approach to Strongly Correlated Electron Systems<br />

b) The next nearest-neighbour hopping process (t') incorporates important correlation <strong>and</strong><br />

b<strong>and</strong> structure effects near half-filling. In particular, the t'-term can be used to reproduce<br />

the FS geometry of LSCO, YBCO, <strong>and</strong> NCCO. Also the NNN hopping provides a possible<br />

origin for the experimentally observed asymmetry in the persistence of AFM order of holedoped<br />

<strong>and</strong> electron-doped systems.<br />

c) The quality of the SRI SB approach was demonstrated in comparison with exact diagonalization<br />

results available for the t-J model on finite square lattices with up to 36 sites. In<br />

this case, the SB method yields an excellent estimate for the quasi-particle b<strong>and</strong> renormalization.<br />

d) Within the saddle-point <strong>and</strong> relaxation time approximation, our SB calculation of the<br />

Hall resistivity in the t-t'-J model provides a reasonable explanation of the experimentally<br />

observed doping dependence of R H on both hole-doped (La 2–x Sr x CuO 4 , YBa 2 Cu 3 O 6+x ) <strong>and</strong><br />

electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides.<br />

e) Using a generalized RPA expression for the spin susceptibility <strong>and</strong> assuming that the<br />

AFM correlations are spatially filtered by the hyperfine form factors, we have calculated the<br />

temperature dependences of spin-lattice <strong>and</strong> spin-spin relaxation rates for planar copper <strong>and</strong><br />

oxygen sites. The results agree qualitatively well with various NMR experiments on YBa 2-<br />

Cu 3 O 6+x . In addition, we can attribute the contrasting ~q-dependence of the magnetic structure<br />

factor S (~q; o† seen in INS experiments for LSCO-type <strong>and</strong> YBCO-type systems to differences<br />

in their fermiology.<br />

Recently, our theory was improved to include (Gaussian) fluctuations beyond the paramagnetic<br />

saddle-point approximation [27, 80]. We derived a concise expression for the spin<br />

susceptibility w s (~q; o† of the t-t '-J model which does not have the st<strong>and</strong>ard RPA form. Then<br />

we were able to show that the instability line obtained from a divergence of w s (~q; 0† is in<br />

agreement with the PM ⇔ spiral state phase boundary in the saddle-point phase diagram,<br />

which in fact proves the consistency of both approaches [27].<br />

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74. M. Takigawa et al.: Physica C, 162, 853 (1989)<br />

75. P.C. Hammel et al.: Phys. Rev. Lett., 63, 1992 (1989)<br />

76. M. Takigawa et al.: Phys. Rev. B, 43, 247 (1991)<br />

77. Y. Itoh et al.: J. Phys. Soc. Jpn., 61, 1287 (1992)<br />

78. M. Deeg: PhD thesis, Universität Bayreuth, (1995)<br />

79. C. Berthier et al.: Physica C, 235–240, 67 (1994)<br />

80. M. Deeg et al., Z. Phys. B, 95, 87 (1994)<br />

81. H. Fehske, Spin Dynamics, Charge Transport, <strong>and</strong> Electron-Phonon Coupling Effects in Strongly<br />

Correlated Electron Systems, Habilitationsschrift, Universität Bayreuth (1995)<br />

112

8 Non-Linear Excitations <strong>and</strong> the Electronic Structure<br />

of Conjugated Polymers<br />

Klaus Fesser<br />

8.1 Introduction<br />

Conjugated polymers have attracted quite substantial research activities during the last 15<br />

years [1]. On one h<strong>and</strong> these materials promise interesting technological applications most<br />

of which are related to the possibility of a reversible charging <strong>and</strong> decharging of these systems.<br />

These include various battery designs as well as storage of (charged) pharmaceuticals<br />

which in turn can be released in a controlled way by application of an electrical current. For<br />

the design of non-linear optical components they play an important role as organic materials<br />

due to their processibility <strong>and</strong> fine tuning of their physical properties via suitable side<br />

groups. Recently light-emitting diodes made from these systems have made conjugated polymers<br />

to possible c<strong>and</strong>idates for the construction of thin displays [2].<br />

On the other h<strong>and</strong> a theoretical description of the whole class of these materials poses<br />

interesting questions which are worth to study on their own right. As essentially quasi onedimensional<br />

systems they give rise to the hope that many of these questions might be answered<br />

analytically. The main problems are the nature of the insulator-metal transition observed<br />

during doping <strong>and</strong> the origin of the intragap states, which are mainly responsible for<br />

the relaxation processes relevant for the light-emitting properties. We have addressed both<br />

aspects within this project <strong>and</strong> this article is organized accordingly. In Section 8.2 we present<br />

the theoretical model stressing the relevant physics <strong>and</strong> discuss the related fundamental symmetries.<br />

Then in Section 8.3 we adopt the view that the doping process mainly introduces<br />

disorder into these systems <strong>and</strong> calculate various properties (density of states, optical absorption)<br />

from this assumption. In Section 8.4 we investigate the non-linear excitations responsible<br />

for the intragap states in more detail <strong>and</strong> close in Section 8.5 with an outlook on<br />

still open problems.<br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1<br />

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8 Non-Linear Excitations <strong>and</strong> the Electronic Structure of Conjugated Polymers<br />

8.2 Models<br />

Common to all conjugated polymers is the existence of a carbon backbone with an alternating<br />

sequence of (short) double <strong>and</strong> (long) single bonds. The various systems, as polyacetylene,<br />

polyparaphenylene, polypyrrole, polythiophene, polyparaphenylenevinylene, etc.,<br />

differ only – from a physcist’s point of view – in the side groups <strong>and</strong> other chemical structures<br />

which are energetically far away from the Fermi energy. Emphasizing the general aspects<br />

of these systems, these chemical details are safely neglected if one restricts to properties<br />

which are mainly due to the electrons around the Fermi energy. Thus the parameters in<br />

the simple model presented below should be regarded as effective parameters including<br />

some aspects of the interactions which are otherwise neglected.<br />

Beside some biological systems such as b-retinol where a finite chain of a conjugated<br />

polymer is present (<strong>and</strong> thus the properties of this polymer might be of relevance for the<br />

biological functions of this material), there may be other, related systems where similar theoretical<br />

concepts are or can be applied. These include the metal-halogen (MX) chains where<br />

instead of one, as in the conjugated polymers, now two electron b<strong>and</strong>s govern the essential<br />

physics. On the same level the polyanilines can be modelled where the phonons of the conjugated<br />

polymers have to be replaced by the librons, i. e. oscillations of the quinoid/benzoid<br />

rings. <strong>Final</strong>ly, a true one-dimensional modification of carbon, carbene with alternating single<br />

<strong>and</strong> triple bonds, can also be understood along these lines.<br />

So far we have only mentioned the construction of adequate models for the investigation<br />

of physical properties. However, there are competing approaches to address the same<br />

set of questions. One of these approaches uses sophisticated quantum-chemical codes [1c,d]<br />

to calculate structure <strong>and</strong> electronic states of these materials. Although such a procedure<br />

may be able to reproduce the observed properties of a specific material quite well it is very<br />

difficult to obtain information about general trends <strong>and</strong> physical mechanisms. Therefore we<br />

did not follow this route. Another well-established method for calculating ground-state properties,<br />

namely the local density functional, has been used to some extent [3]. The drawback<br />

of this method, however, is the finite size of the system which can be calculated within a<br />

reasonable amount of computer resources. Therefore only a few questions have been addressed<br />

via this approach.<br />

Therefore, we shall argue in favor of a simple model which is capable to include the<br />

most essential physics. Assuming the sp 2 -hybridisation of the carbon atom leaves one p-electron<br />

per atom the others being incorporated into the bonds as s-electrons. It is the physical<br />

behaviour of this single p-electron which is responsible for all the interesting effects.<br />

Since there is only this one electron per site the polymer would be metal-like in its<br />

ground state, in contrast to nature where one finds an energy gap of the order of 1 eV. In<br />

one dimension, however, these electrons are unstable when they are coupled to the lattice.<br />

Due to the Peierls effect, scattering off 2 k F phonons, a gap opens right at the Fermi energy<br />

in accordance with the observations. In addition, electron-electron interaction contribute to<br />

the size of this gap [4] stabilizing the semiconductor ground state even further. A single-particle<br />

model along these lines has been put forward by Su et al. [5] in the early stages of conjugate<br />

polymer research. Its parameters although, derived from a simple picture, should<br />

114

8.2 Models<br />

nevertheless be interpreted as effective parameters including parts of the electron-electron<br />

interaction which is otherwise neglected. It has been shown [6] that for some ground-state<br />

properties such redefinition can indeed be performed.<br />

It turns out that there is a typical length scale in this model v F /D (with the Fermi velocity<br />

v F <strong>and</strong> the electronic gap 2 D) which is considerably larger than the interatomic spacing<br />

a. Thus for most physical properties of interest a continuum approximation is valid.<br />

This stresses the generic aspects of the whole class of these materials even more.<br />

In rescaled variables this model now reads<br />

H ˆ X Z<br />

s<br />

dx ‡ s …x† f i 3@ x ‡ 1 …x†g s …x†‡ 1 Z<br />

2<br />

dx 2 …x†<br />

…1†<br />

Here c (x) is a two-component spinor describing electrons moving to the left (right)<br />

along the one-dimensional polymer, s is the spin index which, except for external fields, is<br />

not relevant here. The s 3 term is the kinetic energy originating from the hopping of electrons<br />

between neighbouring sites. D (x) is the lattice order parameter where a constant<br />

D (x) =D 0 describes a uniform dimerization of the lattice. The s 1 term is the electron-phonon<br />

coupling responsible for the Peierls distortion. <strong>Final</strong>ly we have an elastic energy for the<br />

lattice. The kinetic energy is neglected within an adiabatic approximation. All quantities<br />

have been scaled in order to have a single coupling constant (l & 0.2).<br />

At this stage we postpone the non-linear excitations of this model to a subsequent<br />

Chapter, we only discuss some of the symmetries of this model which are of relevance also<br />

for these excitations. First we note that the form of the electronic part of the Hamiltonian<br />

(Eq. 1) has exactly the form of a (relativistic invariant) Dirac operator in one dimension.<br />

This correspondence has been exploited [7] successfully in obtaining solutions of this model<br />

from results known in models of elementary particles. We remark here that similar analogies<br />

can be made for specific forms of the Fermi surface also in higher dimensions. Thus the<br />

connection between solid state physics <strong>and</strong> quantum field theory models can be used for a<br />

better underst<strong>and</strong>ing of both.<br />

In addition there are two symmetries which can be directly related to observable quantities.<br />

The charge-conjugation symmetry, which can be expressed as H being invariant under<br />

c?s 2 Kc (K complex conjugate operator), relates particle <strong>and</strong> hole states <strong>and</strong> thus is responsible<br />

for the single particle spectrum being symmetric with respect to the Fermi energy,<br />

which has been set to zero in H (Eq. 1). A more hidden property is the supersymmetry of<br />

the electronic part H el (Eq. 1),<br />

<br />

0 ˆ H el ;H ‰ el ; 3 Š : …2†<br />

We have shown [8] that this more formal property is responsible for the asymmetry of<br />

the optical absorption peaks of transitions involving intragap states. Both symmetries are absent<br />

in the real systems under consideration, but for most questions this breaking of symmetries<br />

is merely a question of quantity rather than of importance for the existence of these localized<br />

states.<br />

<strong>Final</strong>ly we mention that Eq. 1 has been derived as a model for polyacetylene where<br />

the ground-state order parameters D o <strong>and</strong> –D o yield the same energy. This is not the case for<br />

115

8 Non-Linear Excitations <strong>and</strong> the Electronic Structure of Conjugated Polymers<br />

the other polymers of this class. A simple extension of this model, introduced by Brazovski<br />

<strong>and</strong> Kirova [9], corrects this. As a result all the solutions of Eq. 1 can be carried over [10]<br />

to this more general model where only the location of the intragap states are now parameters<br />

which can be adjusted to a specific material of interest. In consequence the model (Eq. 1)<br />

can be considered as the most simple generic model of conjugated polymers.<br />

8.3 Disorder<br />

A major focus of theoretical investigations is the experimental observation that the electronic<br />

structure <strong>and</strong> consequently the physical properties change drastically upon doping. Here<br />

we restrict ourselves to the question how these properties are affected through a r<strong>and</strong>om<br />

force introduced by these impurities. For simplicity we only consider isoelectronic doping,<br />

i. e. the total number of carriers remains unchanged <strong>and</strong> the dopants introduce only scattering<br />

centers for the electrons.<br />

In consequence two different types (site or bond impurities) of scatterers can be identified.<br />

The first gives rise to a r<strong>and</strong>om contribution to the on-site energy of an electron<br />

whereas the latter modifies the hopping integral between neighbouring sites. In the spirit of<br />

the continuum approximation we are thus lead to consider only backward (Eq. 3) or forward<br />

(Eq. 4) scattering.<br />

P R<br />

H imp ˆ U b dx ‡ …x† 1 …x†…x x j † ; …3†<br />

j<br />

P R<br />

H imp ˆ U s dx ‡ …x†1 …x†…x x j † ; …4†<br />

j<br />

x j denotes the (r<strong>and</strong>om) position of an impurity. Since this problem cannot be solved exactly<br />

we have to redraw to approximate methods.<br />

We have used three main routes.<br />

In the first method we employ the first Born approximation for the impurity self-energy<br />

[11]. This enables us to formulate equations of motion for the full space-dependent<br />

Green functions <strong>and</strong> thus consider the influence of disorder on the non-linear excitations as<br />

well [12] (see next Chapter). Furthermore, the replacement of Eq. 3 <strong>and</strong> Eq. 4 by r<strong>and</strong>om<br />

(Gaussian) fields<br />

Z n<br />

H imp ˆ dx ‡ …x†<br />

1<br />

1<br />

o<br />

V b=s …x† …x† ;<br />

which is correct within the Born approximation, allows the determination of the Green function<br />

<strong>and</strong> higher correlation functions via a functional integral technique [13]. The averaging<br />

procedure is formulated through the introduction of additional Grassmann variables in a<br />

116<br />

…5†

8.4 Non-linear excitations<br />

supersymmetric way. We note that this supersymmetry is not related to the one discussed<br />

earlier. Using the algebraic properties of these variables the average can be performed <strong>and</strong><br />

the resulting functional integral can be calculated via a transfer method in one dimension.<br />

We note that this procedure still works when two coupled chains are considered but cannot<br />

be done exactly in any higher dimension. As result we obtain the density of states <strong>and</strong> also<br />

information about the extension of the (in principle) localized wave functions via the Thouless<br />

formula. It turns out [14] that for moderate disorder a typical realistic chain length of<br />

approximately 200 units is smaller than the localization length of the b<strong>and</strong> states not too<br />

close to the b<strong>and</strong> edgesunverständlich. Thus for such systems these states can indeed be considered<br />

as extended states even in the presence of disorder putting various models which<br />

treat the propagation of electrons along one chain as metal-like on a firmer basis. This<br />

method can easily be extended to higher dimensional systems which means that also the<br />

coupling of polymer chains can be taken into account. Thus we are able to calculate, within<br />

a saddle-point approximation, the optical absorption coefficient for a two-dimensional film.<br />

For an orientation along the chains we find [15] a typical disorder induced broadening of the<br />

absorption edge together with a shoulder on the low-energy side due to neighbouring chains<br />

coupling. Both features agree quite well with experimental findings. Concomitantly, the absorption<br />

perpendicular to the chain direction is featureless.<br />

In the second method, going beyond Born, we examined the density of states within<br />

the coherent potential approximation (CPA) which takes into account multiple scattering processes.<br />

One might think that on this level impurity states are introduced in the gap. However,<br />

we find [16] that the existence of such localized impurity states strongly depends on the relative<br />

strength of site vs. bond impurity. Only states in the gap due to disorder can be found<br />

if the site amplitude |U s | is stronger than the bond amplitude |U b |. Since CPA is an effective<br />

medium theory this result might be questionable in one dimension.<br />

In the third method, in order to check the CPA result, we performed a numerical simulation<br />

[17] where for a given r<strong>and</strong>om distribution the electronic eigenvalues have been calculated<br />

numerically <strong>and</strong> the results been averaged over a large number of realizations. It turned<br />

out that the CPA results could be reproduced quite satisfactorily thus establishing the different<br />

role of both types of impurities.<br />

8.4 Non-linear excitations<br />

Given the single particle states according to Eq. 1 with a uniform order parameter D o one<br />

might expect that the lowest excited state corresponds to the lowest level in the conduction<br />

b<strong>and</strong> being occupied thus requiring the amount 2D o in energy, since the half-filled case considered<br />

here the valence b<strong>and</strong> with E |D o |is<br />

empty. The model (2.1), however, exhibits the property that an additional carrier modifies<br />

the order parameter locally, yielding a non-homogeneous D (x), <strong>and</strong> at the same time creates<br />

through a rearrangement of the b<strong>and</strong> statesadditional state(s) deep in the gap. Mathematically<br />

this behaviour is due to the fact that Eq. 1 is a non-linear model, the non-linearity re-<br />

117

8 Non-Linear Excitations <strong>and</strong> the Electronic Structure of Conjugated Polymers<br />

sulting from the requirement that the total energy functional AH{D (x)}S has to be stationary<br />

with respect to a variation of the order parameter D (x). This gives rise to a self-consistency<br />

equation where this order parameter is governed by the occupied electronic states,<br />

…x† ˆ<br />

P ‡ …x† 1 …x† : …6†<br />

occ<br />

Various exact solutions to this problem are known. The most prominent one is the<br />

kink (soliton) for the case of polyacetylene being,<br />

p<br />

…x† ˆ o thx= 2 :<br />

…7†<br />

As already mentioned, this kink does not exist for the other polymers of this class because<br />

the ground state is non-degenerate in D o . Therefore the simplest non-linear excitation<br />

in a more general sense is the bound kink-antikink pair (polaron) which is characterized by<br />

two localized electronic states in the gap at +o o (e. g. o o /D o & 0.5 for polythiophene). In<br />

addition there exist periodic solutions, e. g. the kink lattice with a periodicity determined by<br />

the concentration of excess charges. Physically these states lower the total energy of the system<br />

through an inhomogeneous order parameter D (x), which raises the energy (cf. Eq. 1)<br />

<strong>and</strong> a much larger compensation through the intragap states, which altogether give a smaller<br />

value of the total energy than the simple single-particle picture. For the polaron this gain in<br />

energy amounts to E p /2D o = 0.98 < 1 for polythiophene. For trans-polyacetylene this number<br />

is 0.90.<br />

One can now envisage the processes which are involved in generating visible light. A<br />

sufficiently strong electric field can promote a single electron into the lowest unoccupied<br />

conduction b<strong>and</strong> state. This state is unstable <strong>and</strong> relaxes on a fast time scale (femtosecond)<br />

into a polaron-like state which then can recombine to the ground state under the emission of<br />

radiation. A full microscopic underst<strong>and</strong>ing of all the processes involved is only possible if<br />

for the dynamics of the lattice the degrees of freedom are fully taken into account as well as<br />

the residual Coulomb interactions.<br />

One step in this direction has been made within this project by Bronold [18]. In his<br />

doctoral thesis he treats on an equal footing electron-electron (exciton) <strong>and</strong> electron-phonon<br />

(polaron) interactions. The coupling of this system to short laser pulses gives rise to characteristic<br />

changes of position <strong>and</strong> shapes of absorption/emission lines in optically stimulated<br />

emission <strong>and</strong> inverse Raman scattering experiments. As these effects have a very short time<br />

scale (femtoseconds), experiments are difficult to perform <strong>and</strong> a comparison with existing<br />

theoretical predictions is not convincing. Nevertheless, one expects that this line of approach<br />

will finally give a detailed underst<strong>and</strong>ing of the functioning of organic light-emitting diodes.<br />

Having gained some insight into the non-linear mechanisms giving rise to localized<br />

electronic (intragap) states, how dopants, mentioned in the previous Chapter, might influence<br />

these states. Two alternatives are feasible:<br />

a) the non-linear aspect dominates, i. e. the picture developed so far is still valid but only<br />

some details are modified due to the disorder. On the basis of the Born approximation we<br />

have indeed calculated [12] the electronic structure of the kink solution in the presence of<br />

impurities <strong>and</strong> found that the spatial extension of this structure is enlarged when the doping<br />

118

8.5 Perspective<br />

concentration is raised. In accordance with the closure of the gap at a critical concentration<br />

we find that the width of the kink tends to infinity at this value. The more interesting case<br />

of the polaron, however, could not be solved satisfactorily due to numerical instabilities,<br />

which could not be avoided. For details see Ref. [12].<br />

b) the more subtle case of the competition between this non-linear mechanism <strong>and</strong> the<br />

multiple scattering processes off the impurities, which is treated in terms of the T-matrix or<br />

the CPA gives also rise to localized states. The experimental observation that the formation<br />

of these states is independent of microscopic details leads to the conclusion that the non-linear<br />

aspect is the dominant one. A fully self-consistent treatment of this problem with an impurity<br />

located at x o <strong>and</strong> (for simplicity) a kink at x 1 gives complicated coupled integral equations<br />

[19] which have not been solved. A recent investigation for the case of a kink lattice<br />

shows both mechanisms working quite independently, however, the method employedwelche<br />

Methode ? does not give a full self-consistent solution..<br />

Summing up this Chapter we note that the non-linear excitations play a dominant role<br />

in various physical applications of conjugated polymers. But a full underst<strong>and</strong>ing of the interplay<br />

of various mechanisms giving rise to these localized states has not been reached yet.<br />

8.5 Perspective<br />

The potential technical applications have stimulated a myriad of experimental <strong>and</strong> theoretical<br />

studies. It is obvious that similar investigations have also been performed, mostly along<br />

different lines, by various groups. The disorder aspect, responsible for the observed metalinsulator<br />

(or semiconductor) transition, in conjunction with a kink (soliton) or polaron lattice<br />

has been treated by many authors [20] in all kinds of approximate approaches. All these studies<br />

resulted in the same prediction that such an M-I transition would occur at the experimentally<br />

observed dopant concentration level. It is now clear from the foregoing Chapter<br />

that only a fully self-consistent treatment will give a satisfactory answer to this question.<br />

But since the applications for conjugated polymers envisaged at present focus on optical<br />

properties rather than electronic transport this question has been lost out of sight. Still, the<br />

nature <strong>and</strong> dynamics of localized electronic states in these materials must be fully understood.<br />

In addition, from an application point of view, these polymers appear to be similar to<br />

conventional semiconductors. The recent proposal [21] that the non-linear excitations actually<br />

modify the conventional picture, e. g. at the interface between a polymer <strong>and</strong> a metal (or<br />

conventional semiconductor) space charge regions (depletion layers) differ from an inorganic<br />

semiconductor, finds renewed interest because transistors made from conjugated polymers<br />

are feasible <strong>and</strong> of interest for the integration of optical <strong>and</strong> electronic components.<br />

From a theoretical point of view the functional integral techniques promise interesting<br />

opportunities to make contacts to other areas of research. Universal properties have been dis-<br />

119

8 Non-Linear Excitations <strong>and</strong> the Electronic Structure of Conjugated Polymers<br />

covered [22] in the absorption spectrum, discussed earlier, as well as in the distribution of<br />

energy levels in certain disordered systems [23]. We propose that there is a close connection<br />

between both aspects. A careful treatment of the underlying correlation functions, including<br />

a more general type of disorder than discussed here, has to be performed. We expect that<br />

the result will lead to new universality classes which technically spoken will show up in different<br />

supersymmetric non-linear s models. Work along this line is in progress.<br />

In summary, conjugated polymers pose interesting problems, both for applications <strong>and</strong><br />

pure theoretical studies. Both aspects have matured during the past 15 years but still questions<br />

of a more general nature are left unanswered.<br />

Acknowledgements<br />

The author is indebted to all his collaborators for enlightening <strong>and</strong> stimulating discussions<br />

as well as fruitful collaborations. Thanks to A.R. Bishop, F. Bronold, H. Büttner, D.K.<br />

Campbell, K. Harigaya, U. Sum, Y. Wada, <strong>and</strong> M. Wolf.<br />

References<br />

1. a) A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su: Rev. Mod. Phys. 60, 781 (1988)<br />

b) T.A. Skotheim (Ed.): H<strong>and</strong>book of Conducting Polymers. Dekker, New York (1986)<br />

c) J.L. Brédas, R. Silbey (Eds.): Conjugated Polymers. Kluwer, Dordrecht (1991)<br />

d) W.R. Salaneck, I. Lundström, B. Ranby (Eds.): Conjugated Polymers <strong>and</strong> Related Materials. Oxford<br />

University Press, Oxford (1993)<br />

e) Proc. Int. Conf. Science Technology of Synthetic Metals: ICSM ’90, Synth. Met. 41–43 (1991);<br />

ICSM ’92, Synth. Met. 55–57 (1993); ICSM ’94, Synth. Met. 69–71 (1995)<br />

2. J.H. Burroughes et al.: Nature 347, 539 (1990)<br />

3. P. Vogl <strong>and</strong> D.K. Campbell: Phys. Rev. Lett. 62, 2012 (1989); Phys. Rev. B 41, 12 797 (1990)<br />

4. For a review see: D. Baeriswyl, D.K. Campbell, S. Mazumdar, in: Conjugated Conducting Polymers,<br />

H. Spiess (Ed.), Springer Series in Solid State Sciences 102, 7 (1992)<br />

5. W.P. Su, J.R. Schrieffer, A.J. Heeger: Phys. Rev. B 22, 2099 (1980)<br />

6. D. Baeriswyl, E. Jeckelmann, in: Electronic Propteries of Polymers, H. Kuzmany, M. Mehring, S.<br />

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7. D.K. Campbell <strong>and</strong> A.R. Bishop: Nucl. Phys. B 200, 297 (1982)<br />

8. U. Sum, K. Fesser, H. Büttner: Ber. Bunsenges. Phys. Chem. 91, 957 (1987)<br />

9. S. Brazovskii <strong>and</strong> N. Kirova: Pis’ma Zh. Eksp. Teor. Fiz 33, 6 (1981) [JETP Lett. 33, 4 (1981)]<br />

10. K. Fesser, A.R. Bishop, D.K. Campbell: Phys. Rev. B 27, 4804 (1983)<br />

11. a) K. Fesser: J. Phys. C 21, 5361 (1988)<br />

b) K. Iwano <strong>and</strong> Y. Wada: J. Phys. Soc. Jpn. 58, 602 (1989)<br />

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References<br />

12. F. Bronold <strong>and</strong> K. Fesser, in: Nonlinear Coherent Structures in Physics <strong>and</strong> Biology, M. Remoissenet<br />

<strong>and</strong> M. Peyrard (Eds.), Springer Lecture Notes in Physics 393, 118 (1991)<br />

13. K.B. Efetov: Adv. in Phys. 32, 53 (1983)<br />

14. M. Wolf <strong>and</strong> K. Fesser: Ann. Physik 1, 288 (1992)<br />

15. M. Wolf <strong>and</strong> K. Fesser: J. Phys. Cond. Matter 5, 7577 (1993)<br />

16. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. Lett. 63. 2401 (1989);<br />

Phys. Rev. B 42, 1268 <strong>and</strong> 1276 (1990)<br />

17. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. B 43, 4141 (1991)<br />

18. F. Bronold: Doct. Thesis, Univ. Bayreuth (1995)<br />

19. K. Fesser: Prog. Theor. Phys. Suppl. 113, 39 (1993)<br />

20. a) E.J. Mele <strong>and</strong> M.J. Rice: Phys. Rev. B 23, 5397 (1981)<br />

b) G.W. Ryant <strong>and</strong> A.J. Glick: Phys. Rev. B 26, 5855 (1982)<br />

c) S.R. Philpott et al.: Phys. Rev. B 35, 7533 (1987)<br />

d) E.M. Conwell, S. Jeyadev: Phys. Rev. Lett. 61, 361 (1988)<br />

21. a) S.A. Brazovskii, N. Kirova: Synth. Met. 55–57, 4385 (1993)<br />

b) G. Paasch <strong>and</strong> T.P.H. Nguyen: unpublished (1995)<br />

22. K. Kim, R.H. Mckenzie, J.W. Wilkins: Phys. Rev. Lett. 71, 4015 (1993)<br />

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9 Diacetylene Single Crystals<br />

Markus Schwoerer, Elmar Dormann, Thomas Vogtmann, <strong>and</strong> Andreas Feldner<br />

9.1 Introduction<br />

Polydiacetylenes can be grown as macroscopic polymer single crystals [1–3]. This property<br />

is unique. They comprise one linear polymer axis <strong>and</strong> can have spatial extensions of up to<br />

several millimetres or more in all three spatial directions (Fig. 9.1). The covalent chemical<br />

bonds along the polymer axis make them mechanically strong along the corresponding crystallographic<br />

axis. Their Young’s modulus is about a quarter of the Young’s modulus of steel<br />

[4, 5] <strong>and</strong> their tensile strength along the polymer axis has been reported to exceed that of<br />

steel [6]. Weak bonds of van der Waals-type perpendicular to the polymer axis are responsible<br />

for extremely low dimensional – generally one-dimensional – macroscopic electronic<br />

properties of these polydiacetylene single crystals. They are insulators, they can become<br />

pyro- or ferroelectric, <strong>and</strong> they show large optical non-linearities. These electronic <strong>and</strong> optical<br />

properties are primarily determined by the p-electron system along the polymer axis.<br />

Most polydiacetylenes differ only in the substituents R <strong>and</strong> R' (Scheme 9.1). But the electro-<br />

Figure 9.1: Photo of macroscopic paratoluylsulfonyloximethylene-diacetylene (TS6) single crystals under<br />

polarized light. Monomer (top), polymer (bottom).<br />

122 Macromolecular Systems: <strong>Microscopic</strong> <strong>Interactions</strong> <strong>and</strong> <strong>Macroscopic</strong> <strong>Properties</strong><br />

Deutsche Forschungsgemeinschaft (DFG)<br />

Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

9.1 Introduction<br />

Scheme 9.1: Polydiacetylene in its isomeric structures: acetylene-type (a) <strong>and</strong> butatriene-type (b). R<br />

<strong>and</strong> R' are the substituents for different diacetylenes (see also Tab. 9.1).<br />

nic structure of their all-trans planar carbon chain is at first approximation of acetylene-type<br />

(a) rather than of butatriene-type (b). Both isomeric structures contain a non-interrupted<br />

p-electron system along the carbon chain.<br />

The term polydiacetylene is somewhat puzzling, at least for a physicist. However, it<br />

becomes quite clear if one takes into account the structure of the diacetylene monomer<br />

(Fig. 9.2). A typical substituent R is paratoluylsulfonyloximethylene (Scheme 9.2).<br />

The diacetylene with R = R' = paratoluylsulfonyloximethylene is termed TS6 (sometimes<br />

TS). It was shown by G. Wegner <strong>and</strong> his co-workers in a series of works, published in<br />

the early 1970s [1, 7], that large molecular crystals can be grown from a solution of TS6,<br />

e. g. in acetone, <strong>and</strong> that these monomer diacetylene crystals can be converted by a topochemical<br />

(or solid state) 1,4-addition reaction to the polydiacetylene single crystals<br />

(Fig. 9.2). The crystal structures of TS before <strong>and</strong> after the reaction have been investigated<br />

in detail by Kobelt <strong>and</strong> Paulus [8], Bloor et al. [8], <strong>and</strong> Enkelmann [9] <strong>and</strong> are sketched in<br />

Fig. 9.3 <strong>and</strong> in Tab. 9.1.<br />

Figure 9.2: The monomer diacetylene crystal is converted by a topochemical or solid state reaction to<br />

the polydiacetylene single crystal.<br />

123

9 Diacetylene Single Crystals<br />

Scheme 9.2: Diacetylene monomer with the substituents R = R' = paratoluylsulfonyloximethylene (TS6).<br />

Figure 9.3: Monomer <strong>and</strong> polymer crystal structure of TS6 deduced from X-ray data [9]. Note the reactive<br />

carbene at the chain end.<br />

Both crystal structures are monoclinic with two monomers or monomer units of different<br />

orientation per unit cell (Fig. 9.3 sketches only one of these two.). The chemical bond<br />

between two carbons of nearest neighbour diacetylenes (1,4-addition) results in the linear<br />

polymer chain <strong>and</strong> the small change in the lattice parameter along the b-axis prevents the<br />

destruction of the macroscopic single crystal during the topochemical reaction. Several surfaces<br />

of diacetylene crystals have been studied by atomic force microscopy (AFM) in order<br />

to investigate both, the single crystal surface structure <strong>and</strong> solid state reactions at the surface<br />

[129, 131].<br />

124

9.1 Introduction<br />

Table 9.1: TS monomer <strong>and</strong> polymer crystals (monoclinic, space group P2 1 /c) [9].<br />

T/K a/Å b/Å c/Å b/ 8 D x<br />

g/cm 3<br />

TS monomer 120 14.61(1) 5.11(1) 25.56(5) 92.0(5) 1.46<br />

TS monomer 295 14.60 5.15 15.02 118.4 1.40<br />

TS monomer 295 14.65(1) 5.178(2) 14.94(1) 118.81(3) 1.40<br />

TS polymer 295 14.993(8) 4.910(3) 14.936(10) 118.14(4) 1.483<br />

TS polymer 120 14.77(1) 4.91(1) 25.34(2) 92.0(5) 1.51<br />

The topochemical reaction can be induced thermally <strong>and</strong>/or photochemically <strong>and</strong>/or<br />

by electron-beam irradiation. For TS the thermal conversion versus time (Fig. 9.4) is<br />

strongly temperature dependent <strong>and</strong> highly non-linear. The thermodynamics of the integral<br />

reaction has been investigated extensively by Bloor et al. [10], Eckhardt et al. [11], Chance<br />

et al. [11], <strong>and</strong> others. The reaction diagram (Fig. 9.5) for TS shows that the dark reaction is<br />

thermally activated <strong>and</strong> has an activation energy of 1 eV per monomer. It is exothermic with<br />

a polymerization enthalpy of 1.6 eV per addition of one monomer. The entire reaction is irreversible<br />

<strong>and</strong> the TS6-polydiacetylene (PTS) crystals are not solvable in ordinary solvents.<br />

During the solid state reaction almost all properties of the diacetylene crystals change<br />

drastically, e. g. the transparent monomer crystals are converted to polymer crystals with<br />

highly dichroic, strongly reflecting surfaces which contain the b-axis. In transmission the<br />

polydiacetylene crystals can only be investigated as thin films (Fig. 9.6). Their absorption<br />

spectrum, e. g. for TS, clearly shows the vibronic spectrum due to the single, double <strong>and</strong> triple<br />

bonds of the polymer chain [12]. Batchelder et al. [13] investigated extensively the optical<br />

absorption, reflection, <strong>and</strong> Raman spectra of TS single crystals.<br />

While these experiments were directed towards the study of the electronic excitations<br />

of the bulk <strong>and</strong> their coupling to the vibrations, Sebastian <strong>and</strong> Weiser [14] investigated the<br />

Figure 9.4: Time conversion curves for the thermal polymerization of PTS at 60 8C (.), 70 8C (#), <strong>and</strong><br />

80 8C (d) [9].<br />

125

9 Diacetylene Single Crystals<br />

Figure 9.5: Reaction diagram for the thermal polymerization of TS (solid curve) <strong>and</strong> photopolymerization<br />

of 4BCMU (dashed curve) [10, 11].<br />

Figure 9.6: Absorption spectra of a thin TS diacetylene single crystal for light polarized parallel <strong>and</strong><br />

perpendicular to the polymer axis b (T = 300 K). Monomer (M) <strong>and</strong> polymer (P) absorption.<br />

defects by electroabsorption. As an example Fig. 9.7 shows the absorption <strong>and</strong> the electroabsorption<br />

spectra, which they have analyzed in great detail.<br />

Since about 1980 <strong>and</strong> especially during our work for the Sonderforschungsbereich 213<br />

we have synthesized several new diacetylenes, the substituents of which are shown in<br />

Tab. 9.2 [15]. For selected diacetylene single crystals we have investigated:<br />

126

9.1 Introduction<br />

Figure 9.7: Absorption a <strong>and</strong> electroabsorption Da of photoproducts in PTS-monomer as a function of<br />

ko. Full curve: experimental spectra, dashed curves: fit by Lorentzian <strong>and</strong> a charge transfer model for<br />

Da [14].<br />

a) in Section 9.2 the elementary steps <strong>and</strong> structures during the solid state photopolymerization<br />

by transient optical spectroscopy, by electron spin resonance (ESR), <strong>and</strong> by electron<br />

nuclear double resonance (ENDOR);<br />

b) in Section 9.3 the application of the photopolymerization of thick diacetylene single<br />

crystals as a very effective holographic storage process;<br />

c) in Section 9.4 the tailoring of diacetylenes as ferro or pyroelectric crystals, which do<br />

not dem<strong>and</strong> considerable efforts for the poling processes <strong>and</strong> which show good thermal stability;<br />

d) in Section 9.5 the optical non-linearity of second <strong>and</strong> of third order <strong>and</strong> their application<br />

for an optical device with femtosecond time resolution.<br />

The present paper is a review of our work with diacetylene single crystals.<br />

127

9 Diacetylene Single Crystals<br />

Table 9.2: A survey of the investigated diacetylenes [15].<br />

Survey of the investigated diacetylenes<br />

R 1 C C C C R 2<br />

DNP<br />

TS<br />

PD-TS<br />

CD -TS<br />

2<br />

FBS<br />

O 2 N<br />

CH 2 O NO<br />

CH 2 O SO 2<br />

D<br />

CD 2 O SO2<br />

D<br />

H<br />

CD2 O SO2<br />

H<br />

CH<br />

2<br />

O SO<br />

2<br />

2<br />

CH 3<br />

D<br />

CD 3<br />

D<br />

H<br />

CH 3<br />

H<br />

F<br />

R 2<br />

ability to polymerize<br />

therm. γ<br />

= R 1<br />

= R 1<br />

= R 1<br />

= R 1<br />

= R 1<br />

+++<br />

+++<br />

+++<br />

+++<br />

+++<br />

-<br />

+++<br />

+++<br />

+++<br />

+++<br />

IPUDO<br />

Name R 1<br />

CH O<br />

O CH 3<br />

(CH 2 ) 4 O C NH CH<br />

= R 1<br />

-<br />

+++<br />

NP/PU<br />

CH 2<br />

O NO2<br />

CH 3<br />

O<br />

2 C NH<br />

-<br />

+<br />

O<br />

NP/4-MPU CH2 O NO<br />

2 CH2 O C NH<br />

CH 3 - +<br />

NP/MBU<br />

DNP/MNP<br />

O 2<br />

O H<br />

CH2 O NO2 CH2 O C NH C + +<br />

CH 3<br />

(-)(S) or (+)(R)<br />

N<br />

O 2 N<br />

CH<br />

NO CH O CH<br />

+++<br />

2 O 2<br />

2<br />

3<br />

DNP/PU<br />

CH<br />

2<br />

N<br />

O<br />

O 2<br />

O<br />

NO2 CH2 O C NH<br />

- -<br />

DNP/4-MPU<br />

O2<br />

N<br />

CH<br />

2 O<br />

O<br />

NO2 CH2 O C NH CH 3<br />

- -<br />

O 2 N<br />

DNP/DMPU CH2 O<br />

NO2<br />

CH<br />

O CH3<br />

O NH<br />

2 C<br />

CH 3<br />

- -<br />

DNP/MPU<br />

O 2 N<br />

CH O<br />

2<br />

O H<br />

NO2 CH2 O C NH C<br />

CH<br />

(-)(S) or (+)(R)<br />

3<br />

- -<br />

TS/FBS CH2 O SO2<br />

CH 3 CH2<br />

O SO 2 F +++<br />

FBS/TFMBS CH2 O SO 2 F CH2 O SO2 CF 3 +++<br />

128

9.2 Photopolymerization<br />

9.2 Photopolymerization<br />

9.2.1 Carbenes<br />

The aim of this Chapter is to review our spectroscopic work towards the analysis of both,<br />

the electronic structures <strong>and</strong> the dynamics of the intermediate reaction products (Fig. 9.2),<br />

during the photopolymerization, i. e. after the excitation of the solid state reaction by light.<br />

For the entirely thermally activated solid state polymerization detailed spectroscopy of the<br />

intermediate states turned out to be difficult or not very efficient. One exception was the<br />

identification of carbenes as reactive species during the thermal solid state polymerization<br />

of TS6 (Fig. 9.8).<br />

In contrast to radicals with one non-bonded electron carbenes have two non-bonded<br />

electrons. It has been shown for the first time by Wassermann et al. [16] that the electronic<br />

ground state of pure methylene (:CH 2 ) is a triplet state, where the total spin quantum number<br />

S is 1. Because of their non-centrosymmetry most molecular triplet states show a splitting<br />

into three components even in the absence of an external field. This splitting is due to the<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

2hν<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

C<br />

C<br />

R<br />

C<br />

C<br />

R<br />

C<br />

C<br />

C<br />

R<br />

kT<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

C<br />

R<br />

C<br />

C<br />

C<br />

kT<br />

R<br />

C<br />

C<br />

C<br />

R C<br />

R<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

C<br />

C<br />

C<br />

R C R<br />

C<br />

C<br />

R C<br />

C R<br />

C<br />

C<br />

R<br />

C<br />

C R<br />

C<br />

C<br />

C<br />

R<br />

kT<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

R C<br />

C<br />

C<br />

C<br />

R C<br />

R<br />

C<br />

C<br />

C<br />

R<br />

Figure 9.8: UV photopolymerization of TS6: The monomer crystal is irradiated with an UV-flash. The<br />

dimer is formed from the monomer by a photoreaction, then a series of thermally activated monomer<br />

addition reactions leads via the diradicals (DR) <strong>and</strong> dicarbenes (DC) to the polymer.<br />

R<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

R C C R<br />

kT<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

R<br />

R C C R<br />

kT<br />

C<br />

R C<br />

R<br />

C<br />

C<br />

C<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

R<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R C R<br />

C<br />

C<br />

R C C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R C<br />

C<br />

R<br />

R<br />

R<br />

R<br />

R<br />

C<br />

C<br />

C<br />

R<br />

kT<br />

R<br />

R<br />

R<br />

R<br />

R<br />

R<br />

R<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

C<br />

R<br />

R<br />

R<br />

R<br />

R<br />

R<br />

C<br />

R C<br />

R<br />

C<br />

C<br />

C<br />

R<br />

129

9 Diacetylene Single Crystals<br />

magnetic dipole-dipole coupling of the two unpaired electrons <strong>and</strong> is called zero-field splitting.<br />

In an external magnetic field the resulting fine structure of the ESR spectra is highly anisotropic,<br />

i. e. it strongly depends on the direction of the external magnetic field with respect<br />

to the orientation of the tensor describing the magnetic dipole-dipole interaction. This interaction<br />

is of course an intramolecular property <strong>and</strong> therefore single crystals are ideal c<strong>and</strong>idates<br />

for the measurement <strong>and</strong> the quantitative analysis of molecular triplet states by ESR. As originally<br />

shown by the work of Hutchison <strong>and</strong> Mangum [17], <strong>and</strong> van der Waals [18] the ESR<br />

spectrum of a molecular triplet state is described by the spin Hamiltonian T H s [19]<br />

T H s ˆ B B o g S^<br />

‡D S^ 2<br />

z ‡ E…S^ 2<br />

x S^ 2<br />

y † ; …1†<br />

where m B is Bohr’s magneton, B 0 the external magnetic field, g the spectroscopic splitting<br />

factor, ^S the spin operator for a spin with total spin quantum number S = 1, ^S u with u = x, y, z<br />

are the components along the principal axes of the magnetic dipole-dipole interaction tensor.<br />

D <strong>and</strong> E represent the two independent values of this tensor, the trace of which is zero. They<br />

usually are called zero-field splitting parameters:<br />

D ˆ o<br />

4p<br />

E ˆ o<br />

4p<br />

3<br />

4 g2 2 z 2<br />

B hr2 r 5 i ; …2†<br />

3<br />

4 g2 2 x 2<br />

B hy2 r 5 i ; …3†<br />

r is the distance of the two unpaired electrons <strong>and</strong> x, y, <strong>and</strong> z are the components of r along<br />

the principal axes of the magnetic dipole-dipole interaction tensor of these two electrons.<br />

Only for systems of cylindrical symmetry (around z) the zero-field splitting parameter E<br />

vanishes (E = 0). And only for spherical symmetry (Ax 2 S = Ay 2 S = Az 2 S = 1/3 Ar 2 S) the entire<br />

fine structure is zero. The values of D <strong>and</strong> E for the triplet carbenes as detected during the<br />

purely thermal solid state polymerization [20] are shown in Tab. 9.3 in comparison with typical<br />

values of different molecular triplet states. By these values <strong>and</strong> especially by the strong<br />

anisotropy of the fine structure in the ESR spectrum these carbenes (as sketched in Fig. 9.8)<br />

are clearly identified as reactive species during the pure thermal solid state polymerization<br />

of TS. They will play a major role in the electronic structures of the intermediate products<br />

during the photopolymerization as described in the following paragraphs.<br />

Table 9.3: Zero-field splitting parameters for the reactive species during the thermal solid state polymerization<br />

of TS [20], pure methylene [16], diphenylmethylene [21], <strong>and</strong> benzene [22] in their first excited triplet<br />

state.<br />

D<br />

hc =cm–1<br />

E<br />

hc =cm–1<br />

TS 0.2731 –0.0048<br />

:CH 2 0.6636 0.0003<br />

:C(C 6 H 5 ) 2 0.39644 –0.01516<br />

C 6 H 6 0.1581 –0.0046<br />

130

9.2 Photopolymerization<br />

9.2.2 Intermediate photoproducts<br />

Further experiments for analyzing the electronic structure of intermediate states were not<br />

very successful, until Sixl et al. [23] <strong>and</strong> Bubeck et al. [24] published their first low-temperature<br />

spectroscopic experiments on partially photopolymerized TS crystals. Thereby, the<br />

monomer crystal is cooled to 4.2 K in the dark. Then it is irradiated with UV light<br />

(l ^ 310 nm) for a short period. After this procedure a large number of different species<br />

show up in both, the optical absorption <strong>and</strong> the ESR spectrum. They persist at helium temperature.<br />

Subsequent annealing in the dark produces further intermediate reaction products<br />

which are also identified by their optical or ESR spectra. <strong>Final</strong>ly, further irradiation with<br />

visible light, which is absorbed only by the intermediate reaction products, produces still<br />

more <strong>and</strong> different reaction products. In a comprehensive series of investigations [25–47]<br />

most of these intermediate products have been identified <strong>and</strong> classified. Moreover, the mechanisms<br />

of their production <strong>and</strong> their reaction kinetics have been analyzed. According to<br />

Sixl, there exist three different series of intermediate products:<br />

1. The diradical-dicarbene series: DR 2 , DR 3 , DR 4 , DR 5 , DR 6 , DC 7 , DC 8 , DC 9 , DC 10 ,<br />

DC 11 … polymer;<br />

2. The asymmetric carbene series (AC);<br />

3. The stable oligomer series (SO).<br />

The DR-DC series leads directly from the monomer to the polymer. It is initiated <strong>and</strong><br />

processed in the following simple <strong>and</strong> clear way. The monomer crystal is irradiated with an<br />

UV flash <strong>and</strong> subsequently rests in the dark. The flash excites the monomers <strong>and</strong> produces<br />

dimers (DR 2 ). These react by a thermally activated step by step addition of monomers, as illustrated<br />

in Fig. 9.8. In the following paragraphs we will show for a few selected examples<br />

how these results have been achieved <strong>and</strong> we will present details of both, the electronic<br />

structures <strong>and</strong> the dynamics.<br />

The AC <strong>and</strong> the SO series are produced by additional irradiation with light, i. e. these<br />

are photoproducts which do not necessarily arise during the solid state polymerization of<br />

TS6. Although they are an important part of the entire variety of structures in partially polymerized<br />

TS6 crystals, we will not treat them in this review. They have been described extensively<br />

by Sixl in his papers cited above <strong>and</strong> in Ref. [48].<br />

9.2.3 Electronic structure of dicarbenes<br />

9.2.3.1 Electron spin resonance of quintet states ( 5 DC n )<br />

As an example for the electronic structure analysis of the intermediate states with ESR, we<br />

will review below the ESR spectra of the dicarbenes DC 7 …DC 13 [28, 32, 42]. They are<br />

characterized by their fine structure <strong>and</strong> temperature dependence. Figure 9.9 shows the ESR<br />

131

9 Diacetylene Single Crystals<br />

ESR-Signal<br />

T<br />

Q<br />

T<br />

T<br />

Q<br />

Q Q T<br />

Q<br />

Q<br />

Q<br />

T<br />

Q<br />

T=10 K<br />

ν =9,46 GHz<br />

T<br />

0 200 400 600<br />

Magnetic Field B 0/mT<br />

Figure 9.9: The ESR spectrum of perdeuterated TS after irradiation for 1000 s. The signals marked<br />

with T arise from triplet states <strong>and</strong> those with Q from quintets. The T-lines are microwave saturated. The<br />

magnetic field B 0 is oriented parallel to the z-axis of the quintet fine structure tensor [28].<br />

spectrum of a perdeuterated TS crystal which has been irradiated with UV light (313 nm) of<br />

a mercury high-pressure arc lamp (HBO 200) at 4.2 K for about 1000 s. All ESR lines in<br />

Fig. 9.9 labelled with Q are due to dicarbenes in their quintet state (S = 2). Prior to the UV<br />

irradiation no ESR signal is observed.<br />

The temperature dependences of the ESR signals (Fig. 9.10) show one common feature:<br />

the ESR intensities vanish for T ? 0, i. e. the ESR signals are thermally activated <strong>and</strong><br />

the ground state is spinless. But the temperatures for the maximum intensities are different<br />

for each ESR signal, indicating different activation energies.<br />

1<br />

10<br />

5<br />

T/K<br />

2<br />

Signal Intensity<br />

0,5<br />

0<br />

0<br />

0,2 0,4 0,6 0,8<br />

1/T/K -1<br />

Figure 9.10: Temperature dependence of the ESR intensities of dicarbenes DC 9 ,DC 10 , <strong>and</strong> DC 11 . The<br />

calculated lines have been fitted by Eq. 9 with the activation energies De* SQ for the quintet states [33, 47].<br />

132

9.2 Photopolymerization<br />

The resonance fields for fixed microwave frequencies are strongly anisotropic.<br />

Figure 9.11 shows as an example this anisotropy for the dicarbenes DC 9 ,DC 10 ,DC 11 , <strong>and</strong><br />

DC 12 , respectively. The external field has been rotated to the polymer backbone plane. Each<br />

anisotropy belongs to one distinct temperature dependence.<br />

The following model (Fig. 9.12 <strong>and</strong> inset Fig. 9.13) describes quantitatively the anisotropic<br />

fine structure <strong>and</strong> the temperature dependence. The ground state (|S>) of each dicarbene<br />

in first order is a singlet state with the total spin quantum number S = 0. The excited<br />

state (|Q>) in first order is a quintet state with S = 2. The excitation energy is De SQ . The<br />

quintet state is built up by the electronic coupling of two triplet carbenes at both ends of the<br />

oligomer via exchange interaction. R 12 is the distance between the two triplet carbenes. If<br />

the singlet-quintet splitting De SQ is large as compared to the magnetic dipole-dipole coupling<br />

(zero field splitting) within the triplet carbenes, the total spin quantum numbers S = 0<br />

<strong>and</strong> S = 2, respectively, are good quantum numbers, i. e. the quintet state is pure. The spin<br />

Hamiltonian for this case has been analyzed by Schwoerer et al. [34].<br />

Figure 9.11: Angular dependence of the resonance fields B 0 of the 4 dicarbene structures DC 9 ,DC 10 ,<br />

DC 11 , <strong>and</strong> DC 12 in perdeuterated TS-diacetylene crystals. The crystal is rotated so that the external<br />

magnetic field B 0 is in the plane of the polymer backbone. The b-axis is the direction of the polymer<br />

chain, y <strong>and</strong> z are the principal axes of the fine structure tensor. The curves are fitted to the experimental<br />

points by computer calculations. A <strong>and</strong> B indicate the two magnetically equivalent directions of the<br />

molecular orientation within the monoclinic unit cell. Dots: experimental values; lines: calculated by<br />

Q H S 0 (Eqs. 4–7) [32, 42].<br />

Figure 9.12: Dicarbene configuration. The dicarbene molecule consists of n diacetylene units with two<br />

identical triplet carbene chain ends. D t <strong>and</strong> E t are the triplet fine structure parameters of the S = 1 carbene<br />

species. j gives the orientation of the triplet fine structure z-axis with respects to the crystal b-<br />

axis.<br />

133

9 Diacetylene Single Crystals<br />

Figure 9.13: Experimental values for the singlet-quintet splitting De SQ for seven different dicarbenes<br />

DC 7 …DC 13 [42].<br />

If De SQ is smaller than or somewhere in the order of the magnetic dipole-dipole coupling<br />

then the singlet <strong>and</strong> quintet states are mixed. The spin Hamiltonian Q H S for this general<br />

case has been derived in the elegant work of Benk <strong>and</strong> Sixl [35]:<br />

Q H S ˆ g B B 0 …^S 1 ‡ ^S 2 †‡ 1 6 " SQ …^S 1 ‡ ^S 2 † 2 ‡H DD<br />

S<br />

…4†<br />

The first term of Eq. 4 represents the electronic Zeeman term, ^S 1 <strong>and</strong> ^S 2 the spin operators<br />

for two triplet carbenes, <strong>and</strong> the second term represents the electronic exchange interaction.<br />

If ^S 1 <strong>and</strong> ^S 2 couple to a quintet state (S = 2), (^S 1 + ^S 2 ) 2 = S 2 = S(S + 1) = 6. If<br />

they couple to a singlet, S = 0. Therefore, this term directly results in the energy level<br />

scheme, indicated in the inset of Fig. 9.13. The pure singlet <strong>and</strong> the pure quintet states are<br />

split by De SQ which turns out to be the characteristic property of each dicarbene. The third<br />

term of Eq. 4 represents the magnetic dipole-dipole coupling of the two triplet carbenes:<br />

H DD<br />

S<br />

^<br />

ˆ D …S 2 1z<br />

^<br />

1<br />

3 S2 1 †‡E…S2 1x<br />

X S^<br />

1x S^<br />

2x ‡ XAS^<br />

1y S^<br />

2y<br />

^<br />

S 2 ^<br />

1y†‡D…S 2 2z<br />

1<br />

^<br />

3 S2 2<br />

^<br />

†‡E…S 2 2x<br />

^<br />

S 2 2y†‡<br />

X…1 ‡ A† S^<br />

1z S^<br />

2z ‡ Xa…S^<br />

1z S^<br />

2y ‡ S^<br />

1y S^<br />

2z† …5†<br />

D <strong>and</strong> E are the fine structure parameters of the identical triplet carbene chain ends;<br />

x, y, <strong>and</strong> z are the principal axes of the corresponding fine structure tensors. The intercarbene<br />

magnetic dipolar interaction is represented by the parameter<br />

134

9.2 Photopolymerization<br />

X ˆ g 2 2 B … 0=4p†R 3<br />

12 : …6†<br />

The geometrical factors A <strong>and</strong> a are only dependent on the orientation j of the fine<br />

structure tensor z-axis with respect to the b-axis of the crystal.<br />

A ˆ 1 3 sin 2 '; ˆ 3 sin 2': …7†<br />

2<br />

The diagonalization Q H S of yields the allowed ESR transitions, their resonance fields,<br />

<strong>and</strong> their dependence on the orientation of the external magnetic field B 0 . At a first sight,<br />

the number of fit parameters seems to be high: D, E, g, the orientation of the triplet fine<br />

structure tensor, R 12 , <strong>and</strong> De SQ . But besides R 12 <strong>and</strong> De SQ these parameters are well-known<br />

from earlier ESR experiments on triplet carbenes [20, 49, 50]. Therefore, R 12 <strong>and</strong> De SQ are<br />

the only free to fit parameters. Figure 9.11 shows the result of the fit. The ESR anisotropy<br />

was calculated by exact diagonalization of Q H S with the fitting parameters R 12 <strong>and</strong> De SQ .In<br />

all cases the fit is almost perfect. Furthermore, it turns out that the intertriplet magnetic dipole-dipole<br />

interaction does not influence the results if R 12 exceeds 12 Å. Therefore, only<br />

the parameter De SQ remains to fit for each dicarbene. The obvious differences between the<br />

anisotropies of the different dicarbenes (Fig. 9.11) are due to De SQ only! The result is shown<br />

in Fig. 9.13 where De SQ is decreasing exponentially with increasing number n of monomer<br />

units:<br />

" SQ ˆ " 0 SQ e R=R 0<br />

; …8†<br />

with R = n1 m (1 m is the length of the monomer unit within the oligomer). The slope in<br />

Fig. 9.13 yields R 0 = 5.4 Å&10 a 0 (a 0 = Bohr radius). The origin of the abscissa in Fig. 9.13<br />

was deduced by Neumann [47]. But even without the knowledge of this origin the value of R 0<br />

as compared to a 0 shows that the triplet carbene is highly delocalized <strong>and</strong> not at all restricted<br />

to the end of the oligomer, as one might think because of Fig. 9.12. The model also gives the<br />

temperature dependencies of the dicarbene signals. Because in first order the ground state is<br />

spinless, the ESR intensities I of the quintet states (beyond Curie’s law) must be thermally activated:<br />

I / 1 T ‰5 ‡ exp…" SQ =kT†Š :<br />

…9†<br />

The lines in Fig. 9.10 are calculated with Eq. 9 by fitting De* SQ . For the dicarbenes<br />

DC 8 ,DC 9 ,DC 10 , <strong>and</strong> DC 11 , both values, De* SQ <strong>and</strong> De SQ , are determined respectively. Within<br />

the experimental uncertainty they are identical [32, 42]. This latter result is an excellent<br />

proof for the dicarbene model because De* SQ <strong>and</strong> De SQ were determined from two completely<br />

different properties: the anisotropy of the fine structure <strong>and</strong> the intensity of the ESR spectra.<br />

We therefore have no doubt that we really did observe dicarbenes (as sketched in Figs. 9.8<br />

<strong>and</strong> 9.12). They are an ideal modelling substance for short linear oligomers of diacetylenes<br />

which are perfectly oriented in the single-crystal lattice. The longest dicarbene (DC 13 ) observed<br />

has according to our model a length of (R =1364,9 Å) 64 Å!<br />

135

9 Diacetylene Single Crystals<br />

9.2.3.2 ENDOR of quintet states<br />

The most important result of the preceding Chapter is the delocalization of the spins S = 1<br />

of the two triplet carbenes which is necessary for their coupling over a distance of up to<br />

64 Å (!) to the well defined quintet states. It was therefore attractive to measure <strong>and</strong> analyze<br />

the ENDOR spectrum of at least one quintet dicarbene. ENDOR should detect at<br />

least the protons of the CH 2 groups of the substituents if the nuclear spins of these protons<br />

are hyperfine coupled with the electron spin of the triplet carbene. As compared to<br />

ENDOR with an electron spin of S = 1/2, the complication is the high anisotropy of the<br />

five electronic Zeeman levels Q u (u = 1 … 5) of the quintet state (Fig. 9.14). Not only<br />

their energy separation, i. e. the ESR transition fields (Fig. 9.11), are strongly dependent<br />

on the direction of the external field. Also their effective spin S eff , i. e. the expectation<br />

value of the spin, is dependent on the direction <strong>and</strong> on the strength of the external field<br />

B 0 . Therefore in this case the orientation quantum number m s is an unsuitable quantum<br />

number not only because of the large zero field splitting, as expressed by D, but also because<br />

of the singlet-triplet mixing, as expressed by De SQ . As described in the preceding<br />

paragraph this problem has been solved with high accuracy, Hartl et al. [36] measured<br />

the ENDOR spectra <strong>and</strong> their anisotropies for one quintet dicarbene ( 5 DC 10 ), which is accessible<br />

most comfortably at T = 4.2 K (Fig. 9.10) <strong>and</strong> for which the total spin quantum<br />

number (S = 2) is a good quantum number. The aim of these experiments was to determine<br />

the hyperfine coupling constants with the above-mentioned protons <strong>and</strong> subsequently<br />

to extract the electron spin density from these values, i. e. the delocalization of<br />

the triplet carbene quantitatively.<br />

Figure 9.14: Quintet state with an external field B o interacting with one proton (I = 1/2). All allowed<br />

ESR transitions (a–d) <strong>and</strong> NMR transitions (1–5) are shown. By ENDOR, in first order, only the two<br />

NMR transitions directly connected to the observed ESR line are detectable [36].<br />

136

9.2 Photopolymerization<br />

The spin Hamiltonian Q H S;i of a quintet dicarbene coupled to one individual proton,<br />

numbered i, is<br />

Q H s;i ˆ QH 0 s ‡ g I K B 0 ^I i ‡ ^SA i I i :<br />

…10†<br />

The first term in Eq. 10 is given by Eq. 4 <strong>and</strong> the second is the nuclear Zeeman energy.<br />

^I i is the nuclear spin operator for nuclear spin 1/2. The third term is the hyperfine interaction<br />

of the individual proton i, as defined by the hyperfine tensor A i . ^S is the total electron<br />

spin operator, ^S ˆ ^S 1 ‡ ^S 2 . As nuclear dipole-dipole interaction can be neglected we<br />

will omit the index i in the following.<br />

The nuclear terms in Eq. 10 are small as compared to the electronic terms. Therefore,<br />

we treat them in first order perturbation theory, taking the solutions of Q H 0 S as basis. Q H 0 S<br />

has five quintet eigenstates |Q u S , u = 1, 2, 3, 4, 5. For very high fields they become the<br />

high field states |Q m S, for which ^S z |Q m s<br />

S = km s |Q m s<br />

S,m s = +2, +1, 0, –1, –2. But for the<br />

fields used in ordinary ESR spectrometers the electronic Zeeman energy is in comparison to<br />

the fine structure not large <strong>and</strong> therefore the electron spin is not quantized along the external<br />

field B 0 [39]. This results in a strong |Q u S-dependence on the direction of B 0 with respect to<br />

the crystal axes.<br />

From Q H 0 S the effective spin<br />

S u eff ˆ<br />

D E<br />

Qu j ^S jQ u<br />

…11†<br />

can be calculated exactly [34]. It is this electron spin which interacts via A with the proton<br />

spin.<br />

The first order perturbation theory of the nuclear terms calculates the shift Dn u of the<br />

individual proton Larmor frequency with respect to the free proton Larmor frequency n F<br />

(ENDOR shift):<br />

hDn u = hn u – g I m K |B 0 | , (12)<br />

(g I m K |B 0 |/h = n F ). The result of the calculation of the Larmor frequencies n u of the hyperfine<br />

coupled protons is [36, 43]:<br />

h u ˆjS u eff A g I K B 0 j : …13†<br />

For a quintet state one should observe 5 different ENDOR lines per proton. This is illustrated<br />

in Fig. 9.14 where the observed NMR transitions in the ENDOR experiment are indicated<br />

by the ciphers (n) = (1), (2), (3), (4), <strong>and</strong> (5); the first order ESR transitions are indicated<br />

by the lower case letters (a), (b), (c), <strong>and</strong> (d).<br />

Fig. 9.15 shows four ENDOR spectra as detected via the ESR transitions (a), (b), (c),<br />

<strong>and</strong> (d), respectively. Four protons i = 1, 2, 3, <strong>and</strong> 4, respectively, are clearly separated from<br />

the free proton frequency n F . In the vicinity of n F a large number of weakly coupled protons<br />

are visible. They also have been resolved by expansion of the NMR frequency scale. For a<br />

few strongly coupled protons we are able to detect all five NMR transitions (1) to (5). This<br />

is a further <strong>and</strong> definite proof that we do observe quintet states.<br />

137

9 Diacetylene Single Crystals<br />

3(2)<br />

2(2)<br />

1(2)<br />

(a)<br />

Bo II X<br />

= 9.570 MHz<br />

ν F<br />

ν F<br />

(c)<br />

Bo II Y<br />

ν F = 14.193 MHz<br />

2(1)<br />

1(4) 2(4) 3(4)<br />

4(4)<br />

0<br />

ν F<br />

3(2) 2(2)<br />

10 20 30<br />

40<br />

0 10 20 30<br />

(b)<br />

Bo II Y<br />

= 10.911 MHz<br />

ν F<br />

2(4)<br />

ν F<br />

ν F<br />

(d)<br />

Bo II X<br />

= 15.553 MHz<br />

ν F<br />

3(4)<br />

4(4)<br />

4(2)<br />

1(2)<br />

0 10 20 30 0 8 12 16 20<br />

NMR Frequency ν/MHz<br />

NMR Frequency ν/MHz<br />

NMR FREQUENCY ν / MHz<br />

Figure 9.15: ENDOR spectra as detected by the ESR transitions (a–d); n F is the free proton frequency.<br />

(1) to (5) label the NMR transitions illustrated in Fig. 9.14, <strong>and</strong> 1, 2, 3 … number the individual protons.<br />

The external field B o is oriented along the y or x-axis of the fine structure tensor [36].<br />

The ENDOR shift anisotropy is shown in Fig. 9.16 for the strongly coupled protons<br />

i = 1, 2, 3, <strong>and</strong> 4. This anisotropy is mainly due to the anisotropy of the effective spin S u eff.<br />

The lines were calculated (via Eqs. 12 <strong>and</strong> 13) by fitting A i . In total we have analyzed<br />

22 protons, the hyperfine tensors A i of which are presented in Tab. 9.4. Two features can be<br />

Figure 9.16: ENDOR shift anisotropies for the four strongest coupled protons i = 1, 2, 3 <strong>and</strong> 4, detected<br />

via the ESR transition (b). x, y <strong>and</strong> z are the principal axes of the fine structure tensor. The experimental<br />

values were taken for the rotation of B 0 in the yz-plane <strong>and</strong> in the zx-plane, respectively. The ENDOR<br />

shifts Dn are calculated for (b)i(2) transitions (drawn out) <strong>and</strong> the (b)i(3) transitions (dashed), respectively<br />

[36].<br />

138

9.2 Photopolymerization<br />

Table 9.4: Complete hyperfine tensors for 22 protons, calculated by fitting the experimental anisotropy<br />

in a least-squares method. The A ij are diagonalized principal values of the fitted tensor, a is the isotropic<br />

coupling constant, B jj the dipolar anisotropic tensor, F, Y, <strong>and</strong> c are Euler angles of the hyperfine tensor<br />

axes relative to the fine structure axes [36].<br />

i A xx A yy A zz a B xx B yy B zz F Y C Assign-<br />

MHz MHz MHz MHz MHz MHz MHz degr. degr. degr. ment<br />

1 16.298 18.953 16.747 17.333 –1.035 1.620 –0.586 13.6 –51.8 2.8 CH2<br />

2 11.870 14.290 10.683 12.281 –0.411 2.009 –1.598 –39.2 76.4 –12.8 CH2<br />

3 2.348 3.737 5.667 3.917 –1.569 –0.180 1.749 28.0 80.5 –6.9 CH2<br />

4 1.905 1.797 1.253 1.652 0.253 0.146 –0.399 74.9 101.2 4.6 CH2<br />

5 1.788 0.710 –0.391 0.702 1.086 0.007 –1.093 56.0 –16.4 –21.8 CH2<br />

6 0.407 0.804 0.703 0.638 –0.231 0.166 0.065 10.3 10.1 –36.9 CH2<br />

7 1.456 0.089 –0.119 0.475 –0.981 –0.387 –0.594 26.3 –25.4 –4.4 CH2<br />

8 0.700 0.620 0.076 0.465 0.234 0.155 –0.389 57.4 –15.3 8.0 CH2<br />

9 –0.791 0.981 0.520 0.237 –1.028 0.744 0.284 –23.4 –80.7 –15.4 CH2<br />

10 –0.992 1.424 0.148 0.194 –1.185 1.230 –0.045 –23.2 1.6 –67.2<br />

11 0.219 –0.079 0.400 0.180 0.039 –0.259 0.220 –67.5 111.6 41.8<br />

12 0.098 0.161 0.271 0.177 –0.079 –0.016 0.094 21.5 90.6 –25.7<br />

13 –0.819 1.321 –0.275 0.076 –0.895 1.245 –0.351 –6.7 –5.7 –6.0 CH2<br />

14 0.075 –0.120 0.239 0.065 0.011 –0.185 0.174 12.6 7.3 –38.9<br />

15 1.969 –1.045 –0.889 0.012 1.958 –1.057 –0.901 –68.2 –7.4 49.5 ARYL<br />

16 –0.066 –0.165 0.235 0.002 –0.067 –0.166 0.234 –24.1 –5.2 38.5<br />

17 –0.456 –0.051 –0.203 –0.203 –0.254 0.254 0.000 91.1 114.5 27.5<br />

18 2.350 –2.533 –0.473 –0.219 2.569 –2.314 –0.254 –20.4 9.5 –53.6 ARYL<br />

19 –0.311 –0.237 –0.442 –0.330 0.019 0.093 –0.112 –23.5 91.3 89.0<br />

20 –0.777 –0.289 –0.143 –0.403 –0.374 0.114 0.260 –36.0 –30.2 18.1 ARYL<br />

21 –2.070 1.303 –1.288 –0.685 –1.385 1.988 –0.603 6.0 –49.3 6.5 CH2<br />

22 –1.072 –0.735 –0.652 –0.820 –0.252 0.085 0.168 4.7 18.6 –27.0 CH2<br />

extracted from Tab. 9.4 immediately: first, the anisotropic part of the hyperfine tensor is<br />

small as compared to the isotropic part for almost all protons, <strong>and</strong> second, several protons<br />

(i = 17, 18, 19, 20, 21, 22) show a negative value of the isotropic coupling constant a.<br />

We are able to assign 6 protons unambiguously: i = 1, 2, 3, 13, 21, <strong>and</strong> 22 (Fig. 9.17).<br />

The analysis of the hyperfine data shows that the spin density at C 1 (Fig. 9.18) is only 11%,<br />

–2,2% at C 2 , 17% at C 3 , –6% at C 4 , 7,9% at C' 1 <strong>and</strong> –2% at C' 2 , i. e. the spin is highly delocalized<br />

within the oligomer (Fig. 9.18). This corresponds with the above-described exchange<br />

coupling of the two carbenes <strong>and</strong> is to our opinion a very impressive demonstration<br />

of the power of ENDOR.<br />

9.2.3.3 ESR <strong>and</strong> ENDOR of triplet dicarbenes 3 DC n<br />

For a long time it was unclear whether the triplet state (S = 1) of the dicarbenes ( 3 DC) does<br />

exist, <strong>and</strong> if it does whether its energy is higher or lower than the energy De SQ of the dicarbene<br />

quintet state ( 5 DC). Müller-Nawrath et al. [51] have shown theoretically <strong>and</strong> experimentally<br />

that the ESR transitions of the triplet states of carbenes ( 3 C) <strong>and</strong> of dicarbenes ( 3 DC),<br />

respectively, are mutually degenerated in diacetylene oligomers of diacetylene crystals if the<br />

139

9 Diacetylene Single Crystals<br />

Figure 9.17: Assignment of 6 hyperfine tensors to methylene protons at the polymerization head. The<br />

arrows indicate the directions of the strongest main value of the respective hyperfine tensors. These 6<br />

tensors have been fitted simultaneously yielding the spin density distribution shown in Tab. 9.4 [36].<br />

20<br />

17<br />

15<br />

10<br />

7,9<br />

11<br />

5<br />

0<br />

-5<br />

-2<br />

-2,2<br />

-6<br />

-10<br />

C2' C1' C4 C3 C2 C1<br />

Figure 9.18: Carbene spin density at the reactive polymer chain end (Fig. 9.17).<br />

oligomers are long, i. e. if the number of added monomers n is more than 7. Therefore 3 C<br />

<strong>and</strong> 3 DC cannot be discriminated by ESR. Their ENDOR spectrum, however, is completely<br />

different. The ENDOR shifts Dn are related by Dn DC = –1/2 Dn C . The existence of several<br />

ENDOR lines, which fulfills the characteristic factor –1/2 in the above-mentioned relation,<br />

unambiguously shows the existence of the triplet state of dicarbenes. Its excitation energy is<br />

lower than the excitation energy De SQ of the quintet states of the same dicarbene [51].<br />

140

9.2 Photopolymerization<br />

9.2.4 Flash photolysis <strong>and</strong> reaction dynamics of diradicals<br />

A single UV laser flash with wavelength l = 308 nm, pulsewidth 15 ns, <strong>and</strong> flash energy<br />

1 mJ initiates photopolymerization by the production of the diradical DR 2 [52]. At low temperature,<br />

i. e. 4.2 K, this photoproduct is stable. It is detected by its absorption spectrum. In<br />

the spectral range the 0-0 transition peaks at 422 nm between monomer <strong>and</strong> polymer absorption<br />

(Fig. 9.20a). Annealing the crystal containing photoproduct DR 2 , prepared as described<br />

above, produces the diradicals DR 3 ,DR 4 ,DR 5 , <strong>and</strong> DR 6 respectively by addition of one<br />

monomer per step (Fig. 9.8). All these diradicals can be detected by optical absorption spectroscopy,<br />

for example by cooling the crystals to low temperature in order to slow down the<br />

dark reaction (Fig. 9.20b). Sixl et al. [48] were the first who detected these optical absorption<br />

spectra [131]. The dark reaction at low annealing temperatures has been investigated extensively<br />

by Gross [38, 46].<br />

If the reaction is photoinitiated by a single UV laser flash at high temperatures<br />

(T > 180 K) the entire time-dependent reaction series, DR 2 ? DR 3 ? DR 4 ?DR 5 ?DR 6 ,<br />

can be observed by monitoring the transient optical absorption of each of the products DR 2<br />

to DR 6 . By this experiment we were able to analyze the reaction kinetics of these thermally<br />

activated steps separately [37, 44]. As an example Fig. 9.19 shows the transient absorption<br />

0.23<br />

T=270 K<br />

DR 422nm<br />

2<br />

0<br />

0.23<br />

DR 514nm<br />

3<br />

∆OD<br />

0.29 0<br />

DR 578nm<br />

4<br />

0.5 0<br />

DR 664nm<br />

5<br />

0<br />

0<br />

10<br />

t/ µ s<br />

20<br />

Figure 9.19: Time sequence of the intermediate products DR 2 ,DR 3 ,DR 4 , <strong>and</strong> DR 5 at T = 270 K after<br />

the UV flash which is indicated by an arrow. DOD is the change of the optical density after the UV<br />

flash [37, 44].<br />

141

9 Diacetylene Single Crystals<br />

Figure 9.20a: Difference of the optical absorption spectra (DOD) of TS6 after <strong>and</strong> before one single<br />

UV pulse (l = 308 nm, t = 15 ns, E = 0.1 mJ, T = 80 K). The peak at 422 nm is due to the absorption<br />

of the Dimer DR 2 .<br />

Figure 9.20b: Appearance of DR n reaction intermediates after a single pulse irradiation at 308 nm <strong>and</strong><br />

additional annealing. (a): 5 K, 0 min; (b): 100 K, 6 min; (c): 100 K, additional 30 min plus 120 K,<br />

36 min; (d): 130 K, 36 min plus 140 K, 16 min [131].<br />

at T = 270 K as detected by the change of the optical density (DOD) after the flash vs. time.<br />

Each intermediate product is detected at the maximum of its optical absorption: DR 2 at<br />

422 nm, DR 3 at 514 nm, DR 4 at 578 nm, <strong>and</strong> DR 5 at 664 nm. The delay in the production<br />

of a subsequent intermediate is clearly demonstrated. The whole reaction passes in a 10 ms<br />

time scale at room temperature.<br />

Assuming that DR 2 is produced by the flash promptly, DR 3 by a dark reaction from<br />

DR 2 ,DR 4 by a subsequent dark reaction from DR 3 , <strong>and</strong> so on, we used a simple kinetic<br />

142

9.2 Photopolymerization<br />

model for the quantitative analysis. This model is described by the following equations for<br />

the concentrations n i <strong>and</strong> the rate constants K i for the DR i , i =1,2,3,4,5:<br />

dn i<br />

dt ˆ K i 1 n i 1 K i n i ; …14 a†<br />

K i ˆ K 0 e …E i=kT† : …14 b†<br />

As an example, the fit of this model to the transient DR 4 at 200 K is shown in<br />

Fig. 9.21. The fit does not show any significant deviation from the experimental curve.<br />

Figure 9.21: Experimental transient <strong>and</strong> model curve according to Eq. 14a for DR 4 in perdeuterated TS<br />

at T = 200 K. The difference between experiment <strong>and</strong> model is plotted around the baseline [37, 44].<br />

One result of these experiments is shown in Fig. 9.22. The addition reactions are thermally<br />

activated, the activation energies being about 0.25 eV per monomer, almost identical<br />

for each step (Tab. 9.5). Figure 9.22 also includes values for a product labelled V which is<br />

presumably due to long polymer chains with reactive carbene chain ends.<br />

A second result of these experiments is an estimation of the polymer yield for the<br />

photoreaction as defined by the number of polymerized monomer molecules per absorbed<br />

UV photon. Q is the quantum yield for the initiation process. Thus, the polymer yield P is<br />

the product of Q <strong>and</strong> the kinetic chain length L, i. e. the number of monomer molecules<br />

which are added to the chain after one initiation process:<br />

P = Q7L = 0.07 ± 0.02 (15)<br />

The polymer yield was determined from the increase of the polymer absorption DOD<br />

due to UV irradiation [53].<br />

If we assume a kinetic chain length of 100, which is a reasonable value, we get a quantum<br />

yield for chain initiation of 7610 –4 . P increases with decreasing temperature from<br />

300 K to 180 K by nearly a factor of five. If TS is perdeuterated P also increases by a factor<br />

of 2.5 [44, 54].<br />

Both, the temperature effect <strong>and</strong> the isotope effect on the polymer, can be explained<br />

qualitatively by an increase of the kinetic chain length due to a longer carbene chain end<br />

lifetime [37, 44, 54].<br />

143

9 Diacetylene Single Crystals<br />

Figure 9.22: Temperature dependencies of the rate constants for the decays of DR 2 ,DR 5 , <strong>and</strong> V. They<br />

can be described by the Arrhenius law in a range of three orders of magnitude [37].<br />

Table 9.5: Activation energies DE i <strong>and</strong> frequency factors K o for the reaction rates of DR i , evaluated from<br />

their Arrhenius plots [37].<br />

K 2 K 3 K 4 K 5<br />

DE i / eV 0.25 ± 0.03 0.26 ± 0.03 0.30 ± 0.03 0.30 ± 0.03<br />

K o /s –1 10 10 ± 1 10 11 ± 1 10 11 ± 1 10 11 ± 1<br />

9.3 Holography<br />

Diacetylenes have been subject to intense work due to their unique ability to undergo topochemical<br />

solid state polymerization, resulting in macroscopic polymer single crystals [1–3,<br />

25, 28, 37]. Whether this reaction takes place depends on the monomer stacking distance<br />

<strong>and</strong> the tilt angle (Fig. 9.3). Both can be influenced by varying the rest groups R. The polymerization<br />

can be initiated by heat, UV radiation, X rays, or g rays <strong>and</strong> is irreversible. The<br />

optical properties, especially the absorption coefficient a <strong>and</strong> the refractive index n, are<br />

known to change dramatically during polymerization.<br />

By means of UV photopolymerization high-efficiency holographic grating on diacetylene<br />

crystals can be recorded, as was first shown by Richter et al. [55, 56]. Utilizing a fre-<br />

144

9.3 Holography<br />

quency-doubled argon laser (l w = 257 nm) Richter et al. obtained surface phase gratings,<br />

due to the low penetration depth of this UV wave (Fig. 9.6). At high exposures higher diffraction<br />

orders up to five has been observed. Because of the 5% stacking distance mismatch<br />

between monomers <strong>and</strong> polymers, Richter et al. often observed a destructive surface peel<br />

off. This problem has been shown to become less important by using longer UV wavelengths<br />

<strong>and</strong>, therefore, higher penetration depths [57].<br />

The first aim within the Collaborative Research Centre 213 was to find a suitable technique<br />

for recording such gratings in an effective <strong>and</strong> reproducible way. Using this technique<br />

we have investigated the most important diffraction characteristics of these gratings: efficiency,<br />

thickness, angular selectivity <strong>and</strong> their dependence on exposure, sample thickness,<br />

<strong>and</strong> prepolymerization. The second aim, however, was to explain these characteristics within<br />

appropriate theoretical approach. Using this knowledge, the chain length of the polymers during<br />

UV polymerization <strong>and</strong> subsequent thermal treatment can be estimated. In the final investigation<br />

we have shown that images <strong>and</strong> even a holographic trick film can be recorded in the<br />

diacetylene crystals at room temperature. The peculiarity of the method is the difference of<br />

recording (UV) <strong>and</strong> reading (VIS) wavelengths. This difference allows prompt readout without<br />

a developing process <strong>and</strong> without perturbation of the hologram by the readout laser.<br />

9.3.1 Theory<br />

Figure 9.23 shows the writing <strong>and</strong> reading beams for recording <strong>and</strong> replay of the simple holographic<br />

grating, respectively. Provided that they are of equal intensity two coherent UV<br />

Figure 9.23: Schematic geometry of writing <strong>and</strong> reading of a holographic grating. Two coherent UV<br />

waves (dashed) form the grating of a spatial periodicity L. A VIS wave (solid) will generally be diffracted<br />

into different orders j.<br />

145

9 Diacetylene Single Crystals<br />

waves, impinging symmetrically the photoactive medium, form an intensity pattern on the<br />

surface of the photoactive material,<br />

I…x† ˆI 0 cos 2 …px=L† :<br />

…16†<br />

L is the grating distance given by<br />

L ˆ w =2 sin w ;<br />

…17†<br />

where l w is the vacuum wavelength <strong>and</strong> y w the angle of incidence outside the medium. This<br />

intensity pattern results in a photoproduct distribution of the same periodicity, represented<br />

by a grating vector K,<br />

K ˆ 2p ^x=L ;<br />

K ˆ jKj ˆ 2p=L ; …18†<br />

<strong>and</strong> results in a UV photopolymerization pattern, the refractive index n, <strong>and</strong> the absorption<br />

coefficient a of which can be described as<br />

n…x† ˆn 0 ‡ P1<br />

n h cos …Khx† ;<br />

hˆ0<br />

a…x† ˆa 0 ‡ P1<br />

a h cos …Khx† :<br />

hˆ0<br />

…19†<br />

The importance of the Fourier coefficients besides h = 1 generally depends on reaction<br />

kinetics, exposure, <strong>and</strong> saturation effects. In general, n <strong>and</strong> a can also vary with the z coordinate.<br />

In principle a readout light beam (vacuum wavelength l r ) will be diffracted by the<br />

grating resulting in different orders j as indicated in Fig. 9.23. Each of them have an amplitude<br />

S j <strong>and</strong> a propagation vector r j . The total electric field inside the medium is a superposition<br />

of all theses waves:<br />

E…z† ˆ<br />

jˆ1 P<br />

jˆ 1<br />

S j …z† ^s j exp… ir j r† : …20†<br />

Here ^s j are the polarization vectors. The wave vectors r j are coupled to the grating<br />

vector K via<br />

r j ˆ r 0 ‡ jK :<br />

…21†<br />

This treatment was used first by Magnusson <strong>and</strong> Gaylord [58] <strong>and</strong> leads to a system<br />

of coupled differential equations for the complex amplitudes S j :<br />

146<br />

uS j<br />

uz ‡ cos …a 0 ‡ i# j †S j ‡<br />

1 X1 <br />

i cos S j h …2pn h = r ia h † ^s j h ^s j ‡ S j‡h …2pn h = r ia h † ^s j‡h ^s j ˆ 0 …22†<br />

2<br />

hˆ1

9.3 Holography<br />

where is given by<br />

# j ˆ jK cos… '† …jK† 2 r =4pn 0 …23†<br />

with a slant angle j between K <strong>and</strong> the x direction; j =908 for the symmetric case shown<br />

in Fig. 9.23. This so-called coupled wave approach is a generalization of the work of Kogelnik<br />

[59], who assumed pure sine gratings read under the Bragg condition where only one<br />

transmitted <strong>and</strong> one diffracted wave is present. Kogelnik [59] gives analytical solutions for<br />

the amplitudes of the reference <strong>and</strong> the signal wave (zeroth <strong>and</strong> first order respectively) –<br />

only for the first ascent period of a growth curve, extending earlier work for the transparency<br />

region. For the efficiency Z j of the transmission grating with the thickness d<br />

j …d† ˆS j …d†S j …d†<br />

…24†<br />

he gets for j = 1 the well-known formula<br />

<br />

…d† ˆ sin 2 …pn 1 d=cos 0 †‡sinh 2 <br />

…a 1 d=2 cos 0 † exp… 2ad=cos 0 † ; …25†<br />

with the replay wavelength l <strong>and</strong> the Bragg angle y 0 . It should be pointed out that this formula<br />

is only valid for the special case of thick (volume) gratings, which show pure Bragg behaviour.<br />

That is they possess neither higher Fourier coefficients nor a modulation amplitude n 1 nor large<br />

enough a 1 , to produce higher diffraction orders, assuming n 1 <strong>and</strong> a 1 do not vary with z.<br />

The thickness of a holographic grating is often described in terms of the Q factor, defined<br />

by<br />

Q ˆ 2pd=L 2 n 0 :<br />

…26†<br />

Gratings with Q^1 are regarded as thin those with Q 610 as thick.<br />

9.3.2 Experimental setup<br />

We used diacetylene single crystal platelets, approx. 20 mm to200mm thick <strong>and</strong> some mm 2<br />

in area, cleaved from a parent TS6 crystal parallel to the (100) surface.<br />

The experimental setup is shown in Fig. 9.24. For writing holographic gratings we utilized<br />

a cw helium-cadmium laser (l w = 325 nm) or a xenon chloride excimer laser (l w = 308 nm).<br />

We gave preference to the recording geometry suggested by Bor et al. [62] <strong>and</strong> not to the wellknown<br />

beamsplitter geometry. The first order diffracted beams of a reflection grating R are reflected<br />

by a pair of parallel mirrors M1 <strong>and</strong> M2 (or pass a biprism instead) <strong>and</strong> are superimposed<br />

on the sample S. This geometry yields four main advantages:<br />

a) Given the fringe distance of the reflection grating, the resulting fringe distance on the<br />

sample is L = D/2 <strong>and</strong> wavelength-independent;<br />

147

9 Diacetylene Single Crystals<br />

b) Both writing beams are superimposed correctly because they have passed the same<br />

number of reflections. This is especially important if non-Gaussian beams are used;<br />

c) The intensities of both beams are equal <strong>and</strong> wavelength-independent;<br />

d) The setup is realizable compactly <strong>and</strong> can easily be adjusted for the use of short coherence<br />

lengths.<br />

Control of exposure is possible using an UV enhanced photodiode (De3). To read the<br />

gratings we used a 3 mW helium-neon laser (l r = 633 nm). At this wavelength the increase<br />

in refractive index can be expected to be high, whereas the absorption should not become<br />

too strong during the induction period of the polymerisation reaction (Fig. 9.4). The sample<br />

holder was mounted on the axis of a stepping motor for varying the angle of incidence. For<br />

analyzing transmitted or diffracted light a preamplifed large-area photodiode (De1) was<br />

mounted on a radius level of a second stepping motor being coaxial to the first one. Both<br />

motors include a reduction gear, giving an angular resolution of 0.06 mrad. Signal improvement<br />

was achieved by chopping the readout beam <strong>and</strong> using a second photodiode (De2) as<br />

an intensity reference. Both photodiode outputs were led to lock-in amplifiers. Two polarizers,<br />

Pol1 <strong>and</strong> Pol2, were used to attain both UV <strong>and</strong> VIS laser polarizations parallel to the<br />

sample’s b-axis.<br />

Figure 9.24: Experimental setup. De1, De2, De3: large area photodiodes, R: reflection grating, S: sample<br />

on sample holder, Sh: beam shutter, BS: beam splitter, M: mirrors, Pol1, Pol2: polarizers (l/2<br />

plates), Ch: chopper.<br />

Two sample orientations relative to the grating fringes were investigated:<br />

a) UV polarization,VIS polarization, <strong>and</strong> polymer axis b lying in the plane of incidence (E<br />

mode), henceforth denoted as b k orientation;<br />

b) UV polarization, VIS polarization, <strong>and</strong> polymer axis b were perpendicular to the plane<br />

of incidence (H mode), called b || orientation.<br />

148

9.3 Holography<br />

The high dichroism of the crystals <strong>and</strong> their well-formed habit can be used to align<br />

their orientation under a polarizing microscope. Of course, the optical anisotropy of the crystals<br />

as well as anisotropic reaction kinetics give rise to birefringence effects.<br />

9.3.3 General characterization<br />

Figure 9.25 shows for typical samples the maximum efficiency Z(y 0 )=Z as a function of<br />

the exposure. These growth curves show that an optimum can be reached between 0.5 <strong>and</strong><br />

1.0 J/cm 2 for l w = 308 nm <strong>and</strong> between 5 <strong>and</strong> 10 J/cm 2 for l w = 325 nm. For TS6 the penetration<br />

depth at 308 nm is about 110 mm, whereas at 325 nm it is about 420 mm. This difference<br />

<strong>and</strong> in addition a presumably lower quantum yield at 325 nm results in much slower<br />

growth curves for this wavelength. The decrease in efficiency is not only an effect of coupling<br />

back intensity from first to zeroth order, but also produced by an increase of absorption.<br />

Sample quality <strong>and</strong> thickness can influence these curves. A destruction of the surface<br />

is only observed if the optimal exposure is exceeded by a factor 2 to 3. This is due to the<br />

lattice mismatch of monomer <strong>and</strong> polymer stacking distance.<br />

The energy needed to reach the maximum is approximately by a factor 5 larger than<br />

at 257 nm [55, 56]. From the growth curves, i. e. from the exposure E which was needed to<br />

produce a certain efficiency Z, the holographic sensitivity S of the material given by<br />

S ˆ<br />

p<br />

=E<br />

…27†<br />

can be estimated. For TS6 <strong>and</strong> depending on sample quality this value can range from 0.4 to<br />

1.4 cm 2 /J. This is in the order of typical sensitivities for photorefractive crystals but far less<br />

0 12.5 25<br />

0.5 0.13<br />

Efficiency η<br />

0.25<br />

0.065<br />

0 0<br />

0<br />

1.5<br />

3.0<br />

Exposure E / Jcm -2<br />

Figure 9.25: Holographic growth curves of TS6 (solid curve: l w = 308 nm, d = 130 mm; dotted curve:<br />

l w = 325 nm, d = 270 mm) <strong>and</strong> IPUDO (dashed curve: l w = 308 nm, d = 260 mm). Orientation b k ,<br />

L = 3.3 mm. Z(y 0 )=Z: efficiency for readout Bragg condition.<br />

149

9 Diacetylene Single Crystals<br />

than for common silver halide materials [63]. Nevertheless, the resolution of TS6 is comparable<br />

to that of common photographic materials. From the experiments of Richter et al. [55]<br />

it is known that a resolution of 2500 lines/mm is achievable. In our electron beam lithography<br />

experiments with a resolution of 5000 lines/mm has been achieved [64].<br />

Supposing the thickness of the sample is equal to the grating thickness, we can calculate<br />

the factor Q for our gratings. At a grating distance of L = 3.3 mm this factor<br />

ranges from 2.5 to 60 for samples of thickness 25–500 mm, whereas at L = 0.8 mm Q<br />

factors of 800 can be achieved. These values are typical for thick phase gratings. Nevertheless,<br />

it should be pointed out that for example a grating with Q = 40 already shows<br />

higher diffraction orders. This confirms the arguments brought by Moharam <strong>and</strong> Young<br />

[60] that the r factor (r = L 2 /L 2 n 1 n 0 ) should be preferred when discussing grating diffraction.<br />

9.3.4 Angular selectivity<br />

The angular selectivity is a measure for diffracted intensity when readout is performed under<br />

off-Bragg conditions or, quantitatively speaking, it is the half-width Dy of the function Z (y),<br />

where y is the angle of incidence. This property is of central interest, when discussing holographic<br />

storage media, because it limits the number of holograms/holographic gratings<br />

which can be stored simultaneously.<br />

Only as a rule of thumb, Kogelnik [59] gives Dy & Ln 0 /d, whereas Magnusson <strong>and</strong><br />

Gaylord [58] do not give any analytical expression for this quantity. Both approaches can easily<br />

be implemented on a computer to simulate diffraction properties. Assuming n h = 0 for<br />

h 62 <strong>and</strong> only for the first ascent period of a growth curve we find a Dy-dependence of the<br />

following simple form<br />

ˆ n 0 =d<br />

…28†<br />

valid only for the first ascent period of a growth curve, because the angular selectivity<br />

curves get split beyond the first maximum [61].<br />

The dependence of Dy on sample thickness d proved to be very well reproducible with<br />

respect to adjustment <strong>and</strong> sample quality. Figure 9.26 shows two typical measurements of<br />

Z (y) representing samples of different thickness. In contrast to the early experiments of<br />

Richter et al. [55] <strong>and</strong> of Niederwald et al. [56] these values represent an improvement of<br />

Dy by up to two orders of magnitude.<br />

Figure 9.27 shows Dy values versus sample thickness for TS6. No significant difference<br />

could be observed with respect to exposure or orientation for this grating distance.<br />

There is also no significant difference between TS6 <strong>and</strong> IPUDO with respect to angular selectivity.<br />

Measurements at L = 0.8 mm are summarized in Fig. 9.28. Within the series the minimum<br />

angular selectivity of 0.188 arose, using a 380 mm thick TS6 platelet. The hyperbolas<br />

plotted in Figs. 9.27 <strong>and</strong> 9.28 are fits of the function Dy = Ln 0 /d to the data points. Averaged<br />

over all exposures n 0 ranges from 1.3 to 1.7.<br />

150

9.3 Holography<br />

1<br />

Rel. Efficiency<br />

0.5<br />

0<br />

-30 0<br />

30<br />

Readout Angle<br />

/ Deg.<br />

Figure 9.26: Dependence of relative efficiency Z as a function of the incidence angle y for two TS6<br />

samples of different thickness d.<br />

Figure 9.27: Angular selectivity Dy as a function of sample thickness d for TS6 (open circles) <strong>and</strong><br />

IPUDO (full circles). L = 3.3 mm, l w = 308 nm [65].<br />

Angular Selectivity ∆Θ / Deg.<br />

2<br />

1<br />

0<br />

0 250<br />

500<br />

Sample Thickness d / µ m<br />

Figure 9.28: Angular selectivity Dy as a function of sample thickness d for TS6. L = 0.8 mm,<br />

l w = 325 nm for both writing geometries, b || (full circles) <strong>and</strong> b k (open circles) [65].<br />

151

9 Diacetylene Single Crystals<br />

9.3.5 Prepolymerized samples<br />

Until now we were assuming that sample thickness <strong>and</strong> hologram thickness are equal. In<br />

fact, for large thickness values d one can observe slight deviations from the hyperbola function<br />

due to the finite penetration depth of the UV light. These deviations should become<br />

more obvious for smaller penetration depths. To observe this some Dy measurements were<br />

carried out using thermally prepolymerized (up to 5 h at 70 8C before recording gratings)<br />

TS6 samples. Penetration depths for these samples varies between 18 to 80 mm at 308 nm<br />

<strong>and</strong> from 16 to 420 mm at 325 nm in the b || orientation.<br />

The effect of prepolymerization on angular selectivity is shown in Fig. 9.29. The values<br />

for samples prepolymerized for 2 h still show for thin samples a weak thickness dependence.<br />

For thick samples Dy does not tend to decrease further. Samples prepolymerized for<br />

5 h do not show any dependence on d, indicating a penetration depth distinctly smaller than<br />

the smallest sample thickness used. Thus, for an increasing polymer concentration the holographic<br />

gratings get thinner <strong>and</strong> thinner. The possibility of producing high efficiency volume<br />

phase gratings is restricted to fresh monomer crystals.<br />

Figure 9.29: Angular selectivity of prepolymerized TS6 samples: 5h (upper open circles), 2 h (full circles),<br />

<strong>and</strong> fresh crystals (lower open circles) [66].<br />

9.3.6 Chain length, polymer profile, <strong>and</strong> grating profiles<br />

A model [66, 68] describing the spatially inhomogeneous reaction kinetics of diacetylenes<br />

must take into account a kinetic chain length L, which depends on the polymer conversion P.<br />

The monomolecular reaction in a simple homogeneous situation then is given by<br />

dP=dt L…P†…1 P† : …27†<br />

For the experiments described so far, L can be assumed constant during the exposure<br />

time, because only very little conversion takes place. Furthermore, for all experiments de-<br />

152

9.3 Holography<br />

scribed, the factor (1 – P) can be dropped, because the experiments were carried out in the<br />

low conversion regime of the induction period <strong>and</strong> because saturation effects affect development<br />

experiments in both orientations. For the two orientations, different situations must be<br />

investigated corresponding to the different interaction of the two characteristic lengths L<br />

<strong>and</strong> L (Fig. 9.30),<br />

b jj : dP=dt L…P† cos 2 …px=L† : …28†<br />

Here a polymer molecule grows parallel to the fringes contributing its whole chain<br />

length L to the polymer growth at point x, where it has been initiated. In the other case, a chain<br />

initiated in x contributes to the polymer growth in the whole interval [x – L/2, x + L/2]:<br />

x‡L=2<br />

b ? : dP=dt R<br />

cos 2 …px 0 =L†dx 0 :<br />

x L=2<br />

…29†<br />

Integrating both Equations we see that in the second case the chain growth smears out<br />

the photoproduct distribution for a certain amount. This effect should be stronger if L/L approaches<br />

1. For the quotient of both polymer modulations we get the ratio d,<br />

ˆ P ?<br />

P jj<br />

ˆ sin…pL=L†<br />

pL=L : …30†<br />

The resulting d values range for fresh samples from 0.85 to 0.95 <strong>and</strong> tend to decrease<br />

with increasing prepolymer content. For samples prepolymerized (5–6 hours, 3–4% polymer)<br />

we find that d is between 0.65 <strong>and</strong> 0.80. These values correspond to chain lengths (L)<br />

of 0.15 to 0.4 mm or 300 to 800 repeat units [66].<br />

Figure 9.30: <strong>Microscopic</strong> model to underst<strong>and</strong> the interaction of the two characteristic lengths L <strong>and</strong> L:<br />

The two geometries investigated change the angle between chain growth direction <strong>and</strong> grating fringes.<br />

153

9 Diacetylene Single Crystals<br />

9.3.7 Multrecording<br />

In additional experiments, we succeeded in recording more than only one grating into one<br />

thick TS6 crystal. After each exposure interval, we rotated the sample for a certain amount<br />

to slant the gratings relatively to each other. Figure 9.31 shows the first order diffraction versus<br />

the angle of incidence for a 310 mm thick sample containing 42 gratings tilted stepwise<br />

by 1.08 8. We observed that gratings already present are almost not influenced by the subsequent<br />

recording processes except by the increase of total absorption, which causes a loss of<br />

efficiency less than 10 %. Thus, we did choose an exposure of only 0.07 J/cm 2 for each grating.<br />

As can be seen from Fig. 9.31, the signal-to-noise ratio is bad for the last written gratings.<br />

It will get worse by decreasing the angular distance of the individual gratings but will<br />

hardly improve by increasing it [67]. A serious diffraction of the UV light by the gratings already<br />

recorded in the sample does not take place. UV diffraction efficiencies are extremely<br />

small in diacetylene crystals.<br />

-3<br />

Efficiency η / 10<br />

8<br />

4<br />

1<br />

2<br />

4<br />

0<br />

-10 20<br />

50<br />

Readout Angle Θ/ Deg.<br />

Figure 9.31: Efficiency of the first order of a multihologramm consisting of 42 single gratings successively<br />

recorded in a 310 mm thick TS6 crystal. The gratings were recorded in the order of the numbers<br />

indicated [69].<br />

9.3.8 Holography<br />

For the reconstruction of a real hologram with different writing <strong>and</strong> reading wavelengths, it<br />

was necessary to use a divergent reference wave. With a test platelet as object a resolution<br />

of 300 lines/mm was achieved in excellent TS6 samples (Fig. 9.32). Holograms up to 32<br />

pictures were written (l w = 325 nm) in one crystal using the angular selectivity of the thick<br />

phase. When rotating the sample the pictures can be reconstructed (l r = 633 nm) successively<br />

(holographic trick film).<br />

The storage density of about 6.7610 9 cm –3 , estimated from the real resolution <strong>and</strong><br />

angular selectivity, is about one to two orders of magnitude lower than the storage density<br />

calculated from the grating distance [70].<br />

154

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

Figure 9.32: Reconstructed image from one of the 32 pictures of a holographic trick film [70].<br />

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

The dielectric properties of diacetylene monomer single crystals are not strikingly different<br />

from those of other organic materials. Typically, they have permittivity values (earlier called<br />

dielectric constant e r ) of about 4–6. Solid state polymerization, however, results in a pronounced<br />

anisotropy of the electric permittivity with maximum values parallel to the polymer<br />

chain direction, due to the large polarizability of the extended p-orbitals. During our TOPO-<br />

MAK activities the change of e r , accompanying solid state polymerization, was analyzed<br />

quantitatively (Section 9.4.1.1). The results of the e r analysis can be applied for the in situ<br />

monitoring of the polymer content (Section 9.4.1.2).<br />

The tailoring of pyro or ferroelectric properties of diacetylenes (Section 9.4.2) is less<br />

straightforward than might be surmised from the well-ordered arrangement of the R, R' substituents<br />

in Scheme 1, <strong>and</strong> from the fact that polar side groups can easily be introduced as<br />

substituents R <strong>and</strong> R' (Tab. 9.2). The difficulties originate from the packing of the individual<br />

polar diacetylene monomer molecules in the elementary cell of the solution-grown single<br />

crystal. This, generally, gives rise to a center of inversion symmetry for a pair of molecules<br />

thus compensating the individual molecular electric dipole moments in an antiferroelectric<br />

arrangement. During solid state polymerization spurious electric polarization has been observed<br />

[71] resulting from intramolecular distortion of originally centrosymmetric monomer<br />

units. Here we want to emphasize that our systematic TOPOMAK investigations have realized<br />

pyroelectric (Section 9.4.2) as well as ferroelectric properties (Section 9.4.3) for appropriately<br />

substituted diacetylenes.<br />

155

9 Diacetylene Single Crystals<br />

9.4.1 Dielectric properties of diacetylenes<br />

Typically, the electric permittivity of monomer single crystals of substituted diacetylenes<br />

ranges from 4 to 6 at room temperature. These e r values are therefore larger than values of<br />

simple non-polar organic polymers, like polytetrafluoroethylene, in agreement with the existence<br />

of polar side groups. As is exemplified in Fig. 9.33, monoclinic crystals of 2,4-hexadiynylene<br />

di-p-toluenesulfonate (TS, see Tab. 9.2) shows generally a weak but non-negligible<br />

anisotropy for e r [72, 75].<br />

ε r (h)<br />

8<br />

7<br />

6<br />

5<br />

TS<br />

T = 60°C<br />

(1)<br />

(2)<br />

4<br />

3<br />

0<br />

10 20 30<br />

polymerization time t(h)<br />

Figure 9.33: Electric permittivity e r (t) as a function of the polymerization time for TS crystals at<br />

60 8C. The electric field was applied parallel to the chain direction of the monoclinic crystals (1) <strong>and</strong> in<br />

the two orthogonal directions (2) <strong>and</strong> (3). The permittivity was measured at 1 kHz for three different<br />

thin parallel-plate single crystal capacitors [75].<br />

(3)<br />

40<br />

9.4.1.1 Correlation of polymer content <strong>and</strong> electric permittivity<br />

Generally, a sigmoid time-conversion curve is observed for thermal solid state polymerization<br />

of diacetylenes. This behaviour was shown for TS in Fig. 9.4 <strong>and</strong> is reported for the unsymmetrically<br />

substituted 6-( p-toluenesulfonyloxy)-2,4-hexadiynyl-p-fluorobenzenesulfonate<br />

(TS/FBS, Tab. 9.2). In Fig. 9.34a the slow conversion of the initial induction period of the<br />

solid state polymerization lasts until a polymer content of about 10 % is achieved. For higher<br />

up to complete conversion, this is followed by an autocatalytic reaction enhancement. The<br />

st<strong>and</strong>ard technique for the derivation of time-conversion curves is the gravimetrical analysis<br />

of a large number of crystals in a point-by-point procedure. After thermal polymerization<br />

for a well-defined period both, the soluble monomer <strong>and</strong> the insoluble polymer portions, are<br />

determined gravimetrically (Fig. 9.34 a). This technique consumes a considerable number of<br />

crystals <strong>and</strong> a substantial amount of time. Furthermore, only averaged time-conversion<br />

curves are obtained.<br />

Figure 9.34b shows that during solid state polymerization of TS/FBS the electric permittivity<br />

parallel to the chain direction, e r|| , increases by a factor of about 2. The permittivity<br />

e r|| is almost a linear function of the polymer content, as is exemplified in Fig. 9.34 c. We<br />

have found comparable behaviour for the substituted diacetylenes TS, FBS, FBS/TFMBS,<br />

<strong>and</strong> DNP [72–75]. Because the reorientation of the side groups during the polymerization is<br />

weak, it does not influence substantially the increase of e r parallel to the chain direction.<br />

156

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

Figure 9.34: Thermal solid state polymerization of TS/FBS (PTS is another acronym for TS).<br />

(a): Time-conversion curve derived by gravimetrical analysis. (b): Time-permittivity curve derived at<br />

1 kHz for a thin parallel-plate single-crystal capacitor oriented with the polymer chain direction (b-axis)<br />

parallel to the electric field. (c): Correlation of conversion <strong>and</strong> electric permittivity (with time as implicit<br />

parameter) obtained by combination of (a) <strong>and</strong> (b) [73].<br />

This proves that only the extended p-electron system of the diacetylene backbone is responsible<br />

for the enhanced polarizability. This conclusion is supported by the experimental analysis<br />

for three orthogonal directions in TS single crystals shown in Fig. 9.33 [75]. The minor<br />

variations for the two orthogonal directions (2) <strong>and</strong> (3) can be explained by the changes of<br />

the lattice parameters. In contrast, the change of the respective lattice parameters with polymer<br />

content does not suffice to explain the change of De r|| .<br />

The linear relation between permittivity <strong>and</strong> polymer content is in principle surprising<br />

because there is a distribution of chain lengths of solid state polymerization, differing between<br />

induction period <strong>and</strong> autocatalytic range. The linearity, shown in Fig. 9.34 c, reveals<br />

that the polarizability of the p-electron system saturates already at chain lengths below the<br />

shortest ones occurring during thermal solid state polymerization.<br />

The experimental range for e r is 1.4 to 2.2 derived for different substituted diacetylenes<br />

by Gruner-Bauer [72–75], which agrees with the observations of other groups [76–<br />

78]. These values compare favourably with the estimate De r & 1.6 obtained for TS by simplified<br />

model calculations [75]. For these theoretical estimates the method of Genkin <strong>and</strong><br />

157

9 Diacetylene Single Crystals<br />

Mednis [79] has been modified by Gruner-Bauer, extending earlier work for the transparency<br />

region [80].<br />

9.4.1.2 Application to topospecifically modified diacetylenes<br />

The linear relation of electric permittivity parallel to the chain direction vs. polymer content<br />

thus established can be used for the control of the polymer content of prepolymerized samples<br />

as well as for the derivation of time-conversion curves of individual single crystals in<br />

situ. We have used this technique to study the solid state polymerization of topospecifically<br />

<strong>and</strong> fully deuterated TS.<br />

Striking differences of their reactivities were reported before by Ch. Kröhnke [54].<br />

The limited amount of samples available was sufficient for the permittivity analysis.<br />

Figure 9.35 shows the behaviour of different topospecifically deuterated derivatives of TS<br />

at T =608C [73]. It should be stressed that the induction periods of single crystals with<br />

nominally the same history did not differ by more than 10% at the same polymerization<br />

temperature. The toposelective modification of these substituted diacetylenes by the deuteration<br />

of the methylene groups close to the triple bonds of the diacetylene monomer<br />

(that are engaged in the crankshaft-type motion of the monomer molecule around its center<br />

of mass during solid state polymerization) evidently has a drastic influence on the solid<br />

state polymerization.<br />

8<br />

7<br />

PTS<br />

T = 60°C<br />

6<br />

ε (t)<br />

5<br />

8<br />

7<br />

6<br />

5<br />

8<br />

7<br />

CD - PTS<br />

T = 60°C<br />

CD - PTS<br />

T = 60°C<br />

2<br />

6<br />

5<br />

0<br />

10<br />

Figure 9.35: Electric permittivity as function of time for solid state polymerization at T =608C for TS,<br />

PD-TS (fully deuterated TS), <strong>and</strong> CD 2 -TS (where only the methylene groups close to the triple bond of<br />

the diacetylene monomer unit are deuterated, see Tab. 9.2 a) [73].<br />

158<br />

20<br />

30<br />

polymerization time t/h<br />

40

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

9.4.1.3 Additional applications<br />

Structural phase transitions accompanying solid state polymerization influence the behaviour<br />

of e r (t). Thus they can not be observed but additional hints concerning the type of structural<br />

changes can be obtained. For example, a structural phase transition for DNP (Tab. 9.2) occurs<br />

at a polymer content above 95% resulting in the loss of order perpendicular to the polymer<br />

chains [81]. The accompanying increase of the side-group mobility results in a distinct<br />

increase of the electric permittivity [75]. Even more dramatic changes of e r were observed<br />

at the transition to a fibrillar structure in polar crystals of DNP/MNP (Tab. 9.2) [74, 75].<br />

Thinking of applications outside fundamental research, the correlation of permittivity<br />

<strong>and</strong> polymer content can also be used for different kinds of ageing control. Since solid state<br />

polymerization is an activated process, the temperature-weighted time at elevated temperature<br />

– like a low-temperature radiation dose – influences the capacity of a diacetylene crystal-plate<br />

capacitor according to a well-defined characteristic history-capacity. Evidently, this<br />

can be adapted to an appropriate electronic ageing control.<br />

9.4.2 Pyroelectric diacetylenes<br />

One fascinating goal of diacetylene materials research is the realization of polar polymer single<br />

crystals without complicated poling procedures. Substituted diacetylenes R 1 –C:–C:C–R 2<br />

incorporating polar side groups R 1 <strong>and</strong> R 2 with large but different electric dipole moments<br />

have to be synthesized. Since solid state polymerization depends on the ability of individual<br />

molecule side groups, which can perform intramolecular torsion <strong>and</strong> giving rise to an overall<br />

crankshaft-like motion, it is difficult – <strong>and</strong> was impossible for us – to predict the packing arrangement<br />

of individual monomer molecules in the solid. Frequently, the unit cell of substituted<br />

diacetylenes was observed to accommodate pairs of formula units in a centrosymmetric<br />

arrangement, thus compensating the net electric polarization. Polymorphism turned out to be<br />

an obstacle to systematic tailoring of dielectric properties of diacetylene single crystals, because<br />

different crystal structures of the same diacetylene derivative could be obtained from<br />

different, <strong>and</strong> occasionally even from the same solvent [82]. Nevertheless, Strohriegl synthesized<br />

during our TOPOMAK activities several non-centrosymmetric diacetylenes, which crystallized<br />

also in a polar phase. Typically, their permittivities were relatively large <strong>and</strong> anisotropic<br />

[83]. We restrict this report to three examples.<br />

9.4.2.1 IPUDO<br />

IPUDO (for the molecular structure see Tab. 9.2) is an example of a symmetrically substituted<br />

diacetylene. For the monomer as well as polymer crystals, non-centrosymmetrical<br />

orthorhombic crystal structures can be found already at room temperature [84]. The c-axis<br />

is the polar axis of the monomer crystal. IPUDO can only be polymerized by g radiation<br />

( 60 Co). For 85% polymerized crystals pyroelectric properties were observed only in b direction.<br />

The permittivity of IPUDO is highly anisotropic, with maximum values of 8.5 for the<br />

159

9 Diacetylene Single Crystals<br />

monomer (parallel to a), or 11.6 for the 85% polymer (parallel to b) <strong>and</strong> with minimum values<br />

(parallel to c) smaller by a factor of 2 (3.5) for the monomer (polymer) [83]. The distortion<br />

of the long side groups by the development of hydrogen bonds between neighbouring<br />

–CO–NH- groups is supposed to be responsible for the non-centrosymmetry [83, 85, 86].<br />

Thus it is not surprising that the variation of the electric polarization of about<br />

3610 –8 Ccm –2 between room temperature <strong>and</strong> 4 K amounts to an unbalancing of only<br />

about 1% of the compensation of oppositely oriented C=O … HN dipole moments.<br />

9.4.2.2 NP/4-MPU<br />

NP/4-MPU (for the molecular structure see Tab. 9.2) is an example of a non-centrosymmetric<br />

diacetylene that forms polar monomer single crystals only, if it is grown from appropriate<br />

solvents, here from 2-propanol [82]. The resulting modification I is orthorhombic<br />

with the polar space group Fdd2 (Z = 16) <strong>and</strong> c as the polar axis [82]. This modification is<br />

not reactive thermally or under X-ray irradiation, because the monomer packing is outside<br />

the favourable range for solid state polymerization. The diacetylene rods make an angle of<br />

678 with the stacking axis c , <strong>and</strong> the stacking distance d = 4.61 Å. The permittivity of this<br />

diacetylene is highly anisotropic <strong>and</strong> shows the largest value of about e r &23 for an electric<br />

field applied in the direction of the polar axis [82, 87].<br />

Figure 9.36 shows the variation of the spontaneous electric polarization DP of NP/4-<br />

MPU with temperatures between 10 K <strong>and</strong> the melting point. DP amounts to about 15% of<br />

the electric polarization, which can be estimated from the volume density of molecular electric<br />

dipole moments of about 3 Debye (10 –29 Cm).<br />

The pyroelectric coefficient p(T) =dP S /dT =8.8610 –10 Ccm –2 K –1 is of the same<br />

order of magnitude (smaller by 1/3) as that of the well-known <strong>and</strong> commercially used pyro<br />

<strong>and</strong> ferroelectric polyvinylidenefluoride. Therefore NP/4-MPU single crystals can be used<br />

for the detection of radiation [89], which we showed by using chopped low-power laser light<br />

as radiation source. The pyroelectric current <strong>and</strong> the total variation of the surface charge<br />

Figure 9.36: Temperature-dependent change of the spontaneous polarization for a NP/4-MPU single<br />

crystal (modification I) along the polar c-axis for temperatures below the melting point T m . The pyroelectric<br />

coefficient P at 300 K is also given [87].<br />

160

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

have been used for the detection. Furthermore, a transversal piezoelectric coefficient of<br />

1018 fCN –1 for NP/4-MPU was derived at room temperature. It is comparable with the value<br />

of a-quartz [88].<br />

9.4.2.3 DNP/MNP<br />

DNP/MNP (for molecular structure see Tab. 9.2) was the most successful one of Strohriegl’s<br />

syntheses [74]. It has a polar crystal structure for monomer <strong>and</strong> polymer crystals (space<br />

group P2 1 ). DNP/MNP polymerizes – thermally or exposed to UV radiation – extremely<br />

fast, because during solid state polymerization the molecules are packed optimally.<br />

The solid obtained by thermal polymerization of monomer crystals exhibits a fibrous<br />

texture, probably due to the large changes in lateral packing of the side groups. The temperature<br />

influence on the spontaneous electric polarization perpendicular to the chain axis c<br />

is shown in Fig. 9.37 for the monomer as well as the polymer crystal. The polarization varies<br />

up to room temperature by 7610 –8 Ccm –2 for the monomer crystal, with a pyroelectric<br />

coefficient of about 3.2610 –10 Ccm –2 K –1 at room temperature.<br />

0.0<br />

-2<br />

∆P(T) / 10 C cm<br />

-7<br />

-0.2<br />

-0.4<br />

-0.6<br />

-0.8<br />

-1.0<br />

0<br />

monomer<br />

DNP / MNP<br />

parallel to polar axis<br />

100 200<br />

T/K<br />

polymer<br />

Figure 9.37: Variation of the spontaneous electric polarization parallel to the polar b axis of DNP/MNP<br />

crystals with temperature. The data of DP (T) =P (T)–P (5 K) were derived by charge integration during<br />

a temperature cycle [74].<br />

300<br />

9.4.2.4 Spurious piezo <strong>and</strong> pyroelectricity of diacetylenes<br />

Sample defects can be another origin of piezo <strong>and</strong> pyroelectric phenomena in substituted<br />

diacetylenes, which generally can be identified via sample dependence <strong>and</strong> smaller size of<br />

these effects [88]. Bloor et al. discussed such spurious pyroelectric effects, which seemed to<br />

be correlated with the occurrence of macroscopic deformations, such as screw dislocations<br />

in TS single crystals [90]. Similarly the analysis of weak polarization, caused by molecular<br />

distortion during solid state polymerization of TS, was reported by Bertault et al. [91]. We<br />

have observed comparable weak pyroelectric phenomena for TS/FBS [87].<br />

161

9 Diacetylene Single Crystals<br />

9.4.3 The ferroelectric diacetylene DNP<br />

Whereas several pyroelectric diacetylenes were identified, uniform ferroelectric phases seem<br />

to be rather the exception, according to our experience with many new substitutions of diacetylenes<br />

[92]. The ferroelectric low-temperature phase of the symmetrically disubstituted<br />

diacetylene DNP, i. e. 1,6-bis(2,4-dinitrophenoxy)-2,4-hexadiyne (Tab. 9.2), turned out to be<br />

one of the rare <strong>and</strong> interesting exceptions [93–98].<br />

On both ends DNP carries polar dinitrophenoxy groups, with an electric dipole moment<br />

of 10 –29 Cm, whose mutual twisting gives rise to the spontaneous electric polarization<br />

of the non-centrosymmetric low-temperature phase (space group P2 1 ) (Fig. 9.38). The DNP<br />

monomer crystal has besides a pyroelectric [93] low-temperature phase a ferroelectric one<br />

for T < T c = 46 K. According to our investigations, the direction of the spontaneous polarization<br />

(Fig. 9.39) can be influenced by an external electric field [97]. Structural defects, occurring<br />

in all DNP crystals, give rise to a distribution of transition temperatures <strong>and</strong> to the<br />

existence of domains, whose polarization can not be inverted with the accessible external<br />

fields – thus behaving like pyroelectrics. Only the domains with the highest transition temperatures<br />

could be poled [97].<br />

b<br />

a<br />

b<br />

a<br />

b<br />

a<br />

a)<br />

b)<br />

c)<br />

Figure 9.38: Packing arrangement of DNP molecules in the monomer crystal viewed along the c-axis<br />

at three temperatures (a): T = 296 K, (b): T = 145 K, (c): T = 5 K [96].<br />

The temperature dependence of the spontaneous polarization (Fig. 9.39) of the monomer<br />

crystal can be described in the framework of L<strong>and</strong>au’s phenomenological theory assuming<br />

a tricritical phase transition [97]. The maximum experimental value of the electric polarization<br />

of 2.4610 –7 Ccm –2 compares favourably with the polarization that was calculated<br />

from the intramolecular twisting angle of the polar dinitrophenoxy groups of 5.18 determined<br />

by X-ray structural analysis at 5 K [96].<br />

The electric permittivity (Fig. 9.40) reaches values of about e r & 150 at the transition<br />

temperature. Its temperature dependence is strongly influenced by defects [97]. Thus, it is<br />

less appropriate for a comparison with theoretical predictions.<br />

In the early days of TOPOMAK, H. Schultes has already observed that the phase transition<br />

of DNP shifts to lower temperatures (Fig. 9.41–43) with increasing polymer content<br />

162

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

3<br />

DNP<br />

-2<br />

P(T) / 10 -7 Ccm<br />

2<br />

1<br />

0<br />

0<br />

10 20 30 40 50 60<br />

T/K<br />

Figure 9.39: Temperature dependence of the spontaneous polarization parallel to the polar b-axis for<br />

different DNP monomer single crystals [97].<br />

75 0.3<br />

DNP<br />

ε r (T)<br />

50<br />

25<br />

0.2<br />

0.1<br />

-1<br />

(χ ferro<br />

(T))<br />

0<br />

0<br />

20<br />

40<br />

Figure 9.40: Temperature dependence of the low-frequency electric permittivity (left axis) <strong>and</strong> the inverse<br />

of the ferroelectric part of the susceptibility (right axis) for a DNP monomer crystal [97].<br />

T/K<br />

60<br />

80 0<br />

-7<br />

polarization / 10 C cm<br />

-2<br />

1,0<br />

0,5<br />

10 h<br />

5h<br />

2h<br />

0h<br />

0<br />

25<br />

Figure 9.41: Variation of the zero-field electric polarization in crystallographic b-direction of DNP single<br />

crystal with duration of thermal polymerization at 130 8C [94] (Fig. 9.34 for conversion curve).<br />

50<br />

T/K<br />

163

9 Diacetylene Single Crystals<br />

ε (T) / ε (293 K)<br />

r<br />

r<br />

8<br />

4<br />

10 h<br />

5h<br />

13 h<br />

14 h<br />

16 h<br />

0<br />

0 25 50 75 100<br />

T/K<br />

Figure 9.42: Influence of duration of thermal solid state polymerization at 130 8C on temperature dependence<br />

of electric permittivity of DNP [94].<br />

50<br />

T c / K<br />

40<br />

30<br />

1,6<br />

1,2<br />

-1<br />

∆S /Jmol K<br />

-1<br />

20<br />

10<br />

0<br />

0<br />

X/%<br />

80<br />

40<br />

0<br />

0<br />

8 16<br />

t/h<br />

4 8<br />

16 0,0<br />

Figure 9.43: Variation of the transition temperature T c (e r maximum) <strong>and</strong> the conversion entropy (transition<br />

entropy change) DS with the duration t p of the thermal solid state polymerization of DNP at<br />

129 8C [83, 87]. The inset shows the typical variation of the polymer content X with t p .<br />

12<br />

t p/h<br />

0,8<br />

0,4<br />

(thermal polymerization) <strong>and</strong> can not be observed in the polymer crystal [94, 98]. This was<br />

a puzzle for our early attempts to underst<strong>and</strong> the ferroelectric phase transition of DNP.<br />

In the microscopic picture of the phase transition, nuclear magnetic resonance spectroscopy<br />

<strong>and</strong> relaxation of the DNP protons gave important additional information [95, 96].<br />

The proton NMR spectrum reflects the orientation of the proton-proton axis of the methylene<br />

groups close to the central diacetylene unit via the nuclear-spin magnetic dipole interaction<br />

in a rather clear-cut way (Fig. 9.44). For fixed crystal orientation <strong>and</strong> varied temperature<br />

the spectrum of the methylene group protons of the DNP monomer proved that both DNP<br />

moieties are twisted around the central C–C single bond of the diacetylene below T c . Since<br />

this degree of freedom is lost during the solid state polymerization the ferroelectric phase<br />

transition is suppressed.<br />

164

9.4 Di-, pyro-, <strong>and</strong> ferroelectricity<br />

r<br />

H0IIb<br />

r<br />

H 0<br />

b<br />

-40<br />

0 40<br />

-40 0 40<br />

Rel. Frequency (kHz)<br />

Figure 9.44: Simulated (left) <strong>and</strong> experimental (right) H NMR spectra as a function of the crystal orientation<br />

with respect to the external magnetic field, recorded at room temperature (n p = 200 MHz). The<br />

crystal was rotated in steps of 58 around its long axis (a-axis), which was oriented perpendicular to the<br />

external field [96].<br />

Additional information on the phase transition was obtained from proton-spin-lattice<br />

relaxation measured as function of the Larmor frequency, temperature, <strong>and</strong> orientation of<br />

the single crystals (Fig. 9.45). The analysis indicated the slowing down of a molecular motion<br />

on approaching the ferroelectric phase transition with the activation energy of about<br />

0.020 eV, which is in the range of known librational <strong>and</strong> torsional modes of diacetylenes<br />

[96].<br />

The phase transition of DNP could further be characterized by specific heat measurements<br />

for monomer <strong>and</strong> thermally polymerized single crystals (Fig. 9.46) [98]. These data<br />

support the description of the phase transition of the monomer crystals as a tricritical transition.<br />

This means it is a borderline case between a first-order <strong>and</strong> second-order phase transition,<br />

with a distribution of transition temperatures. The transition enthalpy was much lower<br />

than the corresponding order-disorder transition, in agreement with results obtained by Bertault<br />

et al. via Raman spectroscopy, which proved the importance of displacive contributions<br />

to the DNP phase transition [99].<br />

165

9 Diacetylene Single Crystals<br />

31 MHz<br />

-3 -1<br />

Spin-lattice relaxation rate (10 s )<br />

90 MHz<br />

200 MHz<br />

Temperature (Kelvin)<br />

Figure 9.45: Spin-lattice relaxation rate of the protons in a DNP monomer single crystal for three Larmor<br />

frequencies. Two rate maxima can be discerned with temperature dependence explained by the<br />

model of Bloembergen, Purcell, <strong>and</strong> Pound (solid line) [96].<br />

30<br />

-1<br />

∆/C/J mol K -1<br />

20<br />

2h<br />

2 p =0h<br />

10<br />

6h<br />

0<br />

30<br />

Figure 9.46: The ferroelectric contribution to the molar heat capacity of solid state polymerized DNP<br />

single crystals for different annealing times t p at 129 8C for [98].<br />

40<br />

T/K<br />

50<br />

9.4.4 Summary<br />

The increase of the electric permittivity for electric fields parallel to the polymer chain direction<br />

during solid state polymerization of diacetylenes can be used for in situ monitoring<br />

the monomer to polymer conversion of individual single crystals. The large librational am-<br />

166

9.5 Non-linear optical properties<br />

plitudes of the diacetylene moiety, required for solid state polymerization, are also the basis<br />

for the occurrence of interesting dielectric phase transitions. The tailoring of diacetylenes as<br />

ferro or pyroelectric crystals, which show good thermal stability <strong>and</strong> do not dem<strong>and</strong> considerable<br />

efforts for the poling process, is a trial-<strong>and</strong>-error process, but has been realized during<br />

our TOPOMAK activities in a number of cases. Thus, material properties useful for applications<br />

of pyro- or piezoelectricity thus have been obtained.<br />

9.5 Non-linear optical properties<br />

9.5.1 Aims of investigation<br />

Polydiacetylenes are polymers showing a one-dimensional semiconducting behaviour. This<br />

one-dimensional structure causes exceptionally high third order non-linearities (w (3) ) [100],<br />

also in off-resonant wavelength regions [101], with extremely short sub-picosecond switching<br />

times [102]. After this discovery it was believed that an optical amplifying switch (optical<br />

transistor) or even an optical computer was close at h<strong>and</strong>.<br />

At the start of our SFB 213 project the initial optimism about the application of the<br />

huge non-linearity of polydiacetylenes in optical switching was already somewhat damped.<br />

It was becoming clear, that the polydiacetylenes’ non-linear optical coefficients, despite belonging<br />

to the largest non-resonant non-linearities, were not sufficient for cascadable, intensity<br />

amplifying switches [103]. There was not much known about the mechanisms leading to<br />

the large non-linearities <strong>and</strong> different, sometimes contradicting theoretical models were proposed<br />

to describe them. On the other side, there was a dem<strong>and</strong> for tailor-made materials<br />

with predictable non-linearities, absorption b<strong>and</strong>s, <strong>and</strong> good optical quality.<br />

In this scenario, polydiacetylenes were nevertheless interesting, as the mechanisms of<br />

their large non-linearity form a good basis to build on. Only a few modifications were well<br />

characterised <strong>and</strong> there was a general lack of measurements with single crystals due to problems<br />

with sample preparation. Therefore, the aim was to characterise systematically the influence<br />

of side groups on the optical properties, preferably in the macroscopically ordered,<br />

stable, <strong>and</strong> reproducible framework of good <strong>and</strong> – later on – thin crystals.<br />

9.5.2 Experimental setup<br />

Mainly two kinds of experiments will be described here, two-beam pump-probe <strong>and</strong> degenerate<br />

four wave mixing (DFWM) measurements.<br />

In a pump-probe experiment, an intense pulsed pump beam <strong>and</strong> a weaker equally<br />

pulsed probe beam of the same wavelength are focused into the sample, <strong>and</strong> the transmission<br />

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9 Diacetylene Single Crystals<br />

of the probe beam is measured. The timing of the two pulses can be shifted, to make it possible<br />

to probe the decay of excitations generated by the pump beam. Two contrary effects<br />

may occur, bleaching <strong>and</strong> induced absorption.<br />

Near-resonant pump-probe lifetime measurements were performed by W. Schmid<br />

[105, 123], using the same picosecond dye laser system as for DFWM, described below.<br />

Th. Fehn further on investigated the subject in the off-resonant wavelength region between<br />

720 nm <strong>and</strong> 820 nm, using a commercial Titan-Sapphire laser system (Coherent) with pulse<br />

lengths of ca. 120 fs, pumped by an argon ion laser (Coherent Mira).<br />

DFWM measurements can be done in a variety of geometries. Here, the forward mixing<br />

geometry is used (Fig. 9.47), where three beams, forming a right angle, are focused into the<br />

sample. This setup allows time-resolved measurements <strong>and</strong>, in contrast to third harmonic generation<br />

(THG) measurements, yields the w (3) (o;–o,o,–o) tensor which is related to an intensity-dependent<br />

refractive index, the interesting quantity for optical switching applications.<br />

Figure 9.47: Beam geometry for DFWM measurements.<br />

The interference pattern of the pump beams 1 <strong>and</strong> 2 forms horizontal stripes in the<br />

medium. As w (3) is directly related to an intensity-dependent refractive index, this interference<br />

pattern generates a refractive index pattern. The third probe beam is partially reflected<br />

on these horizontal planes, generating the signal beam 4. The efficiency of diffraction is related<br />

to |w (3) |. The pulse timing is adjustable, so the decay of the refractive index pattern can<br />

be probed. When the delay between the beams 2 <strong>and</strong> 3 is in the range of the laser’s coherence<br />

length, these pulses generate a diffraction grating for beam 1, resulting in an artificial<br />

raise of the signal, called coherence peak.<br />

All DFWM measurements were done with a commercially available synchronously<br />

pumped dye laser with a cavity dumper (Spectra Physics model 3500). When using Pyridine-1<br />

as radiant dye, the wavelength can be tuned between 670 nm <strong>and</strong> 750 nm. The pulse<br />

width is about 1 ps; the repetition rate can be adjusted between single shot <strong>and</strong> 8 MHz. This<br />

is important to reduce the average heat load absorbed by the crystals.<br />

To suppress stray light signals a well-known modulation technique was used, the modulation<br />

of two pump beams with different frequencies using a chopper blade with a set of<br />

different divisions. As the DFWM signal depends on the product of the input intensities, the<br />

signal can be detected at the sum or difference of the two modulation frequencies [104],<br />

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9.5 Non-linear optical properties<br />

I 3 sig / I 1I 2 I 3 ˆ I 1 …0†…cos ! 1 t ‡ 1†I 2 …0†…cos ! 1 t ‡ 1†I 3 ˆ<br />

1<br />

ˆ I 1 …0†I 2 …0†I 3 ‰<br />

2 cos…! 1 ‡ ! 2 †t ‡ cos…! 1 ! 2 †t ‡ 2 cos ! 1 t ‡ 2 cos ! 2 tŠ: …31†<br />

W. Schmid proposed the separated detection of w (5) effects by an extension of this<br />

modulation method. A w (5) signal without w (3) contributions can be detected at twice the sum<br />

or difference frequency [105], as the w (5) signal contains terms depending on the square of<br />

the product of the modulated beam intensities,<br />

I sig / I 1 I 2 I 3 s k 3 ‡ k 35 … I 1 ‡ I 2 ‡ I 3 †‡k 5 … I 1 ‡ I 2 ‡ I 3 † 2 ; …32†<br />

where k 3 is a function of w (3) , k 5 of w (5) , <strong>and</strong> k 35 depends on both non-linearities <strong>and</strong> their<br />

relative phase.<br />

Although this modulation method permits qualitative measurements of w (5) effects,<br />

especially determination of relaxation times, quantitative evaluation of w (5) is not possible,<br />

due to the infinite number of modulation harmonics with ill-defined relative amplitudes<br />

caused by the trapezoidal modulation form provided by a chopper blade [106].<br />

This problem has been solved by A. Feldner. He developed a new type of modulator,<br />

working with two independently rotating polarizing foils (Fig 9.48). This type of modulator<br />

generates two harmonics in the intensity modulation spectrum with a fixed ratio of 4 : 1<br />

(cos 4 ot) [106].<br />

Figure 9.48: Scheme of intensity modulation by rotating polarizing foils (a). The vertical polarizer (b)<br />

ensures that an anisotropy of the material has no effect on the signal or the modulation spectrum.<br />

For time-resolved pump-probe measurements of shorter relaxation times, a Titan-sapphire<br />

laser pumped by an argon ion laser is employed. In the current configuration, the wavelength<br />

can be tuned between 720 nm <strong>and</strong> 820 nm; the pulse width is about 120 fs.<br />

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9 Diacetylene Single Crystals<br />

9.5.3 Theoretical approaches<br />

The simplest approach is the free electron model [107, 108]. The electrons are treated to<br />

move freely in a one-dimensional box, subject to a potential V 0 cos (px/d) by the ion cores<br />

(d denotes the bond length). The polarizability a <strong>and</strong> hyperpolarizability g is derived from the<br />

2nd <strong>and</strong> 4th order perturbation energies caused by an electrical field along the chain. This<br />

rough model does not account for local field effects nor for the alternating bond length found<br />

along the axis in PDAs. The model yields a static w (3) = 3.0610 –11 esu along the axis.<br />

Agrawal, Cojan et al. [109–111] treated explicitly the linear <strong>and</strong> non-linear optical<br />

properties of PDAs. They computed the electronic energy levels using a Hückel formalism.<br />

They computed a static w (3) = 0.7610 –10 esu for PTS <strong>and</strong> w (3) = 0.25610 –10 esu for<br />

TCDU. This model neglects electron-electron interaction <strong>and</strong> therefore excitonic effects.<br />

The model yields the correct order of magnitude for w (3) , but the wrong sign.<br />

The phase space filling (PSF) model was developed to describe non-resonant NLO<br />

properties in semiconductors [112, 113], especially in quantum well structures [114]. Greene<br />

et al. adapted this model to one-dimensional polymer chains [115, 116]. The model is only<br />

applicable to systems were the low energy absorption b<strong>and</strong> is excitonic, as is the case with<br />

PDAs [117]. Formation of excitons is limited by the number of available electron states that<br />

are necessary to form the exciton. With an increasing number of excitons, the dipole momentum<br />

for forming a new exciton is reduced. The exciton b<strong>and</strong> bleaches.<br />

As the following measurements will show, the PSF model seems to describe best the<br />

non-linear optical behaviour of polydiacetylenes. A very important prediction of this model<br />

is the proportionality of w (3) <strong>and</strong> a in the near-resonant frequency regime [115]. This behaviour<br />

was found in polydiacetylenes, strongly supporting the PSF theory [118].<br />

9.5.4 Sample preparation<br />

For measurements well off the resonance, i. e. with wavelengths larger than 720 nm, p-TS6<br />

crystals were prepared by thermal polymerisation of monomer crystals, grown out of a saturated<br />

solution <strong>and</strong> manually cut using a shaver blade. Thickness of these crystals varies between<br />

40–100 µm.<br />

Later, for measurements closer to resonance, a method has been developed to grow<br />

thin mono-crystalline layers of TS6 <strong>and</strong> 4BCMU between glass substrates. To avoid crystal<br />

strains the monomer crystals have to be removed from the substrate before polymerising.<br />

The resulting crystal thickness can be made as low as 300 nm.<br />

9.5.5 Value <strong>and</strong> phase of the third order susceptibility w (3)<br />

From DFWM measurements |w (3) | was determined for several polydiacetylenes in the nearresonant<br />

to off-resonant wavelength region, 680 nm to 750 nm (Tab. 9.6). Concurrently, the<br />

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9.5 Non-linear optical properties<br />

Table 9.6: w (3) values for some different Polydiacetylenes [118, 119].<br />

Material Modification |w (3) |/esu at wavelength<br />

PTS crystal 2610 –10 720 nm<br />

FBS crystal 2610 –10 720 nm<br />

4BCMU amorphous film 4610 –11 720 nm<br />

4BCMU thin monocrystalline film 3610 –10 720 nm<br />

4BCMU thin monocrystalline film 2610 –9 670 nm<br />

imaginary part of w (3) can be computed from the non-linear absorption coefficient b, obtained<br />

from measurements of the intensity dependence of the sample transmission. It was<br />

found that the real part of w (3) was dominating the imaginary part by a factor of 3 [118].<br />

This is predicted by the PSF model, from the bleaching of the exciton b<strong>and</strong>.<br />

The value of w (3) is nearly identical for p-TS6 <strong>and</strong> p-4BCMU, <strong>and</strong> 4 times lower for<br />

p-IPUDO. These values were reproducible within 30%. For p-FBS, the reproducibility was<br />

only within an order of magnitude. No difference was found between thermally <strong>and</strong> x-ray<br />

polymerised crystals [118].<br />

9.5.6 Relaxation of the singlet exciton<br />

In p-TS6 energy relaxation times could be resolved at wavelengths below 700 nm. As the relaxation<br />

times are of the same order of magnitude as the laser pulse width (1 ps), a model<br />

function has to be fitted to the measured curve to obtain the relaxation (Fig. 9.49).<br />

Figure 9.49: w (3) relaxation in p-TS6 at a wavelength of 680 nm <strong>and</s