Microscopic Interactions and Macroscopic Properties Final Report (1st

Deutsche
Forschungsgemeinschaft
Macromolecular
Systems:
Microscopic
Interactions and
Macroscopic Properties
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

Deutsche
Forschungsgemeinschaft
Macromolecular Systems:
Microscopic Interactions
and Macroscopic Properties
Final report of the
collaborative research centre 213,
``Topospezifische Chemie und Toposelektive
Spektroskopie von MakromolekÏlsystemen:
Mikroskopische Wechselwirkung und
Makroskopische Funktion'', 1984^1995
Edited by
HeinzHoffmann,MarkusSchwoererand
Thomas Vogtmann
Collaborative Research Centres

Deutsche Forschungsgemeinschaft
Kennedyallee 40, D-53175 Bonn, Federal Republic of Germany
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Printed in the Federal Republic of Germany

Contents
Preface ...................................................
List of Contributors ..........................................
XV
XVII
I
Mainly Solids
1 Model Systems for Photoconductive Material ..................... 3
Harald Meyer and Dietrich Haarer
1.1 Introduction . . ............................................. 3
1.2 Experimental techniques . . . ................................... 4
1.3 Transport models ............................................ 5
1.4 Results ................................................... 7
1.4.1 Moleculary doped polymers . ................................... 7
1.4.2 Side-chain polymers ......................................... 9
1.4.3 Conjugated systems .......................................... 10
1.5 Outlook .................................................. 12
References . . . ............................................. 13
2 Novel Photoconductive Polymers ............................... 15
Jörg Bettenhausen and Peter Strohriegl
2.1 Introduction . . ............................................. 15
2.2 Liquid crystalline oxadiazoles and thiadiazoles ...................... 16
2.2.1 The basic idea . ............................................. 16
2.2.2 Monomer synthesis .......................................... 17
2.2.3 Oligo- and polysiloxanes with pendant oxadiazole groups .............. 19
2.2.4 Photoconductivity measurements ................................ 21
2.3 Starburst oxadiazole compounds ................................. 22
2.3.1 Motivation . . . ............................................. 22
2.3.2 Synthesis of starburst oxadiazole compounds ....................... 24
2.3.3 Thermal properties .......................................... 27
References . . . ............................................. 30
V

Contents
3 Theoretical Aspects of Anomalous Diffusion in Complex Systems ...... 31
Alexander Blumen
3.1 General aspects ............................................. 31
3.2 Photoconductivity ........................................... 32
3.3 The Matheron-de-Marsily model ................................ 36
3.4 Polymer chains in MdM flow fields .............................. 38
3.5 Conclusions . . ............................................. 42
Acknowledgements .......................................... 42
References . . . ............................................. 43
4 Low-Temperature Heat Release, Sound Velocity and Attenuation,
Specific Heat and Thermal Conductivity in Polymers ............... 44
Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz,
Junyun Li, and Frank Pobell
4.1 Introduction . . ............................................. 44
4.2 Phenomenological theory for heat release . . ........................ 46
4.2.1 Generalities . . . ............................................. 46
4.2.2 The standard tunneling model with infinite cooling rate . .............. 47
4.2.3 Influence of higher-order tunneling processes and a finite cooling rate . . . . 49
4.2.4 The influence of a constant and thermally activated relaxation rate ....... 52
4.3 Experimental details ......................................... 54
4.4 Experimental results and discussion .............................. 55
4.4.1 Specific heat and thermal conductivity ............................ 55
4.4.2 Internal friction and sound velocity .............................. 58
4.4.3 Heat release . . ............................................. 62
4.5 Conclusions . . ............................................. 65
Acknowledgements .......................................... 66
References . . . ............................................. 66
5 Spectral Diffusion due to Tunneling Processes at very low Temperatures 68
Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer
5.1 Introduction . . ............................................. 68
5.2 The optical cryostat .......................................... 69
5.3 Theoretical considerations . . ................................... 71
5.4 Temperature dependence . . . ................................... 73
5.5 Time dependence ............................................ 74
References . . . ............................................. 76
6 Optically Induced Spectral Diffusion in Polymers Containing Water
Molecules: A TLS Model System ............................... 78
Klaus Barth, Dietrich Haarer, and Wolfgang Richter
6.1 Introduction . . ............................................. 78
6.2 Experimental setup for burning and detecting spectral holes ............ 79
6.3 Reversible line broadening phenomena ............................ 80
6.4 Induced spectral diffusion . . ................................... 83
References . . . ............................................. 87
VI

Contents
7 Slave-Boson Approach to Strongly Correlated Electron Systems ....... 88
Holger Fehske, Martin Deeg, and Helmut Büttner
7.1 Introduction . . ............................................. 88
7.2 Slave-boson theory for the t-t'-J model ............................ 90
7.2.1 SU(2)-invariant slave-particle representation ........................ 90
7.2.2 Functional integral formulation ................................. 93
7.2.3 Saddle-point approximation . ................................... 95
7.2.4 Magnetic phase diagram of the t-t'-J model ........................ 97
7.3 Comparison with experiments .................................. 102
7.3.1 Normal-state transport properties ................................ 102
7.3.2 Magnetic correlations and spin dynamics . . ........................ 105
7.3.3 Inelastic neutron scattering measurements . . ........................ 106
7.4 Summary ................................................. 109
References . . . ............................................. 110
8 Non-Linear Excitations and the Electronic Structure of Conjugated
Polymers ................................................. 113
Klaus Fesser
8.1 Introduction . . ............................................. 113
8.2 Models ................................................... 114
8.3 Disorder .................................................. 116
8.4 Non-linear excitations ........................................ 117
8.5 Perspective . . . ............................................. 119
Acknowledgements .......................................... 120
References . . . ............................................. 120
9 Diacetylene Single Crystals ................................... 122
Markus Schwoerer, Elmar Dormann, Thomas Vogtmann,
and Andreas Feldner
9.1 Introduction . . ............................................. 122
9.2 Photopolymerization ......................................... 129
9.2.1 Carbenes .................................................. 129
9.2.2 Intermediate photoproducts . ................................... 131
9.2.3 Electronic structure of dicarbenes ................................ 131
9.2.3.1 Electron spin resonance of quintet states ( 5 DC n ) .................... 131
9.2.3.2 ENDOR of quintet states . . . ................................... 136
9.2.3.3 ESR and ENDOR of triplet dicarbenes 3 DC n ....................... 139
9.2.4 Flash photolysis and reaction dynamics of diradicals .................. 141
9.3 Holography . . . ............................................. 144
9.3.1 Theory ................................................... 145
9.3.2 Experimental setup .......................................... 147
9.3.3 General characterization . . . ................................... 149
9.3.4 Angular selectivity .......................................... 150
9.3.5 Prepolymerized samples ....................................... 152
9.3.6 Chain length, polymer profile, and grating profiles ................... 152
9.3.7 Multrecording . ............................................. 154
VII

Contents
9.3.8 Holography . . . ............................................. 154
9.4 Di-, pyro-, and ferroelectricity .................................. 155
9.4.1 Dielectric properties of diacetylenes .............................. 156
9.4.1.1 Correlation of polymer content and electric permittivity . .............. 156
9.4.1.2 Application to topospecifically modified diacetylenes . . . .............. 158
9.4.1.3 Additional applications ....................................... 159
9.4.2 Pyroelectric diacetylenes . . . ................................... 159
9.4.2.1 IPUDO ................................................... 159
9.4.2.2 NP/4-MPU . . . ............................................. 160
9.4.2.3 DNP/MNP . . . ............................................. 161
9.4.2.4 Spurious piezo and pyroelectricity of diacetylenes .................... 161
9.4.3 The ferroelectric diacetylene DNP ............................... 162
9.4.4 Summary ................................................. 166
9.5 Non-linear optical properties ................................... 167
9.5.1 Aims of investigation ........................................ 167
9.5.2 Experimental setup .......................................... 167
9.5.3 Theoretical approaches ....................................... 170
9.5.4 Sample preparation .......................................... 170
9.5.5 Value and phase of the third order susceptibility w (3) .................. 170
9.5.6 Relaxation of the singlet exciton ................................ 171
9.5.7 The w (3) tensor components . ................................... 172
9.5.8 Signal saturation ............................................ 173
9.5.9 Spectral dispersion, phase, and relaxation of w (5) ..................... 174
9.5.10 Conclusion . . . ............................................. 176
References . . . ............................................. 177
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids .......... 181
Thomas Giering, Peter Geißinger, Wolfgang Richter, and Dietrich Haarer
10.1 Introduction . . ............................................. 181
10.2 Stochastic theory ............................................ 183
10.3 Rare gases ................................................. 186
10.4 Experimental . . ............................................. 187
10.5 Inhomogeneous absorption lines ................................. 188
10.6 Pressure effects ............................................. 190
10.7 Rare gas mixtures ........................................... 191
10.8 Summary ................................................. 194
Acknowledgements .......................................... 195
References . . . ............................................. 195
II
Mainly Micelles, Polymers, and Liquid Crystals
11 The Micellar Structures and the Macroscopic Properties of Surfactant
Solutions ................................................. 199
Heinz Hoffmann
11.1 General behaviour of surfactants ................................ 199
VIII

Contents
11.2 From globular micelles towards bilayers . . . ........................ 200
11.3 Viscoelastic solutions with entangled rods . ........................ 202
11.3.1 General behaviour ........................................... 202
11.3.2 Viscoelastic systems ......................................... 205
11.3.3 Mechanisms for the different scaling behaviour ..................... 209
11.4 Viscoelastic solutions with multilamellar vesicles .................... 211
11.4.1 The conditions for the existence of vesicles ........................ 211
11.4.2 Freeze fracture electron microscopy .............................. 212
11.4.3 Rheological properties ........................................ 213
11.4.4 Model for the shear modulus ................................... 217
11.5 Ringing gels . . ............................................. 220
11.5.1 Introduction . . ............................................. 220
11.5.2 The aminoxide system ........................................ 221
11.5.3 The bis-(2-ethylhexyl)sulfosuccinate system ........................ 224
11.5.4 PEO-PPO-PEO block copolymers ................................ 226
11.6 Lyotropic mesophases ........................................ 227
11.6.1 Introduction . . ............................................. 227
11.6.2 Nematic phases and their properties .............................. 228
11.6.3 Cholesteric phases and their properties ............................ 232
11.6.4 Vesicle phases and L 3 phases ................................... 233
11.7 Shear induced phenomena . . ................................... 236
11.7.1 General ................................................... 236
11.7.2 Under what conditions do we find drag-reducing surfactants? ........... 236
11.8 SANS measurements on micellar systems . . ........................ 239
11.9 A new rheometer ............................................ 243
References . . . ............................................. 247
12 Photophysics of J Aggregates .................................. 251
Hermann Pschierer, Hauke Wendt, and Josef Friedrich
12.1 Introduction . . ............................................. 251
12.2 Basic aspects of pressure and electric field phenomena in hole burning
spectroscopy of J aggregates ................................... 252
12.3 Experimental . . ............................................. 253
12.4 Results ................................................... 254
12.5 Discussion . . . ............................................. 256
12.5.1 Pressure phenomena ......................................... 256
12.5.2 Electric field-induced phenomena ............................... 258
Acknowledgements .......................................... 258
References . . . ............................................. 259
13 Convection Instabilities in Nematic Liquid Crystals ................ 260
Lorenz Kramer and Werner Pesch
13.1 Introduction . . ............................................. 260
13.2 Basic equations and instability mechanisms ........................ 264
13.2.1 The director equation ......................................... 264
13.2.2 The velocity field ........................................... 266
IX

Contents
13.2.3 Electroconvection ........................................... 267
13.2.3.1 The standard model .......................................... 267
13.2.3.2 The weak electrolyte model . ................................... 268
13.2.4 Rayleigh-Bénard convection . ................................... 269
13.3 Theoretical analysis .......................................... 269
13.4 Rayleigh-Bénard convection . ................................... 275
13.5 Electrohydrodynamic convection ................................ 278
13.5.1 Linear theory and type of bifurcation ............................. 278
13.5.2 Results of Ginzburg-Landau equation ............................. 279
13.5.3 Beyond the Ginzburg-Landau equation ............................ 281
13.5.3.1 Experimental results ......................................... 281
13.5.3.2 Theoretical results and discussion ................................ 282
13.6 Concluding remarks .......................................... 286
Acknowledgements .......................................... 288
Note added . . . ............................................. 289
References . . . ............................................. 290
14 Preparation and Properties of Ionic and Surface Modified
Micronetworks ............................................. 295
Michael Mirke, Ralf Grottenmüller, and Manfred Schmidt
14.1 Introduction . . ............................................. 295
14.2 Polymerization in normal microemulsion . . ........................ 295
14.2.1 Mechanism and size control . ................................... 295
14.2.2 Surface functionalization of microgels ............................ 298
14.3 Polymerization in inverse microemulsion . . ........................ 299
14.3.1 Preparation of ionic microgels .................................. 299
14.3.2 Properties of ionic microgels and interparticle interaction .............. 299
14.4 Conclusion and relevance to future work . . ........................ 303
References . . . ............................................. 304
15 Ferrocene-Containing Polymers ................................ 305
Oskar Nuyken,Volker Burkhardt, Thomas Pöhlmann, Max Herberhold,
Fred Jochen Litterst, and Christian Hübsch
15.1 Introduction . . ............................................. 305
15.2 Addition polymers ........................................... 306
15.2.1 Radical polymerization ....................................... 306
15.2.2 Radical copolymerization . . . ................................... 308
15.2.3 Anionic polymerization of VFc ................................. 308
15.2.3.1 Living polymerization ........................................ 309
15.2.3.2 Block copolymers ........................................... 312
15.2.4 Polymeranalogeous reactions ................................... 314
15.3 Polymers with ferrocene units in the main chain ..................... 315
15.3.1 Polycondensation ............................................ 315
15.3.2 Polymers by addition of dithiols to diolefins ........................ 316
15.3.2.1 Radical reaction ............................................ 316
15.3.2.2 Base catalyzed reactions . . . ................................... 316
X

Contents
15.3.2.3 Acid catalyzed reactions . . . ................................... 318
15.3.3 1,1'-dimercapto-ferrocene as initiator ............................. 319
15.3.4 Reductive coupling .......................................... 320
15.4 Mößbauer studies of polymers containing ferrocene .................. 320
References . . . ............................................. 323
16 Transfer of Vibrational Energy in Dye-Doped Polymers ............. 325
Johannes Baier, Thomas Dahinten, and Alois Seilmeier
16.1 Introduction . . ............................................. 325
16.2 Experimental . . ............................................. 326
16.3 Results and discussion ........................................ 327
16.4 Summary ................................................. 332
References . . . ............................................. 332
17 Picosecond Laser Induced Photophysical Processes of Thiophene
Oligomers ................................................ 333
Dieter Grebner, Matthias Helbig, and Sabine Rentsch
17.1 Introduction . . ............................................. 333
17.2 Experimental . . ............................................. 334
17.3 Spectroscopic properties of oligothiophenes ........................ 337
17.4 Results ................................................... 337
17.4.1 Picosecond-transient spectra of oligothiophenes in solution ............. 337
17.4.2 Time behaviour of transient spectra .............................. 339
17.4.3 Size dependence of spectroscopic properties of oligothiophenes ......... 341
17.4.4 Size dependence of the kinetic behaviour of oligothiophenes ............ 342
17.5 Discussion . . . ............................................. 343
References . . . ............................................. 343
18 Topospecific Chemistry at Surfaces ............................. 344
Hans Ludwig Krauss
18.1 Introduction . . ............................................. 344
18.1.1 The problem . . ............................................. 344
18.1.2 Preparative and analytical methods ............................... 344
18.1.3 Industrial applications ........................................ 345
18.1.4 The standard procedures of the Phillips process ..................... 345
18.1.5 Earlier work . . ............................................. 346
18.2 The support . . . ............................................. 346
18.2.1 Unmodified silica ........................................... 346
18.2.2 Modified silica ............................................. 347
18.2.3 Others .................................................... 348
18.3 Transition metal surface compounds .............................. 348
18.3.1 The metals . . . ............................................. 348
18.3.2 Impregnation and activation . ................................... 349
18.4 Coordinatively unsaturated sites ................................. 350
18.4.1 Reduction of saturated surface compounds . ........................ 350
18.4.2 Elimination of ligands ........................................ 352
XI

Contents
18.5 Physical properties of the coordinatively unsaturated sites .............. 353
18.5.1 Topologically different sites . ................................... 353
18.5.2 Optical and magnetic properties ................................. 353
18.6 Chemical properties of the coordinatively unsaturated sites ............. 355
18.6.1 Survey of catalytic reactions ................................... 355
18.6.2 Olefin polymerization ........................................ 356
18.6.3 Other catalytic reactions . . . ................................... 360
18.7 Deactivation . . ............................................. 361
18.8 Summary and outlook ........................................ 362
References . . . ............................................. 363
III
Biopolymers
19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers 369
Stefan Limmer, Günther Ott, and Mathias Sprinzl
19.1 Introduction . . ............................................. 369
19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes . . 370
19.2.1 Stability of tRNA-derived acceptor stem duplexes .................... 371
19.2.2 Manganese ion binding sites at RNA duplexes ...................... 374
19.2.3 Structural determination of short RNA duplexes by 2D NMR spectroscopy . 377
19.2.4 NMR derived model of the tRNA Ala acceptor arm ................... 379
19.2.5 Chemical shifts and scalar coupling as an indicator of RNA structure in the
vicinity of a G-U pair ........................................ 380
19.2.6 Structure of aminoacyl-tRNA and transacylation of the aminoacyl residue . . 382
19.3 Structure of elongation factor Tu ................................ 383
19.3.1 Sequence of Thermus thermophilus EF-Tu.......................... 384
19.3.2 Crystallization, X-ray analysis, and the tertiary structure . .............. 386
19.3.3 Nucleotide binding and GTPase reaction . . ........................ 387
19.3.4 Mechanism of GTP induced conformational change of EF-Tu ........... 388
19.3.5 Aminoacyl-tRNA in complex with EF-Tu 7 GTP ..................... 389
19.3.6
1 H NMR of yeast Phe-tRNA Phe EF-Tu7GTP complex . . .............. 391
19.3.7
13 C NMR studies of the Val-tRNA Val EF-Tu7GTP ternary complex ....... 393
19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA ..................... 395
19.3.9 EF-Tu interaction with EF-Ts ................................... 395
19. 3.10 Site-directed mutagenesis of EF-Tu ............................... 396
19.4 Summary and conclusions . . ................................... 397
Acknowledgement ........................................... 397
References . . . ............................................. 398
20 Spectroscopic Probes of Surfactant Systems and Biopolymers ......... 401
Alexander Wokaun
20.1 Introduction . . ............................................. 401
20.2 Diffusion in surfactant systems .................................. 402
20.2.1 Structural characteristics of micellar solutions, cubic phases, and multilamellar
vesicles from NMR self-diffusion measurements .............. 402
XII

Contents
20.2.2 Probing of mobilities in multilamellar vesicles by forced Rayleigh scattering 405
20.2.3 Dimensionality of diffusion in lyotropic mesophases from fluorescence
quenching ................................................. 409
20.2.4 Summary of results .......................................... 412
20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides . . 413
20.3.1 Spectroscopic characterization of right and left-helical forms of a hexadecanucleotide
duplex ........................................ 413
20.3.2 SERS spectra of deoxyribonucleotides ............................ 415
20.3.3 Studies of chromophore-DNA interaction by vibrational spectroscopy . . . . . 417
20.3.4 Summary of results .......................................... 418
20.4 Related projects carried out within the framework of the Collaborative
Research Centre ............................................ 419
20.5 Remarks and acknowledgements ................................ 421
References . . . ............................................. 422
21 Energy Transport by Lattice Solitons in a-Helical Proteins ........... 424
Franz-Georg Mertens, Dieter Hochstrasser, and Helmut Büttner
21.1 Introduction . . ............................................. 424
21.2 The model ................................................. 426
21.3 Quasicontinuum approximation ................................. 428
21.4 Velocity range for the quasicontinuum approach ..................... 431
21.5 Solitary waves for realistic parameter values ........................ 432
21.6 Iterative method and stability ................................... 434
21.7 Conclusion . . . ............................................. 437
References . . . ............................................. 438
IV
Appendix
22 Documentation of the Collaborative Research Centre 213 ............ 443
Markus Schwoerer and Heinz Hoffmann
22.1 List of Members ............................................ 443
22.2 Heads of Projects (Teilprojektleiter) .............................. 444
22.2.1 Projektbereich A: Gemeinsame Einrichtungen ...................... 444
22.2.2 Projektbereich B: Festkörper ................................... 444
22.2.3 Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen
und Polymere . ............................................. 445
22.2.4 Projektbereich D: Biopolymere ................................. 446
22.3 Guests .................................................... 447
22.4 Co-workers . . . ............................................. 450
22.5 International Cooperation . . . ................................... 457
22.6 Funding .................................................. 459
XIII

Preface
At the end of 1983, about eight years after the inauguration of the University of Bayreuth
the Deutsche Forschungsgemeinschaft (DFG) agreed to establish the Collaborative Research
Centre 213, TOPOMAC, for the promotion of basic research in chemistry and physics of
macromolecular systems. The title of TOPOMAC in its full length, “Topospezifische Chemie
und Toposelektive Spektroskopie von Makromolekülsystemen: Mikroskopische Wechselwirkung
und Makroskopische Funktion” expessed the intention of the original applicants:
a productive cooperation between physicists, chemists and biochemists across the mutual
borders of their original research fields. Until the end of 1995 TOPOMAC was supported by
the Deutsche Forschungsgemeinschaft with 28 MDM.
The present book documents the achievements of TOPOMAC. It is not a minute addition
of all the results which have been published in periodical journals but rather a survey of
important research fields of members of TOPOMAC. The articles have been written towards
or after the end of the support period as both, review and original publications.
The book covers the fields of:
. Electronic, photoelectric, thermal, dielectric, optical and magnetic properties of macromolecular
solids (polymers and polymer crystals),
. Micellar structures, J-aggregates, liquid crystals, µ-gels, ferrocene-containing polymers
and topospecific chemistry at surfaces and in single crystals, and
. Biopolymers and surfactant systems as studied by site directed spectroscopy and site directed
chemistry and also by the theory of energy transport.
During the period of its support TOPOMAC had 30 members (Teilprojektleiter). Only
ten of them have been members for the entire period, mainly because twenty times a call
from other universities or research institutions reached one of the members of the Collaborative
Research Centre 213. Fourteen of them followed this call and left the Collaborative Research
Centre 213. Less than two years after the end of the period of support some of the remaining
former members of Collaborative Research Centre 213 together with young and
new faculty members began to continue the formal cooperation between chemists and physicists
in the field of macromolecular research. Their actual cooperation in the meantime
never had been terminated.
The editors would like to express the sincere thanks of the members of TOPOMAC to
the foreign guests of TOPOMAC, to the Deutsche Forschungsgemeinschaft, to the University
XV

Preface
of Bayreuth and also to the Freistaat Bayern. Our foreign guests stayed for long or short periods
between one year and one day. They have contributed in an essential and special manner
to the success on our research fields. They also strongly intensified the national and international
scientific relations of both, the members and the research students of TOPO-
MAC. The cooperation of the speakers of TOPOMAC with Dr. Funk from the Deutsche Forschungsgemeinschaft
office throughout the entire period of support was excellent. The help
of the former president of the university, Dr. K. D. Wolff and his chancellor, W. P. Hentschel
as well as the continuous support by the late Ministerialrat G. Grote and by J. Großkreutz
from the Bayerisches Staatsministerium für Unterricht, Kultus, Wissenschaft und Kunst has
been an essential stimulus for the scientific members of TOPOMAC. Last but not least we
thank Doris Buntkowski for her faithful and reliable work as our secretary.
The Editors
XVI

List of Contributors
Alexander Blumen
Theoretische Polymerphysik
Universität Freiburg
Herrmann-Herder-Straße 3
79104 Freiburg
Elmar Dormann
Physikalisches Institut
Universität Karlsruhe
Engesserstr. 7
67131 Karlsruhe
Pablo Esquinazi
Institut für Experimentelle Physik II
Universität Leipzig
Linnestraße 5
04103 Leipzig
Klaus Fesser
Fachbereich Physik
Universität Greifswald
Domstraße 10a
17489 Greifswald
Josef Friedrich
Lehrstuhl für Physik
Technische Universität München
85350 Freising-Weihenstephan
Holger Fehske
Theoretische Physik I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Dietrich Haarer
Experimentalphysik IV
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Heinz Hoffmann
Physikalische Chemie I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Lorenz Kramer
Theoretische Physik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Hans Ludwig Krauss
Heunischstraße 5 b
96049 Bamberg
Franz G. Mertens
Theoretische Physik
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Oskar Nuyken
Lehrstuhl für Makromolekulare Stoffe
Technische Universität München
Lichtenbergstraße 4
85747 München
XVII

List of Contributors
Sabine Rentsch
Institut für Optik und Quantenelektronik
Friedrich Schiller Universität Jena
Max-Wien-Platz 1
07743 Jena
Wolfgang Richter
Experimentalphysik IV
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Paul Rösch
Biopolymere
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Manfred Schmidt
Institut für Physikalische Chemie
Universität Mainz
Welder-Weg 11
55099 Mainz
Markus Schwoerer
Experimentalphysik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Alois Seilmeier
Physikalisches Institut
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Mathias Sprinzl
Biochemie
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Peter Strohriegel
Makromolekulare Chemie I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Thomas Vogtmann
Experimentalphysik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Alexander Wokaun
Bereich F5
Paul Scherrer Institut
CH-5232 Villingen
XVIII

I
Mainly Solids
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

1 Model Systems for Photoconductive Materials
Harald Meyer and Dietrich Haarer
1.1 Introduction
The effect of photoconductivity, i. e. the increase of the electrical conductivity of a material
upon illumination with light of suitable photon energy, was first discovered in selenium by
W. Smith in 1873.
A typical application for these materials is the xerographic process [2, 3], which was
developed by C. F. Carlson in 1942 [1]. Here, a photoconducting film on top of a grounded
electrode is homogeneously charged by a corona discharge. Typical field strengths are up to
10 6 V/m. In a second step the image of the original document is projected onto the film. At
the areas where the film is illuminated charge carriers are generated in the photoconducting
film. One species traverses the film and recombines at the grounded electrode whereas the
oppositely charged species neutralizes the surface charges. With this step the original image
is transferred into an electrostatic image on the photoconductor film. Subsequently, small toner
particles are deposited on the photoconductor. They stick to the charged regions and
thus generate a real image. In the next step, this image is transferred onto paper and fixed
by thermally fusing the toner particles on the paper.
The process for a laser printer is similar except for the fact that the image is written
onto the photoconductor directly by a laser or a diode array. If the toner particles are fused
directly onto the photoconductor instead of transferring them onto paper the photoconductor
can be used as an offset printing master [4].
Potential systems for commercial use have to meet several requirements:
a) sensitivity in the visible region of the spectrum;
b) good charge transport properties, i. e. charge carrier mobility in excess of 10 –7 cm 2 /Vs
[3] and minor trapping effects;
c) good mechanical and dielectrical properties;
d) excellent film forming properties and possibility of manufacturing defect free large area
films;
e) mechanical flexibility for the use in small sized desktop devices.
The last two requirements cannot be met neither by organic nor inorganic crystalline
materials. Therefore both, organic and inorganic amorphous photoconductors, have been de-
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
3

1 Model Systems for Photoconductive Materials
veloped. Inorganic materials like e. g. a-Se, As 2 Se 3 or alloys of Se and Te show good transport
properties with mobilities in the range of 0.1 cm 2 /Vs at room temperature [7], and high
sensitivity for visible light together with moderate mechanical and dielectric properties. One
of their most important disadvantages compared to organic systems is that most of these materials
are highly toxic.
In the past 30 years, since H. Hoegl discovered photoconductivity in the organic polymer
poly(N-vinylcarbazole) (PVK) (Fig. 1.2) [5, 6], organic materials have almost completely
replaced their inorganic counterparts, although the effective mobility in these systems is
typically 5 orders of magnitude lower as compared to a-Se.
Organic photoconductors can be regarded as large or medium bandgap semiconductors.
A typical value for commercially applied systems is 3.5 eV for carbazole derivatives
(Section 1.4). This inherent lack of sensitivity in the visible region has been overcome by
using charge transfer systems [5] or multilayer systems with additional charge generation
layers. For a review see e. g. Ref. [3]. On the other hand, the large bandgap of organic materials
virtually eliminates the influence of thermally generated charge carriers and thus improves
the dielectric properties as compared to low bandgap materials.
Therefore the main target of the works, which will be described in the following, was
to identify the key factors which limit the charge carrier mobilities in organic systems and
to develop new high mobility materials.
1.2 Experimental techniques
Besides the xerographic discharge method [3, 10], the time-of-flight (TOF) technique is generally
used to study the charge carrier transport of thin organic films. Here, the photoconducting
film with a typical thickness of 10 mm is sandwiched between two electrodes (Fig. 1.1). Electron-hole
pairs are generated by the energy hn of a strongly absorbed laser pulse, which is irradiated
through one of the semitransparent electrodes. For most organic materials the charge
carrier generation process can be described by an Onsager model, either in its one or three dimensional
form [11–13].
The wavelength of the laser is chosen to ensure that the penetration depth is considerably
less than the sample thickness. Thus the charge carriers are generated close to the illuminated
surface. Under the influence of an externally applied electrical field the electronhole
pairs are separated and one species, depending on the polarity of the external field, immediately
recombines at the illuminated electrode. The opposite charged species drifts
through the sample, thus giving rise to a time dependent photocurrent I p (t).
For a quantitative analysis the knowledge about the electrical field inside the sample
is necessary. This can be achieved by performing the measurements in the small signal limit,
i. e. disturbations due to space charge effects are negligible and the electric field in the bulk
is determined by the externally applied field. It turns out, that this condition is fulfilled as
long as the generated photocharge Q tot is less than 10 % of the charge Q = CU, which is
4

1.3 Transport models
Figure 1.1: Principle setup for TOF experiments.
stored on the electrodes. With typical experimental parameters (sample thickness d =10mm,
sample area A =4cm 2 ,10 21 monomer units per cm 3 , voltage U = 300 V, sample capacitance
C = 1 nF) the above criterion leads to an upper limit for the photocharges of Q tot

1 Model Systems for Photoconductive Materials
neglected, or in other words, the detrapping always occurs to states at e d . The traps can be
caused by static disorder as well as by self trapping due to polaronic effects or both.
Mathematically equivalent with the MT model is the Continuous Time Random Walk
(CTRW) model [47–49], where the exponential trap distribution density g (e) in the MT
model,
"
g…"† /exp ; …2†
k B T 0
corresponds to the well-known algebraic distribution of hopping times in the CTRW model
[45],
…t† /t 1 ; …3†
where the disorder parameter a in Eq. 3 corresponds to the dimensionless temperature
(a = T/T 0 ).
A variety of experimental data [16, 25, 26, 33, 34] can be described by the empirical
formula,
ef f ˆ 0 exp
p
" 0 E
k B T ef f
; …4†
first proposed by Gill [33], where e 0 is the zero field activation energy, E the electrical field
and b the Poole-Frenkel factor. This equation describes the thermal activated release from
localized traps where the activation energy is lowered by an external field according to the
Poole-Frenkel effect [37].
The effective temperature T eff is related to the physical temperature T by
1
ˆ 1
T ef f T
1
T 0
:
…5†
Initially, the characteristic temperature T 0 simply was an empirical parameter. In
Section 1.4, however, we shall see that in certain cases this parameter can be interpreted microscopically.
An alternative approach [28, 50–54] is based on the assumption that the density of
states can be modelled by a Gaussian distribution. Charge carrier transport occurs via direct
hopping between the localized sites. In general, the differences between the two models are
too small to be detected experimentally. Since our data can be quantitatively explained
within the framework of multiple trapping we will restrict the discussion to this model.
6

1.4 Results
1.4 Results
As stated in Section 1.1, organic photoconductors can be regarded as large or medium bandgap
semiconductors. In contrast to inorganic crystalline semiconductors the electronic coupling between
adjacent sites is weak. Therefore the electronic states in these materials are primarily of
molecular character and the charge carrier transport has to be regarded as an incoherent hopping
process between two neighbouring sites. For the following discussion organic photoconductors
will be divided into three main groups namely molecularly doped polymers (Section
1.4.1), side-chain polymers (Section 1.4.2.), and conjugated systems (Section 1.4.3).
1.4.1 Molecularly doped polymers
In molecularly doped polymers (MDPs) the transport molecules are molecularly dispersed in
an inert matrix. Due to crystallization the maximum concentration of chromophores, which
can be achieved, is typically in the range of 50 mol%.
Typical chemical compounds include oxadiazole derivatives [14], pyrazolines [15–
18], hydrazones [19–22], carbazole derivatives [23–26], triphenylmethane (TPM) derivatives
[27, 28], triphenylamine (TPA) derivatives [29, 30], and TAPC [31], which can be regarded
as a dimer of TPA. The charge carrier mobilities at room temperature are typically
in the range from 10 –6 cm 2 /Vs for N-isopropylcarbazole [25] to 10 –4 cm 2 /Vs for p-diethylaminobenzaldehyde
diphenyl hydrazone (DEH) [22].
In order to study the effect of chemical substitutions of the respective transport molecule
N-isopropylcarbazole (NIPC) and derivatives thereof, with electron donor as well as acceptor
substituents (Fig. 1.2), have been investigated in a polycarbonate host [26].
As compared by 3,6-dibromo-N-isopropylcarbazole (DBr-NIPC) the effective hole
mobility of 3,6-dimethoxy-N-isopropylcarbazole (DMO-NIPC) is slightly decreased. The
reason for this behaviour is the fact that the cation, which is relevant for the transport process,
is stabilized by electron donating substituents like the methoxy group, as can be seen
from the shift of the ionisation potentials [26]. This leads to stronger localization of the
charge within the aromatic rings as compared to derivatives with moderate electron acceptors
like bromine. Therefore the spatial overlap of the electronic states of adjacent molecules,
which are involved in the transport process, is reduced.
With strong acceptors like the nitro groups in 3,6-dinitro-N-isopropylcarbazole (DN-
NIPC) the ionisation potential is already larger than for the host matrix polycarbonate. In
this case even the matrix acts as a trap for holes, thus preventing efficient charge carrier
transport [26].
Since the effective mobility is determined by the spatial overlap of the transport states,
the concentration dependence of the effective mobility can be described by
ef f / r 2 exp
2r
; …6†
r 0
7

1 Model Systems for Photoconductive Materials
N
H 3 CO
N
N
N
N
C 2 H 5
R
R
BTA
R: H: NIPC
Br: DBr-NIPC
OCH 3 : DMO-NIPC
NO 2 : DN-NIPC
Si
Si
O
CH 3
PMPS
n
Ca n
CH CH 2
PVK
(CH 2 ) n
Ca
n
R
Polysiloxan
R
n
Ca:
N
R:
O
DPOP-PPV
Figure 1.2: Chemical structure of photoconducting materials. Abbreviations see text.
H
PPV
where r is the mean distance between adjacent transport molecules and r 0 the wave function
decay parameter. For carbazole derivatives r 0 is typically in the range 1.3–1.6 Å, depending
on the substituted groups [26].
In recent years, also percolation models have been successfully applied to experimental
data [55] as an alternative approach to model the transport properties of doped disordered
systems. For polycarbonate doped with a derivative of benztriazole (BTA) (Fig. 1.2) it has
been shown that the concentration dependence of the effective mobility can be described at
low concentrations by [55]
ef f ˆ 0 …p p c † t …7†
for p>p c , where p is the concentration of transport molecules, p c the percolation threshold,
and t the critical exponent. The experimental values (p c = 0.095 and t = 2.46) are consistent
with 3D-continuum percolation calculations [56].
Since this system shows good transport properties over a wide range of concentrations,
the influence of extrinsic traps could be studied [57]. Here, a sample containing 25 wt% of
BTA in a polycarbonate host was doped with small amounts of the organic dye astrazone
orange (AO). It turned out that the effective mobility is reduced by a factor of 2 when the
8

1.4 Results
concentration of AO is increased to 0.5 wt%. This finding becomes also important when
thinking of new applications like organic electroluminescent devices. One way to increase
the efficiency in these devices is to molecularly disperse the luminescent moiety into a
charge transport material. But there is a trade-off between efficient trapping of the charge
carriers – which is improved by higher concentrations of the luminescent groups – and concentration
quenching of the fluorescence, which can be avoided by keeping the concentration
as low as possible. The above described measurements [57] show that organic dyes can be
very efficient traps for charge carriers. Therefore the concentration of the luminescent moiety
can be as low as 2% or less [61], thus avoiding concentration quenching without loosing
high trapping efficiency.
1.4.2 Side-chain polymers
In side-chain polymers, where the chromophore is covalently bonded to a polymer main
chain, the concentration of transport molecules can be increased as compared to MDPs without
causing crystallization. Because the underlying physical mechanism of charge carrier
transport (nearest neighbour hopping between weakly coupled, localized states) is the same
for both MDPs and side-chain polymers, the transport properties are qualitatively similar.
This finding offers the opportunity to tailor the thermal and mechanical properties of
the photoconductor without seriously affecting the transport properties. This can be achieved
either by variation of the spacer length between the chromophore and the backbone or by variation
of the backbone itself. A typical example for this group is PVK [12, 32, 34, 35]. Here,
the transition from dispersive to non-dispersive transport could be observed [34]. Due to the
fact, that the TOF curves have been measured over 10 decades in time, it was possible by
means of a numerical inverse Laplace transform [34] to calculate from the measured photocurrent
the trapping rate distribution, which is a measure for the density of localized states.
In polysiloxane derivatives with pendant carbazolyl groups [58, 59] (Fig. 1.2) the effect
of a variation of the spacer length was investigated. Starting from a spacer length of 3 or 5
methylene units up to 6 or 11 groups, the glass transition temperature drops from its initial value
of 51 8Cor78Cdownto–58Cor–458C for the C 11 spacer. It turns out, that the zero field
activation energy e act for the hole transport remains unchanged and is the same for NIPC or
PVK, e act = 0.51 eV [58]. This indicates, that the activation energy in side-chain polymers
with pendant carbazolyl groups is dominated by intrinsic properties of the carbazolyl moiety
and is not much affected by other factors like the morphology of the polymer [58].
However, the absolute values for m eff differ by roughly one order of magnitude, where the
compound with the shortest spacer shows an effective mobility in the range of 10 –6 cm 2 /Vs
(T = 300 K, E =3710 5 V/cm). For the materials with a spacer length of 5 and 6 methylene
units the mobility is lower by one order of magnitude and therefore comparable to the mobility
in NIPC or PVK. A striking feature, when comparing NIPC with PVK, is the fact that
polycarbonate doped with 20 wt% of NIPC shows the same hole mobility at room temperature
as compared to PVK where the carbazole concentration amounts to 86 wt%. Obviously,
the covalent attachment of the chromophore to a stiff backbone, like PVK with its glass
transition temperature of 227 8C, induces a mutual orientation of adjacent pendant groups,
9

1 Model Systems for Photoconductive Materials
which reduces the electronic coupling between the two sites. Therefore it is desirable to
induce a well-defined amount of flexibility in both, the spacer and the backbone, to allow
for mutual reorientation of the chromophores. For a polysiloxane backbone with carbazolyl
transport groups the optimum spacer length n is equal to 3.
1.4.3 Conjugated systems
In contrast to the two main groups described above the charge carrier transport in conjugated
systems occurs via the polymer backbone. Various substituents are used to tailor the
mechanical properties and the processibility. For instance, the chain can be a skeleton with
quasi-s-conjugation like in polysilanes [36, 38–42] or polygermylenes [39].
The second group of main chain polymers contains a p-conjugated backbone like
polyacetylene, polythiophene, polypyrrole, polyphenylene, poly(phenylenevinylene) (PPV),
and their derivatives. For a review see [43, 44].
Both types of conjugated systems have been investigated. In the case of poly(methylphenyl
silane) (PMPS), a quasi-s-conjugated polymer (Fig. 1.2), non-dispersive transport
has been found down to a temperature of 250 K [60]. For lower temperatures, the transport
is dispersive, which indicates that the transport in these materials is controlled by traps.
Compared to the materials described above, the depths of the relevant traps are much lower.
The zero field activation energy turns out to be as low as 0.37 eV as compared to 0.5 eV for
materials containing carbazole. This gives a hole mobility at room temperature of roughly
10 –3 cm 2 /Vs, which is 3 orders of magnitude higher than for the materials described above.
The high mobility and the transparency in the visible region makes PMPS also useful for
applications in multilayer electroluminescent devices [61–63].
Besides the s-bonded PMPS, we investigated oligomeric conjugated compounds based
on carbazole [64]. First experiments with a trimer of a carbazole derivative showed that this
material is soluble in most of the common organic solvents. It forms a low molecular weight
glass and shows no tendency to crystallize even after several months. In this material the
p-system has been extended over three monomer units, which results in remarkably high
hole mobilities as compared to the monomer model compound NIPC. First TOF experiments
show an increase in mobility by roughly two orders of magnitude as compared to unconjugated
carbazole systems, which reaches 2 7 10 –4 cm 2 /Vs [64].
The second group of p-conjugated polymers contains one p z -orbital per carbon atom
which stands perpendicular to the plane of the s-bonded skeleton of the main chain. The interaction
between these orbitals leads to electronic states, which are delocalized – at least
partly – along the conjugated chain. As compared to the intrachain coupling, the intrachain
coupling is lower by typically one or two orders of magnitude [66] thus leading to a large
anisotropy of the electronic and optical properties. These materials can be regarded as basically
one-dimensional organic semiconductors with bandgaps in the visible region of the
spectrum, e. g. 1.5 eV (trans-(CH) x ) [43], 2.0 eV (polythiophene), 2.5 eV (polyphenylene vinylene),
and 3.0 eV [poly(para-phenylene)] [67].
We investigated the charge transport properties of the p-conjugated polymer poly
(para-phenylenevinylene) and its soluble substituted derivative DPOP-PPV (Fig. 1.2). In
10

1.4 Results
contrast to e. g. (CH) x , PPV can be prepared in undoped form. If oxygen is carefully heated
out, PPV exhibits excellent film forming properties as well as thermal stability under ambient
conditions. In recent years, PPV has drawn additional attention due to its use in organic
electroluminescent devices [45].
In the following we will show that the charge carrier transport in these materials can
be described by the conventional models mentioned before, which have been developed for
the characterization of disordered molecular systems.
Based on the MT model a method has been developed to calculate the distribution of
capture rates from the measured photocurrent [68, 69]. In contrast to Ref. [34], the method
is based on a Fourier transform technique with better numerical stability as compared to the
inverse Laplace transform used in Ref. [34].
With this approach the capture rate density o c is given by
! c …" ! †k B T ˆ 2
p
Q tot sin
t mic
I…!†
!
! ; …8†
where Q tot is the total photogenerated charge, t mic the microscopic transit time, and I (o) the
Fourier transform of the measured photocurrent, and f the phase shift (Eq. 9). For DPOP-
PPV, t mic = 7.5610 –10 s for the given experimental conditions.
ˆ tan 1 Im I…!†
Re I…!†
…9†
The trap depth e o is related to the frequency o by the expression
! ˆ r 0 exp
" !
; …10†
k B T
where r 0 is the attempt-to-escape frequency (r 0 =10 10 s –1 for DPOP-PPV).
In DPOP-PPV the effective mobility is field dependent and thermally activated according
to Gill’s formula (Eq. 4) [33] with a characteristic temperature T 0 = 465 K (Fig. 1.3).
The calculated trap density g (e) is exponential down to a pronounced cutoff energy e d ,
g…"† /exp
"
: …11†
k B T 0
This cutoff energy corresponds exactly to the measured activation energy of the effective
mobility. Furthermore, the initially simply empirical parameter T 0 in Gill’s formula
(Eq. 4) could be correlated with the decay constant k B T 0 of the trap distribution.
By calculating the total number of trapping events it can be seen that the typical distance
which a charge carrier travels between two trapping events is of the order of 4 Å. This
value is comparable to the intrachain distance and indicates that the transport in these materials
can be best described by conventional hopping between closely neighbouring sites. No
evidence for bandlike transport has been found.
11

µ eff
[cm 2 /Vs]
1 Model Systems for Photoconductive Materials
10 -2
10 -3
10 -4
350 K 300 K 270 K
230 K
500 kV/cm
350 kV/cm
200 kV/cm
75 kV/cm
0kV/cm
10 -5
10 -6
10 -7
10 2
10 0
T 0
= 465K
10 -2
10 -4
10 -6
10 -8
10 -10
0 2 4 6 8
10 -8
2.5 3.0 3.5 4.0 4.5
1/T [1000/K]
Figure 1.3: large frame: effective mobility m eff of DPOP-PPV as a function of temperature for some
electrical fields, data points for zero field were extrapolated from field dependent measurements; inset:
Arrhenius fits for different electrical fields. For details see text. Data from [68, 69].
For the case of PPV the dispersion of the photocurrent is too large for transit time determination.
It can be concluded that even at room temperature the release times from the
deepest traps, which control the transport properties, are in the range of 1 s or less. This corresponds
to an effective mobility of less than 10 –8 cm 2 /Vs [69]. Based on TSC measurements
from different groups [70], we conclude that this behaviour is primarily caused by the
existence of grain boundaries.
It is reasonable to assume that in conjugated polymers with a rigid backbone the preferred
orientation of the main chain will be parallel to the film surface. Therefore, the
charge carrier transport through the film has to occur perpendicular to the backbone, i. e.
perpendicular to the direction with high intrinsic mobility, which would mean a principle
limit for this class of materials for this kind of applications.
1.5 Outlook
As described above, disordered organic materials have been developed with effective hole
mobilities of up to 10 –3 cm 2 /Vs together with good mechanical and dielectric properties.
However it seems that for disordered systems a considerable improvement of this value will
12

References
be difficult to achieve, because for all investigated material groups the charge carrier transport
is limited by the localized nature of the electronic states and by the hopping mechanism
of the transport process.
Based on recent other works [8, 9], we think that for developing even higher mobility
materials it is essential to improve the structural order of the material. This may open new
application fields, e. g. the use of active organic materials in semiconductor devices. One
promising way is the use of highly ordered liquid crystals. With these materials, however,
charge carrier mobilities up to 0.1 cm 2 /Vs or even more are realistic, which would make
them comparable to single crystalline systems [65].
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50. P.M. Borsenberger, L. Pautmeier, H. Bässler: J. Chem. Phys., 94(8), 5447 (1991)
51. G. Schönherr, H. Bässler, M. Silver: Philos. Mag. B, 44(3), 369 (1981)
52. H. Bässler, G. Schönherr, M. Abkowitz, D.M. Pai: Phys. Rev. B, 26(6), 3105 (1982)
53. L. Pautmeier, R. Richert, H. Bässler: Philos. Mag. Lett., 59(6), 325 (1989)
54. L. Pautmeier, R. Richert, H. Bässler: Synth. Met., 37, 271 (1990)
55. H. Domes, R. Leyrer, D. Haarer, A. Blumen: Phys. Rev. B, 36(8), 4522 (1987)
56. D. Stauffer: Introduction to Percolation Theory, Taylor and Francis, London (1985)
57. H. Domes: PhD thesis, Universität Bayreuth (1988)
58. H. Domes, R. Fischer, D. Haarer, R. Strohriegl: Makromol. Chem., 190, 165 (1989)
59. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58(2), 101 (1988)
60. H. Kaul: PhD thesis, Universität Bayreuth (1991)
61. H. Suzuki, H. Meyer, J. Simmerer, J. Yang, D. Haarer: Adv. Mater., 5, 743 (1993)
62. H. Suzuki, H. Meyer, S. Hoshino, D. Haarer: J. Appl. Phys. in press (1995)
63. J. Kido, K. Nagai, Y. Okamoto, T. Skotheim: Appl. Phys. Lett., 59, 2760 (1991)
64. C. Beginn, J.V. Grazulevicius, P. Strohriegl,J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195,
2353 (1994)
65. N. Karl: in: K. Sumino (ed.): Defect Control in Semiconductors, Elsevier Science Publishers,
North Holland (1990)
66. P. Gomes da Costa, R.G. Dandrea, E.M. Conwell: Phys. Rev. B, 47(4), 1800 (1993)
67. G. Leising, K. Pichler, F. Stelzer: in: H. Kuzmany, M. Mehring, S. Roth, (eds.): Springer Series in
Solid-State Sciences, Vol. 91: Electronic Properties of Conjugated Polymers III, Springer, Berlin
(1989)
68. H. Meyer: PhD thesis, Universität Bayreuth (1994)
69. H. Meyer, D. Haarer, H. Naarmann, H.H. Hörhold: Phys. Rev. B, in press (1995)
70. M. Onoda, D.H. Park, K. Yoshino: J. Phys. (London), Condens. Matter, 1(1), 113 (1989)
14

2 Novel Photoconductive Polymers
Jörg Bettenhausen and Peter Strohriegl
2.1 Introduction
Electrophotography is the only area in which the conductivity of sophisticated organic materials
and polymers is exploited in a large scale industrial process today. Photoconductors are
characterized by an increase of electrical conductivity upon irradiation. According to this definition
photoconductive materials are insulators in the dark and become semiconductors if
illuminated. In contrast to electrically conductive compounds photoconductors do not contain
free carriers of charge. In photoconductors these carriers are generated by the action of
light.
The discovery of photoconductivity dates back to 1873 when W. Smith found the effect
in selenium. Based on this discovery C. F. Carlson developed the principles of the xerographic
process already in 1938.
Photoconductivity in organic polymers was first discovered in 1957 by H. Hoegl, who
found that poly(N-vinylcarbazole) (PVK) and charge transfer complexes of PVK with electron
acceptors like 2,4,7-trinitrofluorenone act as photoconductors [1].
Besides the application of photoconductive polymers in photocopiers these materials
are also widely used in laser printers in the last years. The third area in which photoconductors
are applied is the manufacturing of electrophotographic printing plates.
The organic photoconductors used in practice are based on two types of systems. The
first one are polymers in which the photoconductive moiety is part of the polymer, for example
a pendant or in-chain group. The second group involves low molecular weight compounds
imbedded in a polymer matrix. These so-called moleculary doped polymers are
widely used today.
One interesting class of photoconductive materials are oxadiazoles. It is known for a
long time that derivatives of 1,3,4-oxadiazole are good photoconductors. Compound 1 for
example is described in patents and was frequently applied in photocopiers [2].
1 2
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
15

2 Novel Photoconductive Polymers
The oxadiazole 1 is a hole transport material since the electron withdrawing effect of
the oxadiazole group with three electronegative heteroatoms is overcompensated by two
electron donating amino groups.
Within the last years, oxadiazoles like 2-(biphenyl)-5-(4-tert.butylphenyl)-1,3,4-
oxadiazole (PBD) 2 have been frequently applied in organic light emitting diodes [3]. Here
the electron withdrawing oxadiazole unit dominates the electronic properties and the oxadiazole
compounds act as electron injection and transport layers. Furthermore, 2,5-diphenyloxadiazoles
have been used as building blocks in thermostable polymers and they are highly
fluorescent as well.
In this paper the synthesis and characterization of a number of novel low molecular
weight oxadiazole derivatives and polymers is described. The compounds show different molecular
shapes, e. g. rod-like and star-shaped structures have been realized.
In addition to the compounds described here, a series of different main chain and side
group polymers with oxadiazole moieties are presently synthesized in our research group [4].
2.2 Liquid crystalline oxadiazoles and thiadiazoles
2.2.1 The basic idea
The aim of this work is the preparation of photoconductive compounds with high carrier mobilities.
There are a number of indications that the carrier mobility, i. e. the speed of the
charge particles within the sample, depends on the geometric arrangement of the photoconductive
moieties.
At room temperature for instance, the mobility in single crystals of aromatic compounds
like anthracene or perylene is very high, i. e. in the range of 10 –1 cm 2 /Vs [5]. In
amorphous polymers the carrier mobilities are orders of magnitude smaller and typical values
are in the range of 10 –6 and 10 –8 cm 2 /Vs [6].
So we were interested to investigate if the higher order in the liquid crystalline state
leads to higher mobilities in comparison to less ordered amorphous solids. In liquid crystalline
polymers a macroscopic orientation of the photoconductive groups in the mesophase
can be achieved by means of electric or magnetic fields. The orientation can be frozen in by
lowering the temperature below the glass transition. By this, materials with a degree of order
between a perfect single crystal and totally disordered amorphous polymers are obtained.
With this in mind we started the synthesis of a series of liquid crystalline model compounds
and polymers in which the side groups possess both mesogenic and photoconductive
properties.
16

2.2 Liquid crystalline oxadiazoles and thiadiazoles
2.2.2 Monomer synthesis
In the past years we have synthesized a variety of compounds with oxadiazole and thiadiazole
moieties [7]:
3 X=S 4 X=O
The rod-like thiadiazole derivative 3 is liquid crystalline and shows a smectic as well
as a nematic LC phase with the following phase behaviour: c 103 s a 180 n 203 i. An acrylate
monomer was prepared by hydroboration of the terminal double bond and subsequent
esterification with acryloyl chloride and then polymerized. The resulting polymer 5 exhibits
a mesophase which has not yet been identified.
5
Unfortunately the clearing point of the polymer is at 246 8C, where the material starts
to decompose. Therefore orientation in an electric field was not possible.
In contrast to the thiadiazole the oxadiazole 4 is neither liquid crystalline as monomer
nor as polymer. The reason for the lack of mesogenic properties is that the substitution of
the sulphur by oxygen introduces a bend into the molecule which prevents the formation of
a LC phase.
Recently liquid crystalline oxadiazoles (Scheme 2.1) have been described [8]. Here
the oxadiazole is coupled to at least two benzene or cyclohexane rings. Thereby an increase
of the mesophase stability is achieved.
Scheme 2.1: Liquid crystalline oxadiazole compounds [8].
17

2 Novel Photoconductive Polymers
6 7
8
9
10-13
10 R=Ph-OC 6
H 13
12 R = Ph-N(CH 3
) 2
11 R=C 7
H 15
13 R = Ph-N(CH 3
)C 6
H 13
Scheme 2.2: Monomer synthesis.
Our aim was to synthesize a series of different oxadiazole monomers which are functionalized
for the preparation of polymers. Scheme 2.2 shows the synthesis of the monomeric
1,3,4-oxadiazoles. The oxadiazole moiety is formed by a cyclisation reaction of the
diacylhydrazine derivatives 8 with phosphorous oxychloride. The unsymmetrical bishydrazides
8 are prepared by treatment of benzoyl chlorides 6 with the appropriate monohydrazides
7. The last reaction step is an esterification with 4-(5-hexenyloxy)benzoyl chloride.
The esterification reaction is essential in this case because of two reasons. First the rod-like
18

2.2 Liquid crystalline oxadiazoles and thiadiazoles
mesogen is formed, and second a functional group for the preparation of LC polymers is introduced.
Additionally the oxadiazole monomer 14 was synthesized, in which the ester group is
replaced by a biphenyl unit. By this it is possible to investigate the influence of the ester
function on the photoconductive properties of the liquid crystalline compounds.
14
The liquid crystalline behaviour and the phase transitions of the monomeric oxadiazole
derivatives 10–14 were determined by DSC and polarizing microscopy. In Tab. 2.1 the
phase transition temperatures are summarized:
Table 2.1: Phase behaviour of the monomeric oxadiazoles 10–14
Compound
Transition temperature in 8C
10 k 141 n 167 i
11 k 90 s a 116 i
12 k 162 n 179 i
13 k 127 (s a 111) i
14 k 72 s a 77 i
Except compound 13 all monomers are enantiotropic liquid crystalline and show nematic
or smectic mesophases. Only derivative 13 shows a monotropic s a phase.
2.2.3 Oligo and polysiloxanes with pendant oxadiazole groups
The next step was the synthesis of liquid crystalline polysiloxanes with oxadiazole groups in
the mesogenic unit. The polymers were prepared by a polymeranalogous reaction of the
monomers 10 and 11 with poly(hydrogenmethylsiloxane) 15.
Both polymers are liquid crystalline. The transition into the isotropic phase takes place
at about 200 8C and therefore at a much higher temperature than in the monomers. The high
clearing temperature makes the orientation of the polymers difficult which is preferably carried
out near the clearing point. Up to now it was not possible to prepare well-defined polymers
with the monomers 12–14. A possible reason is the inactivation of the Pt catalyst by
the amino groups.
If the cyclic tetrasiloxane 18, which is commercially available in high purity, is used
instead of 15, well-defined monodisperse model compounds are obtained [9]. These com-
19

2 Novel Photoconductive Polymers
10, 11 15
16 R=Ph-OC 6
H 13
17 R=C 7
H 15
Scheme 2.3: Synthesis of polysiloxanes with pendant oxadiazole groups.
pounds can be highly purified, for example by preparative gel permeation chromatography.
This is very important for photoconductors, because it is well-known that even traces of impurities
may reduce the carrier mobility.
18
So the monomeric compounds 10 and 12–14 have been reacted with 18 to yield the
following tetrameric derivatives:
19 R=Ph-OC 6
H 13
,n=1 21 R = Ph-N(CH 3
)C 6
H 13
,n=
20 R = Ph-N(CH 3
) 2
,n=1 22 R = Ph-N(CH 3
)C 6
H 13
,n=
Scheme 2.4: Tetrasiloxanes with pendant oxadiazole groups.
20

2.2 Liquid crystalline oxadiazoles and thiadiazoles
All cyclosiloxanes are liquid crystalline and show several advantages compared to the
polymers. So the cyclic siloxanes form stable glasses but their clearing points are much
lower. For example compounds 21 and 22 possess clearing temperatures at only 102 8C.
The phase behaviour of the tetramers are listed in Tab. 2.2.
Table 2.2: Phase behaviour of the tetrameric siloxanes 19–22.
Comp.
Transition temperature in 8C
19 g 54
a)
m 1 108
a)
m 2 117 s c 155 n 185 i
20 g 67 s a 138 n 181 i
21 g 34 k 66 n 102 i
22 g 38 s a 82 n 102 i
a) Mesophase not identified
2.2.4 Photoconductivity measurements
For the characterization of the novel photoconductive compounds two different experimental
techniques have been used. Some measurements were made by BASF AG using a steady-state
method. These experiments allow a quick characterization, but in contrast to the time-of-flight
(TOF) method no statements about transient photocurrents and carrier mobilities are possible.
The TOF measurements were carried out in the group of D. Haarer, Universität Bayreuth.
The tetrameric cyclosiloxanes are very suitable compounds for physical investigations.
Because of the low viscosity the preparation of samples is much easier than in the case of
the polymers. Additionally the dark currents of the tetramers are very low.
All the compounds are photoconductive, as shown by the steady-state method. For example,
the measurements of the thiadiazole 3 showed a distinct rise of the photocurrent at
the transition from the crystalline to the mesophase, as shown in Fig. 2.1. The decrease of
Figure 2.1: Temperature dependence of the photocurrent of the thiadiazole 3.
21

2 Novel Photoconductive Polymers
the current at the transition from the smectic to the nematic phase is directly correlated with
the loss of order. These results show that the basic idea of this work is correct.
Nevertheless, we were not able to carry out TOF measurements with calamitic monomers,
because the dark currents in the liquid crystalline phases were too high. As mentioned
before, such measurements are possible with the cyclic tetramers. The transient photocurrents
of the tetrasiloxanes 19, 21, 22 illustrate that the carrier transport is totally dispersive,
i. e. dominated by deep traps in which the charge carriers are captured. This is a typical behaviour
of amorphous polymers. No transit time could be detected and no statements about
the carrier mobility can be made for the rod-like mesogens.
In contrast it was shown in the past years that discotic liquid crystals like hexapentyloxytriphenylene
(HPT) 23 have carrier mobilities up to 10 –3 cm 2 /Vs in the liquid crystalline
D ho phase [10]. Here, the disc-shaped molecules are ideally stacked above each other
and therewith allow a fast carrier transport. The discotic hexathioether 24 with a highly ordered
helical columnar (H) phase exhibits mobilities up to 10 –1 cm 2 /Vs, which are almost as
high as in organic single crystals [11].
6
6
23 24
One important observation during the photoconductivity measurements of the tetramer
19 was, that it showed almost the same photocurrent for holes and electrons. This was interesting
because of the lack of electron transporting materials and led us synthesize a variety
of starburst oxadiazole compounds, which are described in the next Chapter.
2.3 Starburst oxadiazole compounds
2.3.1 Motivation
The synthesis of novel materials with high carrier mobilities is one of the major goals in the
field of photoconductive polymers. Within the last years different approaches have been pursued
to reach this goal. First, the photoconductive properties of conjugated polymers like
poly(phenylenevinylene) and poly(methyl phenylsilane) have been investigated [12]. Another
approach are liquid crystals which are the topic of the first Chapter. The third way to realize
22

2.3 Starburst oxadiazole compounds
the goal are glasses of large extended aromatic amines. The best investigated representatives
of this class of compounds are N,N'-diphenyl-N,N'-bis(3-methylphenyl)-[1,1'-biphenyl]-4,4'-
diamine (TPD) 25 from Xerox [13] and 1,1-bis(di-4-tolyl-aminophenyl)cyclohexane (TAPC)
26 from Kodak [14].
Both compounds are derivatives of triphenylamine, a well-known photoconductor. If
thin films of TPD and TAPC are prepared by vacuum evaporation both compounds form
metastable glasses. In such glasses carrier mobilities up to 10 –3 cm 2 /Vs for TPD [13] and
10 –2 cm 2 /Vs for TAPC [14] have been reported. But both TAPC and TPD glasses are metastable
and have a strong tendency to crystallize. If the molecules are imbedded in a polymer
matrix, e. g. polycarbonate or polystyrene, morphologically stable materials are formed, but
the mobilities decrease drastically [13].
25 26
Several attempts have been made to overcome the problems with metastable TPD and
TAPC glasses. Recently we described the synthesis of 3,6-bis[(9-hexyl-3-carbazolyl)ethynyl]-9-hexylcarbazole
27, a trimeric model compound of poly(carbazolylene ethynylene)
[15]. This material shows a glass transition temperature of 41 8C and forms glasses which
are stable for more than a year. Mobilities up to 10 –4 cm 2 /Vs (E =67 10 5 V/cm, T =308C)
have been measured by the time-of-flight technique.
27
Another interesting approach are starburst compounds with high glass transition temperatures.
So 4,4',4@-tris(N-carbazolyl)triphenylamine, which has been published recently
[16], shows a glass transition at 151 8C and forms a stable glass.
Beginning with the work of Thomalia [17] in 1986, starburst molecules (dendrimers)
have achieved enormous interest within the last years. Dendrimers are highly branched regular
molecules, which are usually prepared by stepwise reactions. In many cases the behaviour
of dendritic macromolecules are different from that of linear polymers, e. g. the former
23

2 Novel Photoconductive Polymers
usually show enhanced solubility. The reasons for these differences are the unique three-dimensional
structure and the large number of chain ends in dendrimers.
Two different methods have been developed for the stepwise synthesis of starburst
dendrimers:
a) the divergent approach, where the synthesis starts from a core molecule with two or
more reactive groups;
b) the convergent approach, in which the synthesis starts at the outer sphere of the dendrimer.
In the divergent approach, the reaction of the core molecule with two or more reagents
containing at least two protected branching sites is followed by removal of the protecting
groups and subsequent reaction of the liberated reactive groups which leads to the starburst
molecule of the 1st generation. The process is repeated until the desired size is reached. In
the convergent approach the synthesis starts at what will become the outer surface of the
dendrimer. Step by step large dendrimer arms are prepared and finally the completed arms
are coupled to the core. Both methods produce well-defined large dendritic molecules whose
structures are manifolds of the building blocks. They allow structural as well as functional
group variation.
On the other hand, hyperbranched polymers can be synthesized in a one-step reaction
using highly functionalized monomers of the type A x B, where x is 2 or larger. This method
does not yield polymers of such a well-defined structure, but has the advantage to provide
rapidly large quantities of material. The control of the degree of branching is difficult and
mainly depends on statistics, steric effects, and the reactivity of the functional groups.
The aim of our work is the synthesis of starburst compounds with oxadiazole moieties.
In contrast to the triphenylamine and carbazole derivatives oxadiazoles are strong electron
acceptors. Therefore their transport characteristics can be switched from hole transport materials
like 2,5-(4-diethylaminophenyl)-oxadiazole 1, in which the electron withdrawing effect
of the oxadiazole ring is overcompensated by the two electron donating amino groups,
to electron transport materials like 2-(biphenyl)-5-(4-tert.butylphenyl)-oxadiazole (PBD) 2.
Recently, Saito demonstrated that oxadiazoles in a polycarbonate matrix show electron transport
with mobilities up to 10 –5 cm 2 /Vs [18]. Such electron transport materials are attractive
for the use in copiers and in organic light-emitting diodes.
2.3.2 Synthesis of starburst oxadiazole compounds
Starburst oxadiazole compounds have been mentioned for the first time in a thermodynamic
study of the structure-property-relationship in low molecular weight organic glasses [19,
20]. Upon rapid cooling these compounds form glasses with T g s between 77 and 169 8C.
We have used three different approaches for the preparation of starburst oxadiazole
compounds, which are schematically shown in Fig. 2.2.
The methods A and B can be compared to the divergent synthesis of dendrimers.
Method A starts from a central core and involves the reaction of the acid chloride groups
and a subsequent cyclisation with phosphorous oxychloride to the oxadiazole ring (Chap-
24

O
O
Cl
C
C
Cl
O
C
Cl
Method A
Method B
O
H 2N NH C
O
O
O
O
C
NH
NH
C
C
NH NH C
O
O
C
NH
NH
C
C
N N
N NH
N
N
O
N N O
O
N
N
=core
= shell
Method C
X
Z
+
Y
X
X
Z
Z
X = OH, C CH Y = Br, F Z = O , C C
=core
=shell
Figure 2.2: The different approaches to starburst oxadiazoles.
25

2 Novel Photoconductive Polymers
ter 1). The main difficulty in this case is that three functional groups must react in one step.
Partially incomplete cyclisation causes problems, because the products contain one or two
bishydrazides and are difficult to separate from the target compounds, which have three oxadiazole
groups. Therefore we recently developed a totally different route (method B). The
compounds 31 a–f were prepared by the reaction of benzene tricarbonyl chloride 30 with
the tetrazoles 29. The latter were synthesized from benzonitriles 28 with potassium azide
(Scheme 2.5). With the loss of nitrogen the tetrazole ring is transferred to the oxadiazoles
[21]. This gives an easy access to unsymmetrically substituted oxadiazoles which we used
for the first time in the synthesis of starburst oxadiazole compounds.
In the third approach (method C) which is similar to the convergent synthesis a preformed
oxadiazole precursor is prepared by stepwise synthesis and finally coupled to the
core molecule.
R 1
R 2 CN + NaN 3
NH 4 Cl
R 1
R 2
R 3
R 3
28 29
C
N
N
NH
N
Cl
O
C
O
C
Cl
O
C
Cl
+ 3
R 2
R 1
R 3
C
N
N
NH
N
31a-f
pyridine
2h, reflux
31a-f
30 29
Scheme 2.5: Synthesis of starburst oxadiazoles via tetrazole intermediates.
For the preparation of the novel starburst molecules, different core molecules have
been used: 1,3,5-benzene tricarboxylic acid chloride, 1,3,5-tris(4-benzoyl)benzene, 1,3,5-
triethynylbenzene, and 4,4',4@trihydroxytriphenylamine.
Except benzenetricarboxylic acid which is commercially available, all core molecules
were synthesized following well-known literature procedures [22, 23].
The structures of the starburst oxadiazole compounds are summarized in Tab. 2.3.
31 c and 31g have been described before [19, 20], but no synthetisis procedure was given.
The oxadiazoles 31a–h have been prepared according to method B (Fig. 2.3), starting
from benzene tricarboxylic acid chloride.
Compound 31g with tert.-butyl substituents was synthesized by method A, too. So a
comparison of methods A and B is possible. The tetrazole route (method B) exhibits several
advantages compared to the ring closure with dehydrating agents (method A). One advantage
is the short reaction time. The reaction is finished within 2 hours whereas heating for
1–3 days is necessary if the ring closure is carried out with phosphorous oxychloride. Even
more important is the facile work up procedure. In the case of the dehydration with POCl 3
column chromatography and subsequent recrystallization is necessary to purify the product
which is finally obtained in 33% yield. In contrast only one filtration on a short silica gel
26

2.3 Starburst oxadiazole compounds
Table 2.3: Structures of the starburst oxadiazole compounds 31–34.
Comp. Core a) Shell a)
31
N
O
N
R 1
R 3
R 2
31a R 1 =R 3 =H,R 2 =CH 3
31b R 2 =R 3 =CH 3 ,R 1 =H
31c R 1 =R 3 =CF 3 ,R 2 =H
31d R 1 =R 3 =H,R 2 =C 2 H 5
31e R 1 =R 3 =H,R 2 =OC 2 H 5
31f R 1 =R 3 =H,R 2 = CH(CH 3 ) 2
31g R 1 =R 3 =H,R 2 = C(CH 3 ) 3
31h R 1 =R 3 =H,R 2 = N(C 2 H 5 ) 2
N N
32 R
R = C(CH 3 ) 3
O
C
C
C C
N N
33 R
R = C(CH 3 ) 3
C
C
O
N N
34 N
O
R R = C(CH 3 ) 3
O
a) cf. Figure 2.2.
column is sufficient for the purification of 31 g, prepared by the tetrazole route. In this case
the yield of the pure product is 69%. The starburst oxadiazole 32 has been prepared by
method B too, whereas for 33 and 34 method C (Fig. 2.2) was used. The identity of all compounds
was confirmed by NMR and FTIR spectroscopy.
2.3.3 Thermal properties
The thermal behaviour of the starburst molecules has been investigated by differential scanning
calorimetry and thermogravimetric measurements.
27

2 Novel Photoconductive Polymers
Table 2.4: Thermal properties of the starburst compounds 31–34.
a)
a)
Compound Molecular mass T g T m
[g/mol] [8C] [8C] [8C]
31a 553 – 334 348
31b 595 – 333 358
31c 919 – 250 336
31d 595 – 276 360
31e 643 108 259 337
31f 637 97 225 349
31g 679 142 257 366
31h 724 128 299 329
32 907 165 297 409
33 979 – 320 356
34 1122 137 – 391
a) 3rd heating, determined by DSC with 20 K/min
b) onset of decomposition in nitrogen, thermogravimetric measurement with 10 K/min
T dec
b)
Among the compounds 31 a–h with a small benzene core both crystalline and glass
forming materials exist. So 31 a–d with small methyl, ethyl, or trifluoromethyl substituents
show only melting points in the DSC experiment with cooling rates of 20 K/min. Naito and
Miura reported that it is possible to obtain glasses even with small methyl or trifluoromethyl
substituents if the compounds are rapidly cooled with liquid nitrogen [19]. In contrast, the
compounds 31 e with ethoxy, 31f with iso-propyl, 31g with tert-butyl substituents, and 31h
with diethylamino groups form glasses upon cooling with 20 K/min in the DSC equipment.
When the amorphous samples are heated again the glass transition appears first but on
further heating the samples start to recrystallize and consequently show a melting point at
higher temperatures. Compound 32 with the triphenylbenzene core behaves similar like
31e–h. The most stable glasses are formed by 34 with the triphenylamine core. The DSC
diagram of the novel glass forming oxadiazole compound is shown below. Upon both, heating
and cooling, only a glass transition is observed at 137 8C (Fig. 2.3). In our DSC experi-
Figure 2.3: DSC scan of the starburst oxadiazole compound 34, 3rd heating and cooling, with 20 K/min.
28

2.3 Starburst oxadiazole compounds
ments we never found any evidence for crystallization of 34. Even in the first heating no
melting point is observed up to 350 8C. Consequently, transparent amorphous films are obtained
from the starburst oxadiazole compound 34 by solvent casting.
The thermal behavior of compound 33 is somewhat different. No reproducible DSC
scans are obtained in subsequent heating-cooling cycles. We attribute this to thermal crosslinking
of the triple bonds [24].
The thermal stability has been monitored by thermogravimetric measurements. In
most cases the onset of decomposition is in the range from 330–370 8C. The oxadiazole
compound 32 with a triphenyl benzene core shows a somewhat higher thermal stability up
to 410 8C.
(CH 3 ) 3 C
N
O
N
O
C(CH 3 ) 3
N
N O
O
N
O
N
O
N
C(CH 3 ) 3
34
The starburst oxadiazole compounds are now being tested as electron injection and
transport layer in organic LEDs and as photoconductors. First tests of two layer LEDs with
PPV show that the novel materials possess properties comparable to 2 but have the great advantage
to show no recrystallization if thin films were made by spin-coating. We will report
on these measurements in the near future.
29

2 Novel Photoconductive Polymers
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H. Hoegl: J. Phys. Chem., 69, 755 (1965)
2. W. Wiedemann: Chem.-Ztg., 106, 275 (1982) and references therein
3. A.R. Brown, D.D.C. Bradley, J.H. Burroughes, R.H. Friend, N.C. Green-ham, P.L. Burn, A.B.
Holmes, A. Kraft: Appl. Phys. Lett., 61, 2793 (1992)
4. E. Buchwald, M. Meier, S. Karg, W. Rieß, M. Schwoerer, P. Pösch, H.-W. Schmidt, P. Strohriegl:
Adv. Mater., 7, 839 (1995), M. Greczmiel, P. Pösch, H.-W. Schmidt, P. Strohriegl, E. Buchwald,
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York, (1993)
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199, 345 (1991)
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Phys. Rev. Lett., 70, 457 (1993)
11. D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf,
D. Haarer: Nature, 371, 141 (1994)
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Phil. Mag. B, 61, 25 (1990)
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Carl Hanser Verlag, Munich, 137 (1994)
30

3 Theoretical Aspects of Anomalous Diffusion
in Complex Systems
Alexander Blumen
3.1 General aspects
Many relaxation phenomena in the solid phase depend on diffusion, which is usually treated
in the framework of statistical methods. Regular diffusion, known as Brownian motion, is
characterized by a linear increase of the mean-squared displacement with time. On the other
hand, for a whole series of phenomena this simple relation does not hold; their temporal
evolution of the mean-squared displacement is non-linear and thus obeys at long times:
r 2
…t† t
…1†
with g 0 1. Relation 1 is referred to as anomalous diffusion. In the case that g < 1 one denotes
the behaviour as subdiffusive. A subdiffusive pattern of motion often results from disorder
[1–4]. One has to note, however, that the asymptotic law, Eq. 1, emerges only when
the disorder influences the motion on all scales. In the case that g > 1 the motion is termed
superdiffusive. A classical example for superdiffusive behaviour is furnished by the motion
of particles in a turbulent flow. In this paper we focus on several models for anomalous diffusion
which involve polymeric systems.
Now, the mean-squared displacement is a basic characteristic feature for the motion
but, as an averaged quantity, it can provide only restricted information about the basic microscopic
mechanisms involved. In several works we have also studied the propagator P(r,t),
the probability to be at r at time t having started at the origin at t = 0. Space limitations prevent
us from going into details. Here we focus on Ar 2 (t)S and refer to Klafter et al. and Zumofen
et al. [4, 5] for in-depth analyses of the propagator.
From the beginning, the role of time-dependent aspects in disordered media has to be
emphasized. In usual random walk problems it is often assumed that the disorder is
quenched, so that the dynamics evolves over a static substrate, i. e. the geometrical or the energetical
disorder is frozen in. However, it is also possible that the dynamical processes are
directly under the influence of temporal (possibly medium induced) fluctuations.
First, in Section 3.2 we concentrate on photoconductivity, whose canonical description
involves the continuous time random walk (CTRW) approach [1, 4, 6–8]. Basic ingredients
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
31

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
in CTRWs are waiting time distributions with long time-tails. For photoconductivity one
usually has g 1 shows up.
An interesting dynamical feature is observed when the object to move has itself an internal
structure; that is the case, for instance, for a polymer. In this case, because of the connections
which exist between different segments of the polymer, the motion gets a very rich
structure. The details of this situation will be discussed in Section 3.4.
Finally, we close in Section 3.5 with some conclusions.
3.2 Photoconductivity
A major field of application of scaling ideas in the time domain is photoconductivity [1–3,
7, 12]; here the use of polymeric photoconductors in printing and copying represents one of
the most sophisticated applications for organic materials. Such materials are nowadays
superior to their inorganic counterparts. Now, the issue at stake in amorphous photoconductors
is the appearance of dispersive transport, as contrasted to the familiar diffusive Gaussian
behaviour. One observes experimentally that the motion of the charge carriers becomes
slower and slower with the passage of time, a situation mirrored by Eq. 1 with g t T one finds
I t 1 : …4†
For instance measurements on polysiloxanes with pendant carbazole groups [13] follow
Eqs. 3 and 4 with g = 0.58 very well over four decades in time, see below. In fact this is
32

3.2 Photoconductivity
typical for g significantly lower than one. However, one should note that the behaviour of
I (t) is complex, since for g > 1 in the long-time limit the transport behaviour is non-dispersive.
This is related to the existence of a finite mean waiting-time t = g/(g – 1). It follows
that around g = 1 there is a crossover from dispersive to non-dispersive behaviour [13], as is
also confirmed experimentally.
From the preceding discussion it is obvious that the transport of charge carriers is
much influenced by the waiting-time distributions (WTD) between hops. Physically the
WTD arise from the local disorder in the sample. For large disorder the WTD are not exponential
but decay much more slowly. In line with the preceding arguments one takes for the
WTD expressions c (t) which behave at long times algebraically [7, 12]
…t† t 1 ; …5†
where 0 < g < 1. This choice for c (t) reproduces Eqs. 2–4, see below. Equation 5 is not valid
near the time-origin. Therefore in calculations one prefers to work with functions welldefined
for t 6 0. A suitable choice is for instance the series
…t† ˆ1 a
a
X 1
nˆ1
a n b n exp … b n t† …6†
with a < 1 and b < a –1 [8]. This function is everywhere continuous and finite, and for purely
imaginary t it turns into the Weierstrass function. One verifies readily that for large t Eq. 6
obeys Eq. 5 and that the corresponding g in Eq. 5 is g =lna/ln b. Equation 6 is very useful,
since it allows to vary g freely by a judicious choice of a and b. Another choice for c (t)
which is continuous for t 6 0is
…t† ˆ…1 ‡ t† :
…7†
Here again the long-time behaviour follows Eq. 5.
We now recall the basic ingredients of random walks in continuous time, the so-called
CTRW [6–8]. Let c (r, t) be the probability distribution of making a step of length r in the
time interval t to t +dt. The total transition probability in this time interval is
…t† ˆX
r
…r;t†:
…8†
Furthermore the survival probability at the initial site is
…t† ˆ1
R t
0
…† d;
…9†
so that, switching to the Laplace space (t ? u), one has
…u† ˆ‰1 …u†Š=u: …10†
33

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
The probability density Z(r,t) of just arriving at r in the time interval t to t +dt obeys
the iterative relation
…r;t†ˆP
r 0
R t
0
…r 0 ;† …r r 0 ;t † d ‡ …t† r;0 ; …11†
in which the initial condition of starting at t = 0 from r = 0 is incorporated. One then has
for the probability P (r,t), that the particle is at r at time t,
P …r;t†ˆRt
…r;t † …† d: …12†
0
Now P (r, t) also obeys an iterative relation:
P …r;t†ˆX
r 0
R t
0
P …r 0 ;† …r r 0 ;t † d ‡ …t† r;0 …13†
as may be seen either by inspection or, more formally, by using Eqs. 11 and 12. Clearly, a
description of such convolutions, Eqs. 11–13, is more compact in Fourier-Laplace space.
For P (k, u) one has from Eq. 13:
P …k;u†ˆP …k;u† …k;u†‡ …u†
…14†
with the immediate solution
P …k;u†ˆ …u†=‰ 1 …k;u† Š ˆ 1 …u†
u
1
1 …k;u† : …15†
The last expression generalizes the usual diffusion relation to random media, in which
spatial and temporal aspects are coupled through c (k,u). The analysis is much simplified if
such aspects decouple, which is the case for instance when the disorder is mainly energetic
and the random walker moves over a rather regular lattice. In the decoupled case:
…r;t†ˆ …r† …t† :
…16†
From this it follows immediately that also c (k,u) =l(k) c (u) is decoupled. In the decoupled
scheme P (k, u) takes the form
P …k;u†ˆ1 …u†
u
1
1 …k† …u† ; …17†
where l (k) = P qp(q)e –ik7q is the structure function of the infinite lattice and p (q) denotes
the probability that a step extends over the distance q.
34

3.2 Photoconductivity
Coupling is very important for superlinear behaviour. Thus for the c (r,t) given by
Eq. 16 the mean-squared displacement is either divergent or increases sublinearly or at most
linearly in time. In order to obtain finite Ar 2 (t)S with a superlinear temporal behaviour,
coupled c (r,t) forms have to be used [4]. An example is the WTD
…r;t†ˆAr …r t †; …18†
in which the d-function couples r and t. This WTD leads to Lévy walks.
Let us now focus on the mean-squared displacement. Evidently, one has
r 2 R
…t† ˆ r 2 P …r;t† dr ˆ rk 2 P …k;t† j kˆ0
…19†
from which, in the decoupled scheme, using Eqs. 5 and 17 it follows that in the absence of
any bias Eq. 2 is fulfilled. The procedure is more readily examplified by calculating the current
from the mean displacement of the carrier Ar(t)S in a biasing field [8]. Setting L –1 for
the inverse Laplace transform it follows:
P
r…t† ˆ rP …r;t†ˆir k P …k;t† kˆ0 ˆ L 1 … ir k P …k;u† j kˆ0 †
r
ˆ L 1
@P…k;u†
@
ir k …k† j kˆ0
ˆ1
ˆ L 1 …u†
u ‰ 1 …u† Š
q ;
where in the last line we work in the decoupled scheme and AqS = P q qp (q) is the mean displacement
per hop. In the presence of a bias AqS 0 0. Thus, setting |AqS| = 1, we have for the
current I (t) in an infinite lattice of any dimension,
I …t† ˆ d
r…t†
dt ˆ L 1 …u†
: …21†
1 …u†
For a finite chain of N sites Eq. 21 takes in the Laplace space the form [3, 13]:
I…u† ˆXN
nˆ1
‰ …u† Š n ˆ
…u†
1 …u†
…20†
1 ‰ …u† Š N : …22†
It is precisely this function which was used in Refs. [13, 14] to analyse, together with
the WTD Eq. 6, the time-of-flight currents of photoconductive carriers in polysiloxanes with
pendant carbazole groups. In this work one achieved with g = 0.58 a good agreement with
the experimental findings. The short time behaviour indeed obeys Eq. 3, whereas at long
times the form of Eq. 4 is reproduced fairly well.
35

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
3.3 The Matheron-de-Marsily model
As mentioned in the introduction, superdiffusive (enhanced) behaviour is often found in turbulent
flows. In this Section we adopt a picture different from CTRW. We focus on the influence
of randomly distributed biasing external fields and follow the description of Oshanin
and Blumen [11]. We take a three-dimensional (3D) solvent for which the flow is parallel to
the Y-axis. The direction and magnitude of the flow depend only on the X coordinate of the
position vector [9] so that V Y , the non-vanishing component of the flow field, obeys:
V Y …X;Y;Z†ˆVX ‰ Š: …23†
Here furthermore V[X] is a random function of X. Geometrically the system consists
of parallel layers perpendicular to the X-axis. In each layer the value of V Y is constant but
varies from layer to layer [9].
We assume the random function V[X] of Eq. 23 to be Gaussian with zero mean,
AV[X]S = 0, and with the covariance
VX ‰ 1 ŠVX ‰ 2 Š ˆ … j X1 X 2 j†: …24†
Here the brackets denote configurational averages, which are conveniently expressed
through Fourier integrals,
… jX 1 X 2 j† ˆ R1 1
dwQ…w† exp ‰ iw…X 1 X 2 † Š: …25†
Now many possibilities for Q(w) can be envisaged. For simplicity we take here only a
flat spectrum, Q(w) =W/2p, as in the original Matheron-de-Marsily (MdM) model [9] in
which the flows are delta-correlated,
… jX 1 X 2 j† ˆ W …X 1 X 2 †: …26†
We start from the Langevin dynamics of a single spherical bead subject to the MdM flow.
This allows us to display enhanced (superlinear) diffusion in a simple situation [9, 10, 22]. The
study of the dynamics of Rouse polymers in such flows is deferred to the next Section.
Let R(t) be the position of the center of mass of the bead at time t, and we assume
that R(0) = 0. The components X(t), Y(t) and Z(t) ofR(t) obey the following Langevin equations
m d2 X
dt 2 ˆ dX
dt ‡ f X…t† ;
m d2 Z
dt 2 ˆ dZ
dt ‡ f Z…t† ;
…27†
…28†
36

3.3 The Matheron-de-Marsily model
m d2 Y
dt 2 ˆ dY
dt
VX ‰ Š
‡ f Y …t† : …29†
Here m denotes the mass of the bead and z the friction constant. The terms f X (t), f Y (t),
and f Z (t) give the random (thermal-noise) forces exerted on the bead by the solvent molecules.
These forces are Gaussian, with the moments
f i …t† ˆ0
…30†
and
f i …t† f j …t 0 †ˆ2T i;j …t t 0 † ; …31†
where i,j B {X,Y,Z}. The dash stands for thermal averaging, d i,j is the Kronecker-delta and
the temperature T is measured in units of the Boltzmann constant k B .
Conventionally the acceleration terms in Eqs. 27–29 are neglected, since they are
small relative to the other terms [15]. This leads to
dX
dt ˆ f X…t† ;
dZ
dt ˆ f Z…t† ;
dY
dt ˆ V‰X…t†Š ‡ f Y…t† :
…32†
…33†
…34†
Note that in Eq. 34 the X and Y coordinates are coupled. Equations 32 and 33 are
readily solved,
X…t† ˆ 1 Rt
df X …† ;
Z…t† ˆ 1 Rt
df Z …† :
0
0
…35†
…36†
Thus the bead undergoes a conventional diffusive motion between the layers (along
the X-axis) and in the Z direction. One sees it readily by evaluating, say:
X 2 …t† ˆ 2Rt R
d t
1 d 2 f X … 1 † f X … 2 †ˆ2 1 T Rt R
d t
1 d 2 … 1 2 †ˆ2…T=† t; …37†
0 0
0 0
so that X 2 …t† ˆ2D 1 t, where D 1 = T/z is the diffusion coefficient of a single bead.
37

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
A similar procedure can be performed with respect to the solution of Eq. 34,
Y…t† ˆRt
0
dV‰ X…† Š‡ 1 Rt
df Y …† :
0
…38†
The averaging involves now both, the thermal noise and the configurational disorder:
Y 2 …t† ˆ 2 Rt R t
d 1 d 2 f Y … 1 † f Y … 2 †‡ Rt R t
d 1
0
0
0
0
d 2 V‰X… 1 †Š V‰X… 2 †Š
ˆ 2D 1 t ‡…W=2† Rt R t R 1
d 1 d 2 dw exp ‰iw…X… 1 † X… 2 ††Š; …39†
1
0
0
where in the last line use was made of the representation Eqs. 25 and 26 with the flat spectrum
of the delta-function. The remaining average on the rhs. of Eq. 39 is readily evaluated
by remembering that it is the characteristic functional of the Brownian trajectory
… 1 ; 2 ; w† exp‰iw…X… 1 † X… 2 ††Š ˆ exp w 2 D 1 j 1 2 j
: …40†
Inserting Eq. 40 into Eq. 39 one recovers the following result for the average squared
displacement (ASD) in the direction of the flow field,
Y 2 …t† ˆ 2D1 t ‡ 4W 3
t 3 1=2
: …41†
p D 1
One should remark that at times greater than t c =9pD 1 3 /(4W 2 ) the superlinear growth
hY 2 …t†i t 3=2 in Eq. 41 dominates and the first, diffusive term can be neglected. One has
then a superdiffusive behaviour with an exponent of g = 3/2 in Eq. 1 [9].
We now turn to the analysis of the behaviour of polymers in MdM flow fields.
3.4 Polymer chains in MdM flow fields
The conformational properties and the dynamics of polymers in solutions under various
types of flows have been a subject of considerable interest within the last decades. Much
progress has been gained in the explanation of experimental data for systems in which the
flow velocities are given functions in space and time, see Refs. [16–19]. On the other hand,
the behaviour of polymers in random flows is less understood. In recent works [11] we
(Oshanin and Blumen) succeeded in establishing analytically the behaviour of Rouse polymers
[20] in MdM flow fields. The presentation here follows closely Ref. [11].
38

3.4 Polymer chains in MdM flow fields
In the Rouse model N monomers (beads) are coupled to each other via harmonic
springs [16, 17, 20]. As is well-known, the forces are of entropic origin. It is customary to
revert to a continuous picture in which n, the bead’s running number, takes real values. For a
detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a
polymer in the MdM flow field are
@X n…t†
@t
@Z n…t†
@t
@Y n…t†
@t
ˆ K @2 X n …t†
@n 2 ‡ f x …n; t† ; …42†
ˆ K @2 Z n …t†
@n 2 ‡ f Z …n; t† ; …43†
ˆ K @2 Y n …t†
@n 2 ‡ V‰X N …t†Š ‡ f Y …n; t† ; …44†
see Ref. [11]. Equations 42–44 are the generalization of Eqs. 27–29 to polymers. They are to
be solved subject to the Rouse boundary conditions at the chain’s ends, n = 0 and n = N [17]:
@X n …t†
@n
ˆ @Y n…t†
@n
ˆ @Z n…t†
@n
ˆ 0 :
…45†
As before, the fluctuating forces on the rhs. of Eqs. 42–44 are Gaussian and also
delta-correlated with respect to the running index [11, 17]. For the X and Z components,
which are not subject to the flow, the procedure is standard [17]. Say, for the averaged X
component of the end-to-end vector one has
P 2 X …t† ˆ…X 0…t† X N …t†† 2 ˆ b2 N
; …46†
3
where b is the so-called persistence length.
Furthermore, the X component of the radius of gyration is
X 2 g ˆ 1
2N 2 Z N
0
Z N
0
dndm …X n …t†
X m …t†† 2 ˆ b2 N
18 : …47†
For isotropic situations (in the absence of flow fields) the end-to-end vector P R and
the radius of gyration R g of the polymer are related to P X and X g through P R 2 =3P X 2 and
R g 2 =3X g 2 . Because of the anisotropy one has to consider in the MdM model the different
components separately.
The dynamics of a flexible polymer chain is richer than that of a single bead. In the
Rouse model the dynamics of X n (t) depends essentially on the time of observation t and on
the Rouse time t R ,
R ˆ b2 N 2
3p 2 T : …48†
39

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
Now t R is the largest internal relaxation time of the chain [17, 20]. Exemplarily for
t P t R one finds for the mean-squared displacement of a bead of the chain, say the zeroth one,
…X 0 …t† X 0 …0†† 2 ˆ 2b D 1=2
1t
: …49†
3p
Equation 49 is subdiffusive with g = 1/2 in Eq. 1. This is due to the fact that the trajectory
of a bead in a chain is spatially confined by its neighbours. In the limit t p t R the
chain diffuses as one entity and the bead’s trajectory follows mainly the motion of the
chain’s center of mass. The chain’s center of mass obeys
X 2 …t† ˆ2D R …N† t
…50†
with D R (N) =D 1 /N { T/(zN).
Let us now turn to the Y component of the center of mass for which Eq. 44 leads readily
to
Y…t† 1 N
so that we obtain
Z N
0
dnY n …t† ˆ 1 Z t
d
N
0
Z N
0
dn VX ‰ n …† Š‡ 1 f Y …n; † ; …51†
Y 2 …t† ˆ 2DR …N† t ‡ W Z t Z t Z 1
d 1 d 2
2N
0 0 1
dwg…w; 1 ; 2 †;
…52†
where g(w;t 1 ,t 2 ) denotes the dynamic structure factor of the chain,
g…w; 1 ; 2 †ˆ 1
N
Z N Z N
0
0
dn dm exp‰iw…X n … 1 † X m … 2 ††Š : …53†
It turns out [11], that all these integrations can be performed analytically. In the longtime
limit t p t R one finds
Y 2 …t† ˆ 2DR …N† t ‡ 4W 3
whereas for t P t R the short-time behaviour is given by
40
N 1=2
t 3=2
1 O…t 1=2
† ; …54†
pT
p
Y 2 …t† 2DR …N† t ‡ 3 W
bN 1=2 t2 1 O…t 1=4
† : …55†

3.4 Polymer chains in MdM flow fields
The interpretation of Eq. 54 is that at long times the t 3/2 dynamics dominates the picture;
the Rouse chain behaves like a compact bead. At short times the term t 2 may become
important. This g = 2 case in Eq. 1 is called ballistic; at very short times the center of mass
of the chain hardly moves and it practically does not change the flow pattern to which it is
subjected.
We remark that at short times the motion of the center of mass of the chain and
the motion of a tagged bead are characterized by different dependences on time. The
mean-squared displacement along the Y-axis of a tagged bead, say the zeroth one, grows
in time as
Y0 2…t†
e2DR …N† t ‡ W 1=4
D
b1=2 1
t 7=4 ; …56†
see Ref. [11]. This g = 7/4 dependence is, of course, related to the fact that the segmental
motion at short times is confined, the number of distinct flow layers visited by the bead
growing as t 1/4 . Results of such fractal-type behaviour may be formulated exactly [21, 22].
We now turn to the question of the elongation of the Rouse chain in the MdM flow
and sketch the evaluation of the end-to-end distance along the Y-axis [11].
The solution of Eq. 44 under the boundary conditions Eq. 45 has the form of a Fourier
series,
Y n …t† ˆY…0;t†‡2 X1
pˆ1
cos ppn
Y…p; t† ;
N
where the Y(p,t), p = 0, 1, …, denote the normal coordinates [17]. At t = 0 the chain is assumed
to be in thermal equilibrium, i. e. to have a Gaussian conformation. This can be accounted for
automatically by stipulating it to be subject to the thermal fluctuations since t =–?. Furthermore,
the MdM flow fields are switched on at t = 0. This leads to
…57†
Y…p; t† ˆ 1 Rt 1
d exp… p 2 …t †= R † ~ f Y …p; †
‡ 1 N
Z t
0
d
Z N
0
dn cos ppn
V‰X n …t †Š exp… p 2 = R † ; …58†
N
where the functions ~ f Y denote the Fourier components of the thermal fluctuations [17]. The
Y component of the end-to-end vector follows now from:
P Y …t† ˆY 0 …t†
Y N …t† ˆ2 P1
… 1 … 1† p †Y…p; t† : …59†
pˆ1
After performing the averaging over both, the thermal fluctuations and the realizations
of the flow fields, one obtains from Eqs. 58 and 59 in the long-time limit the equilibrium
value of the end-to-end vector for delta-correlated flows [11],
41

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
P 2 Y …1† b 2
N ˆ 1 ‡ CW2 bN 5=2
3
T 2 ; …60†
where C is a constant. Therefore, a Rouse polymer stretches along the Y-axis under MdM
flows, taking the form of a prolate ellipsoid. The correction term to the Gaussian behaviour
grows as N 7/2 .
It is now interesting to confront this finding with the situation in simple shear flows,
for which the result is [18, 19]:
P 2 Y …t† ˆb2 N
3
1 ‡ 3
6480
_ 2 2 b 4 N 4
T 2
: …61†
In Eq. 61 _ stands for the (constant) shear rate. Because all flow lines point now in
the same direction, the correction term shows a stronger N-dependence and obeys a N 5 law.
3.5 Conclusions
In this review several situations were presented, which lead naturally to the appearance of
anomalous diffusion. Besides the already well-discussed subdiffusive behaviour, which is often
seen in disordered media, we also considered superdiffusive dynamics, such as encountered
in layered random flows. Anomalous diffusion was analysed through several models
and we focussed on the behaviour of polymeric materials under such conditions. The basic
aspect underlying these phenomena is dynamical scaling, which is often encountered experimentally
and theoretically.
On the other hand not all systems scale with time, and care is required in applying the
models presented here; one has to be aware of intrinsic limitations of the scaling range (e. g.
size limitations, other temporal scales involved). Because of this, close cooperations between
experimentalists and theoreticians are much needed for the analysis of systems similar to the
ones described here. Furthermore, despite the success of the now-closing Collaborative Research
Center 213, much work still remains to be done.
Acknowledgements
The research collaboration with Prof. D. Haarer, Prof. J. Klafter, Dr. G. Oshanin, Dr. H.
Schnörer, and Dr. G. Zumofen in our joint work reviewed here was always very helpful and
pleasant. The support of the Deutsche Forschungsgemeinschaft through the Sonder-
42

References
forschungsbereich 213 and Sonderforschungsbereich 60 was fundamental for the whole project.
Additional help was provided by the Fonds der Chemischen Industrie and – in the later
stages – by the PROCOPE-Program of the DAAD.
References
1. A. Blumen, J. Klafter, G. Zumofen: in: I. Zschokke (ed.): Optical Spectroscopy of Glasses, Reidel,
Dordrecht, p. 199 (1986)
2. D. Haarer, A. Blumen: Angew. Chem. Int. Ed. Engl., 27, 1210 (1988)
3. A. Blumen, H. Schnörer: Angew. Chem. Int. Ed. Engl., 29, 113 (1990)
4. G. Zumofen, J. Klafter, A. Blumen: in: R. Richert, A. Blumen. (eds.): Disorder Effects on Relaxational
Processes: Glasses, Polymers, Proteins, Springer, Berlin, p. 251 (1994)
5. J. Klafter, G. Zumofen, A. Blumen: J. Phys., A24, 4835 (1991)
6. E.W. Montroll, G.H. Weiss: J. Math. Phys., 6, 167 (1965)
7. H. Scher, M. Lax: Phys. Rev. B, 7, 4491; 4502 (1973)
8. M.F. Shlesinger: J.Stat. Phys., 36, 639 (1984)
9. G. Matheron, G. de Marsily: Water Resour. Res., 16, 901 (1980)
10. G. Zumofen, J. Klafter, A. Blumen: Phys. Rev. A, 42, 4601 (1990)
11. G. Oshanin, A. Blumen: Macromol. Theory Simul. 4, 87 (1995); G. Oshanin, A. Blumen: Phys.
Rev. E, 49, 4185 (1994)
12. H. Scher, E.W. Montroll: Phys. Rev. B, 12, 2455 (1975)
13. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58, 101 (1988)
14. D. Haarer, H. Schnörer, A. Blumen: Dynamical Processes in Condensed Molecular Systems, in: J.
Klafter, J. Jortner, A. Blumen (eds.): World Scientific, Singapore, p. 107 (1989)
15. M. Fixman: J. Chem. Phys., 42, 3831 (1965)
16. P.G. de Gennes: in: Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, N.Y., (1979)
17. M. Doi, S.F. Edwards: in: The Theory of Polymer Dynamics, Oxford Univ. Press, Oxford, (1986)
18. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager: in: Dynamics of Polymeric Liquids, Vol.2,
2nd Ed. Wiley, New York, (1987)
19. W. Carl, W. Bruns: Macromol. Theory Simul., 3, 295 (1994)
20. P.E. Rouse: J. Chem. Phys., 21, 1273 (1953)
21. J.-U. Sommer, A. Blumen: Croat. Chem. Acta, 69, 793 (1996)
22. G. Zumofen, J. Klafter, A. Blumen: J.Stat. Phys., 65, 991 (1991)
43

4 Low-Temperature Heat Release, Sound Velocity and
Attenuation, Specific Heat and Thermal Conductivity
in Polymers
Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz, Junyun Li,
and Frank Pobell
We have measured the long time (t =5 h to 200 h) heat release of polymethylmethacrylate
(PMMA) and polystyrene (PS) at 0.070 K ^ T ^ 0.300 K. After cooling from a temperature
(the charging temperature) of 80 K the heat release in PMMA shows a t –1 -dependence
in the measured time and temperature ranges in agreement with the tunneling model. In contrast,
for PS we observe strong deviations from a t –1 -dependence and a heat release smaller
than in PMMA in by a factor of ten, in apparent contradiction to specific heat and thermal
conductivity data for PS.
To compare the heat release with other low-temperature properties and to verify the consistency
of the tunneling model we have measured also the acoustical properties (sound velocity
and attenuation), the specific heat and the thermal conductivity of PMMA and PS in the
temperature ranges 0.070 K ^ T ^ 100 K, 0.070 K ^ T ^ 0.200 K and 0.3 K ^ T ^ 4K,
respectively. We show that the anomalous time dependence of the heat release of PS is due to
the thermally activated relaxation of energy states with excitation energies above 15 K.
4.1 Introduction
A disordered material releases heat after cooling it from an equilibrium or charging temperature
T 1 to a measuring temperature T 0 [1]. This heat release _Q (T 1 ,T 0 , t) depends on the
charging temperature T 1 as well as on the temperature T 0 , at which the measurement is performed,
and the elapsed time during and after cooling [2, 3]. The time-dependent heat release,
observed in several disordered systems, is a consequence of the long-time relaxation
of the low-energy excitations, identified as two-level tunneling systems (TS) [4–6]. As a
consequence of the finite relaxation time of the TS through their interaction with thermal
phonons [7] the specific heat depends also on the time scale of the experiment [5], as first
measured by Zimmermann and Weber [1].
In recent publications [5, 6] it was shown that within the tunneling model we can
quantitatively understand the observed temperature and time dependence of the specific heat
44 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

4.1 Introduction
and heat release of vitreous silica (SiO 2 ) over ten orders of magnitude in time. It has also
been pointed out that the measurements of the heat release at workable long times is probably
the only feasible method to obtain at least approximately the low-energy limit of the
tunnel splitting D 0 of the distribution function of TS. This low-energy limit is considered in
the literature by the cut-off parameter u min =(D 0 /E) min of the distribution function (E is the
energy splitting of the TS) and is introduced to keep the number of TS finite, avoiding divergences
in the calculated properties.
The interpretation of the heat release data in terms of the tunneling model is a difficult
task due to the not well-known:
a) temperature and time dependence of the specific heat due to TS at T >3K;
b) influence of relaxation processes of the TS at T > 3 K other than one-phonon tunneling
relaxation, i. e. high-order phonon tunneling and thermally activated relaxation;
c) influence of the cooling procedure.
In a recently published paper Parshin and Sahling [8] showed the complexity of the interpretation
of the heat release data when thermally activated relaxation is taken into account
within the framework of the soft potential model. Further theoretical work on the residual
properties of two-level systems and its dependence on the cooling procedure has been published
by Brey and Prados [9].
In this paper we present a further example of the complexity in the interpretation of
the heat release data and an experimental proof of the influence of the relaxation, probably
by thermally activated processes, of excited states that contribute to the heat release of the
TS at low temperatures. We have studied the long-time heat release of two amorphous polymers
with similar low-temperature specific heats and thermal conductivities, cooled under
similar conditions. In spite of those similarities we have found a large difference in the absolute
value and in the time dependence of the heat release between the two polymers when
cooled from temperatures above 15 K. The similarities and differences in the low-temperature
properties, their interpretation within the tunneling model, and the influence of thermally
activated relaxation are the main scope of this work. Preliminary results were published
in Ref. [10].
The paper is organized in five sections. In Section 4.2 we describe the phenomenological
theory for the heat release, based on the tunneling model and the influence of different
relaxation rates, and the cooling process. In Section 4.3 we briefly describe the experimental
procedures and samples. In Section 4.4 we show and discuss the experimental results. Conclusions
are drawn in Section 4.5.
45

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.2 Phenomenological theory for the heat release
4.2.1 Generalities
New theoretical work for the calculation of the heat release within the soft potential model
has been published recently [8]. In order to simplify the calculations, to minimize the number
of free parameters, and to assure a more transparent interpretation of the results we
decided, however, to interpret our results in terms of the standard tunneling model. According
to Ref. [8] and to our numerical calculations the differences between the soft potential
and standard tunneling model are not significant at the low temperatures of our measurements.
The standard tunneling model, including a thermally activated relaxation of the twolevel
systems, was used successfully to interpret the acoustical properties of vitreous silica
in eight orders of magnitude in phonon frequency from approximately 0.1 K up to room
temperature [11].
For a system of N two-level systems with energy difference E, the difference in the population
of the two levels at a given temperature T and at thermal equilibrium is given by
n 0 ˆ N tanh…E=2k B T† :
…1†
If the thermodynamic equilibrium of the system is slightly perturbed, e. g. the system
is rapidly cooled to a temperature T, the dynamical behaviour of the population difference
can be calculated according to the relaxation time approximation formula:
d…n…t†
n 0 …T††
dt
ˆ n…t† n 0 …T†
…E;T†
; …2†
where t(E,T) is the relaxation time of a tunneling system with energy E at a temperature T.
The heat released by the N two-level systems after the temperature change is given by
_Q ˆ _nE=2 :
…3†
The problem of calculating _Q simplifies to calculating n from Eq. 2. However, the
cooling process has to be taken into account. In this case Eq. 2 must be rewritten as
_n ˆ @n 0
@T
dT
dt
n n 0
: …4†
Equation 4 describes the response of two-level systems during a temperature change
given by the function T(t). In the general case Eq. 4 has to be solved numerically since n 0
and t are temperature-dependent variables.
46

4.2 Phenomenological theory for the heat release
4.2.2 The standard tunneling model with infinite cooling rate
If N two-level systems are in thermal equilibrium at a temperature T 1 and they are cooled to T 0
with infinite cooling rate, i. e. the time to cool the sample from T 1 to T 0 is zero, from Eq. 2 we
obtain
n…t† ˆ…n 0 …T 1 † n 0 …T 0 †† exp… t= …E;T 0 †† ‡ n 0 …T 0 † : …5†
The standard tunneling model assumes that at low temperatures (T < 2 K) the
one-phonon process is the dominating mechanism. The one-phonon relaxation rate is given
by
1
p ˆ 1
pm u2
…6†
with
1
pm ˆ AE3 coth…E=2K B T† ;
…7†
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 and transversal
phonon branches, r is the mass density, g l and g t are the coupling constants between phonons
and TS, and u=D 0 /E (D 0 is the tunneling splitting). For symmetrical TS, i. e. D 0 = E (u = 1),
t p reaches its minimum value t pm .
Furthermore, the standard tunneling model assumes that the distribution function of
TS is constant in terms of two independent variables, namely the asymmetry D and the tunneling
parameter l, i.e.
P…;†dd ˆ Pdd:
…8†
According Ref. [7] the distribution function can be written in terms of the variables E
and u as
P…E;u† ˆ P=u…1 u 2 † 1=2 : …9†
It is also convenient to have the distribution function in terms of the asymmetry and
the barrier height V between the potential wells. Following the work of Tielbürger et al. [11]
and assuming two well-defined harmonic potentials, it can be shown that in a first approximation
l = V/E 0 where E 0 represents the zero-point energy. In this case the distribution function
is
Pdd ˆ
P
E 0
ddV :
…10†
Replacing the total number N of TS with the integrals in E and u, the heat release is
given by
47

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
_Q ˆ PV Z 1 Z
E
E
1
dEE tanh tanh
E 0 2k B T 0 2k B T 1
0
u min
du
p
u 1 u 2
1 …T 0 † exp… t=…T 0 †† ;
(11)
where V is the sample volume and u min is the cut-off in the distribution function at u ?0.
Figure 4.1 shows the heat release at T 0 = 0.1 K as a function of time for different
charging temperatures T 1 following Eq. 11 and taking into account one-phonon relaxation
rate, Eqs. 6 and 7. The calculation was performed with parameters appropriated for vitreous
silica, Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 and u min =5610 –8 [5]. We observe that
at short times and small T 1 the heat release _Q ! t –1 T 2 1. This dependence follows from the
logarithmic time dependence of the specific heat and holds for tPt m /u 2 min. In this limit the
heat release follows the often used approximation from Eq. 11 [1]:
_Q
p2
24 k2 B PV…T 2 1 T 2 0 † 1 t ; …12†
where _Q and P are measured in W and (Jg) –1 . We recognize in Fig. 4.1, however, that at
longer times the theory deviates from the t –1 -dependence. This deviation comes from the
exponential term in Eq. 11 that decreases strongly at long times and is determined by the
product Au 2 min. The smaller this product the larger is the time-range where the approximation
given by Eq. 12 holds. The influence of the cut-off u min can be recognized comparing the
calculated heat release in Fig. 4.1 with that in Fig. 4.2 (dashed lines) calculated with a
smaller u min .
In Fig. 4.1 we note also that at large times the heat release becomes independent of
T 1 . This feature was recognized experimentally in Refs. [2, 3] and was discussed in Ref. [6].
This T 1 -independence within the assumptions described above is a direct consequence of the
finite number of TS given by the cut-off u min . We should note, however, that in SiO 2 a saturation
of the heat release for large charging temperatures (T 1 > 20 K) is observed and still
a t –1 -dependence was measured [2]. This result cannot be explained within the standard tun-
Figure 4.1: Heat release as a function of time t (in s) according to the standard tunneling model and infinite
cooling rate with Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 and u min =5610 –8 at a measuring
temperature of 0.1 K and at different charging temperatures T 1 , bottom: 5 K, top: 80 K.
48

4.2 Phenomenological theory for the heat release
Figure 4.2: Heat release as a function of time t (in s) within the standard tunneling model and infinite
cooling rate. The parameters are the same as in the previous figure but with u min =5610 –9 (dashed
lines) and an additional constant (energy-independent) relaxation time t TA =10 3 s (continuous lines).
neling model. The reason for this discrepancy is the relaxation rate, which is different from
the one-phonon process at higher temperatures and the influence of a finite cooling rate (see
below).
4.2.3 Influence of higher-order tunneling processes and a finite cooling rate
At temperatures above 3 K tunneling processes of higher-order are, in principle, possible. One
particular high-order process was studied theoretically in Ref. [12] and is similar to the optical
Raman process; the relaxation time of the TS is given by an interaction that involves two phonons.
Following Ref. [12] we added this Raman relaxation rate to the one-phonon relaxation
rate (Eq. 6). Following Ref. [12] a new free parameter, i. e. the coupling constant R for the Raman
process, is assumed. Due to its strong temperature dependence (T 7 ), the Raman process
can influence the population difference of the energy states of the TS mainly at high temperatures
(T > 5 K) during the cooling process. We should note that for infinite cooling rate and for
the temperature range of our measurements (T 0^1 K) the contribution of the Raman process
to the time and temperature dependence of the heat release is negligible. This follows from Eq. 5
where only the relaxation of the TS at the measurement temperature T 0 enters.
In order to take into account the cooling process we used the algorithm explained below.
The zero-time point is chosen as the time when the cooling process is started. The function
n (E,u,t) represents the difference in population of the TS energy states at any time t
and n 0 (E,T) means this difference in equilibrium (t ? ?) at a temperature T. Analog to
Eq. 11 the heat release can be written as:
_Q ˆ 1
2 PV
Z 1
0
dEE
Z 1
u min
du
p 1 …T 0 † e 1 …T 0 † t …n 0 …E;T 0 † n…E; u; t ˆ 0†† …13†
u 1 u 2
49

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Note that the population difference n depends on E and u because the relaxation rates,
that influence the relaxation of the TS, depend on these variables. As already described
above, to obtain the function n we need to solve the differential equation given by Eq. 4. In
general this is only possible by numerical methods.
The time dependence of the temperature during the cooling process performed in this
work can be well approximated by a linear function given by
T…t† ˆT 1 ‡ T 0 T 1
t A
: …14†
We assume that the sample is at t =0(t A )atT 1 (T 0 ); t A is the time needed to cool the
sample from T 1 to T 0 . The calculations have been done splitting the function T(t) inN steps
of time Dt (Fig. 4.3). During the time Dt the temperature remains constant and n (E,u,t)
shows an exponential behaviour given by Eq. 5; this is qualitatively shown in Fig. 4.3. To
obtain the population difference at the time Dt cooling from T 1 at t = 0 for a given E and u
we calculate iteratively the value
n…t ˆ 0 ‡ t† ˆn 0 …0 ‡ t† …n 0 …0 ‡ t† n…0† e 1 …T…0‡t††t : …15†
The calculations given in this work have been made taking into account 10 to 200
time-steps or iterations depending on the convergence of the numerical results.
Figure 4.4 shows the heat release as a function of time taking into account one-phonon
(dashed lines) and the Raman process (Rk 7
B = 100 s –1 K –7 ) with a cooling time
t A = 3000 s from T 1 to T 0 = 0.1 K. Taking into account our experimental time scale the
calculations were performed for the time interval 10 4 s ^ t ^ 10 6 s only. For one-phonon
Figure 4.3: Time dependence of the sample temperature during a cooling process and its splitting in N
steps (dashed lines) assumed for numerical calculations. Bottom: qualitative behaviour of the population
difference of the tunneling systems n 0 (T) as a function of time.
50

4.2 Phenomenological theory for the heat release
Figure 4.4: Heat release as a function of time for a finite cooling time T A = 3000 s and with (continuous
lines) or without (dashed lines) Raman processes.
process only, our calculations indicate that a finite cooling rate with t A = 3000 s to 5000 s
has a negligible influence (< 3%) on the heat release in our experimental time scale (compare
the dashed lines in Fig. 4.4 with the dashed lines in Fig. 4.2 obtained with an infinite
cooling rate).
The influence of the Raman process at T 0 = 0.1 K can be well observed if we take
into account a finite cooling rate. In comparison with the results using the one-phonon process
only and for T 1 >20K, _Q is smaller and shows a slightly different time dependence.
Note that the results for T 1 6 20 K resemble those obtained taking into account the one-phonon
process only but with a smaller charging temperature T 1 . We note also that at T 1^ 20 K
the heat release still shows a T 1 2 -dependence but reaches its saturation at a smaller T 1 in
comparison with the results with the one-phonon process only (Fig. 4.4).
We conclude that taking into account Raman processes, that influence the relaxation
rate of TS at temperatures larger than 2 K, it is possible to understand qualitatively the results
of, for example, SiO 2 where _Q reached a saturation at T & 10 K but still shows a t –1 -
dependence at t ^ 3610 5 s [2]. This result is only valid if the product Au 2 min in Eq. 7 remains
small enough.
Because of the uncertainty in the coupling constant R between TS and phonons a
quantitative comparison, however, of the heat release results with the theory taking into account
the contribution of higher-order processes is not useful. Phenomenologically and in order
to simplify the calculations, one can take into account the influence of higher-order processes
by choosing an appropriate smaller charging temperature than the real one. In that
sense charging temperatures T 1 6 10 K can be considered for comparison with theory as
free parameters. From our calculation we learn that it is practically impossible to obtain reliable
information on the density of states of TS from heat release experiments for charging
temperatures in the region of the T 1 -saturation of _Q.
51

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.2.4 The influence of a constant and thermally activated relaxation rate
There are different attempts to link the high-temperature (T > 10 K) with the low-temperature
properties of disordered solids [13, 14]. In particular, it has been proposed that the maximum
in the attenuation of phonons observed in amorphous materials at T > 10 K can be interpreted
assuming a thermally activated relaxation of the two-level systems. In this Section
we discuss the influence of thermally activated relaxation rate on the heat release for a finite
cooling rate.
For a single activation barrier V 0 between the two potential wells and at a measuring
temperature T 0 the thermally activated relaxation rate is given by
1
TA ˆ 1
0 e V 0=k B T 0
: …16†
Taking into account this relaxation rate in addition to quantum tunneling (Eq. 6) and
assuming that both processes are independent, the total relaxation rate can be written as
1 ˆ 1
P ‡ 1
TA :
…17†
Obviously, at a given temperature T 0 and for a single activation barrier we add a constant
relaxation rate to the tunneling process. Figure 4.2 shows the time dependence of the
heat release at different temperatures T 1 and using Eq. 17 with t TA =10 3 s. As expected, at
t P t TA the introduction of an additional constant relaxation rate does not influence the heat
release. At t & t TA the heat release is larger than taking only into account the one-phonon
process (Fig. 4.2). At longer times _Q decreases exponentially with time. This decrease can
be easily understood: the energy levels with long (tunneling) relaxation time have been depopulated
at t & t TA at a larger rate. Since the total amount of heat released by the TS
should be finite a decrease below the standard result (dashed lines in Fig. 4.2) is expected.
To take into account the distribution of potential barriers V and a thermally activated
rate we will follow the approach used by Tielbürger et al. [11] and transform Eq. 11 in a double
integral in the variables D and V using the approximation l = V/E 0 and D 0 & 2E 0 e –l /p:
_Q ˆ PV
2
Z 1
0
d
ZV max
0
dV E
E
tanh
E 0 2k B T 0
E
tanh 1 …T 0 † exp… t=…T 0 †† : …18†
2k B T 1
For the following discussion it is not relevant to introduce in Eq. 18 a specific distribution
of potential barriers V [11]. This is done below for comparison with experimental results
and for computing the acoustic properties. After finding the range of potential barriers
relevant for the heat release we divided the calculations in two regions: T 1 ^ 1 K and
T 1 > 1 K as described below.
In order to find the range of potential barriers relevant for the heat release in our measuring
time and temperature ranges we have calculated it following Eq. 18, splitting the V-
limits of the inner integral from V min = V max –10KtoV max , using only the one-phonon process,
and measuring temperature T 0 = 0.200 K. We have recognized that for V < 130 K and
52

4.2 Phenomenological theory for the heat release
at t&10 3 s the TS are mostly relaxed and do not contribute appreciably to the heat release
in our measured time range (10 4 s ^ t ^ 10 6 s) any more. The TS that relax through the
one-phonon process and are relevant to the heat release are found to be those with potential
barriers 130 K < V < 200 K. The contribution of tunneling systems with V > 200 K to the
heat release is negligible.
At T 1 < 1 K the contribution of a thermally activated rate for TS with potential barriers
130 K < V < 200 K is irrelevant, i. e. using t&10 –13 s (from acoustic measurements,
see below) we obtain a t TA (V = 130 K k B ) of several years. The potential barriers which are
relevant in our time and 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
given time t = 1.5610 5 s taking into account the cooling process and one-phonon process
as well as thermally activated relaxation rate. The numerical results (Fig. 4.5) show clearly a
saturation of _Q for T 1 > 5 K, i. e. at lower temperatures in comparison with the results without
thermally activated relaxation, in qualitative agreement with the results assuming Raman
processes and experimental results [2]. It is also interesting to note that a thermally activated
relaxation rate decreases slightly the exponent in the time dependence of the heat release,
i. e. _Q ! t –a with a < 1. This result is shown in Section 4.4.
Figure 4.5: Heat release at t = 1.5610 5 s as a function of charging temperature T 1 calculated with and
without thermally activated processes and a finite cooling rate.
53

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.3 Experimental details
The experimental setup for measuring the heat release and the specific heat consists of a calorimeter
on which the sample is mounted. The samples were cooled from 80 K to 0.050 K
in about (5–8)610 3 s. The calorimeter is attached to a holder through a thermal resistance
or a superconducting heat switch made of an Al strip. The holder is screwed to the mixing
chamber of a top loading dilution refrigerator.
Two sample holders were used, one from plastic and a second from Teflon. The first
one showed a heat release approximately proportional to t –1 whereas for the second one no
heat release was measured in agreement with previous measurements [2].
We used two different methods to measure the heat release. Firstly, after cooling to
low temperatures (T&0.060 K) the heat switch was opened and the time dependence of the
sample temperature (warmup rate) was measured till about 0.120 K. Then the sample was
cooled again to about 0.050 K before starting another warmup run. The heat release is obtained
from the slope of the temperature in the warmup run dT (t)/dt with the knowledge of
the specific heat of the sample plus calorimeter C(T), i. e. _Q = C(T) dT (t)/dt. The measurements
were completely automated with a personal computer. The background heat leak was
about (0.03–0.50) nW at 0.090 K and at t & 10 5 s depending on the sample holder and the
vibration of the connecting cables. To test the measuring setup and procedure we have produced
well-known heat leaks with a electrical heater, fixed to the sample.
The second method for measuring the heat release is based as before on the measurement
of the time dependence of the sample temperature at constant bath (mixing chamber)
temperature. The difference lies on the selection of a fixed thermal resistance R th between
sample and holder that enables the semiadiabatic measurement of the heat release in a time
scale larger than the intrinsic thermal relaxation time of the arrangement. Approximately
30 minutes after reaching a constant bath temperature T b the decrease of the sample temperature
T s with time provides directly the heat release, i. e. _Q =(T s – T b )/R th (T s ,T b ). The
measurement of the thermal resistance together with the background contribution to the heat
release is made in situ after reaching the time-independent minimum temperature. Within
experimental error (&10 %) both methods show the same results.
The specific heat of both polymers was measured with the heat pulse technique in semiadiabatic
fashion. The thermal conductivity was measured with a top loading 3 He refrigerator
using the standard procedure. The acoustic properties, sound velocity and attenuation, were
measured with the vibrating reed technique [16] in the frequency range (0.2–3) kHz.
Both polymers were prepared following standard procedures. The PMMA sample had
additionally 10 –2 mol% of tetra-4-tert.butyl-phthalocyamin (dye molecule) because it was
used in an early optical hole burning experiment [15]. For the specific heat and heat release
measurements the mass and density of the PMMA sample were determined as 11.97 g and
1.15 g/cm 3 , respectively. We have measured two PS samples prepared from different
batches. The densities of these samples were 1.05 g/cm 3 , and the masses 11.4 g (sample
PS1) and 38.0 g (sample PS2).
For the thermal conductivity measurements two slices were cut from the bulk samples
with length l = 1 cm, width w = 2 mm, and thickness d = 300 mm. For the acoustic measurements
the reeds had the geometry l = (0.7–1) cm, w = (0.1–0.3) cm and d = (100–300) mm.
54

4.4 Experimental results and discussion
4.4 Experimental results and discussion
4.4.1 Specific heat and thermal conductivity
Figure 4.6 shows for both polymers the specific heat devided by the temperature as a square
function of temperature. These measurements were performed only when the heat release is
negligible (t & 10 6 s). For t ?? the specific heat can be written in terms of two contributions:
c (T) =c 1 T + c 3 T 3 with coefficients for the tunneling system c 1 and phonon contribution
c 3 . From Fig. 4.6 we obtain the values c 1 = (3.0+0.3) mJ/gK 2 and c 3 = (93 + 18) mJ/gK 4
for PMMA, and c 1 = (4.6+0.5) mJ/gK 2 and c 3 = (77 +23) mJ/gK 4 for PS. The values of the
linear term are in fair agreement with those published by Stephens [17, 18] c 1 = 4.6 mJ/gK 2
for PMMA and c 1 = 5.1 mJ/gK 2 for [PS], but we obtained larger phonon contribution
(Stephens: c 3 =29mJ/gK 4 for PMMA and c 3 =45mJ/gK 4 for PS). In Figs. 4.7 and 4.8 we
compare our specific heat results with data from Ref. [18]. Within the scatter of the data it
is difficult to conclude whether the c values are really different.
Figure 4.6: Specific heat divided by the temperature as function of the squared temperature for PMMA
(a) and PS (b).
Figure 4.7: Comparison of our specific heat for PMMA with data from Ref. [18].
55

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.8: Comparison of our specific heat for PS with data from Ref. [18].
According to the tunneling model and for t ?? the specific heat due to the TS is given
by [5]:
c…T† ˆp2
6 k2 2
BPV ln… †T ˆ c 1 T:
u min
…19†
If we assume (from heat release data, see below) u min =10 –10 for PMMA and
u min =6610 –9 for PS and from the measured c 1 we obtain a density of states P = 4.0610 38
(7.5610 38 ) 1/Jg for PMMA (PS) in reasonable agreement with the values obtained in earlier
specific heat measurements [17] and also from acoustic measurements for PMMA [24].
Recent optical hole burning experiments on PMMA and PS [15] indicate that the density of
states of TS for PS is about two times larger than for PMMA in reasonable agreement with
our specific heat measurements.
Figure 4.9: Thermal conductivity as a function of temperature for PMMA and PS.
56

4.4 Experimental results and discussion
Figure 4.9 shows the thermal conductivity for PMMA and PS as a function of temperature.
For PMMA and at T < 0.7 K we obtain k =28T 1.84 610 –3 W/mK 2.84 and for PS
k =19T 1.93 610 –3 W/mK 2.93 , in agreement with earlier measurements [17, 19–22], (Figs. 4.10
and 4.11), and with measurements in epoxies [23]. According to the tunneling model and for
T < 1 K the thermal conductivity is given by:
…T† ˆk3 B v
6p 2 … P 2 † 1 T 2 ; …20†
where v is the sound velocity and r the mass density. From Eq. 20 and with the sound velocity
from Ref. [17] we obtain Pg 2 = 9.2610 5 J/m 3 (11.5610 5 J/m 3 ) for PMMA (PS). If we
replace the density of states P from the specific heat results we obtain similar values for the
coupling constant between TS and phonons in both materials g = 0.28 eV (0.27 eV) for
PMMA (PS). The results for the thermal conductivity indicate that the density of states of
TS for the PS sample is about a factor of two larger than for the PMMA sample.
Figure 4.10: Comparison of our thermal conductivity data for PMMA with earlier publications. The
dashed line has been calculated within the tunneling model with parameters taken from our internal friction
data.
Figure 4.11: Comparison of our thermal conductivity data for PS with earlier publications. The dashed
line has been calculated within the tunneling model with parameters taken from our internal friction
data.
57

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.2 Internal friction and sound velocity
Figures 4.12 and 4.13 show the internal friction of PMMA and PS. Below 1 K the internal
friction of PMMA and PS is approximately temperature-independent. According to the tunneling
model the sound absorption should be temperature-independent in the region ot p P1
(fulfilled in our temperature range) where o =2pv is the phonon frequency and t p the relaxation
time of the TS (Eq. 6). In this temperature range a simple relationship is found for
the internal friction [11, 16]
Q 1 ˆ p
2 C …21†
Figure 4.12: Squares: PMMA internal friction as a function of temperature at a frequency of 535 Hz.
Full circles: indicate the attenuation data taken from the ultrasonic measurements at 15 MHz from
Ref. [24]. Solid and dotted lines: the calculated values for v=535 Hz and v=15 MHz following the
modified tunneling model considering a thermally activated relaxation rate. For more details see text.
Figure 4.13: Squares and Circles: PS internal friction as a function of temperature at two different frequencies.
Solid line: the calculated values for v=0.24 kHz following the modified tunneling model described
in the text.
58

4.4 Experimental results and discussion
with the constant C = Pg 2 /ru 2 and the sound velocity u. Equation 21 should hold even if
higher-order processes dominate the relaxation of the TS (if the relaxation rate is proportional
to u 2 !D 0 2 , Eq. 6). From the temperature-independent value of the internal friction
and assuming no background contribution (e. g. due to the clampling) we obtain for PMMA
(PS) C % 2.6610 –4 (8.3610 –4 ). Taking into account the mass density, sound velocity, and
the coupling constant obtained from the thermal conductivity and specific heat results described
above, the values for the parameters C from the internal friction indicate that the
density of states of TS for PS is larger than for PMMA by a factor &2.5. This is slightly
larger than that obtained from the specific heat (about 1.5). This might be attributed to the
unknown clamping contribution to the measured attenuation (which is always present) or to
different u min values (Eq. 19). In Figs. 4.10 and 4.11 we show the calculated thermal conductivity
using Eq. 20 with Pg 2 values from our internal friction at the plateau. Excellent agreement
is obtained for PMMA (Fig. 4.10) but for PS the thermal conductivity is by a factor
of two smaller. This difference might be attributed partially to the background contribution
which is not subtracted from the measurement before computing the parameter C.
The behaviour of the internal friction above about 3 K is qualitatively different for
both polymers. PMMA shows a decrease in the internal friction reaching a minimum at
about 30 K and increasing monotonously to the highest temperature of our experiment,
120 K (not shown in Fig. 4.12) in very good agreement with previous work at similar frequencies
[25]. On the contrary, the internal friction of PS shows the typical temperature dependence
measured for other amorphous materials, for example SiO 2 [16]. It increases,
reaching a frequency-dependent maximum at T&37 K (;40 K), (Fig. 4.13), and increases
again at T > 60 K (70 K) at the frequency 0.24 kHz (3.2 kHz). This internal friction increase
at T > 60 K for PS together with the increase at T > 30 K for PMMA will not be discussed
here. Instead, we will discuss the behaviour of the internal friction at lower temperatures in
terms of an extension of the tunneling model following the procedure described in Ref. [11].
We assume the simplified relation for a thermally activated relaxation rate given by
Eq. 16 and add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials
and within the approximations given by Eq. 10 and with l = V/E 0 , it can be shown that the
internal friction increases linear with temperature just above the temperature-independent region
(plateau) [11]:
Q 1 ˆ pCk BT
E 0
: …22†
This increase is observed for PS, (Fig. 4.13); applying Eq. 22 to the results below
20 K we obtain a zero point energy E 0 =13+ 2 K. In the same temperature region and due
to the influence of the thermally activated relaxation the relative change of the sound velocity
can be written as [11]:
uu
u ˆ Ck BT
E 0
ln…! 0 † : …23†
A nearly linear temperature dependence of uu/u is observed in both polymers at temperatures
about below 50 K (Figs. 4.14 and 4.15). At T ~ 1 K a crossover, due to tunneling
and due to thermally activated relaxation, from the linear T-dependence to the logarithmic
59

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.14: Relative change of sound velocity at v=535 Hz for PMMA. The solid line represents the
linear temperature dependence of the sound velocity at temperatures below 20 K used for the numerical
calculations. Inset: the same data but below 10 K in a semilogarithmic scale.
Figure 4.15: Relative change of sound velocity at two different frequencies for PS. The solid line was
calculated according to the modified tunneling model described in the text. Inset: the same data but below
10 K in a semilogarithmic scale.
T-dependence can be observed for both polymers (insets in Figs. 4.14 and 4.15). Although
the tunneling model with the assumption of thermally activated relaxation of the TS provides
a reasonable fit for the linear temperature dependence, its origin is still controversial. We
note that a linear temperature dependence of the sound velocity above a few Kelvin is a
rather general behaviour observed in several amorphous [26, 27], disordered [28], and polycrystalline
metals [29]. Nava [28] argued recently against an interpretation in terms of thermally
activated relaxation of the TS for the linear T-dependence of the sound velocity. However,
new acoustical results in polycrystalline materials indicate a linear T-dependence of the
sound velocity comparable with those found in amorphous materials [29].
Applying Eq. 23 to the measurements at the two phonon frequencies for PS and with
the value of E 0 from the internal friction we obtain t 0 % (1 +0.5)610 –17 s. As discussed
in Ref. [11] the meaning of the prefactor t 0 is still unknown. The very small value of t 0
found for PS in comparison with the one for SiO 2 (t 0 %10 –13 s) should be taken as an ef-
60

4.4 Experimental results and discussion
fective value until a systematic comparison of the applied model to other experimental data
is available.
To obtain a maximum in the internal friction an upper limit of the potential height
must be assumed. Instead of a cut-off in the distribution P (D, V) and following Ref. [11], a
Gaussian distribution with a width s 0 will be assumed:
P…;V†ˆ
P
E 0
e … V2 =2 2 0 † : …24†
Figures 4.13 and 4.15 show the result of numerical calculations with only the width of
the distribution s as free parameter. The fits were obtained assuming s 0 = 1200 K and with
the values C = 8.3610 –4 , v = 0.24 kHz, t 0 =10 –17 s and E 0 = 13 K using the equations
given in Ref. [11]. A reasonable agreement with the experimental data is achieved for both
acoustic properties in the expected temperature region. The minimum obtained numerically
at T ~ 3 K (Fig. 4.13) has been discussed in Ref. [11] and is attributed to the difference in
the number of TS contributing to the relaxation process in the tunneling or thermally activated
regime.
As pointed out above, the internal friction of PMMA behaves differently from PS
above the plateau. It decreases at T > 3 K and reaches a minimum at T ~ 30 K. The predicted
linear temperature dependence of the internal friction (Eq. 22), at the measured frequency
(v = 535 Hz) is not observed for PMMA. It is tempting to interpret the decrease of
the internal friction results above 3 K assuming an upper bound of the TS density of states
around 15 K. This assumption has been indeed used to interpret the irreversible line broadening
of the optical hole burning experiments [30]. However, ultrasonic attenuation measurements
at 15 MHz for PMMA [24] show clearly a thermally activated maximum at 12 K.
Therefore we have decided to search for a set of parameters that might explain the low and
high frequency results under the assumptions described above and as done for PS.
From the sound velocity data below 20 K (Fig. 4.14) and assuming the validity of
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
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
set of parameters that explain the data. If we take the value for SiO 2 from Ref. [11],
t 0 % 10 –13 s, we obtain E 0 /k B % (20 + 5) K, a value comparable to that for SiO 2 . After several
trials, eventually we have found a set of parameters that fits reasonably the low and
high frequency measurements without the assumption of an energy cut-off.
In Fig. 4.12 we compare the experimental results of the internal friction with the numerical
calculations for frequencies at 535 Hz and at 15 MHz using E 0 /k B =10K,t 0 =10 –13 s,
and a rather small distribution width s 0 = 150 K. The model reproduces fairly well the position
of the maximum in the attenuation at 15 MHz as well as its relative value in comparison
with the plateau [24]. It is worth pointing out, that for PMMA, since the changes produced by
the free parameters E 0 , t 0 and s 0 compete with each other, it is not possible to find another set
of values which can explain the results as well as we can. It is important to note that the internal
friction data for both polymers indicate a much smaller thermally activated contribution to
the phonon attenuation for PMMA than for PS, in spite of the fact that in the quantum tunneling
regime both samples show similar results. This difference may influence the low-temperature
heat release because the number of excited states that are available after the cooling process
will be different.
61

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.3 Heat release
Figure 4.16 shows the heat release as a function of time for the 12 g PMMA sample at
T = 0.090 K and after cooling from 80 K. The heat release follows very well the predicted
t –1 law from the tunneling model (Eq. 12). The same result was obtained in different runs
with slightly different cooling rates and measuring temperatures 0.070 K^T 0 ^0.110 K.
For finite cooling rates and for charging temperatures T 1 p 1 K this independence cooling
rate is to be expected (Section 4.2.3).
As discussed in Section 4.2 the unknown higher-order relaxation processes are taken
into account through an effective charging temperature T 1 . With the tunneling systems density
of states from the specific heat and using Eq. 12 we obtain T 1 % 26 K. As stated in
Section 4.2 for charging temperatures T > 10 K it is not possible to obtain reliable values of
the density of states for TS from heat release measurements.
From the observed t –1 -dependence and using the value for the coupling constant between
TS and phonons from Ref. [24], Ak B 3 = 1.6610 9 K –3 s –1 , we obtain an upper limit for
the parameter u min

4.4 Experimental results and discussion
and (5). Even worse, the computed curve (5) with thermal activation shows much smaller
values for the heat release by almost two orders of magnitude. This result indicates that too
many energy states were depopulated during the cooling process through thermal activation.
This disagreement might be ascribed to the assumed potential distribution function (Eq. 24)
or the approximation l = V/E 0 . Curve (2), which is calculated without thermally activated relaxation,
lies only a factor of two higher than the measured one (Fig. 4.16) and is in qualitative
agreement with the acoustical data where no influence from thermal activation were taken
into account.
For PS, according to the specific heat and acoustical data (at T < 3 K) and due to a larger
density of states of tunneling systems, we expected a larger heat release than for PMMA.
Surprisingly, the opposite is observed. After cooling from 80 K with the same cooling rate,
the heat release for PS measured at the same temperature as before was at least of a factor of
ten smaller than for PMMA (Figure 4.16), and it does not follow the t –1 -dependence.
In order to understand the observed deviations we have measured the heat release of
PS at lower charging temperatures. Figure 4.17 shows the heat release as a function of time
at charging temperatures 0.5 K^T 1 ^1 K. The heat release follows very well the t –1 -dependence
as well as an increase with T 1 2 (Eq. 12, Fig. 4.18). It is interesting to note that a
fair quantitative agreement with the prediction of the tunneling model is obtained. If we calculate
the density of states of tunneling systems P from these data with Eq. 12 (straight line
in Fig. 4.18) we obtain P = 9.0610 –38 J –1 g –1 which is similar to the one obtained from the
specific heat assuming u min =6610 –9 .
Figure 4.17: Heat release as a function of time (in hours) for Polystyrene at a measuring temperature
T 0 = 0.200 K cooling from different T 1 . The straight lines have a t –1 -dependence.
At higher temperatures T 1 we observe the expected saturation of the heat release
(Fig. 4.18), however, no deviation from the t –1 -dependence has been measured within experimental
error. Deviations are observed if we charge the sample at temperatures T 1 >15K
(Fig. 4.16). In this case we have cooled the sample from 80 K and measured the heat release
at 0.090 K (sample PS1) and 0.300 K (sample PS2). According to theoretical estimates the
observed difference in the heat release between the two PS samples is not attributed to the
difference in measuring temperatures. The rather abrupt saturation of the heat release at
T 1 6 7 K is attributed to the depopulation of the excited states through thermally activated
relaxation in the cooling process (Figs. 4.5 and 4.18), as it will become clear below.
63

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.18: Heat release multiplied by the time t as a function of the difference between the squares of
charging temperature T 1 and measuring temperature T 0 for PS. The solid line follows the theoretical
prediction with a density of states of tunneling systems 1.45 larger than the one obtained from the specific
heat results.
The following experiment provides further verification that the non-simple time dependence
of the heat release for PS is due to energy states excited above 15 K and depopulation
during the cooling process, very likely through thermally activated relaxation. We have
charged the sample at 80 K for several hours and cooled it then to 1 K. We have left the sample
for 22 h at 1 K and later continued to 0.3 K, the temperature at which the heat release
was measured (Curve (1) in Fig. 4.19). We observe now a clear deviation from the t –1 law.
That means that even after 22 h at 1 K the energy states excited above 15 K have not relaxed
completely. After 60 h at 0.3 K we warmed the sample to 1 K for 17 h, cooled it to 0.3 K
and measured the heat release (Curve (2) in Fig. 4.19). The heat release follows the theoretical
t –1 -dependence very well.
We have calculated the heat release taking one-phonon,thermally activated processes,
and the experimental cooling process into account. We used the same parameters obtained
from the fit to the acoustical data (Figs. 4.13 and 4.15). With only one free parameter, u min ,
Figure 4.19: Heat release as a function of time for PS. (1): cooled from 80 K to 1 K and leaving the
sample 17 h at this temperature, T 0 = 0.300 K. (2): the sample was warmed to a charging temperature
T 1 = 1 K for 22 h and cooled again to T 0 = 0.300 K. The solid line in (1) is only a guide and the solid
line (2) has a t –1 -dependence.
64

4.5 Conclusions
we can reproduce the measured heat release of PS reasonably well (Fig. 4.16). In this figure
we present the two curves (3) and (4) using the parameter u min =10 –10 and 6610 –9 . This
fit indicates that we can observe the influence of a finite u min through the steeper decrease
of the heat release at long times. Curve (1) in Fig. 4.16 was calculated with parameters for
PS but without thermally activated relaxation.
4.5 Conclusions
We conclude that thermal conductivity, specific heat and acoustical properties at low temperatures
(T < 1 K) can be quantitatively interpreted within the tunneling model for PMMA
and PS. At higher temperatures we have measured very different temperature dependence of
the internal friction for the two polymers. At the used frequencies and at 0.1 K^T^120 K
PMMA shows no thermally activated dissipation peak. PMMA shows a minimum in the internal
friction at T ~ 30 K where PS shows a maximum. Nevertheless, within the tunneling
model and introducing a thermally activated relaxation rate as well as a Gaussian distributon
density of states of tunneling systems, it is possible to understand the acoustical properties.
From the fits we may conclude that the density of states of tunneling systems for PMMA is
restricted to smaller energies, i. e. a smaller width s 0 , than for PS.
Although the thermal conductivity, specific heat, and sound velocity temperature dependence
for both samples are similar, the heat release shows different time dependences as
well as different values in apparent contradiction with the other measured low-temperature
properties. Taking into account the internal friction results, it is tempting to correlate the differences
in the heat release between both samples to the difference in the contribution of
thermally activated processes in the relaxation of the tunneling states during the cooling process.
We have demonstrated that the deviations of the time dependence of the heat release
from the t –1 -dependence predicted by the tunneling model is due to the relaxation of energy
states excited above 15 K. The smaller values of the measured heat release of PS in comparison
with the standard tunneling model can be explained by the depopulation of energy
states relaxing mainly by thermally activated processes during the cooling process.
Without detailed knowledge of the short relaxation times at T 1 > 1 K, a quantitative
fit or even a qualitative understanding of the heat release _Q(T 1 ,T 0 ,t) data for high charging
temperatures seems to be impossible. Therefore, the heat release with high charging temperatures
can be hardly used to obtain information on the density of states of tunneling systems.
Heat release experiments at high measuring temperatures T 0 > 1 K are necessary to
obtain information on the dynamics of the tunneling systems with a relaxation rate not given
by the tunneling one-phonon process. These experiments may clarify the influence of the
two different relaxation mechanisms discussed nowadays in the literature to interpret the
low-temperature properties of amorphous solids above 1 K, namely incoherent tunneling
65

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
[31] and thermal activation within the soft potential model [8]. Both approaches have been
used with impressive success in the last years. It is, however, not yet clear if incoherent tunneling,
a mixture of the two processes (incoherent and thermal activation), or only thermal
activation is the main relaxation mechanism of the TS above 1 K.
Acknowledgements
We wish to acknowledge W. Joy for the sample preparation, and S. Hunklinger, D. Haarer
and J. Friedrich for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft
through the Sonderforschungsbereich 213. A. Nittke was supported by
the “Graduiertenkolleg Po 88/13” of the Deutsche Forschungsgemeinschaft.
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67

5 Spectral Diffusion due to Tunneling Processes
at very low Temperatures
Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer
5.1 Introduction
Pioneered by the work of Zeller and Pohl [1] it was discovered in the early 1970s that amorphous
solids show low-temperature thermal and acoustic properties which are very different
from those observed in crystals. For reviews see for example Refs. [2, 3]. The most wellknown
of these anomalous properties is the specific heat, which is in general considerably
larger than would be expected from the Debye model and varies linear with temperature in
contrast to the Debye T 3 -dependence. Other anomalies are the temperature dependence of
the thermal conductivity, the properties of phonon echoes, and ultrasonic absorption. These
features seem to be quite universal for all kinds of amorphous solids, irrespective of their
chemical composition and structure, i. e. inorganic as well as organic or polymeric.
Very soon after the first experimental evidence for these anomalous low-temperature
excitations Anderson et al. [4] and independently Phillips [5] developed a theoretical description,
called the tunneling model [4, 5]. It is based on the assumption that the low energy
degrees of freedom in amorphous solids arise from tunneling motions of atoms or groups of
atoms between local energetic minima which should exist in any disordered solid. If only
the lowest energy level of each minimum is considered, this leads to two eigenstates. For
this reason, these degrees of freedom are often referred to as two-level system (TLS). An
important consequence of this model is the formal analogy to a system of particles with
spin 1/2, which for example immediately explains the existence of phonon echoes. Another
very striking feature of the tunneling model was the prediction of a time-dependent specific
heat [4]. This was confirmed experimentally several years later [6]. Furthermore, the model
can explain the heat release [7] of amorphous solids, which is a time-dependent non-equilibrium
phenomenon. From these kinds of experiments, it could be concluded that TLS dynamics
occur on time scales extending over many orders of magnitude.
The sensitivity of optical experiments on amorphous solids was hindered, in many instances,
by the large inhomogeneous broadening which arises from the distribution of local
environments of the involved optical transitions. Therefore a very important step was the
discovery of persistent spectral hole burning in 1974 [8, 9]. This method of high-resolution
laser spectroscopy eliminates the inhomogeneous effects induced by the static disorder in
68 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

5.2 The optical cryostat
amorphous hosts. For this purpose dye molecules are embedded in the solid which change
their absorption spectra when irradiated with resonant narrow band laser light. This produces
a dip in the inhomogeneous absorption band, called a spectral hole. A narrow spectral hole
can be regarded as a highly sensitive spectral probe for any kind of distortion of the matrix,
which induces small spectral changes [10]. In this way variations of strain fields caused by
pressure changes of almost some 10 hPa can be detected [11]. Spectral holes are also sensitive
to small changes of other external parameters including electric fields [12, 13].
After observing quite a few anomalous properties of optical transitions in glasses and
attributing them to the dynamics of TLS [14], the tunneling model was adopted by Reinecke
[15] to explain the low-temperature line widths of optical transitions in amorphous solids
using the concept of spectral diffusion. This concept had originally been developed for the
description of spin resonance experiments [16] and had already been applied to the theoretical
treatment of the above mentioned ultrasonic properties of glasses [17]. Soon after this
step, the possibility of a connection between thermal and optical properties of amorphous
solids was supported by the observation of time dependence of spectral hole widths [18].
The application of the tunneling model to the description of spectroscopic properties
proposed an important link suggesting a correlation between the specific heat and the optical
line width. This implies that, besides the mentioned calorimetric and acoustic methods,
optical spectroscopy yields information about the dynamics of the same low energy excitations
of amorphous solids, which dominates the calorimetric experiments. A crucial point in
establishing this connection is the temperature dependence of these physical solid state parameters.
The tunneling model in its original form predicts linear temperature dependence for
the specific heat as well as for the optical line width. This prediction, however, is only valid
at temperatures where the contribution of phonons is of minor importance. Earlier hole burning
measurements on the temperature dependence of optical line widths [19] indicated the
necessity of expanding the temperature range accessible to optical absorption spectroscopy
to values far below 1 Kelvin in order to exclude any other mechanisms except TLS dynamics.
For this reason we have constructed a 3 He/ 4 He dilution refrigerator with optical
windows allowing transmission spectroscopy at temperatures down to 0.025 K, a temperature
regime in which only very few data from optical spectroscopy existed before [20, 21].
In addition, cooling with a 3 He/ 4 He mixture can be performed continuously for very long
times (in principle unlimited) if the necessary care in cryostat design and operation is taken.
In our experiments we have reached operation times up to three months.
5.2 The optical cryostat
In the design of a cryostat for optical spectroscopy two main problems have to be overcome.
In an absorption experiment the sample has to dissipate the absorbed energy in order to
avoid overheat during the measurement. Therefore the best possible thermal contact to the
cold stage is of extreme importance. The second problem is how to guide the laser light to
the sample without creating an intolerably large additional heat leak, for example by infrared
69

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
irradiation from the windows. Since various construction methods for optical cryostats in the
temperature range below 0.500 K did not yield the desired results, special care had to be taken
in the design of our dilution refrigerator. For example it had been reported [22] that for
samples located outside the mixing chamber temperatures below 0.300 K could not be
reached, even if the best thermal contact is realized by pressing the samples to the wall of
the mixing chamber with indium metal as contact material. In another kind of experiment a
glass fibre winding around the cold finger of a dilution refrigerator shows a significant temperature
gradient with respect to the mixing chamber below 0.100 K [20]. For these reasons,
we decided to place the sample directly into the dilute phase of the mixing chamber. This
way optimal thermal contact and a constant temperature of the sample was achieved via the
superfluid 4 He. For performing optical experiments the mixing chamber was designed as a
glass cylinder.
The optical path, used in our cryostat, is shown in Figure 5.1 [23]. Three sets of windows
are mounted on cold copper shields to eliminate most of the room temperature radiation
in three steps: at liquid nitrogen temperature (77 K), liquid helium temperature (4.2 K),
and at approximately 1 K, the latter being achieved by cooling with the pumped 3 He of the
distillation chamber. In contrast to other constructions, the vessel containing the dilution refrigerator
part of the cryostat is in our design not surrounded by the helium tank which has
the advantage that the laser beam is not scattered by boiling liquid helium. We estimate that
the total heat leak caused by room temperature radiation is below 50 nW. The minimum
temperature reached is 0.024 K. The cooling power is about 6 mW at 0.100 K. Although this
is a very low value, it is sufficient since the power of the absorbed laser light was always far
below 1 mW. The light powers used for hole burning in our experiments were on the order of
several nanowatts to several tens of nanowatts depending on the temperature. For hole detection
the power is reduced to the picowatt range. The cooling power of our cryostat and the
thermal conductivity of the involved polymers are high enough to guarantee a uniform temperature
distribution with no significant sample heating during the hole burning process.
4
He - shield mixing - chamber
LN 2 - shield 1-K - shield
sample
Figure 5.1: Alignment of optical windows and sample in the cryostat. The sample is placed inside the
mixing chamber of the dilution refrigerator. The shields are connected to the distillation chamber, the
helium tank, and the nitrogen tank, respectively. The liquid N 2 windows possess an infrared reflective
coating.
70

5.3 Theoretical considerations
The temperature is measured with a RuO 2 thick film resistor (TFR) supplied by Phillips
(type RC-01). The heat capacity and thermal relaxation time of the TFR are equivalent
to those of a carbon resistor but its thermal reproducibility during temperature cycles is better
[24]. The resistor is calibrated against several primary and secondary thermometers including
a NBS fixed point device, CMN and Ge, thermometers. Its accuracy is better than
2% in the temperature range between 4.2 K and 0.025 K. The thermometer is placed into
the dilute phase of the mixing chamber in direct thermal contact to the sample. The resistance
is measured using a four wire ac resistance bridge (AVS-46 RV Elektroniikka, Finland).
At the minimum temperature, the energy dissipated in the resistor is less than 0.5 pW.
5.3 Theoretical considerations
In the tunneling model [4] the TLS was described by two parameters (Fig. 5.2), the asymmetry
parameter D and the overlap parameter l, which contains the barrier height, the distance
of the two minima, and the mass of the tunneling particle. They are related with the total energy
splitting E of the two levels and the tunneling matrix element D 0 by the relations
q
E ˆ 2 ‡ 2 0
and 0 ˆ k exp… † ; …1†
where kO is a typical zero point energy. One of the most important ingredients of the TLS
model is the distribution function characterizing the two parameters. The originally used assumption
for this function, which has also been applied to explain spectral diffusion [15,
17], is a flat distribution in both parameters, namely
P…;†dd ˆ Pdd:
…2†
Figure 5.2: Double minimum potential as a model for a TLS. All relevant parameters are shown in the
figure.
71

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
It is convenient, however, to use experimentally accessible quantities as variables, i. e.
the energy E and a dimensionless relaxation rate R, which can be defined as: R = r/r max =
D 2 0 /E 2 . The transformation to the new variables yields for the distribution function [25]:
1
P…E;R† ˆ P p : …3†
2R 1 R
A very important feature of this distribution is the fact that P (E, R) is independent of
E. In order to keep the total number of TLS finite, some cut-off value for R ? 0 has to be
introduced [3], which is usually denoted by R min .
It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results
in spectral diffusion, which shows up in the experiment as a time and temperature-dependent
Lorentzian line broadening. The width of this Lorentzian line must be calculated by
averaging over the distribution of energies and relaxation rates P(E, R). It can be written as:
…t; T† ˆ2p2
3k C ij
Z Emax
E min
Z 1
dEn…T†
R min
dR E P…E;R†…1 exp‰ tr max RŠ† : …4†
Here AC ij S represents an average coupling constant between the TLS and the optical
transition, n (T) is a thermal occupation factor of the TLS states. For the system which is investigated
in this work, the parameter r max can be estimated from experimental data on ultrasonic
attenuation [26] to be on the order of 10 7 s –1 at a temperature of 0.100 K and even larger
at higher temperatures. Therefore, the condition t 7r max p 1 is well fulfilled on all time
scales exceeding several microseconds. In this limit the rate integration in Eq. 4 can be performed
analytically [27], which leads to the well-known logarithmic time dependence of
spectral diffusion:
!…t; T† ˆ p2
3k C
ij P k B T ln…r max t† : …5†
Equation 5 represents the theoretical prediction of the tunneling model for the time
and temperature-dependent broadening of spectral holes. This result, however, is the result
of a particular form of the density of tunneling states P (E, R), which is based on the a
priori assumption of Eq. 2. A uniform density of states in D is a physically reasonable
choice; the independence of l, however, is difficult to justify, since l consists of several
parameters [28]. The temperature dependence of spectral diffusion is dominated by D the
time evolution stems mainly from l. Therefore, these two predictions from Eq. 5 do not
have the same validity. In our experiments we have investigated time and temperature dependence
separately.
72

5.4 Temperature dependence
5.4 Temperature dependence
In these experiments, we investigated polystyrene (PS) doped with phthalocyanine and polymethylmethacrylate
(PMMA) doped with tetra-4-tert-butyl phthalocyanine. Both sample
materials have also been investigated by heat release and specific heat measurements [29].
The samples had optical densities of about 0.4 at a typical thickness of 3 mm. The samples
were prepared by bulk polymerisation of the solution of the dye in the monomer.
The observed hole shapes are Lorentzian with no detectable deviations in all cases. To
eliminate saturation effects at least 10 holes with different energies were burned at each temperature
and the homogeneous line width was determined by extrapolation to zero burning
fluence [30]. Due to their low relative depths the extrapolation was done assuming a linear
dependence of the hole area on energy [10]. A weak variation of the line width with light
power was observed at our minimum temperature of 0.025 K for PMMA. We attribute this
variation to sample heating.
The temperature dependence of the hole width is shown in Fig. 5.3 [23]. The results are:
a) both systems display a linear temperature dependence over the whole temperature range
from 0.500 K down to 0.025 K and no crossover to a constant line width is seen. The linearity
of the data plot shows that the density of states P (E, R) is indeed independent of the
TLS energy;
b) the relative magnitude of the line width in the two systems correlates with the respective
specific heat [31];
c) the extrapolated line width for T ? 0 is nearly the same for both systems, confirming
that in an amorphous system the line width reaches the lifetime limited value G h = 1/2 pT 1
(Heisenberg limit) of the electronic transition only when T ?0. This limit is reached in our
experiment within 10%!
Figure 5.3: Temperature dependence of the hole burning line width for PS and PMMA between
0.025 K and 0.500 K.
73

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
In the case of PS, where unsubstituted phthalocyanine was used as dye, the result for
the extrapolated line width G (T ? 0) is in very good agreement with the fluorescence lifetime
measurements of this molecule [32]. We are not aware of measurements of the fluorescence
lifetime of substituted phthalocyanine, but our results show that there is no major difference
between both dyes as far as their excited state lifetimes are concerned.
These results imply that in amorphous solids there are dynamical processes of twolevel
systems with an energy spectrum extending down to values as low as k B 70.025 K.
These low-energy excitations dominate the line widths of optical transitions even at low
temperatures and the lifetime limited value can only be reached by extrapolating to zero
temperature.
5.5 Time dependence
After the system has been cooled down spectral holes are burned immediately and after
some delay times, while the sample was at constant temperature, we observe a hole broadening
which depends on this delay time. This behaviour is demonstrated in Fig. 5.4. Dataset A
is the broadening of a hole which is burned immediately after the sample (phthalocyanine in
PMMA) has been cooled from liquid nitrogen temperature to 0.300 K. The curves show the
evolution of holes burned about one day, three days, and one week after cooling down. The
amount of spectral diffusion in a given time interval decreases continuously until a small residual
effect becomes observable, which is independent of the time delay after reaching the
final temperature. This is represented by dataset B in Fig. 5.4.
A
B
Figure 5.4: Time evolution of spectral holes burned at different times after sample cooling. Datasets
corresponding to A and B are investigated in the following.
74

5.5 Time dependence
Investigating the temperature dependence of the time evolution of the hole width [33],
we performed experiments on a PMMA sample doped with phthalocyanine at different temperatures
T, varying from 0.100 K to 1 K. For each of the temperatures T = 0.100, 0.300,
0.500, and 0.700 K, the respective experiments were carried out in a separate run. It took
about one hour to cool the system from 77 K to 4 K. The time for cooling down from 4 K
to the final temperature T depends slightly on T, but is in the range of about (6+1) hours.
Immediately after reaching T, holes are burned and their subsequent broadening is observed
for about one week. These experiments yield data corresponding to set A in Fig. 5.4.
The results of these measurements are shown in Fig. 5.5 [33]. In this plot the time origin for
each run is identical, namely the time at which the cooling down procedure from 77 K was
started. For better visibility, however, the data corresponding to T = 0.500, 0.300, and
0.100 K are shifted by 0.5, 1.0, and 1.5 orders of magnitude along the logarithmic time axis,
respectively. Except for the data taken at 0.500 K the total observation time was 6–10 days.
Figure 5.5: Broadening of holes burned immediately after reaching T. Data shifted for better visibility
(see text).
Keeping the samples at constant temperature new holes are burned after 1–2 weeks
and their broadening is observed for several hours, corresponding to set B in Fig. 5.4. These
data are shown in Fig. 5.6 [33]. Here the observation time is 2–16 hours. Note that the range
of the DG-axis is about 10 times less as compared with Fig. 5.5.
Comparing our data of Figs. 5.5 and 5.6, it is obvious that the measurements which are
started immediately after cooling down show no variation of the observed broadening with the
phonon bath temperature, while the residual behaviour observed many days later exhibits a
rather pronounced dependence on the experimental temperature. It has to be emphasized that
the data shown in Fig. 5.5 and the corresponding data in Fig. 5.6 for the same values of Twere
measured without any change of temperature in between the two subsequent experiments.
In spite of the variation of the final temperature over almost one order of magnitude,
all four datasets in Fig. 5.5 show the same hole broadening behaviour.
We attribute this to the existence of a non-equilibrium state of the TLS ensemble due
to the fast cooling procedure: The large amount of spectral diffusion observed after fast
75

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
Figure 5.6: Broadening of holes burnt after 1–2 weeks at low temperature.
cooling is due to the relaxation of TLS to their thermal equilibrium [34]. This relaxation
process is known to produce the thermal phenomenon of heat release [7]. Thus, it can be
concluded that we have established an important connection between optical and calorimetric
phenomena.
The data of Fig. 5.6 exhibit a pronounced dependence on temperature, as can be expected
from theoretical considerations outlined in Section 5.3 and from the non-time resolved
experiments of Section 5.4. The time evolution, however, shows a distinct non-logarithmic
behaviour in contradiction to the prediction of the tunneling model. For short times
the Fig. 5.6 shows clearly that the hole broadening starts fairly logarithmic. We have performed
transient hole burning experiments on the millisecond time scale, which also yielded
a logarithmic behaviour [35]. At longer times, however, the diffusional broadening increases
faster than logarithmic. This increase occurs later at lower temperatures, which is to be expected
from the temperature dependence of the relaxation rates. These new features are in
qualitative agreement with theoretical work that invokes strong coupling of TLS with phonons
[28, 36].
We believe that our experimental work shows that the method of spectral hole burning
spectroscopy is suitable for studying the low-temperature properties of amorphous solids.
Especially for the investigation of the long-time behaviour of TLS equilibrium dynamics,
where other methods cease to function, it is a very powerful experimental technique.
References
1. R.C. Zeller, R.O. Pohl: Phys. Rev. B, 4, 2029 (1971)
2. Amorphous Solids. Low Temperature Properties, in:W.A. Phillips (ed.): Topics in current Physics,
Vol. 24, Springer, Berlin, (1981)
76

References
3. S. Hunklinger, A. K. Raychaudhuri: in: D. F. Brewer (ed.), Progress in Low Temp. Physics,Vol. IX,
Elsevier Science, Amsterdam, p. 265 (1986)
4. P.W. Anderson, B.I. Halperin, C.M. Varma: Philos. Mag., 25, 1 (1972)
5. W.A. Phillips: J. Low Temp. Phys., 7, 351 (1972)
6. M.T. Loponen, R.C. Dynes,V. Narayanamurti, J.P. Garno: Phys. Rev. Lett., 45, 457 (1980)
7. J. Zimmermann, G. Weber: Phys. Rev. Lett., 46, 661 (1981)
8. B.M. Kharlamov, R.I. Personov, L.A. Bykovskaya: Opt. Commun., 12, 191 (1974)
9. A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane: JETP Lett., 20, 216 (1974)
10. J. Friedrich, D. Haarer: Angew. Chem., 96, 96 (1984); Angew. Chem. Int. Ed. Engl., 23, 113
(1984)
11. Th. Sesselmann, W. Richter, D. Haarer, H. Morawitz: Phys. Rev. B, 36(14), 7601 (1987)
12. A.P. Marchetti, M. Scozzafara, R.H. Young: Chem. Phys. Lett., 51(3), 424 (1977)
13. V.D. Samoilenko, N.V. Rasumova, R.I. Personov: Opt. Spectr., 52(4), 580 (in Russian) (1982)
14. P.M. Selzer, D.L. Huber, D.S. Hamilton, W.M. Yen, M.J. Weber: Phys. Rev. Lett., 36, 813 (1976)
15. T.L. Reinecke: Solid State Commun., 32, 1103 (1979)
16. J.R. Klauder, P.W. Anderson: Phys. Rev., 125(3), 912 (1962)
17. J.L. Black, B. I. Halperin: Phys. Rev. B, 16, 2879 (1977)
18. W. Breinl, J. Friedrich, D. Haarer: J. Chem. Phys., 81(9), 3915 (1984)
19. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88(1), 679 (1988)
20. M.M. Broer, B. Golding, W.H. Haemmerle, J.R. Simpson, D.L. Huber: Phys. Rev. B, 33, 4160
(1986)
21. A. Gorokhovskii,V. Korrovits, V. Palm, M. Trummal: Chem. Phys. Lett., 125, 355 (1986)
22. Korrovits, M. Trummal: Proceedings of the Academy of Sciences of the Estonian SSR, 35(2), 198
(1986)
23. K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991)
24. W.A. Bosch, F. Mathu, H.C. Meijer, R.W. Willekers: Cryogenics, 26, 3 (1985)
25. J. Jäckle: Z. Phys., 257, 212 (1972)
26. G. Federle: PhD thesis, Max-Planck-Institut für Festkörperforschung Stuttgart (1983)
27. S. Hunklinger, M. Schmidt: Z. Phys. B, 54, 93 (1984)
28. K. Kassner: Z. Phys. B, 81, 245 (1990)
29. A. Nittke, M. Scherl, P. Esquinazi, W. Lorenz, Junyun Li, F. Pobell: J. Low Temp. Phys., 98(5/6),
517 (1995)
30. L. Kador, G. Schulte, D. Haarer: J. Phys. Chem., 90, 1264 (1986)
31. K.P. Müller: PhD thesis, Universität Bayreuth (1991)
32. W.H. Chen, K.E. Rieckhoff, E.M. Voigt, L.W. Thewalt: Mol. Phys., 67(6), 1439 (1989)
33. H. Maier, D. Haarer: J. Lum., 64, 87 (1995)
34. S. Jahn, K.-P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)
35. S. Jahn, D. Haarer, B.M. Kharlamov: Chem. Phys. Lett., 181, 31 (1991)
36. K. Kassner, R. Silbey: J. Phys. Condens. Matter, 1, 4599 (1989)
77

6 Optically Induced Spectral Diffusion in Polymers
Containing Water Molecules: A TLS Model System
Klaus Barth, Dietrich Haarer, and Wolfgang Richter
6.1 Introduction
In the past decade much experimental and theoretical work was done on spectral diffusion in
amorphous solids. It turned out that at low temperatures, optical dephasing phenomena [1]
as well as the numerous caloric data [2] can be well described by assuming the presence of
low-energy excitations, the so-called two-level systems (TLS). Photochemical hole burning
[3] and optical echo experiments [4] have provided experimental evidence for different dynamics
of optical transitions in glassy systems as compared to crystalline matrices. For
glassy organic solids the TLS concept was first proposed by Small and co-workers [5, 6]
and was later used by Reinecke [7] to explain low-temperature optical line widths. Especially
the photochemical hole burning data over long observation periods (observation times
longer than three months) at temperatures down to 0.050 K allowed a critical test of the theoretical
approach within the tunneling model [8]. Theoretical and experimental results to
this topic are also given in this book by H. Maier et al.
At temperatures above 1 K tunneling transitions and localized vibrations influence the
physical properties [9]. Both types of mechanisms have recently been incorporated in the
soft potential model [10] which contains the well-known tunneling model as a special case.
A microscopic interpretation of optical line broadening phenomena in terms of e. g.
local excitations of the matrix or switching of optically addressed chemical groups suffered
from the lack of experimental data. In the past, several experimental techniques have been
developed to shed light on the microscopic mechanisms of the phenomena. These techniques
are all based on the generation of phonons in the matrix material, especially in the neighbourhood
of the spectral probe. Heat pulses from external heaters [11] produce a broad distribution
of phonon frequencies inside the chromophore host system and permit the investigation
of the dynamics of barrier crossings and the coupling between the TLS and the dye
molecules [12]. Light-induced phonon generation in hole burning systems can in principle
be achieved either by exciting non-radiative decay processes in appropriate chromophores
[13] or by direct absorption of IR light in the matrix material [13, 14]. The non-radiative decay
method, which has been used to study transient and irreversible spectral diffusion processes,
yields also a broad distribution of phonon frequencies and gives rise to similar experimental
processes as e. g. the heat pulse technique. On the other hand, the IR absorption
78 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

6.2 Experimental setup for burning and detecting spectral holes
method where the sample is illuminated with narrow band IR light allows us to generate
phonons of high density and of a comparatively small energy distribution in the direct neighbourhood
of the optical probe. The contribution of the broad band black-body background
radiation can be minimized by using an appropriate intensity of a narrow band IR source.
For the optical investigation of amorphous systems, photochemical hole burning (PHB)
is a high-resolution technique and a powerful tool to examine local sites in polymers at low
temperatures. Dye molecules embedded in polymers at low concentration exhibit strongly inhomogeneously
broadened absorption bands in a conventional spectroscopic experiment. Selecting
an ensemble of these dye molecules with narrow band laser light, which gives rise to a
photoreaction, leads to a sharp spectral dip at the laser frequency, the so-called spectral hole.
Due to the high resolution, this method is appropriate for detecting small perturbations in the
matrix such as a transient change of the phonon distribution or permanent matrix rearrangements
in the vicinity of the dye molecules. Such rearrangements can be due to a configurational
change of parts of the polymer main chain or due to a reorientation of small molecules
(water), which may be embedded in the matrix in addition to the dye molecules.
In the following, two different experimental conditions are discussed which are based
on the absorption of IR light. Both experimental situations lead to a change of the local interaction
of the optical probe with its local matrix environment. In the first part the phonons
are treated as a time-dependent temperature bath. In the second part experiments are discussed
where local groups of the matrix are selectively addressed by IR radiation of very
low intensity. The observed enhanced spectral diffusion shows a characteristic dependence
on the IR frequency and coincides with only a few of the numerous IR absorption bands. A
quantitative description of this new process together with the identification of the resonant
vibrations is given within a simple kinetic model.
6.2 Experimental setup for burning and detecting spectral holes
The experimental setup for burning and detecting spectral holes with a narrow band laser is
described elsewhere [16]. The experiments were performed at 1.8 K in a bath cryostat in
which the sample was immersed in superfluid helium. As samples we used different polymeric
matrices such as polymethylmethacrylate (PMMA), polyamide (PA), polyethylene, polystyrene,
etc. doped with free base phthalocyanine (H 2 Pc) at low concentration (10 –2 mol%). Their
preparation is described in Ref. [16]. The water content of the samples has been controlled by
heating them under vacuum conditions. Because of its strong hydrophilic nature PMMA has
a high capacity of water absorption at atmospheric conditions. In order to investigate the influence
of the water molecules the experiments, described in Section 6.4, were carried out either
with a nearly water-free PMMA sample or with a PMMA sample containing water molecules
under equilibrium conditions. The optical windows of the cryostat consisted of crystalline
BaF 2 which allowed the study of hole burning spectra in the lowest vibrational band of the
electronic S 0 ? S 1 transition of H 2 Pc at about 0.69 mm as well as under IR illumination between
2 mm and 11 mm. In this setup, the broad band background radiation with an emission
79

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
maximum at 10 mm has an integrated intensity of about 300 mW/cm 2 at the location of the
sample. As IR radiation sources we used a Globar with a maximum emission intensity near
3 mm and different CO 2 laser lines around 10 mm.
An acousto-optic modulator was used for reducing the intensity of the CO 2 laser down
to 1 W/cm 2 and for switching with times some microseconds. The radiation of the Globar
was dispersed in a monochromator and with different dielectric filters to a bandwidth between
0.05 mm and 0.2 mm with typical intensities around 100 mW/cm 2 .
6.3 Reversible line broadening phenomena
In this Section the mechanisms of reversible line broadening phenomena under continuous
or pulsed irradiation conditions are discussed. Phonons are generated by illuminating the
polymeric sample with the light of a single CO 2 laser line. The wavelength of the CO 2 laser
was selected to generate high frequency phonons only within a small penetration depth of
less than 10 mm. The dye molecules, which are homogeneously distributed throughout the
sample thickness of some 100 mm, are a local probe for the increase in the phonon density
during the IR irradiation. Due to the thermalisation processes, phonons show a broad energy
distribution. This change of the phonon distribution gives rise to an increased effective temperature
under cw irradiation conditions. Because of the high cooling rate of the surrounding
superfluid helium, typical irradiation intensities of about 1 W/cm 2 were used to obtain a
significant increase in the phonon density. The influence on the homogeneous line width of
the dye molecules can be investigated in a zero burning fluence experiment, i. e. the limit of
the hole width is given by extrapolating to zero burning fluence. The lower curve in Fig. 6.1,
which was measured without any IR irradiation, yields an extrapolated value for the line
Figure 6.1: Zero burning fluence experiment under different phonon density conditions in polyamide
doped with H 2 Pc (see text).
80

6.3 Reversible line broadening phenomena
width of about 375 MHz. The two upper curves show increased hole widths due to an enhanced
phonon generation either by IR irradiation during the burning process only(triangles)
or during the detection process only(circles).
The profile of a spectral hole can be written as
Z 1 1
A…! ! L †/
nz…! 0 ! L †z…! ! 0 †d! 0 ; …1†
where nz(o' – o L ) is the site distribution function of the molecular line contributing to the
hole spectrum and z(o – o') is the absorption profile of a single molecule.
On the time scale of this experiment (up to some minutes) the spectral hole profiles
are mainly reversible (see also Fig. 6.2 and Fig. 6.3). Therefore, the homogeneous line profile
function z(o) is much more affected by the altered phonon density than the site distribution
function nz(o). Thus, it is obvious that the two upper curves in Fig. 6.1 are similar and
give for the hole width nearly the same zero burning fluence value of 440 MHz. Both curves
are well separated from the lower curve which was obtained without any additional phonon
generation. Applying an effective temperature model the difference between these two zero
burning fluence values corresponds to a temperature change of about 0.3 K within the sample.
Because of the strongly inhomogeneous experimental conditions during phonon generation
and phonon diffusion through the sample, however, each different subensemble of dye
molecules experiences a different distribution in phonon energies and, as a consequence, a
bath description in terms of one single temperature is not appropriate. By using different op-
Figure 6.2: Lower part: Spectral profile of a photochemical hole with and without phonon generation
during the detection process. Upper part: Directly measured difference between these two profiles as
obtained by modulating the IR light at a frequency of 150 Hz.
81

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.3: Decay of the transmission maximum during a laser pulse.
tical energies for the phonon generation, it is in principle possible to study the contribution
of different matrix vibrations on the dephasing of a single absorber function z(o).
The time scale of the hole burning experiments discussed above is several minutes. It
is about 3 orders of magnitude larger than the typical thermal relaxation time of the sample
as given by
ˆ d2 c
; …2†
where d is the dimension (thickness) of the sample, c the specific heat, r the mass density,
and k the thermal conductivity. For the temperature of our experiment (1.8 K) and a typical
sample dimension of 1 mm the thermal relaxation time t is calculated to 10 ms.
A consequence of this short relaxation time is shown in Fig. 6.2 and Fig. 6.3. The two
spectral holes in the lower part of Fig. 6.2 were obtained with and without phonon generation
during the hole detection process. A modulation of the IR light during the detection
process results in a continuous switching between these two hole profiles. In the upper part
of Fig. 6.2, a modulation frequency of 150 Hz was used during the scan of the optical frequency.
The switching time between the two hole profiles is of the same order of magnitude
as the thermal relaxation time. The continuous switching between the two hole profiles is
detected with lock-in technique. The result shows that the phonon generation and relaxation
process is sufficiently fast to follow the 150 Hz modulation of the IR light.
A time-resolved experiment is shown in Fig. 6.3. The optical transmission at the peak
of a hole spectrum was monitored while the sample was being illuminated with a single
CO 2 laser pulse of 600 ms duration. The decay of the transmission signal cannot be described
with a single exponential function. It is well represented by the superposition of two
exponentials with a fast contribution t 1 & 10 ms and a slow contribution t 2 & 150 ms. The
fast contribution is close to the thermal relaxation time at the temperature of the helium
bath. The slow contribution contains all the inhomogeneous experimental conditions as mentioned
above, in particular a temperature gradient inside the sample that gives rise to a variation
of the thermal relaxation time. Since the amplitude of the fast contribution is larger by
a factor of 4, the switching time between the two curves in Fig. 6.2 is mainly determined by
the fast contribution t 1 .
82

6.4 Induced spectral diffusion
The experimental results of this Section show that optically generated phonons can be
used to study the transient broadening of the optical line shape of a single absorber. In principle,
the dependence on the phonon frequency can be studied in such an experiment. In the
same experiment the time-resolved dynamics of phonon diffusion and phonon decay in polymeric
systems can be investigated via the dephasing mechanism of the optical probe.
6.4 Induced spectral diffusion
In this Section we will demonstrate that the resonant absorption of IR light in a polymeric
system can also lead to irreversible line broadening phenomena. Even at a very low irradiation
intensity (a typical value is 100 mW/cm 2 ) the investigation of specific types of relaxation
processes is possible. Such low irradiation intensities do not increase the temperature of
the matrix by more than 0.01 K and hence yield no measurable dephasing effect via a
change of the phonon distribution. Because of the irreversible nature of the optical line
broadening, such phenomena are usually ascribed to spectral diffusion processes. The IR induced
spectral diffusion is caused by addressing specific moieties within the matrix via the
excitation of local vibrations. In the following weakly bound water molecules are studied.
When a change in the spatial configuration takes place during the relaxation process of a
water molecule the optical transition of a neighbouring dye molecule will be affected. Optical
absorbers in glasses are therefore very sensitive probes for local rearrangements and are
suited for a microscopic study of this topic.
Due to the hydrophilic nature of the matrix the investigated guest host system, PMMA
doped with H 2 Pc, contains a comparatively large number of water molecules (up to one volume
percent). The experimental procedure is the following: A spectral hole is burned and its
spectrum is repeatedly recorded while the sample is being exposed to IR radiation of a certain
wavelength at a constant irradiation intensity. Figure 6.4 shows the typical increase of the hole
Figure 6.4: Time evolution of the hole broadening induced by different IR wavelengths. From top to
bottom: 2.80 mm, 2.86 mm, 2.90 mm, 2.96 mm, and 3.04 mm.
83

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.5: Transmission spectra of PMMA without (curve A) and with (curve B) natural water content.
The quotient C = B/A shows the H 2 O absorption lines. Dependence of the hole broadening on the
IR wavelength after t 0 = 35 min (curve D) due to induced spectral diffusion.
width and the strong dependence of this behaviour on the IR wavelength. To illustrate the pronounced
spectral selectivity, one value of each hole broadening curve at t 0 = 35 min is plotted
in Fig. 6.5 (curve D) versus the corresponding IR wavelength. The data points do not simply
reflect the absorption behaviour of the sample. Although the PMMA matrix has many absorption
bands between 2 mm and 11 mm (curve B in Fig. 6.5), there are only two distinct wavelengths
which give rise to induced spectral diffusion processes.
The small concentration of water molecules in the matrix gives rise to non-saturated
absorption bands near 2.8 mm and 6.1 mm. They correspond to the fundamental stretching
and bending vibrations of H 2 O, respectively. The difference spectrum (curve C in Fig. 6.5)
of water free and water saturated PMMA yields the exact position of the H 2 O absorption
bands, which are in good agreement with the resonances found in our experiment. On an expanded
scale in Fig. 6.5, even the asymmetric shape of the inhomogeneous H 2 O band is reflected
by the experimental data. Replacing protonated with deuterated water yields a red
shift of the IR resonances which also agrees very well with the corresponding absorption
spectra of heavy water.
In analogy to the tunneling model, which is based on the assumption of two potential
minima, we assume two stable sites for each water molecule (Fig. 6.6). Since the experimental
results indicate that a permanent change in the matrix occurs transitions between the two
sites are only allowed via the first excited vibrational states. Using such a simple four-level
model for the underlying kinetics, we are able to explain the temporal behaviour of the induced
spectral diffusion in a quantitative fashion.
84

6.4 Induced spectral diffusion
Figure 6.6: Energy level scheme of a H 2 O molecule in a two-site model.
The two ground states are labelled 1 and 2, the first vibrationally excited states 3 and
4, respectively. The IR induced and the spontaneous transition rate for each site are denoted
by the Einstein coefficients B and A, respectively. In our notation, B is proportional to the irradiated
IR intensity. Transitions between the two sites are denoted by the conversion rate k.
The transitions shown in Fig. 6.6 lead to a system of 4 coupled linear rate equations.
Since we used in our experiments very low intensities of the IR light, we consider only the
limit B P A where the population of the excited levels 3 and 4 is negligible at all times. The
result is a time-dependent number of flips between the two ground states n f = n 1?2 + n 2?1 .
Only those flips are taken into account which contribute to a change in the total configuration
of all water molecules with respect to the time t = 0 of the hole burning process.
According to Reinecke [7] the width of the spectral diffusion kernel Do is proportional
to the number of flips n f . Though using the approximation that each flip yields the
same contribution to Do, when taking into account the decrease of the IR intensity across
the sample thickness as well as a random orientation of the H 2 O molecules, the result for
Do can only be calculated numerically. Using a Taylor expansion, however, a very good analytical
approximation of the width of the induced spectral diffusion is given by
!…t† ˆG N
0
4pk 1 1 ‡ 2Bp !
f t b
: …3†
b
The difference between this formula and the numerical result is less than 5%. p f is the
flip probability of a single H 2 O molecule as defined by p f = k /(A + k), b is a dimensionless
geometry parameter of about 2, N 0 is the density of the H 2 O molecules and G is the coupling
constant between dye and water molecules. Equation 3 describes very well the dependence
of the broadening of a spectral hole on the number of absorbed IR photons (Fig. 6.4).
It is possible to perform a test of the model. Equation 3 contains only two fitting parameters,
the flip probability p f , and the coupling constant G. These two parameters can be obtained
from a single experiment.
An example is given in Fig. 6.7. One set of the experimental data (filled circles) is obtained
at an IR intensity of 80 µW/cm 2 . The solid line is a fit according to Eq. 3 and yields
a flip probability of 18% and a coupling constant of G = 3.6 7 10 –44 Jcm 3 .(Note: the slight
difference with respect to the data given in reference [17] is due to the extension of the
model by including the random orientation of the water molecules and the decrease of the
IR intensity according to Beer’s law.) With these two values it is possible to calculate the
85

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Time (s)
Figure 6.7: Test of the model. Hole broadening versus IR irradiation time for three different IR intensities.
The solid line is a fit, the dashed lines are calculated according to Equation 3.
time dependence of the induced spectral diffusion for any other irradiation intensity. The
dotted lines in Fig. 6.7 are calculated with Eq. 3 for the obtained values and the intensities
230 µW/cm 2 and 20 µW/cm 2 . The experimental results under these conditions, represented
by the squares and triangles, are in very good agreement with the theoretical predictions.
The high quantum yield of 18% for the flip of a single water molecule becomes obvious
by a comparison of the number of absorbed IR photons (typically 10 14 s –1 ) with the
total number of water molecules (about 10 18 ) in the sample volume. Since a strong induced
spectral diffusion is observed within several minutes, a local reorientation process with a
high quantum yield must be involved.
The coupling strength between water and dye molecules can be estimated by taking
into account the static dipole moment of the H 2 O molecule. A reorientation of the H 2 O molecules
affects the dye molecules by the concomitant change in the local electric field at the
location of the chromophores. Assuming a mean distance of 15 Å and inserting data from
Stark effect experiments [18], one obtains a value of approximately 2710 –44 Jcm 3 for the
coupling constant which is very close to our fitted value. Together with the resonant phenomena
discussed above, this yields a detailed microscopic understanding of this special
kind of interaction.
We can conclude that spectral diffusion induced by the absorption of IR photons in
the frequency range 2–11 µm is a resonant process. Its microscopic origin is a spatial rearrangement
of weakly bound water molecules after vibrational excitation. Assuming a configurational
model with only two possible sites for each water molecule and applying the theoretical
model of spectral diffusion by Reinecke [7], an analytical description of the resonant
hole broadening behaviour can be obtained. A variation of the IR irradiation intensity shows
that the model description yields excellent agreement with the experimental data.
86

References
References
1. S. Jahn, K. P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)
2. M. Deye, P. Esquinazi: Zeitschr. für Phys. B, 39, 283 (1989)
3. J. Friedrich, D. Haarer: Angew. Chem., Int. Ed. Engl,. 23, 113 (1984)
4. C. A. Walsh, M. Berg, L. R. Narashiman, M. D. Fayer: Chem. Phys. Lett., 130, 6 (1986)
5. J. M. Hayes G. J. Small: Chem. Phys., 27, 151 (1978)
6. G. J. Small: Persistent nonphotochemical hole burning and the dephasing of impurity electronic
transitions in organic glasses, in: V. M. Agranovich R. M. Hochstrasser (eds.): Spectroscopy and
Excitation Dynamics of Condensed Molecular Systems, North-Holland, Amsterdam, p. 515 (1983)
7. T. L. Reinecke: Sol. Stat. Com., 32, 1103 (1979)
8. H. Maier, D. Haarer: J. Lumin., 64, 87 (1995)
9. R. Greenfield, Y. S. Bai, M. D. Fayer: Chem. Phys. Lett., 170, 133 (1990)
10. D. A. Parshin: Phys. Sol. State, 36, 991 (1994)
11. U. Bogner: Phys. Rev. Lett., 37, 909 (1976)
12. T. Attenberger, K. Beck, U. Bogner: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Proc. of the
3rd Int. Conf. on Phonon Physics, p. 555
13. A. A. Gorokhovskii, G. S. Zavt, V. V. Palm: JETP Lett., 48, 369 (1988)
14. W. Richter, Th. Sesselmann, D. Haarer: Chem. Phys. Lett., 159, 235 (1989)
15. W. Richter, M. Lieberth, D. Haarer: JOSA B, 9, 715 (1992)
16. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88, 679 (1988)
17. K. Barth W. Richter: J. of Lumin., 64, 63 (1995)
18. R. B. Altmann, I. Renge, L. Kador, D. Haarer: J. Chem. Phys., 97, 5316 (1992)
87

7 Slave-Boson Approach to Strongly Correlated Electron
Systems
Holger Fehske, Martin Deeg, and Helmut Büttner
7.1 Introduction
The problem of understanding high-temperature superconductivity has been a challenge to
theoreticians from a wide variety of fields. Many theoretical investigations have been carried
out in order to identify the mechanism of this fascinating phenomenon as well as to establish
the canonical model itself [1]. Although excellent progress is being made in deducing
a consistent description of the high-temperature superconductivity systems from a
first-principles theory, at present no microscopic theory can account for their unconventional
normal-phase data in its entirety. The interplay of charge and spin dynamics in the
normal state seems to hold the key to the understanding of the physical mechanism behind
high-temperature superconductivity in the cuprates. Both macroscopic measurements of
transport and magnetic properties as well as microscopic measurements probing the charge
and spin excitation spectra are fundamental in establishing the anomalous normal-state
properties of these materials [1].
The nature of spin excitations of the high-temperature superconductivity cuprates has
been experimentally studied by means of nuclear magnetic/quadrupole resonance (NMR/
NQR) and inelastic neutron scattering (INS) techniques clarifying the persistence of strong
antiferromagnetic (AFM) correlations in the normal and superconducting states [2]. Detailed
investigations of the wave-vector dependence of the low-frequency spin fluctuation spectrum
have revealed remarkable differences between the YBa 2 Cu 3 O 6+x and La 2–x Sr x CuO 4 families
at low doping level x; the dynamic structure factor S (~q, o) keeps its maximum at (p,p) in
the YBCO system, while in LSCO, the peaks are displaced from the commensurate position
to the four incommensurate wave-vectors p(1 +q 0 , 1), p (1,1+q 0 ), where q 0 F2x. On the
other hand, the analysis of the NMR data shows that the relaxation rates on the planar Cu
sites are similar in all materials.
Two striking features are associated with 63 T 1 –1 :
a) as a result of strong local spin fluctuations on 63 Cu sites it is enhanced by one order of
magnitude over the oxygen rate 17 T 1 –1 ; and
b) in sharp contrast to the Korringa-like behaviour at the planar 17 O sites, its temperature
dependence does not follow the Korringa law [2].
88 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

7.1 Introduction
The physical origin of the contrasting ~q-dependence of the spin fluctuation spectrum
is still under discussion. Millis and Monien [3] have argued that the spin dynamics and, in
particular, the temperature dependence of the spin susceptibility w s (T) in LSCO are caused
by a spin density wave instability, whereas in the YBCO family they are due to in-plane
AFM fluctuations and a novel non-Fermi liquid spin singlet pairing of electrons in adjacent
planes. On the other hand, the magnetic properties are intimately related to the energy band
dispersion of the non-interacting system within a Fermi liquid based framework, i. e., in this
way the observed spin dynamics can be attributed to different Fermi surface (FS) geometry
of LSCO-type and YBCO-type, respectively. Along this line, details of the spin fluctuation
spectrum are studied using a nearly antiferromagnetic Fermi liquid approach by Monthoux
and Pines [4]. From a more microscopic point of view, the important effects of FS shape on
the magnetic properties were confirmed by Si et al. [5] within a large Coulomb-U auxiliary
boson scheme and by Fukuyama and co-workers [6] on the basis of a resonating valence
bond (RVB) slave-boson mean-field approach to the extended t-J model. Furthermore, Ito et
al. [7] have recently reported that the charge transport in the CuO 2 plane is determined by
dominant spin scattering, i. e., the spin dynamics are manifest in the extraordinary transport
properties of the high-temperature superconductivity cuprates as well.
Encouraged by these findings it is the aim of this report to study magnetic and transport
phenomena of high-temperature superconductors probably in terms of the most simple
effective one-band model describing both correlation and band structure effects, the socalled
t-t'-J model:
H t t 0 J ˆ t X hi;ji;
~c y i ~c j t 0 X
h
hi;jii;
~c y i ~c j ‡ J X ij h i
~n i ~n j
~S i
~S j
4
: …1†
H t–t'–J acts in a projected Hilbert space without double occupancy, where ~c …y†
c …y†
i …1 ~n i † is the electron annihilation (creation) operator, ~S i ˆ 1
P
0 ~cy i ~ 0 ~c i
i =
0;
2
and
~n i ˆ P ~cy i ~c i. J measures the AFM exchange interaction, t and t' denotes hopping processes
between nearest-neighbour (NN; Ai,jS) and next nearest-neighbour (NNN; AAi,jSS) sites
on a square lattice. Compared to the original t-J model the t'-term incorporates several important
effects near half-filling. Starting from a rather complex three-band Hubbard or Emery
model [8] for the CuO 2 planes, quantum cluster calculations [9] have revealed that the
relative large direct transfer between NN oxygen sites (t pp < t pd /2) leads to a sizeable NNN
hopping t' in the context of an effective one-band description. More recently the t'-term has
been introduced to reproduce the FS geometry observed in ARPES experiments [6, 10–12].
Fitting the quasi-particle dispersion relation
" ~k ˆ 2t…cos k x ‡ cos k y † 4t 0 cos k x cos k y ; …2†
involved in Eq. 1 to experimental and band theory results yields t in the order of 0.3 eV and,
e. g., for the case of YBCO, t'&–0.4 t [4, 5, 13]. Moreover, Tohyama and Maekawa [12]
have emphasized that a t-t'-J model with t' > 0 can be used to describe the electron-doped
systems, e. g. Nd 2–x Ce x CuO 4 (NCCO). In this case one has to shift the momentum ~ k ? ~ k +
(p,p) [12], i. e., within a band-filling scenario one obtains a hole pocket-like FS centred at
(p,p)-point which shrinks with increasing doping [14]. Finally, as pointed out by Lee [15],
89

7 Slave-Boson Approach to Strongly Correlated Electron Systems
in a locally AFM environment doped holes can propagate coherently only on the same sublattice
without disturbing spins. Therefore, the t'-term coupling the same sublattice becomes
crucial for the low-lying magnetic excitations. This clearly is a correlation effect related to
the NNN hopping processes.
7.2 Slave-boson theory for the t-t'-J model
Apart from numerical techniques, the slave-particle methods have been employed extensively
in studying the effects of electron-electron correlation in the (extended) t-J model [6, 16–21].
The two commonly used approaches, the NZA slave-boson [16, 17] and slave-fermion [18]
schemes, however, yield quite different results concerning the (mean-field) ground-state phase
diagram and spin/charge excitations for this model [22]. As yet, the relationship between both
types of slave-field theories is not well understood. Within the NZA formulation, for example,
the Hamiltonian can be solved by a mean-field approximation with the (uniform) RVB order
parameter. A serious difficulty of this approach is the absence of AFM correlations [18, 22],
e. g., in the half-filled case, the lowest energy state, the energy of which is considerable higher
than numerical estimates indicate, does not satisfy the Marshall sign rule and fails to show the
expected long-range Néel order [23]. On the other hand, the mean-field slave-fermion schemes
[18] are known to give reasonable results for the spin susceptibility as well as for the spin correlation
length [21, 22], but also suffer from neglecting important correlation effects, especially
the fermion charge degrees of freedom are not described sufficiently well. In contrast, the fourfield
slave-boson (SB) technique, introduced by Kotliar and Ruckenstein (KR) [24] in the context
of the Hubbard model, has the advantage of treating spin and charge degrees of freedom on
an equal footing. Starting from the scalar KR SB representation of the Hubbard model, one may
generate an intersite exchange interaction via a loop expansion in the coherent state functional
integral [19]. However, the effective t-J Lagrangian, derived in this way, contains only an Ising
interaction term, i. e., important spin-flip exchange processes are neglected. As we shall see below,
to bosonize the complete exchange interaction term one has to use the spin-rotation-invariant
(SRI) extension of the KR SB theory from the beginning [25, 26].
7.2.1 SU(2)-invariant slave-particle representation
For the sake of definiteness, we return to the extended t-J Hamiltonian (Eq. 1), that may be
cast into the form
H t t 0 J ˆ X
t ij
~ y i
i;j
~ j
J
4
X
hiji
~ y i
~ i ~ y
j ~
j : …3†
90

7.2 Slave-boson theory for the t-t'J model
In order to bring out the SU(2) symmetry of the system, we have used in Eq. 3 a spinor
representation, where the one-row [one-column] matrices ~ y i ˆ…~ y i ; ~ i †‰ ~ i Š are built up
by the projected fermion creation [annihilation] operators ~ y i ~cy i ‰ ~ i ~c i Š: ‰ Š denotes
the contravariant [covariant] four-component vector (m =0,x, y, z) of Pauli’s matrices.
To preserve SRI, we apply to H t-J the manifest [SU(2) 6 U(1)] invariant SB scheme [26,
27] based on the SRI SB approach developed for the Hubbard model by Li et al. [28]. Accordingly,
we define scalar boson fields e ({) i and bosonic matrix operators p ({) i (representing
empty and singly occupied sites, respectively) and pseudofermion spinor fields C ({) i in the
following way:
j0 i i ˆ e y i jvaci;
j i i ˆ X y
i py i vac
j i: …4†
The unphysical states in the extended Fock space of pseudofermionic and bosonic
states are eliminated by imposing two sets of local constraints:
C …1†
i
ˆ e y i e i ‡ 2Trp y i p i
1 ˆ 0 ; …5†
expressing the completeness of the bosonic projectors, and
C …2†
i
ˆ i
y
i ‡ 2p y i p i
o ˆ 0 ;
…6†
relating the pseudofermion number to the number of p-type slave-bosons (hereafter underbars
denote a 262 matrix in the spin variables). Obviously, double occupancy has been projected
out. In the transformation of the fermionic spinor fields analogous to [26, 28],
~ i ! z i i ; …7†
the non-linear bosonic hopping operators
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
‰…1 e y i e i † o 2 ~p y i ~p i Š 1=2 …8†
yield a correlation-induced band renormalization where ~p ({)
irr' = rr'p ({)
i,–r',r. Exploiting the
SU(2),O(3) homomorphism, the matrix operators p i
may be decomposed into scalar (singlet)
p o ({) and vector (triplet) ~p i =(p ix , p iy , p iz ) components as
p …y†
i
ˆ 1
2
X
p …y†
i ;
…9†
where the p im obey the usual Bose commutation rules ‰p i; p y j 0Šˆ ij 0. Consequently, the pseudofermions
i ˆ… i; i † T satisfy anticommutation relations of the form f i ; y
j 0gˆ ij 0.
91

7 Slave-Boson Approach to Strongly Correlated Electron Systems
This way the interaction term is converted due to
~ y i ~ i ! 2Trp y i p i
; …10†
where, more explicitly, the locally defined particle number and spin operators are just
~n i ! ~n i …p i †ˆ2Trp y p ˆ P p y p ;
i i i i
~S i ! ~S i …p i †ˆTrp y i
~p i ˆ 1
2 …py io ~p i ‡ ~p y i p io i~p y i ~p i † : …11†
Actually, the components of the bosonized SB spin operator act as generators of rotations
in spin space, i. e., ~S i …p i † satisfies the spin algebra. Since C (1) i and C …2†
i commute with
the SB t-J Hamiltonian, the constraints in Eq. 5 and Eq. 6 can be ensured by introducing the
time-independent Lagrange multipliers …1†
i and …2†
i
ˆ P
…2†
i .1 As a consequence, the
Hamiltonian of the t-J model (Eq. 3) in terms of the slave-boson and pseudofermion operators
has to be replaced by
H SB
t t 0 J ˆ P
t ij
i;j
‡ P i
y
i zy i z j j
… …1†
i C …1†
i
J P
Trp y i p i
Trp y j p j
hiji
‡ Tr …2†
i C …2†
i † : …12†
In the physical subspace, H SB
t–J possesses the same matrix elements for the basis states
(Eq. 4) as the original t-J Hamiltonian (Eq. 3) for the purely fermionic states. To verify this
directly, special attention has to be paid to the bosonization of AFM exchange interaction
term
~ y
i ~ i
~ y j ~
j ˆ 2 P …~c y i ~cy j ~c j ~c i
~c y i ~cy j ~c j ~c i † ; …13†
which includes besides the Ising exchange contributions # i " j $
#i " j the spin-flip processes
# i " j $ "i # j . Let us emphasize that the matrix elements of the spin-flip terms
are not reproduced in the scalar KR SB approach [19]. By contrast, within the SRI SB approach
it is a straightforward exercise to show that these contributions can be expressed in
terms of the p ({) irr' :
X
p y i py j 0 p j p
1
0 i i j ˆ
4
i j :
0
…14†
Thus our SRI SB scheme (Eqs. 4–12) provides a consistent bosonization of the extended
t-J model.
1 Note that additional constraints do not exist. Especially, if the constraint ~p i ~p i ˆ 0 is added [25] the
spin algebra is not satisfied.
92

7.2 Slave-boson theory for the t-t'J model
7.2.2 Functional integral formulation
To proceed further, it is convenient to represent the grand canonical partition function for
the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29]
over Grassmann fermionic and complex bosonic fields as
Z
R
Z ˆ D‰ ; ŠD‰e ;eŠD‰p ; p Š d‰ …1† Š d‰ …2†
dL…†
Š e 0 ; …15†
L…† ˆP
e i …@ ‡ …1†
i †e i ‡ 2Trp T ‰…@ ‡ …1†
i † o …2†T
i
‡ P
Trp ~p i i Trp~p Trp p j j i i Trpp j j
hi;ji
‡ P
i …@ † o ‡ …2†
i i ‡ P t ij
i z i z j j :
i
i;j
i
i
Šp T i
…1†
i
…16†
Apart from the above symmetry considerations the SB functional-integral formalism
reveals additional global and local gauge invariances [25, 30–32].
The action S ˆ R
0
dL…† is invariant under the following site and time-dependent
phase transformations (y i (t)w i
(t)), i. e., under the local symmetry group SU(2) 6U(1),
i ! e ii e i i i ;
e i ! e i e i i
; …17†
p i
! p i
e i i ;
provided that the (five) Lagrange parameters act as time-dependent gauge fields, l (1) i (t) and
…t†, absorbing the time derivatives of the phase factors:
l …2†
i
…1†
i
! …1†
i ‡ i _ i ;
…2†
i
! e i i …2†
i e i i i _ i
‡ i _ i o : …18†
Next, in the continuum limit, the radial gauge is introduced by representing the Bose
fields by modulus and phase,
e i ˆ je i je i'…ei† ;
p i ˆ 1
2
X
p i
e
i …p † i ;
…19†
93

7 Slave-Boson Approach to Strongly Correlated Electron Systems
(for a time-discretized version of this gauge fixing, see [31]). Exploiting the gauge freedom
of the action, we can now fix the five real-valued coefficients y i (t), w im (t) to remove five
phases (j i (t), f im (t)) of the Bose fields e i , p im in the radial gauge. As a consequence, all
the Bose fields {e i , p im } become real, in contrast to the Hubbard model, where one SB field
remains complex [25, 32]. At this point one should notice that the particle number and spin
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
and ~p i now denote the radial parts of the corresponding Bose fields). Then, using the familiar
identity for Gaussian integrals over Grassmann fields [29],
R
D‰ ; Š e ‰ G 1 Š 0
ˆ e Tr ln‰ G 1Š ; …20†
the fermionic degrees of freedom can be integrated out and we obtain the following exact representation
of the grand canonical partition function
Z ˆ R D‰Š e S ef f
…21†
in terms of the real-valued bosonic fields f ia ,
i …† ˆ… i …†† ˆ …e i ;p io ; …2†
io ;…1† i
; p ix ; …2†
ix ; p iy; …2†
iy ; p iz; …2†
iz † :
…22†
In Eq. 21 the effective bosonic action S eff takes the form
(
S ef f ˆ R d P
0 i
…1†
i
e 2 i ‡ P
…1†
i
…2†
io
p 2 i
2p io ~p i ~ …2†
i
…1†
i
‡J P )
1
p io ~p i ~p j p jo
hiji
4 …p2 io ‡ ~p2 i †…p2 jo ‡ ~p2 j †
h
i
Tr ij; 0 ; 0 ln G ij; 1 † ; …23†
where the (inverse) SB Green propagator is given by
h
Gij; 1 0…; 0 †ˆ @ ‡ …2†
io
0
i
~ …2†
i ~ 0 ij … 0 †
t ij …z y i z j† 0 ; 0…1 ij† : …24†
It may be remarked that since the bosons are taken to be real, their kinetic terms,
being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary
conditions imposed on Bose fields (f ia (b) =f ia (0)). Strictly speaking it follows from this
property that all the Bose fields do no longer have dynamics of their own [25].
94

7.2 Slave-boson theory for the t-t'J model
7.2.3 Saddle-point approximation
The evaluation of Eq. 23 proceeds via the saddle-point expansion 2 , where at the first level
of approximation we look for an extremum S … i † of the bosonized action S eff with respect
to the Bose and Lagrange multiplier fields f ia :
@S ef f
@ i
ˆ!
0 ) S ˆ S ef f
iˆ i
: …25†
The physically relevant saddle-point F i is determined to give the lowest free energy
(per site)
f t t 0 J ˆ =N ‡ n ; …26†
where at given mean electron density n =1–d, the chemical potential m is fixed by the requirement
n ˆ 1 @
N @
…27†
( O ˆ S= denotes the grand canonical potential). Obviously, an unrestricted minimization of
the free energy functional is impossible for an infinite system. To keep the problem tractable,
we use the ansatz
~m i ˆ m~u i ; ~u i ˆ…cos ~Q ~R i ; sin ~Q ~R i ; 0† …28†
for the local magnetization ~m i ˆ 2 ~S i ˆ 2p io ~p i . Following earlier analyses of spiral states
for the Hubbard model [34] the unit vector ~u i is chosen as a local spin quantization axis
pointing in opposite directions on different sublattices. Thereby, the order parameter wavevector
~Q is introduced as a new variational parameter to describe several magnetic ordered
states: PM, FM ( ~Q = 0), AFM ( ~Q =(p,p)), and incommensurate (1,1)-spiral ( ~Q =(Q, Q)),
(1,p)-spiral ( ~Q =(Q,p)) and (0,1)-spiral ( ~Q = (0,Q)) states. Note that since fluctuations of
the charge density are not incorporated the scalar Bose fields are homogeneous: e i = e, p io =
p o , l (1) i = l (1) , and l (2) io = l (2) o . The vector fields exhibit the same spatial variation as the magnetization:
~p i ˆ p~u i and l ~ …2†
i ˆ l …2† ~u i . Then transforming Eq. 23 in the ( k; ~ o)-representation
the trace in S eff can be easily performed to give the free energy functional
2 Mean-field approximations to the functional integral are achieved by replacing the bosonic fields by
their time averaged values. For the Hubbard model, a comparison of the paramagnetic and antiferromagnetic
SB solutions with quantum Monte Carlo results shows that such a mean-field like approach
yields an excellent quantitative agreement for local observables [33] and therefore can give a qualitative
correct picture with relatively little effort.
95

7 Slave-Boson Approach to Strongly Correlated Electron Systems
where
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
‡J p 2 1
o p2 …cos Q x ‡ cos Q y †
2 …p2 o ‡ p2 † 2
‡ 1
N
X
ln‰1 n ~k Š ; …29†
~k
n ~k ˆ‰expf…E ~k
†g ‡ 1Š
1
…30†
holds. The renormalized single-particle energies E ~k … ˆ†are obtained by diagonalizing
the kinetic part of Eq. 23 [26]. Requiring that f t-J be stationary with respect to the variation of
the magnetic order vector, the wave-vector ~Q can be obtained from the extremal condition
sin Q x;y ˆ 1
Jp 2 o p2 1
N
X
~k
n ~k
@E ~k
@Q x;y
:
…31†
If one substitutes Eq. 31 together with f a (obtained from the solution of the coupled
self-consistency equations (Eq. 25)) into Eq. 29, the free energy of the t-t'-J model is obtained
as
f t t 0 J ˆ J
4 ‰m2 …cos Q x ‡ cos Q y † 2n 2 Š‡ …2† m ‡… o …2† †n
1 X
ln 1 ‡ e …E k ~ ; …32†
N
~k
where the quasi-particle energy takes on the form
E ~k ˆ …1 ‡ †…" ~k ‡ " ~k
†=y ‡ …2†
~Q o
h i 1=2
2 …" ~k " ~k ~Q †2 =y ‡‰m…" ~k ‡ " ~k ~Q †=y ‡ …2† Š 2
…33†
with m=2 p o p and y=(1+d) 2 – m 2 .
In particular, at the spatially uniform paramagnetic saddle-point, F …PM† =(e, p o , l o (2) ,
l (1) ; 0, 0; 0,0; 0,0) the remaining bosonic fields
96
e 2 ˆ ;
p 2 o ˆ 1 ;
…1† ˆ
…2†
o
ˆ
2 ‡ 3 2
1 2 ~"…0† ;
2
~"…0† …1 †J; …34†
2
…1 ‡ †

7.2 Slave-boson theory for the t-t'J model
are explicitly given in terms of J, d and a single energy parameter ~" (0) defined by
~"…~q† ˆ 2
N
X
" ~k ~q
… E ~k † …35†
~k
(at T=0). Here, the quasi-particle energy E ~k is
E ~k ˆ 2" ~k =…1 ‡ †‡ …2†
0 ; …36†
and m can be determined from d =1– 2 N
free energy becomes simply
P
~k … E ~ k
†. Finally, in this approximation, the
f …PM†
t t 0 J ˆ z2 ~"…0†
1
2 J …1 †2 ‡ …1 † : …37†
7.2.4 Magnetic phase diagram of the t-t'-J model
In the numerical evaluation of the self-consistency loop we proceed as follows: at given
model parameters J and d, we solve the remaining saddle-point equations for m, l (1) , l o
(2)
and l (2) together with the integral equation for m using an iteration technique. Then, in an
outer loop, the order parameter wave-vector ~Q is obtained from the extremal Eq. 31 by
means of a secant method. Convergence is achieved if all quantities are determined with relative
error less than 10 –6 . Note that our numerical procedure allows for the investigation of
different metastable symmetry-broken states corresponding to local minima of the variational
free energy functional (Eq. 32).
At first, let us consider the case t' = 0, i. e., the pure t-J model. The resulting groundstate
phase diagram in the J-d plane is shown in Fig. 7.1. For the case d = 0 (Heisenberg
model), we obtain an AFM ground state. At J=0, the FM is lowest in energy up to a hole concentration
of 0.327, where a first-order transition to the (1, p)-spiral takes place; above
d = 0.39, we find a degenerate ground state with wave-vector (0,p). At d = 0.63 the PM becomes
the lowest in energy state (second-order transition). This coincides with the U ?? SB
results of the Hubbard model (J=4t 2 /U) [35]. The PM-FM instability occurs at d PM-FM = 0.33
[32]. In contrast, the slave-fermion phase diagram [20] exhibits a much larger FM region.
This can be taken as an indication that correlation effects are treated less accurate within the
slave-fermion mean-field scheme [20, 36]. For finite exchange interaction J the (1,1)-spiral
is the ground state at small doping. With increasing d a transition to the FM takes place,
which becomes unstable against the (1,p)-spiral at larger doping concentrations. For
J/t > 0.08, we find a transition from (1,1)-spirals to (1,p)-spirals at about d & 0.2. In
Figure 7.1, the dotted line separates the (1, p)-spiral state from the region, where the (1, p)
and (0, p) states are degenerate. If we admit only homogeneous phases, the phase boundaries
(1,1)-spiral ⇔ (1,p)-spiral at d & 0.2 and (1,p)/(0, p)-phase ⇔ PM at d &0.63 remain
nearly unchanged at larger values of J. However, analyzing the thermodynamic stability of
97

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.1: SB ground-state phase diagram of the t-J model.
the saddle-point solutions (i. e. the curvature of f t J …d), one observes a tendency towards
phase separation into hole-rich and AFM regions above the ‘diagonal’ solid curve in the left
upper part of the J-d phase diagram (see below).
Figure 7.2 displays the variation of the extremal spiral wave-vector ~Q as function of
doping. At d =0, ~Q =(p, p) indicates the AFM order. At low doping the (1,1)-spiral order vector
decreases approximately linear. With decreasing J the AFM exchange is weakened, consequently
the deviation of the order vector from (p, p) increases. The discontinuities reflect the
first-order transition from (1,1)-spirals to (1, p)-spirals. For J/t =0.05 the transition to the FM
state takes place, with Q x jumping down to zero. The (1,p)-spiral wave-vector shows a monotonous
decrease of Q x until at d = 0.63 the transition to the PM (Q x = 0) occurs. Comparing
the magnitude of the theoretical order vector of the spiral solutions to results from inelastic
neutron scattering experiments on La 2–x Sr x CuO 4 [37], we find good agreement for an exchange
interaction strength of J/t =0.4 (which seems to be a reasonable value with respect to
the strong electron correlations observed in the high-temperature superconductors).
Figure 7.2: The x-component of the spiral wave-vector ~Q as a function of doping d (compared with experiments
[_] on LSCO [37]).
98

7.2 Slave-boson theory for the t-t'J model
Next, we investigate the ground-state properties of the t-t'-J model, where we fix
J/t =0.4. Comparing the free energies of several (homogeneous) symmetry-broken states,
we obtain the SB phase diagram shown in the t'/t-d plane in Fig. 7.3. Obviously, we can distinguish
two regions. In the parameter region |t'/t| ^ 0.2, Figure 7.3 resembles the groundstate
phase diagram of the pure t-J model (Fig. 7.1), i. e., we found large regions with incommensurate
spiral order. However, compared to the pure t-J model, the t'-term stabilizes Néel
order in a finite d region near half-filling. The increasing stability of AFM configurations
can be intuitively understood because the t'-term moves electrons without disturbing the
Néel-like background [12]. For larger ratios |t'/t| we have a completely different situation. In
this parameter regime only commensurate states (AFM, FM, PM) occur, where for
t' < t' c = –1.4 t we obtain the AFM state for all d. Note the rather large differences to the value
of t' c obtained within a semiclassical (1/N)-expansion [38]. By varying the exchange coupling
J, the phase boundaries in the t'/t-d plane are not much affected, e. g. for J/t =1 and
t'/t =–0.4, the transitions AFM ⇔ (1,1)-spiral and (1,1)-spiral ⇔ FM take place at d = 0.17
and d = 0.6, respectively. We would like to point out that the main qualitative features of our
SB phase diagram do confirm recent studies of magnetic long-range order in the t-t'-J
model [38, 39].
Figure 7.3: Restricted SB ground-state phase diagram of the 2D t-t'-J model at J/t =0.4.
In Fig. 7.4 we plot the order parameter wave-vector as a function of doping at t'/t
= +0.16 and t'/t =–0.4. The behaviour of Q x reflects a series of transitions AFM ⇔ (1,1)-
spiral ⇔ (1, p)-spiral ⇔ PM. The corresponding (sublattice) magnetization abruptly changes
at the (1,1)-spiral ⇔ (1, p)-spiral first-order transition. From Fig. 7.4 the asymmetry between
hole (t' < 0) and electron doping (t' > 0) becomes evident. In contrast to recent Hartree-
Fock results for the Hubbard model [40] we found the AFM phase near half-filling for both
electron-doped and hole-doped cases (provided t'( 0). In the absence of t ' hopping, for arbitrarily
small doping the AFM is found to be unstable against the (1,1)-spiral phase (cf.
Fig. 7.3). Obviously, the AFM correlations are strongly enhanced by a positive t'-term,
which is also in qualitative agreement with exact diagonalization studies of the t-t'-J model
[12] and confirms the experimental findings for the electron-doped system NCCO [41, 42].
Note that the stability region of the AFM phase agrees surprisingly well with the combined
99

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.4: The x-component of the SB spiral wave-vector ~Q away from half-filling for the negative values
t'/t =–0.16 (solid) and t'/t =–0.4 (long-dashed), i. e., hole doping (d > 0), and for the positive one
t'/t =0.16 (dashed), i. e., electron doping.
phase diagram for La 2–x Sr x CuO 4 and Nd 2–x Sr x CuO 4 obtained from neutron scattering [43]
and muon spin relaxation measurements [44], respectively. For the YBCO parameter t'/t =
–0.4, the AFM disappears around d = 0.1 whereas in the phase diagram of YBCO, determined
by neutron diffraction [45], this transition takes place at about x = 0.4 oxygen content.
However, there exits strong evidence that at least up to x = 0.2 no holes are transferred
from CuO chains to CO 2 planes.
As we have already noted, the phase diagrams in Fig. 7.1 and Fig. 7.3 result from the
relative stability of various homogeneous states. On the other hand, there are arguments for
the existence of inhomogeneous, e. g., phase-separated states in the t-J and related models
[46]. Using very different methods, it was realized by several groups [47–51], that at large
J/t the ground state of the t-J model separates into hole-poor (AFM) and hole-rich regions.
Unfortunately, in the physically interesting regime of small exchange coupling (J/t ~ 0.2–
0.4) and low doping level this point is still controversial. To gain more insight into the phenomenon
of phase separation in t-J-type models of strongly correlated electrons it seems to
be important to investigate the effect of an additional NNN hopping term t' as well. Therefore
we study the free energy as a function of hole density d, where a (concave) convex curvature
indicates local thermodynamic (in)stability implying a (negative) positive inverse isothermal
compressibility k –1 = n 2 @2 f
If k –1 < 0, the domain of the two-phase regime is determined
performing a Maxwell construction for the anomalous increase of the chemical poten-
@n 2 :
tial m by doping.
The results of our analysis of thermodynamic stability are depicted in Figs. 7.5a and
7.5 b for the t-J and t-t'-J model, respectively. The boundary of phase separation for the t-J
model is given by the solid curve in the J-d plane shown in Fig. 7.5 a. As can be seen from
Fig. 7.1, the different phase separated domains are built up by the (AFM) states at half-filling
(d = 0) and the corresponding hole-rich state on the right boundary of the respective region.
At J=0, where we recover the U ?? result of the Hubbard model [32], the free energy
is a convex function Vd, i. e., our SRI SB theory does not support recent arguments
[47] for phase separation in this limit. In the opposite limit of large J, complete charge se-
100

7.2 Slave-boson theory for the t-t'J model
Figure 7.5: Phase diagram of the t-(t')-J model including phase separated states. The phase separation
boundary (solid curve) for the t-J model is shown in the J-d plane (a). We include the transition lines of
Refs. [49] (dashed), [50] (dotted), and [51] (chain dashed). The triangles are the Lanczos results of
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
region consists of AFM and (1,1)-spiral states. For further explanation see text.
paration takes place for J/t > J PS /t=4.0, which seems to be an essentially classical result.
From exact diagonalization (ED) we obtain J PS /t=4.1 +0.1 [52] compared with 3.8 derived
by means of a high-temperature expansion [51]. We note that the homogeneous magnetic
phases are always unstable close to half-filling (provided J/t > 0). This is in qualitative
agreement with results obtained from ED studies [47] as well as from semiclassical [49] or
renormalization-group calculations [50]. Also plotted in Fig. 7.5 a are the results of the hightemperature
expansion method [51], where phase separation may occur only above a critical
exchange J/t =1.2 as d?0, contrary to all the other approaches. The line separating the
two-phase region from the stable states was determined by Marder et al. [49] within a semiclassical
theory to vary as J/t =4 d 2 whereas our theory yields an approximately linear dependence
at small d. We believe that, due to an improved treatment of spin correlations in
our approach, the region of incomplete phase separation is reduced. The instability towards
phase separation at small J can be taken as an indication that charge correlations may play
an important role as well.
The dotted lines of zero inverse compressibility in Fig. 7.5b show that also for |t'/t| >0
there is a finite range of d over which the (1,1)-spiral is locally unstable. Similar results were
recently obtained by Psaltakis and Papanicolaou [38]. But it is important to stress that in our
theory the AFM state is locally stable for both signs of t'. In addition, based on the Maxwell
construction, we can show that near half-filling the AFM state remains also globally stable
against phase separation for the t-t'-J model (cf. Fig. 7.5 b, where the two-phase region is
bounded by the solid lines). At larger values of |t'/t| 6 0.5, the phase-separated region is due
to the first-order nature of the transition AFM ⇔ (1,1)-spiral (dashed curve) [38].
Finally to demonstrate the quality of our SRI SB approach, for the t-J model the expectation
value of the kinetic energy is compared with the results from ED for a finite 16-
site [48] (36-site [52]) lattice in Fig. 7.6. We find an excellent agreement between SB results
and ED data. Obviously, this result does not depend on the interaction strength J, i. e. the SB
101

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.6: Expectation value of the kinetic energy as a function of doping at several interaction
strengths J in comparison to ED results for the 16 (36) site lattice [48, 52].
theory well describes important correlation effects. Note that AH t S t–J /t is directly related to
the effective transfer amplitude of the renormalized quasi-particle band, which is taken as input
for the calculation of transport coefficients in the following section.
7.3 Comparison with experiments
7.3.1 Normal-state transport properties
As one of the main normal-state puzzles of the CuO 2 based high-temperature superconductors,
the anomalous transport properties, in particular the temperature and doping dependence
of the Hall resistivity R H (T,d) [53–56], has been under extensive experimental and
theoretical study. Quasi-particle transport measurements suggest a small density of charge
carriers and hence a small pocket-like Fermi surface (FS) [57]. On the other hand, direct angle-resolved
photoemission (ARPES) probes of the FS [14, 58] yield a large FS which satisfies
Luttinger’s theorem and might be well described by (LDA) band structure calculations
[13, 59]. In principle, this contrasting behaviour is found for hole-doped (La 2–x Sr x CuO 4 ,
YBa 2 Cu 3 O 6+x ) and electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides as well.
Adopting the hypothesis of Trugman [60], most of the normal-state properties of the
cuprates may be explained by the dressing of quasi-particles due to magnetic interactions
and the subsequent modification of their dispersion relation. Then, once the quasi-particle
band E ~k has been obtained, the Hall resistivity R H = s xyz /s xx s yy can be calculated in the relaxation
time approximation, using standard formulas for the transport coefficients:
102

7.3 Comparison with experiments
ˆ
ˆ
e 2 X @n ~k
k 2 u
V k ~ u k ~ ;
@E ~k
~k
e 3 2 X
k 4 cV
~k
u ~ k
" u ~ k
@u ~ k
@k
@n ~k
@E ~k
:
(38)
Here n ~k is given by Eq. 30,V denotes the volume of the unit cell, e lkg is the completely
antisymmetric tensor, and u ~ k ˆ @E ~k =@k . Note that R H does not depend on the relaxation
time t.
To make the discussion more quantitative, let us now consider the doping dependence of
R H (d,T) in terms of the t-t'-J model using the saddle-point and relaxation time approximations,
where FS and correlation effects are involved via the renormalized SB band E ~k (Eq. 33). As we
have pointed out above, in our approach the SB quasi-particle band dispersion E ~k has to be determined
in a self-consistent way at each doping level d. This should be in contrast to the NZA
SB mean-field approach to the t-t'-J model of Chi and Nagi [61] where, in the J ? 0 limit, the
calculation of transport properties is based on the simple replacement " ~k ! ~" ~k =
–2td [(cos k x + cos k y )+2(t'/t)cos k x cos k y ] of the non-interacting band dispersion (Eq. 2).
Figure 7.7 shows the theoretical Hall resistivity as a function of carrier density in
comparison to experiments on LSCO [53], YBCO [54] and NCCO [55, 56]. In the LSCO
and NCCO systems, the concentration of chemically doped charge carriers in the CuO 2
planes (d) definitely agrees with the composition (x) of the substitutes Sr and Ce. This simple
relation, however, no longer holds for YBCO, i. e., the number of holes transferred into
the planes does not increase linearly with the oxygen content. Indeed, the magnetic properties
indicate d & 0uptox=0.2 [45]. In order to compare our theoretical model with the
R H (x) data found on oxygen-doped YBCO, we use the relation d =(x – 0.2)/2 [62].
Figure 7.7: Doping dependence of the Hall resistivity for hole-doped (left panel) and electron-doped
(right panel) systems. The slave-boson results for J/t =0.4 and different ratios t'/t = 0 (solid), –0.16
(dashed), –0.4 (chain dotted) and t'/t =0.16 (dotted) are compared with experiments on LSCO (_) [53]
(at 80 K), YBCO (O) [54] (at 100 K) and NCCO (Z) [55, 56] (at 80 K), respectively. The inset shows
the temperature dependence of R H for t'/t =0atd = 0.1 (short dashed) and d = 0.15 (dotted).
103

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.7 clearly demonstrates the importance of the NNN transfer term t' for a consistent
theoretical description of the experimental Hall data. For J/t =0.4, an excellent agreement
with experiments on LSCO and NCCO, including the sign change of R H (d) at a very similar
value, can be achieved using the parameter values t'/t =0 and t'/t = 0.16, respectively. It
should be noted that in the case of LSCO we obtain t'/t =0 from R H (d) while LDA calculations
yield a ratio t'/t =–0.16 [13, 59]. For YBa 2 Cu 3 O 6+x , where the experiments [54] give
R H > 0 up to x=1, a negative t'-term suffices to give the correct tendency to R H (d). Using
t'/t =–0.4, our theory yields a sign change of R H at d & 0.7. The strong increase (decrease)
of the positive (negative) Hall coefficient as d?0 can be attributed to the formation of
small hole (electron) pockets in the FS, which is a correlation effect. A recent analysis of resistivity
saturation in LSCO [57], based on Boltzmann transport, has been taken as an indication
of a small FS as well, however, the existence of a pocket-like FS is still a subtle and
unresolved issue. We found that the temperature dependence of R H (d,T) is at least in qualitative
agreement with experiments on LSCO (see inset Fig. 7.7 (left panel)). Note that the
quasi-particle dispersion E ~k exhibits extremely flat minima implying the presence of a new
small energy scale D. Therefore, when the temperature becomes comparable to D the hole
pockets are washed out and a sign change of R H occurs (cf. Fig. 7.7 at fixed d (inset)). 3
The FS of the interacting system are shown in Fig. 7.8 for typical ratios t'/t at J/t =0.4
and d = +0.1, where the diagonal (1,1)-spiral phase is lowest in energy. As observed for t-J
and Hubbard models as well [64], we obtain small hole (or electron) pockets with a volume
! |d|. The calculated FS are very anisotropic. As |d| increases, the pockets grow, until the
FS topology changes completely at a critical doping value d c (R H (d c ) = 0 (cf. Fig 7.7), reflecting
the transition from hole to electron carriers for t'/t < 0 and vice versa for t'/t >0.
Figure 7.8: Quasi-particle Fermi surface in the (1,1)-spiral phase at J/t =0.4 for d = 0.1 (hole-doped
system) and d = –0.1 (electron-doped system).
3 We recently learned of a related exact diagonalization study of Dagotto et al. [63], where, similar in
conclusion, the doping and temperature dependence of R H was calculated using a strongly renormalized
flat quasi-particle dispersion.
104

7.3 Comparison with experiments
We want to point out that the renormalization of the quasi-particle band E ~k strongly depends
on both interaction strength J and doping level d [62] which, in fact, calls into question
the frequently used rigid band approximation. Due to the strong coupling of spin and
charge dynamics the characteristic energy scale for the coherent motion of the charge carriers
is J and not t (provided t > J).
7.3.2 Magnetic correlations and spin dynamics
In this Section we try to understand the INS and NMR experiments on the basis of the SRI
SB mean-field theory. Using a generalized RPA expression for the spin susceptibility and assuming
that the AFM correlations are spatially filtered by various ~q-dependent hyperfine
form factors, we focus, in particular, on the spin dynamical properties in the paraphase of
the t-t'-J model. Our starting point is an RPA-like form for the exchange-enhanced spin susceptibility
[5, 6, 65]
o …~q; !†
s …~q; !† ˆ
1 ‡ J 2 …cos q x ‡ cos q y † o …~q; !†
: …39†
The irreducible part 4
o …~q; i! m †ˆ
2 X
N
~k;n
G… ~ k; i! n † G… ~ k ‡ ~q; i! n‡m †
…40†
contains the (dressed) SB Green propagators G… k; ~ i! n †ˆ‰i! n E ~k ‡ ~Š 1 describing noninteracting
electrons with the renormalized band structure E ~k …d† (Eq. 26).
Once the dynamic spin susceptibility has been obtained both, INS measurements and
NMR experiments, can be explored. Probed by INS from the fluctuation-dissipation theorem
the q-dependent and o-dependent spin structure factor is related to the dynamical susceptibility
by
S …~q; !† ˆ 1 1
p 1 exp… k!† Im s…~q; !† …41†
On the other hand, the nuclear spin-lattice relaxation rate a T –1 1a (a =k,k), e. g. for a
field H a applied parallel to the c-axis, given by [66]
a T 1k T 1 k B
1ˆ
2 2 B k lim 1
!!0 N
X
~q
a F 2 ? …~q† Im s…~q; !†
k!
…42†
4 o n = 2np/b [o m = (2m +1)p/b] denote the fermionic [bosonic] Matsubara frequencies.
105

7 Slave-Boson Approach to Strongly Correlated Electron Systems
and the transverse spin-spin relaxation rate, T –1 2G , for the RKKY coupling of the nuclear Cu
spins, given by [67]
T 2
2G ˆ
2
0
123
c 1 X 2
8k 2 …2 B † 4 Fk 2 N
…~q† 1 X
4
s…~q† @ Fk 2 N
…~q† s…~q† A 5 ; …43†
~q
provide local, atomic site (a = {63, 17} specific information. Here m B denotes the Bohr
magneton and the constant c=0.69 is the natural-abundance fraction of the 63 Cu isotope.
The form factors are given for 63 Cu and the planar 17 O nucleus as [68]
63 F …~q† ˆA ‡ 2B…cos q x ‡ cos q y † ; …44†
~q
17 F …~q† ˆ2C cos q x
2 : …45†
Together with the anisotropy of the Cu relaxation rates the measurements of the
Knight shift, a K ˆ 2
a B n k lim ~q!0 a F …~q† s …~q; ! ˆ 0†, have been used to determine the hyperfine
coupling constants A a , B and C on the basis of the Mila-Rice Hamiltonian [69]. Following
Ref. [66] we take A || & –4B, A k & 0.84B, C & 0.87B and B & 3.3610 –7 eV, where
63 gk = 7.5610 –24 erg/G and 17 gk = 3.8610 –24 erg/G.
As we know, the RPA susceptibility contains an unphysical instability of the paramagnetic
phase at some particular wave-vector ~q below a critical doping d c as signaled by the
zero in the denominator of Eq. 39 at o = 0 (Stoner condition). Therefore the use of Eq. 39
only makes sense if the system is far from the magnetic instability, i. e., d > d c , where for
J=0.4 we have d c (t'/t =–0.16) H 0.27 and d c (–0.4) H d c (0) H 0.17.
7.3.3 Inelastic neutron scattering measurements
We begin with a discussion of the RPA dynamical spin structure factor S (~q; o†. The ~qdependence
of S (~q; o† along the main symmetry axis of the Brillouin zone is shown in
Fig. 7.9 at J=0.4 and hole density d = 0.3 for different ratios t'/t. Here the temperature
is T=35 K and the frequency ko = 0.010 eV. For LSCO-type parameters (t'/t =0,
–0.16) we found four pronounced incommensurate peaks located at the points
p (1+q o , 1), p (1,1+q o ). The incommensurate modulation wave-vectors move with increasing
doping level d away from the corner of the Brillouin zone along the directions
(1, p) or(p, 1) (square lattice notation). Note, that the incommensurate peak position obtained
from a three-band RPA calculation of S (~q; o† [5] can be parametrized consistent
with the experimental observation that q 0 H 2 x [37], while all the effective one-band
RPA approaches [6, 10] yield an incommensurability scaling rather as q 0 H d. A more
detailed investigation of the LSCO-type ~q-scans show that the ~q-variation of S(~q; o† is
mainly governed by that of w o …~q; o† and, in accordance with experiments [70], the in-
106

7.3 Comparison with experiments
commensurate peaks considerably broaden when temperature or energy transfer are increased
[71]. By contrast, the same plot for YBCO-type (t'/t =–0.4) parameters shows a
broad and nearly T-independent [71] maximum around the (p, p)-point [72] (Fig. 7.9)
which, due to the flat topology of w o , mainly reflects the ~q-dependence
J (~q) =J (cosq x + cosq y ) (cf. Eq. 39). In this way, our calculations confirm recent arguments
[5, 6, 73] for the importance of band structure (Fermi surface geometry) effects
in explaining the difference between observed LSCO and YBCO spin dynamics. Nevertheless,
whether the incommensurate signals arise from an intrinsic magnetic structure or
whether they result from the formation of domains (charge superstructures) in the LSCO
system remains unanswered by INS [2, 68].
Figure 7.9: Dynamic magnetic structure factor S(~q; o† is plotted along the (1,1) and (1, p)-directions of
the Brillouin zone for different ratios t'/t.
In a next step, we calculate the longitudinal or spin-lattice relaxation rate, T –1 1 , using the
hyperfine form factors (Eq. 45). In Figure 7.10 the temperature dependences of 63 T –1 1k and
17 T –1 1k (inset) are shown for d = 0.35 and t'/t =–0.4 in comparison to experiments [74] on
fully oxygenated YBCO materials (x =1). Although our theory does not succeed in giving
the correct amplitude of a T –1 1k the qualitative features of the NMR data are described surprisingly
well. Obviously, the broad magnetic peak in S (~q; o† at the AFM wave-vector
~Q AFM =(p,p) strongly enhances the relaxation rate on Cu sites while, due to a geometrical
cancellation ( 17 F a (~q ~Q AFM † 0†, the corresponding oxygen rate is rather insensitive to
nearly commensurate AFM fluctuations and therefore is governed by the long wavelength
part ~q 0 of the spin susceptibility [2]. For 63 Cu the nominal Korringa ratio S : (1/T 1 TK 2 S )
(K S denotes the spin part of the Knight shift) is at least one order of magnitude larger. As
can be seen from Fig. 7.10, for YBa 2 Cu 3 O 7 -type parameters, a Korringa (1/T 1 ! T) dependence
(dotted line) holds at both Cu and O sites below T* ~ 120K, demonstrating the existence
of a characteristic temperature T* as well as in all other near-optimum T c compounds
[2, 75]. T* is in good agreement with the coherence energy scale suggested experimentally
[75]. Above T* the 17 O NMR relaxation remains linear whereas the 63 Cu relaxation time
does not follow the Korringa law.
107

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.10: Spin-lattice relaxation rates 63 T –1 1k and 17 T –1 1k (inset) as a function of temperature. SB results
((+); t'/t =–0.4, d = 0.35) are compared with experiments (D) on YBa 2 Cu 3 O 7 [74].
In the oxygen-deficient compound YBa 2 Cu 3 O 6.6 , 1/( 63 T 1k T) shows a broad maximum
at about 150 K (Fig. 7.11), which reflects a strong deviation from the canonical Korringa behaviour.
In the normal-state regime the theoretical results agree even quantitatively with the
experimental 63 Cu NMR data [76]. At this point it is important to stress that the present theory
incorporates considerable band renormalization effects already via w 0 …~q; o†, especially
at low doping level. Thus a rather moderate strength J = 0.4 of the AFM exchange interaction
yields the experimentally observed enhancement of 63 T 1k . In striking contrast to the optimally
doped YBCO system, the Korringa relation is no longer satisfied for the planar 17 O
nucleus sites in the underdoped material (cf. inset Fig. 7.11). Instead, a different behaviour
17 T 1k T 17 K S = const was suggested to hold down to T c [76]. The unconventional T-scaling of
17 T 1k has been taken as a signature for another important feature of the normal-state spin dynamics,
the so-called spin-gap behaviour [2].
Figure 7.11: 63 Cu and planar 17 O relaxation data (^) for underdoped YBa 2 Cu 3 O 6.6 [76] are plotted vs
temperature. Theoretical results (6) are given at t'/t =–0.4, and d = 0.2.
108

7.4 Summary
Complementary measurements of the transverse spin-spin relaxation rate, T –1 2G ,have
provided further insights into the drastic change in the magnetic properties when passing
from the overdoped to the underdoped regime [67, 77]. As experimentally observed, we
found that T –1 2G increases (decreases) with increasing (decreasing) hole doping (temperature)
[78]. In order to detect the opening of a spin-pseudogap as a function of T, a powerful technique
is to measure the ratio T 2G /T 1 T [79] which is nearly constant above ~ 200 K for deoxygenated
YBa 2 Cu 3 O 6.6 [67]. The calculated temperature dependence of this quantity is
shown in Fig. 7.12 together with recent experimental results [67]. Most notably, the opening
of a spin-pseudo gap at T* ~ 135 K [79], i. e. well above T c , is clearly seen as a decrease of
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
is approximately constant above 150 K [2].
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
data (6) att'/t =–0.4, and d = 0.2.
7.4 Summary
In this work we have used a spin-rotation-invariant SB approach to investigate magnetic and
transport properties of the 2D t-t'-J model. Our main results are the following (see also [81]):
a) We present a detailed magnetic ground-state phase diagram of the 2D t-(t')-Jmodel, including
incommensurate magnetic structures and phase separated states. At finite t', a main
feature of the phase diagram, we would like to emphasize, is the existence of an AFM state
away from half-filling, which is locally and also globally stable against phase separation.
This result agrees with the experimentally observed AFM long-range order in the weakly
doped LSCO and YBCO compounds. In contrast, for the simple t-J model we observe no
AFM long-range order at any finite doping due to phase separation.
109

7 Slave-Boson Approach to Strongly Correlated Electron Systems
b) The next nearest-neighbour hopping process (t') incorporates important correlation and
band structure effects near half-filling. In particular, the t'-term can be used to reproduce
the FS geometry of LSCO, YBCO, and NCCO. Also the NNN hopping provides a possible
origin for the experimentally observed asymmetry in the persistence of AFM order of holedoped
and electron-doped systems.
c) The quality of the SRI SB approach was demonstrated in comparison with exact diagonalization
results available for the t-J model on finite square lattices with up to 36 sites. In
this case, the SB method yields an excellent estimate for the quasi-particle band renormalization.
d) Within the saddle-point and relaxation time approximation, our SB calculation of the
Hall resistivity in the t-t'-J model provides a reasonable explanation of the experimentally
observed doping dependence of R H on both hole-doped (La 2–x Sr x CuO 4 , YBa 2 Cu 3 O 6+x ) and
electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides.
e) Using a generalized RPA expression for the spin susceptibility and assuming that the
AFM correlations are spatially filtered by the hyperfine form factors, we have calculated the
temperature dependences of spin-lattice and spin-spin relaxation rates for planar copper and
oxygen sites. The results agree qualitatively well with various NMR experiments on YBa 2-
Cu 3 O 6+x . In addition, we can attribute the contrasting ~q-dependence of the magnetic structure
factor S (~q; o† seen in INS experiments for LSCO-type and YBCO-type systems to differences
in their fermiology.
Recently, our theory was improved to include (Gaussian) fluctuations beyond the paramagnetic
saddle-point approximation [27, 80]. We derived a concise expression for the spin
susceptibility w s (~q; o† of the t-t '-J model which does not have the standard RPA form. Then
we were able to show that the instability line obtained from a divergence of w s (~q; 0† is in
agreement with the PM ⇔ spiral state phase boundary in the saddle-point phase diagram,
which in fact proves the consistency of both approaches [27].
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112

8 Non-Linear Excitations and the Electronic Structure
of Conjugated Polymers
Klaus Fesser
8.1 Introduction
Conjugated polymers have attracted quite substantial research activities during the last 15
years [1]. On one hand these materials promise interesting technological applications most
of which are related to the possibility of a reversible charging and decharging of these systems.
These include various battery designs as well as storage of (charged) pharmaceuticals
which in turn can be released in a controlled way by application of an electrical current. For
the design of non-linear optical components they play an important role as organic materials
due to their processibility and fine tuning of their physical properties via suitable side
groups. Recently light-emitting diodes made from these systems have made conjugated polymers
to possible candidates for the construction of thin displays [2].
On the other hand a theoretical description of the whole class of these materials poses
interesting questions which are worth to study on their own right. As essentially quasi onedimensional
systems they give rise to the hope that many of these questions might be answered
analytically. The main problems are the nature of the insulator-metal transition observed
during doping and the origin of the intragap states, which are mainly responsible for
the relaxation processes relevant for the light-emitting properties. We have addressed both
aspects within this project and this article is organized accordingly. In Section 8.2 we present
the theoretical model stressing the relevant physics and discuss the related fundamental symmetries.
Then in Section 8.3 we adopt the view that the doping process mainly introduces
disorder into these systems and calculate various properties (density of states, optical absorption)
from this assumption. In Section 8.4 we investigate the non-linear excitations responsible
for the intragap states in more detail and close in Section 8.5 with an outlook on
still open problems.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
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8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
8.2 Models
Common to all conjugated polymers is the existence of a carbon backbone with an alternating
sequence of (short) double and (long) single bonds. The various systems, as polyacetylene,
polyparaphenylene, polypyrrole, polythiophene, polyparaphenylenevinylene, etc.,
differ only – from a physcist’s point of view – in the side groups and other chemical structures
which are energetically far away from the Fermi energy. Emphasizing the general aspects
of these systems, these chemical details are safely neglected if one restricts to properties
which are mainly due to the electrons around the Fermi energy. Thus the parameters in
the simple model presented below should be regarded as effective parameters including
some aspects of the interactions which are otherwise neglected.
Beside some biological systems such as b-retinol where a finite chain of a conjugated
polymer is present (and thus the properties of this polymer might be of relevance for the
biological functions of this material), there may be other, related systems where similar theoretical
concepts are or can be applied. These include the metal-halogen (MX) chains where
instead of one, as in the conjugated polymers, now two electron bands govern the essential
physics. On the same level the polyanilines can be modelled where the phonons of the conjugated
polymers have to be replaced by the librons, i. e. oscillations of the quinoid/benzoid
rings. Finally, a true one-dimensional modification of carbon, carbene with alternating single
and triple bonds, can also be understood along these lines.
So far we have only mentioned the construction of adequate models for the investigation
of physical properties. However, there are competing approaches to address the same
set of questions. One of these approaches uses sophisticated quantum-chemical codes [1c,d]
to calculate structure and electronic states of these materials. Although such a procedure
may be able to reproduce the observed properties of a specific material quite well it is very
difficult to obtain information about general trends and physical mechanisms. Therefore we
did not follow this route. Another well-established method for calculating ground-state properties,
namely the local density functional, has been used to some extent [3]. The drawback
of this method, however, is the finite size of the system which can be calculated within a
reasonable amount of computer resources. Therefore only a few questions have been addressed
via this approach.
Therefore, we shall argue in favor of a simple model which is capable to include the
most essential physics. Assuming the sp 2 -hybridisation of the carbon atom leaves one p-electron
per atom the others being incorporated into the bonds as s-electrons. It is the physical
behaviour of this single p-electron which is responsible for all the interesting effects.
Since there is only this one electron per site the polymer would be metal-like in its
ground state, in contrast to nature where one finds an energy gap of the order of 1 eV. In
one dimension, however, these electrons are unstable when they are coupled to the lattice.
Due to the Peierls effect, scattering off 2 k F phonons, a gap opens right at the Fermi energy
in accordance with the observations. In addition, electron-electron interaction contribute to
the size of this gap [4] stabilizing the semiconductor ground state even further. A single-particle
model along these lines has been put forward by Su et al. [5] in the early stages of conjugate
polymer research. Its parameters although, derived from a simple picture, should
114

8.2 Models
nevertheless be interpreted as effective parameters including parts of the electron-electron
interaction which is otherwise neglected. It has been shown [6] that for some ground-state
properties such redefinition can indeed be performed.
It turns out that there is a typical length scale in this model v F /D (with the Fermi velocity
v F and the electronic gap 2 D) which is considerably larger than the interatomic spacing
a. Thus for most physical properties of interest a continuum approximation is valid.
This stresses the generic aspects of the whole class of these materials even more.
In rescaled variables this model now reads
H ˆ X Z
s
dx ‡ s …x† f i 3@ x ‡ 1 …x†g s …x†‡ 1 Z
2
dx 2 …x†
…1†
Here c (x) is a two-component spinor describing electrons moving to the left (right)
along the one-dimensional polymer, s is the spin index which, except for external fields, is
not relevant here. The s 3 term is the kinetic energy originating from the hopping of electrons
between neighbouring sites. D (x) is the lattice order parameter where a constant
D (x) =D 0 describes a uniform dimerization of the lattice. The s 1 term is the electron-phonon
coupling responsible for the Peierls distortion. Finally we have an elastic energy for the
lattice. The kinetic energy is neglected within an adiabatic approximation. All quantities
have been scaled in order to have a single coupling constant (l & 0.2).
At this stage we postpone the non-linear excitations of this model to a subsequent
Chapter, we only discuss some of the symmetries of this model which are of relevance also
for these excitations. First we note that the form of the electronic part of the Hamiltonian
(Eq. 1) has exactly the form of a (relativistic invariant) Dirac operator in one dimension.
This correspondence has been exploited [7] successfully in obtaining solutions of this model
from results known in models of elementary particles. We remark here that similar analogies
can be made for specific forms of the Fermi surface also in higher dimensions. Thus the
connection between solid state physics and quantum field theory models can be used for a
better understanding of both.
In addition there are two symmetries which can be directly related to observable quantities.
The charge-conjugation symmetry, which can be expressed as H being invariant under
c?s 2 Kc (K complex conjugate operator), relates particle and hole states and thus is responsible
for the single particle spectrum being symmetric with respect to the Fermi energy,
which has been set to zero in H (Eq. 1). A more hidden property is the supersymmetry of
the electronic part H el (Eq. 1),
0 ˆ H el ;H ‰ el ; 3 Š : …2†
We have shown [8] that this more formal property is responsible for the asymmetry of
the optical absorption peaks of transitions involving intragap states. Both symmetries are absent
in the real systems under consideration, but for most questions this breaking of symmetries
is merely a question of quantity rather than of importance for the existence of these localized
states.
Finally we mention that Eq. 1 has been derived as a model for polyacetylene where
the ground-state order parameters D o and –D o yield the same energy. This is not the case for
115

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
the other polymers of this class. A simple extension of this model, introduced by Brazovski
and Kirova [9], corrects this. As a result all the solutions of Eq. 1 can be carried over [10]
to this more general model where only the location of the intragap states are now parameters
which can be adjusted to a specific material of interest. In consequence the model (Eq. 1)
can be considered as the most simple generic model of conjugated polymers.
8.3 Disorder
A major focus of theoretical investigations is the experimental observation that the electronic
structure and consequently the physical properties change drastically upon doping. Here
we restrict ourselves to the question how these properties are affected through a random
force introduced by these impurities. For simplicity we only consider isoelectronic doping,
i. e. the total number of carriers remains unchanged and the dopants introduce only scattering
centers for the electrons.
In consequence two different types (site or bond impurities) of scatterers can be identified.
The first gives rise to a random contribution to the on-site energy of an electron
whereas the latter modifies the hopping integral between neighbouring sites. In the spirit of
the continuum approximation we are thus lead to consider only backward (Eq. 3) or forward
(Eq. 4) scattering.
P R
H imp ˆ U b dx ‡ …x† 1 …x†…x x j † ; …3†
j
P R
H imp ˆ U s dx ‡ …x†1 …x†…x x j † ; …4†
j
x j denotes the (random) position of an impurity. Since this problem cannot be solved exactly
we have to redraw to approximate methods.
We have used three main routes.
In the first method we employ the first Born approximation for the impurity self-energy
[11]. This enables us to formulate equations of motion for the full space-dependent
Green functions and thus consider the influence of disorder on the non-linear excitations as
well [12] (see next Chapter). Furthermore, the replacement of Eq. 3 and Eq. 4 by random
(Gaussian) fields
Z n
H imp ˆ dx ‡ …x†
1
1
o
V b=s …x† …x† ;
which is correct within the Born approximation, allows the determination of the Green function
and higher correlation functions via a functional integral technique [13]. The averaging
procedure is formulated through the introduction of additional Grassmann variables in a
116
…5†

8.4 Non-linear excitations
supersymmetric way. We note that this supersymmetry is not related to the one discussed
earlier. Using the algebraic properties of these variables the average can be performed and
the resulting functional integral can be calculated via a transfer method in one dimension.
We note that this procedure still works when two coupled chains are considered but cannot
be done exactly in any higher dimension. As result we obtain the density of states and also
information about the extension of the (in principle) localized wave functions via the Thouless
formula. It turns out [14] that for moderate disorder a typical realistic chain length of
approximately 200 units is smaller than the localization length of the band states not too
close to the band edgesunverständlich. Thus for such systems these states can indeed be considered
as extended states even in the presence of disorder putting various models which
treat the propagation of electrons along one chain as metal-like on a firmer basis. This
method can easily be extended to higher dimensional systems which means that also the
coupling of polymer chains can be taken into account. Thus we are able to calculate, within
a saddle-point approximation, the optical absorption coefficient for a two-dimensional film.
For an orientation along the chains we find [15] a typical disorder induced broadening of the
absorption edge together with a shoulder on the low-energy side due to neighbouring chains
coupling. Both features agree quite well with experimental findings. Concomitantly, the absorption
perpendicular to the chain direction is featureless.
In the second method, going beyond Born, we examined the density of states within
the coherent potential approximation (CPA) which takes into account multiple scattering processes.
One might think that on this level impurity states are introduced in the gap. However,
we find [16] that the existence of such localized impurity states strongly depends on the relative
strength of site vs. bond impurity. Only states in the gap due to disorder can be found
if the site amplitude |U s | is stronger than the bond amplitude |U b |. Since CPA is an effective
medium theory this result might be questionable in one dimension.
In the third method, in order to check the CPA result, we performed a numerical simulation
[17] where for a given random distribution the electronic eigenvalues have been calculated
numerically and the results been averaged over a large number of realizations. It turned
out that the CPA results could be reproduced quite satisfactorily thus establishing the different
role of both types of impurities.
8.4 Non-linear excitations
Given the single particle states according to Eq. 1 with a uniform order parameter D o one
might expect that the lowest excited state corresponds to the lowest level in the conduction
band being occupied thus requiring the amount 2D o in energy, since the half-filled case considered
here the valence band with E |D o |is
empty. The model (2.1), however, exhibits the property that an additional carrier modifies
the order parameter locally, yielding a non-homogeneous D (x), and at the same time creates
through a rearrangement of the band statesadditional state(s) deep in the gap. Mathematically
this behaviour is due to the fact that Eq. 1 is a non-linear model, the non-linearity re-
117

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
sulting from the requirement that the total energy functional AH{D (x)}S has to be stationary
with respect to a variation of the order parameter D (x). This gives rise to a self-consistency
equation where this order parameter is governed by the occupied electronic states,
…x† ˆ
P ‡ …x† 1 …x† : …6†
occ
Various exact solutions to this problem are known. The most prominent one is the
kink (soliton) for the case of polyacetylene being,
p
…x† ˆ o thx= 2 :
…7†
As already mentioned, this kink does not exist for the other polymers of this class because
the ground state is non-degenerate in D o . Therefore the simplest non-linear excitation
in a more general sense is the bound kink-antikink pair (polaron) which is characterized by
two localized electronic states in the gap at +o o (e. g. o o /D o & 0.5 for polythiophene). In
addition there exist periodic solutions, e. g. the kink lattice with a periodicity determined by
the concentration of excess charges. Physically these states lower the total energy of the system
through an inhomogeneous order parameter D (x), which raises the energy (cf. Eq. 1)
and a much larger compensation through the intragap states, which altogether give a smaller
value of the total energy than the simple single-particle picture. For the polaron this gain in
energy amounts to E p /2D o = 0.98 < 1 for polythiophene. For trans-polyacetylene this number
is 0.90.
One can now envisage the processes which are involved in generating visible light. A
sufficiently strong electric field can promote a single electron into the lowest unoccupied
conduction band state. This state is unstable and relaxes on a fast time scale (femtosecond)
into a polaron-like state which then can recombine to the ground state under the emission of
radiation. A full microscopic understanding of all the processes involved is only possible if
for the dynamics of the lattice the degrees of freedom are fully taken into account as well as
the residual Coulomb interactions.
One step in this direction has been made within this project by Bronold [18]. In his
doctoral thesis he treats on an equal footing electron-electron (exciton) and electron-phonon
(polaron) interactions. The coupling of this system to short laser pulses gives rise to characteristic
changes of position and shapes of absorption/emission lines in optically stimulated
emission and inverse Raman scattering experiments. As these effects have a very short time
scale (femtoseconds), experiments are difficult to perform and a comparison with existing
theoretical predictions is not convincing. Nevertheless, one expects that this line of approach
will finally give a detailed understanding of the functioning of organic light-emitting diodes.
Having gained some insight into the non-linear mechanisms giving rise to localized
electronic (intragap) states, how dopants, mentioned in the previous Chapter, might influence
these states. Two alternatives are feasible:
a) the non-linear aspect dominates, i. e. the picture developed so far is still valid but only
some details are modified due to the disorder. On the basis of the Born approximation we
have indeed calculated [12] the electronic structure of the kink solution in the presence of
impurities and found that the spatial extension of this structure is enlarged when the doping
118

8.5 Perspective
concentration is raised. In accordance with the closure of the gap at a critical concentration
we find that the width of the kink tends to infinity at this value. The more interesting case
of the polaron, however, could not be solved satisfactorily due to numerical instabilities,
which could not be avoided. For details see Ref. [12].
b) the more subtle case of the competition between this non-linear mechanism and the
multiple scattering processes off the impurities, which is treated in terms of the T-matrix or
the CPA gives also rise to localized states. The experimental observation that the formation
of these states is independent of microscopic details leads to the conclusion that the non-linear
aspect is the dominant one. A fully self-consistent treatment of this problem with an impurity
located at x o and (for simplicity) a kink at x 1 gives complicated coupled integral equations
[19] which have not been solved. A recent investigation for the case of a kink lattice
shows both mechanisms working quite independently, however, the method employedwelche
Methode ? does not give a full self-consistent solution..
Summing up this Chapter we note that the non-linear excitations play a dominant role
in various physical applications of conjugated polymers. But a full understanding of the interplay
of various mechanisms giving rise to these localized states has not been reached yet.
8.5 Perspective
The potential technical applications have stimulated a myriad of experimental and theoretical
studies. It is obvious that similar investigations have also been performed, mostly along
different lines, by various groups. The disorder aspect, responsible for the observed metalinsulator
(or semiconductor) transition, in conjunction with a kink (soliton) or polaron lattice
has been treated by many authors [20] in all kinds of approximate approaches. All these studies
resulted in the same prediction that such an M-I transition would occur at the experimentally
observed dopant concentration level. It is now clear from the foregoing Chapter
that only a fully self-consistent treatment will give a satisfactory answer to this question.
But since the applications for conjugated polymers envisaged at present focus on optical
properties rather than electronic transport this question has been lost out of sight. Still, the
nature and dynamics of localized electronic states in these materials must be fully understood.
In addition, from an application point of view, these polymers appear to be similar to
conventional semiconductors. The recent proposal [21] that the non-linear excitations actually
modify the conventional picture, e. g. at the interface between a polymer and a metal (or
conventional semiconductor) space charge regions (depletion layers) differ from an inorganic
semiconductor, finds renewed interest because transistors made from conjugated polymers
are feasible and of interest for the integration of optical and electronic components.
From a theoretical point of view the functional integral techniques promise interesting
opportunities to make contacts to other areas of research. Universal properties have been dis-
119

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
covered [22] in the absorption spectrum, discussed earlier, as well as in the distribution of
energy levels in certain disordered systems [23]. We propose that there is a close connection
between both aspects. A careful treatment of the underlying correlation functions, including
a more general type of disorder than discussed here, has to be performed. We expect that
the result will lead to new universality classes which technically spoken will show up in different
supersymmetric non-linear s models. Work along this line is in progress.
In summary, conjugated polymers pose interesting problems, both for applications and
pure theoretical studies. Both aspects have matured during the past 15 years but still questions
of a more general nature are left unanswered.
Acknowledgements
The author is indebted to all his collaborators for enlightening and stimulating discussions
as well as fruitful collaborations. Thanks to A.R. Bishop, F. Bronold, H. Büttner, D.K.
Campbell, K. Harigaya, U. Sum, Y. Wada, and M. Wolf.
References
1. a) A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su: Rev. Mod. Phys. 60, 781 (1988)
b) T.A. Skotheim (Ed.): Handbook of Conducting Polymers. Dekker, New York (1986)
c) J.L. Brédas, R. Silbey (Eds.): Conjugated Polymers. Kluwer, Dordrecht (1991)
d) W.R. Salaneck, I. Lundström, B. Ranby (Eds.): Conjugated Polymers and Related Materials. Oxford
University Press, Oxford (1993)
e) Proc. Int. Conf. Science Technology of Synthetic Metals: ICSM ’90, Synth. Met. 41–43 (1991);
ICSM ’92, Synth. Met. 55–57 (1993); ICSM ’94, Synth. Met. 69–71 (1995)
2. J.H. Burroughes et al.: Nature 347, 539 (1990)
3. P. Vogl and D.K. Campbell: Phys. Rev. Lett. 62, 2012 (1989); Phys. Rev. B 41, 12 797 (1990)
4. For a review see: D. Baeriswyl, D.K. Campbell, S. Mazumdar, in: Conjugated Conducting Polymers,
H. Spiess (Ed.), Springer Series in Solid State Sciences 102, 7 (1992)
5. W.P. Su, J.R. Schrieffer, A.J. Heeger: Phys. Rev. B 22, 2099 (1980)
6. D. Baeriswyl, E. Jeckelmann, in: Electronic Propteries of Polymers, H. Kuzmany, M. Mehring, S.
Roth (Eds.), Springer Series in Solid State Sciences 107, 16 (1992)
7. D.K. Campbell and A.R. Bishop: Nucl. Phys. B 200, 297 (1982)
8. U. Sum, K. Fesser, H. Büttner: Ber. Bunsenges. Phys. Chem. 91, 957 (1987)
9. S. Brazovskii and N. Kirova: Pis’ma Zh. Eksp. Teor. Fiz 33, 6 (1981) [JETP Lett. 33, 4 (1981)]
10. K. Fesser, A.R. Bishop, D.K. Campbell: Phys. Rev. B 27, 4804 (1983)
11. a) K. Fesser: J. Phys. C 21, 5361 (1988)
b) K. Iwano and Y. Wada: J. Phys. Soc. Jpn. 58, 602 (1989)
120

References
12. F. Bronold and K. Fesser, in: Nonlinear Coherent Structures in Physics and Biology, M. Remoissenet
and M. Peyrard (Eds.), Springer Lecture Notes in Physics 393, 118 (1991)
13. K.B. Efetov: Adv. in Phys. 32, 53 (1983)
14. M. Wolf and K. Fesser: Ann. Physik 1, 288 (1992)
15. M. Wolf and K. Fesser: J. Phys. Cond. Matter 5, 7577 (1993)
16. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. Lett. 63. 2401 (1989);
Phys. Rev. B 42, 1268 and 1276 (1990)
17. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. B 43, 4141 (1991)
18. F. Bronold: Doct. Thesis, Univ. Bayreuth (1995)
19. K. Fesser: Prog. Theor. Phys. Suppl. 113, 39 (1993)
20. a) E.J. Mele and M.J. Rice: Phys. Rev. B 23, 5397 (1981)
b) G.W. Ryant and A.J. Glick: Phys. Rev. B 26, 5855 (1982)
c) S.R. Philpott et al.: Phys. Rev. B 35, 7533 (1987)
d) E.M. Conwell, S. Jeyadev: Phys. Rev. Lett. 61, 361 (1988)
21. a) S.A. Brazovskii, N. Kirova: Synth. Met. 55–57, 4385 (1993)
b) G. Paasch and T.P.H. Nguyen: unpublished (1995)
22. K. Kim, R.H. Mckenzie, J.W. Wilkins: Phys. Rev. Lett. 71, 4015 (1993)
23. B.D. Simons and B.L. Altshuler: Phys. Rev. B 48, 5422 (1993)
121

9 Diacetylene Single Crystals
Markus Schwoerer, Elmar Dormann, Thomas Vogtmann, and Andreas Feldner
9.1 Introduction
Polydiacetylenes can be grown as macroscopic polymer single crystals [1–3]. This property
is unique. They comprise one linear polymer axis and can have spatial extensions of up to
several millimetres or more in all three spatial directions (Fig. 9.1). The covalent chemical
bonds along the polymer axis make them mechanically strong along the corresponding crystallographic
axis. Their Young’s modulus is about a quarter of the Young’s modulus of steel
[4, 5] and their tensile strength along the polymer axis has been reported to exceed that of
steel [6]. Weak bonds of van der Waals-type perpendicular to the polymer axis are responsible
for extremely low dimensional – generally one-dimensional – macroscopic electronic
properties of these polydiacetylene single crystals. They are insulators, they can become
pyro- or ferroelectric, and they show large optical non-linearities. These electronic and optical
properties are primarily determined by the p-electron system along the polymer axis.
Most polydiacetylenes differ only in the substituents R and R' (Scheme 9.1). But the electro-
Figure 9.1: Photo of macroscopic paratoluylsulfonyloximethylene-diacetylene (TS6) single crystals under
polarized light. Monomer (top), polymer (bottom).
122 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

9.1 Introduction
Scheme 9.1: Polydiacetylene in its isomeric structures: acetylene-type (a) and butatriene-type (b). R
and R' are the substituents for different diacetylenes (see also Tab. 9.1).
nic structure of their all-trans planar carbon chain is at first approximation of acetylene-type
(a) rather than of butatriene-type (b). Both isomeric structures contain a non-interrupted
p-electron system along the carbon chain.
The term polydiacetylene is somewhat puzzling, at least for a physicist. However, it
becomes quite clear if one takes into account the structure of the diacetylene monomer
(Fig. 9.2). A typical substituent R is paratoluylsulfonyloximethylene (Scheme 9.2).
The diacetylene with R = R' = paratoluylsulfonyloximethylene is termed TS6 (sometimes
TS). It was shown by G. Wegner and his co-workers in a series of works, published in
the early 1970s [1, 7], that large molecular crystals can be grown from a solution of TS6,
e. g. in acetone, and that these monomer diacetylene crystals can be converted by a topochemical
(or solid state) 1,4-addition reaction to the polydiacetylene single crystals
(Fig. 9.2). The crystal structures of TS before and after the reaction have been investigated
in detail by Kobelt and Paulus [8], Bloor et al. [8], and Enkelmann [9] and are sketched in
Fig. 9.3 and in Tab. 9.1.
Figure 9.2: The monomer diacetylene crystal is converted by a topochemical or solid state reaction to
the polydiacetylene single crystal.
123

9 Diacetylene Single Crystals
Scheme 9.2: Diacetylene monomer with the substituents R = R' = paratoluylsulfonyloximethylene (TS6).
Figure 9.3: Monomer and polymer crystal structure of TS6 deduced from X-ray data [9]. Note the reactive
carbene at the chain end.
Both crystal structures are monoclinic with two monomers or monomer units of different
orientation per unit cell (Fig. 9.3 sketches only one of these two.). The chemical bond
between two carbons of nearest neighbour diacetylenes (1,4-addition) results in the linear
polymer chain and the small change in the lattice parameter along the b-axis prevents the
destruction of the macroscopic single crystal during the topochemical reaction. Several surfaces
of diacetylene crystals have been studied by atomic force microscopy (AFM) in order
to investigate both, the single crystal surface structure and solid state reactions at the surface
[129, 131].
124

9.1 Introduction
Table 9.1: TS monomer and polymer crystals (monoclinic, space group P2 1 /c) [9].
T/K a/Å b/Å c/Å b/ 8 D x
g/cm 3
TS monomer 120 14.61(1) 5.11(1) 25.56(5) 92.0(5) 1.46
TS monomer 295 14.60 5.15 15.02 118.4 1.40
TS monomer 295 14.65(1) 5.178(2) 14.94(1) 118.81(3) 1.40
TS polymer 295 14.993(8) 4.910(3) 14.936(10) 118.14(4) 1.483
TS polymer 120 14.77(1) 4.91(1) 25.34(2) 92.0(5) 1.51
The topochemical reaction can be induced thermally and/or photochemically and/or
by electron-beam irradiation. For TS the thermal conversion versus time (Fig. 9.4) is
strongly temperature dependent and highly non-linear. The thermodynamics of the integral
reaction has been investigated extensively by Bloor et al. [10], Eckhardt et al. [11], Chance
et al. [11], and others. The reaction diagram (Fig. 9.5) for TS shows that the dark reaction is
thermally activated and has an activation energy of 1 eV per monomer. It is exothermic with
a polymerization enthalpy of 1.6 eV per addition of one monomer. The entire reaction is irreversible
and the TS6-polydiacetylene (PTS) crystals are not solvable in ordinary solvents.
During the solid state reaction almost all properties of the diacetylene crystals change
drastically, e. g. the transparent monomer crystals are converted to polymer crystals with
highly dichroic, strongly reflecting surfaces which contain the b-axis. In transmission the
polydiacetylene crystals can only be investigated as thin films (Fig. 9.6). Their absorption
spectrum, e. g. for TS, clearly shows the vibronic spectrum due to the single, double and triple
bonds of the polymer chain [12]. Batchelder et al. [13] investigated extensively the optical
absorption, reflection, and Raman spectra of TS single crystals.
While these experiments were directed towards the study of the electronic excitations
of the bulk and their coupling to the vibrations, Sebastian and Weiser [14] investigated the
Figure 9.4: Time conversion curves for the thermal polymerization of PTS at 60 8C (.), 70 8C (#), and
80 8C (d) [9].
125

9 Diacetylene Single Crystals
Figure 9.5: Reaction diagram for the thermal polymerization of TS (solid curve) and photopolymerization
of 4BCMU (dashed curve) [10, 11].
Figure 9.6: Absorption spectra of a thin TS diacetylene single crystal for light polarized parallel and
perpendicular to the polymer axis b (T = 300 K). Monomer (M) and polymer (P) absorption.
defects by electroabsorption. As an example Fig. 9.7 shows the absorption and the electroabsorption
spectra, which they have analyzed in great detail.
Since about 1980 and especially during our work for the Sonderforschungsbereich 213
we have synthesized several new diacetylenes, the substituents of which are shown in
Tab. 9.2 [15]. For selected diacetylene single crystals we have investigated:
126

9.1 Introduction
Figure 9.7: Absorption a and electroabsorption Da of photoproducts in PTS-monomer as a function of
ko. Full curve: experimental spectra, dashed curves: fit by Lorentzian and a charge transfer model for
Da [14].
a) in Section 9.2 the elementary steps and structures during the solid state photopolymerization
by transient optical spectroscopy, by electron spin resonance (ESR), and by electron
nuclear double resonance (ENDOR);
b) in Section 9.3 the application of the photopolymerization of thick diacetylene single
crystals as a very effective holographic storage process;
c) in Section 9.4 the tailoring of diacetylenes as ferro or pyroelectric crystals, which do
not demand considerable efforts for the poling processes and which show good thermal stability;
d) in Section 9.5 the optical non-linearity of second and of third order and their application
for an optical device with femtosecond time resolution.
The present paper is a review of our work with diacetylene single crystals.
127

9 Diacetylene Single Crystals
Table 9.2: A survey of the investigated diacetylenes [15].
Survey of the investigated diacetylenes
R 1 C C C C R 2
DNP
TS
PD-TS
CD -TS
2
FBS
O 2 N
CH 2 O NO
CH 2 O SO 2
D
CD 2 O SO2
D
H
CD2 O SO2
H
CH
2
O SO
2
2
CH 3
D
CD 3
D
H
CH 3
H
F
R 2
ability to polymerize
therm. γ
= R 1
= R 1
= R 1
= R 1
= R 1
+++
+++
+++
+++
+++
-
+++
+++
+++
+++
IPUDO
Name R 1
CH O
O CH 3
(CH 2 ) 4 O C NH CH
= R 1
-
+++
NP/PU
CH 2
O NO2
CH 3
O
2 C NH
-
+
O
NP/4-MPU CH2 O NO
2 CH2 O C NH
CH 3 - +
NP/MBU
DNP/MNP
O 2
O H
CH2 O NO2 CH2 O C NH C + +
CH 3
(-)(S) or (+)(R)
N
O 2 N
CH
NO CH O CH
+++
2 O 2
2
3
DNP/PU
CH
2
N
O
O 2
O
NO2 CH2 O C NH
- -
DNP/4-MPU
O2
N
CH
2 O
O
NO2 CH2 O C NH CH 3
- -
O 2 N
DNP/DMPU CH2 O
NO2
CH
O CH3
O NH
2 C
CH 3
- -
DNP/MPU
O 2 N
CH O
2
O H
NO2 CH2 O C NH C
CH
(-)(S) or (+)(R)
3
- -
TS/FBS CH2 O SO2
CH 3 CH2
O SO 2 F +++
FBS/TFMBS CH2 O SO 2 F CH2 O SO2 CF 3 +++
128

9.2 Photopolymerization
9.2 Photopolymerization
9.2.1 Carbenes
The aim of this Chapter is to review our spectroscopic work towards the analysis of both,
the electronic structures and the dynamics of the intermediate reaction products (Fig. 9.2),
during the photopolymerization, i. e. after the excitation of the solid state reaction by light.
For the entirely thermally activated solid state polymerization detailed spectroscopy of the
intermediate states turned out to be difficult or not very efficient. One exception was the
identification of carbenes as reactive species during the thermal solid state polymerization
of TS6 (Fig. 9.8).
In contrast to radicals with one non-bonded electron carbenes have two non-bonded
electrons. It has been shown for the first time by Wassermann et al. [16] that the electronic
ground state of pure methylene (:CH 2 ) is a triplet state, where the total spin quantum number
S is 1. Because of their non-centrosymmetry most molecular triplet states show a splitting
into three components even in the absence of an external field. This splitting is due to the
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
2hν
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R C R
C
C
R
C
C
R
C
C
C
R
kT
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R C R
C
R
C
C
C
kT
R
C
C
C
R C
R
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R C R
C
C
C
R C R
C
C
R C
C R
C
C
R
C
C R
C
C
C
R
kT
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R C R
R C
C
C
C
R C
R
C
C
C
R
Figure 9.8: UV photopolymerization of TS6: The monomer crystal is irradiated with an UV-flash. The
dimer is formed from the monomer by a photoreaction, then a series of thermally activated monomer
addition reactions leads via the diradicals (DR) and dicarbenes (DC) to the polymer.
R
C
C
C
C
C
C
C
C
C
R
R
R C C R
kT
R
C
C
C
C
R
R
C
C
C
C
R C R
R
R
C
C
C
C
C
R C
C
C
C
C
C
C
C
C
C
C
R
R
R
R C C R
kT
C
R C
R
C
C
C
R
R
C
C
C
C
R C R
R
R
R
C
C
C
C
C
C
C
C
R C R
C
C
R C C
C
C
C
C
C
C
C
C
C
R C
C
R
R
R
R
R
C
C
C
R
kT
R
R
R
R
R
R
R
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
R
R
R
R
R
R
C
R C
R
C
C
C
R
129

9 Diacetylene Single Crystals
magnetic dipole-dipole coupling of the two unpaired electrons and is called zero-field splitting.
In an external magnetic field the resulting fine structure of the ESR spectra is highly anisotropic,
i. e. it strongly depends on the direction of the external magnetic field with respect
to the orientation of the tensor describing the magnetic dipole-dipole interaction. This interaction
is of course an intramolecular property and therefore single crystals are ideal candidates
for the measurement and the quantitative analysis of molecular triplet states by ESR. As originally
shown by the work of Hutchison and Mangum [17], and van der Waals [18] the ESR
spectrum of a molecular triplet state is described by the spin Hamiltonian T H s [19]
T H s ˆ B B o g S^
‡D S^ 2
z ‡ E…S^ 2
x S^ 2
y † ; …1†
where m B is Bohr’s magneton, B 0 the external magnetic field, g the spectroscopic splitting
factor, ^S the spin operator for a spin with total spin quantum number S = 1, ^S u with u = x, y, z
are the components along the principal axes of the magnetic dipole-dipole interaction tensor.
D and E represent the two independent values of this tensor, the trace of which is zero. They
usually are called zero-field splitting parameters:
D ˆ o
4p
E ˆ o
4p
3
4 g2 2 z 2
B hr2 r 5 i ; …2†
3
4 g2 2 x 2
B hy2 r 5 i ; …3†
r is the distance of the two unpaired electrons and x, y, and z are the components of r along
the principal axes of the magnetic dipole-dipole interaction tensor of these two electrons.
Only for systems of cylindrical symmetry (around z) the zero-field splitting parameter E
vanishes (E = 0). And only for spherical symmetry (Ax 2 S = Ay 2 S = Az 2 S = 1/3 Ar 2 S) the entire
fine structure is zero. The values of D and E for the triplet carbenes as detected during the
purely thermal solid state polymerization [20] are shown in Tab. 9.3 in comparison with typical
values of different molecular triplet states. By these values and especially by the strong
anisotropy of the fine structure in the ESR spectrum these carbenes (as sketched in Fig. 9.8)
are clearly identified as reactive species during the pure thermal solid state polymerization
of TS. They will play a major role in the electronic structures of the intermediate products
during the photopolymerization as described in the following paragraphs.
Table 9.3: Zero-field splitting parameters for the reactive species during the thermal solid state polymerization
of TS [20], pure methylene [16], diphenylmethylene [21], and benzene [22] in their first excited triplet
state.
D
hc =cm–1
E
hc =cm–1
TS 0.2731 –0.0048
:CH 2 0.6636 0.0003
:C(C 6 H 5 ) 2 0.39644 –0.01516
C 6 H 6 0.1581 –0.0046
130

9.2 Photopolymerization
9.2.2 Intermediate photoproducts
Further experiments for analyzing the electronic structure of intermediate states were not
very successful, until Sixl et al. [23] and Bubeck et al. [24] published their first low-temperature
spectroscopic experiments on partially photopolymerized TS crystals. Thereby, the
monomer crystal is cooled to 4.2 K in the dark. Then it is irradiated with UV light
(l ^ 310 nm) for a short period. After this procedure a large number of different species
show up in both, the optical absorption and the ESR spectrum. They persist at helium temperature.
Subsequent annealing in the dark produces further intermediate reaction products
which are also identified by their optical or ESR spectra. Finally, further irradiation with
visible light, which is absorbed only by the intermediate reaction products, produces still
more and different reaction products. In a comprehensive series of investigations [25–47]
most of these intermediate products have been identified and classified. Moreover, the mechanisms
of their production and their reaction kinetics have been analyzed. According to
Sixl, there exist three different series of intermediate products:
1. The diradical-dicarbene series: DR 2 , DR 3 , DR 4 , DR 5 , DR 6 , DC 7 , DC 8 , DC 9 , DC 10 ,
DC 11 … polymer;
2. The asymmetric carbene series (AC);
3. The stable oligomer series (SO).
The DR-DC series leads directly from the monomer to the polymer. It is initiated and
processed in the following simple and clear way. The monomer crystal is irradiated with an
UV flash and subsequently rests in the dark. The flash excites the monomers and produces
dimers (DR 2 ). These react by a thermally activated step by step addition of monomers, as illustrated
in Fig. 9.8. In the following paragraphs we will show for a few selected examples
how these results have been achieved and we will present details of both, the electronic
structures and the dynamics.
The AC and the SO series are produced by additional irradiation with light, i. e. these
are photoproducts which do not necessarily arise during the solid state polymerization of
TS6. Although they are an important part of the entire variety of structures in partially polymerized
TS6 crystals, we will not treat them in this review. They have been described extensively
by Sixl in his papers cited above and in Ref. [48].
9.2.3 Electronic structure of dicarbenes
9.2.3.1 Electron spin resonance of quintet states ( 5 DC n )
As an example for the electronic structure analysis of the intermediate states with ESR, we
will review below the ESR spectra of the dicarbenes DC 7 …DC 13 [28, 32, 42]. They are
characterized by their fine structure and temperature dependence. Figure 9.9 shows the ESR
131

9 Diacetylene Single Crystals
ESR-Signal
T
Q
T
T
Q
Q Q T
Q
Q
Q
T
Q
T=10 K
ν =9,46 GHz
T
0 200 400 600
Magnetic Field B 0/mT
Figure 9.9: The ESR spectrum of perdeuterated TS after irradiation for 1000 s. The signals marked
with T arise from triplet states and those with Q from quintets. The T-lines are microwave saturated. The
magnetic field B 0 is oriented parallel to the z-axis of the quintet fine structure tensor [28].
spectrum of a perdeuterated TS crystal which has been irradiated with UV light (313 nm) of
a mercury high-pressure arc lamp (HBO 200) at 4.2 K for about 1000 s. All ESR lines in
Fig. 9.9 labelled with Q are due to dicarbenes in their quintet state (S = 2). Prior to the UV
irradiation no ESR signal is observed.
The temperature dependences of the ESR signals (Fig. 9.10) show one common feature:
the ESR intensities vanish for T ? 0, i. e. the ESR signals are thermally activated and
the ground state is spinless. But the temperatures for the maximum intensities are different
for each ESR signal, indicating different activation energies.
1
10
5
T/K
2
Signal Intensity
0,5
0
0
0,2 0,4 0,6 0,8
1/T/K -1
Figure 9.10: Temperature dependence of the ESR intensities of dicarbenes DC 9 ,DC 10 , and DC 11 . The
calculated lines have been fitted by Eq. 9 with the activation energies De* SQ for the quintet states [33, 47].
132

9.2 Photopolymerization
The resonance fields for fixed microwave frequencies are strongly anisotropic.
Figure 9.11 shows as an example this anisotropy for the dicarbenes DC 9 ,DC 10 ,DC 11 , and
DC 12 , respectively. The external field has been rotated to the polymer backbone plane. Each
anisotropy belongs to one distinct temperature dependence.
The following model (Fig. 9.12 and inset Fig. 9.13) describes quantitatively the anisotropic
fine structure and the temperature dependence. The ground state (|S>) of each dicarbene
in first order is a singlet state with the total spin quantum number S = 0. The excited
state (|Q>) in first order is a quintet state with S = 2. The excitation energy is De SQ . The
quintet state is built up by the electronic coupling of two triplet carbenes at both ends of the
oligomer via exchange interaction. R 12 is the distance between the two triplet carbenes. If
the singlet-quintet splitting De SQ is large as compared to the magnetic dipole-dipole coupling
(zero field splitting) within the triplet carbenes, the total spin quantum numbers S = 0
and S = 2, respectively, are good quantum numbers, i. e. the quintet state is pure. The spin
Hamiltonian for this case has been analyzed by Schwoerer et al. [34].
Figure 9.11: Angular dependence of the resonance fields B 0 of the 4 dicarbene structures DC 9 ,DC 10 ,
DC 11 , and DC 12 in perdeuterated TS-diacetylene crystals. The crystal is rotated so that the external
magnetic field B 0 is in the plane of the polymer backbone. The b-axis is the direction of the polymer
chain, y and z are the principal axes of the fine structure tensor. The curves are fitted to the experimental
points by computer calculations. A and B indicate the two magnetically equivalent directions of the
molecular orientation within the monoclinic unit cell. Dots: experimental values; lines: calculated by
Q H S 0 (Eqs. 4–7) [32, 42].
Figure 9.12: Dicarbene configuration. The dicarbene molecule consists of n diacetylene units with two
identical triplet carbene chain ends. D t and E t are the triplet fine structure parameters of the S = 1 carbene
species. j gives the orientation of the triplet fine structure z-axis with respects to the crystal b-
axis.
133

9 Diacetylene Single Crystals
Figure 9.13: Experimental values for the singlet-quintet splitting De SQ for seven different dicarbenes
DC 7 …DC 13 [42].
If De SQ is smaller than or somewhere in the order of the magnetic dipole-dipole coupling
then the singlet and quintet states are mixed. The spin Hamiltonian Q H S for this general
case has been derived in the elegant work of Benk and Sixl [35]:
Q H S ˆ g B B 0 …^S 1 ‡ ^S 2 †‡ 1 6 " SQ …^S 1 ‡ ^S 2 † 2 ‡H DD
S
…4†
The first term of Eq. 4 represents the electronic Zeeman term, ^S 1 and ^S 2 the spin operators
for two triplet carbenes, and the second term represents the electronic exchange interaction.
If ^S 1 and ^S 2 couple to a quintet state (S = 2), (^S 1 + ^S 2 ) 2 = S 2 = S(S + 1) = 6. If
they couple to a singlet, S = 0. Therefore, this term directly results in the energy level
scheme, indicated in the inset of Fig. 9.13. The pure singlet and the pure quintet states are
split by De SQ which turns out to be the characteristic property of each dicarbene. The third
term of Eq. 4 represents the magnetic dipole-dipole coupling of the two triplet carbenes:
H DD
S
^
ˆ D …S 2 1z
^
1
3 S2 1 †‡E…S2 1x
X S^
1x S^
2x ‡ XAS^
1y S^
2y
^
S 2 ^
1y†‡D…S 2 2z
1
^
3 S2 2
^
†‡E…S 2 2x
^
S 2 2y†‡
X…1 ‡ A† S^
1z S^
2z ‡ Xa…S^
1z S^
2y ‡ S^
1y S^
2z† …5†
D and E are the fine structure parameters of the identical triplet carbene chain ends;
x, y, and z are the principal axes of the corresponding fine structure tensors. The intercarbene
magnetic dipolar interaction is represented by the parameter
134

9.2 Photopolymerization
X ˆ g 2 2 B … 0=4p†R 3
12 : …6†
The geometrical factors A and a are only dependent on the orientation j of the fine
structure tensor z-axis with respect to the b-axis of the crystal.
A ˆ 1 3 sin 2 '; ˆ 3 sin 2': …7†
2
The diagonalization Q H S of yields the allowed ESR transitions, their resonance fields,
and their dependence on the orientation of the external magnetic field B 0 . At a first sight,
the number of fit parameters seems to be high: D, E, g, the orientation of the triplet fine
structure tensor, R 12 , and De SQ . But besides R 12 and De SQ these parameters are well-known
from earlier ESR experiments on triplet carbenes [20, 49, 50]. Therefore, R 12 and De SQ are
the only free to fit parameters. Figure 9.11 shows the result of the fit. The ESR anisotropy
was calculated by exact diagonalization of Q H S with the fitting parameters R 12 and De SQ .In
all cases the fit is almost perfect. Furthermore, it turns out that the intertriplet magnetic dipole-dipole
interaction does not influence the results if R 12 exceeds 12 Å. Therefore, only
the parameter De SQ remains to fit for each dicarbene. The obvious differences between the
anisotropies of the different dicarbenes (Fig. 9.11) are due to De SQ only! The result is shown
in Fig. 9.13 where De SQ is decreasing exponentially with increasing number n of monomer
units:
" SQ ˆ " 0 SQ e R=R 0
; …8†
with R = n1 m (1 m is the length of the monomer unit within the oligomer). The slope in
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
was deduced by Neumann [47]. But even without the knowledge of this origin the value of R 0
as compared to a 0 shows that the triplet carbene is highly delocalized and not at all restricted
to the end of the oligomer, as one might think because of Fig. 9.12. The model also gives the
temperature dependencies of the dicarbene signals. Because in first order the ground state is
spinless, the ESR intensities I of the quintet states (beyond Curie’s law) must be thermally activated:
I / 1 T ‰5 ‡ exp…" SQ =kT†Š :
…9†
The lines in Fig. 9.10 are calculated with Eq. 9 by fitting De* SQ . For the dicarbenes
DC 8 ,DC 9 ,DC 10 , and DC 11 , both values, De* SQ and De SQ , are determined respectively. Within
the experimental uncertainty they are identical [32, 42]. This latter result is an excellent
proof for the dicarbene model because De* SQ and De SQ were determined from two completely
different properties: the anisotropy of the fine structure and the intensity of the ESR spectra.
We therefore have no doubt that we really did observe dicarbenes (as sketched in Figs. 9.8
and 9.12). They are an ideal modelling substance for short linear oligomers of diacetylenes
which are perfectly oriented in the single-crystal lattice. The longest dicarbene (DC 13 ) observed
has according to our model a length of (R =1364,9 Å) 64 Å!
135

9 Diacetylene Single Crystals
9.2.3.2 ENDOR of quintet states
The most important result of the preceding Chapter is the delocalization of the spins S = 1
of the two triplet carbenes which is necessary for their coupling over a distance of up to
64 Å (!) to the well defined quintet states. It was therefore attractive to measure and analyze
the ENDOR spectrum of at least one quintet dicarbene. ENDOR should detect at
least the protons of the CH 2 groups of the substituents if the nuclear spins of these protons
are hyperfine coupled with the electron spin of the triplet carbene. As compared to
ENDOR with an electron spin of S = 1/2, the complication is the high anisotropy of the
five electronic Zeeman levels Q u (u = 1 … 5) of the quintet state (Fig. 9.14). Not only
their energy separation, i. e. the ESR transition fields (Fig. 9.11), are strongly dependent
on the direction of the external field. Also their effective spin S eff , i. e. the expectation
value of the spin, is dependent on the direction and on the strength of the external field
B 0 . Therefore in this case the orientation quantum number m s is an unsuitable quantum
number not only because of the large zero field splitting, as expressed by D, but also because
of the singlet-triplet mixing, as expressed by De SQ . As described in the preceding
paragraph this problem has been solved with high accuracy, Hartl et al. [36] measured
the ENDOR spectra and their anisotropies for one quintet dicarbene ( 5 DC 10 ), which is accessible
most comfortably at T = 4.2 K (Fig. 9.10) and for which the total spin quantum
number (S = 2) is a good quantum number. The aim of these experiments was to determine
the hyperfine coupling constants with the above-mentioned protons and subsequently
to extract the electron spin density from these values, i. e. the delocalization of
the triplet carbene quantitatively.
Figure 9.14: Quintet state with an external field B o interacting with one proton (I = 1/2). All allowed
ESR transitions (a–d) and NMR transitions (1–5) are shown. By ENDOR, in first order, only the two
NMR transitions directly connected to the observed ESR line are detectable [36].
136

9.2 Photopolymerization
The spin Hamiltonian Q H S;i of a quintet dicarbene coupled to one individual proton,
numbered i, is
Q H s;i ˆ QH 0 s ‡ g I K B 0 ^I i ‡ ^SA i I i :
…10†
The first term in Eq. 10 is given by Eq. 4 and the second is the nuclear Zeeman energy.
^I i is the nuclear spin operator for nuclear spin 1/2. The third term is the hyperfine interaction
of the individual proton i, as defined by the hyperfine tensor A i . ^S is the total electron
spin operator, ^S ˆ ^S 1 ‡ ^S 2 . As nuclear dipole-dipole interaction can be neglected we
will omit the index i in the following.
The nuclear terms in Eq. 10 are small as compared to the electronic terms. Therefore,
we treat them in first order perturbation theory, taking the solutions of Q H 0 S as basis. Q H 0 S
has five quintet eigenstates |Q u S , u = 1, 2, 3, 4, 5. For very high fields they become the
high field states |Q m S, for which ^S z |Q m s
S = km s |Q m s
S,m s = +2, +1, 0, –1, –2. But for the
fields used in ordinary ESR spectrometers the electronic Zeeman energy is in comparison to
the fine structure not large and therefore the electron spin is not quantized along the external
field B 0 [39]. This results in a strong |Q u S-dependence on the direction of B 0 with respect to
the crystal axes.
From Q H 0 S the effective spin
S u eff ˆ
D E
Qu j ^S jQ u
…11†
can be calculated exactly [34]. It is this electron spin which interacts via A with the proton
spin.
The first order perturbation theory of the nuclear terms calculates the shift Dn u of the
individual proton Larmor frequency with respect to the free proton Larmor frequency n F
(ENDOR shift):
hDn u = hn u – g I m K |B 0 | , (12)
(g I m K |B 0 |/h = n F ). The result of the calculation of the Larmor frequencies n u of the hyperfine
coupled protons is [36, 43]:
h u ˆjS u eff A g I K B 0 j : …13†
For a quintet state one should observe 5 different ENDOR lines per proton. This is illustrated
in Fig. 9.14 where the observed NMR transitions in the ENDOR experiment are indicated
by the ciphers (n) = (1), (2), (3), (4), and (5); the first order ESR transitions are indicated
by the lower case letters (a), (b), (c), and (d).
Fig. 9.15 shows four ENDOR spectra as detected via the ESR transitions (a), (b), (c),
and (d), respectively. Four protons i = 1, 2, 3, and 4, respectively, are clearly separated from
the free proton frequency n F . In the vicinity of n F a large number of weakly coupled protons
are visible. They also have been resolved by expansion of the NMR frequency scale. For a
few strongly coupled protons we are able to detect all five NMR transitions (1) to (5). This
is a further and definite proof that we do observe quintet states.
137

9 Diacetylene Single Crystals
3(2)
2(2)
1(2)
(a)
Bo II X
= 9.570 MHz
ν F
ν F
(c)
Bo II Y
ν F = 14.193 MHz
2(1)
1(4) 2(4) 3(4)
4(4)
0
ν F
3(2) 2(2)
10 20 30
40
0 10 20 30
(b)
Bo II Y
= 10.911 MHz
ν F
2(4)
ν F
ν F
(d)
Bo II X
= 15.553 MHz
ν F
3(4)
4(4)
4(2)
1(2)
0 10 20 30 0 8 12 16 20
NMR Frequency ν/MHz
NMR Frequency ν/MHz
NMR FREQUENCY ν / MHz
Figure 9.15: ENDOR spectra as detected by the ESR transitions (a–d); n F is the free proton frequency.
(1) to (5) label the NMR transitions illustrated in Fig. 9.14, and 1, 2, 3 … number the individual protons.
The external field B o is oriented along the y or x-axis of the fine structure tensor [36].
The ENDOR shift anisotropy is shown in Fig. 9.16 for the strongly coupled protons
i = 1, 2, 3, and 4. This anisotropy is mainly due to the anisotropy of the effective spin S u eff.
The lines were calculated (via Eqs. 12 and 13) by fitting A i . In total we have analyzed
22 protons, the hyperfine tensors A i of which are presented in Tab. 9.4. Two features can be
Figure 9.16: ENDOR shift anisotropies for the four strongest coupled protons i = 1, 2, 3 and 4, detected
via the ESR transition (b). x, y and z are the principal axes of the fine structure tensor. The experimental
values were taken for the rotation of B 0 in the yz-plane and in the zx-plane, respectively. The ENDOR
shifts Dn are calculated for (b)i(2) transitions (drawn out) and the (b)i(3) transitions (dashed), respectively
[36].
138

9.2 Photopolymerization
Table 9.4: Complete hyperfine tensors for 22 protons, calculated by fitting the experimental anisotropy
in a least-squares method. The A ij are diagonalized principal values of the fitted tensor, a is the isotropic
coupling constant, B jj the dipolar anisotropic tensor, F, Y, and c are Euler angles of the hyperfine tensor
axes relative to the fine structure axes [36].
i A xx A yy A zz a B xx B yy B zz F Y C Assign-
MHz MHz MHz MHz MHz MHz MHz degr. degr. degr. ment
1 16.298 18.953 16.747 17.333 –1.035 1.620 –0.586 13.6 –51.8 2.8 CH2
2 11.870 14.290 10.683 12.281 –0.411 2.009 –1.598 –39.2 76.4 –12.8 CH2
3 2.348 3.737 5.667 3.917 –1.569 –0.180 1.749 28.0 80.5 –6.9 CH2
4 1.905 1.797 1.253 1.652 0.253 0.146 –0.399 74.9 101.2 4.6 CH2
5 1.788 0.710 –0.391 0.702 1.086 0.007 –1.093 56.0 –16.4 –21.8 CH2
6 0.407 0.804 0.703 0.638 –0.231 0.166 0.065 10.3 10.1 –36.9 CH2
7 1.456 0.089 –0.119 0.475 –0.981 –0.387 –0.594 26.3 –25.4 –4.4 CH2
8 0.700 0.620 0.076 0.465 0.234 0.155 –0.389 57.4 –15.3 8.0 CH2
9 –0.791 0.981 0.520 0.237 –1.028 0.744 0.284 –23.4 –80.7 –15.4 CH2
10 –0.992 1.424 0.148 0.194 –1.185 1.230 –0.045 –23.2 1.6 –67.2
11 0.219 –0.079 0.400 0.180 0.039 –0.259 0.220 –67.5 111.6 41.8
12 0.098 0.161 0.271 0.177 –0.079 –0.016 0.094 21.5 90.6 –25.7
13 –0.819 1.321 –0.275 0.076 –0.895 1.245 –0.351 –6.7 –5.7 –6.0 CH2
14 0.075 –0.120 0.239 0.065 0.011 –0.185 0.174 12.6 7.3 –38.9
15 1.969 –1.045 –0.889 0.012 1.958 –1.057 –0.901 –68.2 –7.4 49.5 ARYL
16 –0.066 –0.165 0.235 0.002 –0.067 –0.166 0.234 –24.1 –5.2 38.5
17 –0.456 –0.051 –0.203 –0.203 –0.254 0.254 0.000 91.1 114.5 27.5
18 2.350 –2.533 –0.473 –0.219 2.569 –2.314 –0.254 –20.4 9.5 –53.6 ARYL
19 –0.311 –0.237 –0.442 –0.330 0.019 0.093 –0.112 –23.5 91.3 89.0
20 –0.777 –0.289 –0.143 –0.403 –0.374 0.114 0.260 –36.0 –30.2 18.1 ARYL
21 –2.070 1.303 –1.288 –0.685 –1.385 1.988 –0.603 6.0 –49.3 6.5 CH2
22 –1.072 –0.735 –0.652 –0.820 –0.252 0.085 0.168 4.7 18.6 –27.0 CH2
extracted from Tab. 9.4 immediately: first, the anisotropic part of the hyperfine tensor is
small as compared to the isotropic part for almost all protons, and second, several protons
(i = 17, 18, 19, 20, 21, 22) show a negative value of the isotropic coupling constant a.
We are able to assign 6 protons unambiguously: i = 1, 2, 3, 13, 21, and 22 (Fig. 9.17).
The analysis of the hyperfine data shows that the spin density at C 1 (Fig. 9.18) is only 11%,
–2,2% at C 2 , 17% at C 3 , –6% at C 4 , 7,9% at C' 1 and –2% at C' 2 , i. e. the spin is highly delocalized
within the oligomer (Fig. 9.18). This corresponds with the above-described exchange
coupling of the two carbenes and is to our opinion a very impressive demonstration
of the power of ENDOR.
9.2.3.3 ESR and ENDOR of triplet dicarbenes 3 DC n
For a long time it was unclear whether the triplet state (S = 1) of the dicarbenes ( 3 DC) does
exist, and if it does whether its energy is higher or lower than the energy De SQ of the dicarbene
quintet state ( 5 DC). Müller-Nawrath et al. [51] have shown theoretically and experimentally
that the ESR transitions of the triplet states of carbenes ( 3 C) and of dicarbenes ( 3 DC),
respectively, are mutually degenerated in diacetylene oligomers of diacetylene crystals if the
139

9 Diacetylene Single Crystals
Figure 9.17: Assignment of 6 hyperfine tensors to methylene protons at the polymerization head. The
arrows indicate the directions of the strongest main value of the respective hyperfine tensors. These 6
tensors have been fitted simultaneously yielding the spin density distribution shown in Tab. 9.4 [36].
20
17
15
10
7,9
11
5
0
-5
-2
-2,2
-6
-10
C2' C1' C4 C3 C2 C1
Figure 9.18: Carbene spin density at the reactive polymer chain end (Fig. 9.17).
oligomers are long, i. e. if the number of added monomers n is more than 7. Therefore 3 C
and 3 DC cannot be discriminated by ESR. Their ENDOR spectrum, however, is completely
different. The ENDOR shifts Dn are related by Dn DC = –1/2 Dn C . The existence of several
ENDOR lines, which fulfills the characteristic factor –1/2 in the above-mentioned relation,
unambiguously shows the existence of the triplet state of dicarbenes. Its excitation energy is
lower than the excitation energy De SQ of the quintet states of the same dicarbene [51].
140

9.2 Photopolymerization
9.2.4 Flash photolysis and reaction dynamics of diradicals
A single UV laser flash with wavelength l = 308 nm, pulsewidth 15 ns, and flash energy
1 mJ initiates photopolymerization by the production of the diradical DR 2 [52]. At low temperature,
i. e. 4.2 K, this photoproduct is stable. It is detected by its absorption spectrum. In
the spectral range the 0-0 transition peaks at 422 nm between monomer and polymer absorption
(Fig. 9.20a). Annealing the crystal containing photoproduct DR 2 , prepared as described
above, produces the diradicals DR 3 ,DR 4 ,DR 5 , and DR 6 respectively by addition of one
monomer per step (Fig. 9.8). All these diradicals can be detected by optical absorption spectroscopy,
for example by cooling the crystals to low temperature in order to slow down the
dark reaction (Fig. 9.20b). Sixl et al. [48] were the first who detected these optical absorption
spectra [131]. The dark reaction at low annealing temperatures has been investigated extensively
by Gross [38, 46].
If the reaction is photoinitiated by a single UV laser flash at high temperatures
(T > 180 K) the entire time-dependent reaction series, DR 2 ? DR 3 ? DR 4 ?DR 5 ?DR 6 ,
can be observed by monitoring the transient optical absorption of each of the products DR 2
to DR 6 . By this experiment we were able to analyze the reaction kinetics of these thermally
activated steps separately [37, 44]. As an example Fig. 9.19 shows the transient absorption
0.23
T=270 K
DR 422nm
2
0
0.23
DR 514nm
3
∆OD
0.29 0
DR 578nm
4
0.5 0
DR 664nm
5
0
0
10
t/ µ s
20
Figure 9.19: Time sequence of the intermediate products DR 2 ,DR 3 ,DR 4 , and DR 5 at T = 270 K after
the UV flash which is indicated by an arrow. DOD is the change of the optical density after the UV
flash [37, 44].
141

9 Diacetylene Single Crystals
Figure 9.20a: Difference of the optical absorption spectra (DOD) of TS6 after and before one single
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
of the Dimer DR 2 .
Figure 9.20b: Appearance of DR n reaction intermediates after a single pulse irradiation at 308 nm and
additional annealing. (a): 5 K, 0 min; (b): 100 K, 6 min; (c): 100 K, additional 30 min plus 120 K,
36 min; (d): 130 K, 36 min plus 140 K, 16 min [131].
at T = 270 K as detected by the change of the optical density (DOD) after the flash vs. time.
Each intermediate product is detected at the maximum of its optical absorption: DR 2 at
422 nm, DR 3 at 514 nm, DR 4 at 578 nm, and DR 5 at 664 nm. The delay in the production
of a subsequent intermediate is clearly demonstrated. The whole reaction passes in a 10 ms
time scale at room temperature.
Assuming that DR 2 is produced by the flash promptly, DR 3 by a dark reaction from
DR 2 ,DR 4 by a subsequent dark reaction from DR 3 , and so on, we used a simple kinetic
142

9.2 Photopolymerization
model for the quantitative analysis. This model is described by the following equations for
the concentrations n i and the rate constants K i for the DR i , i =1,2,3,4,5:
dn i
dt ˆ K i 1 n i 1 K i n i ; …14 a†
K i ˆ K 0 e …E i=kT† : …14 b†
As an example, the fit of this model to the transient DR 4 at 200 K is shown in
Fig. 9.21. The fit does not show any significant deviation from the experimental curve.
Figure 9.21: Experimental transient and model curve according to Eq. 14a for DR 4 in perdeuterated TS
at T = 200 K. The difference between experiment and model is plotted around the baseline [37, 44].
One result of these experiments is shown in Fig. 9.22. The addition reactions are thermally
activated, the activation energies being about 0.25 eV per monomer, almost identical
for each step (Tab. 9.5). Figure 9.22 also includes values for a product labelled V which is
presumably due to long polymer chains with reactive carbene chain ends.
A second result of these experiments is an estimation of the polymer yield for the
photoreaction as defined by the number of polymerized monomer molecules per absorbed
UV photon. Q is the quantum yield for the initiation process. Thus, the polymer yield P is
the product of Q and the kinetic chain length L, i. e. the number of monomer molecules
which are added to the chain after one initiation process:
P = Q7L = 0.07 ± 0.02 (15)
The polymer yield was determined from the increase of the polymer absorption DOD
due to UV irradiation [53].
If we assume a kinetic chain length of 100, which is a reasonable value, we get a quantum
yield for chain initiation of 7610 –4 . P increases with decreasing temperature from
300 K to 180 K by nearly a factor of five. If TS is perdeuterated P also increases by a factor
of 2.5 [44, 54].
Both, the temperature effect and the isotope effect on the polymer, can be explained
qualitatively by an increase of the kinetic chain length due to a longer carbene chain end
lifetime [37, 44, 54].
143

9 Diacetylene Single Crystals
Figure 9.22: Temperature dependencies of the rate constants for the decays of DR 2 ,DR 5 , and V. They
can be described by the Arrhenius law in a range of three orders of magnitude [37].
Table 9.5: Activation energies DE i and frequency factors K o for the reaction rates of DR i , evaluated from
their Arrhenius plots [37].
K 2 K 3 K 4 K 5
DE i / eV 0.25 ± 0.03 0.26 ± 0.03 0.30 ± 0.03 0.30 ± 0.03
K o /s –1 10 10 ± 1 10 11 ± 1 10 11 ± 1 10 11 ± 1
9.3 Holography
Diacetylenes have been subject to intense work due to their unique ability to undergo topochemical
solid state polymerization, resulting in macroscopic polymer single crystals [1–3,
25, 28, 37]. Whether this reaction takes place depends on the monomer stacking distance
and the tilt angle (Fig. 9.3). Both can be influenced by varying the rest groups R. The polymerization
can be initiated by heat, UV radiation, X rays, or g rays and is irreversible. The
optical properties, especially the absorption coefficient a and the refractive index n, are
known to change dramatically during polymerization.
By means of UV photopolymerization high-efficiency holographic grating on diacetylene
crystals can be recorded, as was first shown by Richter et al. [55, 56]. Utilizing a fre-
144

9.3 Holography
quency-doubled argon laser (l w = 257 nm) Richter et al. obtained surface phase gratings,
due to the low penetration depth of this UV wave (Fig. 9.6). At high exposures higher diffraction
orders up to five has been observed. Because of the 5% stacking distance mismatch
between monomers and polymers, Richter et al. often observed a destructive surface peel
off. This problem has been shown to become less important by using longer UV wavelengths
and, therefore, higher penetration depths [57].
The first aim within the Collaborative Research Centre 213 was to find a suitable technique
for recording such gratings in an effective and reproducible way. Using this technique
we have investigated the most important diffraction characteristics of these gratings: efficiency,
thickness, angular selectivity and their dependence on exposure, sample thickness,
and prepolymerization. The second aim, however, was to explain these characteristics within
appropriate theoretical approach. Using this knowledge, the chain length of the polymers during
UV polymerization and subsequent thermal treatment can be estimated. In the final investigation
we have shown that images and even a holographic trick film can be recorded in the
diacetylene crystals at room temperature. The peculiarity of the method is the difference of
recording (UV) and reading (VIS) wavelengths. This difference allows prompt readout without
a developing process and without perturbation of the hologram by the readout laser.
9.3.1 Theory
Figure 9.23 shows the writing and reading beams for recording and replay of the simple holographic
grating, respectively. Provided that they are of equal intensity two coherent UV
Figure 9.23: Schematic geometry of writing and reading of a holographic grating. Two coherent UV
waves (dashed) form the grating of a spatial periodicity L. A VIS wave (solid) will generally be diffracted
into different orders j.
145

9 Diacetylene Single Crystals
waves, impinging symmetrically the photoactive medium, form an intensity pattern on the
surface of the photoactive material,
I…x† ˆI 0 cos 2 …px=L† :
…16†
L is the grating distance given by
L ˆ w =2 sin w ;
…17†
where l w is the vacuum wavelength and y w the angle of incidence outside the medium. This
intensity pattern results in a photoproduct distribution of the same periodicity, represented
by a grating vector K,
K ˆ 2p ^x=L ;
K ˆ jKj ˆ 2p=L ; …18†
and results in a UV photopolymerization pattern, the refractive index n, and the absorption
coefficient a of which can be described as
n…x† ˆn 0 ‡ P1
n h cos …Khx† ;
hˆ0
a…x† ˆa 0 ‡ P1
a h cos …Khx† :
hˆ0
…19†
The importance of the Fourier coefficients besides h = 1 generally depends on reaction
kinetics, exposure, and saturation effects. In general, n and a can also vary with the z coordinate.
In principle a readout light beam (vacuum wavelength l r ) will be diffracted by the
grating resulting in different orders j as indicated in Fig. 9.23. Each of them have an amplitude
S j and a propagation vector r j . The total electric field inside the medium is a superposition
of all theses waves:
E…z† ˆ
jˆ1 P
jˆ 1
S j …z† ^s j exp… ir j r† : …20†
Here ^s j are the polarization vectors. The wave vectors r j are coupled to the grating
vector K via
r j ˆ r 0 ‡ jK :
…21†
This treatment was used first by Magnusson and Gaylord [58] and leads to a system
of coupled differential equations for the complex amplitudes S j :
146
uS j
uz ‡ cos …a 0 ‡ i# j †S j ‡
1 X1
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†
2
hˆ1

9.3 Holography
where is given by
# j ˆ jK cos… '† …jK† 2 r =4pn 0 …23†
with a slant angle j between K and the x direction; j =908 for the symmetric case shown
in Fig. 9.23. This so-called coupled wave approach is a generalization of the work of Kogelnik
[59], who assumed pure sine gratings read under the Bragg condition where only one
transmitted and one diffracted wave is present. Kogelnik [59] gives analytical solutions for
the amplitudes of the reference and the signal wave (zeroth and first order respectively) –
only for the first ascent period of a growth curve, extending earlier work for the transparency
region. For the efficiency Z j of the transmission grating with the thickness d
j …d† ˆS j …d†S j …d†
…24†
he gets for j = 1 the well-known formula
…d† ˆ sin 2 …pn 1 d=cos 0 †‡sinh 2
…a 1 d=2 cos 0 † exp… 2ad=cos 0 † ; …25†
with the replay wavelength l and the Bragg angle y 0 . It should be pointed out that this formula
is only valid for the special case of thick (volume) gratings, which show pure Bragg behaviour.
That is they possess neither higher Fourier coefficients nor a modulation amplitude n 1 nor large
enough a 1 , to produce higher diffraction orders, assuming n 1 and a 1 do not vary with z.
The thickness of a holographic grating is often described in terms of the Q factor, defined
by
Q ˆ 2pd=L 2 n 0 :
…26†
Gratings with Q^1 are regarded as thin those with Q 610 as thick.
9.3.2 Experimental setup
We used diacetylene single crystal platelets, approx. 20 mm to200mm thick and some mm 2
in area, cleaved from a parent TS6 crystal parallel to the (100) surface.
The experimental setup is shown in Fig. 9.24. For writing holographic gratings we utilized
a cw helium-cadmium laser (l w = 325 nm) or a xenon chloride excimer laser (l w = 308 nm).
We gave preference to the recording geometry suggested by Bor et al. [62] and not to the wellknown
beamsplitter geometry. The first order diffracted beams of a reflection grating R are reflected
by a pair of parallel mirrors M1 and M2 (or pass a biprism instead) and are superimposed
on the sample S. This geometry yields four main advantages:
a) Given the fringe distance of the reflection grating, the resulting fringe distance on the
sample is L = D/2 and wavelength-independent;
147

9 Diacetylene Single Crystals
b) Both writing beams are superimposed correctly because they have passed the same
number of reflections. This is especially important if non-Gaussian beams are used;
c) The intensities of both beams are equal and wavelength-independent;
d) The setup is realizable compactly and can easily be adjusted for the use of short coherence
lengths.
Control of exposure is possible using an UV enhanced photodiode (De3). To read the
gratings we used a 3 mW helium-neon laser (l r = 633 nm). At this wavelength the increase
in refractive index can be expected to be high, whereas the absorption should not become
too strong during the induction period of the polymerisation reaction (Fig. 9.4). The sample
holder was mounted on the axis of a stepping motor for varying the angle of incidence. For
analyzing transmitted or diffracted light a preamplifed large-area photodiode (De1) was
mounted on a radius level of a second stepping motor being coaxial to the first one. Both
motors include a reduction gear, giving an angular resolution of 0.06 mrad. Signal improvement
was achieved by chopping the readout beam and using a second photodiode (De2) as
an intensity reference. Both photodiode outputs were led to lock-in amplifiers. Two polarizers,
Pol1 and Pol2, were used to attain both UV and VIS laser polarizations parallel to the
sample’s b-axis.
Figure 9.24: Experimental setup. De1, De2, De3: large area photodiodes, R: reflection grating, S: sample
on sample holder, Sh: beam shutter, BS: beam splitter, M: mirrors, Pol1, Pol2: polarizers (l/2
plates), Ch: chopper.
Two sample orientations relative to the grating fringes were investigated:
a) UV polarization,VIS polarization, and polymer axis b lying in the plane of incidence (E
mode), henceforth denoted as b k orientation;
b) UV polarization, VIS polarization, and polymer axis b were perpendicular to the plane
of incidence (H mode), called b || orientation.
148

9.3 Holography
The high dichroism of the crystals and their well-formed habit can be used to align
their orientation under a polarizing microscope. Of course, the optical anisotropy of the crystals
as well as anisotropic reaction kinetics give rise to birefringence effects.
9.3.3 General characterization
Figure 9.25 shows for typical samples the maximum efficiency Z(y 0 )=Z as a function of
the exposure. These growth curves show that an optimum can be reached between 0.5 and
1.0 J/cm 2 for l w = 308 nm and between 5 and 10 J/cm 2 for l w = 325 nm. For TS6 the penetration
depth at 308 nm is about 110 mm, whereas at 325 nm it is about 420 mm. This difference
and in addition a presumably lower quantum yield at 325 nm results in much slower
growth curves for this wavelength. The decrease in efficiency is not only an effect of coupling
back intensity from first to zeroth order, but also produced by an increase of absorption.
Sample quality and thickness can influence these curves. A destruction of the surface
is only observed if the optimal exposure is exceeded by a factor 2 to 3. This is due to the
lattice mismatch of monomer and polymer stacking distance.
The energy needed to reach the maximum is approximately by a factor 5 larger than
at 257 nm [55, 56]. From the growth curves, i. e. from the exposure E which was needed to
produce a certain efficiency Z, the holographic sensitivity S of the material given by
S ˆ
p
=E
…27†
can be estimated. For TS6 and depending on sample quality this value can range from 0.4 to
1.4 cm 2 /J. This is in the order of typical sensitivities for photorefractive crystals but far less
0 12.5 25
0.5 0.13
Efficiency η
0.25
0.065
0 0
0
1.5
3.0
Exposure E / Jcm -2
Figure 9.25: Holographic growth curves of TS6 (solid curve: l w = 308 nm, d = 130 mm; dotted curve:
l w = 325 nm, d = 270 mm) and IPUDO (dashed curve: l w = 308 nm, d = 260 mm). Orientation b k ,
L = 3.3 mm. Z(y 0 )=Z: efficiency for readout Bragg condition.
149

9 Diacetylene Single Crystals
than for common silver halide materials [63]. Nevertheless, the resolution of TS6 is comparable
to that of common photographic materials. From the experiments of Richter et al. [55]
it is known that a resolution of 2500 lines/mm is achievable. In our electron beam lithography
experiments with a resolution of 5000 lines/mm has been achieved [64].
Supposing the thickness of the sample is equal to the grating thickness, we can calculate
the factor Q for our gratings. At a grating distance of L = 3.3 mm this factor
ranges from 2.5 to 60 for samples of thickness 25–500 mm, whereas at L = 0.8 mm Q
factors of 800 can be achieved. These values are typical for thick phase gratings. Nevertheless,
it should be pointed out that for example a grating with Q = 40 already shows
higher diffraction orders. This confirms the arguments brought by Moharam and Young
[60] that the r factor (r = L 2 /L 2 n 1 n 0 ) should be preferred when discussing grating diffraction.
9.3.4 Angular selectivity
The angular selectivity is a measure for diffracted intensity when readout is performed under
off-Bragg conditions or, quantitatively speaking, it is the half-width Dy of the function Z (y),
where y is the angle of incidence. This property is of central interest, when discussing holographic
storage media, because it limits the number of holograms/holographic gratings
which can be stored simultaneously.
Only as a rule of thumb, Kogelnik [59] gives Dy & Ln 0 /d, whereas Magnusson and
Gaylord [58] do not give any analytical expression for this quantity. Both approaches can easily
be implemented on a computer to simulate diffraction properties. Assuming n h = 0 for
h 62 and only for the first ascent period of a growth curve we find a Dy-dependence of the
following simple form
ˆ n 0 =d
…28†
valid only for the first ascent period of a growth curve, because the angular selectivity
curves get split beyond the first maximum [61].
The dependence of Dy on sample thickness d proved to be very well reproducible with
respect to adjustment and sample quality. Figure 9.26 shows two typical measurements of
Z (y) representing samples of different thickness. In contrast to the early experiments of
Richter et al. [55] and of Niederwald et al. [56] these values represent an improvement of
Dy by up to two orders of magnitude.
Figure 9.27 shows Dy values versus sample thickness for TS6. No significant difference
could be observed with respect to exposure or orientation for this grating distance.
There is also no significant difference between TS6 and IPUDO with respect to angular selectivity.
Measurements at L = 0.8 mm are summarized in Fig. 9.28. Within the series the minimum
angular selectivity of 0.188 arose, using a 380 mm thick TS6 platelet. The hyperbolas
plotted in Figs. 9.27 and 9.28 are fits of the function Dy = Ln 0 /d to the data points. Averaged
over all exposures n 0 ranges from 1.3 to 1.7.
150

9.3 Holography
1
Rel. Efficiency
0.5
0
-30 0
30
Readout Angle
/ Deg.
Figure 9.26: Dependence of relative efficiency Z as a function of the incidence angle y for two TS6
samples of different thickness d.
Figure 9.27: Angular selectivity Dy as a function of sample thickness d for TS6 (open circles) and
IPUDO (full circles). L = 3.3 mm, l w = 308 nm [65].
Angular Selectivity ∆Θ / Deg.
2
1
0
0 250
500
Sample Thickness d / µ m
Figure 9.28: Angular selectivity Dy as a function of sample thickness d for TS6. L = 0.8 mm,
l w = 325 nm for both writing geometries, b || (full circles) and b k (open circles) [65].
151

9 Diacetylene Single Crystals
9.3.5 Prepolymerized samples
Until now we were assuming that sample thickness and hologram thickness are equal. In
fact, for large thickness values d one can observe slight deviations from the hyperbola function
due to the finite penetration depth of the UV light. These deviations should become
more obvious for smaller penetration depths. To observe this some Dy measurements were
carried out using thermally prepolymerized (up to 5 h at 70 8C before recording gratings)
TS6 samples. Penetration depths for these samples varies between 18 to 80 mm at 308 nm
and from 16 to 420 mm at 325 nm in the b || orientation.
The effect of prepolymerization on angular selectivity is shown in Fig. 9.29. The values
for samples prepolymerized for 2 h still show for thin samples a weak thickness dependence.
For thick samples Dy does not tend to decrease further. Samples prepolymerized for
5 h do not show any dependence on d, indicating a penetration depth distinctly smaller than
the smallest sample thickness used. Thus, for an increasing polymer concentration the holographic
gratings get thinner and thinner. The possibility of producing high efficiency volume
phase gratings is restricted to fresh monomer crystals.
Figure 9.29: Angular selectivity of prepolymerized TS6 samples: 5h (upper open circles), 2 h (full circles),
and fresh crystals (lower open circles) [66].
9.3.6 Chain length, polymer profile, and grating profiles
A model [66, 68] describing the spatially inhomogeneous reaction kinetics of diacetylenes
must take into account a kinetic chain length L, which depends on the polymer conversion P.
The monomolecular reaction in a simple homogeneous situation then is given by
dP=dt L…P†…1 P† : …27†
For the experiments described so far, L can be assumed constant during the exposure
time, because only very little conversion takes place. Furthermore, for all experiments de-
152

9.3 Holography
scribed, the factor (1 – P) can be dropped, because the experiments were carried out in the
low conversion regime of the induction period and because saturation effects affect development
experiments in both orientations. For the two orientations, different situations must be
investigated corresponding to the different interaction of the two characteristic lengths L
and L (Fig. 9.30),
b jj : dP=dt L…P† cos 2 …px=L† : …28†
Here a polymer molecule grows parallel to the fringes contributing its whole chain
length L to the polymer growth at point x, where it has been initiated. In the other case, a chain
initiated in x contributes to the polymer growth in the whole interval [x – L/2, x + L/2]:
x‡L=2
b ? : dP=dt R
cos 2 …px 0 =L†dx 0 :
x L=2
…29†
Integrating both Equations we see that in the second case the chain growth smears out
the photoproduct distribution for a certain amount. This effect should be stronger if L/L approaches
1. For the quotient of both polymer modulations we get the ratio d,
ˆ P ?
P jj
ˆ sin…pL=L†
pL=L : …30†
The resulting d values range for fresh samples from 0.85 to 0.95 and tend to decrease
with increasing prepolymer content. For samples prepolymerized (5–6 hours, 3–4% polymer)
we find that d is between 0.65 and 0.80. These values correspond to chain lengths (L)
of 0.15 to 0.4 mm or 300 to 800 repeat units [66].
Figure 9.30: Microscopic model to understand the interaction of the two characteristic lengths L and L:
The two geometries investigated change the angle between chain growth direction and grating fringes.
153

9 Diacetylene Single Crystals
9.3.7 Multrecording
In additional experiments, we succeeded in recording more than only one grating into one
thick TS6 crystal. After each exposure interval, we rotated the sample for a certain amount
to slant the gratings relatively to each other. Figure 9.31 shows the first order diffraction versus
the angle of incidence for a 310 mm thick sample containing 42 gratings tilted stepwise
by 1.08 8. We observed that gratings already present are almost not influenced by the subsequent
recording processes except by the increase of total absorption, which causes a loss of
efficiency less than 10 %. Thus, we did choose an exposure of only 0.07 J/cm 2 for each grating.
As can be seen from Fig. 9.31, the signal-to-noise ratio is bad for the last written gratings.
It will get worse by decreasing the angular distance of the individual gratings but will
hardly improve by increasing it [67]. A serious diffraction of the UV light by the gratings already
recorded in the sample does not take place. UV diffraction efficiencies are extremely
small in diacetylene crystals.
-3
Efficiency η / 10
8
4
1
2
4
0
-10 20
50
Readout Angle Θ/ Deg.
Figure 9.31: Efficiency of the first order of a multihologramm consisting of 42 single gratings successively
recorded in a 310 mm thick TS6 crystal. The gratings were recorded in the order of the numbers
indicated [69].
9.3.8 Holography
For the reconstruction of a real hologram with different writing and reading wavelengths, it
was necessary to use a divergent reference wave. With a test platelet as object a resolution
of 300 lines/mm was achieved in excellent TS6 samples (Fig. 9.32). Holograms up to 32
pictures were written (l w = 325 nm) in one crystal using the angular selectivity of the thick
phase. When rotating the sample the pictures can be reconstructed (l r = 633 nm) successively
(holographic trick film).
The storage density of about 6.7610 9 cm –3 , estimated from the real resolution and
angular selectivity, is about one to two orders of magnitude lower than the storage density
calculated from the grating distance [70].
154

9.4 Di-, pyro-, and ferroelectricity
Figure 9.32: Reconstructed image from one of the 32 pictures of a holographic trick film [70].
9.4 Di-, pyro-, and ferroelectricity
The dielectric properties of diacetylene monomer single crystals are not strikingly different
from those of other organic materials. Typically, they have permittivity values (earlier called
dielectric constant e r ) of about 4–6. Solid state polymerization, however, results in a pronounced
anisotropy of the electric permittivity with maximum values parallel to the polymer
chain direction, due to the large polarizability of the extended p-orbitals. During our TOPO-
MAK activities the change of e r , accompanying solid state polymerization, was analyzed
quantitatively (Section 9.4.1.1). The results of the e r analysis can be applied for the in situ
monitoring of the polymer content (Section 9.4.1.2).
The tailoring of pyro or ferroelectric properties of diacetylenes (Section 9.4.2) is less
straightforward than might be surmised from the well-ordered arrangement of the R, R' substituents
in Scheme 1, and from the fact that polar side groups can easily be introduced as
substituents R and R' (Tab. 9.2). The difficulties originate from the packing of the individual
polar diacetylene monomer molecules in the elementary cell of the solution-grown single
crystal. This, generally, gives rise to a center of inversion symmetry for a pair of molecules
thus compensating the individual molecular electric dipole moments in an antiferroelectric
arrangement. During solid state polymerization spurious electric polarization has been observed
[71] resulting from intramolecular distortion of originally centrosymmetric monomer
units. Here we want to emphasize that our systematic TOPOMAK investigations have realized
pyroelectric (Section 9.4.2) as well as ferroelectric properties (Section 9.4.3) for appropriately
substituted diacetylenes.
155

9 Diacetylene Single Crystals
9.4.1 Dielectric properties of diacetylenes
Typically, the electric permittivity of monomer single crystals of substituted diacetylenes
ranges from 4 to 6 at room temperature. These e r values are therefore larger than values of
simple non-polar organic polymers, like polytetrafluoroethylene, in agreement with the existence
of polar side groups. As is exemplified in Fig. 9.33, monoclinic crystals of 2,4-hexadiynylene
di-p-toluenesulfonate (TS, see Tab. 9.2) shows generally a weak but non-negligible
anisotropy for e r [72, 75].
ε r (h)
8
7
6
5
TS
T = 60°C
(1)
(2)
4
3
0
10 20 30
polymerization time t(h)
Figure 9.33: Electric permittivity e r (t) as a function of the polymerization time for TS crystals at
60 8C. The electric field was applied parallel to the chain direction of the monoclinic crystals (1) and in
the two orthogonal directions (2) and (3). The permittivity was measured at 1 kHz for three different
thin parallel-plate single crystal capacitors [75].
(3)
40
9.4.1.1 Correlation of polymer content and electric permittivity
Generally, a sigmoid time-conversion curve is observed for thermal solid state polymerization
of diacetylenes. This behaviour was shown for TS in Fig. 9.4 and is reported for the unsymmetrically
substituted 6-( p-toluenesulfonyloxy)-2,4-hexadiynyl-p-fluorobenzenesulfonate
(TS/FBS, Tab. 9.2). In Fig. 9.34a the slow conversion of the initial induction period of the
solid state polymerization lasts until a polymer content of about 10 % is achieved. For higher
up to complete conversion, this is followed by an autocatalytic reaction enhancement. The
standard technique for the derivation of time-conversion curves is the gravimetrical analysis
of a large number of crystals in a point-by-point procedure. After thermal polymerization
for a well-defined period both, the soluble monomer and the insoluble polymer portions, are
determined gravimetrically (Fig. 9.34 a). This technique consumes a considerable number of
crystals and a substantial amount of time. Furthermore, only averaged time-conversion
curves are obtained.
Figure 9.34b shows that during solid state polymerization of TS/FBS the electric permittivity
parallel to the chain direction, e r|| , increases by a factor of about 2. The permittivity
e r|| is almost a linear function of the polymer content, as is exemplified in Fig. 9.34 c. We
have found comparable behaviour for the substituted diacetylenes TS, FBS, FBS/TFMBS,
and DNP [72–75]. Because the reorientation of the side groups during the polymerization is
weak, it does not influence substantially the increase of e r parallel to the chain direction.
156

9.4 Di-, pyro-, and ferroelectricity
Figure 9.34: Thermal solid state polymerization of TS/FBS (PTS is another acronym for TS).
(a): Time-conversion curve derived by gravimetrical analysis. (b): Time-permittivity curve derived at
1 kHz for a thin parallel-plate single-crystal capacitor oriented with the polymer chain direction (b-axis)
parallel to the electric field. (c): Correlation of conversion and electric permittivity (with time as implicit
parameter) obtained by combination of (a) and (b) [73].
This proves that only the extended p-electron system of the diacetylene backbone is responsible
for the enhanced polarizability. This conclusion is supported by the experimental analysis
for three orthogonal directions in TS single crystals shown in Fig. 9.33 [75]. The minor
variations for the two orthogonal directions (2) and (3) can be explained by the changes of
the lattice parameters. In contrast, the change of the respective lattice parameters with polymer
content does not suffice to explain the change of De r|| .
The linear relation between permittivity and polymer content is in principle surprising
because there is a distribution of chain lengths of solid state polymerization, differing between
induction period and autocatalytic range. The linearity, shown in Fig. 9.34 c, reveals
that the polarizability of the p-electron system saturates already at chain lengths below the
shortest ones occurring during thermal solid state polymerization.
The experimental range for e r is 1.4 to 2.2 derived for different substituted diacetylenes
by Gruner-Bauer [72–75], which agrees with the observations of other groups [76–
78]. These values compare favourably with the estimate De r & 1.6 obtained for TS by simplified
model calculations [75]. For these theoretical estimates the method of Genkin and
157

9 Diacetylene Single Crystals
Mednis [79] has been modified by Gruner-Bauer, extending earlier work for the transparency
region [80].
9.4.1.2 Application to topospecifically modified diacetylenes
The linear relation of electric permittivity parallel to the chain direction vs. polymer content
thus established can be used for the control of the polymer content of prepolymerized samples
as well as for the derivation of time-conversion curves of individual single crystals in
situ. We have used this technique to study the solid state polymerization of topospecifically
and fully deuterated TS.
Striking differences of their reactivities were reported before by Ch. Kröhnke [54].
The limited amount of samples available was sufficient for the permittivity analysis.
Figure 9.35 shows the behaviour of different topospecifically deuterated derivatives of TS
at T =608C [73]. It should be stressed that the induction periods of single crystals with
nominally the same history did not differ by more than 10% at the same polymerization
temperature. The toposelective modification of these substituted diacetylenes by the deuteration
of the methylene groups close to the triple bonds of the diacetylene monomer
(that are engaged in the crankshaft-type motion of the monomer molecule around its center
of mass during solid state polymerization) evidently has a drastic influence on the solid
state polymerization.
8
7
PTS
T = 60°C
6
ε (t)
5
8
7
6
5
8
7
CD - PTS
T = 60°C
CD - PTS
T = 60°C
2
6
5
0
10
Figure 9.35: Electric permittivity as function of time for solid state polymerization at T =608C for TS,
PD-TS (fully deuterated TS), and CD 2 -TS (where only the methylene groups close to the triple bond of
the diacetylene monomer unit are deuterated, see Tab. 9.2 a) [73].
158
20
30
polymerization time t/h
40

9.4 Di-, pyro-, and ferroelectricity
9.4.1.3 Additional applications
Structural phase transitions accompanying solid state polymerization influence the behaviour
of e r (t). Thus they can not be observed but additional hints concerning the type of structural
changes can be obtained. For example, a structural phase transition for DNP (Tab. 9.2) occurs
at a polymer content above 95% resulting in the loss of order perpendicular to the polymer
chains [81]. The accompanying increase of the side-group mobility results in a distinct
increase of the electric permittivity [75]. Even more dramatic changes of e r were observed
at the transition to a fibrillar structure in polar crystals of DNP/MNP (Tab. 9.2) [74, 75].
Thinking of applications outside fundamental research, the correlation of permittivity
and polymer content can also be used for different kinds of ageing control. Since solid state
polymerization is an activated process, the temperature-weighted time at elevated temperature
– like a low-temperature radiation dose – influences the capacity of a diacetylene crystal-plate
capacitor according to a well-defined characteristic history-capacity. Evidently, this
can be adapted to an appropriate electronic ageing control.
9.4.2 Pyroelectric diacetylenes
One fascinating goal of diacetylene materials research is the realization of polar polymer single
crystals without complicated poling procedures. Substituted diacetylenes R 1 –C:–C:C–R 2
incorporating polar side groups R 1 and R 2 with large but different electric dipole moments
have to be synthesized. Since solid state polymerization depends on the ability of individual
molecule side groups, which can perform intramolecular torsion and giving rise to an overall
crankshaft-like motion, it is difficult – and was impossible for us – to predict the packing arrangement
of individual monomer molecules in the solid. Frequently, the unit cell of substituted
diacetylenes was observed to accommodate pairs of formula units in a centrosymmetric
arrangement, thus compensating the net electric polarization. Polymorphism turned out to be
an obstacle to systematic tailoring of dielectric properties of diacetylene single crystals, because
different crystal structures of the same diacetylene derivative could be obtained from
different, and occasionally even from the same solvent [82]. Nevertheless, Strohriegl synthesized
during our TOPOMAK activities several non-centrosymmetric diacetylenes, which crystallized
also in a polar phase. Typically, their permittivities were relatively large and anisotropic
[83]. We restrict this report to three examples.
9.4.2.1 IPUDO
IPUDO (for the molecular structure see Tab. 9.2) is an example of a symmetrically substituted
diacetylene. For the monomer as well as polymer crystals, non-centrosymmetrical
orthorhombic crystal structures can be found already at room temperature [84]. The c-axis
is the polar axis of the monomer crystal. IPUDO can only be polymerized by g radiation
( 60 Co). For 85% polymerized crystals pyroelectric properties were observed only in b direction.
The permittivity of IPUDO is highly anisotropic, with maximum values of 8.5 for the
159

9 Diacetylene Single Crystals
monomer (parallel to a), or 11.6 for the 85% polymer (parallel to b) and with minimum values
(parallel to c) smaller by a factor of 2 (3.5) for the monomer (polymer) [83]. The distortion
of the long side groups by the development of hydrogen bonds between neighbouring
–CO–NH- groups is supposed to be responsible for the non-centrosymmetry [83, 85, 86].
Thus it is not surprising that the variation of the electric polarization of about
3610 –8 Ccm –2 between room temperature and 4 K amounts to an unbalancing of only
about 1% of the compensation of oppositely oriented C=O … HN dipole moments.
9.4.2.2 NP/4-MPU
NP/4-MPU (for the molecular structure see Tab. 9.2) is an example of a non-centrosymmetric
diacetylene that forms polar monomer single crystals only, if it is grown from appropriate
solvents, here from 2-propanol [82]. The resulting modification I is orthorhombic
with the polar space group Fdd2 (Z = 16) and c as the polar axis [82]. This modification is
not reactive thermally or under X-ray irradiation, because the monomer packing is outside
the favourable range for solid state polymerization. The diacetylene rods make an angle of
678 with the stacking axis c , and the stacking distance d = 4.61 Å. The permittivity of this
diacetylene is highly anisotropic and shows the largest value of about e r &23 for an electric
field applied in the direction of the polar axis [82, 87].
Figure 9.36 shows the variation of the spontaneous electric polarization DP of NP/4-
MPU with temperatures between 10 K and the melting point. DP amounts to about 15% of
the electric polarization, which can be estimated from the volume density of molecular electric
dipole moments of about 3 Debye (10 –29 Cm).
The pyroelectric coefficient p(T) =dP S /dT =8.8610 –10 Ccm –2 K –1 is of the same
order of magnitude (smaller by 1/3) as that of the well-known and commercially used pyro
and ferroelectric polyvinylidenefluoride. Therefore NP/4-MPU single crystals can be used
for the detection of radiation [89], which we showed by using chopped low-power laser light
as radiation source. The pyroelectric current and the total variation of the surface charge
Figure 9.36: Temperature-dependent change of the spontaneous polarization for a NP/4-MPU single
crystal (modification I) along the polar c-axis for temperatures below the melting point T m . The pyroelectric
coefficient P at 300 K is also given [87].
160

9.4 Di-, pyro-, and ferroelectricity
have been used for the detection. Furthermore, a transversal piezoelectric coefficient of
1018 fCN –1 for NP/4-MPU was derived at room temperature. It is comparable with the value
of a-quartz [88].
9.4.2.3 DNP/MNP
DNP/MNP (for molecular structure see Tab. 9.2) was the most successful one of Strohriegl’s
syntheses [74]. It has a polar crystal structure for monomer and polymer crystals (space
group P2 1 ). DNP/MNP polymerizes – thermally or exposed to UV radiation – extremely
fast, because during solid state polymerization the molecules are packed optimally.
The solid obtained by thermal polymerization of monomer crystals exhibits a fibrous
texture, probably due to the large changes in lateral packing of the side groups. The temperature
influence on the spontaneous electric polarization perpendicular to the chain axis c
is shown in Fig. 9.37 for the monomer as well as the polymer crystal. The polarization varies
up to room temperature by 7610 –8 Ccm –2 for the monomer crystal, with a pyroelectric
coefficient of about 3.2610 –10 Ccm –2 K –1 at room temperature.
0.0
-2
∆P(T) / 10 C cm
-7
-0.2
-0.4
-0.6
-0.8
-1.0
0
monomer
DNP / MNP
parallel to polar axis
100 200
T/K
polymer
Figure 9.37: Variation of the spontaneous electric polarization parallel to the polar b axis of DNP/MNP
crystals with temperature. The data of DP (T) =P (T)–P (5 K) were derived by charge integration during
a temperature cycle [74].
300
9.4.2.4 Spurious piezo and pyroelectricity of diacetylenes
Sample defects can be another origin of piezo and pyroelectric phenomena in substituted
diacetylenes, which generally can be identified via sample dependence and smaller size of
these effects [88]. Bloor et al. discussed such spurious pyroelectric effects, which seemed to
be correlated with the occurrence of macroscopic deformations, such as screw dislocations
in TS single crystals [90]. Similarly the analysis of weak polarization, caused by molecular
distortion during solid state polymerization of TS, was reported by Bertault et al. [91]. We
have observed comparable weak pyroelectric phenomena for TS/FBS [87].
161

9 Diacetylene Single Crystals
9.4.3 The ferroelectric diacetylene DNP
Whereas several pyroelectric diacetylenes were identified, uniform ferroelectric phases seem
to be rather the exception, according to our experience with many new substitutions of diacetylenes
[92]. The ferroelectric low-temperature phase of the symmetrically disubstituted
diacetylene DNP, i. e. 1,6-bis(2,4-dinitrophenoxy)-2,4-hexadiyne (Tab. 9.2), turned out to be
one of the rare and interesting exceptions [93–98].
On both ends DNP carries polar dinitrophenoxy groups, with an electric dipole moment
of 10 –29 Cm, whose mutual twisting gives rise to the spontaneous electric polarization
of the non-centrosymmetric low-temperature phase (space group P2 1 ) (Fig. 9.38). The DNP
monomer crystal has besides a pyroelectric [93] low-temperature phase a ferroelectric one
for T < T c = 46 K. According to our investigations, the direction of the spontaneous polarization
(Fig. 9.39) can be influenced by an external electric field [97]. Structural defects, occurring
in all DNP crystals, give rise to a distribution of transition temperatures and to the
existence of domains, whose polarization can not be inverted with the accessible external
fields – thus behaving like pyroelectrics. Only the domains with the highest transition temperatures
could be poled [97].
b
a
b
a
b
a
a)
b)
c)
Figure 9.38: Packing arrangement of DNP molecules in the monomer crystal viewed along the c-axis
at three temperatures (a): T = 296 K, (b): T = 145 K, (c): T = 5 K [96].
The temperature dependence of the spontaneous polarization (Fig. 9.39) of the monomer
crystal can be described in the framework of Landau’s phenomenological theory assuming
a tricritical phase transition [97]. The maximum experimental value of the electric polarization
of 2.4610 –7 Ccm –2 compares favourably with the polarization that was calculated
from the intramolecular twisting angle of the polar dinitrophenoxy groups of 5.18 determined
by X-ray structural analysis at 5 K [96].
The electric permittivity (Fig. 9.40) reaches values of about e r & 150 at the transition
temperature. Its temperature dependence is strongly influenced by defects [97]. Thus, it is
less appropriate for a comparison with theoretical predictions.
In the early days of TOPOMAK, H. Schultes has already observed that the phase transition
of DNP shifts to lower temperatures (Fig. 9.41–43) with increasing polymer content
162

9.4 Di-, pyro-, and ferroelectricity
3
DNP
-2
P(T) / 10 -7 Ccm
2
1
0
0
10 20 30 40 50 60
T/K
Figure 9.39: Temperature dependence of the spontaneous polarization parallel to the polar b-axis for
different DNP monomer single crystals [97].
75 0.3
DNP
ε r (T)
50
25
0.2
0.1
-1
(χ ferro
(T))
0
0
20
40
Figure 9.40: Temperature dependence of the low-frequency electric permittivity (left axis) and the inverse
of the ferroelectric part of the susceptibility (right axis) for a DNP monomer crystal [97].
T/K
60
80 0
-7
polarization / 10 C cm
-2
1,0
0,5
10 h
5h
2h
0h
0
25
Figure 9.41: Variation of the zero-field electric polarization in crystallographic b-direction of DNP single
crystal with duration of thermal polymerization at 130 8C [94] (Fig. 9.34 for conversion curve).
50
T/K
163

9 Diacetylene Single Crystals
ε (T) / ε (293 K)
r
r
8
4
10 h
5h
13 h
14 h
16 h
0
0 25 50 75 100
T/K
Figure 9.42: Influence of duration of thermal solid state polymerization at 130 8C on temperature dependence
of electric permittivity of DNP [94].
50
T c / K
40
30
1,6
1,2
-1
∆S /Jmol K
-1
20
10
0
0
X/%
80
40
0
0
8 16
t/h
4 8
16 0,0
Figure 9.43: Variation of the transition temperature T c (e r maximum) and the conversion entropy (transition
entropy change) DS with the duration t p of the thermal solid state polymerization of DNP at
129 8C [83, 87]. The inset shows the typical variation of the polymer content X with t p .
12
t p/h
0,8
0,4
(thermal polymerization) and can not be observed in the polymer crystal [94, 98]. This was
a puzzle for our early attempts to understand the ferroelectric phase transition of DNP.
In the microscopic picture of the phase transition, nuclear magnetic resonance spectroscopy
and relaxation of the DNP protons gave important additional information [95, 96].
The proton NMR spectrum reflects the orientation of the proton-proton axis of the methylene
groups close to the central diacetylene unit via the nuclear-spin magnetic dipole interaction
in a rather clear-cut way (Fig. 9.44). For fixed crystal orientation and varied temperature
the spectrum of the methylene group protons of the DNP monomer proved that both DNP
moieties are twisted around the central C–C single bond of the diacetylene below T c . Since
this degree of freedom is lost during the solid state polymerization the ferroelectric phase
transition is suppressed.
164

9.4 Di-, pyro-, and ferroelectricity
r
H0IIb
r
H 0
b
-40
0 40
-40 0 40
Rel. Frequency (kHz)
Figure 9.44: Simulated (left) and experimental (right) H NMR spectra as a function of the crystal orientation
with respect to the external magnetic field, recorded at room temperature (n p = 200 MHz). The
crystal was rotated in steps of 58 around its long axis (a-axis), which was oriented perpendicular to the
external field [96].
Additional information on the phase transition was obtained from proton-spin-lattice
relaxation measured as function of the Larmor frequency, temperature, and orientation of
the single crystals (Fig. 9.45). The analysis indicated the slowing down of a molecular motion
on approaching the ferroelectric phase transition with the activation energy of about
0.020 eV, which is in the range of known librational and torsional modes of diacetylenes
[96].
The phase transition of DNP could further be characterized by specific heat measurements
for monomer and thermally polymerized single crystals (Fig. 9.46) [98]. These data
support the description of the phase transition of the monomer crystals as a tricritical transition.
This means it is a borderline case between a first-order and second-order phase transition,
with a distribution of transition temperatures. The transition enthalpy was much lower
than the corresponding order-disorder transition, in agreement with results obtained by Bertault
et al. via Raman spectroscopy, which proved the importance of displacive contributions
to the DNP phase transition [99].
165

9 Diacetylene Single Crystals
31 MHz
-3 -1
Spin-lattice relaxation rate (10 s )
90 MHz
200 MHz
Temperature (Kelvin)
Figure 9.45: Spin-lattice relaxation rate of the protons in a DNP monomer single crystal for three Larmor
frequencies. Two rate maxima can be discerned with temperature dependence explained by the
model of Bloembergen, Purcell, and Pound (solid line) [96].
30
-1
∆/C/J mol K -1
20
2h
2 p =0h
10
6h
0
30
Figure 9.46: The ferroelectric contribution to the molar heat capacity of solid state polymerized DNP
single crystals for different annealing times t p at 129 8C for [98].
40
T/K
50
9.4.4 Summary
The increase of the electric permittivity for electric fields parallel to the polymer chain direction
during solid state polymerization of diacetylenes can be used for in situ monitoring
the monomer to polymer conversion of individual single crystals. The large librational am-
166

9.5 Non-linear optical properties
plitudes of the diacetylene moiety, required for solid state polymerization, are also the basis
for the occurrence of interesting dielectric phase transitions. The tailoring of diacetylenes as
ferro or pyroelectric crystals, which show good thermal stability and do not demand considerable
efforts for the poling process, is a trial-and-error process, but has been realized during
our TOPOMAK activities in a number of cases. Thus, material properties useful for applications
of pyro- or piezoelectricity thus have been obtained.
9.5 Non-linear optical properties
9.5.1 Aims of investigation
Polydiacetylenes are polymers showing a one-dimensional semiconducting behaviour. This
one-dimensional structure causes exceptionally high third order non-linearities (w (3) ) [100],
also in off-resonant wavelength regions [101], with extremely short sub-picosecond switching
times [102]. After this discovery it was believed that an optical amplifying switch (optical
transistor) or even an optical computer was close at hand.
At the start of our SFB 213 project the initial optimism about the application of the
huge non-linearity of polydiacetylenes in optical switching was already somewhat damped.
It was becoming clear, that the polydiacetylenes’ non-linear optical coefficients, despite belonging
to the largest non-resonant non-linearities, were not sufficient for cascadable, intensity
amplifying switches [103]. There was not much known about the mechanisms leading to
the large non-linearities and different, sometimes contradicting theoretical models were proposed
to describe them. On the other side, there was a demand for tailor-made materials
with predictable non-linearities, absorption bands, and good optical quality.
In this scenario, polydiacetylenes were nevertheless interesting, as the mechanisms of
their large non-linearity form a good basis to build on. Only a few modifications were well
characterised and there was a general lack of measurements with single crystals due to problems
with sample preparation. Therefore, the aim was to characterise systematically the influence
of side groups on the optical properties, preferably in the macroscopically ordered,
stable, and reproducible framework of good and – later on – thin crystals.
9.5.2 Experimental setup
Mainly two kinds of experiments will be described here, two-beam pump-probe and degenerate
four wave mixing (DFWM) measurements.
In a pump-probe experiment, an intense pulsed pump beam and a weaker equally
pulsed probe beam of the same wavelength are focused into the sample, and the transmission
167

9 Diacetylene Single Crystals
of the probe beam is measured. The timing of the two pulses can be shifted, to make it possible
to probe the decay of excitations generated by the pump beam. Two contrary effects
may occur, bleaching and induced absorption.
Near-resonant pump-probe lifetime measurements were performed by W. Schmid
[105, 123], using the same picosecond dye laser system as for DFWM, described below.
Th. Fehn further on investigated the subject in the off-resonant wavelength region between
720 nm and 820 nm, using a commercial Titan-Sapphire laser system (Coherent) with pulse
lengths of ca. 120 fs, pumped by an argon ion laser (Coherent Mira).
DFWM measurements can be done in a variety of geometries. Here, the forward mixing
geometry is used (Fig. 9.47), where three beams, forming a right angle, are focused into the
sample. This setup allows time-resolved measurements and, in contrast to third harmonic generation
(THG) measurements, yields the w (3) (o;–o,o,–o) tensor which is related to an intensity-dependent
refractive index, the interesting quantity for optical switching applications.
Figure 9.47: Beam geometry for DFWM measurements.
The interference pattern of the pump beams 1 and 2 forms horizontal stripes in the
medium. As w (3) is directly related to an intensity-dependent refractive index, this interference
pattern generates a refractive index pattern. The third probe beam is partially reflected
on these horizontal planes, generating the signal beam 4. The efficiency of diffraction is related
to |w (3) |. The pulse timing is adjustable, so the decay of the refractive index pattern can
be probed. When the delay between the beams 2 and 3 is in the range of the laser’s coherence
length, these pulses generate a diffraction grating for beam 1, resulting in an artificial
raise of the signal, called coherence peak.
All DFWM measurements were done with a commercially available synchronously
pumped dye laser with a cavity dumper (Spectra Physics model 3500). When using Pyridine-1
as radiant dye, the wavelength can be tuned between 670 nm and 750 nm. The pulse
width is about 1 ps; the repetition rate can be adjusted between single shot and 8 MHz. This
is important to reduce the average heat load absorbed by the crystals.
To suppress stray light signals a well-known modulation technique was used, the modulation
of two pump beams with different frequencies using a chopper blade with a set of
different divisions. As the DFWM signal depends on the product of the input intensities, the
signal can be detected at the sum or difference of the two modulation frequencies [104],
168

9.5 Non-linear optical properties
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
1
ˆ I 1 …0†I 2 …0†I 3 ‰
2 cos…! 1 ‡ ! 2 †t ‡ cos…! 1 ! 2 †t ‡ 2 cos ! 1 t ‡ 2 cos ! 2 tŠ: …31†
W. Schmid proposed the separated detection of w (5) effects by an extension of this
modulation method. A w (5) signal without w (3) contributions can be detected at twice the sum
or difference frequency [105], as the w (5) signal contains terms depending on the square of
the product of the modulated beam intensities,
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†
where k 3 is a function of w (3) , k 5 of w (5) , and k 35 depends on both non-linearities and their
relative phase.
Although this modulation method permits qualitative measurements of w (5) effects,
especially determination of relaxation times, quantitative evaluation of w (5) is not possible,
due to the infinite number of modulation harmonics with ill-defined relative amplitudes
caused by the trapezoidal modulation form provided by a chopper blade [106].
This problem has been solved by A. Feldner. He developed a new type of modulator,
working with two independently rotating polarizing foils (Fig 9.48). This type of modulator
generates two harmonics in the intensity modulation spectrum with a fixed ratio of 4 : 1
(cos 4 ot) [106].
Figure 9.48: Scheme of intensity modulation by rotating polarizing foils (a). The vertical polarizer (b)
ensures that an anisotropy of the material has no effect on the signal or the modulation spectrum.
For time-resolved pump-probe measurements of shorter relaxation times, a Titan-sapphire
laser pumped by an argon ion laser is employed. In the current configuration, the wavelength
can be tuned between 720 nm and 820 nm; the pulse width is about 120 fs.
169

9 Diacetylene Single Crystals
9.5.3 Theoretical approaches
The simplest approach is the free electron model [107, 108]. The electrons are treated to
move freely in a one-dimensional box, subject to a potential V 0 cos (px/d) by the ion cores
(d denotes the bond length). The polarizability a and hyperpolarizability g is derived from the
2nd and 4th order perturbation energies caused by an electrical field along the chain. This
rough model does not account for local field effects nor for the alternating bond length found
along the axis in PDAs. The model yields a static w (3) = 3.0610 –11 esu along the axis.
Agrawal, Cojan et al. [109–111] treated explicitly the linear and non-linear optical
properties of PDAs. They computed the electronic energy levels using a Hückel formalism.
They computed a static w (3) = 0.7610 –10 esu for PTS and w (3) = 0.25610 –10 esu for
TCDU. This model neglects electron-electron interaction and therefore excitonic effects.
The model yields the correct order of magnitude for w (3) , but the wrong sign.
The phase space filling (PSF) model was developed to describe non-resonant NLO
properties in semiconductors [112, 113], especially in quantum well structures [114]. Greene
et al. adapted this model to one-dimensional polymer chains [115, 116]. The model is only
applicable to systems were the low energy absorption band is excitonic, as is the case with
PDAs [117]. Formation of excitons is limited by the number of available electron states that
are necessary to form the exciton. With an increasing number of excitons, the dipole momentum
for forming a new exciton is reduced. The exciton band bleaches.
As the following measurements will show, the PSF model seems to describe best the
non-linear optical behaviour of polydiacetylenes. A very important prediction of this model
is the proportionality of w (3) and a in the near-resonant frequency regime [115]. This behaviour
was found in polydiacetylenes, strongly supporting the PSF theory [118].
9.5.4 Sample preparation
For measurements well off the resonance, i. e. with wavelengths larger than 720 nm, p-TS6
crystals were prepared by thermal polymerisation of monomer crystals, grown out of a saturated
solution and manually cut using a shaver blade. Thickness of these crystals varies between
40–100 µm.
Later, for measurements closer to resonance, a method has been developed to grow
thin mono-crystalline layers of TS6 and 4BCMU between glass substrates. To avoid crystal
strains the monomer crystals have to be removed from the substrate before polymerising.
The resulting crystal thickness can be made as low as 300 nm.
9.5.5 Value and phase of the third order susceptibility w (3)
From DFWM measurements |w (3) | was determined for several polydiacetylenes in the nearresonant
to off-resonant wavelength region, 680 nm to 750 nm (Tab. 9.6). Concurrently, the
170

9.5 Non-linear optical properties
Table 9.6: w (3) values for some different Polydiacetylenes [118, 119].
Material Modification |w (3) |/esu at wavelength
PTS crystal 2610 –10 720 nm
FBS crystal 2610 –10 720 nm
4BCMU amorphous film 4610 –11 720 nm
4BCMU thin monocrystalline film 3610 –10 720 nm
4BCMU thin monocrystalline film 2610 –9 670 nm
imaginary part of w (3) can be computed from the non-linear absorption coefficient b, obtained
from measurements of the intensity dependence of the sample transmission. It was
found that the real part of w (3) was dominating the imaginary part by a factor of 3 [118].
This is predicted by the PSF model, from the bleaching of the exciton band.
The value of w (3) is nearly identical for p-TS6 and p-4BCMU, and 4 times lower for
p-IPUDO. These values were reproducible within 30%. For p-FBS, the reproducibility was
only within an order of magnitude. No difference was found between thermally and x-ray
polymerised crystals [118].
9.5.6 Relaxation of the singlet exciton
In p-TS6 energy relaxation times could be resolved at wavelengths below 700 nm. As the relaxation
times are of the same order of magnitude as the laser pulse width (1 ps), a model
function has to be fitted to the measured curve to obtain the relaxation (Fig. 9.49).
Figure 9.49: w (3) relaxation in p-TS6 at a wavelength of 680 nm and</s