Phonons and the Isotopically Induced Mott transition - Physics

**Phonons** **and** **the** **Isotopically** **Induced**

**Mott** **transition**

Greg Freebairn

Honours Thesis

Department of **Physics**

The University of Queensl**and**

Supervisors: Dr. Ben Powell **and** Dr. John Fjaerestad

November 2005

2

Abstract

The study of strongly correlated systems is an active field of research within solid

state physics, especially in light of high temperature superconductivity. We look at

**the** isotopically induced metal-insulator, or **Mott**, **transition** in **the** organic chargetransfer

salts **and** consider whe**the**r phonons are involved in this **transition** by

applying a transformed Hubbard-Holstein Hamiltonian. A general analysis of **the**

applicable parameter range of this Hamiltonian is performed before application to **the**

charge transfer salts. It is found that phonons do not significantly alter **the** onsite

Coulomb repulsion or **the** b**and**width within **the** material **and** we conclude by arguing

that **the**y may in fact drive a metal-insulator **transition** by modifying **the** frustration of

**the** system.

Acknowledgements

First **and** foremost I would like to thank my supervisors, Ben Powell **and** John

Fjaerestad, for your help **and** guidance this year. Your patience in answering **the**

endless stream of questions has been greatly appreciated. Many thanks must go to

my friends **and** colleagues in **the** Honours room. Nothing brings a group of people

toge**the**r like having a **the**sis due in one week with three weeks worth of work to do.

Lastly, but by no means least, many thanks must go to my VERY underst**and**ing wife

Melissa, who has been a widow for **the** past month **and** put up with **the** highs **and**

lows throughout **the** year.

3

4

TABLE OF CONTENTS

1 Introduction ....................................................................................................... 7

2 Overview…… .................................................................................................... 8

2.1 Correlated systems **and** electron-lattice interactions.................................... 8

2.2 The metal-insulator **transition** ....................................................................... 9

2.2.1 The tight binding approach to b**and** **the**ory ............................................... 9

2.2.2 B**and** **the**ory **and** electronic properties .................................................... 10

2.2.3 The Hubbard energy............................................................................... 11

2.3 **Phonons** ..................................................................................................... 12

3 The metal-insulator **transition** in organic charge transfer salts ................ 15

3.1 Evolution of **the** Charge Transfer Salts ...................................................... 15

3.2 Chemical Composition **and** Structure......................................................... 16

3.2.1 Chemical Composition............................................................................ 16

3.2.2 Crystal Structure..................................................................................... 16

3.2.3 The phase diagram................................................................................. 17

3.3 Experimental signatures for phonon involvement in **the** MIT...................... 18

3.3.1 The effects of ‘chemical’ pressure **and** isotopic substitution ................... 18

4 The organics **and** **the** Hubbard model ........................................................ 20

5.5.1 Polaronic behaviour **and** b**and** narrowing ............................................... 42

5.5.2 Exact diagonalization **and** wavefunction overlap .................................... 43

6 Electrons, phonons **and** **the** Hubbard-Holstein model............................... 49

6.1 The Hubbard-Holstein model ..................................................................... 49

6.2 The model analysis (**the** single mode case) ............................................... 50

6.2.1 B**and** narrowing ...................................................................................... 50

6.2.2 The convergence of **the** exact Hamiltonian............................................. 51

6.2.3 Testing **the** approximate wavefunction ................................................... 51

6.2.4 Variation in U-V ...................................................................................... 55

7 Application of **the** polaronic model to **the** ET dimer .................................... 57

7.1 The calculation of dimer parameters with multiple modes.......................... 58

7.1.1 The dimer Coulomb repulsion U d ............................................................ 58

7.1.2 The interdimer hopping (t 1 **and** t 2 )........................................................... 59

8 Ano**the**r possibility - **the** geometrical isotope effect .................................... 63

9 Summary **and** conclusion............................................................................... 64

Appendix A………................................................................................................... 66

Appendix B ............................................................................................................. 75

4.1 The Hubbard model ................................................................................... 20

4.1.1 The Hubbard model **and** **the** tight binding approximation ....................... 20

4.2 The Dimer Hubbard Model ......................................................................... 22

4.3 The Two-Site Hubbard Model .................................................................... 23

4.3.1 Diagonalization of **the** two-site Hubbard model ...................................... 25

4.4 Calculation of **the** Hubbard Parameters (t m , U m **and** V m ) ............................ 26

5 Electron-phonon coupling **and** **the** Holstein Model...................................... 29

5.1 The Holstein model .................................................................................... 29

5.1.1 Polarons ................................................................................................. 29

5.2 Analytic techniques for **the** two-site Holstein model ................................... 31

5.2.1 The adiabatic limit t >> h ω.

..................................................................... 31

5.2.2 The nonadiabatic limit t ≤ h ω ................................................................. 32

5.3 The two-site effective Holstein Hamiltonian................................................ 32

5.4 Determination of Holstein parameters........................................................ 39

5.4.1 Calculation of vibrational modes............................................................. 39

5.4.2 The electron-phonon coupling constant.................................................. 41

5.5 The model analysis (**the** single mode)........................................................ 41

5

6

1 Introduction

Certain classes of materials including **the** **transition** metal oxides, organic (carbon

based) charge transfer salts **and** heavy fermions exhibit properties that can only be

accounted for in terms of strong electron-electron interactions. In general, **the**

mechanism underlying this strongly correlated behaviour are much less well

understood than those underlying behaviour in weakly correlated materials which can

be described by Fermi liquid **the**ory. In this study we will examine an unusual

electrical characteristic unique to strongly correlated materials - a first order metalinsulator

phase **transition** – in a family of charge transfer salts based on **the** BEDT-

TTF molecule. In **the** charge transfer salts this **transition** can be driven by chemical or

isotopic substitution, as well as hydrostatic pressure. Where deuterium is substituted

for hydrogen on **the** BEDT-TTF molecule this material can go from displaying metallic

to insulating properties at ambient pressure. Isotopic substitution is known to change

**the** vibrational spectra of this material which provides a motivation to investigate **the**

role of intermolecular phonons in this **transition** by applying **the** Hubbard-Holstein

minimal model.

We will outline some of **the** general concepts in solid state physics upon which this

work is based such as **the** tight binding model, b**and** **the**ory, phonons **and** how

electron-electron interactions are thought to induce a metal-insulator **transition** before

introducing **the** crystallographic **and** chemical characteristics of **the** organic charge

transfer salts.

Chapters 4 **and** 5 will introduce **the** Hubbard **and** Holstein minimal models which are

often used to investigate strong electron-electron **and** electron-phonon interactions

respectively. By applying a canonical transformation **and** a suitably chosen

variational wavefunction, an effective Hamiltonian (**the** Hubbard-Holstein model)

which accounts for both electron-electron **and** electron-phonon interactions, will be

derived in which **the** phonon degrees of freedom have been averaged out. This is

advantageous because it significantly simplifies **the** analysis. The general parameter

range over which this model is a good approximation will **the**n be explored before

being applied specifically to **the** BEDT-TTF system. Finally numerical results will be

compared with experimental observations **and** future directions discussed.

7

2 Overview

2.1 Correlated systems **and** electron-lattice interactions

One of **the** revolutionary breakthroughs in **the** field of condensed matter physics

occurred with **the** development of Fermi liquid **the**ory by L**and**au in **the** late 1950’s to

explain **the** behaviour of **the** liquid state of 3 He. 11 Within this **the**ory **the** remarkable

success of **the** independent electron approximation in explaining **the** observed

behaviour in many materials (e.g. **the** simple metals) in spite of **the** strength of

electron-electron (e-e) interactions was accounted for by introducing independent

electron like quasiparticles to represent **the** excited states.

In **the** same year that Fermi liquid **the**ory was first published Bardeen, Cooper, **and**

Schrieffer developed a microscopic **the**ory of superconductivity 12 (now referred to as

BCS **the**ory) based upon **the** interaction of electrons with lattice vibrations, or

phonons. Significantly within BCS **the**ory superconductivity could be accounted for by

considering a net attractive ‘pairwise’ interaction between electrons. Although **the**

direct electrostatic interaction is repulsive, **the**y found that ionic motion could

‘overscreen’ **the** Coulomb attraction leading to a net attraction **and** electron pair

formation. From this **the**ory it is evident that **the** addition of phonons into certain

systems can radically alter **the** effective form of **the** e-e interaction.

Through **the** 1980’s new classes of material began to emerge that could not be

accounted for in terms of Fermi liquid **the**ory. One of **the** most important of **the**se was

**the** cuprates, a family of copper oxide based materials that exhibited

superconductivity at remarkably high critical temperatures. It soon became apparent

that this superconductivity could not be explained within **the** BCS framework.

Presently, **the** search for **the** underlying mechanism remains an active area of

research (**and** intense debate) within **the** condensed matter fraternity.

One of **the** key aspects of this search is **the** need to underst**and** **the** state from which

**the** superconductivity arises. 13 The cuprates display unusual metallic properties such

as a highly anisotropic resistivity, bad metallic phases **and** a first order metalinsulator

phase **transition**. These properties, along with o**the**rs displayed by equally

8

unusual materials such as **the** colossal magnetoresistance materials **and** heavy

fermionic materials, can only be accounted for with **the** introduction of strong e-e

interactions. It is **the** need to underst**and** **the** behaviour of **the**se strongly correlated

systems that provides **the** underlying motivation for this work.

b**and** is known as **the** b**and**width W. The formation of this b**and** structure is illustrated

schematically in Fig. 2.1.

We are particularly interested in exploring how, or indeed if, phonons are involved in

some aspects of this behavior. This study will focus specifically on a feature

observed in many strongly correlated materials - a metal-insulator **transition** which is

often referred to as **the** **Mott** **transition**.

2.2 The metal-insulator **transition**

A model widely used to capture **the** basic physics involved in **the** metal-insulator

**transition** within strongly correlated crystalline materials was first formulated by **Mott**

in **the** 1950’s to account for **the** unusual electronic properties observed in some

materials e.g. antiferromagnetic NiO. 3 Based upon **the**ir electronic configurations,

conventional b**and** **the**ory calculations indicated that **the**se materials should display

metallic properties, but instead, **the**y behaved as insulators.

2.2.1 The tight binding approach to b**and** **the**ory

In order to underst**and** this unusual electrical behaviour it is helpful to first look at **the**

expected behaviour in terms of conventional **the**ories. One of **the** earliest methods

used to explain **the** conduction properties of solids was **the** ‘tight binding’ approach.

This takes as its starting point **the** isolated atom (or molecule) for which **the** electronic

eigenstates correspond to a series of discrete energy levels. If we now consider **the**

formation of a monatomic solid by bringing toge**the**r a large number of identical

atoms, **the** atomic energy levels will be split due to **the** interactions between atoms,

as in molecular orbitals in molecules corresponding to bonding **and** antibonding

combinations of wavefunctions **and** **the** closer **the** atoms are forced toge**the**r **the**

greater **the** splitting of **the**se levels. In general, for N closely spaced atoms in **the**

system each energy level is split into N two-fold degenerate (accounting for spin)

sublevels. These sets of sublevels are called b**and**s **and** if N is sufficiently large

22

( N ≈10 )

we can regard all energies within each b**and** as continuously accessible to

**the** electrons. 8 The spread between **the** minimum **and** maximum energies within a

Figure 2.1

(a) Schematic representation of nondegenerate (neglecting spin) electronic levels in an atomic

potential. (b) The energy levels of N such atoms in a periodic array, plotted as a function of mean

inverse interatomic spacing. Then **the** atoms are far apart **the** energy levels are nearly degenerate

but when **the** atoms are closer toge**the**r, **the** levels broaden into b**and**s. 8

2.2.2 B**and** **the**ory **and** electronic properties

Within b**and** **the**ory **the**re is a sharp distinction at absolute zero between crystalline

materials showing metallic conduction, on one h**and**, **and** insulating behaviour, on **the**

o**the**r. a Within this picture metals are materials in which one or more of **the** energy

b**and**s are partially filled. In such materials, at zero temperature, **the** electronic

energy states are filled to a limiting energy E F , known as **the** Fermi energy **and** states

with higher energies are empty. Thus, when a potential is applied electrons are free

to move into **the**se higher states **and** conduction results. Insulators, on **the** o**the**r

h**and**, are materials where all energy b**and**s are completely occupied or completely

empty **and** thus no current can flow. In terms of **the** density of states N(E), this

means that N(E F ) must vanish for an insulator. This principle is illustrated in Figure

2.2.

a At finite temperatures if **the** energy gap (b**and**gap) between **the**se energy b**and**s is sufficiently small

**the**rmal excitation can cause electrons to ‘jump’ into empty conduction b**and**s. This is **the** basis for

conduction in many semiconductors.

9

10

This energy difference U H is known as **the** Hubbard (on site) energy **and** is typically

large (several eV). In **the** condensed state, this onsite e-e repulsion was expressed

by **Mott** as: 3 2

e

φ ( ) φ ( )

where

12

0 12

2 2

UH

= ∫∫ r1 r2 dr1d

r

2

(2.2)

4πε

r

r is **the** inter-electron separation at a site, **and** ( r )

φ is **the** electron

wavefunction. In o**the**r words, in this model **the** e-e repulsion is neglected except for

two electrons on **the** same site. Conventional b**and** **the**ory, which incorporates e-e

interactions within a suitably chosen periodic potential within **the** single particle

Schrödinger equation, cannot take **the** effects of this Hubbard energy into account.

Figure 2.2

The electronic density of states N(E) in a cubic crystalline material where E F denotes **the** Fermi

2.2.3 The Hubbard energy

For particular classes of crystalline materials where electron concentrations are low

**and** hence screening ineffective, 1 as well as in materials with filled d **and** f orbitals,

**the** e-e interactions cannot be neglected. In **the**se materials **the** b**and** picture breaks

down dramatically with **the** occurrence of a first order phase **transition** between

metallic **and** insulating states. To underst**and** this **transition** within **the** framework

developed by Hubbard, 14 **and** later refined by **Mott**, 3 it is instructive to first consider

**the** situation regarding electronic excitations in **the** extreme limit of isolated alkali

atoms. Instead of **the**re being, as in **the** b**and** case, unoccupied electron energy

levels immediately above E F allowing electronic excitation **and** metallic behaviour,

electron hopping between neutral isolated atoms requires a large energy. In order to

remove an electron from an atom, an ionization energy I el must be supplied **and**,

although some energy is recovered as **the** electron affinity χ when this electron is

added to ano**the**r neutral atom to give a negatively charged ion, **the** difference

between **the**se two energies **and** **the**refore **the** net energy required to place two

U = I − χ.

(2.1)

electrons in a given orbital, is given by: 4

H

el

For strongly correlated materials **the** energy cost in placing a second electron on a

site within this model can (for

UH

≥ W

) open a gap in **the** o**the**rwise continuous b**and**

**and** **the**refore make **the** material electrically insulating. This gap is generally not

equal to **the** Hubbard energy because of b**and**width effects. If we consider **the**

2

example of two Ni

+ ions 4 in **the** limit of infinite separation, **the** energy cost in placing

an extra electron on a

2

Ni

+ is equal to U H given by Eq. (2.1). As **the** interatomic

separation is reduced Hubbard b**and**s form, with **the** lower one corresponding to **the**

motion of ‘holes’ among **the** ion sites **and** **the** upper b**and** to **the** motion of electrons.

This is illustrated in Figure 2.3

As **the** b**and**width of **the**se b**and**s increases it becomes easier for electrons **and** holes

to hop between sites **and** **the** eventual overlap of **the** two b**and**s marks **the** **Mott**

**transition** from **the** insulating to **the** metallic state.

2.3 **Phonons**

In **the** same way that **the** energy of an electromagnetic wave is quantized with **the**

photon as **the** quantum of energy, **the** quantum of energy associated with collective

modes of vibration on a lattice is called a phonon. Two types of phonon can be

distinguished depending on **the** nature of **the** mode: ‘acoustic’ **and** ‘optical’. An

acoustic mode is one in which all ions in a primitive cell move essentially as a unit in

phase, **and** **the** dynamics are dominated by interactions between cells: an optical

energy for (a) a normal metal **and** (b) an insulator. 3 12

11

mode, on **the** o**the**r h**and**, arises in crystals that have more than one atom in **the** unit

cell when **the** ions within each primitive cell exhibit what is essentially a molecular

vibratory mode, which is broadened out into a b**and** of frequencies by virtue of **the**

intercellular interactions. 8 The distinction between **the** two is perhaps best illustrated

by considering a diatomic linear chain as shown in Figure 2.4.

Figure 2.3

Schematic illustration of **the** Hubbard b**and**s in NiO as a function of b**and**width W which is

inversely dependent upon **the** interatomic separation. The lower b**and** corresponds to **the** motion

of holes **and** **the** upper b**and** to electron motion. The metal-insulator **transition** occurs when **the**

two b**and**s overlap. 4

Acoustic phonons, as **the** name suggests, correspond to sound waves in **the** lattice

**and** consequently have frequencies that become small at long wavelengths. Optical

phonons, on **the** o**the**r h**and**, will always have some minimum frequency of vibration,

even when **the**ir wavelength is large. They are called "optical" because in ionic

crystals (such as sodium chloride) **the**y are excited very easily by light (in fact,

infrared radiation). This is because **the**y correspond to a mode of vibration where

positive **and** negative ions at adjacent lattice sites swing against each o**the**r, creating

a time-varying electric dipole moment. Optical phonons that interact in this way with

light are called infrared active. Optical phonons that are Raman active can also

interact indirectly with light, through Raman scattering. 15

Figure 2.4

The long wavelength acoustic (a) **and** optical (b) modes in a diatomic linear chain. The primitive

cell contains **the** two ions joined by **the** jagged line. In both cases **the** motion of every primitive cell

is identical, but in **the** acoustic mode **the** ions within **the** cell move toge**the**r, whilst **the**y move 180°

out of phase in **the** optical mode. 8 14

13

3 The metal-insulator **transition** in organic

charge transfer salts

Partly motivated by interest in high temperature superconductivity, 16 **the** MIT has

been extensively studied in **the** **transition** metal oxides, initially in materials such as

(V 1-x Cr x ) 2 O 3 , **and** later in **the** cuprates. In principle, this can be done experimentally

by sweeping **the** hydrostatic pressure continuously through **the** **transition**. However,

ano**the**r class of materials, **the** organic (carbon based) charge transfer salts also

offers an excellent opportunity to study this region because **the**y share many of **the**

same unusual properties with **the** cuprates. Fur**the**rmore, **the**se compounds are

known to display a great sensitivity to variations in hydrostatic pressure 17 **and** **the**y

have **the** added advantage that **the** location of **the** **transition** region on **the**ir phase

diagram can be ‘tuned’ through chemical or isotopic substitution.

3.1 Evolution of **the** Charge Transfer Salts

Most organic materials are electrical insulators (e.g. plastics) but in **the** 1940’s

scientists began to study organic crystals with **the** view of utilizing **the**m as

semiconductors. 3 Of particular interest were **the**ir electronic properties **and** **the**

underlying mechanisms of charge mobility. Following **the** discovery in 1954 of **the**

organic compound perylene bromine 1 which displayed a marked increase in

conductivity over o**the**r materials, **the**re was an explosion in **the** rate of development

of organic metals, a family of organic charge-transfer salts (CT’s) where **the**

constituent intermolecular compounds are stabilized by **the** partial transfer of

electrons between constituent molecules.

The extraordinarily high electrical conductivity of some of **the**se organic conductors

below 60 K led Bill Little of Stanford University to propose **the** possibility of highcritical

temperature organic superconductors in 1964. 18 In 1979 **the** first species of

organic superconductors (TMTSF) 2 X (where X st**and**s for an electron acceptor

molecule), was discovered by Bechgaard, Jerome, Mazaud, **and** Ribault with **the**

syn**the**sis of (TMTSF) 2 PF 6 . 1 Shortly after this breakthrough a second basic

superconducting species was discovered for which **the** BEDT-TTF (bisethylenedithio-

tetrathiafulvalene), or ET molecule, forms **the** basis. It is **the** properties of this second

group that are of interest in this report.

3.2 Chemical Composition **and** Structure

3.2.1 Chemical Composition

The ET molecule consists of carbon **and** sulphur rings with two terminal ethylene

groups as shown in Fig. 3.1.

sulphur

hydrogen

carbon

Figure 3.1

Forming **the** basis for a class of organic charge transfer salts, **the** BEDT-TTF molecule is

composed of carbon **and** sulphur rings along with four terminal ethylene groups.

This molecule, when combined with various monovalent anions, forms compounds

represented by **the** shorth**and** (ET) m X n . However, most of **the** organic

superconductors are of **the** 2:1 composition ratio like (ET) 2 X. 1 In **the** present study X

represents Cu[N(CN) 2 ]Br unless o**the**rwise stated **and** will be abbreviated as Br.

3.2.2 Crystal Structure

The (ET) 2 Br compound can exhibit several different crystal configurations but **the** one

of interest presently is **the** κ-phase. In this configuration ET molecules are paired with

**the**ir central tetrathioethylene planes almost parallel forming ‘dimers’, **and** adjacent

pairs are almost perpendicular to one ano**the**r in **the** bc-plane; this results in a donor

sheet. The Cu[N(CN) 2 ]Br anions are weakly bonded to each o**the**r through **the** Cu-Br

interaction, so as to form planar polymer like structures. This gives a layered

structure with successive ET layers separated by polymerised anion layers with each

dimer having one electron less than **the** full electronic shell due to charge transfer to

**the** anions. This layered structure along with **the** characteristic dimer configuration of

**the** kappa crystal phase is illustrated in Fig. 3.2.

15

16

The observed splitting of proton NMR lines below 26K is evidence of

antiferromagnetic ordering, with **the** magnetic moment estimated from **the** magnitude

of **the** splitting to be in **the** order of ~ µ per dimer. 9 16 The size of antiferromagnetic

B

ordering is indicative of strong e-e correlations **and** indicates that **the** observed metalinsulator

**transition** in **the**se materials can be understood in terms of **the** **Mott** model.

a) b)

Figure 3.2

(a) The kappa crystal phase of **the** ET based charge transfer salt showing conducting

layers of ET molecules s**and**wiched between planar insulating anion sheets 2 **and** b) **the**

pairing of ET molecules in this phase forming a dimer structure. 10 18

The lattice parameters of **the** κ -(ET) 2 Br crystal are given in Table 3.1.

Structure

a

[Å]

κ-(ET) 2 Br Orthorhombic 12.949 30.016 8.539

Table 3.1 Crystallographic data for κ -(ET) 2 Br at room temperature **and** ambient pressure. 1

One particularly important characteristic of this crystal structure, as we will see when

we come to model **the** material, is that it exhibits a very anisotropic conductivity. The

ratio of conductivity in **the** b-c plane (perpendicular to **the** donor sheets) compared

with that in **the** a direction (parallel to **the** donor sheets) has been measured

experimentally to be in **the** order of 1:1000. 1 This implies that **the** material can be

approximated very well as a quasi-2D conductor.

3.2.3 The phase diagram

First suggested by Kanoda, 19 **the** general phase diagram for **the** CT’s shows

insulating, metallic, ‘bad metal’ **and** superconducting phases. Transitions identified

using 1 H NMR, ac susceptibility, magnetization, **the**rmal expansion, **and** resistivity

techniques are given in Fig. 3.3 along with **the** inferred diagram.

b

[Å]

c

[Å]

Figure 3.3

(a) Experimental results for κ-(ET) 2 X showing **the** **transition**s detected using various techniques

(NMR **and** ac conductivity – diamonds, magnetization – pluses, **the**rmal expansion – circles, **and**

resistivity – squares, stars, triangles **and** diamonds depending on **the** **transition** type. Note that **the**

data has been offset to account for “chemical pressure” with P=0 corresponds to **the** position of κ-

(ET) 2 Cl at ambient pressure. (b) shows **the** inferred phase diagram with a first order metalinsulator

phase **transition** between insulating **and** metallic states. 7

3.3 Experimental signatures for phonon involvement in **the** MIT

3.3.1 The effects of ‘chemical’ pressure **and** isotopic substitution

As mentioned earlier one of **the** key behavioral properties of **the** CT’s that makes

**the**m useful in **the** study of **the** MIT is **the**ir ‘tunability’ using isotopic substitution. The

electrical properties of **the** CT’s can be shifted to **the** left on **the** phase diagram by

progressively substituting deuterium for hydrogen on **the** ethylene groups. From Fig.

3.4 we can see that **the** electrical properties of κ-(ET) 2 Br at ambient pressure are

radically changed by **the** substitution of **the** eight terminal hydrogen ions for

deuterium. The line to **the** right on **the** phase diagram represents **the** relative position

of κ-H 8 -(ET) 2 Br (hydrogenated) at ambient pressure. As **the** temperature is reduced

this crystal continues to exhibit metallic properties **and** at sufficiently low

temperatures makes a **transition** into a superconducting state. On **the** o**the**r h**and**, **the**

17

location of κ-D 8 -(ET) 2 Br (fully deuterated) at ambient pressure is illustrated by **the** line

to **the** left on **the** phase diagram. This shows **the** crystal to be an insulator at T=0.

4 The organics **and** **the** Hubbard model

4.1 The Hubbard model

Figure 3.4

The phase diagram for κ-(ET) 2 Br showing **the** shift in **the** electrical properties upon **the** total

substitution of deuterium for hydrogen. At ambient pressure, **the** material goes from exhibiting

metallic properties **and** superconductivity over **the** entire temperature range, to an insulator at

The question we would like to answer is whe**the**r this **transition** is driven by electronphonon

interactions **and** more specifically, by intramolecular (optical) phonons In an

analogy with classical mechanics, where changing **the** relative mass of an oscillating

system changes **the** characteristic modes of vibration, a similar shift should be

observed in **the** vibrational spectra of κ-(ET) 2 Br upon isotopic substitution. The

vibrational spectra of both H 8 -(ET) 2 Br **and** D 8 -BEDT-TTF were investigated in **the** late

1980’s by Kozlov et. al. using infrared **and** Raman spectroscopy. 20 This work was

later exp**and**ed upon by Eldridge et. al. 21 who also considered **the**

C →

12 13

substitution of both **the** central six carbon atoms, **and** **the** two carbon atoms involved

in **the** central double bond, as well as **the**

S → S substitution of all **the** sulphur

32 34

atoms. Both groups did indeed observe a shift in **the** vibrational spectra confirming

**the** isotopic vibrational effect although **the**ir subsequent assignment of normal

vibrational modes is flawed because both groups assume planar symmetry (D 2

symmetry) for **the** molecule. 6

C

One of **the** most useful tools for analysing any system, **and** especially one as

complex as a condensed matter system, is a minimal model. These are models

which have **the** smallest number of possible parameters that (hopefully) capture

enough of **the** essential physics of **the** system to provide at least a qualitative

prediction of its behaviour. One of **the** minimal models often utilized to analyse

strongly correlated systems, **and** particularly **the** organics, is **the** Hubbard model.

This was first developed in **the** 1950’s by Anderson **and** later refined by Hubbard 3 to

study a crystalline array of one-electron atoms:

† †

∑ i, σ i, σ ∑ ( i, σ j,

σ

. ) ∑ .

(4.1)

i i

H = c c − t c c + h c + U n n

µ

↑ ↓

i, σ

ij , σ

i

Within this model µ represents **the** chemical potential, **and** t **the** hopping integral for

an electron hopping from site j to site i. U is **the** on-site Hubbard energy due to

electron-electron interactions on **the** same site,

(creation) operator on site i with spinσ , **and** n

(†)

ci

σ

is **the** electron annihilation

= c c is **the** number operator. The

†

iσ iσ iσ

sum ij runs over pairs of nearest neighbor lattice sites.

4.1.1 The Hubbard model **and** **the** tight binding approximation

The Hubbard model is based upon **the** tight binding approximation which, as we have

seen, takes as its starting point **the** isolated atom. If we assume that **the** electron

wavefunction ( )

φ r corresponding to **the** i th discrete energy level,

i

**the** solution of **the** Schrödinger equation for **the** isolated atom

( ) = E φ ( )

A i i i

E

i

, is known from

H φ r r (4.2)

**the**n **the** overlap between neighbouring atomic wavefunctions is very small from **the**

tight binding approximation, so **the** crystal Hamiltonian can be written as

h

H ≈ H + = − ∇ + V − + −

2

A

2

ν

2me

A n n

( r R ) ν ( r R )

,

(4.3)

most temperatures. 7 20

19

where R n is **the** translation vector for **the** n th site **and** **the** perturbation ν ( − )

r R can

be approximated as a sum over **the** atomic potentials for all sites apart from that (n)

at which **the** electron is localized, i.e.

( −

n ) ≈ ∑VA ( −

j ).

ν r R r R (4.4)

In general **the** eigenstates ψ of this Hamiltonian (4.3) can be written in **the** form

where **the** functions Φ ( − )

i

i

n

∑

j ≠n

( )

ψ ∝

i ⋅

e k Rn

Φ r − R (4.5)

ki i n

n

r R are known as Wannier functions. These have **the**

important property that **the**y are mutually orthogonal. 8

In terms of this formalism **the** chemical potential µ

i

**and** **the** hopping integral t i

are

defined as

**and**

The electron operator

( ) ( ) ( ) ,

µ ∗

i

= −∫ Φ ν

d

i

r − Rn r − Rn Φi r − R

n

r (4.6)

t

(†)

iσ

( ) ν ( ) ( ) d ,

= −∫ Φ r − R r − R Φ r − R r (4.7)

∗

i i m n i n

c annihilates (creates) an electron within a Wannier state

which in **the** case of **the** organics will be assumed to be **the** highest occupied

molecular orbitals (HOMO’s) of **the** molecule. The Hubbard U is given by **the** second

derivative of energy with respect to charge:

( q)

2

δ E0

U = lim ,

δ q→1

2

δq

n

(4.8)

where E 0 (q) denotes **the** groundstate energy for a site **and** q represents **the** number

of electrons in **the** HOMO.

Thus, **the** Hubbard model is based upon two fundamental assumptions: **the** first

being that an adequate description of **the** physics is possible by accounting for only

one orbital per site, **and** **the** second being that **the** shape of **the** orbital is independent

of **the** electronic occupation, i.e. **the** orbital relaxation (**the** variation in **the** shape of

4.2 The Dimer Hubbard Model

The quasi-two dimensionality **and** dimer structure of **the** layered κ-(ET) 2 Br crystal led

McKenzie, 5 following Kino **and** Fukuyama, to propose **the** 2D dimer Hubbard model

on an anisotropic triangular lattice as a minimal microscopic model that may

adequately describe **the** behaviour of this material b :

d † †

d d

∑ σ 1 ∑ ( , σ , σ

. .) 2 ∑ ( , σ , σ

. .)

∑ (4.9)

H = µ

d

ni − t

d

ci c

j

+ h c − t

d

ck cl + h c + Ud n n

i↑ i↓

i ij , σ

{ kl}

, σ

i

where µ

d

is **the** dimer chemical potential, **the** t αd

are hopping amplitudes, Ud

is **the**

(†)

onsite Coulomb repulsion on a single dimer **and** c σ

annihilates (creates) an electron

on dimer i with spin σ . The sum ij runs over pairs of nearest neighbor dimers in

**the** horizontal direction, **and** **the** sum { kl}

runs over pairs along **the** diagonals as

shown in Fig. 4.1. The subscripts m **and** d will be used to represent **the** monomer

**and** dimer respectively.

Figure 4.1

HOMO

i

dimer

monomers

A representation of **the** dimer structure in κ-(ET) 2 Br showing **the** model electronic structure with **the**

relevant hopping terms **and** molecular orbitals. 5

**the** site orbital with **the** charge residing on it) is negligible. 22 22

b This Hamiltonian has been modified slightly to refer to electrons ra**the**r than holes for consistency

throughout **the** report.

21

Using Eq. (4.8) **the** effective Coulomb repulsion between two electrons on a dimer is

given by 5 U E ( ) E ( ) E ( )

= 2 − 2 1 + 0 ,

(4.10)

d d d d

where n is **the** number of electrons in **the** HOMO for a dimer with charge (ET) (2-n)+ 2 .

By applying a dynamical mean-field treatment to **the** Hubbard model, where **the** b**and**

is half filled **and** **the** Coulomb repulsion less than **the** minimum required for an

insulating state (matching **the** metallic phase of **the** κ-(ET) 2 X crystal), Merino et. al.

were able to give a qualitative prediction of some of **the** experimentally observed

transport properties within **the** organics. These include temperature dependent Hall

coefficients, a non-monotonic functional dependence of **the**rmopower on temperature

**and** **the** emergence of a small Drude peak in **the** frequency dependent conductivity at

low temperatures.

Within this model **the** insulating **and** conducting behaviours can be thought of as a

competition between **the** hopping integral **and** **the** onsite Coulomb repulsion. The

hopping term, which represents **the** electronic kinetic energy, allows electrons to

move between sites irrespective of **the** occupancy on that site (without violating **the**

Pauli exclusion principle) , whilst U favours localization with a single electron per site.

The ratio of U d /t d determines whe**the**r **the** system is in **the** insulating or conducting

phase. The larger **the** ratio **the** more localized **the** electrons **and** **the** more likely we

are to see an insulating state.

It can be seen that this model retains **the** same form as **the** dimer Hubbard model

with **the** exception of an additional Coulomb term V m to account for **the** inter-site

Coulomb repulsion due to **the** proximity of **the** second monomer. Within this model,

**the** HOMO of each site now refers to a monomer orbital, each of which can contain

up to two electrons depending upon **the** filling (four in total).

In terms of **the** two-site model, **the** interdimer hopping terms can be expressed as

functions of **the** interdimer molecular hopping terms t b2 , t p **and** t q, which refer to

hopping between monomers on different dimers as illustrated in Fig. 4.2, as well as

**the** two-site parameters t m **and** U m : 5

where

t1 t2

1

= = ( cosθ

− sin θ ),

t t t 2 2

( + )

b2

p q

2 2

U ⎛⎛

⎞ ⎞

m

U

θ = − ⎜

m

tan ⎜ ⎟ + 1 ⎟ .

4t

⎜⎝

4 ⎠ ⎟

m

tm

⎝ ⎠

1

(4.12)

(4.13)

4.3 The Two-Site Hubbard Model

The dimer Hubbard model does not account for **the** internal structure of **the** dimer

**and** **the**refore lacks **the** resolution required to explore **the** role for intramolecular

phonons in **the** MIT. To investigate U d **and** t d (t 1 ,t 2 ) in terms of **the** dimer structure we

can instead consider **the** dimer as consisting of two sites. With this in mind **the** twosite

extended Hubbard model may be a suitable minimal model to account for **the**

effects of pressure variations in **the**se materials:

† †

∑( 1σ 2σ ) ∑( 1σ 2σ 2σ 1σ

) ∑ i i 1 2

(4.11)

H = n + n − t c c + c c + U n n + V n n

H

µ

m m m ↑ ↓ m

σ

σ

i=

1,2

Figure 4.2

Geometrical arrangement of **the** ET molecules in **the** kappa phase showing **the** dominant

intermolecular hopping integrals. Each line represents an ET molecule. 5

If t = t 1 = t 2 **the**n **the** model reduces to **the** Hubbard model on a triangular lattice,

following a rescaling of **the** lattice constants. 5 Although **the**re is no clear consensus

23

24

on **the** phase diagram for this model i it is agreed that in general, for a small U d /t **the**

ground state should be a paramagnetic metal **and** for large U d /t, an insulator.

4.3.1 Diagonalization of **the** two-site Hubbard model

To determine **the** energy eigenvalues **and** eigenstates of this system for different

electron occupancies we choose appropriate bases from **the** monomer molecular

orbital Wannier states. For example, a complete set of states is:

(n=1)

1 = c 0,0 2 = c 0,0

† †

1 1↑

1 1↓

† †

1

= c2 1

= c

↑

2↓

3 0,0 4 0,0

(4.14)

**and** substituting **the** derived energy expressions (4.18) into Eq. (4.19) gives **the**

effective dimer Coulomb repulsion as a function of **the** two-site parameters t , m

Um

**and** V

m

:

2 2

(( ) ) 1 2

1

U ⎛

⎞

d

= ⎜Um + Vm − Um − Vm + 16tm + 4tm

⎟

2 ⎝ ⎠ , (4.20)

which is in agreement with that obtained by McKenzie: 5

for V=0.

2 2

( ( ) )

1 2

1

Ud = 2tm + Um − Um + 16 tm

.

(4.21)

2

(n=2)

(n=3)

(n=4)

1 = c c 0,0 2 = c c 0,0

† † † †

2 1↑ 1↓ 2 1↑ 2↑

† † † †

2

= c c

1 2 2

= c c

↑ ↓ 1↓ 2↓

† † † †

2

= c c

1 2 2

= c c

↓ ↑ 2↑ 2↓

3 0,0 4 0,0

5 0,0 6 0,0

1 = c c c 0,0 2 = c c c 0,0

† † † † † †

3 1↑ 1↓ 2↑ 3 1↑ 1↓ 2↓

† † † † † †

3

= c c c

1 2 2 4

= c c c

↑ ↑ ↓ 1↓ 2↑ 2↓

3 0,0 4 0,0

† † † †

4 1↑ 1↓ 2↑ 2↓

(4.15)

(4.16)

1 = c c c c 0,0

(4.17)

where 0,0 is **the** vacuum state. In Table 4.2 we give **the** complete solution of **the**

two-site extended Hubbard model where **the** groundstate energy eigenvalues E ( n )

of **the** n electron configuration are:

E

0

0

0

( 1)

( )

( )

= µ − t

m

2 2

( )

1 ⎛

E0

( 2) = ⎜4µ

+ U + V − ( U − V ) + 16t

2 ⎝

E 3 = 3µ

+ U + 2V − t

E 4 = 4µ

+ 2U + 4V

m

m m m m m m

m m m m

m m m

Rewriting Eq. (4.10) in terms of **the** two site groundstate energies

( ) ( ) ( )

0 0 0

⎞

⎟

⎠

0

(4.18)

U = E 4 − 2E 3 + E 2 ,

(4.19)

d

4.4 Calculation of **the** Hubbard Parameters (t m , U m **and** V m )

Having derived a **the**oretical expression from **the** two-site Hubbard model for **the**

likely behaviour of **the** system, **the** next challenge in relating this to **the** actual

behaviour of **the** physical system is to obtain accurate estimates of **the** associated

model parameters. As Fortunelli et. al. point out, **the** determination of **the**se

parameters **and**, ultimately, validation with experimental results, presents an ongoing

problem within **the** field. 23

In CT crystals such as ET **the** hopping integral t m is generally estimated from

Extended Huckel (EHT) calculations 22,23 although this approach is only accurate to

about a factor of two. 24 In general, ab initio calculations are expected to give much

more accurate results in **the** calculation of **the** hopping parameter.

In **the** calculation of **the** Coulomb parameters U m **and** V m different groups follow **the**

same basic principle. Using various quantum chemical methods **the** ground state

energies for different charge configurations of **the** monomer are calculated. These

values are **the**n used to calculate U m **and** V m but where each of **the**se groups differs

23 22

is in **the** procedure used to calculate **the** ground state energies. Fortunelli et. al.

25 have based **the**ir calculations on an analysis of ab initio quantum chemistry

calculations using a Restricted Hartree-Fock calculation (RHF-SCF). Castet et. al. 26

estimate **the** relative magnitudes of **the** Coulomb terms U m **and** V m using **the** less

accurate AM1 semi-empirical Hamiltonian **and** a mixed Valence-Bond/Hartree-Fock

25

26

method on a cluster of up to four dimers. Powell et. al. 7 obtain **the**ir value of U m from

an application of density functional **the**ory DTF. These values are summarized along

with **the** associated method in Table 4.1.

Because **the** values of V m are of **the** same order of magnitude as U m **the** offsite

Coulomb repulsion cannot be neglected, as McKenzie has done (c.f. Eq. (4.21)),

when expressing U d as a function of **the** monomer parameters. Choosing reasonable

parameters values of U m = 13t m **and** V m = 8t m (c.f. Table 4.1) in Eq. (4.20) we obtain

≈ 1 + in contrast to **the** 2t 2

m calculated by McKenzie.

a value of U ( U V )

d m m

Parameter Group Method Value (eV)

t m Fortunelli et. al. RHF-SCF 0.272

Fortunelli et. al. Extended Huckel 0.224

T. Komatsu et. al Extended Huckel 0.244

t b2 Fortunelli et. al. RHF-SCF 0.085

Fortunelli et. al. Extended Huckel 0.071

T. Komatsu et. al Extended Huckel 0.092

t q Fortunelli et. al. RHF-SCF 0.040

Fortunelli et. al. Extended Huckel 0.040

T. Komatsu et. al Extended Huckel 0.034

t p Fortunelli et. al. RHF-SCF 0.130

Fortunelli et. al. Extended Huckel 0.094

T. Komatsu et. al Extended Huckel 0.101

U m Castet et. al. AM1 3.90

Fortunelli et. al. RHF-SCF 3.56

Powell et. al. DFT 3.60

V m Ducasse et. al. Mixed Valence Bond/Hartree Fock 2.70

Castet et. al. AM1 ~2.30

Fortunelli et. al. RHF-SCF 3.56

Table 4.1 A summary of inter- **and** intra- dimer hopping terms **and** Hubbard parameters with **the**

associated methods of calculation

Table 4.2 Eigenvalues **and** eigenvectors of **the** two-site Hubbard Hamiltonian for HOMO fillings

of n=1,2,3,4. The labels a, b denote antibonding **and** bonding states respectively, whilst S **and** T

correspond to singlet **and** triplet states **and** tanθ = −4 t / U − V + ( U − V ) + 16t

Filling **and** energy Degeneracy Eigenvectors

(n=1)

E

E

= ε − t

1 0

= ε + t

2 0

(n=2)

0

( ) 2 2

E = 2ε

+ U + V − U − V + 16t

ET

sb

ECT

= 2ε

+ V

0

= 2ε

+ U

0

0

( ) 2 2

E = 2ε

+ U + V + U − V + 16t

sb

(n=3)

E = 3ε

+ U + 2V − t

3b

0

E = 3ε

+ U + 2V + t

3a

0

(n=4)

E = 4ε

+ 2U + 4V

1

4 0

2

2

1

3

1

1

2

2

2 2

( )

† †

( 1↑

2↑

)

1 b, ↑ = 1 c + c 0,0

2

† †

( 1↓

2↓

)

1 b, ↓ = 1 c + c 0,0

2

† †

( −

1↑

2↑

)

1 a, ↑ = 1 c c 0,0

2

† †

( −

1↓

2↓

)

1 a, ↓ = 1 c c 0,0

2

† † † †

( 1↑ 1↓ 2↑ 2↓

)

Sa = 1 ⎡sinθ

c c + c c

2 ⎣

T,1 = c c 0,0

† † † †

( c c c c

1↑ 2↓ 2↑ 1↓

)

− cosθ

+ ⎤

⎦

0,0

† †

1↑

2↑

T, − 1 = c c 0,0

† †

1↓

2↓

† † † †

T,0 = 1 ⎡c c − c c ⎤ 0,0

2↓ 1↑ 1↓ 2↑

2 ⎣

⎦

† † † †

CT = 1 ⎡c c − c c ⎤ 0,0

1↑ 1↓ 2↑ 2↓

2 ⎣ ⎦

† † † †

( 1↑ 1↓ 2↑ 2↓

)

Sb = 1 ⎡cosθ

c c + c c

2 ⎣

† † † †

( c c c c

1↑ 2↓ 2↑ 1↓

)

+ sinθ

+ ⎤

⎦

0,0

† † † † † †

3 b, ↑ = 1 ⎡c c c + c c c ⎤ 0,0

1↑ 1↓ 2↑ 2↑ 2↓ 1↑

2 ⎣

⎦

† † † † † †

3 b, ↓ = 1 ⎡c c c + c c c ⎤ 0,0

1↑ 1↓ 2↓ 2↑ 2↓ 1↓

2 ⎣

⎦

† † † † † †

3 a, ↑ = 1 ⎡c c c − c c c ⎤ 0,0

1↑ 1↓ 2↑ 2↑ 2↓ 1↑

2 ⎣

⎦

† † † † † †

3 a, ↓ = 1 ⎡c c c − c c c ⎤ 0,0

1↑ 1↓ 2↓ 2↑ 2↓ 1↓

2 ⎣

⎦

4 = c c c c 0,0

† † † †

1↑ 1↓ 2↑ 2↓

27

28

5 Electron-phonon coupling **and** **the** Holstein

Model

Having suggested a minimal model to account for **the** electronic component of **the** ET

system we can now extend **the** model to account for **the** ionic degrees of freedom

**and** electron-phonon (e-ph) coupling. Neglecting **the** Hubbard U **and** V for **the**

moment, in materials where conduction electrons couple with lattice vibrations or

‘phonons’, a minimal model often used to describe **the**se interactions is **the** Holstein

model.

The physical properties of **the** polaron differ from those of **the** b**and** electron because,

as **the** ‘polaronic’ electron moves through **the** lattice, it will drag **the** polarization cloud

with it. As a consequence, **the** quasiparticle will behave as if it has a higher inertial

mass **and** hence a lower mobility. Polarons are also characterized by a binding (or

self-) energy

2 2

n λ

Ep

= ,

(5.2)

2hω

which is a measure of **the** depth of **the** potential well induced in **the** lattice.

5.1 The Holstein model

Developed by T. Holstein in 1956, 27 28 in its simplest form **the** Holstein model

consists of one electron hopping term, dispersionless phonons, **and** an interaction

term that couples **the** electron density **and** ionic displacements at a given site:

† † † †

( , σ , σ , σ , σ ) ωµ ( µ µ ) λµ ( µ µ )

∑ ∑ ∑

H = − t c c + c c + h a a + 1 + n a + a

(5.1)

2

Hol m i j j i i i i i i

i, j σ i, µ i,

µ

where ω

µ

is **the** dispersionless phonon frequency of **the** vibrational mode µ ,

**the** phonon annihilation (creation) operator for **the**

i

(†)

a µ

is

th

µ vibrational mode on **the** i th site,

n i is **the** number of electrons on **the** i th site, **and** λ

µ

is **the** electron-phonon coupling

constant for **the**

5.1.1 Polarons

th

µ mode.

A central feature of electron-phonon systems is that **the** interaction induces a

polarizing deformation of **the** lattice around **the** electron. According to general

mechanical principles **the**se induced displacements provide a potential well for **the**

electron **and** if this well is sufficiently deep, **the** electron will occupy a ‘bound’ state,

unable to move unless accompanied by **the** well, that is to say, by **the** induced lattice

deformation. The unit consisting of **the** electron, toge**the**r with its induced lattice

deformation forms a quasiparticle known as a polaron. 4 A schematic illustration of a

large polaron with **the** associated lattice distortion extending over many lattice

constants is given in Figure 5.2.

Figure 5.1

Schematic illustration of a large polaron in a crystal. The electron polarizes its surroundings **and**

**the** carrier plus **the** associated lattice distortion make up **the** polaron. 4

Two types of polaron may be distinguished, depending upon **the** spatial extent of **the**

lattice distortion associated with **the** quasiparticle. When **the** distortion extends over

many lattice constants **the** polaron is considered ‘large’ **and**, conversely, when

restricted to **the** immediate vicinity of **the** electron, **the** quasiparticle is considered

‘small’. 29

Various approaches for studying ‘large’ polarons have been developed depending

upon **the** strength of **the** e-ph coupling. An excellent overview of **the**se approaches is

given by Devreese. 29 In this work, however, we will limit our discussion to small

polarons, which appear as a result of overall short-range Coulomb forces. Small

polarons form when **the** electron-phonon coupling is strong, 30 λ = E / D > 1 where D

is **the** half b**and**width, **and** although **the** parameter range within **the** ET system does

p

29

30

not strictly meet this criterion as we will see in **the** following chapters, we will start

from this assumption **and** explore **the** limits of **the** parameter range over which this is

valid.

5.2 Analytic techniques for **the** two-site Holstein model

5.2.1 The adiabatic limit t >> h ω.

In **the** strong coupling limit all electrons in **the** conduction b**and** are ‘dressed’ by

phonons **and** **the** hopping integral is renormalized. In his original work on small

polaron dynamics using a two-site model Holstein defined two regimes, **the** adiabatic

**and** non-adiabatic, to which different analytical approaches are applied to study this

renormalization. 28

In **the** adiabatic limit it is assumed that **the** lattice ions move much more slowly than

**the** valence electrons. Physically, this corresponds to an electron spending much

less time on a given site (**the** localization time) than required for **the** lattice to deform

into **the** polaronic configuration (**the** relaxation time). Because **the** localization time is

inversely proportional to **the** electronic kinetic energy (represented by **the** hopping

integral t within **the** Holstein framework) **and** **the** relaxation time is of **the** same order

of magnitude as **the** period of **the** lattice vibration,

in terms of **the** Holstein parameters: 30 hω

The two-site Holstein Hamiltonian is: 33

† † †

( ) ( ) {( 1 †

1 2 1σ 2σ 2σ 1σ µ µ µ ) µ ( µ µ )}.

H = µ n + n − t ∑ c c + c c + ∑h ω a a + + g n a + a (5.7)

2

m m i i i i i

σ i,

µ

where ni = n + n . It turns out to be more convenient to work with symmetric (s)

i↑

i↓

**and** antisymmetric (b) combinations

( 1 2 )

s = 1 a + a 2

µ µ µ

( 1 2 )

b 1

µ

= a

µ

− a

µ

.

2

**and** substituting **the**se into Eq. (5.7) allows **the** Hamiltonian to be separated into

symmetric **and** antisymmetric parts ( s b

H H H )

**and**

= + with

† † +

†

( 1 2 ) m∑( 1σ 2σ 2σ 1σ ) ∑h

µ µ ( 1 2 )( µ µ )

µ ω

b

H =

m

n + n − t c c + c c + g n − n b + b

σ µ

∑

( b b )

†

+ hω

1

µ µ µ

+ 2

µ

∑

where g

+ = g / 2 , s% = s + ng + **and**

( % % )

∑

† 2 + 2

µ µ µ µ

µ µ

(5.8)

s

H = hω

s s + 1 − h ω n g

(5.9)

2

% . Physically H

s

represents a

† †

s = s + ng +

displaced oscillator **and** indicates that **the** symmetric phonons couple only with **the**

total number of electrons ( n n n )

= + which is a constant of motion. The last term in

1 2

Eq. (5.9) represents **the** polaronic binding (self) energy

Ep

= ∑h ω n g

(5.10)

µ

2 + 2

µ

**and** corresponds to a lowering of **the** energy achieved through deformations of **the**

lattice sites. The interesting properties of **the** electron-phonon system are contained

within

b

H where antisymmetric phonons couple directly to **the** electronic degrees of

freedom, **and** its solution by any analytic method is a non-trivial problem.

The LF is a perturbative method **and** so it is desirable to transform **the** Hamiltonian

such that **the** major part becomes diagonal. Because **the** symmetric component

is already diagonalized we need only consider **the** antisymmetric component

s

H

b

H .

Lang **and** Firsov 32

transformation

where **the** canonical generator

achieve this by performing a canonical displacement

% (5.11)

b −Rb

b Rb

H = e H e

R

b

is chosen as

+

Rb

= ∑ gµ n − n bµ − bµ

(5.12)

µ

†

( 1 2 )( ).

This diagonalization is exact in **the** limit that t → 0 or g → ∞ **and** it can be shown

that a perturbation expansion of this Hamiltonian yields **the** exponential b**and**-

2

narrowing term t exp( g )

− characteristic of polaronic systems. 34

However, whilst **the** LF perturbation series is expected to converge when hopping is

weak (t ≤ h Ω ) **and** **the** e-ph coupling is relatively strong ( E / D > 1) **the** limit of **the**

coupling strength as a function of hopping, beyond which **the** approach is valid is not

precisely known. 33 34 As hopping increases **and** e-ph coupling strength decreases,

retardation effects between **the** electron **and** **the** lattice deformation become

increasingly significant but this cannot be accounted for within **the** conventional LF

approach. To overcome this limitation **and** **the**refore increase **the** regime over which

it is valid **the** coupling constant in **the** canonical generator (Eq.(5.12)) can be

replaced with a variational parameter for each antisymmetric mode δ

µ

in **the** phonon

basis:

Rb

= ∑ δµ n − n bµ − bµ

(5.13)

µ

†

( 1 2 )( ).

This is **the** basis of **the** modified Lang-Firsov (MLF) method **and** we can see that in

+

**the** limit that δ → g this reverts to **the** conventional LF form.

Applying this canonical transformation to **the** electronic operators in Eq. (5.8) such

that:

p

(†) R (†

b ) −Rb

c% iσ

= e ciσ

e

(5.14)

gives **the** electronic component of **the** transformed Hamiltonian:

⎡

∑( ) ∑⎢∏exp ( 2 ( )) ∏exp ( 2 ( ))

b

† † † †

H% ⎤

el

= µ

m

n1 σ

+ n2 σ

−tm

δµ bµ − bµ c1 σc2σ + − δµ bµ −bµ c2 σc1

σ ⎥.

σ σ ⎣ µ µ

⎦

(5.15)

33

34

A similar transformation of **the** bosonic operators

allows us to transform **the** free phononic H %

**and**

∑

(†) Rb

(†)

Rb

b% −

µ

= e bµ

e ,

(5.16)

0b

ph

**and** interaction

( )( ) ( ) 2

0 b

† † 2

ph

µ

µ

⎣

µ µ µ 1 2 µ µ µ 1 2

b

H % ep

terms of Eq. (5.8):

H% = h ω ⎡b b − δ n − n b + b + δ n − n + 1 ⎤ (5.17)

2 ⎦

∑

†

( )( ) 2 δ ( ) 2

b

H% ep

= h ω ⎡g + µ µ

n1 − n2 bµ + bµ − g +

µ µ

n1 − n ⎤

2

(5.18)

⎣

⎦

µ

Putting **the**se terms toge**the**r we have **the** complete transformed antisymmetric

Hamiltonian:

⎡

∑( ) m∑⎢∏exp ( 2 ( )) ∏exp ( 2 ( ))

% = µ + − δ − + − δ −

⎣

b

† † † †

H

m

n1 σ

n2 σ

t

µ

bµ bµ c1 σc2σ µ

bµ bµ c2 σc1

σ ⎥

σ σ µ µ

∑

+ +

( )( 1 2)( ) ( )( 1 2)

† †

2

+ hω ⎡

µ

bµ bµ + gµ −δµ n − n bµ + bµ + δµ δµ −2gµ

n − n + 1 ⎤.

⎣

⎦

µ

⎤

⎦ (5.19)

( ( b b b ))

† b

†

ϕsq

= ∏ exp α µ µ µ

− µ µ

0 .

(5.22)

ph

µ

has been found to be very effective in lowering **the** total energy of **the** system 35

allowing **the** MLF to give more exact results than LF in intermediate coupling **and**

hopping regimes. 33 In this case α is a variational parameter. A characteristic of such

a state is that **the** expectation values of **the** phonon creation **and** annihilation

operators are zero but **the** expectation values of **the** phonon number operators are

nonzero. 36

The squeezing parameter α has **the** effect of reducing **the** polaronic narrowing effect

**and** consequently enhancing hopping. However, **the** phonon energy increases with

increasing α, **and** **the** competition between phonon **and** hopping energies determines

**the** value of α. In studies on two- **and** four-site Holstein models in intermediate

coupling **and** hopping regimes it has been found that **the** calculated energies are in

very good agreement with exact results. 33

At this point a useful way to proceed is to decouple **the** electron **and** phonon

subsystems by approximating **the** ground state wavefunction as:

where

Ψ ≈ φ ϕ

(5.20)

0 0 0

φ represents **the** electronic ground state wavefunction **and**

0

ϕ is a suitably

0

chosen phonon wavefunction. 35 To determine **the** ground state energy we can take

**the** expectation value over this approximate wavefunction:

where

E

= Ψ H%

Ψ

b

0 0 0

≈ φ ϕ H%

ϕ φ

= φ H%

b

0 0 0 0

φ

b

0 eff 0

(5.21)

b

H % eff

is now **the** effective polaronic Hamiltonian. By averaging out **the** phononic

terms this last expression only contains electronic operators **and** **the** diagonalization

becomes trivial.

The introduction of a two-particle squeezed vacuum state similar to that used in

quantum optics:

The use of this ‘squeezed state’ variational wavefunction forms **the** basis of **the**

modified Lang-Firsov with squeezing (MLFS) approach. We are now in a position to

derive an effective polaronic Hamiltonian by eliminating **the** phonon degrees of

freedom utilizing **the** squeezed state trial wavefunction.

In line with **the** MLFS method we perform an appropriate squeezing transformation

where

H

sb

= (5.23)

S S

e Hbe −

† †

( )

S = ∏ α

µ

bµ bµ − bµ bµ

(5.24)

µ

**the**n take **the** expectation value over **the** phonon vacuum state. Applying (5.23) to **the**

electronic component of **the** transformed Hamiltonian (Eq. (5.15)) we get

**and**

⎛

⎜

⎝

⎞

† † −S

† †

exp ( 2δ

µ ( µ

−

µ )) 1 2 ⎟ = exp ( 2 %

∏ ∏ δ

µ ( µ

−

µ ))

1 2

(5.25)

S

e b b c c e b b c c

µ µ

⎠

⎞

† † −S

† †

exp ( 2δ

µ ( µ

−

µ )) 2σ 1σ ⎟ = exp( −2

%

∏ ∏ δ

µ ( µ

−

µ ))

1σ 2σ

(5.26)

⎛

⎜

⎝

S

e b b c c e b b c c

µ µ

⎠

35

36

with

% δ . (5.27)

−2α

e µ

µ

= δ

µ

Using **the**se expressions, **the** squeezed electronic Hamiltonian becomes:

∑( ) ∑∏exp( 2 ( ))

H% = µ n + n − t % δ b − b c c

b

† †

Sel m 1σ 2σ m

µ µ µ 1σ 2σ

σ σ µ

(

% δ ( ))

−t exp −2 b − b c c .

m

∑∏

σ µ

† †

µ µ µ 2σ 1σ

Note that for α = 0 Eq. (5.28) reverts to **the** MLF electronic Hamiltonian.

(5.28)

To eliminate **the** phonon operators we can now take **the** expectation value over **the**

phonon vacuum state giving **the** electronic term of **the** effective Hamiltonian:

2 −4 α † †

∑( 1σ 2σ ) ∑∏exp ( 2

µ )( 1σ 2σ 2σ 1σ

),

H % = µ n + n − t − δ e c c + c c (5.29)

beff

el m m

σ σ µ

**and** applying similar transformations to **the** phonon operators, we get:

† 1 2 1 3

†

bS

µ

= bµ + 2α µ

bµ + ( 2α µ ) bµ + ( 2 α

µ ) bµ

+ ..., (5.30)

2! 3!

† † 1 2

† 1 3

bS

µ

= bµ + 2α µ

bµ + ( 2α µ ) bµ + ( 2 α

µ ) bµ

+ ... (5.31)

2! 3!

Multiplying **the**se expressions toge**the**r, collecting like terms, **and** contracting **the**

series into exponential form, **the** squeezed phonon number operator become:

cosh ( 2α ) sinh ( 2α ) ⎤

⎦

cosh ( 2α ) sinh ( 2α

)( )

( αµ

)

b b = b b ⎡

⎣

+ + b b + b b

† † 2 2 † †

Sµ Sµ µ µ µ µ µ µ µ µ µ µ

+

2

sinh 2 ,

**and** in a similar way

† †

( Sµ Sµ ) ( Sµ Sµ

)

2α

(5.32)

b + b = b + b e µ

. (5.33)

Using **the**se expressions **the** transformed antisymmetric free phonon **and** interaction

Hamiltonians become:

∑

( ) ( )

H% = hω b b ⎡cosh 2α + sinh 2α

⎤

⎦

0 b

† 2 2

Sph

µ µ µ ⎣

µ µ

µ

∑

† †

( ) ( )( b b b b )

+ hω cosh 2α sinh 2α

+

µ

∑

µ µ µ µ µ µ µ

2 2α

( n1 n2 ) ( n1 n2

)( b b ) e ( )

µ

2 †

+ hω ⎡

1

µ

δ

µ

− − δ

µ

−

µ

+

µ

+ cosh 4 α ⎤ ,

2

µ

⎣

⎦

µ

(5.34)

Once again eliminating **the** phonon operators by taking **the** expectation value of

**the**se Hamiltonians over **the** phonon vacuum state leaves **the** antisymmetric phonon

component of **the** effective Hamiltonian:

∑

+

( ) ( )( ) 2

1 2

% h ω ⎡ α δ δ

⎤ (5.36)

⎦

eff

2

Hph

=

µ

sinh 2

µ

+

µ µ

− 2gµ

n − n

⎣

µ

Combining this with **the** effective electronic component we now have **the** total

effective antisymmetric Hamiltonian

where

2 −4 α † †

∑( 1σ 2σ ) ∑∏exp ( 2

µ )( 1σ 2σ 2σ 1σ

)

H%

= µ n + n − t − δ e c c + c c +

b

eff m m

σ σ µ

∑

µ

+

( )( 1 2 )

2 b

hω ⎡

µ

δµ δ

µ

− 2 gµ

n − n ⎤ + Eph.

⎣

⎦

( α )

(5.37)

= 1 ω cosh 4

2∑h (5.38)

b

E

ph

µ µ

µ

is **the** average phonon energy contribution at zero temperature. From this we can

see that, due to **the** squeezing effect, **the**re is an increase in **the** zero-point b-phonon

energy, as if **the** frequency of **the** latter were renormalized to ω ∗ = ω cosh( 4α

)

. This

frequency renormalization is also a characteristic of **the** Holstein small polaron

approach in **the** adiabatic limit. 30

Although we will not consider **the** symmetric component in **the** proceeding

discussion, for completeness we can consider how **the** symmetric components

contribute to **the** total energy within this approximation. Noting that **the** symmetric

**and** antisymmetric phonon operators commute, **the** transformed symmetric free

phonon Hamiltonian remains unchanged:

∑

( % % )

∑

H = hω

s s + 1 − h ω n g

(5.39)

2

0 s

† 2 + 2

Sph

µ µ µ µ

µ µ

**and**, after taking **the** expectation value over **the** vacuum state, we find that **the** this

term contributes

to **the** total ground state energy.

eff

Es

( )

2 +

n g

2

= ∑h ω 1

µ

−

(5.40)

2

µ

**and**

∑

( )( †

) 2 α

( ) 2

µ

δ

b

H% + +

Sep

= h ω ⎡

µ

gµ n1 − n2 bµ + bµ e − 2 gµ µ

n1 − n ⎤

2

. (5.35)

⎣

⎦

µ

37

38

5.4 Determination of Holstein parameters

5.4.1 Calculation of vibrational modes

The vibrational frequencies of **the** ET molecules can be obtained from **the** analysis of

infrared **and** Raman spectra but in order to calculate **the** electron-phonon coupling

each of **the**se frequencies must be associated with a corresponding normal

coordinate. As previously mentioned, much of **the** experimental work in determining

**the** vibrational spectra of both H 8 -(ET) 2 Br **and** D 8 -(ET) 2 Br was carried out by Kozlov

et. al. 20 **and** later exp**and**ed upon by Eldridge et. al. 21 who also considered of

C → C **and**

12 13

S → S substitutions. To simplify calculations both Kozlov **and**

32 34

Eldridge assumed planar symmetry (D 2 symmetry) for **the** molecule in **the**

subsequent mode calculations which is not strictly true. Experimental observations

by Leung et. al. using X-ray spectroscopy indicate that that **the**re are actually two

conformations that predominate in **the** crystal – staggered **and** eclipsed. These

conformations are shown in Fig. 5.1.

suggests that fast cooling not only increases **the** disorder in **the** sample, but also

reduces **the** “effective chemical pressure” (see Fig. 7.2).

Utilizing a Gaussian based, all electron DFT calculation Powell et. al. 6 calculated **the**

Raman spectra of **the** neutral ET molecule in both conformations giving results (Fig.

5.2) which are in good agreement with **the** experimental spectra observed by

Eldridge et. al. 21

(a)

(b)

Figure 5.1

(a) The staggered conformation **and** (b) **the** eclipsed conformation for **the** ET molecule. Note that

**the** conformational state is primarily dependant upon **the** configurations of **the** terminal ethylene

groups. Both states are energetically separated by 3.4 meV. 6

Figure 5.2

The calculated vibrational spectra for both **the** staggered **and** eclipsed conformations of **the** neutral

ET molecule. These show very good agreement with **the** experimental data obtained by Eldridge et.

al. Taken from Ref. 6

Although **the** former is favoured as **the** groundstate Powell et. al. 6 37 suggest that **the**

‘freezing in’ of both phases may lead to **the** increased disorder in **the** crystal **and**

**the**refore explain **the** changes in critical temperature with cooling rate observed by

Taniguchi et. al. 9 However, for partially deuterated samples dρ / dT < 0 (insulating

behaviour), where ρ is **the** resistivity, just above T c in **the** fast cooled samples. This

We have performed similar calculations on **the** eclipsed **and** staggered conformations

for both **the** hydrogenated **and** deuterated molecules along with **the** 13 C-(ET) 2 Br. The

results for **the** DFT vibrational calculations are summarized in Appendix B.

39

40

5.4.2 The electron-phonon coupling constant

Until recently **the** electron-phonon coupling parameter g was calculated using **the**

frozen phonon method in which **the** e-ph coupling constant is given by 38

g

γ , µ

1 ⎛ ∂λ

⎞

γ

= 2hω

⎜ ∂ ⎟

µ ⎝ Q , (5.41)

µ ⎠

where g γ , µ

describes **the** strength of **the** interaction between **the** vibrational mode µ

represented by **the** dimensionless normal coordinate Q **and** frequency ω , **and** **the**

electronic state λ

γ

. Recently, Powell et. al. developed an alternate method which

greatly simplifies **the** calculation of g by reducing it to an order one problem. 39 By

applying Janak’s **the**orem which states that

∂E

λγ

= (5.42)

∂ n

where E is **the** total energy of **the** system **and** n γ

is **the** electron occupancy of **the**

state γ , **the**y were able to rewrite Eq. (5.41) as

where ( E / Q µ )

x

g

γ , µ

1 1 ⎛ ∂E

⎞

= lim

2hω → n ⎜ ∂Q

⎟

γ

δ n 0

µ γ ⎝ µ ⎠δ

nγ

(5.43)

∂ ∂ indicates that **the** derivative is taken after **the** charge has changed

by x relative to **the** charge of **the** initially optimized geometry. Once **the** electronic

structure is solved in **the** equilibrium geometry of **the** charge neutral system with a

small charge on **the** molecule **the** forces on **the** molecule. Using **the** dynamical matrix

used to calculate **the** phonon spectrum it **the**n becomes a straightforward task to

calculate **the** coupling constants for **the** different modes. We have calculated **the**

coupling constants for both **the** staggered **and** eclipsed conformations of **the**

hydrogenated **and** deuterated molecules as well as 13 C-(ET) 2 Br **and** **the**se are given

in Appendix B.

5.5 The model analysis (**the** single mode)

Having derived an effective Hamiltonian, we can now explore **the** parameter ranges

over which it is a good approximation. For clarity, in **the** following analysis only a

single mode of energy ω will be considered.

5.5.1 Polaronic behaviour **and** b**and** narrowing

Polaronic behaviour is characterised by a reduction in **the** electron hopping (b**and**

narrowing) due to **the** reduction in electron mobility induced by e-ph coupling. For a

single phonon mode in **the** adiabatic limit (i.e. small polaron Holstein model), **the**

2

b**and** narrowing is related to e-ph coupling by t*= t exp( − g )

limit **the** MLFS approach returns **the** variational form

2 −4

( δ µ )

. Within **the** nonadiabatic

∗

t exp 2

m

= tm − e α . (5.44)

To compare how both models behave in **the** intermediate energy

regime( 0.01 ≤ ω / t ≤ 10)

c

(where both **the** LF **and** small polaron Holstein

approximations normally break down) with varying coupling strength, **the** groundstate

wavefunctions from Table 4.1 corresponding to **the** relevant charge configurations

were applied to **the** effective Hamiltonian in order to construct expressions for **the**

ground state energies as functions of both g **and**δ . Because **the** hopping behaviour

will be **the** same for **the** charge configurations n=1 **and** n=3 due to particle-hole

symmetry **and** because **the** electron wavefunction for n=4 does not couple to **the**

phonon degrees of freedom, only **the** two **and** three electron wavefunctions need be

considered.

The value of **the** coupling constant was varied continuously within four different

adiabacity ( ω / t ) regimes **and** at each value **the** energy was minimized for α **and**δ .

These values were **the**n substituted into Eq. (5.44) to obtain a value for **the** effective

hopping integral, which is plotted in Fig. 5.3 along with **the** Holstein small polaron

2

hopping integral t*= t exp( −g

)

for comparison.

These plots indicate that we do not see **the** b**and** narrowing expected within **the** small

polaron Holstein approach with increasing adiabacity. In fact, this narrowing does not

become apparent until g is sufficiently large to satisfy **the** strong coupling

condition( Ep

/ D ~ 1) when **the**re is a sharp decline in electron mobility for both n=2

**and** n=3. Surprisingly, we can see that in **the** nonadiabatic limit ( ω / t ≥ 1) **the**

c From this point onward we will set h =1.

41

42

ehaviour of **the** effective hopping integral actually approaches **the** Holstein result.

Although somewhat unexpected this result is qualitatively similar to **the** behaviour

observed by de Raedt **and** Lagendijk 40 who applied a quantum Monte Carlo method

to **the** analysis of **the** Holstein model in two **and** three dimensions. They too observed

a sharp but continuous **transition** into a polaronic state at a critical value of **the**

coupling strength which varied with **the** dimensionality.

5.5.2.1 The convergence of **the** exact Hamiltonian

To determine **the** exact matrix elements, **the** Hubbard basis sets (Eq.’s (4.11)-(4.15))

for different n are modified to incorporate phonons **and** **the** resulting basis set φ

i,

N

can be written as

φ = e N

(5.45)

i, N i,

where ei

**and** N refer to **the** electronic basis state i **and** **the** number of antisymmetric

phonons respectively. Restricting **the** phonon occupancy in **the** system to a finite

number, N Max , allows **the** antisymmetric Hamiltonian (Eq.(5.8)) to be diagonalized

giving **the** ground state energy **and** corresponding ground state wavefunction ψ .

N Max

To check **the** convergence of this exact solution **the** overlap

Ψ | ψ was

Nmax 100

calculated for varying N Max for different adiabacities **and** coupling strengths as shown

in Figures 5.5 **and** 5.6.

Fig. 5.5 shows that **the** exact wavefunction for both n=2 **and** 3 converges for N Max <

20 over values of g ≤ 1 but breaks down dramatically with strong coupling. This

breakdown be understood by looking at **the** phonon occupancies for **the** various

coupling strengths as shown in Fig. 5.4 with N Max = 100.

Figure 5.3

B**and** narrowing plots for electron occupancies n=2 **and** 3 plotted against coupling strength within

four energy regimes( ω / t = 0.01 ,0.1 ,1 ,10)

. This shows minimal b**and** narrowing occurs until **the**

coupling strength is of **the** order of **the** strong coupling condition upon which **the**re is a sharp but

continuous **transition** into a polaronic state. As **the** energy regime approaches **the** nonadiabatic limit

2

**the** model approaches **the** small polaron result t*/t= exp( g )

− .

5.5.2 Exact diagonalization **and** wavefunction overlap

Having confirmed that **the** effective Hamiltonian displays polaronic behaviour it can

now be compared against an ‘exact’ Hamiltonian to determine how good **the** choice

of **the** trial phonon squeezed state wavefunction is for different adiabacities **and**

coupling regimes.

Figure 5.4

The probable phonon occupancy for N max =100 shows that as **the** coupling strength g increases

**the** occupancy shifts to higher phonon numbers. For strong coupling ~ N max which accounts

for **the** breakdown in our method for obtaining **the** exact solution for large g.

We can see that **the**re is a direct correspondence between g **and** **the** phonon

expectation value N **and** for very large g N approaches N Max . In this region **the**

43

44

exact wavefunction no longer converges but because we will predominantly be

considering values of g60 in all cases. These

results suggest that ψ

100

will provide a good test for **the** approximation.

(a)

n = 2

5.5.2.2 Testing **the** approximate wavefunction

Having settled on a suitable ‘exact’ wavefunction, before being able to compare how

good **the** choice of **the** squeezed state phononic wavefunction is within various

energy **and** coupling regimes, an inverse canonical transformation must first be

applied to **the** effective wavefunction to recover **the** antisymmetric phonon basis

required for comparison with **the** exact wavefunction i.e.

−Rb

−S

%

eff

= e e ψ

eff

(5.46)

ψ

Performing this transformation **and** calculating ψ | 100

ψ%

eff

for various ratios of gω

/ t

we can see from Fig. 5.7 that **the** approximate **and** exact wavefunctions overlap very

well for small gω / t but diverge rapidly as gω / t increases. This behaviour is also

apparent in **the** energy ratios although **the** correspondences in energy are far better

for ω / t ≤ 1 than expected from **the** overlap results. This indicates that **the**

wavefunction overlap is a more rigorous test of **the** approximation than a simple

comparison of **the** respective energies.

(b)

n = 3

Figure 5.5

The wavefunction overlap ψ ψ for (a) n=2 **and** (b) n=3 for various coupling strengths **and**

N max 100

within different e-ph energy regimes where N max is **the** number of phonons involved in **the** exact

diagonalization. In **the** strong coupling region **the** nonconvergence of **the** n=2 exact

wavefunction is probably due to a large .

d There is a strong coupling mode for both **the** hydrogenated **and** deuterated molecules (g ~ 5) at very

low energies (ω~0.01) but within this energy regime **the** overlap remains good. (See Figure 5.5)

45

46

(a)

n = 2

(a)

(b)

n = 3

(b)

Figure 5.6

The energy ratio E Nmax /E 100 for (a) n=2 **and** (b) n=3 for various coupling strengths **and** within

different e-ph energy regimes where N max is **the** number of phonons involved in **the** exact

diagonalization. The energy ratio converges to 1 within all regimes for N max >60.

Figure 5.7

(a) The wavefunction overlap of **the** effective **and** exact Holstein Hamiltonians for n=2 (solid

line) **and** n=3 (dashed line) in various e-ph energy regimes ( ◊ - ω/t = 0.01, o - ω/t=0.1,

× - ω/t=1 , * - ω/t=10 ) . These show that **the** overlap falls away dramatically with both

increasing gω **and** decreasing adiabacity. (b) The ratio of effective to exact energies showing

diverging agreement with increasing gω **and** decreasing adiabacity. INSETS show **the**

behaviour in **the** regime relevant to κ-(ET) 2 Br

47

48

6 Electrons, phonons **and** **the** Hubbard-

Holstein model

6.1 The Hubbard-Holstein model

Having derived an effective Hamiltonian model free of phononic degrees of freedom

from **the** Holstein model electron-electron interactions can now be accounting for by

incorporated **the** Hubbard parameters U m **and** V m . The Hamiltonian combining both

electron-phonon **and** electron-electron which is given by (for a detailed derivation see

Appendix A):

2 −4 α † †

∑( σ σ ) ∑∏exp ( 2

µ )( σ σ σ σ ) ∑

H%

= µ n + n − t − δ e c c + c c + U n n + V n n

b

eff m 1 2 m 1 2 2 1 m i↑

i ↓ m 1 2

σ σ µ

i = 1,2

⎡

+

⎧

⎫⎤

+ ∑hωµ ⎢δ µ ( δ

µ

− 2g µ ) ⎨( n1 + n2 ) + 2∑

n n − 2n i i 1n2

+ E

↑ ↓ ⎬⎥

µ ⎣

⎩

i = 1,2

⎭⎦

which simplifies to

with

∑( σ σ ) ∑( ) ∑

H % n n t c c c c U n n V n n E (6.1)

b * * † † * *

pol

= µ

m 1

+

2

−

m 1 2

+

2 1

+

m

+

i ↑ i↓

m 1 2

+

ph

σ

σ

i = 1,2

*

m

− α

( δµ

)

t = exp −2

e t

* 2 4

m

µ

*

m

*

m

m

+

( 2g

)

µ = µ − hω δ − δ

+

( δ )

U = U − 2 hω δ 2g

−

m

+

( δ )

V = V + 2 hω δ 2g

−

E

b

ph

∏

m

∑

µ

µ

µ

µ µ µ µ

= 1 hω

cosh 4

2

µ

∑

∑

∑

m

µ µ µ µ

µ µ µ µ

( α )

µ µ

b

ph

(6.2)

6.2 The model analysis (**the** single mode case)

6.2.1 B**and** narrowing

To explore how **the** incorporation of e-e interactions impacts upon model behaviour

we will begin by repeating **the** calculations carried out with **the** Holstein model (see

Chapter 5) for **the** HH model.

An inspection of Table 4.1 indicates that U m **and** V m for **the** ET molecule have values

of about 13t **and** 8t respectively **and** **the**se values will be used throughout **the**

analysis unless o**the**rwise specified. Applying **the** b**and** narrowing calculations

detailed in Section 5.5.1 to **the** polaronic Hamiltonian we can see from Fig. 6.1 that

**the** introduction of e-e interactions have little effect in **the** three electron case but

significantly effects **the** behaviour of **the** two electron case when compared to Fig.

Note that this is qualitatively similar to **the** Hubbard model with **the** addition of an

average phonon energy **and** electronic terms that have been renormalized due to **the**

polaronic e-ph coupling.

Figure 6.1

B**and** narrowing plots for electron occupancies n=2 **and** 3 plotted against coupling strength within

four e-ph energy regimes( ω / t = 0.01 ,0.1 ,1 ,10)

for **the** polaronic model with U m = 13t **and** V m =

8t. Comparing this plot with Fig. 5.2 shows that **the** addition of interaction terms does not alter **the**

behaviour of **the** three electron case but it does have a significant impact on **the** two electron case

with a considerable reduction in b**and** narrowing over a larger coupling range as well as a much

sharper **transition** into **the** polaronic state.

49

50

5.3. There is a considerable reduction in b**and** narrowing particularly at lower phonon

energies **and** a much sharper **transition** into a polaronic state.

(a)

n = 2

6.2.2 The convergence of **the** exact Hamiltonian

As with **the** effective Holstein model, **the** exact matrix elements for **the** polaronic

Hamiltonian are constructed using **the** basis sets from Eq. (5.45) **and** **the** overlap **and**

energy ratios compared to **the** N Max = 100 solutions as shown in Figures 6.2 **and** 6.3.

Fig. 6.2 indicates that, as in **the** Holstein case, **the** exact wavefunction for both n=2

**and** 3 converges for N Max < 20 over values of g ≤ 1 but breaks down dramatically for

strong coupling. Once again, this is probably due to large

N in **the** stronger

coupling regimes. Similarly, **the** behaviour of **the** energy in Fig. 6.3 closely resembles

that observed in **the** Holstein case, converging for all values of g with **the** energy

unchanged for N Max >60 in all cases

6.2.3 Testing **the** approximate wavefunction

To determine how accurately **the** squeezed state phononic wavefunction

approximates **the** exact ground state wavefunction with correlations taken into

account **the** same procedure performed in Section 5.5.3 was repeated for **the**

polaronic Hamiltonian with **the** resulting wavefunction overlap **and** energy plots given

in Fig. 6.4. We can see that **the** approximate **and** exact wavefunctions overlap very

well for small values of gω / t but diverge rapidly as with increasing gω

/ t . Similar

behaviour is also evident in **the** energy ratios.

(b)

n = 3

Figure 6.2

The wavefunction overlap ψ ψ

N max 100

for (a) n=2 **and** (b) n=3 with **the** inclusion of e-e

interactions (U m = 13t **and** V m = 8t) for various coupling strengths **and** within different e-ph

energy regimes where N max is **the** number of phonons involved in **the** exact diagonalization. In

**the** strong coupling region **the** nonconvergence of **the** n=2 exact wavefunction is probably due to

a large .

51

52

(a)

n = 2

(a)

(b)

n = 3

(b)

Figure 6.3

The energy ratio E Nmax /E 100 for (a) n=2 **and** (b) n=3 incorporating e-e interactions (U m = 13t **and** V m

= 8t) for various coupling strengths **and** within different e-ph energy regimes where N max is **the**

number of phonons involved in **the** exact diagonalization. As with **the** Holstein Hamiltonian **the**

energy ratio converges to unity within all regimes for N max >60.

Figure 6.4

(a) The wavefunction overlap of **the** polaronic **and** exact H-H Hamiltonians for n=2 (solid line)

**and** n=3 (dashed line) with U* = 13t **and** V* = 8t in various e-ph energy regimes ( ◊ - ω/t = 0.01,

× - ω/t=1 , o - ω/t=0.1 , * - ω/t=10 ) . These show that **the** overlap falls away dramatically with

both increasing gω **and** decreasing adiabacity. (b) The ratio of effective to exact energies

showing diverging agreement with increasing gω **and** decreasing adiabacity. INSETS show **the**

behaviour in **the** regime relevant to κ-(ET) 2 Br

53

54

6.2.4 Variation in U-V

Because **the** polaronic Hamiltonian is **the** Hubbard model representing polaronic

behaviour with a constant average phonon energy E ph , **the** ground state energies for

each of **the** electron occupancies can be expressed as:

n=1

0

1

µ ∗ ∗

m m ph

E = − t + E

(6.3)

of n=3, because

U ∗ m

**and**

V ∗

m

are only additive in Eq. (6.5) **and** we always have

U + V >> t **the**re is not **the** same effect on energy when **the** polaronic **transition**

∗ ∗ ∗

m m m

occurs.

0 ∗ ∗ ∗

∗ ∗

n=2 ( ) 2 ∗ 2

E2 = 2µ

m

+ Um + Vm − U − V + 16t + E

m m m ph

(6.4)

n=3

0

E µ ∗ m

U ∗ m

V ∗ m

t ∗

m

Eph

3

= 3 + + 2 − + (6.5)

n=4

0

E µ ∗ m

U ∗ m

V ∗

m

Eph

4

= 4 + 2 + 4 + (6.6)

As we have seen, **the** behaviour of **the** st**and**ard two-site Hubbard model can be

thought of as a competition between t, which encourages delocalization of **the**

electrons **and** U which encourages localization of a single electron per site. However,

with **the** introduction V this behaviour becomes slightly more complex because **the**

offsite coulomb interaction wants to place two electrons on a single site **and** thus

**the**re is a competition between **the** onsite **and** offsite repulsion terms. In effect, **the**

behaviour of **the** model should **the**refore be characterized by **the** interaction between

t **and** a reduced onsite coulomb repulsion term U eff = (U m -V m ).

To investigate **the** model response both **the** b**and** narrowing **and** **the** groundstate

energies were plotted for n=2 **and** 3 with µ

m

= 0 , ω / t = 1 **and** Um

= 13t

against U eff

as shown in Fig. 6.5. From this plot we can see **the** characteristic sharp b**and**

narrowing for **the** n=2 case at progressively higher coupling strengths with

decreasing U eff . As V m increases in magnitude (i.e U eff gets smaller) greater energy is

required to ‘capture’ an electron in a polaronic state. This energy is supplied by **the**

electron-lattice interaction **and** hence we observe polaronic behaviour at stronger

coupling.

Once polaronic threshold is reached **and** **the**re is substantial b**and** narrowing **the**

energy begins to fall nearly linearly with coupling strength **and** to underst**and** this

behaviour we can consider Eq. (6.4). Once in **the** polaronic state

( ) ( ) ωδ ( δ )

∗ ∗ ∗ +

4t

7 Application of **the** polaronic model to **the**

ET dimer

In order to apply **the** polaronic Hamiltonian to an analysis of interactions between

intramolecular phonons **and** electrons on **the** ET dimer we must first establish

whe**the**r **the** system parameters are within **the** range, established in **the** preceding

chapters, for which **the** model is a good approximation. Adopting representative

monomer electronic values from Table 4.1 (with µ

m

= 0 ), because we are only

interested in relative energy shifts ra**the**r than absolute values,

µ = 0 eV

t

m

U

V

m

= 0.27 eV

m

m

= 3.90 eV

= 2.70 eV

(6.7)

It can be seen from **the** frequency spectra of **the** various isotopic **and** conformational

configurations (Appendix B) that **the** adiabacity of **the** system lay in **the**

range0.01 ≤ ω / ≤ 1. To establish that **the** polaronic approximation is sound within

t m

this parameter range **the** overlaps of **the** variational **and** exact wavefunctions using

**the** above electronic values were plotted for adiabacities of ω / t m

= 0.01 **and** ω / t m

= 1

in Fig. 7.1. We choose **the** staggered conformation as representative of **the**

groundstate of **the** system, **the** locations **and** overlap values of several representative

modes have been indicated.

Figure 7.1

Overlap of **the** polaronic **and** exact wavefunctions for adiabacities of ω / t m

= 0.01 **and**

ω / t m

= 1(indicated by o) showing some characteristic modes (subscripts represent **the** mode

number) **and** **the** corresponding overlap value where it varies significantly from unity. These

indicate that **the** ET system is well within **the** acceptable parameter range for application of **the**

polaronic Hamiltonian. The values used were

Vm

= 2.70 eV .

µ = 0 eV , t = 0.27 eV , U = 3.90 eV **and**

m

m

m

Over **the** plotted parameter range only in n=3 case does **the** overlap of **the**

approximate **and** exact wavefunctions differ noticeably from unity although even this

divergence is very small (~0.01). Comparing **the** overlap with **the** positions of **the**

representative modes we can conclude that **the** ET system is within **the** acceptable

parameter range for **the** application of **the** polaronic Hamiltonian. The same

calculation on **the** deuterated data (not included) gives a similar result.

7.1 The calculation of dimer parameters with multiple modes

7.1.1 The dimer Coulomb repulsion U d

In order to calculate U d **the** renormalized energies calculated from Eq.’s (6.4)-(6.6) for

**the** full 64 vibrational modes can be substituted into **the** two-site form of **the** dimer

Coulomb repulsion

( ) ( ) ( )

U = E 4 − 2E 3 + E 2 .

(6.8)

d

0 0 0

Because **the**re are two variational parameters associated with each mode ( α

µ

**and** δ µ

)

this becomes a 128 parameter minimization problem. Applying **the** method of

steepest descent, we calculated U d for H 8 -(ET) 2 Br **and** D 8 -(ET) 2 Br in both **the**

57

58

eclipsed **and** staggered conformations as well as for 13 C-(ET) 2 Br. These are

summarized in Table 7.1 along with **the** interdimer hopping integrals to be calculated

in **the** next section.

∗

Conformation isotope t1 / t1

Eclipsed

Staggered

Staggered

t / t

∗

2 2

t

∗ 1

/ t

∗ ∗

/

2

UD

H 8 0.7721 0.9120 0.5546 0.9918

D 8 0.7415 0.9120 0.5325 0.9918

H 8 0.8412 0.9180 0.6002 0.9919

D 8 0.8342 0.9160 0.5965 0.9919

13 C 0.8435 0.9178 0.6020 0.9920

Table 7.1 The ratios of **the** renormalized to bare (Hubbard) inter-dimer hopping integrals **and** **the**

onsite dimer Coulomb term calculated for µ = 0 eV , t = 0.27 eV , t = 0.085 eV , t = 0.13 eV ,

2

tq

m

= 0.04 eV U = 3.90 eV **and** V = 0 eV . Using **the** bare parameters t 1 = 0.034 eV, t 2 = 0.068 eV

**and** t 1 / t 2 = 0.5.

m

m

From a comparison of **the** relative shifts in onsite dimer Coulomb repulsion U d for **the**

hydrogenated **and** deuterated configurations we can see that **the** polaronic

renormalization is very nearly **the** same in both cases. From this we can conclude

that **the** intermolecular phonons do not participate in determining this parameter.

A significant result in **the** broader context of **the** study of **the**se systems is that our

calculations show **the** renormalization of U d with **the** inclusion of phonons to be very

small (< 1 %). This is significant because, as we mentioned in Chapter 4, **the**

accurate calculation of model parameters is an ongoing challenge in **the** field. 5 7 23

Being able to neglect e-ph coupling will greatly simplify **the**se calculations.

7.1.2 The interdimer hopping (t 1 **and** t 2 )

Within **the** Hubbard framework **the** MIT is treated as a phase **transition** resulting from

a change in **the** ratio of U d /W where W=8t 2 for t 1 < t 2 . In order to determine how

intramolecular e-ph coupling affects **the** interdimer hopping integrals we can utilize

Eq.’s (4.12) **and** (4.13) with renormalized intramolecular **and** Coulomb terms. Using

**the** polaronic approximation **the** interdimer monomer hopping termst ∗

b2

, t ∗

p

m

b

p

U

D

**and**

p

can be calculated by mapping two monomers from neighboring dimers onto **the** two

site model. We will use **the** bare values (without e-ph interactions) calculated by

Fortunelli et. al. using RHF-SCF which are given in Table 4.1. There are, on

∗

average, three electrons per dimer so t

3

will be used to represent

∗

t

m

in **the**

calculations **and** because of **the** large separation between dimers we can neglect **the**

offsite Coulomb repulsion (V m =0).

To determine **the**se values

0

E

3

(Eq.(6.5)) can be minimized for **the** multiple phonon

modes **and** **the** associated variational parameters substituted into **the** polaronic

expressions in (6.2). Ratios of **the** renormalized to bare (Hubbard) interdimer

molecular hopping integrals are summarized in Table 7.2.

Conformation isotope tb2 / tb2

Eclipsed

Staggered

Staggered

∗ tp

∗ /

∗

tp

tq

/ tq

t

∗ ∗

/ t U / U

H 8 0.7735 0.9120

From a comparison of **the** relative shifts in dimer hopping integrals for **the**

hydrogenated **and** deuterated configurations we can see that **the** inclusion of

intramolecular electron-phonon coupling does cause a significant reduction **the** value

of t 1 **and** a much smaller shift in t 2 upon deuteration in both conformations. This is

certainly in qualitative agreement with what we observe experimentally with

W > W > W > W > W

(6.9)

S S S E E

13

C H D H D

A decrease in interdimer hopping corresponds to an increase in **the** ratio of U d /W.

However this effect seems to be too small to explain **the** observed MIT.

Our results indicate that frustration, ra**the**r than variations in U d or t, may actually be

driving **the** **transition**. For a fixed t **the** critical U at which **the** MIT occurs (U c ) is

increased by frustration **and** **the**refore for fixed U/W (or U/t 2 ) decreasing t 1 /t 2 drives

**the** system closer to **the** **Mott** **transition**. This behaviour is illustrated by an arrow in

Fig. 7.2.

Figure 7.3

The temperature dependence of resistance of κ-(ET) 2 Br salts as a function of d[n,n] (which

represents **the** number of deuterium atoms in each ethylene group} **and** cooling rate. We can see

that increasing **the** cooling rate can induce an insulating state. Modified from Ref. 9

As discussed in Chapter 5, this could be **the** result of **the** ‘freezing in’ of disorder by

increasing **the** ratio of molecules in **the** slightly less energetically favorable eclipsed

conformation as opposed to **the** groundstate staggered conformation. Because **the**

eclipsed conformations show lower frustration we can see that this will result in a

decreased ‘average’ frustration **and** **the**reby allow isotopic substitution to drive **the**

system towards **the** MIT.

Figure 7.2

The results for a resonating valence-bond approximation on **the** Hubbard-Heisenberg model

showing that as frustration decreases on a triangular lattice **the**re is a corresponding decrease in U c .

Modified from Ref 7 .

Using this concept, our results may provide an explanation of **the** changes in

resistivity with cooling rate observed by Taniguchi et. al. 9 From Fig. 7.3 we can see

that increasing **the** cooling rate can result in an increase in resistivity at low

temperatures.

61

62

8 Ano**the**r possibility - **the** geometrical

isotope effect

From **the**se results we can conclude that phonons offer a plausible means to account

for **the** observed MIT upon deuteration in **the** CT’s in **the** form of frustration. However,

throughout this analysis we have assumed that **the** lattice parameters remain

unaffected by isotopic variation but **the**re is some evidence to suggest o**the**rwise. 41

Although we have not dealt with it specifically, this **transition** can also be driven by

hydrostatic pressure **and** **the**se materials are highly sensitive to pressure variations

with insulating, superconducting, **and** metallic phases being observed over a rang of

a few hundred bars. 17 In terms of **the** Hubbard framework increasing pressure

increases **the** HOMO overlap **and** **the**refore W.

Due to this pressure sensitivity, changing **the** lattice parameters by even a small

amount may be enough to drive this **transition**. Watanabe et. al. 41 used X-ray

diffraction to examine how **the** lattice parameters in **the** crystal change with

deuteration. They find that **the**re is up to a 0.11% change that may be associated

with changes in **the** bond lengths on **the** ethylene groups **and** **the** hydrogen bonding

to **the** anionic lig**and**s. These results certainly warrant fur**the**r investigation as a

possible source of **the** isotopically induced MIT.

9 Summary **and** conclusion

In Chapter 2 we introduce **the** concept of strongly correlated systems **and** why we

would like to underst**and** **the** role of phonons in **the** MIT. We **the**n introduced b**and**

**the**ory **and** **the** concept of **the** onsite Coulomb repulsion to account for this **transition**

in terms e-e interactions opening up gaps in o**the**rwise continuous b**and**s.

Chapter 3 introduces a family of quasi-2D organic metals - **the** charge transfer salts –

in which this **transition** can be driven by both isotopic **and** chemical substitution as

well as by hydrostatic pressure. We explain how **the** observed vibrational spectra

change upon isotopic substitution **and** ask **the** question of whe**the**r phonons

participate in **the** MIT.

We **the**n outline **the** dimer Hubbard model as a minimal model that is often applied to

**the**se quasi-2D strongly correlated systems **and** explain how it can be modified to

account for intramolecular phonons through reduction to a two-site model where

each site now represents a monomer. The extended Hubbard Hamiltonian is **the**n

applied to this system. The onsite Coulomb repulsion **and** interdimer hopping

integrals are expressed as functions of **the** intermolecular (intradimer **and** molecular

interdimer) parameters **and** it is noted that **the** offsite Coulomb repulsion changes **the**

value of U d significantly from ~2t m to ~10t m using values for **the** monomer parameters

calculated by o**the**r authors.

In Chapter 5 we introduce **the** Holstein model as a minimal model to account for e-ph

coupling **and** apply a modified Lang Firsov transformation with squeezing to obtain

an effective variational Hamiltonian in which **the** phononic degrees of freedom have

been averaged out by applying a variational squeezed state wavefunction. This

Hamiltonian is **the**n compared to an exact wavefunction to test **the** parameter range

over which this trial wavefunction is a suitable approximation.

Electron-electron interactions are introduced into this effective Hamiltonian in

Chapter 6 with **the** Hubbard-Holstein Hamiltonian. Averaging out **the** phononic

degrees of freedom we obtain what is effectively a polaronic Hubbard model with a

constant phonon energy offset. The parameter range over which this model is tested

63

64

**and** it is shown that **the** ET system lies within this regime. We also explore how **the**

choice of V m affects model behavior **and** conclude that only **the** n=2 state shows a

sensitivity.

In Chapter 7 we apply this model to **the** ET system **and** find that both **the** onsite

Coulomb repulsion **and** **the** b**and**width are unaffected by phonons. This is an

important result because it simplifies **the** task of accurately calculating **the** Hubbard

parameters considerably. However, we do find that intramolecular e-ph interactions

may be involved in **the** MIT by altering frustration. This may account for **the** observed

changes in resistivity introduced with disorder as **the** cooling rate is varied. Finally, in

Chapter 8 we suggest **the** geometrical isotope affect as an alternate mechanism

behind **the** isotopically induced MIT.

10 Appendix A

The Hamiltonian used to model **the** dimer is **the** Hubbard-Holstein Hamiltonian

H = H + H + H

(A.1)

0

HH el ph ep

where H , H **and** H ep

represent **the** electronic, free phonon **and** electron-phonon

el

0

ph

interaction components respectively.

The electronic component can be exp**and**ed as

where

† †

∑( 1σ

2σ

) ∑( 1 2 2 1)

∑ i i 1 2

(A.2)

H n n t c c c c U n n V n n

el

= µ

m

+ −

m

+ +

m

+

↑ ↓ m

σ

σ

i=

1,2

µ

m

is **the** chemical potential,

m

t is **the** hopping amplitude,

m

U is **the** onsite

interaction **and** V m is **the** intersite interaction. Throughout this work **the** subscripts m

(†)

**and** d are used to represent **the** monomer **and** dimer respectively, c σ

is **the** electron

annihilation (creation) operator on site i with spin σ , **and** n is **the** number operator.

∑ ∑ .

†

We have also used **the** notation n = n = c c

i i,

σ iσ iσ

σ

σ

i

The parts of **the** Hamiltonian involving phonons are **the** free-phonon **and** electronphonon

terms

H

∑

( a a )

= hΩ + 1 2

0 †

ph µ iµ iµ

i,

µ

∑

†

( )

H = g hΩ n a + a

ep µ µ i iµ iµ

i,

µ

where Ω

µ

is **the** dispersionless phonon frequency of **the** vibrational mode µ ,

**the** phonon annihilation (creation) operator for **the**

**the** electron-phonon coupling constant for

th

µ mode.

(A.3)

i

(†)

a µ

is

th

µ mode on **the** i th site **and** g µ

is

The free phonon Hamiltonian can be decomposed into symmetric (s) **and**

antisymmetric (b) components

where s µ **and** b µ have been defined as

( 1)

H = H + H = ∑ h Ω b b + s s +

(A.4)

0 0b

0 s

† †

ph ph ph

µ µ µ µ µ

µ

65

( 1 2 )

s 1

µ

= a 2

µ

+ a

µ

b 1

µ

= ( a1µ − a2µ

).

2

The electron-phonon term also has two contributions

where g

+ = g / 2 .

µ µ

∑

µ

µ

†

( 1 2 )( )

s

H = hΩ g n + n s + s

ep

∑

+

µ µ µ µ

†

( 1 2 )( ).

b

H = hΩ g n − n b + b

Displaced oscillator transformation

ep

+

µ µ µ µ

(A.5)

As **the**re are two different boson operators - s **and** b - to perform **the** displaced

oscillator transformation two generators R s **and** R b are required. These are defined

as

**and**

∑

µ

†

( 1 2 )( )

R = δ n − n b − b (A.6)

b

∑

µ

µ µ µ

†

( 1 2 )( )

R = η n + n s − s (A.7)

s

µ µ µ

where ηµ

**and** δ µ

are 2N as yet undetermined parameters where N is **the** total number

if vibrational modes.

Electronic transformation

The electron operators can be transformed using **the** R b generator where we let

†

( )

B = b − b . The transformed operator can be rewritten as

µ µ µ

c

= e c e

Rb

− Rb

1σ

1σ

Using **the** fermionic anticommutator relation { , }

c c = δ gives n c = c ( n − ).

α β α , β

†

1 1σ

1σ

1

1

R

2 2

1 1 {

bj −Rbj

σ

= ⎡

σ

1− µ µ

+

µ µ 1

−

2

+

µ µ 1

−

2

− 1 + ... ⎤

∏ ⎢⎣

⎥⎦

µ

δ δ ( ) 1 ( δ

2 ) ( )

2 2

( 1 2 ) 1 ( δ

2 ) ( 1 2 ) ⎤

⎥}

e c e c B B n n B n n

× ⎡

⎢

1 −δ

µ

Bµ n − n +

µ

Bµ

n − n + ...

⎣

⎦

∏

( δ B )

( )

( δ ) ( ) 1 ( ) ( ) 1 ( )

= c ⎡1 − B + B n − n + n − n − n − n + n − n − 1 + ... ⎤

⎥⎦

2 2 2 2

1σ ⎢

δµ µ µ µ 1 2

2

1 2 1 2

2

1 2

⎣

µ

∏

( δ )

2

= c ⎡1 − δ + + ... ⎤

⎢

B B

⎣

⎥⎦

1σ µ µ µ µ

µ

∏

= c exp −

1σ µ µ

µ

Applying **the** same principle to **the** o**the**r Fermi operators gives

∏

( δ )

c = c exp − B

1σ 1σ µ µ

µ

∏

( δ )

c = c exp B

† †

1σ 1σ µ µ

µ

∏

( δ )

c = c exp B

2σ 2σ µ µ

µ

∏

( δ )

c = c exp − B

† †

2σ 2σ µ µ

µ

(A.8)

These operators can now be applied to **the** terms in (A.2) to give **the** transformed

electronic Hamiltonian. It can be seen from **the** transformed operators in (A.8) that

**the** number operators remain unchanged i.e.

so **the** only noninvariant term in (A.2) is

n = n = c c

(A.9)

†

iσ iσ iσ iσ

c c ⎛

+ c c = c c ∏exp 2 B + c c exp −2

B

⎝

( δ ) ∏ ( δ )

† † † †

1σ 2σ 2σ 1σ ⎜ 1σ 2σ µ µ 2σ 1σ µ µ

µ µ

The transformed electronic component is **the**refore

⎞

⎟

⎠

Using this identity gives

**and** hence

∏

R

e b

c ⎡ B n n B n n ⎤c

( ) 1 ( δ ) ( )

2 2

1σ = 1 + δµ µ 1

−

2

+

1 2

...

2

µ µ

− +

1σ

⎣⎢

⎥⎦

µ

∏

2 2

= c ⎡1 + δ ( − − 1) + 1 ( δ ) ( − − 1 ) + ... ⎤

⎢

B n n B n n

⎣

2

⎥⎦

1σ µ µ 1 2 µ µ 1 2

µ

⎛

⎞

H ( ) ( ) † ( )

†

el

= µ

m∑ n1 σ

+ n2 σ

− tm

⎜∑∏exp 2δ µ

Bµ c1 c2 + ∏exp −2δ

µ

Bµ

c2c1

⎟

σ ⎝ σ µ µ

⎠

+ U n n + V n n

∑

m

i=

1,2

i↑

i↓

m 1 2

Free Phonon **and** electron-phonon transformations

Using R b , R s **and** **the** Campbell-Baker-Hausdorff **the**orem

(A.10)

67

68

X − X

1 1

e Ye = Y + [ X , Y ] + ⎡X ,[ X , Y ] ⎤ ⎡X , ⎡X ,[ X , Y ] ⎤⎤

...

2!

⎣ ⎦ + +

3! ⎣ ⎣ ⎦⎦

(A.11)

allows **the** phonon operators to be transformed as

b

Rb

−Rb

µ

= e bµ

e

1

= bµ + ⎣

⎡Rb , bµ ⎦

⎤ + ⎡Rb , ⎡Rb

, b ⎤⎤

µ

+ ...

2 ⎣ ⎣ ⎦⎦

The commutator relationship is given by

∑

† †

( 1 2 )(( ) ( ))

⎡

⎣R b ⎤ = n − n b − b b − b b − b

b, µ ⎦ δν ν ν µ µ ν ν

ν

∑

ν

µ

† †

( n1 n2

)( b b b b )

= δ − −

ν ν µ µ ν

( n n )

= −δ

−

1 2

As **the** number operators commute with b

† †

,( 1

−

2 ) ⎤⎦

= ∑{ δµ ( 1

−

2 )( µ

−

µ )( 1

−

2 ) −δµ ( 1

−

2 )( 1

−

2 )( µ

−

µ )}

µ

2 2

= ∑δ

(( ) ( ) ( ) ( ))

† †

µ

n − n b 1 2 µ

− bµ − n − n b 1 2 µ

− b

µ

⎡⎣

Rb

n n n n b b n n n n n n b b

µ

= 0

so all higher order commutators vanish giving **the** transformed operators

**and**

µ µ µ

( )

b = b −δ

n − n (A.12)

µ µ µ

1 2

( )

s = s − η n + n (A.13)

1 2 .

Using **the**se transformed operators it is now possible to obtain **the** transformed

Hamiltonians for **the** phonon terms in (A.2). The antisymmetric component is given

by

( )

H + H = e H + H e

0b b Rb

0b b

ph ep ph ep

∑

† †

( b b ) g ( n n )( b b )

1 2

= hΩ ⎡ 1

µ µ µ

+ + 2

µ

−

µ

+ ⎤

µ

⎣

⎦

µ

= ∑ hΩ ⎡

1

µ µ

−

µ

−

µ

−

µ

− + + 2

µ

−

⎣

µ

∑

†

+

(( b δ ( n1 n2 ))( b δ ( n1 n2 ))

) g ( n1 n2

)

†

( bµ bµ 2δ

µ ( n1 n2

))

† + †

+

( b b ) ( g δ )( n n )( b b ) δ ( δ g )( n n )

1 2 1 2

= hΩ ⎡ + + − − + + − −

⎣ 2

µ

− Rb

× + − −

⎤

⎦

2

1

µ µ µ µ µ µ µ µ µ

2

µ

.

⎤

⎦

( ) ( + η )( )( ) ( )( ) 2

1 2

η η

+

1 2

H + H = ∑hΩ ⎡ s s + 1 + g − n + n s + s + − 2 g n + n ⎤.

⎦

0 s s

† †

ph ep

µ

2

µ µ µ µ µ µ µ

⎣

µ

+ ,

Because **the** electronic terms commute with **the** total number of electrons ( n n )

1 2

**the**y are not effected by **the** symmetric generator **and** thus, insensitive to **the** value of

+

η . This means that we can set η = g

µ

to eliminate **the** interaction of **the** electrons

with **the** symmetric phonons from **the** transformed Hamiltonian. Because **the** charge

difference ( n n )

− does not commute with **the** electronic terms **the** same cannot be

1 2

done for δ . Making **the** substitutions gives **the** transformed phonon Hamiltonian

components

∑

(

1 + +

) ( δ )( 1 2 )( ) δ ( δ 2 )( 1 2 )

H + H = hΩ ⎡ b b + + g − n − n b + b + − g n −n

0 b b

† †

ph ep

µ µ µ

2

µ µ µ µ µ µ µ

⎣

µ

∑

+

( ) ( 1 2 )

H + H = hΩ ⎡ s s + 1 − g n + n

2

0 s s

†

ph ep

µ µ µ µ

⎣

µ

2

⎤

⎦

2

⎤

⎦

(A.14)

The effective polaronic Hamiltonian

Electronic Transformation

To obtain **the** electronic term of **the** polaronic Hamiltonian **the** phonon operators must

be eliminated from **the** transformed Hamiltonian. This is achieved by taking **the**

average over **the** squeezed-phonon wave function for each mode

† †

Defining Aµ ( bµ bµ bµ bµ

)

Ψ

† †

−α

( b b b b )

e µ µ µ −

= ∏ µ µ

0

(A.15)

sq

i

µ

= − **and** x = 2δ

we can apply (A.15) to (A.10) **and** consider

µ µ

**the** only term containing phonon operators

⎛ ( 2δµ Bµ ) † ( −2δ

µ B ) ⎞

µ †

Ψ

sq ⎜∏e

c1 σ

c2σ + ∏e

c2 σ

c1

σ ⎟ Ψ

sq

⎝ µ µ

⎠

= 0

αµ ( bµ bµ −b † µ b † µ ) ( 2δ

) µ ( µ µ † µ † µ ) †

e B µ −α b b −b b

e e 0 c c + 0

αµ ( bµ bµ −b † µ b † µ ) ( 2δ

) µ ( µ µ † µ †

µ )

e B µ −α

b b −b b

e e

†

0 c c

∏

1σ 2σ 2σ 1σ

µ µ

∏

∏

∏

−αµ Aµ xµ Bµ αµ Aµ † −αµ Aµ xµ Bµ αµ Aµ

†

e e e c1 σ

c2 σ

e e e c2 σ

c

1σ

µ µ

= 0 0 + 0 0

(A.16)

ph

Similarly, **the** symmetric component is given by

69

70

To calculate **the** commutator relationship between A µ **and** B µ we apply **the** general

†

commutator relation ⎡

⎣b

, b ⎤ µ ν ⎦ = δ

µ ,

giving

ν

† † † † † †

( )( ) ( )( )

⎡

⎣ A , B ⎤

⎦ = b b − b b b − b − b − b b b − b b

† † † † † † † † † † † †

bµ bµ bν bν bµ bµ bµ bµ bν bν bµ bµ bµ bµ bν bν bµ bµ bµ bµ bν bν bµ bµ

= b b b − b b b + b b b − b b b

† † † † † †

ν µ µ µ µ ν ν µ µ µ µ ν

†

2bµ µ , ν

2bµ µ , ν

= 2B

µ ν µ µ µ µ ν ν ν ν µ µ µ µ

= − − + − + + −

= δ −

µ

δ

where **the** third line has been obtained by successive application of **the** commutator

relation. In general

n n−1 n−1

⎡

⎣A , B ⎤

µ ν ⎦ = ⎡

⎣ Aµ , B ⎤

ν ⎦ Bν + B ⎡

ν ⎣ Aµ , B ⎤

ν ⎦

**and** using **the**se relations

= 2 B B δ + B ⎡

⎣

A , B

= 2nB

n−1 n−2

ν ν µ , ν ν µ ν

n

µ

( )

⎤

⎦

⎡A A B ⎤

⎣ ⎣ ⎦⎦

n B

n

2 n

, ⎡

µ µ

, ⎤

ν

= 2

µ

( )

3

⎡

n

n

A , ⎡ A , ⎡ A , B ⎤ ⎤⎤

µ ⎣ µ ⎣ µ ν ⎦⎦

= 2 n Bµ

.

⎣

⎦

(A.17)

(A.18)

Making use of **the** Campbell-Baker-Hausdorff **the**orem again allows **the** exponentials

of (A.16) to be exp**and**ed in **the** form

−α

1 2 1

µ Aµ x B A

3

e e ν αµ µ ⎛ ⎞

ν

e = 1 + xν ⎜ Bν − α

µ

Aµ , Bν αµ Aµ , Aµ , Bν α ⎡ A , A , A , B ⎤

⎣

⎡

⎦

⎤ + ⎡ ...

2!

⎣

⎡

⎦

⎤⎤ − ⎡

µ µ µ

3!

⎣

⎡

µ ν ⎦

⎤⎤

+ ⎟

⎝

⎣ ⎦ ⎣ ⎣ ⎦⎦

⎠

1 2 ⎛

1 2 1 3

⎞

+ xν ⎜ Bν − αµ Aµ , Bν α

µ

Aµ , Aµ , Bν α ⎡

µ

Aµ , Aµ , Aµ , B ⎤

ν

...

2!

⎣

⎡

⎦

⎤ + ⎡

2!

⎣

⎡

⎦

⎤⎤ − ⎡

3!

⎣

⎡

⎦

⎤⎤

+ ⎟

⎝

⎣ ⎦ ⎣ ⎣ ⎦⎦

⎠

1 3 ⎛

1 2 1 3

⎞

+ xν Bν −α

⎡

µ

Aµ , Bν

, , , , , ...

3!

⎜

α

µ

Aµ Aµ Bν α ⎡

µ

Aµ Aµ Aµ B ⎤

⎣

⎤

⎦ + ⎡ ⎡ ⎤⎤ − ⎡ ⎡ ⎤⎤

ν

+

2! ⎣ ⎦ 3!

⎣ ⎦ ⎟

⎝

⎣ ⎦ ⎣ ⎣ ⎦⎦

⎠

+ ...

which can be rewritten using **the** commutator relations from (A.17) as

−α

1 2 2 1

µ Aµ x B A

3 3

e e ( ) ( )

ν αµ µ

⎛

⎞

ν

e = 1+ xµ Bµ ⎜1− 2α µ

+ 2 αµ − 2 α

µ

+ ... ⎟

⎝ 2! 3! ⎠

1 2 2 ⎛ 1 2 2 1 3 3 ⎞

+ xµ Bµ ⎜1− 4α µ

+ ( 4 ) αµ − ( 4 ) α

µ

+ ... ⎟

2! ⎝ 2! 3! ⎠

1 3 3 ⎛ 1 2 2 1 3 3 ⎞

+ xµ Bµ ⎜1− 6α µ

+ ( 6 ) α

µ

− ( 6 ) α

µ

+ ... ⎟

3! ⎝ 2! 3! ⎠

+ ...

−2α 1 2 2 4 1

µ − αµ 3 3 −6αµ

= 1 + xµ Bµ e + xµ Bµ e + xµ Bµ

e + ...

2! 3!

Substituting this relation into equation (A.16) gives

∏

(

−α1A1 x1 B1 α1A1 −α2 A2 x2B2 α2 A2

−α

xν

Bν

α

)

x B A A

Ψ e Ψ = 0 e e e e e e ... e

µ

e e 0

µ µ µ µ µ

µ

⎛

−2α 1 2 2 4 1

µ − αµ 3 3 −6αµ

⎞

= 0 ∏⎜1 + xµ Bµ e + xµ Bµ e + xµ Bµ

e + ... ⎟ 0

µ ⎝

2! 3!

⎠

where **the** operators for different modes have been commuted in **the** first line. By

applying **the** orthogonality relations to **the** resulting wavefunctions this can be

reduced to

x B ⎡ 1 2 4 1 1 4 8 1 1

µ µ − αµ ⎛ ⎞ − αµ ⎛ ⎞ 6 −16α

⎤

µ

Ψ ∏e Ψ = ∏ ⎢1 − xµ e + ⎜ x e x e ...

2 ⎟ µ

− ⎜ 3 ⎟ µ

+

µ µ 2 2! 2 3! 2

⎥

⎣

⎝ ⎠ ⎝ ⎠ ⎦

2 −4α

( δµ

)

µ

= ∏exp −2 e (A.19)

µ

The electronic component of **the** transformed polaronic Hamiltonian can now be

written as

( ) ( 2 4αµ

∑

)( † †

1σ 2σ ∑ ∏exp 2

µ 1σ 2σ 2σ 1σ

)

H = µ n + n − t − δ e −

c c + c c

el m m

σ σ µ

∑

+ U n n + V n n

m i↑

i↓

i=

1,2

m 1 2

(A.20)

Free Phonon **and** electron-phonon transformations

Applying **the** squeezed states to **the** free phonon terms gives

† † † †

−αµ ( bµ bµ −bµ bµ ) ⎛⎛ † 1 ⎞ ⎛ † 1 ⎞⎞

−αµ ( bµ −bµ bµ

)

Eph = Ψ

sq

H

ph

Ψ

sq

= ∑ h Ω

µ

0 e ⎜⎜bµ bµ + ⎟ + ⎜ sµ sµ

+ ⎟⎟e

0

µ

⎝⎝ 2 ⎠ ⎝ 2 ⎠⎠

**and** by **the** above method, **the** antisymmetric term H

b

ph

is

b ⎧ ⎛ † † 1 2 † ⎞ 1 ⎫

Eph

= ∑ ⎨hΩµ 0 ⎜bµ bµ − α ⎡

µ

Aµ , bµ b ⎤

µ

α ⎡

µ

Aµ , ⎡Aµ , bµ b ⎤⎤

⎣ ⎦ +

µ

+ ... ⎟ 0 + Ωµ

⎬

µ ⎝

2! ⎣ ⎣ ⎦

⎩

⎦

h

⎠ 2 ⎭

(A.21)

71

72

where A µ retains **the** previous definition. Taking **the** expectation value of **the**

commutators

**and** in general

( )

⎡

⎣

⎤

⎦ = − + =

† † †

0 α

µ

Aµ , bµ bµ 0 2α

0 bµ bµ bµ bµ

0 0

3 3

1 2 † 2 2 † † 2 2

0 α ⎡A , ⎡A , b b ⎤ ⎤

µ

0 0 ( ) 0

2! ⎣ µ ⎣ µ µ µ ⎦⎦

= α

µ

bµ bµ + bµ bµ = α

µ

2! 2!

5

1 3 † 2 3 † †

0 α ⎡A , ⎡A , ⎡A , b b ⎤ ⎤⎤

µ µ µ µ µ µ

0 = α

µ

0 ( bµ bµ + bµ bµ

) 0 = 0

3! ⎣ ⎣ ⎣ ⎦⎦⎦

3!

7

7

1 4 † 2 4

0 α ⎡

µ

Aµ , ⎡Aµ , Aµ , Aµ , bµ b ⎤⎤

† †

⎡

⎤

2 4

⎡ ⎤

µ

0 = α

µ

0

4! ⎢⎣ ⎣ ⎣ ⎣ ⎦⎦⎦⎥

( bµ bµ + bµ bµ ) 0 = αµ

⎦ 4!

4!

allows (A.21) to be rewritten as

2n−1

⎧2

n

1 n ( 1 ) ( n)

†

α

µ

n even

0 α ⎡A ..., ⎡A , b b ⎤ ⎤ ⎪

µ µ µ µ µ

0 = n!

n!

⎣ ⎣ ⎦⎦

⎨

⎪

⎩0 n odd

1 ⎛

1 ⎞

= ∑ Ω 1+ ( 4α

) + ( 4 α ) + ... ⎟

⎝

⎠

1

= ∑ hΩµ cosh ( 4 α

µ ).

2

b

2 4

E

ph

h

µ ⎜ µ µ

2 µ

4!

µ

(A.22)

From **the** definition of **the** squeezed state **the** expectation value of symmetric

component of **the** free phonon Hamiltonian H is

∑

s

−αµ ( b † µ b † µ −bµ bµ ) ⎛ ⎞ −αµ ( bµ bµ −b † µ b

†

µ )

Eph

= ∑ hΩ µ

0 e ⎜ sµ sµ

+ ⎟e

0

µ

⎝ 2 ⎠

1

= ∑ hΩµ

2

µ

s ph

† 1

(

1

) ( + δ )( 1 2 )( ) δ ( δ 2

+

)( 1 2 )

H + H = hΩ ⎡ b b + + g − n − n b + b + − g n − n

0 b b

† †

ph ep

µ µ µ

2

µ µ µ µ µ µ µ

⎣

µ

∑

+

( ) ( 1 2 )

H + H = hΩ ⎡ s s + 1 − g n + n

2

0 s s

†

ph ep

µ µ µ µ

⎣

µ

2

⎤

⎦

2

(A.23)

Because only **the** antisymmetric phonons couple to **the** electrons we need only

consider **the** antisymmetric terms

The antisymmetric phonon terms of **the** polaronic Hamiltonians can now be written as

+

{ ( δ ( δ 2 )( ) )}

2

µ µ µ µ 1 2

H + H = ∑ hΩ − g n − n + E (A.24)

b b b

ep ph ph

µ

⎤

⎦

Using **the** relationship n

as

2

iσ

= n **the** squared number operator terms can be rewritten

iσ

2

2

( n1 − n2 ) = (( n + n ) − ( n + n

1↑ 1↓ 2↑ 2↓

))

= ( n1 + n2 ) + 2( n n + n n

1 1 2 2 ) − 2( n n + n n + n n + n n

↑ ↓ ↑ ↓ 1↑ 2↑ 1↑ 2↓ 1↓ 2↑ 1↓ 2↓

)

= ( n1 + n2 ) + 2∑

n n − 2n i i 1n

↑ ↓ 2

i=

1,2

Substituting **the**se into (A.24) gives

⎧

⎫

b b b ⎪

+

⎛

⎞⎪

H

ph

+ H

ph

= Eph −∑

⎨hΩµ δ

µ ( δµ − 2gµ

) ⎜( n1 + n2 ) + 2∑

n n − 2n ↑ ↓ 1n i i

2 ⎟⎬

(A.25)

µ ⎪⎩

⎝

i=

1,2

⎠⎪⎭

Combining (A.20) **and** (A.25) gives an effective polaronic Hamiltonian

H = H = H + H + H

b b b b b

pol HH el ph int

⎛

( ) ( 2 −4α ) ( † †

1 2

exp 2

µ

⎞

= µ ∑ n

σ

+ n

σ

− t ⎜∏

− δ

µ

e ⎟∑ c1 c2 + c2c1

) + U ∑ n n

⎝

⎠

m m m i↑

i↓

σ

µ

σ

i=

1,2

⎧

+ +

+ V n n −∑⎨hΩ δ ( δ − 2g )( n + n ) − 2hΩ δ ( δ − 2g ) ∑ n n

⎩

m 1 2 µ µ µ µ 1 2

µ µ µ µ i↑

i↓

µ

i=

1,2

+

( δ ) 1 2}

−2hΩ δ − 2g n n + E

µ µ µ µ

This simplifies to

with

ph

∑( 1σ

2σ

) ∑( 1 2 2 1)

∑ i i

1 2

(A.26)

* * † † * *

pol

= µ

m

+ −

m

+ +

m

+

↑ ↓ m

+

ph

σ

σ

i=

1,2

H n n t c c c c U n n V n n E

+

( 2g

)

*

µ

m

= µ

m

− hΩµ δµ δ

µ

−

µ

µ

*

⎛

2 −4α

⎞

µ

tm

= exp⎜

−2∑δ

µ

e ⎟t

⎝ µ ⎠

*

m

∑

∑

+

( δ )

U = U − 2 hΩ δ − 2g

*

m

m

µ

m

µ µ µ µ

+

( δ )

V = V + 2 hΩ δ − 2 g

E

ph

m

µ

1

= ∑ hΩ

2

µ

∑

µ µ µ µ

( cosh ( 4 α )).

µ µ

73

74

11 Appendix B

75

76

77 78

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