Phonons and the Isotopically Induced Mott transition - Physics

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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 Queensland

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 whether 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 bandwidth within the material and we conclude by arguing

that they 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

together like having a thesis 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 understanding wife

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

lows throughout the year.

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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 band theory ............................................... 9

2.2.2 Band theory 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 band 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 Band 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 Another 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

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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 theory. 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, band theory, 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 then 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 theory by Landau in the late 1950’s to

explain the behaviour of the liquid state of 3 He. 11 Within this theory 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 theory was first published Bardeen, Cooper, and

Schrieffer developed a microscopic theory of superconductivity 12 (now referred to as

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

phonons. Significantly within BCS theory superconductivity could be accounted for by

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

direct electrostatic interaction is repulsive, they found that ionic motion could

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

formation. From this theory 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 theory. One of the most important of these 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 understand 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 others 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 understand the behaviour of these strongly correlated

systems that provides the underlying motivation for this work.

band is known as the bandwidth W. The formation of this band 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 their electronic configurations,

conventional band theory calculations indicated that these materials should display

metallic properties, but instead, they behaved as insulators.

2.2.1 The tight binding approach to band theory

In order to understand this unusual electrical behaviour it is helpful to first look at the

expected behaviour in terms of conventional theories. 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 together 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 together the

greater the splitting of these 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 bands and if N is sufficiently large

22

( N ≈10 )

we can regard all energies within each band 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 together, the levels broaden into bands. 8

2.2.2 Band theory and electronic properties

Within band theory there is a sharp distinction at absolute zero between crystalline

materials showing metallic conduction, on one hand, and insulating behaviour, on the

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

bands 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 these higher states and conduction results. Insulators, on the other

hand, are materials where all energy bands 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 (bandgap) between these energy bands is sufficiently small

thermal excitation can cause electrons to ‘jump’ into empty conduction bands. This is the basis for

conduction in many semiconductors.

9

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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 other words, in this model the e-e repulsion is neglected except for

two electrons on the same site. Conventional band theory, 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 these materials the band picture breaks

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

metallic and insulating states. To understand 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 there being, as in the band 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 another neutral atom to give a negatively charged ion, the difference

between these two energies and therefore 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 otherwise continuous band

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

equal to the Hubbard energy because of bandwidth 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 bands form, with the lower one corresponding to the

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

This is illustrated in Figure 2.3

As the bandwidth of these bands increases it becomes easier for electrons and holes

to hop between sites and the eventual overlap of the two bands 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 other hand, 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 band 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 bands in NiO as a function of bandwidth W which is

inversely dependent upon the interatomic separation. The lower band corresponds to the motion

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

two bands 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 other hand, will always have some minimum frequency of vibration,

even when their wavelength is large. They are called "optical" because in ionic

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

infrared radiation). This is because they correspond to a mode of vibration where

positive and negative ions at adjacent lattice sites swing against each other, 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 together, whilst they move 180°

out of phase in the optical mode. 8 14

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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,

another class of materials, the organic (carbon based) charge transfer salts also

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

same unusual properties with the cuprates. Furthermore, these compounds are

known to display a great sensitivity to variations in hydrostatic pressure 17 and they

have the added advantage that the location of the transition region on their 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 them as

semiconductors. 3 Of particular interest were their 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 other materials, there 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 these 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 stands for an electron acceptor

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

synthesis 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 shorthand (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 otherwise 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

their central tetrathioethylene planes almost parallel forming ‘dimers’, and adjacent

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

sheet. The Cu[N(CN) 2 ]Br anions are weakly bonded to each other 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 these 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 sandwiched 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, thermal 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 transitions detected using various techniques

(NMR and ac conductivity – diamonds, magnetization – pluses, thermal 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

them useful in the study of the MIT is their ‘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 other hand, 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 whether 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 expanded 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 their 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)

then 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 they 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

(†)


( ) 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 rather 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 band

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 thermopower 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 whether 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 therefore 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 these 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 then the model reduces to the Hubbard model on a triangular lattice,

following a rescaling of the lattice constants. 5 Although there 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 theoretical 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 these

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 then used to calculate U m and V m but where each of these groups differs

23 22

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

25 have based their 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 their value of U m from

an application of density functional theory 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 these interactions is the Holstein

model.

The physical properties of the polaron differ from those of the band 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 these 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, together 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 these 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 bandwidth, 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 band 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 these 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 band-

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 therefore 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 these terms together 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

then 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 these 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 these expressions together, 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

)


(5.32)

b + b = b + b e µ

. (5.33)

Using these 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

these 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, there 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 these 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 expanded 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 there 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

therefore 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 theorem 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 , they 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


(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 then 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 these 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 band narrowing

Polaronic behaviour is characterised by a reduction in the electron hopping (band

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

band 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 andδ . 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 andδ .

These values were then 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 band 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 there 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

Band 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 band narrowing occurs until the

coupling strength is of the order of the strong coupling condition upon which there 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 there 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 Band 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 these values will be used throughout the

analysis unless otherwise specified. Applying the band 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

Band 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 band 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 band 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 there 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 standard 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

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

behaviour of the model should therefore 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 band 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 band

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 there is substantial band narrowing the

energy begins to fall nearly linearly with coupling strength and to understand 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

whether 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 rather 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 there 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 these 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 these 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 these 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, rather 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 therefore 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 thereby 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 there 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 Another possibility - the geometrical

isotope effect

From these 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 there is some evidence to suggest otherwise. 41

Although we have not dealt with it specifically, this transition can also be driven by

hydrostatic pressure and these 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 therefore 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 there 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 ligands. These results certainly warrant further 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 understand the role of phonons in the MIT. We then introduced band

theory and the concept of the onsite Coulomb repulsion to account for this transition

in terms e-e interactions opening up gaps in otherwise continuous bands.

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 whether phonons

participate in the MIT.

We then outline the dimer Hubbard model as a minimal model that is often applied to

these 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 then

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 other 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 then 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 bandwidth 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 expanded as

where

† †

∑( 1σ


) ∑( 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 there 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



Using the fermionic anticommutator relation { , }

c c = δ gives n c = c ( n − ).

α β α , β


1 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


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 other 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 therefore




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

− +


⎣⎢

⎥⎦

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


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 theorem

(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 these 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

they 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


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 these 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 theorem again allows the exponentials

of (A.16) to be expanded 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



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


= n the squared number operator terms can be rewritten


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 these 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


) ∑( 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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