kadowaki-woods and kadowaki-woods-like ratios in...

KADOWAKI-WOODS AND KADOWAKI-WOODS-LIKE

RATIOS IN STRONGLY CORRELATED ELECTRON

MATERIALS

by

Yiwei Zhang

BSc, University of Science and Technology of China, 2008

THESIS SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN THE DEPARTMENT

OF

PHYSICS

c© Yiwei ZhangSIMON FRASER UNIVERSITY

Spring 2012

All rights reserved. However, in accordance with the Copyright Act of Canada,this work may be reproduced, without authorization, under the conditions forFair Dealing. Therefore, limited reproduction of this work for the purposes ofprivate study, research, criticism, review, and news reporting is likely to be

in accordance with the law, particularly if cited appropriately.

APPROVAL

Name: Yiwei Zhang

Degree: Master of Science

Title of thesis: Kadowaki-Woods and Kadowaki-Woods-like ratios in strongly

correlated electron materials

Examining Committee: Dr. Eldon Emberly

Associate Professor, Department of Physics (Chair)

Dr. Malcolm Kennett, Senior Supervisor

Associate Professor, Department of Physics

Dr. Igor Herbut, Supervisor

Professor, Department of Physics

Dr. David Broun, Supervisor


Dr. J. Steven Dodge, Internal Examiner


Date Approved: January 13, 2012

ii

Partial Copyright Licence

Abstract

In a Fermi liquid there are well established temperature dependences for the low temper-

ature resistivity and the specific heat. The ratio of these temperature dependences, the

Kadowaki-Woods ratio (KWR), has been found to be roughly constant within families of

strongly correlated electron materials. A recently introduced related ratio [ Jacko et al.

Nature Phys. 5, 422 (2009) ] that takes into account band structure effects, was found to

be roughly constant over a wide range of families of strongly correlated electron materi-

als. Previous theoretical work on these ratios has assumed that the electron self-energy is

momentum independent. We relax this assumption and consider a variety of phenomeno-

logical forms of the self-energy that have been proposed for strongly correlated electron

materials. This leads us to investigate ratios analogous to the KWR for a variety of pro-

posed electron self-energies from both a theoretical and a phenomenological point of view.

In particular, we collate experimental data from heavy fermion compounds that have non-

Fermi liquid phenomenology and investigate the KW-like ratio for those compounds.

iii

To my beloved parents

iv

Acknowledgments

First I would like to express my deepest gratitude to my senior supervisor, Dr. Malcolm

Kennett. His encouragement, guidance and support from the initial to final stages of my

graduate study has helped me to develop the skills and perseverance to not only take on the

problems that come about in the project, but also take on challenges in my life. It is my

great honor to be Malcolm’s student. I am also very grateful to the faculty members who

taught the courses I have taken during my Master’s study, especially to Dr. Igor Herbut,

whose excellent courses have stimulated my interests in condensed matter physics. I would

also like to thank my colleagues Peter Smith, Bitan Roy and Felix Lu for their valuable

suggestions and advice during the course of this work. My appreciation also goes to all

of my friends for their enduring support, especially to Ben Zhu, Shuangxing Dai and Jie

Zhang. And finally, I would like to thank my parents and my siblings for being there when

I needed them, and Brenna Li for her moral support during the writing process.

v

Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgments v

Contents vi

List of Tables ix

List of Figures x

1 Introduction 11.1 Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Kadowaki-Woods ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Motivations for considering self-energies that are different from the Fermi

liquid forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Momentum dependent self-energy in cuprates . . . . . . . . . . . . 9

1.3.2 Marginal Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . 12

1.3.3 Hybrid self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Outline of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Derivation of the Kadowaki-Woods and Kadowaki-Woods-like ratios 162.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

vi

CONTENTS vii

2.1.1 In-plane resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Linear Coefficient of The Heat Capacity . . . . . . . . . . . . . . . 23

2.1.3 Kadowaki-Woods ratio . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26


2.2.2 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3 Generalizations of the KWR for marginal Fermi liquids . . . . . . . 30

2.3 Case III : hybrid self energy . . . . . . . . . . . . . . . . . . . . . . . . . 31


2.3.2 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.3 Kadowaki-Woods and related ratios . . . . . . . . . . . . . . . . . 35

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Kadowaki-Woods and Kadowaki-Woods like ratios for specific bandstructures 383.1 Three dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Momentum independent self-energy and isotropic Fermi surface . . 39

3.1.2 Momentum dependent self-energy and anisotropic Fermi surface . . 40

3.1.3 Numerical calculation of the KWR and KWR-like ratio for three

dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Quasi two dimensional conductors . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Isotropic in-plane dispersion and momentum independent self-energy 51

3.2.2 Momentum dependent self-energy and anisotropic in-plane Fermi

surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.3 Numerical calculation of the KWR and KWR-like ratio for quasi

two dimensional conductors . . . . . . . . . . . . . . . . . . . . . 58

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Marginal Fermi liquid phenomenology and KWR-like ratios 644.1 Examination of the analogy to the KWR for the defined ratio B/ν . . . . . 64

4.2 B/ν and the temperature window of MFL behaviour . . . . . . . . . . . . . 72

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Conclusions 77

CONTENTS viii

A Evaluation of A2(ω,~k) as Σ′′(ω,~k) approaches to zero 79

B Evaluation of the ω integral in Ii j0 at nonzero temperature 81

Bibliography 83

List of Tables

4.1 Data used for the NFL materials studied in the thesis. . . . . . . . . . . . . 66

4.2 Carrier density values for the NFL materials studied in this thesis. . . . . . 68

ix

List of Figures

1.1 Kadowaki-Woods plot for typical heavy Fermion compounds . . . . . . . . 7

1.2 Kadowaki-Woods plot of coefficient A vs γv taking account of unit cell

volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Standard Kadowaki-Woods plot and Kadowaki-Woods plot taking account

of band structure effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Temperature dependences of two different scattering channels for over-

doped thallium cuprate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Doping dependences of two different scattering channels for overdoped

thallium cuprate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Fermi surface for 3D systems . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Coefficients α, α′, α′′

vs chemical potential µ0: 3D systems . . . . . . . . . 48

3.3 Fermi surface and Fermi velocity for quasi two dimensional conductors . . 59

3.4 Coefficients α, α′, α

′′vs chemical potential µ0: Quasi two dimensional

conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 ρ and Cv of YbRh2Si2 at low temperatures . . . . . . . . . . . . . . . . . . 65

4.2 Plot of coefficient B vs ν for heavy fermion materials . . . . . . . . . . . . 67

4.3 Histograms of the KW-like ratio B/ν for heavy fermion materials . . . . . . 68

4.4 Plot of coefficient B vs ν for selected heavy fermion materials . . . . . . . 69

4.5 Plot of coefficient B vs ν′ for selected heavy fermion materials . . . . . . . 70

4.6 Schematic phase diagram near a quantum critical point . . . . . . . . . . . 72

4.7 Plot of coefficient B vs Tmax for investigated heavy fermion materials . . . 73

4.8 Plot of coefficient ν vs Tmax for investigated heavy fermion materials . . . . 74

x

LIST OF FIGURES xi

4.9 Plot of coefficient B vs Tmax/Tmin for investigated heavy fermion materials 75

Chapter 1

Introduction

Metallic compounds based on rare-earth elements with incompletely filled d or f elec-

tron shells have drawn a lot of theoretical and experimental attention in recent decades.

Many interesting physical phenomena, such as the Kondo effect, metal-insulator phase

transitions, heavy fermion systems (which have quasiparticles that behave like electrons,

but with an effective mass up to several hundred times the bare electron mass) and high

temperature superconductivity have been found in these compounds. It has been believed

that strong interactions of d or f electrons with each other and with itinerant electrons is

responsible for the rich and unusual electronic and magnetic properties of these materi-

als. Such materials with strong electron-electron interaction are called strongly correlated

electron materials (SCM). Despite the large amount of work on these materials, a com-

plete theoretical understanding of their various physical properties is lacking. Important

empirical relations including the Kadowaki-Woods ratio and the Wilson ratio, which cor-

relate the magnetic susceptibility, the electron specific heat and the resistivity, have been

found in SCMs. These phenomenological relationships provide quantities to characterize

the properties of SCMs.

As well as exhibiting various interesting physical phenomena, SCMs have the potential

for applications in both scientific research and industries. High temperature superconduc-

tors (HTS) can produce stable and high magnetic fields, which have potential use in mag-

netic resonance imaging and nuclear magnetic resonance. Some complex manganites such

as lanthanum combined with manganese and oxygen (LaMnO3) exhibit the colossal mag-

netoresistance effect, which has been suggested as having possible applications in magnetic

1

CHAPTER 1. INTRODUCTION 2

recording and nonvolatile memories. Many other applications of SCMs are being explored

based on their anomalous physical properties.

SCMs can be hard to describe theoretically because strong Coulomb interactions among

electrons, the energy scale for which is often comparable with the bandwidth for d or

f shell electrons, produce phenomena that can not be understood from a single particle

picture. As a result, traditional approaches for the description of the electronic structures

of many-body systems, such as Hartree-Fock (HF) theory, density functional theory (DFT)

and Fermi liquid (FL) theory, do not work effectively in many of these compounds.

The HF theory provides the starting point for many-body theories of the electron liquid.

It assumes that the exact many-body wavefunction of the system can be approximated by a

single Slater determinant of some one-particle states φ1,...,φN . By requiring the expectation

value of the Hamiltonian to be a minimum, one can derive a set of non-linear equations.

The solutions to these equations, behaving as if each electron is subjected to the mean

field created by other electrons, yield the Hartree-Fock wavefunctions and energy levels

of the system, which are approximations to the exact ones. Correlation effects, which are

effects that stem from deviations from the mean field approximation, are neglected in the

Hartree-Fock theory. However, the electron exchange interaction (which is associated with

the Pauli principle) is treated correctly in this theory [1].

An alternative to the Hartree-Fock theory is density functional theory (DFT), which in-

cludes both electron exchange and correlation effects in an approximate way. The main idea

of density functional theory is to describe an interacting fermionic system via the density

of fermions rather than via many-body wave functions. For N electrons in a solid, the basic

variable of the system depends only on three (the spatial coordinates x, y, and z) rather than

3N degrees of freedom. Practical applications of DFT are based on approximations for the

exchange-correlation potential, which describes the effects of both the Pauli principle and

the Coulomb potential of the electrons. Density functional theory is a successful approach

for describing the electronic properties of many metals, semiconductors and insulators.

The Hartree-Fock and DFT approaches are often effective for simple metals and semi-

conductors, where electron-electron interactions are weak enough that one can get a good

insight into how electrons behave by calculating individual wavefunctions from an effec-

tive single-electron periodic potential. When electron-electron interactions are stronger, a

single particle approach is still viable: in Landau’s Fermi liquid theory, the excitations are


quasiparticles with the same quantum numbers as electrons but with renormalized proper-

ties, such as mass. Finally, when interactions become very strong, Landau’s Fermi liquid

theory can not provide an effective description of the behaviour of electrons in many SCMs

either.

Non-Fermi-liquid phenomenological theories (e.g. the marginal Fermi liquid theory)

have been proposed and developed in order to describe phenomena that are due to strong

electron correlation in SCMs. Given the insights that have been gained from studying

phenomenological quantifies such as the KWR and the Wilson ratio in materials that exhibit

Fermi liquid phenomenology, it is of interest to investigate whether similar approaches may

be applied to materials that exhibit non-Fermi liquid phenomenology.

In this Chapter we provide a general introduction to the Fermi liquid concept and a brief

review of previous work on the Kadowaki-Woods ratio within the framework of Landau’s

Fermi liquid theory. Motivations for considering the effects of non-Fermi-liquid behaviour

on the Kadowaki-Woods ratio are discussed in Sec. 1.2 with emphasis on the marginal

Fermi liquid theory and a phenomenological theory that has recently been proposed for

overdoped cuprates. This Chapter ends with an outline of the thesis.

1.1 Fermi liquid theory

In a free electron system, electrons behave as independent particles. The Hamiltonian for

N non-interacting electrons can be written as

H =N

∑i

Hi =−N

∑i

h2

2m52

i , (1.1)

which is the sum of N identical free electron Hamiltonians. Suppose the N electrons are

confined in a volume V . We may find the ground state of this system by calculating the

energy levels for a single electron and then filling these states up to the Fermi energy.

Respecting the Pauli repulsion principle, there is at most one electron in each eigenstate of

Hi. Thus, eigenstates of the system can be characterized by a set of occupation numbers

N~kσof single-particle eigenstates. ~k and σ label the wave vector and spin of electrons

respectively, and N~kσ= 0 or 1.

At zero temperature all the states with wave vector k < kF are filled, and the rest are left


empty. kF is the Fermi wave vector of the electron system, determined by the condition

2 ∑|~k|<kF

1 = N, (1.2)

where 2 comes from the spin degeneracy. Thus, the occupied states in momentum-space

are confined to lie inside a circle of radius kF for two dimensional systems or a sphere of

radius kF in three dimensional systems. The corresponding energy scale is εF = h2k2F/2m ,

which is the Fermi energy of the system.

This model can qualitatively describe the physical properties of many metals. For ex-

ample, it implies that the specific heat has a linear temperature dependence when the tem-

perature is much smaller than Fermi temperature TF ≡ εF/kB. However, in a metal it is

not well justified because it does not include the mutual Coulomb interactions between

electrons and nuclei. In many metals, Coulomb interaction energies between particles are

comparable to or even larger than kinetic energies, i.e. Ep/Ek ∼ rs/a0 is of order of one,

where rs is the Wigner-Seitz radius (radius of the sphere that encloses, on average, one

electron [2]), and a0 is the Bohr radius. In these metals, Coulomb interaction may not be

safely considered a perturbation, and a more sophisticated approach than the free electron

model is required. Additionally, the occupation number N~kσof a single-particle state with

momentum hk and spin σ will be modified when we turn on the interaction between the

particles. As a result, the appropriate description of the interacting system differs from that

of the non-interacting system.

In the late nineteen fifties, Landau constructed a theory of the low energy excitations

of an interacting Fermi system [3, 4]. He recognized that there is a one to one correspon-

dence between the eigenstates of a non-interacting fermion system and the eigenstates of

an interacting one if the interaction is switched on adiabatically (and is not too large, so

that there are no phase transitions). For example, a state of the non-interacting system with

one particle outside the Fermi surface evolves into a state of the interacting system with

one quasiparticle of the same momentum outside the modified Fermi surface due to the

interaction. The energy spectrum of a Fermi liquid, a system of interacting fermions, is

assumed to be a functional of the quasiparticle distribution function 0 ≤ N~kσ≤ 1, and is

given by [2]

E[N~kσ] = E0 +∑

~kσ

ε~kσδN~kσ

+12 ∑~kσ,~k′σ′

f~kσ,~k′σ′δN~kσ

δN~k′σ′, (1.3)


where E0 is the ground-state energy of the interacting system, f~kσ,~k′σ′is the Landau inter-

action function, ε~kσis the free quasiparticle energy, and δN~kσ

= N~kσ−N(0)

~kσis the deviation

of the quasiparticle distribution function from the Fermi-Dirac distribution at zero temper-

ature.

The remarkable feature of Landau Fermi liquid theory is that the effects of interactions

are captured by a set of Landau parameters. For a three dimensional system which is

invariant under spin and spatial rotations the Landau parameters are defined as

Fs,a` =

2`+12

D0

∫ dΩd

Ωd

f s~k,~k′± f a

~k,~k′

P (cosθ) , (1.4)

where f s~k,~k′

and f a~k,~k′

are the spin symmetric and spin antisymmetric part of Landau in-

teraction function, and D0 is the density of states at the Fermi level. The + and - signs

correspond to Fs` and Fa

` respectively. Quantities such as the electronic compressibility K ,

the spin susceptibility χ and the effective mass m∗ are renormalized and are related to their

non-interacting values via

KK0

=

m∗m0

1+Fs0,

χ

χ0=

m∗m0

1+Fs0,

m∗

m0= 1+

13

Fs1 . (1.5)

Here the subscripts 0 on K0 , χ0 and m0 denote the properties of the non-interacting system.

Note that properties of non-interacting system are not really experimentally accessible since

it is not possible to “ turn off ” interactions in a condensed matter system.

The low temperature properties of Fermi liquids show very specific dependence of

physical properties on temperature. For example, the low temperature specific heat of

a Fermi liquid has a linear temperature dependence [2], that is, CV = γT (where γ =

kFk2Bm∗/3h2 is a function of the effective mass m∗). Meanwhile, the resistivity scales with

temperature as ρ = ρ0+AT 2 [5], where ρ0 arises from disorder scattering and the quadratic

temperature dependence comes from electron-electron scattering.

The electronic self-energy reflects interactions between quasiparticles and the environ-

ment: the real part describes the renormalization of the electron mass which affects the

coefficient γ while the imaginary part can be related to the quasiparticle scattering rate and

hence the coefficient A. For example, in materials with an isotropic energy dispersion and

a momentum independent self-energy, the effective mass is given by m∗/m0 = 1−∂Σ′/∂ω,

where Σ′and ω denote the real part of the electron self-energy and the quasiparticle energy


respectively; the inverse of quasiparticle lifetime is given by 1/τ =−2Z Σ′′

[2], where Σ′′

is

the imaginary part of the electron self-energy and Z is the renormalization constant. Note

that the resistivity can also be related to the self-energy as ρ = m0/nee2τ at the simplest

level, where ne is the conduction electron density.

1.2 Kadowaki-Woods ratio

M. J. Rice observed in 1968 that the ratio A/γ2, where A is the coefficient of the quadratic

term in the temperature dependence of the low temperature resistivity and γ is the coeffi-

cient of the T -linear term of the low temperature specific heat, takes on an universal value

of 0.4µΩcmmol2 K2 J−2 for a number of the transition metals [7]. In 1986 Kadowaki

and Woods found that A/γ2 has a common value of 10µΩcmmol2 K2 J−2 for many heavy

Fermion compounds [8] (Figure 1.1). This ratio, which is the so called Kadowaki-Woods

ratio (KWR), has drawn a lot of attention [8, 9, 10, 11] since it has been believed to be

indicative of the strength of electron-electron correlations [10].

In 2005, N. E. Hussey [11] proposed that much of the differences between KWRs of

different families were due to the differences in their unit cell volume. For example, A/γ2∼50µΩcmmol2 K2 J−2 in La1.7Sr0.3CuO4 [11]. In contrast, A/γ2∼ 500µΩcmmol2 K2 J−2 in

Na0.7CoO2 [12], which is almost 10 times the magnitude of the previous one. If we include

the effect of unit cell volume, then for La1.7Sr0.3CuO4, A/γ2v∼ 0.17µΩcmK2 cm6 J−2 while

for Na0.7CoO2, A/γ2v ∼ 0.29µΩcmK2 cm6 J−2 which differs from the previous value by a

factor of less than two [11]. Hussey introduced γv, proportional to V γ , where V is the unit

cell volume, to remove these unit cell effects. A comparison of Kadowaki-Woods ratios for

a variety of materials with Hussey’s modified Kadowaki-Woods ratio is illustrated in Fig.

1.2 .

Recently, Jacko et al. [9] introduced a modified Kadowaki-Woods ratio which includes

band structure effects, and takes on a roughly constant value across several different fam-

ilies of materials, such as transition metals, transition-metal oxides, heavy fermions and

organic charge transfer salts [9]. Figure 1.3 shows the universality of the modified KWR

across those families of materials.

In their analysis, Jacko et al. [9] assumed that the imaginary part of the self-energy is

momentum independent and takes a conventional Fermi liquid form. For simplicity, they


Figure 1.1: A and γ2 relationship for typical heavy Fermion compounds. (Panel taken from

Ref. [8])

also assumed the Fermi surface to be isotropic in momentum space. They found that the

modified KWR can be written as

A fdx(n)γ2 =

814πhk2

Be2, (1.6)

where fdx(n)≡ nD20⟨v2

0x⟩

ξ2 takes account of band structure effects, and ξ∼ 1 is a constant

[9].


Figure 1.2: Comparison of Kadowaki-Woods ratio plot for a variety of materials with the

plot of A vs γv. (Panel taken from Ref. [11])

1.3 Motivations for considering self-energies that are dif-ferent from the Fermi liquid forms

As mentioned in the previous section, in Jacko et al.’s calculation, it was assumed that the

electronic self-energy is momentum-independent, and that the Fermi surface is isotropic

in momentum space. These assumptions reduce the complexity of the calculation of the

KWR. However, there are several pieces of evidence that Jacko et al’s picture is insufficient

for a number of materials. (i) The scattering rate in the normal state of overdoped cuprates

is found to be anisotropic in momentum space [13, 14]. (ii) The resistivity may be linear in


Figure 1.3: Left. Standard Kadowaki-Woods plot. Right. Comparison of the modified

Kadowaki-Woods ratio introduced by Jacko et al. with experimental data. (Panel taken

from Ref. [9])

temperature over a significant temperature range in a variety of strongly correlated electron

materials including cuprates [15, 16], organic and pnictide superconductors [17], and heavy

fermion compounds [18, 19]. It is hence of interest to revisit the assumptions made in Ref.

[9].

1.3.1 Momentum dependent self-energy in cuprates

We first review evidence that the self-energy in cuprates is momentum dependent. This

evidence comes from three sources: transport measurements, angle resolved photoemission

spectroscopy (ARPES) and theoretical studies.

Abdel-Jawad et al. [14] extracted the transport scattering rate from angular magnetore-

sistance oscillation (AMRO) measurements in overdoped thallium cuprate and found that

deviations from quadratic temperature dependence in the low temperature resistivity were

associated with momentum anisotropy in the scattering rate for electrons. They were able


Figure 1.4: a) Temperature dependence of the isotropic component of (ωcτ)−1(φ,T ),where

ωc is the cyclotron frequency, and τ(T ) is the transport lifetime. The green dashed curve is a

fit to A+BT 2. b) Temperature dependence of the anisotropic component of (ωcτ)−1(φ,T ).

The orange dashed curve is a fit to C+DT . (Panel taken from Ref. [14])

to fit the anomalous temperature dependences of the resistivity by assuming that

1ωcτ(φ,T )

=1+α(T )cos4φ

ω0τ0(T )=

1−α(T )ω0τ0(T )

+2α(T )

ω0τ0(T )cos22φ , (1.7)

where (1−α(T ))/ω0τ0(T ) is isotropic and has quadratic temperature dependence, while

(2α(T )/ω0τ0(T ))cos22φ has the same symmetry as the d-wave superconducting gap and

scales linearly with temperature. Here φ refers to the in-plane angle in momentum space.

The cyclotron frequency ωc was assumed to be isotropic within the basal plane (and equal

to ω0). They interpreted this result as indicating the existence of two different scatter-

ing channels, one isotropic in momentum, the other anisotropic. Additional evidence that

there are two different scattering channels in overdoped thallium cuprates comes from the

doping dependences of (1−α(T ))/ω0τ0(T ) and 2α(T )/ω0τ0(T ) : (1−α(T ))/ω0τ0(T ) is

relatively unaffected by doping, but 2α(T )/ω0τ0(T ) appears to scale with Tc. Similar dop-

ing dependences of scattering channels have also been found in overdoped La2−xSrxCuO4

single crystals [16]. Figure 1.4 shows the temperature dependences of these two scattering

channels and figure 1.5 shows their doping dependences.

More recently, Hussey and collaborators [20] reported similar anisotropy in the scatter-


Figure 1.5: Top: Doping dependence of the isotropic component of (ωcτ)−1(φ,T ),where

ωc is the cyclotron frequency, and τ(T ) is the transport lifetime. Bottom: Doping depen-

dence of the anisotropic component of (ωcτ)−1(φ,T ). (Panel taken from Ref. [13])

ing rate in the heavily overdoped non-superconducting cuprate La1.7Sr0.3CuO4 (LSCO30)

as that in overdoped thallium cuprate [14, 21]. They found that the Hall coefficient in

LSCO30 can only be fitted satisfactorily by including strong in-plane anisotropy in the

transport scattering rate.

Separate evidence for momentum anisotropy of the self-energy comes from ARPES

experiments in cuprates [22, 23]. Xie et al. [22] extracted the self-energy from ARPES

measurements in highly overdoped bismuth cuprate and found that the amplitude of the real

part of the self-energy Σ′ varies with~k while the shape of Σ′ remains nearly~k independent.

ARPES measurements in highly underdoped barium cuprate [23] also show momentum

dependence of Σ′ and find that the shape of Σ′, in contrast to the former system, depends on~k . These results, along with the anisotropic momentum distribution of low energy spectral


weight for cuprates inferred from ARPES [24, 25, 26], provide convincing evidence that

the self-energy in many cuprates is anisotropic in momentum space.

Theoretically, dynamical mean field theory calculations relevant to underdoped cuprates

[27, 28] and Functional Renormalization Group (FRG) calculations relevant to overdoped

thallium cuprate [29] also indicate that there may be strong momentum dependence of

the imaginary part of the self-energy. Ferrero et al. [27] calculated the interplane charge

dynamics in the normal state of cuprates within the valence-bond dynamical mean-field

(VB-DMFT) framework introduced in Ref. [28]. Their results are in good agreement with

spectroscopic and optical experiments and indicate a momentum dependent self-energy.

Ossadnik et al. [29] analyzed the temperature dependence of the quasiparticle scattering

rates in the 2D Hubbard model for a realistic dispersion function using the FRG. They

found that the piece of the scattering rate which is anisotropic in momentum space appears

to have a different temperature dependence than the isotropic piece: linear in temperature

rather than quadratic in temperature.

In light of these results, an evaluation of the effects of momentum anisotropy of the

self-energy on the KWR appears to be in order.

1.3.2 Marginal Fermi liquid theory

The low temperature resistivity of a number of strongly correlated electron systems grows

linearly with temperature, in contrast with the standard Fermi liquid description. R. A.

Cooper et al. [16] measured the low temperature in plane resistivity of several overdoped

La2−xSrxCuO4 cuprates and found that the resistivity can be fitted as having a piece that

varies linearly with temperature over a wide doping range. Similar temperature dependence

of resistivity has been found in the organic superconductor (TMTSF)2PF6 [17], the iron-

pnictide superconductor Ba(Fe1−xCox)2As2 [17] and heavy fermion compounds [18, 19].

The marginal Fermi liquid (MFL) theory is a phenomenological theory that was orig-

inally suggested as a means to describe the normal state properties of cuprates in the

"strange-metal" region [30, 31, 32]. In the original work on MFL theory, it was assumed

that the imaginary part of the complex polarizability has the form [30]

Im P(ω,~q)∼

−N(0)(ω/T ) for |ω|< T,

−N(0)sgnω for |ω|> T.(1.8)


Here N(0) is the unrenormalized density of states at the Fermi level. Equation (1.8) leads

to a logarithmic singularity in the real part of the single-particle self-energy at low energies

which has the form of

Σ(ω,T )∼ g2N2(0)(

ω lnx

ωc− i

π

2x), (1.9)

where x = max(|ω|,T ), g is a coupling constant which could in principle be momentum

dependent, and ωc is an ultraviolet cutoff.

It is of interest to compare the self-energy of a MFL with that of a Fermi liquid to

visualize the differences between them. The low energy self-energy of a Fermi liquid, in

the absence of momentum dependence, can be written as

ΣFL(ω,T )∼ αω− iβ(ω

2 +(πkBT )2) . (1.10)

From Eq. (1.10) one can see that at zero temperature the ratio of the imaginary part of the

self-energy to the real part of the self-energy Σ′′FL(ω)/Σ

′FL(ω) vanishes as ω→ 0 , and the

quasiparticle weight Z = (1− ∂Σ′/∂ω)−1 is finite at the Fermi surface. As a result, the

spectral function has a sharp peak in the vicinity of the Fermi energy, and quasiparticles are

well defined near the Fermi surface. However, in a MFL Σ′′(ω)/Σ

′(ω) diverges logarithmi-

cally as ω approaches zero while the quasiparticle weight vanishes logarithmically. Thus,

there are no well defined quasiparticles in the MFL framework which leads to quite differ-

ent physics from the Fermi liquid theory. For example, at low temperatures, the resistivity

and the specific heat scale with T as ρ ∼ ρ0 +BT and CV ∼ γT + νT lnT for a marginal

Fermi liquid while in a Fermi liquid ρ and CV scale with T as ρ∼ ρ0 +AT 2 and CV ∼ γT .

The MFL theory has been used to account for the anomalous normal state properties of

cuprates, such as the resistivity in the "strange-metal" region [30], the specific heat [33] and

the peak width of the momentum distribution curves [36]. From Eq. (1.9), a contribution

linear in T to the electrical resistivity is obtained, which is consistent with experiments at

optimal doping [38, 39, 40]. In 1999 and 2000, Valla et al. [34, 35] reported that in the

normal state of optimally doped Bi2212 the width of the Lorentzian-like momentum distri-

bution curves at the Fermi energy obtained via ARPES decreased linearly with temperature

for small binding energies, similar to the temperature dependence of its resistivity. Later

Abrahams et al. [36] proposed that this phenomena could be understood within a marginal

Fermi liquid framework.


Recently, Zhu et al. [37] proposed a weakly momentum dependent MFL zero tem-

perature self-energy to address universal features of the energy dispersion and linewidth

of the single-particle spectra for cuprates in the "strange metal" region [41, 42, 43]. The

temperature dependences of the resistivity and specific heat for a marginal Fermi liquid

differ from those of a Fermi liquid, hence we investigate a new ratio that is distinct from

the Kadowaki-Woods ratio in this case.

1.3.3 Hybrid self-energy

As discussed in Sec. 1.3.1, measurements of the in-plane resistivity of overdoped cuprates

reveal that ρab(T ) contains two temperature dependent components, one T -linear, the other

quadratic [16, 44, 45], which suggests that in overdoped cuprates an appropriate phe-

nomenological form for the self-energy may be [46]

Σ′′(ω,~k,T ) =

− h

2τ0− s0

h2ω2+(πkBT )2

h2ω2

c− λ(~k)

2 πkBT, h|ω|< kBT ,

− h2τ0− s0

h2ω2+(πkBT )2

h2ω2

c− λ(~k)

2 πh|ω|, kBT < h|ω|< hωc ,

−(

h2τ0

+ s0

)F(x)− λ(~k)

2 πhωc , |ω|> ωc ,

(1.11)

where x =√

h2ω2+(πkBT )2

hωc. The impurity scattering rate and the bare electron-electron scat-

tering rate are given by h/τ0 and 2s0/h respectively. The function F(x) decreases mono-

tonically as x approaches infinity with the boundary conditions, F(1) = 1, F(∞) = 0. λ(~k)

is a momentum dependent coupling function and ωc is a cutoff. The self-energy intro-

duced here is a combination of a Fermi liquid like piece that is independent of momentum

and an anisotropic marginal Fermi liquid like piece. Recently, Kokalj and McKenzie [46]

found that this type of self-energy can give a consistent quantitative description of results

from AMRO [14], specific heat [47], and ARPES [26] experiments on overdoped Tl2201

materials with the assumption that λ(~k) = λ0 cos2(2φ), where φ refers to in-plane angle in

momentum space. The temperature dependences of the resistivity and the specific heat in

this case are unlike those expected from a Fermi liquid. Besides the standard Kadowaki-

Woods ratio, we introduce a Kadowaki-Woods like ratio similar to the one that we define

for a MFL for this form of the self-energy too.


1.4 Outline of the work

The goal of the work here is to investigate the implications of various electron self-energies

that are not of the conventional Fermi liquid form on the Kadowaki-Woods ratio and ratios

that are similar to the Kadowaki-Woods ratio. In this thesis we study three kinds of momen-

tum dependent self-energy suggested by the experimental and theoretical evidence outlined

above, and discuss some results for several types of materials. In Chapter 2 we recall the

derivation for the standard Kadowaki-Woods ratio and derive Kadowaki-Woods like ratios

for the different self-energies introduced in this Chapter. In Chapter 3 we calculate specific

expressions for different classes of material and study the effects of momentum anisotropy

on the KWR. In Chapter 4 we take experimental data for heavy fermion materials that dis-

play marginal FL phenomenology and test whether relationships similar to the KWR hold

in this case. We present our conclusions in Chapter 5.

The main results of thesis are:

• Momentum anisotropy in the FL piece of the self-energy tends to reduce the value of

the KWR, while momentum anisotropy in the MFL piece of the self-energy tends to

enhance its value.

• Momentum anisotropy in the MFL piece of the self-energy does not have signifi-

cant effect on the renormalization of the KW-like ratio B/ν (which we introduce in

Chapter 2).

• For a reasonably anisotropic energy dispersion, one might hope to see changes in the

KWR as the chemical potential is varied.

• Exploration of the KW-like ratio B/ν using available data on materials with marginal

Fermi liquid phenomenology does not show clear universal behaviour. However,

there does seem to be some connection between this ratio and the size of the Fermi

surface. Further work will be required to explore this connection.

Chapter 2

Derivation of the Kadowaki-Woods andKadowaki-Woods-like ratios

In this Chapter we illustrate how the Kadowaki-Woods ratio and related ratios depend on

assumptions that are made about the form of the electron self-energy. As discussed in

the previous Chapter, experimental and theoretical considerations suggest that there are a

number of materials for which a simple Fermi liquid self-energy gives an inadequate de-

scription of their electronic and thermodynamic properties. This motivates us to look at

different forms of the self-energy allowing for momentum anisotropy and/or non-Fermi-

liquid temperature and energy dependence. There are many possible ways to introduce

momentum anisotropy to the electron self-energy Σ(ω,~k,T ) = Σ′(ω,~k,T ) + iΣ

′′(ω,~k,T )

(we use the standard notation Σ′

and Σ′′

to denote the real and imaginary parts of the self-

energy respectively from now on). Here we introduce three cases that cover both Fermi

liquid and non-Fermi liquid behaviours. Given the momentum dependent Σ′′, we use the

Kramers-Kronig relations [2, 6] to evaluate the real part of the self-energy Σ′.

Case IThe simplest case we consider is that the imaginary part of the self-energy takes the

conventional Fermi liquid form with a small momentum dependent correction (1+ εd(~k))

that multiplies the electron-electron interaction term. We will specify the form of d(~k) later.

The ε = 0 limit in this case corresponds to the self-energy considered by previous authors

discussing the Kadowaki-Woods ratio, e.g. Jacko et al. [9] or Miyake et al. [10]. In this

16

Derivation of the KWR and KWR-like ratios 17

case our expectation is that the temperature dependences of CV and ρ will not be altered,

but we can study how momentum anisotropy renormalizes the coefficients A and γ. We

take the imaginary part of the self-energy to be:

Σ′′(ω,~k,T ) =

− h2τ0− ω2+(πkBT )2

ω∗2s0(1+ εd(~k)), |ω2 +(πkBT )2|< ω∗2,

−(

h2τ0

+ s0(1+ εd(~k)))

F(x), otherwise,(2.1)

where x =

√ω2+(πkBT )2

ω∗ , and F(x) is a monotonically decreasing function of x with the

boundary conditions, F(1) = 1 and F(∞) = 0. The bare electron-electron scattering rate

is assumed to be 2s0/h . The impurity scattering rate is h/τ0 , and ω∗ is the renormalized

inverse density of states at the Fermi-surface which also acts as a cutoff [10]. Note that ω∗

could in principle be momentum dependent, but for simplicity we assume that it is momen-

tum independent.

Case IIThe marginal Fermi liquid (MFL) theory is a phenomenological theory that has been

suggested as a means to describe the normal state properties of cuprates in the "strange-

metal" region of the phase diagram [30, 31, 32]. The original proposal for the self-energy

of a MFL is given by [30]

Σ(ω,T )∼ g2N2(0)(

ω lnx

ωc− i

π

2x), (2.2)

where x = max(|ω|,T ), ωc is an ultraviolet cutoff, and g is a coupling constant.

Recently, Zhu et al. [37] proposed a modified self-energy for MFL which is weakly

momentum dependent. They pointed out that this type of self-energy can describe the

universal features of the dispersion and linewidth of the single-particle spectra in metallic

cuprates. We introduce the temperature dependence into the imaginary part of the self-

energy discussed by Zhu et al. [37] in a similar way to the isotropic in momentum case

shown in Eq. (2.2):

Σ′′(ω,~k,T ) =−λ(~k)

π

2

kBT, h|ω|< kBT,

h|ω|, kBT < h|ω|< hωc ,

hωc , otherwise,

(2.3)


where λ(~k) is a momentum dependent coupling constant.

As mentioned in Sec. 1.3.2, the temperature dependences of the resistivity and specific

heat for a marginal Fermi liquid differ from those of a Fermi liquid. This leads us to define

a new ratio that is distinct from the Kadowaki-Woods ratio in this case.

Case IIIAs discussed in Chapter 1, it has been suggested [46] that an appropriate phenomeno-

logical form for the imaginary part of the self-energy in overdoped cuprates may be

Σ′′(ω,~k,T ) =

− h

2τ0− s0

h2ω2+(πkBT )2

h2ω2

c− λ(~k)

2 πkBT, h|ω|< kBT,

− h2τ0− s0

h2ω2+(πkBT )2

h2ω2

c− λ(~k)


−(

h2τ0

+ s0

)F(x)− λ(~k)

2 πhωc , |ω|> ωc ,

(2.4)

which is a combination of a Fermi liquid like piece that is independent of momentum and

an anisotropic marginal Fermi liquid like piece. We will explore the combined effects of

non-Fermi liquid aspects of the self-energy and momentum anisotropy on the KWR and

the related ratio for MFLs.

In order to obtain the Kadowaki-Woods and related ratios, it is necessary to calculate

the low temperature forms of resistivity and specific heat for each of the self-energies given

above.

2.1 Case I

We now calculate the resistivity and specific heat at low temperatures for the form for

the self-energy specified in Eq. (2.1). This is basically the same calculation as performed

in Jacko et al. [9], but with a momentum dependent self-energy. We also allow for an

anisotropic Fermi surface.

2.1.1 In-plane resistivity

In order to calculate the coefficient A in the resistivity, we first need to calculate the con-

ductivity. To calculate the conductivity we will also need to bear in mind that there may be


anisotropy in the bandstructure. The components of the in-plane conductivity tensor in the

absence of vertex corrections have the form [48]

σdci j = he2

∫ d3~k(2π)3 vi(~k)v j(~k)

∫ dω

2πA2(ω,~k,T )

(−∂ f (ω)

∂ω

), (2.5)

where ~k = (kx,ky,kz) is the momentum and vi(~k) = h−1∂ε0(~k)/∂ki is the i th component

of velocity. A(ω,~k,T ) is the spectral function, which is related to the retarded Green’s

function via [48]

A(ω,~k,T ) =−2Im[GR(ω,~k,T )

]. (2.6)

The retarded Green’s function is given by [48]

GR(ω,~k,T ) =1

ω−ξk−Σ(ω,~k,T ), (2.7)

where Σ(ω,~k,T ) is the self-energy. ξk = ε0(~k)−µ0 is the difference between the unrenor-

malized particle energy ε0(~k) and the bare chemical potential µ0 . Using Eq. (2.7) we can

write the spectral function as

A(ω,~k,T ) =−2Σ

′′(ω,~k,T )(

ω−ξk−Σ′(ω,~k,T )

)2+(

Σ′′(ω,~k,T )2

) . (2.8)

In the vicinity of the Fermi surface, large quasiparticle lifetimes τ~k are expected for a Fermi

liquid, and we can send Σ′′(ω,~k,T ) to zero at low temperatures. In the limit Σ

′′(ω,~k,T )→

0 ,1

A2(ω,~k,T )→− 1

Σ′′(ω,~k,T )

2πZω(~k,T )δ(ω− (εk−µ)) . (2.9)

Here, Zω(~k,T ) is the quasiparticle renormalization factor, given by

Z−1ω (~k,T ) = 1− ∂Σ

′(ω,~k,T )∂ω

∣∣∣∣∣ω=εk−µ

, (2.10)

and εk−µ is the quasi-particle energy measured from chemical potential, which is given by

(εk−µ)−ξk−Σ′(εk−µ,T ) = 0, (2.11)

1See Appendix A for details.


which can be regarded as the defining equation for εk. Differentiating both sides of Eq. (2.11)

with respect to~k implies that [49]

δ(εk−µ) =Zk(~kF ,T )

Zω(~kF ,T )δ(ε0(~k)−µ0) , (2.12)

where

Z−1k (~kF ,T ) =

1hv0

F

∣∣∣∣∣h~v0F +

[∂Σ′(εk−µ,~k,T )

∂~k

]~k=~kF

∣∣∣∣∣' 1+

~v0F

h|v0F |2· ∂Σ

′(εk−µ,~k,T )

∂~k

∣∣∣∣∣~k=~kF

,

~v0F =

1h

∂(ε0(~k)−µ0)

∂~k

∣∣∣∣∣~k=~kF

. (2.13)

With the choice for the imaginary part of the self-energy Eq. (2.1), one may split Σ′′

into a momentum-independent part and a momentum-dependent part

Σ′′(ω,~k,T ) = Σ

′′0(ω,T )+ εΣ

′′1(ω,~k,T ). (2.14)

In the pure limit (τ0→∞) , one can easily find Σ′′1(ω,

~k,T ) = d(~k)Σ′′0(ω,T ) in the frequency

region |ω2+(πkBT )2|< ω∗2 . Noting that ε is a small parameter, we can expand− 1Σ′′(ω,~k,T )

as

− 1

Σ′′(ω,~k,T )

'− 1Σ′′0(ω,T )

+ εΣ′′1(ω,

~k,T )Σ′′20 (ω,T )

, (2.15)

to lowest order in ε.

Thus,

A2(ω,~k)' 2πZω(~k,T )δ(ω− (εk−µ))

(− 1

Σ′′0(ω,T )

+ εΣ′′1(ω,

~k,T )Σ′′20 (ω,T )

). (2.16)

As a result,

σi j = he2∫ d3k

(2π)3 vi(~k)v j(~k)∫ dω

2πA2(ω,~k,T )

(−∂ f (ω)

∂ω

)' he2(Ii j

0 + εIi j1 ), (2.17)


where

Ii j0 =

∫ d3k(2π)3 vi(~k)v j(~k)

∫dωZω(~k,T )δ(ω− (εk−µ))

(− 1

Σ′′0(ω,T )

)(−∂ f (ω)

∂ω

),

(2.18)

Ii j1 =

∫ d3k(2π)3 vi(~k)v j(~k)

∫dωZω(~k,T )δ(ω− (εk−µ))

(Σ′′1(ω,

~k,T )Σ′′20 (ω,T )

)(−∂ f (ω)

∂ω

).

(2.19)

We will evaluate Ii j0 and Ii j

1 separately in the following to determine expressions for σi j and

the coefficient A of the T 2 term in the resistivity. The pure limit, τ0→ ∞ , is used for the

following steps to simplify calculations.

i) Ii j0 :

At low temperatures −∂ f (ω)∂ω

has a sharp peak centered on ω = 0. Hence we replace ω

in the δ-function by 0 and take the δ-function out of the ω-integral. i.e.

Ii j0 '

∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,T )δ(ε0(~k)−µ0)

(∫α

−α

dω+∫ −α

−∞

dω+∫ +∞

α

dω

)×

(− 1

Σ′′0(ω,T )

)(−∂ f (ω)

∂ω

), (2.20)

where α2 +(πkBT )2 = ω∗2 .

At low temperatures the contribution to Ii j0 from the region with |ω| > α is small [9].

We can use the form of Σ′′0(ω,T ) in the region with |ω|< α for all ω with negligible error.

Thus, in the pure limit

Ii j0 '


∫ +∞

−∞

dωω∗2

s0(ω2 +(πkBT )2)

(−∂ f (ω)

∂ω

).

(2.21)

Given 2 ∫ +∞

−∞

dωω∗2

s0(ω2 +(πkBT )2)

(−∂ f (ω)

∂ω

)T 6=0' ω∗2

12s0(kBT )2 , (2.22)

we can approximate Ii j0 as

Ii j0

T 6=0' ω∗2

12s0(kBT )2

∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,T )δ(ε0(~k)−µ0). (2.23)

2See Appendix B for details.


ii) Ii j1 :

We note that the integrand of Ii j1 will be sharply peaked at ω = 0 when T is low just as

for Ii j0 . Hence we may again replace ω in the δ-function by 0 and take the δ-function out of

the ω-integral.

Ii j1 '


∫α

−α

dωd(~k)

Σ′′0(ω,T )

(−∂ f (ω)

∂ω

)

+

(∫ −α

−∞

dω+∫ +∞

α

dω

)(Σ′′1(ω,

~k,T )Σ′′20 (ω,T )

)(−∂ f (ω)

∂ω

). (2.24)

As for Ii j0 , at low temperatures we may ignore the contributions of the integral from high

frequency regions and approximate Ii j1 as

Ii j1

T 6=0' − ω∗2

12s0(kBT )2

∫ d3k(2π)3 vi(~k)v j(~k)d(~k)Zk(~kF ,T )δ(ε0(~k)−µ0). (2.25)

Thus, when T 6= 0

σi j(T )' he2 ω∗2

12s0(kBT )2

∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,T )

(1− εd(~k)

)δ(ε0(~k)−µ0).

When T = 0, −∂ f (ω)∂ω

= δ(ω), then,

σi j(0)' 2e2τ0

∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)

(1− εd(~k)

)δ(ε0(~k)−µ0). (2.26)

where τ0 is the inverse of the impurity scattering rate.

For a Fermi liquid, the low temperature resistivity varies with T as

ρ(T ) =1

σi j(T )= ρ0 +AT 2, (2.27)

where ρ0 arises from quasiparticle impurity scattering at zero temperature. The coefficient

A in the resistivity is then given by

A =ρ(T )−ρ0

T 2 =

1σi j(T )

− 1σi j(0)

T 2 . (2.28)


In the pure limit, τ0→∞ , which would imply σi j(0)→∞ . From our previous calculations

of the conductivity, we may read off that

A =1

σi j(T )T 2

=12s0k2

Bhe2ω∗2

1∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)

(1− εd(~k)

)δ

(ε0(~k)−µ0

) . (2.29)

2.1.2 Linear Coefficient of The Heat Capacity

For a Fermi liquid, the specific heat can be written as

CV =π2k2

BT3V ∑

~kσ

δ(εk−µ)

=2π2k2

BT3

∫ d3k(2π)3

Zk(~kF ,T )

Zω(~kF ,T )δ(ε0(~k)−µ0) . (2.30)

To calculate CV we need to know the real part of the self-energy, which can be evaluated

from Σ′′

by applying the Kramers-Kronig relation [2, 6],

Σ′(ω,~k,T )−Σ

′(∞,~k,T ) =

1π

P∫ +∞

−∞

dω′Σ′′(ω′,~k,T )−Σ

′′(∞,~k,T )

ω′−ω, (2.31)

where P represents the Cauchy principal part of the integral.

In the pure limit (τ0→ ∞)

Σ′′(ω,~k,T ) =

−s(~k)ω2+(πkBT )2

ω∗2, |ω2 +(πkBT )2|< ω∗2,

−s(~k)F(√

ω2+(πkBT )2

ω∗

), otherwise,

(2.32)

where s(~k)≡ s0

(1+ εd(~k)

). The function F is a monotonically decreasing function of ω,

which vanishes as ω→ 0. Hence Σ′′(∞,~k,T ) is equal to zero. Assuming that the singularity


at ω′ = ω is in the region −α≤ ω′ ≤ α, we have

1π

P∫ +∞

−∞

dω′Σ′′(ω′,~k,T )−Σ

′′(∞,~k,T )

ω′−ω=

1π

P∫ +∞

−∞

dω′Σ′′(ω′,~k,T )ω′−ω

= −s(~k)π

P∫ +α

−α

dω′ω′2 +(πkBT )2

ω∗2(ω′−ω)

−s(~k)π

(∫ −α

−∞

dω′+

∫ +∞

+α

dω′) F

(√ω′2+(πkBT )2

ω∗

)ω′−ω

= J1 + J2, (2.33)

where

J1 ≡ − s(~k)πω∗2

P∫ +α

−α

dω′ω′2 +(πkBT )2

ω′−ω,

J2 ≡ −s(~k)π

(∫ −α

−∞

dω′+

∫ +∞

+α

dω′)

F(x)ω′−ω

. (2.34)

The parameter x in J2 is defined as x =√

ω′2+(πkBT )2

ω∗ . In the following we evaluate J1 and

J2 separately to obtain the real part of the self-energy.

i) Based on the definition of Cauchy principal part of the integral, J1 can be evaluated to be

J1 =−s(~k)πω∗2

2ωα+

[ω

2 +(πkBT )2] ln(

α−ω

α+ω

), (2.35)

where α =√

ω∗2− (πkBT )2 .

ii) To evaluate J2 , we first expand 1ω′−ω

as

1ω′−ω

=1ω′

(1+

ω

ω′+(

ω

ω′

)2+ ...

). (2.36)

Noting that the frequency interval is symmetric about ω′ = 0 , we only need to keep even

order terms of ω′ in the Taylor series, which leads to

J2 =−2s(~k)

π

∫ +∞

+α

dω′F(x)

ω′

+∞

∑n=0

(ω

ω′

)2n+1. (2.37)

As a result,

1π

P∫ +∞

−∞

dω′Σ′′(ω′,~k,T )ω′−ω

= − s(~k)πω∗2

2ωα+

[ω

2 +(πkBT )2] ln(

α−ω

α+ω

)−2s(~k)

π

∫ +∞

+α

dω′F(x)

ω′

+∞

∑n=0

(ω

ω′

)2n+1. (2.38)


The real part of self-energy is then given by

Σ′(ω,~k,T ) = Σ

′(∞,~k,T )− s(~k)

πω∗2

2ωα+

[ω

2 +(πkBT )2] ln(

α−ω

α+ω

)−2s(~k)

π

∫ +∞

+α

dω′F(x)

ω′

+∞

∑n=0

(ω

ω′

)2n+1. (2.39)

The renormalization constant Zω(~kF ,T ) and parameter Z−1k (~kF ,T ) can then be written as

Z−1ω (~kF ,T ) = 1− ∂Σ

′(ω, ~kF ,T )

∂ω

∣∣∣∣∣ω=0

= 1+2s(~kF)

πω∗2

(α− (πkBT )2

α

)+

2s(~kF)

π

∫ +∞

+α

dω′F(x)

ω′2(2.40)

Z−1k (~kF ,T ) =

1hv0

F

∣∣∣∣∣h~v0F +

[∂Σ′(εk−µ,~k,T )

∂~k

]~k=~kF

∣∣∣∣∣=

1hv0

F

∣∣∣∣∣h~v0F +

[∂Σ′(∞,~k,T )

∂~k

]~k=~kF

∣∣∣∣∣ . (2.41)

Thus,

CV =2π2k2

BT3

∫ d3k(2π)3 Zk(~kF ,T )δ(ε0(~k)−µ0)

1+

2s(~kF)

πω∗2

(α− (πkBT )2

α

)

+2s(~kF)

π

∫ +∞

+α

dω′F(x)

ω′2

. (2.42)

Finally, the coefficient of the linear term in the temperature dependence of the specific heat

can be written as

γ =2π2k2

B3

∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)

1+

2s(~kF)

πω∗+

2s(~kF)

π

∫ +∞

ω∗dω′F( ω′

ω∗ )

ω′2

.

(2.43)

This may be simplified to

γ =2π2k2

B3

∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)

1+

4ξs0(1+ εd(~kF))

πω∗

, (2.44)

where 2ξ = 1+∫ +∞

1dy

F(y)y2 ≤ 1+

∫ +∞

1dy

1y2 = 2 since F(y) ≤ 1 for y ≥ 1. Noting that

F(y) decreases slowly as y→+∞, we expect ξ≈ 1 for a reasonable form of function F(y) .


2.1.3 Kadowaki-Woods ratio

We have now determined explicit expressions for the coefficients A and γ , we now proceed

to construct the Kadowaki-Woods ratio. Both the expressions for A and γ contain the factor

s0. In strongly correlated systems, s0 ∼ 2n/3πD0, where D0 is the bare density of states at

the Fermi surface and n is the density of conduction electrons [10]. The effective mass m∗

can be much larger than the bare mass m0, in which case s0 ω∗ [9]. As a result

A ' 8nk2B

πhe2ω∗2D0

1∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)

(1− εd(~k)

)δ(ε0(~k)−µ0)

,

γ ' 16nk2Bξ

9ω∗D0

∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)(1+ εd(~kF)). (2.45)

The Kadowaki-Woods ratio becomesAγ2 '

81D0

32πhk2Be2nξ2

1XP2

1+ ε

(YX− 2Q

P

), (2.46)

where we have kept only the lowest order correction to the KWR from momentum anisotropy,

i.e. the O(ε) term, and we have defined

X =∫ d3k

(2π)3 vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0) ,

Y =∫ d3k

(2π)3 vi(~k)v j(~k)Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0) ,

P =∫ d3k

(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0) ,

Q =∫ d3k

(2π)3 Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0) . (2.47)

2.2 Case II

We now turn to consider the second case mentioned earlier, an anisotropic marginal Fermi

liquid. For an anisotropic marginal Fermi liquid, the imaginary part of the momentum

dependent self-energy can be written as [10, 37]

Σ′′(ω,~k,T ) =−λ(~k)

π

2

kBT, h|ω|< kBT ,

h|ω|, kBT < h|ω|< hωc ,

hωc, otherwise .

(2.48)


As mentioned previously, this type of self-energy leads to quite different forms of the in-

plane resistivity and the specific heat to those found for a Fermi liquid. The low temperature

resistivity in this case has a contribution which is proportional to T [30] . The specific heat

at low temperatures has the form of γT +νT lnT as compared to the Fermi liquid form of

γT +δT 3 .

In the following, we will evaluate the coefficient of the T-linear term of the resistivity

and the coefficients (γ and ν) of the specific heat based on the self-energy proposed in

Eq. (2.48) .


Here we evaluate the conductivity first. As in Sec.2.1.1, the components of the in-plane

conductivity tensor have the form

σi j(T ) = he2∫ d3k

(2π)3 viv j

∫ dω

2πA2(ω,~k)

(−∂ f (ω)

∂ω

). (2.49)

Substituting Eq. (2.48) into the expression for the conductivity we get

σi j(T ) =4he2

π

∫ d3k(2π)3 viv jZk(~kF ,T )δ(ε0(~k)−µ0)

1

λ(~k)

1

kBT

∫ kBTh

0dω

(−∂ f (ω)

∂ω

)+∫

ωc

kBTh

dω1

hω

(−∂ f (ω)

∂ω

)+

∫ +∞

ωc

dω1

hωc

(−∂ f (ω)

∂ω

), (2.50)

which we can simplify to

σi j(T ) =he2

πkBT


1

λ(~k)

2(e−1)

e+1

+2kBThωc

[1− tanh

(hωc

2kBT

)]+

∫ hωckBT

1dy

sech2( y2)

y

, (2.51)

where y = hω/kBT .

Since we are interested in temperatures kBT < hωc, the term 2kBThωc

(1− tanh

(hωc

2kBT

))is

negligible compared with the remaining terms and so we may write

σi j(T )'he2

πkBT

(0.924+

∫ hωckBT

1dy

sech2( y2)

y

)∫ d3k(2π)3

vi(~k)v j(~k)Zk(~kF ,T )

λ(~k)δ(ε0(~k)−µ0) .

(2.52)


At low temperatures, we may expand the resistivity to the lowest order in temperature as

ρ(T ) =1

σi j(T )∼ ρ0 +BT +O(T 2) , (2.53)

which leads to

B =πkB

βhe21∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (2.54)

where

β = 0.924+∫ +∞

1dy

sech2( y2)

y∼ 1.523 . (2.55)

2.2.2 Heat Capacity

As discussed in Sec. 2.1.2 , the low temperature specific heat CV can be written as

CV =2π2k2

BT3

∫ d3k(2π)3

Zk(~kF ,T )

Zω(~kF ,T )δ(ε0(~k)−µ0) . (2.56)

As noted previously, the real part of the self-energy can be evaluated by applying the

Kramers-Kronig relations. Noting that Σ′′(∞,~k,T ) has no frequency dependence we have

1π

P∫

∞

∞

dω′Σ′′(∞,~k,T )ω′−ω

= 0 . (2.57)

As a result,

Σ′(ω,~k,T ) = Σ

′(∞,~k,T )+

1π

P∫ +∞

−∞

dω′Σ′′(ω′,~k,T )−Σ

′′(∞,~k,T )

ω′−ω

= I1 + I2, (2.58)

where

I1 = Σ′(∞,~k,T )− λ(~k)h

2P∫

ωc

kBTh

dω′ ω′

ω′−ω,

I2 =λ(~k)h

2

∫ − kBTh

−ωc

dω′ ω′

ω′−ω− λ(~k)hωc

2

(∫ −ωc

−∞

dω′+

∫ +∞

ωc

dω′)

1ω′−ω

−λ(~k)kBT2

∫ kBTh

− kBTh

dω′ 1ω′−ω

. (2.59)


Note that we have assumed the singularity at ω′=ω lies in the region kBT/h<ω<ωc. One

obtains the same result for Σ′(ω,~k,T ) when the singularity lies in the region h|ω|< kBT .

i) Let us evaluate the term I1 first. According to the definition of Cauchy principle value,

we have

I1 = Σ′(∞,~k,T )− λ(~k)h

2limδ→0

(∫ω−δ

kBTh

dω′+

∫ωc

ω+δ

dω′)(

1+ω

ω′−ω

)= Σ

′(∞,~k,T )− λ(~k)h

2

(ωc−

kBTh

)− λ(~k)hω

2ln

∣∣∣∣∣ ωc−ω

ω− kBTh

∣∣∣∣∣ . (2.60)

ii) The term I2 can be evaluated straightforwardly as

I2 =λ(~k)h

2

(ωc−

kBTh

)+

λ(~k)hω

2ln

∣∣∣∣∣ω+ kBTh

ω+ωc

∣∣∣∣∣− λ(~k)hωc

2ln∣∣∣∣ω+ωc

ω−ωc

∣∣∣∣−λ(~k)kBT

2ln∣∣∣∣ hω− kBThω+ kBT

∣∣∣∣ . (2.61)

Thus, the real part of the self-energy has the form

Σ′(ω,~k,T ) = Σ

′(∞,~k,T )− λ(~k)hωc


ω−ωc

∣∣∣∣− λ(~k)kBT2

ln∣∣∣∣ hω− kBThω+ kBT

∣∣∣∣−λ(~k)hω

2ln

∣∣∣∣∣ ω2−ω2c

ω2− (kBTh )2

∣∣∣∣∣ . (2.62)

The renormalization constant Zω(~kF ,T ) is then given by

1

Zω(~kF ,T )= 1− 1

h∂Σ′(ω, ~kF ,T )

∂ω

∣∣∣∣∣ω=0

= 1+λ(~kF) ln(

hωc

kBT

). (2.63)

This leads to

CV =2π2k2

B3

∫ d3k(2π)3 Zk(~kF ,T )δ(ε0(~k)−µ0)

T +λ(~kF)T ln

(hωc

kBT

). (2.64)

Writing CV as

CV = γT +νT ln(

T0

T

)+ ... , (2.65)


we have

γ =2π2k2

B3

∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0),

ν =2π2k2

B3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0) . (2.66)

The coefficient γ here differs from that found in Sec. 2.1.2 , where there is an extra term of

order 4ξs(~k)/πω∗ arising from anisotropic Fermi liquid self-energy. In this case, any mo-

mentum anisotropy in the self-energy enters the coefficient of the correction term T ln(T0/T )

of the low temperature specific heat through the parameter λ(~kF). Also note that T0 =

hωc/KB is a function of the energy cutoff ωc.

2.2.3 Generalizations of the KWR for marginal Fermi liquids

In the section above we determined the coefficient B of the T -linear term of the resistivity,

the coefficient γ of the T -linear term and the coefficient ν of the T ln(T0/T ) term of the

specific heat. In contrast to a Fermi liquid, the low temperature resistivity and specific

heat of a marginal Fermi liquid have extra temperature dependent terms, i.e. BT in the

resistivity and νT ln(T0/T ) in the specific heat. Thus, it is of interest to investigate whether

there exists an ratio, involving one or both of the coefficients of the two extra terms, that

has a similar behaviour as the KWR. Thus, in analogy with the Kadowaki-Woods ratio we

define a new ratio for marginal Fermi liquids of

Bν

=

2πhkBβe2

3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

×∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

−1

. (2.67)

The choice B/ν is suggested by the observation that in the limit of isotropic self-energy, the

ratio is independent of λ, and it has very similar momentum integrals as the KWR for a FL

when λ is momentum independent. In Chapter 4, we use experimental data from materials

with MFL phenomenology to test whether B and ν are related in a manner analogous to the

relationship between A and γ2 via the KWR.


2.3 Case III : hybrid self energy

The self energies considered in Case I and Case II allow for momentum dependence within

the framework of either regular Fermi liquid theory or marginal Fermi liquid theory. A

phenomenological hybrid of these two that combines isotropic Fermi liquid behaviour

and anisotropic marginal Fermi liquid behaviour, was recently suggested by Kokalj and

McKenzie [46]. In the pure limit, Eq. (2.4) becomes

Σ′′(ω,~k,T ) =

−s0

h2ω2+(πkBT )2

h2ω2

c− λ(~k)

2 πkBT, h|ω|< kBT ,

−s0h2

ω2+(πkBT )2

h2ω2

c− λ(~k)


−s0F(x)− λ(~k)2 πhωc, |ω|> ωc ,

(2.68)

where x =√

h2ω2+(πkBT )2

hωcand λ(~k) = λ0 cos2(2φ). Note that φ refers to the in-plane angle

in momentum spaces.


From Sec. 2.1.1 we know the components of the in-plane conductivity tensor have the form

σi j ' he2∫ d3k

(2π)3 viv jZk(~kF ,T )δ(ε0(~k)−µ0)∫

dω

(− 1

Σ′′(ω,~k,T )

)(−∂ f (ω)

∂ω

).

Substituting the expression for Σ′′(ω,~k,T ) into the above equation and letting y = hω

kBT , we

have

σi j =he2

2


∫ 1

0dy

h2ω2

csech2( y2)

s0k2BT 2(y2 +π2)+ λ(~k)

2 πkBT h2ω2

c

+∫ hωc

kBT

1dy

h2ω2

csech2( y2)

s0k2BT 2(y2 +π2)+ λ(~k)

2 πkBT h2ω2

cy+

∫ +∞

hωckBT

dysech2( y

2)

s0F(x)+ λ(~k)2 πhωc

.

(2.69)


Assuming that F(1) = 1;F(∞) = 0, and F(x) decreases sufficiently slowly as x→ ∞, we

have

0 <∫ +∞

hωckBT

dysech2( y

2)

s0F(x)+ λ(~k)2 πhωc

<∫ +∞

hωckBT

dysech2( y

2)

λ(~k)2 πhωc

=4

λ(~k)πhωc

(1− tanh

(hωc

2kBT

)).

(2.70)

When the temperature T is small enough, 4λ(~k)πhωc

(1− tanh

(hωc

2kBT

))→ 0. We may then

drop the term involving F(x) in the expression of σi j to obtain

σi j 'he2

2


∫ 1

0dy

h2ω2

csech2( y2)

s0k2BT 2(y2 +π2)+ λ(~k)

2 πkBT h2ω2

c

+∫ hωc

kBT

1dy

h2ω2

csech2( y2)

s0k2BT 2(y2 +π2)+ λ(~k)

2 πkBTyh2ω2

cy

. (2.71)

We can use this expression to obtain an expansion of the in-plane resistivity in powers of

temperature by assuming that λ(~k)/T is finite as the temperature goes to zero, i.e. the

marginal Fermi liquid piece of the self-energy is comparable to the Fermi liquid one:

ρ(T ) =1

σi j(T )∼ ρ0 +BT +AT 2 + ... , (2.72)

where we have

A =2kB

h3e2ω2cβ2

1(∫ d3k(2π)3

vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

)2

×

s0kBη

∫ d3k(2π)3

vi(~k)v j(~k)

λ2(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

− πh2ω2

cβ

2

∫ d3k(2π)3

vi(~k)v j(~k)

λ(~k)

∂Zk(~kF ,T )∂T

∣∣∣∣∣T=0

δ(ε0(~k)−µ0)

,

B =πkB

βhe21∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (2.73)


with

β =∫ 1

0dysech2

(y2

)+

∫ +∞

1dy

sech2( y2)

y' 1.52 ,

η =∫ 1

0dy(y2 +π

2)sech2(y

2

)+

∫ +∞

1dy(

1+π2

y2

)sech2

(y2

)' 14.28 . (2.74)

Note that the expression obtained for the coefficient A here differs from that found in

Sec. 2.1.1 in that it includes integrals over λ(~k) that are not present in Eq. (2.29).

The expansion of the in-plane resistivity around T = 0, given by Eq. (2.72), is appropri-

ate for temperatures such that λ(~k)/T > 1. However, it is of interest to investigate the co-

efficients A and B when λ(~k)/T vanishes as the temperature goes to zero, i.e. the marginal

Fermi liquid piece of the self-energy is small compared to the Fermi liquid part. From

Eq. (2.71), we find that expanding the in-plane resistivity around λ(~k)/T = 0 leads to

A =2s0k2

B

h3e2ω2cβ′

1∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

2,

B =πkBη′he2β′2

1(∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

)2

×∫ d3k

(2π)3 vi(~k)v j(~k)λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0) , (2.75)

with

β′ =

∫∞

0dy

sech2( y2)

y2 +π2 =16,

η′ =

∫ 1

0dy

sech2( y2)

(y2 +π2)2 +∫

∞

1dyy

sech2( y2)

(y2 +π2)2 ' 0.02 . (2.76)

Note that the expression obtained for the coefficient A here will be exactly the same as that

found in Sec. 2.1.1 when there is no momentum anisotropy in the self-energy in Sec. 2.1.1.


2.3.2 Heat Capacity

As before, we use the Kramers-Kronig relations [2] to evaluate the real part of the self-

energy, focusing on low frequencies h|ω|< kBT , which leads to

Σ′(ω,~k,T ) = Σ

′(∞,~k,T )− λhωc


ω−ωc

∣∣∣∣− λkBT2

ln∣∣∣∣ hω− kBThω+ kBT

∣∣∣∣−λhω

2ln

∣∣∣∣∣ ω2−ω2c

ω2− (kBTh )2

∣∣∣∣∣− 2s0

π

∫ +∞

ωc

dω′F(x)

ω′

+∞

∑n=0

(ω

ω′

)2n+1

− s0

πh2ω2

c

2h2

ωωc +(h2

ω2 +(πkBT )2) ln

∣∣∣∣ω−ωc

ω+ωc

∣∣∣∣ , (2.77)

where x =√

h2ω′2+(πkBT )2

hωc.

Thus, the specific heat CV has the form

CV =2π2k2

BT3

∫ d3k(2π)3 Zk(~kF ,T )δ(ε0(~k)−µ0)

1+λ(~kF) ln

(hωc

kBT

)+

2s0

πhωc

(1− (πkBT )2

h2ω2

c

)+

2s0

πh

∫ +∞

ωc

dω′F(x)

ω′2

, (2.78)

Writing CV in the form

CV = γT +νT ln(

T0

T

)+ ... , (2.79)

we have

γ =2π2k2

B3

(1+

4s0ξ

πhωc

)∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0) ,

ν =2π2k2

B3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0) . (2.80)

where 2ξ = 1+∫ +∞

1dy

F(y)y2 ' 2 . The coefficient γ found here agrees with that found in

Sec. 2.1.2 in the isotropic ε = 0 limit, but differs from that found in the straight anisotropic

marginal Fermi liquid case with a extra term of order 4s0ξ/πhωc arising from the isotropic

Fermi liquid piece of the self-energy. On the other hand, the coefficient ν is exactly the

same as that found in Sec. 2.2.2 .


2.3.3 Kadowaki-Woods and related ratios

In the mixed FL + anisotropic MFL case we have calculated the coefficients of the T 2 term

and the T -linear term of the resistivity to be A and B respectively and the coefficients of the

T -linear term and the T lnT term of the specific heat to be γ and ν respectively.

When the marginal Fermi liquid piece of the self-energy is comparable to the Fermi

liquid one, the contribution of the marginal Fermi liquid piece of the self-energy enters the

coefficient A through λ(~k) even though A is normally viewed as arising from a Fermi liquid.

However, when the marginal Fermi liquid piece of the self-energy acts as a small correction

to the Fermi liquid self-energy, the coefficient A purely arise from the Fermi liquid piece.

In the specific heat, the coefficients γ and ν purely arise from the Fermi liquid piece and the

marginal Fermi liquid piece of the self-energy respectively, which is consistent with results

in Sec. 2.1.2 and Sec. 2.2.2 .

When the two parts of the hybrid self-energy are comparable, we may write down the

ratio

Bν

=

2πhkBβe2

3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

×∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

−1

, (2.81)

which is the same as that introduced in Sec. 2.2.3 . In this case, it seems that this ratio arises

from only the marginal Fermi liquid piece of the self-energy and has nothing to do with the

normal Fermi liquid part of the self-energy. Note that B/ν2 is another possible natural ratio

to investigate as mentioned in Sec.2.2.3 .

As for the Kadowaki-Woods ratio, it can be very different from that found in Sec. 2.1.3.

The KWR seems to be ωc dependent since the two terms in the coefficient A have different

ωc dependence. In strongly correlated systems with large effective mass m∗, the coefficient

γ is approximately equal to

γ =16ξnk2

B9hωcD0

∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0) , (2.82)

with the approach made in Sec. 2.1.3. As a result, the KWR in the mixed FL + anisotropic


MFL case is given by

Aγ2 =

81D20

128he2β2ξ2n2k3B

1(∫ d3k(2π)3

vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

)2

×

2nkBη

3πD0

∫ d3k(2π)3

vi(~k)v j(~k)

λ2(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

− πh2ω2

cβ

2

∫ d3k(2π)3

vi(~k)v j(~k)

λ(~k)

∂Zk(~kF ,T )∂T

∣∣∣∣∣T=0

δ(ε0(~k)−µ0)

× 1(∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)

)2 . (2.83)

However, when the marginal Fermi liquid piece of the self-energy is fairly small compared

to the Fermi liquid part, the KWR is independent of λ(~k). A/γ2 and the ratio B/ν in the

mixed FL + anisotropic MFL case are then given by

Aγ2 =

27D0

64πβ′hne2k2Bξ2

1∫ d3k(2π)3 vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

× 1(∫ d3k(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)

)2 ,

Bν

=3η′

2πhkBe2β′21(∫ d3k

(2π)3 vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

)2

∫ d3k(2π)3 λ(~k)vi(~k)v j(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)∫ d3k

(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (2.84)

with

β′ =

∫∞

0dy

sech2( y2)

y2 +π2 =16

η′ =

∫ 1

0dy

sech2( y2)

(y2 +π2)2 +∫

∞

1dyy

sech2( y2)

(y2 +π2)2 ' 0.02 . (2.85)


2.4 Summary

In this chapter we have calculated the Kadowaki-Woods ratio and KW-like ratios in terms

of integrals over the Fermi surface for three different types of momentum dependent elec-

tron self-energies which are beyond the simple Fermi liquid model. In the next chapter

we will calculate these integrals for several model bandstructures which are appropriate for

layered materials or three dimensional systems. The effects of bandstructures and momen-

tum anisotropy in the self-energy on the KWR and KW-like ratios will be presented in the

next chapter too.

Chapter 3

Kadowaki-Woods and Kadowaki-Woodslike ratios for specific bandstructures

In Chapter 2 we discussed how the Kadowaki-Woods ratio and Kadowaki-Woods-like ra-

tios depend on the form of the electron self-energy and calculated these ratios in terms of

integrals over the Fermi surface. In this chapter we evaluate these Fermi surface integrals

and hence the defined ratios for several example bandstructures, appropriate for materials

with either a quasi two dimensional electronic structure or materials with a three dimen-

sional electronic structure. Our focus is to understand how anisotropy of the self-energy in

momentum space renormalizes these ratios.

We start by considering the KWR and its generalization for a marginal Fermi liquid for

a momentum independent self-energy before studying the effects of momentum anisotropy

on these ratios. We first consider three dimensional systems and then quasi two dimensional

systems. Note that the calculation of the KWR for both three dimensional and quasi two

dimensional materials with momentum independent self-energies and isotropic dispersions

only was performed previously by a number of authors [9, 10, 11]. We introduce these

results for reference purposes.

38

The KWR and KWR-like ratios for specific bandstructures 39

3.1 Three dimensional systems

In this section we investigate the ratios introduced in Chapter 2 for materials with a three

dimensional dispersion first assuming that the dispersion is isotropic and the electronic self-

energy to be momentum independent for cases I and II (noting that momentum anisotropy

in the self-energy in case III is crucial to give a consistent description of various experi-

mental results on overdoped cuprates), and then allowing for an anisotropic dispersion and

momentum anisotropy in the self-energy for cases I to III.

3.1.1 Momentum independent self-energy and isotropic Fermi surface

For a three dimensional material with an isotropic dispersion, the bare density of states at

the Fermi level is given by

D0 = 2∫ d3k

(2π)3 δ

(ε0(~k)−µ0

)=

m0kF

π2h2 , (3.1)

and the in-plane average of vi(~k)v j(~k) over the Fermi surface is

〈vi(~k)v j(~k)〉 =

∫ d3k(2π)3 vi(~k)v j(~k)δ(ε0(~k)−µ0)∫ d3k

(2π)3 δ(ε0(~k)−µ0)

=h2k2

F

3m20. (3.2)

Case I

The expression for the Kadowaki-Woods ratio for three dimensional materials with a mo-

mentum independent self-energy and an isotropic energy dispersion is well known. For a

momentum independent self-energy the factor Zk(~k,T ) is equal to 1. As a result, the KWR


defined in Eq. (2.46) can be written as [9]

Aγ2 =

814πhk2

Be2ξ2

1

〈vi(~k)v j(~k)〉D20n

=35π3h

4k2Bk4

Fe2ξ2n. (3.3)

Case II

Since the self-energy is momentum independent, the ratio B/ν introduced in Sec. 2.2.3 has

the form

Bν

=6

πhkBβe21

D20〈vi(~k)v j(~k)〉

=18π3h

βkBk4Fe2

. (3.4)

Note that the ratio B/ν has the same momentum integral as the KWR for 3 dimensional ma-

terials with a momentum independent self-energy, as inferred from Eq. (3.3) and Eq. (3.4) .

3.1.2 Momentum dependent self-energy and anisotropic Fermi sur-face

In a cubic lattice model, the tight binding energy dispersion including only nearest neigh-

bour hopping has the form

ε0(~k) =−2t1cos(kx)−2t2cos(ky)−2t3cos(kz) , (3.5)

where t1, t2, t3 are the hopping magnitudes along x, y, and z-directions respectively. Note

that the wave numbers are measured in units of the lattice constants. Eq. (3.5) implies

vx =1h

∂ε0(~k)∂kx

=2t1h

sin(kx) ,

vy =1h

∂ε0(~k)∂ky

=2t2h

sin(ky) ,

vz =1h

∂ε0(~k)∂kz

=2t3h

sin(kz) , (3.6)

for the quasiparticle velocities along different directions.


Case I

The KWR introduced in Sec. 2.1.3 has a form

Aγ2 '

81D0

32πhk2Be2nξ2

1XP2

1+ ε

(YX− 2Q

P

), (3.7)

where we have defined

X =∫ d3k


Y =∫ d3k


P =∫ d3k

(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0) ,

Q =∫ d3k

(2π)3 Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0) . (3.8)

When measuring the resistivity along the z-direction, we have

X =∫ d3k

(2π)3 v2z (~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

=1

(2π)3h

∫dkxdky|vz(~k)|Zk(~kF ,0)δ(kz− kzi) , (3.9)

where kzi is the ith root of f (kz) = ε0(~k)−µ0 :

kzi = ±cos−1(−2t1cos(kx)−2t2cos(ky)−µ0

2t3

)= ±cos−1(κ) , (3.10)

with

κ =−2t1cos(kx)−2t2cos(ky)−µ0

2t3. (3.11)

By substituting the expression for kzi into Eq. (3.9) we have

X =4h2

t3(2π)3

∫dkxdky

√1−κ2Zk(kx,ky,k+zi ) , (3.12)

where k+zi = cos−1(κ) .


Following the same procedure we have

Y =4h2

t3(2π)3

∫dkxdky

√1−κ2Zk(kx,ky,k+zi )d(kx,ky,k+zi )

P =1

(2π)3t3

∫dkxdkyZk(kx,ky,k+zi )

1√1−κ2

Q =1

(2π)3t3

∫dkxdkyZk(kx,ky,k+zi )d(kx,ky,k+zi )

1√1−κ2

. (3.13)

When measuring the resistivity along some in-plane direction (e.g. the x-direction) we have

X =∫ d3k

(2π)3 v2x(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

=1

(2π)3t3

∫dkxdkyv2

xZk(kx,ky,k+zi )1√

1−κ2

Y =∫ d3k

(2π)3 v2x(~k)Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0)

=1

(2π)3t3

∫dkxdkyv2

xZk(kx,ky,k+zi )d(kx,ky,k+zi )1√

1−κ2

P =∫ d3k

(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0)

=1

(2π)3t3


1√1−κ2

Q =∫ d3k

(2π)3 Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0)

=1

(2π)3t3


1√1−κ2

. (3.14)

Case II

For a momentum dependent marginal Fermi liquid self-energy, the ratio introduced in

Sec. 2.2.3 is

Bν

=

2πhkBβe2

3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

×∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

−1

, (3.15)


where β ' 1.52 is a constant introduced in Chapter 2. For the tight binding dispersion

specified in Eq. (3.5) we have

ν =2π2k2

B3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

=2π2k2

B3

1(2π)3t3

∫dkxdkyλ(kx,ky,k+zi )Zk(kx,ky,k+zi )

1√1−κ2

, (3.16)

where k+zi = cos−1(κ) and κ = (−2t1cos(kx)−2t2cos(ky)−µ0)/2t3 .

When measuring the resistivity along the z-direction we have

B−1 =βhe2

πkB

∫ d3k(2π)3

v2z (~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

=βhe2

πkB

4h2

t3(2π)3

∫dkxdky

√1−κ2

λ(kx,ky,k+zi )Zk(kx,ky,k+zi ) . (3.17)

The ratio B/ν is then given by(Bν

)−1

=8βπkBhe2

31

(2π)6h2


1√1−κ2

×∫

dkxdky

√1−κ2


When measuring the resistivity along x-direction we have

B−1 =βhe2

πkB

∫ d3k(2π)3

v2x(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

=βhe2

πkB

1(2π)3t3

∫dkxdky

v2x

λ(kx,ky,k+zi )Zk(kx,ky,k+zi )

1√1−κ2

. (3.19)

Thus, (Bν

)−1

=2βπkBhe2

31

(2π)6t23


1√1−κ2

×∫

dkxdkyv2

x


1√1−κ2

. (3.20)


Case III

When the marginal Fermi liquid piece of the self-energy is comparable to the Fermi liquid

part, the ratio B/ν introduced in Sec. 2.3.3 has the same expression as that introduced in

Sec. 2.2.3. Thus when measuring the resistivity along the z-direction we have(Bν

)−1

=8βπkBhe2

31

(2π)6h2


1√1−κ2

×∫

dkxdky

√1−κ2


When measuring the resistivity along x-direction we have(Bν

)−1

=2βπkBhe2

31

(2π)6t23


1√1−κ2

×∫

dkxdkyv2

x(kx,ky,k+zi )


1√1−κ2

, (3.22)

where k+zi and κ are given by

k+zi = cos−1(κ)

κ =−2t1cos(kx)−2t2cos(ky)−µ0

2t3. (3.23)

In this case, the KWR, given by Eq. (2.83), has a very complicated expression. We will

calculate this ratio numerically in the next section.

When the marginal Fermi liquid piece of the self-energy is small compared to the Fermi

liquid one, the ratio B/ν introduced in Sec. 2.3.3 has the form

Bν

=3η′



)2

×


(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (3.24)


with

β′ =

∫∞

0dy

sech2( y2)

y2 +π2 '16,

η′ =

∫ 1

0dy

sech2( y2)

(y2 +π2)2 +∫

∞

1dyy

sech2( y2)

(y2 +π2)2 ' 0.02 . (3.25)

Thus when measuring the resistivity along the z-direction we have

Bν

=3η′

2πhkBe2β′2h2(2π)6

4

∫dkxdkyZk(kx,ky,k+zi )λ(kx,ky,k+zi )

√1−κ2∫

dkxdkyZk(kx,ky,k+zi )1√

1−κ2

× 1(∫dkxdky

√1−κ2Zk(kx,ky,k+zi )

)2 . (3.26)

When measuring the resistivity along x-direction the ratio B/ν has the form

Bν

=3η′(2π)6t2

32πhkBe2β′2

∫dkxdkyv2

xZk(kx,ky,k+zi )λ(kx,ky,k+zi )1√

1−κ2∫dkxdkyZk(kx,ky,k+zi )λ(kx,ky,k+zi )

1√1−κ2

× 1(∫dkxdkyv2


1−κ2

)2 . (3.27)

From Eq. (2.84), in this case the KWR is given by

Aγ2 =

27D0

64πβ′hne2k2Bξ2

1XP2 , (3.28)

where X and P are integrals introduced in Sec. 3.1.2. Thus when measuring the resistivity

along the z-direction we have

Aγ2 =

27D0

64πβ′hne2k2Bξ2

h2(2π)9t34

1∫dkxdky


× 1(∫dkxdkyZk(kx,ky,k+zi )

1√1−κ2

)2 . (3.29)


When measuring the resistivity along x-direction the KWR has the form

Aγ2 =

27D0(2π)9t33

64πβ′hne2k2Bξ2

1∫dkxdkyv2


1−κ2


1√1−κ2

)2 . (3.30)

3.1.3 Numerical calculation of the KWR and KWR-like ratio for threedimensional systems

The integral expressions for A/γ2 and B/ν that we found in Sec. 3.1.2 can not be evaluated

analytically in general. Hence we perform numerical calculations to determine the depen-

dence of these quantities on in-plane momentum anisotropy over a wide range of values

of chemical potential, µ0. In order to simplify our numerical calculations, we assume the

hopping amplitudes are the same in each direction, i.e. t1 = t2 = t3 = t. We also assume

that any momentum anisotropy in the self-energy is confined to the x-y plane. To further

simplify our calculation, we assume the factor Zk(~k) = 1 in all three cases.

Case I

For each value of the chemical potential µ0, we determine the Fermi surface (which gives~kF(kx,ky,kz)) and also the Fermi velocity at each value of kx and ky. As an illustrative

example we plot the Fermi surface and the Fermi velocity for µ0 =−4 t in figure 3.1.

We assume that d(~k) = cos2(2φ), which characterizes the momentum anisotropy in the

self-energy (here φ refers to the in-plane angle in momentum space). To lowest order in ε,

which is the prefactor of the momentum anisotropy in the self-energy in Case I (Eq. 2.1),

we may writeAγ2 =

(Aγ2

)0(1+αε) , (3.31)

where from Eq. (3.7), we know α = (X/Y − 2Q/P). We now proceed to calculate α for

chemical potentials −6 t < µ0 < −2 t. Figure 3.2 shows how the coefficient α, which en-

ters the KWR when there is in-plane momentum anisotropy of the self-energy, varies as a

function of chemical potential when the resistivity is measured along the x-direction.


Figure 3.1: Numerical simulation of the Fermi surface for three dimensional systems with

a tight binding energy dispersion.

From figure 3.2 we find that turning on the momentum anisotropy in the electron self-

energy will reduce the value of the KWR since the coefficient α is negative for chemical

potentials−6 t < µ0 <−2 t. We also find that α decreases slowly as µ0 increases for a large

range of chemical potential (e.g. −6 t < µ0 <−3 t), then decreases more and more rapidly

as µ0 approaches to −2 t, where a square shaped in-plane Fermi surface emerges.

Case II

We assume that the momentum dependent coupling constant in the marginal Fermi liquid

self-energy has the form λ(~k) = λ0(1+ εd(~k)), where d(~k) = cos2(2φ) and ε is a small

parameter (here φ refers to the in-plane angle in momentum space).


Figure 3.2: Three dimensional systems: Renormalization coefficients α, α′, α

′′for the

KWR and KW-like ratio with the given self-energies as a function of chemical potential µ0

(in unit of t). Resistivity is measured along the x-direction.

From Eq. (2.67), to the lowest order in ε, the ratio B/ν can be written as

Bν' 3

2πhkBβe21

PX

(1+ ε

(YX− Q

P

))=

32πhkBβe2

1PX

(1+α

′ε

), (3.32)

where α′= (Y/X −Q/P), and P,Q,X ,Y are integrals as introduced in Sec. 3.1.2 with

Zk(~k) = 1.

The coefficient α′

characterizes the effect of momentum anisotropy of the self-energy

on the ratio B/ν. We calculate its value for chemical potentials−6t < µ0 <−2t. Figure 3.2

shows how the coefficient α′

varies as a function of chemical potential when measured

along the x-direction.


From figure 3.2 we find that the absolute value of the coefficient |α′| ≤ 0.05 for a

large range of chemical potential (e.g. −6 t < µ0 < −3.5 t), which means that momen-

tum anisotropy in the MFL self-energy has little effect on the renormalization of the ratio

B/ν. This phenomena is quite different from what happens to the KWR as we turn on the

momentum anisotropy in the Fermi liquid self-energy. However, as in Case I, this effect

becomes important as the chemical potential increases towards −2 t.

Case III

For the hybrid self-energy, it is of interest to investigate the effects of a small MFL piece

of self-energy (compared to the FL piece) on the KWR and the ratio B/ν. Following the

work by Kokalj and McKenzie [46], we parameterize the momentum dependent coupling

constant in the marginal Fermi liquid piece of the self-energy as λ(~k) = λ0 cos2(2φ) (here

φ refers to the in-plane angle in momentum space). Taking λ0 to be a small parameter, then

from Eq. (2.84), we know that the KWR and the ratio B/ν in this case have the form

Aγ2 =

2716πβ′hne2k2

Bξ2D0

1∫ d3k(2π)3 vi(~k)v j(~k)δ(ε0(~k)−µ0)

Bν

=3η′

2πhkBe2β′21

X2


(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (3.33)

where β′ = 1/6, η′ ' 0.02, and X is the integral introduced in Sec. 3.1.2 with Zk(~k) = 1.

The expression for the KWR introduced here has the same form as the one proposed by

Jacko et al. [9], which indicates that adding a small piece of momentum dependent MFL

self-energy to the isotropic FL part does not change the form of the KWR to first order in

λ(~k). As for the ratio B/ν, we find that it is independent of the small parameter λ0 because

both B and ν are proportional to λ0.

However, when the MFL piece of the self-energy is comparable to the FL part, the

expressions for the KWR and the ratio B/ν are very different from Eq. (3.33). In this case,

we parameterize λ(~k) as λ(~k) = λ0(1+ εd(~k)), where d(~k) = cos2(2φ) and ε is a small

parameter. As mentioned in Sec. 3.1.2, the ratio B/ν has the same form as in Case II.


Hence we only investigate the effect of the anisotropy of the self-energy on the KWR in

this case.

From Eq. (2.83), we may write

Aγ2 ' 27η

16πh2e2β2ξ2k2BnD0

1X

(1+2ε

2

(NX−(

YX

)2))

=

(Aγ2

)0

(1+α

′′ε

2), (3.34)

to the second lowest order in ε. Here X ,Y are the integrals introduced in Sec. 3.1.2, the

coefficient α′′= 2

(NX −

(YX

)2)

, and the parameter N is given by

N =∫ d3k

(2π)3 vi(~k)v j(~k)d2(~k)δ(

ε0(~k)−µ0

). (3.35)

We now calculate the coefficient α′′

for chemical potentials −6t < µ0 < −2t. Figure 3.2

shows how the coefficient α′′, which characterizes the effect of momentum anisotropy of

the self-energy on the KWR, varies as a function of chemical potential when measured

along the x-direction.

From figure 3.2 we find that the coefficient α′′

is positive for chemical potentials−6 t <

µ0 < −2 t, which indicates that introducing momentum anisotropy to the MFL piece of

the self-energy can boost the value of the KWR when the MFL piece and the FL piece of

our investigated hybrid self-energy are comparable. We also notice that, in contrast to the

coefficients α and α′, the coefficient α

′′is fairly constant throughout the whole range of the

chemical potential.

3.2 Quasi two dimensional conductors

Quasi two dimensional conductors can be viewed as stack of two dimensional metallic

layers. In a quasi two dimensional system, there is usually a large anisotropy between the

resistivity parallel to the layers and the resistivity perpendicular to the layers. Quasiparticles

have momentum h~k‖ within the planes and there is hopping between adjacent layers. Many

organic charge-transfer salts [51], transition-metal oxides [52], intercalated graphite [53]

and overdoped thallium cuprate [14, 54] are of this type.


3.2.1 Isotropic in-plane dispersion and momentum independent self-energy

A simple tight-binding energy dispersion for a quasi two dimensional conductor in which

an isotropic in-plane Fermi surface is assumed has the form

ε0(~k) =h2k2‖

2m0−2t⊥cos(ck⊥) , (3.36)

where k‖ is the magnitude projection of the Fermi wave vector along a direction which is

parallel to the layers, k⊥ is the projection perpendicular to the layers, c is the interlayer

spacing and t⊥ is the interlayer hopping integral. The unrenormalized density of states is

then given by

D0 = 2∫ d3k

(2π)3 δ

(h2k2⊥

2m0−2t⊥cos(ck⊥)−µ0

)=

14π3

∫d3k∑

i

1∣∣∣∂ f (k⊥)∂k⊥

∣∣∣k⊥i

δ(k⊥− k⊥i) , (3.37)

where k⊥i is the ith root of f (k⊥) =h2k2‖

2m0−2t⊥cos(ck⊥)−µ0 :

k⊥i =±1c

cos−1

(h2k2‖

4m0t⊥− µ0

2t⊥

). (3.38)

By substituting the expression for k⊥i into Eq. (3.37) we have

D0 =1

2π2t⊥c

∫ k+

k−k‖dk‖

1√1−(

h2k2‖

4m0t⊥− µ0

2t⊥

)2, (3.39)

where k± =√

2m0(µ0±2t⊥)/h .

We now make the substitution

α =h2k2‖

4m0t⊥− µ0

2t⊥, (3.40)

and finally we obtain

D0 =m0

π2h2c

∫ 1

−1dα

1√1−α2

=m0

πh2c, (3.41)


Note that we use a cylindrical coordinate system for all the calculations in the quasi two di-

mensional case. As there is usually a large anisotropy between the resistivity parallel to the

layers and the the resistivity perpendicular to the layers in layered systems, in the following

we calculate the ratios introduced in Chapter 2 for both directions (one is perpendicular to

the layers, the other is parallel to the layers).

Case I

We assume here that the self-energy is momentum independent, hence the factor Zk(~k,T )

is equal to 1 and the KWR defined in Eq. (2.46) has the form

Aγ2 =

814πhk2

Be2ξ2

1

〈vi(~k)v j(~k)〉D20n

, (3.42)

where 〈vi(~k)v j(~k)〉 denotes the average of vi(~k)v j(~k) over the whole Fermi surface, which

has the form

〈vi(~k)v j(~k)〉=2πh2c

m0

∫ d3k(2π)3 vi(~k)v j(~k)δ(ε0(~k)−µ0) . (3.43)

When measuring the resistivity along the perpendicular direction to the layers we have

〈vz(~k)vz(~k)〉= 〈v20⊥(

~k)〉, where v0⊥(~k) is given by

v0⊥(~k) =1h

∂ε0(~k)∂k⊥

=2t⊥c

hsin(ck⊥) . (3.44)

Similarly to the calculation of D0 we have

〈v20⊥(

~k)〉 =2πh2c

m0

∫ d3k(2π)3 v2

0⊥(~k)δ

(ε0(~k)−µ0

)=

2t2⊥c2

h2 . (3.45)

The KWR for resistivity perpendicular to the layers is then given by [9](Aγ2

)⊥=

81πh5

8ξ2k2Be2t2

⊥m20n

. (3.46)

On the other hand, measuring the resistivity along some preferred in-plane direction (with-

out loss of generality, the x-direction) we have

〈vi(~k)v j(~k)〉 = 〈v2x(~k)〉

=12〈v2

0‖(~k)〉 , (3.47)


where the prefactor 12 in Eq. (3.47) comes from the fact that the system is isotropic in the

plane. As a result,

〈v20‖(~k)〉= 〈v2

x(~k)+ v2y(~k)〉= 2〈v2

x(~k)〉 . (3.48)

Given v0‖ = h−1∂ε0(~k)/∂k‖ = hk‖/m0 , we have

〈vi(~k)v j(~k)〉=h2k2

F

2m20, (3.49)

which leads to the KWR for resistivity parallel to the layers as [9](Aγ2

)‖=

81πhc2

2k2Be2k2

Fξ2n. (3.50)

Case II

For a momentum independent marginal Fermi liquid self-energy, the ratio introduced in

Sec. 2.2.3 isBν=

6πhkBβe2

1

D20〈vi(~k)v j(~k)〉

, (3.51)

where D0 is the bare density of states at the Fermi level and β ' 1.52 is a constant intro-

duced in Chapter 2. Note that the ratio B/ν in this case has the same momentum integral

as the KWR introduced in Case I for quasi two dimensional materials with a momentum

independent self-energy.

Using the results for 〈vi(~k)v j(~k)〉 determined in Eq. (3.45) and (3.49) we find(Bν

)⊥=

3πh5

βkBe2m20t2⊥, (3.52)

and (Bν

)‖=

12πhc2

βkBe2k2F. (3.53)

3.2.2 Momentum dependent self-energy and anisotropic in-plane Fermisurface

In a square lattice model, the tight binding energy dispersion for a quasi two dimensional

material, which includes the nearest neighbour interactions, has the from

ε0(~k) =−2t(cos(kx)+ cos(ky))−2t⊥cos(kz) , (3.54)


where the wave numbers are again measured in units of the lattice constants. The quasipar-

ticle velocities along x, y, and z-directions are then given by

vx =1h

∂ε0(~k)∂kx

=2th

sin(kx)

vy =1h

∂ε0(~k)∂ky

=2th

sin(ky)

vz =1h

∂ε0(~k)∂kz

=2t⊥h

sin(kz) . (3.55)

Case I

For a momentum dependent self-energy with the magnitude of momentum anisotropy char-

acterized by ε, to lowest order in ε, the KWR introduced in Sec. 2.1.3 has a form

Aγ2 '

81D0

32πhk2Be2nξ2

1XP2

1+ ε

(YX− 2Q

P

), (3.56)

where

X =∫ d3k


Y =∫ d3k


P =∫ d3k

(2π)3 Zk(~kF ,0)δ(ε0(~k)−µ0) ,

Q =∫ d3k

(2π)3 Zk(~kF ,0)d(~k)δ(ε0(~k)−µ0) . (3.57)


X =1

(2π)3h

∫dk‖dkzdφk‖|vz(~k)|Zk(~k)δ(kz− kzi) , (3.58)

where kzi is the ith root of f (kz) = ε0(~k)−µ0 :

kzi =±cos−1(κ) , (3.59)

with

κ =−2t(cos(kx)+ cos(ky))−µ0

2t⊥. (3.60)


By substituting the expression for kzi into Eq. (3.58) we have

X =4t⊥h2

1(2π)3

∫dkxdky

√1−κ2 Zk(kx,ky,k+zi ) , (3.61)

where k+zi = cos−1(κ) . Following the same procedure we have

Y =4t⊥h2

1(2π)3

∫dkxdky

√1−κ2Zk(kx,ky,k+zi )d(kx,ky,k+zi )

P =1

(2π)3t⊥


1√1−κ2

Q =1

(2π)3t⊥


1√1−κ2

. (3.62)

When measuring the resistivity along some preferred in-plane direction (e.g. the x-direction)

we have

X =1

(2π)3t⊥

∫dkxdkyv2

x(kx,ky)Zk(kx,ky,k+zi )1√

1−κ2

Y =1

(2π)3t⊥

∫dkxdkyv2

x(kx,ky)Zk(kx,ky,k+zi )d(kx,ky,k+zi )1√

1−κ2

P =1

(2π)3t⊥


1√1−κ2

Q =1

(2π)3t⊥


1√1−κ2

. (3.63)

Case II

The ratio introduced in Sec. 2.2.3 has the form

Bν

=

2πhkBβe2

3

∫ d3k(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

×∫ d3k

(2π)3vi(~k)v j(~k)

λ(~k)Zk(~kF ,0)δ(ε0(~k)−µ0)

−1

, (3.64)

where β' 1.52 . For the energy dispersion Eq. (3.54), we have

ν =2π2k2

B3

1(2π)3t⊥


1√1−κ2

, (3.65)


where

k+zi = cos−1(κ)


2t⊥. (3.66)


B−1 =βhe2

πkB

4h2

t⊥(2π)3

∫dkxdky

√1−κ2


Thus,(Bν

)−1

=8βπkBhe2

31

(2π)6h2


1√1−κ2

×

∫dkxdky

√1−κ2


. (3.68)

When measuring the resistivity along some preferred in-plane direction (e.g. the x-direction)

we have

B−1 =βhe2

πkB

1(2π)3t⊥

∫dkxdky

v2x(kx,ky)


1√1−κ2

, (3.69)

which leads to(Bν

)−1

=2βπkBhe2

31

(2π)6t2⊥


1√1−κ2

×

∫dkxdky

v2x(kx,ky)


1√1−κ2

. (3.70)

Case III

When the marginal Fermi liquid piece of the self-energy is comparable to the Fermi liquid

part, the ratio B/ν introduced in Sec. 2.3.3 has the same expression as that introduced in

Sec. 2.2.3. Thus when measuring the resistivity along the z-direction we have(Bν

)−1

=8βπkBhe2

31

(2π)6h2


1√1−κ2

×∫

dkxdky

√1−κ2



When measuring the resistivity along x-direction we have(Bν

)−1

=2βπkBhe2

31

(2π)6t23


1√1−κ2

×∫

dkxdkyv2

x(kx,ky,k+zi )


1√1−κ2

, (3.72)

where k+zi and κ are given by

k+zi = cos−1(κ)


2t⊥. (3.73)

In this case, the KWR, given by Eq. (2.83), has a very complicated expression. We will

calculate this ratio numerically in the next section.

When the marginal Fermi liquid piece of the self-energy is fairly small compared to the

Fermi liquid one, the ratio B/ν introduced in Sec. 2.3.3 has the form

Bν

=3η′



)2

×


(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (3.74)

with

β′ =

∫∞

0dy

sech2( y2)

y2 +π2 =16,

η′ =

∫ 1

0dy

sech2( y2)

(y2 +π2)2 +∫

∞

1dyy

sech2( y2)

(y2 +π2)2 ' 0.02 . (3.75)

Thus when measuring the resistivity along the z-direction we have

Bν

=3η′

2πhkBe2β′2h2(2π)6

4

∫dkxdkyZk(kx,ky,k+zi )λ(kx,ky,k+zi )

√1−κ2∫

dkxdkyZk(kx,ky,k+zi )1√

1−κ2

× 1(∫dkxdky


)2 . (3.76)



Bν

=3η′(2π)6t2

32πhkBe2β′2

∫dkxdkyv2

xZk(kx,ky,k+zi )λ(kx,ky,k+zi )1√

1−κ2∫dkxdkyZk(kx,ky,k+zi )λ(kx,ky,k+zi )

1√1−κ2

× 1(∫dkxdkyv2


1−κ2

)2 . (3.77)

From Eq. (2.84), in this case the KWR is given by

Aγ2 =

27D0

64πβ′hne2k2Bξ2

1XP2 , (3.78)

where X and P are integrals introduced in Sec. 3.2.2. Thus when measuring the resistivity

along the z-direction we have

Aγ2 =

27D0

64πβ′hne2k2Bξ2

h2(2π)9t34

1∫dkxdky



1√1−κ2

)2 . (3.79)


Aγ2 =

27D0(2π)9t33

64πβ′hne2k2Bξ2

1∫dkxdkyv2


1−κ2


1√1−κ2

)2 . (3.80)

3.2.3 Numerical calculation of the KWR and KWR-like ratio for quasitwo dimensional conductors

As in Sec. 3.1.3, we perform numerical calculations to determine the dependence of the

KWR and KWR-like ratio on in-plane momentum anisotropy over a wide range of µ0 for

quasi two dimensional conductors. We assume the ratio of the hopping magnitude along


Figure 3.3: Numerical simulation of the in-plane projection of the Fermi surface for quasi

two dimensional conductors with a tight binding energy dispersion.

the z-direction to the hopping magnitude along the in-plane direction is t⊥/t = 1/20, and

the chemical potential varies between −4.0 t and −0.1 t (i.e. −4.0 t < µ0 < −0.1 t). To

further simplify our calculation, we assume the factor Zk(~k) = 1 in all three cases.

Case I

We determine the Fermi surface (i.e. ~kF(kx,ky)) and the Fermi velocity for each value of

the chemical potential µ0. Figure 3.3 shows an example of the Fermi surface and the Fermi

velocity for µ0 =−3 t.

We assume that d(~k) = cos2(2φ), where φ refers to the in-plane angle in momentum

space. As in Sec. 3.1.3, we may write the KWR as

Aγ2 =

(Aγ2

)0(1+αε) (3.81)


Figure 3.4: Quasi two dimensional conductors: Renormalization coefficients α, α′, α′′

for

the KWR and KW-like ratio with the given self-energies as a function of chemical potential

µ0 when measured along x-direction. The chemical potential µ0 is in unit of t.

to the lowest order of ε, the prefactor of the momentum anisotropy in the self-energy.

The coefficient α = (X/Y − 2Q/P), which characterize the renormalization of the KWR

in the presence of momentum anisotropy in the self-energy in Case I. Figure 3.4 shows

how the coefficient α varies as a function of chemical potential when measured along the

x-direction.

As in Sec. 3.1.3, turning on the momentum anisotropy in the electron self-energy will

reduce the value of the KWR for chemical potentials−4.0 t < µ0 <−0.1 t. From figure 3.4

we find that the coefficient α decreases slowly for small chemical potentials, then more

and more rapidly as µ0 approaches to−0.1 t, where a square shaped in-plane Fermi surface

emerges. We also find that magnitudes of the coefficient α in quasi two dimensional case

are larger than those in three dimensional case, which indicates that momentum anisotropy


in the self-energy has stronger effect on the renormalization of the KWR in quasi two

dimensional conductors than in three dimensional systems.

Figure 3.2 and figure 3.4 suggest that it might be interesting to look at the KWR as a

function of doping in appropriate materials - if the bandstructure is reasonably anisotropic

one might see a reduction of the KWR with increasing chemical potential.

Case II

As in Sec. 3.1.3, we assume that the momentum dependent coupling constant in the marginal

Fermi liquid self-energy has the form λ(~k) = λ0(1+ εd(~k)), where d(~k) = cos2(2φ) and ε

is a small parameter (here φ refers to the in-plane angle in momentum space).

To the lowest order in ε, the ratio B/ν can be written as

Bν' 3

2πhkBβe21

PX

(1+α

′ε

), (3.82)

where α′= (Y/X −Q/P), and P,Q,X ,Y are integrals as introduced in Sec. 3.2.2 with

Zk(~k) = 1.

We now calculate the coefficient α′, which characterizes the effect of momentum anisotr-

opy of the self-energy on the ratio B/ν, for chemical potentials −4t < µ0 < −0.1t. Fig-

ure 3.4 shows how the coefficient α′

varies as a function of chemical potential when mea-

sured along the x-direction.

From figure 3.4 we find that the coefficient α′is negligible when µ0 is small, which indi-

cates that momentum anisotropy in the MFL self-energy has negligible effect on the renor-

malization of the ratio B/ν for small chemical potentials. However, this effect becomes

more and more important as the chemical potential approaches to −0.1 t. This phenomena

is quite different from what happens to the KWR as we turn on the momentum anisotropy

in the Fermi liquid self-energy. We also find that in comparison to the three dimensional

case, the coefficient α′in the quasi two dimensional case has larger values cross the whole

range of chemical potentials, which is analogous to the coefficient α introduced in Case I.

Case III

As in Sec. 3.1.3, we first investigate the effects of small momentum anisotropy of the

self-energy, introduced by the MFL piece, on the KWR and the ratio B/ν. Following the


work by Kokalj and McKenzie [46], we parameterize the momentum dependent coupling

constant in the marginal Fermi liquid piece of the self-energy as λ(~k) = λ0 cos2(2φ) (here φ

refers to the in-plane angle in momentum space). Note that λ0 is a small parameter. From

Eq. (2.84), we know that the KWR and the ratio B/ν in this case have the form

Aγ2 =

2716πβ′hne2k2

Bξ2D0

1∫ d3k(2π)3 vi(~k)v j(~k)δ(ε0(~k)−µ0)

Bν

=3η′

2πhkBe2β′21

X2


(2π)3 λ(~kF)Zk(~kF ,0)δ(ε0(~k)−µ0)

, (3.83)

where β′ ' 1/6, η′ ' 0.02, and X is the integral introduced in Sec. 3.2.2 with Zk(~k) = 1.

From Eq. (3.83), we know that the expression for the KWR in this case is the same as

the one proposed by Jacko et al. [9], i.e. adding a small piece of momentum dependent

MFL self-energy to the isotropic FL part does not change the form of the KWR. And the

ratio B/ν is independent of the small parameter λ0 since both B and ν are proportional to

λ0.

On the other hand, when the MFL piece of the self-energy is comparable to the FL

part, the expressions for the KWR and the ratio B/ν are very different from Eq. (3.83). In

this case, we parameterize λ(~k) as λ(~k) = λ0(1+ εd(~k)), where d(~k) = cos2(2φ) and ε is

a small parameter. As mentioned in Sec. 3.2.2, the ratio B/ν has the same form as that in

Case II. Hence we only investigate the effect of the anisotropy of the self-energy on the

KWR in this case.

According to the Eq. (2.83), to the second lowest order in ε, we may write

Aγ2 ' 27η

16πh2e2β2ξ2k2BnD0

1X

(1+2ε

2

(NX−(

YX

)2))

=

(Aγ2

)0

(1+α

′′ε

2), (3.84)

where the coefficient α′′= 2(

NX −

(YX

)2)

, and X ,Y are the integrals introduced in Sec. 3.1.1.

The parameter N is given by

N =∫ d3k

(2π)3 vi(~k)v j(~k)d2(~k)δ(

ε0(~k)−µ0

). (3.85)


We now calculate the coefficient α′′

for chemical potentials −4t < µ0 <−0.1t. Figure 3.4

shows how the coefficient α′′, which characterize the effect of momentum anisotropy on

the KWR, varies as a function of chemical potential when measured along the x-direction.

The coefficient α′′

is positive for chemical potentials −4 t < µ0 < −0.1 t, which indi-

cates that momentum anisotropy in the MFL piece of the self-energy can boost the value

of the KWR when the MFL piece and the FL piece of our investigated hybrid self-energy

are comparable. We find that, in contrast to the coefficients α and α′, the coefficient α

′′

is fairly constant throughout the whole range of the chemical potential. From figure 3.2

and figure 3.4 we also find that the coefficient α′′

in three dimensional systems has roughly

the same value as in quasi two dimensional conductors, which indicates that the momen-

tum anisotropy in the MFL piece of our investigated hybrid self-energy has roughly the

same effect on the renormalization of the KWR for both three dimensional and quasi two

dimensional materials.

3.3 Summary

In this chapter we have calculated the Kadowaki-Woods ratio and KW-like ratio numeri-

cally for both three dimensional and quasi two dimensional materials with three different

types of momentum dependent self-energy introduced in Chapter 2 and appropriate tight

binding energy dispersions. We find that momentum anisotropy in the MFL self-energy

does not have significant effect on the renormalization of the KW-like ratio B/ν. As for the

KWR, momentum anisotropy in the Fermi liquid piece of the self-energy tends to reduce

the value of the KWR, while momentum anisotropy in the MFL piece of the self-energy

tends to enhance its value. Thus, one might find that the KWR decreases with increasing

chemical potential when the bandstructure of materials is reasonably anisotropic. Our cal-

culation also suggests that momentum anisotropy in the electron self-energies of quasi two

dimensional systems has a qualitatively similar effect on the renormalization of transport

properties to three dimensional systems.

Chapter 4

Marginal Fermi liquid phenomenologyand KWR-like ratios

In Chapter 3 we calculated specific expressions for the KWR and KW-like ratios for sev-

eral different classes of material and studied the effects of anisotropy in the self-energy on

these ratios. In this chapter we follow up on the idea introduced in Chapter 2, that it might

be interesting to investigate ratios analogous to the Kadowaki-Woods ratio for forms of the

self-energy other than the regular Fermi liquid form. In particular, the phenomenology as-

sociated with a marginal Fermi liquid is that there is a linear temperature dependence of the

resistivity, with a coefficient of B, at low temperatures and T lnT dependence of the specific

heat, with a coefficient of ν, at low temperatures. A number of heavy fermion compounds

display non-Fermi liquid behaviour with this phenomenology [19, 55, 57]. Hence, in this

chapter we collate experimental data from heavy fermion systems that have marginal Fermi

liquid behaviours in their specific heat and resistivity and examine whether B and ν allow

for an organization of existing experimental data.

4.1 Examination of the analogy to the KWR for the de-fined ratio B/ν

In order to test the idea that KW-like ratios may be present in non-Fermi liquid systems, we

attempted to collect an exhaustive list of data from materials displaying MFL phenomenol-

64

Marginal Fermi liquid phenomenology and a generalization of the KWR 65

Figure 4.1: Left. Low temperature resistivity of YbRh2Si2 measured along the a-axis at

p = 0 for different magnetic fields applied along c-axis. Right. Low temperature specific

heat of YbRh2Si2, plotted as ∆C/T in a lnT scale at p = 0 for different magnetic fields

applied along a-axis.(Panel taken from Ref. [68])

ogy. For illustrative purposes we display resistivity and specific heat data from the material

YbRh2Si2 in figure 4.1. YbRh2Si2 shows exceptionally clean MFL phenomenology over a

temperature range from a maximum of Tmax ' 10K down to temperatures at least as low as

0.4K in specific heat, and possibly an order of magnitude lower in temperature in resistivity.

For materials such as YbRh2Si2 where the values of B and ν are not published in the liter-

ature, we scanned the data and fitted it to the form ρ = ρ0+BT and Cv/T = γ+ν ln(T0/T )

to obtain B and ν respectively.

In Table 4.1 we report the coefficients B and ν as measured in a variety of heavy fermion

materials with marginal Fermi liquid phenomenology. Data in bold print are reanalyzed

published data that have been scanned and fit to a different temperature dependence than

used in the original papers. Transport measurements with the current perpendicular and

parallel to the basal plane are distinguished by the symbols ‖ and ⊥ respectively.


Materials B ν B/ν Vuc f.u./u.c. Tmin Tmax Refs.

(µΩ cm/K) (J/K2 mol) (µΩ cm mol K/J) (Å3) (K) (K)

U0.1Th0.9Cu2Si2 2.16 3.8 0.6 165.0 8 [55, 56]

CeCoIn5 1.0 0.14 7.1 161.4 1 [57, 58, 59]

CeCoIn4.82Sn0.18 0.9 0.11 8.2 4 [60]

U0.2Y0.8Pd2Al3 0.4 0.34 1.2 311.2 1 5 [61, 62]

CeNi2Ge2 0.26 0.05 5.2 169.5 2 1 4 [63, 64, 65]

CeNi0.7Co0.3Ge2 1.2 0.33 3.6 6 [66]

CePd0.05Ni0.95 0.56 0.042 13.3 0.9 4 [67]

YbRh2Si2 1.8 0.17 10.6 158.3 2 10 [19, 68]

YbRh2(Si0.95Ge0.05)2 1.9 0.15 12.7 0.3 10 [69, 70]

CeCoGe1.5Si1.5 0.8 0.05 16.0 0.3 2 [71]

β−YbAlB4 0.39 0.025 15.6 238.1 4 1 3 [72, 73]

Ce2PdIn8 3.4 0.17 20.0 269.5 1 3 [74]

CeCu5.9Au0.1 27.6 0.6 46.0 421.3 4 1 [75, 76, 77]

CeCu5.8Au0.2 (4.1kbar) 35.8 0.6 59.7 422.7 [76, 78]

CaCu3Ru4O12 1.0 0.016 62.5 407.2 2 0.2 2 [79, 80]

Ce2Co6Al19 4.7 0.035 134.3 4 5 [81]

CeNiGeSi 7.9 0.18 43.9 288.0 4 5 [82, 83]

Ce2NiB9.7 8.9 0.06 148.3 1.4 10 [84]

Ce(Ni0.935Pd0.065)2Ge2 0.24 0.11 2.2 5 [85]

CeRhBi 51.0 0.12 425 280.7 4 8 [86, 87]

CeCu5.2Ag0.8 (2.3T) 26.6 1.1 24.2 440.0 4 0.2 1.5 [88, 89, 90]

U2Pt2In (‖) 1 8.9 0.044 202.3 436.2 4 6 [91, 92]

Ce2IrIn8 (13T) 9.1 0.065 140.0 268.2 1 [93, 94]

Table 4.1: Compilation of B and ν values for non-Fermi liquids displaying marginal Fermi

liquid phenomenology. Vuc is the volume of the unit cell, f.u/u.c is the number of formula

units per unit cell. Tmax and Tmin are the upper and lower limits of the temperature range

where MFL phenomenology was observed. If Tmax (or Tmin) is not displayed, then this

indicates that MFL phenomenology is observed to the highest (or the lowest) available

temperatures.

As mentioned in Chapter 1, the Kadowaki-Woods ratio has a common value of 10µΩcm

mol2 K2 J−2 for many heavy fermion compounds [8]. Thus, it is of interest to test whether

1The in-plane resistivity varies approximately linearly with temperature T, while along the c direction theresistivity at low temperatures, ρc ∼

√T .


Figure 4.2: B and ν relationship for heavy fermion compounds listed in Table 4.1. The

solid and dash lines with different slopes are guides for the eye.

the ratio B/ν has a similar universal behaviour for marginal Fermi liquids. Figure 4.2 shows

the B vs ν relationship for compounds listed in the above table. The plot of B vs ν does not

seem to show a common relationship similar to the KWR. Figure 4.3 shows the histograms

of the KW-like ratio B/ν for heavy fermion compounds listed in Table 4.1. In figure 4.3 a)

we display B/ν values with bins on a linear scale, and in figure 4.3 b) we display the same

data binned on a log scale. Figure 4.3 a) shows that most observed values of B/ν are of or-

der 50 or less, but there is a long tail in the distribution to large values of B/ν; figure 4.3 b)

shows that in fact there seems to be a roughly uniform distribution on a logarithmic scale,

so there is a very wide range of observed B/ν values.

The scatter visible in figure 4.2 is reminiscent of the scatter in the plot of A against γ2

in figure 1.3, in which case the scatter arises from the different values of the KWR in dif-

ferent families of materials. One can see that related MFL compounds (e.g. CeCu5.9Au0.1,

CeCu5.8Au0.2, CeCu5.2Ag0.8) lie in similar regions of figure 4.2, so with appropriate clas-


Figure 4.3: Histograms of the KW-like ratio B/ν for heavy fermion compounds listed in

Table 4.1. a) linear scale and b) logarithmic scale for B/ν. N is the number of compounds

lying in each bin.

sification into families the ratio B/ν may also be roughly constant within a family.

In Chapter 3 we found explicit analytical expressions for the ratio B/ν [Eqs 3.4, 3.52

and 3.53] when the self-energy and the dispersion are isotropic in momentum space. Given

the success of such expressions for the KWR, we compare our expressions to the avail-

able experimental data. Depending on dimensionality, we found B/ν ∼ 1/k4F (in three

dimensions), B/ν∼ 1/k2F (two dimensions: in-plane resistivity) or B/ν∼ 1/m2

0t2⊥ (two di-

mensions: interlayer resistivity), with the rest of the ratio universal constants. The Fermi

wavevector, kF , can be related to the carrier density as kF ∼ n1/d in d dimensions. In the

three dimensional case, B/ν ∼ n−4/3, so we should expect that systems with higher car-

rier density to have smaller B/ν and vice versa. We compared B/ν values with n for all

materials in Table 4.1 that we could obtain carrier densities for, as reported in Table 4.2.

Materials B/ν(µΩcmmolK/J) n(1028 m−3)

U0.1Th0.9Cu2Si2 0.57 4.85

YbRh2Si2 10.6 2.6

U2Pt2In 202 1.22

Table 4.2: Compilation of carrier concentration values for non-Fermi liquids displaying

marginal Fermi liquid phenomenology. n is the carrier density.


Figure 4.4: B and ν relationship for heavy fermion compounds whose Vuc and Z have been

reported in the literature.

We do not find a simple B/ν ∼ n−4/3 relationship. However we do certainly see in

Table 4.2 that the smallest value of B/ν is for the system with largest n and vice versa. This

differs from the KWR in that an isotropic self-energy and dispersion gives a good account

of experimental data. There are a number of possibilities for the disagreement between B/ν

and the simplest expectation based on Eq. (3.4). (i) There is momentum anisotropy in the

coupling constant λ(~k) [37] or the energy dispersion. As shown in Sec. 3.1.3 and Sec. 3.2.3,

momentum anisotropy in λ(~k) leads to diminution of the ratio B/ν. Momentum anisotropy

in the energy dispersion can change the density of states at the Fermi level D0 and the in-

plane average of vi(~k)v j(~k) over the Fermi surface, which leads to change of the value of

B/ν as well. (ii) Our assumption that there is no significant vertex corrections to the con-

ductivity may not be appropriate. It has been reported recently that vertex corrections due

to elastic scattering off either quenched or annealed disorder give a negative contribution


Figure 4.5: Modified ratio B/ν′, which takes account of Vuc and Z for corresponding heavy

fermion compounds.

to the zero temperature conductivity [95]. (iii) Multiple-band effects may mean that the

calculation of B/ν is too naive: bands contribute ‘in series’ to the specific heat, while they

add ‘in parallel’ to the resistivity. (iv) The result Eq. (3.4) was derived assuming a parabolic

dispersion. This may be not necessarily be the case, in which case one expects a different

n dependence for B/ν.

The ratio B/ν compares a volume quantity (B) with a molar quantity (ν). In the context

of the KWR, Hussey [11] pointed out that correcting the KWR for unit cell volume led to a

better collapse of points from different families. Hence, we investigate the effect of unit cell

volume Vuc and the number of formula units per unit cell Z on the ratio B/ν. We introduce a

modified ratio B/ν′, where ν′ is defined as ν′ = Z ν/Vuc, in analogy to γ ′ defined by Hussey

[11]. Figure 4.4 shows the plot of B vs ν for MFL compounds whose Vuc and Z have been

reported in the literature, and figure 4.5 shows the corresponding plot of B vs ν′ taking


account of the effect of the unit cell volume and the number of formula units per unit cell.

Comparing figures 4.4 and 4.5, we see that there is very little qualitative difference in

the plots of B/ν and B/ν′. In trying to fit a line through the data, R2 values of 0.08 and 0.1

are found, confirming this observation.


4.2 B/ν and the temperature window of MFL behaviour

Figure 4.6: Schematic phase diagram near a quantum critical point. Tmax and Tmin, indi-

cated by the arrow, are the upper and lower limits of the temperature range where non-Fermi

liquid phenomenology exits at some specific value of the control parameter. The parameter

along x-axis, which is indicated by C, can be quite general, such as the doping, external

pressure, and the external magnetic field. The dashed line indicates the value of Tmin at

certain value of the parameter C, where a crossover from non-Fermi liquid to Fermi liquid

behaviour occurs.

For the materials presented in Table 4.1, MFL phenomenology is usually only observed

over some temperature window, e.g. Tmin < T < Tmax. The value of Tmax is readily deter-

mined by fits to MFL forms of ρ and Cv, but in many cases Tmin is the lowest accessible

temperature. In Table 4.1 we only display Tmin in the cases where there is a clear de-

parture from MFL phenomenology at temperatures below Tmin. Non-Fermi liquid (NFL)

phenomenology in heavy fermion materials are often thought to be the result of proximity

to a quantum critical point (QCP) [60, 66, 81, 96]. A quantum critical point is associated


Figure 4.7: B vs Tmax relationship for heavy fermions listed in Table 4.1.

with the existence of a phase transition that occurs at zero temperature. In contrast to clas-

sical phase transitions, at nonzero temperatures where thermal fluctuations are important,

quantum phase transitions are driven by a control parameter other than temperature, e.g.,

doping, external pressure, or external magnetic field. If MFL phenomenology is associated

with proximity to a QCP, then one possible schematic scenario involving Tmax, Tmin and

the QCP is sketched in figure 4.6.

One might wonder whether Tmax provides an energy scale that affects either B or ν, or

the ratio B/ν, or whether B, ν, or B/ν depend on Tmax/Tmin, which might serve as a proxy

for proximity to a QCP.

In figures 4.7 and 4.8 we plot B and ν as a function of Tmax for materials listed in

Table 4.1. Figures 4.7 and 4.8 indicate that the coefficients B and ν have no obvious cor-

relation with Tmax. In figure 4.9 we plot B against Tmax/Tmin for the selection of materials

in Table 4.1 that we were able to determine values of both Tmax and Tmin. The data do not


Figure 4.8: ν vs Tmax relationship for heavy fermions listed in Table 4.1.

allow for a meaningful line of best fit, but there does appear to be a tendency that larger

values of B are found as Tmax/Tmin increases, as naively expected. This does not appear to

depend strongly on the nature of the ordered phase that the system is near. For example,

CePd0.05Ni0.95 is close to a ferromagnetic phase, and CeCoGe1.5Si1.5 is near short range

antiferromagnetic order. Neither ν or B/ν appear to show much dependence on Tmax/Tmin.


Figure 4.9: B vs Tmax/Tmin relationship for heavy fermions listed in Table 4.1.


4.3 Summary

In this chapter we collated available data from the literature on materials that show MFL

phenomenology and investigated B, ν and the ratio B/ν. The ratio B/ν does not behave

in the same way as the Kadowaki-Woods ratio in that it does not appear to take a roughly

universal value within families of materials. Figure 4.3 shows that most of the MFL com-

pounds concentrate at low values of B/ν, but there is a long tail in the distribution to large

values of B/ν, which, based on calculations assuming no momentum space anisotropy,

may correspond to low and high carrier concentration respectively. Including the unit cell

volume effect does not lead to a significant collapse of points in the B vs ν plot, which

is different from the KWR [11]. We also find that there are no correlations between the

coefficients B, ν with Tmax, the upper limit of the temperature range where non-Fermi liq-

uid phenomenology exits at some specific heat value of the control parameter. However,

figure 4.9 does suggest some potential correlation between the coefficient B and the ratio

Tmax/Tmin.

In all of these plots, the number of available materials is not necessarily large, and

so caveats related to small number of data points should apply. It will be interesting to see

whether the groupings plotted here provide a helpful way to organize data as more materials

with MFL phenomenology are discovered.

Chapter 5

Conclusions

The ratio of the temperature dependences for the low temperature resistivity and the specific

heat in a Fermi liquid, the Kadowaki-Woods ratio (KWR), has been studied intensively in

past few decades. This ratio was found to take roughly a universal value within families of

strongly correlated electron materials. Until now, momentum anisotropy in quasiparticle

interactions has always been ignored in previous theoretical work on the KWR. In this

thesis we have investigated the effects of momentum anisotropy in the electron self-energy

and the electronic dispersion on the KWR and the KW-like ratio B/ν.

We have calculated the KWR and KW-like ratio B/ν in terms of integrals over the Fermi

surface for three different types of momentum dependent self-energies which are beyond

the simple Fermi liquid model. We calculated these integrals numerically for several model

bandstructures that are appropriate for layered materials and three dimensional systems.

Our results suggest that momentum anisotropy in a marginal Fermi liquid (MFL) piece of

the self-energy does not have significant effect on the renormalization of the ratio B/ν,

while it enhances the values of the KWR. In contrast to this, momentum anisotropy in

a Fermi liquid piece of the self-energy tends to reduce the values of the KWR - if the

bandstructure of materials is reasonably anisotropic one might see a reduction of the KWR

with increasing chemical potential. Our results also suggest that momentum anisotropy

in the electron self-energies of quasi two dimensional systems has a qualitatively similar

effect on the renormalization of transport properties (the resistivity and the specific heat) to

three dimensional systems.

We have also collated experimental data from heavy fermion compounds that exhibit

77

Conclusions 78

marginal Fermi liquid phenomenology and investigated the KW-like ratio B/ν for those

compounds. These MFL materials can be roughly grouped into high and low B/ν classes

according to the plot of B vs ν shown in figure 4.2. However, the ratio B/ν differs from the

KWR in that it does not appear to take a roughly universal value nor to scale as predicted by

a calculation assuming isotropic dispersion and self-energy. Including the unit cell volume

effect does not lead to a significant collapse of the points in the B vs ν plot. In sum, the KW-

like ratio B/ν provides a possible way to group MFL materials. Given that related MFL

compounds lie in similar region of figure 4.2, with appropriate classification into families

the ratio B/ν may also be roughly constant within a family. The limited number of systems

displaying MFL phenomenology limited the number of MFL materials that we were able

to investigate and hence the quantitative results we were able to achieve. As more such

materials are discovered, the status of this ratio may become clearer.

This thesis exhibits the first attempt to include momentum anisotropy in electron self-

energy in the study of the KWR and introduces the KW-like ratio B/ν. In addition to the

anisotropic interactions studied here, the physics of quantum criticality is believed to play

an important role in MFL phenomenology. Multiple band effects are also known to be

relevant for the KWR [11] and may also play a role for the ratio B/ν. Hence a natural

extension of the work here is to include these effects in the investigation of the KWR and

the ratio B/ν, which will hopefully lead to improved understanding of the microscopic

physics of materials that display non-Fermi liquid phenomenology.

Appendix A

Evaluation of A2(ω,~k) as Σ′′(ω,~k)

approaches to zero

To establish the limiting behaviour of the square of the spectral function A2(ω,~k) in the

limit that the imaginary part of the self-energy vanishes, we start off by evaluating the

following integral

∫ +∞

−∞

dy(

2xx2 + y2

)2

= 4∫ +∞

−∞

dyx2

(y+ ix)2(y− ix)2

= 8πiRes(

x2

(y+ ix)2(y− ix)2 , ix)

= 8πi−2x2

(y+ ix)3

∣∣∣∣y=ix

=2π

x, (A.1)

which is evaluated as a contour integral with contour closed in the upper half plane (assum-

ing x > 0). Thus, as x→ 0+,(

2xx2+y2

)2has the limit

(2x

x2 + y2

)2

→ 2π

xδ(y). (A.2)

79

Conclusions 80

Similarly, as −Σ′′ → 0+,

A(ω,~k)2 =

−2Σ′′(ω,~k,T )(

ω−ξk−Σ′(ω,~k,T )

)2+(

Σ′′(ω,~k,T )2

)

2

→ − 1

Σ′′(ω,~k,T )

2πδ(ω−ξk−Σ′(ω,~k,T )) , (A.3)

where ξk = ε0(~k)−µ0 is the free-particle energy measured from the bare chemical potential.

Suppose ω = εk−µ is a solution to the equation of

ω−ξk−Σ′(ω,~k,T ) = 0 , (A.4)

then,

2πδ(ω−ξk−Σ′(ω,~k,T )) = 2πZω(~k,T )δ(ω− (εk−µ)) , (A.5)

where

Z−1ω (~k,T ) = 1− ∂Σ

′(ω,~k,T )∂ω

∣∣∣∣∣ω=εk−µ

. (A.6)

Hence,

A(ω,~k)2 −Σ′′→0+−→ − 1

Σ′′(ω,~k,T )

2πZω(~k,T )δ(ω− (εk−µ)) . (A.7)

Appendix B

Evaluation of the ω integral in Ii j0 at

nonzero temperature

In the pure limit, the impurity scattering lifetime τ0→ ∞. Hence the conventional Fermi

liquid self energy takes the form

Σ′′0(ω,T ) =−s0

ω2 +(πkBT )2

ω∗2, (B.1)

when |ω2 +(πkBT )2| < ω∗2 (i.e. |ω| < α). At low temperatures the contribution to the ω-

integral from region |ω|> α is small [9]. We can use the form of Σ′′0 in region |ω|< α for

all ω.

I ≡∫ +∞

−∞

dω

(− 1

Σ′′0(ω,T )

)(−∂ f (ω)

∂ω

)' (ω∗/kBT )2

s0

∫ +∞

−∞

dω1

(ω/kBT )2 +π2

(−∂ f (ω)

∂ω

). (B.2)

Set x = ω

kBT ⇒dxdω

= 1kBT ⇒ dω = kBT dx , then

∂ f (ω)∂ω

=∂

∂ω

(1+ e

ω

kBT)−1

= − 14kBT

sech2(x

2

), (B.3)

and hence,

I ' (ω∗/kBT )2

4s0

∫ +∞

−∞

dxsech2 x

2x2 +π2 . (B.4)

81

Conclusions 82

To evaluate this integral, consider

J =∫ +∞

−∞

dxsech2 x

2x2 +π2

=1π2

∫ +∞

−∞

dxsech2(x

2

)− 2

π2

∫ +∞

−∞

dxx2

x2 +π2 sech2(x

2

)=

1π2 2tanh

(x2

)∣∣∣+∞

−∞

− 8π2

∫ +∞

−∞

dxx2

x2 +π2ex

(1+ ex)2

=4π2

[1−2

∫ +∞

0dx

x2

x2 +π2ex

(1+ ex)2

]=

4π2

[1+2

∂

∂α

(∫ +∞

0dx

x2

x2 +π21

1+ eαx

)∣∣∣∣α=1

]. (B.5)

Given [50] ∫ +∞

0dx

xx2 +β2

11+ eµx =

12

[ψ

(βµ2π

+12

)− ln

(βµ2π

)], (B.6)

where ψ(x) is the digamma function, we have

J =4π2

[1+

∂

∂α

(ψ

(α+1

2− ln

(α

2

)))∣∣∣∣α=1

]=

4π2

[1+

12

ψ′(1)−1

]=

2π2 ψ

′(1)

=13, (B.7)

where we used ψ′(1) =

∞

∑n=0

1n2 =

π2

6.

Thus, in the pure limit

IT 6=0' ω∗2

12s0(kBT )2 . (B.8)

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