PRESSURE INDUCED QUANTUM PHASE TRANSITIONS IN …...Metal-insulator transitions are studied by means...

PRESSURE INDUCED QUANTUM PHASE TRANSITIONS

IN METALLIC OXIDES AND PNICTIDES

by

Fazel Fallah Tafti

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright © 2011 by Fazel Fallah Tafti

Abstract

PRESSURE INDUCED QUANTUM PHASE TRANSITIONS IN METALLIC

OXIDES AND PNICTIDES

Fazel Fallah Tafti

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2011

Quantum phase transitions occur as a result of competing ground states. The

focus of the present work is to understand quantum criticality and its consequences

when the competition is between insulating and metallic ground states. Metal-

insulator transitions are studied by means of electronic transport measurements and

quantum critical points are approached by applying hydrostatic pressure in two dif-

ferent compounds namely Eu2Ir2O7 and FeCrAs. The former is a ternary metal oxide

and the latter is a ternary metal pnictide.

A major component of this work was the development of the ultra-high pressure

measurements by means of Anvil cells. A novel design is introduced which minimizes

the alignment accessory components hence, making the cell more robust and easier

to use.

Eu2Ir2O7 is a ternary metal oxide and a member of the pyrochlore iridate family.

Resistivity measurements under pressure in moissanite anvil cells show the evolution

of the ground state of the system from insulating to metallic. The quantum phase

ii

transition at Pc ∼ 6 GPa appears to be continuous. A remarkable correspondence

is revealed between the effect of the hydrostatic pressure on Eu2Ir2O7 and the effect

of chemical pressure by changing the R size in the R2Ir2O7 series. This suggests

that in both cases the tuning parameter controls the t2g bandwidth of the iridium

5d electrons. Moreover, hydrostatic pressure unveils a curious cross-over from inco-

herent to conventional metallic behaviour at a T ∗ > 150 K in the neighbourhood

of Pc, suggesting a connection between the high and low temperature phases. The

possibility of a topological semi-metallic ground state, predicted in recent theoretical

studies, is explained.

FeCrAs is a ternary metal pnictide with Fermi liquid specific heat and susceptibil-

ity behaviour but non-metallic non-Fermi liquid resistivity behaviour. Characteristic

properties of the compound are explained and compared to those of superconducting

pnictides. Antiferromagnetic (AFM) order sets in at ∼ 125 K with the magnetic mo-

ments residing on the Cr site. Pressure measurements are carried out in moissanite

and diamond anvil cells in order to suppress the AFM order and resolve the underly-

ing electronic transport properties. While AFM order is destroyed by pressure, the

non-metallic non-Fermi liquid behaviour is shown to be robust against pressure.

iii

Dedication

To my lovely sisters,

Fatemeh and Faezeh

who also are my best friends.

iv

Acknowledgements

It takes more than one person to finish a PhD thesis. In completing the final

stage of my academic education, I have benefited from the help and support of my

dear colleagues, friends, and family to whom I will eternally be grateful.

My PhD adviser, Professor Stephen Julian, has been a great source of inspiration

and insight. I have been very lucky to have the opportunity to work with such a fine

scientist and a wonderful human being. His deep understanding of physics, his kind

manners, and his unwavering support during these years made my graduate life a

fabulous experience.

Members of my supervisory committee, Professor Yong Baek Kim and Professor

Young-June Kim, have considerably helped me over the years. I am grateful to both

of them, for giving me advice and guiding me through my PhD.

I had the pleasure of working with kind and intelligent colleagues in the condensed

matter group including: Dr. Cyrus Turel (Doc), Dr. Patrick Rourke, Dr. Wenlong

Wu, Dr. Alix McCollam, Aaron Sutton, Mingxuan Fu, Di Tian, Dan Sun, Masih

Mehdizadeh, Kevin Grykuliak, Dr. Christianne Beekman, Professor Kenneth Burch,

Christoph Puetter, William Witczak-Krempa, Andreea Lupascu, Vijay Shankar, and

Patrick Morales.

In particular I am very grateful to Mark Aoshima for his invaluable machining

skills and his many good ideas in designing and machining anvil cells. I am deeply

thankful to Dr. Patricia Alireza for all her helpful advice in setting up the transport

measurements in anvil cells.

My family and friends have been the most important sources of emotional sup-

v

port. I would like to express my deep gratitude towards all my friends, in par-

ticular: Elham Farahani, Houfar Daneshvar, Reza Beheshti, Ali Najmei, Shabanaz

Pashapour, Santiago De Lope, Jake Klamka, Alma Bardon, Ettienne Pelletier, Jean-

Michel Menard, Lindsay Leblanc, Marcius Extavour, Reza Mir, Lisa Roach, Mar-

tin Bercx, Cristen Adams, Peter Thompson, Shahidul Islam, Sing-Leung Cheung,

Bin Gao, Xing Xingxing, Girija Dharmaraj, Snezana Prodan, Ylenia Turchiaro and

Miguel De La Bastide.

More than anyone, I am grateful to my sisters, Fatemeh and Faezeh, for their

unconditional love and support all through my life. Without their help, none of this

would be possible. It is to them that I dedicate my thesis.

Fazel Fallah Tafti

vi

Contents

1 Introduction 1

1.1 Metal Insulator Transition . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Herzfeld Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Mott Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Anderson Localization . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Scaling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.5 Pressure Induced Metallization . . . . . . . . . . . . . . . . . 17

1.1.6 Experimental Examples . . . . . . . . . . . . . . . . . . . . . 23

1.2 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.2 Classical Critical Behaviour . . . . . . . . . . . . . . . . . . . 30

1.2.3 Quantum Critical Behaviour . . . . . . . . . . . . . . . . . . . 33

1.2.4 Experimental Examples . . . . . . . . . . . . . . . . . . . . . 38

2 Experimental Development 43

2.1 Anvil Cell Technology . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vii

2.1.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.1.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.3 Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.4 Ruby Fluorescence Technique . . . . . . . . . . . . . . . . . . 63

2.1.5 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.2 1K Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.2.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2.3 LabView Program . . . . . . . . . . . . . . . . . . . . . . . . 77

2.2.4 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Eu2Ir2O7 84

3.1 Material Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 FeCrAs 106

4.1 Material Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . 117

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 Conclusions and Outlook 128

5.1 Anvil Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

viii

5.2 1K probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.3 Eu2Ir2O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4 FeCrAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Bibliography 142

ix

List of Tables

1.1 The evolution of the metallic phase of SmB6 from NFL to FL . . . . 27

2.1 Properties of different anvils . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Comparing 3d to 4,5d orbitals . . . . . . . . . . . . . . . . . . . . . . 89

x

List of Figures

1.1 Mott criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Scaling theory of localization . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Phase Diagram of (V0.989Cr0.011)2O3 . . . . . . . . . . . . . . . . . . . 24

1.5 Pressure dependence of the resistivity of SmB6 . . . . . . . . . . . . . 26

1.6 Quantum critical point . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Level crossing and quantum fluctuations . . . . . . . . . . . . . . . . 34

1.8 Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . . . 37

1.9 QPT in CePd2Si2 and CeIn3 . . . . . . . . . . . . . . . . . . . . . . . 39

1.10 Scaling analysis for the MIT in a 2D electron gas . . . . . . . . . . . 41

2.1 Design of the anvil cell . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Images of the moissanite and diamond anvil cells . . . . . . . . . . . 47

2.3 Anvils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Theory of the gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Thin gasket regime vs. thick gasket regime . . . . . . . . . . . . . . . 56

2.6 Force-pressure plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xi

2.7 Dynamical cross-over between thick and thin gasket regimes . . . . . 62

2.8 Crystal field scheme of ruby . . . . . . . . . . . . . . . . . . . . . . . 65

2.9 Fluorescence spectrometer . . . . . . . . . . . . . . . . . . . . . . . . 66

2.10 Preparing the gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.11 Ruby fluorescence spectroscopy . . . . . . . . . . . . . . . . . . . . . 72

2.12 Design of the 1K probe . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.13 Sample platform on the 1K probe . . . . . . . . . . . . . . . . . . . . 76

2.14 LabView program for the 1K probe . . . . . . . . . . . . . . . . . . . 78

2.15 The portable measurement unit . . . . . . . . . . . . . . . . . . . . . 81

3.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.2 Metal-insulator change-over in the R2Ir2O7 series . . . . . . . . . . . 91

3.3 Resistivity data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4 T∗ cross-over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5 Continuous quantum phase transition in Eu2Ir2O7 . . . . . . . . . . . 98

3.6 Magnetoresistance data . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.7 Phase diagram of Eu2Ir2O7 . . . . . . . . . . . . . . . . . . . . . . . . 103

4.1 Phase diagram of Fe based superconductors . . . . . . . . . . . . . . 107

4.2 Fermi surface of cuprates and pnictides . . . . . . . . . . . . . . . . . 108

4.3 FeCrAs crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 FeCrAs: magnetic susceptibility and specific heat data . . . . . . . . 113

4.5 FeCrAs: resistivity data at ambient pressure . . . . . . . . . . . . . . 115

4.6 FeCrAs: resistivity data under pressure . . . . . . . . . . . . . . . . . 117

xii

4.7 Pressure dependence of TN in FeCrAs . . . . . . . . . . . . . . . . . . 119

4.8 Pressure dependence of ρ(T ) slopes at different pressures . . . . . . . 120

4.9 The incipient Mott insulator state of pnictides . . . . . . . . . . . . . 123

5.1 Daphne oil phase diagram . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2 Argon loading platform . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3 The phase diagram of molybdates . . . . . . . . . . . . . . . . . . . . 135

5.4 Highest pressure data on FeCrAs . . . . . . . . . . . . . . . . . . . . 141

xiii

Chapter 1

Introduction

1.1 Metal Insulator Transition

Electrical conductivity of condensed matter systems varies in a wide range from

∼ 10−22 Ω−1cm−1 in an insulator to ∼ 1010 Ω−1cm−1 in some pure metals (neglecting

superconductors). Despite the plethora of experimental and theoretical surveys over

the last eight decades, the enigmatic metal-insulator phase transition (MIT) remains

one of the most challenging problems of condensed matter physics. The electronic

states responsible for charge transport are spatially extended in a metal but localized

in an insulator. The task of non-perturbatively connecting these two extremes has

proven to be a very difficult one. It is generally agreed that localization can occur by

means of static disorder (Anderson model, [1]), strong electron-electron correlations

(Mott-Hubbard transition, [2]) or strong electron-lattice coupling [3]. Of course in

many cases a combination of these mechanisms is at work and one may reinforce the

1

Chapter 1. Introduction 2

other.

Phenomenologically a metal is a condensed phase of matter with a finite DC con-

ductivity at zero temperature, σ(0) 6= 0, whereas an insulator has zero residual DC

conductivity at zero temperature, σ(0) = 0 or equivalently ρ(0) = ∞. Microscopi-

cally, this is due to the fact that a metal has a Fermi surface with gapless excitations

at zero Kelvin, whereas an insulator has a gapped excitation spectrum. There exist

certain criteria which predict the possibility of a MIT in a solid state system. In this

chapter, I will discuss three criteria for metal to insulator transitions. The first one,

proposed by Herzfeld, explains the transition by approaching it from the insulating

side. The other two, proposed by Mott, approach the MIT from the metallic side.

Mott’s first criterion is almost identical with Herzfeld criterion, however, Mott also

put forward the powerful statement that: “Any MIT at T=0 is discontinuous”. We

will see later in this chapter that his statement was in fact wrong.

1.1.1 Herzfeld Criterion

Perhaps the first serious discussion on the metal-insulator phase transition was the

work of Herzfeld 1 [4]. The Herzfeld criterion approaches the MIT from the insulating

phase where electrons are localized. He showed how electrons could delocalize based

on a polarizability argument. The electric polarizability defined through P = αE

determines the response of valence electrons around nuclei to an external electric field.

By response, I mean a relative charge distribution distortion around each nucleus, as

1Karl Ferdinand Herzfeld (1892-1978) was born in Vienna (Austria) and died in Washington DC(United States).


a result of the applied field. In condensed phases, many body interactions amplify

this effect by a factor [1−(4π/3)(N/V )α0]−1 where N is the Avogadro number and V

is the molar volume [4]. Increasing density (decreasing V) gives rise to a divergence

in this factor which results in the release of the bound electrons and yields a metallic

phase. The Herzfled criterion for such a dielectric catastrophe is given by:(

4π

3

)(

N

V

)

α0 = 1 (1.1)

Eq. 1.1 is sometimes written as R/V = 1 where R ≡ (4π3

)Nα0 is the molar

polarizability. At high densities, R/V ≥ 1, valence electrons delocalize due to the

cooperative polarization effect explained above and the Drude free electron picture

becomes applicable. The Clausius-Mossotti formula relates R/V to the dielectric

constant of insulators through:

R

V=ǫ− 1

ǫ+ 2=n2 − 1

n2 + 2(1.2)

with ǫ and n being the dielectric constant and the refractive index of the medium.

R/V = 1 translates into ǫ − 1 = ǫ + 2 which is valid only when ǫ → ∞ hence the

name dielectric catastrophe. One may take elements from the periodic table and

calculate the R/V ratio for them and show which ones are metallic and which ones

are insulating [5].

1.1.2 Mott Criteria

Mott criteria show how electron localization happens as the MIT is approached from

the metallic side [6]. A metal is made of a lattice of positively charged ions2 plus

2Each ion is made of a nucleus and the valence electrons which are bound to the nucleus.


Figure 1.1: (a) Mott’s first criterion. The dilute system (left) is an insulator and

the dense electron system is a metal (right). a is the interdonor spacing and the

electron density is proportional to 1/a. (b) In a metallic phase, the mean free path

of electrons (l) is larger than their de Broglie wavelength (λF ). The Mott transition

occurs at T = 0, when σ falls below σmin. The blue line shows that passing the Mott

limit results in a first order MIT transition at T = 0 as predicted by Mott. The red

dotted line represents a continuous MIT.


conduction electrons which are free from the ions. At sufficiently high densities, the

itinerant electrons screen the attractive Coulomb potential between electrons and

ions effectively. As a result, conduction electrons will not make bound states with

ions, and the system is a metal with free conduction electrons. If the density of

electrons is gradually decreased by increasing the interparticle distances, the screen-

ing becomes less effective. Eventually, at some critical density (nc) the electron-ion

bound states form and the system becomes an insulator with σ(0) = 0 at T=0

(Fig. 1.1(a)).

Mott argued that at a critical interdonor distance (ac) a first order transition from

metal to insulator would occur. Fig. 1.1(b) compares the first order Mott transition

(blue line) with a second order MIT (red dotted line) at T = 0.

Mott’s first criterion is usually stated in terms of the critical density of donors

nc and the first Bohr radius of the donor nucleus a∗:

n1/3c a∗ ≤ 0.25 (1.3)

Notice that Mott transition occurs at T=0 K, hence the true experimental ver-

ification of Mott’s first order transition is impossible. However, most experimental

data are suggestive of a second order phase transition in both doped semiconduc-

tors and transition metal oxides. The conductivities have been measured down to

a few millikelvin but still one can argue that at even lower temperatures a thermal

activation in σ might set in.

Mott’s first criterion is dimensionally identical to the Herzfeld criterion. Polariz-

ability is proportional to atomic size estimated as a∗ and molar volume is the inverse

of density. Therefore Herzfeld criterion can be phrased as (a∗)3n ∼ 1.


Mott’s second criterion says that there exists a minimum conductivity σmin

for which a material could still be metallic, prior to localization of electrons [7]. His

idea was based upon the beakdown of the Boltzmann theory of electronic transport,

at the limit where the mean free path of electrons becomes less than the lattice

spacing. The conductivity of a metal at T=0 from the Drude model is:

σ(T = 0) =ne2τ

m≈(

e2

h

)

a2−d

2π

l

a= σ0

l

a(1.4)

In deriving the approximation above I used l = vF τ and kF ≈ (dπn)1/d. The electron

density is given by n ≈ a−d where a is the lattice spacing. Disorder enters through

the ratio l/a and if l has a minimum, so does σ. The minimum value of l is the

lattice spacing in Boltzmann transport theory which gives rise to σmin = σ0 from

Eq. 1.4.

Equivalently, minimum conductivity occurs in the limit where the mean free path

of conduction electrons in the lattice is of the order of their de Broglie wave length

λF = 2π/kF . The limit kF l ∼ 1 is referred to as the Ioffe-Regel limit [8].

Ioffe-Regel limit for a 3D metal which has ν conduction electrons per unit cell

can be calculated considering the density of electrons n = ν/a3 where a is the cubic

unit cell dimension. Using the Drude model in three dimensions at the limit l ∼ a

we have:

σ =ne2τ

m∗ =ne2l

hkF

=ne2

h

l

(3π2n)1/3

l∼a→ σIR =e2

h

2πν

(3π2ν)1/3

1

a(1.5)

The final result is written in terms of the quantum of conductance e2/h = 3.9 ×

10−5 Ω−1m−1 and the inverse lattice spacing 1/a. Of course the IR limit of resistivity

can be calculated through ρIR = σ−1IR and the MIT is expected when ρ ≥ ρIR.


The minimum conductivity criterion emphasizes the role of disorder. The mean

free path is controlled by disorder and it was shown by Anderson [1] that the extended

electronic states of a metal can become spatially localized in the limit of strong

disorder. The limit of strong disorder is where the mean free path of the electrons

becomes comparable to the interatomic distance (l ∼ a) which results in σ = σmin.

Furthermore Mott predicted that the resulting MIT is discontinuous which was later

proved to be wrong. The σmin criterion neglects the role of interactions. The density

criterion on the other hand is based on the effect of the Coulomb interactions and

ignores the role of the disorder. Often in the same system, both effects play a role

in the localization of electrons.

1.1.3 Anderson Localization

Anderson theory of localization explains how static but random disorder localizes

electrons in a lattice. Prior to Anderson’s seminal paper [1] the common belief was

that scattering by a random potential causes Bloch waves to loose coherence over a

length scale l known as the mean free path. Nevertheless the wave functions remain

extended. Anderson showed that if the random potential is strong enough it may

actually localize the electronic wave functions by dressing them in an exponentially

decaying envelope:

|Ψ(r)| ∼ e−(r−r0)/ξ (1.6)

This theory applies to the limit of strong disorder where perturbation theory

breaks down hence it is a non-perturbative approach. Anderson used the following

tight binding Hamiltonian to explain the quantum mechanical electronic transport


Figure 1.2: (a) top: Extended Bloch states loose phase coherence over a scattering

length scale l in the presence of a weak random disorder. bottom: In the limit of

strong disorder, Anderson localization causes Bloch functions to decay exponentially

over a localization length scale ξ which determines the spatial extent of the wave

packet. (b) Minimum conductivity can be achieved by fixing the strength of the

disorder potential and varying the energy of the Bloch waves. Once the Fermi energy

is shifted by a mobility edge Ec metallization occurs. This graph is equivalent to

Fig. 1.1(b)


in the presence of a random potential:

H =∑

n

ǫna†nan + V

∑

n,m

a†nam (1.7)

The first term is a sum of the atomic energies of electronic states on each lattice site

n assuming a single orbital. The second term is a NN hopping with the same matrix

element V over the entire system. ǫn belongs to a uniform distribution of width W.

If W=0, a fully ordered phase is achieved where ǫn is the same for all the lattice

sites and electrons form delocalized Bloch states with ballistic transport. Disorder

is introduced into this Hamiltonian by allowing a finite width for the distribution of

orbital energies i.e. W6=0.

If V=0 no hopping is allowed between sites and the system becomes insulating i.e.

σ(T = 0) = 0. One can naively argue that by tuning W/V, a smooth transition can

happen at a critical value (W/V )c, between the limit of strong localization V=0 and

the limit of ballistic transport W/V ≪ 1 . It turns out that this naive idea is wrong.

I will explain in the rest of this chapter that in fact in one and two dimensional

systems, the smallest amount of disorder is enough to immediately localize all the

electronic states and MIT is only truly realized in 3D systems. Even in 3D, it is far

from trivial to see whether the transition is continuous or not.

To study the localization process, Anderson studied the time evolution of wave

functions in the impurity band of the Hamiltonian 1.7. Imagine a particle is located

at site n at time t = 0. Hopping processes will allow the particle to move away from

its original site. Anderson calculated the site-return probability and argued that if

this probability is non-zero, the particle is localized. There is a close connection

between the site-return probability and the diagonal elements of the Green function


[9, 10]. Using the definition of the Green function G(E) = (E − H)−1 where H is

given by Eq. 1.7 we have:

(E − ǫn)Gnm(E) = δnm +∑

n6=l

VnlGlm(E) (1.8)

In the limit of strong localization where V = 0, the second term vanishes and the

energy spectrum is given by the simple poles of the Green function E = ǫm. The

eigenstates of the Green function |φm〉 in this case, reduce to the atomic orbitals |m〉

which form the basis of the tight-binding Hamiltonian ( |φm〉 = |m〉 ).

When V 6= 0, the eigenstates of the Green function will be superposition of the

atomic basis of our tight binding Hamiltonian (Eq. 1.7) i.e. |φm〉 =∑

n cnm|n〉. If

a sufficiently large number of cnm vanish, the resulting eigenstates |φm〉 are char-

acterized by an exponentially decaying envelope, φm ∝ e−r/ξm with ξm being the

localization length of the eigenstate (Fig. 1.2(a)). In this case, Gnn can be expanded

in powers of V :

Gnn(E) =1

(E − ǫn)+∑

l 6=n

VnlVlm

(E − ǫn)(E − ǫl)+ · · · =

1

E − En − Sn(E)(1.9)

The form of the Green function above, is identical to the form of the site-

probability amplitudes. Random walks of all lengths are summed into the site self-

energy Sn(E). The self-energy determines the fate of the localization at long times

(E → 0). In general E can be complex. If Sn(E) is purely real as E approaches

the real axis, the singularities of the Green function are again simple poles and the

eigenstates are localized. However, if ImSn(E) 6= 0 as E approaches the real axis,

the eigenstates of the Green function are extended and their life-times are given by


the inverse of ImSn(E). The critical value (W/V )c can be deduced by requiring

ImSn(E) = 0.

In dimensions d ≤ 2 an infinitesimal amount of disorder in enough to localize the

electrons. In three dimensions forW/V > (W/V )c no extended states exist but below

this limit the extended and the localized states coexist. The separation between the

extended and the localized states is called the mobility edge Ec. An electron in

the localized state requires extra energy Ec to transfer into the conduction state.

To understand the concept of the mobility edge, one can imagine the case where

the strength of the disorder potential is fixed and the Fermi energy lies within the

localized band. In this case σ(T = 0) = 0 and the system is insulating. Now if we

shift ǫF to ǫF + Ec the system becomes metallic so instead of keeping the energy of

the Bloch waves fixed and changing the strength of the disorder potential, one can

fix the latter and change the former by the mobility edge. In this sense, Ec reflects

the strength of the disorder (compare Fig. 1.1(b) and Fig. 1.2(b)).

One can imagine the case W/V < (W/V )c and ask the question; what happens

to conductivity near the mobility edge? Does the system go continuously or abruptly

from metallic to the insulating regime? Both possibilities are shown in Fig. 1.1(b) and

1.2(b). Mott suggested that there is a minimum conductivity below which the system

can no longer support a metallic phase. Mott’s idea was based on the breakdown

of Boltzmann transport theory of electronic conduction in semiconductors at the

limit where the mean free path of electrons is comparable to the lattice spacing.

Experimentally this prediction has not been verified and in fact it was shown later

that Mott’s controversial prediction on the first order nature of MIT was wrong. In


the next section I review a theory which successfully proves this transition to be

continuous in accord with the experimental results[11, 12].

1.1.4 Scaling Theory

The goal of the scaling analysis is to provide a universal relation between the dimen-

sionless versions of an experimentally measurable observable and the tuning parame-

ter such that all experimental data collapse onto the same universal function. In the

scaling theory of localization the dimensionless observable is the so called Thouless

dimensionless conductance g(L) which is the ratio of the experimentally measurable

conductance G(L) divided by the quantum of conductance e2/h. Thouless raised the

question: what happens to the conductivity of a hypercubic system of edge length

L if it is scaled to pL? Here the tuning parameter is the linear dimension of the

hypercubic system whose dimensionless version is the ratio L/ξ with ξ being the

localization length from Eq. 1.6.

In the metallic regime conductance is an extensive quantity, related to the con-

ductivity through Ohm’s law:

G(L) = σLd−2 (1.10)

for the hypercubic system. The insulating phase, however, is characterized by a

non-ohmic relation:

G(L) ∝ e−L/ξ (1.11)

Consider that the linear dimension L of our hypercubic system is scaled to pL. As a

result the energy of an electronic state will shift from E to E + ∆E and the energy

level separation from W to W + ∆W . One can think of the series of hypercubic


systems with different p as one system whose bandwidth ∆E and band separation

∆W can be tuned. Obviously by tuning the ratio ∆E/∆W one can go from the

insulating to the metallic regime and the larger this ratio the larger the conductance.

Thouless argued that the dimensionless universal conductance is in fact given by this

ratio: g = ∆E/∆W [13].

The bandwidth can be estimated as ∆E = h/τ where τ is the diffusion time. In

a cube of edge length L, it takes τ = (L/2)2/D0 for an electron to diffuse from the

centre to a corner. D0 is the diffusion constant which is related to the conductivity

through the Einstein relation: σ = 2e2nD0. Hence we can write:

∆E =h

τ=

h

(L/2)2/D0

=2h

e2σ

nL2(1.12)

The band separation energy is determined by the number of electrons ∆W = 1/N =

1/nLd hence:

g =∆E

∆W=

(2hσ/e2nL2)

1/nLd=

2h

e2σLd−2 (1.13)

and finally using the Ohm’s law (Eq. 1.10) we have:

g(L) =2h

e2G(L) (1.14)

This is how the universal dimensionless conductance is related to the ohmic conduc-

tance in the metallic regime. Notice the importance of the idea that electron motion

in a disordered metal is diffusive instead of ballistic beyond the mean free path. The

phase of an electron’s wave function moving in a disordered medium fluctuates ran-

domly. The distance over which it fluctuates by about 2π defines the mean free path

(l) beyond which the electron motion is diffusive.


The key assumption in the in the scaling theory of localization is that the log-

arithmic derivative of the conductance with respect to the system size:

β =d ln g(L)

d lnL(1.15)

is a monotonic function of the Thouless conductance. If in the limit L→ ∞,

β(g) > 0, then g(L) must diverge which signifies extended states and metallic be-

haviour. If in the same limit, β(g) < 0, then g(L) monotonically vanishes, charac-

terizing localization and insulating behaviour. Hence, the sign of β(g) determines

whether the system is metallic or insulating.

In the regime of high conductance (weak disorder limit, the Ohmic regime), Eq.

1.10 and 1.15 give β(g) = d− 2. In the regime of low conductance (strong disorder

limit), Eq. 1.11 and 1.15 give β(g) = ln g. Our assumption of monotonicity guar-

antees a smooth interpolation between the two limits (Fig. 1.3). According to this

analysis, β(g) < 0 for d ≤ 2 hence all states become localized by the smallest amount

of disorder in 1D and 2D. On the other hand, a 3D system can support a localization

transition at a critical value gc = (∆E/∆W )c where β(gc) = 0.

According to the discussion above and Fig. 1.3 the β function is continuous across

the transition in 3D, hence the scaling analysis does not support Mott’s minimum

conductivity idea. We can in fact derive the actual functional form of β near the

critical point to show the continuous nature of the transition. Near gc, β is continuous

and can be linearized:

β(g) =g − gc

νgc

(1.16)

where ν > 0 determines the slope. Near gc, β varies slowly hence we may approximate


Figure 1.3: (a) The scaling analysis requires the β(g) function to be asymptotic to

d − 2 in the limit of large g, linear as a function of ln g in the limit of small g, and

smoothly connected in the intermediate regime (b) The dimensional dependence of

the scaling function β(g) as a function of ln g. There is no MIT in d=1, 2 and no

discontinuity at the transition (β = 0) in d=3.


it as:

β(g) =d ln g(L)

d lnL≈ 1

gc

dg

d lnL(1.17)

Equating the right hand sides of Eq. 1.16 and Eq. 1.17 and integrating over [L,L0]

we obtain:

g − gc

g0 − gc

=

(

L

L0

)1/ν

(1.18)

where g0 = g(L0). Let us first approach the transition from the localized side where

g0 < gc. One can easily show that in this regime Eq. 1.18 yields:

g = gc

(

1 −(

L

ξ

)1

ν

)

with ξ = L0

(

gc

g0 − gc

)ν

= L0|ǫ|−ν (1.19)

Here |ǫ| measures our dimensionless distance from the critical point. The con-

ductance decreases algebraically with the length of the sample. The fact that the

length scale ξ diverges as we approach the fixed point gc from the insulating side,

establishes it as the localization length.

Let us now approach gc from the metallic side. We know that for a large enough

sample Ohm’s law (Eq. 1.10) must hold. On the other hand, the scaling hypothesis

guarantees that the Thouless conductance is a universal function of L/ξ on all length

scales i.e. g = g(L/ξ). Therefore:

g = g(L/ξ)

G(L) = σLd−2⇒ σ(T = 0) ∝ ξ2−d ∝ |ǫ|(d−2)ν (1.20)

which guarantees that conductivity vanishes continuously at the critical point of an

Anderson localization transition, opposite to the discontinuous drop predicted by

Mott. I conclude this section by stating the main features of the scaling theory:


1. The smallest amount of disorder localizes electronic states in 1D and 2D sys-

tems.

2. 3D systems can support a localization transition.

3. Anderson localization in 3D is continuous.

1.1.5 Pressure Induced Metallization

Generally speaking, pressure favours metallic phases 3. The most obvious effect

of pressure is to compress the lattice and to reduce the unit cell dimensions. The

Heisenberg uncertainty principle [∆x,∆p] = h forces ∆p to increase as ∆x is de-

creased by pressure. Increasing ∆p increases the kinetic energy of electrons and

eventually metallizes the system 4.

The effect of pressure can also be viewed within a band theory picture. The

band theory of solids is based upon the Schrodinger equation for electrons in a weak

periodic potential:

Hψ =

(

− h2

2m∇2 + U(r)

)

ψ = Eψ (1.21)

The potential energy has the periodicity of the Bravais lattice U(r) = U(r + R).

Typically R ∼ 10−10 m is of the order of an electron’s de Broglie wave length hence

quantum mechanical treatment is required. In general U may be a sum over electron-

ion, electron-electron and exchange terms:

U(r) = Uei + Uee + Uex (1.22)

3The exceptions are beyond the scope of this chapter.4Quantum mechanical effects can be quite counter intuitive; in this case, as we confine electrons

to smaller volumes, their kinetic energy increases.


As a first step let us neglect the complexities of the above sum and treat U(r)

as a single potential term which is periodic in R. Bloch’s theorem states that the

eigenstates of the Hamiltonian in Eq. 1.21 are essentially plane waves modulated by

an amplitude function with the periodicity of the Bravais lattice:

ψnk = eik.runk(r) with unk(r) = unk(r + R) (1.23)

The method most widely used to include the details of the potential energy terms of

Eq. 1.22 is the local density approximation. LDA simplifies the many body fermion

problem into a mean field approximation in which each electron moves in the self

consistent average field of all the other electrons and nuclei. The complicated many

body exchange and correlation terms included in the average field U(r) are conven-

tionally approximated by a functional of the local density of electrons hence the name

LDA.

The energy functional has the form E[ρ(r)] = T + U with U given in Eq. 1.22.

E[ρ(r)] can be minimized using a wave function gradient method (∂E/∂φj = 0)

which leads to a set of one-electron Schrodinger equations with the same potential

field for all the electrons:

(

− h2

2m∇2 + Vei(r) + e2

∫

dr′ρ(r′)

|r − r′| +∂Uxc

∂ρ(r)

)

φj(r) = ǫjφj(r) (1.24)

The single electron wavefuncions φj are found self consistently through iterative

minimizations and guessing. ǫj are single electron eigen-energies. Pressure can be

obtained using the virial theorem:

PV =2T

3+U

3(1.25)


where T is the kinetic energy and U is the full potential energy (Eq. 1.22).

To see the effect of pressure, let us rewrite Eq. 1.21 in the momentum space, using

the following Fourier transformations:

ψk(r) =∑

K ck−Kei(k−K).r

U(r) =∑

K UKeiK.r

⇒(

h2

2m(k − K)2 − ε

)

ck−K +∑

K′

UK′−Kck−K′ = 0

(1.26)

K and K′ are reciprocal lattice vectors. The above momentum space Schrodinger

equation gives plane wave solutions when UK′−K = 0 with quadratic dispersions.

In the case of a weak non-zero potential one arrives at Bloch wave solutions whose

eigen-energies can be calculated through perturbation theory. The result of such a

calculation for the energy levels near a single Bragg plane is [14]:

ε =1

2(ε0

q + ε0q−K) ±

[(

ε0q − ε0

q−K

2

)

+ |UK |2]1/2

(1.27)

which shows that including potential energy has two consequences:

1. a gap opens at the Bragg plane q = 12K where the free electron levels ε0

q

and ε0q−K intersect. In other words, the degeneracy between the plane wave

solutions is lifted by a gap of size 2|UK |.

2. the bandwidth is changed.

Pressure can tune the strength of interactions, hence it may change the size of the

band gap and vary the bandwidth. The pressure induced metallization occurs via

increasing the bandwidth and decreasing the band gap. Metallization by “band


closure” is observed in rare gases and alkali halides due to overlap of a filled p-like

band and an empty d-like band. Another example, in SmTe and SmSe, metallization

occurs due to the crossing of a narrow partially filled lanthanide f level with a d-like

conduction band 5 [15].

So far we considered electrons in a metal as nearly free fermions, weakly perturbed

by the periodic potential of the ions. The resulting band theory is constructed

from Bloch states occupying quantized k-levels and the effect of interactions is to

open band gaps and vary the bandwidth. A second approach to construct the band

structure of solids is the tight-binding method which considers the solid to be made

of weakly interacting atoms.

In a tight-binding model, the starting electronic states are the atomic orbitals

as opposed to free electrons. Once these atoms are brought close to each other, the

spatial extent of certain orbitals may become comparable to the interatomic distances

which results in a considerable overlap between the orbitals and allows for electrons

to hop between neighbouring atoms. The tight-binding approximation deals with

the case where the overlap of atomic wave functions is enough to require corrections

to the picture of isolated atoms, but not so much as to render the atomic description

completely irrelevant. This method is in particular applied to the d-shell transition

metal oxides.

The tight binding Hamiltonian is composed of the atomic potential Hat plus all

5Nearly all the divalent chalcogenides of the rare earth metals Sm, Tm, Yb, and Eu undergopressure-induced semiconductor to metal transitions.


corrections (∆U) required to produce the full periodic potential of the crystal:

H = Hat + ∆U(r) where Hatψn = Enψn (1.28)

Tight binding electron wave functions are constructed from a linear superposition of

the atomic wave functions ψn:

ψk(r) =∑

R

eik.Rφ(r − R) with φ(r) =∑

n

bnψn(r) (1.29)

The factor eik.R ensures that the final product ψ obeys Bloch condition ψ(r + R) =

eik.Rψ(r). The energy bands are obtained by solving the full crystal Schrodinger

equation:

Hψ(r) = (Hat + ∆U)ψ(r) = ε(k)ψ(r) (1.30)

The sum in Eq. 1.29 does not run over all atomic orbitals ψn, rather it runs over the

few relevant ones which contribute to the conduction band. s levels give rise to s

bands and d levels give rise to d band. Of course we can have hybridization between

levels to produce mixed bands. The coefficient bn is calculated by multiplying both

sides of Eq. 1.30 by ψ∗ and integrating. This results in overlap integrals which not

only determine both the tight-binding wave function and the band energies. A simple

calculation [14] for the s band gives:

ε(k) = Es − β −∑

NN

γ(R) cosk.R (1.31)

with Es being the single atomic s orbital energy and β and γ being the overlap

integrals:

β = −∫

dr∆U(r)|φ(r)|2 γ(R) = −∫

drφ∗(r)∆Uφ(r − R) (1.32)


This equation reveals the characteristic feature of tight-binding energy bands: The

band width, i.e. the spread between the minimum and the maximum energy of a

band, is proportional to the overlap integrals. The larger the overlap, the wider the

tight-binding bands. Pressure increases the overlap between the neighbouring atomic

orbitals by tuning the lattice parameters, hence increases the bandwidths and drives

the system towards metallization. In the context of the Hubbard model, the hop-

ping integral t is determined by the overlap between neighbouring orbitals. Raising

pressure increases the overlap and increases the t matrix elements and decreases the

U/t ratio in the Hubbard model.

For example in the tight-binding Hamiltonian proposed by Pesin and Balents for

the pyrochlore iridates [16], the band structure calculations give rise to the phase

diagram with tuning parameters U/t and λ/t. Increasing the Hubbard U reduces

the bandwidth and enhances the effect of the spin orbit coupling6.

The effect of pressure on the metal-insulator transition can be summarized as

follows:

1. The Heisenberg uncertainty principle forces ∆p to increase as ∆x is decreased

by pressure. Increasing ∆p increases the kinetic energy of electrons and even-

tually metallizes the system.

2. Shrinking the unit cell dimensions and increasing the density of electrons results

in increasing the band width.

3. Gap closure: usually comes from increasing band widths, so they cross and

6Decreasing t makes the ratio λ/t larger


close the gap.

4. Band mixing: Hybridization of more localized (flatter) bands such as f bands

with more mobile bands such as d, p, s bands.

5. Increasing the overlap in the tight-binding picture which increases the band

width.

1.1.6 Experimental Examples

To show experimental examples of metal insulator transition under pressure, I briefly

review two materials, namely Cr-doped vanadium oxide and samarium hexaboride.

(V1−xCrx)2O3

The stoichiometric compound V2O3 is a paramagnetic metal at high temperatures

and becomes an AFM insulator below 150 K. Doping with chromium makes the

system insulating at high temperatures with no magnetic order [17]. At room tem-

perature the system goes through a metal-insulator transition (MIT) with no sym-

metry breaking as a function of x in (V1−xCrx)2O3. The same MIT is observed in

(V1−xCrx)2O3 with fixed x by applying pressure. Early studies show that in cross-

ing the phase boundary from the insulating to the metallic phase, decreasing the

Cr concentration by ∆x ∼ −0.01 is roughly equivalent to increasing pressure by

∆P ∼ 4 kbar [18, 19].

Limelette et al. showed that pressure induces a first order MIT in (V0.989Cr0.011)2O3

[20]. The first order transition line in the P − T phase diagram ends in a second


Figure 1.4: (A) shows the conductivity data as a function of pressure and temper-

ature. (B) shows the P − T phase diagram. The shaded area is the coexistence

region. PM is the lowest pressure at which a metallic state can be sustained while

decreasing pressure and PI is the the highest pressure at which an insulating state

can be sustained while increasing pressure. (C) shows the global phase diagram of

Cr-doped vanadium oxide. Reprinted figures with permission from [20]. Copyright

(2011) by the Science publications.


order critical end-point (Pc, Tc) = (3738 bar, 457.5 K) (Fig. 1.4(a)). The shaded area

in the P −T phase diagram (Fig. 1.4(b)) shows the coexistence region. By increasing

temperature, the hysteresis loops become smaller and eventually hysteresis vanishes

at Tc = 457.5 K. Varying pressure rather than temperature is essential for a precise

determination of Tc. At the critical temperature, the pressure dependence of (P, Tc)

becomes singular, with a vertical tangent at the critical pressure Pc. For T > Tc,

this singular behaviour is replaced by a continuous variation of the conductivity with

pressure, which defines a cross-over line in the (P, T ) phase diagram (as also depicted

in Fig. 1.4(b)).

A scaling analysis shows that all the conductivity data in the metallic phase at

T > Tc and T < Tc can be collapsed on two respective universal scaling functions

f+ and f− [20]. The scaling analysis shows that the critical exponents of the Mott

transition in (V0.989Cr0.011)2O3 are similar to the mean field values for the liquid-gas

phase transition i.e. both systems belong to the same universality class. The idea of

the universality class and critical exponents will be explained in section 1.2.

SmB6

SmB6 is a strongly correlated electron system with a low concentration of carriers and

a small gap close to the Fermi level. The formation of a small gap in a system which

comprises s-,p- and f -electrons is believed to originate from the hybridization of the

well localized f -states with the broad s-,p-states. Such heavy fermion semiconductors

are commonly referred to as Kondo insulators.

The susceptibility of SmB6 fits to a Curie-Weiss model at T > 100 K, implying the


Figure 1.5: (a) The resistivity data shows a continuous metal to insulator transition

in SmB6 at Pc ≈ 40 kbar. (b) The two-gap structure is shown. Eg represents the

hybridization gap and Ed represents the distance between the bottom of the conduc-

tion band from the in-gap states which are formed within the hybridization gap. (c)

Both Eg and Ed close continuously under pressure at Pc ≈ 40 kbar. Reprinted figures

with permission from [22]. Copyright (2011) by the Amercian Physical Society.

local-moment behaviour. But below 10 K, the susceptibility is Pauli-like indicating

itinerant electron behaviour. The resistivity of the compound at ambient pressure

shows thermal activation with a small gap Eg ∼ 12 meV in the temperature range

15 < T < 70 K. Below 5 K, ρ(T ) levels off and shows no temperature dependence

(see Fig. 1.5(a)) [21].

The saturation of ρ(T ) below 5 K is attributed to a narrow in-gap band which

is separated from the bottom of the conduction band by a direct activation energy

Ed ≈ 5 meV i.e. the Fermi level is pinned inside the hybridization gap (Eg ≈ 12 meV).


Table 1.1: Power law fits to the formula ρ(T ) = ρ0 + AT x in SmB6 show that the

exponent x evolves from 1 to 2 by increasing pressure. The results of the fits are

summarized below.

Pressure 40 kbar 54 kbar 60 kbar 70 kbar

Exponent x=1 x=1.56 x=1.9 x=2

The values of Ed and Eg can be derived from an Ahrenius analysis. Figure 1.5(b)

shows that there are two gaps in the activation energy spectrum of the system. The

higher energy feature in this plot represents Eg which shifts towards lower energies

with increasing pressure. The lower energy feature is Ed which remains unchanged

under pressure [22].

Figure 1.5(a) shows that increasing pressure changes the insulating behaviour at

15 < T < 70 K into a metallic one, continuously, at Pc ≈ 40 kbar. Figure 1.5(b)

shows that Eg shifts to lower temperatures as P is increased and eventually merges

with Ed. Both Eg and Ed vanish continuously at 40 kbar (Fig. 1.5(c)).

Another interesting feature of SmB6 is the evolution of the low temperature re-

sistivity behaviour from non-Fermi liquid (NFL) to Fermi liquid (FL) as the pressure

is increased above Pc. Power law fits of the form ρ(T ) = ρ0 +AT x show an evolution

from x = 1 to x = 2 which is summarized in table 1.1. There is a connection between

the continuous MIT transition in the ground state of the system and the observation

of NFL power law behaviour. Some basic ideas regarding the connection between

quantum phase transitions and unusual power law behaviours at finite temperatures


is discussed in the next section.

1.2 Quantum Phase Transitions

1.2.1 Basic Notions

Continuous Quantum Phase Transitions (QPT) are defined as: continuous phase

transitions which occur at absolute zero of temperature as a result of tuning a non-

thermal parameter [23]. Examples of the tuning parameter are the magnetic field,

pressure, disorder, doping or any physical observable other than temperature. Since

a QPT happens at T=0 it represents a fundamental change in the ground state

of the system. Consider a system defined by the Hamiltonian H(g) where g is a

dimensionless coupling, representing the non-thermal tuning parameter. In order to

study a QPT we monitor the evolution of the ground state of H(g) as we tune g

and look for non-analytic behaviour at a critical value g = gc. This point (T=0,

g = gc) is called the Quantum Critical Point (QCP) and is defined to be: a point of

non-analyticity in the ground state properties of H(g).

As a result of our definition, all finite temperature phase transitions will be re-

garded as classical. The word “classical” does not exclude the involvement of quan-

tum mechanics. For example the order parameter of a superfluid is a complex-valued

quantum mechanical wave function but the superfluid-normal phase transition of 4He

at finite temperature, Tc = 2.17 K, is a classical phase transition (CPT).

Whenever a system crosses the phase boundary of a CPT, a correlation length and

a correlation time diverge. This means that the order parameter fluctuates coherently


over increasing distances and ever more slowly. Since the critical fluctuations show

a diverging correlation time (τ → ∞) close to the critical point, we can assign to

them a frequency ω ∝ 1/τ and assert that as we approach the critical point ω → 0.

Therefore in a close neighbourhood of the classical (finite temperature) critical point

(CCP) we have:

hω ≪ kBTc (1.33)

This means that the critical fluctuations which drive the system to cross the phase

boundary are soft modes in a high temperature regime where all the quantum me-

chanical properties are smeared out, hence they can be treated by classical statistical

mechanics. This is why we call such phase transitions “classical”. A QPT happens

at T = 0 where Eq.(1.33) can no longer be valid. Hence, the critical fluctuations

obey quantum statistics and can not be treated classically. At T = 0 no thermal

energy exists and the QPT is driven by purely quantum critical fluctuations whose

origin lie in Heisenberg’s uncertainty principle [24].

There are two possibilities for a Quantum critical point (QCP) in general. The

first one happens in a system which shows no order at finite temperatures 7 (Fig. 1.6(a)).

Such systems have a QCP at (T = 0, g = gc) but as soon as temperature is raised

above the absolute zero, the quantum ordered phase will disappear for all values of g.

In the second case (Fig. 1.6(b)) there is a whole line of classical critical points (CCP)

at finite temperatures terminating at a QCP at (T = 0, g = gc). In the shaded area

around the CPT line, Eq. 1.33 is valid, meaning that classical statistical mechanics

governs the critical behaviour within the shaded area.

7For example a two dimensional magnet with SU(2) symmetry.


Figure 1.6: Two possible behaviours of a system close to the quantum critical point

(T=0 , g = gc). The full line in (b) shows Tc(g). Equation 1.33 is valid only in the

shaded area in a close neighbourhood of this line where critical fluctuations can be

treated by classical statistical mechanics. The only point where Eq. 1.33 is not valid

is right at T=0.

In the rest of this section, I will first explain the critical behaviour in a classical

phase transitions and then compare with the quantum case. The general features

of continuous phase transitions which are common between QPT and CPT are the

existence of diverging correlations, singular behaviour of the second order derivatives

of the free energy, and the universal behaviour at the scaling limit.

1.2.2 Classical Critical Behaviour

As we have already established, a CCP is a point on the phase diagram which sepa-

rates the ordered phase from the disordered phase at a finite transition temperature

(Tc 6= 0). The disordered phase is characterised by the large amplitude fluctuations

of the order parameter (O) such that 〈O(x, t)〉 = 0. The ordered phase on the other


hand is characterised by two facts: firstly, the appearance of the order parameter,

〈O(x, t)〉 6= 0, and secondly, the divergence of at least one correlation length ξ and

a correlation time τ at the transition. Correlation length is the characteristic length

scale of the exponential decay of the equal time correlation function 〈O(x, t)O(0, t)〉.

Correlation time is the characteristic time scale of the exponential decay of the equilo-

cal correlation function 〈O(x, t)O(x, 0)〉. Close to the critical point, the only relevant

length and time scales of the system are ξ and τ which obey the following relations:

ξ ∝ |T − Tc|−ν (1.34)

τ ∝ ξz −→ τ ∝ |T − Tc|−zν (1.35)

Obviously they both diverge at the critical point (T = Tc). ν is called the critical

exponent and z is called the dynamical critical exponent 8. These exponents have

a universal behaviour at the scaling limit which means that their values are inde-

pendent of the microscopic details of the Hamiltonian (universal behaviour) close to

the critical point where the characteristic length scales of the system are the correla-

tion lengths which are much larger than the microscopic length scales of the system

(scaling limit).

Since ξ and τ diverge at the critical point the system does not “see” its microscopic

length scales such as the lattice parameter (a). In fact, the scaling limit of an

observable is defined as its value when all corrections involving the ratio of the small

lengths (a) to large lengths (ξ) are negligible (a/ξ → 0). In this limit, the physics

of the system is completely determined by the long range fluctuations of the order

8This is often reduced to the “dynamical exponent”.


parameter which happen within the length scale ξ. These fluctuations become slower

and slower as τ grows larger and larger. At the scaling limit, the values of the critical

exponents are determined only by global factors, namely the symmetries of the order

parameter and the dimensionality of the system. All systems which are the same with

regards to these two factors 9 belong to the same universality class which means that

they have the same critical exponents [25]. This means that if we pick the simplest

system in a certain universality class and calculate its critical exponents 10, we can

simply generalize the results to any other system belonging to the same class.

Using ω ∝ 1/τ defined in section 1.2.1 and Eq. 1.35, we realize that the energy of

the critical fluctuations (the only important energy scale at the transition) vanishes

with the power law:

hω ∝ ξ−z ∝ |T − Tc|zν (1.36)

For any finite temperature phase transition (Tc 6= 0), we can always find a neigh-

bourhood of the critical point where the energy of the critical fluctuations (hω) is

much less than the thermal energy at the critical temperature (kBTc). This neigh-

bourhood which is the shaded area of Fig. 1.6(b) can be calculated by solving Eq. 1.33

for T = Tc in a system of units where h = kB = 1. I will use this system of units

throughout this chapter unless otherwise mentioned.

ω ≪ Tc −→ |T − Tc|zν ≪ Tc −→ |T − Tc| ≪ T1

zν

c (1.37)

Since no quantum effects will survive in this neighbourhood, these critical degrees of

9In fact, members of a universality class have three things in common: the symmetry group of theHamiltonian (not the lattice), the dimensionality, and whether or not the forces are short-ranged.

10Notice that z and ν are not the only critical exponents. There are more critical exponents suchas α, β, γ, etc.


freedom obey classical statistical mechanics.

Since a QPT occurs at Tc = 0, one can never solve Eq. 1.37 in a neighbourhood

of a QCP. Hence, quantum critical fluctuations which originate from Heisenberg’s

uncertainty principle will not obey the classical statistics and need to be treated by

quantum statistical mechanics.

1.2.3 Quantum Critical Behaviour

We defined a quantum critical point to be the point of non-analyticity in the ground

state of a system defined through H(g). Most generally, this happens when H(g) =

H0+gH1 where H0 and H1 commute i.e. H1 is a conserved quantity. This means that

H0 and H1 can be diagonalised simultaneously so that eigenfunctions of H(g) are

independent of g whereas their eigenvalues depend on g. If at a certain value of g = gc

a level crossing occurs such that at gc an excited state becomes the ground state, gc

will be a point of non-analyticity of the ground state energy. This singularity may

be a result of an actual level-crossing or the limiting case of an avoided level-crossing

in a large lattice 11 (Fig. 1.7).

Let us characterize the energy of quantum critical fluctuations by ∆. If the

energy spectrum of the system has a gap, ∆ represents the difference between the

ground state energy and the first excited level. In a gapless spectrum, ∆ is defined

to be the energy scale at which there is a qualitative change in the nature of the

frequency spectrum [23]. One characteristic property of a quantum critical system

11The latter is more common.


Figure 1.7: E0, the ground state energy of H(g), is depicted as a function of g and

two possible level-crossings are shown: (a) represents an “avoided” level-crossing and

(b) is an “actual” level-crossing which is usually realized in the large lattice limit.

is the vanishing of ∆ near the QCP:

∆ ∝ |g − gc|zν (1.38)

This is the quantum analogue of Eq. 1.36 for the CPT, considering the distance from

the quantum critical point is |g−gc| in a QPT compared to |T −Tc| in a CPT. z and

ν are similarly called dynamical and critical exponents. ∆ can be associated with a

correlation time τ ∝ 1/∆ which diverges as we cross the QPT boundary

τ ∝ |g − gc|−zν (1.39)

and finally there is always a diverging correlation length which is related to the

correlation time through the dynamical exponent just like the classical case:

τ ∝ ξz −→ ξ ∝ |g − gc|−ν (1.40)

These two equations are the quantum counterparts of equations 1.34 and 1.35.


The concept of universality applies to QPTs as in CPTs. All quantum critical

systems with the same dimensionality and the same symmetries of the order param-

eter have identical critical exponents 12 in the scaling limit (large ξ and τ) i.e. they

belong to the same universality class.

For the first time in 1976 John A. Hertz demonstrated that one can draw an

analogy between certain universal classes of quantum phase transitions and classical

phase transitions [26]. This idea was further developed by A. J. Millis [27] in 1993

and a lot of attention have been drawn to this field afterwards. This analogy which is

often referred to as the QC analogy asserts that: “a QPT in a d dimensional system

shares the same features of a CPT in a d + z dimensional system with z being the

dynamical exponent”.

The extra z dimensions which enter the quantum problem are in fact imaginary

temporal dimensions. In an isotropic system the extra (imaginary) temporal di-

mension is equally weighted as the spatial dimension, i.e. a d dimensional quantum

problem maps into a d + 1 dimensional classical problem. However, when there is

anisotropy between spatial and temporal dimensions, the QC mapping is between d

dimensional quantum and d + z dimensional classical systems. This entanglement

between time and space originates from the quantum mechanical commutation rela-

tions which are encoded in the partition function of a quantum system.

Fig. 1.8(a) is the completed version of Fig. 1.6(a). On the horizontal axis (T = 0),

there is a QCP which separates the ordered ground state from the disordered one.

12By critical exponents, I mean both critical and dynamical exponents. In other words, zν isidentical in all members of a certain universality class.


This diagram represents systems which do not order at finite temperatures. The

T 6= 0 part of this diagram has three different regions, separated by cross-over lines.

For g < gc the system is ordered at T = 0 and disordered at T 6= 0 which means

that only thermal fluctuations are responsible for the disorder, hence we call this

area thermally disordered. For g > gc the system is disordered even at T = 0 hence

we call this part quantum disordered. The most intriguing part of the diagram is

the so called quantum critical area which is a symmetric section around g = gc at

finite temperatures where the amplitude of the thermal and quantum fluctuations

are comparable. The cross-over lines are obtained through:

T ∝ ∆ ∝ |g − gc|zν (1.41)

In the quantum critical area the system “looks” critical although the quantum critical

point is located at T = 0. The critical “look” of the system is manifested by “strange

behaviours” such as unconventional power laws or non-Fermi liquid behaviour which

are considered to be the experimental evidences of the existence of a QCP at T = 0.

It is not possible to achieve T = 0 in the actual experiments but such effects mark

the existence of a QCP at absolute zero temperature at g = gc. Quantum critical

behaviour is cut off at temperatures higher than all the internal energy scales in the

Hamiltonian such as the exchange energies or microscopic couplings. At this stage

the system becomes non-universal 13.

Fig. 1.8(b) is the completed version of Fig. 1.6(b). Here we have a whole line of

13Thermal fluctuations grow with temperature as kBT but quantum fluctuations are temperature-independent, therefore above the non-universal line the quantum effects are all overwhelmed bythermal fluctuations.


Figure 1.8: Generic phase diagrams of quantum critical systems. This figure is the

completed version of Fig. 1.6.

finite temperature CPT terminating at a QCP at T = 0. The shaded area around

this line is the region where the condition ω ≪ T holds, therefore the system obeys

classical statistical mechanics. The borders of this area are defined through Eq. 1.37

which shows that the shaded area becomes narrower as we approach the QCP where

Tc → 0. The rest of the diagram is essentially the same as Fig. 1.8(a). The physics

in the quantum critical region is controlled by the thermal excitations of the quan-

tum critical ground state whose effects can be probed experimentally by tuning g

at sufficiently low temperatures and looking for the strange behaviours mentioned

above.


1.2.4 Experimental Examples

Here, I give two experimental examples of quantum critical behaviour. The first

example, the case of heavy fermion superconductivity, shows a typical case where a

finite temperature transition is tuned to zero by applying pressure. It also establishes

the relation between QCP and the emergence of a new phase. The second example,

the case of 2D metal-insulator transition, shows how scaling analyses can be used to

reveal a QCP.

Heavy Fermion Superconductivity

One of the best places to study quantum criticality and the emergence of new phases

is the case of heavy fermion superconductors. Typical examples are the supercon-

ducting Ce-based heavy fermions, namely CePd2Si2 and CeIn3. Both compounds

support AFM order with TN ≈ 10 K at ambient pressure. The electron spin density

is tuned by applying hydrostatic pressure and TN is suppressed to zero. In a narrow

range near the critical density nc (or equivalently, critical pressure Pc) supercon-

ductivity emerges which is believed to be connected to the quantum critical ground

state.

Superconductivity in these compounds is most probably mediated through mag-

netic interactions [28, 29], especially since the superconducting transition tempera-

ture is in agreement with the solutions of the Eliashberg equations [30] which suggest:

Tc ∼ Γθφ e−(1+λ)/gλ (1.42)

Γ is the bandwidth of the relaxation frequency spectrum (Γq), λ is the enhancement


Figure 1.9: Phase diagram of CePd2Si2 and CeIn3 are shown as examples of quantum

criticality and how it gives rise to an emergent new phase which in these two cases

is superconductivity. The insets show the lattice structure and the spin orientation.

CePd2Si2 has a body centred tetragonal unit cell with AFM wave vector q along

[110]. The nearest neighbour spin coupling along the c-axis is frustrated, hence this

is a 2D AFM. CeIn3 has a simple cubic unit cell with AFM wave vector q along

[111]. Reprinted figures with permission from [31]. Copyright (2011) by the Nature

publications.

of the quasiparticle mass due to magnetic interactions, g is the Lande factor which

depends on the quantum numbers l and s, θ represents damping due to incoherent

inelastic scattering, and φ represents damping due to scattering from impurities.

Figure 1.9 shows that in both cases AFM order is suppressed by means of pressure

and quantum criticality is achieved at Pc ≈ 25 kbar. The resistivity behaviour above

the QCP in the quantum critical region reveals unconventional power law of the


form:

ρ = ρ0 + AT x (1.43)

When x = 2 the above equation describes a Fermi liquid. In CePd2Si2 and CeIn3,

above the superconducting dome, the normal state ρ(T ) shows a non-Fermi liquid

behaviour characterized by x = 1.2 and 1.6 respectively [31]. The difference between

the x-values in CePd2Si2 and CeIn3 originates from different dimensionality [32]. The

former has a tetragonal crystal structure (see the inset of Fig. 1.9) which gives rise

to a quasi-2D AFM and the latter, with a cubic lattice (inset of Fig. 1.9), is a 3D

AFM.

MIT in 2D

As explained in section 1.1.4, a 2D electron gas is predicted to be always localized

at zero temperature no matter how weak the disorder, hence no MIT exists in 2D.

This prediction was defied by a series of experiments on silicon MOSFET 14 which is

essentially a 2D non-interacting electron gas whose density of carriers can be tuned

by changing the bias voltage [33, 34].

The experiments show that the system is insulating below a critical density of

electrons n < nc ∼ 1011 cm−2 as expected from the conventional wisdom discussed

in section 1.1.4. Above the critical density, n > nc, the system is weakly localized

(ρ(T ) ∝ log(T )) at T > 1.5 K, but below ∼ 1.5 K ρ(T ) decreases gradually by an

order of magnitude signalling a metallic ground state. There is no sign of localization

down to 20 mK. It is worth mentioning that such behaviour is observable, only in

14Metal-Oxide-Semiconductor Field-Effect


Figure 1.10: Resistivity data (in units of h/e2) is plotted versus T/T0(n). The inset

shows the density dependence of T0. Reprinted figures with permission from [33].

Copyright (2011) by the American Physical Society.

ultra-pure samples with mobility in excess of 104 cm2/V sec. There is no sign of MIT

in samples of lower quality.

The resistivity data for the insulating and the metallic regimes can be collapsed

on two respective scaling functions for the data at n < nc and n > nc. Figure 1.10 is

the log-log scaling plot of resistivity in units of h/e2 (universal resistance, see section

1.1.4) as a function of T/T0(n) where T0(n) is the MIT temperature at each n > nc

for the metallic phase. The inset shows the density dependence of T0(n).

The existence of a scaling behaviour is the signature of a QCP at T = 0 separating

a metallic ground state (n > nc) from an insulating one (n < nc). This is an example


of a QPT where the density of electrons (n) is the non-thermal tuning parameter.

Chapter 2

Experimental Development

In order to study pressure induced quantum criticality, I had to pressurize my samples

and cool them to very low temperatures. To achieve the high pressure and low

temperature conditions, I developed anvil cell technology and built a dipping 1K

probe.

2.1 Anvil Cell Technology

An anvil cell is a device which provides a rectilinear relative motion along the z

direction for two anvils while limiting their motion in all other directions (x, y, θ and

φ). This chapter explains the design and development of Moissanite and Diamond

anvil cells (MAC/DAC) with a maximum pressure of 100 and 200 kbar1 respectively.

In a nutshell, any anvil cell uses a force driving mechanism to push the two anvils

1 1 bar = 105 Pa, 10 kbar = 1 GPa

43

Chapter 2. Experimental Development 44

against a metal disk called a gasket. The sample is placed in a hole at the centre

of the gasket and the pressure is made hydrostatic by filling the sample hole with a

fluid.

There are several good review articles on the design and operation of anvils cells

[35, 36, 37]. I discuss the construction and the operation of the anvil cell (AC) in

full details. The material used in the cell and the anvil properties are thoroughly

described. A theoretical review on the physics of the gasket is given, followed by

experimental remarks on the operation of the cell. Finally the Ruby fluorescence

technique which is used to measure the pressure inside the sample hole at room

temperature is explained.

2.1.1 Design

The principle mechanism of both MAC and DAC is a simple piston-cylinder assembly

through which one anvil is pressed against the other. My goal was to come up with a

minimal design which would provide a superb alignment comparable to the existing

more complicated designs. Conventionally, alignment screws and backing plates are

used to adjust the x-y and the tilt alignment of the anvils. In my novel design, no

additional alignment accessories are used in the body of the cell which makes the cell

easier to operate and minimizes the structural wear of the cell due to having multiple

components. The design is schematically presented in Figure 2.1. The cell has a fixed

bottom seat and a mobile piston which will be driven towards the bottom seat inside

a cylinder. The pressure is locked by screwing the the piston to the bottom seat.

There are three guide pins on top of the bottom seat cut from a piece of Phosphor


Figure 2.1: MAC design: anvils, gasket and guide pins are shown at the center of

the design. The cylinder can be fixed to the bottom seat by M2 screws. The piston

slides inside the cylinder and presses the top anvil against the bottom one. The long

M3 bolts fix the piston to the bottom seat and lock the pressure.


Bronze wire with 0.67mm diameter. Their purpose is to fix the position of the gasket

relative to the anvils in consecutive trials.

The dimensions of the cell are carefully chosen to render proper alignment, suf-

ficient robustness, enough working space and easy fit into the cryostat. The piston

diameter must be a bit smaller than the cylinder diameter as otherwise it will not

enter or will jam if forced in. The clearance required for optimal functioning is

∼ 10µm. This is a critical dimension in the anvil cell and is achievable by a com-

petent machine worker. The other important dimension is the height of the piston

relative to its width. The larger this ratio the better as it prevents any lateral move-

ments. I used a minimum height/width ratio of 1.3/1 and 1.6/1 in MAC (100 kbar)

and DAC (200 kbar) respectively. During my PhD I made two moissanite anvil cells

(MAC1 and MAC2) and one diamond anvil cell (DAC) with overall sizes of 42×24,

31×19 and 24×13.5 mm2 shown in Fig. 2.2.

Each anvil has a small flat top surface called a “culet” (Fig. 2.3). Once pressed

against each other, the culets of the two anvils meet and if they are properly aligned

optical fringes will appear under microscope (Fig. 2.2(c)). If the culets are not prop-

erly aligned the anvils are certain to break under high loads. To align the anvils in

the x-y direction, I fix one of them to the bottom seat and use a small anvil holder

(Fig. 2.2(a)) to slightly change the position of the other anvil until they align. For

the tilt alignment I use the simple aligning mechanism explained by Eremets and

Timofeev [38]. The piston has 6 holes while the bottom seat has three holes so that

the top anvil can rotate in 60 steps. After aligning the culets in the x-y plane under

the microscope, I turn the piston inside the cylinder in steps of 60 until the Newton


Figure 2.2: (a) Exploded view of the MAC1 (b) Exploded view of DAC (c) Comparing

MAC1 and DAC shows how I miniaturized the diamond anvil cell. The inset shows

the optical fringes (Newton’s rings) indicating perfect tilt and xy alignment of the

moissanite anvils.


Table 2.1: Properties of sapphire, moissanite and diamond single crystals, all used

in anvil cells.

material chemical formula crystal system hardness (knoob scale)

Sapphire Al2O3 trigonal 2000

Moissantie SiC hexagonal 3000

Diamond C cubic 5700

fringes are centered. This completes the alignment procedure.

The maximum pressure achieved by an anvil cell is mainly determined by the

size of the culets. Smaller culet sizes give rise to higher pressures since pressure

is force over area. Nevertheless, using smaller culets bring new challenges such as

dealing with smaller samples and exposing anvils to a higher risk of fracture. I used

moissanite anvils with 800µm culet diameter and diamond anvils with 600µm culet

diameter to reach a maximum of 100 and 200 kbar respectively.

Diamond anvils have been used in ultrahigh pressure measurements since the

1960s [37] due to the extreme hardness of diamond. Synthetic single crystals of

moissanite (Silicon Carbide) have only recently been used in the anvil cell technology

[39]. Table 2.2 compares the physical properties of sapphire, moissanite and diamond,

all used in anvil cells. Figure 2.3 compares the shape and size of the moissanite and

diamond anvils I used in my cells. The culets of the moissanite anvils are bevelled

with a 10 angle and the type Ia 16 sided diamond anvils are double-bevelled with 20

angles. Type Ia diamond anvils have the proper optical properties for the synchroton


Figure 2.3: (a) Moissanite anvil with 800µm culet (b) 16 sided diamond anvil with

600µm culet.

X-ray measurements as well.

2.1.2 Material

The cell is almost entirely made of Beryllium-Copper, a metallic alloy made of a

copper matrix with 3-5% beryllium. Be-Cu has significant metal working advantages

while the addition of beryllium allows it to be heat-treated into a very strong and

durable metal. There are two different types of this alloy: high strength Be-Cu with

higher beryllium content and high conductivity Be-Cu with lower beryllium content.

To make the anvil cells, I used alloy 25 which is a common high strength Be-Cu alloy.

Making the whole cell out of the same material is convenient and avoids the problem

of different thermal expansion coefficients which is important for low temperature

experiments.


I used Phosphor bronze to make the guide pins for the gasket (Fig. 2.1). This is

another common copper alloy which contains 3.5 to 10% tin and up to 1% phospho-

rus. The phosphorus serves as a deoxidizing agent during the melting process and

also improves the fluidity of the molten metal, thereby improving the castability of

the alloy. Phosphor bronze has a low coefficient of friction and a high tensile strength

and is commonly used for springs, musical instrument strings and other applications

where resistance to fatigue, wear and chemical corrosion is required.

It is important to use non-magnetic materials in making anvil cells since it pre-

vents any interactions with the high magnetic fields inside the cryostat. Be-Cu is

negligibly paramagnetic at low temperatures. This is because of a small concentra-

tion of Co (0.5%). Alloys with a lower concentration of magnetic Co atoms would

be preferable, but unfortunately they are poisonous. Apart from acting as a pre-

cipitation hardening impurity in the same way as Be, Cobalt keeps the Be content

below 2.2%. With a Be content above 2.2% the thin oxide layer on Cu-Be contains a

considerable amount of beryllium oxide which is dangerous because its ingestion may

cause Berylliosis. Below 2.2%, the oxide layer is almost pure copper oxide which is

completely safe. For our experiments, the ordinary alloy with 0.5% Co is sufficiently

non-magnetic and we can use it without being worried about the risk of inhaling

beryllium oxide.

Beryllium-Copper is manufactured by dissolving Be in Cu at about 900 . The

material is then quenched. Usually the material is half hardened in the factory which

makes it more machinable. To take it to the full hardness, one needs to heat treat

the material using the following recipe:


1. ramp up the temperature to 320 within 40 minutes.

2. hold the temperature at 320 for 2.5 hours.

3. let the material cool down to room temperature over 6 hours.

Heat treatment forms grain boundaries by pinning dislocations, hence, increases the

tensile strength of the material.2

2.1.3 Gasket

The gasket is the metal disc which sits between the top and the bottom anvils and

undergoes the pressurization process. The hole perforated at the center of the gasket

contains the sample, the feed-through wires and the pressure gauge (ruby chips).

The sample hole is then filled with a liquid to make the pressure hydrostatic. As

the culet faces press against the gasket, hydrostatic pressure will be generated in the

sample hole. In this section the theory of the gasket in anvil cells will be reviewed

briefly, followed by experimental notes. The theoretical explanation is mainly based

on references [40, 41].

• Theory

To understand the theory of gaskets, first we study a simple case: Consider a metal

disck with no central hole 3 is under uniaxial loading from two anvils (Fig. 2.4(a)).

2Heat-treating does not change Young’s modulus [35]3I will explain the more complicated case of a gasket with a central hole later in this section.


Figure 2.4: schematic diagram of (a) a simple metal disk under uniaxial pressure and

(b) a gasket with the central hole in an anvil cell

Let k be the compressive strength 4 of the disk and σr, σθ, σn, and σh be the

radial, circumferential, normal and hydrostatic components of stress inside the disk.

For a homogeneous material the following relations hold to be true:

σr = σθ = σh

σn = σh + k (2.1)

The quantity of interest in the rest of this section (which can be measured experi-

mentally) is the hydrostatic pressure at the centre of the disk, P = σh(0). To find the

pressure one must solve the following differential equation with the proper boundary

conditions:

∂σr

∂r= −2f

t(2.2)

which can be considered in two different regimes. At lower pressures where the

4Compressive strength is the capacity of a material or structure to withstand axially directedpushing forces. When the limit of compressive strength is reached, materials are crushed.


gasket slides between the anvils the friction between gasket and anvils provides the

radial force i.e f = µσn. At higher pressures when µσn exceeds k the plastic flow

will replace the sliding and it has been shown by Schroeder and Webster [42] that

under this condition f = k/√

3. To find an expression for P in both cases we plug

the appropriate form of f in Eq.(2.2) and integrate it by applying the boundary

condition σr(r0) = 0. The result will be the following:

σn =k√3µ

+2k√

3

(

R

t− r

t

)

, 0 ≤ r ≤ R

σn = k exp[

2µ(r0t− r

t

)]

, R ≤ r ≤ r0 (2.3)

This result shows that there exists a cross-over radius R above which the disk slides

over the anvils (low pressure regime) and the pressure rises exponentially, whereas

below R the surface of the disk undergoes a plastic flow (high pressure regime) and

the pressure rises linearly. The cross-over radius is given by:

R = r0 −(

t

2µ

)

log

(

1√3µ

)

(2.4)

and the hydrostatic pressure at the centre (below R) is:

P = σh(0) = σn(0) − k =k√3µ

+2k√

3

(

r0t− 1

2µlog

1√3µ

)

− k (2.5)

Finally, for high values of coefficient of friction µ ≥ 1√3, sliding does not occur at

all and we may use f = k/√

3 for 0 ≤ r ≤ r0 as discussed above and again solve

Eq.(2.2) using the boundary condition σr(r0) = 0. The result will be:

σn = k +2k√

3

(r0t− r

t

)

(2.6)

These solutions show the importance of the ratio r0/t and k. In contrast, the

coefficient of friction between the disk and anvils (traditionally thought to be crucial


to the operation of the anvil cell) has no significant importance at high pressures so

long as it is not too small.

We can also use this model to study a gasket with sample hole of radius rg as

is shown in Fig. 2.4(b). In an anvil cell the gasket is a metal disk with a hole at its

centre which contains the pressure medium and the sample. The pressure medium is

a liquid which ideally has no shear stress i.e. k = 0. Therefore, in the region r ≤ rg,

we have σn = σh = σr from Eq. 2.1. Now we have to solve Eq. 2.2 in three different

regions.

1. Inside the sample hole where k = 0 and f = 0

2. Outside the sample hole, but below the cross-over radius R in the high

pressure regime where f = k/√

3

3. Outside the sample hole and beyond the cross-over radius R in the low

pressure regime where f = µσn

Usually the gasket extends beyond the culet area. This gives support to the material

between the anvils, hence we need to change the boundary condition σr(r0) = 0. If

t = t0 in Fig. 2.4(b) we can simply use σr(r0) = k. If t ≤ t0 we have σr(r0) > k and

in general we can use:

σr(r0) = nk

n = 1 for t = t0

n > 1 for t < t0

(2.7)

Notice that n = 0 recovers the boundary condition σr(r0) = 0 for the metal disk

without the hole. The coefficient n is called the “massive support factor”. In the


case of significant indentation ( t0/t > 5 ) Dunstan proposed [40]:

n = 2 ln(t0/t) + 1 (2.8)

We can solve Eq. 2.2 in the three different regions mentioned above using σr(r0) =

nk and Eq. 2.7. The result for the case 0 < µ < 1√3(n+1)

is:

σn = k

(

1√3µ

− 1

)

+2k√

3

(

R

t− rg

t

)

r < rg

σn =k√3µ

+2k√

3

(

R

t− r

t

)

rg ≤ r ≤ R

σn = (n+ 1)k exp[

2µ(r0t− r

t

)]

R ≤ r ≤ r0 (2.9)

For the case µ ≥ 1√3(n+1)

the result is:

σn = nk +2k√

3

(r0t− rg

t

)

r < rg

σn = (n+ 1)k +2k√

3

(r0t− r

t

)

rg ≤ r ≤ r0 (2.10)

The cross-over radius can be calculated accordingly by equating the last two lines

of Eq. 2.9 at r = R.

R = r0 −t

2µlog

[

1√3(n+ 1)µ

]

(2.11)

Notice that this cross-over occurs only for the case 0 < µ < 1√3(n+1)

. Figure

2.5(a) is a plot of the functions 2.9 and 2.10 for different values of the ratio r0/t and

massive support factor n. By increasing r/r0 the following important features can

be observed in these curves:


Figure 2.5: (a) σn is plotted as a function of radius for aspect ratios r0/t = 20 (a,b)

and r0/t = 5 (c,d) and massive support factors n = 3 (a,c) and n = 0 (b,d). All the

jumps occur at rg and they are all of the size k. The coefficient of friction µ = 0.1

(b) Equation 2.12 is plotted for n = 0 (solid curve) and n = 3 (chain-dotted curve).

The region below the curves corresponds to thin gaskets, the dotted curve shows the

typical behaviour of a gasket in this regime. We can go from the thin to the thick

regime on n = 3 curve by following the arrows. The plots are adopted from reference

[40]


1. within the sample volume the pressure is uniform.

2. there is a discontinuous jump in pressure at the hole boundary with a step of

the size k (shear strength of the gasket). This jump in pressure helps to seal

the sample hole and prevents the pressure medium from leaking out.

3. a cross-over from the linear behaviour at higher pressures into an exponential

behaviour at lower pressures happens for the case 0 < µ < 1/√

3(n+ 1) at

r = R given by Eq. (2.11).

All the theory we discussed so far applies to the case where we start with a gasket

of size t0 and press it in between the anvils. Using Eq. (2.10) for r < rg we get:

t

r0=

2√3(P

k− n)

(1 − rg

r0) (2.12)

which shows that inside the gasket hole where P = σn the thickness is inversely

proportional to pressure. Fig. 2.5(b) shows this behaviour for two different values of

massive support factor. This is called the thick gasket behaviour.

We may imagine a different case with the initial gasket thickness t < t0. If we

put the gasket under stress and preindent it, then repeat the experiment with the

preindented gasket, we are in this new regime. Here, as we increase the pressure,

t/r0 falls below the curves in Fig. 2.5(b), hence we name it the thin gasket regime.

Of course as we increase the pressure we will reach a point where the gasket again

obeys thick behaviour which is described by the set of equations 2.9 and 2.10. As

you can see in Fig. 2.5(b) the thickness of the gasket is constant in the thin regime

and as we move into the thick regime it varies as 1/P (Eq. 2.12).


The plastic deformation of the thin gasket as we increase the pressure occurs as

an inward flow of gasket material into the sample hole which keeps t constant. This

indeed is an advantage since it increases the pressure inside the hole. Remember

that in the thick regime the plastic flow of material is outwards from the hole which

reduces the gasket thickness t as we increase the pressure.

To summarize, in the thick gasket regime:

1. rg remains constant or increases as the pressure is increased.

2. The plastic flow is outwards from the sample hole.

3. Gasket thickness behaves as 1/P according to Eq. (2.12)

In the thin gasket regime:

1. rg decreases as the pressure is increased.

2. The plastic flow is inwards into the sample hole.

3. Gasket thickness does not vary with pressure.

We only discussed increasing the pressure in the cell. Dunstan [40] argued that

reducing the force on the gasket causes the liquid to leak out of the sample hole due

to the reversal of the two conditions in Eq. 2.1 during the reverse process.

σn = σθ = σh

σr = σn + k = P (2.13)

There is no longer an excess pressure σn = P+k between the anvils and the gasket to

seal the pressure and the cell should now leak. Therefore, when we want to perform


measurements in different pressures we must start with low pressure and increase

the pressure at each step.

Figure 2.6: Normalised Force-Pressure plots for (a) thick gaskets and (b) thin gaskets.

(a): Curve (i) represents a simple disk whereas curves (ii) and (iii) represent gaskets

with n = 0 and n = 3 respectively with rg/r = 1/3 in the solid curves and rg/r = 1/2

in the dashed curves. (b): The numerical solutions are given for thin gaskets with an

initial value (at P = 0) of rg/r0 = 1/3 and thickness (i) t = r0/2, (ii) t = r0/6, and

(iii) t = r0/10. This model assumes a compressible pressure medium. The broken

curve is the same as (ii) but with an incompressible liquid. Plots are taken from

reference [40].

Finally, we discuss the relation between the force (F ) exerted on the anvils and

the pressure (P ) generated in the gasket. F can be calculated by integrating any of

the above expressions for σn for all the different cases. In this calculation the forces

on the flanks of the anvils due to an indented gasket and all the frictional forces in


the cell or driving mechanism can be neglected since they are orders of magnitude

smaller than F . As an example let us consider the case of a simple metal disk without

the hole Fig. 2.4(a) while µ ≥ 1/√

3 so that there is no sliding. For this simple metal

disk σn(r) is given by Eq. 2.6 which gives rise to:

F =

∫ r0

0

2πrσn(r) dr = πr20k + (

2k

3√

3t)πr3

0k (2.14)

On the other hand we learned, in the beginning of this section, that P = σh(0) =

σn(0) − k where σn(r) is given in Eq. 2.6. Thus:

P = σh(0) = σn(0) − k =2kr0√

3t(2.15)

Using equations 2.14 and 2.15 we find the relation between P and F .

P = 3F

πr20

− 3k (2.16)

This solution has two main features:

1. P is a linear function of F with the slope dP/dF = 3/πr20

2. There is a threshold F0 = kπr20 required before any pressure is generated. A

more general formula in the case of a non-vanishing massive support factor n

is given by Dunstan [40] to be F0 = (n+ 1)kπr20 .

Both of these features are shown in Fig. 2.6(a). The solid curves show the be-

haviour of the metal disk and the dashed curves show the behaviour of a gasket with

a hole of radius rg. As is shown in this figure, the slope dP/dF is reduced from 3 to

approximately 2 in the case of a gasket with 1/3 ≤ rg/r0 ≤ 1/2. For the thin gasket

regime F is no longer a linear function of P (Fig. 2.6(b)). Here no pressure will be


generated until a force F ′0 sufficient to give extrusion into the gasket hole is reached

(F ′0 > F0). The details of calculating F ′

0 can be found in [40]. The result is:

F ′0 = kπr2

0 +πk

2√

3t(r3

0 − r20rg − r0r

2g + r3

g) (2.17)

The dynamical cross-over between the thin and the thick gasket regimes is shown

in Fig. 2.7 which demonstrates that as the pressure is increased in the thin gasket

from (a) to (d), rg decreases due to the inward extrusion of the gasket material into

the sample hole. At (d) the gasket ceases to be in the thin regime and the hole again

starts to expand, characterizing a thick regime.

Let us list the most important practical conclusions of the this section.

1. The pressure inside the sample hole is not solely generated by squeezing the

gasket between the anvils. There is a major contribution from the massive

support factor. Larger gaskets provide a larger massive support factor (n).

2. Massive support is most effective if the gasket is preindented. For a

significantly indented gasket the massive support factor is given by:

n = 2 ln

(

t0t

)

+ 1 (2.18)

3. It is desirable to make the gasket large in its external diameter, not only to

increase n, but also to keep the bulk of the gasket in the central plane of

anvils and to prevent the gasket from extreme buckling. I used gaskets of 9

mm diameter in the moissanite anvil cells and 6 mm in DAC with the anvil

dimensions given in Fig. 2.3.


Figure 2.7: The dynamical cross-over between the thin and the thick regimes as the

pressure is increased. σn is plotted as a function of radius. The plot is taken from

reference [40]

4. It is highly recommended to work in the thin regime such that the hole

diameter decreases as the pressure increases. Preindentation guarantees that

the cell will operate in the thin regime.

It is important to keep the relative gasket-anvil position fixed in successive trials,

as it prevents anvil breakage, protects the feed-through wires, and helps us to keep

the gasket in the thin regime. This can be easily achieved using the guide pins on

the bottom anvil seat (Fig.2.1).


2.1.4 Ruby Fluorescence Technique

This optical method is used as a quick, precise and fairly easy way to measure the

pressure in the sample hole at room temperature. Ruby is a corundum Al2O3 crystal

just like Sapphire, only it is doped with 0.05 wt.% Cr3+ impurity substituted for Al3+

ions. It is a common pressure gauge in ultrahigh pressure labs these days and was

first introduced by Barnett (1973) [43] and Piermarini (1975) [44] and was calibrated

up to 195 kbar against the Decker equation of state for NaCl [44].

Ruby has two relatively strong fluorescence peaks at wavelengths 6942 A and

6928 A at 25 called R1 and R2 respectively. These lines can be excited by a

simple green 10 mW diode Laser pointer. The linear shift of R1 is used to determine

the pressure. The theory of these transition lines can be found in references [45] and

[46]. A brief review is presented here followed by an explanation of the experimental

technique.

• Theory

R1 fluorescence line originates from the well known excitation-decay process of d-

shell electrons of Cr3+ ions, excited by a Laser beam. In order to study the Ruby

spectrum, one has to consider the effect of crystal field 5 on the 3d3 orbitals of Cr3+

ions embedded inside the corundum lattice.

The symmetry of the octahedral sites is perfect cubic Oh if no distortions occur.

However, the repulsive interactions between neighbouring Al3+ ions create a slight

5The theory of crystal field effect (CFE) also known as ligand field theory was originally devel-oped by Bethe in 1929 [47]


trigonal C3 distortion which wipes out the inversion symmetry of the aluminium

sites. To get the correct electronic states, one can forget about the C3 distortion and

move on with the cubic crystal field theory as a first step. Perfect cubic crystal field

splits the five degenerate d-orbitals of Cr3+ into the eg and t2g levels with the former

being 10Dq higher than the latter in energy.6.

The lowest energy state for Cr3+ in the cubic Oh symmetry is the configuration

with the three d-electrons filling the t2g levels with their spins pointing in the same

direction. The excited states can be constructed by flipping the spins or by sending

one electron to a higher eg level. The trigonal distortion and the spin-orbit interaction

are treated as perturbations which further split the energy levels of the cubic crystal

field. Details of the calculations are explained in reference [45].

The ground state 4A2(t32g) and excited states 2E(t32g),

2T1(t32g),

4T2(t22geg),

2T2(t32g),

4T1(t22geg) of d3 electrons in the Oh symmetry and their further splittings due to the

perturbations mentioned above are demonstrated graphically in Fig. 2.8. Transitions

between these excited states and the ground state 4A2 are denoted by R, R′, U, B,

and Y respectively. Transitions between terms with different electron configurations

from the ground state 4A2(t32g) result in broad bands U and Y while transitions

between terms of the same configuration as the ground state 4A2(t32g) give rise to the

narrow lines R, R′, and B. Figure 2.8 shows that the two narrow fluorescent lines

of our interest i.e. R1 (694.23 nm) and R2 (692.80 nm) originate from a transition

between the excited 2E level and the ground state 4A2. Their splitting comes from

spin-orbit interaction.

6Dq being the cubic crystal-field parameter.


Figure 2.8: Summary of the ambient-pressure electronic structure of ruby in the

visible energy range, adapted from [45].


Figure 2.9: General features of the spectrometer (Ocean optics USB2000). Light

from an optical fiber enters at position (1); goes through the coupling (2) and the

collimator (3); reflects off the collimating mirror (4); is diffracted by the grating (5);

reflects from focusing mirror (6); then finally is read out by the CCD detector (7,8,9).

Picture taken from www.oceanoptics.com .

It was shown for the first time by Barnett [43] that when Ruby is pressurized, the

wavelength of both R lines shift linearly with pressure. This effect is detectable for a

piece of Ruby as small as a 5 µm3. The linear dependence of the R1 fluorescence line

was then calibrated up to 195 kbar against the Decker equation of state for NaCl [44]

with the slope dpdλ

= 2.746 ± 0.014 kbarA−1. This technique is most accurate under

hydrostatic pressure.

• Experiment

To detect the Ruby fluorescence lines I used a USB2000 plug and play miniature spec-

trometer with a grating designed for the spectral range of 620 to 760 nm with groove


density 1800. It is equipped with a long-pass filter which cuts all the wavelengths

below 590 nm. The general anatomy of the spectrometer is depicted in Fig. 2.9; light

from an optical fibre enters at position (1), reflects off the collimating mirror (4),

is diffracted by grating (5), reflects from focus mirror (6) and finally is read out by

the CCD detector (9). To excite the d-shell electrons of Cr3+ ions we use a simple

Green Laser pointer with wavelength 532 nm (which will be cut by our long-pass

filter) with maximum power output 5 mW.

I put Ruby chips7 inside the sample hole to measure the pressure induced shift

in Ruby R1 line using the equations [44]:

dp

dλ= 2.746 kbar/A or

dλ

dp= 0.364 A/kbar (2.19)

The pressure is made hydrostatic by loading the sample hole with an appropriate

fluid pressure medium. The most commonly used pressure media are Daphne oil,

ethanol/methanol mixture (1/4 by volume), liquid Argon and liquid 4He. The first

two are easier to use and do not need any special set up but for the second two, one

we needs to make special settings to load the liquid Ar or He into the sample hole.

I have made an Argon loading platform as explained in chapter 5, however, I never

used it for my measurements. I mainly used Daphne oil as the pressure medium in

my experiments.

7tiny bits of Ruby, made by grinding a piece of Ruby into a fine grain powder


2.1.5 Operation

Preindentation is done by assembling the cell with the gasket between the anvils,

putting it under the hydrolic press and increasing the pressure very slowly. It is

helpful to clean the gasket with a piece of Brasso cloth before preindenting it. In

order to achieve 100 kbar of pressure in the MAC, I use moissanite anvils with culets

of 800 µm diameter and apply 600 psi of pressure to indent Cu-Be gaskets. Gaskets

are cut out of work hardened Cu-Be sheets (alloy 25), using a punching mechanism

made in physics machine shop 8. The pressure is raised slowly and continuously

to prevent premature failure due to anvils breaking. It takes about 40 minutes to

preindent a Cu-Be gasket with initial thickness t0 = 500 µm to a final thickness of

t = 100 µm.

Steel gaskets used in DAC are harder than the Cu-Be ones and I preindent them

even more slowly, over an hour, from t0 = 400 µm to t = 100 µm. Once the

indentation is done I relax the press slowly and measure the thickness of the gasket

using a micrometer. Figure 2.10(a) and (b) show a Cu-Be gasket before and after

preindentaion. For the rest of this section, I discuss the preparation of a Cu-Be

gasket in a Moissantie anvil cell with 800 µm culet size unless otherwise mentioned.

After preindentation is done, I drill a hole at the centre of the preindented area

using a 400 µm drill bit. As a general rule, the diameter of the sample hole should be

at most half the diameter of the preindented area. I drill the sample hole in steps, by

successively using 250, 300, and 400 µm drill bits. The rough margins of the drilled

8I have made three punchers for 10, 9, and 6 mm gaskets, all are kept in the physics machineshop.


Figure 2.10: (a) Gakset before indentation (b) Gasket after indentation (c) Sample

hole is drilled at the center of the indented area (d) Drilling is done under microscope

(e) Gasket is covered with insulation and the wired sample is place in position. (f)

A composite gasket is shown. The gold wires are connected to the 40 and 80 µm

copper wires for the measurement.


hole must be trimmed using scalpel, tweezers, or slightly larger drill bits. Rough

margins can result in cutting or shorting the leads. Figure 2.10(c) shows the gasket

with the sample hole drilled at its center and Fig. 2.10(d) shows how I normally set

up the mini drilling machine and the microscope.

Once the sample hole is drilled, the top side of the gasket will be covered with

an insulating paste to prevent shorts. To do this, a mixture of fine Alumina powder

(Al2O3) with 1 µm grain size and stycast 1266 epoxy is made with the weight ratio

of Al2O3/stycast = 2/1. The mixture is then applied to the gasket and partially

cured at 100 for one hour. Then the cell will be assembled and indented to the

same force used in the preindentation process. The pressure will be locked and the

whole cell will be cured at 100 for two hours. Finally the cell will be opened and

the gasket will be air cured for 12 hours. The sample hole is then re drilled and the

margins are trimmed. Finally a Al2O3/stycast = 2/3 mixture will be applied to the

inside wall of the sample hole and air cured for another 12 hours (Fig. 2.10(e)).

An alternative approach is to prepare a composite gasket which is useful especially

with the DAC which undergoes much higher pressures than the MAC. The idea is

to drill out the indented area of the gasket and replace it with an insulating mixture

to eliminate the possibility of any shorts. I used a fully hardened stainless steel

sheet (grade T301) to make gaskets for the DAC. The gaskets are 6 mm in diameter.

Preindentation is carried out under 700 psi of pressure with gasket final thickness

being t = 90 µm. All of the indented area, which has the same diameter as the

diamond anvil culets (600 µm), is then drilled out with a 700 µm tungsten carbide

drill bit. Normal stainless steel drill bits are not hard enough to drill through the


fully hardened T301 stainless steel gaskets. The hole is then filled with a mixture

of fine diamond powder (0.5 µm grain size) to stycast 1266 epoxy with the ratio

diamond powder/stycast = 3/1 by weight and pre-cured at 100 for 1 hour. Then

the gasket is pressed under 600 psi of pressure, locked, and heat cured for two hours.

The cell is then opened and the gasket is air cured for 12 hours. Finally, the sample

hole is drilled using 250 µm drill bits. Fig. 2.10(f) shows a composite gasket sitting

on top of the diamond anvil. A sample is placed inside the sample hole and the wire

connections are shown 9.

To wire the samples I used 15 and 25 µm gold wires for DAC and MAC respec-

tively. The wires are attached to the sample using Dupont conducting silver paste

model 6838 which needs to be heat cured at 200 for two hours. Usually the sample

needs to be polished to the right size using diamond lapping films of 6 to 1 micron

grain size in several steps. I have also used diamond suspension of 3 to 1 micron

grain size on a Nylon polishing cloth for Eu2Ir2O7 which is more susceptible to get-

ting scratched in the polishing process. Figure 2.10(e) shows the final look of a wired

sample. I usually cover the sample with a drop of 5 minute epoxy to secure the leads.

The sample with the gold wires attached to it is then carefully placed in the sample

hole on the gasket. The gold leads are secured in place by Dupont conducting silver

paste model 4929N. The gold wires are then connected to 40 µm copper wires on

the gasket which in turn are soldered to the heavily insulated 80 µm copper wires

which exit the anvil cell and connect to the electronic devices for the measurement

9I am very grateful to Dr. Patricia Alireza for her kindly giving me invaluable advice regardingthe composite gaskets.


Figure 2.11: (a) A pen laser is used to shine green laser on the ruby chips inside

the anvil cell. The light is collected by a collimator attached to the cell through a

connector. The collimator is connected to the spectrometer through an optical fiber.

(b) Ruby R1 and R2 fluorescent lines are picked up by the spectrometer and shown

on the computer screen.


(Fig. 2.10(f)). The silver paste 4929N needs to be air cured for 12 hours.

Once the sample is in position, I put one or two ruby chips in the sample hole,

add a drop of Daphne oil as the pressure medium, close the cell, and and pressurize

it slowly. At each pressure the cell will be locked tightly and the pressure will be

measured using the ruby fluorescence spectroscopy technique.

To do this, a green 10 mW pen laser is used to excite ruby fluorescent lines. An

optical fiber of 1 mm diameter carries the light from the cell to the spectrometer

(Ocean Optics model USB2000). The optical fiber is connected to a collimator and

the collimator is attached to the bottom of the cell through a connector. Figure

2.11 shows the settings for the ruby spectroscopy. R1 and R2 lines appear on the

computer screen and pressure is measured using the R1 line shift using the formula:

∆p/∆λ = 2.746 kbar/A with ∆λ = λ− 6942.3 A (2.20)

explained in section 2.1.4. This concludes the development and the operation of the

anvil cell. My next major development for our lab was the 1K probe discussed below.

2.2 1K Probe

Dipping probes are commonly used in condensed matter physics labs to study various

properties of materials from room temperature down to about 1 Kelvin, hence they

are referred to as 1K dipping probes [48]. 1K probes can be designed to be versatile

and to be used for various measurements. Compared to other kinds of cryostat,

dipping probes have the advantage of consuming very little liquid 4He which is always

an important issue in a low temperature lab. The lowest temperature that has been


Figure 2.12: Design of the 1K probe.


achieve by a 1K probe is 1.3 K. The probe I made has a base temperature of 1.9 K.

It uses only 2 liters of liquid 4He in each run.

2.2.1 Design

Figure 2.12 shows the anatomy of the continuously operating 1K probe which is

based on a design by E. Swartz [49]. It uses two pump lines; one for the vacuum

chamber (VC) and the other for the 1K pot. Both pump lines go through a manifold

on top which can be clamped to the Dewar’s mouth using a 50 KF flange. The probe

uses a vacuum seal made of a 7 tapered cone and a matching piece at the opening

on the vacuum can. Wire looms go through the VC pump line and connect to two

10 pin Fischer connectors on top. The 1K pot pump line is soldered to the 1K pot

and a capillary system is used to suck liquid 4He from the Dewar into the 1K pot.

For maximum efficiency the capillary has to have the right impedance which can

be calculated using the equation:

Z = ∆P/V η (2.21)

where ∆P is the pressure drop (from 1 bar to 1 mbar) required to cause a volume

flow rate V of a medium with viscosity η. Typically one needs V to be of the order

of 1011 cm−3 [50]. To achieve the right rate of flow, I used 1.1 m of Cu-Ni capillary

with wall thickness 0.004 inch. Figure 2.13(b) shows the VC and the 1K pot pump

lines and the capillary system.

Samples are placed on a sample platform which has two tag boards accommodat-

ing 10 connector pairs shown in Fig. 2.13(a,b). Anvil cells are attached to the sample


Figure 2.13: (a) The sample holder has a Cernox thermometer on it and a heater

circuit and two tag boards each with 5 pairs of connectors. (b) The lower part of

the 1K probe is shown. Wire looms are heat sunk to the 1K pot. Liquid 4He will be

sucked into the 1K pump through a 1.1 m long capillary system.

platform using a M3 connector made of copper. A Cernox thermometer, calibrated

for the temperature range 1.4 to 300 K, is glued on top of the platform with GE

varnish.

2.2.2 Material

The manifold on top, the vacuum chamber, and the 7 tapered cone (Fig. 2.12)

are made of Brass. The 1K pot, the sample platform, and the M3 connector are

made of Copper to stabilize the temperature at the sample platform level by taking

advantage of the high specific heat and thermal conductivity of Copper. Tag boards


are machined out of G7 fibreglass. The VC and the 1K pot pump lines are made of

3/8 × 0.02 inch and 1/2 × 0.01 inch stainless steel tubes 10 (grade 304).

The probe uses two looms, one for the thermometer/heater and the other for the

sample. The two looms are connected to two separate tag boards on the sample

platform (Fig. 2.13). Each loom has five twisted pairs, made of 70 µm Copper wires,

secured between two layers of low temperature yellow tape with the total resistance

of 11 Ω at room temperature. The heater circuit (Fig. 2.13(a)) which is made of

Manganin wires has a total resistance of 690 Ω.

2.2.3 LabView Program

I have written a LabView program for the 1K probe 11 which records the data

through a National Instruments PCI 6289 data acquisition (DAQ) card. The DAQ

card is connected to the measurement electronics through a NI BNC 2120 breakout

box. The user interface of the 1Kprobe.vi program is shown in Fig. 2.14. Once the

program is uploaded one needs to enter the following values for the parameters in

the ACQUISITION panel (Fig. 2.14):

1. Rate : shows the sampling rate of the DAQ card per channel. The optimum

value is 50000.

2. Buffer : shows the buffer size. The optimum value is 20000.

10The numbers represent diameter × wall thickness.11The program is saved on the computer Max in the lab under C:\Documents and Set-

tings\Owner\My Documents\FAZEL\LabV\VIs\1Kprobe.vi


Figure 2.14: The user interface of the LabView program 1Kprobe.vi is shown. A

brief description of the parameters required by the program is written on the left

bottom corner of the user interface. The user has to type the correct values of the

parameters in the ACQUISITION panel as explained in the text. Data are shown

in the DATA panel and plotted against temperature. To store the data, one needs

to press the WRITE TO FILE button. Data files are stored in the .lvm format.


3. Sensitivity : is a parameter which is used by the program to calculate the

temperature by fitting the data from the Cernox thermometer to a

Chebyshev function. The proper value is 100 when the sensitivity of the

thermometry lock-in is set to 10×10 mV (expanded) and 1000 when the

sensitivity of the lock-in is 100 mV. During the course of a measurement, the

lock-in sensitivity must be changed from 10×10 to 100 mV at T = 10 K.

Once the proper values are entered for the above parameters, one can start the

program. The program automatically asks the user where to store the data 12 and

then starts data acquisition. However, it will not record the data to a file unless

the WRITE TO FILE button is pressed. At any time during the measurement this

button can be turned on or off.

There are two real time graphs on the program interface (Fig. 2.14); The Signal

graph shows the signal as a function of temperature. One can toggle between the

real (x) and the imaginary (y) signals. The Temperature Profile panel shows the

temperature as a function of time.

I have also written a heater and a function generator program 13 shown in

Fig. 2.14. To use the heater, one needs to give the T-sweep.vi program the sweep

time. The program automatically sweeps temperature from 4 to 2 K, waits for two

minutes, and sweeps back from 2 to 4 K. Usually I use the built-in function gener-

ator of the SR 830 lock-in so there is no need to use the virtual function generator

12The default address is C:\Documents and Settings\Owner\My Docu-ments\FAZEL\LabV\textfiles

13All my LabView programs are saved in the directory C:\Documents and Settings\Owner\MyDocuments\FAZEL\LabV


however, the FFGen.vi program can be used to create a sinusoidal output of the

desired amplitude and frequency (Fig. 2.14).

I have assembled a portable measurement station with two SR 830 lock-in am-

plifiers to measure the temperature and the signal respectively. The usual set up I

use for a resistivity measurement in the 1K probe is shown in Fig. 2.15. To generate

the current, I use the wave generator of the SR830 lock-in amplifier and output it

to a 2 MΩ resistance to make the current constant. I normally use a current of

amlitude I ∼ 10 µA and frequency f ∼ 20 Hz. The voltage leads from the 1K probe

are connected to the SR830 lock-in and outputted to the transformer (Ithaco 190)

with a 10:1000 turn ratio, then the pre-amplifier (EG&G 165) with gain 20 dB, and

finally to the BNC 2120 breakout box which is connected to the DAQ card on the

computer “Tom”.

Figure 2.15 shows the abovementioned set up on the portable unit to measure

electronic transport using the 1K probe. In this figure, the 1K pot pump line is

connected to a 4He rotary pump and the probe is fully inserted in the liquid 4He

Dewar. The pump and the Dewar are connected to the Helium return line. The

portable unit can be easily moved around to be used for any other measurements in

our lab.

2.2.4 Operation

Before starting to operate the 1K probe, the anvil cell is attached to the bottom of

the sample platform using a M3 screw (Fig. 2.13(a)). The external wires from the

anvil cell are then soldered to the pins on the tag board. The probe is then sealed by


Figure 2.15: The portable measurement unit with two SR 830 lock-in amplifiers, a

EG&G pre-amplifier (model 165), an Ithaco transformer (model 190) and a Tectronix

oscilloscope (model TDS 1002). This is my usual set up to measure the resistivity

of a sample in the anvil cell using the 1K probe.


putting a small amount of vacuum grease on the opening of the VC, inserting it in

place and pumping on it until the pressure falls below 10−4 mbar. It is recommended

to give the VC one turn while pumping, to settle the vacuum grease. After finishing

pumping, a small amount of exchange gas is introduced into the VC to allow for

proper heat exchange.

Before putting the probe into the helium Dewar, pure 4He gas is flowed through

the 1K pot to make sure that the capillary system remains open at low temperatures.

I usually flow Helium gas through the 1K pot pump line with 2-3 psi of pressure for

15 minutes. During this time I start the 1Kprobe.vi program, give it the right value

of parameters mentioned in section 2.2.3, and press the WRITE TO FILE button

to start recording the data to a .lvm file (Fig. 2.14).

The probe is then placed inside the Helium Dewar and lowered slowly. An av-

erage cooling rate of 3-4 K/min, monitored by the Temperature Profile panel, is

used for a typical resistivity measurement. When the temperature is around 10 K,

the sensitivity of the SR 830 lock-in needs to be changed from 10×10 to 100 mV.

Simultaneously, the sensitivity parameter on the 1Kprobe.vi program must be

changed from 100 to 1000. A spike might show up in the data due to this change of

parameters which can simply be eliminated later in the analysis.

To cool the sample down to 4.2 K the probe is lowered through the cold gas

in a liquid Helium Dewar. Once the VC is fully immersed in liquid Helium the

temperature will be 4.22 K. Liquid Helium will then be sucked into the 1K pot

through the capillary system. At this stage, I switch to the “1K mode” by connecting

the 1K pot pump line to a rotary pump (Fig. 2.15) and start pumping on the 1K


pot. This process further cools the 1K pot down to about 2 K, using 4He latent heat

of evaporation. Since there is a continuous flow of liquid He into the 1K pot through

the capillary system, this method is usually referred to as a continuously operating

4He cryostat [50].

To sweep the temperature back and forth between 2 and 4 K one can either use

the T-Sweep.vi program, explained in section 2.2.3, or connect a DC voltage source

to the probe and perform a manual sweep. After the measurement is finished the

probe is warmed up by slowly rising it in the 4He Dewar and eventually taking it out.

Data may be recorded during the warm-up process as well to compare with the data

from the cool-down process. To be more cost effective, I usually warm the probe up

to 80 K in the 4He Dewar, then take it out and use a liquid Nitrogen bucket to warm

it up to room temperature.

In my experience, the vacuum seal is not very reliable, so I pump on the VC

during the warm-up cycle (twice at 20 and 120 K) to make sure that the vacuum

seal remains effective up to room temperature.

Chapter 3

Eu2Ir2O7

Condensed matter physics is the science of studying matter in its condensed form,

most prominently solids and liquids. Of particular interest are the crystalline phases

of matter where broken symmetries confine electrons to certain boundary conditions

which pick certain solutions of the many particle Shrodinger equations [51]. One of

the interesting areas of condensed matter physics is to grow such crystalline phases

to design various platforms in which electrons interact [52]. Modern advances in

material science have made possible the synthesis of crystal structures with tunable

interactions. Electrons may reveal extremely different behaviours depending on the

crystal in which they live and the interactions to which they are exposed. In this

sense, each material may be regarded a new stage for electrons to reveal one of their

many faces [29].

I have studied the electronic transport properties of two crystals, namely Eu2Ir2O7

and CrFeAs, under ultrahigh pressures using anvil cell technology explained in chap-

84

Chapter 3. Eu2Ir2O7 85

ter 2. The former is a rare earth pyrochlore oxide and the latter is a ternary transition

metal pnictide. The physics of both compounds are unified under the theme of metal-

insulator transitions explained in chapter 1. In this chapter I discuss the pressure

tuned insulator to metal quantum phase transition of Eu2Ir2O7.

3.1 Material Background

Ternary metal oxides with the generic formula R2M2O7 produce a large family of

phases isostructural to the mineral (Na,Ca)2Nb2O6(F,OH)1 which turns green upon

ignition, hence named pyrochlore meaning green fire in Greek. Recently the com-

pounds with R being a trivalent rare earth and M being a tetravalent transition metal

have attracted much attention. R2M2O7 comprises both localized f -electrons pro-

vided by R3+ ions and itinerant d-electrons from M4+ ions. Interacting localized and

itinerant electrons in the geometrically frustrated lattice of pyrochlore oxides give rise

to a plethora of interesting condensed matter phenomena such as spin liquid, spin ice,

magnetic monopoles, superconductivity, metal-insulator phase transitions, anoma-

lous Hall effect, order by disorder, topological insulator, etc. [53, 54, 55, 56, 57, 58].

Crystallographically there are two non-equivalent oxygen sites in the pyrochlore

structure, therefore the chemical formula is sometimes written as R2M2O6O′

. Py-

rochlore phases are all cubic structures categorized under the space group Fd3m

with 8 molecules per unit cell (z = 8). R2M2O6O′

is composed of two interpenetrat-

ing R2O′

and M2O6 sublattices each forming a network of corner shared tetrahedra

(Fig. 3.1). The R-cation is 8-coordinated by oxygen anions in a scalenohedral geome-


Figure 3.1: (a) Both the R (blue) and the M (green) sublattices are networks of

corner-shared R or M tetrahedra in a cubic unit cell. (b) In its local geometry, each

M4+ ion is surrounded by 6 oxygen anions in an octahedral fashion (top) and each

R3+ ion is surrounded by 8 oxygen anions in a scalenohedral geometry (bottom).

A scalenohedron is a cube which is compressed along one diagonal. The two O’

oxygen ions are on the poles and the O oxygen ions form a puckered ring around

the R site. (c) Looking along the [111] direction, the pyrochlore lattice is made of

alternate stacking of triangular and Kagome planes. Geometric frustration is an

inherent feature of this structure. Pictures (a) and (c) are taken from reference [59].


try (a distorted cube compressed along one diagonal). In this coordination there are

six equally spaced O2− anions located on a puckered ring around the R3+ ion and

two O′

anions with slightly shorter bond lengths located on the poles. The O′

-R-O′

bond angle is always 180 degrees. The smaller M-cation is 6-coordinated by oxygen

anions in a distorted octahedral cage (Fig. 3.1).

The minimal crystallographic coordinates of each atom in the R2M2O6O′

com-

pounds are R(1/2,1/2,1/2), M(0,0,0), O′

(3/8,3/8,3/8) and O(x,1/8,1/8). The coor-

dination geometry about the R and M sites is controlled by the so called x parameter

[60]. For x = 5/16 = 0.3125 a prefect octahedron forms around the M-site and for

x = 3/8 = 0.3750 a perfect cube forms around the R-site. Usually 0.320 < x < 0.345

and this results in a minor distortion of the M-site ideal coordination (octahedral)

and a major distortion of the R-site ideal coordination (cubic). As a result of these

distortions, the O-M-O bond angle slightly drifts from its ideal θ = 90 value to

80 < θ < 110 but the R site ideal cubic coordination completely changes into a

puckered ring of 6 O anions surrounding R and 2 axial O’ anions normal to the aver-

age O plane (Fig. 3.1(b)). O′

-R-O′

is aligned to the local [111] direction. The R-O′

bond length is typically about 10% shorter than the R-O bond length [59]. This

pronounced axial symmetry has a profound effect on the crystal electric field (CEF)

on the R-site.

The pyrochlore lattice structure is an ideal playground for frustration. Each

octahedral unit is made of four triangles and triangular based structures are most

prone to geometric frustration. Another way to view the pyrochlore crystal structure

is to look at it along the [111] direction of the unit cell. This view reveals a stacking


of alternate Kagome and triangular planes shown in Fig. 3.1(c). The ground state

of a system of Heisenberg spins with nearest neighbour AFM interaction on the

frustrated pyrochlore lattice is theoretically predicted to be a quantum spin liquid

[61]. In a quantum spin liquid (QSL), spin-spin correlations decay exponentially with

distance at all temperatures such that the correlation length is of the order of the

lattice parameter.

Amongst the pyrochlore family, much interest has recently been focused on iri-

dates with the chemical composition R2Ir2O7. Here, the Ir4+ ion has a 5d5 orbital

configuration which in the case of an octahedral crystal field gives rise to a S = 1/2

state with all electrons in the t2g manifold. Therefore, iridates are the prime candi-

dates for the QSL phase mentioned above. While most of the metallic pyrochlores

develop long range magnetic order of some sort at low temperatures, iridates seem

resilient against LRO. Taira et al. measured the magnetic susceptibility of a number

of insulating and metallic iridates under field cooling (FC) and zero field cooling

(ZFC) conditions [62]. None of the samples show LRO in the dc-susceptibility data

but spin freezing occurs at around 120 K in all of them. A recent µSR experiment1

has shown long range commensurate AFM order in Eu2Ir2O7 ruling out the QSL

ground state [65]. So far the only metallic iridate which is believed to be the true

realization of the theorem mentioned above is Pr2Ir2O7.

Recently there has been increasing interest in studying the 4d and 5d systems,

which have different properties from the 3d transition metal oxides. Strong corre-

lations dominate the physics of 3d transition metal oxides (TMO) and give rise to

1muon Spin Rotation (µSR) experiment is a very sensitive probe of magnetism [63, 64].


Table 3.1: Comparing 3d to 4,5d orbitals.

orbital spatial extent coulomb interaction crystal field spin orbit

3d weaker strongly correlated weaker weaker

4d,5d stronger intermediate correlations stronger stronger

interesting phenomena such as high temperature superconductivity [66] and colossal

magnetoresistance [67]. Because 4 and 5d orbitals are spatially more extended than

3d orbitals, they are less strongly correlated and support a stronger electron-lattice

and spin-orbit coupling. Table 3.1 lists the differences between these systems. The

physics of 4 and 5d TMO is not “overwhelmed” by strong correlations. The fine

balance between electron-lattice coupling, spin-orbit coupling and correlation energy

scales gives rise to the interesting phenomena observed in these compounds.

Iridium is a fairly large atom and supports a strong spin-orbit coupling. The spin

orbit Hamiltonian from atomic physics is:

Hso = λS · L with λ =ze2h2

m2c2r3(3.1)

directly proportional to the atomic number z which is much larger for a 5d system

compared to a 3d system. Even though iridium oxides are expected to be less strongly

correlated compared with the less extended orbitals of 3d transition metal oxides,

the strong spin-orbit and electron-lattice couplings can lift the orbital degeneracy of

iridium 5d electrons and narrow their bandwidths. Hence iridates can be delicately

poised near a bandwidth controlled metal insulator transition (MIT).


Successive replacement of the R site in R2Ir2O7 with larger rare earth atoms

causes a change-over from insulating to metallic behaviour [68]. Progressing from

right to left in the Lanthanide series of the periodic table, (Lu, Yb, ..., Gd)2Ir2O7

are all insulators; (Eu, Sm, and Nd)2Ir2O7 are metallic at high temperatures but

turn insulating at low temperatures and Pr2Ir2O7 are metallic down to the lowest

temperatures (Fig. 3.2). Increasing the size of the R atom in R2Ir2O7 makes the

Ir-O-Ir bond angle wider and the Ir-O bond length shorter by pushing the oxygen

anion towards Ir ions in the local geometry (Fig. 3.2(a)). As a result, the iridium

t2g bandwidth increases and eventually passes the metallization threshold beyond a

certain R ionic radius [69].

Thermally driven MIT of (Eu, Sm, and Nd)2Ir2O7 at TMI = 120, 117 and 36

K show the following properties [68, 70]: (1) At high temperatures (T > TMI), the

slopes of the ρ(T ) curves are small and negative for the Eu and Sm compounds

but large and positive for the Nd compound which has the largest R3+ ionic size.

(2) The three compounds have a small and temperature dependent gap (∆ < 10

meV) which is smallest in the Nd compound [70]. (3) The Nd compound has the

smallest RRR value defined as R4K/R300K and the lowest TMI. These properties

indicate that replacing the R site with a larger lanthanide, weakens the insulating

phase at low temperatures and enhances the metallic phase at high temperatures.

Further increase of the R size destroys the insulating phase altogether which is the

case for the metallic Pr2Ir2O7. In this system the RKKY interaction 2 between Pr3+

2RKKY interaction is a magnetic interaction between local moments, mediated through theitinerant electrons.


Figure 3.2: (a: top) Replacing the R site in R2Ir2O7 changes the behaviour of elec-

trons from insulating to metallic. When R = Eu, Sm and Nd a thermally driven

metal - insulator transition is realized. (a: bottom) In the local geometry, each oxy-

gen anion (small blue sphere) is surrounded by two Ir4+ and two R3+ ions. Larger

R ions push the oxygen anion towards Ir ions, making the Ir-O-Ir bond angle larger

and the Ir-O bond length shorter. This results in a bandwidth controlled MIT.


local f -moments mediated through Ir4+ itinerant d-electrons competes against the

Kondo screening of the local f -moments. RKKY prefers a magnetically ordered

ground state but the Kondo effect favours a non-magnetic Fermi liquid ground state.

Since Pr2Ir2O7 is a dilute metal, one normally expects the RKKY interaction to

win the competition. However, geometric frustration suppresses the magnetic order,

enhances the Kondo effect on the Pr3+ underscreened f moments and leads into the

emergence of a metallic spin liquid phase [71].

3.2 Experimental Observations

Electron interactions and correlations of pyrochlore iridates can be tuned either by

changing the chemical composition or by applying hydrostatic pressure. Comparing

the results of chemical to physical pressure may be highly instructive in better un-

derstanding the microscopic mechanisms underlying the physics of these compounds.

Hydrostatic pressure can modify the ground state of the material by fine tuning the

bond angles, lattice dimensions and transfer integrals with the advantage of exclud-

ing any ambiguities due to different sample qualities and stoichiometries. The results

of my experiment show that the MIT properties of R2Ir2O7 with R ≥ Eu which result

from chemical pressure mentioned in section 3.1, can be reproduced by physically

pressurizing Eu2Ir2O7. But an additional advantage is that physical pressure can be

tuned continuously which enabled us to reveal a cross-over energy scale (T∗) which

has not been observed in previous measurements by changing the R size and to access

the metallic ground state at high pressures with a non-magnetic ion on the R-site.


Eu2Ir2O7 single crystals were grown by the Nakatsuji group at ISSP in Japan

[72]. Polycrystalline samples of Eu2Ir2O7 were prepared by solid state reactions

of Eu2O3 with IrO2. Powders were ground and pelletized at regular intervals and

heated at 1273 K for several days. Single crystals were prepared by combining these

polycrystalline samples with KF flux in a ratio of 1:200. The samples were heated

to 1373 K and annealed for 3 to 5 hours. The temperature was then decreased to

1123 K at a rate of 2 K/h.

I pressurized a 150×100×30 µm3 single crystal of Eu2Ir2O7 in the moissanite anvil

cell and measured its resistivity using the four terminal ac method in the pressure

range P = 2 to 12 GPa. I used the 1K dipping probe in the temperature range T =

300 to 2 K and measured the resistivity below 2 K and magnetoresistance using our

Oxford 400 dilution refrigerator.

Figure 3.3(a) shows resistivity data from 2 K to room temperature at nine dif-

ferent pressures from 2.06 to 12.15 GPa. For P < 6 GPa the system is a metal with

negative temperature coefficient of resistivity at high temperatures; it goes through

a metal to insulator phase transition at TMI and at lower temperatures becomes an

insulator with the so called “temperature dependent” gap, but notably the resistiv-

ity does not diverge as T→0. The resistivity of the metallic phase at low pressures

is two orders of magnitude higher than the Ioffe-Regel limit (ρIR

= 0.7 mΩcm).

Further increase of pressure effectively reduces the resistivity values and brings the

system closer to the IR limit ρIR = 0.7 mΩcm (Fig. 3.3(a) and (b)). ρ(T→0) falls

with increasing pressure and collapses between 6.06 and 7.88 GPa. Hence, apart

from lowering the resistivity values, hydrostatic pressure suppresses both the nega-


0 50 100 150 200 250 300T (K)

0

500

1000

1500

2000

ρ (m

Ωcm

)

(a) (b)

(c)

7.88 GPa

10.01 GPa

11.34 GPa

TMI

2.06 GPa2.88 GPa3.49 GPa4.61 GPa6.06 GPa7.88 GPa10.01 GPa11.34 GPa12.15 GPa

0 50 100 150 200 250 300T (K)

0

20

40

ρ (m

Ωcm

)

ρIR

0 5 10 15P (GPa)

80

90

100

110

T (

K)

TMI

Tmin

Figure 3.3: (a) Resistivity as a function of temperature at P = 20.6 to 12.15 GPa.

The approximate location of TMI is indicated by arrows for the three lowest pressure

curves. (b) The higher pressure curves at P ≥ 7.88 GPa. The Ioffe-Regel limit ρIR

is shown with a horizontal line. (c) The evolution of TMI and Tmin as a function of

pressure.


0 50 100 150 200 250 300T (K)

0

100

200

300

400

ρ (m

Ωcm

)

11.34 GPa

12.15 GPa

6.06 GPa7.88 GPa10.01 GPa

0 100 200 300T (K)

0

10

20

30

ρ (m

Ωcm

)

0 50 100 150 200 250 300T (K)

0

5

10

15

20

25

30

ρ (m

Ωcm

)

Tmin

T*

Tmin

7.88 GPa10.01 GPa11.34 GPa12.15 GPa

0 100 200 300T (K)

2.5

3

3.5

4

4.5

ρ (m

Ωcm

)

(a) (b)

T*T

min

T*

Tmin

T*

Tmin

Figure 3.4: (a) ρ(T ) from 6.06 to 10.01 GPa. The inset zooms into these curves to

show T∗ and Tmin more clearly. The resistivity values are shifted to give the reader

a better view on the data. (b) ρ(T ) from 7.88 to 12.15 GPa. Pressure suppresses the

resistivity ratio R4K/R300K . The inset zooms into the two highest pressure points

with the resistivity ratio less than 1.


tive slope of resistivity at high temperatures and the resistivity ratio (R4K/R300K) in

agreement with the effects of chemical pressure [70]. Figure 3.3(b) zooms into the

curves at higher pressures from P = 7.88 to 12.15 GPa. The IR limit is indicated with

a horizontal line. TMI, derived by differentiating the resistivity data with respect to

temperature, is plotted versus pressure in Fig. 3.3(c).

In the P = 6.06 GPa curve, there is a new feature: the negative temperature

coefficient of resistivity changes into a positive one at T∗ = 180 K revealing a cross-

over within the metallic regime (Fig. 3.4(a)). T∗ rises to 270 K at P = 7.88 GPa and

seems to be pushed above room temperature at higher pressures. Since resistivity of

the metallic phase decreases with temperature below T∗, a minimum appears in the

resistivity data at Tmin for P ≥ 6.06 GPa (Fig. 3.4). The pressure dependence of

Tmin and TMI is shown in Fig. 3.3(c).

Thermal activation of the resistivity in the insulating phase can be studied by

plotting log(ρ) as a function of 1/T. Figure 3.5(c) shows that log(ρ) is linear in 1/T

in the temperature range 40K < T < TMI. Below 40 K, log(ρ) is no longer linear

in 1/T and it levels off below 5 K. The value of the insulating gap at each pressure

∆(P ) is extracted from the slope of the linear fit in the temperature range 40K < T

< TMI and plotted against pressure in Fig. 3.5(b) which suggests a continuous closing

of the gap. By extrapolating these lines to zero temperature I extracted the residual

conductivity at each pressure and plotted σ(0) as a function of pressure in Fig. 3.5(a)

suggesting the continuous nature of the phase transition. It should be noted that

at 0 GPa, increasing sample purity leads to decreased σ(0), and it is possible that

an ideally pure sample would show a discontinuous jump to zero conduvtivity across


the MIT.

Even though the insulating phase is collapsed (∆ = 0 and σ(0) 6= 0) above 6.06

Gpa a low temperature rise of resistivity survives up to the highest pressure. Figure

3.5(d) is a semi-logarithmic plot of ρ(T) at P = 10.01 GPa from room temperature

to 100 mK showing the non-exponential rise of resistivity down to the milliKelvin

temperature range. This suggests that the metal to insulator transition at TMI is

replaced by another transition or cross-over between two different metallic regimes

at Tmin for P > 6 GPa. Figure 3.3(c) shows that Tmin is suppressed by pressure

initially, probabely because of the falling insulating resistivity, but at P > 10 GPa it

is clearly enhanced by pressure.

I measured the magnetoresistance (MR) at P = 10.01 GPa by sweeping magnetic

field from 0 to 16 Tesla at 10 different temperatures from 100 mK up to 8 K. Fig-

ure 3.6 shows that the weak positive MR signal observed in our experiment grows

quadratically ( M ∝ H2 ) at low fields, seems to saturate at high fields and becomes

smaller with increasing temperatures. Such behaviour is typically observed in metals

with closed electron orbits on the Fermi surface [73]. In the Drude model of metals

MR is proportional to ωcτ ∝ Bσ0/ene therefore a weak MR signal implies either a

small residual conductivity σ0 or a large density of electrons ne. The former seems

more plausible for our system.


0.01 0.015 0.02 0.025 0.03

T-1

(K-1

)

4

4.5

5

5.5

6

6.5

Log

(ρ)

(a) (b)

(c) (d)

0 2 4 6 8 10 12 14P (GPa)

0

5

10

15

20

25

∆ (m

eV)

0.1 1 10 100T (K)

17

18

19

20

21

22

23

ρ (m

Ωcm

)

10.01 GPa

0 2 4 6 8 10 12 14P (GPa)

0

0.05

0.1

0.15

0.2

0.25

σ (0

) (m

Ω−1

cm-1

)

Pc

2.06 G

Pa

3.49 GPa

6.06 GPa

10.01 GPa

Figure 3.5: (a) The residual conductivity (σ(0)) is plotted as a function of pressure,

suggesting that the pressure induced MIT is continuous. (b) ∆(P ) is continuously

suppressed by pressure. (c) The insulating phase of Eu2Ir2O7 shows a temperature

dependent gap for T < TMI . The insulating gap ∆(P ) is extracted by fitting a single

exponential to the resistivity data at each pressure in the temperature range 40K <

T < TMI. (d) Semi-logarithmic plot of resistivity from room temperature to 100 mK

at P = 10.01 GPa.


0 4 8 12 16H (Tesla)

0

0.01

0.02

0.03

0.04

∆ρ/ρ

0

100 mK200 mK300 mK400 mK800 mK1.2 K2.2 K3.2 K4.2 K8 K

0 4 8 12 16H (T)

0

0.01

0.02

0.03

0.04

P = 10.01 GPa

200 mK

Figure 3.6: Magnetoresistance is plotted as a function of magnetic field at P = 10.01

GPa. Data is taken in a temperature range from 100 mK to 8 K. MR is suppressed

by raising temperature. The inset shows a quadratic fit (black dashed line, M ≈ H2)

to the low field data at T = 200 mK. At high fields MR may be saturating.


3.3 Discussion

The metal-insulator change-over in the R2Ir2O7 series occurs at the boundary be-

tween R = Gd and Eu with Eu2Ir2O7 being the first compound in this series to show

a metallic phase at high temperatures. Metallic phases in the vicinity of localiza-

tion transitions are usually subject to strong fluctuations in the spin, charge and

orbital degrees of freedom. Such metals exhibit quite different behaviours from ordi-

nary metals as observed in transport, optical and magnetic properties. The negative

temperature coefficient of resistivity in the metallic regime (T > TMI) of Eu2Ir2O7

(Fig. 3.3) is likely to be a result of incoherent scattering of electrons off spin and or

charge fluctuations. Since the residual resistivity of this metallic phase at low pres-

sures is two orders of magnitude higher than the Ioffe-Regel limit (ρIR

= 0.7 mΩcm)

the metal-insulator transition at TMI cannot be a purely disorder driven Anderson

localization. In fact disorder wipes out the insulating phase of Eu2Ir2O7 and leaves

the system metallic at all temperatures [68].

The evolution of the slopes of the ρ(T) curves above TMI from negative to positive

with increasing pressure is in accord with the results of chemical pressure [70]. Fur-

thermore, our data unveils a cross-over from negative to positive resistivity slope at

intermediate pressures indicated by T∗ ( Fig. 3.4). The broad peaks at T∗ observed

at P = 6.06 and 7.88 GPa mark a coherent-incoherent cross-over of the quasiparticle

dynamics generic to correlated oxides in proximity to a Mott transition [74, 75].

There is a remarkable similarity in the phenomenology of Eu2Ir2O7 and Gd2Mo2O7

which has Mo3+ ions in the 4d2 state. Gd2Mo2O7 is located at the boundary be-

tween the ferromagnetic metal and the spin glass Mott insulator in the R2Mo2O7


series [76], similar to Eu2Ir2O7 which is located at the boundary between the spin

glass insulator and the spin glass metal in the R2Ir2O7 series [62]. The same pressure

induced coherent-incoherent cross-over is observed in Gd2Mo2O7 at high tempera-

tures (T > 150 K) [77]. T∗ is shifted up by applying pressure in both systems with

a rate ∆T∗/∆P ∼ 50 K/GPa in Eu2Ir2O7 and ∼ 35 K/GPa in Gd2Mo2O7.

My experiment shows that hydrostatic pressure, similar to chemical pressure, in-

duces an insulator to metal quantum phase transition at Pc = 6.0 ± 0.6 GPa. Figure

3.5(a) suggests that the transition is continuous. I could not do a careful scaling

analysis due to the limited number of data points. Similarly, hydrostatic pressure in-

duces a continuous insulator to metal quantum phase transition in Gd2Mo2O7 at Pc

= 2.4 GPa by controlling the t2g bandwidth of Mo 4d orbitals [77] parallel to the ef-

fect of chemical pressure [78]. The major difference between molybdates and iridates

is that the ground state of the former is ferromagnetically ordered on the metallic

side due to double exchange interactions between Mo3+ conduction d electrons [78].

Hydrostatic pressure destroys long range FM order in the metallic molybdates by

tuning the relative strength of double exchange and super exchange interactions.

This order-disorder transition coincides with a change of resistivity behaviour from

a conventional metallic state to a diffusive non-Fermi liquid state whose resistivity

increases with decreasing temperature [79]. I observe the same diffusive transport

below TMI in Eu2Ir2O7 at high pressures. The low temperature rise of resistivity in

both systems may be a manifestation of strong scattering of d electrons off the spin

fluctuations of the frustrated transition metal sublattice. In this scenario, all the

physics occurs in the transition metal sublattice and the rare earth sublattice does


not play a significant role in the electronic transport properties.

It is tempting to attribute the low temperature rise of resistivity of the diffusive

metallic phase to a Kondo-like effect. However, local f moments are quenched in

Eu2Ir2O7 according to the crystal field analysis [80] so I can not draw a direct corre-

spondence between the high pressure metallic phase of Eu2Ir2O7 and the frustrated

Kondo lattice of Pr2Ir2O7 [71]. In the latter system, the two f electrons of Pr3+

fill the ground state doublet which is well separated from an excited singlet, giving

rise to Ising-like moments of ∼ 0.3 µB oriented along the local 〈111〉 direction [80].

According to this CEF scheme, there should not be any local f moments on the

Eu3+ sites in Eu2Ir2O7. Unless distortions of the lattice at high pressures change

the CEF scheme and produce local moments on the Eu site, this diffusive mettlaic

ground state must be distinct from the frustrated Kondo lattice of Pr2Ir2O7.

The concurrence of spin freezing in the iridium sublattice and MIT at roughly the

same temperature in Eu2Ir2O7 [70] plus the fact that metal-insulating change-over in

R2Ir2O7 series is controlled by the Ir-O-Ir bond angle suggest that the Eu sublattice

does not play a major role in determining the ground state of the system. Added

to this, the relatively high values of Tmin and the crystal field induced quench of f

moments convince us to rule out the d-f hybridization mechanism.

Collecting all our data, I present a phase diagram for Eu2Ir2O7 depicted in

Fig. 3.7. T∗ sets the boundary between the ordinary and the incoherent metallic

phases at high temperatures. At low pressures (P < 6 GPa) the system settles in

an insulating ground state characterized by the exponential growth of resistivity at

low temperatures and vanishing residual conductivity (Fig. 3.5). The ground state


Figure 3.7: A phase diagram is constructed for Eu2Ir2O7 from the data presented in

this letter.

goes through a quantum phase transition at Pc = 6.0 ± 0.6 GPa from the insu-

lating to a diffusive metallic state. Our phase diagram shows that the collapse of

the low temperature insulating state occurs just when the high temperature metallic

state begins to show coherent transport suggesting that there may be a connection

between the two phenomena. The resistivity of the diffusive metallic phase at finite

temperatures is above the IR limit (Fig. 3.3(b)). It increases with decreasing tem-

perature and saturates to a finite residual value. One interesting point is that Tmin

and TMI seem to be rather pressure independent and comparable in magnitude.

A recent LSDA+U+SO calculation on the model system Y2Ir2O7 with non-

magnetic R site reveals a Dirac semi-metallic ground state in the regime of interme-

diate d electron correlations [81]. The ground state of this band structure calcula-

tion, which is entirely based on the t2g levels of the iridium sublattice, depends on


the magnetic configuration of the iridium tetrahedra. The non-collinear all-in/out

configuration with zero net magnetic moment has the lowest energy and favours a

metallic ground state at small U (< 1 eV), a Mott insulating ground state at large

U (> 2 eV) and a Dirac semi-metallic ground state at intermediate U (1 < U < 2

eV). The Fermi surface of the Dirac semi-metal is comprised of a set of Dirac points

in the bulk and Fermi arcs on the surface of the Brilloine zone. The low temperature

rise of resistivity of the diffusive metallic phase of Eu2Ir2O7 at high pressures might

be a manifestation of the Dirac semi-metal. However, our observation of a weak MR

signal (Fig. 3.6) is not in favour of a semi-metallic state whose MR is expected to

be large due to small Fermi volume and low density of carriers. The small Fermi

velocity of the Dirac fermions in this theory might partially justify the weak MR

signal. This theory also predicts that a strong magnetic field induces a collinear FM

order which gives rise to metallic bands regardless of the value of U. Usually, FM

metals exhibit a negative MR [73] in contrast to our observation of a positive MR

signal (Fig. 3.6).

The fine balance between correlation, spin orbit and crystal field energy scales

in iridates has lead to the theoretical prediction of their strong topological insulator

ground states [16]. The trigonal distortion of the oxygen octahedron surrounding each

Ir4+ ion which is controlled by oxygen x parameter is believed to be an important

factor in determining whether the ground state of the system is metallic or insulating

[82, 83]. Given that varying the size of the rare earth atom changes the amount of

the trigonal distortion by tuning the x parameter, I believe that hydrostatic pressure

may have a similar effect on Eu2Ir2O7. This raises the possibility that the finite


temperature MIT of Eu2Ir2O7 is connected to a strong topological insulator ground

state via a pressure induced structural distortion. The small value of the gap and its

temperature dependence might be indicative of the TI phase.

3.4 Summary

Eu2Ir2O7 is located on the verge of metal-nonmetal change-over in the R2Ir2O7. I

have studied the evolution of the finite temperature MIT in this compound by means

of electronic transport measurements under pressure up to 12 GPa. Figure 3.7 shows

the phase diagram of Eu2Ir2O7. At high temperature and low pressure the system is

an incoherent metal with high resistivity value. By increasing the pressure, this phase

crosses over into a conventional metallic phase at T∗ which can be revealed only by

pressure and not by chemically substituding the R site with a larger lanthanide.

By decreasing temperature, the incoherent metallic phase goes through a MIT

at TMI and becomes insulating. The conventional metallic phase, however, crosses

over into a diffusive metallic phase at Tmin. TMI and Tmin are comparable in mag-

nitude. Both temperature scales do not show a strong pressure dependence whereas

T∗ strongly depends on pressure.

My experiment shows that the pressure tuned insulator to metal QPT of Eu2Ir2O7

imitates all the properties of the same transition in the R2Ir2O7 series by changing

the R size. All the transport properties of Eu2Ir2O7 can be explained with reference

to the Ir sublattice and the Eu sublattice seems to play a negligible role in the

electronic transport properties.

Chapter 4

FeCrAs

The discovery of Iron based superconductors [84] triggered a lot of research interest

in Iron Pnictides. The critical temperatures (Tc) of these compounds were raised

within a few weeks from 26 K in LaFeAsO1−xFx [84] to 55 K in SmFeAsO1−xFx [85].

The phase diagram of iron pnictide superconductors is similar to that of cuprates.

Antiferromagnetic (AFM) order is destroyed by hole-doping and superconductivity

sets in (Fig. 4.1). However, soon people realized that there are major differences

between the two families of High Tc compounds.

The parent compounds of cuprates are “Mott insulators” due to the large value

of Hubbard U (strong correlations) but the parent compounds of Fe pnictides are

“spin density wave metals” (SDW). AFM order in a Mott insulator occurs because

the spins can lower their kinetic energy if they are antiparallel to their neighbours.

A spin-density wave is a collective effect that typically emerges from an instability of

a paramagnetic Fermi surface. The electronic structure of a cuprate is quasi-2D and

106

Chapter 4. FeCrAs 107

Figure 4.1: (a) Phase diagram of fluorine doped CeFeAsO1−xFx, showing a smooth

(second-order) change from antiferromagnetism (AFM) at low hole-doping to super-

conductivity (SC) at larger dopings. TN is the magnetic transition. Other mem-

bers of the 1111 family have similar phase diagrams. Picture adopted from reference

[86]. (b) Phase diagram of Ba(Fe1−xCox)2As2, showing the structural transition from

tetragonal to orthorhombic at TS slightly above the AFM ordering temperature TN .

In this material, superconductivity and magnetic order coexist below x ≈ 6% Co

doping. Other members of the 122 family have similar phase diagrams. Picture

adopted from reference [87].


Figure 4.2: The Fermi surface of cuprates is made of a single open cylindrical sheet

whose 2D cross section is shown on the left. The Fermi surface of pnictides, shown

on the right, consists of two hole pockets around the center of the Brilloine zone

Γ (0,0), two eliptical electron pockets at (0,π) and (π,0). Some pnictides have an

additional hole poket centered at M (π,π). Picture is adopted from reference [90].

the low energy carriers reside on a single band, hence a single band Hubbard model

is believed to be relevant to cuprates. Pnictides, compared to cuprates, are less 2D

and more 3D 1 which makes the electronic structure less correlated. All the 5 iron d-

bands contribute to the Fermi surface of pnictide hence a single band Hubbard model

is not applicable. Figure 4.2 compares the Fermi surfaces of cuprates and pnictides.

Similar to cuprates, the parent compounds of Fe pnictides show a structural phase

transition from tetragonal to orthorhombic at a TS slightly lower than TN (Fig.4.1)

[87, 88, 89]. Electron or hole doping suppresses both the structural and the magnetic

transitions and superconductivity appears.

1Pnictides do have insulating layers but the interlayer coupling is stronger in pnictides comparedwith cuprates.


FeCrAs shows similar behaviour to Fe pnictides. Above TN = 125K, the system

is a bad metal. Below TN an AFM spin density wave sets in and the system starts

to show a better metallic behaviour. However there are important differences both

in crystal structure and in electronic transport which I will explain. In this chapter,

I will review the basic properties of FeCrAs and discuss the similarities and the

differences of this compound compared with iron pnictides. The crystal growth and

the characterization of the material were done by my colleague Dr. Wenlong Wu

[91]. I studied this compound under pressure using moissanite and diamond anvil

cells to kill the AFM order and realize a quantum critical point in the ground state

of the system with the hope of observing an emergent superconducting phase near

the QCP.

4.1 Material Background

FeCrAs belongs to a large family of ternary transition metal monopnictides, chemi-

cally formulated as MM′Z. In this stoichiometry, M and M′ represent two transition

metals and Z is either phosphorus or arsenic. Depending on which transition metals

and pnictogens are plugged into MM′Z, three different crystal structures may result:

tetragonal, hexagonal, or orthorhombic [92]. The tetragonal phase is the most com-

mon one and the orthorhombic phase is the least common phase. The tetragonal

structure is replaced by the hexagonal and eventually orthorhombic structures as

the lattice parameters decrease. FeCrAs has a hexagonal crystal structure, different

from Fe based pnictides, which have a tetragonal structure.


Figure 4.3: (a) and (b) show the local geometrical coordination around the Fe and

Cr sites. Each Fe site is surrounded by four As (tetrahedral coordination) and each

Cr site is surrounded by five As (pyramidal coordination). (c) shows an array of

3× 3 unit cells corresponding to the magnetic unit cell of the material which has the

ordering wave vector (1/3,1/3,0) (see the text). Picture taken from reference [93].


Figure 4.3 shows the Fe2P hexagonal crystal structure of FeCrAs. Fe2+ ions

shown as green spheres form a 2D triangular lattice of trimers and Cr3+ ions shown

as blue spheres form a distorted Kagome lattice. Such geometries are prone to

magnetic frustration. Each Fe2+ ion is tetrahedrally coordinated with four As5− ions,

similar to the coordination of Fe atoms in pnictide superconductors. The tetrahedral

coordination does not lift the degeneracy of Fe d-orbitals effectively, hence the 3d6

orbital configuration of Fe2+ does not support any magnetic moments in FeCrAs.

This is different from pnictides whose Fe3+ ions do support magnetic moments. Each

Cr3+ is pyramidally coordinated by five As5− anions. Cr3+ is in the 3d3 orbital

configuration and supports a magnetic moment ∼ 2µB as measured by neutron

scattering2.

Of course this is an itinerant system and the ionic picture given above, is not fully

responsible for why the moments are quenched on the Fe site. Another important

factor in quenching the Fe moments is the strong hybridization of the large As anions

with the Fe ions.

Single crystals of FeCrAs are grown by melting stoichiometric quantities of high-

purity Fe, Cr, and As followed by an annealing process at 900 in vacuum for five

to ten days [91, 94]. X-ray powder diffraction is used to check that the samples are

crystallized in the correct structure (space group P 62m).

Figure 4.4(a) shows the magnetic susceptibility of FeCrAs measured by Wu et

al. using a MPMS system [93]. The measurement is done for fields along the a and

c axes in both field cooled (FC) and zero field cooled (ZFC) modes. Both χa (red)

2unpublished data.


and χc (black) show a peak at TN = 125 K indicating AFM ordering. Below TN

the behaviour of the two curves is different: χa goes down but χc goes up. The

overall temperature dependence of χa,c is small and it does not fit to a Curie-Weiss

behaviour as shown in the right inset of Fig. 4.4. Therefore, the susceptibility

of FeCrAs is essentially a temperature independent Pauli-like behaviour which is

typically observed in metals.

Both χa,c curves show spin freezing below Tf = 15 and 7 K for the field along a

and c axis respectively. Tf is strongly sample dependent and the values reported in

Fig. 4.4 are the lowest values for Tf from our “best” samples 3.

A powder neutron scattering measurement 4 confirms the AFM order in the form

of a commensurate SDW with wave vector (1/3,1/3,0) which triples the unit cell.

Magnetic moment 0.6 < µ < 2.2µB resides on the Cr site and no magnetic moments

are detected on the Fe site as expected from theoretical LDA calculations [95]. This

is perhaps the most important difference between FeCrAs and the superconducting

iron pnictides. Magnetism is driven by Cr ions in the former and by Fe ions in the

latter. The superconductivity of pnictides seems to originate from the quasi 2D FeAs

layers which support a SDW on the Fe ions. In contrast, FeCrAs is a 3D system

with quenched moments on the Fe sites. Nevertheless, both systems have the same

Pauli-like susceptibility and roughly similar TN which suggests that there may be

similar physics.

Figure 4.4(b) shows the specific heat data measured with a PPMS system from

3We define our “best” samples to be the ones with the smallest freezing temperatures.4unpublished data.


Figure 4.4: (a) Magnetic susceptibility data with H ‖ a and H ‖ c shown as red and

black curves in both FC and ZFC modes. AFM order sets in at ∼ 125 K. The two

insets show that no hysteresis has been observed (left) and a Curie-Weiss model does

not fit to the data (right). (b) Specific heat data plotted from room temperature

down to 2 K. The low temperature data fits to a Fermi liquid model: C/T = γ+βT 2

with γ = 31.6 mJ/mol K2. Plots taken from reference [91].


300 to 2 K [93]. The low temperature Cv data show Fermi liquid behaviour: C/T =

γ + βT 2. The Summerfeld coefficient γ = 31.6 mJ/mol K2 is rather high for a d-

electron system. The Wilson ratio defined as:

RW =

(

4π2k2B

3g2µ2B

)

χ/γ (4.1)

is between 4 to 5. The Wilson ratio is a measure of the strength of spin-orbit

interactions in frustrated systems. RW = 1 for a free electron gas, 2 for heavy

fermion systems and much larger in FM metals.

Both the magnetic susceptibility and the specific heat data show that FeCrAs

is a metallic system with conventional Fermi liquid behaviour. The resistivity data,

however, give a new twist to the story. Figure 4.5 shows the resistivity along a (red)

and c (black) axes measured in a wide temperature range from 900 K to 80 mK [93].

ρc(T ) increases from about 0.2 to 0.5 mΩ cm with decreasing temperature from 900

K to 125 K. There is no sign of saturation at high or low temperatures. The c-axis

resistivity drops below TN = 125 K signalling the AFM order. Surprisingly, it regains

its rising behaviour below 50 K with a sub-linear power law ρc(T ) = ρc,0 −AT 0.7±0.05

characteristic of a Non Fermi liquid (NFL) metallic phase 5. There is no sign of a gap

or saturation at low temperatures. ρa(T ) shows similar behaviour: it increases with

decreasing temperature with no sign of saturation at low or high T and no sign of a

gap at low T . At the transition ρa(T ) shows a change of slope, but unlike the c-axis

resistivity, it does not drop below TN and continues to rise. The low temperature

behaviour of the a-axis resistivity fits to a NFL power law ρa(T ) = ρa,0 −AT 0.6±0.05.

5The resistivity of a Fermi liquid metallic phase obeys ρ(T ) = ρ0 + AT 2.


Figure 4.5: Resistivity vs. temperature from 900 K to 80 mK along the a (red) and

the c (black) axes. The peak at 125 K on the c-axis curve marks the AFM order.

The low temperature behaviour fits to a sub-linear power law for both ρa(T ) and

ρc(T ). The inset shows the sub-linear rise of ρa(T ) and a weak magnetoresistance

signal.


The most significant feature from ambient pressure measurements of FeCrAs is

the contrast between Fermi liquid like specific heat and susceptibility data and the

non-Fermi liquid behaviour of resistivity at low temperatures. Incoherent transport

at high temperatures has also been observed in the parent compounds of Fe based

superconductors [84], however, conventional metallic behaviour sets in below the

SDW transition. The curious rising resistivity of FeCrAs below ∼ 50 K is in contrast

to what has been observed in superconducting pnictides. This behaviour might be

related to strong spin fluctuations which survive even below TN in the SDW phase.

Geometric frustration might be the underlying mechanism for such fluctuations. Spin

fluctuations seem to be the main driving force behind the NFL behaviour observed

in cuprates, 2D organic materials, and some of the heavy fermions [96].

The NFL behaviour of FeCrAs at low temperatures may be connected to a quan-

tum critical point (QCP) in the ground state of the material. QCP controlled NFL

behaviour is a canonical feature in the phase diagram of cuprates [97]. Quantum

criticality, explained in chapter 1, emerges as a result of competing ground states.

In many cases, a new phase such as superconductivity may emerge as a result of

the competition between the different ground states. The SDW transition at TN can

be suppressed by pressure and once TN goes to zero, a QCP will be realized. A

resistivity measurement under pressure is ideal to establish the connection between

the NFL behaviour and the QCP.

Having developed the anvil cell technology (chapter 2), I embarked on the project

of studying the electronic transport properties of FeCrAs under pressure. The goal

was to suppress TN and look for quantum critical behaviour.


0 50 100 150 200 250 300Temperature (K)

340

360

380

400

420

440

460

480

Res

istiv

ity (

µΩcm

)

8.2 kbar13.3 kbar26.6 kbar40.9 kbar49.1 kbar63.4 kbar75.8 kbar87.9 kbar97.1 kbar

Figure 4.6: Resistivity along the c-axis as a function of temperature, from 300 to 2 K.

Data is taken using the 1K probe. Each curve represents data at a certain pressure.

The position of the peak at each curve marks the AFM transition temperature (TN).

By increasing pressure, TN is shifted to lower values. The low temperature behaviour

fits to a sub-linear power law with nearly the same exponent at all pressures.

4.2 Experimental Observations

Since it is easier to read TN from the c-axis resistivity data, I polished a sample and

wired it along the c-axis. I pressurized a 250×200×30 µm3 single crystal of FeCrAs

in the moissanite anvil cell and measured its resistivity using the four terminal ac

method in the pressure range P = 8 to 100 kbar. I used the 1K dipping probe for

the measurement in the temperature range T = 300 to 2 K.


Figure 4.6 shows the resistivity data which can be studied in reference to three

temperature regimes: T ∼ TN , T < TN , and T > TN .

T ∼ TN : The first prominent feature is the suppression of the AFM order from

∼ 115 K at ambient pressure to 43 K at 97.1 kbar. TN is extracted by finding the

position of the peak in the c-axis resistivity curves at each pressure and plotted as a

function of pressure in Fig. 4.7. Pressure suppresses TN in a seemingly linear fashion

with the rate of 0.7 K/kbar. I extrapolated this line to T = 0 and estimated the

QCP to be located at ∼ 160 kbar. This pressure is achievable with a diamond anvil

cell.

As pressure shifts TN to lower values the AFM peak in the resistivity data, be-

comes smaller. As a result, it is more difficult to extract the TN values at higher

pressures. I used the temperature derivative of resistivity to extract TN more ac-

curately, however, the uncertainty grows larger at higher pressure as shown in Fig.

4.7.

T > TN : Apart from suppressing TN , pressure also decreases the resistivity

values (Fig. 4.6). By fitting the T > TN resistivity data to a line, I compare the

slopes of these ρ(T ) lines at different pressures, to see whether or not pressure is

continuously metallizing the system. If this is true, I expect the slope of the ρ(T )

curves become smaller as the pressure is increased. Figure 4.8 shows that within

the errors of the measurement, the slopes of the ρ(T ) curves at different pressures

are more or less unchanged. This shows the metallic phase at high temperatures is

robust against pressure.

T < TN : The NFL rise of resistivity at low temperatures is also robust against


0 20 40 60 80 100 120 140Pressure (kbar)

0

20

40

60

80

100

120

140

TN

(K)

Pc=157 Kbar

TN

= 114.67 - 0.73 P

Figure 4.7: TN is extracted from the data in Fig. 4.6 and plotted vs. pressure. QCP

is estimated through a linear interpolation to be at Pc ∼ 157 kbar.


0 20 40 60 80 100Pressure (kbar)

0.15

0.2

0.25

0.3

0.35

0.4

Slop

e (µ

Ωcm

/K)

Figure 4.8: In the high temperature regime of Fig. 4.6, T > TN , resistivity data

at each pressure is fitted to a line and the slopes of these linear ρ(T ) fits at each

pressure are plotted vs. pressure. The slopes of the linear ρ(T ) fits seem to be robust

against the pressure.


pressure with roughly the same power law behaviour as observed in the ambient

pressure data ρc(T ) = ρc,0 − AT 0.7±0.1. It seems like pressure changes the magnetic

transition slowly but the electronic transport is not much affected by pressure. This

brings the question that to what extent the magnetism is related to the electronic

transport in FeCrAs.

4.3 Discussion

The characteristic properties of FeCrAs at ambient pressure are:

1. non-metallic resistivity along both a and c axes with a negative coefficient of

resistivity from 80 mK up to 800 K.

2. non-Fermi liquid electronic transport along both a and c axes with the

exponent x ≈ 0.7 at low temperatures.

3. Fermi liquid specific heat with a rather high Summerfeld coefficient γ ∼ 30

mJ/molK2.

4. Pauli like magnetic susceptibility which is usually observed in itinerant

electron systems. The relatively large value of the magnetic susceptibility

yields a Wilson ratio between 4 and 5.

5. magnetic moments reside almost entirely on the Cr site, not the Fe site, in

contrast to the Fe pnictide superconductors.

The main features of my pressure study on this compound reveals the following

features:


1. non-metallic resistivity is robust against pressure (Fig. 4.6).

2. the negative dρ/dT slope at high temperatures (T > TN) does not show a

significant pressure dependence (Fig. 4.8).

3. non-Fermi liquid transport at low temperatures is also robust against the

pressure with the same exponent x ≈ 0.7 observed at ambient pressure.

4. TN is suppressed by pressure with a slow rate ∼ 0.73 K/kbar.

Non-metallic resistivity is a significant feature that FeCrAs shares with the super-

conducting Fe pnictides whose parent compounds are believed to be incipient Mott

insulators. The electron-electron interactions in the parent compounds of pnictide

superconductors are believed to be near but not exceeding the localization thresh-

old. In a theoretical study of Fe pnictide electronic structure, Dai et al. decompose

the electronic excitations of the material into a coherent part near the Fermi sur-

face and an incoherent part further away from the Fermi surface [98]. The latter

comprises incipient lower and upper Hubbard bands, which accommodate localized

Fe moments. A sketch of the electronic structure is shown in Fig. 4.9. This model

supports a magnetic quantum critical point as a result of the competition between

magnetic ordering of the local moments with the mixing of the local moments and the

coherent electrons. The spectral weight of the coherent quasiparticle peak changes

as a result of mixing and once it exceeds a critical value, magnetism will disappear.

In analogy with the model mentioned above, the non-metallic behaviour of Fe-

CrAs may be a manifestation of an incipient Mott insulator. Optical conductivity

can probe the spectral weight of various excitations at different energies, hence it


Figure 4.9: The generic single electron spectrum of the parent compounds of Fe-

based superconductors is composed of a quasiparticle peak at the Fermi level (the

coherent part) and incipient upper and lower Hubbard bands (the incoherent part)

away from the Fermi level. Picture adopted from reference [98].

is a useful probe to investigate the validity of this idea. Within this theoretical

framework, we naively expect that because pressure increases the mixing between

the coherent and incoherent parts of the spectrum, at some critical pressure the mag-

netic order must vanish and it is possible to realize a superconducting dome near the

magnetic QCP similar to pnictides. The size of the local Fe moments in Fe pnictide

parent compounds is usually less than 1µB/Fe [99, 100, 101] which is comparable to

the size of the magnetic moments in FeCrAs (between 0.6 to 2.2 µB/Cr). However,

in pnictides both coherent and incoherent parts of the spectrum originate from Fe

atoms but in FeCrAs local moments reside on the Cr and not the Fe site so that the

incoherent carriers are perhaps released from Cr d-orbitals and the coherent part of

the spectrum most likely stems from Fe d-orbitals hybridized with As p-orbitals.


The analysis of Fig. 4.8 shows that the non-metallic behaviour of FeCrAs is in-

different to the suppression of the magnetic order. The negative dρ/dT slope of the

non-metallic resistivity at high temperatures (T > TN) does not change as pressure

suppresses TN . If the charge carriers in FeCrAs comprise a coherent part and an

incoherent part, this behaviour means that there is not much mixing between the

two parts of the spectrum. If the coherent/incoherent mixing was sizable, we would

expect a change in the resistivity slope as a function of pressure.

Another view is that the localization of electrons in FeCrAs might not be entirely

correlations driven. The resistivity of the material is quite close to the Ioffe-Regel

limit, ρIR

= 0.368mΩ cm. This is comparable to the resistivity values of Fig. 4.5

and 4.6. I calculated ρIR

= 0.368mΩ cm by considering three conduction electrons.

Changing the number of conduction electrons does not change the IR limit consid-

erably. ρIR

= 0.763 and 0.231mΩ cm if I take the number of conduction electrons to

be 1 and 6 respectively.

The suppression of SDW in Fe-based pnictides leads to high temperature super-

conductivity. This certainly is a possibility in FeCrAs. Although the size of the

magnetic moment in FeCrAs is comparable to that of the ferropnictides, the SDW in

FeCrAs is different. One important difference is that magnetic moments in FeCrAs

reside entirely on the Cr site. Apart from this, there also is a difference in the pattern

of magnetic order in FeCrAs and the Fe-based superconductors. FeCrAs has AFM

order with (1/3,1/3,0) wave vector in the ab plane but orders ferromagnetically along

the c-axis. The magnetic order of Fe pnictides is AFM along the a-axis but FM along

the b-axis and the spacer layers do not allow a strong coupling between the layers


[99, 102]. Another important difference is in the relation between the structural

transition and the magnetic ordering. The SDW in ferropnictides is preceded by a

tetragonal to orthorhombic structural phase transition (Fig. 4.1). We have not seen

any evidence of a structural transition in FeCrAs below TN6.

Recent transport measurements associate an electronic nematic order with the

structural phase transition in the superconducting Fe-pnictides [103]. The in-plane

resistivity measurements in the orthorhombic phase of the detwinned crystals of

Ba(Fe1−xCox)As2 show that the resistivity along the shorter b-axis ρb is non-metallic

(similar to that of FeCrAs) while ρa is metallic. This is indeed a surprising result;

Firstly, the orbital overlaps should be stronger along the shorter b-axis and secondly,

the FM order along the b-axis should result in a lower resistance as opposed to the

AFM order along the a-axis. The ratio ρb/ρa is a measure of nematicity. It evolves

with doping showing the following features:

1. The ρb/ρa ratio peaks at ∼ 0.25% near the beginning of the superconducting

dome.

2. It vanishes above 0.8% doping where the system is tetragonal, hence the

nematic phase is a feature of the orthorhombic phase.

3. Weak nematic fluctuations exist well above the structural ordering

temperature.

4. The anisotropy between ρa and ρb terminates above ∼ 200 K.

6unpublished neutron scattering data


These experimental results establish a connection between the crystal symmetry

and the nematicity of the electron liquid. The non-metallic behaviour of ρb is quite

intriguing. At low dopings, below the AFM transition temperature (TN), ρb regains

its metallic behaviour but at higher dopings it keeps increasing as the temperature

is decreased below TN . FeCrAs also shows a non-metallic rise of resistivity at low

temperatures, even below TN . The robustness of the low temperature NFL liquid

phase of FeCrAs against pressure may have roots in the same mechanism which is re-

sponsible for the peculiar behaviour of ρb in Ba(Fe1−xCox)As2. Magnetic fluctuations

seem to be the prime candidate for the scattering processes that lead to non-metallic

resistivity behaviour in both ferropnicitdes and FeCrAs.

Notice that the non-metallic behaviour of Ba(Fe1−xCox)As2 terminates above

∼ 200 K but it lasts in FeCrAs up to 900 K, the highest temperature measured by

Wu et al. [91]. If the magnetic fluctuations are responsible for the non-metallic

resistivity of FeCrAs, we naively expect that the non-metallic transport terminates

above the exchange coupling energy (kBT > J). Typically, the exchange coupling J

can be estimated from TN . The related tetragonal meterial such as Fe2As and Cr2As

order antiferromagnetically at 350 and 395 K respectively [104]. If we take this as

the typical value for J in this family of compounds, it is difficult to see how the non-

metallic resistivity of FeCrAs persists up to 900 K with no sign of saturation. One

possibility is that geometric frustration in FeCrAs enhances magnetic fluctuations

and favours a non-metallic resistivity at higher temperatures.

To understand such non-metallic behaviour in “bad metals” seems to be crucial

in order to solve the mystery of high temperature superconductivity. As a matter of


fact, high Tc material of both cuprate and pnictide families all show poor metallic

resistivity in their normal states. In fact one of the best places to look for a high Tc

superconductor is in the vicinity of a Mott localization transition where the normal

state is a “bad metal” 7.

4.4 Summary

FeCrAs shows a Pauli susceptibility and Fermi liquid specific heat behaviour at low

temperatures but its resistivity is non-metallic from 80 mK up to 800 K. A peak

at TN = 125 K in the susceptibility data shows AFM ordering, below which, ρa

continues to rise but ρc drops. At lower temperatures, ρc rises again. My pressure

study of FeCrAs shows that the non-metallic behaviour at high temperatures and

the non-Fermi liquid transport at low temperatures are robust against pressure.

TN is suppressed with pressure slowly and a magnetic quantum critical point is

expected at around 160 kbar. I have discussed the similarities and the differences

between FeCrAs and the possibility of an emerging superconducting phase around

the quantum critical point.

7This is of course different from what people believed several decades ago, when BCS theoryexplained the “conventional” superconductivity.

Chapter 5

Conclusions and Outlook

In this chapter I will summarize the results of my experiments and discuss the di-

rection for persuing these projects in the future. As I have mentioned in chapter

1, the unifying theme of my research was to study metal-insulator quantum phase

transitions by applying pressure. To pressurize my samples, I developed the anvil

cell technology. In chapter 2, I elaborated on the experimental development of the

anvil cells and the 1K probe. In chapters 3 and 4, I explained the results of my mea-

surements on Eu2Ir2O7 and FeCrAs respectively. In the final chapter of my thesis I

will follow the contents of the previous chapters. I will explain how the experimental

settings may be improved and what future experiments can be done on pyrochlore

iridates and FeCrAs.

128

Chapter 5. Conclusions and Outlook 129

5.1 Anvil Cells

During my PhD, I developed the anvil cell technology from scratch and performed

measurements at ultrahigh pressures. I used both moissanite and diamond anvils.

Moissanite anvils are relatively cheaper yet they tolerate pressures as high as 12

GPa [105]. Moissanite anvils have optical, thermal, electric, and X-ray properties

that rival those of diamond anvils [39]. In designing the anvil cells (section 2.1.1),

I was commited to simplicity and efficiency. I reduced many of the complicated

and unnecessary components used in the previous designs [35, 37]. The advantage

of simplifying the design is two fold: it makes the device more user friendly and it

provides a simple platform on which we can add more complicated components.

I have developed resistivity measurements under pressure. In doing so I overcame

the challenge of wiring very small samples as I explained in section 2.1.5. Although

it is possible to wire very small samples (∼ 100 × 50 × 30µm3), it is preferred to

use more modern techniques such as Focused Ion Beam (FIB) to put wires on the

samples. This technique will allow us to wire even smaller samples and to reduce the

contact resistance. FIB can also be used to micromachine the sample to any desired

shape by means of a beam high energy ions. FIB technique is well explained in a

number of review articles [106, 107, 108, 109].

Another area of improvement for the anvil cell is to use more hydrostatic pressure

media. Anisotropic distribution of pressure in the sample hole can sometimes make

the interpretation of experimental results confusing. Phase transitions are broadened

by non-uniform pressure and in the case of superconductivity, a finite residual resis-

tivity might survive below Tc [110]. For my measurements, I mostly used Daphne oil


Figure 5.1: Daphne oil phase diagram. At room temperature, Daphne oil freezes at

about 2.5 GPa. Plot adopted from reference [112].

7373 , which works very well 1. Other options are isopentane and Fluorinert 77/70.

Isopentane which is a pure chemical substance, may cause serious damage to sam-

ples due to a discontinuous pressure drop associated with solidification on cooling.

Fluorinert 77/70 shows a 30 to 40% pressure deficit from room temperature to liquid

helium temperature. In contrast, Daphne 7373, exhibits almost no discontinuous

pressure drop when it solidifies and only the derivative dp/dT shows a clear anomaly

[111]. This is a very important characteristic of Daphne 7373, namely, it does not

show a non-uniform pressure distribution on passing its freezing point. Figure 5.1

shows the P-T phase diagram of Daphne oil 7373 adopted from [112].

1Daphne oil is a mixture of several olefines which are open chain organic compounds with formulaCnH2n.


The most hydrostatic pressure medium is liquid He, however, highly specialized

equipments are required to load liquid He into the sample hole. Liquid Argon is a

very good candidate since it is relatively easy to load into the sample hole and it

is more hydrostatic than Daphne oil. I have made an Ar liquefying system shown

in Fig. 5.2. This simple liquefier is made of a copper tube and a Styrofoam bucket

which can be filled with liquid nitrogen. The boiling temperatures of nitrogen and

argon are 77 and 87 K respectively. Therefore, by running Ar gas through a copper

tube immersed in liquid nitrogen, we can liquefy argon (Fig. 5.2(a)).

In order to load liquid argon in the sample hole, first I fix the pressure cell in

the brass pot shown in Fig. 5.2. The pot has a lid with inlet and outlet through

which I flow Ar gas before loading liquid Ar (Fig. 5.2(b)). This ensures that air is

not trapped in the sample hole. The cell is loosely assembled in this stage to allow

for the proper flow of Ar gas through the sample hole. Then I fill the container which

surrounds the brass pot with liquid nitrogen and once the brass pot is cold, I open

the lid and slowly pour liquid argon into the pot using a funnel (Fig. 5.2(c)). Once

it is full of liquid argon, I lock the cell using a long Allen key shown (Fig. 5.2(d)).

Pressure is in principle a continuous probe. Nevertheless, we change the pressure

in discrete steps. To obtain data at any certain pressure, we take the cell out of

the cryostat, change the pressure, lock the cell and reinstall it on the cryostat. By

installing an in situ force driving mechanism, one can change the pressure contin-

uously at low temperatures inside the cryostat. One solution is to use a Bowden

cable mechanism where a tube and wire assembly transmit loads limited only by the

tensile strength of the wire. Four stainless steel tubes of about 4 mm descend the


Figure 5.2: (a) Argon liquifier: Ar gas tank is connected to a copper tube inside the

bucket. The copper tube is immersed in liquid nitrogen. (b) The brass pot with Ar

gas inlet and outlet. (c) Pressure cell inside liquid Argon. The brass pot contains

liquid Argon and it is placed in a container filled with liquid Nitrogen to prevent

liquid Argon from evaporating. (d) The bolts will be screwed in using an Allen key.


cryostat. They contain 2 mm wires enabling forces well in excess of 1 tonne to be

transmitted to the pressure cell down in the cryostat [113, 114].

5.2 1K probe

In designing the 1K probe I followed the same principles of simplicity and efficiency.

The wiring, the pump lines, the breakout box and the LabView program are all

designed in a very simple and user friendly yet efficient manner. In particular, I

tried to minimize the parameters required to run the LabView program so that the

future users of this program would not experience any difficulty in running their

measurements and collecting their data 2.

To the 1K probe one can change the vacuum seal with a more reliable indium

seal. To my experience, the vacuum seal is not 100 % reliable so I always pump on

the vacuum chamber before taking it out of the He dewar. To improve it, the vacuum

seal can be replaced by a more reliable indium seal. There are two advantages in

using a vacuum seal; It is easy to use and it provides more experimental space.

Another area of improvement is to write a Proportional Integral Differential (PID)

control program to control the temperature more smoothly. My program is very user-

friendly and only asks for the sweep time and gives a smooth temperature sweep,

but it is not as accurate as a PID controller.

2I gained a lot of insight by studying a much more complicated program, developed by P.M.CRourke, which we use in our lab to run the dilution refrigerator [115].


5.3 Eu2Ir2O7

My research on Eu2Ir2O7 opened fruitful collaborations with theorists and experi-

mentalists 3 and I believe there is room to do much more. To summarize the results

of chapter 3, I showed that the pressure tuned insulator to metal QPT of Eu2Ir2O7

exhibits some common features with the same transition in the R2Ir2O7 series as a

result of changing the R size. In both cases, metallization is bandwidth controlled.

Hydrostatic pressure as a continuous probe enabled us to identify a cross-over at

T∗ which was not observed in the previous chemical pressure experiments. All the

transport properties of Eu2Ir2O7 can be explained with reference to the Ir sublattice

and the Eu sublattice seems not play a significant role.

One interesting point is that pressure is the only probe which is capable of re-

vealing T ∗ in this material. Naively, one can imagine that by doping the Eu site with

larger lanthanides such as Nd or Pr, the same cross-over may be realized. This is not

true because the insulating phase of the pyrochlore iridates proves to be extremely

sensitive to disorder [70] and alloying will kill this phase. To the resolution of our

experiment the QCP in the ground state and the T∗ cross-over above 150 K, appear

simultaneously.

The diffusive metallic phase of Eu2Ir2O7 at high pressures is quite curious. In

chapter 3 I discussed the similarities between iridate and molybdate pyrochlores,

namely the observation of T∗ and the NFL diffusive metallic phase. T ∗ cross-over is

observed in the insulator Gd2Mo2O7 upon metallization at P = 2.4 GPa [77] and the

3I am grateful to Professor Y. B. Kim for helpful discussions and to Professor S. Nakatsuji forproviding Eu2Ir2O7 samples.


Figure 5.3: (a) Increasing the R3+ ionic size metallizes the R2Mo2O7 by reducing

the U/t ratio. The metal-insulator boundary is shown by the vertical black dashed

line. The green and the red lines show the magnetic transitions in the insulating

and the metallic molybdates. The former stays around 20 K for all the insulating

compounds, but the latter varies between 30 to 90 K. (b) Trigonal lattice distortion

splits the t2g levels of Mo4+ ions into a1g and e′g levels. Pictures are adapted from

reference [79].

diffusive metallic phase is observed in the metallic molybdates, (Nd and Sm)2Mo2O7

[79], at high pressures, once the magnetic order is killed.

Figure 5.3(a) shows the phase diagram of the molybdates. The metal-insulator

boundary is between R = Gd and Eu similar to iridates. Unlike the S=1/2 iridates,

the S=1 molybdates order ferromagnetically due to double exchange interactions

amongst Mo d-electrons. The t2g orbitals of Mo4+ ions (4d2 orbital state) further

split into a1g and e′g submanifolds, each containing one electron (Fig. 5.3(b)). The

strong Hund coupling between these two electrons keeps them aligned. Conduction


occurs through the e′g manifold as a1g lies below the Fermi level (Fig. 5.3(b)).

To understand the origin of the diffusive metallic phase one has to first understand

how the paramagnetic (PM) phase is established in the system. There are three en-

ergy scales involved in the problem. The transfer integral t between the conductions

e′g electrons, the antiferromagnetic correlations JAF which characterizes the virtual

hopping of the a1g localized electrons via a superexchange process, and the Hund

coupling JH which keeps these two electrons aligned. Normally JH ≫ t and JAF

which establishes the FM order. Chemical pressure mostly enhances t and physical

pressure affects both t and JAF . This is why physical pressure can destroy the FM

order by tuning the relative strength of super-exchange (SE) and double-exchange

(DE) interactions.

The diffusive NFL metallic regime seems to be a product of strong spin fluctu-

ations on the Mo sublattice as the FM order is suppressed. Iridates are S = 1/2

systems hence there is no SE/DE competition. Nevertheless, the strong spin fluc-

tuations on the frustrated Ir sublattice 4 give rise to the same low temperature rise

of resistivity observed in molybdates. In fact, a S = 1/2 pyrochlore system is the-

oretically predicted not to order at all temperatures [61] due to very strong spin

fluctuations. A recent µSR measurement reveals a commensurate AFM order in

Eu2Ir2O7 [65]. However, pressure may kill the weak AFM order, giving rise to the

diffusive metallic behaviour.

One interesting future project on Eu2Ir2O7 is to look at the T∗ cross-over by

means of optical spectroscopy. Such coherent-incoherent cross-overs are best detected

4The magnetic properties of iridates are briefly reviewed in chapter 3.


by optical transport in the form of spectral weight shifts and broad peaks at high

energies [74]. I intend to pursue this direction of research in the near future to further

study the coherent-incoherent quasiparticle dynamics.

Another interesting idea which needs further investigation is the quantum critical

behaviour in the ground state of Eu2Ir2O7. I have shown that the insulator to metal

QPT of Eu2Ir2O7 at P ∼ 6 GPa is continuous by plotting the residual conductivity

σ(0) vs. pressure in Fig. 3.5. Further evidence to prove the existence of a continuous

QPT in the ground state of a system comes from scaling analyses [116, 117]. I

have not done a scaling analysis due to small number of points, however, I am

interested in taking more data around the critical pressure (6 GPa) and doing a

careful scaling analysis. The purpose of a scaling analysis is to collapse all the data

from insulating and metallic phases respectively and find the critical exponents.

Through such analyses we can figure out the universality class of the transition and

find the universal functions to which the whole set of data can be scaled [118].

To further investigate Eu2Ir2O7 we need to do a Hall effect measurement under

pressure. From the Hall effect data we can find out the mobility, the effective mass

and the sign of the charge carriers. Nakatsuji’s group have already provided us with

high quality samples 5

I am also interested to study the far right side of Eu2Ir2O7 phase diagram (Fig.

3.7) to explore the evolution of the diffusive NFL metallic phase under higher pres-

sures. By pressurizing the system up to 20 GPa which is twice as high as the current

study, we can find out how robust the diffusive transport is against pressure. One

5The new samples have RRR ∼ 1200 compared to the previous samples with RRR ∼ 200.


possibility is that the NFL behaviour turns into a FL behaviour under pressure. The

canonical example of such behaviour is the case of SmB6 which is a narrow band

semiconductor at ambient pressure. The gap is closed by applying hydrostatic pres-

sure with the QCP at Pc = 4 GPa. Above Pc a NFL metal appears characterized by

ρ(T ) = ρ0 + AT x with 1 < x < 2. Far above Pc (∼ 7 GPa) the NFL phase crosses

over to a heavy fermion FL metal where x = 2. Similar behaviour might appear in

Eu2Ir2O7 especially if the Hall measurement gives us a relatively high carrier mass.

It certainly is interesting to pressurize other iridates and study them. The S = 1

molybdates have been studied extensively and it is important to study the S = 1/2

iridtes. Perhaps the next material to study is Nd2Ir2O7 which has quite a small gap.

We can close this gap by relatively lower pressures from what we used for the Eu

compound and study the similarities and the differences between them. Although

Eu is a quenched f -moment system, Nd does have a finite magnetic moment. In

particular, it will be interesting to see if Nd2Ir2O7 also reveals a diffusive metallic

phase.

The metal-insulator boundary of the R2Ir2O7 lies between R = Gd and Eu. While

Eu and Nd compounds are on the metallic side, it is worthwhile to pressurize Gd

from the insulating side. Closing the gap in Gd2Ir2O7 is a more formidable task.

Nevertheless, it is interesting to see if a T ∗ appears in Gd2Ir2O7 similar to Gd2Mo2O7.


5.4 FeCrAs

FeCrAs is an intermetallic d-electron compound. Single crystals of FeCrAs were

grown and characterized in our lab by my colleague, Dr. Wenlong Wu [91]. While

the magnetic susceptibility and the specific heat data suggest that FeCrAs is a Fermi

liquid metal, the resistivity measurement shows a non-metallic non-Fermi liquid be-

haviour. The magnetic susceptibility is Pauli-like along both a and c axes with a

peak at 125 K representing the AFM order. The low temperature specific heat data

fits to C/T = γ + βT 2 with a Summerfeld coefficient γ ≈ 30 mJ/molK2 which is

rather large for a d-electron system. In contrast to both magnetic susceptibility and

specific heat data, resistivity of FeCrAs is non-metallic over a broad temperature

range, from 80 mK up to 800 K. There is a drop in ρc(T ) at 125 K due to AFM order,

but the non-metallic rising resistivity reappears below ∼ 50 K in a non-Fermi liquid

power law fashion (Fig. 4.6).

The goal of my research on FeCrAs was to suppress the magnetic order by pres-

surizing the material in order to reveal the underlying transport properties. My

pressure studies on FeCrAs showed that although AFM transition temperature (TN)

was suppressed by pressure, the non-metallic resistivity at high temperatures and the

non-Fermi liquid resistivity at low temperatures were both robust against pressure

up to 97 kbar (Fig. 4.6 and 4.8).

I extracted the TN value at each pressure from the c-axis resistivity data by finding

the position of the peak on each ρ(T ) curve. At higher pressures, the peak becomes

less prominent, hence the uncertainty in the TN values becomes larger (Fig. 4.7). A

magnetic susceptibility measurement may seem more promising in determining the


accurate values of TN , but since FeCrAs orders antiferromagnetically, the magnetiza-

tion signal is small and a susceptibility measurement in an anvil cell will not give us

a sizeable signal. Even at ambient pressure, we need rather large crystals of FeCrAs

to make the transition visible.

The data which I have presented in chapter 4 clearly shows that pressure sup-

presses the AFM transition. However, the AFM transition was not completely sup-

pressed up to highest pressure of that measurement. At the time of writing this

dissertation, I am collecting new data at higher pressures on another sample of Fe-

CrAs along the a-axis. The non-metallic non-Fermi liquid behaviour seems to be

robust at pressures as high as 173 kbar (Fig 5.4). According to Fig. 4.7 I have passed

the quantum ciritical measurement in this recent measurement. There is no sign

of superconductivity, instead a monotonic non-metallic resistivity persists down to

the lowest temperatures. A closer look at the low temperature data (below 10 K)

shows that the NFL behaviour changes as the system crosses the QCP locate at

Pc ≈ 170 kbar.

Another way to get rid of magnetism in order to understand the strange behaviour

of FeCrAs is to grow a material with the same structure but with a non-magnetic

transition metal on the Cr site. Co seems to be a candidate. The problem is that

in CrCoAs, the Cr and the Co ions will occupy the corresponding Fe and Cr sites of

FeCrAs structure, so the two systems will not be exactly equivalent. The complete

suppression of AFM order is very important especially because it may give rise to a

superconducting phase around the quantum critical point. Pressure seems to be the

best probe to demystify the enigmatic behaviour of FeCrAs.


0 50 100 150 200 250 300T (K)

220

240

260

280

300

320

340

ρ (µ

Ωcm

)

42.075.8136.8173.0

Figure 5.4: ρa plotted as a function of T at P = 136.8 kbar. The measurement

is in progress at the time of writing this thesis. The data shows that non-metallic

non-Fermi liquid behaviour is robust at ultrahigh pressures. It is not easy to tell the

exact value of TN from ρa(T ), but approximately TN ≈ 35 K.

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