Qc Dissertation

7/27/2019 Qc Dissertation

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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

TRANSPORT PROPERTIES IN UNCONVENTIONAL

SUPERCONDUCTORS

By

QINGHONG CUI

A Dissertation submitted to theDepartment of Physics

in partial fulfillment of therequirements for the degree of

Doctor of Philosophy

Degree Awarded:Spring Semester, 2007


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The members of the Committee approve the Dissertation of Qinghong Cui defended on

March 12, 2007.

Kun YangProfessor Directing Dissertation

Naresh S. DalalOutside Committee Member

Nicholas E. BonesteelCommittee Member

Jianming CaoCommittee Member

Jorge PiekarewiczCommittee Member

The Office of Graduate Studies has verified and approved the above named committee members.

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TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Spin quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Andreev reflections and tunneling spectroscopy . . . . . . . . . . . . . . 10

2. UNCONVENTIONAL SUPERCONDUCTORS . . . . . . . . . . . . . . . . . 172.1 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Pairing symmetry in unconventional superconductors . . . . . . . . . . . 192.3 Disordered superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 22

3. SPIN QUANTUM HALL TRANSITIONS IN p-WAVE SUPERCONDUCTORS 273.1 Spin quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4. FULDE-FERRELL-LARKIN-OVCHINNIKOV SUPERCONDUCTOR . . . . 414.1 The Fulde-Ferrell state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Properties of the Fulde-Ferrell state . . . . . . . . . . . . . . . . . . . . . 444.3 Discussion of the Larkin-Ovchinnikov state . . . . . . . . . . . . . . . . . 48

5. CONDUCTANCE CHARACTERISTICS OF NORMAL METAL/SUPERCONDUTORJUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1 Blonder-Tinkham-Klapwijk theory . . . . . . . . . . . . . . . . . . . . . 505.2 The conductance characteristics of the Fulde-Ferrell state case . . . . . . 55

5.3 Andreev bound states in the d-wave Fulde-Ferrell superconductor with(110) junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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LIST OF TABLES

1.1 Symmetry classes of dirty superconductors . . . . . . . . . . . . . . . . . . . 3

3.1 Critical exponents for different cutoff energies with W = 8.0 . . . . . . . . . 35

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LIST OF FIGURES

1.1 Schematic illustration of the Fermi surfaces of spin-up and -down electrons inthe momentum space for a superconductor under Zeeman field. . . . . . . . . 5

1.2 Density of states (DOS) of the 2D electron gas under magnetic field . . . . . 8

1.3 Diagram of metallic loop used in Laughlins argument and large superconduc-

tor ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Schematic drawing of the Andreev reflection and the normal specular reflection 11

1.5 Schematic illustration of the formation of the Andreev bound states . . . . . 12

1.6 Schematic demonstration of the processes in BTK theory . . . . . . . . . . . 14

1.7 The normalized differential conductance vs voltage for a normal metal/s-wavesuperconductor junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 The normalized differential conductance vs voltage for a normal metal/dx2y2-wave superconductor junction . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.9 The normalized conductance spectra taken on the {110} crystal face of YBCOwith a Pt-Ir tip at 4.2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Schematic illustration of the pairing symmetries in conventional and uncon-ventional superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 DOS of superconductors with various pairing symmetries and disorder strengths 24

3.1 DOS and spin Hall conductance for L = 10 and W = 4.0 . . . . . . . . . . . 32

3.2 DOS and density of current carrying states for systems with L = 10 40 and

W = 8.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Percentage of current carrying states vs system size L for W = 8.0 . . . . . . 34

3.4 DOS and Thouless number for systems with L = 40 80 and W = 8.0 . . . . 34

3.5 Area of Thouless number vs system size for different cutoff energy Ecut andW = 8.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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3.6 DOS and Thouless number for systems with L = 40 80 and W = 9.0 . . . . 38

3.7 DOS and Thouless number for systems with L = 40 80 and W = 10.0 . . . 38

3.8 DOS and Thouless number for systems with L = 20 40 and W = 15.0 . . . 39

4.1 Contours of equal pairing potential for an s-wave superconductor underZeeman field at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Same as Fig. 4.1 except that it is now for a d-wave superconductor with qalong the antinodal direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Same as Fig. 4.2 except that q is here along the nodal direction . . . . . . . 45

4.4 Qualitative sketch of the phase diagram of an s-wave superconductor under aZeeman field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Qualitative sketch of the phase diagram of a d-wave superconductor under a

Zeeman field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Representative electron DOS for various Zeeman fields and both s- and d-wavesuperconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Schematic diagram of dispersion relations on the two sides of an NSJ . . . . 52

5.2 The normalized conductance vs voltage for normal-metal/s-wave FF super-conductor junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Same as Fig. 5.2 except it is now for normal-metal/d-wave FF superconductorjunction with (100) contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4 Same as Fig. 5.2 except it is now for normal-metal/d-wave FF superconductorjunction with (110) contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5 The normalized conductance and its two spin components at a large z = 5.0 60

5.6 The normalized conductance vs voltage for the competing uniform BCS states 61

5.7 Comparison of the high-z junction conductance with the un-weighted andcos2 -weighted DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.8 The bias voltage difference of the tunneling conductance peak and the dip atlarge z(=20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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ABSTRACT

This dissertation investigates the transport properties of unconventional superconductors

which differ from the conventional superconductors on two aspects, one is the pairing

symmetry of the order parameter, the other is the net momentum of the Cooper pair. The

former ones are discovered in high-Tc cuprates, heavy-fermion, Sr2RuO4 and so on. The

latter ones can be realized by splitting the Fermi surfaces of spin-up and -down electrons

under Zeeman field and are known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states.

This work is consisted of two parts.

In the first part, we present results of numerical studies of spin quantum Hall transitions

in disordered superconductors, in which the pairing order parameter breaks time-reversal

symmetry. We focus mainly on p-wave superconductors in which one of the spin components

is conserved. The transport properties of the system are studied by numerically diagonalizing

pairing Hamiltonians on a lattice, and by calculating the Chern and Thouless numbers of the

quasiparticle states. We find that in the presence of disorder, (spin-)current carrying states

exist only at discrete critical energies in the thermodynamic limit, and the spin-quantum

Hall transition driven by an external Zeeman field has the same critical behavior as the usual

integer quantum Hall transition of non-interacting electrons. These critical energies merge

and disappear as disorder strength increases, in a manner similar to those in lattice models

for integer quantum Hall transition.

The second part is a proposal of identifying the FFLO state based on its transport

properties in the normal metal/superconductor junction (NSJ). The FFLO state has received

renewed interest recently due to the experimental indication of its presence in CeCoIn5, a

quasi 2-dimensional (2D) d-wave superconductor. However direct evidence of the spatial

variation of the superconducting order parameter, which is the hallmark of the FFLO state,

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does not yet exist. In this work we examine the possibility of detecting the phase structure

of the order parameter directly using conductance spectroscopy through NSJ, which probes

the phase sensitive surface Andreev bound states ofd-wave superconductors. We employ the

Blonder-Tinkham-Klapwijk formalism to calculate the conductance characteristics between

a normal metal and a 2D s- or dx2y2-wave superconductor in the Fulde-Ferrell state, for all

barrier parameter z from the point contact limit (z = 0) to the tunneling limit (z 1). Wefind that the zero-bias conductance peak due to these surface Andreev bound states observed

in the uniform d-wave superconductor is split and shifted in the Fulde-Ferrell state. We also

clarify what weighted bulk density of states is measured by the conductance in the limit of

large z.

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CHAPTER 1

INTRODUCTION

In 1957, Bardeen, Cooper and Schrieffer (BCS) [1] published their famous paper of the

microscopic theory of the superconductivity. In their theory, when the temperature drops

below a critical value (Tc), the phonon-mediated electron-electron interaction leads to the

formation of Cooper pair. Since this interaction is almost isotropic, the pair state has zero

orbital momentum (s-wave symmetry) and is a spin-singlet. This theory also explains the

energy gap existed in the superconductivity state which is the pairing potential. The kind

of superconductors described by the BCS theory are called conventional superconductors.

In an earlier time (1950), Ginzburg and Landau [2] developed the phenomenological theory

of the superconductivity based on the Landau theory of the second-order phase transitions.

In their theory, the order parameter (r) is used to represent the extent of the macroscopic

phase coherence in the superconducting state. The link between the Ginzburg-Landautheory and the BCS theory was established by Gorkov [3]. In the vicinity of Tc, the

Ginzburg-Landau equations could be derived from the BCS theory and the order parameter

is proportional to the pairing potential.

Since the discovery of the high-temperature superconductors in 1986 [4], the internal

structure of the Cooper pairs in the high-Tc superconductors became an active topic in

the condensed matter physics. After years of debate and the phase-sensitive experiments

recently, it is now believed that the pairing symmetry in the high-Tc cuprates is d-wave

type. Besides the cuprates, more exotic superconductors are discovered in the past two

decades, such as heavy-fermion and organic superconductors, and Sr2RuO4. For many of

these superconductors, the pairing symmetry is no longer s-wave type and they are known

as unconventional superconductors.

To understand the nature of the unconventional superconductors better, we will give a

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brief discussion of the symmetry in the superconducting states. As we know from the Landau

theory, the symmetry breaking is often accompanied with a phase transition, which means

when the system undergoes a phase transition by external force or environmental change,

some symmetries possessed by the system before can be lost. For the second-order phase

transition, the symmetry breaking across the transition is continuous and thus the symmetry

group after breaking will be a subgroup of the full symmetry group. In our case here, the

full symmetry group G is the one describing the normal state,

G = GR U(1) T, (1.1)

where G is the point group symmetry of the crystal lattice, R is the symmetry of spin

rotation, U(1) is the one-dimensional global gauge symmetry, and T is the time-reversal

symmetry. The U(1) symmetry is broken spontaneously by the phase coherence in thesuperconducting state, and experimentally we get Meissner effect, flux quantization and so

on. In a conventional superconductor, symmetries other than U(1) are kept, but this is

not the case of unconventional superconductors. By determination of the type of symmetry

breaking besides U(1), we can thus classify the unconventional superconductors and this is

also reflected in the symmetry properties of the order parameter.

A simple classification of the superconductors can be made based on the parity of the

pair state (concerning the space inversion symmetry). Since in the superconducting state,

the electrons need to pair up to form bosons whose total spin (S) is an integer, we thus have

the spin-singlet (S = 0) with even parity or the spin-triplet (S = 1) with odd parity. When

S is fixed, the total orbital angular momentum (L) of the Cooper pair is also restrained by

the Pauli principle. For spin-singlet, L could only be an even integer, while for spin-triplet,

L should be an odd integer. In conventional superconductors, both S and L are zero and

the pairing is known as s-wave in analogy of atomic orbitals. Although we are not quite

clear about the pairing mechanism in the unconventional superconductors, experiments have

found that the paring in the high-Tc cuprate has d-wave symmetry (S = 0 and L = 2) [5, 6],

and Sr2RuO4 favors the p-wave symmetry (S = 1 and L = 1) [7, 8].

In addition to the experimental relevance, unconventional, disordered superconductors

are also of great interest for theoretical reasons, as they represent new symmetry classes

in disordered non-interacting fermion problems that are not realized in metals. In dirty

superconductors, the translational symmetry is broken and the momentum k of a single

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Table 1.1: Symmetry classes of dirty superconductors. (Refer to Ref. [10])

Class Time-reversal Spin-rotationD No No

C No YesDIII Yes NoCI Yes Yes

particle is no longer a good quantum number. The plan-wave eigenfunctions with momentum

k, which are used to form the Cooper pair in the original BCS theory, should be replaced by

position-dependent functions and pairing is between time-reversed states. To find these

functions, we need to set up equations for them. This is achieved by generalizing the

Hartree-Fock equations to include the pairing potential of the superconducting state. The

resulted equations are called Bogoliubov-de Gennes (BdG) equations [9]. These equations

are widely applied to more general situations with order parameter varied in space (such as

the normal metal/superconductor junction discussed later). Since the elementary excitation

(quasiparticle) of superconductors can be viewed as destroying a Cooper pair from the

condensate and creating an electron in the vacancy, the BdG equations are readily used

to describe the bahavior of the quasiparticles in the superconductors. In the meantime the

properties of the dirty superconductor and its classification will be determined by the BdGequations, through which pairing symmetry is reflected. A classification of the symmetry

classes in dirty superconductors have been advanced recently [10, 11]. Depending on the

existence (or the lack) of time-reversal and spin-rotation symmetries, dirty superconductors

can be classified into four symmetry classes, CI, DIII, C, and D in Cartans classification

scheme (Table 1.1). These classes are believed to complete the possible universality

classes [12] in disordered single-particle systems [10, 11].

By now we have only discussed the cases of pairing state in rest (called BCS state below)

and thus the pair has a total momentum of zero. The situation will be changed when we

turn on the Zeeman field (H) and split the Fermi surfaces of the spin-up and -down electrons

apart, which leads to an imbalance of the two electron species (fermions). In this case, we

have the BCS state, the spin polarized state (normal state), and possibly more states to

compete for the ground state. Due to the imbalance of the two electron species, a portion

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of the Cooper pairs will be broken. We begin with a small portion that the energy gap (0)

is not affected1. Accordingly, we gain an energy of 2H from the new spin orientation at

a cost of 20 in breaking a Cooper pair and move toward the spin polarized state. If we

continually increase the number of the broken Cooper pairs, we need H to be larger than

0 to make the spin polarized state eventually energy favored over the BCS state. But,

in fact, the spin polarized state already has a lower energy when H = 1/

20 and this

limit is called Pauli limit [13, 14]. Thus the BCS state with large portion of Cooper pairs

broken is not what we are looking for but an alternative solution is deserved for the ground

state. In the opposite side of the above process, because the Cooper pairs contributing to

the superconducting state mostly are those around the Fermi surface (before the splitting),

we need to boost a spin-up electron to higher kinetic energy so as to pair with the spin-down

electron with opposite momentum [Fig. 1.1(a)]. Again, we find that such Cooper pairs arenot energy favored when more kinetic energy is required (H > 0). Thus it is suggested by

Fulde and Ferrell [15], and Larkin and Ovchinnikov [16] that pairing electrons of opposite

spins located close to their own Fermi surfaces may lower the energy [Fig. 1.1(b)]. Since

the paired electrons have different momenta (k and k), there will be a net momentum

2q = k k in the Cooper pair and it causes the oscillation of the order parameter. Thisstate is now known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state2. It breaks both

translation and rotation symmetries.

Though the FFLO state was studied theoretically in an earlier time, lack of experimentalsupport in the conventional superconductors makes it overlooked for a long time. The

situation has been changed by experimental results suggestive of the FFLO state in heavy-

fermion, quasi-1D organic, or high-Tc superconductors [17, 18, 19, 20, 21, 22, 23, 24, 25].

Recent experimental results in CeCoIn5, a quasi-2D d-wave superconductor, are particularly

encouraging [26, 27, 28, 29, 30, 31, 32]. This subject is also of interest to the nuclear

and particle physics communities because of the possible realization of the FFLO state

in high density quark matter and nuclear matter [33], as well as in cold fermionic atom

systems [34, 35, 36, 37, 38, 39, 40, 41]. On the theory side, more suggestions dealing with

the pairing between unbalanced fermions are also proposed, such as the deformed Fermi

1Here, we consider the s-wave symmetry for simplicity.2More precisely, FF considered the pairing state with only a single momentum q, while LO suggested a

superposition of a series of finite-momentum states.

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(a) (b)

Figure 1.1: Schematic illustration of the Fermi surfaces of spin-up (dash-dotted) and -down(dashed) electrons in the momentum space for a superconductor under Zeeman field. Thesplit from the one (dotted) without Zeeman field is exaggerated to illuminate our argument.In the BCS state (a), the paired electrons have equal and opposite momenta (k and

k),

while in the FFLO state (b), they are located close to their own Fermi surfaces, resulting anet momentum 2q = k k.

surface pairing [42, 43, 44] and the breached pairing states [34, 45, 46]. To classify and

discuss the relation between these different phases, more classification schemes beyond the

Landau theory are necessary and this will serve to enhance our understanding of the quantum

phases and the phase transitions [41, 47, 48, 49].

Besides the pairing symmetry in the unconventional superconductors is unconventional,

the transport properties of the quasi-particles in these superconductors also behave dif-

ferently from the conventional superconductors. There are two kinds of objects in the

superconductor contributing to its transport properties, one is the Cooper pair (condensate),

the other is the quasiparticle. The Cooper pair condensate has no entropy, and its z-

component spin is zero because it is a pair of spin-up and -down electrons. Thus it determines

the charge current properties but does not carry thermal flow or spin current. The latter

two types of flow will be transmitted by the quasiparticles instead. In this way, we can

find different transport properties stemed from the difference of the pairing symmetry. For

example, in the s-wave superconductor, we have a non-zero energy gap and in order to get

the excited states, it needs to absorb an energy of 2 to break the Cooper pair. But for the

dx2y2-wave superconductor, it supports gapless nodal quasi-particle excitations and these

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nodal quasi-particles dominate heat and spin transport at low temperatures. The anisotropic

character of the d-wave pairing also makes difference in the tunneling behavior of the normal

metal/superconductor junctions, which results the zero-bias conductance peak and is used

as a phase-sensitive experiment to detect the d-wave symmetry.

Another interesting difference comes from the disordered superconductors of class C

and D, which break the time-reversal symmetry. For system of class C, the spin-rotation

symmetry is preserved and therefore allows us to consider about the spin current rather than

the charge current. Because of the time-reversal symmetry breaking, it is naturally expected

the Hall conductance of the spin current to be quantized in analogy to the conventional

quantum Hall effect. This effect is called spin quantum Hall effect. This effect can also be

realized in system of class D when the spin-rotation symmetry is not destroyed completely,

such as the px + ipy pairing (S = 1, Sz = 0) which will be studied in our work. In thefollowing two sections, we will discuss more details of the spin quantum Hall effect and the

tunneling conductance which comprise the main topics of this thesis.

1.1 Spin quantum Hall effect

The spin Hall effect that will be discussed in this work is induced by the gradient of a

Zeeman field that couples to spin. With the conservation of the z-component spin in the

Hamiltonian, we can make an analogy to the usual Hall effect. This is different from the

spin Hall effect in semiconductors with spin-orbit coupling, which is driven by an electric

field [50, 51]. There, the spin Hall effect is due to the presence of spin-orbit coupling.

The integer quantum Hall effect (IQHE) was first observed by von Klitzing et al. [52]

in 1980, indicating that the Hall conductance xy of a quasi-two-dimensional electron gas is

quantized to be an integer multiple of e2/h while the longitudinal conductance xx vanishes

at the same time, when the system is subject to a strong magnetic field in low temperature.

This quantization survives over a finite range of physical parameters such as the magnetic

field or the carrier concentration, and it is independent of the macroscopic or microscopic

details. This phenomenon, along with its fractional sibling discovered later, is credited

significance commensurable with superconductivity in condensed matter theory.

As we know from the classical theory, when the system size L is much larger than the

mean free path, the conductance g(L) (a macroscopic quantity) is related to the conductivity

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(a microscopic quantity) by the Ohms law,

g(L) = Ld2, (1.2)

where d is the dimension of the system. From this formula we can see that the two-

dimensional system is a special case that its conductance is independent of the system

size. When taking into account the disorder effect, the scaling theory shows that for a

two-dimensional system at large length scale, even a weak microscopic disorder will drive

the electronic states to be localized. This weak localization could be understood by the

quantum interference effect. In the presence of time-reversal symmetry, there are pairs

of time-reversed paths which start and return to the same point. The two paths of such

pair have the same phase and increase the probability for an electron to return its initial

point. Hence the probability of diffusion away is reduced. Paradoxically, the quantum Halleffect relies on the disorder to destroy the translational invariance, because we can only

get the classical Hall effect otherwise. Fortunately, the magnetic field required in the Hall

effect will break the time-reversal symmetry and the localization properties of the system is

consequently altered.

From quantum mechanics, we know that the 2D electron system under magnetic field is

described by a series of discretized Landau levels (LL). In the presence of disorder the LL will

be broadened into a band, and in large magnetic field limit, there will be no overlapping of

these bands (Fig. 1.2). For each band, only states with energy in the center of the band are

extended states while the states with all other energies are localized. In the noninteracting

limit and by a gauge argument under the situation as shown in Fig. 1.3(a), Laughlin [53] and

Halperin [54] showed that the Hall conductance takes its quantized value e2/h while the

longitudinal conductance xx = 0, when the Fermi level lies in the region of localized states

(mobility gap or energy gap). Here the filling factor , defined as the ratio of the number of

electrons over the number of flux quanta penetrating the sample = N/N, is the number

of LL occupied and thus an integer. Changes of the Fermi level in the mobility gap do not

alter the Hall conductance xy. When the Fermi level moves across the extended state, the

Hall conductance will change by an integer. This explains the step observed in the IQHE.

It is worth noticing the similarity between the quantized Hall effect and the quantized flux

trapped in a superconducting ring [Fig. 1.3(b)] [55, 56], both of which are a consequence of

the gauge invariance.

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(a)

(b)

Figure 1.2: Density of states of the 2D electron gas under magnetic field: (a) withoutdisorder, (b) with disorder (schematic). With disorder, states are localized at energies otherthan the center of each band.

(a) (b)

Figure 1.3: (a) Diagram of metallic loop used in Laughlins argument; (b) Large supercon-ductor ring. The magnetic flux trapped in the ring is quantized.

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So far we have described the physical background of the IQHE. Now we will discuss

the approaches to study the transport properties (localization) in the quantum Hall regime.

One approach that we adopt in this work is first inspired by Thouless et al. [57, 58] in

their topological explanation of the IQHE. They argued that for a state in the mobility

gap, the Hall conductance will be insensitive to the boundary conditions. Therefore, the

conductance can be averaged over the boundary phases and this averaged quantity is proved

to be 2 times an integer, C1(m), the first Chern index, which characterizes the topological

properties of the wave functions. Besides, Arovas et al. [59] showed that for states with

finite C1(m), the zeros of the wave functions can be moved to any position in real space

by replacing the boundary phases, while states with zero C1(m) cannot. This gives us an

efficient criteria to distinguish extended states from localized states unambiguously, even in

a finite-size system. The Chern number method has been very successful in the studies ofquantum Hall transitions, for both integral [60, 61, 62, 63] and fractional effects [64], and

also in other contexts [65, 66, 67, 68].

From the discussion above, we can see that the sensitivity of the wave functions to

the boundary conditions plays an essential role in separating extended states from localized

states. Historically, Edwards and Thouless [69] have argued in an earlier time (1972) that the

sensitivity of the eigenenergies of finite systems to changes in the boundary conditions can

be used to make the separation in the same spirit as Hall conductance [70]. More precisely,

it is shown in the second-order perturbation theory that the longitudinal conductivity xx isapproximately related to the average energy level shift |E| when the boundary conditionchanges from periodic to anti-periodic in one direction by [71]

xx =e2

h

|E|E

e2

hgT(E). (1.3)

Here, Eis the average energy level spacing at the Fermi energy; gT(E) is called the Thouless

number. Since both Chern number and Thouless number are size dependent and thus can

be used as finite-size scaling quantities in the study of localization properties.

Moving back to our study in unconventional superconductors, the possibility of spin (or

thermal) quantum Hall states, as pointed out previously, allows us to draw a close analogy

between the quantum Hall effect and superconductivity. At the same time we can also employ

the numerical methods developed in the study of the IQHE to our problem. In this work, we

report results of numerical studies on a lattice model of disordered p-wave superconductors

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with px+ipy pairing, which conserves the z-component of electron spin. As we will show later,

this is an example of the class D model, and in certain sense the simplest model that supports

a spin quantum Hall phase. We study the localization properties of the quasiparticle states

by calculating the Chern and Thouless numbers of the individual states, in ways similar to

the corresponding studies in the quantum Hall context mentioned above. Physically, the

Chern and Thouless numbers correspond to the Hall and longitudinal spin conductivities in

the present context, respectively, which are also related to the Hall and longitudinal thermal

conductivities. We note that while it has been pointed out earlier that quasiparticle bands

or individual quasiparticle states can be labeled by their topological Chern numbers [72], the

present work represents the first attempt to calculate them numerically and use them to

study the localization properties of the quasiparticle states in the context of unconventional

superconductors.Our main findings are summarized as the following. We find that the p-wave model

we study supports an insulating phase and a spin quantum Hall phase with spin Hall

conductance one in appropriate unit. For relatively weak disorder, there exist two critical

energies at which current-carrying states exist, carrying a total Chern number (or spin Hall

conductance in proper unit) +1 and 1, respectively; they are responsible for the spinquantum Hall phase. Phase transitions between these two phases may be induced either by

changing the disorder strength, or by applying and sweeping a Zeeman field. The field-driven

transition is found to have the same critical behavior as the integer quantum Hall transitionof non-interacting electrons. As disorder strength increases the two critical energies both

move toward E = 0, and annihilate at certain critical disorder strength, resulting in an

insulating phase in which all quasiparticle states are localized. No metallic phase is found

in our model.

1.2 Andreev reflections and tunneling spectroscopy

The Andreev reflection was first introduced by de Gennes and Saint-James in 1963 [73, 74,

75], and was applied to explain the increase of the thermal resistance in the intermediate

state of type-I superconductor successfully by Andreev [76] in 1964. When an electron

(with momentum k) moves in a normal metal (N) toward a superconductor (S), it can be

reflected back with equal momentum (k) to the normal metal as a hole at the interface.This is the Andreev reflection, different from the usual specular reflection (Fig. 1.4). This

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(a) (b)

Figure 1.4: Schematic drawing of the Andreev reflection (a) and the normal specularreflection (b). In the Andreev reflection, the incident electron (solid line) is reflected backas hole (dashed line) with the same momentum.

effect is especially apparent when the incoming electrons have energies below the gap. In

such situation, the electrons cannot enter the superconductor side as quasiparticles because

there are no quasiparticles in the gap. But, by Andreev reflection, they can traverse the

interface and decay into Cooper pair condensate, because there are holes reflected back

which are equivalent as additional charge current (i.e. pairs of 2e across the interface).

Consequently, the Andreev reflection enhances the electrical conductivity of N/S contacts in

the gap. For the thermal transport considered by Andreev, the holes reflected at the domain

walls of the intermediate state will resist the thermal flow and thus explain why the thermal

resistance is higher than in the Meissner state (in which the magnetic field is expelled out).

Unfortunately, the enhancement of the electrical conductivity was less appreciated than thedepression of thermal transport until Zaitsev [77] calculated the enhanced conductance in

1980 and Blonder, Tinkham, and Klapwijk (BTK) [78] gave a complete discussion of this

subject in 1982.

The above discussion of effects of the Andreev reflection on the conductivity of N/S

contacts is only about the conventional superconductor, and the enhancement of conductivity

at bias below the gap is used to measure the gap experimentally. But in high-Tc cuprate,

there exist gapless quasiparticles. Therefore the argument need to be deliberated again. In

1994, Hu [79] considered the situation that a normal metal slab of thickness dN (much smaller

than the mean free path) in contact with a semi-infinite dx2y2-superconductor (Fig. 1.5),

which is similar to the situation considered by Saint-James [74]. As illustrated in Fig. 1.5,

there are two successive Andreev reflections in the normal slab. When the contact interface

is along the nodal line, the two Andreev reflections will experience a phase difference of the

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Figure 1.5: Schematic illustration of the formation of the Andreev bound states in the

structure of a d-wave superconductor coated with a normal metal layer. Two successiveAndreev reflections occur in the normal metal slab and they experience phase change of theorder parameter when the interface is along the nodal line.

order parameter. This causes a zero-energy Andreev bound states (ABSs) (also called the

midgap surface states) to be formed in the metal slab. Hu also proved that this bound states

exist in the limit ofdN 0. However, since the ABSs are a consequence of the phase change

of the order parameter, they will persist when the interface is not exactly along the nodalline, but vanish when the interface is along the antinodal line. Subsequent work showed

that the ABSs will be modified by the presence of an imaginary component in the order

parameter, such as the dx2y2 + is and dx2y2 + idxy symmetries [80]. These features of the

ABSs make it a powerful tool in detecting the order parameter symmetry of unconventional

superconductors [81, 82, 83, 84, 85, 86, 87].

To make the ABSs measurable experimentally, we will calculate the differential con-

ductance [G(V) dI(V)/dV] characteristics of a normal metal/superconductor junction(NSJ). For a general treatment of the interface type, from transparent contact limit to

the strong tunneling limit, BTK used a -function barrier of strength z (from 0 to ) tomodel the interface. When an incident electron with energy E comes to the barrier, it is

Andreev and specular reflected as hole and electron with probabilities ofA(E) and B(E) 3,3Actually, E is the energy of a particle relative to the Fermi surface, thus for the hole in the metal, the

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respectively, and is transmitted into the superconductor as eletron- and hole-like quasiparticle

with probabilities of C(E) and D(E), respectively. This process is illustrated in Fig. 1.6 4.

By solving the Bogoliubov-de Gennes equations and matching the boundary conditions at

the barrier, we can calculate the four probabilities. Then, the differential conductance of a

single channel in 1D can be derived from the Landauer-Buttiker formula [88, 89]

G(E = eV) = 2e2

h

dE[1A(E) + B(E)] f

E, (1.4)

where f is the Fermi function. The result of the s-wave case in the original BTK paper is

shown in Fig. 1.7. There, we can easily see the enhanced conductivity at bias below the

gap in the contact limit (z = 0), which is discussed previously. The BTK theory was later

generalized to the 2D d-wave case with a result shown in Fig. 1.8 [90]. In the tunneling limit

(z = ), we find a peak at the zero-bias [called zero-bias conductance peak (ZBCP)] whenthe interface is off the antinodal line of the dx2y2-wave order parameter, which is the badge

of the ABSs. The quantitative fit of experimental data (e.g. see Fig. 1.9 from Ref. [83]) with

the BTK theory also encourages us to believe that the fermionic description of excitations

by BdG equations is appropriate for high-Tc superconductors.

Since the ABSs are consequences of the phase change of the d-wave order parameter,

their spectra should also be sensitive to the spatial variation of the order parameter. This

inspired us to propose the possibility of using the phase-sensitive experiment based on ABSs

to detect the FFLO phase in the d-wave superconductors. In this work, we will show this

sensitivity by explicit calculations of the conductance characteristics, and from their spectra

detected via conductance spectroscopy through a micro-constriction, one can extract the

momentum of the superconducting order parameter. As we will show below, for a dx2y2-

wave superconductor in the FF state, the ZBCP observed in a (110) contact is split and

shifted by both the Zeeman field and pair momentum; the latter can be determined from

the splitting.

The remainder of this thesis is organized as the following. In Chapter 2 we first provide a

brief review of the mean field thoery describing superconductivity in the lattice model as well

as the classification of the unconventional and dirty superconductors. Chapter 3 is devoted to

the spin quantum Hall transitions in px + ipy-wave superconductors. There, we first describe

allowed transition is at E instead ofE.4It is shown in a more general 2D model instead of the 1D model in BTK for illustration purpose.

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Figure 1.6: Schematic demonstration of the processes occured when an electron falls ontothe interface of a normal metal/superconductor junction. The electron is reflected as electronand hole by specular and Andreev reflections, respectively, and is transimited into thesuperconductor as electron- and hole-like quasiparticles.

Figure 1.7: The normalized differential conductance vs voltage for a normal metal/s-wavesuperconductor junction with various barrier strength z at T = 0. (From Ref. [78])

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Figure 1.8: The normalized differential conductance [(E)] vs voltage (E) for a normalmetal/dx2y2-wave superconductor junction. is the angle between the antinodal line of theorder parameter and the interface of the junction: (a) = 0, (b) = /8, (c) = /4. A,z = 0; B, z = 1; C, z = 5; D, the electron density of states in the superconductor. (FromRef. [90])

the application of Chern and Thouless number methods to our problem and then present

our numerical results and the analysis of the data obtained. Chapters 4 and 5 deal with the

FFLO superconductors, mainly in FF state. In Chapter 4, we calculate the self-consistent

mean field solutions for both s- and dx2y2-wave superconductors under Zeeman field and

identify the FF phase. Some properties of the FF state, such as the density of states and the

current in the superconductor, are discussed. In Chapter 5 we give the scheme to calculate

the conductance characteristics of the NSJ (BTK theory and its generalization), along with

the numerical results of both s- and dx2y2 cases. Further discussion of their relation with

the DOS and ABSs are followed. Finally, this work is summarized in Chapter 6.

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Figure 1.9: The normalized conductance spectra taken on the{

110}

crystal face of YBCOwith a Pt-Ir tip at 4.2K. Main panel is for an STM tunnel junction. Left inset is for apoint-contact junction. The data are given as open circles and the d-wave fits by the solidcurves. Right inset gives the mixed symmetry simulations for the tunnel junction. (FromRef. [83])

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CHAPTER 2

UNCONVENTIONAL SUPERCONDUCTORS

Superconductivity is widely discussed in both continuum and lattice models. The lattice

model, originally the tight-binding model, was introduced by Anderson [91] (1958) to analyse

the localization, and later was developed into more intriguing models, such as the Hubbard

model in studying the metal-insulator transition [92], and its descendant, the t-J model,

applied to the theory of high-Tc cuprates [93, 94]. In our work, we will not get into the

profound physics behind the Hubbard model, but use a simplified one to investigate the

physics we are interested in. In this chapter, we will start from the one-band Hubbard

model and obtain the mean field description in analogy of the continuum model [9]. Then,

a discussion of the pairing symmetries in the unconventional superconductors based on our

model will be followed.

2.1 Hubbard model

The s-wave superconductor on a 2D square lattice can be described by the attractive Hubbard

model, and the Hamiltonian is given by

H = t

cicj i

ni Ui

nini. (2.1)

Here, i and j are site labels and the angle bracket means nearest-neighbor, t is the hopping

term and will be taken as the unit of the energy in our numerical calculations (t = 1),

ni = c

ici is the on-site particle number operator, is the chemical potential, and U > 0 isthe attractive interaction. The lattice spacing is taken to be 1. To obtain the self-consistent

description of the model, we notice from the Wicks theorem that

nini = cicicici= cicicici + cicicici cicicici. (2.2)

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The first term corresponds to the Hartree shift and can be incorporated into the chemical

potential when it is site-independent1, the second term yields the order parameter , and

the third term is zero under the assumption of no magnetic moment. Therefore, by the

variational principle, we find the effective (or called mean field) Hamiltonian

Hm =ij

Hijcicj +i

(icic

i +

i cici), (2.3)

where Hij = ti1,j ij, and i = Ucici is the order parameter (or pair potential).It is interesting to notice that, with , the mean field Hamiltonian (2.3) is symmetricunder the particle-hole transformation

ci eixici, (2.4)

where = (, ) and xi is the position of site i, while this symmetry is absence in the

Hamiltonian (2.1). This is due to the omission of the Hartree shift term.

In order to diagonalize the mean field Hamiltonian (2.3) to be of the form

Hm = Eg +n

Ennn, (2.5)

where Eg and En are the ground state and the excitation energies, respectively, we will use

the Bogoliubov-Valatin transformation

ci =n

(unin vni n). (2.6)

By calculating [Hm, ci] in the same way as in Ref. [9], we finally derive the Bogoliubov-deGennes (BdG) equations2

j

Hij ijji Hji

unjvnj

= En

univni

, (2.7)

where ij = ji = iij for the s-wave superconductor. The BdG equations are especially

useful in studying the behavior of the quasiparticles in the superconductors with order

1In the presence of disorder, this term will be site-dependent and its effect is discussed in Ref. [95]. Inour work, we will neglect this term.

2The formulas adopted here is based on the consideration of including spin-dependent potential, whichis the case of the FFLO state discussed later. The spin index is implied, i.e. the vector (unj , v

nj)

T is in factof a dimension of 4N, where N is the number of the lattice sites.

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parameter varing in space and its application in continuum model can be found in Ref. [ 9].

In order to calculate physical quantities, we need to know the normalization condition of the

wave functions. By calculating {ci, cj } = ij, it is obtained as the following:

n (u

n

iu

n

j

+

v

n

j

v

n

i ) = ij

, (2.8)

which is equivalent to i

(umiuni + v

miv

ni ) = mn. (2.9)

The self-consistent condition of the order parameter is consequently expressed as

i = Un

[univni f(En) univni f(En)], (2.10)

where f is the Fermi function given by

f(E) = 11 + eE

. (2.11)

The free energy of the system in the superconducting state is evaluated to be

H =ij

Hijcicj 1

U

i

|i|2

=n

En[f(En)i

|vni|2] +1

U

i

|i|2. (2.12)

In the limit of zero temperature T = 0, the Fermi function will be replaced by a step function.

2.2 Pairing symmetry in unconventionalsuperconductors

In the last section, we considered a model with only the on-site interation U and it could

be generalized to include the nearest-neighbor interaction. In this section, we will consider

this case and discuss the new pairing from the added interaction. In the lattice model, the

Hamiltonian is written as

H = t c

icj i ni Ui nini V

2

ninj , (2.13)

where V is the nearest-neighbor interaction. Using the same trick (2.2) and neglecting the

Hartree shift, we find the the mean field Hamiltonian to be

Hm =ij

Hijcicj +i

(icic

i +

i cici) +

(ijcic

j +

ijcjci), (2.14)

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where ij = Vcjci. By choosing different parameters (U and V), the self-consistentsolutions will yield different types of unconventional pairing symmetries in the superconduc-

tors, e.g. we can find d-wave symmetry with U < 0 and V > 0 [96]. This problem itself is so

attractive that deserve independent examination. In our work, our interest is to investigate

the behaviors of the quasiparticles resulting from different pairing symmetries and it will

only distract our focus to take into account the self-consistent calculations. Hereinafter, we

will assume the form of the pairing symmetry and use the reduced mean field Hamiltonian3,

H =ij

Hijcicj +

(ijcic

j +

ijcjci), (2.15)

to study the unconventional superconductors. The BdG equations for the quasiparticles are

given by

j

Hij ijji Hij

unjvnj

= En

univni

, (2.16a)

j

Hij jiij Hij

unjvnj

= En

univni

. (2.16b)

For illustration purpose, we will give some examples of the pairing symmetries in the

unconventional superconductors, such as,

dx2y2 pairing (abbr. as d-wave): j,jex =

j,jey = ,

dx2y2 + idxy pairing (d + id-wave): j,jex = j,jey = x2y2, j,j+exey =j,jexey = ixy,

px + ipy pairing (p-wave): j,jex = , j,jey = i,

where ex and ey are unit vectors along the x and y axis, respectively. They are schematically

shown in Fig. 2.1 along with the conventional s-wave symmetry. In order to solve the

spectra of these superconductors in the lattice model, we take the size of the lattice to be

finite (Lx Ly) and put periodic boundary condition on the lattice

cj+Li, = cj,, i = x,y. (2.17)

3We also omit the subscript m standing for mean field.

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Figure 2.1: Schematic illustration of the pairing symmetries in unconventional supercon-ductors defined in Sec. 2.2, along with the s-wave superconductor. In d + id-wave, we use for x2y2 and

for xy.

Instead of consulting to the BdG equations (2.16), we will diagonalize the Hamiltonian (2.15)

in the momentum space directly. With the Fourier transformations

cj =1

LxLy

k

eikxjck, (2.18a)

cj =1

LxLy

k

eikxjck, (2.18b)

we obtain the Hamiltonian

H =k

kckck +k

(kckc

k +

kckck), (2.19)

where k = 2t(cos kx + cos ky) is the single-particle kinetic energy and the orderparameter k is given by

k =

2(cos kx cos ky), d-wave,2x2y2(cos kx cos ky) 4ixy sin kx sin ky, d + id-wave,2i(sin kx + i sin ky), p-wave.

(2.20)

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Immediately, we find the energy spectrum to be

Ek =

2k + |k|2. (2.21)

It can be seen from the expression above that there is no energy gap for d-wave pairing

while there exists a gap for a pure d + id- or p-wave superconductor as the conventional

s-wave superconductor. Now we would like to question how the situation will be when we

add disorder to the system. As we know from the Andersons theorem [ 97], the energy gap

of an s-wave superconductor will survive even with rather strong disorder. But this is not

the case for p-wave and extended s-wave superconductors which will be discussed in the next

section.

2.3 Disordered superconductors

In this section, we will consider the effect of non-magnetic impurities in the superconductors.

In 1958, Anderson [91] introduced a variant of the tight-binding model to investigate the

effect of disorder in electronic systems. The Hamiltonian in this model is

H =i

uicici +

tijcicj, (2.22)

where ui is the on-site energy level. The disorder can be realized by taking ui (diagonal

disorder) or tij (non-diagonal disorder), or both, to be random numbers. Presently, wewill choose tij = t to be fixed and let ui uniformly and independently distributed from

W/2 to W/2. Anderson proved that when W/t is large enough all states will be localized,which means the envelope of the wavefunction decays quickly at large distances from the

localization center.

The effect of disorder in superconductor is rather a different story due to the pairing

interaction. As pointed out by Anderson [97], with weak disorder in conventional super-

conductors, Cooper pair will be formed between time-reversed states whose state density is

not strongly affected by disorder. Thus the transition temperature and the energy gap in

average will not be changed by the weak disorder. But definitly, when the disorder is strong

enough the energy gap will diminish and the superconducting state vanishes. Interestingly,

the energy gap persists with strong disorder even when the off-diagonal correlations are

substantially reduced [98], and this is probably due to the formation of pairs between local

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states without phase coherence [99]. In our work, we will not explore the supercondutor-

insulator transition but study another transition the spin quantum Hall transitions in

unconventional superconductors with time-reversal symmetry broken.

Here, we will use the Anderson model to represent the non-magnetic impurities and the

mean field Hamiltonian is the same as Eq. (2.15) except

Hij = ti1,j + (ui )ij , (2.23)

due to the inclusion of the disorder. As an example to show the different responses of

conventional and unconventional superconductors to disorder, we solve the BdG equations

(2.16) with order parameter defined as in Sec. 2.2 and calculate the density of states (DOS)

(E) =ni

[|ui|2(E En) + |vi|2(E+ En)]. (2.24)

The results are shown in Fig. 2.2. It can be found that the energy gaps of d + id-wave and

p-wave superconductors decrease and finally vanish with increasing the disorder strength W.

For s-wave superconductor it always shows an energy gap, however there is no gap at all for

d-wave superconductor.

In order to deepen our understanding of the behaviors of the dirty superconductor, we

will study its symmetry properties first. Most of the discussion below is based on the work

of Altland and Zirnbauer in Ref. [10].

In Secs. 2.1 and 2.2 we obtained the BdG equations, which govern the dirty supercon-

ductors, from the mean field Hamiltonian by Bogoliubov-Valatin transformation. Here, we

will reach the BdG equations from another point of view when we rewrite the Hamiltonian

(2.15)

H = 12

ij

(Hijcicj + Hijc

icj Hjicicj Hjicicj)

+1

2

(ijcic

j jicicj ijcicj + jicicj)

=1

2

ij

(cicicici)

Hij ijHij jiij Hji

ji Hji

cjcjcjcj

12

(c c)H c

c

. (2.25)

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Figure 2.2: Density of states of superconductors with various pairing symmetries anddisorder strengths (W): (a) s-wave, (b) d-wave, (c) d + id-wave, (d) p-wave. Solid, W = 0;dash, W = 2.0; dot, W = 8.0. In the calculations, we take the chemical potential = 3.0and the lattice to be of finite size (L L) with periodic boundary condition. For s-, d- and

p-wave, = 0.8. For d + id-wave, x2y2 = 0.8, xy = 0.5. For the case W = 0, we chooseL = 5000, while for cases W = 0, we use L = 50 and average over 100 samples.

Thus we see that solving the BdG equations (2.16) is in fact the same as diagonalizing

the 4N 4N matrix H (where N is the number of lattice sites) if we change vj to vj.Generally, the Hamiltonian of a superconductor is of the following form:

H =

(hcc +

1

2c

c

+

1

2cc), (2.26)

where and are indices that label both lattice site and spin of the electron. The matrix

H is thus given by

H =

h hT

. (2.27)

The hermiticity of h requires h = h (or h = h

) and the antisymmetry of fermions

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demands = (or = T). These two conditions are necessary for a superconductorand can be expressed by the following relation:

H = H = xHTx, (2.28)

where

x =

0 12N

12N 0

= x 12N, (2.29)

and x is the Pauli matrix.

Now we will examine the symmetry properties beyond Eq. (2.28). One is the spin-rotation

symmetry. The generators of the spin rotations are, in a 4N 4N form,

Jk =

k

Tk

1N (2.30)

where k = x,y,z. Spin-rotation invariance of the Hamiltonian requires [H, Jk] = 0 for all

ks and this is reduced to h = 1H and = iy , or explicitly,

H =

H 0 0 0 H 00 HT 0

0 0 HT

, (2.31)

where H = (Hij) and = (ij). The condition T = yields T = . Therefore we find

p-wave superconductor does not have spin-rotation symmetry but d-wave and d + id-wave

superconductors have. Physically, it can be easily understood since p-wave is spin-triplet

while d- and d + id-wave are spin-singlets.

Another important symmetry is the time-reversal symmetry. In spin-1/2 systems, the

time-reversal operator can be expressed as K, where = 12 iy 1N and K is thecomplex-conjugate operator. When a Hamiltonian is invariant under time-reversal symmetry,

it means

H = H1. (2.32)

This condition can be simplified to

H = H, = . (2.33)

Accordingly, time-reversal is respected in d-wave superconductor but not in d+id and p-wave

superconductors.

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Thus far, we have derived the conditions for the Hamiltonian to be invariant under spin-

rotation and time-reversal symmetries. Furthermore, Altland and Zirnbauer use the language

of Lie algebra and category the matrix iH into four classes which is listed in Table 1.1. From

the discussion above, we see that p-wave superconductor is an example of class D and we

will study its transport properties in the next chapter.

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CHAPTER 3

SPIN QUANTUM HALL TRANSITIONS IN p-WAVESUPERCONDUCTORS

As discussed in Chapter 1, we can define spin current in a system, whose spin is conserved

in at least one direction, in analogy of the well-studied charge current. Furthermore, when

the time-reversal symmetry is broken, we can expect spin quantum Hall effect induced by

the gradient of a Zeeman field which couples to spin. For dirty superconductors, classes

C and D 1 fall into this category and the latter will be the main subject in this work.

Consequently, methods, such as Chern and Thouless numbers which are used to study the

tranport properties in the usual quantum Hall effect, can be applied to these kinds of systems

in a similar manner. In this chapter, we will first define the spin current in superconductors

and then describe the Chern and Thouless number methods in the context of our problem.

Our numerical results of p-wave superconductors are presented in Sec. 3.2, including resultsof the finite-size scaling analysis of the numerical data. Finally, there is a discussion about

the results.

3.1 Spin quantum Hall effect

In Chapter 2, we introduced the lattice model to describe the superconductors with disorder.

Here, we write down the Hamiltonian as a reference

H = t

c

icj +

(ijc

ic

j +

ijcjci) +i

(ui )c

ici. (3.1)

All notations are the same as in Chapter 2. From the discussion in Sec. 2.3, we know that

p-wave superconductor belongs to class D, which has both spin-rotation and time-reversal

1For class D, we can still define spin current when its spin-rotation symmetry is not completely destroyed,as will be discussed in this chapter.

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symmetries broken. Although the total spin of the system is not conserved for the p-wave

pairing, we note from the Hamiltonian (3.1) that the z-component of the spin is conserved due

to our choice that pairing only occurs between electrons with opposite spin. This becomes

especially clear if we rewrite the Hamiltonian in terms of particle-hole transformed operators

for the electrons with down spins:

di = ci, di = ci, (3.2)

so that

H = (d d)

H HT

dd

d Hd, (3.3)

where H = (Hij) and = (ij). Clearly the number of d particles is conserved, reflectingthe conservation of the z-component of the total electron spin. Thus the corresponding

transport properties of the z-component spin are well-defined. In the following we simply

use the word spin to refer to its z-component, and spin conductances refer to the ratios

between the z-component of the spin current and the gradient of the z-component of the

Zeeman field.

It is useful for us to consider the presence of a uniform Zeeman field

HB = 0Bi (c

ici c

ici)

= 0Bi

(didi + didi) + const, (3.4)

where 0 is the magnetic moment of the electron, B is the Zeeman magnetic field. We note

that the Zeeman field plays a role of the Fermi energy for the (conserved) d particles. More

importantly, its presence changes the symmetry property of the systems. Due to the p-wave

pairing symmetry, we have ij = ji and thus the Hamiltonian obeys

H = xHT

x, x = x 1N, (3.5)beyond the general relation Eq. (2.28). But HB, which is a unit matrix (times 0B) in d

representation, does not possess this additional relation apparently.

In order to carry out numerical calculations in the lattice model, we take the the lattice

to be square and of finite size (LL). Because we will calculate the spin current by taking

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advantage of phase transformation, we use a generalized periodic boundary condition for the

d particles

dj+Li, = eiidj ,

where i = x, y. In terms of c operators, we have

cj+Li = eiicj, cj+Li = e

iicj. (3.6)

The spin Hall conductance of an individual quasiparticle eigenstate |m can be calculatedby the Kubo formula

Sxy(m) =i

A

n=m

m|jSx |nn|jSy |m m|jSy |nn|jSx |m(En Em)2 , (3.7)

where A = L2 is the area of the system,|m,|n

are quasiparticle eigenstates of the

Hamiltonian [Eq. (3.1)] and jS = (jSx , jSy ) the spin current operator. The spin current is

given by

jS =

2

1

i[(x x), H], (3.8)

where

x =i

xicici, (3.9)

and xi is the position of site i.

Following Thouless and co-workers [57, 58], we make the unitary transformation

ci = eixici, (3.10)

where = (x, y) with x = x/L and the same for y. Now, we will work in this

new respresentation c. The generalized periodic boundary condition (3.6) thus becomes the

simple periodic boundary condition. It can also be found that

jS =1

2

H

. (3.11)

Let |n to be the eigenstate in the new representation, we have

H|n = En|n, (3.12)

which yields H

En

|n = (En H) n

, (3.13)

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or

m|H

|n = (En Em)m| n

, m = n. (3.14)

Therefore the Kubo formula (3.7) is reduced to

Sxy(m) = i4A

mx

my m

y

mx

=i

4

m

x

my

m

y

mx

. (3.15)

The spin Hall conductance averaged over boundary conditions is related to a topological

quantum number

Sxy(m) =

8

dxdy

1

2i

m

y

m

x

m

x

m

y

=

8 C1(m), (3.16)

where C1(m) is an integer and known as the first Chern index. As is widely used in quantum

Hall transitions and other contexts [59, 60, 61, 62, 63, 64, 65, 66, 67, 68], C1(m) can be used

to distinguish current carrying states from localized states unambiguously, even in finite-

size systems, thus providing a powerful method to study the localization properties of the

quasiparticle states.

An alternative way to study the localization properties of the states is to calculate the

Thouless number (also known as the Thouless conductance) of the states at a given Fermienergy E, defined as [69, 71]

gT(E) =|E|

E 8

Sxx, (3.17)

where E is the average energy level spacing at energy E, and |E| is the average energylevel shift caused by the change of the boundary condition from periodic to anti-periodic

in one spatial direction. It was argued in the context of electron localization that gT(E) is

proportional to the longitudinal conductance of the system [69, 71]; in the present context we

expect it to provide a measure of the longitudinal spin conductance of the superconductor.

Thouless numbers have also been numerically studied for the conventional integer quantum

Hall effect, in both full [100] and projected [101] lattice models.

In this work we carry out numerical calculations to diagonalize the Hamiltonian H to

obtain the exact quasiparticle eigen wave functions. We calculate their Chern and Thouless

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numbers to study their localization properties, and perform finite-size scaling analysis to

extract critical behavior of the transitions driven by the change of the disorder strength W

or the Zeeman field.

3.2 Numerical resultsIn this work, we choose = 0.5 and the chemical potential = 3.0 to avoid the van Hove

singularity at zero energy in the single electron spectrum. To calculate the Chern number of

each eigenstate, we evaluate the integral in Eq. (3.16) numerically over the boundary phase

space 0 x, y 2. We divide the boundary phase space into M M square gridswith M = 20 80, depending on the system size L = 10 40 to achieve desired precision.

Figure 3.1 shows the density of states (DOS) (per lattice site and spin species) (E) for a

system with L = 10 and W = 4.0. For such a relatively weak disorder, the superconductinggap is still visible. Also shown is the spin Hall conductance Sxy as a function of quasiparticle

Fermi energy E = BB, calculated by summing up Chern number of states below the Fermi

energy. We find that Sxy jumps from zero up by one unit near the (disorder-broadened)

lower band edge, and jumps back to zero above the gap. Therefore, a plateau in Sxy is well

developed around E = 0, clearly indicating the existence of a spin quantum Hall phase. This

phase with topological Chern number equal to one is the simplest possible spin quantum Hall

phase for non-interacting quasiparticles; it is simpler, for example, than the corresponding

phase of an dx2y2 + idxy superconductor, which carries a total Chern number two.

In the following discussion, we focus on cases with disorder strong enough to close the

gap, and look for transitions from the spin quantum Hall phase to other possible phases,

driven by either the disorder strength W or the quasiparticle Fermi energy. In Fig. 3.2,

we plot the total DOS (E) (which is roughly system size independent) and the density of

current carrying states (defined as states with non-zero Chern number) e(E) for systems

with L = 10 40. We find that e(E) has a weak double-peak structure near E = 0 for large

L, whose width shrinks as L increases. This behavior is reminiscent of those seen in the

numerical study of current carrying states in the integer quantum Hall effect [60, 61, 62],

where the current carrying states exist only at discrete critical energies in the thermodynamic

limit and, thus, the width of e(E) peak(s) shrinks to zero as L increases toward infinity.

In the present case the two peaks correspond to two such critical energies, carrying a total

Chern number +1 and 1, respectively, which are responsible for the spin quantum Hall

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Figure 3.1: Density of states (E) (solid line) and spin Hall conductance Sxy(E) (dottedline, in units of /8) for L = 10 and W = 4.0. We average over 500 samples of differentrandom potential realizations.

Figure 3.2: Density of states (DOS) (E) and density of current carrying states (withnonzero Chern number), e(E), for systems with L = 10 40 and W = 8.0.

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plateau when the Fermi energy is between them (so that only the lower critical energy is

below the Fermi energy). According to the scaling theory of localization, e(E) depends on

L only through a dimensionless ratio L/(E) when the system size becomes sufficiently large;

the localization length diverges in the vicinity of a critical energy Ec as (E)

|E

Ec

|.

Therefore, the number of current carrying states Ne(L) behaves as

Ne(L) = 2L2

e(E) dE L21/, (3.18)

from which we can estimate . Assuming we have a similar situation here, we plot Ne(L),

normalized by the total number of states N(L) = 2L2, on a log-log scale in Fig. 3.3. Just as

in the quantum Hall case [60, 61, 62], we can fit the data to a power law (a straight line in

the log-log plot) as in Eq. (3.18) reasonably well, and obtain

= 2.6 0.2.

This is close to the corresponding exponent = 2.3 0.1 for the integer quantum Halltransition. These results suggest that just as in the case of the integer quantum Hall effect,

current carrying states exist at discrete critical energies in the thermodynamic limit, and the

spin quantum Hall transition driven by the Zeeman field (or equivalently, the quasiparticle

Fermi energy) has the same critical behavior as the integer quantum Hall transition. This is

expected on the symmetry ground, because in this case the critical energies are away from

E = 0, and thus can only be reached in the presence of the Zeeman field. As discussed

earlier, the Zeeman field breaks the symmetry of Eq. (3.5) and reduces the symmetry of the

present problem to that of electrons moving in a magnetic field and a random potential.

While the Chern numbers measure the ability of individual states to carry spin Hall

current, we have also calculated the Thouless conductance gT(E), which is a measure of

the longitudinal spin conductance. Unlike the Chern number calculation which requires the

diagonalization of the Hamiltonian for many different boundary conditions, the Thouless

number calculation only needs the diagonalization at two different boundary conditions,

thus allowing us to study larger systems. On the other hand, it is known in the numerical

study of quantum Hall effect that Chern number calculation reaches the scaling behavior

at smaller system sizes. Therefore, these two methods are complementary to each other.

Figure 3.4 shows (E) and gT(E) for systems with L = 40 80, and with W = 8.0. We find

that gT(E) has a similar double-peak structure as e(E) with peaks locating at the same

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Figure 3.3: Percentage of current carrying states Ne/N(L) versus system size L on a log-logscale for W = 8.0. The solid line is a power-law fit of the data.

Figure 3.4: Density of states (E) and Thouless number gT(E) for systems with L = 40 80and W = 8.0.

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Table 3.1: Critical exponents for different cutoff energies Ecut with W = 8.0.

Ecut 3.0 3.2 3.4 3.6 3.8 4.0 2.79 2.73 2.68 2.64 2.58 2.54

0.08 0.08 0.08 0.07 0.07 0.07

energies, and that the peaks become narrower as L increases. In the following we perform

the similar scaling analysis based on the zeroth moment of gT(E) as for the Chern numbers.

Namely, we compute the area A(L) under gT(E) and expect

A(L) =

gT(E)dE L1/. (3.19)

One slight complication is that unlike e(E), gT(E) has long tails extending to the edges of

(E), which clearly has no connection to the critical behavior near the critical energies. To

eliminate the influence of these artificial tails, we introduce a cutoff energy Ecut, and exclude

contributions from |E| > Ecut. Based on the Chern number calculation above (Fig. 3.2), aswell as the gT(E) curves themselves, we can safely choose Ecut between 3.0 and 4.0, beyond

which we find essentially no current carrying states for L 40. In Fig. 3.5, we plot, on alog-log scale, the area A(L) normalized by the area under the DOS curve between Ecut andEcut:

Ncut =

Ecut

Ecut

(E; L)dE

for a series of different Ecut, and list the corresponding in Table ??. We find that has

very weak dependence on the choice of the cutoff energy and its variation between 2 .54 and

2.79 is consistent with the results obtained from the Chern number calculation.

We also studied other disorder strengths. In the case of the integer quantum Hall

transition [61, 62], it is known that as the disorder strength increases, the critical energies

that carry opposite Chern numbers move close together, merge, and disappear at some

critical disorder strength Wc. In the present case, we expect the same to happen and due to

the symmetry of the Hamiltonian, the critical energies can only merge at E = 0. We present

the results for W = 9.0 in Fig. 3.6. In this case we no longer see two split critical energies,

suggesting that the two critical energies that were clearly distinguishable at W = 8.0 either

(i) have moved too close to be distinguishable at the accessible system sizes, or (ii) have

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Figure 3.5: Area A(L) of Thouless number gT(E) normalized by number of states Ncutcounted, versus system size L on a log-log scale for different cutoff energy Ecut and W = 8.0.The lines are power-law fits of the data.

just merged. We believe scenario (i) is much more likely than (ii) based on the following

observations. (a) We find that the peak value of gT(E) is independent of system size and

takes thesame

value as that of W = 8.0. (b) We have performed the same scaling analysisof gT(E) as we did above for W = 8.0 and obtained a similar exponent 2.3 (see insetof Fig. 3.6), which is even closer to the known value of the integer quantum Hall transition.

However, there is another possibility that instead of entering the insulating phase (in which

all quasiparticles states are localized) immediately, the system is in a metallic phase, after

the two critical energies merge so that the system is no longer in the spin quantum Hall

phase. Senthil and Fisher [102] suggested that in this phase both the DOS (E) and the

conductance diverge logarithmically at the band center. Interestingly, we indeed find (E)

to be enhanced at E = 0. We believe, however, this is not associated to the metallic phasefor the following reasons. (i) No such enhancement is seen in the Thouless number, which is

a measure of the longitudinal conductance. (ii) We find (E) to be essentially system size

independent between L = 40 and L = 80, even at E = 0, while one expects [102] (L) log Lin the metallic phase. (iii) We find that (see below) the enhancement of(E) at E = 0 is also

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present at stronger disorder when the system is clearly insulating. Thus it appears unlikely

that the metallic phase is responsible for the single peak in gT(E).

The situation is quite different as W further increases. In Fig. 3.7, we present results for

W = 10.0 and see a very different behavior. Here the peak value of gT(E) systematically

decreases as the system size increases, exhibiting a characteristic insulating behavior.

Combined with results of smaller W, we conclude that in the absence of the Zeeman field (or

when the quasiparticle Fermi energy is at E = 0), the system is driven into the insulating

phase from the spin quantum Hall phase as the disorder strength W increases. The critical

strength Wc is slightly above 9.0 and clearly below 10.0. No evidence has been found for the

existence of an intermediate metallic phase that separates these two phases for our choice of

model parameters ( = 3.0, = 0.5, etc.).

The critical behavior of the transition driven by increasing W is expected to be different

from the one driven by changing the Zeeman field discussed above, due to the additional

symmetry. In order to study the critical property one first needs to determine the critical

disorder strength Wc accurately, which we are unable to do within the accessible system size

in our study. It would be of significant interest to study this transition with more powerful

computers and/or other computational methods.

We give the results of W = 15.0 in Fig. 3.8 as an example of strong disorder,

where all states are clearly localized. Here, the Thouless number drops rapidly as the

system size increases as expected. Interestingly, the enhancement of the DOS at E = 0remains to be quite pronounced, suggesting that it is not associated with possible metallic

behavior discussed above. For comparison, we return to the DOS of a dx2y2 + idxy-

wave superconductor shown in Fig. 2.2 (c). It belongs to class C and has been studied

in considerable detail in Refs. [103, 104, 105, 106]. From the plot, we see the gap vanishes

just like the p-wave case for sufficiently large W. Furthermore, the DOS exhibits a pseudogap

behavior at E = 0 for large W, in the vicinity of which the DOS vanishes in an (apparently

sublinear) power law as predicted [104, 105]. This is a good example that the change of

symmetry profoundly affects the critical behavior as well as other properties of the system.

3.3 Conclusion

In this chapter we have studied the localization properties of the quasiparticle states in

superconductors with spontaneously broken time-reversal symmetry, which support spin

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Figure 3.6: Density of states (E) and Thouless number gT(E) for systems with L = 40 80and W = 9.0. The inset shows the area of Thouless number gT(E) divided by Ncut forEcut = 4.0.

Figure 3.7: Density of states (E) and Thouless number gT(E) for systems with L = 40 80and W = 10.0. The inset is a blow-up of the Thouless number curves near E = 0, whichshows that gT(E = 0) decreases with increasing L.

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Figure 3.8: Density of states (E) and Thouless number gT(E) for systems with L = 20 40and W = 15.0.

quantum Hall phases. Our study is based on the exact diagonalization of microscopic lattice

models and the consequent numerical calculation of the Chern and Thouless numbers of the

quasiparticle states. Our microscopic study is complementary to previous numerical work

on this subject, which have been based almost exclusively on effective network models with

appropriate symmetries [103, 105, 106, 107, 108, 109].

We have focused mostly on a p-wave pairing model in which the time-reversal symmetry

is broken by the (complex) pairing order parameter, while the z-component of the total spin

is conserved so that the transport properties of the z-component of the spin is well defined.

We find the system supports a spin quantum Hall phase with spin Hall conductance one in

appropriate unit, and an insulating phase. Transitions between these two phases may be

induced either by changing the disorder strength, or by applying and sweeping a Zeeman

field. The field-driven transition is found to have the same critical behavior as the integer

quantum Hall transition of non-interacting electrons as expected on symmetry grounds. The

disorder-driven transition in the absence of the Zeeman field is expected to have different

critical properties due to additional symmetry of the Hamiltonian. However, we have not

been able to study the critical behavior of this transition.

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The symmetry properties of the p-wave pairing model in the absence of the Zeeman

field belongs to class D in the classification of general fermion pairing models of Altland and

Zirnbauer [10]. It has been suggested that in addition to the quantum Hall and the insulating

phases, class D models may also support a metallic phase [102], which has logarithmically

divergent density of states and conductance. Such a system can have either a direct transition

between the quantum Hall and the insulating phases, or a metallic phase separating these

two phases. In our model we find a direct transition between the spin quantum Hall and

insulating phases, but no definitive evidence for a metallic phase. This is not unusual as it

is known [108] that specific microscopic models may or may not support the metallic phase.

For comparison, we also look at the density of states of a d-wave superconductor with

dx2y2 + idxy pairing order parameter, which supports a spin quantum Hall phase with

spin Hall conductance two in the same unit. This model has different symmetry propertiesand belongs to class C in the classification of Altland and Zirnbauer. We find that the

density of states vanishes with sublinear power law near E = 0, in agreement with earlier

studies [104, 105, 106, 107]. This is in sharp contrast to the p-wave case in which we

observe an enhanceddensity of states at E = 0 for sufficiently strong disorder, demonstrating

the profound effect of symmetries on the low-energy properties of the system. While this

enhancement is somewhat reminiscent of the divergent density of states of the possible

metallic phase, further analysis suggests this is not the case. The origin of this enhancement

is currently unclear.

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CHAPTER 4

FULDE-FERRELL-LARKIN-OVCHINNIKOVSUPERCONDUCTOR

In this chapter, we will study another type of unconventional superconductors which

distinguishes from those conventional ones by possessing non-zero finite momemtum in

Cooper pairs. In the early 1960s, Fulde and Ferrell [15] and Larkin and Ovchinnikov [16]

proposed the possibility that a superconducting state with a periodic spatial variation of the

order parameter would become stable when a singlet superconductor is subject to a large

Zeeman splitting. The Zeeman splitting could be due to either a strong magnetic field or an

internal exchange field. Under such a strong magnetic or exchange field, there is a splitting

of the Fermi surfaces of spin-up and -down electrons, and condensed pairs of electrons with

opposite spins across the Fermi surface may be formed to lower the free energy from that

of a normal spin-polarized state. These pairs have a non-zero total momentum 2q, whichcauses the phase of the superconducting order parameter to vary spatially with the wave

number 2q. This state is known as the Fulde-Ferrell (FF) state. Larkin and Ovchinnikov

(LO), on the other hand, proposed independently an alternative scenario, in which the order

parameter is real, but varies periodically in space, possibly in more than one directions.

Both types of states are now known (collectively) as the Fulde-Ferrell-Larkin-Ovchinnikov

(FFLO) state. In our work, we focus on the FF state and it can be easily studied in the

continuum model. First we will introduce the model and present the self-consistent mean

field solutions for both s- and dx2y2-wave superconductors in FF phases. Then a discussion

of the properties of FF state is followed. All calculations are performed at zero temperature

T = 0.

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4.1 The Fulde-Ferrell state

We start from the Hamiltonian,

H=

k

(k + 0B)ckck +

k,k,q

Vkkck+qc

k+qck+qck+q, (4.1)

where k is the single-particle kinetic energy relative to the Fermi energy F, B is the

Zeeman magnetic field, 0 is the magnetic moment of the electron. For s-wave SC, we

have Vkk = V0, while for d-wave SC, we have Vkk = V0 cos(2k) cos(2k) (here,|k|, |k| < c F, c is the cutoff, k is the azimuthal angle of k). The mean fieldHamiltonian is,

HMF =k

(k + 0B)ckck k

(kqck+qc

k+q +

kqck+qck+q), (4.2)

where kq is the pairing potential and satisfies the self-consistent condition,

kq = k

Vkkck+qck+q. (4.3)

The mean field Hamiltonian could be rewritten as

HMF =k

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