d-wave superconductors in the vicinity of boundaries · d-wave superconductors in the vicinity of...

D-Wave Superconductors in theVicinity of Boundaries

Dissertation

zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften

an der Mathematisch-Naturwissenschaftlichen Fakultatder Universitat Augsburg

vorgelegt im Oktober 2001

von

Thomas Luckaus Augsburg

Erstgutachter: Prof. Dr. U. Eckern

Zweitgutachter: Prof. Dr. J. Mannhart

Tag der mundlichen Prufung: 30. November 2001

Contents

1 Introduction 5

2 Basic Concepts 11

2.1 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Symmetry, Interaction, and the Order Parameter . . . . . . . . . 15

2.3 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . 19

3 Green’s Functions and the Quasi-Classical Approximation 23

3.1 Green’s Functions Method in Superconductivity . . . . . . . . . . 23

3.2 Quasi-Classical Approximation . . . . . . . . . . . . . . . . . . . . 29

3.3 Superconductors in Thermal Equilibrium . . . . . . . . . . . . . . 33

4 Boundary Conditions for the Quasi-Classical Green’s Functions 37

4.1 Zaitsev’s Boundary Conditions . . . . . . . . . . . . . . . . . . . . 37

4.2 Ideal Tunnel Junctions . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Boundary Conditions according to Shelankov and Ozana . . . . . 45

4.4 Explicit Solution of Zaitsev’s Boundary Conditions . . . . . . . . 51

4.5 Simple Applications to Unconventional Superconductors . . . . . 52

4.5.1 Specular Surface . . . . . . . . . . . . . . . . . . . . . . . 52

4.5.2 Rough Surface . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.3 Ideal Interface . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Rough Surfaces 65

5.1 Surfaces with Disorder . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . 68

5.2 Order Parameters with Subdominant Pairing: dx2−y2 + dxy/s . . . 76


5.3 Rough Surfaces Acting as Beam-Splitters . . . . . . . . . . . . . . 86

4 CONTENTS

5.3.1 S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


6 Rough Interfaces – Josephson Junctions 97

6.1 S-Matrix for Rough Interfaces . . . . . . . . . . . . . . . . . . . . 97

6.2 Asymmetric Junctions . . . . . . . . . . . . . . . . . . . . . . . . 99


6.3 Mirror Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108


7 Conclusions 121

Appendix 125

A Keldysh Green’s Function in Thermal Equilibrium . . . . . . . . . 125

B Bullet-Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C Self-Consistency Equation . . . . . . . . . . . . . . . . . . . . . . 127

D Derivation of the Homogeneous Ginzburg-Landau Equation . . . . 128

E Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . 130

F Current Conservation in the Boundary Conditions . . . . . . . . . 131

Chapter 1

Introduction

Unconventional superconductivity has been a popular field in condensed matter

physics for almost 30 years. It started with the study of superfluid 3He where

the order parameter has p-wave symmetry [1, 2]. Later on several superconduct-

ing heavy fermion compounds such as CeCu2Si2 or UPt3 were found to have

an unconventional symmetry of the order parameter [3, 4]. In recent years the

interest has risen again due to the discovery of d-wave superconductivity in hole-

doped high temperature (high-Tc) superconductors as YBa2Cu3O7−δ (YBCO)

or BiSr2Ca2Cu2O10−δ (BSCCO). Moreover, also the spin-triplet superconductor

Sr2RuO4, which is discussed to exhibit a p-wave or even an f -wave symmetry of

the order parameter, is of growing interest [5, 6].

The determination of the order parameter symmetry in experiments is a chal-

lenging task since many standard methods – for example the measurement of the

specific heat, angle-resolved photo emission spectroscopy (ARPES), or Raman

scattering – are only sensitive to the absolute value of the order parameter [7,8].

Other experimental techniques which can also access the phase of the order pa-

rameter had therefore to be developed. The most striking phase-sensitive exper-

iments are the corner-SQUID experiments [9,10], the tri-crystal experiment [11],

and the π-SQUID experiment [12,13], which established the d-wave symmetry of

the order parameter in YBCO.

In these phase-sensitive experiments, contacts between two superconductors

play a crucial role. This is one reason for us to study the behavior of d-wave

superconductors in the vicinity of boundaries. Moreover, such contacts occur as

grain boundaries in all samples of high-Tc materials. For technological reasons,

the understanding of grain boundaries is of importance, as they reduce the critical

current of the sample drastically [8, 14–16].

6 1 Introduction

Recently, several unusual properties of d-wave superconductors in connection

with boundaries have been discovered. We briefly present those results which are

of interest for the following work; a more comprehensive discussion can be found

in several review articles [7,16–18]. For simplicity, we first concentrate on contacts

consisting of a d-wave superconductor and a normal metal (NIS junctions). Later

on we consider boundaries between two d-wave superconductors (SIS junctions).

The most striking feature of NIS junctions is a zero bias conduction peak

(ZBCP) in the differential conductance if the interface is perpendicular to the

[110]-direction of the d-wave superconductor [19–22]. This very robust property

was first explained by Hu [23] as a consequence of Andreev bound states at the

surface. Such bound states are predicted exactly at zero energy if the sign of the

order parameter changes due to reflection at the surface; this is the case for a

tilted d-wave order parameter.

Later experiments, however, gave rise to various new problems. In some

measurements of the differential conductance the ZBCP splits at low tempera-

tures [24,25]. A possible explanation is the occurrence of a subdominant compo-

nent of the order parameter with a non-trivial phase difference with respect to

the dominant one (e.g. dx2−y2 + idxy or dx2−y2 + is) [26–28]; in such a state the

time-reversal symmetry is broken.

The observation of a ZBCP, even for untilted [100]-surfaces of the supercon-

ductor [24, 29, 30], was also quite puzzling, as it cannot be explained by simple

specular reflection. On the other hand in real samples always large facets (of a

typical scale 10 nm) exist which can lead to a ZBCP as pointed out by Fogelstrom

et al. [27].

In another experiment the roughness of a [110]-interface is varied by ion ir-

radiation [31]. Surprisingly the ZBCP broadens only weakly with increasing

roughness, whereas its height is clearly reduced. Although some theoretical ideas

showing this behavior exist [32–34], a satisfactory solution of the problem has

not yet been found.

For SIS junctions we concentrate on the temperature-dependent critical cur-

rent [35–37] and on the current-phase relation [38–40] of the contact. Due to the

unconventional symmetry of the order parameter, surprising effects are predicted

for particular orientations of the superconductors [17, 18, 41]. However, a com-

plete understanding of the large amount of experimental results has not yet been

reached.

For an asymmetric junction with an untilted superconductor ([100]-direction)

on the one side and a tilted one ([110]-direction) on the other side an additional

7

geometric symmetry occurs which should lead to a vanishing first order contribu-

tion in the tunnel current; i.e. the current-phase relation should show an unusual

sin(2ϕ)-like behavior [41]. This was observed in experiment, but only in some of

the examined samples [39]. Interface roughness or a slight misorientation of the

interface are discussed as possible reasons.

An astonishing temperature dependence of the critical current was theoreti-

cally predicted for so-called mirror junctions [41,42] where both superconductors

are symmetric with respect to a reflection at the boundary: The temperature-

dependent critical current should have a local minimum below the critical tem-

perature, which is related to a π-junction behavior for lower temperatures. This

behavior was recently found in experiment [40]. Many other experiments however

did not show any local minimum [36, 37]. Interface roughness seems to play a

crucial role to explain these observations [42].

Most recently, it was shown that the properties of YBCO grain boundaries can

be controlled by doping with Ca [14,15,43]; in particular it is possible to enhance

the critical current by almost one order of magnitude. This is an interesting

method not only for technological reasons but also for basic studies.

Up to now, we briefly introduced some of the most important experimental

facts about junctions of d-wave (i.e. high-Tc) superconductors. We will now turn

to the theoretical description. For the treatment of a spatially inhomogeneous

superconductor on microscopic scales one has to solve the Bogoliubov-de Gennes

equations, which are second order partial differential equations. In many cases,

a Ginzburg-Landau theory, which is valid near the critical temperature, Tc, is

sufficient to understand experimental results. This approximation however is

too crude for our purposes and we use the technique of quasi-classical Green’s

functions [44–46]: This approach provides more information than the Ginzburg-

Landau theory, but has a simpler mathematical structure than the Bogoliubov-

de Gennes equations since only ordinary differential equations must be solved.

The quasi-classical approximation is valid on scales at least of the supercon-

ducting coherence length at zero temperature, ξ0, which is assumed to be large

compared with the Fermi wave length, ~/pF (pF : Fermi momentum). In ordi-

nary superconductors this condition is fulfilled due to a large coherence length

(ξ0pF/~ & 103); in high-Tc materials the coherence length is much shorter but

still ξ0pF/~ ≈ 10.

In recent years the quasi-classical theory has been applied frequently to bound-

ary problems of d-wave superconductors [26–28, 33, 47–51]. Since the quasi-

classical theory is not directly applicable at boundaries, surfaces or interfaces

8 1 Introduction

must be taken into account by special boundary conditions which have to be de-

rived from the microscopic theory. Depending on the physical situation different

boundary conditions must be applied: Many results [18] can be obtained using

Zaitsev’s boundary conditions [52] which are valid for ideal interfaces with spec-

ular reflection. In most calculations concerning rough interfaces, an approach as

proposed by Ovchinnikov [53] is used, where a clean surface is covered by a thin

dirty layer (thin dirty layer model) [27, 28, 33, 47, 48].

We will apply an alternative approach recently developed by Ozana and She-

lankov [54] which allows us to describe rough interfaces. The properties of an

interface are determined by a scattering matrix which takes into account the

roughness on the microscopic scale. In this method individual realizations of

interfaces can be studied. In particular, for interfaces with random properties

(irregular interfaces) not only averaged quantities can be obtained but also sta-

tistical fluctuations which could be of importance in mesoscopic junctions.

We will begin our work with a chapter on basic theoretical concepts concerning

unconventional spin-singlet superconductors. First of all, the gap equation and

the superconducting density of states are derived in the standard BCS approach

for an arbitrary symmetry of the order parameter. Subsequently, the gap equation

is used to study the relation between the form of the attractive interaction and

the symmetry of the order parameter. Some aspects of a superconductor having

a multi-component order parameter are discussed within the Ginzburg-Landau

theory.

In the third chapter we give a short introduction to the Green’s function

method in superconductivity. We start with the basic definitions and the general

formalism. We present the homogeneous time-independent case as a simple appli-

cation. Afterwards the quasi-classical approximation is introduced. We conclude

with a discussion of the thermal equilibrium situation since it is of particular

importance for our calculations.

In chapter 4 we discuss boundary conditions which are necessary to describe

interfaces in the quasi-classical theory. First we present Zaitsev’s boundary con-

dition [52] and discuss several applications. The second part of the chapter deals

with the boundary conditions of Ozana and Shelankov [54]; we give an idea of

the derivation, and put them in a form convenient for our purposes. Using this

approach, in the next part of the chapter some basic boundary effects of unconven-

tional superconductors are discussed; possible surface bound states are considered

in particular. For simplicity we here neglect a possible modification of the order

parameter at the boundary. The results are useful to interpret those presented

9

in following chapters which are obtained by fully self-consistent calculations.

In the next chapter we discuss various aspects of rough surfaces. We begin by

investigating a d-wave superconductor in the vicinity of an irregular rough surface,

which is described by a random scattering matrix. We calculate the self-consistent

order parameter and the differential conductance for an NIS junction. Our main

interest is the roughness dependence of the ZBCP for various orientations of the

order parameter. Afterwards we will repeat the calculations for a mixed order

parameter (dx2−y2+s/dxy). Moreover, we consider a surface which acts as a beam-

splitter so that an incoming quasi-particle can be reflected into several outgoing

directions. Here the existence of a ZBCP for an untilted order parameter is of

particular interest; its behavior for tilted order parameters is studied as well.

Chapter 6 deals with d-wave superconductors which are linked by an irregular

interface. In section 6.2 we consider an asymmetric configuration of the order

parameter as described above, which in the ideal case exhibits a sin(2ϕ)-like

current-phase relation due to the particular symmetry of the junction. Here

the study of rough interfaces is of interest as roughness destroys this symmetry.

In section 6.3 we treat so-called mirror junctions for various orientations of the

order parameter. As mentioned above, here an unusual temperature dependence

of the critical current occurs in the ideal case. We study the dependence on the

roughness and apply our results to various experimental realizations of mirror

junctions.

We conclude with a summary of our results in chapter 7. Furthermore we

make some suggestions in order to improve future calculations.

10 1 Introduction

Chapter 2

Basic Concepts

In this chapter we give a very short introduction to (unconventional) supercon-

ductivity. For this purpose we use the BCS theory for a translationally invariant

system [8, 55]. As a result of this section, we present the gap equation and the

superconducting density of states for an arbitrary angular dependence of the

attractive interaction. Using the gap equation, we classify the possible symme-

tries of the superconducting phase. We consider superconductors with a one-

component order parameter (e.g. dx2−y2) as well as with mixed order parameters

(dx2−y2 +s/dxy). The latter case is discussed within a Ginzburg-Landau approach.

2.1 BCS Theory

In this section we present the BCS theory for a spatially homogeneous supercon-

ductor [8,55]. We consider arbitrary attractive interactions leading to spin-singlet

superconductivity (i.e. the Cooper-pairs are in a spin-singlet state).

In the BCS theory the Hamiltonian for the electrons is given by a free and an

interacting part

H = H0 +Hi. (2.1)

The free part of the energy is given by

H0 =∑σ=↑↓

∫dp

(2π)d

(p2

2m− µ

)Ψ†

σ(p)Ψσ(p) (2.2)

with the chemical potential µ; for low temperatures (T . Tc EF ) the chemical

potential is given by the Fermi energy, µ ≈ EF . The electronic field operator

12 2 Basic Concepts

−p′ ↓ −p ↓

p′ ↑ p ↑

Figure 2.1: The scattering process which leads to the superconducting state.

Ψ†σ(p) creates an electron with momentum p and spin σ, whereas Ψσ(p) annihi-

lates it. They are defined by the fermionic commutation algebra

Ψσ(p),Ψσ′(p′) =

Ψ†σ(p),Ψ†

σ′(p′)

= 0 (2.3)Ψ†

σ(p),Ψσ′(p′)

= δσσ′δ(p− p′). (2.4)

Note that the units are chosen such that ~ = kB = 1.

In the BCS theory the interaction is assumed to be non-retarded and only

that part which is responsible for superconductivity is taken into account. In the

case of spin-singlet pairing this leads to

Hi =

∫dp

(2π)d

∫dp′

(2π)dV (p,p′)Ψ†

↑(p)Ψ†↓(−p)Ψ↓(−p′)Ψ↑(p

′). (2.5)

This scattering process is illustrated in Fig. 2.1.

The analysis of this Hamiltonian is still non-trivial, and a mean-field theory

is applied for the approximate solution. We use the identity

Ψ↑(p)Ψ↓(−p) = b(p) + [Ψ↓(−p)Ψ↑(p)− b(p)] (2.6)

to separate the operator product into a sum of the mean value b(p) ∈ C and the

fluctuations. Assuming only small fluctuations and neglecting their quadratic

contributions we can approximate the interaction by an expression bilinear in

the field operators

Hi =

∫dp

(2π)d

∫dp′

(2π)dV (p,p′)

[Ψ†↑(p)Ψ†

↓(−p)b(p′)+

+ b∗(p)Ψ↓(−p′)Ψ↑(p′)− b∗(p)b(p′)

].

(2.7)

In the mean-field approximation the parameter b(p) must be determined by using

2.1 BCS Theory 13

the Hamiltonian H = H0 + Hi for the evaluation of the average

b(p) = 〈Ψ↓(−p)Ψ↑(p)〉H . (2.8)

The thermodynamic average of an operator is defined by

〈. . . 〉H ≡ Tr[ρ(H) . . . ], ρ(H) =(Tre−H/T

)−1

e−H/T (2.9)

where T is the temperature of the system. With the self-consistent determination

of b(p) by Eq. (2.8) the problem is closed.

Using the definition

∆(p) = −i∫

dp′

(2π)dV (p,p′)b(p′), (2.10)

the Hamiltonian can be written in a compact form

H =

∫dp

(2π)d

[(Ψ†↑(p)

Ψ↓(−p)

)(ξp i∆(p)

−i∆∗(p) −ξp

)(Ψ↑(p)

Ψ†↓(−p)

)−

− i∆(p)b∗(p) + ξp

] (2.11)

with ξp = p2/2m−µ. It can be diagonalized by a canonical transformation which

is given by

Ψ↑(p) = u∗(p)Φ1(p) + v(p)Φ†2(p) (2.12)

Ψ†↓(−p) = −v∗(p)Φ1(p) + u(p)Φ†

2(p). (2.13)

The eigenvalues of the bilinear part are

E = ±εp ≡ ±√ξ2p + |∆(p)|2, (2.14)

and the coefficients are determined via the relations

i∆∗(p)v(p)

u(p)= εp − ξp, (2.15)

|v(p)|2 = 1− |u(p)|2 =1

2

(1− ξp

εp

). (2.16)

14 2 Basic Concepts

Then, the Hamiltonian in diagonal form reads

H =

∫dp

(2π)d

[εp∑i=1,2

Φ†i (p)Φi(p)− εp + ξp − i∆(p)b∗(p)

]. (2.17)

The first term in the Hamiltonian represents the excitations of the superconduct-

ing state, whereas the last three terms yield the energy difference to the normal

state. It is important to note that the new degrees of freedom given by Φi(p)

are fermions, too; i.e. they fulfill the fermionic commutation relations and their

occupation number in a thermal equilibrium situation is determined by the Fermi

distribution

〈Φ†i (p)Φi(p)〉 =

1

eεp/T + 1. (2.18)

Using the canonical transformation defined in Eqs. (2.12) and (2.13) we obtain the

self-consistency condition for the order parameter ∆(p) from Eqs. (2.8) and (2.10)

∆(p) = i

∫dp′

(2π)dV (p,p′)u∗(p′)v(p′)〈1−

∑i=1,2

Φ†i (p

′)Φi(p′)〉H

=

∫dp′

(2π)dV (p,p′)

∆(p′)

2εp′tanh

( εp′

2T

).

(2.19)

In the BCS theory the interaction is assumed to be independent of the modulus

of the momenta in a small region around the Fermi surface

V (p,p′) =

V (pF ,p

′F ) for |ξp|, |ξp′| < Ec

0 else. (2.20)

The momentum dependence of the interaction is characterized by the orientation

of the Fermi momenta. The cut-off energy, Ec, is assumed to be small compared

to the Fermi energy. In usual superconductors with a phonon mediated attractive

interaction, for example, the cut-off energy is of the order of the Debye energy.

As a consequence of Eq. (2.19), the order parameter is independent of |p| close

to the Fermi surface (i.e. |ξp| < Ec), and we may write ∆(p) = ∆(pF ).

The dispersion of the quasi-particles, εp, exhibits an energy gap which is given

by ∆(pF ). The order parameter ∆(pF ) is therefore referred to as gap function;

the self-consistency equation is called gap equation.

Approximating the density of states in the normal state by a constant, N0, in

2.2 Symmetry, Interaction, and the Order Parameter 15

the vicinity of the Fermi surface, we use the relation∫dp

(2π)d· · · ⇒ N0

∫dξp

∫dpF

SF· · · ≡ N0

∫dξp 〈. . .〉pF

(2.21)

with SF being the area of the Fermi surface; the integral over pF represents

an average over all directions on the Fermi surface. The gap equation can be

rewritten by

∆(pF ) = N0

Ec∫0

dξp′

⟨V (pF ,p

′F )∆(p′F

′)√ξ2p′ + |∆(p′F )|2

tanh

√ξ2p′ + |∆(p′F )|2

2T

⟩

p′F

= N0

Ec∫0

dE tanh

(E

2T

)⟨V (pF ,p

′F )∆(p′F )Θ(E2 − |∆(p′F )|2)√

E2 − |∆(p′F )|2

⟩p′

F

.

(2.22)

The superconducting density of states is defined by

N (E) =

∫dp

(2π)dδ(E − εp). (2.23)

Using Eq. (2.21) this leads to the well-known expression

N (E) = N0

⟨EΘ(E2 − |∆(pF )|2)√

E2 − |∆(pF )|2

⟩pF

. (2.24)

With this result we finish our short review of the BCS theory. In the next section

we will use the gap equation to discuss the relation between the symmetry of the

order parameter and the angular dependence of the attractive interaction.

2.2 Symmetry, Interaction, and the Order Pa-

rameter

In the theory of second order phase transitions the symmetry of the system is of

particular importance, as in the low temperature phase this symmetry is spon-

taneously broken. In conventional superconductors only the gauge symmetry is

broken and the energy gap is often considered as isotropic (∆(pF ) = ∆); if the

16 2 Basic Concepts

gap depends on the direction, but is in agreement with the lattice symmetry,

we will call the superconductors anisotropic (but still conventional). In uncon-

ventional superconductors in addition to the gauge symmetry also the lattice

symmetry is broken. A comprehensive review of this classification is given in

Refs. [2, 4]. We will study the relation between the lattice symmetry of the sys-

tem, the allowed form of the attractive interaction, and the angular dependence

of the order parameter.

In order to describe high-Tc compounds we will assume a tetragonal lattice

symmetry, which can be described by the group D4h; in table 2.1 the even par-

ity basis functions of lowest order for the irreducible representations are given.

Note that the lattice structure of most high-Tc materials has small orthorhombic

distortions which are neglected here [7, 10].

Following these preliminary remarks, we will use the symmetry properties to

construct the form of the attractive interaction in the system. First of all the

interaction V (pF ,p′F ) must be invariant with respect to all symmetry operations.

In other words it must be a basis function of the trivial representation A1g. It

can be shown that the lowest order basis functions of A1g depending on two

variables are given by the products ηi(pF )ηi(p′F ); as we will only consider spin-

singlet superconductors, the order parameters have even parity. The interaction

can therefore be expanded as follows:

V (pF ,p′F ) = −

∑i

Viηi(pF )ηi(p′F ), Vi > 0. (2.25)

It can now be seen from the gap equation (2.22) that also the order parameter

can be expanded in the basis functions

∆(pF ) =∑

i

∆iηi(p′F ) (2.26)

D4h : basis functionsA1g η1(p) = 1, p2

x + p2y, p

2z (anisotropic) s-wave

A2g η2(p) = pxpy(p2x − p2

y)B1g η3(p) = p2

x − p2y dx2−y2-wave (also: d-wave)

B2g η4(p) = pxpy dxy-waveEg η5(p) = (pxpz, pypz)

Table 2.1: Basis functions of lowest order for the group D4h.

2.2 Symmetry, Interaction, and the Order Parameter 17

px

py

pF

#

s-wave: 1 = 1

px

pypF

#

dx2y2

-wave: 3 = cos(2#)

px

pypF

#

dxy-wave: 4 = sin(2#)

Figure 2.2: Angular dependence of three possible order parameters : s-, dx2−y2-,and dxy-wave.

with the self-consistent value of ∆i

∆i

ViN0=

Ec∫0

dE tanh

(E

2T

)⟨ηi(pF )∆(pF )Θ(E2 − |∆(pF )|2)√

E2 − |∆(pF )|2

⟩pF

. (2.27)

As we will see later, this discussion can easily be generalized to spatially varying

situations.

For simplicity we neglect the motion in z-direction; in high-Tc materials this

can be justified by the strong anisotropy (layered structure). The energy disper-

sion reads

ξp =1

2m(p2

x + p2y)− µ (2.28)

which leads to a cylindrical shape of the Fermi surface. Therefore, in a system

homogeneous in z-direction, the Fermi surface average simplifies to

〈. . .〉pF=

π∫−π

dϑ

2π. . . . (2.29)

with the two-dimensional vector pF = pF (cosϑ, sin ϑ), ϑ ∈ [−π, π].

In Fig. 2.2, the most important of the possible order parameters are illus-

trated: The dx2−y2-wave order parameter is applied to high-Tc compounds; we

will also study the possible admixture of a dxy-wave or an s-wave component.

We begin with a discussion of the simplest situation where only one channel

18 2 Basic Concepts

of the interaction is finite: Vi > 0, Vj 6=i = 0. Then the order parameter is given

by

∆(pF ) = ∆ηi(pF ), (2.30)

and the gap equation reads

1

ViN0

=

Ec∫0

dE tanh

(E

2T

)⟨η2(pF )Θ(E2 − |∆|2η2

i (pF ))√E2 − |∆|2η2

i (pF )

⟩pF

. (2.31)

The condition for the critical temperature is given by ∆ = 0 for T → Tc, which

leads to

1

ViN0

=

Ec∫0

dE tanh

(E

2Tc

) 〈η2(pF )〉pF

E. (2.32)

Using the relation

x0∫0

dxtanh(x)

x≈ ln(2.26x0), x0 1 (2.33)

we find the expression

Tc = 1.13Ece−1/ViN0〈η2

i 〉. (2.34)

The limit T → 0 of the gap equation (2.31) yields

1

ViN0

=

Ec∫0

dE

⟨η2

i (pF )Θ(E2 − |∆0|2η2i (pF ))√

E2 − |∆0|2η2i (pF )

⟩pF

, (2.35)

〈η2i (pF )〉pF

〈η2i (pF ) ln |ηi(pF )|〉pF

∆0/Tc

s-wave 1 0 1.76d-wave 1

2−0.0966 2.15

Table 2.2: Angular averages of the basis functions and the resulting relationbetween the zero temperature order parameter and the critical temperature.

2.3 Ginzburg-Landau Theory 19

which results in the zero temperature order parameter

∆0 = 2e−〈η2i ln |ηi|〉/〈η2

i 〉Ece−1/ViN0〈η2

i 〉; (2.36)

here we used the assumption |∆0| Ec. As an example the results for an s-

wave and a d-wave order parameter are presented in table 2.2. In experiments on

high-Tc compounds the ratio 2∆0/Tc & 5 [8,56,57] clearly deviates from the BCS

value. One reason could be the fact that high-Tc materials are no weak-coupling

superconductors; i.e. the approximation of the interaction as in the BCS theory

is too crude [8].

Until now we have considered a situation where only one order parameter

component is finite. The situation becomes more complicated if several inter-

action channels are relevant; in particular, the case of a two-component order

parameter will be discussed in the framework of the Ginzburg-Landau theory in

the next section.

2.3 Ginzburg-Landau Theory

For the study of a two-component order parameter it is convenient to consider

the Ginzburg-Landau theory. The main idea is to find a free energy functional

only depending on the order parameter and the vector potential. The free energy

is expanded with respect to the order parameter which is assumed to be small.

Due to the symmetry properties of the free energy, in the expansion only those

terms may occur which are in agreement with the symmetries of the system.

The physically relevant solution can be found by minimizing the free energy

with respect to the order parameter and the vector potential. This leads to

the Ginzburg-Landau equations. The coefficients of the expansion, which are

not determined by symmetry, can either be obtained from experiments or from

microscopic theories in the limit of a small order parameter.

We will now present the free energy functional for superconductors with two

possible components of the order parameter on a tetragonal lattice. One of them

is assumed to be of the dx2−y2-wave type, and the second has dxy-wave or s-wave

symmetry (see table 2.1):

∆(pF , r) = ∆1(r)η3(pF ) + ∆2(r)η1/4(pF ) (dx2−y2 + s/dxy). (2.37)

Each of the components has its own critical temperature Tc,1/2; i.e. the component

20 2 Basic Concepts

i would become superconducting at temperature Tc,i if the other component were

not present. We assume Tc,1 > Tc,2.

We will first discuss the free energy in a translationally invariant situation.

This part is a polynomial in ∆1(r) and ∆2(r) up to fourth order. Additionally

to the lattice symmetry the free energy must fulfill gauge symmetry,

F [∆1(r),∆2(r)] = F [eiϕ∆1(r), eiϕ∆2(r)], (2.38)

and time-reversal symmetry,

F [∆1(r),∆2(r)] = F [∆∗1(r),∆

∗2(r)]. (2.39)

Under these conditions the free energy is given by the following expression [58,59]:

Fh =

∫V

drf(T ) +

∑i

(ai(T )|∆i|2 + bi|∆i|4

)+

g|∆1|2|∆2|2 + 2dRe[∆2

1∆∗22] (2.40)

where f(T ) is the free energy density of the normal state at temperature T , and

V is the volume of the system. The coefficients ai(T ) change their sign at the

transition temperatures Tc,i: ai(T ) > 0 for T > Tc,i, and ai(T ) < 0 for T < Tc,i.

For an orthorhombic lattice and the case dx2−y2 + s, also a term ∝ Re[∆1∆2]

is allowed by symmetry which leads to a finite admixture of a second order pa-

rameter without phase difference between both components; i.e. in one direction

the lobes of the dx2−y2-wave order parameter will become larger and smaller in

the other direction. As the influence of the orthorhombic distortions is small in

high-Tc materials we will not consider this possibility.

The free energy extremum is determined by the Ginzburg-Landau equations

which read as follows

−a1(T )∆1 = 2b1|∆1|2∆1 + g|∆2|2∆1 + 2d∆22∆

∗1, (2.41)

−a2(T )∆2 = 2b2|∆2|2∆2 + g|∆1|2∆2 + 2d∆21∆

∗2. (2.42)

The coefficients can be determined from an expansion of the gap equation (2.22)

with respect to the order parameter. Following App. D we find bi, g, d > 0 for

a dxy-wave as well as for an s-wave admixture; the coefficients ai(T ) show the

above-mentioned behavior.

2.3 Ginzburg-Landau Theory 21

The phase difference between both order parameter components is solely de-

termined by the term dRe[∆21∆

∗22] in the free energy. As d > 0, the free energy

becomes minimal for a phase difference ∆ϕ = ±π/2; for the considered cases,

dx2−y2 + s and dx2−y2 + dxy, this leads to a finite energy gap on the whole Fermi

surface, provided |∆1|2, |∆2|2 > 0.

For further considerations we choose the gauge such that ∆1 = |∆1|, and

∆2 = ±i|∆2|. From the Ginzburg-Landau equation (2.42) we find the following

expression for the subdominant order parameter

|∆2|2 =−a2(T )

2b2− g − 2d

2b2|∆1|2. (2.43)

As (g − 2d) > 0 (compare App. D), the order parameter ∆2 is suppressed by

the finite order parameter ∆1. This means, that the transition to a phase with

|∆1|2, |∆|2 > 0 takes place at a temperature Tc < Tc,2; the transition temperature

is determined by

−a2(Tc) = (g − 2d)|∆1|2. (2.44)

If Tc,2 Tc,1 it can also be that Tc ≤ 0. This means, that the second order

parameter is totally suppressed in the whole temperature range, although an

attractive interaction exists.

For completeness we also provide the gradient contributions to the free energy

for a dxy-wave [58] and an s-wave admixture [59]. One part of the gradient

contributions occurs in both cases,

F i =

∫V

dr∑

i

ki|∂r∆i|2 + kB2, (2.45)

with the gauge invariant derivative ∂r = [∂r − ieA(r)] and the magnetic field

B(r) = curlA(r). The coefficients ki are positive, which leads to a suppression of

spatial variations of the order parameter; the second term represents the magnetic

field energy. The mixed gradient contributions have a different form in both cases,

F ids = 2k

∫V

drRe[(∂y∆1)∗(∂y∆2)− (∂x∆1)

∗(∂x∆2)], (2.46)

22 2 Basic Concepts

and

F idd = 2k

∫V

drRe[(∂x∆1)∗(∂y∆2)− (∂y∆1)

∗(∂x∆2)]. (2.47)

A detailed discussion of inhomogeneous situations can be found e.g. in Refs. [58–

61].

In the case dx2−y2 + idxy it is useful to integrate the term F idd by parts;

neglecting the boundary contributions we find

F idd = −2ke

∫V

drBzIm[∆∗1∆2]. (2.48)

This means that the mixed order parameter yields a local magnetic moment

∝ Im[∆∗1∆2] in z-direction. Conversely, a magnetic field in z-direction supports

a dxy-wave component with a phase-shift.

The consideration of boundaries in the Ginzburg-Landau theory leads to addi-

tional (boundary) terms in the free energy, as the geometrical symmetry is broken

by the boundary. Up to now some phenomenological treatments of boundaries

exist [58, 59] where some aspects of surfaces (e.g. pair-breaking) are taken into

account. On the other hand, the theory of quasi-classical Green’s function is

more suitable for studying, for example, the dependence of physical properties

on the orientation of the order parameter with respect to a boundary.

In summary, we have shown that a possible admixture of a second order pa-

rameter (dxy-, s-wave) can be totally suppressed by the dominant component

(dx2−y2-wave). Thus it might be that the bulk properties of a system are dom-

inated by only one order parameter component in spite of an interaction which

could lead to a second one. Conversely, a subdominant order parameter (which is

suppressed in the bulk) might become finite in regions where the dominant dx2−y2-

wave component is suppressed e.g. by impurities, vortices or surfaces. The latter

situation will be discussed within the quasi-classical framework in Sec. 5.2.

Chapter 3

Green’s Functions and the

Quasi-Classical Approximation

For the evaluation of many physical quantities a detailed knowledge of the wave

functions is not necessary. The knowledge of particular correlation functions –

so-called Green’s functions – suffices. For these Green’s functions a powerful

perturbation theory was developed during the last decades [62, 63].

In this chapter we discuss the application of this standard method to su-

perconductivity. Moreover we will present the theory of quasi-classical Green’s

functions which is an approximation for slow variations in the system. As an ex-

ample, we will reconsider a homogeneous superconductor in thermal equilibrium

within the Green’s function approach.

3.1 Green’s Functions Method in Superconduc-

tivity

In this section we briefly discuss the Green’s functions approach to superconduc-

tivity using the Keldysh technique [45,46,63]. We derive the equation-of-motion

for the Green’s functions in a BCS approximation (compare Sec. 2.1). As an

application of this theory we rediscuss the homogeneous time-independent situa-

tion.

We begin with the definition of the one-particle Green’s functions. Similar to

the mean-field Hamiltonian (2.11) they exhibit a 2× 2 structure. If the electrons

are described by a Hamiltonian of the form H = H +H ′(t) the Green’s functions

24 3 Green’s Functions and the Quasi-Classical Approximation

read

G>(x1, x2) = −i⟨(

Ψ↑(x1)Ψ†↑(x2) Ψ↑(x1)Ψ↓(x2)

−Ψ†↓(x1)Ψ

†↑(x2) −Ψ†

↓(x1)Ψ↓(x2)

)⟩H

(3.1)

and

G<(x1, x2) = i

⟨(Ψ†↑(x2)Ψ↑(x1) Ψ↓(x2)Ψ↑(x1)

−Ψ†↑(x2)Ψ

†↓(x1) −Ψ↓(x2)Ψ

†↓(x1)

)⟩H

. (3.2)

The fermionic field operators Ψ↑/↓(x) are given in the Heisenberg picture with

respect to full Hamiltonian H, and x = (r, t). The average is taken with respect

to H ; i.e. we assume the system to be in thermal equilibrium at some starting

time t0 with H(t < t0) = 0. The 2 × 2 space is called Nambu space which is

represented by a hatˆ.

All important physical observables can directly be calculated via G<11(x1, x2) =

i〈Ψ†↑(x2)Ψ↑(x1)〉 and G>

22(x1, x2) = i〈Ψ†↓(x1)Ψ↓(x2)〉. The charge density is given

by

ρ(x) = e∑σ=↑↓

〈Ψ†σ(x)Ψσ(x)〉 = −ie

[G<

11(x, x) + G>22(x, x)

](3.3)

with the electronic charge e < 0. The current density is given by the usual

quantum mechanical expression

j(x) =ie

2m

∑σ=↑↓

⟨[∂rΨ

†σ(x)

]Ψσ(x)−Ψ†

σ(x)∂rΨσ(x)⟩

=

=e

2m[∂r′ − ∂r − 2ieA(x)]

(G<

11(x, x′) + G>

22(x′, x)

)x′→x

(3.4)

with the gauge invariant derivative ∂r = [∂r− ieA(x)]; ∂r represents the gradient

with respect to r and A(x) is the vector potential.

It is useful to introduce the retarded (GR), the advanced (GA), and the

Keldysh (GK) Green’s function

GR(x1, x2) =[G>(x1, x2)− G<(x1, x2)

]Θ(t1 − t2), (3.5)

GA(x1, x2) = −[G>(x1, x2)− G<(x1, x2)

]Θ(t2 − t1), and (3.6)

GK(x1, x2) = G>(x1, x2) + G<(x1, x2). (3.7)

3.1 Green’s Functions Method in Superconductivity 25

The inverse relations obviously are

G> =1

2(GK + GR − GA), (3.8)

G< =1

2(GK − GR + GA). (3.9)

Following Ref. [46], the equation-of-motion can be written in a compact form by

building the matrix

ˇG =

(GR GK

0 GA

). (3.10)

This 2 × 2 structure is usually called Keldysh space which is denoted by the

reversed hatˇ. Using these definitions the equation-of-motion (also referred to as

Dyson equation) reads[( ˇG0

)−1 − ˇΣ]

ˇG = ˇ1 (3.11)

which is an abbreviation for∫dx3

[( ˇG0

)−1(x1, x3)− ˇ

Σ(x1, x3)]

ˇG(x3, x2) = ˇ1δ(x1 − x2). (3.12)

The free inverse Green’s function( ˇG0

)−1is given by

( ˇG0

)−1(x1, x2) =

ˇτ 3i∂t2 − 1H0(x2)

δ(x1 − x2) (3.13)

with H0 being the single-particle Hamiltonian

H0(x) = − 1

2m(∂r − ieτ3A(x))2 − µ+ U(x). (3.14)

The matrices ˇτ i are the Pauli matrices in Keldysh space, i.e. ˇτ i = 1τi

ˇτ i =

(τi 0

0 τi

), τ1 =

(0 1

1 0

), τ2 =

(0 −ii 0

), τ3 =

(1 0

0 −1

). (3.15)

The interaction and impurities are taken into account by the self-energyˇΣ which


has the same Keldysh structure as the Green’s function

ˇΣ =

(ΣR ΣK

0 ΣA

). (3.16)

It is now important to note that in this formulation the self-energy can be treated

diagrammaticly. The Feynman rules are discussed in detail by Rammer and

Smith [46]. We will apply this scheme to approximate the attractive interaction

which leads to superconductivity.

As in Sec. 2.1, we assume a non-retarded interaction, but now we start from

a representation in real space

V (x1, x2) = V (r2 − r1)δ(t2 − t1). (3.17)

In the BCS approximation only the the Fock contribution is taken into account

ˇΣ(x1, x2) = x1 x2; (3.18)

the double line represents the full Green’s function, and the wavy line the inter-

action. This diagram leads to the following self-energy as discussed in detail in

Refs. [45, 46]

ˇΣ(x1, x2) =

i

21GK(x1, x2)V (x2, x1). (3.19)

As the interaction is not retarded, the self-energy is diagonal in the Keldysh

space; the retarded and the advanced self-energy are identical.

In the following we will only take into account that part of the self-energy

which is responsible for superconductivity. This means that we neglect the di-

agonal part in Nambu space which only leads to a dispensable energy shift in

the dispersion of the particles. For superconductivity the off-diagonal part is of

importance; this leads to the following definition of the order parameter ∆

∆(x1, x2) ≡(

0 ∆(x1, x2)

∆∗(x1, x2) 0

)≡ i

(0 Σ

R/A12 (x1, x2)

ΣR/A21 (x1, x2) 0

). (3.20)

As an example we discuss the homogeneous and time-independent situation. The

3.1 Green’s Functions Method in Superconductivity 27

Green’s function then has the form

ˇG(x1, x2) =

ˇG(x1 − x2), (3.21)

and the equation-of-motion can be solved by a Fourier transformation with re-

spect to the relative space and time variable

ˇG(r, t) =

∫dp

(2π)d

∫dt

2πe−i(Et−pr) ˇ

G(p, t). (3.22)

This leads to the following expression for the equation-of-motion[ˇτ 3E − ˇ1ξp + i1∆(E,p)

]ˇG(E,p) = ˇ1. (3.23)

The solutions for the retarded and advanced Green’s functions are given by

GR/A(E,p) =1ξp + τ3(E ± iγ) + i∆(pF )

(E ± iγ)2 − [ξ2p + |∆(pF )|2] . (3.24)

To keep the causality of the retarded and advanced Green’s functions, the analytic

properties of the energy dependent Green’s functions must be fixed by E → E±iγwith γ → 0+. In real systems further influence of interaction and impurities,

which is neglected here, can lead to a finite value of γ.

As discussed in detail in App. A, in a thermal equilibrium situation (i.e. the

occupation of the states is determined by the Fermi distribution) the Keldysh

Green’s function is given by

GK(E,p) = tanh

(E

2T

)[GR(E,p)− GA(E,p)

]. (3.25)

This enables us to study thermal properties of the system.

As already mentioned in Sec. 2.1, near the Fermi surface the order parameter

only depends on the direction of the momentum which is represented by the ori-

entation of the Fermi momentum pF . Following App. C, in the current situation

the gap equation (3.19) reads

∆(pF ) = −1

2N0

Ec∫−Ec

dE

2π

⟨V (pF ,p

′F )

∫dξp′GK(E,p′)

⟩p′

F

. (3.26)

For further calculations we evaluate the difference of the retarded and the ad-


vanced Green’s function which reads (for convenience we use the abbreviation

∆ = ∆(pF ))

GR(E,p)− GA(E,p) =

= −iπ(

1ξpE

+ τ3 +i∆

E

)[δ(E −

√ξ2p + |∆|2

)+ δ

(E +

√ξ2p + |∆|2

)]

= −iπ(

1sgn

(E

ξp

)+ τ3

∣∣∣∣Eξp∣∣∣∣ + i∆sgn(E)

|ξp|

)×

×[δ(ξp −

√E2 − |∆|2

)+ δ

(ξp +

√E2 − |∆|2

)]Θ(E2 − |∆|2).

(3.27)

This expression is also needed for the evaluation of the spectral function which

is given by its diagonal part

A(E,p) =i

4πTr[τ3

(GR − GA

)]=

=1

2

(1 +

ξpE

)[δ(E −

√ξ2p + |∆|2

)+ δ

(E +

√ξ2p + |∆|2

)].

(3.28)

Note that the prefactors of the δ-functions are the particle and hole amplitudes

|u(p)|2 and |v(p)|2 which already occurred in our treatment of the BCS theory

in Sec. 2.1.

The density of states which is important for the evaluation of many measurable

quantities is given via the spectral function

N (E) =

∫dp

(2π)dA(E,p). (3.29)

We use Eq. (2.21) which leads to the relation

∞∫−µ

dξp[GR(E,p)− GA(E,p)] =

= −i2π τ3|E|+ isgn(E)∆(pF )√E2 − |∆(pF )|2

Θ(E2 − |∆(pF )|2)

(3.30)

for E µ ≈ EF . The density of states then reads

N (E) = N0

⟨|E|Θ(E2 − |∆(pF )|2)√

E2 − |∆(pF )|2

⟩pF

. (3.31)

3.2 Quasi-Classical Approximation 29

If the order parameter has a non-trivial angle dependence it is reasonable to

define the angle-resolved density of states

N (E,pF ) = N0|E|Θ(E2 − |∆(pF )|2)√

E2 − |∆(pF )|2. (3.32)

Using Eqs. (3.25) and (3.30), we can finally present the gap equation (3.26) in a

homogeneous situation

∆(pF ) = −N0

Ec∫0

dE tanh

(E

2T

)⟨V (pF ,p

′F )

∆(p′F )Θ(E2 − |∆(p′F )|2)√E2 − |∆(p′F )|2

⟩p′

F

(3.33)

which of course is identical to Eq. (2.22).

3.2 Quasi-Classical Approximation

A general treatment of spatially and temporally varying situations is quite a

difficult task. We therefore only consider slow variations. We use the quasi-

classical approximation which allows us to drop the ξp-dependence of the Green’s

function. Finally we present an equation-of-motion for the ξp-integrated Green’s

functions, which suffice for the evaluation of many physical quantities such as the

current density or the order parameter.

To treat space and time dependent situations it is convenient to switch to

center-of-mass and relative coordinates

t =1

2(t1 + t2), r =

1

2(r1 + r2) (3.34)

t′ = t1 − t2, r′ = r1 − r2, (3.35)

and perform the Fourier transformation with respect to the relative coordinates

to reach a mixed representation of any quantity A(x1, x2)

A(E,p; t, r) =

∫dr′∫

dt′ e−i(pr′−Et′)A

(r +

r′

2, t+

t′

2; r− r′

2, t− t′

2

). (3.36)

The variable transformation (t1, r1, t2, r2) → (E,p; t, r) of the convolutions which


occur in the Dyson equation (3.12) reads [46, 64]

(AB)(E,p; t, r) = A •B ≡ei(∂A

r ∂Bp −∂A

p ∂Br )/2e−i(∂A

t ∂BE−∂A

E∂Bt )/2A(E,p; t, r)B(E,p; t, r).

(3.37)

A derivation of this equation for the bullet product A • B is given in App. B.

The transformation of the derivatives with respect to the relative space and time

variables is straightforward: ∂r′ → ip and ∂t′ → −iE. Having this in mind we

find

∂r1A→(ip +

1

2∂r

)A = ip • A, (3.38)

∂2r1A→

(−p2 + ip∂r +

1

4∂2r

)A = −p2 •A, (3.39)

∂t1A→(−iE +

1

2∂t

)A = −iE • A. (3.40)

For the gauge invariant derivative it follows

1

2m

[1∂r1 − ieτ3A(x1)

]2G→ − 1

2m

[1p− eτ3A(t, r)

]2 • G. (3.41)

Then, using the bullet product, the Dyson equation in the mixed representation

is given by

ˇτ 3E −

1

2m1[1p− eτ3A(t, r)

]2+ µ− ˇ1U(r, t)−

− ˇΣ(E,p; t, r)

• ˇG(E,p; t, r) = ˇ1.

(3.42)

As we are only interested in situations with weak spatial and temporal dependence

of physical quantities, this expression is a good starting point for an expansion

in the gradient terms. In superconductors typical spatial variations exist on the

scale of the (zero temperature) coherence length ξ0 (∂r . 1/ξ0), which usually is

larger than the Fermi wave length 1/pF ; moreover we will focus on weak external

perturbations (A, U) varying only on the scale of ξ0 or larger. We therefore only

take into account the leading gradient term p∂R ∼ pF/ξ0. The Dyson equation

then readsˇτ 3E − 1[1p− eτ3A(t, r)]2 + µ+ i

p

2mˇ1∂r − ˇ1U(r, t)−

− ˇΣ(E,p; t, r)

ˇG(E,p; t, r) = ˇ1,

(3.43)

3.2 Quasi-Classical Approximation 31

where we introduced the circle product

A B ≡ e−i(∂At ∂B

E−∂AE∂B

t )/2A(E,p; t, r)B(E,p; t, r). (3.44)

As a consequence the momentum p only occurs as a parameter in the equation-

of-motion.

Interchanging the E- and the p-integral and using Eq. (2.21), we note that

many observables (e.g. the density of states or the order parameter) only depend

on the ξp-integrated Green’s function

ˇg(E,pF ; t, r) =i

πP∫

dξpˇG(E,p; t, r) (3.45)

where the angle-dependence of the quasi-classical Green’s function is given by

pF . AsˇG ∝ 1/ξp for large ξp, the integral must be taken in a principal value

sense:

P∫

dξp · · · ≡ limx→∞

x∫−x

dξp . . . . (3.46)

In a further step we will present an equation-of-motion for ˇg, where the ξp-

dependence is integrated out.

As P∫

dξpξpˇG does not exist, Eq. (3.43) cannot be integrated with respect

to ξp to find an equation which determines ˇg. To circumvent this difficulty we

consider the difference of the Dyson equation and its adjoint equation, with the

result(ˇG−1

0 − ˇΣ

)ˇG− ˇ

G

(ˇG−1

0 − ˇΣ

)= 0. (3.47)

The anomalous derivatives are defined asˇG∂r = −∂r

ˇG, which can be related to

partial integration. In the quasi-classical approximation the strong momentum

dependent part ξpˇG cancels, and we arrive at

[ˇτ 3E +

p

m1( i

21∂r + eτ3A(t, r)

)− ˇ1U(t, r)−

− ˇΣ(E,p; t, r) ,

ˇG(E,p; t, r)

]= 0.

(3.48)

Note that in the commutator the circle product must be used. As the super-


conducting self-energy only depends on the direction of the relative momentum

p and not on its absolute value (see App. C), we are now able to perform the

principal value integration of this equation and we find an equation-of-motion for

the quasi-classical Green’s function

[ˇτ 3E + vF 1

( i21∂r+eτ3A(t, r)

)− ˇ1U(t, r)+

+ i1∆(pF ; t, r) , ˇg(E,pF ; t, r)]

= 0(3.49)

with the Fermi velocity vF = pF/m. This equation is usually referred to as

Eilenberger equation [44].

It is important to note that along classical trajectories, r = r0 + λvF/vF ,

the solution of the Eilenberger equation is determined by an ordinary differential

equation as vF∂r → vF∂λ. Physically speaking, this means that quantum me-

chanical coherence only exists along these trajectories, and neighboring trajecto-

ries are independent. This is the main advantage of the quasi-classical approach

compared to finding the microscopic Green’s functions.

Since the Eilenberger equation is homogeneous and linear, and therefore can-

not determine ˇg completely, an additional condition has to be fulfilled [44]. This

condition must be in agreement with two observations: (i) As the circle product

is associative, if ˇg solves the Eilenberger equation also ˇg ˇg is a solution; because

of the uniqueness of the physical solution this product must be a trivial solution

(i.e. ˇg ˇg = const.). (ii) As can be seen in Eqs. (3.24) and (3.25), in a spatially

homogeneous state which is in thermal equilibrium (then the circle product turns

to the usual product) we find ˇgˇg = ˇ1. As a consequence the physical relevant

solution of the Eilenberger equation is determined by the normalization condition

ˇg ˇg = ˇ1. (3.50)

A comprehensive discussion of this problem can be found in Refs. [44–46]. The

whole procedure presented above is referred to as quasi-classical approximation.

The order parameter ∆ must be determined self-consistently via Eqs. (3.19)

and (3.20). Following App. C we find the gap equation

∆(pF ; t, r) =i

4N0

Ec∫−Ec

dE⟨V (pF ,p

′F )gK(E,p′F ; t, r)

⟩p′

F. (3.51)

3.3 Superconductors in Thermal Equilibrium 33

Other observables can also be expressed via the ξp-integrated Green’s function.

The angle-resolved density of states, for example, is given by

N (E,pF ; t, r) =1

2N0(g

R − gA)11 =1

2N0Re

Tr[τ3g

R]. (3.52)

The evaluation of the charge density ρ and current density j in terms of the

quasi-classical Green’s function is a more subtle issue, as high energy contribu-

tions are also important [46, 64]. We therefore concentrate on the deviations of

the observables from their normal state equilibrium values, so that the trouble-

some contributions cancel; throughout this work we can assume the normal state

equilibrium values to be ρ = const. and j = 0. Using Eq. (3.3) we find the charge

density

ρ(t, r) = −2eN0

1

8

∫dETr

[⟨gK(E,pF ; t, r)

⟩pF

]+ U(t, r)

, (3.53)

where we added the potential term, as it is not captured by the ξp-integrated

Green’s function. With Eq. (3.4), the current density reads

j(t, r) = −1

4eN0

∫dETr

[τ3⟨vF g

K(E,pF ; t, r)⟩pF

]. (3.54)

We have briefly introduced the quasi-classical approximation and will now con-

sider superconductors in thermal equilibrium, on which we focus our attention

during this work.

3.3 Superconductors in Thermal Equilibrium

The solution of a thermal equilibrium situation becomes much simpler, since the

circle product simplifies to the usual product. Moreover, the Keldysh Green’s

function can directly be expressed via the retarded and advanced Green’s func-

tions (see Eq. (3.25))

gK = tanh

(E

2T

)(gR − gA

). (3.55)

For the retarded and advanced Green’s function the Eilenberger equation reads[τ3(E ± iγ) + vF

(i

2∂r + eτ3A(r)

)+ i∆(pF , r), g

R/A(E,pF ; r)

]= 0. (3.56)


Using the normalization condition gRgR = gAgA = 1 the solution for the homo-

geneous case is given by

gR/A(E,pF ) =τ3E + i∆(pF )√

(E ± iγ)2 − |∆(pF )|2(3.57)

where the square root of a complex number z 6∈ R+ is defined by Im√z > 0. The

Keldysh Green’s function reads

gK(E,pF ) = 2τ3|E|+ isgn(E)∆(pF )√

E2 − |∆(pF )|2Θ(E2 − |∆(pF )|2) tanh

(E

2T

)(3.58)

which can also be seen from Eq. (3.30). In thermal equilibrium the advanced

Green’s function is related to the retarded Green’s function via

gA = τ3(gR)†τ3, (3.59)

which can be proved using Eq. (3.56).

For the evaluation of thermodynamic properties we can also use the Mat-

subara technique. As discussed in detail in [62], an analytic continuation of the

retarded or advanced Green’s function with respect to the energy argument has

to be carried out

iEn ↔ E ± iγ. (3.60)

This means that the equation-of-motion must be evaluated at imaginary energies

to get the Matsubara Green’s function gM :

[iτ3En + vF

(i

21∂r + eτ3A(r)

)− 1U(r) + i∆(pF ; r), gM(En,pF ; r)

]= 0.

(3.61)

In addition the normalization condition, (gM)2 = 1, must be fulfilled. This leads

to the solution in a homogeneous situation

gM(En,pF ) =τ3En + ∆(pF )√E2

n + |∆(pF )|2. (3.62)

In the Matsubara approach the integrals containing the distribution function

3.3 Superconductors in Thermal Equilibrium 35

change to sums over the Matsubara energies En = Tπ(2n+ 1) according to

∫dEgK(E,pF ; r) = 4πiT

∞∑n=−∞

gM(En,pF ; r). (3.63)

In particular the self-consistency equation for the order parameter and the current

density assume the following forms:

∆(pF , r) = −πN0T∑

|En|<Ec

⟨V (pF ,p

′F )gM(En,p

′F ; r)

⟩p′

F

, (3.64)

j(r) = −ieπN0T

∞∑n=−∞

Tr[τ3⟨vF g

M(En,pF ; r)⟩pF

]. (3.65)

In summary, we formulated the general theory considering the thermal equilib-

rium situation in particular.

The quasi-classical theory is not directly applicable at boundaries since here

spatial variations of physical quantities exist on microscopic scale. In the quasi-

classical framework interfaces must therefore be incorporated by boundary condi-

tions for the Green’s functions. The boundary conditions are discussed in detail

in the next chapter.

Chapter 4

Boundary Conditions for the

Quasi-Classical Green’s Functions

In the previous chapter we introduced the quasi-classical theory to describe su-

perconductors with slow external perturbations (on the scale of the coherence

length). This approximation is not directly applicable in the vicinity of surfaces

or interfaces, which occur e.g. in Josephson junctions or normal-metal-insulator-

superconductor (NIS) junctions. It is possible however, to treat surfaces and

interfaces by boundary conditions for the quasi-classical Green’s functions, which

is a non-trivial result first found by Zaitsev [52].

In this chapter we will first discuss Zaitsev’s boundary conditions [52] for a

specular reflecting surface and an ideal interface. A generalization, which we

will use later, was recently devised by Ozana and Shelankov [54]. We then con-

sider some simple boundary effects within this framework neglecting the self-

consistency of the order parameter.

4.1 Zaitsev’s Boundary Conditions

The simplest case to consider is a plane surface where the quasi-particles are

scattered specularly, i.e. the momentum parallel to the surface is conserved

pF in,y = pFout,y. (4.1)

38 4 Boundary Conditions for the Quasi-Classical Green’s Functions

px

pF

I SC

y

pFout

pF in

x

pF in pFout

py

#

Figure 4.1: In the case of ideal specular scattering at a surface the parallel mo-mentum is conserved, which is illustrated on the left hand side of the figure.The d-wave order parameter with orientation α presented in momentum spacesymbolizes the superconductor; the in- and the out-direction are shown as well.

Here a classical trajectory is given by

r(λ) = (0, y0) +1

vF

vF inλ λ < 0

vFoutλ λ > 0, (4.2)

with vF in/out = vF (∓ cosϑ, sinϑ); the trajectory meets the surface for λ = 0.

This situation is illustrated in Fig. 4.1 for an order parameter which is tilted

with respect to the surface by an angle α; i.e. for a tilted order parameter its

angle-dependent part given in table 2.2 must be modified, ηi(ϑ) → ηi(ϑ− α).

At the surface the effective boundary conditions require the continuity of the

Green’s functions

ˇg(E,pF in; t, 0) = ˇg(E,pFout; t, 0). (4.3)

As we assume translational invariance in y-direction we drop the y dependence

of the Green’s functions. With the boundary conditions the task is well defined:

One has to find a continuous solution of the Eilenberger equation (3.49) on a

classical trajectory as given by Eq. (4.2); for λ = ±∞ the the homogeneous

solutions of the Green’s functions can be assumed as initial values.

Using the boundary condition one can see that surfaces act pair-breaking on

anisotropic superconductors if the order parameter differs for pF in and pFout:

Assume for example the situation of a d-wave order parameter that is tilted by

4.1 Zaitsev’s Boundary Conditions 39

SC

prF in

plF in

prF in

plFout

I SC

x

y

py py

pxpx

prFout

plF in

plFout

prFout

l

#

r

Figure 4.2: Four trajectories with the same parallel momentum are involved in thescattering process at an ideal interface. The interface separates two superconduc-tors on the left and right side which are represented by d-wave order parameters(in momentum space) with orientation αl/r.

α = 45 with respect to the surface normal:

∆(pF ; t, r) = ∆(t, r) cos[2(ϑ− π/4)]. (4.4)

The self-consistency equation which determines the order parameter at the surface

can now be written as follows:

∆(t, 0) =1

4iN0V

Ec∫−Ec

dE

π∫−π

dϑ′

2πcos[2(ϑ′ − π/4)]gK(E, ϑ′; t, 0). (4.5)

Due to the boundary conditions, the Green’s functions on the in- and out-

trajectories are identical at the surface, i.e. gK(E, ϑ′; t, 0) = gK(E, π − ϑ′; t, 0).

As cos[2(ϑ′ − π/4)] = − cos[2(π − ϑ′ − π/4)] the Fermi surface average is zero

and so is the order parameter at the surface, ∆(t, 0) = 0. We demonstrated

the pair-breaking of surfaces for a special case, but we will see later that the

order parameter generally is suppressed if the scattering connects directions with

different values of ∆(pF ). A related effect is the pair-breaking of anisotropic

superconductors due to non-magnetic impurities [65,66], which does not occur in

isotropic s-wave superconductors (Anderson’s theorem).

We now treat an ideal interface between two superconductors (Fig 4.2); here

we must take into account two in- and two out-trajectories all having the same


parallel momentum

plF in,y = pl

Fout,y = prF in,y = pr

Fout,y. (4.6)

The index l/r indicates the left or right side of the junction. The coupling of both

sides is characterized by the transparency T (or the reflectivity R = 1−T ) of the

junction, which in principle can depend on the direction of incidence. Expressed

by the Green’s functions, the effective boundary conditions can be written in a

suitable form as follows [67]:

ˇgr

a + ˇgl

a = 0, (4.7)

ˇgr

a ˇgr

s ˇgr

s = −1−R1 +R

[ˇg

l

s, ˇg

r

s (

ˇ1− 1

2ˇg

r

a

)](4.8)

with the symmetric and antisymmetric combination of the Green’s functions

ˇgl/r

s (E,pl/rF in; t, 0) = ˇg

l/r(E,p

l/rFout; t, 0) + ˇg

l/r(E,p

l/rF in; t, 0), (4.9)

ˇgl/r

a (E,pl/rF in; t, 0) = ˇg

l/r(E,p

l/rFout; t, 0)− ˇg

l/r(E,p

l/rF in; t, 0). (4.10)

For a specular surface (T = 0) or for a totally transparent interface (T = 1) the

boundary conditions are met by simple continuity conditions

T = 0 → ˇgl/r

(E,pl/rF in; t, 0) = ˇg

l/r(E,p

l/rFout; t, 0), (4.11)

T = 1 → ˇgl/r

(E,pl/rF in; t, 0) = ˇg

r/l(E,p

r/lFout; t, 0). (4.12)

It should be mentioned that Zaitsev’s boundary conditions can also be applied

to describe non-equilibrium situations. Recently Zaitsev’s boundary conditions

were rewritten in a more convenient form for the actual solution of boundary

problems [68]; we will present this formulation for a situation in thermal equilib-

rium in Sec. 4.3 as a particular case of the even more general boundary conditions

developed by Ozana and Shelankov [54].

4.2 Ideal Tunnel Junctions

For arbitrary transmission probability T the boundary conditions are rather dif-

ficult to solve, even numerically. On the other hand this formulation is very

convenient for treating the tunneling limit (T 1), which is of relevance in

4.2 Ideal Tunnel Junctions 41

many experimental setups. We are then able to evaluate the tunnel current to

first order in the transparency.

Expanding the boundary conditions given in Eqs. (4.7) and (4.8) to first order

in T , we find the antisymmetric Green’s function

ˇgr

a,1 = −ˇgl

a,1 = −T8

[ˇg

l

s,0, ˇg

r

s,0

], (4.13)

where ˇgl/r

s,0 are evaluated for two uncoupled superconductors (T = 0). Now the

current density in x-direction (across the interface) to first order in T is deter-

mined by

jx(t, 0) =− eN l0

4

∫dE⟨vl

F,xTr[τ3g

Kla,1(E,p

lF in; t, 0)

]⟩pl

F in

=

=− eN l0

4

∫dE⟨vl

F,xTr[τ3[gRl

s,0(E,plF in; t, 0) , gKr

s,0 (E,prF in; t, 0)

]+

+[gKl

s,0(E,plF in; t, 0) , gAr

s,0(E,prF in; t, 0)

] ]⟩pl

F in

;

(4.14)

i.e. the current density can be expressed by the Green’s functions of two uncou-

pled superconductors.

We consider a different potential on the left and right hand side of the junction

V l/r and a different phase of the order parameter ϕl/r; both sides are assumed

to be in thermal equilibrium. The retarded, advanced, and Keldysh Green’s

functions on the left and right hand side are then given via a gauge transformation

as discussed in App. E

gl/r0 (E,pF ; t, r) =

(g

l/r0 (E − eV l/r,pF ; r) f

l/r0 (E,pF ; r)e−i2χl/r(t)

fl/r0 (E,pF ; r)ei2χl/r(t) g0(E + eV l/r,pF ; r)

)(4.15)

with χl/r(t) = −ϕl/r/2 + eV l/rt; the Green’s function gl/r0 (E,pF ; r) is the time-

independent solution of the thermal equilibrium Eilenberger equation (3.56).

Now we can use Eqs. (3.55) and (3.59) to express the current density solely

in terms of the retarded Green’s function

gRl/r0 (E,pF ; 0) =

(g

Rl/r0 (E,pF ) f

Rl/r0 (E,pF )

fRl/r0 (E,pF ) g

Rl/r0 (E,pF )

). (4.16)

With the phase difference of the order parameter ϕ = ϕl − ϕr and the potential

difference V = V l − V r at the boundary the resulting tunnel current takes the


form [69]

jx(V, t) = j0(V ) + j1(V ) sin(ϕ+ 2eV t) + j2(V ) cos(ϕ+ 2eV t). (4.17)

with

ji(V ) = −eNl0

2

⟨vl

F,x(plF in)T (pl

F in)ji(V,plF in)⟩pl

F in

; (4.18)

T (plF in) is the angle-dependent transparency of the interface.

The quasi-particle contribution j0 is determined by

j0(V,prF in) =

∫dE

[tanh

(E

2T

)− tanh

(E + eV

2T

)]×

Re[gRl0 (E,pl

F in)]Re[gRr0 (E + eV,pr

Fout)];

(4.19)

here one should keep in mind that prFout is determined uniquely by pl

F in. Defining

the term

gR(E, V,plF in) =

1

2fRl

0 (E − eV,plF in)

[fRr

0 (E,prFout)− (fRr

0 (E,prFout))

∗]+1

2fRr

0 (E + eV,prFout)

[fRl

0 (E,plF in)− (fRl

0 (E,plF in))

∗] (4.20)

the current density contributions due to Cooper-pair tunneling are given by

j1(V,plF in) = −

∫dE tanh

(E

2T

)Im[gR(E, V,pl

F in)], (4.21)

j2(V,plF in) =

∫dE tanh

(E

2T

)Re[gR(E, V,pl

F in)]. (4.22)

We will first focus on a tunnel junction of two normal metals. In this case only

the current density j0 is finite; a quantity measured in many experiments is the

differential conductance of a junction

G(T, V ) =dI(T, V )

dV= A

dj0(T, V )

dV(4.23)

where A is the junction area. With the normal state Green’s function on the left

and right hand side of the junction, gRr/l0 (E,pF ) = 1, we recover Ohm’s law

G(T, V ) = R−1N =

⟨Ae2N l

0vlF,x(p

lF in)T (pl

F in)⟩pl

F in

(4.24)

4.2 Ideal Tunnel Junctions 43

with the voltage- and temperature-independent normal state resistance RN .

Next we consider an NIS junction. As before only j0 is finite. After inserting

the normal state Green’s function on the right hand side gRr0 (E,pF ) = 1 we find

G(T, V ) =

∫dE

⟨Ae2vl

F,x(plF in)T (pl

F in)N l(E,plF in)⟩pl

F in

4T cosh2(E + eV/2T ). (4.25)

For low temperatures T → 0 this simplifies to

G(V ) =⟨Ae2vl

F,x(plF in)T (pl

F in)N l(eV,plF in)⟩pl

F in

. (4.26)

The differential conductance of a NIS junction for low temperatures therefore

provides important information on the superconducting density of states at the

surface.

At this stage an important difference between isotropic and anisotropic super-

conductors occurs. The Green’s function (and in particular the density of states)

for an s-wave order parameter is itself isotropic; the product RNG(T, V ) is there-

fore independent of the transparency T (pF ). This is not the case for anisotropic

superconductors and the angular dependence of the transparency is a relevant

ingredient.

Throughout this work we will use the angle-dependent transparency which is

related to a δ-like interface barrier as derived in Ref. [70]. The resulting trans-

parency reads

T (ϑ) =T0 cos2 ϑ

1− T0 sin2 ϑ, T0 ∈ [0, 1] (4.27)

which yields the normal state resistance via Eq. (4.24)

R−1N = Ae2N l

0vlF

2

π

(1− (1− T0)Artanh(

√T0)√

T0

)T0→0→ 4Ae2N l

0vlF

3πT0. (4.28)

Finally, for a contact of two superconductors with finite phase-difference a super-

current occurs (Cooper-pair tunneling). For V = 0, and if the order param-

eters on both sides of the junction can be chosen real, only the contribution

j1 sinϕ exists and j1 is the critical current density of the junction. For spatially

homogeneous order parameters on both sides and a symmetric geometry, i.e.


∆l(plF in) = ∆r(pr

Fout), the critical current density j1 can be evaluated

j1 =π|e|N l

0

2

⟨vl

F,x(plF in)T (pl

F in)∆l(pl

F in) tanh

(∆l(pl

F in)

2T

)⟩pl

F in

. (4.29)

This equation can be applied for an s-wave or an untilted d-wave order parameter

on both sides of the junction, as in these cases no pair-breaking occurs. For

s-wave superconductors with the self-consistent (temperature-dependent) order

parameter ∆l(plF in) = ∆l this leads to the critical current

Ic(T ) = Aj1(T ) =π∆l

2|e|RNtanh

(∆l

2T

)T→0→ π∆0

2|e|RN= 1.57

∆0

|e|RN(4.30)

with the BCS value ∆0 = 1.76Tc (see table 2.2). In the case of an untilted d-wave

superconductor (αl/r = 0), we find for T → 0

Ic(0) =3(3√

2− 2)

10

π∆0

2|e|RN= 1.06

∆0

|e|RN(4.31)

with the self-consistent order parameter ∆0 = 2.15Tc (see table 2.2). In contrast

to the s-wave case the numerical prefactor depends on the angular dependence

of the transparency. It is important to realize that Eq. (4.29) is not applicable

for tilted d-wave order parameters, as the pair-breaking effect of surfaces is not

taken into account.

In summary, we saw that Zaitsev’s boundary conditions are suitable for treat-

ing tunnel junctions to first order in the transparency. Otherwise, if higher order

terms are relevant (e.g. if the first order contribution vanishes or for highly trans-

parent boundaries) it is very difficult to use this formulation of the boundary

conditions.

In the next section we will introduce different boundary conditions which

can be applied to a wider range of physical situations; they allow us to study

arbitrarily rough interfaces. Also the treatment of finite transparencies is easier

than in the formulation presented in this section.

4.3 Boundary Conditions according to Shelankov and Ozana 45

4.3 Boundary Conditions according to She-

lankov and Ozana

We will present the boundary conditions as devised by Shelankov and Ozana [54]

for systems in thermal equilibrium. We will not review the derivation in detail,

but will only give a short description which is sufficient for the further understand-

ing of our calculations. The main idea is to formulate the boundary conditions

for Andreev’s wave equation in terms of a scattering matrix, and translate them

afterwards into the language of quasi-classical Green’s functions.

We start, following Ref. [54], by describing the superconducting state in terms

of the Andreev equation [71]

[ivF∂r +

(E + evFA i∆(pF , r)

i∆∗(pF , r) −E − evFA

)]ψ = 0, ψ =

(u(E,pF ; r)

v(E,pF ; r)

), (4.32)

which is an approximation of the Bogoliubov-de Gennes equation [72] in the

quasi-classical limit (1/pF ξ0). The solution of the Andreev equation, ψ,

describes the slowly varying part (on scales larger than ξ0) of the solution of the

Bogoliubov-de Gennes equation.

For arbitrary boundaries with microscopic roughness the scattering is no

longer ideal as supposed in the previous section and a quasi-particle (wave pack-

age) can be scattered in various directions. In the framework of Andreev’s

wave equation this can be described by a scattering matrix (S-matrix) approach

which is a standard method of formulating boundary conditions for linear wave

equations [73]. For simplicity, we consider only a finite number of directions

pl/rF in/out → p

l/r,iF in/out, i = 1, 2 . . . , n. The boundary conditions read

(ψl,k

out

ψr,kout

)=

n∑k′=1

(Sll

kk′ Slrkk′

Srlkk′ Srr

kk′

)(ψl,k′

in

ψr,k′in

)(4.33)

with (ψl,k

in/out

ψr,kin/out

)=

(ψl(E,pl,k

F in/out, 0)

ψr(E,pr,kF in/out, 0)

)(4.34)

being the Andreev amplitudes at the surface. The (2n× 2n) S-matrix consists of


SC I SC

x

y

py py

pxpx

plF in

prF in

prFout

plFout

plF in

plFout

prFout

prF in

r

l

Figure 4.3: Scattering at a microscopically rough interface where trajectories witharbitrary direction are connected coherently.

four n× n blocks

S =

(Sll Slr

Srl Srr

); (4.35)

these blocks describe the reflection on the left and right side of the junction (Sll

and Srr), and the transmission from the left to the right side and vice versa (Srl

and Slr). To ensure current conservation at the scattering center, the S-matrix

must be unitary

SS† = 1. (4.36)

In this approach several quasi-particle trajectories are connected coherently via

an S-matrix, which is determined by the microscopic structure of the interface. A

rough interface, for example, can lead to a situation as illustrated in Fig. 4.3. In

principle, the S-matrix can spatially vary on a scale larger than the Fermi wave

length. It can be calculated from the microscopic properties of the interface, or

it can be used as a phenomenological input.

To find a translation procedure between the Andreev and the Eilenberger

picture we define the amplitudes

a(E,pF ; r) =u+(E,pF ; r)

v+(E,pF ; r), (4.37)

b(E,pF ; r) =v−(E,pF ; r)

u−(E,pF ; r). (4.38)

Here the solutions Ψ± = (u±, v±) must fulfill different initial conditions on a


classical trajectory, r = r0 + λvF/vF ,

Ψ±(E,pF ;λ→ ±∞) → 0. (4.39)

With the Andreev equation (4.32) the equation-of-motion for a and b yield

vF∂ra = a2∆∗(pF , r) + i2a(E + evFA)−∆(pF , r) (4.40)

vF∂rb = b2∆(pF , r)− i2b(E + evFA)−∆∗(pF , r). (4.41)

These equations are also referred to as Riccati equations.

On the other hand, we can use a particular parameterization of g

g =1

1− ab

(1 + ab −2a

2b −(1 + ab)

)(4.42)

which was first suggested by Maki and Schopohl [74] in the framework of quasi-

classical Green’s functions (details are given in Ref. [75]). The normalization con-

dition (3.50) is fulfilled by construction. Using the Eilenberger equation (3.49)

the equation of motion for a and b can be derived and we recover the Riccati equa-

tions (4.40) and (4.41). To sum up, the translation procedure between the An-

dreev and Eilenberger language via the amplitudes a and b is given by Eqs. (4.37)

and (4.38) on the one side and by Eq. (4.42) on the other side.

Analogous with the Eilenberger and the Andreev equation, the Riccati equa-

tions can be integrated along each classical trajectory, r = r0 + λvF/vF . For

Im[E] > 0 (i.e. for gR or gM with En > 0) the integration of a in vF -direction

and of b in −vF -direction is (numerically) stable; for Im[E] < 0 the direction of

integration must be reversed.

Assuming a homogeneous solution in the bulk of the superconductor

(λ = ±∞) the initial values for the integration of the Riccati equations (4.40)

and (4.41) are given by

a(E,pF ;λ→ −∞) =−i∆−(pF )

E +√E2 − |∆−(pF )|2

, (4.43)

b(E,pF ;λ→ +∞) =i∆∗

+(pF )

E +√E2 − |∆+(pF )|2

(4.44)

where ∆± are the order parameter values for λ = ±∞. If ∆(pF , r) = ∆(pF )

the functions a and b are are constant and given by their initial values (4.43)


and (4.44).

This procedure can be applied to boundaries (see Fig. 4.3): Far away from the

interface the superconductor is homogeneous and therefore we know the initial

values which are given by Eqs. (4.43) and (4.44). Then the Riccati equations

can be integrated towards the boundary, i.e. we know al/r(E,pl/rF in; x) on the

in-trajectories and bl/r(E,pl/rFout; x) on the out-trajectories. To perform the in-

tegration beyond the interface, the boundary conditions must be applied to get

the initial conditions at the boundary al/r(E,pl/rFout; 0) and bl/r(E,p

l/rF in; 0). A fur-

ther integration provides the missing al/r(E,pl/rFout; x) on the out-trajectories and

bl/r(E,pl/rF in; x) on the in-trajectories.

Using the translation procedure between the Andreev approach and the quasi-

classical Green’s functions the boundary conditions (4.33) can be written in terms

of the amplitudes a and b. Following Ref. [54], we define the functions

Ali(β) = det

[1− S

(al 0

0 ar

)S†

(bli(β) 0

0 br

)], (4.45)

Ari (β) = det

[1− S

(al 0

0 ar

)S†

(bl 0

0 bri (β)

)], (4.46)

Bli(α) = det

[1− S

(al

i(α) 0

0 ar

)S†

(bl 0

0 br

)], (4.47)

Bri (α) = det

[1− S

(al 0

0 ari (α)

)S†

(bl 0

0 br

)]. (4.48)

Here we used the diagonal n× n matrices

al/r = diaga

l/r,1in , . . . , a

l/r,nin

, (4.49)

al/ri (α) = diag

a

l/r,1in , . . . , a

l/r,i−1in , α, a

l/r,i+1in , . . . , a

l/r,nin

, (4.50)

bl/r = diagbl/r,1out , . . . , b

l/r,nout

, (4.51)

bl/ri (β) = diag

bl/r,1out , . . . , b

l/r,i−1out , β, b

l/r,i+1out , . . . , b

l/r,nout

(4.52)

where the matrix elements are given by the amplitudes a and b at the boundary

al/r,iin ≡ al/r(E,p

l/r,iF in ; 0), b

l/r,iout ≡ bl/r(E,p

l/r,iFout; 0). (4.53)

From Eq. (4.33) it follows that the boundary conditions for the a’s and b’s are


given by the zeros of the functions Al/ri (β) and B

l/ri (α) in the following way:

Al/ri (β0) = 0 ⇒ a

l/r,iout ≡ al/r(E,p

l/r,iFout, 0) =

1

β0

, (4.54)

Bl/ri (α0) = 0 ⇒ b

l/r,iin ≡ bl/r(E,p

l/r,iF in , 0) =

1

α0. (4.55)

As the determinant is a linear function of each of the matrix elements, the func-

tions Al/ri (β) and B

l/ri (α) are linear in β and α. An explicit solution of the

boundary conditions can therefore be found by calculating Al/ri (β) and B

l/ri (α)

for two arbitrary values of β and α; if we choose α = 0, 1 and β = 0, 1 we get

al/r,iout = 1− A

l/ri (1)

Al/ri (0)

, (4.56)

bl/r,iin = 1− B

l/ri (1)

Bl/ri (0)

(4.57)

as the final form of the boundary conditions. It is worth noting that here, in

contrast to Zaitsev’s formulation of the boundary conditions, the unknown quan-

tities at the interface, al/r,iout and b

l/r,iin , are given explicitly by the quantities a

l/r,iin

and bl/r,iout that are known.

To apply these boundary conditions to interfaces, we must ensure current

conservation across the junction. As discussed in App. F, for a unitary scattering

matrix the presented boundary conditions yield the following conservation law

1

2

n∑i=1

Tr[τ3g(E,p

l,iF in, 0)− τ3g(E,p

l,iFout, 0)

]=

=1

2

n∑i=1

Tr[τ3g(E,p

r,iFout, 0)− τ3g(E,p

r,iF in, 0)

].

(4.58)

This expression resembles Kirchhoff’s law if the trajectories are considered as

wires that are connected at x = 0. On the other hand, the current conservation

perpendicular to the interface is guaranteed by the condition

1

2

⟨vF,xTr

[τ3g(E,p

lF in, 0)− τ3g(E,p

lFout, 0)

]⟩pr

F in=

=1

2〈vF,xTr [τ3g(E,p

rFout, 0)− τ3g(E,p

rF in, 0)]〉pr

F in.

(4.59)

Note that pr/lFout is uniquely determined by p

r/lF in via Eq. (4.6). To ensure current


x

yp

l;iFout

pl;iF in

pr;iFout

pr;iF in

#i

Figure 4.4: Illustration of the in-coming and out-going trajectories for a givenangle ϑi.

conservation across the junction (in x-direction) in our approach, we have to

construct the grid of the discrete directions such that the term vFx = vF cosϑ is

already taken into account. This is guaranteed by the grid (see Fig. 4.4)

pl,iF in = pF

(cos ϑi

sinϑi

), pr,i

F in = pF

(− cosϑi

sinϑi

),

pl,iFout = pF

(− cosϑi

sinϑi

), pr,i

Fout = pF

(cosϑi

sin ϑi

),

sinϑi =2i

n + 1− 1, i = 1, . . . , n.

(4.60)

In other words this grid takes into account that the rate of scattering events is

higher for smaller angles of incidence.

For a given scattering matrix, the scattering probability for a scattering pro-

cess ps,jF in → ps′,i

Fout (s, s′ = l/r) is given by

Ps′s(ϑj → ϑi)∆ϑi = |Ss′sij |2 (4.61)

where P is the probability density, and ∆ϑi ≈ ϑi − ϑi−1 is the weight of the ith

scattering channel. Using the grid, as defined in Eq. (4.60), for a large number

of scattering channels, n 1, we find

Ps′s(ϑj → ϑi) =n

2cosϑi|Ss′s

ij |2; (4.62)

the factor (cosϑi)/2 takes into account the non-equidistant grid of the directions.

To sum up, we found a general procedure to solve boundary problems within

the theory of quasi-classical Green’s functions for a given interface, which is

4.4 Explicit Solution of Zaitsev’s Boundary Conditions 51

described by a scattering matrix:

(i) To find the a’s on the incoming and the b’s on the outgoing trajectories, we

have to integrate the Riccati equations (4.40) and (4.41) starting from the

related initial values given in Eqs. (4.43) and (4.44) in the bulk.

(ii) The boundary conditions (4.56) and (4.57) must be applied to find the

initial values at the interface for the a’s on the out and the b’s on the in

trajectories.

(iii) Then, by integrating Eqs. (4.40) and (4.41) the missing a’s and b’s can be

obtained.

(iv) Finally, the Green’s functions can be constructed from the a’s and b’s via

Eq. (4.42) and the physical quantities can be evaluated.

4.4 Explicit Solution of Zaitsev’s Boundary

Conditions

As already mentioned, the present approach allows the explicit solution of Zait-

sev’s boundary conditions. We will therefore reconsider the situation of an ideal

interface. The scattering matrix is given by

Sll = −Srr = R = diag[√

1− T (ϑi)],

Slr = Slr = T = diag[√

T (ϑi)] (4.63)

with the angle-dependent transparency T (ϑi) as suggested in Eq. (4.27). This

means that for each parallel momentum two in- and two out-trajectories are

connected. Effectively, we have to solve only a 2× 2 problem for each direction.

We obtain

al/rout =

ar/lin T (1− a

l/rin b

r/lout) + a

l/rin R(1− a

r/lin b

r/lout)

T (1− al/rin b

r/lout) +R(1− a

r/lin b

r/lout)

, (4.64)

bl/rin =

br/loutT (1− a

r/lin b

l/rout) + b

l/routR(1− a

r/lin b

r/lout)

T (1− ar/lin b

l/rout) +R(1− a

r/lin b

r/lout)

. (4.65)


Here we used the abbreviations T = T (ϑi), R = 1− T (ϑi) and

al/rin/out = al/r(E,p

l/r,iF in/out; 0), (4.66)

bl/rin/out = bl/r(E,p

l/r,iF in/out; 0). (4.67)

It can be seen by a straight-forward calculation that these expressions are equiv-

alent to Zaitsev’s boundary conditions formulated in Eqs. (4.7) and (4.8) if all

involved momenta have the same parallel component. This formulation of Zait-

sev’s boundary conditions was first found by Eschrig [68].

4.5 Simple Applications to Unconventional Su-

perconductors

In this section we will use the explicit solution of the boundary conditions to

examine some basic properties of anisotropic superconductors. For simplicity we

assume an order parameter which is spatially constant ∆(pF , r) = ∆(pF ). This

means that the results are achieved without a self-consistent evaluation of the

order parameter; possible pair-breaking effects at boundaries are neglected.

In this case the Green’s functions directly at the boundary are easy to evalu-

ate: The a’s on the in-trajectories are given by their initial values, whereas the b’s

are given by their initial values on the out-trajectories. The related values at the

surface on the out- and in-trajectory are determined by the explicit expressions of

the boundary conditions in Eqs. (4.64) and (4.65). The Green’s functions at the

boundary can now be constructed, which is sufficient to evaluate some physical

properties.

Using this procedure, we present some non-trivial results for (unconventional)

superconductors that can be obtained by simple analytical calculations.

4.5.1 Specular Surface

We consider one particular quasi-particle trajectory given by the directions p+F in

and p+Fout. For completeness we also consider the time-reversed path with p−F in =

−p+Fout and p−Fout = −p+

F in (see Fig. 4.5). As T = 0, the functions a and b must

4.5 Simple Applications to Unconventional Superconductors 53

x

y

p+

F in

p+

Fout

p

Fout

p

F in

e+i'=2

ei'=2

Figure 4.5: An ideal surface where the quasi-particles are scattered into anotherorder parameter branch at the surface; due to parity the order parameter is theidentical for time-reversed trajectories.

be continuous at the surface

a(E,p±Fout, 0) = a(E,p±F in)−i∆(p±F in)

E +√E2 − |∆(p±F in)|2

, (4.68)

b(E,p±F in, 0) = b(E,p±Fout) =i∆∗(p±Fout)

E +√E2 − |∆(p±Fout)|2

. (4.69)

We will consider the density of states on the given trajectory at the surface which

is given by the real part of the retarded Green’s function

gR11(E,p

±F in, 0) =

1 + a(ER,p±F in)b(ER,p

±Fout, 0)

1− a(ER,p±F in)b(ER,p

±Fout, 0)

(4.70)

with ER = E + iγ, E ∈ R and γ → 0+. Note that the Green’s functions are

also continuous at the surface: gR(E,p+F in, 0) = gR(E,p+

Fout, 0), gR(E,p−F in, 0) =

gR(E,p−Fout, 0). For an order parameter ∆(p±F in) = ∆(p∓Fout) = ∆e±iϕ/2 (see

Fig. 4.5) the retarded Green’s function reads

gR11(E,p

±F in, 0) =

2ER

√E2

R − |∆|2 ± i|∆|2 sinϕ

2E2R − |∆|2(1 + cosϕ)

. (4.71)

For ∆(p±F in) = ∆(p±Fout) (ϕ = 0) the usual s-wave density of states is recovered

without any sub-gap structure

NN0

=|E|Θ(E2 − |∆|2)√

E2 − |∆|2. (4.72)


This result can be applied to an s-wave order parameter or a d-wave order pa-

rameter, which is not tilted (α = 0).

In the case ∆(p±F in) = −∆(p±Fout) (ϕ = π) a zero energy state appears

NN0

=

√E2 − |∆|2|E| Θ(E2 − |∆|2) + π|∆|δ(E). (4.73)

This situation occurs for a d-wave order parameter, which is tilted by α = 45.

For ϕ 6= 0, we realize a sub-gap structure in the density of states as a pole

exists at E+A for p+

F in and at E−A for p−F in (i.e. on the time-reversed trajectory)

E±A (ϕ) = ±|∆| cos(ϕ/2). (4.74)

This means that the density of states differs for time-reversed paths if ϕ ∈ (0, π);

i.e. the time-reversal symmetry is broken. For the in-trajectories given by the

momentum p±F in, this means that with an increasing phase difference ϕ of the

order parameter a sub-gap state moves from E±A = ±|∆| (ϕ = 0) to E±

A = 0

(ϕ = π). For ϕ ∈ (0, π) one state lies below the Fermi energy whereas the time-

reversed state lies above; i.e. only one of them is occupied. In Fig. 4.10 the

density of states is presented where δ-peaks occur in the gap region (which are

broadened by a finite imaginary part of the energy).

We can find this case, for example, if the order parameter consists of two

components with a relative phase (dx2−y2 + idxy/s). With increasing phase dif-

ference ϕ an increasing amount of spectral weight is shifted into the gap region.

In Sec. 5.2 we will discuss this effect in detail.

The occurrence of the sub-gap structure can be interpreted in terms of An-

dreev reflection. A quasi-particle with pF ↔ vF is scattered at an inhomogeneity

of the order parameter in the following way: The quasi-particle combines with a

time-reversed quasi-particle (−pF ) to a Cooper pair and moves into the super-

conducting region. Due to momentum conservation a quasi-hole with pF ↔ −vF

is reflected. At surfaces this process can be resonant for particular energies and

build Andreev bound states: A quasi-particle with pF in is scattered at the surface

into direction pFout; at the (finite) order parameter it is reflected and moves as

a quasi-hole towards the surface where it is reflected again into direction pF in.

Now, the quasi-hole realizes the order parameter and Andreev reflection turns it

back into a quasi-particle with pF in, which is the starting point again.

It is important to note that the bound states carry a current along the surface

(y-direction) as the quasi-particles and quasi-holes transport the opposite charge


x

yp

1+

F in

p2+

Fout

p2+

F in

p1+

Fout

+

+

Figure 4.6: The sketch of a rough (beam-splitting) surface shows the coherentcoupling of two in- and two out-trajectories with different order parameter values.

with the opposite parallel velocity. For ϕ ∈ (0, π) a quasi-particle current in

p−F in,y-direction shows up, as only the states in one direction lie below the Fermi

energy and are occupied (T ∆). For ϕ = π the currents cancel.

4.5.2 Rough Surface

We will now discuss a simple model of a surface which acts as a beam-splitter: An

incoming quasi-particle with momentum p1+F in is reflected into a direction p1+

Fout

with probability Θ and into another direction p2+Fout with probability 1− Θ; the

quasi-particle with momentum p2+F in is scattered into the same out directions but

with the interchanged probabilities (see Fig. 4.6). The reason for this behavior

could be a particular kind of surface roughness. We describe this behavior by a

2× 2 S-matrix

S =

( √Θ

√1−Θ√

1−Θ −√

Θ

). (4.75)

Formally, this surface can be treated with the boundary conditions given by

Eqs. (4.64) and (4.65) if we make the substitution T → Θ, l/r → 1/2.

We will discuss the situation with a spatially constant order parameter

∆(p1+F in/out) = −∆(p2+

F in/out) = ∆ (see Fig. 4.6). The retarded Green’s function

now reads

gR11(E,p

i+F in/out, 0) =

ER

√E2

R − |∆|2E2

R −Θ|∆|2 . (4.76)

In this case two bound states occur, which are given by the poles of the Green’s


function

EA(Θ) = ±|∆|√

Θ. (4.77)

The density of states is presented in Fig. 4.11; note that it is symmetric and is

identical for the time-reversed trajectory. In contrast to the previous section,

here the time-reversal symmetry is preserved.

For Θ = 1 we recover the case corresponding to an s-wave or untilted d-

wave order parameter at a specular surface (compare ϕ = 0 in Sec 4.5.1). Weak

roughness (Θ . 1) leads to a shift of spectral weight from the continuum (E > ∆)

to sub-gap bound states at EA. The case of a d-wave order parameter which is

rotated by α = 45 at a specular surface corresponds to Θ = 0 (compare ϕ = π in

Sec. 4.5.1); here, weak roughness (Θ & 0) splits the zero energy bound state. We

will see later in Sec. 5.3 that in more general beam-splitting models a zero energy

bound state remains, only some spectral weight is shifted to finite energies.

4.5.3 Ideal Interface

We examine the contribution of one particular direction to the tunnel current

across an ideal interface. We will compare three configurations of the order

parameter, which can occur for junctions between superconductors with a d-wave

order parameter: (i) ∆(pl/rF in/out) = ∆, which is related to the well-known case of

usual s-wave or untilted d-wave superconductors (see Fig. 4.7). (ii) ∆(pl/rF in) = ∆

and ∆(pl/rFout) = −∆ which can occur for tilted d-wave order parameters on the

left and right side (see Fig. 4.8). (iii) ∆(plF in/out) = ∆ and ∆(pr

F in/out) = ±∆ (see

Fig. 4.9); this situation is realized if an untilted d-wave order parameter is present

on the left side of the junction and on the right side it is tilted by αr = 45.

We examine the super-current for a finite phase-difference ϕ = ϕr − ϕl using

the Matsubara technique; i.e. the current is given by Eq. (3.65). This leads to

the following expression for the current on the left side of the junction

jx =⟨vF,xj(p

lF in)⟩pl

F in(4.78)


x

yp

lF in

plFout

prF in

prFout

+ei'

+ei'

+

+

Figure 4.7: At an ideal interface four trajectories are connected coherently. Herethe order parameter in all directions is the same up to a phase shift by ϕ onthe left side. This situation is related to an s-wave or an untilted d-wave orderparameter on both sides.

using the abbreviations

j(plF in) ≡ iπeN0T

∞∑n=−∞

gMl(En,plF in), (4.79)

gMl(En,plF in) ≡

1

2Tr[τ3(gMl(En,p

lF in, 0)− gMl(En,p

lFout, 0)

)]. (4.80)

Below we will work with these newly defined quantities as they already contain

important information about the (qualitative) temperature dependence of the

current and the current-phase relation. For simplicity we only consider the tun-

neling limit; i.e. we expand the current density in the transparency, T 1, and

keep only the first non-vanishing term.

To gain further insight, we also examine the density of states at the interface

for arbitrary transmission.

(i) ∆(pl/rF in/out) = ∆

In this case the expression for the current contribution yields

gMl(En,plF in) =

−i|∆|2E2

n + |∆|2T sinϕ+O(T 2). (4.81)

After the summation over the Matsubara energies this reads

j(plF in) = |e|N0

π|∆|2

tanh

(|∆|2T

)T sinϕ

T→0→ |e|N0π|∆|

2T sinϕ. (4.82)


With the angle dependent transparency given in Eq. (4.27), the Fermi surface

average, defined in Eq. (4.78), leads to the results for jx which are already given

in Eq. (4.30) for ∆(ϑ) = ∆ (s-wave) and in Eq. (4.31) for ∆(ϑ) = ∆ cosϑ (untilted

d-wave).

The density of states at the interface is determined by the real part of the

retarded Green’s function which on the left hand side of the junction reads

gRl11 (E,pl

F in/out, 0) = gRr11 (E,pr

Fout/in, 0) =

=ER

√E2

R − |∆|2 ∓ i2|∆|2T sinϕ

E2R − |∆|2 + T |∆|2 sin2(ϕ

2)

.(4.83)

The density of states is presented in Fig. 4.12 (we used a large transparency

T = 0.4 to make the effect visible), where a sub-gap structure can be seen.

Andreev bound states occur at

EA(T , ϕ) = ±|∆|√

1− T sin2(ϕ/2). (4.84)

Note that the spectral weight of the bound states at each energy is different.

The bound states in this situation can be explained in a way similar to those in

the previous Sec. 4.1. But here, in addition to quasi-particles which are reflected

at the boundary, transmission also occurs. This leads to another scattering pro-

cess which can be resonant: At the interface a quasi-particle moves to the right.

It is then converted to a quasi-hole by Andreev reflection at the finite order pa-

rameter; the quasi-hole moves to the left and passes the boundary. It is now

reflected at the left order parameter and returns to the initial state. The spectral

weight of these bound states grows with the transparency of the surface.

In the present case the bound states for plF in have more spectral weight below

EF (occupied states) than those for plFout. This leads to a current in x-direction;

the current parallel to the junction is canceled by the related time-reversed trajec-

tory, which carries the opposite current in y-, but the same current in x-direction.

From a microscopic point of view, the charge transport across the junction is per-

formed by these current-carrying bound states.

(ii) ∆(pl/rF in) = ∆, ∆(p

l/rFout) = −∆

The expansion of Eq. (4.80) now reads

gMl(En,plF in) =

i|∆|2E2

n

T sinϕ+O(T 2). (4.85)


x

yp

lF in

plFout

prF in

prFout

+ei'

ei'

+

Figure 4.8: The order parameter changes its sign for transmitted as well as forreflected quasi-particles; for ϕ 6= 0 an additional phase for the transmitted quasi-particles exists. Such a situation occurs for tilted d-wave order parameters onboth sides of the junction.

After performing the Matsubara sum this leads to the following current contri-

bution

j(plF in) = −|e|N0

π|∆|24T

T sinϕ. (4.86)

Its properties differ crucially compared to case (i): Obviously, the sign of the

current is changed; this is explained by the sign change of the order parameter

for transmitted quasi-particles. This behavior is related to a free energy minimum

occurring at ϕ = π. As discussed, e.g. in Ref. [8], the free energy of a junction is

connected to the current-phase relation via

I = 2|e|∂F∂ϕ

. (4.87)

Moreover, the current contribution diverges for low temperatures, T → 0, as a

second order pole at En = 0 is present in gMl. This divergence is an artefact of

the expansion in the tunneling limit, T → 0; it is cut off by a finite transparency

but nevertheless the critical current increases drastically for low temperatures.

The time-reversed situation can be obtained by the trivial gauge transformation,

∆ → −∆; the contribution of the time-reversed scattering process to the current

density jx is therefore the same.

The retarded Green’s function is given by

gRl11 (E,pl

F in/out, 0) = gRr11 (E,pr

Fout/in, 0) =

=ER

√E2

R − |∆2|2 ± i2|∆|2T sinϕ

E2R − |∆|2T sin2(ϕ

2)

(4.88)


x

yp

lF in

plFout

prF in

prFout

+ei'

ei'

+

+

Figure 4.9: The situation illustrated in this figure can be related to the case ofan untilted d-wave order parameter on the left and a tilted one on the right side.

and the Andreev bound states occur at

EA(T , ϕ) = ±|∆|√T sin(ϕ/2). (4.89)

The related density of states is presented in Fig. 4.13. For a finite phase difference

the zero energy bound state splits, and bound states at finite energies occur. This

splitting leads to a reduction of the junction energy as only the lowered bound

state contributes; this is another way of explaining the ground state at ϕ = π.

The different spectral weight of the bound states for plF in and pl

Fout leads to a

current across the junction for ϕ ∈ (0, π).

For junctions with tilted d-wave order parameters on each side, both cases (i)

and (ii) can occur simultaneously, each for different directions. At low temper-

atures case (ii) will dominate due to its strongly increasing critical current for

T → 0. But it could be that at a finite temperature Tπ < Tc the contribution of

case (i) will become dominant: The sign of the current then changes for a fixed

phase difference ϕ; i.e. by decreasing the temperature, the free energy minimum

of the junction moves from ϕ = 0 (0-junction) to ϕ = π (π-junction). At Tπ,

the first order contributions in T of the current cancel each other, and we have

to take into account higher order terms. We will discuss this issue in detail in

Sec. 6.3.

(iii) ∆(plF in/out) = ∆, ∆(pr

F in/out) = ±∆

In the third case with a sign change of the order parameter only on the right side,

the situation changes drastically as we now obtain

gMl(En,plF in) =

|∆|2 cosϕ

En

√E2

n + |∆|2T +

i|∆|4 sin(2ϕ)

4E2n(E2

n + |∆|2)T2 +O(T 3). (4.90)


By summing over all Matsubara energies the first order term in T vanishes as it

is antisymmetric in En, and the leading term is given by

j(plF in) = |e|N0

π|∆|2

[tanh

(|∆|2T

)− |∆|

2T

]T 2 sin(2ϕ)

T→0→ −|e|N0π|∆|24T

. (4.91)

which diverges for T → 0 due to a pole of gMl(En,plF in) at En = 0 (compare case

(ii)). As a consequence of the double periodicity of the current-phase relation,

here two degenerate ground states exist at ϕ = π/2, 3π/2.

Due to the asymmetry of this junction, we have to take care of the Green’s

function on the left and right side. They are given by

gRr11 (E,pr

F in/out, 0) =E2

R − |∆|2 + 12T |∆|2(1± cosϕ)

ER

√E2

R − |∆|2 − i2T |∆|2 sinϕ

, (4.92)

gRl11 (E,pl

F in/out, 0) =E2

R − 12T |∆|2(1± cosϕ)

ER

√E2

R − |∆|2 − i2T |∆|2 sinϕ

. (4.93)

The density of states is presented in Fig. (4.14). The Andreev bound states are

at the same energies on the left and the right side of the junction

EA = |∆|√

1

2± 1

2

√1− T 2 sin2(ϕ). (4.94)

Their spectral weight is nevertheless different on each side: On the right side for

T = 0, a zero energy bound state is present, which is shifted to positive energies

for a finite transparency and phase difference. The second bound state on the

right side, near the continuum, has less spectral weight as it occurs only due to

the tunneling and vanishes completely for T = 0. The situation is vice versa on

the left side. The time-reversed process, is given by the gauge transformation

ϕ → ϕ + π, ∆ → −∆; the density of states for the time-reversed situation is

therefore given by Eqs. (4.92) and (4.93) with E → −E.

As can be seen in Eqs. (4.92) and (4.93), for ϕ = π/2 the density of states on

the in- and out-trajectories are identical, which leads to a vanishing current across

the junction. For ϕ ∈ (0, π/2) a difference occurs, and a current in x-direction is

present.

Moreover, for each non-trivial phase difference a current in y-direction occurs

which is carried by the bound states of the time-reversed process: they have

negative energies −EA and are occupied. We will discuss this situation in detail

in Sec. 6.2.


E=

N(E)=N0

21.510.50-0.5-1-1.5-2

3

2.5

2

1.5

1

0.5

0

Figure 4.10: N (p+F in) (short dashes) andN (p−F in) (long dashes) at an ideal surface

for ϕ = π/4, π/2, 3π/4. The solid lines show the density of states for ϕ = 0(divergence at E = ±|∆|) and ϕ = π (peak at E = 0).

E=

N(E)=N0

21.510.50-0.5-1-1.5-2

3

2.5

2

1.5

1

0.5

0

Figure 4.11: N (p1+F in) at a rough surface for Θ = 0.2, 0.4, 0.6, 0.8. The solid

lines show the density of states for Θ = 0 (divergence at E = ±|∆|) and Θ = 1(peak at E = 0).


210-1-2

2

0

-2

E=

N(E)=N0

21.510.50-0.5-1-1.5-2

3

2.5

2

1.5

1

0.5

0

Figure 4.12: Density of states at an ideal interface (case (i)) for T = 0.4 withoutphase-difference for pl

F in/out (solid line) and with phase-difference ϕ = π/2 for

plF in (long dashes) and pl

Fout (short dashes); the inset shows their difference.

210-1-2

2

0

-2

E=

N(E)=N0

21.510.50-0.5-1-1.5-2

3

2.5

2

1.5

1

0.5

0

Figure 4.13: Density of states at an ideal interface (case (ii)) for T = 0.4 withoutphase-difference for pl

F in/out (solid line) and with phase-difference ϕ = π/2 for

plF in (long dashes) and pl

Fout (short dashes); the inset shows their difference.


E=

N(E)=N0

21.510.50-0.5-1-1.5-2

3

2.5

2

1.5

1

0.5

0

Figure 4.14: The density of states at an ideal interface (case (iii)) for T = 0 onthe left (solid line) and right (long dashes) hand side is shown; for ϕ = π/2 andT = 0.4 bound states at finite energies occur on the left (short dashes) and rightside (dotted line).

Chapter 5

Rough Surfaces

In this chapter we examine the behavior of unconventional superconductors in

the vicinity of (microscopically rough) surfaces. We discuss an irregular rough

surface as well as a surface which acts as a beam-splitter. The order parameter is

of the dx2−y2-wave symmetry, and we consider various orientations with respect to

the surface. A possible admixture of the dxy-wave or the s-wave type is discussed

as well.

The behavior of superconductors at surfaces is important for understanding

NIS contacts or scanning tunnel microscopy experiments. Here the differential

conductance, which is defined in Eq. (4.23), is the most important observable;

for low temperatures it is an almost direct measure of the density of states at the

surface, as can be seen in Eq. (4.26).

The existence and the behavior of a ZBCP, respectively, is of main interest

in the following discussions as it is the most striking feature in experiments. In

particular, the roughness dependence of the ZBCP will be examined.

5.1 Surfaces with Disorder

In the first section we study d-wave superconductors in the vicinity of disordered

surfaces; the order parameter has the form ∆(pF , x) = ∆(x) cos[2(ϕ−α)], where

α is the tilting angle of the order parameter with respect to the surface normal.

A physical realization of a disordered surface is assumed to have a microscopic

roughness without any regular structure. In our model, this kind of surface will be

described by a random scattering matrix, so that the weight of specular reflection

is reduced and distributed randomly to all other scattering processes; on the

average the surface is only partially specular and the amount of specular reflection

66 5 Rough Surfaces

px

pF

I SC

y

x

py

pF in

pFout

pFout

pF in

Figure 5.1: At rough surfaces one incoming quasi-particle (solid line) can bescattered into several outgoing directions (dashed lines).

is reduced with increasing disorder. This is illustrated schematicly in Fig. 5.1. In

an initial step we construct an adequate (phenomenological) S-matrix.

5.1.1 S-Matrix

In the case of a surface (Slr = Srl = 0), it is sufficient to use a n×n S-matrix (see

Eq. (4.35): S = Srr). As S must be unitary, we use the exponential representation

S = expiH, H = H†. (5.1)

To simulate the statistical properties of a disordered surface we choose a random

matrix H with Gaussian correlations

〈Hij〉 = 0, 〈H∗ijHi′j′〉 =

τ

nδii′δjj′. (5.2)

The brackets 〈. . . 〉 denote the ensemble average of the disorder. The roughness

of the surface can be varied by the parameter τ . The correlator is normalized by

the number of trajectories, n, which are taken into account; as we will see later,

this ensures that on the average the weight for specular reflection is independent

of n. A similar scattering matrix was also suggested by Yamada et al. [50].

Important information on the surface is contained in the averaged probability

〈|Sij|2〉 for a scattering process j → i, which has a simple behavior: For τ = 0

only |Sii|2 = 1 is finite, and all other elements are zero. If τ increases, the diagonal

elements (responsible for specular reflection) are reduced to 〈|Sii|2〉 ≡ |u(τ)|2 < 1

and the off-diagonal elements become finite 〈|Si6=j|2〉 = |v(τ)|2 . 1/n (Fig. 5.2).

5.1 Surfaces with Disorder 67

= 0:8 = 0:4

i = n=2

n = 100

l

hjSi;i+lj

2i

403020100-10-20-30-40

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 5.2: Averaged scattering proba-bility from the in trajectory i+ l to theout trajectory i = n/2. The specularcontribution is reduced by increasing τ .

ju()j

2

43.532.521.510.50

1

0.8

0.6

0.4

0.2

0

Figure 5.3: Specular scattering weightas a function of τ . For τ = 0 the scat-tering is purely specular; for τ & 3the scattering probability becomes in-dependent of the in- and out-trajectory(|u|2 ≈ |v|2).

Due to the unitarity of S we find

|u(τ)|2 + (n− 1)|v(τ)|2 = 1. (5.3)

The averaged properties of S are therefore completely determined by the proba-

bility of specular reflection |u|2, and we can use it as a measure for the disorder

of the surface; its relation to the parameter τ is shown in Fig. 5.3. The reflection

is partially specular (i.e. |u|2 . 1) if τ is small enough. For very strong disorder,

τ 1, all trajectories are equivalent, and we find |u|2 ≈ |v|2 ≈ 1/n.

For a given disorder strength, τ , the averaged scattering probability density,

which is defined in Eq. (4.62), is given by

〈P (ϑj → ϑi)〉 =1

2cosϑi

[n|u(τ)|2δij + (1− |u(τ)|2)(1− δij)

]; (5.4)

as we are considering a surface, we can drop the left/right index here. In the

strong disorder limit (τ 1) we obtain

〈P (ϑj → ϑi)〉τ1→ 1

2cosϑi. (5.5)

For large disorder all incoming quasi-particles are scattered with the same prob-

ability to a particular out-trajectory, but due to the non-equidistant grid the

scattering probability is anisotropic.

In the continuum limit, n → ∞, we substitute the discrete by continuous

68 5 Rough Surfaces

0.40-0.4

20

10

0 = 0:8#j = 0n = 100

#i=

P(#j!

#i)

0.40.20-0.2-0.4

2

1.5

1

0.5

0

0.40-0.4

2

1

0 = 0:8#j = 0n = 20

#i=

P(#j!

#i)

0.40.20-0.2-0.4

2

1.5

1

0.5

0

Figure 5.4: The probability density of a particular realization of S for differentnumbers of scattering channels, n = 100 (left) and n = 20 (right); the dashedline (∝ cos ϑi) is the averaged value. With growing n, the fluctuations of thescattering matrix are resolved on a smaller angular scale. The inset shows theprobability density on a scale so that also the weight for specular scattering canbe seen.

angles, ϑi → ϑ′ and ϑj → ϑ, which leads to

〈P (ϑ→ ϑ′)〉 = |u(τ)|2δ(ϑ− ϑ′) +1

2(1− |u(τ)|2) cosϑ′. (5.6)

We find a δ-peak in specular direction and a continuous background.

In Fig 5.4 we compare the scattering probability density for different numbers

of scattering channels. For larger n, fluctuations in the scattering probability

density are taken into account on a smaller angular scale. Conversely this means

that we can use the number of trajectories n to adjust the angle on which the

scattering probability density is correlated; for given n the typical angle, up to

which correlations exist, is of the order ϑc ≈ π/n.

5.1.2 Results and Discussion

With the specified S-matrix we evaluate the average order parameter in the

vicinity of the surface and the average conductance of a NIS junction in the

tunnel regime, which is measured in various experiments; the order parameter

is calculated self-consistently for T = 0.1Tc. To evaluate the retarded Green’s

function numerically a small imaginary part γ = 0.01∆0 is added to the energy

E → ER = E + iγ; a γ > 0 mimics finite life-time effects in real systems due to

interaction or impurities.

In order to get a survey of the effects we varied both the orientation of the

surface, α = 0, 24, 45, as well as the roughness from the specular case τ 1

up to a very rough surface τ = 4.


We start the discussion with an untilted order parameter (α = 0). In our

model, no ZBCP can be observed for arbitrary disorder (Fig. 5.6). It is however

worth noting that the conductance at V = 0 increases considerably for τ &0.8 (inset of Fig. 5.6). The divergence at eV = ∆∞ (∆∞: bulk value of the

order parameter) broadens to a hump; its position moves to V . ∆∞ for finite

roughness as the order parameter in the vicinity of the surface is suppressed

(Fig. 5.5).

For α 6= 0 without roughness (τ = 0) a ZBCP is present (Figs. 5.8, 5.10); as

already discussed in Sec. 4.5.1, this can be explained by the existence of trajec-

tories with a changing sign of the order parameter which lead to a zero energy

bound state. In the conductance also a peak (which has broadened to a hump

for α = 45) occurs for eV . ∆∞; this feature can be related to bound states at

finite energies. Compared with α = 0, spectral weight from E ≈ ∆∞ is shifted

into the gap region. The order parameter is suppressed by the surface as already

discussed in Sec. 4.1 (Figs. 5.7, 5.9); for α = 45 the order parameter is fully sup-

pressed at the surface. For growing disorder the height of the ZBCP decreases

whereas its width increases (Figs. 5.8, 5.10). The disorder has also some influence

on the order parameter; in particular, for α = 45, the order parameter becomes

finite at the surface.

For very rough surfaces (τ & 2), the orientation of the surface is less im-

portant: The order parameter is almost independent of the orientation and the

conductance is of the order of the normal state value for all energies; no ZBCP

is present.

Additionally, the angle-resolved density of states is presented for α = 0, 45

and τ = 0.4 (Figs. 5.11, 5.12). The gap-structure for different directions ϑ can

be seen as well as the broad zero energy bound state for α = 45. Here statistical

fluctuations can be seen, since we used only a finite number of samples (. 50) in

the averaging procedure.

In the literature, different models of rough surfaces within the quasi-classical

theory are also discussed. Most of them provide similar results as ours. In

particular, models with a thin dirty layer covering a specular surface, as suggested

first by Ovchinnikov [53], are examined in detail: Using a Born approximation,

the results show the same qualitative behavior as ours [47,48]. On the other hand,

in the unitary limit the ZBCP is only weakly broadened, which is in contrast to

our findings. Good agreement of our results can also be found with those of

the scattering matrix approach of Yamada et al. [50], which was evaluated in a

Born-like approximation.

70 5 Rough Surfaces

In experiments on NIS junctions with a high-Tc compound, a broad ZBCP is

observed for the orientation α = 45 [24,29,76–79]. Comparing the experimental

data with our results (Figs. 5.10) we estimate τ & 0.4 to be a reasonable disorder

value for these samples.

On the other hand, there are some experiments which show a different be-

havior than expected from our results. For example, Aprili et al. [31] observed

an almost constant width of the ZBCP when increasing the surface roughness

by ion-irradiation. This cannot be explained by the particular kind of roughness

specified in our model. In other experiments a ZBCP was observed even for an

untilted surface [24,25,29], which is in contrast to our results. This behavior can

be explained by the existence of facets having a typical size larger than the co-

herence length [80]. Quasi-particles which are scattered at different facets are not

connected coherently. The facets provide an averaging over several orientations

of the surface; this situation can be described by a linear superposition of the

Green’s functions [27]. As we study only disorder which is present on a smaller

scale, we have not taken into account this possibility in our boundary conditions.

Until now, we only considered the mean value of the quantities. But, in

contrast to the other approaches, we are also able to examine their statistical

fluctuations. This might be of importance, for example, in mesoscopic junctions,

where a particular realization of the scattering matrix is measured. Experimental

realizations of mesoscopic junctions could be pin-hole contacts [24] or tunnel-tip

experiments [24, 77–79].

As an example, we studied the statistical fluctuations of the conductance for

α = 45. The standard deviation of the conductance√〈[∆G(V )]2〉 is presented

in Fig. 5.13 for a varying number of trajectories, n. The statistical fluctuations

decrease with an increasing n. This can be understood as follows: The number

of trajectories, n, defines the angular scale, ϑc ≈ π/n, on which the scattering

probability is fluctuating for one particular realization of the disorder (Fig.5.4).

In the evaluation of the differential conductance via Eq. (4.25) an angular aver-

age occurs. If ϑc π (n 1), the fluctuations of the scattering probability are

averaged out very effectively and each realization of the disorder is well-described

by its mean value; for larger ϑc, the the statistical fluctuations e.g. of the con-

ductance increase.

In our model, even for n = 50 the standard deviation of the differential con-

ductance is less than 10%. Therefore, in this situation, the statistical fluctuations

can be assumed to be unimportant; i.e. particular realizations of the disorder are

well described by the average quantities.


= 4 = 2 = 0:8 = 0:4 = 0:08 = 0

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.5: Order parameter for α = 0 at T = 0.1Tc for several values of thedisorder strength. The order parameter is suppressed by the surface disorder.

43210

0.60.40.2

eV=1

hG(V)iRN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.6: Differential conductance for α = 0 at T = 0.1Tc for the same values ofthe disorder strength as in Fig. 5.5. The inset shows the behavior of conductanceat V = 0 as a function of the disorder strength.

72 5 Rough Surfaces

= 4 = 2 = 0:8 = 0:4 = 0:08 = 0

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.7: Order parameter for α = 24 at T = 0.1Tc for several values of thedisorder strength. The order parameter is already suppressed even for τ = 0; inthe presence of disorder it is slightly enhanced.

eV=1

hG(V)iRN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.8: Differential conductance for α = 24 at T = 0.1Tc for the same valuesof the disorder strength as in Fig. 5.7.


= 4 = 2 = 0:8 = 0:4 = 0:08 = 0

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.9: Order parameter for α = 45 at T = 0.1Tc for several values of thedisorder strength. For τ = 0, the order parameter is completely suppressed atthe surface; surface roughness leads to a finite value.

= 2 = 0:8 = 0:4 = 0:08 = 0

eV=1

hG(V)iRN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.10: Differential conductance for α = 45 at T = 0.1Tc for the samevalues of the disorder strength as in Fig. 5.9.

74 5 Rough Surfaces

# = 75# = 60# = 45# = 30# = 15

E=1

hN(E;#;0)i=N0

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.11: Angle-resolved density of states for α = 0 and τ = 0.4.

# = 75# = 60# = 45# = 30# = 15

E=1

hN(E;#;0)i=N0

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.12: Angle-resolved density of states for α = 45 and τ = 0.4.


n = 200n = 150n = 100n = 50 = 0:4

eV=1

qh[G(V)]2i=hG(V)i

1.210.80.60.40.20

0.180.160.140.120.1

0.080.060.040.02

0

n = 200n = 150n = 100n = 50 = 0:8

eV=1

qh[G(V)]2i=hG(V)i

1.210.80.60.40.20

0.180.160.140.120.1

0.080.060.040.02

0

Figure 5.13: Behavior of the relative standard deviation of the conductivity fora varying number of trajectories, n, and α = 45; the standard deviation isdecreasing with increasing n. We present results for different disorder: τ = 0.4and τ = 0.8.

76 5 Rough Surfaces

I SC

y

x

pFout

pFout

px

pFpy

pF in

pF in

Figure 5.14: Dominant dx2−y2 order parameter with sub-dominant dxy or s com-ponent.

5.2 Order Parameters with Subdominant Pair-

ing: dx2−y2 + dxy/s

In this section, we discuss superconductors with a dominant dx2−y2-wave interac-

tion and a weaker interaction in the dxy-wave or s-wave channel. This means that

we consider order parameters of the form ∆(pF , x) = ∆1(x) cos[2(ϕ−α)]+∆2(x)

in the case dx2−y2 + s and ∆(pF , x) = ∆1(x) cos[2(ϕ− α)] + ∆2(x) sin[2(ϕ− α)]

in the case dx2−y2 + dxy. We choose the interaction of both order parameter

components in such a way that only the dominant one (dx2−y2) is finite in the

homogeneous (bulk) situation (compare Sec. 2.3). In our calculations, this is

achieved by the choice of the individual critical temperatures Tc,2 = 0.3Tc,1.

As the dominant order parameter component is suppressed at surfaces a sub-

dominant component can become finite in this region; this leads to a mixed order

parameter consisting of two components as illustrated in Fig. 5.14. In particular,

we consider the effect of disorder on the admixture of a dxy-wave or an s-wave

component.


We start with the dx2−y2 + s case. For the further discussion, it is important to

note, that an s-wave component itself is inert against spatial (non-magnetic) in-

homogeneities of the system; it is only affected (i.e. suppressed) by the dominant

dx2−y2-wave component, as discussed already in terms of the Ginzburg-Landau

theory in Sec. 2.3.

5.2 Order Parameters with Subdominant Pairing: dx2−y2 + dxy/s 77

In the clean case (τ = 0) for α = 45, the dominant dx2−y2-wave order param-

eter is fully suppressed at the surface, and in this region an s-wave component

with a phase-shift of ∆ϕ = ±π/2 becomes finite (Fig. 5.15). In the region where

both components are finite, a gap on the whole Fermi surface is present. The

time-reversal symmetry is spontaneously broken by the admixture of a second

parameter with non-trivial phase-shift, which results in a current in y-direction.

As already discussed in Sec. 4.5.1, the energies of the zero energy Andreev bound

states with current in positive or negative y-direction are shifted; one of them to

positive and one of them to negative energies depending on the sign of ∆ϕ. The

current is shown in Fig. 5.19 for positive phase-shift; it is normalized by

j0 = 2eN0vF ∆0. (5.7)

The shift of the zero energy Andreev bound states can also be seen in the con-

ductance in Fig. 5.16: The ZBCP splits and one part moves to positive energies

whereas another part moves to negative energies.

In the presence of disorder, the suppression of the dominant dx2+y2-wave order

parameter at the surface becomes weaker, which leads to a decrease of the sub-

dominant s-wave component. As a result of the smaller admixture, also the

parallel current and the splitting of the ZBCP decrease; moreover, the peaks are

broadened by the disorder as in the previous Sec. 5.1. For τ = 2, the s-wave

component vanishes as the dominant order parameter is no longer suppressed

sufficiently at the surface; the system then behaves in the same way as in Sec. 5.1.

For α = 24, the dominant component is not fully suppressed even in the clean

case. The admixture of an s-wave component is therefore smaller (Fig. 5.17),

which results in a smaller splitting of the ZBCP and a smaller parallel current

(see Figs. 5.18, 5.20). For τ = 0.8 the dominant order parameter is large enough

to lead to a full suppression of the sub-dominant component. The splitting of

the ZBCP as well as the parallel current therefore decrease with growing disorder

until they vanish completely.

It should be mentioned that the phase-shift of the sub-dominant s-wave com-

ponent for α = 24 differs from ∆ϕ = ±π/2; i.e. also a real part of the s-wave

order parameter is added. Depending on its sign, this leads to an increase in the

positive or negative lobes of the dx2−y2-wave component in the surface region. The

real part takes that sign, which increases the lobes nearer to the surface normal.

This can be explained as follows: No sign change of the order parameter occurs

for glancing (ϑ ≈ ±π/2) and almost perpendicular (ϑ ≈ 0) trajectories; i.e. these

78 5 Rough Surfaces

regions contribute only weakly to the suppression of the order parameter at the

surface. But since the scattering probability given in Eq. (5.4) is larger in the

perpendicular direction, the system reduces the pair-breaking in this direction.

Due to symmetry this effect is negligible for α = 45.

For the case dx2−y2 + dxy, the situation changes crucially as the dxy-wave

component itself is sensitive to spatial inhomogeneities. The effect of disorder is

therefore expected to be stronger than for an s-wave admixture.

For α = 45 and τ = 0,the situation of a sub-dominant dxy-wave component

is similar as discussed in the previous case, since no pair-breaking occurs for the

dxy-wave component. As the dominant component is totally suppressed, a finite

sub-dominant component with a phase-shift ∆ϕ = ±π/2 occurs near the surface

(Fig. 5.21). Time-reversal symmetry is therefore broken and a splitting of the

ZBCP peak (Fig. 5.22), as well as a finite current flowing parallel to the surface,

can be observed (Fig. 5.25). The direction of the current changes inside the

superconductor; this is related to the orbital magnetic moment of a dx2−y2 + dxy

order parameter with a finite phase difference, as mentioned in Sec. 2.3.

In this case, disorder acts in various ways on the sub-dominant order param-

eter. First the dxy-wave component is directly suppressed by the disorder (as a

pure untilted dx2−y2-wave order parameter, see Sec. 5.1). Additionally, the same

process as in the previous case occurs: Due to the disorder the suppression of

the dominant dx2−y2-wave component becomes weaker and, as a result, the sub-

dominant dxy-wave admixture is suppressed more effectively. In the end, this

leads to a drastical reduction of the dxy-wave component with increasing disor-

der. For τ = 0.8 it vanishes completely and the situation is the same as without

any dxy-wave component (compare Sec. 5.1).

For α = 24, no sub-dominant dxy-wave component with phase-shift occurs as

the surface already acts pair-breaking. No splitting of the ZBCP and no current is

therefore present. On the other hand, a real-valued dxy-wave admixture appears

(Fig. 5.23), which leads to an effective rotation of the dx2−y2 component, so that

the pair-breaking effect of the surface is reduced. This results in a shift of spectral

weight from the gap region towards the continuum, which can be seen in Fig. 5.24.

This means that the conductance is enhanced for eV & 0.8∆∞ compared with

the pure dx2−y2-wave case (Fig. 5.8). Due to the additional symmetry for α = 45

there a real admixture is negligible.

Summing up, we studied superconductors with a dx2−y2-wave order parameter

and an additional attractive interaction in the dxy-wave or s-wave channel. In the

bulk only the dominant dx2−y2-wave component is present. As surfaces suppress


the dominant component (α 6= 0) also the sub-dominant components can become

finite near the surface. This leads to a spontaneously broken time-reversal sym-

metry of the surface states. We showed, however, that a dxy-wave as well as an

s-wave admixture is suppressed by surface roughness. For very rough surfaces,

the sub-dominant order parameter vanishes completely and time-reversal sym-

metry is restored. For α ∈ (0, 45), also real-valued admixtures are present in

order to reduce the pair-breaking at the surface.

In the case of a specular surface and a tilting of α = 45 similar results

were also obtained elsewhere [26–28]. In experiment a splitting of the ZBCP was

observed [25], which is believed to be a result of time-reversal symmetry breaking

due to the admixture of a sub-dominant order parameter. Other experiments do

not show such a splitting (see Sec. 5.1); our results suggest that one reason might

be a strong surface roughness.

The statistical fluctuations show a similar behavior as in Sec. 5.1; they are

therefore neglected in the current discussion.

It is worth mentioning that the time-reversal symmetry can also be broken

by an external magnetic field, which also leads to a splitting of the ZBCP [27].

80 5 Rough Surfaces

= 2 = 0:8 = 0:4 = 0:08 = 0

is

dx2y2

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.15: Real dx2−y2-wave and an imaginary s-wave component for α = 45

at T = 0.1Tc.

= 0:8 = 0:4 = 0:08 = 0

eV=1

hG(V)iRN

1.210.80.60.40.20

3

2.5

2

1.5

1

0.5

0

Figure 5.16: Differential conductance for α = 45 and an imaginary s-wave ad-mixture to the order parameter at T = 0.1Tc.


= 0:8 = 0:4 = 0:08 = 0

is

dx2y2

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.17: Real dx2−y2-wave and imaginary s-wave component for α = 24 atT = 0.1Tc.

= 0:8 = 0:4 = 0:08 = 0

eV=1

hG(V)iRN

1.210.80.60.40.20

3

2.5

2

1.5

1

0.5

0

Figure 5.18: Differential conductance for α = 24 and an imaginary s-wave ad-mixture to the order parameter at T = 0.1Tc.

82 5 Rough Surfaces

= 0:8 = 0:4 = 0:08 = 0

x=0

hjy(x)i=j0

876543210

0.25

0.2

0.15

0.1

0.05

0

Figure 5.19: Spontaneous current for a dx2−y2 + is order parameter and α = 45,T = 0.1Tc.

= 0:8 = 0:4 = 0:08 = 0

x=0

hjy(x)i=j0

876543210

0.20.180.160.140.120.1

0.080.060.040.02

0

Figure 5.20: Spontaneous current for a dx2−y2 + is order parameter and α = 24,T = 0.1Tc.


= 0:8 = 0:4 = 0:08 = 0

idxy

dx2y2

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.21: Real dx2−y2-wave and imaginary dxy-wave component for α = 45 atT = 0.1Tc.

= 0:8 = 0:4 = 0:08 = 0

eV=1

hG(V)iRN

1.210.80.60.40.20

3

2.5

2

1.5

1

0.5

0

Figure 5.22: Differential conductance for α = 45 and an imaginary dxy-waveadmixture to the order parameter at T = 0.1Tc.

84 5 Rough Surfaces

= 0:8 = 0:4 = 0:08 = 0

dxy

dx2y2

x=0

h(x)i=1

876543210

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

Figure 5.23: Real dx2−y2-wave and and real dxy-wave component for α = 24 atT = 0.1Tc.

= 0:8 = 0:4 = 0:08 = 0

eV=1

hG(V)iRN

1.210.80.60.40.20

3

2.5

2

1.5

1

0.5

0

Figure 5.24: Differential conductance for α = 24 and a real dxy-wave admixtureto the order parameter at T = 0.1Tc.


= 0:8 = 0:4 = 0:08 = 0

x=0

hjy(x)i=j0

876543210

0.010.005

0-0.005-0.01

-0.015-0.02

-0.025-0.03

-0.035

Figure 5.25: Spontaneous current for a dx2+y2 +idxy order parameter and α = 45,T = 0.1Tc.

86 5 Rough Surfaces

5.3 Rough Surfaces Acting as Beam-Splitters

In the previous sections we have studied rough surfaces without any regular

microscopic structure. If for some reason a regular structure is present (e.g.

some zigzag-shape) the surface could act as a beam-splitter; i.e. an incoming

wave splits into several outgoing waves with distinct directions. We construct

a simple S-matrix for beam-splitting surfaces and examine its influence on a d-

wave superconductor. The behavior of Andreev bound states at such surfaces is

of particular interest.

5.3.1 S-Matrix

In a simple model of a beam-splitting surface an incoming particle is reflected

into m possible directions. We assume that specular reflection occurs with weight

|u|2 in the S-matrix and reflection in any of the (m−1) other directions with the

constant weight |v|2. For simplicity we take the number of channels, n, to be a

multiple of m, which leads to a simpler form of the S-matrix but has no physical

implications for m n. The scattering matrix describing such a surface has a

block matrix form

S =

u v · · · v

v. . .

. . ....

.... . .

. . . v

v · · · v u

, (5.8)

where u = u1n/m and v = v1n/m (1k: k × k unity matrix).

As the scattering matrix must be unitary the amplitudes u and v must fulfill

the conditions

|u|2 + (m− 1)|v|2 = 1, (5.9)

uv∗ + u∗v + (m− 2)|v|2 = 0. (5.10)

It can be shown that these conditions are satisfied by a parameterization with a

real variable τ

u(τ) = 1 +eimτ − 1

m, (5.11)

v(τ) =eimτ − 1

m. (5.12)

5.3 Rough Surfaces Acting as Beam-Splitters 87

juj2 = 0:5m = 5

#j = 0n = 100

#i=

P(#j!

#i)

0.40.20-0.2-0.4

25

20

15

10

5

0

Figure 5.26: For m = 5, the scatteringprobability is finite for m = 5 distinctdirections.

This leads to the weight for specular reflection

|u(τ)|2 =1

m2[(m− 1)2 + 1 + 2(m− 1) cos(mτ)]. (5.13)

For fixed m the scattering probability density, which is defined in Eq. (4.62),

reads

P (ϑj → ϑi) =n

2cosϑi

|u(τ)|2δij + |v(τ)|2)

m∑|l|=1

δi,kjl

(5.14)

with kjl = j + ln/m; it is important to note, that for a given in-channel, j,

only m − 1 terms in the sum can contribute as the out-channels are given by

i = 1, . . . , n. As an example, the probability density is presented in Fig. 5.26 for

n = 100 and m = 5. In the continuum limit we find

P (ϑ→ ϑ′) = |u(τ)|2δ(ϑ′ − ϑ) +(1− |u(τ)|2)

m− 1

m∑|l|=1

δ(ϑ′ − ϑl(ϑ)) (5.15)

with

sin(ϑl(ϑ)) = sinϑ+2l

m. (5.16)

This means, that for each in-coming trajectory with direction ϑ, the scattering

probability is non-zero only in m distinct directions.

88 5 Rough Surfaces


In this section, two issues are of particular interest both concerning the ZBCP:

First we examine untilted order parameters (α = 0), to decide whether or not

beam-splitting surfaces can create a ZBCP. Second we consider the tilting α =

45 in order to study the behavior of the ZBCP at a beam-splitting surface; in

Sec. 4.5.2 we already observed that a ZBCP of a d-wave superconductor can split

due to particular surface roughness.

We start with the simplest case, m = 2, where an incoming wave is split into

two outgoing waves. The scattering matrix has the form

S =

(u v

v u

)(5.17)

with u = u1n/2 and v = v1n/2; the functions u and v are given by

u(τ) = cos τ, v(τ) = sin τ. (5.18)

For α = 0 and |u|2 < 1 the order parameter is suppressed due to the rough-

ness (see Fig. 5.27). In the differential conductance (see Fig. 5.28) bound states at

finite energies are observed. Their energies are near the continuum (EA . ∆∞)

for almost specular reflection; for decreasing |u|2, they move towards EA = 0,

which is reached for |u|2 = 0.

For α = 45 the situation is reversed: as a result of the roughness, the order

parameter at the surface is not totally suppressed; the non-specular reflection

leads to a splitting of the ZBCP, which grows with decreasing |u|2. These obser-

vations are in agreement with the non-self-consistent calculation of Sec. 4.5.2.

In a next step, we consider the case m = 3: An incoming wave splits into

three outgoing waves. The S-matrix is given by

S =

u v v

v u v

v v u

(5.19)

with u = u1n/3 and v = v1n/3. It can be seen in Eq. (5.13) that in the present

model the probability for specular reflection can only be chosen in the interval

|u|2 ∈ [1/9, 1].

For α = 0, bound states move from the continuum to lower energies with

decreasing probability of specular reflection (Fig. 5.32). A ZBCP is not created


for any value of |u|2. For α = 45, the result is qualitatively different compared

to m = 2: A ZBCP can be observed for arbitrary |u|2 (Fig. 5.34), i.e. only

some spectral weight of the zero energy bound state is shifted to positive and

negative energies, and another part stays at EA = 0. The order parameter shows

qualitatively the same behavior as for m = 2 (Figs. 5.31, 5.33).

We also considered the case m = 4 (Figs. 5.35-5.38). Here, the weight for

specular scattering can only be chosen in the interval |u|2 ∈ [1/4, 1]. The main

properties of the differential conductance and the order parameter are in quali-

tative agreement with m = 3.

Summing up, we find two general features of beam-splitting surfaces: (i) No

ZBCP is created for α = 0. (ii) For α = 45 the weight of the ZBCP is reduced,

and new peaks in the conductance are created at finite voltages (the case m = 2

must be considered separately).

We compare our results with those found for a surface with a microscopic zig-

zag shape within the Bogoliubov-deGennes approach [81, 82]. Those results are

qualitatively in agreement with ours, since there surface bound states appear at

zero energy as well as at finite energies. We showed that such surfaces can also be

examined within the quasi-classical theory using a phenomenological scattering

matrix.

The ZBCP found in some experiments also for untilted order parameter [24,

25, 29] can neither be explained by beam-splitting surfaces nor by disorder, as

discussed in Sec. 5.1; most probably this ZBCP is a result of facets larger than

the coherence length [27] which are not taken into account here.

90 5 Rough Surfaces

juj2 = 0juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0

Figure 5.27: Order parameter at T = 0.1Tc for α = 0, m = 2.

eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.28: Differential conductance at T = 0.1Tc for m = 2, α = 0 and thesame values of |u|2 as in Fig. 5.27.


juj2 = 0juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0


eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0


92 5 Rough Surfaces

juj2 = 0:11juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0


eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.32: Differential conductanceat T = 0.1Tc for m = 3, α = 0 and thesame values of |u|2 as in Fig. 5.31.


juj2 = 0:11juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0


eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0


94 5 Rough Surfaces

juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0


eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0

Figure 5.36: Differential conductanceat T = 0.1Tc for m = 4, α = 0 and thesame values of |u|2 as in Fig. 5.35.


juj2 = 0:25juj2 = 0:5juj2 = 0:75juj2 = 1

x=0

(x)=1

876543210

1

0.8

0.6

0.4

0.2

0


eV=1

G(V)RN

1.210.80.60.40.20

2

1.5

1

0.5

0


96 5 Rough Surfaces

Chapter 6

Rough Interfaces – Josephson

Junctions

In this chapter we consider interfaces between two d-wave superconductors. The

order parameter on the left and right hand side, respectively, has the form

∆l/r(pF , x) = ∆l/r(x) cos[2(ϑ − αl/r)]. We focus in particular on two junction

geometries: (i) the asymmetric junction with αl = 0 and αr = 45, and (ii) mir-

ror junctions with αl/r = ±α for various angles α. In the clean case both junction

types show some unusual behavior (see Sec. 4.5.3): The asymmetric junction has

a π-periodic current-phase relation; the mirror junction shows a transition from

a 0- to a π-junction with decreasing temperature. Our main concern is the sta-

bility of this behavior with respect to interface disorder, which usually is present

in experiments.

We begin by constructing a scattering matrix for irregular interfaces which al-

lows us to control the roughness and the transparency independently. Afterwards

we study the asymmetric and mirror junctions. For a given phase difference of

the self-consistently determined order parameter we evaluate the supercurrent

across the junction. From the current-phase relations we extract the critical cur-

rent which is defined as the maximum current that can pass the interface without

voltage drop. Additionally we study possible currents parallel to the interface.

6.1 S-Matrix for Rough Interfaces

In this section a model for rough (irregular) interfaces is presented. The roughness

and the transparency are implemented independently via the following procedure:

98 6 Rough Interfaces – Josephson Junctions

We construct a S-matrix for a rough interface by

S =

(U1 0

0 U2

)(T R

R −T

)(U3 0

0 U4

)(6.1)

with T = diag√T (ϑi), R = diag

√1− T (ϑi) and four arbitrary unitary

matrices Uk, k = 1, 2, 3, 4 as suggested in Ref. [83]; the scattering matrix for

an ideal interface as given in Eq. (4.63) can be obtained for Uk = 1. The

transparency of the interface is purely given by the function T (ϑi) as defined in

Eq. (4.27), whereas the roughness is determined by the unitary matrices Uk.

We will focus on interfaces without regular structure on the microscopic

scale constructing the matrices Uk in the same way as for disordered surfaces

in Sec. 5.1.1; we choose

Uk = expiHk (6.2)

where each Hk is a random matrix with Gaussian correlations

〈Hkij〉 = 0, 〈Hk

ij

∗Hk′

i′j′〉 =τ

2nδii′δjj′δkk′. (6.3)

The matrices Uk show the same average properties as the scattering matrix de-

fined in Eqs. (5.1) and (5.2). We can therefore define 〈|Uk,ii|2〉 ≡ |u(τ)| and

〈|Uk,i6=j|2〉 ≡ |v(τ)|; the unitarity leads to the relation |u| + (n − 1)|v| = 1. Due

to the additional factor 1/2 in the correlator, for small τ , the |u| defined here is

identical to that defined in Sec. 5.1.1, which is shown in Fig. 5.3. This means

that for small τ and T (ϑi) = 0 the results obtained by the scattering matrix,

defined in Eq. (6.1), and those found in Sec. 5.1 are identical.

Using the Eqs. (4.62) and (4.27), the averaged scattering probability density

reads (n 1)

〈Plr(ϑj → ϑi)〉 = 〈Prl(ϑj → ϑi)〉 =1

2cosϑi

n|u|2T (ϑi)δij+

+ |u|(1− |u|)[T (ϑi) + T (ϑj)](1− δij) + κ(1− |u|)2,

(6.4)

with

κ =1

n

∑i

T (ϑi) →

T0/2 for T0 1

1 for T0 = 1. (6.5)

6.2 Asymmetric Junctions 99

SC

plF in

prF in

plFout

I SC

x

y

py py

pxpx

prFout

prF in

plF in

plFout

prFout

45Æ

Figure 6.1: An asymmetric 45-junction consists of two superconductors with anuntilted d-wave order parameter on the left side, and a d-wave order parameterwhich is tilted by αr = 45 on the right side.

The probabilities 〈Prr〉 = 〈Pll〉 can easily be obtained via the substitution T →(1− T ) and κ→ (1− κ) in Eq. (6.4).

The continuum limit leads to

〈Plr(ϑ→ ϑ′)〉 = 〈Prl(ϑ→ ϑ′)〉 = |u|2T (ϑ′)δ(ϑ′ − ϑ)+

+1

2cosϑ′

|u|(1− |u|)[T (ϑ′) + T (ϑ)] + κ(1− |u|)2

.

(6.6)

For the evaluation of physical quantities in this chapter, we will use n = 40

directions on each side of the junction; this defines the typical angle ϑc = π/40

up to which the scattering probability can be assumed to be correlated. The

influence of a finite number of scattering channels, n, on physical quantities, in

particular on their statistical fluctuations, is the same as discussed for surfaces

in Sec. 5.1.

6.2 Asymmetric Junctions

We will first consider the asymmetric junction with αl = 0 and αr = 45,

which is well-known for its unusual current-phase relation, as discussed already

in Sec. 4.5.3.

In general, the current-phase relation of any junction existing of d-wave su-

perconductors on both sides is given by the expansion

Ix(ϕ) = I1 sin(ϕ) + I2 sin(2ϕ) + . . . (6.7)


with the phase difference, ϕ, between the left and the right hand side. For tunnel

junctions (T0 1), the current contributions scale as

Ik ∝ T k0 . (6.8)

Here only the linear term in the transparency, I1, is relevant. For larger trans-

parency higher order terms must also be taken into account.

In the present situation, I1 vanishes due to the particular geometrical sym-

metry of the system: The symmetry operation y → −y leads to a phase-shift

ϕ→ ϕ+ π; on the other hand, the current in x-direction must be invariant. We

therefore find the following condition for the current

Ix(ϕ) = Ix(ϕ+ π) ⇔ I2k−1 = 0, k ∈ N (6.9)

and the current-phase relation takes the form

Ix(ϕ) = I2 sin(2ϕ) + I4 sin(4ϕ) + . . . . (6.10)

The dominant sin(2ϕ)-behavior was already shown in Eq. (4.91) in a non-self-

consistent calculation. Asymmetric junctions therefore have two degenerate

ground states at ϕ = π/2, 3π/2. For this reason such junctions are discussed

as possible realizations of qubits [84, 85].

In general, for a rough interface, the subtle symmetry responsible for the

sin(2ϕ)-behavior is broken and a finite current I1 exists. We therefore ask which

conditions guarantee the unusual current-phase relation for a rough surface.


We will examine the average values of the currents I1 and I2 as well as their

standard deviations. Since, in our model, the symmetry y → −y is still present on

average, it follows immediately that 〈I1〉 = 0; on the other hand the fluctuations

can be finite√〈(∆I1)2〉 > 0.

The results for small transparency (T0 = 0.01) are presented in Figs. 6.2

and 6.3: In the clean case I2 < 0 is finite whereas I1 = 0 vanishes; i.e. the current-

phase relation is purely sin(2ϕ)-like. With increasing roughness the averaged

contribution 〈I2〉 is slightly suppressed, whereas fluctuations of I1 become finite,

i.e.√〈(∆I1)2〉 > 0. For strong disorder the fluctuations can be of the order of

〈I2〉 or bigger (τ = 2).


The temperature dependence (Figs. 6.2 and 6.3) of both contributions is

also quite different: 〈I2〉 grows rapidly for decreasing temperatures (compare

Sec. 4.5.3), whereas√〈(∆I1)2〉 stays almost constant for T → 0. This can also be

seen in the current-phase relation of a typical realization, as presented in Figs. 6.4

and 6.5: For T = 0.1Tc, the influence of the I2 contribution is more pronounced

than for T = 0.5Tc. Depending on the roughness, three different scenarios can be

observed: (i) For small roughness (τ . 0.08), the sin(2ϕ)-behavior is present al-

most up to the critical temperature. (ii) For medium roughness (τ ≈ 0.4), the I2contribution dominates for low temperatures, and the I1 contribution for higher

temperatures; a cross-over occurs at finite temperatures. (iii) For large roughness

(τ & 0.8), the usual sinϕ-like current-phase relation exists for all temperatures

T < Tc.

For higher transparencies, the influence of the I2 contribution increases, which

can be seen for T0 = 0.1 in Figs. 6.6 and 6.7; this is in agreement with the trans-

parency dependence of the coefficients Ik as given in Eq. (6.8). The dependence

on the roughness and the temperature of the examined quantities is qualitatively

the same as for T0 = 0.01. But here the temperature and the roughness must be

larger in order to destroy the sin(2ϕ)-behavior of the junction: For T = 0.1Tc the

current-phase relations for typical realizations of the roughness is sin(2ϕ)-like up

to τ = 0.4 (Fig. 6.8). For T = 0.5Tc this behavior vanishes for τ = 0.4 or higher

(Fig. 6.9).

We now concentrate on junctions with a sin(2ϕ)-like current-phase relation

(i.e. the roughness and/or the temperature are small enough). In particular, we

examine the properties of the ground state, which has a finite phase difference

ϕ ≈ π/2, 3π/2 between the left and right hand side of the junction.

In the ground state a current parallel to the junction exists whereas no current

in x-direction is present. This is shown in Figs. 6.10 and 6.11 for ϕ = π/2 (for

ϕ = 3π/2 the direction of the current is reversed); the parallel current scales with

the transparency T0. Surface roughness leads to a suppression of the current on

the side with the untilted order parameter, whereas on the other side the current

grows with growing roughness.

This can be explained as follows: At a totally reflecting interface no bound

state exists on the left hand side and a zero energy bound state is present on

the right hand side of the junction (where the order parameter is tilted). For a

finite transparency, a bound state is also induced on the left hand side (compare

Eq. (4.94)); its spectral weight grows with the transparency (compare Eq. (4.93)).

The finite phase difference leads to a shift of the zero energy bound states on both


sides, so that some of them lie below the Fermi energy and are occupied. These

Andreev bound states carry the parallel current. The current on the right side is

much larger than that on the left side, as the spectral weight of the bound states

on this side is also larger.

Interface roughness has a different effect on the current on the left and right

side. On the left side (with an untilted order parameter) the induced bound

state is suppressed by the growing disorder and the current decreases. On the

right side the bound state is more stable against disorder. An additional effect

leads to a growing current with increasing roughness: For an ideal surface the

transparency of glancing trajectories (ϑ ≈ ±90) as given in Eq. (4.27) is very

small; due to disorder these trajectories are coupled more effectively to the left

side of the junction. Their contribution to the current therefore increases. As

these trajectories have a large y-component, the current on the side of the tilted

order parameter is enhanced. This effect should be smaller if the transparency

in the clean case is already large, which is in agreement with our results.

Asymmetric junctions were also considered in the literature [41,86,87]. There,

however, the fluctuations of the current I1 have not been studied. The average

quantities obtained within our approach are in good agreement with those re-

sults. Note that the presented junction has the same qualitative properties if the

untilted d-wave superconductor (left side) is substituted by an s-wave supercon-

ductor.

In experiment the sin(2ϕ)-like behavior was observed by Il’ichev et al. [39].

However, only some of their samples showed the sin(2ϕ)-like current-phase rela-

tion. Obviously in these samples the symmetry responsible for I1 = 0 is broken.

As mentioned above, one reason could be microscopic roughness; another possi-

bility is the uncertainty of the orientation of the order parameter, αr = 45± 1.

As already discussed in Sec. 5.1, the statistical fluctuations of the current

contribution I1 might be important in mesoscopic realizations of such junctions,

where only particular realizations of the disorder are measured. We showed that

in such junctions the statistical fluctuations can destroy the sin(2ϕ)-like behavior

if the temperature and/or the interface disorder are high enough. For large

junctions, however, the influence of the fluctuations becomes irrelevant as they

average out.


= 2 = 0:8 = 0:4 = 0:08 = 0

T=Tc

ehI 2iRN

=0

10.80.60.40.20

0.005

0.004

0.003

0.002

0.001

0

Figure 6.2: Average value of the current I2 for T0 = 0.01 and varying strength ofthe roughness; interface roughness suppresses the current I2.

= 2 = 0:8 = 0:4 = 0:08

T=Tc

eqh(I 1)2iRN

=0

10.80.60.40.20

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

Figure 6.3: Standard deviation of the current I1 for T0 = 0.01 and varyingstrength of the roughness; this current contribution is increasing with the rough-ness. We used n = 40 scattering channels.


'=

eIxRN

=0

10.80.60.40.20

0.016

0.012

0.008

0.004

0

-0.004

Figure 6.4: Current-phase relation for a typical realization of the disorder forT = 0.1Tc; the roughness and the transparency are the same as in Fig. 6.2.

'=

eIxRN

=0

10.80.60.40.20

0.005

0.004

0.003

0.002

0.001

0



= 2 = 0:8 = 0:4 = 0:08 = 0

T=Tc

ehI 2iRN

=0

10.80.60.40.20

0.05

0.04

0.03

0.02

0.01

0

Figure 6.6: Average value of the current I2 for T0 = 0.1 and varying strength ofthe roughness; interface roughness suppresses the current I2. Compared with theresults for T0 = 0.01 (Fig. 6.2) the scale is enhanced by a factor 10.

= 2 = 0:8 = 0:4 = 0:08

T=Tc

eqh(I 1)2iRN

=0

10.80.60.40.20

0.006

0.004

0.002

0

Figure 6.7: Standard deviation of the current I1 for T0 = 0.1 and varying strengthof the roughness; this current contribution is increasing with the roughness. Weused n = 40 scattering channels.


'=

eIxRN

=0

10.80.60.40.20

0.02

0.01

0

-0.01

-0.02


'=

eIxRN

=0

10.80.60.40.20

0.004

0.003

0.002

0.001

0

-0.001



= 2 = 0:8 = 0:4 = 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.0016

0.0012

0.0008

0.0004

0

Figure 6.10: Averaged current density parallel to the junction for T0 = 0.01 andvarying strength of the roughness.

= 2 = 0:8 = 0:4 = 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.016

0.012

0.008

0.004

0

Figure 6.11: Averaged current density parallel to the junction for T0 = 0.1 andvarying strength of the roughness. Compared with the results for T0 = 0.01(Fig. 6.10) the scale is enhanced by a factor 10.


6.3 Mirror Junctions

In this section we will consider mirror junctions, which are illustrated in Fig. 6.12.

For an ideal junction, a particular symmetry is present, which reads

∆l(plF in) = ∆r(pr

F in),

∆l(plFout) = ∆r(pr

Fout).(6.11)

This means that, for a given incoming quasi-particle, the order parameter is iden-

tical for the transmitted and the reflected quasi-particle. All scattering processes

can therefore be divided into two classes: For some directions the sign of the

order parameter is the same for all involved trajectories (∆l(plF in)∆

l(plFout) > 1).

This s-wave-like case was already studied non-self-consistently in Sec. 4.5.3. As

can be seen in Eq. (4.81), for fixed phase difference the current contribution of

these directions is positive; the current is carried by bound states near the con-

tinuum (see Eq. (4.84)). For other directions the sign changes for the in- and

out-trajectories (∆l(plF in)∆

l(plFout) < 0), which is also studied in Sec. 4.5.3. As

can be seen in Eq. (4.86), this leads to a negative current contribution which is

rapidly increasing for low temperatures; these contributions are carried by bound

states near zero energy (see Eq. (4.89)). Altogether the s-wave-like contributions

dominate for large temperatures, which, for a fixed phase difference, leads to a

positive current, whereas for low T the anomalous contributions are enhanced,

which results in a negative current.

From another point of view, the ground state of the junction shifts from

ϕ = 0 (0-junction) to ϕ = π (π-junction) for decreasing temperatures. This

transition occurs at the temperature Tπ, where both contributions cancel; at

this temperature, the situation is comparable to that of Sec. 6.2 as the leading

sinϕ-contribution vanishes and the junction is dominated by higher order terms.

As the amount of directions preferring a π-junction increases with an growing

angle α, also the temperature Tπ increases (Tπ = 0/Tc for α = 0/45).

In the following we will discuss the influence of interface roughness on the

temperature dependence of the critical current. In particular the transition to a

π-junction is examined.


We calculate the current contributions I1 and I2 as defined in Eq. (6.7), and

evaluate the critical current, Ic, which is defined as the maximum current that

6.3 Mirror Junctions 109

SC I SC

x

y

py py

pxpx

plF in

prF in

plFout p

rF in p

rFout

plF in p

lFout

prFout

Figure 6.12: In mirror junction the order parameters on both sides are tilted bythe same amount, α, but in opposite directions. For an ideal interface, the currentcontribution of some directions is positive (dashed lines), whereas other directionshave negative contribution (solid lines), as the sign of the order parameter changesat the interface.

can pass the junction without voltage drop (V = 0). A negative value of Ic in the

graphs indicates a π-junction behavior. The temperature dependence is presented

for several orientations and varying roughness in the range from the clean case

to the very rough limit (τ ∈ [0, 4]). Moreover we compare the critical current for

high and low transparency: The results for T0 = 0.01 (tunnel-limit) are shown in

Figs. 6.13-6.16, and those for T0 = 0.2 in Figs. 6.17-6.20. At first glance, we see

only a small difference of the product IcRN for different transparencies. We will

therefore mainly discuss the dependence on the orientation α and the roughness

of the interface.

For α = 0 (Figs. 6.13 and 6.17), an Ambegaokar-Baratoff-like behavior [88]

is found for τ = 0. For growing roughness the critical current is reduced. The

main reason is the suppression of the d-wave order parameter in the vicinity of a

rough surface/interface (see Sec. 5.1).

The situation becomes more interesting if we consider non-trivial mirror junc-

tions with α > 0 (Figs. 6.14-6.16, and 6.18-6.20). We will concentrate on the clean

case first. There, as mentioned above, the directions preferring a π-junction and

those preferring a 0-junction compete: We find π-junction behavior below a spe-

cific temperature Tπ (see table 6.1), whereas for high temperatures the usual

0-junction state is favored. This can be seen in the leading current contribu-

tion I1 which changes its sign at Tπ. As already discussed, the temperature Tπ

increases with the angle α as a larger amount of directions prefer a π-junction.


α τ T0 Tπ/Tc T−π /Tc T+

π /Tc

22.5 0 0.01 0.358 0.357 0.3590.2 0.368 0.355 0.382

22.5 0.08 0.01 0.322 0.321 0.3230.2 0.332 0.315 0.349

22.5 0.4 0.01 0.223 0.223 0.2240.2 0.225 0.209 0.241

18 0 0.01 0.151 0.151 0.1520.2 0.132 0.097 0.150

18 0.08 0.01 0.129 0.128 0.1300.2 0.110 0.082 0.128

Table 6.1: Temperature of the transition to a π-junction, Tπ, and temperatureregion (T−

π , T+π ) with sin(2ϕ)-like current-phase relation.

For decreasing temperatures near Tπ, the shift of the ground state from ϕ = 0

to ϕ = π can be seen in the current-phase relation, which is shown in Figs. 6.21,

6.22 for α = 22.5 and T0 = 0.01, 0.2. In particular a finite critical current can be

observed near Tπ, since the contribution I2 stays finite, whereas I1 = 0 vanishes

at Tπ. This leads to a degenerate ground state at Tπ, as the current-phase relation

is sin(2ϕ)-like.

In contrast to our findings for asymmetric junctions in the previous section,

the sin(2ϕ)-contribution dominates the behavior of the junction only in a small

temperature region around Tπ. This means that for T ∈ (T−π , T

+π ) the condition

|I1| < |I2| is obeyed (see table 6.1), and the ground state can be found at a non-

trivial phase difference ϕ 6= 0, π. This leads to a ground state current parallel to

the junction. However, in contrast to the asymmetric junction, here the currents

on both sides have opposite directions, and identical absolute value, which scales

with the transparency. The parallel current is presented in Figs. 6.23-6.26.

Obviously the roughness suppresses the π-junction behavior. With increasing

τ , the temperature Tπ decreases until the transition vanishes for very rough in-

terfaces. This can be understood as follows: The negative current contributions

are carried by the zero energy bound states; as they are broadened by disorder

the strong increase of the negative contribution to I1 for T → 0 becomes weaker.

Due to this suppression at rough interfaces, the normal 0-junction contribution

becomes dominating also for lower temperatures, and we can find I1 > 0 in the

whole temperature range. The π-junction contribution can also be suppressed by


α ARN(Ωcm2) T0 RNIc(mV) RNIc(mV)Ref. [37] estimated Ref. [37] calculated

12 5.4× 10−9 0.024 1.3 3.618 1.5× 10−8 0.008 0.75 2.622.5 1.2× 10−8 0.011 0.13 1.9

Table 6.2: Comparison of experimental results for IcRN and calculated values;for the calculated values we choose T0 = 0.01, T = 0.05Tc, and τ = 4.

a large transparency, which leads to a splitting of the zero energy bound state

(see Eq. (4.89)): For α = 12 and T0 = 0.2 (Fig. 6.18), the π-junction behavior

vanishes even in the clean case.

If the π-junction is completely suppressed, the critical current is enhanced

with growing roughness until the negative contributions to I1 have vanished.

Then, the temperature dependence of the critical current is almost universal;

i.e. Ic(T )/Ic(0) depends only weakly on the orientation α. If the roughness is

increased further, the critical current is suppressed in the usual way as for α = 0.

The parallel current in the ground state (Figs. 6.23-6.26) increases rapidly

with growing disorder, which can be understood as follows: As discussed in the

previous section the transmission probability for glancing trajectories is enhanced

by finite roughness. As these trajectories have a large y-component the parallel

current increases strongly.

We also studied the relative fluctuations of the currents I1 and I2, which are

found to be less then 15% for n = 40 scattering channels. This means that the

values of the critical current or of Tπ can differ slightly for a particular realization

of the disorder, but the qualitative behavior is well described by their average

values.

In the following, we will compare our results for the critical current with

experimental data of Refs. [16, 36, 37, 89]. There an almost universal behavior

of the quantity I(T )/I(0) was reported, independent of the orientation of the d-

wave order parameter; the critical current grows monotonically with decreasing

temperature. This is in agreement with our observations for very rough interfaces.

In table 6.2 we compare the numerical results for the product RNIc with the

measurements of Ref. [37] at T = 4.2K. For YBCO (Tc ≈ 90K), we used the

values vF = 4.34 × 106cm/s and N0 = 2.13 × 1022/cm3eV [90] to estimate the

transparency T0 from the resistance RN . We find that the junctions are in the

tunnel-limit (T0 ≈ 0.01). Therefore, we evaluate IcRN at T = 0.05Tc for T0 = 0.01


Ca-conc. ARN(Ωcm2) T0 RNIc(mV) Tc(K) RNIc(mV)Ref. [43] estimated Ref. [43] Ref. [43] calculated

0 1× 10−8 0.005 3.3 92 3.70.04 5× 10−9 0.01 2.8 87 3.50.15 1.5× 10−9 0.04 1.75 79 3.20.2 1× 10−9 0.05 1.75 80 3.30.3 6× 10−10 0.09 1.25 79 3.30.4 1× 10−9 0.05 1 75 3.2

Table 6.3: Comparison of measurements of IcRN for an α = 12-junction withour results where we used T0 = 0.01, 0.1, T = 0.05Tc, and τ = 4.

and τ = 4 (in principle τ could be used as a fitting parameter, which was not

possible due to limitations of computing time).

We realize that the calculated critical currents are large when compared to

the measurements, and the agreement becomes worse with increasing angle α.

One reason might be that we neglected the influence of facets on a µm-scale [80];

in other words each junction is effectively an average over all facets with distinct

orientations. Some of these facets show a π-junction behavior, i.e. their current

contribution is negative. The strong decrease of the IcRN product for growing α

in the experiments can be explained by the existence of such π facets, as their

number is increasing. Another possible explanation could be that the microscopic

roughness increases with growing angle α, whereas our results were calculated for

constant τ .

As reported recently [43], it is also possible to modify the properties of the

junctions without varying the orientation, α. The main idea is to alter the charge

carrier density in the vicinity of the interface. In YBCO this can be achieved by

doping with Ca-atoms. As can be seen in table 6.3, in our model this leads to an

increase of the effective transparency of the contact. Comparing our results for a

constant roughness, τ = 4, with the experimental results of Ref. [43] a qualitative

difference can be seen: In the experiment the product IcRN decreases faster than

in our calculations with increasing transparency. A reason for this discrepancy

could be an increase of the microscopic interface disorder with increasing doping.

Summing up, our results have the right order of magnitude compared with

experiments. But it is not possible to understand all aspects within our simplified

model. In particular, it is unclear how the interface roughness is modified if the

orientation, α, or the doping with Ca-atoms are changed.

Very recently, for a mesoscopic junction with α = 22.5 a non-monotonic


behavior of the temperature-dependent critical current has been reported [40],

which appears due to the transition to a π-junction for low temperatures (note

that in experiment the absolute values of Ic are measured). The minimum of

the critical current was found for Tπ = 12K = 0.13Tc. The transparency is quite

high, as can be seen from the large sin(2ϕ)-contribution. The measurement is in

qualitative agreement with our calculations for T0 & 0.2 and τ & 0.4. It is also

worth mentioning that this transition was observed only for some of the samples;

this might be due to the sensitivity of mesoscopic junctions to fluctuations of the

interface properties. Note that here facets are of minor importance, as the width

of the junction (≈ 0.5µm) is of the same order of magnitude as the typical facet

size (≈ 0.1µm); so it is possible to have junctions with well-defined orientation.

In summary, we showed, that the behavior of mirror junctions can be modified

drastically by interface roughness: If there is only little roughness and the angle

α is large enough, a transition to a π-junction at small temperatures can be

observed; near the transition temperature Tπ, the junction has a sin(2ϕ)-like

current-phase relation. For larger roughness the π-junction behavior is destroyed

and a universal behavior of the quantity Ic(T )/Ic(0) is observed. When comparing

with experimental data, we have to realize that certain aspects of mirror junctions

cannot be explained within our model of interface roughness. In particular, large

facets should be taken into account.


= 4 = 2

= 0:8 = 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

1.2

1

0.8

0.6

0.4

0.2

0

Figure 6.13: Average critical current for T0 = 0.01 and α = 0.

= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.6

0.5

0.4

0.3

0.2

0.1

0



= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1


= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.15

0.1

0.05

0

-0.05

-0.1

Figure 6.16: Average critical current for T0 = 0.01 and α = 22.5.


= 4 = 2

= 0:8 = 0:4 = 0:08 = 0

T=Tc

hIciRN

=0

10.80.60.40.20

1.2

1

0.8

0.6

0.4

0.2

0


= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.6

0.5

0.4

0.3

0.2

0.1

0



= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1


= 4 = 2 = 0:8

= 0:4 = 0:08 = 0

T=Tc

ehI ciRN

=0

10.80.60.40.20

0.15

0.1

0.05

0

-0.05

-0.1

Figure 6.20: Average critical current for T0 = 0.2 and α = 22.5.


'=

eIxRN

=0

10.80.60.40.20

0.0015

0.001

0.0005

0

-0.0005

-0.001

-0.0015

-0.002

Figure 6.21: Current-phase relation of an ideal α = 22.5 junction with T0 = 0.01at T ≈ Tπ ± k0.001Tc (k = 0, 1, 2, 3; increasing T from bottom to top).

'=

eIxRN

=0

10.80.60.40.20

0.010.0080.0060.0040.002

0-0.002-0.004-0.006-0.008-0.01

-0.012

Figure 6.22: Current-phase relation of an ideal α = 22.5 junction with T0 = 0.2at T ≈ Tπ ± k0.005Tc (k = 0, 1, 2, 3; increasing T from bottom to top).


= 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.0001

8e-05

6e-05

4e-05

2e-05

0

-2e-05

-4e-05

-6e-05

-8e-05

-0.0001

Figure 6.23: Parallel current for α = 18, T0 = 0.01, and T = Tπ (see table 6.1).

= 0:4 = 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.00015

0.0001

5e-05

0

-5e-05

-0.0001

-0.00015

Figure 6.24: Parallel current for α = 22.5, T0 = 0.01, and T = Tπ (see table 6.1).


= 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.0025

0.002

0.0015

0.001

0.0005

0

-0.0005

-0.001

-0.0015

-0.002

-0.0025

Figure 6.25: Parallel current for α = 18, T0 = 0.2, and T = Tπ (see table 6.1).

= 0:4 = 0:08 = 0

x=0

hjyi=j 0

86420-2-4-6-8

0.003

0.002

0.001

0

-0.001

-0.002

-0.003

Figure 6.26: Parallel current for α = 22.5, T0 = 0.2, and T = Tπ (see table 6.1).

Chapter 7

Conclusions

In this work we examined d-wave superconductors in the vicinity of rough inter-

faces. For this purpose we applied a novel method to treat boundaries within the

quasi-classical theory of superconductors.

First we introduced the boundary conditions recently developed by Shelankov

and Ozana, and adapted them to the problem of surfaces and interfaces in such

a way that the current conservation across the junction is guaranteed. We fi-

nally presented a universal scheme for the treatment of junctions. The main

advantage compared to other approaches is the possibility to implement arbi-

trary boundaries. The physical properties of these interfaces are represented by

a unitary scattering matrix, which is determined by the microscopic structure

of the interface. We suggested different phenomenological scattering matrices to

describe beam-splitting or disordered interfaces. In the latter case, it is possible

to investigate single realizations of the disorder as well; i.e. we also consider the

statistical fluctuations of the physical quantities, which might be of importance

in mesoscopic systems.

In order to study disordered surfaces, we used a random scattering matrix,

which leads to partially specular scattering. For tilted d-wave order parame-

ters we find a zero bias conductance peak (ZBCP). We showed that the width

of the ZBCP becomes broader with increasing disorder. We also discussed the

influence of statistical fluctuations, which are found to be less important for

surfaces. Furthermore we examined d-wave superconductors with a possible ad-

mixture of a sub-dominant order parameter at a rough surface; we discussed the

case dx2−y2 + dxy as well as dx2−y2 + s. Without disorder there is a finite phase

difference between the dominant and the sub-dominant component of the order

parameter, which leads to a breaking of the time-reversal symmetry of the sur-

122 7 Conclusions

face states. We showed that the time-reversal symmetry is restored for strong

enough roughness both for a dxy-wave and for an s-wave admixture. Also an

admixture without phase difference can occur, which reduces the pair-breaking

at the surface.

Moreover we constructed a scattering matrix for a beam-splitting surface,

which could be related to a surface with microscopically regular roughness (e.g.

a zig-zag shape). We found that such microscopic roughness does not lead to a

ZBCP for an untilted order parameter; for a tilted order parameter, some fraction

of the ZBCP is shifted to finite energies.

For the treatment of disordered interfaces, we constructed the scattering ma-

trices in analogy to those for disordered surfaces. We studied two junction geome-

tries. For an ideal asymmetric junction, the leading contribution of the tunnel

current vanishes due to the particular geometrical symmetry of the system; this

leads to an unusual sin(2ϕ)-behavior of the current-phase relation. We considered

individual realizations of the disorder, where this symmetry no longer exists. We

showed that the statistical fluctuations lead to a non-vanishing sinϕ-contribution

to the tunnel-current, which can restore the sinϕ-like current-phase relation for

strong disorder.

Mirror junctions also show an unusual behavior for ideal scattering at the

interface: We observe a π-junction for low temperatures and a usual 0-junction

for higher temperatures. As a result the temperature dependent critical current

has a local minimum at a finite temperature Tπ < Tc. We found that the π-

junction behavior is suppressed by disorder; i.e. Tπ is reduced. For strong disorder

no π-junction occurs, and we observe a universal temperature dependence of the

critical current (i.e. it does not depend on the orientation of the order parameter),

which has no local minimum at at any temperature T < Tc. When comparing our

results with experimental data [37, 43] (without a minimum in the temperature

dependent critical current) we found reasonable agreement for strong disorder,

although we are not able to explain all details. In a recent experiment [40] a

local minimum in the temperature dependent critical current has been observed,

which is consistent with our findings for small disorder.

To summerize, we demonstrated that the boundary conditions of Shelankov

and Ozana can be successfully applied to describe interfaces and surfaces of su-

perconductors. In particular we considered d-wave superconductivity, which is

relevant for high-Tc materials. We found a variety of interesting new results for

rough interfaces. At the same time we have to admit that our description of an

interface by just two parameters – the transparency and the roughness – is too

123

crude for a detailed understanding of all experimental data. In future calculation

facets should quantitatively be taken into account. The description of interfaces

should also include modifications of the electronic structure at surfaces of high-Tc

materials. Moreover, a thorough adaption of our method to mesoscopic systems

would be promising.

124 7 Conclusions

Appendix

A Keldysh Green’s Function in Thermal Equi-

librium

In order to find the Keldysh Green’s function in thermal equilibrium we use its

definition (3.7)

GK = G> + G<. (7.1)

The correlation functions of two operators A(t) and B(0) in thermal equilibrium

is given by

〈A(t)B(0)〉 =1

Tr [e−H/T ]Tr[e−H/T eiHtAe−iHtB

]. (7.2)

Due to the cyclic invariance of the trace the following relation can be found

〈A(t1 − t2)B(0)〉 = 〈B(t2)A(t1 + iβ)〉. (7.3)

This can be applied to the Green’s function G>/< which are defined in Eqs. (3.2)

and (3.1)

1

−iG>(t) =

1

iG(t+ iβ). (7.4)

with t = t1− t2; here we dropped the spatial dependence. After Fourier transfor-

mation this reads

G>(E) = −eβEG<(E) (7.5)

126 Appendix

and

GK(E) = (1− eβE)G<(E). (7.6)

Using the definitions (3.5)-(3.7), G< can be expressed by

2G< = GK − GR + GA. (7.7)

The two latter equations result in

GK(E) = tanh

(E

2T

)[GR(E)− GA(E)

]. (7.8)

B Bullet-Product

We first concentrate on the time dependence of the functions and make a trans-

formation of variables from (t = 12(t1 + t3), t

′ = t1 − t3) to (t, E). This leads to

the bullet product:

(AB)(E, t) ≡∫

dt′eiEt′∫

dt2A(t1, t2)B(t2, t3) =∫dt′eiEt′

∫dt2

∫dE ′

2πe−iE′(t1−t2)A(E ′, (t1 + t2)/2)×∫

dE ′′

2πe−iE′′(t2−t3)B (E ′′, (t2 + t3)/2) .

(7.9)

We now introduce a formal representation of the Taylor expansion A(t + ∆t) =

e∆t∂tA(t) and arrive at

∫∫dE ′

2π

dE ′′

2π

∫dt2

∫dt′eiEt′e−iE′(t1−t2)−iE′′(t2−t3)×

e12(t2−t3)∂A

t + 12(t2−t1)∂B

t A(E ′, t)B(E ′′, t) =∫∫dE ′

2π

dE ′′

2π

∫dt′δ

(−E ′ + E ′′ + i∂A

t /2 + i∂Bt /2

)×

eiEt′e−iE′t1+iE′′t3− 12t3∂A

t − 12t1∂B

t A(E ′, t)B(E ′′, t) =∫dE ′′

2π

∫dt′eit′(E−E′′− i

2∂A

x )A(E ′′ + i∂A

t /2 + i∂Bt /2, t

)B(E ′′, t) =∫

dE ′′

2πδ(−E ′′ + E − i∂A

t /2)A(E ′′ + i∂A

t /2 + i∂Bt /2, t

)B(E ′′, t) =

A(E + i∂B

t /2, t)B(E − i∂A

t /2, t)

= ei(∂AE ∂B

t −∂At ∂B

E )/2A(E, t)B(E, t).

(7.10)

C Self-Consistency Equation 127

In the Fourier transformation of the spatial arguments an additional sign occurs;

this leads to a sign change in the exponent of the final result and the bullet

product is given by

A •B ≡ (AB)(E,p, t, r) = ei(∂Ar ∂B

p −∂Ap ∂B

r )/2e−i(∂At ∂B

E−∂AE ∂B

t )/2AB (7.11)

C Self-Consistency Equation

Starting with Eq. (3.19)

ΣR/AS (x1, x2) =

i

2GKV (x1 − x2) (7.12)

we transform the superconducting self-energy to a form more suitable for further

calculations. With the relative and the center-of-mass coordinates, (t′,p′) and

(t,p), as defined in Eqs. (3.34) and (3.35) we find

ΣR/AS (E,p, t, r) =

i

2

∫dr′∫

dt′e−i(pr′−Et′)×∫dp′

(2π)d

∫dE ′

2πei(p′r′−E′t′)V (p′)

∫dp′′

(2π)d

∫dE ′′

2πei(p′r′−E′t′)G(E ′′,p′′; t, r) =

=i

2

∫dp′

(2π)d

∫dE ′

2πV (p′)×

×∫

dp′′

(2π)d

∫dE ′′

2πG(E ′′,p′′; t′, r′)δ(p− p′ − p′′)δ(E − E ′ − E ′′) =

=i

2

∫dp′′

(2π)d

∫dE ′′

2πV (p− p′′)G(E ′′,p′′; t, r).

(7.13)

In the BCS approximation the interaction is assumed to be constant near the

Fermi surface and vanishes for large momenta (compare Eq. (2.20)): V (p−p′) →V (pF ,p

′F ); the interaction is cut off at an energy Ec. With Eq. (2.21) and the

definition ΣR/AS = −i∆ we find

∆(p; t, r) = −1

2N0

Ec∫−Ec

dE′

2π

∫dp′FSd

V(pF,p′F)

∫dξpG

K(E′,p′; t, r). (7.14)

In real systems this cut-off is provided by the non-trivial time dependence of the

interaction V (x1, x2) which is in general not δ-like, but retarded (in conventional

128 Appendix

superconductors with a phonon mediated interaction the maximum energy can

be estimated by the Debye frequency). Near the Fermi surface (ξp Ec) the

order parameter independent of the modulus of p and we can write ∆(p; t, r) =

∆(pF ; t, r).

With the definition of the quasi-classical Green’s function (3.45) we arrive at

the final form of the self-consistency equation

∆(pF , t, r) =i

4N0

∫dp′FSd

V (pF ,p′F )

Ec∫−Ec

dE ′gK(E ′,p′F ; t, r). (7.15)

D Derivation of the Homogeneous Ginzburg-

Landau Equation

In this section we derive the Ginzburg-Landau equations from the quasi-classical

theory for a two-component order parameter ∆(pF ) = ∆1η1(pF ) + ∆2η2(pF )

which is assumed to be small; here we restrict ourselves to the homogenous sit-

uation. Using the Matsubara Green’s function given in Eq. (3.62), we start by

expanding the gap equation (3.64) with respect to the order parameter

∆i

Vi

= πN0T∑

|En|<Ec

⟨ηi(pF )∆(pF )√E2

n + |∆(pF )|2

⟩pF

=

= πN0T∑

|En|<Ec

⟨ηi(pF )∆(pF )

[1

|En|− |∆(pF )|2

2|En|3

]⟩p′

F

.

(7.16)

Initially, we consider the sums over the Matsubara energies. Assuming T Ec

the first sum yields

S1 = πN0T∑

|En|<Ec

1

|En|= N0 ln

(1.13Ec

T

). (7.17)

In the other sum the restriction to |En| < Ec can be neglected as this part

converges, and we find

S3 = πN0T∑En

1

2|En|3= N0

7ζ(3)

8π2T 2. (7.18)

D Derivation of the Homogeneous Ginzburg-Landau Equation 129

α1 α2 2b1 2b2 g 2dN0〈η2

1〉 N0〈η22〉 S3〈η4

1〉 S3〈η41〉 2S3〈η2

1η22〉 S3〈η2

1η22〉

dx2−y2 + s 12N0 N0

38S3 S3 S3

12S3

dx2−y2 + dxy12N0

12N0

38S3

38S3

14S3

18S3

Table 7.1: Coefficients of the Ginzburg-Landau free energy (2.40) found by anexpansion of the gap equation.

Here the expansion of the gap equation leads to the Ginzburg-Landau equations

0 = ∆1

(1

V1

− S1

⟨η2

1(pF )⟩pF

)+ |∆1|2∆1S3

⟨η4

1(pF )⟩pF

(7.19)

+ 2|∆2|2∆1S3

⟨η2

1(pF )η22(pF )

⟩pF

+ ∆∗1∆

22S3

⟨η2

1(pF )η22(pF )

⟩pF

0 = ∆2

(1

V2

− S1

⟨η2

2(pF )⟩pF

)+ |∆2|2∆2S3

⟨η4

2(pF )⟩pF

(7.20)

+ 2|∆1|2∆2S3

⟨η2

2(pF )η21(pF )

⟩pF

+ ∆∗2∆

21S3

⟨η2

2(pF )η21(pF )

⟩pF.

Using Eq. (2.32) we find

ai(T ) =1

Vi− S1

⟨η2

i (pF )⟩pF

=

= N0

⟨η2

i (pF )⟩pF

ln

(T

Tc,i

)≡ αi ln

(T

Tc,i

).

(7.21)

Note that ai(T ) is positive for T > Tc,i, and negative for T < Tc,i. Comparing the

other terms of Eqs. (7.19), (7.20) and Eqs. (2.41), (2.42) we find the coefficients

as given in table D.

To study a one-component order parameter (∆2 = 0) near the critical tem-

perature Tc,1 we can make the approximation

a1(T ) ≈ α1T − Tc,1

Tc,1

; (7.22)

the expression S3 must be evaluated at Tc,1.

In order to examine a situation with a (finite) second order parameter near

130 Appendix

Tc,2, the following expansion is useful:

a1(T ) ≈ α1 ln

(Tc,2

Tc,1

)+ α1

T − Tc,2

Tc,2

, (7.23)

a2(T ) ≈ α2T − Tc,2

Tc,2

; (7.24)

now, S3 must be evaluated at Tc,2. As the expansion with respect to the order

parameters is only reasonable if they are small, the condition Tc,1 & Tc,2 should

be fulfilled.

E Gauge Transformation

We consider the particular unitary transformation given by

ˇS = expiˇτ 3χ(t) = ˇ1 cosχ(t) + ˇτ 3 sinχ(t). (7.25)

Using the Eilenberger equation (3.49) we find the following relations

ˇg(E,pF ; t, r) → S† ˇg(E,pF ; t, r) SU → U + χ(t)

∆ → ∆e−i2χ(t).

(7.26)

In thermal equilibrium always a time independent solution ˇg(E,pF ; r) can be

found for given potential, U , and order parameter ∆. Then the solution for

the potential U − eV and the order parameter ∆ expi(ϕ − 2eV t) is given by

the transformation (7.26) with χ(t) = −ϕ/2 + eV t; the Green’s function can be

expressed via the time-independent solution of the original problem

ˇg(E,pF ; t, r) =S† ˇg(E,pF ; r) S

=

(g(E − eV,pF ; r) f(E,pF ; r)e−i2χ(t)

f(E,pF ; r)ei2χ(t) g(E + eV,pF ; r)

).

(7.27)

The transformation presented here is a particular case of a more general gauge

transformation discussed in Ref. [64].

F Current Conservation in the Boundary Conditions 131

F Current Conservation in the Boundary Con-

ditions

We will show that the following relation is obeyed

2n∑q=0

1 + aqinb

qin

1− aqinb

qin

=2n∑

q=0

1 + aqoutb

qout

1− aqoutb

qout

(7.28)

with

aiin/out = ali

in/out, an+iin/out = ari

in/out

biin/out = bliin/out, bn+iin/out = bri

in/out

(7.29)

which is equivalent to (4.58). We introduce the determinant

D = det1− SaS†b (7.30)

with the matrices as defined in Sec. 4.3 (the left/right indices can be neglected

here). As D is linear in each aqin and bqout we can make the expansion for each q

D = D(aqin = 0) + aq

in

∂

∂aqin

D ≡ Dq0 + aq

inDq1 = Bq(a

qin) (7.31)

D = D(bqout = 0) + bqin∂

∂bqout

D ≡ Dq0 + bqoutDq

1 = Aq(bqout). (7.32)

The boundary conditions (4.54) and (4.55) can alternatively be written

1

aqout

= −Dq0

Dq1

, (7.33)

1

bqin= −D

q0

Dq1

, (7.34)

and we can evaluate the terms occurring in Eq. (7.28) with the help of the latter

relations

1 + aqinb

qin

1− aqinb

qin

= −1− 2Dq

0

D (7.35)

1 + aqoutb

qout

1− aqoutb

qout

= −1− 2Dq

0

D . (7.36)

132 Appendix

That means we only have to show

2n∑q=1

Dq0 =

2n∑q=1

Dq0. (7.37)

Since the matrix elements have the form (1 − SaS†b)qq′ = δqq′ − apinb

q′outSqpS

∗q′p,

the determinant is a sum with addends of the form aq1

in . . . aqkin b

q′1

out . . . bq′k

out. Each

term vanishes k times in the left hand sum (aqj

in = 0) and k times in the right

hand sum (bq′j

out = 0). Finally, we showed the current conservation as formulated

in Eq. (4.58).

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Acknowledgements

I would like to express my thanks to all the people who supported me whilst

I was working on this thesis, starting with my supervisor, Prof. Ulrich Eck-

ern, who gave me the possibility of graduating in his group and taught me the

mysteries of superconductivity. I am particularly grateful for his helpful advice

on all problems which arose over the last years. I would also like to thank Prof.

Jochen Mannhart, the second referee of this thesis, for his kind support, especially

concerning experimental questions. I am particularly indebted to Prof. Andrei

Shelankov (University of Umea and A.F. Ioffe Institute) for a very fruitful col-

laboration. In many interesting discussions he disclosed the secrets of boundary

conditions in the quasi-classical theory.

I express my particular gratitude to Michael Dzierzawa (thanks for all the

passes when playing soccer! ) and Peter Schwab (thanks for patiently answering

all my silly questions! ) for many extensive discussions and helpful comments on

various problems. I am also much obliged to Dierk Bormann (thanks for the nice

stays in Umea and Neuchatel! ), who initiated the DAAD-project with the Umea

group, for all his kind support during the last years. Furthermore I would like to

thank Marek Ozana (thanks for your bike and all the interesting conversations! )

together with Andrei Shelankov and Jørgen Rammer for their kind hospitality

during my stays in Sweden.

I am indebted to Cosima Schuster and Ralf Utermann for their patient help

in the sensitive issue concerning computers. I am also grateful to Colleen Wunsch

(thanks for all the nice chatting! ) for putting the English into readable form. I

thank all members of our group for the congenial atmosphere during the last five

years.

Last, but not least, I want to thank Kerstin for her love and her patience with

me, sometimes 24 hours a day.

This thesis is dedicated to my parents and grandparents who lovingly supported

me during the last 30 years.

Curriculum Vitae

of Thomas Luck

Date of Birth: 8 September 1971

Place of Birth: Augsburg, Germany

Nationality: German

Education: September 1978: Elementary school in Augsburg

September 1982: Grammar school in Konigsbrunn

July 1991: High School Graduation

Military Service: October 1991 - September 1992

Studies: October 1992: Study of Physics at the University of Augs-

burg

November 1994: Preliminary diploma

February 1998: Diploma in Physics

(Diploma Thesis on the BKT-transition in

the presence of disorder. Supervisor: Prof.

Dr. U. Eckern, University of Augsburg.)

Graduation: March 1998: PhD student at the University of Augs-

burg.

Supervisor: Prof. Dr. U. Eckern, Uni-

versity of Augsburg.

Rigorosum: 30. November 2001

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