cms.uni-konstanz.de€¦ · Contents 1 Introduction 5 2 Fundamentals 7 2.1 Superconductivity . . ....

State density in superconductor/normalmetal contacts

Diploma thesis

submitted on March 21st 2012

by

Johannes Reutlinger

at the

University of Constance

to the

Departement of Physics

First assessor: Prof. Dr. W. BelzigSecond assessor: Prof. Dr. Yu. V. Nazarov

Contents

1 Introduction 5

2 Fundamentals 72.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 The Bogoliubov-de Gennes equations . . . . . . . . . . . . . . . . . . . . . . 142.4 Andreev scattering and Andreev bound states . . . . . . . . . . . . . . . . . 192.5 Application of Green’s function methods to superconductivity . . . . . . . . 27

2.5.1 General properties of Green’s functions . . . . . . . . . . . . . . . . . 272.5.2 Equation of motion for a superconducting system . . . . . . . . . . . 342.5.3 Incorporation of random impurity potentials . . . . . . . . . . . . . . 412.5.4 Quasiclassical approximation and Eilenberger equation . . . . . . . . 452.5.5 Dirty limit and Usadel equation . . . . . . . . . . . . . . . . . . . . . 50

2.6 Quantum Circuit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Application of Quantum Circuit Theory to SNS junctions 573.1 Derivation of the main equation for symmetric contacts . . . . . . . . . . . . 583.2 Infinite and zero Thouless energy . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Tunnel contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4 Ballistic contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Equal transmission eigenvalues 6= 1 . . . . . . . . . . . . . . . . . . . . . . . 723.6 Differently weighted discrete transmission eigenvalues . . . . . . . . . . . . . 743.7 Dirty contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.8 Diffusive connector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.9 Symmetric ballistic double contacts . . . . . . . . . . . . . . . . . . . . . . . 793.10 Asymmetric ballistic double contacts . . . . . . . . . . . . . . . . . . . . . . 813.11 Analytical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.12 Power law distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Summary, Conclusion and Outlook 91

Chapter 1

Introduction

In this diploma thesis I want to investigate the superconducting proximity effect, which de-scribes the modification of physical properties in a normal metal caused by the vicinity of asuperconductor. In particular I analyze this effect in a nanostructure consisting of a normalmetal connected to two superconductors through arbitrary symmetric contacts (SNS struc-ture). In this context symmetric means identical transmission properties of both contacts.The proximity effect comes about through induction of superconducting correlations on thenormal side of the interface providing a finite value of the pair amplitude 〈Ψα(~r)Ψβ(~r)〉st 1,whereas it is suppressed on the superconducting side due to Cooper pair leakage (inverseproximity effect). The suppression inside the superconductor as well as the generation of apair amplitude on the normal side decay exponentially with the distance from the boundarylayer. The characteristic length scale is the superconducting coherence length ξ ∼ vF/|∆|,which is the typical variation length of the superconducting order parameter ∆(~r). Theorder parameter is a complex quantity which is not only characterized by its absolute valuebut has furthermore a phase ϕ. ∆(~r) depends on temperature and vanishes at the criticaltemperature Tc. The microscopic mechanism for the proximity effect is Andreev reflection.This process describes reflection of an electron approaching the boundary from the normalside as a hole or vice versa, while a Cooper pair enters/leaves the superconductor. As theCooper pair is correlated, the hole on the normal side is also correlated in respect of theincoming electron. This process can thus be interpreted as the creation of a correlated elec-tron pair on the superconducting side and a correlated hole pair on the normal side. Dueto the missing attractive electron-phonon-interaction on the normal side, such correlationsdecay with increasing distance from the superconductor.

The observable that I’m in particular interested in is the local density of states (LDOS)in the normal region, since it is strongly influenced by the proximity effect and furthermoredirectly accessible to experiments. A typical property of superconductors is a gap in thedensity of states symmetrically around the Fermi energy of the normal state. In a bulksuperconductor the width of this gap is given by twice the absolute value of the orderparameter and is thus directly related to the superconducting correlations. One of themost striking consequences of the proximity effect is the induction of a gap in the LDOSof the normal region called minigap. The existence of the minigap is not new and it hasbeen studied theoretically in detail in many publications (see for example [14] and [16]).

1α and β denote the two possible spin directions of spin-1/2-particles

6 CHAPTER 1. INTRODUCTION

Experimental access to this gap however requires very low temperatures due to its smallsize (∼ 10−3 − 10−4eV ) and was thus not possible for years. Significant progress in coolingtechniques in recent years allowed for experimental access to this proximity induced minigap.Since the LDOS is directly proportional to the differential conductance dI/dV between thetunneling tip of a scanning tunneling microscope (STM) and the analyzed sample, suchmeasurements can be used in order to determine the LDOS ([17], [23] and [24]).

In my theoretical calculations I found for certain contacts another gap in the LDOS, whichis about two orders of magnitude smaller than the usual minigap and strongly depends on thecontact properties as well as on the other system parameters. This gap is not yet mentionedor analyzed in other publications. The aim of this work is the complete characterization ofthis secondary gap and if possible to give a criterion determining its existence.

In chapter 2 I begin by giving an introduction to the theoretical background of super-conductors in general and to the methods that are used in this work. In order to describemesoscopic structures without dealing with the microscopic details on the order of the Fermiwavelength, but still under regard of quantum coherence, I use the method of quasiclassicalGreen’s functions for my analysis. The definitions and properties of the different Green’sfunctions as well as the derivation of the equation of motion, which determines them in thewhole system, is given in chapter 2.5. There I also explain how random impurity potentialscan be incorporated in the following calculations. In a next step the quasiclassical approxi-mation is explained and I show how the quasiclassical Green’s functions are related to theusual ones and to the desired observables, essentially to the LDOS. From the equation ofmotion for the usual Green’s functions, which is called Gor’kov equation, I then derive atransport-like equation of motion for the quasiclassical Green’s functions. This equationis called Eilenberger equation [26]. In general it has to be solved self-consistently with aposition-dependent order parameter ∆(~r). In the diffusive (dirty) limit the coherence lengthis assumed to be much larger than the mean free path of elastic scattering. In this limit theGreen’s function is approximately isotropic and averaging of the Eilenberger equation overangles yields a diffusion-like equation for the isotropic component. This equation is calledUsadel equation [15]. It is a non-linear differential equation, which is in general difficult tosolve for arbitrary geometries. In this work I use a method called Quantum Circuit Theory(QCT) [12] [19] [4]. This method transforms the nonlinear differential Usadel equation bydiscretization into a set of algebraic equations with a much simpler structure.

In chapter 3 I solve these algebraic equations for different symmetric contact types andcalculate the corresponding LDOS in the normal region. The volume of the normal metalenters through a characteristic energy scale called Thouless energy ETh [22]. This energy isrelated through the usual uncertainty principle to the time an excitation needs to diffuse fromthe normal region to either superconductor. ETh is inversely proportional to the volume.The transport properties of the contacts are described by a characteristic distribution oftransmission eigenvalues ρ(T ) in the interval [0, 1]. It turns out that the results for theLDOS strongly depend on the volume of the normal metal in form of ETh, the transmissiondistribution ρ(T ) and the phase difference ϕ between the two superconductors. Effects likethe existence of gaps can be different if only one of them is changed while the others arekept constant.

Chapter 2

Fundamentals

2.1 Superconductivity

In 1911 Heike Kamerlingh Onnes experimented with mercury at low temperatures andobserved the complete disappearance of electric resistivity below a critical temperatureTc. This critical temperature is for mercury approximately Tc = 4.15K. Due to thischaracteristic property, this phenomenon was called superconductivity, although it turnedout that a superconductor is not only a perfect conductor, but also a perfect diamagnet.The difference is that in the presence of a magnetic field, the field in the interior of a mereperfect conductor depends on the order of cooling below Tc and switching on the magneticfield. In a superconductor below the transition temperature however the magnetic field isalways repelled from the interior independently of this order. This effect is known as theMeissner-Ochsenfeld-effect (Figure 2.1), named after its discoverers Walther Meissner andRobert Ochsenfeld, who discovered it in 1933.

In the following years this phenomenon was observed for most metals as well as for manykinds of alloys. However none of these superconductors had a transition temperature ofmore than 23 K and thus liquid helium is needed for cooling. Therefore cooling of metallicsuperconductors is complicated and expensive.

The theoretical description of superconductivity was unsolved for a long time. It tookalmost 40 years until W. L. Ginsburg and L. D. Landau developed a phenomenological theoryof superconductivity, which describes the phenomenon as a second order phase transition.After further ten years John Bardeen, Leon N. Cooper and John R. Schrieffer published theirtheoretical description of superconductivity on a microscopic level. This theory is called BCStheory. It succeeds in describing superconductivity in metallic superconductors. In 1986the situation became complicated again when J. G. Bednorz and K. A. Mueller found a newclass of ceramic superconductors with transition temperatures above 23 K. BCS theory isnot able to explain superconductivity in these high temperature superconductors and themicroscopic mechanism behind this effect is still an open question. In my work I deal onlywith metallic superconductors, which are well described by the BCS theory.

The basic assumption in this theory is the existence of an attractive interaction betweenelectrons in a metal. In view of the repulsive Coulomb interaction this assumption first seemsa bit contradictory. But in fact, electrons in a metal do not only interact through Coulombrepulsion, but also through attractive phonon exchange, which can overcompensate Coulomb

8 Superconductivity

Figure 2.1: Repulsion of the magnetic eld from the interior of a superconductor below thecritical temperature Tc. The order of switching on the magnetic eld and cooling below Tcdoesn t inuence the result. This eect is called Meissner-Ochsenfeld-eect [27].

repulsion. Below Tc the attractive phonon exchange leads to an instability of the normalmetal ground state (Cooper instability). Electrons couple to Cooper pairs, which allows fora decrease of total energy compared to the normal ground state without interaction. Thisdecrease in total energy is possible no matter how weak the attraction is. The normal statethus becomes unstable with respect to pairing of electrons, when the transition temperatureis reached. This becomes clear by considering a simple model of two fermions togetherwith a non-interacting homogeneous Fermi sea background at zero temperature, where allsingle particle states inside the Fermi sphere are occupied and all states outside are empty.This model is drafted in Figure 2.2. The two Fermions are assumed to interact attractivelywith each other but not with the Fermi sea background, except through the Pauli exclusionprinciple. Due to the exclusion principle they cannot occupy states inside the Fermi sphere.In the ground state without interaction they thus occupy empty single particle states directlyat the Fermi level. In the following it is shown that the attractive interaction lowers thenon-interacting ground state energy by creation of a Cooper pair ([1] [2] [3]). The interaction

generates scattering with a transition from initial states χk1,χk2 to nal states χk1,χk2 with the

condition that the total momentum must be conserved: χK = χk1 +χk2 = χk1 +χk2. The spatialcomponent of the pair wave function can thus be written as a linear combination of stateswith xed total momentum:

(χr1, χr2) =

χk1+χk2= χK

g(χk1,χk2) · eiχk1χr1eiχk2χ.r2.

In BCS theory the interaction is approximated by a pointlike -interaction in a thin layerof thickness D above the Fermi surface, where D denotes the Debye-frequency. Thisapproximation is justied by the fact, that D is the maximum phonon energy in a crystal

Superconductivity 9

Figure 2.2: Ground state of a non-interacting Fermi gas with all single particle states insidethe Fermi sphere occupied and two additional Fermions, which interact attractively witheach other but not with the Fermi sea background, besides through the Pauli exclusionprinciple. In the ground state of a non-interacting system they occupy states directly at theFermi surface.

and thus the maximum energy that can be exchanged by the interacting electrons. Theeffective interaction is therefore approximated by an isotropic s-wave interaction, which de-pends only on the absolute values of the participating momenta and not on their directions.It is given by

V~k,~k′ =

−V/L3, for EF ≤ ~2k2

2m≤ EF + ~ωD and EF ≤ ~2k′2

2m≤ EF + ~ωD

0, otherwise. (2.1)

Due to momentum conservation, for a given ~k1 and ~k2 and a finite ~K = ~k1 + ~k2, scatteringcan in general proceed only to a small number of states (gray deposited overlap in Figure2.3). For antiparallel momenta the overlap volume becomes maximal and comprises thewhole interaction layer. Electrons with zero total momentum thus have the largest varietyof exchangeable phonons and the energy decrease due to this interaction is maximal. Inthe following pairing is always considered between states with antiparallel momenta. Thespatial wave function of the two interacting fermions becomes

Ψ(~r1, ~r2) =∑~k

g(~k) · ei·~k·(~r1−~r2). (2.2)

The probability amplitude g(~k) is zero for all momenta below the Fermi surface and has afinite value for pair states, which are coupled through the interaction. Normalization of thepair wave function and the minimization condition on the ground state energy fix the values

10 Superconductivity

Figure 2.3: Momentum conservation allows for finite ~K only for scattering into a smallnumber of states (gray deposited area). For antiparallel momenta this area becomes maximaland comprises the whole interaction layer [13].

of g(~k). Besides the spatial component the total wave function contains a spin componentand has to be antisymmetric under particle exchange:

|Ψσ1,σ2(~r1, ~r2)〉 = Ψ(~r1, ~r2) · |χ(σ1, σ2)〉.

The antisymmetry can either be carried by the spatial component of the wave functionΨ(~r1, ~r2) or by the spin component |χ(σ1, σ2)〉. For a symmetric spatial component (2.2)

becomes a sum of cos(~k(~r1 − ~r2)

)and the two electrons are in an antisymmetric singlet

spin state of the form (|α1〉|β2〉 − |β1〉|α2〉)/√

2. Here |α〉 and |β〉 denote the two possiblesingle particle spin states of spin-1/2-fermions and the index denotes the particle. For

an antisymmetric spatial wave function (2.2) becomes a sum of sin(~k(~r1 − ~r2)

)with a

symmetric triplet spin component, which can have the three possible forms:

|α1〉|α2〉; |β1〉|β2〉; (|α1〉|β2〉+ |β2〉|α1〉)/√

2.

In principle both cases are possible, but for metallic superconductors and probably also forhigh temperature superconductors Cooper pairs are found to be in the singlet state. Forsuperfluid He-3 triplet pairing was detected unambiguously and there is some evidence fortriplet pairing in heavy fermion superconductors. An explanation for the preference of thesinglet state is that the interaction is attractive and the singlet state with cosine spatialdependence has a larger probability amplitude at small particle distances. The pair statehas the form:

Superconductivity 11

|Ψα,β(~r1, ~r2)〉 =∑~k

g(~k) cos(~k (~r1 − ~r2)

) 1√2

(|α1〉|β2〉 − |β1〉|α2〉) . (2.3)

Solution of the stationary Schrodinger equation for the wave function (2.2) provides the pairenergy: (

− ~2

2m(∇2

1 +∇22) + V (~r1 − ~r2)

)Ψ(~r1, ~r2) =

(E +

~2k2F

m

)Ψ(~r1, ~r2).

Here E denotes the energy of the pair counted from twice the Fermi energy. Fourier trans-formation provides an equation for the probability amplitudes g(~k):

~2k2

mg(~k) +

∑~k′

g(~k′)V~k,~k′ = (E + 2EF )g(~k). (2.4)

With the BCS approximation for the interaction (2.1) equation (2.4) becomes(−~2k2

m+ E + 2EF

)g(~k) = − V

L3

∑~k′

g(~k′) = c.

Here the sum is taken only over states inside the interaction layer, where the Fourier com-ponents of the interaction are non-zero. Solving this equation for g and inserting it into

c = − VL3

∑~k′

g(~k′)

yields a self-consistency equation of the form:

1 = − VL3

∑~k′

1

−~2k′2

m+ E + 2EF

. (2.5)

In the last step the equation was divided by c. This is only valid for c 6= 0. But asc ∼

∑~k′ g(~k′) and Ψ(~r1 − ~r2 = 0) =

∑~k′ g(~k′) = 1 in the singlet state this is actually valid

for singlet pairing. The summation over all states inside the layer can be replaced by anintegral from EF to EF + ~ωD multiplied with the density of states, which is approximatelyequal to the density of states at the Fermi level N0 in the whole interval. After integrationthe equation can be solved for E. The result is given by

E = −2~ωDe−2/N0V .

This relation shows, that no matter how weak the attraction is, it provides a decrease inenergy compared to the non-interacting ground state. Therefore this state becomes unstablewith respect to electron pairing as soon as the effective interaction becomes attractive. Thisis in contrast to usual two body interaction, where there are no bound states below acertain threshold attraction. It is the effect of the non-interacting Fermi sea background,which allows for a decrease in energy even for weak attraction. This example gives quitea good idea of what happens at low temperatures in a metal in presence of an attractiveinteraction. The pair-wise occupied states are called Cooper pairs after Leon N. Cooper,who first discovered the drop in energy due to attractive interaction and electron pairing.

12 The BCS ground state

2.2 The BCS ground state

In order to describe the superconducting ground state, interactions between all particles haveto be taken into account. Therefore expression (2.3) for the pair state must be generalizedto the whole system. The most general ansatz for a n-particle ground state with regard ofCooper pairing between two particles respectively can be written as

|Ψn(~r1...~rn)〉 = AΨ(~r1 − ~r2) ·Ψ(~r3 − ~r4) · ... ·Ψ(~rn−1 − ~rn) · |α1〉|β2〉 · ... · |αn−1〉|βn〉. (2.6)

Here the operator A assures the antisymmetry of the fermionic many particle wave function.This state is only defined for an even number of electrons n. In the case of an odd number,the extra electron has to be placed in an empty state. For large particle numbers the effectof this extra electron on the properties of the system is negligible and it is sufficient to workwith wave functions of type (2.6) in the following. Using expression (2.2) for the pair wavefunctions in (2.6) yields

|Ψn(~r1...~rn)〉 =∑~k1

...∑~kn/2

g(~k1)...g(~kn/2)Ae~k1(~r1−~r2)...e

~kn/2(~rn−1−~rn) · |α1〉|β2〉 · ... · |αn−1〉|βn〉.

The antisymmetric state is the Slater determinant of the states:

(~k1, α), (−~k1, β), ..., (~kn/2, α), (−~kn/2, β).

This determinant can be expressed in a more manageable form through the method ofsecond quantization:

|Ψn(~r1...~rn)〉 =∑~k1

...∑~kn/2

g(~k1)...g(~kn/2)a†~k1,αa†−~k1,β

...a†~kn/2,αa†−~kn/2,β

|0〉. (2.7)

The operators a†~k,σ are the usual creation operators, which create an electron in the single

particle momentum eigenstate ~k with spin σ. Their adjoints a~k,σ annihilate a particle inthe corresponding state. In order to reproduce all properties of the Slater determinant in astate like (2.7), the operators have to obey fermionic anticommutation relations:

[a†~k,σ, a~k′,σ′

]+

= δ~k,~k′δσ,σ′ ;[a~k,σ, a~k′,σ′

]+

=[a†~k,σ, a

†~k′,σ′

]+

= 0.

The structure of (2.7) is still quite complicated to handle, since there is a huge variety of

possibilities, how n particles can be distributed on different ~k-states. In a state like (2.7) theparticle number is fixed and the occupancy of each state is determined by the occupancies ofall other states. It was the idea of Bardeen, Cooper and Schrieffer to work with statisticalaverages and to replace the exact occupancy of each state by a mean occupancy, which

The BCS ground state 13

depends only on the mean occupancies of all other ~k-states. The resulting wave functioncan be expressed as follows:

|ΨBCS〉 =∏~k′

(u~k′ + v~k′ a

†~k′αa†−~k′β

)|0〉. (2.8)

Here v~k = |v~k|eiϕ is the probability amplitude that the pair state ((~k, α), (−~k, β)) is occupied

and u~k is the probability amplitude that this state is empty. The amplitudes have a ~k-independent phase difference ϕ, which turns out to be the phase of the superconductingorder parameter. Both amplitudes have to obey the condition:

|v~k|2 + |u~k|

2 = 1.

Obviously the BCS ground state (2.8) is no longer an eigenstate of particle number. Inorder to replace exact occupancies by mean occupancies the requirement of a fixed particlenumber had to be dropped. (2.8) is a phase coherent linear combination of states with fixedparticle numbers of the type (2.7), which differ by an even number of particles, respectivelyby the number of occupied pair states:

|ΨBCS〉 =∑n

λn · |Ψn〉.

Since ϕ is ~k-independent the phase of each term |Ψn+2〉 is shifted by ϕ compared to theprevious term |Ψn〉. In [2] it is shown that the step from (2.7) to (2.8) is precisely to fix thisphase, which makes particle number uncertain. Conversely a fixed particle number impliesa totally uncertain ϕ. This phase-coherence turns out to be important for the generalizedmean field approach to a Hamiltonian, which is related to a ground state of the form (2.8).

It can be shown that the uncertainty of the particle number in the BCS ground statedoesn’t influence the physical properties of the system if it is large enough. Such systemscomprise huge particle numbers and the pair amplitude λn has a sharp maximum with arelative variance that approaches zero for large particle numbers.

14 The Bogoliubov-de Gennes equations

2.3 The Bogoliubov-de Gennes equations

The ground state of a homogeneous superconducting system is described by the BCS state(2.8). The method presented in this chapter describes excitations above the ground statetaking into account external potentials, magnetic fields and inhomogeneous order parame-ters. Furthermore the equation of motion for these excitations is derived. The Hamiltonianof a superconducting system in BCS approximation contains besides the usual kinetic en-ergy an attractive, isotropic δ-interaction of the form (2.1). It has a constant value in athin layer above the Fermi surface and is zero everywhere else. In general the Hamiltoniancontains in addition a vector potential ~A(~r) and a scalar external potential U0(~r). A pos-sibility to account for a scalar external potential U0(~r) in a ground state like (2.8) can bereached by defining this state with solutions of the stationary single particle Schrodingerequation. General time-reversed eigenstates are occupied pair-wise instead of momentumeigenstates with reverse momenta 1. This method however doesn’t account for magneticfields or a position-dependent order parameter. This is however necessary to describe con-tacts between normal metals and superconductors. Instead, the problem is solved by usingSchrodinger field operators and application of a generalized mean field approach to theHamiltonian, which can then be diagonalized. The diagonalization-condition provides anequation of motion for excitations in a superconductor.

The previously described Hamiltonian of a homogeneous system is given by

H =∑~k,σ

E~k,σa†~k,σa~k,σ −

V

2

∑~k,~k′,~q,σ,σ′

a†~k+~q,σa†~k′−~q,σ′ a~k′,σ′ a~k,σ.

In a next step momentum creation and annihilation operators are replaced by Schrodingerfield operators. They are defined through

Ψσ(~r) =∑~k

ei~k~ra~k,σ,

Ψ†σ(~r) =∑~k

e−i~k~ra†~k,σ,

and obey the usual anticommutation relations for fermionic field operators. The Hamiltoniantransforms to

H =∑σ

∫d3rΨ†σ(~r)

(−~2∆

2m

)Ψσ(~r)− V

2

∑σ,σ′

∫d3rΨ†σ(~r)Ψ†σ′(~r)Ψσ′(~r)Ψσ(~r). (2.9)

In this expression the external potential U0(~r) and the vector potential ~A(~r) can be incor-porated by the principal of minimal coupling. The total Hamiltonian is then given by

1The single particle momentum eigenstates with reverse momenta and antiparallel spin are the time-reversed states in the absence of an external potential U0

The Bogoliubov-de Gennes equations 15

H = H0 + HI ;

H0 =∑σ

∫d3rΨ†σ(~r)

(1

2m(~p− e

c~A)2 + U0(~r)

)Ψσ(~r);

HI = −V2

∑σ,σ′

∫d3rΨ†σ(~r)Ψ†σ′(~r)Ψσ′(~r)Ψσ(~r).

For a homogeneous system excitations above the ground state (2.8) can be reached by adding

a further unpaired electron of momentum ~k and spin σ to the ground state:

|Ψexc,1〉 = a†~k,σ|ΨBCS〉.

The application of the creation operator produces zero for all components of |ΨBCS〉 with

the pair ((~k, σ), (−~k,−σ)) occupied, and a state with only (~k, σ) occupied and (−~k,−σ)unoccupied for all other components. This state is thus orthogonal to the ground state.The situation is more complicated if a further particle is added to the system:

|Ψexc,2〉 = a†~k′,σ′ a†~k,σ|ΨBCS〉.

In the case where ~k′ = −~k and σ′ = −σ this state is not orthogonal to the ground state. Inorder to make sure that all excited states are orthogonal to the ground state and to eachother further considerations are necessary. This is in principle possible but very cumbersome.In the following a more powerful method is presented, which is able to describe excitedstates in a superconductor taking into account external potentials, magnetic fields, andinhomogeneous order parameters [1]. The interaction term HI contains four operators andis a sum over all two-particle interactions in the system. It is replaced in the following by amean field, which is generated by all particles and acts on each single particle. It thereforecontains only two operators. The mean field method approximates

− V

2

∑σ,σ′

∫d3rΨ†σ(~r)Ψ†σ′(~r)Ψσ′(~r)Ψσ(~r)

≈∑σ

∫d3rU(~r)Ψ†σ(~r)Ψσ(~r)−∆(~r)Ψ†α(~r)Ψ†β(~r)−∆∗(~r)Ψβ(~r)Ψα(~r).

α and β denote the two possible single particle spin states of spin-1/2-fermions (α = ↑,β = ↓). The fields U(~r) and ∆(~r) are defined as statistical averages of the form:

U(~r) = −V⟨

Ψ†α(~r)Ψα(~r)⟩st

= −V⟨

Ψ†β(~r)Ψβ(~r)⟩st

; (2.10)

∆(~r) = −V⟨

Ψα(~r)Ψβ(~r)⟩st

= V⟨

Ψβ(~r)Ψα(~r)⟩st. (2.11)

In contrast to a mean field approach in a normal metal, expectation values in a supercon-ductor are calculated with states like (2.8). Such states are coherent superpositions of many


particle states, which differ by the number of occupied pair states. Expectation values oftwo annihilation operators as in (2.11) are non-zero for such states. This pairing potentialvanishes in the normal state, but gives a finite contribution in the superconducting statedue to the phase coherence of this state. In this approximation an effective grand canonicalHamiltonian can be defined, which is given by the difference of the mean field Hamiltonianand the chemical potential times the particle number N :

Heff = H − EF · N

=

∫d3r∑σ

Ψ†σ(~r)

(1

2m(~p− e

c~A)2 + U0(~r)− EF

)Ψσ(~r)+

+

∫d3r∑σ

U(~r)Ψ†σ(~r)Ψσ(~r)−∆(~r)Ψ†α(~r)Ψ†β(~r)−∆∗(~r)Ψβ(~r)Ψα(~r). (2.12)

Here the particle number operator

N =

∫d3r∑σ

Ψ†σ(~r)Ψσ(~r)

was introduced. Subtracting the particle number times the chemical potential fixes the meanparticle number as usual in grand-canonical calculations. This is equivalent to choosing thezero of kinetic energy to be EF . The effective Hamiltonian can be diagonalized in thefollowing by defining new fermionic operators γ†~k,σ through a Bogoliubov transformation.

This transformation is quite intuitive, if one thinks about all ways of creating an excitation(~k, σ) in a homogeneous system. This can not only be reached by applying a creation-

operator a†~k,σ, but also by annihilating an electron with (−~k,−σ) through a−~k,−σ. This is of

course only possible if the corresponding pair state is occupied. In both cases this producesthe same state, where only (~k, σ) is occupied by a single electron and (−~k,−σ) is empty. Ageneral expression γ†~k,σ of an operator creating such an excitation must thus contain both

of these possibilities. In the homogeneous case the transformation is therefore defined by:

γ†~k,α = u~ka†~k,α− v~ka−~k,β ; (2.13)

γ†~k,β = u~ka†~k,β

+ v~ka−~k,α . (2.14)

The sign in front of the amplitudes are fixed by the condition that so defined annihilationoperators applied to the ground state (2.8) produce zero. Otherwise (2.8) is not the groundstate without excitations. This transformation can be generalized to inhomogeneous systemsby expressing the operators a~k,σ through field operators Ψσ(~r). Inversion of (2.13) and (2.14)using the definitions of the field operators provides an expression for the field operatorsthrough γ operators:

Ψα(~r) =∑m

(um(~r)γm,α − v∗m(~r)γ†m,β

); (2.15)

Ψβ(~r) =∑m

(um(~r)γm,β + v∗m(~r)γ†m,α

). (2.16)

The Bogoliubov-de Gennes equations 17

Here ~k, which is a good quantum number only for homogeneous systems, was replaced by amore general quantum number m. The amplitudes um and vm gain a position dependencein a inhomogeneous system. The new operators γ have to obey fermionic anticommutation-relations:

[γ†m,σ, γm′,σ′

]+ = δm,m′δσ,σ′ ;

[γm,σ, γm′,σ′ ]+ =[γ†m,σ, γ

†m′,σ′

]+

= 0 .

The condition that the Hamiltonian is required to be diagonal, if expressed through γoperators determines un(~r) and vn(~r). Heff must have the form

Heff = Eg +∑m,σ

Emγ†m,σγm,σ. (2.17)

Eg denotes the energy of the ground state. By commuting Heff with γm,α and γ†m,α, thediagonalization condition becomes

[Heff , γm,σ

]−

= −Emγm,σ; (2.18)[Heff , γ

†m,σ

]−

= Emγ†m,σ. (2.19)

To obtain equations for um(~r) and vm(~r), the commutator between the field operators andHeff must be calculated:

[Ψα(~r), Heff

]−

=

(1

2m(~p− e

c~A)2 + U0(~r)− EF + U(~r)

)Ψα(~r)−∆(~r)Ψ†β(~r);[

Ψβ(~r), Heff

]−

=

(1

2m(~p− e

c~A)2 + U0(~r)− EF + U(~r)

)Ψβ(~r) + ∆∗(~r)Ψ†α(~r).

Expressing the field operators through γ operators by (2.15), (2.16) and using the diag-onalization conditions (2.18), (2.19) provides two equations for um(~r) and vm(~r). Theseequations describing excitations in a superconductor are called the Bogoliubov-de Gennesequations (BdG equations):

Emum(~r) =

(1

2m

(~p− e

c~A)2

+ U0(~r)− EF + U(~r)

)um(~r)−∆(~r)vm(~r);

Emvm(~r) = −(

1

2m

(~p∗ − e

c~A)2

+ U0(~r)− EF + U(~r)

)vm(~r)−∆∗(~r)um(~r).

They can be coupled to one vector equation in Nambu space of electrons and holes of theform

Em

(um(~r)−vm(~r)

)=

(He + U(~r) ∆(~r)

∆∗(~r) −(H∗e + U(~r))

)·(um(~r)−vm(~r)

). (2.20)


Here He is introduced as

He =1

2m

(−i~~∇− e

c~A)2

+ U0(~r)− EF .

The reason to write this vector equation with a - sign in front of vm(~r) becomes clearlater, this can for example be avoided by a redefinition of the phase ϕ in (2.8). (2.20) isa generalization of the stationary Schrodinger equation to the case of superconductivity.Excitations in a superconductor are linear combinations of electron and hole states due tothe finite value of the pair potential ∆(~r), which couples electrons and holes. Their timeevolution is determined by the time dependent form of equation (2.20). In the normal statethe superconducting order parameter ∆(~r) is zero. The vector equation (2.20) decouplesinto two separate equations: the Schrodinger equation describing electron excitations, andthe time-reversed Schrodinger equation describing holes. Because electron solutions are alsocontained in the time reversed equation as solutions with negative excitation energies andhole solutions are contained in the usual Schrodinger equation as solutions with negativeexcitation energies, the BdG equation provides a double set of solutions. By droppingsolutions with negative excitation energies this double counting is avoided.

(2.20) is an equation for um(~r) and vm(~r), depending on the mean fields U(~r) and ∆(~r).These fields themselves depend on um(~r) and vm(~r) through (2.10) and (2.11) and thereforehave to be determined self-consistently. The field U(~r) is already present in the normalstate. Here it plays the role of an external potential, which can be neglected in the followingsince it does not contribute to the superconducting phenomena. The important quantity isthe field ∆(~r), which is absent in the normal state. It is a complex quantity of the form

∆(~r) = |∆(~r)| · eiϕ.

Dealing with structures where only one superconductor is involved the complex phase ϕ isnot important and can be ignored, but it gets important in structures involving multiplesuperconductors. For SNS junctions in particular the phase difference ϕ1 − ϕ2 between thesuperconducting order parameters is important and must not be neglected.

Andreev scattering and Andreev bound states 19

2.4 Andreev scattering and Andreev bound states

The BdG equation (2.20) provides a microscopically correct description of excitations ina superconductor, including the inhomogeneous case. Similar to the Schrodinger equationthe excitation spectrum is given by the eigenvalue spectrum of (2.20). In the homogeneouscase, where ∆(~r) = ∆ = const. in the whole volume (for example deep inside a bulk

superconductor) and furthermore ~A = 0 and U0 = 0, the solutions are plane waves:(u~k(~r)−v~k(~r)

)=

(cucv

)· ei~k~r.

The excitation energies are defined by the determinant:

det

(~2k2

2m− EF − E ∆

∆∗ −~2k2

2m+ EF − E

)= det

(ξ~k − E ∆

∆∗ −ξ~k − E

)!

= 0.

Hence the excitation energies are:

E =

√(~2k2

2m− EF

)2

+ |∆|2 =√ξ2~k

+ |∆|2, (2.21)

where the quantity ξ~k denotes the difference between the kinetic energy connected to ~k andthe Fermi energy EF :

ξ~k =~2~k2

2m− EF .

Since solutions with negative excitation energies are dropped, all wave vectors of electron-like excitations are above the Fermi surface and those of hole-like excitations are below. Foreach excitation energy E > |∆| and each momentum direction there are two quasiparticlestates of different momenta with such an excitation energy, one above and one below theFermi surface.

(2.21) suggests, that there are no excitations with energies smaller than |∆|. Even fora momentum directly at the Fermi surface, the corresponding energy is |∆|. Consequentlythe excitation spectrum of a superconductor exhibits a gap of two times the absolute valueof the order parameter. Due to the temperature dependence of ∆, the width of the gap alsodepends on temperature. For zero temperature the energy gap of metallic superconductorsis of the order of 10−3− 10−4eV . This is much less than the energy gap of a semiconductor,which is of the order of eV, for example 1.1 eV for silicon.

In order to describe scattering at contacts between superconductors and normal metalsit is necessary to apply the Nambu matrix structure artificially also to the normal metal.Then the BdG equation can be solved with a position-dependent order parameter ∆(~r) forarbitrary initial conditions taking into account specific boundary conditions at the contactlayer [10]. These boundary conditions depend on the scattering properties of the contact.They are similar to those already known from solving the Schrodinger equation for one-dimensional potentials. For an ideal contact without scattering, the wave function and thefirst derivative must be continuous at the boundary. An example of a special initial condition

20 Andreev scattering and Andreev bound states

Figure 2.4: All possible scattering processes for an incoming electron with E > |∆|: Normalreflection, Andreev reflection, Transmission with/without branch-crossing. For E < |∆|the excitation can’t enter the superconductor and the only possible processes are normalreflection and Andreev reflection [10].

is an electron excitation coming from the normal metal towards the boundary. The ansatz forthe outgoing excitation includes all possible processes that can occur with certain probabilityamplitudes. The amplitude of each process is determined by the boundary conditions. In thespecial case of only one transport channel, the possible scattering processes for an incomingelectron from the normal side towards the boundary with an excitation energy E > |∆| are:normal reflection as an electron, Andreev reflection as a hole, transmission with or withoutbranch crossing. All these processes are depicted in Figure 2.4. In each of them momentumand energy must be conserved. Since there are no quasiparticle states in the superconductorbelow E = |∆| no transmission is possible in this energy range. The important process forthis work is Andreev reflection at energies below |∆|, because in SNS junctions this leadsto Andreev bound states, which determine the local density of states (LDOS) in the normalregion for E < |∆|.

The general self-consistent solution of equation (2.20) with a position dependent orderparameter ∆(~r) is a quite difficult task. Fortunately, in many cases there is a possibility,how the BdG equation can be highly simplified with little influence on the results. Thismethod is called quasiclassical approximation [4] [8] [11]. The condition that determines itsvalidity is that the absolute values of the order parameter must be small compared to theFermi energy:

|∆(~r)| EF . (2.22)

In many cases, especially for most metallic superconductors this condition is well satisfied.


Relevant excitations are close to the Fermi surface, and thus their wave vectors can bewritten in the general inhomogeneous case as

~k = ~kF + ~q(~r).

Since relevant excitations are assumed close to the Fermi surface ξ~k is of the order of |∆(~r)|and thus |~q(~r)| ∼ |∆(~r)|/~vF . For the general solution of the BdG equation the followingansatz is used: (

um(~r)−vm(~r)

)=

(um(~r)vm(~r)

)· ei~kF~r. (2.23)

The second factor in (2.23) describes fast oscillations connected to ~kF , the first factor is anenvelope function related to ~q(~r), which varies at length scales of the order of the coherencelength ξ. This is the characteristic variation length of the order parameter ∆(~r). It isdirectly related to the order parameter and the Fermi velocity by

ξ ∼ vF|∆|

.

Typically the coherence length is of the order of 0.1− 1µm. Using this ansatz for the BdGequation (2.20) and neglecting quadratic terms in the variation of the envelope functiontransforms (2.20) into a first order differential equation for the envelope function:

Em

(um(~r)vm(~r)

)=

(−i~~vF ~∇ ∆(~r)

∆∗(~r) i~~vF ~∇

)·(um(~r)vm(~r)

). (2.24)

The justification for neglecting quadratic terms in the variation of the envelope functioncompared to the order parameter ∆(~r), can be seen from

1

2mξ2∼ |∆|

2

2mv2F

∼ |∆|EF︸︷︷︸1

|∆| |∆|.

Here the condition (2.22) determining the validity of the quasiclassical approximation wasused. (2.24) is the BdG equation in quasiclassical approximation. It describes spatialvariation of the envelope function without fast oscillations for inhomogeneous systems. Inthe following it is solved for the one-dimensional case of an ideal contact between a normalmetal and a superconductor with the simplifying assumption of a constant order parameterin both regions. Since ∆(~r) is constant in both half spaces, the solutions are plane waveswith momenta q: (

u(x)v(x)

)=

(cucv

)eiqx

In the simplest case there is no barrier at the contact, which causes mixing betweendifferent transport channels. Without mixing it is sufficient to consider only one transportchannel. Furthermore the process of normal reflection is forbidden due to momentum con-servation. Andreev reflection is allowed, because the momentum of a reflected hole has the


same direction as the momentum of the incoming electron with only a slightly smaller ab-solute value. In the ansatz for the wave function in the normal metal it is thus necessary toaccount for an incoming electron and a reflected hole, weighted with a probability amplituderA: (

u(~r)v(~r)

)n

=

(eix E

~vF

rAe−ix E

~vF

).

rA is a complex quantity, which also accounts for possible phase shifts. For electron energiesE < |∆| the excitation cannot be transmited into the superconductor. q becomes imaginaryand the wave function decays exponentially inside the superconductor:

q =

√E2 − |∆|2~vF

=i√|∆|2 − E2

~vF.

The solution of the exponentially decaying wave function inside the superconductor has thefollowing form: (

u(~r)v(~r)

)s

=

(fefh

)e−x√|∆|2−E2

~vF . (2.25)

The phase relation between fe and fh can be found by inserting (2.25) into (2.24):

fh = feE − i

√|∆|2 − E2

|∆|· e−iϕ. (2.26)

As (2.24) is a first order differential equation only the wave function itself has to be con-tinuous at the ideal boundary (x=0). This provides two conditions for the probabilityamplitudes:

1 = fe ; rA = fh.

With (2.26) this becomes

rA =E − i

√|∆|2 − E2

|∆|· e−iϕ !

= eiχ. (2.27)

Here the phase χ was introduced. It is defined through

χ = − arccos

(E

|∆|

)− ϕ.

The probability for Andreev reflection is given by RA = |rA|2 = 1. As expected Andreevreflection is the only possible process and excitations are fully Andreev reflected in thisenergy range. Similar calculation with a hole coming from the normal metal towards theboundary leads to the following probability amplitude for an Andreev reflected electron:

r′A = eiχ,

with


χ = − arccos

(E

|∆|

)+ ϕ.

In this calculation a perfect boundary layer without scattering was assumed. The situa-tion can be more generalized by including a potential barrier at the boundary, describedthrough a scattering matrix se(E) for electrons and sh(E) for holes [4] [18]. These matricesare in general energy dependent. Because hole excitations with energy E are related toelectron excitations with energy −E, the scattering matrix sh(E) is connected to se(−E)[4]. Furthermore, a hole with a certain velocity direction has a momentum pointing in op-posite direction. Thus it is related to an electron moving in the opposite direction. Thetime evolution of holes is described by the time reversed Schrodinger equation. This canbe taken into account by transposing the scattering matrix of electrons. Altogether, therelation between the scattering matrices of electrons and holes is given by

sh(E) =(s−1e (−E)

)T= s∗e(−E). (2.28)

For the last equality the general unitary condition s† = (s∗)T = s−1 of scattering matriceswas used. If channel mixing is neglected, it is sufficient to consider only one transportchannel with a 2× 2 scattering matrix

se(E) =

(re t′ete r′e

)=

(r(E) t′(E)t(E) r′(E)

).

t(E) and r(E) describe the energy dependent transmission/reflection amplitudes for excita-tions propagating to the barrier from the left. t′(E) and r′(E) describe these amplitudes forexcitations propagating from the right to the barrier. Due to relation (2.28), the scatteringmatrix of holes becomes

sh(E) =

(rh t′hth r′h

)=

(r∗(−E) t′∗(−E)t∗(−E) r′∗(−E)

).

Considering an incoming electron from the normal side with an excitation energy E < |∆|,the total amplitude of Andreev reflection can be calculated by summing up all sub-processesthat convert an incoming electron into an outgoing hole. The simplest sub-process is given bythe product of the following probability amplitudes: Electron transmits through the barrier,is Andreev reflection as a hole at the superconductor and the hole transmits through thebarrier. The probability amplitude of this first order sub-process is thus

rA,1 = te · rA · t′h,

where rA is determined by (2.27). All higher order processes involve further reflections of theexcitation inside the barrier. In a second order process for example, the barrier reflects theAndreev reflected hole back to the superconductor. At the superconductor it is transformedinto an electron through Andreev reflection, which is again reflected by the barrier. At thethird Andreev process the electron is transformed back into a hole, which finally transmitsthrough the barrier. All higher order sub-processes just differ by the number of reflections.Their probability amplitude is given by


rA,n = te · rA · t′h · (r′hr′Ar′erA)n−1.

The total amplitude of Andreev reflection at the barrier is obtained by summing up allsub-processes. To account for quantum coherence it is important to sum up probabilityamplitudes and not absolute probabilities:

rA =∞∑n=1

te · rA · t′h · (r′hr′Ar′erA)n−1 = terAt′h

1

1− r′hr′Ar′erA=

tet′heiχ

1− r′hr′eei(χ+χ).

The scattering matrix varies on an energy scale which is called Thouless energy of the barrier[22]. This energy scale is related to the dwell time of excitations inside the barrier throughETh = ~/tdwell. If it is much larger than the involved energies 2, this energy dependencecan be neglected. By assuming small excitation energies E |∆|, the denominator in theprevious expression can be simplified as follows:

χ+ χ = −2 arccos(E

|∆|) ≈ −2 arccos(0) = −π.

The total probability of Andreev reflection is thus given by

RA = |rA|2 =T 2

(2− T )2.

It depends only on the properties of the barrier, in the case of one single transport channelon the transport eigenvalue of this channel. For a perfect contact without barrier (T=1),the previous result RA = 1 is reproduced. For excitation energies below |∆| no transmissioninto the superconductor is possible, since there exist no excitations in this energy range. Theonly possible process besides Andreev reflection is thus normal reflection. The probabilityof normal reflection is given by RN = 1−RA.

In this work I am interested in the LDOS in SNS junctions, therefore the next stepis to look at the effect of Andreev reflection on the LDOS in a normal region, placedbetween two superconductors. In this case the phase difference ϕ of the involved orderparameters becomes important. I consider identical superconductors with equal absolutevalues of the order parameter: |∆1| = |∆2| = |∆|. The normal region together with bothboundaries can be described by one single scattering matrix. In order to be able to neglectthe energy dependence of this matrix, the normal structure must be sufficiently small tohave a Thouless energy much larger than the other characteristic energy scales of the system.As before, only one single transport channel with an energy E below the gap edge insidethe superconductors |∆| is considered. Thus an excitation (electron or hole) can leave thenormal region to neither side, and the only possible process at the boundaries is Andreevreflection. Due to the finite size of the normal metal region this leads to bound states. Theyprovide the essential contribution to the LDOS in the normal region below E = |∆|. Eachbound state at a discrete energy is related to one transport channel. In the limit of an infinitenumber of transport channels there is a continuous local density of states below E = |∆|.The scattering matrix of the normal region relates incoming and outgoing excitations in thefollowing way:

2Involved energies are the difference in Potential energy eV and the width of the gap |∆|


Figure 2.5: Visualization of the two equations (2.29) and (2.30) relating outgoing quasipar-ticles from the normal metal with incoming quasiparticles from both sides and includingAndreev reection processes on both sides. The rst line of equation (2.29), which statesthat an electron reaching the left superconductor has its origin either in a reected electroncoming from the left or in a transmitted electron coming from the right becomes immediatelyclear.

⟩⟩⟨bLebRebLhbRh

=

⟩⟩⟨re te 0 0te re 0 00 0 re te0 0 te te

·⟩⟩⟨aLeaReaLhaRh

. (2.29)

The amplitudes aL and aR describe incoming excitations from the particular superconductortowards the normal structure and the amplitudes bL and bR describe outgoing excitationsfrom the normal structure towards the superconductors. For example, the rst line ofequation (2.29) states, that an electron approaching towards the left superconductor has itsorigin either in a reected electron coming from the left or in a transmitted electron comingfrom the right.

Furthermore, outgoing and incoming excitations are related via the probability ampli-tudes of Andreev reection:⟩⟩⟨

aLeaReaLhaRh

=

⟩⟩⟨0 0 ei L 00 0 0 ei R

eiL 0 0 00 eiR 0 0

·⟩⟩⟨bLebRebLhbRh

. (2.30)

The relation between outgoing quasiparticles from the normal metal to either side andincoming quasiparticles together with the particular probability amplitudes of transmissionand reection as well as the Anreev reection processes on both sides are illustrated inFigure 2.5. Inserting (2.30) into (2.29) provides the condition, that the matrix product of


Figure 2.6: Energy of an Andreev bound state in dependence of ϕ for different transmissioneigenvalues T of the corresponding transport channel. For ϕ = 0 the energy of the boundstate is for all transmissions at E = |∆|. In this case the LDOS should be zero belowE = |∆| in the normal metal as well [4].

the two 4× 4-matrices has an eigenvalue of one. This condition can be converted to providea relation between the energy of a bound state E and the transmission eigenvalue of thecorresponding channel:

E = |∆|√

1− T sin2(ϕ/2).

E depends on the transmission eigenvalue of the corresponding transport channel and ismodulated by the phase difference of the order parameters. In Figure 2.6 the energy of anAndreev bound states is is plotted for various values of T in dependence of ϕ. For ϕ = 0 theenergy of the Andreev bound state is E = |∆| for all transport channels and thus the LDOSshould be zero below the superconducting gap edge in the normal metal as well. For finitephase differences ϕ 6= 0 the LDOS depends on the distribution of transmission eigenvaluesof the analyzed structure and on the phase difference ϕ. The aim of this work is to calculatethe LDOS for various contacts with different distributions of transmission eigenvalues.

Application of Green’s function methods to superconductivity 27

2.5 Application of Green’s function methods to super-

conductivity

2.5.1 General properties of Green’s functions

The microscopic description, which was used in the previous chapter to describe boundariesbetween superconductors and normal metals is of course also valid in the realistic case withmany transport channels. With an increasing number of transport channels however thestructure of scattering matrizes gets more and more complicated and a detailed microscopicdescription is no longer manageable. In order to treat such systems the Green’s functionsmethod is used [5] [6]. Green’s functions are defined as statistical averages of products ofHeisenberg field operators. The Heisenberg picture is one possible formulation of quantummechanics, where the time dependence is carried by the operators and the states are timeindependent. The time evolution of an operator in this picture is described by the Heisenbergequation of motion:

d

dtOH(t) =

i

~

[H, OH(t)

]−

+

(∂O(t)

∂t

)H

. (2.31)

These operators can be derived directly from the operators in the Schrodinger picture. Inthe most general case of an explicitly time-dependent Hamiltonian the Schrodinger equationcan formally be integrated, which leads to

|Ψ(t)〉 = |Ψ(t0)〉 − i

~

∫ t

t0

dt1H(t1)|Ψ(t1)〉.

Reinserting this equation into itself provides a series representation of the wave function foran arbitrary time t:

|Ψ(t)〉 =

(1 +

∞∑n=1

(−i~

)n∫ t

t0

dt1

∫ t1

t0

dt2 . . .

∫ tn−1

t0

dtnH(t1)H(t2) . . . H(tn)

)︸︷︷︸

=S(t,t0)

|Ψ(t0)〉, (2.32)

with (t ≥ t1 ≥ t2 ≥ . . . ≥ tn ≥ t0).

The order of the Hamilton operators in the time evolution operator S(t, t0) is essential, asthey do not necessarily commute for different times. Considering the first two integrals withthe condition (t1 ≥ t2), they can be rewritten in the following way:

∫ t

t0

dt1

∫ t1

t0

dt2H(t1)H(t2) =

∫ t

t0

dt2

∫ t

t2

dt1H(t1)H(t2) =

∫ t

t0

dt1

∫ t

t1

dt2H(t2)H(t1). (2.33)

These equalities become clearer by considering Figure 2.7. In the second step of (2.33) theindices were renamed. Using (2.33), the two integrals can be rewritten as

28 Application of Green’s function methods to superconductivity

Figure 2.7: Visualization of conversion (2.33). [5]

∫ t

t0

dt1

∫ t1

t0

dt2H(t1)H(t2) =1

2

∫ t

t0

dt1

∫ t

t0

dt2(H(t1)H(t2)Θ(t1 − t2) + H(t2)H(t1)Θ(t2 − t1)).

By incorporation of all integrals, this expression can be generalized and the time evolutionoperator S(t, t0) can be rewritten as a time ordered exponential function:

S(t, t0) = 1 +∞∑n=1

1

n!(−i~

)n∫ t

t0

dt1

∫ t

t0

dt2 . . .

∫ t

t0

dtnTt(H(t1)H(t2) . . . H(tn))

= Tte− i

~∫ tt0dt′H(t′)

.

The operator Tt denotes the time ordering of the integral, beginning with the largest timet and ending with the time t0. Consideration of the expectation value of an observable O,which obviously must be the same in the Heisenberg picture and in the Schrodinger picture,provides a general expression for the Heisenberg operator OH(t):

〈O〉(t) = 〈ΨH(0)|OH(t)|ΨH(0)〉 != 〈Ψ(t)|O(0)|Ψ(t)〉 = 〈Ψ(0)| S−1(t)O(0)S(t)︸︷︷︸

OH(t)

|Ψ(0)〉.

Here the property S∗ = S−1 of the time evolution operator, with a time ordering in theopposite direction, was used. In the following the inverse time ordering is described by the

operator ˆTt. Another important property of the time evolution operator is:


S(t1, 0) · S−1(t2, 0) = S(t1, t2), if t1 > t2. (2.34)

With the previous results, the Heisenberg field operators can be most generally defined as:

ΨH,σ(~r, t) = S−1(t)Ψσ(~r, 0)S(t) =(

ˆTtei~∫ t0 dt′H(t′)

)Ψσ(~r, 0)

(Tte− i

~∫ t0 dt′H(t′)

); (2.35)

Ψ†H,σ(~r, t) = S−1(t)Ψ†σ(~r, 0)S(t) =(

ˆTtei~∫ t0 dt′H(t′)

)Ψ†σ(~r, 0)

(Tte− i

~∫ t0 dt′H(t′)

). (2.36)

The single particle Green’s functions are defined as statistical averages of products of theseoperators. In most applications calculations are performed in a grand-canonical ensemble.Thus the Hamiltonian H in the definitions (2.35) and (2.36) must be replaced by a grand-canonical Hamiltonian H − µN . In the following definitions of Green’s functions and thederivation of their properties the substitution ~ = 1 is used, ~ is resubstituted when appro-priate. Depending on the physical situation, four Green’s functions have turned out to bemost useful. The first one is the retarded Green’s function. It is related to the first orderresponse of a system to an external perturbation. The definition of the retarded Green’sfunction is

GRσσ′(~r1, t1, ~r2, t2) = −iZ−1 Tr

(e−β(H−µN)

[ΨH,σ(~r1, t1), Ψ†H,σ′(~r2, t2)

]±

)Θ(t1 − t2).

[. . .]± denotes a commutator for Bose fields and an anticommutator in the case of Fermifields. Quite similar but less intuitive is the advanced Green’s function, which is defined by

GAσσ′(~r1, t1, ~r2, t2) = iZ−1 Tr

(e−β(H−µN)

[ΨH,σ(~r1, t1), Ψ†H,σ′(~r2, t2)

]±

)Θ(t2 − t1).

The third type of Green’s function is the causal Green’s function. It is used in the descriptionof systems at zero temperature. It is given by

GCσσ′(~r1, t1, ~r2, t2) =− iZ−1 Tr

(e−β(H−µN)Tt

(ΨH,σ(~r1, t1)Ψ†H,σ′(~r2, t2)

))=− iZ−1 Tr

(e−β(H−µN)ΨH,σ(~r1, t1)Ψ†H,σ′(~r2, t2)

)Θ(t1 − t2)

± iZ−1 Tr(e−β(H−µN)Ψ†H,σ′(~r2, t2)ΨH,σ(~r1, t1)

)Θ(t2 − t1).

The meaning of Tt is the same as before. It orders the product of field operators beginningwith late times. For fermionic operators a - sign must be incorporated for t2 > t1, whichgenerates a + sign in the third line. The - sign is for bosonic operators. Whenever thereare two signs in the following, the upper one refers to fermionic systems, the lower one tobosonic systems. The causal Green’s function is directly related to the last type of Green’sfunction, which is called the temperature Green’s function. It is obtained by replacing thereal-valued time t with an imaginary time τ = it. This corresponds to a rotation in the


plane of the complex variable t by 90 degrees. It is called a Wick rotation. The timeordering operator Tt is replaced by the operator Tτ which orders in imaginary time τ . Theimaginary time Heisenberg annihilation operator is no longer the adjoint of the creationoperator. Accordingly a new notation for the imaginary time Heisenberg field operators isintroduced:

ΨH,σ(~r, τ) = S−1(τ)Ψσ(~r, 0)S(τ) = ( ˆTτe1~∫ τ0 dτ ′H(τ ′))Ψσ(~r, 0)(Tτe

− 1~∫ τ0 dτ ′H(τ ′)); (2.37)

ˆΨH,σ(~r, τ) = S−1(τ)Ψ†σ(~r, 0)S(τ) = ( ˆTτe1~∫ τ0 dτ ′H(τ ′))Ψ†σ(~r, 0)(Tτe

− 1~∫ τ0 dτ ′H(τ ′)). (2.38)

With these imaginary time field operators, the temperature Green’s function is defined inthe interval −1/T < τ1, τ2 < 1/T by

GTσσ′(~r1, τ1, ~r2, τ2) =− Z−1 Tr

(e−β(H−µN)Tτ

(ΨH,σ(~r1, τ1) ˆΨH,σ′(~r2, τ2)

))=− Z−1 Tr

(e−β(H−µN)ΨH,σ(~r1, τ1) ˆΨH,σ′(~r2, τ2)

)Θ(τ1 − τ2)

± Z−1 Tr(e−β(H−µN) ˆΨH,σ′(~r2, τ2)ΨH,σ(~r1, τ1)

)Θ(τ2 − τ1).

The approach in the following is first to derive the equation of motion for these singleparticle Green’s functions in a superconducting system, and then to solve it for specificboundary conditions. These boundary conditions are determined by the properties of theanalyzed structure. As soon as the Green’s functions are known, all observables can beexpressed through them. One important property of all four Green’s functions is, thatalthough they in principle depend on two time arguments, under certain conditions thisdependence is actually only a dependence on the time difference, which is effectively onlyone time argument. If the Hamiltonian of the system is not explicitly time-dependent, theBoltzmann factor from the statistical average commutes with the time evolution operator.Correlations of the system can thus be converted in the following way:

⟨ΨH,σ(~r1, τ1) ˆΨH,σ′(~r2, τ2)

⟩st

=

⟨S−1(τ1, 0)Ψσ(~r1, 0) S(τ1, 0)S−1(τ2, 0)︸︷︷︸

S(τ1,τ2)

Ψ†σ′(~r2, 0)S(τ2, 0)

⟩st

=

⟨S(τ2, 0)S−1(τ1, 0)︸︷︷︸

S−1(τ1,τ2)

Ψσ(~r1, 0)S(τ1, τ2)Ψ†σ′(~r2, 0)

⟩st

=⟨

ΨH,σ(~r1, τ1 − τ2)Ψ†σ′(~r2, 0)⟩st.

This conversion is also valid for Green’s functions, since Green’s functions are superpositionsof correlations. In this derivation property (2.34) and the cyclic invariance of the trace wereused. Furthermore τ1 > τ2 was assumed, similar considerations are possible for τ1 < τ2. Thisconversion holds true also for real time Green’s functions. For such systems it is convenientto work with Fourier transformed Green’s functions with respect to the (imaginary) timedifference of their arguments t = t1− t2 (τ = τ1− τ2). It turns out that the Fourier integral


for GR(t) only converges in the upper half-plane of the complex variable ω and the Fouriertransform of GA(t) exists only in the lower half plane of complex ω, but the function G(ω)to which they converge is the same in both cases. The function G(ω) is analytic in the wholecomplex plane except on the real axis. For real energies E the retarded and the advancedfunctions can be expressed through it as follows:

GRσσ′(~r1, ~r2, E) = lim

δ→0Gσσ′(~r1, ~r2, E + iδ); (2.39)

GAσσ′(~r1, ~r2, E) = lim

δ→0Gσσ′(~r1, ~r2, E − iδ). (2.40)

An important property of Green’s functions is that they are not independent of each other.Not only GR(E) and GA(E) are different limits of the same function G(ω). It can be shownthat G(ω), as well as the frequency representations of the other Green’s functions can bewritten in a spectral representation, where all of them only depend on one function A(x),which is called the spectral function. As soon the spectral function is known all of theGreen’s functions can be calculated immediately. It is given by

Aσσ′(~r1, ~r2, x) = Z−1(1± e−βx

)∑i,j

e−βEi(Ψσ(~r1))ij(Ψ†σ′(~r2))jiδ(x− Ej + Ei). (2.41)

(...)ij denotes the matrix element of the particular operator with respect to the eigenstates

|i〉, |j〉 of the Hamiltonian with eigenvalues Ei and Ej, for example(Ψσ(~r))ij = 〈i|Ψσ(~r)|j〉.The spectral representation of G(ω) is given by

Gσσ′(~r1, ~r2, ω) =

∫ ∞−∞

dxAσσ′(~r1, ~r2, x)

ω − x. (2.42)

Because the temperature Green’s function is defined only in the interval [−1/T, 1/T ], itsFourier representation is only defined at discrete frequencies ωn, which are called the Mat-subara frequencies. The Green’s function is called the Matsubara Green’s function. Itsrelation to the spectral function is given by

GMσσ′(~r1, ~r2, ωn) =

∫ ∞−∞


iωn − x. (2.43)

A similar representation can be found for GCσσ′(~r1, ~r2, ω), but since it doesn’t play a role in

this work, it is omitted here. It can be found in [6] as well as the detailed derivations of theprevious results. (2.42) and (2.43) provide a relation between GM(ωn) and G(ω). GM(ωn)can be analytically continued to the whole complex plane and is then directly related to thefunction G(ω): 3

GMσσ′(~r1, ~r2, ω) = Gσσ′(~r1, ~r2, iω). (2.44)

3This relation is valid if Gσσ′(~r1, ~r2, iω) is known, and GMσσ′(~r1, ~r2, ωn) is to be determined. The approachin the following is reverse. Gσσ′(~r1, ~r2, iω) is determined from GMσσ′(~r1, ~r2, ωn). In that case a boundarycondition for Gσσ′(~r1, ~r2, iω) is necessary to determine it unambiguously. In what follows this boundarycondition is automatically fulfilled. More on this topic can be found in [6].


Due to this relation it is not necessary to solve an equation of motion for each Green’sfunction separately. The solution for one of them directly determines the others. In thefollowing the equation of motion is derived for the temperature Green’s function, becausethis function has certain properties, which make it very manageable. Furthermore (2.44)states that as soon as GM(ωn) is found, the function G(ω) can be derived directly byperforming a rotation in the plane of complex ω. And G(ω) determines GR(E) and GA(E)through (2.39) and (2.40). To determine the spectral function is more complicated, becauseit is not well-defined, if only GM(ωn) is known. Multiple spectral functions are imaginable,which result in the same GM(ωn). Hence, further conditions are necessary to determine A(x)unambiguously. In this work it is sufficient to find GR(E), because it is directly related tothe LDOS.

It follows from (2.39) and (2.42), that the retarded Green’s function for real-valued Ebecomes

GRσσ′(~r1, ~r2, E) = lim

δ→0Gσσ′(~r1, ~r2, E + iδ) = lim

δ→0

∫ ∞−∞


E + iδ − x

= −iπAσσ′(~r1, ~r2, E) + P

(∫ ∞−∞


E − x

).

The second term in the second line symbolizes the Cauchy principal value of the integral.Since the spectral function (2.41) is real for coinciding spatial arguments ~r1 = ~r2 and equalspins σ = σ′, its diagonal component is related to the imaginary part of the retarded Green’sfunction as follows:

− 1

πImGR

σσ(~r, ~r, E) = Aσσ(~r, ~r, E). (2.45)

In homogeneous systems the Green’s functions depend only on the difference of spatialcoordinates. It is thus advantageous to work with spatially Fourier transformed Green’sfunctions. For the temperature Green’s function this transformation is defined by

GTσσ′(

~k, τ) =

∫d3rei

~k~ρGTσσ′(~ρ = ~r1 − ~r2, τ).

To find the expression for GT (~k, τ), the Heisenberg field operators are expanded in momen-tum eigenstate creation operators:

ΨH,σ(~r, τ) =1√V

∑~k

aH,~kσ(τ)ei~k~r;

ˆΨH,σ(~r, τ) =1√V

∑~k

ˆaH,~kσ(τ)e−i~k~r.

Inserting these expansions and requiring momentum conservation provides an expression forthe spatially Fourier transformed temperature Green’s function:

GTσσ′(

~k, τ) = −⟨Tτ (aH,~k,σ(τ)ˆaH,~k,σ′(0))

⟩st.


For a not explicitly time-dependent Hamiltonian time Fourier representation can again beused for GT

σσ′(~k, τ). Similar to (2.43) it is given as an integral over the spectral function

Aσσ′(~k, x), which results from (2.41) by expanding field operators in momentum eigenstatecreation operators:

Aσσ′(~k, x) = Z−1(1± e−βx)∑i,j

e−βEi(a~kσ)ij(a†~kσ′

)jiδ(x− Ej + Ei). (2.46)

For coinciding spins σ′ = σ this spectral function is again real-valued and directly relatedto the imaginary part of the retarded Green’s function GR

σσ(~k, E):

− 1

πImGR

σσ(~k, E) = Aσσ(~k, E). (2.47)

This relation is important for this work, as the spectral function Aσσ(~k,E) is related to the

density of states. This property becomes clear by considering A(~k, E) at zero temperature.(2.46) provides

Aσσ(~k,E) =∑m

|〈m|a†~k,σ|0〉|2δ(E − Em). (2.48)

For fixed momentum ~k and spin σ and thus a fixed momentum creation operator a†~k,σ the

sum counts for a given energy E the states |m〉, which result from the vacuum state by

application of a†~k,σ. By virtue of the δ-function Aσσ(~k,E) is exactly the density of states at

a certain energy for given momentum and spin. The total density of states is given by thesum over all momenta and spins:

N(E) =1

V

∑~kσ

Aσσ(~k,E) =1

V

∑~kσ

− 1

πImGR

σσ(~k,E) = − 2

πImGR

αα(~r, E)|~r=0.

The factor 2 arising in the last step accounts for the spin degeneracy of each state. Allthese calculations can be generalized to a system, which is not translationally invariant.A position dependent local density of states can then be defined through Aσσ(~r, ~r, E). Itsinterpretation is similar to the homogeneous case:

N(~r, E) =∑σ

Aσσ(~r, ~r, E) = − 2

πImGR

σσ(~r, ~r, E).

The approach is to determine the retarded Green’s function for a system consisting ofcontacts between superconductors and normal metals. By virtue of the previous relationthe LDOS is given by the imaginary part of the retarded Green’s function. Due to therelation between the retarded Green’s function and the Matsubara Green’s function, it isequivalent to derive the equation of motion for the Matsubara Green’s function.


2.5.2 Equation of motion for a superconducting system

All properties of Green’s functions that were depicted so far are not specific for supercon-ductors and are valid for any general many particle system, containing bosonic or fermionicparticles. In the derivation of the equation of motion for the temperature Green’s functionin a superconducting system, the special properties of such a system have to be taken intoaccount [8]. In chapter 2.3 it was already shown, that expectation values of the type (2.11)are non-zero in the superconducting ground state (2.8). In the derivation of the equationof motion this property brings new Green’s functions into play, which don’t emerge in thenormal state, but play an important role in superconducting systems. They contain twocreation or annihilation operators and are called Gor’kov Green’s functions. Because elec-trons are fermions, in the following the Green’s functions from the last chapter are usedwith the sign for fermionic field operators.

To derive the equation of motion for the temperature Green’s function, the first step isto consider the equation of motion for the imaginary time Heisenberg field operators (2.37)

and (2.38). Using the Hamiltonian (2.9) in the absence of magnetic fields ( ~A = 0) andexternal potentials (U0 = 0) (2.31) yields:

∂

∂τΨH,σ(~r, τ) =

(∇2

2m+ µ

)ΨH,σ(~r, τ) + V

∑γ

ˆΨH,γ(~r, τ)ΨH,γ(~r, τ)ΨH,σ(~r, τ); (2.49)

∂

∂τˆΨH,σ(~r, τ) = −

(∇2

2m+ µ

)ˆΨH,σ(~r, τ)− V

∑γ

ˆΨH,σ(~r, τ) ˆΨH,γ(~r, τ)ΨH,γ(~r, τ). (2.50)

The temperature Green’s function is not analytic at τ = 0. This can be seen from

limτ→0+

(GTσσ′(~r1, ~r2, τ)−GT

σσ′(~r1, ~r2,−τ))

= − limτ→0+

⟨ΨH,σ(~r1, τ) ˆΨH,σ′(~r2, 0) + ˆΨH,σ(~r1, 0)ΨH,σ′(~r2,−τ)

⟩st

= −δσσ′δ(~r1 − ~r2).

This non-analytic property has to be taken into account in the calculation of the timederivative of GT

σσ′(~r1, ~r2, τ1, τ2). The detailed derivation can be found in [8]. The result isgiven by:

∂

∂τ1

GTσσ′(~r1, ~r2, τ1, τ2)

=− δσσ′δ(~r1 − ~r2)δ(τ1 − τ2) +

(∇2

1

2m+ µ

)GTσσ′(~r1, ~r2, τ1, τ2)

− V∑γ

⟨Tτ

ˆΨH,γ(~r1, τ1)ΨH,γ(~r1, τ1)ΨH,σ(~r1, τ1) ˆΨH,σ′(~r2, τ2)⟩st︸︷︷︸

(∗)

. (2.51)

The time derivative of a single particle Green’s function is obviously related to a two-particle Green’s function (∗). The statistical average in (∗) is an average with respect


to the Hamiltonian (2.9), which contains amongst others an interaction term with fouroperators. This makes Wick’s theorem not applicable here. Wick’s theorem is only validfor non-interacting systems. An exact treatment requires a perturbative expansion of theinteraction term in the Boltzmann factor in the interaction strength V and a calculation ofFeynman diagrams. But like before in the derivation of the Bogoliubov-de Gennes equations,it has proved valid in the description of superconductivity to use a generalized HartreeFock approximation by application of Wick’s theorem to (∗). This is only valid for weaklycorrelated systems. It approximates

⟨Tτ

ˆΨH,γ(x1)ΨH,γ(x1)ΨH,σ(x1) ˆΨH,σ′(x2)⟩st

≈ −⟨Tτ ΨH,γ(x1) ˆΨH,γ(x1)

⟩St

⟨Tτ ΨH,σ(x1) ˆΨH,σ′(x2)

⟩st

+⟨Tτ ΨH,σ(x1) ˆΨH,γ(x1)

⟩St

⟨Tτ ΨH,γ(x1) ˆΨH,σ′(x2)

⟩st

−⟨Tτ ΨH,σ(x1)ΨH,γ(x1)

⟩St

⟨Tτ

ˆΨH,γ(x1) ˆΨH,σ′(x2)⟩st.

Here the abbreviation (~r1, τ1) = (x1) was introduced. In the following two types of in-teractions have to be distinguished. The first two terms have their origin in interactions,which are already present in the normal state and don’t contribute to the superconductingproperties of the system. They lead to a renormalization of the chemical potential andcan bring temporal decay of quasiparticles into play. The third average however describesinteractions, which are responsible for the superconducting properties. They vanish in thenormal state. In the following it is sufficient to account only for this type of interaction,and to disregard the rest. The equation of motion for GT

σ,σ′(x1, x2) (2.51) becomes

(∂

∂τ1

− ∇21

2m− µ

)GTσσ′(x1, x2)−

∑γ

∆σγ(x1)F Tγσ′(x1, x2) = −δσσ′δ(x1 − x2), (2.52)

where a new type of Green’s functions was introduced. They are called Gor’kov Green’sfunctions and are defined by

F Tσσ′(x1, x2) = −

⟨Tτ ΨH,σ(x1)ΨH,σ′(x2)

⟩st

;

F Tσσ′(x1, x2) = −

⟨Tτ

ˆΨH,σ(x1) ˆΨH,σ′(x2))⟩st.

The pairing potential ∆σ,σ′(x) is directly related to them through

∆σσ′(x) = V F Tσσ′(x, x). (2.53)

As noted in chapter 2.1, for most conventional superconductors the antisymmetry of thepair wave function is carried by the spin component. The two electrons of a Cooper pair arethus in a singlet spin state and the wave function is symmetric in spatial coordinates. Theinteraction causing this singlet pairing must have the same symmetry (s-wave-interaction,


d-wave-interaction,...). The pair amplitude F Tσσ′(x1, x2) and the pair potential ∆σσ′(x) are

thus antisymmetric in spin indices:

F Tσσ′(x1, x2) =− F T

σ′σ(x1, x2);

∆σσ′(x) =−∆σ′σ(x).

Obviously the diagonal components with σ = σ′ are zero. Pair potential and Gor’kovGreen’s functions in spin space can thus be defined as follows:

∆σσ′(x) = iτ(2)σσ′∆(x);

∆†σσ′(x) = −iτ (2)σσ′∆

∗(x);

F Tσσ′(x1, x2) = iτ

(2)σσ′F

T (x1, x2);

F Tσ′σ(x1, x2) = −iτ (2)

σσ′FT (x1, x2).

Here τ(2)σσ′ is the Pauli spin matrix:

τ (2) =

(0 −ii 0

).

The relation between ∆(x), ∆∗(x) and F T (x1, x2), F T (x1, x2) is given by

∆(x) = V F T (x, x);

∆∗(x) = V F T (x, x).

This definition of the pair potential ∆σσ′(x) is consistent with the definition (2.11) from themicroscopic approach. The temperature Green’s function GT

σσ′(x1, x2) is proportional to theunit matrix in spin indices. Due to these symmetries, equation (2.52) can be rewritten as(

∂

∂τ1

− ∇21

2m− µ

)GT (x1, x2)−∆(x1)F T (x1, x2) = −δ(x1 − x2). (2.54)

For F T (x1, x2) a similar equation of motion can be found:(∂

∂τ1

+∇2

1

2m+ µ

)F T (x1, x2)−∆∗(x1)GT (x1, x2) = 0. (2.55)

Analog equations exist for F T (x1, x2) and a function GT (x1, x2), which is defined as

GTσσ′(x1, x2) =

⟨Tτ

ˆΨH,σ(x1)ΨH,σ′(x2))⟩St

= δσ,σ′GT (x1, x2).

The two equations of motion have the following form:

−(∂

∂τ1

+∇2

1

2m+ µ

)GT (x1, x2)−∆∗(x1)F T (x1, x2) = −δ(x1 − x2); (2.56)(

− ∂

∂τ1

+∇2

1

2m+ µ

)F T (x1, x2)−∆(x1)GT (x1, x2) = 0. (2.57)


Equations (2.54), (2.55), (2.56) and (2.57) can be combined to one single 2 × 2 matrixequation in Nambu space of electrons and holes. This equation describes the time evolutionof the matrix Green’s function GT (x1, x2), which is given by

GT (x1, x2) =

(GT (x1, x2) F T (x1, x2)−F T (x1, x2) GT (x1, x2)

). (2.58)

Furthermore the matrix operator

(GT )−1 = −τ (3) ∂

∂τ1

− H

is introduced, where

H =

(−∇

21

2m− µ ∆

−∆∗ −∇21

2m− µ

)and τ (3) is the Pauli matrix in Nambu space:

τ (3) =

(1 00 −1

).

Altogether the equation of motion for the matrix Green’s function GT (x1, x2) is given by

(GT (x1))−1GT (x1, x2) = 1δ(x1 − x2). (2.59)

This equation of motion for the temperature Green’s function in a superconducting systemis called Gor’kov equation. Similar to the Green’s function in a normal metal, which isrelated to the Schrodinger equation, the Gor’kov equation is related to the Bogoliubov-deGennes equation (2.20). Making calculations consistent in the whole work was the reasonto define the two component wave function in (2.20) with a - sign in front of vm(~r). If thelower component is multiplied with (-1) and all is brought to the same side (2.20) becomes((

Em 00 −Em

)︸︷︷︸

−τ (3) ∂∂τ

−(He + U(~r) ∆(~r)

−∆∗(~r) (H∗e + U(~r))

)︸︷︷︸

=H

)(um(~r)−vm(~r)

)︸︷︷︸→GT (x1,x2)

= 0︸︷︷︸→1δ(x1−x2)

.

Replacing the two component wave function with the matrix Green’s function (2.58) andincorporation of a δ-function on the right side directly yields (2.59). The Gor’kov equa-tion depends like the Bogoliubov-de Gennes equation on the pairing potential ∆, whichconversely depends on the function F through equation (2.53) and has to be determinedself-consistently.

A similar equation can be derived for the dependence on the second pair of coordinates(~r2, τ2). It has the form

GT (x1, x2) ¯GT

(x2)−1 = 1δ(x1 − x2),

where the inverse matrix operator ¯GT

(x2)−1 is defined by


¯GT

(x2)−1 = τ (3) ∂

∂τ2

− ¯H.

As long as no magnetic field is included ¯H = H, otherwise the sign in front of the vectorpotential ~A(~r) changes.

In chapter 2.5.1 it was shown that for a not explicitly time-dependent Hamiltonian theGreen’s function depends only on the time difference of its arguments and it is advantageousto work with temporal Fourier transformed Green’s functions:

GM(~r1, ~r2, ωn) =

∫ 1/T

0

eiωnτ GT (~r1, ~r2, τ)dτ. (2.60)

For Fermions the discrete Matsubara frequencies are ωn = (2n+1)πT . The Gor’kov equationin frequency representation becomes

(iωnτ(3) − H)GM(~r1, ~r2, ωn) = 1δ(~r1 − ~r2). (2.61)

The self-consistency equation for the pairing potential ∆(~r) transforms to

∆(~r)

V= T

∑n

FM(~r, ~r, ωn). (2.62)

In a similar way the dependence on spatial coordinates can be replaced by a Fourier transfor-mation to momentum space. In general, Green’s functions depend on two spatial arguments.It is thus necessary to perform two Fourier transformations, one for each argument. Themomentum representation is defined by

GM(~r1, ~r2, ωn) =

∫d3k

(2π)3

d3k′

(2π)3GM(~k,~k′, ωn)ei

~k(~r1−~r2)+i~k′(~r1+~r2)/2. (2.63)

The Fourier transform was carried out in respect of the relative coordinate ~r1 − ~r2 and thecenter of mass coordinate (~r1+~r2)/2. For general inhomogeneous systems this representationis not advantageous, because in order to determine the Gor’kov equation in momentumspace, it is necessary to transform a product of two quantities, which both depend onspatial coordinates. The Fourier transform of such a product is a convolution, which isquite complicated to handle compared to a linear differential equation. It is more favorableto work in this case with a mixed Fourier representation, where only the relative coordinate~r1 − ~r2 is transformed to momentum space and the center of mass coordinate (~r1 + ~r2)/2 iskept in position space. The matrix operator depends only on this coordinate.

In the special case of a homogeneous system Green’s functions depend only on the relativecoordinate ~r1 − ~r2 and the matrix operator (GT )−1 is not position-dependent. Thus noconvolutions come into play and it is sufficient so perform one transformation in respect ofthe relative coordinate. The transformation has the form

GM(~r1, ~r2, ωn) =

∫d3k

(2π)3GM(~k, ωn)ei

~k(~r1−~r2) (2.64)

and the Gor’kov equation is no longer a differential equation:


(iωnτ

(3) −(

ξ~k ∆−∆∗ ξ~k

))GM(~k, ωn) = 1. (2.65)

The definition of ξ~k is the same as in chapter 2.4. The order parameter equation becomes

∆

V= T

∑n

∫d3k

(2π)3FM(~k, ωn). (2.66)

Equation (2.65) is algebraic and can directly be solved. The order parameter ∆ is constantin the whole volume and the Gor’kov equation splits into two sets of two coupled algebraicequations:

(iωn − ξ~k)GM + ∆FM = 1;

(iωn + ξ~k)FM + ∆∗GM = 0;

(iωn − ξ~k)FM −∆GM = 0;

−(iωn + ξ~k)GM + ∆∗FM = 1.

These equations can be solved by inserting them into each other. The results are thestationary Green’s functions in a bulk superconductor:

GM = −ξ~k + iωn

ξ2~k

+ |∆|2 + ω2n

; FM =∆

ξ2~k

+ |∆|2 + ω2n

;

GM = −ξ~k − iωn

ξ2~k

+ |∆|2 + ω2n

; FM =∆∗

ξ2~k

+ |∆|2 + ω2n

.

By application of (2.39),(2.40) and (2.44), the retarded and advanced Green’s functions canbe determined through analytic continuation and the rotation ωn = −iE:

GR(A)(E) =

(GR(A) FR(A)

−FR(A) GR(A)

)=− lim

δ→0+

(ξ~k + E −∆

∆∗ ξ~k − E

)× 1

(ξ~k + i√|∆|2 − (E ± iδ)2)(ξ~k − i

√|∆|2 − (E ± iδ)2)

.

This result can also be obtained by solving the Gor’kov equation for the retarded andadvanced Green’s functions, which can be found from (2.61) or (2.65) respectively by theusual rotation. From the retarded Green’s function the density of states can be immediatelycalculated:

N(E) = 1/V∑~k,σ

(− 1

πIm(GR(~k, E)

))≈ − 2

π

∫d3k

1

(2π)3Im(GR(~k,E)

)≈ −N0

πIm

(∫dξ~kG

R(ξ~k, E)

).


Figure 2.8: BCS density of states in a bulk superconductor with a symmetric gap of width2|∆| around the Fermi energy of the normal state. At the gap edge the density of statesdiverges.

The calculation of this expression is not trivial, because the integral consists of a componentwith ξ~k in the enumerator, which diverges for large ξ~k. For the density of states, thisdivergence is not relevant, as it doesn’t contribute to the imaginary part of the integral andcan thus be neglected. The imaginary part becomes

Im

(∫dξ~kG

R(ξ~k, E)

)= −πRe

(E√

(E + iδ)2 − |∆|2

).

Hence the density of states in a bulk superconductor is given by

N(E) = N0 · Re

(E√

(E + iδ)2 − |∆|2

). (2.67)

This result is illustrated in Figure 2.8. The characteristic is a symmetric gap around theFermi energy of the normal state of width 2|∆|. At the gap edge the density of statesdiverges. The previously noted diverging integral in the derivation of the density of stateswas no problem, because it didn’t contribute to the imaginary part of the integral. But itcontributes to other observables, like for example the particle density. These are observables,which have already a finite expectation value in the normal state. For them, the observablemust be split into a normal part and a superconducting contribution, which vanishes in thenormal state. This is explained in more detail in the chapter about quasiclassical Green’sfunctions 2.5.4. Another detail was suppressed in this derivation. Figure 2.8 shows, thatthe density of states is an even function of energy, while from expression (2.67) an unevenfunction is expected. This has got to do with the sign of the complex square root in the


denominator, which has to be chosen in the right way. The right choice of this sign makes(2.67) an even function of energy. It is fixed by the infinitesimal imaginary parts. This isexplained in more detail in the chapter about quasiclassical Green’s functions.

2.5.3 Incorporation of random impurity potentials

A real superconductor, as every real crystal, contains many impurities and crystal defects,which are randomly distributed. Green’s function methods allow for a possibility how ran-dom impurity potentials can be taken into account. This makes the calculated observablesmore realistic and more comparable to experimental results. The method is called crossdiagram technique [6] [7] [8]. The basic assumption is, that the physical properties of thesystem do not depend on the exact realization of the impurity distribution and can thusbe obtained by averaging over all possible realizations. Furthermore, the impurity potentialU(~r) is assumed to be a small perturbation compared to the unperturbed Hamiltonian:

G−1 = G−10 − U(~r) ; U(~r) G−1

0 . (2.68)

The full Green’s function can thus be expanded in powers of this perturbation:

G = G(0) + G(1) + G(2) + . . . . (2.69)

Gor’kov equations for the unperturbed and the perturbed Green’s functions are given by

G−1G = 1δ(x1 − x2);

G−10 G(0) = 1δ(x1 − x2).

Inserting (2.68) and the expansion (2.69) into the equation for the perturbed system andsubtracting the equation for the unperturbed system provides by comparing equal powersin perturbation a relation between the different order corrections:

G−10 G(k) − UG(k−1) = 0.

Multiplication by G(0) from the left and integration over x relates the k-th order correctionto the k-1-th order correction. Together with (2.69) this results in a Dyson equation forthe Green’s function of the perturbed system. This equation is equivalent to the Gor’kovequation, since the Gor’kov equation was used for its derivation:

G(x, x′) =G(0)(x, x′) +

∫d4x1G

(0)(x, x1)U(x1)G(0)(x1, x′)

+

∫ ∫d4x1d

4x2G(0)(x, x1)U(x1)G(0)(x1, x2)U(x2)G(0)(x2, x

′) + . . .

=G(0)(x, x′) +

∫d4x1G

(0)(x, x1)U(x1)G(x1, x′). (2.70)

This equation can be graphically presented by Feynman diagrams. This is shown in Figure2.9. The perturbation potential due to impurities is given by


Figure 2.9: Diagrammatic representation of the Dyson equation (2.70). The full Green’sfunction is symbolized by a double line, a single line symbolizes the unperturbed Green’sfunction [8].

U(~r) =∑a

u(~r − ~ra).

u(~r − ~ra) is the potential of a single impurity atom. The total perturbation is a sumover all these potentials. Inserting this potential into the Dyson equation (2.70) and usingmomentum representation of the Green’s functions (2.64) provides

G(~r, ~r′) =

∫d3k

(2π)3G(0)(~k)ei

~k(~r−~r′)

+∑a

∫d3kd3k′

(2π)6ei(

~k~r−~k′~r′)e−i(~k−~k′)~raG(0)(~k)u(~k − ~k′)G(0)(~k′)

+∑a,b

∫d3kd3k′d3k1

(2π)9ei(

~k~r−~k′~r′)e−i(~k−~k1)~rae−i(

~k1−~k′)~rb

· G(0)(~k)u(~k − ~k1)G(0)(~k1)u(~k1 − ~k′)G(0)(~k′) + . . . . (2.71)

Here a homogeneous system was assumed for simplicity, but generalization to a inhomoge-neous system can be performed in the same way. Due to the assumption of this methodthe particular realization of the impurity potential U(~r) is not important. (2.71) is thusaveraged over all possible impurity configurations. This average is equivalent to replacingthe sum over impurity positions by an integral over all possible positions multiplied withthe density of impurity atoms: ∑

a

→ nimp

∫d3ra.

Application of Green s function methods to superconductivity 43

Figure 2.10: Diagrammatic representation of the impurity contribution, if scattering occursat dierent impurities ra = rb [8].

Figure 2.11: a) Diagram where scattering occurs two times at each impurity without crossingscattering lines b) Diagram with two scattering events per impurity atom with crossingscattering lines [8].

The integrals generate -functions. It is important to distinguish two cases when averaging(2.71). In the rst case, scattering occurs at dierent impurity atoms χra = χrb. Thenexpression (2.71) becomes a Dyson equation for a constant diagonal perturbation of theform

H1 =

nimpu(0) 0

0 nimpu(0)

.

Such terms lead to a renormalization of the chemical potential and can be ignored. Thediagrams related to this case are depicted in Figure 2.10. In the second case scatteringoccurs at the same atom. This leads to diagrams as depicted in Figure 2.11. Equation(2.71) becomes


Figure 2.12: Dyson equation, containing all relevant diagrams without crossing interactionlines [8].

G(~k) = G(0)(~k) + G(0)(~k)Σ(1)(~k)G(0)(~k) + G(0)(~k)Σ(2)(~k)G(0)(~k) + . . . ,

where the first order self-energy Σ(1) as well as higher order self-energies were introduced.They are defined by

Σ(1)(~k) = nimp

∫d3k1

(2π)3|u(~k − ~k1)|2G(0)(~k1);

Σ(2)(~k) = nimp

∫d3k1

(2π)3

d3k2

(2π)3u(~k − ~k1)u(~k1 − ~k2)u(~k2 − ~k)G(0)(~k1)G(0)(~k2);

. . . .

It can be shown that diagrams, where scattering occurs more than twice at the same atom areby a factor u/EF , which was assumed small, less important than those with two scatteringevents per atom. Furthermore, the momentum argument of each Green’s function in thediagram must be close to the Fermi surface to give a relevant contribution. Diagrams withcrossed scattering lines require more restrictions on the momentum integrations than thosewithout crossed interaction lines and are thus less important. The exact derivations of thesetwo statements as well as the exact calculations of the previously mentioned properties canbe found in [8]. It is thus sufficient to sum up only diagrams with two scattering eventsat the same atom without crossed interaction lines. With these approximations equation(2.71) becomes

G(~k) = G(0)(~k) + G(0)(~k)Σ(~k)G(~k). (2.72)

This expression contains the full Green’s function on the right side. The full self energyΣ(~k) is defined by

Σ(~k) = nimp

∫d3k1

(2π)3|u(~k − ~k1)|2G(~k1). (2.73)

This Dyson equation can be represented by the diagram in Figure 2.12. The Gor’kovequation, which is related to (2.72), is given by

(iωnτ(3) − H − Σ)G(~r1, ~r2, ωn) = 1δ(~r1 − ~r2). (2.74)


In this approximation the only effect of impurities is that the self-energy Σ(~k) has to beincorporated into the Gor’kov equation.

2.5.4 Quasiclassical approximation and Eilenberger equation

The aim of Green’s function methods is to solve the equation of motion for a given systemwith fixed boundary conditions and then to calculate the desired observables, which canbe expressed through Green’s functions. For this purpose the Green’s functions in positionspace representation must be considered in the limit ~r1 → ~r2, multiplied with a specificpower of the quasiparticle momentum. This specific power depends on the desired observ-able. In momentum representation this limit annihilates the exponential factor and whatremains is the Green’s function, integrated over momentum space. All relevant quasiparticlestates have energies close to the Fermi surface in a layer of thickness of several |∆| and thecontributions to the integral come from the poles of the Green’s function at these energies.The relative change of momentum in this layer is given by

δp

pF∼ δξ~pvFpF

∼ |∆|v2Fm∼ |∆|EF∼ (ξ0pF )−1.

If the quasiclassical condition (2.22) is fulfilled, the quasiparticle momenta in these integralshardly change and can be replaced by the Fermi momentum. The integrals can then berewritten as an integral over all energies, multiplied with the density of states and thespecific power of the Fermi momentum ~kF . An example for such an observable is theelectron density. The specific momentum power is zero. It is thus given through

n = 2T limτ→0−

∑n

GM(~r, ~r, ωn)e−iωnτ

= 2T limτ→0−

∑n

e−iωnτ∫d3k d3k′


~k′~r

= 2T limτ→0−

∑n

e−iωnτ∫

d3k′

(2π)3ei~k′~r

∫dξ~kN(ξ~k)G

M(~k,~k′, ωn)︸︷︷︸(∗∗)

. (2.75)

The factor two arises due to the two possible spin directions of electrons. For all otherobservables similar expressions can be found. The idea is thus to use right from the beginningGreen’s function of the type (∗∗), which are already integrated over energy, and find anequation of motion for them [8] [25]. These functions are called quasiclassical Green’sfunctions.

If the quasiclassical condition (2.22) applies, the spatial dependence of the usual Green’sfunctions becomes

GM(~r1, ~r2, ωn) =

∫d3k d3k′


~k(~r1−~r2)ei~k′(~r1+~r2)/2

≈∫d3k d3k′


~kF (~r1−~r2)ei~k′(~r1+~r2)/2.


Figure 2.13: Fast oscillations of the Green’s functions drop out in quasiclassical approxi-mation. Quasiclassical Green’s functions vary smoothly on length scales of the order of thecoherence length of the system

The usual Green’s function are thus rapidly oscillating functions, which vary on lengthscales of the order k−1

F . By considering the limit ~r1 → ~r2 and performing the energyintegration these fast fluctuations drop out, and what remains is the spatial dependence onthe center of mass coordinate, which is a smooth variation on length scales of the order ofthe coherence length of the system. This is similar to chapter 2.4, where fast oscillations ofthe two component wave function dropped out of the Bogoliubov-de Gennes equation afterapplication of the quasiclassical approximation. This property of the quasiclassical Green’sfunctions is sketched in Figure 2.13. The quasiclassical Green’s functions are defined as 4

fMωn(k, ~k′) =

∮dξ~kiπFMωn(~k,~k′) ; fMωn(k, ~k′) =

∮dξ~kiπFMωn(~k,~k′); (2.76)

gMωn(k, ~k′) =

∮dξ~kiπGMωn(~k,~k′) ; gMωn(k, ~k′) =

∮dξ~kiπGMωn(~k,~k′). (2.77)

Similar to the usual Green’s function they can be combined to a matrix Green’s function inNambu space:

gMωn(k, ~k′) =

(gMωn fMωn−fMωn gMωn

). (2.78)

4The sign of these definitions is different to the sign in most textbooks (for example to the sign in [25]).I used the definition of the quasiclassical Green’s functions from [8]. There the usual Green’s functionswere defined with the opposite sign compared to most textbooks (for example [6]). In [8] this definitionthus provides the same sign for the quasiclassical Green’s functions. I defined the usual Green’s functionlike in [6] and thus end up with a different sign in the quasiclassical Green’s functions compared to mosttextbooks. This must be kept in mind in the following calculations. In my work the LDOS is given by−Re(gR), whereas it is Re(gR) in most textbooks.


The definition of the off-diagonal components is quite clear. As the Green’s functions F andF decay for large energies ∼ ξ−2

~k, the integrals converge. This can be seen from the solutions

for the Green’s functions in the homogeneous case, which were calculated in chapter 2.5.2.For the diagonal components the situation is more complicated, because they already existin the normal state, and the integrals over G and G diverge. This problem was alreadymentioned in the derivation of the density of states in a bulk superconductor. To solve it,the Green’s functions have to be separated into a contribution, which already exists in thenormal state and a contribution arising in the superconducting state. The quasiclassicalGreen’s functions are defined as integrals over the superconducting component. This issymbolized by

∮, which accounts only for quasiparticles close to the Fermi surface. Observ-

ables, which already exist in the normal state, like for example the electron density, must besplit into a normal component and a superconducting contribution. The superconductingcontribution can be expressed through quasiclassical Green’s functions. A more detaileddescription, how this difficulty is managed, can be found for example in [8].

A very useful representation of quasiclassical Green’s functions is the previously men-tioned mixed Fourier representation. In this representation the quasiclassical Green’s func-tions depend on the direction of quasiparticle momentum and the center of mass coordinate:

gMωn(k, ~r) =

∫d3k′

(2π)3ei~k′~rgMωn(k, ~k′). (2.79)

From the results for the Green’s functions in the homogeneous case, the quasiclassicalGreen’s functions of a homogeneous system can be immediately calculated:

fMωn =

∮dξ~kiπ

∆

(ξ~k + i√ω2n + |∆|2)(ξ~k − i

√ω2n + |∆|2)

=∆

i√ω2n + |∆|2

; (2.80)

fMωn =

∮dξ~kiπ

∆∗

(ξ~k + i√ω2n + |∆|2)(ξ~k − i

√ω2n + |∆|2)

=∆∗

i√ω2n + |∆|2

; (2.81)

gMωn = −∮dξ~kiπ

ξ~k + iωn

(ξ~k + i√ω2n + |∆|2)(ξ~k − i

√ω2n + |∆|2)

= − ωn√ω2n + |∆|2

; (2.82)

gMωn = −∮dξ~kiπ

ξ~k − iωn(ξ~k + i

√ω2n + |∆|2)(ξ~k − i

√ω2n + |∆|2)

=ωn√

ω2n + |∆|2

. (2.83)

An important property of quasiclassical Green’s functions is the normalization condition

gg =

(gg − ff (g + g)f(g + g)f gg − ff

)=

(1 00 1

).

For the homogeneous case its validity can be immediately checked by insertion of (2.80)-(2.83).

From the quasiclassical Matsubara Green’s function it is possible to determine the re-tarded and advanced quasiclassical Green’s functions by performing the usual Wick rotationand analytical continuation from the positive/negative imaginary axis to the particular com-plex half plane:


gRiωn = gωn for ωn > 0;

gAiωn = gωn for ωn < 0.

This is equivalent to replacing ωn = −iE in (2.80)-(2.83). A difficulty arises from thecomplex square root, respectively from the sign of the root, which is not well-defined. Tomake the definitions unambiguous, the retarded and advanced functions in the normal statewith |∆| = 0 are fixed by defining the Matsubara Green’s function as

gM(n)ωn = −sgn(ωn). (2.84)

There is no physical significance for defining the functions in the normal state with this sign,but only by defining it this way, observables like the density of states have the expected signin the following. With definition (2.84), the retarded and advanced functions in the normalstate become

gRE = −gAE = −1.

This definition fixes the sign of the complex square root, because in the limit |∆| = 0 theprevious relation must be fulfilled. The retarded and advanced functions are given by:

gRE =iE√

−(E + iδ)2 + |∆|2; (2.85)

gAE =iE√

−(E − iδ)2 + |∆|2. (2.86)

The infinitesimal imaginary parts iδ are important as they determine the analytical prop-erties of the functions. Taking into account these imaginary parts it becomes obvious thatfor E < |∆| the retarded and advanced functions are odd in E, whereas they are even inE for E > |∆|. The other components of the quasiclassical retarded and advanced matrixfunctions can be determined similarly. In particular, the square root sign for a given energyis the same for all functions, also for the off-diagonal components, which are absent in thenormal state.

The LDOS is related to the quasiclassical Green’s functions through

N(E,~r = (~r1 + ~r2)/2) = −N0

πIm

(∫dξ~kG

R(E, ξ~k, ~r)

)= −N0 Re

(gRE(k, ~r)

).

The next step is to find an equation of motion for quasiclassical Green’s functions. Itis derived in mixed Fourier representation in order to avoid convolutions. In frequencyrepresentation with spatial coordinates expressed through relative coordinate ~ρ = ~r1 − ~r2

and center of mass coordinate ~r = (~r1 + ~r2)/2 the Gor’kov equation has the form

(iωnτ(3) − H − Σωn)GM(~r1 = ~r +

~ρ

2, ~r2 = ~r − ~ρ

2, ωn) = 1δ(~r1 − ~r2). (2.87)


Here the elastic scattering self-energy (2.73) is already incorporated. The matrix Hamilto-nian H in these coordinates is defined through

H =

(− (~∇~ρ+~∇~r/2)2

2m− µ ∆(~r)

−∆∗(~r) − (~∇~ρ+~∇~r/2)2

2m− µ

).

In the following the spatial Fourier representation of GMωn(~r, ~ρ) with respect to the relative

coordinate ~ρ = ~r1 − ~r2 is used:

GM(~r, ~ρ, ωn) =

∫d3k

(2π)3ei~k~ρGM(~k, ~r, ωn). (2.88)

Inserting this expression into (2.87) provides

(iωnτ

(3) −

(− (i~k+~∇~r/2)2

2m− µ ∆(~r)

−∆∗(~r) − (i~k+~∇~r/2)2

2m− µ

)− Σ

)GM(~k, ~r, ωn) = 1. (2.89)

Quadratic terms in the variation of the Green’s function with the center of mass coordinatecan be neglected, because in this coordinate it varies on a length scale of the order of thecoherence length ξ, which is much larger than k−1

F due to (2.22). The right-sided version ofequation (2.89) is defined by

GM(~k, ~r, ωn)

(iωnτ

(3) −

(− (−i~k+~∇~r/2)2

2m− µ ∆(~r)

−∆∗(~r) − (−i~k+~∇~r/2)2

2m− µ

)− Σ

)= 1.

Calculation of the difference of both equations yields

−i~k

m∇~rGM(~k, ~r, ωn) +

[−iωnτ (3) + Σ +

(0 ∆(~r)

−∆∗(~r) 0

), GM(~k, ~r, ωn)

]−

= 0.

Integration over energy near the Fermi surface provides an equation of motion for the qua-siclassical Matsubara Green’s function:

−i~vF∇~rgMωn(k, ~r) +

[(−iωn ∆(~r)−∆∗(~r) iωn

)︸︷︷︸

H0

+Σ, gMωn(k, ~r)

]−

= 0. (2.90)

This equation is called Eilenberger equation [26]. Its solution is determined only up to aconstant factor by the boundary conditions. A further condition is necessary so solve itunambiguously. This condition was already introduced in the homogeneous case and iscalled normalization condition for quasiclassical Green’s functions:

(gMωn)2 = 1. (2.91)

Its validity in the homogeneous case is obvious, and it can be shown that this doesn’tchange under the influence of the Eilenberger equation. A proof can be found in [7] and [8].


The Eilenberger equation for fixed boundary conditions and the normalization conditiondetermine the Green’s function unambiguously.

The expression for the self energy Σ in dependence of the quasiclassical Green’s functionscan be found from (2.73). It becomes

Σωn(k, ~r) = nimp

∫d3k′

(2π)3|u(k − k′)|2GM(~k′, ~r, ωn)

= nimpN0πi

∫dΩ~k′

4π|u(k − k′)|2gMωn(k′, ~r) + nimp1P

∫|u(k − k′)|2k

′2dk′

2π2

1

ξ~k′︸︷︷︸(∗∗∗)

.

Since all involved momenta are close to the Fermi surface, the momentum dependencies ofthe scattering potential in (2.73) were replaced by the directions of the particular momenta.The contribution (∗ ∗ ∗) is proportional to the unit matrix and can thus be incorporated inthe chemical potential. In the case of an isotropic scattering potential u(k− k′) = u = const.the previous expression can be further simplified and is then given by

Σωn(k, ~r) = nimp|u|2N0π︸︷︷︸(2τimp)−1

i〈gMωn(k, ~r)〉Ω, (2.92)

where the scattering mean free time τimp was introduced.In the next chapter equation (2.90) is considered in the dirty (diffusive) limit. In this limit

a large number of impurities is assumed and thus the mean free path of elastic scattering isassumed small compared to the other characteristic length scales of the system.

2.5.5 Dirty limit and Usadel equation

For s-wave superconductors, the pairing potential ∆(~k) is independent of the momentum di-rection k. The considerations in this chapter are valid only for this type of superconductors.In the dirty limit the length scale of interest is much larger than the mean scattering freepath. On these length scales each excitation experiences multiple scattering events, whichchange the initial momentum direction randomly. Because impurity scattering is elastic,only the direction of momentum changes, not the absolute value. The Eilenberger equationcan then be averaged over momentum directions to derive the equation of motion for ho-mogeneous Green’s functions. For this purpose the Green’s function gMωn(k, ~r) is written as

an isotropic main part gM(0)ωn (~r) with a small anisotropic contribution g

M(1)ωn (k, ~r):

gωn(k, ~r) = g(0)ωn (~r) + g(1)

ωn (k, ~r) , g(1)ωn g(0)

ωn . (2.93)

To avoid to many indices, the index M for the Matsubara Green’s functions was dropped.The aim is to derive the equation of motion for the isotropic component. With the previousansatz the normalization condition can be rewritten as

(g(0)ωn (~r) + g(1)

ωn (k, ~r))2

=(g(0)ωn (~r)

)2︸︷︷︸=1

+(g(0)ωn (~r)g(1)

ωn (k, ~r) + g(1)ωn (k, ~r)g(0)

ωn (~r))

+(g(1)ωn (k, ~r)

)2

︸︷︷︸≈0

= 1.


The condition becomes

g(0)ωn (~r)g(1)

ωn (k, ~r) + g(1)ωn (k, ~r)g(0)

ωn (~r)!

= 0.

Substitution of the ansatz (2.93) into the Eilenberger equation (2.90) and angle averagingprovides

e2N0

[H0, g

(0)ωn (~r)

]− − i∇~r

⟨e2N0~vF g

(1)ωn (k, ~r)

⟩Ω︸︷︷︸

~j

= 0.

The scattering term containing the commutator between Σ and gωn(k, ~r) drops out by av-

eraging. Furthermore the matrix current density ~j was introduced, where the prefactor was

chosen such, that the full current ~I has the unit of conductivity. It still depends on thenon-isotropic component g

(1)ωn (k, ~r). As the aim is to find an equation of motion for the

isotropic part, g(1)ωn (k, ~r) must be expressed through g

(0)ωn (~r). To achieve this, the angle aver-

aged equation is subtracted from the Eilenberger equation (2.90). Using the normalizationcondition and neglecting H0, which is assumed small compared to the scattering self energyΣ, and in the case of an isotropic scattering potential, this leads to

g(1)ωn (k, ~r) = τimp~vF g

(0)ωn (~r)∇~rg(0)

ωn (~r).

Inserting this expression into the matrix current results in an equation for the homogeneouspart g

(0)ωn (~r), which no longer depends on the inhomogeneous component g

(1)ωn (k, ~r):

e2N0

[H0, g

(0)ωn (~r)

]− i∇~r

⟨e2N0~vF g

(1)ωn (k, ~r)

⟩Ω

= e2N0

[H0, g

(0)ωn (~r)

]− i∇~r

⟨e2N0~v

2F τimpg

(0)ωn (~r)∇~rg(0)

ωn (~r)⟩

Ω

= e2N0

[H0, g

(0)ωn (~r)

]− i∇~re2N0

~v2F

3τimp︸︷︷︸D

g(0)ωn (~r)∇~rg(0)

ωn (~r) = 0.

Finally the equation of motion for the isotropic component g(0)ωn (~r) becomes

−ie2N0

[H0, g

(0)ωn (~r)

]− +∇~r~j(~r) = 0, with ~j(~r) = − e2N0D︸︷︷︸

σ(~r)

g(0)ωn (~r)∇~rg(0)

ωn (~r). (2.94)

This equation is called Usadel equation for the Matsubara quasiclassical Green’s functionin a diffusive conductor [15]. It is valid on length scales larger than the mean scatteringfree path. Equations for the retarded and advanced Green’s functions have the same formand can be obtained from (2.94) through the usual rotation. (2.94) has the form of a

diffusion equation with a diffusion constant D. As the matrix current density ~j(~r) containsa product of Green’s functions, the Usadel equation is a non-linear differential equation,which is in general difficult to solve. In the following a method is presented, which allows forapproximative solution of this equation by transforming it into a set of algebraic equations

52 Quantum Circuit Theory

through discretization. This method is called Quantum Circuit Theory (QCT) [4] [12] [19].Besides the simpler structure of algebraic equations compared to differential ones, anotheradvantage of this method is that it provides a possibility to incorporate boundary conditionsin a very general way.

2.6 Quantum Circuit Theory

The main idea of QCT is to treat equation (2.94) in a similar way like the continuity equationis treated in electrostatics. The equation for the current density in an arbitrary conductorhas a similar form as equation (2.94). Is is given by

~∇~j(~r) = 0; with ~j(~r) = −σ(~r)~∇V. (2.95)

It describes charge conservation with a current density, that is related to the voltage dis-tribution in the particular conductor through the conductivity σ(~r), which is in generalposition-dependent. Although this is a differential equation, it is not necessary to solve itexactly in order to get an idea of voltages and currents in an electric circuit. This is aquite difficult task, because (2.95) is of second order with complicated boundary conditionsfor arbitrary shape. Instead, the electric circuit can be subdivided into contacts with fixedvoltages, connectors and nodes. Then Kirchhoff’s rules are applied, which indicate that thetotal current at each node has to be zero and that the current through each connector isrelated to the voltages at the ends of the connector through Ohm’s law. These conditionsprovide a set of algebraic equations, which are relatively simple to solve. The structure ofthe Usadel equation is very similar to the structure of equation (2.95), although the matrix

current in the Usadel equation is not conserved, due to the term −ie2N0

[H0, g

(0)ωn (~r)

]−

. But

it is a diffusion-like equation for a matrix current density, which is related to a position-dependent quantity g

(0)ωn (~r), similar to the voltage in the case of an electric circuit. The

matrix structure of this equation brings special properties into play, which have to be con-sidered additionally. The idea of QCT is to discretize the given structure, similar to thesubdivision of a conductor in connectors and nodes, and to find rules, similar to Kirchhoff’srules, which allow for the calculation of the Green’s function at discrete points. Due to thematrix structure of Green’s functions and current densities the theory has a 2 × 2 matrixstructure in Nambu space. Furthermore the Green’s functions play the role of a matrix volt-age in this theory. The finite element approach is similar to the electric circuit. Regions inwhich the Green’s function hardly changes are described as nodes. Regions, which connectthese nodes, with big change in the Green’s function are described as connectors. Reservoirswith fixed values of the Green’s function constitute the boundary conditions, depending onthe physical properties (superconductor, normal metal, ...). For example, inside a hugereservoir in the superconducting state, the order parameter is approximately constant andthe Green’s function is given by the solution for the homogeneous superconducting state.The fineness of the subdivision depends on the desired exactness of the results. The finerthe subdivision is, the larger is the computation effort in the following, but the exacter arethe results. It has to be kept in mind, that the Usadel equation is valid only on lengthscales larger than the mean scattering free path. This restriction sets a lower limit to the

Quantum Circuit Theory 53

fineness of the subdivision. One difficulty is to account for the not conserved current densityin equation (2.94). And furthermore the relation between the matrix current in a connectorand the Green’s functions at its ends has to be found. In fact current conservation consti-tutes not a real problem, because it can be artificially fulfilled by defining a leakage currentdensity of the following form:

jlc = −ie2N0

[H0, g

(0)]− . (2.96)

It is important to note that this leakage current does not describe a leakage of electrons,but is rather a leakage of coherence. It has its origin on the one hand in slightly differingmomenta of electron and hole contributions to a quasiparticle of a given energy and on theother hand in the conversion of quasiparticles into Cooper pairs, triggered by the pairingpotential ∆(~r). Requiring current conservation for an arbitrary volume, including leakagecurrent, provides directly the Usadel equation:

Itot =

∫dS(~j~n) +

∫dV jlc =

∫dV ~∇~r~j +

∫dV jlc =

∫dV(~∇~r~j − ie2N0

[H0, g

(0)]−

)︸︷︷︸

(2.94)

= 0.

Here the theorem of Gauss was used. This shows that by including leakage currents theUsadel equation follows directly from current conservation. To include leakage currentsin a discretized sample, at each node an imaginary reservoir has to be applied, to whichthe leakage current flows. In contrast to the previous matrix current density, which is acurrent per unit area, the leakage current density is a current per unit volume. The leakagecurrent density thus has to be integrated over the volume of a node in order to get the totalleakage current. The bigger the volume is, the larger is the leakage current. Intuitively onemight explain it with the fact that the time which an excitation spends in a big structureis larger than the time in a small structure and so decoherence and temporal decay play amore important role. The strength of the leakage current in a node can be described by acharacteristic energy scale called Thouless energy ETh. It is related to the time an excitationneeds to diffuse out of a structure (here: out of a node) as follows: ETh = ~/tdwell = ~D/L2

[22]. D denotes the diffusion constant and L is the charcteristic length of the structure.Since tdwell is larger for bigger volumes, the Thouless energy is inversely proportional to thevolume, hence large Thouless energies correspond to small volumes and vice versa. Therelation for ETh can be rewritten as

ETh =1

πN0Vnode

∑c

Gc

GQ

. (2.97)

In this definition the sum runs over all contacts connected to a node with conductances Gc.The conductances are given by the Landauer formula

Gc = GQ

∑i

Tc,i, (2.98)

where the sum runs over all transport channels of the particular connector. GQ is called theconductance quantum. It is given by e2/(π~). Using the Thouless energy, the total leakagecurrent of a node can be expressed as


Ilc = −GQi

ETh

(∑c,i

Tc,i

)[H0, g

(0)]− .

In order to apply current conservation to each node, the current through each connectorhas to be expressed in terms of the Green’s functions at its ends. In an electric circuit thisrelation is given by Ohm’s law, which is just a linearized form of the relation between thecurrent density and the voltage distribution. In the QCT this relation is more complex toderive since the dependence of the current density on the Green’s function is non-linear andfurthermore the matrix structure of the whole theory has to be taken into account. TheQCT is not only able to provide a solution for a discretized diffusive conductor, but canadditionally account for boundaries between diffusive regions. The transport properties ofthe connectors thus play an important role. In the solution of the Usadel equation suchboundaries imply specific boundary conditions, depending on the scattering potential atthe contact. In the QCT such boundaries are described through connectors with specifictransmission eigenvalues, one for each transport channel. Each has a value between zeroand one. As derived in chapter 2.4, these transmission eigenvalues are determined by thescattering matrix of the contact and contain information about the boundary conditions forthe Green’s function. In the case of an infinite number of transport channels, the connectoris characterized by a distribution of these transport eigenvalues in the interval 0 to 1. Thematrix current depends in general not only on the Green’s functions at the ends, but alsoon the transmission eigenvalues of the particular connector. The detailed derivation of therelation for the matrix current can be found in [12]. The result is given by

I1,2 = GQ

∑i

Ti(g(0)1 g

(0)2 − g

(0)2 g

(0)1 )

2 + Ti/2(g(0)1 g

(0)2 + g

(0)2 g

(0)1 − 2)

, (2.99)

where g(0)1 and g

(0)2 denote the Green’s functions at the ends of particular connector. For

some special types of connector this expression can still be simplified. An important exampleis a tunnel contact with all transmission eigenvalues close to zero. In that case the secondterm in the denominator, proportional to Ti, can be neglected compared to 2 and equation(2.99) becomes

IT1,2 =GQ

2

∑i

Ti

[g

(0)1 , g

(0)2

]−. (2.100)

In [4] it is shown that linearization of the Usadel equation (2.94) provides a current of thetunnel type (2.100) between neighboring nodes. Linearization of a diffusive conductor thusmeans to describe it as a set of nodes, connected through tunnel contacts.

Another important connector type is the ballistic contact with all transmission eigenval-ues either one or zero. The matrix current through a ballistic connector becomes

IB1,2 = GQ

∑open channels

(g(0)1 g

(0)2 − g

(0)2 g

(0)1 )

1 + (g(0)1 g

(0)2 + g

(0)2 g

(0)1 )/2

. (2.101)

With these relations the procedure is straightforward. As soon as the observed structure

Quantum Circuit Theory 55

Figure 2.14: Example for a quantum circuit, consisting of three nodes with leakage currents,three reservoirs and five connectors. Each connector is characterized through a specific setof transmission eigenvalues, which contains information about boundary conditions for theGreen’s functions between the nodes. Current conservation at each node provides threealgebraic matrix equations for three unknown Green’s functions. The Green’s functionsin the three reservoirs have fixed values, which depend on the physical properties of thesereservoirs [4].

is discretized with the desired accuracy, for each node the total current including leakagecurrent is required to be zero:

Itot =∑

connectors

I!

= 0.

All currents are expressed in terms of Green’s functions at the ends of the connectors through(2.99). These conditions provide a set of n algebraic equations for n nodes. The boundaryconditions are given by the Green’s functions in the reservoirs. The Green’s functions in thenodes are determined unambiguously by these equations and the normalization condition(2.91). An example for such a quantum circuit is illustrated in Figure 2.14. This circuitconsists of three reservoirs, where the Green’s functions are fixed. Furthermore the structureis subdivided into three nodes, which are connected through five connectors. At the nodescurrent conservation has to be demanded, which results in three equations, that have to befulfilled. The difficulty lies in the exact properties of the contacts, which can in general havearbitrary transmission eigenvalues. This makes it necessary to use in general expression(2.99) for the matrix current. Furthermore, the transmission eigenvalues, or equivalentlythe eigenvalue distributions, for each contact can in general be different.

Chapter 3

Application of Quantum CircuitTheory to SNS junctions

In this chapter the previously presented method is applied to a structure, where a normalmetal is connected symmetrically to two superconductors. Such a junction can be describedas one node in the normal state with leakage current, connected to two superconductingreservoirs. The discretized description is depicted in Figure 3.1. Current conservation at thenormal node provides one matrix equation, which determines the Green s function in thisnode. The following calculations are performed for two identical superconductors, whoseorder parameters have equal absolute values | 1 | = | 2 | = | | , but in general a phasedierence τ = τ 1 τ 2. The Green s functions in the reservoirs are those of a homogeneoussuperconductor, given by (2.80)-(2.83). The aim is to determine the Green s function in thenode taking into account the normalization condition and then to calculate the LDOS forvarious symmetric contacts between the normal node and the superconducting reservoirs independence of the phase dierence and the size of the normal metal. As mentioned before,the size of the normal metal is characterized by the Thouless energy (2.97). The sum overall connectors contains in this case only the two symmetric connectors and thus provides afactor 2 in the denition of ETh.

Figure 3.1: Finite element description of a SNS junction with one normal node connectedsymmetrically to two superconducting reservoirs with identical superconductors. The orderparameters have equal absolute values, but in general a phase dierence τ = τ 1τ 2. Currentconservation and the normalization condition (2.91) determine the Green s function in thenormal node.

58 Derivation of the main equation for symmetric contacts

3.1 Derivation of the main equation for symmetric

contacts

In this chapter the main equation determining the Green’s function in the normal node forarbitrary contacts, Thouless energies and phase differences is derived. The calculations areperformed following [9]. The order parameters of the two superconducting reservoirs canhave in general two different phases ϕ1 and ϕ2. Since it is not possible to measure theindividual phases of the superconductors, but only the phase-difference ϕ = ϕ1 − ϕ2, theycan be chosen symmetrically: ϕ1 = ϕ/2 ; ϕ2 = −ϕ/2. With this choice the phase in thenormal node is zero due to the symmetry of the system. For |E| > |∆|, the homogeneousretarded Green’s functions in the reservoirs are given by

gR1 =

(− cosh(Θ) − sinh(Θ)eiϕ/2

sinh(Θ)e−iϕ/2 cosh(Θ)

);

gR2 =

(− cosh(Θ) − sinh(Θ)e−iϕ/2

sinh(Θ)eiϕ/2 cosh(Θ)

). (3.1)

Here the parameter Θ is defined by: coth(Θ) = E|∆| . Using of the trigonometric properties

cosh(coth−1(x)) =1√

1− 1/x2;

sinh(coth−1(x)) =1

x√

1− 1/x2,

(3.1) is equivalent to (2.85) and the according expressions for the other components. For|E| < |∆| the Green’s function can be defined as

gR1 =

(i sinh(Θ′) i cosh(Θ′)eiϕ/2

−i cosh(Θ′)e−iϕ/2 −i sinh(Θ′)

);

gR2 =

(i sinh(Θ′) i cosh(Θ′)e−iϕ/2

−i cosh(Θ′)eiϕ/2 −i sinh(Θ′)

), (3.2)

where Θ′ is given by tanh(Θ′) = E∆

. This is also consistent with (2.85), if the followingtrigonometric properties are used:

cosh(tanh−1(x)) =1√

1− x2;

sinh(tanh−1(x)) =x√

1− x2.

For symmetric contacts, the sum in the definition of the Thouless energy (2.97) provides afactor 2 and it becomes:

Derivation of the main equation for symmetric contacts 59

ETh =2

πN0Vnode

∑i

Ti.

Requiring current conservation at the normal node provides the following condition:

−i EETh

[σ3, g

Rn

]− +

I1(gRn , gR1 )

2GQ

∑i Ti︸︷︷︸

F (gRn ,gR1 )

+I2(gRn , g

R2 )

2GQ

∑i Ti︸︷︷︸

F (gRn ,gR2 )

= 0. (3.3)

Here gRn denotes the Green’s function in the normal node. The general definition of thematrix current (2.99) suggests that F (gRn , g

R2 ) actually depends only on the product of the

two matrices gRn gR2 , because

gR2 gRn = (gR2 )−1(gRn )−1 = (gRn g

R2 )−1

and (2.99) doesn’t contain other terms. The function F can be rewritten through thefollowing general property of 2× 2-matrices:

F (gRn gR2 ) = F

(1

2

([gRn , g

R2

]+

+[gRn , g

R2

]−

))=

1

2

(F (a+ γ) + F (a− γ)

)+

1

2

[gRn , g

R2

]−F (a+ γ)− F (a− γ)

2γ, (3.4)

where the matrix γ is a diagonal matrix with the eigenvalues of 12

[gRn , g

R2

]− on its diagonal

and the matrix a is also diagonal and defined by a = 12

[gRn , g

R2

]+

. The commutator of both

matrizes is thus zero: [a, γ]− = 0. Due to the symmetric contacts and the special choice ofthe phases in the reservoirs a general ansatz for the Green’s function gRn in the node can bemade, for which the phase of the off-diagonal components is zero:

gRn =

(g3 g1

−g1 −g3

). (3.5)

Due to this ansatz, there are two unknown quantities g1 and g3 in the following calcula-tions, which are connected through the normalization condition (2.91). Current conservationthus yields one scalar algebraic equation for one unknown. Mathematically the problem isreduced to the search of a one-dimensional root. The situation is more complicated forasymmetric contacts. In that case another unknown, i.e. the phase of the off-diagonal com-ponents in the node, comes into play. Then current conservation provides two equationsfor two unknowns and the task is to find a two-dimensional root, which is mathematicallymuch more complex.

By inserting gRn and gR2 , the matrices a and γ can immediately be calculated. The matrixa is a diagonal matrix of the form

a =1

2

[gRn , g

R2

]+

=1

2

[(g3 g1

−g1 −g3

),

(c se−iϕ/2

−seiϕ/2 −c

)]+

= (cg3 − sg1 cos(ϕ/2))1. (3.6)


The definitions of the quantities c and s are related to the infinitesimal imaginary part in(2.85). For |E| > |∆| they are given by:

c = − 1√1− (|∆|2/E2)

; s = − 1√1− (|∆|2/E2)(E/|∆|)

.

For |E| < |∆| the definitions are:

c =iE√

|∆|2 − E2; s =

i|∆|√|∆|2 − E2

.

In each definition the square roots are the principal square roots with positive real parts. Theexact definition of the principal square root is given in [28]. Incorporation of an infinitesimalimaginary part like in (2.85) and taking into account the definition of the retarded Green’sfunction in the normal state automatically fixes the sign choice of the square roots for bothcases and renders the previous distinction of cases unnecessary.

In order to find the matrix γ, the following commutator has to be calculated first:

1

2

[gRn , g

R2

]− =

1

2

[(g3 g1

−g1 −g3

),

(c se−iϕ/2

−seiϕ/2 −c

)]−

=

(−ig1s sin(ϕ/2) g3se

−iϕ/2 − g1cg3se

iϕ/2 − g1c ig1s sin(ϕ/2)

). (3.7)

The eigenvalues of (3.7) can be determined in the usual way. They are given by:

λ2 = −2g1g3cs cos(ϕ/2)− g21s

2 sin2(ϕ/2) + g23s

2 + g21c

2.

γ2 is thus given by the following expression:

γ2 =

(λ2 00 λ2

)=(−2g1g3cs cos(ϕ/2)− g2

1s2 sin2(ϕ/2) + g2

3s2 + g2

1c2)

1. (3.8)

By application of the results (3.6) and (3.8) and taking into account the normalizationconditions (gRn )2 = 1 and (gR2 )2 = 1, it can be shown immediately, that

a2 − γ2 = 1. (3.9)

Since a and γ are proportional to the unit matrix they commute and (3.9) is equivalent to

(a− γ)(a+ γ) = 1.

This means that

(a− γ) = (a+ γ)−1;

(a+ γ) = (a− γ)−1.

With these two properties the function F (gRn gR2 ) can be calculated. Using the definition

(3.3) and (2.99) the first term in (3.4) becomes:


(F (a+ γ) + F (a− γ)

)2

=1

8∑

i Ti

(∑i

Ti((a+ γ)−

(a−γ)︷︸︸︷(a+ γ)−1)

1 + Ti/4((a+ γ) + (a+ γ)−1︸︷︷︸(a−γ)

−2)

+∑i

Ti((a− γ)−

(a+γ)︷︸︸︷(a− γ)−1)

1 + Ti/4((a− γ) + (a− γ)−1︸︷︷︸(a+γ)

−2)

)

=1

8∑

i Ti

(∑i

2Tiγ

1 + Ti/2(a− 1)+∑i

−2Tiγ

1 + Ti/2(a− 1)

)= 0.

A similar calculation yields for the second term:

(F (a+ γ)− F (a− γ)

)2γ

=1

2γ

1

4∑

i Ti

(∑i

Ti((a+ γ)−

(a−γ)︷︸︸︷(a+ γ)−1)

1 + Ti/4((a+ γ) + (a+ γ)−1︸︷︷︸(a−γ)

−2)

−∑i

Ti((a− γ)−

(a+γ)︷︸︸︷(a− γ)−1)

1 + Ti/4((a− γ) + (a− γ)−1︸︷︷︸(a+γ)

−2)

)

=1

2γ

1

4∑

i Ti

(∑i

2Tiγ

1 + Ti/2(a− 1)−∑i

−2Tiγ

1 + Ti/2(a− 1)

)

=

(∑i

Ti1 + Ti/2(a− 1)

)/

(2∑i

Ti

)=X−1(a)

2.

Here the characteristic function X(a) was introduced, which is defined by (3.12) and (3.13).Due to the ansatz (3.5), which contains only contributions proportional to σ3 and σ2 andthe commutation relations for Pauli matrices, the commutator in (3.3) is proportional tothe σ1 Pauli matrix. It is thus sufficient to consider only the σ1 component of the equation.The σ1 component of (3.7) is given by(

1

2

[gRn , g

R2

]−

)σ1

= g3s cos(ϕ/2)− g1c.


(F (gRn g

R2 ))σ1

thus becomes

(F (gRn g

R2 ))σ1

= (g3s cos(ϕ/2)− g1c)X−1(a)

2. (3.10)

Due to the symmetric setup(F (gRn g

R1 ))σ1

can be found directly from (3.10) by the replace-

ment ϕ → −ϕ. It is equivalent to(F (gRn g

R2 ))σ1

due to the ϕ symmetry of (3.10). Matrixcurrent conservation is thus accounted for by the equation:

−2iE

EThg1X(a) + g3s cos(ϕ/2)− g1c = 0 (3.11)

Information about the contacts is contained in the characteristic function X(a) in form ofthe particular transmission eigenvalues. For discrete transmission eigenvalues it has theform

X(a) =

(∑i

Ti

)/

(∑i

Ti1 + Ti/2(a− 1)

), (3.12)

for a continuous transmission distribution ρ(T ) it is given by

X(a) =

(∫ 1

0

ρ(T )TdT

)/

(∫ 1

0

ρ(T )T

1 + T/2(a− 1)dT

). (3.13)

The distribution function ρ(T ) is usually normalized in such a way, that integration of Tover all transmissions, weighted with this distribution, yields the conductance of the contactin the normal state in units of the conductance quantum GQ:

GQ

∫ 1

0

ρ(T )TdT!

= GN .

Equation (3.11) still contains two unknowns g1 and g3. However they are related throughthe normalization condition

g23 − g2

1 = 1. (3.14)

As I am interested in the density of states, which is related to g3, the normalization conditionis used to eliminate g1 in (3.11). Solving (3.14) for g1 provides

g1 = −√g2

3 − 1. (3.15)

This square root is again the principal square root. The choice of the sign must be consistentwith the definition of the Green’s function in the reservoirs, where for |E| > |∆|

s = −√c2 − 1.

Eliminating g1 in (3.11) and dividing by s provides

2iE

ETh

√g2

3 − 1

sX(a) + g3 cos(ϕ/2) +

√g2

3 − 1c

s= 0.


For the numerical calculations c and s are replaced by their general expressions includinginfinitesimal imaginary parts:

c =− 1√1− |∆|2/(E + iδ)2

;

s =− |∆|√1− |∆|2/(E + iδ)2(E + iδ)

.

The equation becomes:

−2i(E + iδ)2

ETh|∆|√

1− |∆|2/(E + iδ)2

√g2

3 − 1X(a)

+ g3 cos(ϕ/2) +√g2

3 − 1(E + iδ)

|∆|= 0. (3.16)

For computational calculations dimensionless variables are required. They are defined by

ε =E

|∆|; εTh =

ETh|∆|

.

Equation (3.16) becomes

−2i(ε+ iδ)2

εTh

√1− 1/(ε+ iδ)2

√g2

3 − 1X(a) + g3 cos(ϕ/2) +√g2

3 − 1(ε+ iδ) = 0. (3.17)

The expression for a (3.6) also depends on g1, which has to be eliminated. After insertingc and s it becomes

a = − |∆|√1− |∆|2/(E + iδ)2(E + iδ)

(g3E + iδ

|∆|+√g2

3 − 1 cos(ϕ/2)

). (3.18)

And with dimensionless variables:

a = − 1√1− 1/(ε+ iδ)2(ε+ iδ)

(g3(ε+ iδ) +

√g2

3 − 1 cos(ϕ/2)

). (3.19)

In the following X(a) is calculated for various contact types and then equation (3.17) issolved for g3, which contains the desired information about the LDOS in the normal node.In some cases these calculations can be performed analytically, but for most contacts (3.17)must be solved numerically. One example where it can be solved analytically is in the limitof an infinite Thouless energy.

64 Infinite and zero Thouless energy

3.2 Infinite and zero Thouless energy

Equations (3.11), (3.16) and (3.17) are valid for arbitrary contacts and Thouless energies.The only condition that was set on the contacts is that they have to be symmetric. Allinformation about their transport properties are contained in the characteristic functionX(a). This function can be arbitrarily complicated, depending on the exact form of thetransmission distribution. Only for some special distributions the integrations can be per-formed analytically. But even if X(a) is known explicitly, in most cases equations (3.11),(3.16) and (3.17) can only be solved numerically. A limiting case, where an analytical so-lution is possible, is for an infinite Thouless energy. This corresponds to an infinitesimallythin normal layer between the two superconductors. The first term in (3.11) is proportionalto the inverse Thouless energy and drops out in the considered limit. (3.11) becomes

g1

g3

=s

ccos(ϕ/2) =

|∆|E

cos(ϕ/2). (3.20)

(3.20) doesn’t contain a dependence on the function X(a) anymore. Thus the result isindependent of the transport properties of the contacts. The LDOS of the normal node isrelated to g3 via

N(E)

N0

= −Re(g3). (3.21)

Using the normalization condition (3.14) provides

g23 − g2

1 = 1 ⇒ g23(1− g2

1

g23

) = 1 ⇒ g3 = − 1√1− |∆|2

E2 cos2(ϕ/2).

The sign in front of the principal square root is fixed by the usual condition. Insertion of(3.20) provides an analytic expression for the LDOS of the normal node in the limit of aninfinite Thouless energy:

N(E) = N0 Re

1√1− |∆|2

E2 cos2(ϕ/2)

. (3.22)

The phase and energy dependence is illustrated in Figure 3.2. For zero phase differenceof the two superconductors the result is equivalent to the result in Figure 2.8 for bulksuperconductors with a gap in the density of states of width 2|∆|. This gap has beeninvestigated in detail theoretically as well as experimentally (see for example [14], [16], [17],[23] and [24]). For non-zero phase difference Figure 3.2 states that the width of the minigapgets narrower until it vanishes for a phase difference of ϕ = ±π. The result exhibits a 2π-periodicity in phase ϕ due to the cos2(ϕ/2) dependence in (3.22). Since this result doesn’tdepend on the contacts, in all following calculations for different contact types, the LDOSmust approach this limit for large Thouless energies, i.e. small volumes.

Similarly the influence of the superconductors and thus the proximity effect are weakfor large volumes and thus small Thouless energies. In this limit the LDOS is expected toapproach the constant value N0 of the normal state. This behavior again doesn’t depend

Tunnel contacts 65

Figure 3.2: LDOS in the normal node for infinite Thouless energy in dependence of thesuperconducting phase difference ϕ and energy E. For ϕ = 0 the density of states of a bulksuperconductor is reproduced. With increasing phase the gap diminishes until it vanishesfor phase differences of ϕ = ±π.

on the contact type for sufficiently small ETh. In the intermediate Thouless energy rangethe transmission properties of the contacts and thus the function X(a) are expected to beimportant and to strongly influence the LDOS in the normal region.

3.3 Tunnel contacts

In this and the following chapters the LDOS is analyzed not in some limiting case of theThouless energy, but in the intermediate Thouless energy range, for which the results areexpected to depend strongly on the contact properties. For this purpose the QCT equation issolved taking into account the special contact properties. At first, symmetric tunnel contactsare investigated, where all transmission eigenvalues are close to zero. In this case the generalexpression for the matrix current (2.99) can be simplified to expression (2.100). (2.100)has a relatively simple structure, because the transport eigenvalues enter the calculationonly through the conductance of the contacts. And actually it turns out that for suchcontacts equation (3.11) can be solved analytically. The first step however is to calculatethe characteristic function X(a). As all transmission eigenvalues are close to zero the secondterm in the denominator of the general definition (3.12) can be neglected compared to one.The characteristic function thus becomes X(a) = 1 and equation (3.11) yields

g1

g3

=s cos(ϕ/2)

c+ 2iE/ETh. (3.23)

66 Tunnel contacts

Figure 3.3: LDOS for a finite Thouless energy ETh = 2|∆| in dependence of the phasedifference ϕ and energy E. Compared to the result for an infinite Thouless energy thewidth of the minigap is decreased for all values of ϕ. For ϕ = ±π it disappears as before.

For ETh =∞ this is equivalent to the result (3.20). The limiting behavior is thus consistentwith the previous result. For finite ETh, insertion of c and s into (3.23) and usage of (3.21)provide an analytical expression for the LDOS in dependence of the order parameter phasedifference ϕ, the Thouless energy ETh and energy E:

N(E) = N0 Re

1− cos2(ϕ/2)

(E+iδ)2

|∆|2

(1− 2iE/ETh

√1− |∆|2

(E+iδ)2

)2

− 1

2

. (3.24)

This result is illustrated in Figure 3.3 and Figure 3.4. Due to the dependence of the LDOSon three variables, in these figures one variable is fixed respectively. Figure 3.3 shows (3.24)with a fixed value of the Thouless energy of ETh = 2|∆|. The LDOS is plotted in dependenceof the phase difference ϕ and energy E. Compared to an infinite Thouless energy, the widthof the minigap has decreased and is even for ϕ = 0 smaller than 2|∆|. Similar to the previousresult the width of the gap decreases with the phase approaching ±π and again the resultexhibits a 2π-periodicity. Around |E| = |∆| and ϕ = 0 the LDOS is increased. For allphases the usual minigap is the only gap. Figure 3.4 shows the LDOS in dependence of theThouless energy ETh and energy E for a fixed phase difference ϕ = 0. At this phase thewidth of the minigap has its maximum value. It can be seen that for large Thouless energiesthe LDOS approaches the limiting result of the previous chapter. For the Thouless energyapproaching zero, the LDOS converges towards the constant value of the normal state N0.This behavior is thus exactly as expected before. For all Thouless energies there appears

Ballistic contacts 67

Figure 3.4: LDOS for a constant phase difference ϕ = 0. With decreasing Thouless energythe minigap becomes narrower and approaches the constant value of the normal state forETh approaching zero. For large Thouless energies the density of states converges to thelimit of the previous chapter with a gap width of 2|∆| for ϕ = 0.

no further gap. This is important because it turns out in the following that this is not thecase for all contact types.

3.4 Ballistic contacts

In this chapter the other extreme of possible contact types is studied, i.e contacts whereall transmission eigenvalues are one. Such contacts are called ballistic contacts. In thiscase the general expression for the matrix current can be brought in the form (2.101). Itturns out that for ballistic contacts the QCT equation can no longer be solved analytically.For a solution with arbitrary values of the variables, it has to be analyzed numerically.Analytical considerations are however possible in some parameter ranges, for which certainapproximations can be made [9]. To be able to solve equation (3.16) numerically, thefirst task however is to find the characteristic function X(a) for these contacts. Sinceall transmission eigenvalues are one the sums in the denominator and in the enumeratorprovide only a prefactor with the number of transport channels, which cancels out. Thecharacteristic function becomes

X(a) =a+ 1

2.

Compared to the characteristic function in the previous chapter, which was independent ofg3, this characteristic function contains the quantity a, which depends on g3. This property

68 Ballistic contacts

Figure 3.5: LDOS in the normal node for symmetric ballistic contacts in dependence of Efor τ = 0 and ETh = 2 | | . The width of the usual minigap is reduced compared to thelimit of an innite Thouless energy similar to the tunnel case. Slightly below E = | | thereappears another gap in the LDOS, which is about two magnitudes smaller than the usualminigap. This gap was not present for tunnel contacts. Inset: More detailed illustration ofthe secondary gap, which indicates that this is not only a suppression of the LDOS, but areal gap with nite width.

makes QCT equations for such contacts much more complex and it is thus not astonishingthat they are no longer solvable analytically. Inserting X(a) in equation (3.16) togetherwith the denition of a (3.18) yields

i(E + i)

ETh

(E + i)

| |g3 +

]g2

3 1 cos(τ /2) (E + i)

| |

[1 | | 2

(E + i)2

) ]g2

3 1

+ g3 cos(τ /2) +]g2

3 1(E + i)

| |= 0.

The task is to nd the root of this expression for dierent values of the variables E, ETh andτ . The following numerical calculations are performed with the computer algebra systemWolfram Mathematica. In Mathematica the function FindRoot is used, which applies aNewton method in order to nd a numerical solution for the root of a function. As usualfor Newton methods a start value is necessary, which must be chosen as close as possible tothe real solution in order to assure convergence. For a xed starting set of the variables E,ETh and τ the estimation of the rst starting value is not clear, but as soon as the solutionfor the rst set is found, it can be taken as a starting value for the next run, for which oneor more variables are slightly dierent. The variation step size of the variables should notbe too large in order to assure that the starting value is always close to the real solution.


As soon as the equation is solved in the desired variable range, the LDOS can immediatelybe calculated using relation (3.21).

Figure 3.5 shows a 2D-Plot of the results, in which the energy E is varied and the valuesof the phase and the Thouless energy are fixed at ϕ = 0 and ETh = 2|∆|. Similar to theresults for tunnel contacts, the width of the minigap is reduced compared to an infiniteThouless energy. However another property of the results, which can be seen in Figure 3.5,is even more interesting. In the region around |E| = |∆| there seems to be a suppressionof the LDOS, which did not appear for tunnel contacts. By closer examination it becomesclear, that this is not only a suppression, but there appears another gap besides the usualminigap in the LDOS slightly below/above |E| = |∆| 1. This can be seen in the inset ofFigure 3.5. Its width is about two orders of magnitude smaller compared to the width ofthe usual minigap. The variation of the secondary gap with ϕ and ETh is depicted in Figure3.6 and Figure 3.7. It can be seen that it doesn’t exist for all phases and its width exhibits aphase dependence. It has its maximum value for ϕ = 0 and diminishes as phase approachesϕ = ±π. This behavior is similar to that of the usual minigap with the difference thatthe usual minigap vanishes totally only for ϕ = ±π. The secondary gap however existsonly in a smaller ϕ-range around ϕ = 0, which depends on ETh. Another difference is thatthe width of the usual minigap grows with increasing Thouless energy, whereas the widthof the secondary gap begins to shrink above a specific value of ETh, which is for ballisticcontacts approximately ETh = |∆|. It is however interesting to note, that although the gapdecreases, it persists to large Thouless energies. For each finite value of ETh, the gap canbe found, although its width converges to zero as ETh approaches infinity. In Figure 3.6 a)the secondary gap is for example shown for ETh = 100|∆|.

For decreasing Thouless energy, the behavior of the gap is depicted in Figure 3.7. Thewidth has its maximum value for all phases at approximately ETh = |∆|, for smaller Thoulessenergies the gap begins to shrink. In contrast to the behavior for large Thouless energiesthere is a minimum Thouless energy between ETh = 0.7|∆| and ETh = 0.6|∆| for which thegap disappears. For smaller ETh the LDOS approaches as usual the constant value of thenormal state. Another interesting property is that whereas the upper gap edge is always atE = |∆| for Thouless energies larger than ETh = |∆|, it moves to smaller energies for EThbetween |∆| and 0.6|∆| until it reaches the lower gap edge and the gap disappears. Theevolution of the LDOS in the gap region with increasing Thouless energy in the intervalbetween 0.5|∆| and 1.5|∆| is illustrated in the annexed video Single_Ballistic_E_Th_05_

15.mp4.So far I studied the two limiting cases of possible contact types: For tunnel contacts all

transmissions are close to zero and there are no further gaps besides the usual minigap. Forballistic contacts all transmissions are one and there appeared a secondary gap in the inter-mediate Thouless energy range, which disappeared for infinite and zero Thouless energy. Inthe following I study contacts between these two extrema, who have discrete transmissioneigenvalues between zero and one or are characterized by a continuous distribution in theinterval [0, 1]. The aim is to find out if the existence of the secondary gap is only a propertyof the ballistic limit, or if this gap also appears for other contacts. At first I consider con-tacts, which are characterized similar to ballistic contacts by a single discrete transmission

1In what follows I restrict myself to the LDOS in the the energy range E > 0. Due to symmetry betweenelectron and hole excitations all these results are also valid for E < 0

70 Ballistic contacts

Figure 3.6: Evolution of the secondary gap with decreasing Thouless energy for large Thou-less energies > |∆|: a) ETh = 100|∆|, b) ETh = 50|∆|, c) ETh = 10|∆|, d) ETh = 2|∆|.Energy and phase ranges are adjusted to the phase and energy width of the gap. By con-sidering axis labels it becomes clear that the width of the gap grows as well as the phaserange where it appears with decreasing Thouless energy. It can be seen that the gap persistseven at large Thouless energies (ETh = 100|∆|) and that the upper gap edge is in all fourexamples at E = |∆|.


Figure 3.7: Evolution of the secondary gap with decreasing Thouless energy for small Thou-less energies ≤ |∆|: a) ETh = |∆|, b) ETh = 0.9|∆|, c) ETh = 0.8|∆|, d) ETh = 0.7|∆|. Atabout ETh = |∆| the width of the gap has its maximum, below it begins to shrink until ittotally disappears between ETh = 0.7|∆| and ETh = 0.6|∆|. Below ETh = |∆| the uppergap edge begins to move towards smaller energies. Although the lower gap edge also movesto smaller energies the motion of the upper edge is faster and the gap disappears belowsome critical value of ETh.

72 Equal transmission eigenvalues 6= 1

eigenvalue, but with the extension that it can be an arbitrary value between zero and one.

3.5 Equal transmission eigenvalues 6= 1

In this chapter the LDOS is analyzed in the intermediate Thouless energy range for contacts,which are characterized by a single discrete transmission eigenvalue between zero and one.The main interest concerns the question if the secondary gap, which was found in the LDOSfor symmetric ballistic contacts, can also be found for other contact types. The approach isthe same as in the previous two chapters. At first the characteristic function X(a) has tobe determined. Then the QCT equation can be solved numerically. For equal transmissioneigenvalues Tconst the characteristic function becomes

X(a) = 1 +Tconst

2(a− 1).

Like in the ballistic limit, the function X(a) depends explicitly on a and thus the QCTequation has to be treated numerically. By inserting X(a) and the expression (3.18) for ainto (3.16), it becomes

i(E + iδ)2

ETh|∆|

√g2

3 − 1

×

(−2

√1− |∆|2

(E + iδ)2+ Tconst

(g3 +

|∆|(E + iδ)

√g2

3 − 1 cos(ϕ/2) +

√1− |∆|2

(E + iδ)2

))

+ g3 cos(ϕ/2) +√g2

3 − 1(E + iδ)

|∆|= 0.

This equation again is solved numerically with Wolfram Mathematica. The results in de-pendence of phase and energy are depicted in Figure 3.8 for different values of the discretetransmission Tconst. It turns out that for each value of Tconst, there is a Thouless energyrange for which the secondary gap appears. Tconst can be chosen arbitrarily close to zero aslong as Tconst 6= 0. Similar to the ballistic case the gap width is maximal for ϕ = 0. Theevolution of the secondary gap with variation of ETh is similar to the ballistic case, with thedifference that the maximum gap width is shifted to smaller Thouless energies for smallertransmissions. For Tconst = 0.5 for example the maximum gap width is reached at approx-imately ETh = 0.5|∆|. In addition it can be observed that for all phases the maximumgap width decreases with decreasing Tconst, while the phase range in which the gap appearsdoesn’t depend on Tconst (if ETh/T is kept constant). Figure 3.8 shows the secondary gapfor different values of Tconst, where the Thouless energy is adapted such that the gap adoptsapproximately its maximum width (ETh/T is constant in all four plots). The property thatthe gap width shrinks while the phase range in which the gap appears is constant can thusbe seen quite well.

Equal transmission eigenvalues 6= 1 73

Figure 3.8: Secondary gap for different values of the single transmission eigenvalue Tconstwith adapted Thouless energy: a) T=0.5, ETh = 0.5|∆|, b) T=0.1, ETh = 0.1|∆|, c) T=0.05,ETh = 0.05|∆|, d) T=0.01, ETh = 0.01|∆|. The maximum gap width decreases with Tconstapproaching 0. The ϕ-range in which the gap appears however doesn’t depend on Tconst.The dependence on Thouless energy is similar to ballistic contacts, besides the shift of themaximum gap width to smaller Thouless energies.

74 Differently weighted discrete transmission eigenvalues

3.6 Differently weighted discrete transmission eigen-

values

A generalization of the calculations in the previous chapter can be achieved by allowing notonly for one transport eigenvalue Tconst, but for multiple transport eigenvalues Ti, weightedwith the number of transport channels with the particular eigenvalue. For such contacts thecharacteristic function (3.12) can’t be much further simplified, as the sums in the enumeratorand the denominator provide not only a prefactor, which chancels out, but contain differenttransmission eigenvalues. X(a) becomes

X(a) =

(∑i

wiTi

)/

(∑i

wiTi1 + Ti/2(a− 1)

). (3.25)

There is a big variety of possibilities, how the values Ti can be chosen and weighted. Twosets of more or less arbitrarily chosen transmission eigenvalues, weights and the correspond-ing LDOS are presented in this chapter. For the first example the following transmissioneigenvalues and weights were used:

Ti 0.1 0.5 1wi 40 100 100

The task is to solve the usual equation (3.16), respectively (3.17), with the particularX(a), which is given for general transmissions Ti and weights wi by (3.25), and then tocalculate the LDOS (3.21). The numerical results are presented in Figure 3.9 and Figure3.10.

Figure 3.9: LDOS for symmetric contacts, which are characterized through three differentlyweighted discrete transmission eigenvalues, for ETh = 2|∆|. The previously found secondarygap splits up in two subgaps with nonzero LDOS in an energy range in between.

Differently weighted discrete transmission eigenvalues 75

Figure 3.10: LDOS for ϕ = 0 and ETh = 2|∆| with split up secondary gap.

Figure 3.9 shows a 3D plot and a contour plot in the same energy and phase range, which ischosen in the secondary gap region. The Thouless energy in both plots is ETh = 2|∆|. Themost striking difference to the previous two chapters with only one discrete transmissioneigenvalue is that for several transmission eigenvalues the secondary gap even splits up inmultiple subgaps with nonzero LDOS in an energy range in between. This splitting howeverdepends on the values of Ti and wi. In the two examples, which are presented in this chapter,these values are adapted such that the splitting can be observed. For other choices it comesabout that there appears only one secondary gap. In the first example with three differentlyweighted transmission eigenvalues the secondary gap splits up in two subgaps. In the twoplots of Figure 3.9 the smaller gap below E = |∆| is difficult to identify as a real gap. Thisbecomes clearer from the 2D plot in Figure 3.10.

As a second example I consider a contact with even more differently weighted discretetransmission eigenvalues. They are chosen in the following way:

Ti 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1wi 10 5 5 2 3 20 50 70 90 200

It turns out that for more discrete transmission eigenvalues, the minigap even splits upin a larger number of subgaps, which causes some kind of band structure. The numericalresults for the LDOS are presented in Figure 3.11 again as a 3D plot and a contour plotfor ETh = 2|∆|. For this value of ETh the secondary gap is split up into four subgaps. Thenumber of subgaps however depends on the value of ETh. With increasing ETh, these gapsdon’t vanish altogether, but one by one disappears. An example is given in Figure 3.12.There the multiply gapped region is shown for ϕ = 0 for two different values of ETh. Whilethere exist four subgaps for ETh = 2|∆|, there remain only three subgaps for ETh = |∆|.

The two examples in this chapter were chosen quite arbitrarily, they have no immediatephysical significance. However they show that the phenomenon of a secondary gap in theLDOS is not bound to one single gap, but there can appear multiple gaps in this energy andphase range. It is not excluded that such a multiply gapped LDOS can be found for more

76 Dirty contacts

Figure 3.11: Multiply gapped LDOS for contacts with ten discrete differently weightedtransmission eigenvalues for ETh = 2|∆|. The previously found splitting into subgaps canbe observed again with even a larger number of subgaps.

realistic contacts, even though I did not find an example in this work, where this actuallyhappens.

3.7 Dirty contacts

So far all calculations were restricted to contacts with discrete transmission eigenvalues.However transmission matrices of real contacts are more often characterized by a continu-ous transmission distribution. In the following chapters the LDOS is thus investigated forcontacts with a continuous distribution of transmission eigenvalues ρ(T ). The first contacttype are dirty connectors, where the transmission distribution is given by

Figure 3.12: The number of gaps varies with ETh. For ETh = 2|∆| there appear four subgapsin the LDOS, whereas for ETh = |∆| only three subgaps can be observed. Both plots arefor ϕ = 0 with an adapted energy range.

Dirty contacts 77

Figure 3.13: Transmission distributions for dierent contact types: Double ballistic contact,diusive contact, dirty contact (symmetric double tunnel contact). The three distributionsare not normalized to the same conductance in this plot.

Figure 3.14: QCT representation of a normal node connected symmetrically to two super-conductors through general double tunnel contacts. A dirty connector can be modeled as acompound connector consisting of two identical tunnel contacts. In this picture symmetricdirty connectors correspond to G1 = G2.

σdirty(T ) =G

ρGQ

1

T 3/2

1 T. (3.26)

The prefactor is chosen such, that the distribution is normalized to the total conductanceG. This distribution is depicted in Figure 3.13 (yellow curve). A dirty contact can bemodeled as a compound connector consisting of two identical tunnel contacts in series,which is characterized through the same transmission distribution (3.26). A derivation ofthis property can be found for example in [4]. The discretized QCT approach to this systemis drafted in Figure 3.14. Compared to the other transmission distributions illustrated inFigure 3.13, which are discussed in detail in the following chapters, the dirty connector hasthe strongest weight at small transmissions. The detailed calculation of the transmissiondistribution and especially of the characteristic function X(a) of a compound connector is

78 Diffusive connector

Figure 3.15: LDOS in the normal node for symmetric dirty contacts for an order parameterphase difference ϕ = 0 and different values of ETh. For all analyzed Thouless energies thereappears no secondary gap, however the LDOS is weakly suppressed around E = |∆| forETh = 0.3|∆|.

described in [4]. I restrict myself to the result here, which is for example given in [20]. Thecharacteristic function X(a) for a symmetric double tunnel contact is given by

X(a) =

(∫ 1

0

ρdirty(T )TdT

)/

(∫ 1

0

ρdirty(T )T

1 + T/2(a− 1)dT

)=

√1 + a√

2.

My main interest concerns the question, if the previously found secondary gap also appearsfor such realistic contacts with continuous transmission distributions. Therefore I analyzethe LDOS for ϕ = 0, which was in the previous examples the phase with maximal gap width.If the secondary gap appears, it is expected to appear for this value of ϕ. The results arepresented in Figure 3.15. For all analyzed values of ETh there appears no secondary gapin the energy range below E = |∆|. There can be observed a weak suppression of theLDOS in this region for Thouless energies smaller than ETh = |∆|. However the LDOS isnot suppressed to zero, much less there appears a gap in it. In the limit of large Thoulessenergies the LDOS approaches the result of section 3.2 and for small Thouless energies thewidth of the minigap decreases and the LDOS approaches the constant value N0 of thenormal state. For other values of ϕ there doesn’t appear a secondary gap either.

3.8 Diffusive connector

Another contact type with a continuous distribution of transmission eigenvalues is the dif-fusive connector. As mentioned in chapter 2.6 and shown in [4] discretizing a diffusivesuperconductor in QCT means to describe it through a set of tunnel contacts. A diffusiveconnector is thus characterized by the same transmission distribution like a compound con-nector, consisting of many tunnel contacts in series. The analyzed system is depicted inFigure 3.16. The distribution of transmission eigenvalues for a diffusive contact is given by

Symmetric ballistic double contacts 79

Figure 3.16: Normal node connected symmetrically to two superconductors through diusiveconnectors. The discretized representation of a diusive connector is given by a set ofidentical tunnel contacts.

σdiffusive(T ) =1

2

G

GQ

1

T

1 T. (3.27)

The prefactor is again chosen such, that the distribution is normalized to the total conduc-tance of the connector G. It is illustrated in Figure 3.13 by the red line. Compared to thedirty connector it has less weight at small transmissions and thus larger transmissions playa more important role 2. The LDOS is thus expected to be more similar to the ballistic casecompared to the LDOS of the dirty connector. In particular, the suppression of the LDOSaround E = | | is expected to be stronger than it was for dirty contacts. The characteristicfunction, related to the transmission distribution (3.27) is given in [20] and has the form

X(a) =

1 a2

arccos(a).

The numerical results for the LDOS are illustrated in Figure 3.17. Calculations were againperformed for τ = 0 due to the reasons that were mentioned in the previous chapter. Itcan be seen that the suppression of the LDOS is essentially stronger than it was for dirtyconnectors, but still there is no secondary gap in this region. Another dierence is thatthe suppression can already be observed at larger Thouless energies. For ETh = | | therewas not yet a suppression visible in the dirty case, whereas for diusive connectors thesuppression can already be observed.

3.9 Symmetric ballistic double contacts

The third distribution in Figure 3.13 (blue line) belongs to a symmetric double ballisticcontact, where both partial contacts have the same conductance. The general setup witha normal node connected through identical ballistic double contacts is drafted in Figure3.18. In this chapter I analyze the symmetric case with G1 = G2. The asymmetric case isconsidered in the next chapter 3.10. Symmetric ballistic double contacts are described by atransmission distribution of the form

2The three distributions in Figure 3.13 are not normalized to the same conductance

80 Symmetric ballistic double contacts

Figure 3.17: LDOS in the normal node for diffusive contacts for an order parameter phasedifference ϕ = 0 and four different values of ETh. The suppression, which was alreadyobserved for symmetric dirty contacts, is even stronger, however there doesn’t appear asecondary gap either. Due to the form of the distribution function, which has a strongerweight at larger transmissions, the amplification of the suppression around E = |∆| is notastonishing and was already expected.

ρdiffusive(T ) = 2G

GQ

1√T√

1− T(3.28)

and the corresponding characteristic function is given by

X(a) =1

2

(1 + a

2+

√1 + a√

2

).

From the three distributions given in Figure 3.13 (3.28) has the weakest weight at smalltransmissions. The suppression of the LDOS at E = |∆| is thus expected to be the strongestfor these contacts. The numerical solutions of the QCT equation are illustrated in Figure3.19. The suppression is stronger than for the other distributions, but similarly there isno secondary gap in the suppression region. From the main plot in Figure 3.19 one mightget the impression that there could exist a secondary gap with finite width, as the densityof states is suppressed to zero. By more detailed examination of the suppression region itbecomes obvious that there is no gap. This is shown in the inset of Figure 3.19.

Asymmetric ballistic double contacts 81

Figure 3.18: Normal node connected symmetrically to two superconductors through doubleballistic contacts. In general the conductances of the two ballistic contacts can be dierent.In chapter 3.9 the symmetric case with G1 = G2 is analyzed, in chapter 3.10 the QCTequation is solved for G1 = G2.

Figure 3.19: LDOS in the normal node for symmetric double ballistic SNS junctions for aorder parameter phase dierence of τ = 0 and dierent Thouless energies. The suppressionis the strongest of all three distributions in Figure 3.13 since the symmetric double ballisticcontact has the strongest weight at large transmissions. However there is no gap with anite width for any analyzed Thouless energy. Inset: Closer look at the suppression region,which makes clear that there is no secondary gap.

3.10 Asymmetric ballistic double contacts

In order to avoid confusion, at rst the meaning of asymmetric in the topic of this chaptermust be claried. It refers to the conductances G1 and G2 of the two contacts composingeach particular connector (Figure 3.18). In the previous chapter 3.9 the symmetric case with

82 Asymmetric ballistic double contacts

G1 = G2 was investigated, in this chapter I analyze the asymmetric case with G1 6= G2.The whole setup however is still symmetric. Equations (3.11), (3.16) and (3.17) are onlyvalid for a symmetric setup. In the previous chapters it turned out that for continuoustransmission distributions covering the whole interval [0, 1] the LDOS exhibits no secondarygap, only a suppression of the LDOS around E = |∆| can be observed in a certain Thoulessenergy range. For asymmetric ballistic double contacts, the distribution does no longercover the whole interval [0, 1], but there is a minimum transmission, which is determined bythe proportion of the two conductances:

Tmin =

(G1 −G2

G1 +G2

)2

. (3.29)

It is zero for equal conductances and approaches one if one conductance is much larger thanthe other. This is quite intuitive because if one conductance is much larger than the other,it doesn’t contribute to the transport properties of the double contact and what remains isa single ballistic contact with all transmission eigenvalues equal to one. The characteristicfunction X(a) is given by

X(a) =G1G2

(G1 +G2)2(a− 1)

1

1−√

1− 4G1G2

(G1+G2)2a−1a+1

.

The approach in the derivation of this expression is similar to the derivations of the pre-viously used characteristic functions for double contacts. It is drafted in [4]. Because theweight of the distribution in the asymmetric case is shifted towards higher transmissionscompared to the symmetric case and since in the symmetric case the LDOS is already sup-pressed to zero (Figure 3.19), the assumption is nearby that for asymmetric contacts theLDOS exhibits a secondary gap of finite width. Transmission distributions together with thecorresponding numerical results for the LDOS for ϕ = 0, ETh = 2|∆| and different valuesof the proportion of conductances G2/G1 are presented in Figure 3.20. In Figure 3.20 b)the LDOS for ballistic contacts is added as a reference. It can be seen that for increasingproportion of conductances G2/G1 there appears a gap in the LDOS which approaches theballistic limit for G2/G1 →∞.

The evolution of the secondary gap is illustrated in two annexed videos. They showthe contour view of the secondary gap range with increasing proportion of conductancesbeginning with G2/G1 = 1. The first one Asymmetric_Ballistic_E_Th_08.mp4 is for aThouless energy of ETh = 0.8|∆|, the second one Asymmetric_Ballistic_E_Th_2.mp4 isfor ETh = 2|∆|.

Asymmetric ballistic double contacts 83

Figure 3.20: a) Transmission distributions for general ballistic double contacts. For equalconductances the distribution covers the whole interval [0, 1] and diverges for T = 0 (blueplot). For asymmetric contacts the distribution is shifted to higher transmissions and hasa minimum transmission Tmin, given by (3.29). The illustrated distributions are again notnormalized to the same conductance. b) LDOS for ϕ = 0 and ETh = 2|∆| for differentvalues of the proportion G2/G1. For G2/G1 > 1 there appears a secondary gap in theLDOS, which approaches the ballistic limit (light blue) as G2/G1 increases.

84 Analytical calculations

3.11 Analytical calculations

Summarizing the results of my analysis so far: The existence of the secondary gap in theLDOS of ballistic contacts, doesn’t depend on the value of the discrete transmission and canalso be found for Tconst 6= 1. The width of the gap however decreases with decreasing Tconstand in the tunnel limit, where Tconst is infinitesimally close to zero, no secondary gap canbe observed. For continuous distributions covering the whole transmission interval [0, 1],the gap doesn’t appear either (dirty contact, diffusive connector, symmetric ballistic doublecontact). For some of them however a suppression of the LDOS in a certain Thouless energyrange can be found. A continuous distribution which doesn’t cover the whole transmissioninterval can be reached through asymmetric ballistic double contacts and as expected theLDOS of such contacts exhibits a secondary gap. These results suggest that the existence ofthe secondary gap is strongly related to the transmission distribution at small transmissions,i.e. in a small transmission range around T = 0. In this chapter a general criterion thatdetermines whether the LDOS exhibits a secondary gap or not is derived.

The conjecture, that I want to prove in this chapter states, that this gap appears whenevera contact is characterized by a transmission distribution which doesn’t have a contributionat small transmissions independently of the form of the distribution in the rest of thetransmission interval. In order to prove this conjecture for arbitrary distributions, equation(3.11) is analyzed analytically in the energy range below E = |∆| for ϕ = 0. What is notshown however is the reverse, i. e. that the gap doesn’t appear if there is a contributionin the transmission distribution around T = 0. This question I consider more closely in thelast chapter 3.12.

At first equation (3.11) is rewritten as

g3

g1

=c

s cos(ϕ/2)+

2iE/EThs cos(ϕ/2)

X(a). (3.30)

The normalization condition (3.15) relates g1 to g3 through

g1 = −√g2

3 − 1.

If g3 is assumed large, the left side of (3.30) can be expanded in 1/g3 and becomes

g3

g1

= − g3√g2

3 − 1=

1√1− 1/g2

3

≈ 1 +1

2g23

.

Here special attention must be applied to the sign after the second equality. Due to thedefinition of the principal square root (see [28]) and of the diagonal component g3 withRe(g3) ≤ 0 the following relation holds:

g3 = −√g2

3.

Because in all numerical calculations the width of the secondary gap was maximal for ϕ = 0,equation (3.30) is analyzed for zero phase difference of the two superconductors and thusfor cos(ϕ) = 1. Inserting the definitions for c and s for E < |∆| provides

Analytical calculations 85

c

s=

E

|∆|= −δ + 1;

1

s=

√|∆|2 − E2

i|∆|=

√(|∆| − E)(|∆|+ E)

i|∆|≈ |∆|

√2δ

i|∆|= −i

√2δ.

Here the quantity δ was introduced, which denotes the relative energy difference to E = |∆|:

δ =|∆| − E|∆|

.

In the following I am interested in energies below but close to E = |∆|, which correspondsto considering small δ. As I account only for first order contributions in δ the prefactor inthe second term on the right side of (3.30) can be approximated as

2iE

ETh≈ 2i|∆|

ETh.

Altogether equation (3.30) for a sufficiently small energy range below E = |∆| becomes

1

2g23

= −δ +2|∆|ETh

√2δX(a). (3.31)

For a general function X(a), the expression for a must also be expanded around δ = 0.Expanding (3.6) provides

a = cg3 − sg1 = g1s

(c

s︸︷︷︸−δ+1

g3

g1︸︷︷︸≈1+ 1

2g23

−1

)≈ g1︸︷︷︸≈g3

s

(−δ +

1

2g23

)≈ i√

2δ

(−δg3 +

1

2g3

).

Equation (3.31) becomes

1

2g23

= −δ +2|∆|ETh

√2δX

(i√2δ

(−δg3 +

1

2g3

)).

In order to get an equation which doesn’t depend on ETh and |∆|, δ and g3 are rescaledwith the following substitutions:

δ = x

(|∆|

2ETh

)2

;

g3 = y

(|∆|

2ETh

)−1

. (3.32)

This provides

a =i√2x

(−xy +

1

2y

).

86 Analytical calculations

and

1

2y2= −x+ 4

√2xX

(i√2x

(−xy +

1

2y

)). (3.33)

Since the LDOS is related to the real part of g3, the task is to show that g3 or equivalentlyy is purely imaginary for small δ, respectively small x. To achieve this, at first the functionX(a) is considered. (3.13) suggests, that the enumerator of X(a) constitutes only a constantfactor given by the conductance of the contact. The denominator D(a) of (3.13) is given by

D(a) =

∫ 1

0

ρ(T )T

1 + T/2(a− 1)dT =

∫ 1

0

ρ(T )1

1T

+ 12

(i√2x

(−xy + 1

2y

)− 1)dT.

The distribution function is assumed to have no contribution in a finite range around T = 0.The integral from zero to one can thus be replaced by an integral from a minimum valueTmin with Tmin 6= 0 to one. Furthermore I make the assumption that −xy can be disregardedcompared to 1/2y for x → 0. This assumption is only valid if the solution y(x) satisfieslimx→0 y(x) ∼ xα with an exponent α > −1/2. The result in the end must show that thisassumption was justified and that the solution is consistent with this approximation. Thedenominator becomes

D(a) ≈∫ 1

Tmin

ρ(T )1

1T

+ 12

(i

2√

2xy− 1)dT.

Since I am interested in limx→0 and due to the behavior of the solution y(x) in this limit, −1can be neglected compared to i

2√

2xy. Furthermore 1/T also produces only finite contributions

due to Tmin 6= 0 and can thus also be disregarded if the considered energy range is closeenough to x = 0. The range in which this approximation is valid depends on Tmin. Thelarger Tmin is, the smaller is this range, because the smaller must be x in order to be ableto neglect 1/Tmin compared to i

4√

2xy. D(a) becomes

D(a) ≈ 4√

2xy

i

∫ 1

Tmin

ρ(T )dT.

And the function X(a) becomes in this limit

X(a) =

∫ 1

Tminρ(T )TdT∫ 1

Tminρ(T )dT︸︷︷︸

=const.=c

i

4√

2xy, (3.34)

where the real valued constant c was introduced. The definition of c suggests that it canonly adopt values between zero and one. In the integral in the enumerator the factor Tcan be interpreted as a weighting factor of the function ρ(T ) and since both integrals runfrom Tmin to one with 0 < Tmin ≤ 1 the value of the integral in the enumerator is smallerthan or equal to the value of the integral in the denominator. Since the integral in theenumerator provides the conductance, the condition on a general transmission distribution

Analytical calculations 87

Figure 3.21: The plot shows the absolute values of the two solutions |y1,2| for c = 1/2 togetherwith the reference function f(x) = x−1/2. Since the absolute value of the solution y1 divergesfaster than x−1/2 this solution is not consistent with the previously made assumption. Theabsolute value of y2 however diverges slower than x−1/2 (it is even finite for x = 0). y2 isthus the desired consistent solution.

is that this integral doesn’t diverge. Because a minimal transmission Tmin 6= 0 was assumed,the integral in the denominator cannot diverge either. In this consideration a value c = 0 isthus not possible and c has a value in the interval 0 < c ≤ 1. The value c = 1 correspondsto the ballistic limit. With (3.34) equation (3.33) becomes

1

2y2= −x+

i

yc. (3.35)

The problem is reduced to the solution of the following quadratic equation:

y2 − ic

xy +

1

2x= 0.

The two solutions of this equation are given by

y1,2 =i

2

(c

x±√c2

x2+

2

x

).

First of all it can be seen that both solutions are purely imaginary for x > 0 and thusboth solutions provide a LDOS, which is zero for all x inside the validity range of theapproximation. The behavior of the solutions for x → 0 must still be checked in order toassure that the previously made approximation is justified. For this purpose the absolutevalues of the two solutions are illustrated in Figure 3.21 for c = 1/2 together with thereference function f(x) = x−1/2. It can be seen that the solution |y1| diverges for x → 0faster than x−1/2. It thus doesn’t comply with the made assumption that for x → 0 theterm −xy can be disregarded in comparison to 1/2y. The solution |y2| however is conformwith this condition and is thus the desired consistent solution. This is of course only valid

88 Power law distributions

Figure 3.22: Solution |y2| divided by x−1/2 for c = 1; 3/4; 1/2; 1/10. For all values of c thesolution y2 stays finite for x = 0 and constitutes a consistent solution. The only problematicvalue c = 1 was already excluded from these considerations previously.

for sufficiently small x. To show that there is a consistent solution not only for c = 1/2Figure 3.22 shows the solution |y2| divided by x−1/2 for different values of c. For x→ 0 thefunction |y2(c)|/x−1/2 approaches zero for all values of c and thus the solution y2 divergesslower than x−1/2. The only problematic value is c = 0, since for this value the solution y2

diverges exactly as x−1/2 and thus the made approximation is not valid. The case c = 0however was already excluded in the previous argumentation.

Another important aspect is that in all numerical calculations the secondary gap vanishedbelow some critical value of ETh. In the derivation of equation (3.33) the condition of largeETh was not used, which suggests that the solution y2 signifies a gap for all values of ETh,which is obviously wrong. In order to obtain equation (3.33) it was however assumed thatg3 is large. The substitution relation (3.32) between g3 and y and the finite value of y2 forx = 0 require the Thouless energy to be sufficiently large compared to |∆|, in order to makethis assumption reasonable. How large ETh must be to make the performed calculationsvalid depends on the value of y2 for x→ 0 and thus on the value of c. c is determined by theexact form of the transmission distribution in the whole transmission interval [0, 1]. SinceI am only interested in the question on the existence of the gap and not in the Thoulessenergy range where it appears, the Thouless energy can always be chosen sufficiently largeto make the previous calculation valid.

3.12 Power law distributions

In the previous chapter it was proved that there is a Thouless energy range with a secondarygap, whenever the transmission distribution has no contribution in a finite range aroundT = 0. The reverse however was not shown, i.e. that there appears no gap if the transmissiondistribution provides a contribution around T = 0. In the chapters 3.7, 3.8 and 3.9 all

Power law distributions 89

Figure 3.23: Transmission distributions of the form ρ(T ) = Tα with α ≥ 0 for a wide rangeof different exponents α.

contacts were characterized through transmission distributions, which diverged at T = 0.The existence of the secondary gap might be related to the power law dependence of thetransmission distribution around T = 0 and exist not only for zero contribution, but for alldistributions where the exponent exceeds some critical value. This statement is not refutedanalytically in this chapter, but the LDOS is calculated numerically for a wide range ofpower law dependencies of ρ(T ). The analyzed transmission distributions are presented inFigure 3.23.

The reason to consider only distributions of the form ρ(T ) = Tα is that the results ofthe analysis so far suggest that only small transmissions determine the existence of thesecondary gap. Around T = 0 the transmission distribution is assumed to be described by apower law of this type. Since for α = 0 the LDOS exhibits no secondary gap, it is sufficientto restrict calculations to α ≥ 0. (3.13) determines the particular characteristic functions.

The numerical results for the LDOS in the energy range directly below E = |∆| for ϕ = 0and ETh = |∆| are illustrated in Figure 3.24. They show that the analyzed transmissiondistributions exhibit no secondary gap in the LDOS. With increasing α the weight at smalltransmissions decreases and the LDOS in the secondary gap region is strongly suppressed.Since the transmission distribution approaches the ballistic case with increasing α, theLDOS also converges towards this limit. But apparently without generation of a gap. Thisbecomes clear from the logarithmic plot in Figure 3.24 b). Whereas from the linear plot inFigure 3.24 a) it is not obvious to decide if there is a gap in the LDOS or a mere suppression,Figure 3.24 b) shows that there is no gap of finite width in this energy range. The resultsfor the distributions that are shown in Figure 3.24 a) but are omitted in Figure 3.24 b) areso strongly suppressed that the calculation precision I used for my numerical calculationswas insufficient. I expect however the same behavior for them but with even a strongersuppression.

Since there is a finite even though small contribution close to zero for all analyzed distri-butions, the observations support the conjecture that there is no secondary gap in the LDOS

90 Power law distributions

if the transmission distribution has a contribution around T = 0. However, the results ofthis chapter show the validity only for the analyzed power law distributions. This is nogeneral proof for an arbitrary transmission distribution, but strongly suggests that a gap inthe transmission distribution around T = 0 might be necessary for a secondary gap in theLDOS.

Figure 3.24: LDOS for ϕ = 0 and ETh = |∆| for the power law transmission distributionsillustrated in Figure 3.23. The LDOS is strongly suppressed in the secondary gap regionand approaches the ballistic limit with increasing exponent. The numerical results showthat this happens without generation of a secondary gap.

Chapter 4

Summary, Conclusion and Outlook

In this diploma thesis I analyzed the proximity effect in a normal metal, which is connectedsymmetrically to two superconductors (SNS structure), for various contact types. In orderto calculate the local density of states (LDOS) I used quasiclassical Green’s functions. Inthe diffusive regime these functions are determined by the non-linear differential Usadelequation. An approximate solution of the Usadel equation is provided by the QuantumCircuit Theory (QCT), which discretizes this differential equation to obtain a set of algebraicequations.

As I analyzed the proximity effect only for symmetric contacts, these symmetries could beused to reduce the set of algebraic equations to one single scalar equation, which determinesthe quasiclassical Green’s function in the normal metal. The LDOS is directly related tothe diagonal components of the matrix Green’s function in Nambu space. In chapter 3.2I investigated the limit of an infinite Thouless energy. It turned out that the transmissionproperties of the contacts drop out of the calculations and thus the results are independentof contact types. The same is true for very small Thouless energies, which corresponds to alarge volume of the normal metal, hence the influence of the superconductors is weak. Thislimiting behavior was found in all following calculations. In the intermediate Thouless energyrange however the LDOS in the normal metal depends strongly on the contact properties,which entered the calculations through a characteristic function X(a). In chapter 3.3 Ianalyzed symmetric tunnel contacts with all transmission eigenvalues close to zero. Due tothe proximity of the superconductors the LDOS exhibits a gap symmetrically around theFermi energy of the normal state. The existence of this gap is not new and was alreadyexpected ([14], [16]). So far all calculations could be performed analytically. In a next step(chapter 3.4) I considered the other limit of possible contacts, i.e. ballistic contacts withall transmission eigenvalues equal to one. In this case the main result is the appearanceof another gap in the LDOS, which doesn’t exist in the tunnel limit. The existence ofthis gap was totally unexpected and the aim of the following work was to study this gapmore intensively. One particular question was, if this gap only arises in the case of ballisticcontacts or if it is a more general property of SNS junctions. It turned out that the gapappeared whenever contacts without a contribution at small transmissions were regarded.This was the case for contacts characterized by a single transmission eigenvalue 6= 1 (chapter3.5) as well as for the more realistic example of asymmetric double ballistic contacts (chapter3.10). For continuous distributions in the whole transmission interval the gap didn’t appear

92 CHAPTER 4. SUMMARY, CONCLUSION AND OUTLOOK

(chapters 3.7, 3.8, 3.9). These observations suggested the thesis that the existence of this gapis only determined by small transmissions and it appears whenever symmetric contacts arecharacterized through a transmission distribution without a contribution in an arbitrarilysmall range around T = 0. With the analytical calculations of chapter 3.11 I showed thevalidity of this assumption for general transmission distributions with the only constraint,that they have no contribution around T = 0. Thus it was shown that this secondarygap is a general property of SNS junctions, which is not confined to the limit of ballisticcontacts. What was not shown however is that the gap doesn’t appear if the transmissiondistribution provides a contribution around T = 0. In the last chapter 3.12 I assumedtransmission distributions which can be described through a power law around T = 0,i.e. weak but nonzero contribution at small transmissions. For such contacts the LDOSin the normal node is strongly suppressed and converges towards the ballistic limit withincreasing exponent in the transmission distribution. The calculated results however showthat this happens without creation of a gap. This was not proved analytically in this workfor arbitrary transmission distributions.

An important question for theoretical predictions is how these results can be verifiedexperimentally. One possibility is to use STM measurements similar to those used in [16] or[23] to investigate the proximity induced minigap. The width of the secondary gap is at leasttwo orders of magnitude smaller than the width of the usual minigap. Hence it requestsspecial needs on the energy resolution of an STM. From Figure 3.5 the maximum width ofthe secondary gap can be estimated as ∼ 0.005|∆|. In usual metallic superconductors atzero temperature |∆| is of the order of 10−3 − 10−4eV . The width of the secondary gap isthus of the order of 10−6− 10−7eV . As the resolution of an STM is approximately given bykBT , a detection of the secondary gap requires a temperature of less than 10−2 K. This isquite below the operating temperature of 50mK of the STM in [16] or even 270mK in [22].Even if measurements at such temperatures can be arranged, it is still difficult to distinguishbetween a real gap and a mere suppression in the LDOS. In Figure 3.24 for example, thesuppression looks very similar to a real gap.

Another difficulty might be the fact that the gap appears only above a critical Thoulessenergy, which is determined by the exact form of the transmission distribution. Since largeThouless energies correspond to small volumes, the normal layer has to be chosen sufficientlythin. In order to identify the desired thickness information about the transport propertiesof the contacts are necessary in advance. Furthermore the volume mustn’t be chosen toothin because for large Thouless energies the width of the secondary gap decreases again.In an experimental sample it is not possible to vary ETh as easily as it can be done in mytheoretical calculations.

Generalizations of the calculations I performed here are possible, for example by investi-gating the secondary gap for asymmetric contacts. This generalization however makes thecalculation effort considerably larger, as the phase in the ansatz for the Green’s function inthe normal metal (3.5) can no longer be chosen to zero. The QCT thus provides a systemof three equations with three unknowns. In general only the normalization condition canbe used to eliminate one of them. Thus a set of two equations must be treated numeri-cally. Further generalization can be reached by introducing additional superconductors. Ifall contacts are symmetric the phase in the normal metal can also be chosen to zero. TheQCT then provides one scalar equation for one unknown quantity. However there appear

CHAPTER 4. SUMMARY, CONCLUSION AND OUTLOOK 93

several phase differences, which can influence the existence of the secondary gap.

94

Zusammenfassung

In meiner Diplomarbeit beschaftige ich mich mit dem Proximity-Effekt, der die Veranderungder physikalischen Eigenschaften eines Normalmetalls in der Nahe eines Supraleiters be-schreibt. Im Speziellen untersuche ich die lokale Zustandsdichte im Bereich der Fermi-Energie eines Normalmetalls, das uber symmetrische Kontakte mit zwei identischen Supra-leitern verbunden ist (SNS-Struktur). Unter symmetrischen Kontakten verstehe ich hier-bei identische Transmissionseigenschaften. Wahrend die Zustandsdichte eines Normalme-tall im Bereich um die Fermienergie naherungsweise konstant ist, besitzen Supraleiter ei-ne Lucke ohne besetzbare Anregungszustande. Die Große dieser Lucke ist weit im Inne-ren des Supraleiters durch den Betrag des komplexwertigen Ordnungsparameters gegeben(Abbildung 2.8). Die Ordnungsparameter der hier betrachteten Supraleiter unterscheidensich betragmaßig nicht, konnen im Allgemeinen jedoch einen relativen Phasenunterschiedaufweisen. Zur Berechnung der Zustandsdichte wird der Formalismus der quasiklassischenGreenschen Funktionen in Nambu-Darstellung verwendet, deren Ortsabhangigkeit durch dietransportartige Eilenberger-Bewegungsgleichung [26] beschrieben ist. Ein spezieller Grenz-fall der Eilenberger-Gleichung ist der diffusive Transport im schmutzigen Grenzfall. Dabeiwird die mittlere freie Weglange der elastischen Streuung als sehr klein gegenuber allenanderen charakteristischen Langenskalen, im Wesentlichen gegenuber der Korrelationslangedes Ordnungsparamters, angenommen. Fur die in diesem Limit naherungsweise isotropeGreen’sche Funktion kann die Eilenberger-Gleichung uber alle Richtungen gemittelt werden,was eine diffusionsartige Bewegungsgleichung fur die isotrope Komponente der GreenschenFunktion liefert, die Usadel-Gleichung genannt wird [15]. Die lokale Zustandsdichte ist jenach Definition der Greenschen Funktion durch den Real-, bzw. den Imaginarteil der Diago-nalkomponenten der retardierten Greenschen Funktion gegeben. Um diese in jedem Punktdes zusammengesetzten Gesamtsystems zu berechnen, musste die Usadel-Gleichung unterBerucksichtigung aller Randbedingungen selbstkonsistent gelost werden. Da die Usadel-Gleichung eine nichtlineare Differentialgleichung ist, ist dies fur beliebige Geometrien nurunter großem Aufwand moglich. In meiner Arbeit verwende ich daher die so genannte Quan-tum Circuit Theorie (QCT) [12] [19] [4], die durch Diskretisierung das ursprungliche Pro-blem auf die Losung einer algebraischen Gleichung reduziert, im symmetrischen Fall aufdie Suche nach einer eindimensionalen Nullstelle. In Abbildung 3.1 ist die hier untersuchteSNS-Struktur, diskretisiert im Sinne der QCT, dargestellt. Die Form der lokalen Zustands-dichte im Normalmetall hangt neben dem relativen Phasenunterschied noch von der Großedes normalen Bereichs und den Transport-Eigenschaften der Kontakte ab. Die Große desNormalmetalls kann durch eine charakteristische Energieskala, die so genannte Thouless-Energie ETh, beschrieben werden [22]. Diese ist uber die bekannte Unscharferelation mitder Zeit verknupft, die eine Anregung braucht um das Normalmetall durch Diffusion inRichtung eines Supraleiters zu verlassen. Je kleiner die Thouless-Energie also ist, destogroßer ist das Normalmetall. Die Streuung von Anregungen in den Kontakten und damitdie Transporteigenschaften der Kontakte geht uber eine spezifische Verteilung von Trans-missionseigenwerten in die Rechnung ein.

Fur einige wenige Sonderfalle gelingt die Nullstellensuche analytisch, im Allgemeinenmussen die Gleichungen der QCT jedoch numerisch gelost werden. Ein analytisches Beispielwird in Kapitel 3.2 behandelt. Dabei betrachte ich den Grenzfall unendlich großer Thouless-

95

Energie und damit infinitesimal kleinen Volumens des Normalmetalls. In diesem Limes hangtdas Ergebnis (Abbildung 3.2) nicht von den Transport-Eigenschaften der Kontakte ab undist identisch fur alle Kontaktarten. Als ersten speziellen Kontakttyp betrachte ich in Kapi-tel 3.3 symmetrische Tunnelkontakte, fur die die Nullstelle ebenfalls analytisch berechnetwerden kann. Je nach Phasenunterschied und Thouless-Energie ergibt sich eine mehr oderweniger weit ausgedehnte Lucke in der lokalen Zustandsdichte im Normalmetalls (Abbildung3.3, Abbildung 3.4). Die Existenz dieser so genannten Minilucke ist nicht neu und sowohltheoretisch wie auch experimentell detailliert untersucht (siehe beispielsweise [14], [16], [17],[23] und [24]). Fur ballistische Kontakte (Kapitel 3.4), fur die alle Transmissionseigenwerteeins sind, kann bereits analytisch die Existenz einer weiteren kleineren Lucke in der Zu-standsdichte gezeigt werden [9], die sich in etwa um den Betrag des Ordnungsparametersober- bzw. unterhalb der Fermi-Energie befindet und deren Große und Existenz von derThouless-Energie abhangen. Nach oben bleibt die Lucke fur beliebig große jedoch endlicheWerte von ETh bestehen, im Limit einer unendlich großen Thouless-Energie verschwindet siejedoch und das Ergebnis aus Kapitel 3.2 wird reproduziert. Fur kleine Thouless Energien gibtes einen Wert, bei dem die Ausdehnung der Lucke ihr Maximum erreicht hat und unterhalbdessen sie sich wieder schließt bis sie bei einem endlichen Wert von ETh ganz verschwindet.In Abbildung 3.6 und Abbildung 3.7 sind die von mir numerisch berechneten Zustandsdich-ten im Bereich der sekundaren Energielucke fur verschiedene Werte der Thouless-Energiedargestellt. Im Folgenden untersuche ich diese sekundare Energielucke auch fur andere Kon-taktarten um herauszufinden ob es sich dabei um ein Phanomen handelt, das lediglich aufballistische Kontakte beschrankt ist, oder ob es sich um einer allgemeinere Eigenschaft vonSNS-Strukturen handelt. Eine Verallgemeinerung der ballistischen Rechnung auf Kontaktemit einem konstanten Transmissionseigenwert 6= 1 liefert ebenfalls eine Zustandsdichte mitsekundarer Energielucke. Die Ergebnisse dieser Rechnungen sind in Figure 3.8 dargestellt.Bisher wurden ausschließlich Kontakte mit konstanten diskreten Transmissionseigenwertenbetrachtet. Reale Kontakte sind jedoch eher durch eine kontinuierliche Transmissionvertei-lung im Intervall [0, 1] beschrieben. In den folgenden Kapiteln (3.7, 3.8, 3.9, 3.10) werdendie Zustandsdichten solcher realistischer Kontakte untersucht. Dabei stellt sich heraus, dassfur alle Verteilungen, die Beitrage auf dem gesamte Transmissions-Interval [0, 1] liefern, kei-ne sekundare Energielucke auftaucht. Lediglich eine Unterdruckung der Zustandsdichte imjeweiligen Bereich lasst sich beobachten. Fur Verteilungen hingegen, die keinen Beitrag ineinem endlichen Interval um T = 0 liefern, ist wiederum eine sekundare Lucke vorhanden.Diese Beobachtungen fuhren zu der Vermutung, dass die Existenz dieser Lucke ausschließ-lich von den Transmissionseigenschaften um T = 0 bestimmt ist. Im folgenden Kapitel 3.11analysiere ich die zu losende Gleichung analytisch fur den relevanten Parameterbereich. Da-bei zeige ich, dass fur ausreichend große ETh immer dann eine sekundare Lucke auftaucht,wenn ein Kontakt durch eine Transmissionsverteilung charakterisiert ist, die keinen Beitragin einem endlichen Interval um T = 0 liefert. Das Verschwinden der sekundaren Lucke furTransmissionsverteilungen, die einen Beitrag um T = 0 liefern, zeige ich nicht analytisch.Die numerischen Rechnungen im letzten Abschnitt 3.12 legen jedoch die Richtigkeit dieserAussage nahe.

Bibliography

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