Topology, “Smile”-Gaps and Level Fluctuations in the Density of States of Superconducting Proximity Systems

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Topology, “Smile”-Gaps and Level Fluctuations in the

Density of States of

superconducting Proximity Systems

Dissertation zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften: Doctor rerum

naturalium (Dr.rer.nat.)

vorgelegt von

Reutlinger, Johannes

an der

Mathematisch-Naturwissenschaftliche Sektion Fachbereich Physik

Tag der mündlichen Prüfung: 29. Juni 2016 1. Referent: Prof. Dr. Wolfgang Belzig

2. Referent: Prof. Dr. Guido Burkard

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-348999

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Preface

This dissertation contains the work which was done during my PhD in the Quantum Transport Group at the University of Konstanz. It contains already published results (chapters3 and 4), results to be published soon (chapters5 and6), as well as some additional material in the appendix, which might be helpful for the understanding of some calculations in the main part.

All of these results were obtained under the supervision of Prof. Wolfgang Belzig and thus contain a lot of input from him. Furthermore, all of the work which I did during my PhD, was done within collaborations with Prof. Yuli Nazarov from the TU Delft and in parts with Prof. Leonid Glazman from Yale University, who contributed with many discussions and often indicated, on which points to focus. The results of chapter 6 were obtained during a period of direct coworking with Dr. T. Yokoyama and thus contain also input from him, which cannot be separated from the work I did. Here I present only those parts which were mainly obtained together with me. Some further sections which will be contained in a publication currently in preparation, were mainly done by Dr. T. Yokoyama and are thus not contained here. These parts are not mandatory for the understanding of chapter 6. One thing to mention is that the code used for the numerics of chapters 6.4.2 and 6.5.1 was mainly written by Dr. T. Yokoyama and only to a small degree adapted by myself.

Chapter 2 gives a short summary of the theoretical methods and formulas used in this dissertation. It does not contain exact derivations of any of those equations. These can be found in the references mentioned in this chapter.

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Zusammenfassung in deutscher Sprache

Diese Arbeit beschäftigt sich mit dem „Proximity Effekt“ in einem chaotischen Normalmetall, der durch Kontakt mit einem oder mehreren Supraleitern in einem nicht supraleitenden Material auftritt. Es ist bekannt, dass in solchen Systemen in der mittlere Zustandsdichte des Normalmetalls eine Energielücke auftaucht, welche sich symmetrisch um die Fermi-Energie erstreckt und unter dem Namen „Minigap“ bekannt ist. Seit der ersten Entdeckung dieser En- ergielücke in chaotischen Systemen, erweckte sie großes Interesse, sowohl in the- oretischer Forschung als auch in experimentellen Untersuchungen. Theoretisch wurde die Zustandsdichte für vielerlei mögliche Systemgeometrien und System- parameter berechnet. Fortschritte in den Methoden der Experimentalphysik führten in den letzten Jahren dazu, dass das „Minigap“ auch experimentellen Untersuchungen zugänglich wurde, vor allem durch STM-Spektroskopie.

Die Hauptentdeckung dieser Dissertation ist eine weitere Energielücke in einer speziellen Klasse von „Cavity“-Geometrien, die an zwei Supraleiter gekoppelt sind, welche bisher übersehen wurde. Aufgrund seiner charakteristischen Form als Funktion des Phasenunterschieds der beteiligten Supraleiter, wird die Bezeichnung „Smile“-Gap für diese Lücke eingeführt.

Diese Arbeit ist folgendermaßen aufgebaut: Kapitel 2 beginnt mit einer kurzen Einleitung der im Folgenden zur Berechnung der mittleren Zustands- dichte im quasiklassischen Limit und der Fluktuationen einzelner Andreev- Zustände verwendeten theoretischen Methoden. Außerdem gibt dieses Kapitel eine kurze Einführung in den „Proximity Effekt“ und die mikroskopischen Prozesse die dazu führen, dass sich eine Energielücke bilden kann. Für die Rechnungen in den Kapiteln 3 und 4 werden die Grundlagen der Theorie quasiklassischer Green’scher Funktionen eingeführt, welche eine mikroskopische Theorie basierend auf den Gesetzen der Quantenmechanik ist. Diese Methode erlaubt es uns die quasiklassische mittlere Zustandsdichte zu berechnen, die in diesem Fall durch ein kontinuierliches Spektrum von Andreev-Zuständen

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Im quasiklassischen Limit liefern beide Methoden äquivalente Ergebnisse, der Vorteil der „Random Matrix Theorie“ ist jedoch, dass damit auch Fluktuatio- nen, die über das mittlere Verhalten hinausgehen, berechnet werden können.

Diese Methode kommt in den Kapiteln 5und 6zum Einsatz.

Das in den Kapiteln3 und4betrachtete System, besteht aus einer einzelnen chaotischen „Cavity“, welche an zwei Supraleitern gekoppelt ist. Die Hauptent- deckung dieser Arbeit ist eine sekundäre Energielücke in der Zustandsdichte der

„Cavity“ direkt unterhalb oder in der Nähe der supraleitenden Lücke∆, je nach Wahl der System-Parameter. Eine detaillierte Beschreibung dieser Lücke für ballistische Kontakte wird im Kapitel3vorgestellt, das Kapitel4erweitert diese Untersuchungen auf allgemeinere Kontakte. Hierbei wird auch ein allgemeines Kriterium für das Auftauchen einer solchen Energielücke hergeleitet.

Um auch Informationen die über die mittlere Zustandsdichte hinausgehen zu erhalten, wird im Kapitel 5 eine Beschreibung durch die „Random Ma- trix Theorie“ für dieses System verwendet und damit die Fluktuationen des

„Smile“-Gaps berechnet. Im Parameterbereich großer Thouless-Energien stellen sich neben dem universellen Verhalten der mittleren Zustandsdichte auch die Fluktuationen als universell heraus. Die Verteilungsfunktion in diesem Limit scheint dieselbe Verteilung zu sein die bereits für das „Minigap“ gefunden wurde. Es weist also einiges darauf hin, dass es sich hierbei um eine Verteilung mit ähnlich universellem Charakter für Rand-Niveaus handelt, wie die Wigner- Dyson-Verteilung dies für Energieniveaus im Inneren eines kontinuierlichen Spektrums ist. Es stellt sich außerdem heraus, dass die gesamte Anzahl der Energieniveaus zwischen dem „Minigap“ und dem „Smile“-Gap immer genau der Anzahl der Transportkanäle zwischen der „Cavity“ und den Supraleitern entspricht. Level-Sprünge von einzelnen Energieniveaus durch das „Smile“-Gap sind daher sehr unwahrscheinlich und werden exponentiell unterdrückt.

In Kapitel 6wird ein System untersucht, bei dem vier Supraleiter beteiligt sind, ein so genanntes „Multi-Terminal Josephson System“. Für die spezielle Realisierung, die hier betrachtet wird, wird die Bezeichnung „4T-Ring“ einge- führt. Ausgehend von einem Parameterbereich, bei dem Andreev-Zustände als entartete Bündel ausschließlich zwischen zwei benachbarten Supraleitern lokalisiert sind, werden die Systemparameter so geändert, dass einzelne Andreev- Niveaus mehr als nur zwei Supraleiter „sehen“ und daher auch von mehr als nur einem supraleitenden Phasenunterschied abhängen. Die besondere Aufmerk- samkeit gilt hierbei den Kreuzungspunkten von Bündeln, wobei sich anhand

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numerischer Ergebnisse zwei Arten von Kreuzungspunkten unterscheiden lassen.

Unter Anwendung entarteter Störungsrechnung kann der Unterschied der beiden Kreuzungstypen auf die unterschiedliche Rolle der ersten Ordnung Korrektur- Terme zurückgeführt werden: Während im einen Fall die erste Ordnung Ko- rrektur verschwindet, liefert sie im anderen Fall den führenden Beitrag zur Korrektur der Energiniveaus. Zwischen den fast entarteten Bündeln befinden sich große „Smile“-Gaps, die auf die topologischen Eigenschaften dieses Systems zurückzuführen sind.

Ein kurzes Fazit in Kapitel 7schließt die Arbeit ab.

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Appendix 125

A Universal quasiclassical regime at constant T 125 A.1 Detachment of the secondary gap from E =∆ . . . 125

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A.2 Analytical properties in the universal regime ETh ∫∆ . . . 126

A.2.1 Properties of the gap edge atÏ = 0 . . . 126

A.2.2 Critical phaseÏ_c atE =∆ . . . 127

B Quasiclassical average results from RMT 129 C Effective Hamiltonian description at ETh ∫ ∆ and single level model 133 C.1 Definition of the effective Hamiltonian . . . .133

C.2 Single level model . . . 135

C.2.1 Expansion for energies E .∆ . . . .136

C.2.2 Correct expansion of the determinant . . . .137

D Reduction of the matrix-dimension 139 E Level crossings 143 E.1 Crossings mediated by mesoscopic fluctuations . . . 143

E.2 Level-crossings by variation of single transmission eigenvalues . . 147

F Root finding 149

G Derivation of the second moment m₂ 153

Bibliography 157

Publications 169

Curriculum Vitae 171

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Contents v

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CHAPTER 1 Abstract and Motivation

This work is motivated by the amazing effects that arise in non-superconducting materials by coupling them to a superconductor. Such superconducting heterostructures can consist of superconductors coupled to normal metals, ferromagnets, semiconductors or insulators. One of the most popular ones is a normal metal or an insulating barrier between two superconductors, which is called a Josephson junction. It has the peculiar property of giving rise to a dissipationless supercurrent in equilibrium. But also coupling to exotic materials like topological insulators, 2-dimensional systems like graphene or 2- dimensional electron gases or even 1-dimensional systems like carbon nanotubes are possible and lead to a huge variety of interesting new effects and applica- tions. These comprise for example thermoelectric elements, superconducting qubits for quantum computation, cooling elements for mechanical oscillators or topological superconductors, which can host Mayorana fermions, to name just some, which have recently become very popular.

But even in a simple metal grain, the coupling to a superconductor has interesting effects. Whereas the chaotic nature of an uncoupled grain cannot be determined from the average density of states, but only from the level statistics of the grain, coupling to a superconductor reveals this hidden property by inducing a gap in the density of states of a chaotic grain symmetrically around the Fermi energy. This gap is known as the minigap. It appears not only in a chaotic metal grain but also in dirty normal metals with a layered structure of finite thickness. Since the first reports on the existence of this gap lots of interest was attributed to it theoretically, as well as experimentally. It was calculated for many geometries and regimes. Advances in experimental techniques in recent years made it also accessible to experiments using STM

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spectroscopy.

The main finding of this dissertation is another secondary gap, which exists in grain- or cavity-like geometries coupled to two superconductors, which has been overseen so far. Due to its characteristic shape as a function of superconducting phase differenceÏ, it is named “Smile”-gap.

The work is structured as follows: Chapter 2gives a short introduction to the methods used in the following chapters to calculate the average density of states in the quasiclassical regime, as well as the fluctuations of single Andreev levels. Furthermore this chapter gives a short introduction to the field of proximity effect and the processes that give rise to the formation of the minigap.

For the calculations in chapters 3 and4 the method of quasiclassical Green’s functions is introduced, which is a theory based on the microscopic laws of quantum mechanics. It is used to calculate the average density of states in the system, corresponding to a continuous distribution of Andreev levels. The second method introduced in chapter2 is a phenomenological theory known as

“Random Matrix Theory”. This method gives equivalent average results in the quasiclassical regime like the quasiclassical Green’s function method, but has the advantage that it can also be used to investigate mesoscopic fluctuations beyond the average. This method is used in chapters 5 and 6.

The system considered in chapters3 and 4consist of a chaotic cavity, which is coupled to two superconducting reservoirs. The main finding of this work is a secondary “Smile”-gap in the density of states of the cavity, which is situated directly below or close to the superconducting gap edge ∆, depending on the choice of the system parameters. A detailed characterization of this gap for ballistic contacts is given in chapter 3, chapter 4extends this analysis to more general contacts, including a derivation of a general criterion for its existence.

In order to go beyond quasiclassical average results Random Matrix Theory is used in chapter 5 to calculate the fluctuations of the “Smile”-gap. In the regime of large Thouless energies a universal behavior is found not only for the average density of states, but also for the fluctuations of the gap. The universal distribution of the gap seems to be the same like the one found for the minigap. This distribution seems to be a similar universal function at the edge of a continuous spectrum, like the Wigner-Dyson distribution is inside the continuous part of the spectrum. It turns out that the number of levels between the minigap and the “Smile”-gap is always equal to the number of transport channels towards the superconductors. Crossings of single levels through the “Smile”-gap are thus very improbable events, which are exponentially suppressed.

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3 A structure involving four superconductors, a so-called Multi-Terminal Josephson junction, is considered in chapter 6. The special realization considered in this chapter is nicknamed a “4-T ring”. The analysis starts with a regime where Andreev levels are located only between neighboring superconducting reservoirs and come in degenerate bunches. This limit is called the extreme open regime in the following. By changing the parameters the system is turned into a regime where Andreev levels “see” more than two superconductors and thus depend on more than one phase difference. Special attention is concen- trated on the crossings of bunches. From the numerical results two different crossing-types can be distinguished qualitatively. Using perturbation theory the difference between the two crossing types can be related to the role of first order corrections, which vanish for one type of crossing and are the dominant contribution for the other. Between the bunches there are secondary gaps, which are related to the topological properties of this system.

A short conclusion in chapter7 completes this work.

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CHAPTER 2 Introduction

2.1 Superconductivity

This work deals with the so-called proximity effect, which appears in a normal metal in proximity to a superconductor. Throughout this work the superconductors are assumed to be conventional superconductors, more precisely speaking superconductors describable by the BCS theory [Bar57]. A characteristic property of BCS superconductors is a complete suppression of the density of states (DOS) at the Fermi energy, leading to a gap, which has a width of twice the absolute value of the superconducting order parameter ∆[DeG99; Sch64;

Tin04]. The levels inside the gap are shifted to higher/smaller energies, leading to a characteristicsqrt-divergence in the DOS at E =±∆ [Tin04]. A plot of the DOS around the Fermi energy is shown in Fig. 2.1. A consequence of this gap is that no excitations can enter the superconductor in this energy range.

Thus any excitation coming from a normal metal adjacent to a superconductor is necessarily reflected. More on the possible reflection processes atE <∆and the peculiar phenomenon of Andreev reflection [And64] is described in sections 2.4.1 and 2.4.2.

The origin of the gap and superconductivity in general is an effective phonon- mediated attraction between electrons at low temperatures, which couples them to pairs, so called Cooper pairs [Coo56]. These pairs condense into to a common ground-state. This condensate is a macroscopic quantum state, which is characterized by a complex order parameter, which has besides its absolute value ∆ also a phase. Cooper pairs are strongly correlated and can diffuse from a superconductor into a neighboring normal metal without immediately losing their correlations [Deu69]. In this way superconducting properties can

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- 2 - 1 0 1 2 0

1 2 3 4

E /Δ N ( E )/ N 0

Figure 2.1: Density of states in a BCS-superconductor with a gap symmetrically around the Fermi energy of width 2∆ and characteristic sqrt-divergency at E =±∆.

be induced into a neighboring normal metal, which is known as the proximity effect.

2.2 Green’s functions

Green’s function methods play an important role in the theoretical description of transport through mesoscopic quantum systems and the calculation of observables. In nanoscale systems particle interactions play a crucial role and cannot be neglected. One example, which nicely demonstrates this, is the blockade of the electrical current through a quantum dot if it is charged due to its small capacitance. This effect is known as the Coulomb blockade [Dit98;

Naz09]. The interaction between charged particles effectively shifts the energy levels of the dot, which can block the electrical transport through the system.

In general, systems of interacting particles cannot be solved exactly. In their description there is thus a need for methods that allow for an approximative, systematic treatment of interactions, which covers the relevant contributions to certain physical effects. Green’s functions offer such possibilities. The advantage of these methods is that they allow for a perturbative treatment of single-particle scattering as well as many-particle interactions [Ric80]. Since the Green’s functions themselves are related to single particle propagators of

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2.2 Green’s functions 7 the system, the perturbative treatment can nicely be visualized by Feynman- diagrams [Fey49]. In many cases it turns out that certain classes of diagrams can be related to a physical effect. In this work Green’s functions are used in chapters 3and 4. This section gives a short overview of their properties and a description of their application to the system under consideration.

2.2.1 Definition

Depending on the application, different types of Green’s functions have turned out to be useful. The causal single-particle Green’s function contains information on the occupation of states. It is defined as the statistical average of the time-ordered product of one Heisenberg creation operator and one Heisenberg annihilation operator:

G^C_‡‡Õ(r, t,r^Õ, t^Õ) = ≠iD

TtŒ‡(r, t)Œ_‡^†Õ(r^Õ, t^Õ)E

. (2.1)

‡ and ‡^Õ indicate the spin, Tt is the time-ordering operator. The spectral properties of the system are related to the retarded Green’s function

G^R_‡‡Õ(r, t,r^Õ, t^Õ) = ≠iD

[Œ‡(r, t),Œ_‡^†Õ(r^Õ, t^Õ)]E

«(t^Õ≠t), (2.2) where [...] denotes the commutator and «(t) is the step function. È...Í is the statistical and quantum average. Since all calculations in this work are diagonal in spin space, the‡-indices are not explicitly written. The two coordinatesr andt are combined to a single variable x= (r, t). The following properties are valid for all Green’s functions, causal and retarded Green’s functions are thus not distinguished.

2.2.2 Single particle potential

For a system of non-interacting particles, the Hamiltonian has the formH = H₀+H_pot, where H₀ is the Hamiltonian of free particles and H_pot describes scattering at an external potential.

H₀ =ˆ

d³rŒ^†(r)H0(r)Œ(r) (2.3) H_pot=ˆ

d³rŒ^†(r)U(r)Œ(r) (2.4)

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The Green’s function is a solution to the differential equation

(≠iˆ_t+H₀(r) +U(r)≠µ)G(x, x^Õ) =≠”(x≠x^Õ), (2.5) µdenoting the chemical potential. Using the solution to this equation for free particles, the full Green’s function Gcan be expressed through the free particle Green’s functionG⁽⁰⁾ via

G(x, x^Õ) = G⁽⁰⁾(x, x^Õ) +ˆ

dx^ÕÕG⁽⁰⁾(x, x^ÕÕ)U(x^ÕÕ)G(x^ÕÕ, x^Õ). (2.6) This equation can be reinserted into itself iteratively, which leads to an infinite series of terms. Each of these terms can be visualized by a Feynman diagram, which gives a very intuitive interpretation of the Green’s function as a propagator. Eq. (2.6) can be rewritten as

G(x, x^Õ) =G⁽⁰⁾(x, x^Õ) +ˆ

dx^ÕÕG⁽⁰⁾(x, x^ÕÕ)Àpot(x^ÕÕ)G(x^ÕÕ, x^Õ), (2.7) where À_pot is the irreducible potential scattering self energy. Such an equation is called Dyson equation [Dys49].

2.2.3 Interacting particles

For interacting particles, the Hamiltonian contains an additional term of four field operators

H_int =ˆ

d³rd³r^ÕŒ^†(r)Œ^†(r^Õ)V(r ≠r^Õ)Œ(r^Õ)Œ(r). (2.8) Due to this term there exists no closed equation for the single particle Green’s function like previously Eq. (2.5), but in the equation of the single particle Green’s function there appears a coupling to the two-particle Green’s function:

(≠iˆt+H₀(r) +U(r)≠µ)G(x, x^Õ) =

≠”(x≠x^Õ) +ˆ

d³r^ÕÕV(r ≠r^ÕÕ) lim

t2æt1æt t₂>t₁>t

G₂(r^ÕÕ, t₁,r, t,r^Õ, t^Õ,r^ÕÕ, t₂). (2.9) G₂ denotes the two-particle Green’s function. Similar couplings appear in the equations for higher order Green’s functions, leading to an infinite set of coupled equations. Expressing the single particle Green’s function instead via field

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2.2 Green’s functions 9 operators in the interaction picture, the statistical averageÈ...Í with respect to an interacting Hamiltonian can be rewritten as a series of time-ordered terms, which are averaged with respect to a quadratic Hamiltonian, for which Wick’s theorem is applicable [Wic50]. All higher order Green’s functions in this series thus can be expressed via products of single particle Green’s functions.

Each term in this infinite sum again has a visual interpretation in terms of Feynman diagrams. A set of rules relates these diagrams to mathematical operations. Like before, the problem can be reduced to a Dyson equation and the calculation of an interaction self energyÀ_int:

G(x, x^Õ) = G⁽⁰⁾(x, x^Õ) +ˆ

dx^ÕÕdx^ÕÕÕG⁽⁰⁾(x, x^ÕÕÕ)Àint(x^ÕÕÕ, x^ÕÕ)G(x^ÕÕ, x^Õ). (2.10) In translational invariant systems, which are homogeneous in time (the Hamilto- nian does not explicitly depend on time), everything depends only on differences x≠x^Õ. It is convenient to change variables via Fourier transformations with respect to space and time coordinates: (∆r,∆t)æ(k, E). The Dyson equations (2.7) and (2.10) reduce to algebraic equations

G=G⁽⁰⁾+G⁽⁰⁾ÀG. (2.11)

All quantities appearing are functions of energy and momentum, their arguments must fulfill energy and momentum conservation. In general, the self energyÀ cannot be calculated exactly, but is approximated by certain classes of diagrams, which cover the relevant physical effect, for which the method is used.

2.2.4 Observables

Once the Green’s functions are known, all observables can be expressed through them. The expectation value for the particle density for example can be expressed through the causal Green’s function. It becomes

n(r) =i lim

xæx^Õ x>x^Õ

G^C(x, x^Õ). (2.12) The density of states is related to the retarded Green’s function:

ﬂ(r, E) = ≠1 ﬁIm⇥

G^R(r, E+i”⁺)⇤

. (2.13)

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2.2.5 Green’s functions in BCS superconductors

This formalism can straightforwardly be applied to conventional BCS superconductors [Bar57]. Due to a point-like”-interactionU(r≠r^Õ) = ≠(V /2)”(r≠r^Õ) in this model, electrons couple to Cooper pairs, which merge to a superconducting condensate. The special form of the BCS ground state [Tin04], which is no eigenstate of the particle number operator, leads to a coupling of electrons and holes. This coupling makes it impossible to treat electrons and holes independently, but instead a combined formalism in Nambu-space is necessary.

Creation and annihilation operators acquire a spinor form [Bel99]:

Œˆ = (Œ_¿,Œ_ø^†)^T, Œˆ^†= (Œ_¿^†,Œ_ø).

The Green’s functions must be defined accordingly and acquire a matrix structure [Gor58; Kop01]:

G(x, xˆ ^Õ) = ✓ G(x, x^Õ) F(x, x^Õ)

≠F^†(x, x^Õ) G^†(x, x^Õ)

◆

. (2.14)

The off-diagonal components, which vanish in the normal state are called Gor’kov, or anomalous Green’s functions. They contain expectation values of two creation or annihilation operators. Equations of motion analogous to Eq.

(2.5) for this matrix Green’s function are [Bel99; Gor58; Kop01]

h(iˆ·3ˆt+ 1

2mˆ1Ò²r+ ˆ∆(x) + ˆ1µ(x))”(x≠x^ÕÕ)≠À(x, xˆ ^ÕÕ)i

¢G(xˆ ^ÕÕ, x^Õ)

= ˆ1”(x≠x^Õ), (2.15) G(x, xˆ ^ÕÕ)¢h

(iˆ·₃ˆt^Õ + 1

2mˆ1Ò²r^Õ+ ˆ∆(x^Õ) + ˆ1µ(x^Õ))”(x^Õ ≠x^ÕÕ)≠À(xˆ ^ÕÕ, x^Õ)i

= ˆ1”(x≠x^Õ), (2.16)

where the pairing potential

∆(x) =ˆ ✓ 0 ≠∆(x)

∆^ú(x) 0

◆ with ∆(x) = V lim

x1æx2+F^C(x1, x₂)

was introduced. ·ˆi is the i-th Pauli Matrix in Nambu space. ¢ denotes the matrix product together with an integral over the internal variable x^ÕÕ. In Eq.

(2.16) the operators act to the left on the second argument ofG. Eq. (2.15) andˆ (2.16) are for the most general form of a self-energy, which involves a convolution

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2.2 Green’s functions 11 ofÀˆ with G. The elastic scattering self-energy in Born approximation [Kop01],ˆ which is used in this work, depends only on the center of mass coordinate. It has the formÀ(x, xˆ ^ÕÕ) = ˆÀ(x)”(x≠x^Õ), which makes the convolution trivial.

This allows us to use the so-called Wigner representation [Bel99; Kop01] by introducing relative- and center-of-mass coordinates

ﬂ=r ≠r^Õ, rcm = (r+r^Õ)/2.

The spatial derivatives must be replaced as well:

Òr =Ò^ﬂ+Ò^rcm/2, Òr^Õ =≠Ò^ﬂ+Ò^rcm/2.

After a Fourier transform with respect toﬂandt and by subtracting Eq. (2.16) from (2.15) the equation of motion acquires a very compact form [Bel99]:

≠ik

mÒ^rcmG(k,ˆ rcm, E) = h

E·ˆ₃≠À(rˆ cm, E) + ˆ∆(rcm),G(k,ˆ rcm, E)i

. (2.17) k is the conjugate momentum to the relative coordinate ﬂ. In the derivation of Eq. (2.17) second order derivatives with respect to rcm were neglected, because in this variable Gˆ varies on large length scales of the order of the superconducting coherence length ›, which is much larger than the Fermi wavelength⁄_F. It was already used thatÀˆ depends only onrcm. The Fourier transformation with respect toﬂ is more elaborate in the general case, where a convolution is involved, which is not needed for our purposes. More details on this topic can be found in [Bel99]. In the following, the notation for the center of mass coordinate is changedrcm ær. To incorporate isotropic, elastic scattering at random impurities, the self-energy for this type of scattering must be calculated. If the scattering is weak, the Born method is a good approximation. In this approximation the self-energy is given by [Kop01]

À(r, E) =ˆ 1 2ﬁ·imp

ˆ

d›_k^ÕD

G(rˆ ,k^Õ, E)E

œ, (2.18)

where ›_k =k²/(2m)≠µand ·_imp is the elastic scattering mean free time. The averageÈ...Íœ denotes an average over momentum directions.

2.2.6 Quasiclassical approximation

In this section the quasiclassical approximation is introduced. The idea behind this approximation is to replace the usual Green’s function, which depends on

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two spatial coordinates, by a function which depends only on a single coordinate.

The usual Green’s function fluctuates strongly on length scales of the order of the Fermi wavelength ⁄F as a function of the relative coordinate ﬂ. Since observables vary on much larger length scales, these fluctuations can have no influence on them and thus be averaged out. The length scale of variations with the center of mass coordinate r is much larger. This fact was already used in the derivation of Eq. (2.17), when second order derivatives with respect to this variable were neglected. A common way to define the quasiclassical Green’s functions is thus [Bel99]

ˆ

g(r,e_k, E) = i ﬁ

˛

d›_kG(rˆ ,k, E), (2.19) where ¸

indicates that only poles of Gˆ close to the Fermi surface are taken into account. Only the absolute value of k is integrated out, which means that the result still depends on the direction ek. More on the details of the definition of ˆ

g can be found in [Kop01]. Especially the diagonal components which have a normal contribution, which is not related to superconductivity, require some care. Integrating Eq. (2.17) and using that all momenta are close to kF it becomes [Eil68; Lar69]

≠ivFÒrˆg(ek,r, E) = h

E·ˆ₃≠À(rˆ , E) + ˆ∆(r),g(eˆ k,r, E)i

. (2.20) This equation of motion for quasiclassical Green’s function is called Eilenberger equation. The quasiclassical Green’s function (2.19) fulfills a normalization condition

ˆ

g² = 1. (2.21)

Its validity follows directly from the solution of Eq. (2.20) for a homogeneous system. Furthermore, one can show [Kop01] that this doesn’t change under the influence of the Eilenberger equation (2.19) and is thus a general property of quasiclassical Green’s functions. The elastic scattering self-energy (2.18) can be expressed through g:ˆ

À(r, E) =ˆ ≠i

2·impÈg(rˆ ,ek, E)Íœ. (2.22)

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2.2 Green’s functions 13

2.2.7 Dirty limit

If the superconducting coherence length › exceeds the elastic scattering mean free path, a self-averaging over momentum directions of the Green’s function takes place caused by multiple scattering before it changes due to a variation of other parameters. This self-averaging can already be taken into account at the level of the equation of motion [Usa70]. In order to derive an equation of motion for the averaged Green’s function, the quasiclassical Green’s function is split into an isotropic main component and a small non-isotropic part:

ˆ

g(r,ek, E) = ˆg⁽⁰⁾(r, E) + ˆg⁽¹⁾(r,ek, E) with g⁽¹⁾ πg⁽⁰⁾. (2.23) Following the steps described in [Kop01;Naz09] leads to an equation of motion for the isotropic component g⁽⁰⁾:

ie²N₀h

E·ˆ₃+ ˆ∆(r),ˆg⁽⁰⁾(r, E)i

+Òrˆj(r, E) = 0, (2.24) where N0 is the density of states at the Fermi energy in the normal state. The matrix currentˆj is defined as

ˆj(r) = ‡(r)ˆg⁽⁰⁾(r, E)Òrˆg⁽⁰⁾(r, E) (2.25) with the conductivity ‡(r) =e²N₀D(r), where D=·_impv_F²/3 is the diffusion constant, which can in general depend onr. Eq. (2.24) is called Usadel equation [Usa70]. The matrix current (2.25) is not conserved, due to the commutator term appearing in Eq. (2.24). In the description of realistic devices it has turned out, that the assumption of fast isotropization due to impurity scattering is well satisfied in many cases and thus a description using isotropic Green’s functions is possible (see for example [Sue08], where a numerical solution of Eq. (2.24) is compared with STM measurements on a dirty wire).

Throughout this work all calculations involving Green’s functions treat this regime. Because of the Green’s function product in the matrix current (2.25), Eq. (2.24) is a non-linear differential equation, which requires special methods for its solution. In the following the abbreviation ˆg⁽⁰⁾(r, E)æˆg(r, E) is used.

2.2.8 Quantum Circuit Theory

The Quantum Circuit Theory (QCT) [Naz99;Naz09] is a method to solve the Usadel Eq. (2.24) by transforming it into a set of algebraic equations. To achieve this, the structure is separated into parts where the Green’s function is

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almost constant and which are described as nodes in the following. Typically, the upper limit for the size of a node is the superconducting coherence length

›. These nodes are connected via connectors, along which strong changes in the Green’s functions occur, described in terms of matrix currents. These currents depend on the Green’s functions in the nodes they connect and vanish if the Green’s functions are equal. The advantage of this method is that it can not only be applied to a single superconducting element, but also allows for a description of heterostructures including normal metals or ferromagnets, since the matrix currents represent the boundary conditions, which have to be imposed at the interfaces between different elements.

Integrating Eq. (2.24) over the volume of a node Vn where the Green’s function and ∆ˆare approximately constant and applying Gauss’ theorem to the gradient part, Eq. (2.24) transforms to

iVne²N0

h

E·ˆ3+ ˆ∆n,gˆn(E)i +X

j

Iˆjn. (2.26)

The sum runs over all connectors attached to this node. Introducing the Thouless energy [Tho77]

ETh = (ﬁN0Vn)^≠1GÀ/GQ, (2.27) where GÀ is the sum over all attached conductances, a leakage current for this node can be defined [Naz99; Naz09]:

I_{leak, n} = iGÀ

ETh

hE·ˆ₃+ ˆ∆_n,ˆg_n(E)i

. (2.28)

The matrix current through each connector is expressed via the Green’s functions at its ends and the transmission eigenvalues of the channels [Naz99;

Naz09]:

Iˆij = 2GQ

X

n

T_nⁱ(ˆgjgˆi≠gˆiˆgj)

4 +T_nⁱ(ˆgjgˆi+ ˆgiˆgj ≠2). (2.29) Including leakage currents, the overall current at each node must vanish.

Geometrical details of the nodes, like for example the shape of the boundary, are not important. Only the volume enters the value of E_Th via Eq. (2.27).

Instead of solving a differential equation, the problem is reduced to a coupled set of algebraic equations. Depending on the discretization, this problem can

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2.3 Random Matrix Theory 15 as well become arbitrarily complicated, however in many cases a discretization with a small number of nodes provides already a suitable description, for which this method is a considerable simplification. In chapters 3 and 4 a chaotic cavity is considered, which is described by a single normal node, coupled to two superconducting reservoirs. In order to simulate a system, where the spatial dependence in the normal region is important, in section4.4.6the normal metal is discretized using three nodes which are coupled to two superconductors. A finer subdivision is not necessary for our purposes.

2.3 Random Matrix Theory

A completely different, phenomenological theory of transport in mesoscopic systems is given by the so-called Random Matrix Theory (RMT) [Bee97].

The idea behind this method is the fact that in chaotic systems the matrices, determining their properties, are unknown, because they depend on microscopic details, like for example the exact shape of the boundaries or the distribution of impurities in a diffusive system. These matrices are either the Hamiltonian in case of a closed system, or the scattering matrix in case of an open system.

On the other hand, it is known that these microscopic details should not be important if the systems are large enough, similar as for quasiclassical Green’s functions. Thus the idea of RMT is to start from an ensemble of random matrices and to use as input only the fundamental symmetries of the system [Dys62; Zir11]. These symmetries are time-reversal symmetry (TRS), which can be broken by a magentic field and spin-rotation symmetry (SRS), which is broken by spin-orbit interaction.

2.3.1 General properties

Starting from a matrix with randomly distributed elements, RMT makes predictions on how the eigenvalues and eigenvectors, or in the case of scattering matrices, the transmission eigenvalues, of these matrices are distributed. In most cases, the main interest concerns the eigenvalues (transmission eigenvalues in the case of scattering matrices), because they determine the values of physical observables. In general, they are correlated and their distribution is described by correlation functions. Depending on the particular observable, it might be sufficient to know the average distribution of eigenvalues. Observables for which this is the case are called linear statistics, they are just a sum of functions of single eigenvalues, not necessarily a linear function like the name might suggest. The correlation functions give access not only to the average values of observables, but determine all their statistical properties and can also be used

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to calculate their fluctuations.

One of the main assumptions of RMT is that correlations of eigenvalues are purely “geometrical”[Bee97], which means that they are a purely mathematical effect, which arises due to the parametrization of a matrix in terms of eigenvalues and eigenvectors. Physical properties do not generate correlations between eigenvalues themselves. This is the reason why the distribution of eigenvalue spacings is a universal function, which is known as the Wigner- Dyson-distribution, and why many systems which are describable by RMT show universal behavior in all kinds of respects. More details on this topic can be found in [Bee97].

2.3.2 Random Matrix Theory of random Hamiltonians

The original problem which led to the development of RMT was the need for a description of energy levels in heavy nuclei, about whom very little knowledge exists, besides the fundamental symmetries that their Hamiltonians must fulfill.

Using these symmetries as the only input led Wigner and Dyson to study the ensemble

P(H)≥e^≠^—^Tr(V^(H⁾⁾, (2.30) which is known as the Wigner-Dyson-ensemble [Bee97]. Later it turned out that this ensemble is not only applicable to atomic nuclei, but to chaotic systems in general [Efe82; Efe83]. Such systems are called “non-integrable”. It doesn’t matter whether the chaotic character in these systems is due to impurity scattering in a diffusive system, or due to chaotic boundary scattering in a ballistic system. The symmetry index — is 1 in the presence of TRS and SRS, which is the case considered throughout this work. The Hamiltonian is a real, symmetric matrix in this case. V is a general function, which must have the property of preventing eigenvalues from escaping to infinity. Its exact functional form is related to microscopic details of the system. ForV(H)≥H² the matrix elements are independently distributed, which makes some calculations easier.

The probability distribution of energy eigenvalues related to this ensemble is given by [Bee97]

P({En})≥exp

≠—✓ X

i<j

≠ln(|Ei≠Ej|) +X

i

V(Ei)◆

, (2.31)

which has the form of a Gibbs distribution at inverse temperature— for particles in an external potentialV, including a logarithmic repulsion term. A logarithmic

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2.3 Random Matrix Theory 17 repulsion is realized in nature by two line charges, which repell each other electrostatically. This analogy leads to the fact that the repulsion of energy eigenvalues is commonly compared with a Coulomb gas of charged particles.

This analogy is sketched in Fig. 2.2. According to Eq. (2.31) no correlations arise due to the functionV, containing information on the microscopic details of the system. This independence of correlations of this function, is what is meant with “geometrical correlations” in section 2.3.1. In fact, it can be shown that this logarithmic repulsion of energy levels is true only for energy separations smaller than the Thouless energy of the system, setting a limit to the applicability of RMT [Alt86].

2.3.3 Random Matrix Theory of quantum transport

In order to be able to describe transport through open systems, a similar method of random matrices was developed for scattering matrices of mesoscopic systems. This theory is known as RMT of quantum transport. Instead of ensembles of random Hamiltonians it uses ensembles of random scattering matrices to describe transport through chaotic systems. Maybe the most important difference compared to the RMT of random Hamiltonians is that in the case of scattering matrices the interest concerns not directly the eigenvalues of the random matrix, but rather the transmission eigenvalues of the matrix.

These are the eigenvalues of the transmission matrix, multiplied with its hermitian conjugate. The transmission matrix is a submatrix of the original random scattering matrix. Another difference is that the random matrices are unitary instead of hermitian. For TRS and SRS present in the system, a further constraint besides unitarity is that the matrix has to be symmetric. More details on additional properties for the other symmetry classes can be found in [Bee97]. Since the volume of the space of unitary matrices with respect to an appropriately defined Haar measure is finite [Zyc94], no confining potential V is needed in order to define an ensemble of random scattering matrices. A generalization of the Wigner-Dyson ensemble to scattering matrices is

P(S)≥e^≠^—^Tr[V^(tt^†^)], (2.32) t being the transmission matrix. This ensemble leads to a logarithmic transmission eigenvalue repulsion, similar to Eq. (2.31). Microscopic details of the system are related to the functionV and generate no correlations.

There are two physical systems for which the statistics of transmission eigenvalues are known: A chaotic cavity and a diffusive connector. Both cases

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Figure 2.2: Schematic representation of line charges at positionsE_i interacting logarithmically and being forced into a finite region by a confining potential V. Taken from [Bee97].

are discussed in the following. The chaotic cavity in more detail, since this is the system relevant for this work.

Chaotic cavity

The ensemble of scattering matrices for a chaotic cavity can be derived from the Wigner-Dyson-Ensemble of Hamiltonians by coupling the discrete energy levels of the isolated cavity to the transport modes entering and leaving the cavity. Depending on the transmission properties of the leads, one finds different ensembles of scattering matrices. For point contacts with perfect transmission the ensemble of scattering matrices is known as the circular ensemble. This name results from the fact that in this ensemble scattering phase-shifts are uniformly distributed in the interval [0,2ﬁ], which corresponds to a uniform distribution of phase factors on the unit circle in the complex plane. This ensemble is described by a constant probability distribution in the space of unitary matrices:

P(S) = const. (2.33)

It corresponds to the ensemble (2.32) withV = 0. The transmission eigenvalues of the circular ensemble are distributed according to

P({Tn})≥ Y

n<m

|Tn≠Tm|^—Y

k

T_k^—(|N²^≠N¹^{|+1≠2/—)/2}, (2.34)

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2.3 Random Matrix Theory 19 and thus have a logarithmic repulsion. N1 andN2 are the numbers of transport channels in the two leads. A generalization to non-ideal contacts is given by the so-called Poisson-Kernel [Bee97]

P(S) =|det(1≠S¯^†S)|^≠—(N¹^+N²^≠1+2/—), (2.35) where S¯is a sub-unitary matrix given by

S¯=✓ r₁ 0

0 r₂

◆ .

r₁ and r₂ are the reflection matrices of the two leads respectively. The circular ensemble (2.33) follows from Eq. (2.35) forS¯= 0. The validity of Eq. (2.35) is shown in [Bro95], where the transfer matrix method is used to derive the distribution of the full scattering matrix of three scatterers in series. The scattering matrices of the leads are constant and the central scattering matrix of the cavity is assumed to be distributed according to the circular ensemble (2.33). The ensembles (2.33) and (2.35) are very useful if the the energy dependence of S(E) is not needed or in regimes where it is so weak that it can be neglected. In problems where the energy dependence ofS(E) is essential, these ensembles can not be used, but instead one has to use the underlying Hamiltonian ensemble of the cavity. The Hamiltonian contains information on correlations of scattering matrices at different energies. The energy-dependent scattering matrix can be constructed from the Hamiltonian via

S(E) = 1≠2ﬁiW^†⇥

E≠H+iﬁW W^†⇤≠1

W, (2.36)

where W(E)is a M ◊N coupling matrix describing the coupling of M levels inside the cavity to N = N₁ + N₂ modes in the leads. The N non-zero eigenvalues ofW W^† are given by [Bee97]

w_n = M”s

ﬁ²Tn

(2≠T_n±2p

1≠T_n). (2.37)

Tnare the channel transmissions in the leads,”sis the level-spacing of the cavity.

The Hamiltonian H is distributed according to the Wigner-Dyson ensemble (2.30) with V(H) = ﬁ²/(4M”_s²)H², which is thus called Gaussian ensemble.

The sign in (2.37) is arbitrary, the final results do not depend on it.

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Disordered wire

The second system for which the distribution of transmission eigenvalues is known, is the disordered wire. There are two ways to approach this problem.

The first is by considering a system of many ballistically coupled chaotic cavities in series, where for each the ensemble of scattering matrices is known.

A second approach is to consider many weakly scattering segments in series and to treat the effect of an additional segment perturbatively. This leads to a differential equation for the probability distribution P, known as the Dorokhov-Mello-Pereyra-Kumar equation [Dor82; Mel88]:

limpˆLP({⁄n}, L) = 2

—N + 2≠— XN n=1

ˆ⁄n⁄n(1 +⁄n)J({⁄n})ˆ⁄n

P({⁄n}, L) J({⁄n}) .

(2.38) Here the variables ⁄n = (1≠Tn)/Tn were introduced, limp is the scattering mean free path and J({⁄_n}) =Q

i<j|⁄_i≠⁄_j|^—. Both approaches lead to equal results at large length scales. It turns out that the eigenvalue repulsion is non-logarithmic, thus the ensemble of scattering matrices cannot be of the form of Eq. (2.32).

2.4 The proximity effect

The name proximity effect denotes effects, which arise in a non-superconducting material, for example a normal metal or a semiconductor, due to the proximity of a superconductor. The reason for this effect is the coupling of electrons inside a superconductor to Cooper pairs, which leads to the generation of a superconducting condensate. The electrons, which build up a Cooper pair, are correlated and if they leave the superconductor and enter the normal metal these corellations are not lost immediately, but decay slowly. The length scale, which determines this decay is the superconducting coherence length ›.

Microscopically the process of Andreev reflection, which is described in section 2.4.1, is responsible for this transfer of correlations. If the dimensions of the normal metal are of the order of ›, the proximity effect becomes especially pronounced. One of its manifestations concerns normal metals, which have a classically chaotic dynamics. This can be either due to elastic impurity scattering in the normal part or due to chaotic scattering at the boundaries in a clean system. In such systems, the density of states is suppressed around the Fermi-energy. This gap is known as the “Minigap”. It is discussed in section 2.4.4. Other effects are the possibility of supercurrents through the normal

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2.4 The proximity effect 21 part in systems, which involve two or more superconductors. These systems are known as Josephson-junctions [Jos62]. Recently a special class of proximity systems, where a BCS-superconductor is in proximity to a semiconducting nanowire, has attracted attention. It has been predicted [Kit01] that such systems can host Majorana Fermions, an exotic type of fermions which are their own antiparticle, and recent experiments[Den12; Mou12] seem to confirm these predictions.

2.4.1 Andreev reflection

At interfaces between superconductors and normal metals a peculiar reflection process appears, which converts incoming electrons into reflected holes and vice versa. This process is called Andreev reflection [And64]. During this process a Cooper pair is created or annihilated inside the superconductor. The probability amplitude of Andreev reflection depends on the scattering properties of the contact and on the energy of the incoming excitation. It can be calculated by matching two-component Bogoliubov wave-functions in Nambu-space [DeG99]

at the interface [Blo82]. Especially in the regimeE <∆and for clean interfaces, Andreev reflection becomes important, because neither normal reflection, nor transmission into the superconductor are possible. The probability of Andreev reflection becomesRA= 1, however the excitation collects an energy-dependent phase during the reflection process [Naz09]:

S_A^e^æ^h = exp [≠i(Ï+ arccos(E/∆))], (2.39) S_A^h^æ^e = exp [≠i(≠Ï+ arccos(E/∆))]. (2.40) Ï is the phase of the superconducting order parameter ∆.

2.4.2 Andreev bound states

Repetitive Andreev reflection processes lead to bound states inside the normal region which decay exponentially into the superconductor. If more than one superconductor is involved the phases Ïi are important, for a single superconductor this phase can be chosen toÏ = 0. The energies of these bound states can be calculated by describing electronic scattering inside the normal metal by a scattering matrixS^e(E). Due to symmetry the scattering matrix for holes is given byS^h(E) = S^e^ú(≠E). Combining all processes leads to Beenakker’s determinant condition for the energies of Andreev bound states [Bee92a;Bee91;

Hei13]:

det⇥

1≠S_A^hæeS^h(E)S_A^eæhS^e(E)⇤ = 0 (2.41)

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Figure 2.3: Schematic representation of the generation of Andreev bound states in a system with two superconductors by repeated Andreev reflections and a general energy-dependent scattering matrixS(E) in the normal part.

The generation of Andreev bound states is sketched in Fig. 2.3 for the case of two superconductors, Eq. (2.41) however is more general and holds for an arbitrary number of superconductors. A general solution of Eq. (2.41) is not possible, because the energy-dependence of S(E)depends on the properties of the normal region, like shape and size. In the short junction regime however, this energy dependence is weak and can be neglected. For a single transport channel and two superconductors the energy of the Andreev level is determined by the transmissionT of this channel [Hei13; Naz09]:

E =∆

q1≠T sin²(∆Ï/2) (2.42) Experimental realizations with measurements and manipulation of single channel Andreev bound states predicted by Eq. (2.42) have been realized using carbon nanotubes [Pil10] or atomic contacts [Bre13; Jan15] between superconducting terminals. Phase differences of the order parameters are usually controlled by a magnetic flux through a superconducting ring geometry.

2.4.3 Andreev billiards

Andreev billiards consist of a chaotic normal region, which is coupled to a superconductor by a ballistic point contact [Kos95]. Chaoticity is not caused by impurity scattering, but due to chaotic scattering at the boundaries. Between two scattering events, excitations follow ballistic paths. Without coupling to a superconductor such systems are called non-integrable, which means that classical paths don’t follow closed loops. The role of the superconductor is to

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2.4 The proximity effect 23 convert electrons into holes and vice-versa by Andreev-reflection. The reflected holes/electrons follow the time-reversed path of the incoming particle leading to closed loops independent of the shape of the billiard. In a billiard without coupling to a superconductor, the chaotic nature is not revealed from the average density of states. Only the distribution of level-spacings, which has the form of a Poisson-distribution in the integrable case and the form of a Wigner-Dyson distribution in the non-integrable case, distinguishes integrable from non-integrable systems [Bee05]. This is different in Andreev billiards.

Due to the proximity to a superconductor a gap appears in the average density of states in the chaotic case, whereas there is no gap in an integrable system [Lod98].

Andreev spectrum from RMT

In order to calculate the spectrum of an Andreev billiard, a straightforward method is to combine the RMT description of the scattering matrix of a chaotic cavity (Sec. 2.3) with the process of Andreev-reflection described in Sec. 2.4.1.

Replacing the two leads with a single lead, the ensemble (2.33) describes scattering in the normal cavity, relating outgoing electronic modes towards the superconductor to the incoming modes coming from the same lead. Application of Eq. (2.41) requires a knowledge of the energy-dependence of S(E), which is related to the Hamiltonian via Eq. (2.36). Instead of using the scattering matrix ensemble (2.33), the Hamiltonian ensemble (2.30) has to be used. Eq.

(2.41) with Eq. (2.36) becomes [Bee05; Fra96]

deth

Eˆ≠Hˆ + ˆW(E,Ï)i

= 0. (2.43)

The matricesH and W are defined as [Bro97]

Hˆ =H·ˆ₃, Wˆ(E,Ï) = ﬁ

Ô∆²≠E²

✓ EW W^† ∆W eⁱ^Ï^ˆW^†

∆W e^≠i^Ï^ˆW^† EW W^†

◆ . ˆ

Ï is a diagonal matrix with the phases of the involved superconductors on the diagonal. The disadvantage of Eq. (2.43) is clearly the fact that the dimension of the problem is2M (M being the number of levels in the cavity), whereas the dimension in Eq. (2.41) is2N (N being the number of electronic modes between the superconductor and the cavity). In these systems the number of levels typically is much larger than the number of modes: M ∫N. This makes

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Figure 2.4: Sketch of an Andreev billiard. Chaoticity is due to random scattering at the walls of the billiard. Between two scattering events the motion is ballistic. Andreev reflection at the superconductor converts electrons into holes, which follow time-reversed paths.

Eq. (2.43) from calculational aspects more costly than Eq. (2.41).

Effective Hamiltonian

For small energies E π ∆ Eq. (2.43) reduces to an eigenvalue equation of an effective Hamiltonian [Fra96]. The diagonal components in W(E)ˆ can be neglected and Eq. (2.43) reduces to

det Eˆ≠

✓ H ≠ﬁW W^†

≠ﬁW W^† ≠H

◆

| {z }

Hˆ_{ef f}

= 0. (2.44)

This approximation is only valid in a regime with solutions at small energies E π∆. The relevant parameter is the Thouless energy

E_Th = N≈ ”s

2ﬁ . (2.45)

It determines the ensembles of Hamiltonians through ”s in V(H)in Eq. (2.30).

Only for ETh π ∆ there are energy levels with E π ∆. Thus only in this regime the effective Hamiltonian description (2.44) is valid.