Odd-frequency superconducting pairing in Kitaev-based junctions

Odd-frequency superconducting pairing in Kitaev-based junctions

Athanasios Tsintzis

Supervisor: Dr. Jorge Luis Cayao Díaz

Local examiner: Prof. Annica M. Black-Schaffer

Department of Physics and Astronomy, Materials Theory

Abstract

In this Master Thesis we study the emergence of odd-frequency (odd-ω) superconducting pairing in one- dimensional Kitaev-based junctions. In particular, we examine the relationship between odd-ω pairing and the topological phase of the Kitaev model, which harbors Majorana Bound States (MBSs). For our analysis we use the advanced and retarded Green’s Functions (GFs), which we construct from the scattering states.

In the first Chapter we introduce basic notions regarding superconducting systems, odd-ω pairing and

the topological phases of the Kitaev model. In the second Chapter we present the GF formalism we

use. In the third Chapter we calculate the GFs for a normal metal-superconductor (NS) junction and

we study the symmetry properties of the pairing functions through the anomalous GF’s components, for

various chemical potential and frequency regimes. We find that the odd-ω component (OTE) magnitude is

enhanced in the topological regime and for MBS frequencies (ω = 0), yet is in general finite for ω > 0 and

in the trivial regime. In the fourth Chapter we consider a S’S junction and we discuss the dependence of the

pairing functions on the phase difference between the two superconductors, which is introduced through

the scattering coefficients. We find that both the OTE and ETO (even-ω) components’ magnitudes reflect

the Andreev Bound States (ABS) spectrum. Finally, in the fifth Chapter we consider a long SNS junction

(finite normal region length L _N ) and we examine how this parameter affects the ABS spectrum and thus

the pairing functions.

Sammanfattning

I denna magisteravhandling kommer vi undersöka av uddafrekvens-supraledningsparning och uppkomsten av Majorana fermioner Kitaev-baserade övergångar.

Termen “uddafrekvens-parning” tillämpas till vilket system vilket beståndsdelar visar en korrelation mellan icke lokal tid. Vadim Berezinskii var den första att förslå ett supraledning system, i 1971, men det var senare där man insåg uddafrekvens-supraledningsparning kan tillbringas i strukturer som är kombine- rad med supraledare,dvs supraledare med metaller, halvledare och ferromagneter.

Ettore Majorana förutspådde, i 1937, Majorana fermioner som ett av de elementarpartikel men att den- na iakttagelse har ännu inte skett. Majoranas var försagt som ett noll-energi lokaliserad excitation i topolo- gisk supraledare. Kitaev modelen beskriver den enkleste supraledaren som kan generara Majorana fermio- ner. Modelen beskriver en normal metal-supraledare (NS) och en supraledare-normal metal-supraledare (SNS) korsningar sattes upp för att uppfatta om uddafrekvens-parning uppstår endast från Majoranas.

Från resultaten vi har fått uddafrekvens-supraledningsparning är ett större fenomen vilket förstärks

med närvaron av Majoranas, men det betyder att det kan uppsåt utan dem.

Table of Contents

1 Introduction 3

1.1 Overview . . . . 3

1.2 BCS theory of superconductivity . . . . 4

1.3 Odd-frequency superconducting pairing . . . . 6

1.4 The Kitaev model . . . . 7

1.5 In this Thesis . . . . 12

2 The Green’s functions formalism 15 2.1 Green’s functions for a one-dimensional wire . . . . 15

2.2 Green’s functions from scattering states . . . . 16

3 Normal metal - Superconductor junction 19 3.1 Construction of the scattering states . . . . 19

3.2 Green’s functions for the NS junction . . . . 30

3.3 Analysis of the pairing functions . . . . 34

4 Superconductor - Superconductor junction 41 4.1 Construction of the scattering states . . . . 42

4.2 Green’s functions for the S’S junction . . . . 43

5 Superconductor - Normal metal - Superconductor junction 49 5.1 Construction of the scattering states . . . . 50

5.2 Green’s function for the SNS junction . . . . 52

6 Conclusions - Discussion 59

A Appendix for Chapter 1 61

B Appendix for Chapter 2 66

C Appendix for Chapter 3 68

D Appendix for Chapter 4 73

E Appendix for Chapter 5 77

References 81

Acknowledgements

I would first like to thank Professor Annica Black-Schaffer for giving me the opportunity to work on my Master Thesis in a research group and for providing precious feedback and advice not only on my Thesis but also on how the academic world functions.

I would also like to thank my supervisor and thus the person I collaborated most closely with, Doctor Jorge Luis Cayao Díaz. Without his ideas and guidance the completion of this project would not be possible. He was always there for me, available for countless hours of discussions which helped with advancing with my work and with building a character and perspective on how scientific research is conducted in a more general context.

Special thanks should go to my good friends Ali Abbas and Patrik Posluk who helped me with writing the Abstract of this Thesis in Swedish.

For the support during the past year I cannot thank enough my partner Dimitra Daniil who was always present at my downs and ups while working on this Thesis and whose encouragement kept me going, day by day, until this project’s completion.

Since I wrote the last part of this Thesis at my hometown in Greece, I finally want to thank my mother

Christina and my sister Jenny for their support during this last month, which I especially value since

everything gets a little bit harder when one approaches finishing a project.

Chapter 1 Introduction

1.1 Overview

The term “odd-frequency pairing” applies to any physical system which can be described using correlation (or pairing) functions to account for the interference between the two parts constituting the “pair”. As the name suggests, these functions are odd in frequency (and thus odd in time, as we will see shortly).

This implies that the interactions between the two parts of the pair are “non-local in time” [1], as their correlation function is zero for zero relative time (t = t ₁ −t ₂ = 0) and obtains a finite value for finite relative time (t 6= 0), where with t 1 (t ₂ ) we denote the time coordinate of part 1(2). There are several physical systems which could theoretically exhibit odd-frequency (odd-ω) pairing, such as superconductors [2], ultracold Fermi gases [3] and Bose-Einstein condensates [4], as pointed out in [1]. Among them, one could argue that superconducting systems offer the most natural platforms to investigate odd-ω pairing, as this possibility is allowed by the total antisymmetry condition on the “Cooper pair” wave function [5], which is imposed by the Fermi-Dirac statistics. This observation is credited to Berezinskii [6], as the original formulation of the BCS theory of superconductivity [7] (named after John Bardeen, Leon Cooper and John Robert Schrieffer) suggested an even in time pairing. Although Berezinskii’s initial proposal concerned an odd in time superconducting order parameter ∆ [6] and thus superconductors which would exhibit odd-ω pairing in the bulk, the whole field took a turn in 2001 [1], when it was realized that odd-ω pairing could explain the “long-range proximity effect” in superconductor/ferromagnet (S/F) heterostructures [2, 9]. Therefore, the theoretical description and the search for experimental signatures of odd-ω pairing in the superconducting bulk [10–12] and in heterostructures containing superconductors [13–16] have been particularly active areas of research during the past fifteen years [1].

The present study focuses on the connection between odd-ω pairing and Majorana excitations in con-

densed matter systems. Majorana fermions were first predicted in 1937 by Ettore Majorana [17, 18] as

special solutions of the Dirac equation [19]. In the context of second quantization, these particles are de-

scribed by real creation operators, a fact that is interpreted as Majoranas being their own antiparticles. In

condensed matter systems Majoranas are predicted as localized zero-energy excitations which are usually

referred to as Majorana Bound States (MBSs) [20]. The solid state systems that have been suggested to

host MBSs involve topological superconductors [21, 22]. The prototype one-dimensional system is the

Kitaev model [21], where Majoranas emerge as bound edge states. This model has received a lot of at-

tention due to its theoretical simplicity and relatively accessible experimental realization using nanowires

with Rashba spin-orbit coupling [23, 24]. Apart from the interest for observing Majoranas as a verifi-

cation of a theoretical prediction, MBSs exhibit non-Abelian statistics (as opposed to the high energy physics Majoranas which are fermions), constituting thus a potential topological quantum computation platform [25, 26].

It is quite natural to consider the interplay between MBSs and odd-ω pairing, since the Majorana propa- gator can be shown to be odd in ω [1, 27]. It has also been argued that “odd-ω pairs exist wherever the MBSs stay” [27]. In this Master Thesis we investigate the emergence of odd-ω pairing in junctions made of Kitaev chains, which host MBSs in the topological phase [21]. In particular, we want to find out how tightly intertwined these two concepts (odd-ω and MBSs) really are. Can the emergence of odd-ω pairing be attributed solely to the emergence of Majoranas in a Kitaev junction (as suggested in [27])? Or is odd-ω pairing an ubiquitous, broader phenomenon which is just enhanced by the presence of the odd-ω MBSs?

These are the main questions we set out to answer in the present study.

1.2 BCS theory of superconductivity

Although superconducting phenomena had been known and studied since 1911 [28] their proper theo- retical description was missing, as superconducting systems could not be described using approximation methods (e.g. RPA [29–32]). Several experimental findings (e.g. [33]) were pointing at the direction that the superconducting condensate was a coherent system comprised of the conduction electrons [34].

But since electrons are fermions, a mechanism that would allow them to condense was needed, as such a condensation is prohibited by the Fermi-Dirac statistics. Pairs of electrons with integer total spin could exhibit such a behavior. A possible pairing mechanism was provided by Bardeen and Pines [35], who considered the system of electrons moving in a metal as a whole, so that the positive ions could accom- modate a net attraction between electrons through the electron-phonon interaction. The discovery of the Isotope Effect [33] further supported their findings, as it directly connected the ions’ masses with the crit- ical superconducting transition temperatures (T _c ). Leon Cooper first showed [5] that this picture could not fit in the Landau-Fermi liquid framework [36]. In this framework, one initially considers a Fermi gas of non-interacting electrons and adiabatically turns on the interactions in order to obtain the interacting state from the non-interacting one, using perturbation theory [37]. Cooper showed that, if an attractive interaction between electrons is considered, the scattering of two electrons with opposite momenta and spins would cause the perturbation series to diverge [5]. Thus, perturbation theory cannot work and a totally new framework had to be invented to describe this scattering, a realization which led to the birth of the BCS theory of superconductivity [7] shortly after Cooper’s discovery. We note that the mentioned discovery is referred to as “Cooper instability” and the two electrons involved in the attractive interaction are referred as a “Cooper pair”.

1.2.1 The BCS Hamiltonian in the mean field approach

In this section we follow [38] and [39]. The BCS mean field Hamiltonian reads:

H _MFA = ∑

kσ

ξ _k c ^† _kσ c _kσ + ∑

k

∆ _k c _−k↓ c _k↑ + ∑

k

∆ ^∗ _k c ^† _k↑ c ^† _−k↓ (1.1)

The quantity ξ _k = ^¯h _2m

^k

− µ describes free electrons with parabolic dispersion, µ is the chemical potential,

∆ _k = ∑ k

V _kk

hc ^† _k↑ c ^† _−k↓ i is known as the pairing potential and hc ^† _k↑ c ^† _−k↓ i as the pairing amplitude. ∆ k includes

the interaction kernel V _kk

, while hc ^† _k↑ c ^† _−k↓ i describes the pairing between electrons with opposite momenta and spin. Acting on a given state |ψi, c ^† _kσ (c _kσ ) create (annihilate) electrons of momentum k and spin σ . Thus, c ^† _kσ |ψi = N + |ψ, e _k,σ i and c _kσ |ψ, e _k,σ i = N ₋ |ψi, where with |e _k,σ i we denote the state of one electron with momentum k and spin σ and we have also included the normalization factors N ₊ , N ₋ . We also note here that, any annihilation operator annihilates the ground state c |0i = 0 and that the same thing happens when a creation operator adds an electron with the same quantum numbers as an already existing in |ψi electron, because of the Pauli exclusion principle. Thus, to be concrete, c ^† _k,σ |ψi = 0, if

|ψi = |X, e _k,σ i where the state |Xi stands for any other electron in |ψi. Since they are fermionic operators they obey the anti-commutation relations:

{c ^† _kσ , c ^† _k

σ

} = {c _kσ , c _k

_σ

} = 0 and {c ^† _kσ , c _k

_σ

} = δ _kk

δ _{σ σ}

. (1.2) The terms from the first sum in Eq. (1.1) include the number operators ˆ N = c ^† _kσ c _kσ which count the number of electrons in the system with momentum k and spin σ . Notice that in this BCS description the pairing amplitude and the pairing potential are even in time, as there is no explicit time dependence.

The purpose now is to diagonalize H _MFA in order to obtain information on the superconducting spectrum.

The Bogoliubov-de Gennes formalism [40] is a convenient, although seemingly redundant way, to diag- onalize this Hamiltonian and to discuss quasiparticle excitations in this theory. We have to re-write the sums in the above Hamiltonian in a more suggestive form, in order to treat electrons and holes equally (see Appendix A). H _MFA can then be written as:

H _MFA = 1 2 ∑

k

Ψ ^† _k H _BdG Ψ _k , (1.3)

where we have defined the Nambu spinor Ψ _k = 

c _k↑ c _k↓ c ^† _−k↑ c ^† _−k↓

 T

[41] and the Hamiltonian:

H _BdG =









ξ _k 0 0 ∆ _k

0 ξ _k −∆ k 0

0 ∆ ^∗ _k −ξ −k 0

∆ ^∗ _k 0 0 −ξ _−k









, (1.4)

which we refer to as the BdG Hamiltonian for obvious reasons. Solving the eigenvalue problem H _BdG Ψ = EΨ we can find the energy-momentum relation,

E = ± q

ξ _k ² + ∆ ² , (1.5)

where ξ _k is the energy dispersion in the system for ∆ = 0, and ∆ = |∆ _k | is the superconducting gap.

ξ _k = ξ _−k has also been used. We notice that the energy momentum relation is symmetric around the zero energy axis [37]. This is due to the electron-hole symmetry, as we can look at the two lower components of a Nambu spinor as annihilation operators of holes instead of creation operators of electrons [39], since the holes can be considered as the electrons’ antiparticles. The eigenstates must thus come in pairs with energy ±E [39]. The quasi-particle eigenstates will be addressed in later chapters.

In Fig.1.1 we plot the energy-momentum relation for an s-wave superconductor given by Eq. (1.5) for

different values of the chemical potential µ. First, for ∆ = 0 (dashed lines in Fig. 1.1(a)) the spectrum

±p _F = ± √

2µm. A finite ∆ opens a gap in the spectrum mixing electrons and holes (continuous lines).

For µ > 0 the positive branch develops two minima at p = ±p _F and one local maximum at p = 0 which correspond to energies E = ∆ and E = E _s = (µ ² + ∆ ² ) ^1/2 respectively. For µ = 0 (Fig. 1.1(b)) we only have a minimum E = ∆ for p = 0. For µ < 0 (Fig. 1.1(c)) the minimum becomes E = E _s . We see that for a finite value of ∆ (∆ 6= 0) the energy spectrum is gapped for any value of µ. This renders the system topologically trivial, in contrast to the p-wave case which will be examined in Section 1.4.

(b) E

μ=0

p Δ

(c) E

p μ<0 E

(a) _{E μ>0}

E

Δ Δ

-p

p

p

e

h

Figure 1.1: Energy-momentum relation in an s-wave superconductor for different values of the chemical potential µ. In (a) we also depict the normal metal dispersion for ∆ = 0 (dashed). The energy spectrum is always gapped for ∆ 6= 0. E S = (µ ² + ∆ ² ) ^1/2 and p _F = ± √

2mµ.

1.3 Odd-frequency superconducting pairing

In this part we follow [1] and [2]. A Cooper pair is composed of two electrons with quantum numbers that include spin, position and time, as depicted in Fig. 1.2 and it is a central concept when one discusses superconducting systems. It is thus important to investigate the symmetry properties of the Cooper pair’s wavefunction, studying quantities which measure the correlation between the electrons of the pair, such as the anomalous Green’s function (GF) [1]:

f _σ

_σ

(r ₁ , r ₂ ;t ₁ ,t ₂ ) = T hc σ

(r ₁ ;t ₁ )c _σ

(r ₂ ;t ₂ )i. (1.6) The action of the time-ordering operator is defined as: T c(t 1 )c(t ₂ ) = c(t ₁ )c(t ₂ ) if t ₁ < t ₂ and T c(t 1 )c(t2) =

−c(t ₂ )c(t ₁ ) if t ₂ < t ₁ . Indices σ i , r i and t i refer to spin, position and time coordinates of the fields consid- ered. We note that we have explicitly included the time coordinates which are usually omitted in the BCS theory.

According to Fermi-Dirac statistics, the correlation function must be antisymmetric under the exchange of all the indices, meaning that:

f _σ

_σ

(r ₂ , r ₁ ;t ₂ ,t ₁ ) = − f _σ

_σ

(r ₁ , r ₂ ;t ₁ ,t ₂ ). (1.7)

Exchanging these indices can be viewed as the Parity (P), Time reversal (T ) and Spin permutation (S) op-

erators acting at the same time, as discussed in [1]. We can use the relative and center of mass coordinates

r = r ₁ − r ₂ , t = t ₁ − t ₂ , R = (r ₁ + r ₂ )/2 and T = (t ₁ + t ₂ )/2 and omit the two latter, since we only focus

Electron 1 c _σ1 (r ₁ ,t ₁ )

Electron 2 c _σ2 (r ₂ ,t ₂ ) Cooper pair

Figure 1.2: A Cooper pair with the quantum numbers of spin (σ i ), position (r _i ) and time (t _i ) for each electron depicted (i = 1, 2).

Class Frequency Spin Parity Total

ESE +1 -1 +1 -1

OSO -1 -1 -1 -1

ETO +1 +1 -1 -1

OTE -1 +1 +1 -1

Table 1.1: The allowed symmetry configurations. The odd-frequency classes OSO and OT E are highlighted. Table adapted from [1].

on the dependence on the relative ones [1]. The GF can be then Fourier transformed into the frequency domain:

f _σ

_σ

(r; ω) = Z

dte ^iωt f _σ

_σ

(r;t), (1.8)

and therefore f _σ

_σ

(r; ω) determines the frequency dependent superconducting correlations. We will use the frequency (ω) expression from now on. Now, the only possibilities in agreement with Fermi-Dirac statistics are [2, 42]:

(i) even in frequency - spin singlet - even in parity (ESE), (ii) odd in frequency - spin singlet - odd in parity (OSO), (iii) even in frequency - spin triplet - odd in parity (ETO), (iv) odd in frequency - spin triplet - even in parity (OTE),

and they are also depicted in Table 1.1. Then, in order to have odd-ω pairing, the pairing amplitude must obey f (r, −ω) = − f (r, ω). This means that the Cooper pair’s correlation function has a zero value for ω (or t) = 0 and obtains a finite value for ω (or t) 6= 0. These properties seem peculiar and it was initially believed that it would be hard to find systems that could harbor odd-frequency superconducting pairing [1]. As it turned out, it can be found, at least theoretically, even in normal metal - superconductor (NS) junctions [13, 16, 43], even though more involved heterostructures had been initially considered [9].

1.4 The Kitaev model

We will now describe and analyze the simplest model that realizes topological superconductivity in one

dimension, originally proposed by Kitaev [21], following the analysis of [39] . We consider a supercon-

be either 1 or 0. That is because we consider the spin degree of freedom “frozen” and due to the Pauli exclusion principle we cannot have a second fermion on the same site. The tight-binding Hamiltonian of such a system is:

H = −µ

N

∑ j=1

(c ^† _j c _j − 1 2 ) +

N−1 j=1 ∑

 − t(c ^† _j c _j+1 + c ^† _j+1 c _j ) + ∆c _j c _j+1 + ∆ ^∗ c ^† _j+1 c ^† _j , (1.9)

where the operators c _j (c ^† _j ) are creation (annihilation) operators, acting on site space in this case. The anti-commutaion relations (1.2) are valid here as well if we disregard spin (σ ) and replace k, k ⁰ → j, i.

The first term containing µ counts the number of fermions we have in our system since it also contains the number operator ˆ N = c ^† _j c _j , and µ can be interpreted as the energy to be taken into account for each site if it is occupied. The t term accounts for the nearest-neighbor hopping, since the operators involved destroy a fermion on one site and create a fermion on a neighboring site. Finally, the ∆ and ∆ ^∗ terms account for annihilation (c _j c _j+1 ) and creation (c ^† _j+1 c ^† _j ) respectively of Cooper pairs. Notice that this kind of pairing potential is different to the one discussed previously within conventional BCS theory, as this pairing takes place between fermions with equal spins. This has certain implications when one considers the transformation of Eq. (1.9) to momentum space (see section 1.4.3 ), as the pairing function acquires a linear dependence on momentum, meaning that it can be written in the form ∆ = ∆ ⁰ k. Such a pairing potential is explicitly odd under k → −k and we refer to it as p−wave pairing. This is in contrast to the superconductor described by Fig. 1.1, as there ∆ is k−independent and we refer to that kind of pairing as s−wave pairing.

In order for the topological properties of the model to come to light, we will now define a new set of operators, decomposing the fermionic operators on each site j in terms of the so called Majorana operators γ ^A(B) _j :

c _j = 1

2 γ ^A _j + iγ ^B _j
, c ^† _j = 1

2 γ ^A _j − iγ ^B _j
, (γ ^A(B) _j ) ^† = γ ^A(B) _j , (1.10) as schematically shown in Fig. 1.3(b). Majorana operators γ ^A _j and γ ^B _j on each site compose a complex fermionic operator as seen in Fig. 1.3. It is worth noting here that such a decomposition of fermionic operators into Majorana operators is always possible but usually not as useful as in this case. The inverse transformation is

γ ^A _j = c ^† _j + c _j , γ ^B _j = i(c ^† _j − c _j ), (1.11) under the light of which the hermiticity of the Majorana operators becomes apparent. Combining (1.2) and (1.10) we arrive at the following relations:

{γ _i ^A , γ ^B _j } = 2δ i j δ _AB , (γ j ) ² = (γ ^† _j ) ² = 1. (1.12) The Kitaev Hamiltonian (1.9) in the Majorana basis given by (1.10) is (see App. A):

H = − iµ 2

N j=1 ∑

γ ^A _j γ ^B _j + i 2

N−1 j=1 ∑

h

(∆ + t)

| {z }

ω

γ ^B _j γ ^A _j+1 + (∆ − t)

| {z }

ω

γ ^A _j γ ^B _j+1 i

, (1.13)

where the first term containing µ describes hopping between the two Majoranas on the same site, the term

containing ω ₊ = ∆ + t describes hopping between the second Majorana of one site and the first Majorana

of the next site and the term containing ω ₋ = ∆ − t describes hopping between the first Majorana of one

site and the second Majorana on the next site. These remarks are illuminated in Fig. 1.4(a). According to

the choices we make for the values of µ, ∆ and t, the system can be found in different phases.

γ

γ

γ

γ

γ

γ

γ

γ

(b)

-μ -μ -μ -μ

c

c

c

c

Δ Δ Δ Δ Δ

-t -t -t -t -t

(a)

Figure 1.3: A pictorial representation of the Kitaev chain (a) Each operator c _j corresponds to one site (b) Two Majorana operators correspond to each site constituting one complex fermionic operator. Figure adapted from [39]

1.4.1 Trivial phase

For t = ∆ = 0 and µ 6= 0 the Kitaev Hamiltonian (1.13) becomes

H = − iµ 2

N

∑ j=1

γ ^A _j γ ^B _j . (1.14)

The above Hamiltonian describes a state in which the only possible pairing is between Majoranas from the same site. There are no hopping terms between different sites so the fermions on each site are localized.

The energy of the system depends on the number of sites occupied alone and since nothing interesting happens we say that the system is in the trivial phase (Fig. 1.4(b)).

γ ₁ ^Α γ ₁ ^Β γ j Α γ j B γ ^A _j+1 γ ^B _j+1 γ _N ^Α γ _N ^B

(a) _{-μ ω}

_{=Δ+t ω}

_=Δ-t

γ ₁ ^Α γ ₁ ^Β γ _j ^Α γ _j ^B γ ^A _j+1 γ ^B j+1 γ _N ^Α γ _N ^B

(b) _Δ=t=0

γ ₁ ^Α γ ₁ ^Β γ _j ^Α γ _j ^B γ ^A _j+1 γ ^B _j+1 γ _N ^Α γ _N ^B

(c) _{μ=0, t=Δ}

Figure 1.4: Topological phases of the Kitaev model. (a) The Kitaev model in the Majorana basis. The parameters µ, ω + and ω −

have the role of hopping amplitudes pairing the Majorana fermions as shown in the picture. (b) Trivial phase. Only Majoranas

that belong to the same site can be paired. (c) Topological phase: The only possible pairing is of the form γ ^B _j γ ^A _j+1 leaving γ ₁ ^A

and γ _N ^B unpaired at the two ends of the chain(black circles). Figure adapted from [39].

1.4.2 Topological phase

For ∆ = t and µ = 0 the Hamiltonian (1.13) becomes:

H = it

N−1 j=1 ∑

γ ^B _j γ ^A _j+1 . (1.15)

We see that in this case γ ₁ ^A and γ _N ^B are not present in the Hamiltonian (Fig. 1.4(c)) and therefore [γ ₁ ^A , H] = [γ _N ^B , H] = 0. This suggests that localized Majoranas with zero energy emerge at the ends of the chain [21, 39], implying that:

Hγ ₁ ^A |GSi = γ ₁ ^A H |GSi = E ₀ γ ₁ ^A |GSi ,

Hγ _N ^B |GSi = γ _N ^B H |GSi = E ₀ γ _N ^B |GSi . (1.16) Thus adding a Majorana at each end of the chain does not change the energy of the ground state (|GSi), which means that zero energy is required to add them [21, 39]. This is why we refer to these Majoranas as Majorana Zero Modes (MZMs). Since they are localized at the ends of the chain we also refer to them as Majorana Bound States (MBSs). The emergence of these MBSs characterizes the topological phase.

We note here that this localization at the ends of the wire is an idealized picture, corresponding to reality more as the length of the wire increases. In practice, the wavefunctions of the two MBSs spread across the wire and they overlap [39] (the wavefunctions’ probability densities are still greater at the two ends of the wire).

The fact that we have zero energy excitations in the topological phase constitutes a crucial difference between p-wave and s-wave superconductors. In s-wave superconductors there are no quasi-particle (qp) excitations for energies below the gap (|E| < ∆). As a result, the qp local density of states (LDOS) is zero for energies below the gap and has a finite value for energies above the gap, exhibiting a peak for energies close to ±∆. For p-wave superconductors in the topological regime, MBSs emerge at E = 0 and this emergence manifests as a peak in the LDOS. These points are depicted schematically in Fig. 1.5.

0 1 2 3

-1 0 1

0 2 3

-1 0 1

1

N

/N

(0) N

/N

(0)

E/Δ E/Δ

(a) (b)

MBS

qp qp

4 4

Figure 1.5: (a) Quasi-particle LDOS (N _S (E)) for an s-wave superconductor. N N (0) is the LDOS in the normal state (b) For p-wave superconductors in the topoloical regime there exists a Majorana peak (red) at E = 0. Figure adapted from [37].

We note that in order for the MBS to appear we need to have a finite size system and if we were to consider

an infinite size superconductor no such excitation at E = 0 would arise.

To summarize, in this part we have demonstrated that the Kitaev model exhibits a topological phase which is characterized by the emergence of MBSs, one at each end.

1.4.3 Closed boundary conditions

We now want to study the bulk of the Kitaev chain and we consider periodic boundary conditions [39], meaning that the sites now form a closed chain. In order to account for this change we need to make an addition to the Hamiltonian that will allow hopping and superconducting pairing between the sites 1 and N as well. We achieve that by letting the sum of the second term in the original Hamiltonian (1.9) go up to N [39]:

H = −µ

N

∑

j=1

(c ^† _j c _j − 1 2 ) +

N

∑

j=1

 − t(c ^† _j c _j+1 + c ^† _j+1 c _j ) + ∆(c j c _j+1 + c ^† _j+1 c ^† _j ), (1.17) where we have again considered ∆ to be real. Of course the sites j and N + j coincide. In order to write the above Hamiltonian in momentum space, we will use the Fourier transformations of c _j and c ^† _j :

c _j = 1

√ N ∑

k

e ^−ikx

c _k , c ^† _j = 1

√ N ∑

k

e ^ikx

c ^† _k . (1.18)

The Hamiltonian (1.17) in momentum space is then given by (see Appendix A):

H = ∑

k

ξ _k



c ^† _k c _k − 1 2

 + 1

2 ∑

k

(−2t cos ka) + ∑

k

∆



c _−k c _k e ^−ika + c ^† _k c ^† _−k e ^ika



, (1.19)

where ξ _k = −µ − 2t cos ka and a is the distance between neighboring sites. This Hamiltonian can be brought to BdG form ( App. A):

H = 1 2 ∑

k

Ψ ^† _k H _BdG Ψ _k , (1.20)

where:

H _BdG =

 ξ _k 2i∆ sin ka

−2i∆ sin ka −ξ k



, Ψ _k =  c _k c ^† _−k



. (1.21)

Diagonalizing H _BdG we find the energy eigenvalues:

E = ± q

(2t cos ka + µ) ² + 4∆ ² sin ² ka . (1.22) Before going further we recall that two gapped systems are considered to be topologically equivalent if the one can be transformed into the other by slowly varying the Hamiltonian’s parameters, without having a gap closing [39, 44]. If during this process a gap closing occurs, we have a topological phase transition.

This classification has its roots in the “Adiabatic Theorem” [45] which states that “two Hamiltonians are adiabatically connected if during the transition from the one to the other the system remains in its ground state in all the intermediate steps.” If the energy gap during this transition closes, nothing ensures that the system will remain in its ground state. In conclusion, having a finite gap during a transition renders the transition topologically trivial.

Considering Eq. (1.22), in order for the energy gap to close we need to have E = 0. This can only happen

for the values k = 0, µ = 2t and k = ^π , µ = −2t. According to our discussion in the previous paragraph,

these values define topological transition points. For |µ| < 2t we are in the topological phase and for

|µ| > 2t we are in the trivial phase. We also have to stress here that in order to even be able to talk about a phase transition and a gap closing we need to actually have a gap, so we need to have ∆ 6= 0. Now, we want to solve the eigenvalue problem H _BdG Ψ = EΨ but before going there, we will implement certain modifications to H _BdG . For a ' 0 → ka ' 0 we can Taylor expand sin ka and cos ka and write H _BdG in a more familiar form (going this way to the continuous limit [44]). We can also rotate our system back so that the superconducting phase appears explicitly again. We have:

H _BdG =

 ε _p ∆ ⁰ p

∆ ^0∗ p −ε p



, (1.23)

where we have now named ε _p = _2m ^p

− µ ⁰ , µ ⁰ = µ + 2t, m ^∗ = ^¯h

2ta

, ∆ ⁰ = ^2∆a _¯h e ^iϕ and have used p = ¯hk.

We note that defined this way, ∆ ⁰ has units of velocity and that now the topological and the trivial regime correspond to µ ⁰ > 0 and µ ⁰ < 0 respectively . In the next Chapters we drop the star index from the mass and the prime index from ∆ and µ, since it will be the continuum H _BdG version (1.23) we will use and there will be no instance of confusion.

Diagonalizing (1.23) we obtain the energy momentum relation E = ±

q

ε ² _p + p ² ∆ ² ₀ . (1.24)

From dE/d p = 0 we find that the energy-momentum relation has always a critical point for p = 0 corresponding to energy E = |µ|. It is only for the case µ > m∆ ² ₀ that the energy develops a “dou- ble well” form [46]. In this case E = |µ| is a local maximum and we also have two energy minima E ₁ = ∆ 0

√ 2m(µ − ∆ ² ₀ m/2) ^1/2 for p = ± √

2m(µ − m∆ ² ₀ ) ^1/2 . In the case µ ≤ m∆ ² ₀ , E = |µ| is the only minimum of energy. In order to visualize this analysis, we plot the energy-momentum relation for the different values of µ in Fig. 1.6. The striking difference from the s-wave case is that here we have a gap closing for µ = 0, a point that marks a topological transition.

There is a way to clearly show that the µ < 0 and µ > 0 regimes are the non-topological and the topological respectively [44]. If we consider the normal metal (∆ = 0) in the µ < 0 regime we are in the gapped insulating phase. Introducing superconductivity (∆ 6= 0) the system passes to a gapped quasi-particle state and the two states are connected adiabatically. In contrast, if we consider the normal metal in the µ > 0 regime we are in a gapless metallic state. Introducing superconductivity the system passes to a gapped quasi-particle state. We cannot have an adiabatic system transition from this last state to the trivial insulating phase (µ < 0, ∆ = 0) without having a gap closing. It is thus the µ > 0 case which is topologically non-trivial, as it is separated from the topologically trivial insulating phase. In the above analysis we have silently considered that the superconductor for µ < 0 is topologically trivial. This can be seen considering the Majorana number [21, 39] which is a topological invariant defined by M = (−1) ⁿ , where n is the number of pairs of Fermi points for ∆ = 0. A topologically trivial system is defined as one having n = even, (n = 0 for 1.7(a), while a topologically non-trivial system as one having n = odd, (n = 1 for 1.7(b)). Thus, the µ < 0 case is the topologically trivial one, topologically different to the µ > 0 case as they are separated by a gap closing, as we have explained above.

1.5 In this Thesis

After having introduced odd-ω superconducting pairing and having analyzed the topological properties

of the Kitaev model in Chapter 1, we are now ready to study how these two concepts are related. In

E E

E

p

p μ

μ

>m|Δ|

μ

<m|Δ|

(a) _{E E}

p

μ

<0

|μ

|

(b)

E

p

μ

μ

=0

Figure 1.6: Energy-momentum relation of the Kitaev model in the continuum limit. (a) Topological regime (p ₁ = √ 2m(µ − m∆ ² ₀ ) ^1/2 , E ₁ = ∆ 0

√ 2m(µ − ∆ ² ₀ m/2) ^1/2 ). (b) Topological phase transition on the left and non-topological regime on the right.

We notice that the gap closes at µ = 0

E E

E

Δ

p p

e

h

qp

μ μ

(b)

μ>0 Topological

h e h

E E

Normal metal Superconductor

Δ

p p

e

h

|μ| |μ| qp

μ<0 (a)

Trivial

Figure 1.7: The passing from the normal to the superconducting state in the trivial (a) and the topological (b) regimes. In order

to have a transition from the (µ < 0) insulating phase ((a), left) to the µ > 0 superconducting phase ((b), right), we need to have

a gap closing.

Chapter 2 we introduce the Green’s functions (GF) formalism and describe how information about the superconducting pairing can be extracted from the anomalous GF’s elements. In Chapter 3 we consider a Normal metal - Superconductor (NS) junction. After having obtained the eigenfunctions of H _BdG in the normal metal and the superconductor, we construct the scattering states and the GFs in these two regions.

Finally we analyze the anomalous GF’s elements. We go on with the same analysis considering a short (vanishing normal region length L _N ) SNS (S’S) junction in Chapter 4 and an SNS junction with finite L _N in Chapter 5. Our main results are summarized in Chapter 6. The steps we follow in our analysis are depicted in Fig. 1.8

BdG equation

Construction of scattering states

Construction of Green's functions

Off-diagonal (anomalous) elements

Matching conditions Boundary

conditons

Normal metal Superconductor

Superconductor Normal metal

Pairing functions

Figure 1.8: Description of the course followed in Chapters 3, 4 and 5.

Chapter 2

The Green’s functions formalism

In this part we follow [47] for the one-dimensional quantum wire example in Section 2.1. The tools we will use to analyze the superconducting pairing in the junctions fall under the Green’s function (GF) formalism [47]. In general, the GF of a system is defined as G = [E − H] ⁻¹ , where E is the energy of the system and H is its Hamiltonian. Defined this way, the GF must satisfy the equation of motion (EOM) [47]:

E − HG(r, r ⁰ , E) = δ (r − r ⁰ ), (2.1)

where r stands for the postion and δ (r − r ⁰ ) is the Dirac δ function in the three-dimensional space. The above relation is similar to the Schrödinger equation E − HΨ = 0 and we can interpret the GF as the wavefunction of a system at point r as a result of the transmission of a unit excitation (represented by the δ function) at point r ⁰ [47]. The GF is thus a propagator connecting the points r and r ⁰ .

2.1 Green’s functions for a one-dimensional wire

In order to illuminate the GF’s physical meaning, we consider a one-dimensional quantum wire with Hamiltonian H = − _2m ^¯h

^∂

∂ x

+ V . Following [47], we integrate the EOM (2.1) around x = x ⁰ , and obtain the matching conditions:

G(x > x ⁰ ) 

x=x

=G(x < x ⁰ ) 

x=x

, h ∂ G(x > x ⁰ )

∂ x i

x=x

− h ∂ G(x < x ⁰ )

∂ x i

x=x

= 2m

¯h ² . (2.2)

Solving the EOM and using the matching conditions we obtain two solutions:

G ^r (x, x ⁰ , E) = − i

¯hv exp ik|x − x ⁰ |, G ^a (x, x ⁰ , E) = + i

¯hv exp − ik|x − x ⁰ |, (2.3)

where v = ¯hk/m and k = 2m(E − V ) ^1/2 /¯h. The first solution describes waves propagating away from

the excitation at x = x ⁰ , as for x > x ⁰ the wave propagates to the right and for x < x ⁰ the wave propagates to

the left. We refer to this solution as the retarded (r) GF. The second solution describes waves propagating

towards the excitation, as for x > x ⁰ the wave propagates to the left and for x < x ⁰ the wave propagates to

the right. We refer to this solution as the advanced (a) GF. The different natures of these two solutions are

x=x' x

(a) Retarded GF

x=x' x

(b) Advanced GF

Figure 2.1: (a) Retarded GF for the 1-D wire. The solutions propagate away from the source (vertical arrow). (b) Advanced GF for the same system. The solutions propagate towards the source. Figure adapted from [47].

2.2 Green’s functions from scattering states

Given the eigenvectors Ψ of a Bogoliubov-de Gennes Hamiltonian H _BdG (which can be calculated from the eigenvalue problem H _BdG Ψ = EΨ) we can construct the retarded GF’s of the system using the scattering states [48]. To be concrete, let’s consider a Normal metal - Superconductor junction with the interface at x = 0 (Fig. 2.2 ). In order to be consistent with the Kitaev model we disregard spin. A right-moving electron hitting the interface from the left side is the excitation (or source) in this case. The scattering outcomes are left-moving electrons or holes back in the normal region (x < 0) or right-moving quasi- electron or quasi-holes in the superconducting region (x > 0) and they are waves propagating away from the excitation at x = 0. We can then define the scattering states:

Ψ ^N ₁ = Ψ in1 + Ψ ^N _out1 and Ψ ^S ₁ = Ψ ^S _out1 , (2.4) which are depicted in Fig. 2.2. Similarly we can define the states Ψ 2 , Ψ 3 and Ψ 4 in the normal and in the superconducting region considering the scattering of a right-moving hole, a left-moving quasi-electron and a left-moving quasi-hole respectively. We stress that we can only study the GF separately either in the normal or in the superconducting region.

The exact form of the scattering states Ψ _i will be discussed in Chapter 3. The GF must be a linear combination of every possible product between states including incoming to the interfcace particles and states including only outgoing from the interface particles, considering thus every possible excitation together with any possible outcome [50]. The relevant combinations are then Ψ ₁ Ψ ₃ , Ψ ₁ Ψ ₄ , Ψ ₂ Ψ ₃ , Ψ ₂ Ψ ₄ according to Fig. 2.2. Since the scattering wave functions are Hilbert space elements, the proper way to form such bilinear quantities is the direct product Ψ _j Ψ ^† _i . We have checked that Ψ ^† _j = f Ψ _j

T , where f Ψ _j are the states we obtain using the eigenvectors of the conjugated Hamiltonian e H _BdG (p) = H _BdG ^∗ (−p) = H _BdG ^T (−p). Keeping the above in mind, we can then define the retarded GF [42, 49, 50] :

G ^r (x, x ⁰ , ω) =



 

 

 

 

a ₁ Ψ 1 (x) e Ψ ^T ₃ (x ⁰ ) + a ₂ Ψ 1 (x) e Ψ ^T ₄ (x ⁰ )

+a ₃ Ψ 2 (x) e Ψ ^T ₃ (x ⁰ ) + a ₄ Ψ 2 (x) e Ψ ^T ₄ (x ⁰ ), x > x ⁰ b ₁ Ψ 3 (x) e Ψ ^T ₁ (x ⁰ ) + b ₂ Ψ 4 (x) e Ψ ^T ₁ (x ⁰ )

+b ₃ Ψ 3 (x) e Ψ ^T ₂ (x ⁰ ) + b ₄ Ψ 4 (x) e Ψ ^T ₂ (x ⁰ ), x < x ⁰ ,

(2.5)

We prefer to define the GF using the frequency ω instead of the energy E. Defined this way, the GF (2.5) has the following form [42]:

G ^r (x, x ⁰ , ω) =  G ^r _ee (x, x ⁰ , ω) G ^r _eh (x, x ⁰ , ω) G ^r _he (x, x ⁰ , ω) G ^r _hh (x, x ⁰ , ω)



. (2.6)

N S

Ψ _in1 =Ψ _e

N S

Ψ _in2 =Ψ _h

Ψ _out2 ^S Ψ _out1 ^N Ψ _out1 ^S Ψ _out2 ^N

N S

Ψ _out3 ^N Ψ _out3 ^S

N S

Ψ _out4 ^N Ψ _out4 ^S Ψ _in3 =Ψ _qe Ψ _in4 =Ψ _qh

x=0 x=0

Ψ ₁ ^Ν Ψ ₁ ^S Ψ ₂ ^Ν Ψ ₂ ^S

Ψ ₃ ^Ν Ψ ₃ ^S Ψ ₄ ^Ν Ψ ₄ ^S

(1) (2)

(3) (4)

Figure 2.2: NS junction with the interface at x = 0. The scattering processes of a right-moving electron, a right-moving hole, a left-moving quasi-electron and a left-moving quasi-hole are depicted in (1), (2), (3) and (4) respectively. With Ψ ^N,S _i , i = 1, 4, we denote the sum of the outgoing and incoming waves that appear in each region.

From the diagonal, normal electron-electron and hole-hole components (G ^r _ee , G ^r _hh ), one can extract infor- mation about, e.g., the local density of states and the local spin density. We will focus on the off-diagonal (or anomalous) electron-hole and hole-electron components (G ^r _eh , G ^r _he ) which give information about the superconducting pairing correlations.

The coefficients a _i , b _i in (2.5) can be calculated by considering the matching conditions at x = x ⁰ . Integrat- ing the GF EOM:

[ω − H BdG ]G ^r (x, x ⁰ , ω) = δ (x − x ⁰ ) (2.7) around x = x ⁰ we obtain for the specific H _BdG of our system (see appendix B):

G ^r (x > x ⁰ ) 

x=x

=G ^r (x < x ⁰ ) 

x=x

, h ∂ G ^r (x > x ⁰ )

∂ x i

x=x

− h ∂ G ^r (x < x ⁰ )

∂ x i

x=x

= 2m

¯h ² σ _z . (2.8)

The above conditions constitute a system of eight equations with the coefficients a _i , b _i as unknowns.

The GF’s expressions for the normal and the superconducting region are different so we will study these regions separately.

Once the GF is fully determined, we calculate the pairing amplitudes which in this case are directly given by G ^r _eh . Then we decompose the anomalous GF into even and odd parity components:

f _E(O) ^r,P (x, x ⁰ , ω) = f ^r (x, x ⁰ , ω) ± f ^r (x ⁰ , x, ω)

2 , (2.9)

where we have adopted the notation G ^r _eh = f ^r . Finally we note here that we can retrieve the advanced GF

0 0 †

odd frequency components, the passing to negative frequency requires changing to the advanced GF [42]:

f _E ^r,ω _(O) (x, x ⁰ , ω) = f ^r (x, x ⁰ , ω) ± f ^a (x, x ⁰ , −ω)

2 . (2.10)

After having introduced the tools required to construct GFs from scattering states, we are now ready to

investigate the superconducting pairing for various junctions in the following chapters.

Chapter 3

Normal metal - Superconductor junction

In this Chapter we study the superconducting pairing correlations in a normal metal-p-wave superconduc- tor (NS) junction (Fig. 3.1). First, we solve the BdG equations in the normal metal and in the supercon- ductor. Next, we use the BdG eigenstates to construct the scattering states for the NS junction. Using the matching conditions at the interface we obtain the scattering coefficients. With the scattering states known, we move on to obtain the retarded Green’s functions (GFs) as we described in Chapter 2. Finally, we isolate the anomalous GF’s components and study their symmetry properties in various superconductor chemical potential (µ _p ) and frequency (ω) regimes.

N S,Δ

e

x=0

Figure 3.1: NS junction with the interface at x = 0

3.1 Construction of the scattering states

3.1.1 Solving the BdG eigensystem

As we have shown in Chapter 1, the Hamiltonian of a p-wave one-dimensional superconductor is given by:

H = ∑

p

1

2 Ψ ^† _p H _BdG Ψ _p , H _BdG =

 ε _p ∆ p

∆ ^∗ p −ε _p



, (3.1)

where ε _p = p ² /2m − µ is the kinetic term and Ψ p = (c _p , c ^† _−p ) ^T is the Nambu spinor [41]. One can

explicitly see the linear momentum dependence of the pairing potential, which is expected for a p-wave

superconductor. We will solve for the eigenvalues and the eigenvectors in the normal (∆ = 0) and the

superconducting (∆ 6= 0) regions going to position space where p = −i¯h _{∂ x} ^∂ .

3.1.1.a Normal region (∆ = 0)

At zero pairing potential the BdG equation using Eq. (3.1) reads:

− ^¯h _2m

^∂

− µ N 0 0 ^¯h _2m

^∂

+ µ N

!

ψ (x) = Eψ (x). (3.2)

We look for solutions in term of plane waves:

u(x) v(x)



= A exp ^ikx U _N V _N



, (3.3)

where A, U _N and V _N are to be determined. Inserting the solution (3.3) into (3.2) we get:

p

2m − µ N 0

0 − _2m ^p

+ µ N

! U _N V _N



= E U _N V _N



, (3.4)

where p = ¯hk. The energy eigenvalues are E = ±(p ² /2m − µ N ) (Fig. 3.2). Calculating the group velocity we find v _g = dE/d p = ±p/m. By definition, electrons (e) have a positive group velocity for positive momenta whereas holes (h) have a negative group velocity for positive momenta. We conclude that the E = p ² /2m − µ N energy eigenvalue refers to electrons and the E = −p ² /2m + µ N refers to holes. Solving for the momenta we get:

p = ± √ 2m p

µ _N + E = ±p _e , for electrons, p = ± √

2m p

µ _N − E = ±p _h , for holes. (3.5)

E

p

μ _N

-μ _N

p _F -p _F

μ _N >0

e

h

Figure 3.2: Energy-momentum relation for the normal region. The normal line corresponds to electrons and the dashed line to holes. p _F = √

2mµ N .

Since we are going to examine scattering processes and the energy-momentum relation is symmetric for negative and positive energies, we will fix the energy to E ≥ 0 and we will consider only the upper part of Fig. 3.2. In Fig. 3.3 it becomes clear that for electrons, positive group velocities correspond to positive momenta and negative group velocities correspond to negative momenta. So, a right moving e will have p > 0 and a left moving e will have p < 0. The exact opposite is true for holes. In order to find the

E

p μ _N

-p _F p _F

-p _e -p _h p _h p _e

Figure 3.3: Energy-momentum relation in the normal region for E ≥ 0. White circles correspond to e and black circles to holes.

The arrows point at the direction of the group velocity.

eigenvectors that correspond to the eigenvalues E = ±ε = ±(p ² /2m − µ N ) we need to solve Eq. (3.4) with the now known energy eigenvalues. Using also the normalization condition |U 0 | ² + |V ₀ | ² = 1, we obtain:

U _N V _N



= 1 0



= u _e , for electrons,

U _N V _N



= 0 1



= u _h , for holes.

(3.6)

Taking Fig. 3.3 into account as well, we have:

ψ e+ = Au _e e ^ik

^x , right moving e, ψ e− = Bu _e e ^−ik

^x , left moving e, ψ _h− = Cu _h e ^−ik

^x , right moving h, ψ _h+ = Du _h e ^ik

^x , left moving h,

(3.7)

where we have included the normalization constants A, B, C, D.

3.1.1.b Superconducting region (∆ 6= 0) The BdG equations given by Eq. (3.1) read:

− ^¯h _2m

^∂

− µ p ∆ 0 e ^iϕ (−i¯h∂ x )

∆ ₀ e ^−iϕ (−i¯h∂ x ) ^¯h _2m

^∂

+ µ p

!

ψ (x) = Eψ (x), (3.8)

where we have considered a different chemical potential for the superconducting region (µ _p ). Now we look for solutions of the form:

u(x) v(x)



= ae ^ikx  U _S e ^iϕ/2 V _S e ^−iϕ/2



, (3.9)

and we end up with the eigenvalue problem

p

2m − µ p ∆ ₀ p

∆ ₀ p − _2m ^p

+ µ p

! U _S V _S



= E U _S V _S



. (3.10)

The energy eigenvalues we find are:

E = ± q

ε ² _p + p ² ∆ ² ₀ , (3.11)

where - similarly to the N case - ε _p = p ² /2m − µ p is the kinetic term. Since we have a scattering problem we will work with the positive energy eigenvalues E = +(ε ² _p + p ² ∆ ² ₀ ) ^1/2 . Solving the energy-momentum relation for the momenta we obtain

p = ± √ 2m

r

µ p − m∆ ² ₀ ± q

E ² − E ₁ ² = ±p _qe,qh , (3.12)

where E 1 = ∆ 0 [2m(µ p − ∆ ² ₀ m/2)] ^1/2 . In order to figure out which momenta correspond to electrons and which to holes we have to calculate the group velocity v _g = dE/d p = ±p(E ² − E ₁ ² ) ^1/2 /(Em). We see that, if we want to have a positive group velocity for a positive momentum and a negative group velocity for a negative momentum we need to choose the momentum solution with +(E ² − E ₁ ² ) ^1/2 from Eq. (3.12). So it is this solution that corresponds to electrons. The exact opposite is true for holes, which explains why we chose to name the momenta solutions as in Eq. (3.12). Schematically, this can be seen in Fig. 3.4.

The expressions for the momenta for different values of µ _p and E are depicted in Table 3.1. From the eigenvalue problem (3.10) we obtain two equations

ε p U _S + ∆ 0 pV _S = EU _S ,

∆ ₀ pU _S − ε _p V _S = EV _S . (3.13)

The above two equations combined give the energy-momentum relation. Solving the second one for U _S we obtain U _S = ^E+ε

∆

p V _S . Using the normalization condition |U _S | ² + |V _S | ² = 1 we obtain:

V _S =

r E − ε p

2E , U _S = E + ε p

∆ ₀ p

r E − ε p

2E . (3.14)

µ _p E p _qe , p _qh

µ _p ≥ m∆ ² ₀

E ≥ |µ p | p _qe = √ 2m 

µ _p − m∆ ² ₀ + q

E ² − E ₁ ²  1/2

p _qh = i √ 2m 

− µ p + m∆ ² ₀ + q

E ² − E ₁ ²  1/2

E ₁ ≤ E ≤ |µ p | p _qe,qh = √ 2m



µ p − m∆ ² ₀ ± q

E ² − E ₁ ²  1/2

0 ≤ E ≤ E 1 p _qe,qh = √ 2m



µ p − m∆ ² ₀ ± i q

E ₁ ² − E ²  1/2

m∆

2

2 ≤ µ p ≤ m∆ ² ₀

E ≥ |µ p | p _qe = √ 2m 

µ _p − m∆ ² ₀ + q

E ² − E ₁ ²  1/2

p _qh = i √ 2m 

− µ p + m∆ ² ₀ + q

E ² − E ₁ ²  1/2

E ₁ ≤ E ≤ |µ p | p _qe,qh = i √ 2m

 − µ p + m∆ ² ₀ ∓ q

E ² − E ₁ ²  1/2

0 ≤ E ≤ E 1 p _qe,qh = √ 2m



µ p − m∆ ² ₀ ± i q

E ₁ ² − E ²  1/2

µ _p ≤ ^m∆ ₂

E ≥ |µ p | p _qe = √ 2m 

µ _p − m∆ ² ₀ + q

E ² − E ₁ ²  1/2

p _qh = i √ 2m 

− µ p + m∆ ² ₀ + q

E ² − E ₁ ²  1/2

0 ≤ E ≤ |µ p | p _qe,qh = i √ 2m

 − µ p + m∆ ² ₀ ∓ q

E ² − E ₁ ²  1/2

Table 3.1: Expressions for the momenta p _qe , p _qh for different values of E, µ p . E ₁ = √ 2m∆ 0

q

µ − ∆ ² ₀ m/2.

|μ _P |

E ₁ p ₁ p _qe p _qh

-p _qh -p _qe

E ₁

- p

E _scat E

p

Figure 3.4: Energy-momentum relation in the superconducting region. For a given energy E ≥ E ₁ we have quasiparticle

excitations which are denoted by the grey circles. The arrows are at the direction of the group velocity.

Depending on the quasi-particle eigenstate we want to describe (qe, qh, right or left moving), we have to substitute different momenta in expressions (3.14) and thus we have different expressions for U _S and V _S . Noticing also that V _S is contained in U _S we adopt a different notation:

U _S = E + ε _qe,qh

±∆ 0 p _qe,qh ·

r E − ε _qe,qh

2E = ±U _qe,qh ·V _qe,qh , V _S =

r E − ε _qe,qh

2E = V _qe,qh .

(3.15)

With the help of Fig. 3.4 and Eq. (3.9) and (3.15) we can now write down the wavefunctions of the quasiparticles in the superconducting region:

ψ _qe ⁺ =a e ^iϕ/2 U _qe V _qe e ^−iϕ/2 V _qe



e ^ik

^x = a · u ⁺ _qe · e ^ik

^x , right-moving qe (p > 0),

ψ _qe ⁻ =b

 −e ^iϕ/2 U _qe V _qe e ^−iϕ/2 V _qe



e ^−ik

^x = b · u ⁻ _qe · e ^−ik

^x , left-moving qe (p < 0),

ψ _qh ⁻ =c

 −e ^iϕ/2 U _qh V _qh e ^−iϕ/2 V _qh



e ^−ik

^x = c · u ⁻ _qh · e ^−ik

^x , right-moving qh (p < 0),

ψ _qh ⁺ =d e ^iϕ/2 U _qh V _qh e ^−iϕ/2 V _qh



e ^ik

^x = d · u ⁺ _qh · e ^ik

^x , left-moving qh (p > 0).

(3.16)

We are now ready to construct the scattering states (scattering of right-moving electron, right-moving hole, left-moving quasi-electron, left-moving quasi-hole, Fig. 3.6), using the BTK (Blonder, Tinkham and Klapwijk) formalism [48].

3.1.2 The scattering states

3.1.2.a Scattering of right moving electron

An electron moving towards the interface in the normal region must be a right moving one. This electron can be scattered in four different ways. It can either be reflected and remain in the normal region as a left moving electron (normal reflection) or a left moving hole (Andreev reflection [51]) or be transmitted to the superconducting region as a right moving quasi-electron or a right moving quasi-hole. In order to account for all of these possibilities the scattering of a right moving electron is properly described by the following scattering wave function:

Ψ _scat,e =



 

 



 

 



Au _e e ^ik

^x

| {z }

right moving e

+r _ee Bu _e e ^−ik

^x

| {z }

left moving e

+r _eh Du _h e ^ik

^x

| {z }

left moving h

, x < 0 t _eqe au ⁺ _qe e ^ik

^x

| {z }

right moving qe

+t _eqh cu ⁻ _qh e ^−ik

^x

| {z }

right moving qh

, x > 0 (3.17)

where the r _xx and t _xx are the reflection and transmission amplitudes from right moving electron to any other

state. We note that, for example, in this notation t eqe means a transmission from a right moving electron to

a right moving quasi-electron. The scattering process (3.17) is depicted in Fig. 3.5.