Transport in hybrid superconducting systems with chiral majorana modes

BACHELOR’S THESIS

TRANSPORT IN HYBRID SUPERCONDUCTING SYSTEMS WITH CHIRAL MAJORANA MODES

Antònia Verdera Garau

Degree in Physics Faculty of Science

Academic Year 2019-20

TRANSPORT IN HYBRID SUPERCONDUCTING SYSTEMS WITH CHIRAL MAJORANA MODES

Antònia Verdera Garau

Bachelor’s Thesis Faculty of Science

University of the Balearic Islands

Academic Year 2019-20

Key words:

Topology, transport, superconductor, junction, reflection, transmission, modes

Thesis Supervisor’s Name: Llorenç Serra Crespí

The University is hereby authorized to include this project in its institutional repository for its open consultation and online dissemination, for academic and research purposes only.

Author

Supervisor Yes No Yes No

☒ ☐ ☒ ☐

Contents

1 Introduction 1

1.1 Theoretical introduction . . . 1

1.1.1 Quantum-anomalous Hall effect . . . 2

1.1.2 Majorana modes . . . 2

1.1.3 Klein effect . . . 3

1.2 Background . . . 4

1.2.1 Motivation: NS junction . . . 5

2 Theory and model 9 2.1 Hamiltonian of the system . . . 9

2.1.1 Symmetries of the system . . . 11

2.2 Method . . . 11

2.2.1 Conductance calculation . . . 13

3 Results and analysis 15 3.1 Phase diagram . . . 16

3.2 Conductance of NN’S juction . . . 17

3.2.1 Variation with Zeeman parameter . . . 18

3.2.2 Variation with energy . . . 20

3.3 Orbital effects . . . 21

4 Conclusions 25

v

Chapter 1

Introduction

Topological transport systems are characterized by the presence of current-carrying states along the boundaries and edges of the material. This work addresses a theoretical descrip- tion of topological transport in a class of materials, quantum-anomalous Hall insulators (QAHI) with induced superconductivity. A QAHI is a hybrid superconducting system that presents chiral Majorana modes.

1.1 Theoretical introduction

Before addressing the objectives and results achieved in this work, this section provides a brief explanation about topological systems and their relevance in current physics. Con- densed matter physics is interested in the classification of different states of matter, and that is the reason why there are topologies. Topology is a branch of mathematics that studies the spatial properties with continuous deformations of space. This allocation method can be understood by considering geometrical objects, two objects belong to the same topological class if they can be deformed into each other without creating or destroying any holes. The two main properties of the topologies are connectivity and compactness. Here small details can be forgotten, only fundamental differences between the shapes are important.

This bachelor thesis is about quantum-anomalous Hall insulators with induced super- conductivity, materials that could have many implications for quantum computing and are used in many recent studies. To understand what is a QAHI we need to describe the quantum-anomalous Hall effect happening inside them and the states that are possibly formed within them, Majorana modes. Also, we are going to briefly describe the Klein effect, an effect related to the transmission modes, since a similar phenomenon will occur in the system presented in our study.

1

1.1.1 Quantum-anomalous Hall effect

The quantum Hall effect is a quantum version of the classical Hall effect. It occurs in two-dimensional electron systems with electrons confined in a semiconductor, subjected to low temperatures and intense magnetic fields, in which the electrical conductivity σ takes quantized values: σ=νe²/h(whereeis the fundamental charge of the electron and h is Planck’s constant). The factorν is known as the filling factor and can take integer or fractional values, depending on which, we refer to it as the integer or fractional quantum Hall effect, respectively.

The discovery of the quantum Hall effect has shown that topology is an essential ele- ment for the quantum description of Condensed Matter systems. This effect was the first to exhibit that topological properties were essential for the description of some character- istics intrinsic to the material. It causes the electrical conductance of a material under special conditions to be quantized, besides, when we have this effect the current is only carried along the edge of the material sample, not through the bulk. The number of edge state channels was linked to the Chern number, a topological invariant of the occupied bands. When we have a specific topology we have an associated Chern number to it.

Actually, in the system of our study, the contributing effect is the quantum-anomalous Hall effect. While the Hall effect requires the presence of a perpendicular magnetic field to generate a finite Hall voltage, the anomalous Hall effect generates this voltage without a magnetic field due to the combination of the material magnetization and spin-orbit coupling. The anomalous Hall conductances are quantized to integer multiples of the quantum conductance, similarly to the ”normal” quantum Hall effect in this sense. But the integer here is the Chern number, mentioned before, which arises from the topological properties of the material in its band structure. These effects are seen in our system, called QAHI or Chern insulators.

In the last decade, we have seen the discovery of topological insulators [1], materials that are insulating although they present conductive surface states. Moreover, in these topological insulators, the topological protection of the surface states makes them robust against disorder and impurities. Therefore, topological insulators are very relevant for laboratory and commercial applications nowadays.

1.1.2 Majorana modes

Superconductivity is the intrinsic capacity of certain materials to drive an electric current with null resistance in certain conditions. In this work, we discuss the transport properties in systems having low-energy topological excitations, with currents attached to edges or interfaces, and hybrid superconductivity induced by the proximity of a material with a superconductor. The term hybrid superconductivity is referred to the possibility that one of the layers of the material has superconductive properties and the other layer has normal features. The hybrid superconductivity allows the existence of non-trivial topological phases with unpaired Majorana modes.

1.1. THEORETICAL INTRODUCTION 3 Majorana particles or Majorana fermions, proposed by Ettore Majorana in 1937 [2], are particles that are their own antiparticles. The existence of elementary Majorana fermions is not entirely clear, but it does seem clear that they can exist as low energy excitations (called quasiparticles) in certain systems. A Majorana fermion would then be a quasiparticle that is its anti-quasiparticle (superposition of equal parts of quasiparticle and anti-quasiparticle). The interest in these strange fermions lies in their exotic statistical physics. A normal fermion can be expressed as an overlap of two Majorana fermions. This property, together with the robustness of the topological states, can have applications in quantum computing. Majorana modes are composite quantum mechanical states, with distinct and perhaps even more intriguing properties. The fundamental aspects of the Majorana fermion qubits is fault-tolerant quantum computation. The Majorana zero mode is a charge-neutral bound state that exists strictly at zero energy.

1.1.3 Klein effect

The physicist Oskar Klein obtained a shocking result in 1929 [3] by applying Dirac’s equation to the familiar problem of electron scattering from a potential barrier. In non- relativistic quantum mechanics, the electron tunnel into a barrier is observed, with expo- nential damping. Nevertheless, Klein’s result showed that if the potential is of the order of the electron mass,V ∼mc², the barrier is almost transparent. Besides, as the potential approaches infinity, the reflection decreases, and the electron is always transmitted.

We can explain this problem in more detail, assuming that an electron with energy E is classically confined in the regionx <0 under a potential step of finite heightV > E (we suppose that initially V −E < mc²). The quantum description given by Dirac’s equation shows that an evanescent wave exists in space x > 0, penetrating with a slight thickness into the classically forbidden region, with an exponentially decreasing amplitude. But, the relativistic solution shows the opposite. When V > E +mc² the amplitude of the wave for x > 0 is constant. The wave is no longer evanescent, the electron is no longer confined to the left. More interesting still, the amplitude of the reflected wave is greater than the amplitude of the incident wave. For this reason, we can say that we have a perfect transmission.

This paradox is resolved by abandoning the description of a single particle in the problem. The correct interpretation comes from quantum field theory when V exceeds E+mc² and electron-positron pairs are produced.

As we can see in Ref. [4] the Klein paradox can be shown in a conceptually simple experiment using electrostatic barriers in graphene. Due to the chiral nature of their quasi-particles, the quantum tunnel in these materials becomes almost relativistic. Mass- less Dirac fermions in (single-layer) graphene allow a closer performance of the Klein experiment due to the their crossing spectrum. While massive chiral fermions in bilayer graphene is an illuminating system that allows us to understand the basic physics involved, thanks to their band diagram.

1.2 Background

To have a better understanding of the study developed in this thesis, other works of the current state of art have been considered. In theoretical physics we are interested in describing the electronic properties on the two-dimensional boundary of these materials, so we are going to focus on our QAHI system and its conductance.

In the recent experiment of Ref. [5] it was argued that conductance 0.5e²/h of a topo- logical NSN double junction when the central part hosts a single chiral Majorana mode is a clear evidence of the presence of such a Majorana mode. Particularly the authors of this article investigated the conductance of 2D strips having normal contacts and a central superconducting region; i.e., a normal-superconducting-normal (NSN) double junction in different topological phases of each region.

Figure 1.1: Sketch of the model of a strip (orange) in contact with a superconductor bar (blue). The presence of chiral edge modes in the different regions is indicated by the lines with arrows. The phase transitions with the corresponding conductances for varying magnetic fields is indicated by the red curve of the upper plot. In particular, in the second panel from the left a single chiral Majorana mode is present in the intermediate region and the corresponding conductance is 0.5e²/h. Figure reproduced from [6].

Figure 1.1 shows the device of this experiment [5]. We can see that in the presence of a single Majorana mode an electron incident from the left is half transmitted and half reflected and this leads to the halved conductance. This interpretation relies on the full quantum-coherence of the nanostructure, especially with the Majorana mode keeping its wave-function in the whole central part. When there are two Majorana modes, which are equivalent to a Fermion mode, the incident electron is fully transmitted and the conductance takes the value e²/h, the reflection process does not take place. As we have indicated in the diagram 1.1 in [5] the variation of the magnetic fieldB is what determined the topology to which the NSN system is. The topology of a QAHI system can change by varying parameters such as the external magnetic field or the potential (energy).

1.2. BACKGROUND 5 Although the mentioned article seemed to have great impact at the time, it has been thoroughly commented on and analyzed. In Refs. [7] and [8] it is discussed how this quantization of conductance could occur in other cases, not only in the case of the presence of a single Majorana mode. These articles argue that when transport coherence is lost in the central part of the system the above scenario of a 0.5 conductance is to be expected with more generality. That is, the halved conductance can also be obtained in the presence of pairs of Majorana modes (or Fermion modes) and this cannot be interpreted as proof of the presence of single Majorana mode.

Quite recently Ref. [9] measurements of the NSN conductance have indeed found 0.5e²/h in general, independently on the state, in many devices under different condi- tions. This experiment provides a more complete understanding of the superconducting proximity effect observed in QHAI devices and shows that the half-quantized conductance plateau is unlikely to be induced by chiral Majorana fermions in samples with a highly transparent interface. This conclusion refers to the rather large, mm-sized, junctions of the Ref. [7] where quantum coherence is more difficult to be achieved than in smaller devices.

1.2.1 Motivation: NS junction

Based on the aforementioned articles we want to analyze what happens in our system when there is a quantum disconnect within the superconducting material itself. To make things easier, we could consider that we go from having an NSN union to having NS-SN.

So we are going to focus on the discussion of the NS junction. We want to show that if we take into account two systems NS disconnected from each other, we will always obtain a conductance quantification of 0.5e²/hof the complete system NSN. This will be correct if the conductance in NS is always equal to unity in any case.

The transport phenomena in a point contact junction between a superconductor and normal metal are dominated by Andreev and Normal reflection denoted as R_eh and R_ee respectively. The normal reflection is a process that transfers an electron into another electron. But the Andreev reflection is a process that transforms an electron into a hole that retraces the path of an incoming particle. This reflection process is detailed in Ref. [10] for the specific system that we are considering. In a qualitative sense, the Andreev process causes us to have two electrons instead of one, and the Normal process causes them to cancel out.

The theoretical description of the conductance of this system is given by Blonder, Tinkham, and Klapwijk, in a theory that is usually referred to as BTK [11]. In the NS junction, the conductance is determined from the number of incident electron channels N_e, the electron-electron reflection R_ee, and the electron-hole reflectionR_eh.

In Fig.1.2 we have detailed the transmission probabilities of the propagating modes.

We have indicated neutral states in the hybrid superconducting part, but not electric currents since we know that single Majorana modes are not current-carrying. By contrast, the Fermionic states of the N part have charge and they carry electric current. When

the current flowing through this device passes to the superconducting material, it is transformed into a supercurrent, and this is the reason why it seems that we do not have continuity in the current. What we indicate in this diagram and in all the drawings that we are going to present in this section is the probability of quasi-particle propagation. We indicate in green and pink the incident electrons and reflected holes, respectively. Besides, the probability of propagation of these is indicated withγ₁andγ₂for the Majorana modes.

In the sketch of the left, Fig.1.2a, we have drawn an NS junction, where the supercon- ductor part has an associated Chern number equal to 1. Thus, here the transmission is by a single Majorana mode. Due to this fact, we see that the probability of transmission is halved and the rest is reflected, both as Andreev and Normal reflection. In the right sketch Fig.1.2b, the superconductor part has associated a Chern number equal to 2. This means that the transmission is by perfectly coupled fermionic states, and we see that the transmission has no reflection probability in this case.

a) b)

Figure 1.2: NS junction in a DC bias. The conductance ise²/hfor both cases: (a) A single Majorana mode and (b) Two Majorana modes.

To clarify the physics that occur in devices used in Ref. [5], we have sketched a diagram in Fig. 1.3. In this diagram, we use the transmission probabilities that we have explained in Fig.1.2. Using a known formula for the conductance we see how this takes different values for each of the presented topologies because now we are not considering quantum disconnected systems. In other words, quantum coherence is preserved in all parts of the device.

The left panel is for the case of a single Majorana mode in the central part, Fig.1.3a, while the right panel is for the case of two Majorana modes. Analyzing the left panel we can say that now when moving from normal material with an electron incident to super- conducting material we have Normal and Andreev reflection, but then when moving from superconducting material to normal the process seems to be inverted and that is why we arrive at the result of half-conductance. Notice that two Majorana modes are equivalent to a single Fermion mode and, therefore, the incident Fermion is fully transmitted in the right panel, as shown in Fig.1.3b. However, when comparing with the left panel we do not have reflections in the normals materials, so we directly have the conductance quantified

1.2. BACKGROUND 7 to the unity. Therefore, we conclude that the quantum coherent results for one and two Majorana are different as we already mentioned above.

a) b)

Figure 1.3: NSN double junction in a DC bias. Left panel (a) is for the case of a single Majorana mode in the central part, while right panel (b) is for the case of two Majorana modes.

If we consider that the system is disconnected, Fig.1.4, the diagrams completely change and we see that the dependence on topology is no longer present in the measure of con- ductance. We can calculate the conductance as the serial combination of two independent conductances:

G= G_aG_b

G_a+G_b = 0.5e²

h , (1.1)

where Ga and Gb are the conductances associated with NS junctions and are equal to 1e²/h in any topology case. So we can see in diagram 1.4 it will not be necessary to consider a specific topology for our system for obtaining a conductance equal to 0.5e²/h, thus demonstrating that, as already mentioned in Ref. [9], the quantization of conductance is not a clear proof of the existence of Majorana chiral modes.

In all the presented diagrams we have the hybrid system referenced to the ground potential since this configuration is the one used in devices that have been characterized experimentally. In these devices, the potential is provided from a part of normal material to a superconducting part. The voltage differences are measured to have a value of the energy that passes through the system. There is a potential drop V /2 in each junction and the central part is grounded but no current is flowing from the ground due to the compensation of currents from the two junctions.

In the showed disconnected diagram, 1.4, we can see that the half conductance does not allow us to distinguish the cases of a single Majorana mode and two Majorana modes. The two junctions contribute a unit of conductance and this leads to a 0.5 total conductance due to the series combination. That is why our discussion will be focused on the NS junction. We will explore how other differences can emerge in single NS junctions with

different topology. In more detail, we will investigate the sensitivity to an interface barrier and the presence of a perpendicular magnetic field of the NS junction hosting one and two Majorana modes.

Figure 1.4: Conductance of an NSN system described as two independent NS junctions. Upper diagram (a) is the case of single Majorana mode and the lower one (b) is associated to two Majorana modes.

Chapter 2

Theory and model

Having presented the basics of the NS junction physics, in order to have a deeper knowl- edge of our system, we investigated the conductance of 2D strips having a normal part and a superconducting region, a normal-superconducting (NS) junction for different topo- logical phases of each region. For this reason, in this chapter we will focus on our specific model, emphasizing first what is the hamiltonian that defines the QAHI materials and then explaining the resolution method we have used.

2.1 Hamiltonian of the system

We are going to characterize our regions with the following Hamiltonian, which is de- scribed in Refs. [12] and [13]. We use the model of a double-layer 2D hybrid QAHI strip, to which it corresponds the following Hamiltonian:

H =

m₀+m₁ p²_x+p²_y τ_zλ_x+ ∆_Zσ_z+ ∆_pτ_x + ∆_mτ_xλ_z− α

~ (p_xσ_y−p_yσ_x) τ_zλ_z , (2.1) in this hamiltonian we are using vectors of Pauli matrices for variables representing usual spin by ~σ, isospin (charge) by ~τ and pseudospin (layer) by ~λ, in a representation called Nambu spinorial representation, the field operators are grouped and the Hamiltonian becomes a multiple-block matrix.

The parameters ∆_p and ∆_m in Eq. 2.1 are indicators of the superconductivity on the two layers of the material, given by the pairing gap energies ∆_1,2. Actually, those two parameters are defined by

∆_p,m= ∆1±∆2

2 . (2.2)

where each of the ∆_1,2 is associated with a layer. In the normal region, we obviously have

∆1,2 = 0, while they take constant values in the hybrid superconducting region, in our case, ∆₁ = 1 meV and ∆₂ = 0.1 meV. By setting these two parameters at different values,

9

it is possible to obtain different topologies from our superconducting system. Moreover,

∆_Z represents the Zeeman term due to the magnetization of the material and α the spin- orbit coupling. Parametersm₀ and m₁ describe the coupling between two layers, in a low order expansion.

It is important to assume realistic values for the parameters that appear in the Hamil- tonian. The strip confinement along the lateral coordinate (y) is obtained by assuming that m₀ takes a large value for y 6∈[−L_y/2, L_y/2], effectively forcing the wave functions to vanish on the lateral edges. In our calculations we take α = 0.26 meV, m0 = 1 meV and m₁ = 10⁻³m⁻¹_U (were m_U = 7.6×10⁻⁵m_e is our mass unit). The energy and the length scales are meV and µm.

Introducing this first Hamiltonian, the orbital effects of the magnetic field are not taken into account. If we want to consider them, we simply must make the minimal substitutionp_x →p_x−~y/l_z²τ_z, wherel_z is the magnetic length defined asl_z =p

~c/eB_z. When we consider a magnetization in a particular material we have two effects, the already considered Zeeman effect (quantified with parameter ∆_Z) and the orbital effects (quantified with a parameter called magnetic length l_z). These two depend on each other because they both come from the magnetic field. It is interesting to discuss what the results are with and without orbital effects to understand the physics of each term. To consider orbital effects we should add the following terms:

H_orb= α

l²_z yσ_yλ_z+m₁ ~²

l⁴_z y²τ_z−2~² l²_z yp_x

λ_x , (2.3)

the first contribution in Equation 2.3 originates in the spin-orbit-like α term, and the latter ones originate in the m₁ quadratic-in-momentum term and they are familiar from the kinetic contributions of the standard quantum Hall effect.

In these materials, the Zeeman-type magnetic effects and the orbital ones are not the same because there is an intrinsic magnetization of the material. Meaning, the Zeeman contains the intrinsic magnetization and the magnetic field, while the orbital effect only contains the magnetic field. Magnetic interactions are usually considered to be dominated by intrinsic magnetization and the orbital effects, are neglected. A large g factor is also indicating small orbital effects.

From a theoretical point of view, we can assume that the two parameters can be adjusted independently and study the two effects separately (Zeeman and orbital). In reality, one will usually depend on the other. We will consider an independent study of the two effects to favor a better understanding. Therefore, throughout the work, we will consider as Hamiltonian the one presented in the first place 2.1 without the orbital effects.

It will only be in the last section of the results where we will take into account this last orbital term and therefore use the complete Hamiltonian: H⁰ =H+Horb, where Horbcan be discussed as a perturbation of the original Hamiltionan.

2.2. METHOD 11

2.1.1 Symmetries of the system

Symmetries in quantum mechanics describe features of spacetime and particles that re- main unchanged under some transformation. To see what properties our system fulfills, we will briefly describe the three symmetries it meets.

Particle-hole symmetry indicates that the spectrum of eigenstates of the Hamiltonian (2.1) always appears in pairs at energiesE and−E, representing particle and antiparticle;

each one being the particle-hole conjugate of the other. This symmetry is associated with the antiunitary operator Q = τ_yσ_yK, where K stands for complex conjugation. The particle-hole symmetry has a deep influence on the topology, and it plays a central role in superconducting systems.

Time reversal symmetry is described by another operator Θ = −iσ_yK. But it is not an exact symmetry in this case because it is broken by the Zeeman term as it includes a single vector that changes sign under time reversal.

Finally, there is another symmetry, ”Chiral” symmetry. The word ”chiral” is written in quotation marks as it is used here to refer to a general extra symmetry, not necessarily related to the chirality of the propagating edge states discussed above. With the particle- hole operator given by Q = τ_yσ_yK, the ”chiral” symmetry operator is C = τ_yσ_y. Since, Q =CK. The chiral symmetry operator C is a usual linear unitary operator that fulfills the self-inverse requirement C² = 1. This symmetry changes the topological properties of a system dramatically. When we do not have this symmetry, there is no degeneracy in eigenmodes, so we can have none or at most one zero-mode associated with null energy.

Those considerations can be applied in our case since chiral symmetry is not fulfilled, but we have to take into account that we have two systems superposed (bottom and top surfaces). Therefore, we can have zero, one, or two modes associated with null energy, so we can have single and coupled Majorana modes in the systems described by this Hamiltonian 2.1.

2.2 Method

In Ref. [14] a review of the application of the complex band structure is made to describe 1 or 2-dimensional states that appear near the edges or interfaces between different types of topological materials. Furthermore, it is provided the resolution method that we are using to solve the equations of our system. Unlike our case, in that article localized Majorana modes are always considered

In very basic terms, we solve the Bogoliubov–de Gennes equation of the homogeneous systems, and then we superpose the solutions. The confined condition on the junction edge is obtained by a set of grid equations connecting the normal and the superconductor sides. We are using wavenumber analysis which has a high spatial resolution at a low computational cost. These high resolutions allow us to calculate the local currents at the NS junction and, more importantly, to relate the characteristics of the current distribution

to the precise value of the conductance under different conditions that allow different topologies. The method is also used to calculate the conductance.

The first step is to find the homogeneous solution for this Hamiltonian, 2.1. We can calculate the topological phase transition using the band structure of an infinite system.

Thus, the solutions of the infinite 2D system with a given wavenumber k present the following structure:

Ψ^(k)(x, y, η_σ, η_τ) = X

sσ,sτ

Ψ^(k)_s

σsτ(y)e^ikxχ_sσ(η_σ)χ_sτ(η_τ). (2.4) The equation that gives the full band structure, where h(k) is a k-dependent effective 1D Hamiltonian, is now:

X

s⁰_σ,s⁰_τ

hs_σs_τ|h(k)|s⁰_σs⁰_τiΨ^(k)_s0

σs⁰_τ(y) =EΨ^(k)_s

σsτ(y). (2.5)

In Ref. [14] we can see that this problem has two analytic solutions, one in the regime of the longitudinal magnetic field and another in the regime of strong orbital effects, but in our case, we want to solve that in a general way. To solve it we use a wave number analysis in combination with a y-matching numerical method.

In our system we consider two semi-infinite contacts separated by a junction (denoted by C). The right contact (denoted by R) is a superconductor, while the left contact is a normal semiconductor material (denoted by L). The junction is defined in a central region of interface between the cables, so the junction equations are for the interface. The solutions of each of the two contacts are a linear superposition of the infinite wave functions of homogeneous nanowires with well-defined k’s. Formally, both sides of the junction are described with the Hamiltonian equation, 2.1, but with a zero coupling superconductor on the left side of the junction, a finite value on the right, and a position-dependent value on the central region. Asymptotic solutions read

Ψ(x, y, η_σ, η_τ) = X

α nα

d^(α,c)_nα e^{ik(α,c)nα x}φ^(α,c)_nα (y, η_σ, η_τ), (2.6) where α = i, o represents the input(i) and output(o) modes for contact c = L, C, R (Left, central or right). In this case, we have ingoing and outgoing propagating modes of real wavenumber k^(α,c)nα . The 3-fold (α, n_α, c) labels are for the mode type, its ordering number, and the contact (L, R), respectively. The output amplitudes d^(o,c)no determine the asymptotic solutions that can be arbitrarily extended in the leads since there the x-dependence is analytic. Non-propagating modes can be labeled also as output modes.

Now as in typical scattering problems, the algorithm has to reproduce the set of output amplitudes in terms of the input ones, for a given energy E, from

(H_BdG−E) Ψ(x, y, η_σ, η_τ) = 0. (2.7) In Ref. [14] we have this method developed with more details. But the essence is to solve this equation with thek-complex method and obtain the equations for the junction region.

The resolution is fully characterized by obtaining the input and output amplitudes d^(α,c)nα , as the usual scattering problems.

2.2. METHOD 13

2.2.1 Conductance calculation

When presenting the background of the study that we are going to carry out, in the first chapter, we have already indicated what are the formulas for conductance depending on Normal or Andreev reflection. Now we are going to give a more detailed analysis of how this physical quantity is obtained since this is the specific object of our study.

The output coefficients appearing in 2.6 are required to compute the global conduc- tance G. With a given energy E it is described by:

G(E) = dI(E) dV = e²

h[N(E)−R_ee(E) +R_eh(E)], (2.8) where N are the number of propagating modes and we describe the Andreev (electron- hole) and the Normal(electron-electron) reflection probabilities, as

R_ee(E) = X

nonσ

|d^(o,L)_no | Z

dy|φ^(o,L)_no (y, η_σ,⇑)|², (2.9)

R_eh(E) = X

nonσ

|d^(o,L)_no | Z

dy|φ^(o,L)_no (y, η_σ,⇓)|². (2.10) To summarize, the calculation of conductance and distribution of currents of an NS junction requires; first, the eigensolutions of each of the sides, normal and superconductor;

second, obtain the asymptotic mode amplitudes and the wave function on a small grid around the junction. Finally, knowing the wave function and asymptotic amplitudes at energy E the conductance is really easy calculable.

Chapter 3

Results and analysis

Our study will be restricted to the union of a superconductor and a normal material. Both are defined as QAHI materials, so both follow the Hamiltonian of the system presented in Chapter 2. And we can calculate the equivalent conductance of the system with the complex wavenumber method explained. This calculation is subject to the choice of parameters like energy, external magnetic field, or dimensions of the system. In all cases, we will choose parameters that make physical sense with the experimental articles seen [5, 9].

To characterize this study what we will do is introduce a ”barrier” that affects quantum scattering between the superconducting and normal regions. And through the thickness of the barrier variation and the variation of some essential parameters of the complete system such as energy, the Zeeman effect (∆Z), and the magnetic orbital effects (magnetic length), we will see which topological phases give us a conductance quantified at e²/h and which not. The aim of introducing this barrier material is to see if when we have quantum scattering disconnection between the superconductor and the normal material, the topology of the system plays an essential role to determine the conductance. In addition, we want to see if the introduction of a barrier can be a good experimental method to determine the topology of the NS junction.

This ”barrier” has the same parameters as the normal material but we have changed one, the only factor that we change is m0, a measure of the coupling between the two layers of the system. In the normal material m₀ = 1 meV and in the barrier material m₀ = 1.5 meV. We have chosen this value for the barrier material so that it presents a higher energy band than the normal material.

In our practical case, we are going to study the NN’S union, where N’ is the barrier material. In the previous chapter, we have explained how we must solve the NS junction problem. Following this procedure, what we do now as solve the union problem twice.

Once for NN’ and once for N’S.

15

3.1 Phase diagram

To understand what happens in the coupled system, first, we are going to analyze the physics in the homogeneous superconducting system. The band diagram for the super- conducting material with which we are working is the one we can see indicated in many articles, such as Ref. [13]. It is a diagonal cross E(k) dispersion, what means that there is no existing energy gap. In this system, we can indeed have Majorana modes, because we have energy zero when the wavenumber is zero. Moreover, these states are charge-neutral states, so theoretically we can have them inside this system.

Analyzing the bandgap, one could easily understand that by increasing the energy of the system we can change its topology. Simply, we could say that when we have a higher energetic system, more propagating modes are activated and this means that we have one associated topology or another. It is important to take into account that in our system the change of topology is going to cause a change in the band structure.

In this work, we want to see which topology is more robust, and what happens if we probe the robustness of the conductance quantification in different systems. To study the different topologies we will present a phase diagram, like in Ref. [15]. The phase boundaries are determined by the bulk BdG gap closing in the hamiltonian presented.

This diagram shows the transitions between the different topologies due to the energy and the Zeeman parameter variation ∆_Z. The entire study will be without considering orbital effects until otherwise indicated.

E

Δ

(meV)

(meV)

Figure 3.1: Phase diagram of superconducting system withLy = 2µm. Supercon- ductor parameters: m₀ = 1 meV,m₁= 0.001m⁻_U1, ∆₁ = 1 meV and ∆₂= 0.1 meV.

In the legend, we have indicated the number of active states or Chern number. For even values we will have states associated with paired Majorana modes, and for odd values we will have single modes.

3.2. CONDUCTANCE OF NN’S JUCTION 17 In Fig. 3.1 we have indicated the Chern number in the different regions. This number indicates how many modes are transmitted inside the superconducting system. In the first zone, we have non-propagating modes N = 0. But in the regions where N = 1 we have single Majorana modes configuration and in the region where N = 2 we have the coupled one. For the divisions of N = 3 or 4, we have more complex modes of excitation, which explains the lack of good quantification when calculating its conductance.

We can see that if we consider a fixed Zeeman parameter, as we increase the energy, more and more transmission modes are activated. A predictable fact viewing the band diagrams. On the other hand, if we consider constant energy, the Zeeman factor gives symmetrical results independently of its sign, since the sign does not affect the number of active modes. We see that as the absolute value of energy increases, the more active modes we have.

If we were to graph the homogeneous normal system we would not have the possibility of uncoupled modes, since all Chern numbers are even. That can only exist inside a superconductor with ∆₁ 6= ∆₂. If we see the band structure of the normal material we can see that now we have double axes overlapping diagonal crosses, so the number of propagative modes is always even because we know that the hybrid superconductivity allows non-trivial topological phases existence with unpaired Majorana modes.

3.2 Conductance of NN’S juction

The study of the conductance gives us an overview of the change of a physics parameter that can be measured experimentally. Considering the NN’S junction, where N’ is the barrier material. To give an overview of the problem we are going to show four results, from a thick barrier material to a small barrier thickness.

In this section, we will analyze what happens to conductance when we change the Zeeman parameter or the energy. However, rather than evaluating the physical effects caused by the Zeeman parameter, the main objective will be to study the effects of having different topologies. We will not focus on the details, but we will give an overview to show that having a quantized conductance does not necessarily mean that this fact is linked to a specific topology, in absence of barrier. Remarkably, in the presence of barriers we find the conductance depends on topology.

For our study, where we investigate the energy and Zeeman variation separately, we are going to consider the parameters E = 0.07 meV, and ∆_Z = 1.5 meV because they are realistic parameters that pass over different topologies. In the case of energy, we select this low value, E = 0.07 meV, to sweep through the topologies on which we focus our study, N = 0, 1, and 2. If we chose a higher energy we would enter modes that are not of our interest. On the other hand, for the Zeeman factor, ∆_Z = 1.5 meV is an interesting value since it makes it possible to pass across the regimes N = 1 and N = 2 in the first low energies.

3.2.1 Variation with Zeeman parameter

The ∆_Z measure indicates how strong the intrinsic magnetization of the field is, having a big effect of this when the parameter is greater. In this case, we have considered a fixed low energy of 0.07 meV, and sweep the parameter ∆Z. So, in the following diagrams, we expect that three regions can be distinguished, thanks to the presented phase diagram 3.1. The first associated with N = 0 where there will be no transmission since there are no propagating modes, it is in the range ∆Z ∈ [0,1.2] meV. The second region will be identified as N = 1, a single Majorana mode is transmitted, in the interval

∆_Z ∈[1.3,1.9] meV. And the last one is the related to coupled Majorana modes, N = 2, in the range ∆Z ∈[1.9,3] meV. We will indicate with vertical lines the separation between these phases in Fig.3.2.

a) b)

c) d)

Δ_𝑍

Δ_𝑍(meV) (meV)

𝒩 = 0

𝒩 = 1

𝒩 = 2 𝒩 = 2

𝒩 = 2 𝒩 = 2

𝒩 = 1

𝒩 = 1

𝒩 = 1

𝒩 = 0 𝒩 = 0

𝒩 = 0

Figure 3.2: Conductance of NN’S junction, Normal and Andreev reflection for different values of the thickness of the barrier material: (a)Lx = 0.2µm, (b)Lx = 0.5µm, (c) L_x = 1µm and (d) L_x = 2µm. Units of all the parameters are meV except conductance G, which is in e²/h. E = 0.07 meV. The different topological phases are indicated by vertical lines.

Notice that the results in Fig. 3.2 are sensitive to the barrier only for the range

∆_Z ∈ [1.2,1.9] meV. As we have pointed this range corresponds to the phase with one Majorana mode in the superconductor part. For ∆_Z >1.9 meV the superconductor lead has two Majorana modes and the results are then not sensitive to the used barrier. From a general perspective, we see that the conductance is moving away from the quantified

3.2. CONDUCTANCE OF NN’S JUCTION 19 value as we increase the size of the barrier and that the Andreev reflection process grows remarkably. In this diagram, we have only represented the range ∆_Z ∈[0.75,2.5] meV so that the area of interest (N = 1) can be appreciated well, although in our calculations we have reached ∆_Z values from 0 to 4 meV to check that there are no sudden changes.

When we have a thin barrier material we can not distinguish the topological phase.

Paying attention to the thin barrier case we have represented Fig. 3.2a, we note that in the region corresponding to phase N = 1, the conductance is quantized at 1 thanks to the fact that the Normal and Andreev reflection compensate each other, Eq.2.8, and therefore the barrier is not affecting the transmission. In the case of no barrier, we remember that for this topological configuration a Majorana mode must be transmitted with a probability of 0.5 and the rest must be reflected (0.25 Andreev and 0.25 Normal reflection). Approximately this is what happens here, the probabilities of reflection have a plateau around 0.25.

By increasing the barrier and having a thickness of 0.5µm, Fig. 3.2b, we can say that in the region of interest the conductance is not so constant but we still see that its predominant value is 1. In this case for the first Zeeman parameters of the area, Andreev reflection contributes more than Normal causing the conductance not to reach the value of 1. But for higher ∆_Z within the same zone, the reflections end up starting as happened with the previous case.

In both Figs. 3.2c and 3.2d we have sizeable barriers, and we obtain a conductance that is approximately linear with the ∆_Z value in the N = 1 region. It has no constant value or plateau, we have no quantification for conductance in these cases. We state that when we have a thicker barrier material the normal reflection is reduced and the Andreev one increasing, the incident electron modes are reflected to a greater extent as holes.

The two reflections are not compensated until the point where they fall into the range of N = 2.

When we have the topology of two-Majorana-phase, the conductance is always quan- tized in one unit due to the Klein effect. In Fig. 3.2 for ∆_Z >1.9 meV we have perfect transmission in any barrier thickness. The Klein effect, as we have explained in the first chapter, is the penetration of relativistic particles though high and wide potential bar- riers. Now we do not have exactly the theoretical case but we do not have an energy gap and the movement of the band energy does not affect the transmission. Specifically, what is happening is that the barrier material N’ will have the same band diagram as the material N but will have it displaced, allowing a perfect transmission passing inside the superconductor since the superconductor band diagram is double-crossed when we have coupled Majorana modes. So when we have a ∆_Z which allowed a topology associated with N = 2, and a low energy value (in this case E = 0.07 meV) we will have a perfect conductance quantization for the NS junction. We do not have any reflection and the conductance has a quantized value of the unit.

3.2.2 Variation with energy

In this section, we set the Zeeman parameter to a value of 1.5 meV. Therefore, looking at the diagram presented in Fig.3.1 it is expected that in the calculation of the conductance we will see different topologies, which we have represented in 3.3 with purple vertical lines.

Firstly we have a short non-transmission zone (N = 0) in E ∈[0,0.01] meV. Practically at the beginning of the diagram, we enter the region ofN = 1, a single Majorana mode for the rangeE ∈[0.01,0.27] meV. Then we have the zone ofN = 2 forE ∈[0.27,0.65] meV.

And finally, we have more excitation modes, which will give a behavior that we will not analyze.

a) b)

c) d)

E

E(meV) (meV)

𝒩 = 1

𝒩 = 1 𝒩 = 1

𝒩 = 1 𝒩 = 2

𝒩 = 2 𝒩 = 2

𝒩 = 2

Figure 3.3: Conductance of NN’S junction, Normal and Andreev reflection for different values of the thickness of the barrier material: (a)L_x = 0.2µm, (b)L_x = 0.5µm, (c) Lx = 1µm and (d) Lx = 2µm. Units of all the parameters are meV except conductance G, which is in e²/h. ∆_Z = 1.5 meV. The different topological phases are indicated by vertical lines.

In Fig. 3.3 we can see that the conductance changes shape a lot depending on the thickness of the barrier material. Unlike the case in which we varied the Zeeman parameter now the different topological regions cannot be appreciated in such a clear way, although we can appreciate changes between them. Now there is no perfect transmission in the region of two active modes, the Klein effect is still the justification for this to happen but now not in such a clean way.

3.3. ORBITAL EFFECTS 21 For the minimum value of L_x for the barrier material Fig. 3.3a, we have practically no distinction between the two topological phases of our interest. In the region of N = 1 Normal and Andreev reflection cancel each other out and in the region ofN = 2 reflection probabilities are directly equal to zero.

When this barrier increases a bit, Fig. 3.3b, we have strange behavior. The conduc- tance in the zone of two Majorana modes is not exactly quantified in the unit but it is highly above this value, this fact again can be explained with the Klein effect that not only argue of perfect transmission but also of the transmission of a higher energy than the initially had. In this region, the reflection is in essence void. For the area ofN = 1 we ob- tain an approximate value to the unity for conductance because the reflection practically cancels each other.

In Fig. 3.3c, for N = 1 we have an accidental plateau around the value of 0.5, and for the next zone, we have a practically stable value at 1, with marked initial and final peaks due to the Andreev reflection. For the case of thicker barrier material that we have represented, Fig. 3.3d, in the first zone Andreev’s reflection is reaching to a value close to unity, it is for this reason that the conductance almost vanishes. For the area ofN = 2 we have an initial peak which is due to an Andreev reflection peak and then an approximate equilibrium to unity, where the normal reflection is not strictly zero, but it is a negligible contribution.

Both, in this case of energy variation, Fig. 3.3, and in the Zeeman parameter variation, Fig. 3.2, the conclusion we draw is the same. In a first region associated with unpaired Majorana modes we have a strong dependence on the conductance with the thickness of the barrier material and in the area associated with paired Majorana modes this depen- dence disappears, thanks to the Klein effect we do not care whether or not we have this barrier material. Therefore the topology N = 2 is more robust than the topology N = 1, against the presence of a barrier in an NN’S system.

3.3 Orbital effects

∆_Z andl_z are two parameters that depend on the magnetic field. But they are not equal because the Zeeman effect depends on the gyromagnetic parameter and the intrinsic magnetization, which is controlled by the external magnetic field, while the effects of the orbital motion are only controlled by the external magnetic field. In all the discussions that we have made, we do not contemplate l_z. Now recalculating the NN’S junction conductance we are going to explore what happens with this parameter.

Considering the case of ∆_Z = 1.5 meV and E = 0.07 meV, since these are the cases we have shown in the last section, we are going to compute conductance and reflections probabilities for different values of l_z⁻². In all the thesis the value that we have considered isl⁻²_z = 0 since we have not paid attention to the orbital effects. In the case of considering E and ∆Z constants, we can calculate the Chern number for differentlz values, as we did in the phase diagram section. When we do this calculation we find that for a range of l_z⁻²

from 0 to 6 we have two possible topologies, N = 1 in l⁻²_z ∈ [0,1.7]µm⁻² and N = 2 in l⁻²_z ∈[1.7,6]µm⁻². So, we expect that the conductance changes the behaviour at about l²_z = 1.7µm⁻², like we are indicating in Fig. 3.4 with a vertical purple line.

In this section, we have not considered the same values for the thickness of the barrier material N’ than in the previous sections, because for low values of Lx we do not find a specific behavior, but when we increase the barrier we obtain more interesting results, as we show in Fig. 3.4. Calculations have been made for barriers of up to 20µm.

a) b)

c) d)

𝑙_𝑧⁻²

𝑙_𝑧⁻²(𝜇𝑚⁻²) (𝜇𝑚⁻²)

𝒩 = 2 𝒩 = 1

𝒩 = 2 𝒩 = 1

𝒩 = 2 𝒩 = 2 𝒩 = 1

𝒩 = 1

Figure 3.4: Conductance of NN’S junction, Normal and Andreev reflection for different values of the thickness of the barrier material: (a) L_x = 1µm, (b) L_x = 2µm, (c)Lx= 10µm and (d)Lx= 20µm. Units of reflections are meV, conductance Gis ine²/hand the magnetic length is in µm. ∆_Z = 1.5 meV andE = 0.070 meV.

The different topological phases are indicated by vertical lines.

In Fig. 3.4 we see two opposed behaviors. In the region associated with single Majo- rana modesN = 1, we have a non-oscillatory behavior, in Figs. 3.4a and 3.4b conductance have variable values depending on the magnetic length. But, when the length of the bar- rier material increases the conductance is strictly zero, for a thick barrier we do not have transmission. In any case, the Andreev reflection is void and the conductance value of the conductance is only a sample of the Normal reflection probability. For the thinnest barrier, it has a non constant value distant from unity, but as we increase the thickness of the barrier material it is approaching one. In the case of not having a barrier, the con- ductance would give us quantified in the unit for this region, but with the introduction of a fine barrier material, we already see a great change in it.

3.3. ORBITAL EFFECTS 23 In the region related to two Majorana modes N = 2, we have an oscillating behavior around the unit for conductance. When the barrier increases, the oscillatory character of the conductance increases. This is the reason why we now consider barrier widths larger than in the previous sections to show these oscillations in a way they can be appreciable.

Figs 3.4c and 3.4d show that oscillation maxima are equally spaced and therefore these could be quantified concerning the width given to a barrier. This fact is verifiable to the naked eye, but we have also calculated numerically the relationship between the maxima and we have seen that when approximately we double increase barrier length the maxima are twice as together as before. Approximately we would have a maximum in (1/l²_z)_max ∝n/L²_x where n is an integer.

To explain these oscillations in a very simple way, we could say that considering an electron that travels through the normal material, it will bounce inside the barrier material allowing an oscillatoryl_z-dependent normal reflection process to occur. For this to bounce, we will need L_x to be long enough for these bounces to occur, a fact that happens in the two last cases of length Figs. 3.4c and 3.4d. In the first cases, the length of the barrier is not sufficiently large to make this process visible.

To conclude, we can say that for these oscillations to occur we have two options. On the one hand, applying a non-intense magnetic field (with a large l_z associated) where it will be necessary for the width of the barrier material to be thick for bounces and oscillations can occur. Or on the other hand, applying an intense magnetic field to our system (with a small l_z associated) where it will not be necessary to have a thick barrier material, with values smaller for L_x we will be able to appreciate these oscillations. Obviously what we have represented is with the first consideration, of a non-intense magnetic field.

Chapter 4

Conclusions

This work is focused on analyzing the robustness of the different topologies that a hybrid superconducting material can present when it is in an NS junction, formed by a normal material and a superconducting material. This analysis has been approached in this way motivated by some important current articles discussing whether the quantification of the conductance demonstrates the existence of chiral Majorana modes or not. We have focused on the NS junction to see when it has a quantum conductance in the unit and if this is so we can say that the NSN junction with quantum disconnection of the superconductor has a conductance of 0.5e²/h. It makes sense to talk about a quantum disconnection inside a superconductor because the perfect connection of this would be very difficult to obtain experimentally.

To make the analysis of the junction and test the robustness of the topologies we have introduced a barrier material N’ between the superconducting material and the normal one. And we have studied how this affects the transmission processes and the value we obtain of conductance. The barrier material N’ has a variable length and the same properties of N but a different parameter m₀.

We have focused on the variation of fundamental parameters, the Zeeman parameter, as a parameter that can be variable by changing the applied magnetic field, and the energy, as a parameter that is easily measurable as voltage. In both cases, the results we found are consistent. For the case in which the Chern number associated with the system is zero, there is no transmission and therefore there is no conductance. When we have a topology in Majorana’s single-mode hybrid superconducting material, the barrier affects the transmission and therefore the values obtained from the conductance. When we have a non-existent or thin barrier it will have a quantified value in the unit because the reflection processes are compensating each other or are void. But when the barrier material increases in thickness we cannot say that the conductance has a quantified value forN = 1. Whereas in the case of having a topology associated with two paired Majorana modes the conductance is equal to unity regardless of the thickness we give to the barrier material, this fact is analog to the Klein effect.

25

Finally, we analyzed the junction where we considered the orbital effects, which we had neglected up to this point. When we consider these effects due only to the external magnetic field applied to our system, we see that when the barrier material vanishes the conductance is not well quantified although it has values close to unity for any topology.

But when we increase the barrier significantly, we have a first region N = 1 where the conductance is zero and a second N = 2 where it oscillates around the unit. This conductance oscillation with the magnetic field could be the subject of future studies.

In this study we have considered the variation of a single parameter with respect to the materials N and N’, the results would be different in the case of having made more distinctions between the two materials or having considered a higher value of m₀ for N’.

The effect analogous to the Klein effect that we have observed, in the case of considering a different barrier material would not have to have occurred since as we have explained the transmission is perfect because the band diagrams allow us to do so. Changing the properties of N’ has changed its band diagram.

On the other hand, we have considered significantly small barrier lengths, although in the calculations we have tested many different values for the barrier thickness and we have represented these because they are a significant sample of the physics of the problem.

In short, the most important and significant conclusion of this work and that can be applied experimentally is that by introducing a barrier material, we can know in which topological phase is our system NN’S in a simple way by measuring the conductance, while when there is no such barrier material we have no distinction between phases. In the event that this study can be implemented experimentally, we would come to the conclusion that the topology does not affect the conductance in the case of considering an NS junction without a barrier. And therefore if we consider quantum disconnection in the NSN system, the measurement of the conductance at 0.5e²/h would not be proof of the existence of single chiral Majorana modes.

BIBLIOGRAPHY 27

Bibliography

[1] C. Kane and J. Moore, “Topological insulators,”Physics World, vol. 24, no. 02, p. 32, 2011.

[2] E. Majorana, “Teoria simmetrica dell’elettrone e del positrone,” Il Nuovo Cimento, vol. 14, no. 4, pp. 171–184, apr 1937.

[3] O. Klein, “Die Reflexion von Elektronen an einem Potentialsprung nach der relativis- tischen Dynamik von Dirac,”Zeitschrift f¨ur Physik, vol. 53, no. 3-4, pp. 157–165, mar 1929.

[4] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nature Physics, vol. 2, no. 9, pp. 620–625, sep 2006.

[5] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.- C. Zhang, K. Liu, J. Xia, and K. L. Wang, “Chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure,” Science, vol. 357, no.

6348, pp. 294–299, 2017.

[6] V. S. Pribiag, “A twist on the Majorana fermion,” Science, vol. 357, no. 6348, pp.

252–253, 2017.

[7] W. Ji and X.-G. Wen, “¹₂(e²/h) conductance plateau without 1d chiral Majorana fermions,” Phys. Rev. Lett., vol. 120, p. 107002, Mar 2018.

[8] Y. Huang, F. Setiawan, and J. D. Sau, “Disorder-induced half-integer quantized con- ductance plateau in quantum anomalous Hall insulator-superconductor structures,”

Phys. Rev. B, vol. 97, p. 100501, Mar 2018.

[9] M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang, F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fijalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li, L. W.

Molenkamp, M. H. W. Chan, N. Samarth, and C.-Z. Chang, “Absence of evidence for chiral majorana modes in quantum anomalous hall-superconductor devices,”Science, vol. 367, no. 6473, pp. 64–67, 2020.

[10] M. Diez, J. P. Dahlhaus, M. Wimmer, and C. W. J. Beenakker, “Andreev reflection from a topological superconductor with chiral symmetry,” Phys. Rev. B, vol. 86, p.

094501, Sep 2012.

[11] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, “Transition from metallic to tunnel- ing regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion,” Phys. Rev. B, vol. 25, pp. 4515–4532, Apr 1982.

[12] J. Osca and L. Serra, “Conductance oscillations and speed of chiral Majorana mode in a quantum anomalous hall two-dimensional strip,”Phys. Rev. B, vol. 98, p. 121407, Sep 2018.

[13] ——, “Magnetic orbital motion and 0.5e²/hconductance of quantum-anomalous-Hall hybrid strips,” Applied Physics Letters, vol. 114, no. 13, p. 133105, Apr 2019.

[14] ——, “Complex band-structure analysis and topological physics of Majorana nanowires,” The European Physical Journal B 2019 92:5, vol. 92, no. 5, pp. 1–19, may 2019.

[15] J. Wang, Q. Zhou, B. Lian, and S. C. Zhang, “Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition,”

Physical Review B - Condensed Matter and Materials Physics, vol. 92, no. 6, p.

064520, aug 2015.