LOW-TEMPERATURE TRANSPORTIN

HYBRID INAS NANOWIRES

a bachelor thesis byMaksim Borovkov

supervisor: Prof. Georgios Katsaros (IST Austria)scientific advisor: Prof. Pavel Ioselevich (HSE University)

MOSCOW INSTITUTE OF PHYSICS AND TECHNOLOGYLandau School of Physics and Research

JUNE 2020

ACKNOWLEDGMENTS

First and foremost, I would like to express my deepest gratitude to my supervisor Georgios

Katsaros, for this invaluable opportunity to join his group for working on a bachelor thesis.

I genuinely appreciate encouraging discussions with him, his readiness to answer any of my

questions, and his relentless support since we first met in September 2018. Back to Russia,

I would like to thank my scientific advisor Pavel Ioselevich for his guidance through the

peculiarities of condensed matter theory. I am also grateful to Galina Tsirlina for her help

with chemistry-related questions as well as for her support during my junior and senior years

at MIPT. I am also deeply indebted to Marco Valentini and Andrea Hofmann. If there is

something I know how to do in the lab or the cleanroom it is mainly because of the time

they have patiently spent teaching me. Many thanks to the whole Nanoelectricians team.

Alessandro, Lada, Marian, Daniel, Josip, Luka, Kushagra, Stavroula, Frederico, and Marin,

it is incredibly fun to work together with you. This thesis was done in a fruitful collaboration

with the group of Peter Krogstrup at the University of Copenhagen who provided wonderful

nanowires. I also had the great pleasure of working with the NFF staff at the IST Austria

cleanroom. In unprecedented times of global pandemic, I received enormous support from

the IST Austria administration. Thus, I am extremely grateful to Vlad Cozac for his help

with immigration issues. Finally, I thank my family and friends for their care and love.

2

LOW-TEMPERATURE TRANSPORT IN HYBRID INAS NANOWIRES

Abstract

by Maksim Borovkov,Moscow Institute of Physics and Technology

June 2020

supervisor: Prof. Georgios Katsaros, IST Austriascientific advisor: Prof. Pavel Ioselevich, HSE University

Recent advances in molecular beam epitaxy techniques enabled the growth of hybrid nanowire

devices with a remarkable alignment of the crystal lattices of the device compounds [1].

The created clean interfaces among materials with different mesoscopic properties lead to

enhanced proximity effects, which are of great importance for the realization of a robust

platform for topological quantum computation [2, 3]. Already in hybrid InAs/Al nanowires

signatures of p-wave superconductivity and non-Abelian excitations have been demonstrated

[4, 5]. However, these experimental schemes require substantial external magnetic fields that

would prevent the realization of braiding protocols [6]. The problem can be resolved by

invoking exchange interaction among semiconductor electrons and magnetic moments of a

ferromagnetic insulator [7]. In the present work, we investigate the properties of this ex-

change interaction in hybrid InAs/EuS nanowires via single-electron transport techniques.

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TABLE OF CONTENTS

Page

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Topological Phenomena in 1D Channels . . . . . . . . . . . . . . . . . . . . . . 7Kitaev Chain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Physical Implementation of the Kitaev Model . . . . . . . . . . . . . . . . . . 16Towards Braiding in the Nanowire Networks . . . . . . . . . . . . . . . . . . . 20

Single-Electron Transport in Quantum Dots . . . . . . . . . . . . . . . . . . . 22Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Single-Electron Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Experimental Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Exchange Field in InAs/EuS Hybrid Devices . . . . . . . . . . . . . . . . . . . 28Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Measurements and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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Introduction

In the past 30 years condensed matter physics has been moving in two at the first glance dis-tinct directions. The first one manifests an interest in defining and manipulating individualquantum systems. This research paradigm has already proved itself to be quite successfuland the ongoing work on performing universal quantum computation protocols on individualsystems embraces a wide range of platforms. To mention some, these are superconductingqubits [8], hole spin [9] and electron spin [10] qubits in semiconductors, trapped ion qubits[11] and etc.

Meanwhile, scientific attention has been also drawn to the concept of emergent andstrongly-correlated phenomena. The hallmarks of these endeavors are the experimentalobservations of the fractional quantum Hall effect [12], high-temperature superconductivity[13], and topological states of matter [14], alongside theoretical advancement of topologicalfield theories. A significant outcome of this area of research is the development of notions ofnon-Abelian excitations - a special type of quasiparticles that follow neither fermionic norbosonic statistics.

These two fields are now merging together thus opening new pathways for controlling andmanipulating the properties of the many-body excitations. Due to the non-Abelian natureof the later, creating and manipulating many-body excitations holds a considerable promisefor constructing a fault-tolerant quantum computer [15].

One of the most intriguing topological states of matter that is currently intensively stud-ied around the world is p-wave superconductivity. Two seminal theoretical works predictedthat such systems would host non-Abelian excitations which are usually called MajoranaZero Modes [16, 17] and are suitable for performing topological quantum computation pro-tocols [6]. Later on, a more detailed theoretical analysis suggested using several platformsfor realizing these modes experimentally. These are, for instance, a topological insulator cou-pled to an s-wave superconductor [18], ferromagnetic atomic chains [19], and semiconductornanowires with strong spin-orbit coupling [2, 3].

The nanowire platform is one of the most promising ones due to its high tunability.Although, numerous experimental data [4, 20, 5] since 2012 leaves much for a debate [21,22] whether Majorana Zero Modes have been really observed or not, the field has definitelyspurred on the technological development of hybrid nanowires based on InAs [23, 1, 7, 24].This in turn paves the way for new fundamental physics not necessarily related to topologicalquantum phenomena. Moreover, since creating and manipulating non-Abelian excitationsin the nanowires is a very demanding experiment one has to invest some effort into thesystematic studies of the underlying physical properties of such nanowires.

Given this motivation, in the present work, we investigate the transport properties of

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hybrid semiconductor/ferromagnetic insulator nanowires to assess its potential for hostingMajorana Zero Modes detection and braiding experiments. The thesis is organized in thefollowing way. In the first section, we provide a pedagogical introduction to topologicalphenomena in one-dimensional channels. We study the main properties of the Kitaev modelas well as the spin-orbit nanowire proposals and conclude with a review of the nanowirenetworks proposals and technical challenges they raise. In the second section, we elucidatesingle-electron transport in quantum dots which is our major tool for experiments. In theexperimental section, we outline the fabrication recipe of the devices, describe our experi-mental setup, and finish with providing the measurements data and its analysis. Finally, wemake a conclusion and suggestions for future work.

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Topological Phenomena in 1D Channels

Kitaev Chain ModelWith the growing complexity of emergent phenomena in condensed matter systems, nowa-days toy models have become an indispensable part of any condensed matter theorist’stoolkit. It indeed seems much more appealing to design a minimal model that capturesmain physical features instead of performing complex and computationally demanding abinitio calculations. An increasing interest in non-singlet pairing superconductivity and Ma-jorana quasiparticle excitations at the end of the 20th century created a call for simple modelsclarifying such phenomena. The first comprehensive study was performed by Reed and Greenin 1999 [17] where the authors investigated properties of p-wave superconductivity in twodimensions. A year later its one-dimensional counterpart was considered by Kitaev [16].Non-physical at first glance, Kitaev’s model is simple yet extremely rich in terms of physicalpredictions and today can be used as a pedagogical introduction to the field of topologicalcondensed matter physics. In this section we review Kitaev’s toy model for one-dimensionalp-wave superconductivity and elucidate the most common notions in physics of topologicalphenomena, namely topological phase transitions, topological indices, Majorana Zero Modes,Bulk-Boundary Correspondence, and Domain Walls.

The Kitaev toy model is a tight-binding model for spinless electrons described by theHamiltonian

H = −µN∑x=1

c†xcx −1

2

N−1∑x=1

(tc†xcx+1 + ∆eiφcxcx+1 + h.c.

), (1)

where µ is the chemical potential, t ≥ 0 is the nearest-neighbor hopping amplitude, ∆eiφ

(∆ ≥ 0) is the BCS mean-field pairing potential. The words spinless electrons already mightappear rather confusing. By saying so we imply electrons of the same spin species and assumethat there exists a mechanism to disregard electrons of the opposite spin. Next, fermionicoperators c†x (cx) denote creation (annihilation) of an electron on the site x ∈ [1, . . . , N ] andtherefore obey fermionic commutation relation

cx′ , c

†x

= δxx′ . To simplify math we also

put the lattice constant to unity.First, we would like to understand the emergence of physically disparate quantum phases

and define what we mean by saying that one phase is topologically different from anotherone. In order to do so, we explore the bulk properties of the system by imposing periodicboundary conditions, i.e. closing the chain into the ring: c†N+1 = c†1. In this way, the crystalmomentum k ∈ [−π, π] is a good quantum number and enables us to pass to momentumspace. Upon the Fourier Transformation cx =

∑k e

ikxck, the Hamiltonian in momentum

7

space up to a constant reads

H = −∑k

(µ+ t cos k)c†kck + ∆∑k

(c†kc†−ke

ik + c−kcke−ik). (2)

The easiest way to diagonalize the Hamiltonian above is to rewrite it in a single-excitationform. Therefore, we introduce a redundant degree of freedom that accounts for the creation(annihilation) of holes of the momentum k with the corresponding operators c−k (c†−k)

1. Inthis extended Hilbert space we define a new basis Ψ†k = (c†k, c−k) which in literature is usuallycalled the Nambu basis. This technical step enables us to recast Hamiltonian (2) in a moreconvenient form:

H =1

2

∑k

Ψ†kHBdG(k)Ψk, (3)

where HBdG(k) is the so-called Bogoliubov-de Gennes Hamiltonian:

HBdG(k) =

(−µ− t cos k i∆e−iφ sin k−i∆eiφ sin k µ+ t cos k

). (4)

Hamiltonian (3) is then diagonalized by the Bogoliubov transformation. More precisely, weconsider an electron-hole excitation (Bogoliubov’s quaiparticle) of the form

ak = ukck + vkc†−k, (5)

which is an eigenvector of the BdG Hamiltonian and reformulate the problem in terms ofthese excitations:

H =∑k

E(k)a†kak. (6)

As long as there exists the particle-hole symmetry which comes merely as an artifact ofworking in the Nambu basis, a Bogoliubov’s excitation a†k with the energy E(k) is physicallyindistinguishable from an excitation a−k with the energy −E(−k). This is a first hint thatparticles which are their own antiparticles - Majorana excitations - appear naturally insuperconducting systems [25]. Indeed, as discussed before, in the language of Bogoliubov’sexcitations the difference between particles and anti-particles is blurred and thus one mightbe expecting to find a special case when an excitation satisfies exactly the γ† = γ Majoranacondition. Moreover, this line of thought allows us to consider only the positive eigenvalueof the matrix (4):

E(k) =

√(−µ− t cos k)2 + ∆2 sin2 k. (7)

Let us now carefully analyze the dispersion relation (7). Firstly, we notice that the p-wavesuperconductivity leads to the ∆2 sin2 k term. This in turn means that the system can

1If we take an electron of the momentum +k out of the filled Fermi sea the resulting momentum of thesystem will become −k. Since annihilation of an electron ck is the same as creating a hole, it implies thatthe hole must possess −k momentum.

8

exhibit a gapless excitation spectrum in contrary to s-wave superconductivity. Indeed, whenthe chemical potential is precisely µ = ±t the conduction band (Fig 1a) is either emptyor fully occupied, i.e. the Fermi surface is k = 0 or k = ±π respectively. At these pointshowever Cooper pairing is prohibited (∆ sin k = 0) and thus the system stays gapless.

The borderlines µ = ±t separate two different regimes, namely the |µ| < t weak-pairingregime and the |µ| > t strong-pairing regime. The physical difference between them can beeasily grasped when neglecting the pairing parameter. Indeed, the former regime correspondsto a metallic state where the chemical potential lays inside the conduction band while thelatter regime is a band insulator. When the pairing parameter is present, the formation of theCooper pairs undergoes in different ways. It can be demonstrated that in the weak-pairingregime a Cooper pair is infinite in size, while in the strong-pairing it is spatially bounded[26]. Although these regimes are physically different, within the Landau order parameterformalism no phase transition occurs since the two phases share the same symmetries andnone of them breaks. This implies that one has to establish a new theoretical framework toredefine the notions of the order parameter and phase transition.

To begin with, we note that the properties described above obviously hold upon smoothadiabatic transformations of the Hamiltonian. Therefore, instead of taking care of the sym-metry breaking, one could rather construct an order parameter which would be invariant fordifferent states connected by smooth transformations and changes ones the system appearsin a topologically distinct phase. Our current system is relatively simple and therefore wewill construct a topological order parameter in an illustrative manner, albeit we acknowledgethere are more mathematically rigorous and general strategies to proceed [16].

We rewrite the BdG Hamiltonian (4) in a slightly different way invoking Pauli matrices−→τ = (τx, τy, τz) acting in the particle-hole space:

HBdG = (∆ sinφ sin k)τx + (∆ cosφ sin k)τy + (−µ− t cos k)τz =−→h (k) · −→τ , (8)

where we introduced a vector−→h (k) = (∆ sinφ sin k,∆ cosφ sin k,−µ − t cos k)T . Suppose

we work in the gapped regime, i.e. µ 6= ±t, thus we are able to define a unit vectorh(k) = h(k)/|h(k)| which can be regarded as a map from the Brillouin zone to a unit sphere:

h(k) : BZ = [−π, π] ∼= S1 → S2. (9)

Since in the Brillouin zone points k = ±π are identical, the map (9) in fact defines a loop onthe unit sphere. Our goal now is to prove that there are only two topologically equivalentclasses of such loops, and therefore for each of these classes we can assign a topological index.In what follows we concentrate on cases µ < −t and |µ| < t since µ > t is particle-holesymmetrical to the former and hence would share the same topological index.

Firstly, we point out several restrictions imposed on the loops. That is, let us calculatethe unit vector h(k) at k = 0,±π:

case 1 : µ < −t⇒ h(0) = +1 · z; h(±π) = +1 · zcase 2 : |µ| < t⇒ h(0) = −1 · z; h(±π) = +1 · z (10)

Therefore, we see that every loop starts and ends up at the North Pole of the sphere. Alongthe way, however, there are two options how to go. When k = 0 a loop must go either

9

a)

N

S

M=-1

M=+1

b)

M=+1

M=-1

Topological

Trivial

c)

Figure 1 a) Kinetic energy dispersion relation in the Kitaev model. Dashed linesat µ = ±t distinguish two phases, namely topological |µ| < t (shaded) and trivial|µ| > t. b) Representative loops from the topological classes M = −1 (blue) andM = +1 (red). While the former passes through the South Pole at k = 0, the laterreturns to the North Pole. c) Topological phase diagram of the Kitaev chain model.

back to the North Pole or it has to pass through the South Pole instead (Fig 1b). Thisobservation divides the set of all loops governed by map (9) into two equivalence classes.All the loops within one equivalence class can be connected via smooth transformations,however there is no transoformation that could connect loops from different classes. We willcall the equivalence class of loops that go back to the North Pole trivial since by means ofsmooth transformations every loop in the class can be contracted to a point. Therefore, thephase µ < −t is called trivial as well. In contrary, loops that are pinned to both North andSouth Poles of the sphere cannot be smoothly contracted to a point and thus are topologicallydifferent from the trivial class. In this way, the phase |µ| < t is called topological. The onlything left is to define a topological order parameter which would mathematically distinguishthese two phases. The easiest way is to define an index M = sign(h(0)h(π)), which is +1in the trivial regime and −1 in the topological. With this we can finally draw a topologicalphase diagram (Fig 1c).

While the discussion above is mathematically elegant and gives an insight into a newparadigm in theoretical physics, so far we have not yet introduced an observable feature thatmanifests the topological phase transition and can be measured in a real physical experiment.Yet, this feature exists and usually is referred to as an edge state - a collective electronicexcitation which is localized by an edge of the system. Now we will elucidate what thatexactly means.

Let us get back to the lattice Hamiltonian (1) and, following Kitaev, introduce a ’changeof variables’. More precisely, we decompose the fermionic operators c†x and cx at the site xin terms of two new operators γA,x and γB,x positioned at the same site:

cx =e−iφ/2

2(γA,x + iγB,x) . (11)

From the fermionic statistics of c†x and cx operators it immediately follows that these oper-

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ators obey the following commutation relations:

γα,x, γα′,x′ = 2δα′αδx′x, γα,x = γ†α,x. (12)

The latter condition is usually referred to as the Majorana condition. However, we find itmisleading to call operators γA,x and γB,x Majorana fermions, since their statistics is non-abelian rather than fermionic as we will show in the next section. Hereafter we call theseoperators as Majorana Modes and next rewrite the Hamiltonian (1) in the following form:

H = −µ2

N∑x=1

(1 + iγB,xγA,x)−i

4

N−1∑x=1

[(∆ + t)γB,xγA,x+1 + (∆− t)γA,xγB,x+1] (13)

In order to get an intuition on possible fermionic states in the system we will first considertwo limiting cases corresponding to the topological phases discussed before (Fig 1c).

Trivial phase: µ < 0, t = ∆ = 0. In this case the Hamiltonian (13) reads

H = −µ2

N∑x=1

(1 + iγB,xγA,x), (14)

where no coupling between the lattice sites’ is present and Majorana modes are coupledin a trivial way (Fig 2a). This entails that the ground state of the system is simply thenon-degenerate vacuum and putting an electron would cost an energy |µ|. Therefore, it isa band insulator with an energy gap in accordance to the ring’s geometry picture, implyingthat in this regime edges do not play a significant role.

Topological phase: µ = 0, t = ∆ 6= 0. For this set of parameters the Hamiltonian (13)transforms into

H = −i t2

N−1∑x=1

γB,xγA,x+1, (15)

where coupling of Majorana modes is happening at the adjacent cites (Fig 2b). In orderto turn back to the fermionic picture of the system we redifine fermionic operators in thefollowing way:

dx =1

2(γA,x+1 + iγB,x), f =

1

2(γA,1 + iγB,N). (16)

Thus we recast the Hamiltonian back to the diagonal form of the electronic excitations:

H = tN−1∑x=1

(d†d− 1

2

)+ 0 · f †f. (17)

This Hamiltonian implies that there is a zero-energy non-local fermion consisting of twoMajorana modes γA,1 and γB,N residing at the edges of the lattice and thus dubbed edgestates. This in turn implies that the ground state of the system is two-fold degenerate,meaning that |1〉 = f † |0〉 is also the ground state if |0〉 is the vacuum.

11

Although the examples above bring in some intuition on the physical properties of thesystem, they seem to be quite artificial. In fact, one might reasonably ask why the zero-energyedge states appear in the system for any Hamiltonian parameters configuration correspondingto the topological regime and would not be present in case of the trivial phase. This questionis answered by the Bulk-Boundary Correspondence principle which as a conjecture we nowattempt to prove.

Essentially, we would like to explicitly demonstrate the existence of zero-energy statesexponentially localized by the edges of the lattice in the topological phase and their absencein the trivial phase. This said, we try an ansatz of the following form:

|Φ〉 =∑x

e−qxc†x |0〉 , (18)

which has to satisfy the Schrödinger equation

H |Φ〉 = 0 · |Φ〉 , (19)

Thus, the state |Φ〉 describes a zero-energy solution of the Hamiltonian (1) which exponen-tially decays into the bulk of the wire. Our first task is to find an expression for the q factoras a function of parameters µ, t and ∆. To simplify math from now on we assume that thepairing potential ∆ is real.

In the language of the second quantization |Φ〉 defines creation and annihilation oper-ators denoted as Φ† =

∑x e−qxc†x and Φ =

∑x e−qxcx respectively. These operators are

’decaying’ counterparts of propagating wave operators ck =∑

x e−ikxcx in which basis the

BdG Hamiltonian (4) was written. Therefore, we can think of a decaying wave e−qx as apropagating wave eikx with an imaginary wave-number k = iq. This observation justifiesthe following math trick: we will use the BdG Hamiltonian (4) and substitute all k-termswith iq. Moreover, we are again shifting to the particle-hole space and therefore we will besearching for a decaying Bogoliubov’s excitation

Γ = uΦ + vΦ†, (20)

which diagonalizes the BdG Hamiltonian for the decaying wave. Putting everything together,we rewrite the Schrödinger equation (18) as follows:

HBdG(k → iq)

(uv

)=

(−µ− t cosh q ∆ sinh q−∆ sinh q µ+ t cosh q

)(uv

)= 0 ·

(uv

). (21)

Now what remains is simple linear algebra. Since the eigenvalue is known (E = 0), theeigenvalue equation determines q factor which we are interested in:

− (µ+ t cosh q)2 + ∆2 sinh2 q = 0⇒ µ+ t cosh q = ±∆ sinh q. (22)

Solving the eigenvector problem we find that for the ± sign in equation (22) the corre-sponding spinor takes the form (u, v)T = (1,±1)T . For the reasons that will become clearlater, we will denote these spinors as ΓA and ΓB respectively.

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For the q it suffices to consider only the ′+′ case, since the ′−′ would immediately followfrom the former upon the substitution q → −q due to the parity properties of the hyperbolicfunctions.

Solving the quadratic equation obtained from (22)

t+ ∆

2e−2q + µe−q +

t−∆

2= 0 (23)

with respect to e−q, we arrive to

e−q± =−µ±

√µ2 − t2 + ∆2

t+ ∆. (24)

In this way, we obtained two linearly independent decaying waves e−q± and thus a generaldecaying solution will be a linear combination of these exponents:

α+e−q+ + α−e

−q− , (25)

where coefficients α± are determined from the boundary conditions. Next, we recall thatthis e−q± were calculated for spinor ΓA = (1, 1)T . Therefore, the excitation (20) is given by:

ΓA = Φ + Φ† =∑x

(αA+e

−q+x + αA−e−q−x

) (cx + c†x

)︸ ︷︷ ︸γA,x

=∑x

(αA+e

−q+x + αA−e−q−x

)γA,x. (26)

Analogously for spinor ΓB, where q± have to be substituted by −q±:

ΓB = Φ− Φ† =∑x

(αB+e

q+x + αB−eq−x) (cx − c†x

)︸ ︷︷ ︸iγB,x

=∑x

(αB+e

q+x + αB−eq−x)γB,x, (27)

where we put i in the coefficients on the way. To summarize, we obtained a general expressionof zero-energy excitations localized at the edges of the lattice:

ΓA =∑

x

(αA+e

−q+x + αA−e−q−x

)γA,x

ΓB =∑

x

(αB+e

q+x + αB−eq−x)γB,x

(28)

From γA,x = γ†A,x and γB,x = γ†B,x obviously follows that ΓA = Γ†A and ΓB = Γ†B andhence these are indeed Majorana Zero Modes. Moreover, from the form of the exponents weimmediately realize that ΓA sets the mode localized at the left edge, whereas ΓB correspondsto the right one.

Equations (28) are the general expression for decaying states, where the unknown coef-ficients are to be determined by the boundary conditions. Now we have to prove that forany set of parameters in the trivial regime one cannot find αA+, αA−, αB+, αB− s.t. the boundaryconditions are satisified. On contrary, we will show that for parameters corresponding to thetopological phase such coefficients exist and therefore MZMs described by (28) appear.

For defining open boundary conditions let us artificially extend the lattice model byputting extra sites x = 0 and x = N + 1. Since these sites do not belong to the wire,

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...

c1

γ1,A γ1,B

c2

γ2,A γ2,B

c3

γ3,A γ3,B

cN

γN,A γN,B

...γ1,A γ1,B γ2,A γ2,B γ3,A γ3,B γN,A γN,B

ΓA ΓB

...

c1

γ1,A γ1,B

c2

γ2,A γ2,B

c3

γ3,A γ3,B

cN

γN,A γN,B

a)

b)

c)

Figure 2 a) Trivial pairing in the Kitaev chain corresponding to the Hamiltonian(14). b) Topological pairing for the specific case µ = 0, t = ∆ 6= 0 given by theHamiltonian (15). Two decoupled Majorana Zero Modes at the edges constitute adelocalized fermion. c) Topological pairing for an arbitrary choice of µ, t, ∆ withinthe topological phase. In this case, Majorana Zero Modes ΓA, ΓB are describedby collective excitations (28) of all Majorana Modes γA,x, γB,x in the chain. Theresulting wavefunctions are localized by the edges and exponentially decay into thebulk.

there is no electron density, therefore the electronic wavefunction must be zero at x = 0 andx = N + 1. In terms of operators this means that ΓA should not comprise a term at x = 0,whereas ΓB at x = N + 1.

Consider first the trivial phase, i.e. |µ| > t. Then, from (24) it follows that either|e−q+| > 1, |e−q−| < 1 or |e−q+| < 1, |e−q− | > 1. Without loss of generality, assume, forinstance, |e−q+| > 1, |e−q− | < 1 and consider the Majorana Zero Mode ΓA:

ΓA =∑x

(αA+e

−q+x + αA−e−q−x

)γA,x. (29)

We see that while the term e−q−x vanishes in the limit x 1, the term e−q+x exponentiallydiverges. Therefore, if the corresponding coefficient αA+ is non-zero, the wavefunction of ΓAis not normalizable due to the exponential divergence. This implies that αA+ must be zero.But then, another coefficient αA− must be also zero in order to satisfy the boundary conditionfor ΓA at the left edge of the chain: αA+ + αA− = 0. Therefore, the total solution is identicalto zero, which is non-physical. The case of ΓB can be considered in the same way. Hence,we conclude that in the trivial regime no zero-energy edge state can exist.

In contrast, for the topological phase |µ| < t and ∆ 6= 0 from (24) we have |e−q+| <

14

1, |e−q−| < 1. The boundary conditions can be then satisfied for both ΓA and ΓB at theleft and right edges respectively by setting αA+ + αA− = 0 and αB+eq+(N+1) + αB−e

q−(N+1) = 0.Therefore, in the topological phase a localized zero-energy solution always exists and thiscompletes the proof.

In this way, we have demonstrated that for any set of parameters in the topological phasethere would be always Majorana Zero Modes (MZMs) localized at the edges. Apart from thelimiting case µ = 0, t = ∆ 6= 0 those states would exponentially decay into the bulk of thewire. In the proof above we implicitly assumed that N →∞ and thus MZMs are decoupled.In principle, however, they can be interacting with each other via an effective term in theHamiltonian:

Heff ∝i

2e−|q|NΓAΓB, (30)

where |q| is taken as the smallest of |q+| and |q−|. This term should be understood fromthe phenomenological point of view. Indeed, if the Majorana Modes are coupled to eachother we would expect to have a term in the Hamiltonian of the same form as in (15).The only question is what the corresponding coupling parameter t should be. Since thecoupling is proportional to the overlap of the left and right Majorana Zero Modes, it can becharacterised by the damping factor e−|q|N of the MZMs wavefunction. From this, we writedown an effective coupling term as in (30). However, we note that for a rigorous derivationone has to directly diagonalize the Kitaev Hamiltonian (1) either analytically or numerically.The term (30) in the Hamiltonian shifts the MZMs from the zero energy, yet upon sufficientwire length N , this shift is exponentially negligible.

In the end of our discussion of the Kitaev model, we will study the concept of topologicalphase transition in more detail and elaborate on the notion of Domain Walls which separateregions of different topological phases in real space. We will explicitly demonstrate that anemergence of a domain wall implies emergence of a Majorana Zero Mode pinned to the wall.

To begin with, let us consider the system close to the phase transition point µ = −t.At this transition point from the dispersion relation (7) it follows that the gap is closed atthe momentum k = 0. Therefore, the physics of the phase transition can be understood inthe vicinity of k = 0. By invoking the BdG Hamiltonian (4) and expanding it for k ∼ 0we arrive to a Hamiltonian of the following form (for simplicity we assume that the pairingpotential ∆ is real: φ = 0):

HBdG(k) =

(−µ− t cos k i∆ sin k−i∆ sin k µ+ t cos k

)k→0−−→

(−µ− t i∆k−i∆k µ+ t

)(31)

This type of Hamiltonian is usually called Dirac-like because it is linear in momentum andresembles the Dirac Hamiltonian for the relativistic electron. To move further with thisanalogy, we also denote −µ − t as the Dirac mass m. Therefore, a negative mass m < 0corresponds to the trivial phase of the Kitaev model (M = +1), a positive mass m > 0corresponds to the topological phase (M = −1), while m = 0 is the singularity point of thephase transition where the topological indexM is ill-defined.

Assume now, that m can vary monotonically in space and changes the sign at x = 0.More specifically, let us define m as a function m(x) with the following properties:

15

limx→±∞

m(x) = ±M, m(0) = 0, M > 0. (32)

Physically this means, that for x < 0 the system is in the trivial regime, for x > 0 thesystem is in the topological regime and at x = 0 a domain wall is formed. Now, we wouldlike to demonstrate that there exists a zero-energy state localised at the domain wall. Let usfind this bound state by recasting the BdG Hamiltonian (31) in real space k → −i∂x. TheSchrödinger equation for a zero-energy solution then reads:

HΨ(x) = 0 ·Ψ(x) ⇐⇒ (−∆τyi∂x +m(x)τz) Ψ(x) = 0. (33)

This is an equation that we need to solve. In order to simplify it, we multiply the equationby τy from both sides and arrive to the first-order differential equation

∂xΨ(x) =m(x)

∆τxΨ(x), (34)

which can be easily integrated via the matrix exponential. The resulting two linearly inde-pendent solutions are then given by

Ψ(x) = exp

±∫ x

0

m(x′)

∆dx′(

1±1

). (35)

Since m is changing the sign when crossing zero from minus to plus, we pick up the solutionwith the − sign to fulfil the normalizability requirement. In this way we found an exponen-tially localized solution pinned to the domain wall. Therefore, we proved the statement thatif there are topologically distinct phases connected to each other, at the boundary there willbe always a Majorana Zero Mode.

Physical Implementation of the Kitaev ModelIn the previous section we discussed the Kitaev model which introduces the notion of topo-logical phenomena and at least qualitatively predicts the emergence of a specific type ofquasiparticles - Majorana Zero Modes. The whole discussion, however, was substantiallybased on the fact that somehow one is able to pull out only one species of electrons andinduce the Cooper pairing among it. This requirement, while being straightforward, cannotbe trivially implemented in real systems. Firstly, there is no consensus on whether mate-rials with intrinsic triplet pairing are found in nature, albeit some candidates are currentlystudied [27]. Secondly, an artificial platform for hosting p-wave Cooper pairs should be eas-ily tuned in experiments. It took almost a decade since the Kitaev proposal was releasedbefore several research theory groups realized how to implement p-wave superconductivityin technologically available materials. It turned out that a semiconductor nanowire with asufficient spin-orbit interaction proximitized by a standard s-wave superconductor leads toan effective p-wave superconductivity and therefore could host MZMs (Fig. 3a).

In general, III-V semiconductor nanowires enables one-dimensional ballistic transport,however the main reason to use them is their intrinsic spin-orbit coupling and substantial

16

g-factors. The former is crucial for creating a pseudo-spin which spin-degeneracy can belifted via the Zeeman field without destroying the Cooper pairing. The later allows applyingminimal magnetic fields that do not affect the parent superconductor which induces theCooper pairing inside the Nanowire. In this section, following original works [2, 3], wereview theoretical models describing the system and demonstrate emergence of MajoranaZero Modes in physically realistic systems.

We start with the effective Hamiltonian for the Rashba semiconductor nanowire writtenin the second quantization formalism (~ = 1):

HSM =

∫dxΨ†(x)

[− ∂2x

2m− µ− iα∂xσy + VZσ

x

]Ψ(x), (36)

where Ψ(x) = (ψ↑(x), ψ↓(x)) is an electron spinor in the σz basis, m is the effective electronmass, µ is the chemical potential. α is the strength of the spin-orbit interaction which favorsaligning of the electron spin along the y-axis, while VZ is the Zeeman splitting energy arisingfrom the parallel magnetic field and orienting the spin along the x-axis. The band dispersionis then given by plugging plain-wave solutions Ψ(x) = φ(k)eikx in (36) and diagonalizing theHamiltonian:

ε±(k) =k2

2m− µ±

√(αk)2 + V 2

Z . (37)

The corresponding spinor components then read:

φ±(k) =1√2

(±γk

1

), (38)

where γk = (iαk+VZ)√V 2Z+α2k2

. In the limit VZ = 0 the energy dispersion consists of two parabolas

centered at ±kSO = ±mα (Fig. 3b). Although we manage to effectively lock the electronspin to the momentum, the Fermi surface is still degenerate in spin and therefore no p-wavesuperconductivity is possible. However, this degeneracy is lifted upon setting VZ 6= 0 (Fig.3c). In this way, we construct two energetic bands in each of those the spin orientation islocked to the momentum k. When the chemical potential is tuned to lay inside the gap atk = 0 only the lower band is occupied and hence the system is effectively spinless.

Assume now that the interface between the semiconductor nanowire and the parent s-wave superconductor is clean enough so that the order parameter ∆ penetrates into the coreof the nanowire. Thus, we can write down the corresponding s-wave coupling within themean-field BCS formalism:

HSC =

∫dx[∆ψ†↑ψ

†↓ + h.c.

](39)

In order to explicitly demonstrate emergent p-wave superconductivity we will recast the totalHamiltonian HSM + HSC in the basis of the helical states (38). Thus, we introduce fieldoperators ψ†±(k) which create an electron at momentum k with the spinor φ±(k) and energyε±(k). In this basis the total Hamiltonian reads

17

s-wave superconductor, Δ

x

z

1D Rashba semiconductor, α

Bx

γ1 γ2

a)

−kSO +kSO

µ

k

E

b)

2VZ

+ _

c)

Figure 3 a) Schematics of the Rashba nanowire proposal. A one-dimensional ballis-tic nanowire is proximitized by an s-wave superconductor. The Zeemen effect liftsthe spin degeneracy and thus effective p-wave superconductivity is created mani-festing the emergense of two Majorana Zero Modes by the edges of the nanowire.b) Dispersion relation of the Rashba nanowire for α 6= 0, VZ = 0. c) Dispersionrelation of the Rashba nanowire for α 6= 0, VZ 6= 0

H = HSM +HSC =

∫dk

2π

[ε+(k)ψ†+(k)ψ+(k) + ε−(k)ψ†−(k)ψ−(k)

]+

+

[∆p

+(k)

2ψ†+(k)ψ†+(−k) +

∆(p)− (k)

2ψ†−(k)ψ†−(−k) + ∆(s)(k)ψ†+(k)ψ†−(−k) + h.c.

],

(40)

where

∆p±(k) =

∓iαk∆√V 2Z + α2k2

, ∆s(k) =VZ∆√

V 2Z + α2k2

(41)

are the intraband and interband Cooper coupling respectively. We note that ∆p(k) ∝ ∆kand therefore it is an odd function in momentum k, thus indeed establishing the p-wavecoupling. The Hamiltonian (40) should be again diagonilized and hence we, as in the previoussection, have to invoke the Nambu basis Ψ(k) = (ψ+(k), ψ−(k), ψ†+(−k), ψ†−(−k))T . Thecorresponding BdG Hamiltonian is then given by the following matrix:

HBdG(k) =

ε+(k) 0 ∆p

+(k) ∆s(k)0 ε−(k) −∆s(k) ∆p

−(k)

∆p†− (k) −∆s†(k) −ε+(−k) 0

∆s†(k) ∆p†+ (k) 0 −ε−(−k)

. (42)

Since it is a 4 × 4 matrix, it has four eigenenergies corresponding to two electron-like andtwo hole-like Bogoliubov’s excitations. The eigenenergies are given by

18

electrons

holes

a) α 6= 0, VZ = 0, ∆ = 0

k

E(k

)

b) α 6= 0, VZ 6= 0, ∆ = 0

k

E(k

)

c) α 6= 0, VZ 6= 0, ∆ 6= 0

Figure 4 The dispersion relation (43) plotted for different sets of the parameters.Because of the particle-hole symmetry the bands are artificially doubled and can beregarded as electron-like (blue) and hole-like (red).

E2±(k) = |∆p

+(k)|2 + ∆s(k)2 +ε2+(k) + ε2−(k)

2±|ε+(k)− ε−(k)|

√∆s(k)2 +

(ε+(k)− ε−(k)

2

)2

.

(43)If we now plot the eigenenergies in the limit ∆ = 0 and for ∆ 6= 0 we will clearly see that thepairing potential lifts the degeneracy at the Fermi points (Fig. 4). However there emergesa competition between the Zeeman splitting energy VZ and the pairing potential ∆ whichmanifests itself in closing the gap at the momentum k = 0. From the dispersion relation(43) we obtain the condition for closing the gap at the momentum k = 0:

VZ,critical =√

∆2 + µ2, (44)

which defines the critical magnetic field and therefore the topological phase transition. In-deed, as we realized in the previous section any closure of the gap highlights the singularityof the phase transition, however as long as the symmetries of the system are preserved thephase transition must be topological. Let us show that the condition VZ > VZ,critical impliesthe emergence of the MZMs and therefore defines the topological regime. More specifically,we need to show that under this condition the system is smoothly connected to the Kitaevtoy model in the continuum limit, i.e. when the lattice constant a→ 0.

Indeed, in the regime VZ > VZ,critical we can pick the magnetic field VZ s.t. VZ >>VZ,critical ∼ mα2, ∆. Therefore, the upper band is projected away from the lower-energyproblem. In this way, neglecting high-energy excitations ψ†+(k), we arrive to the followingeffective Hamiltonian:

Heff =

∫dk

2π

[ε−(k)ψ†−(k)ψ−(k) +

∆p−(k)

2ψ†−(k)ψ†−(−k) + h.c.

]. (45)

The BdG Hamiltonian for this problem is then given by

19

HBdG =

(ε−(k) ∆p

−(k)

∆p†− (k) −ε−(−k)

), (46)

which is exactly the Kitaev Hamiltonian (4) in the continuum limit (cos k → 1− k2

2, sin k →

k). As we explicitly demonstrated in the previous section, under certain parameters the Ki-taev chain model hosts MZMs and therefore the Rashba nanowire model in the lower-energylimit should host them as well. Since the emergence of MZMs is a topological phenomena,their existence shall be extended to the whole VZ >

√∆2 + µ2 domain.

This asymptotic connection to the Kitaev model turns out to be quite useful. Indeed, nowwe can use the fact that at the boundary between distinct topological phases in the nanowirethe edge states emerge. This observation lies at the root of hardware and computationalprotocols of a potential topological qubit.

Towards Braiding in the Nanowire NetworksSince the beginning of the century, Majorana Zero Modes arouse significant interest in thecondensed matter community not only because of their topological protection and fractionalnature but also because of plausible applications in the field of quantum computing. Thelater stems from yet another surprising property of MZMs, namely non-Abelian statistics,i.e. a non-trivial change of the wavefunction via the spatial exchange of the Majorana Modes.

First, the non-Abelian statistics of Majorana Modes was proven in two dimensions [28],where the MZMs pinned to the vortices in a topological superconductor revolve around eachother thereby changing the overall wavefunction in a non-Abelian manner. Although theproposal is rather simple, the technical implementation of the two-dimensional topologicalsuperconductivity so far seems to be unrealistic. The Fractional Quantum Hall platforms atthe filling factor ν = 5/2 are very sensitive to the electron doping and therefore cannot beeasily tuned. On the contrary, one-dimensional nanowires demonstrate high tunability andone might expect more flexibility in operating MZMs in such platforms. However, being one-dimensional, a nanowire itself cannot be used for performing an exchange, in other words,braiding, of the MZMs, since on the way the Majorana Modes would inevitably meet eachother and thus annihilate to a normal electron.

This obstacle was overcome with the proposal of the T-junction geometry [6]. Assumethere are electrostatic gates beneath the nanowire, thus one can tune the chemical potentialand move the boundary of the distinct topological phases (Fig. 5a). Hence, the MajoranaZero Mode will be also shifting. The braiding protocol could be then written down in thefollowing way (Fig. 5b). We start with two Majoranas positioned in the horizontal partof the T-junction. Upon changing the electrostatic potential, one shifts the left Majoranato the center. Then, by switching off the topological regime in the left arm and graduallyswitching it on in the middle arm, one brings the left Majorana down. In the same way, theright Majorana Mode is transferred to the left and the down Majorana can be positioned atthe right end.

This braiding protocol majorly relies on the fact that we can precisely control the emer-gence of the topological phase in T-junctions. We recall from the Rashba nanowire model,

20

γ1 γ2

γ1

γ1

γ1

γ1

γ1γ2

γ2

γ2

γ2

γ2a)

b) c)

d) e)

Figure 5 a) Keyboards of gates beneath Rashba nanowires, upon locally tuningthe chemical potential at each gate, can change the position of the domain walland therefore shift MZMs in a controllable manner. b) Protocol for exchangingMajorana Zero Modes in T-geometry. The figure is adapted from [6].

however, that this requires inducing the Zeeman energy inside the core of the nanowire.Keeping in mind, that the parent s-wave superconductor must be placed in the proximity tothe nanowire, applying perpendicular magnetic fields would destroy the superconductivity,and therefore only parallel magnetic fields are allowed. Therefore, a non-trivial experimen-tal challenge arises - how to simultaneously apply magnetic fields in mutually orthogonaldirections. This problem motivates the present work.

21

Single-Electron Transport in QuantumDots

In this section we review basic notions related to quantum dots that we subsequently use inour measurements and analysis of the data. We start with the constant interaction modelthrough which we describe the transport phenomenology for the first-order transitions andintroduce the important concept of Coulomb blockade. Next, we briefly discuss a qualitativepicture of the higher-order processes and particularly the Kondo effect.

Quantum DotAn electron quantum dot in a broad perspective is a zero-dimensional region in space wherean electron can be trapped. By zero-dimensionality we imply that the corresponding length-scales of the system are of the order of the electron’s de Broglie wavelength. Therefore, theconfinement of electrons inside a quantum dot entails two physically distinct phenomena.The first one has a purely quantum origin, that is, a confined electron would have quantizedsingle-particle energetic states. The second, however, stems solely from the classical electro-dynamics reasons - there is always Coulomb repulsion between charged particles and thus,being trapped in close proximity, electrons repulse from each other.

In general, the problem of strongly-correlated electrons is extremely complicated andinvolves advanced theoretical techniques. In some cases, however, an effective model can beintroduced. The most famous one is the constant interaction model for a single quantumdot [29] which we now will review.

The model can be effectively constructed in the following way. First, we characterizethe interactions among electrons inside the dot as well as between electrons in the dot andthe outer environment by a constant capacitance term C = CS + CD + CG, where CS, CD,and CG are capacitances between the dot and the source, drain and gate respectively (Fig6a). Second, we assume that the single-electron energy-level spectrum is not affected by thepresence of other electrons. Therefore, the total energy U(N) of the dot is given by thefollowing phenomenological expression:

U(N) =(−|e|(N −N0) + CSVS + CDVD + CGVG)2

2C+

N∑n=1

En(B), (47)

where N0 represents the screening charge density that compensates the environmental pos-itive charge; CSVS, CDVD and CGVG account for induced electrostatic charges in the leads,

22

Figure 6 a) Constant interaction model schematics. A quantum dot is capacitivelycoupled to the source (S), drain (D) and gate (G) with a total capacitance C. Thefigure is adapted from [30]. b) Ladder of electrochemical potentials in the Coulombblockade regime. c) Ladder of electrochemical potentials in the electron transportregime d) Coulomb peaks. The figure is adapted from [29].

and the last term is the the total energy of the occupied single-electron energetic states.Since we are primarily interested in transport properties, it is more convenient to pass

to description in terms of electrochemical potentials µ(N). Indeed, in order to transfer anelectron through a quantum dot we need to pay an energy difference given by:

µ(N) = U(N)−U(N − 1) =

(N −N0 −

1

2

)EC −

EC|e| (CSVS +CDVD +CGVG) +EN , (48)

where EC = e2/C is the charging energy of the quantum dot and EN is the chemical potential.In this way, we have demonstrated that a quantum dot within the constant interaction modelcan be effectively described by a ladder of electrochemical potentials (Fig 6b,c).

Single-Electron TransportNow we would like to investigate the features of electron transport through a quantumdot. Assume we apply a bias voltage VSD which shifts the electrochemical potentials in thesource and drain leads such that µS − µD = VSD. In this way, an electron can tunnel onlyif µS ≥ µ(N) ≥ µD (Fig 6c), which results in a peak of the current usually dubbed as aCoulomb peak. The precise shape of the Coulomb peak depends on the transparency ΓS, ΓDof the leads. When there is no electrochemical potential level within the bias window, nocurrent can pass through and the system is said to be in the Coulomb blockade regime (Fig6b). As long as we can tune the ladder of electrochemical potentials by applying the gatevoltage VG, we can successively increase or decrease the number N of electrons inside the dot

23

Figure 7 Stability diagram. The shaded area correspond to dI/dV = 0, while redand blue lines denote the border with regions of the current through the excitedstates. The figure is adapted from [31].

(Fig 6d). The difference between successive electrochemical potentials, and thus successiveCoulomb peaks, is given by:

Eadd(N) = µ(N + 1)− µ(N) = EC + ∆E, (49)

where ∆E depends on the degeneracy of the single-electron energy level. More specifically,it is 0, when the N + 1 electron can occupy the same state as the N electron, and nonzeroelse.

Upon opening the bias window, we eventually will come to the point when −e|VSD| >Eadd. This implies that at any gate voltage VG there would be current flowing and thereforethe Coulomb blockade regime is lifted. If we now plot the current I as a function of VGand VSD we obtain a stability diagram for the quantum dot under consideration. Mostof the times it is more convenient to plot the differential conductance dI

dVinstead of the

current I. In this way, the stability diagram looks like in Fig. 7. The shaded regions,called Coloumb Diamonds, denote the absence of the conductance and therefore a definednumber of electrons can be assigned to each of these domains. Since a diamond closes atVSD = Eadd

e, the height of the diamond would tell us the energy difference between successive

electrochemical potentials. Moreover, the Eadd can be also retracted from the voltage spacingbetween adjacent Coulomb peaks ∆VG by

Eadd

e= α∆VG, (50)

where α is the level-arm between the gate and the quantum dot and denotes coupling prop-erties.

Finally, since it often occurs that EC > Eorb, electron transport through an excited statein the dot is possible. This transport is denoted by a characteristic line in the stability

24

diagram which is parallel to the corresponding Coulomb diamond and indicate a transitionto the region with an enhanced current. From this, an orbital energy Eorb can be extracted(Fig 7).

Kondo EffectIn the last part of the section we qualitatively discuss higher-order cotunneling processeswhich result in emerging conductance through a quantum dot in the Coulomb blockaderegime. There are two possibilities of how an electron can tunnel through a quantum dotin the blockade regime, namely without dissipation (elastic scattering) and with dissipation(inelastic scattering). Here we concentrate on elastic scattering [32].

Within elastic scattering events there are two distinct phenomena that occur during thecotunneling. Both of them are depicted in Fig 8. The first one is related to the creationof a virtual state of a lifetime τ in which an electron can tunnel out from the quantum dotaccording to the Heisenberg uncertainty principle. More precisely, if an electron occupies astate of the energy ε0 below the Fermi level of the leads for the time τ = ~/ε0 it can overcomethe energetic barrier.

Figure 8 Higher-order tunneling process resulting in the enhanced conductance atthe zero-bias window. a) Tunneling in case of the even occupancy of the dot. b)Tunneling in case of the odd occupancy of the dot. The figure is adapted from [32].

25

Figure 9 a) Conductance for odd and even quantum dot occupancies in differenttemperature regimes. b) Zero-bias Kondo peak with a characteristic width TK . Thefigure is adapted from [32].

Therefore, elastic cotunneling can take place for both even and odd quantum dot occu-pancies (Fig 8a,b). For temperatures higher than a characteristic temperature TK , whichphysical sense we will discuss later, the conductance through even and odd occupancies is thesame. The difference between these two cases becomes prominent when we begin decreasingthe temperature of the system (Fig 9a). In the limit of low-temperatures (Kondo regime)T TK the cotunneling within an even occupancy is strongly suppressed whereas for anodd occupancy it is enhanced.

The reason behind this behavior can be intuitively explained in the following way. Foreven occupancies, the density of electronic channels participating in the tunneling processesis proportional to the temperature ∼ kBT since only these electrons can be taken from theleads. Therefore, upon decreasing the temperature the number of the conduction channelsis reducing which leads to suppressing the overall conductance.

The case with an odd occupancy is, however, strikingly different. An unbalanced spinin the quantum dot creates a screening cloud of spins of the opposite direction in the leadswhich manifests the emergence of a strongly-correlated many-body state. The screeningitself implies occurrence of spin-flipping processes through the quantum dot similar to thosein metals with magnetic impurities. According to the original Kondo problem in metals [33],the spin-flipping processes contribute to new resonant levels which enhance the scatteringcross-section and lead to a logarithmic divergence of the resistance. In a quantum dot, on theother hand, the same spin-flipping resonant levels imply an enhancement in the conductance.Indeed, while there were no scattering channels in the quantum dot in the Coulomb blockaderegime, with the emergence of the Kondo many-body state new transmission channels werecreated. This leads to similar logarithmic divergence of the conductance which, however,should saturate to the quantum conductivity G0 = 2e2/h.

As mentioned before there is a characteristic temperature TK below which the processdiscussed above can take place. From the spectroscopic point of view the Kondo effect leads

26

to the formation of a zero-bias peak. It can be demonstrated [34] that a width of this peakat half maximum is exactly the characteristic temperature TK , which in literature is usuallycalled the Kondo temperature.

27

Experimental Section

In the present chapter, we discuss experimental work performed at the Institute of Scienceand Technology Austria during January-May 2020. Firstly, we discuss the material we workwith and thereby present specific motivation for the studies of transport in ferromagneticinsulator (FI) - semiconductor (SM) hybrid devices. Secondly, we provide the full nanodevicefabrication recipe. Next, we outline the major components of the experimental setup thatwe use for low-noise cryogenic transport measurements. Finally, we share our data on thecharacterization of the transport in the devices and analyze it.

Exchange Field in InAs/EuS Hybrid DevicesIn previous sections we described nanowire networks schemes for performing non-Abelianexchange protocols. The major experimental obstacle in the realization of such schemesis the necessity of applying mutually orthogonal magnetic fields, which inevitably wouldalter the topological regime in the corresponding nanowires. Moreover, it is generally morepreferable to have a platform which is intrinsically topological and cannot be easily tuned outto the trivial regime due to the material properties. This line of thought, therefore, motivatesexploiting local magnetic fields that emerge as the result of the proximity to ferromagneticmaterials [35].

While usually a ferromagnet is a metal, recent technological advances in fabricating FIsopened a pathway for coupling it to a semiconductor/superconductor without shortening.Assuming a clean interface, the resulting exchange interaction could appear quite strong. Forinstance, the magnetic proximity effect at the interface has been reported to reach effectivefields more than 10 T in graphene [36].

In our experiments we use InAs nanowires grown via the vapour-liquid-solid method [23]with a molecular beam epitaxy grown layer of EuS [7]. The average length of the nanowiresis 10 µm, the InAs core has a hexagonal cross-section with a diameter of 100nm, and thethickness of the EuS facets is around 10nm (Fig 10a). A less than 1% mismatch betweenthe InAs and EuS lattices leads to a clean interface which is important for the enhancedproximity effects. In the beginning we considered two batches of the nanowires - one with afull EuS shell and one which was partially covered. Since the later did not demonstrate anygate-tunability, we concentrated on the full-shell batch.

The ferromagnetic properties of the EuS shell have been already studied at the Universityof Copenhagen via the SQUID magnetometry technique [7]. Imaging the stray field ofthe nanowire shell revealed a weak field close to the middle of the nanowire and strong

28

dipole-like field at the edges. This measurement indicates a single ferromagnetic domainstructure, where the aggregated magnetic moment is aligned with the VLS growth axis. Thetemperature dependence measurements demonstrated the formation of a ferromagnetic orderat 19 K (higher than the Curie temperature of the bulk EuS: TC ∼ 16− 17 K).

Since the ferromagnetic order of EuS can penetrate through the material interface dueto proximity effects, the EuS layer creates an effective magnetic field, which in literatureis usually dubbed as the exchange field. The exchange field originating from the EuS layerhas been investigated in both InAs and Al films via the X-ray Magnetic Circular Dichroism(XMCD) [24] and tunneling spectroscopy [37] methods, respectively. The XMCD measure-ments, which indicate the ratio between scattering amplitudes of spin-polarized neutrons, didnot show any induced magnetic order inside the InAs core within detection limits. In con-trast, the tunneling spectroscopy revealed substantial spin-splitting in the density of statesof Al resulting in the effective exchange field Beff ∼ 1T.

Although tricrystal systems InAs/EuS/Al such as in [37] exhibit signatures of topologi-cal superconductivity and Majorana Zero Modes, which implies the presence of the Zeemaneffect in the InAs core, the mechanism of the magnetic field penetration into the core re-mains unknown. As a speculative hypothesis it is now believed that the EuS layer inducesspin-splitting in the Al layer, which in turn proximitizes the InAs core with the spin-split su-perconductivity. We note, however, that there is currently not enough experimental evidencesupporting or refuting this hypothesis.

Moreover, the XMCD method does not account for conduction electrons due to theirlow-density compared to the total density of all electrons inside the InAs core. Therefore,even in the case of strong exchange interaction between the conduction electrons of InAsand the magnetic moments of EuS this coupling will be averaged out in XMCD. Thus,a systematic study of the exchange field inside the InAs core via transport measurementscurrently remains an actual and unresolved problem.

This said, the goal of the present work is the following. We aim to create a quantum dotinside an InAs/EuS nanowire and investigate the spin-slitting properties due to the exchangeinteraction with the EuS layer via single-electron transport techniques.

Device FabricationSince InAs/EuS full-shell nanowires are grown vertically on the substrate, we have to transferthem on a prepatterned Si/SiO2 basechip for the subsequent EuS etching and metal depo-sition. The chip comprises a layer of 500µm thick n-doped Si with a 285nm cap of SiO2

and has the dimensions of 6 mm× 6 mm. Being degenerately doped, the Si layer is usuallyused as a back gate. Cr(15nm)/Au(70nm) bonding pads, connection links and markers aredeposited on the chip via standard e-beam lithography techniques discussed below. Theinter-marker distance used initially is 20µm (Fig. 10b), while the more recent design hasmarkers separated by 5µm. Before transferring the nanowires, a prepatterned chip is cleanedby ultrasonic agitation in acetone for 2 minutes with subsequent rinsing in isopropanol (IPA).Finally, the residual dirt is removed via a Diener Pico RF CE low-pressure plasma asher.

We transfer the nanowires by a 0.1µm-diameter tungsten needle which drags the nanowires

29

InAs

EuS

100nm

10nm

a)

bonding pads

markersfield

20μm

b)

10μm

c)

Figure 10 a) Full-shell InAs/EuS nanowire. b) Design of the prepatterned markers,connections and bonding pads. c) SEM micrograph of a nanowire on a chip. Theposition of the nanowire is given with respect to the markers

via the Van der Waals forces. The needle can be moved with a micrometer precision by anEppendorf nanomanipulator, while the overall localization is monitored by a NIKON Micro-scope L 200N. After the nanowires are positioned on the chip, Scanning Electron Microscopy(SEM) imaging at a FEI TENEO SEM machine is performed (Fig 10c). In this way, we ob-tain precise positioning of the nanowires and can prepare a design for the gates and etchingwindows.

Before an e-beam lithography session a 220nm - thick layer of polymethyl methacrylate(PMMA) 950K 4% resist is put on top. It is done by spinning a drop of the resist for 60seconds at the speed of 6000rpm. The resist layer is afterwards baked at the temperature of115C for 5 minutes. For the e-beam lithography we use a RAITH EBPG5150 machine andfirst lithographically write the areas from which EuS should be etched away.

The chip is afterwards developed by immersing it in 1:3 methyl isobutyl ketone (MIBK):IPA solution for 30 seconds with a subsequent cleaning by IPA for 1 minute. For etching theEuS layer we dip the chip into deionized water for 40 seconds and then clean with IPA for30 seconds. Finally, in order to remove the resist we use acetone for 130 seconds and IPAfor 30 seconds. A micrograph of a nanowire after etching is given in Fig. 11a.

We note that extensive calibration tests indicate that the current etching recipe is notreproducible. More specifically, we speculate that during the etching process the deionizedwater creeps beneath the resist and removes extra parts of the EuS layer. The creep rates varydramatically depending on the position of the nanowire on the chip and the total nanowiredensity. It turns out, that for well separated nanowires (average distance between NWs -60µm) along the edges of the marker field the creeping rate is 165nm/40sec ± 20nm/40sec,while for a dense arrangement (average distance between NWs - 25µm) in the center of thechip the creeping rate is 100nm/40sec ± 14nm/40sec. However, even this varies from deviceto device. Since the imaging of 250-400nm of the EuS layer that correspond to a formationof the quantum dot requires close exposure of the nanowire to the electron beam in SEMwhich can contaminate the insulating surface by trapping the electron charges, we did notdo the imaging after the etching. This in turn implied that during the fabrication we did

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not know the actual size of the EuS layer left. Therefore, in most of the cases we undertookelectrical measurements on bare InAs without any EuS layer on top, which was revealedonly during the post SEM imaging. Since the EuS layer should be etched precisely, with anacceptable error of ± 50nm from one side, one inevitably has to perform SEM imaging rightafter the etching, picking up suitable NWs.

After the etching step, we lithographically write the gates, source/drain leads, and con-nection wires on the 220nm thick resist layer (PMMA 950K 4%) and subsequently developit in the same way as for the etching windows. Then, we deposit 5nm of Ti and 180nm of Auvia the electron-beam evaporation system Plassys MEB550S, after firstly cleaning out oxidelayers by 150 seconds of Argon milling at 250 V. The lift off is done afterwards by leavingthe chip in acetone hot bath at a temperature of 50 C for 15 min and subsequently rinsingit in IPA. A micrograph of the resulting device is demonstrated in Fig. 11b.

Experimental SetupIn order to detect single-electron transport the thermal excitations in the system shouldbe maximally reduced. In our experiments we used both a BlueFors Dilution Refrigeratorand an H3Oxford HelioxVL refrigerator, which can reach base temperatures of 13 mK and280 mK, respectively. For loading a chip into a refrigerator, it is firstly glued to a printedcircuit board (PCB) by means of silver paste and next connected to the PCB pads via aF&S Bondtec 5330 wire bonder.

The measurement scheme used in our experiments is the following. We apply a biasvoltage Vsource to the Source contact of a nanowire and receive the output current I at theDrain. A current to voltage amplifier is then used and the resulting voltage is measured with adigital multimeter (DMM). We note, that the actual voltage drop Vbias at the nanowire differs

596 nm

a) b)

Figure 11 a) Zoom in on the remaining part of the EuS layer (magnification50000X). b) Complete nanowire device after the fabrication.

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from the applied bias voltage Vsource, since the PCB, electronic lines and I/V amplifier havetheir own resistances. The total resistance in the presented measurements was Rext = 32Ω.Therefore, data correction is performed in the following way:

Vbias = Vsource −RextI (51)

For all the electronic measurements we use a battery-powered IVVI rack developed at DelftUniversity. The scheme above was usually used for current measurements and rough charac-terization of devices. For the low-noise fine measurements, however, we employed an MFLIlock-in amplifier by Zurich Instruments. An AC excitation signal Vext is mixed into the DCvoltage Vsource which are both send to the sample. The measured current is again transformedvia the IV-converter into a voltage which is afterwards sent to the DMM and the lock-in.The lock-in finally gives out the device’s differential response dVlock to the excitation signal.The differential conductance can be then computed by the following formula:

dI

dV=dIlockdVreal

=1

A

dVlockVexc −RextdIlock

=1

A

dVlock

Vexc −RextdVlockA

, (52)

where A = dVlock/dIlock is the amplification factor of the IV-converter.

Measurements and AnalysisThroughout the present work we have fabricated and characterized 7 chips, each of thosehosting 7-10 InAs nanowires with a full-shell of EuS. For the first two devices only dryetching before the metal deposition was applied, which resulted in poor contact between thesemiconductor nanowire and the source/drain leads. Therefore, wet etching was consideredand, being inspired by the fabrication recipe of InAs/Al devices [5], for the third device weperformed etching with an Aluminum etchant type D by TRANSENE. Although the EuSlayer was completely removed, the transport measurements of the bare InAs core indicatedpromising single-electron transport signatures. Thus, we decided to keep wet etching andtried out various etchants that can remove parts of the EuS layer in a controllable manner.Eventually, we realized that deionized water can be used for controllable etching and hencewe fabricated four more chips aiming to create quantum dots covered by the EuS layer ofthe different lengths. However, since we do not undertake SEM imaging of the nanowiresright after the etching, we realized that three out of the four chips have no EuS layer onlyafter the transport measurements were conducted. Yet, these studies are important sincethey reveal main features of the transport in the bare InAs core and can be used for futureplanning of experiments. For the final device with 150 nm of the EuS layer we observed anunexpected transport behaviour, which we are currently unable to explain. In the presentsection, we report and analyze the data for two devices without the EuS layer as well as forthe device with the EuS layer.

In the Figure 12a we present a stability diagram for a quantum dot formed inside theInAs nanowire due to the Schottky band bending mechanism at semiconductor-metal in-terfaces. The distance between the Ti/Au contacts is 200 nm (Fig 12b, however, shows adifferent device with a quantum dot of 300 nm length, since the original device exploded

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electrostatically during unloading). Therefore, we estimate the length of the quantum dotto be about 200 nm. The source-drain voltage VSD is applied to one of the leads and thecurrent I is measured as discussed in the previous section at the opposite lead. We tunethe electrochemical potential inside the quantum dot via the potential VSG applied to theside gate positioned 50nms away from the nanowire. The measurements are conducted atthe base temperature of 280 mK and the data correction from the equation (51) was appliedafterwards.

First, from the stability diagram we observe the formation of the Coulomb diamondswith different closing bias voltages VSD. In order to get a rough estimation, we can assumethat EC = eVSD. Therefore, we conclude that the charging energy is decreasing with anincreasing number of electrons N in the dot. It starts from EC > 10meV at N and dropsdown to EC = 2.84meV at N + 15. From this we calculate the corresponding total dotcapacitance C = e2

ECand obtain values ranging from 16aF to 57aF. The measurements

indicate that the side gate can tune the barrier couplings at the edges of the quantum dot.Since the total capacitance of the dot C is not constant the simple constant interaction modelis not valid here.

Next, we can estimate the level arm α from the derivative of the slope α = EC/e∆VSG,where VSG is the voltage difference between adjacent Coulomb diamond apexes. In this way,we obtain a level arm α = 0.06 which enables us to calculate the gate capacitance Cg. Giventhat Cg = e/α∆VSG, we estimate the gate capacitance to be ranging from 7aF to 18aF.

As long as we are interested in features that can reveal the presence of magnetic field, it isimportant to estimate the orbital excitation energies of bare InAs. From the stability diagramwe conclude that Eorb = 1.9meV , which is smaller than the charging energy EC . Therefore,in the presence of magnetic field these additional conductance lines can be mixed up withthose coming from the Zeeman splitting which makes the subsequent analysis difficult.

Finally, we note that from the stability diagram it follows that the tunnel barriers at theedges of the quantum dot are of high transparency which makes it possible to observe secondand higher order transport properties such as inelastic cotunneling and the Kondo effect. Tomake it more pronounced we plot traces taken from the stability diagram at fixed side-gatevoltages VSG = −7.13V (cyan) and VSG = −7.29V (green). The resulting figure (Fig. 12c)clearly indicates a zero-bias peak and cotunneling steps. Since the zero-bias Kondo peakshould be split in the presence of magnetic field, we note that the present quantum dotconfiguration can be convenient for measuring magnetic fields.

In the next device (Fig. 13) we investigated properties of the Kondo peak in more detail.The device is again a 200 nm length quantum dot with the design similar to one depictedin Fig. 12b. We performed the stability diagram measurement at T = 280 mK in order tolocalize the diamond where the zero-bias peak emerges and then undertook a spectroscopymeasurement in the vicinity of the Kondo peak via the MFLI lock-in amplifier (Fig. 13b-right). As an AC excitation we send a signal with an amplitude of Vexc = 2.5µV and notethat in our case this is the minimal signal that still enables discerning the zero-bias peak.We calculate the conductance from the output lock-in signal according to the equation (52).

In order to prove that the detected zero-bias peak is indeed the Kondo peak, we take tracesat the fixed side-gate voltage VSG = −6.505 for different temperatures and observe a gradualdisappearance of the peak (Fig 13b-left). The peak is completely fused at a temperature

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Figure 12 A quantum dot measured in a H3 Oxford HelioxVL refrigerator at abase temperature of 280 mK. a) Stability diagram: the Coulomb diamonds with thecharging energy EC and orbital excitation energies Eorb. b) False-color micrograph,yellow color indicates Ti/Au contacts, while red color indicates InAs nanowire. c)Traces taken from the stability diagram at voltages VSG = −7.13V (cyan) andVSG = −7.29V (green) demonstrate features of the higher order tunneling processes.

around 1300 mK. From the full width at half maximum (FWHM) we can estimate a Kondotemperature [38] TK ∼ 3.43K. We also note that since the minimal excitation signal is 2.5µVthe minimal magnetic field that can be detected by measuring the splitting of the Kondopeak below the Kondo temperature can be estimated from the relation:

2µBgBeff = eVexc, (53)

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Figure 13 A quantum dot measured in a H3 Oxford HelioxVL refrigerator at abase temperature of 280 mK. a) Stability diagram with a prominent Kondo zero-bias peak. b) A zoom-in into the region of the Kondo peak and a temperaturedependence measurement at a fixed side-gate voltage VSG = −6.505V

where the g-factor can be roughly taken as 10. From this we obtain that the lowest detectablefield Beff is approximately 2.2 mT. Therefore, if the EuS layer induces in the InAs coreeffective fields comparable to those in Al, we conclude that in this scheme we would be ableto detect them.

Finally, we discuss the device with a layer of EuS. The SEM image of the device is givenin Fig. 14b. The quantum dot has a length of 400 nm and the EuS layer has dimensions of150 nm. The device was measured in a BlueFors dilution refrigerator at a base temperature

35

Figure 14 A 400 nm InAs island covered by 150nm of the EuS layer. a) Stabilitydiagram with a characteristic beating. b) False color micrograph c) Current vs sidegate voltage curves.

of 14 mK.The current stability diagram is given in Fig 14a (no data correction is undertaken).

We note that the resulting shape of the Coulomb diamonds resemble a beating betweentwo charge tunneling processes. There are two energy scales in the problem. The first onedescribes the envelope of the Coulomb diamonds and has a Coulomb diamond-like regionsperiodicity of ∆V1 = 600mV. The second corresponds to the beating signal with a peakperiodicity of ∆V2 = 50mV . Although, a physical explanation of this phenomena is currentlymissing, we speculate that we created multiple dots inside the nanowire. For instance, a band

36

bending is possible at the interface InAs/Eus and therefore quantum dots can be formedbeneath the EuS layer as well as to the right and level sides of it. The interplay betweencharge transferring through the dots might lead to the observed beating. This implies thatfor future devices we should consider reducing the dimensions of uncovered InAs and beprecise on the length of the remaining EuS layer.

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Conclusions

In the present work we studied transport properties of hybrid InAs/EuS nanowires. Afabrication recipe for quantum dot devices has been developed, yet at the moment it stillhas crucial flaws. More specifically, the dimensions of the remaining EuS layer after etchingare hard to anticipate due to the irreproducibility of the current recipe. Therefore, anintermediate step with SEM imaging seems to be unavoidable. To implement this we areplanning to change the design of the markers’ patterns on the Si/SiO2 chips. This wouldallow localization of the remaining layer of EuS with high precision and therefore a quantumdot can be formed in a controllable manner.

We note that it is also important to form a single quantum dot in order to avoid aninterplay between transport through multiple dots. A possible solution to this problem canbe achieved by depositing source-drain leads with a small overlap with the EuS layer leadingto formation of a quantum dot just below the EuS.

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