From Andreev to Majorana bound states in hybrid superconductor-semiconductornanowires

Elsa Prada1, Pablo San-Jose2, Michiel W. A. de Moor3, Attila Geresdi3, Eduardo J. H. Lee1,

Jelena Klinovaja4, Daniel Loss4, Jesper Nygard5, Ramon Aguado2, Leo P. Kouwenhoven3,6

1Departamento de Fısica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),Universidad Autonoma de Madrid, Madrid, Spain

2Instituto de Ciencia de Materiales de Madrid (ICMM),Consejo Superior de Investigaciones Cientıficas (CSIC), Madrid, Spain

3QuTech and Kavli Institute of Nanoscience, Delft University of Technology, Delft, Netherlands4Department of Physics, University of Basel, Basel, Switzerland

5Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, Copenhagen, Denmark

6Microsoft Station Q at Delft University of Technology, Delft, Netherlands.(Dated: October 6, 2020)

Electronic excitations above the ground state must overcome an energy gap in superconductorswith spatially-homogeneous s-wave pairing. In contrast, inhomogeneous superconductors such asthose with magnetic impurities or weak links, or heterojunctions containing normal metals or quan-tum dots, can host subgap electronic excitations that are generically known as Andreev bound states(ABSs). With the advent of topological superconductivity, a new kind of ABS with exotic quali-ties, known as Majorana bound state (MBS), has been discovered. We review the main propertiesof ABSs and MBSs, and the state-of-the-art techniques for their detection. We focus on hybridsuperconductor-semiconductor nanowires, possibly coupled to quantum dots, as one of the mostflexible and promising experimental platforms. We discuss how the combined effect of spin-orbitcoupling and Zeeman field in these wires triggers the transition from ABSs into MBSs. We showtheoretical progress beyond minimal models in understanding experiments, including the possibilityof different types of robust zero modes that may emerge without a band-topological transition. Weexamine the role of spatial non-locality, a special property of MBS wavefunctions that, togetherwith non-Abelian braiding, is the key to realizing topological quantum computation.

I. INTRODUCTION

Ever since Kamerlingh Onnes discovered the “zero re-sistance state” of metals at very low temperatures in 1911[1, 2], the superconducting state of matter [3, 4] has fas-cinated physicists. In the last century, the understand-ing of superconductivity has evolved extraordinarily andhas garnered eight Nobel prizes, turning it into one ofthe most iconic topics in condensed matter physics [5].As described by the seminal Bardeen-Cooper-Schrieffer(BCS) theory of superconductivity [6], the characteristicfeature of superconductors (SCs) is the macroscopic oc-cupation of bound pairs of electrons, known as Cooperpairs [7], in the same quantum-coherent ground state.The condensation of Cooper pairs into such ground stateis associated with a superconducting complex order pa-rameter ∆ = ∆eiϕ [8, 9], where ϕ is the conjugate of thenumber of Cooper pairs. In a homogeneous s-wave BCSSC, the spectrum of single-particle excitations above theground state develops an energy gap ∆. These gappedexcitations are propagating superpositions of electronsand holes with different energy-dependent weights. How-ever, if the order parameter –also called the pair potential[3]– varies in space, ∆(r), lower energy (‘subgap’) exci-tations may develop. Such is the case of states trappedin magnetic flux vortices (so-called Caroli-Matricon-DeGennes states [10]), at magnetic domains or impurities(Yu-Shiba-Rusinov states [11–13]), at weak links between

SCs or at normal metal-superconductor (NS) contacts[14], to name a few. Collectively, these subgap states aredubbed Andreev bound states (ABSs), and are the focusof numerous theoretical and experimental works, as wellas the basis of promising emerging quantum technologies,see Fig. 1.

The core physical mechanism behind the formation ofABSs in inhomogeneous systems with ∆(r) is a remark-able scattering process, predicted by Andreev [15, 16],in which an incoming particle-like excitation can convertinto an outgoing hole-like one and viceversa, see centralrow of Fig. 1. Many of such Andreev scattering eventscoherently concatenated lead to the formation of subgapABSs [17, 18] that are localized near the region where thepair potential has strong spatial variations (for a recentreview see [19]).

In the last decade, a new twist in the possibilities af-forded by the superconducting pairing of electrons hasbeen possible with the advent of topological materials[20, 21]. Inspired by notions of topology [22], severalauthors have predicted the existence of new states ofmatter known collectively as topological superconduct-ing phases, see Refs. [23–28] for reviews. These arisein particular in so-called p-wave SCs, which possess arare triplet-like pair potential (an exotic form of super-conductivity involving only a single spin band [29–33]).Topological SC phases are characterized by the emer-gence of a rather special type of subgap bound state oc-curring at topological defects such as vortices, bound-

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As opposed to standard ABSs, which can be pushedout of the gap by continuous deformations of the Hamil-tonian, Majorana bound states (MBSs) cannot be re-moved from zero energy by any local perturbation or localnoise that does not close the gap. This robust zero-energypinning is a consequence of the bulk-boundary correspon-dence principle of band topology [37], which predicts thatat the boundaries between materials with different topo-logical indices, edge states must appear that are pro-tected against perturbations by the topology of the bulk.Quite remarkably, MBSs do not follow fermion statis-tics, unlike the original particles predicted by Majorana[34], but rather possess non-Abelian exchange statistics.Upon exchange of two MBSs (braiding), a non-trivialunitary operation will be performed on them. This prop-erty, together with their topological protection against lo-cal noise, holds promise for applications in fault-tolerantquantum computing [38, 39].

The interesting connection between Dirac physics, su-perconductivity and Majorana zero modes was exploitedby Fu and Kane in 2008 [40], who put forward the con-ceptual breakthrough of effectively creating p-wave su-perconductivity and MBSs out of standard s-wave SCsby virtue of the proximity effect acting onto the helicaledge states of topological insulators (propagating edgestates with spin-momentum locking). The possibility ofcombining different materials to engineer the topologicalsuperconducting state has spurred immense interest inthe physics of Majorana states in hybrid systems.

Fu and Kane’s idea was soon extended to other ma-terials with helical states produced by strong spin-orbit(SO) coupling, but different from topological insulators[33]. A popular practical proposal was put forward inde-pendently by two groups in 2010 (Lutchyn et al. [41] andOreg et al. [42]), that realizes the conceptual model forone-dimensional (1D) p-wave superconductivity proposedby Kitaev in 2001 [32]. It was based on 1D low-densitysemiconducting nanowires under an external magneticfield B, which readily allowed its implementation in ex-periments. The combination of the SO interaction andthe Zeeman field VZ = gµBB/2 associated to B gener-ates, for a small chemical potential µ in the nanowire,a helical phase similar to that of topological insulatorsbut with broken time-reversal symmetry [43]. By cover-ing the nanowire with a conventional SC, its spectrumbecomes gapped by the proximity effect. In this device,sometimes dubbed a Majorana nanowire, a topologicaltransition in the form of a band inversion was predictedto occur at a critical Zeeman energy V cZ of the order of

the induced superconducting gap (Box A). The mate-rial properties necessary to realize this proposal in thelab can be achieved by using e.g. InAs or InSb semi-conducting nanowires [44, 45]. Hybrid superconducting-semiconducting devices based on such nanowires can betuned to the topological phase by increasing B and de-pleting the wires by means of gate voltages. In finite,but sufficiently long wires, zero energy MBSs emerge inpairs for VZ > V cZ , one localized at either end. One pairof Majorana states forms a non-local fermion. The occu-pation of two such fermions defines the elementary qubitin proposals of topological quantum computers [32].

In this work we review the formation and properties ofgeneral subgap bound states in nanowires and nanowirejunctions, as they evolve from conventional ABSs in high-density nanowires (Sec. II) to topological Majorana zeromodes at low-densities and finite magnetic fields (Sec.III). We summarize the main experimental approachescurrently used for their detection and characterization[46–48], including ABSs in nanowire quantum dots (QDs)[49–54]. Going beyond, we discuss in Sec. IV variousphysical extensions of the minimal description of Majo-rana nanowires. These include multimode effects [55–58],renormalized g-factors and SO couplings due to strongproximity effect with the parent SC [59–65], effects ofthe charge density distribution across the wire sectionand of the electrostatic environment [63, 66–72] or den-sity and pairing inhomogeneities [73–77]. Such gener-alized nanowires have been predicted to sometimes de-velop robust zero modes [73–89] that cannot be classi-fied using the band-topological concepts of uniform Ma-jorana nanowires. They allow nevertheless a classifica-tion within the more general context of non-Hermitiantopology [87]. We review the open questions that remainas to their nature (e.g. their location within the wire[75, 77, 80, 81, 83, 85, 88, 89], their degree of fermionicnon-locality [80, 90, 91], their decay into external leads[87, 92], their resilience to perturbations [93–106]) andthe conditions for their emergence. Understanding thesezero modes without a clear relation to bulk topology isparticularly important currently, in view of the many ob-servations of robust zero bias anomalies reported in re-cent experiments [107–109].

Although we focus on semiconductor nanowires inthis review, we note that MBSs are also investigatedin other material platforms, including atomic chains[110, 111], monolayer islands [112–114], topological insu-lators [115, 116] and planar semiconductor heterostruc-tures [108, 117], while ABSs were studied in e.g. atomicpoint contacts [118, 119], carbon nanotubes [120, 121],graphene [122], and nanoparticles [123].

3

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Andreev QubitsAndreev qubits Topological electronics3

Figure 1. Examples of systems allowing implementation of a Kitaev chain.(a) A chain of QDs in a 2DEG. The QDs are connected to each other, andto superconductors (labeled SC), by means of quantum point contacts (QPCs).The first and the last dots are also coupled to external leads. The normal stateconductance of QPCs between adjacent dots or between the end dots and theleads is Gk, and of the QPCs linking a dot to a superconductor is G?. Theconfinement energy inside each QD can be controlled by varying the potentialVgate. (b) Realization of the same setup using a nanowire, with the difference thateach dot is coupled to two superconductors in order to control the strength of thesuperconducting proximity effect without the use of QPCs.

separated by gate-controlled tunnel barriers, and all the tuning can be done by gates, exceptfor the coupling to a superconductor. This coupling, in turn, can be controlled by coupling twosuperconductors to each dot and applying a phase difference to these superconductors. Thelayout of a nanowire implementation of our proposal is shown in figure 1(b).

This geometry has the advantage of eliminating many of the problems mentioned above.By using single-level QDs, and also quantum point contacts (QPCs) in the tunneling regime,we solve issues related to multiple transmitting modes. Additional problems, such as accidentalclosings of the induced superconducting gap due to disorder, are solved because our setup allowsus to tune the system to a point where the topological phase is most robust, as we will show.

We present a step-by-step tuning procedure which follows the behavior of the system inparallel to that expected for the Kitaev chain. As feedback required to control every step weuse the resonant Andreev conductance, which allows us to track the evolution of the system’senergy levels. We expect that the step-by-step structure of the tuning algorithm should eliminatethe large number of non-Majorana explanations of the zero bias peaks.

New Journal of Physics 15 (2013) 045020 (http://www.njp.org/)

Quantum simulation

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The broadening totG stems fromdecay and dephasing of the double dot.MBQ readout is nowpossible either byobserving a peak in the amplitude of the transmitted photon spectrum (I A Iout

20w = w( ) ∣ ∣ ) at frequency zw = W

determined byminimizing Re z0w w c- +∣( ) ( )∣ in equation (6),figure 2(e), or bymeasuring the z-dependentphase shift of the transmitted signal ( Aargf wD = - w( ) ( )), figure 2(f).

As a variant of the quantum-dot-based readout proposed here, wemention the possibility of using theregimewhere the tunneling through the reference arm (t0) ismuch stronger than the (co-)tunneling through theMBQ (t1). In this limit, the two dots are effectively hybridized into a single dot tunnel coupled to twoMajoranaoperators, say 2g and 3g . The energy shift of theQDdepends on z i 2 3g g=ˆ which therefore can be read out by ameasurement of the dot charge [25] or the quantum capacitance [31].

At this point, it is worth stressing that all the above readout schemes are topologically protected in the sensethat imperfections thatmay reduce the readout fidelity (which can be compensated for by longer integrationtimes) do not change the projection caused by themeasurement. This is because themeasured operator isuniquely defined by the dots or leads being addressed. The robustness of the projection is a consequence of thenon-local and fractionalized nature of theMBQquantum spin.

So farwe discussed readout and preparation of z-eigenstates. Using the three-dot devicewith an interferencelink infigure 3(a), the z-measurement is readily generalized to readout of all three Pauli operators (x y z, ,ˆ ˆ ˆ).Here, a phase-coherent reference arm connecting far ends of the box is needed, e.g., between 1g and 2g . For thispurpose, afloating TSwire (top) acts as a single fermion level stretched out over the entire wire length [26, 27].Thereby, readout andmanipulations along the far side of theMBQbecome possible. Figure 3(b) lists thecorresponding dot pairs to access all Pauli operators. This simple geometry allows for non-trivial testexperiments, e.g., tofirst prepare an eigenstate in one basis, and thenmeasure a different Pauli operator.

Similar protocols allow tomanipulate arbitraryMBQ states yñ∣ . For instance, consider an electron transferfromdot 2 3 infigure 3(a), implemented by ramping the detuning parameter ε.With interference linksturned off (t 00 = ), the tunneling amplitude is t z1 ˆ, seeequation (4). The protocol begins with an electron on dot2, 0 2dyY ñ = ñ Ä ñ∣ ( ) ∣ ∣ . Assuming that a latermeasurement detects an electron on dot 3, the final state is

z3 3 T e 0 3 . 7f d d tHdt

di

t

0ò yY ñ = ñá Y ñ = ñ Ä ñ- ¢⎛⎝⎜

⎡⎣⎢

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⎞⎠⎟∣ ∣ ∣ ∣ ( ) ( ˆ∣ ) ∣ ( )

In effect, the Pauli- z operator has thus been applied, zy yñ ñ∣ ˆ∣ . Equation (7) holds because all odd-in-t1 termsare proportional to z and because the finalmeasurement has confirmed the transfer 2 3. This protocol worksbeyond the adiabatic regime [15, 16] and allows for fast high-fidelity operations.Moreover, after a failed transferattempt, t2 2 0f d dY¢ ñ = ñá Y ñ = Y ñ∣ ∣ ∣(∣ ( ) ) ∣ ( ) , one can simply retry. Likewise, other Pauli operators are accessible,

Figure 3. Single- and two-qubit devices. (a)MBQwith three quantumdots and an interference link for readout of all Pauli operatorsand full one-qubit control. Dark squares indicate either a charge sensor or a resonator system, seefigure 2. (b)Possible combinationsof active dot pairs addressing particular Pauli operators, see equation (1). (c)Device with twoMBQs a and b connected by dots 4 and 5,allowing for readout of their joint parity via theMBQproduct operator z za bˆ ˆ . The other dots serve to read andmanipulate qubitsindividually.

4

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covered with superconductor is much less effective due to efficient screening. The number of occupied subbands in this part is unknown, but it is most likely multi-subband. As shown in figs. S9 and S11 of (20) we do have to tune gate 1 and the tunnel barrier to the right regime in order to observe the ZBP.

We have measured in total several hundred panels sweeping various gates on different devices. Our main observations (20) are (i) ZBP exists over a substantial voltage range for every gate starting from the barrier gate until gate 4, (ii) we can occasionally split the ZBP in two peaks located symmetrically around zero, and (iii) we can never move the peak away from zero to finite bias. Data sets such as those in Figs. 2 and 3 demonstrate that the ZBP remains stuck to zero energy over considerable changes in B and gate voltage Vg.

Figure 3D shows the temperature dependence of the ZBP. We find

that the peak disappears at around ~300 mK, providing a thermal energy scale of kBT ~ 30 ȝeV. The full-width at half-maximum at the lowest temperature is ~20 ȝeV, which we believe is a consequence of thermal broadening as 3.5·kBT(60 mK) = 18 ȝeV.

Next we verify explicitly that all the required ingredients in the theo-retical Majorana proposals (Fig. 1A) are indeed essential for observing the ZBP. We have already verified that a nonzero B-field is needed. Now, we test if spin-orbit interaction is crucial for the absence or pres-ence of the ZBP. Theory requires that the external B has a component perpendicular to Bso. We have measured a second device in a different setup containing a 3D vector magnet such that we can sweep the B field in arbitrary directions. In Fig. 4 we show dI/dV versus V while varying the angle for a constant field magnitude. In Fig. 4A the plane of rotation is approximately equal to the plane of the substrate. We clearly observe that the ZBP comes and goes with angle. The ZBP is completely absent around ʌ/2, which thereby we deduce as the direction of Bso. In Fig. 4B the plane of rotation is perpendicular to Bso. Indeed we observe that the ZBP is now present for all angles, because B is now always perpendicu-lar to Bso. These observations are in full agreement with expectations for the spin-orbit direction in our samples (17, 31). We have further verified that this angle dependence is not a result of the specific magnitude of B or a variation in g-factor (20).

As a last check we have fabricated and measured a device of identi-cal design but with the superconductor replaced by a normal Au contact (i.e., a N-NW-N geometry). In this sample we have not found any signa-ture of a peak that sticks to zero bias while changing both B and Vg (20).

Fig. 3. Gate voltage dependence. (A) 2D color plot of dI/dV versus V and voltage on gate 2 at 175 mT and 60 mK. An-dreev bound states cross through zero bias, for example near -5 V (dotted lines). The ZBP is visible from –10 to ~5 V (although in this color setting it is not equally visible every-where). Split peaks are observed in the range of 7.5 to 10 V (20). In (B) and (C) we compare voltage sweeps on gate 4 for 0 and 200 mT with the zero bias peak absent and pre-sent, respectively. Temperature is 50 mK. [Note that in (C) the peak extends all the way to –10 V (19).] (D) Temperature dependence. dI/dV versus V at 150 mT. Traces have an off-set for clarity (except for the lowest trace). Traces are taken at different temperatures (from bottom to top: 60, 100, 125, 150, 175, 200, 225, 250, and 300 mK). dI/dV outside ZBP at V = 100 ȝeV is 0.12 ± 0.01·2e2/h for all temperatures. A full-width at half-maximum of 20 ȝeV is measured between ar-rows. All data in this figure are from device 1.

Fig. 2. Magnetic field dependent spectroscopy. (A) dI/dV versus V at 70 mK taken at different B-fields (from 0 to 490 mT in 10 mT steps; traces are offset for clarity, except for the lowest trace at B = 0). Data from device 1. (B) Color scale plot of dI/dV versus V and B. The zero-bias peak is highlight-ed by a dashed oval. Dashed lines indicate the gap edges. At ~0.6 T a non-Majorana state is crossing zero bias with a slope equal to ~3 meV/T (indicated by sloped dotted lines). Traces in (A) are extracted from (B).

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FIG. 1. Andreev bound states (ABSs) in semiconducting nanowires. ABSs are at the heart of several physicalmechanisms (central row), experimental techniques (lower row) and applications (upper row) in condensed matter physics.Central row: Andreev reflection at an NS junction, see central panel, is the retro-reflection of an electron into a hole (orviceversa) of opposite spin and velocity, with the addition (or removal) of a Cooper pair to the SC condensate. In contrast,normal (specular) reflection leaves the particle and spin quantum numbers unchanged. The probability of each (RA vs. 1−RA

below the gap) depends on normal-state transparency TN and energy of the incident electron E. The central-right panel showsthe Andreev reflection probability versus E at an NS junction of TN in the high density limit (chemical potential µ muchlarger than superconducting gap ∆). Multiple coherent Andreev reflections in a short SNS Josephson junction produce an ABSconfined to the N region with energy E(ϕ) below the gap. This energy depends on TN and the phase difference ϕ between thetwo SCs, see central-left panel. Lower row: Several experimental techniques have been developed to probe the ABS spectrum,amongst which we highlight Josephson spectroscopy using the AC Josephson effect of a capacitively coupled tunnel junction,microwave spectroscopy through the dispersive shift of a planar resonator, and tunneling spectroscopy using the differentialconductance into the nanowire through an opaque barrier. In the bottom-left panel, a nanowire Josephson junction with a gatevoltage Vg is embedded in a SQUID loop, which sets a phase bias of ϕ = 2πΦ/Φ0, where Φ is the applied flux and Φ0 = h/2ethe superconducting flux quantum. The Andreev level excitation frequency f is set by Vs = hf/2e, the spectrometer biasvoltage. Nearby we show the measured excitation spectrum in the single channel regime, where the phase-dependent Andreevlevel (upper line) and the Josephson plasma oscillations (lower line) contribute to the signal. Experimental data are reproducedfrom Ref. [47]. The bottom-central panel showcases experiments using the dispersive shift ∆f of an inductively coupled planarsuperconducting microwave resonator. The data, as a function of the excitation frequency fexc and phase bias ϕ, are reproducedfrom Ref. [48]. The bottom-right panel shows the setup and an experimental dataset for voltage bias spectroscopy, reproducedfrom Ref. [46], where the differential conductance dI/dV is measured as a function of the voltage bias V . A tunnel barrier iscreated by depleting a section of the nanowire by the local gate voltage Vtunnel. Upper row: Potential application domains ofABSs in quantum technologies include single spin readout [124], Andreev quantum bits [125], topological quantum electronics[126] and hybrid quantum simulators [127].

4

II. ABSs IN HIGH-DENSITY NANOWIRESAND QDs

A. Formation of ABSs

ABSs arise in superconducting systems as the resultof an unusual form of quantum confinement caused byso-called Andreev reflection [15, 16]. In a metallic sys-tem in its normal phase, electrons become specularly re-flected at planar interfaces with vacuum or insulatingmaterials. This is known as normal reflection. However,at an NS boundary [14, 128], an incoming electron fromthe N side may transform into an outgoing hole withinverted spin and wave vector. This hole is said to beretro-reflected since both the parallel and normal veloc-ity components to the interface change sign, whereas ina normal reflection the parallel component remains thesame. This process is known as Andreev reflection, andis accompanied by the injection of a Cooper pair intothe SC. If the interface is highly transparent, below thegap such Andreev process dominates with high proba-bility RA ≈ 1, whereas in the opposite limit the elec-tron becomes normal-reflected (1− RA ≈ 1), see centralpanel of Fig. 1. Experimentally, RA can be character-ized based on the finite subgap conductance of the NSinterface [109]. The bias-dependent conductance of shortnanowire segments between two SC leads was also usedto extract the magnitude of RA, close to one, in varioussemiconductor nanowires, including SiGe [129, 130], InAs[131–134] and InSb [135, 136].

Consider now an electron in a normal metal betweentwo or more insulating interfaces. When the metallic re-gion is small, multiple coherent normal reflections on theboundaries leads to the formation of electronic states ofquantized energy. A similar process takes place whensome or all of the confining insulators are replaced by SCboundaries [16]. This leads to the formation of ABSs,which are the superconducting analogue of the aboveparticle-in-a-box states of quantum mechanics.

The formation of ABSs becomes particularly simplein the common case of high density SCs with negligibleSO coupling and zero magnetic field. In such systemsthe superconducting gap is much smaller than the chem-ical potential, ∆ � µ, a condition known as the An-dreev limit [137]. The NS Andreev reflection probabilityRA then exhibits a simple dependence with normal-statejunction transparency TN and energy E (relative to theSC chemical potential), see Fig. 1 central-right panel. Itreaches RA = 1 at E = ∆ from RA ≈ T 2

N/(2 − TN )2 atE = 0 [138]. This result assumes a step-like pair poten-tial at the interface, a common approximation known asthe rigid boundary-condition [139].

We now combine two such NS interfaces into a 1DSNS junction with normal length LN . A computationin the Andreev limit of the energy E(ϕ) of ABSs below∆ for fully transparent interfaces, and as a function ofSC phase difference ϕ across the junction, yields the fol-

lowing quantization condition [18]

ϕ− 2 arccos[E(ϕ)/∆]− 2E(ϕ)

∆

LNξ

= 2πn, (1)

where n is an integer and ξ is the superconducting coher-ence length. A generalization to a multimode junction offinite transparency yields, in the short junction LN � ξlimit [18, 137, 140–143], an explicit E(ϕ) solution formode i

Ei(ϕ) = ±∆

√1− T iN sin2 ϕ

2, (2)

where T iN is the normal transmission for each indepen-dent scattering-matrix eigenmode i in the normal phase[144]. The presence of such bound states has importantconsequences for transport, since, as argued by Kulik[18], it implies that a normal metal can carry a dissipa-tionless supercurrent IA(ϕ) between two SCs over arbi-trarily long lengths, provided that transport is coherent.This is the celebrated dc Josephson effect [145, 146]. Atzero temperature and neglecting the contribution to thesupercurrent coming from the continuum of states above∆, IA(ϕ) = −(2e/~)

∑i ∂Ei(ϕ)/∂ϕ (where the factor 2

accounts for spin degeneracy).Figure 1 central-left panel illustrates the solution E(ϕ)

for different TN in a single-channel junction. Near ϕ =0, 2π the ABSs touch the continuum of single quasiparti-cle states above ∆ while they reach their minimum valueat ϕ = π, with an E(π) that decreases with increas-ing transparency until reaching an accidental zero-energycrossing as TN → 1 (assuming ∆ � µ). It is impor-tant to realize that, since |E(ϕ)| < ∆, the ABS wave-functions are confined to the junction, and exponentiallydecay into the bulk of the SC leads on a length scaleξABS = ξ/(

√TN | sin(ϕ/2)|).

Deviations from the Andreev limit, relevant in low-density nanowires, introduce important corrections to theAndreev reflection RA and ABS energies Ei(ϕ), and willbe discussed in Sec. III.

B. ABS spectroscopy

Several measurement techniques have been developedto obtain information about ABSs in nanowire Josephsonjunctions. Here we focus on three broad classes: Joseph-son spectroscopy, microwave spectroscopy and tunnelingspectroscopy, see bottom row of Fig. 1.

In a Josephson junction, parity-conserving transitionsbetween the ground and excited states with an addi-tion energy of 2E(ϕ) [see Eq. (2)] can be created byan incident photon with a frequency of f = 2E(ϕ)/h,where h is Planck’s constant. Note that the SC gap∆ ≈ 180µeV of Al corresponds to a frequency rangeof 2∆/h ≈ 90 GHz. A precise treatment of the pair tran-sition leads to an effective microwave impedance Z(f)associated with the transition [147]. It can be detected

5

via the inelastic Cooper-pair tunneling [148] in a capaci-tively coupled auxiliary Josephson junction [149], whichis sensitive to the environmental impedance seen by thisspectrometer junction. The probing frequency f can beset by applying a voltage bias of Vs = hf/2e (Fig. 1lower-left panel). Measurements of this type confirmedthe applicability of the short junction formula Eq. (2) ina wide range of excitation energies in InAs semiconductorchannels with epitaxial Al leads and demonstrated thatfew-channel configurations of high channel transparencycan be attained [47].

The Andreev two-level system [Eq. (2)] can also becharacterized and manipulated by the well-establishedtoolbox of circuit quantum electrodynamics [150], basedon the coupling between a resonator with frequency frand the junction hosting the Andreev level. In the low-est order, this coupling is described by the HamiltonianHc = MIAIr, where M is the mutual inductance (Fig. 1

lower-central panel), and IA, Ir are the current opera-tors of the Andreev level (see Sec. II A) and the res-onator, respectively. It is instructive to note that thesupercurrent IA changes sign between the ground andexcited state. Furthermore, the odd parity state withan unpaired quasiparticle yields IA = 0. These threestates can then be distinguished by the dispersive fre-quency shift of the coupled resonator, enabling a realtime tracking of the junction charge parity [125]. Thecharacteristic parity lifetimes are measured to be in ex-cess of 100µs in InAs nanowire Josephson junctions. Inthe same experiment, typical relaxation times ranging upto ∼ 10µs allowed for the coherent manipulation of thenanowire-based Andreev level quantum bit.

Direct quasiparticle tunneling into the ABSs can alsoprobe the ABS spectrum (Fig. 1 lower-right panel).These experiments utilize a gate-defined depleted sectionof the nanowire [46] or an in-situ grown axial tunnel bar-rier [54, 151] as the opaque probe junction. This measure-ment geometry allows for the characterization of energyspectra in proximitized semiconductor segments [152] orquantum dots [50, 52], and makes non-local correlationexperiments possible [153]. However mesoscopic interfer-ence effects in the leads may yield additional features inthe differential conductance [53].

It is worth noting that the ABS spectrum can indi-rectly be characterized via the measurement of the phase-dependent supercurrent IA(ϕ) ∼ dE/dϕ, which was per-formed by an inductively coupled SQUID loop [154, 155].These experiments yielded strongly skewed current-phaserelations, the signature of highly transparent channels inan InAs nanowire with Al superconducting leads. Simi-larly, the Josephson inductance, L−1

J ∼ dIA(ϕ)/dϕ couldserve as another probe of the anharmonicity in the cur-rent phase relationship [156]. Finally, external tunnelbarriers, typically AlOx of a few atomic layers, attachedto a metallic probe also became an established techniqueto detect ABSs in other systems, such as carbon nan-otubes [120] and graphene flakes [122].

C. ABSs in QDs

For the junctions above, it was assumed that the chan-nel connecting the SC leads allowed for coherent trans-port through a ballistic nanowire segment. By contrast,in QDs, charges localize in the channel and the effect ofa finite electrostatic charging energy U must be takeninto account. QDs can be formed in a nanowire by e.g.inducing barriers with electrostatic gates [Fig. 2(a)]. Atlow temperatures and for low bias voltages, transport isblocked by the large U and the system is in the so-calledCoulomb blockade regime with a well defined number ofelectrons n. Current flow is only possible at discrete de-generacy points where the energies of the n and n + 1charge states become degenerate. Given the strong con-finement in nanoscale QDs, U can easily exceed ∆ in theelectrodes, resulting in an interesting interplay betweensingle-electron charge transport, localized spins and su-perconductivity [160].

The formation of ABSs can be understood by consider-ing a single QD level coupled to a superconducting elec-trode. If the level is singly occupied, it holds an un-paired spin, i.e. a spin-doublet ground state [Fig. 2 (b)].Conceptually, this scenario is identical to having an iso-lated magnetic impurity in a superconducting host. Asshown by Yu, Shiba and Rusinov (YSR) in the 1960s[11–13], the magnetic impurity induces localized boundstates within the SC gap. At a critical exchange couplingthe system undergoes a quantum phase transition to amagnetically screened, spin-singlet ground state. Con-versely, at weaker coupling, the system maintains its orig-inal doublet state. While the above YSR picture appliesfor classical magnetic impurities, a full quantum treat-ment naturally leads to the physics of the Kondo effect[161] where, despite the absence of screening electronswithin ∆ of the electrodes, the localized spin can still bescreened by the above-gap quasiparticles in the SC. Asin normal metals, Kondo physics sets in below a charac-teristic temperature TK , which results in singlet-doublettransitions occurring at kBTK/∆ ∼ 0.3. Early work onhybrid dots indicated the importance of Kondo-like corre-lations [162, 163], while more recent experimental workhas provided precise boundaries for the transition [51].Figure 2(b) shows the generic phase diagram of a hybridQD as a function of dot parameters [50, 51, 159].

ABSs in QDs can be detected by transport spec-troscopy [50, 120, 121, 123, 157, 158, 164–167], wherebydI/dV is measured as a function of bias voltage V . Thesub-gap transport reflects resonant Andreev reflectionprocesses at voltages matching the energy difference EBSbetween the ground and the excited state of the QD [Figs.2 (c) and (d)]. This results in dI/dV peaks located sym-metrically around V = 0, corresponding to ABS reso-nances at energies ±EBS . Figure 2 (e) shows a typi-cal transport spectrum, where ABSs are visible as ridgesbelow the gap. As the charge state, and thereby theparity, of the dot is tuned, the ground state switchesbetween the singlet and doublet states, as reflected by

6

FIG. 2. ABSs in hybrid quantum dots (QDs). (a) Sketch of a semiconductor nanowire contacted by a normal metal (N)and a superconductor (SC). Local gates can be used to confine a QD, and to tune the dot-electrode tunnel couplings and the dotoccupation/parity [52, 157]. QDs can also form unintentionally in a nanowire, e.g. by barriers at interfaces [50, 51, 107, 152, 158].(b) Top panel: charge stability diagram of a normal QD as a function of the bias voltage, V , and the gate voltage, Vg. Thedot occupation (0, 1 or 2) is well-defined inside the Coulomb diamonds. Bottom panel: phase diagram of a hybrid QD as afunction of Vg and the QD-SC coupling, ΓS , normalized to the charging energy, U = e2/2C (e being the electron charge and Cthe QD capacitance). In the weak coupling limit, ΓS/U � 1, the ground state is a spin-doublet when the dot is occupied by anodd number of electrons. Conversely, for ΓS/U � 1, the ground state is a spin-singlet irrespective of the dot occupancy. Theprecise boundary between both states can be obtained by experimentally tuning the ratio ΓS/U [51] in very good agreementwith theoretical results obtained by a superconducting analog of the Anderson model [159]. (c) Transport spectroscopy of ABSsformed within the SC gap by the Yu-Shiba-Rusinov mechanism, where the confined spin (impurity) is screened by itinerantquasiparticles. Resonant dI/dV peaks are observed when the chemical potential of the N probe matches the bound state energy,±EBS , which represents the excitation energy from the ground state of the QD-SC system to an excited state. The transportcycle first involves the tunneling of an electron (hole) to the QD-SC system, changing its parity, followed by an Andreevreflection process whereby a Cooper pair is formed (broken) in SC and a hole (electron) is reflected to the probe. (d) Diagramof the possible transitions between ground and excited states of a hybrid QD. An external magnetic field, B, splits the doubletstate by the Zeeman energy, 2VZ . Top panel: when the ground state is the doublet, the bound state energy increases withB (green arrow). The transition between the two spin-polarized states is not visible by tunneling spectroscopy (red arrow).

Bottom panel: when the ground state is the singlet, both transitions to the spin-polarized excited states are visible, E↑BS and

E↓BS . (e) Subgap spectrum of a QD with a single SC electrode as a function of Vg at B = 0. Crossings of the bound stateresonances at V = 0 signal transitions between singlet and doublet ground states. Data reproduced from Ref. [157]. (f) Thebound states only split in the presence of an external B when the ground state is the singlet. (g) Zeeman splitting of the boundstates as a function of B. The dashed vertical line underscores a quantum phase transition (QPT) whereby the ground stateof the system turns from the singlet to a spin-polarized state. Data reproduced from Ref. [50].

the ABS crossings at zero bias. Remarkably, the groundstate remains a singlet in some odd-occupancy regionsdue to the strong screening discussed above, which leadsto avoided ABS crossings in the spectra. The experi-mental phase diagram of the QD-S system has been ex-

plored [51, 157, 166], finding excellent quantitative agree-ment with theory [159, 168]. In some cases, however, oneneeds to go beyond the bulk treatment of the SC above(to include soft gaps, finite-length effects, etc) in orderto understand the complex ABS spectra of finite-length

7

proximitized nanowires [53, 54]. Transport spectroscopyof ABSs can also be performed by replacing the N probeby a weakly coupled superconductor. Here, all spectro-scopical features are shifted by ∆ [49, 165]. ABSs existalso in coupled hybrid dot systems [52, 169] where onecan observe YSR screening of higher spin states and amore intricate phase diagram than Fig. 2 (b) [52, 170].We note that YSR states have also been studied in STMexperiments as reviewed e.g. in Ref. [171].

In an external magnetic field, the Zeeman effect liftsthe spin degeneracy of the doublet state. This stronglyimpacts the transport spectra of the ABSs [Fig. 2(d)]. Incase of a singlet ground state, two (parity-changing) tran-sitions are allowed, thanks to the splitting of the exciteddoublet state. In contrast, when the ground state is adoublet, only one transition remains accessible indepen-dent of B. As a result, the ABSs shift to higher energiesbut do not split. Figure 2 (f) depicts these two distinctbehaviors of the ABSs at finite B [50]. Interestingly, forhigh enough fields, the lowest-energy, spin-split ABSs cancross the Fermi level, denoting a quantum phase transi-tion from the singlet ground state to a spin-polarizedstate [50, 157]. This transition represents a parity cross-ing and appears as a zero-bias peak at the critical field[Fig. 2 (g)]. While the transition is a true crossing, thepeak can persist at V = 0 for a wider range of B owingto the broadening of ABS resonances or to repulsion withother states or the gap edge [50, 157, 172].

In addition to the above ABS spectroscopy, the physicsof a hybrid QD can also be captured by measurements ofthe Josephson supercurrent in a S-QD-S geometry [173–176]. Notably, QDs have also been used to investigateMBSs in various device configurations [91, 107, 136, 177].

III. LOW-DENSITY NANOWIRES AND MBSs

A. ABSs in trivial SNS junctions with SO couplingand Zeeman field

As the Fermi energy µ of a nanowire SNS junction isreduced (low density regime), it may become compara-ble to other energy scales in the problem, such as the SOenergy ESO = m∗α2/2~2 (where α is the SO couplingand m∗ the effective mass), the Zeeman energy VZ at thejunction, or the gap ∆ of the SCs at either side, see BoxFig. 7. The Andreev reflection at a low-density NS in-terface deviates considerably from the standard picturedescribed in Sec. II A. Figure 3 (a-c) shows the typicaldependence of RA with energy for a single channel con-tact when both N and SC sides have a common Fermienergy, SO coupling and Zeeman. Similarly, the Andreevspectrum of the corresponding low-density SNS nanowirejunction is no longer well described by the conventionalEq. (2), even in the short junction limit, see Fig. 3 (e-g). Note in particular that the parity crossing presentat ϕ = π in high-density transparent junctions becomesan anticrossing even at TN = 1 as soon as the Andreev

limit ∆� µ is not satisfied. This contrasts with the pro-tected (TN -independent) ϕ = π crossing in topologicalSNS junctions, as we will see.

To understand the main low-density corrections weconsider first the case of an SNS junction in which µbecomes comparable to the SO energy µ ∼ ESO, whilestill remaining in the Andreev limit ∆ � µ. We furtherconsider the realistic complication that the SO couplingα and the Zeeman field VZ are largely confined to thenormal part of the nanowire. The Fermi energy µN inN is also assumed to differ from that of the SC contactsµS . The corresponding bandstructures will thus exhibita Fermi momentum mismatch, which reduces Andreevreflection and affects the resulting ABS spectrum. In anominally perfect, single mode SNS junction of nanowirelength LN with VZ = 0, Eq. (2) can be generalized to[178]

E(ϕ) = ∆

√1− sin2(ϕ/2)

1 + κ sin2(k0LN ), (3)

where κ = [(kSF )2 − k20]/(2kSF k0) captures the effect of

momentum mismatch acting as an effective barrier ateach interface, with a transmission TN = 1

1+κ sin2(k0LN )

that is smaller than 1, except at resonant values of thenanowire length k0LN = nπ, n ∈ Z. Here the SC

and N Fermi wavevectors are kS,NF =√

2m∗µS,N/~, and

k0 =√

(kNF )2 + 4k2SO. This k0 depends also on the SO

momentum kSO = m∗α/~2, that captures the momen-tum band shift of the two spin sectors in the nanowire(see Box A). E(ϕ) of Eq. (3) remains doubly degeneratefor all ϕ despite the shift kSO of the two spin sectors;electron-hole pairs can still form in a similar manner asfor a spin-degenerate single parabolic dispersion, see Fig.7. While Eq. (3) still yields a zero energy crossing atTN = 1 and ϕ = π, it captures the fact that TN < 1 evenwith nominally perfect contacts due to the momentummismatch.

In the absence of a Zeeman field, spin splitting of theABS spectrum can be achieved by a nonzero ϕ in atwo-subband model with intersubband coupling. Specifi-cally, mixing between the two lowest transverse subbandsin a low density regime may produce a strongly spin-

dependent Fermi velocity v↑F 6= v↓F , and hence coher-

ence lengths ξ↑/↓ =~v↑/↓F

∆ , which leads to spin-dependentquantization conditions according to Eq. (1). For TN =1, and assuming λi = LN/ξi � 1 or Ei � ∆, the ABSscan be written as [179]

Ei(ϕ) = ±∆cos(ϕ/2)

1 + λi sin(ϕ/2). (4)

The spin splitting between ABSs reads

E↑(ϕ)− E↓(ϕ) =∆(λ↑ − λ↓) sin(ϕ)

2[1 + λ↑ sin(ϕ/2)][1 + λ↓ sin(ϕ/2)].

(5)

8

��� ��� ��� ��� ���

���

���

���

���

���

���

�/Δ

��

��� �������

ϵ = μ

μ/Δ = 1.8VZ/Δ = 0α = 0

a

��� ��� ��� ��� ���

�/Δ

������

ϵ = μ - B

μ/Δ = 2VZ/Δ = 0.15

α = 0

b

��� ��� ��� ��� ���

�/Δ

������+���

ϵ = μ - B

μ/Δ = 2.4VZ/Δ = 0.7

α > 0

c

��� ��� ��� ��� ���

�/Δ

�����������

μ/Δ = 2.4VZ/Δ = 3.12

α > 0

dN

S

Andreevreflection

TN10.70.3

� π �π

-���

-���

���

���

���

φ

�(φ)/Δ μ/Δ = 1.8

VZ/Δ = 0α = 0

e

� π �π

φ

μ/Δ = 2.VZ/Δ = 0.15

α = 0

f

� π �π

φ

μ/Δ = 2.4VZ/Δ = 0.7

α > 0

g

� π �π

φ

μ/Δ = 2.4VZ/VZc = 1.2

α > 0

h

SN

S

Andreevbound

states

FIG. 3. Theory of Andreev reflection and bound state formation in low-density NS and SNS nanowire junctions:Andreev reflection probability RA (top row) and ABS energy E(ϕ) (bottom row) computed within a tight-binding approach inconstant µ ∼ ∆ low-density NS and short SNS junctions of varying normal-state transparency TN , respectively. Energies arenormalized to the SC gap ∆ at Zeeman energy VZ = 0. Panels (a,e) correspond to zero Zeeman and spin-orbit (SO) coupling α.Note that at low-densities the Andreev limit ∆� µ is not satisfied. Hence, the {ϕ = π, TN = 1} ABS crossing at zero energyexpected from Eq. (2) becomes an anticrossing due to ∆-induced normal reflection (in contrast to Fig. 1). In panels (b,f) wesee that a small VZ splits the gap edge into two sectors ∆±, with their respective quasiparticle continuum colored in differentshades of gray. All curves in (a,b,e) are double, one per incident spin channel (a,b) or level (e). The addition of α in (c,g)breaks the remaining spin-symmetry of RA around the Zeeman field direction (along the wire). This is visible as a splitting

of same color curves. As one cranks up VZ , ∆− goes to zero and at V cZ ≡

√∆2 + µ2 undergoes a band inversion. Upon its

reopening for VZ > V cZ , a zero-energy scattering resonance arises in the Andreev reflection of the NS junction (panel d). The

resonance manifests as a universal RA = 1 at the Fermi level E = 0, regardless of junction transparency TN . It is the result ofthe emergence of a MBS at the NS junction. In a short SNS geometry, two such MBSs emerge that hybridize into an ABS. Itsenergy E(ϕ) is detached from the continuum at ϕ = 0, 2π for TN < 1 and exhibits a protected zero-energy parity crossing atϕ = π (panel h), yielding a 4π-periodic state at fixed parity. Note that subgap curves in (d,h) are effectively spinless.

This phase-dependent spin splitting is finite for ϕ 6= 0, π,and comes from the difference in coherence lengths andFermi velocities. Spin-degeneracy at ϕ = 0 and ϕ = πis protected by time-reversal symmetry. The combinedeffect of Zeeman and SO coupling on the Andreev levelspectra of single channel nanowires has been studied inRef. [64, 180]. Among others, an important consequenceof the interplay of VZ and α is the strong suppressionof the g-factor owing to SO coupling and/or high elec-tron density. This g-factor renormalization drasticallychanges the spin splitting of Andreev levels for increas-ing magnetic fields.

The theory of spin-split ABS formation outlinedabove has been confirmed by recent experiments in acircuit quantum electrodynamics geometry using InAsnanowires [48, 181].

B. Emergence of MBSs

In Fig. 3 (e-g) we have illustrated the strong effect ofSO coupling and Zeeman fields in the ABS spectrum ofa low-density SNS nanowire junction. When the SC con-tacts are taken as low-density proximitized nanowires,the Oreg-Lutchyn minimal model predicts that a suf-ficiently strong VZ > V cZ will make them undergo atopological phase transition, with MBSs at each inter-face. Their presence results in a topologically protectedRA = 1 Andreev reflection amplitude at E = 0, seeFig. 3 (d), and a protected ϕ = π parity crossingof SNS ABSs for all transparencies, (h). The paritycrossing is robust regardless of the microscopic chan-nel configuration of the junction, and ideally gives riseto the topological Josephson effect, characterized by 4π-periodic supercurrents as a function of ϕ at fixed par-ity [32, 41, 42, 182]. The 2π-periodic E(ϕ) solution inthe trivial phase, Eq. (2), transforms in the topologi-cal regime into E(ϕ) ≈ ±√TN∆ cos(ϕ/2), with different

9

signs for opposite parities [182].Figure 4 shows a complementary picture of the topo-

logical transition in a low-density, isolated ISI uniformnanowire of length L (where I stands for ‘insulator’),both in the long (a-c) and short (d-f) nanowire regime.As VZ > V cZ , a MBS appears localized at each end ofthe SC region, with zero energy in the large L limit, orwith characteristic Majorana oscillations around zero forshorter L, resulting from their hybridization into conven-tional fermions due to their finite overlap. The lowest en-ergy level Emin [red in (a,d)] clearly traces the topologicalphase diagram for large L, panel (c). It is interesting tonote the role of finite L in the topological Josephson effect[compare panels (g,h)]. Due to the overlap of the ‘inner’MBSs in the junction and the ‘outer’ MBSs at the oppo-site ends of the nanowires, the 4π Josephson periodicity isdestroyed under an adiabatic ϕ(t), and a non-topological2π-periodic Josephson effect is restored [183, 184]. A sim-ilar effect is expected from quasiparticle poisoning (ex-change of quasiparticles with the junction’s environmentwhich breaks parity conservation) and by higher-energyquasiparticle excitation [184].

The role of SO coupling is crucial for the physics ofMBSs. For α = 0 and VZ larger than ∆ the spectrum isgapless (the magnetic field just kills superconductivity),so that no localized MBSs emerge, while for VZ < ∆the system has a gap. The addition of SO coupling rad-ically transforms this picture, and enables a topologicalminigap to emerge at VZ > V cZ . The minigap can beshown to be effectively p-wave, and hence topologicallynon-trivial. The Majorana zero modes at the ends of aVZ > V cZ nanowire are in fact a manifestation of the bulk-boundary correspondence of this topological gap. Theyare thus topologically protected states. The extension ofthe Majorana wavefunction is the coherence length cor-responding to the minigap (also known as the Majoranalength ξM [185, 186]) and is hence smaller for stronger SOcoupling. The Majorana oscillatory hybridization is thusexponentially suppressed by both a strong SO (minigap)and nanowire length. In both limits, an exact Majoranatopological protected zero mode is recovered at each endof the nanowire.

C. MBS spectroscopy

In this section, we outline the experimental techniquesused to probe potential Majorana zero modes. First,tunneling spectroscopy as described in Sec. II B hasbeen performed extensively in nanowires in devices ofthe kind shown in Fig. 5 (a), with the chemical poten-tial in the nanowire controlled by the purple gate voltageVS . The gate-defined tunnel barrier (red gates) allowsthe conductance through the junction to probe the localdensity of states at the left end of the hybrid nanowire(green), typically exhibiting a roughly BCS-like gap [seethe orange linecut in Fig. 5 (d)]. In these experiments[46, 107, 172, 187–190], the expected signature of a Ma-

jorana zero mode is a zero bias conductance peak abovea threshold magnetic field [see Fig. 5 (c)], resulting fromthe resonant Andreev reflection on the MBS at the junc-tion. This zero bias conductance peak is broadened dueto both coupling to the normal lead (tunneling broaden-ing Γ) and temperature (thermal broadening kBT ). Inthe tunneling dominated regime kBT � Γ, theory pre-dicts that the peak should exhibit a universal quantizedvalue of 2e2/h [191–193]. While most experiments yieldmuch lower conductance values (see e.g. Fig. 5 (c,d)[107]), consistent with the thermally broadened regimekBT & Γ [73, 194], some experiments have reported scal-ing with the ratio kBT/Γ and saturation values close tothe ideal 2e2/h limit at low temperatures [108]. Furthercomparison with theory [Fig. 4 (c)] can be performed bymapping the presence of the zero bias conductance peakas a function of the magnetic field B and the gate voltageVS to create a phase diagram (see Fig. 5 (b) [188, 189]).

Another class of experiments targets superconduct-ing islands, i.e., proximitized nanowires in a floatingisland geometry and characterized by combined super-conducting and Coulomb blockade phenomenology. TheCoulomb peak periodicity of the islands is found to tran-sition from approximately 2e- to e-periodic under a finiteZeeman field due to the appearance of subgap near-zeromodes [195–197]. The peak positions as a function of gatevoltage show deviations from perfect periodicity, whichare interpreted as energy splittings of the subgap states.The splittings were shown to oscillate around zero energywith Zeeman field, with an overall oscillation amplitudethat decreases exponentially with increasing island lengthL (Fig. 5 (e) [195, 197]). The oscillations and their cutofflength ξM ∼ 260 nm have been interpreted as resultingfrom Majorana splittings, see Fig. 4 (b,e).

Apart from tunneling spectroscopy measurements, inorder to detect MBSs one can also explore dynamical de-tection techniques in SNS junctions. In Sec. II A we dis-cussed the ABS spectrum and concluded that in a finite-transparency, high-density, short junction, they exhibitan avoided crossing at ϕ = π, while in Sec. III B wesaw that in the topological phase, E(ϕ) has a protectedcrossing at π, leading to the 4π-periodic Josephson ef-fect. At first glance, a tempting experimental detectionof the topological Josephson junction is to directly mea-sure the gapless nature of the ABSs, E(ϕ), or the cor-responding current-phase relation, IA(ϕ) ∼ dE(ϕ)/dϕ.However, as mentioned in the preceding section, in a fi-nite length system of length L, the overlap between the‘inner’ and ‘outer’ Majorana wavefunctions restores theavoided crossing with an energy scale ∼ exp(−L/ξM )[183, 184, 200]. In addition, the tunneling of unpairednon-equilibrium quasiparticles enables relaxation to theparity ground state, resulting in a trivial, 2π-periodicbehavior on timescales much longer than the parity poi-soning time of the system [41, 201].

Due to these challenges, the experimental detection ef-forts of the 4π-periodic Andreev levels have typically fo-cused on dynamical detection techniques based on the

10

FIG. 4. Theory of MBS formation and length-dependence in finite nanowires and Josephson junctions: Panels(a-f) correspond to an ideal, uniform proximitized nanowire of length L = 3µm (a-c) and L = 0.6µm (d-f) distance between thetwo insulators (possibly vacuum). When the nanowire length L is long as compared to the SC coherence length, the low energyspectrum as a function of Zeeman splitting VZ has the characteristic shape shown in (a). For VZ < V c

Z the system is triviallygapped, but this gap decreases until it reaches zero at the topological phase transition at V c

Z . From the latter, two Majoranazero modes around the ends of the nanowire emerge, whose exponentially decaying wave functions in the Majorana basis (seeBox B) are shown in (b). For shorter lengths, these MBSs hybridize into fermions of oscillatory energy around zero (d), with

overlapping wave functions (e). The energy of the lowest excitation clearly traces the topological transition at V cZ ≡

√∆2 + µ2,

producing the phase diagrams of (c,f) as a function of Zeeman energy VZ and chemical potential µ. The two Majoranas ateither side of a VZ > V c

Z short TS-N-TS junction combine into the characteristic low energy ABS spectrum E(ϕ) ∼ cos(ϕ/2)of a topological Josephson junction that vanishes at ϕ = π (g). This parity crossing leads to a 4π-periodic ground state whenfixing quasiparticle parity, and to the so-called topological Josephson effect. The exact zero modes correspond to the two outerMBSs, decoupled from the two ϕ-dependent junction (inner) MBSs for long L. The parity crossing at ϕ = π becomes lifted byfour-Majorana overlaps for short TS nanowires, that couples inner and outer MBSs, thereby destroying the 4π periodicity (h).In the schematics at the left and right edges of the figure, I stands for insulator, S for trivial SC, TS for topological SC andthe red circles symbolize the presence of MBSs at the junctions.

ac Josephson effect [145], Fig. 5 (h). In conventional2π-periodic Josephson junctions in the tunneling limitTN � 1 a junction bias V produces an oscillating su-percurrent I(t) = IC sin (2πft), with f/V = 2e/h ≈486 MHz/µV [202]. In the topological Josephson ef-fect, the 4π-periodicity of subgap states translate intoa halving of the frequency f/V = e/h. This halving be-comes visible in Shapiro step measurements [203], wherethe junction is irradiated at a frequency f in the mi-crowave domain. The dc component of I(V ) developsdiscrete voltage steps with a spacing of V2π = hf/2eand V4π = hf/e for the trivial and topological state,respectively [201, 204, 205]. While the disappearanceof the first voltage step was repeatedly observed, Fig.5 (g), higher odd steps typically persist in experiments[199, 206]. It has been argued that the interpretation ofthe measurements needs to include the deviations fromthe tunnel junction behavior, such as non-sinusoidal su-percurrents [204], overheating effects [207, 208], capac-itive shunting [209], Landau-Zener tunneling betweenthe Andreev bands and to the quasiparticle continuum[183, 205, 210]. Furthermore, the addition of several non-

topological ABSs has a non-trivial effect on the observedShapiro steps [204, 211].

Another class of experiments rely on the direct spectro-scopical detection of the Josephson radiation of voltage-biased junctions, which is expected to be centered atf2π = 2eV/h or at f4π = eV/h [184]. This transitionhas been observed in InAs/Al nanowire Josephson junc-tions integrated with an on-chip SIS microwave detector(Fig. 5 (f), [198]), and by using a conventional microwaveamplifier chain [206].

It should be noted that additional measurementschemes were proposed to observe the 4π-periodicJosephson effects as a probe for topological superconduc-tivity. These utilize Shapiro steps in the low-frequencyregime [205], Andreev level pair excitations in long junc-tions [212], critical current measurements [213–215], orthe shape of switching current histograms [216].

11

B (T)

0.30.080.00

-0.320 1

dI/dVbias (e2/h)

dI/dVbias(e

2 /h)

0 T

1 T

1.5 T

V bias(m

V)

-0.3 0 0.30.00

0.12

0.5 T

Vbias (mV)

1

10

100

A(μeV

)

1.51.00.5L (μm)

Amplitude

No zero bias peak

B(T)

0

1

2

0

Zero biaspeak

0

1

2

-10 -8 -6 -4 -2VS (V)

E Z=½

� Bg I

nSbB

(meV

)

500nm

c

e fba

d g h

V

hf=e*V

B (mT)

-24

-12

0

12

24B=0 1.0T 1.6T 2.1T 2.5T

V(µV)

L

1.5

2.0

2.5

0.5

1.0e*/e

3000 100 200

200-200I (nA)

200-200I (nA)

200-200I (nA)

200-200I (nA)

200-200I (nA)

I

VS

Vbias

VTunnel

A

B

FIG. 5. Experimental signatures in the search for MBSs. (a) A false color scanning electron micrograph of a device builtfor zero bias conductance peak (ZBP) measurements between the proximitized segment (nanowire covered by the SC, in green)and the normal metal ohmic contact (in yellow). The red electrostatic gates tune the transparency of the tunnel barrier and thepurple gates change the electrochemical potential of the proximitized segment. The magnetic field B points along the nanowire.(b) A ZBP phase diagram measured on an InSb nanowire covered by a NbTiN superconductor, reproduced from Ref. [189].(c) Experimental data of a robust ZBP taken on an InAs nanowire with epitaxial aluminum leads, reproduced from Ref. [107].(d) Linecuts taken at B-fields indicated by colored labels in panel (c). (e) The amplitude of lowest-energy-level oscillations in afinite-sized island device of length L, decreasing exponentially with increasing L [195]. (f, g) AC Josephson effect experimentsusing the geometry sketched in (h). The characteristic frequency is proportional to the applied bias f = e?V/h with e? = 2efor the conventional Josephson effect and e? = e for a topological Josephson junction. The halving of e? and the 4π periodicityof the topological Josephson effect was demonstrated as a function of B by a frequency-sensitive measurement of the Josephsonradiation of an InAs/Al nanowire junction [(f) taken from Ref. [198]] and by missing odd Shapiro steps in etched InSb/Nbjunctions [(g) taken from Ref. [199]].

IV. MBSs BEYOND THE MINIMAL MODEL

A. Extensions of the minimal model

The minimal Oreg-Lutchyn model has proven to bea first useful guide to investigate the physics of Majo-rana nanowires. However, discrepancies between its pre-dictions and experimental observations have motivatedextensions that provide a more complete understandingof the experimental system. A natural extension of the1D single band model is to allow for multiple subbandsin the nanowire [55–58]. This results in a more compli-cated phase diagram, depending on the number of occu-pied bands and their relative energies. Additionally, theorbital effects of the magnetic field (i.e. the magneticflux across the nanowire section) may become relevant,especially when the number of occupied subbands is in-creased [217]. They have been shown to dramaticallymodify the topological phase diagram [72, 217] [see Fig.6 (b)] and the dispersion of states in the nanowire, lead-ing to large effective g-factors [218, 219] and suppressed

topological gaps [217]. Although numerical simulationsof multiband wires can shed additional light on the ex-perimental results, they tend to depend strongly on de-tails such as the geometry and effective parameter valueswhich are not always experimentally accessible.

While initial experiments generally suffered from un-wanted quasiparticle states inside the superconductinggap [referred to as “soft gap” [46, 72, 220], see Fig. 6(c)], clean superconducting gaps comparable to the bulkgap of the parent SC have since been achieved [109, 152]by engineering epitaxial interfaces between the two ma-terial systems [221, 222]. Both the “soft gap” issue [75]and the large gaps measured in later experiments ignitedinterest in a more complete description of the supercon-ducting proximity effect in these systems. This includespair breaking effects that suppress superconductivity be-yond a critical value of the magnetic field, or a moreaccurate model for the induced pairing in the form of anenergy-dependent anomalous self-energy. The latter ex-tends the regime of weak coupling between the semicon-ductor and the SC, wherein the induced superconduct-ing gap is simply proportional to the coupling strength

12

between the two systems. It was found that in the oppo-site, strong coupling regime, the band structure of thenanowire is significantly altered, resulting in a strongrenormalization of model parameters [59]. It has alsobeen demonstrated that the proximity effect can stronglydepend on the thickness of the SC film [60, 61, 86]. TheSC-semiconductor coupling has furthermore been foundto depend on the details of the electrostatic environ-ment [63, 68], resulting in gate voltage dependent effec-tive parameters such as the g-factor [62, 65], the SO cou-pling [223] and the induced gap [69].

A notable disagreement between most experiments andthe minimal model revolves around the Majorana oscil-lations. The oscillatory energy splittings are predictedto be regular and grow with Zeeman field [73, 224–227],while in most experiments robust zero-bias peaks appearwithout oscillations [107, 109]. Several model extensionshave been explored that predict a reduction or suppres-sion of oscillations, such as interactions with a dielectricenvironment or among carriers [67, 71, 225], orbital ef-fects [228], dissipation [87, 229, 230] or non-uniform po-tentials [80, 227], pairing [77] or SO coupling [231]. A fur-ther common disagreement is a lack of visible bandgap-closing and reopening in some experiments [46, 62, 107],which is a key feature of the model’s topological transi-tion. This has been explained as the result of poor vis-ibility resulting from tunnel probe smoothness [73, 232]and even by a lack of bulk transition altogether [233], aswill be discussed in Sec. IV C.

The topological phase transitions in these extendedmodels are generally calculated using the chemical po-tential µ and the Zeeman energy VZ . However, the con-trol parameters used in experiments are gate voltages andmagnetic fields. Calculating the phase diagram in termsof gate voltages requires a self-consistent treatment of theelectrostatics [66]. While some progress has been madein self-consistent Schrodinger-Poisson calculation for 3Ddevice geometries [63, 70–72] [see Fig. 6 (a)], this remainsa difficult problem to solve reliably. In addition to elec-trostatic modifications of the phase diagram, interactioneffects have been demonstrated to play a role in the lowenergy spectrum of Majorana nanowires [66, 67, 71].

An immediate effect of a self-consistent description ofnanowire junctions, both for electrostatics and the prox-imity effect, is a smoothening of the pairing and Fermienergy profiles [73], which can no longer be assumedpiecewise-constant as in the minimal model. Smooth∆(r), µ(r) at a junction have been shown to give rise tonear-zero modes without the need of a topological bulk.We devote the next subsections to these and other typesof non-topological zero modes.

B. Zero energy pinning with a topologically trivialbulk

The combination of multiband wires with disorder hasbeen shown [226, 234, 235] to produce topologically triv-

ial zero energy states in class D Hamiltonians [236, 237].Since the advent of cleaner experiments, it has becomepossible to distinguish disorder-based mechanisms fromzero bias peaks of different origin, as the former are asso-ciated to specific observable features (e.g. soft gap, lowtransport peak heights) that have been optimized away.Strong interband coupling in multimode clean wires witha single short-range potential inhomogeneity have alsobeen shown to conspire to produce approximate zero en-ergy states of non-topological states [89, 172].

Near-zero bound states can also be generically presentin a tunneling spectroscopy nanowire setup if there is anon-superconducting section between the tunnel barrierand the superconducting wire [61, 73, 79–81, 84, 85, 87,88, 238–240] [Fig. 6 (d)]. Such an N region can hostABSs that become spin-polarized under a Zeeman fieldand may thus be tuned to zero energy, much like theShiba states, possibly with a strongly renormalized g-factor due to SO coupling [64]. In the simplest situation,these are readily distinguished because their zero energyresults from fine tuning parameters such as B to spe-cific values, unlike for topological MBSs. Under somecircumstances, however, these modes can become pinnedto zero or near-zero energy for an extended range in mag-netic field and other control parameters, resembling thebehavior expected from MBSs, but with the SC in thetopologically trivial phase [73–75, 79–85, 87, 88].

In the case of isolated NS nanowire junctions, we candistinguish two main mechanisms for zero-energy pin-ning of a non-topological zero mode: smooth confinement[80, 83, 85, 87, 88] and SO-induced pinning [81]. Both ef-fects ultimately cause an enhanced Andreev reflection ofa normal electron on the trivial SC. In the case of a junc-tion with spatially smooth parameters, the momentumtransfer required for normal reflection at the junctionis suppressed, and hence Andreev reflection dominates.Under these conditions, states at the junctions decou-ple into two sectors around different Fermi wavevectorand spin (due to the SO coupling and the Zeeman field)[74, 80, 85], each of which behaves as an independenttopological p-wave SC that gives rise to a zero-energyMBS decoupled from its partner. The Majorana wave-function corresponding to the two wavevectors are cen-tered at different positions along the wire and exhibit dif-ferent spatial profiles (oscillatory exponential and smoothgaussian, respectively [80, 88]), see Fig. 6 (e).

A similar pinning effect can be caused by SO coupling.For large SO, the effective g-factor is strongly renormal-ized and ABSs can become largely insensitive to magneticfields [64]. When the length of the N section is furthertuned to an approximately odd-integer multiple of theSO length, an ABS will appear pinned near zero energyrespect to VZ [81]. This SO-induced pinning does notrequire junction smoothness, but the above Fabry-Perotresonance condition on length must be satisfied.

A third route towards stabilising zero modes belongingto a nominally trivial bulk has been proposed in topolog-ically trivial nanowires open to fermion reservoirs (which

13

Subtype Bulk topologySpatial Overlap of

Majorana componentsSpatial extension of

Majorana components Zero energy pinningNon-Abelian

braiding

Standard: SO=0, VZ=0 trivial completespread across junction/

normal region no no

SO≠0, VZ<VZc trivial partial

spread across junction/normal region no no

strong SO, VZ<VZc trivial partial

spread across junction/normal region

only vs. VZ,rest fine-tuned no

coupled multiband + short-range inhomogeneity trivial high

spread across inhomogeneity approximate no

Shiba state trivial complete localized to impurity no no

Long (L≫ξM, VZ>VZc) nontrivial

exponentially suppressed localized to edges yes yes

Short (L≲ξM, VZ>VZc) nontrivial partial

localized to edges but overlapping

no(Majorana oscillations) no

Smoothly confined S trivial partial localized to smooth edge yesyes

(parametric)

Smooth S'S/NS junction(nontopological MBS/ps-MBS/

quasi-MBS/EP-MBS)trivial partial localized to smooth

junctionyes yes

(parametric)

Type

MB

SsA

BSs

Smoo

thze

ro m

odes

TABLE I. Classification of near-zero-energy subgap states in proximitized nanowire systems. We divide thepossible near-zero-energy subgap states in three main types: ABSs, topological MBSs and zero modes produced by smoothinhomogeneities in a trivial nanowire. Each of these is distinguished by the band-topology of the nanowire bulk, the amountof spatial overlap between the Majorana components of the state, their spatial location and extension, whether the state’senergy remains pinned to zero as system parameters are perturbed, and whether they are expected to exhibit spatial and/orparametric non-Abelian braiding statistics. See Sec. IV for a discussion.

is a standard geometry in NS junctions used to performtransport spectroscopy). When such a nanowire becomescoupled to the reservoir, it can develop an ‘exceptionalpoint’ (EP) bifurcation in its complex (non-Hermitian)spectrum, see Fig. 6 (f), where the real part of the lowestquasibound Bogoliubov mode becomes robustly pinnedto zero energy as the imaginary part bifurcates. Thiskind of non-Hermitian topological transition stabilizes acouple of quasibound states at the contact with differ-ent decay rates. One of the two may become essentiallynon-decaying after the exceptional point bifurcation, thusbecoming a stable Majorana zero mode without the needof a bulk topological transition. An EP requires a finitecoupling asymmetry of the two Majorana components tothe reservoir. Sources of asymmetry include finite length[87, 241], smooth potentials [87], spin-polarized leads[92], etc. Research into Majorana states in open sys-tems for quantum computation purposes is still in itsearly stages. The field is advancing rapidly, however,with e.g. new non-Hermitian topological classificationtheories being developed recently [242–245] that extendband-topological concepts to open systems where thesedo not strictly apply.

C. The MBS vs. ABS controversy

As studies began to unveil the above phenomenologybeyond the minimal model, it became clear that many

experimental hints of Majoranas could easily be mistak-ing zero modes of non-topological origin with MBSs re-sulting from a non-trivial bulk topology. Notable exam-ples include experiments showing zero bias peaks robustagainst magnetic field variations, or even conductancevalues close to the ideal quantized 2e2/h value. Untilrecently, both cases were considered strong signatures ofemergent Majoranas after a bulk topological transition,but a growing body of literature shows that this is notnecessarily the case [84, 85, 87, 88, 246]. A prominentreason is the possibility of robust but trivial zero modesarising at smooth inhomogeneities [73–85, 87, 88]. In-stead of emerging from a band inversion at a criticalV cZ , these subgap states are predicted to emerge as alone ABS that detaches from the continuum as VZ in-creases, and gradually becomes pinned to zero energywith no intervening bulk topological transition or bandinversion [73, 80, 83, 85, 247], see Fig. 6 (d). This tell-tale feature is often observed in experiments, see e.g.Fig. 5 (c), and should be taken as a strong hint thatthe zero mode might not be the result of an underlyingbulk topological transition. This type of zero mode hasbeen dubbed a quasi-MBS [85], partially-separated MBS(ps-MBS) [83], or non-topological MBS [87]. Notably,these states are not localized at opposite edges of thenanowire but are instead confined to the inhomogeneityneighborhood, whose location is often uncontrolled. Asshown in Fig. 6 (e), subpanel 2, they typically exhibit asubstantial spatial overlap.

14

Quasi-MBSs could also complicate the interpretationof superconducting Coulomb islands experiments [195–197]. The Coulomb peak spacing technique used thereallows to extract the energy splitting of nanowire zero-bias anomalies with high precision, see Sec. ??. Thesemeasurements show oscillatory splittings that decay ex-ponentially with nanowire length. The decay is compati-ble with MBSs spatially separated by a gapped topolog-ical bulk, see Fig. 5 (e). However, the observations mayalso be compatible with pairs of quasi-MBSs, as the split-tings of the latter can become exponentially suppresseddue to reasons other than spatial separation [248]. Theoscillations, moreover, are found to decay with magneticfield, contrary to the behaviour expected from topologicalMBSs [249], and more in line with that of quasi-MBSs.

This kind of interpretation loophole is likely unavoid-able in almost any type of experiment using purely localprobes, which explains the longevity of the MBS vs ABScontroversy. Other than finding a quantitative matchwith theoretical models across a large portion of param-eter space, it seems that the only way out of these ambi-guities will require truly non-local experimental schemes,see Sec. IV D. In view of this, it becomes crucial to con-sider all possible types of zero modes when interpretingcurrent and future experiments.

A summary of the main types of zero modes and theirdefining properties is given in Table I. We distinguishbroadly between (a) conventional, topologically trivialABSs and variations thereof, (b) MBSs of topological ori-gin, and (c) zero modes produced by some form of smoothinhomogeneity with a trivial bulk. The first group in-cludes the SNS ABSs of Figs. 1 and 3 (e-g), with or with-out SO coupling and Zeeman field. These states can befine-tuned to zero energy by e.g a phase difference ϕ ∼ πacross the junction or an adequate Zeeman field VZ . Theanalogous ABSs in INS junctions, magnetic impuritiesor proximitized quantum dots are grouped under Shibastates, see Fig. 2 (g). ABSs usually show no pinning tozero energy. However, ABSs INS junctions with strongSO coupling [81] or with short-range inhomogeneities andinterband coupling [89, 172] may exhibit approximatezero-energy pinning as a function of some system param-eters, as mentioned in Sec. IV B. In general all theseABSs have a high degree of spatial overlap of their Ma-jorana components. In the second group we consider thetopological MBSs in nanowires with a topological bulk,both for short and long nanowires. The latter case cor-responds to MBSs with exponentially small overlaps, theparadigmatic case highlighted by minimal models. In thelast group we include all zero modes produced by suffi-ciently smooth inhomogeneities. We distinguish statesin smoothly confined SC nanowires [74] and the vari-ous forms of topologically trivial zero modes in smoothNS or S’S junctions [73, 75–85, 88], including the excep-tional point MBS (EP-MBS) generalization in open sys-tems [87, 92]. The distinction between these subclasses ismostly historical, however, as the underlying mechanismfor their formation is the same. All these states are char-

acterized by a strong pinning as smoothness is increased,and partially overlapping wavefunctions.

The debate on the interpretation of experimental sig-natures has often been framed in terms of “true” and“fake” MBSs, or actual MBSs (of the topological classabove) and conventional zero energy ABSs. Such di-chotomy has had the unfortunate side effect of establish-ing an imprecise terminology in some of the literature,whereby the term “Majorana” is used as a synonym ofnon-trivial band topology, instead of its original mean-ing of a self-conjugate zero-energy eigenstate. In truth,any zero energy fermionic eigenstate c of a hybrid sys-tem, regardless of its origin, can be formally expressedas the sum of two self-conjugate Majorana eigenstatesc = γ1 + iγ2 [32], see Box. B. This includes zero-energytrivial-bulk quasi- or ps-MBSs.

As we expand upon in the next section, what makesMBSs of topological origin special is the exponential sup-pression of the spatial overlap between the γ1 and γ2

wavefunctions (i.e. the degree of non-locality of c). Thissuppression, however, requires nanowires longer than theMajorana length, L � ξM , and of sufficiently unifor-mity so that MBSs are truly located at their ends (i.e.no additional zero modes appear in the bulk at uncon-trolled inhomogeneities or defects). In real, finite-lengthnanowires these stringent conditions need not be sat-isfied. In a generic case, a useful formulation of theMBSs vs ABSs debate is based on whether a given robustzero-energy mode, regardless of its trivial or non-trivialbulk topology, has a sufficiently small Majorana over-lap for a given application. Some requirements, such asresilience to arbitrary local noise or the possibility of spa-tial braiding, demand exponentially suppressed overlaps,while others, such as parametric non-Abelian braiding(i.e. relative phase manipulations without spatial dis-placements), merely need that a local probe may be se-lectively coupled to a single Majorana [85, 87, 92]. Insuch cases, ps-MBSs may be good enough.

D. Protection against errors and MBS overlaps

Topological quantum computation was proposed as away to achieve scalability through the hardware-level re-silience of Majorana-based qubits to arbitrary local noise,in principle guaranteed by spatial non-locality. The Ma-jorana qubit is defined in terms of the occupation of non-local fermion states such as c = γ1 + iγ2 [32], see Box B.As efforts develop towards realising this promise, differ-ent error-inducing mechanisms have been identified andstudied for topological MBS qubits, such as those cre-ated by a coupling to ungapped [94] or gapped [93, 95]fermionic baths (quasiparticle poisoning), as well as tofluctuating bosonic fields [100] (e.g. phonons [103, 106],photons [96, 98, 101], thermal fluctuations of a gate po-tential [97, 104, 105], or electromagnetic environments[103]). One must also consider the errors induced byqubit manipulation, such as unwanted excitations cre-

15

The zero-bias peak,measured experimentally in [8–12], is a non-specific signature ofMajoranas, sincesimilar features arise due toKondo physics or weak anti-localization [13, 14]. To help distinguishingMajoranasignatures from these alternatives, we focus on the parametric dependence of twoMajorana properties: theshape of the topological phase boundary [15, 16] and the oscillations in the coupling energy of twoMajoranamodes [17–21].

Both phenomena depend on the response of the chemical potential to amagnetic field, and hence onelectrostatic effects.Majorana oscillations were analyzed theoretically in two extreme limits for the electrostaticeffects: constant chemical potential [19–21] and constant density [20] (see appendix A for a summary of thesetwo limits). In particular [20], found different behavior ofMajorana oscillations in these two extreme limits.Weshow that the actual behavior of the nanowire is somewhere in between, and depends strongly on theelectrostatics.

2. Setup andmethods

2.1. The Schrödinger–Poisson problemWediscuss electrostatic effects in a device design as used byMourik et al [8], however ourmethods arestraightforward to adapt to similar layouts (see appendix B for a calculation using a different geometry). Sinceweare interested in the bulk properties, we require that the potential and theHamiltonian terms are translationallyinvariant along thewire axis andwe consider a 2D cross section, shown infigure 1. The device consists of ananowirewith a hexagonal cross section of diameter W 100 nm= on a dielectric layer with thicknessd 30 nmdielectric = . A superconductor with thickness d 187 nmSC = covers half of thewire. The nanowire has adielectric constant 17.7r� = (InSb), the dielectric layer has a dielectric constant 8r� = (Si3N4). The device hastwo electrostatic boundary conditions: afixed gate potentialVG set by the gate electrode along the lower edge ofthe dielectric layer and afixed potential VSC in the superconductor, whichwemodel as a groundedmetallic gate.We set this potential to eitherV 0 VSC = , disregarding awork function difference between theNbTiNsuperconductor and the nanowire, or we assume a small work function difference [22, 23] resultinginV 0.2 VSC = .

Wemodel the electrostatics of this setup using the Schrödinger–Poisson equation.We split theHamiltonianinto transverse and longitudinal parts. The transverseHamiltonian T reads

m x ye x y

E

2,

2, 1T

2 2

2

2

2

gap⎛⎝⎜

⎞⎠⎟*

� f= -

¶¶

+¶¶

- +( ) ( )

with x y, the transverse directions, m m0.014 e* = the effective electronmass in InSb (withme the electronmass), e- the electron charge, andf the electrostatic potential.We assume that in the absence of electric fieldthe Fermi level EF in the nanowire is in themiddle of the semiconducting gap Egap, with E 0.2 eVgap = for InSb(see figure 2(a).We choose the Fermi level EF as the reference energy such that E 0F º .

The longitudinalHamiltonian L reads

m z zE

2i , 2y zL

2 2

2 Z*

� a s s= -

¶¶

-¶¶

+ ( )

with z the direction along thewire axis,α the spin–orbit coupling strength, EZ the Zeeman energy and s thePaulimatrices. The orientation of themagnetic field is along thewire in the zdirection. In this separation, wehave assumed that the spin–orbit length l mSO

2 *� a= ( ) is larger or comparable to thewire diameter,

Figure 1. Schematic cross section of theMajorana device. It consists of a nanowire (red hexagon) lying on a dielectric layer (bluerectangle)which covers a global back gate. A superconducting lead (yellow region) covers half of the nanowire.

2

New J. Phys. 18 (2016) 033013 AVuik et al

QUANTIFYING WAVE-FUNCTION OVERLAPS IN … PHYSICAL REVIEW B 98, 235406 (2018)

dissect P (η,!s ) into partial probability densities for increas-ing degree of Fermi energy inhomogeneity "µ. We find thatfor inhomogeneities "µ < 1 meV, the estimator preserves ahigh r = 0.95 correlation with !s (red subpanel), but increas-ing "µ (green, blue subpanels) suppresses r , though the effectis not drastic, with r ≈ 0.9 still. This remains true regardlessof the maximum nanowire density considered.

V. SMOOTH NS NANOWIRES

We now consider the second type of inhomogeneousnanowire, wherein the pairing, like φ(x), is also positiondependent, "(x). We again consider a simple profile thatinterpolates between a left side and a right side. The left side isalways normal in this case, with "N = 0, so that the nanowirecontains a smooth NS interface centered at x = LN ,

φ(x) = φN + (φS − φN )θζ (x − LN ),

"(x) = "Sθζ (x − LN ). (16)

This model is relevant to many devices explored in recentexperiments. Nanowires are often made superconducting bygrowing an epitaxial superconductor on their surface. Often,the epitaxial coverage of the nanowire is incomplete, so itis natural to assume a suppressed pairing in the exposed

portions. Like in the S′S nanowire, a thorough microscopicvalidation of this model would require a detailed characteri-zation of the device in question.

The fundamental interest of the Lutchyn-Oreg model witha smooth NS interface is particularly high because of the factthat, perhaps surprisingly, it can also host near-zero modesat finite Zeeman field B, much like the smooth S′S, despitenot developing a topological gap on the normal side. Thisis shown in Fig. 4, which is the NS version of Fig. 3. Thesuppressed pairing gives rise to Andreev levels in the normalregion. Depending on the normal length LN , their level spac-ing δϵ can be much smaller than the induced gap ", whichresults is many subgap levels (unlike the S′S case, where onlya lone level, detached from the quasicontinuum appears). Afinite B field Zeeman splits all these subgap levels that evolve,avoiding each other due to spin-orbit coupling. This is true forall except the lowest two excitations (blue), which convergeto zero energy with a finite slope at low B fields [48] (thisis unlike in the S′S case, where the lone detached level startsoff flat at B = 0) [69]. Despite the superficial resemblanceto Zeeman-induced parity crossings in quantum dots [6,69](see Fig. 6), near-perfect Andreev reflection of N electrons onthe smooth NS interface stabilises this low-lying subgap levelnear zero energy for B > δϵ, but still well before BS

c .

(a)

(c) (d)

(b)

FIG. 4. Smooth NS nanowires. Equivalent to Figs. 2 and 3, with identical model and sampling parameters as in the latter, except for azero pairing "N = 0 on the left side and finite "S = 0.5 meV on the right side of the smooth junction. Note the similar wave functions of thesmooth junction Majoranas as compared to the S′S case of Fig. 3.

235406-9

UNIFIED NUMERICAL APPROACH TO TOPOLOGICAL … PHYSICAL REVIEW B 99, 245408 (2019)

In Fig. 11(a) we show the phase diagram of the three-facet device without orbital effects. The phase diagram lookssimilar to earlier findings of multisubband wires [23,26,89],although we find a strong dependence of the semiconductor-superconductor coupling on the subband, resulting in a largevariation of minimal critical magnetic fields corresponding tothe phase transition. As has also been pointed out in Ref. [26],the lever arm of chemical potential vs gate voltage is signifi-cantly larger at positive or small negative gate voltages than atlarge negative ones. Consequently, the density of topologicalphases is higher in VG for small negative gate voltages inFig. 11. The reason for this is twofold: First, the electronstates localized near the gate are more easily tuned by the backgate than the states close to the superconductor. Second, thescreening effect of the holes decreases the lever arm furtherfor large negative gate voltage. In general, not taking orbitaleffect into account leads often to magnetic fields, at whichthe topological phase transitions, being large compared toexperiments.

Turning the orbital effect on in Fig. 11(b) changes theshapes of the phase boundaries dramatically. For small neg-ative gate voltages the phase diagram is dominated by theorbital effect of magnetic field. This becomes apparent due tothe small magnetic fields at which the topological transitionoccur and the very nonparabolic shape of the phase bound-aries. In this regime one often finds two topological regionsemerging close in gate voltage at similar magnetic fields, thatseparate from each other, one drifting to larger gate voltagesand the other to smaller gate voltages. These result from twosubbands that are near angular momentum eigenstates, withapproximately opposite angular momentum [38]. One of the

-3

-4

-5

-6

VG

(V)

(a) (b)

0 1 2

B (T)

0.0

-0.5

-1.0

-1.5

VG

(V)

0 1 2

B (T)

0

5

10

15

20E

g(µ

eV)

FIG. 11. Topological phase diagram of the three-facet devicewith 7 nm Al shell and ρacc = 2 × 1019 e/cm3 for (a) without orbitaleffects, (b) with orbital effects. For case (b) the topological gap isoverlaid in selected regions (it has not been calculated in the grayregions).

−0.04 −0.02 0.00 0.02 0.04

kz (nm−1)

−50

0

50

E(µ

eV)

(a)

−0.04 −0.02 0.00 0.02 0.04

kz (nm−1)

(b)

FIG. 12. (a) [(b)] Band structures in the topological phase for thethree-facet [two-facet] device with the parameters VG = −4.15 V andB = 0.375 T (VG = −1.22 V and B = 0.575 T).

reasons why the orbital effect is so strong is the high electrondensity, which is a result of the large band-offset of InAs/Aland the accumulation layer resulting in about ten occupiedsubbands in InAs. High subbands have high orbital quantumnumbers coupling strongly to magnetic field [38]. At largenegative gate voltage the orbital effect is suppressed and thephase boundaries look closer to the ones without orbital effect,although the influence of the orbital effect is still stronglypresent.

From Fig. 11(b) it becomes apparent that only topologicalphases with appreciable negative back-gate voltage have a siz-able topological gap. We find that the maximum topologicalgap is only slightly larger than 20 µeV. While this seemslike a small value we emphasize that it is proportional to thestrength of the Rashba spin-orbit coupling. In our calculation,the value of spin-orbit coupling is conservative since we takeonly electrostatic origin of spin-orbit coupling into account.The value of α we obtain from Eq. (10) is typically about10 meV nm, whereas experiments report values in the range of10 to 30 meV nm [90,91] which would result in a significantlylarger topological gap.

D. Effect of broken mirror symmetry in the two-facet device

In terms of symmetries, the most significant differencebetween the three- and two-facet devices is the vertical mirrorsymmetry in the (y, z)-plane Myz. Additionally consideringthe particle-hole symmetry PH (k)P−1 = −H∗(−k), P2 =+1, which protects the MZMs, and the time-reversal symme-try T H (k)T −1 = H∗(−k), T 2 = −1 it can be shown that thecombination of the three symmetries create a chiral symmetry

CH (k)C−1 = −H (k), C2 = +1 (11)

that survives at finite magnetic field parallel to the (y, z) mirrorplane. For the specific case of our Hamiltonian Eq. (6) thechiral symmetry is given by C = τyσzδ(x + x′) [with δ(x + x′)being the real-space reflection operator taking x to −x]. Notethat the Rashba term αx breaks this chiral symmetry.

One particular consequence of the chiral and particle-holesymmetry is that the band structure is line-reflection symmet-ric around the k = 0 and E = 0 axes, see Fig. 12(a). In thetwo-facet device the chiral symmetry is broken because of themissing mirror symmetry Myz. Therefore, the band structureis only point-inversion symmetric around the (E = 0, k = 0)point, as dictated by the particle-hole symmetry. At finiteB this generically leads to a tilting of the band structure

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UNIFIED NUMERICAL APPROACH TO TOPOLOGICAL … PHYSICAL REVIEW B 99, 245408 (2019)

In Fig. 11(a) we show the phase diagram of the three-facet device without orbital effects. The phase diagram lookssimilar to earlier findings of multisubband wires [23,26,89],although we find a strong dependence of the semiconductor-superconductor coupling on the subband, resulting in a largevariation of minimal critical magnetic fields corresponding tothe phase transition. As has also been pointed out in Ref. [26],the lever arm of chemical potential vs gate voltage is signifi-cantly larger at positive or small negative gate voltages than atlarge negative ones. Consequently, the density of topologicalphases is higher in VG for small negative gate voltages inFig. 11. The reason for this is twofold: First, the electronstates localized near the gate are more easily tuned by the backgate than the states close to the superconductor. Second, thescreening effect of the holes decreases the lever arm furtherfor large negative gate voltage. In general, not taking orbitaleffect into account leads often to magnetic fields, at whichthe topological phase transitions, being large compared toexperiments.

Turning the orbital effect on in Fig. 11(b) changes theshapes of the phase boundaries dramatically. For small neg-ative gate voltages the phase diagram is dominated by theorbital effect of magnetic field. This becomes apparent due tothe small magnetic fields at which the topological transitionoccur and the very nonparabolic shape of the phase bound-aries. In this regime one often finds two topological regionsemerging close in gate voltage at similar magnetic fields, thatseparate from each other, one drifting to larger gate voltagesand the other to smaller gate voltages. These result from twosubbands that are near angular momentum eigenstates, withapproximately opposite angular momentum [38]. One of the

-3

-4

-5

-6

VG

(V)

(a) (b)

0 1 2

B (T)

0.0

-0.5

-1.0

-1.5

VG

(V)

0 1 2

B (T)

0

5

10

15

20

Eg

(µeV

)

FIG. 11. Topological phase diagram of the three-facet devicewith 7 nm Al shell and ρacc = 2 × 1019 e/cm3 for (a) without orbitaleffects, (b) with orbital effects. For case (b) the topological gap isoverlaid in selected regions (it has not been calculated in the grayregions).

−0.04 −0.02 0.00 0.02 0.04

kz (nm−1)

−50

0

50

E(µ

eV)

(a)

−0.04 −0.02 0.00 0.02 0.04

kz (nm−1)

(b)

FIG. 12. (a) [(b)] Band structures in the topological phase for thethree-facet [two-facet] device with the parameters VG = −4.15 V andB = 0.375 T (VG = −1.22 V and B = 0.575 T).

reasons why the orbital effect is so strong is the high electrondensity, which is a result of the large band-offset of InAs/Aland the accumulation layer resulting in about ten occupiedsubbands in InAs. High subbands have high orbital quantumnumbers coupling strongly to magnetic field [38]. At largenegative gate voltage the orbital effect is suppressed and thephase boundaries look closer to the ones without orbital effect,although the influence of the orbital effect is still stronglypresent.

From Fig. 11(b) it becomes apparent that only topologicalphases with appreciable negative back-gate voltage have a siz-able topological gap. We find that the maximum topologicalgap is only slightly larger than 20 µeV. While this seemslike a small value we emphasize that it is proportional to thestrength of the Rashba spin-orbit coupling. In our calculation,the value of spin-orbit coupling is conservative since we takeonly electrostatic origin of spin-orbit coupling into account.The value of α we obtain from Eq. (10) is typically about10 meV nm, whereas experiments report values in the range of10 to 30 meV nm [90,91] which would result in a significantlylarger topological gap.

D. Effect of broken mirror symmetry in the two-facet device

In terms of symmetries, the most significant differencebetween the three- and two-facet devices is the vertical mirrorsymmetry in the (y, z)-plane Myz. Additionally consideringthe particle-hole symmetry PH (k)P−1 = −H∗(−k), P2 =+1, which protects the MZMs, and the time-reversal symme-try T H (k)T −1 = H∗(−k), T 2 = −1 it can be shown that thecombination of the three symmetries create a chiral symmetry

CH (k)C−1 = −H (k), C2 = +1 (11)

that survives at finite magnetic field parallel to the (y, z) mirrorplane. For the specific case of our Hamiltonian Eq. (6) thechiral symmetry is given by C = τyσzδ(x + x′) [with δ(x + x′)being the real-space reflection operator taking x to −x]. Notethat the Rashba term αx breaks this chiral symmetry.

One particular consequence of the chiral and particle-holesymmetry is that the band structure is line-reflection symmet-ric around the k = 0 and E = 0 axes, see Fig. 12(a). In thetwo-facet device the chiral symmetry is broken because of themissing mirror symmetry Myz. Therefore, the band structureis only point-inversion symmetric around the (E = 0, k = 0)point, as dictated by the particle-hole symmetry. At finiteB this generically leads to a tilting of the band structure

245408-9

b

f

d

QUANTIFYING WAVE-FUNCTION OVERLAPS IN … PHYSICAL REVIEW B 98, 235406 (2018)

dissect P (η,!s ) into partial probability densities for increas-ing degree of Fermi energy inhomogeneity "µ. We find thatfor inhomogeneities "µ < 1 meV, the estimator preserves ahigh r = 0.95 correlation with !s (red subpanel), but increas-ing "µ (green, blue subpanels) suppresses r , though the effectis not drastic, with r ≈ 0.9 still. This remains true regardlessof the maximum nanowire density considered.

V. SMOOTH NS NANOWIRES

We now consider the second type of inhomogeneousnanowire, wherein the pairing, like φ(x), is also positiondependent, "(x). We again consider a simple profile thatinterpolates between a left side and a right side. The left side isalways normal in this case, with "N = 0, so that the nanowirecontains a smooth NS interface centered at x = LN ,

φ(x) = φN + (φS − φN )θζ (x − LN ),

"(x) = "Sθζ (x − LN ). (16)

This model is relevant to many devices explored in recentexperiments. Nanowires are often made superconducting bygrowing an epitaxial superconductor on their surface. Often,the epitaxial coverage of the nanowire is incomplete, so itis natural to assume a suppressed pairing in the exposed

portions. Like in the S′S nanowire, a thorough microscopicvalidation of this model would require a detailed characteri-zation of the device in question.

The fundamental interest of the Lutchyn-Oreg model witha smooth NS interface is particularly high because of the factthat, perhaps surprisingly, it can also host near-zero modesat finite Zeeman field B, much like the smooth S′S, despitenot developing a topological gap on the normal side. Thisis shown in Fig. 4, which is the NS version of Fig. 3. Thesuppressed pairing gives rise to Andreev levels in the normalregion. Depending on the normal length LN , their level spac-ing δϵ can be much smaller than the induced gap ", whichresults is many subgap levels (unlike the S′S case, where onlya lone level, detached from the quasicontinuum appears). Afinite B field Zeeman splits all these subgap levels that evolve,avoiding each other due to spin-orbit coupling. This is true forall except the lowest two excitations (blue), which convergeto zero energy with a finite slope at low B fields [48] (thisis unlike in the S′S case, where the lone detached level startsoff flat at B = 0) [69]. Despite the superficial resemblanceto Zeeman-induced parity crossings in quantum dots [6,69](see Fig. 6), near-perfect Andreev reflection of N electrons onthe smooth NS interface stabilises this low-lying subgap levelnear zero energy for B > δϵ, but still well before BS

c .

(a)

(c) (d)

(b)

FIG. 4. Smooth NS nanowires. Equivalent to Figs. 2 and 3, with identical model and sampling parameters as in the latter, except for azero pairing "N = 0 on the left side and finite "S = 0.5 meV on the right side of the smooth junction. Note the similar wave functions of thesmooth junction Majoranas as compared to the S′S case of Fig. 3.

235406-9

QUANTIFYING WAVE-FUNCTION OVERLAPS IN … PHYSICAL REVIEW B 98, 235406 (2018)

dissect P (η,!s ) into partial probability densities for increas-ing degree of Fermi energy inhomogeneity "µ. We find thatfor inhomogeneities "µ < 1 meV, the estimator preserves ahigh r = 0.95 correlation with !s (red subpanel), but increas-ing "µ (green, blue subpanels) suppresses r , though the effectis not drastic, with r ≈ 0.9 still. This remains true regardlessof the maximum nanowire density considered.

V. SMOOTH NS NANOWIRES

We now consider the second type of inhomogeneousnanowire, wherein the pairing, like φ(x), is also positiondependent, "(x). We again consider a simple profile thatinterpolates between a left side and a right side. The left side isalways normal in this case, with "N = 0, so that the nanowirecontains a smooth NS interface centered at x = LN ,

φ(x) = φN + (φS − φN )θζ (x − LN ),

"(x) = "Sθζ (x − LN ). (16)

This model is relevant to many devices explored in recentexperiments. Nanowires are often made superconducting bygrowing an epitaxial superconductor on their surface. Often,the epitaxial coverage of the nanowire is incomplete, so itis natural to assume a suppressed pairing in the exposed

portions. Like in the S′S nanowire, a thorough microscopicvalidation of this model would require a detailed characteri-zation of the device in question.

The fundamental interest of the Lutchyn-Oreg model witha smooth NS interface is particularly high because of the factthat, perhaps surprisingly, it can also host near-zero modesat finite Zeeman field B, much like the smooth S′S, despitenot developing a topological gap on the normal side. Thisis shown in Fig. 4, which is the NS version of Fig. 3. Thesuppressed pairing gives rise to Andreev levels in the normalregion. Depending on the normal length LN , their level spac-ing δϵ can be much smaller than the induced gap ", whichresults is many subgap levels (unlike the S′S case, where onlya lone level, detached from the quasicontinuum appears). Afinite B field Zeeman splits all these subgap levels that evolve,avoiding each other due to spin-orbit coupling. This is true forall except the lowest two excitations (blue), which convergeto zero energy with a finite slope at low B fields [48] (thisis unlike in the S′S case, where the lone detached level startsoff flat at B = 0) [69]. Despite the superficial resemblanceto Zeeman-induced parity crossings in quantum dots [6,69](see Fig. 6), near-perfect Andreev reflection of N electrons onthe smooth NS interface stabilises this low-lying subgap levelnear zero energy for B > δϵ, but still well before BS

c .

(a)

(c) (d)

(b)

FIG. 4. Smooth NS nanowires. Equivalent to Figs. 2 and 3, with identical model and sampling parameters as in the latter, except for azero pairing "N = 0 on the left side and finite "S = 0.5 meV on the right side of the smooth junction. Note the similar wave functions of thesmooth junction Majoranas as compared to the S′S case of Fig. 3.

235406-9

x/L

QUANTIFYING WAVE-FUNCTION OVERLAPS IN … PHYSICAL REVIEW B 98, 235406 (2018)

dissect P (η,!s ) into partial probability densities for increas-ing degree of Fermi energy inhomogeneity "µ. We find thatfor inhomogeneities "µ < 1 meV, the estimator preserves ahigh r = 0.95 correlation with !s (red subpanel), but increas-ing "µ (green, blue subpanels) suppresses r , though the effectis not drastic, with r ≈ 0.9 still. This remains true regardlessof the maximum nanowire density considered.

V. SMOOTH NS NANOWIRES

We now consider the second type of inhomogeneousnanowire, wherein the pairing, like φ(x), is also positiondependent, "(x). We again consider a simple profile thatinterpolates between a left side and a right side. The left side isalways normal in this case, with "N = 0, so that the nanowirecontains a smooth NS interface centered at x = LN ,

φ(x) = φN + (φS − φN )θζ (x − LN ),

"(x) = "Sθζ (x − LN ). (16)

This model is relevant to many devices explored in recentexperiments. Nanowires are often made superconducting bygrowing an epitaxial superconductor on their surface. Often,the epitaxial coverage of the nanowire is incomplete, so itis natural to assume a suppressed pairing in the exposed

portions. Like in the S′S nanowire, a thorough microscopicvalidation of this model would require a detailed characteri-zation of the device in question.

The fundamental interest of the Lutchyn-Oreg model witha smooth NS interface is particularly high because of the factthat, perhaps surprisingly, it can also host near-zero modesat finite Zeeman field B, much like the smooth S′S, despitenot developing a topological gap on the normal side. Thisis shown in Fig. 4, which is the NS version of Fig. 3. Thesuppressed pairing gives rise to Andreev levels in the normalregion. Depending on the normal length LN , their level spac-ing δϵ can be much smaller than the induced gap ", whichresults is many subgap levels (unlike the S′S case, where onlya lone level, detached from the quasicontinuum appears). Afinite B field Zeeman splits all these subgap levels that evolve,avoiding each other due to spin-orbit coupling. This is true forall except the lowest two excitations (blue), which convergeto zero energy with a finite slope at low B fields [48] (thisis unlike in the S′S case, where the lone detached level startsoff flat at B = 0) [69]. Despite the superficial resemblanceto Zeeman-induced parity crossings in quantum dots [6,69](see Fig. 6), near-perfect Andreev reflection of N electrons onthe smooth NS interface stabilises this low-lying subgap levelnear zero energy for B > δϵ, but still well before BS

c .

(a)

(c) (d)

(b)

FIG. 4. Smooth NS nanowires. Equivalent to Figs. 2 and 3, with identical model and sampling parameters as in the latter, except for azero pairing "N = 0 on the left side and finite "S = 0.5 meV on the right side of the smooth junction. Note the similar wave functions of thesmooth junction Majoranas as compared to the S′S case of Fig. 3.

235406-9

RAPID COMMUNICATIONS

ELSA PRADA, PABLO SAN-JOSE, AND RAMON AGUADO PHYSICAL REVIEW B 86, 180503(R) (2012)

FIG. 1. (Color online) Schematics of the nanowire junction inthe NdSdS (a) and NSdS (b) setups, and spatial variation ofsuperconducting gap and potential profiles (c). Gate Vd depletes thewire, while Vp creates a tunnel contact (I) to the left (normal) reservoir.One (red) or two (red and yellow spheres) Majorana bound states mayappear at the edges of the depleted region depending on the Zeemanfield and gate voltage Vd . (d) Transport regimes for a transparent NS

junction (Vd,p = 0, µ = 4!) in the Zeeman-field–bias plane.

a length scale L! ≡ h/√

m! ≡ 142 nm. Strong SO coupling,representative of InSb wires,20 is α = 20 meV nm, with SOlength LSO = h2/(mα) = 200 nm = 1.4L!.21

Scales. A localized MBS is formed at the boundary of atrivially gapped and a TS portion of the wire. At a point x thewire will be in the TS phase if !(x) > 0 and

B >!

[µ −U (x)]2 + !(x)2. (1)

The asymptotic value of the critical field is the proper (bulk)critical field Bc. Apart from Bc, several other Zeeman scalesdictate the junction’s transport properties. The first one is theTS critical field in the depleted part of the superconductingwire, Bd

c ≡!

(µ −Ud )2 + !2, which is smaller than Bc, as isthe purpose of the depletion gate. It should be noted, however,that the depleted Sd region has a finite length, which cruciallyaffects Majorana modes for Bd

c < B < Bc, as discussed later,while the S portion is assumed infinite. Second, there is thefield above which the normal side of the wire becomes ahelical liquid (momentum and spin become correlated). In theNSdS case (normal side not depleted), this is Bh ≡ µ, which istypically slightly smaller than Bc, but bigger than both Bd

c andthe corresponding helical field in the NdSdS case, namely,Bd

h ≡ |µ −Ud | < Bdc . Finally, there is the superconducting

gap itself, B! ≡ !, whose significance will become clear later.All these scales (B! plus Bd

c < Bh < Bc in the NSdS case, orBd

h < Bdc < Bc in the NdSdS), control different aspects of the

junction’s differential conductance in the B-V plane.Differential conductance. The dI/dV of a NS junction may

be related to the intrinsic conductance at zero temperature bythe expression22

dI (V )dV

= e2

h

"N − Tr(r†

eeree) + Tr(r†ehreh)

#ϵ=V

.

FIG. 2. (Color online) Density plots of the dI/dV in the NdSdS

junction (µ = 4!, Ud = 3.25!, Up = 25!, δ = 0) for LSO = 1.4L!

as a function of bias voltage V and Zeeman field B with atunnel pinch-off barrier and a depletion region of length LNd + LSd ,Fig. 1(a). Different columns feature increasing values of LSd fromleft to right, whereas different rows feature increasing length LNd

from top to bottom.

Here, N is the number of propagating channels in the normalside at energy ϵ = V , and ree and reh are their normal andAndreev reflection matrices. These matrices can be computedin a number of ways. The most flexible is the recursive NambuGreen’s function approach, employed here (for full details, seeRef. 23).

Before considering the effect of U (x), we show the transportphase diagram [see Fig. 1(c)] in the simple NS transparentlimit, i.e., in a regime where the concepts of MBSs and ZBAsno longer hold. We observe different transport regions inthe B-V plane characterized by an integer dI/dV ≈ ne2/h,with n = 0,1,2,3,4. Such is the case of Cooper pair transport(region I, n = 4) or single quasiparticle transport (region III,n = 2). The latter is a TS regime, whose topology becomesevident in the dI/dV despite the fact that the associatedMajorana fermion is completely smeared out due to thegapless spectrum for x < 0.24–27 Between these two regions,the helical regime is characterized by a fully suppressedzero-bias conductance (region II, n = 0). These results extendthe concept of half-integer conductance quantization24 beyondlinear response.

We now consider the NdSdS junction with the full U (x).Its dI/dV response (with LSO = 1.4L!) is plotted in Fig. 2.Different panels cover different ratios LNd /L! and LSd /L!.The tunnel barrier Up is tuned in each case to yield spectro-scopic resolution in the transport response. A wide range ofbehaviors becomes apparent, which reflects the local density ofstates (DOS) at the pinch-off gate. The most paradigmatic oneis probably the one in the top-left panel. It reflects the closingof the effective superconducting gap (marked by the gap-edge

180503-2

RAPID COMMUNICATIONS

ELSA PRADA, PABLO SAN-JOSE, AND RAMON AGUADO PHYSICAL REVIEW B 86, 180503(R) (2012)

FIG. 1. (Color online) Schematics of the nanowire junction inthe NdSdS (a) and NSdS (b) setups, and spatial variation ofsuperconducting gap and potential profiles (c). Gate Vd depletes thewire, while Vp creates a tunnel contact (I) to the left (normal) reservoir.One (red) or two (red and yellow spheres) Majorana bound states mayappear at the edges of the depleted region depending on the Zeemanfield and gate voltage Vd . (d) Transport regimes for a transparent NS

junction (Vd,p = 0, µ = 4!) in the Zeeman-field–bias plane.

a length scale L! ≡ h/√

m! ≡ 142 nm. Strong SO coupling,representative of InSb wires,20 is α = 20 meV nm, with SOlength LSO = h2/(mα) = 200 nm = 1.4L!.21

Scales. A localized MBS is formed at the boundary of atrivially gapped and a TS portion of the wire. At a point x thewire will be in the TS phase if !(x) > 0 and

B >!

[µ −U (x)]2 + !(x)2. (1)

The asymptotic value of the critical field is the proper (bulk)critical field Bc. Apart from Bc, several other Zeeman scalesdictate the junction’s transport properties. The first one is theTS critical field in the depleted part of the superconductingwire, Bd

c ≡!

(µ −Ud )2 + !2, which is smaller than Bc, as isthe purpose of the depletion gate. It should be noted, however,that the depleted Sd region has a finite length, which cruciallyaffects Majorana modes for Bd

c < B < Bc, as discussed later,while the S portion is assumed infinite. Second, there is thefield above which the normal side of the wire becomes ahelical liquid (momentum and spin become correlated). In theNSdS case (normal side not depleted), this is Bh ≡ µ, which istypically slightly smaller than Bc, but bigger than both Bd

c andthe corresponding helical field in the NdSdS case, namely,Bd

h ≡ |µ −Ud | < Bdc . Finally, there is the superconducting

gap itself, B! ≡ !, whose significance will become clear later.All these scales (B! plus Bd

c < Bh < Bc in the NSdS case, orBd

h < Bdc < Bc in the NdSdS), control different aspects of the

junction’s differential conductance in the B-V plane.Differential conductance. The dI/dV of a NS junction may

be related to the intrinsic conductance at zero temperature bythe expression22

dI (V )dV

= e2

h

"N − Tr(r†

eeree) + Tr(r†ehreh)

#ϵ=V

.

FIG. 2. (Color online) Density plots of the dI/dV in the NdSdS

junction (µ = 4!, Ud = 3.25!, Up = 25!, δ = 0) for LSO = 1.4L!

as a function of bias voltage V and Zeeman field B with atunnel pinch-off barrier and a depletion region of length LNd + LSd ,Fig. 1(a). Different columns feature increasing values of LSd fromleft to right, whereas different rows feature increasing length LNd

from top to bottom.

Here, N is the number of propagating channels in the normalside at energy ϵ = V , and ree and reh are their normal andAndreev reflection matrices. These matrices can be computedin a number of ways. The most flexible is the recursive NambuGreen’s function approach, employed here (for full details, seeRef. 23).

Before considering the effect of U (x), we show the transportphase diagram [see Fig. 1(c)] in the simple NS transparentlimit, i.e., in a regime where the concepts of MBSs and ZBAsno longer hold. We observe different transport regions inthe B-V plane characterized by an integer dI/dV ≈ ne2/h,with n = 0,1,2,3,4. Such is the case of Cooper pair transport(region I, n = 4) or single quasiparticle transport (region III,n = 2). The latter is a TS regime, whose topology becomesevident in the dI/dV despite the fact that the associatedMajorana fermion is completely smeared out due to thegapless spectrum for x < 0.24–27 Between these two regions,the helical regime is characterized by a fully suppressedzero-bias conductance (region II, n = 0). These results extendthe concept of half-integer conductance quantization24 beyondlinear response.

We now consider the NdSdS junction with the full U (x).Its dI/dV response (with LSO = 1.4L!) is plotted in Fig. 2.Different panels cover different ratios LNd /L! and LSd /L!.The tunnel barrier Up is tuned in each case to yield spectro-scopic resolution in the transport response. A wide range ofbehaviors becomes apparent, which reflects the local density ofstates (DOS) at the pinch-off gate. The most paradigmatic oneis probably the one in the top-left panel. It reflects the closingof the effective superconducting gap (marked by the gap-edge

180503-2

e

Reservoir µ(x)∆ (x)

LN LS

Γ—

Γ+

a

Non-Hermitian topology: a unifying framework for the Andreev versus Majoranastates controversy

J. Avila,1, ⇤ F. Penaranda,2, ⇤ E. Prada,2 P. San-Jose,1, † and R. Aguado1, ‡

1Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM),Consejo Superior de Investigaciones Cientıficas (CSIC),

Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain2Departamento de Fısica de la Materia Condensada,

Condensed Matter Physics Center (IFIMAC) and Instituto Nicolas Cabrera,Universidad Autonoma de Madrid, E-28049 Madrid, Spain

(Dated: July 13, 2018)

Andreev bound states (ABSs) in hybrid semiconductor-superconductor nanowires can have near-zero energy in parameter regions where band topology predicts trivial phases. This surprising facthas been used to challenge the interpretation of a number of transport experiments in terms of non-trivial topology with Majorana zero modes (MZMs). We show that this ongoing ABS versus MZMcontroversy is fully clarified when framed in the language of non-Hermitian topology, the naturaldescription for open quantum systems. This change of paradigm allows us to understand topologicaltransitions and the emergence of pairs of zero modes more broadly, in terms of exceptional point(EP) bifurcations of system eigenvalue pairs in the complex plane. Within this framework, we showthat some zero energy ABSs are actually non-trivial, and share all the properties of conventionalMZMs, such as the recently observed 2e2/h conductance quantization. From this point of view,any distinction between such ABS zero modes and conventional MZMs becomes artificial. The keyfeature that underlies their common non-trivial properties is an asymmetric coupling of Majoranacomponents to the reservoir, which triggers the EP bifurcation.

Introduction—Since the remarkable prediction [1, 2]that a hybrid semiconductor-superconductor nanowirecan be tuned into a topological superconductor phasewith MZMs [3], there have been a number of papersreporting experimental data in the form of a zero-biasanomaly (ZBA) in the di↵erential conductance (dI/dV )for increasing Zeeman fields [4–10]. This behavior is con-sistent with tunneling into a MZM that emerges after thesystem undergoes a topological phase transition.

This Majorana interpretation has recently been chal-lenged since an alternative explanation in terms of ABSswith near-zero energy in the topological trivial phase,namely for Zeeman fields smaller than the critical fieldpredicted by band topology B < Bc, reproduces all theexpected phenomenology in transport. Following earlycalculations that proved that smooth confinement poten-tials inevitably lead to near-zero energy ABSs [11, 12], anumber of papers [13–18] have reported numerical ob-servations that systematically demonstrate that, indeed,ABSs in the trivial regime mimic Majoranas. This nag-ging ABS-versus-MZM question is compounded by therecent observation of 2e2/h conductance quantization[19], which can be also reproduced by ABSs [16], andthus is considered a serious objection in the field.

In this letter, we argue that the above question is illposed, and is the result of a viewpoint, that of bandtopology, only truly applicable to semi-infinite systems.We argue that a more general framework, relevant to theexperimental setup and rigorously well defined for finitesamples, allows us to precisely distinguish trivial fromnon-trivial zero modes. Among the latter are the MZMsfrom conventional band topology theory, but also a large

Re Im

-+

B+E

-E

EP

2

FIG. 1. Exceptional points. (a) The bifurcation, asa function of some external parameter B, of two complexGreen’s function poles across an exceptional point (EP) esta-bilises quasi-bound Majorana zero modes (red). Inset showsthe evolution of real and imaginary pole energies across theEP. (b) Sketch of the normal-superconductor (NS) junctionformed when a proximitized nanowire with inhomogeneouschemical potential and pairing, µ(x) and �(x), is coupled toa reservoir. Such junction is a natural host for Majorana zeromodes even below the critical field B < Bc that emerge fromEP bifurcations around parity crossings when their couplingis asymmetric �+

0 > ��0 due to spatial non-locality.

subset of ABSs zero modes. From this point of view bothkinds of states are really one and the same, which ex-plains why they cannot be distinguished. The key idea tounderstand this claim is to realize that, instead of conven-tional band topology, the natural language to describe anormal-superconductor (NS) junction, the geometry rel-evant to transport experiments, is that of open quantumsystems. In particular, we consider the non-Hermitiantopology defined in terms of the complex poles ✏p of theretarded Green’s function (or, equivalently, of the scat-

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Non-Hermitian topology: a unifying framework for the Andreev versus Majoranastates controversy

J. Avila,1, ⇤ F. Penaranda,2, ⇤ E. Prada,2 P. San-Jose,1, † and R. Aguado1, ‡

1Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM),Consejo Superior de Investigaciones Cientıficas (CSIC),

Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain2Departamento de Fısica de la Materia Condensada,

Condensed Matter Physics Center (IFIMAC) and Instituto Nicolas Cabrera,Universidad Autonoma de Madrid, E-28049 Madrid, Spain

(Dated: July 13, 2018)

Andreev bound states (ABSs) in hybrid semiconductor-superconductor nanowires can have near-zero energy in parameter regions where band topology predicts trivial phases. This surprising facthas been used to challenge the interpretation of a number of transport experiments in terms of non-trivial topology with Majorana zero modes (MZMs). We show that this ongoing ABS versus MZMcontroversy is fully clarified when framed in the language of non-Hermitian topology, the naturaldescription for open quantum systems. This change of paradigm allows us to understand topologicaltransitions and the emergence of pairs of zero modes more broadly, in terms of exceptional point(EP) bifurcations of system eigenvalue pairs in the complex plane. Within this framework, we showthat some zero energy ABSs are actually non-trivial, and share all the properties of conventionalMZMs, such as the recently observed 2e2/h conductance quantization. From this point of view,any distinction between such ABS zero modes and conventional MZMs becomes artificial. The keyfeature that underlies their common non-trivial properties is an asymmetric coupling of Majoranacomponents to the reservoir, which triggers the EP bifurcation.

Introduction—Since the remarkable prediction [1, 2]that a hybrid semiconductor-superconductor nanowirecan be tuned into a topological superconductor phasewith MZMs [3], there have been a number of papersreporting experimental data in the form of a zero-biasanomaly (ZBA) in the di↵erential conductance (dI/dV )for increasing Zeeman fields [4–10]. This behavior is con-sistent with tunneling into a MZM that emerges after thesystem undergoes a topological phase transition.

This Majorana interpretation has recently been chal-lenged since an alternative explanation in terms of ABSswith near-zero energy in the topological trivial phase,namely for Zeeman fields smaller than the critical fieldpredicted by band topology B < Bc, reproduces all theexpected phenomenology in transport. Following earlycalculations that proved that smooth confinement poten-tials inevitably lead to near-zero energy ABSs [11, 12], anumber of papers [13–18] have reported numerical ob-servations that systematically demonstrate that, indeed,ABSs in the trivial regime mimic Majoranas. This nag-ging ABS-versus-MZM question is compounded by therecent observation of 2e2/h conductance quantization[19], which can be also reproduced by ABSs [16], andthus is considered a serious objection in the field.

In this letter, we argue that the above question is illposed, and is the result of a viewpoint, that of bandtopology, only truly applicable to semi-infinite systems.We argue that a more general framework, relevant to theexperimental setup and rigorously well defined for finitesamples, allows us to precisely distinguish trivial fromnon-trivial zero modes. Among the latter are the MZMsfrom conventional band topology theory, but also a large

Re

FIG. 1. Exceptional points. (a) The bifurcation, asa function of some external parameter B, of two complexGreen’s function poles across an exceptional point (EP) esta-bilises quasi-bound Majorana zero modes (red). Inset showsthe evolution of real and imaginary pole energies across theEP. (b) Sketch of the normal-superconductor (NS) junctionformed when a proximitized nanowire with inhomogeneouschemical potential and pairing, µ(x) and �(x), is coupled toa reservoir. Such junction is a natural host for Majorana zeromodes even below the critical field B < Bc that emerge fromEP bifurcations around parity crossings when their couplingis asymmetric �+

0 > ��0 due to spatial non-locality.

subset of ABSs zero modes. From this point of view bothkinds of states are really one and the same, which ex-plains why they cannot be distinguished. The key idea tounderstand this claim is to realize that, instead of conven-tional band topology, the natural language to describe anormal-superconductor (NS) junction, the geometry rel-evant to transport experiments, is that of open quantumsystems. In particular, we consider the non-Hermitiantopology defined in terms of the complex poles ✏p of theretarded Green’s function (or, equivalently, of the scat-

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Non-Hermitian topology: a unifying framework for the Andreev versus Majoranastates controversy

J. Avila,1, ⇤ F. Penaranda,2, ⇤ E. Prada,2 P. San-Jose,1, † and R. Aguado1, ‡

1Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM),Consejo Superior de Investigaciones Cientıficas (CSIC),

Sor Juana Ines de la Cruz 3, 28049 Madrid, Spain2Departamento de Fısica de la Materia Condensada,

Condensed Matter Physics Center (IFIMAC) and Instituto Nicolas Cabrera,Universidad Autonoma de Madrid, E-28049 Madrid, Spain

(Dated: July 13, 2018)

Andreev bound states (ABSs) in hybrid semiconductor-superconductor nanowires can have near-zero energy in parameter regions where band topology predicts trivial phases. This surprising facthas been used to challenge the interpretation of a number of transport experiments in terms of non-trivial topology with Majorana zero modes (MZMs). We show that this ongoing ABS versus MZMcontroversy is fully clarified when framed in the language of non-Hermitian topology, the naturaldescription for open quantum systems. This change of paradigm allows us to understand topologicaltransitions and the emergence of pairs of zero modes more broadly, in terms of exceptional point(EP) bifurcations of system eigenvalue pairs in the complex plane. Within this framework, we showthat some zero energy ABSs are actually non-trivial, and share all the properties of conventionalMZMs, such as the recently observed 2e2/h conductance quantization. From this point of view,any distinction between such ABS zero modes and conventional MZMs becomes artificial. The keyfeature that underlies their common non-trivial properties is an asymmetric coupling of Majoranacomponents to the reservoir, which triggers the EP bifurcation.

Introduction—Since the remarkable prediction [1, 2]that a hybrid semiconductor-superconductor nanowirecan be tuned into a topological superconductor phasewith MZMs [3], there have been a number of papersreporting experimental data in the form of a zero-biasanomaly (ZBA) in the di↵erential conductance (dI/dV )for increasing Zeeman fields [4–10]. This behavior is con-sistent with tunneling into a MZM that emerges after thesystem undergoes a topological phase transition.

This Majorana interpretation has recently been chal-lenged since an alternative explanation in terms of ABSswith near-zero energy in the topological trivial phase,namely for Zeeman fields smaller than the critical fieldpredicted by band topology B < Bc, reproduces all theexpected phenomenology in transport. Following earlycalculations that proved that smooth confinement poten-tials inevitably lead to near-zero energy ABSs [11, 12], anumber of papers [13–18] have reported numerical ob-servations that systematically demonstrate that, indeed,ABSs in the trivial regime mimic Majoranas. This nag-ging ABS-versus-MZM question is compounded by therecent observation of 2e2/h conductance quantization[19], which can be also reproduced by ABSs [16], andthus is considered a serious objection in the field.

In this letter, we argue that the above question is illposed, and is the result of a viewpoint, that of bandtopology, only truly applicable to semi-infinite systems.We argue that a more general framework, relevant to theexperimental setup and rigorously well defined for finitesamples, allows us to precisely distinguish trivial fromnon-trivial zero modes. Among the latter are the MZMsfrom conventional band topology theory, but also a large

e Im

FIG. 1. Exceptional points. (a) The bifurcation, asa function of some external parameter B, of two complexGreen’s function poles across an exceptional point (EP) esta-bilises quasi-bound Majorana zero modes (red). Inset showsthe evolution of real and imaginary pole energies across theEP. (b) Sketch of the normal-superconductor (NS) junctionformed when a proximitized nanowire with inhomogeneouschemical potential and pairing, µ(x) and �(x), is coupled toa reservoir. Such junction is a natural host for Majorana zeromodes even below the critical field B < Bc that emerge fromEP bifurcations around parity crossings when their couplingis asymmetric �+

0 > ��0 due to spatial non-locality.

subset of ABSs zero modes. From this point of view bothkinds of states are really one and the same, which ex-plains why they cannot be distinguished. The key idea tounderstand this claim is to realize that, instead of conven-tional band topology, the natural language to describe anormal-superconductor (NS) junction, the geometry rel-evant to transport experiments, is that of open quantumsystems. In particular, we consider the non-Hermitiantopology defined in terms of the complex poles ✏p of theretarded Green’s function (or, equivalently, of the scat-

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c

UNIFIED NUMERICAL APPROACH TO TOPOLOGICAL … PHYSICAL REVIEW B 99, 245408 (2019)

−6 −4 −2 0

VG (V)

0.0

0.2

0.4

0.6

0.8

∆m

in/∆

(a)(a)

−6 −4 −2 0

VG (V)

(b)(b)

FIG. 4. Induced gap as a function of back-gate voltage for thedifferent devices and configurations. (a) Comparison of three-facetdevices with 7 nm Al shell for the different accumulation layerstrengths. (b) Same as (a) but for a two-facet device with a 10 nmAl shell.

semiconductor is often found to be of the same order as thesuperconductor gap [17].

The combined results for the minimal induced gap areshown in Fig. 4. All devices have large induced gap forappropriate gate voltages. We now discuss the four differentelectrostatic configurations presented in Fig. 2 and the effectsof disorder in detail.

A. Three-facet device

In Fig. 5 we show the energy spectrum and DOS in theInAs region for different back-gate voltages in the three-facetwire. For all back-gate voltages we find semiconductorlikestates that have a strong hybridization with the superconduc-tor. These states appear below the Al gap but have a stronghybridization with Al as indicated by the color in Fig. 5. TheDOS in the semiconductor is obtained by integrating the bandstructure over momentum, multiplying each eigenstate by itsweight in the semiconductor [76]. A temperature broadeningof the energy levels of 50 mK is assumed. The calculated

−0.25

0.00

0.25

E(m

eV)

VG = +1V

−0.25

0.00

0.25

E(m

eV)

VG = +0V

−0.25

0.00

0.25

E(m

eV)

VG = −1V

0.0 0.1 0.2 0.3

kz (nm−1)

−0.25

0.00

0.25

E(m

eV)

VG = −2V

0 10

DOS (a. u.)0.0 0.1 0.2 0.3

kz (nm−1)0 10

DOS (a. u.)

Al

InAs(a) (b)

FIG. 5. (a) Induced gap in the three-facet nanowire with 7 nmAl shell and ρacc = 2 × 1019 e/cm3, for different values of the back-gate voltage. The one-dimensional energy spectrum and integratedone-dimensional DOS in the InAs region is shown. The color scaleindicates the weight of the wave functions in the InAs and Al regions.For the dispersion and k integration the point spacing is dk = 2 ×10−4 nm−1. (b) Same as (a) but for ρacc = 5 × 1018 e/cm3.

DOS is consistent with experiment [18,49]: For negative gatevoltages we typically find a hard gap, with the position of thecoherence peaks showing little dependence on gate voltage.For positive gate voltages nonsuperconducting states enter thegap.

At positive back-gate voltages, we find accumulation ofelectrons near the back gate, on the opposite side of Al. Thesestates live almost completely in the InAs region and havenegligible hybridization with the superconductor and thusno, or very small, induced gap. They contribute to a subgapconductance for VG > 0.

Around VG ≈ 0 the electron density is distributed along thesurface of the semiconductor (see Fig. 2). In this regime allstates have nonzero hybridization with the superconductor anda hard gap opens up.

For sufficiently negative back-gate voltage VG < 0 the onlystates left are in close proximity to the superconductor. Theseare characterized by strong hybridization and induced gap onthe order of the superconductor gap. Note that a single stateat k ≈ 0.25 nm−1 has significantly smaller hybridization andinduced gap than the other states in Figs. 5(a) and 5(b) forVG ! −1 V. Furthermore, note that our Schrödinger solveronly includes the electrons, hence no hole states show up inthe DOS and band structures of Fig. 5. According to Fig. 2hole accumulation would be expected for VG ! −2 V forFig. 5(a) [VG ! −1 V for 5(b)].

B. Effect of disorder on the induced gap

Some previous attempts at simulating the superconductingproximity effect by treating the semiconductor and super-conductor on equal footing often found an induced gap thatis strongly dependent on geometric and microscopic detailsand significantly smaller than the one reported in experiments[27,32]. This is a consequence of the constraint imposedby momentum conservation at a smooth interface betweensemiconductor and superconductor. In such a case, tunnelingbetween the two subsystems is suppressed due to energy andmomentum constraints. Indeed, in this case the tunneling rate,which is relevant for the proximity effect [23], is effectivelyproportional to one-dimensional DOS and decreases withEF . Since EF is large in metals such as aluminum, at anygiven parallel momentum the phase space which satisfiesboth constraints is small. In other words, the level spacingcoming from one-dimensional subband quantization in thesuperconductor is several orders of magnitude larger than thesuperconducting gap for Al films with a thickness of 10 nm.This results in a strong and nonmonotonic dependence ofthe induced gap on the thickness of the superconductor [27].This dependence on the thickness of the superconductor isnot observed in experiment and is an artifact of a parallel-momentum-conserving approximation at the surface. In fact,experimentally the opposite effect is observed, that the gap isenhanced for thinner Al thicknesses [77,78].

Reference [26] demonstrated that disorder in the supercon-ductor enhances the induced gap dramatically and, providedit is sufficiently strong, removes the nonmonotonic depen-dence on the thickness of the superconducting layer. Sincea fully three-dimensional simulation of a semiconductor-superconductor heterostructure would be extremely challeng-

245408-5

UNIFIED NUMERICAL APPROACH TO TOPOLOGICAL … PHYSICAL REVIEW B 99, 245408 (2019)

−6 −4 −2 0

VG (V)

0.0

0.2

0.4

0.6

0.8

∆m

in/∆

(a)(a)

−6 −4 −2 0

VG (V)

(b)(b)

FIG. 4. Induced gap as a function of back-gate voltage for thedifferent devices and configurations. (a) Comparison of three-facetdevices with 7 nm Al shell for the different accumulation layerstrengths. (b) Same as (a) but for a two-facet device with a 10 nmAl shell.

semiconductor is often found to be of the same order as thesuperconductor gap [17].

The combined results for the minimal induced gap areshown in Fig. 4. All devices have large induced gap forappropriate gate voltages. We now discuss the four differentelectrostatic configurations presented in Fig. 2 and the effectsof disorder in detail.

A. Three-facet device

In Fig. 5 we show the energy spectrum and DOS in theInAs region for different back-gate voltages in the three-facetwire. For all back-gate voltages we find semiconductorlikestates that have a strong hybridization with the superconduc-tor. These states appear below the Al gap but have a stronghybridization with Al as indicated by the color in Fig. 5. TheDOS in the semiconductor is obtained by integrating the bandstructure over momentum, multiplying each eigenstate by itsweight in the semiconductor [76]. A temperature broadeningof the energy levels of 50 mK is assumed. The calculated

−0.25

0.00

0.25

E(m

eV)

VG = +1V

−0.25

0.00

0.25

E(m

eV)

VG = +0V

−0.25

0.00

0.25

E(m

eV)

VG = −1V

0.0 0.1 0.2 0.3

kz (nm−1)

−0.25

0.00

0.25

E(m

eV)

VG = −2V

0 10

DOS (a. u.)0.0 0.1 0.2 0.3

kz (nm−1)0 10

DOS (a. u.)

Al

InAs(a) (b)

FIG. 5. (a) Induced gap in the three-facet nanowire with 7 nmAl shell and ρacc = 2 × 1019 e/cm3, for different values of the back-gate voltage. The one-dimensional energy spectrum and integratedone-dimensional DOS in the InAs region is shown. The color scaleindicates the weight of the wave functions in the InAs and Al regions.For the dispersion and k integration the point spacing is dk = 2 ×10−4 nm−1. (b) Same as (a) but for ρacc = 5 × 1018 e/cm3.

DOS is consistent with experiment [18,49]: For negative gatevoltages we typically find a hard gap, with the position of thecoherence peaks showing little dependence on gate voltage.For positive gate voltages nonsuperconducting states enter thegap.

At positive back-gate voltages, we find accumulation ofelectrons near the back gate, on the opposite side of Al. Thesestates live almost completely in the InAs region and havenegligible hybridization with the superconductor and thusno, or very small, induced gap. They contribute to a subgapconductance for VG > 0.

Around VG ≈ 0 the electron density is distributed along thesurface of the semiconductor (see Fig. 2). In this regime allstates have nonzero hybridization with the superconductor anda hard gap opens up.

For sufficiently negative back-gate voltage VG < 0 the onlystates left are in close proximity to the superconductor. Theseare characterized by strong hybridization and induced gap onthe order of the superconductor gap. Note that a single stateat k ≈ 0.25 nm−1 has significantly smaller hybridization andinduced gap than the other states in Figs. 5(a) and 5(b) forVG ! −1 V. Furthermore, note that our Schrödinger solveronly includes the electrons, hence no hole states show up inthe DOS and band structures of Fig. 5. According to Fig. 2hole accumulation would be expected for VG ! −2 V forFig. 5(a) [VG ! −1 V for 5(b)].

B. Effect of disorder on the induced gap

Some previous attempts at simulating the superconductingproximity effect by treating the semiconductor and super-conductor on equal footing often found an induced gap thatis strongly dependent on geometric and microscopic detailsand significantly smaller than the one reported in experiments[27,32]. This is a consequence of the constraint imposedby momentum conservation at a smooth interface betweensemiconductor and superconductor. In such a case, tunnelingbetween the two subsystems is suppressed due to energy andmomentum constraints. Indeed, in this case the tunneling rate,which is relevant for the proximity effect [23], is effectivelyproportional to one-dimensional DOS and decreases withEF . Since EF is large in metals such as aluminum, at anygiven parallel momentum the phase space which satisfiesboth constraints is small. In other words, the level spacingcoming from one-dimensional subband quantization in thesuperconductor is several orders of magnitude larger than thesuperconducting gap for Al films with a thickness of 10 nm.This results in a strong and nonmonotonic dependence ofthe induced gap on the thickness of the superconductor [27].This dependence on the thickness of the superconductor isnot observed in experiment and is an artifact of a parallel-momentum-conserving approximation at the surface. In fact,experimentally the opposite effect is observed, that the gap isenhanced for thinner Al thicknesses [77,78].

Reference [26] demonstrated that disorder in the supercon-ductor enhances the induced gap dramatically and, providedit is sufficiently strong, removes the nonmonotonic depen-dence on the thickness of the superconducting layer. Sincea fully three-dimensional simulation of a semiconductor-superconductor heterostructure would be extremely challeng-

245408-5

UNIFIED NUMERICAL APPROACH TO TOPOLOGICAL … PHYSICAL REVIEW B 99, 245408 (2019)

−6 −4 −2 0

VG (V)

0.0

0.2

0.4

0.6

0.8

∆ min/∆

(a)(a)

−6 −4 −2 0

VG (V)

(b)(b)

FIG. 4. Induced gap as a function of back-gate voltage for thedifferent devices and configurations. (a) Comparison of three-facetdevices with 7 nm Al shell for the different accumulation layerstrengths. (b) Same as (a) but for a two-facet device with a 10 nmAl shell.

semiconductor is often found to be of the same order as thesuperconductor gap [17].

The combined results for the minimal induced gap areshown in Fig. 4. All devices have large induced gap forappropriate gate voltages. We now discuss the four differentelectrostatic configurations presented in Fig. 2 and the effectsof disorder in detail.

A. Three-facet device

In Fig. 5 we show the energy spectrum and DOS in theInAs region for different back-gate voltages in the three-facetwire. For all back-gate voltages we find semiconductorlikestates that have a strong hybridization with the superconduc-tor. These states appear below the Al gap but have a stronghybridization with Al as indicated by the color in Fig. 5. TheDOS in the semiconductor is obtained by integrating the bandstructure over momentum, multiplying each eigenstate by itsweight in the semiconductor [76]. A temperature broadeningof the energy levels of 50 mK is assumed. The calculated

−0.25

0.00

0.25

E(m

eV)

VG = +1V

−0.25

0.00

0.25

E(m

eV)

VG = +0V

−0.25

0.00

0.25

E(m

eV)

VG = −1V

0.0 0.1 0.2 0.3

kz (nm−1)

−0.25

0.00

0.25

E(m

eV)

VG = −2V

0 10

DOS (a. u.)0.0 0.1 0.2 0.3

kz (nm−1)0 10

DOS (a. u.)

Al

InAs(a) (b)

FIG. 5. (a) Induced gap in the three-facet nanowire with 7 nmAl shell and ρacc = 2 × 1019 e/cm3, for different values of the back-gate voltage. The one-dimensional energy spectrum and integratedone-dimensional DOS in the InAs region is shown. The color scaleindicates the weight of the wave functions in the InAs and Al regions.For the dispersion and k integration the point spacing is dk = 2 ×10−4 nm−1. (b) Same as (a) but for ρacc = 5 × 1018 e/cm3.

DOS is consistent with experiment [18,49]: For negative gatevoltages we typically find a hard gap, with the position of thecoherence peaks showing little dependence on gate voltage.For positive gate voltages nonsuperconducting states enter thegap.

At positive back-gate voltages, we find accumulation ofelectrons near the back gate, on the opposite side of Al. Thesestates live almost completely in the InAs region and havenegligible hybridization with the superconductor and thusno, or very small, induced gap. They contribute to a subgapconductance for VG > 0.

Around VG ≈ 0 the electron density is distributed along thesurface of the semiconductor (see Fig. 2). In this regime allstates have nonzero hybridization with the superconductor anda hard gap opens up.

For sufficiently negative back-gate voltage VG < 0 the onlystates left are in close proximity to the superconductor. Theseare characterized by strong hybridization and induced gap onthe order of the superconductor gap. Note that a single stateat k ≈ 0.25 nm−1 has significantly smaller hybridization andinduced gap than the other states in Figs. 5(a) and 5(b) forVG ! −1 V. Furthermore, note that our Schrödinger solveronly includes the electrons, hence no hole states show up inthe DOS and band structures of Fig. 5. According to Fig. 2hole accumulation would be expected for VG ! −2 V forFig. 5(a) [VG ! −1 V for 5(b)].

B. Effect of disorder on the induced gap

Some previous attempts at simulating the superconductingproximity effect by treating the semiconductor and super-conductor on equal footing often found an induced gap thatis strongly dependent on geometric and microscopic detailsand significantly smaller than the one reported in experiments[27,32]. This is a consequence of the constraint imposedby momentum conservation at a smooth interface betweensemiconductor and superconductor. In such a case, tunnelingbetween the two subsystems is suppressed due to energy andmomentum constraints. Indeed, in this case the tunneling rate,which is relevant for the proximity effect [23], is effectivelyproportional to one-dimensional DOS and decreases withEF . Since EF is large in metals such as aluminum, at anygiven parallel momentum the phase space which satisfiesboth constraints is small. In other words, the level spacingcoming from one-dimensional subband quantization in thesuperconductor is several orders of magnitude larger than thesuperconducting gap for Al films with a thickness of 10 nm.This results in a strong and nonmonotonic dependence ofthe induced gap on the thickness of the superconductor [27].This dependence on the thickness of the superconductor isnot observed in experiment and is an artifact of a parallel-momentum-conserving approximation at the surface. In fact,experimentally the opposite effect is observed, that the gap isenhanced for thinner Al thicknesses [77,78].

Reference [26] demonstrated that disorder in the supercon-ductor enhances the induced gap dramatically and, providedit is sufficiently strong, removes the nonmonotonic depen-dence on the thickness of the superconducting layer. Sincea fully three-dimensional simulation of a semiconductor-superconductor heterostructure would be extremely challeng-

245408-5

The zero-bias peak,measured experimentally in [8–12], is a non-specific signature ofMajoranas, sincesimilar features arise due toKondo physics or weak anti-localization [13, 14]. To help distinguishingMajoranasignatures from these alternatives, we focus on the parametric dependence of twoMajorana properties: theshape of the topological phase boundary [15, 16] and the oscillations in the coupling energy of twoMajoranamodes [17–21].

Both phenomena depend on the response of the chemical potential to amagnetic field, and hence onelectrostatic effects.Majorana oscillations were analyzed theoretically in two extreme limits for the electrostaticeffects: constant chemical potential [19–21] and constant density [20] (see appendix A for a summary of thesetwo limits). In particular [20], found different behavior ofMajorana oscillations in these two extreme limits.Weshow that the actual behavior of the nanowire is somewhere in between, and depends strongly on theelectrostatics.

2. Setup andmethods

2.1. The Schrödinger–Poisson problemWediscuss electrostatic effects in a device design as used byMourik et al [8], however ourmethods arestraightforward to adapt to similar layouts (see appendix B for a calculation using a different geometry). Sinceweare interested in the bulk properties, we require that the potential and theHamiltonian terms are translationallyinvariant along thewire axis andwe consider a 2D cross section, shown infigure 1. The device consists of ananowirewith a hexagonal cross section of diameter W 100 nm= on a dielectric layer with thicknessd 30 nmdielectric = . A superconductor with thickness d 187 nmSC = covers half of thewire. The nanowire has adielectric constant 17.7r� = (InSb), the dielectric layer has a dielectric constant 8r� = (Si3N4). The device hastwo electrostatic boundary conditions: afixed gate potentialVG set by the gate electrode along the lower edge ofthe dielectric layer and afixed potential VSC in the superconductor, whichwemodel as a groundedmetallic gate.We set this potential to eitherV 0 VSC = , disregarding awork function difference between theNbTiNsuperconductor and the nanowire, or we assume a small work function difference [22, 23] resultinginV 0.2 VSC = .

Wemodel the electrostatics of this setup using the Schrödinger–Poisson equation.We split theHamiltonianinto transverse and longitudinal parts. The transverseHamiltonian T reads

m x ye x y

E

2,

2, 1T

2 2

2

2

2

gap⎛⎝⎜

⎞⎠⎟*

� f= -

¶¶

+¶¶

- +( ) ( )

with x y, the transverse directions, m m0.014 e* = the effective electronmass in InSb (withme the electronmass), e- the electron charge, andf the electrostatic potential.We assume that in the absence of electric fieldthe Fermi level EF in the nanowire is in themiddle of the semiconducting gap Egap, with E 0.2 eVgap = for InSb(see figure 2(a).We choose the Fermi level EF as the reference energy such that E 0F º .

The longitudinalHamiltonian L reads

m z zE

2i , 2y zL

2 2

2 Z*

� a s s= -

¶¶

-¶¶

+ ( )

with z the direction along thewire axis,α the spin–orbit coupling strength, EZ the Zeeman energy and s thePaulimatrices. The orientation of themagnetic field is along thewire in the zdirection. In this separation, wehave assumed that the spin–orbit length l mSO

2 *� a= ( ) is larger or comparable to thewire diameter,

Figure 1. Schematic cross section of theMajorana device. It consists of a nanowire (red hexagon) lying on a dielectric layer (bluerectangle)which covers a global back gate. A superconducting lead (yellow region) covers half of the nanowire.

2

New J. Phys. 18 (2016) 033013 AVuik et al

The zero-bias peak,measured experimentally in [8–12], is a non-specific signature ofMajoranas, sincesimilar features arise due toKondo physics or weak anti-localization [13, 14]. To help distinguishingMajoranasignatures from these alternatives, we focus on the parametric dependence of twoMajorana properties: theshape of the topological phase boundary [15, 16] and the oscillations in the coupling energy of twoMajoranamodes [17–21].

Both phenomena depend on the response of the chemical potential to amagnetic field, and hence onelectrostatic effects.Majorana oscillations were analyzed theoretically in two extreme limits for the electrostaticeffects: constant chemical potential [19–21] and constant density [20] (see appendix A for a summary of thesetwo limits). In particular [20], found different behavior ofMajorana oscillations in these two extreme limits.Weshow that the actual behavior of the nanowire is somewhere in between, and depends strongly on theelectrostatics.

2. Setup andmethods

2.1. The Schrödinger–Poisson problemWediscuss electrostatic effects in a device design as used byMourik et al [8], however ourmethods arestraightforward to adapt to similar layouts (see appendix B for a calculation using a different geometry). Sinceweare interested in the bulk properties, we require that the potential and theHamiltonian terms are translationallyinvariant along thewire axis andwe consider a 2D cross section, shown infigure 1. The device consists of ananowirewith a hexagonal cross section of diameter W 100 nm= on a dielectric layer with thicknessd 30 nmdielectric = . A superconductor with thickness d 187 nmSC = covers half of thewire. The nanowire has adielectric constant 17.7r� = (InSb), the dielectric layer has a dielectric constant 8r� = (Si3N4). The device hastwo electrostatic boundary conditions: afixed gate potentialVG set by the gate electrode along the lower edge ofthe dielectric layer and afixed potential VSC in the superconductor, whichwemodel as a groundedmetallic gate.We set this potential to eitherV 0 VSC = , disregarding awork function difference between theNbTiNsuperconductor and the nanowire, or we assume a small work function difference [22, 23] resultinginV 0.2 VSC = .

Wemodel the electrostatics of this setup using the Schrödinger–Poisson equation.We split theHamiltonianinto transverse and longitudinal parts. The transverseHamiltonian T reads

m x ye x y

E

2,

2, 1T

2 2

2

2

2

gap⎛⎝⎜

⎞⎠⎟*

� f= -

¶¶

+¶¶

- +( ) ( )

with x y, the transverse directions, m m0.014 e* = the effective electronmass in InSb (withme the electronmass), e- the electron charge, andf the electrostatic potential.We assume that in the absence of electric fieldthe Fermi level EF in the nanowire is in themiddle of the semiconducting gap Egap, with E 0.2 eVgap = for InSb(see figure 2(a).We choose the Fermi level EF as the reference energy such that E 0F º .

The longitudinalHamiltonian L reads

m z zE

2i , 2y zL

2 2

2 Z*

� a s s= -

¶¶

-¶¶

+ ( )

with z the direction along thewire axis,α the spin–orbit coupling strength, EZ the Zeeman energy and s thePaulimatrices. The orientation of themagnetic field is along thewire in the zdirection. In this separation, wehave assumed that the spin–orbit length l mSO

2 *� a= ( ) is larger or comparable to thewire diameter,

Figure 1. Schematic cross section of theMajorana device. It consists of a nanowire (red hexagon) lying on a dielectric layer (bluerectangle)which covers a global back gate. A superconducting lead (yellow region) covers half of the nanowire.

2

New J. Phys. 18 (2016) 033013 AVuik et al

The zero-bias peak,measured experimentally in [8–12], is a non-specific signature ofMajoranas, sincesimilar features arise due toKondo physics or weak anti-localization [13, 14]. To help distinguishingMajoranasignatures from these alternatives, we focus on the parametric dependence of twoMajorana properties: theshape of the topological phase boundary [15, 16] and the oscillations in the coupling energy of twoMajoranamodes [17–21].

Both phenomena depend on the response of the chemical potential to amagnetic field, and hence onelectrostatic effects.Majorana oscillations were analyzed theoretically in two extreme limits for the electrostaticeffects: constant chemical potential [19–21] and constant density [20] (see appendix A for a summary of thesetwo limits). In particular [20], found different behavior ofMajorana oscillations in these two extreme limits.Weshow that the actual behavior of the nanowire is somewhere in between, and depends strongly on theelectrostatics.

2. Setup andmethods

2.1. The Schrödinger–Poisson problemWediscuss electrostatic effects in a device design as used byMourik et al [8], however ourmethods arestraightforward to adapt to similar layouts (see appendix B for a calculation using a different geometry). Sinceweare interested in the bulk properties, we require that the potential and theHamiltonian terms are translationallyinvariant along thewire axis andwe consider a 2D cross section, shown infigure 1. The device consists of ananowirewith a hexagonal cross section of diameter W 100 nm= on a dielectric layer with thicknessd 30 nmdielectric = . A superconductor with thickness d 187 nmSC = covers half of thewire. The nanowire has adielectric constant 17.7r� = (InSb), the dielectric layer has a dielectric constant 8r� = (Si3N4). The device hastwo electrostatic boundary conditions: afixed gate potentialVG set by the gate electrode along the lower edge ofthe dielectric layer and afixed potential VSC in the superconductor, whichwemodel as a groundedmetallic gate.We set this potential to eitherV 0 VSC = , disregarding awork function difference between theNbTiNsuperconductor and the nanowire, or we assume a small work function difference [22, 23] resultinginV 0.2 VSC = .

Wemodel the electrostatics of this setup using the Schrödinger–Poisson equation.We split theHamiltonianinto transverse and longitudinal parts. The transverseHamiltonian T reads

m x ye x y

E

2,

2, 1T

2 2

2

2

2

gap⎛⎝⎜

⎞⎠⎟*

� f= -

¶¶

+¶¶

- +( ) ( )

with x y, the transverse directions, m m0.014 e* = the effective electronmass in InSb (withme the electronmass), e- the electron charge, andf the electrostatic potential.We assume that in the absence of electric fieldthe Fermi level EF in the nanowire is in themiddle of the semiconducting gap Egap, with E 0.2 eVgap = for InSb(see figure 2(a).We choose the Fermi level EF as the reference energy such that E 0F º .

The longitudinalHamiltonian L reads

m z zE

2i , 2y zL

2 2

2 Z*

� a s s= -

¶¶

-¶¶

+ ( )

with z the direction along thewire axis,α the spin–orbit coupling strength, EZ the Zeeman energy and s thePaulimatrices. The orientation of themagnetic field is along thewire in the zdirection. In this separation, wehave assumed that the spin–orbit length l mSO

2 *� a= ( ) is larger or comparable to thewire diameter,

Figure 1. Schematic cross section of theMajorana device. It consists of a nanowire (red hexagon) lying on a dielectric layer (bluerectangle)which covers a global back gate. A superconducting lead (yellow region) covers half of the nanowire.

2

New J. Phys. 18 (2016) 033013 AVuik et al

FIG. 6. Illustrative examples of theoretical results beyond the minimal model. (a) Schrodinger-Poisson computationsin a realistic nanowire set-up including the electrostatic environment yield a non-homogeneous charge density distribution andelectric field in the wire’s cross section [66]. (b) Topological phase diagram of a nanowire covered by an epitaxial SC shellon three of its facets [72]. The presence of several filled subbands and the orbital effects due to the magnetic flux along thenanowire dramatically alter the phase diagram [compare to Fig. 4 (c)]. The topological minigap is overlaid in selected regions(it has not been calculated in gray areas). (c) Effect of the backgate voltage on the induced gap. For more negative voltage(bottom panel), the wave function is pushed closer to the SC, resulting in an increased coupling and a hard gap. In contrast, amore positive voltage (bottom panel) results in a decreased coupling and a softer gap [72]. (d) A normal region in a proximitizednanowire, bounded by a smooth (screened) confinement potential and pairing can explain the emergence of non-topologicalzero modes in the trivial VZ < V c

Z phase, a reduced visibility of the band inversion at the critical VZ = V cZ and the appearance

of additional finite-energy subgap states, all apparent in dI/dV simulations [73]. (e) Low energy spectrum vs Zeeman splittingVZ at an NS smooth junction in a finite length nanowire. At VZ = 0 there exist several ABSs below the induced gap (0.3 meVin this simulation) located in the normal region. As VZ increases, the lowest energy mode (in blue) approaches zero energyand remains pinned to zero for an extended VZ range before entering the bulk topological phase at V c

Z . After this point atopological minigap opens and MBSs oscillations ensue. The wave functions (in the Majorana basis) along the nanowire ofthe lowest energy mode are shown to the right, at three values of VZ (numbered circles). In 1 and 2 the entire wire is in thetrivial phase, while in 3 the right part has become topological. The wavefunctions of the non-topological robust zero modesin 2 exhibit partial spatial overlap and different (fast and slow) wavevector components (red and blue, respectively) [80]. Notethat the blue wavefunction in 2 is centered at the smooth NS junction, while it shifts to the right end after the topologicaltransition. (f) Opening a (trivial or topological) nanowire to a fermionic reservoir may induce a decoupling of the two Majoranacomponents of a low-energy fermionic state (blue and purple), which then acquire complex energies with distinct imaginaryparts Γ±, and real energies exactly pinned to zero. This happens at a so-called exceptional point (EP), which constitutes anon-Hermitian topological transition [87]. The evolution of Γ± and the real energies with magnetic field B across the EP areshown in the inset (solid and dashed lines). The bifurcated Γ± represent different decay rates of the two Majorana componentsinto the reservoir. The decoupling requires partial separation of the two Majorana wavefunctions in the isolated wire. Afterdecoupling, the two Majorana components become zero energy quasibound states dubbed EP-MBSs. One of the two EP-MBSs(in blue) may develop a vanishing decay rate Γ− → 0, thus becoming a non-decaying and robust zero energy eigenstate [92].

ated by nonadiabatic manipulation [99, 102], which arelargely controlled by the superconducting minigap.

Given the likely ubiquity of non-topological zero modesin realistic devices [172], particularly when including QDsand screened barriers as basic elements of many proposedschemes for topological quantum computing [126, 250–

252], it has become important to understand whether theprotection of topological MBSs applies in some form alsoto non-topological zero modes. A precise answer requiresquantifying the Majorana overlap of a given zero modein a sample. Traditional experimental schemes to mea-sure the subgap spectrum of nanowires, such as tunneling

16

spectroscopy, rely on local probes, so that they do notdirectly access the degree of non-locality of a given zerobias anomaly. An alternative, though still local schemehas been proposed to extract a quantitative estimate ofthe degree of MBS overlap [80, 90, 253, 254]. It con-sists of measuring tunneling spectroscopy into the endof the nanowire through a QD in series, which revealsthe asymmetric coupling of spin-polarized states in theQD with the two spatially separated Majorana compo-nents of the zero mode. Such a scheme was implementedin a recent experiment [91] that demonstrated a vary-ing degree of wavefunction overlap in otherwise similarzero modes. Multiple tunnel probes have also been em-ployed recently in search of clearer and less ambiguousevidence of zero-mode non-locality and non-trivial topol-ogy [153, 190, 255, 256].

V. SUMMARY AND OUTLOOK

We have reviewed the remarkable advances towardscharacterizing the detailed structure of ABSs in hybridnanowires with strong SO coupling, and related systems.In ideal nanowires, ABSs evolve with magnetic fields andgates, developing a spatial separation of their wavefunc-tion components and ultimately transforming into ro-bust, non-local, topological Majorana zero energy modesbeyond a critical Zeeman field. Experimental obser-vations, however, deviate from the predictions of min-imal models, making evident the need to incorporateextensions, such as the electrostatic environment, mul-timode and orbital effects, spatial inhomogeneities andmomentum mixing. These additions introduce a com-plex, non-universal phenomenology. Theoretical workhas identified alternative routes towards stabilizing zeromodes, different from a bulk topological phase transi-tion. Such routes include using smooth and/or spin-dependent confinement, or exceptional point bifurcationsin open (non-Hermitian) systems. However, the wave-function non-locality obtained by such means is gener-ally poor. Given the importance of non-locality for theprotection of MBSs, their resilience against noise and thepossibility of carrying out braiding operations on them,it has become a major focus point in current experimentsto detect and quantify the degree of Majorana overlap.First results in this area have been obtained using purelylocal probes. These have intrinsic limitations, unfortu-nately, and can only suggest, not demonstrate, Majorananon-locality. Experiments are underway, nevertheless, toexploit truly non-local measurements in more complexnanowire setups without such limitations [257–259]. Theultimate demonstration, non-Abelian braiding, remainsan open challenge.

Nonlocality and braiding are the cornerstones behindthe original promise of Majorana applications, namely, toharness the hardware-level resilience of Majorana qubitsto solve the scalability problem of quantum computers.This is an ambitious long-term endeavour that will re-

quire solving important challenges to finally bring Ma-joranas out of the lab. Majorana nanowires will likelyhave an important role in this journey, as they rep-resent the most readily accessible system of this kindand the most explored topological superconductor sofar. However, they also have some disadvantages suchas the unwanted presence of nontopological in-gap statesor the necessity to subject the wire to strong magneticfields. So far, partially-covered nanowires have been in-tensively explored, but other alternatives with some in-teresting advantages are beginning to be studied, suchas full-shell nanowires [197, 260] and ferromagnetic hy-brid structures [261]. Additionally, the recent demon-stration of gate-tunable nanowire-based superconductingqubits, so-called gatemons [262, 263], which include full-shell nanowires [264] and junctions with quantum dots[265, 266], opens new possibilities for studying Majoranaphysics in hybrid architectures that use more maturetechnologies such as circuit QED and transmon qubits[267–271].

Nevertheless, it is entirely possible that radically dif-ferent Majorana platforms will be discovered that exhibitcleaner and larger gaps, improved Majorana non-locality,easier braiding, and crucially, are more amenable to scal-able fabrication and integration. Leading explorationsinclude experiments on shallow quantum wells that canbe proximitized by epitaxial superconductors, while be-ing compatible with lithographic patterning techniques[117]. On such platforms crossed-Andreev reflection hasbeen proposed as an alternative way to stabilize Majo-ranas [272, 273]. Also, the use of two-dimensional van derWaals crystals has received attention recently [274, 275],where the SO coupling, required for generating MBSs,could be replaced by carrier chirality and intrinsic in-teractions [276]. Solving the material-science side of thechallenge is crucial before tackling the more applied prob-lems of engineering, operating and integrating coherentensembles of topological quantum gates. We expect theseefforts to continue fertilizing condensed-matter researchwith novel and exciting possibilities.

VI. AUTHOR CONTRIBUTIONS

L. P. K. initiated this review and E. P. coordinatedthe project. All authors discussed the general structureof the manuscript. M. W. A. M. and A. G. wrote “ABSspectroscopy”, “MBS spectroscopy” and contributed to“Extensions of the minimal model”. E. J. H. L., J. N. andR. A. wrote “ABSs in QDs”. J. K. and D. L. contributedto “Zero-energy pinning with a topologically trivial bulk”and ‘Protection against errors and MBS overlaps”. E.P., P. S.-J. and R. A. wrote everything else. All authorsreviewed and polished the manuscript.

17

Non-trivial

Trivial

a

b

FIG. 7. Band structure and subgap Andreev scattering pro-cesses of a semiconducting nanowire in contact with a SC.The nanowire SO coupling is responsible for the 2kSO andESO shifts, and a longitudinal Zeeman field is responsible forthe 2VZ splitting at k = 0. Depending on whether VZ < V c

Z

(a) or VZ > V cZ (b), the proximitized nanowire is a topolog-

ically trivial (s-wave) or non-trivial SC (effectively spinlessp-wave).

ACKNOWLEDGMENTS

Research supported by the Spanish Ministry ofScience, Innovation and Universities through grantsFIS2015-65706-P, FIS2015-64654-P, FIS2016-80434-P,FIS2017-84860-R, PCI2018-093026 and PGC2018-097018-B-I00 (AEI/FEDER, EU), the Ramon y Cajalprogramme grant RYC-2011-09345 and RYC-2015-17973, the Marıa de Maeztu Programme for Unitsof Excellence in R&D (MDM-2014-0377), the Euro-pean Union’s Horizon 2020 research and innovationprogramme under grant agreements Nos 828948 (FE-TOPEN AndQC), 127900 (Quantera SuperTOP), theEuropean Research Council (ERC) Starting Grantagreements 716559 (TOPOQDot), 757725 (ETOPEX)and 804988 (SiMS), the Netherlands Organizationfor Scientific Research (NWO), Microsoft, the DanishNational Research Foundation, the Carlsberg Founda-tion, and the Swiss National Science Foundation andNCCR QSIT. We also acknowledge support from CSICResearch Platform on Quantum Technologies PTI-001.

Appendix A: Box A – Proximitized nanowire model

The starting point of the proximitized nanowire model(R. M. Lutchyn et al [41] and Y. Oreg et al [42]) is aHamiltonian describing a 1D semiconducting nanowire

with Rashba spin-orbit (SO) interaction and in the pres-ence of an external magnetic field B perpendicular to theRashba field (here, we assume that B is applied parallelto the nanowire axis x):

Hw =1

2

∫dxΨ†(x)H(x)Ψ(x),

with

Hw(x) =

(−~2∂2

x

2m∗− µ− iα∂xσy

)τz + VZσx, (A1)

where m∗ is the effective mass of the semiconductor, µ itschemical potential, α the SO coupling and VZ = 1

2gµBBthe Zeeman energy produced by B, given in terms of thenanowire’s g-factor (with µB being the Bohr’s magne-

ton). Ψ(x) = (ψ†↑, ψ†↓, ψ↓,−ψ↑) are Nambu spinors and

σ and τ Pauli matrices in spin and particle-hole space,respectively.

Solving the above Hamiltonian in reciprocal space

yields a dispersion relation of the form Ek,± = ~2k2

2m∗ −µ±√V 2Z + α2k2. In the absence of Zeeman field, VZ = 0,

the Rashba term removes the spin degeneracy of the 1Dparabolic band and gives rise to two parabolas shiftedrelative to each other along the momentum axis (eachby an amount kSO = m∗α/~2) and displaced down inenergy by an amount ESO = m∗α2/2~2, where ~ the re-duced Planck’s constant, see Fig. 7 (a). These parabolascorrespond to spin up and spin down projections alongthe spin-quantization axis fixed by the Rashba coupling(here σy). On the other hand, a finite Zeeman VZ 6= 0mixes both spins and hence removes the spin degener-acy at k = 0 by opening up a gap of size 2VZ , Fig. 7(b). These split bands define two sectors + and - of op-posite helicity at large and small momenta, respectively.If |µ| < |VZ |, only the + sector is present around theFermi energy (helical regime). Projecting a standard

s-wave pairing term Hs =∑k ∆ψ†↑,kψ

†↓,−k + H.c. onto

the helical basis one obtains intraband (spinless) pairing

terms of the form Hp =∑k

∑i=±∆i

kψ†i,kψ

†i,−k + H.c.,

with ∆±k = ∓iαk∆

2√V 2Z+α2k2

having so-called p-wave symme-

try ∆k = −∆−k. In the helical regime, the minimalHamiltonian H = Hw +Hs is thus a realistic implemen-tation of the Kitaev model for 1D p-wave superconduc-tivity [32]. When the applied Zeeman field is larger than

the critical value V cZ =√

∆2 + µ2, the 1D SC becomestopological and hosts MBSs at its ends.

To implement this proposal in experimentally realiz-able systems, one needs semiconductors with large g-factors in order to achieve a large VZ under moderateexternal magnetic fields B below the critical field of theSC. A good proximity effect with conventional SCs anda large Rashba energy are also necessary. Last but notleast, one needs to be able to keep the chemical potentialµ of the nanowire close to zero (in order to reach the he-lical regime with spin-momentum locking for moderateB), despite the proximity to the SC.

18

|��

�+|��

� Trivial� Non-trivial�

��� ��� ��� ��� ��� ���

�/�

|�����

�

�

��� ��� ��� ��� ��� ���

�/�

�

FIG. 8. Wavefunction of the lowest energy state ψ0 in a uni-form L = 1µm Majorana nanowire for a trivial VZ = 0.5V c

Z (a)and a non-trivial VZ = 1.4V c

Z (b) cases. (c,d) Wavefunctionsof the corresponding Majorana components uM

1,2(x) in the Ma-jorana basis. In the trivial regime the left (red) and right(blue) Majorana components strongly overlap (c), whereas inthe topological regime they move apart and concentrate atthe ends of the nanowire (d).

Appendix B: Box B – Majorana basis

A Bogoliubov-de Gennes eigenstate in a superconduct-ing system is an excitation |ψn〉 = ψ†n|BCS〉 of energy εnover its ground state |BCS〉 that consists of a superposi-tion of one electron and one hole quasiparticles,

ψn =

∫dx∑σ

[unσ(x)ψσ(x) + vnσ(x)ψ†σ(x)

].

Here ψ†σ(x) and ψσ(x) create and destroy a quasiparticleof spin σ perfectly localized at point x, respectively, andunσ(x), vnσ(x) are electron/hole wavefunctions.

If the energy of a given eigenstate |ψ0〉 = ψ†0|BCS〉becomes negligibly small ε0 ≈ 0 as in the case of atopological Majorana nanowire, the eigenstates |BCS〉and |ψ0〉 are both degenerate ground states, of evenand odd fermionic parity, respectively. If we denote|ψeven〉 ≡ |BCS〉 and |ψodd〉 ≡ |ψ0〉, we find that ψ0 and

ψ†0 switch between the two(|ψeven〉|ψodd〉

)=

(0 ψ0

ψ†0 0

)(|ψeven〉|ψodd〉

).

The matrix elements of ψ0, ψ†0 in this subspace are

therefore 〈ψeven,odd|ψ0|ψeven,odd〉 = (σ1 + iσ2)/2 and

〈ψeven,odd|ψ†0|ψeven,odd〉 = (σ1−iσ2)/2, where σi are Paulimatrices.

By performing a unitary rotation to the so-called Ma-

jorana basis, the eigenstate operators ψ0, ψ†0 can be de-

composed into two Majorana operators that satisfy self-

conjugation, γ1 = γ†1 and γ2 = γ†2, so that

ψ0 = (γ1 + iγ2)/2, ψ†0 = (γ1 − iγ2)/2;

γ1 = ψ†0 + ψ0, γ2 = i(ψ†0 − ψ0).(B1)

Each Majorana operator corresponds to a fermionic

eigenstate, in the sense that {γ†i , γj} = 2δij , but the Ma-jorana reality property also implies that, unlike a conven-tional fermion, γ2

i = 1. The matrix elements of γi in theground state subspace are 〈ψeven,odd|γi|ψeven,odd〉 = σi.

The Majorana states created by γ1 and γ2 are some-times intuitively described as half-fermions, as they al-ways come in pairs and any two in a system can be com-bined to create a conventional fermion as above. In atopological Majorana nanowire, the wavefunction uMi,σ(x)of

γi =

∫dx∑σ

[uMiσ (x)ψσ(x) + uMiσ

∗(x)ψ†σ(x)

]is localized at either end of the nanowire, unlikeu0,σ(x), v0,σ(x) of ψ0, that occupies both ends, see Fig.8. The latter is hence called a non-local fermion.

The above transformations, Eqs. (B1), can be appliedto an arbitrary ABS ψn of finite energy. In such case, theresulting Majorana states are not eigenstates. However,the decomposition still allows to examine the degree ofMajorana non-locality of ψn by computing the overlapbetween the corresponding uM1,σ(x) and uM2,σ(x).

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