Majorana Fermions in Semiconductor Nanowires: Fundamentals, Modeling, and Experiment

Tudor D. Stanescu1 and Sumanta Tewari21Department of Physics, West Virginia University, Morgantown, WV 26506.

2Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

After a recent series of rapid and exciting developments, the long search for the Majorana fermion - theelusive quantum entity at the border between particles and antiparticles - has produced the first positive ex-perimental results, but is not over yet. Originally proposed by E. Majorana in the context of particle physics,Majorana fermions have a condensed matter analog in the zero–energy bound states emerging in topologicalsuperconductors. A promising route to engineering topological superconductors capable of hosting Majoranazero modes consists of proximity coupling semiconductor thin films or nanowires with strong spin–orbit inter-action to conventional s–wave superconductors in the presence of an external Zeeman field. The Majorana zeromode is predicted to emerge above a certain critical Zeeman field as a zero–energy state localized near the orderparameter defects, viz., vortices for thin films and wire-ends for the nanowire. These Majorana bound states areexpected to manifest non–Abelian quantum statistics, which makes them ideal building blocks for fault–toleranttopological quantum computation. This review provides an update on current status of the search for Majoranafermions in semiconductor nanowires by focusing on the recent developments, in particular the period followingthe first reports of experimental signatures consistent with the realization of Majorana bound states in semicon-ductor nanowire–superconductor hybrid structures. We start with a discussion of the fundamental aspects ofthe subject, followed by considerations on the realistic modeling which is a critical bridge between theoreticalpredictions based on idealized conditions and the real world, as probed experimentally. The last part is dedicatedto a few intriguing issues that were brought to the fore by the recent encouraging experimental advances.

CONTENTS

I. Introduction 1

II. Theoretical background 3A. Spinless (px + ipy) superconductor/superfluid 3B. Semiconductor–superconductor heterostructure 4C. Topological class 5D. Topological invariant 5E. Minigap 6F. Phase diagram of semiconductor Majorana wire,

sweet spots, and approximate chirality symmetry 6G. Probing Majorana fermions I: zero bias

conductance peak and fractional AC Josephsoneffect 8

H. Probing Majorana fermions II: quantumnon-locality, transconductance, and interference 9

III. Realistic modeling of semiconductor Majoranawires 10A. Tight–binding models for semiconductor

nanowires 11B. Proximity effect in semiconductor

nanowire–superconductor hybrid structures 13C. Effective low–energy Bogoliubov–de Gennes

Hamiltonian for the Majorana wire 14D. Low–energy physics in Majorana wires with

disorder 15

IV. Recent experiments and theoretical interpretations 16A. Experimental signatures of Majorana fermions in

hybrid superconductor–semiconductor nanowiredevices 16

B. Suppression of the gap–closing signature at thetopological quantum phase transition 18

C. Discriminating between Majorana bound statesand garden variety low–energy states 18

D. Majorana physics in finite–size wires: A “smokinggun” for the existence of the Majorana mode 19

V. Conclusions 21

A. Inter–band pairing and the “sweet spot” 22

B. Projection of the effective tight–binding Hamiltonianonto a low–energy subspace 23

C. Proximity effect in hole–doped semiconductors 23

D. Proximity effect in multi–band systems and thecollapse of the induced gap 24

E. Low–energy states in the presence of disorder 25

References 27

I. INTRODUCTION

In relativistic quantum mechanics spin–1/2 fermions are de-scribed by the solutions of the Dirac equation [1],

(iγµ∂µ −m)ψ = 0. (1)

Here, γµ (µ = 0, 1, 2, 3) is a set of 4 × 4 matrices satisfy-ing the anti–commutation relations {γµ, γν} = 2gµν , wheregµν represents the Minkowski metric and γ0γµγ0 = ㆵ [2].In general, the matrices γµ have complex elements, makingEq. (1) a set of coupled differential equations with complexcoefficients.Thus, the general solution ψ(x) of Eq. (1) repre-senting the fermion field is a complex bi–spinor that is notan eigenstate of the charge conjugation (CC) operator. Since

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charge conjugation, ψ → ψ∗, maps a particle into its anti–particle [3], a complex solution of Eq. (1) represents a fermionthat has a distinct anti–fermion, with the same mass and spinbut opposite charge and magnetic moment, as its counter-part. Therefore, the field of a relativistic fermion that coin-cides with its own antiparticle, should it exist, is necessarilyan eigenstate of CC that must be described by a real solu-tion χ(x) of Eq. (1). Real solutions of the Dirac equation arepossible, provided one can find a suitable representation ofthe matrices γµcharacterized by purely imaginary non–zeromatrix elements. Such a representation of the γµ matricesthat renders Eq. (1) purely real was found by E. Majoranain 1937 [4]. The real solutions χ(x) correspond to charge–neutral fermions, known as Majorana fermions (MFs), rep-resenting particles that are their own antiparticles. Chargeneutrality and the identification of the particle with its ownanti–particle are not uncommon features among bosons - pho-tons and π0–mesons being two standard examples - but theyare quite special among fermions. The neutron, for exam-ple, which is a charge–neutral fermion, has an anti–particle(the anti–neutron) that is distinguished from the neutron bythe sign of its magnetic moment. Also, neutrinos produced inbeta-decay are thought to be charge–neutral and have a smallbut non–zero rest mass (so that they cannot be Weyl fermions[5], which are massless), but whether they are Dirac or Majo-rana fermions is still an unsettled question in particle physics.

While Majorana fermions have remained undetected inhigh energy physics for almost 70 years, in the last decadethey have made an emphatic entrance into the realm of con-densed matter physics [6–10]. In this context, the term Ma-jorana fermion does not refer to an elementary particle, butrather to quasiparticles corresponding to collective excitationsof an underlying complicated many–body ground state. Asshown by N. Read and D. Green [11], in the so–called weak–pairing phase of a two–dimensional (2D) spinless px + ipysuperconductor (or superfluid), the quasiparticles satisfy anequation - the Bogoliubov–de Gennes (BdG) equation - thatis similar to the Majorana form of the Dirac equation. Thesequasiparticles represent the solid state analog of the Majo-rana fermions from high–energy physics and, in general, arecharacterized by a finite mass and have finite energy. How-ever, in the context of the recent developments, the term “Ma-jorana fermion” has a slightly different meaning. In a su-perconductor, the spinless fermionic Bogoliubov quasiparti-cles satisfy particle–hole symmetry, γ†E = γ−E , and, con-sequently, the zero–energy quasiparticles can be viewed asfermions that are identical with their own anti–quasiparticles,γ†0 = γ0. These BdG quasiparticles with zero excitation en-ergy emerge as localized states bound to defects in the su-perconductor, where the order parameter amplitude vanishes,e.g., vortices and sample edges [11]. Furthermore, in twodimensions the quantum quasiparticles associated with thesezero–energy bound states were shown to obey a form of quan-tum statistics known as non-Abelian statistics [11–15]. Ex-changing two particles that obey non-Abelian statistics rep-resents a non–commutative operation. Recently, the interestin the properties and the possible realization of non-Abelianzero–energy quasiparticles has increased dramatically, after

they have been proposed [16] as possible building blocks forfault tolerant topological quantum computation (TQC) [6].

In this review, the terms “Majorana fermion”, “zero–energyMajorana state”, or “Majorana bound state” refer to local-ized, charge–neutral, zero–energy states that occur at defectsand boundaries in topological superconductors. The creationoperator for such a zero energy state is a hermitian secondquantized operator γ† = γ that anti–commutes with otherfermion operators and satisfy the relation γ2 = 1 . In a su-perconductor, a non–degenerate localized zero–energy eigen-state enjoys a form of topological protection that makes itimmune to any local perturbation that does not close the su-perconducting gap. Such perturbations cannot move the stateaway from zero energy because of the particle–hole symmetryand the non–degeneracy condition. Since small perturbationsto the BdG differential equation are not expected to changethe total number of solutions, it necessarily follows that localperturbations (i.e., perturbations that do not couple pairs ofMFs) leave the non–degenerate zero energy eigenvalues un-perturbed. This argument also implies that the zero–energysolutions are characterized by vanishing expectation valuesfor any local physical observable, such as charge, mass, andspin. Otherwise, in the presence of a non–zero average, a lo-cal field that couples to the corresponding operator would shiftthe energy of the state. Thus, our use of the term “Majoranafermion” is more restrictive than that in the particle physicsliterature: the MFs in this review are charge–less, mass–less,spin–less, i.e., free of any internal quantum number, and obeynon–Abelian statistics, while the MFs in particle physics sat-isfy standard Fermi statistics and, although charge–neutral,since they are eigenstates of the charge–conjugation opera-tor, can be massive and spin-full . Finally, as zero energyMFs in solid state systems are topologically protected, theycan be removed from zero energy only by driving the systemthrough a topological quantum phase transition (TQPT) [17]characterized by the closing of the energy gap at the topologi-cal critical point, where the MFs become entangled with othergapless states [11].

Majorana fermions have recently been proposed in lowtemperature solid–state systems in the context of fractionalquantum Hall (FQH) effect [11–13, 18–22], spinless chiral p-wave superconductors/superfluids [11, 14, 21, 23], in particu-lar in strontium ruthenate, via the realization of the so–calledhalf–quantum vortices [24–26], heterostructures of topologi-cal insulators (TI) and superconductors [27–32], metallic sur-face states [33], metallic ferromagnet–cuprate high-Tc super-conductor [34], non–centro–symmetric superconductors [35–37], superconductors with odd–frequency pairing [38, 39], he-lical magnets [40], carbon nanotubes and graphene [41–44],spin–orbit–coupled ferromagnetic Josephson junctions [45],chains of magnetic atoms on a superconductor [46], and coldfermion systems with p–wave Feshbach resonance [47–50],or with s–wave Feshbach resonance plus artificially gener-ated spin–orbit coupling and Zeeman splitting [51, 52]. It hasalso been shown that MFs can exist as quasiparticles localizedin the topological defects and at the boundaries of a spin–orbit (SO) coupled electron–doped semiconductor 2D thinfilm [53, 54] or 1D nanowire [54–56] with proximity induced

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s–wave superconductivity and an externally induced Zeemansplitting. More recently, it was shown that similar mecha-nisms can lead to the emergence of MFs in hole–doped semi-conductor structures [57, 58]. Although the physics responsi-ble for the emergence of the zero energy MFs is identical inboth 2D and 1D systems (in 3D a similar mechanism producesa so–called Weyl superconductor/superfluid [59, 60] char-acterized by gapless, topologically protected, Weyl fermionnodes in the bulk), the nanowire setting has some experimen-tal advantages, such as the option of generating the Zeemansplitting by using a parallel magnetic field [55, 56] and a sig-nificantly enhanced gap (the so–called mini–gap) that protectsthe end–localized MFs from thermal effects [61]. In addition,the nanowires can be arranged in networks that allow the ma-niputation of MFs for TQC [62, 63].

The 1D version of the semiconductor–superconductor het-erostructure [54–56] – the so–called semiconductor Majoranananowire – is a realization of the 1D topological supercon-ductor (TS) model first proposed by Kitaev [23] in the contextof TQC. In the presence of a small Zeeman splitting Γ, theMajorana nanowire with proximity induced s-wave supercon-ductivity is in a conventional superconducting state with noMFs, while for Γ larger than a critical value Γc, localized MFsexist at the ends of the quantum wire. We emphasize that, inthe proposals for the realization of Majorana fermions usingsemiconductor–superconductor hybrid structures, the Zeemansplitting plays the key role of lifting the degeneracy associatedwith the fermion doubling problem [64, 65]. In practice, therequired Zeeman splitting can be obtained either by applyingan external magnetic field, or by proximity to a ferromagneticinsulator. We note that the realization of Majorana fermionsin a 3D topological insulator–superconductor heterostructuredoes not require a Zeeman field, since the fermion doublingproblem is avoided by the spatial separation between the sur-face states localized on opposite surfaces. Furthermore, theZeeman splitting does not represent the only solution to thefermion doubling problem [65]. For example, in the case ofa topological insulator nanowire proximity coupled to a su-perconductor [32, 66] the cure of the fermion doubling prob-lem is ensured by the orbital effect of a magnetic field ap-plied along the wire, while the corresponding Zeeman split-ting is negligible. More generally, each proposal for realizingMFs in solid state systems has to contain a specific solution tothe fermion doubling problem. Finally, we note that the MFsemerging in a Majorana nanowire can be detected by measur-ing the zero–bias conductance peak (ZBCP) associated withtunneling into the end of the wire [54, 67–71], or by detect-ing the predicted characteristic fractional AC Josephson ef-fect [23, 28, 55, 56, 72, 73]. While these techniques havealready been implemented experimentally, measurements in-volving the direct observation of non–Abelian Majorana inter-ference will ultimately be required to validate the existence ofthe zero–energy Majorana bound states. In the last year, thesemiconductor Majorana wire, which is the focus of this re-view, has attracted considerable attention as a result of the re-cently reported experimental evidence for both the ZBCP [74–78] and the fractional AC Josephson effect in the form of dou-bled Shapiro steps [79].

II. THEORETICAL BACKGROUND

A. Spinless (px + ipy) superconductor/superfluid

The spinless (px + ipy) superconductor (superfluid) is thecanonical system that supports zero energy MFs localized atthe defects of the order parameter, such as vortices and sam-ple edges [11]. In 2D the mean field Hamiltonian for such asystem is given by,

Hp2D =

∑p

ξpc†pcp + ∆0

∑p

[(px + ipy)c†pc

†−p + h.c.

],

(2)where ξp = p2/2m− εF , with εF the Fermi energy, and spinindices are omitted because the system is considered spinless(or spin-polarized). The lowest–energy solution of the BdGequations Hp

2DΨ(r) = EΨ(r) near a vortex or near the sam-ple edges (where the order parameter ∆0 vanishes), calculatedunder appropriate conditions, is non–degenerate and has zeroenergy (E = 0), while the corresponding second–quantizedoperator (the creation operator for the Bogoliubov state) ishermitian, γ† = γ. In 1D the zero energy MF solutions occurnear the ends of the wire and, as argued by Kitaev [23], shouldbe observable in a fractional AC Josephson effect–type exper-iment. Although a spinless p–wave superconductor does notexist in nature, this example makes it clear that the three mainingredients that are important for realizing zero–energy MFsin a condensed matter system are superconductivity, chirality(i.e., the px + ipy orbital form of the order parameter), andthe spinless or spin–polarized nature of the system (which en-sures that the zero energy solution localized at a defect of theorder parameter is non–degenerate).

Despite the possibilities opened by superconducting stron-tium ruthenate [80] and cold fermion systems in the pres-ence of a p–wave Feshbach resonance [47–50] of realizingthe physical conditions necessary for the emergence of MFs,actually realizing and observing the MFs in these systemsare challenging tasks. In strontium ruthenate, even if therequired half–quantum vortices can be realized, the mini–gap ∼ ∆2/εF ∼ 0.1mK (with εF the Fermi energy and ∆the magnitude of the p–wave superconducting order parame-ter) that separates the zero energy MF bound states from thehigher energy regular BdG excitations also localized at thevortex cores is unrealistically small. On the other hand, incold fermion systems with p–wave Feshbach resonance in theunitary limit, even if the mini–gap ∼ ∆2/εF ∼ εF can berelatively large, the short lifetimes of the p–wave pairs andmolecules represents a major experimental challenge.

The first major attempt to reproduce the essential physics ofspinless p–wave superconductors using only s–wave pairingwas made in Ref. [27], by using a heterostructure consistingof a 3D strong topological insulator and an s–wave supercon-ductor. In the cold fermion context, Ref. [51] showed that(also see Ref. [52]) an artificial, laser–induced, Rashba spin–orbit coupling and a Zeeman field, in conjunction with s–wavepairing interactions, can reproduce the topological superfluid-ity and MFs of spinless chiral p–wave superfluids using an s–wave Feshbach resonance. Later, the same approach that was

4

used in the cold fermion context was applied to a solid stateheterostructure where the spin–orbit coupling, Zeeman field,and s–wave superconductivity are provided by a semiconduc-tor, a magnetic field or the proximity to a magnetic insulator,and the proximity to an s–wave superconductor, respectively.The proposed setups are shown in Fig. 1. As discussed be-low, in the semiconductor–superconductor heterostructure thetwo main ingredients (in addition to superconductivity) thatare essential for realizing MFs, i.e., chirality and spinlessness, are provided by a Rashba–type spin–orbit coupling and byrestricting the low–energy physics of the semiconductor to asingle (or an odd number of) relevant Fermi surfaces.

B. Semiconductor–superconductor heterostructure

The Rashba SO–coupled semiconductor (e.g., InSb, InAs)with proximity induced s–wave superconductivity and Zee-man splitting is mathematically described by the followingBardeen-Cooper-Schrieffer (BCS)-type Hamiltonian:

H = (ηk2 − µ)τz + Γn · σ +α

2(k × σ) · zτz + ∆τx, (3)

where η = 1/m∗, with m∗ being the effective mass of thecharge–carriers, µ is the chemical potential measured fromthe bottom of a pair of spin sub–bands (in this review by“band”, we always mean a pair of a spin–split sub–bands,the band themselves being separated by the energy gaps dueto lateral confinement), α is the Rashba spin–orbit couplingconstant, and ∆ is the s–wave superconducting pair–potentialin the semiconductor, which is assumed to be proximity in-duced from an adjacent superconductor. In addition, n is asuitably chosen direction of the applied Zeeman spin splittinggiven by Γ = 1

2gµBB with g the effective Lande g-factor,B the applied magnetic field and µB the Bohr magneton.Note that H is written in terms of the 4–component Nambuspinor (u↑(r), u↓(r), v↓(r),−v↑(r)), and that the Pauli ma-trices σx,y,z, τx,y,z act on the spin and particle-hole spaces, re-spectively. For a Zeeman splitting Γ larger that a critical value,H describes a 2D topological superconductor when the direc-tion of the Zeeman field is perpendicular to the plane contain-ing the vector k = (kx, ky), i.e., for n = z. A 1D topologicalsuperconductor with k = kx obtains for either n = x orn = z. In general, a finite topological superconducting gapdevelops only if the component of the Zeeman field perpen-dicular to the effective k–dependent Rashba spin–orbit field islarger than a critical value.

The Hamiltonian in Eq. (3) has recently been studied exten-sively in the context of 2D, 1D and quasi–1D (multichannel)semiconductors [51–56, 81–87]. Two setups proposed for theexperimental realization of the Majorana physics described bythis Hamiltonian are shown in Fig. 1. The system is charac-terized by a topological quantum critical point (TQCP) thatcorresponds to the critical Zeeman field Γc =

√∆2 + µ2,

where the quantity C0 = (∆2 + µ2 − Γ2) changes sign. ForC0 > 0, the (low–Γ) state is an ordinary, non–topological su-perconductor that includes perturbative effects from the Zee-man field and tha spin–orbit coupling but does not host MFs.

(a)

(c)

(b)

FIG. 1. (Color online) Proposed setups for realizing MFs in semi-conductors. (a) Two–dimensional semiconductor with Rashba SO–coupling proximity coupled to a superconductor and a magnetic insu-lator. The MFs are predicted to occur at the order parameter defectssuch as vortices. The required Zeeman field oriented perpendicularto the semiconductor (n = z in Eq. (3)) it proximity–induced bycoupling the semiconductor to a magnetic insulator. If the semicon-ductor has both Rashba and Dresselhaus SO couplings, the requiredZeeman splitting can be parallel to the surface and can be induced bya parallel magnetic field [83]. (b) One dimensional setup with Zee-man splitting parallel to the nanowire (n = x in Eq. (3)) generatedby an external magnetic field. The end–of–wire MFs can be probedas zero bias conductance peaks in a local tunneling measurement.(c) The spectrum of the spin–split sub–bands (or the two top–mostoccupied sub–bands in a quasi–1D system) in the presence of Zee-man splitting. Figure adapted from Ref. [53] (panel a) and Ref. [54](panels b and c).

However, for C0 < 0, the (high–Γ) state has non–perturbativeeffects from α and can support zero–energy MF states local-ized at the defects of the pair–potential ∆. Interestingly, theproximity–induced pair–potential ∆ itself remains non–zeroand continuous across the TQCP [54, 88], and, consequently,the two superconducting states break exactly the same sym-metries, namely the gauge and time–reversal symmetries. Inthe absence of topological defects and boundaries, the singleparticle spectrum of the high–Γ phase is similar to that of thelow–Γ phase, as both are completely gapped in the bulk. How-ever, the high–Γ topological state can be distinguished fromthe non–topological superconductor at Γ < Γc by probing thetopological defects and the boundaries, as they host MFs onlyfor Γ > Γc.

The topological quantum critical point Γc is marked by thevanishing of the single–particle minimum excitation gap E0.By diagonalizing the Hamiltonian in Eq. (3) one obtains the

5

lower–branch of the quasiparticle excitation spectrum,

E2k = ∆2 + ε2 + r2

k − 2√

Γ2∆2 + ε2r2k, (4)

where ε = ηk2 − µ and r2k = Γ2 + α2k2. For Γ near Γc,

the minimum of Ek is at k = 0, which corresponds to theminimum quasiparticle gap E0,

E0 = |Γ−√

∆2 + µ2|. (5)

Note that, E0 vanishes exactly at C0 = 0, marking the loca-tion of the TQCP as a function of Γ, ∆, or µ. Furthermore,it has been shown that the quantity C0 is the Pfaffian of theBdG Hamiltonian at k = 0, (C0 = Pf(H(k = 0)σyτy))[36]. The sign of C0, which determines whether the systemhas 0 or 1 MFs, is related to the Pfaffian topological invariantQ (see Eq. (11)) given by the product of the Pfaffians of theBdG Hamiltonian at k = 0 and k = π [23]. Since the sign ofthe Pfaffian of the BdG Hamiltonian at k = π is genericallypositive, the sign of C0 by itself determines whether the stateof the semiconductor hosts 0 or 1 MFs (or, in general, an evenor odd number of Majorana bound states). The topological in-variant of the semiconductor-superconductor heterostructureis discussed in more detail in section II D.

C. Topological class

Recent work [89–91] has established that the quadraticHamiltonians describing gapped topological insulators andsuperconductors can be classified into 10 distinct topologi-cal classes that can be characterized by certain topologicalinvariants. The 2D spinless chiral (px + ipy) superconduc-tor in the weak pairing phase (µ > 0) belongs to the topo-logical class D, whish is characterized by an integer Z top-logical invariant. This implies that, under appropriate condi-tions, the system can support Z number of chiral Majoranaedge modes that remain protected against small perturbations.The analogous system of a 2D Rashba–coupled semiconduc-tor with Γ > Γc is also characterized by a Z invariant and sup-ports Z gapless chiral Majorana edge modes with dispersionEedge(k) = (∆/kF )k. Since the 2D system is in class D, doesit automatically imply that the corresponding 1D or quasi-1Dsystem is also in class D? The topological class of the 1Dnanowire can be guessed from a heuristic dimensional reduc-tion argument. The MF mode in a 1D wire (say, along thex-axis) can be viewed as the dimensionally–reduced versionof the chiral Majorana edge mode of a 2D system on an edgeparallel to the y axis. From the mathematical equivalence,H2DBdG(ky = 0) = H1D

BdG, where H1DBdG is the BdG Hamilto-

nian near an end of the wire, and the propertyEedge(0) = 0, itfollows that the end of the wire supports a zero energy eigen-state. Since in 2D there are Z allowed chiral edge modes,it follows that in 1D there must be, under appropriate condi-tions, Z zero energy MFs. Thus, the nanowire should alsobe characterized by a Z (not Z2) invariant, and, consequently,the purely 1D Majorana nanowire should belong to the BDItopological class.

To reveal the appropriate BDI classification for thenanowire and the associated Z invariant, it is necessary toidentify a hidden chirality symmetry of the 1D system[61, 92].In 1D, the Hamiltonian in Eq. (3) anticommutes with a unitaryoperator S = τx, where the chirality symmetry operator S canbe written as the product of an artificial ‘time reversal’ opera-tor K and a particle–hole transformation operator Λ = τx · K.Here, K is the complex conjugation operator. The existenceof all three symmetries - ‘time reversal’, particle-hole, andchirality - ensures that the Hamiltonian is in the BDI sym-metry class [89–91] characterized by an integer topologicalinvariant. Under invariance of this symmetry, the strictly 1Dnanowire can support an arbitrary integer number N (= Z)of protected zero energy MFs at each end. The situation issimilar to that of a 1D spinless p–wave superconductor thatwas shown [93] to support any integer number of MFs at eachend. However, for the semiconductor Majorana wire, the ex-perimentally realistic case of a quasi–1D system retains onlyan approximate chirality symmetry (the chirality symmetry isweakly broken by the inter–band Rashba coupling, which nolonger commutes with S [61, 94]) and allows multiple nearzero energy modes at each end for values of the Zeeman cou-pling close to the confinement energy gap and above. The ex-istence of the near zero modes suppresses the gap protectingthe MF end modes in certain parameter regimes characterizedby Γ larger than the confinement energy [61, 94]. Furtherconsequences of the chirality symmetry for the electrical con-ductance are discussed in Ref. [95].

D. Topological invariant

The topological class and topological invariant of thesemiconductor-superconductor heterostructure is analogousto that of a spinless px + ipy superconductor. For the topo-logical invariant of the 2D spinless px + ipy superconductor(which is in class D), we rewrite the Hamiltonian in Eq. (2)inthe particle–hole basis (c†k, c−k) as,

Hp2D = ξkτz + ∆xkxτx −∆ykyτy, (6)

where we have allowed for different pair potentials ∆x,∆y

along the x, y directions. Writing the Hamiltonian in termsof the Anderson pseudo–spin vector [96] ~d(k) as Hp

2D(k) =~d(k).~τ , one can observe that in D = 2 all three componentsof ~d are non-zero. The topological invariant is an integer, as Zis the relevant homotopy group π2(S2) of the mapping fromthe 2D k space to the 2–sphere of the 3–component unit vec-tor d = ~d/|~d| [11, 17]. On the other hand, in D = 1, thecorresponding Hamiltonian can be made purely real (e.g., ∆y

drops out from Eq. (6) if the system is along the x–axis) andthe vector ~d has only two components. Now since the k-spaceis also one–dimensional, the topological invariant must againbe in Z (class BDI) since π1(S1) = Z. This invariant is sim-ply the winding number,

N =1

2π

∫ 2π

0

dθ(k), (7)

6

where θ(k) is the angle the unit vector d makes with, say, thez-axis in the x − z plane. The winding number counts thenumber of times the 2–component vector d makes a completecircle as k varies in the 1D Brillouin zone.

The topological invariant for the semiconductor-superconductor heterostructure can be defined in a similarway to the spinless chiral p–wave superconductor. Oneneeds, however, to account for the higher dimensionalityof the Hamiltonian matrix in Eq. (3) and the consequentgeneralization of the winding number invariant. It is clearthat in D = 2 the Hamiltonian in Eq. (3) cannot be madereal because of the complex Rashba term. In contrast, inD = 1 H can be made purely real, but the components ofthe ~d-vector in the 4 × 4 Hamiltonian are themselves 2 × 2matrices. More generally, the BdG Hamiltonian of a TSsystem in D = 1, despite being real (thus preserving thechiral symmetry given by the operator S = τx), can be alarge 2N × 2N square matrix. Using the general frameworkfor chiral symmetric systems [97, 98], the integer topologicalinvariant for this system can still be defined [61, 92] bygeneralizing the concept of the ~d-vector winding number forarbitrary dimensional matrices. Since, as discussed in thelast subsection, H anticommutes with the chirality operatorS = K · Λ = τx, H can be expressed as a block off–diagonalmatrix in a basis that diagonalizes the unitary operator S:

UH(k)U† =

(0 A(k)

AT (−k) 0

). (8)

Following Ref. [61 and 92] one can now define the variable,

z(k) = exp(iθ(k)) = Det(A(k))/|Det(A(k))|, (9)

and calculate the topological invariant,

W =−iπ

∫ k=π

k=0

dz(k)

z(k), (10)

which is an integer, including zero (W ∈ Z).If the chirality symmetry is broken (it is weakly broken for

quasi–1D multi–band wires with transverse Rashba couplingαy 6= 0), the number of exact zero energy MF at each end ofthe wire goes back to 0 or 1. The corresponding Z2 topologi-cal invariant is

Q = sgn {Pf [H(k = 0)σyτy]× Pf [H(k = π)σyτy]} ,(11)

and the topological class for the quasi-1D system reducesfrom BDI to D. Q = +1(−1) represents the topologicallytrivial (non-trivial) state with 0 (1) MF at each end of thewire. In the presence of the chirality invariance in 1D, itcan be shown that Q gives the parity of the integer topolog-ical invariant W , in analogy to the 2D case where Q for thesemiconductor–superconductor heterostructure gives the par-ity [36] of the first Chern number topological invariant, whichis an integer.

E. Minigap

The minigap for topological superconductors is defined asthe energy of the first excited regular fermion state (above thezero energy MF state) in either the bulk or the edge of thesystem. This gap is responsible for the thermal protection ofthe MFs and, consequently, for the non–trivial physics associ-ated with them, such as non–Abelian statistics, by protectingthe Majorana bound state from mixing with regular fermionstates above the minigap. In naturally occurring quasi–2Dchiral–p wave superconductors, such as strontium ruthenate,even if the MFs are realized in the cores of the half–quantumvortices, the minigap δ ∼ ∆2/εF ∼ 0.1mK is unrealisticallysmall. In chiral p-wave superfluids, potentially realizable us-ing cold fermions with p–wave Feshbach resonance, this isless of a problem because in this case ∆ ∼ εF and the mini-gap ∆2/εF is of order ∆. In solid state hybrid structures, theminigap has to be large enough so that the experiments can ac-cess the MF physics at realistically achievable temperatures.This is one important experimental aspect in which the semi-conductor nanowire heterostructure has significant advantageover the corresponding 2D systems. In the 1D system, theminigap naturally scales with the induced superconductinggap (see equation (5)), while this is generically not the casefor the 2D systems (see, however, Ref. [ 81]).

The reason why the minigap in the nanowires is greatly en-hanced over the 2D heterostructures can be understood fromthe dimensional reduction arguments. In going from the 2Dplane to the 1D wire, as the width Ly in the y-direction is re-duced, the energies of the quantized chiral edge modes scaleas 1/Ly (recall that E(ky) = (∆/kF )ky), i.e. as the inverseof the confinement size of the wire in the transverse direction.Since 1/Ly diverges in the strict 1D limit Ly → 0, it impliesthat the minigap becomes arbitrarily large, in fact of the orderof the bulk quasiparticle gap E0. In principle, the end–stateMFs protected by such a large minigap can be probed in localtunneling experiments without having to worry about otherlow energy states with energies comparable to experimentaltemperatures.

In practice, for clean quasi–1D nanowires (Lx � Ly >Lz) with hard–wall confinement, the lowest excited stateabove the MF end state (for Γ > Γc, but less than the con-finement energy gap) has an energy that is a significant frac-tion of E0 [86]. In quasi–1D wires with the Zeeman splittingcomparable to the confinement energy gap, the minigap canbe drastically suppressed due to the existence of multiple nearzero energy modes at the same end, which is a manifestationof the approximate hidden chirality symmetry [61].

F. Phase diagram of semiconductor Majorana wire, sweetspots, and approximate chirality symmetry

In a strictly 1D wire with proximity–induced superconduc-tivity the Majorana–supporting topological phase emerges atZeeman fields higher than the critical value [52, 54]

Γc =√

∆2 + µ2. (12)

7

FIG. 2. (Color online) Phase diagram as a function of chemical po-tential and Zeeman field oriented perpendicular to the SM–SC in-terface. White regions correspond to the topologically trivial phase,while colored regions indicate topologically non–trivial SC phases.The dashed lines show the location of the normal state SM sub–bands. Note the characteristic anti–crossings between the sub–bandsof the transverse ny bands with opposite parity, e.g., between ny andny+1. For a Zeeman field oriented along the wire (see Fig. 3), thesebands cross at the “sweet spots” (for example, the regions marked byred circles). Figure adapted from Ref. [87].

Consequently, for Γ < Γc the superconducting phase is topo-logically trivial, while for Γ > Γc the topological phase isrealized. How is this simple picture modified in a quasi–1Dsystem with multiple occupied confinement bands? If the ap-plied Zeeman field is much smaller than the spacings betweenthe confinement bands, nothing much is changed, except thatµ in Eq. (12) has to be understood as the chemical poten-tial measured from the bottom of the top–most occupied band(also called the Majorana band). However, the phase diagramis non–trivially modified when the Zeeman field is compara-ble with or higher than the spacings between the confinementbands.

Consider a finite–width wire oriented along the x directionand proximity–coupled to an s–wave superconductor, with theinterface perpendicular to the z direction. The effective spin–orbit field generated by Rashba coupling is oriented along they direction. Two phase diagrams of this quasi–1D Majoranawire are shown in figures 2 and 3. The main difference be-tween the two diagrams is due to the different orientations ofthe Zeeman field: in Fig. 2 the field is perpendicular to thesemiconductor-superconductor interface, while in Fig. 3 it isoriented along the wire. In both cases the Zeeman field is per-pendicular to the effective spin–orbit field generated by theRashba coupling. Note that, when the magnetic field is paral-lel to the z axis, the normal state spectrum is characterized byanti–crossings of the sub–bands corresponding to transverseny modes with opposite parity (see Fig. 2, red circles). Bycontrast, when the magnetic field is parallel to the wire, thesebands cross at the “sweet spots” [85], critical points where(in the absence of inter–band pairing) two topologically triv-ial and two topologically nontrivial phases meet. We empha-size that inter–band pairing stabilizes the topological phase inthe vicinity of the sweet spots (see Fig. 3; more details aboutsweet spot physics are provided in Appendix A). By contrast,

FIG. 3. (Color online) Phase diagram of the multiband nanowire withthe Zeeman field oriented along the wire. Superconducting phasescharacterized by an odd (even) number N of nearly–zero energymodes are topologically nontrivial (trivial). The phase boundariescorrespond to topological quantum phase transitions characterized bythe vanishing of the bulk quasiparticle gap. The energy unit is givenby the spin–orbit characteristic energy Eα = meffα

2R. Nonuniform

SM-SC coupling induces inter–band pairing and stabilizes the topo-logical phase near the sweet spots. Figure adapted from Ref. [86].

rotating the field away from the x direction favors the topo-logically trivial SC phase.

The phase diagram in Fig. 3 reveals that, upon increasingthe Zeeman field, the system undergoes a series of topologicalquantum phase transitions as it traverses successive topolog-ically trivial and nontrivial phases. These phases are charac-terized by a certain numberN of low–energy modes localizedat each end of the wire. In the thin wire limit, Ly → 0, thesemodes become zero–energy Majorana bound states. In a fi-nite width wire, for N even, the low–energy modes acquirea small non–zero gap, while for N odd, N − 1 modes be-come gaped and one mode (the Majorana mode) remains gap-less. The opening of the small low–energy gap is due to theinter–band Rashba coupling [61, 94], (see, for example, theinter–band Rashba term qnyn′

ygiven by Eq. (B4 in Appendix

B), that beaks the hidden chirality symmetry of the purely 1Dmodel [61, 92].

Finally, we note that in the limit of weak semiconductor–superconductor (SM–SC) coupling, the location of the phaseboundaries is closely related to the non–superconducting SMspectrum [99]. However, the locations of the phase bound-aries (in particular the boundary between the N = 0 and N =1 phases) depends strongly on the coupling strength [86].Specifically, from Eq. (12) the minimal Zeeman field nec-essary for reaching the topological phase in the single bandcase is Γc0 = ∆. This critical field corresponds to the tran-sition between the N = 0 and N = 1 phases at low val-ues of the chemical potential. For a multi–band system, we

8

have to take into account the band dependence of the inducedgap ∆nn′ and the proximity–induced energy renormalizationZnn′ (see section III C for details). Within the decoupled–band static approximation (valid when the effective SM–SCcoupling is much weaker than the inter–band spacing and theSC bulk gap), we find that the critical Zeeman field requiredfor entering the topological phase with N = 1 is character-ized by multiple minima Γcn = γnn corresponding to valuesof the chemical potential at the bottom of each band n. Here,γnn is the effective SM–SC coupling corresponding to band n.Consequently, increasing the coupling strength will move thatphase boundary toward higher Zeeman fields. In addition, thepresence of proximity–induced inter–band pairing widens thetopological phase near the sweet spots and further reduces thedependence of Γc on µ [86]. Experimentally, this may resultsin an apparent insensitivity of the critical magnetic field asso-ciated with the emergence of the Majorana bound state on thechemical potential. In addition, the inter–band pairing aris-ing from a non uniform coupling with the bulk superconduc-tor results in a phase diagram characterized by a non-simplyconnected structure [86], which allows topological adiabaticpumping [100].

G. Probing Majorana fermions I: zero bias conductance peakand fractional AC Josephson effect

The simplest way to establish the possible presence of aMajorana bound state at the end of a spin–orbit coupled semi-conductor wire with proximity– induced superconductivity isby performing a tunneling spectroscopy measurement. Sincethe MF state is a zero energy bound state localized at the endof the wire, it is expected to produce a zero bias conductancepeak in the tunneling conductance dI/dV , similar to otherzero energy boundary states in superconductors [101, 102].Here, I is the current from the lead to the Majorana wire andV is the applied voltage difference. The zero bias conductancepeak associated with tunneling into MFs is a result of reso-nant local Andreev reflection of the lead electrons at the lead–nanowire interface [69]. In a local Andreev process, the MFlocalized at the other end of the wire plays no role, providedthe wire is long enough so that the two MF wavefunctions donot overlap. If there is a significant overlap between the MFwavefunctions, two other processes can also contribute to thetransport current. One process corresponds to the direct trans-port of electrons via the conventional (Dirac) fermion stateformed by the two overlapping MFs [68, 103, 104]. The sec-ond process is the crossed Andreev reflection, in which anincident electron from a lead coupled to one end of the wire isfollowed by a hole ejected into the lead coupled to the otherend [105]. These two processes contribute to the non–localconductance or transconductance through the Majorana wirethat we discuss in the next subsection. We note that the localAndreev reflection process typically gives rise to a non–zerosub–gap conductance even in the absence of MFs at the wireends [106]. To suppress this background conductance, a gate–induced barrier potential may be applied at the lead–nanowireinterface.

Dif

fere

nti

al c

on

du

ctan

ce dI/

dV

(a.u

.)

Bias voltage eV/Ea

FIG. 4. (Color online) Predicted differential conductance for tunnel-ing into the end of a semiconductor nanowire coupled to a supercon-ductor. The lines represent cuts (shifted for clarity) corresponding todifferent values of the Zeeman field. As the field is increased, thesuperconducting quasiparticle gap closes at the critical Zeeman fieldΓc that marks the topological quantum phase transition. By furtherincreasing the Zeeman field, the presence of a zero energy MF at thewire end is revealed as a pronounced zero bias conductance peak.In this calculation, the chemical potential is set near the middle ofan inter–band gap, close to a sweet spot (see Fig. 3). For values ofthe chemical potential close to the bottom of a band, the signatureassociated with the gap closing at the TQPT is strongly suppressed,although the presence of a zero bias peak still reveals the MF in thetopological phase (for details see Sec. IV B, in particular Fig. 9) .Figure adapted from Ref. [86].

The presence of a zero bias conductance peak representsa necessary, but not sufficient, condition for the existence ofMFs that was theoretically proposed [54, 67–71, 86, 107–109] (see Fig. 4) and experimentally observed in recent mea-surements on quasi–1D semiconductor–superconductor hy-brid structures [74–76]. The main problem concerning theunambiguous identification of such a zero bias peak with end–localized MFs is that other sources of zero– or low–energystates localized at the wire–boundary [110, 111] may producesimilar zero–bias peaks. A more conclusive signature of MFsin a charge transport measurement is the quantized value ofthe T = 0 zero bias conductance [69]. This value should be2e2/h for tunneling into a non–degnerate MF at the wire end,zero for the wire in the topologically trivial phase with no lowenergy end state, and 4e2/h in the presence of a conventional(non–Majorana) zero energy state at the wire end. Such a con-

9

ductance quantization, which can be taken as a smoking gunsignature of MFs in semiconductor Majorana wires, was notyet observed and may be difficult to obtain under realistic ex-perimental conditions [112, 113].

Another proposal for detecting the existence of MFs insemiconductor wires involves the Josephson effect. In aJosephson junction (JJ) between two topologically trivial s–wave superconductors maintained at a phase difference φ, theJosephson supercurrent I is related to φ as I = Ic sinφ, whereIc is the Josephson critical current. In a Josephson junc-tion between topological superconducting wires with end–state MFs, this relation changes to I = Ic sinφ/2 [23, 28, 72].Consequently, if the phase difference φ is controlled by amagnetic flux threaded through a loop containing the Joseph-son junction, the periodicity of the Josephson current changesfrom 2π to 4π as the semiconductor Majorana wire is tunedthrough the TQPT [55, 56]. In practice, the doubling of theperiod of the Josephson effect can be measured by applyinga small voltage V across the junction and measuring the ACJosephson effect. The frequency of the AC Josephson effectin the TS phase, eV/h, is half the conventional Josephson fre-quency, 2eV/h, obtained in the topologically trivial supercon-ducting phase. Note that the use of the AC Josephson effectis necessary to avoid the inelastic relaxation between the twobranches of the local Andreev bound states involved in theJosephson effect[72].

There are several ways to understand the change in the pe-riodicity of the AC Josephson effect in the presence of MFs.Mathematically, it can be understood as a change in the de-pendence of the energies of the Andreev bound states at thejunction on the phase difference φ. While in the absenceof the MFs the energy of the Andreev bound state goes asE ∝ cosφ, with MFs this formula changes to E ∝ cosφ/2.Since the Josephson supercurrent I is related to the Andreevbound state energy as I = (2e/h)dE/dφ, it follows that theperiod of the supercurrent, I ∝ sinφ/2, is doubled in thepresence of the MFs. In order for this period doubling to beobservable, it is necessary for the two branches of the Andreevbound states, ±E, to maintain their occupation number as φis varied adiabatically. For DC Josephson effect this is impos-sible at thermodynamic equilibrium due to quasiparticle poi-soning effect (the higher (lower) energy state is unoccupied(occupied)) [72]. In AC Josephson effect, provided the periodof the Josephson oscillations is shorter than the inelastic re-laxation between the two branches ±E(φ), the Josephson pe-riod doubling may be observable via the fractional frequencyeV/h.

The doubling of the Josephson period in φ can also be un-derstood from purely topological considerations. The fermionparity of the ground state of the topological superconductor(which is in the form of a ring with a weak link in the Joseph-son set up) flips as the threaded flux changes by 2π. With a 2πflux change the superconductor is thus in an excited state withan electron from the ground state ejected to the junction andheld between the two MF states. Only with a total flux changeof 4π the ground state Fermion parity flips back and the elec-tron can leave the state formed by the MFs. This makes thejunction return to the original state and consequently doubles

the period of the Josephson effect from 2π to 4π [11, 23].Even though the fractional Josephson effect is a robust sig-

nature of MFs, such an effect cannot be ruled out in ballisticS–N–S junctions and in JJs made of 1D p–wave supercon-ducting wires such as the quasi–1D organic superconductors[72]. However, in fermion parity protected superconductors,the MF mediated fractional Josephson effect is topologicallyprotected while there is no such robustness (for instance, todisorder in the ballistic S–N–S junctions, or to magnetic fieldsin p–wave wires) in the other two systems. Nevertheless, thefractional Josephson effect in semiconductor Majorana wiresis susceptible to the quasiparticle poisoning effect [72], aswell as to the non–adiabaticity effects [114]. The latter effect,in particular, shows that the fractional Josephson effect cannotbe taken as an unambiguous signature of MFs because half orother fractional frequencies are in principle possible even injunctions between conventional superconductors. For recentwork on the fractional Josephson effect in the limits of strongtunneling and long junctions, see Refs. [115] and [116]. Ex-perimental evidence of fractional AC Josephson effect in theform of doubled Shapiro steps has been reported in semicon-ductor Majorana wires [79]. In this review we will not furtherdiscuss this effect, but will focus instead on experiments tar-geting the zero bias conductance peak.

H. Probing Majorana fermions II: quantum non-locality,transconductance, and interference

Neither the non–quantized zero bias conductance peak, northe fractional AC Josephson effect could constitute a suf-ficient proof for the existence of MFs in semiconductor–superconductor heterostructures, as in principle they can ariseeven in the absence of MFs. However, a conclusive experi-mental proof could be obtained by making use of the intrin-sic non–local properties of these states. In MFs, non–localitystems from the absence of an occupation number associatedwith them individually. To define the electron occupationnumber in terms of spatially separated MFs, one must con-sider a pair of MFs, γa, γb, and define the second quantizedelectron creation operator as d† = γa + iγb. The quantumstate of the system is then determined by the eigenvalues ofthe electron occupation number operator nd = d†d = 0, 1.Since nd is related to the MF operators by

nd =1 + iγaγb

2, (13)

it follows that the state of the whole system is determined bynon–local correlations between the spatially separated MFs γaand γb. An idea to probe this non-locality involves injectingan electron into one end of the Majorana wire and retrievingit at the opposite end. By connecting leads to the left andthe right ends of a wire that hosts MF states localized nearthe two ends, one could imagine that an electron injected intothe end a flips the occupation number nd from nd = 0 tond = 1. The injected electron can then escape from the endb, flipping the occupation number of the nanowire state backto nd = 0. Such a process, where an electron can enter from

10

one end and exit at the other end as an electron, can be viewedas Majorana–assisted electron transfer. As has been discussedbefore [68, 103, 104], such a transfer should not violate thelocality and causality principles. In other words, in the ab-sence of an overlap between the two MF wavefunctions, theprobability of an electron appearing at end b becomes com-pletely independent of the lead–wire voltage difference ap-plied at the opposite and [68]. Hence, there is no violationof causality. The non–local conductance (transconductance),given by dIb/dVa, can only be non–zero when there is a fi-nite overlap between the two MF wavefunctions. However, inaddition to the Majorana–assisted electron transfer, a secondsource [105] for non–zero transconductance is represented bythe crossed Andreev reflection (CAR), also known as Cooperpair splitting. In this process an electron injection into a Ma-jorana bound state at one end is followed by the emission ofa hole from a second Majorana state at the other end. Thenet result is the injection of a Cooper pair in the topologicalsuperconductor nanowire. The CAR contribution to transcon-ductance, which is opposite in sign to the contribution aris-ing from Majorana assisted electron transfer, is also non–zeroonly when the wavefunctions of the two MF states overlap[105]. It has been shown that [117], in the case of symmetrictunneling between the two leads and the MFs (ta = tb = t),the transconductance in the semiconductor Majorana wire isgiven by,

dIbdVa

= δ32Va

16Γ2 + (δ2 − V 2a )2 + 8Γ2(δ2 + V 2

a ). (14)

Here, Va is the voltage at end a, Ib is the current at end b,Γ ∝ t2 is the lead–induced broadening of the MF level, andδ is the overlap integral between the two MF wavefunctions.Note that the transconductance vanishes in the limit δ → 0,which is consistent with earlier results [68, 104, 105]. Evenwhen the direct wavefunction overlap of the MFs is vanish-ingly small, an effective coupling between the MFs, and con-sequently a non-zero Majorana assisted electron transfer am-plitude, may be present if the topological superconductor hasan appreciable charging energy [118]. In this case, both theMajorana assisted electron transfer and CAR can result in anon–zero transconductance between the leads. Recently, theshot noise and the current–current correlations due to CARhave been proposed as possible experimental signatures of endstate MFs [119].

Even the MF induced transconductance, while interestingand nontrivial, cannot be considered as a definitive signatureof MF modes because conventional near–zero energy states(such as those produced by localized impurities) trapped nearthe contacts with the leads can also produce such non–localsignatures in the presence of superconductivity [117]. On theother hand, a non–local tunneling spectroscopy interferenceexperiment, similar to earlier interference based proposals intopological insulators and superconductors [29, 30, 118, 120],has recently been suggested as capable to provide a directverification of the non–local physics of 1D wires containingend MFs [117]. The effect requires non–local fermion par-ity [121], which is unique to topological systems and can-not be emulated by conventional near–zero–energy Andreev

states or any other local excitations near the wire ends. Theproposed scheme, where the tunneling amplitude is measuredas a flux dependence of an energy level and the fermion–parityis fixed by a superconducting single–electron transistor con-figuration, is suitable for unambiguous experimental testingof the presence of MFs in semiconductor Majorana wires. Wenote that other methods for detecting MFs have also been pro-posed [46, 122–128].

III. REALISTIC MODELING OF SEMICONDUCTORMAJORANA WIRES

The construction of the effective low–energy model for aMajorana hybrid structure involves three main steps: i) devel-oping a tight–binding model for the component that providesspin–orbit coupling (e.g., semiconductor, topological insula-tor, etc.) and projecting onto a reduced low–energy subspace,ii) incorporating the superconducting proximity effect, and iii)defining an effective Hamiltonian based on a linear approxi-mation for the frequency–dependent proximity–induced self–energy. Below, we provide the relevant details and point outthe main approximations involved in this construction. In ad-dition, we discus several aspects of Majorana–supporting hy-brid nanostructures that are not captured by simple modelsof ideal 1D systems, but represent critical components of theexperimental realizations of Majorana nanowires, such as thepresence of disorder and smooth confining potentials.

The existence of Majorana fermions in an ideal topologi-cal SC system does not depend on the details of the Hamilto-nian. However, under realistic laboratory conditions, the sta-bility of the Majorana mode and the low–energy phenomenol-ogy of the heterostructure depend critically on a large num-ber of specific parameters, including details of the electronicband structure, nanostructure size and geometry, coupling atthe interface, type and strength of disorder, and applied gatepotentials. To account for the complex phenomenology of aMajorana–supporting structure and make connection with ex-perimental observations, it is key to incorporate these factorsin the theoretical model. The general form of a Hamiltonianthat describes a semiconductor–superconductor (SM–SC) hy-brid structure is

Htot = HSM +Hint +HZ +HV +HSC +HSM−SC, (15)

where HSM is a non–interacting model for the semiconductorcomponent (or, in general, other spin–orbit coupled material,e.g., topological insulator), Hint contains many–body elec-tronic interactions, HZ describes the applied Zeeman field,HV contains terms that account for disorder and gate poten-tials, HSC is the Hamiltonian for the superconductor, andHSM−SC describes the semiconductor–superconductor cou-pling. Here, we will not address the problem of electron–electron interaction and will assume that Hint = 0. However,we emphasize that Coulomb interaction plays an importantrole in low–dimensional systems, as it generates a density–dependent renormalization of the electrostatic potential. Con-sequently, the Zeeman field Γ and the chemical potential µof a Majorana wire are not independent variables[129] and,

11

for a given configuration of external gate potentials, the sys-tem is characterized by a specific function µ = µ(Γ). Thishas important experimentally observable consequences[129],as we will discuss in Section IV. At the Hartree level, theeffects of Coulomb interaction can be incorporated using aself–consistent scheme similar to that described in Ref. [130].Below, we derive the low–energy effective model for the het-erostructure described by Eq. (15) under the assumptionHint = 0.

A. Tight–binding models for semiconductor nanowires

The first element in the development of a low–energy ef-fective theory for a Majorana–supporting hybrid structure is atight–binding model for the spin–orbit coupled component ofthe system (e.g., electron–doped SM, hole–doped SM, topo-logical insulator, etc.) corresponding to HSM in Eq. (15). Ingeneral, when choosing the model, one has to strike a balancebetween accuracy and simplicity and one has to consider twokey aspects: i) the dimensional reduction from three dimen-sions (3D) to quasi–2D or quasi–1D, and ii) the projectiononto a reduced low–energy subspace. To illustrate some ofthe possible issues, we consider the case of electron–dopedand hole–doped semiconductors.

The simplest tight–binding model for an electron–dopedwire (or a thin film) is a two–band model with nearest neigh-bor hopping. The natural starting point for constructing sucha model is a 3D Hamiltonian for conduction electrons in theeffective mass approximation,

H0(k) =∑k,σ

(h2k2

2m∗− µ

)c†kσckσ, (16)

where ckσ is the annihilation operator for a particle with wavevector k and spin σ, m∗ the effective mass of the conductionband, and µ is the chemical potential. In a thin film, the mo-tion in the transverse direction is quantized and the transversemodes can be obtained by solving the quantum problem in-volving the single–particle Hamiltonian corresponding to Eq.(16) with the substitution kz → −i∂z and the confining po-tential V (z). In the presence of a transverse field that breaksinversion symmetry, the spin and orbital degrees of freedomare coupled and the effective Rashba–type spin–orbit interac-tion (SOI) is described by

HSOI(k) = αR∑k

c†k(kyσx − kxσy)ck, (17)

where k = (kx, ky), αR is the Rashpa coefficient, σi are Paulimatrices, we have used the spinor notation c†k = (c†k↑, c

†k↓),

and we have assumed that only one transverse mode is rele-vant to the low–energy physics. The physics described in thelong wavelength limit by the HamiltonianH0 +HSOI +HV,where HV represents the contribution from the confining po-tential, can be also determined using a tight–binding modeldefined on a lattice. For example, considering a simple cubiclattice with lattice constant a and nearest–neighbor hopping,

the semiconductor Hamiltonian reads

HSM = H0 +HSOI = −t0∑i,δ,σ

c†i+δσcjσ − µ∑i,σ

c†iσciσ

+iα

2

∑i,δ

[c†i+δxσyci − c

†i+δy

σxci + h.c.], (18)

where H0 includes the first two terms and describes nearest–neighbor hopping on a simple cubic lattice and the last termrepresents the Rashba spin-orbit interaction. Here, i =(ix, iy, iz) labels the lattice sites, δ ∈ {δx, δy, δz} arenearest–neighbor position vectors. The confining potential in-volves the additional local term HV =

∑i,σ V (i)c†iσciσ . The

parameters of the tight–binding Hamiltonian (18) are deter-mined by the condition that HSM and H0 + HSOI describethe same physics in the long–wavelength limit k → 0. Con-sequently, the hopping parameter in (18) is t0 = h2a−2/2m∗

and the Rashba coupling is α = αR/a.For hole–doped SMs, an effective tight–binding Hamil-

tonian can be obtained using a similar approach and start-ing from the Luttinger 4–band model [131, 132], H0(k) =∑k c†k[hL(k) − µ]ck, where ck and c†k are four–component

spinors. The single–particle Luttinger Hamiltonian is

hL(k) =−h2

m0

[2γ1 + 5γ2

4k2 (19)

− γ2

∑i

k2i J

2i − γ3

∑i6=j

kikjJiJj

,where m0 is the free electron mass, k = (kx, ky, kz) is thewave–vector, J = (Jx, Jy, Jz) is a set of 4 × 4 matrices rep-resenting angular momentum 3/2, and γ1, γ2, and γ3 are theLuttinger parameters. We assume that the dominant contribu-tion to the valence band comes from p–like orbitals, |X〉, |Y 〉,and |Z〉, and that in the presence of SOI the top valence bandcorresponds to the eigenstates of the total angular momentummomentum J with j = 3/2,

Φ± 12

=1√6

(∓|X〉 − i|Y 〉)χ∓ +

√2

3|Z〉χ±,

Φ± 32

=1√2

(∓|X〉 − i|Y 〉)χ±, (20)

where m = ±1/2,±3/2 are the eigenvalues of Jz and χ± arespin eigenstates with Sz = ±1/2. The effective tight–bindingmodel is constructed using the eigenstates (20) as a basis.More specifically, we consider an fcc lattice and nearest–neighbor hopping between the states Φm(i) and Φm′(j),where i and j are nearest neighbor sites, with hopping matrixelement tmm

′

ij . The tight–binding Hamiltonian has the form

H0 =∑m,m′

∑i,j

tmm′

ij c†imcjm′ − µ∑i,m

c†imcim, (21)

where c†im is the creation operator for the state Φm(i). If weconsidering the xy plane perpendicular to the (0, 0, 1) crys-tal axis, each site has four in–plane and eight out–of–plane

12

Ener

gy

(e

V)

G L X

FIG. 5. (Color online) Comparison of 3D spectra corresponding tothe 8–band Kane–type model (blue lines), the 4–band Luttinger–typemodel given by Eq. (21) (orange), and the 2–band model describedby H0 in Eq. (18) (yellow). Each band is double spin–degenerate.Note that the top valence band (heavy hole band) has the same disper-sion in the 8–band and 4–bad models, while quantitative agreementbetween the light–hole bands and between the conduction bands ispresent only in the vicinity of the Γ point, i.e., for k → 0. Thenumerical values of model parameters correspond to InSb.

nearest–neighbors. The corresponding hopping matrix ele-ments can be expressed in terms of three independent parame-ters, t1, t2, and t3. For example, the in–plane diagonal compo-nents are t±3/2±3/2

in = t1/2+2t2 and t±1/2±1/2in = t1/2−2t2,

while the diagonal out–of–plane hoppings can be written ast±3/2±3/2out = t1/2 − t2, and t±1/2±1/2

out = t1/2 + t2. Finally,the values of the independent parameters are determined bythe condition that the low–energy, long–wavelength spectrumof the lattice Hamiltonian (21) be identical with the spectrumof the Luttinger model. Explicitly, we have ti = γih

2/m0a2,

i = 1, 2, 3, where a is the lattice constant. For a hole–dopedthin film with structural inversion asymmetry, the Rashba–type spin–orbit coupling is modeled phenomenologically byadding a term similar to HSOI in Eq. (18), but for spin 3/2,i.e., with σ → J . This SOI term induces a splitting of thetop valence band (the heavy hole band) that is proportional tok3 in the limit k → 0. This is contrast with the linear split-ting that characterizes the Rashba splitting of the conductionband modeled by the Hamiltonian HSM from Eq. (18). Notethat for both the 2–band model (18) and the 4–band Luttinger–type model, the effective Rashba coefficient α is an indepen-dent parameter that does not depend on the confining potentialV (i). Typically, in the numerical calculations V (i) is a hard–wall potential ( zero inside a finite region and infinite other-wise) that does not beak the structural inversion symmetry.

Expanding the basis (20) to include the eigenstates of thetotal angular momentum momentum J with j = 1/2 (corre-sponding to the split–off band), as well as s–type states (cor-responding to the conduction band), allows to construct an 8–band tight–binding model for both electron–doped and hole–doped SMs that is equivalent in the long–wavelength limitto the 8–band Kane model [133]. There are two key differ-ences between this model and the simpler models describedabove. First, the Rashba–type spin–orbit coupling in the 8–

Ener

gy

(e

V)

L X k (1/A)

LZ=50nm

FIG. 6. (Color online) Comparison between the conduction bandspectra of a 50nm InSb film obtained using the 8–band (blue) andthe 2–band (yellow) models. Note that both models predict thesame value of the effective mass for the lowest energy confinement–induced sub–band, but the effective masses of higher energy sub–bands, as well as the values of the energy gaps between sub–bands,are significantly different.

band model is determined by implicitly by the asymmetricconfining potential V (i) and does not involve any additionalindependent parameter. Consequently, this model capturesthe correlation between the strength of the spin–orbit cou-pling and the transverse profile of the wave function. In turn,this profile plays a key role in the SC proximity effect, as wewill show in section III B. By contrast, these correlations arenot captured by the simplified models. The second key dif-ference becomes manifest when describing low–dimensionalnanostructures. In the large length scale limit, the 2–band and4–band models described above generate conduction and va-lence band spectra, respectively, that agree quantitatively withthe results obtained using the 8–band model. This is illus-trated in Fig. 5, which shows a comparison between the 3Dspectra obtained using these models in the absence of spin–orbit coupling. The large discrepancies at wave vectors awayfrom the Γ point suggest that for systems with reduced di-mensionality theses models will generate significantly differ-ent results. This behavior is illustrated in Fig. 6, which showsa comparison between the conduction band spectra of a SMfilm of thickness Lz = 50nm obtained using the 8–band andthe 2–band models. In essence, the spectrum of the 8–bandmodel is characterized by strong non–parabolicity and by asub–band–dependent effective mass. Also, the quasi–2D ef-fective mass predicted by the 8–band model is strongly de-pendent on the film thickness. These features cannot be re-produced by the simplified 2–band model. Moreover, similardiscrepancies characterize the valence bands obtained usingthe 8–band and 4–band models. All these differences becomeeven more pregnant in quasi–1D nanostructures. Nonetheless,since Majorana physics is mainly controlled by a relativelysmall number of low–energy states, the simplified models canprovide a reasonably accurate description of the semiconduc-tor system with a proper choice of effective model parame-ters and within a limited range for the control parameters, i.e.,chemical potential and Zeeman splitting.

An important aspect that has to be addressed when solving

13

numerically a specific model for the Majorana wire is rep-resented by the large number of degrees of freedom in theproblem. We emphasize that Majorana fermions are zero–energy bound states localized near the ends of a SM wire thatis proximity–coupled to a superconductor, hence we are inter-ested in modeling a finite quasi–1D system with no particularsymmetries, in the presence of disorder and external poten-tials. For a typical electron–doped SM nanowires modeledusing the 2–band model (18) the number of degrees of free-dom is of the order 107–109. One possible solution is to usea coarse–grained lattice model with an effective lattice con-stant aeff much larger the actual SM lattice constant. For ex-ample, choosing aeff = 40a reduces the number of degreesof freedom to about 103–104. However, while this is numer-ically convenient, it becomes difficult to address the short–range properties of the system, for example the effects of cer-tain types of disorder. More importantly, the properties as-sociated with Majorana physics are basically controlled by areduced number of low–energy degrees of freedom, hence itis more natural to project the Hamiltonian onto the relevantlow–energy sub–space than to perform an overall energy–independent reduction of the Hilbert space. In addition, thesimple models used in the calculations are expected to behighly inaccurate at high energy. The technical details associ-ated with the projection onto a low–energy subspace are pre-sented in Appendix B for the case of the 2–band tight–bindingmodel.

B. Proximity effect in semiconductornanowire–superconductor hybrid structures

A critical ingredient of any recipe for realizing Majoranafermions in a solid state hybrid structure is the proximity–induced superconductivity. In essence, the electrons from theSM nanowire acquire SC correlations by partly penetrating inthe nearby bulk s–wave superconductor. In conjunction withspin–orbit coupling, these correlations generate effective p–type induced superconductivity in the SM nanowire. Morespecifically, each double degenerate confinement–inducedband generates a combination of px + ipy and px − ipy su-perconductivity. Further, the Zeeman splitting breaks time–reversal symmetry and selects one of these two combinations,making the SM nanowire a direct physical realization of Ki-taev’s toy model for a 1D spinless p–wave superconductor[23]. While induced pairing is the most prominent aspect ofthe proximity to the bulk superconductor, another effect is therenormalization of the energy scale for the nanowire. Quali-tatively, this renormalization can be understood as resultingfrom the reduction of the quasiparticle weight of the low–energy SM states due to the partial penetration of the cor-responding wave functions into the SC. As a consequence,the low–energy effective model for the nanowire will con-tain rescaled values of the original parameters (e.g., hoppingparameters, spin–orbit couplings, Zeeman field, etc.), as weshow below in section III C.

To address quantitative aspects of the SC proximity ef-fect, one has to consider specific models for the relevant

terms in the total Hamiltonian (15): HSC, which describes s–wave bulk superconductor, andHSM−SC, representing the SMnanowire–superconductor coupling. At this point, one possi-ble approach is to treat the pairing problem self–consistently[134–136] in order to account for spatial variations of the or-der parameter near the interface and effects due to interactionsinside the nanowire [137–140] and the presence of a mag-netic field. In this work, we will not discuss this aspect ofthe proximity effect and will model the bulk semiconductor atthe mean–field level using a simple tight–binding Hamiltoniancharacterized by a constant pairing amplitude ∆0. Explicitly,we have

HSC =∑i,j,σ

(tscij − µscδij

)a†iσajσ+∆0

∑i

(a†i↑a†i↓+ai↓ai↑),

(22)where i and j label SC lattice sites, a†iσ is the creation operatorcorresponding to a single–particle state with spin σ localizednear site i, and µsc is the chemical potential. Similarly, wecan write the SM–SC coupling as

HSM−SC =∑i0,j0

∑m,σ

[tmσi0j0c

†i0m

aj0σ + h.c.], (23)

where i0 = (ix, iy, i0z) and j0 = (jx, jy, j0z) label latticesites near the interface in the SM and SC regions, respec-tively, m is the quantum number that labels the SM states(e.g., if the SM is described by a 2–band model m ≡ σ), andtmσi0j0 are coupling matrix elements between SM and SC localstates. Since we are interested in the low–energy physics ofthe SM nanowire, it is convenient to integrate out the SC de-grees of freedom and define an effective action for the wire.In the Green function formalism, this amounts to includingof a surface self–energy contribution in the SM Green func-tion [54, 87, 141]. The basic structure of this self–energy con-tribution is illustrated by the local term

Σi0i0(ω) = −|t|2νF

[ω + ∆0σyτy√

∆20 − ω2

+ ζτz

], (24)

where |t| represents a measure of the SM–SC coupling, νFis the surface density of states of the SC metal at the Fermienergy, σλ and τλ are Pauli matrices associated with thespin and Nambu spaces, respectively, and ζ is a proximity–induced shift of the chemical potential. Note that 24 con-tains an anomalous term proportional to τy that describes theproximity–induced pairing. Also, the diagonal term that islinear in frequency in the limit ω → 0 is responsible for thereduced quasiparticle weight and the corresponding energyrenormalization. Since the basic aspects have been addressedby several authors [54, 81, 87, 141–144], we focus here ontwo problems that are critical to understanding the proximityeffect in real SM–SC hybrid structures: i) the role of multi-band physics, and ii) the dependence on specific features ofthe SM–SC coupling matrix elements.

Proximity effect in multiband nanowires

The key feature that differentiates the multiband case fromits single band counterpart is the emergence of proximity–

14

induced inter–band pairing. In addition, the proximity–induced renormalization of the SM energy scales has a ma-trix structure, rather than being described by an overall factor.To illustrate these features, we consider an electron–dopedSM nanowire with rectangular cross section and dimensionsLx � Ly ∼ LZ in contact with an s–wave SC. The nanowireis modeled using the 2–band model (18) written in the basisψnσ(i), as described by equations (B1–B6), and the SM–SCcoupling is given by Eq. (23). For simplicity, we assume lat-tice matching between the SM and the SC and nearest neigh-bor hopping across the interface, tmσi0j0 = t(iy)δi0+d j0 , whered = (0, 0, d) is a nearest–neighbor position vector. The de-pendence of t = t(iy) reflects the possibility of nonuniformSM–SC coupling across the nanowire, as in the recent exper-iments by Mourik et al. [74]. After integrating out the SCdegrees of freedom, the effective surface self–energy can bewritten in the spinor basis ψn = (ψn↑, ψn↓, ψn↑, ψn↓)

T as

Σnn′ =∑ix,iy

∑i′x,i

′y

ψTn(i0)t(iy)GSC(ω, j0, j′0)t(i′y)ψn′(i′0),

(25)where j0 = i0 + d and j′0 = i′0 + d and GSC is a matrixthat contains both normal and anomalous terns that representsthe Green function of the superconductor at the interface. Wenote that there are three sources of proximity–induced inter–band coupling in Eq. (25): i) the position dependence ofGSC ,ii) the non–vanishing coupling between states with arbitraryvalues of nz and n′z , and iii) the position dependence of t.

The first source has not yet been explored, but could havesignificant effects when the SC itself has small character-istic length scales. To account for these effects, the SCGreen function has to be calculated explicitly by taking intoaccount all relevant details, including the size and geome-try of the system. On the other hand, assuming a largesuperconductor with a planar surface, the SC Green func-tion becomes GSC = GSC(ω, ix − i′x, iy − i′y) and canbe expressed [86, 87, 141] in terms of its Fourier transformGSC(ω,k||) ≈ −νF [(ω+∆0σyτy)/

√∆2

0 − ω2+ζτz]. In thiscase, since the dependence of GSC(ω,k||) on the in–planewave–vector k|| is very weak [9, 86, 141], the surface self–energy contribution is practically local and does not representan additional source of inter–band coupling.

The second mechanism, which couples differentconfinement–induced nz bands, is due to the presenceof the interface and and does require any in–plane spatial in-homogeneity. Qualitatively, this coupling can be understoodin terms of virtual processes in which a electron occupyinga state from the nz band tunnels into the superconductor,then returns to the SM wire into a state from a differentband, n′z . Assuming uniform hopping t across the interface, the effective coupling due to such processes depends onthe values of the wave function at the interface and canbe expressed as γnzn′

z= νF |t|2φnz (i0z)φn′

z(i0z), where

φnz is given by Eq. (B1). We note that the effect of thisinter–band coupling becomes negligible in the limit of strongconfinement, when γnzn′

zis much smaller than the inter–band

gap.The third source of proximity–induced inter–band coupling

is due to non–homogeneous interface hopping, t = t(iy). Thismechanism is discussed in detail in Ref. [86]. The effectiveSM–SC coupling of a SM nanowire with rectangular crosssection and nonuniform interface tunneling can be written as

γnn′ = δnxn′xνFφnz (i0z)φn′

z(i0z)

Ny∑iy=1

|t(iy)|2φny (iy)φn′y(iy).

(26)Neglecting the spatial dependence of the SC Green functionat the interface, the proximity–induced effective self–energybecomes

Σnn′ = −γnn′

[ω + ∆0σyτy√

∆20 − ω2

+ ζτz

], (27)

where γnn′ is given by Eq. (26) and ζ is a constant that de-pends on the details of the SC band structure. Note that Eq.(27) is valid for frequencies inside the SC gap, |ω| < ∆0.Also, we emphasize that the discussion leading to Eq. (27)was based on the 2–band tight–binding model for the SMwire, which is constructed s–type localized orbitals using asa basis, with spin being the only internal degree of free-dom. In more complex models, the local states are labeledby a quantum number m 6= σ, or by a set of quantum num-bers. The hopping matrix elements across the interface, whichparametrize the coupling Hamiltonian (23), depend explic-itly on these quantum numbers and, in turn, the proximity–induced surface self–energy will depend on the details of thiscoupling. An example illustrating the the proximity effect forhole–doped SM nanowires is given in Appendix C.

C. Effective low–energy Bogoliubov–de Gennes Hamiltonianfor the Majorana wire

In the presence of proximity–induced superconductivity,the low–energy physics of the SM nanowire is described bythe Green function matrix Gnn′(ω) that includes nonzeroanomalous terms. Using the results of sections III A and III B,one can write the inverse of the Green function matrix as

[G−1]nn′(ω) = ω −Hnn′ − Σnn′(ω), (28)

where Hnn′ is the effective low–energy Hamiltonian forthe SM nanowire in the Nambu space and Σnn′(ω) is theproximity–induced self–energy. For an electron–doped wiredescribed using the 2–band tight–binding model, the self–energy is given by Eq. (27), while the low–energy Hamil-tonian can be obtained by expanding Eq. (B6) to include bothparticle and hole sectors. Explicitly, we have

Hnn′ = [εn + Γσx] τzδnn′

+ iαδnzn′z

[qnxn′

xσyδnyn′

y− qnyn′

yσxτzδnxn′

x

], (29)

where the relevant quantities are the same as in Eq. (B6), τzis a Pauli matrix in the Nambu space, and the identity ma-trices in the spin and Nambu spaces, σ0 and τ0, respectively,

15

have been omitted for simplicity. The eigenvalues of the low–energy states can be obtained by solving the Bogoliubov–deGennes (BdG) equation

det[G−1(ω)] = 0, (30)

where the frequency is restricted to values inside the bulk SCgap, |ω| < ∆0, and the Green function is given by equations(28), (27), and (29).

Solving Eq. (30) numerically can be rather demanding, butone can further simplify the problem by noticing that the rele-vant energy scale for majorana physics (e.g., the induced pairpotential ∆) is typically much smaller than the bulk SC gap∆0. Consequently, one can focus on the the low–frequencylimit |ω| � ∆0 and consider the self–energy within the staticapproximation

√∆2

0 − ω2 ≈ ∆0. In the static approxima-tion, Eq. (28) becomes [G−1]nn′ = ωQnn′ − Hnn′ +γnn′(σyτy + ζ/∆0τz), where Qnn′ = δnn′ + γnn′/∆0 ac-counts for the proximity–induced normalization of the SM en-ergy scales. Since Qnn′ is a positive definite matrix, the BdGequation (30) can be rewritten in the static approximation asdet[ω − Heff ] = 0, where Heff is a frequency–independentquantity that can be viewed as an effective BdG Hamiltonianfor the SM nanowire with proximity–induced superconductiv-ity. Explicitly, the effective Hamiltonian has the form

Heffnn′ = ZnmHmm′Zm′n′ −∆nn′σyτy − δµnn′τz. (31)

The first term in the right hand side of Eq. (31) representsthe SM Hamiltonian renormalized by the proximity effect.Note that, in general, this proximity–induced renormalizationis described by a matrix, rather than an overall factor Z repre-senting the reduced quasiparticle weight. The renormalizationmatrix is the solution of the equation

Znm

(δmm′ +

γmm′

∆0

)Zm′n′ = δnn′ , (32)

where γmm′ is the effective SM–SC coupling matrix. In thesingle–band limit, we have Z2 ≡ Z = (1 + γ/∆0)−1. Notethat the renormalized Hamiltonian is written in the symmetricform ZHZ, rather than ZH , to ensure its hermiticity in thegeneral, multi–band case. The second term Eq. (31) containsthe proximity–induced pairing matrix

∆nn′ = Znmγmm′Zm′n′ . (33)

In the single–band limit, this reduces to the induced SC pair–potential ∆ = γ∆0/(γ+∆0). Finally, the last term in (31) de-scribes proximity–induced energy shifts and inter–band cou-plings that have only a rather limited quantitative relevance,δµnn′ = ζ∆nn′/∆0.

We emphasize here a key aspect of the proximity ef-fect in multi–band SM-SC hybrid structure: the proximity–induced inter–band coupling and the induced inter–band pair-ing. These effects depend on both the details of the SM–SCcoupling at the interface and the dimensions of the nanosys-tem. More specifically, considering a SM wire with rect-angular cross section Ly × Lz , the inter–band effects be-come critical whenever the gaps ∆Enn′ separating differ-ent confinement–induced bands n = (ny, nz) are compara-ble with the bulk SC gap ∆0. This situation occurs in wide

wires and in the “sweet spot” regime [85, 86], when two spinsub–band become degenerate at finite Zeeman field. By con-trast, in the limit ∆Enn′ � ∆0 the inter–band effects arenegligible and one can treat the proximity effect in the de-coupled band approximation, i.e., γnn′ ≈ 0 if n 6= n′. Inthis approximation, the matrices describing the proximity ef-fect become diagonal, e.g., γnn′ = γnδnn′ , and we haveZn = 1/

√1 + γn/∆0 and ∆n = γn∆0/(γn + ∆0). Also,

we note that there is a key difference between the proximity–induced inter–band effects associated with confinement in thedirections normal and parallel to the interface, respectively.Specifically, assuming nz = n′z , i.e., strong confinement inthe normal direction (so that the low–energy physics is con-trolled by a single nz mode), and a weaker confinement inthe y–direction, along with with non–uniform SM–SC hop-ping t(iy), results in an effective SM–SC coupling γnn′ =γnyn′

ythat contains non vanishing off–diagonal terms. How-

ever, for typical SM–SC couplings, γnyn′y

becomes negligi-ble when the difference |ny − n′y| is large, as one can in-fer from Eq. (26), and only neighboring ny bands are sig-nificantly coupled. Consequently, when constructing the ef-fective Hamiltonian (31) it is enough to consider a reducednumber of ny bands to obtain an accurate description of thelow–energy physics. This scenario was investigated in de-tail in Ref. [86]. On the other hand, when the confinementin the z–direction is weak, bands with arbitrary nz becomecoupled, as γnzn′

z∝ φnz (i0z)φn′

z(i0z). In this case, it is crit-

ical to include the high–energy nz bands in the calculation,as they re–normalize the low–energy physics via proximity–induced virtual processes. The role of the proximity–inducedinter–band coupling is studied in detail in Ref. [145]. It isfound that the proximity induced gap is strongly suppressedin the intermediate and strong tunnel coupling regimes when-ever the SM thickness exceeds a characteristic crossover valuedetermined by the band parameters of the SM [145]. Further-more, the strong coupling regime is characterized by a smallinduced gap that decreases weakly with the SM–SC couplingstrength [145], in sharp contrast with expectations based onthe decoupled band approximation, which predicts an inducedgap of the order of the bulk SC gap ∆0. An example of low–energy spectrum for a SM thin film – SC slab heterostructurewith uniform coupling across a planar interface illustrating thetheoretical scheme described above is presented in AppendixD.

D. Low–energy physics in Majorana wires with disorder

There are several types of disorder that may play a role inthe low–energy physics of semiconductor–supraconductor hy-brid structures [86]: impurities inside the bulk s-wave SC, dis-order in the SM nanowire, and disorder induced by randomSM–SC coupling. Below, we mention a few representativesources that could generate these types of disorder and brieflydiscuss some of their main features. As a general remark, wepoint out that the theoretical treatment of disorder in semi-conductor Majorana wires can be done along two differentdirections: calculations of disorder–averaged quantities and

16

calculations involving specific disorder realizations. How-ever, the low–energy physics of the nanowire (on a scale ofthe order of the induced gap) is controlled by a small numberof quantum states, typically less than 100 (see, for example,Figs. 18 and 19, upper panels, in Appendix E) having mostof their spectral weight inside the SM wire. When disorderis located in the nanowire or at the interface, the energies andthe wave functions characterizing these states depend signifi-cantly on the specific disorder realization and the relevance ofthe disorder–averaged quantities is questionable. The reasonfor this dependence can be understood qualitatively by notic-ing that features of the disorder potential with length scalessmaller than 1/kF , where kF is the Fermi wave vector of theMajorana band, are irrelevant, since they are averaged overby the low–energy SM states. Typically, 1/kF is of the order102nm, hence the effective disorder potential in a nanowireof length Lx ∼ 1µm is characterized by a small number ofscattering centers and, consequently, the details of the disor-der potential become relevant. Disorder–averaged quantitiesprovide a good description of the low-energy physics (for ex-ample, the value of the induced SC gap) in long wires [144],but do not capture the specific properties of a given small seg-ment of that wire, or those of a short wire of similar length.

Scattering off impurities inside a disordered bulk supercon-ductor has a negligible effect on the topological SC phase ofthe nanowire [146, 147]. In essence, this behavior is due thefact that the SM effective impurity scattering rate involveshigher–order SM–SC tunneling processes and is suppressedby the destructive quantum interference of quasi–particle andquasi–hole trajectories [146]. Consequently, static disorder inthe superconductor does not suppress the proximity inducedtopological superconductivity in the semiconductor.

By contrast, disorder in the SM nanowire or at the SM–SCinterface can strongly affect the stability of the topologicalSC phase. Some of the generic features of the low–energystates in the presence of disorder are illustrated in AppendixE. Possible sources of disorder are the random variations ofthe width of the SM wire and, in general, surface roughness,and random potentials created by charged impurities locatedon or near the surface of the wire [86]. Another possiblesource of disorder is represented by the random coupling atthe semiconductor–superconductor interface [86, 148]. Wenote that this random coupling has a twofold manifestation: afluctuating induced pairing potential and a random proximity–induced renormalization of the SM Hamiltonian (see Sec.III C). In general, the suppression of topological superconduc-tivity by disorder represents a serious challenge to the exper-imental realization of MFs. A recent proposal for optimiz-ing the stability of the topological phase against disorder in-volves replacing the semiconductor wire by a chain of quan-tum dots connected by s–wave superconductors [149]. In ad-dition to the adverse effect on the stability of the topologicalSC phase, the presence of disorder impacts the low–energyphysics [150] of the SM nanowire–SC hybrid structures in anumber of other ways, some of them being investigated theo-retically in several recent studies. For example, it was shownthat the topological quantum phase transition is characterizedby a quantized thermal conductance and electrical shot noise

power that are independent of the degree of disorder [151].The robustness of the topological phase against disorder wasshown to depend non-monotonically on the Zeeman field ap-plied to the wire [152]. The interplay between disorder and in-teraction in one-dimensional topological superconductors wasaddressed in Ref. [153]. Also, it was shown that in systemswith disorder located at the end of the wire the weight of thecharacteristic Majorana–induced ZBCP in the differential tun-neling conductance is strongly enhanced by mixing of sub–bands [154]. On the other hand, the presence of disorder cangenerate a ZBCP even when the superconducting wire is topo-logically trivial due to the proliferation of disorder–inducedlow–energy states [111, 155], or due to the weak antilocal-ization resulting from random quantum interference by dis-order [156]. A similar ZBCP can occur in the topologicallytrivial phase as a result of fermionic end states with exponen-tially small energy that emerge if the confinement potential atthe end of the wire is smooth [110]. In the light of the recentexperiments on semiconductor nanowire–superconductor hy-brid structures, ruling out the alternative mechanisms for theconductance peak represents a serious challenge.

IV. RECENT EXPERIMENTS AND THEORETICALINTERPRETATIONS

In this section we briefly summarize several recent obser-vations of experimental signatures consistent with the realiza-tion of Majorana bound states in semiconductor nanowire–superconductor structures. We also discuss a number of ap-parent discrepancies between the observed features and thetheoretical predictions based on simple model calculations forthe Majorana wire.

A. Experimental signatures of Majorana fermions in hybridsuperconductor–semiconductor nanowire devices

The observation of a zero bias conductance peak in lo-cal tunneling conductance measurements on semiconductornanowires coupled to an s-wave superconductor has been re-cently reported in Ref. [74]. This observation, which is con-sistent with the theoretical predictions, may represent the firstexperimental evidence of Majorana fermions in a condensedmatter system. Soon after, observations of similar ZBCPshave been reported by two other groups [75, 76] and, recently,by two more groups [77, 78]. Since there are significant dif-ferences among the experimental setups, establishing conclu-sively that the ZBCPs observed in different systems are dueto the same mechanism remains a critical open question. Be-low, we briefly summarize the main results of Ref. [74]. Wemention that a measurement of the fractional a.c. Josephsoneffect has also been reported in Ref. [79]. The observationsuggests the presence of a Shapiro step with a height twicelarger than the value expected for conventional superconduc-tor junctions [79], which would be consistent with the pres-ence of MFs.

As discussed in the previous sections, an optimal hybrid

17

FIG. 7. (Color online) Top panel: Schematic representation of theexperimental setup showing a cross–section through the device mea-sured in Ref. [74]. Bottom panel: Differential conductance dI/dVversus bias potential V at a temperature T = 70mK and for differentvalues of the magnetic field from 0 to 490 mT in of 10 mT. Tracesare offset for clarity. Figure adapted from Ref. [74].

system capable of hosting Majorana bound states and of mea-suring the expected ZBCP should posses certain characteris-tics, such as i) a large SC gap, to protect the MF and to allowtuning the Zeeman field in a wide range without destroyingthe superconducting phase, ii) a long–enough wire length, toaccommodate Majorana bound states at the wire ends withoutlarge overlap, iii) low disorder, iv) the ability to control thechemical potential, and v) the ability to tunnel from a normallead, while minimizing the perturbation on the topological SCstate. The basic experimental setup for the measurements re-ported in Ref. [74] consists of a InSb nanowire in contact witha superconductor (NbTiN) and a metallic (Au) lead, as shownschematically in the top panel of Fig. 7. To measure the differ-ential conductance dI/dV , a bias voltage is applied betweenthe normal lead and the superconductor (SC) and a tunnel bar-rier is created in the region between the metallic lead and theSC segment of the nanowire by applying a negative voltage toa narrow gate. The InSb nanowire is characterized by a highg-factor (g ≈ 50), as well as a strong spin–orbit (SO) cou-pling (the Rashba parameter is α ≈ 0.2eV·A), making it agood candidate for the realization of MFs in the presence ofproximity–induced superconductivity. The barrier potentialsuppresses the background conductance due to Andreev pre-cesses and helps to reveal the ZBCP generated by the possiblepresence of MFs that could emerge for values of the magneticfield above a certain critical value. The dependence of the dif-ferential tunneling conductance on the bias voltage for differ-ent values of the magnetic field is shown in the bottom panelof Fig. 7. Note that a ZBCP emerges when the applied mag-netic field exceeds about 100 mT. The ZBCP is robust to fur-

FIG. 8. (Color online) Dependence of dI/dV on the magnetic fieldorientation. (A) Rotation in the plane of the substrate for a magneticfield |B| = 200 mT. Note that the ZBCP vanishes whenB is perpen-dicular to the wire, i.e., parallel to the spin–orbit field. (B) Rotationin the plane perpendicular to the spin–orbit field for |B| = 150 mT.Note that a ZBCP is present for all angles. The top panels showline–cuts at angles marked with corresponding colors in (A) and (B).Figure adapted from Ref. [74].

ther increasing the magnetic field until about 400 mT, abovewhich the single peak seems to split into a two–peak structure.These observations are consistent with the MF interpretationof the tunneling conductance, as shown, for example, by thetheoretical predictions illustrated in Fig. 4.

An important consistency check on the MF interpretation ofthe ZBCP is the requirement that the peak must disappear ifthe angle between the applied magnetic field and the directionof the effective spin–orbit field vanishes. The likely orienta-tion of the spin–orbit field is along the direction perpendicularto the wire in a plane that is roughly parallel to the nanowire–SC and nanowire–substrate interfaces. The angle dependenceof the ZBCP was measured in Ref. [74] (see Fig. 8) and foundto be consistent with the Majorana scenario, thus strengthen-ing the identification of the observed ZBCP with the presenceof a MF bound state at the normal lead–nanowire interface.

The main observations reported in Ref [74], in particu-lar the emergence of a ZBCP at finite magnetic field, havebeen subsequently confirmed by other groups [75, 76]. Thesimilarities are rather surprising, considering that many ofthe relevant parameters that characterize the devices mea-sured in these experiments are very different from those ofRef [74]. For example, the experiments reported in Ref. 76use InAs nanowires in proximity to superconducting Al (in-stead of InSb and NbTiN). More importantly, the segment ofthe nanowire that is proximity coupled to the superconductoris very short (approximately 150−200nm, which is compara-ble to the SC coherence length). Is it possible to realize Majo-rana bound states in such a short nanowire? A possible answeris provided in Ref. 157, which shows that, in a short wire, the

18

lowest energy state represents a pseudo–Majorana mode thatevolves continuously into a true zero–energy Majorana modeas the wire length increases.

In the recently reported experiments, there are several strik-ing features, the most prominent being the absence of a sig-nature associated with the closing of the quasiparticle gap atthe topological quantum phase transition (TQPT) that sepa-rates the topologically trivial and nontrivial SC phases. The-oretically, this vanishing of the gap at the critical magneticfield must precede the emergence of the Majorana–inducedZBCP [11, 54, 86]. This can be clearly seen in the simulationshown in Fig. 4, but is absent in the experimentally measuredtunneling conductance (see Fig. 8). Another striking featureis the “soft” nature of the induced SC gap, as consistentlyrevealed by the differential conductance measured in the re-cent experiments [74–76]. Furthermore, the observed ZBCPis more than an order of magnitude weaker than the quantizedvalue of 2e2/h predicted theoretically. Subsequent theoreti-cal work [112] has attributed this discrepancy to a finite tem-perature effect in conjunction with a hybridization–inducedsplitting of the Majorana mode in finite wires. In additionto these unexpected experimental features, a number of alter-native scenarios for the emergence of ZBCPs, which do notinvolve the presence of MFs but more conventional mecha-nisms involving strong disorder [111, 156, 158], smooth con-finement [110], or Kondo physics [159], have been recentlyproposed theoretically, making this problem an exciting but,at the same time, a rather fluid and confusing subject. Below,we discuss several theoretical proposals that provide possi-ble explanations for the experimentally observed features andsuggest further tests that could better reveal the nature of theZBCPs observed in charge transport measurements on semi-conductor nanowire–superconductor structures. We empha-size that the recent developments in this field reveal that keyobservable features are strongly dependent on various detailsof the system. Significant understanding of the relevant phys-ical mechanisms responsible for these features, at least at aqualitative level, can be achieved by realistically modeling thesemiconductor–superconductor structures.

B. Suppression of the gap–closing signature at the topologicalquantum phase transition

While the observations [74] of a ZBCP in conductancemeasurements on semiconductor nanowires coupled to super-conductors may represent the first experimental evidence ofMFs, the absence of any signature associated with the closingof the superconducting gap (see Fig. 7) at the critical mag-netic field asscociated with the topological quantum phasetransition (TQPT) casts serious doubt on the MF interpreta-tion of the ZBCP. The closing of the quasiparticle gap at theTQPT is a fundamental theoretical requirement. The questionis whether or not this gap closure has a visible signature whenthe system is probed experimentally. Ref. [160] has offereda possible explanation for the observed non–closure of thegap at the quantum phase transition between the trivial super-conductor and the topological superconductor supporting the

MFs in the end–of–wire tunneling experiments. By solvingnumerically an effective tight–binding model for multibandnanowires with realistic parameters, it has been shown [160]that, in the vicinity of the topological transition, the amplitudeof the low–energy states at positions near the ends of the wiremay be orders of magnitude smaller than the amplitudes of thelocalized MFs. Consequently, if the chemical potential of thesystem is located near the bottom of a confinement–inducedsemiconductor band, the contributions of these low–energystates to the end–of–wire local density of states (LDOS), andhence to the tunneling conductance, are essentially invisible(see Fig. 9, middle panel). This behavior results in an appar-ent non–closure of the gap, as reflected by these local quan-tities, even though the magnetic field approaches the criticalvalue and the system is driven through a TQPT. By contrast,the closing of the gap mandated by the topological transitionis clearly revealed by other quantities, such as the total den-sity of states (DOS) (see Fig. 9, top panel) and the LDOSnear the middle of the wire (Fig. 9, bottom panel). A defi-nite prediction of Ref. [160] is that a tunneling measurementnear the middle of the wire should clearly reveal the closingof the gap at the critical field Γc, but the zero–bias peak asso-ciated with the presence of Majorana bound states should beabsent. Correlated with an end–of–wire measurement charac-terized by a zero–bias peak for Γ > Γc, this would constitute apowerful argument for the presence of zero–energy Majoranabound states. In addition, the model calculations have shownthat the non–closure of the gap, as revealed by the end–of–wire LDOS, is a non–universal phenomenon. For example, ifa local measurement is performed in a regime characterizedby a value of chemical potential that is not close to the bot-tom of a semiconductor band (i.e., ∆µ � ∆), the closing ofthe gap should be visible. This regime is illustrated in Fig. 4.Finally, we note that the dominant feature in the end–of–wireLDOS (see Fig. 9, middle panel) are due to states associatedwith low–energy occupied bands that have significant ampli-tudes near the ends of the wire. In certain conditions (e.g.,in the presence of disorder or in systems with smooth con-finement) these states can have energies lower than the bulkquasiparticle gap and, consequently, can provide substantialcontributions to the in–gap LDOS.

C. Discriminating between Majorana bound states and gardenvariety low–energy states

Despite their conceptual simplicity, the zero bias conduc-tance peak experiments do not constitute a sufficient prooffor the existence of Majorana bound states in semiconductor–superconductor hybrid structures. In a recent theoreticalwork [110], it has been shown that a non–quantized nearlyzero bias peak, such as that observed in the recent experi-ments [74–76], can arise even without end state MFs, pro-vided the confinement potential at the wire end is smooth. Anon–quantized nearly zero energy peak at the wire ends hasalso been shown to occur due to strong disorder effects [111],even when the nanowire is in the topologically trivial phase.In essence, these nearly–zero bias peaks occur at finite val-

19

B / Bc

Ener

gy (m

eV)

Ener

gy

(m

eV)

Ener

gy

(m

eV)

LDOS (x=L/2)

LDOS (x=0)

DOS

FIG. 9. (Color online) Top: Density of states (DOS) as a functionof B for a wire with four partially occupied bands (seven spin sub–bands) and a chemical potential near the bottom of the fourth band(for B = 0). At the TQPT characterized by Bc ≈ 0.2T the bulk gapcloses. Middle: Local density of states (LDOS) at the end of the wireas a function of magnetic field. The strong finite energy features havea weak dependence on B. For B > Bc the Majorana peak is clealyvisible. Note that there is no visible signature of the bulk gap closingat the TQPT. Bottom: LDOS at the middle of the wire. Note theclosure of the gap at the TQPT and the absence of the zero–energyMajorana peak. Figure adapted from Ref. [160].

ues of the Zeeman field due to the proliferation of low–energystates in the presence of disorder (see Sec. III D) or in wireswith smooth confinement (seeSec. IV D) . To discriminate be-tween different possible mechanisms responsible for the zerobias conductance occurring in a tunneling experiment, a di-agnostic signature for the Majorana–induced ZBCP has beenrecently proposed [161]. It was shown that, for smooth con-finement at the ends of the wire, the emergence of the nearZBCPs is necessarily accompanied by a signature similar tothe closing of a gap in the end–of–wire local density of state.This signature occurs even though there is no correspondingquantum phase transition, as the system stays in the topolog-ically trivial phase, and traces the Zeeman field dependenceof the nearly–zero energy states. In the absence of such a gapclosing signature, a ZBCP is unlikely to result from the softconfinement effect [110]. Similarly, when the ZBCP appearsat the wire ends from disorder effects (but without MFs), theemergence of the zero conductance is preceded by a signaturesimilar to the closing of the gap [111, 161]. So far, amongall the scenarios that have been considered, the topologicalphase transition scenario involving the emergence of the MFs

is the only one consistent with a ZBCP that occurs beyond acertain critical magnetic field and the apparent non–closureof the quasiparticle gap before the emergence of the ZBCP.Since this is precisely what is observed in the experiments inRef. [74], these theoretical results strengthen the identificationof the observed ZBCP with topological Majorana bound stateslocalized at the ends of the wire.

We note that the presence of low–energy sub–gap states insystems with disorder and smooth confinement may have an-other important consequence with dramatic experimentally–observable implications: the existence of a soft superconduct-ing gap, i.e., a SC gap characterized by a non–vanishing den-sity of states and a v–shaped sub–gap tunneling conductanceeven at very low temperatures. This feature is manifestlypresent in the experimental data [74] (see, for example, Fig. 7)and represents one of the most important open issues in thisfield. There are several critical aspects that need to be clar-ified, such as the origin of the in–gap spectral weight, whythese in–gap contributions occur as a smooth background,rather than sharply defined peaks in the LDOS, and whetheror not a Majorana–induced ZBCP can be well–defined in asystem with soft gap. Recently, disorder induced by interfacefluctuations was identified [162] as the likely source of thein–gap states responsible for the soft gap. Another theoreti-cal work [161] has shown that there are two key ingredientsthat may explain the emergence of the soft gap in weakly con-fined wires: i) A finite potential barrier allows states with largespectral weight near the end to hybridize with metallic statesfrom the leads. ii) States from lower–energy bands can pene-trate through the barrier, hybridize strongly with the metallicstates, and generate broad contribution to the LDOS. By con-trast, the Majorana mode, which is associated with the topoccupied band, couples weakly to the lead, hence it is weaklybroadened and still generates a well defined ZBCP on top ofthe smooth background.

D. Majorana physics in finite–size wires: A “smoking gun” forthe existence of the Majorana mode

It has recently been proposed [129] that direct observationof the splitting of the zero bias conductance peak could serveas a “smoking gun” evidence for the existence of the Majo-rana mode. In essence, the Majorana bound states come al-ways in pairs [23] that are localized near the ends of the wire,if the system is clean enough. In any finite wire, the wavefunctions of the two Majoranas overlap, leading to a split-ting [112, 157, 163, 164] of the Majorana mode, which isa pure zero–energy mode only in the infinite wire limit. Atfixed chemical potential, this hybridization–induced energysplitting is characterized by an oscillatory behavior that de-pends on the Fermi wave vector of the top occupied band andthe length of the wire. By contrast, when the particle den-sity is constant, the oscillations can be suppressed [129], butthe splitting of the Majorana mode is still a generic feature.Furthermore, regardless of conditions, two independent tun-neling measurements at the opposite ends of a wire shouldobserve exactly the same splitting of the ZBCP [129], as long

20

DV=2meV

a V

(x)

(meV

) |Y

(x)|

2 (

a.u.)

x (mm)

1 3 5

2

b

c

d

FIG. 10. (Color online) Semiconductor nanowire with four partiallyoccupied bands in the presence of a nonuniform potential V (x). Theposition-dependence of V is shown in panel (a). The profiles of sev-eral low–energy states Ψn with |E1| ≤ |E2| ≤ . . . correspond-ing to different values of the Zeeman field are shown in panels (b)Γ = 0.25meV (trivial SC phase) , (c) Γ = 0.4meV (topological SCphase), and (d) Γ = 0.8meV. Note that the wire is effectively sep-arated into two segments for states with a small characteristic wavevector. States corresponding to large kF values penetrate throughthe potential barrier ∆V . With increasing Zeeman field, low–energystates associater with the low–energy bands emerge because of thesmooth confinement, e.g., the state n = 3 (green line) in panel (d).

as: i) the peak is due to Majorana splitting, and ii) a singlepair of Majorana bound states exists in the system. We em-phasize that features other than the splitting itself observedin local measurements at the two ends of the wire may becompletely different. If confirmed experimentally, this wouldconstitute strong evidence for the existence of the elusive Ma-jorana mode in semiconductor–superconductor structures.

In the presence of disorder, several states with energieslower than the induced SC gap will emerge (see Sec. III D).The energies of these sub–gap states depend on the strengthof the disorder and the values of the Zeeman field. How-ever, the Majorana mode is still protected, as long as the dis-order strength does not exceed a certain critical value, and,consequently, the hallmark signature described above shouldstill be observable. Nonetheless, strong disorder can effec-tively cut the nanowire into two or more segments that caneach host a pair of Majorana bound states. A natural ques-tion concerns the fate of the splitting oscillations generated bythese multiple Majorana bound states that emerge in the pres-ence of strong scattering centers. To address this question,we consider we consider a wire with rectangular cross sec-tion Ly × Lz = 100 × 40 nm, four partially occupied bands,and smooth confinement in the presence of a strong scatteringcenter consisting in a potential barrier of hight ∆V located atdistances Lx1 ≈ 0.6µm and Lx2 ≈ 1.2µm from the two endsof the wire, respectively. The confinement and the barrier areprovided by the position–dependent potential V (x) shown in

DV=1meV

a

V(x

) (m

eV)

|Y(x

)| 2

(a.

u.)

x (mm)

1 3 5

2

b

c

d

FIG. 11. (Color online) Same as in Fig. 10, but for a lower value ofthe potential barrier ∆V . Note that, unlike Fig. 10, the lowest energystate (n = 1, red line), corresponding to two overlapping Majoranabound states, is peaked near the ends of the wire for Zeeman fieldsassociated with the topological SC phase.

the upper panels of Fig. 10 and Fig. 11. When Γ > Γc = 0.35meV and ∆V = 0, two Majorana bound states are localizednear the ends of the wire (i.e., x ≈ 0.2µm and x ≈ 2.2µm,respectively), while for large ∆V the wire is split into twodisconnected segments, each of them hosting a pair of Majo-ranas. The behavior of the low–energy states in the interme-diate regime is illustrated in figures 10 and 11. As a generalfeature, we note that states corresponding to small character-istic wave vectors (i.e., long wavelength oscillations) tend tobe contained in one of the two segments of the wire separatedby ∆V . By contrast, states with large characteristic wave vec-tors (rapid oscillations) can more easily penetrate through thefinite barrier. Furthermore, we note that Fermi k–vector kFassociated with the top occupied band increases with the Zee-man field, while the Fermi wave vectors corresponding to thelower energy bands are always larger than kF . Consequently,in the presence of a large barrier, the two pairs of Majoranasare characterized by a strong intra–pair hybridization and aweak inter–pair hybridization, which leads to two low–energymodes mostly localized inside the Lx1 and Lx2 segments, re-spectively. The corresponding wave functions (states n = 1 -red line - and n = 2 - yellow) are shown in Fig. 10. Reducing∆V (or increasing the Zeeman field) increases the inter–pairhybridization, which results in a lowest energy state with max-ima near the ends of the wire (see Fig. 11).

The hybridization–induced splitting oscillations of the Ma-jorana mode can be clearly seen in the field dependence ofthe density of states (DOS), as shown in Fig. 12. Wewant to emphasize two features. First, regardless of thenumber of coupled Majorana pairs, only one mode exhibitszero–energy crossings at discrete values of the Zeeman field,which is a characteristic signature of Majorana physics infinite nanowires[157]. By contrast, the other low–energymode is characterized by minima that vanish only in the limit

21

Ener

gy

(m

eV)

Ener

gy

(m

eV)

Zeeman field (meV)

FIG. 12. (Color online) Density of states as a function of the Zeemanfield in the presence of a nonuniform potential with ∆V = 1meV(top) and ∆V = 2meV (bottom). Note the closing of the bulk gap atΓc ≈ 0.35meV and the oscillations of the lowest energy modes forΓ > Γc. The corresponding LDOS at the opposite ends of the wireis shown in Fig. 13.

∆V → ∞, i.e., vanishing inter–pair tunneling. Second, theperiod and the amplitude of the oscillations induce by intra–pair hybridization increase with decreasing wire length ap-proximately as 1/Lx [129]. Since Lx2 ≈ 2Lx1, the ratiobetween the corresponding periods and amplitudes is approx-imately two, as evident from the lower panel of Fig. 12. Wenote that, in very short wires (quantum dots), the period of theoscillations may be large, so that practically only one zero–energy crossing is accessible [157]. Also, in such quantumdot–superconductor structures the Coulomb interaction is ex-pected to play an important role [165].

The characteristic signatures of the splitting oscillations ina set of two independent local measurements at the oppositeends of the wire, as reflected by the corresponding local den-sity of states (LDOS), are shown in Fig. 13. Several featuresneed to be emphasized. As discussed in Sec. IV B, the sup-pression of the gap closing signature stems from the spatialproperties of various low–energy states. In short wires, thereis no qualitative difference between extended states and stateslocalized near the wire ends, and consequently, a gap closingsignature should be visible. This signature should be strongerat the left end of the wire, since the segment Lx1 is shorter.However, the corresponding features are rather weak (see Fig.13) and could be very hard to resolve when on top of an in-coherent in–gap background, such as that responsible for thesoft gap observed experimentally [74–76]. In an actual exper-iment, the low–energy states couple to metallic states from theleads, while the states associated with low–energy bands areexpected to have significant weight outside the superconduct-ing region of the wire [161]. This leads to strong broadeningand could explain the the soft gap observed in the experimentswhile hiding all the weak features that are present in Fig. 13.Consequently, we expect the splitting oscillations measured atthe two ends of a wire containing a strong scattering center to

En

erg

y (m

eV)

En

erg

y (m

eV)

En

erg

y (m

eV)

En

erg

y (m

eV)

Zeeman field (meV)

L, DV=1

R, DV=1

L, DV=2

R, DV=2

FIG. 13. (Color online) Splitting oscillations of the ZBCP as re-flected by the LDOS at the left (L) and right (R) ends of the wire.The top two panels correspond to the parameters of Fig. 11, whilethe lower two panel are for the system illustrated in Fig. 10.

be uncorrelated, as shown by the dominant low–energy fea-tures corresponding to ∆V = 2meV in Fig. 13. By contrast,in the presence of a weak scattering center (∆V = 1meV inFig. 13), the dominant low–energy features reveal the sameZBCP slitting at both ends of the wire. However, the ampli-tude of the splitting is strongly reduced because of the largerLx and the oscillations may not be visible at finite energy res-olution. Therefore, we conclude that the optimal experimentalsetup for observing correlated Majorana–induced splitting os-cillations should use high quality wires with Lx < 1µm.

V. CONCLUSIONS

The emergence of a zero–bias conductance peak abovea certain critical value of the Zeeman field represents thenecessary condition for the existence of zero–energy Ma-jorana bound states in spin-orbit-coupled semiconductor-superconductor hybrid structures. Strong experimental evi-dence for signatures consistent with Majorana physics wererecently reported in charge transport measurements on suchnanowire heterostructures. These encouraging experimental

22

developments have also raised a number of rather unexpectedquestions that need to be clarified before claiming victory.Nonetheless, corroborating these observations by performinga “smoking gun” measurement, such as the proposed obser-vation of the correlated splitting of the zero–bias peak, couldbe a step towards unambiguously establishing the existence ofMajorana bound states in semiconductor nanowires. This goalcould be achieved in the near future. In addition, a fractionalAC Josephson effect, 2e2/h quantized conductance througha MF state, or a signature of a Zeeman-tuned TQPT, say, bytunneling to a region away from the wire ends, can decisivelyestablish the existence of MFs in semiconductor nanowire het-erostructures.

However, the search for the non-Abelian Majorana fermionzero modes in semiconductor heterostructures is far from over.Future experiments have to address the sufficient conditionsfor the existence of the Majorana mode, which may consista tunneling based interference measurement such as that dis-cussed in Sec. IIH or the direct observation of non–Abelianbraiding statistics in some form. This direction will involvechallenging experimental problems regarding the controlledengineering of complex devices, as well as basic aspects re-lated to quantum decoherence and the manipulation of Majo-rana bound states. An important lesson provided by the re-cent developments is that, in real systems, key observable fea-tures are determined by various details of the structure, despitethe topological nature of the superconducting state predictedto host the Majorana bound states. Therefore, to be able todiscriminate between alternative scenarios and to clarify themechanisms responsible for various features observed in theexperiments, it is critical to develop realistic models of theheterostructures as discussed in Sec. III.

Acknowledgment: We would like to thank our collaboratorsJ. D. Sau, R. M. Lutchyn, C. Zhang, and S. Das Sarma. Thiswork was supported in part by WV HEPC/dsr.12.29, DARPA-MTO (FA9550-10-1-0497), and NSF (PHY-1104527).

Appendix A: Inter–band pairing and the “sweet spot”

In the ’standard model’ of the Majorana nanowire, whichinvolves a semiconductor (SM) wire with spin–orbit coupling,proximity–induced superconductivity, and Zeeman spin split-ting, the emergence of zero energy Majorana bound states re-quires values of the chemical potential consistent with an oddnumber of partially occupied sub–bands. As an example, letus consider the case illustrated in Fig. 14 involving two bands,n = 1 and n = 2, and four values of the Zeeman field Γ. Thesplitting between the pairs of sub–bands with opposite helicity(i.e., n− and n+) is proportional to the applied field and, forΓ = Γ0, the sub–bands 1+ and 2− are degenerate at kx = 0,ε1+(0) = ε2−(0) = µ2. The condition for the existence ofa topological superconducting (SC) phase is satisfied for allfour values of Γ when the chemical potential is µ = µ1 (onepartially occupied sub–band) or µ = µ3 (three partially occu-pied sub–bands), but the system is topologically trivial whenµ = µ2 (two occupied sub–bands). This picture holds as longas the inter–band paring is zero. However, when ∆12 6= 0,

m1

m2

m3

0.4G0 0.8G0 1.2G0 1.6G0

2+

2-

1+

1-

FIG. 14. (Color online) Schematic representation of the evolutionof the SM nanowire spectrum with the applied Zeeman field. Thecurves represent the energy as function of the wave vector kx forfour different values of the Zeeman field Γ. For Γ = Γ0 (not shown)the sub–bands 1+ and 2− are degenerate at kx = 0. In the presenceof induced superconductivity, a system with a value of the chemicalpotential µ = µ1 or µ = µ3 is in a topological SC state, whilefor µ = µ2 the SC state is topologically trivial. However, in thepresence of inter–band paring, ∆12 6= 0, the SC state correspondingto µ = µ2 becomes topologically nontrivial in the vicinity of Γ = Γ0

(the sweet spot).

the number of partially occupied sub–bands does not repre-sent a good criterion for establishing the topological natureof a given SC phase. More specifically, a quasi-1D nanowirewith inter–band pairing and control parameters in the vicinityof the “sweet spot” Γ − Γ0 and µ = µ2 is in the topologi-cally nontrivial SC phase provided the inter sub–band spacing|ε1+−ε2−| is smaller that the off-diagonal coupling ∆12 [85].In the limit of strong inter–band mixing, the sweet spot regimeis characterized by a topological state that is robust againstchemical potential fluctuations, such as those created by dis-order. This stronger immunity can be understood in terms ofthe range of chemical potentials consistent with a topologicalSC state for a given value of the Zeeman field. Consideringexample shown in Fig. 14, for Γ = 0.4Γ0 there are two nar-row topological regions (for chemical potentials in the vicinityof µ1 and µ3, respectively), separated by a topologically triv-ial phase. By contrast, for Γ sufficiently close to Γ0 the rangeof µ consistent with a topological SC state extends from thebottom of sub–band 1− to the bottom of 2+, without passingthrough any topological quantum phase transition.

How can inter–band mixing occur in SM nanowires withproximity–induced superconductivity? In essence, non–uniform interface tunneling generically induces inter–bandcoupling. Consider, for example, a hybrid structure contain-ing a SC and a long SM nanowire with rectangular cross sec-tion with Ly � Lz . The low–energy physics is controlledby the lowest energy transverse mode, nz = 1, but , in gen-eral, involves several ny bands. Now let us assume that onlyhalf of the nanowire is covered by the superconductor, so that

23

t(y) = t for y < Ly/2 and t(y) = 0 for y > Ly/2. Using Eq.(26), the effective SM–SC coupling becomes

γnyn′y

= γ0

Ny/2∑iy=1

φny (iy)φn′y(iy), (A1)

where γ0 = νF |t|2[φ1(i0z)]2. The diagonal coupling is

γnyny = γ0/2, but, in addition, each band couples to allother bands of opposite parity. The inter–band couplings forneighboring bands, n′y = ny ± 1, are comparable to the di-agonal value, e.g., γ12 ≈ 0.85γ11. As described in SectionIII C, the off–diagonal SM–SC coupling generates proximity–induced inter–band pairing, ∆nyn′

y, which leads to the sweet

spot physics described above. Finally, we note that the exper-imental realization of non–uniform SM–SC coupling by par-tially covering the SM nanowire with superconductor [74] canprovide additional benefits, such as reducing SC–generatedscreening and allowing gate–voltage control of the chemicalpotential.

Appendix B: Projection of the effective tight–bindingHamiltonian onto a low–energy subspace

To realize the projection of the effective tight–bindingHamiltonian onto a low–energy subspace, one needs to iden-tify a convenient low–energy basis. While, in general, thisproblem has to be addressed using a combination of analyt-ical and numerical tools, the 2–band lattice model describedabove has a simple analytical solution. Specifically, for anelectron–doped nanowire with rectangular cross section anddimensions Lx � Ly ∼ Lz , the single particle quantumproblem corresponding to H0 from Eq. (18) has eigenstatesψnσ(i) =

∏3λ=1 φnλ(iλ)χσ , where n = (nx, ny, nz) with

1 ≤ nλ ≤ Nλ, χσ is an eigenstate of the σz spin operator, and

φnλ(iλ) =

√2

Nλ + 1sin

πnλiλNλ + 1

. (B1)

In Eq. (B1) λ = x, y, z and Lλ = aNλ, where a is the latticeconstant. The eigenvalues corresponding to ψnσ are

εn=−2t0

(cos

πnxNx+1

+cosπnyNy+1

+cosπnzNz+1

− 3

)−µ,

(B2)where µ is the chemical potential.

Next, we assume that only a few bands are occupied, andthat the low–energy subspace is defined by the eigenstates sat-isfying the condition εn < εmax, where the cutoff energy εmax

is typically of the order 100meV. Using this low-energy basis,the matrix elements of the total Hamiltonian can be writtenexplicitly, as described in Ref. [86]. The matrix elements ofthe SOI Hamiltonian from Eq. (18) are

〈ψnσ|HSOI|ψn′σ′〉 = αδnzn′z

{qnxn′

x(iσy)σσ′δnyn′

y

− qnyn′y(iσx)σσ′δnxn′

x

}, (B3)

where

qnλn′λ

=1− (−1)nλ+n′

λ

Nλ + 1

sin πnλNλ+1 sin

πn′λ

Nλ+1

cos πnλNλ+1 − cos

πn′λ

Nλ+1

. (B4)

The first term in Eq. (B3) represents the intra–band Rashbaspin–orbit interaction, while the second term couples differ-ent confinement–induced bands. Similarly, assuming that theZeeman splitting Γ is generated by a magnetic field orientedalong the wire (i.e., along the x-axis), Γ = g∗µBBx/2, whereg∗ is the effective g–factor for the SM nanowire, the matrixelements for the corresponding term in Eq. (15) are

〈ψnσ|HZ|ψn′σ′〉 = Γδnn′δσσ′ , (B5)

where σ = −σ. Adding together these contributions, the ef-fective Hamiltonian describing the low–energy physics of thesemiconductor nanowire in the presence of a Zeeman field be-comes Hnn′ = 〈ψn|HSM + HZ|ψ′n〉, with ψn representingthe spinor (ψn↑, ψn↓). Explicitly, we have

Hnn′ = [εn + Γσx] δnn′

+ iαδnzn′z

[qnxn′

xσyδnyn′

y− qnyn′

yσxδnxn′

x

], (B6)

where n = (nx, ny, nz), εn is given by Eq. (B2) and qnλn′λ

by Eq. (B4). Note that a similar effective low–energy Hamil-tonian can be written for an infinite quasi–1D wire. In thelimit Lx → ∞, the wave vector kx becomes a good quantumnumber and we have

Hnn′(kx) = [εn(kx)+αRkxσy+Γσx]δnn′−iαqnyn′yσxδnzn′

z,

(B7)where αR = αa and n = (ny, nz) labels the confinement–induced bands with energy

εn(kx)=h2k2

x

2m∗− 2t0

(cos

πnyNy+1

+cosπnzNz+1

− 2

)−µ.

(B8)

Appendix C: Proximity effect in hole–doped semiconductors

To illustrate this the dependence of the SC proximity effecton the details of the SM–SC coupling, we consider the caseof hole–doped SM nanowires. We use the 4–band Luttinger–type model defined by equations (20) and (21) to describe ahole–doped SM nanowire with rectangular cross section prox-imity coupled an s-wave SC. We assume that the z axis isperpendicular to the interface and corresponds to the (0, 0, 1)crystal axis of the underlying fcc lattice. The specific form ofthe SM–SC coupling Hamiltonian (23) depends on symmetryof the localized states that define the effective SC Hamiltonian(22). Assuming that µsc lies within a band with s-type charac-ter, we notice that the orbitals that are responsible for the cou-pling across the interface are the |Z〉 orbitals. Consequently,only the Φ± 1

2eigenstates couple to the SC. The corresponding

matrix elements in equation (23) can be written as

ts12σ

i0j0=

√2

3t δi0+d j0δsσ, (C1)

24

Ly/Lz

Eff

ecti

ve

cou

pli

ng

(a.

u.)

Lz ≈ 8nm

Lz ≈ 12nm

FIG. 15. (Color online) Dependence of the effective SM–SC cou-pling of the top valence band in a hole–doped SM wire with rectan-gular cross section Ly × Lz on the Ly/Lz ratio. The strength of thecoupling decreases with increasing the wire thickness Lz but, in con-trast with electron–doped wires, it shows a strong dependence on thewire width Ly . In the limit Ly →∞ the coupling vanishes becausethe top valence band has purely heavy–hole character, i.e., ψ(s)

± 12

= 0.

where s 12 = ± 1

2 and t is a constant, while the coupling of

the 3/2 states vanishes, t s32σ

i0j0= 0. We address the follow-

ing question: What is the consequence of this selective cou-pling on the strength of the proximity effect in hole–dopednanowires with rectangular cross section? Specifically, we fo-cus on the effective SM–SC coupling γv for the top valenceband.

To determine the effective coupling, we calculate numer-ically the wave functions Ψ± corresponding to the dou-ble degenerate top valence band at kx = 0. These statescan be expressed as four–component spinors, Ψs(iy, iz) =

[ψ(s)32

, ψ(s)12

, ψ(s)

− 12

, ψ(s)

− 32

]T , and we have

γc =2

3νF |t|2

Ny∑iy=1

|ψ(s)

± 12

(iy, i0z)|2. (C2)

For comparison, in the case of an electron–doped wire withuniform tunneling across the SM–SC interface, the effectivecoupling of the lowest conduction band can be obtained fromEq. (26) for t(iy) = t and n = n′ = (1, 1, 1). We haveγc = νF |t|2φ2

1(i0z) ≈ 2π2νF |t|2/(Nz + 1)3. Note that γcis independent of the wire width Ly and decreases approxi-mately as 1/L3

z with increasing wire thickness Lz , due to thedecrease of the wave function amplitude at the interface. Bycontrast, the effective coupling of the top valence band is char-acterized by a strong dependence on Ly for a fixed wire thick-ness Lz . The numerical results are shown in Fig. 15. Notethat in the quasi–2D limit, Ly → ∞, the effective couplingof the top valence band to the bulk SC vanishes. This can beunderstood by noticing that the top valence band of a SM slabis a purely heavy–hole band and, for a (0, 0, 1) surface orien-tation, the corresponding states have vanishing ψ(±)

± 12

compo-nents. Consequently, we conclude that no proximity–inducedsuperconductivity can occur in the top valence band of planar

(k.a) E

ner

gy

(m

eV)

En

erg

y (

meV

)

Nz =40

FIG. 16. (Color online) Top panel: Spectrum of a SM slab withNz = 40 layers along an arbitrary direction in the (kx, ky) plane.The SM slab is described by a tight–binding model with nearest–neighbor hopping t = 1eV on a simple cubic lattice with latticeconstant a. The energy is measured with respect to the chemicalpotential. Bottom panel: Comparison between the BdG spectrumobtained using Eq. (30) (orange lines) and the effective Hamilto-nian (31) (blue circles). The bulk SC gap is ∆0 = 1meV and theeffective SM–SC coupling corresponds to γ4 = ∆0/3 for the topoccupied band. The minima of the BdG spectrum occur in the vicin-ity of the Fermi k–vectors corresponding to different nz bands (onlythe minima corresponding to nz = 4 and nz = 3 are shown) and aredetermined by the corresponding induced SC gap (∆4 = 0.25meVand ∆3 ≈ 0.14meV).

SM–SC heterostructures with a (0, 0, 1) interface. This re-sult holds for different interface orientations. Nonetheless, inquasi–1D nanowires, the top valence band is a superpositionof heavy–hole and light–hole states and, consequently, the ef-fective coupling to the SC is nonzero, as shown in Fig. 15.Note that the amplitude of the of the light–hole componentvaries non–monotonically with Ly and has a sharp disconti-nuity for a width–to–thickness ration Ly/Lz ≈ 2. A quan-titative description of the SC proximity effect in hole–dopednanowires, requires a more detailed modeling of the wire (e.g.,using the 8–band model).

Appendix D: Proximity effect in multi–band systems and thecollapse of the induced gap

We illustrate the role of multi–band physics in proximity–coupled finite size systems by calculating the low–energyspectrum of a SM thin film – SC slab heterostructure with uni-form coupling across the planar interface. This allows us to fo-

25

(k.a)

En

erg

y (

meV

) E

ner

gy

(m

eV)

Nz =120

FIG. 17. (Color online) Top panel: Normal spectrum of a SM slabwith Nz = 120. The other parameters are the same as in Fig. 16.Bottom panel: Collapse of the induced SC gap due to proximity–generated inter–band coupling. All parameters (except Nz) are thesame as in Fig. 16. The solution obtained using Eq. (30) (orangelines) includes inter–band coupling, while the effective Hamiltonian(31) (blue circles) is constructed in the decoupled band approxima-tion, γnzn′

z∝ δnzn′

z.

cus on the proximity–induced inter–band coupling involvingbands with different nz quantum numbers, a problem that wasnot previously investigated in the literature. First, we considerthe limit ∆Enn′ � ∆0 and we test the accuracy of the staticapproximation that allows us to define the effective Hamilto-nian (31). For simplicity, the SM film is described using the2–band model with nearest neighbor hopping t = 1eV, noRashba spin–orbit coupling (α = 0), and no Zeeman field(Γ = 0). The normal state spectrum of a slab containingNz = 40 layers is shown in the upper panel of Fig. 16. Thechemical potential is near the bottom the fourth band, nz = 4,and intersects the bands with nz ≤ 4 at different Fermi wavevectors kFnz . The SM slab is coupled to a SC with a bulk gap∆0 = 1meV. The effective SM–SC coupling has the formγnzn′

z= γ0φnz (i0z)φn′

z(i0z), where φnz (i0z) is given by

Eq. (B1), and the constant γ0 is chosen so that the effectivecoupling of the fourth band be γ4 ≡ γ44 = ∆0/3. Since∆Enn′ � ∆0, the effective Hamiltonian is constructed us-ing the decoupled band approximation and the correspondinglow–energy spectrum is compared with the solution of the fullBdG equation (28). The results are shown in the lower panelof Fig. 16. Note that the effective Hamiltonian description ishighly accurate at energies within the SC gap, except near thegap edge where the static approximation

√∆2

0 − ω2 ≈ ∆0

manifestly breaks down.The role of proximity–induced inter–band coupling is illus-

Ener

gy

(m

eV)

Ener

gy

(m

eV)

Disorder strength (meV)

State index (n)

G=0.6meV

Dm=0

Vd=0

Vd=5meV

FIG. 18. (Color online) Top: BdG spectrum of a SM wire withproximity–induced superconductivity and Zeeman splitting Γ =0.6meV. The chemical potential relative to the bottom of the thirdband is ∆µ = 0. In a clean wire (Vd = 0) the Majorana bound states(red diamonds) are protected by a gap of the order 130µeV. This gapcollapses in the presence of strong disorder. Bottom: Dependence ofthe non–zero lowest–energy modes on the strength Vd of the disorderpotential.

trated in Fig. 17, which shows the spectra corresponding toa SM film with Nz = 120 layers. The other parameters areidentical with those in Fig. 16, including the effective SM–SCcoupling γnzn′

z. Note that the decoupled band approximation

fails, as the inter–band gaps are now comparable with ∆0 (seethe upper panel of Fig. 17). In particular, inter–band couplingresults in a collapse of the induced SC gap. A detailed analy-sis of this effect is presented in Ref. [145]. We emphasize thatthis effect is not due to the decrease of the wave function am-plitude at the interface with increasing Nz , as this is compen-sated for by increasing the transparency of the interface, i.e.,γ0, but stems from the off–diagonal elements of the effectivecoupling matrix γnzn′

z. Also, we note that in this regime the

construction of the effective Hamiltonian (31) has to involvethe high–energy nz bands, as they are intrinsically coupled tothe low–energy bands and renormalize them strongly, whichultimately leads to the collapse of the induced SC gap.

Appendix E: Low–energy states in the presence of disorder

We illustrate some of the generic features of low–energyBdG spectrum of a disordered SM–SC hybrid system by con-sidering a SM nanowire of length lx = 3µm and rectangu-lar cross section with Ly = 80nm and Lz = 40nm in thepresence of a disorder potential V(r) = VdfV (r), where the

26

Ener

gy

(m

eV)

Ener

gy

(m

eV)

Disorder strength (meV)

State index (n)

G=0.6meV

Dm=3meV

Vd=0

Vd=10meV

FIG. 19. (Color online) Top: BdG spectrum of a SM wire with pa-rameters corresponding to the topologically trivial phase. Note thatthere are no zero–energy states. The quasiparticle gap collapses inthe presence of strong disorder. Bottom: Dependence of the lowest–energy modes on the strength Vd of the disorder potential. Note thatthe proliferation of disorder–induced low–energy states requires astronger disorder potential that in the topological SC phase (see Fig.18).

disorder profile is described by the random function fV with|fV (r)| ≤ 1 and Vd represents the amplitude of the disorderpotential. The low–energy spectrum is calculated using a 2–band model, as described in Sec. III A. All the calculationare done for a specific disorder realization, i.e., a fixed profilefV , but for variable disorder strength Vd. First, we considera system with three partially occupied bands (five spin sub–bands) in the presence of a Zeeman splitting Γ = 0.6meV anda fixed chemical potential ∆µ = 0, as measured relative tothe bottom of the third band in the absence of Zeeman split-ting. The clean wire (Vd = 0) is in the topological SC phasethat supports zero–energy Majorana bound states. The corre-sponding BdG spectrum is shown in the top panel of Fig. 18(black circles and red diamonds for the Majorana states). Inthe presence of disorder (Vd 6= 0), the mini–gap that protectsthe Majorana bound state becomes smaller and, eventually,collapses. The dependence of the non–zero lowest–energymodes on the strength of the disorder potential is shown inthe bottom panel. We note that, in the presence of disorder,multiple zero–energy modes are possible as disorder effec-tively cuts the wire into disconnected topologically nontriv-ial segments supporting Majorana bounds states at their ends.Typically, these segments are relatively short and the states lo-calized at the ends of each segment (which can be viewed asrepresenting a Majorana chain) overlap significantly and arecharacterized by nearly–zero energies that oscillate with the

x (mm)

|Y(x

)|2

(a.u

.)

Dm=0

Dm=0

Dm=3

Dm=3

Vd=5

Vd=10

Vd=0

Vd=0

FIG. 20. (Color online) Position dependence of the amplitude of thelowest–energy states for a wire with Lx = 3µm and Γ = 0.6meV.The top panel corresponds to a clean wire in the topological SCphase. The red line with maxima near the ends of the wire repre-sents the Majorana bound states. In the presence of disorder, all thestates become localized in various regions of the wire. The wavefunction of the lowest energy state of clean wire (Vd = 0) in thetopological trivial state is characterized by an envelope with max-ima near the ends of the wire and oscillations corresponding to acertain value of kF . The lowest–energy states associated with thelow–energy occupied bands (not shown) have similar characteristics,but higher values of kF . For given values of the disorder strength andZeeman splitting, states with larger characteristic Fermi wave vectors(shorter oscillation period) exhibit weaker localization.

chemical potential and the Zeeman splitting.A similar proliferation of low–energy states with increas-

ing the strength of the disorder can be seen in a system withparameters corresponding to the trivial SC phase, as shown inFig. 19. Note that in this case the decrease of the quasiparticlegap with Vd is slower than that corresponding to ∆µ = 0. Thisbehavior can be understood qualitatively by noting that disor-der tends to localize the low–energy states and that this effectdepends of the characteristic Fermi wave vector associatedwith those states. Increasing ∆µ corresponds to larger valuesof kF and to a weaker localization. This behavior is illustratedby the profiles of the low–energy states shown in Fig. 20. Thelow–energy states corresponding to ∆µ = 0 exhibit strongerlocalization that those for ∆µ = 3meV, in spite of the smalleramplitude of the disorder potential. We note that the low–energy states associated with the low–energy bands are char-acterized by large values of kF and, consequently, are harder

27

to localize. Finally, we note that the other key parameter thatcontrols the effect of disorder on the low–energy states is theZeeman splitting. Increasing Γ, which breaks time–reversalsymmetry, facilitates the collapse of the SC gap. Again, thiseffect is stronger for states with low values of the characteris-tic kF , which typically corresponds to the top occupied band,although significant band mixing is possible for certain typesof disorder. Consequently, in a wire with hard confinementthe nearly–zero energy states with most of the spectral weight

coming from low–energy bands require higher values of theZeeman splitting than the disorder–induced nearly–zero en-ergy states associated with the top band. On the other hand,the energy of these states can become exponentially smalleven in the absence of disorder if the confinement potentialis smooth [110]. Furthermore, the states associated with low–energy bands can penetrate through finite barrier potentialsand extend into the normal section of the wire [161].

[1] Dirac P A M. 1928 Proceedings of the Royal Society A: Math-ematical, Physical and Engineering Sciences 117 610.

[2] Peskin M and Schroeder D. 1995 An Introduction to QuantumField Theory, Westview Press.

[3] Gell-Mann G and Pais A. 1955 Phys. Rev. 97 1387.[4] Majorana E. 1937 Nuovo Cimento 14 171.[5] Weyl H. 1929 Z. Phys. 56 330.[6] Nayak C, Simon S H, Stern A, Freedman M, and Das Sarma

S. 2008 Rev. Mod. Phys. 80 1083.[7] Wilczek F. 2009 Nature Phys. 5 614.[8] Beenakker C W J. 2011 e-print arXiv:1112.1950.[9] Alicea J. 2012 Rep. Prog. Phys. 75 076501.

[10] Leijnse M and Flensberg K. 2012 Semicond. Sci. Technol. 27124003.

[11] Read N and Green D. 2000 Phys. Rev. B 61, 10267.[12] Moore G and Read N. 1991 Nuclear Physics B 360 362.[13] Nayak C and Wilczek F. 1996 Nucl. Phys. B 479 529.[14] Ivanov D A. 2001 Phys. Rev. Lett. 86 268.[15] Stern A, Oppen F v, and Mariani E. 2004 Phys. Rev. B 70

205338.[16] Kitaev A Y. 2003 Annals. Phys. 303 2.[17] Volovik G E. 1988 Zh. Eksp. Teor. Fiz. (Sov. Phys. JETP) 94

(67) 123 (1804).[18] Wen X G. 1993 Phys. Rev. Lett. 70 355.[19] Read N and Moore G. 1992 Prog. Theor. Phys. (Kyoto) Supp.

107 157.[20] Das Sarma S, Freedman M, and Nayak C. 2005 Phys. Rev.

Lett. 94 166802.[21] Stern A and Halperin B I. 2006 Phys. Rev. Lett. 96 016802.[22] Bonderson P, Kitaev A, and Shtengel K. 2006 Phys. Rev. Lett.

96 016803.[23] Kitaev A Y. 2001 Physics-Uspekhi 44 131.[24] Das Sarma S, Nayak C, and Tewari S. 2006 Phys. Rev. B 73

220502(R).[25] Chung S B, Bluhm H, and Kim E-A. 2007 Phys. Rev. Lett. 99

197002.[26] Jang J, Ferguson D G, Vakaryuk V, Budakian R, Chung S B,

Goldbart P M, and Maeno Y. 2011 Science 331 186.[27] Fu L and Kane C L. 2008 Phys. Rev. Lett 100 096407.[28] Fu L and Kane C L. 2009 Phys. Rev. B 79 161408 (R).[29] Fu L and Kane C L. 2009 Phys. Rev. Lett. 102 216403.[30] Akhmerov A R and Nilsson J Beenakker C W J. 2009 Phys.

Rev. Lett. 102 216404.[31] Linder J, Tanaka Y, Yokoyama T, Sudbo A, and Nagaosa N.

2010 Phys. Rev. Lett. 104 067001.[32] Cook A and Franz M. 2011 Phys. Rev. B (R) 84 201105.[33] Potter A C and Lee P A. 2012 Phys. Rev. B 85 094516.[34] Takei S, Fregoso B M, Galitski V, and Das Sarma S. 2013

Phys. Rev. B 87 014504.[35] Sato M and Fujimoto S. 2009 Phys. Rev. Lett 79 094504.

[36] Ghosh P, Sau J, Tewari S, and Das Sarma S. 2010 Phys. Rev.B 82 184525.

[37] Tanaka Y, Mizuno Y, Yokoyama T, Yada K, and Sato M. 2010Phys. Rev. Lett. 105 097002.

[38] Tanaka Y, Sato M, and Nagaosa N. 2012 J. Phys. Soc. Jpn. 81011013.

[39] Asano Y and Tanaka Y. 2012 e-print arXiv:1204.4226.[40] Martin I and Morpurgo A F. 2012 Phys. Rev. B 85 144505.[41] Sau J D and Tewari S. 2011 eprint arXiv:1111.5622.[42] Klinovaja J, Gangadharaiah S, and Loss D. 2012 Phys. Rev.

Lett. 108 196804.[43] Klinovaja J, Ferreira G J, and Loss D. 2012 Phys. Rev. B 86

235416.[44] Klinovaja J and Loss D. 2013 Phys. Rev. X 3 011008.[45] Black-Schaffer A M and Linder J. 2011 Phys. Rev. B 84(R)

180509.[46] Nadj-Perge S, Drozdov I K, Bernevig B A, and Yazdani A.

2013 e-print arXiv:1303.6363.[47] Regal C A, Ticknor C, Bohn J L, and Jin D S. 2003 Phys. Rev.

Lett. 90, 053201.[48] Ticknor C, Regal C A, Bohn J L, and Jin D S. 2004 Phys. Rev.

A 69 042712.[49] Schunck C H, Zwierlein M W, Stan C A, Raupach S M F, Ket-

terle W, Simoni A, Tiesinga E, Williams C J, and Julienne PS. 2005 Phys. Rev. A 71 045601.

[50] Tewari S, Das Sarma S, Nayak C, Zhang C, and Zoller P. 2007Phys. Rev. Lett. 98 010506.

[51] Zhang C, Tewari S, Lutchyn R, and Das Sarma S. 2008 Phys.Rev. Lett 101 160401.

[52] Sato M, Takahashi Y, and Fujimoto S. 2009 Phys. Rev. Lett103 020401.

[53] Sau J D, Lutchyn R M, Tewari S, and Das Sarma S. 2010Phys. Rev. Lett 104 040502,http://link.aps.org/doi/10.1103/PhysRevLett.104.040502.

[54] Sau J D, Tewari S, Lutchyn R M, Stanescu T D, and Das SarmaS. 2010 Phys. Rev. B 82 214509,http://link.aps.org/doi/10.1103/PhysRevB.82.214509.

[55] Lutchyn R M, Sau J, and Das Sarma S. 2010 Phys. Rev. Lett.105 077001.

[56] Oreg Y, Refael G, and Oppen F V. 2010 Phys. Rev. Lett. 105177002.

[57] Mao L, Shi J, Niu Q, and Zhang C. 2011 Phys. Rev. Lett. 106157003.

[58] Mao L, Gong M, Dumitrescu E, Tewari S, and Zhang C. 2012Phys. Rev. Lett. 108 177001.

[59] Gong M, Tewari S, and Zhang C. 2011 Phys. Rev. Lett. 107195303.

[60] Sau J D and Tewari S. 2012 Phys. Rev. B 86 104509.[61] Tewari S, Stanescu T D, Sau J D, and Das Sarma S. 2012

Phys. Rev. B 86 024504.

28

[62] Alicea J, Oreg Y, Refael G, Oppen F v, and Fisher M P A.2011 Nature Physics 7 412.

[63] Halperin B I, Oreg Y, Stern A, Refael G, Alicea J, and vonOppen F. 2012 Phys. Rev. B 85 144501.

[64] Nielssen H and Ninomiya N. 1983 Phys. Lett. B 130 389.[65] Santos L, Ryu S, Chamon C, and Mudry C. 2010 Phys. Rev. B

82, 165101.[66] Cook A, Vazifeh M M, and Franz M. 2012 Phys. Rev. B 86

155431.[67] Sengupta K, Zutic I, Kwon H J, and Das Sarma S Yakovenko

V M. 2001 Phys. Rev. B 63 144531.[68] Tewari S, Zhang C, Das Sarma S, Nayak C, and Lee D H. 2008

New Journal of Physics 100 027001.[69] Law K T, Lee P A, and Ng T K. 2009 Phys. Rev. Lett. 103

237001.[70] Flensberg K. 2010 Phys. Rev. B 82 180516.[71] Wimmer M, Akhmerov A R, Dahlhaus J P, and Beenakker

C W J. 2011 New Journal of Physics 13 053016.[72] Kwon H J, Sengupta K, and Yakovenko V M. 2004 The Euro-

pean Physical Journal B 37 349.[73] Jiang L, Pekker D, Alicea J, Refael G, Oreg Y, and von Op-

pen F. 2011 Phys. Rev. Lett. 107 236401.[74] Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P A

M, and Kouwenhoven L P. 2012 Science 336, 1003,http://www.sciencemag.org/content/336/6084/1003.full.pdf.

[75] Deng M T, Yu C L, Huang G Y, Larsson M, Caroff P, and Xu HQ. 2012 Nano Lett. 12 6414.

[76] Das A, Ronen Y, Most Y, Oreg Y, Heiblum M, and ShtrikmanH. 2012 Nature Physics 8 887.

[77] Churchill H O H, Fatemi V, Grove-Rasmussen G, Deng MT, Caroff P, Xu H Q, and Marcus C M. 2013 e-printarXiv:1303.2407.

[78] Finck A D K, Van Harlingen D J, Mohseni P K, Jung K, andLi X. 2005 Phys. Rev. Lett. 110 126406.

[79] Rokhinson L P, Liu X, and Furdyna J K. 2012 Nature Physics8, 795.

[80] Mackenzie A P and Maeno Y. 2003 Rev. Mod. Phys. 75 657.[81] Sau J D, Lutchyn R M, Tewari S, and Das Sarma S. 2010

Phys. Rev. B 82, 094522.[82] Tewari S, Sau J, and Das Sarma S. 2010 Annals of Physics

325 219.[83] Alicea J. 2010 Phys. Rev. B 81 125318.[84] Potter A C and Lee P A. 2010 Phys. Rev. Lett. 105 227003.[85] Lutchyn R M, Stanescu T D, and Das Sarma S. 2011 Phys.

Rev. Lett. 106 127001.[86] Stanescu T D, Lutchyn R M, and Das Sarma S. 2011 Phys.

Rev. B 84 144522,http://link.aps.org/doi/10.1103/PhysRevB.84.144522.

[87] Potter A C and Lee P A. 2011 Phys. Rev. B 83, 094525,http://link.aps.org/doi/10.1103/PhysRevB.83.094525.

[88] Tewari S, Sau J, Scarola V, Zhang C, and Das Sarma S. 2012Phys. Rev. B 85 155302.

[89] Schnyder A P, Ryu S, Furusaki A, and Ludwig A W W. 2008Phys. Rev. B 78 195125.

[90] Kitaev A Y. 2009 AIP Conf. Proc. 1134 22.[91] Ryu S, Schnyder A, Furusaki A, and Ludwig A W W. 2010

New J. Phys. 12, 065010.[92] Tewari S and Sau J D. 2012 Phys. Rev. Lett. 109 150408.[93] Niu Y, Chung S B, Hsu C H, Mandal I, Raghu S, and

Chakravarty S. 2012 Phys. Rev. B 85 035110.[94] Lim J S, Serra L, Lopez R, and Aguado R. 2012 Phys. Rev. B

86 121103.[95] Diez M, Dahlhaus J P, Wimmer M, and Beenakker C W J.[96] Anderson P W. 1958 Phys. Rev. B 110 827.

[97] Ryu S and Hatsugai Y. 2002 Phys. Rev. Lett. 89, 077002.[98] Zak J. 1989 Phys. Rev. Lett. 62 2747.[99] Gibertini M, Taddei F, Polini M, and Fazio R. 2012 Phys. Rev.

B 85 144525.[100] Gibertini M, Fazio R, Polini M, and Taddei F. 2013 e-print

arXiv:1302.2736.[101] Ekin J W, Xu Y, Mao S, Venkatesan T, Face D W, Eddy M,

and Wolf S A. 1997 Phys. Rev. B 56 13746.[102] Wagenknecht M, Koelle D, Kleiner R, Graser S, Schopohl N,

Chesca B, Tsukada A, Goennenwein S T B, and Gross R. 2008Phys. Rev. Lett. 100 227001.

[103] Semenoff G W and Sodano P. 2006 e-print arXiv:0601261.[104] Bolech C J and Demler E. 2007 Phys. Rev. Lett. 98 237002.[105] Nilsson J, Akhmerov A R, and Beenakker C W J. 2008 Phys.

Rev. Lett. 101 120403.[106] Blonder G E, Tinkham M, and Klapwijk T M. 1982 Phys. Rev.

B 25 4515.[107] Qu C, Zhang Y, Mao L, and Zhang C. 2011 e-print

arXiv:1109.4108.[108] Fidkowski L, Alicea J, Lindner N, Lutchyn R M, and Fisher

M P A. 2012 Phys. Rev. B 85 245121.[109] Sau J D, Halperin B I, Flensberg K, and Das Sarma S. 2011

Phys. Rev. B 84 144509.[110] Kells G, Meidan D, and Brouwer P W. 2012 Phys. Rev. B (R)

86 100503.[111] Liu J, Potter A C, Law K T, and Lee P A. 2012 Phys. Rev.

Lett. 109 267002.[112] Lin C-H, Sau J D, and Das Sarma S. 2012 Phys. Rev. B 86

224511.[113] Lutchyn R M and Skrabacz J H. 2013 e-print

arXiv:1302.0289.[114] Sau J D, Berg E, and Halperin B I. 2012 eprint

arXiv:1206.4596.[115] Nogueira F S and Eremin I. 2012 J. Phys.: Condens. Matter

24 325701.[116] Beenakker C W J, Pikulin D I, Hyart T, Schomerus H, and

Dahlhaus J P. 2013 Phys. Rev. Lett. 110 017003.[117] Sau J D, Swingle B, and Tewari S. 2012 eprint

arXiv:1210.5514.[118] Fu L. 2010 Phys. Rev. Lett. 104 056402.[119] Liu J, Zhang F-C, and Law K T. 2012 e-print

arXiv:1212.5879.[120] Sau J D, Tewari S, and Das Sarma S. 2011 Phys. Rev. B 84

085109.[121] Turner A, Pollman P, and Berg E. 2011 Phys. Rev. B 83

075102.[122] Bermudez A, Patane D, Amico L, and Martin-Delgado M A.

2009 Phys. Rev. Lett. 102 135702.[123] Bermudez A, Amico L, and Martin-Delgado M A. 2010 New

J. Phys. 12 055014.[124] Sticlet D, Bena C, and Simon P. 2012 Phys. Rev. Lett. 2012

096802.[125] Chevallier D, Sticlet D, Simon P, and Bena C. 2012 Phys. Rev.

B 85 235307.[126] Mei F, Shan C-J, Feng X-L, Zhu S-L, Zhang Z-M, Kwek L C,

Wilkowski D, and Oh C H. 2012 e-print arXiv:1204.3974.[127] Appelbaum I. 2013 e-print arXiv:1303.4719.[128] Pachos J K, Alba E, Lahtinen V, and Garcia-Ripoll J J. 2013

e-print arXiv:1209.5115.[129] Das Sarma S, Sau J D, and Stanescu T D. 2012 Phys. Rev. B

86 220506(R).[130] Galanakis D and Stanescu T S. 2012 e-print arXiv:1206.2043.[131] Luttinger J M and Kohn W. 1955 Phys. Rev. 97 869.[132] Luttinger J M. 1956 Phys. Rev. 102 1030.

29

[133] Kane E O. 1957 J. Phys. Chem. Solids 1 249.[134] Black-Schaffer A M. 2011 Phys. Rev. B 83 060504.[135] Lababidi M and Zhao E. 2011 Phys. Rev. B 83 184511.[136] Tewari S, Stanescu T D, Sau J D, and Das Sarma S. 2011 New

Journal of Physics 13 065004.[137] Lutchyn R M and Fisher M P A. 2011 Phys. Rev. B 84 214528.[138] Gangadharaiah S, Braunecker B, Simon P, and Loss D. 2011

Phys. Rev. Lett 83 036801.[139] Sela E, Altland A, and Rosch A. 2011 Phys. Rev. B 84 085114.[140] Stoudenmire E M, Alicea J, Starykh O A, and Fisher M P A.

2011 Phys. Rev. B 84 014503.[141] Stanescu T D, Sau J D, Lutchyn R M, and Das Sarma S. 2010

Phys. Rev. B 81 241310.[142] Khaymovich I M, Chtchelkatchev N M, and Vinokur V M.

2011 Phys. Rev. B 84 075142.[143] Grein R, Michelsen J, and Eschrig M. 2011 e-print

arXiv:1111.0445.[144] Sau J D, Tewari S, and Das Sarma S. 2012 Phys. Rev. B 85

064512.[145] Stanescu T D and Das Sarma S. 2013 e-print

arXiv:1303.1187.[146] Lutchyn R M, Stanescu T D, and Das Sarma S. 2012 Phys.

Rev. B 85 140513(R).[147] Potter A C and Lee P A. 2011 Phys. Rev. B 83 184520; Phys.

Rev. B 84 059906(E).[148] Chevallier D, Simon P, and Bena C. 2013 e-print

arXiv:1301.7420.[149] Sau J D and Das Sarma S. 2012 Nature Communications 3

964.[150] Brouwer P W, Duckheim M, Romito A, and von Oppen F.

2011 Phys. Rev. Lett. 107 196804.[151] Akhmerov A R, Dahlhaus J P, Hassler F, Wimmer M, and

Beenakker C W J. 2011 Phys. Rev. Lett. 106 057001.[152] Brouwer P W, Duckheim M, Romito A, and von Oppen F.

2011 Phys. Rev. B 84 144526.[153] Lobos A M, Lutchyn R M, and Das Sarma S. 2012 Phys. Rev.

Lett. 109 146403.[154] Pientka F, Kells G, Romito A, Brouwer P W, and von Oppen F.

2012 Phys. Rev. Lett. 109 227006.[155] Rainis D, Trifunovic L, Klinovaja J, and Loss D. 2013 Phys.

Rev. B 87 024515.[156] Pikulin D I, Dahlhaus J P, Wimmer M, Schomerus H, and

Beenakker C W J. 2012 New J. Phys 14 125011.[157] Stanescu T D, Lutchyn R M, and Das Sarma S. 2013 Phys.

Rev. B 87 094518.[158] Bagrets D and Altland A. 2012 e-print arXiv:1206.0434.[159] Lee E J H, Jiang X, Aguado R, Katsaros G, Lieber C M, and

De Franceschi S. 2012 Phys. Rev. Lett. 109 186802.[160] Stanescu T D, Tewari S, Sau J D, and Das Sarma S. 2012

Phys. Rev. Lett. 109 266402.[161] Stanescu T D and Tewari S. 2012 e-print arXiv:1208.6298.[162] Takei S, Fregoso B M, Hui H-Y, Lobos A M, and Das Sarma

S. 2012 e-print arXiv:1211.1029.[163] Cheng M, Lutchyn R M, Galitski V, and Das Sarma S. 2009

Phys. Rev. Lett. 103 107001.[164] Prada E, San-Jose P, and Aguado R. 2012 Phys. Rev. B 86

180503(R).[165] Lee E J H, Jiang X, Houzet M, Aguado R, Lieber C M, and

De Franceschi S. 2013 e-print arXiv:1302.2611.