Majorana Fermions in Chiral Topological Ferromagnetic Nanowires

Eugene Dumitrescu1, Brenden Roberts1, Sumanta Tewari1, Jay D. Sau2, and S. Das Sarma21Department of Physics and Astronomy, Clemson University, Clemson, SC 29634

2 Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742

Motivated by a recent experiment in which zero-bias peaks have been observed in scanning tunneling mi-croscopy (STM) experiments performed on chains of magnetic atoms on a superconductor, we show, by gener-alizing earlier work, that a multichannel ferromagnetic wire deposited on a spin-orbit coupled superconductingsubstrate can realize a non-trivial chiral topological superconducting state with Majorana bound states localizedat the wire ends. The non-trivial topological state occurs for generic parameters requiring no fine tuning, atleast for very large exchange spin splitting in the wire. We theoretically obtain the signatures which appear inthe presence of an arbitrary number of Majorana modes in multi-wire systems incorporating the role of finitetemperature, finite potential barrier at the STM tip, and finite wire length. These signatures are presented interms of spatial profiles of STM differential conductance which clearly reveal zero energy Majorana end modesand the prediction of a multiple Majorana based fractional Josephson effect. A substantial part of this workis devoted to a detailed critical comparison between our theory and the recent STM experiment claiming theobservation of Majorana fermions in ferromagnetic atomic chains on a superconductor. The conclusion of thisdetailed comparison is that although the experimental observations are not manifestly inconsistent with ourtheoretical findings, the very small topological superconducting gap and the very high temperature of the exper-iment make it impossible to decisively verify the existence of a localized Majorana zero mode, as the spectralweight of the Majorana mode is necessarily spread over a very broad energy regime exceeding the size of thegap. Such an extremely broad (and extremely weak) conductance peak could easily arise from any sub-gapstates existing in the rather complex system studied experimentally and may or may not have anything to dowith a putative Majorana zero mode as discussed in the first half of our paper. Thus, although the experimentalfindings are indeed consistent with a highly broadened and weakened Majorana zero bias peak, much lowerexperimental temperatures (and/or much larger experimental topological superconducting gaps) are necessaryfor any definitive conclusion.

I. INTRODUCTION

In contrast to ordinary charged Dirac fermions (e.g. elec-trons, positrons) which come in oppositely charged parti-cle anti-particle pairs, Majorana fermions are their own anti-particles1,2. Although Majorana fermions, which are a spe-cial kind of neutral Dirac fermions, were first predicted inthe context of high energy physics1 (specifically, as a hypo-thetical model for neutrinos, which may or may not be Ma-jorana fermions), a condensed matter analog of them has re-cently been proposed to exist as localized zero-energy quasi-particles bound to order parameter defects in topologicallyordered systems3,4. In low dimensional systems (d 2),these localized defect-bound zero-energy Majorana quasipar-ticles obey exotic non-Abelian quantum statistics35 (and aretherefore not any kind of Dirac fermions at all). Due totheir non-Abelian braiding statistics and non-local topologi-cal nature, these zero-energy Majorana bound states (MBS)can be used as the building blocks of a fault-tolerant quan-tum computer4,6. Recently, realistic materials such as chiralp-wave superconductors e.g., strontium ruthenate7, topologi-cal insulator-superconductor interfaces8, fermionic cold atomgases9,10, spin-orbit coupled semiconducting thin films1113

and nanowires1315 in proximity to conventional supercon-ductors, all realizing an analog of topological spinless p-wave superconductors3,4, have been proposed as hosts forMBS. Recent experimental observations in semiconductor-superconductor heterostructures1621 seem to support the pres-ence of MBS in condensed matter systems, although any con-clusive evidence for non-Abelian statistics is still lacking22.

In concurrent formal theoretical developments, recentwork2326 established that the quadratic Hamiltonians de-scribing gapped topological insulators and topological su-perconductors (TS) can be classified into ten distinct topo-logical symmetry classes, and that each is characterizedby a topological invariant counting the number of topo-logically protected edge modes. According to this clas-sification, the experimentally investigated semiconductor-superconductor heterostructures1121 belong to the topologicalclass D in which MBS are protected by the superconductingparticle-hole (PH) symmetry. Additionally, one-dimensionaltopological superconductors belonging to the time reversalclass DIII2736 and BDI3742 have recently been proposed.DIII topological superconductivity is indexed by a Z2 topo-logical invariant indicating the presence or absence of a Ma-jorana Kramers pair, while in BDI topological supercon-ductivity, for instance as recently proposed43 in the contextof putative spin-triplet ferromagnetic superconductors suchas the organic superconductors and lithium purple bronze(Li0.9Mo6O17), a Z invariant counts the number of MBS.

Very recent experimental work44 suggests that atomic scaleferromagnetic Fe nanowires on the [110] surface of super-conducting Pb may support Majorana modes and topologi-cal superconductivity. The theoretical prediction that prox-imity induced superconductivity in ferromagnetic (or half-metallic a half-metal is a perfect ferromagnet with com-plete spin-polarization at the Fermi level) wires might leadto a topological superconducting phase with the associated lo-calized Majorana zero energy modes was made some yearsago by several groups4548. There was even an experimen-tal report of the observation of long-range superconducting

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2proximity effect through a ferromagnetic nanowire49. Veryrecently, two theoretical papers50,51 have predicted the pos-sible existence of topological superconductivity in ferromag-netic nanowires lying on superconducting substrates usingShiba chain50 and a one-dimensional nanowire51 model, re-spectively, closely mimicking the experimental system (i.e.Fe atoms on superconducting Pb) studied in Ref. 44. Ex-tensive recent theoretical work by different groups has sug-gested several different mechanisms which could lead toMBS-carrying TS in magnetic nanowires placed on supercon-ducting substrates. The earliest such mechanisms5254 mod-eled the nanowire as a chain of magnetic impurities in a spin-spiral phase. The spin-spiral, following previous work55, isused to mimic an effective spin-orbit coupling that would inturn lead to an effective triplet pairing superconducting prox-imity effect from the singlet superconducting substrate, ex-actly as in the existing semiconductor nanowire models oftopological superconductivity1315. The magnetic impuritieswere suggested to generate an array of Yu-Shiba-Rusinov(YSR)56 bound states in the superconductor. The combinedeffect of the superconductivity and spin-texture leads to an ef-fective Kitaev chain model4 that can support Majorana boundstates under appropriate conditions5254. However, the theo-retical plausibility of creating such a spin-spiral phase57,58 wasdebated, and it was shown that such spin spirals are unstabletoward the formation of purely ferromagnetic or antiferromag-netic phases59.

The absence of a spin-spiral in the experimental systemhas led to the conjecture of an alternative mechanism involv-ing the strong spin-orbit coupling of the Pb superconduct-ing substrate itself contributing to topological superconduc-tivity in the magnetic nanowire59. The basic model, whichhas been studied in this context by several authors44,50,51, pro-poses that only the spin-triplet component of Cooper pair-ing, if any, may be proximity-induced in a ferromagnetic wirefrom a spin-orbit coupled superconductor. This mechanismhas previously been proposed as an approach to topologi-cal superconductivity46 and also been invoked48 to explainthe long-range proximity-effect observed through ferromag-netic nanowires49. The mechanism of triplet proximity ef-fect on a ferromagnetic wire arising from a spin-orbit cou-pled superconducting substrate has been studied in detail bythree of us recently and shown to potentially support MBS-carrying topological superconductivity in the BDI chiral sym-metry class51. Such a symmetry would suggest unsplit Majo-rana modes whenever the effective chemical potential in theferromagnetic wire is positive.

In this context, it may be useful, particularly for laterdiscussion of the experimental results44, to distinguish twocomplementary and distinct theoretical models for topo-logical superconductivity and Majorana bound states in aone-dimensional (or quasi-one-dimensional) ferromagneticnanowire (Fe in Ref. 44) lying on a two-dimensional sur-face of an underlying bulk superconductor (Pb in Ref. 44).One model, which we refer to as the Shiba model (or Shibachain model), discussed in Refs. 53 and 50 respectively forthe helical and the ferromagnetic magnetic order in the wire,describes the magnetic atoms (Fe in Ref. 44) as essentially

independent quenched classical magnetic impurities with lit-tle direct inter-atomic hopping along the chain, i.e. the one-dimensional band width of the chain is basically zero (orequivalently, vanishing inter-atomic hopping amplitude t).The other model, which we refer to as the nanowire (or sim-ply, the wire) model, introduced in Ref. 51 for the ferromag-netic order in the magnetic chain, describes the ferromagneticchain as strongly directly tunnel-coupled along the chain withconsiderable inter-atom hopping leading to one-dimensionalbands of fairly large band-widths (or equivalently, large inter-atomic hopping amplitude t). These two models have beenrecently introduced and studied in the context of a ferromag-netic chain on a superconductor (i.e. the experimental sys-tem of Ref. 44) in Refs. 50 and 51 respectively, where it hasbeen explicitly pointed out that the models are complemen-tary, and depending on the nature of the ferromagnetic chain(i.e. whether there is or is not considerable hopping whichis defined simply by whether the inter-atomic hopping energyalong the chain is smaller or larger than the superconductinggap in the substrate superconductor) one or the other modelwill apply, and by definition, there cannot be a situation whereboth models apply simultaneously. The accompanying bandstructure calculations for the Fe chain on Pb presented in theexperimental work44 clearly show that the hoping term on thechain t is of the order of eV whereas the superconducting gapin Pb is of course of the order of meV. Therefore, the systemstudied in Ref. 44 is deep inside the ferromagnetic nanowireregime very far from the Shiba chain model. To apply theShiba model to the experimental situation of Ref. 44, one willhave to assume absurd superconducting gaps of eV size (withsuperconducting critical temperatures which would be > 104

Kelvins!). Thus, we present all our results using the nanowiremodel introduced in Ref. 51, and not the Shiba chain modeldiscussed in Refs. 53 and 50. The reason we are emphasiz-ing this seemingly simple and obvious conceptual point is thatRef. 44 has a confusing interpretation of the experimental datapresented therein, where depending on different aspects of thedata, the theory used in Ref. 44 has randomly varied betweenthe Shiba model and the nanowire model, which are, as de-scribed above, completely incompatible with each other. Ob-viously, the Shiba chain model has no place in the analysis ofthe experiment in Ref. 44 since the inter-atomic hopping en-ergy along the Fe chain is thousands of times larger than thesuperconducting gap of Pb. As we discuss later in this work,this fundamental inconsistency between different aspects ofthe results presented in Ref. 44 remains unresolved with thereported induced topological gap being 104eV and the ob-served strong lattice-level, < 5 nm, localization of the Ma-jorana mode requiring an estimated superconducting gap of 1eV . This basic incompatible dichotomy must be resolvedbefore the observations of Ref. 44 can be considered to be ev-idence for the existence of localized MBS in the Fe/Pb hybridsystem.

Another extremely important physical parameter in the ex-periment of Ref. 44 is the very high experimental temperature(T 1.2K) which is comparable to the estimated inducedtopological superconducting gap ( .12 meV ) extractedin Ref. 44. The fact that kbT in the experiment makes

3any discussion of a precise zero-energy Majorana mode quitemeaningless because at such high temperatures, the Majoranaresponse will be broadened over the whole subgap regime (orperhaps even above the gap) with the Majorana signal indistin-guishable from any ordinary fermionic subgap state. We alsopoint out that the experiment of Ref. 44 does not actually ob-serve any obvious superconducting gap in the ferromagneticnanowire, and the evidence for the existence of any topologi-cal gap is indirect. In fact, the instrumental energy resolutionin Ref. 44 is also of the order of the topological gap and thetemperature, making any discussion of possible weak subgapfeatures as representing anything definitive somewhat prema-ture.

The current work is an extension of Ref. 51 carried outin the context of the putative experimental MBS observationclaimed in Ref. 44 in order to provide a detailed critical com-parison between theory and experiment, which is necessarysince the rather strong claim of a direct observation of Ma-jorana modes must be thoroughly validated from all possibleperspectives. It may be useful in this context to emphasizethat the recent spurt in the experimental MBS activity, includ-ing both the earlier work on semiconductor (InSb and InAs)nanowires1621 and the very recent work on Fe nanowires44, iscompletely dependent on theoretical predictions and analysesfor its validation (i.e. these are not independent serendipi-tous experimental discoveries happening on their own spon-taneously) since the observations themselves involving tinyzero-bias tunneling conductance peaks at low temperaturesin rather complex hybrid systems are remarkably unremark-able, becoming noteworthy only because theories specificallypredicted that such zero-bias tunneling peaks should exist inthese specific hybrid structures as MBS signatures. In partic-ular, Ref. 13 not only predicted the existence of the Majoranabound states in semiconductor-superconductor hybrid struc-tures, specifically laying out the type of structures (and thematerials) experiments should use, but also carried out realis-tic calculations showing that the resulting MBS-induced zero-bias tunneling peaks should have a small height (because offinite temperatures, tunnel barrier heights, and wire lengths)compared to the expected quantized value60 associated withthe perfect Andreev reflection anticipated for MBS. This earlypaper13 also specifically suggested the use of STM in order tolook for topological zero energy Majorana excitations in hy-brid systems as has eventually been accomplished in Ref. 44following the later suggestion in Ref. 52 of using an STM cou-pled specifically with a magnetic chain on a superconductor.

While topological superconductivity in the chiral symmetryclass has been established for ferromagnetic wires with a sin-gle spatial orbital per atom, the number of Majorana modesarising from such a model is limited to two. On the otherhand, the band-structure calculation for the experimentally re-alistic system44 suggests that the number of channels in thewire can be significantly enhanced by the presence of mul-tiple orbitals per atom and multiple atoms along the diame-ter of the chain. In this work, we consider a multichannelgeneralization of the FM heterostructure and its topologicalproperties starting from the nanowire model of Ref. 51. Non-trivial zero-bias phenomena appear across a broad range of

parameters in contrast to the fine tuning necessary for a non-trivial topological phase in class D1115 or DIII systems2736.Within this framework, manipulating the system width (i.e.coupling parallel magnetic chains) enhances or reduces thezero-bias conductance peak (ZBCP) height accordingly. Ma-nipulation of the zero-bias conductivity tuned by the widthof the magnetic chain would be a direct signature of the chi-ral class BDI topological superconductors. Additionally, wecalculate spatially resolved scanning tunneling conductanceprofiles including effects of finite temperature and finite sizeof the wire (as well as the finite tunnel barrier effects) whichare experimentally accessible by STM. Finally, we show thatthe fractional Josephson effect maintains its 4pi periodicityin phases supporting multiple spatially overlapping MBS andcomment on how the Josephson current may be enhanced inthe presence of Majorana multiplets4,14 for a definitive con-clusion regarding the existence of MBS in the ferromagneticnanowire system of Ref. 44.

It may be useful to point out the connection between MBSin much-studied semiconductor nanowire systems with that inthe new platform of interest in Ref. 44 involving ferromag-netic nanowires. Although it may appear at first sight that thetwo systems are completely distinct, from a theoretical per-spective the MBS in the ferromagnetic wires are described byessentially the same theory as developed earlier for the semi-conductor nanowires in Refs. 1315, provided that one is inthe nanowire limit of large inter-atomic hopping along thechain (and not in the Shiba limit), and that one is in the limit ofthe spin splitting (induced in the semiconductor case by an ex-ternal magnetic field or by a proximate exchange splitting) be-ing very large (much larger than the other energy scales in theproblem including the spin-orbit coupling energy, the Fermienergy, and the superconducting gap in the ferromagnetic wirecase). In this large spin-splitting limit, the semiconductor sys-tem is also essentially an effective half-metallic ferromag-net exactly as the Fe wire studied in Ref. 44 is claimed tobe. In the semiconductor nanowire case also, the topologicalsuperconducting phase will be generic in this very large spin-splitting limit since the chemical potential would by definitionbe in the single spin polarized subband, with the supercon-ducting gap being smaller than the spin splitting. Thus, thedistinction made between semiconductor nanowires and mag-netic nanowires with respect to topological superconductivityis a distinction without much difference, since one can takethe existing theory for the semiconductor nanowire and ob-tain all the necessary formula for the ferromagnetic wire caseby assuming the spin-splitting to be by far the largest energyscale. (We provide the details on this connection to the semi-conductor nanowire system in Appendix A). We emphasize,however, the obvious fact that although the ferromagnetic wirecase can be thought to be a limiting situation (i.e. very largespin-splitting limit) of the semiconducting nanowire Majo-rana theory, the two experimental systems (namely, the semi-conductor nanowire in the presence of an external magneticfield inducing spin-splitting and the ferromagnetic nanowirewith its spontaneous exchange-driven spin-splitting) are, ofcourse, completely different physical platforms from an ex-perimental perspective utilizing totally different materials and

4measurement techniques. The emergence of this ferromag-netic nanowire platform44, in addition to the already existingsemiconductor nanowire platforms1621, is therefore an excit-ing new development in the search for MBS and topologicalquantum computation.

II. EXPERIMENTAL SETUP AND THEORETICALMODEL

Taken very close to a sample surface, an STM can be usedas an electrode to measure transport properties (Fig. 1). Amovable scanning point contact tunneling experiment is po-tentially very useful in investigating the edge character of Ma-jorana zero modes since the STM is particularly well-suitedin measuring the local density of states. This idea, origi-nally proposed in Ref. 13, is rather impressively implementedin the highly demanding spin-polarized STM measurementspresented in Ref. 44. Provided the electrical contact is goodbetween the ferromagnet (Fe nanowire) and the supercon-ducting substrate (Pb), Cooper pairs will leak into the ferro-magnet, thereby proximity inducing superconductivity in thenanowire. We model a finite quantum wire with dimensionsLy Lx L by considering a Nx Ny site square latticewith unit spacing. The effective Hamiltonian for the topolog-ical superconductor is HTS = Ht + HS=0 + H

S=1 + HZ

where

Ht =ij

t[cicj + H.c.

]j

jcjcj

HS=0 =j

scjcj + H.c

HS=1 =j

ip(cjcj+1 cjcj+1) + H.c

HZ =j

cj(V )cj

(1)

Here cj is the electronic creation operator for site j, ij in-dicates nearest neighbor sites, = (x, y, z) is the vectorof Pauli matrices, t is the hopping amplitude in the nanowire,and is the chemical potential. The superconducting pair-ing is a mixture of singlet and triplet terms. For the tripletpair potential term we have taken a Cooper pairing with spinprojection Sx = 0, which is the familiar equal-spin-pairing = . The Zeeman spin-splitting due to an internalmagnetization in the ferromagnet is M is V = gBM =(Vx, Vy, Vz) where g and B are the Lande g-factor and Bohrmagneton respectively. We note that our effective Hamilto-nian, as given in Eq. 1, describes the TS phase of the fer-romagnetic nanowire assuming that the degrees of freedomof the underlying superconducting substrate (Pb in Ref. 44)have been integrated away with Eq. 1 now describing only theelectrons in the Fe magnetic wire. We refer to Ref. 51 forthe details on how to obtain Eq. 1 which is our starting pointin the current work. We note that in this context our effectivemodel, derived from Ref. 51 which should be consulted for thedetails, describes only the ferromagnetic nanowire, hiding all

PbW

L

STM

V

xy

z

FIG. 1. (Color online) a) Schematic diagram of the proposedheterostructure involving a series of ferromagnetic quantum wires(gray), with large intrinsic magnetization M, deposited on top ofa spin-orbit coupled s-wave superconductor such as Pb (blue sub-strate). Spin singlet and triplet pairing potentials are proximity in-duced in the FM wires due to the strong spin orbit coupling and inter-orbital mixing in the superconductor. A STM probe, at coordinatex, measures the spatial dependence of the differential conductancealong the longitudinal axis.

information about the underlying superconducting substratewith the parameters for the spin-orbit coupling, the bulk su-perconducting gap of the substrate, the hopping amplitude ofCooper pairs between the substrate and the nanowire induc-ing the singlet and triplet proximity effect, etc. being implic-itly contained in the induced superconducting pair potentialss and p, which we use as phenomenological parameters tobe obtained from the experimental measurements themselves.Our goal here is to obtain the phenomenological consequencesof the minimal topological nanowire model (i.e. Eq. 1) for theferromagnet/superconductor hybrid system to make observ-able predictions and to carry out comparison with the existingdata. We also ignore all nonessential complications such asthe number of orbitals per Fe atom and the effective width ofthe wire, and so on which can be absorbed in the multichan-nel generalization we consider below (i.e. the W-parameterdenoting the number of active wire channels as described be-low). Our goal here is to utilize the minimal model and workout its implications in great details. Our Eq. 1 serves as theminimal model for the experimental system of Ref. 44 in thecurrent work.

Throughout this work we fix all of our parameters rela-tive to the hopping integral t in the nanowire. To begin andto establish our general results, we use s = p = t/10,V = Vz = 2.0t, L = 100 while W (the number of transversechannels) and the chemical potential are allowed to vary.For simplicity and numerical convenience, we will chooses = p. Choosing two such values s = p = 0.1tand s = p = 0.01t (both of which are orders of magni-tude larger than reported in Ref. 44 assuming t 2 eV asgiven in Ref. 44) will allow us to estimate the order of mag-nitude of parameters such as the Majorana decay length (seeFig. 5). We will present our numerical results for a few val-ues of L,, and T including T = 0 results (for the sake of

5comparison). A discussion concerning experimentally realis-tic parameters and their effect on the measured tunneling con-ductance is left to a later section. We note that this choice ofgeneric parameters incorporates the half-metallic character ofthe ferromagnetic wire since only one spin subband is occu-pied for a large range of chemical potential values keeping thesystem in the topological phase without any additional fine-tuning of parameters. Solving Eq. 1 directly numerically wefind zero energy Majorana states which are localized at eachend of the wire. The evolution of the low energy spectrum asa function of the chemical potential, as well as a function ofthe number of zero energy modes, is presented in the bottompanel of Fig. 2. To understand how Eq. 1 realizes an integernumber of Majorana zero modes we analyze the topologicalproperties of this model in the next section.

III. TOPOLOGICAL PROPERTIES AND QUANTUMPHASE TRANSITIONS

According to the Altland-Zirnbauer classificationscheme23, free fermion systems are characterized bytheir dimensionality as well as by the presence and the sign ofanti-unitary symmetries. There are ten topological classes intotal and five of them are non-trivial (i.e. a non-trivial topo-logical invariant can be defined) for a given dimension. Thetwo anti-unitary symmetries used are time-reversal symmetry(TRS) and particle-hole symmetry (PHS), with the latter oftenbeing referred to as the charge conjugation symmetry. Denot-ing the TRS and the PHS operators by and respectively,the anti-unitary symmetries are present when the followingreality conditions are satisfied: H1 = UHU

= +H

and H1 = UHU = H . Here U, denote the

unitary part of the TR and PH operators. A system is chiralinvariant (or sublattice symmetric) when both TR and PHare present and is given by the unitary operator S = .The classification triplet (T,C, S) = (2,2,S2) is used toindex each symmetry class, where TR and PH operators cansquare to 1 and the chiral operator is restricted to S2 = +1.We write O2 = 0 if an operator is not present.

Invariants are generally formulated in terms of the bulkHamiltonians topology, so we now look at a strictly 1D ver-sion of HTS . Fourier transforming Eq. 1 with a single spa-tial channel (i.e. no transverse hopping), the momentumspace Bogoliubov-de Gennes (BdG) Hamiltonian becomesH =

k kH(k)k where

H(k) = (2t cos(k) )0z (2)+ [s0 + p sin(k)d ] x+ V 0.

Here k kx is the one-dimensional crystal momentumand k = (ck, ck, c

k,ck)T is our four component

Nambu spinor which acts on the particle-hole ( ) and spinspaces (). In our calculations we use d = (1, 0, 0),V =(0, 1, 0) but leave d,V in Eq. 3 to highlight the generic prop-erties of the various symmetry classes. Given our choice ofbasis, the anti-unitary TR and PH operators have the matrix

Energ

y

1

2

3

4

1

0

1

0

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5/t

0.4

0.2

0.0

0.2

0.4

E/t 0

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5/t

0.4

0.2

0.0

0.2

0.4

E/t 0

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5/t

0.4

0.2

0.0

0.2

0.4

E/t 0

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5/t

0.4

0.2

0.0

0.2

0.4

E/t 0 1 2 3 4 6

b)a)

c)

FIG. 2. (Color online) Comparison of the topological properties ofthe class D spin-orbit coupled semiconducting nanowire (red) andour ferromagnetic system (blue). In panel (a) an externally appliedmagnetic field induces a Zeeman splitting between the originally de-generate spin bands. This system is topological non-trivial if an oddnumber of bands is occupied and Cooper pairs are supplied by anearby superconductor. When the Fermi energy lies in the shadedregion, the class D Z2 invariant non-trivial and a single Majoranabound state emerges at each end. b) Normal state band structure fora multichannel ferromagnetic wire (see Fig. 1). In the presence ofproximity induced p-wave pairing, the FM is promoted to the topo-logical class BDI, which is characterized by a non-trivial Z invariantfor a generic band occupancy. Additionally, the large intrinsic mag-netization provides a broad non-trivial parameter regime in which anon-trivial topological state persists even if the chiral symmetry isbroken, say, by a second Zeeman field perpendicular to the magne-tization (BDI D). In this case, the shaded regions with an odd Zinvariant remain non-trivial while the others become trivial. (c) Lowenergy quasiparticle spectrum as a function of the chemical potentialin the ferromagnetic wire. As increases, the number of Majoranazero modes at each end of the FM wire increases by one followingeach gap closing.

structure = iy0K and = yyK where K is the com-plex conjugation operator.

In momentum space, the reality conditions for BlochHamiltonians are61

H(k)1 = +H(k) (3)H(k)1 = H(k).

PH symmetry emerges from the BCS mean-field theory andis intrinsic to all BdG Hamiltonians, so in the absence ofany additional symmetries (T,C, S) = (0, 1, 0). This tripletcorresponds to the topological class D, which is charac-terized by a Z2 topological invariant in d = 1. Recall,Z2 Z/2Z is the cyclic quotient group with two ele-ments {0, 1}. The topological invariant for class D sys-tems is given by Kitaevs Majorana number, which is de-

6fined as4 M = sgn[Pf(A(0))Pf(A(pi))

]where A is the

momentum space Hamiltonian written in a skew symmet-ric form, determines when the system is topologically non-trivial. The MBS-carrying spin-orbit coupled semiconductor-superconductor heterostructure proposal1115 belongs to thetopological class D for the most general types of spin-orbitcoupling. However, the specific models studied in the originalproposals1115 assumed that the spin-orbit direction was per-pendicular to the Zeeman coupling. These models are there-fore in a more restricted BDI class that will be discussed fur-ther at the end of this section because of its relevance to theferromagnetic wire model. In this system, a Zeeman fieldsplits degenerate spin-orbit coupled bands. The goal of thissplitting is to remove a single Fermi surface thus renderingthe system effectively spinless. Typically, the Zeeman split-ting is small compared to the bandwidth, resulting in a smallnon-trivial topological parameter range (see Fig. 2). In thiscase, the difficult task of fine tuning the chemical potential,by using gate electrodes for example1621, may be necessaryto achieve a non-trivial topological state if the chemical po-tential lies near half filling for a subband. As emphasizedat the end of the Introduction, however, we are free to takethe very large spin-splitting (i.e. very large Vz) limit of thesemiconductor model (although this would not be a particu-larly physically relevant model for semiconductors per se, itis a perfectly allowed theoretical limit), which then coincideswith the current ferromagnetic wire situation of interest to theexperimental system in Ref. 44 (see Appendix A for the de-tails).

If Zeeman splitting is absent in Eq. 3, then the first real-ity condition from Eq. 3 is satisfied. Using = iy0K wesee that T = 1 so a class DIII TR invariant system is char-acterized by the triplet (T,C, S) = (1, 1, 1). Class DIII ischaracterized by a Z2 topological index which is related to aKramers polarization62 similar to the way the class D invariantis related to the electric polarization of the wire61. If H be-longs to this class, any hybridization between time-reversedMajorana zero modes is forbidden by Kramers degeneracy,and each end of the wire hosts a perfectly degenerate Majo-rana Kramers pair.

In addition to the two classes discussed above, supercon-ducting systems can belong to the topological class BDI. Notethat there exists the chiral operator S = d y which anti-commutes with the Hamiltonian in Eq. 3; note in the TR in-variant case the chiral operator is SDIII = 0y . This unitarychiral operator must be a product of two anti-unitary opera-tors, one of which is . By simple algebra, one can showthat our missing operator is O = (d y + i(d y) )Kso that O2 = 1 (i.e. T = +1) and that H(k) satisfiesOH(k)O1 = +H(k). We continue to call this operatorO, even though it leads to the same reality condition as ,in order to distinguish it from the usual time reversal sym-metry. A crucial difference between classes BDI and D/DIIIis that the former is characterized by an integer Z invariant.Because the invariant can take any integer value, multiplespatially overlapping MBS can coexist in contrast to class Dsystems where localized zero-energy anyonic MBS hybridize

into conventional finite-energy fermionic quasiparticle states.As illustrated in Fig. 2 panel (b), a BDI chiral system is non-trivial for a generic parameter range. We numerically diago-nalize and plot the low energy quasiparticle spectrum forHTSas a function of the chemical potential in Fig2 panel (c). TheMajorana occupancy grows when increases and successivehigher energy bands are occupied. Therefore for any genericchemical potential one expects a non-trivial topological statewith end-localized zero energy MBS.

Composing the two reality conditions in Eq. 3 we seethat the chiral operator satisfies {S, H(k)} = 0. This anti-commutation relation implies that in the eigenbasis of S theHamiltonian is off-diagonal,

H (k) =(

0 A(k)A(k) 0

). (4)

Here we have used U to represent the unitary transformationmatrix between the original and the chiral basis. For a sin-gle channel A(k) is a 2 2 complex Hermitian matrix whosedeterminant D(k) Det(A(k)) is generally complex. Obvi-ously the complex phase exp[i(k)] = D(k)/|D(k)| lies onthe unit circle and we have established a mapping from theBrillouin zone (S1 in 1D) to U(1) . The fundamental grouppi1(U(1)) = Z is well defined here so that we may write thetopological winding invariant as37,

W = 12pi

2pi0

argD(k)dk. (5)

The integerW counts the number of times the complex argu-ment (k) winds about the origin in the complex plane andis invariant under smooth deformations. In other words, Wcan change only if the winding curveD(k) passes through theorigin. However, by looking at the from of Eq. 4 we knowthat the k-point where D(k) vanishes constitutes a gap clos-ing with a concomitant topological quantum phase transition.Note that the DIII chiral operator is odd under time-reversalsymmetry, {SDIII ,} = 0 which implies that the end modeswith chiral charge +1 are compensated by an equal number ofmodes with charge 1. Therefore, while this procedure maybe mathematically well defined, it is trivial in the sense thatthe net DIII chiral topological charge always vanishes.

The winding number defined in Eq. 5 can also be usedto calculate the chiral topological invariant for multichannelwires63. The quasi-one-dimensional Hamiltonian used in thisprocedure is one in which a Fourier transform has been per-formed along the longitudinal x-direction, but not along they-direction. Using l, l [0,W ] to indicate the y-coordinate,we write HTS =

kll

kl(H(k)l,l + H

l,l)kl where

Hl,l = t0z(l,l+1 + l,l1). We use the procedure out-lined above, whereA(k) is now a 2W 2W dimensional ma-trix and multiple Bloch bands can now be mapped to U(1) bythe determinant function. The result is sketched in Fig. 2 panel(b). As the chemical potential increases and higher bands arefilled, the gap closing occurs in the spectrum of HTS andthe corresponding topological invariantW increases by unity.We emphasize that when the chemical potential is in the low-est spin-split band (Fig. 2), the topological phase is generi-

7cally present in this half-metallic FM situation since the spin-splitting is much larger than the induced superconducting gap.We shall now discuss the experimental signatures which are aconsequence of our model.

IV. SCANNING TUNNELING DIFFERENTIALCONDUCTANCE

Consider an STM brought close to the surface of the multi-channel FM wire described by Eq. 1 (see Fig. 1) . The STMtip weakly couples to FM wire orbitals through a small hop-ping integral HSTM =

t(csds + H.c.). Here d annihi-

lates electrons at the STM tip which we take to be three siteswide and centered the x-coordinate s = (x 1, x, x+ 1). Wewill parametrize the tunneling barrier at the STM tip (whichdetermines the size of the zero bias tunneling peak at finitetemperatures13,60) by the single parameter t for simplicitytypically t t in the STM set up of Ref. 44. A poten-tial difference V is now applied between STM and drain (i.e.the grounded superconductor on which the FM has been de-posited). We will now set up a scattering matrix formalismto calculate the differential conductance through the FM wire,in order to experimentally detect the MBS. Within this ap-proach we model the STM, which is the first scattering lead,as a normal metal electron reservoir biased at a variable elec-trochemical potential N + eV measured relative to the su-perconducting Fermi energy. Our quasi-one-dimensional FMwire (Eq. 1) acts as the scattering region and the second leadis the grounded electron drain which is held at chemical po-tential N . We adopt a BTK perspective64 in assuming thatequilibrium Fermi distribution functions determine the incom-ing quasiparticle occupancy levels. Here, in = (Sin,

Din)

T

are plane waves originating deep within the semi-infinite leadSTM and drain leads which are described by the Fermi func-tions f(E eV ) and f(E) respectively. Note, in generalS,(D) is an N(M) component spinor given N(M) occupiedchannels in the STM (drain) lead. For a quantum coherentprocess we can relate the outgoing modes to the incomingmodes by the scattering matrix out = Sin where

S =

(r tt r

). (6)

Here r is a 4N 4N matrix consisting of complex reflectioncoefficients between all the occupied incoming STM chan-nels. Likewise r is the reflection matrix for the drain andt, t are the transmission coefficient matrices connecting thetwo leads. (Note that we use the same notations t, t to de-note the transmission matrix elements for the leads as whatwere used to define the tunneling amplitudes in defining thebasic Hamiltonian, but there is no scope for any confusionhere since the transmission matrix elements t, t only appearin Eq. 6 above in defining the S-matrix and in our numericalwork and nowhere else in the text below.)

In the presence of a proximity induced superconductinggap, single electrons cannot tunnel from the STM to the FMfor low bias-voltages V . As a result, all of the flowingcurrent in the subgap regime is generated through the Andreev

reflection process in which excess Cooper pairs are createdand simultaneously the incident electrons are converted intoholes. The reflection matrix can be written as

r =

(ree rehrhe rhh

). (7)

where ree (reh) refers to the normal (Andreev) reflection sub-matrix. At low bias voltages the differential conductance, pro-portional to the transmission probability at a given energy E,is expressed in terms of the STM reflection matrices as65

dI(V )

dV=e2

h

[N Tr(reeree) + Tr(rehreh)

]E=V

. (8)

We generate the scattering coefficients numerically using theKwant66 numerical package.

A. Results

We know from the topological properties discussion inSec. III that MBS appear as soon as any FM bands, withina normal state picture, become occupied. Setting V = 0 andusing Eq. 8 we see a peak in the zero-bias conductance, quan-tized in units of 2e2/h, abruptly appearing at the critical valueof the chemical potential when the first band becomes occu-pied ( 0.7t) as shown in panel (a) of Fig. 3. As in-creases, higher sub-bands are filled while the Majorana oc-cupancy increases, and each MBS contributes its own factorof 2e2/h to the total zero-bias differential conductance. Thezero-bias peak is a direct probe of the Majorana occupancyas the plot for W = 6 in Fig. 3 (a) clearly mirrors the thezero-energy excitation spectrum given in Fig. 2. In realisticexperiments, because of disorder and electrical contact com-plications, it is difficult to increase the chemical potential uni-formly across an entire sample in order to induce a topologicalphase transition. It is for this reason that we instead proposemanipulating the system width, i.e. tightly packing parallelmagnetic atomic chains, as an experimental test of the chiraltopological state. Fig. 3 panel (a) illustrates the zero-bias sig-nal behavior for various values of W . Samples with differentwidths are expected to have a similar chemical potential, butthe strength of the zero-bias peak at that should increase (reddashed line for example) as a function of W . It is also impor-tant to note that while finite temperature effects generally sup-press the ZBCP height (as seen below), this effect is uniformand transitioning from W = 2 to W = 4 at correspondingto the dashed line, would still double the zero-bias signal. Theobservation of such jumps in the ZBCP height with increas-ing the number of wires or channels will be a strong indicationthat the ZBCP is indeed arising from the localized MBS in theferromagnetic wires in the BDI class.

Typical differential conductance profiles over finite voltagerange are presented in Fig. 3 panel (b). The green, red and bluelines are dI/dV profiles generated for values correspondingto topologically distinct phases indexed by an integer windinginvariant |W | = (0, 1, 2). A superconducting gap devoid ofsubgap states (V .1t) is characteristic of the trivial regime

80.3 0.2 0.1 0.0 0.1 0.2 0.3V/t

0.0

0.1

0.2

0.3

0.4

0.5

dI/d

V(e

2/h

)

= 0.6t = 0.0t

= 0.7t

0.3 0.1 0.1V/t

0

2

4

0.3

1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0/t

0

2

4

6

8

10

12dI/dV(e2/h

)| V=0

W = 2

W = 4

W = 6

W = 8

a)

b)

c)

0.2 0.1 0.0 0.1 0.2V/t

0.00

0.05

0.10

0.15

0.20

0.25

dI/d

V(e

2/h

)

t = 0.4tt = 0.3tt = 0.2tt = 0.1t

0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20V/t

0.0

0.5

1.0

1.5

2.0

dI/d

V(e

2/h

)

T = 0

T = 0.005t

T = 0.01t

T = 0.02t

d)

FIG. 3. (Color online) (a) The zero-bias differential conductance asa function of chemical potential for various wire widths W usinga topological gap = 0.1t and a nanowire length of L = 200sites. Since each Majorana modes contributes a factor of 2e2/h tothe zero-bias signal, this measurement directly probes the number ofMajorana states present. For a given geometry the maximum possi-ble conductance is Gmax = 4We2/h (not shown here). (b) Rep-resentative dI/dV curves from parameter regimes with an integertopological invariant |W | = (0, 1, 2) are given by the green, blueand red curves respectively. Inset shows the quantized peak heightfor the blue (2e2/h) and red (4e2/h) curves. (c) Finite temperaturethermally broadens the zero-bias conductance peak width, therebyreducing the peak height to well below the quantized value of 2e2/h.(d) Weak STM - FM nanowire coupling, i.e. small t, in conjunctionwith finite temperature (T = 0.05t) further reduces the peak height.Note that the abscissa corresponds to an energy range much largerthan the topological gap given by = 0.1t.

(green curve) while a quantized zero-bias signal appears in thenon-trivial regimes.

The conductance at finite temperature T is given by

dI(V, T )

dV=

dV dI(V , T = 0)

dV d

dVf(V, T ), (9)

where f(E, V, T ) = (exp [(E eV )/T ] + 1)1 is thefermi function. In this paper all zero temperature resultswill be assumed to be smeared by an infinitesimal tempera-ture T = 105t. The finite temperature is crucial to avoidanomalies that depend on exponentially small coupling be-tween Majorana modes which must exist in any finite lengthsystem no matter how long the wire is. As seen from previouscalculations67 the zero-bias conductance vanishes at strictlyzero temperature even for a topological system. However,this anomaly reduces to the usual result of a quantized con-ductance at temperature T larger than the exponentially smallMajorana splitting energy, but smaller than the tunneling en-ergy between the Majorana mode and the lead. Strictly speak-ing, the tunnel coupling t between the STM tip and the Fenanowire is unknown in the experiment of Ref. 44 except thatit is known to be very small. On the other hand, the exper-imental temperature in Ref. 44 is very high, > 1K, so thecondition t > T is probably not satisfied in Ref. 44. For-tunately, this does not cause any qualitative problem in thetheoretical analyses where most of the experimental param-eters, except for the temperature, are not precisely known.Thermally smeared differential conductance curves are plot-ted in Fig. 3 panel (c) for various temperatures. Similar tothe zero-bias phenomena observed in recent semiconductorexperiments, where the peaks are generally an order of mag-nitude smaller than 2e2/h1618,20,21, thermal effects smear ourzero-bias peaks to well below its quantized value as was al-ready pointed out in Ref. 13. Furthermore, the very weak cou-pling between the STM and the ferromagnetic nanowire, i.e.t t, in conjunction with finite temperature further reducesthe ZBCP height. Choosing a temperature of T = 0.02t weillustrate this phenomena in Fig. 3 panel (d).

By varying the STM coordinate x, we now simulate thetunneling spectra which would result from spatially sweepingthe STM probe across the length of the sample, which hasrecently been experimentally achieved44. MBS are localizedat each end of the wire, and we expect the ZBCP to vanishas the STM reaches the wire midpoint. Fig. 4 a (b) showsthe zero (finite) temperature differential conductance spatialprofile. The signal due to tunneling into quasiparticle statesabove the superconducting gap remains approximately con-stant as the probe position varies, in contrast to the zero-biassignal which disappears in the bulk. A zero-bias spatial pro-file displayed in Fig. 4 (c) illustrates the exponential decay ofthe zero modes away from the edges as well as the end local-ization scaling with the characteristic length (see discussionin next section, Fig. 5). The features shown in Figs. 3 and4 are generally consistent with the experimental findings inRef. 44, providing some level of confidence that the experi-mentally observed ZBCP may indeed be arising from MBS-related physics (although the model parameters used in thesefigures are not realistic representations of the Fe/Pb systemused in Ref. 44). It is interesting to note that the very high tem-perature (T ) used in the experiment of Ref. 44 along with

90 20 40 60 80 100x

0.0

0.5

1.0

1.5

2.0

dI/d

V(e

2/h

) T = 0

T = 0.005t

T = 0.010t

T = 0.015t

20 40 60 80 100x

0.3

0.2

0.1

0.0

0.1

0.2

0.3

V/t

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36

x

20 40 60 80 100x

0.3

0.2

0.1

0.0

0.1

0.2

0.3

V/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

a)

b)

c)

FIG. 4. (Color online) Spatially resolved differential conductanceprofiles from the left edge to the middle of the nanowire. We use atopological pair potential = 0.1t and the length of the nanowireis L = 200 sites. Panel a) shows the zero temperature signal, whilepanel b) illustrates the effect of thermal smearing by a temperaturesmaller than the topological gap (T = /10 = 0.01t). Note thatfor panels a,b) the energy is swept across the range 3 to 3 andthat the color scales in the top panels differ by an order of magnitude.Additionally panel c) highlights the spatial structure of the zero-biassignal for various temperatures. See discussion below Eq. 9 for anote regarding the T = 0 result.

the very small tunnel coupling (t) to the STM tip not onlyleads to a very strong suppression of the ZBCP strength fromits quantized Majorana value of 2e2/h, but also suppresses therange of x values (indicating how far from the x = 0 end pointof the wire) over which the Majorana effectively resides asobserved in the STM conductance measurement. The fact thatthe experimental Majorana observation seems to be localizednear the wire end could just be a feature of the very high ex-perimental temperatures. Thus, it is imperative that additionalexperimental data is obtained with higher values of /T (ei-

ther by increasing the effective topological superconductinggap or by lowing the experimental temperature) before onecan reach a definitive conclusion regarding the existence ornot of Majorana fermions in the experiment of Ref. 44. Wenote that the experimental values of /T in Ref. 44 are muchlower than those used in Figs. 3 and 4, making it difficult,if not impossible, to reach any conclusion about the possibleexistence of Majorana fermions in the system.

In the next two sections, we provide a more detailed com-parison between our numerical results and the experimentaldata of Ref. 44.

V. EXPERIMENTAL IMPLICATIONS

Having established generic features of our model, we nowturn our attention to a comprehensive comparison with a re-cent experiment44 which shares many, but unfortunately notall, features with our theoretical results. Our focus here ismainly on comparing the qualitative phenomenological prop-erties of the experimentally observed ZBCP and our theoret-ical results. In addition to analyzing the height and widthof the ZBCP as a function of temperature, wire length, andthe STM tunnel barrier, we will also closely examine the spa-tial structure of the differential conductance profile, which canbe directly calculated from our STM simulation (see Fig. 4).To begin with, we first recapitulate the system parameters asquoted in Ref. 44 and then set up our numerical parametersaccordingly for comparison. The ferromagnetic splitting isestimated to be J = 2.4 eV , which is much greater thanthe estimated hopping parameter t = 1 eV (which in turnis much larger than the superconducting gap 1 meV in thesubstrate, thus allowing us to use the half-metallic ferromag-netic nanowire model for the theoretical description). Addi-tionally, the superconducting gap in the underlying substrateis s = 1.36 meV while the induced p-wave gap is estimatedto be p = 100 eV , although no direct nanowire supercon-ducting gap with well-developed coherence peaks is visible atall in the experimental data presented44. Measurements weremade on atomic chains between 515 nm in length at a tem-perature T = 1.4K which corresponds to 100 eV in energy(roughly equal to the topological gap). We mention here thatthese ferromagnetic nanowires are extremely short in length,containing only 10-50 Fe atoms only these wire lengths areby far the shortest lengths in the problem, being even shorterthan the superconducting coherence length ( 80 nm) of Pb,the substrate superconducting material. The topological co-herence length in the Fe wire ( 1000 nm) is more than anorder of magnitude larger than the length of the wires them-selves. The estimate for the coherence length in the Fe wireobtained here assumes the induced topological superconduct-ing gap to be 100 200 eV as provided in Ref. 44. Thisexperimentally used parameter regime is obviously a non-ideal regime for studying topological superconductivity sincethe lowest energy scale is the topological gap in the system,which is the same as the temperature of the system. Tempera-ture would therefore be expected to suppress any signatures ofthe topological gap, including the Majorana zero-mode, which

10

0 20 40 60 80 100 x0.0050.0100.0150.0200.0250.030

2

20 40 60 80 100 x0.050.100.15

2

100 200 300 400 500 600 700 x0.0050.0100.0150.020

2

100 200 300 400 500 600 700 x0.050.100.15

2a)

b)

c)

d)

=0.1t

=0.01t

=0.1t

=0.01t

500 1000 1500L

35302520151050

log(

E)

50 100 150 200 250L

35302520151050

log(

E)

= 99

= 10=0.1t

=0.01t

e)

f)

FIG. 5. (Color online) Majorana wavefunctions for small ( =0.01t), and large ( = 0.1t), pairing regimes as calculated for short(L = 100) and long (L = 700) systems. For both lengths consid-ered in the large pairing regime, i.e. panels a,b), the Majorana decaylength is much smaller than the system length ( L) and the zeroenergy excitations are heavily localized to the FM wire endpoints.Panels c,d) illustrate how reducing the pair potential to = 0.01tsubstantially increases the Majorana decay length (by an order ofmagnitude) and while a 700 site system still hosts Majorana states,considerable wavefunction overlap in the 100 site system hybridizesthe end modes into non-zero energy, conventional delocalized quasi-particle states. Panels e,f) show the logarithm of the energy splittingE between the two Majorana modes. As the wire length L in-creases E falls off exponentially. Majorana decay lengths of of 10 sites, in the large pairing regime, and 99 sites, for thesmall pairing regime, are extracted from the black linear fits in panelse,f) respectively.

would merge with the bulk states.

To compare our results with the experiment we introducetwo parameter regimes, each characterized by different Ma-jorana decay length scales, and then compare the results be-tween the two regimes. In the large pairing regime we take themagnitude of the superconducting pair potential to be p =0.1t while in the small pairing regime we use p = 0.01t.(To be clear, the system is always in a topological state andthis is not to be confused with the weak/strong pairing regimesof Ref 3 which describe topologically non-trivial and trivialphases.) In both cases we choose s = p and simplify ournotation by referring to this quantity simply as (keeping inmind that p is responsible for the topological properties). Inboth parameter sets we use = .65t, W = 2 and Vy = 2t.We also typically choose very small values of t( t) to sim-ulate the very large tunnel barriers occurring at the STM tipcontact with the nanowire. (Very small values of t are es-sential for obtaining extremely weak zero-bias signals for theMajorana modes as observed experimentally.)

In a finite system, the localized MBS wavefunctions expo-nentially decay into the bulk with the characteristic supercon-ducting coherence length vF /, thus acquiring a finiteenergy due to wavefunction overlap from the two end MBS ontwo sides (true zero modes only occur in the L limit).Fig. 5 panels (a,b) show the Majorana amplitude ||2 on sys-tems composed of L = 700, 100 sites in the large pairingregime. The Majorana decay length is clearly much shorterthan the wire length for both cases, so zero-energy Majoranabound states are localized at each end of the wire and with||2 being negligible near the midpoint. Panels (c,d) illus-trate the wavefunction amplitude in the small pairing regime,in which the Majorana decay length is comparable to the sys-tem size for the L = 100 case. The end modes appear unaf-fected in the L = 700 wire, however, the wavefunctions forMajorana modes bound to opposite ends of the wire overlapsignificantly in the L = 100 case, and as we discuss later, thishas important ramifications for the zero-bias signal. Note thatthe small gap used in the small pairing regime ( = 0.01t) iscloser to the experimentally quoted parameters which wouldindicate a minuscule value of = 104t (since the exper-imental system has t 1eV and 100eV ). We donot use even smaller due to the prohibitive computationalresources which would be required; however, the physics isgeneric and the topological pair potentials we have chosen arealready sufficiently small to illustrate our point. In fact, ourtheory is strongly over-emphasizing the topological aspects ofthe experimental systems all topological signatures will beweaker in the experiment compared with our results since theinduced gap is smaller in Ref. 44 than our chosen theoreticalvalue.

E, the MBS splitting, can be directly captured usingan effective Hamiltonian spanning the zero-energy Majoranasubspace, Heff = i(f/2)LR where f exp (L/). InFig. 5 panels (e,f) we numerically calculate E as a func-tion of length in order to determine the decay length. Plottedon a logarithmic scale, the red circles represent the raw datawhile the black linear regression has been fit to the data. Tak-ing f = exp (L/) we extract coherence (or equivalently,MBS localization) lengths of = 10, 100 sites in the large and

11

0.05 0.10 0.15 0.20t/t

0.02

0.04

0.06

0.08

0.10T/t

76543210

ln(dI/d

V| V=

0)

0.05 0.10 0.15 0.20t/t

0.02

0.04

0.06

0.08

0.10

T/t

12.010.59.07.56.04.53.0

ln(dI/d

V| V=

0)

0.05 0.10 0.15 0.20t/t

0.02

0.04

0.06

0.08

0.10

T/t

9876543210

ln(dI/d

V| V=

0)

0.05 0.10 0.15 0.20t/t

0.02

0.04

0.06

0.08

0.10

T/t

6.45.64.84.03.22.41.60.80.0

ln(dI/d

V| V=

0)

a) b)

c) d)

Lx=700, =0.1t

Lx=100, =0.01tL

x=700, =0.01t

Lx=100, =0.1t

FIG. 6. (Color online) Zero-bias differential conductance peak height(panels correspond to same parameter values as used in Fig. 5 panelsa-d), at one end of the wire (x = 0) as a function of STM-FM cou-pling t and temperature T . When the Majorana decay length is sig-nificantly shorter than L (i.e. panels a-c) a quantized zero-bias signalof 2e2/h (dark red on logarithmic color scale) is seen for zero tem-perature and t > 0.03t (note that the STM-FM coupling in Ref. 44is expected to be smaller than this critical value of t). The quantizedsignal decays rapidly by introducing finite temperature or decreasingt. Panel d) No zero-bias signal is present near zero temperature (seebelow Eq. 9 for details) due to Majorana hybridization which splitsthe zero-bias peak into two separate finite bias signals. The cause ofthis effect is finite temperature, which smears two finite-bias peakstogether for an effective zero bias signal (see to Fig. 9 for details).

small pairing regimes respectively. Note that the beading ontop of the exponential decay is due to constructive and de-structive interference between the MBS, and the length scaleof these oscillations goes as 1/kF 68.

Next, we consider the roles of STM-FM coupling t andfinite temperature T in the quantitative suppression of theZBCP strength in both parameter regimes. As we have alreadynoted in Fig.3 a small t reduces the ZBCP, which in conjunc-tion with thermal smearing effects, significantly reduces thezero-bias signal. As seen in Fig. 6 panels (a,b), the T = 0zero-bias signal in the large pair potential regime, where bothwire lengths support MBS, saturates to the quantized valueof 2e2/h as t becomes large. Increasing temperature or de-creasing t both monotonically reduces the magnitude of thezero-bias signal, and similar behavior is found in panel (d)(long wire in small pairing regime), which also hosts well de-fined Majorana excitations. Interestingly, as seen in panel (b),the short wire in the small pairing regime, i.e. one in whichMBS have hybridized due to the small wire length ( L),displays a finite temperature zero-bias signal comparable inmagnitude to the finite temperature signal seen in the otherpanels. Moving upward from the zero-temperature x-axis to-wards higher temperature, this signal grows until some criticalvalue, after which the zero-bias signal decays like in the otherpanels. As discussed later in this section, the source of thisunusual zero-bias peak behavior increasing with temperatureis thermal smearing between a pair of split Majorana statesnear zero energy.

Having established that a strongly suppressed ZBCP is ageneric feature of the experimental parameter regime (i.e.

0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08V/t

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

dI/d

V(e

2/h

)

t = 0.01tt = 0.005t

0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08V/t

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

dI/d

V(e

2/h

)

t = 0.01tt = 0.005t

x= L/4

x= 0

0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08V/t

0.00000.00010.00020.00030.00040.00050.00060.00070.00080.0009

dI/d

V(e

2/h

)

t = 0.01tt = 0.005t

x= L/2

a)

b)

c)

FIG. 7. (Color online) Differential conductance calculated using anSTM coordinate x = 0, L/4, , L/2 for a long wire L = 700 inthe small pairing regime ( = 0.01t) using an STM-FM couplingt = 0.01t. Solid and dashed lines are at temperature T = 0, 0.01trespectively, and green (red) lines correspond to a STM-FM couplingof t = 0.01t (0.005t). Panel a) shows a clear zero-bias signaturewhich is visible at the wire endpoints for both zero and finite tem-perature. The green dashed line (T = ) displays a peak height(103e2/h 40nS) and width ( ) which are comparable tothe experimentally reported values. Panels b,c) The zero temperatureMajorana peak decays as the STM moves into the FM nanowire bulk.The peak completely vanishes at the midpoint, and is not visible atzero or finite temperature signal. Note that the energy scale for theabscissa is much larger than the topological gap in these figures.

small t, large T , and small topological gap), observable withor without the existence of zero energy Majorana excitations,we now analyze the spatial profile of the ZBCP, which can inprinciple be used to distinguish between a signal originatingfrom zero energy or finite-energy split quasi-MBS. Focusingon the small pairing parameter regime first, i.e., the param-eters which are closer to those reported in Ref. 44, we plotthe differential conductance measured at three STM positionsx = 0, L/4, L/2 along a wire of length 700 sites (see Fig. 7).In panel (a) the conductance is measured from the wire end-point (i.e. x = 0) and we observe that finite bias quasipar-ticle states are separated from the Majorana signal by a gapwhich is comparable in magnitude to the pair potential (recall, = 0.01t). Note the zero-bias Majorana signal (green and

12

0.100.050.00 0.05 0.10V/t

50

100

150

200

250

300

x

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

dI/d

V(e

2 /h)

0.100.05 0.00 0.05 0.10V/t

50

100

150

200

250

300

x

12.010.59.07.56.04.53.0

log[dI/d

V](e2/h

)

a) b)

FIG. 8. (Color online) Differential conductance spatial profile asa function of the STM coordinate x for a long wire L = 700 inthe small pairing regime ( = 0.01t) using an STM-FM couplingt = 0.01t. Panels a,b) correspond to temperature T = 0, 0.01trespectively, and we present the zero temperature data on a logarith-mic scale for improved visibility. At zero temperature the spatiallylocalized Majorana mode (the localization length appears longer ona logarithmic scale) resides within a well defined superconductinggap which is not visible at finite temperature due to thermal smear-ing effects. Note that the true spatial extent of the Majorana mode isrevealed by the finite temperature result which is plotted on a linearscale and that the energy scale for the abscissa is much larger thanthe topological gap in these plots.

red solid lines are almost completely superimposed and there-fore not discernible) is delta function shaped as a consequenceof the small coupling parameter t. The dashed lines indicatethe signal at a finite temperature of T = = 0.01t, which isthe case in Ref. 44. The thermally broadened peak height forthe green dashed line is 103e2/h 40nS which is compara-ble to that reported in the experiment. Additionally, the peakwidth at half maxima is , which is also consistent with ex-perimental results. Panel (b) shows how by moving the STMtip into the bulk of the wire (x = L/4) the zero-bias signaldrastically falls off, to the point where it is of the same orderof magnitude as the conventional background thermal quasi-particle signal. Due to the large separation between the zero-and finite-bias signals, a valley, centered around V = 0, ap-pears in the thermally broadened conductance profile (dashedline). Lastly, in panel (c) we see that, as expected, the Majo-rana peak is completely absent at the wire midpoint x = L/2.All these features appear to be qualitatively consistent withthe experimental data reported in Ref. 44.

A detailed spatial profile of the differential conductanceshould reveal the highly localized Majorana wavefunctionfrom Fig. 5. In order to numerically reveal the localized natureof these Majorana wavefunctions, we smoothly vary the STMtip position x and plot the zero temperature differential con-ductance at each point in Fig. 8 panel (a). Along most of thewire, the spatially resolved dI/dV indicates a well formedsuperconducting gap separating the single Majorana peak atzero energy from the finite energy quasiparticles. Note that, inthis plot, the spatial extension of the Majorana wavefunctionis exaggerated due to the logarithmic scale which has beenused to increase the visibility of the data. Panel (b) shows thespatially resolved conductance at finite temperature (T = ),which reveals the true spatial extent of the Majorana mode.

0.10 0.05 0.00 0.05 0.10V/t

10

20

30

40

x

16.515.013.512.010.59.07.56.04.5

log[dI

/dV](e2/h

)

0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08V/t

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

dI/d

V(e

2 /h)

0.10 0.05 0.00 0.05 0.10V/t

10

20

30

40

x

0.00004

0.00008

0.00012

0.00016

0.00020

0.00024

0.00028

0.00032

dI/d

V(e

2 /h)

a)

b) c)

FIG. 9. (Color online) When the ferromagnetic nanowire is short (i.e.L ) Majorana bounds states at opposite ends hybridize as theirwavefunctions overlap significantly in the bulk. Panel a) Using thesame parameters from the Fig. 8 except L = 100, we see that thezero-bias Majorana signature splits into two distinct peaks centerednear V = 0.01t. The green, red, and blue curves correspond to thedifferential conductance calculated at x = 0, L/4, L/2 respectively,and the solid (dashed) lines denote the zero (finite T = = 0.01t )temperature. Panels b,c) Spatially resolved differential conductancealong the longitudinal axis for zero (on a log scale) and finite tem-perature (linear scale). In panel c) we see that the split Majoranaconductance peaks are indiscernible at finite temperature due to ther-mal smearing (signal at V 0). This finite temperature signal isflanked on both sides by broadened conventional quasiparticle peaks.Solid and dashed lines are used to indicate the STM position for thedI/dV curves presented in panel a). Note that the energy scale forthe abscissa is much larger than the topological gap ( = 0.01t) inthese plots.

We pause to note that while the model parameters used hereare similar those quoted in the experiment, the localizationlength seen in panel (b) is significantly larger than reported inRef. 44.

Since small pair potentials and short wire lengths are quotedin Ref. 44, we now investigate the small pair potential regime( = 0.01t, = 99) on the L = 100 system, i.e. parame-ters which should be most applicable to the experiment. Re-member that for these parameters the Majorana modes gen-erally hybridize (see Fig. 5) and therefore the Majorana split-ting should be visible. Using an STM-FM coupling strengtht = 0.01t, the green, red and blue solid lines in Fig. 9 panel(a) show the differential conductance calculated at STM posi-tions x = 0, L/4, L/2. We immediately note that, due to finitesize effects, the zero-bias signal has split into two peaks cen-tered around V = 0, with an estimated energy splitting com-parable to the gap energy (E = 0.01t). Also note, that whilethe tunneling signal into these two finite energy quasiparticlestates may be the largest at x = 0, the signal persists well intothe bulk of the wire (x = L/4, L/2). At finite temperature

13

T = , the splitting between the peaks is no longer visuallyresolvable since temperature has thermally broadened the sig-nal across a range greater than the original Majorana splittingE (dashed green line). Panels (b,c) illustrate the differen-tial conductance spatial profile for temperatures T = 0, 0.01t,where again we have presented the zero temperature data on alogarithmic plot. In these bottom panels we see that the splitquasi-Majorana modes are spatially extended across the entirelength of the wire. Green, red and dark blue solid and dashedlines superimposed on the spatial profiles correspond to thedI/dV curves presented in panel (a). Thus, in the context ofa small pairing potential present in a short wire, we see a fi-nite temperate zero-bias signal extending into the bulk of thewire, with no measurable decay. For reasons which remainunclear at this stage, this theoretically expected splitting ofthe Majorana mode and the associated spatial delocalizationof quasi-Majorana modes are again not observed in the exper-iment, but, see the next section for a possible resolution of thispuzzle.

An additional problem in the interpretation of the experi-mental data of Ref. 44 is the issue of disorder which shouldvery strongly suppress the induced p-wave superconductivity.Given that the induced gap is 0.1 meV and the typical elec-tronic energy scales in the ferromagnetic chain (i.e. hoppingenergy, chemical potential, exchange energy) are all in the eVrange, one expects the slightest static fluctuations in the sys-tem (e.g. 0.1 % variation in the locations of the Fe atomsor the presence of any neighboring random impurities nearthe chain) to completely destroy the topological superconduc-tivity in the system since the p-wave pairing is not protectedagainst disorder by Andersons theorem69,70. One simple andapproximate way to estimate disorder effects here is to askabout the amount of elastic scattering which would be neces-sary to completely suppress the reported 100 eV topologicalgap in Ref. 44. Equating the reported p-wave gap to a disor-der induced collisional level broadening of 100 eV in the Fechain and using the band parameters estimated in Ref. 44 forthe system, it is easy to conclude that the electronic mean freepath along the Fe chain must be longer than 100nm for thedisordered system to manifest any topological gap (assumingthe clean system gap to be 100 eV ). This is of course incon-sistent with the observation of a topological gap in chains ofvariable lengths between 5 nm and 15 nm as reported44 sincethe wire length serves as a cut off for the maximum possiblemean free path in the system. One could of course assumethat the measured gap already incorporates the disorder effect(starting from a much larger clean topological gap), but thiswould imply very strong dependence of the measured topo-logical gap on the wire length, not reported in Ref. 44.

In the next section, we discuss in detail some possibili-ties for the reconciliation of the experimental observations ofRef. 44 with the theoretical Majorana interpretation. In par-ticular, several specific ambiguities regarding the Majorana in-terpretation of the experimental observations described in thecurrent section are shown to become less severe once certainadditional elements of physics, which might be playing a rolein the measurements of Ref. 44, are taken into account.

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(a)

(b)

(c)

FIG. 10. Evolution of the splitting of the Majorana conductancepeak, plotted across V = (1.5, 1.5), for systems of lengthL = 300(a), 200(b), 100(c). We have taken the pairing potential tobe = 0.01t and have calculated the conductance from the end ofthe wire (STM positioned at x = 0 with an STM nanowire couplingt = 0.01t). The solid (dashed) curves show the differential conduc-tance at zero (finite) temperature. Because the three lengths used arecomparable to the superconducting coherence length ( = 100 sites),the Majorana modes hybridize into finite energy quasiparticles withenergies E. This hybridization splits the ZBCP into two peakswhich are indiscernible at finite temperature due to thermal smearingeffects.

VI. COMPARISONWITH A RECENT EXPERIMENT

The most compelling qualitative features of the experimen-tal STM results presented in Ref. 44, providing evidence forzero-energy Majorana modes in Fe chains lying on supercon-ducting Pb substrates, are the existence of a (weak and broad)zero-bias differential tunneling conductance peak spatially lo-calized near the ends of the chains which seems to disappearas the STM tip probes the middle regions of the chains awayfrom the ends. This observation of a zero-bias peak seem-ingly spatially localized at the wire ends while being a neces-sary signature for Majorana modes as pointed out a long timeago60 is, however, not sufficient (see the next section for thechiral fractional Josephson effect which could serve as a suf-ficient condition).

The puzzling features of the experimental observation44 aremany: (i) the lack of any obvious superconducting gap mani-festing in the STM data on the Fe nanowire although a strikinggap, in precise quantitative agreement with the BCS theory,

14

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030dI/d

V(e

2/h

) T = 0

T = /2

T =

(a)

(b)

(c)

FIG. 11. Evolution of the splitting of the Majorana conductancepeak, plotted across V = (1.5, 1.5), for systems of lengthL = 300(a), 200(b), 100(c). Here we use a smaller pairing po-tential of = 0.005t and have again calculated the conductancefrom the end of the wire (STM positioned at x = 0 with an STMnanowire coupling t = 0.01t). The solid (dashed) curves showthe differential conductance at zero (finite) temperature. Because thethree lengths used are comparable to the superconducting coherencelength ( = 200 sites), the Majorana modes hybridize into finite en-ergy quasiparticles with energies E. Again, this hybridizationsplits the ZBCP into two peaks (distinct at zero temperature) whichare indiscernible at finite temperature due to thermal smearing ef-fects.

shows up for the Pb substrate itself; (ii) the observed zero-biaspeak is minuscule, being reduced by a factor of 103 to 104from the canonical 2e2/h zero-bias conductance associatedwith the perfect Andreev reflection from the Majorana mode;(iii) the zero-bias peak is exceptionally broad, being compa-rable to the estimated topological superconducting gap in thenanowire; (iv) the experimental temperature is comparable tothe estimated topological superconducting gap which makesit very difficult, if not impossible, to discuss features associ-ated with Majorana modes; (v) the very small topological gapimplies very long coherence length, and consequently, longMajorana localization length ( 1), which makes theexperimental observation of the spatial localization of the Ma-jorana modes to atomic sharpness ( 15 nm) and the shortlengths of the nanowire (5 15 nm) used in the experimentimply that considerable Majorana splitting oscillations shouldmanifest themselves in the experiment (since the coherencelength is greater than the wire length), which are, however, notseen; (ix) no obvious Majorana peak (even split peaks) is seenin the very short (< 5 nm) wires; (x) the STM gap signaturefor superconductivity which manifests itself strikingly in thePb substrate itself away from the nanowire, disappears com-pletely on the nanowire without any obvious signature for theShiba subgap states in the Pb superconducting gap as shouldbe expected for Fe, a magnetic impurity, on Pb, an ordinarys-wave superconductor (unless, of course, the weak peaks be-ing identified as Majorana modes are really the subgap Shibastates induced by Fe in Pb).

Some of the problems with the Majorana interpretation ofthe data in Ref. 44 have already been mentioned in the earliersections of this paper, but we summarize all of them tougherright in the beginning of this section because we will providedetailed numerical results in this section establishing that, inspite of all these problems the observations in Ref. 44 are notinconsistent with the Majorana interpretation, but any defini-tive conclusion would necessitate much lower measurementtemperatures and much higher instrumental resolution, as wellas a system exhibiting a more definitive topological gap.

First, the high measurement temperature (kBT > p) sup-

15

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(a) (b) (c)

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(d) (e) (f)

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(g) (h) (i)

FIG. 13. Evolution of the splitting of the Majorana conductance peak, plotted across V = (1.5, 1.5), for systems of length L =300(left column), 200(middle column), 100(right column). We have taken the pairing potential to be = 0.01t and the weakly coupled(t = 0.01t) STM is now positioned away from the system edge at x = L/4 for panels (a-c), at x = L/4 + 2 for panels (d-f), and atx = L/4 + 3 for panels (g-i). The solid (dashed) curves show the differential conductance at zero (finite) temperature. Because the threelengths used are comparable to the superconducting coherence length ( = 100 sites), the Majorana modes hybridize into finite energyquasiparticles with energies E. This hybridization splits the ZBCP into two peaks which are indiscernible at finite temperature due tothermal smearing.

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(e) (f)(d)

(b) (c)(a)

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t0.00000.00050.00100.00150.00200.00250.0030dI/dV(e2 /h) T = 0T = /2T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

(h) (i)(g)

FIG. 14. Evolution of the splitting of the Majorana conductance peak, plotted across V = (1.5, 1.5), for systems of length L =300(left column), 200(middle column), 100(right column). We have taken the pairing potential to be = 0.005t and the weakly coupled(t = 0.01t) STM positioned at x = L/4 for panels (a-c), at x = L/4 + 2 for panels (d-f), and at x = L/4 + 3 for panels (g-i). Thesolid (dashed) curves show the differential conductance at zero (finite) temperature. Because the three lengths used are comparable to thesuperconducting coherence length ( = 200 sites), the Majorana modes hybridize into finite energy quasiparticles with energies E. Thishybridization splits the ZBCP into two peaks which are indiscernible at finite temperature due to thermal smearing.

presses and broadens all features associated with any inducedsuperconducting gap and all associated subgap features (Ma-jorana or non-Majorana), thus making it difficult to observeany Majorana energy splitting oscillations since the featuresare simply too weak. To make this point explicit, we show inFigs. 10-12 our calculated STM tunneling spectra on the su-perconducting ferromagnetic nanowire at the wire end (x = 0)for three different temperatures (T = 0,/2,) , two dif-ferent topological gap values ( = 0.01t, 0.005t), and fourdifferent wire lengths (L = 700, 300, 200, 100) with a typi-cal coherence lengths being 1 = 100 (for = 0.01t) and2 = 200 (for = 0.005t). We have chosen the tempera-ture, the topological gap, the wire length and the STM tun-nel barrier strength at the tip (t = 0.01t) to be qualitativelyconsistent with the experimental results, and, by construction,

the voltage bias region (1.5) chosen along the abscissa fo-cuses entirely on the possible Majorana physics in the topo-logical gap of the ferromagnetic wire. We note that since ourmodel has, by construction, no subgap states other than theMajorana zero modes (which may very well be split due tohybridization in the finite wire), any subgap structure mani-festing itself in our results, by definition, arises from the Ma-jorana fermions. The most prominent feature of Figs. 10-12 isthat, although the T = 0 Majorana peaks are sharp and maymanifest energy splitting (the magnitude of the splitting de-creases with increasing wire length) as expected theoretically,the finite temperature differential conductance at x = 0 for allfour wire lengths shows very broad features (with the broad-ening being comparable to or even larger than the topologicalgap itself) with a weak smooth peak at zero energy. For the

16

0.006 0.004 0.002 0.000 0.002 0.004 0.006V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

dI/d

V(e

2/h

) T = 0

T = /2

T =

0.015 0.010 0.005 0.000 0.005 0.010 0.015V/t

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030dI/d

V(e

2/h

) T = 0

T = /2

T =

(a)

(b)

FIG. 15. Differential conductance tunneling spectrum, plotted acrossV = (1.5, 1.5), for systems of length L = 700. The weaklycoupled (t = 0.01t) STM is positioned at at x = L/4 and weuse a pair potential of = 0.01t (corresponding coherence length = 100 sites) and = 0.005t ( = 200 sites) in panels (a,b)respectively. The solid (dashed) curves show the differential conduc-tance at zero (finite) temperature. The system length is much greater(L ) than the superconducting coherence length and no Majo-rana splitting is observed.

smallest gap ( = 0.005t in Fig. 11) and for the shorter wires(L = 100, 200), the peak is actually asymmetric and in factgoes out of the gap region (see Fig. 11) due to the Majoranaoverlap. These features are all consistent with the experimentwhere a very weak and very broad zero bias peak is seen onlyin the longer wires (at the wire ends), and for shorter wires,the peak is not observed.

To understand the absence of any split Majorana peaks orMajorana oscillations away from the ends of the Fe wire inthe experiment, we show in Figs. 13-15 the calculated Majo-rana conductance peaks around the spatial location x = L/4for L = 100, 200, 300, 700 again for T = 0,/2, and for = 0.01t and 0.005t. The interesting (and somewhat sur-prising) point to note is that essentially no Majorana peaks oroscillations are discernible around x = L/4 (i.e. Figs. 13-15) for finite temperature although the T = 0 situation doesreflect split peaks. In fact, all traces of Majorana modes,have disappeared completely around x = L/4 in the finitetemperature results in agreement with the experimental find-ing, although the superconducting coherence length ( L,certainly > L/4) is long enough that one would expect(as found at T = 0) signatures of Majorana oscillations atx = L/4. Again, the high experimental temperature com-pared with the induced superconducting gap in the nanowireis responsible for the suppression of Majorana oscillations.