Majorana splitting from critical currents in Josephson junctions

Jorge Cayao1, Pablo San-Jose2, Annica M. Black-Schaffer1, Ramon Aguado2, Elsa Prada31Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden2Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain

3Departamento de Fısica de la Materia Condensada,Condensed Matter Physics Center (IFIMAC) & Instituto Nicolas Cabrera,

Universidad Autonoma de Madrid, E-28049 Madrid, Spain(Dated: November 23, 2017)

A semiconducting nanowire with strong Rashba spin-orbit coupling and coupled to a supercon-ductor can be tuned by an external Zeeman field into a topological phase with Majorana zeromodes. Here we theoretically investigate how this exotic topological superconductor phase mani-fests in Josephson junctions based on such proximitized nanowires. In particular, we focus on criticalcurrents in the short junction limit (LN � ξ, where LN is the junction length and ξ is the super-conducting coherence length) and show that they contain important information about nontrivialtopology and Majoranas. This includes signatures of the gap inversion at the topological transitionand a unique oscillatory pattern that originates from Majorana interference. Interestingly, this pat-tern can be modified by tuning the transmission across the junction, thus providing complementaryevidence of Majoranas and their energy splittings beyond standard tunnel spectroscopy experiments,while offering further tunability by virtue of the Josephson effect.

I. INTRODUCTION

The theoretical proposals1,2 for artificially creat-ing topological superconductors3 out of semiconductingnanowires with strong spin-orbit (SO) coupling and prox-imitized with conventional s-wave superconductors havespurred an unprecedented experimental effort towards re-alizing such exotic phase with mundane materials4. Thetopological superconductor phase is expected to occurwhen an external Zeeman field exceeds a critical fieldBc ≡

√∆2 + µ2, defined in terms of the induced su-

perconducting gap ∆ and the chemical potential µ ofthe wire. Since the electron density can be modifiedby external electrostatic gates and the external Zeemanfield can be larger than the induced superconductinggap, the topological phase can in principle be tuned insitu. Such nontrivial phase is characterized by the emer-gence of non-Abelian Majorana bound states (MBSs) lo-calized at the nanowire ends and at zero energy. Theemergence of MBSs is predicted to give rise to perfectAndreev reflection, and thus perfectly quantized con-ductance G = 2e2/h at zero bias voltage, in normal-superconductor junctions.5

During the last few years, this idea has led toa vast number of important experiments in hybridsuperconductor-semiconductor systems looking for suchzero-bias anomaly (ZBA) in the differential conductanceas the Zeeman field increases.6–11 However, and de-spite the initial excitement, the interpretation of theseexperiments in terms of Majoranas has been hotlydebated owing to the difficulty in ruling out alter-native explanations such as parity crossings of An-dreev levels,11–13 Kondo physics,9,10,12,13 disorder,14–18

or smooth confinement.19,20 The nagging question ofwhether MBSs have been observed in nanowires and theso-called soft gap problem6–11 have resulted in a sec-ond generation of experiments, with improved devices,

which show more convincing signatures of Majorana zeromodes.21–23 Despite these improvements, the topologi-cal origin of the ZBA is still under debate since subtlereffects, such as the sticking of Andreev levels owing tonon-homogeneous chemical potentials,24 can mimic Ma-joranas. Therefore, it would be very useful to study othergeometries and signatures25 beyond ZBAs in normal-superconductor junctions.

The direct measurement of a 4π-periodic Joseph-son effect in nanowire-based superconductor-normal-superconductor (SNS) junctions may provide one of suchstrong signatures. Unfortunately, the 4π effect, whichappears as a result of protected fermionic parity cross-ings as a function of the superconducting phase difference(φ) in the junction, is very hard to observe experimen-tally. First, the unavoidable presence of two additionalMBSs at the wire ends (outer Majoranas), and their over-lap with MBSs at the junction (inner Majoranas), ren-ders the Josephson effect 2π-periodic.26 A second prob-lem is stochastic quasiparticle poisoning which spoilsfermion parity conservation. The 4π-periodic Josephsoneffect may still be detected by out-of-equilibrium mea-surements, such as ac Josephson radiation or noise,26–28

as demonstrated recently in Josephson junctions fabri-cated with quantum spin Hall edges.29,30 Another routeis offered by the multiple Andreev reflection mecha-nism in a voltage-biased SNS junction31 which in cur-rent experiments32,33 could provide stronger evidence ofMBSs.

In this work we take an arguably easier route and focuson equilibrium supercurrents in SNS junctions which, de-spite the overall 2π-periodicity in the ground state, stillcontain useful information about the nontrivial topol-ogy and MBSs in the junction. In particular, we studyshort junctions with LN � ξ, where LN is the normalregion length and ξ is the superconducting coherencelength. Our main findings are summarized in Figs. 4

arX

iv:1

707.

0511

7v2

[co

nd-m

at.m

es-h

all]

21

Nov

201

7

2

and 5. We find that the critical current Ic through ashort junction between two finite length superconduc-tors [two proximitized regions, each of length LS, seeFig. 1(a)] and tunable normal transmission TN, showsa distinct feature at the gap inversion for the criticalZeeman field B = Bc [Fig. 4(a)]. For B < Bc, thecritical current does not depend on the superconduct-ing region length LS, as expected for a trivial ballisticjunction in the short junction regime.34 In contrast, thecritical current in the topological regime increases withLS. This unique length-dependence in a short SNS junc-tion is a direct consequence of the Majorana overlap oneach S region of the junction, with one MBS at each endof S. For long enough wires (LS � 2ξM with ξM beingthe Majorana localization length) and small transmissionTN � 1, we obtain a re-entrant behavior of Ic as a func-tion of Zeeman field. This reentrance results from anenhanced Majorana-mediated critical current Ic ∼

√TN

(for B > Bc) relative to that of conventional Andreevlevels Ic ∼ TN (for B < Bc), as also discussed in Refs.[35 and 36].

On the other hand, we find that the critical currentfor B > Bc exhibits oscillations derived from the oscil-latory energy splitting of overlapping Majoranas aroundzero energy. The oscillations with B exhibit a surpris-ing period doubling effect as the junction transparencyis reduced [Fig. 5], which can be understood from thehybridisation properties of the four Majoranas in the sys-tem. For small TN, the oscillatory pattern is that of twonearly decoupled finite-length topological wires with twooverlapping Majoranas at the ends of each S region. Asthe transparency increases, the oscillations evolve intoa more complex pattern which, at perfect transparencyTN → 1, results in the competing hybridization of thefour Majoranas, with two energy splittings oscillatingout-of phase. All these findings remain robust under fi-nite temperatures (below the induced gap), as well asunder the presence of disorder. Our results constitute acomprehensive picture, extending previous studies35,36,of the effects of Majoranas on the Ic of short SNS junc-tions. We include the dependence of Ic with Zeeman fieldB, superconductor length LS and transparency TN.

We conclude that critical currents reveal signatures ofMajorana splittings and, despite the finite lengths in-volved in the geometry of the junction, for B > Bc theycontain distinct signatures of the bulk topological transi-tion, derived from the specific four-Majorana hybridiza-tion pattern. These findings hold relevance in view ofrecent proposals for topological quantum computationthat crucially rely on controlling the Josephson couplingin nanowire-based Josephson junctions.37,38

The paper is organized as follows. In Sec. II we de-scribe the model for SNS junctions based on nanowireswith SO. Then, in Sec. III we investigate the Zeeman fielddependence of critical currents and discuss the emergenceof oscillations as a result of MBSs overlaps. In Sec. IV wepresent our conclusions. In Appendix A we discuss therobustness of our findings under finite temperature and

disorder. For completeness, we also include a short dis-cussion about the Majorana localization length and itstypical values for the parameters used in this work.

II. SNS JUNCTIONS MADE OFSEMICONDUCTING NANOWIRES

A. Model

We consider a single channel semiconducting nanowirewith Rashba SO whose Hamiltonian is given by

H0 =p2x2m− µ− αR

~σypx +Bσx , (1)

where px = −i~∂x is the momentum operator, µ isthe chemical potential that determines the filling of thenanowire, αR represents the strength of Rashba SO

(which results in a SO energy ESO =mα2

R

2~2 ), B = gµBB/2is the Zeeman energy resulting from the applied mag-netic field B, g is the wire’s g-factor and µB the Bohrmagneton. Notice that the Zeeman and the SO axes areperpendicular to each other. The Hamiltonian given inEq. (1) is discretized into a tight-binding lattice followingstandard methods.39 In all our calculations we considertypical values for InSb nanowires6 that include the elec-tron’s effective mass m = 0.015me, with me the massof the electron, and the SO strength αR = 20 meV nm,unless otherwise stated.

In order to model the SNS junction, we now assumethat the left and right regions of the nanowire are ingood contact with two parent s-wave superconductors.Due to the proximity effect, this results in a nanowirecontaining left (SL) and right (SR) regions of length LS

with induced superconducting pairing potential ∆SL,R,

leaving the central region, of length LN, in the normalstate (N). This model is relevant in recent experimentswhere hard gaps are proximity-induced into nanowires.40

The full Hamiltonian describing the SNS junction isgiven in Nambu space as39

HSNS =

(hSNS ∆SNS

∆†SNS −h∗SNS

), (2)

where hSNS and ∆SNS are matrices in the junction spaceand read

hSNS =

HSLHSLN 0

H†SLNHN HNSR

0 H†NSRHSR

,

∆SNS =

∆SL 0 00 0 00 0 ∆SR

.

(3)

All the elements in the diagonal of hSNS have the struc-ture of H0 given by Eq. (1), with chemical potentialµSL,R

for HSL,Rand µN for HN. The induced pairing

potentials in the left and right superconducting regions

3

FIG. 1. (Color online) Sketch of a SNS junction with Sand N regions of finite length LS and LN. The transmis-sion across the junction can be controlled by the parametersτL(R) ∈ [0, 1], which parametrize the coupling between neigh-boring sites at the interfaces. (b) For B > Bc, the S regionsbecome topological and the system hosts four MBSs closeto zero energy at φ = π: two outer γ1,4 and two inner γ2,3ones. The MBSs are localized at the ends of the S regions andtheir wavefunctions exhibit an oscillatory exponential decayinto the bulk of S with a localization length ξM. (c) Rela-tion between normal transmission TN and coupling parameterτL(R) = τ for different values of the Zeeman field. Parameters:∆ = 0 and µ = 0.5 meV, LN = 20 nm and αR = 20 meVnm.

are, respectively, ∆SL = ∆e−iφ/2 and ∆SR = ∆eiφ/2,where ∆ = iσy∆ . HSLN is the Hamiltonian that cou-ples the superconducting region SL to the normal regionN, while HNSR

couples N to SR. The elements of thesecoupling matrices are non-zero only for adjacent sitesthat lie at the interfaces of the S and N regions. Thisis a good model when the barrier length is much shorterthan the Fermi energy, otherwise see Ref. [41]. This bar-rier coupling is parametrized by a hopping τL,Rv, whereτL,R ∈ [0, 1] and v is the spin-resolved hopping matrixin the wire. In general, we will asume a symmetric casewhere τL,R = τ.

In order to show that τ indeed controls the transmis-sion, we calculate the normal transmission TN acrossthe junction following the standard Green’s functionstechnique39 assuming ∆ = 0 and µ = 0.5 meV. Thisis shown in Fig. 1(c) for a short N region and differentvalues of the Zeeman field. For B = 0, TN exhibits anon-linear dependence on τ that quickly becomes zerofor τ ≤ 0.6, which signals the tunneling regime. Thetransparent regime is reached when τ ≈ 1, as expected.Thus, the range τ ∈ [0.6, 1] allows us to explore the fullrange of transparencies from the tunnel to the transpar-ent regime. By increasing B, the maximum transmissionin the full transparent regime is halved, namely TN = 0.5,due to the emergence of the helical state in the N regionfor B > µ. Notice that the tunnel regime remains almost

0 1 2Phase difference φ/π

-1

0

1

Ener

gy ε

/∆

LS<2ξΜ

0 1 2Phase difference φ/π

LS>>2ξM

(a) (b)

FIG. 2. (Color online) Low-energy spectrum as a functionof the superconducting phase difference in a short SNS junc-tion for LS < 2ξM (a) and LS � 2ξM (b) at B = 2Bc. Thetwo lowest levels, almost insensitive to phase, correspond tothe two outer Majoranas γ1,4. The next two levels below thetopological minigap (green dashed line), are strongly disper-sive with phase, and come from the hybridization of the innerMajoranas γ2,3 which anticross the former at φ = π. Param-eters: LN = 20 nm, (a) LS = 2000 nm, (b) LS = 10000 nm,αR = 20 meVnm, µ = 0.5 meV, ∆ = 0.25meV.

unaltered even when the Zeeman field is large.

B. Phase-dependent Andreev spectrum

At finite magnetic field the S regions develop two gaps∆1,2 at low and high-momentum, respectively, and theyexhibit a different behavior with Zeeman field B. As Bincreases beyond Bc, each S region undergoes a topo-logical phase transition, determined by the inversion of∆1 at zero momentum, and enters into a topological su-perconducting phase.1,2 The topological gap, sometimescalled minigap, is then given by ∆2 for B � Bc, which iscontrolled by the SO coupling. In this topological phase,MBSs emerge localized at each end of SL,R. Their wave-functions decay in an exponentially and oscillatory fash-ion within a Majorana localization length ξM into thebulk of the superconducting regions,19,42,43 see Fig. 1(b).As long as Majoranas do not overlap spatially, i.e. in longenough superconducting regions LS � 2ξM, they remainzero-energy eigenstates. A finite overlap, however, en-ables the hybridization of any two given Majoranas intoa fermionic state of finite energy. When the left and righttopological regions in the Josephson junction becomecoupled by a finite TN, the two inner Majoranas γ2 andγ3 also hybridise, although in a phase-dependent manner.This gives rise to a distinctive phase-dependent Andreevspectrum, as first discussed in Refs. 26, 28, 39, and 44.

In Fig. 2 we present this low-energy spectrum for LS <

4

0 1 2Phase difference φ/π

Supe

rcur

rent

I/I 0

LS<2ξΜ

0 1 2Phase difference φ/π

10.90.80.70.6

LS>>2ξΜ

(a) (b)τ

0.2

Finite temperature κΒΤ=0.02meV

-0.2

0

FIG. 3. (Color online) Supercurrent in a short SNS junc-tion as a function of φ at B = 2Bc for LS < 2ξM (a) andLS � 2ξM (b). Different curves correspond to different valuesof τ. A sawtooth profile in (b) is developed when MBS wave-functions within each superconducting section do not over-lap, Fig. 1(b). This feature remains robust as τ is reduced,but not so with finite temperature (see dashed purple curve,with κBT = 0.02 meV and τ = 1). Parameters: LN = 20 nm,(a) LS = 2000 nm, (b) LS = 10000 nm, αR = 20 meVnm,µ = 0.5 meV, ∆ = 0.25meV and I0 = e∆/~.

2ξM (a) and LS � 2ξM (b). It consists of a discreteset of levels below the topological minigap (green dashedline), originated from the hybridization of the four Ma-joranas at the junction, and a quasi-continuum above.The lowest two levels are (almost) insensitive to φ andcome from the outer MBSs, γ1,4. The other two energylevels below the minigap originate from the inner Ma-joranas, γ2,3, strongly disperse with φ and anticross atφ = π with the outer ones. For LS < 2ξM, Fig. 2(a), aconsiderable splitting around φ = π appears, which arisesdue to the finite spatial overlap between the MBSs wave-functions within each S region. A true crossing is indeedreached when considering semi-infinite superconductingleads, LS → ∞. In such geometry the outer MBSs arenot present in the description and the crossing at φ = πis protected by conservation of the total fermion parity.

The splitting of the four MBSs at φ = π can be reducedeither by increasing the SO strength and therefore reduc-ing the Majorana localization length,39 or by increasingthe length of the superconducting regions LS, as shownin Fig. 2(b). In the LS � 2ξM limit, the outer MBSs aretrue Majorana zero modes, while the inner ones approachzero energy only at φ = π. As we argue in what follows,the critical current across the junction is very sensitive tothe φ = π anticrossing structure and can, therefore, beused to extract useful information about the Majoranahybridizations in the system.

C. Current-phase characteristics

The supercurrent can be obtained from the free energyof the junction as

I(φ) =1

Φ0

dF

dϕ, (4)

where Φ0 = ~2e is the reduced (superconducting) flux

quantum. Taking care about the double counting inher-ent to the BdG description, the above expression can berewritten as

I(φ) = − e~∑εp>0

tanh( εp

2κBT

)dεpdϕ

, (5)

where κB is the Boltzman constant, T is the temperatureand εp are the energy levels, obtained by exact diagonal-ization of the Hamiltonian in Eq. (2). Importantly, thesum must be performed over all degeneracies (spin in ourcase) explicitly.45 In what follows, we focus on zero tem-perature calculations, unless otherwise stated, and shortjunctions. The full phenomenology of current-phase re-lationships, including the long junction limit, will be dis-cussed elsewhere.46

Fig. 3 shows the phase-dependent supercurrents corre-sponding to the Andreev spectra of Fig. 2. The finitesplitting at φ = π, observed for LS < 2ξM, gives rise tosupercurrents with a sine-like behavior as a function ofφ, see Fig. 3(a). The opposite case of LS � 2ξM thatcorresponds to an almost perfect crossing at φ = π re-sults in a sawtooth profile, Fig. 3(b). Note how the over-all shape in both limits remains unaltered irrespectiveof the transmission across the normal region all the wayfrom fully transparent junctions τ = 1 to the tunnel limitτ = 0.6. As expected, finite temperature (dashed ma-genta curves), however, washes out the sawtooth profile.Thus, current-phase curves cannot be used to clearly dis-tinguish the trivial and topological phases based on thesawtooth profile at φ = π. Despite this, the critical cur-rent still contains useful information in the topologicalphase, as we discuss now.

III. CRITICAL CURRENTS

The critical current Ic is the maximum supercurrentthat can flow through the junction. We calculate it bymaximizing the supercurrent I(φ) with respect to thesuperconducting phase difference φ,

Ic = maxφ[I(φ)] . (6)

In Fig. 4 we show Ic for short and transparent SNSjunctions as the Zeeman field B increases from the trivial(B < Bc) to the topological phase (B > Bc). Differentcurves in Fig. 4(a) correspond to different values of thesuperconducting region length LS. The critical current ismaximum at zero field and decreases as the Zeeman field

5

0 1 2Zeeman field B/Bc

0

0.2

0.4

0.6

0.8

1C

ritic

al c

urre

nt I c/I 0

Eq.(7)∞6000500040002500200015001000

0 1 2Zeeman field B/Bc

1004020

(a) (b)

αR [meVnm]

LS [nm]LS=2000nm

FIG. 4. (Color online) Critical current Ic in a short and trans-parent SNS junction as a function of the Zeeman field B. In(a) different curves represent different values of the super-conducting region length LS. As B increases towards Bc, Icdecreases, approximately following the evolution of the gap.At the gap closing B = Bc (red vertical dashed line), Ic ex-hibits a non-trivial feature but remains finite. For B > Bc,the critical current develops an oscillatory behavior that isrelated to the finite inner-outer Majorana splitting and dis-appear as the length LS is increased. Notice that the magni-tude of Ic for B > Bc increases with increasing LS while itsvalue for B < Bc remains unchanged. This unconventionaldependence is a direct signature of nontrivial topology of thejunction. (b) Ic for fixed LS = 2000 nm and increasing SOstrength αR. Observe that at large fields and in the verystrong SO regime the critical current saturates to 1/2 of itsB = 0 value. Parameters: LN = 20 nm, αR = 20 meVnm,µ = 0.5 meV, ∆ = 0.25meV and I0 = e∆/~.

B increases, tracing the reduction of the low-momentumgap ∆1.

Remarkably, the energy gap closes at B = Bc but thecritical current remains finite and develops a robust kink-like feature, as can be seen in Fig. 4(a). At B = Bc, Icis roughly halved with respect to its B = 0 value, asa result of the system losing one channel (the nanowirebands change from spinful to spinless). As the Zeemanfield is increased further, the critical current develops os-cillations when LS . 2ξM. This structure arises fromthe anticrossing of the four MBSs around φ = π. Theoscillations disappear as the length LS is increased, andthe outer Majoranas decouple from the junction, as canbe seen in Fig. 4(a). Another interesting feature is thatfor B < Bc, Ic remains almost unaltered with LS, asexpected of a conventional ballistic junction. However,for B > Bc, Ic increases as LS becomes longer, an ef-fect that can be traced back to the protected crossingat φ = π of the topological Andreev spectrum, see Fig.2(b). Ultimately, the increase in critical current is there-fore related to the Majorana origin of the low-energyAndreev bound states in the junction. This remarkable

0 1 2Zeeman field B/Bc

0

0.2

0.4

0.6

Crit

ical

cur

rent

I/I 0

τ=0.6

τ=1

LS= 2000nm

FIG. 5. (Color online) Ic for fixed LS = 2000 nm and in-creasing values of τ ∈ [0.6, 1], in steps of 0.01 (indicated byvertical orange arrow). In the trivial phase, B < Bc, the criti-cal current is quickly suppressed as τ is decreased towards thetunneling limit. In the topological phase the critical currentis suppressed much more slowly with τ, which results in a re-entrant Ic at Bc for small τ. For B > Bc, the period of theIc oscillations is doubled as the transparency of the junctionis reduced (magenta and blue dashed lines). Parameters arethe same as in Fig. 3.

length-dependence in a short and transparent junctionwhen B > Bc is thus a direct signature of the nontriv-ial topology and provides a way to distinguish the effectsdescribed here from other trivial explanations such as π-junction behavior or oscillatory critical currents due toorbital effects.47

In the asymptotic limit LS → ∞ the outer MBSs arenot involved in the supercurrent and therefore only theinner ones determine the critical current. This limitis shown by the dot-dashed curve in Fig. 4(a), whichmatches quite accurately a simplified model for Ic com-ing purely from the inner Majoranas, see black dashedline. This model, valid in the transparent limit, yields31

Ic ≈I02

∆1 + ∆2

∆, B < Bc

Ic ≈I02

∆2

∆, B > Bc,

(7)

with ∆1 = |B −Bc| and

∆2 ≈2∆ESO√

ESO(2ESO +√B2 + 4E2

SO),

the low and high-momentum gaps introduced before, re-spectively. Here, I0 = e∆/~ is the maximum critical cur-rent supported by an open channel. The critical currentin this simple model results from the contribution of twoAndreev bound states within ∆1 and ∆2 for B < Bc, andonly one bound state within ∆2 for B > Bc. As in the

6

-1

0

1E

ner

gy ε

/∆

0 1 2Zeeman field B/B

c

-1

0

1

Ener

gy ε

/∆

0 1 2Zeeman field B/B

c

τ=1 τ=0.9

τ=0.6τ=0.7

(a) (b)

(d)(c)

FIG. 6. (Color online) Low-energy Andreev spectrum as afunction of the Zeeman field in a short SNS junction at φ = πfor different values of the coupling between N and S regionsτ. Note that upon decreasing τ the low-energy zero-energycrossings for B > Bc merge in pairs, which results in a dou-bling of the period of oscillation in Ic. Parameters same as inFig. 3 and LS = 2000 nm.

exact numerical calculation, the ratio Ic/I0 decreases asthe Zeeman field increases, which reflects the transitionfrom spinful to spinless bands at B = Bc.

31

Fig. 4(b) shows the dependence of Ic(B) with SO cou-pling. In the strong SO limit (dashed curve), the outergap becomes ∆2 ≈ ∆ and thus the ratio between thecritical currents at zero and high Zeeman fields reaches1/2, which is yet another strong signature of a topologi-cal phase. Note, however, that the required SO couplingto reach this ratio may be unrealistically large in mostnanowire systems.

The preceding results correspond to transparent junc-tions. In Fig. 5 we show the critical current as a functionof B for different values of τ, or junction transparency.As expected, Ic in the trivial regime B < Bc gets reducedas τ decreases, until it becomes negligible in the tunnel-ing regime. The kink-like feature in Ic for a transparentτ ≈ 1 junction associated to the closing of the gap atB = Bc evolves into a step-like increase as τ is reduced.This causes a re-entrant critical current in the tunnelingregime, where Ic is essentially zero below Bc but then in-creases suddenly at the topological phase transition. There-entrance of the low-transparency Ic as B exceeds Bc

can be understood from the fact that the critical currentcarried by the inner Majoranas scales as Ic ∼

√TN for

B > Bc. At small TN this current is much larger thanthe critical current Ic ∼ TN associated to conventionalAndreev levels for B < Bc.

35,36 Importantly, we also findthat the critical current oscillations for B > Bc show acharacteristic period doubling as τ is reduced, see Fig. 5.This period doubling is highlighted by the magenta andblue vertical dashed lines.

0

0.2

0.4

0.6

Crit

ical

cur

rent

I c/I 0

0 1 2Zeeman field B/Bc

0

0.2

0.4

0.6

Crit

ical

cur

rent

I c/I 0

0 1 2Zeeman field B/Bc

τR=1 LSL=0.5LSRτL=1

LSL=0.75LSR

(a) (b)

(d)(c)τR=0.9

τL=0.6

τ=1

τL=1τ=1

τ=0.6

τ=0.6τL=0.6

FIG. 7. (Color online) Critical currents as a function of theZeeman field in a short SNS junction. (a,c) Asymmetry in thecouplings to the left and right superconducting leads, τL(R),for fixed LS. The coupling to the right S region τR is fixed andthe one for the left one varies from tunnel to the full trans-parent regime in steps of 0.1. This situation is expected toarise in experiments. Observe that the oscillations for B > Bc

remain robust with a clear period doubling for decreasing τL.(b,d) Asymmetry in the length of the superconducting regionsLS = 2000 nm for different values of the coupling parameterτL(R) = τ from tunnel to the full transparent regime in steps of0.1. Parameters: LN = 20 nm, αR = 20 meVnm, µ = 0.5 meVand ∆ = 0.25 meV.

-1

0

1

Ener

gy ε

/∆

0 1 2Phase difference φ/π

-1

0

1

Ener

gy ε

/∆

0 1 2Phase difference φ/π

τ=1 τ=0.9

τ=0.6τ=0.7

(a) (b)

(c) (d)

FIG. 8. (Color online) Low-energy Andreev spectrum as afunction of the superconducting phase difference φ atB = 2Bc

for different values of the coupling between N and S regionsτ. Note that upon decreasing τ the outer (two lowest) MBSsacquire a finite energy at φ = 2πn, for n = 0, 1, 2, . . . , whiledeveloping a crossing at φ = π with their inner counterparts.This is the limit of two identical decoupled superconductingnanowires with overlapping Majoranas at each end. Parame-ters same as in Fig. 3 and LS = 2000 nm.

7

The period doubling in the topological regime B < Bc

can be understood in terms of changes in the structureof the φ = π anticrossing of the four hybridised MBSs inthe system as τ is reduced is presented in Fig. 6. Aroundthis point the four states are close to zero and have amaximal slope with φ. Hence the supercurrent I(φ) fromthe MBSs reaches its maximum value close to φ = π aswe saw in Figs. 2 and 3. The structure of the low-energyφ = π anticrossing therefore controls Ic. The anticrossingresults from the B-dependent hybridization of the fourMBSs, and exhibits an oscillatory pattern with B. Theoscillatory structure of the resulting φ = π spectrum in afully transparent junction (τ = 1) is shown in Fig. 6(a).As the coupling τ is reduced, Fig. 6(b-c), two consecutivezero-energy crossings (magenta vertical dashed lines inpanel a) tend to approach each other and finally fuse intoone single crossing in the tunneling regime (blue verticaldashed lines in Fig. 6(d)). In this limit, the periodicityis that of overlapping Majoranas in essentially decoupledwires, namely two independent pairs of Majoranas whichdo not couple through the normal region.

We now analyze the robustness of the period-doublingeffect against various asymmetries of the system. In Fig.7 we plot Ic as a function of B for asymmetries in (a,c)the parameter τL,R and (b,d) the length of the S re-gions. Remarkably, by fixing τR and varying τL the pe-riod doubling remains robust indicating that this effectmay be observable in real experiments, as asymmetricjunctions is the natural experimental situation. Althoughthe asymmetry of the length of the S regions suppressesthe visibility of the period doubling, the effect is still ap-preciable.

The period doubling effect can be also seen in thephase-dependent Andreev spectra. This is shown in Fig.8, where different panels correspond to different values ofτ. Observe that as τ is reduced the inner and the outerMBSs are strongly affected. In fact, the inner MBSs re-duce their energies at φ = 0 but the outer ones slightlyincrease, while keeping almost a constant value aroundφ = π. In the tunneling regime, one outer and one in-ner MBS, which correspond to the same S region, getpaired with a small dependence on φ. Therefore, the Icoscillations and the period doubling upon reducing thetransparency of the junction (with e.g. a central gate)can be understood in terms of Majorana hybridisationsin the system.

IV. CONCLUSIONS

We have theoretically investigated how Majoranaphysics manifests in Josephson junctions based onRashba nanowires proximitized with superconductors.We have demonstrated that the critical current in shortjunctions between two finite length superconductors andtunable normal transmission TN contains important in-formation about nontrivial topology and MBSs. ForB < Bc, Ic does not depend on the superconducting re-

gion length, as expected for a trivial ballistic junction inthe short junction regime. In contrast, the critical cur-rent in the topological regime increases with LS. Thisanomalous length-dependence in a short SNS junctionis a direct consequence of Majorana overlap on each Sregion of the junction. We have observed a re-entranteffect in Ic in the tunneling regime which results froman enhanced Majorana-mediated critical current. Ic isessentially zero below Bc but increases abruptly at thetopological phase transition. Importantly, we have alsofound that the critical current for B > Bc exhibits os-cillations derived from the oscillatory splitting of over-lapping Majoranas around zero energy, which are robustagainst temperature and changes in TN. Tuning the lat-ter, in particular, gives rise to a period doubling effect inthe oscillations which can be unmistakably attributed tooverlapping MBSs.

Interestingly, these findings are robust under finitetemperatures (below the induced gap), as well as un-der finite disorder. Our findings complement previ-ous studies35,36 in SNS junctions and, quite remarkably,based on a recent experiment, we believe that our pro-posal is experimentally reachable.36 Moreover, all thesefeatures can be modified by tuning the normal transmis-sion across the junction, thus providing further evidenceof Majoranas and their energy splittings beyond tunnelspectroscopy.

ACKNOWLEDGEMENTS

We acknowledge financial support from the Span-ish Ministry of Economy and Competitiveness throughGrant No. FIS2015-65706-P (MINECO/FEDER)(P. S.-J), FIS2015-64654-P (R. A.), FIS2016-80434-P(AEI/FEDER, EU) (E. P.) and the Ramon y Cajal pro-gramme through grant No. RYC-2011-09345 (E. P).J.C. and A.B.S. acknowledge financial support from theSwedish Research Council (Vetenskapsradet), the GoranGustafsson Foundation, the Swedish Foundation forStrategic Research (SSF), and the Knut and Alice Wal-lenberg Foundation through the Wallenberg AcademyFellows program.

Appendix A: Finite temperature, disorder andMajorana localization length

We have shown in the main text that at zero temper-ature the critical current exhibits a reentrant behaviorin the tunneling regime at B = Bc, oscillations in thetopological phase B > Bc and a unique period doublingeffect. In this part, we present additional calculationsthat support the robustness of these findings.

First, we discuss the effect of finite temperature on thecritical current Ic, where we employ Eq. (5). The resultsare presented in Fig. 9. In Fig. 9(a) we show the criticalcurrent for different temperature values in a transparent

8

junction τ = 1. In Fig. 9(b-d) different panels correspondto different values of temperature for different values ofthe normal transmission τ. We observe that at finite, butsmall, temperature the critical current remains robust.In particular, one notices that the reentrant behavior atB = Bc of the critical current in the tunneling regime isvisible, as well as the oscillations in the topological phaseB > Bc. Remarkably, the period doubling effect of thecritical current oscillations in the topological phase alsoremain visible at finite temperature. As expected, hightemperature values tend to wash out the oscillations inthe critical current, as one can indeed see in the threelowest curves of Fig. 9(a).

Another important effect that we discuss here is therole of on-site disorder, which introduces random fluctua-tions in the chemical potential of the system µ. It is intro-duced as a random onsite potential Vi in the tight-bindingform of Hamiltonian given by Eq. (1), where the valuesof Vi lie within [−w,w], being w the disorder strength.Fig. 10(a) presents the critical current as a function ofthe Zeeman field for different disorder strengths, whereeach curve corresponds to 20 disorder realizations. A re-markable feature here is that the oscillations of the crit-ical current in the topological phase persist even for wslightly stronger than the chemical potential µ (red, yel-low and green curves). For very strong disorder, however,

0

0.2

0.4

0.6

Crit

ical

cur

rent

I c/I 0

0 1 2Zeeman field B/Bc

0

0.2

0.4

0.6

Crit

ical

cur

rent

I c/I 0

0 1 2Zeeman field B/Bc

τ=1

κBT=0.04∆

κBT=0.02∆

κBT=0.08∆

(a) (b)

(d)(c)

κBTτ=1

τ=1 τ=1

τ=0.6

τ=0.6τ=0.6

FIG. 9. (Color online) Effect of finite temperature on thecritical currents as a function of the Zeeman field in ashort SNS junction. (a) Full transparent junction τ = 1and increasing temperature (top-to-bottom curves). Fromtop to bottom: red, yellow, green, blue, indigo, violet,dashed black and dash-dot black curves correspond to κBT =(0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.2, 0.4)∆. (b-d) Different pan-els correspond to a different temperature value for differentcases of τ. Observe that the reentrant behavior, oscillationsfor B > Bc and period doubling effect remain robust at finitetemperatures, but much less than the superconducting gap.Parameters: LN = 20 nm, αR = 20 meVnm, µ = 0.5 meV and∆ = 0.25 meV.

the visibility of the oscillations is considerable degraded(blue and indigo curves). This, together with the discus-sion about finite temperature, therefore suggests that ourproposal represents an experimental feasible route for thesearch of MBSs.

For completeness we also show the dependence of theMajorana localization length ξM on the Zeeman field fordifferent values of the SOC strength. It is obtainedafter solving numerically the polynomial equation2,39

k2 + 4(µ+Cα2R)Ck2 + 8λC2∆αRk + 4C0C

2 = 0 , whereC = m/~2 and C0 = µ2 + ∆2 − B2. This equation isobtained from Eq. (1) with induced superconducting po-tential ∆. We define the Majorana localization lengthas ξM = max[−1/ksol], where {ksol} are solutions to theprevious polynomial equation. Observe that for realis-tic parameters (dashed line) ξM is large implying thatone needs very long wires to avoid the energy splittingof MBSs. These results are indeed in good agreementwith the conditions for the ratio between the supercon-ducting region lengths and Majorana localization lengthpresented in the main text when discussing Majoranasplitting in the SNS junction setup.

0 1 2Zeeman field B/Bc

0

0.2

0.4

0.6

Criti

cal c

urre

nt I c/I 0

01246

1 2Zeeman field B/Bc

0

1

2

3ξ M

/100

0 [n

m]

1020100

(a) (b)αR [meVnm]w/µ

FIG. 10. (Color online) (a) Effect of disorder on the criticalcurrents as a function of the Zeeman field in a short SNSjunction. Notice that the oscillations for B > Bc persist evenwhen the strength of disorder is comparable to the value ofthe chemical potential µ. (b) Majorana localization lengthas a function of B for different values of the SOC strength.Parameters: LN = 20 nm, αR = 20 meVnm, µ = 0.5 meV and∆ = 0.25 meV.

9

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