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Topological suppression of magnetoconductance oscillations in NS junctions Javier Osca1,∗ and Lloren¸c Serra1,2 1Institut de F´ısica Interdisciplinària i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain 2Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain (Dated: January 15, 2017) We show that the magnetoconductance oscillations of laterally-confined 2D NS junctions are completely suppressed when the superconductor side enters a topological phase. This suppression 7 canbeattributedtothemodificationofthevortexstructureoflocalcurrentsatthejunctioncaused 1 bythetopologicaltransitionofthesuperconductor. Thetworegimes(withandwithoutoscillations) 0 could be seen in a semiconductor 2D junction with a cleaved-edge geometry, one of the junction 2 arms having proximitized superconductivity. We predict similar oscillations and suppression as a n functionoftheRashbacoupling. Theoscillationsuppressionisrobustagainstdifferencesinchemical a potential and phases of lateral superconductors. J 5 PACSnumbers: 73.63.Nm,74.45.+c 1 ] I. INTRODUCTION l l a h Magnetoconductanceoscillationsarea centraltopicof - the field of quantum transport in nanostructures. Fa- s mous examples are the celebrated Aharonov-Bohm os- e m cillations in a quantum ring and the Shubnikov-deHass oscillations in the quantum Hall effect [1]. In a general . t sense,theaccumulationofcomplexphases(angles)inthe a m wave function during orbital motion in a perpendicular magneticfieldisthe basiccausebehindthemagnetocon- - d ductance oscillations. Inpresence ofa superconductivity n pairing gap, transport can be described as the propa- o gation of electron and hole quasiparticles, with Andreev FIG.1. Sketchesof cleaved-edge2D wires in a uniform mag- c processesallowingthetransformationofonetypeintothe netic field. One of the arms is proximity coupled with an [ other. Andreev reflections in a normal-superconductor s-wave superconductor. The magnetic field is perpendicular 2 (NS) junction are affected by a magnetic field acting on and parallel to thenormal andhybrid2D leads, respectively. v the normal side, the field thus modifying the magneto- 2 conductance of the junction. 8 The interplay of Andreev reflection and magnetic or- nating electron-hole and hole-electron reflections at the 0 bital effects has received a long lasting attention [2– transverse boundary of the 2D junction. Therefore the 3 0 11]. In particular, Takagaki [3] studied the magneto- resultingconductancedependsonthespatialstructureof . conductance of a 2D NS junction, laterally confined to thesereflectionswithrespecttothe transverseboundary 1 a width L and with the N terminal being a semicon- of the junction. This scenario of magnetoconductance 0 y 7 ductor. As the magnetic field is increased the magne- oscillations has been discussed in detail in Refs. [3–15] 1 toconductance tends to decrease in a stepwise manner and it has been experimentally confirmed in Ref. [7]. : due to the depopulation of the active Landau bands of In this work we investigate the fate of Takagaki oscil- v the semiconductor. It was predicted, however, that for lations when the trivial superconductor lead is replaced i X high enough fields conspicuous magnetoconductance os- by a topological superconductor lead. We consider a 2D r cillationssuperimposedtothegeneralstepwisereduction junction of semiconductor wires with effective topolog- a would be present. Maxima are related to (electron-hole) ical superconductivity in one of the junction sides in- Andreev reflection while minima are due to enhanced duced by proximity with an s-wave superconductor. It normal (electron-electron) reflection. Andreev reflection hasbeenmuchstudied recentlyhowthe resulting hybrid suppression or enhancement appears because of the spa- semiconductor-superconductorsystemcanbedriveninto tialseparationbetweentheelectronandholeedgestates, a topological phase by the combined action of a parallel each one attached to a different edge of the N lead. In magneticfieldandRashbaspin-orbitinteraction[16–18]. presenceoforbitalmagneticeffects the only waythe two In hybrid nanowires the characteristic of the topological edgechannelscanbeconnectedisthroughmultiplealter- phase is the emergence of zero-energy Majorana modes attached to possible potential barriers or wire ends [19– 23]. In this work, we provide evidence that in nanowire junctions the nature of the Andreev reflections changes ∗ javier@ifisc.uib-csic.es fromthetrivialtothetopologicalphasesuchthatmagne- 2 toconductance oscillations are completely suppressed in proximity and a more clever arrangement is needed in the topological phase. The abrupt magnetoconductance order to avoid orbital effects in the gapped region of the change of behavior across the phase transition is thus hybrid superconducting section, such as in Fig. 1. anotherclearsignalofthe topologicalsuperconductivity. The system we have in mind, sketched in Fig. 1, is a 2D semiconductor wire in a cleaved-edge-like geometry A. The model [24], with one of the two arms proximitized with a stan- dard s-wave superconductor. The magnetic field is such A 2D junction, with x and y the longitudinal (parallel that it is perpendicular and parallel to the normal and to transport) and transverse coordinates, respectively, is proximitizedarms,respectively. Thiswaywemaysimul- describedby the followingBogoliubov-deGennesHamil- taneouslyachievewithauniformfieldtherequiredquan- tonian tum Hall behavior on the N arm and the possibility to induce the topological transition on the hybrid S (prox- H =H +H +H +H , (1) imitized)arm. Thecleaved-edgedevicesuggestedbyFig. BdG W Z R 0 1 assumes a uniform magnetic field, a realistic approxi- whereH containsthekineticandconfinementpotential mation in view of the small size of the nanostructure. W terms In principle, non uniform fields could also be created by attaching micromagnets [25], but this would possibly be p2 +p2 a technically more involved alternative. It should also H = x y +V(y)−µ τ , (2) W z 2m be stressed that our simplified model only considers the ! different relative orientations of the magnetic field with respect to the quasiparticle motions, while other effects HZ is the Zeeman term, of the cleavage such as confinement inhomogeneities af- fecting the motion at the bending are disregarded in a HZ =∆B(x)~n(x)·~σ , (3) first approximation [26]. The work is organized as follows. Section II gives the HR is the Rashba spin-orbit term, details of the theoretical model and formalism. In a first stage(Sec.III)theanalysisofresultsisperformedassum- α(x) H = (p σ −p σ )τ , (4) R ~ x y y x z ing the field canbe tuned independently in both armsof the junction. This allows a more clear understanding of and, finally, H is the superconductor pairing term the physical behavior. The uniform (homogenous) field 0 is thenconsideredinSec.IVfor varyingwirewidths and H =∆ (x)τ . (5) Rashba strengths. The robustness of the oscillation sup- 0 0 x pression for finite biases and considering lateral super- The potential V(y) models the transversal confinement, conductorsofdifferent pairing-gapphasesis discussedin with zero value inside the nanowire and infinite out- Sec. V. Finally, Sec. VI summarizes the conclusions of side. In the Zeeman term, ~n gives the orientation of the the work. field, along z and x for the N and S sides, respectively. The intensities of Zeeman, Rashba and pairing interac- tions are given by {∆ (x),α(x),∆ (x)} and they may II. THEORETICAL MODEL AND FORMALISM B 0 take different constant values in the N and S sides, i.e., {∆(N),α(N),∆(N)}and{∆(S),α(S),∆(S)}. Orbitalmag- Weconsidernanowirescontainingalltheingredientsto B 0 B 0 neticeffectsareincludedforperpendicularfieldsthrough createtopologicalphasesandmagnetoconductanceoscil- the substitution p → p −~y/l2 as, e.g., in Ref. [27], x x z lations. Hybridnanowiresthatcombines-wavesupercon- with the magnetic length defined by the perpendicular ductivity, Rashba interaction, and an external magnetic component of the field B as l2 = ~c/eB . The Zeeman z z z fieldinaparallelorientationareknowntosustainatopo- parameter∆ is relatedto the modulus of the magnetic B logical phase if a critical magnetic field is exceeded [17]. ∗ ∗ field B as ∆ = g µ B/2, with g the effective gyro- B B s-wavehybridsuperconductivityisachievedbyproximity magnetic factor and µ the Bohr magneton. B coupling a semiconductor nanowire with a superconduc- We shall obviously have ∆(N) =0, i.e., no pairing gap torwhileRashbainteractionisapropertyofthesemicon- 0 in the normal side, and the model also assumes a pri- ductornanostructuredue tothe confinementasymmetry (N) (S) ori that the two Zeeman intensities ∆ and ∆ can ofthenanowire. Ontheotherhand,formagnetoconduc- B B be varied independently. The latter is only for discus- tanceoscillationsweneedatwo-dimensionalNSjunction sion purposes since a more realistic situation with a uni- with orbital effects in the normal side but not in the superconductor side. This condition is automatically met formfieldrequires∆(N) =∆(S) (Sec.IV).Also,different B B withtruemetallic superconductorsthatdonotallowthe Rashba intensities (α(N), α(S)) represent a modification penetration of magnetic fields to its interior. However, ofthe verticalasymmetry,e.g.,withagate,ofoneofthe with hybrid nanowires superconductivity is obtained by junction sides with respect to the other. 3 B. Resolution method For a given energy E we study the solutions of the equation (H −E) Ψ(x,y,η ,η )=0, (6) BdG σ τ where η ,η are the spin and isospin (electron-hole) dis- σ τ crete variables. The solution is obtained by means of the numerical method presented in [28]. The algorithm is basedonthe quantum-transmitting-boundarymethod and,inessence,providesawayofmatchingtwodifferent sets of asymptotic solutions characterized as superposi- tions of complex-k plane waves. Advantages of this ap- proach are the high spatial resolution and the inclusion of large numbers of evanescent modes without requiring large 2D grids, i.e, without exceedingly large computa- tional costs. In the leads, at both sides of the junction, the wave function can be expressed as a superposition of plane waves,where the wavenumber k may be real or complex FIG. 2. Magnetoconductances as a function of ∆(N) for a B and is a characteristic of the fully translationally invari- fixed ∆(S) (blue dots). The number of propagating modes B antwire[29]. Foreachlead,labeledas(contact)c=N,S in the N wire is given by the dashed line. Panel a) corre- the wavefunction thus reads sponds to a trivial superconductor (∆(BS) = 6EU) and panel Ψ(x,y,ησ,ητ)= d(nαα,c)eikn(αα,c)xφ(nαα,c)(y,ησ,ητ), mb)aitnoinagtoppaoralomgeictaelrssuapreercEon=duc0t,orα((S∆)(BS=) =α(8NE) U=).2TEhUeLrUe-, αX,nα ∆(0S) =0.7EU,µ=70EU. Theratio∆(0S)/µissmallenough (7) (≈ 0.01) to ensure the validity of the so-called BTK regime where α = i,o labels the input/output condition of [30]. The smallest critical magnetic field of the hybrid wire, the mode and nα = 1,2,... the mode number. The Eq.(11), is ∆(Bc,)4 ≈7EU. wavenumbers k(α,c) and the transversal components nα φ(α,c) are obtained solving a transformed version of the nα BdGequationforthecontacts,recastasanon-Hermitian III. INDEPENDENTLY TUNABLE FIELDS eigenvalue problem for the wavenumbers [28]. It is as- sumed that a large set of them is known. The coeffi- The resultsdiscussedbelow aregivenin the same unit (α,c) cients dnα , the channel amplitudes, are obtained from systemofRef.[27],characterizedbyalengthunitLU and the matching equations. Note that the energy is used a corresponding energy unit E = ~2/mL2. A natural U U here as a parameter that is fixed selecting the proper in- choice is L =L , the transversewidth of the 2D stripe U y put channel. Finally, the conductance is calculated from although below we will also use in some specific cases the wavefunction as different values for L and L . With L = 150 nm and U y U m = 0.033m , typical with InAs, it is E = 0.10 meV. e U dI e2 ∗ From ∆ =g µ B/2, with µ the Bohr magneton and (E)= N(E)−P (E)+P (E) , (8) B B B dV h ee eh g∗ =15 (gyromagneticfactor), the magnetic field modu- (cid:2) (cid:3) luscorrespondingtoagiven∆ isB =0.23(∆ /E )T. B B U where 2 2 P (E)= d(o,N) dy φ(o,N)(y,η ,⇑) , (9) ee no no σ A. Conductance in trivial and topological phases nXoησ(cid:12) (cid:12) Z (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Peh(E)= (cid:12)dn(oo,N)(cid:12)2 dy (cid:12)φn(oo,N)(y,ησ,⇓)(cid:12)2 (10) Firstly,westudythe magnetoconductanceofthejunc- nXoησ(cid:12) (cid:12) Z (cid:12) (cid:12) tion model of Sec. II assuming that the magnetic field (cid:12) (cid:12) (cid:12) (cid:12) in the two junction sides (the two arms of the cleaved- (cid:12) (cid:12) (cid:12) (cid:12) are the normal (electron-electron) and Andreev edge nanowire) can be tuned independently. Although (electron-hole) reflection probabilities. Note that this is not very realistic, it is a good starting point from (o,N) (o,N) d φ (y,η ,η ) is the spinor amplitude in a a theoretical point of view as it allows controlling the no no σ τ particularelectron(η =⇑)orhole(η =⇓)channelwith topological and trivial phases of the (hybrid) supercon- τ τ spin (η =↑,↓)andcorrespondingto output towardsthe ductor without changing the parameters for the normal σ N lead. side. Thisway,anydifferencebetweenthetwosituations 4 can be ascribed to the superconductor phase condition. Figures 2a and 2b show the magnetoconductance as a function of the magnetic field on the normal side of the junction ∆(N), while the superconductor side of the B junctions remains under a constant field ∆(S). The fig- B urealsoshowstheevolutionofthenumberofpropagating modes in the normal side (dashed line). Note that, this number decreases with the magnetic field in a stepwise manner, with steps of one unit. This behavior is due to our normal contact including a non zero Zeeman term FIG. 3. Pattern of probability currents for the trivial (a) (cf. Ref. [3]). We consider a nanowire made of an ho- and topological (b) phases of the superconductor lead. The mogeneousmaterial,hence with aconstantRashbaterm parameters (for each panel) are the same as in Fig. 2. The throughout. Together, Rashba and Zeeman terms split incidentandreflectedcurrentsinb)aredifficulttodistinguish and mix the spin degrees of freedom in a way that full because of the vortex scale dominating the normalization of pure Andreev reflection is not achievedfor low magnetic theplot. fields, but there is some normalreflection too. As a con- sequence, the conductance in this limit is less than two times the number of active modes. Full Andreev reflec- one channel can be attached to the topological mode of tionatvanishingfieldsisrecoveredwhendisregardingthe the superconductor while the rest undergo Andreev or Zeeman and Rashba terms (but maintaining the orbital normal reflection by means of the usual physics of the effects). trivial superconductor. InFig.2athesuperconductoriskeptinatrivialphase (S) with ∆ below a critical value, while in Fig. 2b it is B in a topological phase because of the larger ∆(S). While B. Chemical potential dependence B both figures are similar for weak magnetic fields, clear conductance oscillations arise for strong fields when the In Fig. 4 we show magnetoconductances of the NS superconductoris inthe trivialphase. These oscillations junction whenthe chemicalpotentials ofthe normaland are of the same kind and share the same origin of those superconductor regions differ. The superconductor re- discussedinRef.[3]. Inasemiclassicalimagetheincident gion is kept in the topological phase using the same pa- electrons are reflected as holes by the junction but the rameters of Fig. 2b and the changes in chemical poten- orbital effect bends their trajectories towards the junc- tial are only in the normal region. The main effect of tion again. This process is repeatedalongthe transverse increasing the N chemical potential is the displacement boundary creating a pattern that may enhance or sup- to higher magnetic fields of the plateau of two incident press Andreev reflection against normal reflection. This modes. Therefore the region of conductance oscillation effect can be seen as causing a complicated vortexstruc- suppression is consistently displaced to stronger mag- ture in the junction (Fig. 3a). In the particular case of netic fields too. Furthermore, for high enough N chemi- Fig. 2a the oscillations are seen only in the plateau of cal potentials oscillations appear in regions where more two active modes; however, higher chemical potentials than two incident modes are active. This is the usual µ allowing more active modes for strong magnetic fields physics of magnetoconductance oscillations with trivial wouldenableoscillationsatdifferentincidentconditions. superconductors and, as discussed above, the oscillation On the other hand, when the superconductor is in the suppressionaffects only the channel that attaches to the topologicalphase (Fig. 2b) the oscillations are fully sup- topological mode of the superconductor. pressed in the two-mode plateau. This occurs because thevortexstructureinthejunctionisdramaticallymod- ified with respect to the trivial case. In the topological IV. UNIFORM MAGNETIC FIELDS phase a single dominant vortex is formed at the junction(Fig.3b)irrespectiveofthestrengthofthemagnetic field, eliminating this way any transverse structure that After having analyzed in the preceding section the may enhance or suppress Andreev reflection. Note, too, topological suppression of oscillations with two indepen- that in the topological phase only one channel is open dent tunable fields, we study now the more realistic case to Andreev reflection while the other one is reflected as of the cleaved-edge nanowire in a uniform field (Fig. 1). normal electron-electron back scattering. The reason is Our model neglects the presence of localized states at thatthetopologicalnumberofthesuperconductorisonly the cleavage but it is enough to study the magnetocon- zero or one, larger values being forbidden in the D sym- ductance oscillations. The modulus of the field may be metry class of the Hamiltonian. As mentioned above, aboveorbelowthecriticalvaluefortheSlead. Thiscrit- the complete suppression of magnetoconductance oscil- icalvalue in presence of a 2D Rashba field was discussed lations is only achieved in the two-mode plateau. Only in Ref. [27], generalizing previously known expressions 5 FIG. 4. Magnetoconductances for different chemical potentials of the normal and superconductor leads (µ(N),µ(S)). We have used fixed values ∆B(S) = 8EU and µ(S) = 70EU (topological superconductor),whileforeachpanelµ(N) is: a) 70.5EU, b) 105EU, c) 140EU and d) 250EU. Dashed line and rest of parameters as in Fig. 2. [17, 18, 31]. It reads mα2 2 ∆(c) = µ−ǫ + +∆2 (11) B,n s n 2~2 o (cid:18) (cid:19) where ǫ = ~2π2n2/2mL2 with n = 1,2,... are the 1D n y FIG. 5. Magnetoconductances of the cleaved-edge wire in square well eigenenergies. uniform field (Fig. 1), using ∆B =∆(BS) =∆(BN), for selected wirewidthsLy: a)0.7LU,b)0.73LU,c)1.0LU. Dashedline and rest of parameters as in Fig. 2. A. Width dependence From Eq. (11) it is clear that we can adjust the critical field by changing the width of the nanowire. In this manner we can choose in which phase of the hybrid su- α can be tuned using external electric gates [32]. If the perconductor we find the plateau of two incident modes, superconductorisinatrivialphaseweobservevariations wherewe expectto recoverthe behaviordiscussedinthe in the conductance when changing the Rashba coupling preceding section. For instance, in Fig. 5a the nanowire in the normalside of the junction (Fig. 6a). On the con- is narrow enough to find the superconductor only in its trary,ifthesuperconductorisinthetopologicalphasethe trivialphase,whileinFig.5cthenanowireiswideenough conductance remains stuck at the quantized value pro- to find the plateau of two incident modes already in the vided that only two incident modes are active, as shown topological region. We can also find an intermediate in Fig. 6b. We are assuming the existence of a gate that width in which oscillations are present at the beginning allow us to tune the Rashba coupling in the normal side of the plateau and perfect suppression is seen in the rest ofthejunctionindependentlyofthesuperconductorside. of the plateau (Fig. 5b). Both regimes are separated in With large enough α’s additional incident modes are ac- an abrupt way by the critical magnetic field. This is yet tive,thus generatingthe magnetoconductancevariations another proof of the topological suppression of magne- observed at the right end of Fig. 6b. toconductance oscillations as it appears exactly at the expected critical value. Besides the two-mode restriction, it is important to also take into account that the value of the critical field, B. Dependence on Rashba coupling Eq. (11), changes with α and the chemical potential in the superconductor side. In particular in Fig. 6 we have Another option to avoid using spatial variations of changed the overall chemical potential in the nanowire the magnetic field strength is tuning the intensity of the to control its phase while maintaining the homogeneous Rashba coupling. It is well known from spintronics that external magnetic field fixed. 6 FIG. 7. Magnetoconductance when the hybrid nanowire is proximity coupled to two different superconductors with phases φ and φ (phase difference φ ≡ φ −φ ) sketched 1 2 1 2 in the inset to panel b. Each superconducting region has a FIG. 6. Magnetoconductance as a function of the Rashba width 0.2LU, while the width of the intermediate region is coupling in the normal side α(N). We have used fixed values 0.6LU. The different panels are for a) ∆(BS) = 6EU (trivial ∆B = ∆(BN) = ∆B(S) = 8EU, α(S) = 2EULU and ∆(0S) = phase), φ=0, b) ∆(BS) =8EU (topological phase), φ=0, c) 0.7EU. The superconductoris always in the trivial phase for ∆(BS) =6EU (trivialphase),φ=πandd)∆(BS) =8EU (topo- panel a) where µ(S) = 60EU while it is in the topological logical phase),φ=π. Dashedlineandrestofparametersare phaseforpanelb)withµ(S) =70EU. Therestofparameters as in Fig. 2. are thesame of Fig. 2. p-wave. Andreev reflection is only found in the topo- V. ROBUSTNESS AND GENERALITY logical phase and therefore Takagaki-like oscillations are never found with p-wave superconductivity. A relevant question is how robust is the suppression effect discussed in this work. Suppression of magnetoconductance oscillations occurs around zero energy, cor- A. Lateral S phases responding to vanishing potential bias on the NS junction. Moving away from zero energy (non zero bias) will eventually give rise to oscillations again. The suppres- We next ask ourselves what happens to the magneto- sion, however, is not a single-energy phenomenon fading conductance oscillations if the hybrid nanowire is later- awaywithinfinitesimalenergiesbutsurvivesforenergies allycoupledtotwodifferentsuperconductorswithdiffer- within a finite range. The region around zero in which ent phases. One superconductor will be proximity cou- the suppression occurs gets wider with an increasing su- pled at one side of the wire while the other is in con- perconducting gap. On the other hand, it is known that tact with the other side in the transverse direction. The trivialsuperconductorsstopprovidingAndreevreflection middle region of the hybrid nanowire will remain nor- forlargegaps,andthereforeanyrelatedmagnetoconduc- mal. The resulting transverse structure is sketched in tance oscillations are also quenched for large gaps. Nev- Fig. 7b. As shown in Fig. 7a, it is not needed that the ertheless,Andreev reflectionwith oscillationsuppression superconductorpairingextendsthroughoutthenanowire ismorerobustandeasiertofindwithlargesuperconduc- to obtain magnetoconductanceoscillations,providedthe tor gaps in the topological phase of the superconductor. superconductors are in the trivial phase. Furthermore, We believe magnetoconductance suppression is a gen- if the superconductors are in the topological phase we eral property of any kind of topological superconductor. recover the same kind of oscillation suppression already We have actually checked that with a p-wave supercon- seenfor the full superconducting nanowire. Remarkably, ductor in the topological phase [33] the same kind of inFig.7cwecanseehowaphasedifferencebetweenboth suppression of magnetoconductance variations is found. superconductors changes the shape of the conductance However,themaindifferencebetweenp-waveands-wave oscillations in the trivial phase, displacing the positions cases is that with p-wavesuperconductorsit is no longer of maxima and minima. Despite these changes in the appropriatetothinkoftopologicaloscillationsuppression trivialphase, in panel 7d we cancheck againthe robust- since oscillations are never present [34]. This is because nessoftheoscillationsuppression,thatisnowresilientto no Andreev reflection is found in the trivial phase of the superconductor phase changes in the two-mode plateau. 7 VI. CONCLUSIONS pressed) tuning the Rashba coupling with an external electric field. With p-wave superconductors the same In this work we have shown how a superconductor in physics is found in the topological phase, although mag- topologicalphasesuppressesthemagnetoconductanceos- netoconductance oscillations are absent in the trivial cillations of NS junctions when the N wire sustains only phase and one can not properly speak of oscillationsup- two active modes. These oscillations are present under pression. Finally, we have seen how the oscillations de- the same conditions but for the superconductor in the pend on the phase difference between two trivial super- trivialphase. Thisresultisduetotheformationofalarge conductors proximity coupled with the nanowire in the vortex of probability current at the junction when the S transversedirection. Inthiscasetoo,whenthesupercon- wire enters the topological phase. We have analyzed the ductorsareintopologicalphasethesuppressionisrobust phenomenon from a theoretical point of view, assuming and independent of their relative phase difference. independently tunable fields on both junction sides, and we also studied more realistic scenarios of cleaved-edge wiresin uniformfields. Inthe latter caseit is possible to ACKNOWLEDGMENTS tune the suppression of the oscillations by changing the width of the nanowire. We have proved the robustness of the oscillation sup- This work was funded by MINECO-Spain (grant pression provided the superconductor topological phase FIS2014-52564), CAIB-Spain (Conselleria d’Educacio´, is achieved. Similar oscillations can be found (or sup- Cultura i Universitats) and FEDER. [1] T. Ihn,Semiconductor nanostructures (Oxford, 2010). [18] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, [2] C. W. J. Beenakker, Phys. Rev.B 46, 12841 (1992). Phys. Rev.Lett. 105, 077001 (2010). [3] Y.Takagaki, Phys.Rev. B 57, 4009 (1998). [19] J. 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