SNS junctions in nanowires with spin-orbit coupling: role of confinement and helicity on the sub-gap spectrum
We study normal transport and the sub-gap spectrum of superconductor-normal-superconductor (SNS) junctions made of semiconducting nanowires with strong Rashba spin-orbit coupling. We focus, in particular, on the role of confinement effects in long ballistic junctions. In the normal regime, scattering at the two contacts gives rise to two distinct features in conductance, Fabry-Perot resonances and Fano dips. The latter arise in the presence of a strong Zeeman field that removes a spin sector in the leads (helical leads), but not in the central region. Conversely, a helical central region between non-helical leads exhibits helical gaps of half-quantum conductance, with superimposed helical Fabry-Perot oscillations. These normal features translate into distinct subgap states when the leads become superconducting. In particular, Fabry-Perot resonances within the helical gap become parity-protected zero-energy states (parity crossings), well below the critical field at which the superconducting leads become topological. As a function of Zeeman field or Fermi energy, these zero-modes oscillate around zero energy, forming characteristic loops, which evolve continuously into Majorana bound states as exceeds . The relation with the physics of parity crossings of Yu-Shiba-Rusinov bound states is discussed.
Majorana fermions, particles that are their own antiparticles, have been the subject of intense research over the past decades in the context of particle physics and cosmologymajorana; Avignone:RMP08. During the last few years, this interest extended to the condensed matter arena where Majorana fermions are intensely studied nowadays wilcek; Elliott-Franz. This state of affairs has been driven by the key observation that emergent quasiparticles in superconductors can be described as Majorana fermions Alicea:RPP12; Leijnse:SSAT12; Beenakker:ARCMP13; Stanescu:JOPCM13; Elliott:14. This, together with the recent advances in the field of topological materials Hasan:RMP10; Qi:RMP11, has spurred an intense search for condensed matter realizations of Majorana fermions. Most of these realizations focus on zero-energy modes inside the gap of topological superconductors. These zero modes are Majoranas from the point of view of particle-antiparticle conjugation, but they do not obey fermionic exchange statistics 111In fact they obey non-Abelian exchange statistics which might have potential applications in fault-tolerant quantum computation. See, Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-abelian anyons and topological quantum computation”, Rev. Mod. Phys. 80, 1083–1159 (2008). Thus, instead of Majorana fermions, they are now more precisely referred to as Majorana bound states (MBSs) or Majorana zero modes. 222Recently, it has been argued that Bogoliubov quasiparticles in conventional superconductors are true Majorana fermions. See, C, Chamon, R. Jackiw, Y. Nishida, S.-Y. Pi and L. Santos, “Quantizing Majorana fermions in a superconductor”, Phys. Rev. B 81, 224515, (2010). Their Majorana fermion nature can be revealed by annihilation processes, see C. Beenakker, “Annihilation of Colliding Bogoliubov Quasiparticles Reveals their Majorana Nature”, Phys. Rev. Lett. 112, 070604 (2014).
Early proposals suggested that MBSs can emerge in exotic superconductors, such as p-wave, since they realize topological phases that support edge excitations with Majorana fermion character PhysRevB.44.9667; volovik; PhysRevB.61.9690; PhysRevB.61.10267; kitaev; PhysRevB.73.220502. Even though p-wave pairing is not robust against disorder and thus scarce in nature, one can engineer systems to mimic such non trivial superconductivity. These are based on the proximity effect between a conventional s-wave superconductor and a topological insulator Fu:PRL08, or a semiconductor nanowire (NW) with strong spin-orbit (SO) coupling Sato:PRL09; Sau:PRL10; Alicea:PRB10; Lutchyn:PRL10; Oreg:PRL10. For the latter case it has been shown Lutchyn:PRL10; Oreg:PRL10 that if an external Zeeman field , orthogonal to the SO axis, exceeds a critical value , where is the Fermi energy and the induced s-wave pairing, zero energy MBSs emerge at the nanowire ends signaling a topologically non-trivial phase.
Unfortunately, the outcome of the simplest detection protocol for MBSs in NW devices Sengupta:PRB01; Bolech:PRL07; Law:PRL09, detection of subgap zero modes through zero-bias anomalies in transport Mourik:S12; Deng:NL12; Das:NP12; Finck:PRL13; Churchill:PRB13; Lee:13, can be obscured, or even mimicked, by other effects Lee:PRL12; Pientka:PRL12; Bagrets:PRL12; Liu:PRL12; Sau:13; Prada:PRB12; Kells:PRB12; Zitkoetal. As a result, there is no clear consensus yet on whether MBSs have been observed or not in NWs 333 Quite recently, further evidence of zero-bias anomalies related to Majoranas have been reported in a different setup consisting of a ferromagnetic atomic chain on top of a superconducting substrate. S. Nadj-Perge, et al, “Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor”, Science, 346, 602 (2014)..
Thus, the time seems right to move beyond zero-bias anomaly experiments and study more complex geometries such as Superconductor-Normal-Superconductor (SNS) junctions pablo; PhysRevLett.112.137001; Flensberg-Xu. This geometry has a number of advantages including the possibility of studying supercurrents Deng:NL12; Doh:S05; Nishio:N11; Nilsson:NL12; Gunel:JOAP12, or direct spectroscopy of Andreev bound states (ABS) PhysRevLett.110.217005; nphys1811; PhysRevLett.104.076805; PhysRevB.88.045101; PhysRevB.89.045422; PhysRevX.3.041034; Levy; Lee:13; PhysRevLett.85.170. As we shall discuss, this latter technique can be used, in principle, to directly monitor the detailed evolution from the trivial to the nontrivial regime. Previous papers have mostly focused on short junctionspablo; Bena1; gilbertini2012; Bena2; stefano2014, while detailed studies of ABS in other relevant geometries, including long and intermediate-length junctions, remain largely unaddressed. In particular, the role of Fabry-Perot resonances occurring in normal transport as the middle NW finite-lenght section of the junction is depleted has never been studied to the best of our knowledge. In this work we fill this void and present detailed calculations of the normal conductance and Andreev spectra in such geometries. We emphasize here that all nanowire experiments should ideally belong to the category studied here, as confinement effects should be present when a ballistic quasi-one dimensional conductor is contacted between leads, especially when the normal part of the NW (in our geometry, the region of length not directly in contact with leads) is gated. This electrical gating naturally creates quantum wells (or barriers) with their associated confined quantum levels in the middle region of the NW.
In the first half of this work, we discuss normal transport across a finite length ballistic NW. We show how bandstructure details in the presence of strong Rashba SO coupling and Zeeman fields may dominate transport, and give rise to distinct features associated to helical phases (defined by singly-degenerate subbands at the Fermi level with spin locked to momentum) known as helical gaps (Fig. 1). Likewise, finite contact resistance induces confinement resonances in conductance as quasibound states develop in the NW. In the simplest case of non-interacting electrons 444Coulomb blockade effects will be discussed elsewhere., we find that confinement generates two types of resonances: Fabry-Perot resonances and helical Fano dips. Fabry-Perot resonances for a spinful mode Kretinin:NL10 will give conductance oscillations with a ceiling of , unless the central NW is depleted into its helical regime, in which case one may observe helical Fabry-Perot resonances with a half-quantum ceiling. For long enough junctions, many helical Fabry-Perot resonances may occur. We discuss that, while confinement effects may mask the helical gap, the characteristic reentrance of helical Fabry-Perot resonances with Zeeman field or gate voltage contains valuable information about non-trivial helical transport through the NW. The second kind of resonances are sharp Fano dips when the central section of the NW is gated to form a quantum well (non-helical) and the NW sections below the contacts (the leads) are helical. Therefore, both types of resonant features in normal transport may signal the helical regime in different sections (central or below the contacts) of the NW. In the presence of superconducting leads, the two lead to distinct effects.
In the second half of this work we consider the connection of this phenomenology to transport in the superconducting regime. Each helical Fano dip in the normal phase translates, in an SNS geometry, into a single subgap state that crosses zero energy as a function of external parameters (Fermi energy or Zeeman field). Such a crossing is often known as a parity crossing, since it is protected by conservation of number parity in the junction. As we discuss, these parity crossings are made possible by the nontrivial topology in the underlying effective p-wave superconductor for . Similar bound states originated from nonmagnetic impurities in topological superconductors and superfluids have been recently discussed in Refs. Sau-Demler; Huetal and can be considered the p-wave counterparts of Yu-Shiba-Rusinov bound states subgap1; subgap2; subgap3; subgap4; subgap5 in standard s-wave superconductors with magnetic impurities. A more direct analogy with standard Yu-Shiba-Rusinov magnetic bound states actually applies in the non-topological phase . In this situation, helical Fabry-Perot resonances in normal conductance translate, in the superconducting regime, into loops around zero energy in the ABS spectrum as a function of external parameters. For long junctions, many of these loops are visible, each separated by a parity crossing at zero energy. As a result, the subgap spectrum contains near-zero energy subgap states that oscillate as a function of Fermi energy or Zeeman field when the N region of the junction is helical. Interestingly, we find that these oscillating near-zero subgap states in the trivial regime are smoothly connected to MBS when Zeeman is increased beyond .
This paper is organized as follows. In section II we describe the Hamiltonian model employed in our work. Section III focuses on the normal conductance and how the two types of resonances, helical Fabry-Perot and helical Fano dips, appear in the system. The rest of the paper is devoted to analysing the consequences of these resonant levels in the sub-gap spectrum in the superconducting regime. After a brief discussion on how the SNS junction is modeled, as well as a discussion about the relevant length scales of the problem, section IV presents a systematic study of the subgap spectrum of SNS junctions, including its dependence on the superconducting phase difference across the junction, . We discuss in detail how the presence of confined levels within the central region affect the ABS and lead to parity crossings in the topological phase. The dependence of the ABS on phase difference, Fermi energy of the normal region and Zeeman field is discussed for both short and long junctions in subsections IV.2 and IV.3, respectively. Our conclusions are presented in Section V. In appendix A we describe in detail how we model SNS junctions by using a tight-binding version of the model resented in Section II. Appendix B discusses an effective model that fully explains the phenomenology behind helical Fano resonances.
Ii Nanowire Model
We present the model for a nanowire with Rashba SOC and in the presence of an external Zeeman. We restrict ourselves to the strictly one dimensional (single-mode) case for simplicity. Generalisations to multimode nanowires are relatively straightforward. The model Hamiltonian reads
where is the momentum operator, is the effective electron mass, the Rashba SOC strength, the Fermi energy and the spin Pauli matrices. An external magnetic field along the wire produces a Zeeman splitting , where is the Bohr magneton and the wire -factor. The Rashba coupling defines a typical length, the spin-orbit length , with the spin-orbit energy defined as . For typical InSb values , with the electron mass and eV Å, the spin-orbit energy is which gives SO lengths of the order of nm.
Note that the Rashba and Zeeman fields in Eq. (1) are perpendicular. As a result, the two spinful bands (shifted by SO) become mixed by the Zeeman term and the zero-field crossing point at zero momentum becomes an anticrossing of size . When the chemical potential lies within this anti crossing gap, the system has two Fermi points, as opposed to four Fermi points for above or below this gap. This window is a helical gap, since the two fermi points correspond to counter propagating states with different spins (the spin projection is locked to momentum) helical, see inset in Fig. 1.
Iii The normal conductance
Before discussing the sub-gap Andreev spectrum of a NW coupled to superconducting leads, we characterize the normal regime in the presence of a Zeeman field. We are interested in particular in the normal conductance as the Fermi energy () in the middle section of the NW (length ) varies with respect to the one in the left and right leads . Such situation models a NW contacted between normal electrodes and with a Fermi energy tuned by a central gate, see e. g. Ref. Churchill:PRB13. For simplicity in the discussion, we model the gate-induced electrostatic potential with an abrupt profile (the role of smooth gate potentials has been recently discussed in Ref. PhysRevB.90.235415).
For computations purposes we discretize Eq. (1) into a tight-binding lattice. The momentum operator introduces hopping elements between nearest-neighbor sites. The transparency of the left and right contacts is parameterised by a factor , introduced in the hopping matrix across the two interfaces, see Appendix A. is calculated by means of the Greens function techniqueCaroli; Haug:07,
where is the full retarded Green’s function. The bare Green’s function of the normal region without the presence of the leads is . The hamiltonian corresponds to in Eq. 1 with . The leads are taken into account through the self-energies , where stands for the left/right lead’s propagator, when decoupled from the system. In this case, corresponds to in Eq. (1) with . Finally, . In practice, is computed recursively with the boundary conditions imposed by the leads.
To set the stage, we first consider an NW-N junction between a semi infinite nanowire and a good metal, which will allow us to discuss deviations when we consider confinement effects. Fig. 1 shows the expected conductance profile as a function of the NW Fermi energy , for different values of the Fermi energy in the metal. At finite magnetic fields, the normal conductance exhibits a gap (with ) of size . As we explained, this gap is a direct consequence of the combined action of Zeeman effect and strong SO coupling and reflects the presence of helical transport, namely spin-polarized counter propagating states helical. As discussed in Ref. PhysRevB.90.235415, the visibility of this helical gap depends on various factors which, importantly, include the actual value of the SO energy. Indeed, as the ratio is made larger, the visibility of the gap in is rapidly degraded (see lower curves in Fig. 1).
We now consider the confined N-NW-N junction geometry. Due to the confinement of the central NW section, Fabry-Perot resonances are expected. Fig. 2 shows the extreme case of a very short central region with only one resonant quasibound state in the junction. As expected, the conductance without external Zeeman field (solid curve) has a Lorentzian shape and reaches its maximum value when (vertical dashed guideline). Similar results are found for small Zeeman fields (dashed). When , however, the leads becomes spin-polarized (or helical, to be precise) and hence the maximum conductance is halved to (red curve).
We consider first the situation with . This regime is characterised by strong Fano dips that appear when is positive, namely when the junction is gated to create a quantum dot instead of a barrier, see Eq. (1). At these Fano dips destructive interference is maximum and . Moreover, the position of these Fano resonances moves to higher as increases (Fig. 2, insets). The Fano dips can be understood by noticing that the system develops a truly bound state at an energy below as increases (Fig. 3a). While for this level lies far below the chemical potential of the leads and cannot significantly affect , in the case at hand, the situation is markedly different. At such high fields, one spin sector in the leads is removed away from the chemical potential, and the leads become helical. Similarly, the bound state below is Zeeman-split, such that the component corresponding to the removed spin sector may then cross the Fermi level at a given (Fig. 3b-d). This results in one spin projection strongly coupled to the continuum (the sector that is not removed), while the other spin projection remains weakly coupled to this helical continuum through the split bound state (dashed circles), owing to the small spin canting induced by SO. This configuration mimics the physics of a Fano resonance, as we explicitly demonstrate in Appendix B with an effective model. Note that SO is essential to mimic the physics of the Fano effect (two channels with very different coupling to the continuum). Indeed, we have checked that for (namely a fully spin-polarized system without canting) the effect disappears (not shown). The general behaviour is related to the so-called Fano-Rashba effect in systems with inhomogeneous Rashba couplings PhysRevB.74.153313; Fano-Rashba-B although in our case the bound states originate from the Fermi energy inhomogeneity, which is probably more realistic for NWs with gates. For intermediate lengths, the system can accommodate many of the above resonances but the helical gap is not visible (not shown).
Consider now the regime complementary to the preceding discussion. In this situation, there exist two propagating channels in the leads, and conductance may reach at Fabry-Perot maxima, as long as the central NW is likewise non-helical (). Otherwise, for long enough junctions ( for the realistic NW parameters in our simulation) a helical gap develops in conductance, such that . As central is tuned into and out of the helical regime, conductance exhibits a reentrant behavior, switching from to and back to . This reentrance can be resolved across multiple resonant helical Fabry-Perot oscillations. This is illustrated in Fig. 4 where we plot the conductance for a 4-long nanowire as a function of the central Fermi energy . Note the reentrant conductance, and the helical Fabry-Perot resonances with an ceiling, signalling helical transport in the junction. The visibility of the conductance reentrance and the helical gap is lost for fields , see bandstructure inset in Fig. 1. At such fields, the helical gap becomes an extended half-plateau (potentially with superimposed Fano resonances if also exceeds ) that emerges directly from pinch-off . Note that the regime with helical Fano dips in the normal conductance is quite relevant towards reaching topological superconductivity: the NW under the contacts can become a non-trivial topological superconductor in the presence of pairing as long as it can be depleted and made helical in the normal phase. Hence our prediction of helical Fano dips superimposed on a half-plateau of constitutes a strong signature of helical behaviour as precursor of non-trivial superconductivity.
Similar phenomenology is obtained for conductance at fixed magnetic fields and increasing (Fig. 5). As expected, the Fano dips disappear as soon as while the gap coming from helicity in the central section in the NW is much more robust. Increasing results in well defined Fabry-Perot resonances in the helical gap region. The normal conductance as a function of magnetic field is shown in Fig. 6. Here, a change from irregular behavior to regular oscillations as a function of magnetic field signals the helical regime when PhysRevB.90.235415.
Having in mind that there exists no conclusive experimental evidence of the helical regime in nanowires in the literaturenphys610; VanWeperen, the nontrivial resonant effects in finite-length junctions that we have described, both helical Fabry-Perot resonances and helical Fano dips, could be used as an interesting option for detecting such helical transport regime in long junctions. Even more significant, these helical resonant features give rise to a non-trivial subgap spectrum when the leads become superconducting, as we discuss in what follows.
Iv Subgap levels in SNS junctions
iv.1 SNS junction model and relevant length scales
To model a SNS junction we assume that the outer parts of the wire are coupled to an s-wave superconductor (with bulk values and pairing ), while the central is not (see Fig. 7). Superconducting correlations are induced by proximity effect into the nanowire. For good enough contact between the NW and the superconductor, the Cooper pair amplitude is finite inside the NW regions below the superconductor. In most papers in the literature, this situation is modeled by including by hand a pairing term, , in the hamiltonian of such NW regions. While, rigorously speaking, this is incorrect (the superconducting coupling constant is zero inside the NW), it is well known that it provides a good description of the proximity effect for large enough gaps (in such cases, the parameter is essentially the low frequency limit of a tunneling self-energy and is given by the tunnel coupling between the normal and superconducting parts, see e.g. Bena2). Therefore, we adopt this approximation here for simplicity (we have checked that all our conclusions remain unaltered irrespective of whether we use this simplified model or a full NW + SC coupling model, see appendix A.3). In cases where the interface transparencies are small, extra Fabry-Perot resonances coming from insulating layers could complicate our analysis, see Ref. Galaktionov-Zaikin.
In particular, we model the regions of the nanowire below the superconducting contacts as regions with Fermi energy and pairing potential on the left (L) and right (R) contact given by and , with . The region in the middle of the nanowire without superconducting correlations is the normal region (N) with Fermi energy denoted by as before. At high enough magnetic fields, the regions of the NW below the superconductors (S regions of the junction) can be driven into a topological superconducting phase when . Owing to the finite length , this results in a SNS junction with four Majorana bound states for a phase difference of between the superconductors: two inner Majorana bound states, labeled , that form inside the junction, and two outer Majorana bound states, , see Fig. 7. On the other hand, for a zero phase difference, only the outer MBSs are present.
SNS Josephson junctions are classified in two types, depending on the relationship between the length of the normal region (i.e. distance between the superconducting contacts) and the coherence length , where is the Fermi velocity. Short junctions are characterized by , whereas in long junctions. Such classification can be also given in terms of natural energy scales of the problem, the Thouless energy, , and the induced superconducting pair potential , being the Fermi velocity, and the length of the normal region. The above conditions, in terms of these energy scales, are for short junctions and for long ones. The significance of this classification is related to the typical number of Andreev subgap states of the junction, in addition to the MBSs at zero energy.
The MBSs wave functions decay from both ends of the topological superconducting leads. The inner and outer MBSs may feel their mutual presence if their wave functions exhibit a non zero spacial overlap. The relevant decay distance characterizing this overlap is the Majorana localization length (appendix C). For finite the overlap between MBSs is significant and therefore they are no longer true zero modes.
In what follows, we discuss the subgap spectrum of short SNS junctions in the topological regime as well as the subgap spectrum of long SNS junctions as one goes from the helical junction regime to the topological one. The helical junction regime is defined by a central region depleted into the helical regime, while the S regions remain non-topological, namely by , and .
iv.2 Short junctions
For very short junctions, the ABS spectrum at and does not contain sub-gap states (Figs. 8a and b). This is expected for a short junction with . The spectrum (Figs. 8c and d), on the other hand, is much more interesting. It contains the expected subgap state near zero energy for all (coming from the weakly coupled outer Majoranas for , the inner MBS at are strongly hybridized and form standard ABS at energy ). Notably, this MBS coexists with a bound state that crosses zero energy for a given (dashed line). This bound state originates from the single resonance that the junction accommodates for increasing (see Fig. 3), which we discussed in connection to Fano resonances. If we interpret this resonant state as an impurity level, our results for are consistent with Anderson’s theorem which prevents the existence of bound states inside the gap of an s-wave superconductor for non-magnetic impurities Anderson-theorem. The reason is that the zero-enery crossing appears for , such that the superconductor is effectively p-wave. Therefore, the emergence of these subgap states crossing zero energy should be understood as a direct consequence of nontrivial topology in the junction Sau-Demler; Huetal. The precise condition for the level crossing coincides with the condition for having a Fano dip. As we discussed in section III, this is the condition in the normal regime for having a single resonant state which interferes destructively with a helical contact; the latter condition is here fulfilled because . These subgap states and zero-energy crossings should be understood as the p-wave counterparts of so-called Yu-Shiba-Rusinov sub-gap states subgap1; subgap2; subgap3; subgap4 and their corresponding parity crossings subgap5 in s-wave superconductors with magnetic impurities.
Further insight comes from the magnetic field dependence at fixed (Fig. 9). After the closing of the gap across the topological phase transition at , the spectrum of the junction exhibits a perfect zero-energy state accompanied by a zero-energy crossing (dashed line) similar to the one discussed in Fig. 8. Note here that, despite the finite length of the central NW, the zero energy state for does not oscillate as a function of Zeeman field, unlike what is typical of overlapping MBSs Lim:PRB12; Prada:PRB12; Rainis:PRB13; DasSarma:PRB12. This can be easily understood as this state comes from the outer MBSs which at are effectively decoupled across the junction, since we assume .
We now analyse in more detail the full phase dependence in the topological phase for different values of . The low-energy sector is characteristic of a short junction: two almost -independent levels near zero energy coming from outer MBSs and two dispersive levels coming from hybridization of inner MBSs across the junction. The anti crossings near are only visible for finite . For (Fig. 10a), the zero-energy levels are flat and the anti crossing at becomes negligible 555In limit, the outer Majoranas are no longer involved in transport while the levels at exactly cross (not shown) giving rise to anomalous -periodic spectrum and Josephson currents if fermionic parity is conserved. In the following, we refer to the dispersive ABS with almost perfect crossings at as Majorana ABSs. As increases, an extra bound state emerges from the continuum as an almost dispersionless subgap state and interacts very weakly with the Majorana ABSs (Fig. 10b). Importantly, after crossing zero energy (Fig. 10c) and reemerging at finite energy (Fig. 10d), the anti crossing with the Majorana ABS is considerably larger, indicating that the bound state has changed its parity character.
iv.3 Long junctions
The ABS spectrum of long junctions at small magnetic fields differs considerably from the one of short junctions. Even for (Fig. 11a), the spectrum is very sensitive to the sharp increase of conductance at small negative , when the junction goes rapidly from pinch-off to fully transmitting (solid black line in Fig. 4). This is reflected in a feature that resembles the closing and reopening of a gap (but, of course, is related to the central region becoming metallic, rather than with a gap closing). The emergence of Fabry-Perot resonances in the normal phase is translated into the appearance of level pairs at finite energies, or loops, that oscillate with system parameters in the superconducting phase. A distinct change in the loop structure takes place as is increased within a window . This, recall, corresponds to the helical regime of the normal region, characterised in normal transport by a helical gap and helical Fabry-Perot oscillations. The loops inside said window reconnect, and give rise to new loops around zero-energy, separated by parity crossings (Fig. 11b). Each of these crossings corresponds to a helical Fabry-Perot resonance in the normal regime. For larger Zeeman energies, supporting many helical Fabry-Perot resonances within the helical gap, correspondingly many consecutive zero-energy loops become visible in the superconducting regime. As soon as the normal side ceases to be helical (), the spectrum does no longer show loops around zero energy. Since depleting the normal section of the NW junction should be much easier than gating the proximized region, we expect that said near-zero loops and parity crossings should be ubiquitous for finite size junctions near depletion 666intermediate junctions also show the same behaviour, not shown and constitute yet another alternative scheme to detect the helical regime.
Each loop in the helical regime (see e.g. Fig. 11b) is similar to the ones expected for magnetic impurities subgap1; subgap2; subgap3; subgap4, or quantum dots in the Coulomb blockade regime Lee:13; PhysRevLett.110.217005 coupled to superconductors (we emphasize here that our junction is noninteracting). This result again suggests an interesting analogy with the physics of Yu-Shiba-Rusinov states in superconductors with magnetic impurities. Here, the combined action of Zeeman-induced spin-polarization and depletion is crucial.
Consecutive loops around zero energy, resemble the oscillatory behavior expected from overlapping MBSs in finite length NWs. However, since the helical gap condition does not involve , which may be large, the zero-energy loops may exist while the proximized S regions are still in the topologically trivial regime (Fig. 11 c and d). Remarkably, there exists a profound connection between zero-loops and MBSs. We find that the former actually evolve continuously into outer MBSs as is increased beyond . To illustrate this key idea, we compare in Fig. 12, a situation without near-zero energy loops at low B fields (, panel a) with another with loops at very low B coming from a helical normal region (, panel b). While the MBSs in the first configuration emerge from a situation without zero energy states/crossings at low fields, the ones corresponding to the second configuration are clearly evolving from the low B-field loops around zero energy. We emphasize here that both configurations correspond to the same physical nanowire junction with the sole difference of a depletion in the normal part of the junction in the second case. Fig. 12 nicely illustrates two of our main results: 1) long loops with parity crossings in the ABS spectrum can be used to identify the helical regime in a Rashba NW and 2) such helical loops, coming from depletion in the normal side of the junction, continuously evolve into MBS for large enough magnetic fields.
To obtain more precise information about the nature of this interesting connection between near-zero loops and MBS states, we study their evolution for increasing SO coupling (Fig. 13). For (Fig. 13a), Zeeman-induced depairing closes the superconducting gap and the spectrum becomes a dense quasi-continuum (the full junction is in the normal regime), as expected. Any removes all finite energy crossings while preserving the parity-protected crossings at zero energy. As a result, the spectrum is still gapped after the first parity crossing (the Zeeman field is no longer fully depairing) and many parity crossings are possible. This important observation is illustrated in Fig. 13(b,c) (see also Fig. 12b). For finite , the low-energy spectrum remains gapped after the first crossing and also after subsequent crossings. Another interesting conclusion that we can draw from our results is that a clear distinction between the near-zero states in the and regions can no longer be made. The only difference is quantitative, in that the amplitude of MBS oscillations in the topological regime become smaller for increasing , unlike for . (The SO length becomes much shorter and, hence ). However, other spectral properties, such as the mini gap separating the near-zero modes from the first excited states, is roughly the same in both the trivial and non-trivial phases.
To finish, we consider the phase dependence of the subgap spectrum. While topological SNS junctions with are -periodic as a function of phase difference due to the characteristic parity-protected crossing at (see e.g. Fig. 10a), in finite junctions (Fig. 14a), said crossing is avoided, and splits by a small energy due to the hybridization of MBSs at the junction (inner) and MBSs at the far ends of each S region (outer), which leads to a more conventional -periodicity San-Jose:PRL12a. Interestingly, the subgap spectrum at (Fig. 14b) shows essentially the same phase-dependence which further confirms the deep connection between the and parity crossings. Note that the resulting Josephson current Cheng:PRB12, which only depends on the Andreev spectrum, would be effectively the same (not shown).
We have studied the normal transport and the sub-gap spectrum of SNS junctions based on semiconducting nanowires with strong Rashba spin-orbit coupling. In particular, we have focused on the role of confinement effects in ballistic finite-length junctions and analyzed the distinct properties of the ABS for short and long junctions as different sections of the underlying NW (N or S or both) become helical. For , confined levels in the normal section give rise to bound subgap states, as expected from the effective p-wave nature of the topological superconductor. In normal transport, such bound states give rise to helical Fano dips. Perhaps more strikingly, we have found that a long junction with a helical normal section, but still in the topologically trivial regime with , supports a low-energy subgap spectrum consisting of multiple-loop structures and parity crossings. Such states are derived from helical Fabry-Perot resonances in the normal regime. We have argued that such multiple loop structure in the ABS spectrum could be used to unambiguously identify the helical regime in NWs. Interestingly, these multiple loops smoothly evolve towards Majorana bound states as the Zeeman field exceeds the critical value. This suggests an interesting connection between subgap parity crossings in helical junctions with and Majorana bound states in topological ones with . A recent study of fully open helical-N/trivial-S contactsSan-Jose:14b further confirms the profound connection between subgap states in the helical regime and Majorana physics.
We acknowledge the support of the European Research Council and the Spanish Ministry of Economy and Innovation through the JAE-Predoc Program (J. C.), Grants No. FIS2011-23713 (P. S.-J), FIS2012-33521 (R. A.), FIS2013-47328 (E. P.) and the Ramón y Cajal Program (E. P).
Appendix A The SNS junction model
In this appendix we describe the model we use for SNS junctions.
a.1 Tight-binding discretisation
For computation purposes, we consider a discretisation of the continuum model Eq. (1) for the Rashba nanowire into a tight-binding lattice with a small lattice spacing . Thus reads,
where the symbol means that couples nearest-neighbor sites. This discretisation transforms Eq. (1), in terms containing on-site energy and into nearest-neighbor hopping matrices which arise from the momentum operator ,
are matrices in spin space and .
a.2 The SNS junction model
The Hamiltonian of the full system considering the proximized NW regions as left and right superconducting leads (see discussion at the beginning of section IVA) is given by
where is the Hamiltonian of the superconducting lead that we consider to be the same, the Hamiltonian that couples the superconducting lead to the normal region, while the Hamiltonian that couples the normal region to the lead . These coupling matrices are non-zero for adjacent sites that lie at the interfaces of the superconducting leads and of the normal region, only. This coupling is parametrized by a hopping matrix between the sites that define the interfaces of the SNS junction, where . A tunnel junction can be modelled by considering , while a full transparent junction with . All the elements in the diagonal of matrix Eq. (5) have the structure of given by Eq. (1) taking into account that the superconducting lead regions have a Fermi energy (or for the normal transport study), while this is for the normal region. It is important to point out here that the matrix of Eq. (5) is of finite size since we are dealing with a fine size system.
Effects of superconductivity are induced by the pairing potential , thus leading to the Nambu description where the new Hamiltonian reads,
The superconducting pairing potential in the previous Hamiltonian equation, that corresponds to the full system, must have the same structure as the SNS Hamiltonian, , thus
where since in the normal region the superconducting correlations are absent.
Superconductivity is induced by an s-wave pairing potential that couples particles of different spin and momenta. So that, is given by
a.3 Induced superconducting pairing
A more realistic model consists on the following description. The full NW is divided in three sections: a central normal region (N) and two normal regions (M). See Fig. 15. Each of the sections describe NW regions coupled to a superconductor which, to distinguish from the previous notation, we denote as . As opposed to the previous subsection, the full NW is now a normal system and the proximity effect comes now from the tunneling coupling between the superconductors and the normal parts of the NW.
In this case, the problem is described by the following Hamiltonian
where is a normal region of the same dimension as the superconducting one and is a diagonal matrix in site space that couples the superconductor with the normal lead . This coupling can be parametrized by the parameter . The superconducting pairing is then written in the same basis as ,
where with describe the bulk s-wave superconducting leads.
As described in the main text, the approximate description of the proximity effect in the previous subsection (Eq. 7) is a good approximation provided that we are in a large gap limit and that the contact transparency is good. We have benchmarked the approximate solution of the previous subsection against the full proximity model in various relevant cases and always found good agreement in the correct parameter range. We here illustrate this point by showing a calculation using the full proximity effect model of Eq. 9 instead of the approximate model of Eq. 7. In Fig. 16 we show results corresponding to the same physical situation we presented in Fig. 11c in the main text, the only difference being that the bulk gap in is much larger than the induced gap used in the calculations of Fig. 11c (). The overall behaviour of the subgap states in Fig. 16 is the same as in Fig. 11c (including the loops in the helical region described in the main text), demonstrating that the simplified model is indeed justified when the bulk gap is the largest energy scale. Importantly, note the rescaled axis which explicitly shows that the relevant energy scale is not the original bulk gap included in the calculation but the smaller value , in agreement with our previous claim.
Appendix B Model for the conductance
In this part we make use of an effective model to describe the physics of Fano resonances. An effective spinless model based on Green’s functions is constructed where two semi-infinite tight-binding chains (leads) are coupled through to a central region , formed by one site, that is additionally weakly coupled through to a resonant level , being a fixed parameter that represents the separation between the quantum dot level and the resonant level (in principle this parameter mimics the role of the Zeeman splitting in our numerics). Consider that is the lattice constant and the hopping between sites in the leads. The normal transmission, , through a central system formed by one site can be calculated by using the Caroli’s formula,
where is the retarded full system Green’s function, and
takes into account the influence of the leads on the central system through the left(right) L(R) self-energies . The full system Green’s function can be calculated by using the Dyson’s relation,
where is the retarded Green’s function of the isolated central region (this central region can for instance be a quantum dot) without the influence of the leads and without the influence of the resonant level. It reads
where is the onsite energy of the central region.
The self-energy ,
contain the effect of the left and right leads as well as the influence of the resonant level , respectively. Such self-energies are defined as follows,
where is the retarded semi-infinite left (right) lead Green’s functions. In principle such lead’s Green’s functions can be computed considering a recursive approach,
is the onsite energy in the leads. From previous equation one has,
Adding a convergence factor to frequency, , one finds the retarded or advanced Green’s function. We have the following properties of ,
where for the first case the density of states is zero, while in the second case it exhibits a non zero value. These results allow us to obtain . The impurity self-energy reads,
where is the coupling of the resonant level to the system. With these expressions for the different self-energies, we may compute ,
The normal conductance is calculated from the transmission as,
where by construction we have already in a spinless channel. Since we are interested in low temperature physics, , and . Therefore,
where is the Fermi energy which is the zero of energy in our calculations.
The aim of this part was to construct an effective model that contains the whole physics of our numerics where a resonance in the trivial phase and a dip in the helical phase the transmission develops. Indeed, by plugging previous equations in the expression for the transmission and conductance, one ends up with the desired result that is plotted in Figs. 17, and 18.
In such plots, we consider a strong hopping between sites in the leads in comparison to the couplings and . For weak coupling between leads and the central region a resonant tunnelling peak is obtained at the energy of the central region . Upon increasing the coupling between the leads and the central region the resonant peak at becomes broader and a sharp Fano feature emerges at the resonant impurity . The new feature has the typical Fano structure of a zero followed by a peak, and arises from the interference of the two possible paths for the carriers, through the very broadened (strongly coupled) site at , and through the weakly coupled resonant level at . For strong enough coupling , the contributes with a uniform background to conductance, while the Fano feature becomes a pure dip to zero.
In conclusion, we have developed an effective model that contains the physics involved in our numerics where a resonance peak is present at the energy of the quantum dot for weakly coupled system. By increasing the coupling of the quantum dot to the leads a Fano feature (dip to zero followed by a peak) appears in conductance at the energy of the resonant level.
Appendix C Majorana localization length
The calculation of is carried out by solving the polynomial equation for the wave vector , where and . Here, we point out that although the previous equation was derived in Ref. Lutchyn:PRL10 for a semiinfinite case, it gives reasonable values for the Majorana localization length. Indeed, in Fig. 19 one observes that linearly increases as one increases for realistic SOC (dashed line), while it acquires smaller values and remains roughly constant for stronger SOC (solid curve).