Andreev spectrum and supercurrents in nanowire-based SNS junctions containing Majorana bound states

  1. Jorge Cayao1,
  2. Annica M. Black-Schaffer1,
  3. Elsa Prada2 and
  4. Ramón Aguado3ORCID Logo

1Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
2Departamento de Física de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) & Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
3Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain

  1. Corresponding author email

This article is part of the Thematic Series "Topological materials".

Associate Editor: J. M. van Ruitenbeek
Beilstein J. Nanotechnol. 2018, 9, 1339–1357. doi:10.3762/bjnano.9.127
Received 03 Jan 2018, Accepted 04 Apr 2018, Published 03 May 2018

Abstract

Hybrid superconductor–semiconductor nanowires with Rashba spin–orbit coupling are arguably becoming the leading platform for the search of Majorana bound states (MBSs) in engineered topological superconductors. We perform a systematic numerical study of the low-energy Andreev spectrum and supercurrents in short and long superconductor–normal–superconductor junctions made of nanowires with strong Rashba spin–orbit coupling, where an external Zeeman field is applied perpendicular to the spin–orbit axis. In particular, we investigate the detailed evolution of the Andreev bound states from the trivial into the topological phase and their relation with the emergence of MBSs. Due to the finite length, the system hosts four MBSs, two at the inner part of the junction and two at the outer one. They hybridize and give rise to a finite energy splitting at a superconducting phase difference of π, a well-visible effect that can be traced back to the evolution of the energy spectrum with the Zeeman field: from the trivial phase with Andreev bound states into the topological phase with MBSs. Similarly, we carry out a detailed study of supercurrents for short and long junctions from the trivial to the topological phases. The supercurrent, calculated from the Andreev spectrum, is 2π-periodic in the trivial and topological phases. In the latter it exhibits a clear sawtooth profile at a phase difference of π when the energy splitting is negligible, signalling a strong dependence of current–phase curves on the length of the superconducting regions. Effects of temperature, scalar disorder and reduction of normal transmission on supercurrents are also discussed. Further, we identify the individual contribution of MBSs. In short junctions the MBSs determine the current–phase curves, while in long junctions the spectrum above the gap (quasi-continuum) introduces an important contribution.

Keywords: hybrid superconductor–semiconductor nanowire–superconductor junctions; Josephson effect; Majorana bound states; nanowires; spin–orbit coupling; Zeeman interaction

Introduction

A semiconducting nanowire with strong Rashba spin–orbit coupling (SOC) with proximity-induced s-wave superconducting correlations can be tuned into a topological superconductor by means of an external Zeeman field [1-3]. This topological phase is characterized by the emergence of zero-energy quasiparticles with Majorana character localized at the nanowire ends. These Majorana bound states (MBSs) are attracting a great deal of attention owing to their potential for topological, fault-tolerant quantum computation [4-6]. Tunneling into such zero-energy MBSs results in a zero-bias peak of high 2e2/h in the tunnelling conductance in normal–superconductor (NS) junctions due to perfect Andreev reflection into a particle–hole symmetric state [7]. Early tunnelling experiments in NS junctions [8-12] reported zero-bias peak values much smaller than the predicted 2e2/h. This deviation from the ideal prediction, together with alternative explanations of the zero-bias peak, resulted in controversy regarding the interpretation. Recent experiments have reported significant fabrication improvements and high-quality semiconductor–superconductor interfaces [13-16] with an overall improvement on tunnelling data that strongly supports the observation of MBS [17-21].

Given this experimental state-of-the-art [22], new geometries and signatures beyond zero-bias peaks in NS junctions will likely be explored in the near future. Among them, nanowire-based superconductor–normal–superconductor (SNS) junctions are very promising since they are expected to host an exotic fractional 4π-periodic Josephson effect [4,23,24], signalling the presence of MBSs in the junction. While this prediction has spurred a great deal of theoretical activity [25-32], experiments are still scarce [33], arguably due to the lack of good junctions until recently. The situation is now different and, since achieving high-quality interfaces is no longer an issue, Andreev-level spectroscopy and phase-biased supercurrents should provide additional signatures for the unambiguous detection of MBSs in nanowire SNS junctions. Similarly, multiple Andreev reflection transport in voltage-biased SNS junctions [34,35] is another promising tool to provide further evidence of MBSs [36].

Motivated by this, we here present a detailed numerical investigation of the formation of Andreev bound states (ABSs) and their evolution into MBSs in nanowire-based short and long SNS junctions biased by a superconducting phase difference [Graphic 1]. Armed with this information, we also perform a systematic study of the phase-dependent supercurrents in the short- and long-junction limits. Due to finite length, the junction always hosts four MBSs in the topological regime. Apart from the MBSs located at the junction (inner MBSs), two extra MBSs are located at the nanowire ends (outer MBSs). Despite the early predictions [4,23,24] of a 4π-periodic Josephson effect in superconducting junctions containing MBSs, in general we demonstrate that the unavoidable overlap of these MBSs renders the equilibrium Josephson effect 2π-periodic [26,27] in short and long junctions, since they hybridize either through the normal region or through the superconducting regions giving rise to a finite energy splitting at phase difference [Graphic 2] = π. As an example, our calculations show that, for typical InSb parameters, one needs to consider junctions with long superconducting segments of the order of LS ≥ 4μm, where LS is the length of the S regions, in order to have negligible energy splittings.

In particular, we show that in short junctions with [Graphic 3], where [Graphic 4] is the normal region length and ξ is the superconducting coherence length, the four MBSs (inner and outer) are the only levels within the induced gap. On the contrary, the four MBSs coexist with additional levels in long junctions with [Graphic 5], which affect their phase dependence. Despite this difference, we demonstrate that the supercurrents in both limits exhibits a clear sawtooth profile when the energy splitting near [Graphic 6] = π is small, therefore signalling the presence of weakly overlapping MBSs. We find that while this sawtooth profile is robust against variations in the normal transmission and scalar disorder, it smooths out when temperature effects are included, making it a fragile, yet useful, signature of MBSs.

We identify that in short junctions the current–phase curves are mainly determined by the levels within the gap and by the four MBSs, with only very little quasi-continuum contribution. In long junctions, however, all the levels within the gap, the MBSs and the additional levels due to longer normal region together with the quasi-continuum determine the current–phase curves. In this situation, the additional levels that arise within the gap disperse almost linearly with [Graphic 7] and therefore affect the features of the supercurrents carried by MBSs only.

Another important feature we find is that the current–phase curves do not depend on LS in the trivial phase (for both short and long junctions), while they strongly depend on LS in the topological phase. Our results demonstrate that this effect is purely connected to the splitting of MBSs at [Graphic 8] = π, indicating another unique feature connected with the presence of MBSs in the junction. The maximum of such current–phase curves in the topological phase increases as the splitting is reduced, saturating when the splitting is completely suppressed. This and the sawtooth profile in current–phase curves are the main findings of this work. Results presented here therefore strongly complement our previous study on critical currents [37] and should provide useful insight for future experiments looking for Majorana-based signatures in nanowire-based SNS junctions.

The paper is organized as follows. In section “Nanowire model” we describe the model for semiconducting nanowires with SOC, where we show that only the right combination of Rashba SOC, a Zeeman field perpendicular to the spin–orbit axis and s-wave superconductivity leads to the emergence of MBSs. Similar results have been presented elsewhere but we include them here for the sake of readability of the next sections. In section “Results and Discussion” we discuss how nanowire-based SNS junctions can be readily modeled using the tools of section “Nanowire model”. Then, we describe the low-energy Andreev spectrum and its evolution from the trivial into the topological phase with the emergence of MBSs. In the same section, we report results on the supercurrent, which exhibits a sawtooth profile at [Graphic 9] = π as a signature of the emergence of MBSs. In section “Conclusion” we present our conclusions. For the sake of completeness, we also show wavefunction localization and exponential decay as well as homogeneous charge oscillations of the MBSs in wires and SNS junctions in Supporting Information File 1.

Nanowire model

The aim of this part is to properly describe the emergence of MBSs in semiconducting nanowires with SOC. We consider a single-channel nanowire in one-dimension with SOC and Zeeman interactions, the model Hamiltonian of which is given by [38-43]

[2190-4286-9-127-i1]
(1)

where [Graphic 10] is the momentum operator, μ the chemical potential that determines the filling of the nanowire, αR represents the strength of Rashba spin–orbit coupling, [Graphic 11] is the Zeeman energy as a result of the applied magnetic field [Graphic 12] in the x-direction along the wire, g is the g-factor of teh wire and μB the Bohr magneton. Parameters for InSb nanowires include [8]: the effective mass of the electron, m = 0.015me, with me being the mass of the electron, and the spin–orbit strength αR = 20 meV·nm.

We consider a semiconducting nanowire placed in contact with an s-wave superconductor with pairing potential ΔS′ (which is in general complex) as schematically shown in Figure 1. Electrons in such a nanowire experience an effective superconducting pairing potential as a result of the so-called proximity effect [44,45]. In order to have a good proximity effect, a highly transmissive interface between the nanowire and the superconductor is required, so that electrons can tunnel between these two systems [13-16]. This results in a superconducting nanowire, with a well-defined induced hard gap (namely, without residual quasiparticle density of states inside the induced superconducting gap). The model describing such a proximitized nanowire can be written in the basis ([Graphic 13]) as

[2190-4286-9-127-i2]
(2)

where ΔS < ΔS′. Since the superconducting correlations are of s-wave type, the pairing potential is given by

[2190-4286-9-127-i3]
(3)

where [Graphic 14] is the superconducting phase. We set [Graphic 15] = 0 when discussing superconducting nanowires, while the SNS geometry of course allows a finite phase difference [Graphic 16] ≠ 0 across the junction.

[2190-4286-9-127-1]

Figure 1: A semiconducting nanowire with Rashba SOC is placed on a s-wave superconductor (S’) with pairing potential ΔS′ and it is subjected to an external magnetic field [Graphic 17] (denoted by the black arrow). Superconducting correlations are induced into the nanowire via proximity effect, thus becoming superconducting with the induced pairing potential ΔS < ΔS′.

It was shown [1,2,46] that the nanowire with Rashba SOC and in proximity to an s-wave superconductor, described by Equation 2, contains a topological phase characterized by the emergence of MBSs localized at the ends of the wire. This can be understood as follows: The interplay of all these ingredients generates two intraband p-wave pairing order parameters

[Graphic 18]

and one interband s-wave

[Graphic 19]

where + and − denote the Rashba bands of H0. The gaps associated with the ± Bogoliubov–de Gennes (BdG) spectrum are different and correspond to the inner and outer part of the spectrum, denoted by Δ1,2 at low and high momentum, respectively. These gaps depend in a different way on the Zeeman field. Indeed, as the Zeeman field B increases, the gap Δ1, referred to as the inner gap, is reduced while Δ2, referred to as the outer gap, is slightly reduced although for strong SOC it remains roughly constant. The inner gap Δ1 closes at B = Bc and reopens for B > Bc giving rise to the topological phase, while the outer gap remains finite. The topological phase is effectively reached due to the generation of an effective p-wave superconductor, which is the result of projecting the system Hamiltonian onto the lower band (−) keeping only the intraband p-wave pairing Δ−− [1,2]. Deep in the topological phase B > Bc, the lowest gap is Δ2.

In order to elucidate and visualize the topological transition, we first analyze the low-energy spectrum of the superconducting nanowire. This spectrum can be numerically obtained by discretising the Hamiltonian given by Equation 1 into a tight-binding lattice:

[2190-4286-9-127-i4]
(4)

where the symbol [Graphic 20] means that v couples the nearest-neighbor sites i, j; h = (2t − μ)σ0 + Bσx and v = −tσ0 + itSOσy are matrices in spin space, [Graphic 21] is the hopping parameter and tSOC = αR/(2a) is the SOC hopping. The dimension of H0 is set by the number of sites of the wire. Then, it is written in Nambu space as given by Equation 2. Such a Hamiltonian is then diagonalized numerically with its dimensions given by the number of sites NS of the wire. Since this description accounts for wires of finite length, it is appropriate for investigating the overlap of MBSs. The length of the superconducting wire is LS = NSa, where NS is the number of sites and a is the lattice spacing. As mentioned before, the superconducting phase in the order parameter is assumed to be zero as it is only relevant when investigating Andreev bound states in SNS junctions.

In Figure 2 we present the low-energy spectrum for a superconducting nanowire as a function of the Zeeman field at a fixed wire length LS. Figure 2a shows the case of zero superconducting pairing and finite SOC (Δ = 0, αR ≠ 0), while Figure 2b shows a situation of finite pairing but with zero SOC (Δ ≠ 0, αR = 0). These two extreme cases are very helpful in order to understand how a topological transition occurs when the missing ingredient (either superconducting pairing of finite SO) is included. This is illustrated in the bottom panels, which correspond to both finite SOC and superconducting pairing for LS <M and LS >M, respectively. Here, ξM represents the Majorana localization length, which can be calculated from Equation 2[1,31],

[Graphic 22]

where [Graphic 23] and C0 = μ2 + Δ2B2. The Majorana localization length is defined as ξM = max[−1/ksol].

[2190-4286-9-127-2]

Figure 2: Low-energy spectrum of a superconducting nanowire as function of the Zeeman field B. At zero superconducting pairing with finite SOC the spectrum is gapless and becomes spin-polarized at B = μ as indicated by the green dashed line (a), while a finite superconducting pairing with zero SOC induces a gap for low values of B (b). As B increases, the induced gap is reduced and closed at B = Δ (vertical magenta dash-dot line). The bottom panels correspond to both finite superconducting pairing and SOC for LS = 4000 nm (c) and LS = 10000 nm (d). Note that as the Zeeman field increases the spectrum exhibits the closing of the gap at B = Bc. While in the trivial phase, B < Bc, there are no levels within the induced gap (c,d), in the topological phase for B > Bc, the two lowest levels develop an oscillatory behaviour around zero energy (c). These lowest levels are the sought-for MBSs. For sufficiently long wires the amplitude of the oscillations is reduced (d) and these levels acquire zero energy. Solid red, green and dashed cyan curves indicate the induced gaps Δ1,2 and min(Δ1, Δ2). Parameters: α0 = 20 meV·nm, μ = 0.5 meV, Δ = 0.25 meV and LS = 4000 nm (a,b).

For the sake of the explanation, we plot the spectrum in the normal state (Δ = 0), Figure 2a, which is, of course, gapless. As the Zeeman field increases, the energy levels split and, within the weak Zeeman phase, B < μ, the spectrum contains energy levels with both spin components. In the strong Zeeman phase, B > μ, one spin sector is completely removed giving rise to a spin-polarized spectrum at low energies as one can indeed observe in Figure 2a. The transition point from weak to strong Zeeman phases is marked by the chemical potential B = μ (green dashed line). Figure 2b shows the low-energy spectrum at finite superconducting pairing, Δ ≠ 0, and zero SOC, αR = 0. Firstly, we notice, in comparison with Figure 2a, that the superconducting pairing induces a gap with no levels for energies below Δ at B = 0, being in agreement with Anderson’s theorem [47]. A finite magnetic field induces a so-called Zeeman depairing, which results in a complete closing of the induced superconducting gap when B exceeds Δ. This is indeed observed in Figure 2b (magenta dash-dot line). Further increasing of the Zeeman field in this normal state gives rise to a region for Δ < B < Bc, which depends on the finite value of the chemical potential (between red and magenta lines) where the energy levels contain both spin components (for μ = 0 the magenta dash-dot and the red dashed line coincide, not shown). Note that [Graphic 24]. For B > Bc, one spin sector is removed and the energy levels are spin-polarized, giving rise to a set of Zeeman crossings that are not protected. Remarkably, when αR ≠ 0, the low-energy spectrum undergoes a number of important changes, Figure 2c,d. First, the gap closing changes from Δ, Figure 2b, to [Graphic 25] (bottom panels). Second, a clear closing of the induced gap at B =Bc and reopening for B > Bc is observed as the Zeeman field increases. This can be seen by plotting the induced gaps Δ1,2, which are finite only at finite Zeeman fields. In Figure 2d, the red, green and dashed cyan curves correspond to Δ1, Δ2 and min(Δ1, Δ2). Remarkably, the closing and reopening of the induced gap in the spectrum follows exactly the gaps Δ1,2 derived from the continuum (up to some finite-size corrections). Third, the spin-polarized energy spectrum shown in Figure 2b at zero SOC for B > Bc is washed out, keeping only the crossings around zero energy of the two lowest levels. This kind of closing and reopening of the spectrum at the critical field Bc indicates a topological transition where the two remaining lowest-energy levels for B > Bc are the well-known MBSs. Owing to the finite length LS, the MBSs exhibit the expected oscillatory behaviour due to their finite spatial overlap [48-51]. For sufficiently long wires [Graphic 26], the amplitude of the oscillations is considerably reduced (even negligible), which pins the MBSs to zero energy. Fourth, the SOC introduces a finite energy separation between the two lowest levels (crossings around zero) and the rest of the low-energy spectrum denoted here as “topological minigap”. Note that the value of this minigap, related to the high momentum gap Δ2, remains finite and roughly constant for strong SOC. In the case of weak SOC the minigap is reduced and for high Zeeman field it might acquire very small values, affecting the topological protection of the MBSs.

To complement this introductory part, calculations of the wavefunctions and charge density associated with the lowest levels of the topological superconducting nanowire spectrum are presented in the Supporting Information File 1.

Results and Discussion

Nanowire SNS junctions

In this part, we concentrate on SNS junctions based on the proximitized nanowires that we discussed in the previous section. The basic geometry contains left (SL) and right (SR) superconducting regions of length LS separated by a central normal (N) region of length LN, as shown in Figure 3. The regions N and SL(R) are described by the tight-binding Hamiltonian H0 given by Equation 4 with their respective chemical potentials, μN and [Graphic 27]. The Hamiltonian describing the SNS junction without superconductivity is then given by

[2190-4286-9-127-i5]
(5)

where [Graphic 28] with i = L/R and HN are the Hamiltonians of the superconducting and normal regions, respectively, [Graphic 29] and [Graphic 30] are the ones that couple Si to the normal region N. The elements of these coupling matrices are non-zero only for adjacent sites that lie at the interfaces of the S regions and of the N region, while zero everywhere else. This coupling is parametrized between the interface sites by a hopping matrix v0 = τv, where [Graphic 31], providing a good control of the normal transmission TN. The parameter τ controls the normal transmission that ranges from fully transparent (τ = 1) to tunnel (τ ≤ 0.6), as discussed in [37] for short junctions, being also valid for long junctions.

[2190-4286-9-127-3]

Figure 3: Schematic of SNS junctions based on Rashba nanowires. Top: A nanowire with Rashba SOC of length L = LS + LN + LS placed on top of two s-wave superconductors (S’) with pairing potentials ΔS′ and subjected to an external magnetic field [Graphic 32] (denoted by the black arrow). Superconducting correlations are induced into the nanowire through the proximity effect. Bottom: Left and right regions of the nanowire become superconducting, denoted by SL and SR, with induced pairing potentials [Graphic 33] and chemical potentials [Graphic 34], while the central region remains in the normal state with ΔN = 0 and chemical potential μN. This results in a superconductor–normal–superconductor (SNS) junction.

The superconducting regions of the nanowire are characterized by chemical potential [Graphic 35] and the uniform superconducting pairing potentials [52,53] [Graphic 36] and [Graphic 37], where Δ < ΔS′ and [Graphic 38]. The central region of the nanowire is in the normal state without superconductivity, ΔN = 0, and with chemical potential μN. Thus, the pairing potential matrix in the junction space reads

[2190-4286-9-127-i6]
(6)

Next, we define the phase difference across the junction as [Graphic 39]. Thus, the Hamiltonian for the full SNS junction reads in Nambu space [31,37]

[2190-4286-9-127-i7]
(7)

In what follows, we discuss short ([Graphic 40]) and long ([Graphic 41]) SNS junctions, where LN is the length of the normal region and [Graphic 42] is the superconducting coherence length [52]. The previous Hamiltonian is diagonalized numerically and in our calculations we consider realistic system parameters for InSb as described previously.

Low-energy Andreev spectrum

Now, we are in a position to investigate the low-energy Andreev spectrum in short and long SNS junctions. In particular, we discuss the formation of Andreev bound states and their evolution from the trivial (B < Bc) into the topological phases (B > Bc). For this purpose we focus on the phase and the Zeeman-dependent low-energy spectrum in short and long junctions, presented in Figure 4 and Figure 5 for LS ≤ 2ξM. For completeness we also present the case of [Graphic 43] in Figure 6 and Figure 7.

[2190-4286-9-127-4]

Figure 4: Low-energy Andreev spectrum as a function of the superconducting phase difference [Graphic 44] in a short SNS junction with LN = 20 nm and LS = 2000 nm. Different panels show the evolution with the Zeeman field: trivial phase for B < Bc (a–c), topological transition at B = Bc (d), and in the topological phase for B > Bc (e,f). The energy spectrum exhibits the two different gaps that appear in the system for finite Zeeman field (marked by red and green dashed horizontal lines). Note that after the gap inversion at B = Bc, two MBSs emerge at the ends of the junction as almost dispersionless levels (outer MBSs), while two additional MBSs appear at [Graphic 45] = π (inner MBSs). Parameters: αR = 20 meV·nm, μN = μS = 0.5 meV and Δ = 0.25 meV.

[2190-4286-9-127-5]

Figure 5: Same as in Figure 4 for a long junction with LN = 2000 nm and LS = 2000 nm. Note that, unlike short junctions, in this case the four lowest states for B > Bc coexist with additional levels within the induced gap which arise because LN is longer.

[2190-4286-9-127-6]

Figure 6: Same as in Figure 4 for a short junction with LN = 20 nm and LS = 10000 nm. Note that in this case, the emergent outer MBSs are dispersionless with [Graphic 46], while the inner ones touch zero at [Graphic 47] = π acquiring Majorana character.

[2190-4286-9-127-7]

Figure 7: Same as in Figure 4 for a long junction with LN = 2000 nm and LS = 10000 nm. The four lowest levels coexist with additional levels. The outer MBSs lie at zero energy and the inner ones reach zero at [Graphic 48] = π acquiring Majorana character.

We first discuss short junctions with LS ≤ 2ξM. In this regime, at B = 0 two degenerate ABSs appear within Δ as solutions to the BdG equations described by Equation 7, see Figure 4a. It is interesting to point out that within standard theory for a transparent channel the ABS energies reach zero at [Graphic 49] = π in the Andreev approximation [Graphic 50] [54]. Figure 4a, however, shows that in general the ABS energies do not reach zero at [Graphic 51] = π. The dense amount of levels above |εp| > Δ represents the quasi-continuum of states, which consists of a discrete set of levels due to the finite length of the N and S regions. Moreover, it is worth to point out that the detachment (the space between the ABSs and quasi-continuum) of the quasi-continuum at [Graphic 52] = 0 and 2π is not zero. It strongly depends on the finite length of the S regions (see Figure 6).

For a non-zero Zeeman field, Figure 4b and Figure 4c, the ABSs split and the two different gaps Δ1 and Δ2, discussed in section ‘Nanowire model’, emerge indicated by the dash-dot red and dashed green lines, respectively. By increasing the Zeeman field, the low-momentum gap Δ1 gets reduced (dash-dot red line), as expected, while the gap Δ2 (dashed green line) remains finite although it gets slightly reduced (Figure 4b and Figure 4c). For stronger, but unrealistic values of SOC we have checked that Δ2 is indeed constant. The two lowest levels in this regime, within Δ1 (dash-dot red line), develop a loop with two crossings that are independent of the length of the S region but exhibit a strong dependence on SOC, Zeeman field and chemical potential. We have checked that they appear due to the interplay of SOC and Zeeman field and disappear when μ acquires very large values, namely, in the Andreev approximation.

At B = Bc, the energy spectrum exhibits the closing of the low-momentum gap Δ1, as indicated by red dash-dot line in Figure 4d. This indicates the topological phase transition, since two gapped topologically different phases can only be connected through a closing gap. By further increasing the Zeeman field, Figure 4e,f, B > Bc, the inner gap Δ1 acquires a finite value again. This reopening of Δ1 indicates that the system enters into the topological phase and the superconducting regions denoted by SL(R) become topological, while the N region remains in the normal state. Thus, MBSs are expected to appear for B > Bc at the ends of the two topological superconducting sectors, since they define interfaces between topologically different regions.

This is what we indeed observe for B > Bc in Figure 4e and Figure 4f, where the low-energy spectrum has Majorana properties. In fact, for B > Bc, the topological phase is characterized by the emergence of two (almost) dispersionless levels with [Graphic 53], which represent the outer MBSs γ1,4 formed at the ends of the junction. Additionally, the next two energy levels strongly depend on [Graphic 54] and tend towards zero at [Graphic 55] = π, representing the inner MBSs γ2,3 formed inside the junction. For sufficiently strong fields, B = 2Bc, the lowest gap is Δ2 indicated by the green dashed line, which in principle bounds the MBSs. The quasi-continuum in this case corresponds to the discrete spectrum above and below Δ2, where Δ2 is the high-momentum gap marked by the green dashed horizontal line in Figure 4e,f.

The four MBSs develop a large splitting around [Graphic 56] = π, which arises when the wave-functions of the MBSs have a finite spatial overlap LS ≤ 2ξM. Since the avoided crossing between the dispersionless levels (belonging to γ1,4) and the dispersive levels (belonging to γ2,3) around [Graphic 57] = π depends on the overlap of MBSs on each topological segment. It can be used to quantify the degree of Majorana non-locality (a variant of this idea using quantum-dot parity crossings has been discussed in [55,56]). This can be explicitly checked by considering SNS junctions with longer superconducting regions, where the condition [Graphic 58] is fulfilled such that the energy splitting at [Graphic 59] = π is reduced.

As an example, we present in Figure 6 the energy levels as a function of the phase difference for [Graphic 60], where the low-energy spectrum undergoes some important changes. First, we notice in Figure 6 that the energy spectrum at B = 0 for |εp| > Δ, exhibits a visibly denser spectrum than that in Figure 4 signaling the quasi-continuum of states. Notice that in the topological phase, B > Bc, the lowest two energy levels, associated to the outer MBSs, are insensitive to [Graphic 61] remaining at zero energy. Thus, they can be considered as truly zero modes. On the other hand, the inner MBSs are truly bound within Δ2, unlike in Figure 4, and for [Graphic 62] = 0 and [Graphic 63] = 2π they merge with the quasi-continuum at ±Δ. Thus, an increase in the length of the superconducting regions favors the reduction of the detachment between the discrete spectrum and the quasi-continuum at 0 and 2π, as it should be for a ballistic junction [23,24]. Moreover, the energy splitting at [Graphic 64] = π is considerably reduced, even negligible. However, it will be always non-zero, though not visible to the naked eye, due to the finite length and, thus, due to the presence of the outer MBSs.

The low-energy spectrum as a function of the superconducting phase difference for different values of the Zeeman field in long SNS junctions is presented in Figure 5 for LS ≤ 2ξM. Additionally, we show in Figure 7 the case for [Graphic 65].

As expected, long junctions contain more levels within the energy gap Δ, see Figure 5a and Figure 7a, than short junctions. As we shall discuss, this eventually affects the signatures of Majorana origin in the supercurrents for B B c, namely, the ones corresponding to the lowest four levels.

The above discussion can be clarified by considering the dependence of the low-energy spectrum on the Zeeman field at fixed phase difference [Graphic 66] = 0 and [Graphic 67] = π. This is shown in Figure 8 (short junction limit), Figure 9 (intermediate junction limit) and Figure 10 (long junction limit) for LS ≤ 2ξM (panels a and c in each figure) and [Graphic 68] (panels b and d in each figure). In panels a and b, the gaps Δ1, Δ2 and min(Δ12) are also plotted as solid red, solid green and dashed cyan lines to visualize the gap closing and reopening discussed in the previous section. In all cases, it is clear that MBS smoothly evolve from the lowest ABS either following the closing of the induced gap Δ1, indicated by the solid red curve, at [Graphic 69] = 0 or evolving from the lowest detached levels at [Graphic 70] = π. The latter first cross zero energy, owing to Zeeman splitting, and eventually become four low-energy levels oscillating out of phase around zero energy (Figure 8c). The emergence of these oscillatory low-energy levels, separated by a minigap Δ2, indicated by the solid green curve, from the quasi-continuum characterizes the topological phase of the SNS junction. As expected, the oscillations become reduced for [Graphic 71] and the four levels at [Graphic 72] = π become degenerate at zero energy, see Figure 8b,d.

[2190-4286-9-127-8]

Figure 8: Low-energy Andreev spectrum as a function of the Zeeman field in a short SNS junction at [Graphic 73] = 0 (a,b) and [Graphic 74] = π (c,d) with LS = 2000 nm (a,c) and LS = 10000 nm (b,d). The low-energy spectrum traces the gap closing and reopening by the solid red curve that corresponds to Δ1, while for B > Bc we observe the emergence of two MBSs at [Graphic 75] = 0 (a) and four MBSs at [Graphic 76] = π (c), which oscillate around zero energy with B due to LS ≤ 2ξM within a minigap defined by Δ2 (solid green curve). For [Graphic 77] the MBSs are truly zero modes (b,d). Parameters: LN = 20 nm, αR = 20 meV·nm, μ = 0.5 meV and Δ = 0.25 meV.

[2190-4286-9-127-9]

Figure 9: Same as in Figure 8 for an intermediate junction with LN = 400 nm.

[2190-4286-9-127-10]

Figure 10: Same as in Figure 8 for a long junction with LN = 2000 nm.

An increase in the length of the normal section results in an increase of the amount of subgap levels as observed in Figure 9 and Figure 10. In both cases, in the topological phase, B > Bc, these additional levels reduce the minigap with respect to short junctions and also slightly reduce the amplitude of the oscillations of the energy levels around zero as seen Figure 9a and Figure 9b as well as Figure 10a and Figure 10b. Also, the minigaps for [Graphic 78] = 0 and [Graphic 79] = π are different, in contrast to short junctions. In fact, the minigap at [Graphic 80] = 0 is almost half of the value at [Graphic 81] = π for the chosen parameters. This can be understood from the phase dispersion of the long junction ABS spectra such as the ones in Figure 5 and Figure 7. For longer N regions, this difference can be even larger.

From the above discussion it is clear that the energy spectrum of SNS nanowire junctions offers the possibility to directly monitor the ABSs that trace the gap inversion and eventually evolve into MBSs.

Supercurrents

After having established in detail the energy spectrum in short and long SNS junctions, we now turn our attention to the corresponding phase-dependent supercurrents. They can be calculated directly from the discrete Andreev spectrum εp as [37,54,57]:

[2190-4286-9-127-i8]
(8)

where krmB is the Boltzmann constant, T is the temperature and the summation is performed over positive eigenvalues εp. By construction, our junctions have finite length, which implies that Equation 8 exactly includes the discrete quasi-continuum contribution.

In Figure 11 and Figure 12 we present supercurrents as a function of the superconducting phase difference I([Graphic 82]) for different values of the Zeeman field in short and long SNS junctions, respectively. Panels a and c correspond to LS ≤ 2ξM, while panels b and d correspond to [Graphic 83].

[2190-4286-9-127-11]

Figure 11: Supercurrent as a function of the superconducting phase difference in a short SNS junction, I([Graphic 84]), for LS = 2000 nm ≤ 2ξM (a,c) and LS = 10000 nm [Graphic 85]M (b,d). Panels a and b show the Josephson current in the trivial phase for different values of the Zeeman field, B < Bc, while panels c and d correspond to different values of the Zeeman field in the topological phase, BBc. Note the sawtooth feature at [Graphic 86] = π for [Graphic 87]. Parameters: αR=20 meV·nm, μ = 0.5 meV, Δ = 0.25 meV and [Graphic 88].

[2190-4286-9-127-12]

Figure 12: Supercurrent as a function of the superconducting phase difference in a long SNS junction with LN = 2000 nm, for LS = 2000 nm ≤ 2ξM (a,c) and LS = 10000 nm [Graphic 89]M (b,d). Panels a and b show the Josephson current in the trivial phase for different values of the Zeeman field, B < Bc, while panels c and d correspond to different values of the Zeeman field in the topological phase, BBc. In this case the magnitude of the supercurrent is reduced, an effect caused by the length of the normal section.

First, we discuss the short junction regime presented in Figure 11. At B = 0 the supercurrent I([Graphic 90]) has a sine-like dependence on [Graphic 91], with a relative fast change of sign around [Graphic 92] = π that is determined by the derivative of the lowest-energy spectrum profile around [Graphic 93] = π. This result is different from usual ballistic full transparent supercurrents in trivial SNS junctions [54], where the supercurrent is proportional to sin([Graphic 94]/2) being maximum at [Graphic 95] = π. This difference from the standard ballistic limit can be ascribed to deviations from the ideal Andreev approximation, see also the discussion of Figure 4a, owing to the relatively low chemical potential needed to reach the helical limit and, eventually, the topological regime as the Zeeman field increases. At finite values of the Zeeman field B, but still in the trivial phase B < Bc (dashed and dash-dot curves), I([Graphic 96]) undergoes changes. First, the maximum value of I([Graphic 97]) is reduced due to the reduction of the induced gap that is caused by the Zeeman effect. Second, I([Graphic 98]) develops a zig-zag profile (just before and after [Graphic 99] = π) signalling a 0–π transition in the supercurrent. This transition arises from the zero-energy crossings in the low-energy spectrum, see Figure 4b,c. As discussed above, these level crossings result from the combined action of both SOC and Zeeman field at low μ, and introduce discontinuities in the derivatives with respect to [Graphic 100]. Besides these features, all the supercurrent curves for B < Bc, for both LS ≤ 2ξM and [Graphic 101], exhibit a similar behavior, see Figure 11. Interestingly, the system is gapless at the topological transition, B = Bc, but the supercurrent remains finite, see red curve in Figure 11c.

For B > Bc, the S regions of the SNS junction become topological and MBSs emerge at their ends, as described in the previous subsection. Despite the presence of MBSs, the supercurrent I([Graphic 102]) remains 2π-periodic, i.e., I([Graphic 103]) = I([Graphic 104] + 2π). This results from the fact that we sum up positive levels only, as we deal with an equilibrium situation. Since the supercurrent is a ground state property, transitions between the negative and positive energies are not allowed, because of an energy gap originating from Majorana overlaps. Strategies to detect the presence of MBSs beyond the equilibrium supercurrents described here include the ac Josephson effect, noise measurements, switching-current measurements, microwave spectroscopy and dynamical susceptibility measurements [25-30].

As the Zeeman field is further increased in the topological phase, B > Bc, the supercurrent tends to decrease due to the finite Majorana overlaps when LS ≤ 2ξM, see dotted and dashed blue curves in Figure 11d. On the other hand, as the length of S becomes larger such that [Graphic 105] the overlap is reduced, which is reflected in a clear sawtooth profile at [Graphic 106] = π, see dotted and dashed blue curves in Figure 11d. This discontinuity in I([Graphic 107]) depends on LS and results from the profile of the lowest-energy spectrum at [Graphic 108] = π, as shown in Figure 6d. The sawtooth profile thus indicates the emergence of well-localized MBSs at the ends of S and represents one of our main findings.

As discussed above, the supercurrent for B < Bc, Figure 11a and Figure 11b, does not depend on LS. In contrast, supercurrents in the topological phase, Figure 11c and Figure 11d, do strongly depend on LS owing to the emergence of MBSs.

In Figure 12 we present I([Graphic 109]) for long junctions [Graphic 110] at different values of the Zeeman field. Panels a and c correspond to LS ≤ 2ξM and panels b and d correspond to [Graphic 111]. Even though the maximum value of the current is reduced in this regime, the overall behavior is very similar to the short-junction regime for both B < Bc and B > Bc. The only important difference with respect to the short junction case is that I([Graphic 112]) in the long-junction regime does not exhibit the zig-zag profile due to 0–π transitions.

As the system enters into the topological phase for B > Bc and LS ≤ 2ξM, Figure 12c, the maximum supercurrent decreases, indicating the non-zero splitting at [Graphic 113] = π in the low-energy spectrum. Deep in the topological phase, the supercurrent exhibits a slow (slower than in the trivial phase Figure 12a) sign change around [Graphic 114] = π, and its dependence on [Graphic 115] tends to approach a sine function. However, for [Graphic 116], shown in Figure 12d, the supercurrent acquires an almost constant value as B increases and develops a clear sawtooth profile at [Graphic 117] = π due to the zero energy splitting in the low-energy spectrum at [Graphic 118] = π, similarly to the short-junction case. It is worth to point out that, although the maximum supercurrent is slightly reduced, deep in the topological phase (dashed and dotted blue curves) its maximum value is approximately close to the maximum value in the trivial phase, which is different from what we found in the short-junction case. This is a clear consequence of the emergence of additional levels within the induced gap due to the increase of LN. These additional levels exhibit a strong dependence on the superconducting phase, very similar to the inner MBSs as one can see in Figure 5e,f.

In order analyze the individual contribution of outer and inner MBSs with respect to the quasi-continuum we calculate and identify supercurrents for such situations. This is presented in Figure 13 for short junctions (Figure 13a,b) and for long junctions (Figure 13c,d). In this regime the lowest gap is the high-momentum gap Δ2, and the levels outside this gap constitute the quasi-continuum.

[2190-4286-9-127-13]

Figure 13: Supercurrent as a function of [Graphic 119] at B = 1.5Bc. Contributions to the supercurrent for (a,b) short and (c,d) long junctions. (a,c) LS ≤ 2ξM and (b,d) [Graphic 120]. The different curves in (a,b) correspond to individual contributions to I([Graphic 121]) from outer, inner, and outer + inner (levels within the lowest induced gap Δ2), quasi-continuum (levels above the lowest gap Δ2) and total levels. In (c,d), the additional magenta curve corresponds to all levels within Δ2. In long junctions the number of levels within Δ2 exceeds the number of MBSs. MBSs coexist with additional levels within Δ2. Parameters: αR = 20 meV·nm, μ = 0.5 meV, Δ = 0.25 meV and [Graphic 122].

Firstly, we discuss short junctions. The supercurrent due to outer MBSs for LS ≤ 2ξM is finite only around [Graphic 123] = π, exhibiting an odd dependence on [Graphic 124] around π. However, away from this point it is approximately zero and independent of [Graphic 125] (see blue curve in Figure 13a). When [Graphic 126], the outer MBSs are very far apart and their contribution to I([Graphic 127]) is zero (see blue curve in Figure 13b). On the other hand, the contribution of the inner MBSs to I([Graphic 128]) is enormous and the outer part only slightly changes the shape of the maximum supercurrent when LS ≤ 2ξM, while for [Graphic 129] the outer MBSs do not contribute, as shown by the dashed brown curve in Figure 13a,b. Moreover, the inner contribution exhibits roughly the same dependence on [Graphic 130] as the contribution of the whole energy spectrum shown by the black curve in Figure 13a,b. As described in the previous subsection, the quasi-continuum is considered to be the discrete energy spectrum above |εp| > Δ2. The quasi-continuum contribution to I([Graphic 131]) is finite and odd in [Graphic 132] around π, as shown by green curves in Figure 13a,b. The quasi-continuum contribution to the total supercurrent I([Graphic 133]) far away from [Graphic 134] = π is appreciable mainly when the MBSs exhibit a finite energy splitting as seen in Figure 13a. Interestingly, the outer and in particular the inner MBSs (levels within Δ2) are the main source when such overlap is completely reduced and determine the profile of I([Graphic 135]), as shown in Figure 13b.

In long junctions the situation is different, mainly because more levels emerge within Δ in the trivial phase. For B > Bc within Δ2 these additional levels coexist with the inner and outer MBSs, see Figure 13c,d. The contribution of the outer MBSs to I([Graphic 136]) exhibits roughly similar behaviour as for short junctions although smoother around [Graphic 137] = π , shown by the blue curve in Figure 13c,d. The inner MBSs, however, have a strong dependence on [Graphic 138] and develop their maximum value close to [Graphic 139] = 2πn with n = 0,1,… (see red curve). The outer MBSs almost do not affect the overall shape of the I([Graphic 140]) curve (see dashed brown curve). Since a long junction hosts more levels, we also show by the dash-dot magenta curve the contribution of all the levels within Δ2, including also the four MBSs. This contribution is considerably large only close to [Graphic 141] = π, with a minimum and maximum value before and after [Graphic 142] = π for LS ≤ 2ξM, respectively. This is indeed the reason why the supercurrent is reduced as B increases in the topological phase for LS ≤ 2ξM, see dotted and dashed blue curves in Figure 12c. For [Graphic 143] the contribution of all the levels within Δ2 exhibits a sawtooth profile at [Graphic 144] = π, which, instead of reducing the quasi-continuum contribution (green curve), increases the maximum value of I([Graphic 145]) at [Graphic 146] = π resulting in the solid black curve. Importantly, unlike in short junctions, in long junctions the quasi-continuum modifies I([Graphic 147]) around [Graphic 148] = π. Thus, a zero-temperature current-phase measurement in an SNS junction setup could indeed reveal the presence of MBSs by observing the reduction of the maximum supercurrent. In particular, well-localized MBSs are revealed in the sawtooth profile of I([Graphic 149]) at [Graphic 150] = π. In what follows we analyze the effect of temperature, variation of normal transmission and random disorder on the sawtooth profile at [Graphic 151] = π of the supercurrent.

Temperature effects

In this part, we analyze the effect of temperature on supercurrents in the topological phase. In Figure 14 we present the supercurrent as a function of the superconducting phase difference, I([Graphic 152]), in the topological phase B = 1.5Bc at different temperature values for LS ≤ 2ξM (Figure 14a) and [Graphic 153] (Figure 14b). At zero temperature, for LS ≤ 2ξM, shown by the black solid curve in Figure 14a, the dependence of the supercurrent on [Graphic 154] approximately corresponds to a sine-like function. A small increase in temperature kBT = 0.01 meV (magenta dashed curve) slightly modifies the profile of the maximum supercurrent. However, for [Graphic 155] (Figure 14b), the same temperature (dashed curve) value has a detrimental effect on the sawtooth profile of I([Graphic 156]) at [Graphic 157] = π, which reduces the maximum value and smooths the curve out due to the thermal population of ABSs. We have checked that smaller temperature values than the ones presented in Figure 14 also smooth out the sawtooth profile but the fast sign change around [Graphic 158] = π is still visible. This effect remains as long as [Graphic 159]. As the temperature increases, I([Graphic 160]) smoothly acquires a true sine shape, as seen in Figure 14a. Although the sawtooth profile might be hard to observe in real experiments, the maximum value of I([Graphic 161]), which is finite in the topological phase and almost halved with respect to the trivial phase in short junctions [37], still provides a measure to distinguish it from I([Graphic 162]) in trivial junctions.

[2190-4286-9-127-14]

Figure 14: Finite temperature effect on the supercurrent, I([Graphic 163]), in (a,b) a short and (c,d) a long junction. (a,c) LS = 2000 nm ≤ 2ξM and (b,d) LS = 10000 nm [Graphic 164]M. Different curves correspond to different values of kBT. The sawtooth profile smooths out at finite temperature. Parameters: LN = 20 nm for short and LN = 2000 nm for long junctions, αR = 20 meV·nm, μ = 0.5 meV, Δ = 0.25 meV and [Graphic 165].

Normal transmission effects

The assumption of perfect coupling between N and S regions in previous calculations is indeed a good approximation because of the enormous advances in fabrication of hybrid systems. However, it is also relevant to study whether the sawtooth profile of I([Graphic 166]) is preserved or not when the normal transmission TN, described by τ, is varied.

Figure 15 shows the supercurrent I([Graphic 167]) in short junctions at B = 1.5Bc for different values of τ for LS ≤ 2ξM (Figure 15a) and [Graphic 168] (Figure 15b). When τ is reduced, the supercurrent I([Graphic 169]) is also reduced. However, for LS ≤ 2ξM, there is a transition from a sudden sign change around [Graphic 170] = π to a true sine function with reducing τ, very similar to the effect of temperature discussed above. Notice that in the tunnel regime, τ = 0.6, I([Graphic 171]) is approximately zero. For [Graphic 172] the sawtooth profile at [Graphic 173] = π is preserved and robust when τ is reduced from the fully transparent to the tunnel regime, as seen in Figure 15b. Quite remarkably, in the tunneling regime, I([Graphic 174]) is finite away from nπ for n = 0,1,…. The finite value of the supercurrent could serve as another indicator of the non-trivial topology and, thus, of the emergence of MBSs in the junction.

[2190-4286-9-127-15]

Figure 15: Effect of normal transmission through the coupling parameter τ on the supercurrent, I([Graphic 175]), in (a,b) a short and (c,d) a long SNS junction. (a,c) LS = 2000 nm ≤ 2ξM and (b,d) LS = 10000 nm [Graphic 176]M. Although after decreasing τ the magnitude of the supercurrent at [Graphic 177] = π decreases, the sawtooth profile is preserved. Parameters: LN = 20 nm for short and LN = 2000 nm for long junctions, αR = 20meV·nm, μ = 0.5 meV, Δ = 0.25 meV and [Graphic 178].

Disorder effects

Now we analyze the sawtooth profile of I([Graphic 179]) for B > Bc in the presence of disorder. Disorder is introduced as a random on-site potential Vi in the tight-binding Hamiltonian given by Equation 4. The values of Vi lie within [−w, w], with w being the disorder strength. When considering this kind of disorder, the chemical potential undergoes random fluctuations. Hence, values of w do not include [Graphic 180].

In Figure 16(a,b) we present I([Graphic 181]) in short junctions at B = 1.5Bc for 20 disorder realizations and different values of the disorder strength w. Disorder of the order of the chemical potential μ has little effect on I([Graphic 182]) as shown by dashed curves in Figure 16a,b. The behavior of I([Graphic 183]) is approximately the same as without disorder. This reflects the robustness of the topological phase, and thus of MBSs, against fluctuations in the chemical potential [58,59]. Stronger disorder (dotted and dash-dot curves) reduce the maximum value of I([Graphic 184]) although its general behavior is preserved. The sawtooth profile at [Graphic 185] = π in Figure 16b is robust against moderate values of disorder strength. We have confirmed that these conclusions are still valid even when we consider disorder of the order of 5μ (not shown).

[2190-4286-9-127-16]

Figure 16: Effect of random on-site scalar disorder on the supercurrent I([Graphic 186]) in (a,b) a short and (c,d) a long SNS junction at B = 1.5Bc. (a,c) LS = 2000 nm ≤ 2ξM and (b,d) LS = 10000 nm [Graphic 187]M. Each curve corresponds to 20 realizations of disorder, where w is the disorder strength. For small values of w of the order of the chemical potential, the sawtooth profile at [Graphic 188] = π is preserved (see right panel). Parameters: LN = 20 nm for short and LN = 2000 nm for long junctions, αR = 20 meV·nm, μ = 0.5 meV, Δ = 0.25 meV and [Graphic 189].

Conclusion

In this numerical work we have performed a detailed investigation of the low-energy spectrum and supercurrents in short ([Graphic 190]) and long ([Graphic 191]) SNS junctions based on nanowires with Rashba SOC and in the presence of a Zeeman field.

In the first part, we have studied the evolution of the low-energy Andreev spectrum from the trivial phase into the topological phase and the emergence of MBSs in short and long SNS junctions. We have shown that the topological phase is characterized by the emergence of four MBSs in the junction (two at the outer part of the junction and two at the inner part) with important consequences to the equilibrium supercurrent. In fact, the outer MBSs are almost dispersionless with respect to superconducting phase [Graphic 192], while the inner ones disperse and tend to reach zero at [Graphic 193] = π. A finite energy splitting at [Graphic 194] = π occurs when the length of the superconducting nanowire regions, LS, is comparable to or less than 2ξM. Although in principle such energy splitting can be reduced by making the S regions longer, we conclude that in a system of finite length the current–phase curves are 2π-periodic and the splitting always spoils the so-called 4π-periodic fractional Josephson effect in an equilibrium situation.

In short junctions the four MBSs are truly bound within Δ only when [Graphic 195], while in long junctions the four MBSs coexist with additional levels, which profoundly affects phase-biased transport. As the Zeeman field increases in the trivial phase B < Bc, the supercurrent I([Graphic 196]) is reduced due to the reduction of the induced gap. In this case, the supercurrents I([Graphic 197]) are independent of the length of the superconducting regions, LS, an effect preserved in both short and long junctions.

In short junctions in the topological phase with B > Bc the contribution of the four MBSs levels within the gap determines the shape of the current–phase curve I([Graphic 198]) with only little contribution from the quasi-continuum. For LS <M, the overlap of MBS wavefunctions at each S region is finite, and the quasi-continuum contribution is appreciable and of the opposite sign than the contribution of the bound states. This induces a reduction of the maximum supercurrent in the topological phase. For [Graphic 199], when both the spatial overlap between MBSs and the splitting at [Graphic 200] = π are negligible, the quasi-continuum contribution is very small and the supercurrent I([Graphic 201]) is dominated by the inner MBSs. Remarkably, we have demonstrated that the current–phase curve I([Graphic 202]) develops a clear sawtooth profile at [Graphic 203] = π, which is independent of the quasi-continuum contribution and represents a robust signature of MBSs.

In the case of long junctions we have found that the additional levels that emerge within the gap affect the contribution of the individual MBSs. Here, it is the combined contribution of the levels within the gap and the quasi-continuum that determine the full current–phase curve I([Graphic 204]), unlike in short junctions. The maximum supercurrent in long junctions is reduced in comparison to short junctions, as expected. Our results also show that the maximum value of the supercurrent in the topological phase depends on LS, acquiring larger values for [Graphic 205] than for LS ≤ 2ξM.

Finally, we have analyzed the robustness of the characteristic sawtooth profile in the topological phase against temperature, changes in transmission across the junction and random on-site scalar disorder. We found that a small finite temperature smooths it out due to thermal population of ABSs. We demonstrated that, although this might be a fragile indicator of MBSs, the fast sign change around [Graphic 206] = π could help to distinguish the emergence of MBSs from trivial ABSs. Remarkably, the sawtooth profile is preserved against changes in transmission, i.e., it is preserved even in the tunneling regime. And finally, we showed that reasonable fluctuations in the chemical potential μ (up to 5μ) do not affect the sawtooth profile of I([Graphic 207]) at [Graphic 208] = π.

Our main contribution are summarized as follows. In short and long SNS junctions of finite length four MBSs emerge, two at the inner part of junction and two at the outer ends. The unavoidable overlap of the four MBSs gives rise to a finite energy splitting at [Graphic 209] = π, thus rendering the equilibrium Josephson effect 2π-periodic in both short and long junctions. Current–phase curves of short and long junctions exhibit a clear sawtooth profile when the energy splitting near [Graphic 210] = π is small, which indicates the presence of weakly overlapping MBSs. Remarkably, the current–phase curves do not depend on LS in the trivial phase for both short and long junctions, while they strongly depend on LS in the topological phase. This effect is solely connected to the splitting of MBSs at [Graphic 211] = π, indicating a unique feature of the topological phase and therefore of the presence of MBSs in the junction.

Supporting Information

Supporting Information File 1: Majorana wavefunction and charge density in SNS junctions.
Format: PDF Size: 1.9 MB Download

Acknowledgements

J.C. thanks O. A. Awoga, K. Björnson, M. Benito and S. Pradhan for motivating and helpful discussions. J.C. and A.B.S. acknowledge financial support from the Swedish Research Council (Vetenskapsrådet), the Göran Gustafsson Foundation, the Swedish Foundation for Strategic Research (SSF), and the Knut and Alice Wallenberg Foundation through the Wallenberg Academy Fellows program. We also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Grant No. FIS2015-64654-P (R. A.), FIS2016-80434-P (AEI/FEDER, EU) (E. P.) and the Ramón y Cajal programme through grant No. RYC-2011-09345 (E. P).

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