Creation of spin-triplet Cooper pairs in the absence of magnetic ordering

# Creation of spin-triplet Cooper pairs in the absence of magnetic ordering

Daniel Breunig Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany    Pablo Burset Department of Applied Physics, Aalto University, FIN-00076 Aalto, Finland    Björn Trauzettel Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
July 14, 2019
###### Abstract

In superconducting spintronics, it is essential to generate spin-triplet Cooper pairs on demand. Up to now, proposals to do so concentrate on hybrid structures in which a superconductor (SC) is combined with a magnetically ordered material (or an external magnetic field). We, instead, identify a novel way to create and isolate spin-triplet Cooper pairs in the absence of any magnetic ordering. This achievement is only possible because we drive a system with strong spin-orbit interaction–the Dirac surface states of a strong topological insulator (TI)–out of equilibrium. In particular, we consider a bipolar TI-SC-TI junction, where the electrochemical potentials in the outer leads differ in their overall sign. As a result, we find that nonlocal singlet pairing across the junction is completely suppressed for any excitation energy. Hence, this junction acts as a perfect spin triplet filter across the SC generating equal-spin Cooper pairs via crossed Andreev reflection.

Introduction.—In spintronics, it is desirable to achieve spin manipulation in the absence of magnetic fields because electronic switching processes can be done much faster than their magnetic counterparts. The prime example of such a device is the famous Datta-Das transistor based on materials with strong spin-orbit coupling Datta and Das (1990). The situation is similar in superconducting spintronics Eschrig (2010); Linder and Robinson (2015). However, to the best of our knowledge, no device proposal has been made so far that allows for a performance in the absence of magnetic ordering [Proposalsforsuperconductngspinvalvesrequiretheuseofmagneticelements.See; forexample; ][]Fominov_2010; *Buechner_2012; *Jiang_2013; *Aarts_2014; *Robinson_2014. Specifically, it would be exciting to create equal-spin Cooper pairs on demand in a device in which spin-orbit coupling is the crucial feature for its application in spintronics. For this vision to come true, the surface states of three-dimensional (3D) topological insulators (TIs) are promising building blocks because they mimic a truly relativistic spin-orbit coupling in a condensed matter setting. We have identified a bipolar device based on two areas of TI surface states connected to each other via a common central superconductor (SC), i.e. a TI-SC-TI junction, that can act as a generator for equal-spin Cooper pairs on demand, cf. Fig. 1. In fact, the working principle of the device is based on crossed Andreev reflection–generated out of equilibrium–which enables a transfer of Cooper pairs (with a finite net spin) from the two TI regions into the SC.

The bipolar TI-SC-TI setup is inspired by a seminal work by Cayssol where a similar junction has been studied in the context of graphene Cayssol (2008). In that work, the spin degree of freedom played no role because of the weak spin-orbit coupling in graphene. In TIs, instead, strong spin-orbit coupling in combination with superconducting and/or magnetic ordering gives rise to intriguing physics Linder et al. (2010), for instance, the emergence of Majorana bound states Fu and Kane (2008); Tanaka et al. (2009) or odd-frequency SC pairing Yokoyama (2012); Black-Schaffer and Balatsky (2012); Crépin et al. (2015); Burset et al. (2015). The underlying reason is that spin-rotational invariance is broken by the spin-orbit coupling and fundamentally different gaps, including a spin triplet state Tkachov and Hankiewicz (2013), can be induced in TI surface states by proximity to an -wave SC and/or a magnetic insulator [Tripletsuperconductivitynaturallyarisesinmaterialsorproximitizedstructureswithbrokenspin-rotationsymmetry:][]Frigeri_2004; *Frigeri_2004b; *Burset_2014; *Maslov_2015; *Bergeret_2015; *Bergeret_2016. From the experimental side, it seems to be more feasible to induce superconducting order into TI surface states than magnetic order, although both tasks have been recently achieved Wang et al. (2012); Zareapour et al. (2012); Sacépé et al. (2012); Maier et al. (2012); Veldhorst et al. (2012); Williams et al. (2012); Cho et al. (2013); Oostinga et al. (2013); Sochnikov et al. (2015); Wiedenmann et al. (2016); Lee et al. (2016). This observation implies that our setup should be directly realizable in the current generation of hybrid structures on TI surfaces.

In this proposal, we assume that an -type region of the Dirac surface states (of a 3D TI) is connected to a -type region via a superconducting electrode, see Fig. 1 for a schematic. We also want the two TI regions to be biased by separate voltage sources (corresponding to biases and applied to the left () and the right () TI, respectively) and the SC to be grounded. In such a bipolar TI-SC-TI junction, Andreev reflection in the same lead and electron tunneling between leads can be efficiently turned off, as previously proposed for graphene Cayssol (2008) and ordinary semiconductors Veldhorst and Brinkman (2010). We show below that, for the case of TIs, spin singlet pairing is strongly suppressed, making this device a perfect spin triplet filter where crossed Andreev reflection processes pump equal-spin Cooper pairs into the SC.

Model.—To model the TI-SC-TI device, we introduce the basis , where is the creation operator of an electron with momentum and spin . In this basis, we can determine the corresponding Bogoliubov-de Gennes (BdG) Hamiltonian

 ^HBdG=(H0(k)iΔ(x)σy−iΔ(x)σy−H∗0(−k)), (1)

where the electron Bloch Hamiltonian reads as

 H0(k)= vF(^kxσx+^kyσy)−μ(x)σ0 (2) ^= vF(^kxσx+kyσy)−μ(x)σ0. (3)

Here, is the Fermi velocity, are the Pauli matrices in spin space, are the momentum operators in position basis, and define spatially dependent SC pairing and electrochemical potentials, respectively. We set in what follows. In Eq. (3), we reduce the Bloch Hamiltonian to a quasi-1D operator, where is now a parameter defining the angle of incidence. This is possible due to the choice of the coordinate system in Fig. 1 and the perfect alignment of the interfaces (between TI and SC) at and along the -axis. Experimentally, it is very difficult to resolve the –dependence. Hence, we will average subsequent characteristics/observables with respect to . In our TI-SC-TI-junction, we assume the electrochemical potentials to be constant in each domain. In particular, we define , with the Heaviside function. The superconducting pairing is chosen to be finite only underneath the SC, i.e. .

Setting up the scattering problem for the TI-SC-TI junction, see the supplemental material (SM Sup ()), we obtain the electron and hole wave vectors in the normal leads as , with and the excitation energy. Here, defines if the particle stems from the valence or the conduction band and is the angle of incidence for electrons/holes. In such a junction, four transport channels exist: (i) normal reflection (NR); (ii) local Andreev reflection (LAR); (iii) electron cotunneling (CO); and (iv) crossed Andreev reflection (CAR).

We obtain a bipolar system by choosing the electrochemical potentials in the TIs to have the same modulus, but different signs, . Due to the nodal dispersion relation, this choice allows us to completely suppress two out of four transport channels: LAR and CO. As illustrated in Fig. 2, this complete suppression is achieved by applying a voltage to such that the corresponding excitation energy coincides with the electrochemical potential, i.e. . Evidently, by this particular choice we exactly hit the Dirac point of the hole dispersion in and the electron dispersion in . Since these states have zero momentum, , they do not contribute to transport, completely suppressing LAR and CO.

Superconducting pairing.—Next, we study the symmetry of the superconducting pairing potential that is proximity-induced into the TI-SC-TI device. To do so, we analytically compute the retarded Green function using a scattering state approach, see Sup (). In the basis defined above, this Green function can be written as with

 Gr(x,x′,ky,ω) =(GreeGrehGrheGrhh), (4) Grnm(x,x′,ky,ω) =(Gr,↑↑nmGr,↑↓nmGr,↓↑nmGr,↓↓nm). (5)

Dependencies inside the matrices are omitted for ease of notation. Here, is the energy shifted infinitesimally into the positive complex plane to impose outgoing boundary conditions Datta (2007).

For convenience, we define the anomalous Green function in a rotated basis as

 Fr(x,x′,ky,ω)≡−iGrehσy=(Gr,↑↓eh−Gr,↑↑ehGr,↓↓eh−Gr,↓↑eh), (6)

which we further decompose into singlet/triplet parts

 Fr(x,x′,ky,ω)=∑i∈{0,x,y,z}fri(x,x′,ky,ω)σi. (7)

In this equation, is the singlet, and as well as are the triplet amplitudes.

Before we discuss specific results on the pairing amplitude for our setup, we state more general arguments based on symmetry. It is straightforward to show that and are odd in , independently of the choice of and . Since we later on average all quantities with respect to , they do not contribute to the (averaged) triplet pairing amplitudes Burset et al. (2015). This implies that and only differ in their sign. The remaining amplitudes, and , are even functions of . Hence, they can (in principle) remain finite after averaging.

One of our main results is that this bipolar TI-SC-TI-junction acts as a perfect filter for nonlocal triplet pairing in the superconductor because of the helicity of the TI surface states. To substantiate this claim, we first present our results on the nonlocal singlet pairing amplitude. Labeling as () any point in region (), we find 222The expressions in Eqs. (8)-(10) are derived under the condition . They capture the spin amplitudes correctly as we approach the bipolar regime.

 fr0(xL,xR)=sin(θeL−θhR2)g↑↓eh(xL,xR), (8)

with

 g↑↓eh(xL,xR)=exp(iθeL+θhR2)cos(θeL)e−i(keLxL−khRxR)d1. (9)

For the nonlocal equal-spin triplet amplitude we get

 fr↑↑(xL,xR) =−i2ei(θeL+θhR)cos(θeL)ei(khRxR−keLxL)d1. (10)

In these equations, denotes the scattering amplitude related to an electron incident from being transmitted into as a hole, see Sup (). Similar equations to Eqs. (8)-(10) also apply to the choice . In the bipolar regime, , we immediately deduce from the definition of the angles of incidence that . As a consequence, the nonlocal spin singlet amplitude in Eq. (8) evaluates to zero, making this pairing absent across the junction independently of frequency and mode index 111The local spin singlet, where the two electrons forming the Cooper pair are taken from the same region, is proportional to the LAR amplitude. It is thus also suppressed at the bipolar junction. . In contrast, the nonlocal spin triplet amplitude in Eq. (10) remains finite, see Fig. 3.

Transport properties.—After the identification of pure triplet pairing across the junction, we want to use this particular property for an application in superconducting spintronics. To achieve this goal, we need to look at non-equilibrium transport properties of the TI-SC-TI junction by means of the (extended) Blonder-Tinkham-Klapwijk (BTK) formalism Blonder et al. (1982); Anantram and Datta (1996); Lambert and Raimondi (1998); Falci et al. (2001); Linder and Sudbø (2008). This allows for the determination of the probability amplitudes of the scattering processes, in general, and the local and nonlocal conductance, in particular. A detailed evaluation of the probability amplitudes, see Sup (), confirms the physical picture we elaborated in Fig. 2. For an electron excited in , both the amplitudes for LAR () and CO () vanish at , while those for NR () and CAR () remain finite in the bipolar setup. This property has a striking effect on the non-linear conductance at zero temperature . Changing only and setting , the non-linear conductance reduces to distinct local and nonlocal parts,

 ∂IL∂VL =2e2h[1−Ree1(eVL,ky)+Reh1(eVL,ky)], (11) ∂IR∂VL =2e2h[Tee1(eVL,ky)−Teh1(eVL,ky)]. (12)

The unitarity of the scattering matrix, i.e. , yields and thus

 ∂IL∂VL(eVL=μ,ky) =−∂IR∂VL(eVL=μ,ky). (13)

At this particular point, corresponding to the choice of bias in our setup, the local and the nonlocal conductance coincide in their absolute value, see Fig. 4. This property is very unusual in superconducting devices because, in general, the local conductance dominates its nonlocal counterpart. Thus, we do not only observe a particular behavior in the nonlocal superconducting pairing, but we are also able to pinpoint a sweet spot at with striking features in the transport that we can connect to the pumped spin into the SC.

Spin injection.—To do so, we quantify the non-equilibrium spin pumped into the SC via Andreev reflection processes introducing the quantity , which we denote the non-equilibrium net spin pumped into the SC,

 (14)

with the local and nonlocal contributions

 (15)

Here, is the Cooper pair spin operator (see Sup ()) and , and are the spin expectation values of the incident, the Andreev reflected, and the crossed Andreev reflected particle, respectively. Let us explain why this quantity is a good measure for the spin pumped into the system through triplet Cooper pairs. If we consider, for instance, an electron incident from moving towards the interface at , then there are two processes resulting in the emergence of non-equilibrium Cooper pairs in the SC, (i) LAR with probability , where the spins of the incident electron and the reflected hole, i.e. , are transferred to the SC; and (ii) CAR with probability , where the spin of the transmitted hole is added to that of the electron, i.e. . Thus, a good estimate of the net spin of the Cooper pairs pumped into the SC via Andreev reflection is the sum of both contributions, and , weighted with their respective probability amplitudes. Explicitly, we obtain (under the choice )

 sgn(ε+μL)2(ζeLcosθeL,sinθeL,0)T, (16a) sgn(ε−μL)2(−ζhLcosθhL,sinθhL,0)T, (16b) sgn(ε−μR)2(ζhRcosθhR,sinθhR,0)T. (16c)

The norm of these quantities is always , as expected for fermions, and their -component vanishes. Moreover, their -component is odd under , such that its average with respect to will vanish. Thus, only the -component of the net spin (pumped in the SC after averaging over all angles of incidence) remains finite. We can therefore focus on this part of Eqs. (16). Indeed, we find a bound for this component of the net spin

 0≤|Sx|≤RLAR+TCAR≤1, (17)

which corresponds to the maximal angular momentum transferred per scattering event. If we plot this quantity as a function of both and , see Fig. 5, we find that, in the bipolar setup and for our choice of the SC length , we pump the largest amount of non-equilibrium net spin into the SC by tuning the bias in the vicinity of the sweet spot .

Summary.—The breaking of spin rotational invariance in materials with strong spin-orbit locking reveals striking physics. We found that, by connecting a -type to an -type TI via a SC domain, this bipolar TI-SC-TI junction acts as an effective nonlocal spin triplet filter, where nonlocal singlet pairing is completely suppressed. Due to the helicity of the TI states, the non-equilibrium triplet Cooper pairs pumped into the SC carry a finite net spin. We thus propose this setup as an all-electric nanostructure for applications in spintronics. The effect of the spin pumping is strongest in the vicinity of the sweet spot , where we estimate a net spin of, depending on the bias range, approximately per unit volume (see Sup ()) for a SC length of . This sweet spot is experimentally easy to detect, since the local and the nonlocal conductivities for a bias applied to the left TI coincide in their moduli. Interestingly, pumped equal-spin Cooper pairs out of equilibrium into a superconductor could lead to long-range spin accumulation Silaev et al. (2015); Bergeret et al. (2017). Connecting a second superconductor to the central region of the bipolar junction, the spin accumulation could be measured through local SQUID or Hall probes Björnsson et al. (2005); Kirtley et al. (2007); Sengupta and Yakovenko (2008).
We thank Y. Asano, F. S. Bergeret, F. Crepin, F. Dominguez, J. Linder, J. Pekola, B. Scharf, S. Zhang and N. Traverso Ziani for interesting discussions. Financial support by the DFG (SPP1666 and SFB1170 "ToCoTronics"), the Helmholtz Foundation (VITI), and the ENB Graduate School on "Topological Insulators" is gratefully acknowledged. P.B. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 743884.

## Supplemental material

In this supplementary material, we introduce the definitions for the eigenstates and scattering states used to derive all the quantities in the main text. The methods to obtain the Green function as well as the transport properties are illustrated and some significant results are presented explicitly.

## Appendix A Preliminary calculations and definitions

Starting from Schrödinger’s equation

 ^HBdGΨ(x,y)=εΨ(x,y) (A.1)

with the Bogoliubov–de Gennes (BdG) and Bloch Hamiltonian introduced in Eqs. (1) and (2) of the main text,

 ^HBdG=(H0(k)iΔ(x)σy−iΔ(x)σy−H∗0(−k)), H0(k) =vF(^kxσx+^kyσy)−μ(x)σ0, k=(^kx,^ky)T, (A.2)

and the definitions of the electrochemical and superconducting (SC) pairing potential,

 μ(x)=μLΘ(−x)+μSΘ(x)Θ(LS−x)+μRΘ(x−LS), Δ(x)=Δ0Θ(x)Θ(LS−x), (A.3)

we make use of the translational invariance along the –axis and perform a partial Fourier transformation in this variable,

 Ψ(x,y)=∞∫−∞ψ(x,ky)eikyydky. (A.4)

As a result, reduces to a quasi–1D operator, which simplifies further calculations. From here, we obtain the eigenstates of each domain by solving Eq. (1) in its quasi–1D representation. This yields right and left moving states

 ψL\lx@stackrel→e/h(x) =1√2⎛⎝¯ke/hL∓ikyε±μL^e1/3+^e2/4⎞⎠Tei¯ke/hLx, ψL\lx@stackrel←e/h(x)=1√2⎛⎝ke/hL±ikyε±μL^e1/3−^e2/4⎞⎠Te−ike/hLx, (A.5)

in the left () topological insulator (TI) and, correspondingly,

 ψR\lx@stackrel→e/h(x) =1√2⎛⎝ke/hR∓ikyε±μR^e1/3+^e2/4⎞⎠Teike/hRx, ψR\lx@stackrel←e/h(x)=1√2⎛⎝¯ke/hR±ikyε±μR^e1/3−^e2/4⎞⎠Te−i¯ke/hRx, (A.6)

in the right () TI. In the superconductor (SC), we find

 ψS1/2(x) =Neq/hq⎛⎜ ⎜ ⎜ ⎜⎝(ε±Ω)(keq/hq−iky)(ε±Ω)(μS±Ω)−(μS±Ω)Δ0(keq/hq−iky)Δ0⎞⎟ ⎟ ⎟ ⎟⎠eikeq/hqx, ψS3/4(x)=Neq/hq⎛⎜ ⎜ ⎜ ⎜⎝(ε±Ω)(keq/hq+iky)−(ε±Ω)(μS±Ω)(μS±Ω)Δ0(keq/hq+iky)Δ0⎞⎟ ⎟ ⎟ ⎟⎠e−ikeq/hqx (A.7)

with the normalization factors

 N−1eq=2√ε(ε+Ω)(μS+Ω), N−1hq=2√ε(ε−Ω)(μS−Ω). (A.8)

To obtain a complete solution, each of these states needs to be multiplied by the plane wave . Here, the alphabetical indices distinguish electrons () from holes () in the TIs and electron–like () from hole–like () quasi–particles in the SC, while the arrows imply the direction of motion with respect to the –axis. is the complex conjugate of the quantity and is the complete, orthonormal set of the standard basis vectors of . The wave vectors in the TIs and the SC read

 ke/hi=ζe,hi√(ε±μi)2−k2y, keq/hq=ζeq/hq√(μS±Ω)2−k2y, (A.9)

with

 ζe,hi=sgn(ε±μi+|ky|), ζeq/hq=sgn(ε±√(μS+|ky|)2+Δ20), Ω=⎧⎪⎨⎪⎩sgn(ε)√ε2−Δ20,|ε|≥Δ0i√Δ20−ε2,|ε|<Δ0. (A.10)

Note that, in the main text, we have used an angular representation for argumentative reasons. Definitions of the angles can be found there. We intentionally do not use such a representation here, since this renders the –integration as well as the determination of the retarded Green function more apparent.

We now define the scattering states for the TI–SC–TI nanostructure introduced in the main text, i.e.,

 ϕ1/2(x) =⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩ψL\lx@stackrel→e/h(x)+a1/2ψL\lx@stackrel←h/e(x)+b1/2ψL\lx@stackrel←e/h(x),x<04∑i=1si1/2ψsi(x),0LS, ϕ3/4(x) =⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩c3/4ψL\lx@stackrel←e/h(x)+d3/4ψL\lx@stackrel←h/e(x),x<04∑i=1si3/4ψsi(x),0LS.

Here, describes the scattering processes based on an electron(hole) incident from the left, while describes the same for an electron(hole) incident from the right. is the amplitude for an incident particle being Andreev reflected at the interface (LAR), the one for normal reflection (NR). Electron or hole co–tunneling (CO) is linked to the amplitude , while crossed Andreev reflection (CAR) goes with . These scattering coefficients are determined by matching the wave functions at the interfaces,

 ϕl(x=0−0+)=ϕl(x=0+0+), ϕl(x=LS−0+)=ϕl(x=LS+0+). (A.12)

To calculate the retarded Green function, we need to find as well the eigenstates of the transposed McMillan (1968); Kashiwaya and Tanaka (2000) BdG Hamiltonian in Eq. (A.2), which reads as

 ^HTBdG=(H∗0(−k)iΔ(x)σy−iΔ(x)σy−H0(k)). (A.13)

In this context, we need to define the partial Fourier transform as

 ~Ψ(x,y)=∞∫−∞~ψ(x,ky)e−ikyydky. (A.14)

We denote as the transposed eigenstates. Explicitly, they are given by

 ~ψL\lx@stackrel→e/h(x) =1√2⎛⎝¯ke/hL∓ikyε±μL^e1/3−^e2/4⎞⎠Tei¯ke/hLx, ~ψL\lx@stackrel←e/h(x)=1√2⎛⎝ke/hL±ikyε±μL^e1/3+^e2/4⎞⎠Te−ike/hLx, (A.15) ~ψR\lx@stackrel→e/h(x) =1√2⎛⎝ke/hR∓ikyε±μR^e1/3−^e2/4⎞⎠Teike/hRx, ~ψR\lx@stackrel←e/h(x)=1√2⎛⎝¯ke/hR±ikyε±μR^e1/3+^e2/4⎞⎠Te−i¯ke/hRx, (A.16)

in and , respectively, and

 ~ψS1/2(x) =Neq/hq⎛⎜ ⎜ ⎜ ⎜⎝(ε±Ω)(keq/hq−iky)−(ε±Ω)(μS±Ω)(μS±Ω)Δ0(keq/hq−iky)Δ0⎞⎟ ⎟ ⎟ ⎟⎠eikeq/hqx, (A.17)

in the SC. With this, we define the transposed scattering states by replacing by and by in Eq. (LABEL:eq:A6), with . Note that these scattering coefficients can easily be related to each other by means of the Wronskian determinant Datta (2007) and Liouville’s formula so to obtain

 ~al=−al, ~bl=bl, ~cl=cl, ~dl=−dl, ~sil=(−1)lsil. (A.18)

For a given excitation energy , we average an arbitrary quantity by means of the integral

 (A.19)

Here, is the modulus of the wave vector of the incident particle at energy and , i.e., . Intuitively, the limits of the integral should be and , however, one needs to take the Fermi wave vector mismatch (cf. Kashiwaya et al. (1996); Linder and Sudbø (2008)) into account, which eventually reduces the latter values to a critical wave vector (corresponding to a critical angle Cayssol (2008)) .

## Appendix B Green function

We are now able to express the retarded Green function by means of outer products of the scattering states,

 GR(x,x′)=⎧⎪ ⎪⎨⎪ ⎪⎩α1ϕ3(x)~ϕT1(x′)+α2ϕ3(x)~ϕT2(x′)+α3ϕ3(x)~ϕT1(x′)+α4ϕ3(x)~ϕT2(x′),xx′. (A.20)

The coefficients and are determined by requiring the discontinuity of at ,

 GR(x,x′)∣∣x=x′+0+−GR(x,x′)∣∣x=x′−0+ =−i(τ0⊗σx) (A.21)

 α1 =β1=−iω+μLRe(keL)c4c3c4−d3d4=−iω+μRRe(keR)c2c1c2−d1d2, (A.22) α2 =−β2=iω−μLRe(khL)d4c3c4−d3d4=−iω+μRRe(keR)d1c1c2−d1d2, (A.23) α3 =−β3=iω+μLRe(keL)d3c3c4−d3d4=−iω−μRRe(khR)d2c1c2−d1d2, (A.24) α4 =β4=−iω−μLRe(khL)c3c3c4−d3d4=−iω−μRRe(khR)c1c1c2−d1d2. (A.25)

The first expression is obtained by solving Eq. (A.21) in the domain , the second in the domain . The coefficients and take the same value along the junction. However, it is advantageous to know their explicit analytical expressions in terms of the parameters of each domain. With this, we obtain the nonlocal, retarded anomalous Green functions as

 GReh(x<0,x′>LS) (A.26) GReh(x>LS,x′<0) (A.27)

In angular representation, we have

 ke/hi+ikyω±μi=ζe/hisgn(ω±μi)eiζe/hiθe/hi, ke/hi−ikyω±μi=ζe/hisgn(ω±μi)e−iζe/hiθe/hi, (A.28)

with such that we obtain the pairing amplitudes in Eqs. (8)-(10) of the main text under the condition .

## Appendix C Transport properties

The probability amplitudes for each of the scattering processes are determined from the probability current density operator in the –direction,

 jx=∂^kx^HBdG=(τ0⊗σx) (A.29)

and read, normalized to the incident current, as

• electron excited in moving towards the interface at

 Ree1=|b1|2, Reh1=ε+μLε−μLRe(khL)keL|a1|2, Tee1=ε+μLε+μRRe(keR)keL|c1|2, Teh1=ε+μLε−μRRe(khR)keL|d1|2. (A.30)
• electron excited in moving towards the interface at