Introduction

The search for topological superconductors that support Majorana fermions (MFs)1 has attracted much theoretical and experimental studies in recent years2,3,4,5,6,7,8. These studies are strongly motivated by the fact that MFs in topological superconductors are non-Abelian particles and have potential applications in fault-tolerant quantum computations9,10. Recent studies have pointed out that one of the most promising ways to engineer topological superconductors is by inducing s-wave superconductivity on semiconductor wires with Rashba spin-orbit coupling in the presence of external magnetic fields11,12,13,14,15,16,17. This results in so-called D class topological superconductors that break time-reversal symmetry and support a single MF end state at each end of a superconducting wire18,19. These D class topological superconductors also exhibit a number of interesting transport properties such as fractional Josephson effects20,21, resonant Andreev reflections22,23 and enhanced crossed Andreev reflections24,25. So far, the search for D class topological superconductors has been one of the most important areas in the study of topological superconductors.

However, according to the Altland–Zirnbauer symmetry classification scheme18, there exist other topological superconductors that belong to different symmetry classes. Many aspects of the physical properties and potential applications of various types of topological superconductors have yet to be explored.

Here we demonstrate that inducing s-wave superconductivity on a AIII class topological insulator18,19, which respects a chiral symmetry and supports fermionic end states as illustrated in Fig. 1a, will result in a new type of superconductor. The resulting superconductor is in the BDI class, which respects a time-reversal-like symmetry and the particle hole symmetry, and it is classified by an integer topological invariant NBDI19,26,27,28. This BDI class superconductor supports two distinct topological phases distinguished by one (NBDI=1) and two (NBDI=2) MF end states at each end of the wire, respectively. While the superconductor in the NBDI=1 phase has almost identical transport properties as a D class topological superconductor, the NBDI=2 phase exhibits several transport anomalies. Particularly, in the NBDI=2 phase, local Andreev reflections (ARs) are completely suppressed at the normal lead/topological superconductor interface at zero bias due to the destructive interference between the AR amplitudes induced by the two MFs. When two normal leads are attached to the two ends of the superconductor, resonant-crossed AR processes can happen, causing an electron from one normal lead to be reflected as a hole in the other lead with probability of unity. In reverse processes, when a current is driven from the superconductor to the leads, Cooper pairs can split into two spatially separated leads and form correlated electron pairs with perfect efficiency. We call this phenomenon resonant Cooper pair splitting. Remarkably, the outgoing currents of the two leads are correlated and spin-polarized. Importantly, we show that these unique transport properties of BDI class topological superconductors can be experimentally realized using quantum anomalous Hall insulators in proximity to an s-wave superconductor.

Figure 1: From a AIII class topological insulator to a BDI topological superconductor.
figure 1

(a) AIII topological insulator with fermionic end states Ψ1 and Ψ2 located at the ends of the wire. (b) By inducing superconductivity on the AIII class topological insulator, each fermionic end state becomes two MFs and the system becomes a BDI class topological superconductor in the NBDI=2 phase. As the pairing strength Δ increases, one of the Majorana fermions at each end merges into the bulk. (c) After the bulk gap is closed by increasing Δ, only one MF end state is left at each end and we have the NBDI=1 phase. (d) Phase diagram of the BDI class topological superconductor characterized by the topological invariant NBDI as functions of the pairing strength Δ and Γz−2tso.

Results

From class AIII to class BDI

In this section, we first point out how to obtain a BDI class topological superconductor from an AIII class topological insulator. The properties of the MF end states are also studied. Second, we study the local AR properties of the BDI class topological superconductor by attaching a normal lead to one end of the topological superconductor. Third, we examine the effects of resonant-crossed ARs and resonant Cooper pair splitting induced by the double MF end states in the NBDI=2 phase. The generation of correlated spin currents using these novel phenomena is also discussed. Lastly, we discuss the relation between the BDI class topological superconductor and quantum anomalous Hall insulators.

An AIII class topological insulator is a one-dimensional system that respects a chiral symmetry and supports fermionic end states19,29. A simple AIII class Hamiltonian, which can be topologically non-trivial in the basis of (ck, ck), can be written as29

Here ck (ck) denotes a spin up (down) fermionic operator, ts is the hopping amplitude, Γz is the Zeeman term and tso is the hopping amplitude with spin flip. For simplicity and without loss of generality, we assume ts, tso and Γz to be positive real numbers. Since the Hamiltonian contains only the σy and σz terms, H(k) respects the chiral symmetry σxH(kx =−H(k) and H(k) belongs to AIII class according to symmetry classifications18. In the regime where |Γz| <2ts, H(k) is topologically non-trivial. For a topologically non-trivial AIII class wire with open boundaries, the wire supports a single fermionic end state at each end of the wire29 as depicted in Fig. 1a.

Interestingly, the AIII class topological insulator becomes a BDI class topological superconductor when superconducting s-wave pairing terms Δ0ckck + H.c. are added. On the Nambu basis (ck, ck, ), the Hamiltonian is:

where σi and τi are Pauli matrices acting on spin and particle-hole space, respectively.

In the presence of the pairing terms, the symmetry class of the Hamiltonian is changed from AIII to BDI. In particular, we note that the Hamiltonian satisfies a time-reversal-like symmetry and a particle-hole symmetry , where , and is the complex conjugate operator. Since , there is no Kramer's degeneracy associated with . As a result of and symmetries, we have , where . Therefore, HBDI(k) is in the BDI class18,19. It has been shown that a BDI class topological superconductor is classified by an integer topological invariant NBDI18,19,26,27,28, which denotes the number of topologically protected MF end states at each end of the superconducting wire.

The topological invariant NBDI can be easily evaluated27 and the phase diagram of HBDI, as functions of Γz−2ts and Δ, is depicted in Fig. 1d. It is evident that there are two topological phases with NBDI=2 and NBDI=1, respectively. The phase boundaries are the two lines Γz−2tsΔ, on which the energy gap of HBDI closes.

For a semi-infinite BDI class wire occupying the space with y≥0, the zero energy end states in the topological regime can be found in the continuum limit by solving HBDI(k →−iy)γ(y)=0. In the regime with NBDI=2 where 2ts−Γz>Δ, there are two solutions γ1(y)=[1, 1, 1, 1] and γ2(y)=i[1, 1, −1, −1] . Here λl±=. Note that the zero energy solutions satisfy the conditions γi =, so that the end states are MFs. Moreover, under the time-reversal symmetry like operation , we have =γi. As a result, the coupling between the two MF end states, which can be written as 1γ2, breaks the symmetry. This term is not allowed so long as is respected. Therefore, the two MF end states do not couple to each other, which is a feature of the BDI class topological superconductor. This is in sharp contrast to D class topological superconductors where an even number of MFs can couple to each other and the MFs are lifted to finite energy.

It is interesting to note that as we approach the phase boundary between the NBDI=2 and NBDI=1 phases, where Γz−2ts =−Δ, we have λ2−→ 0 and γ2 is no longer localized at the end of the wire. The process of approaching the phase boundary is depicted in Fig. 1b,c. In the regime where NBDI=1, only one MF end state γ1 remains. In the regime where NBDI=0, there are no zero energy end state solutions.

For a long wire with length L and neglecting the coupling between the left and right MFs, there are two more MF solutions γ3(y)=i[1, −1, −1, 1] and γ4(y)=[1, −1, 1, −1] as depicted in Fig. 1b. It is worthwhile to note that the forms of the MF wavefunctions are important for determining the transport properties of the superconductor as shown below. Moreover, we note that besides the set of symmetries discussed above, the Hamiltonian HBDI(k) respects another time-reversal-like symmetry such that =HBDI(−k). The four MF end states transform under as =γ1/4 and =−γ2/3. On the other hand, when the wire is finite, the MFs from the two ends of the wire can couple to each other. While the interaction terms 1γ4 and 2γ3 break the symmetry and are not allowed, the coupling terms 1γ3 and 2γ4 are allowed.

Local Andreev Reflections

It has been shown in previous studies22,23 that a single MF end state induces resonant local ARs at a normal lead/topological superconductor junction where an incoming electron is reflected as a hole in the same lead with probability of unity. The resonant local ARs result in zero bias conductance (ZBC) peaks of height 2e2/h in transport measurements at zero temperature. It has also been shown that 1D DIII class topological superconductors, which respect time-reversal symmetry and particle-hole symmetry, support two MF end states at one end of the wire19,27,30,31,32,33. The two MF end states can induce a ZBC peak of height 4e2/h27. Therefore, one may expect that the BDI class topological superconductor in the phases with NBDI=1 and NBDI=2 can both induce ZBC peaks in tunnelling experiments. Surprisingly, we find that, while the single MF end state in the NBDI=1 phase can induce ZBC peaks, the two MFs in the NBDI=2 phase completely suppress local ARs at zero bias and cause a conductance dip at low voltages.

The experimental setup for the BDI topological superconductor attached to a normal lead is depicted in Fig. 2a. To calculate the tunnelling spectroscopy of the BDI topological superconductor at different phases, we first write down a real space tight-binding model, which corresponds to HBDI(k) as described in the Methods section. A semi-infinite normal metal lead is attached to the left end of the topological superconductor. The zero temperature conductance of the normal metal/topological superconductor junction can be calculated from the reflection matrix Rhe of the junction:

Figure 2: Transport properties of a normal lead/BDI class topological superconductor junction.
figure 2

(a) A normal lead is attached to one end of a semi-infinitely long BDI class topological superconductor in the NBDI=2 phase. At zero voltage bias, the electrons are totally reflected due to the destructive interference of the AR amplitudes induced by the two MF end states. (b) ZBC of the setup depicted in (a) as functions of Δ and Γz. It is evident that the ZBC is quantized at 2e2/h in the NBDI=1 phase and zero otherwise. The details of the tight-binding model used is given in the Methods section. (ce) The conductance as a function of voltages at points A1, A2, B1, B2, C1 and C2 denoted in a, respectively. The conductance peaks at high voltages are due to bulk states and they appear only when the voltage bias is larger than the energy gap of the superconductor.

where Rhe(E)ij denotes the local AR amplitude of an electron with energy E at channel j to be reflected as a hole in channel i, which is calculated using the recursive Green’s function approach25,34,35,36.

The ZBC as a function of Δ and Γz is shown in Fig 2b. As expected, in the phase with NBDI=0, the ZBC is strongly suppressed. When NBDI=1, the ZBC is quantized at 2 due to the MF-induced resonant ARs22,23. Surprisingly, in the NBDI=2 phase, the ZBC is zero even though there are two zero energy MFs at the end of the topological superconductor. The conductance at finite voltages are shown in Fig. 2c–e. It is evident from Fig. 2e that there is a ZBC dip at the NBDI=2 phase instead of a ZBC peak. In the following, we construct an effective Hamiltonian of the normal lead/topological superconductor junction and show that the ZBC dip at NBDI=2 is due to destructive inference between the local AR amplitudes caused by the two MFs.

For voltage bias smaller than the pairing gap, we expect the transport properties of the junction to be described by an effective Hamiltonian

Here HL is the effective Hamiltonian for the left lead and vF is the Fermi velocity of the lead. We note that, in general, one should consider a metal lead with electrons carrying spin pointing to the positive x direction and electrons carrying spin pointing to the negative x direction . However, it can be shown that using the form of the wavefunctions of γ1 and γ2 that only electrons can couple to the MF end states and are decoupled from the superconductor. The form of the effective coupling term HLM is crucial for the study of the transport properties. The coupling between the left lead and the two MF end states of the topological superconductor is described by HLM and ωi are the coupling amplitudes.

With H1eff, the scattering matrix can be easily calculated using the equation of motion approach22. It can be shown that the local AR amplitudes for an incoming electron with energy E is Rhe=, where ζ1,2+iEvF/2. Therefore, at E=0, Rhe(E=0)=0 as the two local AR amplitudes caused by the two MFs have opposite signs and they cancel each other out, as long as both ω1 and ω2 are finite. In other words, the suppression of the local ARs at zero bias is caused by the destructive interference of AR amplitudes caused by the two MF end states. This is in sharp contrast to the resonant ARs caused by a single MF end state in the D class case.

From the wavefunctions of the end states studied in Section IIA, we note that as Δ increases, γ1 remains localized at the end and γ2 merges into the bulk gradually. Then ω2 reduces to zero as Δ approaches the phase transition line Γz−2ts=−Δ. Further increasing Δ would change the phase from NBDI=2 to the NBDI=1 phase. When ω2=0 in the NBDI=1 phase, we have |Rhe(E=0)| =1 and the resulting ZBC is 2e2/h according to equation (3) as expected22,23.

To understand the transport properties at finite voltages, we note that when ω2<<ω1, the local AR amplitudes become significant when the energy of the incoming electrons reaches |E|≈2/vF. As a result, the width of the ZBC dip becomes narrower as Δ increases, as shown in Fig. 2e, and the ZBC dip disappears when ω2 goes to zero.

Resonant-crossed ARs

In the above sections, it is shown that local AR processes are suppressed at a normal lead/topological superconductor junction for the NBDI=2 phase. Owing to the suppression of the local AR amplitudes and the conservation of probability, we expect that other tunnelling processes can become more important. In this section, we show that the two MF end states in the NBDI=2 phase can strongly enhance the crossed AR processes in a normal lead/topological superconductor/normal lead junction, provided that the length of the superconducting wire is comparable to the localization lengths of the MF end states such that the MFs from the two ends can couple to each other. In a crossed AR process, an electron from one lead is reflected as a hole in the other lead. As a result, two electrons from the two leads form a Cooper pair and get injected into the superconductor, as depicted in Fig. 3a.

Figure 3: Transport properties of a normal lead/topological superconductor/normal lead junction.
figure 3

(a) Two normal leads are attached to the two ends of a wire with finite length. In a crossed AR process, an electron from the left lead is reflected as a hole in the right lead. As a result, two electrons are injected into the superconductor to form a Cooper pair. For our setup, all the electrons on the left (right) lead have spin pointing to the positive (negative) x direction. (b) In a Cooper pair splitting process, a Cooper pair is split into two electrons with opposite spins. One electron is injected into each lead. (c) Local AR for a short wire with length L=20a. In this case, the local ARs are still strongly suppressed but they can deviate from zero. (d) Crossed AR amplitudes at zero bias of the wire. The crossed AR amplitudes can be close to one in the NBDI=2 phase for a large phase space. (e) Elastic co-tunnelling amplitudes at zero bias. (f) Normal reflection amplitudes at zero bias. It is important to note that the normal reflection amplitudes have a minimal value of one. This is due to the fact that one of the spin channels of the normal lead is completely decoupled from the superconductor and all the electrons of that spin channel are reflected. (g) The ZBC (blue line) and the crossed AR amplitude (green line) as functions of Γz for parameters denoted by the horizontal dashed line in d. The vertical red dashed lines separate the three phases with different NBDI.

To calculate the transport properties of the superconductor, we attach two normal leads to the superconductor as depicted in Fig. 3a. The superconductor is described by a tight-binding model presented in the Methods section. The length of the superconductor is L=20a, which is comparable to the localization length of the MF end states. Here a is the lattice constant of the tight-binding model and the parameters of the model is given in the Methods section. Focusing on the transport properties of the left normal lead, the local AR amplitudes, the crossed AR amplitudes, the elastic electron co-tunnelling amplitudes and the electron normal reflection amplitudes for the three different phases at zero bias are shown in Fig. 3. It is surprising that, in the NBDI=2 phase, there are parameter regimes where the crossed AR amplitude is unity. When this happens, all other tunnelling amplitudes for the electrons, including the elastic co-tunnelling amplitudes for which electrons tunnel directly from the left lead to the right lead, vanish.

On the other hand, crossed AR amplitudes in the NBDI=1 phase have similar properties as the cases of D class topological superconductors24,25. In this phase, there are regimes where local AR processes are suppressed and the crossed AR processes dominate. However, crossed AR amplitudes are always equal to the elastic co-tunnelling processes in the NBDI=1 phase24,25. Therefore, the crossed AR cannot reach unity. As shown in Fig. 3e, the maximal crossed AR amplitude is in general much smaller than unity in the NBDI=1 phase. Therefore, the possibility of inducing resonant-crossed ARs is a unique signature of the NBDI=2 phase.

To understand the numerical results, we expect the transport properties for voltage bias smaller than the superconducting pairing gap to be well described by an effective Hamiltonian, which includes the coupling between the MFs with the two leads as well as the coupling among the four MF end states. The Hamiltonian reads:

The Hamiltonian of the left lead HL and the coupling between the left lead and the MFs HLM have been discussed above. Here HR describes the right normal lead and denotes an annihilation operator of an electron with spin pointing to the negative x direction. It is important to note that for the right lead, only electrons that are spin-polarized along the negative x direction are coupled to the MFs due to the form of the MF wavefunctions γ3 and γ4. HRM describes the coupling between the right lead and the MFs. The coupling between the four MF end states is described by HM, where E13 and E24 are real numbers denoting the coupling strength between the MFs from the opposite ends of the wire. As discussed above, the coupling terms such as 1γ4 and 2γ3 are not allowed by symmetry.

For the effective Hamiltonian H2eff, the scattering matrix can be found and the crossed AR amplitudes from one lead to another lead at E=0 is −ω1ω3E13vF/ω2ω4E24vF/. Crossed AR processes are depicted in Fig. 3a. When both the conditions E13/vF=ω1ω3 and E24/vF=ω2ω4 are satisfied, the crossed AR amplitude is unity and all other tunnelling amplitudes are zero. We call this phenomenon resonant-crossed ARs. As shown in Fig. 3d, there is a sizeable phase space in which the crossed AR amplitudes are close to one. The oscillating behaviour of the tunnelling amplitudes in the phases with MFs is due to the fact that the coupling strengths of the MFs oscillate as a function of Δ and Γz25.

As depicted in Fig 3b, the reverse processes of the crossed ARs are the Cooper pair splitting processes. When a current is driven from the superconductor to the two leads, a Cooper pair from the superconductor can be split into two spatially separated but correlated electrons and one electron is injected into each of the two leads. In the language of scattering matrix, the Cooper pair splitting amplitude is equivalent to the amplitude for an incoming hole from the left lead to be reflected as an electron in the right lead. One can show that the Cooper pair splitting amplitude equals the crossed AR amplitude. As a result, when a current is driven from the superconductor to the leads, we can have resonant Cooper pair splitting.

Remarkably, for the left lead, only electrons with spin pointing to the positive x direction are coupled to the superconductor and for the right lead, only electrons with spin pointing to the negative x direction are coupled to the superconductor due to symmetry constraints. Therefore, the current of the left (right) lead is spin-polarized to the positive (negative) x direction. Moreover, due to the resonant-crossed ARs, the conductance of each normal lead is G =2e2/h and the current is spin-polarized. The ZBC of the left lead, with parameters corresponding to the horizontal dashed line in Fig. 3d, is shown in Fig. 3g.

In Fig. 3g, the ZBC is denoted by the blue line and the crossed AR amplitudes are denoted by the green line. As Δ is fixed and Γz increases, all the three phases with NBDI=2, 1 and 0 can be reached. In the NBDI=2 phase, it is clear that the conductance is almost solely determined by the crossed AR amplitude as the local AR amplitudes are strongly suppressed as shown in Fig. 3c. When the crossed AR amplitude approaches unity, the ZBC approaches 2e2/h. In the NBDI=1 phase, the conductance can reach 2e2/h due to local ARs. In the NBDI=0 phase, the ZBC goes to zero. Since the currents out of the left and right leads are spin-polarized, and the fact that there are no spin-orbit coupling in the normal lead, the normal lead can sustain a spin current. Therefore, the BDI class topological superconductor in the NBDI=2 phase can be a novel source of conserved spin currents for spintronic applications.

Realistic Cooper pair splitters

In this section, we point out that the anomalous transport properties of BDI class topological superconductor discussed above can be experimentally realized using anomalous Hall insulators in proximity to an s-wave superconductor.

A quantum anomalous Hall insulator (QAHI) is an insulator with gapless chiral fermionic edge states in the absence of an external magnetic field, which has been experimentally discovered recently37. Interestingly, it was shown by Qi et al.38 that in proximity with an s-wave superconductor, a QAHI can be turned into a topological superconductor, which supports one or two branches of chiral MF edge states, as depicted in Fig. 4a. The topological superconducting phases can be classified by Chern numbers NChern with NChern denoting the number of branches of MF edge states. The Hamiltonian of a QAHI in the presence of superconducting pairing and in the Nambu basis {φk↑, φk↓, } can be written as:

Figure 4: Transport properties of a QAHI in proximity to an s-wave superconductor.
figure 4

(a) In the NChern=2 phase of a QAHI in proximity to an s-wave superconductor, there are two branches of MF edge states localized at the edge of the system. The MF edge states are denoted by γi. (b) Two normal leads are attached to the edges of a QAHI. A Cooper pair is split into two electrons. The electrons injected to the left lead and the right lead have definite spin polarizations. This results in spin-polarized currents leaving the normal leads. (c) The momentum-resolved conductance from the normal lead to a wide strip of superconducting QAHI in the NChern=1 phase. The tight-binding model used is described in the Methods section. The width of the QAHI is Ly=200a. Periodic boundary conditions in x direction is assumed. The strong local AR amplitudes at energy within the gap of the superconducting QAHI is due to the chiral MF edge state. (d) The local AR amplitudes in the NChern=2 phase with Ly=200a. The local AR amplitudes are strongly suppressed at low voltages even in the presence of two chiral MF edge states as depicted in a. (e) The momentum-resolved local AR amplitudes in the NChern=2 phase for a narrow strip of superconducting QAHI with Ly=20a. The distance between the two edges in the y direction is comparable to the localization length of the chiral edge states. (f) The momentum-resolved crossed AR amplitudes of a narrow strip of superconducting QAHI in the NChern=2 phase with Ly=20a. The crossed AR amplitudes can be close to one at low voltages near kx=0.

Here, and are real numbers characterizing the model38. For general momentum k, the Hamiltonian is in the D class that respects only the particle-hole symmetry. The time-reversal-like symmetries and are broken by the sinkxτ0σx term. However, for kx=0, HQAHI+S is equivalent to HBDI in equation (2). As a result, the kx=0 component of HQAHI+S is a BDI class topological superconductor. Moreover, the NChern=1 (NChern=2) phase in the quantum Anomalous Hall system corresponds to the NBDI=1 (NBDI=2) phase of the BDI class topological superconductor.

A strip of QAHI in proximity to a superconductor and attached to two metal leads is depicted in Fig. 4b. The tight-binding model used to describe HQAHI+S is presented in the Methods section. The momentum-resolved local AR amplitudes from the left normal lead to the QAHI in the NChern=1 and NChern=2 phases are shown in Fig. 4c,d, respectively. The width of the QAHI in this case is Ly=200a, which is much longer than the localization length of the MF edge states. Focusing on the transport properties at kx=0, we note that the local AR resonates at zero bias for the NChern=1 phase but is suppressed for the NChern=2 phase. Similar results were obtained by Ii et al.39 while the reasons of the transport anomalies were not given. By establishing the correspondence between BDI class topological superconductor and HQAHI+S, we have shown with the effective tunnelling Hamiltonian approach that the suppression of the local AR at the NChern=2 phase is a consequence of the destructive interference of the AR amplitudes induced by the two MFs with kx=0 at the edge of the QAHI.

Owing to the strong suppression of the local AR amplitudes near kx=0 in the NChern=2 phase, we expect that the crossed AR amplitudes can be enhanced near kx=0 when the width of the QAHI is reduced. The local AR and the crossed AR amplitudes for a narrow strip of QAHI with width Ly=20a is presented in Fig. 4e,f, respectively. From Fig. 4f, it is shown that the crossed AR amplitudes can reach almost unity for kx≈0 at low voltage bias. At the same time, the local AR amplitudes in the NChern=2 phase is strongly suppressed for this narrow strip of QAHI. As a result, similar to the case of the BDI topological superconductor in the NBDI=2 phase, when a current is driven from the superconductor to the lead, the QAHI can split the Cooper pairs effectively and result in correlated spin-polarized currents leaving the two normal leads.

Discussion

In this work, we show that the BDI class topological superconductor in the NBDI=2 phase can be used as an efficient Cooper pair splitter, whereby the Cooper pairs can be split into two streams of spin-polarized currents. Two important results are used to reach these conclusions, namely, the suppression of local ARs and the fact that the MF end states only couple to electrons with fixed spin polarizations of the leads. In this section, we argue that these results can be understood easily in the regime with small paring amplitudes.

First, since the local ARs compete with crossed ARs due to conservation of probability, the MFs should not induce strong local ARs, as in the case of D class topological superconductors. Otherwise, the crossed AR amplitudes would be small. For a BDI class topological superconductor in the NBDI=2 phase obtained by inducing superconductivity on a AIII class topological insulator with fermionic end states, the suppression of local ARs at zero bias is indeed quite natural.

Suppose that the AIII class topological insulator is in the non-trivial phase with a fermionic end state, adding a small superconducting pairing term does not close the energy gap and there is no topological phase transition. In this case, the fermionic end state can be regarded as two MF end states. Therefore, we have a BDI class topological superconductor with NBDI=2. However, when the pairing terms are zero, there cannot be any local ARs since the system is simply an insulator. Consequently, one may expect that the local AR amplitudes are strongly suppressed when the pairing amplitudes are small. The suppression of the local AR amplitudes opens up the possibility for the crossed AR amplitudes to be enhanced in the presence of finite Δ.

It is important to note that the suppression of local ARs in the NBDI=2 phase does not contradict the results of Diez et al.40 who predicted that the conductance at zero bias should be NBDI at a normal lead/BDI topological superconductor junction. The reason is that the results obtained in Diez et al.40 would apply only if NBDI is calculated using the chiral symmetry where is the particle-hole symmetry operator and and is the complex conjugate operator. This chiral symmetry is respected by HBDI of equation (2). Using this set of symmetries, one would find that the topological invariant equals to zero in the parameter regimes where NBDI=2 and NBDI=0. Moreover, =1 in the regime where NBDI=1. Therefore, the ZBC should be zero in both the NBDI=0 and NBDI=2 phases. This is consistent with the results by Diez et al.40. However, the symmetry arguments alone are not enough to understand the conductance at finite voltages. In short, the NBDI=2 topological phase in this work is different from the =2 phases found in previous works26,27 as a different set of symmetry operators were used to calculate the topological invariants. The suppression of local ARs caused by usual fermionic Andreev bound states was also studied by Ioselevich and Feigelman41. However, resonant-crossed ARs cannot happen in trivial superconductors due to the lack of symmetry constraints to restrict the form of the interactions among different Andreev bound states.

Second, by definition, a AIII class topological insulator respects a chiral symmetry. As a result, a non-degenerate zero energy fermionic end state at one end of the system has to be an eigenstate of the chiral symmetry operator. For the AIII class model used in equation (1), the chiral operator is σx. Therefore, the two end states at opposite ends of the wire are eigenstates of σx with opposite eigenvalues29. If there are no spin flip terms in the leads, the end states can only couple to electrons that have the same spin as the end states. Using the form of the MF wavefunctions, one can show that this is true even in the presence of the pairing terms. As a result, in the effective Hamiltonian H2eff, one can regard the left and the right normal leads as having opposite spin. This result is important for obtaining spin-polarized currents in the leads by splitting Cooper pairs.

Moreover, experiments on the efficient splitting of Cooper pairs using Coulomb blockade effect42 have been reported43,44. However, the currents leaving the superconductors are not spin-polarized and it is not known whether the electrons on different leads are correlated45. Therefore, being able to generate correlated spin currents by splitting Cooper pairs is a very unique property of the BDI class topological superconductor.

Finally, we discuss the stability of the topological phases that support two zero energy Majorana modes on each edge of the system. As discussed above, the NBDI=2 phase of the BDI class topological superconductor is protected by the chiral symmetry =σxτ0. Therefore, terms such as σxτz can break the chiral symmetry and make the superconductor topologically trivial. However, the NChern=2 phase of the superconducting QAHI is in the D class18 and the zero energy Majorana modes on the edge are robust against perturbations as long as the bulk gap is not closed. Therefore, one can always extract an effective 1D Hamiltonian from the two dimensional (2D) superconducting QAHI, which supports two Majorana modes at each edge. The coupling of this 1D Hamiltonian with normal leads can be described by equation (4). Therefore, the suppression of local ARs and the appearance of almost resonant-crossed ARs in the QAHI case are robust against perturbations.

Methods

Tight-binding models

For the calculations of the momentum-resolved transport properties of the QAHI with superconducting pairing terms, we apply periodic boundary conditions in the x direction and open boundary conditions in the y direction. Spinful normal leads are attached to the two edges parallel to the x directions. The tight-binding model for a strip of QAHI with superconducting pairing terms can be written as:

Here denotes an electron operator at site i along the y direction and has momentum quantum number kx along the x direction and spin up (spin down) with respect to the z direction. In all the figures in Fig. 4, the parameters are: =1, =10. For Fig. 4c,e, =40 and Δ=1 so that the system is in the NChern=1 phase. In Fig. 4d,f, =36.4 and Δ=1 so that the system is in the NChern=2 phase.

The same tight-binding model HQAHI+S(kx), with kx=0, can be used to describe the BDI class topological superconductor HBDI(k) of equation (2) with parameters =tso, =ts and =Γz+2ts. In Figs 2 and 3, ts =10 and the number of sites in the y direction is L=200a and L=20a, respectively, where a is the lattice constant.

Additional information

How to cite this article: He, J. J. et al. Correlated spin currents generated by resonant-crossed Andreev reflections in topological superconductors. Nat. Commun. 5:3232 doi: 10.1038/ncomms4232 (2014).