Abstract
Topological superconductors, which support Majorana fermion excitations, have been the subject of intense studies due to their novel transport properties and their potential applications in faulttolerant quantum computations. Here we propose a new type of topological superconductors that can be used as a novel source of correlated spin currents. We show that inducing superconductivity on a AIII class topological insulator wire, which respects a chiral symmetry and supports protected fermionic end states, will result in a topological superconductor. This topological superconductor supports two topological phases with one or two Majorana fermion end states, respectively. In the phase with two Majorana fermions, the superconductor can split Cooper pairs efficiently into electrons in two spatially separated leads due to Majoranainduced resonantcrossed Andreev reflections. The resulting currents in the leads are correlated and spinpolarized. Importantly, the proposed topological superconductors can be realized using quantum anomalous Hall insulators in proximity to superconductors.
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Introduction
The search for topological superconductors that support Majorana fermions (MFs)^{1} has attracted much theoretical and experimental studies in recent years^{2,3,4,5,6,7,8}. These studies are strongly motivated by the fact that MFs in topological superconductors are nonAbelian particles and have potential applications in faulttolerant quantum computations^{9,10}. Recent studies have pointed out that one of the most promising ways to engineer topological superconductors is by inducing swave superconductivity on semiconductor wires with Rashba spinorbit coupling in the presence of external magnetic fields^{11,12,13,14,15,16,17}. This results in socalled D class topological superconductors that break timereversal symmetry and support a single MF end state at each end of a superconducting wire^{18,19}. These D class topological superconductors also exhibit a number of interesting transport properties such as fractional Josephson effects^{20,21}, resonant Andreev reflections^{22,23} and enhanced crossed Andreev reflections^{24,25}. 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 scheme^{18}, 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 swave superconductivity on a AIII class topological insulator^{18,19}, 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 timereversallike symmetry and the particle hole symmetry, and it is classified by an integer topological invariant N_{BDI}^{19,26,27,28}. This BDI class superconductor supports two distinct topological phases distinguished by one (N_{BDI}=1) and two (N_{BDI}=2) MF end states at each end of the wire, respectively. While the superconductor in the N_{BDI}=1 phase has almost identical transport properties as a D class topological superconductor, the N_{BDI}=2 phase exhibits several transport anomalies. Particularly, in the N_{BDI}=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, resonantcrossed 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 spinpolarized. 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 swave superconductor.
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 resonantcrossed ARs and resonant Cooper pair splitting induced by the double MF end states in the N_{BDI}=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 onedimensional system that respects a chiral symmetry and supports fermionic end states^{19,29}. A simple AIII class Hamiltonian, which can be topologically nontrivial in the basis of (c_{k↑}, c_{k↓}), can be written as^{29}
Here c_{k↑} (c_{k↓}) denotes a spin up (down) fermionic operator, t_{s} is the hopping amplitude, Γ_{z} is the Zeeman term and t_{so} is the hopping amplitude with spin flip. For simplicity and without loss of generality, we assume t_{s}, t_{so} and Γ_{z} to be positive real numbers. Since the Hamiltonian contains only the σ_{y} and σ_{z} terms, H(k) respects the chiral symmetry σ_{x}H(k)σ_{x} =−H(k) and H(k) belongs to AIII class according to symmetry classifications^{18}. In the regime where Γ_{z} <2t_{s}, H(k) is topologically nontrivial. For a topologically nontrivial AIII class wire with open boundaries, the wire supports a single fermionic end state at each end of the wire^{29} as depicted in Fig. 1a.
Interestingly, the AIII class topological insulator becomes a BDI class topological superconductor when superconducting swave pairing terms Δ_{0}c_{k↑}c_{−k↓} + H.c. are added. On the Nambu basis (c_{k↑}, c_{k↓}, ), the Hamiltonian is:
where σ_{i} and τ_{i} are Pauli matrices acting on spin and particlehole 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 timereversallike symmetry and a particlehole 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, H_{BDI}(k) is in the BDI class^{18,19}. It has been shown that a BDI class topological superconductor is classified by an integer topological invariant N_{BDI}^{18,19,26,27,28}, which denotes the number of topologically protected MF end states at each end of the superconducting wire.
The topological invariant N_{BDI} can be easily evaluated^{27} and the phase diagram of H_{BDI}, as functions of Γ_{z}−2t_{s} and Δ, is depicted in Fig. 1d. It is evident that there are two topological phases with N_{BDI}=2 and N_{BDI}=1, respectively. The phase boundaries are the two lines Γ_{z}−2t_{s}=±Δ, on which the energy gap of H_{BDI} closes.
For a semiinfinite 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 H_{BDI}(k →−i∂_{y})γ(y)=0. In the regime with N_{BDI}=2 where 2t_{s}−Γ_{z}>Δ, there are two solutions γ_{1}(y)=[1, 1, 1, 1] and γ_{2}(y)=i[1, 1, −1, −1] . Here λ_{l±}=. Note that the zero energy solutions satisfy the conditions γ_{i} =, so that the end states are MFs. Moreover, under the timereversal symmetry like operation , we have =γ_{i}. As a result, the coupling between the two MF end states, which can be written as iγ_{1}γ_{2}, 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 N_{BDI}=2 and N_{BDI}=1 phases, where Γ_{z}−2t_{s} =−Δ, we have λ_{2−}→ 0 and γ_{2} 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 N_{BDI}=1, only one MF end state γ_{1} remains. In the regime where N_{BDI}=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 γ_{3}(y)=i[1, −1, −1, 1] and γ_{4}(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 H_{BDI}(k) respects another timereversallike symmetry such that =H_{BDI}(−k). The four MF end states transform under as =γ_{1/4} and =−γ_{2/3}. 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 iγ_{1}γ_{4} and iγ_{2}γ_{3} break the symmetry and are not allowed, the coupling terms iγ_{1}γ_{3} and iγ_{2}γ_{4} are allowed.
Local Andreev Reflections
It has been shown in previous studies^{22,23} 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 2e^{2}/h in transport measurements at zero temperature. It has also been shown that 1D DIII class topological superconductors, which respect timereversal symmetry and particlehole symmetry, support two MF end states at one end of the wire^{19,27,30,31,32,33}. The two MF end states can induce a ZBC peak of height 4e^{2}/h^{27}. Therefore, one may expect that the BDI class topological superconductor in the phases with N_{BDI}=1 and N_{BDI}=2 can both induce ZBC peaks in tunnelling experiments. Surprisingly, we find that, while the single MF end state in the N_{BDI}=1 phase can induce ZBC peaks, the two MFs in the N_{BDI}=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 tightbinding model, which corresponds to H_{BDI}(k) as described in the Methods section. A semiinfinite 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 R_{he} of the junction:
where R_{he}(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 approach^{25,34,35,36}.
The ZBC as a function of Δ and Γ_{z} is shown in Fig 2b. As expected, in the phase with N_{BDI}=0, the ZBC is strongly suppressed. When N_{BDI}=1, the ZBC is quantized at 2 due to the MFinduced resonant ARs^{22,23}. Surprisingly, in the N_{BDI}=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 N_{BDI}=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 N_{BDI}=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 H_{L} is the effective Hamiltonian for the left lead and v_{F} 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 γ_{1} and γ_{2} that only electrons can couple to the MF end states and are decoupled from the superconductor. The form of the effective coupling term H_{LM} 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 H_{LM} and ω_{i} are the coupling amplitudes.
With H_{1eff}, the scattering matrix can be easily calculated using the equation of motion approach^{22}. It can be shown that the local AR amplitudes for an incoming electron with energy E is R_{he}=, where ζ_{1,2}≡+iEv_{F}/2. Therefore, at E=0, R_{he}(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 ω_{1} and ω_{2} 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, γ_{1} remains localized at the end and γ_{2} merges into the bulk gradually. Then ω_{2} reduces to zero as Δ approaches the phase transition line Γ_{z}−2t_{s}=−Δ. Further increasing Δ would change the phase from N_{BDI}=2 to the N_{BDI}=1 phase. When ω_{2}=0 in the N_{BDI}=1 phase, we have R_{he}(E=0) =1 and the resulting ZBC is 2e^{2}/h according to equation (3) as expected^{22,23}.
To understand the transport properties at finite voltages, we note that when ω_{2}<<ω_{1}, the local AR amplitudes become significant when the energy of the incoming electrons reaches E≈2/v_{F}. 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 ω_{2} goes to zero.
Resonantcrossed ARs
In the above sections, it is shown that local AR processes are suppressed at a normal lead/topological superconductor junction for the N_{BDI}=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 N_{BDI}=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.
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 tightbinding 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 tightbinding 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 cotunnelling 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 N_{BDI}=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 cotunnelling amplitudes for which electrons tunnel directly from the left lead to the right lead, vanish.
On the other hand, crossed AR amplitudes in the N_{BDI}=1 phase have similar properties as the cases of D class topological superconductors^{24,25}. 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 cotunnelling processes in the N_{BDI}=1 phase^{24,25}. 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 N_{BDI}=1 phase. Therefore, the possibility of inducing resonantcrossed ARs is a unique signature of the N_{BDI}=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 H_{L} and the coupling between the left lead and the MFs H_{LM} have been discussed above. Here H_{R} 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 spinpolarized along the negative x direction are coupled to the MFs due to the form of the MF wavefunctions γ_{3} and γ_{4}. H_{RM} describes the coupling between the right lead and the MFs. The coupling between the four MF end states is described by H_{M}, where E_{13} and E_{24} 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 iγ_{1}γ_{4} and iγ_{2}γ_{3} are not allowed by symmetry.
For the effective Hamiltonian H_{2eff}, the scattering matrix can be found and the crossed AR amplitudes from one lead to another lead at E=0 is −ω_{1}ω_{3}E_{13}v_{F}/−ω_{2}ω_{4}E_{24}v_{F}/. Crossed AR processes are depicted in Fig. 3a. When both the conditions E_{13}/v_{F}=ω_{1}ω_{3} and E_{24}/v_{F}=ω_{2}ω_{4} are satisfied, the crossed AR amplitude is unity and all other tunnelling amplitudes are zero. We call this phenomenon resonantcrossed 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 Γ_{z}^{25}.
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 spinpolarized to the positive (negative) x direction. Moreover, due to the resonantcrossed ARs, the conductance of each normal lead is G =2e^{2}/h and the current is spinpolarized. 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 Γ_{z} increases, all the three phases with N_{BDI}=2, 1 and 0 can be reached. In the N_{BDI}=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 2e^{2}/h. In the N_{BDI}=1 phase, the conductance can reach 2e^{2}/h due to local ARs. In the N_{BDI}=0 phase, the ZBC goes to zero. Since the currents out of the left and right leads are spinpolarized, and the fact that there are no spinorbit coupling in the normal lead, the normal lead can sustain a spin current. Therefore, the BDI class topological superconductor in the N_{BDI}=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 swave 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 recently^{37}. Interestingly, it was shown by Qi et al.^{38} that in proximity with an swave 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 N_{Chern} with N_{Chern} 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 {φ_{k↑}, φ_{k↓}, } can be written as:
Here, and are real numbers characterizing the model^{38}. For general momentum k, the Hamiltonian is in the D class that respects only the particlehole symmetry. The timereversallike symmetries and are broken by the sink_{x}τ_{0}σ_{x} term. However, for k_{x}=0, H_{QAHI+S} is equivalent to H_{BDI} in equation (2). As a result, the k_{x}=0 component of H_{QAHI+S} is a BDI class topological superconductor. Moreover, the N_{Chern}=1 (N_{Chern}=2) phase in the quantum Anomalous Hall system corresponds to the N_{BDI}=1 (N_{BDI}=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 tightbinding model used to describe H_{QAHI+S} is presented in the Methods section. The momentumresolved local AR amplitudes from the left normal lead to the QAHI in the N_{Chern}=1 and N_{Chern}=2 phases are shown in Fig. 4c,d, respectively. The width of the QAHI in this case is L_{y}=200a, which is much longer than the localization length of the MF edge states. Focusing on the transport properties at k_{x}=0, we note that the local AR resonates at zero bias for the N_{Chern}=1 phase but is suppressed for the N_{Chern}=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 H_{QAHI+S}, we have shown with the effective tunnelling Hamiltonian approach that the suppression of the local AR at the N_{Chern}=2 phase is a consequence of the destructive interference of the AR amplitudes induced by the two MFs with k_{x}=0 at the edge of the QAHI.
Owing to the strong suppression of the local AR amplitudes near k_{x}=0 in the N_{Chern}=2 phase, we expect that the crossed AR amplitudes can be enhanced near k_{x}=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 L_{y}=20a is presented in Fig. 4e,f, respectively. From Fig. 4f, it is shown that the crossed AR amplitudes can reach almost unity for k_{x}≈0 at low voltage bias. At the same time, the local AR amplitudes in the N_{Chern}=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 N_{BDI}=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 spinpolarized currents leaving the two normal leads.
Discussion
In this work, we show that the BDI class topological superconductor in the N_{BDI}=2 phase can be used as an efficient Cooper pair splitter, whereby the Cooper pairs can be split into two streams of spinpolarized 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 N_{BDI}=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 nontrivial 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 N_{BDI}=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 N_{BDI}=2 phase does not contradict the results of Diez et al.^{40} who predicted that the conductance at zero bias should be N_{BDI} at a normal lead/BDI topological superconductor junction. The reason is that the results obtained in Diez et al.^{40} would apply only if N_{BDI} is calculated using the chiral symmetry where is the particlehole symmetry operator and and is the complex conjugate operator. This chiral symmetry is respected by H_{BDI} of equation (2). Using this set of symmetries, one would find that the topological invariant equals to zero in the parameter regimes where N_{BDI}=2 and N_{BDI}=0. Moreover, =1 in the regime where N_{BDI}=1. Therefore, the ZBC should be zero in both the N_{BDI}=0 and N_{BDI}=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 N_{BDI}=2 topological phase in this work is different from the =2 phases found in previous works^{26,27} 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 Feigelman^{41}. However, resonantcrossed 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 nondegenerate 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 σ_{x}. Therefore, the two end states at opposite ends of the wire are eigenstates of σ_{x} with opposite eigenvalues^{29}. 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 H_{2eff}, one can regard the left and the right normal leads as having opposite spin. This result is important for obtaining spinpolarized currents in the leads by splitting Cooper pairs.
Moreover, experiments on the efficient splitting of Cooper pairs using Coulomb blockade effect^{42} have been reported^{43,44}. However, the currents leaving the superconductors are not spinpolarized and it is not known whether the electrons on different leads are correlated^{45}. 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 N_{BDI}=2 phase of the BDI class topological superconductor is protected by the chiral symmetry =σ_{x}τ_{0}. Therefore, terms such as σ_{x}τ_{z} can break the chiral symmetry and make the superconductor topologically trivial. However, the N_{Chern}=2 phase of the superconducting QAHI is in the D class^{18} 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 resonantcrossed ARs in the QAHI case are robust against perturbations.
Methods
Tightbinding models
For the calculations of the momentumresolved 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 tightbinding 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 k_{x} 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 N_{Chern}=1 phase. In Fig. 4d,f, =36.4 and Δ=1 so that the system is in the N_{Chern}=2 phase.
The same tightbinding model H_{QAHI+S}(k_{x}), with k_{x}=0, can be used to describe the BDI class topological superconductor H_{BDI}(k) of equation (2) with parameters =t_{so}, =t_{s} and =Γ_{z}+2t_{s}. In Figs 2 and 3, t_{s} =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 resonantcrossed Andreev reflections in topological superconductors. Nat. Commun. 5:3232 doi: 10.1038/ncomms4232 (2014).
References
Wilczek, F. Majorana returns. Nat. Phys. 5, 614–618 (2009).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010).
Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).
Beenakker, C. Search for Majorana fermions in superconductors. Annu. Rev. Con. Mat. Phys. 4, 113–136 (2013).
Franz, M. Majorana's wires. Nat. Nanotechnol. 8, 149–152 (2013).
Stanescu, T. D. & Tewari, S. Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment. J. Phys.: Condens. Matter 25, 233201 (2013).
Kitaev, A. Faulttolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. NonAbelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).
Sato, M., Takahashi, Y. & Fujimoto, S. NonAbelian topological order in swave superfluids of ultracold fermionic atoms. Phys. Rev. Lett. 103, 020401 (2009).
Sau, J. D., Lutchyn, R. M., Tewari, S. & Das Sarma, S. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).
Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductorsuperconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).
Alicea, J. Majorana fermions in a tunable semiconductor device. Phys. Rev. B 81, 125318 (2010).
Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
Brouwer, P. W., Duckheim, M., Romito, A. & von Oppen, F. Topological superconducting phases in disordered quantum wires with strong spinorbit coupling. Phys. Rev. B 84, 144526 (2011).
Potter, A. C. & Lee, P. A. Majorana end states in multiband microstructures with Rashba spinorbit coupling. Phys. Rev. B 83, 094525 (2011).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Teo, J. C. Y. & Kane, C. L. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010).
Kitaev, A. Unpaired Majorana fermions in quantum wires. Phys.Uspekhi 44, 131 (2001).
Kwon, H., Sengupta, K. & Yakovenko, V. Fractional ac Josephson effect in p and dwave superconductors. Eur. Phys. J. B 37, 349 (2004).
Law, K. T., Lee, P. A. & Ng, T. K. Majorana fermion induced resonant Andreev reflection. Phys. Rev. Lett. 103, 237001 (2009).
Wimmer, M., Akhmerov, A. R., Dahlhaus, J. P. & Beenakker, C. W. J. Quantum point contact as a probe of a topological superconductor. New J. Phys. 13, 053016 (2011).
Nilsson, J., Akhmerov, A. R. & Beenakker, C. W. J. Splitting of a Cooper pair by a pair of Majorana bound states. Phys. Rev. Lett. 101, 120403 (2008).
Liu, J., Zhang, F.C. & Law, K. T. Majorana fermion induced nonlocal current correlations in spinorbit coupled superconducting wires. Phys. Rev. B 88, 064509 (2013).
Tewari, S. & Sau, J. Topological invariants for spinorbit coupled superconductor nanowires. Phys. Rev. Lett. 109, 150408 (2012).
Wong, C. L. M. & Law, K. T. Majorana Kramers doublets in d x2y2 wave superconductors with Rashba spinorbit coupling. Phys. Rev. B 86, 184516 (2012).
Sato, M., Tanaka, Y., Yada, K. & Yokoyama, T. Topology of Andreev bound states with flat dispersion. Phys. Rev. B 83, 224511 (2011).
Liu, X.J., Liu, Z.X. & Cheng, M. Manipulating topological edge spins in onedimensional optical lattice. Phys. Rev. Lett. 110, 076401 (2013).
Nakosai, S., Budich, J. C., Tanaka, Y., Trauzettel, B. & Nagaosa, N. Majorana bound states and nonlocal spin correlations in a quantum wire on an unconventional superconductor. Phys. Rev. Lett. 110, 117002 (2013).
Zhang, F. a. n., Kane, C. L. & Mele, E. J. Time reversal invariant topological superconductivity and Majorana Kramers pairs. Phys. Rev. Lett. 111, 056402 (2013).
Liu, X.J., Wong, C. L. M. & Law, K. T. NonAbelian Majorana Doublets in TimeReversal Invariant Topological Superconductor. http://arxiv.org/abs/1304.3765 (2013).
Keselman, A., Fu, L., Stern, A. & Berg, E. Inducing time reversal invariant topological superconductivity and fermion parity pumping in quantum wires. Phys. Rev. Lett. 111, 116402 (2013).
Lee, P. A. & Fisher, D. S. Anderson localization in two dimensions. Phys. Rev. Lett. 47, 882 (1981).
Fisher, D. S. & Lee, P. A. Relation between conductivity and transmission matrix. Phys. Rev. B 23, 6851 (1981).
Sun, Q. F. & Xie, X. C. Quantum transport through a graphene nanoribbonsuperconductor junction. J. Phys. Condens. Matter. 21, 344204 (2009).
Chang, C.Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
Qi, X.L., Hughes, T. L. & Zhang, S. C. Chiral topological superconductor from the quantum Hall state. Phys. Rev. B 82, 184516 (2010).
Ii, A., Yamakage, A., Yada, K., Sato, M. & Tanaka, Y. Theory of tunneling spectroscopy for chiral topological superconductors. Phys. Rev. B 86, 174512 (2012).
Diez, M., Dahlhaus, J. P., Wimmer, M. & Beenakker, C. W. J. Andreev reflection from a topological superconductor with chiral symmetry. Phys. Rev. B 86, 094501 (2012).
Ioselevich, P. A. & Feigel'man, M. V. Tunneling conductance due to a discrete spectrum of Andreev states. New J. Phys. 15, 055011 (2013).
Recher, P., Sukhorukov, E. V. & Loss, D. Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spinentangled electrons. Phys. Rev. B 63, 165314 (2001).
Schindele, J., Baumgartner, A. & Schoenberger, C. Nearunity Cooper pair splitting efficiency. Phys. Rev. Lett. 109, 157002 (2012).
Das, A., Ronen, Y., Heiblum, M., Mahalu, D., Kretinin, A. V. & Shtrikman, H. Highefficiency Cooper pair splitting demonstrated by twoparticle conductance resonance and positive noise crosscorrelation. Nat. Commun. 3, 1165 (2012).
Braunecker, B., Burset, P. & Yeyati, A. Entanglement detection from conductance measurements in carbon nanotube Cooper pair splitters. Phys. Rev. Lett. 111, 136806 (2013).
Yamakage, A. & Sato, M. Interference of Majorana fermions in NS junctions. Physica E 55, 13–19 (2014).
Acknowledgements
We thank Masatoshi Sato, Keiji Yada and Yi Zhou for discussion. K.T.L. is indebted to Tai Kai Ng for insightful discussions and his encouragements throughout this project. K.T.L. acknowledges the support of HKRGC through Grant 605512, Grant 602813 and HKUST3/CRF09. Y.T. thanks the support of MEXT of Japan through Grant 22103005 and Grant 2065403. After the submission of this work, we note that there is an independent work by A. Yamakage and M. Sato on the study of the suppression of the local ARs of the N_{Chern}=2 phase^{46}.
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J.J.H. and J.W. are involved in the analytic and numerical calculations. T.P.C. and X.J.L. are involved in analysing the models. Y.T. and K.T.L. initiated and supervised the project. K.T.L. conceived the ideas of this paper and prepared the manuscript with contributions from all the authors.
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He, J., Wu, J., Choy, TP. et al. Correlated spin currents generated by resonantcrossed Andreev reflections in topological superconductors. Nat Commun 5, 3232 (2014). https://doi.org/10.1038/ncomms4232
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DOI: https://doi.org/10.1038/ncomms4232
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