Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
We study the proximity effect between an -wave superconductor and the surface states of a strong topological insulator. The resulting two dimensional state resembles a spinless superconductor, but does not break time reversal symmetry. This state supports Majorana bound states at vortices. We show that linear junctions between superconductors mediated by the topological insulator form a non chiral 1 dimensional wire for Majorana fermions, and that circuits formed from these junctions provide a method for creating, manipulating and fusing Majorana bound states.
pacs:71.10.Pm, 74.45.+c, 03.67.Lx, 74.90.+n
Excitations with non-Abelian statisticsmooreread () are the basis for the intriguing proposal of topological quantum computationkitaev (). The simplest non-Abelian excitation is the zero energy Majorana bound state (MBS) associated with a vortex in a spinless superconductorreadgreen (); ivanov (); stern (); stone (). The presence of vortices leads to a fold ground state degeneracy. Braiding processes, in which the vortices are adiabatically rearranged, perform non trivial operations in that degenerate space. Though MBSs do not have the structure necessary to construct a universal quantum computerfriedman (), the quantum information encoded in their degenerate states is topologically protected from local sources of decoherencekitaev3 ().
MBSs have been proposed to exist as quasiparticle excitations of the quantum Hall effectmooreread (); readgreen (), in the cores of vortices in the -wave superconductor SrRuOdassarma () and in cold atomsgurarie (); tewari (). In this paper we show that the proximity effect between an ordinary -wave superconductor and the surface of a strong topological insulator (TI)fkm (); moore (); roy (); fukane () leads to a state which hosts MBSs at vortices. We then show that a linear superconductor - TI - superconductor (STIS) junction forms a non chiral 1D wire for Majorana fermions. Such junctions can be combined into circuits, which allow for the creation, manipulation and fusion of MBSs.
A strong TI is a material with an insulating time reversal invariant bandstructure for which strong spin orbit interactions lead to an inversion of the band gap at an odd number of time reversed pairs of points in the Brillouin zone. Candidate materials include the semiconducting alloy BiSb, as well as HgTe and -Sn under uniaxial strainfukane (). Strong TIs are distinguished from ordinary insulators by the presence of surface states, whose Fermi arc encloses an odd number of Dirac points and is associated with a Berry’s phase of . In the simplest case, there is a single non degenerate Fermi arc described by the time reversal invariant Hamiltonian
Here are electron field operators, are Pauli spin matrices and is the chemical potential. can only exist on a surface because it violates the fermion doubling theoremnielson (). The topological metal is essentially half of an ordinary 2D electron gas.
Suppose that an s-wave superconductor is deposited on the surface. Due to the proximity effect, Cooper pairs can tunnel into the surface states. This can be described by adding to , where depends on the phase of the superconductor and the nature of the interfacevolkov (). The states of the surface can then be described by , where in the Nambu notation and
are Pauli matrices that mix the and blocks of . Time reversal invariance follows from , where and is complex conjugation. Particle hole symmetry is expressed by , which satisfies . When is spatially homogeneous, the excitation spectrum is . For , the low energy spectrum resembles that of a spinless superconductor. This analogy can be made precise by defining for and . The projected Hamiltonian is then . Though this is formally equivalant to a spinless superconductor there is an important difference: respects time reversal symmetry, while the superconductor does not.
It is well known that a vortex in a superconductor leads to a MBSreadgreen (). This suggests that for a similar bound state should exist for (2). The bound states at a vortex are determined by solving the Bogoliubov de Gennes (BdG) equation in polar coordinates with . A zero energy solution exists for any . The algebra is simplest for , where the zero mode has the form
with and .
Another feature of superconductors is the presence of chiral edge states readgreen (); buchholtz (); sigrist (). With time reversal symmetry, chiral edge states can not occur in our system. The surface - which itself is the boundary of a three dimensional crystal - can not have a boundary. By breaking time reversal symmetry, however, a Zeeman field can introduce a mass term into (1,2) which can open an insulating gap in the surface state spectrum. By solving (2) we find that the interface between this insulating state and the superconducting state has chiral Majorana edge states. This could possibly be realized by depositing superconducting and insulating magnetic materials on the surface to form a superconductor-TI-magnet (STIM) junction. It is interesting to note that for spinless electrons the superconductor violates time reversal, while the vacuum does not. For our surface states it is the insulator which violates time reversal. A related effect could also occur at the edge of a two dimensional TIkm (); murakami (); bhz (), which is described by (1,2) restricted to one spatial dimension. At the boundary between a region with superconducting gap and a region with insulating gap we find a MBS, analogous to the end states discussed in Refs. kitaev2, ; semenoff, . In the following we will focus on STIS junctions, which can lead to non chiral one dimensional Majorana fermions, as well as MBSs.
Consider a line junction of width and length between two superconductors with phases and in contact with TI surface states. We analyze the Andreev bound states in the surface state channel between the superconductors by solving the BdG equation with for , for and otherwise. The calculation is similar to Titov, Ossipov and Beenakker’sbeenakker () analysis of graphene SNS junctions, except for the important difference that graphene has four independent Dirac points, while we only have one. For there are two branches of bound states, which disperse with the momentum in the direction. For we find
For the spectrum is gapless. It is useful to construct a low energy theory, for and . Finite and can then easily be included. We first solve the BdG equation for the two modes at and . It is useful to choose them to satisfy . Up to a normalization they may be written
We next evaluate and to obtain the “” Hamiltonian,
where and . The Pauli matrices act on and are different from those in (2). In this basis and . resembles the Su Schrieffer Heeger (SSH) modelssh (). However, unlike that model, the states are not independent, and the corresponding Bogoliubov quasiparticle operators satisfy . The system is thus half a regular 1D Fermi gas, or a non chiral “Majorana quantum wire”.
Below it will be useful to consider junctions that bend and close. When a line junction makes an angle with the axis the basis vectors (5) are modified according to . , however, is unchanged even when varies. On a circle, changes sign when advances by . Therefore, eigenstates of must obey antiperiodic boundary conditions, .
Next consider a tri-junction, where three superconductors separated by line junctions meet at a point, as in Fig. 1c. When is in the shaded region of Fig. 1d, a MBS exists at the junction. Though the general BdG equation cannot be solved analytically, this phase diagram can be deduced by solving special limits. When there is no bound state. Another solvable limit is when three line junctions with are oriented at 120, and . This is a discrete analog of a vortex with symmetry, and is indicated by the circles in Fig. 1d. For we find a MBS identical to (3) with the exponent replaced by . Here is a constant unit vector in each superconductor that bisects the angle between neighboring junctions. The MBS can not disappear when are changed continuously unless the energy gap closes. The phase boundaries indicated in Fig. 1d therefore follow from the solution of the line junction, and occur when the phase difference between neighboring superconductors is .
It is instructive to consider the limit where two of the lines entering the tri-junction are nearly gapless. For and Fig. 1d predicts a MBS when . This can be understood with Eq. 6, which describes the lower two line junctions, which have masses . When changes sign, leading to the well known midgap state of the SSH modelssh (); jackiw (), which in the present context is a MBS.
A line junction terminated by two tri-junctions allows MBSs to be created, manipulated and fused. When passes through MBSs appear or disappear at both ends. To model this we assume the phases of the superconductors on either side of the line junction are and , and that the superconductors at the left (right) ends have phases , which are not close to or . This allows us to model the ends using a hard wall boundary condition , where the sign at each end is . It is straightforward to solve (6) to determine the spectrum as a function of for a line of length using this boundary condition. There are two cases depending on the sign of .
For either zero or a pair of MBSs are expected. The spectrum, shown in Fig. 2a, may be written , where are solutions to . Midgap states are present for . For a pair of zero energy states are localized at each end with wavefunctions
where are given in (5). For finite the eigenstates are , with energies . These define Bogoliubov quasiparticle operators, . Since it follows that where are Majorana operators. The pair thus define a two state Hilbert space indexed by . The splitting between then characterizes the interaction between the MBSs,
The case is similar. Eq. 8 applies to both cases, provided is associated with the vortex.
This provides a method for both creating and fusing pairs of MBSs. Suppose we begin in the ground state at with no MBSs present. Upon adiabatically decreasing through MBSs appear in the state . Next suppose that initially , and a pair of MBSs are present in the state . When is adiabatically increased through the system will remain in , which will either evolve to the ground state or to a state with one extra fermion. The difference between the two states can be probed by measuring the current flowing across the linear junction, which depends on whether the Andreev bound state is occupied. The measured current will be , where the current carried by is for . For meV nA.
Finally, consider the case , in which a MBS is at one end or the other, as in Fig. 2b. There are plane wave solutions with energy for , along with a single state with wavefunction
Depending on the sign of , is exponentially localized at one end or the other. When changes sign, the MBS smoothly switches sides. This provides a method for transporting a MBS from one node to another.
We now discuss simple circuits built from STIS junctions. First, consider Fig. 3a and a process in which the phase of the central island is adiabatically advanced from to . For there are no MBSs. At two pairs of MBSs are created at the top and bottom line junctions. At the MBSs are fused at the left and right line junctions. If the system begins in the ground state , then when we findkitaev3 ()
Thus, after the cycle, the left and right segments are in an entangled state. The currents measured across the left and right junctions will be with 50% probability and will be perfectly correlated.
Eq. 10 can be understood in two ways. First, the cycle effectively creates two pairs of MBS’s, interchanges a pair (say 2 and 4) and brings the pairs back together. As shown by Ivanov ivanov (), this corresponds to the operator , which leads directly to (10). Alternatively, this result can be derived from (6,8). From (8), the Hamiltonian for is . For it becomes . Here the minus sign arises because, as explained after Eq. 6, the closed 1D circuit must have antiperiodic boundary conditions. Thus, one of the line junctions must have a cut where the wavefunction changes sign. We chose the cut to be on the junction between 2 and 3. It is then straightforward to express the groundstate of in terms of the eigenstates of , which leads directly to (10).
Fig. 3b gives a geometry for interchanging MBSs without fusing them. For MBSs are located as shown. When advances by the MBSs hop counterclockwise three times and are interchanged. Ivanov’s rulesivanov () predict , . Again the minus sign can be understood in terms of the cut due to antiperiodic boundary conditions. One can imagine larger arrays, where this process performs elementary braiding operations.
The experimental implementation of this proposal will require progress on many fronts. The first is to find a strong TI with a robust gap. Bi Sb and strained HgTe can have gaps of order 30 meV fukane (). The next is to interface with an appropriate superconductor. depends on the quality of the interface, Schottky barriers and the mismatch in the Fermi wavelengthsvolkov (). If these can be optimized, can be comparable to the gap of the bulk superconductor chrestin ().
The simplest experimental geometry would be to consider a single line junction with . For .1 meV and eVÅ this could be achieved with m. This should be similar to a graphene SNS junctionbeenakker (). A signature of the Majorana character of the junction could be probed by measuring the thermal conductance along the channel for . For the central charge of the gapless Majorana modes leads to a quantized Landauer thermal conductance . By constructing a pair of tri-junctions as in Fig. 2 the presence of MBSs can be controlled. It would then be interesting to perform tests of the non locality of MBSs envisioned in Refs. semenoff, ; demler, .
Manipulating and fusing MBSs places more stringent requirements on the energy gaps. The junctions should be sufficiently short that , but sufficiently long that the MBSs are well localized. The good news is that varies as a power of , while the MBS overlap is exponential, so at low temperature both criteria can be achieved.
If the process of varying to manipulate the MBSs is non adiabatic or then additional quasiparticles could be excited. If those quasiparticles escape and interact with other MBSs, then the state of the MBSs will be disturbed. However, if the excited quasiparticles will be confined to the segment in which they were created. If is turned up so that , and the system relaxes back to its ground state, then the state of the MBSs will remain intact. Thus, if there is sufficient dynamic range between and , the system can tolerate these excitations.
We thank Sankar das Sarma and Steve Simon for helpful discussions. This work was supported by NSF grant DMR-0605066, and by ACS PRF grant 44776-AC10.
- (1) G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).
- (2) A. Kitaev, Ann. Phys. 303, 2 (2003).
- (3) N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
- (4) D.A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).
- (5) A. Stern, F. von Oppen, E. Mariani, Phys. Rev. B 70, 205338 (2004).
- (6) M. Stone and S.B. Chung, Phys. Rev. B 73, 014505 (2006).
- (7) M.H. Freedman, M. Larsen and Z. Wang, Commun. Math. Phys. 227, 605 (2002).
- (8) A. Kitaev, Ann. Phys. 321, 2 (2006).
- (9) S. Das Sarma, C. Nayak, S. Tewari, Phys. Rev. B 73, 220502(R) (2006).
- (10) V. Gurarie, L. Radzihovsky and A.V. Andreev, Phys. Rev. Lett. 94, 230403 (2005).
- (11) S. Tewari, et al., Phys. Rev. Lett. 98, 010506 (2007).
- (12) L. Fu, C.L. Kane and E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
- (13) J.E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) (2007).
- (14) R. Roy, arXiv:cond-mat/0607531 (unpublished).
- (15) L. Fu and C.L. Kane, Phys. Rev. B 76, 045302 (2007).
- (16) H. Nielssen and N. Ninomiya, Phys. Lett. 130B, 389 (1983).
- (17) A.F. Volkov et al., Physica C 242, 261 (1995).
- (18) L.J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 (1981).
- (19) M. Matsumoto and M. Sigrist, J. Phys. Soc. Jpn. 68, 994 (1999).
- (20) C.L. Kane and E.J. Mele Phys. Rev. Lett. 95 226801 (2005); ibid 95 146802 (2005).
- (21) S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).
- (22) A. Bernevig, T. Hughes and S.C. Zhang, Science 314, 1757 (2006).
- (23) A. Kitaev, ArXiv:cond-mat/0010440 (unpublished).
- (24) G. W. Semenoff and P. Sodano, ArXiv:cond-mat/0601261 (unpublished).
- (25) M. Titov and C.W.J. Beenakker, Phys. Rev. B 74, 041401(R) (2006); M. Titov, A. Ossipov and C.W.J. Beenakker, Phys. Rev. B 75, 045417 (2007).
- (26) W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. B 22, 2099 (1980).
- (27) R. Jackiw and C. Rebbi, Phys Rev. D 13, 3398 (1976).
- (28) A. Chrestin, T. Matsuyama and U. Merkt, Phys. Rev. B 55, 8457 (1997).
- (29) C. J. Bolech and E. Demler, Phys. Rev. Lett. 98, 237002 (2007).