Topological Superconductivity and Surface Andreev Bound States in Doped Semiconductors: Application to CuxBi2Se3
TTopological Superconductivity and Surface Andreev Bound States in DopedSemiconductors: Application to Cu x Bi Se Timothy H. Hsieh
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
Liang Fu
Department of Physics, Harvard University, Cambridge, MA 02138
The recently discovered superconductor Cu x Bi Se [1] is a candidate for three-dimensional time-reversal-invariant topological superconductors[2], which are predicted to have robust surface Andreevbound states hosting massless Majorana fermions. In this work, we present an analytical andnumerical study of the surface Andreev bound state wavefunction and dispersion. We find thetopologically protected Majorana fermions at k = 0, as well as a new type of surface Andreev boundstates at finite k . We relate our results to a recent point-contact spectroscopy experiment[3]. PACS numbers: 74.20.Rp, 73.43.-f, 74.20.Mn, 74.45.+c
The discovery of topological insulators has generatedmuch interest in not only understanding their proper-ties and potential applications to spintronics and thermo-electrics but also searching for related topological phasesin new directions. A particularly exciting avenue istopological superconductivity[2, 4–13], in which uncon-ventional pairing symmetries generate nontrivial super-conducting gaps in a similar way that spin-orbit cou-pling generates inverted band gaps for topological insu-lators. The hallmark of a topological superconductor isthe existence of gapless surface Andreev bound states(SABS) hosting itinerant neutral Bogoliubov quasiparti-cles, which are the analog of massless Majorana fermionsin high energy physics.There is currently an intensive search for topologi-cal superconductors. In particular, a recently discov-ered superconductor Cu x Bi Se with T c ∼ K has at-tracted much attention[1]. It was proposed that thestrong spin-orbit coupled band structure of Cu x Bi Se may favor an unconventional odd-parity pairing symme-try, which naturally leads to a time-reversal-invarianttopological superconductor[2]. Subsequently, many ex-perimental and theoretical efforts[15–19] have been madetowards understanding superconductivity in Cu x Bi Se .In a very recent point-contact spectroscopy experiment,Sasaki et al. [3] have observed a zero-bias conductancepeak which is attributed to SABS and seems to signifyunconventional pairing[14].Motivated by this finding, in this Letter we study thephase diagram of odd-parity topological superconductiv-ity and the resulting surface Andreev bound states indoped semiconductors with strong spin-orbit coupling, ofwhich Cu x Bi Se is a prime candidate. We start from a k · p Hamiltonian which captures the essential features ofits band structure near the Fermi surface. By studyingthe phase diagram of the k · p Hamiltonian as a func-tion of band gap, pairing potential, and doping, we es-tablish three gapped phases: topological superconductor
FIG. 1: a) Side view of a semi-infinite crystal of Bi Se . Thetwo relevant p z orbitals are shown in the zoom-in view of theQL unit cell. b) Bulk and surface bands of the tight-bindingmodel for Bi Se . µ and µ denote two chemical potentialswhere the surface states have, respectively, not merged andmerged into the bulk bands. (TSC), topological insulator (TI), and normal band in-sulator (BI). We characterize these phases in a unifiedway by introducing a topological invariant—a general-ized mirror Chern number. We find that the odd-paritytopological superconductivity in both doped BI and TIgives rise to surface Majorana fermions with a linear dis-persion at k = 0, as expected from the bulk-boundarycorrespondence. However, the SABS of the two becomequite different at large k . In particular, we infer fromthe mirror Chern number that the SABS in a supercon-ducting doped TI must become gapless again near theFermi momentum. This results in a new type of zero-energy SABS. To support these findings, we constructa two-orbital tight-binding model to calculate the SABSdispersion numerically. Finally we relate these results tothe recent experiment[3]. Band Structure:
We begin by reviewing the crystalstructure of Bi Se and the k · p Hamiltonian for its bandstructure. Bi Se is a rhombohedral crystal in which each a r X i v : . [ c ond - m a t . s up r- c on ] O c t unit cell is a quintuple layer (QL) of the form Se(A)-Bi(B)-Se(C)-Bi(A)-Se(B). Here A, B, C denote three tri-angular lattices stacked with offset along the (111) direc-tion, which we will define as the z -axis. Undoped Bi Se is a topological insulator. Cu doping introduces elec-tron carriers into the conduction band with a density of ∼ cm − , which results in a small 3D Fermi surfacecentered at Γ. The band structure near Γ is described bya k · p Hamiltonian[2, 20]: H ( k ) = mσ x + v z k z σ y + vσ z ( k x s y − k y s x ) . (1)Here σ z = ± p z orbitals on the upper and lowerpart of the QL unit cell respectively (see Fig.1), and eachhas a two-fold spin degeneracy labeled by s z = ±
1. Asexplained in Ref.[2], the form of H ( k ) can be deducedentirely by time reversal and crystal symmetry consider-ations. The physical origin of various terms in (1) willbecome clear from an explicit tight-binding Hamiltonianlater. While the magnitudes of the parameters m, v z and v in H ( k ) have been obtained by fitting the band disper-sion with the angle-resolved photoemission spectroscopymeasurement[15], their signs turn out to be particularlyimportant in this work. Phase Diagram:
Now consider the superconductingCu x Bi Se with the spin triplet, orbital singlet pairingproposed in Ref.[2]. The mean-field Hamiltonian is givenby H MF = (cid:90) d k [ c † k , ¯ c − k ] H BdG ( k ) (cid:20) c k ¯ c †− k (cid:21) ,c † k = ( c † k ↑ , c † k ↓ ) , ¯ c k = ( c k ↓ , − c k ↑ ) H BdG ( k ) = ( H ( k ) − µ ) τ z + ∆ σ y s z τ x . (2)Here H BdG is the Bogoliubov-de Gennes (BdG) Hamil-tonian; τ x,z are Pauli matrices in Nambu space; ∆ is thepairing potential; µ is chemical potential.The Hamiltonian (2) exhibits three topologically dis-tinct gapped phases as a function of the band gap, pairingpotential and doping. At zero doping ( µ = 0) and in theabsence of superconductivity (∆ = 0), the system is ei-ther an ordinary BI or a TI, depending on the sign of m . At finite electron doping, the chemical potential liesinside the conduction band: µ >
0. When the odd-paritypairing ∆ occurs in such a doped BI or TI, the system be-comes a fully gapped TSC. For the sake of our argument,it is useful to note that the TSC phase is adiabaticallyconnected to the µ = 0 and ∆ > | m | limit. Fig.2 showsthe BI, TI and TSC phases in the µ = 0 phase diagram asa function of m and ∆. The topological phase transitionbetween TSC and BI/TI occurs at ∆ = ± m .To characterize these three phases in a unified way,we introduce a common topological invariant—a gener-alized “mirror Chern number” n M . Recall that reflection M with respect to the yz mirror plane is a symmetry el- m Δ TI BITSC ( n M = 0)( n M = 2)( n M = 1) FIG. 2: Phase diagram of fully-gapped odd-parity supercon-ductivity in doped semiconductors as a function of band gap m and pairing potential ∆, showing three gapped phases:band insulator, topological insulator and topological super-conductor. They are topologically distinguished by the mirrorChern numbers n M . ement of the Bi Se crystal point group D d . Under mir-ror reflection, the band structure (1) is invariant, whereasthe pairing order parameter in (2) changes sign. (Mirrorreflection flips s y and s z .) This sign change can be com-pensated by a gauge transformation ∆ → − ∆, so thatthe BdG Hamiltonian is invariant under a generalizedmirror operation ˜ M = M τ z , M = − is x : H BdG ( k x , k y , k z ) = ˜ M H
BdG ( − k x , k y , k z ) ˜ M − (3)In particular, since H BdG in the k x = 0 plane commuteswith ˜ M , a mirror Chern number n M can be definedin the same way as for an insulator[21]. Specifically, h ≡ H BdG ( k x = 0) is a direct sum of two subsystems h ± with mirror eigenvalues ± i respectively: h ± = P ± h ,where P ± ≡ (1 ∓ i ˜ M ) /
2. The Chern number n ± for h ± satisfies n + + n − = 0 required by time reversal symmetry.However, the difference defines the mirror Chern number: n M ≡ ( n + − n − ) / n + . Using n M (BI) = 0 as a reference, we obtain the mirrorChern number for the TI and TSC by calculating thechange of n M across the phase transition to the BI. Dueto the double counting of electrons and holes, the mirrorChern number of an insulator defined in Nambu spaceis twice the value of that defined previously[21]. As aresult, a direct transition from TI to BI at ∆ = 0 changes n M by two. For ∆ (cid:54) = 0, this transition is split into twotransitions with an intermediate TSC phase, so that eachtransition changes n M by one. Therefore we have n M (TI) = 2 n M (TSC) . (4)The fact that TI and TSC have mirror Chern numbersof the same sign will play a key role in our analysis ofgapless excitations on their surfaces. Surface States : Consider a semi-infinite Bi Se crys-tal occupying z <
0, which is naturally cleaved betweenQLs (see Fig.1). The boundary condition correspondingto such a termination in k · p theory is σ z ψ ( z = 0) = ψ ( z = 0) . (5)As explained in Ref[2], this boundary condition reflectsthe vanishing of the wavefunction on the bottom layer( σ z = −
1) at z = 0, whereas the other termination withinthe QL (not yet reported in experiments) corresponds toa different boundary condition σ z ψ = − ψ at z = 0[22].First we study surface states of Bi Se by solving thedifferential equation H ( k x , k y , − i∂ z ) ψ = Eψ subject tothe boundary condition (5). We obtain two sets of exactsolutions ψ ± ( k (cid:107) , z ) with the energy-momentum disper-sion E ± ( k ) = ± vk . ψ ± ( k (cid:107) , z ) = e z/l (1 , σ ⊗ (1 , ± ie iφ ) s , (6)where l = − v z /m is the decay length; φ is the azimuthalangle of k (cid:107) ; the subscripts σ and s denote the orbital σ z and spin s z basis. In order for the surface states to existin a TI, we must have decaying solutions in the − z di-rection, which implies v z m <
0. The spin polarization of ψ ± is locked to its momentum, forming a two-dimensionalDirac cone. In contrast, a BI has v z m > x Bi Se . We start by solving the BdG equation at k (cid:107) = 0. A Kramers pair of zero-energy eigenstates ψ α = ± ( z ) with mirror eigenvalues ˜ M = i · α is expectedfrom the topology and symmetry of H BdG ( k (cid:107) = 0). ψ α ( z ) satisfies a reduced 4-component equation:[( mσ x − iv z σ y ∂ z − µ ) τ z + ∆ σ y τ x ] ψ α ( z ) = 0 . (7)By multiplying both sides of Eq.(7) with τ z , it becomesclear that ψ α is an eigenstate of τ y . The correspondingeigenvalue is given by sgn( v z ) in order to have a decay-ing solution. Eq.(7) then reduces to a two-componentequation in orbital space, which has two independent so-lutions: ξ ± ( z ) = (1 , e ± iθ ) σ · e ± ik F z + κz . (8)Here k F is the Fermi momentum in the z direction, givenby k F = (cid:112) µ − m /v z ; κ is the inverse decay lengthgiven by κ = ∆ / | v z | > θ is an angle defined by e iθ = ( m + i (cid:112) µ − m ) /µ . We now choose a suitablelinear combination of ξ + and ξ − to satisfy the boundarycondition (5) and obtain the wavefunction of SABS: ψ α ( z ) ∝ e κz (sin( k F z − θ ) , sin( k F z )) σ ⊗ [(1 , − α ) s ⊗ (1 , τ + i sgn( v z )(1 , α ) s ⊗ (0 , τ ] , where τ = ± ψ s ( z ) isparticle-hole symmetric: Ξ ψ s ( z ) = ψ s ( z ) up to an overallphase (Ξ = s y τ y K ), and therefore represents a Majoranafermion. In the limit m = 0, ψ α agrees with the resultfrom Ref.[2].Away from k = 0, the Kramers doublet ψ and ψ − issplit by spin-orbit coupling term in (1). To lowest orderin k , the SABS dispersion (cid:15) α ( k ) is linear and given by (cid:15) α ( k ) = α ˜ vk . The velocity ˜ v is obtained from first-orderperturbation theory:˜ v = v ∆ + sgn( v z )∆ m ∆ + sgn( v z )∆ m + µ (9)Since ∆ (cid:28) | m | < µ in weak-coupling superconductors(provided m is not vanishingly small), (9) simplifies to˜ v (cid:39) v · sgn( v z )∆ m/µ . (10)While the SABS in previous studies of single-band su-perconductors can often be obtained by quasi-classicalmethods from the geometry of the Fermi surface alone,this is not the case for multi-orbital systems such asCu x Bi Se . Here both the wavefunction and dispersionof SABS depend on the dimensionless factor m/µ , whicharises from the orbital character of electron wavefunc-tions on the Fermi surface.To deduce the behavior of SABS dispersion for large k , we make use of the mirror Chern number introducedearlier. According to the principle of bulk-boundary cor-respondence, the sign of mirror Chern number n M deter-mines the helicity of surface states as follows. n M < k x = 0 with mir-ror eigenvalues ∓ i (i.e., s x = ±
1) moves (anti-)clockwisewith respect to + x axis at the edge of the yz plane, whichmeans that the electron’s spin and angular momentumfrom circulation at the edge are parallel. Such a helic-ity forces the dispersion of the s x = +1 surface bandto eventually merge into the E > k .Given this correspondence between the mirror Chernnumber and the helicity, the SABS in the TSC must havethe same helicity as the TI surface states, because themirror Chern numbers (4) of the two phases have thesame sign. Moreover, the helicity of Bi Se surface statesis directly given by the sign of Dirac velocity v (since thedispersion is monotonic). Comparing the SABS velocity˜ v in (10) with v , we find that sgn(˜ v ) = sgn( v ) for TSCin a doped BI ( v z m > v ) = − sgn( v ) forTSC in a doped TI ( v z m < E = 0 once again at some finite k (or an odd number oftimes). As long as perturbations do not change the signof ˜ v at k = 0, the second crossing is robust due to themirror Chern number argument. Similar reasoning waspreviously applied to surface states in TIs[21, 23, 24].To demonstrate the SABS explicitly, we construct atwo-orbital tight-binding model in the rhombohedral lat-tice shown in Fig.1 and calculate the SABS dispersionnumerically. The Hamiltonian is defined as follows: H = H + H + H soc + H (cid:48) . (11) H = (cid:80)
0, Fig.3a-b) and doped BI ( m >
0, Fig.3c). TheSABS in both cases have a branch of linearly dispers-ing Majorana fermion at k = 0, which signifies a three-dimensional topological superconductor. Moreover, theSABS in doped TI has a “twisted” dispersion with a sec-ond crossing near Fermi momentum, as anticipated fromour earlier mirror Chern number argument.The tight-binding calculation sheds light on the originof twisted SABS. The presence of the second crossing isa remnant of the TI surface states in the normal state.This can be easily understood in the case where electronsurface states remain well-defined at the Fermi energyin the normal state, e.g, at the chemical potential µ in Fig.1c. The surface states at k and − k have oppo-site mirror eigenvalues, whereas ∆ only pairs states withthe same mirror eigenvalues. Due to this symmetry in-compatibility, surface states must remain gapless even inthe presence of such an odd-parity pairing. Therefore,the Majorana fermion SABS at k = 0 smoothly evolvesinto TI surface states near Fermi momentum. Indeed wefind that the decay length of SABS at and beyond thesecond crossing is comparable to that of the electron sur-face state given by | v z | / | m | , which is much shorter thanthe typical decay length of SABS given by | v z | / ∆. Thisis further confirmed by an analytic calculation using the k · p BdG Hamiltonian (2). We search for an E = 0 solu-tion at a nonzero k and find k is the nontrivial solutionof 2Re( W ) + mE F ( − | W | ) = 0 , (12)where W ≡ vk − i (∆ + iE F ) (cid:112) ( vk ) + (∆ + iE F ) . (13)For ∆ = 0 and m <
0, the solution is given by k = µ/v ,which represents the TI surface state at Fermi energy.For ∆ (cid:54) = 0, the solution is perturbed: k = µv (1 − ∆ m )to leading order in ∆. No solution exists for m > no surface states at theFermi energy in the normal state of a doped TI? Forexample, at the chemical potential µ in Fig.1c, the TIsurface states have merged into the bulk[25]. Remark-ably, after turning on the odd-parity pairing (Fig.3b),the SABS still has the second crossing, as required bythe mirror Chern number. The resulting gapless SABSnear Fermi momentum have significant particle-hole mix-ing and therefore cannot be interpreted as unpaired TIsurface states. This represents a new type of surface An-dreev bound state, which originates from the interplaybetween band structure and unconventional supercon-ductivity. Such SABS defy a conventional quasi-classicaldescription and are worth further study.Finally, we relate our findings to the recent point-contact spectroscopy experiment on Cu x Bi Se [3]. Azero-bias conductance peak with a width of 0 . . k along all in-plane directionsmake the density of states at zero energy finite, in sharpcontrast to the vanishing density of states at k = 0 char-acteristic of the linear dispersion in 2D as found in adoped BI (Fig. 3c). Moreover, for small m/µ the SABSdispersion is quite flat as shown in Fig.3b and 3d. Thesefeatures may naturally explain the experimental obser-vations, and will be studied in a future work.While the main focus of this Letter is Cu x Bi Se , weend by discussing the applicability of our findings toa large class of superconducting doped semiconductorswith inversion symmetry. Potential candidates includeBi Te [30] and TlBiTe [31] under pressure, PbTe[32],SnTe[33], and GeTe[34]. Provided that the Fermi sur-face is centered at time-reversal-invariant momenta, theDirac-type relativistic k · p Hamiltonian gives a valid low-energy description of its band structure[26]. Moreover,if the pairing is odd under spatial inversion and fullygapped, the system is (almost) guaranteed to be a topo-logical superconductor according to our criterion[2, 29].In that case, our findings of unusual surface Andreevbound states apply directly. In addition, if the pairingsymmetry of noncentrosymmetric superconductors (suchas YPtBi[35]) has a dominant odd-parity component,their properties are often inherited from the centrosym-metric limit. We hope this work will encourage furtherexplorations of unconventional surface Andreev boundstates and stimulate the search for topological supercon-ductors.
Note:
Two recent studies [3, 19] have calculated thesurface spectral function numerically in Cu x Bi Se tight-binding models different from ours. The connection toour work on Majorana fermion surface Andreev boundstate remains to be understood. Acknowledgement : We thank Erez Berg and Yang Qifor helpful discussions, as well as Yoichi Ando, AntonAkhmerov, and David Vanderbilt for helpful commentson the manuscript. TH is supported by the U.S. De-partment of Energy under cooperative research agree-ment Contract Number DE-FG02-05ER41360 and theNational Science Foundation Graduate Research Fellow-ship under Grant No. 0645960. LF would like tothank Physics Department at MIT, Institute of Physicsin China, and Institute of Advanced Study at TsinghuaUniversity for generous hosting, as well as the Harvard Society of Fellows for support. [1] Y. Hor et al , Phys. Rev. Lett. , 057001 (2010)[2] L. Fu and E. Berg, Phys. Rev. Lett. , 097001 (2010).[3] S. Sasaki et. al. , arXiv:1108.1101[4] A. Schynder, S. Ryu, A. Furusaki and A. Ludwig, Phys.Rev. B , 195125 (2008).[5] A. Kitaev, arXiv:0901.2686[6] N. Read and D. Green, Phys. Rev. B , 10267 (2000).[7] R. Roy, arXiv:0803.2868[8] X. L. Qi, T. L. Hughes, S. C. Zhang, Phys. Rev.Lett. , 187001 (2009); Phys. Rev. B , 134508 (2010).[9] M. M. Salomaa and G. E. Volovik, Phys. Rev. B , 9298(1988); M. A. Silaev, G. E. Volovik, J. Low Temp. Phys., , 460 (2010).[10] M. Sato, Phys. Rev. B , 220504(R) (2010).[11] S. K. Yip, J. Low Temp. Phys., , 12 (2010).[12] S. Ryu, J. E. Moore and A. Ludwig, arXiv:1010.0936[13] K. Nomura, S. Ryu, A. Furusaki, and N. Nagaosa,arXiv:1108.5054[14] For reviews on surface Andreev bound states in un-conventional superconductors, see S. Kashiwaya and Y.Tanaka, Rep. Prog. Phys. , 1641 (2000); G. Deutscher,Rev. Mod. Phys. , 109 (2005).[15] L. A. Wray et al , Nature Physics, , 855 (2010); Phys.Rev. B, , 224516 (2011).[16] M. Kriener et al , Phys. Rev. Lett. , 127004 (2011).[17] M. Kriener, et al , Phys. Rev. B , 054513 (2011).[18] P. Das et al , Phys. Rev. B , 220513(R) 2011).[19] L. Hao and T. K. Lee, Phys. Rev. B , 134516 (2011)[20] C. X. Liu et al , Phys. Rev. B , 045122 (2010). The k · p Hamiltonian in this work takes the same form as Ref.[2]after a change of basis which interchanges σ x and σ z .[21] J. C. Y. Teo, L. Fu, and C. L. Kane. Phys Rev. B, ,045426. (2008).[22] In contrast, the boundary condition in Ref.[20] does notdifferentiate the two surface terminations.[23] D. Hsieh, et al , Science , 919 (2009).[24] R. Takahashi and S. Murakami, arXiv:1105.5209[25] Surface states merge into the bulk at some large k forwhich the k.p approximation is no longer valid.[26] L. Fu and C. L. Kane, Phys. Rev. B , 045302 (2007).[27] For a general discussion of boundary conditions for Diracmaterials, see A. R. Akhmerov and C. W. J. Beenakker,Phys. Rev. B , 085423 (2008).[28] The tight-binding model in its present form has a defi-ciency at k (cid:29) k F . We thank David Vanderbilt for point-ing this out.[29] Y. Qi and L. Fu, to be published[30] J. L. Zhang et al , PNAS, , 24 (2011).[31] R. A. Hein and E. M. Swiggard, Phys. Rev. Lett. , 53(1970).[32] Y. Matsushita et al , Phys. Rev. B , 134512 (2006).[33] R. Hein, Physics Letters , 435 (1966).[34] R. A. Hein et al , Phys. Rev. Lett. , 320 (1964).[35] N. P. Butch et alet al