In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an s-WaveSuperconductor Heterostructure

Ya-Jie Wu,1,2 Junpeng Hou,1 Yun-Mei Li,1 Xi-Wang Luo,1,* Xiaoyan Shi,1 and Chuanwei Zhang1,†1Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA

2School of Science, Xi’an Technological University, Xi’an 710032, China

(Received 6 July 2019; accepted 18 May 2020; published 2 June 2020)

Second-order topological superconductors host Majorana corner and hinge modes in contrast toconventional edge and surface modes in two and three dimensions. However, the realization of suchsecond-order corner modes usually demands unconventional superconducting pairing or complicatedjunctions or layered structures. Here we show that Majorana corner modes could be realized using a 2Dquantum spin Hall insulator in proximity contact with an s-wave superconductor and subject to an in-planeZeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac massesbetween adjacent boundaries, leading to one Majorana mode at each corner. A similar paradigm alsoapplies to 3D topological insulators with the emergence of Majorana hinge states. Avoiding complexsuperconductor pairing and material structure, our scheme provides an experimentally realistic platform forimplementing Majorana corner and hinge states.

DOI: 10.1103/PhysRevLett.124.227001

Introduction.—Majorana zero energy modes in topo-logical superconductors and superfluids [1–4] haveattracted great interest in the past two decades becauseof their non-Abelian exchange statistics and potentialapplications in topological quantum computation [5,6].A range of physical platforms [7–18] in both solid stateand ultracold atomic systems have been proposed to hostMajorana modes. In particular, remarkably experimentalprogress has been made recently to observe Majorana zeroenergy modes in s-wave superconductors in proximitycontact with materials with strong spin-orbit coupling,such as semiconductor thin films and nanowires, topologi-cal insulators, etc., [19–24]. In such topological super-conductors and superfluids, Majorana zero energy modesusually localize at 2D vortex cores or 1D edges, where theDirac mass in the low-energy Hamiltonian changes sign.Recently, a new class of topological superconductors,

dubbed as higher-order topological superconductors, hasbeen proposed [25–38]. In contrast to conventional topo-logical superconductors, rth-order (r ≥ 2) topologicalsuperconductors in d dimensions host (d − r)-dimensionalMajorana bound states, rather than d − 1-dimensionalgapless Majorana excitations. For example, in 2D sec-ond-order topological superconductors, the edge modesmanifest themselves as 0D Majorana excitations localizedat the corners, instead of 1D edges, giving rise to Majoranacorner modes (MCMs). A variety of schemes have beenproposed recently to implement MCMs, such as p-wavesuperconductors under magnetic field [26], 2D topologicalinsulators in proximity to high temperature superconduc-tors (d-wave or s�-wave pairing) [29–31], π-junctionRashba layers [34] in contact with s-wave superconductors.

However, those schemes demand either unconventionalsuperconducting pairings or complicated junction or latticestructures, which are difficult to implement with currentexperimental technologies.In this Letter, we propose that MCMs can be realized

with a simple and experimentally already realized hetero-structure [39–42] composing of an s-wave superconductorin proximity contact with a quantum spin Hall insulator(QSHI) and subject to an in-plane Zeeman field, assketched in Fig. 1. Here we consider a simple squarelattice. At each edge of the 2D QSHI, there are two helicaledge states with opposite spins and momenta, which thussupport a proximity-induced s-wave superconducting pair-ing, resulting in a quasiparticle band gap for the helicaledge mode spectrum [39–42].

FIG. 1. Illustration of a heterostructure composing of a quan-tum spin Hall insulator on top of an s-wave superconductor andsubject to an in-plane Zeeman field. The spheres at four cornersrepresent four Majorana zero energy modes.

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Because of different spin-orbit coupling at adjacentedges, an in-plane Zeeman field induces quite differenteffects on adjacent edges. Across a critical Zeeman field,the quasiparticle band gap along one edge first closes thenreopens, indicating a topological phase transition, butremains unaffected for adjacent edges. Before the phasetransition, the Dirac mass term in the low-energy effectiveHamiltonian for the helical edge states has the same sign fortwo adjacent edges. After the topological phase transition,the Dirac mass term reverses its sign at the cornerconnecting two edges, resulting in MCMs. In contrast,the corner Dirac mass sign change in previous schemesoriginates from the sign change of the pairing order throughunconventional superconducting pairing. There is only oneMCM at each corner due to the time-reversal symmetrybreaking, instead of Majorana Kramers pairs [29,30].Applying similar physics to three dimensions, we find

that second-order topological superconductor can be imple-mented in a 3D strong topological insulator, where theinterplay between s-wave pairing and Zeeman field (notnecessarily in-plane) gives rise to four domain walls on theedges between two neighboring surfaces, yieldingMajoranahinge modes.Physical system and model Hamiltonian.—Consider a

QSHI in proximity contact with an s-wave superconductorand subject to a Zeeman field h (see Fig. 1). The four edgesof a square sample are labeled by i, ii, iii, iv. The physics ofthe heterostructure can be described by an effectiveHamiltonian [29]

HðkÞ ¼ 2λx sin kxσxszτz þ 2λy sin kyσyτz

þ ðξkσz − μÞτz þ Δ0τx þ h · s; ð1Þ

under the basis Ck ¼ ðck;−isyc†−kÞT with ck ¼ ðck;a;↑;ck;b;↑; ck;a;↓; ck;b;↓ÞT . Here λi is the spin-orbit couplingstrength, Δ0 denotes s-wave superconducting order param-eter induced by proximity effect, ξk ¼ ϵ0 − 2tx cos kx −2ty cos ky with 2ϵ0 being the crystal-field splitting energyand ti the hopping strength on the square lattice, and μ is thechemical potential. Three Pauli matrices σ, s and τ act onorbital (a, b), spin (↑, ↓) and particle-hole degrees offreedom, respectively. For simplicity of the presentation,we focus on the μ ¼ 0 case, where simple analytic resultsfor edge modes can be obtained.In the absence of superconducting pairing and Zeeman

field, the Hamiltonian (1) is invariant under the time-reversalT ¼ isyK and space-inversion I ¼ σz operations, where Kis the complex-conjugation operator. Here the band topologycan be characterized by a Z2 topological index protectedby T symmetry or an equivalent Z index for the spin Chernnumber [43]. The system is a QSHI in the band invertedregion ½ϵ20 − ð2tx þ 2tyÞ2�½ϵ20 − ð2tx − 2tyÞ2� < 0. With theopen boundary condition, there are two helical edge states

with opposite spins and momenta propagating along eachedge in the QSHI phase [1,2].Topological phase diagram and MCMs.—In the pres-

ence ofΔ0, a finite quasiparticle energy gap is opened in theedge spectrum for two helical edge states due to the s-wavepairing. The in-plane Zeeman field hx has different effectson the single particle edge spectra (i.e., Δ0 ¼ 0) along the xand y directions: it can (cannot) open the gap along the kx(ky) direction [44]. Such anisotropic effect of hx leads tovery different physics whenΔ0 ≠ 0. Along the kx direction,the quasiparticle band gap first closes [Fig. 2(a)] at thecritical point hxc ¼ Δ0 and then reopens with increasing hx,indicating a topological phase transition. While along theky direction, the quasiparticle band gap does not close [44].The difference between the edge spectra drives the hetero-structure to a second-order topological superconductor.The emergence of MCMs after the topological phase

transition is confirmed by the numerical simulation ofcorresponding lattice tight-binding model in real space, asshown in Figs. 2(b), 2(c). Before the topological phasetransition [Fig. 2(b)], there are no zero energy boundedstates localized at edges. After the in-plane Zeeman fieldexceeds the critical point hxc, four zero energy MCMsemerge at each corner of the square sample [Fig. 2(c)]. Theemergence of MCMs is independent of the underlyinggeometry of the sample. For example, in Fig. 2(d), a similarresult is observed in a equilateral right triangular sample.

0

0

2

4

2

4

(a)

(c)

y

x

E

(b)

E

y

(d)

-4

-2

0

2

4

0kx

6

12

18

24

30

36

2 0

y

1

6

12

18

24

30

36

xx1 6 12 18 4 3 36 1 6 12 18 24 30 36

FIG. 2. (a) Quasiparticle bands with edge spectrum (red lines)for open boundary conditions along the y direction. The gap foredge spectrum closes at hx ¼ Δ0 ¼ 0.4. (b) Density distributionsof the edge bound states (red dots in the inset) for a trivialsuperconductor, where we have chosen hx ¼ 0.0, Δ0 ¼ 0.4.(c)–(d) Density distributions of MCMs in different geometries.The radii of the blue disks are proportional to local density. Theinsets show the energy levels for hx ¼ 0.8 and Δ0 ¼ 0.4. In bothFig. 2 and Fig. 3, tx ¼ ty ¼ λx ¼ λy ¼ 1.0 and ϵ0 ¼ 1.0.

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In this case, there are only two MCMs at two left cornersdue to the orientation of the hypotenuse edge that leads todifferent effects of the in-plane Zeeman field.To examine the topological characterization of MCMs,

we further calculate the Majorana edge polarizations pedge;yx

and pedge;xy using the Wilson loops on a cylindrical

geometry [45,46]. Majorana edge polarization at the

y-normal edge is defined by pedge;yx ¼ PNy=2

iy¼1 pxðiyÞ, whereNy is the number of unit cells along y, and the polarization

distribution is pxðiyÞ ¼ ð1=NxÞP

j;kx;β;n j½unkx �iy;β½νjkx�nj2νjx.

Here, ½νjkx �n represents the nth component of the jth eigen-

vector corresponding to theWannier center νjx of theWannierHamiltonian HWx

¼ −i lnWx with Wx the Wilson loopoperator [44]. ½unkx �iy;β is the ðiy; βÞ-th component of occupiedstate junkxiwith iy and β being the site index and the internaldegrees of freedom, respectively. Similarly, we can defineMajorana edge polarization pedge;x

y . In the MCM phase, onlythe Wannier spectra νx contain two half-quantized Wanniervalues, as shown in Fig. 3(a), implying that the edgepolarizations occur only along the y-normal edges but notthe x-normal edges. Such an observation has been numeri-cally verified by distributions of localized edge polarizationalong y [see Fig. 3(b)], and zero edge polarization distribu-tions along x. This further leads to half quantization ofpedge;y

x

and vanishing pedge;xy , as demonstrated in Fig. 3(c).

We remark that the above topological characterizationsshow that the MCM phase in our system falls into the classof extrinsic higher-order topological phases distinguishedby gap closings of the edge spectra [45] on a cylindricalgeometry, instead of bulk spectra on a torus geometry forintrinsic higher-order phases. However, the MCMs cannotbe annihilated by perturbations without closing the edgeenergy gap [44].Low-energy theory on edges.—All above numerical

results can be explained by developing an effectivelow-energy theory on edges. With both Δ0 and hx, theHamiltonian HðkÞ possesses both inversion symmetryand particle-hole symmetry PHðkÞP−1 ¼ −Hð−kÞ, butbreaks the time-reversal symmetry, where P ¼ τxK.Without loss of generality, we assume a positive in-plane Zeeman field applied along x direction, i.e.,hx > 0 and hy ¼ hz ¼ 0. The eigenenergies of HðkÞare EðkÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2λx sin kxÞ2 þ ðς� hxÞ2

p, where ς ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ξ2k þ ð2λy sin kyÞ2 þ Δ20

qand each of them is twofold

degenerate. For large hx, the system must be a normalsuperconductor, which becomes gapless for moderate hx.When hx is small, the low-energy effective Hamiltonian

can be obtained through the lowest order expansion withrespect to k at Γ point

HeffðkÞ ¼ ðϵþ txk2x þ tyk2yÞσzτz þ 2λxkxσxszτz

þ 2λykyσyτz þ Δ0τx þ hxsx; ð2Þ

where ϵ ¼ ϵ0 − 2tx − 2ty < 0 is assumed for topologicallynontrivial QSHI.Assuming an open-boundary condition along the x direc-

tion for edge i, we can replace kx with −i∂x and rewriteHeffðkÞ ¼ H0ð−i∂xÞ þHpðkyÞ with H0¼ðϵ−tx∂2

xÞσzτz−2iλx∂xσxszτz, and Hp¼tyk2yσzτzþ2λykyσyτzþΔ0τxþhxsx.When Δ0 is small comparing to the energy gap, we cantreatHp as a perturbation and solveH0 to derive the effectiveedgeHamiltonian for edge i. Assume thatΨa is a zero energysolution for H0 bounded at edge i, σyszτzΨa is also theeigenstate forH0 due tofH0; σyszg ¼ 0.We choose thebasisvector ζβ for Ψa satisfying σysz ζβ ¼ −ζβ, where ζ1 ¼j−;þ;þi, ζ2 ¼ jþ;−;þi, ζ3 ¼ j−;þ;−i, ζ4 ¼ jþ;−;−iare eigenstates of σyszτz. Under this basis, the effective low-energy Hamiltonian for the edge becomes Hedge;i ¼2iλyszτz∂y þ Δ0τx with the topology characterized by a Zinvariant [47]. Similarly, we obtain the low-energyHamiltonian for every edge

Hedge;j ¼ −iλjszτz∂lj þ Δ0τx þ hjsx: ð3Þ

Here the parameters are λj ¼ f−2λy; 2λx; 2λy;−2λxg, lj ¼fy; x; y; xg, and hj ¼ f0; hx; 0; hxg for j ¼ i–iv edges.

0.5

0

-0.5

-10 0.2 0.4 0.6 0.8

hx

hx

hx

5

(d)(c)

(a)

0.2 0.4 0.6 0.80

6

1

0

Δ1

1

px(iy)

0.5

0.2

1 10 20 30 40iy

(b)

0

FIG. 3. (a) Wannier spectra νx versus hx. The inset showcasesWannier centers for different state indexes with hx ¼ 0.8.(b) Majorana polarization distribution pxðiyÞ versus lattice indexiy. (c) Majorana edge polarizations pedge;y

x and pedge;xy along y-

normal and x-normal edges, respectively. In (a)–(c), Δ0 ¼ 0.4and μ ¼ 0.0 are used. (d) Phase diagram with μ ¼ 0.2. NSCdenotes a trivial superconductor, and GSC denotes a gaplesssuperconductor. MCM and NSC phases are separated by thephase boundary hx ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ2 þ Δ2

0

p.

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From the effective edge Hamiltonian (3), we see that thesuperconducting order induces quasiparticle gaps for allhelical edge states regardless of Zeeman fields sincefszτz; τxg ¼ 0. On the other hand, Eq. (3) indicates thatthe in-plane Zeeman field hx only opens a gap on twoparallel edges (ii and iv), but keeps two perpendicular edges(i and iii) untouched [44].When Δ0 ¼ 0, the low-energy edge Hamiltonian pos-

sesses two zero-energy bound states on edge i: Ψ1ðxÞ ¼A1ðsin αxÞe−ðλx=txÞxðζ1 þ ζ2Þ and Ψ2ðxÞ ¼ A2ðsin αxÞe−ðλx=txÞxðζ3 þ ζ4Þ, where α ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−ðλ2x=t2x þ hx=tx þ ϵ=txÞ

pand A1ðA2Þ is the normalization constant. Similarly, thereare two zero energy bound states localized at edge iii,which are confirmed by real space numerical simula-tion [44].After a unitary transformationU ¼ 1 ⊕ ð−isyÞ, the edge

Hamiltonian reads

H0edge;j ¼ −iλjsz∂lj þ Δ0sxτz þ hjsx; ð4Þ

on the rotated basis χ1 ¼ j þ 1ij þ 1i, χ2 ¼ j þ 1ij − 1i,χ3 ¼ j − 1ij þ 1i, χ4 ¼ j − 1ij − 1i, which are eigenstatesof szτy. For edge i, the Hamiltonian H0

edge;i has twodecoupled diagonal blocks with Dirac masses Δ0 þ hxand hx − Δ0, respectively. While for edge ii, Dirac massesare the same Δ0 for two blocks. When ðΔ0 − hxÞΔ0 < 0(i.e., hx > Δ0), the Dirac masses on edges i and ii haveopposite signs, leading to the emergence of a localizedmode at the intersection of two edges, which is the MCMobserved numerically in Fig. 2(c). At the corner betweenedges i and ii, the MCM can be obtained from the zero-energy wave function

Φðx; yÞ ∝�e−ðjΔ0−hxj=2λyÞjy−y0jðχ3 − iχ4Þ ðedge iÞ;e−ðΔ0=2λxÞjx−x0jðχ3 − iχ4Þ ðedge iiÞ;

ð5Þ

where the corner locates at ðx0; y0Þ. We see that MCMscould have different density distributions along differentdirections when jΔ0 − hxj=λy ≠ Δ0=λx.For the triangle geometry in Fig. 2(d), the effect of the in-

plane Zeeman field on the hypotenuse edge can be studiedby projecting it to the direction of the Zeeman field, whichshows that the Zeeman field acts uniformly on thehypotenuse and upper edges. Consequently, there is nokink of Dirac mass at that corner, i.e., no MCM. Generally,such an argument applies to all geometric configurationswith odd edges (e.g., a square with a small right triangleremoved at a corner), which is consistent with bulk spectrabecause the particle-hole symmetry demands that zero-energy modes must be lifted pairwise.For a general form of the in-plane Zeeman field, MCMs

emerge in the regionffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2x þ h2y

q> Δ0. However, an out-of-

plane Zeeman field (hzsz term) does not induce MCMsbecause the helical edge states of QSHIs remain gapless for

any hz and hz affects each edge in the same way [44].Finally, for a nonzero chemical potential μ ≠ 0, thespectrum is more complicated [44], and MCMs still existfor hx >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiμ2 þ Δ2

0

pwith bulk spectrum being gapped. The

phase diagram with a finite μ is shown in Fig. 3(d).Majorana hinge modes in three dimensions.—Similar

physics also applies to three dimensions. Consider a 3Dtopological insulator described by theHamiltonianHTðkÞ ¼ξ0kσzs0 þ

Pi λi sin kiσxsi with ξ0k ¼ m0 þ

Pi ti cos ki,

which respects both time reversal and inversion symmetries.For 1 < jm0j < 3, HTðkÞ represents a 3D topologicalinsulator that possesses surface Dirac cones with gappedbulk spectrum protected by I and T symmetries. In thepresence of an s-wave superconducting order Δ0 and aZeeman field,

H3DðkÞ ¼ ξ0kσzτz þ λx sin kxσxsxτz þ λy sin kyσxsyτz

þ λz sin kzσxszτz þ Δ0τx þ h · s: ð6Þ

For Δ0 ≠ 0, jhj ¼ 0, the surface states are gapped and thesystem is a trivial superconductor. When hx > 0,hy ¼ hz ¼ 0, the in-plane Zeeman field hx breaks the timereversal symmetry in the x direction, generating a class-Dsuperconductor. Tuning hx > Δ0, we observe the gaplesschiral Majorana hinge modes propagating along the zdirection as shown in Fig. 4. Such a 3D second-ordertopological superconductor can be characterized by a Zinvariant [47].Figure 4(a) shows the energy spectrum with open

boundary conditions along x and y directions, where thechiral Majorana hinge modes (each twofold degenerate)emerge in the bulk energy gap. The combination of theZeeman field and the superconductor order gives rise tofour domain walls at which the Dirac mass sign changes.Because of the inversion symmetry, the chiral modes atdiagonal hinges propagate along opposite directions, asillustrated in Fig 4(b). We remark that the requirement ofin-plane Zeeman field can be released in three dimensions

x

y

zE

(b)

0 2

0

1

1

(a)

kz

FIG. 4. (a) Quasiparticle spectrum along kz with open boundaryconditions along the x and y directions. (b) Majorana hingeexcitations in a 3D second-order topological superconductor.Parameters are tx ¼ ty ¼ tz ¼ λx ¼ λy ¼ λz ¼ 1.0, Δ0 ¼ 0.3,hx ¼ 0.6, and m0 ¼ 2.0.

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and the direction of the Zeeman field can be used to controlthe directionality of the hinge modes. Specifically, whenthe Zeeman field lies along the y (z) direction and hy > Δ0

(hz > Δ0), the chiral Majorana hinge modes propagatealong the xðyÞ direction with periodic boundary conditions.Discussion and conclusion.—InAs/GaSb quantum wells

are 2D Z2 QSHIs with large bulk insulating gaps up to∼50 meV, and significant experimental progress has beenmade [48–51] recently to observe their helical edge states.Superconducting proximity effects in InAs/GaSb quantumwells were also observed in experiments [39–41]. Inparticular, edge-mode superconductivity due to proximitycontact with an s-wave superconductor has been detectedthrough transport measurement [40], and giant supercurrentstates have been observed [41]. The in-plane Zeeman fieldcould be realized using an in-plane magnetic field due tothe relatively large g factor for InAs/GaSb quantum wells[52]. By engineering a suitable quantum device, zero-biaspeaks for MCMs should be observable in transport or STMtypes of experiments.Another potential material is the monolayer WTe2 [53]

that has been confirmed as a QSHI in recent experiments[54,55]. When proximate to superconductors, a proximity-induced superconducting gap of the order of ∼0.7 meV[42] emerges. To achieve a comparable spin Zeemansplitting, an in-plane magnetic field H ∼ 0.3–3 T isrequired, given the Lande g factor ranges from 4.5[54,56] to larger than 44 [57] in WTe2, depending onthe direction of the applied fields. Based on the afore-mentioned parameters, s-wave superconductor NbN can beused for the device fabrication, given its both hightransition temperature, Tc ∼ 12 K, and high critical field,for example, Hc > 12 T at 0.5 K [58]. For Majorana hingemodes in three dimensions, effective Zeeman fields couldbe induced by doping magnetic impurities into 3D topo-logical insulators.In conclusion, we have shown that a heterostructure

composing of QSHI=s-wave superconductor can become asecond-order topological superconductor with MCMs inthe presence of an in-plane Zeeman field. Because neitherexotic superconducting pairings nor complex junctionstructures are required, our scheme provides a simpleand realistic platform for the experimental study of thenon-Abelian Majorana corner and hinge modes.

This work is supported by Air Force Office of ScientificResearch (FA9550-16-1-0387), National ScienceFoundation (PHY-1806227), and Army Research Office(W911NF-17-1-0128). The work by C. Z. was performedin part at the Aspen Center for Physics, which is supportedby National Science Foundation Grant No. PHY-1607611.This work is also supported in part by NSFC under theGrant No. 11504285, and the Scientific Research ProgramFunded by the Natural Science Basic Research Plan inShaanxi Province of China (Program No. 2018JQ1058),the Scientific Research Program Funded by Shaanxi

Provincial Education Department under the GrantNo. 18JK0397, and the scholarship from ChinaScholarship Council (CSC) (Program No. 201708615072).

*xiwang.luo@utdallas.edu†chuanwei.zhang@utdallas.edu

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