PHYSICAL REVIEW B 102, 205402 (2020)

Multiple Majorana edge modes in magnetic topological insulator–superconductor heterostructures

Qingming Li,1,2 Yulei Han ,2 Kunhua Zhang,2,3 Ying-Tao Zhang,1,* Jian-Jun Liu,1,4,† and Zhenhua Qiao 2,‡

1College of Physics, Hebei Normal University, Shijiazhuang, Hebei 050024, China2ICQD, Hefei National Laboratory for Physical Sciences at Microscale, Chinese Academy of Sciences Key Laboratory of Strongly-Coupled

Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China3School of Physics and Technology, Nanjing Normal University, Nanjing, 210023, China

4Department of Physics, Shijiazhuang University, Shijiazhuang, Hebei 050035, China

(Received 16 August 2020; accepted 12 October 2020; published 3 November 2020)

We numerically investigate the electronic transport properties (i.e., electron tunneling and Andreev reflection)of a topological superconductor composed of a magnetic topological insulator and superconductors. A phasediagram is provided to distinguish various topological phases and their corresponding distinct Majorana edgemodes. When superconductors are proximity coupled with the top and bottom surfaces of a magnetic topologicalinsulator thin film, a quantum phase transition from topological insulator to quantum anomalous Hall effectpasses through the regime possessing both chiral and helical edge modes. The hallmark feature is that thecoefficient of electron tunneling is quantized to be 5/4 and the remaining scattering processes exhibit anidentical probability with a magnitude of 1/4 in the coexisting states with a Chern number of N = ±1. Whenthe superconductors are proximity coupled with a nonmagnetic topological insulator thin film, we find thatthe perfectly quantized electron tunneling or crossed Andreev reflection can alternately appear via tuning thechemical potential.

DOI: 10.1103/PhysRevB.102.205402

I. INTRODUCTION

Majorana fermions, being their own antiparticles, canbe realized as quasiparticles of topological states of quan-tum matter in condensed matter physics [1–5]. Becauseof the non-Abelian statistics and the nonlocality charac-teristic of Majorana fermions, the braiding of Majoranafermions is considered as the basic building block for fault-tolerant topological quantum computations [6–9]. So far,several proposed material systems were raised to realize suchstates, e.g., semiconductor-superconductor heterostructures[10,11], magnetic-atomic chains on top of superconductors[12,13], and topological insulators proximity coupled withsuperconductors [14–16]. Due to the superconducting prox-imity effect to the topological insulators, the correspondingheterostructures are also named after topological supercon-ductors (TSCs). Depending on the time-reversal symmetry ofthe gapless boundary modes, topological superconductors canbe classified into two different categories.

One is the “chiral” topological superconductor, in whichthe time-reversal symmetry is broken [17–20]. It possessestopologically protected chiral Majorana edge modes that canbe analogous to the gapless edge modes of the quantumanomalous Hall effect [21–24], which is associated with anonzero Chern number C. From the view of topology, thequantum anomalous Hall effect with a Chern number ofC = ±1 is equivalent to the chiral topological superconduc-

*zhangyt@mail.hebtu.edu.cn†liujj@mail.hebtu.edu.cn‡qiao@ustc.edu.cn

tor with a Chern number of N = ±2. And the N = ±1chiral topological superconductor can be realized when thesystem undergoes a topological phase transition from thequantum anomalous Hall effect to the normal Anderson in-sulator. The half-integer conductance plateau observed in arecent experiment had been used to be the evidence supportingMajorana fermions [25]. However, the subsequent theoreticaland experimental interpretations imply that the half-integerconductance plateau is not sufficient evidence for the exis-tence of a single chiral Majorana edge mode [26–28].

The other is the “helical” topological superconductor, inwhich the time-reversal symmetry is preserved [29–34]. It ischaracterized by a Z2 topological index and possesses gaplesshelical Majorana edge modes, which are composed of twochiral Majorana edge modes with opposite chiralities, there-fore also called the Majorana Kramers pair. By comparing thetopological properties between the quantum anomalous Halleffect and chiral topological superconductor, it is natural togeneralize the helical edge modes of the quantum spin-Hallinsulator with twofold degrees of freedom to correspond tothe helical topological superconducting edge states. Differentfrom the chiral topological superconductor, the helical topo-logical superconductor has yet to be observed in experiment.

Andreev reflection (AR) is an electron/hole transport pro-cess that occurs at the interface of a superconductor [35]. Thelocal Andreev reflection converts an incident electron into ahole back to the same terminal, and a Cooper pair is createdinside the superconductor [36–38]. The crossed Andreev re-flection (CAR), also known as non-local Andreev reflection,is a nonlocal process describing the process of converting anelectron incoming from one terminal into an outgoing holeto another spatially separated terminal [39–42]. Based on

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LI, HAN, ZHANG, ZHANG, LIU, AND QIAO PHYSICAL REVIEW B 102, 205402 (2020)

crossed Andreev reflection, a Cooper pair in the superconduc-tor can be split into two electrons propagating at two spatiallyseparated terminals while keeping their spin and momentumentangled. These spatially separated entangled electrons arethe key building blocks for solid-state Bell-inequality experi-ments, quantum teleportation, and quantum computation. Theprobability of crossed Andreev reflection is usually smallerthan that of the local Andreev reflection. Therefore, howto increase the efficiency of Cooper-pair splitting (probabil-ity of crossed Andreev reflection) is crucial in engineeringentangled electron states. With the discovery of topologicalmaterials, an intensive theoretical effort has been made inexploring the electron/hole transport properties in topologi-cal insulator–superconductor junctions, where high-efficiencyCooper-pair splitters were proposed to greatly enhance thecrossed Andreev reflection and suppress other scattering pro-cesses [43,44]. However, these theoretical proposals mostlyfocused on the transport properties of the chiral topologicalsuperconducting mode, and are rarely devoted to the othertopological superconducting modes.

In this paper, we investigate the topological superconductorwith a Chern number of N = ±1 to propose a topologicalstate with coexisting chiral and helical Majorana edge modes.Compared to the quantum anomalous Hall effect, the quantumspin-Hall effect is topologically equivalent to a topological su-perconductor with two pairs of helical Majorana edge modes.Thus, by coupling the quantum anomalous Hall effect to thes-wave superconductors, the topological phase transition fromthe quantum anomalous Hall effect to the quantum spin-Halleffect may undergo a region with both chiral and helical Majo-rana edge states. In order to clearly understand the topologicalphase transition, we construct a system setup as displayed inFig. 1 to study the electron tunneling (ET), AR, and CAR inthe topological superconductor junction, where the left/rightterminal is the quantum spin-Hall insulator, and the centralscattering region is either the quantum anomalous Hall effector quantum spin-Hall effect. We find that for the quantumanomalous Hall effect there exists a region with coexistingchiral and helical edge states, where the transport coefficientsare, respectively, TET = 5/4 and TAR = TCAR = 1/4; for the

FIG. 1. Schematic plot of a two-terminal transport setup: In thecentral region, two superconductors are placed on top and bottomof a quantum spin-Hall insulator, while the left and right terminalsare exactly extended from the quantum spin-Hall insulator. Notethat in our paper the quantum spin-Hall insulator can be tuned tobe the quantum anomalous Hall effect by introducing time-reversalbreaking ferromagnetism. Black arrows denote the edge states, redcircles indicate electrons, and upper/lower arrows represent the spin-up/-down states.

quantum spin-Hall effect, the quantized crossed Andreev re-flection may appear by tuning the chemical potential viaelectric gating.

II. SYSTEM MODEL HAMILTONIAN

In our paper, we adopt the system of Bi2Se3 Z2 topologicalinsulator thin films [45–47], and the corresponding quantumanomalous Hall insulator (or magnetic topological insulator)can be realized by introducing the ferromagnetic Cr and Vatoms. Its low-energy effective Hamiltonian of the surfacestates near the Dirac point of the magnetic topological insu-lator thin film can be expressed as follows on the basis ofψk = (ct↑, ct↓, cb↑, cb↓)T :

H0(k) = υF(kyσxτz − kxσyτz ) + m(k)τx + λσz, (1)

where t/b represents the top/bottom surface state; ↑/↓ repre-sents the spin-up/-down state; σx,y,z and τx,y,z are, respectively,Pauli matrices in spin space and layers; and υF is the Fermivelocity. The first term corresponds to the kinetic energy. Thesecond term couples the top and the bottom surface states withm(k) = m0 + m1(k2

x + k2y ), where m0 and m1 are, respectively,

the hybridization gap and the parabolic band component. Thelast term describes the exchange field along the z axis to breakthe time-reversal symmetry.

When the s-wave superconductors are proximately coupledwith the topological insulator, a topological superconductor isproduced and pairing potentials are induced. The correspond-ing Bogoliubov–de Gennes (BdG) Hamiltonian can be writtenas [18,33,48]

HBdG =∑

k

�†k HBdG�k/2, (2)

where �k = [(ctk↑, ct

k↓, cbk↑, cb

k↑), (ct†−k↑, ct†

−k↓, cb†−k↑, cb†

−k↑)]T

and

HBdG =(

H0(k) − μs �k

�†k −H∗

0 (−k) + μs

), (3)

�k =(

i�1σy 00 i�2σy

), (4)

where μs is the potential energy, and �1/2 are, respectively,the pairing gap functions on the top/bottom surface, respec-tively. To preserve the time-reversal symmetry, we set μs = 0and �1 = −�2 = �, and the BdG Hamiltonian can be rewrit-ten in a block-diagonal form:

HBdG =(

H+(k) 00 H−(k)

), (5)

where H±(k) = vF (kyσx ∓ kxσy) + [m(k) ± λ]σzςz ∓ �σyςy

with ςx,y,z being Pauli matrices in the Nambu space. TheBdG Hamiltonian can therefore be decoupled into two partswith opposite chiralities, and the new basis in Eq. (5) be-comes (ct

k↑ + cbk↑, ct

k↓ − cbk↓, ct†

−k↑ + cb†−k↑, ct†

−k↓ − cb†−k↓)T /

√2

for H+(k) and (ctk↓ + cb

k↓, ctk↑ − cb

k↑, ct†−k↓ + cb†

−k↓, ct†−k↑ −

cb†−k↑)T /

√2 for H−(k).

Note that the number of edge modes of the topologicalsuperconductor is determined by the Chern numbers N+ andN− of the block-diagonalized Hamiltonians rather than the

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FIG. 2. (a) Phase diagram of the magnetic topological insulator thin films in the plane of exchange field λ and mass m0. (b) Phase diagramof the proximity coupled magnetic topological insulator thin films in the plane of exchange field λ and mass m0 at fixed �1 = −�2 = � andμs = 0. The phase boundaries intersect at four points: (0,�), (0,−�), (�, 0), and (−�, 0) in the (λ, m0 ) plane. (c–h) Corresponding bandstructures (or energy spectra) of the I-VI regions as displayed in panel (b). The parameters are set to be m0 = 0.5 in panels (c), (d), and (g) andm0 = −0.5 in panels (e), (f), and (h); � = 1 in panels (d) and (f)–(h); � = 0.3 in panels (c) and (e); λ = 0 in panels (c), (e), and (f); λ = 1 inpanels (g) and (h); and λ = 2 in panel (d). The red and blue lines represent the direction of propagation of Majorana fermions at the systemboundary, respectively.

total Chern number of N = N+ + N−, where N+/− is theChern number of H+/−(k). For example, the topological statewith a total Chern number of N = 1 can be composed ofN+/− = 0/1 or −1/ + 2. For the former case, the topologi-cal superconductor possesses a single chiral Majorana edgemode, while for the latter case the Chern number N− = 2from H−(k) corresponds to a pair of edge modes with samechirality, while the Chern number N+ = −1 from H+(k) cor-responds to a single edge mode with the opposite chirality. Forconcise and accurate description, the edge modes with Chernnumbers N+/− = −1/ + 2 are named after the coexistence ofchiral and helical Majorana edge modes.

When an electron comes from the left terminal, the elec-tronic transport properties, i.e., the direct electron-tunnelingcoefficient TET, the Andreev reflection coefficient TAR, and thecrossed Andreev reflection coefficient TCAR, can be numeri-cally evaluated by [49]

TET = Tr[R

eeGree

LeeGa

ee

],

TAR = Tr[L

eeGreh

LhhGa

he

],

TCAR = Tr[R

eeGreh

LhhGa

he

],

where e/h represent the electron and hole, respectively.L/R(E ) = i[�r

L/R − �aL/R] are the linewidth functions cou-

pling the left/right terminals with the central scattering re-gion. Gr (E ) = [E − H − �r

L − �rR]−1 is the retarded Green’s

function and H is the BdG Hamiltonian in the tight-bindingrepresentation.

III. ENERGY SPECTRA AND PHASE DIAGRAMS

Let us first study the phase diagram of the magnetictopological system described in Eq. (1). Since the topo-logical phase transition usually occurs when the bulk bandgap closes and reopens, one can determine the phaseboundaries by solving the band-crossing condition E (k) =±

√υ2

Fk2 ± [m(k) ± λ]2 = 0. As displayed in Fig. 2(a), thephase diagram in the (λ, m0) plane can be divided intofour topologically different regimes by two-phase boundariesm0 = ±λ. For the case of |m0| < |λ|, regions II and II’ are,respectively, the quantum anomalous Hall phases with Chernnumbers of C = ±1. For the case of |m0| > |λ|, the Chernnumber is C = 0, with region I being a trivial insulator andregion III being a quantum spin-Hall insulator [50].

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Next, we consider the hybrid system with the s-wave su-perconductor proximity coupled with a magnetic topologicalinsulator. At fixed μs = 0, m1 = 1, and �1 = −�2 = �, theenergy dispersion is E (k) = ±

√υ2

Fk2 + {λ ± [m(k) ± �]}2 ,and the band gap closes at the point for � ± (m0 ± λ) = 0.Figure 2(b) displays the phase diagram of the topological su-perconducting system. The four-phase boundaries separate the(λ, m0) plane into nine gapped regions, including six differenttopological phases. The intersections of all phase boundaries,falling into the coordinate axis, yield the multicritical pointsthat are closely related to �. To confirm the phase dia-gram from the bulk-edge correspondence, we investigate theband spectra of the superconducting nanoribbon with periodicboundary condition along the x direction and open boundarycondition along the y direction. The topological phases withthe corresponding band spectra are, respectively, character-ized by the following.

(i) For m0 > |λ| + � as shown in region I, it is the normalsuperconducting phase with a Chern number of N = 0. Thissuperconducting phase is adiabatically connected with thetrivial insulator that is shown in region I of Fig. 2(a). In thecorresponding band structure displayed in Fig. 2(c), a finiteband gap opens without gapless edge states.

(ii) For |m0| < |λ| − �, when the superconducting prox-imity effect is infinitesimal, the weak pairing strength drivesthe quantum anomalous Hall phase with the Chern number ofC = ±1 [corresponding to the II (II’) region in Fig. 2(a)] intothe superconducting phase with a Chern number of N = ±2,as displayed in the II (II’) region of Fig. 2(b). From thecorresponding band structure shown in Fig. 2(d), one can seethat two pairs of Majorana edge modes with the same chiral-ity emerge at both sides of the topological superconductingribbon.

(iii) For m0 < −|λ| − � as displayed in region III, it is aN = 0 superconducting phase, that is adiabatically connectedwith the quantum spin-Hall phase [corresponding to region IIIin Fig. 2(a)]. As discussed above, a quantum anomalous Hallstate is topologically equivalent to two pairs of Majorana edgemodes with the same chirality. Therefore, in the presence ofsuperconducting proximity effect, the quantum spin-Hall statecan be naturally considered as a superconductor with two pairsof helical Majorana edge states. Due to even pairs of helicalMajorana edge modes, according to the classification of Z2

topological invariants, the class of DIII superconductor withZ2 index ν = 0 is topologically trivial. The band structures ofthe two pairs of helical states can be observed in Fig. 2(e).

(iv) For |m0| < � − |λ| as displayed in region IV, due tothe s-wave superconducting proximity coupling, the transitionfrom the normal insulator to the quantum spin-Hall insulatorshould pass through a helical topological superconductor withthe Chern number N = 0. In the special case of λ = 0, twoMajorana edge modes with opposite chiralities localizing atone edge of the topological superconductor form a Kramerspair, which is protected by the time-reversal symmetry. In thecase of λ �= 0 in region IV, although weak magnetic dopingbreaks the time-reversal symmetry, the system is still adiabat-ically connected with the helical topological superconductorthat is protected by time-reversal symmetry. It is known thatthe phase transition from the time-reversal-invariant phase

to a trivial superconducting phase cannot occur without theband gap closing and reopening [51]. According to theclassification of Z2 topological invariants, the class of DIII su-perconductor with Z2 index ν = 1 is topologically nontrivial.The band structure of the helical topological superconductoris displayed in Fig. 2(f). One can see that the top and bottombands are separated by a gap and two pairs of counterprop-agating edge modes with opposite chirality emerge at bothsides of the ribbon.

(v) For ||λ| − �| < m0 < |λ| + � shown in region V (V’),a N = ±1 chiral topological superconducting phase emergesduring the topological phase transition from a quantumanomalous Hall insulator to a trivial insulator. The corre-sponding band structure is shown in Fig. 2(g). One can findthat a pair of chiral gapless edge state modes traverses acrossthe point, which holds a chiral Majorana mode.

(vi) For −|λ| − � < m0 < −||λ| − �| shown in region VI(VI’), there exist chiral and helical Majorana edge states withChern number N = ±1. Physically, the quantum spin-Hallinsulator is topologically equivalent to the superconductorwith two pairs of helical Majorana edge states, and the chi-ral topological superconductor with Chern number N = 2is topologically equivalent to the quantum anomalous Hallinsulator. When considering the topological phase transitionbetween quantum spin-Hall insulator and quantum anomalousHall insulator, the superconducting proximity effect annihi-lates one chiral edge state and gives rise to the coexistenceof chiral and helical Majorana edge states. The band structureof such coexisting states is displayed in Fig. 2(h). One cansee that three pairs of gapless edge states traverse acrossthe point, propagating along both sides of the topologicalsuperconductor.

IV. ELECTRONIC TRANSPORTAND NUMERICAL ANALYSIS

To identify the various types of Majorana edge modes inour topological superconducting system, we investigate thequantum tunneling and Andreev reflection in the topologicalsuperconductor composed of quantum spin-Hall insulator andsuperconductor. The central region of our system setup isa quantum spin-Hall insulator proximity coupled with twos-wave superconductors with opposite signs of the pairingfunctions at the top and bottom surfaces, and the left/rightterminals are the exact extension of the quantum spin-Hallinsulator region, as displayed in Fig. 1. In order to clearlyillustrate the electron/hole transport processes, the edge stateconfigurations of the topological superconductor are plottedin Fig. 3. Due to the topological equivalence, the helical edgemodes of the quantum spin-Hall insulator are split into fourMajorana edge modes at the left and right terminals. Half ofthe Majorana edge modes with the same chirality from theupper block of Eq. (5) propagate along one direction, whilethe remaining Majorana edge states from the lower block ofEq. (5) propagate along the opposite direction. As discussedin the previous section, the variation of the exchange field λ orthe hybridization gap m0 can result in a series of topologicalphase transitions, which can be uniquely reflected from theedge state transport features.

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FIG. 3. Schematic plot of edge mode transport in the setup shown in Fig. 1. The central TSC region is set to be (a) the N = 0 normalsuperconductor, (b) the N = 1 topological superconducting edge mode, (c) N = 2 topological superconducting edge modes, (d) N = 0 helicaltopological superconducting edge modes, (e) coexisting chiral and helical edge modes, and (f) N = 0 two-pair helical edge modes. Blue andred arrows indicate, respectively, Majorana edge modes with opposite chirality.

In Fig. 4(a), at fixed � = 0.3 and m0 = 0.5, the Chernnumber of the topological superconductor is N = ±2 for|λ| > 0.8, N = ±1 for 0.2 < |λ| < 0.8, and N = 0 for |λ| <

0.2. When N = ±2 is chosen in the central region of ourconsidered system, two chiral Majorana edge modes propa-gate along the system boundaries; i.e., in Fig. 3(c) one pairof Majorana edge modes propagates along the boundariesof the central region with the same chirality, while otheredge modes are scattered back at the interfaces between thecentral region and terminals. Due to the fact that two Ma-jorana edge modes with the same chirality can be combinedinto one quantum anomalous Hall state, such configuration istopologically equivalent to the quantum spin-Hall insulator–quantum anomalous Hall insulator junction. Therefore, theelectron-tunneling coefficient is TET = 1, and other transportcoefficients are zero.

When the exchange field decreases, a topological phasetransition occurs, leading to N = ±1 with one Majorana edgestate being destroyed. As displayed in Fig. 3(b), one can seethat two edge modes, the propagating direction of which isopposite to the chiral edge mode in the central region, cannottransit through the central scattering region, while the othertwo edge modes, the propagating direction of which is thesame as that of the chiral edge modes in the central region,

are split into two chiral Majorana modes with one branch ofchiral Majorana edge modes being perfectly transmitted butthe other branch being perfectly reflected. Since the singlechiral Majorana edge mode can be considered as half of theidentical copies of the quantum anomalous Hall edge modeand contributes equally to the electron transport and Andreevscattering, one can get TET = TLAR = TCAR = 1/4 [17,18].

When the exchange field further decreases, the central re-gion is driven into the N = 0 normal superconducting phase.Figure 3(a) displays that all the edge modes are completelyreflected at the interface between the central region and the leftterminal. As a result, all the electron tunneling TET, Andreevreflection TAR, and crossed Andreev reflection are vanishing.

In addition to the exchange field, the hybridization gap(m0) is also crucial to control the topological phase transi-tions. In the case of m0 = −0.5, although the Chern numberof topological superconductors in the central scattering re-gion has not been changed, the corresponding number ofedge states and the transmission coefficients are significantlydifferent from the case of m0 = 0.5. Figure 4(b) plots thetransmission coefficients as functions of the exchange field λ

at fixed � = 0.3 and m0 = −0.5. When |λ| < 0.2, two pairsof helical Majorana edge states are present in the topologi-cal superconducting region, as displayed in Fig. 3(f). Since

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FIG. 4. Transmission coefficients of electron tunneling TET,Andreev reflection TAR, and crossed Andreev reflection TCAR vsexchange field λ in the topological superconductor with the energyE = 0 of the incident electron. �1 = −�2 = 0.3 for panels (a) and(b); �1 = −�2 = 1 for panels (c) and (d). m0 = 0.5 for panels(a) and (c); m0 = −0.5 for panels (b) and (d). The length of thecentral TSC is L = 30, and the ribbon width is N = 120.

these edge modes are topologically equivalent to the quantumspin-Hall edge modes, all the incoming edge modes fromthe terminals are perfectly transmitted, giving rise to TET = 2and TAR = TCAR = 0. To be specific, as the exchange fieldis within the range of 0.2 < |λ| < 0.8, the superconductingregion experiences a topological phase transition with oneMajorana edge state emerging into the bulk, consequentlyleaving three Majorana edge states with odd Chern numberN = ±1. When such edge states exist in the central topolog-ical superconducting region, one can see that two Majoranaedge modes with the same propagation direction combinedwith another Majorana edge mode propagating along the op-posite direction form coexisting edge states appearing at theinterface between the topological superconductor and vac-uum, as displayed in Fig. 3(e). The two edge channels withthe same chirality pass through the central scattering regionand contribute to the tunneling coefficient with a value of 1,while another Majorana edge current in the opposite direc-tion also passes through the central scattering region, leadingto TET = TAR = TCAR = 1/4. Therefore, the total electronictransmission coefficient TET is 5/4, and the AR coefficient TAR

and the CAR coefficient TCAR are equal to 1/4.Figures 4(c) and 4(d) plot the dependence of transmission

coefficients on the exchange field λ for different hybridizationgaps m0 = 0.5 and −0.5, at the fixed pairing gap � = 1. Onecan see that the Chern number of the topological superconduc-tor is N = ±2 for |λ| > 1.5, N = ±1 for 0.5 < |λ| < 1.5,and N = 0 for |λ| < 0.5. For |λ| < 0.5, the N = 0 topologi-cal superconducting phase exhibits counterpropagating helicalMajorana edge states. This helical topological superconductoris analogous to the quantum spin-Hall insulator. As shownin Fig. 3(d), one can find that one-half of the Majoranaedge states with opposite chiralities in the quantum spin-Hallterminal region goes through the central scattering region, andthe other half of the edge modes are perfectly reflected. Since

FIG. 5. (a) Phase diagram of the superconducting system for λ =0 in the plane of the chemical potential μs and the pairing gap �.(b–d) Transmission coefficients of electron tunneling TET, Andreevreflection TAR, and crossed Andreev reflection TCAR as a function ofchemical potential μs. m0 = 0.5 and � = 0.7 for panel (b), m0 = 0.5and � = 0.2 for panel (c), and m0 = −0.5 and � = 0.2 for panel(d). The length of the central TSC is L = 20, and the ribbon width isN = 120.

the scattering coefficient of the helical topological supercon-ductor is double that of the chiral topological superconductorwith Chern number N = ±1, we then have TET = TAR =TCAR = 1/2.

In Refs. [52,53], it was shown that a quantized crossedAndreev reflection of TAR = 1 can occur in a quantum anoma-lous Hall insulator proximity coupled with a superconductor.Since the helically propagating edge states of the quantumspin-Hall insulator can be regarded as two copies of the quan-tum anomalous Hall effect with opposite chiralities, one caneasily predict that the crossed Andreev reflection coefficientcould be doubled in the system with the quantum spin-Hallinsulator proximity coupled with superconductors. To furtherconfirm that the crossed Andreev reflection coefficient is ex-actly equal to 2, λ = 0 and μs �= 0 are chosen in Eq. (3).The time-reversal-invariant topological superconductor can be

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classified by Z2 invariant ν. As shown in Fig. 5(a), one can seethat two quantum phases in the (μs,�) plane are separatedby the boundary of �2 + μ2

s = m20. In the region of |m0| >√

�2 + μ2s , one can get ν = 1, indicating a helical Majorana

edge state. In the region of |m0| <√

�2 + μ2s with m0 > 0,

one can get ν = 0, suggesting a trivial superconducting phase.Furthermore, one can obtain a trivial superconducting phasewith two pairs of helical edge states in the region of |m0| <√

�2 + μ2s with m0 < 0. As shown in Figs. 5(b)–5(d), TET,

TAR, and TCAR are dependent on the potential energy μs inthe quantum spin-Hall insulator–topological superconductorjunction. In a narrower topological superconducting region,Majorana fermions can also tunnel between the left and rightinterfaces of the topological superconductor in addition to thenormal propagation along the boundaries of the topologicalsuperconductor. To be specific, by increasing μs, the phaseof the tunneling Majorana fermions can be regulated betweenzero and π . As a result, the boundary-propagating Majoranafermions and the tunneling Majorana fermions combine to-gether to form electrons or holes, which can emit from thedevice terminals. Because of the helical nature of the Majo-rana edge states in our system, the perfect electron tunnelingand crossed Andreev reflection can alternately occur withTET = 2 and TCAR = 2.

V. CONCLUSIONS

We numerically investigated topological phases of thetopological superconductor realized by a magnetic topolog-ical insulator thin film proximity coupled with two s-wavesuperconductors on the top and bottom surfaces with opposite

pairing gap functions. By adjusting the system parameters, thetopological superconductor exhibits rich topological phaseswhich correspond to various Majorana edge modes. In par-ticular, a topological phase with both chiral and helical edgemodes can be achieved through a topological phase transitionbetween quantum spin-Hall effect and quantum anomalousHall effect. We further propose several transport experimentsto detect multiple Majorana edge modes. One unique transportsignature of the coexisting states with a Chern number ofN = ±1 is that the coefficient of electron tunneling is quan-tized to be 5/4 and the remaining scattering probabilities arequantized into 1/4. When the superconductors are proximitycoupled with a nonmagnetic topological insulator thin film, byadjusting the chemical potential, the perfectly quantized elec-tron tunneling or crossed Andreev reflection can alternatelyappear.

ACKNOWLEDGMENTS

This work was financially supported by the NationalKey Research and Development Program (Grant No.2017YFB0405703), the National Natural Science Founda-tion of China (Gants No. 11974327, No. 11474084, No.11704366 and No. 12074097), Natural Science Foundation ofHebei Province (Grant No. A2020205013), The Fundamen-tal Research Funds for the Central Universities (Grant No.WK2340000087), and Anhui Initiative in Quantum Informa-tion Technologies. We also thank the supercomputing serviceof AM-HPC and the Supercomputing Center of Universityof Science and Technology of China for providing the high-performance computing resources.

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