Mass-like gap creation by mixed singlet and triplet state in superconducting topological insulator
MMass-like gap creation by mixed singlet and triplet state insuperconducting topological insulator
M. Khezerlou (cid:3)1;2 , and H. Goudarzi y11
Department of Physics, Faculty of Science, Urmia University, P.O.Box: 165, Urmia, Iran National Elites Foundation, Iran
Abstract
We investigate proximity-induced mixed spin-singlet and spin-triplet superconducting state onthe surface states of a topological insulator. Such hybrid structure features fundamentally distinctelectron-hole excitations and resulting effective superconducting subgap. Studying the particle-holeand time-reversal symmetry properties of the mixed state Dirac-Bogoliubov-de Gennes effectiveHamiltonian gives rise to manifesting possible topological phase exchange of surface states, sincethe mixed-spin channels leads to appearance of a band gap on the surface states. This is verified bydetermining topological invariant winding number for chiral eigenstates, which is achieved by intro-ducing a chiral symmetry operator. We interestingly find the role of mixed superconducting state ascreating a mass-like gap in topological insulator by means of introducing new mixed-spin channels (cid:1)1 and (cid:1)2 . The interplay between superconducting spin-singlet and triplet correlations actually re-sults in gaped surface states, where the size of gap can be controlled by tuning the relative s and p -waves pairing potentials. We show that the system is in different topology classes by means ofchiral and no-chiral spin-triplet symmetry. In addition, the resulting effective superconductor sub-gap manipulated at the Fermi surface presents a complicated dependency on mixed-spin channels.Furthermore, we investigate the resulting subgap tunneling conductance in N/S and Josephson cur-rent in S/I/S junctions to unveil the influence of effective symmetry of mixed superconducting gap.The results can pave the way to realize the effective superconducting gap in noncentrosymmetricsuperconductors with mixed-spin state. PACS : 73.20.-r; 74.45.+c
Keywords : mixed-state superconductivity; topological insulator; chiral symmetry; Andreev subgap con-ductance; Josephson current
I INTRODUCTION
Topological insulators (TIs), as an interesting topologically nontrivial phase of condensed matter rep-resent distinct electronic properties comparing to the conventional band insulators. On the surface of athree-dimensional topological insulator (3DTI), topologically protected quantum channels are formed ina manner that the charge carriers obey from massless Dirac-like fermions. These gapless surface statesare protected by time-reversal (TR) symmetry and are robust against disorder and perturbations. Thereexist odd number of Dirac cones in the Brillouin zone, resulted from inversion symmetry breaking owingto the Rashba-type spin-orbit interaction [1, 2]. These peculiar features enable TIs to be potentially usedto spintronics [3, 4, 5] and topological quantum information applications [6, 7, 8]. Moreover, supercon-ductivity induction by proximity-effect on the surface states of a 3DTI has been of noticeable importanceduring the last decade. Several experimental probes [9, 10, 11, 12, 13, 14, 15] have evidenced existenceof spin-singlet and spin-triplet pairing states in the hybrid structure of a 3DTI and a superconductor. Oneof key findings in this topic is the manipulation of Majorana fermions in the Andreev bound states (ABS)established at the 3DTI ferromagnet-superconductor (FS) interface [16, 17, 18]. Merging the spin-singlet (cid:3) [email protected] y [email protected] p -wave symmetry. Actually, this leads tosuppression of Andreev process for energy excitations lower than superconducting effective gap [19].Unconventional superconductivity in 2D Dirac materials plays an important role [19, 20, 21, 22].However, regarding the inversion symmetry breaking in TIs, it will not be (at least from the dynamicalsymmetry point of view) ungraceful to take into mixed spin-singlet and spin-triplet superconductingstate contribution to the quasiparticle excitations, since the ( s + p )-wave state is also found to breakthe inversion symmetry. The symmetry of Cooper pair states in new systems with broken inversionsymmetry, such as noncentrosymmetric (NCS) superconductors can not be classified based on orbital andspin parts. The Cooper pair in these systems is, therefore, a mixture of singlet and triplet spin states. Anew type of NCS superconductor Zr3Ir has been very recently reported [23]. It is noted that, in cuprats,which have no inversion center in their crystal structure, the inversion symmetry is broken, leading toappearance of robust asymmetric spin-orbit interaction. Therefore, the superconducting pair potentialmixes singlet and triplet states [24, 25]. As a noticeable result, the spin-polarized current has beenpredicted to appear on the surface of a superconductor with mixed singlet and chiral triplet state [26, 27].In addition, in these materials, there is an even number of Majorana fermions in two-dimensional (2D)TR symmetric superconducting bound states [28, 29].Moreover, the exact pairing potentials describing many of superconductors with mixed singlet andchiral triplet states remain unknown. An updated list of recently discovered such superconductors can befound in Ref. [30]. As mentioned, 2D materials with spin-orbit coupling, such as 3DTI, is expected tohost a triplet state using a conventional s -wave superconductor, for example see, Ref. [31]. Interestingly,the surface state of superconducting 3DTI is strongly related to the 2D unconventional superconductors(such as p -wave symmetry) with broken inversion symmetry [28]. Hence, the outcomes of transport ofcharge carriers on the surface of a 3DTI hybrid (with mixed singlet and triplet superconducting states)contact can pave the way to unveil the effective dynamics of superconductor pairing state.Therefore, we proceed, in this paper, to investigate particularly a newly appeared distinct manifesta-tion of presence of mixed s and p -wave symmetries on the surface states (see Fig. 1(a)). From this pointof view, the interplay between inversion, PH and TR symmetries in the 3DTI mixed superconductorHamiltonian is expected to introduce a distinct scenario for quasiparticle excitation, where we find it topresent a mass-like gap opening in Dirac point by the varying comparative magnitudes of singlet s -waveand triplet p -wave pairing potentials. We try to show the possible topological property of band structurevia calculating the winding number, which is related to the Berry phase reflecting the topological struc-ture of wavefunction [32]. Specifically, electro-hole conversion at the normal metal-superconductor (NS)interface, called Andreev reflection (AR), and appearance of chiral Majorana state at the ferromagnet-superconductor (FS) interface can be considered as essential phenomena, which is directly influencedby the interplay between the spin-singlet and spin-triplet states on top of a 3DTI [33, 34, 35]. In lightof the above attributes, further treatment to study mixed-state 3DTI can be more impressive to evaluateits dynamical and transport properties. We show that the effective superconductor gap at the interfacehas a more complicated dependency on the magnitude of s - and p -wave pair potentials ( (cid:1)s and (cid:1)p ).The Dirac-point gapless band structure with s -wave symmetry is converted to the gaped states with themixed ( s + p )-wave symmetry via Z2 topological invariant. In Ref. [36], the authors have just pointedout the energy eigenvalue of hybrid 3DTI mixed superconductor Hamiltonian. Here, we have succeededto present analytical expression of the corresponding electron(hole) wavefunction in order to capture itstopological nature, the resulting Andreev subgap tunneling conductance (Fig. 1(b)) and, of course, the( )-current-phase relation in a Josephson junction (Fig. 1(c)). These results have been achieved in areally cumbersome analytical procedure.This paper is organized as follows. Section II is devoted to describe the discrete symmetry proper-ties of 3DTI Hamiltonian in the presence of mixed superconducting state. The chiral symmetry of thesystem is investigated. Next, the effective superconducting gap and electron-hole energy excitations areintroduced. The winding number for electron-hole pairing in a closed Berry connection in Brillouin zone2s studied by using the analytically obtained chiral eigenstates. In Sec. IIIA, we represent the explicitexpressions of normal and Andreev reflection amplitudes in a corresponding NS junction. The numericalresults of subgap tunneling conductance are presented along with a discussion of main characteristics ofsystem. The Josephson junction is considered in Sec. IIIB in order to investigate the property of Andreevbound state (ABS) and resulting supercurrent-phase exhibition. Finally, a brief discussion is given inSec. IV. II THEORETICAL FORMALISM
A Discrete symmetries of mixed state 3DTI
We begin by setting up a topological insulator-based model for the proximity effect, that the pairingpotential contains both spin-singlet and spin-triplet states. The order parameter for a mixture of such stateadopts the general form ^(cid:1)m( k ) = iei' [(cid:1)s( k )^(cid:27)0 + d ( k ) (cid:1) ^(cid:27)℄ ^(cid:27)2 , where the Pauli matrices ^(cid:27)i acting onthe spin space and ' indicates the superconducting phase. The spin-singlet component is an even functionof the wave vector, and we assume that the pairing potential (cid:1)s( k ) = j(cid:1)sj to be constant and real. Theorder parameter of spin-triplet pairing is described by an odd vector function d ( k ) of the momentum.For the chiral spin-triplet pairing, d ( k ) may then be written in the form d ( k ) = j(cid:1)pj [ os (cid:18) + i(cid:31) sin (cid:18)℄ ^z ,where j(cid:1)pj measures the amplitude of the triplet order parameter and (cid:31) labels the orientation of theangular momentum of the Cooper pair (featuring the chirality).The real and positive parameter (cid:1)m is introduced to quantify the energy scale of the superconductinggap. Throughout present work, the singlet (cid:1)s and triplrt (cid:1)p pair potential parameters are normalized by (cid:1)m . We employ the DBdG Hamiltonian H( k ) = (cid:18) ^hT I ( k ) ^(cid:1)m( k )^(cid:4) ^(cid:1)m( k )^(cid:4)(cid:0)1 ^(cid:4)^hT I ( k )^(cid:4)(cid:0)1 (cid:19) (1)in Nambu space for the surface states of a topological insulator to obtain the energy dispersion relationunder the influence of superconducting proximity effect. The gapless surface states are described by the2D linear Hamiltonian ^hT I ( k ) = vF (^(cid:27)1kx + ^(cid:27)2ky) (cid:0) (cid:22)s , ( ~ = 1 ), where vF and (cid:22)s denote velocityof charge carriers and chemical potential, respectively. PH symmetry operator ^(cid:4) is involved by an an-tiunitary operator, which may act on Dirac Hamiltonian and superconductor order parameter. By actingthe PH symmetry operator and defining two complex pair potentials as (cid:1)1;2 = (cid:1)s (cid:6) (cid:1)p ei(cid:31)(cid:18) , the Hamiltonian of mixed superconducting topological insulator hybrid yields: H( k ) = 0BB (cid:0)(cid:22)s vF jkj e(cid:0)i(cid:18) 0 (cid:1)1vF jkj ei(cid:18) (cid:0)(cid:22)s (cid:0)(cid:1)2 00 (cid:0)(cid:1)(cid:3)2 (cid:22)s vF jkj ei(cid:18)(cid:1)(cid:3)1 0 vF jkj e(cid:0)i(cid:18) (cid:22)s 1CCA : (2)Spin-singlet and spin-triplet admixture gives rise to two new spin channels (cid:1)1 and (cid:1)2 . The effectivemixed pairing potential depends on the angle (cid:18) , where, only for (cid:1)2 channel, there exist the possibility tobe zero. This case occurs when spin-triplet contribution is dominated, j(cid:1)pj (cid:21) j(cid:1)sj . Both spin channels (cid:1)1 and (cid:1)2 have no zero value for every angle (cid:0)(cid:25)=2 (cid:20) (cid:18) (cid:20) (cid:25)=2 , when spin-singlet potential is dominant.The effective two mixed-spin pair potentials is demonstrated, in detail, in Fig. 1(d).Let now unveil the topological symmetry properties of this given state. The resulting effective Hamil-tonian (2) satisfies the PH symmetry relation, which is (cid:0)H(cid:3)((cid:0) k ) = ^(cid:4)H( k )^(cid:4)(cid:0)1 , when ^(cid:4) = (^(cid:28)1 (cid:10) ^(cid:27)0) ^C:^C is the complex conjugation operator. The operator ^(cid:28)1 is the Pauli matrix in particle-hole space. In thiscase, the needed PH symmetry of mixed superconductor gap is provided in the surface states of 3DTI.The square PH symmetry operator is found ^(cid:4)2 = +1 . It is noted, that this symmetry may prove the spin3egeneracy of the Fermi surface to be lifted, and consequently it allows for exotic chiral Majorana modes[37]. On the other hand, the TR symmetry operator can be given by ^(cid:2) = (^(cid:28)0 (cid:10) i^(cid:27)2) ^C (with ^(cid:28)0 being in particle-hole space), under which the Hamiltonian H( k ) is related to H(cid:3)((cid:0) k ) . Notethat, the presence of chiral spin-triplet pairing causes TR symmetry breaking.By means of specific topological invariant, we remember that, in each spatial dimension, there existfive distinct classes of topological insulators, three of which are characterized by an integral Z topologicalnumber, while the remaining two possess a binary Z2 topological quantity. Regarding the particle-holeand chirality symmetries of matrices associated with the proposed Hamiltonian, one can determine thetopology class. According given topological classification in Ref. [32], the Hamiltonian (2) is foundto be placed in topologically nontrivial symmetry class D . Meanwhile, for two other no chiral case ofspin-triplet p -wave symmetry ( d ( k ) = (cid:1)p os (cid:18)^z and (cid:1)p sin (cid:18)^z ), we find Hamiltonian to commute withTR symmetry operator. Hence, the new topology class can be possible, and the system is classified intopology class DIII [32].
B Mass-like gap
The energy dispersion relation for superconducting excitations can be obtained by diagonalizing the Eq.(2). It is instructive to diagonalize the Hamiltonian H upon a unitary transformation H0 = ^U H ^U y . Weintroduce a unitary matrix to do this goal ^U = 1p2 (cid:18) ^(cid:27)0 ^(cid:27)1^(cid:27)1 (cid:0)^(cid:27)0 (cid:19) ; (3)under which H0 is transformed to block-diagonal form H0( k ) = 0BB (cid:1)1 vF jkj e(cid:0)i(cid:18) 0 0vF jkj ei(cid:18) (cid:0)(cid:1)2 0 00 0 (cid:1)2 vF jkj ei(cid:18)0 0 vF jkj e(cid:0)i(cid:18) (cid:0)(cid:1)1 1CCA : The presence of mixed two spin channels (cid:1)1 and (cid:1)2 in diagonal elements implies appearance of bandgap on the surface states of 3DTI. Hence, the energy eigenvalue dependency on the mixed spin channelsis easily given by
Emix((cid:1)1; (cid:1)2) = (cid:16)q(vF jkj)2 + ~(cid:1)1 + (cid:22)2s + (cid:29)p"R ; (4)where "R = ~(cid:1)22 + (2vF jkj(cid:22)s)2 + (vF jkj)2 j(cid:1)1 (cid:0) (cid:1)2j2 denotes the renormalized excitation energy re-lated to the mixed state. (cid:16) = (cid:6)1 refers the electron-like and hole-like excitations, and (cid:29) = (cid:6)1 dis-tinguishes between the conduction and valence bands. The parameters ~(cid:1)1 and ~(cid:1)2 are defined ~(cid:1)1;2 =12 (cid:16)j(cid:1)1j2 (cid:6) j(cid:1)2j2(cid:17) as new normalized mixed spin channels.Simple inspection of the electron-hole excitation spectrum in NCS superconductors indicates, thatthere is an essential physical distinction in surface states of topological insulators with mixed pairingstate. The Hamiltonian of two-dimensional NCS superconductors is decoupled into two spin channels (cid:1)1 and (cid:1)2 with different energies. The exchange between two energies is provided by the sign ofelectron wave vector [25]. Hence, the pairing potential is only related to the direction of motion (i.e. (cid:6)jkj ). In the presence of a topological insulator, we see that the energy dispersion is affected by two (cid:1)1 and (cid:1)2 spin channels in a fundamentally distinct manner. With the alone singlet or triplet pairing state,what that makes energy spectrum electronically interesting is the fact that the conduction and valencebands touch each other at Dirac point. Whereas, no strikingly say, the band topology of mixed state3DTI undergoes a change, and a sizeable energy gap is manipulated at Dirac point. This gap can becontrolled by tuning the relative magnitude of singlet j(cid:1)sj and triplet j(cid:1)pj pair potentials. It seems,that the correlations between the spin-singlet and spin-triplet plays the role of effective Dirac mass in4he surface states of topological insulator. This can be of an interesting feature of mixed-spin statesuperconductors in proximity with Dirac-like materials. It is noted that when two spin channels becomeequal (cid:1)1 = (cid:1)2 = 1 , the superconducting excitation reduces to s -wave-like one.However, mass-like gap in Dirac point of surface states can be clearly presented by vanishing quasi-particle wavevector. Consequently, the energy in Dirac point is separated into two parts correspondingtwo mixed spin channels Emix(jkj = 0) = (cid:18) (cid:6)p(cid:22)2s + j(cid:1)1j2(cid:6)p(cid:22)2s + j(cid:1)2j2 (cid:19) :
When we set j(cid:1)1j = j(cid:1)s + (cid:1)pj = 1 , the size of mass-like gap dependency on the mixed pair potentialcan be clearly obtained, as shown in Fig. 2. For (cid:1)2 = (cid:6)1 , we see the energy gap to be closed. Thisis in agreement with s and p -waves superconducting excitations in topological insulators. In the caseof (cid:0)1 < (cid:1)2 < +1 , the gap is immediately opened, and has a maximum when the singlet and tripletstate contributions are equal. In the next, we proceed to investigate the dynamical property of suchDirac gap opening via the calculating winding number topological invariant, since it can be of significantimportance in topological phase exchange point of view. Moreover, we are interested in zero energysuperconductor excitations. Solving Eq. (4) for Emix: = 0 , which gives zero subgap energy, results insuperconducting gapless state only in the absence of spin-singlet state ( (cid:1)s = 0 ). To present, in detail,the superconducting zero-energy, we plot in Fig. 3 dispersion energy as a function of kx and ky for threecases of singlet, triplet and mixed symmetries of pairing order. When, (cid:1)2 = (cid:0)1 , which means puretriplet case, the zero-energy occurs. Hence, triplet superconductor components in 3DTI Hamiltonianmay give rise to gapless states in Fermi wavevectors kF = p(cid:22)2s + j(cid:1)pj2 (see, Ref. [36]), as shown inFig. 3(c). C Topological nature of system
In this section, we proceed to investigate the possible topological properties of mixed superconductorstate 3DTI in order to achieve an answer for question whether the creation of Dirac mass-like gap isaccompanied by topological invariant of band phase exchange. Reaching this goal in our proposed systemcan be feasible by determining the Berry phase, including the non-trivial topological structure of thewavefunction. Using the wavefunction jn( ~R)i of a system, where ~R is the space set of parameters thequantities so-called Berry connection A ~R and Berry curvature B ~R are given by A ~R = (cid:0)= Dn( ~R) (cid:12)(cid:12)r ~R(cid:12)(cid:12) n( ~R)E ; B ~R = r ~R (cid:2) A ~R When a system moves along a close path C in the space, the resulting Berry phase (cid:13) acquired in thewavefunction is (cid:13) = (cid:0) IC d ~R (cid:1) A ~R = (cid:0) ZS ~dS (cid:1) B ~R Here, S represents area in the parameter space, enclosed by the contour C . In our system, the parameterset is specified by momentum ~k , and the Berry phase is also called the Zak phase [38]. Therefore, thetopological invariant related to the Zak phase is winding number ! = (cid:0)12(cid:25) ZBZ dk T r: (cid:2)^q(cid:0)1(k) k ^q(k)(cid:3) : (5)The integration is performed over a closed path including wavevectors belonging to the first Brillouinzone. The spectral projection operator ^q(k) defines a map from the reciprocal space in Brillouin zone tothe space of unitary matrices belonging to the symmetry group. The ^q(k) matrix is determined via theseveral constraints concerning to the discrete symmetries imposed on Hamiltonian.The chiral symmetric Hamiltonian is a needed condition to calculate the winding number, so that itneeds to be in block off-diagonal form. Therefore, we may construct formal chiral symmetry operator5ia the TR and PH symmetry operators (given in the previous section) as following ^ C = ^(cid:28)1 (cid:10) i^(cid:27)2 = 0BB 0 0 0 (cid:0)i0 0 i 00 (cid:0)i 0 0i 0 0 0 1CCA : Now, we are able to introduce an unitary transformation ^U , using the eigenvectors of above chiral sym-metry matrix ^U = 0BB 1 0 1 00 1 0 10 (cid:0)i 0 ii 0 (cid:0)i 0 1CCA ; under which the original mixed superconducting DBdG Hamiltonian (2) is transformed to block off-diagonal chiral form ^U H( k ) ^U (cid:0)1 = 0BB 0 0 (cid:0)i(cid:1)1 vF jkje(cid:0)i(cid:18)0 0 vF jkjei(cid:18) (cid:0)i(cid:1)2i(cid:1)1 vF jkje(cid:0)i(cid:18) 0 0vF jkjei(cid:18) i(cid:1)2 0 0 1CCA : (6)The energy eigenvalue of filled states is given by eE1;2 = (cid:0)rv2F jkj2 + ~(cid:1)1 (cid:6) q ~(cid:1)22 + v2F jkj2((cid:1)1 (cid:0) (cid:1)2)2: The corresponding chiral eigenstates of Hamiltonian (6) are easily given by j u
1i = 0BB vF jkj eE(cid:0)11 a1e(cid:0)i(cid:18)i eE(cid:0)11 b1ivF jkj 1e(cid:0)i(cid:18)1 1CCA ; j u
2i = 0BB i eE(cid:0)11 b2vF jkj eE(cid:0)11 a2ei(cid:18)1ivF jkj 1ei(cid:18) 1CCA ;j u
3i = 0BB vF jkj eE(cid:0)12 a01e(cid:0)i(cid:18)i eE(cid:0)12 b01ivF jkj 2e(cid:0)i(cid:18)1 1CCA ; j u
4i = 0BB i eE(cid:0)12 b02vF jkj eE(cid:0)12 a02ei(cid:18)1ivF jkj 2ei(cid:18) 1CCA ; (7)where a1(2) = 1 + (cid:1)1(2) 1; b1(2) = (cid:0)(cid:1)2(1) + v2F jkj2 1;a01(2) = 1 + (cid:1)1(2) 2; b01(2) = (cid:0)(cid:1)2(1) + v2F jkj2 2; 1(2) = (cid:1)2 (cid:0) (cid:1)1A1(2) ; A1(2) = (cid:1)21 + v2F jkj2 (cid:0) eE21(2):
To facilitate the calculation, it helps to introduce the projection operator ^p(k) = Pi2filled j u ii h u ij .For what follows, it is convenient to introduce the Q matrix by ^Q(k) = 2^p(k) (cid:0) ^1 . Corresponding to theblock-off-diagonal chiral symmetric Hamiltonian (6), the Q(k) matrix is also block-off-diagonal: ^Q(k) = (cid:18) 0 ^q(k)^qy(k) 0 (cid:19) :
Here is a zero matrix with by dimension and q(k) is the off-diagonal component of Q matrix.There can be a topological invariant, which is obtained only in the presence of a symmetry. Indeed, thechiral symmetry gives rise to result in winding number topological invariant. We are now set to cal-culate the topological invariant winding number via the ^q(k) matrix. Having more complicated chiraleigenfunctions (7), we try to find a huge expression for ^q(k) matrix, and neglect to write it here. In-evitably, the numerical method may be used to find the winding number. The analytical expression forthe off-diagonal block of spectral projector matrix is unwieldy, and further treatment about evaluatingthe topological invariant of this system are processing now.6 Effective subgap
To more clarify the mixed superconducting state exhibition, we focus on superconductor effective gaporiginated from singlet and triplet correlations. Actually, magnitude of mixed effective gap dependson the relative amplitude between the singlet and triplet components, which can control the height offorming subgap at the interface, playing a crucial role in hole-reflection for incident electrons. In orderto derive the exact form of effective gap, we need to refer energy spectra of topological insulator inproximity of a s -wave and p -wave superconductor, separately [36] (cid:15)2(s(cid:0)wave) (cid:0) j(cid:1)sj2 = (vF jkj (cid:7) (cid:22)s)2 ;(cid:15)(p(cid:0)wave) = (cid:6)vF jkj (cid:7) q(cid:22)2s + j(cid:1)pj2: We reconstruct the energy spectra of Eq. (4) as
E2mix: (cid:0) j(cid:1)eff j2 = (cid:18)vF jkj (cid:6) p"R2vF jkj (cid:19)2 ; in order to exploit an exact expression for effective mixed gap as following: j(cid:1)eff j = vuut ~(cid:1)1 (cid:0) j(cid:1)1 (cid:0) (cid:1)2j24 (cid:0) ~(cid:1)22(cid:22)0s + q(cid:22)02s + 4 ~(cid:1)22 : (8)The normalized chemical potential (cid:22)0s = 2(cid:22)2s + 12 j(cid:1)1 (cid:0) (cid:1)2j2 indicates mixed spin channels.The position of superconducting gap in (cid:0) point corresponds to the relation
12 ((cid:22)0s + q(cid:22)02s + 4 ~(cid:1)22)1=2:
It should be noted that in the limit of alone singlet or triplet case, the (cid:0) point only depends on (cid:22) , whilein the case of mixing potential, existence of two mixed components (cid:1)1 and (cid:1)2 causes to shift theposition of superconducting gap. We find (cid:1)eff to become zero in the absence of spin-singlet contributionachieving by (cid:1)1 = (cid:0)(cid:1)2 . Interestingly, in the lake of spin-triplet contribution, which is obtained by (cid:1)1 = (cid:1)2 , the effective gap is clearly reduced to the isotropic order parameter. This is completelyin agreement with previously reported results, that the former corresponds to the gapless topologicalinsulator superconductor state, and the latter means conventional s -wave superconducting excitationsone. The behavior of these cases is shown in Fig. 4. III TRANSPORT PROPERTIES
A Andreev tunneling conductance
In this section, we will focus on the transport properties of the simplest hybrid normal/superconductorstructure deposited on top of a topological insulator in order to investigate how Andreev reflection andconductance spectroscopy are influenced by the superconducting mixed order parameter. The uncon-ventional mixed superconductivity in TIs should manifest itself in the observable phenomena at theboundaries of a hybrid structure. We analyze Andreev reflection probability in the surface states byemploying a scattering matrix formulation along the lines of Blonder-Tinkham-Klapwijk (BTK) theory.To this end, let us now proceed to introduce the eigenstates of Hamiltonian (2). The wave function inthe topological insulator mixed superconducting is achieved from a set of coupled matrix equa-tions. Here, there are four unknowns to derive the eigenfunction in the electron-hole basis (Nambu basis), mix: = h k"; k . The normalization condition, j k"j2 + j k is used to conserve the intensity of the edge states. From equation
H mix: = Emix: mix: , we, aftercumbersome analytical calculations, express eigenfunction of a electron(hole)-like quasiparticle states interms of following equation: mix: = r 2A 0BB M1vF jkjM2ei(cid:18)vF jkjM3ei(cid:18)M4 1CCA ; (9)7here A is the normalization constant and M1 = (cid:1)1 j(cid:1)2j2 + (cid:1)2v2F jkj2 + (cid:1)1((cid:22)2s (cid:0) E2mix:);M2 = (cid:1)1((cid:22)s (cid:0) Emix:) + (cid:1)2((cid:22)s + Emix:);M3 = (cid:1)1(cid:1)(cid:3)2 + v2F jkj2 (cid:0) ((cid:22)s + Emix:)2;M4 = ((cid:22)s + Emix:) (cid:16)j(cid:1)2j2 + (cid:22)2s (cid:0) E2mix: + v2F jkj2(cid:17) (cid:0) 2(cid:22)sv2F jkj2:
Due to relativistic dynamics, two independent spin channels (cid:1)1 and (cid:1)2 are simultaneously appearedin the wave function. Because the motion of quasiparticles is determined by incidence angle (cid:18) , theresulting wave function is related to the direction of motion. If we define angle (cid:18) for right movers, thenleft movers is described by (cid:25) (cid:0)(cid:18) . Accordingly, pairing potentials spatially depend on direction of motion.Therefore, two spin channels in Eq. (9) are defined only for right movers, and we can replace them forleft movers by (cid:1)1(2) ! (cid:1)2(1) (see Fig. 1(a)). Also, the explicit wavevector of quasiparticles in termsof superconducting excitation energy and mixed-spin channels is given by jkj = E2mix: + (cid:22)2s (cid:0) (cid:1)1(cid:1)2 (cid:0) q4E2mix:(cid:22)2s + E2mix: j(cid:1)1 (cid:0) (cid:1)2j2 (cid:0) (cid:22)2s j(cid:1)1 + (cid:1)2j2:
To accommodate superconductivity by means of the proximity effect experimentally, it is necessary torealize the condition (cid:22)s (cid:29) j(cid:1)1;2j to have a sufficiently large density of states. In this way, a supercon-ductor electrode deposited on top of the topological insulator would be suitable experimentally, as Fig.1(b).The total wave function in the normal region of junction ( x < 0 ) by regarding two possible fatesupon scattering, normal and Andreev reflections of an incident electron, may then be written as: (cid:9)N = eikyN y (cid:16)eikeN x (cid:2)1 ei(cid:11) 0 0(cid:3)T +re(cid:0)ikeN x (cid:2)1 (cid:0) e(cid:0)i(cid:11) 0 0(cid:3)T + rAeikhN x (cid:2)0 0 1 (cid:0) e(cid:0)i(cid:11)h (cid:3)T (cid:17) ; (10)where (cid:11) and (cid:11)h denote the electron and hole angles of incidence, while r and rA are the normal andAndreev scattering coefficients, respectively. Due to the broken translational symmetry, the x -componentof the momentum in normal region ( kxN ) is non-conserved, whereas y -component ( kyN ) is conserved, andcan be acquired from normal region eigenstate. The Fermi momentum in the normal and superconductingpart of the system can be controlled by means of chemical potential in each region. Setting up thescattering wavefunctions and utilizing appropriate boundary condition, (cid:9)N = (cid:9)S at x = 0 , where (cid:9)S = te emix: + th hmix: , one is able to extract the normal and Andreev reflection coefficients, whichdepend on the angle of incidence and the mixed state channels excitation energy. We find followingsolutions for normal and Andreev reflection coefficients: r = (cid:0) hMe1(cid:17)4 (cid:0) Mh1 (cid:17)3i (cid:0) 1; (11-a) rA = (cid:0) hMe3(cid:17)4 (cid:0) Mh3 (cid:17)3i ; (11-b)where we have introduced (cid:0) = 2 os (cid:11)(cid:17)1(cid:17)4 (cid:0) (cid:17)2(cid:17)3 ;(cid:17)1(3) = Me2(4) + Me1(3)e(cid:0)i(cid:11)((cid:11)h);(cid:17)2(4) = Mh2(4) + Mh1(3)e(cid:0)i(cid:11)((cid:11)h): It follows, according to the BTK formalism [39], the normalized conductance (
G=G0 ) can be calculated,
G=G0 = Z (cid:25)=2(cid:0)(cid:25)=2 d(cid:11) os (cid:11) h1 + jrAj2 (cid:0) jrj2i ; (12)8nd the normalization constant is chosen as
G0 (cid:25) N (EF )we2=(cid:25)~2 , where
N (EF ) (cid:25) EF =2(cid:25)(~v2F ) isdensity of state with w being width of the junction.As we show now, the effect of the two distinct spin channels can be nicely seen in the experimen-tally accessible electrical conductance. In Fig. 5(a), we plot the subgap conductance spectra of the NSstructure resulting from the Andreev process, calculated with superconductor and normal region chem-ical potentials (cid:22)s = 10(cid:1)m and (cid:22)N = 1(cid:1)m , respectively. The maximum suppression of conductancehappens for the case of opposite spin channels, (cid:1)1 = (cid:0)(cid:1)2 . In this case, however, it seems that theappearance of unconventional superconductivity is manifested by an enhancement of the zero-bias con-ductance peak.Importantly, in the mixed state range , the two coherence conductance peaks existat Emix: = (cid:1)eff , and a transition of conductance peak into the zero-bias conductance can also beachieved by increasing the amplitude of (cid:1)2 . By focusing on the effective gap relation, Eq. (8), onecan find (cid:1)eff (cid:24)= (cid:1)s for high values of chemical potential of superconductor. Therefore, in orderto preserve mixed state subgap effect, we may apply minimum possible doping, where the condition j(cid:1)mj (cid:28) (cid:22)s; EF is still satisfied. Stehno et al have reported the experimental implementation of thisscenario [40]. The presence of s -wave pairing with subdominant p -wave admixture order parameterhas been predicted on N b /topological insulator/ Au devices, where the topological insulator is either al-loyed Bi1:5Sb0:5T e1:7Se1:3 or BiSbT eSe2 . Indeed, the conductance dips at the induced-gap value andthe increased conductance near zero energy in above both spectra of samples, can be explained by thedominant triplet superconducting components in 3DTI [40]. In analogy, in NCS superconductors withbroken inversion symmetry, the transport signatures in N/S junction depend on the degree of mixing ofsinglet and triplet pair potentials. In Ref. [25], Burset et al have analyzed tunneling conductance ofnormal/noncentrosymmetric superconductor junction, and reported a zero-bias conductance peak for thecase (cid:1)s < (cid:1)p , analogous to our finding, here.In Fig. 5(b), we present the signature of doping level of N region in resulting normal conductanceand formation of zero-bias conductance peak. A sharp conductance peak in zero-bias can be nicelyseen in a low doping, whereas the zero dip of conductance is appeared by increasing the normal regiondoping. It is interesting to note that the conductance peaks can also be controlled by changing the pairingpotential admixture. For comparison, we have included in Fig. 5(c) the conductance of junction withtwo possible p -wave symmetry functions. For d ( k ) = (cid:1)p os (cid:18)^z , the conductance peaks located at theeffective mixed gap is smaller than that for d ( k ) = (cid:1)p sin (cid:18)^z . This scenario becomes completely viceversa for resulting zero-bias conductance. To more clarify the signature of two mixed-state channels inconductance peak displacement, we present, in Fig. 5(d), subgap conductance curve in terms of (cid:1)2 andbias energy. This figure clearly demonstrates conductance peak displacement towards zero-bias with theincrease of magnitude of triplet pair potential. B Andreev Bound States in Josephson junction
We now consider the strictly one-dimensional superconductor/insulator/superconductor (S/I/S) Joseph-son junction in the x -direction on the surface of 3D topological insulator, as sketched in Fig. 1(c). Themeasurement of the supercurrent which is carried by Cooper pairs can be one of the useful tools to revealeffective symmetry manipulated by inducing an actual superconductivity. The mixed superconductiv-ity in topological insulator particularly manifests itself in the Josephson effect. The pairing potentialvanishes in the insulator middle region and is nonzero in the two superconductor terminals. The orderparameter is assumed to have different phases and the same amplitude in the left and right superconduc-tors. The insulator region length L (distance between two superconductor terminals) is assumed to bemuch smaller than the superconducting coherence length (cid:24) = ~vF =(cid:1) . For make contact with experi-mental parameters, the junction length should be smaller than . We introduce a gate potential U0 for insulator region for the possibility of electron scattering in the junction. In this condition, the maininteresting Klein tunneling effect takes place between the terminals that is independent of the barriershape.The wave function in the insulator region remains the same to normal region while we re-label (cid:9)S !(cid:9)rS and concomitantly (cid:8)te; th(cid:9) ! (cid:8)ter; thr(cid:9) for the right superconductor region x > L . The pairing9otential is a combination of singlet and triplet states which adopts the following form for each left andright S regions (cid:1)1;2 = (cid:26) (cid:0)(cid:1)s (cid:6) (cid:1)p ei(cid:18)(cid:1) ei'r ; x > L(cid:0)(cid:1)s (cid:7) (cid:1)p ei(cid:18)(cid:1) ei'l ; x < 0 : (13)The pairing potential is assumed to have different phases in the left and right regions, and the currentflowing the Josephson junction depends on the phase difference (cid:1)' = 'r (cid:0) 'l . It, then, remains tointroduce the wave function for the left superconductor region ( x < 0 ), which reads (cid:9)lS = tel emix: +thl hmix: . To identify the energy spectrum for the Andreev bound state, we match the wave functionsaround x = 0 , which yields (cid:9)lSx!0 = (cid:18) 1 + iZ(cid:27)1 00 1 (cid:0) iZ(cid:27)1 (cid:19) (cid:9)rSx!0; (14)where the barrier strength is defined as dimensionless parameter Z . By inserting the superconductingwave functions into Eq. (14), we arrive at four linear algebraic equations for the four constants ter , thr , tel and thl . For the case of (cid:1)p = 0 and (cid:1)s = 1 , which we have no longer mixed state, the ABS solutionsarrive at the well known previously reported equation [16]. When the spin state is mixed, finding theanalytical expression for ABS becomes impossible. The cumbersome and time-consuming analyticalcalculations has been done in this relation, and finally, from Eq. (14), we obtain an equation e(cid:0)2i'G1 + G2 = 0; where G1 and G2 are more complicated functions of bound energy, barrier parameter and incidence angle.We can numerically obtain ABS spectrum as a function of superconducting phase difference (cid:1)' andpropagation angle (cid:18) .We show that the same outcomes similar to those previously obtained for Josephson effect in topo-logical insulator with alone s -wave symmetry are achieved [41]. The -periodic gapless bound energiesin normal incidence (cid:18) = 0 are appeared, which are protected by the TR symmetry (see, Fig. 6(a)). Also,these states correspond to the chiral Majorana bound energy modes, so that the energy curves of electronand hole are continuously connected. The range of superconductor state admixture is controlled by themagnitude of spin channel (cid:1)2 , where we take the other mixed spin channel to be unit, (cid:1)1 = 1 . Hence,when (cid:1)2 is varied from to (cid:0)1 , the mixed state level is continuously changed from s -wave symmetryto p -wave one. Independent of admixture level tuned by (cid:1)2 , the ABS spectra exhibits zero energy andmaximum slope for superconductor phase difference (cid:1)' = (2n + 1)(cid:25) ( n is integer number). Whereas, (cid:1)' = 2n(cid:25) results in flat energy curve. It is noticed, that the amplitude of ABS oscillations significantlydiminishes in the mixed spin state. These features are presented in Fig. 6(a), where ABS plot are givenas a function of phase difference for the superconductor chemical potential and middle region insulatorstrength parameter magnitudes (cid:22)s = 15 and Z = 0:5 , respectively.For the critical case of mixed spin channel (cid:1)2 = (cid:0)1 , which our Josephson junction will be in purespin-triplet symmetric state, the ABS curvature goes to flattening. These behaviors of mixed supercon-ducting ABS can be originated from Dirac band gap creation and strongly effective subgap decreasingin the system. Furthermore, in Fig. 6(b), we plot bound state energy for finite angle of incidence as afunction of phase difference. As expected, the signature of nonzero incidences of quasiparticles to thesuperconductor/insulator interface is observed as vanishing chiral Majorana mode via the opening a largegap in ABS. Consequently, the period of ABS oscillations becomes in the presence of a momentummismatch, which is due to finite backscattering. The decrease of the amplitude of ABS is determined bythe mixing level and the angle of incidence. It should, however, be noted, that the change of amplitude ofABS curves with the incidence angle strongly depends on the magnitude of (cid:1)2 . We show increasing theangle of incidence in the range from to enhances the value of the bulk gap from to for themixing state ( (cid:1)2 = 0:2 ), whereas for (cid:1)2 = 1 ( s -wave superconductivity dominant case), it takes placefrom to . 10 supercurrent After the ABS spectrum is found, we numerically calculate the angle-averaged supercurrent that is givenby the phase difference dispersion of bound state energy
E((cid:1)') . The normalized Josephson current inthe short junction case can be calculated according to the standard expression [42],
I=I0 = Z (cid:25)=2(cid:0)(cid:25)=2 d(cid:18) os (cid:18) tanh (cid:18) E((cid:1)')2KBT (cid:19) dE((cid:1)')d(cid:1)' (15)where
I0 = ejkjW (cid:1)m(cid:25)~ is the normal current in a sheet of TI of width W , KB and T are the Boltzmannconstant and temperature, respectively. In Fig. 7(a), the Josephson current as a function of superconduct-ing phase difference is demonstrated for several magnitudes of (cid:1)2 . As a usual result in similar systems,the -periodic current-phase curve is found for every admixture level, in spite of the presence of thespin-triplet component of the pair potential. The main difference between the mixed-spin channel andpure spin-singlet one ( (cid:1)2 = 1 ), as shown in Fig. 7(a), is that the Josephson supercurrent is stronglysuppressed as the amplitude of the spin-triplet contribution grows upto (cid:1)p = 0:8 . In Fig. 7(b), we repeatthe previous calculation of Josephson current for different values of insulator barrier strengths Z . Here,it occurs an interesting scenario, where the exact sinusoidal curve of supercurrent is achieved in the caseof large Z . According previous specific work [43], the abrupt crossover phase-current curve originatedfrom the gapless ABS is observed in the low value of barrier strength. Finally, to clarify the signature ofmixed superconductivity on the critical current, which is defined as the maximum of Josephson current,we analyze numerically and plot the barrier strength dependence of critical current. Figure 7(c) showsthe normalized critical current I =I0 for different magnitudes of mixed-spin characteristic parameter (cid:1)2 .We show that the critical current strongly decreases with the increase of spin-triplet contribution. Thereason of this effect may be described by decreasing the effective superconducting gap in the mixed state.
IV SUMMARY AND CONCLUSIONS
In summary, from a more fundamental perspective, the distinction between the energy spectrum in themixed-spin state superconductors and surface states of topological insulators teaches us something newabout the interplay between mixed state of superconductivity and topologically protected by time-reversalsymmetry Dirac-like fermions. In one hand, the inversion symmetry breaking in a noncentrosymmetricsuperconductor, and gapless surface state resulted from spin-orbit coupling on the other hand, can bestrongly inter-correlated to capture the new effects in spin magnetization and spin transportation. Magne-toelectric effect caused by supercurrent in NCS superconductors has been reported, recently [44]. Thereis a delicate point, that the Hamiltonian of two-dimensional NCS superconductors is decoupled into twospin channels (cid:1)1 and (cid:1)2 with different energies. Whereas, in the presence of topological insulator,two spin channels are strongly coupled with the same energy, and both right-moving and left-movingelectron(hole) quasiparticles may experience the two spin channels.In this paper, we have analyzed the effect of proximity-induced mixed spin-singlet and spin-tripletsymmetry on the surface states of a topological insulator. The particle-hole and chiral symmetric prop-erties of Dirac-Bogoliubov-de Gennes Hamiltonian has been investigated to capture the topology class.We have introduced the new spin channels (cid:1)1 and (cid:1)2 for mixed state in the presence of topological insu-lator. Particularly, we have found a transformation matrix, under which the Hamiltonian is diagonalized,and, interestingly, the new mixed-spin channels were located at the diagonal elements. Consequently, itis formally expected to appear a Dirac mass-like gap in the surface states. This can be considered as akey feature of the present structure. It is noticed that there exist similar situation, when a magnetizationin z -direction mz is induced to 3DTI [16]. Therefore, one can report that mixed-state superconductivityinduction may play simultaneous role of magnetic field appearance in topological insulators. This isconsidered a significant feature, particularly in NCS superconductors [23, 45, 46, 47]. Next, we havefurther tried to clarify possible phase transition from original gapless in conventional superconductingto gaped surface states in unconventional mixed one via the evaluating the topological invariant windingnumber for the chiral eigenstates. To this end, we have constructed a chiral unitary matrix, under which11he Hamiltonian is transformed to its block-off-diagonal form. Because the spectral projection matrix ^q(k) has been obtained in an unwieldy analytical expression, then the winding number will be presentedin another work.Regarding the fact that superconducting electron-hole excitation in topological insulator is gaplesswith p -wave pairing symmetry, it was necessary to reveal the effective subgap in the mixed state case,which is identified to have a complicated dependency on mixed spin channels. However, we see a sizablesubgap on the Fermi surface, and it diminishes when the p -wave symmetry contribution is dominated.We have thus systematically proceeded to investigate the characteristic transport properties for subgaptunneling in N/S and Josephson S/I/S junctions. Our proposal has clear advantages in experimentalaccessibility. The Josephson current on the surface of the 3DTI has been experimentally observed, whereJosephson junction N b /epitaxial (Bi0:5Sb0:5)2T e3 / N b in the two steps have been fabricated and good
I (cid:0) V characteristics presented [11]. Also, tunneling conductance spectroscopy has been performedacross hetero-
N b /topological insulator/ Au , recently [40]. Acknowledgements
The authors would like to thank Vice-presidency and also National Elites foundation of I.R. of Iranunder grant number 15/295 for supporting the present work and post-doctorate course of MK at the UrmiaUniversity.
References [1] C. L. Kane, E. J. Mele, Phys. Rev. Lett. (2005) 146802.[2] L. Fu, C. L. Kane, E. J. Mele, Phys. Rev. Lett. (2007) 106803.[3] A. A. Baker, A. I. Figueroa, L. J. Collins-McIntyre, G. van der Laan, T. Hesjedal, Sci. Rep. (2015) 7907.[4] M. H. Fischer, A. Vaezi, A. Manchon, and E.-A. Kim, Phys. Rev. B (2016) 125303.[5] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A.Kim, N. Samarth et al., Nature (London) (2014) 449.[6] Z.-Z. Li, F.-C. Zhang, Q.-H. Wang, Sci. Rep. (2014) 6363.[7] A. M. Cook, M. M. Vazifeh, M. Franz, Phys. Rev. B (2012) 155431.[8] A. Cook, M. Franz, Phys. Rev. B (2011) 201105.[9] M-X. Wang, C. Liu, J-P. Xu, F. Yang, L. Miao, M-Y. Yao, C. L. Gao, C. Shen, X. Ma, X. Chen et al., Science (2012) 52.[10] J-P. Xu, C. Liu, M-X. Wang, J. Ge, Z-L. Liu, X. Yang, Y. Chen, Y. Liu, Z-A. Xu, C-L. Gao et al., Phys. Rev.Lett. (2014) 217001.[11] H. Zhang, X. Ma, L. Li, D. Langenberg, C-G. Zeng, G-X. Miao, J. Mater. Res. (2018) 2423.[12] R. Klett, J. Schonle, A. Becker, D. Dyck, K. Borisov, K. Rott, D. Ramermann, B. Buker, J. Haskenhoff, J.Krieft et al., Nano Lett. (2018) 1264.[13] M. Kayyalha, A. Kazakov, I. Miotkowski, S. Khlebnikov, L. P. Rokhinson, Y. P. Chen, arXiv:1812.00499.[14] J-P. Xu, M-X. Wang, Z. L. Liu, J-F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian et al., Phys.Rev. Lett. (2015) 017001.[15] J. Wiedenmann, E. Liebhaber, J. Kubert, E. Bocquillon, P. Burset, C. Ames, H. Buhmann, T. M. Klapwijk,L. W. Molenkamp, Phys. Rev. B (2017) 165302.[16] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa, Phys. Rev. B (2010) 184525.[17] Y. Tanaka, T. Yokoyama, N. Nagaosa, Phys. Rev. Lett. (2009) 107002.[18] J-P. Xu, M-X. Wang, Z. L. Liu, J-F. Ge, X. Yang, C. Liu, Z.A. Xu, D. Guan, C.L. Gao, D. Qian et al., Phys.Rev. Lett. (2015) 017001.
19] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa, Phys. Rev. Lett. (2010) 067001.[20] A. Yamakage, M. Sato, K. Yada, S. Kashiwaya, Y. Tanaka, Phys. Rev. B (2013) 100510(R).[21] M. Khezerlou, H. Goudarzi, Phys. Rev. B (2016) 115406.[22] J. Linder, A. Sudbo, Phys. Rev. B (2008) 064507.[23] T. Shang, S. K. Ghosh, L-J. Chang, C. Baines, M. K. Lee, J. Z. Zhao, J. A. T. Verezhak, D. J. Gawryluk, E.Pomjakushina, M. Shi et al., arXiv:1901.01414[24] P. A. Frigeri, D. F. Agterberg, A. Koga, M. Sigrist, Phys. Rev. Lett. (2004) 097001; (2004) 099903(E).[25] P. Burset, F. Keidel, Y. Tanaka, N. Nagaosa, B. Trauzettel, Phys. Rev. B (2014) 085438.[26] A. B. Vorontsov, I. Vekhter, M. Eschrig, Phys. Rev. Lett. (2008) 127003.[27] Y. Tanaka, T. Yokoyama, A. V. Balatsky, N. Nagaosa, Phys. Rev. B (2009) 060505.[28] L. Santos, T. Neupert, C. Chamon, C. Mudry, Phys. Rev. B (2010) 184502.[29] A. P. Schnyder, P. M. R. Brydon, C. Timm, Phys. Rev. B (2012) 024522.[30] E. Bauer, M. Sigrist, Non-Centrosymmetric Superconductors (Springer-Verlag, Berlin, 2012).[31] Jorge Cayao and Annica M. Black-Schaffer, Phys. Rev. B (2017) 155426.[32] A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W. Ludwig, Phys. Rev. B (2008) 195125.[33] M. Amundsen, H. G. Hugdal, A. Sudbo, J. Linder, Phys. Rev. B (2018) 144505.[34] M. Khezerlou, H. Goudarzi, S. Asgarifar, Phys. Lett. A (2018) 351.[35] K. Zhang, Q. Cheng, Supercond. Sci. Technol. (2018) 075001.[36] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa, Phys. Rev. B (2010) 184525.[37] M. Sato, S. Fujimoto, Phys. Rev. B (2009) 094504.[38] J. Zak, Phys. Rev. Lett. (1989) 2747.[39] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B (1982) 4515.[40] M. P. Stehno, N. W. Hendrickx, M. Snelder, T. Scholten, Y. K. Huang, M. S. Golden, A. Brinkman, Semicond.Sci. Technol. (2017) 094001.[41] M. Snelder, M. Veldhorst, A. A. Golubov, A. Brinkman, Phys. Rev. B (2013) 104507.[42] C. W. J. Beenakker, in Transport Phenomena in Mesoscopic Systems, edited by H. Fukuyama, T. Ando(Springer, Berlin, 1992)[43] G. Tkachov, E. M. Hankiewicz, Phys. Rev. B (2013) 075401.[44] W-Y. He, K. T. Law, arXiv:1902.02514; L. S. Levitov, Y. Nazarov, G. M. Eliashberg, JETP (1985) 445;V. M. Edelstein, Phys. Rev. Lett. (1995) 2004.[45] B. Fak, M. Enderle, G. Lapertot, J. Phys.: Conf. Ser. (2017) 012006.[46] D. Singh, K. P. Sajilesh, S. Marik, A. D. Hillier, R. P. Singh, Supercond. Sci. Technol. (2017) 125003.[47] M. X. Wang, Y. Xu, L. P. He, J. Zhang, X. C. Hong, P. L. Cai, Z. B. Wang, J. K. Dong, S. Y. Li, Phys. Rev.B (2016) 020503(R). igure captionsFigure 1(a), (b), (c), (d) (color online) (a) A schematic of two coupled mixed-spin channels (cid:1)1 and (cid:1)2 for right-left-moving quasiparticles, (b) and (c) sketch of the topological insulator-based N/S subgaptunneling and S/I/S Josephson junctions with mixed-spin state superconductivity, (d) A polar plot of twospin channels (cid:1)1 and (cid:1)2 with the incidence angle is shown for three different values of (cid:1)s and (cid:1)p .The solid lines denote the (cid:1)1 , while the crossed lines correspond to the (cid:1)2 . Black curves representpair potential for j(cid:1)sj = 0:8; j(cid:1)pj = 0:2 , violet curves for j(cid:1)sj = j(cid:1)pj = 0:5 and blue curves for j(cid:1)sj = 0:2; j(cid:1)pj = 0:8 . Figure 2 (color online) The excitation spectra in superconductor topological insulator, calculated fromEq. (4). The mixed superconducting state features a mass-like gap in topological insulator.
Figure 3(a), (b), (c) (color online) Contour plot of superconducting excitation spectra on the surfacestate of topological insulator for (a) j(cid:1)2j = 0 , (b) j(cid:1)2j = 1 and (c) j(cid:1)2j = (cid:0)1 . It is seen that thesuperconducting zero energy only occurs in j(cid:1)2j = (cid:0)1 . Figure 4 (color online) Effective band gap resulting from Eq. (8) with (cid:22)s = 0:1(cid:1)m as a function of (cid:1)1 and (cid:1)2 . Figure 5(a), (b), (c), (d) (color online) Normalized tunneling conductance versus bias voltage in N/Sjunction. The plots in (a) show the results for different values of (cid:1)2 and the effect of the pairing potentialis indicated, (b) show the results for different values of (cid:22)N and (c) indicate results for two differentsymmetry of p -wave functions. The tunneling conductance versus bias voltage and spin channel (cid:1)2 isplotted in (d). Figure 6(a), (b) (color online) (a) Plot of the Andreev bound state energy versus phase difference when (cid:18) = 0 . The black solid line corresponds to j(cid:1)2j = 1 , blue solid line to j(cid:1)2j = 0:6 , red solid line to j(cid:1)2j = 0:2 while the green dashed line corresponds to j(cid:1)2j = 0 . Also, the pink dashed line correspondsto j(cid:1)2j = (cid:0)0:2 , light blue dashed line to j(cid:1)2j = (cid:0)0:6 and light green dashed line to j(cid:1)2j = (cid:0)1 , (b)Plot of the Andreev bound state energy versus phase difference for various values of (cid:18) . The black curvescorrespond to j(cid:1)2j = 1 and pink curves to j(cid:1)2j = 0:2 . Dependence of the Andreev bound state energyfor diferent values (cid:18) = 0 (pink solid line), (cid:18) = 0:1(cid:25) (pink dashed line), (cid:18) = 0:2(cid:25) (pink dash-dotted line)and (cid:18) = 0:25(cid:25) (pink dotted line) when (cid:22)s = 15(cid:1)m and
Z = 0:5 is shown.
Figure 7(a), (b), (c) (color online) Plot of the normalized Josephson supercurrent as a function of thephase difference with respect to varying (a) (cid:1)2 and (b) barrier parameter Z when (cid:22)s = 10(cid:1)m . We haveset Z = 0:25 for (a) and j(cid:1)2j = 0:6 for (b). Plot (c) represents the critical current versus barrier strengthwhen (cid:22)s = 10(cid:1)m . 14 (d)
Figure 1: (a), (b), (c), (d) -2 -1 0 1 2 |k| S up e r c ondu c t i ng E xc i t a t i on s s =1 p =0 s =0.6 p =0.4 s =0.4 p =0.6 s =0 p =1 Dirac gap
Figure 2:15 k x -2-1012 k y (a) -2 -1 0 1 2 k x -2-1012 k y (b) -2 -1 0 1 2 k x -2-1012 k y (c) Figure 3: (a), (b), (c) E ff ec t i ve G a p Figure 4:16 mix / m G / G =1 =0.6 =0.2 =0 =-0.2 =-0.6 =-1 (a) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 mix / m G / G N =1 mN =3 mN =5 mN =7 mN =9 m =0.2(b) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2mix / m0.50.7511.251.5 G / G d(k)= p cos s d(k)= p sin s (c) m i x / m (d) Figure 5: (a), (b), (c), (d) / -1-0.500.51 A nd r eev B ound S t a t es (a) / -1-0.500.51 A nd r eev B ound S t a t es (b) Figure 6: (a), (b) / -0.6-0.4-0.200.20.40.6 J o se ph s on c u rr e n t =1 =0.6 =0.2 =0 =-0.2 =-0.6 (a) / -0.500.5 J o se ph s on C u rr e n t Z=1Z=0.5Z=0.1 (b) Z/ I C / I =1 =0.6 =0.2 =-0.2 (c) Figure 7: (a), (b), (c)(a), (b), (c)