Anisotropic Andreev reflection in semi-Dirac materials
AAnisotropic Andreev reflection in semi-Dirac materials
Hai Li, ∗ Xiang Hu,
1, 2 and Gang Ouyang † Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,Key Laboratory for Matter Microstructure and Function of Hunan Province,School of Physics and Electronics, Hunan Normal University, Changsha 410081, China Department of Physics, William & Mary, Williamsburg, VA 23187, USA (Dated: January 6, 2021)In the framework of Bogoliubov-de Gennes equation, we theoretically study the Andreev reflec-tion in normal-superconducting junctions based on semi-Dirac materials. Owing to the intrinsicanisotropy of semi-Dirac materials, the configuration of Andreev reflection and differential conduc-tance are strongly orientation-dependent. For the transport along the linear dispersion direction,the differential conductance exhibits a clear crossover from retro Andreev reflection to specularAndreev reflection with an increasing bias-voltage, and the differential conductance oscillates with-out a decaying profile when the interfacial barrier strength increases. However, for the transportalong the quadratic dispersion direction, the boundary between the retro Andreev reflection andspecular Andreev reflection is ambiguous, and the differential conductance decays with increasingthe momentum mismatch or the interfacial barrier strength. We illustrate the pseudo-spin texturesto reveal the underling physics behind the anisotropic coherent transport properties. These resultsenrich the understanding of the superconducting coherent transport in semi-Dirac materials.
I. INTRODUCTION
Semi-Dirac material (SDM) has recently been exper-imentally realized in black phosphorus with in situ de-position of K [1] or Rb [2] atoms, offering a promisingplayground for further exploring the exotic attributes ofSDMs. Unlike most Dirac materials that possess linerdispersions in all momentum-space directions [3–5], inSDMs the low-energy excitations disperse quadraticallyin one direction but linearly along the orthogonal direc-tion [6–10]. The unique band structures of SDMs areresponsible for a series of novel phenomena [9–24], in-cluding the consequences of anisotropic aspect in the su-perconducting order parameter correlations [25–27]. Re-cent theoretical efforts have demonstrated that the su-perconductivity in SDMs can be induced by arbitrar-ily weak attractions in the present of random chemi-cal potential [25]. Resorting to the mean-field calcula-tion [26] and renormalization group analysis [27], it isrevealed that the s-wave superconductivity is more fa-vorable in SDMs. More strikingly, owing to the intrinsicanisotropy, the stiffness of superconducting order param-eter and the divergence behavior of correlation length arehighly orientation-dependent [26, 27]. These progresses,together with the developments in the materialization ofSDMs, provide foundations for exploring coherent trans-port properties in SDM-based normal-superconducting(NS) junctions.In a NS junction with ideal contacts, the transportproperties are dominated by the Andreev reflection (AR)in the subgap energy regime of E ≤ ∆ , with E the inci-dent energy and ∆ the superconducting gap [28, 29]. Inmost conventional-metal-based NS junctions, the chem-ical potential in the N region satisfies µ N (cid:29) ∆ , andthe AR is a intra-band phase-coherent scattering process,during which an incident electron from the N region isreto-reflected as a hole [28–31]. While in the NS junctions based on Dirac materials, µ N can be continuously tunedto the subgap regime satisfying µ N < E [3, 4, 32]. Con-sequently, a conduction-band electron incident from theN region is specularly reflected back as a valence-bandhole, leading to a inter-band phase-coherent scatteringprocess known as specular-AR [32–38]. Remarkably, byincreasing E within the subgap regime, a crossover fromreto-AR to specular-AR occurs, manifesting itself as adip at E = µ N in the E -dependent conductance spec-trum. This signature has been experimentally observedin bilayer-graphene-based NS junctions [39]. Moreover,in Dirac-material-based NS junction, due to the novelmomentum-spin/pseudo-spin textures of Dirac fermions,the subgap differential conductance oscillates with theinterfacial barrier strength without a decay profile [34].Although the scenarios of AR in the systems with purequadratic [28–31] or linear [32–38] dispersions have trig-gered extensive studies, the AR and related subgap con-ductance in SDM-based NS junctions have received noattention to date. Since the low-energy excitations inSDMs host unique dispersions intermediate between thequadratic and linear energy spectra, it is natural to askhow the intrinsic anisotropy manifests itself in the su-perconducting coherent transport. In this paper we in-vestigate the subgap transport properties in SDM-basedNS junctions. The manifestations of the anisotropic dis-persion in the subgap transport can be summarized astwo points. First, we find a clear crossover from retro-AR to specular-AR when the transport along the linearlydispersion direction, while for the transport along thequadratical dispersion direction, the boundary betweenretro-AR and specular-AR is ambiguous. Second, theinfluences of momentum-mismatch and interfacial bar-rier on the subgap transport are strongly orientation-dependent. For the transport along the quadratic dis-persion direction, the subgap differential conductancerapidly decays with increasing the interfacial barrier a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n (a)N S y x (b) FIG. 1. (Color online) Schematic plots of SDM-based NSjunctions extending along (a) the x -direction and (b) the y -direction, and the low-energy excitations in the N regions dis-perse linearly and quadratically along the x - and y -directions,respectively. We assume the chemical potential in the N andS regions could be tuned independently. In the N region, E + e denotes the electron-like conduction band, and E +( − ) h repre-sents the hole-like valence (conduction) band. In the S re-gion, E + − S indicates one branch of Bogoliubov quasiparticledispersion. The solid (open) circles denote electron (hole)-likequasiparticles and the arrows indicate the directions of groupvelocities. strength or momentum-mismatch between the N andS regions. Whereas the transport in the linear disper-sion direction is insensitive to the momentum-mismatch,and the subgap differential conductance periodically os-cillates with the interfacial barrier strength without adecaying profile. We illustrate the pseudo-spin texturesof semi-Dirac fermions to understand our findings.This paper is organized as follows. We present themodel and method in Sec. II. In Sec. III, we give thenumerical results and concentrate on the manifestationsof intrinsic anisotropy of SDMs in the subgap transportproperties. The conclusions are briefly drawn in Sec. IV. II. MODEL AND METHOD
To address the effects of the intrinsic anisotropy onthe subgap transport properties, we introduce two repre-sentative SDM-based NS junctions extending along the x - and y - axes, respectively. As schematically shownin Fig. 1, the low-energy dispersion is linear (quadratic)in the N region of the NS junction extending along the x ( y )-axis. We consider the transport properties alongthe x ( y )-direction in the NS junction extending alongthe x ( y )-axis, and assume that the translational symme-try in the y ( x )-direction is preserved, so that the trans-verse momentum k y ( k x ) can be treated as a good quan-tum number [32–38]. Moreover, under this assumptionthe influences of boundary effects on the transport canbe rationally neglected. In the S region, we take theintra-sublattice/orbit s-wave pairing, as proposed by re-cent theoretical work [26, 27]. In practice, the super-conductivity in the S region can be induced by a s-wavesuperconductor via the proximity effect, as implementedin similar NS junctions based on graphene [39, 40] andWeyl semimetals [41, 42]. Under these lines, the Bogoliubov–de Gennes (BdG)Hamiltonian describing the low-lying physics is given by[26, 27] H BdG = (cid:18) h ( k ) − µ ( r ) ∆( r )∆ † ( r ) − h ( k ) + µ ( r ) (cid:19) , (1)acting on the pseudo-spin ⊗ Nambu space. The single-particle effective Hamiltonian h ( k ) = (cid:126) vk x σ x + ηk y σ y ,where σ x,y label the Pauli matrices operating on thepseudo-spin space, v represents the Fermi velocity, and η ≡ (cid:126) / (2 m ) with m the effective mass. The chemi-cal potential µ ( r ) = µ N Θ( − r ) + µ S Θ( r ), where Θ( r )is the Heaviside step function and µ S ( µ N ) denotes thechemical potential in the S (N) region. In this paper, weassume that the relation of µ S (cid:29) µ N is satisfied, so thatthe leakage of Cooper pairs from the S to N regions canbe safely neglected [33–38] . In doing so, the pair poten-tial can be effectively modeled by a step function, i.e.,∆( r ) = ∆ σ e iφ Θ( r ), with φ the superconducting phaseand σ a 2 × E νλS = ν (cid:114)(cid:16)(cid:113) (cid:126) v k x + η k y + λµ S (cid:17) + ∆ , (2)where ν = ± and λ = ± indicate the different branches.To ensure the validity of mean-field approximation, therelation of ∆ (cid:28) µ S should be satisfied. Therefore, thebranches E ν + S are far away from the subgap regime andonly E ν − S bands need to be considered. The schematicplots of E + − S as functions of k x and k y are shown in Fig. 1(a) and (b), respectively. In the N region, the low-energyspectrum can be formulated as E ± e ( h ) = ± (cid:113) (cid:126) v k x + η k y − (+) µ N , (3)where the subscripts e and h denote the electronlike andholelike excitation spectra, respectively.According to Eq. (3), the group velocities in the Nregion can be parameterized as v e ( h ) ,x = (cid:126) v k x ε +( − ) , v e ( h ) ,y = 2 η k y (cid:126) ε +( − ) , (4)where ε ± = E ± µ N . The components v e ( h ) ,x and v e ( h ) ,y exhibit distinct behaviors with respect to the momentum,reflecting the anisotropic aspect of SDMs. We note thatdue to the intrinsic anisotropy, in the scattering issuesthe scattering angle should be defined as one betweenthe directions of associated group velocity and current,instead of the angle between the directions of momentumand current. Resorting to Eq. (4), for a NS junction ex-tending along the y -axis, the scattering angle should be α ye ( h ) ≡ arctan( v e ( h ) ,x /v e ( h ) ,y ) = (cid:126) v k x / (2 η k y ), differ-ing from the angle of arctan( k x /k y ) in isotropic systems[33–36]. For a NS junction extending along the x -axis,the scattering angle α xe ( h ) = π/ − α ye ( h ) . A. NS junction extending along the x -axis We now turn to the scattering problem in a NS junctionextending along the x -axis. In practice, at the bound-ary between the N and S regions, the defects and latticemismatch are inevitable during the fabrication of device,which may profoundly influence the transport properties.To take into account the proposed effects, we introducea interfacial barrier U ( x ) ≡ U Θ( x )Θ( d − x ), and takethe limit of U → ∞ and d → U d/ ( (cid:126) v ) ≡ Z being finite. We note that in the current direction, thelow-energy excitations in SDMs are similar as that inDirac materials, thus the scattering amplitudes can becalculated by the standard procedure employed in relatedDirac-material-based NS junctions [32–38]. By solving t eq ψ + eq + t hq ψ + hq = M ( ψ + e + r ee ψ − e + r he ψ − h ) (5)at the boundary of x = 0, we can obtain r he ( ee ) and t eq ( hq ) , which are the reflection amplitude for the AR(normal reflection) and the transmission amplitude forthe electron(hole)-like quasiparticle, respectively. Thedetails of basis scattering states ψ + eq ( hq ) and ψ ± e,h are, re-spectively, given by Eqs. (B1) and (B3). The transfermatrix M = e iσ x τ Z , with τ a 2 × µ S (cid:29) max( µ N , E, ∆ ), the re-flection amplitudes are, respectively, given by r ee = (cid:126) v ( k − e ε − − k + h ε + ) cos β + i ( (cid:126) v k − e k + h − ε + ε − ) sin β (cid:126) v ( k + h ε + + k + e ε − ) cos β + i ( (cid:126) v k + e k + h + ε + ε − ) sin β , (6a) r he = 2 (cid:126) vk e ε + e − iφ (cid:126) v ( k + h ε + + k + e ε − ) cos β + i ( (cid:126) v k + e k + h + ε + ε − ) sin β , (6b)where k ± e ( h ) = k e ( h ) ± iηk y / ( (cid:126) v ), k e ( h ) =sgn[ ε +( − ) ] (cid:113) ε − ) − η k y / ( (cid:126) v ), and β =cos − ( E/ ∆ )Θ(∆ − E ) − i cosh − ( E/ ∆ )Θ( E − ∆ ).As can be seen, both r ee and r he are independent of Z , implying that in the limit of µ S (cid:29) max( µ N , E, ∆ ),the transport properties along the x -axis direction areinsensitive to the interfacial barrier.Resorting to the reflection amplitudes, the probabili-ties for the normal reflection (NR) and AR processes canbe, respectively, defined as R ee = (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ψ − e |J x | ψ − e (cid:105)(cid:104) ψ + e |J x | ψ + e (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) | r ee | , (7a) R he = (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ψ − h |J x | ψ − h (cid:105)(cid:104) ψ + e |J x | ψ + e (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) | r he | , (7b) where the particle current density operator J x ≡ − i (cid:126) [ x, H BdG ] = vσ x τ z , with τ z the Pauli matrix operatingon the Nambu space.Taking advantage of the Blonder-Tinkham-Klapwijk(BTK) formula [31], the zero-temperature differentialconductance along the x -direction is given by G x ( eV ) = 2 e Wh (cid:90) dk y π [1 − R ee ( eV, k y )+ R he ( eV, k y )] , (8)where V indicates the bias voltage and W is the widthalong the y -direction of the junction. To normalize theconductance, we define G x ( eV ) = e Wπh (cid:112) | eV + µ N | /η ,which is the conductance along the x -direction of a SDM-based NN junction in the ballistic limit. B. NS junction along the y -axis In a NS junction along the y -axis, the low-energy ex-citations of the N region disperse quadratically in thecurrent direction. For electron(hole)-like excitations with | k x | ≤ | E +( − ) µ N | / ( (cid:126) v ), Eq. (3) determines four possiblescattering modes, including two propagating modes with k y = ± q e ( h ) and two evanescent ones with k y = ± iκ e ( h ) .The related parameters are given by q e ( h ) = s e ( h ) (cid:114)(cid:113) ε − ) − (cid:126) v k x /η, (9a) κ e ( h ) = (cid:114)(cid:113) ε − ) − (cid:126) v k x /η, (9b)with s e ( h ) = sgn[ ε +( − ) ]. Although the evanescent modesdo not contribute to the current, they need to be includedin the wave functions to get correct boundary conditions.With this in mind, for a propagating electron-like inci-dent mode, the wave function in the N region should beparameterized asΦ N = ϕ + e, + ˜ r ee, ϕ − e, + ˜ r ee, ϕ − e, + ˜ r he, ϕ − h, + ˜ r he, ϕ − h, , (10)with ˜ r ee ( he ) , and ˜ r ee ( he ) , the reflection amplitudes ofpropagating and evanescent modes in the NR(AR) pro-cesses, respectively. The detailed structures of relatedbasis scattering states are given by Eq. (B5). In the Sregion, the wave function takes the form ofΦ S = ˜ t eq, ϕ + eq, +˜ t eq, ϕ + eq, +˜ t hq, ϕ + hq, +˜ t hq, ϕ + hq, , (11)where the associated basis scattering states are shownin Eq. (B6). The coefficients ˜ t eq ( hq ) , and ˜ t eq ( hq ) , , re-spectively, represent the transmission amplitudes for thepropagating and evanescent electron(hole)-like quasipar-ticles.To model the effects resulting from the interfacial de-fects and lattice mismatch, we introduce a interfacial bar-rier modeled by ˜ U ( y ) ≡ ˜ U δ ( y ), with δ ( y ) being the Deltafunction. In doing so, the boundary conditions can beformulated as Φ S | y =0 + = Φ N | y =0 − , (12a) ∂ y Φ S | y =0 + = ˜ M Φ N | y =0 − , (12b)where ˜ M = ∂ y − ˜ U η σ y τ . For brevity of notation, weintroduce a parameter χ ≡ ˜ U ηq FS to characterize the in-terfacial barrier strength, with q FS = (cid:112) µ S /η . The prob-abilities for the NR and AR processes are, respectively,defined as˜ R ee, = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ϕ − e, |J y | ϕ − e, (cid:105)(cid:104) ϕ + e, |J y | ϕ + e, (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ r ee, | , (13a)˜ R he, = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) ϕ − h, |J y | ϕ − h, (cid:105)(cid:104) ϕ + e, |J y | ϕ + e, (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ r he, | , (13b)with the particle current density operator J y ≡ − i (cid:126) [ y, H BdG ] = − iη (cid:126) σ y τ z ∂y . We note that the probabil-ities of evanescent modes ˜ R ee, and ˜ R he, are vanishingand do not contribute to the current. Therefore, in accor-dance with the BTK formula [31], the zero-temperaturedifferential conductance along the y -direction can bewritten as G y ( eV ) = 2 e Lh (cid:90) dk x π (cid:104) − ˜ R ee, ( eV, k x )+ ˜ R he, ( eV, k x ) (cid:105) , (14)where L is the width along the x -direction of the junc-tion. It is convenient to normalize the conductance by G y ( eV ) = e Lπh | eV + µ N | (cid:126) v , which represents the conduc-tance in the y -direction of a SDM-based NN junction inthe ballistic limit. III. RESULTS AND DISCUSSIONA. Pseudo-spin Textures
To understand the underlying physics behind thetransport properties, it is instructive to reveal thepseudo-spin textures in the N regions of SDMs-based NSjunctions. To do so, we define the pseudo-spin vector as P ≡ (cid:104) σ ⊗ τ (cid:105) , which is a unit vector consisting of theexpectation values of operator σ ⊗ τ in normalized basisscattering states, where σ ≡ ( σ x , σ y ) [43–46]. Specifi-cally, the pseudo-spin carried by a propagating electron-like mode can be formulated as P e ( k x , k y ) = (cid:32) (cid:126) vk x ε + , ηk y ε + (cid:33) . (15)Accordingly, the pseudo-spin components P e,x and P e,y depend, respectively, linearly and quadratically on the k x k y NRX IN k x k y INNRY (a) (b)
FIG. 2. (Color online) Schematic plots of pseudo-spin textureof an electron-like conduction band in the N region of a SDM-based NS junction, where the arrow denotes the pseudo-spinand the solid curve depicts the iso-energy contour. The sym-bol + ( − ) in (a) and (b), respectively, denotes the directionof v e,x and v e,y is parallel (anti-parallel) to the current direc-tion. The notations IN and NRX (NRY) indicate the k -spaceregions of incidence and normal reflection in the NS junctionextending along the x ( y )-axis. momentum, inheriting the intrinsic anisotropy of SDMs.Fig. 2 gives the schematic plots for the pseudo-spin tex-tures of electron-like conduction band. As can be seen, P e,y is always along the positive y -direction, regardless ofthe sign of k y , this scenario is profoundly different fromthat in Dirac materials [43–49]. As will be proposed inSec. III B, the unique pseudo-spin textures provide illu-minating elucidations for the subgap transport propertiesin SDMs-based NS junctions. B. Differential conductance
In this subsection, we proceed to analyze the numericalresults and discuss the differential conductance in thesubgap regime. What we mainly concentrate on are themanifestations of the intrinsic anisotropy in the subgaptransport properties, implemented by comparing the ARconfigurations and subgap differential conductance alongthe x - and y -axes. To ensure the validity of the mean-field approximation, we assume the S regions are heavilydoped to the regime of µ S (cid:29) ∆ , and we set µ S = 500∆ in the numerical calculation for definiteness.As a starting point, we focus on the bias-voltage-dependent subgap differential conductance, and analyzethe results in terms of the unique pseudo-spin textures.In a NS junction extending along the x -axis, the zero-biasconductance is insensitive to the momentum-mismatch(or alternatively, the ratio of µ N /µ S ). As shown inFig. 3(a), the differential conductance always exhibits azero-bias peak, regardless of the value of µ N . Remark-ably, for any nonvanishing µ N , G x (0) /G x (0) takes thesame value of . On the contrary, the G y (0) /G y (0)strongly depends on the momentum-mismatch in theNS junction extending along the y -axis. As can be N / = 0 0.1 0.5 1 5 10 G x / G x eV/ (a) (b) eV/ G y / G y N / = 0 0.1 0.5 1 5 10 eV/ G y / G y FIG. 3. (Color online) Bias-voltage-dependent differential conductance for NS junctions extending along the (a) x -directionand (b) y -direction, without interfacial barriers. The inset in panel (b) is the zoom-in of sub-gap conductance for µ N / ∆ = 0,0 .
1, 0 .
5, and 1. seen in Fig. 3(b), the subgap differential conductancealmost vanishes in the presence of large momentum-mismatch (e.g., µ N / ∆ = 0 , . , . , x -direction, for a small k y (i.e., with a small incident angle), the pseudo-spin car-ried by an electron-like incident mode is nearly oppo-site to that of the normally reflected one. Consequently,when k y is small enough, the conservation of pseudo-spin results in the prohibition of NR, and thus leads tothe enhancement of AR in the subgap regime, even inthe presence of strong momentum-mismatch. Moreover,since the conductance is mainly contributed by the modeswith small k y , the zero-bias conductance is insensitive tothe momentum-mismatch. However, in the NS junctionextending along the y -direction, a couple of incident andnormally reflected modes possess the same pseudo-spin,so that the back scattering is enabled. Therefore, the sub-gap differential conductance in the y -direction is stronglysuppressed by increasing the momentum-mismatch.Parenthetically, although the configuration of G x ( eV ) /G x ( eV ) shown in Fig. 3(a) is similar asthat in associated NS junctions based on graphene[33] and silicene [35, 36], there is a significant dif-ference on the value of G x (0) /G x (0). For eV = 0,according to Eqs. (6) and (7), the reflection prob-abilities reduces into R ee | eV =0 = η k y /µ N and R he | eV =0 = ( µ N − η k y ) /µ N , respectively. Sub-stituting the results into Eq. (8), we arrive at G x (0) /G x (0) = (cid:113) ηµ N (cid:82) √ µ N /η (2 − η k y /µ N ) dk y = ,differing from the value of in graphene- and silicene-based NS junctions [33, 35, 36]. This consequence canbe ascribed to the unique band structure of SDMsintermediate the linear and quadratic spectra. The anisotropic aspect of the subgap differential con-ductance also manifests itself in the distinct crossoverbehaviors from the retro-AR to specular-AR in the tworepresentative NS junctions. In the NS junction witha weakly doped N region satisfying µ N < ∆ , the sub-gap differential conductance vanishes at eV = µ N , dueto the absence of AR for any incident angles. Further-more, in the regimes of eV < µ N and eV > µ N , the sub-gap differential conductance is dominated by the retro-AR and specular-AR, respectively. Therefore, the pointof eV = µ N serves as the crossover boundary betweenthe retro-AR and specular-AR in the subgap regime. Inthe NS junction along the x -axis, as the bias-voltageincreases, the subgap differential conductance exhibitsa clear crossover from retro-AR to specular-AR, as de-picted by Fig. 4(a). However, as illustrated in Fig. 4(b),the boundary between retro-AR and specular-AR is am-biguous in the NS junction extending along the y -axis.This scenario can be understood from the unique pseudo-spin textures. In the NS junction extending in the y -axis, the incident and normally reflected modes carry thesame pseudo-spin, thus the NR is favorable, especiallyin the presence of large momentum-mismatch. Conse-quently, as illustrated in Fig. 3(b) and the inset therein,for µ N ≤ ∆ (cid:28) µ S , the AR process is strongly sup-pressed so that G y ( eV ) /G y ( eV ) ∼ G y ( µ N ) /G y ( µ N ) = 0in the subgap regime, thus blurring the boundary be-tween the retro-AR and specular-AR.We now turn to the effects of the interfacial barrierson the subgap differential conductance. For the transportalong the x -axis, the pseudo-spin carried by the incidentand normally reflected modes are nearly opposite to eachother when the incident angle is small enough, resultingin the prohibition of NR. As a consequence, the zero-biasdifferential conductance just oscillates with Z without adecaying profile, as charted out by Fig. 5(a). In the caseof the NS junction extending along the y -axis, the backscattering is available since the incident and normally re- Specular-AR eV/ N / (a)Retro-AR Specular-ARRetro-AR (b) eV/ N / FIG. 4. (Color online) Panels (a) and (b), respectively, present the contour plots of differential conductance G x ( eV ) /G x ( eV )and G y ( eV ) /G y ( eV ), without interfacial barriers. The differential conductance vanishes at the line of eV = µ N , indicating thecrossover boundary between retro-AR and specular-AR. Z/ N / (a) (b) / N / FIG. 5. (Color online) Contour plots of the zero-bias differential conductance (a) G x (0) /G x (0) and (b) G y (0) /G y (0), wherethe starting value of µ N is chosen as 10 − ∆ . flected modes host the same pseudo-spin. Therefore, theAR is strongly suppressed by increasing the interfacialbarrier strength, leading to the exponential decaying pro-file in the χ -dependent G y (0) /G y (0) shown in Fig. 5(b).In this regard, the influences of the interfacial barrier onthe subgap transport are orientation-dependent, reflect-ing the anisotropic aspect of SDMs. IV. CONCLUSION
In conclusion, we have studied the subgap transportproperties in SDMs-based NS junctions in the frame-work of BdG equation. In terms of the tight-bindingapproach, the BdG Hamiltonian has been derived froma graphene-like system proximitized to a s-wave super-conductor. We have figured out the manifestations ofthe intrinsic anisotropy of SDMs in the AR configura-tions and subgap differential conductance. For the trans-port along the linear dispersion direction, the subgapdifferential conductance exhibits a clear crossover from retro-AR to specular-AR by enhancing the bias-voltage,and the zero-bias differential conductance is insensitive tothe momentum mismatch. Moreover, when the interfa-cial barrier strength increases, the zero-bias conductanceexhibits an oscillating configuration without a decayingprofile. However, for the transport along the quadraticdispersion direction, the boundary between the retro-ARand specular-AR is ambiguous, and the the zero-bias dif-ferential conductance rapidly drops as the momentummismatch or the interfacial barrier strength increases.These results would provide some intriguing insights forthe coherent transport in SDMs-based superconductinghybrid structures, and we anticipate more interesting re-sults for the Andreev bound states and supercurrents inSDMs-based Josephson junctions.
ACKNOWLEDGMENTS
H. L. and X. H. would like to thank E. Rossi and R.Wang for helpful discussions. This work was supportedby the National Natural Science Foundation of China(Grants No 11804091 and No. 91833302), the Science andTechnology Planning Project of Hunan Province (GrantNo. 2019RS2033), the Hunan Provincial Natural Science Foundation of China (Grant No. 2019JJ50380), and theexcellent youth fund of the Hunan Provincial EducationDepartment (Grant No. 18B014).
Appendix A: Derivation of the BdG Hamiltonian
Physically, the semi-Dirac dispersion can be realized in a graphene-like systems with breaking the hexagonal sym-metry [7, 17, 20]. Thus we take an artificial honeycomb lattice model with anisotropic nearest-neighbor (NN) hoppingas a prototype. The NN tight-binding model containing an intro-sublattice/orbit Bardeen-Cooper-Schrieffer super-conducting pairing term is given by H = (cid:88) l ,α [( tB † l + m ,α A l ,α + tB † l + m ,α A l ,α + t (cid:48) B † l ,α A l ,α )+ h.c. ] − µ (cid:88) l ,α ( A † l ,α A l ,α + B † l ,α B l ,α )+ (cid:88) l ,α [ s ∆( A † l ,α A † l , − α + B † l ,α B † l , − α )+ h.c. ](A1)where the primitive lattice vectors m = a (3 , √
3) and m = a (3 , −√
3) with a the interatomic distance, and s = +( − ) for α = ↑ ( ↓ ). Performing Fourier-transformations according to a k ,α = √ N (cid:80) l A A k ,α e − i k · l A and b k ,α = √ N (cid:80) l B B k ,α e − i k · l B , we arrive at H = t (cid:88) k ,α [ γ k b † k ,α a k ,α + γ ∗ k a † k ,α b k ,α ] − µ (cid:88) k ,α ( a † k ,α a k ,α + b † k ,α b k ,α )+ (cid:88) k ,α [ s ∆( a † k ,α a †− k , − α + b † k ,α b †− k , − α )+ s ∆ ∗ ( a − k , − α a k ,α + b − k , − α b k ,α )] , (A2)where the parameter γ k is given by γ k = e − i k · δ + e − i k · δ + t (cid:48) t e − i k · δ = 2 cos( √ k y a/ e − ik x a/ + t (cid:48) t e ik x a , (A3)with the NN lattice vectors being defined as δ = a (1 , √ δ = a (1 , −√ δ = a ( − , (cid:15) = ±| γ k | .For the critical case of t (cid:48) = 2 t , (cid:15) vanishes at M = ( π a , H for a small k around the M point by setting k x → k x + π a and k y → k y , with k x a (cid:28) k y a (cid:28) k , we arrive at γ M + k (cid:39) (3 iak x e iπ/ + 34 a k y e iπ/ ) , (A4a) γ ∗ M + k (cid:39) ( − iak x e − iπ/ + 34 a k y e − iπ/ ) , (A4b) γ − M − k (cid:39) ( − iak x e − iπ/ + 34 a k y e − iπ/ ) , (A4c) γ ∗− M − k (cid:39) (3 iak x e iπ/ + 34 a k y e iπ/ ) . (A4d)Substituting Eq. (A4) into Eq. (A2), we obtain the Hamiltonian at q = M + k point as˜ H q = t (cid:88) q ,α [(3 iak x + 3 a k y ) e i π b † q ,α a q ,α + ( − iak x + 3 a k y ) e − i π a † q ,α b q ,α + ( − iak x + 3 a k y ) e − iπ b †− q ,α a − q ,α +(3 iak x + 3 a k y ) e iπ a †− q ,α b − q ,α ] − µ (cid:88) q ,α [ a † q ,α a q ,α + b † q ,α b q ,α + a †− q ,α a − q ,α + b †− q ,α b − q ,α ]+ (cid:88) q ,α [ s ∆( a † q ,α a †− q , − α + b † q ,α b †− q , − α ) + s ∆ ∗ ( a − q , − α a q ,α + b − q , − α b q ,α )] . (A5)where the relation of (cid:80) k = (cid:80) − k has been used. Rewrite ˜ H q in the form of ˜ H q = (cid:80) q ψ † q H q ψ q , with the basis ψ † q = [ a † q ↑ , b † q ↑ , a − q ↓ , b − q ↓ , a † q ↓ , b † q ↓ , a − q ↑ , b − q ↑ ] , (A6)and H q = (cid:18) H ( k , ∆) 00 H ( k , − ∆) (cid:19) . (A7)By defining v ≡ at (cid:126) , η ≡ a t , and µ/ → µ , the upper block can be formulated as H ( k , ∆) = − µ ( − i (cid:126) vk x + ηk y ) e − iπ ∆ 0( i (cid:126) vk x + ηk y ) e iπ − µ † µ ( i (cid:126) vk x − ηk y ) e − iπ † ( − i (cid:126) vk x − ηk y ) e iπ µ , (A8)where the anticommutation relation has been employed. Taking a unitary transformation H BdG ≡ U † HU with U = e − iπ ie iπ e − iπ ie iπ . (A9)The BdG Hamiltonian can be compactly written as H BdG = ( (cid:126) vk x σ x + ηk y σ y − µσ ) τ z + ∆ (cos φσ τ x − sin φσ τ y ) , (A10)where we rewrite ∆ ≡ ∆ e iφ with ∆ the amplitude of pairing potential and φ the superconducting phase, σ is a2 × σ x,y and τ x,y,z act on the pseudo-spin and Nambu spaces, respectively. Appendix B: Calculation of the basis scatteringstates in NS junctions based on semi-Dirac materials
In this appendix we present necessary calculation de-tails regarding the wave functions and related quantitiesin SDMs-based NS junctions.
1. NS junction extending along the x -axis For the NS junction extending along the x -axis, we as-sume the translational symmetry is preserved in the y di-rection, thus the transverse momentum k y can be treatedas a good quantum number. In the S region, solving theBdG equation H BdG ( − i∂ x , k y ) ψ = Eψ straightforwardlyyields ψ ± eq = Λ ± eq Γ eq Λ ± eq e − i ( β + φ ) Γ eq e − i ( β + φ ) e ± ik eq x + ik y y , (B1a) ψ ± hq = Λ ± hq Γ hq Λ ± hq e i ( β − φ ) Γ hq e i ( β − φ ) e ± ik hq x + ik y y , (B1b) where the related parameters are defined by (cid:126) vk eq ( hq ) = +( − ) (cid:113) [ µ S + ( − )Ω] − η k y , (B2a)Λ ± eq ( hq ) = ± (cid:126) vk eq ( hq ) − iηk y , (B2b)Γ eq ( hq ) = (cid:113) (cid:126) v k eq ( hq ) + η k y , (B2c)Ω = (cid:113) E − ∆ , (B2d)In the N region, the basis scattering states can be for-mulated as ψ ± e = ± (cid:126) vk e − iηk y ε + e ± ik e x + ik y y , (B3a) ψ ± h = ∓ (cid:126) vk h + iηk y ε − e ± ik h x + ik y y . (B3b)In the interfacial barrier region, the x -components ofmomenta k Be ( h ) = sgn[ ε +( − ) ] (cid:113) ε − ) − η k y / ( (cid:126) v ). Bysubstituting µ N and k e ( h ) , respectviely, with U and k Be ( h ) into Eq. (B3), we obtain the basis scattering states ψ B ± e,h in the interfacial barrier region. The related wave func-tions can be expressed asΨ B = (cid:88) m = e,h (cid:88) n = ± b nm ψ Bnm , (B4)where b nm label the scattering amplitudes.
2. NS junction along the y -axis In the NS junction along the y -direction, we assume thetransverse momentum k x is preserved. In the N region,solving the BdG equation H BdG ( k x , − i∂ y ) ϕ = Eϕ givesthe basis states ϕ ± e, = (cid:126) vk x − iηq e ε + e ± iq e y + ik x x , (B5a) ϕ ± e, = (cid:126) vk x + iηκ e ε + e ∓ κ e y + ik x x , (B5b) ϕ ± h, = − (cid:126) vk x + iηq h ε − e ± iq h y + ik x x , (B5c) ϕ ± h, = − (cid:126) vk x − iηκ h ε − e ∓ κ h y + ik x x . (B5d) In the S region, the basis states can be formulated as ϕ ± eq, = λ eq, γ eq, λ eq, e − i ( β + φ ) γ eq, e − i ( β + φ ) e ± iq eq, y + ik x x , (B6a) ϕ ± hq, = λ hq, γ hq, λ hq, e i ( β − φ ) γ hq, e i ( β − φ ) e ± iq hq, y + ik x x , (B6b)with the associated parameters being defined as q eq ( hq ) , = +( − ) (cid:115) (cid:112) [ µ S + ( − )Ω] − (cid:126) v k x η , (B7a) q eq ( hq ) , = i (cid:115) (cid:112) [ µ S + ( − )Ω] − (cid:126) v k x η , (B7b) λ eq ( hq ) , = (cid:126) vk x − iηq eq ( hq ) , , (B7c) λ eq ( hq ) , = (cid:126) vk x − iηq eq ( hq ) , , (B7d) γ eq ( hq ) , = (cid:113) (cid:126) v k x + η q eq ( hq ) , , (B7e) γ eq ( hq ) , = (cid:113) (cid:126) v k x + η q eq ( hq ) , . (B7f)We note that for E > ∆ , ϕ ± eq ( hq ) , denote theelectron(hole)-like scattering states propagating alongthe ± y -directions, while ϕ ± eq ( hq ) , represent the evanes-cent ones decaying exponentially as y → ±∞ . In thecase of E < ∆ , all scattering states given by Eq. 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