Double Andreev reflections and double electron transmissions in a normal-superconductor-normal junction based on type-II Weyl semimetal

We study the quantum transport behavior of a normal-superconductor-normal junction based on type-II Weyl semimetal, which is arranged in the tilting direction of the Weyl semimetal. We find that both the crossed Andreev reflection and normal reflection are forbidden, while there will be double Andreev reflections and double electron transmissions for the incident electron from the semimetal side. Andreev reflections and transmissions occur both in the retro and specular directions simultaneously, symmetric about the normal of the interface but with different amplitudes, depending on the angle and energy of incident electrons. These transport processes make the junction here quite different from that based on the normal metal or graphene. In addition, the differential conductance is studied for experimental signatures. We find that the conductance is almost unaffected by the chemical potential and it is enhanced with increasing junction length.

We study the quantum transport behavior of a normal-superconductor-normal junction based on type-II Weyl semimetal, which is arranged in the tilting direction of the Weyl semimetal. We find that both the crossed Andreev reflection and normal reflection are forbidden, while there will be double Andreev reflections and double electron transmissions for the incident electron from the semimetal side. Andreev reflections and transmissions occur both in the retro and specular directions simultaneously, symmetric about the normal of the interface but with different amplitudes, depending on the angle and energy of incident electrons. These transport processes make the junction here quite different from that based on the normal metal or graphene. In addition, the differential conductance is studied for experimental signatures. We find that the conductance is almost unaffected by the chemical potential and it is enhanced with increasing junction length.

I. INTRODUCTION
The interface between a normal and a superconducting material has been a widely used platform to realize exotic transport behaviors, especially in recent years when plenty of new materials with massless linear excitations or nontrivial topological properties are predicted and used to fabricate these interfaces [1][2][3][4][5][6][7][8] . In a conventional normal metal-superconductor (NS) junction, there exists not only a normal reflection (NR) process but also an Andreev reflection (AR) process, in which an incident electron from the normal side is reflected back as a positive charged hole, and a Cooper pair forms in the superconductor (SC) 9 . It is also known as retro AR, since the hole retraces almost the path of the incident electron. Following the successful fabrication of some new lowdimensional materials, novel AR phenomena have been reported from various aspects. One typical result is the specular AR, which has been discovered with the hole reflected along a specular path of the incident electron in the NS junction based on the graphene-like materials 1,10 . And the perfect AR has been proposed and discovered in the NS junction of topological insulator 11,12 . Besides, in the SNS junction of topological SCs, the fractional Josephson effect is allowed to come into being 13,14 .
Very recently, the type-II Weyl semimetal (WSM) has been predicted and observed in several materials, e.g., MoTe 2 15,16 , LaAlGe 17 , WTe 2 18,19 and Mo x W 1−x Te 2 20,21 . Like the type-I WSM, the low-energy excitations can be described by Weyl equation, but with a tilted anisotropic energy spectrum. For the type-I WSM, the conduction and valence bands intersect at several Weyl nodes and thus the Fermi surface is point-like and the spectra around the nodes are coniclike 2 . However, the spectra of the type-II WSM are tilted by rotating around the Weyl nodes and there will exist electron and hole pockets near the line-like Fermi surface with a large density of states [22][23][24] . Therefore, the type-II WSMs will show a lot of interesting properties, in comparison with the type-I WSMs, such as the anomalous Hall effect, chi-ral anomaly and other intriguing properties [25][26][27][28][29][30] . Besides, the tilted energy bands of type-II WSM have opportunities to bring new transport mechanisms. According to the previous works, the phenomenon of double ARs has been predicted at the interface between the type-II WSM and its-based SC 31 , where the retro and specular ARs coexist accompanied by the forbidden NR.
In view of the AR result contributed by the type-II WSM, one can be sure that novel transport behavior will emerge in a normal-superconductor-normal (NSN) junction based on type-II WSM. Motivated by such a topic, in the present work we concentrate on the NSN junction arranged in the tilting direction of the type-II WSM and perform the discussion the AR properties, with the help of the scattering matrix method. As is known, in a conventional NSN junction of normal metal, four transport processes coexist, i.e., NR, normal electron transmission (ET), AR, and crossed AR, as shown in Fig. 1 (a) [32][33][34][35][36][37] , where the crossed AR process refer to the incident electron and the hole come from the different metal on both sides of the SC. However, our study indicates that the NR and crossed AR are forbidden in the NSN junction based on the type-II WSM. Instead, the double ARs and double ETs happen simultaneously, including retro Andreev reflected (A 1 ), specular Andreev reflected (A 2 ), normal transmission (T 1 ), and the abnormal specular transmission (T 2 ), just as shown in Fig. 1 (b). It shows that two ARs (ETs) have the same reflection (refraction) angle but with different amplitudes in general. In addition, the relationships between the four scattering processes and some controllable variables, e.g., the chemical potential, the incident angle and the junction length, are exhaustedly investigated in this paper. Also, the differential conductance is studied, and it has been found to have a larger magnitude considering the large momentum mismatch between the normal and SC region. Moreover, it is almost unaffected by the chemical potential and enhanced with the increase of junction length, which can be viewed as the obvious signatures for experimental observation.
The rest of the paper is organized as follows. In Sec. II, we give the model Hamiltonian and introduce the phenomenon of double ARs and double ETs. In Sec. III, we calculate the amplitudes of these four transport processes with the scattering matrix method. In Sec. IV, we study the differential conductance. In Sec. V, we give a brief conclusion.

A. Model
The schematic of the NSN junction based on the type-II WSM is shown in Fig. 1 (b). We model the type-II WSM with a most simple low-energy effective Hamiltonian which respects time reversal symmetry while breaks spatial inversion symmetry 38,39 . Near Weyl point K 0 , the Hamiltonian is written as 31 in which k is the wave vector measured from K 0 . σ = (σ x , σ y , σ z ) denotes the Pauli matrix, σ 0 is the identity matrix. v 2 describes the Fermi velocity and v 1 determines the tilt along x direction. It is a type-II WSM for |v 1 | > v 2 while a type-I WSM for |v 1 | < v 2 . The Hamiltonian near the −K 0 node is related with that near K 0 via time reversal symmetry, and it takes the form In the SC region, Cooper pairing occurs between the electrons near ±K 0 . We consider the BCS pairing for simplicity. We use the electron and hole representations for quasiparticles near ±K 0 respectively. Then the Bogoliubov-de Gennes (BdG) Hamiltonian of the whole junction in real place can be written as 40 where the wave vector has been replaced with k = −i∇ r , since translation invariance is broken along x-direction. µ(r) and ∆(r) are the chemical potential and superconducting order parameter, respectively, which can be expressed as The energy dispersion for quasiparticles in the SC region is written as The dispersion for eigenstates with k y = 3, k z = 0 is plotted in Fig. 2 (b) and it indicates that there are two rightmoving and two left-moving modes for a fixed energy. In the WSM region, superconducting order ∆ vanishes and there are two electron modes and two hole modes with energy dispersions given as in which k = k 2 The z-component, which is missed here, shares a similar form as the y-component since the system is rotation invariant about x-axis. It is clear that both electron modes are right-moving while both hole modes are left-moving for type-II WSM with |v 1 | > v 2 , as is shown in Fig. 2 (a) and 2 (c).

B. Double ARs and double ETs
Now we consider an incident electron from the left WSM region in the channel ψ e− with energy E and wave vectors k y and k z , shown in Fig. 2 (a). Because of the translation invariance, k y and k z are good quantum numbers and keep invariant in the transport process. Double ARs occur at the interface between the left WSM and SC regions. The incident electron will be retro reflected (A 1 ) or specular reflected (A 2 ) as a hole of modes ψ h+ or ψ h− , respectively. These two Andreev modes are symmetric about the normal of the interface in the incident plane with the same reflection an- ) but different amplitudes depending on the incident angle and energy. Double ETs occur at the interface between the right WSM and SC regions. The incident electron transmits into the intraband and interband modes ψ e− and ψ e+ , respectively. The intraband/interband ET corresponds to the normal/specular ET with amplitude T 1 /T 2 . The intraband mode is identical to the incident mode while the specular ET is symmetric with the normal ET, with the same angle θ e = arctan( ). On the other hand, we note that the NR and crossed AR are forbidden. These novel transport behaviors make this junction distinguished from the junction based on normal metal or graphene 32-37 .

III. AMPLITUDES OF DOUBLE ANDREEV REFLECTIONS AND DOUBLE ELECTRON TRANSMISSIONS
In this section, we study in detail the amplitudes of each transport process of the double ARs and double ETs via the scattering matrix method. The effect of chemical potential and junction length is also investigated. In the numerical calculations, we set v 1 = 2, v 2 = 1, ∆ = 1, U = 100 and µ = 0.5 unless otherwise stated.

A. Formalisms
The amplitude of each transport process and the differential conductance can be solved by the scattering matrix method. Since this system is rotation invariant about xaxis, we set k z = 0 in the following discussion. Consider an electron incidents with energy E and wave vector k y . The wave-functions in the three regions can be expressed as Ψ I (r) = Ψ e− (r) + r 1 Ψ h+ (r) + r 2 Ψ h− (r), Ψ II (r) = aΨ a (r) + bΨ b (r) + cΨ c (r) + dΨ d (r), Ψ III (r) = t 1 Ψ e− (r) + t 2 Ψ e+ (r), where I, II, III denotes the left, central, and right region, respectively. ψ e± ( ψ h± ) are the eigenvectors of two electron (hole)-modes and Ψ a,b,c,d are corresponding modes in the SC region. The expressions of these eigenvectors are given in the Appendix. r 1/2 is the retro/specular AR coefficient, and t 1/2 is the normal/specular ET coefficients. a, b, c, and d are coefficients of the qusiparticle modes in the SC region. These coefficients are determined by the boundary conditions at the two interfaces, The current density operator in the normal WSM region can be derived via where τ z is the z-component of Pauli matrix which acts in the electron-hole space. Then the retro and specular AR coefficients A 1 and A 2 are evaluated by And the normal and specular ET coefficients T 1 and T 2 are evaluated by Because of the conservation of current, we have A 1 +A 2 + T 1 + T 2 = 1.

B. Numerical results
In Fig. 3, we calculate the amplitudes of double ARs and double ETs, respectively. Fig. 3 (a) show the normal incidence case with junction length L = ξ =h(v 1 −v 2 )/∆. For the chemical potential, it is taken to be µ = 0.5, which can be tuned by adjusting the gate voltage in experiment. One can find that the specular ET is forbidden, i.e., T 2 = 0, and the incident electron is Andreev reflected in the A 1 or A 2 mode for E < µ and E > µ, respectively. However, the retro and specular modes coincide in the normal incident case to contribute identically to the quantum transport processes. Next, with the increase of incident energy, the ETs are enhanced while ARs are weakened monotonically in the energy regime E < 2∆. In the oblique incidence case shown in Fig. 3 (b)-(c), the four transport processes co-contribute to the transport through this junction. The retro AR coefficient A 1 decreases monotonically when the incident energy increases in the SC-gap regime E < ∆, while the specular AR A 2 and normal ET coefficient T 1 exhibit increments in this regime, for junctions with length L = ξ and 2ξ [See Fig. 3 (b)-(c)]. However, the specular ET coefficient T 2 can decrease or increase in the cases of L = ξ [ Fig. 3 (b)] and L = 2ξ [ Fig. 3 (c)] in the low-energy regime E < ∆. In the regime E > ∆, all these scattering processes exhibit an oscillatory behaviors due to the coherent tunnelling determined by the standing-wave condition. For the specular ET T 2 , the valley (peak) of the oscillation emerges outside the gap regime leading to the increase (decrease) behavior in the gap regime for L = ξ and 2ξ, respectively. In Fig. 3 (d), we present the total ET coefficient T = T 1 + T 2 , the specular ET coefficient T 2 and the specular AR coefficient A 2 as functions of the incident energy for chemical potentials µ = 0.5 and 5. It can be clearly found that the change of chemical potential does not affect the ET coefficients T and T 1 , but only alters the relative weight of retro and specular ARs.
In what follows, we calculate the dependence of both double AR and double ET coefficients on incident angle θ and energy E, as shown in Fig. 4. The incident angle is related to the group velocity by the formula tanθ = vy vx . By the velocity expressions in Eq. (8), we know that θ has a upper limit θ c = arctan v2 √ v 2 1 −v 2 2 = 0.17π, since the equi-energy surface is a hyperbola in the WSM region. Next, in Fig. 4 (a) it shows that the retro AR A 1 decreases or increases monotonically with the increment of incident angle θ for electrons with incident energy E < µ and E > µ, respectively. However, compared with A 1 , the specular AR A 2 exhibits an opposite dependence on θ, as displayed in Fig. 4 (b). With respect to the double ETs, the results in Fig. 4  decrease or increase monotonically if θ is increased in the whole energy regime 0 < E < 2∆. Therefore, according to these results, we can find that the retro AR A 1 contributes mainly in the E < µ regime while specular AR A 2 mainly in the E > µ regime. This suggests that the AR can be tuned via a gate voltage. Alternatively, the intraband ET T 1 contributes almost for all incident energy and angles, while the specular ET T 2 occurs mainly for large incident angles.
In Fig. 5, we study the dependence of AR and ET coefficients on the length of the SC region. Both the normal and oblique incidences [See Fig. 5 (b)] are considered, by taking the incident energy to be E = 0.6. Firstly, in Fig. 5 (a) one can find that in the normal incident case, only one AR mode and one ET mode appear, respectively, i.e., A 2 and T 1 . This is exactly consistent with the result in Fig. 3 (a) where both A 1 and T 2 modes vanish for E > µ. In the short-junction limit L → 0, only the intraband ET T 1 is allowed with its amplitude T 1 = 1, just as expected, since no scattering potential exists. With the increase of L, ET is weakened gradually while AR enhanced. However, in the long-junction limit, ET is suppressed and the complete AR takes place, in accordance with the result of type-II WSM-SC junction 31 . Next, in the oblique incidence case, both retro and specular AR coefficients, A 1 /A 2 , increase monotonically until reach the saturation value, with the increase of junction length. In contrast, the ET coefficients will decrease accompanied by oscillation. As a consequence, the peak of the normal-ET curve meets the specular-ET valley, in the short junction limit L → 0. The two ETs share an identical oscillation period, which can be approximated by L = 2π |kx1−kx2| determined from the standing wave con- dition. k x1 (k x2 ) is the real component of the wave vector of left (right) moving mode in the SC region. Most interestingly, there exists a phase shift of π between normal and specular ETs, which vanishes the oscillation of the total transmission coefficient T = T 1 + T 2 .

IV. CONDUCTANCE
Following the previous analysis of double ARs and double ETs, we investigate the properties of differential conductance in this section. We apply a bias voltage to the left WSM region, while the SC region and the right WSM region are grounded. Thus, the differential conductance can be calculated by the well-known BTK formula 31,41 : (14) where S is the cross-sectional area of the junction, m = ± is to distinguish the contribution of transport processes due to incident modes ψ e± , and A (m) = A m 1 + A m 2 is the total AR coefficient. The integration over the wave vectors is constrained near the Weyl node with a cut-off q m , i.e., (k 2 y + k 2 z ) < q m . The contributions of ETs and ARs to the conductance, i.e., G T and G A , can be written as GT (eV ) = 2e 2 S π 2 h dkydkzT (ky, kz, eV ), GA(eV ) = 2e 2 S π 2 h dkydkz2A(ky, kz, eV ). (15) If we denote G N (eV ) = 2e 2 S π 2 h · πq 2 m , the conductance of a normal junction, there will be This indicates that it is impossible to enhance both the total conductance and ET conductance at the same time.
The conductance G and its ET component G T as functions of bias for several chemical potential µ and junction length L are shown in Fig. 6 (a) and (b), respectively. It can be obviously seen that G and G T decrease or increase with the enhancement of the bias voltage, respectively. This is consistent with the result in Eq. 16 and the previous analyses of ET coefficients, which can be increased by the enlargement of incident energy. Besides, it shows that the conductance is independent of chemical potential µ, such a result can shown by understood by observing the properties of scattering coefficients A and T . On the other hand, the conductance can be affected by the junction length to a great extent. Namely, the longer junction will induce the weaker ET conductance while the larger AR conductance and therefore the larger total conductance, since the ET (AR) coefficients decreases (increases) in this process [See Fig. 5]. In the long-junction limit, G T will vanish, leading to the occurrence of the perfect AR in the SC-gap regime E < µ. One can then find the result that G/G N = 2. If the bias is further increased, G (G T ) decreases (increases) gradually with an oscillation, due to the coherent tunnelling determined by the standing-wave condition. However in the short-junction limit, both G and G T are approximately equal to G N in the SC-gap regime, because of the vanishing of AR. In addition, we notice that the conductance here has a large magnitude, the reason is the consideration of the large momentum is mismatched between the normal and SC region, which are identical to an effective scattering potential. In fact, this is exactly the forbidden of NR in our model leads to this phenomenon.

V. SUMMARY
In Summary, we have studied the transport property of the NSN junction based on type-II WSM, and predicted the novel phenomenon of double ARs and double ETs. It means that four scattering processes for incident electrons coexist in this system, i.e., the retro and specular ARs, the normal and specular ETs. However, the NR and crossed AR are forbiden. The retro and specular AR modes are symmetric about the normal of the interface but with different amplitudes, which is also the case for normal and specular ET processes. The dependences of the amplitudes on the incident angle and energy have been studied in detail with the effect of chemical potential and junction length considered. In addition, we have studied the differential conductance. It is found that the conductance is independent of chemical potential of the WSM region, suggesting its robustness. The normalized conductance will increase as enlarging junction length. The conductance has been formed to have a large value considering the scattering potential due to the momentum mismatch between the normal and SC region.
In this appendix, we give the eigenvectors in the WSM and SC regions.
The BdG equation in WSM region is represented as Two incident and two reflected modes with energy E and wave vector k y and k z can be given as 31 In Eq.(A2), the wave vectors and their x-and ycomponents of each energy band are respectively given as , k± = k 2 x± + k 2 y + k 2 z , k ± = k 2 x± + k 2 y + k 2 z , where k x± and (k x± ) are corresponding x-component of wave vectors, shown in Fig. 2.
The BdG equation of the SC region is given in Eq. 3. In the large-U limit 1 , the four excited modes with energy E > 0 and wave vector k y and k z can be approximated as, 31 Ψ b/d (Ψ a/c ) are right (left) moving modes with positive slope as shown in Fig. 2. And the parameters are where k xL± and (k xR± ) are corresponding to the xcomponent of wave vectors, shown in Fig. 2.