Andreev reflection adjusting in the multi-terminal device with the kink states Lin Zhang1;3, Chao Wang∗1;2, Peipei Zhang1, and Yu-Xian Li∗1 1College of Physics and Hebei Advanced Thin Film Laboratory, Hebei Normal University, Shijiazhuang 050024, People's Republic of China 2College of Physics, Shijiazhuang University, Shijiazhuang 050035, People's Republic of China 3Shool of Mathematics and Science, Hebei GEO University, Shijiazhuang 050031, People's Republic of China (Dated: June 14, 2021) At the domain wall between two regions with the opposite Chern number, there should be the one-dimensional chiral states, which are called as the kink states. The kink states are robust for the lattice deformations. We design a multi-terminal device with the kink states to study the local Andreev reflection and the crossed Andreev reflection. In the three-terminal device the local Andreev reflection can be suppressed completely for either "0 = 0:5∆s and V0 = 0:1t or "0 = −0:5∆s and V0 = −0:1t, where "0 is the on-site energy of the graphene terminals and V0 is the stagger energy of the center region. The coefficient of the crossed Andreev reflection can reach 1 in the four-terminal device. Besides adjusting the phase difference between superconductors, the local Andreev reflection and the crossed Andreev reflection can be controlled by changing the on-site energy and the stagger energy in the four-terminal device. Our results give some new ideas to design the quantum device in the future. I. INTRODUCTION terminal 2 A S B Topological insulator[1] has been one of the no- + terminal 3 ticeable advanced materials due to the novel physical − properties[2,3], such as quantum Hall effect[4,5] and W quantum spin Hall effect[6,7]. It is of great significance to terminal 1 study the properties of topological insulator for develop- + ing a new generation of quantum components. Recently, y − the quantum anomalous Hall effect has been identified in o x S the three-dimensional magnetic topological insulator[8{ terminal 4 11], which opens new possibilities for chiral-edge-state- L based devices in zero external magnetic field[12, 13]. The chiral states can also appear at the domain walls be- FIG. 1. (Color online) The schematic diagram of the four- tween two regions with the opposite Chern number[14]. terminal device with the kink states. The terminal 1 and 3 are For example, the graphene can be gapped by the sublat- the graphene ribbons. The width and the length of the central tice symmetry breaking staggered on-site energy, which region are W = 8 and L = 28, respectively. The terminal 2 and 4 are the superconductors. In the center region marked is showed in Fig.1. There are one-dimensional states[15{ 0+0(green), the on-site energy is " = V and " = −V , 18], which are referred to as kink states below, pre- A 0 B 0 where V0 is called as the stagger energy. In the region marked 0 0 senting at the boundaries between regions with different − (orange), "A = −V0 and "B = V0. The kink states are at quantized Hall conductances in the graphene nanorib- the interface between two regions with the opposite stagger bon. A Majorana fermion, which is its own antipar- energy. ticle, has the potential for quantum computing. Over . the last two decades it has been realized that Majorana fermions can emerge at zero-energy modes in topologi- cal superconductors[19{21]. After the chiral Majorana dreev reflection come from the different valleys in the edge modes in the quantum anomalous Hall insulator- graphene-superconductor device[30{32]. superconductor structure is realized in the experiment, In this paper, we calculate the transmission coefficients there are more researches focused on the topological su- in a two-terminal device, where there are lattice deforma- perconductor device[23{29]. tions around the domain wall. The transmission coeffi- At the interface between a superconductor and a nor- cients are close to 1 in the two-terminal device, which mal conductor, an incident electron from the normal con- proves the robustness of the kink states[15{17]. Then ductor can be reflected as a hole, which is called Andreev we study the Andreev reflection in the four-terminal de- arXiv:2106.06146v1 [cond-mat.mes-hall] 11 Jun 2021 reflection. When the bias voltage is lower than the su- vice with the topological kink states showed in Fig.1. perconductor gap, the conductance of the superconduc- Through calculating the coefficients of the Andreev re- tor hybrid device is mainly determined by the progress flection in the four-terminal device, we find that the of the Andreev reflection. Keeping to the time-reversal progress of the crossed Andreev reflection can be con- symmetry, the electron and the hole taking part in An- trolled by adjusting the on-site energy and the stagger 2 energy. We now turn to analyze the process that an inci- The rest of this paper is arranged as follows. In Sec.II, dent electron from the graphene terminal is reflected the model Hamiltonian for the system is presented into a hole with a Cooper pair emerging in the su- and the formulas for calculating the Andreev reflection perconductor terminal. Using nonequilibrium Green's coefficients are derived. Our main results are shown function method, we can calculate the retarded and ad- r a y and discussed in Sec.III. Finally, a brief conclusion is vanced Green's function G (E) = [G ] = 1=(EI HC P r − − presented in Sec.IV. α Σα), where HC is the Hamiltonian of the center re- gion in the Nambu representation and I is the unit ma- r r trix with the same dimension as HC . Σα = tcgα(E)tc is the retarded self-energy due to the coupling to the r II. MODEL AND METHOD terminal α, where gα(E) is the surface Green's func- tion of the terminal α. We can numerically calcu- The four-terminal device with the kink states is showed late the surface Green's function of the graphene termi- in Fig.1, where the terminal 1 and 3 are the normal nals. For superconductor terminals, the surface Green's graphene ribbons and the terminal 2 and 4 are super- function[23, 25, 27] in real space is conductors. The center region is the colour region. The 1 ∆=E kink states are at the interface between the regions with gr (E) = iπρβ(E)J [k (x x )] ; (5) α,ij 0 f i j ∆∗=E 1 different colour. Along the x direction and the y direc- − tion they are zigzag and armchair, respectively. The total where α = 2; 4 and ρ is the density of normal Hamiltonian of this junction can be represented as electron states. J0[kf (xi xj)] is the Bessel func- − tion of the first kind with the Fermi wavevector kf . H = HC + HG + HS + HT ; (1) p 2 2 β(E) = iE= ∆s E for E < ∆s and β(E) = p 2 − 2 − j j whereH , H , H and H are the Hamiltonian of the E = E ∆ for E > ∆s. C G S T j j − s j j center region, the graphene nanoribbons, the supercon- The Andreev reflection coefficients for the incident ductor terminals and the coupling between the center electron coming from the graphene terminal 1 can be region and the superconductor terminals, respectively. obtained[23] by In the tight-binding representation, HC and HG are r a TA;11(E) = Tr Γ1;""G Γ1;##G ; given by f "# #"g r a TA;13(E) = Tr Γ1;""G Γ3;##G ; (6) X y X y f "# #"g HC=G = "0anan [tanam + H:c:]; (2) n − <m;n> where the subscripts , , and represent the 11, 22, 12 and 21 matrix"" ## elements,"# respectively,#" in the y where an and an are the creation operator and annihila- Nambu representation. The linewidth function Γα is r r y tion operators of the nth point. " is the on-site energy in defined with the aid of self-energy as Γα = i[Σ (Σ ) ]. 0 α − α the graphene terminals and the center region. In the cen- TA;11 and TA;13 represent the coefficients of the local ter region, "0 = V0 for the different partition, where V0 Andreev reflection and the crossed Andreev reflection, is the stagger energy± of A and B sublattice. The second respectively. Because the Andreev reflection from an term in Eq. (2) stands for the nearest-neighbor hopping electron to a hole is equivalent to that from a hole to an Hamiltonian. Considering that the center region is di- electron under particle-hole symmetry, in this work we rectly coupled to the superconductor terminals, we use only consider the Andreev reflection, where an incident the BCS Hamiltonian for the superconductor terminals electron is reflected as a hole. described by a continuum model, X y X ∗ y y HS = " C C + (∆C C + ∆ C C ); (3) k k σ k σ k # −k " −k " k # k ,σ k III. NUMERICAL RESULTS AND ANALYSIS iθ where ∆ = ∆se . Here ∆s is the superconductor gap In numerical calculations, we set the nearest-neighbour and θ is the superconductor phase. The coupling between hopping energy t = 2:75eV . The length of the nearest- superconductor terminals and graphene is described by neighbor C-C bond is set to be a0 = 0:142 nm as in a X y real graphene sample. The superconductor gap is set to HT = tcan,σCσ(xi) + H:c: (4) −1 − be ∆s = 0:02t and the Fermi wavevector kf = 10 nm . n,σ For the convenience of discussing the influence of the kink Here xi and n represent the positions of the coupling states on Andreev reflection, The Fermi energy Ef is set atoms on the interface of superconductor and the center to be zeros in our calculations for the convenient.