PDF《Pure spin current in graphene NS》

2 tonian H± = c(σxpx ± σypy) + U(x), (1) where c is the Fermi velocity and p = ?i ?. In regions of constant U this equation de?nes a conical energy band, or valley, ε ? U = ±c|p|. The Pauli matrices σx,y per- mute electrons between two triangular sublattices of the honeycomb lattice. The two signs in Eq. (1) (+) and (?) correspond to the two valleys K and K′. We consider a graphene sheet in the x ? y plane, with the region x <

0 covered by a conventional supercon- ductor (Fig. 1). Following Ref. [8] we assume that in this con?guration a pair potential ?(x) = ?0Θ(?x) can be induced [16] in the graphene sheet by the proximity e?ect, accompanied by a su?ciently strong shift in the scalar potential U(x) = ?U0Θ(?x), so that U0 ? ?0. The re?ection at the NS interface (x = 0) is described by a separate four-dimensional Dirac-Bogoliubov-de Gennes equation [8] for each valley H± ? EF ?(x) ?(x) EF ? H± u v = ε u v , (2) where the two spinors u and v represent the electron and hole components of the φ± wavefunctions for H±, respectively. In addition to carrying pseudo-spin, electron-hole, and valley indices, the quasiparticle wave function should also describe the usual spin. Since for ε <

?0 no spin can be injected into the superconductor, the spin of the incident electron is transferred to the re?ected particle. For several decades it was considered a basic feature of Andreev re?ection [9] that the hole produced upon electron-hole conversion retraces the incident electron'

s trajectory. Even in the traditional materials, however, this repetition of trajectories is exact only for zero exci- tation energy ε in Eq. (2). At ?nite excitation energy the two trajectories do not quite coincide, leading to inter- esting e?ects in semiclassical dynamics [17] and the spec- trum [18] of Andreev billiards. In graphene, the Andreev re?ection described by Eq. (2) has the standard form only if ε ? EF . Because of the possibility to tune the Fermi energy close to the Dirac point, one may reach in graphene the regime ε ? EF , so that an incident electron in the conduction band produces an Andreev re?ected hole in the valence band (opposite side of the cone in the Dirac excitation spectrum). This valence band hole has the same velocity along the NS interface as the incident electron, and consequently is specularly re?ected [8]. For vanishing Fermi energy, both the electron and hole com- ponents of the re?ected wave follow precisely the same trajectory. We now restrict ourselves to this most inter- esting case of EF = 0. The transformation of a quantum superposition of in- cident electron and hole into the corresponding super- position of outgoing waves upon Andreev re?ection is described by a

2 *

2 re?ection matrix, RA, u v out = RA u v in = r rA rA r u v in . (3) With the notation α = arctan |py/px|, ε = c p2 x + p2 y, and ξ = ?2

0 ? ε2 one has [19] r = ?iε sinα ε + iξ cos α , rA = ?0 cos α ε + iξ cos α . (4) III. NS INTERFACE IN V-JUNCTION GEOMETRY Characteristic of specular Andreev re?ection in graphene is the spatial separation of incident and re- ?ected electron-hole beams. The simplest device that makes use of this property is a V-junction, as shown in Fig. 1. Suppose one can inject a spin-polarized collimated monoenergetic beam of electrons through one arm of the junction. [Monoenergetic beams may potentially be pro- duced by resonant transport through a graphene quan- tum dot (QD) [20], as was done for semiconductor QD'

s e.g. in Ref. [21].] The injected electron will be either nor- mally or Andreev re?ecte........