Spin-orbit effects in a graphene bipolar pn junction

A graphene pn junction is studied theoretically in the presence of both intrinsic and Rashba spin-orbit couplings. We show that a crossover from perfect reflection to perfect transmission is achieved at normal incidence by tuning perpendicular electric field. By further studying angular dependent transmission, we demonstrate that perfect reflection at normal incidence can be clearly distinguished from trivial band gap effects. Introduction. – There has been recent interest in a novel class of band insulators, called topological insulators (TIs) [1]. TIs are characterized by a bulk gap and spin filtered edge states at the boundary. These gapless edge states originate from the lattice spin-orbit (SO) effect and are protected by time-reversal symmetry from moderate disorder and interaction. In their seminal paper [2], Kane and Mele demonstrated that a graphene monolayer may become a TI when the intrinsic SO coupling dominates over the extrinsic Rashba coupling. Then the graphene layer is characterized by a Z2 topological number [3] and has spin-filtered edge states that disappear if the Rashba coupling becomes too large, or if Coulomb interactions exceed a certain threshold [4]. Therefore the existence of the topological phase in graphene depends crucially on the values of the intrinsic and extrinsic (Rashba) SO couplings. In the concluding section, we shall discuss recent estimates of these SO interactions which allows for considering the transition between the topological (∆ ≥ λR) and ordinary (λR ≥ ∆) phases. Since the insulating regime is rather difficult to access experimentally, we propose to characterize this crossover by transport properties in the doped regime. In particular we stress that transport through a bipolar pn junction should differ strongly in the distinct cases ∆ ≥ λR and λR ≥ ∆ respectively. Quasi-relativistic Klein tunneling [5] was demonstrated experimentally [6–12] by using local gating techniques, and the corresponding theory have received a great deal of attention [13–16] in the absence of SO coupling (∆ = λR = 0). In this Letter, we obtain the angular dependent transmission of a pn junction in the presence of both intrinsic and Rashba SO couplings. At normal incidence, we show that the pn junction transmission exhibits a crossover from perfect reflection to perfect transmission when the Rashba coupling is tuned by the perpendicular electric field. Similar topological phases have also been predicted [17] and now observed [18] in materials with larger SO interactions such as HgTe/CdTe quantum wells. The protected edge states appear when the width of the HgTe layer exceeds a critical value. We choose to study graphene since the Kane-Mele is the simplest possible model that have four spin-split bands, which is the minimum required for the nontrivial phase to exist [19, 20]. Moreover our predictions might be compared to the studies of Klein tunneling in graphene performed in the absence of SO coupling (∆ = λR = 0). Our results should pertain to twodimensional HgTe as well, at least at the qualitative level. Kane-Mele model and single valley approximation. – The Kane-Mele model describes the low-energy dynamics of quasiparticles near the K and K ′ points of graphene in the presence of spin-orbit effects [2]. The corresponding Hamiltonian HKM = H0 + HSO + HR acts on the slowly varying envelop ψ(x, y) of electronic Bloch wavefunctions, which are indexed by real spin (Pauli matrices si, i = x, y, z), lattice isospin (σi) and valley isospin (τ i) quantum numbers. The kinetic Hamiltonian H0 = −i~vFψ(σxτ z∂x + σy∂y)ψ (1) describes massless Dirac fermions and is spin-independent. In the following we shall use units with ~ = vF = 1.