Chiral interface states in graphene p−n junctions

Abstract

We present a theoretical analysis of unidirectional interface states which form near p-n junctions in a graphene monolayer subject to a homogeneous magnetic field. The semiclassical limit of these states corresponds to trajectories propagating along the p-n interface by a combined skippingsnaking motion. Studying the two-dimensional Dirac equation with a magnetic field and an electrostatic potential step, we provide and discuss the exact and essentially analytical solution of the quantum-mechanical eigenproblem for both a straight and a circularly shaped junction. The spectrum consists of localized Landau-like and unidirectional snaking-skipping interface states, where we always find at least one chiral interface state. For a straight junction and at energies near the Dirac point, when increasing the potential step height, the group velocity of this state interpolates in an oscillatory manner between the classical drift velocity in a crossed electromagnetic field and the semiclassical value expected for a purely snaking motion. Away from the Dirac point, chiral interface states instead resemble the conventional skipping (edge-type) motion found also in the corresponding Schrödinger case. We also investigate the circular geometry, where chiral interface states are predicted to induce sizeable equilibrium ring currents.

Coexistence of classical snake states and Aharonov-Bohm oscillations along graphene p−n junctions

“…The factor k∆L in Equation 9 accounts for a possible path-difference between the edge states, where k is replaced by k = k F + eV /( v F β). The parameter β (0 ≤ β ≤ 1) was introduced to account for the renormalized edge state velocity compared to the Fermi velocity [50]. We have found that the second term alone (k∆L at α = 0) cannot lead to substantial bias dependence if the parameters are chosen realistically (∆L ∼20 nm and β = 1).…”

Spin-orbit interaction and snake states in a graphene p-n junction

“…This value does not depend on graphene's Fermi velocity. Since V 0 /e B can be interpreted as the electric field E x induced by the potential step in the x-direction and 2 B ∝ 1/B, V 0 B / corresponds to the classical drift velocity of a charged particle in crossed magnetic and electric fields, cE x /B [42]. The effect of the Rashba SOI is a renormalization factor which decreases for increasing λ from 2/ √ π (for λ = 0) to 1 √ π (for λ ω c ) and thus tends to slightly reduce the velocities.…”

“…For λ = 0 the Hamiltonian (1) is block-diagonal in spin. Each block coincides with the Hamiltonian studied in [42] (up to the intrinsic SOI term and the Zeeman term, which can be easily incorporated) and we can borrow their results. The wave functions can be expressed as…”

Gradient-index electron optics in graphene pn junctions

“…Most of the work on electron optics in graphene pn junctions involves interfaces where the electrostatic potential (and hence the refractive index) changes abruptly [8,29,35,[66][67][68]. Recent experiments have demonstrated that such abrupt junctions can indeed be realized [13].…”