Abstract
We discuss the electronic properties of graphene and graphene nanoribbons including ‘pseudo-Rashba’ spin–orbit coupling. After summarizing the bulk properties of massless and massive Dirac particles, we first analyze the scattering behavior close to an infinite mass and zigzag boundary. For low energies, we observe strong deviations from the usual spin-conserving behavior at high energies such as reflection acting as a spin polarizer or switch. This results in spin polarization along the direction of the boundary due to the appearance of evanescent modes in the case of non-equilibrium or when there is no coherence between the two one-particle branches. We then discuss the spin and density distribution of graphene nanoribbons.
Highlights
We discuss the electronic properties of graphene and graphene nanoribbons including ‘pseudo-Rashba’ spin–orbit coupling
We show the results for an incident plane wave of type I and type II with an infinite mass
If there is no coherence between the incident plane waves of types I and II, e.g. due to temperature, time-reversal symmetry is effectively broken and we find a net polarization in the x-direction by adding the two contributions ρ I and ρ II
Summary
The single-particle Hamiltonian of monolayer graphene with ‘pseudo-Rashba’ spin–orbit interaction can be formulated as [4]–[6], [24]. Where, among standard notation, λ is the spin–orbit coupling parameter, and the Pauli matrices τ , σ describe the sublattice and the electron spin degree of freedom, respectively. For a given wave vector k, this Hamiltonian reads explicitly as. It is straightforward to obtain the full eigensystem: We find a gaped pair of eigenvalues ε1,± = ± (hvF k)2 + λ2 + |λ|. With gV = 2 being valley degeneracy, the corresponding density of states reads ρ1(ε) gV 2π(hvF ). The other pair of dispersion branches does not exhibit a gap, ε2,± = ± (hvF k)2 + λ2 − |λ| ,. Φ is the usual azimuthal angle of the wave vector, k = k(cos φ, sin φ). Where for sin θ < 1 the sublattice and electron spin degree of freedom are entangled with each other
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