Chiral Bloch states in single-layer graphene with Rashba spin-orbit coupling: Equilibrium spin current

Focusing on equilibrium spin current density tensor, we analyze the spin physics of electrons in single layer graphene subject to a one-dimensional periodic Kronig-Penney potential u(x) and uniform Rashba spin-orbit coupling of strength λ. The combination of Dirac theory of massless two-dimensional electrons, Klein paradox, spin-orbit coupling, and the Bloch theorem yields peculiar features relevant to graphene spintronics. For transverse wave number ky=0, we show that: (1) The charge current and the spin density vector vanish. (2) Yet, the components Jxy,Jyx,Jzy of the equilibrium spin current density tensor are finite, oscillating in space, with amplitudes that increase quadratically for small λ and then linearly with λ. Quite remarkably, Jzy(x)≠0, that is, the spin is polarized along z, perpendicular to the graphene plane. (3) Due to the continuity equation, the space dependence of the spin current density Jyx(x) is associated with a finite spin torque density Ty(x)=∂Jyx(x)∂x.