Complete spin and valley polarization by total external reflection from potential barriers in bilayer graphene and monolayer transition metal dichalcogenides

Abstract

It is shown that potential barriers in bilayer graphene (BLG) and monolayer transition metal dichalcogenides (TMDs) can split a valley unpolarized incident current into reflected and transmitted currents with opposite valley polarization. Valley asymmetric transmission inevitably occurs because of the low symmetry of the total Hamiltonian and when total external reflection occurs the transmission is 100% valley polarized in BLG and 100% spin and valley polarized in TMDs, except for exponentially small corrections. By adjusting the potential, 100% polarization can be obtained regardless of the crystallographic orientation of the barrier. A valley polarizer can be realized by arranging for a collimated beam of carriers to be incident on a barrier within the range of angles for total external reflection. The transmission coefficients of barriers with a relative rotation of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/3$ are related by symmetry. This allows two barriers to be used to demonstrate that the current is valley polarized. A soft-walled potential is used to model the barrier and the method used to find the transmission coefficients is explained. In the case of monolayer TMDs, a four-band $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian is used and the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ parameters are obtained by fitting to ab initio band structures.