Valley-dependent band structure and valley polarization in periodically modulated graphene

Abstract

The valley-dependent energy band and transport property of graphene under a periodic magnetic-strained field are studied, where the time-reversal symmetry is broken and the valley degeneracy is lifted. The considered superlattice is composed of two different barriers, providing more degrees of freedom for engineering the electronic structure. The electrons near the $K$ and ${K}^{\ensuremath{'}}$ valleys are dominated by different effective superlattices. It is found that the energy bands for both valleys are symmetric with respect to ${k}_{y}=\ensuremath{-}({A}_{M}+\ensuremath{\xi}{A}_{S})/4$ under the symmetric superlattices. More finite-energy Dirac points, more prominent collimation behavior, and new crossing points are found for ${K}^{\ensuremath{'}}$ valley. The degenerate miniband near the $K$ valley splits into two subminibands and produces a new band gap under the asymmetric superlattices. The velocity for the ${K}^{\ensuremath{'}}$ valley is greatly renormalized compared with the $K$ valley, and so we can achieve a finite velocity for the $K$ valley while the velocity for the ${K}^{\ensuremath{'}}$ valley is zero. Especially, the miniband and band gap could be manipulated independently, leading to an increase of the conductance. The characteristics of the band structure are reflected in the transmission spectra. The Dirac points and the crossing points appear as pronounced peaks in transmission. A remarkable valley polarization is obtained which is robust to the disorder and can be controlled by the strain, the period, and the voltage.