  Modulation of electronic states in two-dimensional (2D) materials can be
achieved by using in-plane variations of the band gap or the average potential
in lateral quantum structures. In the atomic configurations with hexagonal
symmetry, this approach makes it possible to tailor the valleytronic properties
for potential device applications. In this work, we present a multi-band theory
to calculate the valley-dependent electron transport in graphene-based lateral
quantum structures. As an example, we consider the structures with a single
interface that exhibits an energy gap or potential discontinuity. The
theoretical formalism proceeds within the tight-binding description, by first
deriving the local bulk complex band structures in the regions of a constant
gap or potential and, next, joining the local wave functions across the
interface via a cell-averaged current operator to ensure the current
continuity. The theory is applied to the study of electron reflection off and
transmission through an interface. Both reflection and transmission are found
to exhibit valley-contrast behavior that can be used to generate
valley-polarized electron sources. The results vary with the type of
interfaces, as well as between monolayer and bilayer graphene based structures.
In the monolayer case, the valley contrast originates from the band warping and
only becomes sizable for incident carriers of high energy; whereas in
AB-stacked bilayer graphene, the vertical interlayer coupling emerges as an
additional important cause for valley contrast, and the favorable carrier
energy is also found to be drastically lower. Our numerical results clearly
demonstrate the propitious valleytronic properties of bilayer graphene
structures.
