Graphene Superlattices & Kronig-Penney model

Main Question or Discussion Point

Numerically, Kronig-Penney model in graphene superlattices (SL) is drastically different from those in semiconductor SL. In semiconductors, transverse momentum k// can be isolated from the longitudinal momentum q, even if the superlattices potential is complex. In graphene, however, $$k_y$$ cannot be isolated from $$k_x$$. So Kronig-Penney model is numerically a 1D problem in the calculation of the density of states, while KP model in graphene is a 2D problems which is very difficult due to the failure in interpolation. Let me give a more detailed explaination:

in semiconductor: $$\cos(q \ell)=f(q,E-\frac{\hbar^2 k^2_{//}}{2m^*})$$
in graphene: $$\cos(k_x \ell)=f(k_y,E(k_x,k_y))$$
$$f$$ is a function from the trace of the transfer matrix, (actually is a complex expression which can not be written in analytical form.)

I search the keywords "graphene, kronig-penney" using google, but can not find a numerical methods in the calculation of density of states in graphene superlattices. Who can help?

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