Graphene Superlattices & Kronig-Penney model

[PRB147]

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 explanation:

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?

 

[eljose79]

Hello,

Thank you for bringing up this interesting topic. I am a scientist who specializes in graphene research, and I would be happy to help you understand the numerical methods used in calculating the density of states in graphene superlattices.

Firstly, you are correct in stating that the Kronig-Penney model in graphene superlattices is a 2D problem, unlike in semiconductors where it can be treated as a 1D problem. This is because in graphene, the transverse momentum k_y cannot be isolated from k_x, making it a more complex system to study.

In terms of numerical methods, one commonly used approach is the transfer matrix method, which involves calculating the transfer matrix for each layer in the superlattice and then using it to determine the overall transfer matrix for the entire structure. This method takes into account both the periodic potential of the superlattice and the energy dispersion of graphene.

Another method that has been used is the recursive Green's function method, which involves solving for the Green's function of the superlattice using a recursive algorithm. This method has been shown to accurately capture the band structure and density of states in graphene superlattices.

I would also like to mention that there have been recent developments in using machine learning algorithms to calculate the density of states in graphene superlattices. These methods have shown promising results and could potentially provide a more efficient and accurate way to study these systems in the future.

I hope this helps answer your question and provides some insight into the numerical methods used in studying graphene superlattices. If you have any further questions, please do not hesitate to ask.