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Time dynamics of electron waves in graphene superlattices

Fernandes, D. E. ; Rodrigues , M. ; Falcão, G. ; Silveirinha, M. G.

Time dynamics of electron waves in graphene superlattices, Proc Theory, Modelling and Computational Methods for Semiconductors - European Session, Granada, Spain, Vol. 1, pp. 1 - 1, January, 2015.

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Abstract
Graphene is a two-dimensional material with an unusual electronic “relativistic” spectrum. It may be possible to gain some control over the propagation of charge carriers in graphene by applying an electrostatic potential on graphene’s surface –using, for instance, periodically patterned gates – creating in this manner a superlattice. For example, it has been suggested that graphene superlattices (GSLs) may collimate the electron flow with virtually no lateral spreading or diffraction. In general, the proper tailoring of the microscopic potential may allow controlling the transport properties of electrons in superlattices.

It was recently proposed that the low energy electronic states in GSLs can be characterized using an effective medium framework . Within this approach, the microscopic details of the superlattice are homogenized and the structure is regarded as a continuous medium described by some effective parameters. This theory was successfully applied to the study of scattering problems , demonstrating interesting phenomena such as “wormholes” for electron waves, where the electrons are perfectly transmitted across a GSL slab as if it was non-existent.

Here, we develop a Finite-Difference-Time-Domain (FDTD) algorithm to characterize time evolution problems in GSLs. Using both effective medium and microscopic approaches, we study the propagation of initial electronic states, proving explicitly that the effective Hamiltonian describes the time evolution of macroscopic states, i.e. of states that vary slowly on the scale of the spatial period. The results confirm that GSLs allow for the propagation of electrons in regimes of extreme anisotropy. Moreover, we study problems with interfaces between different superlattices, and show that within the effective medium approach the wave function satisfies a non-trivial boundary condition at the interfaces. Examples of the FDTD algorithm applied to more complex GSLs structures will be presented at the conference.