Abstract
The time evolution of electron waves in graphene superlattices is studied using both microscopic and “effective medium” formalisms. The numerical simulations reveal that in a wide range of physical scenarios it is possible to neglect the granularity of the superlattice and characterize the electron transport using a simple effective Hamiltonian. It is verified that as general rule the continuum approximation is rather accurate when the initial state is less localized than the characteristic spatial period of the superlattice. This property holds even when the microsocopic electric potential has a strong spatial modulation or in presence of interfaces between different superlattices. Detailed examples are given both of the time evolution of initial electronic states and of the propagation of stationary states in the context of wave scattering. The theory also confirms that electrons propagating in tailored graphene superlattices with extreme anisotropy experience virtually no diffraction.
Highlights
Graphene is a carbon-based material where the atoms are arranged in a honeycomb lattice.[1,2,3,4,5,6,7,8] This genuinely two-dimensional material is characterized by unusual and remarkable electronic properties, including a “relativistic”-type spectrum, such that the propagation of low-energy electrons in graphene is described by the massless Dirac equation.[3]
We develop a finite-difference time-domain (FDTD) algorithm to characterize the propagation of electron waves in superlattices using both microscopic and macroscopic formalisms
It is underlined that the theory developed in Refs. 30 and 31 cannot be directly applied to the propagation of electron waves in the context of the effective medium model considered here, and this is the reason why we develop our own numerical scheme to solve the modified Dirac equation
Summary
Graphene is a carbon-based material where the atoms are arranged in a honeycomb lattice.[1,2,3,4,5,6,7,8] This genuinely two-dimensional material is characterized by unusual and remarkable electronic properties, including a “relativistic”-type spectrum, such that the propagation of low-energy electrons in graphene is described by the massless Dirac equation.[3]. We are unaware of similar studies in related physical platforms To this end, a FDTD algorithm based on a leapfrog update scheme[40] is developed and applied to the characterization of graphene superlattices using both the macroscopic and microscopic models. 30 and 31 cannot be directly applied to the propagation of electron waves in the context of the effective medium model considered here, and this is the reason why we develop our own numerical scheme to solve the modified Dirac equation.
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