Ab-initio wave packet dynamical simulation of defects in 2D materials

Geza I Mark,Peter Vancso

doi:10.3390/asec2020-07760

Abstract

Materials consisting of a single layer of atoms, often called two dimensional (2D) materials have many promising applications, due to their extraordinary physical properties. These properties, however, depend on the presence, density, and kind of structural defects present in the perfect 2D crystalline lattice. Electrons with energies falling into the allowed band, described by Bloch waves in quantum mechanics, propagate freely in a perfect crystal, but defects act as scattering centers for the Bloch waves. We studied the influence of different structural defects on the transport properties of a graphene lattice by calculating the scattering of electronic wave packets. We applied and compared two different methods. Within the first method, we describe the atomic lattice and the electronic structure of graphene by an atomic pseudopotential, then calculate the Bloch functions and corresponding E(kBloch) energies, where kBloch is the Bloch wave vector. The defect is represented by a local potential, then we compute the scattering by the time development of a wave packet composed of the Bloch waves. In the second method, however, we don’t need to calculate the wave functions, thus we also don’t need the graphene potential, because we incorporate the E(k) dispersion relation directly into the kinetic energy operator and the defect is still represented by a local potential. The dispersion relation can be a simple tight-binding (TB) dispersion relation, but for a more accurate representation of the electronic structure, we can utilize E(k) relations from an ab-initio DFT calculation.

Highlights

In order to design nanoelectronic devices, it is important to precisely understand the dynamics of electrons in these structures
The physical system is described by a Hamiltonian and the initial conditions are given by an initial wave function
We presented an extension to the wave packet dynamical method, where the initial wave packet is constructed from the numerically calculated Bloch states of the pseudopotential representing the physical system

Summary

Introduction

In order to design nanoelectronic devices, it is important to precisely understand the dynamics of electrons in these structures. This was calculated by numerically solving the time dependent Schrödinger equation with the splitoperator FFT method [12], [13], [14] using the Ĥ = Kfree + V Hamilton operator, where Kfree is the free space kinetic energy operator and V = V(r⃗) is the graphene pseudopotential [5]. This time evolution – i.e. that for the infinite lattice – can be analytically calculated by inserting the Exp[−iE(k⃗⃗Bloch)t] time propagator into the kernel of Eq., where E(k⃗⃗Bloch) is the graphene dispersion relation. We can see that the WP is scattered on the defect and the angular distribution of the scattered WP has a hexagonal symmetry

Band structure governed wave packet dynamics

Conclusions