Semiclassical approach to 2d impurity bound states in Dirac systems

TL;DR

This paper introduces a semiclassical method for analyzing impurity bound states in 2D systems with parabolic or Dirac-like bands, simplifying the problem by reducing it to effective 1D models.

Contribution

The paper develops a novel semiclassical approach that simplifies the analysis of impurity bound states in 2D Dirac systems by reducing the problem to 1D effective models.

Findings

Method successfully applied to models with parabolic dispersion

Method effectively applied to Dirac-like dispersion relevant to topological insulators

Provides an intuitive tool for impurity state analysis in 2D materials

Abstract

The goal of this paper is to provide an intuitive and useful tool for analyzing the impurity bound state problem. We develop a semiclassical approach and apply it to an impurity in two dimensional systems with parabolic or Dirac like bands. Our method consists of reducing a higher dimensional problem into a sum of one dimensional ones using the two dimensional Green functions as a guide. We then analyze the one dimensional effective systems in the spirit of the wave function matching method as in the standard 1d quantum model. We demonstrate our method on two dimensional models with parabolic and Dirac-like dispersion, with the later specifically relevant to topological insulators.