A low-dimensional analysis of bound states for impurities in a lattice using delta potentials and Green functions

Abstract

In solid state physics, impurities in a lattice of atoms can 'capture' an electron in bound states. When we treat the lattice as continuous space, we can describe these impurities by using delta potentials. In this thesis we study the bound states and bound state energies from the time-independent Schrödinger equation adding three types of delta potentials, which we try to find using the Green function technique. We also apply a numerical approximation based on the central-difference method and compare the results from the two methods. First, we study an impurity in a one-dimensional lattice, secondly, two impurities in a one-dimensional lattice and finally, an impurity in a two-dimensional lattice.