Signatures of Impurities in 1- and 2-dimensional Lattice and Continuum Models for Metals

Abstract

This thesis develops a method of quantitatively describing the effects of inserting an impurity into a one- or two-dimensional crystal. Firstly, the Green's function, a mathematical function necessary to find the density of states for such a system, is introduced. From there on, the thesis will proceed to finding the eigenenergies for the pure crystals using an exact and an approximate method. The impurity is then inserted in the form of a Dirac delta potential and the results of both methods are compared. For low energies the two methods reinforce each other and the results indicate that particles wish to scatter of with specific momenta and some positions are unfavorable to be occupied. For higher energies the approximation is no longer viable and results differ from the exact results.