Treating disorder in introductory solid state physics

Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.