Calculation of Bulk Material Electronic Band Structure Using Matrix Mechanics: A Pedagogical Approach

Abstract

We present a method, intended primarily for pedagogical purposes, to extend the one-dimensional Kronig-Penney model (the standard starting point for solid-state physics) to the case of arbitrary potential shapes and to higher dimensions using matrix mechanics. We generate, either analytically or numerically, the matrix elements for a unit cell of some potential which can then be diagonalized to give the energies and eigenstates. Bloch’s theorem can be introduced as purely additive non-potential-dependent terms on the main diagonal which allows us to “sweep out” the band structure at virtually no additional cost. In one-dimension our results correspond exactly with the known analytical solutions to the Kronig-Penney model, and in higher dimensions our results match the usual shallow and deep well limits, namely the nearly free electron model and the tight binding model respectively. Our method has the advantage of using only concepts familiar to a senior quantum mechanics student, lending it conceptual simplicity and clarity. This provides a tool for senior undergraduates and beginning graduate students to generate their own band structures, allowing for a tight coupling between computational exploration and intuition formation.