The relation between the effective band mass in a solid and the free electron mass

Abstract

The Kronig–Penney model is a model used to study properties of electrons confined in a periodic potential. It is a useful model since the electronic properties can be studied analytically with the aid of Bloch’s theorem. An important concept that emerges from the resultant electron band theory is the electron effective mass. Through the use of the Kronig–Penney model we use the so-called ‘effective mass theorem’, that allows us to ask and answer the question of how the electron effective mass depends on both the free electron mass and the characteristics of the periodic potential. We illustrate the crossover from the case of a weak periodic potential to a strong periodic potential using this theorem and explicit calculations with the Kronig–Penney model. The special case of the Dirac comb model is also treated in this work. Through the use of the ‘effective mass theorem’, we show, among other properties, that an electron–hole effective mass asymmetry is generally expected, even without considering contributions from electron–electron interactions.