Rigorous Perturbation Theory for Bloch Electrons in a Uniform Electric Field

Abstract

The quantum mechanics for an electron in an electrified Kronig—Penney lattice of infinite extent is solved rigorously to first order in the electric field strength. The existence of Stark ladders in the energy spectrum is confirmed. However, contrary to popular opinion, it appears that each energy band of the unperturbed crystal splits into not one, but many such ladders, the number increasing with diminishing electric field. Their interplay leads to an interesting "lattice effect" which should be observable in Franz-Keldysh absorption. The existence of a critical field ${\mathcal{E}}_c$ is demonstrated, above which only one Stark ladder per band survives, in agreement with conventional theory. ${\mathcal{E}}_c$ is typically about ${10}^7$ V/cm. The wave functions are also studied. The stationary states are found to have spatial extent inversely proportional to the electric field intensity. From them generalized Bloch states are constructed which tend to the field-free Bloch functions in the zero-field limit. Finally, the results are reevaluated for finite crystals. Many of the infinite-crystal solutions are found to be extraneous; those applicable to finite crystals are reliable only above a certain impressed electric potential, typically on the order of 0.1 V. The transition from energy band to Stark ladder is discussed. It is argued that multiple Stark ladders in the spectrum are necessary to make the zero-field limit intelligible.