Time evolution of Bloch electrons in a homogeneous electric field

Abstract

The theory of a Bloch electron moving in the presence of a homogeneous electric field is reviewed and objections to the conventional derivations are discussed. A new derivation of the time development of a Bloch electron moving in a homogeneous, but time-dependent, electric field is presented using a vector potential to describe the field rather than the usual scalar potential. This new treatment avoids all the basic assumptions of the conventional derivations and demonstrates that a Bloch electron will oscillate in a single band with the Bloch period if a homogeneous electric field is abruptly turned on, with a tunneling probability into other bands given by the conventional expression. It is also shown that the calculated optical absorption will have the same ladderlike structure that would be obtained if Wannier-Stark quantized energy levels are assumed, although the present calculation makes no such assumption. The previous objections to the existence of Bloch oscillations for electrons in a perfect periodic potential are examined and found to be irrelevant provided the tunneling probability per Bloch oscillation period is much less than one, a condition that is generally satisfied for typical elemental and compound semiconductors for electric fields smaller than ${10}^6$ V/cm.