Evolution operator for bloch electrons in external electric field

Abstract

The s tudy of mot ion of Bloeh electrons ira the presence of au externa l e lectr ic field cons t i tu tes an impor t an t area in sol id-s ta te physics. The m~iden work in the a rena is t ha t of Zener(~), concerning wi th wha t is t e rmed as ~ Zencr tunncling~). A large number of theore t ica l papers came up dur ing m a n y past years regard ing var ious aspects of the motion of Bloch electrons in the presence of an ex te rna l electr ic field. Bulk of these works (~..4) have widely deal t wi th Zener tunne l ing as well as t he exci t ing phenomenon , named as, (, %~annier Stark L~ddcrs ~; in such inves t iga t ions , one is requi red to use a wave fune{ion, which describes the behav iour of t im electron under the combined a(.t ion of the periodic crys ta l po ten t i a l and the externa l e lectr ic field. IIousTO~(~) was the first worker to propose a w~ve funct ion for th is purpose, mak ing use of the fa.ct t ha t the wave vec tor var ies l inear ly wi th t ime in presence of the externa l e lectr ic field. Houston ' s wave funct iou has since been widely used in s tudying phenomen~ like Zoner tunneling, s tark laddel 's etc. (.z.~.6.~). ]n recent years, the ide~ of evolu t ion opera tor has been explo i ted by many workers in s tudying the behav iour of Bloch electrons in electr ic fields. The cases inves t iga ted nre, however , ve ry m u c h res t r ic ted (7.10) and a general form of such an evolu t ion operator , does not seem to h~ve been yet formlf la ted undeba tab ly (10). In the present art icle, we repor t the der iva t ion of a general form of the a fore-ment ioned evolu t ion operator. W e also show tha t the evolu t ion opera tor der ivcd by us, when ~pprox ima ted to first order in ex terna l field, yields Hous ton ' s wave funct ion. Fur the r , we show t h a t the equa l i ty of the ex te rna l field to the first-order t ime de r iva t ive of the k-vector , follows from qui te general considerat ions, ~ 'hen one makes use of the evolu t ion opera tor der ived in this art icle.