Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field

Abstract

Following up an earlier communication, wave functions are constructed which satisfy the Schr\"odinger equation for a potential which is a sum of a periodic and a uniform field term. The wave functions are Houston modifications of Bloch type functions; the Bloch functions form an orthogonal set whose members are fully determined except for phase. The theory exhibits them in the form of power series in the field strength; the unmodified Bloch band functions form the zero order term of that series. The solutions themselves do not allow for a Zener effect, but the fact that they are only given as power series in $E$ may imply that there is a remainder term causing interband transitions; it would have to be asymptotically smaller than any power of $E$. Instead of constructing time dependent solutions of the Schr\"odinger equation one can take the time independent functions to construct an effective Hamiltonian for electrons in one band; it has the form (16). Certain indeterminacies are attached to this form of representation; it is shown, however, that final physical answers are unique. The study furnishes an incidental proof that k-space is a finite space consisting in its entirety of what is customarily called the first Brillouin zone. An appendix treats the case of degenerate bands; such bands have singularities in k-space even in the absence of a field. The difficulty is circumvented by working with a set which is not yet diagonalized but free of singularities; these intermediate functions can be continued as power series in $E$ in the same way as nondegenerate band functions.