Theory of the effective Hamiltonian for degenerate bands in an electric field

Recent experiments have generated a renewed interest in the properties of the degenerate valence states in semiconductors under the influence of a uniform external electric field. In response, a number of authors have proposed that the standard Luttinger-Kohn effective-mass Hamiltonian should be modified to include the energy of interaction between the electric field and the dipole matrix elements of the relevant zone-centre Bloch functions. This article examines these proposals by comparing the proposed dipole interaction with rigorous derivations of the field-dependent Hamiltonian in the Bloch and Luttinger-Kohn representations. It is shown that the dipole matrix element of a unit cell is not a suitable foundation for a Hamiltonian because it depends on the choice of unit cell, which is arbitrary. Moreover, the correct Luttinger-Kohn Hamiltonian has no wave-vector-independent matrix elements that are linear in the applied field ε, except for small terms of relativistic origin. Therefore, the proposed modifications to the Luttinger-Kohn theory are not valid. The correct form of the Luttinger-Kohn Hamiltonian is derived here through terms of order k3, kε, and ε2, along with the momentum matrix to first order in k and ε. In addition, the recent discovery of field-induced mixing at the Brillouin zone boundary in the Luttinger-Kohn representation is studied in detail.