Theory of degenerate1sexcitons in zinc-blende-type crystals in a magnetic field: Exchange interaction and cubic anisotropy

Abstract

The magneto-optical behavior of the eightfold $1s$-exciton states arising from the ${\Gamma }_6$ conduction and ${\Gamma }_8$ valence-band pair is studied. The effective Hamiltonian for these states is obtained by symmetry considerations as the sum of invariant terms with appropriate coefficients. The explicit forms of the coefficients are described in terms of the band parameters by using the perturbational scheme proposed by Altarelli and Lipari. The most general forms for the ${\Gamma }_6$ and ${\Gamma }_8$ band edges are considered as well as the electron-hole exchange interaction. By separating the problem into a spherical part and an anisotropic part of cubic symmetry, a clear-cut view is obtained with respect to the dependence of the optical spectrum on the direction of the magnetic field. The result is expressed in matrix forms for the magnetic field along the $〈001〉$, $〈111〉$, and $〈110〉$ crystal axes.