Cubic anisotropy of hole Zeeman splitting in semiconductor nanocrystals

We study theoretically cubic anisotropy of Zeeman splitting of a hole confined in a semiconductor nanocrystal. This anisotropy originates from three contributions: crystallographic cubically symmetric spin and kinetic energy terms in the bulk Luttinger Hamiltonian and the spatial wave function distribution in a cube-shaped nanocrystal. From symmetry considerations, an effective Zeeman Hamiltonian for the hole's lowest even state is introduced, containing a spherically symmetric and a cubically symmetric term. The values of these terms are calculated numerically for spherical and cube-shaped nanocrystals as functions of the Luttinger Hamiltonian parameters. We demonstrate that the cubic shape of the nanocrystal and the cubic anisotropy of hole kinetic energy (so-called valence band warping) significantly affect effective g factors of hole states. In both cases, the effect comes from the cubic symmetry of the hole wave functions in a zero magnetic field. Estimations for the effective g factor values in several semiconductors with zinc-blende crystal lattices are made. Possible experimental manifestations and potential methods for measuring of the cubic anisotropy of the hole Zeeman splitting are suggested.