Holes localized in nanostructures in an external magnetic field: $g$-factor and mixing of states

The energy spectrum and wave functions of holes in the valence band in semiconductor nanosystems, including quantum wells, quantum wires, and quantum dots, in an external magnetic field are theoretically investigated. The dependence of Zeeman splitting of the hole ground state upon variation in the size-quantization parameters with regard to the complex structure of the valence band and magnetic field-induced mixing of hole states is traced. Analytical formulas for describing the Zeeman effect in the valence band in the limiting cases of a quantum disk, spherically symmetric quantum dot, and quantum wire are presented. It is demonstrated that the $g$-factor of a hole is extremely sensitive to the hole-state composition (heavy or light hole) and, consequently, to the geometry of the size-quantization potential.