We formulate the multiband $k·p$ theory of hyperfine interactions for semiconductor nanostructures in the envelope function approximation. We apply this theoretical description to the fluctuations of the longitudinal and transverse Overhauser field experienced by a hole for a range of InGaAs quantum dots of various compositions and geometries. We find that for a wide range of values of $d$-shell admixture to atomic states forming the top of the valence band, the transverse Overhauser field caused by this admixture is of the same order of magnitude as the longitudinal one, and band mixing adds only a minor correction to this result. In consequence, the $k·p$ results are well reproduced by a simple box model with the effective number of ions determined by the wave-function participation number, as long as the hole is confined in the compositionally uniform volume of the dot, which holds in a wide range of parameters, excluding very flat dots.

• Received 25 February 2019
• Revised 1 August 2019

DOI:https://doi.org/10.1103/PhysRevB.100.085305