LAUSR

We formulate the multi-band 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, in agreement with previous estimates, that the transverse Overhauser field is of the same order of magnitude as the longitudinal one and is dominated by the $d$-shell admixture to atomic states with only a minor correction from band mixing in all the cases. 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.