$p$-type $\delta$-doping quantum wells and superlattices in Si: Self-consistent hole potentials and band structures

The hole-subband and -miniband structures of periodically acceptor $\delta$-doped quantum wells and superlattices (SL’s) in silicon are calculated self-consistently within the effective-mass theory and the local-density approximation. The full six-band Luttinger-Kohn effective-mass equations are solved, together with Poisson equation, in a plane-wave representation. Nonparabolicities due to couplings between heavy, light, and spin-orbit split bands are fully taken into consideration. To account for exchange and correlation (XC) effects within the multicomponent hole gas, a parametrized expression for the XC potential energy is adopted. Hole band structures, Fermi levels, and potentials are presented for a series of p-type $\delta$-doping SL’s, varying the acceptor doping concentrations, periods, and doping spreads. The inclusion of the spin-orbit split band is reflected essentially in nonparabolicities, and it starts to play an important role already for intermediate concentrations. For acceptor doping concentrations above $1.1\times {10}^{14}\hspace{0.5em}{\mathrm{cm}}^{-2},$ the split-off band is populated for SL periods in both SL and isolated well regimes. A comparison with the available experimental data shows fairly good agreement. Particularly, the data reported on admittance and infrared spectroscopies can be reasonably interpreted if one assumes indirect transitions between subbands, as is the case in p-type $\delta$-doped GaAs.