Theoretical study of interacting hole gas in $p$-doped bulk III-V semiconductors

We study the homogeneous interacting hole gas in $p$-doped bulk III-V semiconductors. The structure of the valence band is modeled by Luttinger’s Hamiltonian in the spherical approximation, giving rise to heavy and light hole dispersion branches, and the Coulomb repulsion is taken into account via a self-consistent Hartree-Fock treatment. As a nontrivial feature of the model, the self-consistent solutions of the Hartree-Fock equations can be found in an almost purely analytical fashion, which is not the case for other types of effective spin-orbit coupling terms. In particular, the Coulomb interaction renormalizes the Fermi wave numbers for heavy and light holes. As a consequence, the ground state energy found in the self-consistent Hartree-Fock approach and the result from lowest-order perturbation theory do not agree. We discuss the consequences of our observations for ferromagnetic semiconductors, and for the possible observation of the spin Hall effect in bulk $p$-doped semiconductors. Finally, we also investigate elementary properties of the dielectric function in such systems.

Figure 1

The ratios of renormalized and unrenormalized Fermi wave numbers, $q_{h∕l}$ and $k_{h∕l}$, respectively, as a function of hole density for different III-V semiconductors. The renormalization of Fermi wave numbers affects primarily the light hole wave number at low densities. The inset shows $q_h$ and $q_l$ as a function of density for $\mathrm{GaAs}$.

Figure 2

The Hartree-Fock dispersions ${\epsilon }_{h∕l}^{HF}(k;q_h,q_l)$ for $\mathrm{GaAs}$ at a hole density of $n=5\times {10}^{-4}{\mathrm{nm}}^{-3}$. The solid lines show the dispersion including Coulomb exchange for renormalized Fermi wave number $q_{h∕l}$, while the dashed lines represent the dispersions of the free hole gas in the absence of interactions. The singularities in ${\epsilon }_{h∕l}^{HF}(k;q_h,q_l)$ at $k=q_{h∕l}$ are clearly pronounced while the singularities at $k=q_{l∕h}$ are weaker and hardly visible in the plot.

Figure 3

The ground state energy per particle $E_{tot}(q_h,q_l)∕N$ from the self-consistent Hartree-Fock treatment (with renormalized Fermi wave numbers) and the result $E_{tot}(k_h,k_l)∕N$ from lowest-order perturbation theory (with unrenormalized Fermi wave numbers) as a function of the density for $\mathrm{GaAs}$. The inset shows the same data as the main panel but as a function of the density parameter $r_s$.

Figure 4

The pair distribution function $g\left(r\right)$ for $\mathrm{GaAs}$ at three different densities.

Figure 5

The spin Hall conductivity ${\sigma }^S$ (converted to units of charge transport) as a function of hole density $n$ for $\mathrm{GaAs}$.