Dielectric function of the semiconductor hole liquid: Full frequency and wave-vector dependence

We study the dielectric function of the homogeneous semiconductor hole liquid of $p$-doped bulk III-V zinc-blende semiconductors within random-phase approximation. The single-particle physics of the hole system is modeled by Luttinger’s four-band Hamiltonian in its spherical approximation. Regarding the Coulomb-interacting hole liquid, the full dependence of the zero-temperature dielectric function on wave vector and frequency is explored. The imaginary part of the dielectric function is analytically obtained in terms of complicated but fully elementary expressions, while in the result for the real part nonelementary one-dimensional integrations remain to be performed. The correctness of these two independent calculations is checked via Kramers-Kronig relations. The mass difference between heavy and light holes, along with variations in the background dielectric constant, leads to dramatic alternations in the plasmon excitation pattern, and, generically, two plasmon branches can be identified. These findings are the result of the evaluation of the full dielectric function and are not accessible via a high-frequency expansion. In the static limit a beating of Friedel oscillations between the Fermi wave numbers of heavy and light holes occurs.

Figure 1

Regions of nonvanishing contributions to (a) $I_{hh}(\vec{q},\omega )$, (b) $I_{ll}(\vec{q},\omega )$, (c) $I_{hl}^{+}(\vec{q},\omega )$ (solid lines) and $I_{hl}^{-}(\vec{q},\omega )$ (dashed lines), and (d) $I_{lh}^{+}(\vec{q},\omega )$ (solid lines) and $I_{lh}^{-}(\vec{q},\omega )$ (dashed lines); cf. Tables . We have chosen the mass parameters of GaAs, $m_h=0.5m_0$, $m_l=0.08m_0$, and a hole density of $n=0.01{\mathrm{nm}}^{-3}$.

Figure 2

The modulus $\left|\mathrm{Re}\right[{\varepsilon }^{\text{RPA}}(\vec{q},\omega )\left]\right|$ of the real part of the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for a model system with ${\varepsilon }_r=1$. The ratio of heavy and light mass is varied at constant $m_H+m_l=m_0$, and the total hole density is $n=0.01{\mathrm{nm}}^{-3}$.

Figure 3

The imaginary part $\mathrm{Im}\left[{\varepsilon }^{\text{RPA}}(\vec{q},\omega )\right]$ of the the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for a model system with ${\varepsilon }_r=1$. The ratio of heavy and light mass is varied at constant $m_H+m_l=m_0$, and the total hole density is $n=0.01{\mathrm{nm}}^{-3}$.

Figure 4

The modulus $|{\varepsilon }^{\text{RPA}}(\vec{q},\omega )|$ of the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for a model system with ${\varepsilon }_r=1$. The ratio of heavy and light mass is varied at constant $m_H+m_l=m_0$, and the total hole density is $n=0.01{\mathrm{nm}}^{-3}$. The dark areas indicate zeros of the dielectric function corresponding to plasmon excitations.

Figure 5

The real part of the free polarizability ${\chi }_0(\vec{q},\omega )$ as a function of frequency at different wave vectors for the same choice of heavy and light hole masses as in the previous figures.

Figure 6

The imaginary part of the free polarizability ${\chi }_0(\vec{q},\omega )$ as a function of frequency at different wave vectors for the same choice of heavy and light hole masses as in the previous figures.

Figure 7

The modulus $\left|\mathrm{Re}[\right({\varepsilon }^{\text{RPA}}(\vec{q},\omega )\left]\right|$ of the real part of the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for various semiconductor systems at a total hole density of $n=0.35{\mathrm{nm}}^{-3}$.

Figure 8

The imaginary part $\mathrm{Im}\left[{\varepsilon }^{\text{RPA}}(\vec{q},\omega )\right]$ of the the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for various semiconductor systems at a total hole density of $n=0.35{\mathrm{nm}}^{-3}$.

Figure 9

The modulus $|{\varepsilon }^{\text{RPA}}(\vec{q},\omega )|$ of the RPA dielectric function as a function of wave number $q$ and energy $\hslash \omega$ for various semiconductor systems at a total hole density of $n=0.35{\mathrm{nm}}^{-3}$. The dark areas indicate zeros of the dielectric function corresponding to plasmon excitations.

Figure 10

The real (top panels) and imaginary (bottom panels) part of the free polarizability ${\chi }_0(\vec{q},\omega )$ as a function of frequency at different wave vectors for the same semiconductor systems as before (cf. Table ).

Figure 11

(Left panel) The static free polarizability ${\chi }_0(\vec{q},0)$ for the same choice of parameters as in Figs. 2, 3, 4, 5, 6. (Right panels) ${\chi }_0(\vec{q},0)$ and ${\varepsilon }^{\text{RPA}}(\vec{q},0),$ the same III-V semiconductor systems as in Figs. 7, 8, 9, 10 (cf. Table ).