Interplay between phonon and impurity scattering in two-dimensional hole transport

We investigate temperature-dependent transport properties of two-dimensional $p$-GaAs systems taking into account both hole-phonon and hole-impurity scattering effects. By analyzing the hole mobility data of $p$-GaAs in the temperature range $10\text{K}<T<100$ K, we estimate the value of the appropriate deformation potential for hole-phonon coupling. Due to the interplay between hole-phonon and hole-impurity scattering the calculated temperature-dependent resistivity shows interesting nonmonotonic behavior. In particular, we find that there is a temperature range (typically $2\text{K}<T<10$ K) in which the calculated resistivity becomes independent of temperature due to a subtle cancellation between the temperature-dependent resistive scattering contributions arising from impurities and phonons. This resistivity saturation regime appears at low carrier densities when the increasing resistivity due to phonon scattering compensates for the decreasing resistivity due to the nondegeneracy effect. This temperature-independent flat resistivity regime is experimentally accessible and may have already been observed in a recent experiment.

Figure 1

(a) Schematic resistivity behavior in the presence of both hole-impurity scattering and acoustic-phonon scattering as a function of temperature. Typically $T_1\sim T_{\mathrm{F}}$ and $T_2\sim 5–10$ K depending on the carrier density.

Figure 2

Mobility as a function of temperature for several values of $D$ with $n=2\times {10}^{11}$ cm${}^{-2}$, $n_{\mathrm{imp}}=1.22\times {10}^{10}$ cm${}^{-2}$, $n_{\mathrm{depl}}=0$ and $d_{\mathrm{imp}}=0$. Black dots represent the experimental data (Ref. 14).

Figure 3

Fitted deformation potential $D$ as a function of depletion density $n_{\mathrm{depl}}$ for a sample with $n=2\times {10}^{11}$ cm${}^{-2}$ and $d_{\mathrm{imp}}=0$.

Figure 4

The relaxation rate (solid line) and scattering rate (dashed line) as a function of hole energy $E$ for (a) $T=10$ K and (b) $T=1$ K with $n={10}^{11}$ cm${}^{-2}$, $n_{\mathrm{imp}}=0$, $n_{\mathrm{depl}}=0$, and $D=12.7$ eV. DP and PE represent the deformation potential and piezoelectric potential contributions, respectively.

Figure 5

(a) Acoustic-phonon-limited resistivity of 2DHS as a function of temperature for $n={10}^8$, ${10}^9$, ${10}^{10}$, ${10}^{11}$, ${10}^{12}$ cm${}^{-2}$ and (b) the calculated exponent $a$ in $\rho \left(T\right)\propto T^a$ which is obtained from logarithmic derivatives of (a).

Figure 6

(a) Acoustic-phonon-limited mobility of 2DHS as a function of temperature for $n={10}^8$, ${10}^9$, ${10}^{10}$, ${10}^{11}$, ${10}^{12}$ cm${}^{-2}$. (b) Density dependence of coefficient $\alpha$ for $n_{\mathrm{imp}}=0$ and $5\times {10}^9$ cm${}^{-2}$, where $1/\mu =1/{\mu }_0+\alpha T$ in $10\text{K}<T<60$ K range.

Figure 7

(a) Resistivity of 2DHS as a function of temperature for $n_{\mathrm{imp}}=0,1,2,3,5,7,10\times {10}^9$ cm${}^{-2}$ with $n={10}^{10}$ cm${}^{-2}$, $n_{\mathrm{depl}}=0$, and $d_{\mathrm{imp}}=0$. Dotted lines indicate the calculated resistivity due to the charged impurity scattering alone. (b) Same as (a) but rescaled by ${\rho }_0=\rho (T=0.1$ K).

Figure 8

(a) Resistivity of 2DHS as a function of temperature for $d_{\mathrm{imp}}=0,5,10,15,20,25,30$ nm with $n={10}^{10}$ cm${}^{-2}$, $n_{\mathrm{imp}}=5\times {10}^9$ cm${}^{-2}$, and $n_{\mathrm{depl}}=0$. Dotted lines indicate the calculated resistivity due to the charged impurity scattering alone. (b) Same as (a) but rescaled by ${\rho }_0=\rho (T=0.1$ K).