Spin-orbit splitting and effective masses in $p$-type GaAs two-dimensional hole gases

We present magnetotransport measurements performed on two-dimensional hole gases embedded in carbon doped $p$-type GaAs/AlGaAs heterostructures grown on [001] oriented substrates. A pronounced beating pattern in the Shubnikov–de Haas oscillations proves the presence of strong spin-orbit interaction in the device under study. We estimate the effective masses of spin-orbit-split subbands by measuring the temperature dependence of the Shubnikov–de Haas oscillations at different hole densities. While the lighter heavy-hole effective mass is not energy dependent, the heavier heavy-hole effective mass has a prominent energy dependence, indicating a strong spin-orbit induced nonparabolicity of the valence band. The measured effective masses show qualitative agreement with self-consistent numerical calculations.

Figure 1

(a) Longitudinal resistivity of the ungated sample for various temperatures, from $80\mathrm{mK}$ (blue line) to $800\mathrm{mK}$ (red line). (b) Power spectrum of the low temperature magnetoresistance (as a function of $1/B$) shown in (a). (c) Zero-field SO splitting as a function of carrier density for the ungated (square) and gated (dots) sample.

Figure 2

Analysis using method A. (a) Power spectrum of the low temperature magnetoresistance at density $n=3.0\times {10}^{15}$ m${}^{-2}$ together with the filters used to extract the different components. The power spectrum has been normalized in order to compare it with the filters. No windowing is performed. (b) SdH oscillations obtained after inverse Fourier transforming the filtered spectrum; the oscillations have been vertically offset for clarity. (c) Effective masses deduced from the filtered oscillations as a function of magnetic field. $m_1$ and $m_2$ are the effective masses of the two SO-split subbands; $m_3$ and $m_4$ are fictitious effective masses describing the temperature dependencies of the $n_2+n_1$ and $2n_1$ peaks, respectively. (d) Quantum scattering times of the two SO-split subbands.

Figure 3

Analysis using method B. (a) Temperature dependence of the resistivity power spectrum in the ungated device. Dots and squares indicate the height of the $n_1$ and $n_2$ peaks, respectively. (b) Peak heights as a function of temperature together with a fit (line). (c) and (d) The same as in (a) and (b) for the gated sample. The density was $2.1\times {10}^{-15}{\mathrm{m}}^{-2}$ and the SO splitting $13\%$. The insets are zoom-ins of the $n_2$ peak.

Figure 4

Effective masses as a function of density. Comparison of the results for $m_1$, $m_2$, and $m_{1,2}$ obtained using method A, method B, and self-consistent calculations.