Strong spin-orbit interactions and weak antilocalization in carbon-doped $p$-type $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructures

We present a comprehensive study of the low-field magnetoresistance in carbon-doped $p$-type $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ heterostructures aiming at the investigation of spin-orbit interaction effects. The following signatures of exceptionally strong spin-orbit interactions are simultaneously observed: a beating in the Shubnikov–de Haas oscillations, a classical positive magnetoresistance due to the presence of the two spin-split subbands, and a weak antilocalization dip in the magnetoresistance. The spin-orbit-induced splitting of the heavy hole subband at the Fermi level is determined to be around 30% of the total Fermi energy. The phase-coherence length of holes of around $2.5\mu \mathrm{m}$ at a temperature of $70\mathrm{mK}$, extracted from weak antilocalization measurements, is promising for the fabrication of phase-coherent $p$-type nanodevices.

Figure 1

(a) Shubnikov–de Haas oscillations in the magnetoresistance, with a top gate set to $V_{\mathrm{TG}}=0\mathrm{V}$ and the total density $3.0\times {10}^{11}{\mathrm{cm}}^{-2}$. Inset: Fourier transform of the shown magnetoresistance, taken in the $\mathrm{B}$-field range (0.4, $1.5\mathrm{T}$) displaying three peaks. (b) Spin-orbit splitting energy of the heavy hole subband at the Fermi level as a function of the Fermi energy in the system and the total density.

Figure 2

Left column: Fit of the low-field magnetoresistivity with the two-band theory (Ref. 25) (black lines represent the measurement and thicker gray lines are fitted lines) in the following gate configurations: (a1) $V_{\mathrm{TG}}=0$, $N=3.0\times {10}^{11}{\mathrm{cm}}^{-2}$; (b1) $V_{\mathrm{TG}}=1\mathrm{V}$, $N=2.3\times {10}^{11}{\mathrm{cm}}^{-2}$. Values for $K_1$, $K_2$, and $K_{12}$ are given in units 1/s. Right column: Nonlinearity in the low-field Hall resistivity (black lines are measured data and gray lines are calculated curves; see text for detailed explanations) in the following gate configurations: (a2) $V_{\mathrm{TG}}=0$, $N=3.0\times {10}^{11}{\mathrm{cm}}^{-2}$; (b2) $V_{\mathrm{TG}}=1\mathrm{V}$, $N=2.3\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 3

Left column: Raw magnetoresistivity data (symmetrized) at $T=65\mathrm{mK}$ in the following gate configurations: (a1) $V_{\mathrm{TG}}=0$, $N=3\times {10}^{11}{\mathrm{cm}}^{-2}$, $\mu =160000{\mathrm{cm}}^{-2}∕\mathrm{V}\mathrm{s}$; (b1) $V_{\mathrm{TG}}=1\mathrm{V}$, $N=2.3\times {10}^{11}\mathrm{cm}$, $\mu =130000{\mathrm{cm}}^{-2}∕\mathrm{V}\mathrm{s}$. Right column: Quantum correction of the resistivity obtained after subtraction of the classical two-band positive magnetoresistivity for (a2) $V_{\mathrm{TG}}=0$ and (b2) $V_{\mathrm{TG}}=1\mathrm{V}$.

Figure 4

Fit of the antilocalization conductance peak with the HLN theory [Eq. (4)]—full lines are fitted curves and points are experimental data for the top-gate configurations $V_{\mathrm{TG}}=1\mathrm{V}$, $k_F{l}_m=120$ (gray) and $V_{\mathrm{TG}}=0$, $k_F{l}_m=200$ (black).

Figure 5

(a) Temperature dependence of the antilocalization resistivity minimum in the top-gate configuration $V_{\mathrm{TG}}=1\mathrm{V}$, $k_F{l}_m=120$. (b) Temperature dependence of the inverse phase-coherence time of holes. Inset: Fit of the antilocalization conductance peak with the HLN theory [Eq. (4)] for temperatures of $70\mathrm{mK}$ (black) and $190\mathrm{mK}$ (gray)—full lines are fitted curves and points are experimental data.

Figure 6

(a) Fits of the weak antilocalization conductance peak with the HLN theory including SOI [Eq. (5)] in the range $\mid B\mid <0.5\mathrm{mT}$ (dark gray line) and in the range $\mid B\mid <5\mathrm{mT}$ (light gray line). (b) Sensitivity of the fits in the range $\mid B\mid <5\mathrm{mT}$ to the change of ${\tau }_{\mathrm{SO}}$ [the full gray line represents the fit obtained in (a), while dashed lines correspond to different values of ${\tau }_{\mathrm{SO}}$, as quoted in the figure].