Characterization of Spin-Orbit Interactions of GaAs Heavy Holes Using a Quantum Point Contact

We present transport experiments performed in high-quality quantum point contacts embedded in a GaAs two-dimensional hole gas. The strong spin-orbit interaction results in peculiar transport phenomena, including the previously observed anisotropic Zeeman splitting and level-dependent effective $g$ factors. Here we find additional effects, namely, the crossing and the anticrossing of spin-split levels depending on subband index and magnetic field direction. Our experimental observations are reconciled in a heavy-hole effective spin-orbit Hamiltonian where cubic- and quadratic-in-momentum terms appear. The spin-orbit components, being of great importance for quantum computing applications, are characterized in terms of magnitude and spin structure. In light of our results, we explain the level-dependent effective $g$ factor in an in-plane field. Through a tilted magnetic field analysis, we show that the quantum point contact out-of-plane $g$ factor saturates around the predicted 7.2 bulk value.

Figure 1

(a) QPC linear conductance for $\mathbf{B}=\mathbf{0}$ as a function of gate voltage; a gate-dependent contact resistance is subtracted from the raw data. (b) Transconductance as a function of gate voltage and $B_{\perp }$. The arrows point to three examples of anticrossing (red) and crossing (black). (c) Transconductance as a function of gate voltage and $B_{\parallel }$, with $B_{\parallel }$ aligned with the QPC axis. (d) Transconductance as a function of gate voltage and $B_{\parallel }$, with $B_{\parallel }$ aligned perpendicular to the QPC axis.

Figure 2

(a) Crossing fields $B_n^{(i)}$ extracted from Fig. 1 (dots) together with a fit of Eq. (2) assuming harmonic confinement (solid lines) or hard wall confinement (dotted lines). (b) Calculated $k_x=0$, $B_{\parallel }=0$ eigenvalues of $H_0+H_{\text{SO}}^{(3)}$. The red arrows indicate anticrossing points between $n$ and $n+1$ states; the black arrows indicate crossing points between $n$ and $n+2$ states. We used $\hslash \omega =0.4\mathrm{meV}$, $g_{\perp }=7.2$, $m=0.3m_0$, and ${\hslash }^3{\alpha }_{2,3}=0.08{\gamma }_{2,3}\mathrm{eV}{\mathrm{nm}}^3$ (with ${\gamma }_{2,3}$ Luttinger parameters), close to the numbers used in Refs. [6, 19].

Figure 3

QPC transconductance as a function of gate voltage and magnetic field for different tilt angles $\theta$. (a) $\theta =86.7°$. (b) $\theta =78°$. (c) $\theta =70°$. (d) $\theta =60°$. The red and black arrows indicate two features that gradually turn from crossing in (a) to anticrossing in (d).

Figure 4

Calculated subband energies as in Fig. 2 for a finite tilt angle $\theta =86.7°$ (a) and $\theta =78°$ (b). We used an isotropic $H_{\text{SO}}^{(2)}$ with a coupling parameter $\gamma$ obtained from $B_0=33\mathrm{T}$.

Figure 5

(a) Measured $g_{\parallel }$ as a function of $n$. (b) Measured $g_{\perp }$ as a function of $n$. The predicted bulk value [5] of $g_{\perp }=7.2$ is indicated by the dashed line.