Cubic Rashba Spin-Orbit Interaction of a Two-Dimensional Hole Gas in a Strained-$\mathrm{Ge}/\mathrm{SiGe}$ Quantum Well

The spin-orbit interaction (SOI) of a two-dimensional hole gas in the inversion symmetric semiconductor Ge is studied in a strained-$\mathrm{Ge}/\mathrm{SiGe}$ quantum well structure. We observe weak antilocalization (WAL) in the magnetoconductivity measurement, revealing that the WAL feature can be fully described by the $k$-cubic Rashba SOI theory. Furthermore, we demonstrate electric field control of the Rashba SOI. Our findings reveal that the heavy hole (HH) in strained Ge is a purely cubic Rashba system, which is consistent with the spin angular momentum $m_j=\pm 3/2$ nature of the HH wave function.

Figure 1

(a),(b) The effective magnetic field direction with respect to the in-plane wave vector $k$ for (a) the linear-Rashba term, ${\mathbf{\Omega }}_1$ and (b) the cubic-Rashba term, ${\mathbf{\Omega }}_3$. (c) Schematic illustration of the valence band structure for strained Ge. The HH, LH, and SO bands are indicated.

Figure 2

(a),(b) The $V_G$ dependence of (a) sheet resistance $R_{\text{sheet}}$ and hole concentration $n_{\text{hole}}$ and (b) the hole effective mass $m_{\text{hole}}$. (c) The change of magnetoconductivity $\mathrm{\Delta }\sigma (B)$ measured at 1.6 K at various $V_G$.

Figure 3

The $V_G$ dependence of the change of magnetoconductivity $\mathrm{\Delta }\sigma (B)$ shown with the fitting results from the ILP model. Black open circles indicate experimental data, and the red solid line and blue dashed lines show calculations using the ILP model for the cubic- and linear-Rashba terms, respectively.

Figure 4

(a) The $V_G$ dependence of the characteristic magnetic fields for the $k$-cubic spin-orbit coupling $B_{\mathrm{SO}3}$ and the phase coherence $B_{\varphi }$. (b) Hole concentration dependence of the spin relaxation time ${\tau }_{\mathrm{SO}}$ and momentum scattering time ${\tau }_{\text{tr}}$. (c) Spin-splitting energy $\mathrm{\Delta }$ versus the hole concentration. (d) The change of Rashba spin-orbit coefficient ${\alpha }_3{E}_z$ with respect to the hole concentration.