Why experiments fail to detect the finite linear Rashba spin-orbit coupling of two-dimensional holes in semiconductor quantum wells: The case of Ge/SiGe quantum wells

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The magnetotransport experiments based on the weak antilocalization (WAL) effect have confirmed the common belief that the Rashba spin-orbit coupling (SOC) of two-dimensional (2D) holes in semiconductor quantum wells (QWs) is a $k$-cubic term as the lowest order with negligible linear terms. However, an emerging finite linear Rashba SOC was recently found in 2D holes by semiempirical pseudopotential method (SEPM) due to the direct dipolar coupling of an external electric field to the valence subbands in the presence of heavy-hole--light-hole (HH-LH) mixing. Here, we resolve this discrepancy by illustrating that the hole densities in the experiments are so high that the emerging linear term becomes undetectable since its strength declines substantially as increasing the wavevector $k$. Taking the example of a strained $\mathrm{Ge}/{\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ QW utilized in the experiment [Phys. Rev. Lett. 113, 086601 (2014)], we demonstrate that the hole density must be reduced by a factor of 5 to below $2.1\ifmmode\times\else\texttimes\fi{}{10}^{11}{\text{cm}}^{\ensuremath{-}2}$ in order to probe the $k$-linear term. We also evaluate the possibility to achieve WAL at low hole densities in order to measure SOC. These findings shed new light on the experimental measurement of Rashba SOC.