Beating of Friedel oscillations induced by spin-orbit interaction

By exploiting our recently derived exact formula for the Lindhard polarization function in the presence of Bychkov-Rashba (BR) and Dresselhaus (D) spin-orbit interaction (SOI), we show that the interplay of different SOI mechanisms induces highly anisotropic modifications of the static dielectric function. We find that under certain circumstances the polarization function exhibits doubly singular behavior. It leads to an intriguing phenomenon, beating of Friedel oscillations, which can be controlled by external fields. This effect is a general feature of systems with $\text{BR}+\text{D}$ SOI and should be observed in structures with a sufficiently strong SOI.

Figure 1

(Color online) (a) The Fermi contour in the presence of $\text{BR}+\text{D}$ SOI. Arrows indicate directions of spin. The thick solid and dashed lines are diameters along the $\left[1\overline{1}0\right]$ direction, connecting maximally distant points on the different chirality subbands and on the outer subband, respectively. The thick dashed-dotted line is the intersubband diameter in [110] direction. The BR and D strengths are related as $\alpha \approx 1.21\beta$ with $\rho =0.2k_F$ and the axes are in units of $k_F$. (b) The lengths of respective diameters of the Fermi contour in units of $2k_F$ as a function of the parameter $\theta$ for $\rho =0.2k_F$.

Figure 2

(Color online) The static polarization function ${\Pi }^0\left(\vec{q}\right)$ as a function of the momentum $q$ for its two orientations: (a) ${\varphi }_{\mathbf{q}}=3\pi /4$ and (b) $\pi /4$. The different curves correspond to different values of $\theta$, shown on the graph legends for $r=0.1$.

Figure 3

(a) The doubly singular polarization function for $\alpha =1.21\beta$ with $\rho =0.2k_F$ (solid line). Inset shows the height of cusp vs $\theta$. The dotted and dashed curves represent ${\Pi }^0\left(\vec{q}\right)$ for $\alpha =\beta$ with $\rho =0.2k_F$ and $\rho =0$, respectively. (b) Beating of Friedel oscillations, induced by the doubly singular behavior of ${\Pi }^0\left(\vec{q}\right)$ in (a). The electron density $n=2\times {10}^{16}{\text{m}}^{-2}$.