Screening in the two-dimensional electron gas with spin-orbit coupling

We study screening properties of the two-dimensional electron gas with Rashba spin-orbit coupling. Calculating the dielectric function within the random phase approximation, we describe the new features of screening induced by spin-orbit coupling, which are the extension of the region of particle-hole excitations and the spin-orbit-induced suppression of collective modes. The required polarization operator is calculated in an analytic form without any approximations. Carefully deriving its static limit, we prove the absence of a small-$q$ anomaly at zero frequency. On the basis of our results at finite frequencies we establish the new boundaries of the particle-hole continuum and calculate the SO-induced lifetime of collective modes such as plasmons and longitudinal optical phonons. According to our estimates, these effects can be resolved in inelastic Raman scattering. We evaluate the experimentally measurable dynamic structure factor and establish the range of parameters where the described phenomena are mostly pronounced.

Figure 1

The domains $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$ [see Eqs. (43, 44, 45, 46)] corresponding to the different orderings of the roots (37).

Figure 2

SO-induced extension of the particle-hole continuum shown for $y=0.1$. It is bounded from above by parabola $w_1\left(z\right)$ [Eq. (49)] and bounded from below by the parabola $w_4\left(z\right)$ [Eq. (52)] and by the parabola $w_0\left(z\right)=z^2+z$ of the conventional particle-hole (Landau damping) region. The small darkened triangle indicates the region where the approximation $\sigma (z,w)\approx \sigma (0,w)$ is applicable.

Figure 3

The real part of conductivity for the values of $z_c=0.0$, 0.025, 0.005, 0.01, 0.02, 0.03, 0.04 calculated at $y=0.1$. The unit step corresponds to $z_c=0$. The dashed line corresponds to $z_c=y^2=0.01$, which is the limiting value for the applicability of the approximation $\sigma (z,w)\approx \sigma (0,w)$.

Figure 4

The behavior of $-\mathrm{Re}\Pi (z,0)∕\nu$ near $z=1$ for $y=0.1$ (solid line) and $y=0$ (dotted line). The difference between the two curves is shown in the inset.

Figure 5

The behavior of $-\mathrm{Re}\Pi (z,0)∕\nu$ (solid line) and $-\mathrm{Re}{\Pi }^I(z,0)∕\nu$ (dashed line) near $z=y$ for $y=0.1$.

Figure 6

Contour plots of $S(z,w)$ showing the SO-induced wedge-shaped damping region (bounded by the dashed lines). The plasmon mode is depicted by the bold line. Insets show the cross sections $S(z=0.1,w)$ as a function of $w$. Upper panel: $y=0.07$, $r_s=0.2$. Lower panel: $y=0.04$, $r_s=0.6$.

Figure 7

The cross section $S(z=0.1,w)$ (solid line) compared to the approximation $S(z=0.1,w{)}_{SO\text{−}damppl}$ (dashed line). Parameters: $y=0.07$, $r_s=0.2$.

Figure 8

The plane $(y,r_s)$ is divided into the domains I and II, where the plasmon has one or two undamped pieces, respectively (cf. Fig. 6). Insets show SO-induced plasmon width $\overline{\gamma }\left(z\right)=\gamma \left(z\right)\times {10}^4$ for the same parameters as in Fig. 6.

Figure 9

The plasmon contribution to the energy-loss function at $y=0$ (dashed line) compared with the SO-damped plasmon contribution at $y=0.07$ (solid line). For the both curves $r_s=0.2$.

Figure 10

SO-induced damping of the LO phonon ${\overline{\gamma }}_{LO}\left(z\right)={\gamma }_{LO}\left(z\right)\times {10}^4$ for $w_0=0.07$, $r_s=0.4$, ${\epsilon }_{\infty }=12$, and ${\epsilon }_0=15$. Solid and dashed lines correspond to the cases $y=0.07$ $(\mid w_0-y\mid <y^2)$ and $y=0.06$ $(\mid w_0-y\mid >y^2)$, respectively. Insets show the location of the phonon spectrum (solid line) relative to the SO-induced damping region (dashed lines) and to the strip $y-y^2<w<y+y^2$ (dotted lines) for $y=0.07$ (left) and $y=0.06$ (right).