Dielectric function, screening, and plasmons of graphene in the presence of spin-orbit interactions

We study the dielectric properties of graphene in the presence of Rashba and intrinsic spin-orbit interactions in their most general form, i.e., for arbitrary frequency, wave vector, doping, and spin-orbit coupling (SOC) parameters. The main result consists in the derivation of closed analytical expressions for the imaginary as well as for the real part of the polarization function. Several limiting cases, e.g., the case of purely Rashba or purely intrinsic SOC, and the case of equally large Rashba and intrinsic coupling parameters are discussed. In the static limit the asymptotic behavior of the screened potential due to charged impurities is derived. In the opposite limit ($q=0$, $\omega \to 0$), an analytical expression for the plasmon dispersion is obtained and afterwards compared to the numerical result. Our result can also be applied to related systems such as bilayer graphene or topological insulators.

Figure 1

Energy dispersion in units of ${\lambda }_R$ for $E_{+\pm }$ (solid lines) and $E_{-\pm }$ (dashed lines): (a) ${\lambda }_I=2{\lambda }_R$, (b) ${\lambda }_I={\lambda }_R$, (c) ${\lambda }_I={\lambda }_R/2$.

Figure 2

Single-particle continuum (dark area) for the particular choice of ${\lambda }_R=2{\lambda }_I=0.3\mu$. Analytical expressions for the boundaries of the distinct regions I, II, and III can be found in Sec. 3b.

Figure 3

Static real part of the charge susceptibility for (a) undoped graphene with fixed ${\lambda }_{+}$ for ${\lambda }_R=2{\lambda }_I$ (dotted), ${\lambda }_R={\lambda }_I$ (dot-dashed), ${\lambda }_I=2{\lambda }_R$ (dashed), ${\lambda }_R={\lambda }_I=0$ (straight), and doped graphene with (b) ${\lambda }_R={\lambda }_I=0.3\mu$, (c) ${\lambda }_R=2{\lambda }_I=0.3\mu$, and (d) $2{\lambda }_R={\lambda }_I=0.3\mu$ in units of the density of states $D\left(0\right)=g_v\mu /\pi$.

Figure 4

Asymptotic screened potential (in units of $Q{\lambda }_{+}/{\epsilon }_0$) of undoped graphene for fixed ${\lambda }_{+}={\lambda }_R+{\lambda }_I$ and different spin-orbit coupling parameters. (a) ${\lambda }_R={\lambda }_I$ (straight line), ${\lambda }_I=2{\lambda }_R$ (dashed), ${\lambda }_R=0$ (dotted). Also shown is the noninteracting case ${\lambda }_R={\lambda }_I=0$ (dot-dashed). (b) ${\lambda }_I=0$ (straight), ${\lambda }_R=2{\lambda }_I$ (dashed).

Figure 5

Asymptotic screened potential (in units of $Q\mu /{\epsilon }_0$) of doped graphene for various spin-orbit coupling parameters: (a) $({\lambda }_R/\mu ,{\lambda }_I/\mu )=(0,0.3)$, (b) $(0.3,0.15)$, (c) $(0.15,0.3)$, and (d) $(0.3,0.3)$.

Figure 6

Density plot of $\mathrm{Im}\left\{-1/\varepsilon (q,\omega )\right\}$ for various spin-obit coupling parameters: (a) $({\lambda }_R/\mu ,{\lambda }_I/\mu )=(0,0)$, (b) $(0,0.5)$, (c) $(0.25,0.25)$, (d) $(0.5,0)$. Straight lines correspond to the numerically calculated zeros of the real part of the dielectric function while dashed lines represent the long-wavelength result of Eq. (19).

Figure 7

Energy loss function $\mathrm{Im}\left\{-1/\varepsilon (q,\omega +i0)\right\}$ for fixed $q=0.1\mu$ with (a) ${\lambda }_R=0$, ${\lambda }_I=0.5\mu$ and (b) ${\lambda }_R=0.5\mu$, ${\lambda }_I=0$.

Figure 8

Energy loss function $\mathrm{Im}\left\{-1/\varepsilon (q,\omega +i0)\right\}$ for (a) $({\lambda }_R/\mu ,{\lambda }_I/\mu )=(0.25,0)$, (b) $(0.25,0.25)$, (c) $(0.25,0.5)$, and (d) $(0.25,0.75)$. The straight blue lines show the undamped plasmon modes. The black lines indicate the boundaries of the single-particle continuum (see Sec. 3b).

Figure 9

Top: Energy loss function $\mathrm{Im}\{-1/\varepsilon (q,\omega +i0\left)\right\}$ for ${\lambda }_R={\lambda }_I=0.25\mu$ and various wave vectors $q$. Bottom: The same for ${\lambda }_R=0.25\mu$ and ${\lambda }_I=0$.