Plasmon mass and Drude weight in strongly spin-orbit-coupled two-dimensional electron gases

Spin-orbit-coupled two-dimensional electron gases (2DEGs) are a textbook example of helical Fermi liquids, i.e., quantum liquids in which spin (or pseudospin) and momentum degrees of freedom at the Fermi surface have a well-defined correlation. Here we study the long-wavelength plasmon dispersion and the Drude weight of archetypical spin-orbit-coupled 2DEGs. We first show that these measurable quantities are sensitive to electron-electron interactions due to broken Galilean invariance and then discuss in detail why the popular random phase approximation is not capable of describing the collective dynamics of these systems even at very long wavelengths. This work is focused on presenting approximate microscopic calculations of these quantities based on the minimal theoretical scheme that captures the basic physics correctly, i.e., the time-dependent Hartree-Fock approximation. We find that interactions enhance the “plasmon mass” and suppress the Drude weight. Our findings can be tested by inelastic light scattering, electron energy loss, and far-infrared optical-absorption measurements.

Figure 1

The quantity $\mathcal{A}$ in Eq. (100) as a function of $r_s$ for two values of the Rashba SOC strength $\alpha$. Notice that $\mathcal{A}>0$ and thus ${\chi }_{\sigma {}^y\sigma {}^y}/\chi {}_{\sigma {}^y\sigma {}^y}^{(0)}>1$.

Figure 2

The renormalized Hartree-Fock energy bands $E_{\mathrm{HF},\pm }(k)$ (in units of ${\varepsilon }_{\mathrm{F},0}$) as functions of $k$ (in units of $k_{\mathrm{F}}$) for $r_s=1$ and $\alpha =5\times {10}^{-11}\hspace{0.5em}\mathrm{eV}\hspace{0.5em}\mathrm{m}$ for the case of unscreened ($\xi =0$) Coulomb interactions. The dashed (blue) line denotes the minority band [$E_{\mathrm{HF},+}(k)$] while the solid (red) line denotes the majority band [$E_{\mathrm{HF},-}(k)$]. The thin black lines are the energy bands for the noninteracting case. The vertical lines denote the location of $\pm k_{\mathrm{F},\pm }^{(0)}$.

Figure 3

The Hartree-Fock self-energies ${\Sigma }_0(k)$ (solid line) and ${\Sigma }_1(k)$ (dashed line) (in units of ${\varepsilon }_{\mathrm{F},0}$) as functions of momentum $k$ (in units of $k_{\mathrm{F}}$) for $r_s=1$ and $\alpha =5\times {10}^{-11}\hspace{0.5em}\mathrm{eV}\hspace{0.5em}\mathrm{m}$. The vertical lines denote the location of $\pm k_{\mathrm{F},\pm }^{(0)}$.

Figure 4

The minority (a) and majority (b) quasiparticle effective masses $m_{\pm }^{\star }$ (in units of the bare mass $m_{\mathrm{b}}$) as functions of $r_{\mathrm{s}}$ for different values of $\alpha$. These results have been obtained by using the definition (77) and fully screened Thomas-Fermi interactions ($\xi =1$). The solid line represents the classic result by Janak (Ref. 54): our results for $\alpha =0$ (filled circles) are in excellent agreement with the analytical expression (16) in Ref. 54. The inset to (b) illustrates $\Delta v_{\pm }^{\star }\equiv v_{\pm }^{\star }-{v_{\pm }^{\star }|}_{\alpha =0}$ (in units of $v_{\mathrm{F}}\equiv k_{\mathrm{F}}/m_{\mathrm{b}}$) as functions of $\mathrm{\alpha ̄}$ and for $r_s=0.25$ (dashed and dotted lines). The solid line is the weak-SOC analytical result (74) in Ref. 25. Notice that our numerical results extend up to a large value of the SOC constant since $\mathrm{\alpha ̄}=0.7$ corresponds to $\alpha ~38\times {10}^{-11}\hspace{0.5em}\mathrm{eV}\hspace{0.5em}\mathrm{m}$.

Figure 5

The vertex correction $\delta B_{\mathrm{T},1}(k)$ [in units of $B_{\mathrm{ext}}=\mathit{eA}\alpha /(\hslash c)$] as a function of $k$ (in units of $k_{\mathrm{F}}$) for $r_{\mathrm{s}}=1$ and different values of $\alpha$. (a) Results for unscreened Coulomb interactions ($\xi =0$). (b) Results for fully screened Thomas-Fermi interactions ($\xi =1$). Note that when expressing $\delta B_{\mathrm{T},1}(k)$ in units of $B_{\mathrm{ext}}$ a factor $\alpha$ is extracted. Also note that for $\alpha =5\times {10}^{-11}\hspace{0.5em}\mathrm{eV}\hspace{0.5em}\mathrm{m}$, $k_{\mathrm{F},+}^{(0)}/k_{\mathrm{F}}\simeq 0.6$ and $k_{\mathrm{F},-}^{(0)}/k_{\mathrm{F}}\simeq 1.3$. For $\alpha ={10}^{-11}\hspace{0.5em}\mathrm{eV}\hspace{0.5em}\mathrm{m}$, $k_{\mathrm{F},+}^{(0)}/k_{\mathrm{F}}\simeq 0.9$ and $k_{\mathrm{F},-}^{(0)}/k_{\mathrm{F}}=1.1$.

Figure 6

The in-plane spin susceptibility ${\chi }_{\sigma {}^y\sigma {}^y}$ (in units of the noninteracting value $\chi {}_{\sigma {}^y\sigma {}^y}^{(0)}={\nu }_0/2$) as a function of $r_s$ and for different values of $\alpha$. (a) Results for unscreened Coulomb interactions ($\xi =0$). (b) Results for fully screened Thomas-Fermi interactions ($\xi =1$).

Figure 7

The Drude weight $\mathcal{D}$ [in units of the noninteracting value ${\mathcal{D}}_0=\pi e^2(n/m_{\mathrm{b}}-\alpha {}^2{\nu }_0/2)$], calculated from Eqs. (48) and (95), as a function of $r_s$ and for different values of $\alpha$. (a) Results for unscreened Coulomb interactions ($\xi =0$). (b) Results for fully screened Thomas-Fermi interactions ($\xi =1$).

Figure 8

Renormalized SOC coupling strengths ${\mathrm{\alpha ̃}}_{\pm }$ (in units of the bare value $\alpha$) for the minority ($+$) and majority ($-$) bands as functions of $r_s$ and for different values of $\alpha$. These results refer to unscreened Coulomb interactions ($\xi =0$).

Figure 9

As Fig. 8 but for $\xi =1$. The thin solid (black) line in both panels labels the result of Ref. 21 [see Eq. (119)], which is asymptotically exact in the weak-SOC limit.

Figure 10

The renormalized SOC constants ${\mathrm{\alpha ̃}}_{\pm }$ (in units of the noninteracting value $\alpha$) as functions of the logarithm of density (expressed in units of ${\mathrm{cm}}^{-2}$). In this figure we have taken into account the density dependence of the bare SOC coupling $\alpha$ in the simple approximation given in Eq. (125). (a) Results for unscreened Coulomb interactions ($\xi =0$). (b) Results for fully screened Thomas-Fermi interactions ($\xi =1$).

Figure 11

The spin susceptibility enhancement ${\chi }_{\sigma {}^y\sigma {}^y}/\chi {}_{\sigma {}^y\sigma {}^y}^{(0)}$ as a function of the logarithm of density (expressed in units of ${\mathrm{cm}}^{-2}$). In this figure we have taken into account the density dependence of the bare SOC coupling $\alpha$ in the simple approximation given in Eq. (125).