Retardation effects on plasma waves in graphene, topological insulators, and quantum wires

Retardation effects (REs) are known to cause a crossover from linear to sublinear behaviors in the dispersion relation of two-dimensional (2D) plasma waves at long wavelengths. In the present work, we systematically analyze REs on plasma waves in both 2D and 1D electron gases, and we clarify the experimental conditions for observing the crossover. We show that graphene is the only material that might gratify the conditions with current technology. Although it closely imitates graphene, the surface of 3D topological insulators is found to be unsuitable for this purpose due to strong electronic scattering by intrinsic disorder. We also show that the crossover, though theoretically possible, virtually never occurs in 1D plasma. In addition, the relevance of our results to terahertz plasmonics is discussed.

Figure 1

(a) A sheet of electron gas is placed in the $x-y$ plane, in which plasma waves are incited to propagate along the $x$ direction. (b) Sketch of a quantum wire with lateral dimension $\sim r_0$. Plasma waves are propagating along the wire ($z$ direction). Throughout the paper, we use $\vec{x}=(x,y,z)$ and reserve $\vec{r}=(x,y)$ for points in the $x-y$ plane. Colors schematically encode the charge density.

Figure 2

2D plasma dispersion relation ${\omega }_p^{2\text{D}}\left(q\right)$. The red solid curve is given by Eq. (16), which reduces to the green solid curve for $q>q^{*}$ but becomes degenerate with the light line (dashed curve) for $q<q^{*}$. The crossover is controlled by the single parameter $q^{*}$, which varies significantly from one system to another. In units of ${\omega }^{*}=c{q}^{*}$ and $q^{*}, {\omega }_p^{2\text{D}}$ assumes a form that is independent of the materials.

Figure 3

Electron-energy-loss spectra (EELS) evaluated by $\mathrm{Im}-({\epsilon }^{-1}(\omega ,q))$, with $\epsilon (\omega ,q)$ given by the left-hand side of Eq. (15). EELS is useful in unveiling the effects of electronic scattering (scattering rate = ${\tau }^{-1}$) on plasma dispersion ${\omega }_p^{2\text{D}}\left(q\right)$ and lifetime, which is the inverse of the broadening factor ${\gamma }_p^{2\text{D}}\left(q\right)$. The profile of EELS depends critically on the value of ${\omega }^{*}\tau$. As shown in panel (a), if ${\omega }^{*}\tau >1$, the plasma disperses according to ${\omega }_p^{2\text{D}}$, with ${\gamma }_p^{2\text{D}}$ increasing as $q$ increases. However, as displayed in (b), if ${\omega }^{*}\tau <1$, a nondispersive central peak is seen for $q$ around $q^{*}$. In (c), we plot the frequency scan for three values of $q$. For $q\sim q^{*}$ (curves with $q=0.5q^{*}$ and $q=1.5q^{*}$), central peaks are revealed, as in (b). For very large $q$ (e.g., $q=200q^{*}$), very weak dispersion remains. For the sake of comparison, the line labeled $q/q^{*}=200$ has been multiplied by 20, and $\omega$ has been rescaled by a factor 10. To observe the crossover phenomenon, it is necessary that ${\omega }^{*}\tau >1$.

Figure 4

1D plasma dispersion relation, ${\omega }_p^{1\text{D}}\left(q\right)$. In contrast to ${\omega }_p^{2\text{D}}, {\omega }_p^{1\text{D}}$ depends sensitively on the parameter $\lambda ={(c/c^{*})}^2$, where $c^{*}$ depends on the materials. The red solid curve is given by Eq. (A7). It goes on top of the green solid curve [upper line of Eq. (A11)] for $q>q^{*}$ but collapses to the solid black line [lower line of Eq. (A11)] for $q<q^{*}$. Here $q^{*}\sim exp\left(-\sqrt{\lambda }\right)$ relies completely on $\lambda$, which makes quantum quasi-1D electron gases essentially immune from retardation effects (REs).