Universal self-amplification channel for surface plasma waves

We present a theory of surface plasma waves (SPWs) in metals with arbitrary electronic collision rate ${\tau }^{-1}$. We show that there exists a universal intrinsic amplification channel for these waves, subsequent to the interplay between ballistic motions and the metal surface. We evaluate the corresponding amplification rate ${\gamma }_0$, which turns out to be a sizable fraction of the SPW frequency ${\omega }_s$. We also find that the value of ${\omega }_s$ depends on surface scattering properties, in contrast with the conventional theory.

Figure 1

Numerically calculated SPW frequency ${\omega }_s$ and its intrinsic amplification rate ${\gamma }_0$ as a function of the Fuchs parameter $p$, by Eq. (18). $k/k_s=0.1$ and $q_c/k_s=1.5$. The approximate analytical solutions are also shown.

Figure 2

Possible instability caused by the self-amplification channel upon cooling down the system. At the critical temperature $T^{*},{\gamma }_0=1/\tau$. Here the sketch assumes that $1/\tau$ is dominated by phonon and impurity scattering.

Figure 3

Sketch of SPWs supported on the surface of a semi-infinite metal. $k/k_s=0.1$ and $p=1$. (a) Charge density map (color) and electric field map (arrows). (b) Plots of $\rho \left(z\right)$ and $E_{\mu }\left(z\right)$, where $\mu =x,z$. $\rho \left(z\right)$ is calculated by Eq. (16) while $E_{\mu }\left(z\right)$ by (2) and (3). Both $\rho \left(z\right)$ and $E_{\mu }\left(z\right)$ have been normalized.

Figure 4

Snapshots of (a) diffusive current density ${\mathbf{J}}_D(\mathbf{x},t)$ and (b) surface-ballistic current density ${\mathbf{J}}_B(\mathbf{x},t)$ in SPWs. These two current densities are not discriminated by the value of $\tau$, but by their dependence on the surface. ${\mathbf{J}}_B$ signifies genuine surface effects and would totally disappear without the surface, while ${\mathbf{J}}_D$ is a bulk property. $k/k_s=0.1$ and $p=1$. $\mathrm{Im}\left(\overline{\omega }\right)$ has been neglected to emphasize the differences.