Thermoelectric and viscous contributions to the hydrodynamic ratchet effect

We study thermoelectric and viscous contributions to the ratchet effect, i.e. radiation-induced generation of the direct electric current, $J_{\mathrm{r}\mathrm{a}\mathrm{t}},$ in asymmetric dual-grating gate structure without inversion center. Previously [E.Mönch et al, Phys. Rev. B {\bf 105}, 045404 (2022)], it was demonstrated that frequency dependence of the $J_{\mathrm{r}\mathrm{a}\mathrm{t}}$ is essentially different within hydrodynamic (HD) and drift-diffusion (DD) regimes of the electron transport: $J_{\mathrm{r}\mathrm{a}\mathrm{t}}^{\mathrm{H}\mathrm{D}}\propto 1\mathrm{/}{\omega }^6$ and $J_{\mathrm{r}\mathrm{a}\mathrm{t}}^{\mathrm{D}\mathrm{D}}\propto 1\mathrm{/}{\omega }^2$ for $\omega \to \mathrm{\infty }\mathrm{.}$ Here we analyze previously neglected thermoelectric contribution and find that it yields high-frequency asymptotic $1\mathrm{/}{\omega }^2$ even in the HD regime and can change sign of the response. Account of the finite viscosity of the electron liquid yields contribution which scales at high frequency as $1\mathrm{/}{\omega }^4\mathrm{.}$ We also find plasmonic resonances in the $J_{\mathrm{r}\mathrm{a}\mathrm{t}},$ and demonstrate that asymmetry of the structure allows for excitation of the so-called directional travelling plasmons.