Microwave resonance and weak pinning in two-dimensional hole systems at high magnetic fields

Microwave frequency conductivity $\mathrm{Re}\left({\sigma }_{\mathrm{xx}}\right)$ of high quality two-dimensional hole systems (2DHS) in a large perpendicular magnetic field (B) is measured with the carrier density ${(n}_s)$ of the 2DHS controlled by a backgate bias. The high-$B$ insulating phase of the 2DHS exhibits a microwave resonance that remains well defined, but shifts to higher peak frequency ${(f}_{\mathrm{pk}})$ as $n_s$ is reduced. In different regimes, $f_{\mathrm{pk}}$ vs $n_s$ can be fit to $f_{\mathrm{pk}}\propto n_s^{-1/2}$ or to $f_{\mathrm{pk}}\propto n_s^{-3/2}.$ The data clearly indicate that both carrier-carrier interactions and disorder are indispensable in determining the dynamics of the insulator. The $n_s$ dependence of $f_{\mathrm{pk}}$ is consistent with a weakly pinned Wigner crystal in which domain size increases with $n_s,$ due to larger carrier-carrier interaction.