Correlation Lengths of the Wigner-Crystal Order in a Two-Dimensional Electron System at High Magnetic Fields

The insulator terminating the fractional quantum Hall series at low Landau level filling $\nu$ is generally taken to be a pinned Wigner crystal (WC), and exhibits a microwave resonance that is interpreted as a WC pinning mode. For a high quality sample at several densities, $n$, we find maxima in resonance peak frequency, $f_{pk}$, vs magnetic field, $B$. $L$, the correlation length of WC order, is calculated from $f_{pk}$. For each $n$, $L$ vs $\nu$ tends at low $\nu$ toward a line with positive intercept; the fit is accurate over as much as a factor of 5 range of $\nu$. The linear behavior is interpreted as due to $B$ compressing the electron wave functions, to alter the effective electron-impurity interaction.

Figure 1

Real part of diagonal conductivity $\mathrm{R}\mathrm{e}({\sigma }_{xx})$ vs magnetic field, $B$, for the sample at three different carrier densities $n$, filling factors $\nu$ marked by arrows. (a) $n=1.8\times {10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$. Inset shows top view of transmission line, where black indicates metal film. (b) $n=4.0\times {10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$ and $n=5.3\times {10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$.

Figure 2

Resonance spectra, real part of diagonal conductivity, Re$({\sigma }_{xx})$, vs frequency $f$. (a) Density $n=1.8\times {10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$, various magnetic fields $B$. Successive curves offset in steps of $2.5\mu \mathrm{S}$, for clarity; the 4 T curve is not offset. (b) Density $n=5.3\times {10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$, various $B$. Successive curves offset in steps of $2.0\mu \mathrm{S}$, for clarity. (c) Various $n$, for fixed $B=18\mathrm{T}$.

Figure 3

(a) Peak frequency $f_{pk}$ vs Landau filling $\nu$ for different carrier densities, $n$. (b) Full width at half-maximum linewidth, $\Delta f$, vs $\nu$. Symbols are the same as in (a).

Figure 4

Calculated domain sizes $L$ vs Landau filling $\nu$ for various carrier densities $n$. (a) Successive curves are offset from each other by $0.05\mu \mathrm{m}$. Curves are labeled with $n$ (${10}^{10}{\mathrm{c}\mathrm{m}}^{-2}$). Heavy lines are linear least squares fits to $L=\beta \nu +L_{\infty }$, extending over ranges of $\nu$ that were fit. (b) Same data as in (a), but not offset. Symbols for the $n$’s are the same as in (a). Inset shows fit parameters, $\beta$ and $L_{\infty }$, vs $n$.