Pinning Modes and Interlayer Correlation in High-Magnetic-Field Bilayer Wigner Solids

We report studies of pinning mode resonances in the low total Landau filling ($\nu$) Wigner solid of a series of bilayer hole samples with negligible interlayer tunneling and with varying interlayer separation $d$. Comparison of states with equal layer densities $(p,p)$ to single layer states $(p,0)$ produced in situ by biasing, indicates that there is interlayer quantum correlation in the solid at small $d$. Also, the resonance frequency at small $d$ is decreased just near $\nu =1/2$ and $2/3$, indicating the importance in the solid of correlations related to those in the fractional quantum Hall effects.

Figure 1

(a) Magnified cross section, not to scale, through the transmission line and sample. (b) Real part of diagonal conductivity $\mathrm{Re}({\sigma }_{xx})$ vs magnetic field $B$ for M465, and M433, in their balanced states. Total Landau filling factors $\nu$ are marked. (c) $\mathrm{Re}({\sigma }_{xx})$ vs frequency $f$ for M465 in its balanced state, for many $B$ (or $\nu$). Spectra are vertically displaced proportional to $\nu$ and shaded downward to their zeros. (d) Peak frequency $f_{pp}$ and $Q$ vs $\nu$ ($Q$ is $f_{pp}$ divided by full width at half maximum).

Figure 2

(a)–(d) For four samples, the spectrum in the balanced state with layer densities $(p,p)$ (solid line) is plotted along with the spectrum [with the measured $\mathrm{Re}({\sigma }_{xx})$ multiplied by two] of the single layer state $(p,0)$ (dotted line), all at magnetic field $B=10\mathrm{T}$. (e) $\eta =f_{pp}/f_{p0}$ at 10 T vs $\tilde{d}=d(4\pi p{)}^{1/2}$, where $f_{pp}$, $f_{p0}$ are the peak frequencies for the $(p,p)$ and $(p,0)$ states, respectively, and $d$ is layer separation. Inset: $\eta$ vs $\nu$, the total filling in the bilayer $(p,p)$ state, for five samples.