Origin of Pinning Disorder in Magnetic-Field-Induced Wigner Solids

At low Landau level filling factors ($\nu$), Wigner solid phases of two-dimensional electron systems in GaAs are pinned by disorder and exhibit a pinning mode, whose frequency is a measure of the disorder that pins the Wigner solid. Despite numerous studies spanning the past three decades, the origin of the disorder that causes the pinning and determines the pinning mode frequency remains unknown. Here, we present a study of the pinning mode resonance in the low-$\nu$ Wigner solid phases of a series of ultralow-disorder GaAs quantum wells which are similar except for their varying well widths $d$. The pinning mode frequencies $f_p$ decrease strongly as $d$ increases, with the widest well exhibiting $f_p$ as low as $\simeq 35\mathrm{MHz}$. The amount of reduction of $f_p$ with increasing $d$ can be explained remarkably well by tails of the wave function impinging into the alloy-disordered ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ barriers that contain the electrons. However, it is imperative that the model for the confinement and wave function includes the Coulomb repulsion in the growth direction between the electrons as they occupy the quantum well.

Figure 1

Real diagonal conductivity $\mathrm{Re}({\sigma }_{\mathrm{xx}})$ vs magnetic field $B$ for the 30-nm QW. Inset: schematic of measurement setup, with the metal film of coplanar waveguide shown as black.

Figure 2

Color-scale plots of real diagonal conductivity $\mathrm{Re}({\sigma }_{\mathrm{xx}})$, on frequency-filling factor ($f\text{−}\nu$) axes, for the four QWs, marked with well widths $d$. The graphs are shown with larger $\nu$ at the bottom, so magnetic field $B$ increases going upward. The color scale is in percentages of full scales, which are 50, 210, 120, and $100\mathrm{\mu }\mathrm{S}$, respectively, for (a)–(d). For the 70-nm QW of (d) only, $\mathrm{Re}({\sigma }_{\mathrm{xx}})$ is multiplied by a factor of 5 for $\nu >0.20$, to enhance visibility in that range, which has small $\mathrm{Re}({\sigma }_{\mathrm{xx}})$.

Figure 3

Density-corrected resonance frequency $\alpha f_p$ (see the text) vs QW width $d$. Measured $\alpha f_p$ for the four QWs at $\nu =0.15$ are shown as black dots. For $d=30$, 40, 50, and 70 nm, respectively, $\alpha =1.0$, 1.0, 0.90, and 1.07. Curves for the models discussed in the text are as shown in the legend. The one-particle infinite QW (interface roughness) and finite QW (roughness and alloy) curves are multiplied by scale factors to intercept $(d,\alpha f_p)=(45.3\mathrm{nm},0.125\mathrm{GHz})$, the centroid of the measured $\alpha f_p$ data points on the log-log plot. The curves marked “alloy” are the calculated $P_B^2$, and all “roughness” curves are the calculated $(\partial E_s/\partial d{)}^2$; see the text for details. The $P_B^2$ curves generated for the Hartree and LDA self-consistent calculations are scaled to least-squares fit the $\alpha f_p$ data. The fits are done on the logged $\alpha f_p$ data.

Figure 4

(a) Normalized charge density ${\rho }_c$ (red lines) and self-consistent potential $V$ (black lines) for QWs of widths $d=30$, 50, and 70 nm. These are from Poisson-Schrödinger, self-consistent (Hartree) calculations for $B=12.4\mathrm{T}$. (b) Hartree (solid line) and LDA (dashed line) calculated charge distribution tails within the barriers, for the four QWs. The edge of the barrier is the origin of the horizontal axis.