Anisotropic Two-Dimensional Disordered Wigner Solid

The interplay between the Fermi sea anisotropy, electron-electron interaction, and localization phenomena can give rise to exotic many-body phases. An exciting example is an anisotropic two-dimensional (2D) Wigner solid (WS), where electrons form an ordered array with an anisotropic lattice structure. Such a state has eluded experiments up to now as its realization is extremely demanding: First, a WS entails very low densities where the Coulomb interaction dominates over the kinetic (Fermi) energy. Attaining such low densities while keeping the disorder low is very challenging. Second, the low-density requirement has to be fulfilled in a material that hosts an anisotropic Fermi sea. Here, we report transport measurements in a clean (low-disorder) 2D electron system with anisotropic effective mass and Fermi sea. The data reveal that at extremely low electron densities, when the $r_s$ parameter, the ratio of the Coulomb to the Fermi energy, exceeds $\simeq 38$, the current-voltage characteristics become strongly nonlinear at small dc biases. Several key features of the nonlinear characteristics, including their anisotropic voltage thresholds, are consistent with the formation of a disordered, anisotropic WS pinned by the ubiquitous disorder potential.

Figure 1

Overview of our sample’s geometry and its transport data. (a) Top view of our sample, an AlAs quantum well, in which the 2D electrons can occupy one or two anisotropic conduction-band valleys. Projections of the anisotropic Fermi seas of the $X$ and $Y$ valleys are shown in blue and red; these valleys have their major axes along the [100] and [010] crystallographic directions, respectively. The sample has a $\simeq 1.5\times 1.5{\mathrm{mm}}^2$ van der Pauw geometry, and contacts to the sample are denoted by 1–8. For measuring $R_{[100]}$, we pass current from contact 8 to 2 and measure the voltage between contacts 6 and 4. For $R_{[010]}$, the current is passed from contact 8 to 6 and we measure the voltage between contacts 2 and 4. (b) Magnetoresistance traces taken along [100] as a function of the perpendicular magnetic field at different densities $n$, as indicated (in units of ${10}^{10}{\mathrm{cm}}^{-2}$), highlighting the extremely high quality of the 2DES at very low densities and large $r_s$. Traces are shifted vertically, and are plotted vs $1/\nu$. In all these traces, all the 2D electrons occupy the $Y$ valley. (c) Resistances along [100] and [010] directions at zero strain as a function of density. $R_{[100]}$ and $R_{[010]}$ suddenly split when a spontaneous valley transition occurs at $n_V\simeq 6.3$. Also, two clear “kinks” (marked with vertical lines) are seen at $n\simeq 3.5$ and 2.0 that closely correlate with the onset of the metal-insulator transition ($n_{\mathrm{MIT}}$) and the spontaneous spin transition, respectively. At even lower densities, below $n_{\mathrm{WS}}\simeq 1.80$, we start to see nonlinear $I-V$ characteristics, suggestive of a pinned WS; this is the main subject of our work presented here. (d),(e) Arrhenius plots of $R_{[100]}$ and $R_{[010]}$. Both $R_{[100]}$ and $R_{[010]}$ exhibit metallic transport for $n\gtrsim 3$, and an insulating behavior for smaller $n$. $R_{[100]}$ and $R_{[010]}$ exhibit approximately an activated behavior at low $T$ ($\lesssim 0.6\mathrm{K}$), $R\propto e^{E_A/2k_{B}T}$, for $n\lesssim 3$. (f) shows the deduced $E_A$ along [100] and [010] vs density.

Figure 2

(a) $dV/dI$ data at $T=0.30\mathrm{K}$ along the [100] (black traces) and [010] (red traces) crystallographic directions, plotted as a function of the dc voltage drop along the sample; data are shown for $n=1.10$, 1.20, 1.30, and 1.50 (in units of ${10}^{10}{\mathrm{cm}}^{-2}$). Along both directions $dV/dI$ traces show a reasonably abrupt drop above a threshold voltage ($V_{\mathrm{th}}$). The vertical arrows (placed symmetrically with respect to $V_{\mathrm{dc}}=0$) mark $V_{\mathrm{th}}$, and are based on the average value of $V_{\mathrm{th}}$ for $+V_{\mathrm{dc}}$ and $-V_{\mathrm{dc}}$. (b) Extracted $V_{\mathrm{th}}$ and (c) pinning gap (${\mathrm{\Delta }}_p$) plotted vs $n$, highlighting the increase in $V_{\mathrm{th}}$ and ${\mathrm{\Delta }}_p$ as $n$ is lowered.

Figure 3

Temperature dependence of $dV/dI$ vs $V_{\mathrm{dc}}$ at (a) $n=1.20$ and (b) $n=1.70$, measured along [100] and [010] directions. (c) Critical temperature ($T_c$), below which the $I-V$ characteristic becomes nonlinear, plotted vs electron density. The data suggest a drop in $T_c$ with increasing density, extrapolating to 0 K as the onset density for the pinned WS formation ($n\simeq 1.80$) is reached. The dashed curve is a guide to the eye. The inset displays schematically an anisotropic WS; the shaded regions represent the electron charge distribution with anisotropic Bohr radii, reflecting the anisotropic electron effective mass.