# Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions
## 6 Transport Evidence for a Sliding Quantum Electron Solid
Two threshold voltages are observed at electron densities below \(\approx 6\times 10^{10}\) cm\({}^{-2}\): with increasing applied voltage, the current stays near zero up to a first voltage threshold \(V_{\text{th1}}\), then increases sharply until a second threshold voltage \(V_{\text{th2}}\) is reached, above which the slope of the \(V-I\) curve is significantly reduced and the \(V-I\) curve is linear although not ohmic (see also the top inset to Figure 6). As the electron density is increased, the value of \(V_{\text{th1}}\) decreases, while the second threshold becomes less pronounced and eventually disappears. The authors observed no hysteresis in the range of electron densities studied. They pointed out that the observed behavior is quite distinct from the behavior reported in the insulating state in amorphous InO films, where the current was found to jump at the threshold voltage by as much as several orders of magnitude and the \(V-I\) curves exhibited hysteresis consistent with bistability and electron overheating [69; 70]. It is also worth noting that the occurrence of the double threshold cannot be explained within the percolation picture, in which case, a single threshold is expected [6]. Thus, the double threshold behavior cannot be described within existing traditional models. Two remarks are due here. First, at electron density \(n_{\text{s}}\approx 6\times 10^{10}\) cm\({}^{-2}\), below which the double-threshold behavior is evident, the interaction parameter is equal to \(\approx\)22. This is below the critical value \(r_{\text{s}}=37\pm 5\) at which the quantum Wigner crystallization in a disorder-free 2D system is expected to happen in zero magnetic field, according to the numerical simulations [64]. However, in  
Figure 6: \(V-I\) curves are shown for different electron densities in the insulating state of a silicon MOSFET at a temperature of 60 mK. The dashed lines are fits to the data using Equation (6). The top inset shows the \(V-I\) curve for \(n_{\text{s}}=5.20\times 10^{10}\) cm\({}^{-2}\) on an expanded scale; also shown are the threshold voltages \(V_{\text{th1}}\) and \(V_{\text{th2}}\), the static threshold \(V_{\text{s}}=V_{\text{th2}}\), and the dynamic threshold \(V_{\text{d}}\) that is obtained by the extrapolation of the linear region of the \(V-I\) curve to zero current. Bottom inset: activation energy \(U_{\text{c}}\) vs. electron density. Vertical error bars represent standard deviations in the determination of \(U_{\text{c}}\) from the fits to the data using Equation (6). The dashed line is a linear fit. From [67]. subsequent simulations, it was shown that the crystallization should occur at lower values of \(r_{\rm s}\) when disorder is present [71]. Second, the disorder-independent divergence of the effective mass and the disorder-dependent formation of the quantum electron solid immediately imply the existence of an intermediate phase preceding the electron solidification. Indeed, the existence of such a phase was predicted in [46; 65; 72]. Figure 7a shows the \(V-I\) characteristics for \(n_{\rm s}=5.36\times 10^{10}\) cm\({}^{-2}\) at different temperatures. As the temperature, \(T\), is increased, the second threshold \(V_{\rm th2}\) becomes less pronounced, and the threshold behavior of the \(V-I\) curves eventually smears out due to the shrinkage of the zero-current interval. The measured broad-band noise is shown as a function of voltage in Figure 7b for different temperatures at electron density \(n_{\rm s}=5.36\times 10^{10}\) cm\({}^{-2}\). A large increase in the noise is observed between the thresholds \(V_{\rm th1}\) and \(V_{\rm th2}\) at the lowest temperature. This large noise decreases rapidly with increasing temperature in agreement with the two-threshold behavior of the \(V-I\) curves of Figure 7a. The frequency dependence of the noise amplitude is the \(1/f^{\alpha}\) law with \(\alpha=0.6\pm 0.1\), which is close to unity. The authors of [67] analyzed their results in light of a phenomenological theory based on pinned elastic structures.