# Recent Developments in the Field of the Metal-Insulator Transition in Two Dimensions
## 6 Transport Evidence for a Sliding Quantum Electron Solid
Indeed, there is a striking similarity between the double-threshold \(V-I\) dependences in the low-\(n_{\rm s}\) state of Si MOSFETs and those (with the voltage and current axes interchanged) known for the collective depinning of the vortex lattice in Type-II superconductors (see, e.g., [68]). The physics of the vortex lattice in the latter system, in which the existence of two thresholds is well documented, can be adapted for the case of an electron solid. In a superconductor, current flows for zero voltage; the depinning of the vortex lattice occurs when a non-zero voltage appears. In the authors' case, the situation is reciprocal: a finite voltage is applied, but at first, the current is not flowing in the limit of zero temperature; the depinning of the electron solid is indicated by the appearance of a non-zero current. The intermediate regime between the dynamic (\(V_{\rm d}\)) and static (\(V_{\rm s}\)) thresholds corresponds to the collective pinning of the solid. In this region, the pinning occurs at the centers with different energies, and the current is thermally activated:  
Figure 7: (a) \(V-I\) characteristics at \(n_{\rm s}=5.36\times 10^{10}\) cm\({}^{-2}\) in a silicon MOSFET for different temperatures. The dashed lines are fits to the data using Equation (6). (b) The broad-band noise as a function of voltage for the same electron density and temperatures. The three upper curves are shifted vertically for clarity. From [67]. \[I\propto\exp\left[-\frac{U(V)}{k_{\rm B}T}\right], \tag{2}\]  
where \(U(V)\) is the activation energy. The static threshold \(V_{\rm s}=V_{\rm th2}\) signals the onset of the regime of solid motion with friction. This corresponds to the condition:  
\[eEL=U_{\rm c} \tag{3}\]  
(here, \(E\) is the electric field and \(L\) is the characteristic distance between the pinning centers with maximal activation energy \(U_{\rm c}\)). The balance of the electric, pinning, and friction forces in the regime of solid motion with friction yields a linear \(V-I\) characteristic that is offset by the threshold \(V_{\rm d}\) corresponding to the pinning force:  
\[I=\sigma_{0}(V-V_{\rm d}), \tag{4}\]  
where \(\sigma_{0}\) is a coefficient (near \(V_{\rm d}\), one can in general expect a power-law behavior of \((V-V_{\rm d})\)). Assuming that the activation energy for the electron solid is equal to:  
\[U(V)=U_{\rm c}-eEL=U_{\rm c}(1-V/V_{\rm s}), \tag{5}\]  
the authors obtained the expression for the current:  
\[I=\left\{ \begin{array}{c}\sigma_{0}(V-V_{\rm d})\ {\rm if}\ V>V_{\rm s}\\ \sigma_{0}(V-V_{\rm d})\exp\left[-\frac{U_{\rm c}(1-V/V_{\rm s})}{k_{\rm B}T} \right]\ {\rm if}\ V_{\rm d}<V\leq V_{\rm s}.\end{array} \right. \tag{6}\]  
The fits to the data using Equation (6) are shown by dashed lines in Figures 6 and 7a. Equation (6) successfully describes the experimentally observed two-threshold \(V-I\) characteristics. The value of \(U_{\rm c}\) decreases approximately linearly with electron density and reaches zero at \(n_{\rm s}\approx 6\times 10^{10}\ {\rm cm}^{-2}\) (the bottom inset to Figure 6) (note that this is in contrast to the vanishing activation energy of electron-hole pairs at \(n_{\rm c}\) obtained by measurements of the resistance in the limit of zero voltages/currents [17]). Probably, the vanishing \(U_{\rm c}\) is related to the minimum number of the strong pinning centers for which the collective pinning is still possible. The coefficient \(\sigma_{0}\) is approximately constant (\(\sigma_{0}\approx 1.6\times 10^{-7}\ {\rm Ohm}^{-1}\)), and this indicates that the solid motion with friction is controlled by weak pinning centers [68]. The authors of [67] argue that the large noise in the regime of the collective pinning of the solid between \(V_{\rm d}\) and \(V_{\rm s}\) should be suppressed in the regime of solid motion with friction at \(V>V_{\rm s}\).