Dynamic localization of two-dimensional electrons at mesoscopic length scales

We have investigated the local magnetotransport in high quality two-dimensional electron systems at low carrier densities. The positive magnetoresistance in perpendicular magnetic field in the strongly insulating regime has been measured to evaluate the spatial concentration of electron trap sites within a mesoscopic region of the samples. A simultaneous measurement of the electron density within the same region shows an unexpected correspondence between the density of electrons in the metallic regime and that of the localizing sites in the insulating phase. We have shown this to imply a self-localized many-body ground state in the interacting regime, stabilized by optimal disorder.

Figure 1

(a) A schematic of the device structure. Contacts 1 and 2 are for current probes, while 3 and 4 are used for voltage probes. (b) Illustration of gate voltage $\left(V_g\right)$ dependence of the four-probe resistance $R_{12,34}$ at a constant magnetic field $(B=0.742\mathrm{T})$ corresponding to an integral filling factor ${\nu }_0(=10)$ in sample A78. The dashed lines are the expected values of $R_{12,34}$ at different filling factors $\nu$ within the gated region (see text). (c) $n_s=eB\nu ∕h$, at values of $V_g$’s corresponding to the center of the plateaus in $R_{12,34}$. The horizontal error bars represent the width of the plateaus in $V_g$. The solid lines are fits to the linear part of the data. For both samples, different symbols correspond to different integral value of ${\nu }_0$ at which $R_{12,34}-V_g$ sweeps were recorded.

Figure 2

Temperature dependence of resistivity $\rho$ in A78 for different values of gate voltage $\left(V_g\right)$ recorded at $B=1\mathrm{T}$. Inset: gate-voltage dependence of the pre-exponential factor ${\rho }_3$ at three magnetic fields.

Figure 3

Dependence of resistivity $\rho$ on $B^2$ for different values of gate voltage $\left(V_g\right)$ recorded at $T=0.3\mathrm{K}$. Inset: $V_g$ dependence of $\rho$ at zero and finite $B$. (a) sample A07, and (b) sample A78.

Figure 4

Gate voltage $\left(V_g\right)$ dependence of the slope $\left(\alpha \right)$ of the linear region of $\ln \left(\rho \right)-B^2$ data shown in Fig. 3. The data were obtained at $T=0.3\mathrm{K}$. The fit uncertainty increases, and hence the error bar on $\alpha$, as $\rho$ decreases to $\sim 2h∕e^2$. (a) sample A07, and (b) sample A78. Inset (a), (b): dependence of ${\alpha }^{-2∕3}$ on $V_g$. The solid lines depict the approximately linear behavior.

Figure 5

Correspondence between $n_L$ and $n_s$. For clarity, $n_L$, $n_s-\Delta V_g$ are plotted in log-log scale, where $\Delta V_g=V_g-V_{gL}^0$ for $n_L$ (empty symbols) and $V_g-V_{gs}^0$ for $n_s$ (filled symbols). For A07, the dashed and dotted lines represent estimates of $n_L$ assuming $A_L=0.083$ and 0.143, respectively. Inset: temperature dependence $\alpha$ at three gate voltages for A78.