Figure 1

(a) Optical microscope picture of a typical device with mesa width $W=8\mu \mathrm{m}$. The gates labeled 1–3 have lengths $L=0.5$, 1, and $2\mu \mathrm{m}$, respectively. The other gates were not used. (b) Schematic representation of long- and short-range disorder in mesoscopic devices. For devices smaller than the length scale of long-range disorder, short-range fluctuations of $O[\delta ]$ are dominant. (c)–(e) Comparison of temperature dependence of resistivity for two mesoscopic devices ($W=8\mu \mathrm{m}$, $L=2$, $0.5\mu \mathrm{m}$ for device I, II) with a macroscopic one ($100\mu \mathrm{m}\times 900\mu \mathrm{m}$) from wafers with the same spacer (40 nm). While the macroscopic device shows an exponential increase in resistivity to lowest $T$, a saturation or downturn occurs in the mesoscopic devices. Solid lines are fits of the form $\rho ={\rho }_0\exp [E_0/k_{B}T]$. The inset to (c) shows the same data on a $\log (\rho )$ against $1/T$ scale, highlighting the exponential behavior.Reuse & Permissions

Figure 2

Resistivity as a function of inverse temperature $1/T$ at $B=0\mathrm{T}$ (symbols). At all densities, the strongly insulating $T$ dependence at higher temperatures is followed by a decrease in resistance at low $T$. Device dimensions are $W\times L=8\mu \mathrm{m}\times 0.5\mu \mathrm{m}$, spacer $\delta =40\mathrm{nm}$. Electron densities are indicated by arrows in the inset to (a). Solid lines represent a fit of Eq. (1) to the data. Inset to (a): Resistivity as a function of electron density at $T=60\mathrm{mK}$, 500 mK, 4 K. Inset to (d): $\rho$ as function of $1/T$ at the same density as (d) but at $B_{\perp }=1.5\mathrm{T}$.Reuse & Permissions

Figure 3

Comparison of metallicity (defined as $\frac{d\rho (T)/dT}{\rho (T)}$ at $T=300\mathrm{mK}$) for four devices from three different wafers with spacer $\delta =20–60\mathrm{nm}$. A metallic state shows positive values, and a saturated behavior negative values. The mesa width is $W=8\mu \mathrm{m}$ for all devices, while the gate lengths $L$ are indicated in the graph. Device III is shown in Fig. 2 and is discussed extensively in the text.Reuse & Permissions

Figure 4

Physical quantities extracted from data fits to Eq. (1): (a) ${\rho }_0$ is the resistivity extrapolated to zero $T$. Note that ${\rho }_0\gg h/e^2$ but finite for all $n_s$. (b) The temperature exponent is $\gamma \approx 2$ over a wide density range. (c) Temperature $T_{\theta }$ at which the resistivity is maximal. It shows a nonmonotonic density dependence with a maximum value $T_{\theta }\approx 1.6\mathrm{K}$. $T_{\theta }$ marks the transition between the insulating and the metallic phase and, hence, defines a phase diagram in $T-n_s$ space.Reuse & Permissions