Critical Behavior of a Strongly Disordered 2D Electron System: The Cases of Long-Range and Screened Coulomb Interactions

A study of the temperature ($T$) and density ($n_s$) dependence of conductivity $\sigma (n_s,T)$ of a highly disordered, two-dimensional (2D) electron system in Si demonstrates scaling behavior consistent with the existence of a metal-insulator transition (MIT). The same critical exponents are found when the Coulomb interaction is screened by the metallic gate and when it is unscreened or long range. The results strongly suggest the existence of a disorder-dominated 2D MIT, which is not directly affected by the range of the Coulomb interactions.

Figure 1

Sample $B_{\text{thin}}$. (a) Conductivity $⟨\sigma ⟩$ vs $T$ for different $n_s$, as shown. $n_s$ was varied at high $T\approx 20\mathrm{K}$. Solid lines guide the eye. All data are in the regime of $T\ll T_F$ ($T_F$—Fermi temperature [25]) and $k_{F}l\ll 1$ ($k_F$—Fermi wave vector, $l$—mean free path). (b) $⟨\sigma ⟩$ vs $T^{-1/3}$ for several $n_s$ in the insulating regime. The solid lines are fits to $⟨\sigma ⟩\propto \exp [-(T_0/T{)}^{1/3}]$. Inset: $T_0$ vs $n_s$ with a linear fit, and an arrow showing $n_c$. Only $n_s$ with the activation energies $E_A(T)=T_0^{1/3}{T}^{2/3}\gtrsim 0.6\mathrm{K}$ were used in the fit. In both (a) and (b), the error bars show the size of the fluctuations with time.

Figure 2

Sample $B_{\text{thin}}$. (a) $⟨\sigma ⟩$ vs $T^{1.5}$ for a few $n_s\ge n_c$, as shown. The solid lines are linear fits. For $n_s=4.26\times {10}^{11}{\mathrm{cm}}^2$, $⟨\sigma (T=0)⟩=0$, i.e., $⟨\sigma (n_c,T)⟩\propto T^x$ with $x=1.5\pm 0.1$. (b) $⟨\sigma (n_s,T=0)⟩$ vs $n_s$. The dashed line guides the eye. (c) $⟨\sigma (n_s,T=0)⟩$ vs ${\delta }_n=(n_s-n_c)/n_c$, the distance from the MIT. The solid line is a fit with the slope equal to the critical exponent $\mu =2.7\pm 0.3$.

Figure 3

Scaling of $⟨\sigma ⟩/⟨{\sigma }_c⟩\propto ⟨\sigma ⟩/T^x$, $x=1.5$, with $T$ for sample $B_{\text{thin}}$ ($d_{\mathrm{ox}}=6.9\mathrm{nm}$). Different symbols correspond to $n_s$ from $3.40\times {10}^{11}{\mathrm{cm}}^{-2}$ to $6.70\times {10}^{11}{\mathrm{cm}}^{-2}$; $n_c=4.26\times {10}^{11}{\mathrm{cm}}^2$. It was possible to scale the data below about 1.5 K. Inset: $T_0$ vs ${\delta }_n$. The lines are fits with slopes $z\nu =1.98\pm 0.03$ and $z\nu =1.91\pm 0.02$ on the insulating and metallic sides, respectively.

Figure 4

Scaling of $⟨\sigma ⟩/⟨{\sigma }_c⟩\propto ⟨\sigma ⟩/T^x$, $x=1.5$, with $T$ for sample $A1$ ($d_{\mathrm{ox}}=50\mathrm{nm}$). Different symbols correspond to $n_s$ from $3.45\times {10}^{11}{\mathrm{cm}}^{-2}$ to $8.17\times {10}^{11}{\mathrm{cm}}^{-2}$; $n_c=5.22\times {10}^{11}{\mathrm{cm}}^2$. It was possible to scale the data below $\sim 0.3\mathrm{K}$ down to the lowest $T=0.13\mathrm{K}$. Inset: $T_0$ vs ${\delta }_n$. The lines are fits with slopes $z\nu =2.13\pm 0.01$ and $z\nu =2.13\pm 0.03$ on the insulating and metallic sides, respectively.