Metal-Insulator Transition in Two Dimensions: Experimental Test of the Two-Parameter Scaling

We report a detailed scaling analysis of resistivity $\rho (T,n)$ measured for several high-mobility 2D electron systems in the vicinity of the 2D metal-insulator transition. We analyzed the data using the two-parameter scaling approach and general scaling ideas. This enables us to determine the critical electron density, two critical indices, and temperature dependence for the separatrix in the self-consistent manner. In addition, we reconstruct the empirical scaling function describing a two-parameter surface which fits well the $\rho (T,n)$ data.

Figure 1

Temperature dependences $\rho (T)$: (a) for sample Si2, the densities increase in steps of 0.224 starting from 6.72 (topmost curve); (b) for sample Si43, the densities are (from top to bottom) 7.16, 7.34, 7.52, 7.70, 7.88, 8.06, 8.51, 8.96, 9.41, 9.86, 10.31, 10.76, 11.6, 12.5, 13.4, 14.9. The inset shows $T_{\max }$ versus electron density: squares are the data and the line is the theoretical curve (see below). Density values are quoted in units of ${10}^{10}{\mathrm{cm}}^{-2}$.

Figure 2

The density dependences of $\ln \rho (n)$ for various temperatures (from the top to the bottom, in K): (a) 1, 1.23, 1.43, 1.6, 1.8, 2.1, 2.35, 2.64, 2.95, 3.31, 3.71, 4, 4.33 for sample Si43; (b) 1.4, 1.6, 1.8, 2.2, 2.5, 3, 3.6, 4.2 for sample Si2. The density is given in units of ${10}^{11}{\mathrm{cm}}^{-2}$. The data are shown in a limited density range, $X<0.7$. The insets illustrate the procedure of extracting $\kappa$ and $T_0$ using Eq. (3). The bars display the range of intersections for all curves.

Figure 3

Scaling of the ${\rho }^{*}(T)/{\rho }_c^0$ data for densities $n=8.74$, 8.96, 9.18, 9.41, $9.63\times {10}^{10}{\mathrm{cm}}^{-2}$. The lower inset demonstrates the quality of the scaling for large $X\sim 0.5$. The upper inset shows a fit of the separatrix using Eq. (3) with three parameters $\zeta$, $T_1$, and ${\rho }_c^0$. Sample Si2.

Figure 4

The empirical two-parameter scaling function $\mathcal{R}(X,Y)$ for five samples (dots) and the model surface $\mathcal{F}=\tilde{X}\tilde{Y}$ (grid). The straight line $\rho /{\rho }_c^0=\tilde{Y}$ at $\tilde{X}=0$ represents the separatrix.