Universal scaling in disordered systems and nonuniversal exponents

The effect of an electric field on conduction in a disordered system is an old but largely unsolved problem. Experiments cover a wide variety of systems—amorphous/doped semiconductors, conducting polymers, organic crystals, manganites, composites, metallic alloys, double perovskites—ranging from strongly to weakly localized systems and from strongly to weakly correlated ones. Theories have singularly failed to predict any universal trend resulting in separate theories for separate systems. Here, we discuss a one-parameter scaling that has recently been found to give a systematic account of the field-dependent conductance in two diverse, strongly localized systems of conducting polymers and manganites. Except for a limited number of systems which are described by the hot electron models, the vast majority of different systems in various disorder regimes in two (2D)- and three (2D)-dimensions obey the scaling. The nonlinearity exponent $x$ associated with the scaling was found to be nonuniversal and exhibiting a structure. For 2D weakly localized systems, the nonlinearity exponent $x$ is $\ge$7 and is roughly inversely proportional to the sheet resistance. The existing theories of weak localization prove to be adequate and a complete scaling function is derived. In a 2D strongly localized system, a temperature-induced scaling-nonscaling transition (SNST) is revealed. For 3D strongly localized systems, the exponent lies between $-$1 and 1, and surprisingly is quantized ($x\approx 0.08n$). This poses a serious theoretical challenge. Various results are compared with predictions of the existing theories.

Figure 1

Scaling $(\epsilon \sim -0.005)$ in 2D weakly localized metallic thin films of AuPd. Plots of resistance vs electric bias at different temperatures are shown in the inset (a) (data adapted from Ref. [23]). Scaling of the same data leads to its collapse as shown in the main panel. $R_o\left(T\right)$ and $V_o\left(T\right)$ are the Ohmic resistance and the onset bias, respectively, at temperature T. The solid line is a fit to ${\Phi }_{\mathrm{WL}}={(1+0.9q^2)}^{1/130}(q=V/V_o)$. Inset (b) shows log-log plots of $V_o$ vs $R_o$ (solid symbols) and T (open symbols). The solid lines are linear fits to the data with the slopes as mentioned.

Figure 2

Effect of film thickness on the nonlinearity exponent in 2D metallic states. The exponent is plotted against sheet resistance for various materials and film thicknesses. One set of data (open symbols) belongs to thin films ($\diamond$-AuPd [23], $\square$-Pt [27], $▵$-2DEG [7], $\times$-Au [28]), and $\circ$-Cu [29]). See text for explanation on error bars. Another set of data (closed symbols) is from thick films (Ref. [30]). The vertical arrow schematically indicates progress towards an incipient thickness-induced 2D-3D crossover. The solid line is a least-square fit and the dashed line is drawn parallel to it. The vertical line corresponds to the maximum metallic resistance of 30 k$\Omega$.

Figure 3

Scaling $(\epsilon \sim 0.05)$ of the bias-dependent conductivity at different temperatures in a metallic thick film of GeAu (data adapted from Ref. [30]). The solid line is a fit to ${\Phi }_{\mathrm{WL}}={(1+1.18q^2)}^{1/15.8}$. Inset shows log-log plots of $F_o$ vs ${\sigma }_o$ (solid symbols) and T (open symbols). The solid lines are linear fits to the data with the slopes as mentioned.

Figure 4

Evidence for a temperature-induced transition in the 2DSL regime. In a 2DES sample, the bias-dependent conductivity data scale only at lower temperatures $(\epsilon \sim -0.1)$. The data are adapted from Ref. [7]. The solid line is a fit to ${\Phi }_H^{0.16}-1/{\Phi }_H={\left(0.227q\right)}^2$ and the dotted line to $\Phi =exp(-0.105q)$. Insets show log-log plots of $V_o$ vs $R_o{T}^{4.5}$ (a) and $R_o$ (b). The solid lines are least-square fits to the selected data with the slope as shown.

Figure 5

$r_d$ vs T for the scaling-nonscaling transition (SNST) in Fig. 4. The curve is a fit to $r_d=1.02{(T-0.111)}^{0.27}$.

Figure 6

Scaling $(\epsilon \sim -0.1)$ of the bias-dependent conduction data at different temperatures in a $n$-type Si:P:B sample on the insulating side of the MIT. The data $a$ (shifted upwards by a factor 1.2 for clarity) and $b$ are adapted from Refs. [15, 16], respectively. The solid line is a fit to ${\Phi }_H^{-1}=1-{\left(0.354q\right)}^2$ and the dotted line to $\Phi =exp(-0.105q)$. Inset shows log-log plots of $V_o$ vs $R_o{T}^{5.5}$ corresponding to data $a$ (closed symbols) and data $b$ (open symbols). The solid line is a least-square fit to the data (closed symbols) with the slope as shown.

Figure 7

Scaling $(\epsilon \sim 1)$ of the bias-dependent conductivity data at different temperatures in a doped Si:P sample on the insulating side of the MIT. The data are adapted from Ref. [34]. The solid line is a fit to the SGM expression [12]: $\Phi =1+0.9q^{4/3}+0.06q^{18/5}+2.2\times {10}^{-5}{q}^{27/4}$, dashed line to ${\Phi }_H^{0.3}-1/{\Phi }_H={\left(0.855q\right)}^2$ and the dotted line to $\Phi =exp\left(0.67q\right)$. Inset shows two log-log plots of $F_o$ vs ${\sigma }_o$ (closed symbols) and T (open symbols). The solid lines are least-square fits to the data with the slopes as shown.

Figure 8

Scaling $(\epsilon \sim 1)$ of the bias-dependent conductance data in a correlated, charge-ordered organic salt at ten different temperatures (0.29–2 K) (data from Ref. [35]). The line mode for data representation is used to highlight the quality of data collapse. The solid line is a fit to the SGM expression [12]: $\Phi =1+0.97q^{4/3}+0.0025q^{14/3}+1.5\times {10}^{-8}{q}^{70/9}$ and the dotted line to $\Phi =exp\left(0.67q\right)$. Inset shows two log-log plots of $V_o$ vs ${\Sigma }_o$ (closed symbols) and T (open symbols). The solid line is a linear fit with the slope as shown.

Figure 9

The nonlinearity exponents $x_T$ in several (a) uncorrelated and (b) correlated classes of materials. The exponents are quantized and given by $x_M\approx 0.08n$. This is highlighted by plotting $x_M/0.08$ against $n$ with lines dropped off the data points to $y$ axes. Each panel shows a line passing through the origin with a slope of unity. Data points either lie on, or are very close to, the lines. The inset in (a) displays the histogram of the exponent $x_T$ in Table 1 only.

Figure 10

Plot of $x_T$ vs $T_o$ in VRH systems (Tables 1 and 2). Horizontal bands, each 0.02 wide, are drawn at $x_T=0.08n$ where $n$ is an integer. Data from Table 1 (purple closed symbols) including one in 2D (diamond) and Table 2 (magenta half-open symbols) are included. The vertical line roughly separates points according to its $m$ values. Dashed lines are only guides to the eye. Most of the points are seen to lie within the bands.

Figure 11

Summary of the nonlinearity exponent $x_T$ in various regimes in the $T=0$ scaling diagram of Abrahams et al. [3]. The solid point represents the metal-insulator transition (MIT). The star on the 2D curve (corresponding to $R_{\square }\approx 30\mathrm{k}\Omega$) represents the WL-SL crossover. Both the MIT and WL-SL crossover are accompanied by the change in the exponent value: $x_T>1$ on the metallic side and $x_T<1$ on the insulation side. The dashed line schematically denotes crossover from 2D to 3D with increase of thickness.