Resolution of the exponent puzzle for the Anderson transition in doped semiconductors

The Anderson metal-insulator transition (MIT) is central to our understanding of the quantum mechanical nature of disordered materials. Despite extensive efforts by theory and experiment, there is still no agreement on the value of the critical exponent $\nu$ describing the universality of the transition—the so-called “exponent puzzle.” In this Rapid Communication, going beyond the standard Anderson model, we employ ab initio methods to study the MIT in a realistic model of a doped semiconductor. We use linear-scaling density functional theory to simulate prototypes of sulfur-doped silicon (Si:S). From these we build larger tight-binding models close to the critical concentration of the MIT. When the dopant concentration is increased, an impurity band forms and eventually delocalizes. We characterize the MIT via multifractal finite-size scaling, obtaining the phase diagram and estimates of $\nu$. Our results suggest an explanation of the long-standing exponent puzzle, which we link to the hybridization of conduction and impurity bands.

Figure 1

Critical concentrations $n_{\text{c}}$ and exponents $\nu$ as a function of the energy $\varepsilon$ from the Fermi level ${\varepsilon }_{\text{F}}$, for $q=0$ (red circles) and $q=1$ (blue triangles). Solid and open symbols denote, respectively, the results for $\lambda =1/4$ and $\lambda =1/2$ coarse grainings. The error bars, shown only for $\lambda =1/4$ and if larger than the symbol size, represent the 95% confidence level on the fit parameters. The error bars for $\lambda =1/2$ are of the same order of magnitude as for $\lambda =1/4$ and are omitted for clarity.

Figure 2

Schematic description of the workflow. (a) represents the catalog of prototypes. For clarity, we show a projection on the $xy$ plane and distances in units of $a$, the Si lattice parameter. The upper plot depicts one impurity (yellow with orange border) and the neighboring Si atoms (green); the lower plot denotes two impurities at distance $a$ and their Si neighbors (dark green). Gray sites indicate Si atoms unaffected by the impurity potential. In (b) we indicate how we build an effective tight-binding model of 4096 atoms with 29 impurities. The color code is the same as in (a) and indicates which catalog is used. Due to the projection on the $xy$ plane some impurities appear closer than they are. Their three-dimensional (3D) distribution is shown in (c) and (d), where we plot (c) a localized state deep in the IB and (d) an extended state above ${\varepsilon }_{\text{F}}$. We represent the 90% largest wave-function values with spheres of volume proportional to ${\left|\psi \right|}^2$. Opacity and color are proportional to $-{log}_L{\left|\psi \right|}^2$, with $L=16$ here, so that lower (higher) values are in red transparent (violet solid). The box size is as in (b).

Figure 3

DOS of the IB for 4096 atoms at three different concentrations. The shading indicates energies where states are on average delocalized (in the $L\to \infty$ limit), according to Fig. 1. The delocalized CB states, separated by a vertical dashed line at ${\varepsilon }_{\text{F}}$, are also shaded. Crossed shading indicates states that might be delocalized, but are outside our concentration range.

Figure 4

Distribution of the moments ${\alpha }_0$ as a function of $\varepsilon$ shifted with ${\varepsilon }_{\text{F}}$ (vertical dashed line). For $N_{\text{S}}=40$ we show the density plot of the distribution (from blue for low to red for high density—see color scale) and the contour lines enclosing 68% (white) and 95% (black) of the ${\alpha }_0$'s. For $N_{\text{S}}=100$ we indicate the same contours (red, dashed). As in Fig. 3, the shading denotes the delocalized region (in the $L\to \infty$ limit) according to Fig. 1.

Figure 5

Moments ${\alpha }_0(\varepsilon ,n)$ for $N_{\text{S}}=40$ indicated by colored bars as given in the color scale. In addition, we show the estimated $n_c$ from Fig. 1 for $\lambda =1/4$ with $q=0$ ($\circ$) and 1 ($▿$). The horizontal gray line at $n\lesssim 3$ indicates the lack of enough low-doping S concentrations to allow scaling fits for $q=0$. Regions of ${\alpha }_0\sim 3$, indicating much less-localized states, begin to extend below the Fermi level at concentrations above where the gap closes. Inset: For easier comparison, we plot the estimated $\nu$ values for $\lambda =1/4$ with $q=0$ ($\circ$) and 1 ($▿$) as in Fig. 1.