Resonance width distribution for high-dimensional random media

We study the distribution of resonance widths $\mathcal{P}\left(\Gamma \right)$ for three-dimensional (3D) random scattering media and analyze how it changes as a function of the randomness strength. We are able to identify in $\mathcal{P}\left(\Gamma \right)$ the system-inherent fingerprints of the metallic, localized, and critical regimes. Based on the properties of resonance widths, we also suggest a criterion for determining and analyzing the metal-insulator transition. Our theoretical predictions are verified numerically for the prototypical 3D tight-binding Anderson model.

Figure 1

Scattering setup. The sample is a cubic lattice of linear length $L$. To each of the $M=L^2$ sites of the layer $n_x=1$ semi-infinite single mode leads are attached.

Figure 2

The distribution of resonance widths, plotted as $\mathcal{P}(1∕\Gamma )$ vs ${\ln }^3(1∕\Gamma )$, for $\Gamma <\Delta$ in the diffusive regime. $L=16$ and $W=10$, 12, and 14 (from left to right). The dashed lines are (shifted) linear fittings to the distributions.

Figure 3

The resonance width distribution $\mathcal{P}\left(\tilde{\Gamma }\right)$ for $L=16$ and various scattering configurations (from up to down): $M=L^2$ leads attached to the boundary, one lead attached to the boundary, and one lead attached to a site in the bulk of the sample. The dashed lines are the corresponding theoretical predictions for $\Gamma \gtrsim {\Gamma }_{\mathrm{Th}}$ given by Eqs. (10, 11, 12), see main text.

Figure 4

$\mathcal{P}\left(\tilde{\Gamma }\right)$ in the localized regime for various combinations of $W$ and $L$ in the range $\tilde{\Gamma }⩽1$. The log-normal decay is highlighted by Gaussian fits (full curves) whose maximum decreases with increasing strength of disorder and also shifts towards smaller values of $\tilde{\Gamma }$. When keeping the ratio $l_{\infty }∕L\approx 0.136$ fixed, similar distributions were obtained for different combinations of $L$ and $W$ (filled circles and open squares). (b) For $\tilde{\Gamma }⩾1$ the power-law decay $\mathcal{P}\left(\tilde{\Gamma }\right)\sim 1∕\tilde{\Gamma }$ is observed (dashed line) which becomes more robust for increasing strength of disorder.

Figure 5

Universal behavior of $\mathcal{P}\left(\tilde{\Gamma }\right)$ at the MIT (reported here as $\mathcal{P}\left[\ln \left(\tilde{\Gamma }\right)\right]$) for various sample sizes $L$ and potential distributions.

Figure 6

(a) The integrated distribution ${\mathcal{P}}_{\mathrm{int}}\left(\tilde{\Gamma }\right)$ in the case of perfect coupling, $w=1$, for various sample sizes $L$ and potential distributions. The dashed line is the theoretical prediction ${\mathcal{P}}_{\mathrm{int}}\left(\tilde{\Gamma }\right)\sim {\tilde{\Gamma }}^{-0.333}$, see Eq. (15). (b) ${\mathcal{P}}_{\mathrm{int}}\left(\tilde{\Gamma }\right)$ for a box distribution, $L=10$, and different coupling strengths $w=0.001$, 0.01, and 0.5 from left to right.

Figure 7

${\mathcal{P}}_{\mathrm{int}}^0(W,L)$ as a function of $W$ for different system sizes $L$ provides a means to determine the critical point $W_c$ of the MIT (vertical line at $W=16.5$).

Figure 8

The one-parameter scaling of ${\mathcal{P}}_{\mathrm{int}}^0(W,L)$ [Eq. (16)] is confirmed for various system sizes $L$ and disorder strengths $W$ using the box distribution.