Conductance distribution in strongly disordered mesoscopic systems in three dimensions

Recent numerical simulations have shown that the distribution of conductances $P\left(g\right)$ in three-dimensional strongly localized systems differs significantly from the expected log normal distribution. To understand the origin of this difference analytically, we use a generalized Dorokhov-Mello-Pereyra-Kumar (DMPK) equation for the joint probability distribution of the transmission eigenvalues which includes a phenomenological (disorder and dimensionality dependent) matrix $K$ containing certain correlations of the transfer matrices. We first of all examine the assumptions made in the derivation of the generalized DMPK equation and find that to a good approximation they remain valid in three dimensions (3D). We then evaluate the matrix $K$ numerically for various strengths of disorder and various system sizes. In the strong disorder limit we find that $K$ can be described by a simple model which, for a cubic system, depends on a single parameter. We use this phenomenological model to analytically evaluate the full distribution $P\left(g\right)$ for Anderson insulators in 3D. The analytic results allow us to develop an intuitive understanding of the entire distribution, which differs qualitatively from the log-normal distribution of a Q1D wire. We also show that our method could be applicable in the critical regime of the Anderson transition.

Figure 1

(Color online) Probability distribution of normalized $k_{ab}$ for Q1D systems. (엯: $a=b=1$, ◇: $a=1$, $b=2$, ▿: $a=b=2$). Full symbols: $W=2$, $L=10$, $L_z=16L$; open symbols: $W=2$, $L=10$, $L_z=24L$. Lines with symbols: $W=2$, $L=14$, $L_z=8L$. Data confirm that the distribution becomes narrower when $L$ increases. Inset shows the same for $W=4$ and $L=18$, $L_z=4L$. As expected, distributions are broader.

Figure 2

(Color online) The distribution of normalized $k_{ab}$ for 3D systems with $W=4$ and various system sizes. (critical disorder $W_c\approx 9$.) Open symbols: $a=b=1$; full symbols: $a=1$, $b=2$. Inset shows the $L$-dependence of ${\Delta }_{ab}=\sqrt{\mathrm{var}{k}_{ab}}∕K_{ab}$ which decreases when $L$ increases. Data confirm that distribution is self-averaging (it becomes narrower and ${\Delta }_{ab}\to 0$ when $L$ increases) although it is broader than in the Q1D case.

Figure 3

(Color online) Probability distribution of normalized matrix elements $k_{ab}$. (a) $P(k_{aa}∕K_{aa})$ ($a=1,2$, and 3) at the critical point $W=W_c\approx 9$. Distribution is $L$-independent (apart from the exponential tail which is broader for larger $L$ since mean value $K_{aa}\sim 1∕L$). Note the logarithmic scale on the $y$ axis. (b) Distribution of off-diagonal elements $k_{12}$ and $k_{23}$ possess sharp maxima close to zero, and long exponential tails. Insets show standard deviations of distributions as a function of mean values for $8⩽L⩽18$. These data also show the accuracy of our estimate of the critical point since we expect both the mean and the standard deviation to be $L$-independent at $W=W_c$. The distributions for Q1D systems $L\times L\times 8L$ are almost identical to those for cubes (data not shown).

Figure 4

(Color online) Probability distribution $P\left(k_{11}\right)$ in the insulating regime $(W=16)$. Although distribution becomes narrower when system size $L$ increases, it is not self-averaging. Inset shows the size dependence of the ${\Delta }_{11}=\sqrt{\mathrm{var}{k}_{11}}∕K_{11}$ for $W=16$ and $W=20$. ${\Delta }_{11}$ converges to a nonzero constant when $L\to \infty$.

Figure 5

(Color online) At the critical point, which depends on the anisotropy parameter $t$, both $L{K}_{11}$ and $L{K}_{12}$ converge to $L_z$-independent values when $L_z∕L\to \infty$. The limiting value $L{K}_{11}$ depends on $t$. Shown are also data for the insulating regime $W=20$, $t=0.4$. All data for $K_{12}$ almost coincide so that they are not distinguishable in the figure.

Figure 6

(Color online) $1∕L^2$ dependence of $\kappa =(N+1)K_{11}∕2$ (open symbols) and $(N+1)K_{12}$ (full symbols) in the metallic regime. Data confirms that $\kappa$ depends on $L$. This agrees with data in Fig. 12. Also, $2K_{12}$ differs from $K_{11}$ for small $L$. This agrees with data in Fig. 13. To estimate limiting behavior of $K_{11}$, we also considered Q1D systems $L\times L\times 4L$. $\kappa$ is close to 1 only for very weak disorder $W=2$. For $W=4$ we obtained $\kappa \approx 1.36$. As this value does not depend on $L$ for $8⩽L⩽18$ (◇), we expect that 3D data for $W=4$ will converge to the same value when $L\to \infty$.

Figure 7

(Color online) Disorder dependence of $K_{11}$ and ${\gamma }_{12}$ for various system sizes. Note the common crossing point at $W=W_c\approx 9$. Note also that ${\gamma }_{12}\to 1$ for $W<W_c$ and $L\to \infty$, as expected from DMPK, but ${\gamma }_{12}$ decreases with the system size for $W>W_c$.

Figure 8

(Color online) ${\gamma }_{12}$ as a function of the system size for various strengths of disorder. In the metallic regime, ${\gamma }_{12}$ converges to 1 when $L\to \infty$. At the critical point, ${\gamma }_{12}$ is $L$-independent, and in the insulating regime ${\gamma }_{12}$ decreases when $L$ increases. Solid lines are linear fits ${\gamma }_{12}=a+b∕L$ with $a\sim {10}^{-3}$ for $W⩽40$.

Figure 9

(Color online) System size dependence of $K_{11}$ for various values of disorder. Solid lines are linear fits. Data in legend presents limiting values ${\tilde{K}}_{11}={\lim }_{L\to \infty }{K}_{11}\left(L\right)$. Shown are also data for the metallic regime $(W=4)$ for which $K_{11}\sim 1∕L^2$ (Table ). While ${\tilde{K}}_{11}=0$ in metal and at the critical point $(W=9)$, it is nonzero in the insulating regime. Thus, it may be used as the order parameter of the Anderson transition.

Figure 10

(Color online) System size dependence of $K_{12}$ for various values of disorder in the insulating regime. Data confirm that $K_{12}\sim 1∕L$ and that ${\lim }_{L\to \infty }{K}_{12}=0$.

Figure 11

(Color online) Estimation of the critical point from the mean conductance $⟨g⟩$ and from $L{K}_{11}$. Both parameters $⟨g⟩$ and $L{K}_{11}$ are $L$-independent at $W=W_c$. $L{K}_{11}\propto L$ $(\propto 1∕L)$ in the insulating (metallic) regime, respectively. From data we conclude that $9<W_c<9.5$ and use $W_c=9$ throughout the paper.

Figure 12

(Color online) $a$-dependence of $K_{aa}\times (N+1)∕2$ for 3D and Q1D systems with $W=2$, (mean free path $l\approx 9.2$) and for $W=4$ $(l\approx 1.8)$. $K_{aa}$ are larger than is predicted by the DMPK theory. Good agreement with DMPK is observed only for Q1D systems with very small disorder $(W=2)$.

Figure 13

(Color online) $a$-dependence of ${\gamma }_{1a}$ for the metallic regime $(W=4)$. Finite size effects are clearly visible so that ${\gamma }_{12}=1$ only in the limit of large system size. As expected, convergence is better for Q1D systems than for cubes. Nevertheless, data confirm that ${\gamma }_{1a}$ converge to 1 in the limit $L\to \infty$ even when $K_{11}$ and $K_{12}$ do not converge to values assumed by DMPK given in Eq. (10).

Figure 14

(Color online) Critical point: We plot $L{a}^{3∕2}{K}_{aa}$ to show that $L{K}_{aa}\propto a^{-1∕2}$ are $L$-independent for all $a⩽20$.

Figure 15

(Color online) Critical point. From the linear dependence $(a^{3∕2}∕L){\gamma }_{1a}\propto a∕N$ we conclude that ${\gamma }_{1a}\propto 1∕a^{1∕2}L$. $(N=L^2)$.

Figure 16

(Color online) Insulating regime: $a$-dependence of $K_{aa}$ for 3D systems with $W=30$. To see the $a$-dependence more clearly, we plot the difference $[K_{aa}-K_{11}]\sqrt{a}$ which behaves as $\sim -a$ for small $a$. Solid line is a linear fit, from which we have $K_{aa}=K_{11}+0.05-0.04∕\sqrt{a}$ for $L=18$ and $a⩽18$ $(K_{11}=0.44)$. Right inset presents data for $L=14$ and various strength of the disorder. Left inset shows $K_{aa}$ for $L=18$.

Figure 17

(Color online) Insulating regime $(W=30)$: $a$-dependence of ${\gamma }_{1a}$ ($L=6$, 10, and 14). Similar to the critical point, data shows that $a^{3∕2}∕L\times {\gamma }_{12}\sim a∕N$ so that ${\gamma }_{1a}\sim 1∕\left(\sqrt{a}L\right)$. Inset shows data for $L=14$ and various strength of the disorder.

Figure 18

(Color online) Insulating regime: Estimation of the localized length $\xi \left(W\right)$ from the linear $L$ dependence $x_1\left(L\right)=\mathrm{const}+L∕\xi$. $[{\lambda }_1={\sinh }^2\left(x_1\right)]$. Values of $\xi$ are given in the legend. $g\approx {\cosh }^{-2}{x}_1\approx \exp -2L∕\xi$.

Figure 19

(Color online) (a) shows how $L{K}_{11}$ depends on $x_1$ for various disorder in the localized regime. Data confirm linear relation $L{K}_{11}\propto x_1$. (b) shows how limiting values of $K_{11}$, obtained from the $L$-dependence of $K_{11}\left(L\right)$ (Fig. 9), depend on values of $1∕\xi$ (Fig. 18). Data confirms that there is an unambiguous $\xi$-dependence of $K_{11}$.

Figure 20

(Color online) $L{\gamma }_{12}$ defined by Eq. (18) as a function of $\xi$. Data confirms that there is an unambiguous $\xi$-dependence of ${\gamma }_{12}$. This is important for the formulation of the single parameter scaling theory. Dashed line is the linear dependence ${\gamma }_{12}=\xi ∕2L$, considered in Ref. 15.

Figure 21

Ratio $\Gamma ∕{\gamma }_{12}=1∕2L{K}_{12}l$, given by Eq. (21) for a cubic system, as a function of disorder for 3D systems in the insulating regime $W>W_c$.

Figure 22

(Color online) Mean value $⟨\ln g⟩$ and variance $\mathrm{var}\ln g$ as a function of ${\gamma }_{12}$ in the strongly insulating regime, calculated from analytical distribution (83). As expected, $\mathrm{var}\ln g\approx -⟨\ln g⟩$ for ${\gamma }_{12}⪡1$ but $\mathrm{var}\ln g\approx -2⟨\ln g⟩$ for ${\gamma }_{12}=1$. Note that both mean and variance depends only weakly on ${\gamma }_{12}$ when ${\gamma }_{12}$ is small (which is always the case when the system size $L$ is large).

Figure 23

(Color online) Skewness as a function of ${\gamma }_{12}$ in the strongly insulating regime. As expected, skewness is zero for ${\gamma }_{12}=1$ (Q1D limit).

Figure 24

(Color online) $P\left(\ln g\right)$ obtained from analytical formulas [Eqs. (82, 83, 84)] for $\Gamma =0.03$ and various values of ${\gamma }_{12}$.

Figure 25

(Color online) Conductance distribution for 3D insulators obtained from direct numerical simulation for $W=16$, $L=26$ (circles), and from Eq. (83) for $\Gamma =0.063$ and ${\gamma }_{12}=\Gamma ∕2$ (solid line). Both have the same mean value $⟨\ln g⟩\approx -10.6$. Dashed and dotted lines show Eq. (86) with $x_{2\min }=1∕2\Gamma +3∕4$ and $x_{2\min }=1∕\Gamma$, respectively, with $\Gamma =0.063$. Shown are also numerical data for $W=26$ and $L=10$ (triangles). Similar agreement is obtained for other values of $⟨\ln g⟩$ if $\Gamma$ is used as a free parameter; see Ref. 15 for the case $⟨\ln g⟩\approx -12.6$ fitted with $\Gamma =0.054$ 29.

Figure 26

(Color online) Conductance distribution for insulating samples of the size $L^2\times L_z$ $(L=10)$ In contrast to 3D system (Fig. 25), parameter $\Gamma ∕{\gamma }_{12}<1$. From numerical data, we have $\Gamma =0.3$, ${\gamma }_{12}=0.22$ for $W=16$, and $\Gamma =0.133\left(0.102\right)$, ${\gamma }_{12}=0.32\left(0.36\right)$ for systems with $W=14$ and $L_z=30$ (40), respectively. Solid lines show analytical result, Eq. (83) with ${\gamma }_{12}$ and $⟨\ln g⟩$ as given form numerical data.

Figure 27

(Color online) Mean value $⟨\ln g⟩$ as a function of $1∕\Gamma$. Open symbols: analytical result, Eq. (83). Data confirm linear dependence $⟨\ln g⟩\propto 1∕\Gamma$ for small $\Gamma$ (strong disorder). For smaller disorder (larger $\Gamma$) analytical model is less accurate. Data confirm that $⟨\ln g⟩$ does not depend on ${\gamma }_{12}$ provided that both $\Gamma$ and ${\gamma }_{12}$ are small. Full symbols: numerical data with ${\Gamma }^{-1}=L{K}_{11}∕\ell$. We estimate $\Gamma$ for a given disorder using the mean free path $\ell$ obtained from data in Fig. 19a [$\ell \left(W\right)$ is shown in inset] and the relation $K_{11}L=x_1\ell$. Numerical $⟨\ln g⟩$ is an unambiguous function of $\Gamma$, as required in the analytical model. Small deviations are due to finite size effects which affect actual values of parameters $x_1$ and $K_{11}$.

Figure 28

(Color online) (a) Probability distribution $P\left(\ln g\right)$ for $\Gamma \sim 1$ where critical regime is expected. The distribution agrees qualitatively with numerical data for the critical regime. (b) shows detail of the distribution shown for $\Gamma =0.35$, ${\gamma }_{12}=\Gamma ∕2$ and for $\Gamma =0.6$, ${\gamma }_{12}=1$. It shows that there is indeed a nonanalyticity in the distribution close to $\ln g=0$ with position of the nonanalyticity at $\ln g>0$, in agreement with analytical results (Ref. 6).

Figure 29

(Color online) Analytic result for the metallic regime ($\Gamma >1$ and ${\gamma }_{12}=1$). Data confirm that the distribution $P\left(g\right)$ is Gaussian with variance $\mathrm{var}g=0.064$ for $\Gamma =1.5$ and $\mathrm{var}g=0.09$ for $\Gamma =3.5$ which is comparable to the UCF value.

Figure 30

(Color online) Typical form of the distribution $p\left(\ln g\right)$ in 2D disordered systems compared with Gaussian distribution with the same mean value and variance. $⟨\ln g⟩=-20.6$, $\mathrm{var}\ln g=23.5$ and skewness is 0.251. The figure indicates that the deviation from the Gaussian distribution exists in 2D as well.