Universal conductance fluctuation of mesoscopic systems in the metal-insulator crossover regime

We report a theoretical investigation on conductance fluctuation of mesoscopic systems. Extensive numerical simulations on quasi-one-dimensional, two-dimensional, and quantum dot systems with different symmetries [circular orthogonal ensemble, circular unitary ensemble (CUE), and circular symplectic ensemble (CSE)] indicate that the conductance fluctuation can reach a universal value in the crossover regime for systems with CUE and CSE symmetries. The conductance distribution is found to be a universal function from diffusive to localized regimes that depends only on the average conductance, dimensionality, and symmetry of the system. The numerical solution of DMPK equation agrees with our result in quasi-one dimension. Our numerical results in two dimensions suggest that this universal conductance fluctuation is related to the metal-insulator transition. In the localized regime with average conductance $⟨G⟩<0.3$, the conductance distribution seems to be superuniversal independent of dimensionality and symmetry.

Figure 1

(Color online) [(a)–(c)] Average conductance and [(d)–(f)] its fluctuation vs disorder strength $W$ for different symmetry index $\beta$ in quasi-1D systems. Insets: localization length vs $W$ for $\beta =2,4$.

Figure 2

(Color online) Conductance fluctuation vs average conductance for [(a) and (b)] quasi-1D systems, (c) QD systems, and (d) 2D systems.

Figure 3

(Color online) [(a) and (c)] Average conductance and its fluctuation vs s for quasi-1D systems and [(b) and (d)] compared to results of DMPK equation for different $N=4,6,\dots ,18$. Arrow points the direction of increasing $N$.

Figure 4

(Color online) Average conductance and its fluctuation vs $W$ for QD systems with $\beta =1,2,4$.

Figure 5

(Color online) Localization length, average conductance, and conductance fluctuation of 2D symplectic systems for different $L$ with $E=2.3$ and $t_{so}=0.4$. (a) $\xi /L_1$ vs $W$. (b) $\text{rms}\left(G\right)$ vs $⟨G⟩$ for different $L=60,70,\dots ,120$. (c) $\text{rms}\left(G\right)$ and $⟨G⟩$ vs $W$.

Figure 6

(Color online) The conductance distribution $P\left(G\right)$ vs $G$ at fixed $⟨G⟩=0.5,0.8,2.0$ for 2D systems with $\beta =1,2,4$. (a)–(c) correspond to $⟨G⟩=0.5$ and $\beta =1,2,4$. (d)–(f) correspond to $⟨G⟩=0.8$ and $\beta =1,2,4$. (g)–(i) correspond to $⟨G⟩=2.0$ and $\beta =1,2,4$.

Figure 7

(Color online) The conductance fluctuation $\text{rms}\left(G\right)$, its third and fourth moments ${\gamma }_1$ and ${\gamma }_2$ vs $⟨G⟩$ for [(a)–(c)] QD systems panel and (d)–(f) quasi-1D systems panel as well as (g)–(i) 2D systems panel with $\beta =1,2,4$. In each panel, different symmetries $\text{with}=1,2,4$ are denoted by symbols with red, blue, and green colors, respectively. Different symbols such as square, circle, star, etc. are used to represent different parameters.

Figure 8

(Color online) The conductance fluctuation and conductance distribution are depicted for all data from quasi-1D, QD, 2D systems $\text{with}=1,2,4$. (a) Conductance fluctuation. [(b) and (c)] Conductance distributions at fixed $⟨G⟩=0.2$ and 0.6, respectively.