Quantum percolation and ballistic conductance on a lattice of wires.

Abstract

Motivated by concepts of classical electrical percolation theory, we study the quantum-mechanical electrical conductance of a lattice of wires as a function of the bond-occupation probability p. In the ordered or ballistic case (p=1), we obtain an analytic expression for the energy dispersion relation of the Bloch electrons, which couples all the transverse momenta. We also get closed-form expressions for the conductance ${\mathit{g}}_{\mathit{N}\mathit{L}}$ of a finite system of transverse dimension ${\mathit{N}}^{\mathit{d}\mathrm{\ensuremath{-}}1}$ and length L (with d=2 or 3). In the limit L\ensuremath{\rightarrow}\ensuremath{\infty}, the conductance is quantized similarly to what is found for the conductance of narrow constrictions. We also obtain a closed-form expression for the conductance of a Bethe lattice of wires and find that it has a band whose width shrinks as the coordination number increases. In the disordered case (p1), we find, in d=3 dimensions, a percolation transition at a quantum-mechanical threshold ${\mathit{p}}_{\mathit{q}}$ that is energy dependent but is always larger than the classical percolation threshold ${\mathit{p}}_{\mathit{c}}$. Near ${\mathit{p}}_{\mathit{q}}$ (namely, for small values of \ensuremath{\Vert}\ensuremath{\Delta}\ensuremath{\Vert}==\ensuremath{\Vert}p-${\mathit{p}}_{\mathit{q}}$\ensuremath{\Vert}), the mean quantum-mechanical conductance 〈${\mathit{g}}_{\mathit{L}}$〉 of a cube of length L follows the finite-size-scaling form 〈${\mathit{g}}_{\mathit{L}}$(p)〉\ensuremath{\approxeq}${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{-}}\mathit{t}/\ensuremath{\nu}}$F(\ensuremath{\Delta}${\mathit{L}}^{1/\ensuremath{\nu}}$), where the scaling function F and the critical exponent \ensuremath{\nu} are different from their classical analogues.Our numerical estimate of the critical exponents is \ensuremath{\nu}=0.75\ifmmode\pm\else\textpm\fi{}0.1 and t=\ensuremath{\nu} in accordance with results of nonlinear \ensuremath{\sigma} models of localization. The distribution of the conductance undergoes a substantial change at threshold. The conductance in the diffusive (metallic) regime in d=3 dimensions follows Ohm's law (it is proportional to L). As p\ensuremath{\rightarrow}1, the crossover between the metallic and the ballistic regimes is governed by the scaling law 〈${\mathit{g}}_{\mathit{L}}$(p)〉\ensuremath{\approxeq}${\mathit{L}}^{2}$K(L(1-p)). No percolation transition is found for d=2 but as p\ensuremath{\rightarrow}1, the crossover between the quasimetallic and the ballistic regimes is governed by a similar scaling law.