Finite-temperature conductance in one dimension

Abstract

The Landauer formula and its finite-temperature extension are used to obtain the temperature-dependent conductance $G\left(T\right)\mathrm{}$ of a one-dimensional system. Large resonant structure in $G$ as a function of Fermi energy, a manifestation of a zero-temperature quantum-mechanical eigenstate tunneling phenomenon, persists to finite temperature. The resulting statistical properties of $G$, $\ln G$, $R\equiv \frac{1}{G}$, and $\ln R$ are studied. At intermediate temperatures, the mean dependence of $\ln G$ on $T$ follows the one-dimensional $T^{\frac{-1\mathrm{}}{2}}$ Mott law, with $\ln G$ displaying large nonthermodynamic fluctuations about this mean behavior. We verify these observations with numerical simulations on a one-dimensional Kronig-Penney model. The mean Mott-like behavior is reproduced, and the relationship between the Mott temperature $T_0$ and the localization length $L_0$ is verified. At higher temperature we show that the energy dependence of $L_0$ or of the density of states ${\rho }_s$ can cause $G\left(T\right)\mathrm{}$ to deviate from the $T^{\frac{-1\mathrm{}}{2}}$ behavior, and we derive expressions for the modified temperature dependence of $G$.