Abnormal Lattice Thermal Conductivity of a One-Dimensional, Harmonic, Isotopically Disordered Crystal

Abstract

Energy transport is investigated in a model system for which exact analytic results can be obtained. The system is an infinite, one-dimensional harmonic crystal which is perfect everywhere except in a finite segment which contains N isotopic defects. Initially, the momenta and displacements of all atoms to the left of the defect region are canonically distributed at a temperature T, and the right half of the crystal is at a lower temperature. This initial nonequilibrium state evolves according to the equations of motion, and ultimately a steady state is established in the vicinity of the region containing the defects. The thermal conductivity is calculated from exact expressions for the steady state energy flux and thermal gradient. For a crystal in which the N isotopic defects are distributed at random but in which the overall defect concentration is fixed, we demonstrate that the thermal conductivity approaches infinity as least as fast as N1/2. A Monte Carlo evaluation of the thermal conductivity for a given defect-to-host mass ratio and concentration is carried out for a series of random configurations of N defects for N in the range, 25 ≤ N ≤ 600. The thermal conductivity is proportional to N1/2 within the statistical uncertainty except for slight deviations at the smallest values of N.