Heat conduction and phonon localization in disordered harmonic crystals

¹  Raman Research Institute - C.V. Raman Avenue, Bangalore 560080, India
²  Department of Physics and Astronomy, University of Sheffield - Sheffield S3 7RH, UK, EU
³  Department of Physics, University of California-San Diego - La Jolla, CA 92093, USA
⁴  Departments of Mathematics and Physics, Rutgers University - Piscataway, NJ 08854, USA
⁵  Zentrum Mathematik, Technische Universität München - D-85747 Garching, Germany, EU

Corresponding author: anupam@rri.res.in

Received: 29 January 2010
Accepted: 6 May 2010

Abstract

We investigate the steady-state heat current in two- and three-dimensional isotopically disordered harmonic lattices. Using localization theory as well as kinetic theory we estimate the system size dependence of the current. These estimates are compared with numerical results obtained using an exact formula for the current given in terms of a phonon transmission function, as well as by direct nonequilibrium simulations. We find that heat conduction by high frequency modes is suppressed by localization while low frequency modes are strongly affected by boundary conditions. Our heuristic arguments show that Fourier's law is valid in a three-dimensional disordered solid except for special boundary conditions. We also study the pinned case relevant to localization in quantum systems and often used as a model system to study the validity of Fourier's law. Here we provide the first numerical verification of Fourier's law in three dimensions. In the two-dimensional pinned case we find that localization of phonon modes leads to a heat insulator.