Thermal conductivity of disordered harmonic solids

As a crystal is disordered, a point may be reached where the typical mean free path of phonons is so short that the wavelength and mean free path are no longer sharp concepts, and the textbook phonon-gas model for thermal conductivity breaks down. This paper proposes an alternate theory for the disordered regime, and the subsequent paper implements the theory for a realistic model of amorphous silicon. The idea is that the dominant scattering is correctly described by a harmonic Hamiltonian, which is, in principle, transformable into a one-body problem of decoupled oscillators. From this the thermal conductivity can be exactly calculated by an analog of the Kubo-Greenwood formula for electrical conductivity of disordered metals. Anderson localization is correctly contained in this theory; localized states contribute no currents in harmonic approximation. What is required is an atomistic model with a large unit cell and periodic boundary conditions (to avoid undesired surface effects). The linear size of the model should be larger than the mean free paths of the dominant phonons. A Kubo formula and then a Kubo-Greenwood-type formula are derived for this problem. A ``mode diffusivity'' ${\mathit{D}}_{\mathit{i}}$ for the ith exact oscillator state is defined. The heat is carried by off-diagonal elements of the heat current operator, which have a nonzero contribution because the temperature gradient introduces nonzero off-diagonal elements of the density matrix 〈${\mathit{a}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{a}}_{\mathit{j}}$〉. An effort is made to interpret these results physically. Schemes for implementing this formalism are discussed.