Lattice Thermal Conductivity

The lattice thermal conductivity for monatomic crystals is discussed for high temperatures, above and around the Debye temperature. While this problem cannot be treated in a suitable way by conventional methods for calculating transport coefficients, it represents an excellent example of the use of correlation-function techniques. By splitting the total anharmonic interatomic lattice potential into its diagonal and nondiagonal contributions, $V={V}_{\mathrm{d}}+{V}_{\mathrm{nd}}$, the effect of the equilibrium and nonequilibrium properties of the crystal lattice on the thermal conductivity can be studied separately. The current-current correlation function is calculated to second order in $V$ in a single-phonon-lifetime approximation. While ${V}_{\mathrm{nd}}$ only contributes in this approximation to the space-time-dependent part of the correlation function, ${V}_{\mathrm{d}}$ contributes as well to the space-time-independent part. The latter contribution ${{V}_{\mathrm{d}}}^{0}$ represents a temperature-dependent Hartree approximation of the lattice potential and determines the equilibrium properties of the crystal lattice. At constant pressure ${{V}_{\mathrm{d}}}^{0}$ gives rise to the thermal expansion of the system, which causes a decrease of the Debye frequency with increasing temperature, resulting in a depression of the conductivity below the $\frac{1}{T}$ behavior. Even in the case of constant volume, the temperature dependence of ${{V}_{\mathrm{d}}}^{0}$ causes a decrease of the phonon frequency with increasing temperature. This again gives rise to a depression of the conductivity below the $\frac{1}{T}$ behavior and leads to a ${T}^{2}$ term in the resistivity in the region around the Debye temperature. This latter effect is studied by taking into account the entire anharmonicity of the lattice potential which leads to ${{V}_{\mathrm{d}}}^{0}$ and calculating the space-time-dependent part of the correlation function in lowest order in the atomic displacements. Thus Peierls' expression for the conductivity is rederived in terms of the renormalized phonon frequency and group velocity due to ${{V}_{\mathrm{d}}}^{0}$, rather than the pure harmonic approximation of the lattice potential. The entire dissipative mechanism of the system, which is given by the space-time-dependent part of the correlation function, calculated to second order in $V$, cancels partially the effect of ${{V}_{\mathrm{d}}}^{0}$ on the conductivity. For temperatures around and just above the Debye temperature the temperature dependence of the lattice thermal conductivity is given by $\frac{1}{T}(1+\ensuremath{\alpha}T)$. With increasing temperature the conductivity reaches a minimum, which is followed by a steep rise as one approaches a dynamical instability of the system.

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