Improved Callaway model for lattice thermal conductivity

In developing the phonon quasiparticle picture, Peierls discovered that, in a perfect crystal, without anharmonic umklapp ($U$) events, a current-carrying distribution can never relax to a zero-current distribution. Callaway introduced a simplified approximate model version of the Peierls-Boltzmann equation, retaining its ability to deal separately with normal ($N$) and $U$ events. This paper clarifies and improves the Callaway model, and shows that Callaway underestimated the suppression of $N$ processes in relaxing thermal current. The new result should improve computations of thermal conductivity from relaxation-time studies.

Figure 1

Ratio of Callaway's old solution (${\kappa }_C^{*}$) to the new solution (${\kappa }_C$) of the Callaway model in Debye approximation, with quadratic behavior $1/{\tau }_N\left(\omega \right)={\gamma }_N{(\omega /{\omega }_D)}^2$. The dashed curves use the Herring quadratic behavior also for $U$, with ratio ${\tau }_N\left(\omega \right)/{\tau }_U\left(\omega \right)={\gamma }_U/{\gamma }_N=g$ and $g=A\exp (-{\Theta }_D/3T)$. The solid curves have $p_U=4$ or ${\tau }_N\left(\omega \right)/{\tau }_U\left(\omega \right)=g{(\omega /{\omega }_D)}^2$, and the same form for $g$. In both cases, the four curves, from lowest to highest, are for values of $A$ set to 1, 2, 4, and 10.

Figure 2

Ratio of relaxation-time approximation (${\kappa }_{\mathrm{RTA}}$) to full solution ${\kappa }_C$ for the Callaway model in Debye approximation with quadratic $\omega$ dependence for $1/{\tau }_N$ and quartic $\omega$ dependence for $1/{\tau }_U$. The four solid curves use the same parameters as the solid curves of Fig. 1. The dashed curves have $A=10$ (as does the top solid curve), but also include Rayleigh-type impurity scattering [$1/{\tau }_{\mathrm{imp}}={\gamma }_{\mathrm{imp}}{(\omega /{\omega }_D)}^4$], a momentum-nonconserving event which adds to $1/{\tau }_U$ without the low-$T$ thermal suppression. The strength of the impurity term is ${\gamma }_{\mathrm{imp}}/{\gamma }_N=1,2,4,10$ (from the lowest dashed curve to the highest).