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The thermal conductivities ($\kappa$) of bulk and thin-film$\alpha$-Al$_2$O$_3$ are calculated from first principles using both the localdensity approximation (LDA), and the generalized gradient approximation (GGA)to exchange and correlation. The room temperature single crystal LDA value$\sim39~$W/m$~$K agrees well with the experimental values $\sim35-39~$W/m$~$K,whereas the GGA values are much smaller $\sim$26$~$W/m$~$K. Throughout thetemperature range, LDA is found to slightly overestimate $\kappa$ whereas GGAstrongly underestimates it. We calculate the $\kappa$ of crystalline$\alpha$-Al$\mathrm{_2}$O$\mathrm{_3}$ thin films and observe a maximum of79$\%$ reduction for $10~$nm thickness.

Ab initio lattice thermal conductivity of bulk and thin-film $\alpha$-Al$\mathrm{_2}$O$\mathrm{_3}$