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We calculate the lattice-driven in-plane $(\kappa_{\parallel})$ andout-of-plane $(\kappa_{\perp})$ thermal conductivities of Bi$_2$Se$_3$ bulk,and of films of different thicknesses, using the Boltzmann equation with phononscattering times obtained from anharmonic third order density functionalperturbation theory.  We compare our results for the lattice component of the thermal conductivitywith published data for $\kappa_{\parallel}$ on bulk samples and with ourroom-temperature thermoreflectance measurements of $\kappa_{\perp}$ on films ofthickness (L) ranging from 18~nm to 191~nm, where the lattice component hasbeen extracted via the Wiedemann-Franz law. Ab-initio theoretical calculationson bulk samples, including an effective model to account for finite samplethickness and defect scattering, compare favorably both for the bulk case (fromliterature) and thin films (new measurements). In the low-T limit thetheoretical in-plane lattice thermal conductivity of bulk Bi$_2$Se$_3$ agreeswith previous measurements by assuming the occurrence of intercalated Bi$_2$layer defects. The measured thermal conductivity monotonically decreases byreducing $L$, its value is $\kappa_{\perp}\approx 0.39\pm 0.08$~W/m$\cdot$K for$L=18$ nm and $\kappa_{\perp}=0.68\pm0.14$~W/m$\cdot$K for $L=191$ nm. We showthat the decrease of room-temperature $\kappa_{\perp}$ in Bi$_2$Se$_3$ thinfilms as a function of sample thickness can be explained by the incoherentscattering of out-of-plane momentum phonons with the film surface. Our workoutlines the crucial role of sample thinning in reducing the out-of-planethermal conductivity.

Thermal conductivity of Bi$_2$Se$_3$ from bulk to thin films: theory and experiment