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In spite of their relatively high lattice thermal conductivity$\kappa_{\ell}$, the XNiSn (X=Ti, Zr or Hf) half-Heusler compounds are goodthermoelectric materials. Previous studies have shown that $\kappa_{\ell}$ canbe reduced by sublattice-alloying on the X-site. To cast light on how the alloycomposition affects $\kappa_\ell$, we study this system using the phononBoltzmann-transport equation within the relaxation time approximation inconjunction with density functional theory.The effect of alloying throughmass-disorder scattering is explored using the virtual crystal approximation toscreen the entire ternary Ti$_x$Zr$_{y}$Hf$_{1-x-y}$NiSn phase diagram. Thelowest lattice thermal conductivity is found for the Ti$_x$Hf$_{1-x}$NiSncompositions; in particular, there is a shallow minimum centered atTi$_{0.5}$Hf$_{0.5}$NiSn with $\kappa_l$ taking values between 3.2 and 4.1 W/mKwhen the Ti content varies between 20 and 80\%. Interestingly, the overallbehavior of mass-disorder scattering in this system can only be understood froma combination of the nature of the phonon modes and the magnitude of the massvariance. Mass-disorder scattering is not effective at scattering acousticphonons of low energy. By using a simple model of grain boundary scattering, wefind that nanostructuring these compounds can scatter such phonons effectivelyand thus further reduce the lattice thermal conductivity; for instance,Ti$_{0.5}$Hf$_{0.5}$NiSn with a grain size of $L= 100$ nm experiences a 42\%reduction of $\kappa_{\ell}$ compared to that of the single crystal.

physical review. b./physical review. b

Lattice thermal conductivity of Ti$_x$Zr$_y$Hf$_{1-x-y}$NiSn  half-Heusler alloys calculated from first principles: Key role of nature of  phonon modes