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

In spite of their relatively high lattice thermal conductivity ${\kappa }_{\ell }$, the $X\mathrm{NiSn}$ ($X=\text{Ti}$, Zr, or Hf) half-Heusler compounds are good thermoelectric materials. Previous studies have shown that ${\kappa }_{\ell }$ can be reduced by sublattice alloying on the $X$ site. To cast light on how the alloy composition affects ${\kappa }_{\ell }$, we study this system using the phonon Boltzmann-transport equation within the relaxation time approximation in conjunction with density functional theory. The effect of alloying through mass-disorder scattering is explored using the virtual crystal approximation to screen the entire ternary ${\mathrm{Ti}}_x{\mathrm{Zr}}_y{\mathrm{Hf}}_{1-x-y}\mathrm{NiSn}$ phase diagram. The lowest lattice thermal conductivity is found for the ${\mathrm{Ti}}_x{\mathrm{Hf}}_{1-x}\mathrm{NiSn}$ compositions; in particular, there is a shallow minimum centered at ${\mathrm{Ti}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$ with ${\kappa }_{\ell }$ taking values between 3.2 and 4.1 W/mK when the Ti content varies between 20% and 80%. Interestingly, the overall behavior of mass-disorder scattering in this system can only be understood from a combination of the nature of the phonon modes and the magnitude of the mass variance. Mass-disorder scattering is not effective at scattering acoustic phonons of low energy. By using a simple model of grain boundary scattering, we find that nanostructuring these compounds can scatter such phonons effectively and thus further reduce the lattice thermal conductivity; for instance, ${\mathrm{Ti}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$ with a grain size of $L=100$ nm experiences a 42% reduction of ${\kappa }_{\ell }$ compared to that of the single crystal.

Figure 1

A color map of the $X$-site mass variance parameter $M_{\mathrm{var}}$ of ${\mathrm{Ti}}_x{\mathrm{Zr}}_y{\mathrm{Hf}}_{1-x-y}\mathrm{NiSn}$.

Figure 2

The phonon dispersion for (a) TiNiSn, (b) ZrNiSn, and (c) HfNiSn. The curves for the three compounds are similar, but with a decreasing frequency of the three lower optical-phonon bands when going from TiNiSn to HfNiSn.

Figure 3

The full and partial phonon density of states (DOS) for (a) TiNiSn, (b) ZrNiSn and (c) HfNiSn, as well as that of the alloys (d) ${\mathrm{Ti}}_{0.5}{\mathrm{Zr}}_{0.5}\mathrm{NiSn}$, (e) ${\mathrm{Ti}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$, and (f) ${\mathrm{Zr}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$ obtained in the VCA. The total DOS is shown by the gray, filled area. The partial DOS of the $X$ site (Ti, Zr, Hf, or their mixtures) is given by the red curve, whereas Ni and Sn are shown by the green and blue curves.

Figure 4

The lattice thermal conductivity ${\kappa }_{\ell }$ of the binary HH mixtures ${\mathrm{Ti}}_x{\mathrm{Hf}}_{1-x}\mathrm{NiSn}$ (a) and ${\mathrm{Zr}}_x{\mathrm{Hf}}_{1-x}\mathrm{NiSn}$ (b) as a function of composition. The present theoretical results are shown as squares connected with solid lines. Circles connected by dotted lines represent experimental results for ${\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{Hf}}_{\mathrm{x}}\mathrm{NiSn}$ (a) by Katayama et al. [66] and ${\mathrm{Zr}}_{1-\mathrm{x}}{\mathrm{Hf}}_{\mathrm{x}}\mathrm{NiSn}$ (b) by Liu et al. [56]. *The latter experiment was performed on a sample doped with 1.5% Sb on the Sn site. The lines are a guide to the eye. The filled (unfilled) markers correspond to a temperature of $300\left(600\right)\mathrm{K}$.

Figure 5

A ternary map of ${\kappa }_{\ell }$ for the composition ${\mathrm{Ti}}_x{\mathrm{Zr}}_y{\mathrm{Hf}}_{1-x-y}\mathrm{NiSn}$ at 300 K based on the virtual crystal approximation. The bottom right corner corresponds to TiNiSn, the top to ZrNiSn, and the bottom left to HfNiSn. Panel (a) shows the entire ${\kappa }_{\ell }$, including anharmonic phonon-phonon scattering and mass-disorder scattering. Panel (b) shows ${\kappa }_{\ell }$ when only the anharmonic phonon-phonon scattering is included through ${\kappa }_{\ell }^{\mathrm{anh}}$, and mass-disorder scattering is neglected.

Figure 6

The total calculated thermal conductivity with grain-boundary scattering included ${\kappa }_{\ell }^{\mathrm{GB}}$, as function of average grain size $L$ for TiNiSn at temperatures 300 and 600 K (red dashed and dotted) and ${\mathrm{Ti}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$ at corresponding temperatures (green solid and dashed-dotted). The inset shows the lattice thermal conductivity scaled by the bulk (single crystal) value ${\kappa }_{\ell }^{\mathrm{GB}}\left(L\right)/{\kappa }_{\ell }$ for the four cases.

Figure 7

The cumulative ${\kappa }_{\ell }$ at 300 K as a function of phonon frequency given in (a) and its derivative in (b). Pure TiNiSn with and without grain-boundary scattering is shown as red squares and circles, respectively. The composition ${\mathrm{Ti}}_{0.5}{\mathrm{Hf}}_{0.5}\mathrm{NiSn}$ with and without grain-boundary scattering is shown as green squares and circles.