Phase separation of full-Heusler nanostructures in half-Heusler thermoelectrics and vibrational properties from first-principles calculations

Previous studies have indicated that the figure of merit ($\mathit{ZT}$) of half-Heusler (HH) alloys with composition $M\mathrm{NiSn}$ ($M=\mathrm{Ti}$, Zr, or Hf) is greatly enhanced when the alloys contain a nano-scale full-Heusler (FH) $\mathrm{MN}{\mathrm{i}}_2\mathrm{Sn}$ second phase. However, the formation mechanism of the FHnanostructures in the HH matrix and their vibrational properties are still not well understood. We report on first-principles studies of thermodynamic phase equilibria in the MNiSn-$\mathrm{MN}{\mathrm{i}}_2\mathrm{Sn}$ pseudobinary system as well as HH and FH vibrational properties. Thermodynamic phase diagrams as functions of temperature and Ni concentration were developed using density functional theory (DFT) combined with a cluster expansion and Monte Carlo simulations. The phase diagrams show very low excess Ni solubility in HH alloys even at high temperatures, which indicates that any Ni excess will decompose into a two-phase mixture of HH and FH compounds. Vibrational properties of HH and FH alloys are compared. Imaginary vibrational modes in the calculated phonon dispersion diagram of $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ indicate a dynamical instability with respect to cubic [001] transverse acoustic modulations. Displacing atoms along unstable vibrational modes in cubic $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ reveals lower-energy structures with monoclinic symmetry. The energy of the monoclinic structures is found to depend strongly on the lattice parameter. The origin of the instability in cubic $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ and its absence in cubic $\mathrm{ZrN}{\mathrm{i}}_2\mathrm{Sn}$ and $\mathrm{HfN}{\mathrm{i}}_2\mathrm{Sn}$ is attributed to the small size of the Ti $3d$ shells compared to those of Zr and Hf atoms. Lattice constants and heat capacities calculated by DFT agree well with experiment.