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.

Figure 1

Conventional unit cell of HH compounds with composition $M\mathrm{NiSn}$, where $M=\mathrm{Ti}$, Zr, or Hf. The vacancy site (Va) in the HH phase can be filled with Ni to create the FH structure, $\mathrm{MN}{\mathrm{i}}_2\mathrm{Sn}$.

Figure 2

Formation energy per formula unit cell relative to the ground states for $\mathrm{TiN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (a), $\mathrm{ZrN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (b), and $\mathrm{HfN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (c). DFT calculated energies are shown as blue diamonds. The CE predicted energies for the DFT structures and further predictions for configurations up to $x=0.1$ and $x=0.9$ are shown as red dots. The dashed circle in (b) indicates specific configurations discussed in the text.

Figure 3

Temperature-concentration pseudobinary phase diagrams of $\mathrm{TiN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (a), $\mathrm{ZrN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (b), and $\mathrm{HfN}{\mathrm{i}}_{1+x}\mathrm{Sn}$ (c). The small black diamonds represent calculated points along the phase boundary and the horizontal dashed line indicates experimental melting points for the HH compound.

Figure 4

Phonon dispersion curves of TiNiSn (a), ZrNiSn (b), and HfNiSn (c) calculated with DFT shown along high-symmetry paths. Optical bands are shown in red and acoustic in blue.

Figure 5

Phonon dispersion curves of $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ (a), $\mathrm{ZrN}{\mathrm{i}}_2\mathrm{Sn}$ (b), and $\mathrm{HfN}{\mathrm{i}}_2\mathrm{Sn}$ (c) calculated with DFT shown along high-symmetry paths. Optical bands are shown in red and acoustic in blue. Imaginary frequencies are shown as negative values.

Figure 6

Total density of states (DOS) and partial density of states (PDOS) calculated with DFT. Total DOS is shown in black and contributions from each atom are shown in color, green for $M=\mathrm{Ti}$, Zr, or Hf, red for Ni, and blue for Sn. Parts (a) through (f) show the DOS for TiNiSn, ZrNiSn, HfNiSn, $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}, \mathrm{ZrN}{\mathrm{i}}_2\mathrm{Sn}$, and $\mathrm{HfN}{\mathrm{i}}_2\mathrm{Sn}$, respectively.

Figure 7

Heat capacity per atom calculated using DFT. TiNiSn results are compared to experimental data [45].

Figure 8

Atomic motions of the $X$-point TA phonon mode shown for each atom of the $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ structure. The phonon mode travels in the cubic [001] direction, out of the page. Black arrows indicate atomic motion with amplitudes magnified by 20 times.

Figure 9

Energy per formula unit of $2\times 2\times 2$ supercells is shown as a function of phonon mode displacement amplitude for (a) $X$-point, (b) $K$-point, and (c) $U$-point modes for $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$. $\mathrm{ZrN}{\mathrm{i}}_2\mathrm{Sn}$ and $\mathrm{HfN}{\mathrm{i}}_2\mathrm{Sn}$ energies are shown in (a) for comparison and have positive curvature along the distortion path, whereas $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ has negative curvature with energies that drop 1.1 meV below that of cubic $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$. $K$ and $U$ points show no instabilities along their paths. The horizontal axis measures the displacement of Ni atoms from their equilibrium position.

Figure 10

Contour plot shows the change in formation energy of $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ structure as a function of $X$-TA phonon mode amplitudes ${\varepsilon }_1$ and ${\varepsilon }_2$.

Figure 11

Formation energies of TA-$X$ mode distorted $\mathrm{TiN}{\mathrm{i}}_2\mathrm{Sn}$ structures relative to the fcc structure shown for cells of varying lattice parameter $a$, where $a=d*a_0, d=0.98$, 0.99, 1.00, 1.01, 1.02, 1.04, and $a_0=6.116\AA$.

Figure 12

Schematic of the fcc structure (A) and the $X$-TA mode displaced structure (B), viewed along the cubic [001] direction. The distance between atoms is shown next to each bond. Black arrows centered on atoms show the direction of their displacement relative to the fcc structure.