Origin of the Tetragonal Ground State of Heusler Compounds

We describe the general mechanism of tetragonal distortion in Heusler compounds $X_{2}YZ$. From 286 compounds studied using density-functional theory, 62% are found to be tetragonal at zero temperature. Such a large share of compounds with tetragonal distortions can be explained by the peak-and-valley character of the density of states (DOS) of these compounds in the cubic phase (arising from localized $d$ bands and van Hove singularities) in conjunction with a smooth shift of peaky DOS structure relative to the Fermi energy, $E_F$, when valence electrons are added to the system. A shift of the DOS in the $Y$ or $Z$ series leads to an alternation of stable and nonstable cubic phases that depends on the value of the DOS at $E_F$ in the cubic phase. Groups of compounds with a large share of tetragonal distortions are identified and explained.

Figure 1

(a) Regular and (c) inverse cubic Heusler structure. (b) Regular and (d) inverse tetragonal Heusler structure.

Figure 2

Number of (a) cubic and (b) tetragonal Heusler compounds in bins corresponding to different values of $\mathrm{DOS}(\mathrm{cub},E_F)$. Cyan corresponds to 286 compounds and blue corresponds to a subset of 220 stable compounds that have $|E_{ct}|\ge 0.1\mathrm{eV}$ (which includes cubic compounds that do not have a metastable tetragonal phase).

Figure 3

(a) Probability of tetragonal distortion calculated using the data presented in Fig. 2 for corresponding $\mathrm{DOS}(\mathrm{cub},E_F)$ bins, estimated from 286 compounds (cyan diamonds) and a subset of 220 stable compounds that have $|E_{ct}|\ge 0.1\mathrm{eV}$ (blue dots). DOS of bcc Fe (b), fcc Co (c), and fcc Ni (d).

Figure 4

$\mathrm{DOS}(\mathrm{tet},E_F)$ as a function of $\mathrm{DOS}(\mathrm{cub},E_F)$ for 191 Heusler compounds that have a tetragonal phase (see the text for details).

Figure 5

DOS for majority ($\mathrm{DOS}>0$) and minority ($\mathrm{DOS}<0$) electron bands of cubic (indicated by “cub,” left panels) and tetragonal (indicated by “tet,” right panels) inverse ${\mathrm{Mn}}_2\mathrm{YGa}$ compounds with $Y=\mathrm{Mn}$ (a,f), Fe (b,g), Co (c,h), Ni (d,i), and Cu (e). $E_{ct}$ values are shown in the right (tetragonal) panels. The black curves show the total DOS for each electron spin. Red, orange, and cyan curves show the partial DOS projected into the $d$ orbitals of tetrahedrally coordinated (by Ga) Mn, octahedrally coordinated (by Ga) Mn, and $Y$ atoms, respectively.