Competing structural ordering tendencies in Heusler-type alloys with high Curie temperatures: ${\text{Fe}}_2{\text{CoGa}}_{1-x}{\text{Zn}}_x$ studied by first-principles calculations

The influence of Zn substitution on the structural, magnetic, and electronic properties and lattice vibrations of ferromagnetic ${\text{Fe}}_2{\text{CoGa}}_{1-x}{\text{Zn}}_x$ alloys in the conventional $X_{2}YZ$ and inverse $\left(XY\right)XZ$ Heusler structures is investigated, by means of ab initio and Monte Carlo calculations, which predict strong ferromagnetic coupling and high Curie temperatures between 770 and 925 K. In the Ga-rich systems the inverse Heusler structure is energetically favored but no indication for a structural instability is found in contrast to Fe-Co-Ga-Zn alloys in the conventional Heusler structure. The origin of the remarkably strong preference of the cubic $\left(c/a=1\right)$ inverse phase is believed to originate from the bcc-like environment of the two inequivalent Fe atoms and their stronger hybridization with the Co states compared to the conventional structure. In the quaternary compounds, ${\text{Fe}}_2{\text{CoGa}}_{1-x}{\text{Zn}}_x$, substitution of Ga by Zn reduces the energetic preference of the inverse structure caused by weakening of the Co-Fe hybridization. Simultaneously, Zn leads to higher magnetic moments and Curie temperatures because of localization effects. In addition, since Zn weakens the miscibility of the alloys, we propose that the composition of ${\text{Fe}}_2{\text{CoGa}}_{1-x}{\text{Zn}}_x$ alloys has to be carefully chosen in order to yield an interesting future ferromagnetic shape-memory alloy system.

Figure 1

(Color online) (a) The cubic unit cell of ${\text{Fe}}_2\text{CoGa}$ in the conventional Heusler structure (denoted by $X_{2}YZ$) in which iron atoms ${\text{Fe}}_{\text{A}}$ and ${\text{Fe}}_{\text{B}}$ (light spheres) occupy the $X$ sites and Co (dark spheres) and Ga atoms (medium gray spheres) share one cubic sublattice and occupy $Y$ and $Z$ sites, respectively. (b) Inverse Heusler [denoted by $\left(XY\right)XZ$] structure in which the Co and one Fe atom exchange their sites. One of the two $X$ sites is now occupied by Co. The interchanged Fe atom is marked as ${\text{Fe}}_{\text{C}}$. The remaining half of the Fe atoms stays at their original sites.

Figure 2

(Color online) Total energy $E-E_0$ (lower panel) with respect to the energy of the cubic inverse structure for each alloy system and total magnetic moment $M$ per f.u. (upper panel) of ${\text{Fe}}_2\text{CoGa}$ (diamonds), ${\text{Fe}}_2{\text{CoGa}}_{0.75}{\text{Zn}}_{0.25}$ (circles), ${\text{Fe}}_2{\text{CoZn}}_{0.75}{\text{Ga}}_{0.25}$ (triangles), and ${\text{Fe}}_2\text{CoZn}$ (squares) for both crystal structure types as a function of the tetragonal distortion, $c/a$. Open and filled symbols distinguish between the conventional and inverse Heusler structures, respectively. The conventional Heusler structures are characterized by two energy minima while the inverse structures show only one energy minimum in the cubic state $\left(c/a=1\right)$. With increasing Zn concentration the energy of the conventional Heusler structure is considerably lowered approaching the energy curve of the corresponding inverse structure while the magnetic moments increase.

Figure 3

(Color online) Magnetic-exchange parameters $J_{ij}$ as a function of the distance $d$ between the atoms in units of the equilibrium lattice constant $a_0$ for the cubic cases $\left(c/a=1\right)$ and in units of $a$ for the tetragonal structures for (a) $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$ $\left(c/a=1\right)$, (b) $\text{L}{1}_0$ ${\text{Fe}}_2\text{CoGa}$ $\left(c/a=1.46\right)$, (c) inverse $\text{F}\overline{4}3\text{m}$ (FeCo)FeGa $\left(c/a=1\right)$, (d) $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoZn}$ $\left(c/a=1\right)$, (e) $\text{L}{1}_0$ ${\text{Fe}}_2\text{CoZn}$ $\left(c/a=1.40\right)$, and (f) inverse $\text{F}\overline{4}3\text{m}$ (FeCo)FeZn $\left(c/a=1\right)$. The distance is given with respect to the first atom in the figure legend. For the ${\text{Fe}}_x{\text{-Fe}}_y$ interaction $\left(x,y=\text{A},\text{B},\text{C}\right)$ there is always an ${\text{Fe}}_x$ atom at the origin, for the Co-Co interactions a Co-atom occupies the origin. Squares (blue) mark the interaction between two Co atoms, stars (green) define the ${\text{Fe}}_{\text{A}}{\text{-Fe}}_{\text{A}}$ interactions, filled rightward triangles (magenta) belong to the ${\text{Fe}}_{\text{A}}{\text{-Fe}}_{\text{B}}$ interactions, and open circles (black) mark the interactions between ${\text{Fe}}_{\text{A}}$ and Co. Open upward triangles (magenta) indicate the ${\text{Fe}}_{\text{C}}{\text{-Fe}}_{\text{C}}$ interaction, filled diamonds (indigo) show the ${\text{Fe}}_{\text{C}}\text{-Co}$ interactions, and downward triangles (orange) the ${\text{Fe}}_{\text{C}}{\text{-Fe}}_{\text{A}}$ interactions. The strongest interactions are ferromagnetic and appear between Fe atoms and Fe and Co atoms.

Figure 4

(Color online) The influence of addition of Zn to ${\text{Fe}}_2\text{CoGa}$ in the conventional tetragonally distorted Heusler structure $\left(\text{L}{1}_0\right)$ on the Curie temperature $T_{\text{C}}$ (left axis, lower panel), the energy difference between the cubic phase at $c/a=1$ and the absolute energy minimum at $c/a>1$, $\Delta E_{c/a}$ (right axis, lower panel), the mixing energy, $E_{\text{mix}}$ (left axis, upper panel), and the energy difference between the absolute energy minima of the conventional and the inverse phases, $E_{\text{order}}$ (right axis, upper panel), as a function of the valence-electron concentration $e/a$. All functional behavior depends linearly on $e/a$.

Figure 5

(Color) Phonon-dispersion relations of the inverse cubic Heusler compounds (FeCo)FeGa (left) and (FeCo)FeZn (right). For the inverse structure involving zinc, a tendency for phonon softening is observed regarding the ${\text{TA}}_2$ mode in [110] direction but no imaginary phonons are apparent, contrary to the conventional cubic systems ${\text{Fe}}_2\text{Co}\left(\text{Ga},\text{Zn}\right)$ (not shown). For the latter compounds the negative curvature of $E\left(c/a\right)$ at $c/a=1$ causes instabilities in the [110]-${\text{TA}}_2$ phonon branch.

Figure 6

(Color online) The two figures on the left show the total and element-specific DOS of the cubic $\text{L}{2}_1$ phases of (a) ${\text{Fe}}_2\text{CoGa}$ and (b) ${\text{Fe}}_2\text{CoZn}$ while the two figures on the right show the corresponding total and element-specific DOS of the tetragonal $\text{L}{1}_0$ phases of (c) ${\text{Fe}}_2\text{CoGa}$ and (d) ${\text{Fe}}_2\text{CoZn}$. The high peak at the Fermi level in the spin-down channel in case of ${\text{Fe}}_2\text{CoGa}$ is of $\text{Co}{e}_g$ origin and mainly responsible for the instability of the ordered conventional phase and may also favor the tetragonal distortion (band-Jahn-Teller effect) thereby lowering the energy of the $\text{L}{1}_0$ phase with respect to the $\text{L}{2}_1$ phase. For ${\text{Fe}}_2\text{CoZn}$ a peak of mainly $\text{Fe}{e}_g$ and $\text{Fe}{t}_{2g}$ lies right below $E_{\text{F}}$ (as in case of $\text{L}{2}_1$ ${\text{Ni}}_2\text{MnGa}$) which in this case is responsible for the instability. Note that for the $\text{L}{1}_0$ phase of both compounds the gaplike region of the cubic phase below $E_{\text{F}}$ is now filled with states of Co and Fe. The thicker solid (black) lines show the total DOS, thinner (blue) lines belong to Fe (for $\text{L}{1}_0$ this is the sum of contributions from the two inequivalent atoms ${\text{Fe}}_{\text{A}}$ and ${\text{Fe}}_{\text{C}}$), dashed (red) lines mark the Co states, dashed-dotted (green) lines represent Ga and Zn (maroon), respectively. For better visibility the DOS of Ga and Zn is enhanced by a factor of 5.

Figure 7

(Color online) The DOS of the inverse structures for the cubic case for (a) (FeCo)FeGa and (b) (FeCo)FeZn. The individual contributions of both inequivalent Fe are shown in blue and violet. There are two pseudogap structures below $\left(-0.8\text{eV}\right)$ and above (0.5 eV) the Fermi level. The DOS of the minority-spin appears rather compressed and the peak structure at $E_{\text{F}}$ has disappeared compared with the conventional case.

Figure 8

(Color online) The $e_g$ and $t_{2g}$ decomposed DOS of conventional (a) ${\text{Fe}}_2\text{CoGa}$ and (b) ${\text{Fe}}_2\text{CoZn}$ showing the peak structure near the Fermi level which is responsible for the martensitic instability in both systems. The peak structure is less pronounced in case of ${\text{Fe}}_2\text{CoZn}$. Note also the extra spectral weight of Fe at energies above $E_{\text{F}}$ and the extra spectral weight of Co at 2 eV below $E_{\text{F}}$ in the minority-spin states which bears features of covalent magnetism as in CoFe, see Ref. 18. Figs. 8c and 8d show the $e_g$ and $t_{2g}$ decomposed DOS of the corresponding inverse Heusler compounds. The peak structure at $E_{\text{F}}$ has vanished and the DOS is rather different compared with the conventional structures.

Figure 9

(Color online) (a) Majority-spin density of states of $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$ as obtained by self-consistent ab initio calculations showing that the DOS per atom of Fe and Co are very similar. (b) Minority-spin DOS of the rigid band picture of the Stoner model in case of $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$ compared with (c) the actual minority-spin DOS of $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$. The influence of covalent interaction (hybridization) in the minority-spin channel in case of (c) is clearly different from the rigid-bandlike picture in (b) and is the cause of covalent magnetism similar to the case of FeCo discussed in Ref. 18. We have, however, to keep in mind that the different crystal fields acting on Fe (octahedral) and on Co (tetrahedral) have also influence on the detailed shape of the minority-spin DOS of $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$.

Figure 10

(Color) Calculated spin moments per f.u. (highlighted in red for Ga-based and orange for Zn-based alloys) for the cubic conventional (open symbols) and the cubic inverse structures (filled symbols) as a function of the total number of valence electrons per f.u. $Z$ in comparison to data found in literature. Please note that $Z$ is always given for the Heusler structure, thus $Z$ of Co in bcc structure is here $4\times 9=36$. The half-metallic Heusler alloys (open, black diamonds) and our Fe-Co-Ga-Zn systems follow linear increasing functions indicating a fixed number of minority-spin states or reflecting an increasing percentage of Fe atoms in the compound, respectively. The linear decreasing slopes of the Ni-Mn-(Ga,Sn,Sb) systems originate from a constant number of majority spins or from increasing antiferromagnetic tendencies with increasing Mn concentration. The data of the stoichiometric ${\text{Fe}}_2\text{Co}\left(\text{Ga},\text{Zn}\right)$ systems are perfectly in line with the results for Ni-Mn-Ga. This and the overall linear character of the spin moments imply that despite hybridization effects the magnetic moments possess localized character.

Figure 11

(Color online) Total energy of a few previously unconsidered alloys relative to the energy of the respective cubic inverse phase. As in Fig. 2, open and filled symbols distinguish between conventional and inverse structures, respectively. For comparison the corresponding curves of all alloys discussed above in the inverse structure and of $\text{L}{2}_1$ ${\text{Fe}}_2\text{CoGa}$ (from Fig. 2) are shown by gray symbols. In addition, results for two representative alloys, ${\text{Fe}}_7{\text{Co}}_4{\text{Zn}}_5$ (black squares) and ${\text{Fe}}_2\text{CoCu}$ (red, rightward triangles) are also shown. For the latter two systems, clear martensitic tendencies are observed.