Ni-based Heusler compounds: How to tune the magnetocrystalline anisotropy

Tailoring and controlling magnetic properties is an important factor for materials design. Here, we present a case study for Ni-based Heusler compounds of the type ${\mathrm{Ni}}_{2}YZ$ with $Y=$ Mn, Fe, Co and $Z=$ B, Al, Ga, In, Si, Ge, Sn based on first-principles electronic structure calculations. These compounds are interesting since the materials properties can be quite easily tuned by composition and many of them possess a noncubic ground state being a prerequisite for a finite magnetocrystalline anisotropy (MAE). We discuss systematically the influence of doping at the $Y$ and $Z$ sublattices as well as the effect of lattice deformation on the MAE. We show that in case of ${\mathrm{Ni}}_2\mathrm{Co}Z$ the phase stability and the MAE can be improved using quaternary systems with elements from main group III and IV on the $Z$ sublattice whereas changing the $Y$ sublattice occupation by adding Fe does not lead to an increase of the MAE. Furthermore, we studied the influence of the lattice ratio on the MAE. Showing that small deviations can lead to a doubling of the MAE as in case of ${\mathrm{Ni}}_2\mathrm{FeGe}$. Even though we demonstrate this for a limited set of systems, the findings may carry over to other related systems.

Figure 1

Computational design of magnetic materials: Basic properties which can be obtained from first-principles calculations $(T=0$ K) and temperature-dependent quantities which can be determined from additional atomistic modeling.

Figure 2

(a) Unit cell of the cubic $\mathrm{L}{2}_1$ Heusler structure and the corresponding primitive cell (b). In ordinary Heusler alloys with space group 225, the sublattices A and C are occupied by material $X$, B hosts the $Y$ metal, and the main group element $Z$ is located on the D sublattice. For the inverse Heusler structure $\left({\mathrm{Hg}}_2\mathrm{CuTi}\right)$, the occupation of the sublattices B and C is inverted.

Figure 3

Energy as function of the $c/a$ ratio for ${\mathrm{Ni}}_{2}YZ$ Heusler alloys for $Z$ elements with three (B, Al, Ga, In) and four (Si, Ge, Sn) valence electrons. Filled symbols denote normal Heusler structure $X_{2}YZ$. Systems which crystallize in the inverse ${\mathrm{Hg}}_2\mathrm{CuTi}$ structure are marked by open symbols. Please note the different scales on the $y$ axis for $Y=$ Mn (a). The inset in (b) shows the $E(c/a)$ for $\mathrm{L}{2}_1$ and inverse-ordered ${\mathrm{Ni}}_2\mathrm{FeGe}$ in the vicinity of the global minimum.

Figure 4

(a) Calculated energy difference between the $\mathrm{L}{2}_1$ and the inverse-ordered phase for ${\mathrm{Ni}}_2\mathrm{Fe}Z$ (red squares) and ${\mathrm{Ni}}_2\mathrm{Co}Z$ (blue triangles) compounds. For the Mn compounds (brown circles) only the values for the systems with noncubic ground state are shown. (b) Energy of formation for ${\mathrm{Ni}}_{2}YZ$ Heusler compounds in with $Y=$ Fe (red squares) and Co (blue triangles) as obtained from Eq. (1). Negative values denote systems stable against decomposition. The formation energy of ${\mathrm{Pd}}_2\mathrm{FeGe}$ and ${\left(\mathrm{NiPd}\right)}_2\mathrm{FeGe}$ is given by hatched squares with vertical and diagonal stripes, respectively. Lines should only be viewed as guide to the eye.

Figure 5

Dependence of the magnetocrystalline anisotropy on the lattice ratio $c/a$ for ${\mathrm{Ni}}_2\mathrm{FeGe}$ (red diamonds) and ${\mathrm{Ni}}_2\mathrm{CoGa}$ (blue triangles) as obtained from rspt. In case of ${\mathrm{Ni}}_2\mathrm{FeGe}$, no sign change, i.e., rotation of the MAE from easy axis to planar is observed depending on the $c/a$ ratio, whereas ${\mathrm{Ni}}_2\mathrm{CoGa}$ undergoes the typical change from planar (negative values) to uniaxial when the $c/a$ changes from $c/a>1$ to $c/a<1$.

Figure 6

Calculated orbital resolved density of states of ${\mathrm{Ni}}_2\mathrm{FeGe}$ (a)–(f) and ${\mathrm{Ni}}_2\mathrm{CoGa}$ (g)–(h). The upper row shows the Ni $d$ states of the systems, whereas the Fe and Co $d$ states are given in the bottom row. In (a) and (b), the DOS of the ground-state configuration of ${\mathrm{Ni}}_2\mathrm{FeGe}\left(c/a=1.35\right)$ is given, (c) and (d) denote the DOS of the system with the highest MAE $\left(c/a=1.25\right)$. The DOS for the ground state of the inverse-ordered $({\mathrm{Hg}}_2\mathrm{CuTi}$ structure) system ${\mathrm{Ni}}_2\mathrm{FeGe}\left(c/a_i=1.35\right)$ is given in (e) and (f). For ${\mathrm{Ni}}_2\mathrm{CoGa}$, only the ground-state configuration with $c/a=1.38$ is given [see (g) and (h)].

Figure 7

Calculated magnetic moments per formula unit (a) and magnetocrystalline anisotropy (b) depending on the impurity concentration, i.e., the valence electron number $e/a$ for ${\mathrm{Ni}}_2{\mathrm{CoIn}}_{1-x}{Z}_x$ with $Z=$ Ge (squares) and Si (circles). The inset at the top left in (b) gives the formation energy and stabilization of the system with increasing Ge and Si content, respectively. The inset on the bottom of the right-hand side shows the supercell (16 atoms) used for the quaternary systems, here 50% of In have been substituted by Ge. The hatched circle corresponds to the MAE of the local minima of ${\mathrm{Ni}}_2{\mathrm{CoIn}}_{0.5}{\mathrm{Si}}_{0.5}\left(c/a=1.2\right)$.

Figure 8

Total magnetic moments of the ground state. The value of the cubic $\left(c/a=1.0\right)$ structure is given in brackets. All values are in ${\mu }_{\mathrm{B}}/\mathrm{f}.\mathrm{u}.$ The dashed line marks the 34-$N_v$ rule for shape memory alloys as suggested by Dannenberg et al. [45] whereas the magnetic moment for the inverse-ordered systems with the $Z$ element being from group IV follows shows a 35-$N_v$ behavior (dashed-dotted line). Inverse-ordered systems including the quaternary ones (hatched symbols) follow the dashed line since their moment is reduced due to the reversed NN positions.