Magnetocrystalline anisotropy of Laves phase Fe$_2$Ta$_{1-x}$W$_x$ from first principles - the effect of 3d-5d hybridisation

Abstract: The magnetic properties of Fe$2$Ta and Fe$_2$W in the hexagonal Laves phase are computed using density functional theory in the generalised gradient approximation, with the full potential linearised augmented plane wave method. The alloy Fe$_2$Ta${1-x}$W$x$ is studied using the virtual crystal approximation to treat disorder. Fe$_2$Ta is found to be ferromagnetic with a saturation magnetization of $\mu_0 M\text{s} = 0.66~\mathrm{T}$ while, in contrast to earlier computational work, Fe$2$W is found to be ferrimagnetic with $\mu_0 M\text{s} = 0.35~\mathrm{T}$. The transition from the ferri- to the ferromagnetic state occurs for $x \leq 0.1$. The magnetocrystalline anisotropy energy (MAE) is calculated to $1.25~\mathrm{MJ/m^3}$ for Fe$2$Ta and $0.87~\mathrm{MJ/m^3}$ for Fe$_2$W. The MAE is found to be smaller for all values $x$ in Fe$_2$Ta${1-x}$W$_x$ than for the end compounds and it is negative (in-plane anisotropy) for $0.1 \leq x \leq 0.9$. The MAE is carefully analysed in terms of the electronic structure. Even though there are weak 5d contributions to the density of states at the Fermi energy in both end compounds, a reciprocal space analysis, using the magnetic force theorem, reveals that the MAE originates mainly from regions of the Brillouin zone with strong 3d-5d hybridisation near the Fermi energy. Perturbation theory and its applicability in relation to the MAE is discussed.