Ab initio calculations of the phase behavior and subsequent magnetostriction of ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ within the disordered local moment picture

A holistic approach for studying both the nature of atomic order and finite-temperature magnetostrictive behavior in the binary alloy Galfenol (${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x, 0\le x\le 0.25$) is presented. The phase behavior is studied via atomistic modeling with inputs from ab initio calculations, and the ordered phases of interest at nonstoichiometric concentrations are verified to exhibit $B2$- and ${D0}_3$-like order. The finite-temperature magnetoelasticity of these phases, in particular the magnetoelastic constant $B_1$, is obtained within the same ab initio framework using disordered local moment theory. Our results provide an explanation for the origin of the experimentally observed peak and subsequent fall in the material's magnetostriction at $x\sim 0.19$, which has been disputed. In addition, we show that it is possible to enhance the magnetostriction of ${D0}_3-{\mathrm{Fe}}_3\mathrm{Ga}$ by removing a small fraction of electrons from the system, suggesting that a Fe-Ga-Cu or Fe-Ga-Zn alloy could exhibit greater magnetostrictive properties than Galfenol.

Figure 1

Diagrams of the nonstoichiometric $A2, B2$-like, and ${D0}_3$-like phases of ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$. Dashed lines denote unit vectors. In the ${D0}_3$ case, we have opted to show only one of the two 8c sites.

Figure 2

Computed site occupancies at 1200 and 800 K. The three dashed lines are indicative of what site occupancies would be in a pure $A2$ phase, what 4a and 4b sites would be in a pure $B2$-like phase, and what 4a site occupancies would be in a pure ${D0}_3$-like phase. At low concentrations of gallium, the system is in an $A2$ state, with the Ga spread homogeneously across the lattice. At 1200 K the system first transitions to $B2$-like order before further transitioning to a mixed $B2-D{0}_3$ state at approximately 14% Ga. At 800 K the transition is much sharper, and the system transitions to an almost pure ${D0}_3$-like state at approximately 8% Ga.

Figure 3

Average radial Ga-Ga densities for Galfenol obtained via Monte Carlo simulations. The interchange parameters used were those computed for the ferromagnetic state, and the simulation temperature was 800 K. The black dashed lines indicate the expected radial density for a completely disordered ($A2$) state, while the red crosses indicate the expected peaks for ${D0}_3$-like order.

Figure 4

Total and site-resolved values of the magnetoelastic constant $-B_1$ as a function of Ga content in nonstoichiometric $B2 {\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ for lattice parameters between 5.35 and 5.50 a.u. (a) $-B_1$ for the total system and experimental measurements at room temperature [5]; (b) site-resolved $-B_1$ at site 1a (see Fig. 1); (c) site 1b.

Figure 5

Total and site-resolved values of the magnetoelastic constant $-B_1$ as a function of Ga content in nonstoichiometric ${D0}_3$-like ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ for lattice parameters between 5.35 and 5.50 a.u. (a) $-B_1$ for the total system and experimental measurements at room temperature [5]; (b) combined contribution from sites 4a and 4b (see Fig. 1); (c) combined contribution of the two 8c sites.

Figure 6

Comparisons of the site-resolved magnetoelastic Ga concentration dependence at sites 1b in the $B2$-like phase and 8c in the ${D0}_3$-like phase, for different lattice parameters: (a) 5.35 a.u., (b) 5.40 a.u., (c) 5.45 a.u., and (d) 5.50 a.u.

Figure 7

The scalar-relativistic density of states of ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ in the $B2$-like phase ($x=0$, 0.15, and 0.25) at the 1b site, where zero energy is defined to be the Fermi energy. Also shown are the orbital-resolved density of states, separated into $e_g$ and $t_{2g}$ states.

Figure 8

The scalar-relativistic density of states of ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ in the ${D0}_3$-like phase ($x=0$, 0.15, and 0.25) at the 8c site in the ${D0}_3$ phase, where zero energy is defined to be the Fermi energy. Also shown are the orbital-resolved density of states, separated into $e_g$ and $t_{2g}$ states.

Figure 9

Calculated Fermi level dependence of the site-resolved MCA in tetragonally deformed (${\varepsilon }_{zz}=0.2\%$) ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ ($x=0$, 0.10, 0.15, 0.20, and 0.25) for the 1b site of the $B2$ phase (left) and the 8c site of the ${D0}_3$ phase (right).

Figure 10

The strain-induced change in the majority channel of the state-projected density of states $[\mathrm{\Delta }\text{DOS}=\text{DOS}({\varepsilon }_{zz}=0.2\%)-\text{DOS}({\varepsilon }_{zz}=0\%)]$ in ${\mathrm{Fe}}_{0.75}{\mathrm{Ga}}_{0.25}$. The top panel shows the 1b site of the partially ordered $B2$-like phase, and the bottom panel shows the 8c site of the ${D0}_3$-like phase.

Figure 11

$B_1$ as a function of magnetic order parameter $m$, calculated in the $B2$-like phase at Ga concentrations of 15%, 20%, and 25% (purple circles, green up-triangles, and blue down-triangles, respectively) and the ${D0}_3$-like phase at 20% and 25% (orange diamonds and yellow pentagons). The lower panel shows additional experimental measurements from Ref. [5], where an asterisk denotes the use of interpolation to determine experimental values of $c^{\prime }$ in the determination of $B_1$ (see the main text).