Ab initio calculations of temperature-dependent magnetostriction of Fe and $A2{\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ within the disordered local moment picture

The fully relativistic disordered local moment (DLM) theory is used to perform calculations of the magnetic torque of tetragonally distorted Fe and fully disordered $\left(A2\right){\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x\left(0\le x\le 0.2\right)$ alloys to describe the temperature dependence of their magnetoelasticity. The finite-temperature magnetoelasticity, in particular the magnetoelastic constant $B_1$, is obtained within DLM theory by studying the response of the magnetic torque generated by the magnetocrystalline anisotropy to the application of a tetragonal strain. Calculations of $B_1$ have been performed on bcc Fe across its ferromagnetic temperature range. Our results show good qualitative agreement with experiment, in particular reproducing the anomalous, nonmonotonic thermal behavior of bcc Fe's magnetostriction, which has been largely unexplained for more than 50 years. The method has also been used to calculate the finite-temperature magnetoelasticity of the $A2$ phase of ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ alloys as a starting point for further investigations into the giant magnetostriction of Galfenol alloys. Our calculations show that homogeneously doping bcc Fe with Ga does not produce an enhancement in magnetostriction and that the nonmonotonic temperature dependence and significant volume dependence are suppressed by increasing Ga content.

Figure 1

Torques $T_{\theta }$ calculated for bcc Fe magnetized along the direction $\widehat{\mathit{n}}=(1,0,1)/\sqrt{2}$ with a strain applied along the [0 0 1] direction, for different magnetic order parameters $m$.

Figure 2

Top: The variation in the (negative) magnetoelastic constant $B_1$ with respect to reduced magnetization $m$ for lattice parameters between 5.20 a.u. and 5.50 a.u. in bcc Fe. Bottom: The experimentally measured magnetoelastic constant $B_1$ of bcc Fe, extracted from magnetostriction [13, 17] and elastic constant [62, 63] data and plotted in terms of the reduced magnetization $m=M\left(T\right)/M\left(0\right)$ (cf. Appendix pp1). Blue triangles correspond to magnetostriction data from Ref. [17] and red circles from Ref. [13].

Figure 3

The magnetoelastic constant $B_1$ calculated at the experimental lattice parameters taking thermal expansion into account as described in the text (red circles). The values of $B_1$ calculated at fixed lattice constants 5.40, 5.45, 5.50 a.u. (cf. Fig. 2) are also shown as gray symbols.

Figure 4

The high-temperature variation in magnetoelastic constant $B_1$ with respect to reduced magnetization for lattice parameters between 5.20 a.u. and 5.50 a.u. for bcc Fe, plotted on a logarithmic scale and fitted to a power law $A{x}^{\gamma }$.

Figure 5

The density of states for the majority and minority spin channels (positive/negative scales, respectively) in bcc Fe for $a=5.30$ (red, solid), 5.40 (blue, dashed), and 5.50 a.u. (green, dotted), where zero energy corresponds to the Fermi energy.

Figure 6

The variation in zero-temperature magnetoelastic constant $B_1$ with respect to change in lattice parameter from $a=5.20$ a.u. (red line and axes) and shift in Fermi energy (blue line and axes). The arrows indicate which axes the data belong to.

Figure 7

The magnetoelastic constant $B_1$ calculated for fully disordered $\left(A2\right){\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x$ as a function of Ga concentration, for different lattice parameters.

Figure 8

The scalar-relativistic density of states projected onto the Fe atoms in $A2{\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x\left(x=0,0.1,0.2\right)$ for $a=5.30$ (red, solid), 5.40 (blue, dashed), and 5.50 a.u. (green, dotted). The energy zero corresponds to the Fermi energy.

Figure 9

The variation in $B_1$ with respect to reduced magnetization $m\left(T\right)=M\left(T\right)/M\left(0\right)$ for lattice parameters between 5.20 a.u. and 5.55 a.u. in ${\mathrm{Fe}}_{1-x}{\mathrm{Ga}}_x\left(x=0.05,0.10,0.15,0.20\right)$. The black, dashed curve on the $x=0.2$ graph is a plot of $B_1\left(T\right)=B_1(T=0)m^3$, as predicted by single-ion theory. The inset in the bottom right plot is a log-log plot of $B_1$ against $m$ for $x=0.20$ at 5.50 a.u., demonstrating the $m^{2.2}$ dependence at low temperatures.

Figure 10

Experimentally measured values of (a) reduced magnetization (Ref. [64]), (b) elastic constants (Ref. [62] for 0–300 K and Ref. [63] for $>300\mathrm{K})$, and (c) magnetostriction ${\lambda }_{001}$ (Ref. [13], red circles; Ref. [17], blue triangles). The line connecting the magnetization data in (a) is the function introduced in Ref. [65] as described in the text. The magnetoelastic constants calculated using Eq. (4), the elastic constants, and the two magnetostriction data sets are shown in (d).

Figure 11

Experimentally measured lattice constants of bcc Fe reported in Ref. [61] as a function of (a) temperature and (b) reduced magnetization.