Spin Orientation and Magnetostriction of ${\mathrm{Tb}}_{1-x}{\mathrm{Dy}}_x{\mathrm{Fe}}_2$ from First Principles

The optimal amount of dysprosium in the highly magnetostrictive rare-earth compounds ${\mathrm{Tb}}_{1-x}{\mathrm{Dy}}_x{\mathrm{Fe}}_2$ for room-temperature applications has long been known to be $x=0.73$ (Terfenol-D). Here, we derive this value from first principles by calculating the easy magnetization direction and magnetostriction as a function of composition and temperature. We use crystal-field coefficients obtained within density-functional theory to construct phenomenological anisotropy and magnetoelastic constants. The temperature dependence of these constants is obtained from disordered-local-moment calculations of the rare-earth magnetic order parameter. Our calculations find the critical $\mathrm{Dy}$ concentration required to switch the magnetization direction at room temperature to be $x_c=0.78$, with magnetostrictions ${\lambda }_{111}=2700$ and ${\lambda }_{100}=-430$ ppm, close to the Terfenol-D values.

Figure 1

The local $T_d$ environment of the RE atom in the cubic Laves phase, showing nearest-neighbor ($\mathrm{Fe}$, gray) and next-nearest-neighbor (RE, purple) atoms. The RE—RE bonds are oriented along the $⟨111⟩$ directions.

Figure 2

The change in the CF coefficients $\mathrm{\Delta }{B}_{lm}$ for ${\mathrm{Tb}\mathrm{Fe}}_2$ for different $(l,m)$ with a shear strain ${\epsilon }_S$ (blue) or a tetragonal strain ${\epsilon }_T$ (red) applied. The inset shows $\mathrm{\Delta }{B}_{lm}$ for a larger variation in ${\epsilon }_S$. The straight lines are fits to the calculations.

Figure 3

The easy direction of magnetization of ${\mathrm{Tb}}_{1-x}{\mathrm{Dy}}_x{\mathrm{Fe}}_2$, calculated by minimizing $E(\hat{\mathbf{n}},\mathit{\epsilon },x,T)$ (red- and blue-shaded regions). The symbols are experimental measurements of the easy direction using Mössbauer spectroscopy [20, 56]. The dotted lines mark the boundaries between different magnetization directions extracted from torque magnetometry [57], where above 150 K the boundary is between [111] and [100] and below encloses a region of intermediate magnetization direction. The dashed line is the [111]-[100] boundary obtained by minimizing $E_{\mathrm{MCA}}$ only.

Figure 4

The spin-orientation diagram of ${\mathrm{Tb}}_{1-x}{\mathrm{Dy}}_x{\mathrm{Fe}}_2$ calculated with different sets of elastic constants. The same experimental data are shown as in Fig. 3. The diagonal lines represent the boundaries between [111] and [100] directions of magnetization for the different sets of elastic constants listed in Table 2.