Systematic study of magnetodynamic properties at finite temperatures in doped permalloy from first-principles calculations

By means of first-principles calculations, we have systematically investigated how the magnetodynamic properties Gilbert damping, magnetization, and exchange stiffness are affected when permalloy (Py) (${\mathrm{Fe}}_{0.19}{\mathrm{Ni}}_{0.81}$) is doped with $4d$ or $5d$ transition metal impurities. We find that the trends in the Gilbert damping can be understood from relatively few basic parameters such as the density of states at the Fermi level, the spin-orbit coupling, and the impurity concentration. The temperature dependence of the Gilbert damping is found to be very weak which we relate to the lack of intraband transitions in alloys. Doping with $4d$ elements has no major impact on the studied Gilbert damping, apart from diluting the host. However, the $5d$ elements have a profound effect on the damping and allow it to be tuned over a large interval while maintaining the magnetization and exchange stiffness. As regards the spin stiffness, doping with early transition metals results in considerable softening, whereas late transition metals have a minor impact. Our result agree well with earlier calculations where available. In comparison to experiments, the computed Gilbert damping appears slightly underestimated, whereas the spin stiffness shows a general good agreement.

Figure 1

Calculated equilibrium volumes of Py-M, where M stands for a $4d$ (left) or $5d$ (right) transition metal. Values for $10\%$ and $15\%$ doping concentrations are shown. Reference value of pure Py is displayed with a dashed line.

Figure 2

(Upper) Total magnetic moment (spin and orbital) for different impurities and concentrations. Reference value for pure Py marked with a dashed line. (Lower) Local impurity magnetic moment for ${\mathrm{Py}}_{0.95}{\mathrm{M}}_{0.05}$.

Figure 3

Bloch spectral function $A(E,\mathbf{k})$ of Py (upper panel) and Py doped with $20\%$ Pt impurities (lower panel). The Fermi level is indicated with a horizontal black line at zero energy.

Figure 4

(Upper) Calculated Gilbert damping parameter for Py+M in different concentrations of $4d$ and $5d$ transition metal M at low temperatures ($T=10\text{K}$). Experimental results from Ref. [5] measured at room temperature are displayed by solid squares and dashed lines indicate the reference value for pure Py. (Lower) Total (blue) and impurity (black) density of states at the Fermi level $E_F$ for 10% impurities in Py.

Figure 5

Upper: the spin-orbit parameter of $d$ electrons of the impurity atoms. Lower: qualitative comparison between our calculations and the torque correlation (TC) model for the damping with 10% impurity concentration.

Figure 6

Calculated Gilbert damping as a function of Os, Ag and vacancy (Vac) concentration in Py. Open red circle: calculation from Ref. [19]; solid red circle: calculation from Ref. [20]; red solid square: experimental data from Ref. [5].

Figure 7

Gilbert damping parameter including temperature effects from both atomic displacements and spin fluctuations (upper panel). The effect of spin fluctuations on the Gilbert damping (lower panel); see text in details.

Figure 8

Spin-wave stiffness $D$ of Py-M in the ground state (top) and exchange stiffness constant $A$ at room temperature $T=300\text{K}$ (bottom) as a function of doping concentration. The horizontal dashed and solid lines show the reference values of pure Py from calculation and experiments, respectively. The scattered dots indicate the experimental data for Py+15%M (Ag/Pt/Au) from Ref. [9].