Temperature dependence of magnetic anisotropy: An ab initio approach

We present a first-principles theory of the variation of magnetic anisotropy, $K$, with temperature, $T$, in metallic ferromagnets. It is based on relativistic electronic structure theory and calculation of magnetic torque. Thermally induced local moment magnetic fluctuations are described within the relativistic generalization of the disordered local moment theory from which the $T$ dependence of the magnetization, $m$, is found. We apply the theory to a uniaxial magnetic material with tetragonal crystal symmetry, $L{1}_0$-ordered FePd, and find its uniaxial $K$ consistent with a magnetic easy axis perpendicular to the $\mathrm{Fe}∕\mathrm{Pd}$ layers for all $m$ and proportional to $m^2$ for a broad range of values of $m$. This is the same trend that we have previously found in $L{1}_0$-ordered FePt and which agrees with experiment. We also study a magnetically soft cubic magnet, the ${\mathrm{Fe}}_{50}{\mathrm{Pt}}_{50}$ solid solution, and find that its small magnetic anisotropy constant $K_1$ rapidly diminishes from $8\mu \mathrm{eV}$ to zero. $K_1$ evolves from being proportional to $m^7$ at low $T$ to $m^4$ near the Curie temperature. The accounts of both the tetragonal and cubic itinerant electron magnets differ from those extracted from single ion anisotropy models and instead receive clear interpretations in terms of two ion anisotropic exchange.

Figure 1

(Color online) The magnetization of FePt versus temperature. The filled circles refer to a magnetization along $\widehat{n}=(1,0,1)$. $T_c$ is at $1105\mathrm{K}$ with the easy axis, (0,0,1). The full line shows the mean field approximation to a classical Heisenberg model for comparison.

Figure 2

(Color online) The magnetic anisotropy of FePd as a function of the square of magnetization. The filled circles show the calculations from the ab initio theory, the full line $K_0{[m\left(T\right)∕m\left(0\right)]}^2$, and the dashed line the single-ion model function $K_0{⟨g_2\left(\widehat{e}\right)⟩}_T∕{⟨g_2\left(\widehat{e}\right)⟩}_0$ with $K_0=-0.335\mathrm{meV}$.

Figure 3

(Color online) The magnetic anisotropy constants $K_2$, $K_4$ of FePd as a function of the square of magnetization. The filled circles and full line show the calculations from the ab initio theory of the sum, the dashed line with filled diamonds describes $K_2$ and the dotted line with squares shows $K_4$.

Figure 4

(Color online) The magnetic anisotropy energy $K$ calculated in a mean field approximation for a model of a uniaxial magnet which has both anisotropic exchange and single ion anisotropy. The full line shows results with single ion anisotropy only, $k=0.002J^{\perp }$ and $J^{\perp }-J^{\Vert }=0$. The dashed line shows results for the same $k$ and $J^{\perp }-J^{\Vert }=0.01J^{\perp }$.

Figure 5

(Color online) The magnetic anisotropy constant $K_1$ of the cubic magnet ${\mathrm{Fe}}_{50}{\mathrm{Pt}}_{50}$ as a function of the fourth power of the magnetization, $m^4$. The filled circles show the calculations from the ab initio theory, the dashed line from the single-ion anisotropy model $k{\sum }_i(e_{x,i}^2{e}_{y,i}^2+e_{y,i}^2{e}_{z,i}^2+e_{z,i}^2{e}_{y,i}^2)$ and the dotted-dashed line from the anisotropic exchange two ion model $\frac{1}{2}\Delta J{\sum }_{i,j}(e_{x,i}^2{e}_{y,j}^2+e_{y,i}^2{e}_{z,j}^2+e_{z,i}^2{e}_{x,j}^2)$ with $k=\Delta J=8.4\mu \mathrm{eV}$.