Spin-Fluctuation Mechanism of Anomalous Temperature Dependence of Magnetocrystalline Anisotropy in Itinerant Magnets

The origins of the anomalous temperature dependence of magnetocrystalline anisotropy in $({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2\mathrm{B}$ alloys are elucidated using first-principles calculations within the disordered local moment model. Excellent agreement with experimental data is obtained. The anomalies are associated with the changes in band occupations due to Stoner-like band shifts and with the selective suppression of spin-orbit “hot spots” by thermal spin fluctuations. Under certain conditions, the anisotropy can increase, rather than decrease, with decreasing magnetization due to these peculiar electronic mechanisms, which contrast starkly with those assumed in existing models.

Figure 1

Calculated temperature dependencies of MCA energy $K$ in $({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2\mathrm{B}$ and $({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}{)}_2\mathrm{B}$ alloys.

Figure 2

Partial minority-spin spectral function for the transition-metal site in $({\mathrm{Fe}}_{0.05}{\mathrm{Co}}_{0.95}{)}_2\mathrm{B}$ at (a) $T=0$, and (b) $T/T_C=0.7$. SOC is included, $\mathbf{M}\parallel z$, and the energy is in eV. Color encodes the orbital character of the states. The intensities of the red, blue, and green color channels are proportional to the sum of $m=\pm 2$ ($xy$ and $x^2-y^2$), sum of $m=\pm 1$ ($xz$ and $yz$), and $m=0$ ($z^2$) character, respectively.

Figure 3

MCA in $({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2\mathrm{B}$ and $({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x{)}_2\mathrm{B}$ alloys calculated within VCA (empty circles) compared with CPA (filled circles). The spin decomposition is given for VCA.

Figure 4

Spectral functions at the $\mathrm{\Gamma }$ point at (a) $x=1$, (b) $x=0.9$, (c) $x=0.8$. Solid lines: $\mathbf{M}\parallel z$. Dashed lines: $\mathbf{M}\parallel x$. A small imaginary part is added to energy to resolve the bands in panel (a).

Figure 5

Contributions to $K$ in (a) ${\mathrm{Fe}}_2\mathrm{B}$ and (b) $({\mathrm{Fe}}_{0.05}{\mathrm{Co}}_{0.95}{)}_2\mathrm{B}$ from different spins ($K_{\uparrow }$ and $K_{\downarrow }$). $K_{\sigma }^{+}$ and $K_{\sigma }^{-}$ in panel (b): total positive and negative contributions to $K_{\sigma }$ coming from different $\mathbf{k}$ points. (Dotted lines show $K_{\uparrow }^{+}$, $K_{\uparrow }^{-}$.)

Figure 6

Wave vector-resolved $K_{\downarrow }$ (units of $\mathrm{meV}{a}_0^3$, where $a_0$ is the Bohr radius) on the $\mathrm{\Gamma }MX$ plane in the $({\mathrm{Fe}}_{0.05}{\mathrm{Co}}_{0.95}{)}_2\mathrm{B}$ alloy at $T=0$ (upper left) and $T/T_C=0.7$ (lower right).