Microscopic Origin of Heisenberg and Non-Heisenberg Exchange Interactions in Ferromagnetic bcc Fe

By means of first principles calculations, we investigate the nature of exchange coupling in ferromagnetic bcc Fe on a microscopic level. Analyzing the basic electronic structure reveals a drastic difference between the $3d$ orbitals of $E_g$ and $T_{2g}$ symmetries. The latter ones define the shape of the Fermi surface, while the former ones form weakly interacting impurity levels. We demonstrate that, as a result of this, in Fe the $T_{2g}$ orbitals participate in exchange interactions, which are only weakly dependent on the configuration of the spin moments and thus can be classified as Heisenberg-like. These couplings are shown to be driven by Fermi surface nesting. In contrast, for the $E_g$ states, the Heisenberg picture breaks down since the corresponding contribution to the exchange interactions is shown to strongly depend on the reference state they are extracted from. Our analysis of the nearest-neighbor coupling indicates that the interactions among $E_g$ states are mainly proportional to the corresponding hopping integral and thus can be attributed to be of double-exchange origin. By making a comparison to other magnetic transition metals, we put the results of bcc Fe into context and argue that iron has a unique behavior when it comes to magnetic exchange interactions.

Figure 1

Orbitally decomposed NN exchange interaction in elemental $3d$ metals in the bcc structure.

Figure 2

(Bottom panel) $J_{ij}{R}_{ij}^3$ with the neighbors selected to lie along the (111) direction in bcc Fe. $R_{ij}$ is the intersite distance. (Top panel) The same data plotted together with an analytical function $y=A_0\sin (1.3R_{ij}+{\varphi }_0)$ (the dashed curve), whose period was obtained from the FS analysis. $A_0$ and ${\varphi }_0$ were adjusted to give the best agreement with the DFT results.

Figure 3

Distance dependence of the $J_1^{E_g\text{−}{E}_g}$ and $J_1^{E_g\text{−}{T}_{2g}}$ interactions in bcc Fe. $a_0$ corresponds to the equilibrium lattice constant (2.86 Å). To facilitate the analysis, the magnetic moment was constrained to the value obtained self-consistently for $a=a_0$, which is $2.2{\mu }_B$.