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

Abstract

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 Eg and T2g 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 T2g 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 Eg 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-neighbour coupling indicates that the interactions among Eg states are mainly proportional to the corresponding hopping integral and thus can be attributed to be of double-exchange origin.Iron is one of the most abundant elements in the universe. Its elemental phase has several polymorphs and among those the most stable crystal structure at ambient conditions is a body-centered cubic (bcc) one. Ferromagnetism is known to play a decisive role in defining the stability of this structure.1,2 The bcc phase is ferromagnetic (FM) up to critical temperature (T c ) of 1045 K. Quite importantly, above the Curie point, the bcc structure is preserved in a certain temperature range before it undergoes a transition to the fcc phase. This fact implies that the local magnetic moments exist in the paramagnetic (PM) phase, where strong short-ranged magnetic order was proposed.3 The magnetism of Fe is of mixed itinerant and localised nature and it is a matter of debate, which model describes it the best. Both the T c and the magnetic excitation spectra at low temperatures can be well described by means of the Heisenberg Hamiltonian (HH), parameterised by ab initio calculations.4-10 However, in several works 11,12 it was argued that in order to describe a large palette of magnetic states, higher-order (biquadratic) exchange interactions have to be taken into account. The results of the self-consistent spin spiral calculations also indicate that the magnitude of the magnetic moment in bcc Fe can differ by almost 30% in various states, 13,14 which in principle disagrees with the assumptions of the Heisenberg picture. Indeed, it was previously pointed out that the parameterisation of HH for bcc Fe depends on the magnetic configuration they are extracted from. 15-17The strong correlation effects are known to be important for bcc Fe at finite temperatures, as was shown in Ref. 18 by means of density functional theory plus dynamical mean field theory (DFT+DMFT) calculations.Earlier, it was suggested from a qualitative analysis of the electronic structure that the E g electrons in iron are much more correlated than the T 2 g ones.19 This statement was qua...

Theory of noncollinear interactions beyond Heisenberg exchange: Applications to bcc Fe

“…4 where the J H and J N H parameters are shown not only for the first and second but also for a third, fourth, fifth and sixth neighbouring pairs in bcc Fe. Within an agreement with the previous finding [2,5,11,18], the first and second neighbor J ij 's dominate the J ij 's calculated for further neighbors. This conclusion holds in the noncollinear framework too.…”

“…These findings are in a very good agreement with the conclusions of Ref. [2] based on the LKAG formalism. We also note that the nearest neighbour angles between the atomistic spins in bcc Fe are usually small at low temperatures (where the background is strongly spin polarized), hence a Heisenberg model with the LKAG exchange formula can give a good approximation.…”

“…We show, however, that an E g -T 2g symmetry analysis based on the orbital decomposition of the J ij leads to a similar conclusion for the origin of Heisenberg and non-Heisenberg terms as was published in Ref. [2] within the LKAG approach. cluding four-spin exchange has been successfully applied to simulate the magnetic phase diagram of heavy rare earth elements [24].…”

“…This means that there is a pronounced antiferromagnetic coupling coming from to the T 2g orbitals (T 2g channel) that was shown to be related to Fermi-surface mechanisms (like RKKY oscillations calculated in the asymptotic regime) by calculating the exchange parameters for further neighbour pairs in a given direction. However, in the E g and mixed channels the microscopic origin is mainly the double-exchange mechanism [2] that requires the presence of the (ferromagnetic) host. Note that we calculate A αβ ij in Eq.…”

“…The microscopic origin of the exchange interactions in ferromagnetic bcc Fe (as well as in metallic magnets in general) is a part of ongoing scientific discussions [1][2][3], in spite of the fact that iron is probably the best-known magnet. Especially at finite temperature when the atomistic spin moments deviate from a collinear and uniform direction, i.e., a global magnetization axis cannot be easily identified, the dependence of the interatomic exchange parameters on the underlying spin configuration could become more significant [3].…”