Mapping the magnetic exchange interactions from first principles: Anisotropy anomaly and application to Fe, Ni, and Co

Abstract

Mapping the magnetic exchange interactions from model Hamiltonian to density-functional theory is a crucial step in multiscale modeling calculations. Considering the usual magnetic force theorem but with arbitrary rotational angles of the spin moments, a spurious anisotropy in the standard mapping procedure is shown to occur provided by bilinear like contributions of high-order spin interactions. The evaluation of this anisotropy gives a hint on the strength of nonbilinear terms characterizing the system under investigation. DOI: 10.1103/PhysRevB.82.180404 PACS number͑s͒: 75.30.Ϫm, 73.22.Ϫf, 75.70.Ϫi Multiscale modeling approaches are extremely important for describing huge magnetic systems, e.g., at the micrometer scale which would be impossible with only densityfunctional theory ͑DFT͒. In magnetism, usually the multiscale approach is performed after mapping the magnetic exchange interactions ͑MEI͒ of a classical Heisenberg model to the DFT counterparts. This is a crucial task which can lead to wrong results if not done carefully. The simple model is described bywhere J ij describes the pairwise ͑two-spin͒ MEI between spins at lattice sites i and j while e ជ i ͑1, , ͒ defines the direction of the local moment M ជ i . Sometimes, higher-order terms such as the four-spin or the biquadratic MEI are introduced in the previous Hamiltonian for a better mapping of the DFT results. 1,2Once the MEI extracted, the investigation of magnetism of several type of systems can be performed going from molecules, 3 transition metals alloys, 4,5 and surfaces, 6,7 diluted magnetic semiconductors, 8,9 to clusters, 10-13 and even for strongly correlated systems.14 Thermodynamical properties are then easily accessible such as Curie temperatures, specific heat or magnetic excitation spectra and spin waves stiffness in multidimensional systems.An elegant method to extract the MEI is based on a Greens-function technique which has been derived 20 years ago by Liechtenstein et al. 15 ͑noted in the text LKAG͒. Instead of calculating several magnetic configurations, this method, based on the magnetic force theorem ͑MFT͒, 16,17 allows the evaluation of the MEI from one collinear configuration which is usually ferromagnetic. Computationally, this method is thus very attractive.Assuming infinitesimal rotation angles of the magnetic moments ͑limit of infinite magnon wavelength͒ is necessary to get the final LKAG formula for the MEI. However, one should note that this formalism is used for arbitrary big rotation angles ͑finite magnon wavelength͒ as well. Thus, many improvements of the formalism have been proposed recently: Bruno 18 proposed a renormalized MFT using the constrained DFT ͑Ref. 19͒ leading to unrealistic high local density approximation ͑LDA͒ Curie temperature ͑T c ͒ for fcc Ni. The same effect has been observed using the proposal of Antropov. 20 Katsnelson and Lichtenstein 21 proposed in their recent publication a reconciliation between the old formalism 15 and the new renormalized theories. 18,20 They have shown that the impr...

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

“…Then we recover the results for δE two ij published in Ref. [25] where the anisotropic type term was mapped onto a four-spin Hamiltonian. Note that Eq.…”

“…It can be shown that the non-collinear pairwise energy variation formula recovers the pairwise energy formula published in Ref. [25] for a collinear case by keeping the higher (fourth) order terms in the infinitesimal angle variation, i.e., an anisotropic term is present even in the collinear limit, which was found to be numerically significant in bcc Fe [18]. Here, we will show that in a non-equilibrium case, when one spin is rotated by a finite angle in a ferromagnetic background (single spin rotation shown in Fig.…”

“…The source of the anisotropic term (which was reported already in Ref. [25]) is induced by the spinpolarized host due to double exchange mechanism. Note that in a paramagnetic phase the A µµ ij terms (µ = x, y, z) are the same in statistical sense, i.e., we cannot speak of terms that are induced by the symmetry broken host [18].…”

Exchange interactions and local-moment fluctuation corrections in ferromagnets at finite temperatures based on noncollinear density-functional calculations

“…71 Using a multiple-scattering approach, they consider the energy expansion as a function of the angles between moments. As they find, at high angles corrections are necessary to the phenomenological Hamiltonian (e.g., biquadratic or four-spin terms).…”

Thickness-dependent magnetic structure of ultrathin Fe/Ir(001) films: From spin-spiral states toward ferromagnetic order

“…[1][2][3][4] Moreover, higher order interaction terms (multiple spin interactions) also have to be considered in many cases [5][6][7] to give an accurate description of magnetism. Very recent studies indicate that such interactions may even lead to the formation of exotic states like magnetic skyrmion lattices.…”