Linear response based theories for Dzyaloshinskii-Moriya interactions

Abstract

The linear response is a perturbation theory establishing the relationship between given physical variable and the external field inducing this variable. A wellknown example of the linear response theory in magnetism is the susceptibility relating the magnetization with the magnetic field. In 1987, Liechtenstein et al came up with the idea to formulate the problem of interatomic exchange interactions, which would describe the energy change caused by the infinitesimal rotations of spins, in terms of this susceptibility. The formulation appears to be very generic and, for isotropic systems, expresses the energy change in the form of the Heisenberg model, irrespectively on which microscopic mechanism stands behind the interaction parameters. Moreover, this approach establishes the relationship between the exchange interactions and the electronic structure obtained, for instance, in the first-principles calculations based on the density functional theory. The purpose of this review is to elaborate basic ideas of the linear response theories for the exchange interactions as well as more resent developments. The special attention is paid to the approximations underlying the original method of Liechtenstein et al in comparison with its more recent and more rigorous extensions, the roles of the on-site Coulomb interactions and the ligand states, and calculations of antisymmetric Dzyaloshinskii-Moriya interactions, which can be performed alongside with the isotropic exchange, within one computational scheme. The abilities of the linear response theories as well as many theoretical nuances, which may arise in the analysis of interatomic exchange interactions, are illustrated on magnetic van der Walls materials CrX 3 (X= Cl, I), half-metallic ferromagnet CrO 2 , ferromagnetic Weyl semimetal Co 3 Sn 2 S 2 , and orthorhombic manganites AMnO 3 (A= La, Ho), known for the peculiar interplay of the lattice distortion, spin, and orbital ordering.

Linear response theories for interatomic exchange interactions

“…However, such agreement is probably fortuitous because the models are very different 7 . The DM interactions in CrCl 3 are considerably weaker (|d z | ∼ 0.02 meV [53]). However, this is to be expected and the order of magnitude difference of the DM interactions in CrCl 3 and CrI 3 well correlates with the strength of the SO coupling on the ligand sites, which differs by the same order of magnitude: ξ Cl = 95 meV versus ξ I = 881 meV.…”

“…This expression was obtained in [53] basically heuristically, by the analogy with isotropic interactions and similar expression formulated in terms of the xc fields, which will be considered in section 2.8. Here, we have provided a more rigorous proof of equation (21).…”

“…⟩ denotes the expectation value in the ground state. The strong-coupling limit for the DM interaction (31) results in the spin-current model [58,70], which can be viewed as the relativistic counterpart of the double exchange mechanism [53]. The expression for RKKY interactions can be also derived starting from equation (30), but using slightly different philosophy [9].…”

“…In this section, we briefly discuss the behavior of DM interactions in CrI 3 , which are driven by the strong SO coupling of the heavy I atoms (so as other magnetic interactions of the relativistic origin [86]). The DM interactions in CrCl 3 are considerably weaker [49,53] and not significant [83]. In addition to d z , two other components of the DM vector, d x and d y , can be computed by rotating the coordinate frame and applying the same equation ( 28 Therefore, all intersublattice interactions will identically vanish [6,7].…”

“…In LSDA, these parameters depend on spin. Furthermore, Ĥσ = [H σ ia,jb ] can include spin-diagonal part of the SO coupling, which is needed to calculate DM interactions [15,53]. In this case, Ĥσ also depends on spin, even in the nonmagnetic LDA, where it holds: Ĥ↓ = Ĥ↑ * .…”