Plane wave implementation of the magnetic force theorem for magnetic exchange constants: application to bulk Fe, Co and Ni

Abstract

We present a plane wave implementation of the magnetic force theorem, which provides a first principles framework for extracting exchange constants parameterizing a classical Heisenberg model description of magnetic materials. It is shown that the full microscopic exchange tensor may be expressed in terms of the static Kohn-Sham susceptibility tensor and the exchange-correlation magnetic field. This formulation allows one to define arbitrary magnetic sites localized to predefined spatial regions, hence rendering the problem of finding Heisenberg parameters independent of any orbital decomposition of the problem. The susceptibility is calculated in a plane wave basis, which allows for systematic convergence with respect to unoccupied bands and spatial representation. We then apply the method to the well-studied problem of calculating adiabatic spin wave spectra for bulk Fe, Co and Ni, finding good agreement with previous calculations. In particular, we utilize the freedom of defining magnetic sites to show that the calculated Heisenberg parameters are robust towards changes in the definition of magnetic sites. This demonstrates that the magnetic sites can be regarded as well defined and thus asserts the relevance of the Heisenberg model description despite the itinerant nature of the magnetic state.

Antiferromagnetism in two-dimensional materials: progress and computational challenges

“…The latter approach is known as the RPA and is expected to work well for low temperatures, but probably not in the vicinity of the critical temperature. Nevertheless, it has been applied extensively to estimate Curie temperatures of ferromagnets with exchange constants obtained from first principles [92,100,101]. In the case of uniaxial single-ion anisotropy, the RPA (like linear spinwave theory) predicts a spinwave gap of 2AS rather than A(2S − 1), which obviously represents a severe shortcoming for 2D materials.…”

“…This allows for a perturbative treatment and the exchange interactions may be obtained directly from the linear transverse Kohn-Sham susceptibility evaluated on the ground state. For a collinear system the microscopic expression is given by [101]…”

“…calculations to a site-based model, but it has been shown that magnon energies are typically insensitive to the choice of magnetic sites as long as these contain the majority of the local moments [101]. Alternatively, the approach can be based on Wannier functions centered on the magnetic sites and in that case the exchange couplings in a site basis are extracted directly [113].…”

Type II multiferroic order in two-dimensional transition metal halides from first principles spin-spiral calculations

“…This error is directly reflected by the variation of the magnitude of moments in figure 5. The true exchange parameters can only be obtained either by mapping to eigenstates of the model [78] or by considering infinitesimal rotations of the spin, which may be handled non-selfconsistently using the magnetic force theorem [37,[79][80][81][82]. Nevertheless, the deviations between exchange parameters obtained from classical and quantum mechanical energy mapping typically deviates by less than 5% [78] and for insulators it is a good approximation to extract the magnon energies from planar spiral calculations although the mapping is only strictly valid in the limit of small q.…”

Type II multiferroic order in two-dimensional transition metal halides from first principles spin-spiral calculations

“…This error is directly reflected by the variation of the magnitude of moments in figure 4. The true exchange parameters can only be obtained either by mapping to eigenstates of the model [74] or by considering infinitesimal rotations of the spin, which may be handled non-selfconsistently using the magnetic force theorem [37,[75][76][77][78]. Nevertheless, the deviations between exchange parameters obtained from classical and quantum mechanical energy mapping typically deviates by less than 5 % [74] and for insulators it is a good approximation to extract the magnon energies from planar spiral calculations although the mapping is only strictly valid in the limit of small q.…”