Wannier-based calculation of the orbital magnetization in crystals

Abstract

We present a first-principles scheme that allows the orbital magnetization of a magnetic crystal to be evaluated accurately and efficiently even in the presence of complex Fermi surfaces. Starting from an initial electronic-structure calculation with a coarse ab initio k-point mesh, maximally localized Wannier functions are constructed and used to interpolate the necessary k-space quantities on a fine mesh, in parallel to a previously-developed formalism for the anomalous Hall conductivity [X. Wang, J. Yates, I. Souza, and D. Vanderbilt, Phys. Rev. B 74, 195118 (2006)]. We formulate our new approach in a manifestly gauge-invariant manner, expressing the orbital magnetization in terms of traces over matrices in Wannier space. Since only a few (e.g., of the order of 20) Wannier functions are typically needed to describe the occupied and partially occupied bands, these Wannier matrices are small, which makes the interpolation itself very efficient. The method has been used to calculate the orbital magnetization of bcc Fe, hcp Co, and fcc Ni. Unlike an approximate calculation based on integrating orbital currents inside atomic spheres, our results nicely reproduce the experimentally measured ordering of the orbital magnetization in these three materials.

How disorder affects the Berry-phase anomalous Hall conductivity: A reciprocal-space analysis

“…The expression for the unfolded Berry curvature involves several other matrix objects, which we now define borrowing the notation from Ref. [23]. The two basic objects are (omitting orbital indices)…”

“…where f NM is the occupation matrix [23]. For insulatorsP = With these definitions, the Berry curvatures in the original SBZ and unfolded onto the NBZ via Eq.…”

“…Essentially, we combine two Wannier-based methodologies: that of Refs. [10] and [23] for computing the Berry curvature and that of Ref. [4] for BZ unfolding.…”

Higher-dimensional Wannier Interpolation for the Modern Theory of the Dzyaloshinskii–Moriya Interaction: Application to Co-based Trilayers

“…In strong formal analogy to the derivations in Ref. 34 for the orbital magnetization, we arrive at the following expressions for the phase-space objects that underlie the DMI spiralization:…”

“…A gauge-invariant formulation of the Wannier interpolation of orbital magnetization at zero temperature exists, 34) and we extend this scheme here to evaluate Eq. (5) based on HDWFs.…”