Relativistic magnetic interactions from nonorthogonal basis sets

We propose a method to determine the magnetic exchange interaction and onsite anisotropy tensors of extended Heisenberg spin models from density functional theory including relativistic effects. The method is based on the Liechtenstein-Katsnelson-Antropov-Gubanov torque formalism, whereby energy variations upon infinitesimal rotations are performed. We assume that the Kohn-Sham Hamiltonian is expanded in a nonorthogonal basis set of pseudoatomic orbitals. We define local operators that are both Hermitian and satisfy relevant sum rules. We demonstrate that in the presence of spin-orbit coupling a correct mapping from the density functional total energy to a spin model that relies on the rotation of the exchange field part of the Hamiltonian can not be accounted for by transforming the full Hamiltonian. We derive a set of sum rules that pose stringent validity tests on any specific calculation. We showcase the flexibility and accuracy of the method by computing the exchange and anisotropy tensors of both well-studied magnetic nanostructures and of recently synthesized two-dimensional magnets. Specifically, we benchmark our approach against the established Korringa-Kohn-Rostoker Green's function method and show that they agree well. Finally, we demonstrate how the application of biaxial strain on the two-dimensional magnet $\mathrm{T}\text{−}{\mathrm{CrTe}}_2$ can trigger a magnetic phase transition.

Figure 1

Sum rule for the global spin rotation of a dimer system: the sum of the energy of the local spin rotation of both atoms plus the interaction energy is zero in the absence of spin-orbit interaction.

Figure 2

(a) Illustrates the vectors appearing in the classical (left) and the KS Hamiltonian (right). It is also indicated by curved arrows which of those should be transformed upon rotations in order to achieve a faithful DFT2S mapping. ${\mathbf{V}}_{\text{XC}}$ is parallel to the atomic magnetization ${\mathbf{e}}_A$ by definition, while, quite generally, ${\mathbf{V}}_{\text{SO}}$ is not. (b) Shows graphically the impact of a rotation about the quantization axis on the classical Hamiltonian (left) and the KS Hamiltonian (right). The classical Hamiltonian is left invariant, while the KS Hamiltonian changes if ${\mathbf{V}}_{\text{SO}}$ is rotated, thus invalidating the DFT2S mapping. We will show in Sec. 5a and Table 2 how this reasoning is applied in the case of a platinum dimer.

Figure 3

Schematic drawings of the simulated chromium trimers. (a) The isolated trimer, where the convention for the coordinate axes is indicated; [(b)–(d)] top and side views of the Cr trimer/Au(111) heterostructure for three different heights $h=2.36$, 2.83, and 2.06 Å. Red- and gold-colored spheres denote the Cr and the surface Au atoms, respectively. The figure also shows the DM vectors in green arrows. The DM magnitudes are rescaled among the three figures to allow for a better visualization of the vectors.

Figure 4

(a) Top view of the simulated Mn monolayer deposited onto a W(110) slab. Orange (gray) spheres correspond to Mn (W) atoms. Note that the W atoms below the Mn atoms occupy sites in the subsurface W layer. The DM vectors referred to the central Mn atom (labeled by the letter c) are depicted in green color. All the DM vectors lie within the surface plane. Their moduli from the first to the fifth nearest neighbor are 3.7, 2.5, 1, 2, and 0.7 meV. (b) PDOS projected onto the Mn atom, obtained from a collinear spin calculation. The top (bottom) panels refer to spin-up (-down) projections, respectively. Black, red, and blue lines refer to results from the codes KKR (${\ell }_{\max }=3$), siesta, and vasp.

Figure 5

(a) Top and side views of the $\mathrm{T}\text{−}{\mathrm{CrTe}}_2$ crystal structure. Red (gray) colors correspond to Cr (Te) atoms. First, second and third neighbors relative to the central c atom are marked as 1, 2, and 3, respectively. (b) Isotropic exchange coupling $J_H$ for first, second, and third neighbors as a function of the lattice constant are plotted with black, red, and blue solid (dotted) lines. corresponding to the local (onsite) projection, respectively. The angle $\alpha$ subtended by the Cr-Te-Cr bonds is plotted at the top $x$ axis.

Figure 6

(a) Top and (b) side views of the lower monolayer of a $\mathrm{T}\text{−}{\mathrm{CrTe}}_2$ bilayer, where only the Cr atoms are drawn for clarity. We show the DM vectors up to the third neighbor. The largest moduli correspond to the first neighbors and are of the order of 0.2 meV.