Constituents of magnetic anisotropy and a screening of spin–orbit coupling in solids

Abstract

Using quantum mechanical perturbation theory (PT) we analyze how the energy of perturbation of different orders is renormalized in solids. We test the validity of PT analysis by considering a specific case of spin-orbit coupling as a perturbation. We further compare the relativistic energy and the magnetic anisotropy from the PT approach with direct density functional calculations in FePt, CoPt, FePd, MnAl, MnGa, FeNi, and tetragonally strained FeCo. In addition using decomposition of anisotropy into contributions from individual sites and different spin components we explain the microscopic origin of high anisotropy in FePt and CoPt magnets. The magnetocrystalline anisotropy is a central magnetic property for both fundamental and practical reasons.1-3 It can depend sensitively on many quantities such as dopants or small changes in lattice constant.4 While control of this sensitive quantity can be crucial in many applications, e.g. permanent magnetism 5 , magnetooptics 6 and magnetoresistive random-access memory devices 7 , it is often unclear what mechanisms are responsible for these anisotropy variations, even from a fundamental point of view. It was understood long ago 8,9 that the magnetic anisotropy energy (MAE) K in bulk materials is a result of simultaneous action of spin-orbit coupling (SOC) and crystal field (CF). While in general this statement is still valid, existing microscopic methods do not accurately describe K in the majority of materials. One can calculate MAE using ab-initio electronic structure methods based on density functional theory, however quantitative agreement is often rather poor. In any case such methods are usually not well equipped to resolve it into components that yield an intuitive understanding, to enable its manipulation and control. Sometimes K is analyzed in terms of SOC matrix elements of ξl·s, where ξ is the SOC constant. However, this perturbation also induces changes in other terms contributing the total energy, which can affect the MAE as well. Below we show how the actual atomic SOC is 'screened' in crystals and study spin decomposition of SOC and MAE in real world magnets.Let us write the total Hamiltonian of magnetic electronic system aswhere H 0 is the non-relativistic Hamiltonian (sum of kinetic and potential energies of electrons) and V = ξl·s is * Materials of this paper have been presented at the 58th Annual the SOC Hamiltonian. We assume that ξ is small relative to CF and spin splittings. The change in the total energy of the system when SOC is added (below we call it relativistic part of the total energy) can be written aswhere E so is the matrix element of SOC with full perturbed wavefunction and ∆E 0 is the induced energy change of the scalar-relativistic Hamiltonian ( sum of kinetic and potential energies) due to the SOC perturbation. Using standard quantum mechanical perturbation theory (PT) each quantity |φ = |n , E = E (n) and E so = V (n) (wave function, total energy and perturbation V ) can be expressed as a sum over orders n: V (n) is proportional to...

Tunable dimensional crossover and magnetocrystalline anisotropy in Fe2P -based alloys

“…To identify the mechanisms of MCA and its dependence on band filling, we first consider the site and spin decomposition of the anisotropy of the SOC energy [19,31]. We have verified that, as expected, this quantity closely follows the concentration dependence of K in these alloys.…”

Origin of magnetic anisotropy in doped Ce2Co17 alloys

“…According to second-order perturbation theory [19,20], K ≈ i K so (i), where i indicates the atomic sites. Unlike K, which is calculated from the total energy difference, K so is localized and can be decomposed into sites, spins, and subband pairs [19,20].…”

“…By evaluating the on-site spin-orbit coupling (SOC) energy [19,20], we resolved anisotropy into contributions from atomic sites, spins, and orbital pairs. Furthermore, we explained the electronic-structure origin of MAE enhancement.…”

Structures and magnetic properties of Co-Zr-B magnets studied by first-principles calculations

“…We also calculated the MAE of the rhombohedral Co 5 Zr structure by carrying out all-electron calculations using the full-potential (FP) LMTO method to check VASP calculation results. In addition, by evaluating the SOC matrix elements, the anisotropy of orbital moment and MAE was resolved into sites, spins and orbital pairs [27] to identify their contribution to the magnetic properties. Curie temperature (T c ) is checked for selected structures using mean-field approximation and more details can be found in Ref.…”