Understanding magnetocrystalline anisotropy based on orbital and quadrupole moments

Abstract

Understanding magnetocrystalline anisotropy (MCA) is fundamentally important for developing novel magnetic materials. Therefore, clarifying the relationship between MCA and local physical quantities observed by spectroscopic measurements, such as the orbital and quadrupole moments, is necessary. In this review, we discuss MCA and the distortion effects in magnetic materials with transition metals (TMs) based on the orbital and quadrupole moments, which are related to the spin-conserving and spin-flip terms in the second-order perturbation calculations, respectively. We revealed that orbital moment stabilized the spin moment in the direction of the larger orbital moment, while the quadrupole moment stabilized the spin moment along the longitudinal direction of the spin-density distribution. The MCA of the magnetic materials with TMs and their interfaces can be determined from the competition between these two contributions. We showed that the perpendicular MCA of the facecentered cubic (fcc) Ni with tensile tetragonal distortion arose from the orbital moment anisotropy, whereas that of Mn-Ga alloys originated from the quadrupole moment of spin density. In contrast, in the Co/Pd(111) multilayer and Fe/MgO(001), both the orbital moment anisotropy and quadrupole moment of spin density at the interfaces contributed to the perpendicular MCA. Understanding the MCA of magnetic materials and interfaces based on orbital and quadrupole moments is essential to design MCA of novel magnetic applications.

Impact of a rubrene buffer layer on the dynamic magnetic behavior of nickel layers on Si(100)

“…By the discovery of perpendicular magnetic anisotropy (PMA), researchers have been prompted to readdress magnetic anisotropy with second-order anisotropy parameter ( K 2 ) owing to the contribution of uniaxial anisotropy. 10,11 There has been increased interest in magnetic multilayers due to the spatial fluctuations 10,12 that could contribute to the second-ordered anisotropy energy. 13 In both theory 14 and experiment, 15 K 2 signifies the easy axis along the surface normal to the film, but there are numerous other states that can be established by stabilizing the angle subtended between magnetization and the surface normal.…”

Understanding voltage-controlled magnetic anisotropy effect at Co/oxide interface

“…Based on Bruno’s theory, the anisotropy of m orb corresponds to the stabilisation of PMA 32 . Interface PMA in 3 d TM ferromagnets is often discussed based on Bruno’s theory 3 , 10 , 33 – 35 . Thus, the increase in the interface PMA and interfacial m orb observed in Pt/Co/AlO x trilayers may also be linked to the large VCMA effect obtained at surface-oxidised Co/oxide interfaces.…”

Strain-induced magnetic anisotropy in Heusler alloys studied from first principles

“…10) On the basis of second-order perturbation theory, the MCA energy can be decomposed in terms of atomic-site, spin, and orbital contributions. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Bruno proposed that the spin-conserving terms between the occupied and unoccupied minority-spin states (↓ ↓ ) are related to the orbital magnetic moment observed experimentally. 11) Wang, Wu, and Freeman also showed that the spin-flip term between the occupied majority-spin states and unoccupied minority-spin states (↑ ↓ ) is connected with the quadrupole moment of the spin density.…”

“…, where Q zz is the quadrupole moment of the electron-density distribution, S z is the z component of the spin angular momentum S, μ B is the Bohr magneton, ℏ is the reduced Planck constant, L is the orbital angular momentum, and L z is its z component. 13,14,16,17,23) Together with the orbital magnetic moment m orb , E MCA can be approximated as 16,23) ( )…”

“…( 1), while the spin-flip term E MCA  is the main component of the quadrupole term. 11,12,14,15,21,23) To examine these relations and assumptions, we analyze the coupling-resolved E MCA from second-order perturbations as shown in Fig. 3.…”