Magnetocrystalline Anisotropy Energy Research Articles

First- and second-order magneto-optical effects and intrinsically anomalous transport in the two-dimensional van der Waals layered magnets CrXY (X=S,Se,Te;Y=Cl,Br,I)

Recently, the two-dimensional magnetic semiconductor CrSBr has attracted considerable attention due to its excellent air-stable property and high magnetic critical temperature. Here, we systematically investigate the electronic structure, magnetocrystalline anisotropy energy, first-order magneto-optical effects (Kerr and Faraday effects), and second-order magneto-optical effects (Sch\"afer-Hubert and Voigt effects) as well as intrinsically anomalous transport properties (anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects) of two-dimensional van der Waals layered magnets $\mathrm{Cr}\mathit{XY} (X=\mathrm{S},\mathrm{Se},\mathrm{Te};Y=\mathrm{Cl},\mathrm{Br},\mathrm{I})$ by using first-principles calculations. Our results show that monolayer and bilayer $\mathrm{Cr}\mathit{XY} (X=\mathrm{S},\mathrm{Se})$ are narrow-band-gap semiconductors, whereas monolayer and bilayer $\mathrm{CrTe}Y$ are multiband metals. The magnetic ground states of bilayer $\mathrm{Cr}\mathit{XY}$ and the easy magnetization axis of monolayer and bilayer $\mathrm{Cr}\mathit{XY}$ are confirmed by the magnetocrystalline anisotropy energy calculations. Utilizing magnetic group theory analysis, the first-order magneto-optical effects as well as anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects are identified to exist in a ferromagnetic state with out-of-plane magnetization. The second-order magneto-optical effects are not restricted by the above symmetry requirements, and therefore they can arise in ferromagnetic and antiferromagnetic states with in-plane magnetization. The calculated results are compared with the available theoretical and experimental data of other two-dimensional magnets and some conventional ferromagnets. The present work reveals that monolayer and bilayer $\mathrm{Cr}\mathit{XY}$ with superior magneto-optical responses and anomalous transport properties provide an excellent material platform for the promising applications of magneto-optical devices, spintronics, and spin caloritronics.

DFT calculation of intrinsic properties of magnetically hard phase L10 FePt

Due to its strong magnetocrystalline anisotropy, FePt L1$\mathrm{_0}$ phase is considered as a promising magnetic recording media material. Although the magnetic properties of this phase have already been analyzed many times using density functional theory (DFT), we decided to study it again, emphasizing on full potential methods, including spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) and full-potential local-orbital (FPLO) scheme. In addition to the determination of exact values of the magnetocrystalline anisotropy constants K$\mathrm{_1}$ and K$\mathrm{_2}$, the magnetic moments (m), the Curie temperature, and the magnetostriction coefficient, we focused on the investigation of the magnetocrystalline anisotropy energy (MAE) dependence on the magnetic moment values using the fully relativistic fixed spin moment (FSM) method with various exchange-correlation potentials. We present nearly identical MAE(m) curves near the equilibrium point, along with different equilibrium values of MAE and magnetic moments. For a magnetic moment reduced by about 10%, we determined a theoretical MAE maximum in the ground state (0 K) equal to about 20.3 MJ m$\mathrm{^{-3}}$ and independent of the choice of the exchange-correlation potential form. These calculations allow us to understand the discrepancies between the previous MAE results for different exchange-correlation potentials.

Pressure dependent magnetic properties on bulk CrBr3 single crystals

The van der Waals class of materials offer an approach to two-dimensional magnetism as their spin fluctuations can be tuned upon exfoliation of layers. Moreover, it has recently been shown that spin-lattice coupling and long-range magnetic ordering can be modified with pressure in van der Waals materials. In this work, the magnetic properties of quasi two-dimensional CrBr 3 are reported applying hydrostatic pressure. The application of pressure up to 0.844 GPa shows a 1.77% decrease in saturation magnetization with a decrease in the Curie temperature from 33.05 to 30.41 K. Density functional theory calculations with pressure up to 1 GPa show a reduction in volume and interplanar distance as pressure increases. To further understand the magnetic properties with applied pressure, the magnetocrystalline anisotropy energy (MAE) and exchange coupling parameter(s) ( J ) are calculated. There is small decrease in MAE and the first nearest neighbor interaction ( J 1 ) ( U = 2.7 eV and J = 0.7 eV) is increasing with respect to increasing pressure. Overall, CrBr 3 displays ferromagnetic interlayer coupling and the calculated exchange coupling and MAE parameters match well with the observations from the experimental work. • Hydrostatic pressure decreases saturation magnetization and curie temperature. • Magnetocrystalline anisotropy energy decreases with increasing pressure. • Super exchange coupling increases with increasing pressure. • Pressure decreases the volume and lattice parameters of the system. • The bond angle between Cr-Br-Cr is decreasing up to 1 GPa.

Electronic and magnetic properties of iridium-based novel Heusler alloys

Half-metallicity and magnetism including exchange splitting are the most significant physical parameters to predict and design a candidate material for spintronic applications. We report here an ab-initio investigation on chemical formation and dynamical stability along with electronic structure and magnetic properties of Ir2Cr (Si, Ge) and IrRhCr (Si, Ge) Heusler alloys. The negative formation and cohesive energies with positive phonon dispersions confirm the stabilities of these alloys. Electronic structure calculations reveal that Ir2Cr (Si, Ge) and IrRhCrSi alloys are half-metallic ferromagnets with unprecedented exchange splitting. In addition, IrRhCrGe also shows semi-metallic nature. All of these materials follow Slater Pauling rule with large magnetic moments and 100% spin-polarization. With Cr bearing the majority of the local magnetic moment and exchange splitting, a ferromagnetic state is more stable than a nonmagnetic state. The electronic charge distribution and population analysis confirm mixed ionic and covalent bonding. The magnetocrystalline anisotropy energy, with the easy magnetization along the [1 1 1] direction, is significantly high in Ir2CrGe. Elastic constants such as shear (G), bulk (B), Young’s moduli, and Poisson’s ratio indicate that the IrRhCrSi and IrRhCrGe alloys are mechanically stable, and Ir2CrSi and Ir2CrGe are mechanically unstable. The Pugh’s (B/G) and Poisson’s ratios confirm that the stable alloys are ductile.

Electronic and Vibrational Properties of Fe₂NiAl and Co₂NiAl Full Heusler Alloys: A First-Principles Comparison

We provide a comparative first-principles investigation of the structural, electronic, vibrational, and thermodynamic properties of the full Heusler compounds, cubic Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NiAl, and tetragonally distorted Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NiAl. In both cases, we find the inverse Heusler structure to be in close competition with a layered arrangement of the elements, which breaks cubic symmetry and is accompanied by significant absolute values for the magnetocrystalline anisotropy energy. While for Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NiAl the layered arrangement is predicted as a new ground state, we show evidence that it might become dynamically stable around room temperatures also for Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NiAl. We identify in both systems subtle differences in the electronic and vibrational density of states between the different structures, which might be related to their particular stability.

The role of Os substitution on structural, magnetic, and optical features of LuFeO3

The powders of LuFeO3 and LuFe1-xOsxO3 (x = 0.05 and 0.10) were fabricated by using solid-state technique. The presence of particle agglomeration as well as the porosity of the analyzed materials have been shown via scanning electron microscope (SEM) studies. The oxidation states of the constituted elements in the fabricated samples have been determined with X-ray photoelectron spectroscopy (XPS) studies. It has been realized that the Lu has 3+, Fe accommodates both 2+ and 3+ valence stated in three samples. In addition, the oxidation state of Os has been identified as 4+ in the Os doped samples. It has been demonstrated that the Raman peak intensity of the Os substituted samples declines with Os presence. Furthermore, it has been seen that the peaks shift to lower wavenumber values in the Os doped specimens. Magnetic studies performed via vibrating sample magnetometer (VSM) found that the Os doping increases not only Hc but also Mr values at 10 K and 100 K, yet Hc and Mr values for undoped LFO at 300 K are higher than the Os doped samples. This rise in Hc and Mr values might be attributed to 1) deviations in the tilted angles of Fe3+ moments and 2) the evolution of energy of magnetocrystalline anisotropy. Moreover, the influence of Os doping on the optical band gap of LFO was determined. It exhibited that the optical band gap of LFO increases as the Os doping ratio elevates from 2.19 eV to 2.22 eV.

Electron mediation enhanced magnetocrystalline anisotropy and Curie temperature of FeCl2 monolayer by an electride substrate

Two-dimensional magnetic materials offer an ideal platform to study the fundamental research and potential applications in spintronics. However, it is still a challenging issue to raise their magnetic anisotropy energy and Curie temperature. Based on the density functional theory and the Monte Carlo method, we investigate the electron-mediated electronic structures and magnetic properties in a FeCl2/Ca2N heterostructure. It is found that a Ca2N monolayer devotes about 0.46 electrons/f.u. to a FeCl2 monolayer due to the low work function. This provides a high electron doping density of 4.28 × 1014 cm−2 and avoids the disorder effect of the traditional doping methods simultaneously such as the intercalation, adsorption, and substitution. Moreover, the magnetocrystalline anisotropy energy and Curie temperature of the FeCl2/Ca2N heterostructure enhance 66% and 100% compared with the FeCl2 monolayer, respectively. Our findings provide a practical and effective avenue to modulate the magnetic properties of two-dimensional magnetic materials.

Free energy of twisting spins in Mn3Sn

The magnetic free energy is usually quadratic in magnetic field and depends on the mutual orientation of the magnetic field and the crystalline axes. Tiny in magnitude, this magnetocrystalline anisotropy energy (MAE) is nevertheless indispensable for the existence of permanent magnets. Here, we show that in Mn$_3$Sn, a non-collinear antiferromagnet attracting much attention following the discovery of its large anomalous Hall effect, the free energy of spins has superquadratic components, which drive the MAE. We experimentally demonstrate that the thermodynamic free energy includes terms odd in magnetic field ($\mathcal{O}(H^3)+\mathcal{O}(H^5)$) and generating sixfold and twelve-fold angular oscillations in the torque response. We show that they are quantitatively explained by theory, which can be used to quantify relevant energy scales (Heisenberg, Dzyaloshinskii-Moriya, Zeeman and single-ion anisotropy) of the system. Based on the theory, we conclude that, in contrast with common magnets, what drives the MAE in Mn$_3$Sn is the field-induced deformation of the spin texture.

First-principles study on magnetocrystalline anisotropy of cobalt films: hcp vs fcc

In this study, density functional calculations are performed on magnetocrystalline anisotropy (MCA) of hcp and fcc Co films to reveal influence of symmetry, thickness, and surface. First, MCA of bulk is revisited. MCA energy (EMCA) of hcp is 15.61 μeV/atom, two orders of magnitude larger than 0.53 μeV/atom of fcc Co. Smaller EMCA of fcc Co originates from the negative contributions from the highly peaked unoccupied minority |m|=1 states at Fermi level (EF). In Co films, EMCA is significantly enhanced to 175.90 and 211.48 μeV/atom at both hcp and fcc surfaces, resulting in the same order of magnitude for both the films. Substantial EMCA enhancement stems from the increased occupied and unoccupied minority |m|=2 states located very close to EF. Strong EMCA oscillation with thickness is observed for less than 10 monolayers, whose origin comes from the quantum well |m|=1 states at EF.

Strain engineering magnetocrystalline anisotropy in strongly correlated VTe2 with room-temperature ferromagnetism

Two-dimensional (2D) magnetic transition metal dichalcogenides have attracted great interest in various fields, including information storage, logic devices, and high-frequency detectors. However, the currently reported 2D magnets are limited by their low Curie temperatures. Based on the density-functional theory calculation, we predicted the metastable $H$ phase $\mathrm{V}{\mathrm{Te}}_{2}\phantom{\rule{4pt}{0ex}}(\mathrm{H}\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2})$ is a promising room-temperature 2D in-plane ferromagnetic semiconductor without charge-density wave (CDW) transition. To achieve the $H\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2}$ from the $T$ phase or CDW-phase $\mathrm{V}{\mathrm{Te}}_{2}$, electron doping is proposed as a feasible strategy in practice. Moreover, the significant modulation effect of external strain fields on the magnetocrystalline anisotropy energy and magnetic ordering of monolayer $H\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2}$ are systematically studied. The electron correlation included from a $\mathrm{Hubbard}\text{\ensuremath{-}}U$ term has a profound influence on the ground state of magnetic order of $H\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2}$, as well as the direction of the easy-magnetization axis. These results not only provide an in-depth understanding of the modulation mechanism of the magnetic $H\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2}$, but also shed light on the potential applications of $H\text{\ensuremath{-}}\mathrm{V}{\mathrm{Te}}_{2}$ in spintronics and sensors.

Electric-field-induced modulation of giant perpendicular magnetic anisotropy obtained by insertion of an Ir layer at the Fe/MgO interface: a first-principles study

First-principles calculations were performed to study the effect of the insertion of magnetic layers composed of 3d, 4d, and 5d elements at the Fe/MgO interface on the magnetocrystalline anisotropy energy (MAE) and its modulation by an external electric-field. We found that the insertion of an Ir layer on the MgO substrate increases the MAE. In particular, the insertion of a Fe/Co/Ir magnetic layer leads to a large perpendicular magnetic anisotropy (PMA). The largest PMA and the largest voltage-controlled MA coefficient is obtained for the Au/Fe/Co/Ir/MgO system.

Electric-field driven stability control of skyrmions in an ultrathin transition-metal film

AbstractTo realize future spintronic applications with magnetic skyrmions—topologically nontrivial swirling spin structures—it is essential to achieve efficient writing and deleting capabilities of these quasi-particles. Electric-field assisted nucleation and annihilation is a promising route, however, the understanding of the underlying microscopic mechanisms is still limited. Here, we show how the stability of individual magnetic skyrmions in an ultrathin transition-metal film can be controlled via external electric fields. We demonstrate based on density functional theory that it is important to consider the changes of all interactions with electric field, i.e., the pair-wise exchange, the Dzyaloshinskii–Moriya interaction, the magnetocrystalline anisotropy energy, and the higher-order exchange interactions. The energy barriers for electric-field assisted skyrmion writing and deleting obtained via atomistic spin simulations vary by up to a factor of three more than the variations of the interactions calculated from first-principles. This surprising effect originates from the electric-field dependent size of metastable skyrmions at a fixed magnetic field. The large changes in lifetimes allow the possibility of electric-field assisted thermally activated writing and deleting of skyrmions.

Phase field-finite element analysis of magnetic-induced deformation in ferromagnetic shape memory alloy

A phase field theory based model considering non-uniform demagnetizing field within ferromagnetic shape memory alloy and free space is established and realized by finite element method in a two-dimensional setting to save huge storage spaces and computational times. The martensite reorientation (contain nucleation and growth) from the stress field-favored martensite variant to magnetic field-favored martensite variant is numerically simulated. The simulation reasonably depicts the essence of magnetic-induced martensite reorientation and magnetization change for NiMnGa ferromagnetic shape memory alloy and their dependence on the loading level. Furthermore, the evolution and interaction of martensite domain structure, magnetic domain structure, magnetic potential and demagnetizing field are analyzed. And the dominant driving force generated by minimization of total Helmholtz free energy for loading level dependent martensite reorientation and magnetization change are revealed by addressing the influence of loading level. The simulation results show that the local demagnetizing field within ferromagnetic shape memory alloy and free space is the key to the local nucleation of new martensite variant. The dominant driving force of local nucleation and growth of new martensite variant are completely different: the local nucleation of new martensite variant is driven by local high magnetocrystalline anisotropy energy caused by local demagnetizing field; the local high Landau-type double well potential energy caused by local nucleation of new martensite variant has the largest impact on growth of the new martensite variant. The local high magnetocrystalline anisotropy energy caused by local nucleation and growth of new martensite variant has the largest impact on local rapid magnetization change within new martensite variant domain. When it comes to high loading level, the inhibition of martensite reorientation is attributed to the elastic strain energy, and the approximately uniform magnetization change is driven by Zeeman energy.

Ferroelectric polarization tailored interfacial charge distribution to modify magnetic properties of two-dimensional Janus FeBrI/In2S3 heterostructures

The discovery of two-dimensional magnetic and ferroelectric materials provides a feasible opportunity to realize the nonvolatile electrical control of magnetic properties in van der Waals multiferroic heterostructures. Here, based on the density of functional theory, we study the electronic structures and magnetic properties of the Janus FeBrI/In2S3 multiferroic heterostructure. The Curie temperature and magnetic anisotropy energy (MAE) can be modulated by the ferroelectric polarization due to the interfacial charge redistribution. According to the orbital-resolved magnetocrystalline anisotropy energy (EMCA) and the second-order perturbation theory, the difference of EMCA mainly stems from the change in electronic states near the Fermi level, which is associated with interfacial charge redistribution in different ferroelectric polarization directions. These findings not only reveal the physical mechanism of the change of MAE in detail but also suggest a feasible strategy to modulate the magnetic properties.

Effect of magnetocrystalline anisotropy on magnetocaloric properties of an AlFe2B2 compound

It is well known that the temperature dependence of the effective magnetocrystalline anisotropy energy obeys the $l(l+1)/2$ power law of magnetization in the Callen-Callen theory. Therefore, according to the Callen-Callen theory, the magnetocrystalline anisotropy energy is assumed to be zero at the critical temperature where the magnetization is approximately zero. This study estimates the temperature dependence of the magnetocrystalline anisotropy energy by integrating the magnetization versus magnetic field ($M\ensuremath{-}H$) curves, and found that the magnetocrystalline anisotropy is still finite even above the Curie temperature in the uniaxial anisotropy, whereas this does not appear in the cubic anisotropy case. The origin is the fast reduction of the anisotropy field, which is the magnetic field required to saturate the magnetization along the hard axis, in the case of cubic anisotropy. Therefore, the magnetization anisotropy and anisotropic magnetic susceptibility, which are the key factors of magnetic anisotropy, could not be established in the case of cubic anisotropy. In addition, the effect of magnetocrystalline anisotropy on magnetocaloric properties, as the difference between the entropy change curves of ${\mathrm{AlFe}}_{2}{\mathrm{B}}_{2}$ appears above the Curie temperature, is in good agreement with a previous experimental study. This is proof of magnetic anisotropy at slightly above Curie temperature.

Effect of transition metal doping on magnetic hardness of CeFe[formula omitted]-based compounds

ThMn12-type ternary cerium alloys with tetragonal structure (Pearson symbol tI26, space group I4/mmm) are considered as promising materials for permanent magnets. In this work, compositions of CeFe11X (s.g. Pmmn, No. 59) and CeFe10X2 (s.g. P4/mmm, No. 123) with all 3d, 4d, and 5d transition metal substitutions are considered. Since many previous studies have focused on the CeFe11Ti compound, this particular case became the starting point of our considerations and we gave it special attention. We first determined the optimal symmetry of the simplest CeFe11Ti structure model. We then observed that the calculated magnetocrystalline anisotropy energy (MAE) correlates with the magnetic moment, which in turn strongly depends on the choice of the exchange–correlation potential. MAE, magnetic moments, and magnetic hardness were determined for all compositions considered. Moreover, the calculated dependence of the MAE on the spin magnetic moment allowed us to predict the upper limits of the MAE. We also showed that it does not depend on the choice of the exchange–correlation potential form. The economically justifiable compositions with the highest magnetic hardness values are CeFe11W, CeFe10W2, CeFe11Mn, CeFe10Mn2, CeFe11Mo, CeFe10Mo2, and CeFe10Nb2. However, calculations suggest that, like CeFe12, these compounds are not chemically stable and could require additional treatments to stabilize the composition. Further alloying of the selected compositions with light elements embedded in interstitial positions confirms the positive effect of such dopants on hard magnetic properties. Subsequent calculations performed for comparison for selected isostructural La-based compounds lead to similar MAE results as for Ce-based compounds, suggesting a secondary effect of 4f electrons. Our preliminary results obtained using the intra-atomic Hubbard repulsion term showed a relatively small difference for CeFe12 compared to the results without this correction. Calculations were performed using the full-potential local-orbital electronic structure code FPLO18, whose unique fully relativistic implementation of the fixed spin moment method allowed us to calculate the MAE dependence of the magnetic moment.

Detection and manipulation of the antiferromagnetic Néel vector in Cr2O3

Detection and manipulation of the N\'eel vector in antiferromagnetic materials are promising for more stable, faster, and much higher-density spintronic devices. However, the electrical manipulation of the antiferromagnetic N\'eel vector remains challenging and controversial due to the difficulty in the detection of the zero net magnetization and the unavoidable complications from thermal artifacts. In this work, by utilizing the uniaxial antiferromagnet (AFM) ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$, we demonstrate the detection and manipulation of the antiferromagnetic N\'eel vector. We reveal unambiguously the spin-dependent electrical responses of the coherent ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ N\'eel vector switching, where a symmetric Hall signal and a fourfold angular-dependent magnetoresistance are captured. We also demonstrate the in-plane arbitrary manipulation of the ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ N\'eel vector when the magnetocrystalline anisotropy energy is compensated. Our work for detecting and manipulating N\'eel vectors offers a critical guide for antiferromagnetic-based N\'eel vector switching exploration.

Insight into anisotropic magnetocaloric effect of CrI[formula omitted

CrI3 is considered to be a promising candidate for spintronic devices and data storage. We derived the Heisenberg Hamiltonian for CrI3 from density functional calculations using the Liechtenstein formula. Moreover, the Monte–Carlo simulations with the Sucksmith–Thompson method were performed to analyze the effect of magnetic anisotropy energy on the thermodynamic properties. Our method successfully reproduced the negative sign of isothermal magnetic entropy changes when a magnetic field was applied along the hard plane. We found that the temperature dependence of the magnetocrystalline anisotropy energy is not negligible at temperatures slightly above the Curie temperature. We clarified that the origin of this phenomenon is attributed to anisotropic magnetic susceptibility and magnetization anisotropy. The difference between the entropy change of the easy axis and the hard plane is proportional to the temperature dependence of the magnetic anisotropy energy, implying that the anisotropic entropy term is the main source of the temperature dependence of the free energy difference when magnetizing in a specific direction other than the easy axis. We also investigated the magnetic susceptibility that can be used for the characterization of the negative sign of the entropy change in the case of a hard plane. The competition of the magnetocrystalline anisotropy energy and external magnetic field at a low temperature and low magnetic field causes a high magnetic susceptibility as the magnetization fluctuates. Meanwhile, the anisotropy energy is suppressed as a sufficient magnetic field is applied along the hard axis, and the magnetization is fully rotated to the direction of the external magnetic field.

Lattice dynamics and its effects on magnetocrystalline anisotropy energy of pristine and hole-doped YCo5 from first principles

We study the lattice dynamics effects on the phase stability and magnetocrystalline anisotropy (MCA) energy of CaCu$_5$-type YCo$_5$ at finite temperatures using first-principles calculations based on density functional theory (DFT). Harmonic lattice dynamics (HLD) calculations indicate that YCo$_5$ with 56 full valance electrons is dynamically unstable and this instability can be cured by reducing the number of electrons ($N_e$). Crystal orbital Hamilton population analysis reveals that the observed phonon instability originates from the large population of antibonding states near the Fermi level, which is dominated by the Co atoms in the honeycomb layer. The antibonding state depopulates with decreasing $N_e$, resulting in stable phonons for hole-doped YCo$_5$ with $N_e$ $\leq$ 55. We then evaluate the temperature-dependent MCA energy using both HLD and $ab$ $initio$ molecular dynamics (AIMD) methods. For the pristine YCo$_5$, we observe a very weak temperature decay of the MCA energy, indicating little effect of lattice dynamics. Also, the MCA energies evaluated with AIMD at all target temperatures are larger than that of the static hexagonal lattice at 0 K, which is mainly attributed to the structural distortion driven by soft phonon modes. In the hole-doped YCo$_5$, where the distortion is suppressed, a considerable temperature decay in MCA energy is obtained both in HLD and AIMD methods, showing that lattice dynamics effects on MCA energy are non-negligible.

Magnetic anisotropy and orbital magnetic moment in Co films and Co/X bilayers ( X=Pd and Pt)

Magnetocrystalline anisotropy (MCA) energy and the anisotropy of the orbital magnetic moment (AOM), relations between them, and methods of their calculation are investigated for the (001) fcc Co film and $\mathrm{Co}/X$ bilayers ($X=\mathrm{Pd}$ and Pt), using a realistic tight-binding model. The relations derived by Bruno [Phys. Rev. B 39, 865 (1989)] and van der Laan [J. Phys.: Condens. Matter 10, 3239 (1998)], including the AOM of Co only, are found to give largely incorrect mean values of the MCA energy but reproduce its oscillatory variation versus the Co thickness, with scaling factors needed for the bilayers. The presently proposed extension of the Bruno relation, with the AOM of both Co and Pd or Pt included, predicts the correct sign of the MCA energy for both bilayers and reproduces its magnitude and oscillation pattern very well, without extra scaling, for the Co/Pd bilayer, though this relation is not satisfied locally, by the MCA and AOM layer terms. A similarly extended van der Laan relation fails to reproduce the MCA energy, its sign, and the oscillation pattern, and largely overestimates its magnitude. For all investigated systems, the Co orbital moment oscillates versus the Co thickness with the 2 monolayer (ML) period for the out-of-plane direction of magnetization and the 5 ML period for the in-plane direction while the oscillations of the Pd and Pt orbital moments are more complex and similar for both magnetization directions. The exact and approximate MCA energies obtained with the force theorem and the perturbation theory (PT), respectively, are close to each other for the Co film and the Co/Pd bilayer. For the Co/Pt bilayer, only the mean value of the MCA energy is well approximated by the PT while its oscillation amplitude is overestimated a few times due to the large SOC in Pt. It is also shown that the MCA energy includes the intraband term, usually neglected, but, in fact, finite and vital for systems without the inversion symmetry as its magnitude is comparable to that of the interband term for the Co/Pd bilayer.