Local Moment Picture Research Articles

Persistent flat band splitting and strong selective band renormalization in a kagome magnet thin film.

Magnetic kagome materials provide a fascinating playground for exploring the interplay of magnetism, correlation and topology. Many magnetic kagome systems have been reported including the binary FemXn (X = Sn, Ge; m:n = 3:1, 3:2, 1:1) family and the rare earth RMn6Sn6 (R = rare earth) family, where their kagome flat bands are calculated to be near the Fermi level in the paramagnetic phase. While partially filling a kagome flat band is predicted to give rise to a Stoner-type ferromagnetism, experimental visualization of the magnetic splitting across the ordering temperature has not been reported for any of these systems due to the high ordering temperatures, hence leaving the nature of magnetism in kagome magnets an open question. Here, we probe the electronic structure with angle-resolved photoemission spectroscopy in a kagome magnet thin film FeSn synthesized using molecular beam epitaxy. We identify the exchange-split kagome flat bands, whose splitting persists above the magnetic ordering temperature, indicative of a local moment picture. Such local moments in the presence of the topological flat band are consistent with the compact molecular orbitals predicted in theory. We further observe a large spin-orbital selective band renormalization in the Fe spin majority channel reminiscent of the orbital selective correlation effects in the iron-based superconductors. Our discovery of the coexistence of local moments with topological flat bands in a kagome system echoes similar findings in magic-angle twisted bilayer graphene, and provides a basis for theoretical effort towards modeling correlation effects in magnetic flat band systems.

Quantification of electronic and magnetoelastic mechanisms of first-order magnetic phase transitions from first principles: application to caloric effects in La(FeSi)13

and derived quaternary compounds are well-known for their giant, tunable, magneto- and barocaloric responses around a first-order paramagnetic-ferromagnetic transition near room temperature with low hysteresis. Remarkably, such a transition shows a large spontaneous volume change together with itinerant electron metamagnetic features. While magnetovolume effects are well-established mechanisms driving first-order transitions, purely electronic sources have a long, subtle history and remain poorly understood. Here we apply a disordered local moment picture to quantify electronic and magnetoelastic effects at finite temperature in from first-principles. We obtain results in very good agreement with experiment and demonstrate that the magnetoelastic coupling, rather than purely electronic mechanisms, drives the first-order character and causes at the same time a huge electronic entropy contribution to the caloric response.

Layered polymorph of titanium triiodide

A previously unreported layered spin 1/2 triangular lattice polymorph of TiI3 is described, synthesized under 6 GPa of applied pressure at 900 C, but stable at atmospheric pressure. This air-sensitive material has a CdI2-type layered structure (P-3m1 (#164), a = 4.012 A and c = 6.641 A at 120 K, Z = 1 of Ti0.667I2) with an in-plane triangular lattice, related to that of TiI4 (Ti0.5I2). Although the TiI3 formula is consistent with expectations for a layered honeycomb lattice of spin 1/2 Ti(III), there appears to be disorder in the crystal structure. Magnetic susceptibility and heat capacity measurements suggest that the material undergoes several low temperature phase transitions.

Quantifying temperature-induced magnetic disorder effects on the anisotropy of MnBi

MnBi is remarkable for having a magnetocrystalline anisotropy (MCA) that increases with temperature. This unusual behavior has been attributed to the thermal expansion of the lattice and, more recently, to an anisotropic vibrational free energy. However, the effect of magnetic fluctuations on the MCA has not yet been quantified. Here, first-principles density-functional theory calculations based on the disordered local moment picture (DFT-DLM) are used to calculate the MCA of MnBi in the presence of magnetic disorder. The MCA is obtained from the magnetic torque, calculated as a function of magnetization angle and temperature $T$. At fixed ionic positions, the MCA decays monotonically with increasing $T$. The DFT-DLM torques provide access to the individual anisotropy constants ${\ensuremath{\kappa}}_{l}$ (which parametrize the relation between magnetic energy and magnetization angle), and their dependence on order parameter $m$. The lowest order constant ${\ensuremath{\kappa}}_{2}$ follows single-ion-like behavior at low $T$ but decays as ${m}^{4}$ as $T$ increases, while the higher-order constants deviate strongly from single-ion predictions. Zero-$T$ calculations show that a spin reorientation transition can be triggered by removing 0.1 electrons, resulting in $c$-axis magnetization once 0.25 electrons have been removed. The zero-temperature calculations are cross-validated using an alternative implementation of DFT based on wave functions, plane waves, and pseudopotentials.

MARMOT: magnetism, anisotropy, and more, using the relativistic disordered local moment picture at finite temperature

We present MARMOT, a hybrid Python/FORTRAN implementation of the disordered local moment picture within multiple scattering density-functional theory. MARMOT takes atom-centred, scalar-relativistic potentials and constructs an effective medium (within the coherent potential approximation) to describe the disordered magnetic moment orientations at finite temperature. By solving the single-site scattering problem fully relativistically, spin–orbit effects are included, allowing the magnetocrystalline anisotropy to be calculated. Magnetic transition temperatures, spin and orbital moments, the density-of-states, and analytical parameterizations of the magnetic potential energy surface can also be calculated. Here, we describe the theory and practical implementation of MARMOT, and demonstrate its use by calculating Curie temperatures, magnetizations and anisotropies of bcc Fe, GdFe2 and YCo5.

Ab initiocalculation of the magnetic Gibbs free energy of materials using magnetically constrained supercells

We present a first-principles approach for the computation of the magnetic Gibbs free energy of materials using magnetically constrained supercell calculations. Our approach is based on an adiabatic approximation of slowly varying local moment orientations, the so-called finite-temperature disordered local moment picture. It describes magnetic phase transitions and how electronic and/or magnetostructural mechanisms generate a discontinuous (first-order) character. We demonstrate that the statistical mechanics of the local moment orientations can be described by an affordable number of supercell calculations containing noncollinear magnetic configurations. The applicability of our approach is illustrated by firstly studying the ferromagnetic state in bcc Fe. We then investigate the temperature-dependent properties of a triangular antiferromagnetic state stabilizing in two antiperovskite systems Mn$_3$AN (A = Ga, Ni). Our calculations provide the negative volume expansion of these materials as well as the ab initio origin of the discontinuous character of the phase transitions, electronic and/or magnetostructural, in good agreement with experiment.

First-principles calculations for Gilbert damping constant at finite temperature

We propose a first-principles calculation method for the Gilbert damping constants α at finite temperature. α is described by the torque correlation model in which the electronic structure is computed by the tight-binding linear muffin-tin orbital method. We include the finite-temperature effect as the transverse spin fluctuation in the disordered local moment picture within the coherent potential approximation. Applying the present method to bcc-Fe and L10-FePt, we demonstrate this temperature-dependent α. By comparing our calculated results with experimental results, we find the calculated values are less than half of the experimental values, reflecting the characteristics of the torque correlation model.

Theoretical Study of Temperature Dependence of Spin Susceptibility in Anisotropic Itinerant Ferromagnets

We developed a framework for directly calculating the spin susceptibility of anisotropic itinerant ferromagnets in the full temperature range within the coherent potential approximation in the disordered local moment picture. As a test of our formulation, we demonstrate the computation for the temperature ($T$) dependence of the spin susceptibility for Rashba-type ferromagnets with the Curie temperature $T_\mathrm{C}$. In a certain parameter, we found that the inverse transverse susceptibility $1/\chi_\perp\simeq\mathrm{const.}$ for $T<T_\mathrm{C}$ and $1/\chi_\perp\propto T-T_\mathrm{C}$ for $T>T_\mathrm{C}$, which reflects characteristics of itinerant ferromagnets including a spin-orbit interaction.

Nature of the magnetic moment of cobalt in ordered FeCo alloy

The magnets are typically classified into Stoner and Heisenberg type, depending on the itinerant or localized nature of the constituent magnetic moments. In this work, we investigate theoretically the behaviour of the magnetic moments of iron and cobalt in their B2-ordered alloy. The results based on local spin density approximation for the density functional theory (DFT) suggest that the Co magnetic moment strongly depends on the directions of the surrounding magnetic moments, which usually indicates the Stoner-type mechanism of magnetism. This is consistent with the disordered local moment picture of the paramagnetic state, where the magnetic moment of cobalt gets substantially suppressed. We argue that this is due to the lack of strong on-site electron correlations, which we take into account by employing a combination of DFT and dynamical mean-field theory (DMFT). Within LDA + DMFT, we find a substantial quasiparticle mass renormalization and a non Fermi-liquid behaviour of Fe–3d orbitals. The resulting spectral functions are in very good agreement with measured spin-resolved photoemission spectra. Our results suggest that local correlations play an essential role in stabilizing a robust local moment on Co in the absence of magnetic order at high temperatures.

Paramagnetic phases of two-dimensional magnetic materials

The magnetic long-range ordering states of van der Waals magnets at low temperatures have gained much interest, yet their high-temperature paramagnetic states remain largely unexplored. In this paper, we apply the disordered local moment picture for describing the paramagnetic states of two-dimensional magnetic semiconductors. We calculate the electronic structure of paramagnetic phases of ${\mathrm{CrI}}_{3},\phantom{\rule{0.28em}{0ex}}{\mathrm{CrSiTe}}_{3}$, and ${\mathrm{NiPS}}_{3}$ using the density functional theory with static Hubbard corrections. The semiconducting electronic structures are successfully reproduced without any inclusion of dynamical correlation effects. The local electronic structure of magnetic ions in the paramagnetic phase resembles those in the magnetically ordered phase. The band structures of the paramagnetic phases are also analyzed. The itinerant ferromagnet ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$ is also briefly discussed where the disordered local moment picture is not applicable due to the coexistence of itinerant and local magnetism.

Ferrimagnetism in EuFe4As12 revealed by Eu153 NMR and As75 NQR measurements

Filled skutterudite compound EuFe$_4$As$_{12}$ shows the highest magnetic ordering temperature of $T_{\rm C}$ = 154 K among Eu-based skutterudite compounds, but its magnetic ground state has not been determined yet. Here, we performed $^{153}$Eu nuclear magnetic resonance (NMR) and $^{75}$As nuclear quadrupole resonance (NQR) measurements on EuFe$_4$As$_{12}$ to reveal its magnetic ground state as well as the physical properties from a microscopic point of view. From the temperature and magnetic field dependence of $^{153}$Eu NMR spectrum in the magnetically ordered state, we found that the Eu ions are in Eu$^{2+}$ state with a nearly 7 $\mu_{\rm B}$ corresponding to $S$ = 7/2 spins. Combined with the magnetization measurements which show the reduced saturation moments of 4.5 $\mu_{\rm B}$/f.u., we determined the ground magnetic structure in EuFe$_4$As$_{12}$ to be ferrimagnetic where the Eu$^{2+}$ 4$f$ and the Fe 3$d$ ordered moments are ferromagnetically aligned in each sublattice but the moments between the sublattices are antiferromagnetically aligned. We also found the local distortion at the Eu site from the cubic symmetry in the magnetically ordered state. The relationship between the rattling motion of Eu atoms and the local symmetry of the Eu ions is discussed. From the $^{75}$As NQR nuclear spin-lattice relaxation time measurements as well as $^{153}$Eu NMR measurements, we found that the 4$f$ electrons of the Eu ions are well described by the local moment picture in both the magnetic and paramagnetic metallic states.

Torque magnetometry study of the spin reorientation transition and temperature-dependent magnetocrystalline anisotropy in NdCo5

We present the results of torque magnetometry and magnetic susceptibility measurements to study in detail the spin reorientation transition (SRT) and magnetic anisotropy in the permanent magnet NdCo5. We further show simulations of the measurements using first-principles calculations based on density-functional theory and the disordered local moment picture of magnetism at finite temperatures. The good agreement between theory and experimental data leads to a detailed description of the physics underpinning the SRT. In particular we are able to resolve the magnetization of, and to reveal a canting between, the Nd and Co sublattices. The torque measurements carried out in the ac and ab planes near the easy direction allow us to estimate the anisotropy constants, K1, K2 and K4 and their temperature dependences. Torque curves, τ(γ) recorded by varying the direction of a constant magnetic field in the crystallographic ac plane show a reversal in the polarity as the temperature is changed across the SRT (240 < T < 285 K). Within this domain, τ(γ) exhibits unusual features different to those observed above and below the transition. The single crystals of NdCo5 were grown using the optical floating zone technique.

Temperature-dependent magnetocrystalline anisotropy of rare earth/transition metal permanent magnets from first principles: The light RCo5 (R=Y, La-Gd) intermetallics

Computational design of more efficient rare earth/transition metal (RE-TM) permanent magnets requires accurately calculating the magnetocrystalline anisotropy (MCA) at finite temperature, since this property places an upper bound on the coercivity. Here, we present a first-principles methodology to calculate the MCA of RE-TM magnets which fully accounts for the effects of temperature on the underlying electrons. The itinerant electron TM magnetism is described within the disordered local moment picture, and the localized RE-4f magnetism is described within crystal field theory. We use our model, which is free of adjustable parameters, to calculate the MCA of the RCo$_5$ (R=Y, La-Gd) magnet family for temperatures 0--600 K. We correctly find a huge uniaxial anisotropy for SmCo$_5$ (21.3 MJm$^{-3}$ at 300 K) and two finite temperature spin reorientation transitions for NdCo$_5$. The calculations also demonstrate dramatic valency effects in CeCo$_5$ and PrCo$_5$. Our calculations provide quantitative, first-principles insight into several decades of RE-TM experimental studies.

Structural and magnetic properties ofGdCo5−xNix

GdCo$_5$ may be considered as two sublattices - one of Gd and one of Co - whose magnetizations are in antiparallel alignment, forming a ferrimagnet. Substitution of nickel in the cobalt sublattice of GdCo$_5$ has been investigated to gain insight into how the magnetic properties of this prototype rare-earth/transition-metal magnet are affected by changes in the transition metal sublattice. Polycrystalline samples of GdCo$_{5-x}$Ni$_x$ for 0 $ \leq x \leq $ 5 were synthesized by arc melting. Structural characterization was carried out by powder x-ray diffraction and optical and scanning electron microscope imaging of metallographic slides, the latter revealing a low concentration of Gd$_2$(Co, Ni)$_7$ lamellae for $x \leq 2.5$. Compensation - i.e. the cancellation of the opposing Gd and transition metal moments is observed for $1 \leq x \leq 3$ at a temperature which increases with Ni content; for larger $x$, no compensation is observed below 360 K. A peak in the coercivity is seen at $x \approx 1$ at 10K coinciding with a minimum in the saturation magnetization. Density-functional theory calculations within the disordered local moment picture reproduce the dependence of the magnetization on Ni content and temperature. The calculations also show a peak in the magnetocrystalline anisotropy at similar Ni concentrations to the experimentally observed coercivity maximum.

Ab initio calculations of temperature-dependent magnetostriction of Fe and A2Fe1−xGax within the disordered local moment picture

The fully relativistic disordered local moment (DLM) theory is used to perform calculations of the magnetic torque of tetragonally distorted Fe and fully disordered (A2) Fe1−xGax (0≤x≤0.2) alloys to describe the temperature-dependence of their magnetoelasticity. The finite temperature magnetoelasticity, in particular the magnetoelastic constant B1, is obtained within DLM theory by studying the response of the magnetic torque generated by the magnetocrystalline anisotropy to the application of a tetragonal strain. Calculations of B1 have been performed on bcc Fe across its ferromagnetic temperature range. Our results show good qualitative agreement with experiment, in particular reproducing the anomalous, non-monotonic thermal behavior of bcc Fe’s magnetostriction, which has been largely unexplained for more than 50 years. The method has also been used to calculate the finite temperature magnetoelasticity of the A2 phase of Fe1−xGax alloys as a starting point for further investigations into the giant magnetostriction of Galfenol alloys. Our calculations show that homogeneously doping bcc Fe with Ga does not produce an enhancement in magnetostriction and that the non-monotonic temperature dependence and significant volume dependence are suppressed by increasing Ga content.

First-principles calculations of the magnetocrystalline anisotropy of the prototype 2:17 cell boundary phaseYCo1-x-yFexCuy5

We present a computational study of the compound Y(Co$_{1-x-y}$Fe$_x$Cu$_y$)$_5$ for 0 $\leq x,y \leq 0.2$. This compound was chosen as a prototype for investigating the cell boundary phase believed to play a key role in establishing the high coercivity of commercial Sm-Co 2:17 magnets. Using density-functional theory, we have calculated the magnetization and magnetocrystalline anisotropy at zero temperature for a range of compositions, modeling the doped compounds within the coherent potential approximation. We have also performed finite temperature calculations for YCo$_5$, Y(Co$_{0.838}$Cu$_{0.162}$)$_5$ and Y(Co$_{0.838}$Fe$_{0.081}$Cu$_{0.081}$)$_5$ within the disordered local moment picture. Our calculations find that substituting Co with small amounts of either Fe or Cu boosts the magnetocrystalline anisotropy $K$, but the change in $K$ depends strongly on the location of the dopants. Furthermore, the calculations do not show a particularly large difference between the magnetic properties of Cu-rich Y(Co$_{0.838}$Cu$_{0.162}$)$_5$ and equal Fe-Cu Y(Co$_{0.838}$Fe$_{0.081}$Cu$_{0.081}$)$_5$, despite these two compositions showing different coercivity behavior when found in the cell boundary phase of 2:17 magnets. Our study lays the groundwork for studying the rare earth contribution to the anisotropy of Sm(Co$_{1-x-y}$Fe$_x$Cu$_y$)$_5$, and also shows how a small amount of transition metal substitution can boost the anisotropy field of YCo$_5$.

Magnetic properties of the itinerant A- type antiferromagnet CaCo2P2 studied by Co59 and P31 nuclear magnetic resonance

$^{59}$Co and $^{31}$P nuclear magnetic resonance (NMR) measurements in external magnetic and zero magnetic fields have been performed to investigate the magnetic properties of the A-type antiferromagnetic (AFM) CaCo$_2$P$_2$. NMR data, especially, the nuclear spin lattice relaxation rates 1/$T_1$ exhibiting a clear peak, provide clear evidence for the AFM transition at a N\'eel temperature of $T_{\rm N}\sim$110~K. The magnetic fluctuations in the paramagnetic state were found to be three-dimensional ferromagnetic, suggesting ferromagnetic interaction between Co spins in the ${\it ab}$ plane characterize the spin correlations in the paramagnetic state. In the AFM state below $T_{\rm N}$, we have observed $^{59}$Co and $^{31}$P NMR signals under zero magnetic field. From $^{59}$Co NMR data, the ordered magnetic moments of Co are found to be in $ab$ plane and are estimated to be 0.35 $\mu_{\rm B}$ at 4.2 K. Furthermore, the external field dependence of $^{59}$Co NMR spectrum in the AFM state suggests a very weak magnetic anisotropy of the Co ions and also provides microscopic evidence of canting the Co ordered moments along the external magnetic field directions. The magnetic state of the Co ions in CaCo$_2$P$_2$ is well explained by the local moment picture in the AFM state, although the system is metallic as seen by $1/T_1T$ = constant behavior.

Lattice relaxations in disordered Fe-based materials in the paramagnetic state from first principles

In this work we propose a method for the structural relaxation of magnetic materials in the paramagnetic regime, in an adiabatic fast-magnetism approximation within the disordered local moment (DLM) picture in the framework of density functional theory (DFT). The method is straight forward to implement using any $ab$ $initio$ code that allows for structural relaxations. We illustrate the importance of considering the disordered magnetic state during lattice relaxations by calculating formation energies and geometries for an Fe vacancy and C insterstitial atom in bcc Fe as well as bcc Fe$_{1-x}$Cr$_x$ random alloys in the paramagnetic state. In the vacancy case, the nearest neighbors to the vacancy relax towards the vacancy of 0.16 {\AA} (-5% of the ideal bcc nearest neighbor distance), which is twice as large as the relaxation in the ferromagnetic case. The vacancy formation energy calculated in the DLM state on these positions is 1.60 eV, which corresponds to a reduction of about 0.1 eV compared to the formation energy calculated using DLM but on ferromagnetic-relaxed positions. The carbon interstitial formation energy is found to be 0.41 eV when the DLM relaxed positions are used, as compared to 0.59 eV when the FM-relaxed positions are employed. For bcc Fe$_{0.5}$Cr$_{0.5}$ alloys, the mixing enthalpy is reduced by 5 meV/atom, or about 10%, when the DLM state relaxation is considered, as compared to positions relaxed in the ferromagnetic state.

Rare-earth/transition-metal magnets at finite temperature: Self-interaction-corrected relativistic density functional theory in the disordered local moment picture

Atomic-scale computational modeling of technologically relevant permanent magnetic materials faces two key challenges. First, a material's magnetic properties depend sensitively on temperature, so the calculations must account for thermally induced magnetic disorder. Second, the most widely-used permanent magnets are based on rare-earth elements, whose highly localized 4$f$ electrons are poorly described by standard electronic structure methods. Here, we take two established theories, the disordered local moment picture of thermally induced magnetic disorder and self-interaction-corrected density functional theory, and devise a computational framework to overcome these challenges. Using the new approach, we calculate magnetic moments and Curie temperatures of the rare-earth cobalt (RECo$_5$) family for RE=Y--Lu. The calculations correctly reproduce the experimentally measured trends across the series and confirm that, apart from the hypothetical compound EuCo$_5$, SmCo$_5$ has the strongest magnetic properties at high temperature. An order-parameter analysis demonstrates that varying the RE has a surprisingly strong effect on the Co--Co magnetic interactions determining the Curie temperature, even when the lattice parameters are kept fixed. We propose the origin of this behavior is a small contribution to the density from $f$-character electrons located close to the Fermi level.

Effect of the lattice dynamics on the electronic structure of paramagnetic NiO within the disordered local moment picture

Using the disordered local moments approach in combination with the ab initio molecular dynamics method, we simulate the behavior of a paramagnetic phase of NiO at finite temperatures to investigate the effect of magnetic disorder, thermal expansion, and lattice vibrations on its electronic structure. In addition, we study its lattice dynamics. We verify the reliability of our theoretical scheme via comparison of our results with available experiment and earlier theoretical studies carried out within static approximations. We present the phonon dispersion relations for the paramagnetic rock-salt (B1) phase of NiO and demonstrate that it is dynamically stable. We observe that including the magnetic disorder to simulate the paramagnetic phase has a small yet visible effect on the band gap. The amplitude of the local magnetic moment of Ni ions from our calculations for both antiferromagnetic and paramagnetic phases agree well with other theoretical and experimental values. We demonstrate that the increase of temperature up to 1000 K does not affect the electronic structure strongly. Taking into account the lattice vibrations and thermal expansion at higher temperatures have a major impact on the electronic structure, reducing the band gap from $\ensuremath{\sim}3.5$ eV at 600 K to $\ensuremath{\sim}2.5$ eV at 2000 K. We conclude that static lattice approximations can be safely employed in simulations of the paramagnetic state of NiO up to relatively high temperatures $(\ensuremath{\sim}1000$ K), but as we get closer to the melting temperature vibrational effects become quite large and therefore should be included in the calculations.