Theory of Defect-Induced Crystal Field Perturbations in Rare-Earth Magnets.

We present a theory describing the single-ion anisotropy of rare-earth (RE) magnets in the presence of point defects. Taking the RE-lean 1∶12 magnet class as a prototype, we use first-principles calculations to show how the introduction of Ti substitutions into SmFe_{12} perturbs the crystal field, generating new coefficients due to the lower symmetry of the RE environment. We then demonstrate that these perturbations can be described extremely efficiently using a screened point charge model. We provide analytical expressions for the anisotropy energy that can be straightforwardly implemented in atomistic spin dynamics simulations, meaning that such simulations can be carried out for an arbitrary arrangement of point defects. The significant crystal field perturbations calculated here demonstrate that a sample that is single phase from a structural point of view can nonetheless have a dramatically varying anisotropy profile at the atomistic level if there is compositional disorder, which may influence localized magnetic objects like domain walls or skyrmions.

Fermi-Surface Origin of Skyrmion Lattices in Centrosymmetric Rare-Earth Intermetallics.

We show from first principles that barrel-shaped structures within the Fermi surface of the centrosymmetric intermetallic compounds GdRu_{2}Si_{2} and Gd_{2}PdSi_{3} give rise to Fermi surface nesting, which determines the strength and sign of quasi-two-dimensional Ruderman-Kittel-Kasuya-Yosida pairwise exchange interactions between the Gd moments. This is the principal mechanism leading to their helical single-q spin-spiral ground states, providing transition temperatures and magnetic periods in good agreement with experiment. Using atomistic spin-dynamic simulations, we draw a direct line between the subtleties of the three-dimensional Fermi surface topology and the stabilization of a square skyrmion lattice in GdRu_{2}Si_{2} at applied magnetic fields as observed in experiment.

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, K 1, K 2 and K 4 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.

Crystal field coefficients for yttrium analogues of rare-earth/transition-metal magnets using density-functional theory in the projector-augmented wave formalism.

We present a method of calculating crystal field coefficients of rare-earth/transition-metal (RE-TM) magnets within density-functional theory (DFT). The principal idea of the method is to calculate the crystal field potential of the yttrium analogue ('Y-analogue') of the RE-TM magnet, i.e. the material where the lanthanide elements have been substituted with yttrium. The advantage of dealing with Y-analogues is that the methodological and conceptual difficulties associated with treating the highly-localized 4f  electrons in DFT are avoided, whilst the nominal valence electronic structure principally responsible for the crystal field is preserved. In order to correctly describe the crystal field potential in the core region of the atoms we use the projector-augmented wave formalism of DFT, which allows the reconstruction of the full charge density and electrostatic potential. The Y-analogue crystal field potentials are combined with radial 4f  charge densities obtained in self-interaction-corrected calculations on the lanthanides to obtain crystal field coefficients. We demonstrate our method on a test set of ten materials comprising nine RE-TM magnets and elemental Tb. We show that the calculated easy directions of magnetization agree with experimental observations, including a correct description of the anisotropy within the basal plane of Tb and NdCo5. We further show that the Y-analogue calculations generally agree quantitatively with previous calculations using the open-core approximation to treat the 4f  electrons, and argue that our simple approach may be useful for large-scale computational screening of new magnetic materials.

Field-induced canting of magnetic moments in GdCo5 at finite temperature: first-principles calculations and high-field measurements.

We present calculations and experimental measurements of the temperature-dependent magnetization of a single crystal of GdCo5 in magnetic fields of order 60 T. At zero temperature the calculations, based on density-functional theory in the disordered-local-moment picture, predict a field-induced transition from an antiferromagnetic to a canted alignment of Gd and Co moments at 46.1 T. At higher temperatures the calculations find this critical field to increase along with the zero-field magnetization. The experimental measurements observe this transition to occur between 44-48 T at 1.4 K. Up to temperatures of at least 100 K, the experiments continue to observe the transition; however, at variance with the calculations, no strong temperature dependence of the critical field is apparent. We assign this difference to the inaccurate description of the zero-field magnetization of the calculations at low temperatures, due to the use of classical statistical mechanics. Correcting for this effect, we recover a consistent description of the high-field magnetization of GdCo5 from theory and experiment.

Calculating the Magnetic Anisotropy of Rare-Earth-Transition-Metal Ferrimagnets.

Magnetocrystalline anisotropy, the microscopic origin of permanent magnetism, is often explained in terms of ferromagnets. However, the best performing permanent magnets based on rare earths and transition metals (RE-TM) are in fact ferrimagnets, consisting of a number of magnetic sublattices. Here we show how a naive calculation of the magnetocrystalline anisotropy of the classic RE-TM ferrimagnet GdCo_{5} gives numbers that are too large at 0 K and exhibit the wrong temperature dependence. We solve this problem by introducing a first-principles approach to calculate temperature-dependent magnetization versus field (FPMVB) curves, mirroring the experiments actually used to determine the anisotropy. We pair our calculations with measurements on a recently grown single crystal of GdCo_{5}, and find excellent agreement. The FPMVB approach demonstrates a new level of sophistication in the use of first-principles calculations to understand RE-TM magnets.

Theory of Magnetic Ordering in the Heavy Rare Earths: Ab Initio Electronic Origin of Pair- and Four-Spin Interactions.

We describe a disordered local moment theory for long-period magnetic phases and investigate the temperature and magnetic field dependence of the magnetic states in the heavy rare earth elements (HREs), namely, paramagnetic, conical and helical antiferromagnetic (HAFM), fan, and ferromagnetic (FM) states. We obtain a generic HRE magnetic phase diagram which is consequent on the response of the common HRE valence electronic structure to f-electron magnetic moment ordering. The theory directly links the first-order HAFM-FM transition to the loss of Fermi surface nesting, induced by this magnetic ordering, as well as provides a template for analyzing the other phases and exposing where f-electron correlation effects are particularly intricate. Gadolinium, for a range of hexagonal, close-packed lattice constants c and a, is the prototype, described ab initio, and applications to other HREs are made straightforwardly by scaling the effective pair and quartic local moment interactions that emerge naturally from the theory with de Gennes factors and choosing appropriate lanthanide-contracted c and a values.

Verification of Anderson Superexchange in MnO via Magnetic Pair Distribution Function Analysis and ab initio Theory.

We present a temperature-dependent atomic and magnetic pair distribution function (PDF) analysis of neutron total scattering measurements of antiferromagnetic MnO, an archetypal strongly correlated transition-metal oxide. The known antiferromagnetic ground-state structure fits the low-temperature data closely with refined parameters that agree with conventional techniques, confirming the reliability of the newly developed magnetic PDF method. The measurements performed in the paramagnetic phase reveal significant short-range magnetic correlations on a ∼1 nm length scale that differ substantially from the low-temperature long-range spin arrangement. Ab initio calculations using a self-interaction-corrected local spin density approximation of density functional theory predict magnetic interactions dominated by Anderson superexchange and reproduce the measured short-range magnetic correlations to a high degree of accuracy. Further calculations simulating an additional contribution from a direct exchange interaction show much worse agreement with the data. The Anderson superexchange model for MnO is thus verified by experimentation and confirmed by ab initio theory.

Using density functional theory to describe slowly varying fluctuations at finite temperatures: local magnetic moments in Gd and the 'not so local' moments of Ni.

We briefly describe the density functional theory (DFT)-based 'disordered local moment' (DLM) picture for magnetism at finite temperatures. It shows how relatively slowly fluctuating local moments can emerge from the interacting electrons of many materials. Such entities have rigid magnitudes and fluctuate their orientations from atomic site to atomic site on a timescale long compared to other electronic times. We illustrate this theory with calculations of the magnetocaloric effect in Gd where we find excellent agreement with experiments. Fluctuating moments do not appear to establish naturally over such small regions for some other materials. We show how the DFT-DLM theory can be extended to these materials with the use of the Korringa-Kohn-Rostoker nonlocal coherent potential approximation (KKR-NLCPA) to allow for more extensive, slow magnetic fluctuations. We present the first application of this approach by revisiting the description of the magnetic fluctuations prevalent in the paramagnetic state of nickel. We find that local moments can emerge above Tc and that these form coherently over small clumps of atomic sites (4-8 sites).