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A scheme is presented to calculate on a first-principles level the spin-lattice coupling (SLC) parameters needed to perform combined molecular-spin dynamics (MSD) simulations. By treating changes to the spin configuration and atomic positions on the same level, closed expressions for the atomic SLC parameters could be derived in a coherent way up to any order. The properties of the SLC parameters are discussed considering separately the symmetric and antisymmetric parts of the SLC tensor. The changes due to atomic displacements of the spin-spin exchange coupling (SSC) parameters estimated using the SLC parameters are compared with the SSC parameters calculated for an embedded cluster with the central atom displaced, demonstrating good agreement of these results. Moreover, this allows to study the impact of different SLC contributions, linear and quadratic with respect to displacements, on the properties of the modified SSC parameters. In addition, we represent an approach to calculate the site-diagonal SLC parameters characterizing local magnetic anisotropy induced by a lattice distortion, which is a counterpart of the approach based on magnetic torque used for the investigations of magneto-crystalline anisotropy (MCA) as well as for calculations of the MCA constants. In particular, the dependence of the induced magnetic torque on different types of atomic displacements is analyzed., Comment: 16 pages

The transfer and control of angular momentum is a key aspect for spintronic applications. Only recently, it was shown that it is possible to transfer angular momentum from the spin system to the lattice on ultrashort time scales. In an attempt to contribute to the understanding of angular momentum transfer between spin and lattice degrees of freedom we present a scheme to calculate fully-relativistic spin-lattice coupling parameters from first-principles. By treating changes in the spin configuration and atomic positions at the same level, closed expressions for the atomic spin-lattice coupling parameters can be derived in a coherent manner up to any order. Analyzing the properties of these parameters, in particular their dependence on spin-orbit coupling, we find that even in bcc Fe the leading term for the angular momentum exchange between the spin system and the lattice is a Dzyaloshiskii-Moriya-type interaction, which is due to the symmetry breaking distortion of the lattice., Comment: 4 figures

In the spirit of multi-scale modeling, we develop a theoretical framework for spin-lattice coupling that connects, on the one hand, to ab initio calculations of spin-lattice coupling parameters and, on the other hand, to the magneto-elastic continuum theory. The derived Hamiltonian describes a closed system of spin and lattice degrees of freedom and explicitly conserves the total momentum, angular momentum and energy. Using a new numerical implementation that corrects earlier Suzuki-Trotter decompositions we perform simulations on the basis of the resulting equations of motion to investigate the combined magnetic and mechanical motion of a ferromagnetic nanoparticle, thereby validating our developed method. In addition to the ferromagnetic resonance mode of the spin system we find another low-frequency mechanical response and a rotation of the particle according to the Einstein-de-Haas effect. The framework developed herein will enable the use of multi-scale modeling for investigating and understanding a broad range of magneto-mechanical phenomena from slow to ultrafast time scales.

We develop a formulation of the coherent potential approximation (CPA) on the basis of the Wannier representation to develop a computationally efficient method for the treatment of homogeneous random alloys that is independent on the applied first-principles electric structure code. To verify the performance of this CPA implementation within the Wannier representation, we examine the Bloch spectral function, the density of states (DOS), and the magnetic moment in Fe-based transition-metal alloys Fe-X (X = V, Co, Ni, and Cu), and compare the results with those of the well-established CPA implementation based on the KKR Green's function method. The Wannier-CPA and the KKR-CPA lead to results very close to each other. The presented Wannier-CPA method has a wide potential applicability to other physical quantities and large compound systems because of its low computational effort required., 10 pages, 7 figures

While the ground state of magnetic materials is in general well described on the basis of spin density functional theory (SDFT), the theoretical description of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. Time-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are computationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin Hamiltonian and SDFT-based electronic structure calculations, giving access to the dynamical and finite-temperature properties for example via spin-dynamics simulations using the Landau–Lifshitz–Gilbert (LLG) equation or Monte Carlo simulations. We present an overview on the various methods to calculate the parameters of the various phenomenological Hamiltonians with an emphasis on the KKR Green function method as one of the most flexible band structure methods giving access to practically all relevant parameters. Concerning these, it is crucial to account for the spin–orbit coupling (SOC) by performing relativistic SDFT-based calculations as it plays a key role for magnetic anisotropy and chiral exchange interactions represented by the DMI parameters in the spin Hamiltonian. This concerns also the Gilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral multispin interaction parameters of the extended Heisenberg Hamiltonian, as well as spin–lattice interaction parameters describing the interplay of spin and lattice dynamics processes, for which an efficient computational scheme has been developed recently by the present authors.

The mechanism behind the three-spin chiral interaction (TCI) included in the extended Heisenberg Hamiltonian and represented by an expression worked out recently (Phys.\ Rev.\ B, {\bf 101}, 174401 (2020)) is discussed. It is stressed that this approach provides a unique set of the multispin exchange parameters which are independent of each other either due to their different order of perturbation or due to different symmetry. This ensures in particular the specific properties of the TCI that were demonstrated previously via fully relativistic first principles calculations, and that result from the common influence of several issues not explicitly seen from the expression for the TCI parameters. Therefore, an interpretation of the TCI is suggested, showing explicitly its dependence on the relativistic spin-orbit coupling and on the topological orbital susceptibility (TOS). This is based on an expression for the TOS that is worked out on the same footing as the expression for the TCI. Using first-principles calculations we demonstrate in addition numerically the common topological properties of the TCI and TOS. To demonstrate the role of the relativistic spin-orbit coupling (SOC) for the TCI, a so-called topological' spin susceptibility (TSS) is introduced. This quantity characterizes the SOC induced spin magnetic moment on the atom in the presence of non-collinear magnetic structure, giving a connection between the TOS and TCI. Numerical results again support our conclusions., 11 pages, 6 figures

Based on first principles calculations we investigate the electronic and magnetic properties of Pt layers in Pd$(001)$/Co/Pt thin film structures exposed to an external electric field. Due to the Co underlayer, the surface Pt layers have induced moments that are modified by an external electric field. The field induced changes can be explained by the modified spin-dependent orbital hybridization that varies non-linearly with the field strength. We calculate the x-ray absorption and the x-ray magnetic circular dichroism spectra for an applied external electric field and examine its impact on the spectra in the Pt layer around the L$_{2}$ and L$_{3}$ edges. We also determine the layer dependent magneto-crystalline anisotropy and show that the anisotropy can be tuned easily in the different layers by the external electric field., 9 pages, 11 figures

We demonstrate the low temperature suppression of the platinum (Pt) spin Nernst angle in bilayers consisting of the antiferromagnetic insulator hematite ($\alpha$-Fe$_2$O$_3$) and Pt upon measuring the transverse spin Nernst magnetothermopower (TSNM). We show that the observed signal stems from the interplay between the interfacial spin accumulation in Pt originating from the spin Nernst effect and the orientation of the N\'eel vector of $\alpha$-Fe$_2$O$_3$, rather than its net magnetization. Since the latter is negligible in an antiferromagnet, our device is superior to ferromagnetic structures, allowing to unambiguously distinguish the TSNM from thermally excited magnon transport (TMT), which usually dominates in ferri/ferromagnets due to their non-zero magnetization. Evaluating the temperature dependence of the effect, we observe a vanishing TSNM below ~100 K. We compare these results with theoretical calculations of the temperature dependent spin Nernst conductivity and find excellent agreement. This provides evidence for a vanishing spin Nernst angle of Pt at low temperatures and the dominance of extrinsic contributions to the spin Nernst effect., Comment: 4 pages, 4 figures

An approach to account for the effect of thermal lattice vibrations when calculating exchange coupling parameters is presented on the basis of the Korringa-Kohn-Rostoker Green function method making use of the alloy analogy model. Using several representative systems, it is shown that depending on the material the effect of thermal lattice vibrations can have a significant impact on the isotropic exchange as well as the Dzyaloshinskii-Moriya interactions (DMI). This should lead in turn to an additional contribution to the temperature dependence of the magnetic properties of solids, which cannot be neglected in the general case. As an example, we discuss such an influence on the critical temperature of various magnetic phase transitions. In particular, in the case of skyrmion hosting materials, a strong impact of lattice vibrations on the DMI is an additional source for the temperature dependent skyrmion size and stability, which should be taken into consideration. The present approach gives also access to features of magnetic properties that are associated with static atomic displacements as, for example, in random high-entropy alloys.

Ab initio calculations of the spin wave stiffness constant $D$ for elemental Fe and Ni performed by different groups in the past have led to values with a considerable spread of 50-100 %. We present results for the stiffness constant $D$ of Fe, Ni, and permalloy Fe$_{0.19}$Ni$_{0.81}$ obtained by three different approaches: (i) by finding the quadratic term coefficient of the power expansion of the spin wave energy dispersion, (ii) by a damped real-space summation of weighted exchange coupling constants, and (iii) by integrating the appropriate expression in reciprocal space. All approaches are implemented by means of the same Korringa-Kohn-Rostoker (KKR) Green function formalism. We demonstrate that if properly converged, all procedures yield comparable values, with uncertainties of 5-10 % remaining. By a careful analysis of the influence of various technical parameters we estimate the margin of errors for the stiffness constants evaluated by different approaches and suggest procedures to minimize the risk of getting incorrect results., Comment: 11 pages, 7 figures, 12 tables

Taking the noncollinear antiferromagnetic hexagonal Heusler compound ${\mathrm{Mn}}_{3}\mathrm{Ge}$ as a reference system, the contributions to linear response phenomena arising solely from the chiral coplanar and noncoplanar spin configurations are investigated. Orbital moments, x-ray absorption, and anomalous and spin Hall effects, as well as corresponding spin-orbit torques and Edelstein polarizations, are studied depending on a continuous variation of the polar angle relative to the kagome planes of corner-sharing triangles between the noncollinear antiferromagnetic and the ferromagnetic limits. By scaling the speed of light from the relativistic Dirac case to the nonrelativistic limit the chirality-induced or topological contributions can be identified by suppressing the spin-orbit coupling.

Previous studies on the anomalous Hall effect in coplanar noncollinear antiferromagnets are revisited and extended to magneto-optic properties, namely magneto-optic Kerr effect (MOKE) and x-ray magnetic dichroism (XMCD). Starting from group-theoretical considerations the shape of the frequency-dependent conductivity tensor for various actual and hypothetical spin configurations in cubic and hexagonal ${\mathrm{Mn}}_{3}X$ compounds is determined. Calculated MOKE and x-ray dichroism spectra are used to confirm these findings and to give estimates of the size of the effects. For ${\mathrm{Mn}}_{3}\mathrm{IrPt}$ and ${\mathrm{Mn}}_{3}\mathrm{PtRh}$ alloys the concentration dependence of the anomalous and spin Hall conductivity is studied in addition.

An extension of the Heisenberg Hamiltonian is discussed that allows us to go beyond the standard bilinear spin Hamiltonian taking into account various contributions due to multispin interactions having both chiral and nonchiral character. The parameters of the extended Hamiltonian are calculated from first principles within the framework of the multiple-scattering Green's function formalism giving access to an explicit representation of these parameters in real space. The discussions are focused on the chiral interactions, i.e., biquadratic and three-spin Dzyaloshinskii-Moriya-like vector interactions ${\stackrel{P\vec}{\mathcal{D}}}_{ijij}$ (BDMI) and ${\stackrel{P\vec}{\mathcal{D}}}_{ijkj}$ (TDMI), respectively, as well as the three-spin chiral interaction ${J}_{ijk}$. Although all parameters are driven by spin-orbit coupling (SOC), some differences in their properties are demonstrated by calculations for real materials. In particular it is shown that the three-spin chiral interactions ${J}_{ijk}$ may be topology as well as SOC induced, while the TDMI is associated only with the SOC. As the magnitude of the chiral interactions can be quite sizable, they can lead to a stabilization of a noncollinear magnetic texture in some materials that is absent when these interactions are neglected.

Transport properties of permalloy doped with V, Co, Pt, and Au are explored via ab-initio calculations. The Kubo-Bastin formula is evaluated within the fully relativistic Korringa-Kohn-Rostoker Green function formalism. Finite temperature effects are treated by means of the alloy analogy model. It is shown that the fact that the host is disordered and not crystalline has a profound effect on how the conductivities characterizing the anomalous Hall effect and the spin Hall effect depend on the dopant concentration. Several relationships between quantities characterizing charge and spin transport are highlighted. The decrease of the longitudinal charge conductivity with increasing doping depends on the dopant type, following the sequence Co-Au-Pt-V. The dependence of the anomalous Hall and spin Hall conductivities on the dopant concentration is found to be non-monotonic. Introducing a finite temperature changes the overall trends significantly. The theoretical results are compared with available experimental data., Comment: 29 pages, 15 figures, 2 tables

An accurate scheme to calculate the interatomic Dzyaloshinskii-Moriya interaction (DMI) parameters is presented, which is based on the fully relativistic Korringa-Kohn-Rostoker Green function (KKR-GF) technique. Corresponding numerical results are compared with those obtained using other schemes reported in the literature. The differences found can be attributed primarily to the different reference states used in the various approaches. The advantage of the present approach is that the DMI vectors can in a natural way be connected with the DMI-related torques on local magnetic moments. In addition an expression for the DMI parameters formulated for a micromagnetic model Hamiltonian is presented that provides a connection to the DMI parameters calculated for atomistic Hamiltonians. This formulation also allows the discussion of the DMI in terms of specific features of the electronic band structure.

The temperature dependent stability of the magnetic phases of FeRh were investigated by means of total energy calculations with magnetic disorder treated within the uncompensated disordered local moment (uDLM) approach. In addition, Monte Carlo simulations based on the extended Heisenberg model have been performed, using exchange coupling parameters obtained rom first principles. The crucial role and interplay of two factors in the metamagnetic transition in FeRh has been revealed, namely the dependence of the Fe-Fe exchange coupling parameters on the temperature-governed degree of magnetic disorder in the system and the stabilizing nature of the induced magnetic moment on Rh-sites. An important observation is the temperature dependence of these two competing factors.

A scheme is presented that is based on the alloy analogy model and allows one to account for thermal lattice vibrations as well as spin fluctuations when calculating response quantities in solids. Various models to deal with spin fluctuations are discussed concerning their impact on the resulting temperature-dependent magnetic moment, longitudinal conductivity, and Gilbert damping parameter. It is demonstrated that, by using the Monte Carlo (MC) spin configuration as input, the alloy analogy model is capable of reproducing the results of MC simulations on the average magnetic moment within all spin fluctuation models under discussion. On the other hand, the response quantities are much more sensitive to the spin fluctuation model. Separate calculations accounting for the thermal effect due to either lattice vibrations or spin fluctuations show that they give comparable contributions to the electrical conductivity and Gilbert damping. However, comparison to results accounting for both thermal effects demonstrates violation of Matthiessen's rule, showing the nonadditive effect of lattice vibrations and spin fluctuations. The results obtained for bcc Fe and fcc Ni are compared with the experimental data, showing rather good agreement for the temperature-dependent electrical conductivity and the Gilbert damping parameter.

We investigate to what extent the magnetic properties of deposited nanostructures can be influenced by selecting as a support different surfaces of the same substrate material. Fully relativistic ab initio calculations were performed for Co monolayers and adatoms on Pd(100), Pd(111), and Pd(110) surfaces. Changing the crystallographic orientation of the surface has a moderate effect on the spin magnetic moment, a larger effect on the orbital magnetic moment, but sometimes a dramatic effect on the magnetocrystalline anisotropy energy and on the magnetic dipole term ${T}_{\ensuremath{\alpha}}$. The dependence of ${T}_{\ensuremath{\alpha}}$ on the magnetization direction $\ensuremath{\alpha}$ can lead to a strong apparent anisotropy of the spin magnetic moment as deduced from the x-ray magnetic circular dichroism sum rules. For systems in which the spin-orbit coupling is not very strong, the ${T}_{\ensuremath{\alpha}}$ term can be understood as arising from the differences between components of the spin magnetic moment associated with different magnetic quantum numbers $m$.

We present a detailed theoretical investigation on the magnetic properties of small single-layered Fe, Co, and Ni clusters deposited on Ir(111), Pt(111), and Au(111). For this, a fully relativistic ab initio scheme based on density functional theory has been used. We analyze the element, size, and geometry specific variations of the atomic magnetic moments and their mutual exchange interactions as well as the magnetic anisotropy energy in these systems. Our results show that the atomic spin magnetic moments in the Fe and Co clusters decrease almost linearly with increasing coordination number on all three substrates, while the corresponding orbital magnetic moments appear to be much more sensitive to the local atomic environment. The isotropic exchange interaction among the cluster atoms is always very strong for Fe and Co exceeding the values for bulk bcc Fe and hcp Co, whereas the anisotropic Dzyaloshinski-Moriya interaction is, in general, one or two orders of magnitude smaller when compared to the isotropic one. For the magnetic properties of Ni clusters, the magnetic properties can show quite a different behavior, and we find in this case a strong tendency towards noncollinear magnetism.

A Kubo-Greenwood-like equation for the Gilbert damping parameter $\alpha$ is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure method in combination with Coherent Potential Approximation (CPA) alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for a series of alloys of permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of $\alpha$ when small amounts of substitutional Cu are introduced.

Recently, the interplay between spin and lattice degrees of freedom has gained a lot of attention due to its importance for various fundamental phenomena as well as for spintronic and magnonic applications. Examples are ultrafast angular momentum transfer between the spin and lattice subsystems during ultrafast demagnetization, frustration driven by structural distortions in transition-metal oxides, or in acoustically driven spin-wave resonances. In this work, we provide a systematic analysis of spin-lattice interactions for ferro- and antiferromagnetic materials and focus on the role of lattice symmetries and dimensions, magnetic order, and the relevance of spin-lattice interactions for angular momentum transfer as well as magnetic frustration. For this purpose, we use a recently developed scheme, which allows an efficient calculation of spin-lattice interaction tensors from first principles. In addition to that, we provide a more accurate and self-consistent scheme to calculate ab initio spin-lattice interactions by using embedded clusters, which allows us to benchmark the performance of the scheme introduced previously. published