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A second-order perturbation (2PT) approach to the spin-orbit interaction (SOI) is implemented within a density-functional theory framework. Its performance is examined by applying it to the calculation of the magnetocrystalline anisotropy energies (MAE) of benchmark systems, and its efficiency and accuracy are compared with the popular force theorem method. The case studies are tetragonal FeMe alloys (Me=Co, Cu, Pd, Pt, Au), as well as FeMe (Me=Co, Pt) bilayers with (111) and (100) symmetry, which cover a wide range of SOI strength and electronic band structures. The 2PT approach is found to provide a very accurate description for 3d and 4d metals and, moreover, this methodology is robust enough to predict easy axis switching under doping conditions. In all cases, the details of the bandstructure, including states far from the Fermi level, are responsible for the finally observed MAE value, sometimes overruling the effect of the SOI strength. From a technical point of view, it is confirmed that accuracy in the MAE calculations is subject to the accuracy of the Fermi level determination.

Magnetic topological insulators are narrow-gap semiconductor materials that combine non-trivial band topology and magnetic order 1 . Unlike their nonmagnetic counterparts, magnetic topological insulators may have some of the surfaces gapped, which enables a number of exotic phenomena that have potential applications in spintronics 1 , such as the quantum anomalous Hall effect 2 and chiral Majorana fermions 3 . So far, magnetic topological insulators have only been created by means of doping nonmagnetic topological insulators with 3 d transition-metal elements; however, such an approach leads to strongly inhomogeneous magnetic 4 and electronic 5 properties of these materials, restricting the observation of important effects to very low temperatures 2 , 3 . An intrinsic magnetic topological insulator—a stoichiometric well ordered magnetic compound—could be an ideal solution to these problems, but no such material has been observed so far. Here we predict by ab initio calculations and further confirm using various experimental techniques the realization of an antiferromagnetic topological insulator in the layered van der Waals compound MnBi 2 Te 4 . The antiferromagnetic ordering  that MnBi 2 Te 4  shows makes it invariant with respect to the combination of the time-reversal and primitive-lattice translation symmetries, giving rise to a ℤ 2 topological classification; ℤ 2  = 1 for MnBi 2 Te 4 , confirming its topologically nontrivial nature. Our experiments indicate that the symmetry-breaking (0001) surface of MnBi 2 Te 4 exhibits a large bandgap in the topological surface state. We expect this property to eventually enable the observation of a number of fundamental phenomena, among them quantized magnetoelectric coupling 6 – 8 and axion electrodynamics 9 , 10 . Other exotic phenomena could become accessible at much higher temperatures than those reached so far, such as the quantum anomalous Hall effect 2 and chiral Majorana fermions 3 . An intrinsic antiferromagnetic topological insulator, MnBi 2 Te 4 , is theoretically predicted and then realized experimentally, with implications for the study of exotic quantum phenomena.

We use the constrained random phase approximation (cRPA) method to calculate the Hubbard \\(U\\) parameter in four one-dimensional magnetic transition metal atom oxides of composition XO\\(_2\\) (X = Mn, Fe, Co, Ni) on Ir(100). In addition to the expected screening of the oxide, i.e., a significant reduction of the \\(U\\) value by the presence of the metal substrate, we find a strong dependence on the electronic configuration (multiplet) of the X(\\(d\\)) orbital. Each particular electronic configuration attained by atom X is dictated by the O ligands, as well as by the charge transfer and hybridization with the Ir(100) substrate. We find that MnO\\(_2\\) and NiO\\(_2\\) chains exhibit two different screening regimes, while the case of CoO\\(_2\\) is somewhere in between. The electronic structure of the MnO\\(_2\\) chain remains almost unchanged upon adsorption. Therefore, in this regime, the additional screening is predominantly generated by the electrons of the neighboring metal surface atoms. The screening strength for NiO\\(_2\\)/Ir(100) is found to depend on the Ni(\\(d\\)) configuration in the adsorbed state. The case of FeO\\(_2\\) shows an exceptional behavior, as it is the only insulating system in the absence of metallic substrate and, thus, it has the largest \\(U\\) value. However, this value is significantly reduced by the two mentioned screening effects after adsorption.

DFT calculations within the generalized Bloch theorem approach show that interfacial Dzyaloshinskii-Moriya interactions (DMI) at both interfaces of Graphene/Co\\(_n\\)/Pt(111) multilayer heterostructures are decoupled for \\(n \\geq 3\\). Unlike the property of magnetocrystalline anisotropy for this system, DMI is not affected by stacking defects in the Co layer. The effect of Graphene (Gr) is to invert the chirality of the vaccum/Co interfacial DMI, overall reducing the DMI of the heterostructure, which is nevertheless dominated by the strong spin-orbit coupling (SOC) of Pt. A spectral analysis in the reciprocal space shows that DMI at both the Gr/Co and Co/Pt interfaces have the same nature, namely SOC-split hybrid bands of \\(d\\)-orbital character. This proves that a DMI model based on a single band, such the Rashba DMI model, is insuficient to describe the behaviour of this family of Gr-capped \\(3d/5d\\) metal heterostructures.

In intermetallic compounds with zero-orbital momentum (\\(L=0\\)) the magnetic anisotropy and the electronic band structure are interconnected. Here, we investigate this connection on divalent Eu and trivalent Gd intermetallic compounds. We find by X-ray magnetic circular dichroism an out-of-plane easy magetization axis in 2D atom-thick EuAu\\(_2\\). Angle-resolved photoemission and density-functional theory prove that this is due to strong \\(f-d\\) band hybridization and Eu\\(^{2+}\\) valence. In contrast, the easy in-plane magnetization of the structurally-equivalent GdAu\\(_2\\) is ruled by spin-orbit-split \\(d\\)-bands, notably Weyl nodal lines, occupied in the Gd\\(^{3+}\\) state. Regardless of the \\(L\\) value, we predict a similar itinerant electron contribution to the anisotropy of analogous compounds.

First-principles calculations are used to gauge different levels of approximation to calculate the magnetocrystalline anisotropy energies (MAE) of five \\(L1_0\\) FeMe alloys (Me=Co, Cu, Pd, Pt, Au). We find that a second-order perturbation (2PT) treatment of the spin-orbit interaction (SOI) breaks down for the alloys containing heavier ions, while it provides a very accurate description of the MAE behaviour of FeCo, FeCu, and FePd. Moreover, the robustness of the 2PT approximation is such that in these cases it accounts for the MAE of highly-non-neutral alloys and, thus, it can be used to predict their performance when dopants are present or when they are subject to applied gate bias, which are typical conditions in working magnetoelectric devices. We also observe that switching of the easy axis direction can be induced in some of these alloys by addition or removal of, at least, one electron per cell. In all cases, the details of the bandstructure are responsible for the finally observed MAE value and, therefore, suggest a limited predicting power of models based on the expected orbital moment values and bandwidths. Finally, we have confirmed the importance of various calculation parameters to obtain converged MAE values, in particular, those related to the accuracy of the Fermi level determination.

A combination of theoretical modelling and experiments reveals the origin of the large perpendicular magnetic anisotropy (PMA) that appears in nanometer-thick epitaxial Co films intercalated between graphene (Gr) and a heavy metal (HM) substrate, as a function of the Co thickness. High quality epitaxial Gr/Co\\n/HM(111) (HM=Pt,Ir) heterostructures are grown by intercalation below graphene, which acts as a surfactant that kinetically stabilizes the pseudomorphic growth of highly perfect Co face-centered tetragonal (\\(fct\\)) films, with a reduced number of stacking faults as the only structural defect observable by high resolution scanning transmission electron microscopy (HR-STEM). Magneto-optic Kerr effect (MOKE) measurements show that such heterostructures present PMA up to large Co critical thicknesses of about 4~nm (20~ML) and 2~nm (10~ML) for Pt and Ir substrates, respectively, while X-ray magnetic circular dichroism (XMCD) measurements show an inverse power law of the anistropy of the orbital moment with Co thickness, reflecting its interfacial nature, that changes sign at about the same critical values. First principles calculations show that, regardless of the presence of graphene, ideal Co \\(fct\\) films on HM buffers do not sustain PMAs beyond around 6~MLs due to the in-plane contribution of the inner bulk-like Co layers. The large experimental critical thicknesses sustaining PMA can only be retrieved by the inclusion of structural defects that promote a local \\(hcp\\) stacking such as twin boundaries or stacking faults. Remarkably, a layer resolved analysis of the orbital momentum anisotropy reproduces its interfacial nature, and reveals that the Gr/Co interface contribution is comparable to that of the Co/Pt(Ir).

The magnetic anisotropy and exchange coupling between spins localized at the positions of 3d transition metal atoms forming two-dimensional metal–organic coordination networks (MOCNs) grown on a Au(111) metal surface are studied. In particular, we consider MOCNs made of Ni or Mn metal centers linked by 7,7,8,8-tetracyanoquinodimethane (TCNQ) organic ligands, which form rectangular networks with 1:1 stoichiometry. Based on the analysis of X-ray magnetic circular dichroism (XMCD) data taken at T = 2.5 K, we find that Ni atoms in the Ni–TCNQ MOCNs are coupled ferromagnetically and do not show any significant magnetic anisotropy, while Mn atoms in the Mn–TCNQ MOCNs are coupled antiferromagnetically and do show a weak magnetic anisotropy with in-plane magnetization. We explain these observations using both a model Hamiltonian based on mean-field Weiss theory and density functional theory calculations that include spin–orbit coupling. Our main conclusion is that the antiferromagnetic coupling between Mn spins and the in-plane magnetization of the Mn spins can be explained by neglecting effects due to the presence of the Au(111) surface, while for Ni–TCNQ the metal surface plays a role in determining the absence of magnetic anisotropy in the system.

Quantum states of matter combining non-trivial topology and magnetism attract a lot of attention nowadays; the special focus is on magnetic topological insulators (MTIs) featuring quantum anomalous Hall and axion insulator phases. Feasibility of many novel phenomena that \\emph{intrinsic} magnetic TIs may host depends crucially on our ability to engineer and efficiently tune their electronic and magnetic structures. Here, using angle- and spin-resolved photoemission spectroscopy along with \\emph{ab initio} calculations we report on a large family of intrinsic magnetic TIs in the homologous series of the van der Waals compounds (MnBi\\(_2\\)Te\\(_4\\))(Bi\\(_2\\)Te\\(_3\\))\\(_m\\) with \\(m=0, ..., 6\\). Magnetic, electronic and, consequently, topological properties of these materials depend strongly on the \\(m\\) value and are thus highly tunable. The antiferromagnetic (AFM) coupling between the neighboring Mn layers strongly weakens on moving from MnBi2Te4 (m=0) to MnBi4Te7 (m=1), changes to ferromagnetic (FM) one in MnBi6Te10 (m=2) and disappears with further increase in m. In this way, the AFM and FM TI states are respectively realized in the \\(m=0,1\\) and \\(m=2\\) cases, while for \\(m \\ge 3\\) a novel and hitherto-unknown topologically-nontrivial phase arises, in which below the corresponding critical temperature the magnetizations of the non-interacting 2D ferromagnets, formed by the \\MBT\\, building blocks, are disordered along the third direction. The variety of intrinsic magnetic TI phases in (MnBi\\(_2\\)Te\\(_4\\))(Bi\\(_2\\)Te\\(_3\\))\\(_m\\) allows efficient engineering of functional van der Waals heterostructures for topological quantum computation, as well as antiferromagnetic and 2D spintronics.

Sykes {\\it et al.} [Proc. Natl. Acad. Sci. {\\bf 102}, 17907 (2005)] have reported how electrons injected from a scanning tunneling microscope modify the diffusion rates of H buried beneath Pd(111). A key point in that experiment is the symmetry between positive and negative voltages for H extraction, which is difficult to explain in view of the large asymmetry in Pd between the electron and hole densities of states. Combining concepts from the theory of ballistic electron microscopy and electron-phonon scattering we show that H diffusion is driven by the \\(s\\)-band electrons only, which explains the observed symmetry.