Universal distribution of magnetic anisotropy of impurities in ordered and disordered nano-grains

We examine the distribution of the magnetic anisotropy (MA) experienced by a magnetic impurity embedded in a metallic nano-grain. As an example of a generic magnetic impurity with partially filled $d$-shell, we study the case of $d^{1}$ impurities imbedded into ordered and disordered Au nano-grains, described in terms of a realistic band structure. Confinement of the electrons induces a magnetic anisotropy that is large, and can be characterized by 5 real parameters, coupling to the quadrupolar moments of the spin. In ordered (spherical) nano-grains, these parameters exhibit symmetrical structures and reflect the symmetry of the underlying lattice, while for disordered grains they are randomly distributed and, - for stronger disorder, - their distribution is found to be characterized by random matrix theory. As a result, the probability of having small magnetic anisotropies $K_L$ is suppressed below a characteristic scale $\Delta_E$, which we predict to scale with the number of atoms $N$ as $\Delta_E\sim 1/N^{3/2}$. This gives rise to anomalies in the specific heat and the susceptibility at temperatures $T\sim \Delta_E$ and produces distinct structures in the magnetic excitation spectrum of the clusters, that should be possible to detect experimentally

Friedel oscillations induced surface magnetic anisotropy

We present detailed numerical studies of the magnetic anisotropy energy of a magnetic impurity near the surface of metallic hosts (Au and Cu), that we describe in terms of a realistic tight-binding surface Green's function technique. We study the case when spin-orbit coupling originates from the d-band of the host material and we also investigate the case of a strong local spin-orbit coupling on the impurity itself. The splitting of the impurity's spin-states is calculated to leading order in the exchange interaction between the impurity and the host atoms using a diagrammatic Green's function technique. The magnetic anisotropy constant is an oscillating function of the separation d from the surface: it asymptotically decays as 1/d2 and its oscillation period is determined by the extremal vectors of the host's Fermi Surface. Our results clearly show that the host-induced magnetic anisotropy energy is by several orders of magnitude smaller than the anisotropy induced by the local mechanism, which provides sufficiently large anisotropy values to explain the size dependence of the Kondo resistance observed experimentally.Comment: 11 pages, 7 figures, submitted to PR

Theory of anisotropic Rashba splitting of surface states

We investigate the surface Rashba effect for a surface of reduced in-plane symmetry. Formulating a k.p perturbation theory, we show that the Rashba splitting is anisotropic, in agreement with symmetry-based considerations. We show that the anisotropic Rashba splitting is due to the admixture of bulk states of different symmetry to the surface state, and it cannot be explained within the standard theoretical picture supposing just a normal-to-surface variation of the crystal potential. Performing relativistic ab initio calculations we find a remarkably large Rashba anisotropy for an unreconstructed Au(110) surface that is in the experimentally accessible range.Comment: 4 pages, 5 figure

Microscopic origin of Heisenberg and non-Heisenberg exchange interactions in ferromagnetic bcc Fe

By means of first principles calculations we investigate the nature of exchange coupling in ferromagnetic bcc Fe on a microscopic level. Analyzing the basic electronic structure reveals a drastic difference between the $3d$ orbitals of $E_g$ and $T_{2g}$ symmetries. The latter ones define the shape of the Fermi surface, while the former ones form weakly-interacting impurity levels. We demonstrate that, as a result of this, in Fe the $T_{2g}$ orbitals participate in exchange interactions, which are only weakly dependent on the configuration of the spin moments and thus can be classified as Heisenberg-like. These couplings are shown to be driven by Fermi surface nesting. In contrast, for the $E_g$ states the Heisenberg picture breaks down, since the corresponding contribution to the exchange interactions is shown to strongly depend on the reference state they are extracted from. Our analysis of the nearest-neighbour coupling indicates that the interactions among $E_g$ states are mainly proportional to the corresponding hopping integral and thus can be attributed to be of double-exchange origin.Comment: 5 pages, 4 figure

Exchange interaction between magnetic adatoms on surfaces of noble metals

We present first-principles calculations of the exchange interactions between magnetic impurities deposited on (001), (110), and (111) surfaces of Cu and Au and analyze them, in particular, in the asymptotic regime. For the (110) and the (111) surfaces, we demonstrate that the interaction shows an oscillatory behavior as a function of the distance, R, of the impurities and that the amplitude of the oscillations decays as 1/R(2). Furthermore, the frequency of the oscillations is closely related to the length of the Fermi vector of the surface states existing on these surfaces. Due to the asymmetry of the surface-states dispersion, the frequency of the oscillations becomes also asymmetric on the (110) surfaces, while on the Au(111) surface two distinct frequencies are found in the oscillations as a consequence of the Bychkov-Rashba splitting of the surface states. Remarkably, no long-range oscillations of the exchange interaction are observed for the (001) surfaces where the surface states are unoccupied. When burying the impurities beneath the surface layer, oscillations mediated by the bulk states become visible

Higher-order contributions to the Rashba-Bychkov effect with application to Bi/Ag(111) surface alloy

In order to explain the anisotropic Rashba-Bychkov effect observed in several metallic surface-state systems, we use k.p perturbation theory with a simple group-theoretical analysis and construct effective Rashba Hamiltonians for different point groups up to third order in the wavenumber. We perform relativistic ab initio calculations for the Bi/Ag(111) ordered surface alloy and from the calculated splitting of the band dispersion we find evidence of the predicted third-order terms. Furthermore, we derive expressions for the corresponding third-order Rashba parameters to provide a simple explanation to the qualitative difference concerning the Rashba-Bychkov splitting of the surface states at Au(111) and Bi/Ag(111).Comment: 7 pages, 3 figure

Theory of noncollinear interactions beyond Heisenberg exchange: Applications to bcc Fe

We show for a simple noncollinear configuration of the atomistic spins (in particular, where one spin is rotated by a finite angle in a ferromagnetic background) that the pairwise energy variation computed in terms of multiple-scattering formalism cannot be fully mapped onto a bilinear Heisenberg spin model even in the absence of spin-orbit coupling. The non-Heisenberg terms induced by the spin-polarized host appear in leading orders in the expansion of the infinitesimal angle variations. However, an E g − T 2 g symmetry analysis based on the orbital decomposition of the exchange parameters in bcc Fe leads to the conclusion that the nearest-neighbor exchange parameters related to the T 2 g orbitals are essentially Heisenberg-like: they do not depend on the spin configuration, and can, in this case, be mapped onto a Heisenberg spin model even in extreme noncollinear cases.Supports from the Swedish Research Council (VR), the KAW foundation (grants 2012.0031 and 2013.0020) and eSSENCE are acknowledged.The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC). A. Bergman acknowledges support from CEA-Enhanced Eurotalents, co-funded by FP7 Marie Sk lodowska-Curie COFUND Programme (Grant Agreement n 600382). P. F. Bessarab acknowledges support from the Icelandic Research Fund (Grant No. 163048-052) and the mega-grant of the Ministry of Education and Science of the Russian Federation (grant no. 14.Y26.31.0015). R. Cardias, D. C. M. Rodrigues, and A. B. Klautau acknowledge financial support from CAPES and CNPq, Brazil.Peer Reviewe