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The orbital Hall effect has been theoretically predicted but its direct observation is a challenge. Here, we report the magneto-optical detection of current-induced orbital accumulation at the surface of a light 3$d$ transition metal, Cr. The orbital polarization is in-plane, transverse to the current direction, and scales linearly with current density, consistent with the orbital Hall effect. Comparing the thickness-dependent magneto-optical measurements with $\textit{ab initio}$ calculations, we estimate an orbital diffusion length in Cr of $6.6\pm 0.6$ nm., Comment: 6 pages, 5 figures

Identifying the microscopic nature of non-equilibrium energy transfer mechanisms among electronic, spin and lattice degrees of freedom is central for understanding ultrafast phenomena such as manipulating magnetism on the femtosecond timescale. Here we use time and angle-resolved photoemission spectroscopy to go beyond the often-employed ensemble-averaged view of non-equilibrium dynamics in terms of quasiparticle temperature evolutions. We show for ferromagnetic Ni that the non-equilibrium electron and spin dynamics display pronounced variations with electron momentum whereas the magnetic exchange interaction remains isotropic. This highlights the influence of lattice-mediated scattering processes and opens a pathway towards unraveling the still elusive microscopic mechanism of spin-lattice angular momentum transfer.

Atomistic spin dynamics (ASD) is a standard tool to model the magnetization dynamics of a variety of materials. The fundamental dynamical model underlying ASD is entirely classical. In this paper, we present two approaches to effectively incorporate quantum effects into ASD simulations, thus enhancing their low temperature predictions. The first allows to simulate the magnetic behavior of a quantum spin system by solving the equations of motion of a classical spin system at an effective temperature relative to the critical temperature. This effective temperature is determined a priori from the microscopic properties of the system. The second approach is based on a \semi model where classical spins interact with an environment with a quantum-like power spectrum. The parameters that characterize this model can be calculated ab initio or extracted from experiments. This semi-classical model quantitatively reproduces the absolute temperature behavior of a magnetic system, thus accounting for the quantum mechanical aspects of its dynamics, even at low temperature. The methods presented here can be readily implemented in current ASD simulations with no additional complexity cost., Comment: 9 pages, 5 figures

In optimal covariance cleaning theory, minimizing the Frobenius norm between the true population covariance matrix and a rotational invariant estimator is a key step. This estimator can be obtained asymptotically for large covariance matrices, without knowledge of the true covariance matrix. In this study, we demonstrate that this minimization problem is equivalent to minimizing the loss of information between the true population covariance and the rotational invariant estimator for normal multivariate variables. However, for Student's t distributions, the minimal Frobenius norm does not necessarily minimize the information loss in finite-sized matrices. Nevertheless, such deviations vanish in the asymptotic regime of large matrices, which might extend the applicability of random matrix theory results to Student's t distributions. These distributions are characterized by heavy tails and are frequently encountered in real-world applications such as finance, turbulence, or nuclear physics. Therefore, our work establishes a connection between statistical random matrix theory and estimation theory in physics, which is predominantly based on information theory.

The equilibrium properties of nanoscale systems can deviate significantly from standard thermodynamics due to their coupling to an environment. For the generalised $\theta$-angled spin-boson model, we first derive a compact and general form of the classical equilibrium state including environmental corrections to all orders. Secondly, for the quantum spin-boson model we prove, by carefully taking a large spin limit, that Bohr's quantum-classical correspondence persists at all coupling strengths. This correspondence gives insight into the conditions for a coupled quantum spin to be well-approximated by a coupled classical spin-vector. Thirdly, we demonstrate that previously identified environment-induced 'coherences' in the equilibrium state of weakly coupled quantum spins, do not disappear in the classical case. Finally, we provide the first classification of the coupling parameter regimes for the spin-boson model, from weak to ultrastrong, both for the quantum case and the classical setting. Our results shed light on the interplay of quantum and mean force corrections in equilibrium states of the spin-boson model, and will help draw the quantum to classical boundary in a range of fields, such as magnetism and exciton dynamics., Comment: Expanded proof details in main text; reduces supplementary material

We explore the possibility of ultrafast, coherent all-optical magnetization switching in antiferromagnets by studying the action of the inverse Faraday effect in CrPt, an easy-plane antiferromagnet. Using a combination of density functional theory and atomistic spin dynamics simulations, we show how a circularly polarized laser pulse can switch the order parameter of the antiferromagnet within a few hundred femtoseconds. This nonthermal switching takes place on an elliptical path, driven by the staggered magnetic moments induced by the inverse Faraday effect and leading to reliable switching between two perpendicular magnetic states., Comment: 6 pages, 6 figures

The anomalous Hall effect (AHE) is a fundamental spintronic charge-to-charge-current conversion phenomenon and closely related to spin-to-charge-current conversion by the spin Hall effect. Future high-speed spintronic devices will crucially rely on such conversion effects at terahertz (THz) frequencies. Here, we reveal that the AHE remains operative from DC up to 40 THz with a flat frequency response in thin films of three technologically relevant magnetic materials: DyCo$_{5}$, Co$_{32}$Fe$_{68}$ and Gd$_{27}$Fe$_{73}$. We measure the frequency-dependent conductivity-tensor elements ${\sigma}_{xx}$ and ${\sigma}_{yx}$ and find good agreement with DC measurements. Our experimental findings are fully consistent with ab-initio calculations of ${\sigma}_{yx}$ for CoFe and highlight the role of the large Drude scattering rate (~100 THz) of metal thin films, which smears out any sharp spectral features of the THz AHE. Finally, we find that the intrinsic contribution to the THz AHE dominates over the extrinsic mechanisms for the Co$_{32}$Fe$_{68}$ sample. The results imply that the AHE and related effects such as the spin Hall effect are highly promising ingredients of future THz spintronic devices reliably operating from DC to 40 THz and beyond.

Electrical control of magnetization is of crucial importance for integrated spintronics devices. Spin-orbit torques (SOT) in heavy-metal/ferromagnetic heterostructures have emerged as promising tool to achieve efficiently current-induced magnetization reversal. However, the microscopic origin of the SOT is being debated,with the spin Hall effect (SHE) due to nonlocal spin currents and the spin Rashba-Edelstein effect (SREE) due to local spin polarization at the interface being the primary candidates. We investigate the electrically induced out-of-equilibrium spin and orbital polarizations in pure Pt films and in Pt/3d-metal (Co, Ni, Cu) bilayer films using ab initio electronic structure methods and linear-response theory. We compute atom-resolved response quantities that allow us to identify the induced spin-polarization contributions that lead to fieldlike SOTs, mostly associated with the SREE, and dampinglike (DL) SOTs, mostly associated with the SHE, and compare their relative magnitude, dependence on the magnetization direction, as well as their Pt-layer thickness dependence. We find that both the FL and DL components contribute to the resulting SOT at the Pt/Co and Pt/Ni interfaces, with the former contributions being larger at the Pt interface layer and the latter larger in the Co or Ni layers. Our calculations show that the electrically-induced transverse orbital polarization is exceedingly larger than the induced spin polarization and present even without spin-orbit coupling, in contrast to the spin polarization., Comment: 21 pages, 15 figures

Amorphous Tb$_{x}$Co$_{100-x}$ magnetic alloys exhibit a list of intriguing properties, such as perpendicular magnetic anisotropy, high magneto-optical activity and magnetization switching using ultrashort optical pulses. Varying the Tb:Co ratio in these alloys allows for tuning properties such as the saturation magnetic moment, coercive field and the performance of the light-induced magnetization switching. In this work, we investigate the magnetic, optical and magneto-optical properties of various Tb$_{x}$Co$_{100-x}$ thin film alloy compositions. We report on the effect the choice of different seeding layers has on the structural and magnetic properties of Tb$_{x}$Co$_{100-x}$ layers. We also demonstrate that for a range of alloys, deposited on fused silica substrates, with Tb content of 24-30 at.$\%$, helicity dependent all-optical switching of magnetization can be achieved, albeit in a multi-shot framework. We explain this property to arise from the helicity-dependent laser induced magnetization on the Co sublattice due to the inverse Faraday effect. Our study provides an insight into material aspects for future potential hybrid magneto-plasmonic TbCo-based architectures., Comment: 17 pages, 18 figures

Efficient manipulation of magnetic order with electric current pulses is desirable for achieving fast spintronic devices. The Rashba-Edelstein effect, wherein a spin polarization is electrically induced in noncentrosymmetric systems, provides a mean to achieve current-induced staggered spin-orbit torques. Initially predicted for spin, the orbital counterpart of this effect has been disregarded up to now. Here, we present a generalized Rashba-Edelstein effect, which generates not only spin polarization but also orbital polarization, which we find to be far from being negligible and could play a crucial role in the magnetization dynamics. We show that the orbital Rashba-Edelstein effect does not require spin-orbit coupling to exist. We present first-principles calculations of the frequency-dependent spin and orbital Rashba-Edelstein susceptibility tensors for the noncentrosymmetric antiferromagnets CuMnAs and Mn2Au. We show that the electrically induced local magnetization has both staggered in-plane components and non-staggered out-of-plane components, and can exhibit Rashba-like or Dresselhaus-like symmetries, depending on the magnetic configuration. Furthermore, there is an induced local magnetization on the nonmagnetic atoms as well, that is smaller in Mn2Au than in CuMnAs. We compute sizable induced magnetizations at optical frequencies, which suggest that electric-field driven switching could be achieved at much higher frequencies., Comment: 10 pages, 5 figures

Time-resolved coherent X-ray diffraction is used to measure the spatially resolved magnetization structure within FePt nanoparticles during laser-induced ultrafast demagnetization. The momentum-dependent X-ray magnetic diffraction shows that demagnetization proceeds at different rates at different X-ray momentum transfer. We show that the observed momentum-dependent scattering has the signature of inhomogeneous demagnetization within the nanoparticles, with the demagnetization proceeding more rapidly at the boundary of the nanoparticle. A shell region of reduced magnetization forms and moves inwards at a supermagnonic velocity. Spin-transport calculations show that the shell formation is driven by superdiffusive spin flux mainly leaving the nanoparticle into the surrounding carbon. Quantifying this non-local contribution to the demagnetization allows us to separate it from the local demagnetization., Comment: 6 pages

Spin torques play a crucial role in operative properties of modern spintronic devices. To study current-driven magnetization dynamics, spin-torque terms providing the action of spin-polarized currents have previously often been added in a phenomenological way to the Landau-Lifshitz-Gilbert equation describing the local spin dynamics, yet without derivation from fundamental principles. Here, starting from the Dirac-Kohn-Sham theory and incorporating nonlocal spin transport we rigorously derive the various spin-torque terms that appear in current-driven magnetization dynamics. In particular we obtain an extended magnetization dynamics equation that precisely contains the nonrelativistic adiabatic and relativistic nonadiabatic spin-transfer torques (STTs) of the Berger and Zhang-Li forms as well as relativistic spin-orbit torques (SOTs). We derive in addition a previously unnoticed relativistic spin-torque term and moreover show that the various obtained spin-torque terms do not appear in the same mathematical form in both the Landau-Lifshitz and Landau-Lifshitz-Gilbert equations of spin dynamics., Comment: 13 pages, 1 figure

The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to first-order relativistic effects. To include magnetic inertial effects the LLG equation has previously been extended with a supplemental inertia term and the arising inertial dynamics has been related to second-order relativistic effects. Here we start from the relativistic Dirac equation and, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that contains relativistic correction terms to any higher order. Using the Heisenberg equation of spin motion we derive general relativistic expressions for the tensorial Gilbert damping and magnetic inertia parameters, and show that these tensors can be expressed as series of higher-order relativistic correction terms. We further show that, in the case of a harmonic external driving field, these series can be summed and we provide closed analytical expressions for the Gilbert and inertial parameters that are functions of the frequency of the driving field., Comment: 18 pages, 1 figure

Here we report the measurement of the interfacial spin accumulation induced by the spin Hall effect in Pt and W thin films using magneto-optical Kerr microscopy. We show that the Kerr rotation has opposite sign in Pt and W and scales linearly with current density. By comparing the experimental results with ab-initio calculations of the spin Hall and magneto-optical Kerr effects, we quantitatively determine the current-induced spin accumulation at the Pt interface as $5*10^{-12} \mu_B$A$^{-1}$cm$^2$ per atom. From thickness-dependent measurements, we determine the spin diffusion length in a single Pt film to be $11 \pm 3$ nm, which is significantly larger compared to that of Pt adjacent to a magnetic layer., Comment: 16 pages including Supplemental Material

The influence of possible magnetic inertia effects has recently drawn attention in ultrafast magnetization dynamics and switching. Here we derive rigorously a description of inertia in the Landau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-Wouthuysen transformation up to the order of $1/c^4$ gives the intrinsic inertia of a pure system through the 2$^{\rm nd}$ order time-derivative of magnetization in the dynamical equation of motion. Thus, the inertial damping $\mathcal{I}$ is a higher order spin-orbit coupling effect, $\sim 1/c^4$, as compared to the Gilbert damping $\Gamma$ that is of order $1/c^2$. Inertia is therefore expected to play a role only on ultrashort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping are related to one another through the imaginary and real parts of the magnetic susceptibility tensor respectively., Comment: 8 pages, 1 figure

Femtosecond magneto-optical pump-probe measurements of ultrafast demagnetization show an intriguing difference in the first 100 fs of the magneto-optical Kerr response depending on whether the polarization of the pump and probe beams are in parallel or perpendicular configuration [Bigot et al., Nature Phys. 5, 515 (2009)]. Starting from a most general relativistic Hamiltonian we focus on the ultra-relativistic light-spin interaction and show that this coupling term leads to different light-induced opto-magnetic fields when pump and probe polarization are parallel and perpendicular to each other, providing thus an explanation for the measurements. We also analyze other pump-probe configurations where the pump laser is circularly polarized and the employed probe contains only linearly polarized light and show that similar opto-magnetic effects can be anticipated., Comment: 16 pages, 1 figure

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment., Comment: 11 pages

We present the first materials specific ab initio theory of the magnetization induced by circularly polarized laser light in metals. Our calculations are based on non-linear density matrix theory and include the effect of absorption. We show that the induced magnetization, commonly referred to as inverse Faraday effect, is strongly materials and frequency dependent, and demonstrate the existence of both spin and orbital induced magnetizations which exhibit a surprisingly different behavior. We show that for nonmagnetic metals (as Cu, Au, Pd, Pt) and antiferromagnetic metals the induced magnetization is antisymmetric in the light's helicity, whereas for ferromagnetic metals (Fe, Co, Ni, FePt) the imparted magnetization is only asymmetric in the helicity. We compute effective optomagnetic fields that correspond to the induced magnetizations and provide guidelines for achieving all-optical helicity-dependent switching., Comment: 5 pages, 5 figures

Excitation of a metallic ferromagnet such as Ni with an intensive femtosecond laser pulse causes an ultrafast demagnetization within approximately 300 fs. It was proposed that the ultrafast demagnetization measured in femtosecond magneto-optical experiments could be due to relativistic light-induced processes. We perform an ab initio investigation of the influence of relativistic effects on the magneto-optical response of Ni. To this end, we develop, first, a response theory formulation of the additional appearing ultra-relativistic terms in the Foldy-Wouthuysen transformed Dirac Hamiltonian due to the electromagnetic field, and, second, compute the influence of relativistic light-induced spin-flip transitions on the magneto-optics. Our ab initio calculations of relativistic spin-flip optical excitations predict that these can give only a very small contribution ($\leq 0.1$%) to the laser-induced magnetization change in Ni., Comment: 10 pages, 3 figures, Accepted for publication in Physics Review B