Session N16: Focus Session: Spin and Magnetization Dynamics

{\it Ab-initio} calculation of electron-phonon coupling for spin relaxation in metals.

Spin-electronic devices have motivated an important effort in understanding the mechanisms for spin-relaxation, because the operation of such devices requires long spin-diffusion lenghts. Two main factors contribute to spin relaxation: (i) spin-orbit interaction, which mixes the spin-up and spin-down components of the electronic wavefunction, and (ii) electron scattering from defects or phonons. In metals, the phonon-mediated Elliot-Yafet mechanism is believed to be dominant. Realistic calculations are computationally demanding,\footnote{J. Fabian and S. Das Sarma, {\it Phys. Rev. Lett.} {\bf 83}, 1211 (1999).} requiring an accurate description of the electronic states near the Fermi surface and their coupling to the lattice (phonons). Here we use a Density Functional Perturbation Theory implementation to calculate from first-principles the electron-phonon interaction in systems with spin-orbit coupling. Combined with recently-developed Wannier-interpolation methods for sampling efficiently the Brillouin zone, this will allow for a fully {\it ab-initio} calculation of the spin relaxation in metals.   

Gilbert damping and spin Coulomb drag in a magnetized electron liquid with spin-orbit interaction.

We present a microscopic calculation of the Gilbert damping constant for the magnetization of a two-dimensional spin- polarized electron liquid in the presence of intrinsic spin- orbit interaction. First we show that the Gilbert constant can be expressed in terms of the auto-correlation function of the spin-orbit induced torque. Then we specialize to the case of the Rashba spin-orbit interaction and we show that the Gilbert constant in this model is related to the spin-channel conductivity. This allows us to study the Gilbert damping constant in different physical regimes, characterized by different orderings of the relevant energy scales -- spin-orbit coupling, Zeeman coupling, disorder, $e-e$ interaction, spin precession frequency -- and to discuss its behavior in various limits. Particular attention is paid to interaction effects, which enter the spin conductivity via the spin Coulomb drag coefficient.   

Wannier interpolation study of the Elliot-Yafet spin relaxation in metals

Energy states of nonmagnetic metals may be chosen to be purely spin up and down in the absence of spin-orbit coupling. Spin-orbit coupling mixes the two states by a small amount $b^2$. A spin-conserving interaction (e.g. electron-phonon) causes transitions between the two states, and flips the electron's spin. Some insight into this Elliot-Yafet spin relaxation mechanism can be obtained by averaging $b^2$ over the Fermi surface. In trivalent metals, such as aluminum, $b^2\ll 1$ almost everywhere on the Fermi surface, except at small ``hot spot'' regions. \footnote{J. Fabian and S. Das Sarma, Phys. Rev. Lett. {\bf 81}, 5624 (1998).} Although the small regions of large $b^2$ dominate the spin relaxation process, they are difficult to capture numerically. We describe a Wannier interpolation strategy \footnote{X. Wang, J. Yates, I. Souza, and D. Vanderbilt, Phys.\ Rev.\ B, in press (cond-mat/0608257).} to compute $\langle b^2\rangle$. We validate it by performing {\it ab initio} calculations on aluminum, finding good agreement with previous results.$^1$ We also discuss interpolating {\it ab initio} electron-phonon matrix elements to compute the spin relaxation rate.   

Berry phase Chern number spin Hamiltonians for nanomagnets using DFT techniques

We will present a formalism capable of describing the low-energy spin dynamics of ferromagnetic metal nanoclusters consisting of up to a few tens of atoms[1]. Our procedure is based on a quantum action with a single magnetization-orientation degree of freedom corresponding to the direction of the Kohn-Sham spin- density functional theory wave-function. Besides the magnetic anisotropy energy functional, the action contains a Berry phase term arising when the fast electronic degrees of freedom are integrated out. The associated Berry curvature has a nontrivial dependence on magnetization orientation when spin-orbit interactions are included; its average over all magnetization directions is a topological invariant known as Chern number, which can only be a multiple of half integers. From the magnetic anisotropy energy and Berry curvature functionals, it is possible to construct an effective quantum Hamiltonian for the nanomagnet, in terms of a single giant-spin degree of freedom whose magnitude is equal to the Berry phase Chern number. We illustrate this procedure by computing within DFT the anisotropy energy and Berry curvature for small clusters of transition metal atoms, from which we extract the corresponding spin Hamiltonians. We show that the Berry phase term can profoundly alter the dynamics of the spin degree of freedom. Our approach can address the spin dynamics of small nanomagnets, which is now accessible experimentally in STM-engineered magnetic clusters[2]. [1] C.M. Canali, A. Cehovin and A.H. MacDonald, Phys. Rev. Lett. {\bf 91}, 046805 (2003); [2] C.F. Hirjibehedin et al., Science {\bf 312}, 1021 (2006); D. Kitchen et al., Nature {\bf 442}, 436 (2006).   

Gauge fields, the Berry phase, motive forces and the dynamics of domain walls etc.

The theory of the dynamics of domain walls and spin valves is described within the Stoner model. Using principally domain walls as examples, to be outlined are issues which arise from the requirements of energy conservation and the nature of relaxation within such a simple model. While they are not currently common currency for those working in this field, emphasized are the importance of certain vector potentials which reflect angular momentum transfer and energy conservation and which lie beyond the traditional single electron approach to this simplest model. The (majority/minority electron) spin derived forces $\vec f^\pm_s$ which arise from such dynamics are given by \begin{equation} \vec f^\pm_s = - \frac{\hbar}{2}\frac{ \partial \vec A^\pm_s}{\partial t} - \vec \nabla_{\vec r} \varphi_s^\pm. \label{force} \end{equation} where the vector potential $\vec A^\pm_s$, introduced here, reflects the Berry phase and corresponds to a ``no name'' non-conservative spin forces. The {\it usual\/} ``Stern-Gerlach'' forces correspond to the second term. This and a second gauge field $\vec A^t_s$ are required if the dynamical version of the Stoner theory is to conserve energy and angular momentum. The effects are {\it not\/} small and have significant experimental consequences and device applications.   

Topology of composite domain walls in magnetic nanostrips

We discuss the internal structure of domain walls in thin magnetic nanostrips of submicron width. The walls are composite objects made from elementary topological defects. These defects are characterized by two topological charges: the O(2) vortex winding number [1] and the O(3) skyrmion number. The defects are ordinary vortices and antivortices in the bulk and fractional vortices with half-integer winding numbers at the edge. Topology and energetics restrict the allowed compositions of a domain wall to a halfvortex and an antihalfvortex (a transverse wall) or a vortex and two antihalfvortices (a vortex wall). We present a variational model [2] that reproduces quite well the major features of a vortex wall. Despite the apparent complexity, the wall has a rigid structure. Its main degrees of freedom are the location of the vortex core and the out-of-plane magnetization of the core, which is related to the skyrmion number of the vortex. [1] O. Tchernyshyov and G.-W. Chern, Phys. Rev. Lett. \textbf{95}, 197204 (2005). [2] H. Youk \textit{et al.}, J. Appl. Phys. \textbf{99}, 08B101 (2006).   

Dissipative dynamics of composite domain walls in magnetic nanostrips

We describe the dynamics of domain walls in thin magnetic nanostrips of submicron width under the action of magnetic field. Once the fast precession of magnetization is averaged out, the dynamics reduces to purely dissipative motion where the system follows the direction of the local energy gradient (Glauber's model A) [1]. We then apply the method of collective coordinates [2] to our variational model of the domain wall [3] reducing the dynamics to the evolution of two collective coordinates (the location of the vortex core). In weak magnetic fields the wall moves steadily. The calculated velocity is in good agreement with the results of numerical simulations (no adjustable parameters were used). In higher fields the steady motion breaks down and acquires an oscillatory character caused by periodic creation and annihilation of topological defects comprising the domain wall [3]. Numerical simulations uncover at least two different modes of oscillation. [1] C. J. Garc\'{\i}a-Cervera and W. E, J. Appl. Phys. \textbf{90}, 370 (2001). [2] A. S\'anchez and A. R. Bishop, SIAM Rev. {\bf 40}, 579 (1998). [3] Preceding talk by O. Tchernyshyov.   

First-principles laser-driven magnetic switching scenario in NiO

The dispersionless discrete intragap d-character levels of the (001) surface and the bulk of NiO can be selectively addressed by laser pulses and thus serve as intermediate levels for a Lambda-based all-optical magnetic switching scenario [1]. To this goal the existence of spin-mixing terms in the Hamiltonian of the system is essential, in our case it is the spin-orbit coupling term in combination with a static external magnetic field. We compute from first principles the aforementioned intragap levels with high-level correlated quantum chemistry on a doubly embedded cluster model [2] and we propagate the population in time under the influence of the laser field. The polarization, duration, shape and geometrical dependences on the laser pulse as well as the influence of the static magnetic field are shown, and the importance of going beyond the electric dipole approximation is discussed. \newline \newline [1] R. G\'{o}mez-Abal, O. Ney, K. Satitkovitchai and W. H\"{u}bner, Phys. Rev. Lett. 92, 227402 (2004) \newline [2] G. Lefkidis and W. H\"{u}bner, Phys. Rev. Lett. 95, 77401 (2005).   

Atomistic simulations of domain wall dynamics in magnetic wires

The dynamical interplay between the conduction electrons and magnetization in mesoscopic magnetic structures generates interesting new physics. For instance, there is the possibility of a domain wall (DW) motion, driven by a spin-polarized electron flux. Here we address computationally the reverse phenomenon, namely, the generation of an electromotive force (emf) by the motion of a domain wall. We describe a one-dimensional magnetic wire within the \textit{s-d} model, where conduction electrons are locally exchange coupled to classical magnetizations. For this closed quantum-classical spin-polarized system we have developed an Ehrenfest Molecular Dynamics simulation, which allows us to study the spatial and temporal evolution of any observables, characterizing the system. We have studied the motion of DWs in magnetic field as function of their thicknesses (ranging from the physical limit of one atomic spacing to two orders of magnitude thicker). For all of those we have systematically found charge redistribution along the wire, governed by the DW motion, which is a signiture of an emf.   

Magnetization induced by an acoustic wave

We predict that in a semiconductor with a Rashba-type spin-orbit coupling to strain, an acoustic wave will induce a wave of magnetization. We study the effect in the ballistic and diffusive regime, and we estimate its magnitude.   

Effects of surface waves on crystals of molecular magnets: Semi-classical approach.

The effect of surface waves on the spin-state of a molecular magnet is theoretically investigated. As it was recently noted, the anisotropy axis of a molecular magnet is locally defined, so that its direction is modified by local distortions of the lattice. Therefore, its spin-Hamiltonian must be generally written as $\mathcal{H} = \exp[-\imath{\bf S}\cdot \delta {\bf {\phi}} ]\mathcal{H}_A \exp[\imath {\bf S}\cdot \delta {\bf {\phi}}] + \mathcal{H}_Z $, where $\delta {\bf \phi} = \frac{1}{2}\nabla \times {\bf u}({\bf r})$ is the angle of the local rotation induced by the displacement field ${\bf u}({\bf r})$, $\mathcal{H}_A$ is the anisotropy Hamiltonian and $\mathcal{H}_Z$ is the Zeeman term. Based on this idea we obtain the Hamiltonian describing the interaction between spin and the distortion of the lattice produced by the surface waves. We then analyze the spin-dynamics of a single nanomagnet by employing a semi-classical approach: the displacement field ${\bf u}({\bf r})$ is treated as a continuous classical field, whereas the spin-state of the nanomagnet is described quantum-mechanically. Analytical formulas for the spin-dynamics are given for certain geometrical arrangements.   

Ferromagnetism and current-controlled magnetization of nanomagnets with giant magnetic anisotropy

Because the giant uniaxial magnetic anisotropy is advantageous in keeping the spins stable for practical applications in information processing and storage, we study ferromagnetism of nanomagnets with giant uniaxial magnetic anisotropy and how to control their magnetization by injecting a spin-polarized current. The giant anisotropy leads to a barrier for reversing a spin. We use kinetic Monte Carlo method to simulate the spin dynamics. We obtain the experimental ferromagnetism and its temperature dependence with experimental parameters. The ferromagnetism is formed because the nanomagnets are limited in space and the experimental duration is finite in time. Furthermore, we design a special nanomagnet and study its magnetization reversal under applied spin-polarized currents. We observe a hysteresis loop against the current. Starting from whatever value, the magnetization can be controlled by the spin-polarized current. Y Li and B-G Liu: Phys. Rev. Lett. 96, 217201 (2006); Phys. Rev. B 73, 174418 (2006).   

Strain and Stress in nano-structure spintronics devices due to spin transfer torque.

There have been many interests of the effect of magnetization reversal induced by current in spintronics, namely, spin transfer or spin torque effect, firstly predicted by Slonczewski and Berger in 1996. Because of the conservation of angular momentum in the spin transfer process, an additional lattice angular momentum has to be brought to balance the redundant angular momentum of the spin transfer torque. The lattice angular momentum introduces strain and stress to the nano structure of a spintronics device. In this theoretical work, we calculate the strain and stress tensors due to spin transfer in two kinds of structure: a giant magnetoresisteance (GMR) sandwich structure and a ferromagnetic nanowire. When high-density current (above some threshold value) is through them and then the named spin transfer effect occurs, the strain and stress in both longitudinal and transverse direction of the structure appear. We obtain the relationship between the strain tensors and the spin polarized current density, and sketch the diagram of the strain of the nano structures. The stain and stress produced by the spin transfer torque may introduce destructive force in spintronics devices.