Donald G. Porter

Domain wall traps for low-field switching of submicron elements

In magnetic random access memory, power consumption depends on the coercivity of the magnetic elements in the memory cells. In this article a new method is described that uses a “domain wall trap” element shape to reduce both the coercivity and the dependence of coercivity on element size in submicron magnetic elements. Micromagnetic simulations of a shaped permalloy element show coercivity less than one tenth the coercivity calculated for a rectangular permalloy element of the same size. The switching times for the domain wall traps are shown to be comparable to those of rectangular elements.

Velocity of transverse domain wall motion along thin, narrow strips

Micromagnetic simulation of domain wall motion in thin, narrow strips leads to a simplified analytical model. The model accurately predicts the same domain wall velocity as full micromagnetic calculations, including dependence on strip width, thickness, and magnitude of applied field pulse. Domain wall momentum and retrograde domain wall motion are both observed and explained by the analytical model.

Behavior of μMAG standard problem No. 2 in the small particle limit

For a uniformly magnetized rectangular particle with dimensions in the ratio 5 : 1 : 0.1, the coercive and switching fields in the (1,1,1) direction are determined to be Hc/Ms=0.057069478 and Hs/Ms=0.057142805. Previous micromagnetic computations of coercive and switching fields that did not approach these values for small particles are analyzed. It is shown that the disagreement was primarily due to a disparity in the method of calculating demagnetization energy. Corrected simulations are shown to agree with analytically determined values.

Comparison of magnetostatic field calculation methods on two-dimensional square grids as applied to a micromagnetic standard problem

Magnetization reversal modes and coercivities were calculated for a magnetic particle with thickness : width : length aspect ratios 0.1 : 1 : 5 as a function of the reduced particle width d/lex, where d is the particle width and lex is the intrinsic magnetostatic exchange length. With only exchange energy and magnetostatic energy included, the particle corresponds to μMAG standard problem No. 2. The problem is modeled with two-dimensional grids of three-dimensional spins, and the results are compared for two methods of calculating magnetostatic energies, the “constant magnetization” method and the “constant charge” method. For both magnetostatic computational methods, the coercivity decreases from Hc/Ms=0.06±0.003 to 0.014±0.003 over the range 3<d/lex<80, where the uncertainties reflect the field step size. Also over this interval, as d/lex increases, the magnetization exhibits three modes of reversal: nearly uniform rotation, transverse switching of end domains followed by propagation of head-to-head dom...

Switching dynamics and critical behavior of standard problem No. 4

We report results for μMAG standard problem No. 4, a 500 nm×125 nm×3 nm rectangle of material with properties to mimic Permalloy. Switching dynamics are calculated for fields applied instantaneously to an initial s state: Field 1 at 170° and Field 2 at 190° (−170°) from the positive long axis. Reversal in Field 1 proceeds by propagation of end domains toward the sample center. Reversal in Field 2 involves rotation of the end domains in one direction while the center of the particle rotates in the opposite direction, resulting in collapsing 360° walls with complex dynamics on fine length scales. Approaching the static coercivity, Hc, in small field steps, we find that the ring down frequency, f, and susceptibility, χ, are in approximate agreement with a single spin model that predicts f∝(Hc−H)1/4 and χ∝(Hc−H)−1/2. We show a correlation between the modes of oscillation that become unstable at the critical field and the switching behavior.

Precession axis modification to a semianalytical Landau–Lifshitz solution technique

A recent article [Van de Wiele et al., IEEE Trans. Magn. 43, 2917 (2007)] presents a semianalytical method to solve the Landau–Lifshitz (LL) equation. Spin motion is computed analytically as precession about the effective field H, where H is assumed fixed over the time step. However, the exchange field dominates at short range and varies at the time scale of neighbor spin precessions, undermining the fixed field assumption. We present an axis corrected version of this algorithm. We add a scalar multiple of m to H (preserving torque and hence the LL solution) to produce a more stable precession axis parallel to the cross product of the torques m×H at two closely spaced time steps. We build a predictor-corrector solver on this foundation. The second order convergence of the solver enables calculation of adjustable time steps to meet a desired error magnitude.

Analysis of switching in uniformly magnetized bodies

Summary form only given. When defining standard problems for verifying micromagnetic calculations, one important limiting case is uniform magnetization. Exchange energy vanishes and the total energy can be simplified to the expression, E=(/spl mu//sub 0//2)M/sup T/DM-/spl mu//sub 0/H/sup T/M, where H=Hh is the applied field and by choice of coordinates the diagonal matrix D=diag(D/sub x/,D/sub y/,D/sub z/) accounts for self-magnetostatic and uniaxial anisotropy energy. This is the 3D analog to the familiar 2D Stoner-Wohlfarth model. Lagrange multiplier analysis defines M/sub /spl nu//= h/sub /spl nu///(D/sub /spl nu//-/spl lambda/) for /spl nu/= x, y, z and the constraint |M|= M/sub s/ leads to an equation to be solved for the n stationary points of total energy. Depending on the value of H, n may vary between 2 and 6. With the solutions /spl lambda//sub 1/ to /spl lambda//sub n/, in increasing order, our complete analysis shows that /spl lambda//sub 1/ corresponds to the global energy minimum and /spl lambda//sub n/ the global energy maximum. Examination of the solutions leads to an iterative calculation for the critical switching fields and a closed form calculation for the coercive field when it differs from the switching field.

Generalization of a two-dimensional micromagnetic model to nonuniform thickness

A two-dimensional micromagnetic model is extended to support simulation of films with nonuniform thickness. Zeeman and crystalline anisotropy energies of each cell scale with the cell thickness, while the exchange energy of a pair of neighbor cells scales by a weight dependent on the thicknesses of both cells. The self-magnetostatic energy is computed by scaling the moment of each cell by its thickness, and adding a local correction to the out-of-plane field. The calculation of the magnetostatic field for a 10×10×1 oblate spheroid is shown to be more accurate by the nonuniform thickness model than by a uniform thickness model. With the extended model a 530×130×10 nm film in the shape of a truncated pyramid with tapering over the 15 nm nearest the edges is shown to have smaller switching field and different reversal mechanism compared with uniform thickness films of similar size and shape.