Michael J. Donahue: Selected Abstracts

See also my publication title page, presentation page, and the Magnetic Materials Group publication list.

Periodic boundary conditions for demagnetization interactions in micromagnetic simulations
K.M. Lebecki, M.J. Donahue, and M.W. Gutowski
Journal of Physics D-Applied Physics, 41, 175005 (Sep 2008).
Abstract: A new method for the introduction of periodic boundary conditions to the self-magnetostatic (demagnetization) term in micromagnetic simulations is described, using an Ewald-like summation method in real space. The long-range character of the dipolar interactions is included without any distance cut-offs. The accumulated errors are carefully monitored to provide easy control of the quality of the results. This allows the calculations to be either accurate up to floating point limitations or less precise when computational speed requirements dominate. This method is incorporated into a full micromagnetic program, and comparisons are made to analytic results.
Title page.
Reversal mechanisms in perpendicularly magnetized nanostructures
J.M. Shaw, S.E. Russek, T. Thomson, M.J. Donahue, B.D. Terris, O. Hellwig, E Dobisz, and M. Schneider
Physical Review B, 78, 024414 (Jul 2008).
Abstract: We demonstrate that magnetic reversal in perpendicularly magnetized nanostructures is highly dependent on the nature and condition of the edges. To understand the impact of edge damage, we compare nanostructures created by ion milling to those prepared on prepatterned substrates. The size- and temperature-dependent reversal properties of 25 nm - 1 µm diameter nanodots show that reversal in prepatterned nanostructures is controlled by nucleation within the interior, whereas ion milling results in an edge nucleation process with an unpredicted temperature dependence of the reversal field.
Title page.
Precession axis modification to a semianalytical Landau-Lifshitz solution technique
D.G. Porter and M.J. Donahue
Journal of Applied Physics, 103, 07D920 (Feb 2008).
Abstract: 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.
Title page, Presentation page.
Numerical micromagnetics: finite difference methods
J. Miltat and M.J. Donahue
in Handbook of magnetism and advanced magnetic materials, edited by H. Kronmüller and S. Parkin (Wiley-Interscience, Chichester), Vol. 2, pp. 742-764 (Sep 2007).
Abstract: Micromagnetics is based on the one hand on a continuum approximation of exchange interactions, including boundary conditions, on the other hand on Maxwell equations in the non-propagative (static) limit for the evaluation of the demagnetizing field. The micromagnetic energy is most often restricted to the sum of the exchange, demagnetizing or (self-)magnetostatic, Zeeman and anisotropy energies. When supplemented with a time evolution equation, including field induced magnetization precession, damping and possibly additional torque sources, micromagnetics allows for a precise description of magnetization distributions within finite bodies both in space and time. Analytical solutions are, however, rarely available. Numerical micromagnetics enables the exploration of complexity in small size magnetic bodies. Finite difference methods are here applied to numerical micromagnetics in two variants for the description of both exchange interactions/boundary conditions and demagnetizing field evaluation. Accuracy in the time domain is also discussed and a simple tool provided in order to monitor time integration accuracy. A specific example involving large angle precession, domain wall motion as well as vortex/antivortex creation and annihilation allows for a fine comparison between two discretization schemes with as a net result, the necessity for mesh sizes well below the exchange length in order to reach adequate convergence.
Title page.
Adiabatic domain wall motion and Landau-Lifshitz damping
M.D. Stiles, W.M. Saslow, M.J. Donahue, and A. Zangwill
Physical Review B, 75, 214423 (Jun 2007).
Abstract: Recent theory and measurements of the velocity of current-driven domain walls in magnetic nanowires have reopened the unresolved question of whether Landau-Lifshitz damping or Gilbert damping provides the more natural description of dissipative magnetization dynamics. In this paper, we argue that (as in the past) experiment cannot distinguish the two, but that Landau-Lifshitz damping, nevertheless, provides the most physically sensible interpretation of the equation of motion. From this perspective, (i) adiabatic spin-transfer torque dominates the dynamics with small corrections from nonadiabatic effects, (ii) the damping always decreases the magnetic free energy, and (iii) microscopic calculations of damping become consistent with general statistical and thermodynamic considerations.
Title page.
Micromagnetics on curved geometries using rectangular cells: error correction and analysis
M.J. Donahue and R.D. McMichael
IEEE Transactions on Magnetics, 43, 2878-2880 (Jun 2007).
Abstract: This paper presents an edge field correction for micromagnetic computations of arbitrarily shaped objects on rectangular grids. The correction is compatible with FFT techniques and involves factors that are precomputed using the standard self-magnetostatic algorithms applied on a local, refined mesh. To evaluate this correction, we introduce a quantitative measure that is based on calculating an edge mode resonance for different orientations of an edge with respect to the rectangular mesh. Applied to a 350 nm Ni₈₀Fe₂₀ square, we find up to a 50% frequency shift for the uncorrected approach, but less than a 5% shift using the proposed method. We also study vortex expulsion in a 220 nm Ni₈₀Fe₂₀ square, and again find that the proposed correction significantly reduces the dependence of the expulsion field on the orientation angle of the sample square with the mesh.
Title page, Presentation page.
High-Frequency Domain-Wall Motion and Magnetization Rotation of Patterned Permalloy Films under External Magnetic Field Excitation
Syed Azeemuddin, Axel Hoffmann, Ralu Divan, Michael J Donahue, Seok Hwan Chung, and Pingshan Wang
in Sixth IEEE Conference on Nanotechnology, 853-856 (June 2006).
Abstract: The incorporation of ferromagnetic materials into integrated microwave devices is a promising approach for the development of on-chip high-performance circuit components. Therefore, high-frequency domain-wall motion and magnetization rotation, which yield permeability, are of primary interest. However, so far it has not been attempted to physically separate high-frequency domain-wall motion and magnetization rotation that are under high-frequency magnetic field excitation. Nor have there been attempts for the corresponding characterizations. In this work, patterned permalloy films are integrated with on-chip microstrip lines. Domain-wall motion and magnetization rotation are separated through aspect ratio and dimension control. The measured results show that high-frequency-field driven domain-wall motion is fast and different from current driven domain-wall motion. It is also shown that coupling effects are not important when the distance between two adjacent permalloy films is around 1 micrometer despite their large lateral dimensions. The experimental results agree with simulation results.
Title page.
Vortex head-to-head domain walls and their formation in onion-state ring elements
M. H. Park, Y. K. Hong, B. C. Choi, M. J. Donahue, H. Han, and S. H. Gee
Physical Review B, 73, 094424 (Mar 2006).
Abstract: Magnetization configuration of vortex head-to-head HTH domain walls and the wall-formation process in Ni₈₀Fe₂₀ ring elements were investigated using magnetic force microscopy MFM and micromagnetic simulation. At remanence, two types of vortex HTH domain walls were observed to be stable in the onion configuration, depending on the film thickness: single- and double-vortex HTH domain walls for 40 and 65 nm thick ring elements, respectively. As the vortex core nucleated during formation of the HTH domain wall, exchange energy began to decrease, accompanied by an increase in the width of the wall. Vortex nucleation in the 65 nm thick ring was found to be much faster than in the 40 nm thick ring element. This effect can be attributed to the higher initial magnetostatic energy density in the thicker ring.
Title page.
Micromagnetic calculations of eddy currents with time-varying fields
L. Yanik, E. Della Torre, and M. J. Donahue
Physica B-Condensed Matter, 372, 290-293 (Feb 2006).
Abstract: This paper extends a recently presented program for solving the eddy current problem in a cylindrical geometry, by investigating the effect of time-varying fields. When the applied field is turned off, wall motion slows by several orders of magnitude, but since the wall energy can be reduced by reducing the length of the wall, it continues to move, albeit much more slowly. Reversing the applied field has the effect of nucleating the opposite kind of wall which propagates inward and eventually annihilates the previous wall.
Title page.
Manipulation and sorting of magnetic particles by a magnetic force microscope on a microfluidic magnetic trap platform
Elizabeth Mirowski, John Moreland, Arthur Zhang, Stephen E. Russek, and Michael J. Donahue
Applied Physics Letters, 86, 243901 (Jun 2005).
Abstract: We have integrated a microfluidic magnetic trap platform with an external magnetic force microscope MFM cantilever. The MFM cantilever tip serves as a magnetorobotic arm that provides a translatable local magnetic field gradient to capture and move magnetic particles with nanometer precision. The MFM electronics have been programmed to sort an initially random distribution of particles by moving them within an array of magnetic trapping elements. We measured the maximum velocity at which the particles can be translated to be 2.2 mm/s ± 0.1 mm/s, which can potentially permit a sorting rate of approximately 5500 particles/min. We determined a magnetic force of 35.3 ± 2.0 pN acting on a 1 µm diameter particle by measuring the hydrodynamic drag force necessary to free the particle. Release of the particles from the MFM tip is made possible by a nitride membrane that separates the arm and magnetic trap elements from the particle solution. This platform has potential applications for magnetic-based sorting, manipulation, and probing of biological molecules in a constant-displacement or a constant-force mode.
Title page.
Micromagnetic Eddy Currents in Conducting Cylinders
L. Yanik, E. Della Torre, M. J. Donahue, and E. Cardelli
Journal of Applied Physics, 97, 10E308 (2005).
Abstract: The inclusion of eddy currents into micromagnetic programs is important for the proper analysis of dynamic effects in conducting magnetic media. This paper introduces a numerical implementation for eddy current calculations, in a limited geometry, for a thick domain wall. In the special case of a zero-thickness wall, our results are directly comparable with an analytical model previously presented. Our calculations provide some computational results for testing more complex programs.
Title page.
Dynamic Susceptibility of Nanopillars
N. Dao, M. J. Donahue, I. Dumitru, L. Spinu, S. L. Whittenburg, and J. C. Lodder
Nanotechnology, 15, S634-S638 (2004).
Abstract: We have calculated the dynamic susceptibility of patterned cobalt and Permalloy pillars with a diameter of 50 nm and different pillar heights using micromagnetic simulations. The resonance modes obtained from these simulations are compared to the results obtained from an analytical solution of Kittel's equation for spheroids. We also compared directly to Kittel's equation with the simulation of cobalt spheroids.
Title page.
Velocity of Transverse Domain Wall Motion Along Thin, Narrow Strips
D. G. Porter and M. J. Donahue
Journal of Applied Physics, 95, 6729-6731 (2004).
Abstract: 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.
Title page, Presentation page.
Differential Equation Model for Accommodation Magnetization
Edward Della Torre, Levent Yanik, A. Emre Yarimbiyik, and Michael J. Donahue
IEEE Transactions on Magnetics, 40, 1499-1505 (2004).
Abstract: We use the differential equation method of com- puting the accommodation magnetization in a modified Preisach model. We present the properties of this model for a Gaussian medium, and show that the resulting model has neither the congruency property nor the deletion property.
Title page.
Integrated Microfluidic Isolation Platform for Magnetic Particle Manipulation in Biological Systems
E. Mirowski, J. Moreland, S. E. Russek, and M. J. Donahue
Applied Physics Letters, 84, 1786-1788 (2004).
Abstract: We have developed a micromachined fluid-cell platform that consists of patterned magnetic thin-film elements supported on a thin silicon-nitride membrane. In the presence of an external magnetic field, the field gradients near the magnetic elements are sufficiently large to trap magnetic particles that are separated from the patterned films by a 200 nm thick nitride membrane. The two main applications of this fluid-cell platform are to provide a means to control and position magnetic microparticles, which can be tethered to biological molecules, and also to sort superparamagnetic microparticles based on their size and magnetic susceptibility. We determine the characteristic trapping forces of each trap in the array by measuring the Brownian motion of the microparticle as a function of applied external field. Typical force constants and forces on the superparamagnetic particles are 4.8×10^-4±0.7×10^-4 N/m and 97±15 pN, respectively.
Title page.
Exchange Energy Formulations for 3D Micromagnetics
M. J. Donahue and D. G. Porter
Physica B, 343, 177-183 (2004).
Abstract: Exchange energy is especially sensitive to the numerical representation selected. We compare three discretized exchange energy formulations for 3D numerical micromagnetics on rectangular grids. Explicit formulae are provided for both Neumann and Dirichlet boundary conditions. Results illustrate the convergence order of these methods as a function of discretization cell size and the effect of cell size on vortex pinning.
Title page, Presentation page.
A Test Bed for a FDTD Micromagnetic Program with Eddy Currents
L. Yanik, E. Della Torre, and M. J. Donahue
Physica B, 343, 216-221 (2004).
Abstract: The inclusion of eddy currents into micromagnetic programs is important for the proper analysis of dynamic effects in conducting magnetic media. This subject has received little attention in the past although it can cause significant errors in device calculations. This paper introduces a computational test bed for eddy current calculations and discusses some interesting analytic cases in this simplified geometry.
Title page, Presentation page.
Defect Related Switching Field Reduction in Small Magnetic Particle Arrays
M. J. Donahue, G. Vértesy, and M. Pardavi-Horvath
Journal of Applied Physics, 93, 7038-7040 (2003).
Abstract: An array of 42 µm square, 3 µm thick garnet particles has been studied. The strong crystalline uniaxial anisotropy of these particles results in the stable remanent state being single domain with magnetization parallel to the film normal. Magnetooptic measurements of individual particles provide distribution statistics for the easy-axis switching field H_sw, and the in-plane hard-axis effective anisotropy field, H_eff, which induces the formation of a metastable stripe domain structure. Both H_sw and H_eff are much smaller than the crystalline anisotropy field. Micromagnetic simulations show that the small H_sw cannot be attributed to shape anisotropy, but is consistent with smooth, localized reductions in the crystalline anisotropy caused by defects in either the particles or the substrate.
Title page, Presentation page.
Micromagnetics Simulation of Asymmetric Pseudo-Spin Valve Dots
N. Dao, F. J. Castano, C. A. Ross, M. J. Donahue, and S. L. Whittenburg
Materials Research Society Symposia Proceedings, 731, 291-296 (2002).
Abstract: We present simulation results for Ni₇₉Fe₂₁ (5 nm)/Cu (3 nm)/Co (4nm) pseudo-spin valves. These simulations have been conducted on several different aspect ratios of rectangular dots. Distinct switches of the two magnetic layers were observed. At smaller aspect ratios, magnetization reversal proceeds through a leaf state in the soft layer and a flower state in the hard layer. For larger aspect ratios, reversal proceeds by nucleation and annihilation of domain walls. Our simulations show a reasonable agreement with the experimental results. Differences between the experimental and simulation results are discussed.
Title page.
Analysis of Switching in Uniformly Magnetized Bodies
M. J. Donahue and D. G. Porter
IEEE Trans. Mag., 38, 2468-2470 (2002).
Abstract: A full analysis of magnetization reversal of a uniformly magnetized particle by coherent rotation is presented. The magnetization energy of the particle in the presence of an applied field H is modeled as E=(½)µ₀ M^TDM - µ₀ H^TM. This model includes shape anisotropy, any number of uniaxial anisotropies, and any energy that can be represented by an effective field that is a linear function of the uniform magnetization M. The model is a generalization to three dimensions of the Stoner-Wohlfarth model. Lagrange multiplier analysis leads to quadratically convergent iterative algorithms for computation of switching field, coercive field, and the stable magnetization(s) of the particle in the presence of any applied field. Magnetization dynamics are examined as the applied field magnitude |H| approaches the switching field H_s, and it is found that the precession frequency is proportional to (H_s-|H|)^¼ and the susceptibility is proportional to (H_s-|H|)^-½.
Title page, Presentation page.
Generalization of a Two-Dimensional Micromagnetic Model to Non-Uniform Thickness
(Original title: Non-uniform Thickness in Two-dimensional Micromagnetic Simulation)
D. G. Porter and M. J. Donahue
Journal of Applied Physics, 89, 7257-7259 (2001).
Abstract: A two-dimensional micromagnetic model is extended to support simulation of films with non-uniform 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 x 10 x 1 oblate spheroid is shown to be more accurate by the non-uniform thickness model than by a uniform thickness model. With the extended model a 530 x 130 x 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.
Title page, Presentation page.
Micromagnetic Calculation of the High Frequency Dynamics of Nano-Size Rectangular Ferromagnetic Stripes
O. Gérardin, H. Le Gall, M. J. Donahue and N. Vukadinovic
Journal of Applied Physics, 89, 7012-7014 (2001).
Abstract: Nano-size ferromagnetic dots, wires and stripes are of strong interest for future high speed magnetic sensors and ultra high density magnetic storage. High frequency dynamic excitation is one way to investigate the time scale of the magnetization reversal in submicron particles with lateral nanometer dimension. Macroscopic models like the Landau-Lifshitz (LL) model are often used to describe the switching process. However, these models do not take into account the non uniformity of the magnetization structure. In this paper dynamic micromagnetic calculations are used in determining the high frequency susceptibility of a 1 µm x 50 nm x 5 nm Permalloy stripe. The studied structure exhibits two resonance modes. The higher, primary peak is around 10 GHz, and can be identified with the uniform resonance mode predicted by the macroscopic LL model. The low frequency peak is attributed to the splay of the magnetization distribution near the end of the stripe.
Title page, Presentation page.
Mathematics and Measurement
R. F. Boisvert, M. J. Donahue, D. W. Lozier, R. D. McMichael, and B. W. Rust
Journal of Research of the National Institute of Standards and Technology, 106, 293-313 (2001).
Abstract: In this paper we describe the role that mathematics plays in measurement science at NIST. We first survey the history behind NIST's current work in this area, starting with the NBS Math Tables project of the 1930s. We then provide examples of more recent efforts in the application of mathematics to measurement science, including the solution of ill-posed inverse problems, characterization of the accuracy of software for micromagnetic modeling, and in the development and dissemination of mathematical reference data. Finally, we comment on emerging issues in measurement science to which mathematicians will devote their energies in coming years.
Title page.
Switching Dynamics and Critical Behavior of Standard Problem No. 4
R. D. McMichael, M. J. Donahue, D. G. Porter and Jason Eicke
Journal of Applied Physics, 89, 7603-7605 (2001).
Abstract: We report results for muMAG standard problem no. 4, a 500 nm x 125 nm x 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 directions 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, H_c , in small field steps, we find that the ring down frequency, f, and the susceptibility are in approximate agreement with with a single-spin model that predicts that f is proportional to (H_c - H)^¼ and susceptibility is proportional to (H_c - H)^-½. We show a correlation between the modes of oscillation that become unstable at the critical field and the switching behavior.
Title page.
Micromagnetics of the Dynamic Susceptibility for Coupled Permalloy Stripes
O. Gérardin, J. Ben Youssef, H. Le Gall, N. Vukadinovic, P. M. Jacquart and M. J. Donahue
Journal of Applied Physics, 88, 5899-5903 (2000).
Abstract: The dynamic susceptibility of arrays of narrow permalloy stripes (9 mm by several microns by 200 nm) has been investigated using a single-coil broadband susceptibility spectrometer. Disagreement is observed between experimental results and the macroscopic Landau-Lifshitz (LL) model. This model does not take into account the dipolar interaction between magnetic stripes. We have performed micromagnetic calculations that include these dipolar interactions, and have found the resulting frequency dependence of the dynamic susceptibility in the linear regime to be in good agreement with our experimental data.
Title page.
Behavior of muMAG Standard Problem No. 2 in the Small Particle Limit
M. J. Donahue, D. G. Porter, R. D. McMichael and J. Eicke
Journal of Applied Physics, 87, 5520-5522 (2000).
Abstract: 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 = 0.057142806. 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.
Title page.
Domain Wall Traps for Low-Field Switching of Sub-Micron Elements
R. D. McMichael, J. Eicke, M. J. Donahue and D. G. Porter
Journal of Applied Physics, 87, 7058-7060 (2000).
Abstract: In magnetic random access memory (MRAM), power consumption depends on the coercivity of the magnetic elements in the memory cells. In this paper 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.
Title page.
High-Speed Dynamics, Damping, and Relaxation Times in Submicrometer Spin-Valve Devices
S. E. Russek, S. Kaka and M. J. Donahue
Journal of Applied Physics, 87, 7070-7072 (2000).
Abstract: The dynamical response of spin-valve devices with line widths of 0.8 µm has been measured after excitation with 160 ps magnetic impulses. The devices show resonant frequencies of 2 to 4 GHz which determine the upper limit of their operation frequency. The dynamical response can be fit with Landau-Lifshitz models to extract an effective uniform-mode damping constant, a_um. The measured values of a_um were between 0.04 and 0.01 depending on the magnitude of the longitudinal bias field. The appropriate damping coefficient for use in micromagnetic modeling, a_mm, was extracted from the dynamical response with large longitudinal bias field. This value was used to model the switching of a 0.1 µm x 1.0 µm magnetoresistive random access memory (MRAM) cell. The micromagnetic model included shape disorder that is expected to be found in real devices. The simulations showed that, while the magnetization reverses rapidly (< 0.5 ns), it took several nanoseconds for the energy to be removed from the magnetic system. The switching energy was stored in short wavelength magnetic fluctuations that could dramatically affect the re-reversal process 1 to 2 ns after the first reversal.
Title page.
OOMMF User's Guide, Version 1.0
M. J. Donahue and D. G. Porter
NISTIR 6376, 87 pages (Sept 1999).
Abstract: This manual describes OOMMF (Object Oriented Micromagnetic Framework), a public domain micromagnetics program developed at the National Institute of Standards and Technology. The program is designed to be portable, flexible, and extensible, with a user-friendly graphical interface. The code is written in C++ and Tcl/Tk. Target systems include a wide range of Unix platforms, Windows NT, and Windows 95/98.
Title page.
Comparison of Magnetostatic Field Calculation Methods on 2-D Square Grids as Applied to a Micromagnetic Standard Problem
R. D. McMichael, M. J. Donahue, D. G. Porter and J. Eicke
Journal of Applied Physics, 85, 5816-5818 (1999).
Abstract: 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/l_ex, where d is the particle width and l_ex is the intrinsic magnetostatic exchange length. With only exchange energy and magnetostatic energy included, the particle corresponds to muMAG standard problem #2. The problem is modeled with 2D grids of 3D 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 H_c/M_s = 0.06 ± 0.003 to 0.014 ± 0.003 over the range 3 < d/l_ex < 80, where the uncertainties reflect the field step size. Also over this interval, as d/l_ex increases, the magnetization exhibits three modes of reversal: nearly uniform rotation, transverse switching of end domains followed by propagation of head-to-head domain walls from the ends to the center of the particle, and nucleation and propagation of vortices accompanied by more complex domain structures.
Title page.
A Variational Approach to Exchange Energy Calculations in Micromagnetics
M. J. Donahue
Journal of Applied Physics, 83, 6491-6493 (1998).
Abstract: This paper presents a magnetization interpolation method for micromagnetic exchange energy calculations using a variational procedure to relax spins on a supplemental (refined) lattice. The approximations implicit in standard micromagnetic discretization schemes fail when angles between neighboring spins in the model become large, but the described interpolation effectively reduces the angle between neighboring spins, alleviating many of the associated problems. Moreover, this method does not introduce excessive discretization-induced vortex pinning observed with some large angle exchange energy formulations. This paper includes details on proper post-interpolation exchange torque calculation, bounds on nearest neighbor angles for interpolated lattices, a simple model predicting discretization induced Néel wall collapse, and an example of an apparently collapsed (1 cell wide) domain wall that can be restored by the proposed technique.
Title page.
Head to head domain wall structures in thin magnetic strips
R. D. McMichael and M. J. Donahue
IEEE Trans. Mag., 33, 4167-4169 (1997).
Abstract: We present calculations of head to head domain wall structures in magnetic strips of permalloy with widths, w, ranging from 75 to 500 nm and thicknesses, t, from 1 to 64 nm. Neglecting magnetocrystalline and magnetostrictive anisotropy energies, minimization of exchange and magnetostatic energy leads to one of two types of domain wall structures: "transverse" walls with magnetization at the center of the wall directed transverse to the strip axis and "vortex" walls where the magnetization forms a vortex at the center of the wall. Calculation of the domain wall energies leads to a proposed phase diagram for head to head domain walls where transverse walls have lower energy when dimensions are less than t_c w_c = 130 A/µ₀ M_s²
Title page.
Spin Reorientation Transitions and Domain Structure in Magnetic Multilayers
V. I. Nikitenko, L. M. Dedukh, V. S. Gornakov, Yu. P. Kabanov, L. H. Bennett, M. J. Donahue, L. J. Swartzendruber, A. J. Shapiro, H. J. Brown
IEEE Trans. Mag., 33, 3661-3663 (1997).
Abstract: Spin reorientation phase transitions and evolution of magnetic phase states arising during magnetization reversal of antiferromagnetic CoNiCu/Cu superlattice are studied using a magneto-optical indicator film technique, accompanied by vector VSM hysteresis loop measurements. Spin-flop and nonsymmetric angle phases have been found and analyzed. It is shown that they arise due to the nucleation and growth of domains.
Title page.
Exchange Energy Representations in Computational Micromagnetics
M. J. Donahue and R. D. McMichael
Physica B, 233, 272-278 (1997).
Abstract: In order to solve Brown's equations, which describe a continuous medium, computational micromagnetic modeling requires a discrete representation of the magnetization M(r), and a discrete representation of the derivatives of M(r) must be chosen. This choice may be made through an explicit choice of interpolation or through the choice of numerical representation of Brown's equations. In this paper we describe some alternative representations of the exchange energy on a square 2-D grid, and test these representations though comparison with analytical results for magnetization spirals and with simulations testing vortex and domain wall mobility.
Keywords: Micromagnetics, exchange, hysteresis
Title page.
Investigation of Domain Wall Formation and Motion in Magnetic Multilayers
L. H. Bennett, M. J. Donahue, A. J. Shapiro, H. J. Brown, V. S.Gornakov and V. I. Nikitenko
Physica B, 233, 356-364 (1997).
Abstract: The magnetization reversal processes in two electrodeposited [Co₆₄Ni₃₁Cu₅ 2 nm/Cu]₂₀₀ multilayers are investigated using an advanced magneto-optical indicator film (MOIF) technique together with SQUID and vector vibrating sample magnetometry. The nonmagnetic Cu spacers are 1 nm thick in one specimen leading to predominantly antiferromagnetic exchange coupling between the ferromagnetic Co₆₄Ni₃₁Cu₅ layers, and 3 nm in the other, with ferromagnetic coupling. The hysteresis loop of the ferromagnetic multilayer is conventional, indicating the stages of domain wall formation, motion and saturation. Nucleation and movement of domain walls in different layers proceed in a partially uncorrelated manner, and are determined by defects near the surface edge and inside the multilayer. As a result, the front of the magnetization reversal has a staggered configuration. The antiferromagnetic multilayer has an atypical loop, first with one susceptibility, then a step to a new value, then another susceptibility, and with non-symmetrical behavior about the field axis. Narrow and nonstaggered domain wall images in antiferromagnetically coupled layers are observed. The MOIF technique is used to provide a portrait of the vertical component of the magnetostatic field intensity, helping to elucidate the spin-flip and/or spin-flop processes which are apparently responsible for the hysteresis behavior.
Title page.
Rates of Convex Approximation in Non-Hilbert Spaces
M. J. Donahue, L. Gurvits, C. Darken, and E. Sontag
Constructive Approximation, 13, 187-220 (1997).
Abstract: This paper deals with sparse approximations by means of convex combinations of elements from a predetermined "basis" subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, for L_p spaces with 1<p<\infty, the O(n^-1/2) bounds of Barron and Jones are recovered when p=2.
One motivation for the questions studied here arises from the area of "artificial neural networks," where the problem can be stated in terms of the growth in the number of "neurons" (the elements of S) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more "robust" (resistant to exemplar noise) L_p, 1<=p<2 norms.
The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.
AMS classification: 41A25, 46B09, 68T05.
Title page.
Experimental Study of Magnetization Reversal Processes in Nonsymmetric Spin Valve
V. S. Gornakov, V. I. Nikitenko, L. H. Bennett, H. J. Brown, M. J. Donahue, W. F. Egelhoff, R. D. McMichael, A. J. Shapiro
Journal of Applied Physics, 81, 5215-5217 (1997).
Abstract: We have investigated a nonsymmetric bottom giant magnetoresistance spin valve with the structure Si/NiO/Co/Cu/Co/Ta, as well as single ferromagnetic Co layers on antiferromagnetic NiO, with or without a nonmagnetic Cu spacer. Magnetic hysteresis loops have been measured by SQUID magnetometry, and magnetic domain structures have been imaged using an advanced magneto-optical indicator film (MOIF) technique. The MOIF technique demonstrated that the first stage of magnetization reversal is characterized by nucleation of many microdomains. With increasing reversed field, the domain walls move over small distances (5-20 microns) until annihilation. The domain size was observed to increase with the thickness of the Co layer. When an alternating magnetic field was applied, the domain structure was dramatically changed.
Title page.
Sparse Representations for Image Decomposition with Occlusions
Mike Donahue, Davi Geiger, Robert Hummel and Tyng-Luh Liu
Computer Vision and Pattern Recognition --- CVPR '96 (San Fransisco, 7-12 (June 1996).
Abstract: We study the problem of how to detect "interesting objects" appearing in a given image, I. Our approach is to treat it as a function approximation problem based on an over-redundant basis, and also account for occlusions, where the basis superposition principle is no longer valid. Since the basis (a library of image templates) is over-redundant, there are infinitely many ways to decompose I. We are motivated to select a sparse/compact representation of I, and to account for occlusions and noise. We then study a greedy and iterative "Weighted L^p Matching Pursuit" strategy, with 0<p<1. We use an L^p result to compute a solution and select the best template at each stage of the pursuit.
Title page.
Image Recognition with Occlusions
Tyng-Luh Liu, Mike Donahue, Davi Geiger, and Robert Hummel
Proc. 4th European Conf. on Computer Vision --- ECCV '96 (London), 556-565 (April 1996).
Abstract: We study the problem of how to detect "interesting objects" appearing in a given image, I. Our approach is to treat it as a function approximation problem based on an over-redundant basis. Since the basis (a library of image templates) is over-redundant, there are infinitely many ways to decompose I. To select the "best" decomposition we first propose a global optimization procedure that considers a concave cost function derived from a "weighted L^p norm" with 0<p<=1. This concave cost function selects as few coefficients as possible, producing a sparse representation of the image and handling occlusions. However, it contains multiple local minima. We identify all local minima so that a global optimization is possible by visiting all of them. Secondly, because the number of local minima grows exponentially with the number of templates, we investigate a greedy "L^p Matching Pursuit" strategy.
Title page.
Magneto-Optical Indicator Film Study of the Magnetization of a Symmetric Spin Valve
V. I. Nikitenko, V. S. Gornakov, L. M. Dedukh, Yu. P. Kabanov, A. F. Khapikov, L. H. Bennett, P. J. Chen, R. D. McMichael, M. J. Donahue, L. J. Swartzendruber, A. J. Shapiro, H. J. Brown, W. F. Egelhoff, Jr.
IEEE Trans. Mag., 32, 4639-4641 (1996).
Abstract: A magneto-optical indicator film (MOIF) technique is used for direct experimental study of the magnetization reversal process in a symmetric NiO/Co/Cu/NiFe/Cu/Co/NiO spin valve. It is shown for the first time that the reversal of the free center layer proceeds by nonuniform magnetization rotation. The observed switching mechanism is presumed to be associated with the influence of the nonuniform magnetostatic field that follows from surface roughness and the polycrystalline structure of the magnetic layers.
Title page.
Complementary Imaging of Granular Co-Ag Films with Magneto-Optical Indicator Film Technique and Magnetic Force Microscopy
M. J. Donahue, L. H. Bennett, R. D. McMichael, L. J. Swartzendruber, A. J. Shapiro, V. I. Nikitenko, V. S. Gornakov, L. M. Dedukh, A. F. Khapikov, V. N. Matveev, V. I. Levashov
Journal of Applied Physics, 79, 5315-5317 (1996).
Abstract: A Magneto-optical indicator film (MOIF) technique and magnetic force microscopy were used for visualization and direct real-time experimental study of the magnetization processes of magnetic Co₉₀Ag₁₀ granular films. It is shown that the magnetization reversal of the as-deposited films follows a specific two-step course. The first stage is characterized by gradual spin rotation to large angles up to 90° without domain formation. Further magnetization reversal proceeds by the nucleation and motion of zigzag-shaped domain walls. The dendritic structure of the domain walls was observed using both techniques. Also tracks of magnetic inhomogeneities remaining behind moving zigzag-shaped domain walls were revealed by MOIF.
PACS: 75.60.Ch, 61.16.Ch, 75.60.Lr
Title page.
Artifacts in Magnetic Resonance Imaging from Metals
L.H. Bennett, P.S. Wang, and M.J. Donahue
Journal of Applied Physics, 79, (1996).
Abstract: Metallic biomedical implants, such as aneurysm clips, endoprostheses, and internal orthopedic devices give rise to artifacts in the magnetic resonance image (MRI) of patients. Such artifacts impair the information contained in the image in precisely the region of most interest, namely near the metallic device. Ferromagnetic materials are contraindicated because of the hazards associated with their movement during the MRI procedure. In less-magnetic metals, it has been suggested that the extent of the artifact is related to the magnetic susceptibility of the the metal, but no systematic data appear to be available. When the susceptibility is sufficiently small, an additional artifact due to electrical conductivity is observed. We present an initial systematic study of MRI artifacts produced by two low susceptibility metals, titanium (relative permeability µ_r approx. 1.0002) and copper (µ_r approx. 0.99998), including experimental, theoretical, and computer simulation results.
PACS: 87.59.Pw, 75.20.En
Title page, Presentation page.
On the Use of Level Curves in Image-Analysis
M.J. Donahue and S.I. Rokhlin
CVGIP-Image Understanding, 57, 185-203 (1993).
Abstract: A digitized image is viewed as a surface over the xy-plane. The level curves of this surface provide information about edge directions and feature locations. This paper presents algorithms for the extraction of tangent directions and curvatures of these level curves. The tangent direction is determined by a least-squares minimization over the surface normals (calculated for each 2 x 2 pixel neighborhood) in an averaging window. The curvature calculation, unlike most previous work on this topic, does not require a parameterized curve, but works instead directly on the tangents across adjacent level curves. The curvature is found by fitting concentric circles to the tangent directions via least-squares minimization. The stability of these algorithms with respect to noise is studied via controlled tests on computer generated data corrupted by simulated noise. Examples on real images are given which show application of these algorithms for directional enhancement, and feature point detection.
Title page.
A Method for Automatic Inspection of Printed-Circuit Boards
A.P. Sprague, M.J. Donahue, and S.I. Rokhlin
CVGIP-Image Understanding, 54, 401-415 (1991).
Abstract: An algorithm for automatic inspection of printed circuits is discussed which includes elements of the reference-comparison and the design-rule methods. First a directed graph, called the segment graph, is created from the image of the PCB. Construction of this graph involves comparison of adjacent rows, and hence each pixel is examined only twice in this construction. Feature points, including wire intersections, wire ends, and circuit defects, which uniquely represent the PCB image are extracted from the segment graph. In this way a very compact representation of the image is extracted. The feature points of the board being tested are then matched against the feature points from a reference board. Simultaneous with feature point extraction, the design-rule checking algorithm performs wire width and spacing inspection, and marks the unsatisfactory regions. Results of the algorithm implementation are demonstrated.
Title page.

michael.donahue@nist.gov
17-Oct-2008