2058 IEEE TRANSACTIONS ON MAGNETICS, VOL. 37, NO. 4, JULY 2001 Dynamic Micromagnetic Simulation of the Configurational Anisotropy of Nanoelements Josef Fidler, Thomas Schrefl, Dieter Süss, and Werner Scholz Abstract-A finite element method was used to simulate the nonellipsoidal particles have been rigorously studied applying magnetization reversal of nanostructured Ni80Fe20 elements with finite difference [5], [6] or finite element techniques [7]. The zero anisotropy. The numerical results show a strong influence of numerical results clearly show that strong stray fields, which the size of the cubic and platelet shaped (square and triangular) elements on the switching field. The calculated switching fields cause the magnetization to become inhomogeneously arranged, range from influence the reversal process drastically [8]­[10]. 0 = 0 002 to 0 6 T. Differences of the demag- netizing field which arise when the field is applied in different Differences in switching fields for various directions of the directions, lead to configurational anisotropy effects. Platelet applied magnetic field are predicted from our micromagnetic shaped elements show identical switching behavior in different model calculations for cubic and platelet shaped (squared and directions within the platelet plane. Inhomogeneous magnetization reversal processes become dominant with increasing element size triangular) nanoelements neglecting thermally activated magne- 100 nm and strongly influence the switching behavior. tization reversal processes and surface anisotropy effects. The influence of the element size on the magnetic switching be- Index Terms-Configurational anisotropy, nanomagnets, numerical micromagnetics, switching field. havior determined only by strayfield and exchange energy will be shown and possible mechanisms for the magnetization re- versal will be discussed. I. INTRODUCTION OVERRECENTyearstheinvestigationofthemagnetic II. MICROMAGNETIC SIMULATION switching behavior of nanostructured permalloy elements A. Micromagnetic and Numerical Background [1], [2] has become more advanced due to improvements in nu- merical micromagnetic methods on the theoretical side and high Micromagnetism starts from the total magnetic Gibb's free accuracy fabrication methods, such as electron beam lithog- energy, , of a ferromagnetic system, which is the sum of the raphy and focused ion beam techniques. The worldwide interest exchange energy, the Zeeman energy, the magnetostatic energy, in these elements is their potential for possible future application and neglecting the magneto-crystalline anisotropy energy [11]. in high density magnetic data storage and microsensor appli- cations. Shape and magnetocrystalline anisotropy determine (1) the magnetization reversal properties. In mesoscopic or nanos- tructured magnets with the switching fields can be varied Here denotes the ferromagnetic exchange constant, is by the choice of the geometric shape of the magnets and the ori- the magnetic polarization. denotes the external field. When entation of the applied field. This phenomenon is described as the components of the polarization vector are approximated by configurational anisotropy [3]. Numerical micromagnetic mod- piecewise linear functions on the finite element mesh, the en- eling using the finite difference or finite element method reveals ergy functional (1) reduces to an energy function with the nodal the correlation between the local arrangement of the magnetic values of the vector components as unknowns. Its minimization moments and the microstructural features on a length scale of with respect to the at the nodal points, subject to the constraint several nanometers and gives a quantitative treatment of the in- , provides an equilibrium distribution of the polariza- fluence of the shape of mesoscopic or nanostructured magnets tion. To satisfy the constraint, the polarization is represented by on the magnetization reversal and switching. polar coordinates. Traditional investigations of magnetization reversal in small The crucial part of the micromagnetic simulation of nanoele- ferromagnetic particles assume spherical or ellipsoidal particles ments is the accurate calculation of the magnetic stray field. uniformly magnetized along the easy direction for zero applied In finite element field calculation, micromagnetic simulations field. At the nucleation field the magnetization starts to deviate introduce a magnetic scalar or magnetic vector potential to from the equilibrium state according to the preferred magne- calculate the demagnetizing field. For the calculation of the tization mode [4]. The magnetization reversal mechanisms in demagnetizing field of mesoscopic or nanostructured magnets the magnetic scalar potential was calculated using a hybrid finite element/boundary element technique, which was origi- Manuscript received October 15, 2000. This work was supported by the Austrian Science Fund Projects P13260-TEC nally proposed by Fredkin and Koehler [12]. The numerical and Y132-PHY. integration of the Landau Lifshitz-Gilbert equation of motion The authors are with the Institute of Applied and Technical Physics, Vienna provides the time resolved magnetization patterns during University of Technology, Wiedner Hauptstr. 8-14, A-1040 Wien, Austria (e-mail: fidler@tuwien.ac.at). the reversal process. A Runge-Kutta method optimized for Publisher Item Identifier S 0018-9464(01)06171-4. mildly-stiff differential equations [13] proved to be effective 0018­9464/01$10.00 © 2001 IEEE FIDLER et al.: DYNAMIC MICROMAGNETIC SIMULATION OF THE CONFIGURATIONAL ANISOTROPY OF NANOELEMENTS 2059 TABLE I TABLE II NUMBER OF SURFACE / VOLUME ELEMENTS CALCULATED COERCIVE (SWITCHING) FIELD  H [T ] Fig. 1. Schematic drawing of the shape of the simulated elements showing the different applied field directions. for the simulation using a regular finite element mesh and a Gilbert damping constant . However, for an irregular mesh as required for triangular nanoelements and a Fig. 2. Numerically calculated demagnetization curves of a triangular platelet time step smaller than 10 fs is required to obtain an accurate with 40 40 8 nm . The external field was applied parallel to the [001], [010] solution with the Runge-Kutta method. In this highly stiff and [100] directions. regime, backward difference schemes allow much larger time steps and thus the required CPU time remains considerably Fig. 1 presents the cubic, squared and triangular nanoele- smaller than with the Runge-Kutta method. Since the stiffness ments and shows the magnetic field directions. The magnetic arises mainly from the exchange term, the demagnetising field field was applied in certain crystallographic directions, for can be treated explicitly and thus is updated after a time interval cubes parallel to [001], [101], [111], for square platelets par- . During the time interval the Gilbert equation is integrated allel to [001], [010], [110] and for triangular platelets parallel with a fixed demagnetising field using a higher order backward to [001], [010], [100]. The calculations were started after difference method. is taken to be inversely proportional to the saturation. The field was reduced in steps of T maximum torque acting over the finite element mesh. starting from T. B. Finite Element Model and Intrinsic Properties III. NUMERICAL RESULTS The nanomagnets were in the size range 10, 20, 40 and A 3D micromagnetic FE simulation based on the Landau 100 nm for the edge length and in the thickness range 2, 4, Lifshitz-Gilbert equation of motion has been used to system- 8 and 20 nm for the platelet shaped geometries (square and atically compare the influence of cubic, cylindrical, disk and triangular). The aspect ratio between edge length and thickness platelet (square and triangular) shaped nanoelements on the was kept constant. In order to avoid the influence of the switching field behavior. Table II shows the dependence of magnetocrystalline anisotropy effects the following material the coercive or switching field with a reversed field applied parameters were chosen: T, pJ/m and parallel to the [001], [101] and [111] directions. Depending on the orientation of the field it is clearly visible that a switching The nanomagnets were discretized into tetrahedral finite el- behavior occurs for small elements. ements with a constant edge length of 5 nm and 2.5 nm for Nanomagnets nm show an inhomogeneous vortex-like 10 nm edge length, respectively. The total number of the ele- magnetization structure during the reversal process. The ments varied from 88 (10 nm cube) to 44 800 (100 nm cube). resulting switching fields become independent of the direction Table I summarizes the number of surface and volume elements of the magnetic field. In 100 nm cubic elements the clear after discretization of the nanoelements. switching characteristic is replaced by inhomogeneous rotation 2060 IEEE TRANSACTIONS ON MAGNETICS, VOL. 37, NO. 4, JULY 2001 Fig. 5. Comparison of the transient magnetization states during the reversal of triangular elements of the size 20 20 4 nm and 100 100 20 nm with zero magnetocrystalline anisotropy under the influence of a constant reversed field of  H = 0:002 T parallel to the [010] direction. IV. SUMMARY Using the hybrid finite element/boundary element method we Fig. 3. Numerically calculated demagnetization curves of a square platelet with 100 100 20 nm . The external field was applied parallel to the [001], investigated the influence of size and shape on the switching [010] and [110] directions. dynamics of mesoscopic or nanostructured Ni Fe elements with . * The numerical results show a strong influence of the ele- ment size on the switching behavior. * Configurational anisotropy effects were only observed in platelet shaped elements with magnetic field directions perpendicular to the platelet plane. * Inhomogenous magnetization reversal processes become dominant with increasing element size nm. 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