IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5, SEPTEMBER 2002 1967 Finite Element Simulation of Discrete Media With Granular Structure R. Dittrich, T. Schrefl, Member, IEEE, H. Forster, D. Suess, W. Scholz, J. Fidler, Member, IEEE, and V. Tsiantos, Member, IEEE Abstract—Discrete media show great potential for future ultra- high density magnetic recording. A hybrid finite element/boundary element method is used to compare the magnetization reversal process in a perpendicular granular film, a patterned media, and a single magnetic island. The results show that the influence of magnetostatic interactions on the switching field is comparable with the spread of the nucleation field due to the dispersion of the magnetic easy axes. For CoCrPt, this value is about 75 kA/m. Index Terms—Boundary element method, finite element simula- tion, patterned granular media, treecode. I. INTRODUCTION T HE term discrete media is used to refer to media that con- sist of arrays of discrete elements, for example ion-beam patterned magnetic elements [1], each of which can store one bit of data. Ideally, the storage density is then equal to the surface density of the elements. In patterned media, each discrete ele- ment is exchange-isolated from other elements, but inside each element polycrystalline grains are strongly exchange-coupled, behaving more like a larger single magnetic grain. Nevertheless, the micromagnetic simulations presented in Section III show that a single island reverses incoherently by the expansion of a small reversed nucleus. Section II of this paper describes a novel algorithm to treat the magnetostatic interactions between the islands using a hierarchical method. Section III compares the hysteresis behavior of a single island with a continuous film and a patterned media. In addition, we analyze the effect of the magnetostatic interactions on the coercive fields and calculate the dynamic coercivity as a function of the Gilbert damping con- stant. II. MICROMAGNETICS A. Method The dynamic response of a magnetic particle to an applied field follows from the Gilbert equation of motion [2] (1) Manuscript received February 12, 2002; revised June 3, 2002. This work was supported by the Austrian Science Fund Y132PHY. The authors are with the Vienna Universityof Technology, A-1040 Vienna, Austria (e-mail: rok.dittrich@tuwien.ac.at). Digital Object Identifier 10.1109/TMAG.2002.802786. The effective field is obtained from the variational derivate of the total Gibbs free energy (2) We apply the finite element method [3] and backward differ- entiation scheme [4] to discretize the partial differential equa- tion (1). For the calculation of strayfield, , a novel numerical method is used which is explained in the next sec- tion. B. Calculations of the Stray Field The stray field is obtained from a boundary value problem (3) A hybrid finite-element/boundary element method [5] is used to treat the magnetostatic interactions between the islands and to apply the boundary condition at infinity. The advan- tage of this method is that no finite elements are needed outside of the magnetic particle. For the solution of (3), we split into two parts, .The potential is 0 outside of the mag- netic particles and the solution of the Poisson equation with the boundary condition . Then the potential is solution of the Laplace equation with the boundary condition [6] (4) Here, is the surface of the magnetic particles, and is the solid angle. The direct evaluation of (4) requires a matrix vector product with a fully populated matrix. Especially for thin films as in the case of patterned media, the number of sur- face nodes can get very high, since most nodes are located at the boundaries. The following method is more efficient. The first term of the right-hand side of (4) is the potential of a dipole sheet with the dipole density [6]. Therefore, the surface in- tegral over the surface can be approximated by a sum over 0018-9464/02$17.00 © 2002 IEEE