3240 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004 Computing Solenoidal Fields in Micromagnetic Simulations E. Martinez, L. Torres, Member, IEEE, and L. Lopez-Diaz Abstract—In finite-difference micromagnetic simulations, the electric field originated from time variations of magnetic induction is evaluated by means of a discrete version of Faraday’s law. The electric field can be then calculated as the convolution of a tensor and the time derivative of the magnetic induction. This paper presents an analytical expression for the tensor. The paper also reports on a quantitative test of the tensor that simulates the electric field of an oscillating magnetic point dipole. Index Terms—Ampere field, eddy currents, micromagnetics, solenoidal fields. I. INTRODUCTION M ICROMAGNETIC simulations are an efficient tool that are widely used to get a better understanding of the dynamic processes taking place in the mesoscopic scale. Dy- namic processes in micromagnetic simulations require solving the Landau–Lifshitz–Gilbert equation coupled to Maxwell equations. Within a finite-difference scheme, the ferromagnetic samples are discretized in a three-dimensional (3-D) regular cubic mesh. The magnetization is assumed to be uniform in each computational cell, and it is allowed to rotate in 3-D. Each computational cell experiences an effective magnetic field deriving from the total free energy of the system. From a fundamental point of view, three contributions to the total free energy are usually taken into account: exchange, anisotropy, and classical Maxwell contribution. As it is well known, the first two terms are contributions based on quantum principles, whereas the other is the mesoscopic classical contribution deriving from Maxwell equations. In most of the literature on micromagnetic calculations [1], magnetostatic and Zeemann interactions are the only two con- tributions considered in the classical Maxwell term. In this case, the two relevant Maxwell equations inside the sample are (1) (2) where , with being the external field created by external sources, and the de- magnetizing field, whose source is the magnetization distribu- Manuscript received January 21, 2004; revised May 12, 2004. This work was supported in part by the Spanish Ministerio de Ciencia y Tecnología under Project MAT2002-03094, and by the Junta de Castilla y León under Project SA056/02. The authors are with the Departamento de Física Aplicada, University of Sala- manca, E-37008 Salamanca, Spain (e-mail: a2577@usal.es). Digital Object Identifier 10.1109/TMAG.2004.832758 tion . The solution of (1) and (2) can be found by direct integration over the ferromagnetic body [2] (3) where , and and are the surface and volume of the ferromagnetic body. When calculating the demagnetizing field in discrete problems, various approximations can be con- sidered, but in all cases the demagnetizing field can be expressed as the convolution of a tensor and its sources. Different expres- sions of the demagnetizing tensor can be found in the literature [3]–[5]. In the case of ferromagnetic materials with appreciable con- ductivity and when fast magnetization dynamics is to be analyzed, the electrical effects need to be included in the micro- magnetic formalism [6]. The Maxwell equations including the electric field are Gauss and Faraday’s laws (4.1) (4.2) where is the volume density electric charge. If we as- sume local neutrality over the volume of the sample, , the only source of electric field is the time derivative of magnetic induction . From an electrical point of view, we will also assume that our ferromagnetic body is a linear, isotropic, and homogeneous media characterized by the electrical permitivity . According to Faraday’s law (4.2), when magnetic induction varies in time, an electric field is induced. In conducting media, this electric field produces a current inside the sample which is obtained by the Ohm’s law. Therefore, (2) must be re- placed by (5) where now , with being the contribution due to the electric effects (eddy currents). is the conductivity of the material. The fields and are irrotational so that (5) can be written (6) 0018-9464/04$20.00 © 2004 IEEE