Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
Finite Element Approach for 2D Micromagnetic Systems H. Szambolics *1, L. Buda-Prejbeanu2, J. C. Toussaint1, 2 and O. Fruchart1 1 Laboratoire Louis Néel, CNRS-INPG-UJF, 25 rue des Martyrs, BP 166, 38042 Grenoble cedex 9, France 2 Laboratoire SPINTEC, CEA-CNRS-INPG-UJF, 17 rue des Martyrs, 38054 Grenoble cedex 9, France * Corresponding author:
[email protected]
Abstract: A finite element formalism (FEM) is proposed to solve the micromagnetic equations. Two bidimensional test problems are treated to estimate the validity and the accuracy of this FEM approach. Keywords: finite elements, micromagnetism, stripe domains.
1. Introduction The imbalance between the four fundamental magnetic interactions (the magneto-crystalline anisotropy, the exchange and magnetostatic interactions, and the Zeeman coupling) is responsible of a very large diversity of magnetic behaviors that the researchers try to explore and to use for technical applications (data storage, sensors, memories, medical imaging …). A large spectrum of shapes and sizes must be studied in order to develop a device or to optimize its performances. Thanks to the high-resolution fabrication techniques (lithography, patterning, self-assembly), it became quite easy to obtain diverse submicron magnetic systems [1]. Instead difficulties are met in the understanding of their complex behavior and that is why the experimental studies and numerical modeling must be combined. Nowadays most of the micromagnetic softwares are based on the finite difference (FD) approximation. The most important advantage of this method is its speed, the periodic discretization allowing the use of Fast Fourier Transforms [2]. Furthermore, specific integration schemes were developed to integrate the LandauLifshitz-Gilbert (noted LLG hereafter) equation describing the magnetization dynamics [3]. These integration schemes take into account the constraint on the magnetization norm implicitly. Unfortunately the regular discretization induces a numerical roughness at the sample surfaces as shown in figure 1. Thus only the systems bounded exclusively by planar surfaces parallel to the main axes of the grid, can be in principle reliably computed [4].
Figure 1. Real circular disk and its FD numerical approximation.
A possible way of overcoming these undesired mesh effects consists in treating the micromagnetic problem by applying the finite element method [5], used from a long time to describe engineering problems with complex geometries. This alternative approach consists in projecting the micromagnetic equations on so called test functions. Thus one manages to avoid the unwanted numerical roughness but, unfortunately, the mathematical background is more complex than in the case of the FD approach. In the present paper we show the steps we followed when building up our FEM approach for micromagnetism and the results obtained for two 2D magnetic test cases are presented.
2. Micromagnetic equations The micromagnetic theory is used for the description of ferromagnetic and ferrimagnetic continuous media. There are two basic hypotheses: a) The individual magnetic moments are replaced by a continuous distribution of magnetization, and consequently the fields and the energies are all continuous functions of position and time. b) Since the micromagnetism deals with ferromagnetic materials, the amplitude of the magnetization vector is considered to be constant everywhere in the sample. ⎧⎪M (r, t ) = M S m(r, t ) ⎨ ⎪⎩ m(r, t ) = 1
(1)
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
Ms denotes the mean magnetic moment per unit volume and is considered to be constant [3]. The aim of micromagnetism is to find the equilibrium state of a magnetic system. The equilibrium configuration is obtained by minimizing the total energy with respect to the local orientation of the magnetization:
[
]
Etot (m ) = ∫ Aex (∇m ) dV + ∫ K1 1 − (u k ⋅ m ) dV − 2
V
∫μ
V
2
V
0
1 M S m ⋅ H app dV − ∫ μ 0 M S m ⋅ H m dV 2 V
(2)
To describe the relaxation of the magnetization towards the equilibrium one must integrate the LLG dynamic equations: ∂m ∂m ⎞ ⎛ = − μ 0γ (m × H eff ) + αμ 0γ ⎜ m × ⎟ ∂t ∂t ⎠ ⎝
(3)
with the constraint |m|=1, where γ is the gyromagnetic ratio of the free electron and α is the dimensionless damping parameter. Heff is the effective field obtained by variational derivation of the total energy Etot with respect to m(r). Following (2) the effective field can be written as the sum of four fields: exchange field Hex, anisotropy field Hani, magnetostatic field Hm and applied field Happ [3]. Solving the micromagnetic problem by FEM means passing to the weak form of the equations involved.
3. Weak formulations 3.1 Weak form of the magnetostatic equations The evaluation of Hm is the most difficult issue because of its long-range character. Let us consider two of Maxwell’s equations in magnetostatics: ∇⋅B = 0
∇ × Hm = 0
(4)
the magnetization being related to the magnetic induction B and to the demagnetizing field Hm by: (5) B = μ 0 (H m + M ) There are two possible ways to solve the magnetostatic problem: the magnetic vector potential (applicable only in 2D) or the magnetic scalar potential approach. The starting point in the magnetic vector potential approach is the solenoidal nature of the B vector. This means that B can be written as the curl of the magnetic vector potential A.
This way the magnetostatic field becomes: Hm =
1
μ0
B−M =
1
μ0
∇×A −M
(6)
Moreover, a Dirichlet condition for the magnetic vector potential A is applied at infinity to ensure the uniqueness of the solution: (7) A( r → ∞ ) → 0 To derive its weak form, the magnetostatic equation (6) is multiplied by a vector weighting function v and then integrated over the whole space Ω: (8) ∫ v ⋅ ∇ × H m dΩ = 0 Ω
By using:
div (v × H m ) = H m ⋅ ∇ × v − v ⋅ ∇ × H m
(9) and the continuity condition of the tangential component of Hm at free surfaces, the weak form becomes: (10) ∫ ∇ × v ⋅ ∇ × A dΩ = μ0 ∫ M ⋅ ∇ × v dΩ Ω
Ω
The invariance of the studied system along the Oz direction indicates that only the z component of the vector potential A must be considered. Since v=(0, 0, v) the final form of the weak formulation is: ∫ ∂ x v(∂ x Az + μ0 M y ) +∂ y v(∂ y Az + μ0 M x ) = 0 (11) For the treatment of the condition at infinity a method based on spatial transformations was used. In this method the infinite exterior that must be considered for this problem is converted into a finite domain, so the “open boundary problem” becomes a “closed boundary problem” [6]. In order to apply the transformation the 2D system is modified: the upper and the lower semi-infinite regions are converted in two finite domains bounded by straight lines: -Y∞
