Nonlinear conjugate gradient methods in micromagnetics

Abstract

Conjugate gradient methods for energy minimization in micromagnetics are compared. When the step length in the line search is controlled, conjugate gradient techniques are a fast and reliable way to compute the hysteresis properties of permanent magnets. The method is applied to investigate demagnetizing effects in NdFe12 based permanent magnets. The reduction of the coercive field by demagnetizing effects is µ0H = 1.4 T at 450 K. I. INTRODUCTIONThe computation of hysteresis properties of large ferromagnetic systems such as sensor elements or permanent magnets require fast and reliable solvers. Hysteresis simulations are based on the theory of micromagnetics "Brown (1963)". The primary purpose of these simulations is to understand the influence of the microstructure on magnetization reversal. In this work we are focusing on the role of demagnetizing fields in platelet shaped grains of permanent magnets. We also describe the key elements of a micromagnetic solver suitable for simulating large magnetic systems.After discretization of the total Gibbs free energy with finite elements or finite differences the states along the demagnetization curve can be computed by subsequent minimization of the energy for decreasing applied field as outlined in "Kinderlehrer (1997)". The system is in a metastable state. A change of the applied field shifts the position of the local energy minimum. At a critical field, the magnetization becomes unstable. An irreversible switching occurs which is seen as a kink in the demagnetization curve. Then the system either accesses a different metastable state or if fully reversed the magnetization is in a stable state. A reliable numerical method for energy minimization must track all local minima along the demagnetization curve. The resulting algebraic minimization problem is large. Typically the number of unknowns is in the order of 10 to 50 million for a model magnet consisting of around 10 grains. Therefore fast numerical methods are required to obtain results in a Electronic

Micromagnetics of rare-earth efficient permanent magnets

“…We gather these vectors into the vector x which has the dimension 3n, where n is the number of nodes or cells. Then the Gibbs free energy may be written as [14] E…”

“…The vectors x, h d (x), and h ext hold the unit vectors of the magnetization, the demagnetizing field, and the external field at the nodes of the finite element mesh or the cells of a finite difference grid, respectively. For computing the demagnetizating field, equations (12) to (15) can be solved using an algebraic multigrid method on the finite element mesh [14,44,45].…”

Micromagnetics and spintronics: models and numerical methods

“…which suggests a strong coupling of the electric potential u with m and s. If the charge current distribution j e is known, the spin-diffusion dynamics (138) can be solved by inserting (140) into (139) via the gradient of the electric potential ∇u, which results in the spin-current definitioñ…”

“…13 (a), while the conversion of spin currents into charge currents is referred to as inverse spin-Hall effect, see 13 (b). Incorporating these effects into the spin-diffusion model is done by extending the original current definitions (140) and (139) according to Dyakonov [63]. The extended current definitions j 1 e andj 1 s are defined in terms of the original current definitions and read…”

“…For the solution of the self-consistent spin-diffusion model given by the current definitions (140), (139) and their respective source equations (143), (143), the function space for both the solution and the test functions has to be extended. The solution of the self-consistent model comprises both the spin accumulation s and the electric potential u.…”