Numerical Micromagnetics: Finite Difference Methods

Abstract

Micromagnetics is based on the one hand on a continuum approximation of exchange interactions, including boundary conditions, on the other hand on Maxwell equations in the nonpropagative (static) limit for the evaluation of the demagnetizing field. The micromagnetic energy is most often restricted to the sum of the exchange, demagnetizing or self‐magnetostatic, Zeeman, and anisotropy energies. When supplemented with a time evolution equation, including field induced magnetization precession, damping and possibly additional torque sources, micromagnetics allows for a precise description of magnetization distributions within finite bodies both in space and time. Analytical solutions are, however, rarely available. Numerical micromagnetics enables the exploration of complexity in small size magnetic bodies. Finite difference methods are here applied to numerical micromagnetics in two variants for the description of both exchange interactions/boundary conditions and demagnetizing field evaluation. Accuracy in the time domain is also discussed and a simple tool provided in order to monitor time integration accuracy. A specific example involving large angle precession, domain wall motion as well as vortex/antivortex creation and annihilation allows for a fine comparison between two discretization schemes with as a net result, the necessity for mesh sizes well below the exchange length in order to reach adequate convergence.

Size dependence of spin-wave modes in Ni80Fe20 nanodisks

“…This is possibly due to the gridding associated with finite difference method based simulations involving curved geometries. 36 Thus, hereafter, the radius of the nodal ring refers to the mean of the maximum and the minimum distances from its centroid to its periphery. The radius of mode (2, 1) for diameters d = 250 nm, 450 nm, and 1000 nm are 0.31d, 0.27d, and 0.26d, respectively.…”

Influence of magnetization variations in the free layer on a non-volatile magnetic flip flop

“…They interact with the local free layer's magnetization and excite local precessions. The resulting driving force acts only in the region where the current flows, but through the exchange coupling of the local magnetic moments with their neighbors the precessions start to move out of the polarizer stack overlapping region and travel through the free layer, until they hit the other end, get reflected, move back, get pushed out again and so on [13]. During this kind of oscillating motion, the localized precessions of the magnetic moments in the common free layer are excited and start to build up, until the magnetization eventually passes the energy barrier separating its two stable states and relaxes into the other stable state fast.…”

“…variations from grain to grain). The device is modeled with the Landau-Lifshitz-Gilbert equation [16,17]:…”

Controlling vortex chirality and polarity by geometry in magnetic nanodots

“…In our own-coded micromagnetic simulations, the soft ferromagnetic material is discretized into a uniform array of cubic cells (micromagnetic cells) in which the effective field H eff (the sum of applied, stray, and exchange fields) 20,21 is calculated by a finite-difference method. [21][22][23] The stray field is obtained directly using volume (micromagnetic cell) averaged expressions.…”

“…[21][22][23] The stray field is obtained directly using volume (micromagnetic cell) averaged expressions. 24 Calculation time is reduced by computing the stray field from farther cells as the field due to a larger effective cell whose magnetization is the average of the small contained ones.…”