Micromagnetic theory of ferromagnets

The micromagnetic approach to modeling magnetic systems is similar to the phase field method in the sense that a continuum description of an order parameter is provided. In ferromagnets, below the critical temperature, the spins order forming a nonzero and net spin magnetization - see Stöhr and Siegmann (2006). Additionally, the spins respond to a specific symmetry characterized by the presence (or absence) of magnetocrystalline anisotropy. This drives the tendency to form ordered domains but competitions from the ferromagnetic exchange or the demagnetizing fields (internal or external) can favor other spin textures such as vortices, spin cycloids, or skymrions. To predict the magnetic ground states (and/or their dynamics), the Landau-Lifshitz-Gilbert (LLG) equation is used,

(1)

where is the electron gyromagnetic ratio and is a phenomenological (dimensionless) Gilbert damping coefficient. The vector is the effective field which arises from different physics of the magnet (exchange, anisotropy, spin-torque transfer, stray demagnetizing fields,...). The first term in the LLG equation is conservative and leads to a precession about the effective field . The second term in the governing equation drives the system towards its energy minima (or system equilibrium) in the presence of nonzero Gilbert damping . In contrast to ferroelectrics where the magnitude of the electric dipole moment can stretch or shrink, the magnetization is expected to stay on the unit sphere during the LLG time evolution.

There are two possible numerical resources within the FERRET/MOOSE ecosystem to use to enforce this constraint. One possibility is to renormalize at the conclusion of every time step in the evolution of Eq. (1). Another option is to evolve the Landau-Liftshitz-Bloch (LLB) equation

(2)

in which the latter term provides a longitudinal restoring force to mapping it back onto the unit sphere. See the references of Garanin and Chubykalo-Fesenko (2004) for an overview of the LLB equation. We have demonstrated that both methods are essentially equivalent (they also can be used together for stricter convergence criteria).

Micromagnetic theory for antiferromagnets

If the material is an antiferromagnet, then the spin system forms two sublattices whose interaction is mediated by a short-range (typically nearest- or next-nearest-neighbor) exchange. To simulate an antiferromagnet at the micromagnetic level of theory, we have the two-sublattice LLG equation,

where . In a collinear antiferromagnet, the total net magnetization, , is zero indicating that the spin sublattices perfectly compensating each other. However, under a field, this symmetry is broken leading to resonance phenomena due to the conservative term. As mentioned before, one can choose to use the LLB equation or let the modulus be renormalized at every time step for by a MOOSE UserObject.