Scipy-based Landau-Lifshitz-Gilbert (LLG) equation solver for a magnetic system, described by a classicalmodel including Heisenberg exchange interaction between nearest-neighbor spins, current-driven exchange interaction between lattice spins and nonequilibrium electrons, various anisotropy terms, Dzyaloshinskii-Moriya interaction, and an external field.

The main equation to be integrated is the Landau-Lifshitz-Gilbert equation (LLG)

which governs the time-dependent evolution of a spin on site .
The effective field is computed by

with the classical Hamiltonian capturing all dominant interactions

Here, denotes pairs of nearest neighbors, and the sum over it captures the lattice geometry and the boundary conditions. Within this module, following boundary conditions are implemented:

• Periodic in both x and y direction
• Periodic in x only, y open
• Periodic in y only, x open
• Open boundaries in x and y direction

on a square lattice of dimensions X,Y.

The parameters of this Hamiltonian are

• Nearest-neighbor exchange interaction between neighboring spins in the lattice
• Exchange interaction between conducting electrons and localized magnetic moments
• Dzyaloshinskii-Moriya interaction
• magnetic anisotropies and
• Demagnetization
• External field
• and is the magnitude of magnetic moment of the localized magnetic moments in the lattice.