Micromagnetism

MagTense solves the Landau-Lifshitz (LL) equation on a mesh of tetrahedral or rectangular prism cells, using the same fully analytical demagnetization tensor that is used by the magnetostatic framework. The micromagnetic model is implemented in Fortran (sub-project MagTenseMicroMag) and is driven either from Matlab through the MagTenseLandauLifshitzSolver_mex MEX-file, or from Python through the magtense.micromag.MicromagProblem class.

This part of the manual describes the input parameters and features of the micromagnetic model. The pages below go through the model one topic at a time; Micromagnetic parameter reference lists every parameter with its Matlab name, its Python name and its default value.

• Setting up a micromagnetic problem
  • Matlab problem setup
  • Python problem setup
  • Units and conventions
• Grids and meshes
  • Uniform grid
  • Unstructured prisms
  • Tetrahedral grid
  • Passing an externally computed exchange matrix
• Material parameters
  • Magnetocrystalline anisotropy
  • Time-dependent damping
• Exchange interaction
  • Exchange on a uniform grid
  • Exchange on unstructured meshes
  • Exchange across a material interface
  • Supplying the exchange matrix directly
  • Returning the exchange matrix
• Demagnetization field
  • Switching the demagnetization off
  • Cell-averaged versus point-evaluated tensor
  • Approximating the tensor
  • Adding noise to the demagnetization field
  • Storing and reusing the tensor
  • Parallelism and hardware
• Solvers and time integration
  • Solver type
  • Time grid and applied field
  • ODE solver settings
  • Adaptive hysteresis
• Periodic boundaries and macrogeometry
  • Macrogeometry
  • Sample shape correction
  • Periodic exchange boundaries
  • Periodic boundaries on a tetrahedral mesh
• Thermal fluctuations
  • The stochastic field
  • Reproducible random numbers
  • Validation
• Micromagnetic output
  • Matlab solution struct
  • Python result list
  • Dry runs
• Micromagnetic parameter reference
  • Grid and geometry
  • Material and anisotropy
  • Solver, time and applied field
  • Adaptive hysteresis parameters
  • Exchange
  • Demagnetization
  • Macrogeometry and shape correction
  • Thermal parameters
  • FMM demagnetization
  • Output, tracing and timing
  • Matlab-only convenience fields
• Micromagnetic examples and validation
  • muMag standard problems
  • Feature tests
  • Other examples
  • Running the test suites

The equation that is solved

MagTense integrates the Landau-Lifshitz equation in the form

\[\frac{\partial \mathbf{m}}{\partial t} = -\gamma \, \mathbf{m} \times \mathbf{H}_\mathrm{eff} - \alpha \, \mathbf{m} \times \left( \mathbf{m} \times \mathbf{H}_\mathrm{eff} \right),\]

where \(\mathbf{m} = \mathbf{M}/M_s\) is the reduced magnetization of a cell, i.e. a unit vector. Both the precession constant \(\gamma\) and the damping constant \(\alpha\) are given in units of \(\mathrm{m/(A\,s)}\), i.e. this is the Landau-Lifshitz form and not the Gilbert form. They are related to the dimensionless Gilbert damping \(\alpha_\mathrm{G}\) by

\[\alpha = \frac{\gamma \, \alpha_\mathrm{G}}{1 + \alpha_\mathrm{G}^2}.\]

Note

Setting \(\gamma = 0\) removes the precession term. Several of the quasi-static examples do this - among them the relaxation stage of standard problem 4 and the adaptive hysteresis example - since only the relaxed state is of interest there and dropping the precession makes the relaxation much faster.

The effective field is the sum of five contributions, all in \(\mathrm{A/m}\),

\[\mathbf{H}_\mathrm{eff} = \mathbf{H}_\mathrm{ext} + \mathbf{H}_\mathrm{exc} + \mathbf{H}_\mathrm{dem} + \mathbf{H}_\mathrm{ani} + \mathbf{H}_\mathrm{th},\]

being the applied (external) field, the exchange field, the demagnetization field, the magnetocrystalline anisotropy field and the stochastic thermal field. Each term is described on its own page below. The thermal term is only included when a non-zero temperature is specified, see Thermal fluctuations.

All quantities are in SI units. Note in particular that fields and magnetizations are in \(\mathrm{A/m}\) and not in tesla; multiply by \(\mu_0\) to convert.