In micromagnetics, magnetisation dynamics are described using the LLG equation. For non-zero temperature simulations we can use the LLB equation which contains an additional longitudinal susceptibility effective field contribution, as well as a longitudinal damping term. Other equations available in Boris include stochastic versions of these equations, as well as magnetisation dynamics equations complemented by spin-transfer torques and spin-orbit torques. Additionally Boris allows calculations of various spin torques in single and multi-layered geometries using a self-consistent spin transport solver (see Spin Transport).
The LLG equation is given as (for definitions of parameters refer to the manual):
In addition to external magnetic fields, a number of contributions can be included in the effective field Heff, the most commonly used are given below.
Direct exchange interaction:
Dzyaloshinskii-Moriya bulk exchange interaction:
Interfacial Dzyaloshinskii-Moriya exchange interaction:
Surface exchange in multi-layered geometries:
Uniaxial magneto-crystalline anisotropy:
Cubic magneto-crystalline anisotropy: