Dynamics terms and dynamics equation

There are several different dynamics terms that are implemented in micromagneticmodel. Here, we will provide a short list of them, together with some basic properties.

Dynamics terms

1. Precession

The main parameter required for the precession term is the gyrotropic ratio gamma. Optionally, name can be given to the dynamics term.

import micromagneticmodel as mm

precession = mm.Precession(gamma0=2.211e5)

The values of its arguments are

precession.gamma0

221100.0

precession.name

'precession'

String and LaTeX representations are

repr(precession)

'Precession(gamma0=221100.0)'

precession

$-\frac{\gamma_{0}}{1 + \alpha^{2}} \mathbf{m} \times \mathbf{H}_\text{eff}$

2. Damping

Damping dynamics term requires Gilbert damping \(\alpha\) to be provided. Optionally, name can be given as well.

damping = mm.Damping(alpha=0.1)

The values of attributes are

damping.alpha

0.1

damping.name

'damping'

String and LaTeX representations are

repr(damping)

'Damping(alpha=0.1)'

damping

$-\frac{\gamma_{0} \alpha}{1 + \alpha^{2}} \mathbf{m} \times (\mathbf{m} \times \mathbf{H}_\text{eff})$

3. Zhang-Li term

This dynamics term requires the non-adiabatic factor \(\beta\) and velocity vector \(\mathbf{u}\) to be passed. As before, name is optional as well.

zhangli = mm.ZhangLi(u=1e6, beta=0.5)

The values of attributes are

zhangli.u

1000000.0

zhangli.beta

0.5

String and LaTeX representations are

repr(zhangli)

'ZhangLi(u=1000000.0, beta=0.5)'

zhangli

$-\frac{1+\alpha\beta}{1+\alpha^{2}} \mathbf{m} \times (\mathbf{m} \times (\mathbf{u} \cdot \boldsymbol\nabla)\mathbf{m}) - \frac{\beta - \alpha}{1+\alpha^{2}} \mathbf{m} \times (\mathbf{u} \cdot \boldsymbol\nabla)\mathbf{m}$

For time-dependent current terms please refer to this notebook.

Dynamics equation

Dynamics equation of the micromagnetic system is the sum of dynamics terms. For instance, if we sum two dynamics terms, we get:

type(precession + damping)

micromagneticmodel.dynamics.dynamics.Dynamics

If we assign this value to a separate variable, we can explore some of its properties.

dynamics = precession + damping

The string representation is:

repr(dynamics)

'Precession(gamma0=221100.0) + Damping(alpha=0.1)'

Similarly, the LaTeX representation is

dynamics

$-\frac{\gamma_{0}}{1 + \alpha^{2}} \mathbf{m} \times \mathbf{H}_\text{eff}-\frac{\gamma_{0} \alpha}{1 + \alpha^{2}} \mathbf{m} \times (\mathbf{m} \times \mathbf{H}_\text{eff})$

This dynamics equation consists of two dynamics term. To add another term to it += operator can be used.

dynamics += zhangli

Dynamics equation is now

dynamics

$-\frac{\gamma_{0}}{1 + \alpha^{2}} \mathbf{m} \times \mathbf{H}_\text{eff}-\frac{\gamma_{0} \alpha}{1 + \alpha^{2}} \mathbf{m} \times (\mathbf{m} \times \mathbf{H}_\text{eff})-\frac{1+\alpha\beta}{1+\alpha^{2}} \mathbf{m} \times (\mathbf{m} \times (\mathbf{u} \cdot \boldsymbol\nabla)\mathbf{m}) - \frac{\beta - \alpha}{1+\alpha^{2}} \mathbf{m} \times (\mathbf{u} \cdot \boldsymbol\nabla)\mathbf{m}$

Accesing individual dynamics terms from the dynamics equation

There are two ways of retrieving an individual dynamics term from the dynamics equation. Let us say we want to change the value of the Gilbert damping constant \(\alpha\).

If a dynamics term with name mydynamicsterm was added to the Hamiltonian, that term can be accessed by typing hamiltonian.nydynamicsterm.

dynamics.damping

$-\frac{\gamma_{0} \alpha}{1 + \alpha^{2}} \mathbf{m} \times (\mathbf{m} \times \mathbf{H}_\text{eff})$

dynamics.damping.alpha

0.1

dynamics.damping.alpha = 5e-3

dynamics.damping.alpha

0.005

Similarly, the precession term is

dynamics.precession

$-\frac{\gamma_{0}}{1 + \alpha^{2}} \mathbf{m} \times \mathbf{H}_\text{eff}$