Core equations

Landau-Lifshitz-Gilbert (LLG) equation

The dynamic of magnetic moment \(\vec{\mu}_i\) is governed by LLG equation,

\[\frac{\partial \vec{S}_i}{\partial t} = -\frac{\gamma}{(1+\alpha^2)} \vec{S}_i \times (\vec{H}_i + \alpha \vec{S}_i \times \vec{H}_i) ]\]

where \(\vec{S}_i\) is the unit vector of magnetic moment,

\[\vec{S}_i=\frac{\vec{\mu}_i}{\mu_s},\]

and \(\vec{\mu}_s = |\vec{\mu}_i|\), the effective field \(\vec{H}_i\) is defined as

\[\vec{H}_i = - \frac{1}{\mu_s} \frac{\partial \mathcal{H}}{\partial \vec{S}_i}.\]

Interactions

In the level of atomic moments, \(\vec{\mu}_s\), originated from the angular momentum of electrons in atoms, could be employed as the base unit in simulations. There are several typical interactions between magnetic moments. The total Hamiltonian is the summation of them.

\[\mathcal{H} = \mathcal{H}_{ex} + \mathcal{H}_{an} + \mathcal{H}_d + \mathcal{H}_{ext}\]

Exchange interaction

The classical Heisenberg Hamiltonian with the nearest-neighbor exchange interaction,

\[\begin{split}\mathcal{H}_{ex} = -J \sum_{<i,j>}\vec{S}_i \cdot \vec{S}_j\end{split}\]

where the summation is taken only once for each pair, so the effective field is

\[\begin{split}\vec{H}_{i,ex} = \frac{J}{\mu_s} \sum_{<i,j>} \vec{S}_j\end{split}\]

In the continuum limit the exchange energy could be written,

\[E_{ex} = \int_\Omega A (\nabla \vec{m})^2 dx\]

so the corresponding effective field is

\[\vec{H} = \frac{2 A}{\mu_0 M_s} \nabla^2 \vec{m}\]

Once we implemented the Heisenberg exchange interaction, the effective field in the continuum case can be computed by the same codes with

\[J_x = 2A \frac{\Delta y \Delta z}{\Delta x}\]

Note that we needs the factor of \(\mu_0\) to convert the units from T to A/m.

Anisotropy

The Hamiltonian for uniaxial anisotropy with easy axis in x direction is expressed as,

\[\mathcal{H}_{an} = - D \sum_i (\vec{S}_{x,i})^2\]

and the corresponding field is

\[\vec{H}_{i,an} = \frac{2 D}{\mu_s} S_{x,i} \vec{e}_x\]

UniaxialAnisotropy

The UniaxialAnisotropy energy of the magnetic system is defined as

\[E_{anis} = \int_\Omega K [ 1 - (\vec{m} \cdot \vec{u})^2 ] dx\]

the the effective field is,

\[\vec{H}=\frac{2 K}{\mu_0 M_s} (\vec{m} \cdot \vec{u}) \vec{u}\]

Dipolar interaction

The Hamiltonian for dipolar interaction is,

\[\begin{split}\mathcal{H}_{d}=-\frac{\mu_0}{4\pi}\sum_{i<j}\frac{3 (\vec{\mu}_i\cdot \vec{e}_{ij})(\vec{\mu}_j\cdot \vec{e}_{ij}) - \vec{\mu}_i \cdot \vec{\mu}_j}{r_{ij}^3}\end{split}\]

(1)\[\vec{H}_{i,d} =\frac{\mu_0}{4\pi}\sum_{i \neq j}\frac{3 \vec{e}_{ij} (\vec{\mu}_j\cdot \vec{e}_{ij}) - \vec{\mu}_j}{r_{ij}^3}\]

Dzyaloshinskii-Moriya interaction (DMI)

DMI is an antisymmetric, anisotropic exchange coupling beteen spins (magnetic moments),

\[\begin{split}\mathcal{H}_{dmi}= \sum_{<i,j>} \vec{D}_{ij}\cdot [\vec{S}_i \times \vec{S}_j]\end{split}\]

Note that \(\vec{a}\cdot(\vec{b}\times\vec{c})=(\vec{a}\times\vec{b})\cdot\vec{c}\), the effective field can be computed by

\[\begin{split}\vec{H}_i = - \frac{1}{\mu_s} \frac{\partial \mathcal{H}}{\partial \vec{S}_i} = \frac{1}{\mu_s} \sum_{<i,j>} \vec{D}_{ij}\times\vec{S}_j\end{split}\]

For bulk materials \(\vec{D}_{ij} = D \vec{r}_{ij}\) and for interfacial DMI one has \(\vec{D}_{ij} = D \vec{r}_{ij} \times \vec{e}_z\), in both cases the vector \(\vec{D}_{ij}\) such that \(\vec{D}_{ij}=-\vec{D}_{ji}\).

In the continuum limit the bulk DMI energy could be written,

\[E_{dmi} = \int_\Omega D_a \vec{m} \cdot (\nabla \times \vec{m}) dx\]

where \(D_a = -D/a^2\) and the effective field is

\[\vec{H}=-\frac{2 D_a}{\mu_0 M_s} (\nabla \times \vec{m})\]

For the interfacial case, the effective field thus becomes,

\[\vec{H}=\frac{2 D}{M_s a^2} (\vec{e}_x \times \frac{\partial \vec{m}}{\partial y} - \vec{e}_y \times \frac{\partial \vec{m}}{\partial x} )\]

Compared with the effective field [PRB 88 184422]

\[\vec{H}=\frac{2 D_a}{\mu_0 M_s} ((\nabla \cdot \vec{m}) \vec{e}_z - \nabla m_z)\]

we have \(D_a = D/a^2\), note that there is no negative sign for the interfacial case.

Zeeman energy

The zeeman energy is,

\[\mathcal{H}_{dmi}= - \sum_{i} \mu_s \vec{H}_{ext}\cdot \vec{S}_i\]

Basically, we will follow the above equations to write codes.