3.1. Energy Contributions

The total energy of a system is the sum of several energy parts, some of whom are listed below. Each energy listed in this section is a functional \(E_i: \Mcal \to \Rbb\), that from each magnetization returns a real number.

It is important to notice that depending on the situation, different energies might have completely different scales and this leads to simplifications by ignoring the negligible energies.

3.1.1. Zeeman energy

Favors alignment with an external field \(\Hbf^{\mathsf{zee}}\).

\[ E^{\mathsf{zee}} (\mbf) = - \mu_0 \int_{\Omega} \Ms \ \mbf \cdot \Hbf^{\mathsf{zee}} \dxbf \]

3.1.2. Exchange energy

Favors parallel spin alignment for neighbors.

\[ E^{\mathsf{ex}} (\mbf) = \int_{\Omega} A \left( \nabla \mbf \right)^2 \dxbf \]

\(A \in \Rbb\) is the exchange coupling constant and

\[ (\nabla \mbf)^2 = \sum_{i,j} \left(\frac{\partial m_i}{\partial x_j}\right)^2. \]

3.1.3. Demagnetization energy

Favors locally demagnetized states and generates vortices. Also called magnetostatic energy and stray-field energy.

\[ E^{\mathsf{dem}} (\mbf) = - \frac{\mu_0}{2} \int_{\Omega} \Ms \mbf \cdot \Hbf^{\mathsf{dem}} (\mbf) \dxbf \]

where \(\Hbf^{\mathsf{dem}} (\mbf)\) is the (conservative) magnetic field satisfying \(\Hbf^{\mathsf{dem}} = - \nabla u\), and \(u\) solves

\[ \nabla \cdot (- \nabla u + \Mbf) = 0 \]

in \(\Rbb^3\) with open boundary condition: \(u(\xbf) = \mathcal{O} \left( 1 / \|\xbf\| \right)\) for \(\|\xbf\| \to \infty\).

3.1.4. Anisotropy energy

Arises from the crystal structure.

Uniaxial anisotropy typically occurs in hexagonal or tetragonal crystal structure. In this case a single easy axis \(\ebf_\mathsf{u}\) is favored:

\[ E^{\mathsf{aniu}} (\mbf) = - \int_{\Omega} \left[ K_{\mathsf{u}1} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^2 + K_{\mathsf{u}2} \left( \mbf \cdot \ebf_{\mathsf{u}} \right)^4 + \mathcal{O} (\|\mbf\|^6) \right] \dxbf \]

with scalar (uniaxial) anisotropy constants \(K_{\mathsf{u}1}, K_{\mathsf{u}2} \in \Rbb\). In the case of negative anisotropy constants, uniaxial anisotropy favors an easy plane.

Cubic anisotropy typically occurs in crystals with cubic symmetry. It involves three (pairwise orthogonale) easy axes \(\ebf_i\) for \(i=1,2,3\):

\[ E^{\mathsf{anic}} (\mbf) = \int_{\Omega} \left[ K_{\mathsf{c}1} \left( m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2 \right)^2 + K_{\mathsf{c}2} m_1^2 m_2^2 m_3^2 \right] \dxbf \]

with scalar (cubic) anisotropy constants \(K_{\mathsf{c}1}, K_{\mathsf{c}2} \in \Rbb\). In the case of negative anisotropy constants, cubic anisotropy favors four easy axes.

3.1.5. DMI energy

The Dzyaloshinkii-Moriya interaction (DMI) is an asymmetric exchange interaction that arises in several applications. It counteracts the exchange energy and penalizes domain walls in favor of configuration where the magnetization changes continuously, such as the skyrmion.

One example is a magnetic layer with an interface to a heavy-metal layer. In this case

\[ E^{\mathsf{dmii}} (\mbf) = \int_{\Omega} D_\mathsf{i} \left[ \mbf \cdot \nabla ( \ebf_\mathsf{d} \cdot \mbf ) - (\nabla \cdot \mbf) (\ebf_\mathsf{d} \cdot \mbf) \right] \dxbf \]

where \(D_\mathsf{i} \in \Rbb\) is a coupling constant and \(\ebf_\mathsf{d}\) is the interface normal.

Another example is a magnetic bulk material lacking inversion symmetry. In this case

\[ E^{\mathsf{dmib}} (\mbf) = \int_{\Omega} D_\mathsf{b} \ \mbf \cdot ( \nabla \times \mbf ) \dxbf \]

where \(D_\mathsf{b} \in \Rbb\) is a coupling constant.