Micromagnetic Energy Contributions

Intuition

A ferromagnet is a battleground. Exchange wants every spin parallel to its neighbor. Magnetostatic energy wants the material to look “neutral” from the outside, with no stray field. The external field (Zeeman) wants every spin pointing its way. And the crystal lattice (anisotropy) wants spins along its preferred axes. None of these factions can fully win — the compromise they reach is what gives ferromagnets their rich phenomenology: domains, domain walls, hysteresis loops, skyrmions, and the shape-dependence of switching fields.

Micromagnetism is, at its core, the variational calculus of this competition: minimize the total energy functional $W[\vec{m}]$ over the unit magnetization vector field $\vec{m}(\vec{r})=\vec{M}/M_s$.

Formal Definition

The total micromagnetic energy is a functional of $\vec{m}(\vec{r})$:

$$W[\vec{m}]=W^{\text{ex}}+W^{\text{ms}}+W^Z+W^{\text{anis}}.$$

Its functional derivative defines the effective field that drives the dynamics in the LLG equation:

$${\vec{H}}^{\text{eff}}=-\frac{1}{{\mu }_0{M}_s}\frac{\delta W}{\delta \vec{m}}.$$

Equilibrium configurations satisfy $\vec{m}\times {\vec{H}}^{\text{eff}}=\vec{0}$ everywhere — i.e. $\vec{m}$ is locally parallel to the effective field generated by all the other terms.

Key Results

1. Exchange energy — favors uniform alignment {#exchange}

The exchange interaction is a quantum-mechanical effect (Pauli + Coulomb) that lowers the energy of parallel neighboring spins. It deserves a moment of explanation, because nothing classical predicts its size.

Why exchange and not dipole–dipole? The magnetic dipole–dipole energy between two neighbouring atomic moments is

$$E_{\text{dd}}\sim \frac{{\mu }_0{\mu }_B^2}{4\pi a^3}\sim k_B\cdot 1\text{K},$$

which would predict Curie temperatures of order one Kelvin — three orders of magnitude too small to explain the $T_C\sim 1000$ K of Fe, Co, Ni. The mechanism that actually aligns spins is electrostatic in origin: the Pauli exclusion principle forbids two electrons of the same spin from occupying the same orbital, which spatially separates them and lowers their Coulomb repulsion. The total energy of two neighbouring electrons therefore depends on the relative orientation of their spins, even though the interaction itself contains no spin operator. This is exchange.

Heisenberg Hamiltonian — the discrete localized version of exchange:

$${\mathcal{H}}_{\text{ex}}=-J\sum _{⟨i,j⟩}{\vec{S}}_i\cdot {\vec{S}}_j,$$

where $J$ is the exchange integral between neighbours. The sign of $J$ decides the magnetic order entirely:

• $J>0$: parallel alignment is favoured → ferromagnetism (Fe, Co, Ni).
• $J<0$: antiparallel alignment is favoured → antiferromagnetism or ferrimagnetism (Cr, MnO, magnetite).

In the continuum limit, slow gradients of $\vec{m}(\vec{r})$ give

$$W^{\text{ex}}={\int }_{V}A(\nabla \vec{m}{)}^{2}dV,$$

where $A$ (in J/m) is the exchange stiffness constant, related to $J$ by $A\sim J{S}^2/a$ for a cubic lattice of spacing $a$.

Picture: a spring between neighbouring spins. Bending $\vec{m}$ in space stretches the spring and costs energy — which is why domain walls have a finite width set by the balance between exchange and anisotropy.

Localized vs itinerant. The Heisenberg picture assumes integer spins fixed on atoms; it works for insulating magnets (rare earths, oxides). For the 3d transition metals the spins live in delocalized bands and the right starting point is the Stoner model, where the same exchange physics surfaces as a band splitting.

2. Magnetostatic (demagnetizing) energy — favors flux closure

Each magnetized region generates its own stray field ${\vec{H}}^{\text{ms}}$. The self-energy of $\vec{M}$ in this field is

$$W^{\text{ms}}=-\frac{{\mu }_0}{2}{\int }_V\vec{M}\cdot {\vec{H}}^{\text{ms}}dV,$$

with ${\vec{H}}^{\text{ms}}$ satisfying the magnetostatic Maxwell equations

$$\nabla \times {\vec{H}}^{\text{ms}}=0,\nabla \cdot ({\mu }_0(\vec{M}+{\vec{H}}^{\text{ms}}))=0.$$

Picture: the field that “leaks out” of a magnet costs energy. A thin film prefers in-plane magnetization (small stray field) over out-of-plane (large stray field). Large uniform domains develop high $W^{\text{ms}}$ and break up into smaller domains to short-circuit their own flux — this is the origin of domain structure.

Magnetostatic energy is nonlocal: ${\vec{H}}^{\text{ms}}(\vec{r})$ depends on $\vec{M}$ everywhere in the sample, which makes it by far the most expensive term to evaluate in micromagnetic simulations.

3. Zeeman energy — favors alignment with the applied field

$$W^Z=-{\mu }_0{\int }_V\vec{M}\cdot {\vec{H}}^{\text{ext}}dV.$$

Picture: a bar magnet in a field rotates to align with it. This is the only term that contains an external driving parameter — it’s the knob the experimentalist turns when sweeping ${\vec{H}}^{\text{ext}}$ to trace out a hysteresis loop.

4. Anisotropy energy — favors crystallographic easy axes

The crystal lattice breaks rotational symmetry: spin–orbit coupling makes some directions of $\vec{m}$ cheaper than others. Generically,

$$W^{\text{anis}}={\int }_{V}K{f}^{\text{anis}}(\vec{m})dV,$$

where $K$ is an anisotropy constant and $f^{\text{anis}}(\vec{m})$ encodes the crystal symmetry. Two common cases:

• Uniaxial (e.g. hcp Co, $\hat{u}$ the easy axis):

$$f_{\text{uni}}^{\text{anis}}(\vec{m})=1-(\vec{m}\cdot \hat{u}{)}^2.$$

• Cubic (e.g. bcc Fe):

$$f_{\text{cub}}^{\text{anis}}(\vec{m})=m_x^2{m}_y^2+m_y^2{m}_z^2+m_z^2{m}_x^2.$$

Picture: in iron, magnetization along the cube edges $⟨100⟩$ is energetically preferred; the diagonals $⟨111⟩$ are the “hard” directions.

5. Other contributions (omitted here)

For completeness — neglected in this basic treatment but important in specific contexts:

• Magnetoelastic energy: strain couples to magnetization (relevant for magnetostriction and strain-mediated switching).
• Thermal energy: randomizes spins at finite $T$ (relevant for superparamagnetism and stochastic switching).
• Dzyaloshinskii–Moriya interaction (DMI): an antisymmetric exchange that favors chiral spin textures — essential for skyrmions and chiral domain walls in interfacial systems.

Summary

Energy	Symbol	Favors	Range
Exchange	$W^{\text{ex}}$	uniform alignment of neighboring spins	local
Magnetostatic	$W^{\text{ms}}$	flux closure / domain formation	nonlocal
Zeeman	$W^Z$	alignment with ${\vec{H}}^{\text{ext}}$	local
Anisotropy	$W^{\text{anis}}$	alignment along easy axes	local

The equilibrium configuration is the compromise these four terms strike. The characteristic length scales of this competition are:

$${\ell }_{\text{ex}}=\sqrt{\frac{2A}{{\mu }_0{M}_s^2}}\text{(exchange length)},{\ell }_K=\sqrt{\frac{A}{K}}\text{(domain-wall width parameter)}.$$

These set the smallest features (vortex cores, domain walls, skyrmion radii) that micromagnetic structures can resolve.

Connections

• effective-field — ${\vec{H}}^{\text{eff}}=-({\mu }_0{M}_s{)}^{-1}\delta W/\delta \vec{m}$
• llg-equation — the dynamics driven by the energy gradient
• magnetic-domains — domains and walls as the equilibrium of these energies
• exchange-interaction — deep dive on $W^{\text{ex}}$ (stub)
• magnetostatic-energy — deep dive on $W^{\text{ms}}$ (stub)
• zeeman-energy — deep dive on $W^Z$ (stub)
• magnetocrystalline-anisotropy — deep dive on $W^{\text{anis}}$ (stub)

References

• W. F. Brown Jr., Micromagnetics (Interscience, 1963).
• A. Hubert & R. Schäfer, Magnetic Domains (Springer, 1998).
• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010).