Stoner–Wohlfarth Model

Intuition

How small must a ferromagnetic particle be before it stops splitting into domains and behaves as one rigid macrospin? And once it does, how does its magnetization respond to a field?

The Stoner–Wohlfarth model (1948) is the minimal answer: take a single uniaxial nano-particle, assume it stays uniformly magnetized during reversal (no domain wall, no buckling, no curling), and add one easy axis plus one external field. Two energy terms, one angle. From this toy problem comes the whole language of switching fields, coercivity versus angle, and the astroid that engineers still use to design magnetic-recording media.

Formal definition

Consider an ellipsoidal single-domain particle of volume $V$, saturation magnetization $M_s$, and uniaxial anisotropy constant $K_u>0$. Let $\varphi$ be the angle between the easy axis and the applied field $\vec{H}$, and $\theta$ the angle between $\vec{M}$ and $\vec{H}$. The Stoner–Wohlfarth energy is then a single function of $\theta$:

$$E(\theta ,\varphi )=V[K_u{\sin }^2(\theta -\varphi )-{\mu }_0{M}_{s}H\cos \theta ].$$

The first term is the uniaxial anisotropy energy (minimal when $\vec{M}$ lies along the easy axis); the second is the Zeeman energy (minimal when $\vec{M}$ is parallel to $\vec{H}$).

Key results

1. Two stable states, one bit of memory

At $H=0$, the energy has two minima at $\theta =\varphi$ and $\theta =\varphi +\pi$ — the two ends of the easy axis. The barrier between them is

$$\Delta E=K_{u}V,$$

setting the lifetime of each magnetization state through the Arrhenius law $\tau \propto \exp (K_{u}V/k_{B}T)$. This is why magnetic-recording media must have large $K_{u}V$ to keep written bits stable for years.

2. Switching field and coercivity

For the field aligned with the easy axis ($\varphi =0$), one barrier collapses when

$$\boxed{H_{\text{sw}}(\varphi =0)=H_K=\frac{2K_u}{{\mu }_0{M}_s}}.$$

This is the anisotropy field — the upper bound on the coercivity of a single-domain particle, and the natural scale of every hysteresis loop in this wiki.

For general $\varphi$, the switching field is given by the Stoner–Wohlfarth astroid:

$$H_{\text{sw}}(\varphi )=\frac{H_K}{{\left({\sin }^{2/3}\varphi +{\cos }^{2/3}\varphi \right)}^{3/2}},$$

an astroid-shaped curve when plotted in the $(H_{\parallel },H_{\perp })$ plane. The lowest switching field, $H_{\text{sw}}=H_K/2$, occurs at $\varphi =45^{\circ }$ — counter-intuitive but crucial for engineering realistic write fields in MRAM and hard-disk technology.

3. The single-particle hysteresis loop

Sweep $H$ from $+\infty$ to $-\infty$ at fixed $\varphi$:

• For $\varphi =0$ the magnetization stays rigidly along the easy axis and flips abruptly at $H=-H_K$. The loop is a perfect rectangle of width $2H_K$ and remanence $M_s$.
• For $0<\varphi <90^{\circ }$ the loop becomes sheared: $\vec{M}$ rotates reversibly toward $\vec{H}$, then jumps when one minimum disappears.
• For $\varphi =90^{\circ }$ no jump occurs — the loop closes into a straight line. Reversal is fully reversible and the coercivity vanishes.

The set of single-particle loops over $\varphi$ generates the textbook “family of S-shapes” plotted on every Stoner–Wohlfarth figure.

4. From single particle to real magnets

Real ferromagnets are made of many grains with different anisotropy axes. Averaging the Stoner–Wohlfarth loop over a random orientation distribution gives the Stoner–Wohlfarth coercivity:

$$H_c^{\text{SW}}\approx 0.48H_K=0.96\frac{K_u}{{\mu }_0{M}_s}.$$

Measured coercivities in bulk magnets are almost always lower than this — by factors of 10–100 — because domain walls, nucleation defects and thermal activation provide easier reversal pathways. The gap between $H_c^{\text{SW}}$ and measured $H_c$ is known as Brown’s paradox and motivates much of modern magnetism research.

Application: magnetic recording {#magnetic-recording}

Every bit on a hard disk is, to first approximation, a Stoner–Wohlfarth particle:

• $M_s$ sets the read signal — bigger is better.
• $K_{u}V$ sets the thermal stability — bigger is better.
• $H_K=2K_u/({\mu }_0{M}_s)$ sets the write field — smaller is better.

These three demands pull in opposite directions, and the resulting trilemma is the central design problem of magnetic-recording media. Heat-assisted recording (HAMR) sidesteps it by temporarily reducing $K_u$ during writing.

Summary

Two energy terms, one angle: the Stoner–Wohlfarth model is the minimal nano-magnet. It predicts:

• two stable states with a barrier $K_{u}V$,
• a switching field $H_K=2K_u/({\mu }_0{M}_s)$ on the easy axis,
• an astroid switching surface in the $(H_{\parallel },H_{\perp })$ plane,
• a polycrystalline coercivity $\approx 0.48H_K$.

Modern micromagnetism builds on the same picture but lets $\vec{m}(\vec{r},t)$ vary in space and time — recovering domain walls, vortices and the LLG dynamics that the Stoner–Wohlfarth ansatz forbids.

Connections

• magnetocrystalline-anisotropy — where $K_u$ comes from
• hysteresis — what $H_c$, $M_r$, and $M_s$ mean macroscopically
• micromagnetic-energy — the energy terms that survive when $\vec{m}$ is allowed to vary
• magnetic-domains — how the single-domain assumption breaks down
• llg-equation — dynamics that interpolate between SW stable states

References

• E. C. Stoner & E. P. Wohlfarth, Phil. Trans. Roy. Soc. A 240, 599 (1948).
• A. Hubert & R. Schäfer, Magnetic Domains (Springer, 1998), Ch. 3.
• R. Skomski, Simple Models of Magnetism (Oxford, 2008), Ch. 4.