Scales in Magnetism

Intuition

A piece of iron has $\sim 10^{23}$ electrons. No simulation will ever follow them all. So magnetism is a multiscale subject: each scale keeps only the degrees of freedom that matter at that resolution, and hands the parameters it produces up to the next scale.

Three scales dominate:

• Ab-initio (DFT) — every electron explicit, on the scale of angstroms and femtoseconds.
• Atomistic spin — one classical vector per atom, on the scale of nanometers and picoseconds.
• Micromagnetics — a continuous field $\vec{M}(\vec{r},t)$, on the scale of $10$–$10^4$ nm and nanoseconds to microseconds.

Each scale is the coarse-graining of the previous one. The boundary between them is set by the exchange length ${\ell }_{\text{ex}}=\sqrt{2A/({\mu }_0{M}_s^2)}\sim 5$–$10$ nm: below it, the spin field doesn’t really vary, so a continuum is fine; above it, even atomistic models become wasteful.

Formal hierarchy

Scale	Method	Length	Time	DoF	What it computes
Quantum	DFT	$\sim 0.1$ nm	fs	electrons ${\psi }_i(\vec{r})$	atomic moments, $A$, $K_u$, exchange integrals $J$
Atomic	atomistic spin	$\sim 1$ nm	ps	one ${\vec{S}}_i$ per atom	thermodynamics at $T\to T_C$, ultrafast switching
Continuum	micromagnetics	$5$–$10^4$ nm	ns–µs	$\vec{m}(\vec{r},t)$	domain walls, vortices, devices, hysteresis

Each upper level inherits parameters from the level below: DFT gives the exchange stiffness $A$, anisotropy $K_u$, and atomic moment $\mu$ that the atomistic spin model uses; the atomistic model in turn provides the saturation magnetization $M_s(T)$ and damping ${\alpha }_G$ used by LLG micromagnetic simulations.

Key results

1. Why a continuum is allowed at all

The continuum approximation rests on a single observation: in a ferromagnet, exchange enforces parallel alignment over a length ${\ell }_{\text{ex}}$. On scales much shorter than that, the spin field is essentially uniform — there is nothing for a finer description to resolve. On scales much longer, atomic granularity disappears entirely. The “useful” window where $\vec{m}(\vec{r})$ varies smoothly but appreciably is exactly the regime where micromagnetics works.

This is the same physics that makes the Navier–Stokes equation a useful description of water even though water is made of molecules: the molecular mean free path is the analogue of ${\ell }_{\text{ex}}$.

2. Where each scale wins

• DFT is essential when the atomic moment itself matters — determining whether an interface is ferro- or antiferromagnetic, computing $K_u$ of an interface, predicting tunneling spin polarizations.
• Atomistic spin is the right tool for finite-temperature thermodynamics ($T_C$, critical exponents) and for ultrafast laser-driven demagnetization (sub-picosecond dynamics where the continuum picture is suspect).
• Micromagnetics is the workhorse for devices: hard-disk read heads, MRAM cells, spin-torque oscillators, domain-wall race-track memories. Tools: OOMMF, MuMax3, MagPar, Fidimag.

3. Multiscale handoff

The standard recipe to design a new spintronic stack is:

1. DFT the stack — get $M_s$, $A$, $K_u$, interfacial $P$, $\Delta E_F^{\uparrow }-\Delta E_F^{\downarrow }$.
2. Feed those parameters into a micromagnetic simulation of the device geometry, with the LLG-Slonczewski equation if currents matter.
3. Compare to measured switching curves, FMR spectra, R(H) loops.
4. Iterate by adjusting the stack composition.

The big computational bottleneck is step 2: the magnetostatic field ${\vec{H}}^{\text{ms}}$ is nonlocal and dominates simulation cost. State-of-the-art codes accelerate it with FFT, FMM, or tensor methods on GPUs.

Summary

Magnetism is a three-scale subject:

• DFT (electrons, fs, 0.1 nm) — what an atom’s moment is and how it talks to its neighbors.
• Atomistic spin (one $\vec{S}$ per atom, ps, nm) — finite-$T$ thermodynamics, ultrafast switching.
• Micromagnetics ($\vec{m}(\vec{r},t)$, ns, $10$–$10^4$ nm) — domain walls, vortices, devices.

The boundary that sets the continuum approximation is the exchange length ${\ell }_{\text{ex}}\sim 5$–$10$ nm. The rest of this wiki lives at the third scale.

Connections

• micromagnetic-energy — the continuum energy functional itself
• llg-equation — the dynamics equation solved at the micromagnetic scale
• magnetic-domains — sub-µm structures naturally captured by micromagnetics
• stoner-model — itinerant-electron picture that lives at the DFT scale

References

• A. Aharoni, Introduction to the Theory of Ferromagnetism (Oxford, 2000), Ch. 9.
• R. F. L. Evans et al., Atomistic Spin Model Simulations of Magnetic Nanomaterials, J. Phys. Condens. Matter 26, 103202 (2014).
• A. Vansteenkiste et al., The design and verification of MuMax3, AIP Adv. 4, 107133 (2014).