Micromagnetism

Welcome

This is a digital garden on micromagnetism — the continuum theory that describes how the magnetization $\vec{M}(\vec{r},t)$ of a ferromagnet evolves in space and time. It sits between the atomistic world of individual spins and the macroscopic world of bulk magnetic materials, and it is the language used to model everything from magnetic domain walls and skyrmions to the switching dynamics inside the MRAM cell in your computer.

Pages here are written for two readers at once: a researcher who wants rigorous math, and a newcomer who wants to feel the physics first. Each note opens with intuition, then formal definitions, then key results — so you can read as shallow or as deep as you need.

Why micromagnetism?

A piece of ferromagnetic iron contains $\sim 10^{23}$ spins. Treating each one quantum-mechanically is hopeless. But on scales larger than a few nanometers, the magnetization varies smoothly, and we can replace the discrete lattice of spins with a continuous unit vector field

Micromagnetism is the variational theory built on this field — balancing exchange, anisotropy, magnetostatic, and Zeeman energies — and its time evolution is governed by the Landau–Lifshitz–Gilbert equation.

Table of contents

Foundations

• magnetic-atom — What makes an atom magnetic in the first place
• magnetic-moment — The atomic-scale seed of magnetism
• magnetization — The vector field $\vec{M}(\vec{r})$ at the heart of it all
• magnetic-materials — Diamagnetic, paramagnetic, ferromagnetic
• modeling-scales — DFT, atomistic spin, and the micromagnetic continuum
• micromagnetic-energy — Exchange, magnetostatic, Zeeman, anisotropy
• magnetocrystalline-anisotropy — Spin-orbit coupling and easy axes
• magnetic-domains — Why ferromagnets break into domains

Ferromagnetism

• stoner-model — Itinerant ferromagnetism and the Stoner criterion
• stoner-wohlfarth — The minimal single-domain hysteretic nano-magnet
• hysteresis — The $M(H)$ loop, $M_s$, $M_r$, $H_c$, soft vs hard

Magnetization dynamics

• llg-equation — Prprocecession, damping, and spin-transfer torque
• larmor-precession — How $\vec{m}$ orbits the effective field
• effective-field — The driver of dynamics: ${\vec{H}}^{\text{eff}}$

Vortex dynamics

• thiele-equation — Collective-coordinate equation for the vortex core

Spintronics

• spin-valve — Magnetoresistive FM/NM/FM stack; origin of GMR
• spin-polarized-current — Currents that carry net spin angular momentum
• spin-transfer-torque — How a spin current torques a magnet (LLGS, STT-MRAM, STNO)
• magnetic-tunnel-junction — The MTJ, Jullière TMR, and MRAM

Beyond ferromagnetism

• superconductivity — Meissner, BCS, Type I vs II, Abrikosov vortices, applications

How to navigate

Every page cross-links to its neighbors via [[wikilinks]]. There is no single “right” reading order — follow the links as a trail. If you are new, start here, then read llg-equation, and branch out from there. The graph view in the right sidebar shows the wiki’s structure at a glance.