About Magnetic Domains
A ferromagnet below its Curie temperature spontaneously breaks into magnetic domains — regions where all atomic spins are aligned — separated by narrow domain walls roughly 10–100 nm wide. The Ising model captures the essential physics: each lattice spin interacts with its neighbours via an exchange coupling J, favouring parallel alignment, while thermal fluctuations (parameterised by kBT) compete against order. At the critical temperature a second-order phase transition occurs, and long-range order disappears. The equilibrium domain pattern minimises total energy, balancing exchange, magnetostatic, and anisotropy terms.
This simulation uses the Metropolis Monte Carlo algorithm on a 2D Ising lattice. You can tune temperature relative to the Curie point, apply an external field to watch domain walls sweep across the grid, and observe how the animated B-H hysteresis loop evolves as the field cycles.
Frequently Asked Questions
Why does a ferromagnet split into domains instead of being uniformly magnetised?
Uniform magnetisation would produce large stray magnetic fields outside the material, storing enormous magnetostatic energy. The material reduces total energy by forming domains with opposite magnetisation directions that largely cancel externally. The trade-off is the energy stored in domain walls themselves (~10⁻³ J m⁻²), so the equilibrium domain size balances wall energy against magnetostatic energy.
What is the Ising model and how does it model ferromagnetism?
The Ising model places a spin variable si = ±1 on each lattice site and assigns energy E = −J Σ sisj − H Σ si, where J is the exchange coupling and H the applied field. For J > 0 (ferromagnetic coupling) and temperatures below the critical Tc, the system spontaneously magnetises. Lars Onsager solved the 2D version exactly in 1944, one of the landmark results of statistical mechanics.
How does the Metropolis algorithm simulate thermal fluctuations?
At each step a random spin is selected and its energy change ΔE calculated if it were flipped. If ΔE < 0 the flip is accepted; if ΔE > 0 it is accepted with probability exp(−ΔE/kBT). This satisfies detailed balance and samples the Boltzmann distribution correctly. Running many such steps allows the lattice to thermally equilibrate, producing realistic domain patterns without solving any differential equations.
What happens at the Curie temperature?
At T = Tc the ferromagnet undergoes a continuous (second-order) phase transition. The spontaneous magnetisation M falls to zero as M ∝ (Tc−T)^β with β ≈ 0.326 in 3D. Fluctuations diverge in both length scale (correlation length ξ → ∞) and time (critical slowing-down), producing fractal-like spin patterns visible in the simulation as T approaches Tc.
What is a domain wall and how thick is it?
A Bloch domain wall is a gradual rotation of spins from one domain orientation to the other, spread over many lattice spacings. The wall width δ = π√(A/K), where A is the exchange stiffness and K the magnetocrystalline anisotropy. In iron δ ≈ 40 nm (roughly 140 atomic layers); in hard magnets like SmCo₅ with large K, walls are only ~4 nm wide, which is why such materials pin walls so effectively.
How are magnetic domains observed experimentally?
Several techniques image domains directly: Bitter pattern decoration (fine magnetic particles collect at domain walls), magneto-optical Kerr effect (MOKE) microscopy (polarisation rotation reveals domain contrast), magnetic force microscopy (MFM) with nanometre resolution, and X-ray magnetic circular dichroism (XMCD) for element-specific imaging. These are essential tools for developing magnetic recording media and spintronic devices.
Why does applying a field make domains move rather than flip uniformly?
Domain-wall motion requires overcoming local energy barriers at defects (pinning sites) — much less energy than coherently rotating all spins simultaneously (the Stoner-Wohlfarth mechanism). So at small applied fields, walls move first; only in very fine particles (single-domain grains below ~50 nm for iron) where no walls can form does coherent rotation govern switching, giving much higher coercivity.
What is exchange coupling between layers in thin films?
In magnetic multilayer stacks (e.g., Fe/Cr/Fe), quantum-mechanical RKKY coupling mediated by conduction electrons oscillates between ferromagnetic and antiferromagnetic as the non-magnetic spacer thickness changes. The antiferromagnetic configuration is the basis of giant magnetoresistance (GMR), which enabled the hard-disk read heads that triggered the data-storage revolution — recognised by the 2007 Nobel Prize in Physics.
What is superparamagnetism?
When a ferromagnetic particle is smaller than a single-domain critical size (~10–20 nm for Fe), thermal energy is sufficient to randomly flip the entire particle's moment between easy axes. The particle behaves like a giant paramagnetic atom — zero remanence at zero field, but a large saturable magnetisation. This is the basis of magnetic nanoparticle contrast agents in MRI and is also the fundamental data-density limit in hard drives (the "superparamagnetic limit").
How do domain patterns change under mechanical stress?
Magnetostriction couples magnetic and mechanical degrees of freedom: domains elongate or contract along the magnetisation direction, and conversely applied stress rotates domain orientations (the Villari effect). In nickel magnetostriction is negative (domains contract along M); in iron it is positive. This coupling enables magnetostrictive actuators in sonar transducers and precision positioning, and it causes the audible hum in transformer cores.
What is skyrmion and how does it relate to domains?
A magnetic skyrmion is a topologically protected swirling spin texture — a vortex-like object where spins point in all directions and wrap around a sphere once. Unlike ordinary domain walls, skyrmions cannot be continuously deformed into a uniform state (topological protection), making them highly stable. Nanometre-scale skyrmions in thin films are candidates for ultra-dense, low-energy magnetic memory bits, where a single skyrmion could represent one data bit.