Explaining micromagnetics in Chinese

We need to explain a physics concept in Chinese regarding micromagnetics, focusing on exchange length and domain size. The relationship involves exchange interaction vs anisotropy, scale of magnetization variation, and domain wall width. Domain size is influenced by magnetostatic energy, and equilibrium values are described by formulas like the Kittel law. Both exchange and magnetostatic energies contribute to determining the domain structure.

Explaining domain size and exchange length

The exchange length determines local gradients and domain wall width, not the domain size. Domains span multiple exchange lengths because the cost of creating numerous walls outweighs stray field reduction until a certain scale is reached. Domain size results from a balance of wall energy and magnetostatic forces.

Clarifying domain wall and period size

The width of the domain wall, δ, is proportional to the square root of the anisotropy to exchange energy ratio. The domain period connects to the wall width via a constant. For stripe ferromagnets, the domain period follows Kittel's law. There's a relationship between wall energy and magnetic properties affecting width.

Explaining domain width in thin films

For thin films with up/down stripes, the domain width $w$ is proportional to the square root of the product of wall energy and film thickness, divided by magnetization squared. With specific material properties (like the exchange constant $A$ and anisotropy constant $K$), domain size increases significantly compared to exchange length.

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Explaining anisotropies and domain walls

In zero anisotropy, exchange length l_K diverges, but other lengths, like l_d, remain finite. Even without anisotropy, uniform magnetization still leads to domain formation. Domain walls become diffuse, with energy density approaching zero, but their width grows infinitely. However, demagnetizing energy still influences domain structure at typical lengths of 5-10 nm, balancing exchange and dipolar effects.

Calculating exchange length

With a typical Ms around 1e6 A/m, the exchange length l_d computes to about 4 nm. This sets the vortex/core size and Néel caps, but not domain size, which depends on sample geometry and competing energy considerations. As anisotropy K approaches zero, wall energy diminishes, but exchange effects still play a role.

Explaining the ground state with K=0

When K=0, exchange alone can't form well-defined walls; instead, the magnetization spreads, reducing energy density. The result is a vortex or flux closure, minimizing magnetostatic energy. Domain sizes depend on sample geometry — thinner films create in-plane magnetization due to shape anisotropy, influencing domain structures, not just exchange length.

Considering domain wall behavior with weak K

When crystalline anisotropy K approaches zero, the exchange length l_K grows large, affecting domain wall width. As K weakens, domain walls become broader, leading to an increase in domain sizes. However, if K is very small, wall definition blurs, and domain sizes may increase due to weaker wall energy penalties.

Clarifying domain wall width formulas

There’s confusion about the dependence of domain wall width w on anisotropy K. Using Kittel's law, w should increase with K, meaning w ∝ K^{1/4}. However, there's a contradiction with another formula, where w seems to decrease with K. The δ used in Kittel's law should not be the Bloch wall width, but an effective exchange length.

Clarifying domain behavior with weak anisotropy

When anisotropy is weak or absent, domain sizes do not shrink to the exchange length. In fact, the ground state may exhibit magnetization curling or closure domains, where the domain size can be influenced by sample geometry and other factors, not just the exchange length. Domains are often single-domain in-plane when shape anisotropy dominates.

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