Magnetic anisotropy and domain walls

Let the magnetization have approximately fixed magnitude Ms and direction n. A continuum model is useful when n varies over distances much greater than the atomic spacing and well below scales that would obscure a wall. For a uniaxial material with easy axis z, take an anisotropy density $K\sin^2\theta$ with K positive, where theta is the angle to that axis. Both plus and minus z are minima, separated by orientations that cost anisotropy energy.

For a flat one-dimensional wall rotating in a fixed plane, exchange gives a density proportional to the square of the angular gradient. The exchange stiffness A has units joules per metre, so this density has joules per cubic metre. Integrating across the wall leaves energy per wall area. This simplified functional excludes magnetostatic charges, applied fields, chiral interactions, and variations of the magnetization magnitude. Those exclusions are appropriate for deriving a basic uniaxial wall, but not for predicting every thin-film texture.

The Euler–Lagrange equation has a first integral in which exchange and anisotropy densities are equal along the minimum-energy profile. A nonzero integration constant would conflict with the uniform zero-energy domains at both infinities. Choosing the increasing-angle branch gives an angular gradient proportional to sine theta. Separating variables produces an exponential in position, centered at an arbitrary wall position because this uniform model has translation symmetry.

The profile length is the square root of A divided by K. A larger exchange stiffness favors a broader, gentler rotation, while larger anisotropy penalizes the interior and narrows the wall. A widely used conventional Bloch-wall width is pi times that length, so reported widths can differ by pi without disagreeing physically. Substituting the optimum profile into the energy gives four times the square root of the product AK. Increasing anisotropy therefore narrows the wall while increasing its surface cost.

Uniform magnetization can create a stray magnetic field whose energy extends outside the sample. For a fixed magnetization distribution, the demagnetizing field satisfies magnetostatics, and its energy can be written as $\mu_0\int_{\mathbb R^3}|\mathbf H_d|^2dV/2$. Breaking a large specimen into appropriately arranged domains may reduce this energy enough to pay for walls. Exchange alone would prefer a uniform orientation and cannot explain why an otherwise homogeneous ferromagnet splits into many oppositely magnetized regions.

Shape matters because surface magnetic charges depend on the component of magnetization normal to the boundary. For a uniformly magnetized ellipsoid, the SI demagnetizing tensor gives $\mathbf H_d=-\mathsf N\mathbf M$, with its principal factors summing to one. This convention differs from systems that use factors summing to four pi. Domain patterns, pinning, and hysteresis also depend on defects and preparation; a local minimum reached experimentally need not be the globally lowest-energy pattern. A wall-length estimate is therefore not a complete domain-size prediction.