Physics of Magnetism

SECTION 12.3. DOMAINS AND DOMAIN WALLS 111

if

written as

For two atoms with an angle between their spin moments, the variable part of the

exchange energy is and would be equal to

the magnetization would change abruptly over radians. Let us consider the case that the

directional change would be realized more gradually, and would involve N equal steps

which is a much smaller value. This result

the presence of the magnetocrystalline anisotropy energy, favoring collinear spin moments,
being oriented in one of the two opposing easy directions. The actual width of the wall is
determined by a competition between both energies.

with equal angles between neighboring spins. The total energy would then be only

shows that the exchange energy favors a large wall width. However, the width is limited by

A crude estimate of the energy and width of a domain wall can easily be obtained if
we neglect the demagnetizing energy. Consider a 180° wall of width W in a simple cubic
material extending along a given [100] direction in which the moment direction gradually
changes from the positive to the negative [001] direction. Let us further assume that any
deviation from the [001] direction involves an anisotropy energy given by
Within the wall, the moment directions largely deviate from the easy direction and the total
anisotropy energy involved is roughly proportional to the wall width. If a is the lattice
constant and if the wall extends over N lattice spacings, one obtains a rough estimate of the
where W = Na

ic lattice, the number of rows per unit area of wall is The exchange energy

total anisotropy energy as is the width of the wall.
The increase in exchange energy for one row of atoms in the wall is For
a simple cub
per unit area of wall is therefore The total energy per
unit area associated with the wall is then

A minimum with respect to W is obtained when

This leads to the following expression for the wall width W:

where A is the average exchange energy. Substitution of Eq. (12.3.6) into (12.3.4) leads to

the following expression for the wall energy per unit area of a wall with width W: