111
SECTION 12.3. DOMAINS AND DOMAIN WALLS
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:
SECTION 12.3. DOMAINS AND DOMAIN WALLS
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: