Read Back Signals in Magnetic Recording - Research Group Fidler

Basics
Here J ij represents the exchange integral. In case of ferromagnetism the exchange integral is
positive and leads to the observed strong coupling and parallel alignment of spins.
Considering only nearest neighbor (n.n.) angular momentums and assuming a constant
exchange integral Jij � J lead to
N n. n.
i j
Eexch =−J ∑∑S ⋅S
. (2.40)
i= 1 j≠i The assumption of a constant exchange integral is justified, because experiments show only
small differences when measuring along different crystallographic directions.
As angular moments are described by a continuous function J = J() r in micromagnetics, the
sum can be approached by a volume integral,
A
E = ( ∇ J ) + ( ∇ J ) + ( ∇J
) dV
2 ∫ . (2.41)
2 2 2
exch
J s V
x y z
Here J s is the spontaneous polarization, and the exchange constant A [J/m] is proportional to
the exchange integral J. The value of A has been found experimentally for a lot of magnetic
materials.
2.5.4 Magnetocrystalline Anisotropy Energy
In general a ferromagnetic material does not show isotropic magnetization behavior.
Experimentally certain directions can be found, which favor magnetization (easy axes) and
others which do not (hard axes). This leads to a new energy term.
ani ani
V
( )
E = ∫ f J dV
(2.42)
Here f ani denotes a direction dependent function. In case of hexagonal systems (e.g. Co) a
uniaxial anisotropy can be assumed. Therefore f ani can be expressed as series expansion
f ( θ ) = K + K sin θ+ K sin θ+ ... . (2.43)
ani
2 4
0 1 2
θ denotes the angle between J and the c-axis of the hexagonal system and 0 K , 1 K , 2
are the so called anisotropy constants. In practice all higher terms are neglected.
K , etc.
In case of cubic systems, their cubic symmetry can be taken into account, if f ani is written as
17