Ab initio investigations of magnetic properties of ultrathin transition ...

3.4 Magnetic Anisotropy 45
where α is the fine structure constant, e 2 /�c/, and c is the speed of light. The negative
sign means that the shape anisotropy is pulling the magnetization into the film plane or
along the wire axis. The shape anisotropy is the most important for bulk samples, thick
films, patterned nanostructures and wires. For thin films of few atomic layers and wires and
magnetic structures that are not FM, the assumption that the magnetization can be treated
by continuous magnetic medium is no longer valid. Then the magnetic dipole-dipole energy
has to be evaluated. In transition-metals, the magnetization distribution around the atom
is almost spherical and is thus treated to a good approximation as a collection of discrete
magnetic dipoles, which are regularly arranged on a crystalline lattice. The dipolar energy
Edip per atom experienced by a dipole at site i due to the presence of ferromagnetically
aligned dipoles on all other sites j can then be expressed as
K (i)
dip (θ) =K(i)
dipcos2 (θ) = 2
c2 �
j(j�=i)
mimj
R3 (1 − 3cos
i,j
2 θij) (3.52)
θij is the angle between the direction of the magnetic moment m of the dipoles at sites i or j
given in units of Bohr magneton and the vector Ri,j connecting atoms i and j, it denotes the
relative distance between these dipoles or atoms, respectively. The θ-dependence expresses
the fact that the dipole-dipole interaction contributes to the magnetic anisotropy. In thin
films and wires the anisotropy energy depends on the position of the atom i normal to the
surface or wire axis, and on the film thickness or wire diameter (in difference to Kshape where
all atoms have the same value). For crystalline thin wires and films the sum in equation
(3.52) can be evaluated straight forwardly with fast converging summation techniques
[102, 103]. Draaisma and de Jonge[93] worked out in detail the layer dependent dipolar
anisotropy K (i)
dip . In general, the outer atoms experience values of Kdip that are smaller
than those of the inner layers which finally approach Kshape. The inner atoms reach 95%
of Kshape after about 15 ˚Abelow the surface. The exact details depend on the crystal
structure and surface orientation, e.g. a reduction between 25% and 45% of Kshape was
reported for a (001) oriented fcc or bcc monolayer. The deviation of Kdip from Kshape gives
the dipolar contribution to the MCA K dip
MCA
in equation (3.50), which occurs here due to
the presence of a surface or interface and is sometimes also called the surface contribution of
the dipolar anisotropy. If the MAE is expressed in terms of energy densities ɛ, EMAE = Vɛ.
This K dip
MCA is expressed in terms of an areal density. The dipolar energy contributes also
to the MCA of bulk systems or thick films or wires, if the underlying lattice structure
has a twofold symmetry. For this three-dimensional case more sophisticated summation
techniques such as the Ewald summation method[104] is required to obtain reliable results
for equation (3.52).
The spin-orbit interaction, treated typically by a Pauli-type addition to the Hamiltonian
as:
Hsoc = μB
μB
σ · (E(r) × p) = σ · (▽V (r) × p) (3.53)
2mc 2mc
provides the essential contribution to the MCA. This Pauli approximation is derived from