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

3.4 Magnetic Anisotropy 43
A- and B-coefficients depend on the local and the global spin and are obtained from the
boundary conditions
e i(k+G)r χσ = � � �
A μG
Lσσα(k)uαlσα(r)+BμG Lσσα(k)˙uα lσα(r) �
YL(ˆr)χ αg
σα. (3.42)
σ α
L
The local can be transformed to the global spin-coordinate frame S g by a rotation R α gl,
given by the Euler angles (α, β). In this case, the Euler angles are equivalent to the polar
angles of the local quantization axis in the global frame, α = ϕ, β = θ. The magnetization
density and the magnetic field, seen from the global frame, m αg (r) and B αg (r), are related
to the same quantities seen from the local frame by
m αg (r) = R αgl m αl (r)
B αg (r) = R αgl B αl (r). (3.43)
where the index α indicates, that this corresponds to quantities inside the muffin-tin of
atom type α. The Pauli spinors transform according to
where
χ αl
↑ =
χ αg = U αgl χ αl , (3.44)
� �
1
, χ
0
αl
↓ =
� �
0
1
(3.45)
is their representation in the local spin frame. The matrices R αgl and U αgl are given with
⎛
cos ϕ cos θ − sin ϕ cos ϕ sin θ
sin ϕ cos θ cos ϕ sin ϕ sin θ
R αgl = ⎝
− sin θ 0 cos θ
U αgl �
−i e
=
ϕ
2 cos( θ)
2
ϕ
−e−i 2 sin( θ
2 )
ϕ
i e 2 sin( θ
3.4 Magnetic Anisotropy
2
) ei ϕ
2 cos( θ
2 )
⎞
⎠ , (3.46)
�
. (3.47)
In an isotropic material, all physical properties are identical for different special directions.
When the physical property depends on the crystallographic directions, like velocity of
sound or elastic properties, then the material is no more isotropic. Magnetic crystals can
be magnetized with minimum energy in a certain direction, called easy axis. On the other
hand, it is hard to magnetize the crystal in another certain direction where maximum
energy is needed, called the hard axis. The energy needed to change the magnetization
direction from the hard to the easy axis is called the magnetic anisotropy energy (MAE).