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

48 3 Magnetism of low dimensional systems
Figure 3.6: d-level splittings shown in the left figure at a given k-point due to a crystal
field in a square monolayer results in a density of states, shown on the right[86].
moments. In the literature several methods have been discussed how this deficiency can be
overcome[106, 94, 107]. For example, the orbital moments of the bulk magnets are about
twice the value if Brooks’ orbital polarization is applied[95, 108]. The effect of OP is much
more drastic in low dimensions[109].
Stör introduced a simple model to to interpret ab initio results on thin films using unsupported
(001) oriented d-metal monolayer[110]. Assume that the d-band is substantially
exchange split and more than half filled, so that only the (partially filled) minority band
has to be considered. The d-orbitals at each atom site experience in the a crystal field V
in the monolayer plane. This leads to a splitting of these levels. If the surface normal is
assumed to be in z-direction, the dxy and d x 2 −y 2 levels will experience a stronger field than
the out-of-plane directed dzx, dyz and d z 2 orbitals. The crystal field leads to a splitting
of 2V� for the in-plane oriented orbitals and 2V⊥ for the out-of-plane oriented ones. In
band-picture, these splittings can be translated into band widths W , which will then be
twice as large (cf. Figure 3.6). Normally, V� will be larger than V⊥, so that R = V⊥/V� < 1.
However, if the monolayer is sandwiched between two slabs of nonmagnetic material the
situation could be changed. If the minority band is half filled -like in the case of Co-; the
dxy and d x 2 −y 2 states will split symmetrically by ±V� around the Fermi level, the (dzx,dyz)
and d z 2 states by ±V⊥. In a band-picture, these splittings will of course depend on the
considered � K� point. Now we can use perturbation theory equation (3.55) to calculate the
orbital moments
m �
� �
ξμB 3 2
l = +
2V� R R +1
and m ⊥ l =4 ξμB
2V⊥
from this equation, only the in-plane orbital moments, m �
l
out-of-plane oriented states, where the out-of-plane moment, m⊥ l
in-plane crystal field V�. This is clear since m ⊥ l
(3.57)
, depends on the splitting of the
, is only quenched by the
corresponds to an in-plane motion of the
electron, i.e. hopping between the dxy and d x 2 −y 2 states that are separated by V�. Using
equation 3.56, the magnetocrystalline anisotropy energy becomes:
KMCA = E �
MCA − E⊥ MCA = ξ
4μB
(m �
l − m⊥l )=− ξ2
� �
3 2
+ − 4
8V� R R +1
(3.58)