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

46 3 Magnetism of low dimensional systems
the Dirac equation, and is normally sufficient for treating relativistic effects in transitionmetal
magnets. For a radially symmetric potential, equation (3.53) can be rewritten as:
Hsoc = μB
1 dV (r)
μB 1 dV (r)
σ · (r × p) = (σ · L) =
2mc r dr 2mc r dr
μB
ξ(r)σ · L (3.54)
2mc
where � L is the angular momentum operator. Since the radial derivative of the potential in
a crystal will be largest in the vicinity of a nucleus, then the major contribution to the spinorbit
interaction will come from this region. For small r the potential will be Coulomb-like
(V = − Z ), therefore the radial expectation value of ξ(�r) leads to a material dependent
r
spin-orbit coupling constant ξ, which is proportional to Z2 . The MCA dominates in low
dimensions over the shape anisotropy and the anisotropy depends crucially on the symmetry
of the system. In a solid, the symmetry is determined by the crystal field, then the
spin-orbit coupling will introduce orbital moments and magnetocrystalline anisotropy by
coupling the states which do not carry orbital moments such that the combination form
an orbital moment, eg. a dxy + dx2−y2 → m = ±z.
In second-order perturbation theory, the expectation value of the orbital moment operator
μBL can be written as:
ml = μB
� � �
�L = μB
i,j
〈ψi|L|ψj〉〈ψj|Hsoc|ψi〉
f(ɛi)[1 − f(ɛj)] (3.55)
ɛi − ɛj
where f is the Fermi function insuring that the wavefunction ψi is occupied and ψj is
unoccupied. In metals, several bands crossing the Fermi level EF . The sum of all contributions
from bands near EF determine the orbital moment. Van der Laan[105] showed that
when the majority and minority bands are well separated by the exchange interaction, the
spin-orbit coupling changes the total energy of the system in second-order perturbation
theory as:
δE = �
i,j
〈ψi|Hsoc|ψj〉〈ψj|Hsoc|ψi〉
ɛi − ɛj
f(ɛi)[1 − f(ɛj)] ≈− ξ
4πμB
�ms · [m ↓
l
− m↑
l ] (3.56)
where �ms is the direction of the spin moment, and m ↓
l (m↓ l ) is the orbital moment vector
for spin-down (up) bands. If the spin-up band is completely filled, then the energy change
δE is proportional to the size of the orbital moment and the magnetocrystalline anisotropy
energy. This means that the difference of δE for two different magnetization directions
will be proportional to the difference in the orbital moments. This relation between orbital
moment anisotropy and MCA was first derived by Bruno[55].
The spin moment increases in low-dimensional systems because of reduced coordination
number (see subsec. 3.1.1). This increase will enable the formation of large orbital
moments, as can be seen from most atoms. In figure 3.5 some representative values of