Exploring the microscopic origin of magnetic anisotropies with X-ray ...

J. Sto( hr / Journal of Magnetism and Magnetic Materials 200 (1999) 470}497 495
The spin}orbit (s}o) interaction within the d
shell
H "L ) S"(¸ S #¸ S #¸ S ), (C.1)
has the e!ect of mixing di!erent d orbitals and the
spin-up and spin-down states. If we choose the
spin quantization axis z along the magnetization
direction then the components (S , S , S ) of
the spin S in the crystal frame can be expressed in
terms of the components (S , S , S ) in the
rotated spin frame (x , y , z ) as discussed in Appendix
B. This gives the following expressions for
H , zx, y or z
H x: H "(¸ S #¸ S !¸ S ), (C.2)
H y: H "(!¸ S #¸ S !¸ S ), (C.3)
H z: H "(¸ S #¸ S #¸ S ). (C.4)
The angle-dependent orbital moment
m "!¸ / is calculated by use of the second-order
perturbation theory expression [11]
¸ "
2
(k)¸ (k)
S
"¸!¸, (C.5)
where the sum extends over "lled states n and
empty states m within the spin-up and spin-down
manifolds (index j) and (k) denotes a zeroth-order
band state associated with spin function , where
S "$1/2. Matrix elements d ¸ d
are given by Ballhausen (see p. 70 in Ref. [96]).
Note that the coupling between "lled pairs of states
or empty pairs of states does not need to be considered
since the spin}orbit induced terms cancel
each other for any pair. Also, to "rst-order m does
not depend on the mixing of spin-up and spindown
states by the spin}orbit interaction, since the
relevant matrix elements d ¸ d "0.
Thus there are no spin-#ip contributions to
the orbital moment. According to Eq. (C.5)
the orbital momentum is the sum of contributions
from all "lled states in the spin-up and spin-down
subbands. If a subband is "lled its contribution
vanishes.
The angle-dependent spin}orbit energy is given
by the second-order expression
H "
4
(k)¸ (k)
k
#
k
(k)H
(k)
"E #E ,
(C.6)
where the terms E and E represent the contribu-
tions from states of the same and opposite spin,
respectively, and the sums extend over "lled states
(n, j) and empty states (m, j) and (l, j).
It is seen that for E "0 we obtain
H "E "
4 (¸ #¸), (C.7)
showing the direct correlation between the orbital
moments of the spin-up and spin-down manifolds
and the spin}orbit energy. Note that the contributions
of the spin-down states, ¸, enter with
opposite signs in the expressions for the orbital
moment (Eq. (C.5)) and the spin}orbit energy (Eq.
(C.7)). In general, we therefore obtain a direct proportionality
between the orbital moment and the
spin}orbit energy only if ¸"0, i.e. if the spin-
down band is full. In the limit of a vanishing exchange
splitting the orbital moment vanishes
(¸"¸), and so does the spin}orbit energy
(E "!E ).
With the sign convention of Eq. (11) and footnote
4 the magnetocrystalline anisotropy energy is given
by
E "H !H "H !H, . (C.8)
References
[1] A.J. Freeman, R. Wu, J. Magn. Magn. Mater. 100 (1991)
497.
[2] O. Eriksson et al., Solid State Commun. 78 (1991) 801.
[3] O. Eriksson et al., Phys. Rev. B 45 (1992) 2868.
[4] Y. Wu, J. StoK hr, B.D. Hermsmeier, M.G. Samant, D. Weller,
Phys. Rev. Lett. 69 (1992) 2307.
[5] R. Wu, D. Wang, A.J. Freeman, Phys. Rev. Lett. 71 (1993)
3581.
[6] R. Wu, A.J. Freeman, Phys. Rev. Lett. 73 (1994) 1994.
[7] D. Weller, Y. Wu, J. StoK hr, M.G. Samant, B.D. Hermsmeier,
C. Chappert, Phys. Rev. B 49 (1994) 12888.