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

24 2 The FLAPW method
Evac is the vacuum energy parameter and V0(z) is the planar averaged part of the vacuum
potential. As in the case of ˙ul in the muffin-tins, the function ˙uG � (k�,z) is calculated
from a Schrödinger-like equation, which can be obtained by deriving (2.28) with respect
to the energy.
�
− 1 ∂
2
2
∂z2 + V0(z) − Evac + 1
2 (G� + k�) 2
�
The resulting basis functions have the form
ϕG�G⊥ (k�,
�
r) =
AG �G⊥ (k�)uG � (k�,z)+BG �G⊥ (k�)˙uG � (k�,z)
˙uG � (k�,z)=uG � (k�,z) (2.29)
�
e i(G �+k �)r � (2.30)
The coefficients AG �G⊥ (k�) and BG �G⊥ (k�) are determined by requiring that the functions
are continuous and differentiable at the vacuum boundary. Instead of the energy parameter
Evac, a whole series of G⊥-dependent energy parameters, E i vac = E G⊥
vac = Evac − 1
2 G2 ⊥ can
be used to increase the variational freedom in the vacuum basis functions [80].
Finally, the basis set used for thin film calculations with the FLAPW method has the form
ϕG�G⊥ (k�,
⎧
e
⎪⎨
r) =
i(G�+k�)r� iG⊥z
e
�
AG�G⊥ Int.
(k�)uG (k�,z)
� �
(2.31)
+BG �G⊥ (k�)˙uG � (k�,z)
⎪⎩ [ �
L
e i(G �+k �)r � Vac.
A μG
μG
L (k)ul(r)+BL (k)˙ul(r)]YL(ˆr) MT μ.
2.4 The Kohn-Sham-Dirac Equation
Close to the nucleus, where the kinetic energy is large, the relativistic effects are significant.
This means the electrons inside the muffin-tins should be treated relativistically, where it is
reasonable to treat the interstitial region and the vacuum non-relativistically. This makes
the Kohn-Sham equation to be as single particle Dirac equation
� � 2 eff
cα · p +(β − 1)mc + V (r) Ψ = EΨ (2.32)
˜α =
�� 0 σx
σx 0
� �
0 σy
,
σy 0
� �
0 σz
,
σz 0
� �
I2 0
β =
0 −I2 ��tr
=
�
0
�
˜σ
˜σ 0
(2.33)
(2.34)