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

3.3 Non-Collinear Magnetism 41
⇒ e ik·Rn
�
i(k−q/2)r e α(k, r + Rn)
ei(k+q/2)r �
= e
β(k, r + Rn)
ik·Rn
�
i(k−q/2)r e α(k, r)
ei(k+q/2)r �
β(k, r)
⇒ α(k, r + Rn) =α(k, r), β(k, r + Rn) =β(k, r). (3.39)
The fact that α and β are periodic functions is very important for the implementation
of the generalized Bloch theorem into the FLAPW[38] and many other (plane-wave based)
methods.
3.3.4 Non-Collinear Magnetism in FLAPW
The first implementation of non-collinear magnetism in the ab-initio calculations [98, 90,
89, 99, 91, 34, 100], allowed only one direction of magnetization per atom, i.e. the direction
of the magnetization density ˆm is not allowed to change within one sphere 1 , but varies
only from sphere to sphere (so-called the atomic sphere approximation for the direction of
magnetization). This agrees with the intuitive picture that an atom carries a magnetic moment
of a certain size and only the direction of these moments differs between the atoms.
Such methods describe only the inter-atomic non-collinearity. However, in general the direction
of the magnetization changes continuously from site to site, though, in many cases,
the deviations from the main atomic direction are only significant in a region between the
atom, where the magnitude of the magnetization is rather small. The first calculation that
treated the magnetization as a continuous vector quantity was published by Nordström et
al. [101]. They followed the most general approach allowing the magnetization to change
magnitude and direction continuously, i.e. even within an atom. Thus, their implementation,
that is based on the FLAPW method, allows them to also investigate the intra-atomic
non-collinearity, which is important for actinides like Pu.
Our method uses a “hybrid” approach (Fig.3.4) where the magnetization is treated as
a continuous vector field in the interstitial and in the vacuum regions, while inside each
muffin-tin sphere we only allow for one direction of magnetization.
Inside the muffin-tins, like in the collinear case, it is still possible to work with V↑ and
V↓ in the non-collinear case, since we restrict the magnetization to the local quantization
quantization axis. Therefore, a local spin-space coordinate-frame is introduced with the
z-axis parallel to the local quantization axis. V↑ and V↓ are now spin-up and -down with
respect to the local axis. Since both, the potential and the basis functions, are set up
in terms of the local spin-coordinate frame, the determination of the basis functions and
calculation of the integrals of these functions with the Hamiltonian inside the muffintins
is completely unchanged. The changes come in, when the basis functions inside the
muffin-tins are matched to the plane waves in the interstitial region, because the local
spin-coordinate frame S α is rotated with respect to the global frame S g .
The FLAPW method uses augmented plane waves as basis functions. Therefore, each
basis function for a given k-vector, � k, can be uniquely identified by is wave vector G and
the spin direction. The basis functions in the interstitial region are:
1 Within the muffin-tin spheres, however, magnetization can vary in magnitude