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

30 3 Magnetism of low dimensional systems
3.1 Stoner Model
Figure 3.1: Graphical solution of (3.10).
The one–particle nature of the Kohn–Sham equation makes it possible to derive a Stoner
like theory for ferromagnetism [51, 52, 53, 54]. Within the spin-density functional theory,
the magnetization density of solids is usually defined to be the difference between the
majority and minority spin densities, |m(r)| = n ↑ −n ↓ . It is small compared to the electron
density, n(r) =n ↑ + n ↓ . Expanding the exchange correlation energy Exc(n(r),m(r)) into
a Taylor series in terms of the parameter ξ = m/n yields
Exc(n, ξ) =Exc(n, 0) + 1
2 E′′ xc(n, 0)ξ 2 + ··· (3.1)
On taking the derivative of the exchange-correlation energy with respect to spin-up and
spin-down densities,
n ↑ = (n + m)/2 � n(1 + ξ)/2
n ↓ = (n−m)/2� n(1 − ξ)/2,
the exchange-correlation potential for the two spin directions becomes
(with (+) for ↑ and (−) for ↓), where
(3.2)
V ↑(↓)
xc (r) =V ±
xc(r) =V 0
xc(r) ∓ ˜ Vxc(r)m(r) (3.3)