Association

12 2. Theoretical background
such as superexchange, 37 and is thus of a different origin than the itinerant electron ferromagnetism
of the elemental 3d ferromagnets. In EuO, the strong ferromagnetic coupling is
directly related to unique crystal features, the large ionic radius (see Fig. 2.7), combined with
a high density of magnetic Eu 2+ ions (inset) both ensure a strong long-range ferromagnetic
order. In the following, we briefly discuss the microscopic origin of the magnetic coupling
in EuO, including competing exchange interactions. Finally, we show up tuning possibilities
for T C and the benefits of EuO for an application as a spin filter tunnel barrier.
Weiss’ mean field model
We begin with a simple model for the magnetic order of EuO, in which the molecular mean
field is proportional to the macroscopic magnetization. This mean field model considers a
local magnetic moment interacting with the mean magnetic field of the crystal. In EuO,
the Eu 2+ ions carry a magnetic moment of 7μ B (see Ch. 2.2.1) which is interacting with the
effective magnetic field of the EuO crystal. The reduced magnetization σ can be expressed in
a simple form,
σ ≡ m(T )
m(T =0) ∝ (T C − T ) β , where β = 1 2 . (2.4)
The Weiss molecular field is responsible for the long-range interatomic magnetic order and its
magnitude determines the Curie temperature. 54 However, experimentally β =0.37 instead
of 1/2 was found. 55 For a better description of the magnetic coupling between the localized
Eu 2+ ions, the Heisenberg model is applied.
Heisenberg models for the magnetization of EuO
The Heisenberg model suggests that a spontaneous magnetization arises from the exchange
interactions between spins moments of neighboring atoms. In EuO, the 4f 7 spin magnetic
moment of a Eu 2+ ion interacts with the 4f 7 spin moments of its nearest neighbors. Since
the 4f 7 orbital has maximum spin multiplicity S = 8 and L = 0 (term symbol 8 S7/2) inthe
ferromagnetic ground state, † it is of spherical s-like symmetry. Thereby, the exchange is considered
isotropic, and one effective Hamiltonian is sufficient to describe the nearest neighbor
interaction:
∑
H = −J ex S i ·S j (2.5)
H denotes the Heisenberg Hamiltonian which sums up the spins over nearest neighbors,
where J ex is the positive exchange energy describing a pure ferromagnetic interaction. Although
there is no exact solution of eq. (2.5), the Bloch T 3/2 law was found, 54
σ ≡ m(T )
( ) 3 1 sc structure,
0.0587 kB 2
T
⎧⎪ ⎨
=1− , Q = 2 bcc structure,
m(T =0) S ·Q 2J ex S ⎪ ⎩ 4 fcc structure.
(2.6)
Remarkably, the density of Eu 2+ ions in fcc EuO is 44% larger than in bcc Eu metal, and comparable to the
density of Gd ions in ferromagnetic Gd metal. 22
† The occupation of the 4f 7 orbital is described in Ch. 2.2.1.