Association

2.3. Magnetic properties of EuO 13
which fits the experimental data very well. The three-dimensional Heisenberg model provides
a power law for temperatures below the magnetic phase transition: 56
σ ∝ (T C − T ) β , where β =0.36. (2.7)
For EuO an experimental value β =0.37 was found, 55,57 thus EuO is considered as a clear
realization of a 3D Heisenberg ferromagnet. In the Weiss-Heisenberg mean field model, the
most-used temperature dependence of the spontaneous magnetization σ follows a Brillouin
function B J (x). 58 This behavior can be derived from the mean field theory for ideal paramagnets
by substituting the external field B ext by the Weiss molecular field βH Weiss . Then, at
constant H Weiss , the temperature dependence is
(
σ ∝Bσ
3J
J +1
)
T C
. (2.8)
T
A more complex Heisenberg model has been proposed by Mauger and Godart (1986) including
two exchange interactions J 1 and J 2 for the EuO magnetic order, which are the nearest
neighbor and next-nearest neighbor Eu 2+ –Eu 2+ interactions, respectively. 18 The nearest
neighbors interaction is ferromagnetic with an experimental value of J 1 = ∼5.3 × 10 −5 eV. 59
The next-nearest neighbor exchange J 2 is more complex and still under debate, one experimental
value being J 2 ∼ 1.05 × 10 −5 eV. 59 The ordering temperature T C is expressed via the
mean field approach
T C =2S(S +1)·12J 1 +6J 2
3k B
, (2.9)
where S is the total spin of the EuO magnetic shell (4f 7 ). The dominant interaction J 1 acts
between the 4f orbitals of the 12 nearest neighbors via a virtual exchange involving the
unoccupied 5d conduction states for transfer. Between the six next-nearest neighbors, the
J 2 interaction is of mainly super-exchange nature and involves different mechanisms. Until
now, the nature of the J 2 interaction is not clear – is it antiferromagnetic or ferromagnetic in
EuO? Several mechanisms are currently discussed in order to describe the J 1 and J 2 parts of
the magnetic coupling in EuO, and we summarize the most established ones in the following.
Competing exchange interactions in EuO
The microscopic origin of the exchange interaction is still under discussion. Proposed by
Kasuya (1993), 42 several competing exchange mechanisms exist in EuO:
Nearest neighbor exchange A4f electron is excited to the 5d band where the electrons are
itinerant, is then influenced by the exchange interaction of the 4f spins of the nearest
neighbors, and falls back to the initial state. This indirect interaction leads to ferromagnetism.
This idea is first proposed by Mauger and Godart (1986). 18
The Kramers-Anderson superexchange An f electron is transferred via oxygen to an f orbital
of a neighboring atom. This exchange is antiferromagnetic and very small.
Superexchange via the d–f interaction Oxygen electrons are transferred to the d orbitals
of neighboring Eu atoms, where they influence the 4f spins via the d–f exchange. This
leads to an antiferromagnetic exchange. 60–62