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