Some of experimental and theoretical studies have been published on LCMO/YBCO heterostructures. However, a limited section of them is mainly discussed here in order to introduce some key concepts that are used later in this work to explain our model.
First of all, P. Fulde, R.A. Ferrell, A.I. Larkin and Y.N. Ovchinnikov presented a theory
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for a superconductor in the presence of a strong, spatially homogeneous magnetic exchange field H [15,16]. Their model is based on the Bardeen – Cooper – Schrieffer (BCS) theory [17], where the electrons are form so-called Cooper pairs. These quasi-particles consist of two electrons which are spin singlets and have equal energy EF (Fermi energy), but posses opposite momentum
kF kF
. Therefore, the total momentum of the Cooper pair vanishes:
kCooper kF kF 0
.The Zeeman splittingEexof the energies corresponding to the spin down and spin up electron states has been exhibited under the presence of a ferromagnetic exchange field, H.
Subsequently, the properties of the Cooper pairs become modified: Although, electrons remained the same energyE , but the momentum of the spin up electron is just F k1/ 2Eex, while the one of the spin down electron is increased by 1/ 2
Eex
k . Since the momentums of electron directions are opposite, the Cooper pairs obtain a finite momentum given by 2 1/ 2
Cooper Eex
k k .
Due to this momentum, the superconducting order parameter becomes spatially modulated on a length scale of 2 transition temperature Tsc on the ferromagnetic layer of thickness dFM[18, 19]. In that case, the characteristic length scale over which the superconducting order parameter decays into that of the ferromagnet is given by
where DFM is the diffusion coefficient in the ferromagnet and Eexthe exchange energy of the ferromagnet (Zeeman splitting of the spin up and spin down conduction band’s energies due to the magnetic exchange field H). Because the exchange energy favours one of the spin orientations, it acts as a pair breaker for the spin singlet of the Cooper pairs and reduces the value ofFM. In a normal metal, where there is no such exchange energy, the corresponding length scale over which the superconducting order parameter decays is provided by
2
where D is the diffusion coefficient in the normal metal. N
In the theoretical study of Z. Radovi´c et al., the superconducting order parameter is also assumed to be depressed on the superconductor side of the interface. For the bulk superconductor, this characteristic length scale is given by
0 temperature. Since the Tsc0will be reduced in a thin layer in proximity to ferromagnetism, Z.
Radovi´c et al. used a corresponding length scale scwhich depends on the reduced transition temperatureTscred: superconductivity occurs: Tscredvanishes if the thickness of the superconductor d is smaller sc than twice the length sc over which the superconducting order parameter changes
dsc 2sc
. In the opposite case, Tscred is finite. If Tscredis finite and the ferromagnetic layer thickness is of the same order as the coherence length of the superconducting order parameter in the ferromagnetic layer
dFM /FM 1
, an oscillatory behaviour of Tscred is expected in superconductor / ferromagnet / superconductor heterostructures: In the limit of dFM FM, the phase of the superconducting order parameter remains the same in the two superconducting layers. In this limit, the exchange energy Eex acting on the superconductor increases with increasingdFM . Therefore, Tscred decreases with increasingdFM . If dFM is about the same asFM, it becomes more favourable for the superconducting order parameter to introduce a π-phase shift from one superconducting layer to the next one. This reduces the pair-breaking effect of the exchange energy Eexon the superconducting order parameter.Therefore, Tscredis enhanced even though the thickness of the ferromagnetic layer has been increased. With further increase in dFM, Tscredwill again be reduced, as the phase of the superconducting order parameter changes over a length scale of FM . This oscillatory behaviour of Tscredas a function of dFM has been confirmed in several experiments on LCMO/YBCO heterostructures based on conventional, non-oxide materials [20, 21, 22].
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Third, I. Baladi´e and A.I. Buzdin had carried out more rigorous calculations considering the thermodynamic properties of ferromagnet / superconductor / ferromagnet nanostructures as a function of thickness of the ferromagnetic layer dFMand the interface transparency [23].
They assumed the superconducting layer thickness dsc to be smaller than the superconducting coherence length scand assumed the dirty limit for all layers in order to use the Usadel’s equations [19] same as that used by Z. Radovi´c et al. in Ref. [18]. For the limit of a high interface transparency, they calculatedTscredas
0 explained the surprisingly high conductance observed in metallic ferromagnets in proximity to a superconductor in the superconducting state [24] with a spin-triplet contribution to the superconducting order parameter. They assumed a small value of the anomalous quasi-classical Green’s function (low interface transparency) in order to linearise the Usadel’s equations. They showed that an inhomogeneity in the magnetisation at an interface can induce such a triplet component of the superconducting order parameter that corresponds to Cooper pairs with parallel electron spins [25,26,27]. The penetration depth of this triplet component into the ferromagnetic layer is actually much larger than the one of the singlet oneFM:
4
The penetration depth of the singlets component of a normal metal and the length FM are of the same order (see Equation 5.2). Following the idea of a triplet component of the superconducting order parameter, they calculated the influence of the conduction electrons on the magnetization of the ferromagnet and on the magnetic moment induced in the superconductor. In Ref. [26] they used a simple mean field approximation model, where they assume the ferromagnetic exchange energy Eex to be smaller than the Fermi energy and a
low interface transparency. They concluded that the magnetization in the ferromagnet can be reduced and that a magnetic moment aligned antiparallel to the one in the ferromagnet can be induced in the superconductor over the length scale of the superconducting coherence length
sc[26,27]. In an extremely simplified picture, one can imagine Cooper pair singlets of which one of the electrons penetrates into the ferromagnetic layer, while the second one is more localized in the superconductor. The electron in the ferromagnet will align its spin along the local magnetic field. Subsequently, the spin of the second electron has to bealigned antiparallel, in order to sustain the singlet state of the Cooper pair (see Figure 5.1). F.S.
Bergeret, A.F. Volkov and K.B. Efetov called this effect the inverse proximity effect because there is a magnetic moment induced in the superconductor, which is antiparallelly aligned to the ferromagnetic moment.
Figure 5.1 Inverse proximity effect: One electron of a Cooper pair which takes a place mainly in the ferromagnet aligns its spin parallel to the ferromagnetic moment. However, the second one of the same Cooper pair aligns its spin antiparallel to conserve the Cooper pair’s singlet state. This figure was taken from [26].
Moreover, Z. Sefrioui and co-workers reported a new proximity effect [28]. They observed by transport (resistance) and by SQUID measurements that superconductivity survives even in 3.5 nm thick YBCO layers that are adjacent to a ferromagnetic layer. These are considerably thinner superconducting layers than achievable with conventional superconductors. They also observed that T changes with varying ferromagnetic layer sc thickness in heterostructures with a ferromagnetic layer thickness of up to 100 nm. Moreover, the former observation can be explained by the short superconducting coherence length in the superconducting YBCO of 0.1 – 0.3 nm (along the c-axis), while the latter remains subject to
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speculations. According to the theory of Z.Radovi´c et al. [18], the interaction between the superconducting and ferromagnetic layers should only be possible if the ferromagnetic layer is thinner than two times the decay length of the superconducting order parameter into the ferromagnet
dFM 2FM
. T should then only be dependent on the thickness of the sc ferromagnetic layers. The length scale observation is special landmark to study superconductivity, since the exchange energy in LCMO is very large (estimate about 3 eV [29]) and therefore FM small. Z. Sefrioui and co–workers proposed that a reduced magnetic moment in the LCMO layers and a high interface transparency are the main reasons for the unusually large value of FM. A different explanation which has not been discussed by Z.Sefrioui and co-workers would be a triplet component of the T as described by F.S.Bergeret, sc A.F.Volkov and K.B.Efetov [25]: The value of FM can be considerably larger than the one of FMbecause the ferromagnetic exchange coupling in the ferromagnetic layers does not give rise to a pair breaking of a triplet.
V.Pe˜na et al. (2005): V.Pe˜na and co-workers measured an unconventional giant magnetoresistance effect in LCMO/YBCO/ LCMO trilayers in the superconducting state [30].
If the magnetic moments were in the layer plane and the temperature was close to T , they sc found a maximum magnetoresistance R R/
RmaxRmin
/Rmin of LCMO up to 1600%, which was decreasing exponentially when getting closer to T . The only precondition for sc this effect was a working temperature belowT . The most important difference compared to a sc conventional CMR effect was, that they measured the highest resistance for an antiparallel alignment of the magnetic moments in the LCMO layers and the lowest one for a parallel alignment. This is opposite to the systems with conventional superconductors that are discussed by I.Baladi´e and A.I.Buzdin in Ref. [31]. The effect observed by V.Pe˜na and co-workers occurred in heterostructures with a YBCO layer thickness of up to 30 nm, which is considerably larger than the fraction of a nanometer of the superconducting coherence length scin YBCO along the c-axis. This is opposite to the assumption made by I.Baladi´e and A.I.Buzdin, where the thickness of the superconducting layer was smaller thansc. It seems therefore, that there is an additional length scale which has to be considered in order to explain the observed physical phenomena. They gave an explanation without focusing on an additional length scale. They argued with the injection of spin-polarised carriers into the YBCO layer: In case of an antiparallel alignment, the injected charge carriers find a highpotential barrier to leave the superconductor at the interface to the second ferromagnetic layer and therefore accumulate in the YBCO layer. The superconducting current density can subsequently be reduced by the accumulated spins. In a parallel case, this spin accumulation does not take place and the resistance through the layers remains very low. An alternative explanation could be the formation of a spin density wave in the YBCO layer that is similar to the one which can be induced by the applying an external magnetic field in underdoped La2−xSrxCuO4 single crystals [32,33,34]. Such a spin density wave may couple the ferromagnetic layers through longer distances than superconductivity.