Superconducting crossed correlations in ferromagnets: implications for thermodynamics and quantum transport

It is demonstrated that non local Cooper pairs can propagate in ferromagnetic electrodes having an opposite spin orientation. In the presence of such crossed correlations, the superconducting gap is found to depend explicitly on the relative orientation of the ferromagnetic electrodes. Non local Cooper pairs can in principle be probed with dc-transport. With two ferromagnetic electrodes, we propose a ``quantum switch'' that can be used to detect correlated pairs of electrons. With three or more ferromagnetic electrodes, the Cooper pair-like state is a linear superposition of Cooper pairs which could be detected in dc-transport. The effect also induces an enhancement of the ferromagnetic proximity effect on the basis of crossed superconducting correlations propagating along domain walls.Comment: 4 pages, RevTe

Local density of states in superconductor-strong ferromagnet structures

We study the dependence of the local density of states (LDOS) on coordinates for a superconductor-ferromagnet (S/F) bilayer and a S/F/S structure assuming that the exchange energy h in the ferromagnet is sufficiently large: $>>1,$ where $\tau$ is the elastic relaxation time. This limit cannot be described by the Usadel equation and we solve the more general Eilenberger equation. We demonstrate that, in the main approximation in the parameter $(h\tau)^{-1}$, the proximity effect does not lead to a modification of the LDOS in the S/F system and a non-trivial dependence on coordinates shows up in next orders in $(h\tau) ^{-1}.$ In the S/F/S sandwich the correction to the LDOS is nonzero in the main approximation and depends on the phase difference between the superconductors. We also calculate the superconducting critical temperature $T_{c}$ for the bilayered system and show that it does not depend on the exchange energy of the ferromagnet in the limit of large h and a thick F layer.Comment: 9 pages, 5 figure

Manifestation of triplet superconductivity in superconductor-ferromagnet structures

We study proximity effects in a multilayered superconductor/ferromagnet (S/F) structure with arbitrary relative directions of the magnetization ${\bf M}$. If the magnetizations of different layers are collinear the superconducting condensate function induced in the F layers has only a singlet component and a triplet one with a zero projection of the total magnetic moment of the Cooper pairs on the ${\bf M}$ direction. In this case the condensate penetrates the F layers over a short length $\xi_J$ determined by the exchange energy $J$. If the magnetizations ${\bf M}$ are not collinear the triplet component has, in addition to the zero projection, the projections $\pm1$. The latter component is even in the momentum, odd in the Matsubara frequency and penetrates the F layers over a long distance that increases with decreasing temperature and does not depend on $J$ (spin-orbit interaction limits this length). If the thickness of the F layers is much larger than $\xi_J$, the Josephson coupling between neighboring S layers is provided only by the triplet component, so that a new type of superconductivity arises in the transverse direction of the structure. The Josephson critical current is positive (negative) for the case of a positive (negative) chirality of the vector ${\bf M}$. We demonstrate that this type of the triplet condensate can be detected also by measuring the density of states in F/S/F structures.Comment: 14 pages; 9 figures. Final version, to be published in Phys. Rev.

Josephson current in superconductor-ferromagnet structures with a nonhomogeneous magnetization

We calculate the dc Josephson current $I_J$ for two types of superconductor-ferromagnet (S/F) Josephson junctions. The junction of the first type is a S/F/S junction. On the basis of the Eilenberger equation, the Josephson current is calculated for an arbitrary impurity concentration. If $h\tau\ll1$ the expression for the Josephson critical current $I_c$ is reduced to that which can be obtained from the Usadel equation ($h$ is the exchange energy, $\tau$ is the momentum relaxation time). In the opposite limit $h\tau\gg1$ the superconducting condensate oscillates with period $$ and penetrates into the F region over distances of the order of the mean free path $l$. For this kind of junctions we also calculate $I_J$ in the case when the F layer presents a nonhomogeneous (spiral) magnetic structure with the period $2\pi /Q$. It is shown that for not too low temperatures, the $\pi$-state which occurs in the case of a homogeneous magnetization (Q=0) may disappear even at small values of $Q$. In this nonhomogeneous case, the superconducting condensate has a nonzero triplet component and can penetrate into the F layer over a long distance of the order of $\xi_{T}=$. The junction of the second type consists of two S/F bilayers separated by a thin insulating film. It is shown that the critical Josephson current $I_{c}$ depends on the relative orientation of the effective exchange field $h$ of the bilayers. In the case of an antiparallel orientation, $I_{c}$ increases with increasing $h$. We establish also that in the F film deposited on a superconductor, the Meissner current created by the internal magnetic field may be both diamagnetic or paramagnetic.Comment: 13 pages, 11 figures. To be published in Phys. Rev.

Proximity effects and characteristic lengths in ferromagnet-superconductor structures

We present an extensive theoretical investigation of the proximity effects that occur in Ferromagnet/Superconductor ($F/S$) systems. We use a numerical method to solve self consistently the Bogoliubov-de Gennes equations in the continuum. We obtain the pair amplitude and the local density of states (DOS), and use these results to extract the relevant lengths characterizing the leakage of superconductivity into the magnet and to study spin splitting into the superconductor. These phenomena are investigated as a function of parameters such as temperature, magnet polarization, interfacial scattering, sample size and Fermi wavevector mismatch, all of which turn out to have important influence on the results. These comprehensive results should help characterize and analyze future data and are shown to be in agreement with existing experiments.Comment: 24 pages, including 26 figure

Josephson effect in double-barrier superconductor-ferromagnet junctions

We study the Josephson effect in ballistic double-barrier SIFIS planar junctions, consisting of bulk superconductors (S), a clean metallic ferromagnet (F), and insulating interfaces (I). We solve the scattering problem based on the Bogoliubov--de Gennes equations and derive a general expression for the dc Josephson current, valid for arbitrary interfacial transparency and Fermi wave vectors mismatch (FWVM). We consider the coherent regime in which quasiparticle transmission resonances contribute significantly to the Andreev process. The Josephson current is calculated for various parameters of the junction, and the influence of both interfacial transparency and FWVM is analyzed. For thin layers of strong ferromagnet and finite interfacial transparency, we find that coherent (geometrical) oscillations of the maximum Josephson current are superimposed on the oscillations related to the crossover between 0 and $\pi$ states. For the same case we find that the temperature-induced $0-\pi$ transition occurs if the junction is very close to the crossovers at zero temperature.Comment: 13 pages, 6 figure

Symmetry and Topology in Superconductors - Odd-frequency pairing and edge states -

Superconductivity is a phenomenon where the macroscopic quantum coherence appears due to the pairing of electrons. This offers a fascinating arena to study the physics of broken gauge symmetry. However, the important symmetries in superconductors are not only the gauge invariance. Especially, the symmetry properties of the pairing, i.e., the parity and spin-singlet/spin-triplet, determine the physical properties of the superconducting state. Recently it has been recognized that there is the important third symmetry of the pair amplitude, i.e., even or odd parity with respect to the frequency. The conventional uniform superconducting states correspond to the even-frequency pairing, but the recent finding is that the odd-frequency pair amplitude arises in the spatially non-uniform situation quite ubiquitously. Especially, this is the case in the Andreev bound state (ABS) appearing at the surface/interface of the sample. The other important recent development is on the nontrivial topological aspects of superconductors. As the band insulators are classified by topological indices into (i) conventional insulator, (ii) quantum Hall insulator, and (iii) topological insulator, also are the gapped superconductors. The influence of the nontrivial topology of the bulk states appears as the edge or surface of the sample. In the superconductors, this leads to the formation of zero energy ABS (ZEABS). Therefore, the ABSs of the superconductors are the place where the symmetry and topology meet each other which offer the stage of rich physics. In this review, we discuss the physics of ABS from the viewpoint of the odd-frequency pairing, the topological bulk-edge correspondence, and the interplay of these two issues. It is described how the symmetry of the pairing and topological indices determines the absence/presence of the ZEABS, its energy dispersion, and properties as the Majorana fermions.Comment: 91 pages, 38 figures, Review article, references adde

Thermodynamic properties of ferromagnetic/superconductor/ferromagnetic nanostructures

The theoretical description of the thermodynamic properties of ferromagnetic/superconductor/ferromagnetic (F/S/F) systems of nanoscopic scale is proposed. Their superconducting characteristics strongly depend on the mutual orientation of the ferromagnetic layers. In addition, depending on the transparency of S/F interfaces, the superconducting critical temperature can exhibit four different types of dependences on the thickness of the F-layer. The obtained results permit to give some practical recommendations for the spin-valve effect experimental observation. In this spin-valve sandwich, we also expect a spontaneous transition from parallel to anti-parallel ferromagnetic moment orientation, due to the gain in the superconducting condensation energy

Vortex trapping by tilted columnar defects

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Local quasiparticle density of states in ferromagnet/superconductor nanostructures

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