Equal Spin Triplet Pairing Research Articles

Josephson current induced by spin-mixing Cooper pairs

Based on the Bogoliubov-de Gennes equations, we investigate the transport of the Josephson current in a one-dimensional S/F<sub>L</sub>-F-F<sub>R</sub>/S junction, where S and F are superconductor and ferromagnet, and F<sub>L,R</sub> are the left and right interfaces with noncollinear magnetizations. It is found that the F<sub>L</sub> and F<sub>R</sub> interfaces can induce spin-mixing and spin-flip effects, which can transform a part of spin-singlet pairs in the S into equal-spin triplet pairs in the F. For the short S/F<sub>L</sub>-F-F<sub>R</sub>/S junction, the spin-singlet pairs and the equal-spin triplet pairs can survive in the F layer. Therefore, with the increase of the ferromagnetic exchange field and the angle difference of interface magnetization rotation, the critical current oscillates on a base level. If the F is transformed into half-metal, only the equal-spin triple pairs exist in the F layer, and the oscillation characteristic of critical current disappears. In addition, the F<sub>L</sub> and F<sub>R</sub> interfaces can work as conventional potential barriers. As a result, the critical current exhibits double oscillation behaviors with the increase of ferromagnetic thickness, in which the long-wave oscillation arises from the phase change of the spin-singlet pairs in the ferromagnetic layer, and the short-wave oscillation is caused by the resonant tunneling effect when the spin-singlet pairs and the equal-spin triplet pairs pass through two interfacial barriers.

Tunneling spectroscopy of few-monolayerNbSe2in high magnetic fields: Triplet superconductivity and Ising protection

Equal spin triplet pairs (ESTPs) of nuclei form the ground state of superfluid ${}^{3}$He. ESTPs of electrons have long been sought in superconducting systems. In layered transition metal dichalcogenides with 2$H$ structure, such as NbSe${}_{2}$, ESTP correlations have been predicted to arise from the interaction of the Ising spin-orbit field, which pins the internal spin axis of conventional singlet pairs out-of-plane, and an applied in-plane magnetic field. The tunneling spectroscopy study done here of bilayer NbSe${}_{2}$ shows experimental signatures of these ESTPs.

Signature of odd-frequency equal-spin triplet pairing in the Josephson current on the surface of Weyl nodal loop semimetals

We theoretically predict proximity-induced odd-frequency (odd-$\omega$) pairing on the surface of a Weyl nodal loop semimetal, characterized by a nodal loop Fermi surface and drumhead-like surface states (DSSs), attached to conventional spin-singlet $s$-wave superconducting leads. Due to the complete spin polarization of the DSS, odd-$\omega$ equal-spin triplet pairing is present, and we show that it gives rise to a finite Josephson current. Placing an additional ferromagnet in the junction can also generate odd-$\omega$ mixed-spin triplet pairing, but the pairing and current is not affected if the magnetization is orthogonal to the DSS spin-polarization, which further confirms the equal-spin structure of the pairing.

Supercurrent in ferromagnetic Josephson junctions with heavy-metal interlayers. II. Canted magnetization

It has been suggested by theoretical works that equal spin-triplet Cooper pairs can be generated in Josephson junctions containing both a ferromagnet and a source of spin-orbit coupling. Our recent experimental work suggested that spin-triplet Cooper pairs were not generated by a Pt spin-orbit coupling layer when the ferromagnetic weak link had entirely in-plane anisotropy (N. Satchell and N.O. Birge, Phys. Rev. B 97, 214509 (2018)). Here, we revisit the experiment using Pt again as a source for spin-orbit coupling and a [Co(0.4 nm)/Ni(0.4 nm)]$_{\times8}$/Co(0.4 nm) ferromagnetic weak link with both in-plane and out-of-plane magnetization components (canted magnetization). The canted magnetization more closely matches theoretical predictions than our previous experimental work. Our results suggest that there is no supercurrent contribution in our junctions from equal spin-triplets. In addition, this work includes the first systematic study of supercurrent dependence on Cu interlayer thickness, a common additional layer used to buffer the growth of the ferromagnet and which for Co may significantly improve the growth morphology. We report that the supercurrent in the [Co(0.4 nm)/Ni(0.4 nm)]$_{\times8}$/Co(0.4 nm) ferromagnetic weak links can be enhanced by over two orders of magnitude by tuning the Cu interlayer thickness. This result has important application in superconducting spintronics, where large critical currents are desirable for devices.

Creation of equal-spin triplet superconductivity at the Al/EuS interface

In conventional superconductors, electrons of opposite spins are bound into Cooper pairs. However, when the superconductor is in contact with a non-uniformly ordered ferromagnet, an exotic type of superconductivity can appear at the interface, with electrons bound into three possible spin-triplet states. Triplet pairs with equal spin play a vital role in low-dissipation spintronics. Despite the observation of supercurrents through ferromagnets, spectroscopic evidence for the existence of equal-spin triplet pairs is still missing. Here we show a theoretical model that reveals a characteristic gap structure in the quasiparticle density of states which provides a unique signature for the presence of equal-spin triplet pairs. By scanning tunnelling spectroscopy we measure the local density of states to reveal the spin configuration of triplet pairs. We demonstrate that the Al/EuS interface causes strong and tunable spin-mixing by virtue of its spin-dependent transmission.

Spin transport in half-metallic ferromagnet-superconductor junctions

We investigate the charge and spin transport in half-metallic ferromagnet ($F$) and superconductor ($S$) nanojunctions. We utilize a self-consistent microscopic method that can accommodate the broad range of energy scales present, and ensures proximity effects that account for the interactions at the interfaces are accurately determined. Two experimentally relevant half-metallic junction types are considered: The first is a $F_1 F_2 S$ structure, where a half-metallic ferromagnet $F_1$ adjoins a weaker conventional ferromagnet $F_2$. The current is injected through the $F_1$ layer by means of an applied bias voltage. The second configuration involves a $S F_1 F_2 F_3 S$ Josephson junction whereby a phase difference $\Delta\varphi$ between the two superconducting electrodes generates the supercurrent flow. In this case, the central half-metallic $F_2$ layer is surrounded by two weak ferromagnets $F_1$ and $F_3$. By placing a ferromagnet with a weak exchange field adjacent to an $S$ layer, we are able to optimize the conversion process in which opposite-spin triplet pairs are converted into equal-spin triplet pairs that propagate deep into the half-metallic regions in both junction types. For the tunnel junctions, we study the bias-induced local magnetization, spin currents, and spin transfer torques for various orientations of the relative magnetization angle $\theta$ in the $F$ layers. We find that the bias-induced equal-spin triplet pairs are maximized in the half-metal for $\theta\approx90^\circ$ and as part of the conversion process, are anticorrelated with the opposite-spin pairs. We show that the charge current density is maximized, corresponding to the occurrence of a large amplitude of equal-spin triplet pairs, when the exchange interaction of the weak ferromagnet is about $0.1E_F.$

Theory of type-II superconductivity in ferromagnetic metals with triplet pairing

The superconducting state in uranium compounds UGe2, URhGe and UCoGe is formed at temperatures far below the Curie temperature pointing on nonconventional nature of superconductivity in these materials - namely the superconductivity with triplet pairing. The emergence of superconductivity is accompanied by the slight magnetization expulsion typical for the type-II superconductors. Following classic Abrikosov paper I develop the theory of type-II superconductivity in application to two-band ferromagnetic metal with equal spin triplet pairing.

Nonlocal Andreev entanglements and triplet correlations in graphene with spin-orbit coupling

Using a wavefunction Dirac Bogoliubov-de Gennes method, we demonstrate that the tunable Fermi level of a graphene layer in the presence of Rashba spin orbit coupling (RSOC) allows for producing an anomalous nonlocal Andreev reflection and equal spin superconducting triplet pairing. We consider a graphene junction of a ferromagnet-RSOC-superconductor-ferromagnet configuration and study scattering processes, the appearance of spin triplet correlations, and charge conductance in this structure. We show that the anomalous crossed Andreev reflection is linked to the equal spin triplet pairing. Moreover, by calculating current cross-correlations, our results reveal that this phenomenon causes negative charge conductance at weak voltages and can be revealed in a spectroscopy experiment, and may provide a tool for detecting the entanglement of the equal spin superconducting pair correlations in hybrid structures.

Long-range supercurrents induced by the interference effect of opposite-spin triplet state in clean superconductor-ferromagnet structures

By now it is known that in an s-wave superconductor-ferromagnet-superconductor (SFS) structure the supercurrent induced by spin singlet pairs can only transmit a short distance of the order of magnetic coherence length. The long-range supercurrent, taking place on the length scale of the normal metal coherence length, will be maintained by equal-spin triplet pairs, which can be generated by magnetic inhomogeneities in the system. In this paper, we have shown an unusual long-range supercurrent, which can take place in clean junction with non-parallel orientation of magnetic moments. The mechanism behind the enhancement of Josephson current is provided by the interference of the opposite-spin triplet states deriving from and interfaces when both ferromagnetic layers have the same values of the length and exchange field. This discovery can offer a natural explanation for recent experiments (Robinson et al 2010 Phys. Rev. Lett. 104 207001; Baek et al 2014 Nat. Commun. 5 3888).

Peculiar long-range supercurrent in superconductor-ferromagnet-superconductor junction containing a noncollinear magnetic domain in the ferromagnetic region

We study the supercurrent in clean superconductor-ferromagnet-superconductor heterostructure containing a noncollinear magnetic domain in the ferromagnetic region. It is demonstrated that the magnetic domain can lead to a spin-flip scattering process, which reverses the spin orientations of the singlet Cooper pair and simultaneously changes the sign of the corresponding electronic momentum. If the ferromagnetic layers on both sides of magnetic domain have the same features, the long-range proximity effect will take place. That is because the singlet Cooper pair will create an exact phase-cancellation effect and gets an additional π phase shift as it passes through the entire ferromagnetic region. Then, the equal spin triplet pair only exists in the magnetic domain region and can not diffuse into the other two ferromagnetic layers. So, the supercurrent mostly arises from the singlet Cooper pairs, and the equal spin triplet pairs are not involved. This result can provide a approach for generating the long-range supercurrent.

Evidence for spin selectivity of triplet pairs in superconducting spin valves

Spin selectivity in a ferromagnet results from a difference in the density of up- and down-spin electrons at the Fermi energy as a consequence of which the scattering rates depend on the spin orientation of the electrons. This property is utilized in spintronics to control the flow of electrons by ferromagnets in a ferromagnet (F1)/normal metal (N)/ferromagnet (F2) spin valve, where F1 acts as the polarizer and F2 the analyser. The feasibility of superconducting spintronics depends on the spin sensitivity of ferromagnets to the spin of the equal spin-triplet Cooper pairs, which arise in superconductor (S)-ferromagnet (F) heterostructures with magnetic inhomogeneity at the S-F interface. Here we report a critical temperature dependence on magnetic configuration in current-in-plane F-S-F spin valves with a holmium spin mixer at the S-F interface providing evidence of a spin selectivity of the ferromagnets to the spin of the triplet Cooper pairs.

Anomalous spin-triplet current in Ho/Co/Ho-based Josephson junctions

The supercurrent through a holmium/cobalt/holmium-based Josephson junction is investigated by using the extended the Blonder-Tinkham-Klapwijk (BTK) approach. It is found that the spiral magnetism of the Ho layer can induce both spin-split and spin-flip scatterings, which control the conversion between the spin-singlet and equal spin-triplet pairing correlations, generating a tunable long-range spin-triplet pair ordering. Spin-triplet current sharp peaks variations with the Ho layer thickness as well as a slow decay of the current amplitude with the Co layer thickness are revealed. The theoretical results are in good agreement with the relevant experimental observations (Robinson J. W. A. et al., Science, 329 (2010) 59).

Proximity effects and triplet correlations in ferromagnet/ferromagnet/superconductor nanostructures

We report the results of a study of superconducting proximity effects in clean Ferromagnet/Ferromagnet/Superconductor (${\rm F_1F_2S}$) heterostructures, where the pairing state in S is a conventional singlet s-wave. We numerically find the self-consistent solutions of the Bogoliubov-de Gennes (BdG) equations and use these solutions to calculate the relevant physical quantities. By linearizing the BdG equations, we obtain the superconducting transition temperatures $T_c$ as a function of the angle $\alpha$ between the exchange fields in $\rm F_1$ and $\rm F_2$. We find that the results for $T_c(\alpha)$ in ${\rm F_1F_2S}$ systems are clearly different from those in ${\rm F_1 S F_2}$ systems, where $T_c$ monotonically increases with $\alpha$ and is highest for antiparallel magnetizations. Here, $T_c(\alpha)$ is in general a non-monotonic function, and often has a minimum near $\alpha \approx 80^{\circ}$. For certain values of the exchange field and layer thicknesses, the system exhibits reentrant superconductivity with $\alpha$: it transitions from superconducting to normal, and then returns to a superconducting state again with increasing $\alpha$. This phenomenon is substantiated by a calculation of the condensation energy. We compute, in addition to the ordinary singlet pair amplitude, the induced odd triplet pairing amplitudes. The results indicate a connection between equal-spin triplet pairing and the singlet pairing state that characterizes $T_c$. We find also that the induced triplet amplitudes can be very long-ranged in both the S and F sides and characterize their range. We discuss the average density of states for both the magnetic and the S regions, and its relation to the pairing amplitudes and $T_c$. The local magnetization vector, which exhibits reverse proximity effects, is also investigated.

Equal-spin Andreev reflection and long-range coherent transport in high-temperature superconductor/half-metallic ferromagnet junctions

The penetration of a superconducting current from a superconductor into a half-metallic ferromagnet is usually forbidden. Resonances in the conductance spectra of superconductor/half-metal heterostructures suggest this restriction is lifted by the occurrence of unconventional equal-spin Andreev reflection. Conventional superconductivity is incompatible with ferromagnetism, because the magnetic exchange field tends to spin-polarize electrons and breaks apart the opposite-spin singlet Cooper pairs1. Yet, the possibility of a long-range penetration of superconducting correlations into strong ferromagnets has been evinced by experiments that found Josephson coupling between superconducting electrodes separated afar by a ferromagnetic spacer2,3,4,5,6,7. This is considered a proof of the emergence at the superconductor/ferromagnetic (S/F) interfaces of equal-spin triplet pairing, which is immune to the exchange field and can therefore propagate over long distances into the F (ref. 8). This effect bears much fundamental interest and potential for spintronic applications9. However, a spectroscopic signature of the underlying microscopic mechanisms has remained elusive. Here we do show this type of evidence, notably in a S/F system for which the possible appearance of equal-spin triplet pairing is controversial10,11,12: heterostructures that combine a half-metallic F (La0.7Ca0.3MnO3) with a d-wave S (YBa2Cu3O7). We found quasiparticle and electron interference effects in the conductance across the S/F interfaces that directly demonstrate the long-range propagation across La0.7Ca0.3MnO3 of superconducting correlations, and imply the occurrence of unconventional equal-spin Andreev reflection. This allows for an understanding of the unusual proximity behaviour observed in this type of heterostructures12,13.

Coexistence of Ferromagnetism with Spin Triplet Superconductivity

The experimental results for ZrZn2, URhGe, and in some pressure ranges also for UGe2, showed that the ferromagnetic superconductors are weak itinerant ferromagnets. Guided by these results we describe the phenomenon of coexistence between equal spin triplet pairing superconductivity and ferromagnetism using the extended Stoner model, which includes in Hamiltonian the on-site Coulomb interaction, U , and occupation dependent hopping integral. We use the Hartree–Fock approximation and the Green functions technique. In the Hartree–Fock approximation the on-site Coulomb interaction plays the role of the on-site exchange (Hund’s) field. All inter-site

Odd-frequency pairs and Josephson current through a strong ferromagnet

We study Josephson current in superconductor/diffusive ferromagnet/superconductor junctions by using the recursive Green function method. When the exchange potential in a ferromagnet is sufficiently large compared to the pair potential in a superconductor, an ensemble average of Josephson current is much smaller than its mesoscopic fluctuations. The Josephson current vanishes when the exchange potential is extremely large so that a ferromagnet is half-metallic. Spin-flip scattering at junction interfaces drastically changes the characteristic behavior of Josephson current. In addition to spin-singlet Cooper pairs, equal-spin triplet pairs penetrate into a half metal. Such equal-spin pairs have an unusual symmetry property called odd-frequency symmetry and carry the Josephson current through a half metal. The penetration of odd-frequency pairs into a half metal enhances the low energy quasiparticle density of states, which could be detected experimentally by scanning tunneling spectroscopy. We will also show that odd-frequency pairs in a half metal cause a nonmonotonic temperature dependence of the critical Josephson current.

Phase fluctuations and vortex lattice melting in triplet quasi-one-dimensional superconductors at high magnetic fields

Assuming that the order parameter corresponds to an equal spin triplet pairing symmetry state, we calculate the effect of phase fluctuations in quasi-one-dimensional superconductors at high magnetic fields applied along the y ${(b}^{\ensuremath{'}})$ axis. We show that phase fluctuations can destroy the theoretically predicted triplet reentrant superconducting state, and that they are responsible for melting the magnetic-field induced Josephson vortex lattice, above a magnetic-field dependent melting temperature ${T}_{m}.$