Odd-frequency Triplet Research Articles

Spin-singlet to triplet Cooper pair converter interface

Combining magnetic and superconducting functionalities enables lower energy spin transfer and magnetic switching in quantum computing and information storage, owing to the dissipationless nature of quasi-particle mediated supercurrents. Here, we put forward a system where emergent spin-ordering and diffusion of Cooper pairs are achieved at a non-intrinsically magnetic nor superconducting metallo-molecular interface. Electron transport, magnetometry and low-energy muon spin rotation are used to probe time-reversal symmetry breaking in these structures. By comparing the Meissner expulsion in a system including a Cu/C60 spin-converter interface to one without, we observe a paramagnetic contribution that can be explained due to the conversion of spin-singlet Cooper pair states into odd-frequency triplet states. These results demonstrate the potential of metallo-molecular interfaces to achieve singlet to triplet Cooper pair conversion, a capability not present in either metal or molecule separately that could be used in the generation and controlled diffusion of spin polarised dissipationless currents.

Dynamical torque from Shiba states ins-wave superconductors

Magnetic impurities inserted in a $s$-wave superconductor give rise to spin-polarized in-gap states called Shiba states. We study the back-action of these induced states on the dynamics of the classical moments. We show that the Shiba state pertains to both reactive and dissipative torques acting on the precessing classical spin that can be detected through ferromagnetic resonance measurements. Moreover, we highlight the influence of the bulk states as well as the effect of the finite linewidth of the Shiba state on the magnetization dynamics. Finally, we demonstrate that the torques are a direct measure of the even and odd frequency triplet pairings generated by the dynamics of the magnetic impurity. Our approach offers non-invasive alternative to the STM techniques used to probe the Shiba states.

Superconducting vortices in half-metals

When the impurity mean free path is short, only spin-polarized Cooper pairs which are non-locally and antisymmetrically correlated in time may exist in a half-metallic ferromagnet. As a consequence, the half-metal acts as an odd-frequency superconducting condensate. We demonstrate both analytically and numerically that quantum vortices can emerge in half-metals despite the complete absence of conventional superconducting correlations. Because these metals are conducting in only one spin band, we show that a circulating spin supercurrent accompanies these vortices. Moreover, we demonstrate that magnetic disorder at the interfaces with the superconductor influences the position at which the vortices nucleate. This insight can be used to help determine the effective interfacial misalignment angles for the magnetization in hybrid structures, since the vortex position is experimentally observable via STM-measurements. We also give a brief discussion regarding which superconducting order parameter to use for odd-frequency triplet Cooper pairs in the quasiclassical theory.

Electrically controllable spin filtering based on superconducting helical states

The magnetoelectric effects in the surface states of the three-dimensional (3D) topological insulator (TI) are extremely strong due to the full spin-momentum locking. Here the microscopic theory of S/3D TI bilayer structures in terms of quasiclassical Green's functions is developed. On the basis of the developed formalism it is shown that the DOS in the S/TI bilayer manifests giant magnetoelectric behavior and, as a result, S/3D TI heterostructures can work as nonmagnetic fully electrically controllable spin filters. It is shown that due to the full spin-momentum locking the amplitudes of the odd-frequency singlet and triplet components of the condensate wave function are equal. The same is valid for the even frequency singlet and triplet components. We unveil the connection between the odd-frequency pairing in S/3D TI heterostructures and magnetoelectric effects in the DOS.

Tunable odd-frequency triplet pairing states and skyrmion modes in chiral p-wave superconductor

Bogliubov-de Gennes equations are solved self-consistently to investigate the properties of bound states in chiral p-wave superconductive disks. It shows that either an s-wave or the mixed d- and s-wave state with odd-frequency and spin-triplet symmetry is induced at the vortex core, depending both on the chirality of the pairing states and on the vortex topology. It is also found that the odd-frequency triplet even parity (OTE) bound state can be manipulated with a local non-magnetic potential. Interestingly, with an appropriate potential amplitude, the zero-energy OTE bound state can be stabilized at a distance from the vortex core and from the local potential. Possible existences of the Majorana fermion modes are expected if the particle-hole symmetry property is applied to the zero-energy OTE bound state. Moreover, skyrmion modes with an integer topological charge have been found to exist.

Long-range proximity effect in Nb-based heterostructures induced by a magnetically inhomogeneous permalloy layer

Odd-frequency triplet Cooper pairs are believed to be the carriers of long-range superconducting correlations in ferromagnets. Such triplet pairs are generated by an inhomogeneous magnetism at the interface between a superconductor (S) and a ferromagnet (F). So far, reproducible long-range effects were reported only in complex layered structures designed to provide the magnetic inhomogeneity. Here we show that spin triplet pair formation can be found in simple unstructured Nb/permalloy ()/Nb trilayers and Nb/Py bilayers, but only when the thickness of the ferromagnetic layer ranges between 140 and 250 nm. The effect is related to the emergence of an intrinsically inhomogeneous magnetic state, which is a precursor of the well-known stripe regime in Py that in our samples sets in at thickness larger than 300 nm.

Pure odd-frequency superconductivity at the cores of proximity vortices

After more than a decade, direct observation of the odd frequency triplet pairing state in super- conducting hybrid structures remains elusive. We propose an experimentally feasible setup that can unambiguously reveal the zero energy peak due to proximity-induced equal spin superconducting triplet correlations. We theoretically investigate a two dimensional Josephson junction in the diffu- sive regime. The nanostructure consists of a normal metal sandwiched between two ferromagnetic layers with spiral magnetization patterns. By applying an external magnetic field perpendicular to the junction plane, vortices nucleate in the normal metal. The calculated energy and spatially resolved density of states, along with the pair potential, reveal that remarkably, only triplet Cooper pairs survive in the vortex cores. These isolated odd frequency triplet correlations result in well defined zero energy peaks in the local density of states that can be identified through tunneling spectroscopy experiments. Moreover, the diffusive regime considered here rules out the possibility of Andreev bound states in the vortex core as contributors to the zero energy peaks.

Superconducting proximity effect in three-dimensional topological insulators in the presence of a magnetic field

The proximity induced pair potential in a topological insulator-superconductor hybrid features an interesting superposition of a conventional spin-singlet component from the superconductor and a spin-triplet one induced by the surface state of the topological insulator. This singlet-triplet superposition can be altered by the presence of a magnetic field. We study the interplay between topological order and superconducting correlations performing a symmetry analysis of the induced pair potential, using Green functions techniques to theoretically describe ballistic junctions between superconductors and topological insulators under magnetic fields. We relate a change in the conductance from a gapped profile into one with a zero-energy peak with the transition into a topologically nontrivial regime where the odd-frequency triplet pairing becomes the dominant component in the pair potential. The nontrivial regime, which provides a signature of odd-frequency triplet superconductivity, is reached for an out-of-plane effective magnetization with strength comparable to the chemical potential of the superconductor or for an in-plane one, parallel to the normal-superconductor interface, with strength of the order of the superconducting gap. Strikingly, in the latter case, a misalignment with the interface yields an asymmetry with the energy in the conductance unless the total contribution of the topological surface state is considered.

Odd-frequency triplet superconductivity at the helical edge of a topological insulator

Non-local pairing processes at the edge of a two-dimensional topological insulator in proximity to an $s$-wave superconductor are usually suppressed by helicity. However, additional proximity of a ferromagnetic insulator can substantially influence the helical constraint and therefore open a new conduction channel by allowing for crossed Andreev reflection (CAR) processes. We show a one-to-one correspondence between CAR and the emergence of odd-frequency triplet superconductivity. Hence, non-local transport experiments that identify CAR in helical liquids yield smoking-gun evidence for unconventional superconductivity. Interestingly, we identify a setup -- composed of a superconductor flanked by two ferromagnetic insulators -- that allows us to favor CAR over electron cotunneling which is known to be a difficult but essential task to be able to measure CAR.

Effect of a spin-active interface on proximity-induced superconductivity in topological insulators

We examine the effect of a spin-active interface on the symmetry of proximity-induced superconducting pairing amplitudes in topological insulators. We develop a model to investigate the leading-order contribution to the pairing amplitude considering three different kinds of spin-active interfaces: (i) those that induce spin-dependent scattering phases, (ii) those that flip the spin of incident electrons, and (iii) interfaces that both induce spin-dependent phases and flip the spins of incident electrons. We find that in cases (i) and (iii) odd-frequency triplet pairing is induced in the TI while for case (ii) no odd-frequency pairing is induced to leading order. We compare our results to those for normal metals and ferromagnetic materials finding that the nontrivial spin structure of the TI leads to qualitatively different behavior.

Meissner effect probing of odd-frequency triplet pairing in superconducting spin valves

Superconducting correlations which are long ranged in magnetic systems have attracted much attention due to their spin-polarization properties and potential use in spintronic devices. Whereas experiments have demonstrated the slow decay of such correlations, it has proven more difficult to obtain a smoking gun signature of their odd-frequency character which is responsible, e.g., for their gapless behavior. Here we demonstrate that the magnetic susceptibility response of a normal metal in contact with a superconducting spin valve provides precisely this signature, namely, in the form of an anomalous positive Meissner effect, which may be tuned back to a conventional negative Meissner response simply by altering the magnetization configuration of the spin valve.

Vortex contribution to the critical temperature oscillations in hybrid superconductor-ferromagnet coaxial cylinders

The s-wave triplet Usadel equations which include the odd frequency triplet condensate are employed to study the oscillatory behavior of the superconducting critical temperature Tc in multiply connected hybrid systems e.g., a ferromagnetic core surrounded by a superconducting shell. This configuration allows us to treat the angular momentum number L of the Cooper pair as the vorticity parameter. The spiral exchange field in the ferromagnet characterized by a spiral wave vector Q, which rotates in the z-plane and its direction varies within an in-plane of the ferromagnetic core. The induced superconductivity in the ferromagnet core is controlled by the electrical contact at the boundary for which we are interested in the switching of the superconducting states among the various vorticity numbers. The peculiar superconducting states with L≠0 bear resemblance to the pi-phase state in the bilayer systems. We demonstrate the switching phenomena by solving the Usadel equations together with the self-consistent order parameter subject to the Kupriyanov-Lukichev boundary conditions at the contact surface to obtain the secular equation in the multimode method to determine the oscillatory behavior of Tc in the ferromagnet core. The Abrikosov–Gorkov like-formula is also obtained within the single mode approximation.

Field modulation of the critical current in magnetic Josephson junctions

The dependence of the critical current of a simple Josephson junction on the applied magnetic field is well known and, for a rectangular junction, gives rise to the classic ‘Fraunhofer’ modulation with periodic zeros at the fields that introduce a flux quantum into the junction region. Much recent work has been performed on Josephson junctions that contain magnetic layers. The magnetization of such layers introduces additional flux into the junction and, for large junction areas or strong magnetic materials, can significantly distort the modulation of the critical current and strongly suppress the maximum critical current. The growing interest in junctions that induce odd-frequency triplet pairing in a ferromagnet, and the need to make quantitative comparisons with theory, mean that a full understanding of the role of magnetic barriers in controlling the critical current is necessary. This paper analyses the effect of magnetism and various magnetic configurations on Josephson critical currents; the overall treatment applies to junctions of general shape, but the specific cases of square and rectangular junctions are considered.

Stationary Josephson effect in a short multiterminal junction

We study the dc Josephson effect and the density of states in a multiterminal structure of cross-type geometry that consists of four superconducting electrodes connected by one-dimensional normal or ferromagnetic wires. We find that the Josephson current ${I}_{Jz}$ has a sinusoidal dependence on the phase difference ${\ensuremath{\phi}}_{z}$ between the superconductors in the horizontal wire with a critical current ${I}_{c}$ that can be varied by changing the phase difference ${\ensuremath{\phi}}_{y}$ between the superconductors in the vertical wire. The period of the function ${I}_{Jz}({\ensuremath{\phi}}_{z})$ depends on the ratio of the interface resistances ${R}_{Sn,z}/{R}_{Sn,y}$ being equal to $2\ensuremath{\pi}$ if this ratio is small and equal to $4\ensuremath{\pi}$ in the opposite limit. We also calculate the amplitudes of both the singlet and the odd-frequency triplet components in the system under consideration. The triplet component amplitude may be significantly larger than the amplitude of the singlet component.

Josephson and proximity effects on the surface of a topological insulator

We investigate Josephson and proximity effects on the surface of a topological insulator on which superconductors and a ferromagnet are deposited. The superconducting regions are described by the conventional BCS Hamiltonian, rather than the superconducting Dirac Hamiltonian. Junction interfaces are assumed to be dirty. We obtain analytical expressions of the Josephson current and the proximity-induced anomalous Green's function on the topological insulator. The dependence of the Josephson effect on the junction length, the temperature, the chemical potential, and the magnetization is discussed. It is also shown that the proximity-induced pairing on the surface of a topological insulator includes even- and odd-frequency triplet pairings as well as a conventional $s$-wave pairing.

Properties of Magnetic-Superconducting Proximity Systems

We discuss properties of proximity systems made of superconducting, ferromagnetic, and graphite nanoscale thin films. In superconducting-graphite proximity systems, we show how size quantization results in oscillations of the density of states with increasing graphite film thickness. In superconducting-ferromagnetic multilayers, we discuss the critical temperature variations of the proximity system as a function of ferromagnetic layer thickness and magnetic inhomogeneity. We also review the Josephson effect through a superconductor–ferromagnet–superconductor multilayer. In this context, we discuss the presence of a long-range odd-frequency triplet order parameter.

Strong-coupling superconductivity with mixed even- and odd-frequency pairing

We investigate general structure of Landau free energy for stabilizing a novel superconducting state with both even-frequency and odd-frequency components in gap function. On the basis of the Luttinger-Ward functional, we elucidate an emergent mixing between different "parities" in time. The simplest case of the conventional s-wave singlet mixed with the odd-frequency triplet state under broken time-reversal symmetry is examined to demonstrate the anomalous structure of the free-energy functional. The induced odd-frequency component alters behaviors of physical quantities from those obtained by neglecting the odd-frequency component. The novel mixed state may also be relevant to strong-coupling superconductivity coexisting with ferromagnetism.

강자성 초전도체의 연구동향과 전망

강자성 초전도체의 발견은 상극으로 알려진 강자성과 초전도성이 어떻게 상호작용하여 조화롭게 공존하는지에 대한 학문적인 연구뿐만 아니라 새로운 기술적인 응용을 위한 광범위한 탐구를 이끌고 있다. 본 해설논문에서는 강자성 초전도체에 대한 이해를 돕기 위하여, 먼저 초전도체의 쿠퍼쌍을 깨뜨리는 강한 자기장의 궤도 효과와 상자성 효과에 대하여 설명한다. 자기장의 이러한 효과 이외에도 초전도체/강자성체 복합 구조의 계면에서 발생하는 근접 효과에 의해 단일상 쿠퍼쌍은 강자성체를 지나가는 동안 불안정하여 아주 짧은 침투깊이를 가진다. 그러나 쿠퍼쌍이 홀-진동수 삼중상인 경우 안정되고 긴 유효길이를 가지게 되는데, 새로운 스핀 전자소자로서의 개발을 위해 그 연구의 중요성이 높아지고 있다. 마지막으로 다양한 강자성 초전도체와 양자구속효과에 의해 두 성질이 공존하는 저차원의 물질들을 소개한다. The findings of ferromagnetic superconductor have attracted much attention not only for fundamental research to investigate how the antagonistic properties of ferromagnetism and superconductivity coexist peacefully but also for potential technological applications. Firstly, in order to help for understanding the ferromagnetic superconductor, I have explained the orbital and paramagnetic pair-breaking effects of magnetic field, which breaks the superconducting Cooper pairs. In addition to such effects of magnetic field, the singlet Cooper pairs become unstable upon going through the ferromagnetic materials by the proximity effect. The proximity effect occurs at the interface of thin films composing of superconductor and ferromagnet and leads to have very short penetration depth of Cooper pairs. However, a type of odd-frequency triplet in comparison with the singlet could be very stable and has a longer effective depth. It needs to be explored for the innovative spintronic devices. Finally, various ferromagnetic superconductors coexist and the lower-dimensional materials under the Quantum confinement effect have been introduced.

Inducing Odd-Frequency Triplet Superconducting Correlations in a Normal Metal

This work discusses theoretically the interplay between the superconducting and ferromagnetic proximity effects, in a diffusive normal metal strip in contact with a superconductor and a nonuniformly magnetized ferromagnetic insulator. The quasiparticle density of states of the normal metal shows clear qualitative signatures of triplet correlations with spin one (TCS1). When one goes away from the superconducting contact, TCS1 focus at zero energy under the form of a peak surrounded by dips, which show a typical spatial scaling behavior. This effect can coexist with a focusing of singlet correlations and triplet correlations with spin zero at finite but subgap energies. The simultaneous observation of both effects would enable an unambiguous characterization of TCS1.

Density of states spectra in ferromagnet/superconductor/ferromagnet heterostructures

We investigate density of states (DOS) spectra of a ferromagnet/superconductor/ferromagnet structure by including the interface spin-flip scattering potential into the existing mean-field Hamiltonian. We show that the spin-flip scattering can lead to electron spin mixing in the vicinity of an FS interface, enabling s-wave singlet and odd-frequency triplet pairs conversion and the existence of Andreev bound states with subgap energy ($\ifmmode\pm\else\textpm\fi{}{\ensuremath{\varepsilon}}_{b}$) via the intra- and interband Andreev reflections. This adds spin-resolved subgap peaks to the DOS spectrum. The relevant elastic co-tunneling at $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\varepsilon}}_{b}$ enhances the local conductance, while its competition with the crossed Andreev reflection will lead to a sign change in the nonlocal differential conductance with the bias energy.