We have measured the dynamics of individual magnetic fluxoids entering and leaving thin film, photolithographically patterned rings of underdoped high-temperature superconductor BSCCO using a variable sample temperature scanning SQUID microscope. These measurements can be understood within a phenomenological model in which the fluxoid number changes by thermal activation of a Pearl vortex in the ring wall. We place upper limits on the "vison" binding energy in these samples from these measurements.
We have observed spontaneous fluxoid generation in thin-film rings of the amorphous superconductor Mo$_3$Si, cooled through the normal-superconducting transition, as a function of quench rate and externally applied magnetic field, using a variable sample temperature scanning SQUID microscope. Our results can be explained using a model of freezout of thermally activated fluxoids, mediated by the transport of bulk vortices across the ring walls. This mechanism is complementary to a mechanism proposed by Kibble and Zurek, which only relies on causality to produce a freezout of order parameter flu…
We have measured the dynamics of individual magnetic fluxoids entering and leaving photolithographically patterned thin film rings of the underdoped high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$, using a variable sample temperature scanning SQUID microscope. These results can be qualitatively described using a model in which the fluxoid number changes by thermally activated nucleation of a Pearl vortex in, and transport of the Pearl vortex across, the ring wall.
The authors report magnetic measurements of fluxoid transitions in mesoscopic, superconducting aluminum rings. The transitions are induced by applying a flux to the ring so that the induced supercurrent approaches the critical current. In a temperature range near $T_c$, only a single fluxoid enters or leaves at a time, leading to a final state above the ground state. Upon lowering the temperature, several fluxoids enter or leave at once, and the final state approaches the ground state, which can be reached below approximately 0.5 $T_c$. A model based on the widely used time dependent Ginzburg-…
Fluxoid solitons in superconducting tapered tubes and bottlenecks
University of Jyväskylä（于韦斯屈莱大学）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（马德里材料科学研究所，西班牙国家科学研究委员会）；Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU（材料物理中心，西班牙国家科学研究委员会-巴斯克国立大学联合中心）
Rings are a model system for studying phase coherence in one dimension. Superconducting rings have states with uniform phase windings that are integer multiples of 2$π$ called fluxoid states. When the energy difference between these fluxoid states is of order the temperature so that phase slips are energetically accessible, several states contribute to the ring's magnetic response to a flux threading the ring in thermal equilibrium and cause a suppression or downturn in the ring's magnetic susceptibility as a function of temperature. We review the theoretical framework for superconducting fluc…
In a micro ring of a superconductor with a spin-triplet equal-spin pairing state, a fluxoid, a combined object of magnetic flux and circulating supercurrent, can penetrate as half-integer multiples of the flux quantum. A candidate material to investigate such half-quantum fluxoids is Sr$_\mathsf{2}$RuO$_\mathsf{4}$. We fabricated Sr$_\mathsf{2}$RuO$_\mathsf{4}$ micro rings using single crystals and measured their resistance behavior under magnetic fields controlled with a three-axes vector magnet. Proper Little-Parks oscillations in the magnetovoltage as a function of an axially applied field,…
We report on the first experimental verification of the Zurek-Kibble scenario in an isolated superconducting ring over a wide parameter range. The probability of creating a single flux quantum spontaneously during the fast normal-superconducting phase transition of a wide Nb loop clearly follows an allometric dependence on the quenching time $τ_{Q}$, as one would expect if the transition took place as fast as causality permits. However, the observed Zurek-Kibble scaling exponent $σ= 0.62\pm0.15$ is two times larger than anticipated for large loops. Assuming Gaussian winding number densities we…
We evaluate the influence of thermal fluctuations on superconducting rings that enclose a magnetic flux, using the time-dependent Ginzburg--Landau (TDGL) or the Kramer--Watts-Tobin (KWT) model, while thermal fluctuations are accounted for by means of Langevin terms. This method is applicable in situations where previous methods are not, such as nonuniform loops, rings with large width to radius ratio and loops with large coherence length to perimeter ratio. We evaluate persistent currents, position and statistical behavior of flux-induced vortices and lifetime of metastable fluxoid states. The…
It is shown that currents in thin-film superconducting rings irradiated by coherent microwave fields, can discretely decay with photons emission. These quantized jumps of the supercurrent correspond to destruction one or several magnetic flux quanta trapped in the ring. The ring thickness should be less than the skin-depth of the low-frequency field. The probability of the microwave field-induced single-photon decay of the supercurrent states in these rings is obtained. The angle distributions of the photons emitted by superconducting rings, and the current state lifetimes, depending on the ri…
The global phase coherence in doubly-connected superconductors leads to fluxoid quantization, allowing the superfluid velocity vs to be controlled by an applied magnetic flux. In ultrasmall samples this quantization requirement leads, surprisingly, to the destruction of the phase coherence itself around half-integer flux quanta, because of the sample-size-induced growth in vs, as predicted by de Gennes. We report observations of the predicted phenomenon in ultrathin Al and Au0.7In0.3 cylinders, and the corresponding phase diagram for ultrathin superconducting cylinders. The new phase diagram f…
We present a scanning probe technique for measuring the dynamics of individual fluxoid transitions in multiply connected superconducting structures. In these measurements, a small magnetic particle attached to the tip of a silicon cantilever is scanned over a micron-size superconducting ring fabricated from a thin aluminum film. We find that near the superconducting transition temperature of the aluminum, the dissipation and frequency of the cantilever changes significantly at particular locations where the tip-induced magnetic flux penetrating the ring causes the two lowest-energy fluxoid sta…
When a thin-film ring with supercurrent is irradiated by coherent microwave field, the field induces coherent oscillations of all the Cooper pairs involved in the supercurrent. The oscillating charged condensate will emit photons with energies determined by the difference of quantized energy levels of the superconducting ring. This effect can be observed only in the rings, the thickness of which is less than both the field skin-depth and the London penetration depth. The probability of the microwave field-induced single-photon and two-photon emission is calculated. The energy - angle distribut…
A phase-bias tool, based on a trapped fluxoid in a ring, is proposed and demonstrated. It can provide arbitrary phase values and is simple to fabricate. The phase bias has been realized in two superconducting quantum interference devices, where the critical current versus magnetic flux is shown to be shifted by a π/2 and π.
The distributions of electrical current and magnetic field in a thin-film superconductor ring is calculated by solving the London equation. The maximum amount of flux trapped by the hole, the fluxoid saturation number, is obtained by limiting the current density by the depairing current. The results are compare it with similar results derived for the bulk case of a long hollow cylinder [Nordborg & Vinokur, Phys. Rev. B 62, 12408 (2000)]. In the limit of small holes our result reduces to the Pearl solution for an isolated vortex in a thin film. For large hole radius, the ratio between saturatio…
We present the first measurements of thermal signatures of the Little-Parks effect using a highly sensitive nanocalorimeter. Small variations of the heat capacity $C\_p$ of 2.5 millions of non interacting micrometer-sized superconducting rings threaded by a magnetic flux $Φ$ have been measured by attojoule calorimetry. This non-invasive method allows the measurement of thermodynamic properties -- and hence the probing of the energy levels -- of nanosystems without perturbing them electrically. It is observed that $C\_p$ is strongly influenced by the fluxoid quantization (Little-Parks effect) n…
Consistent with the predictions of Kibble and Zurek, scaling behaviour has been seen in the production of fluxoids during temperature quenches of superconducting rings. However, deviations from the canonical behaviour arise because of finite-size effects and stray external fields. Technical developments, including laser heating and the use of long Josephson tunnel junctions, have improved the quality of data that can be obtained. With new experiments in mind we perform large-scale 3D simulations of quenches of small, thin rings of various geometries with fully dynamical electromagnetic fields,…
Recently Babaei Brojeny and Clem [Phys. Rev. B 68, 174514 (2003)] considered superconducting thin-film rings in perpendicular magnetic fields in the ideal Meissner state with negligibly small magnetic penetration depth and presented useful analytical limiting expressions and numerical results for the magnetic-field and sheet-current profiles, trapped magnetic flux, self-inductance, magnetic moment, and focusing of magnetic flux into the hole when no net current flows in the ring. The present paper generalizes all these results to rings with arbitrary values of the two-dimensional effective pen…
We describe measurements of the thermally-activated transitions between fluxoid states of a single isolated superconducting ring. We compare these measurements with theoretical predictions in which all of the relevant parameters are determined via independent characterization of the same ring. This no-free-parameters comparison shows qualitative agreement over a wide range of temperatures. We discuss possible origins for the remaining discrepancies between the data and theory, in particular the choice of model for the superconducting order parameter's damping.
We study a mesoscopic superconducting ring threaded by a magnetic flux when the single particle level spacing is not negligible. It is shown that, for a superconducting ring with even parity, the behavior of persistent current is equivalent to what is expected in a bulk superconducting ring. On the other hand, we find that a ring with odd parity shows anomalous behavior such as fluxoid quantization at half-integral multiples of the flux quantum and paramagnetic response at low temperature. We also discuss how the parity effect in the persistent current disappears as the temperature is raised o…
Superconducting current instability in thin rings, irradiated by a coherent microwave electromagnetic field, is studied. On the regular circular motion of the Cooper pairs in the supercurrent states the field imposes coherent oscillations of the condensate as a whole. The oscillating condensate should emit photons with the energies determined mainly by the discrete values of the superconducting ring energy. For the first time the probability of the supercurrent decay accompanied by the two- photon emission, is derived. The ring thickness is less than both the field skin-depth and the London pe…
The magnetic response and fluxoid transitions of superconducting aluminum rings of various sizes, deposited under conditions likely to generate a layered structure, show good agreement with a two-order-parameter Ginzburg-Landau model. For intermediate couplings, we find metastable states that have different phase winding numbers around the ring in each of the two order parameters. Those states, previously theoretically predicted, are analogous to fractional vortices in singly connected samples with two-order-parameter superconductivity. Larger coupling locks the relative phase so that the two …
Using the nonlinear Ginzburg-Landau theory we investigated the dependence of the magnetic coupling between two concentric mesoscopic superconducting rings on their thickness. The size of this magnetic coupling increases with the thickness of the rings.
The differential cross section for the inelastic magnetic scattering of neutrons by superconducting rings is derived theoretically taking account of the interaction of the neutron magnetic moment with magnetic field created by the superconducting current. In this scattering process the neutron kinetic energy increases discretely and, respectively, the number of the magnetic flux quanta trapped in the ring, reduces. Quantitative calculations of the scattering cross section for cold neutrons is carried out for thin film rings from type-II superconductors with the film thickness $d<λ$($λ$ is the …
A formula of the spin current in mesoscopic superconductors is derived from the mean-field theory of superconductivity. The spin flow is generated by the spatial fluctuations of $\vec{d}$ which represents a spin state of spin-triplet superconductors. We discuss a possibility of the circulating spin current in isolated p-wave superconducting rings at the zero magnetic field. The direction of the spin current depends on topological numbers which characterize the spatial configuration of $\vec{d}$ on the ring.
Superconducting thin-film rings smaller than the film penetration depth (the Pearl length) are considered. The current distribution, magnetic moment, and thermodynamic potential ${\cal F}(H,N,v)$ for a flat, washer-shaped annular ring in a uniform applied field $H$ perpendicular to the film are solved analytically within the London approach for a state with winding number $N$ and a vortex at radius $v$ between the inner and outer radii.
The transitions between the different vortex states of thin mesoscopic superconducting disks and rings are studied using the non-linear Ginzburg-Landau functional. They are saddle points of the free energy representing the energy barrier which has to be overcome for transition between the different vortex states. In small superconducting disks and rings the saddle point state between two giant vortex states, and in larger systems the saddle point state between a multivortex state and a giant vortex state and between two multivortex states is obtained. The shape and the height of the nucleation…
A closed superconducting circuit containing an odd number of $π$-junctions, a $π$-ring, has a finite current in the ground state. We explicitly construct such rings for d-wave superconductors and demonstrate the existence of spontaneous currents by direct self-consistent solutions to the Bogoliubov de Gennes equations. We show that the current has a topological origin due to the frustration of the d-wave order parameter, which is only partially explained by the Sigrist-Rice tunneling formula.
The spectrum of excitations of the chiral superconducting ring with internal and external radii, comparable with coherence length, trapping a unit flux is calculated. We find within the Bogoliubov-deGennes approach that there exists a pair of precisely zero energy states. They are not protected by topology, but are stable under certain deformations of the system. We discuss the ways to tune the system so that it grows into such a "Majorana disk". This condition has a character of a resonance phenomenon.
Magnetic dependencies of the critical current of superconducting, aluminium rings, with homogeneous section and with small narrow spot, are measured. Only permitted state at each magnetic flux value is observed in the conventional ring because of strong discreteness of its spectrum. But two states are observed because of the small narrow spot.
An experimental and theoretical investigation was made of flux jumps and irreversible magnetization curves of mesoscopic Al superconducting rings. In the small magnetic field region the change of vorticity with magnetic field can be larger than unity. This behavior is connected with the existence of several metastable states of different vorticity. The intentional introduction of a defect in the ring has a large effect on the size of the flux jumps. Calculations based on the time-dependent Ginzburg-Landau model allows us to explain the experimental results semi-quantitatively.
The states of two phase-coupled superconducting rings have been investigated. Multiple current states have been revealed in the dependence of the critical current on the magnetic field. The performed calculations of the critical currents and energy states in a magnetic field have made it possible to interpret the experiment as the measurement of energy states into which the system comes with different probabilities because of the equilibrium and non-equilibrium noises upon the transition from the resistive state to the superconducting state during the measurement of the critical current
Experimental determination of the magnetic moment of the Cooper pairs can shed light on the pairing symmetry in cuprates. We argue that the intrinsic magnetic moments of the Cooper pairs can be detected in experiments with superconducting thin-film rings. At sufficiently low temperatures the magnetic field generated by the supercurrent, can cause the ordering of these magnetic moments that produces changes in the supercurrent and magnetic induction distributions, the heat capacity and magnetic moment of the ring. It is shown that the intrinsic magnetic moment of the Cooper pairs can be extract…
At a superconductor interface with a ferromagnetic insulator (FI), the FI acts to induce a local exchange field within the S layer, which in the presence of spin-orbit interaction promotes a phase modulated superconducting state. Here we demonstrate that within a thin superconducting loop that is partially proximitized by a FI, spontaneous currents form with a magnetization-orientation-dependent chirality with sizable shifts in Little-Parks oscillations. Furthermore, the critical temperature of the loop is also magnetization-orientation-dependent and conversely, the superconducting transition …
We theoretically investigate the temperature-to-phase conversion (TPC) process occurring in dc superconducting quantum interferometers based on superconductor--normal metal--superconductor (SNS) mesoscopic Josephson junctions. In particular, we predict the temperature-driven rearrangement of the phase gradients in the interferometer under the fixed constraints of fluxoid quantization and supercurrent conservation. This allows sizeable phase variations across the junctions for suitable structure parameters and temperatures. We show that the TPC can be a basis for sensitive single-photon sensors…
Magneto-optical imaging was used to observe dendritic flux avalanches connecting the outer and inner edges of a ring-shaped superconducting MgB2 film. Such avalanches create heated channels across the entire width of the ring, and inject large amounts of flux into the central hole. By measuring the injected flux and the corresponding reduction of current, which is typically 15%, we estimate the maximum temperature in the channel to be 100 K, and the duration of the process to be on the order of a microsecond. Flux creep simulations reproduce all the observed features in the current density bef…
Applying the time-dependent Ginzburg-Landau equations, transitions between metastable states of a superconducting ring are investigated in the presence of an external magnetic field. It is shown that if the ring exhibits several metastable states at a particular magnetic field, the transition from one metastable state to another one is governed by both the relaxation time of the absolute value of the order parameter tau_{|psi|} and the relaxation time of the phase of the order parameter tau_{phi}. We found that the larger the ratio tau_{|psi|}tau_{phi} the closer the final state will be to the…
It was established experimentally that a critical value of the diamagnetic current, excited by an external magnetic field in a superconducting ring (with an inductance of about ~ 10^-6 H) with a Nb-Nb clamping point contact having the Josephson contact properties is a strictly periodic function of field strength, despite the complex microstructure of the clamping contact. The reasons of the periodic dependence are discussed on a basis of the interference model of diamagnetic current and quantized values of the circulating current in the microinterferometer formed by the clamping contact.
We show how to calculate the magnetic-field and sheet-current distributions for a thin-film superconducting annular ring (inner radius a, outer radius b, and thickness d<<a) when either the penetration depth obeys lambda < d/2 or, if lambda > d/2, the two-dimensional screening length obeys Lambda = 2 lambda^2/d << a for the following cases: (a) magnetic flux trapped in the hole in the absence of an applied magnetic field, (b) zero magnetic flux in the hole when the ring is subjected to an applied magnetic field, and (c) focusing of magnetic flux into the hole when a magnetic field is applied b…
Thermal fluctuations of the Josephson current $I$ induced by the magnetic flux through a ring of $N$ superconducting grains are studied. When a half-fluxon is threading the ring, $I$ exhibits incoherent transitions between the two degenerate states due to thermal phase slips. We propose a new numerical method to deal with both equilibrium and dynamic properties of Josephson systems. Computed transition rate has the form $Γ= A(N)\exp[-B(N)/T]$, where $B(N)$ agrees with the analytical result derived for the energy barrier associated with phase slips. In the non-degenerate case (e.g., at a quarte…
We study the ratchet effect in a narrow pinning-free superconductive ring based on time-dependent Ginzburg-Landau (TDGL) equations. Voltage responses to external dc an ac currents at various magnetic fields are studied. Due to asymmetric barriers for flux penetration and flux exit in the ring-shaped superconductor, the critical current above which the flux-flow state is reached, as well as the critical current for the transition to the normal state, are different for the two directions of applied current. These effects cooperatively cause ratchet signal reversal at high magnetic fields, which …
Magnetic field dependences of critical current, resistance, and rectified voltage of asymmetric (half circles of different widths) and symmetrical (half circles of equal widths) aluminum rings were measured in the temperature region close to the superconducting transition. All these dependences are periodic magnetic field functions with periods corresponding to the flux quantum in the ring. The periodic dependences of critical current measured in opposite directions were found to be close to each other for symmetrical rings and shifted with respect to each other by half the flux quantum in asy…
We report experiments in which one dimensional (1D) and two dimensional (2D) arrays of YBa2Cu3O7-x-Nb pi-rings are cooled through the superconducting transition temperature of the Nb in various magnetic fields. These pi-rings have degenerate ground states with either clockwise or counter-clockwise spontaneous circulating supercurrents. The final flux state of each ring in the arrays was determined using scanning SQUID microscopy. In the 1D arrays, fabricated as a single junction with facets alternating between alignment parallel to a [100] axis of the YBCO and rotated 90 degrees to that axis, …
Superconductivity is shown to be completely destroyed in thin mesoscopic or nanoscopic rings closed by the junction with a noticeable interfacial pair breaking and/or a Josephson coupling, if a ring's radius r is less than the minimum radius r_min. The quantity r_min depends on the phase difference across the junction, or on the magnetic flux that controls the phase difference in the flux-biased ring. It also depends on the Josephson and interfacial effective coupling constants, and in particular, on whether the ring is closed by 0 or pi junction. The current-phase relation is substantially mo…
The quantum dynamics of quasi-one-dimensional ring with varying electron filling factor is investigated in presence of external electric field. The system is modeled within Hubbard Hamiltonian with attractive Coulomb correlation, which results in superconducting ground state when away from half-filling. The electric field is induced by applying time-dependent Aharonov-Bohm flux in the perpendicular direction. To explore the non-equilibrium phenomena arising from the field, we adopt exact diagonalization and Crank-Nicolson numerical method. With increase in electric field strength, the electron…
Magneto-optical imaging was used to observe flux dendrites with opposite polarities simultaneously penetrate superconducting, ring-shaped MgB$_2$ films. By applying a perpendicular magnetic field, branching dendritic structures nucleate at the outer edge and abruptly propagate deep into the rings. When these structures reach close to the inner edge, where flux with opposite polarity has penetrated the superconductor, they occasionally trigger anti-flux dendrites. These anti-dendrites do not branch, but instead trace the triggering dendrite in the backward direction. Two trigger mechanisms, a n…
Inspired by the seminal, ground-breaking work of Abrikosov in 1957, we developed a new approximation to the interaction between two widely separated superconducting vortices. In contrast with Abrikosov's, we take into account the finite size of the vortices and their internal magnetic profile. We consider the vortices to be embedded within a superconducting, infinitely long hollow cylinder, in order to simplify the symmetry and boundary conditions for the mathematical analysis. We study this system in the context of a magnetic Ginzburg-Landau functional theory, by solving for the magnetic fiel…
We analyze the influence of noise on magnetic properties of a su- perconducting loop which contains three Josephson junctions. This circuit is a classical analog of a persistent current (flux) qubit. A loop supercurrent induced by external magnetic field in the presence of thermal fluctuations is calculated. In order to get connection with experiment we calculate the impedance of the low-frequency tank cir- cuit which is inductively coupled with a loop of interest. We compare obtained results with the results in quantum mode - when the three junction loop exhibits quantum tunneling of the magn…
We investigate the behavior of a mesoscopic one-dimensional ring in an external magnetic field by simulating the time dependent Ginzburg-Landau equations with periodic boundary conditions. We analyze the stability and the different possible evolutions for the phase slip phenomena starting from a metastable state. We find a stability condition relating the winding number of the initial solution and the number of flux quanta penetrating the ring. The analysis of multiple phase slips solutions is based on analytical results and simulations. The role of the ratio of two characteristic times u is s…
As a function of the magnetic flux threading the object, the Little-Parks oscillation in the critical temperature of a large-radius, thin-walled superconducting ring or hollow cylinder has a period given by $h/2e$, due to the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillation in the resistance or persistent current for a clean (i.e. ballistic) normal-state system having the same topological structure has a period given by $h/e$. A basic question is whether the Little-Parks oscillation changes its character, as the radius of the superconduct…
The giant vortex states of a multiply connected superconductor, with radius comparable to the penetration depth and the coherence length, are theoretically investigated based on the nonlinear Ginzburg-Landau theory, in which the induced magnetic field by the super-currents is accurately taken into account. The solutions of Ginzburg-Landau equations are found to be actually independent of the angular momentum L in a gauge invariant point of view, provided that the hole is in the center. Different cases with the paramagnetic current, the diamagnetic current, and the coexistence of the above two,…
Existing concepts of reversible superconducting circuits as well as demonstrated adiabatic circuits require three-phase bias/clock signals generated by room temperature sources. A while ago, we suggested that a multi-phase bias/clock could be provided by a local Josephson junction-based generator. The generator looks like a long annular Josephson junction, only composed of discreet elements - junctions and inductors, and closed into a ring via a flux pump to inject any required number of vortices into the ring. A steady motion of the vortices forced by a uniformly distributed dc bias current a…
Shift of the maxima of the critical currents of different polarity relative to the zero magnetic flux along the flux axis in a superconducting asymmetric aluminum ring
Institute of Microelectronics Technology and High Purity Materials, Russian Academy of Sciences（俄罗斯科学院微电子技术与高纯材料研究所）
The quantum oscillations in magnetic field of the critical current of asymmetric superconducting rings with different widths of the half-rings are shifted to opposite sides for measurement in the opposite direction. The value of this shift found before equal half of the flux quantum for single ring with radius 2 μm has smaller value at lower critical current and for system of the rings with smaller radius.
The periodic response of a metallic or a superconducting ring to an external magnetic flux is one of the most evident manifestations of quantum mechanics. It is generally understood that the oscillation period hc/2e in the superconducting state is half the period hc/e in the metallic state, because the supercurrent is carried by Cooper pairs with a charge 2e. On the basis of the Bardeen-Cooper-Schrieffer theory we discuss, in which cases this simple interpretation is valid and when a more careful analysis is needed. In fact, the knowledge of the oscillation period of the current in the ring pr…
We have measured mesoscopic superconducting Au$_{0.7}$In$_{0.3}$ rings prepared by e-beam lithography and sequential deposition of Au and In at room temperature followed by a standard lift-off procedure. In samples showing no Little-Parks resistance oscillations, highly unusual double resistance anomalies, two resistance peaks found near the onset of superconductivity, were observed. Although resistance anomaly featuring a single resistance peak has been seen in various mesoscopic superconducting samples, double resistance anomalies have never been observed previously. The dynamical resistance…
