The celebrated Little-Parks effect in mesoscopic superconducting rings has recently gained great attention due to its potential to probe half-quantum vortices in spin-triplet superconductors. However, despite the large number of works reporting anomalous Little-Parks measurements attributed to unconventional superconductivity, the general signatures of spin-triplet pairing in the Little-Parks effect have not yet been systematically investigated. Here we use Ginzburg-Landau theory to study the Little-Parks effect in a spin-triplet superconducting ring that supports half-quantum vortices; we calculate the field-induced Little-Parks oscillations of both the critical temperature itself and the residual resistance resulting from thermal vortex tunneling below the critical temperature. We observe two separate critical temperatures with a single-spin superconducting state in between and find that, due to the existence of half-quantum vortices, each minimum in the upper critical temperature splits into two minima for the lower critical temperature. From a rigorous calculation of the residual resistance, we confirm that these two minima in the lower critical temperature translate into two maxima in the residual resistance below and establish the general conditions under which the two maxima can be practically resolved. In particular, we identify a fundamental trade-off between sharpening each maximum and keeping the overall magnitude of the resistance large. Our results will guide experimental efforts in designing mesoscopic ring geometries for probing half-quantum vortices in spin-triplet candidate materials on the device scale.
Little-Parks effect names the oscillations in the superconducting critical temperature as a function of the magnetic field. This effect is related to the geometry of the sample. In this work, we show that this effect can be enhanced and manipulated by the inclusion of magnetic nanostructures with perpendicular magnetization. These magnetic nanodots generate stray fields with enough strength to produce superconducting vortex-antivortex pairs. So that, the L-P effect deviation from the usual geometrical constrictions is due to the interplay between local magnetic stray fields and superconducting…
Little-Parks oscillation of a Möbius strip (or ring equivalently) made of a superconductor is studied based on Ginzburg-Landau theory. It is shown that, if the strip is wide enough, a novel state appears when the number of magnetic flux quanta threading the ring is close to a half odd integer. As a result the shape of Little-Parks oscillation of critical temperature is modified. We estimate the free energy of this state and give the phase diagram of the superconducting Möbius strip in a magnetic field.
As is known, Little-Parks effect concerned with oscillations of the critical temperature of a superconducting transition was one of the first effects which suggested the existence of Cooper pairing in conventional superconductors. It is shown that in high-temperature superconductors (HTSC) the Little-Parks effect based on bipolaron mechanism which in HTSC is the equivalent of Cooper pairing can be anomalously high. The results obtained can be used as a new method for detecting soft phonon modes in nanostructures on the basis of HTSC. The results can be also used to enhance the critical tempera…
Within the phenomenological Ginzburg-Landau theory we investigate the phase diagram of a thin superconducting film with ferromagnetic nanoparticles. We study the oscillatory dependence of the critical temperature on an external magnetic field similar to the Little-Parks effect and formation of multiquantum vortex structures. The structure of a superconducting state is studied both analytically and numerically.
We theoretically investigate the critical bias current $j_c$ for a superconducting (SC) ring in a magnetic field. Based on the Ginzburg-Landau theory, we show that $j_c$ exhibits a Little-Parks (LP) oscillation as a function of the magnetic flux passing through the ring, similarly to the LP oscillation in the SC transition temperature. It is also found that for a spin-triplet SC ring, the ${\bf d}$-vector rotates to yield a larger $j_c$, forming a texture along the circumference of the ring. Experimental implications of our result are discussed.
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…
On the basis of of linearized Usadel equations we consider superconductivity nucleation in multiply connected mesoscopic superconductor/ferromagnet hybrids such as thin-walled superconducting cylinders placed in electrical contact with a ferromagnetic metal. We study the interplay between the oscillations of $T_c$ due to the Little--Parks effect and the oscillations due to the exchange field. We demonstrate that the exchange field provokes the switching between the superconducting states with different vorticities and this may result in the increase the critical temperature of the superconduct…
We present results of measurements obtained from a mesoscopic ring of a highly disordered superconductor. Superimposed on a smooth magnetoresistance background we find periodic oscillations with a period that is independent of the strength of the magnetic field. The period of the oscillations is consistent with charge transport by Cooper pairs. The oscillations persist unabated for more than 90 periods, through the transition to the insulating phase, up to our highest field of 12 T.
It is demonstrated that a thin-film Fabry-Perot superconducting resonator can be used to reveal the Little-Parks (LP) effect even at temperatures much lower than the critical temperature. A pair of parallel nanowires is incorporated into the resonator at the point of a supercurrent antinode. As magnetic field is ramped, the Meissner current develops, changing the kinetic inductance of the wires and, correspondingly, the resonance frequency of the resonator, which can be detected. The LP oscillation are revealed as a periodic set of distorted parabolas observed in the transmission of the resona…
We investigate the Little-Parks oscillations caused by a persistent current loop set on the top edge of a mesoscopic superconducting thin-walled cylinder with a finite height. For a short cylinder the Little-Parks oscillations are approximately the same ones as the standard effect, as there is only one magnetic flux piercing the cylinder. For a tall cylinder the inhomogeneity of the magnetic field makes different magnetic fluxes pierce the cylinder at distinct heights and we show here that this produces two distinct Little-Parks oscillatory regimes according to the persistent current loop. We …
The destruction of superconducting phase coherence by quantum fluctuations and the control of these fluctuations have been a problem of long-standing interest, with recent impetus provided by its relevance to the pursuit of very high temperature superconductivity. Building on the work of Little and Parks, de Gennes predicted more than three decades ago that superconductivity could be destroyed near half-integer-flux quanta in ultrasmall loops with a side branch, resulting in a destructive regime. We report the experimental observation of this Little-Parks-de Gennes effect in Al loops prepared …
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,…
The properties of single submicron high-temperature superconductor (HTS) rings are investigated. The Little-Parks effect is observed and is accompanied by an anomalous behavior of the magnetic dependence of the resistance, which we ascribe to non-uniform vorticity (superfluid angular momentum) within the ring arms. This effect is linked to the peculiar HTS-relationship between the values of the coherence length and the London penetration depth.
In superconductors, the condensation of Cooper pairs gives rise to fluxoid quantization in discrete units of $Φ_0 = hc / 2e$. The denominator of $2e$ is the signature of electron pairing, which is evidenced by a number of macroscopic quantum phenomena, such as the Little-Parks effect and the Josephson effect, where the critical temperature or the critical current oscillates in the period of $Φ_0$. Here we report the observation of fractional Little-Parks effect in mesoscopic rings of epitaxial $β$-Bi$_2$Pd, a topological superconductor. Besides $Φ_0$, novel Little-Parks oscillation periodiciti…
Little-Parks oscillations of a hollow superconducting cylinder are of interest for flux-driven topological superconductivity in single Rashba nanowires. The oscillations are typically symmetric in the orientation of the applied magnetic flux. Using double InAs nanowires coated by an epitaxial superconducting Al shell which, despite the non-centro-symmetric geometry, behaves effectively as one hollow cylinder, we demonstrate that a small misalignment of the applied parallel field with respect to the axis of the nanowires can produce field-asymmetric Little-Parks oscillations. These are revealed…
Little-Parks oscillations are observed in a system of 110 series-connected aluminum rings 2000 nm in diameter with the use of measuring currents from 10 nA to 1000 nA. The measurements show that the amplitude and character of the oscillations are independent of the relation between the measuring current and the amplitude of the persistent current. By using asymmetric rings, it is demonstrated that the persistent current has clockwise or contra-clockwise direction. This means that the total current in one of the semi-rings may be directed against the electric field at measurement of the Little-…
We have studied the nucleation of superconductivity in a mesoscopic Al loop, enclosing magnetic dot with perpendicular magnetization. The superconducting phase boundary Tc(B), determined from transport measurements, is asymmetric with respect to the polarity of an applied magnetic field. The maximum critical temperature has been found for a finite applied magnetic field, which is antiparallel to the magnetization of the dot. Theoretical phase boundary shows a good agreement with the experimental data.
A semiconductor transmon with an epitaxial Al shell fully surrounding an InAs nanowire core is investigated in the low $E_J/E_C$ regime. Little-Parks oscillations as a function of flux along the hybrid wire axis are destructive, creating lobes of reentrant superconductivity separated by a metallic state at a half-quantum of applied flux. In the first lobe, phase winding around the shell can induce topological superconductivity in the core. Coherent qubit operation is observed in both the zeroth and first lobes. Splitting of parity bands by coherent single-electron coupling across the junction …
Resistance of disordered superconducting Au$_{0.7}$In$_{0.3}$ cylindrical films was measured as a function of applied magnetic field. In the high-temperature part of the superconducting transition regime, the resistance oscillated with a period of $h/2e$ in the unit of the enclosed magnetic flux. However, at lower temperatures, the resistance peaks split. We argue that this splitting is due to the emergence of an oscillation with a period of $h/4e$, half of the flux quantum for paired electrons. The possible physical origin of the $h/4e$ resistance oscillation is discussed in the context of ne…
A prime category of superconducting materials in which to look for spin-triplet pairing and topological superconductivity are superconductors without inversion symmetry. It is predicted that the breaking of parity symmetry gives rise to an admixture of spin-singlet / spin-triplet pairing states; a triplet pairing component, being substantial, seems all but guaranteed. However, the experimental confirmation of pair mixing in any particular material remains elusive. In this work, we perform phase-sensitive experiment to examine the pairing state of noncentrosymmetric superconductor $α-$BiPd. The…
An inhomogeneous superconducting ring placed in a magnetic field is considered. It is shown that the superconducting transition of the section with lowest critical temperature may be a first order phase transition if the magnetic flux contained within the ring is not divisible by the flux quantum. In the visinity of this transition, thermal fluctuations can induce the direct voltage in the ring with rather small sizes. The voltage value depends in a periodic manner on the magnetic flux contained within the ring. According to this result, the entropy value can be decreased in the considered qua…
Applying the linearized Usadel equations, we consider the nucleation of superconductivity in multiply connected mesoscopic superconductor/ferromagnet (S/F) hybrids such as a thin superconducting ring on a ferromagnet with a uniform in-plane magnetization M and a spin-active S/F interface. We demonstrate that the exchange field in F provokes a switching between superconducting states with different vorticities which may increase the critical temperature ( Tc ) of the superconductor in a magnetic field. We study the interplay between oscillations in Tc due to the Little--Parks effect and oscilla…
This is an analytic study of the problem of transitions between normal and superconducting phases for a sample which encloses a magnetic flux. A preliminary study of this problem, based on numerical minimization of the free energy for a particular form of the thickness of the sample, was published in Phys. Rev. Lett. {\bf 75}, 320 (1995). For a sample of uniform thickness the order parameter is uniform, but even infinitesimal deviations from uniform thickness give rise to a singly connected state in which the order parameter vanishes at a suitable layer, so that the superconducting part does n…
Unlike the more common local conductance spectroscopy, nonlocal conductance can differentiate between nontopological zero-energy modes localized around inhomogeneities, and true Majorana edge modes in the topological phase. In particular, negative nonlocal conductance is dominated by the crossed Andreev reflection. In graphene, the Andreev reflection and the inter-band Klein tunneling couple electron-like and hole-like states through the action of either a superconducting (SC) pair potential or an electrostatic potential. We are here probing quantum phenomena in modified graphitic samples. Fou…
We report results of our low-temperature magneto electric transport measurements on micron-sized short cylinders of odd-parity, spin-triplet superconductor Sr$_2$RuO$_4$ with the cylinder axis along the $c$ axis. The in-plane magnetic field and measurement current dependent magnetoresistance oscillations were found to feature an amplitude much larger than that expected from the conventional Little-Parks effect, suggesting that the magnetoresistance oscillations originate from vortex crossing. The free-energy barrier that controls the vortex crossing was modulated by the magnetic flux enclosed …
Understanding the rich and competing electronic orders in cuprate superconductors may provide important insight into the mechanism of high-temperature superconductivity. Here, by measuring Bi2Sr2CaCu2O8+x in the extremely underdoped regime, we obtain evidence for a distinct type of ordering, which manifests itself as resistance oscillations at low magnetic fields (<10 T) and at temperatures around the superconducting transition. By tuning the doping level p continuously, we reveal that these low-field oscillations occur only when p<0.1. The oscillation amplitude increases with decreasing p but…
We show in the case of a superconducting Nb ladder that a mesoscopic superconductor typically exhibits magnetoresistance oscillations whose amplitude and temperature dependence are different from those stemming from the Little-Parks effect. We demonstrate that these large resistance oscillations (as well as the monotonic background on which they are superimposed) are due to {\it current-excited moving vortices}, where the applied current in competition with the oscillating Meissner currents imposes/removes the barriers for vortex motion in increasing magnetic field. Due to the ever present cur…
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…
Odd-parity, spin-triplet superconductor Sr2RuO4 has been found to feature exotic vortex physics including half-flux quanta trapped in a doubly connected sample and the formation of vortex lattices at low fields. The consequences of these vortex states on the low-temperature magnetoresistive behavior of mesoscopic samples of Sr2RuO4 were investigated in this work using ring device fabricated on mechanically exfoliated single crystals of Sr2RuO4 by photolithography and focused ion beam. With the magnetic field applied perpendicular to the in-plane direction, thin-wall rings of Sr2RuO4 were found…
We report measurements of Little-Parks oscillation on the hierarchical honeycomb-superconducting network for investigating possible effects of hierarchical structure in terms of spatial symmetry, parity and duality. We observed an asymmetric Little-Parks oscillation about $Φ/Φ_0 = 1/2$, although spatial symmetry was kept in the network. In comparison with a regular honeycomb network, the asymmetric oscillation is attributed to hierarchy which induces mixture of commensurate and incommensurate regions. The asymmetric oscillation is found to indicate breaking of the duality of vortex configurati…
The Little-Parks oscillations of the resistance and the quantum oscillations of the rectified voltage observed for asymmetric superconducting Al rings give experimental evidence of two degenerate energy states at the magnetic flux $Φ_{0}$/2. The quantum oscillations of the critical current as a function of magnetic field have also been measured. On the one hand, these oscillations confirm that the quantum oscillations of the rectified voltage are a consequence of periodical dependence of the asymmetry of the current-voltage curves and, on the other hand, comparison of the oscillations with Lit…
Half-quantum vortices in spin-triplet superconductors are predicted to host Majorana zero modes and may provide a viable platform for topological quantum computation. Recent works also suggested that, in thin mesoscopic rings, the superconducting pairing symmetry can be probed via Little-Parks-like magnetoresistance oscillations of periodicity $Φ_0 = h / 2e$ that persist below the critical temperature. Here we use the London limit of Ginzburg-Landau theory to study these magnetoresistance oscillations resulting from thermal vortex tunneling in spin-triplet superconducting rings. For a range of…
An inhomogeneous magnetic exchange field at a superconductor/ferromagnet interface converts spin-singlet Cooper pairs to a spin-aligned (i.e. spin-polarized) triplet state. Although the decay envelope of such triplet pairs within ferromagnetic materials is well studied, little is known about their decay in non-magnetic metals and superconductors, and in particular in the presence of spin-orbit coupling (SOC). Here we investigate devices in which triplet supercurrents are injected into the s-wave superconductor Nb. In the normal state of Nb, triplet supercurrents decay over a distance of 5 nm, …
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…
Recently, the ``$(p+h)-$wave'' form of pairing symmetry has been proposed for the superconductivity in $PrOs_{4}Sb_{12}$ compound [Parker D, Maki K and Haas S, \textbf{cond-mat/0407254}]. In the present paper, a stationary Josephson junction as a weak-link between $PrOs_{4}Sb_{12}$ triplet superconductors is theoretically investigated. The quasiclassical Eilenberger equations are analytically solved. The spin and charge current-phase diagrams are plotted and the effect of misorientation between crystals on the spin current, and spontaneous and Josephson currents is studied. It is found that su…
We study the Little-Parks effect for mesoscopic loops with very nonuniform thickness. The results follow the trend of the phase diagram obtained for almost uniform thickness. In particular, the singly-connected state is stable on a line segment delimited by two critical points. Most of this study considers loops with piecewise constant thickness; in this case the Euler-Lagrange equation can be integrated analytically. Under appropriate conditions, the temperature range where the singly-connected state is stable is proportional to the square of the ratio between the maximal and the minimal thic…
Nb films containing extended arrays of holes with 45-nm diameter and 100-nm spacing have been fabricated using anodized aluminum oxide (AAO) as substrate. Pronounced matching effects in the magnetization and Little-Parks oscillations of the superconducting critical temperature have been observed in fields up to 9 kOe. Flux pinning in the patterned samples is enhanced by two orders of magnitude as compared to unpatterned reference samples in applied fields exceeding 5 kOe. Matching effects are a dominant contribution to vortex pinning at temperatures as low as 4.2 K due to the extremely small s…
Measurements of the Little-Parks oscillations at measuring current much lower than the persistent current give unambiguous evidence of the dc current flowing against the force of the dc electric field because of the Aharonov-Bohm effect. This result can assume that an additional force is needed for description of the Aharonov-Bohm effect observed in semiconductor, normal metal and superconductor nanostructures in contrast to the experimental result obtained recently for the case of the two-slit interference experiment.
It was experimentally found that quantum magnetoresistive hc/2e periodic oscillations of the Little-Parks type in a superconducting mesoscopic ring with decreasing temperature and increasing applied dc current are modified to the sum of harmonic (hc/2e)/m periodic oscillations. Multiple Andreev reflection can be a possible cause of this effect.
We study the magnetic interaction between two superconducting concentric mesoscopic Al loops, close to the superconducting/normal phase transition. The phase boundary is measured resistively for the two-loop structure as well as for a reference single loop. In both systems Little-Parks oscillations, periodic in field are observed in the critical temperature Tc versus applied magnetic field H. In the Fourier spectrum of the Tc(H) oscillations, a weak 'low frequency' response shows up, which can be attributed to the inner loop supercurrent magnetic coupling to the flux of the outer loop. The amp…
Low frequency plasma oscillations in superconducting tubes are considered. The emergence of two different dimensionality regimes of plasma oscillations in tubes, exhibiting a crossover from one-dimensional to two-dimensional behavior, depending on whether $k R\ll 1$ or $k R\gg 1$, where $k$ is the plasmon wave vector and $R$ is the radius of the tube, is discussed. The Aharonov-Bohm effect pertaining to plasma oscillations in superconducting tubes and rings, resulting in an oscillatory behavior of the plasmon frequency as a function of the magnetic flux, with a flux quantum period $hc/2e$ (ana…
Field Theory Models for a Holographic Superconductor in Two Dimensions
University of Pisa（比萨大学）；INFN, Sezione di Pisa（意大利国家核物理研究所，比萨分部）；Università Cattolica del Sacro Cuore（天主教大学）；INFN Sezione di Perugia（意大利国家核物理研究所，佩鲁贾分部）
