We study heat transport and thermoelectric effects in two-dimensional superconductors in a magnetic field. These are modeled as granular Josephson-junction arrays, forming either regular or random lattices. We employ two different models for the dynamics, relaxational model-A dynamics or resistively and capacitively shunted Josephson junction (RCSJ) dynamics. We derive expressions for the heat current in these models, which are then used in numerical simulations to calculate the heat conductivity and the Nernst coefficient for different temperatures and magnetic fields. At low temperatures and zero magnetic field the heat conductivity in the RCSJ model is calculated analytically from a spin wave approximation, and is seen to have an anomalous logarithmic dependence on the system size, and also to diverge in the completely overdamped limit C -> 0. From our simulations we find at low magnetic fields that the Nernst signal displays a characteristic "tilted hill" profile similar to experiments and a non-monotonic temperature dependence of the heat conductivity. We also investigate the effects of granularity and randomness, which become important for higher magnetic fields. In this regime geometric frustration strongly influences the results in both regular and random systems and leads to highly non-trivial magnetic field dependencies of the studied transport coefficients.
For almost a century thermoelectricity in superconductors has been one of the most intriguing topics of physics. At the early stages in the 1920s, the mere existence of thermoelectric effects in superconductors was questioned. Theoretical breakthrough came in the 1970s, when the generation of a measurable thermoelectric magnetic flux in superconducting loops was predicted; however a major crisis developed when experiments showed a gross discrepancy with the theory. Moreover, different experiments disagreed with each other by orders of magnitude. This led to a stalemate in bringing theory and e…
We report thermal transport measurement of KxFe2-ySe2 superconducting single crystal. Significant peak anomaly in thermal conductivity is observed at nearly TC/2 indicating a large phonon mean-free-path in the superconducting state. The zero-temperature extrapolated thermoelectric power is smaller than the value in typical strongly correlated superconductors, implying large normalized Fermi temperature. In contrast to other iron superconductors, thermoelectric power in our sample does not exhibit significant anomalies. These findings indicate that KxFe2-ySe2 is a weakly or intermediately corre…
We show that a huge thermoelectric effect can be observed by contacting a superconductor whose density of states is spin-split by a Zeeman field with a ferromagnet with a non-zero polarization. The resulting thermopower exceeds $k_B/e$ by a large factor, and the thermoelectric figure of merit $ZT$ can far exceed unity, leading to heat engine efficiencies close to the Carnot limit. We also show that spin-polarized currents can be generated in the superconductor by applying a temperature bias.
We consider a mesoscopic four-terminal superconductor/normal metal (S/N) structure in the presence of a temperature gradient along the N wire. A thermoemf arises in this system even in the absence of the thermoelectric quasiparticle current if the phase difference between the superconductors is not zero. We show that the thermoemf is not small in the case of a negligible Josephson coupling between two superconductors. It is also shown that the thermoelectric voltage has two maxima: one at a low temperature and another at a temperature close to the critical temperature. The obtained temperature…
It is widely believed that temperature gradient does not induce electric field in the superconductor and consequently that thermoelectric effects do not exist, or are negligible in these materials. This statement is correct only as far as effective electric field or gradient of the electrochemical potential is concerned. In normal metals temperature gradient generates effective electric field, which nulls out thermally induced diffusion current. In superconductor the diffusion current of quasiparticles is canceled a counterflow of supercurrent. Superconducting current induces the true electric…
We present analytic and numerical results for the thermoelectric effect in unconventional superconductors with a dilute random distribution of impurities, each scattering isotropically but with a phase shift intermediate between the Born and unitary limits. The thermoelectric response function has a linear temperature dependence at low temperatures, with a slope that depends on the impurity concentration and phase shift. Although the thermoelectric effect vanishes identically in the strict Born and unitary limits, even a small deviation of the phase shift from these limits leads to a large res…
We suggest a new type of an ultrasensitive detector of electromagnetic fields exploiting the giant thermoelectric effect recently found in superconductor/ferromagnet hybrid structures. Compared to other types of superconducting detectors where the detected signal is based on variations of the detector impedance, the thermoelectric detector has the advantage of requiring no external driving fields. This becomes especially relevant in multi-pixel detectors where the number of bias lines and the heating induced by them becomes an issue. We propose different material combinations to implement the …
We have calculated the thermopower of the Bi2Sr2CuO6 and Bi2Sr2CaCu2O8 superconductors using an ARPES-derived dispersion, with a model pseudogap, and a marginal-Fermi liquid scattering rate that has a minimum with respect to energy at the van Hove singularity (vHs). Good fits with data are achieved across the entire phase diagram, thus confirming the dispersions, the locations of the vHs and the dominance of the diffusion thermopower over the phonon drag contribution.
We demonstrate that superconductors which break time-reversal symmetry can exhibit thermoelectric properties, which are entirely different from the Ginzburg mechanism. As an example, we show that in the $s+is$ superconducting state there is a reversible contribution to thermally induced supercurrent, whose direction is not invariant under time-reversal operation. Moreover in contrast to Ginzburg's mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. The effect can be used to confirm or rule out the $s+is$ state, which is widely expected to be realized…
We show that superconductors with broken time-reversal symmetry have very specific magnetic and electric responses to inhomogeneous heating. A local heating of such superconductors induces a magnetic field with a profile that is sensitive to the presence of domain walls and crystalline anisotropy of superconducting states. A nonstationary heating process produces an electric field and a charge imbalance in different bands. These effects can be measured and used to distinguish $s+is$ and $s+id$ superconducting states in the candidate materials such as Ba$_{1-x}$K$_x$Fe$_2$As$_2$.
The superconductor-ferromagnet thermoelectric detector (SFTED) is a novel ultrasensitive radiation detector based on the giant thermoelectric effect in superconductorferromagnet tunnel junctions. We demonstrate analytical models and solutions in the time domain for a SFTED operated as a microcalorimeter (pulse excitation), in the linear smallsignal limit. Based on these solutions, the signal current and temperature pulse response were studied for two different electrical circuit models, providing design conditions for stable and non-oscillatory response.
We studied the effects of multiband electronic structure on the thermoelectric and electrical transport properties in the normal state of kagome superconductors $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs). In all three members, the multiband nature is manifested by sign changes in the temperature dependence of the Seebeck and Hall resistivity, together with sublinear response of the isothermal Nernst and Hall effects to external magnetic fields in the charge ordered state. Moreover, ambipolar transport effects appear ubiquitously in all three systems, giving rise to sizable Nernst signal. Finally, possib…
It is shown that the electric charge of vortices can result in a helical instability of straight vortex lines in layered superconductors, particularly Bi-based cuprates or organic superconductors. This instability may result in a phase transition to a uniformly twisted vortex state, which could be detected by torque magnetometry, neutron diffraction, electromagnetic or calorimetric measurements.
Thermoelectric effects result from the coupling of charge and heat transport, and can be used for thermometry, cooling and harvesting of thermal energy. The microscopic origin of thermoelectric effects is a broken electron-hole symmetry, which is usually quite small in metal structures, and vanishes at low temperatures. We report on a combined experimental and theoretical investigation of thermoelectric effects in superconductor/ferromagnet hybrid structures. We investigate the depencence of thermoelectric currents on the thermal excitation, as well as on the presence of a dc bias voltage acro…
We have extended Brandt's method for accurate, efficient calculations within Ginzburg-Landau theory for periodic vortex lattices at arbitrary mean induction to lattices of "doubly quantized" vortices.
We investigate the dynamics of magnetic vortices in type II superconductors with normal state pinning sites using the Ginzburg-Landau equations. Simulation results demonstrate hopping of vortices between pinning sites, influenced by external magnetic fields and external currents. The system is highly nonlinear and the vortices show complex nonlinear dynamical behaviour.
We report controlled local manipulation of single vortices by low temperature magnetic force microscope (MFM) in a thin film of superconducting Nb. We are able to position the vortices in arbitrary configurations and to measure the distribution of local depinning forces. This technique opens up new possibilities for the characterization and use of vortices in superconductors.
We report on the experimental observation of nonlocal spin-dependent thermoelectric effects in superconductor-ferromagnet multiterminal structures. Our samples consist of a thin superconducting aluminum wire with several ferromagnetic tunnel junctions attached to it. When a thermal excitation is applied to one of the junctions in the presence of a Zeeman splitting of the density of states of the superconductor, a thermoelectric current is observed in remote junctions at distances exceeding $10~\mathrm{μm}$. The results can be explained by recent theories of coupled spin and heat transport in h…
Observability of half-quantum vortices and skyrmions in p-wave superconductors is an outstanding open question. Under the most common conditions, fractional flux vortices are not thermodynamically stable in bulk samples. Here we show that in chiral p-wave superconductors, there is a regime where, in contrast lattices of integer flux vortices are not thermodynamically stable. Instead skyrmions made of spatially separated half-quantum vortices are the topological defects produced by an applied external field.
We derive the exact modified London equation for the two-gap superconductor, compare it with its single-gap counterpart. We show that the vortices in the two-gap superconductor are soft (or continuous) core vortices. In particular, we discuss the topological structure of the finite energy vortices (Abrikosov-like vortices), and find that they can be viewed as the incarnation of the baby skyrmion stretched in the third direction. Besides, we point out that the knot soliton in the two-gap superconductor is the twisted Abrikosov-like vortex with its two periodic ends connected smoothly. The relat…
We study the pinning of vortices in thin film superconductors by magnetic dots in the London approximation. A single dot is in general able to pin multiple field-induced vortices, up to a saturation number n_s, which can be much larger than one. However, the magnetic field of the dot also creates intrinsic vortices and anti-vortices, which must be accounted for. In a ferromagnetic dot array, the intrinsic anti-vortices are pinned only interstitially. Much stronger pinning effect is expected of an antiferromagnetic dot array. Possible realizations of various magnetic configurations are discusse…
We report on the experimental observation of thermoelectric currents in superconductor-ferromagnet tunnel junctions in high magnetic fields. The thermoelectric signals are due to a spin-dependent lifting of particle-hole symmetry, and are found to be in excellent agreement with recent theoretical predictions. The maximum Seebeck coefficient inferred from the data is about $-100~\mathrm{μV/K}$, much larger than commonly found in metallic structures. Our results directly prove the coupling of spin and heat transport in high-field superconductors.
We report on large thermoelectric effects in superconductor-ferromagnet tunnel junctions in proximity contact with the ferromagnetic insulator europium sulfide. The combination of a spin-splitting field and spin-polarized tunnel conductance in these systems breaks the electron-hole symmetry and leads to spin-dependent thermoelectric currents. We show that the exchange splitting induced by the europium sulfide boosts the thermoelectric effect in small applied fields and can therefore eliminate the need to apply large magnetic fields, which might otherwise impede applications in thermometry or c…
We study the effects of spin-splitting and spin-flip scattering in a superconductor (S) on the thermoelectric properties of a tunneling contact to a metallic ferromagnet (F) using the Green's function method. A giant thermopower has been theoretically predicted and experimentally observed in such structures. This is attributed to the spin-dependent particle-hole asymmetry in the tunneling density of states in the S/F heterostructure. Here, we evaluate the S density of states and thermopower for a range of temperatures, Zeeman-splitting, and spin-flip scattering. In contrast to the naive expect…
Studies of the copper-based superconductors demonstrate how their phase diagram becomes more complex as experimental probes improve, able to distinguish among subtly different electronic phases. One of those phases, nematicity, has become the matter of great interest also in the iron-based superconductors, where it is detected deep in the tetragonal state. Here we present the evolution of the in-plane Nernst effect anisotropy in the strain detwinned Ca(Fe1-xCox)2As2 single crystals, whose behaviour can be explained within the approach developed to describe the nematic order parameter in liquid…
In systems combining type-II superconductivity and magnetism the non-stationary magnetic field of moving Abrikosov vortices may excite spin waves, or magnons. This effect leads to the appearance of an additional damping force acting on the vortices. By solving the London and Landau-Lifshitz-Gilbert equations we calculate the magnetic moment induced force acting on vortices in ferromagnetic superconductors and superconductor/ferromagnet superlattices. If the vortices are driven by a dc force, magnon generation due to the Cherenkov resonance starts as the vortex velocity exceeds some threshold v…
Magnetic flux structures in single crystals of the layered spin triplet superconductor Sr$\_{2}$RuO$\_{4}$ are studied by scanning micro SQUID Force microscopy. Vortex chains appear as the applied field is tilted along the in-plane direction of the superconductor. The vortex chains align along the direction of the in-plane component of the applied magnetic field. The decoration of in-plane vortices by crossing Abrikosov vortices is observed: two vortex orientations are apparent simultaneously, one along the layers and the other perpendicular to the layers. The crossing vortices appear preferen…
We theoretically describe a thermoelectric effect that is entirely due to Andreev processes involving the formation of Cooper pairs through the coupling of electrons and holes. The Andreev thermoelectric effect can occur in ballistic ferromagnet-superconductor junctions with a dominant superconducting proximity effect on the ferromagnet, and it is very sensitive to surface states emerging in unconventional superconductors. We consider hybrid junctions in two and three dimensions to demonstrate that the thermoelectric current is always reversed in the presence of low-energy Andreev bound states…
We discuss the main features of Hall tunneling of pancake vortices in superclean high-$T_{c}$ superconductors. The general formalism for the calculation of the lifetime of a vortex pinned in a metastable configuration is described. The results are applied to the problem of quantum tunneling of a pancake vortex from a columnar defect in the limit of a small driving current.
By using the time-dependent Ginzburg-Landau theory, we show that extremely diverse experimental data on flux-flow resistivity in multiband superconductors can be qualitatively explained by a composite nature of Abrikosov vortices consisting of elementary fractional vortices in different bands. In composite vortices, the ratio of a core size to electric field relaxation length is found to vary in wide limits depending on system parameters. As a result, the flux-flow magneto-resistance can strongly exceed the single-component Bardeen-Stephen estimation provided that moving vortices generate elec…
We take account of normal currents that emerge when vortices move. Moving Abrikosov vortices in the bulk and Pearl vortices in thin films are considered. Velocity dependent distributions of both normal and persistent currents are studied in the frame of time-dependent London equations. In thin films near the Pearl vortex core, these distributions are intriguing in particular.
Polycrystalline sample of the new layered superconductor Bi4O4S3 is successfully synthesized by solid-state reaction method by using Bi, S and Bi2O3 powders with one step reaction. The superconducting transition temperature (Tconset=4.5 K), the zero resistance transition temperature (Tc0=4.07 K) and the diamagnetic transition temperature (4.02 K at H=10 Oe) were confirmed by electrical transport and magnetic measurements. Also, our results indicate a typical type II-superconductor behavior. In addition, a large thermoelectric effect was observed with a dimensionless thermoelectric figure of me…
We discuss the problem of broken time reversal symmetry near grain boundaries in a d-wave superconductor based on a Ginzburg-Landau theory. It is shown that such a state can lead to fractional vortices on the grain boundary. Both analytical and numerical results show the structure of this type of state.
We use large-scale Monte-Carlo simulations to study thermal fluctuations in chiral $p$-wave superconductors in an applied magnetic field in three dimensions. We consider the thermal stability of previously predicted unusual double-quanta flux-line lattice ground states in such superconductors. In previous works it was shown that, neglecting thermal fluctuations, a chiral $p$-wave superconductor forms an hexagonal lattice of doubly-quantized vortices, except extremely close to the vicinity of $H_{c2}$ where double-quanta vortices split apart. We find dissociation of double-quanta vortices drive…
We have measured temperature dependence of thermoelectric power (TEP) on MgB$_2$ superconductor under hydrostatic pressure. The sign and temperature dependence of TEP shows metallic hole carriers are dominant with activation type contribution at higher temperature. TEP increases with pressure while $T_c$ decrease with ratio -0.136 K/kbar. The data are discussed in consideration of carriers from different bands and anisotropy of compressibility.
A key prediction of the SO(5) theory is the antiferromagnetic vortex state. Recent neutron scattering experiment on LSCO superconductors revealed enhanced antiferromagnetic order in the vortex state. Here we review theoretical progress since the original proposal and present a theory of static and dynamic antiferromanetic vortices in LSCO superconductors. It is shown that the antiferromagnetic region induced by the vortices can be greater than the coherence length, due to the light effective mass of the dynamic antiferromagnetic fluctuations at optimal doping, and close promixity to the antife…
We demonstrate here real-time imaging of individual vortices in a NbSe2 single crystal using polarized light microscopy. A new high-sensitivity magneto-optical (MO) imaging system enables observation of the static vortex lattice as well as single vortex motion at low flux densities.
It is shown that the order parameter $Δ$ induced in the normal part of superconductor-normal-superconductor proximity system is modulated in the magnetic field differently from vortices in bulk superconductors. Whereas $Δ$ turns zero at vortex centers, the magnetic structure of these vortices differs from that of Abrikosov's.
We revise the applicability of the theory of self-organized criticality (SOC) to the process of magnetic relaxation in type-II superconductors. The driving parameter of self-organization of vortices is the energy barrier for flux creep and not the current density. The power spectrum of the magnetic noise due to vortex avalanches is calculated and is predicted to vary with time during relaxation.
The theory of thermoelectric effects in superconductors is discussed in connection to the recent publication by Marinescu and Overhauser, Phys. Rev. B 55, 11637 (1997). We argue that the charge on-conservation arguments by Marinescu and Overhauser do not require any revision of the Boltzmann transport equation in superconductors, and show that their ``electron-conserving transport theory'' contradicts thermodynamics. Possible mechanisms responsible for the discrepancy between some experimental data and the theory by Galperin, Gurevich, and Kozub, JETP Lett. 17, 476 (1973) are discussed.
We consider the one-dimensional (1D) topological superconductor that may form in a planar superconductor-metal-superconductor Josephson junction in which the metal is is subjected to spin orbit coupling and to an in-plane magnetic field. This 1D topological superconductor has been the subject of recent theoretical and experimental attention. We examine the effect of perpendicular magnetic field and a supercurrent driven across the junction on the position and structure of the Majorana zero modes that are associated with the topological superconductor. In particular, we show that under certain …
The intervortex interaction is investigated in very anisotropic layered superconductors in tilted magnetic field. In such a case, the crossing lattice of Abrikosov vortices (AVs) and Josephson vortices (JVs) appears. The interaction between pancakes vortices (PVs), forming the AVs, and JVs produces the deformation of the AV line. It is demonstrated that in the result of this deformation a long range attraction between AVs is induced. This phenomenon is responsible for the dense vortex chains formation. The vortex structure in weak perpendicular magnetic field is the vortex chain phase, while i…
The voltage shift RfIc with respect to the flux flow ohmic behavior of the current-voltage characteristic in type II superconductors is ascribed to the irreversible processes occurring when a vortex crosses defects. We include irreversible effects of the vortices-defects interaction into an effective law of motion. The obtained current-voltage characteristic at finite temperature is in agreement with experimental data.
In the framework of London theory we study the novel magnetic state in two-component superconductors with finite density of fractional flux vortices stabilized near the surface. We show that the process of vortex entry into the two-component superconductor consists of several steps, while the external magnetic field increases from zero. At the first stage only vortices in one of the order parameter components penetrate and sit at the equilibrium position near the surface. When the magnetic field is increased further vortices in the second order parameter component eventually enter the supercon…
The dynamics of magnetic vortices in the mixed state of d-wave superconductors is affected by interaction with quasiparticles near the gap nodes. We study this effect by computing the overlap of the ground state wave functions of the nodal quasiparticles in a two-dimensional d-wave superconductor as the ground state changes in response to the motion of the vortex. We find that the overlap is strongly suppressed. This orthogonality catastrophe is specific to a half-quantum vortex and disappears in the case of a doubly-quantized vortex. This implies strong inhibition of the motion of half-quantu…
Using the Bogoliubov de-Gennes formalism, we investigate the charge and spin-dependent thermoelectric effects in superconductor graphene junctions. Results demonstrate that despite normal-superconductor junctions, there is a temperature-dependent spin thermopower both in the graphene-based ferromagnetic-superconductor (F-S) and ferromagnetic-Rashba spin-orbit region-superconductor (F-RSO-S) junctions. It is also shown that in the presence of Rashba spin-orbit interaction, the charge and spin-dependent Seebeck coefficients can reach to their maximum up to 3.5$k_B/e$ and 2.5$k_B/e$, respectively…
Strains in superconductors due to moving vortices and vortex lattices are discussed. It is shown that the energy stored in elastic strains increases with vortex velocity. For moving vortex lattices, the elastic energy depends on velocity orientation relative to the lattice having minimum for the velocity directed along one of the unit cell vectors of the moving lattice. It is shown that for supersonic motion, the vortex induced stress field has a shape similar to supersonic shock waves.
Since the experimental formation of a vortex with a fractional quantized magnetic flux in a thin superconducting $s$-wave bi-layer was recently reported, we calculate the local density of states (LDOS) around two half quantum vortices in a $s$-wave superconductor on the basis of the quasiclassical Eilenberger theory to investigate the effect of the phase discontinuity. We show that the LDOS pattern around fractional vortices is qualitatively different from that around conventional vortex. We find that the novel bow-tie-shaped bound states appear in the high energy region, which can be an evide…
In multiband superconductors, each superconducting condensate supports vortices with fractional quantum flux. In the ground state, vortices in different bands are spatially bounded together to form a composite vortex, carrying one quantum flux Φ_0. Here we predict dissociation of the composite vortices lattice in the flux flow state due to the disparity of the vortex viscosity and flux of the vortex in different bands. For a small driving current, composite vortices start to deform, but the constituting vortices in different bands move with the same velocity. For a large current, composite vor…
Using numerical simulations we show that novel transport phenomena can occur for vortices moving in periodic pinning arrays when two external perpendicular ac drives are applied. In particular, we find a ratchet effect where the vortices can have a net dc drift even in the absence of a dc drive. This ratchet effect can occur for ac drives which create orbits that break one or more reflection symmetries.
We have investigated vortex states in chiral helimagnet/superconductor bilayer systems under an applied external magnetic field Happl, using the Ginzburg-Landau equations. Effect of the chiral helimagnet on the superconductor is taken as a magnetic field HCHM, which is perpendicular to the superconductor and oscillates spatially. For Happl = 0 and weak HCHM, there appear pairs of up-and down-vortices. Increasing Happl, down-vortices gradually disappear, and number of up-vortices increases in the large magnetic field region. Then, up-vortices form parallel, triangular, or square structures.
We study vortex clustering in type II Superconductors. We demonstrate that the ``second peak'' observed in magnetisation loops may be a dynamical effect associated with a density driven instability of the vortex system. At the microscopic level the instability shows up as the clustering of individual vortices at (rare) preferential regions of the pinning potential. In the limit of quasi-static ramping the instability is related to a phase transition in the equilibrium vortex system.
The quantization of magnetic flux in superconductors is usually seen as vortices penetrating the sample. While vortices are unstable in bulk type I superconductors, restricting the superconductor causes a variety of vortex structures to appear. We present a systematic study of giant vortex states in type I superconductors obtained by numerically solving the Ginzburg-Landau equations. The size of the vortices is seen to increase with decreasing film thickness. In type I superconductors, giant vortices appear at intermediate thicknesses but they do not form a well-defined vortex lattice. In the …
We use magnetic force microscopy to both image and manipulate individual vortex lines threading single crystalline YBa$_2$Cu$_3$O$_{6.4}$, a layered superconductor. We find that when we pull the top of a pinned vortex, it may not tilt smoothly. Sometimes, we observe a vortex to break into discrete segments that can be described as short stacks of pancake vortices, similar to the "kinked" structure proposed by Benkraouda and Clem. Quantitative analysis gives an estimate of the pinning force and the coupling between the stacks. Our measurements highlight the discrete nature of stacks of pancake …
We explore the role of proximity-induced odd-frequency pairing in the thermoelectricity of a ferromagnet when coupled to a conventional $s$-wave spin-singlet superconductor through a spin-active interface. By varying both the polarization and its direction in the ferromagnet and the interfacial spin-orbit interaction strength, we analyze the behavior of all proximity-induced pair amplitudes in the ferromagnet and their contributions to the thermoelectric coefficients. Based on our results for the Seebeck coefficient, we predict that odd-frequency spin-triplet Cooper pairs are much efficient th…
We study a current-biased 0-π-0 Josephson junction made by high-T_c superconductors, theoretically. When a length of the πjunction is large enough, this junction contains a vortex-antivortex pair at both ends of the πjunction. Magnetic flux carried by the vortices is calculated using the sine-Gordon equation. The result shows that the magnetic flux of the vortices is suppressed to zero as the distance between the vortices is reduced. By applying an external current, the orientation of the vortices is reversed, and a voltage pulse is generated. The current needed for this transition and generat…
Magnetic relaxation in a MgB$_2$ superconductor was measured. The temperature dependence of the normalized relaxation rate was determined for three different magnetic fields. By extrapolating these rates to T=0 K, we find that these extrapolations do not approach zero, indicating quantum tunneling of vortices in the superconductor. A quantum correction of the relaxation rate, followed by the correction of the magnetic moment, is proposed. Using the quantum correction, we find that $U_0$ increases with decreasing temperature and approaches a maximum at T=0.
