Abstract
The response of superconducting devices to electromagnetic radiation is a core concept implemented in diverse applications, ranging from the currently used voltage standard to single photon detectors in astronomy. Suprisingly, a sufficiently high power subgap radiation may stimulate superconductivity itself. The possibility of stimulating type II superconductors, in which the radiation may interact also with vortex cores, remains however unclear. Here we report on superconductivity enhanced by GHz radiation in type II superconducting Pb films in the presence of vortices. The stimulation effect is more clearly observed in the upper critical field and less pronounced in the critical temperature. The magnetic field dependence of the vortex related microwave losses in a film with periodic pinning reveals a reduced dissipation of mobile vortices in the stimulated regime due to a reduction of the core size. Results of numerical simulations support the validy of this conclusion. Our findings may have intriguing connections with holographic superconductors in which the possibility of stimulation is under current debate.
Introduction
Microwave (mw) irradiation has been used to control the quantum properties of different systems, from supercurrents in superconductors to mechanical oscillators^{1,2,3,4}. Using nonequilibrium pumping for cooling is currently a hot topic^{5,6}. In 1966 microwave stimulated superconductivity (MSSC) was discovered^{7} in superconducting bridges and later confirmed for different type I superconducting systems such as films^{8,9}, tunnel junctions^{10} or cylinders^{11,12}. This counterintuitive effect was explained by Eliashberg^{13} as a consequence of an irradiationinduced redistribution of quasiparticles away from the gap edge. Very recently MSSC has been observed in transient regimes (on ps time scales) in NbN films^{14} and was demonstrated to improve the quality factor of superconducting mw resonators^{15}.
In type II superconductors with magnetic field penetrating in form of quantized flux (vortices)^{16}, the reduced inelastic relaxation time could suppress or modify some signatures of MSSC. A renewed interest in the type II SCs is related with the proposal of holographic superconductors (HS)^{17}, mapping solutions of astrophysics problems to scalar condensates. Just like as type II SCs in solid state, HSs can exhibit vortex configurations^{18} and can be phenomenologically described by the time dependent GinzburgLandau equation (TDGL)^{19} in the proximity of the critical temperature. The possibility of stimulated SC in the HSs is under current debate^{20,21}. Clearly, the experimental verification of stimulated superconductivity in type II SCs in the vortex state could therefore have important implications both inside and outside condensed matter physics community, paving the way for further progress in modelling physics of black holes and gravity through HSs.
A periodic mw pump of sufficient amplitude induces the motion of vortices that results in dissipation^{22,23,24,25}. Though dynamics of vortices was extensively addressed^{26}, the possibility of MSSC in the vortex state is not fully understood. One can speculate that the energy balance in microwavedriven vortices should depend on a competition between frictioninduced heating of quasiparticles in the vortex cores^{27,28,29} and energy pumping outside the core at large vortex velocities^{30}. However, the full picture of nonlinear electromagnetic response of the vortex matter in the proximity to the critical temperature remains unsettled.
Our paper reports on broadband nonlinear response to mw radiation in the GHz range in type II superconducting Pb films. We observe experimentally MSSC in an enhancement of the critical temperature, while much larger effects are seen in the second critical field. In order to investigate stimulation under varying pinning strength (i.e. vortex motion amplitude), we have carried detailed studies of MSSC in the Pb films with periodic pinning centers (PbPCC) and with applied DC field inclined about 4° off the film plane. Such configuration has been chosen because the most pronounced effect has been observed with it. Besides, the small perpendicular field component H_{⊥} creates vortices, permitting us, by properly chosing the magnetic field intensity, to achieve the situations where the number of vortices is an integer multiple of the number of pinning centers (matching conditions). In that case, the vortex lattice rearranges itself in a specially stable manner, and vortex motion becomes restricted. Therefore, the commensurability between the vortex lattice and the periodic pinning centers facilitates the investigation of the complex vortex dynamics under periodically varying pinning conditions. The experiments unambiguously reveal a reduction of the dissipation of microwave driven mobile superconducting vortices at moderate frequencies and powers. Supported by TDGL simulations, we relate this unexpected behaviour to the reduction of the vortex core size at large vortex velocities, predicted by Larkin and Ovchinnikov (LO)^{30} and seen indirectly when the flux is driven with a DC current^{31}.
Results
Stimulation of critical temperature and upper critical field
A description of the samples and measurement details are provided in the Methods section and in the Supplementary material. Figure 1b shows the real and imaginary parts of the microwave permeability parameter U, defined in the Methods section, measured in PbPPC at small mw powers. Measurements are done in a temperature (T) sweep for different magnetic fields, at a fixed frequency (f) and are in accordance with the CoffeyClem model^{33}. The dependence of the response on the mw power (P) for the same PbPPC sample is seen in the contour plot for U′ in the plane P − T for magnetic field H = 0 and f = 6 GHz, (Fig. 1c). To characterize the shift of the transition as a function of P, we introduce an effective critical temperature, , (Supplementary Fig. S3), determined with an error around 1 mK. Figure 1d presents a 3D plot of in the coordinates P − H in the PbPPC at f = 6 GHz. One observes a nonmonotonic dependence of , increasing at small P and decreasing at large P. Though the does not have the meaning of critical temperature of the superconducting transition, the observed increase of can be interpreted as an indication of MSSC.
To investigate the nonlinear response in the vortex state as a function of pinning strength, we have carried out measurements of U′ and U″ as functions of H, P, f and T in PbPPC. Figure 2a shows a 3D plot of U′ for the PbPCC sample at T = 7.19 K in the coordinates H − P. Dark red tones correspond to the normal state. As H is lowered, the samples become superconducting and the magnetic permeability changes in agreement with expectations^{33}. Figure 2b shows a typical set of crosssections of U′ at fixed magnetic fields. While mw power above 5 dBm destroys superconductivity, for intermediate values the superconducting response is the most intense. We will refer to the applied P that yields the strongest superconducting response as “optimum power” (P_{O}). The dashed line indicates how P_{O} changes with H. The response (inset of Fig. 2b) shows directly a transition between linear to nonlinear vortex response regimes. At lowest powers it attenuates and is noisier, since the relative noise is larger compared to the weaker signal received in port 2 of the network analyzer.
Similarly to , we introduce an effective critical magnetic field, . Figure 2c shows a plot of in the PbPPC sample (see Suplementary material for the method of finding , and the results for the plain film). For each temperature, has been normalized by its value at the lowest P, so relative values of can be compared. As T → T_{c}, the relative increase of under mw at P = P_{O} becomes larger. Figure 2d compares the normalized as a function of reduced temperature in both types of samples (with and without pinning centers), showing that MSSC effects in are stronger in the PbPPC sample. Figure 3a shows that P = P_{O} increases with mw frequency with a maximum value that saturates around 15 GHz. These frequencies are well below those corresponding to the superconducting gap.
Reduced dissipation of microwave driven mobile vortex
The changes in P_{O} indicate that the “cooling” effectiveness of mw radiation depends on the pinning strength through the applied field which changes the number of vortices per pinning center and correspondingly their mobility. In Fig. 3b this fact is exposed for f = 6 GHz, at different values of T. Solid lines show P_{O} vs. magnetic field with matching conditions indicated by vertical dotted lines. Red circles represent a typical measurement of U′(f, P, H) at fixed values of f and P, in which the same matching anomalies appear in the mw permeability. The lowest value of P_{O} for every temperature is found always at zero field, and decreases locally in matching conditions. This hints the relevance of the vortex mobility for the value of P_{O}: the enhanced vortex mobility out of matching conditions provides relatively larger (respect to matching) P_{O} values and correspondingly larger “cooling” efficiency.
This counterintuitive result has been corroborated through a set of independent experiments investigating the H and P dependencies of U″. At low f, when MSSC is not yet pronounced, the dissipation (U″) at matching conditions shows (as expected) dips in a broad range of P (Supplementary Fig. 7). However, for higher f the dips of losses at matching conditions convert into peaks. The same effect can be observed as a function of power, in Fig. 3c. In other words, vortices moving with higher average velocities out of matching conditions manage to dissipate less than pinned in matching conditions. These observations indicate a qualitative change in the microwave response of superconducting vortices at high mw frequencies.
Discussion
Mechanism of nonlinear vortex response and modelling
Mechanisms of nonlinear response of vortices to microwave radiation are far from being fully understood. LO theory^{30} predicts a nonlinear response at sufficiently large electric fields, that induces a high speed DC motion of the vortices. If their speed exceeds some critical value, v_{c}, much lower than the critical velocity for breaking Cooper pairs, the current decreases with increasing voltage. This is a consequence of an electronic instability of the nonequilibrium distribution of quasiparticles at high velocities, leading to a reduction of the vortex core size. A further increase of vortex velocity leads to an abrupt switching into a state with higher electric resistivity. On the other hand, the nonequilibrium quasiparticle distribution close to the energy gap where the density of states is maximal can cause stimulation of superconductivity^{13}. One can anticipate an interplay between the above mechanisms in a mixed state of mw driven typeII superconductors. However, we are not aware of a theory quantitatively interpreting our experimental results.
To understand the nonlinear response of mw driven vortices we have simulated the ac response of vortices using the time dependent GinzburgLandau equations (see Suplementary material for details). The time derivative of the order parameter modulus Ψ shows that vortices oscillate about their equilibrium positions, especially at lower frequencies, when they are able to follow the external ac field without delay (Supplementary Fig. S9 b). As the ac field amplitude increases (which is equivalent to increasing the mw power of our measurements) the order parameter oscillates throughout the entire sample, being weaker at the maxima of amplitude of mw field h_{rf}, as expected. This effect is specially pronounced in the outer part of vortex cores, as can be seen in Fig. 4b, c. When one compares the radius of a vortex (at a given value of, for example, Ψ = 0.5, see Fig. 4c) for different moments of an oscillation period, vortices are narrower at zero ac field amplitude than at its maximum ac. As happens in vortex cores displacement, the higher the frequency of the ac field, the more difficult is for a vortex core radius to change size (see Fig. 4d). A transition from linear to nonlinear response regimes is observed, leading to a substantial reduction of the average vortex size at high mw drives as a function of f. The reason for the different ranges of frequencies considered in the experiment and simulation are commented in the methods section.
The oscillations of the vortex core size under mw radiation are always present, and more notorious for higher ac field amplitudes, which is in qualitative agreement with the DC model by LO^{30}. The above confirmation of LOtype mechanism in ac conditions agrees with simulations of DC driven vortices^{34} and does not exclude electron overheating in the vortex core as an additional factor contributing anomalous velocity dependence of vortex viscosity^{29}.
The vortex velocity can be limited by the critical value v_{c} for the LO instability^{30,31,35}. Assuming that the maximum mwinduced shift of a vortex is limited by the interdot distance and that the dependence of P_{O} on f starts to saturate at we estimate the geometrically restricted maximal vortex velocity as v_{max} = f_{sat} · a ≈ 6–10 km/s. This is 2–3 times larger than the values of v_{c} reported for Nb and highT_{c} superconducting films^{25,36}.
Summary and conclusions
The experimental observation of stimulated superconductivity in type II superconductors has been used to quantify relative changes in vortex dissipation as a function of mobility (pinning). At high enough mw power and/or low enough mw frequencies (Fig. 3c) when MSSC is not effective, the vortex matching effects are clearly observed as periodic dips in the mw losses when H_{⊥} = nΦ_{0}/a^{2} (n is an integer number and Φ_{0} the magnetic flux quantum). In contrast to that, in a broad range of mw powers (sufficiently below limiting values which heat the sample) and at high enough frequencies (above about 0.6 GHz) mobile (offmatching) vortices dissipate less than pinned vortices. One clearly observes peaks in the vortex dissipation in matching conditions. The higher the frequency, the broader the mw power range where matching anomalies are seen as peaks in losses. Microwave stimulation changes from dips at matching fields at the lowest frequencies to peaks at frequencies exceeding a few GHz, in agreement with TDGL simulations, that indicate a transition to a nonlinear regime when mobile (interstitial) vortices dissipate less than pinned ones. The observed effects (transition from peaks to dips) remain qualitatively unchanged for the range up to 3 vortices per pinning center, meaning that intervortex interaction has a weak influence on our resuts. The supplementary video 1 shows a simulation of the vortex response to a mw magnetic field. One clearly observes the changes of the vortex core radius and (through differential analysis of the modulus of the order parameter) the vortices motion. We find vortex deformation to be minimum because its displacement at mw frequencies is small in comparison with radius oscillations.
We point out that stray fields of Py dots do not play an essential role for the effects we observe. Unlike previous simulations^{37}, our dots are in the magnetic vortex state with minimum stray fields^{38} and with ferromagnetic resonance (FMR) suppressed^{39}, i.e., the dots are not saturated. Besides, there should be a strong structural pinning profile due to the fact that the SC film covers the array of dots and not vice versa as in Refs. 37. A qualitative similarity in the microwave losses measured with perpendicular or with inclined nearly parallel magnetic fields (Supplementary Fig. 8) further confirms that the observed effects are not induced by the presence of an inplain component of magnetic field.
In conclusion, we have observed experimental signatures of stimulated superconductivity in type II superconductors in the vortex state including an enhancement of the upper critical fields and a somewhat less noticable increase of the critical temperature. Moreover, we have found experimentally and supported by simulations the unique fingerprint of MSSC in vortex dynamics the reduced dissipation of microwave driven vortices due to a reduction of the vortex core size. Besides significance for condensed matter physics, our results may have implications for the current controversy on the possibility of stimulated superconductivity in holographic superconductors^{20,21}.
Methods
We investigated two types of samples: plain 60 nm thick Pb films and 60 nm thick Pb films deposited over a square array of periodic pinning centers (PbPPC), consisting of circular Py dots (see Suplementary material for further details). All figures (except Fig. 2d and Supplementary Fig. S5) refer to the PbPPC sample.
The broadband measurements were done with a Vector Network Analyzer (VNA) connected to a coplanar waveguide (CPW) situated inside a cryostat with a superconducting magnet (see Suplementary material for details). The VNA signal excites the sample, placed on the CPW (Fig. 1a). The complex mw permeability, U ≡ U′ + iU″, is determined as the VNA transmission parameter S_{21}, dependent on microwave power (P), frequency (f), temperature (T) and magnetic field (H), normalized by S_{21} at a reference H or T, in the normal state (Suplementary material for more details). Experimental figures corresponds to the estimated values of power waves travelling through the waveguide, but not absorbed by the vortex system.
To understand better the individual behavior of superconducting vortices under the influence of an in plane ac magnetic field, we have simulated the TDGL equation in 3D. Simulations allow to include a DC field perpendicular to the film to create vortices, and a sinusoidal field parallel to the plane that represents the mw field generated by the CPW. Both field components are introduced through the appropriate boundary conditions (see Suplementary material). Our simulations are based on the finite difference approach used several times in the past in 2D (see for example^{32}). The GinzburgLandau parameter used is κ = 2. The temperature has been fixed far from T_{c} = 7.2 K (T = 4 K) because vortices are better observed, being the results obtained still valid (although less vissible) at higher temperatures. A mismatch between the frequency range presented in the measurements and that of the simulations is due to the absence of precise knowledge of the characteristic time scales of the normal and superconducting parts of the GL simulation. We use our simulation just for qualitative confirmation of existence of LO mechanism for the microwave driven vortex. Future work could also try to analyze numerically possible coupled magnetic dotsuperconducting vortex dynamics. This task, however, presents great challenge because of the need to include dynamics of magnetic pinning centers.
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Acknowledgements
Authors gratefully acknowledge Yu. Galperin, T.M. Klapwijk and V. Vinokur for insightful discussions and A. Awad for experimental help on the initial stages. This work has been supported in parts by Spanish MINECO (MAT201232743), and Comunidad de Madrid (NANOFRONTMAGCM S2013/MIT2850) and NANOSC COSTAction MP1201. A. Lara thanks UAM for FPIUAM fellowship. Authors thank CCCUAM (SVORTEX) for computational capabilities. The work of A.V.S. was partially supported by Mandat “d'Impulsion Scientifique” of the F.R.S.FNRS.
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Affiliations
Dpto. Física Materia Condensada C03, Instituto Nicolas Cabrera (INC), Condensed Matter Physics Institute (IFIMAC), Universidad Autoenoma de Madrid, 28049, Madrid, Spain
 Antonio Lara
 & Farkhad G. Aliev
INPAC Katholieke Universiteit Leuven, Celestijnenlaan 200D, B3001, Leuven, Belgium and Departement de Physique, Université de Liège, B4000 Sart Tilman, Belgium
 Alejandro V. Silhanek
INPAC Katholieke Universiteit Leuven, Celestijnenlaan 200D, B3001, Leuven, Belgium
 Victor V. Moshchalkov
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Contributions
All authors discussed the results and commented on the manuscript. F.G.A. managed the project, A.L. performed microwave measurements and simulations, A.S. grew and characterized the samples, F.G.A. and A.L. wrote the manuscript with input from A.S. and V.V.M.
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The authors declare no competing financial interests.
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Correspondence to Farkhad G. Aliev.
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