3640 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007 Probing the Nonlinearities Arising in the Microwave Response of Superconductors by Intermodulation Distortion A. Andreone, G. Cifariello, E. Di Gennaro, G. Lamura, N. Emery, C. Hérold, J. F. Marêché, P. Lagrange, P. Orgiani, X. X. Xi, and J. C. Villégier Abstract—We present a systematic study of microwave nonlin- earity, undertaken on different types of superconductors (Nb, NbN, , ), in thin film form. Experiments are performed in a dielectrically loaded cavity operating at 7 GHz. The dependence of the surface impedance and of the third order in- termodulation (IMD) products on the power feeding the cavity is analysed, with the aim of shedding a light on the primary mech- anisms of nonlinearity. Data from different superconductors are quantitatively compared. Index Terms—Electrodynamics, intermodulation distortion, mi- crowave measurements, superconducting materials. I. INTRODUCTION T HE main limit to the performance of superconducting devices and systems operating in the GHz region is the nonlinear response to an applied r.f. field. This nonlinear be- havior, besides being characterized by an increase of the surface impedance , introduces harmonic generation and mixing of high frequency signals at neighboring frequencies. Usually, fre- quency transformation appears at much lower field values than change in . All these effects are detrimental for applications such as resonant cavities for particle accelerators or passive filters in the field of telecommunications, and should be fully understood. In fact, despite many efforts spent by a number of researchers worldwide in the last several years, the nature of microwave nonlinearities in superconductors is still under debate. In particular, it should be determined unambiguously Manuscript received August 29, 2006. This work was supported in part by MIUR under PRIN 2004 at the University of Naples “Federico II”, in part by the NSF under Grant DMR-0306746, and in part by the ONR under Grant N00014-00-1-0294 at Pennsylvania State University. A. Andreone, G. Cifariello, and E. Di Gennaro are with CNISM and also with the Department of Physics, University of Naples “Federico II,” 80125 Naples, Italy (e-mail: andreone@unina.it). G. Lamura is with CNR-INFM Coherentia and also with the Department of Physics, University of Naples “Federico II,” 80125 Naples, Italy. N. Emery, C. Hérold, J. F. Marêché, and P. Lagrange are with the Laboratoire de Chimie du Solide Minéral-UMR 7555, Université Henri Poincaré Nancy I, 54506 Vandœuvre-lès-Nancy Cedex, France. P. Orgiani was with the Department of Physics, Pennsylvania State Univer- sity, State College, PA 16802 USA. He is now with CNR-INFM SuperMat and also with the Department of Physics “E. R. Caianiello,” University of Salerno, 84081 Baronissi (Sa), Italy. X. X. Xi is with the Department of Physics, Pennsylvania State University, State College, PA 16802 USA. J. C. Villégier is with CEA-Grenoble SPSMS/LCP, F38054 Grenoble Cedex 9, France. Digital Object Identifier 10.1109/TASC.2007.900044 if and in which materials the intrinsic limit has been reached. A clear answer to this question, besides its importance for a better comprehension of the superconducting electrodynamics, is a preliminary requisite to understand if any further improve- ment in terms of performance is possible. The combination of nonlinear surface impedance and intermodulation distortion measurements is presently the most powerful probe to discern among different sources of dissipation. Nonlinearities can be roughly classified as having two possible origins: extrinsic, due to the presence of grains and grain boundaries [1], and intrinsic, because of the nonlinear Meissner effect [2], [3]. In the extrinsic case, the Josephson coupling between grains lowers the first vortex penetration field, thus increasing the level of nonlinearities [1], [4]. This effect is expected to not work for single crystals and epitaxial thin films, where nonlinearities might have only an intrinsic origin: the backflow of excited quasiparticles at finite temperatures well deep in the Meissner state. It has been shown [5] that at low temperatures and small fields the presence of quasiparticles determines the temperature behavior of the IMD amplitudes in a way that depends strictly on the symmetry of the gap function . This behavior can be affected also by the presence of multiple gaps, as in the case of magnesium diboride [6], and/or by gap anisotropies and strong coupling effects [7]. In particular, it has been found that the IMD power arising from the surface of a superconductor and being irradiated from a resonant structure can be described using the general relation [8], [9]: (1) valid for weak input signals only. Here (dB) is the insertion loss, is the loaded quality factor, is a factor depending on the resonator parameters (type of resonator, mode of excitation, power conversion factor, effec- tive area covered by the superconducting surface), and is the power circulating inside the cavity. The nonlinear contribu- tion due to the quasiparticle backflow is represented by the pa- rameter whose temperature behavior depends on the sym- metry of . is a factor related exclusively to the parameters of the material under test, strongly dependent for example on the zero temperature penetration depth [7], [9]. Dividing (1) by , the result becomes independent of the energy level of the electromagnetic excitation feeding the cavity. If one assumes that has the same dependence as the theoret- ical model, it is possible to rescale the experimental data on the theoretical curves. This procedure has been successfully used 1051-8223/$25.00 © 2007 IEEE