New aspects of microwave properties of Nb in the mixed state
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t New aspects of microwave properties of Nb in the mixed state
N. Pompeo a , ∗ , E. Silva a , b , S. Sarti c , C. Attanasio d , C. Cirillo d a Dipartimento di Fisica “E. Amaldi” and Unit`a CNISM, Universit`a Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy b Laboratorio Regionale SuperMat, CNR-INFM Salerno, I-84081 Baronissi, Italy c Dipartimento di Fisica, Universit`a ”La Sapienza”, 00185 Roma, Italy d Laboratorio Regionale SuperMat, CNR-INFM Salerno and Dipartimento di Fisica “E. R. Caianiello”, Universit`a degli Studidi Salerno, I-84081 Baronissi, Salerno, Italy
Abstract
We present a study of the frequency dependence of the vortex dynamics in a conventional superconductor. We haveemployed a swept-frequency, Corbino-disk technique to investigate the temperature (3.6K- T c ) and high-field (from H c / H c ) microwave complex resistivity in Nb thin (20-40 nm) films as a function of the frequency (1-20 GHz).We have found several previously unnoticed features: (i) a field-dependent depinning frequency in the GHz range; (ii)deviations from the accepted frequency dependence [1], that can be ascribed to some kind of vortex creep; (iii) thepresence of switching phenomena, reminiscent of vortex instabilities. We discuss the possible origin of the featureshere reported. Key words:
Nb, Corbino disk, surface impedance, vortex dynamics
PACS:
1. Introduction
The microwave response of superconductors in themixed state is a particularly suitable probe to in-vestigate the short-range vortex dynamics. Histor-ically, vortices in conventional superconductors arethought to follow the well-established Gittleman-Rosenblum predictions [1], based on single-vortexresponse to an alternating current without thermaleffects. Surprisingly, there are only a few experi-mental reports on this behaviour, in particular asa function of frequency, temperature and magneticfield. The advancement on knowledge about highfrequency vortex dynamics in the recent years [2],triggered by the discovery of high- T c superconduc-tors, and the revamped interest in conventional su- ∗ Corresponding author.
Email address: [email protected] (N. Pompeo). perconductors as potential components of devicesbased on superconductor/ferromagnet multilayers[3], stimulates a direct, careful experimental deter-mination of the vortex dynamics in conventional su-perconductors.When vortices are set in motion by time-varyingcurrents, they experience heterogeneous forceswhich include a viscous drag force, phenomenologi-cal representation of vortex motion-induced powerdissipation, pinning forces, arising from interactionwith material defects and hindering vortex move-ment, and stochastic forces of thermal origin whichpromote vortex detachment from pinning sites.The electrodynamic high frequency response aris-ing from the interaction of the fluxon system withmicrowave currents has been modeled by many au-thors [1,2,4,5]. On very general grounds, all modelscan be represented through a universal expressionfor the complex vortex resistivity [6]:
Preprint submitted to Elsevier 25 August 2018 vm = ρ vm, + i ρ vm, = ρ ff ε + i ( ν/ν )1 + i ( ν/ν ) (1)where the resistivity ρ ff represents the free fluxflow value, reached in the high frequency limit inwhich vortices experiences only the dissipative vis-cous drag, ν is a characteristic frequency and thedimensionless parameter 0 ≤ ε ≤ ε = [ I ( u )] − and ν = ν p (1 − ε ) − ( I ( u ) /I ( u )).Here, u = U ( T, B ) / (2 K B T ) is the normalized en-ergy barrier height within the assumption of a uni-form periodic pinning potential of height U , ν p isthe so-called pinning frequency, which marks thecrossover between the vortex pinned motion andthe purely dissipative motion, arising at ν ≫ ν p , I n is the n -order modified Bessel functions of thefirst kind. The CC model reverts to the simplerGittleman and Rosemblum (GR) expression ρ vm = ρ ff (1 + i ( ν p /ν )) / (cid:16) ν p /ν ) (cid:17) [1] for vanishingcreep ( ε → ν → ν p . In the latter, thepinning frequency can be immediately derived as ν p /ν = ρ vm, /ρ vm, .The exact determination of the three vortex pa-rameters comes directly from comparison to themeasured frequency dependence, while only esti-mates can be obtained from single-frequency mea-surements resorting to an accurate analysis [6].Neglecting the weight of some of the parametersin order to simplify the analysis may significantlyaffect the estimates of the parameters.In the following, we illustrate microwave measure-ments performed on Nb thin films by means of theCorbino disk technique, which allows wide band,frequency dependent determinations of the complexresistivity.
2. Experimental results and discussion
We measure the swept frequency ( ν =1-20 GHz)microwave response of Nb samples, at low and con-stant input power ( < . ≤ T ≤ T c (stability < .
001 K)through a Helium flow cryostat. A static magneticfield H c / ≤ H ≤ H c is applied perpendicularlyto the sample surface.The studied Nb thin films have thicknesses d =20-40 nm and were grown on 0.5 mm thick, 5 × square Al O substrates in a ultra high vacuumdc diode magnetron sputtering system (10 − mbar base pressure; 10 − mbar sputtering Argon pres-sure). The fabrication was realized at room tempera-ture. The deposition rate of 0.3 nm/s was controlledby a quartz crystal monitor calibrated by low-angleX-Ray reflectivity measurements performed usinga Philips X-Pert MRD high resolution diffractome-ter. The sample thickness was determined by fittingthe reflectivity profile of the sample with a simula-tion curve obtained following the Parrat and Nevot–Croce formalism [7,8]. The fit revealed the forma-tion of a Nb O oxide layer of the order of 2–3 nmon the top of film which causes a reduction of theeffective Nb layer thickness. This in turn results in adepression of T c , which is strongly thickness depen-dent for low thickness values such as ours [9]. Fourcontact measurements of the dc resistivity on a typi-cal sample 20 nm thick, similar to the one here stud-ied, yielded a constant normal state ρ n =22 µ Ωcmfor temperatures T c < T <
30 K. The critical tem-perature T c = 6 .
37 K, evaluated as the midpoint ofthe resistive transition.The microwave response is measured througha Corbino disk [10] setup: a swept frequency mi-crowave radiation, generated by a Vector NetworkAnalyser (VNA) system, is fed in a coaxial cableshort-circuited on the sample. We use a launcherwith a custom-made, spring loaded central pin inorder to ensure good contact with the Nb film.We measured the (complex) reflection coefficientΓ( ν ) of the electromagnetic wave impinging uponthe sample surface. The modulus | Γ | , boundedwithin [0,1], is a good indication of power dissipa-tion of the (super)conducting sample. In particular,a vanishingly low dissipation (as in the Meissnerstate) would yield | Γ | = 1, whereas by increasingdissipation | Γ | would decrease below 1. A quantita-tive analysis requires the extraction of the complexeffective surface impedance Z eff ( ν ) of the sampleaccording to the well-known expression [11]: Z eff ( ν ) = Z ν )1 − Γ( ν ) (2)where Z is the characteristic impedance of the coax-ial line. The small thickness of our samples allows theuse of the thin film approximation in which Z eff =˜ ρ/d , where ˜ ρ is the complex resistivity of the su-perconductor. We did not observe, in the [ T , H , ν ]region here explored, resonances such as the wellknown substrate resonances occurring in thin films[12].The actually measured reflection coefficientΓ m ( ν ), determined by the VNA at its input, include2he sample response Γ( ν ) as well as the interposedcoaxial line response. The calibration of the linecontribution is a critical issue, as described in Ref.[10]. Here, we used as reference the normal state sothat all results are presented as normalized resistiv-ities ˜ ρ/ρ n .In the following we consider a Nb sample 20 nmthick (Nb20). Raw data are presented in Fig. 1 todemonstrate the effect of a magnetic field and of themeasuring frequency on the superconducting tran-sition. The black circles correspond to single fre- | G | Fig. 1. | Γ | vs T at selected fields and frequencies, sampleNb20. Circles: ν = 5 GHz; squares: ν = 17 GHz. quency data taken at ν = 5 GHz and µ H = 0 Tat various T . A finite width of the transition is dueto finite quasiparticle contribution that can be ob-served at high frequency.The application of a magnetic field (white cir-cles in Fig. 1) determines an evident and progres-sive broadening of the transition, signature of theincreased dissipation caused by the motion of thevortices penetrated in the sample.In a magnetic field, dissipation increases at largerfrequency ( ν = 17 GHz, squares in Fig. 1), directlyshowing that the vortex characteristic frequency liesin our measuring frequency range.If, as customarily performed, we rely on single fre-quency measurements and derive the characteristicfrequency neglecting creep (GR model, Ref. [1]), we obtain different values by using data at differentfrequencies: 2.1 GHz using data at ν , and 3.8 GHz ν p can be computed directly from the complex Γ,given the straightforward derivation ρ vm, ( ν )) /ρ vm, =2 | Γ | sin( φ ) / (1 − | Γ | ) where φ = arg(Γ). using data at ν . Clearly, such a procedure is unsuit-able, and wide band measurements are required.A sample measurement in the full 1-20 GHz rangeis reported in Fig. 2, taken at T = 3 .
66 K and µ H =1 . n (GHz) r / r n , Dr / r n T=3.66 K m H=1.3 T
Fig. 2. Normalized ρ and ∆ ρ (black and grey symbols,respectively) vs ν , sample Nb20. Thin continuous line: fitaccording to CC model (see text). of switching-like phenomena appearing simultane-ously on both the real, ρ , and imaginary, ∆ ρ = ρ ( H ) − ρ (0), parts of the complex resistivity. Mostlikely the phenomenon originates from vortex insta-bilities, as we will discuss in the following. Here weonly stress that it can be thought as superimposingto an underlying, unperturbed response, representedby the lower/upper envelopes of the real and imag-inary parts, respectively, of the measured ˜ ρ . Themeasured (unperturbed) ρ + i∆ ρ shows no signsof quasi-particle/superfluid pairbreaking contribu-tions (as can be expected not too close to B c ( T )): ρ increases and then saturates at high ν whereas∆ ρ presents a maximum. These features are clearsignatures of single vortex dynamics, so that it canbe safely taken ρ + i∆ ρ = ρ vm .Therefore, a fit of ρ +i∆ ρ can be done by follow-ing the CC model. Fit results are reported as thincontinuous lines in Fig. 2: their quality is remark-able. The obtained fitting parameters are ρ ff /ρ n =0 . ν p = 7 . ε = 0 . ν p dif-fers significantly from the simple GR estimate previ-ously calculated. This fact is connected to the pres-ence of significant creep processes ( ε = 0 .
41 upona theoretical maximum value of 1), which evidently3xcludes the use of simple, GR-like models. This isan important result of this work: we found that sig-nificant creep is present even at low (3.66 K) temper-atures. It is worth stressing again that these resultsare made possible by the use of wide band measure-ments, which demonstrates to be essential for thecomprehensive study of vortex dynamics.Moreover, a reliable determination of vortex pa-rameters enables also any successive comment ontheir eventual field dependence. By increasing thefield, we observe an increase of creep and a decreaseof ν p , which can be explained by a weakening of vor-tex pinning by approaching the H c ( T ) line.Finally, the obtained ρ ff yields, within theBardeen Stephen model ( ρ ff = ρ n B/B c ) [14], avalue of B c ∼ . B c ∼ . B ≈ µ H ), deter-mined as the field at which the superconductingsignal vanishes within the sensitivity of our system.We now come back to the switching phenomenon.This is the second main result of this work. In-deed, to the best of our knowledge, this is the firsttime that similar phenomena are observed in themicrowave range. Our whole set of measurementsshows that this phenomenon appears for H & H c / H → H c . Moreover,switches occur upward on ρ and downward for ρ :this feature points to commutations of the vortexsystem forth to and back from a higher dissipationstate. In order to get additional insight, we per-formed continuous wave, fixed frequencies measure-ments. A sample of them is reported in Fig. 3, interms of (uncalibrated) | Γ m | measured as a functionof time in a sample = 30 nm thick ( T c ∼ . G m | T=7.26 K n =3 GHz Fig. 3. | Γ m | vs time in residual field µ H ∼
30 mT, Nbsample 30 nm thick. time-dependent nature and the pseudo-periodicityof the switches is clearly apparent, as well as the“large” time scales involved (pseudo-period=1.2–1.5 s). Continuous wave data also show no apparent de-pendence on both the frequency and the power ofthe microwave stimulus. Finally, the phenomenonhas been observed in all the samples and resultedindependent from the instrumentation used.Although the origin of this switching is at the mo-ment unclear, two main factors are relevant. First,the noted correlation to the vicinity of the H c ( T )line suggests some form of thermally-driven, fluc-tuation enhanced metastability involving the vor-tex system [15]. Second, the Corbino disk geometrylikely plays a role, since the strong velocity gradi-ents it induces on the fluxon lattice are consideredthe source of peculiar and unconventional vortex dy-namics regimes [16,17].
3. Summary
We have studied the frequency dependence of thevortex dynamics in Nb thin films at different tem-peratures and magnetic fields in the range 1-20 GHzthrough the Corbino disk technique. The resultingsingle vortex dynamics was well modelized withinframeworks such the Coffey Clem one. In particular,all the three main vortex parameters were clearlydetermined, showing the presence of significantcreep contribution even at our lowest (3.66 K) tem-peratures. In addition we observed time-dependentswitching phenomena likely related to metastabili-ties involving the fluxon system. This phenomenonis probably linked to thermal and fluctuation effectsinvolving the vortex system, as well as to the pecu-liar fluxon velocity pattern imposed by the Corbinodisk geometry. Further studies will be necessary toascertain the exact nature of the phenomenon.This work has been partially supported by an Ital-ian MIUR-PRIN 2007 project.References [1] J. Gittleman and B. Rosenblum, Phys. Rev. Lett. 16(1966) 734.[2] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67 (1991)386.[3] L. R. Tagirov, Phys. Rev. Lett. 83 (1999) 2058; B. L.Ioffe et al., Nature 398 (1999) 697; V. V. Ryazanov etal., Phys. Rev. Lett. 86 (2001) 2427.[4] E. H. Brandt, Phys. Rev. Lett. 67 (1991) 2219.[5] T. Hocquet et al., Phys. Rev. B 46 (1992) 1061; B.Placais et al., Phys. Rev. B 54 (1996) 13083.
6] N. Pompeo and E. Silva, Phys. Rev. B 78 (2008) 094503.[7] L. G. Parrat, Phys. Rev. 95 (1954) 359.[8] L. Nevot and P. Croce, Rev. Phys. Appl. 15 (1980) 761.[9] C. Cirillo et al., Phys. Rev. B 72 (2005) 144511.[10] S. Sarti, C. Amabile and E. Silva,arXiv:cond-mat/0406313 (2004).[11] R. E. Collin
Foundation for Microwave Engineering ,McGraw-Hill International Editions (1992)[12] E. Silva, M. Lanucara, and R. Marcon, Physica C 276(1997) 84; N. Pompeo et al., Supercond. Sci. Technol.20 (2007) 1002.[13] E. Silva, M. Lanucara, R. Marcon, Supercond. Sci.Technol. 9 (1996) 934.[14] J. Bardeen and M. J. Stephen, Phys. Rev. 140 (1965)A1197.[15] S. Okuma, S. Morishima, and M. Kamada, Phys. Rev.B 76 (2007) 224521.[16] D. L´opez et al., Phys. Rev. Lett. 82 (1999) 1277.[17] N. S. Lin, V. R. Misko, and F. M. Peeters, Phys. Rev.Lett. 102 (2009) 197003.,McGraw-Hill International Editions (1992)[12] E. Silva, M. Lanucara, and R. Marcon, Physica C 276(1997) 84; N. Pompeo et al., Supercond. Sci. Technol.20 (2007) 1002.[13] E. Silva, M. Lanucara, R. Marcon, Supercond. Sci.Technol. 9 (1996) 934.[14] J. Bardeen and M. J. Stephen, Phys. Rev. 140 (1965)A1197.[15] S. Okuma, S. Morishima, and M. Kamada, Phys. Rev.B 76 (2007) 224521.[16] D. L´opez et al., Phys. Rev. Lett. 82 (1999) 1277.[17] N. S. Lin, V. R. Misko, and F. M. Peeters, Phys. Rev.Lett. 102 (2009) 197003.