1 Observation of Bifurcations and Hysteresis in Nonlinear NbN Superconducting Microwave Resonators 5 0 0 2 Baleegh Abdo, Student Member, IEEE, Eran Segev, Oleg Shtempluck, and Eyal Buks n a J 0 Abstract—In this paper we report some extraordinary non- 1 linear dynamics measured in the resonance curve of NbN superconducting stripline microwave resonators. Among the ] n nonlinearities observed: aburpt bifurcations in the resonance o response at relatively low input powers, asymmetric resonances, c multiple jumps within the resonance band, resonance frequency - drift, frequency hysteresis, hysteresis loops changing direction r p and critical couplingphenomenon.Weak linksin theNbN grain u structure are hypothesized as the source of the nonlinearities. s IndexTerms—Bifurcations,nonlineareffects,hysteresis,NbN, Fig.1. Stripline geometry. . t microwave resonators. a m - I. INTRODUCTION YBCO stripline resonator, and by [16] in their YBCO thin d film dielectric cavity. All three studies attributed the observed n NONLINEAReffectsinsuperconductorsinthemicrowave nonlinear behavior to abrupt changes in the resistive loss of o regime have been the subject of a large number of c weak links, thermal quenching and weak link switching to intensive studies in recent years. Most of the attention is [ normal state. focused on studying one or more of the following issues: in- In this study, being interested in the behavior of nonlinear 2 vestigatingthe originsof nonlineareffects in superconductors v resonances, we have fabricated different NbN superconduct- [1],[2], introducing theoretical models that explain nonlinear 4 ing microwave resonators exhibiting some unusual nonlinear 1 behavior [3],[4], identifying the dominant factors that mani- effects, which to the best of our knowledge, have not been 1 fest these effects [5],[6], find ways to control and minimize reported before in the literature. We study the dependence of 1 nonlinear effects [7],[8] such as, harmonic generation and theseresonatorsontheinjectedinputpowerlevel,andexamine 0 intermodulation distortions, which degrade the performance 5 the resonance curve behavior under different scan directions of promising superconductingmicrowave applications mainly 0 showing interesting features. To account for our results, we / in the telecommunication area [9]. consider briefly some possible physical mechanisms that may t a Among the nonlinear effects reported in the literature as- be responsible for the observed effects. m sociated with resonance curves, one can find the commonly - knownDuffingnonlinearitywhich is characterizedby skewed d II. RESONATORS DESIGN resonance curves above certain power level, appearance of n infinite slope in the resonance lineshape, pronounced shift of A. Resonator Geometries o c theresonancefrequencyandhystereticbehavior[10],[11],[12]. The resonators were designed in the standard stripline : To account for this effect, associated with the rise of kinetic geometry, which consists of five layers as shown in the cross v i inductance of superconductors, both thermal [13], and weak section illustration depicted in Fig. 1. X link [10] explanations have been successfully applied. Other The superconductingresonatorwas dc-magnetronsputtered r nonlinear effects were reported by Portis et al. [14], where ononeofthesapphiresubstrates,whereasthesuperconducting a they observed notches that develop on both sides of the ground planes were sputtered on the inner covers of a gold frequency response of their HTS microstrip patch antenna, plated Oxygen Free High Conductivity(OFHC) Cupper Fara- accompanied with hysteresis and frequency shift, as they day package that was employed to house the resonators. The have driven their antenna into the nonlinear regime. Similar dimensions of the sapphire substrates were 34mm X 30mm results were reported also by Hedges et al. [15], in their X 1mm. The resonator geometries implemented, which we will refer to them, for simplicity, by the names B1, B2, B3, This work has been submitted to the IEEE for possible publication. are presented in the insets of Figs. 3, 4, 5 respectively. The This work was supported by the German Israel Foundation under grant 1-2038.1114.07, the Israel Science Foundation under grant 1380021, the width of the feedlines and the thin part of the resonators DeborahFoundation andPoznanskiFoundation. was set to 0.4mm to obtain characteristic impedance of 50Ω. The authers are with the Department of Electrical Engineering and The gap between the feedline and the resonators was set to Microelectronics Research Center, Technion, Haifa 32000, Israel (e-mail: [email protected]). 0.4mm in the B1, B3 cases, and to 0.5mm in B2 case. The 2 TABLEI SPUTTERINGPARAMETERS Process parameter B1 B2 B3 Partial flow ratios (Ar,N2) (87.5%,12.5%) (75%,25%) (70%,30%) Base temperature 11◦C 11◦C 13◦C Total pressure 6.9·10−3 torr 8.1·10−3 torr 5.7·10−3 torr Discharge current 0.36A 0.55A 0.36A Discharge voltage 351V 348V 348V Discharge power 121W 185W 133W Deposition rate 6 A˚ 7.8 A˚ 3.8 A˚ sec sec sec Thickness (t) 2200A˚ 3000A˚ 2000A˚ Base pressure 3.1·10−8 torr 7.3·10−8 torr 8·10−8 torr Target-substrate distance 80mm 90mm 90mm frequency modes of B1, B2, B3 resonators were theoretically calculated using a simple transmission line model, presented in appendix A, and were also experimentally measured using vectornetworkanalyzer(NA).Thetheoreticalcalculationwas generallyfoundtobeingoodagreementwiththemeasurement results, as discussed in Appendix A. III. FABRICATION PROCESS The sputtering of the NbN films was done using a dc- magnetron sputtering system. All of the resonators reported here were deposited near room temperature [17],[18],[19], wherenoexternalheatingwasapplied.Thesystemwasusually pumped down prior to sputtering to 3 8 10−8torr base − · pressure (achieved overnight). The sputtering was done in Ar/N atmosphere under current stabilization condition [20]. 2 The relative flow ratio of the two gases into the chamber and thetotalpressureofthemixturewerecontrolledbymassflow meters. The sputtering usually started with a two minute pre- Fig.2. Dischargecurrentvs.dischargevoltageofthesputteringsystemdis- sputteringintheselectedambientbeforeremovaloftheshutter playingcurrent-voltagekneefordifferentpercentagesofArgonandNitrogen, atambienttemperature of20◦C. and deposition on the substrate. The sputtering parameters of the three resonatorsare summarized in table 1. Following the NbN deposition, the resonator features were patterned using standard photolithography process, whereas the NbN etching inthepresenceofN gasrelativetothevaluemeasuredinpure 2 was done using Ar ion-milling. argonat the same dischargecurrent,is also pointedout in the The fabricated resonators were characterized by relatively figure,correspondingtodifferentcurrentsandN percentages. 2 lowT forNbNandrelativelyhighnormalresistivityρ,thatis c ingoodagreementwithRef.[18].Tc measuredforB1,B2and IV. MEASUREMENTRESULTS B3 was 10.7, 6.8, 8.9 [K] respectively, whereas ρ measured for B2 and B3 was 348 and 500 [µΩcm] respectively. All measurements presented in this paper have been con- To obtain resonators with reproducible physical properties ducted at liquid helium temperature. wehaveusedthesputteringmethoddiscussedin[20],whereit wasclaimedthatreproducibleparametersoffilmsareassured, A. S Measurements by keeping the difference between the discharge voltage in 11 a gas mixture, and in pure argon, constant, for the same The resonance response of the resonators was measured discharge current. In Fig. 2 we show one of the characteri- using the reflection parameter S of a vector NA. The res- 11 zation measurements applied to our dc-magnetron sputtering onance response obtained for the third mode of B1 resonator system, exhibiting a knee-shapegraph of discharge currentas 8.26GHz at low input powers, between 23 dBm and 18 ∼ − − a function of discharge voltage. The knee-shape graph was dBm in steps of 0.05 dBm, is shown in Fig. 3. A small offset obtained for different Ar/N mixtures at room temperature as was applied between the sequential graphs, corresponding 2 showninthefigure.Thedischargevoltagedifferencemeasured to different input powers, for clarity, and to emphasize the 3 Fig. 4. S11measurement ofB2 resonance at ∼ 4.385GHz with 30MHz Fig. 3. S11measurement of B1 resonance at ∼ 8.26GHz with 10MHz span,atlowinputpowers,exhibitingstrongnonlineareffects.Theresonance span, at low input powers, exhibiting extraordinary nonlinear effects. The curves corresponding to different input powers were shifted by a constant resonance curves corresponding to different input powers were shifted by a offsetforclarity. Theinsetshowstheresonatorgeometry. constant offsetforclarity. Theinsetshowstheresonatorgeometry. nonlinear evolution of the resonance response as the input power is increased. The interesting characteristics of this nonlinear evolution could be summarized as follows: 1) In the power range between 23.5 dBm and 23.25 − − dBm, the resonance is symmetrical and broad. 2) At input power level of 23.25 dBm, a sudden jump of − about 15 dB occurs in the resonance curve at the minima − where the slope of the resonance response is small. 3) As the input power is increased in steps of 0.05 dBm the resonance becomesasymmetrical, and the left jump shifts towards the lower frequencies gradually. 4) As we continue to increase the input power, the jumps decrease their height but the resonance curve following the jumps becomes more symmetrical and deeper, and at certain Fig.5. NonlinearresponseofB3resonanceat∼1.6GHz,correspondingto input power level we even witness a critical coupling phe- inputpowerlevels increasing in0.01dBm.Atinputpowerof1.49dBmwe nomenonwhereS11(ω)atresonanceisalmostzero,nopower observe two obvious bifurcations in the resonance band, and another small reflection is present. one at the right side, marked with circles. The different resonance curves were shifted by a constant offset for clarity. The inset shows the resonator 5) The resonance becomes symmetrical again and broader geometry. and the bifurcations disappear. 6) All previously listed effects occur within a frequency spanof10MHz,powerrangeofabout5dBm,andpowerstep and 18 dBm, in B2 case they happen between 9.5 dBm of 0.05 dBm. − − and 1.5 dBm and in B3 case they happen around 1 dBm. Moreover, in order to estimate how narrow this resonance − 2) In Fig. 5 corresponding to B3 resonator we witness two could be in the vicinity of critical coupling, a separate S 11 apparent bifurcations within the resonance band as indicated measurement has been applied near critical coupling power, by circles on the figure, a feature that we did not encounter using1601measurementpointsandfrequencyspanof2MHz. in Figs. 3 and 4. The bandwidth of the resonance curves measured, according to the +3 dB method,was about0.25 10−5GHz, whereasthe ratio f/∆f 3.3 106. · B. Verifications ∼ · Similar behavior to that exhibited by the nonlinear third In order to verify that the bifurcation feature, previously mode of resonator B1 can be clearly seen in Fig. 4 and Fig. measured using NA S parameter, is not a measurement 11 5, which show the nonlinear dynamicevolution of the second artifact, we applied a different measurement configuration, mode of resonator B2 and the first of B3 respectively. The shown in Fig. 6, where we scanned the frequency axis with main differences between the figures are: CWmodeofanAgilentsynthesizerandmeasuredthereflected 1) The power levels at which these nonlinear effects take powerfromtheresonatorbyapowerdiodeandvoltagemeter. place. Whereas in B1 case they happen between 23 dBm The load that appears in Fig. 6 following the diode is an − 4 Fig.6. Asetupthatwasusedtoverifytheoccurrenceofthebifuractionsat Fig.8. (a)AforwardCWmodescanusingNA,ina12MHzspanaround thenonlinear resonance(B1thirdmode),measuredpreviously byNA. the left jump of the nonlinear resonance of B2. The scan includes 20,000 frequencies,whichisequivalent toafrequencystepof∼600Hzbetweenthe datapoints.Inspiteofthissmallfrequencystep,thejumpstilloccursbetween justtwosequentialfrequencies.(b)AbackwardCWmodescanusingNA,in a 10MHzspan around the left jump of the nonlinear resonance ofB2. The scan includes 4000 frequencies, which is equivalent to a frequency step of ∼2.5kHzbetweenthedatapoints.Alsoforthiscasethejumpoccursbetween justtwosequential frequencies. D. Frequency Hysteresis Applyingforward and backward frequencysweeps to these resonators, reveals a very interesting hysteretic behavior. In Fig. 9 we show a representative frequencyscan of B2 second mode, applied in both directions, featuring the following nonlinear dynamic behavior: 1) At low input powers 8.05 dBm and 8.04 dBm, the − − resonance is symmetrical and there is no hysteresis. 2) As the power is increased by 0.01 dBm to 8.03 dBm, Fig.7. NonlinearresponseofB1thirdmodemeasuredusingtheconfiguration − twobifurcationsoccuratbothsidesoftheresonanceresponse showninFig.6.Theredlinerepresentsaforwardscanwhereastheblueline represents a backward scan. Two abrupt jumps appear at both sides of the and hysteresis loops form at the bistable regions. resonance curve andsmall hysteresis loops are present atthe vicinity ofthe 3) As we continue to increase the input power gradu- jumps. The resonance curves corresponding to different input powers were ally, the hysteresis loop, associated with the right bifurca- shiftedbyaconstant offsetforclarity. tion, changes direction. At first it circulates counterclockwise between 8.03 dBm and 7.99 dBm, at 7.98 dBm the − − − two opposed bifurcations, at the right side, meet and no Agilent load used in order to extend the linear regime of the hysteresis is detected, as we increase the power further the power diode. The results of this measurement configuration right hysteresis loop appears again, circulating, this time, in areshownin Fig.7. Thefrequencyscanaroundtheresonance the opposite direction, clockwise. was done using 201 points in each direction (forward and Furthermore,itisworthmentioningthatthehysteresisloops backward). A small hysteresis loop can be seen around the changing direction are not unique to this resonator, or to the two bifurcations. bifurcation occurring on the right side of the resonance. It appearsalso in the modesof B1, and it occursat the left side bifurcation as well, but at different power level. C. Abrupt Bifurcations E. Multiple Bifurcations In attempt to find out whether the resonance curve of these Frequency sweep applied to B3 first resonance in both nonlinear resonances changes its form along two or more directions, exhibits yet another feature, in addition to the two frequency points, further measurements where carried out bifurcations at the sides of the resonance curve, which we usingNA,wherewescannedthefrequencyaxisinthevicinity have seen earlier, there are another two smaller bifurcations ofthe jumpwith highfrequencyresolution.Themeasurement accompanied with hysteresis within the resonance lineshape, resultscorrespondingto frequencystep of 600Hz and 2.5 adding up to 4 bifurcations in each scan direction, as can be ∼ ∼ kHz are presented in Fig. 8 plot (a) and (b) respectively, seen in Fig. 10. This feature may have a special significance indicating abrupt transition between two bistable states. in explaining the physical origin of these nonlinearities. 5 Fig.11. Resonator generalgeometrymodel. tions, hammerhead resonance lineshape and similar effects, suggest similar physical mechanism and further support the former hypothesis. Fig. 9. Forward and backward scan measurement, performed using NA, measuring B2 second mode nonlinear resonance. The red line represents a forward scan whereas the blue line represents a backward scan. The graphs exhibit clear hysteresis loops forming at the vicinity of the bifurcations, VI. CONCLUSION and hysteresis loop changing direction as the input power is increased. The In the course of this experimentalwork we have fabricated resonance curves corresponding to different input powers were shifted by a several stripline NbN resonators dc-magnetron sputtered on constant offsetforclarity. sapphire substrates at room temperature implementing dif- ferent geometries. The resonators have exhibited similar and unusual nonlinear effects in their resonance response curves. The onset of the nonlinear effects in these NbN resonators varied between the different resonators, but usually occurred at relatively low powers, typically 2-3 orders of magnitude lower than Nb for example. Among the nonlinear effects observed: abrupt and multiple bifurcations in the resonance curve, power dependent resonance frequency shift, hysteresis loops in the vicinity of the bifurcations, hysteresis loops changing direction, and critical coupling phenomenon. Weak linksformingintheNbNfilmsarehypothesizedasthesource ofthenonlinearities.Furtherstudyoftheseeffectsunderother modes of operation and measurement conditions would be carriedinthefuture,inordertosubstantiateourunderstanding of these extraordinary effects. Fig. 10. S11 parameter measurement ofthe firstresonance of B3at input APPENDIX I powers 1.52 dBm through 1.58 dBm in steps of 0.02 dBm using forward and backward CW mode scan of NA employing 2000 measurement points RESONANCE FREQUENCY CALCULATION OF B1, B2, B3 in each direction. The graph shows clearly 4 jumps within the band of the RESONATORS resonanceineachdirection. Theresonancecurvescorrespondingtodifferent inputpowerswereshifted byaconstant offsetforclarity. The calculation process of the resonance frequenciesof B1 andB2 makesuse of oppositetravelingvoltage-currentwaves method [26], [27]. For this purpose we model B1 and B2 resonators as a straight transmission line extending in the z- V. DISCUSSION direction with two characteristic impedance regions Z and 1 This unusual dynamic behavior of our NbN films, demon- Z2 as shown in Fig. 11. strated earlier, highly suggest built-in Josephson junctions, Theequivalentvoltagealongtheresonatortransmissionline forming at the boundaries of the granular NbN columnar wouldbegiven,in general,by a standingwavesexpressionin structure [21],[22], as the underlying physical mechanism re- the form: sponsiblefortheobservednonlinearities[1].Anotherphysical mechanism that may be considered as a strong candidate for A cosβ(z a)+B sinβ(z a) zǫ(a,b) + + explaining the effects, is the local heating mechanism which V (z)= Aco−sβz+Bsinβz − zǫ( a,a) − was hypothesized as the source of notches and switching A−cosβ(z+a)+B−sinβ(z+a) zǫ( b, a) effects observed in HTS films [14],[15],[16]. Nevertheless − −(1) recentmeasurementsdoneonSQUID ringcontaininga single where β =2πf√εr/c is the propagation constant along (cid:0) (cid:1) Josephson junction inductively coupled to a radio frequency the transmission line, and A+,B+,A,B,A−,B− are con- resonant circuit [23], [24], [25] exhibiting opposed bifurca- stants that can be determined using boundary conditions and 6 applied power amplitude. However due to the symmetry of The antisymmetric case: the problem z z, we expect the solutions to have From the antisymmetric V ( b)= V ( b) and the conti- ←→ − − − − defined parity, where V (z)=V ( z) for symmetric solution nuity V (b)= V ( b) conditions, we get V (b)= V ( b)= − − − and V (z) = V ( z) for antisymmetric solution. Thus by 0, which yields: − − taking advantage of this property and demanding that V (z) be continuous at z =a and z = a, one gets: − sin(βa)cos(β(b a))+ηcos(βa)sin(β(b a))=0 (3) cosβacosβ(z a) − − − zǫ(a,b) Vsym(z)= +Bsscionsββ(zz−a) zǫ(−a,a) 0.5S,ul1bsti=tutinag th=e fo1l3lomwmin,gl2nu=meribca−l vaalue=s η6=.5mZ2m/Z1in=to cosβacosβ(z+a) the above resonance frequency conditions and solving for zǫ( b, a) −sinBβsasicnoβsβ(z(+z a)a) − − f(r2e.q5u0e3n5cGieHsz,b5el.o6w97G1H0Gz,H8z.,16y4i7eGldHs zt)heto fthoelloswyminmgetsroicluctiaosnes, − zǫ(a,b) Vanti(z)= +Bassiinnββ(zz−a) zǫ(−a,a) amnedtri(c2.c9a8s0e4,GwHithz,d5o.u1b7l8y6dGeHgezn,e8ra.1te64m7oGdHeza)t 8to.16th4e7GanHtizs.yBmy- sinβacosβ(z+a) comparing these calculated resonances to the directly mea- − zǫ( b, a) where Vsym(z) stan+dsBfaosrinthβe(szym+mae)tric solut−ion −whereas sSu1r1edmreeassounraenmceensto(f2B.5181re2sGonHazt,o5r,.6o3b0ta4iGneHdzu,s8i.n4g18a8bGroHazd)b,awnde Vanti(z)fortheantisymmetricsolution.Tocalculatethevalue findthattheexcitedresonancescorrespondtothesymmetrical of the new constants Bs,Ba , we require that the equivalent caseonly.Theantisymmetricmodesdonotgetexcitedbecause current I(z) along the transmission line, which is given they have a voltage node at the feeding line position. by I(z) = (i/βZ )dV/dz where Z is the characteristic i i impedance of the line, be continuous at z = a and z = a. B. B2 Resonator − Following this requirement one gets B = ηsin(βa) and s − Since the resonator ends are open-circuited we demand B = ηcos(βa), where η = Z /Z . The symmetric and a 2 1 I(b)=I( b)=0. antisymmetric solutions of V (z) and I(z) are given by: − The symmetric case: cosβacosβ(z a) − zǫ(a,b) We require that the current associated with the symmetric ηsinβasinβ(z a) Vsym(z)= − cosβz − zǫ(−a,a) voltage, vanishes: cosβacosβ(z+a) zǫ( b, a) +ηssininββaacsoisnββ((zz+aa)) − − cos(βa)sin[β(b−a)]+ηsin(βa)cos[β(b−a)]=0 (4) − zǫ(a,b) The antisymmetric case: +ηcosβasinβ(z a) Vanti(z)= sin(βz) − zǫ(−a,a) ricWveolrtaegqeu,irveatnhiasthethse: currentassociatedwith theantisymmet- sin(βa)cosβ(z+a) − zǫ( b, a) +(−ηiccoossβ(βasain)sβi(nz−βa()z+a) − − sin(βa)sin(β(b a))+ηcos(βa)cosβ(b a)=0 (5) Isym(z)= −(−iηisci−onsβiβasaiZZcnZso22i(1nsβββz()(zz−+aa))) zzǫ(ǫ−(aa,,ba)) 4in9−tS.o9u/bt1hs0eti.t4uabt=ionvg4e.t7hr9ee,slof1on−l=alonwcaei=nfgr1en1qu.u9me7nemcriymcac,lol2vnad=liutiebos−nηs−a=a=ndZ62s./4oZ3lvm1inm=g +(−iηissiinnββaaZZcso22insββ((zz+−aa))) zǫ(−b,−a) (f2o.r6f4r8e6qGueHnczi,e6s.0b2e8lo8wGH10zG,8H.5z5,8y8iGeldHsz)thteotfhoellosywminmgestroilcuctiaosnes, Z2 zǫ(a,b) and (1.1763GHz, 4.3698GHz, 7.4597GHz, 9.7778GHz) to Ianti(z)= +iηcosiβcaoZZcs1o2βszβ(z−a)) zǫ(−a,a) nthaencaenstistoymthmeedtriircecctalysem. BeaysucroemdpraersionnganthceesseocfalBc2ularteesdonraetsoor-, +(iηiscionsββaasZZcino22sββ((zz++aa))) zǫ(−b,−a) 4om.b4etn2ait5nGbeedHtwzu,esei6nn.g3t8ha0e6btGwrooHadzre,bsa8un.ld1ts7.S6T1G1hHemzm)e,iasswsuienregfimrneednsotan(a2gn.o5co1eds52daGogrHneoez-,t get excited apparentlybecause of the coupling location of the A. B1 Resonator feedline relative to the resonator. Since the resonator ends are shorted we demand V (b) = V ( b). C. B3 resonator − The symmetric case: B3 resonator, in contrast, showed some larger discrepancy In this case we either have maximum or minimum at V (b) between the measured value for the first mode 1.6 GHz thuswe get I(b)=0,yieldingthe followingconditionon the ∼ (seen in Fig. 5) and the theoretical value f = 2.4462 GHz resonance frequencies: 1 calculated according to the approximated equation: nc f = (6) cos(βa)sin(β(b−a))+ηsin(βa)cos(β(b−a))=0 (2) n 2l√εr 7 where n is the mode number,c is the lightvelocity,l is the [18] D. D. Bacon, A.T. English, S. 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[7] T.Dahm,andD.J.Scalapino,”Theoryofintermodulationinasupercon- ductingmicrostripresonator,”J.Appl.Phys.,vol.81,p.2002,February 1997. Baleegh Abdo (S’2002) was born in Haifa, Israel in 1979. He received [8] D.E.Oates,S.-H.Park,M.A.Hein,P.J.Hirst,andR.G.Humphreys, the B.Sc. degree in computer engineering, in 2002, and the M.Sc. degree ”Intermodulation distortion and third-harmonic generation in YBCO in electrical engineering in 2004, both from the Technion–Israel Institute films of varying oxygen content,” IEEE Trans. Appl. Supercond., vol. of Technology, Haifa, Israel. Currently he is pursuing the Ph.D. degree in 13,p.311,June2003. electrical engineering at the Technion. His graduate research interests are [9] T. Dahm, and D. J. Scalapino, ” Analysis and Optimization of inter- nonlinear effects in superconducting resonators in the microwave regime, modulationinHighTcsuperconductingmicrowavefilterdesign,”IEEE resonatorcoupling andquantum computation. Trans.Appl.Supercond.,vol.8,p.149,December1998. [10] C. C. Chen, D. E. Oates, G. Dresselhaus, and M. S. Dresselhaus, ”Nonlinearelectrodynamics ofsuperconductingNbNandNbthinfilms atmicrowavefrequencies,”Phys.Rev.B.,vol.45,p.4788,March1992. [11] J. Wosik, L.-M. Xie, J. H. Miller, Jr., S. A. Long, and K. Nesteruk, ”Thermally-induced nonlinearities in the surface impedance of super- conducting YBCO thin films,” IEEE Trans. Appl. Supercond. , vol. 7, p.1470,June1997. Eran Segev-Arbel was born in Haifa, Israel in 1975. He received the [12] B. A. Willemsen, J. S. Derov, J. H. Silva, and S. Sridhar, ”Nonlinear B.Sc. degree in electrical engineering from the Technion–Israel Institute of Technology,Haifa,Israel,in2002.HeiscurrentelyworkingtowardtheMS.c. response of suspended high temperature superconducting thin film microwave resonators,” IEEETrans.Appl.Supercond., vol. 5,p.1753, inelectricalengineeringattheTechnion.Hisresearchisfocusedonparametric June1995. gaininsuperconducting microwave resonators. [13] L.F.Cohen,A.L.Cowie,A.Purnell,N.A.Lindop,S.Thiess,andJC Gallop, ”Thermally induced nonlinear behavior of HTS films at high microwave power,”Supercond. Sci.andTechnol.,vol.15,p.559,2002. [14] A.M.Portis,H.Chaloupka, M.Jeck,andA.Pischke, ”Power-induced switching of an HTS microstrip patch antenna,” Superconduct. Sci. & Technol.,vol.4(9),p.436,September 1991. [15] S. J. Hedges, M. J. Adams, and B. F. Nicholson, ”Power dependent Oleg Shtempluck was born in Moldova in 1949. He received the M.Sc. effects observedforasuperconducting stripline resonator,” Elect. Lett., degreeinelectronicengineering fromthephysicaldepartmentofChernovtsy vol.26,no.14, July1990. StateUniversity,SovietUnion,in1978.Hisresearchconcernedsemiconduc- [16] J. Wosik, L.-M. Xie, R. Grabovickic, T. Hogan, and S. A. Long, tersanddielectrics. From1983to1992,hewasateamleaderinthedivision ”Microwave power handling capability of HTS superconducting thin of design engineering in Electronmash factory, and from 1992 to 1999 he films:weak links andthermal effects induced limitation,” IEEETrans. workedasstampandmoulddesignengineerinIkarcompany,bothinUkraine. Appl.Supercond., vol.9,p.2456,June1999. CurrentlyheisworkingasalaboratoryengineerinMicroelectronicsResearch [17] Z.Wang,A.Kawakami,Y.Uzawa,andB.Komiyama,”Superconducting Center, Technion- IsraelInstitute ofTechnology, Haifa, Israel. properties and crystal structures of single-crystal niobium nitride thin films deposited at ambient substrate temperature,” J. Appl. Phys., vol. 79,p.7837,February1996. 8 Eyal Buks received the B.Sc. degree in mathematics and physics from the Tel-Aviv University, Tel-Aviv, Israel, in 1991 and the M.Sc. and Ph.D. degrees in physics from the Weizmann Institute of Science, Israel, in 1994 and 1998, respectively. His graduate work concentrated on interference and dephasing in mesoscopic systems. From 1998 to 2002, he worked at the California Institute of Technology (Caltech), Pasadena, as a Postdoctoral Scholar studying experimentally nanomachining devices. He is currently a Senior Lecturer at the Technion—Israel Institute of Technology, Haifa. His currentresearch isfocusedonnanomachining andmesoscopicphysics.