Benzing, Walter

Jutzi, Wilhelm

Lochschmied, Rainer

KITopen

Abstract--The average microwave losses of interferometers withtwo and four strongly damped Josephson junctions biased byan microwave current source at 10 GHz much below theplasma frequency have been simulated as a function of anexternal magnetic field. Microwave losses appear periodicallywithin distinct ranges of the magnetic field, with microwaveamplitudes below and above threshold currents. The mainperiod corresponds to one flux quantum in everyinterferometer loop. The absorption rate increases withmicrowave power in qualitative agreement with measurements.As the microwave has a finite penetration depth from one endof a long interferometer with many Josephson junctions, theabsorption lines  decrease within the main period.I. INTRODUCTIONOne dimensional arrays of Josephson junctions can beused to explain electrical and magnetical effects insuperconductor devices. The magnetic sensitivity of themicrowave absorption in a single YBa2Cu3O7-δ crystal can bemodeled by an interferometer with many Josephsonjunctions. The absorption of microwave power wascalculated in [1] using a superconductor loop with manyjunctions. Another method was proposed in [2], based on amodel of weakly coupled superconducting loops.In this work the microwave losses have been simulatedwith the circuit analysis program SPICE. A simpleRCSJ-model (McCumber) with a nonlinear quasiparticlecurrent, has been compared with a more complex simulationmodel, based on the theory of Werthamer (WA-model) [3].The interferometer simulations assume a critical currentdensity of jmax = 0.52 kA / cm2. The junction capacitance isCF = 5 µF / cm2, corresponding to a plasma frequency= 89.5GHz.fp = jmax2pi⋅Φ0⋅CFThe junction area of a single point junction is A = 20 µm2,the critical current I0 = 0.1 mA and the capacitance of apoint junction CJ = 1 pF. The Josephson junctions arestrongly damped with a shunt resistor R to get anonhysteretic I,V-characteristic similar to a SNS junction.The McCumber parameter of a Josephson junction isβ = 2piΦ0 ⋅ RF2 ⋅ CF ⋅ jmax = 1.3with R = 2 Ω and RF = R .A = 40 Ωµm2. Both simulation models have the same quasiparticleI,V-curve in a sufficient large magnetic field. The averagepower dissipation versus  frequency has been calculated.With the above parameters, no essential difference betweenthe results of WA- and RCSJ-model at 4.2 K and a frequencyof fLO = 10 GHz has been found, where the microwaveabsorption in a single YBa2Cu3O7-δ crystal has been measured[4]-[6]. Since the WA-model needs more computation time,the RCSJ-model has been used to get the results of thispaper. The resistors are assumed to have no Johnson noise. II. MICROWAVE LOSSES IN A  SQUIDAn interferometer with two identical strongly dampedJosephson junctions is chosen as a simple model to describeapproximately microwave losses in a single  YBa2Cu3O7-δcrystal. With the mentioned parameters and a inductanceL = 4.97 pH the characteristic phase is .λ = 2piΦ0 ⋅ L ⋅ I0 = 0.1piThe microwave is injected in the center of the inductance.The control current Ic = ΦC / L represents the DC magneticfield. The circuit diagram of the simulated SQUID issketched in Fig. 1.The static SQUID characteristics [7],[8] with different fluxmodes, corresponding to 0, 1, 2 and 3 flux quanta in theloop, are plotted in Fig. 2a. The static SQUID characteristicis periodical with Φ0. The flux modes are stable inside theclosed curves. Outside the threshold characteristic theSQUID switches into the voltage state.The injected microwave current has a sinusoidal shape at a frequency fLO = 10 GHz far belowI = ILO ⋅ sin(2pifLOt)the plasma frequency fp. The control current  is kept constantManuscript received October 16, 1994, ASC '94 BostonFig. 1. Circuit diagram of the simulated SQUID. Normalized parameters areλ = 0.1pi and β = 1.3. The microwave current is symmetrically injected at thegate. The magnetic field is modeled by the DC-current ΦC / L.R I0λ = 0.1piL/2RL/2I0ILOΦ / LCIC IG~R. Lochschmied, W. Benzing and W. JutziInstitut für Elektrotechnische Grundlagen der Informatik, University of Karlsruhe, D-76187 KarlsruheMicrowave Losses of Interferometers with many Josephson Junctionsversus DC Magnetic Fieldfor every calculation of the power losses. The average of theinstantaneous powerp = U(t) ⋅ ILO ⋅ sin (2pifLOt)is numerical taken over ten microwave periods. Thecomputation of the average starts, after the system hasreached the steady state. Each absorption curve in Fig. 2b-dcomprises 600 simulated points for ΦC. The average power isnormalized with the power dissipation in the shunt resistors pN = 12 ⋅RN ⋅ ILO2The normalized power dissipation of the SQUID  forp / p2different normalized microwave amplitudes is plottedILO/I0in Fig. 2. For  the microwave current does notILO/I0 = 0.06exceed the threshold characteristic from a stable flux mode tothe voltage state at any point of ΦC. Nevertheless, maxima ofpower dissipation appear periodically, when the microwavecurrent gets close to the threshold current.The first derivation s = dp / dΦC calculated with the resultsin Fig. 2b,c and d reminds microwave absorptionmeasurements at single YBa2Cu3O7-δ crystals [4]-[6]. Thelinewidth ΦB is introduced as the difference of two values ΦCbetween the highest positive and negative slope, i. e. .smaxFor microwave currents larger than the thresholds in Fig. 2athe SQUID switches into the "voltage state", as shown inFig. 2c, where the phases rotate in one direction during onehalf of the microwave period, and dissipates considerablymore power than in the "non voltage state". Simulations withdifferent amplitudes  and  illustrateILO = 0.5 ⋅ I0 ILO = 1.5 ⋅ I0that the linewidth broadens with increasing microwaveamplitude.If the amplitude increases further, the SQUID will remainin the quasi stable voltage state at any value of ΦC as shownin Fig. 2d for  and . With increasingILO = 2.3 ⋅ I0 ILO = 3I0microwave current, the dependence on ΦC changes fromsharp peaks to a smooth sinusoidal function.The normalized linewidth ΦB / Φ0 versus microwaveamplitude is shown in Fig. 3. If the microwave is below thethreshold current, the linewidth ΦB is extremely small. Athreshold of  could be defined. With increasingILO / I0amplitude the SQUID switches into the voltage state. If  , the linewidh increases almost linearly. AtILO < 2I0 the normalized linewidth increases rapidly to itsILO > 2I0maximum, then it falls to its saturation value ofΦB / Φ0 = 0.5. The increase of the linewidth is in qualitativeagreement with measurements [5],[9].III. DYNAMIC EFFECTS IN LONG JOSEPHSONJUNCTIONSA single SQUID is an unsuitable model to describe themicrowave penetration into a single YBa2Cu3O7-δ crystal. Auniform long junction might give a better insight of basicproperties. To get a simple approximation, small variationsof  ϕ in the nonstationary Sine Gordon equation withdamping are assumed for a replacement of the  sin(ϕ)-termby ϕ. With the complex microwave phase  ϕ = ϕLO ⋅ ej2pifLOtand after some algebraic transformations the simpledifferential equation is obtained∂2ϕ∂x2 −1λJ21 − fLOfP 2+ j fLOfc ⋅ ϕ ≈ 0The Josephson penetration depth is,λJ = Φ02piµ0 ⋅d⋅jmaxFig. 3. Normalized linewidth of the SQUID ΦB / Φ0  versus normalizedmicrowave amplitude. 0.0 0.5 1.0 1.5 2.0 2.5 3.00.000.200.400.600.801.001.20Microwave Amplitude (I   / I )0LOAbsorption Rate(Φ  / Φ ) 0BFig. 2. normalized power dissipation of  a SQUID with λ = 0.1pi versus flux.a) Static interferometer characteristics  of a SQUID. plotted  are the fluxmodes 0, 1, 2 and 3.Average power dissipation at the microwave amplitude b)                               c)                       and  d)                       and 0.0 0.5 1.0 1.5 2.0 2.5 3.0-3-2-10123±0.0625-0.5-1.5-2.3-30.51.52.330 0.5 1 1.5 2 2.5 300.20.40.60.810 0.5 1 1.5 2 2.5 300.20.40.60.810 0.5 1 1.5 2 2.5 300.20.40.60.81Φ  / ΦB 0I    / I  = 0.06LO 0I   / ILO 0I    / I  = 0.5LO 0 I    / I  = 1.5LO 0I    / I  = 2.3LO 0 I    / I  = 3LO 0Φ  / ΦB 00 1 2 3Gate Current    (I   / I )G0Power Dissipation (p / p  ) 2(Φ  / Φ )C 0Normalized Fluxa)b)c)d)ILO = 0.06 ⋅ I0 ILO = 0.5 ⋅ I0 ILO = 1.5 ⋅ I0ILO = 2.3 ⋅ I0 ILO = 3 ⋅ I0with d = tox + λL1 + λL2. The London penetration depths ofthe junction electrodes are λL1 and λL2. tox is the layerthickness between both electrodes, and fp is the plasmafrequency of the junction. The characteristic frequency is.fc = I0 ⋅RΦ0With fp >> fLO  fc the differential equation ≈∂2ϕ∂x2 −1λJ2⋅ (1 + j) ⋅ ϕ ≈ 0indicates, that microwave losses modify the penetration ofthe magnetic field into the long junction. For the investigatedcase the characteristic frequency is fc = 9.7 GHz  fLO. A≈microwave penetration depth must be taken into account.IV. FOUR JUNCTION INTERFEROMETERA long Josephson junction can be approximated by aninterferometer with many point junctions. The circuitdiagram of a long interferometer is sketched in Fig. 4. Themicrowave current is fed asymmetrically. A simple approximation of a long Josephson junction is afour junction interferometer. Static gate characteristics ofinterferometers with many Josephson junctions are found in[10]. The static characteristic for a uniform interferometerwith λ = 0.1pi and N = 4 is plotted in Fig. 5a.  Only the fluxmodes M = 000, 010, 101 and 111 exist for 0 < ΦC < Φ0. Thecharacteristic phase corresponds to an interferometerinductance of L = 4.97 pH. The interferometer approximatesa single long Josephson  junction for ΦC  Φ0 / 2 with its≤normalized length,lλJ = N ⋅ (N − 1) ⋅ λwhere N is the number of Josephson Junctions in theinterferometer. As the normalized length  is l / λJ = 1.9, theinterferometer approximates a long junction, where the selffield effects can no more be neglected. Each curve 5b-dcomprises 1200 points between ΦC = 0 and ΦC = 3Φ0.A microwave with an amplitude  is below theILO = 0.2 ⋅ I0threshold characteristic at every value of ΦC. The peaksappear, if the microwave nearly reaches the thresholdcurrent. Three peaks are in each period m of .ΦC = m ⋅ Φ0Equally large absorption maxima could be expected at the1. and 3. peak in regarding only the static characteristic inFig. 5a. However, the 3. peak is lower than the 1., becausethe microwave does not fully penetrate the interferometer. Atthe 2. peak the microwave current does not reach thethreshold current as close as in the 3. peak, therefore themicrowave absorption is smaller.A microwave amplitude   exceeds theILO = 0.75 ⋅ I0threshold current. The simulation results are shown inFig. 5c.  The interferometer switches into the voltage stateand the linewidth increases, too. In the voltage state theimpedance of Josephson junctions changes and themicrowave current can much better penetrate to the end ofthe interferometer. In this case all absorption peaks have thesame maximum absorption. The influence of the flux modesM = 010 and 101 on the dissipation decreases withincreasing microwave current as shown in Fig. 5d for . A further increase of the amplitude yields aILO = 2I0sinusoidal shape of the power dissipation versus flux.To demonstrate a large microwave shielding effect, theinductance of an interferometer loop is increased toL = 20 pH corresponding to λ = 0.4pi and l / λJ = 3.9. Thestatic threshold characteristics are plotted in Fig. 6a, wherein contrast to the characteristic Fig. 5a the additional modesM = 001, 100, 011 and 110 are stable. They are within themodes 000, 010 and 101, so that the interferometer does notswitch into the additional modes. The microwave current Fig. 4. Equivalent circuit of al long uniform interferometer with manyequal Josephson-junctions. The microwave current is injectedasymmetrically at the left end. The  magnetic field is represented by aDC-current ΦC / L.I0 I0 RLRΦ / LCI0 RLI0 RLILOICIG~Fig. 5. Power dissipation of a four junction interferometer with λ = 0.1pi,l / λJ = 1.9 and β = 1.3.a) Static interferometer characteristics. The flux modes 000, 010 and 101are stable.Average power dissipation at the microwave amplitudesb)                          c)                              d) Gate Current    (I   / I )G0Power Dissipation (p / p  ) 4(Φ  / Φ )C 0Normalized Flux0.0 0.5 1.0 1.5 2.0 2.5 3.0-4-3-2-1012342-0.75±0.20.75-20 0.5 1 1.5 2 2.5 300.511.50 0.5 1 1.5 2 2.5 300.511.50 0.5 1 1.5 2 2.5 300.511.5I   / ILO 0I    / I  = 0.2LO 0I    / I  = 0.75LO 0I    / I  = 2LO 00001010101111. Peak2. Peak3. Peaka)b)c)d)ILO = 0.2 ⋅ I0 ILO = 0.75 ⋅ I0 ILO = 2 ⋅ I0I has been chosen below the static threshold.The simulated result is shown in Fig. 6b. Peaks appear, ifthe microwave current switches the interferometer betweentwo flux modes or gets close to the threshold of a flux mode.Similarly to the simulations with λ  = 0.1pi, a group of threepeaks appears periodically with . The peakΦC = m ⋅ Φ0maximum within the group decreases with increasingnormalized flux, because the microwave current is shieldedby the interferometer inductance L. To check the frequencydependence, the microwave frequency has been decreased to1 GHz, while all other circuit parameters remainedunchanged. At fLO = 1 GHz all peaks have the same maxima.Absorption lines occur, when the phase of one or moreJosephson junctions is close to ϕ = pi / 2, i.e. the microwavecurrent reaches the threshold. The closer the phase to pi / 2the higher will be the microwave absorption. The equivalentlarge nonlinear inductance of the Josephson junctions causesa high current through the damping resistor R. If there is noflux quantum in the interferometer, the microwave currentswitches the junctions at the left end first. If there are moreflux quanta trapped, the power dissipation depends on theinner junctions. If the microwave does not fully penetrate theinterferometer, the junctions within the interferometer do notapproach the threshold as far as for fLO << fc . They stronglychange, but do not switch into another mode, as in thequasistatic case.For  the interferometer enters the voltage stateILO / I0 = 2within each half cycle. Since the excursion of each Josephsonjunction phases gets larger than 2pi and since thecharacteristic phase is relatively large, each phase can have amore complicated dependence on time than for . Therefore the absorption versus external fieldILO / I0 = 0.75has a different character in Fig. 6b and c especially incomparison with Fig. 5d. New lines appear in Fig. 6c. Thesenew lines need further investigations.CONCLUSIONMicrowave absorption in a one dimensional array ofstrongly Josephson junctions and lumped inductances dependin a complicated way on the characteristic phase λ, the staticthreshold characteristic, the microwave current amplitude,the microwave and the characteristic frequency. Extremelysmall ranges of applied field exist, where the absorption hasa sharp peak. The basic behavior of linewidth broadening isin agreement with measurements.ACKNOWLEDGEMENTThe authors would like to thank C. Kessler, B. Nebendahland  Prof. M. Mehring for useful discussions.REFERENCES[1] H. Vichery, F. Beuneu, P. Lejay, "Microwave Absorption in SingleCrystall of YBaCuO at Low Magnetic Field,"Physica C 159 (1989),pp 823-830[2] T. K. Xia, D. Stroud, "Nonlinear electrodynamics and resonantmicrowave absorption in ceramic superconductors,"Phys. Rev. B,Vol. 39, No. 7, March 1989, pp 4792.4795[3] H. Kratz, W. Jutzi, "Microscopic Simulation Model of JosephsonJunctions for Standard Circuit Analysis Programs," IEEE Trans.Mag. Vol.-MAG 23, No. 2, March 1987, pp 731-734[4] K. W. Blazey, A. M. Portis, K. A. Müller, J. G. Bednorz,F. H. Holtzberg, "Spin Glass Microwave Study in High-TCSuperconductors," Intern. Conf. on High-Temp. Supercond. andMaterials and Mechanism of Superconductivity, Interlaken, 1988[5] K. W. Blazey, A. M. Portis, K. A. Müller, F. H. Holtzberg,"Macroscopic Flux Quantisation and Microwave Excitation in SingleCrystal YBa2Cu3O7-δ," Europhysics Letters, March 1989[6] Y. Maniwa, A. Grupp, M. Mehring, "Mechanism of DirectMicrowave Absorption in YBa2Cu3O7-x," Physica C 156 (1988),pp 785-760[7] H. Beha, "Asymetric 2 Josephson-Junction Interferometer as a LogicGate," Electronics Letters, Vol. 13, No. 7, March 1977, pp 216-218[8] J. Wunsch, W. Jutzi, E. Crocoll, "Parameter Evaluation ofAsymmetric Interferometers with two Josephson-Junctions," IEEETrans. Mag. , Vol. MAG-18, No. 2, March 1982, pp735-737[9] C. Kessler, B. Nebendahl, A. Dulcic, Th. Wolf, M. Mehring, "Originof discrete microwave absorptions lines in YBa2Cu3O7-x singlecrystalls," Physica C 192 (1992), pp 79-84[10] R. Simons, W. Jutzi, H. Beha, "Static Interferometer Characteristicswith many equal Josephson junctions," IEEE Trans. Mag. Vol.MAG-19, No. 1, January 1983, pp 44-50Fig. 6. Power Dissipation of a four junction interferometer with λ = 0.4pi,l / λJ = 3.9 and β = 1.3a) Static interferometer characteristics, all possible flux modes are stable.Average power dissipation at the microwave amplitude b)                                              c)                  .ILO = 0.75 ⋅ I0(Φ  / Φ )C 0Normalized FluxGate Current    (I   / I )G0Power Dissipation(p / p  ) 4a)b)0 0.5 1 1.5 2 2.5 300.511.5-0.750.750.0 0.5 1.0 1.5 2.0 2.5 3.0-4-3-2-1012340 0.5 1 1.5 2 2.5 300.511.5I   / ILO 0I    / I  = 0.75LO 0I    / I  = 2LO 0-22000111101011100 110001010c)ILO = 2 I0

Microwave losses of interferometers with many Josephson junctions versus DC magnetic field

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Microwave losses of interferometers with many Josephson junctions versus DC magnetic field

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