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178 R. GROSS AND A. MARX Chapter 4II cΦI camplifierlock-indetectorintegratorL foscillator≈R fV inI fFigure 4.13: The modulation and feedback circuit of a dc-SQUID.from the condition |δΦ f | = |δΦ|δV in = R fk 2 L fδΦ . (4.1.57)We see that the output signal increases with increasing feedback resistance. Furthermore, the outputsignal is linear with δΦ even if the flux change is several Φ 0 .The typical modulation frequency of the flux-locked loop circuit is from 100 kHz to several MHz. Usingsuitable electronics a very high dynamic range above 140 dB and a signal bandwidth of up to 100 kHzcan be achieved. An important quantity is the slew rate, which gives the speed at which the feedbackcircuit can compensate for rapid flux changes at the input. State of the art SQUID electronics has slewrates of up to 10 7 Φ 0 /s.Bias Current Reversal: For dc-SQUIDs based on high temperature superconductors often the biascurrent is modulated in addition to the flux. The reason for that is that Josephson junctions basedon high temperature superconductors usually show large low-frequency fluctuations of the critical current.40,41,42,43 These fluctuations can be eliminated by a periodic reversal of the bias current. 44,45 Sincethe fluctuations of the critical current of the two junctions are independent, they can be separated in asymmetric and antisymmetric part. The symmetric part results in a shift of the V (Φ ext ) curves alongthe voltage axis. This shift is eliminated by the flux modulation technique discussed above, since onlythe modulated signal at f mod is detected by the lock-in amplifier. In contrast, the asymmetric part correspondsto a circulating current, which induces magnetic flux fluctuations and hence results in a shift ofthe V (Φ ext ) curve along the flux axis. By a periodic reversal of the bias current at a frequency f rev muchlarger than the low-frequency I c -fluctuations one obtains a periodic shift of the V (Φ ext ) curve in oppositedirections. Then, by averaging over several periods also the asymmetric fluctuations can be eliminated.40 R. Gross, P. Chaudhari, M. Kawasaki, A. Gupta, M. B. Ketchen, IEEE Trans. Magn. MAG-27, 2565 (1991).41 R. Gross, Grain Boundary Josephson Junctions in the High Temperature Superconductors, in Interfaces in High-T c SuperconductingSystems, S. L. Shinde and D. A. Rudman eds., Springer Verlag, New York (1994), pp. 176-21042 A. Marx, L. Alff, R. Gross, IEEE Trans. Appl. Supercond. 7, 2719 (1997).43 A. Marx, R. Gross, Appl. Phys. Lett. 70, 120 (1997).44 R.H. Koch, J. Clarke, W. M. Goubau, J. M. Martinis, C. M. Pegrum, and D. J. Van Harlingen, J. Low Temp. Phys. 51, 207(1983).45 V. Foglietti, W. J. Gallagher, M. B. Ketchen, A. W. Kleinsasser, R. H. Koch, S. I. Raider, and R. L. Sandstrom, Appl. Phys.Lett. 49, 1393 (1986).© Walther-Meißner-Institut
Section 4.1 APPLIED SUPERCONDUCTIVITY 179The application of both flux modulation and bias current reversal is called double modulation technique.It can reduce the low-frequency 1/ f -noise of SQUIDs based on high temperature superconductors byseveral orders of magnitude. 46,47Additional Positive Feedback: An important reason for the use of the flux modulation technique is thefact that the voltage changes δV (Φ ext ) (typically less than 100 µV/Φ 0 ) and the SQUID impedance (typicallya few Ω) are small. This is inadequate for semiconductor devices. Applying the flux modulation,the SQUID impedance can be increased by a step-up transformer and matched to the room temperaturesemiconductor electronics. An alternative way is to use the additional positive feedback (APF) technique,in which part of the bias current is used to obtain an asymmetric V (Φ ext ) dependence with a steepslope and hence larger value for ∂V /∂Φ. In this case a direct read-out of the SQUID signal with lownoise room temperature semiconductor electronics is possible. 48,49Additional Topic:Digital Read-Out SchemesFujimaki et al. 50 and Drung et al. 51,52 have developed schemes in which the output from the SQUIDis digitized and fed back to the SQUID as an analog signal to flux-lock the loop. Fujimaki et al. usedJosephson digital circuits to integrate their feedback system on the same chip as the SQUID. Drunget al. obtained a flux resolution of about 10 −6 Φ 0 / √ Hz in a 50 pH SQUID. They were also able toreduce the 1/ f noise by using a modified bias current modulation scheme. In general, the cryogenicdigital feedback schemes have the advantage that they are compact, offer wide flux-locked bandwidthand produce digitized output signals for transmission to room temperature.Additional Topic:The Relaxation Oscillation SchemeMück and Heiden have operated a dc SQUID with hysteretic junctions in a relaxation oscillator. 53 Here,the SQUID is shunted by a series connection of an inductor and resistor. The circuit performs relaxationoscillations at a frequency depending on the flux in the SQUID. The oscillation frequency has a minimumfor (n + 1 2 )Φ 0 and a maximum for nΦ 0 with a typical frequency modulation of about 100 kHz at anoscillation frequency of about 10 MHz. The advantage of this scheme is that it produces a large voltageacross the SQUID so that no matching network to the room temperature electronics is required. The roomtemperature electronics is allowed to be simple and compact. A flux resolution of about 10 −5 Φ 0 / √ Hzfor a 80 pH SQUID operated at 4.2 K has been achieved. The so-called double relaxation oscillationSQUID (DROS) is briefly addressed in section 4.3.1.46 R.H. Koch, W. Eidelloth, B. Oh, R. P. Robertazzi, S. A. Andrek, and W. J. Gallagher, Appl. Phys. Lett. 60, 507 (1992).47 A.H. Micklich, D. Koelle, E. Dantsker, D. T. Nemeth, J. J. Kingston, R. F. Kroman, and J. Clarke, IEEE Trans. Appl.Supercond. 3, 2434 (1993).48 D. Drung, Physica C 368, 134 (2001).49 D. Drung, in SQUID Sensors: Fundamentals, Fabrication and Applications,NATO Science Series E: Applied Sciences, Vol. 329, Kluwer Academic Publishers, Dordrecht, Boston, London (1996).50 N. Fujimaki, H. Tamura, T. Imamura, S. Hasuo, ISSCC San Francisco, (1988), pp. 40-41.51 D. Drung, Cryogenics 26, 623-627 (1986).52 D. Drung, E. Crocoll, R. Herwig, M. Neuhaus, W. Jutzi, IEEE Trans. Magn. MAG-25, 1034-1037 (1989).53 M. Mück, C. Heiden, IEEE Trans. Magn. MAG-25, 1151-1153 (1989).2005
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