Applied Superconductivity - Walther Meißner Institut - Bayerische ...

162 R. GROSS AND A. MARX Chapter 4I m s / I c(a)2.01.61.20.80.40.0-3 -2 -1 0 1 2 3Φ / Φ 03210-1-2-3-3 -2 -1 0 1 2 3Φ ext/ Φ 0 (b)Φ ext/ Φ 0Figure 4.2: (a) The maximum supercurrent Ism plotted versus the applied magnetic flux Φ ext for a dc-SQUIDwith two identical Josephson junctions in the limit β L ≪ 1. In (b) the flux threading the SQUID loop is plottedversus the applied flux Φ ext .Negligible Screening: β L ≪ 1In the case β L ≪ 1 the flux generated by the circulating current is small compared to the flux quantumand therefore can be neglected compared to Φ ext . At a given Φ ext the maximum supercurrent of thedc-SQUID is found by maximizing (4.1.9) with respect to ϕ 1 . From the condition dI s /dϕ 1 = 0 we obtain(cos ϕ 1 + π Φ )extΦ 0= 0 . (4.1.18)( )Thus, at the maximum we have sin ϕ 1 + π Φ extΦ 0= ±1 and the maximum value of the supercurrent isfound by taking the sign of the sine term. That is, we obtain the result∣ ( ∣∣∣Is m ≃ 2I c cos π Φ )∣ext ∣∣∣, (4.1.19)Φ 0which is of course equivalent to (4.1.10). As shown in Fig. 4.2, Ism is a periodic function of the externalflux. Note that for a loop area of 2 mm 2 an applied field of 1 nT results in Φ ext = Φ 0 , that is, theperiodicity of the curve corresponds to the very small field of 1 nT, which is more than four orders ofmagnitude smaller than the earth magnetic field.Large Screening: β L ≫ 1For large inductance L we have LI c ≫ Φ 0 and the circulating current tends to compensate the applied flux.The loop of the SQUID looks more and more like the single loop formed by a superconducting wire. Thissituation was discussed already in section 1.2 when we discussed flux quantization in multiply connectedsuperconductors. Consequently, the total flux in the loop will tend to be quantized:Φ = Φ ext + LI cir ≃ nΦ 0 . (4.1.20)Let us consider the case of large screening a bit more closely. The transport supercurrent through theSQUID is the sum of the currents passing junction 1 and 2:I s = I c sinϕ 1 + I c sinϕ 2 . (4.1.21)© Walther-Meißner-Institut