Applied Superconductivity - Walther Meißner Institut - Bayerische ...

Section 4.1 APPLIED SUPERCONDUCTIVITY 16343Φ / Φ 0210-1β L = 2/πβ L = 1β L = 0.2-2-3-4β L = 2β L = 6-4 -3 -2 -1 0 1 2 3 4Φ ext/ Φ 0Figure 4.3: The total magnetic flux Φ plotted versus the applied magnetic flux Φ ext for a dc-SQUID with twoidentical Josephson junctions for different values of the screening parameter β L .On the other hand, the circulating screening current is given byI cir = I c2 (sinϕ 1 − sinϕ 2 ) . (4.1.22)Both (4.1.21) and (4.1.22) are constraint by the conditionϕ 2 − ϕ 1 = 2πΦΦ 0. (4.1.23)Note that here the magnetic flux is the sum of the external flux Φ ext and the flux Φ cir = LI cir due to thescreening current. Given the applied current I and the total flux Φ we have two equations for the twophase differences ϕ 1,2 and hence can solve for them and finally for Φ cir and Φ ext . For example, if I ≃ 0,we have sinϕ 1 ≃ −sinϕ 2 and obtainΦ ext = Φ + LI c sin(π Φ )Φ 0Φ ext= Φ + β (LΦ 0 Φ 0 2 sin π Φ )Φ 0or. (4.1.24)This relationship of course can be inverted to obtain Φ as a function of Φ ext as shown in Fig. 4.3.An interesting case occurs for Φ = nΦ 0 , for which ϕ 1 = ϕ 2 + n2π, so that I cir = 0 and Φ = Φ ext . We seethat the SQUID response to Φ ext in integer multiples of Φ 0 is not affected by the screening. However,for practical applications it is often required that the relation between Φ and Φ ext is single-valued andnon-hysteretic. As shown by Fig. 4.3 this is possible only for small values of the screening parameterβ L . This results from the fact that the maximum possible value of Φ cir is LI c . Since roughly speakinga multivalued relationship between Φ and Φ ext can be avoided only for |Φ cir | ≤ Φ 0 /2, we immediatelysee that this is equivalent to LI c ≤ Φ 0 /2 or β L = 2LI c /Φ 0 ≤ 1. A more detailed analysis shows that ahysteretic Φ(Φ ext ) dependence can be avoided for β L ≤ 2/π.2005