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180 R. GROSS AND A. MARX Chapter 44.2 Additional Topic:The rf-SQUIDIn contrast to the dc SQUID the rf-SQUID is formed by a superconducting loop containing only a singleJosephson junction. Although it is still widely used today, it has seen less development in recent yearscompared to the dc SQUID. Whereas the dc SQUID is operated by applying a dc current and measuringthe time-averaged voltage, the rf-SQUID is operated by applying an rf current via a tank circuit inductivelycoupled to the SQUID loop and measuring the time-averaged rf-voltage of the tank circuit. Thismakes the choice of names obvious. The advantage of the rf-SQUID compared to the dc SQUID is thefact that it requires only a single Josephson junction and no dc current has to be applied. Therefore, nocurrent leads have to be attached which guarantees safe operation and good protection against currentspikes. However, as we will see below, at 4.2 K the energy resolution of the rf-SQUID is limited by theread-out electronics and therefore is worse than that of dc-SQUIDs. However, at 77 K, the commonlyused operation temperature of SQUIDs based on high temperature superconductors, this problem is relaxed.Although less sensitive than the dc-SQUID, the rf-SQUID is entirely adequate for a large varietyof applications and is until today more widely used than the dc SQUID.4.2.1 The Zero Voltage StateWe consider the rf-SQUID shown in Fig. 4.14. In the same way as for the dc SQUID the total phasechange along a closed contour line is 2πn:∮C∇θ · dl = 2π n = (θ Qb − θ Qa ) + (θ Qa − θ Qb ) + 2π n (4.2.1)I s =I c sin ϕQ aQ bBI cirFigure 4.14: The rf-SQUID formed by a single Josephson junction intersecting a superconducting loop. Thebroken line indicates the closed contour path of the integration.© <strong>Walther</strong>-Meißner-<strong>Institut</strong>
Section 4.2 APPLIED SUPERCONDUCTIVITY 181Using ∇θ = 2πΦ 0(ΛJ s + A) (compare (2.2.2)) and ϕ = θ 2 − θ 1 − 2πΦ 0θ Qb − θ Qa = +ϕ + 2π ∫Q bΦ 0θ Qa − θ Qb =∫Q bQ a2∫A · dl (compare (2.2.3)) we obtain1Q aA · dl (4.2.2)∇θ · dl = + 2π ∫Q bΦ 0Substitution these expression into (4.2.1) yieldsϕ = − 2πΦ 0∮CA · dl − 2π ∫Q bΦ 0Q aΛJ s · dl + 2π∫Q bΦ 0Q aA · dl . (4.2.3)Q aΛJ s · dl . (4.2.4)The integration of A around the close contour is equal to the total flux Φ enclosed by the superconductingloop. The integration of J s follows the same contour C but excludes the integration over the insulatingbarrier. Furthermore, if the superconducting loop consists of a superconducting material with a thicknesslarge compared to the London penetration depth λ L , the integration path can be taken deep inside thesuperconducting material where the current density is negligible. Therefore, the integral involving thecurrent density can be omitted and we obtainϕ = − 2πΦΦ 0. (4.2.5)The phase difference across the junction determines the supercurrent(I s = I c sin − 2πΦ )Φ 0= −I c sin 2πΦΦ 0. (4.2.6)The total flux Φ threading the loop is given by the sum of the flux Φ ext due to an externally appliedmagnetic field B ext and the flux LI cir due to the circulating current I cir . With I cir = I s we obtainΦ = Φ ext + LI cir = Φ ext − LI c sin 2πΦΦ 0. (4.2.7)Introducing the screening parameter β L,rf = 2πLI c /Φ 0 we can rewrite this expression and obtain 54Φ= Φ ext− β (L,rfΦ 0 Φ 0 2π sin 2π Φ )Φ 0. (4.2.8)The variation of Φ with Φ ext is shown in Fig. 4.15 for two different values of β L,rf . The regions withpositive slope are stable, whereas those with negative slope are not. Regions with negative slope onlyappear for β L,rf > 1. Most practical rf-SQUIDs are operated in this regime.54 Note that the screening parameter deviates from that defined for the dc SQUID by a factor of π.2005
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