Applied Superconductivity - Walther Meißner Institut - Bayerische ...

192 R. GROSS AND A. MARX Chapter 44.4.1 MagnetometersThe probably most simple and straightforward SQUID instrument is the SQUID magnetometer. Here, apick-up loop with inductance L p is connected to the input coil of the SQUID forming a superconductingflux transformer. That is, a small flux change δΦ p = N p A p δB ext applied to the pick-up loop is causinga shielding current I sh flowing through both the pick-up and the input coil. Here, A p and N p are the areaand turn number of the pick-up loop. The current through the input coil generates a magnetic flux that iscoupled into the SQUID loop. Flux quantization requires thatδΦ p + (L i + L p )I sh = N p A p δB ext + (L i + L p )I sh = 0 . (4.4.2)We have neglected the effects of the SQUID on the input circuit. The flux coupled into the SQUIDoperated in the flux locked loop isδΦ = M i |I sh | = M iδΦ pL i + L p= α√ L i LL i + L pδΦ p = α√ L i LL i + L pN p A p δB ext , (4.4.3)where M i = α √ L i L is the mutual inductance between L i and L.In order to find the minimum detectable value of δΦ p , we equate δΦ to the equivalent flux noise of theSQUID. Defining S p Φas the spectral density of the flux noise referred to the pick-up loop, we findS p Φ = (L i + L p ) 2M 2 iS Φ = (L i + L p ) 2α 2 L i LS Φ . (4.4.4)Introducing the equivalent noise energy referred to the pick-up loop, we obtainε p =Sp Φ= (L i + L p ) 22L p L i L pS Φ2α 2 L = (L i + L p ) 2L i L pεα 2 . (4.4.5)Analyzing (4.4.5) we see that it has the minimum valueε p ( f ) = 4ε( f )α 2 (4.4.6)for L i = L p . Thus, a maximum fraction α 2 /4 of the energy in the pick-up loop is transferred to theSQUID, if we match L p and L i . Here, we have neglected the noise currents in the input circuit and thefact that the input circuit reduces the SQUID inductance.With the optimum flux resolution for L p = L i we can give the corresponding magnetic field resolutionS p B ( f ) = Sp Φ ( f )/(πr2 p) 2 , where r p is the radius of the pick-up loop. With S p Φ = 8εL p/α 2 we obtainS p B ( f ) = 8L pα 2 (πr 2 ε( f ) . (4.4.7)2p)The inductance of the superconducting pick-up coil made from a wire with radius r 0 is given byL p = µ 0 r p [ln(8r p /r 0 ) − 2] and can be approximated by L p ≃ 5µ 0 r p over a wide range of values r p /r 0 .Therefore, we obtain S p B ( f ) ≈ 4µ 0ε/α 2 r 3 p ∝ 1/A 3/2p . This shows that we can increase the magnetic fieldresolution by increasing the radius of the pick-up loop while keeping L p = L i . In practice, of coursethere is a limitation due to the finite size of the cryostat used for cooling down the system. Furthermore,a spatially varying signal is averaged over the area of the pick-up loop. Taking ε ≃ 10 −28 J/Hz, α = 1and r p = 25 mm, we calculate √ S p B ≃ 5 × 10−15 T/ √ Hz. This is a much better value than that achievedwith non-superconducting magnetometers.© Walther-Meißner-Institut