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Superconducting Quantum Interference Devices Superconducting Quantum Interference Devices John Clarke University of California, Berkeley Wallenberg Centre for Quantum Technology Summer School Säröhus, Sweden 22 – 26 August 2019 Superconducting Quantum Interference Devices • History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise • Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue SQUID Applications • Brief Topics • Cosmology • Shedding Light on Dark Energy • Cold Dark Matter: The Hunt for the Axion • Ultra Low Field Magnetic Resonance Imaging • Epilogue Discussion Superconducting Quantum Interference Devices • History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise • Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue A Little Personal History: How Did I Get Into SQUIDs? King’s College Chapel, Cambridge English Gothic 1446 - 1515 St. Bene’t’s Church Anglo-Saxon 1000 – 1050 AD The Perse School The Perse School was founded in 1615 by Dr Stephen Perse who left money in his will to educate 100 boys from Cambridge and nearby villages at no cost. The school was originally located on “Free School Lane”. Perse Outside the shop that was once my grandfather’s picture-framing shop Grandad’s shop Perse Entrance to the Cavendish Laboratory Through the gate… The Royal Society Mond Laboratory Grandad’s shop Mond Perse 1 October 1964 Eric Gill 1933 Thesis advisor: Brian Pippard Royal Society Mond Laboratory Thesis research required Cavendish Laboratory measuring voltages of Cambridge University 10−12 to10−13 volt State-of-the-art 10−9 volt Brian Josephson Explains Josephson Tunneling November 1964 Courtesy Brian Josephson The Discovery of Superconductivity (1911) Hg ) Ω Resistance ( Resistance Heike Kamerlingh Onnes 4.0 4.2 4.4 Temperature (K) Leiden, The Netherlands Resistance vanishes below the transition (or critical) temperature Tc The Theory of Superconductivity (1957) Bardeen-Cooper-Schrieffer Theory (1957) Cooper pairs condensed into a single, macrosopic quantum state I I Supercurrent is carried by Cooper pairs of electrons with charge -2e BCS theory: electron-phonon interaction water bed The Birth of Superconducting Electronics: 1961 - 1964 Flux Quantization Φ = nΦ0 Φ = nΦ0 (n = 0, ±1, ±2, ...) -15 2 where Φ0 ≡ h/2e ≈ 2 x 10 Tm J is the flux quantum Experimental observation 1961: Deaver and Fairbank Doll and Näbauer Josephson Tunneling Insulating barrier Superconductor 1 Superconductor 2 I I ~ 20 Å V I = I0sinδ δ = φ1 – φ2 dδ/dt = 2eV/ħ = 2πV/Φ0 Josephson 1962 Josephson Tunneling Insulating barrier Superconductor 1 Superconductor 2 Sn-SnOx-Pb I I Tunnel junction ~ 20 Å 1.5 K V I = I0sinδ δ = φ1 – φ2 0.006 G dδ/dt = 2eV/ħ 0.4 G = 2πV/Φ0 Josephson 1962 Anderson and Rowell 1963 Bell Labs Birth of the Superconducting Quantum Interference Device (SQUID) I I0 Φ0 Φ V Φ Sn-SnOx-Sn junctions • Critical current versus applied magnetic field for two different junction spacings • Rapid oscillations due to interference, slow oscillations due to diffraction • Essential physics analogous to two-slit interference in optics • Sensitive detector of magnetic field Jaklevic, Lambe, Silver and Mercereau (Ford Motor Company Laboratory) 1964 Brian Josephson Explains Josephson Tunneling November 1964 Courtesy Brian Josephson The Very Next Day Brian: “John, How would you like a voltmeter with a resolution of 2 × 10−15 V in 1 second?” Brian Pippard November 1964 Brian’s Idea R I M = L V L L V in out τ = L/R M IΦ0 Digital voltmeter: IΦ0 = Φ0/M = Φ0/L -15 Voltage resolution: Vin = IΦ0R = (Φ0/L)R = Φ0/τ = 2 × 10 V for τ = 1 s Six order of magnitude improvement over the state of the art! Sir Brian Pippard Serves Tea to Lady Bragg Courtesy Cavendish Laboratory At Tea • Paul Wraight, a fellow research student, pointed out that Nb has a surface oxide layer and PbSn solder is a superconductor. At Tea • Paul Wraight, a fellow research student, pointed out that Nb has a surface oxide layer and PbSn solder is a superconductor. Why don’t you put a blob of solder on a piece of Nb wire? Niobium Before dinner I 4.2 K I Copper SnPb solder V 5 5mm mm The Day After Niobium I Brian Pippard: “It looks as though a slug crawled through the window last night and I Copper expired on your desk!” SnPb solder V 5 mm SLUG The SLUG Superconducting Low-Inductance Undulatory Galvanometer Niobium IB I I Voltage Copper SnPb solder Current IB in niobium wire V 5 mm I B February 1965 The SLUG as a Voltmeter V Niobium δV 1 µΩ δΦ Φ 0 1 2 Φo Flux bias where V vs. Copper V Φ is steepest Solder Voltage noise 10 fVHz-1/2 (10-14 VHz-1/2) 5 mm Analog voltmeter Why Does IB Modulate the Critical Current? IB IB • Blue region indicates penetration depths in Nb wire and solder • Current IB couples flux into this region • Why are there typically only 2 or 3 junctions? Other Early SQUIDs Zimmerman and Silver 1966 John Wires Up a SLUG One has to be lucky! In January 1968 I moved to the University of California, Berkeley for a 1-year postdoctoral appointment. Apart from sabbatical leaves at various institutions in Europe, I have been there ever since. Thin-Film Cylindrical SQUID Shadow masks 5 mm 10-14 THz-1/2 (10 fTHz-1/2) JC, Goubau, Ketchen (1974) Geophysical Prospecting Superconducting Quantum Interference Devices • History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise • Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue Josephson Tunnel Junction I Superconducting film Superconducting film + oxide I V • Junction has intrinsic capacitance C Resistively-Shunted Junction (RSJ) Model U(δ) I I N 0 I < I . 0 < δ! > = 0 δ V/R + I0sin δ + C V ! = I + IN(t) ! Neglect IN(t), set V = "δ/2e: !C ! 2e ∂U U(δ) "δ"+ δ" + = 0, < δ! > = 0 2e 2eR ! ∂δ δ δ Φ0 where U(δ) = − (Iδ + I0 cosδ). 2π δ! > 0 • This differential equation δ represents the motion of a particle of mass ∝ C in a tilted washboard Stewart (1968) potential, U, with damping ∝ 1/R McCumber (1968) Effects of Damping in the RSJ Model !C ! 2e ∂U "δ"+ δ" + = 0, 2e 2eR ! ∂δ Underdamped Case: !δ! term dominates δ! term βc≡ (2πI0R/Φ0)(RC) = ωJRC >> 1 • When I is reduced to below I0 the kinetic energy of the particle keeps it rolling: Hence hysteresis in the I-V characteristic. Overdamped Case: δ! term dominates !δ! term I βc≡ (2πI0R/Φ0)(RC) = ωJRC << 1 • When I is reduced to below I0 the V = R(I2-I 2)1/2 damping stops the motion of the 0 particle: Hence no hysteresis in the I-V characteristic. V We are concerned only with the Overdamped Case Thermal Noise in the Overdamped RSJ (βc << 1) "C " 4k BT Langevin Equation !δ! + δ! + I0 sin δ = I + I N (t), SI (f ) = 2e 2eR R U I > I0 δ Noise rounding (Ambegaokar and Halperin 1969) I I < I0 V(t) I0 t V Voltage Noise in the Overdamped RSJ: Thermal (Likharev and Semenov 1972) 2 ⎡ 4k T 1 I 4k T ⎤ I B ⎛ 0 ⎞ B 2 (I > I ) S v (f m ) = ⎢ + ⎜ ⎟ ⎥ R D 0 ⎣⎢ R 2⎝ I ⎠ R ⎦⎥ I0 “Straight through” “Mixed down” Dynamic noise noise resistance V Measurement frequency fm << Josephson frequency fJ = 2eV/h Voltage Noise in the Overdamped RSJ: Quantum Quantum Langevin equation (ħC/2e)! δ! + (ħ/2eR) δ! + I0sinδ = I + IN(t) SI(f) = (2hf/R)coth(hf/2kBT) −1 = (4hf/R){[exp(hf/kBT) − 1] + ½} Planck Zero distribution point fluctuations Setting hfJ = 2eV: 2 2 Sv(fm) = [4kBT/R + (2eV/R)(I0/I) coth(eV/kBT)]RD (I > I0) “Straight “Mixed through” down” noise noise 2 In the quantum limit eV(I0/I) >> 2kBT: 2 2 Sv(fm) = (2eV/R)(I0/I) RD Likharev and Semenov (1972) Koch, Van Harlingen & JC (1980) Quantum Noise: Experiment fm = 70, 106, 183 kHz Immersed in liquid helium helium liquid in Immersed Rc Nyquist noise calibration resistor Voltage noise across tank circuit: 2 2 2 2 Q Sv(fm) = ω Lt [Sv(fm)/RD ] (Q = ωLt/RD) 2 2 Here, Sv(fm)/RD = [4kBT/R + (2eV/R)(I0/I) coth(eV/kBT)] “Straight “Mixed through” down” noise noise Koch, Van Harlingen & JC (1981) Current Noise in Shunt Resistor No fitted parameters • Quantum fluctuations With zero must dominate point term to achieve a quantum limited SQUID amplifier Planck −1 (4hν/R){[exp(hν/kBT) − 1] + ½} (4hν/R){[exp(hν/k T) − 1]−1} Koch, Van Harlingen B & JC (1981) Superconducting Quantum Interference Devices • History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise • Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue The DC Superconducting Quantum Interference Device • dc SQUID Two Josephson junctions on a superconducting ring I V • Current-voltage (I-V) characteristic modulated by magnetic flux Φ: -15 2 Period one flux quantum Φo = h/2e = 2 x 10 T m I V nΦo Ib δV (n+1/2)Φo V δΦ Φ ΔV 0 1 2 Φo The DC SQUID: Important Parameters Flux-to-voltage transfer coefficient: VΦ = (∂V/∂Φ)I Voltage noise VN(t) Equivalent flux noise ΦN(t) = VN(t)/VΦ Voltage noise spectral density SV(f) 2 Equivalent flux noise spectral density SΦ(f) = SV(f)/VΦ Thermal Noise Theory for the dc SQUID 1 2 ( C/ 2e)!δ! ( / 2eR)δ! I sin δ I (I/ 2) J " 1 + " 1 + 0 1 + N1 = − SI (f ) = 4k BT/ R ( C/ 2e) !δ! ( / 2eR)δ! I sin δ I (I/ 2) J " 2 + " 2 + 0 2 + N 2 = + δ1 − δ2 = (2π / Φ0 )(Φ + LJ) δ"1 + δ" 2 = 4eV/ ! Tesche & Clarke 1976 Thermal Fluctuations: Constraints on dc SQUID • Josephson coupling energy much greater than thermal energy: I0Φ0/2π >> kBT (Γ ≡ 2πkBT/I0Φ0 << 1) ⎧0.17µA at 4.2K or I0 >> 2πkBT/Φ0 ≈ ⎨ ⎩3.3µA at 77K • Energy of a flux quantum in the SQUID loop much greater than thermal energy: 2 Φ 0 / 2L >> k BT 2 ⎧5.6nH at 4.2K or L << Φ 0 / 2 k BT ≈ ⎨ ⎩0.33nH at 77K • At least a factor of 5 greater is required for the inequalities.
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