1 Fundamentals of Normal Metal and Superconductor Electrodynamics Steven M. Anlage Center for Nanophysics and Advanced Materials Physics Department University of Maryland College Park, MD USA

2 Outline High Frequency Electrodynamics of Superconductors Experimental High Frequency Superconductivity Further Reading

3 The Three Hallmarks of Superconductivity Zero Resistance I V DC Resistance Temperature TcTc 0 Complete Diamagnetism Magnetic Induction Temperature TcTc 0 T>T c T<T c Macroscopic Quantum Effects Flux  Flux quantization  = n  0 Josephson Effects

4 Zero Resistance R = 0 only at  = 0 (DC) R > 0 for  > 0 E Quasiparticles Cooper Pairs  0 The Kamerlingh Onnes resistance measurement of mercury. At 4.15K the resistance suddenly dropped to zero Energy Gap

5 Perfect Diamagnetism Magnetic Fields and Superconductors are not generally compatible The Meissner Effect The Yamanashi MLX01 MagLev test vehicle achieved a speed of 361 mph (581 kph) in 2003 Super- conductor T>T c T<T c Spontaneous exclusion of magnetic flux T (T) (0) TcTc vacuum superconductor is independent of frequency (  < 2  ) magnetic penetration depth  surface screening currents

6 High Frequency Electrodynamics of Superconductors Why are Superconductors so Useful at High Frequencies? Normal Metal Electrodynamics The Two-Fluid Model London Equations BCS Electrodynamics Nonlinear Surface Impedance

7 Why are Superconductors so Useful at High Frequencies? Low Losses: Filters have low insertion loss  Better S/N, filters can be made small High Q  Filters have steep skirts, good out-of-band rejection NMR/MRI SC RF pickup coils  x10 improvement in speed of spectrometer Low Dispersion: SC transmission lines can carry short pulses with little distortion RSFQ logic pulses – 1 ps long, ~2 mV in amplitude:

8 Normal Metal Electrodynamics Consider a TEM wave incident normally on a metal half-space Metal Constitutive equations for metal Ohm’s law (local limit) LIH media Continuity Equation So ~ (1  cm)(8.85 x F/m) ~ s Hence we can ignore free charge in the conductor In reality free charge dissipates at the collision time scale,  c ~ – s

9 Normal Metal Electrodynamics Take the curl of the curl equations These are wave equations with a  dissipative term Ansatz with One finds The waves oscillate and decay as they enter the metal Define the skin depth For a metal with  = 1  -cm at 2.5 GHz,

10 Animated version Normal Metal Electrodynamics Griffiths, Electrodynamics Phase difference between E, B:

11 Electrodynamics of Superconductors in the Meissner State E Quasiparticles (Normal Fluid) Cooper Pairs (Super Fluid)  0 LsLs nn  Superfluid channel Normal Fluid channel Energy Gap  =  n – i  2 J = J s + J n JsJs JnJn Current-carrying superconductor J =  E  n = n n e 2  /m  2 = n s e 2 /m  n n = number of QPs n s = number of SC electrons  = QP momentum relaxation time m = carrier mass  = frequency T 0 n n n (T) n s (T) TcTc J n = n s (T) + n n (T)

1()1() 2()2() 1()1()  2 (  ) ~ 1/      i   Superconductor Electrodynamics “binding energy” of Cooper pair ( 100 GHz ~ few THz) T = 0 ideal s-wave Surface Impedance (  > 0) Normal State Superconducting State (  < 2  ) Penetration depth (0) ~ 20 – 200 nm  n s e   m  Finite-temperature: X s (T) =  L =   (T) → ∞ as T →T c (and  ps (T) → 0) Narrow wire or thin film of thickness t : L(T) =   (T) coth(t/ (T)) →  0 2 (T)/t Kinetic Inductance Superfluid density 2 ~ m/n s ~ 1/  ps 2 T n s (T) TcTc 0 0 Normal State (T > T c ) (Drude Model) 1/ 

13 Surface Impedance x -z y J E H Surface Resistance R s : Measure of Ohmic power dissipation RsRs conductor Local Limit Surface Reactance X s : Measure of stored energy per period XsXs X s =  L s =  L geo L kinetic

14 Two-Fluid Surface Impedance Because R s ~  2 : The advantage of HTS over Cu diminishes with increasing frequency R s crossover at f ~ 100 GHz at 77 K LsLs nn Superfluid channel Normal Fluid channel M. Hein, Wuppertal R s ~  2 R n ~  1/2

15 The London Equations Newton’s 2 nd Law for a charge carrier Superconductor: 1/   0 1 st London Equation  = momentum relaxation time J s = n s e v s 1 st London Eq. and (Faraday) yield: London surmise 2 nd London Equation These equations yield the Meissner screening vacuum superconductor L L ~ 20 – 200 nm L is frequency independent (  < 2  )

16 The London Equations continued Normal metalSuperconductor B is the source of J s, spontaneous flux exclusion E is the source of J n E=0: J s goes on forever Lenz’s Law 1 st London Equation  E is required to maintain an ac current in a SC Cooper pair has finite inertia  QPs are accelerated and dissipation occurs

17 BCS Microwave Electrodynamics Low Microwave Dissipation Full energy gap → R s can be made arbitrarily small for T < T c /3 in a fully-gapped SC R s,residual ~  at 1.5 GHz in Nb M. Hein, Wuppertal Filled Fermi Sea ss kxkx kyky HTS materials have nodes in the energy gap. This leads to power-law behavior of (T) and R s (T) and residual losses R s,residual ~  at 10 GHz in YBa 2 Cu 3 O 7-  Filled Fermi Sea dd node kxkx kyky

18 Nonlinear Surface Impedance of Superconductors YBa 2 Cu 3 O 7-  Thin Film Made by Pulsed Laser Deposition and sputtering 19 GHz The surface resistance and reactance values depend on the rf current level flowing in the superconductor Data from M. Hein, Wuppertal Similar results for X s (B s )

19 How can Superconductors become Nonlinear? Superconducting grains Josephson weak links Granularity Small  ~ grain boundary thickness Intrinsic Nonlinear Meissner Effect rf currents cause de-pairing – convert superfluid into normal fluid Nonlinearities are generally strongest near T c and weaken at lower temperatures J NL (T) calculated by theory (Dahm+Scalapino) JJs have a strongly nonlinear impedance McDonald + Clem PRB 56, (1997) Edge-Current Buildup J rf 2 (a.u.) Microstrip (Longitudinal view) Scanning Laser Microscope image YBCO strip at T = 79 K f = GHz, Laser Spot Size = 1  m See Appl. Phys. Lett. 88, (2006) + Vortex Entry and Flow Heating

20 How to Model Superconducting Nonlinearity? (1) Taylor series expansion of nonlinear I-V curve (Z. Y. Shen) = 0 if I(-V) = -I(V) 3 rd order term dominates V = V 0 sin(  t) input yields ~ V 0 3 sin(3  t) + … output 1/R linear term I V (2) Nonlinear transmission line model (Dahm and Scalapino) I C RL V L and R are nonlinear: additional terms 3 rd harmonics and 3 rd order IMD result

21 Experimental High Frequency Superconductivity Resonators Cavity Perturbation Measurements of Nonlinearity Topics of Current Interest Microwave Microscopy

22 Resonators … the building block of superconducting applications … Microwave surface impedance measurements Cavity Quantum Electrodynamics of Qubits Superconducting RF Accelerators Metamaterials (  eff < 0 ‘atoms’) etc. P out co-planar waveguide resonator P in f f0f0 |S 21 (f)| 2 resonator transmission Port 1 Port 2

23 Resonators (continued) T = 79 K P = - 10 dBm f = GHz YBCO/LaAlO 3 CPW Resonator Excited in Fundamental Mode Imaged by Laser Scanning Microscopy* W strip = 500  m 1 x 8 mm scan *A. P. Zhuravel, et al., J. Supercond. 19, 625 (2006)

24 Transmission Line Unit Cell Transmission Line Resonators Transmission Line Model Transmission Line Resonator Model C coupling

25 Resonators (continued)

26 Cavity Perturbation Objective: determine R s, X s (or  1,  2 ) from f 0 and Q measurements of a resonant cavity containing the sample of interest Sample at Temperature T Microwave Resonator InputOutput frequency transmission f0f0 ff f0’f0’  f’  f = f 0 ’ – f 0   (Stored Energy)  (1/2Q)   (Dissipated Energy) Quality Factor ~ microwave wavelength T1T1 T2T2 B Cavity perturbation means  f << f 0  is the sample/cavity geometry factor

27 Measurement of Nonlinearities Intermodulation is a practical problem signal frequency 2f 1 -f 2 f 1 f 2 2f 2 -f 1 2f 1 P out (dB) P in (dB) linear  –5 –10 –15 –20 –25 –15–10– rd -order intercept Point (TOI) 2  1 -  2, 2  2 -  1 Bandwidth of passband Intermodulation harmonic generation Nonlinear (i. e., signal strength dependent) microwave response induces undesirable signals within the passband by intermodulation. M. Hein, Wuppertal SC Device P in P out

28 Topics of Current Interest In Microwave Superconductivity Research Identifying and eliminating the microscopic sources of extrinsic nonlinearity Increase device yield Allows further miniaturization of devices Allow development of ILC Nb cavities with BCS-limited properties Superconducting Metamaterials: J. Opt. 13, (2011) Low-loss, compact, tunable metamaterial ‘atoms’ Controlling de-coherence in superconducting qubits Identify and eliminate two-level systems in dielectrics

29 Microwave Microscopy of Superconductors Use near-field optics techniques to obtain super-resolution images of: 1) Materials Properties: Nonlinear response 2) RF fields in operating devices 200  m loop probe 500Å YBCO film SrTiO 3 J 30° misorientation Bi-crystal grain boundary Laser Scanning Microscopy Image J rf 2 (x,y) in an operating superconducting microwave device Image J IMD reflectance J rf 2 20  m Phys. Rev. B 72, (2005) IEEE Trans. Appl. Supercond. 17, 902 (2007) See: 2MY-08 5MZ-03

30 Superconducting Metamaterials Build artificial ‘atoms’ with tailored electric and magnetic response An array of these sub-wavelength ‘atoms’ are described by  eff,  eff

31 Transmission Negative Index Passband with a Superconducting All-Nb Metamaterial Increasing temperature See: 3EZ-01 4EB-05 5EPG-05 arXiv: Appl. Phys. Lett. 87, (2005)

32 References and Further Reading Z. Y. Shen, “High-Temperature Superconducting Microwave Circuits,” Artech House, Boston, M. J. Lancaster, “Passive Microwave Device Applications,” Cambridge University Press, Cambridge, M. A. Hein, “HTS Thin Films at Microwave Frequencies,” Springer Tracts of Modern Physics 155, Springer, Berlin, “Microwave Superconductivity,” NATO- ASI series, ed. by H. Weinstock and M. Nisenoff, Kluwer, T. VanDuzer and C. W. Turner, “Principles of Superconductive Devices and Circuits,” Elsevier, T. P. Orlando and K. A. Delin, “Fundamentals of Applied Superconductivity,” Addison-Wesley, R. E. Matick, “Transmission Lines for Digital and Communication Networks,” IEEE Press, 1995; Chapter 6. Alan M. Portis, “Electrodynamcis of High-Temperature Superconductors,” World Scientific, Singapore, 1993.

33 Graduate course on Superconductivity (Anlage) Superconductivity Links Gallery of Abrikosov Vortex Lattices Wikipedia article on Superconductivity Superconductor Information for the Beginner YouTube videos of Superconductivity (Alfred Leitner)

34 Please Ask Questions!