SRF FUNDAMENTALS. Jean Delayen. First Mexican Particle Accelerator School Guanajuato. 26 Sept 3 Oct 2011

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1 First Mexican Particle Accelerator School Guanajuato 6 Sept 3 Oct 011 SRF FUNDAMENTALS Jean Delayen Center for Accelerator Science Old Dominion University and Thomas Jefferson National Accelerator Facility Page 1

2 Historical Overview Page

3 Perfect Conductivity Kamerlingh Onnes and van der Waals in Leiden with the helium 'liquefactor' (1908) Page 3

4 Perfect Conductivity Persistent current experiments on rings have measured s s 10 s > n 15 Resistivity < 10-3 Ω.cm Decay time > 10 5 years Perfect conductivity is not superconductivity Superconductivity is a phase transition A perfect conductor has an infinite relaxation time L/R Page 4

5 Perfect Diamagnetism (Meissner & Ochsenfeld 1933) Perfect conductor Superconductor B = t 0 B = 0 Page 5

6 Penetration Depth in Thin Films Very thin films Very thick films Page 6

7 Critical Field (Type I) Superconductivity is destroyed by the application of a magnetic field È ÊT ˆ Hc( T) Hc(0) Í1- Á ËT ÍÎ c Type I or soft superconductors Page 7

8 Critical Field (Type II or hard superconductors) Expulsion of the magnetic field is complete up to H c1, and partial up to H c Between H c1 and H c the field penetrates in the form if quantized vortices or fluxoids p f 0 = e Page 8

9 Thermodynamic Properties Entropy Specific Heat Energy Free Energy Page 9

10 Thermodynamic Properties st When T < T phase transition at H = H ( T) is of 1 order fi latent heat c c nd At T = T transition is of order fi no latent heat c jump in specific heat C es ( T ) 3 C ( T ) c en c C ( T) = g T en C ( T) ª at es 3 electronic specific heat reasonable fit to experimental data Page 10

11 Thermodynamic Properties At T : S ( T ) = S ( T ) c s c n c The entropy is continuous S C Recall: S(0) = 0 and = T T Tc c at gt 3g T fi dt = dt Æ a = C = 3g Ú 0 3 T 3 es T 0 T Tc Tc 3 T T Ss( T ) = g Sn( T ) = g T T 3 c Ú c For T < T S ( T) < S ( T) c s n superconducting state is more ordered than normal state A better fit for the electron specific heat in superconducting state is bt - c T C = ag T e with a ª 9, bª 1.5 for T T es c c Page 11

12 Energy Difference Between Normal and Superconducting State at U ( T ) = U ( T ) n c s c Εnergy is continuous 3 g g U T U T C C dt T T T T ( ) Tc 4 4 n ( )- s ( ) = Ú ( es - en) = ( ) c - - c - T 4 Tc 1 H H c c n s g c ( ) ( ) T= 0 U 0 - U 0 = T = 4 8p H c at T π 0, is the free energy difference 8p Hc ( T) 1 È ÊT ˆ =D F = ( Un -Us) -T( Sn - Sc) = g Tc Í1-8p 4 Á T ÍÎ Ë c 1 È ÊT ˆ Hc( T) = ( pg ) TcÍ1- Á T ÍÎ Ë c is thecondensationenergy 8p The quadratic dependence of critical field on T is related to the cubic dependence of specific heat Page 1

13 Isotope Effect (Maxwell 1950) The critical temperature and the critical field at 0K are dependent on the mass of the isotope (0) - T with c Hc M a a 0.5 Page 13

14 Energy Gap (1950s) At very low temperature the specific heat exhibits an exponential behavior - bt c / c T s μ e with b 1.5 Electromagnetic absorption shows a threshold Tunneling between superconductors separated by a thin oxide film shows the presence of a gap Page 14

15 Two Fundamental Lengths London penetration depth λ Distance over which magnetic fields decay in superconductors Pippard coherence length ξ Distance over which the superconducting state decays Page 15

16 Two Types of Superconductors London superconductors (Type II) λ>> ξ Impure metals Alloys Local electrodynamics Pippard superconductors (Type I) ξ >> λ Pure metals Nonlocal electrodynamics Page 16

17 Material Parameters for Some Superconductors Page 17

18 Phenomenological Models (1930s to 1950s) Phenomenological model: Purely descriptive Everything behaves as though.. A finite fraction of the electrons form some kind of condensate that behaves as a macroscopic system (similar to superfluidity) At 0K, condensation is complete At T c the condensate disappears Page 18

19 Two Fluid Model Gorter and Casimir T < T x = c fractionof "normal" electrons (1- x) : fractionof "condensed" electrons (zero entropy) Assume: Minimizationof FT x f T + -x f T 1/ ( )= n( ) (1 ) s( ) 1 fn ( T) =- g T 1 fs( T) =- b =- gt 4 ( ) F T T fi Ces = 3g T c Ê T gives x = Á ËT 3 C C free energy independent of temperature È 1/ Ê T ˆ fi FT ( ) = x fn( T) + (1- x) fs( T) = - b Í1+ Á T ÍÎ Ë C ˆ 4 4 Page 19

20 Two Fluid Model Gorter and Casimir Superconducting state: Normal state: È 1/ Ê T ˆ FT ( ) = x fn( T) + (1- x) fs( T) = - b Í1+ Á T ÍÎ Ë C g Ê T ˆ FT ( ) = fn ( T) = - T = -b Á ËT C 4 H c Recall = difference in free energy between normal and 8p superconducting state È Ê T ˆ Hc( T) Ê T ˆ = b Í1 -Á 1 T fi = - C Hc(0) ÁT ÍÎ Ë Ë C The Gorter-Casimir model is an ad hoc model (there is no physical basis for the assumed expression for the free energy) but provides a fairly accurate representation of experimental results Page 0

21 Model of F & H London (1935) Proposed a -fluid model with a normal fluid and superfluid components n s : density of the superfluid component of velocity v s n n : density of the normal component of velocity v n u m =-ee superelectrons are accelerated by E t J =-en u s s J t s = nse E m superelectrons J n = s E n normal electrons Page 1

22 Model of F & H London (1935) J t s = nse E m B Maxwell: E =- t Ê m ˆ m fi J + B = 0 fi Js + B=Constant tá Ën e n e s s s F&H London postulated: m ne s J + B =0 s Page

23 Model of F & H London (1935) combine with B= m0js m ne - 0 m 0 s B B = ( ) = o exp [- / ll] B x B x l 1 È m L = Í m0ne s Î The magnetic field, and the current, decay exponentially over a distance λ (a few 10s of nm) Page 3

24 Page ( ) (0) 1 L s s C L L C m ne T n T T T T l m l l È = Í Î È Ê ˆ Í μ - Á Ë Í Î = È Ê ˆ - Í Á Ë Í Î From Gorter and Casimir two-fluid model Model of F & H London (1935)

25 Model of F & H London (1935) B London Equation: l Js = - =-H m0 A= H choose i A= 0, A = 0 n on sample surface (London gauge) J s 1 =- A l Note: Local relationship between J and s A Page 5

26 Penetration Depth in Thin Films Very thin films Very thick films Page 6

27 Quantum Mechanical Basis for London Equation ÂÚ Ï e e J () r = Ì Èy y - y y - A( r ) y y d( r-r ) dr -dr n mi Î Ó mc * * * n n n n 1 n In zero field A= 0 J( r) = 0, y = y 0 Assume y is "rigid", ie the field has no effect on wave function r() re J() r =- A() r me r() r = n Page 7

28 Pippard s Extension of London s Model Observations: -Penetration depth increased with reduced mean free path -H c and T c did not change -Need for a positive surface energy over 10-4 cm to explain existence of normal and superconducting phase in intermediate state Non-local modification of London equation Local: Non local: 1 J = - A cl J () r = = + x x 0 3s 4px lc 0 Ú RÈÎR i A( r ) e 4 R R - x du Page 8

29 London and Pippard Kernels Apply Fourier transform to relationship between c J () r and A() r : J() k =- K() k A() k 4p Specular: l eff Effective penetration depth dk = Ú Diffuse: l = p o Kk ( ) eff + k Kk ( ) Ú È ln Í 1+ o k Î p dk Page 9

30 London Electrodynamics Linear London equations J s E 1 =- H - H = t l m l 0 0 together with Maxwell equations H H = Js E =-m 0 t describe the electrodynamics of superconductors at all T if: The superfluid density n s is spatially uniform The current density J s is small Page 30

31 Ginzburg-Landau Theory Many important phenomena in superconductivity occur because n s is not uniform Interfaces between normal and superconductors Trapped flux Intermediate state London model does not provide an explanation for the surface energy (which can be positive or negative) GL is a generalization of the London model but it still retain the local approximation of the electrodynamics Page 31

32 Ginzburg-Landau Theory Ginzburg-Landau theory is a particular case of Landau s theory of second order phase transition Formulated in 1950, before BCS Masterpiece of physical intuition Grounded in thermodynamics Even after BCS it still is very fruitful in analyzing the behavior of superconductors and is still one of the most widely used theory of superconductivity Page 3

33 Ginzburg-Landau Theory Theory of second order phase transition is based on an order parameter which is zero above the transition temperature and non-zero below For superconductors, GL use a complex order parameter Ψ(r) such that Ψ(r) represents the density of superelectrons The Ginzburg-Landau theory is valid close to T c Page 33

34 Ginzburg-Landau Equation for Free Energy Assume that Ψ(r) is small and varies slowly in space Expand the free energy in powers of Ψ(r) and its derivative f * b 4 1 Ê e ˆ h = fn0 + ay + y + y * m Á - A + Ë i c 8p Page 34

35 Field-Free Uniform Case f - f n0 f - f n 0 = ay + b y 4 f - f n0 a > 0 a < 0 a y =- b y y y Near T c we must have b > 0 a( t) = a ( t-1) At the minimum Hc f - f = - a n0 y and Hc (1 t) 8p = - b fi μ - Page 35

36 Field-Free Uniform Case f - f n = ay + 0 b y 4 a y =- b b > 0 a( t) = a ( t-1) fiy μ(1 -t) It is consistent with correlating Ψ(r) with the density of superelectrons ns μl - μ - (1 t) near Tc At the minimum a Hc f - fn0 = - = - (definition of Hc) b 8p fi H μ(1 -t) c which is consistent with H = H -t (1 ) c c0 Page 36

37 Field-Free Uniform Case Identify the order parameter with the density of superelectrons n s 1 l (0) Y( T ) L 1 a ( T ) L( T) L( T) Y(0) n =Y fi = =- l l b since a T Hc ( T) = b 8p 1 ( ) H ( T) l ( T) H ( T) l ( T) na( T) =- n = 4 (0) 4 (0) 4 c L c L and b 4 p ll p ll Page 37

38 Field-Free Nonuniform Case Equation of motion in the absence of electromagnetic field 1 m - y + a T y + b y y = * ( ) 0 Look at solutions close to the constant one a ( T ) y = y + d where y = - b To first order: 1 4 m a ( T) d - d = * 0 Which leads to d ª e - r/ x ( T) Page 38

39 Field-Free Nonuniform Case - r / x ( T ) 1 p n l (0) d ª e where x( T) = = L * * m a ( T) mhc( T) ll( T) is the Ginzburg-Landau coherence length. It is different from, but related to, the Pippard coherence length. x ( T ) GL parameter: ll ( ) k ( T ) = T x ( T ) x 0 ( 1- t ) 1/ Both l ( T) and x( T) diverge as T Æ T but their ratio remains finite L c k ( T ) is almost constant over the whole temperature range Page 39

40 Fundamental Lengths London penetration depth: length over which magnetic field decay l ( T ) L * 1/ b ˆ Tc e a Tc T Ê m = Á Ë - Coherence length: scale of spatial variation of the order parameter (superconducting electron density) x ( T ) 1/ Ê ˆ Tc = Á * Ë4m a T - T c The critical field is directly related to those parameters f0 Hc ( T) = x( T) l ( T) L Page 40

41 Surface Energy 1 s ÈHcx H l 8p Î - H l : Energy that can be gained by letting the fields penetrate 8p Hcx : Energy lost by "damaging" superconductor 8p Page 41

42 Surface Energy 1 s Hcx H l 8p È Î - Interface is stable if s >0 If x >> l s>0 Superconducting up to H where superconductivity is destroyed globally c Æ x If l >> x s<0 for H > H c l Advantageous to create small areas of normal state with large quantized fluxoids area to volume ratio More exact calculation (from Ginzburg-Landau): l 1 k = < x : Type I l 1 k = > x : Type II Page 4

43 Magnetization Curves Page 43

44 Intermediate State At the center of each vortex is a normal region of flux h/e Vortex lines in Pb.98 In.0 Page 44

45 Critical Fields Even though it is more energetically favorable for a type I superconductor to revert to the normal state at H c, the surface energy is still positive up to a superheating field H sh >H c metastable superheating region in which the material may remain superconducting for short times. Type I H H c sh Thermodynamic critical field Hc Superheating critical field k Field at which surface energy is 0 Type II H c Thermodynamic critical field H H c c1 = H H k c c H c 1 (ln k +.008) H k c (for k 1) Page 45

46 Superheating Field Ginsburg-Landau: H sh 0.9Hc k for k <<1 1. H for k 1 c 0.75 H for k >> 1 c The exact nature of the rf critical field of superconductors is still an open question Page 46

47 Material Parameters for Some Superconductors Page 47

48 BCS What needed to be explained and what were the clues? Energy gap (exponential dependence of specific heat) Isotope effect (the lattice is involved) Meissner effect Page 48

Cooper Pairs Assumption: Phonon-mediated attraction between electron of equal and opposite momenta located within of Fermi surface w D Moving electron distorts lattice and leaves behind a trail of

49 Cooper Pairs Assumption: Phonon-mediated attraction between electron of equal and opposite momenta located within of Fermi surface w D Moving electron distorts lattice and leaves behind a trail of positive charge that attracts another electron moving in opposite direction Fermi ground state is unstable Electron pairs can form bound states of lower energy Bose condensation of overlapping Cooper pairs into a coherent Superconducting state Page 49

50 Cooper Pairs One electron moving through the lattice attracts the positive ions. Because of their inertia the maximum displacement will take place behind. Page 50

51 BCS The size of the Cooper pairs is much larger than their spacing They form a coherent state Page 51

52 BCS and BEC Page 5

53 BCS Theory 0,1 :states where pairs ( q,-q) are unoccupied, occupied q q a, b : probabilites that pair ( q,- q) is unoccupied, occupied q q BCS ground state P q ( a 0 1 ) q b q q q Y = + Assume interaction between pairs q and k V qk = - V if xq wd and xk wd = 0 otherwise Page 53

54 Hamiltonian H = e n + c c k * q Â k BCS * * k k qk q -q k -k qk destroys an electron of momentum k creates an electron of momentum k * n = c c number of electrons of momentum k k k k Â V cc cc Ground state wave function P q * * ( q q q -q) 0 Y = a + b c c f Page 54

55 BCS The BCS model is an extremely simplified model of reality The Coulomb interaction between single electrons is ignored Only the term representing the scattering of pairs is retained The interaction term is assumed to be constant over a thin layer at the Fermi surface and 0 everywhere else The Fermi surface is assumed to be spherical Nevertheless, the BCS results (which include only a very few adjustable parameters) are amazingly close to the real world Page 55

56 BCS Is there a state of lower energy than the normal state? a a = 0, b = 1 for x < 0 q q q = 1, b = 0 for x > 0 q q q yes: bq 1 = - x x q +D q 0 where D = 0 w D È 1 sinh Í Îr(0) V w D e 1 - r (0) V Page 56

57 BCS Critical temperature È 1 ktc = 1.14 w D exp Í - VN( EF ) Î D 0 = 1.76 kt ( ) c Coherence length (the size of the Cooper pairs) uf x 0 =.18 kt c Page 57

58 Page 58 BCS Condensation Energy (0) 8 / 10 / 10 s n F V E E H N k K k K r e p e D - = - Ê ˆ D - D = Á Ë D F Condensation energy:

59 BCS Energy Gap At finite temperature: Implicit equation for the temperature dependence of the gap: w 1 D 1 tanh È( e +D ) / kt = Î 1 d e V r(0) ( e +D ) Ú 0 Page 59

60 BCS Excited States Energy of excited states: ek = x k +D 0 Page 60

61 BCS Specific Heat Specific heat C es Ê D ˆ T exp Á- for T< c Ë kt 10 Page 61

62 Electrodynamics and Surface Impedance in BCS Model H ex f H0f + Hex f = i t e Hex = Â A( ri, t) pi mc is treated as a small perturbation H rf << H c There is, at present, no model for superconducting surface resistance at high rf field RR [ AI ] ( w, RTe, ) J μ Ú dr 4 R c J( k) =- K( k) A( k) 4p K(0) π 0: Meissner effect - R l similar to Pippard's model Page 6

63 Penetration Depth l = p Ú dk Kk ( ) + k dk ( specular) 1 Represented accurately by l near 4 ÊTc ˆ 1- Á Ë T T c Page 63

64 Surface Resistance Temperature dependence -close to T c : dominated by change in () t 4 3 (1- t ) Tc - for T < : dominated by density of excited states e A Ê D ˆ Rs w exp Á- T Ë kt Frequency dependence w is a good approximation l t -D kt Page 64

65 Surface Resistance Page 65

66 Surface Resistance Page 66

67 Surface Resistance Page 67

68 Surface Impedance - Definitions The electromagnetic response of a metal, whether normal or superconducting, is described by a complex surface impedance, Z=R+iX R : Surface resistance X : Surface reactance Both R and X are real Page 68

69 Definitions For a semi- infinite slab: Z = Ú x 0 E J x (0) ( z) dz Definition E (0) E (0) = = iwm H E z z x x 0 y(0) x( ) / z= 0 + From Maxwell Page 69

70 Definitions The surface resistance is also related to the power flow into the conductor Z = Z S(0 ) / S(0 ) ( m ) e 1/ Z0 S = E H = 0 W Impedance of vacuum Poynting vector and to the power dissipated inside the conductor P = R H - 1 (0 ) Page 70

71 Normal Conductors (local limit) Maxwell equations are not sufficient to model the behavior of electromagnetic fields in materials. Need an additional equation to describe material properties J J s s + = 0 E fi s ( w) = t t t 1-iwt For Cu at 300 K, t = sec so for wavelengths longer than infrared J = s E Page 71

72 Normal Conductors (local limit) In the local limit J ( z) = s E( z) The fields decay with a characteristic length (skin depth) Ê ˆ d = Á Ëmws 0 1/ E ( z) = E (0) e e x x 0 -z/ d -iz/ d (1 - i) Hy( z) = Ex( z) mwd Z E (0) (1 + i) (1 + i) mwd Êmwˆ x 0 = = 0 = = (1 + i) H y (0) sd Á Ë s 1/ Page 7

73 Normal Conductors (anomalous limit) At low temperature, experiments show that the surface resistance becomes independent of the conductivity As the temperature decreases, the conductivity σ increases 1/ The skin depth decreases Ê ˆ d = Á Ë mws The skin depth (the distance over which fields vary) can become less then the mean free path of the electrons (the distance they travel before being scattered) The electrons do not experience a constant electric field over a mean free path The local relationship between field and current is not valid J ( z) π s E( z) 0 Page 73

74 Normal Conductors (anomalous limit) Introduce a new relationship where the current is related to the electric field over a volume of the size of the mean free path (l) 3s RÈ ÎR E( r, t-r/ v ) 4p l R F - Rl / J ( rt, ) = Ú dr e with R= r -r 4 V Specular reflection: Boundaries act as perfect mirrors Diffuse reflection: Electrons forget everything Page 74

75 Normal Conductors (anomalous limit) In the extreme anomalous limit Ê3 l Á Ë d cl ˆ /3 Ê 3 m0 w l ˆ Zp= 1 = Zp= 0 = Á + i 16ps Ë ( 1 3) p : fraction of electrons specularly scattered at surface 1 - p : fraction of electrons diffusively scattered Page 75

76 Normal Conductors (anomalous limit) 1/3 5 /3 l - Ê ˆ Rl ( Æ ) = w Á Ë s For Cu: l / s = W m R(4. K,500 MHz) R(73 K,500 MHz) 1/3-5 /3Ê l ˆ w Á Ë s = ª mw 0 s 0.1 Does not compensate for the Carnot efficiency Page 76

77 Surface Resistance of Superconductors Superconductors are free of power dissipation in static fields. In microwave fields, the time-dependent magnetic field in the penetration depth will generate an electric field. B E =- t The electric field will induce oscillations in the normal electrons, which will lead to power dissipation Page 77

78 Surface Impedance in the Two-Fluid Model In a superconductor, a time-dependent current will be carried by the Copper pairs (superfluid component) and by the unpaired electrons (normal component) J = J + J J n n n 0 s = s E e -iw t c -iwt -iwt s = 0 e c = - 0 mew ( Ohm's law for normal electrons) ne J i E e ( m v ee e ) J = s E e 0 -iw t ne s = s + is with s = = n s s c 1 mew 0 L m l w Page 78

79 Surface Impedance in the Two-Fluid Model For normal conductors 1 For superconductors R s = sd R s È 1 1 sn 1 sn = Í = ÎlL sn ss ll sn ss ll ss ( + i ) + The superconducting state surface resistance is proportional to the normal state conductivity Page 79

80 Surface Impedance in the Two-Fluid Model s n R s 1 s l s nel n È D( T ) 1 l exp s mv Í s e F Î kt 0 L = μ - = L n s m lw R s 3 È μll w l exp Í - Î D( T ) kt This assumes that the mean free path is much larger than the coherence length Page 80

81 Surface Impedance in the Two-Fluid Model For niobium we need to replace the London penetration depth with L= l 1 + x / l L As a result, the surface resistance shows a minimum when x ª l Page 81

82 Surface Resistance of Niobium Page 8

83 Electrodynamics and Surface Impedance in BCS Model f H0f + Hex f = i t e Hex = Â A( ri, t) pi mc H is treated as a small perturbation H << H ex rf c There is, at present, no model for superconducting surface resistance at high rf field J [ ] ( w,, ) R R A I R T e μ Ú dr 4 R c J( k) =- K( k) A( k) 4p K 0 π 0: Meissner effect ( ) - R l similar to Pippard's model Page 83

84 Surface Resistance of Superconductors Temperature dependence -close to T c : dominated by change in Tc - for T < : A w exp T () t 4 ( 1- t ) 3 dominated by density of excited states e R s Ê D ˆ Á- Ë kt l t -D kt Frequency dependence w is a good approximation A reasonable formula for the BCS surface resistance of niobium is R BCS ( GHz) 5 f Tc = exp Ê 1.83 ˆ Á - T Ë T Page 84

85 Surface Resistance of Superconductors The surface resistance of superconductors depends on the frequency, the temperature, and a few material parameters Transition temperature Energy gap Coherence length Penetration depth Mean free path A good approximation for T<T c / and ω<<δ/h is R s A w exp T Ê D ˆ Á- + Ë kt R res Page 85

86 Surface Resistance of Superconductors R s A w exp T Ê D ˆ Á- + Ë kt R res In the dirty limit l x R μ l 0 BCS -1/ In the clean limit l x0 RBCS μ l R res : Residual surface resistance No clear temperature dependence No clear frequency dependence Depends on trapped flux, impurities, grain boundaries, Page 86

87 Surface Resistance of Superconductors Page 87

88 Surface Resistance of Niobium Page 88

89 Surface Resistance of Niobium Page 89

90 Super and Normal Conductors Normal Conductors Skin depth proportional to ω -1/ Surface resistance proportional to ω1/ /3 Surface resistance independent of temperature (at low T) For Cu at 300K and 1 GHz, R s =8.3 mω Superconductors Penetration depth independent of ω Surface resistance proportional to ω Surface resistance strongly dependent of temperature For Nb at K and 1 GHz, R s 7 nω However: do not forget Carnot Page 90

FUNDAMENTALS. SUPERCONDUCTIVITY SURFACE RESISTANCE RF and CAVITIES MICROPHONICS. Jean Delayen

FUNDAMENTALS. SUPERCONDUCTIVITY SURFACE RESISTANCE RF and CAVITIES MICROPHONICS. Jean Delayen LINEAR COLLIDER SCHOOL Villars-sur-Ollon 5 Oct.-5 Nov. 010 FUNDAMENTALS SUPERCONDUCTIVITY SURFACE RESISTANCE RF and CAVITIES MICROPHONICS Jean Delayen Center for Accelerator Science Old Dominion University