Chapter Phenomenological Models of Superconductivity

Transcription

1 TT1-Chap2-1 Chapter 3 3. Phenomenological Models of Superconductivity 3.1 London Theory The London Equations 3.2 Macroscopic Quantum Model of Superconductivity Derivation of the London Equations Fluxoid Quantization Josephson Effect 3.3 Ginzburg-Landau Theory Type-I and Type-II Superconductors Type-II Superconductors: Upper and Lower Critical Field Type-II Superconductors: Shubnikov Phase and Flux Line Lattice Type-II Superconductors: Flux Lines

2 3.1 London Theory * 7 March 1900 in Breslau 30 March 1954 in Durham, North Carolina, USA study: Bonn, Frankfurt, Göttingen, Munich and Paris. Ph.D.: 1921 in Munich : Göttingen and Munich 1926/27: Assistent of Paul Peter Ewald at Stuttgart, studies at Zurich and Berlin with Erwin Schrödinger. 1928: Habilitation at Berlin : London : Paris 1939: Emigration to USA, Duke Universität at Durham Fritz Wolfgang London ( ) TT1-Chap2-2

3 TT1-Chap London Theory Heinz and Fritz London

4 3.1 London Theory 1935 Fritz and Heinz London describe the Meißner-Ochsenfeld effect and perfect conductivity within phenomenological model homogeneous pair condensate London Equations starting point is equation of motion of charged particles with mass m and charge q (t = momentum relaxation time) two-fluid model: - normal conducting electrons with charge q n and density n n - superconducting electrons with charge q s density n s normal state: n n = n, n s = 0 superconducting state n n = 0, n s = max (for T 0), t, J s = n s q s v s 1 st London equation London coefficient BCS theory: m s = 2m e, q s = -2e n s = n/2 TT1-Chap2-4

5 3.1.1 London Equations take the curl of 1 st London equation and use flux through an arbitrary area inside a sample with infinite conductivity stays constant e.g. flux trapping when switching off the external magnetic field Meißner-Ochsenfeld effect tells us: not only but itself must be zero expression in brackets must be zero 2 nd London equation use Maxwell s equation with London penetration depth TT1-Chap2-5

6 TT1-Chap London Equations example: B = B z B z solution: z y B z decays exponentially with characteristic decay length l L B z,0 T dependence of l L empirical relation: l L x

7 TT1-Chap London Equations with 2 nd London equation we obtain for J s : z B z J y,0 y l L x

8 TT1-Chap London Equations example: thin superconducting sheet of thickness d with B ǁ sheet Ansatz: boundary conditions: z y solution: B ext,z B ext,z l L x d

9 TT1-Chap London Equations 1 st London equation London coefficient 2 nd London equation London penetration depth remarks to the London model: 1. normal component is completely neglected not allowed at finite frequencies! 2. we have assumed a local relation between J s, E and B - J s is determined by the local fields for every position r - this is problematic since l for t nonlocal extension of London theory by A.B. Pippard (1953)

10 TT1-Chap SC as Macroscopic Quantum Phenomenon more solid derivation of London equations by assuming that superconductor can be desrcribed by a macroscopic wave function - Fritz London (> 1948) - derived London equations from basic quantum mechanical concepts basic assumption of macroscopic quantum model of superconductivity complete entity of all superconducting electrons can be described by macroscopic wave function amplitude phase hypothesis can be proven by BCS theory (discussed later) normalization condition: volume integral over y ² is equal to the number N s of superconducting electrons

11 TT1-Chap SC as Macroscopic Quantum Phenomenon general relations in electrodynamics: electric field: flux density: A = vector potential f = scalar potential electrical current is driven by gradient of electrochemical potential: canonical momentum: kinematic momentum:

12 TT1-Chap SC as Macroscopic Quantum Phenomenon Schrödinger equation for charged particle: electro-chemical potential insert macroscopic wave-function (Madelung transformation) replacements: Y y, q q s, m m s calculation yields after splitting up into real and imaginary part two fundamental equations - current-phase relation: connects supercurrent density with gauge invariant phase gradient - energy-phase gradient: connects energy with time derivative of the phase

13 TT1-Chap Special Topic: Madelung Transformation we start from Schrödinger equation: electro-chemical potential we use the definition and obtain with

14 TT1-Chap Special Topic: Madelung Transformation

15 3.2 Special Topic: Madelung Transformation equation for real part: energy-phase relation (term of order ²n s is usually neglected) TT1-Chap2-15

16 TT1-Chap Special Topic: Madelung Transformation interpretation of energy-phase relation corresponds to action in the quasi-classical limes the Hamilton-Jacobi equation the energy-phase-relation becomes

17 TT1-Chap Special Topic: Madelung Transformation equation for imaginary part: conservation law for probability density continuity equation for probability density and probability current density

18 TT1-Chap Special Topic: Madelung Transformation we define supercurrent density J s by multiplying J r with charge q s of superconducting electrons : gauge invariant phase gradient current-phase relation v s J s = n s q s v s canonical momentum: p = m s v s + q s A p = m s ħ m s θ r, t q s m s A r, t + q s A p = ħ θ r, t v s zero total momentum state for vanishing phase gradient: Cooper pairs k, k

19 3.2 SC as Macroscopic Quantum Phenomenon energy-phase relation 1 supercurrent density-phase relation (London parameter) 2 equations (1) and (2) have general validity for charged and uncharged superfluids (i) superconductor with Cooper pairs of charge q s = -2e (ii) (iii) neutral Bose superfluid, e.g. 4 He neutral Fermi superfluid, e.g. 3 He we use equations (1) and (2) to derive London equations note: k drops out!! TT1-Chap2-19

20 TT1-Chap Derivation of London Equations 2 nd London equation and the Meißner-Ochsenfeld effect: take the curl second London equation: with Maxwell s equations: which leads to: describes Meißner-Ochsenfeld effect: applied field decays exponentially inside superconductor, decay length: (London penetration depth)

21 3.2.1 Derivation of London Equations 1 st London equation and perfect conductivity: take the time derivative inserting and substituting yields first London equation: neglecting 2 nd term: (linearized 1 st London equation) interpretation: for a time-independent supercurrent the electric field inside the superconductor vanishes dissipationless dc current TT1-Chap2-21

22 3.2.1 London Equations - Summary energy-phase relation 1 supercurrent density-phase relation (London parameter) 2 2 nd London equation and the Meißner-Ochsenfeld effect: take the curl which leads to: 2 nd London equation 1 st London equation and perfect conductivity: take the time derivative which leads to: 1 st London equation TT1-Chap2-22

23 TT1-Chap Derivation of London Equations the assumption that the superconducting state can be described by a macroscopic wave function leads to a general expression for the supercurrent density J s London equations can be directly derived from the general expression for the supercurrent density J s London equations together with Maxwell s equations describe the behavior of superconductors in electric and magnetic fields London equations cannot be used for the description of spatially inhomogeneous situations Ginzburg-Landau theory London equations can be used for the description of time-dependent situations Josephson equations

24 TT1-Chap Derivation of London Equations Processes that could cause a decay of J s (plausibility consideration): example: Fermi sphere in two dimensions in the k x k y plane T = 0: all states inside the Fermi circle are occupied current in x-direction shift of Fermi circle along k x by ±k x normal state: superconducting state: normal state relaxation into states with lower energy (obeying Pauli principle) centered Fermi sphere, current relaxes Cooper pairs with the same center of mass moment (discussion later) only scattering around the sphere no decay of supercurrent superconducting state

25 TT1-Chap Additional Topic: Linearized 1. London Equation usually, 1. London equation is linearized: allowed if E >> v s B condition is satisfied in most cases kinetic energy of superelectrons in order to discuss the origin of the extra term (nonlinearity) we use the vector identity to write then, by using the second London equation, we can rewrite the 1. London equation as with and we obtain (Lorentz law)

26 3.2.1 Additional Topic: Linearized 1. London Equation we can conclude: the nonlinear first London equation results from the Lorentz's law and the second London equation exact form of the expression describing the phenomenon of zero dc resistance in superconductors the first London equation is derived using the second London equation Meißner-Ochsenfeld effect is the more fundamental property of superconductors than the vanishing dc resistance we can neglect the nonlinear term if as variations of J s occur on l L, we have and obtain the condition with 2. London equation: typically, v s < 1 m/s even at very high current densities of the order of 10 6 A/cm² due to the large values of n s TT1-Chap2-26

27 TT1-Chap Additional Topic: The London Gauge in some cases it is convenient to choose a special gauge often used: The London Gauge if the macroscopic wavefunction is single valued (this is the case for a simply connected superconductor containing no flux) we can choose such that everywhere frequently, we have no conversion of J s in J n or no supercurrent flow throuh sample surface a vector potential that satisfies is said to be in the London gauge 1. London equation:

28 TT1-Chap Fluxoid Quantization integration of expression for supercurrent density around a closed contour Stoke s theorem (path C in simply or multiply connected region): integral of gradient: if r 1 r 2 (closed path) integral to zero but: phase is only specified within modulo 2π of its principle value π, π : θ n = θ 0 + 2πn

29 TT1-Chap Fluxoid Quantization then: fluxoid is quantized in units of F 0 flux quantum: quantization condition holds for all contour lines including contour that can be shrunk to single point r 1 = r 2 : r1 r 2 θ dl = 0 contour line can no longer be shrunk to single point inclusion of nonsuperconducting region in contour r 1 = r 2 : we have built in memory in integration path n 0 possible, r1 r 2 θ dl = 2πn

30 TT1-Chap Fluxoid Quantization H ext magnetic flux J s J s r r screening current on inner wall screening current on outer wall

31 TT1-Chap Fluxoid Quantization Fluxoid Quantization: total flux = external applied flux + flux generated by induced supercurrent must have discrete values Flux Quantization: - superconducting cylinder, wall much thicker than λ L - application of small magnetic field at T < T c screening currents, no flux inside application of B cool during cool down: screening current on outer and inner wall amount of flux trapped in cylinder: satisfies fluxoid quantization condition wall thickness λ L : closed contour deep inside with J s = 0 then: flux quantization remove field after cooling down trapped flux = integer multiple of flux quantum

32 TT1-Chap Fluxoid Quantization Flux Trapping: why is flux not expelled after switching off external field J s t = 0 according to 1st London equation: E = 0 deep inside SC (supercurrent flow only on surface within λ L ) with and we get: Φ: magnetic flux enclosed in loop contour deep inside the superconductor: E = 0 and therefore flux enclosed in cylinder stays constant

33 3.2.2 Fluxoid Quantization 1961 Doll/Näbauer at Munich, Deaver/Fairbanks at Stanford quantization of magnetic flux in a hollow cylinder Cooper pairs with q s = 2e cylinder with wall thickness λ L different amounts of flux are frozen in during cooling down in B cool measure amount of trapped flux required: large relative changes of magnetic flux small fields and small diameter d for d = 10 μm, we need B cool = T for one flux quantum measurement of (very small) torque D = ¹ x B p due to probe field B p resonance method amplitude of rotary oscillation exciting torque B p quartz thread Pb quartz cylinder Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, (2011) d 10 µm TT1-Chap2-33

34 3.2.2 Fluxoid Quantization resonance amplitude (mm/gauss) (a) F B cool (Gauss) trapped magnetic flux (h/2e) (b) B cool (Gauss) R. Doll, M. Näbauer Phys. Rev. Lett. 7, 51 (1961) B.S. Deaver, W.M. Fairbank Phys. Rev. Lett. 7, 43 (1961) Paarweise im Fluss D. Einzel, R. Gross, Physik Journal 10, No. 6, (2011) prediction by F. London: h/e experimental proof for existence of Cooper pairs q s = 2e TT1-Chap2-34

35 TT1-Chap Josephson Effect (1962) Brian David Josephson (born 1940)

36 3.2.3 Josephson Effect what happens if we weakly couple two superconductors? - coupling by tunneling barriers, point contacts, normal conducting layers, etc. - do they form a bound state such as a molecule? - if yes, what is the binding energy? B.D. Josephson in 1962 (nobel prize with Esaki and Giaever in 1973) Cooper pairs can tunnel through thin insulating barrier expectation: - tunneling probability for pairs T 2 2 extremely small Josephson: - tunneling probability for pairs T 2 - coherent tunneling of pairs ( tunneling of macroscopic wave function ) finite supercurrent at zero applied voltage oscillation of supercurrent at constant applied voltage Josephson effects finite binding energy of coupled SCs = Josephson coupling energy TT1-Chap2-36

37 TT1-Chap Josephson Effect coupling is weak supercurrent density is small ψ 2 = n s is not changed in S 1 and S 2 supercurrent density depends on gauge invariant phase gradient: simplifying assumptions: - current density is homogeneous - γ varies negligibly in electrodes - J s same in electrodes and junction area γ in superconducting electrodes much smaller than in insulator I then: - replace gauge invariant phase gradient γ by gauge invariant phase difference:

38 3.2.3 Josephson Effect first Josephson equation: we expect: for J s = 0: phase difference must be zero: therefore: J c : critical/maximum Josephson current density general formulation of 1 st Josephson equation: current-phase relation in most cases: keep only 1 st term (especially for weak coupling): 1. Josephson equation: generalization to spatially inhomogeneous supercurrent density: derived by Josephson for SIS junctions supercurrent density varies sinusoidally with φ = θ 2 θ 1 w/o external potentials TT1-Chap2-38

39 TT1-Chap Josephson Effect other argument why there are only sin contributions to Josephson current time reversal symmetry if we reverse time, the Josephson current should flow in opposite direction - t t, J s J s the time evolution of the macroscopic wave functions is exp iθ t = exp iωt - if we reverse time, we have t t if the Josephson effect stays unchanged under time reversal, we have to demand satisfied only by sin-terms

40 voltage drop TT1-Chap Josephson Effect second Josephson equation: time derivative of the gauge invariant phase difference: substitution of the energy-phase relation gives: supercurrent density across the junction is continuous (J s (1) = J s (2)): (term in parentheses = electric field) 2. Josephson equation: voltage phase relation

41 TT1-Chap Josephson Effect for a constant voltage across the junction: I s is oscillating at the Josephson frequency n = V/F 0 : voltage controlled oscillator applications: - Josephson voltage standard - microwave sources

42 3.2 Summary TT1-Chap2-42 Macroscopic wave function ψ: describes ensemble of macroscopic number of superconducting electrons ψ 2 = n s is given by density of superconducting electrons Current density in a superconductor: Gauge invariant phase gradient: Phenomenological London equations: flux/fluxoid quantization:

43 3.2 Summary TT1-Chap2-43 Josephson equations: maximum Josephson current density J c : wave matching method more detail in chapter 5

44 3.3 Ginzburg-Landau Theory London theory: suitable for situations where n s = const. how to treat spatially inhomogeneous systems? example: step-like change of wave function at surfaces and interfaces associated with large energy gradual change on characteristic length scale expected Vitaly Lasarevich Ginzburg and Lew Davidovich Landau (1950) description of superconductor by - complex, spatially varying order parameter ψ r with ψ r 2 = n s r (density of superconducting electrons) - no time dependence (cannot describe Josephson effect) based on Landau theory of 2 nd order phase transitions Alexei Alexeyevich Abrikosov (1957) prediction of flux line lattice for type-ii superconductors Lev Petrovich Gor'kov (1959) GL-theory can be inferred from BCS theory for T T c Ginzburg-Landau- Abrikosov-Gor'kov (GLAG) theory TT1-Chap2-44

45 3.3 Ginzburg-Landau Theory TT1-Chap2-45 A: spatially homogeneous superconductor in zero magnetic field develop free enthalpy density g s of superconductor in power series of Ψ 2 free entalpy density of normal state higher order terms can be neglected for T T c as ψ small discussion of coefficients α and β: - β > 0, otherwise large Ψ always results in g s < g n minimum of g s always for Ψ - α changes sign at phase transition T > T c : α > 0, since g s > g n T < T c : α < 0, since g s < g n Ansatz:

46 3.3 Ginzburg-Landau Theory TT1-Chap2-46 g s g n for T < T c and T > T c 4 3 (a) a > 0 T > T c 4 3 (b) a < 0 T < T c g s - g n (arb. units) g s - g n (arb. units) B cth2 / 2µ 0 ψ Y (arb. units) Y (arb. units)

47 3.3 Ginzburg-Landau Theory TT1-Chap2-47 equilibrium condition yields describes homogeneous equilibrium state at T physical meaning of coefficients: g s g n corresponds to condensation energy which, in turn, is given by solve for α and β: = (g s -g n )/(n s /2) condensation energy per Cooper pair

48 3.3 Ginzburg-Landau Theory TT1-Chap2-48 Temperature dependence of B cth : GLAG theory prediction: experimental observation: with reasonable agreement for T close to T c GLAG expressions only valid for T close to T c

49 3.3 Ginzburg-Landau Theory B: spatially inhomogeneous superconductor in external magnetic field μ 0 H ext additional terms in free enthalpy density for B ext = μ 0 H ext, b = μ 0 H ext + M work required for field expulsion (B ext b)² with takes into account spatial variations of local flux density b are associated with J s according to 2 nd London eq. additional kinetic energy prevents spatial variations of ψ stiffness of order parameter v s ² gradient of amplitude gradient of phase TT1-Chap2-49

50 3.3 Ginzburg-Landau Theory TT1-Chap2-51 minimization of free enthalpy G s : integration of enthalpy density g s over whole volume V of superconductor divergence theorem boundary condition during minimization of free enthalpy no current density through surface of superconductor = 0 v s replace by for superconductor/normal conductor interface

51 3.3 Ginzburg-Landau Theory TT1-Chap2-54 minimization of free enthalpy: minimization of G s with respect to variations = 0, since equation must be satisfied for all 1. Ginzburg-Landau equation

52 3.3 Ginzburg-Landau Theory TT1-Chap2-55 minimization of G s with respect to variation

53 3.3 Ginzburg-Landau Theory TT1-Chap2-56 integral of last term: = 0, since equation must be satisfied for all with London gauge we obtain from Maxwell equation : 2. Ginzburg-Landau equation

54 3.3 Ginzburg-Landau Theory TT1-Chap2-57 Summary: minimization of free enthalpy G s = V g s dv with yields Ginzburg-Landau equations: 1. GL-equation 2. GL-equation

55 TT1-Chap GL-Theory vs. Macroscopic Quantum Model Ginzburg-Landau theory: 1. GL-equation 2. GL-equation Macroscopic quantum model: energy-phase relation (London parameter) supercurrent density-phase relation Note: - GL theory can well describe spatially inhomogeneous situations - but cannot account for time-dependent phenomena (there is no time derivative) - macroscopic quantum model cannot account for spatially inhomogeneous situations - but can describe time-dependent phenomena (Josephson effect)

56 3.3 Ginzburg-Landau Theory Characteristic length scales: 2 nd GL equation: for n s = α/β expression for supercurrent density 1 st and 2 nd London equation characteristic screening length for B ext : 1 st GL equation: London penetration depth normalization and use of 2 nd characteristic length scale GL coherence length ξ GL TT1-Chap2-60

57 TT1-Chap Ginzburg-Landau Theory Temperature dependence of characteristic length scales: Ansatz for α and β: with and experimental T-dependence: both length scales diverge for T T c reasonable agreement for T close to T c

58 3.3 Ginzburg-Landau Theory Ginzburg-Landau parameter: solve for B cth relation between GL and BCS coherence length: - α = condensation energy / (n s /2) - BCS: average condensation energy per electron pair α/2 corresponds to 3Δ 2 (0)/4E F correct BCS result: TT1-Chap2-62

59 3.3 Ginzburg-Landau Theory TT1-Chap2-63 Superconductor-normal metal interface: solve for x > 0 assumptions: no magnetic field (A = 0), ψ x = 0 = 0 superconductor extends at x > 0 boundary conditions: and solution:

60 3.3.1 Type-I and Type-II Superconductors TT1-Chap2-64 experimental facts: type-i superconductors: expel magnetic field until B cth only Meißner phase single critical field B cth type-ii superconductors: partial field penetration above B c1 B i > 0 for B ext > B c1 Shubnikov phase between B c1 B ext B c2 upper and lower critical fields B c1 and B c2

61 3.3.1 Type-I and Type-II Superconductors TT1-Chap2-65 thermodynamic critical field defined as: (for type-i and type-ii superconductors) condensation energy area under M H ext curve is the same for type-i and type-ii superconductor with the same condensation energy:

62 3.3.1 Type-I and Type-II Superconductors difference between type-i and type-ii superconductors: NS boundary energy savings in field expulsion energy (per area) loss in condensation energy (per area) resulting boundary energy always positive for λ L < ξ GL up to B ext = B cth can be negative for λ L > ξ GL for B ext < B cth type-ii SC: κ = λ L /ξ GL > 1/ 2 (numerical result) - boundary energy negative for B ext < B cth - formation of boundary favorable formation of NS boundary lowers energy already for B ext < B cth TT1-Chap2-66

63 3.3.1 Type-I and Type-II Superconductors TT1-Chap2-67

64 TT1-Chap Type-I and Type-II Superconductors: Demagnetization Effects and Intermediate State ideal B i (H ext ) dependence valid only for vanishing demagnetization effects - e.g. long cylinder of slab for finite demagnetization with (perfect diamagnetism) sphere: H lok = 1.5 H equator µ 0 H lok = B µ 0 H ext = 2/3 B cth long cylinder: N 0 H lok H ext flat disk: N 1 H lok sphere: N = 1/3 H lok = 1.5 H ext intermediate state can have complex structure

65 3.3.1 Type-I and Type-II Superconductors: Demagnetization Effects and Intermediate State magneto-optical image of the intermediate state of In bright: normal regions (from Buckel) TT1-Chap2-69

66 TT1-Chap Type-II Superconductors: Upper and Lower Critical Field Task: derive expressions for B c1 and B c2 from GLAG-equations 1 st GL-equation: (a) high magnetic fields B c2 approximation: y is small for B ext close to B c2 neglect β ψ 2 ψ term in 1 st GL-eq. linearized 1 st GL-equation: further approximations: B = µ 0 H ext, since M 0 for µ 0 H ext B c2 H ext = H z A = (0,A y,0) with A y = µ 0 H z x = B z x corresponds to Schrödinger equation of charged particle in magnetic field Ansatz:

67 3.3.2 Type-II Superconductors: Upper and Lower Critical Field differential equation for u(x): spring constant with: energy eigenvalues of harmonic oscillator: and solve for B z : maximum B z for n = 0 and k z = 0: B c2 > B cth for κ > 1/ 2 TT1-Chap2-71

68 TT1-Chap Type-II Superconductors: Upper and Lower Critical Field

69 TT1-Chap Type-II Superconductors: Upper and Lower Critical Field B cth and λ L of type-i superconductors B c2 and λ L of type-ii superconductors

70 TT1-Chap Type-II Superconductors: Upper and Lower Critical Field B c2 of type-ii superconductors

71 TT1-Chap Type-II Superconductors: Upper and Lower Critical Field (b) low magnetic fields B c1 derivation of lower critical field B c1 is more difficult (no linearization of GL equations) we use simple argument, that flux generated by B c1 in area πλ L 2 must be equal to Φ 0 B c1 given by the fact that the minimum amount of flux through normal area is F 0 here, we have assumed (London approximation) more precise result based on GL theory:

72 3.3.3 Type-II Superconductors: Shubnikov Phase and flux Line Lattice solution of the GL-equations in intermediate field regime complicated linearization of GL-equations is no longer a good approximation qualitative discussion How is the magnetic flux arranged in Shubnikov phase above B c1? - maximize NS boundary, since boundary energy is negative many small normal conducting N regions - upper limit is set by flux quantization: flux content in a N region cannot be smaller than F 0 - flux lines can be viewed as small bar magnets which repel each other arrange flux lines with maximum distance regular lattice of flux lines with flux content F 0 Abrikosov flux line lattice hexagonal lattice TT1-Chap2-76

73 3.3.3 Type-II Superconductors: Shubnikov Phase and flux Line Lattice (a) (b) µ 0 H ext flux line, vortex (c) NbSe 2 (d) contour lines of n s k = 2, 5, 20 B(r) / B c1 n s, B / B c n s (r) 200 nm r / l L H.F. Hess et al., Phys. Rev. Lett. 62, 214 (1989) E.H. Brandt, Phys. Rev. Lett. 78, 2208 (1997) TT1-Chap2-77

74 3.3.3 Type-II Superconductors: Shubnikov Phase and flux Line Lattice distance between flux lines is maximum in hexagonal lattice energetically most favorable state square lattice also often observed, since other effects (e.g. Fermi surface topology) play a significant role TT1-Chap2-78

75 3.3.3 Type-II Superconductors: Shubnikov Phase and flux Line Lattice 200 nm NbSe 2 : flux-line lattice of non-irradiated single crystal at 1 T distortion of ideal flux-line lattice by defects flux-line pinning Right: STM-images showing the flux line lattice of ion irradiated NbSe 2 (T=3 K, I=40 pa, U=0.5 mv) taken during increasing the applied magnetic filed to 70, 100, 200, 300 mt. The images always show the same sample area of 2 x 2 µm. (source: University of Basel) TT1-Chap2-79

76 TT1-Chap Type-II Superconductors: Shubnikov Phase and flux Line Lattice Bitter technique: decoration of flux-line lattice by Fe smoke imaging by SEM U.Essmann, H.Träuble (1968) MPI Metallforschung Nb, T = 4 K disk: 1mm thick, 4 mm ø B ext = 985 G, a =170 nm D. Bishop, P. Gammel (1987 ) AT&T Bell Labs YBCO, T = 77 K B ext = 20 G, a = 1200 nm similar: - L. Ya. Vinnikov, ISSP Moscow - G. J. Dolan, IBM NY

77 3.3.4 Type-II Superconductors: Flux Lines TT1-Chap2-82 in flux line lattice: how does n s r and b r look like across a flux line? radial dependence of ψ: we consider isolated flux line and use the Ansatz insertion into nonlinear GL equations yields expression for f r : with c 1 ψ r 2 = f 2 (r)

78 3.3.4 Type-II Superconductors: Flux Lines TT1-Chap2-83 radial dependence of b (London vortex, we use approximation f r = δ 2 r ): we use 2 nd London equation accounts for the presence of vortex core interpretation with Maxwell eqn. we obtain λ L 2 b + b = z Φ 0 δ 2 r integration over circular area S with r λ L perpendicular to z yields 2 b ds + λ L S Φ ( b) dl = z Φ 0 Φ = Φ 0 S 0 since b = μ 0 J s and J s 0 for r λ L

79 3.3.4 Type-II Superconductors: Flux Lines TT1-Chap2-84 solution of is exact result only if we assume ξ GL 0 0 th order modified Bessel function of 2 nd kind

80 3.3.4 Type-II Superconductors: Flux Lines TT1-Chap2-85 solution of exact result if we assume finite ξ GL

81 3.3 Summary The Ginzburg-Landau Theory explains: all London results type-ii superconductivity (Shubnikov or vortex state): ξ GL < λ L behavior at surface of superconductors and interfaces to non-superconducting materials The Ginzburg-Landau Theory does not explain: q s = 2e microscopic origin of superconductivity not applicable for T << T c non-local effects Literature: P.G. De Gennes, Superconductivity of Metals and Alloys M. Tinkham, Introduction to Superconductivity N.R. Werthamer in Superconductivity, edited by R.D. Parks TT1-Chap2-86