Ginzburg-Landau Theory There are two main theories in superconductivity: i Microscopic theory describes why materials are superconducting Prof. Damian Hampshire Durham University ii Ginzburg-Landau Theory describes the properties of superconductors in magnetic fields Cambridge Winter School January 7 1 Ginzburg-Landau theory This is a phenomenological theory, unlike the microscopic BCS theory. Based on a so-called phenomenological order parameter. Not strictly an ab initio theory, but essential for problems concerning superconductors in magnetic fields. G-L was the first theory to explain the difference between Type I and Type II superconductivity, and enable the calculation of two critical fields H c1 and H c. Outline of the Lecture Ginzburg-Landau theory The two G-L equations The basic phenomenology Type I and Type II superconductors Time dependent G-L theory Ginzburg-Landau and Pinning Conclusions 3 4
Ginzburg-Landau Theory Ginzburg and Landau (G-L postulated a Helmholtz energy density for superconductors of the form: The two Ginzburg-Landau Equations Functional differentiation w.r.t. ψ and A gives the two G-L equations (coupled partial differential equations: 1 4 1 f = αψ + βψ + (-i - e A ψ + Hd B m G-L I 1 m (- i ea Ψ + αψ + β Ψ Ψ = where α and β are constants and ψ is the wavefunction. α is of the form α (T-T C which changes sign at T C G-L II J = i e * * 4e ( A m Ψ Ψ Ψ Ψ m Ψ 5 6 London Theory Type I superconductors The London brothers - E = μλ t B = μλ where * m λl = * μ n e L J s B = λ L J s s Substituting the second London equation into one of Maxwell s equations we get the Meissner state: 1 L B 7 Magnetisation of superconductors M -H c H H c1 H c H c Type I Type II H T H c Mixed H c H c1 Meissner Normal The magnetisation (M of a type I and type II superconductor, with the same H c as a function of applied magnetic field (H o. Inset: The phase diagrams of type I and type II superconductors. 8 T c
The structure of the fluxon The Mixed state μ H c1 B z (r J θ B z ψ ψ(r J θ (r ξ λ The order parameter, magnetic field (B and supercurrent density (J C as a function of distance (r from the centre of an isolated vortex (κ 8. The order parameter squared is proportional to the density of superelectrons. r 9 λ: depth of the current flow d: fluxon fluxon spacing If B increases, d decreases. 1 The Mixed State in NbSe The Mixed State in Nb (a (b The hexagonal Abrikosov lattice: (a contour diagram of the order parameter from Kleiner et al.; the lines also represent contours of B and streamlines of J. (b Scanning-tunnellingmicroscope image of the flux lattice in NbSe (1 T, 1.8 K from Hess et al. the vortex spacing is ~479Å. 11 Vortex lattice in niobium the triangular layout can clearly be seen. (The normal regions are preferentially dark because of the ferromagnetic powder. 1
Flux Quantisation By considering the persistent current as a wave, phase coherence demands (n: integer - leads to flux quantisation. k: momentum of the superelectrons. 13 G-L Theory Type I and Type II superconductors Ginzburg-Landau theory predicts that a superconductor should have two characteristic lengths: Coherence length Penetration depth The Ginzburg-Landau parameter ξ = λ = m e 4e m e α β μ α λ κ = ξ This ratio, κ, distinguishes Type-I superconductors, for which κ 1/, from Type-II superconductors which have higher κ values. 14 Thermodynamic Critical Field, H C In Type I superconductors H C is the critical field at which superconductivity is destroyed ψ drops abruptly to zero in a first-order phase transition. At the thermodynamic critical field H C, the Gibbs free energies for superconducting and normal phases are equal. For the superconducting phase B =, and for the normal phase ψ = H c = α μβ Lower and upper critical fields HC Hc1 ln κ κ H c μ = α μβ φ H c = πξ H κh κ H c C C1 15 16
Flux entering a superconductor Time evolution of order parameter Time evolution of order parameter This simulation is simply applying an instantaneous fields of.5h c. 17 18 Time evolution of order parameter Time evolution of order parameter 19
Time evolution of order parameter Time evolution of order parameter 1 Flux nucleation and entry into a superconductor 5 3 1-1 -3-5 -6-4 - 4 6 Contour plot of a superconductor bounded by an insulating outer surface. The applied magnetic field was increased to above the initial vortex penetration field (i.e. to Hp +.1Hc 3 Other Ginzburg-Landau predictions The magnetization of in the mixed state near B c is given by: μ c M = ( B B ( κ 1 β A β A = ψ 4 ( ψ Surface barriers C3 C Depairing current density: B = 1.69B J D Hc Hc = =.385 3 3κξ κξ 4
Critical Current Density (H c /κ ξ Critical Current Density (H c /κ ξ.1.8.6.4. Josephson diffraction Junction Thickness.5ξ 3x1-3...1..3.4.5 Applied Field (H c Junction Thickness 4.5ξ x1-3 1x1-3..1..3.4.5 Applied Field (H c Critical Current Density (H c /κ ξ Critical Current Density (H c /κ ξ.3.5..15.1.5 Junction Thickness 1.5ξ 4x1-4...1..3.4.5 Applied Field (H c Junction Thickness 9.5ξ 3x1-4 x1-4 1x1-4..1..3.4.5 Applied Field (H c J c computed for an ρ N = 1ρ S junction in a 3ξwide κ = 5 superconductor. 5 Ginzburg-Landau predictions - restricted dimensionality behaviour Behaviour of thin films - A thin film has a much higher critical field (if the field lines are parallel to the film, than a bulk superconductor. This is predicted by Ginzburg- Landau theory. Anisotropic Ginzburg-Landau theory - It is possible to extend Ginzburg-Landau theory to anisotropic systems (eg high-t c superconductors. This shows that H c is isotropic, but H c is not. Lawrence-Doniach theory - D version of GL theory. Predicts that H c is higher for layered superconductors, and that at low T s, fluxons are locked parallel to planes (a kind of transverse Meissner effect 6 Flux Pinning using G-L theory Model Pinning at surfaces 5 (Dew-Hughes 1974 Pinning at surfaces 7 (Dew-Hughes 1987 Force F p =J C.B per unit volume B 1 c = b ( 1 b 4μκD π 3Bc = b( 1 b 16μκD Reversible Magnetic Properties of Perfect Superconductors Flux shear past point pinning sites 3 (Kramer, Labusch 31 C 66 Flux shear past point pinning sites 3 (Kramer, Brandt 33 C 66 Maximum shear strength of lattice 7 (Dew-Hughes 1987 Flux shear along grain boundaries in D 15 (Pruymboom 1988 Collective pinning model 34 (Larkin-Ovchinnikov =.8 1 5 B c π 1 b μκ φ ( 1 a d ( 1 b 1 4 5 3 4 Bc π 1.9b b 1 = 3.9 1 exp b ( 1 b μκ φ 3 ( 1 a κb d Bc b 1 = exp ( 1.9 ( 1 b b b 8πμ κ ( D a 3κ b 5 3 Bc π 1.9 1.5 1 b ( 1 b μκ φ nf = 16aC C 4 p p 3 44 66 7 Below Hc, Type I superconductors are in the Meissner state: current flows in a thin layer around the edge of the superconductor, and there is no magnetic flux in the bulk of the superconductor. (Hc : Thermodynamic Critical Field. In Type II superconductors, between the lower critical field (Hc1, and the upper critical field (Hc, magnetic flux penetrates into the sample, giving a mixed state. 8
Reversible Magnetization Loop The reversible response of a superconductor Magnetizaion M (H c..1..4.6.8 1 -.1 ( ΗC H -. M =, (κ 1 β μ H SC C A -.3 -.4 -.5 -.6 -.7 μ H C1 Applied field H (H c 1ξ 8ξ, κ = 5 M vs. H for a Superconductor Coated With a Normal Metal, κ = 5 Magnetizaion M (H c..1 -.1 -. -.3 -.4 -.5 -.6 -.7 -.8 H =.5 H c Applied field H (H c..4.6.8 1 1ξ 8ξ, κ = 5 The material is in the Meissner state -.8 9 3 Magnetization Loop Magnetization Loop Magnetizaion M (H c..1 -.1 -. -.3 -.4 -.5 -.6 -.7 -.8 Applied field H (H c..4.6.8 1 1ξ 8ξ, κ = 5 Magnetizaion M (H c..1 -.1 -. -.3 -.4 -.5 -.6 -.7 -.8 Applied field H (H c..4.6.8 1 1ξ 8ξ, κ = 5 H =.1 H c The material is in the Meissner state H =.15 H c The material is in the mixed state 31 3
Magnetization Loop Magnetization Loop...1 Applied field H (H c.1 Applied field H (H c Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 -.6 -.6 -.7 -.7 -.8 -.8 H =.4 H c H =.7 H c Note the nucleation of fluxons at the superconductor-normal boundary In the reversible region, one can determine κ 33 34 Magnetization Loop Magnetization Loop...1 Applied field H (H c.1 Applied field H (H c Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 -.6 -.6 -.7 -.7 -.8 -.8 H =.9 H c H = 1. H c The core of the fluxons overlap and the average value of the order parameter drops Eventually the superconductivity is destroyed 35 36
Magnetization Loop Magnetization Loop...1 Applied field H (H c.1 Applied field H (H c Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 Magnetizaion M (H c -.1 -. -.3 -.4 -.5..4.6.8 1 1ξ 8ξ, κ = 5 -.6 -.6 -.7 -.7 -.8 -.8 H =.5 H c H =. H c Note the Abrikosov fluxline-lattice with hexagonal symmetry A few fluxons remain as the inter-fluxon repulsion is lower than the surface pinning 37 38 Time-dependent Ginzburg-Landau equations e 1 eϕ Δ 1 Δ+ A Δ+ + i Δ = ξ D t 1 T T c i J e 1 * A = Re e ϕ Δ A Δ σ + eμλ i t These equations were postulated by Schmid (1966, and then derived using microscopic theory in the gapless case by Gor kov and Eliashberg (1968 39 Uncertainty/Complexity in G-L theory Thermodynamics is seductive but very complex. There are number of different approaches to deriving the G-L equations (DeGennes, Campbell, Clem, Pippard, Jackson and Landau and Lifshitz There is confusion in the literature about the length scales on which G-L applies TDGL for real systems has only been possible in the last 3-5 years 4
Model for a polycrystalline superconductor A collection of truncated octahedra Avoids the problematic straight channels of a simple cubic system. The Bean critical state model is used to obtain J c. A symmetrical straightline Bean profile is fitted to the calculated B-field data The E-field is calculated from the ramp rate by Maxwell s equations. Obtaining bulk J c Local Field (B c.31.3.9 DIffused region Down ramp at.3h c Up ramp at.98h c region over which J is calculated DIffused region.8-6 -4-4 6 y-coordinate (ξ 41 4 Branching test runs up, fast 1.45 Layout of granular system Current Density (1 3 H c /κ ξ 1 8 6 4 Applied Field Main-line Data Branch Data E max = +5.33 1 4 H c ρ S /κ ξ Diffusive resistivity 31ρ S.4.35.3.5 Applied Field (H c y b a x Superconductor Weak superconductor Normal metal Edges matched by periodic boundary conditions 3 35 4 45 5 55. 6 Time (t 43 Show the -field movie! 44
Flux motion in low fields (.43 HC along grain boundaries 15 15 b a Time dependant Ginzburg-Landau theory provides the framework for understanding flux pinning 1 1 5 5 1 1^-3 1^-4 -.1-3 1^-5 1^-6-4.1-5 3 4-5 1^-7-1 -1-5 5 (a Order parameter 45 Order Parameter at.43 Bc -5 5 (b Normal current 46 Normal Current at.43 Hc The motion of flux through the system takes place predominantly along the grain boundaries. 47 The normal current movie shows that dissipation above Jc occurs principally in the grain boundaries. 48
Flux motion in high fields into the interior of grain boundaries Order Parameter at.94 Hc 15 15 b a 1 1 5 5 1 1^-3-1^-4.1-3 1^-5-4.1-5 1^-6 3 4-5 1^-7-1 -1-5 5 (a Order parameter -5 5 (b Normal current 49 Normal Current at.94 Hc 5 Conclusion Ginzburg-Landau theory is a triumph of physical intuition. It is the central theory for understanding the properties of superconductors in magnetic fields Bibliography/electronic version of the talk will be available at: http://www.dur.ac.uk/superconductivity.durham/ 51 5