Vortices in superconductors: I. Introduction

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1 Tutorial BEC and quantized vortices in superfluidity and superconductivity 6-77 December 007 Institute for Mathematical Sciences National University of Singapore Vortices in superconductors: I. Introduction François Peeters Literature Superconductivity of Metals and Alloys, P. G. De Gennes (Addison- Wesley, New York, 1989). Introduction to superconductivity, M. Tinkham (Mc. Graw Hill, New York, 1966). Vortices in unconventional superconductors and superfluids, Eds. R.P. Huebener, N. Schopohl and G.E. Volovik (Springer, Berlin, 00). Fundamentals of the theory of metals, A. A. Abrikosov (North- Holland, Amsterdam, 1988). Quantum theory of solids, C. Kittel (John Wiley & Sons, N.Y., 1987). Theory of superconductivity, J.R. Schrieffer (Benjamin/Cummings Publ. Comp., London, 1964). 1

2 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 3 I.1. Historical survey 1911: Discovery of superconductivity 1933: Meissner effect 1935: London theory 1950: Ginzburg-Landau theory 1957: BCS theory 1986: Discovery of high T c superconductors 1991: Superconducting fullerenes 000: MgB 4

resistance (Ω) Nobel prize: 1913 temperature (K) 5 Meissner effect

3 Discovery of superconductivity 1911: H. Kamerlingh Onnes (Leiden) First basic property: Perfect conductivity resistance (Ω) Nobel prize: 1913 temperature (K) 5 Meissner effect 1933: Meissner and Ochsenfeld Magnetic levitation Second basic property: Perfect diamagnetism 6 3

Ginzburg-Landau theory 1950: Ginzburg and Landau They introduced the order parameter Ψ: Ψ = 0 when T > T c Ψ 0 when T < T c Ψ is

4 London theory F. London and H. London Proc. R. Soc. London Ser. A 149, 71 (1935). First attempt to develop a theory to describe the electrodynamics of a superconductor: absence of resistance Meissner effect 7 Ginzburg-Landau theory 1950: Ginzburg and Landau They introduced the order parameter Ψ: Ψ = 0 when T > T c Ψ 0 when T < T c Ψ is related to the density of superconducting electrons n s as Ψ =n s Ψ is in general a spatial varying function GL theory was able to explain the distinction between type-i and type-ii superconductors 8 4

5 BCS theory 1956: Cooper Superconducting electrons are coupled two by two and such a pair has a lower energy than two individual electrons. Cooper-pairs 1957: Bardeen, Cooper and Schrieffer Microscopic theory Nobel prize: Evolution of T c T c (K) Nb 0 3 Sn NbC NbN Nb 3 Ge Pb V 3 Si Hg 0 Nb Year of discovery 10 5

140 Evolution of T c 10 100 T c (K) 80 60 40 T c = -558.4 + 0.

6 140 Evolution of T c T c (K) T c = year Nb 0 3 Sn NbC NbN Nb 3 Ge Pb V 3 Si Hg 0 Nb Year of discovery 11 High T c superconductors J.G. Bednorz and K.A. Müller, Z. Phys. B 64, 189 (1986) IBM Zurich Research Laboratory, Rüschlikon, Switzerland. Nobel prize:

High T c superconductors 1973 : Nb 3 Ge T c = 3.

CuO-layers Ba: hole doping Spring 1986 : La-Ba-Cu-O Bednorz and Müller

Liquid N T c (K) 140 10 100 80 60 fluorinated Hg-13 HgBa Ca Cu 3 O 8

7 High T c superconductors 1973 : Nb 3 Ge T c = 3.K Intermetallic material with A15 structure Y: separates the CuO-layers Ba: hole doping Spring 1986 : La-Ba-Cu-O Bednorz and Müller T c = 36K; about 10K higher than any previously known superconductor. November 1986 : Y-Ba-Cu-O P.C.W. Chu (University of Houston) T c > 90K 13 High T c superconductors Liquid N T c (K) fluorinated Hg-13 HgBa Ca Cu 3 O 8 Tl Ba Ca Cu 3 O 10 YBa Cu 3 O 7 Liquid He 40 LaBaCuO 4 Nb 0 3 Sn NbC NbN Nb 3 Ge Pb V 3 Si Hg 0 Nb Year of discovery 14 7

Fullerenes 140 10 fluorinated Hg-13 HgBa Ca Cu 3 O 8 Tl Ba Ca Cu 3 O 10 T c (K) 100 80 YBa Cu 3 O 7 60 Cs 3 C 60 @1.

8 Fullerenes fluorinated Hg-13 HgBa Ca Cu 3 O 8 Tl Ba Ca Cu 3 O 10 T c (K) YBa Cu 3 O 7 60 Cs 3 C 40 LaBaCuO 4 RbCs 60 Nb 0 3 Sn NbC NbN Nb 3 Ge Pb V 3 Si Hg 0 Nb K 3 Cs Year of discovery 15 Discovery of MgB I. Nagamatsu et al., Nature 410, 63 (001). T c = 39K Hexagonal AlB -type crystal structure. B-atoms form graphite like sheets which are separated by hexagonals layers of Mg atoms. 16 8

9 MgB fluorinated Hg-13 HgBa Ca Cu 3 O 8 Tl Ba Ca Cu 3 O 10 T c (K) YBa Cu 3 O 7 60 Cs 3 C 40 LaBaCuO MgB 4 RbCs 60 Nb 0 3 Sn NbC NbN Nb 3 Ge Pb V 3 Si Hg 0 Nb K 3 Cs Year of discovery 17 Transition temperatures 18 9

10 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 19 I.. Basic properties Perfect conductivity: zero resistance Meissner effect: perfect diamagnetism 0 10

11 London equation F. London and H. London, Proc. R. Soc. London Ser. A 149, 71 (1935). First attempt to develop a theory to describe the electrodynamics of a superconductor: absence of resistance Meissner effect dv A free electron in an electric field: m ee. dt = j = n ev with n e the electron density. e London equation: Λ = Λ = t n e m ( j) E with rot Λ j = rot E t 1 B (Maxwell eq.) c t 1 Λ + = 0 or (because( because B=rotA rota): rot j B c e Zero resistance! 1 rot Λ j + B = 0 t c 1 j = A c Λ 1 London equation Next we will show that London s equation gives that inside a superconductor: j = 0 and B = rot π π B = j (M axwell equation) rot rot B = rot j c c = grad ( div B ) ΔB 1 rot Λ j + B = 0 : London equation c 1 mc ΔB B = 0 with λl= the London penetration depth λ 4πne L e This equation accounts for the Meissner effect. Example: for a superconductor which occupies the half-space x < 0: with B 0 the magnetic field at the surface. From the London equation: ( ) = cb jy x 0 x / λl e 4πλL ( ) B x = B e B 0 x / λl 0 B 0 x 11

12 London equation It was assumed that all electrons participate in the superconducting current λ L depends on temperature n e (T). Different form for London equation: B = rot A rot ( Λ j) + B = 0 rot Λ j + A = 0 j = A c c Λc 1 B 1 A 1 A rot E = rot E + = 0 E = c t c t c t Equalities are valid up to a gradient-term, i.e. rot (grad) = 0. In fact it corresponds to the following gauge choice: div A = 0 Theory is valid in the limit ξ 0 or extreme type-ii materials. 3 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 4 1

Ginzburg-Landau theory Developed by Ginzburg and Landau in 1950. Based on the ideas of Landau s theory of second-order phase transitions.

Landau 5 GL-theory: derivation Order parameter: wave function of Cooper-pairs in the Bose condensate ( r) : characterizes all particles of the condensate Near the critical temperature T ~T

13 Ginzburg-Landau theory Developed by Ginzburg and Landau in Based on the ideas of Landau s theory of second-order phase transitions. Gor kov showed in 1959 that the GL equations can be derived from the BCS theory in the limit T c T << T c. V.L. Ginzburg L.D. Landau 5 GL-theory: derivation Order parameter: wave function of Cooper-pairs in the Bose condensate ( r) : characterizes all particles of the condensate Near the critical temperature T ~T c Ψ r is small Ψ ( ) F = F n + a Ψ + ½b Ψ 4 + and a and b can be expanded in τ = (T-T c )/T c. F-F n a,b > 0 a < 0 b > 0 For T > T c : Ψ = 0 and thus we must have a > 0. For T < T c : Ψ # 0 and consequently a < 0. Therefore to first order in τ : a = ατwith α > 0. b > 0 because Ψ must be finite. Ψ 6 13

14 Equilibrium: GL-theory: Equilibrium & Thermodynamics ( ατ b ) F = 0 Ψ + Ψ = 0 Ψ ( ) ατ H Ψ = 0 ; T > T ατ Ψ = = Ψ 0 ; T < T b c Fn Fs = = b 8π π thermodynamic critical field of a bulk superconductor: H c = α b Thermodynamics: F specific heat: C = T T * pm F C s Cn = T 3 T α 3 Cn Cs = Electron Fermi liquid T b c c 4 Tc T T c c T c T 7 GL-theory: external magnetic field When an external magnetic field is applied: H() r because of the Meissner effect. Ψ() r order parameter is a spatial varying function. Energy: H : energy of magnetic field per unit volume. 8π Ψ : energy due to spatial variation of Ψ (if variation is slow). e i i A : because Ψ is wavefunction of Cooper-pairs. c H = 0 b 4 1 e H n ατ Fs = F + dv Ψ + Ψ + i A Ψ + 4m c 8π 8 14

15 Boundary condition n i A = boundary b i ( ) ψ ψ boundary 9 GL-theory: comments All constants appearing in the GL theory have a microscopic interpretation (from the derivation from BCS): α b= n e : total electron density ( 1 7 ( 3) )( mt c pf) α = π ζ Ψ = Δ : Δ r =superconducting gap 1/ ( r) ( ζ ( ) ne π ) ( ( r) Tc) ( ) 30 15

16 GL-theory: region of validity pure superconductors: T T T Tc EF 1 c c dirty superconductors: T T T T τ E 1 c c c tr F 3 4 T c /E F : thermal fluctuations (~ 10-3 ) Thus: GL theory is valid up to T c. This condition is less stringent, but it also practically rules out the fluctuation regime. Thus, fluctuations are unimportant in the thermodynamics of bulk superconductors. But in high-t c ceramics the fluctuation regime is relatively large. This can be due to the small correlation length (~ 30Å) or the relatively large value of T c /E F. In thin films and filaments the role of fluctuations increases. 31 Ginzburg-Landau formalism condensation energy (<0) interaction energy (>0) α Fs = Fn + dr β 4 Ψ ( r) + Ψ( r) 1 * + Ψ m* internal magnetic field rot A( r) = h( r) * e 1 () r i A() r Ψ () r + (() h r H) c 8π kinetic energy operator for Cooper pairs: m*=m, e*=e applied magnetic field α = mξ o T (1 ), length scales: T o ξ = / m * α coherence length λ = c m β 4 e π α penetration length 3 16

17 F s Ginzburg Landau equations δ F δ F s s Ψ, A = 0, = 0 δ Ψ δ A 1 m e i A Ψ = αψ βψ Ψ c e ( i A) Ψ = 0 c n 4π A = j c e * * j = ( Ψ Ψ Ψ Ψ ) Ψ im mc 4 e A 33 Dual point of the GL equations Dimensionless free energy: 1 F = dv B + κ ( 1 Ψ ) + ( ia) Ψ The GL equations are obtained by minimizing F with respect to Ψ and B = A ( ia) Ψ= κ ( ) Ψ 1 Ψ & B = j * where the current is given by j = Im( Ψ Ψ) Ψ A, and ia Ψ = the boundary condition at the superconducting/insulater interface: ( ) 0. The GL equations are coupled nonlinear second-order differential equations. κ = 1 Dual point of the GL equations. Separates type-i and type-i superconductors. It was identified by Bogomol nyi [Sov. J. Nucl. Phys. 4, 449 (1997)] in the general context of stability and integrability of classical solutions of some quantum field theories. n 34 17

18 Dual point For κ = 1 the GL equations can be reduced to first-order differential equations. Introducing the operator D = x + i y i( Ax + iay) one can write the GL equations at κ = 1 as the two Bogomol nyi equations: DΨ = 0 B = 1 Ψ These two equations can be decoupled and one obtains that Ψ is a solution of the ln Ψ = Ψ 1. second-order nonlinear equation ( ) Relation to other fields: the equation is related to the Liouville equation. it appears in other situations, eg. Higgs, Yang-Mills and Chern-Simons field theories. It was proven that these equations admit families of vortex solutions. See e.g. E. Akkermans and K. Mallick, J. Phys. A: Math Gen. 3, 7133 (1999). 35 Gross-Pitaevskii equation Nonlinear Schrödinger equation. Bose-Einstein condensation of dilute gasses: Ψ ( rt, ) i = H0Ψ rt, + NU 0 0 Ψ rt, Ψ rt, t ( ) ( ) ( ) Potential due to the mean field Ψ ( rt, ) : condensate wave function of all the other atoms. : number of particles in the condensate N 0 Interaction between the atoms modeled by a zero-range potential whose strength is given by the s-wave scattering length a U 0 = 4πħ a/m. H 0 = H kin + V trap 36 18

Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4.

19 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 37 Bulk superconductors Two characteristic length scales: - Coherence length ξ: typical length scale over which the size of the Cooper-pair density can vary. - Penetration depth λ: typical length scale over which the magnetic field can vary. Ginzburg-Landau parameter κ determines type of superconductivity: λ κ = < ξ λ κ = ξ > 1 1 Type-I Type-II λ ξ 38 19

20 Properties of some superconductors Type I High T c 39 Type-I type-ii superconductors The difference between the two types of superconductors refers to the way the superconducting state reacts to an applied magnetic field. 40 0

21 Type-I superconductors H 0 H (T) c Meissner State 0 T c0 T Meissner State 41 Type-I B = H + 4π M -M - Inside the superconductor: 1 B = 0 M = H 4π The superconductor is an ideal diamagnet: 1 χ = 4π Meissner State Hc H0 4 1

22 Type-II Two critical magnetic fields. H < H c1 : material behaves like a type-i superconductor. H > H c : normal superconductor. -M H c1 < H < H c : Mixed state: the magnetic field lines (i.e. vortices) penetrate the sample. At the core of the vortices the material is in the normal state. Meissner State Mixed State 0 H c1 H c H 0 43 Type-II superconductors H 0 H c3(t) Surface superconductivity Mixed State H c(t) Meissner State H c1(t) 0 T c0 T Mixed state: Abrikosov vortex lattice 44

23 First vortex creation first critical field Magnetic field penetrates an area πλ. Flux = flux quantum: H Type-II φ πλ φ H πλ 0 c1 0 c1 Maximum number of vortices second critical field Vortices overlap almost uniform penetration of magnetic field Area of superconducting material between vortices ~ πξ φ0 Stabilization energy of a vortex. Hcπξ φ0 Hc πξ H c 8π Thermodynamic condensation energy H c Fcore πξ Energy cost due to the decreased amount of superconducting phase. 8π Ba Fmag πλ Energy gain because an amount of field does not have to be expelled. 8π 1 F Fcore + Fmag = ( Hcξ Ba λ ) 8π 45 1 F F + F = H B 8π ( ξ λ ) core mag c a Type-II The vortex is stable when F < 0. F = B H ξ = H λ 0 a c c1 First critical field H c c1 H H H c Inverse relation between H c1 and H c. H c1 c 46 3

$superconductors A.A. Abrikosov, Soviet Phys. JETP 5, 1174 (1957) First experimental observation: D. Cribier et al, Phys. Lett. 9, 106 (1964) using neutron diffraction Earliest visualization: U.$

24 1 M = B H 4π Magnetization ( 0 ) -M -M Meissner State Meissner State Mixed State Hc H 0 H c1 H c H 0 (a) Type-I (b) Type-II 47 Mixed state: vortex lattice Abrikosov lattice in type II (hard) superconductors A.A. Abrikosov, Soviet Phys. JETP 5, 1174 (1957) First experimental observation: D. Cribier et al, Phys. Lett. 9, 106 (1964) using neutron diffraction Earliest visualization: U. Essmann and H. Träube, Phys. Lett. A 4, 56 (1967) using Bitter patterning 48 4

Imaging of vortices STM image of Abrikosov vortices in MgB M.R. Eskildsen et al, Phys. Rev. Lett.

14,, 79 (001) Vortices in superconducting films with arrays of artificial pinning ing sites Scanning Hall microscopy S.B.

Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3.

25 Imaging of vortices STM image of Abrikosov vortices in MgB M.R. Eskildsen et al, Phys. Rev. Lett. 89,, (00) Magneto-optical optical image of vortex lattice in NbSe P.E. Goa et al, Supercond. Sci. Technol. 14,, 79 (001) Vortices in superconducting films with arrays of artificial pinning ing sites Scanning Hall microscopy S.B. Field et al, Phys. Rev. Lett. 88,, (00) Lorentz microscopy K. Harada et al, Science 74,, 1167 (1996) 49 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 50 5

26 Vortices in Confined Systems 51 Vortices in confined systems London approximation 5 6

27 Overview I. Introduction to superconductivity 1. Historical introduction. Basic properties & early phenomenological theories 3. Phase transitions and the GL theory 4. Type-I and type-ii superconductors 5. Molecular dynamics: London approach 6. Numerical solution of GL-equations 53 Theoretical model Dimensionless Ginzburg-Landau equations: i A ψ = ψ ψ ψ A ( ) 1 1 * * ( A) = ( ψ ψ ψ ψ ) ψ A κ i Boundary conditions: = A 0 i n ( i A) ψ = ψ boundary b boundary length scale: ξ = / m * α order parameter: ψ 0 = α / β vector potential: c /eξ magnetic field scale: Hc = κ Hc = c /eξ 54 7

28 D formalism axial symmetry and fixed vorticity L: il Ψ ρφ, = f ρ e φ ( ) ( ) Restriction to thin disks: d < ξ GL equations: 1 F L ρ + A f = f ( 1 f ) 1D ρ ρ ρ ρ 1 ρa A L ρ z κ + = A f θ θ ρρ ρ z ρ R d D 55 D formalism Boundary conditions: - Vector potential: - Order parameter: A f ρ r 1 = eh φ 0ρ f = 0 ρ = 0 ρ ρ= R ρ=

29 D formalism Magnetic field created by supercurrents has a H 1/r 3 dependence. finite simulation region with size R s and thickness d s. 1 A( z, ρ = Rs ) = H0Rs 1 A( z = ds, ρ ) = H0ρ 57 Finite difference representation: - Non-uniform space grid ρ i, z j - In every grid point (i,j): Order parameter: f i Vector potential: A i,j - Iteration step k D formalism 58 9

30 η ρ f f ρ f f η A a k k k k k i+ 1 i i i 1 f fi i+ 1/ i 1/ ρi+ 1/ ρi 1/ ρi+ 1 ρi ρi ρi 1 D formalism L + A f f + f f = f + f ρ i k k k k k κ ρi+ 1 Aji, + 1 ρiaji, ρiaji, ρi 1 Aji, 1 j, i ρi+ 1/ ρ i 1/ ρi+ 1 ρi ρi ρi ( ) η ( ) k k k k k k 3 i i 3 i i f i i A A A A L = z z z z z z k k k k κ j+ 1, i ji, ji, j 1, i k k k 1 Aj, i ( fi ) ηaa j, i j+ 1/ j 1/ j+ 1 j j j 1 ρi 59 3D formalism General solution Restriction to thin disks d < λ and ξ GL equations: ( i A) ψ = ψ ψ ψ d Δ 3D A = δ ( z) j D κ 1 * * where j D = ( ψ Dψ ψ Dψ ) ψ A i 60 30

31 3D formalism Finite difference representation: - Cartesian grid (x,y) - In grid point j: Order parameter Ψ j l Vector potential A j a y - Link variable approach - Time derivatives to find i steady-state solutions of the GL equations a x m j n k f g h 61 First GL equation: 3D formalism Ψ 1 U U U U Ψ 4Ψ = t 1 a a a a a kj ij mj gj Ψk Ψi Ψm g j x x x x x + Ψ Ψ + ( 1 T)( 1) f () t j j j where U is the link variable: r r 1, r Uμ = exp Aμ ( r) dμ ( μ = x, y) r1 6 31

32 3D formalism Second GL equation * 1 * 1 1 * j xy. = Ψ A, A, Ψ+Ψ Ψ i x, y xy i x, y xy where the link variable is used: kj 1 U Ψ Ψ Ax Ψ i i x a x k j x From the supercurrents one calculates the vector potential: d Δ 3D A= δ ( z) jd κ 63 Expansion method β 4 * GL functional: G G = n + dr α Ψ + Ψ +Ψ LΨ * 1 e * * L = i A * m c Kinetic energy operator with e = e; m = m Orthonormal eigenfunctions of the kinetic energy operator Expansion of the order parameter: Ψ = N i C Ψ i i Lψ i i = εψ i i * β ij * * GL functional: G Gn = ( α + εi) CC i i + ACCCC kl i j k l ij * * Akl = drψ with matrix elements iψψψ j k l numerically calculated See: V.A. Schweigert and F.M. Peeters,, Phys. Rev. Lett. 83,, 409 (1999) 64 3

33 Expansion method The sample geometry enters in the calculations only through the eigenenergies ε i and the eigenfunctions ψ i, through the boundary condition. Circular symmetry: ψ i= (, l n) = exp( ilφ) fn( ρ) l: angular momentum n: Landau level Expansion of order parameter over all eigenfunctions with ε i < ε * (cut-off). Thus, the superconducting state is mapped in a D cluster of N particles (x,y) (Re(C), Im(C)), which is governed by the Hamiltonian * β ij * * G Gn = ( α + εi) CC i i + ACCCC kl i j k l. No distortion of the magnetic field: No second GL equation. Magnetic field = externel field everywhere (λ-large, i.e. κ=λ/ξ>>1) Improve by: solve LGL with effective H eff which is taken as a varational parameter See: J.J. Palacios, F.M. Peeters and B.J. Baelus,, Phys. Rev. B 64,, (001) 65 The end part I 66 33

Superconductivity and Superfluidity

Superconductivity and Superfluidity Contemporary physics, Spring 2015 Partially from: Kazimierz Conder Laboratory for Developments and Methods, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Resistivity