Mathematical Models and Numerical Simulations of Superconductivity: an Introduction

Transcription

1 Mathematical Models and Numerical Simulations of Superconductivity: an Introduction Qiang Du Dept of Math, NUS 1

2 Outline of the tutorial Superconductivity and vortex state a brief account of the phenomena and history Mathematical models derivation, analysis, discretization and simulation Future challenges and questions possible research topics Disclaimer: materials taken from various articles, talks and web resources intended for beginning students/post-docs NUS

3 Thanks Inst of Math Sci at NUS Dr.Weizhu Bao and other members of the program committee Support from various agencies: CAS, US NSF, US DOE, HKRGC, British Council Collaborators over the years NUS 3

4 In collaboration with: A. Aftalion, CNRS & Univ. of Paris VI W. Bao National Singapore Univ. J. Chapman, Oxford University Z. Chen, Chinese Academy of Sciences J. Deang, Lockheed Martin Corp S. Ding, South China Normal Univ. P. Gray, University of N. Iowa M. Gunzburger, Florida State Univ. L. Ju, IMA, UMN and Univ. of S. Carolina C. Liu, Penn State Univ. F. Lin, Courant Institute, NYU R. Nicolaides, Carnegie Mellon Univ. J. Peterson, Florida State Univ. Y. Pomeau ENS & Univ. of Arizona J. Remski, University of Michigan-Dearborn X. Wu, Hong Kong Baptist Univ. Y. Yang, Polytechnic Univ., NY P. Zhang, Chinese Academy of Sciences NUS 4

5 Mathematical models: a brief introduction Lecture 1 : History, Phenomenon, Theory NUS 5

6 What is it? Superconductivity a phenomenon occurring in certain materials at low temperatures, characterized by the complete absence of electrical resistance and the damping of the interior magnetic field NUS 6

7 Superconductivity Why is superconductivity so fascinating? the fascination with superconductivity is associated with the words : perfect, infinite, & zero --- B. Maple NUS 7

8 A brief history 1911 Kamerlingh-Onnes Nobel prize winner (1913), for: liquefaction of helium in 1908, bringing the temperature of the helium down to 0.9K also noted: a momentous discovery (1911) was that of the superconductivity of pure metals such as mercury, tin and lead at very low temperatures, & following from this the observation of persisting currents NUS 8

9 A brief history Perfect conductivity below a critical T c, electrical resistance becomes zero (infinite conductance) ρ ρ T Tc T Normal metal Superconductor NUS 9

10 A brief history Critical applied magnetic field 1913 H < H c, ρ = 0 H > H c, ρ > 0 Phase transition between normal and superconducting state Type-I superconductor NUS 10

11 Type-I superconductors Pure elements with the coldest transition temperature, exhibiting sharp transitions to superconducting state: Lead (Pb) 7.0K Lanthanum (La) 4.88K Tantalum (Ta) 4.47K Mercury (Hg) 4.15K Tin (Sn) 3.7K Indium (In) 3.41K Thallium (Tl).38K Rhenium (Re) 1.70K Protactinium (Pa) 1.40K Thorium (Th) 1.38K Aluminum (Al) 1.18K Gallium (Ga) 1.08K Other elements with Tc below 1.0K: Molybdenum (Mo) Osmium (Os) Zinc (Zn) Americium (Am) Zirconium (Zr) Cadmium (Cd) Ruthenium (Ru) Titanium (Ti) Uranium (U) Hafnium (Hf) Beryllium (Be) Iridium (Ir) Tungsten (W) Platinum (Pt) Rhodium (Rh), NUS 11

12 Pure superconducting elements NUS 1

13 A brief history 1933 Meissner and Ochsenfeld Perfect diamagnetism: magnetic flux being expelled from the sample below T c NUS 13

14 A brief history Meissner-Ochsenfeld effect: independence on the history of preparation, thus a true thermodynamic state Magnetic levitation: Meissner effect is so strong that a magnet can actually be levitated over a superconductor NUS 14

15 A brief history Earlier theoretical work 1935 London & London: Long range order, London equation - λ h + h = H λ : penetration depth H h Sample interior NUS 15

16 A brief history Discovery of the type-ii superconductor 1930 dehaas and Voogd (not recognized until 1933) Shubnikov identified two critical magnetic fields. type-ii superconductors change from a normal state to a superconducting state in a more gradual process across a region of mixed state behavior. M NUS 16

17 A brief history Theory of superconductivity (low Tc) 1950 Ginzburg & Landau phenomenological theory complex wave function Ψ= ρ e iθ ρ : density of superconducting carriers 195 Abrikosov vortex lattice theory Nobel prize NUS 17

18 A brief history Mixed state in type-ii superconductor The appearance of quantized vortices (by now, a well-known signature of superfluidity) Flux quantum.07e-7gs/cm Supercurrents Abrikosov lattice (worked out in 195, published in 1957, experimentally confirmed in 1967) Superconducting Normal NUS 18

19 A brief history Type-I vs. type-ii superconductors More refined phase diagrams can be constructed (for instance, may include H c3 - surface superconductivity) NUS 19

20 A brief history Theory of superconductivity (low T c ) 1957 Bardeen, Cooper & Schrieffer (Nobel prize 1973) energy gap Cooper pairing NUS 0

21 A brief history 196 Josephson: tunneling effect 1986 Muller&Bednorz: high-tc superconductor Tc Record? Hg 0.8 Tl 0. Ba Ca Cu 3 O K NUS 1

22 A brief history "High T c " Superconductors YBCO : Layered Cu-O ceramic superconductor made of Yttrium, Barium, Copper & Oxygen, T c =9K LaSrCuO HgBaCaCuO NUS

23 A brief history Superconducting transition temperature (K) Hg Pb NbC Nb NbN Nb 3 Sn V 3 Si HgBa Ca Cu 3 O 9 (under pressure) HgBa Ca Cu 3 O 9 TlBaCaCuO BiCaSrCuO YBa Cu 3 O 7 (LaBaCuO Nb 3 Ge Liquid Nitrogen temperature (77K) Lecture NUS 3

24 A brief history Other Discoveries Nano-scale superconductors Doped C 60 (Buckminster fullerene) single-walled carbon nanotubes NUS 4

25 A brief history Other Discoveries MgB a conventional BCS superconductor (H c ~ 3.5 Tesla in carbon-doped MgB ) MgCNi 3 - between conventional and cuprate NUS 5

26 Superconductivity New definition after discovering HTS How much has changed over time: Webster s New World Pre-1987: The phenomenon, exhibited by certain metals and alloys, of continuously conducting electrical current without resistance when cooled to temperatures near absolute zero. Now: An electronic state of matter characterised by zero resistance, perfect diamagnetism, and long-range quantum mechanical order NUS 6

27 Superconductivity Applications energy: transmission lines, generators electronics: switches, high-speed chips medical: MR devices, detectors NUS 7

28 Superconductivity Applications science: SQUID, SFFT transportation: MAGLEV, submarine communication: antenna NUS 8

29 Mathematical models: a brief introduction Lecture : BCS model (basic idea) Ginzburg-Landau model (more later) (mostly for conventional superconductors) NUS 9

30 The world of superconductivity research Science Technology Experiments Device Development Theory in our lectures this week Microscopic Macroscopic Mezoscale NUS 30

31 Mathematical Models Microscopic Mezoscopic Macroscopic Hubbard London Bean t-j, SO(5) Pippard Critical State BCS Ginzburg Landau Vortex density Quantum Mechanical Macroscopic vast range of scales / multiscale NUS 31

32 Type of mathematical problems Variational problems: non-convex, gauge Elliptic systems: semi-linear to nonlinear Parabolic systems: degenerate, nonlinear Hyperbolic systems: vorticity transport Integral equations: BCS, current sheet Stochastic equations: thermal fluctuation - plenty problems, different variety! NUS 3

33 Superconductivity references Books: M. Tinkham Introduction to Superconductivity P. DeGennes Superconductivity of Metals and Alloys J. Schrieffer Theory of Superconductivity Paper: Du-Gunzburger-Peterson SIAM Review NUS 33

34 Microscopic theory of superconductivity 1957 Bardeen, Cooper & Schrieffer (Nobel prize 1973) Fundamental contribution to the understanding of the mechanism for Low T c superconductivity energy gap Cooper pairing NUS 34

35 MICROSCOPIC BCS THEORY Instability of the Fermi sea in normal metal due to Cooper pairing Phonon mediated electron attraction Superconducting ground state and the existence of energy gap Zero resistance NUS 35

36 Mechanism of Cooper pairing In the presence of an arbitrarily weak attraction and a filled Fermi sea, two electrons are always unstable w.r.t. to the formation of a bound state - known as a Cooper pair -a set of two electrons traveling together about nm apart, manages to find a path through the atoms, forms a inward trough through the lattice, and generates phonons, which create a positive charge helping push the electrons through the elements with no resistance NUS 36

37 MICROSCOPIC BCS THEORY The BCS gap equation: V -1 =N (λ + ) -1/ tanh[((λ + ) -1/ /k Β Τ)] dλ Critical temperature =0 for T=Tc V -1 =N λ -1 tanh(λ/k Β Τc) dλ k B T c =1.14 h ω C e -/NV NUS 37

38 MICROSCOPIC BCS THEORY Some related mathematical works: Existence of nontrivial positive solutions to the BCS gap equation: Y. Yang 1990 Monotone schemes and estimates of critical temperature: Q. Du and Y. Yang 199 Generalization to multi-gap: Y. Yang NUS 38

39 BCS THEORY - a short summary In a superconductor all the electrons are in the same quantum state - superconductivity is a macroscopic quantum state There exists an energy spectrum gap - energy to break a Cooper pair For pure elements, BCS theory is incredibly accurate (everything seemed settled and understood, before the discovery of HTS) Not applicable to HTS - Open Problem! NUS 39

40 GINZBURG-LANDAU THEORY 1950 Benefited from Gorter & Casimir: two fluids electrons in superconducting condensate normal electrons London: equations for magnetic field & supercurrent, macroscopic quantum phenomenon NUS 40

41 GINZBURG-LANDAU THEORY 1950 Based on the general theory of second order phase transition developed by Landau Formally derived from Microscopic BCS Variations for high Tc superconductors Nobel Physics Prize in 003 Remained as a standard initial approach Du-Gunzburger-Peterson SIAM Rev NUS 41

42 GINZBURG-LANDAU THEORY The choice of order parameter was not obvious, but a complex scalar field Ψ was a natural choice because of the analogy with liquid He : ψ is known to represent the superfluid density (a quantum mechanical density should be a complex wave function squared) NUS 4

43 GINZBURG-LANDAU THEORY The biggest leap of GL was to specify correctly how to couple electromagnetic fields (no obvious analog in superfluid He). They exploited the similarity of the formalism to ordinary quantum mechanics, coupled the fields in the usual way to charges e * associated with particles of mass m * : (h - i e * A/c)ψ /(m * ) NUS 43

44 GINZBURG-LANDAU THEORY 1950 Basic hypothesis: Introduction of a complex wave function ψ with ψ =n s, the density of superconducting carriers Free energy can be expanded as a suitable power series of ψ The coefficients in the series expansion are regular functions of the temperature valid near the critical transition temperature NUS 44

45 GINZBURG-LANDAU THEORY The homogeneous, isotropic, stationary model Ω : cross-section of an infinite cylinder occupied by the superconducting sample H : a constant applied field perpendicular to Ω The free energy density without applied field: α= α c (T-T c ), β=β c >0, α c >0 (near T c ) NUS 45

46 GINZBURG-LANDAU THEORY The dependence of free energy density on T α= α c (T-T c ), β=β c >0, α c >0 (near T c ) NUS 46

47 GINZBURG-LANDAU THEORY The kinetic energy with applied magnetic field velocity NUS 47

48 GINZBURG-LANDAU THEORY Energy density with an applied field H = µ 0 or NUS 48

49 GINZBURG-LANDAU THEORY Energy density with an applied field H Again: α= α c (T-Tc), β=β c >0, α c >0 (near Tc) H Ω Variational Problem: Min Ω f NUS 49

50 GINZBURG-LANDAU MODEL G-L equations NUS 50

51 GINZBURG-LANDAU MODEL Boundary conditions: Continuity of tangential magnetic field Zero flux: DeGennes proximity effect: - α ψ NUS 51

52 GINZBURG-LANDAU MODEL G-L equations Boundary conditions NUS 5

53 GINZBURG-LANDAU MODEL Supercurrent j=n * v NUS 53

54 G-L MODEL Coherence length ξ: Penetration depth λ: NUS 54

55 Physical length scales Penetration depth ~ A BCS Material Pure elements Organic Cuprates Coherence length ~100 nm >> 10 A ~10 nm ~ 10 A ~1 nm ~ 1 A Cuprates? BEC NUS 55

56 GINZBURG-LANDAU THEORY Thermodynamic critical field Normal state f =0 Meissner state f =-α /β+h /8π H c =(8πα /β) 1/ =(8πα c /β c ) 1/ (T c -T) NUS 56

57 Mathematical references on G-L models keyword: Ginzburg Landau Model Du-Gunzburger-Peterson, SIAM REVIEW, 199 physical background, mathematical formulation, basic theory, finite element approximation Chapman-Ockendon-Howison, SIAM REV, 199 type-i, phase transition, free boundary Chapman, SIAM REVIEW, 001 Rubinstein, 6 lectures on superconductivity F-H Lin, Barret lecture notes on G-L equations Books: Brezis-Bethuel-Helein, Hoffmann-Tang, Serfaty-Sandier? NUS 57

58 Mathematical models: a brief introduction Lecture : Summary BCS model: phonon mediated Cooper pairing existence of energy gap Ginzburg-Landau model: state described by minimizers of the free energy functional, competition of kinetic energy, potential energy and magnetic energy NUS 58

59 Mathematical models: a brief introduction Lecture 3: Ginzburg-Landau models and their variants Quantized vortices NUS 59

60 GINZBURG-LANDAU THEORY Energy density with an applied field H Again: α= α c (T-Tc), β=β c >0, α c >0 (near Tc) H Ω Variational Problem: Min Ω f NUS 60

61 GINZBURG-LANDAU THEORY Primary variables A: vector-valued magnetic potential Ψ: complex-valued order parameter Φ: scalar-valued electric potential Physical variables Ψ curla : density of superconducting carriers : induced magnetic field NUS 61

62 G-L MODEL Below Tc : β>0, α <0 Normal state interface curl A = H Ψ = 0 Meissner state Ψ = α curl A = 0 β NUS 6

63 G-L MODEL Ginzburg-Landau parameter κ =λ/ξ=m * (β c /8π) 1/ /e * h a material constant, independent of T. Nondimensionalization: κ has a critical value: 1, determined by the sign of an interfacial energy (G-L) NUS 63

64 G-L Model Min G ( Ψ, A) gauge invariance: for any function ρ G (Ψ, A) = G ( Ψ e i κρ, A + ρ) gauge fixing: H ( Ω,div) H ( Ω) { diva = 0 in Ω, A n = 0 on Ω} 1 n constrained minimization NUS 64

65 G-L Model Du,Gunzburger,Peterson SIAM Rev 9 Min G ( Ψ, A) gauge invariance Min G A H ( Ψ,A) 1 n ( Ω,div) No divergencefree constraint! (ideal in computation) F ( Ψ, A) A H 1 ( Ω ) Min n F ( Ψ, A) = G ( Ψ, A) + div A NUS 65

66 Time-dependent G-L Model φ: electric potential, κ,η,σ:parameters η Electric field NUS 66

67 Time-dependent G-L Model Gauge invariant gradient flow for the energy η( t Ψ + i Φ Ψ ) = δ δψ G ( Ψ, A) σ( A + Φ t Boundary condition ) δ = G ( Ψ, A) δa E n = J a Applied current curl A n = H n NUS 67

68 Time-dependent G-L Well-posedness Du 9, 94, Tang-Wang 96, Long-time behavior (J a =0) Lin-Du 97, Takac 99 Simplified equation: Du-Gray 96 η Ψ t + i ( Φ, A given) 0 Φ 0 Ψ = ( ia Compare with the semilinear equation: ) Ψ + ε (1 η 1 Ψ t = Ψ + ε (1 Ψ ) Ψ Ψ ) Ψ NUS 68

69 GINZBURG-LANDAU MODEL Qualitative behavior of solutions Effect of applied magnetic field (flux penetration) Effect of applied current (resistance) Effect of sample properties (parameters, temperature, anisotropy, inhomogeneity, geometry, topology) NUS 69

70 Effect of κ: two types of superconductors Type-I κ small positive interfacial energy cannot exclude large field Type-II κ large negative interfacial energy superconducting in larger field What prevents the creation of interface at arbitrarily small scales? Flux quantization NUS 70

71 Flux quantization What mechanism prevents the creation of interface at arbitrarily small scales? Draw a contour inside superconducting region, V = 0 thus v dl=0 Flux = Adl = (ch/e * ) φ dl = nπch/e*= nφ 0 n : integer, Φ 0 : unit flux quantum Flux quantization: current is the result of a phase gradient NUS 71

72 Effect of applied magnetic field For largeκ, large H, solutions of G-L have isolated vortices { x Ψ(x)=0} (Abrikosov, 1957) field penetrates sample through vortex core v Hc1 lower critical Hc upper critical much higher than Hc NUS 7

73 Ginzburg-Landau vortices Basic profile Ψ curl A r Stability: n=1 stable, n>1, unstable, Chen-Berger, Lin, Liu, Jimbo,, Sternberg, Ovchinnikov-Segal, Segal, NUS 73

74 Vortices in type-ii superconductors Vortex nucleation - applied magnetic field creates isolated vortices, giant vortex Vortex interaction - vortices with like-sign repel, opposite-sign attract, stability Vortex motion applied current (or voltage) induces vortex motion Critical fields magnetic field & current All could be explained through G-L theory, and much have been rigorously proved! NUS 74

75 Vortex solution Stationary solution of G-L model in a d box Η=0.5 κ=5 Ω=15ξ X 15ξ Surface and shadow plot of Ψ NUS 75

76 Steady state vortex solution Ψ j Ω=0ξ X 0ξ Η=0.5 κ=5 Finite element code, Du,Gunzburger,Peterson, PRB 9, NUS 76

77 Steady state vortex solutions Parameters affecting the solution branches in a d square domain : d: diameter of the domain, κ: Ginzburg-Landau parameter, H: applied field Effect of H: applied field NUS 77

78 Effect of applied field Some rigorous analysis in the large κ limit estimates of critical applied field [ ] ( ia ) 1 (1 ) curla-h Ψ + Ψ + Ω ε 1997 Lin & Du hysteresis near Hc Bauman, Phillips & Tang giant vortex Hc 1999 Pan & Lu surface superconductivity at Hc Serfaty, Sandier characterization of Hc1 expansion (in terms of ε) of the vortex core energy + the interaction energy + magnetic energy NUS 78

79 Effect of κ & d, bifurcation diagrams Aftalion-Du, 000, Physica D Steady state G-L L solutions Stability? Aspect ratio? NUS 79

80 Vortex interaction In d, vortices of the same sign repel, opposite signs attract ample evidence: asymptotics-numerics but analysis still incomplete d vortices may be viewed as crosssections of 3d vortex lines NUS 80

81 G-L MODEL in 3D Ω [ ] ( iκa) Ψ + 1(1 Ψ ) + R 3 curla -H dx d Ω Reduction to bounded domain via ABC Du - Wu SIAM Num Anal 1999 introduce artificial boundary, transfer energy in the unbounded domain to the boundary NUS 81

82 G-L MODEL in 3D Iso-surface plot of the vortex tubes Du-Gary SISC NUS 8

83 3D G-L: isolated vortex lines Self-induced motion: asymptotics Vortex lines in superconductor Vortex lines in incompressible fluid Velocity ~curvature Velocity ~curvature Velocity // - normal Velocity // - binormal NUS 83

84 Interaction of 3d vortex lines Parallel vortex lines in superconductor Parallel vortex lines in fluid NUS 84

85 Effect of applied current -TDGL Due to Lorentz force, vortices move in a direction perpendicular to applied current Du-Gray, 96, Du 0 time applied current Ja Solving time dependent GL with an applied current: we observe the motion of vortices (carrying quantized magnetic flux) generates induced electric current NUS 85

86 Mathematical models: a brief introduction Lecture 3: Summary Ginzburg-Landau models steady state model time-dependent model Quantized vortices isolated zeros of order parameter nucleated by applied field NUS 86

87 Mathematical models: a brief introduction Lecture 4: Time dependent Ginzburg-Landau models Dynamics of quantized vortices NUS 87

88 Time-dependent G-L Model Gauge invariant gradient flow for the energy η( t Ψ + i Φ Ψ ) = δ δψ G ( Ψ, A) σ( A + Φ t Boundary condition ) δ = G ( Ψ, A) δa E n = J a Applied current curl A n = H n NUS 88

89 Effect of applied current -TDGL Due to Lorentz force, vortices move in a direction perpendicular to applied current Du-Gray, 96, Du 0 time applied current Ja Solving time dependent GL with an applied current: we observe the motion of vortices (carrying quantized magnetic flux) generates induced electric current NUS 89

90 Effect of applied current Ψ 1 t + i Φ0Ψ = ( ia0) Ψ + ( ia 0 Ψ = Ψ 0 at t = ε ) Ψ n = 0 on Ω 0 (1 Ψ Φ curla ) Ψ 0 0 = J = H Existence of critical applied current? Consider a simpler problem: find steady states Steady state equation: non-variational form Small J, perturbation to a variational problem! NUS 90

91 Critical current Ψ 1 t + i Φ0Ψ = ( ia0) Ψ + ε (1 Ψ ) Ψ A perturbation result: Du-Liu (00) at least for unit disk, small ε, H near H c1, there exist steady states with one vortex for J < J 1! J=0 J< J 1 Main stability of vortex solution with J=0 ingredients: + implicit function thm + energy upper/lower bounds on solutions for J < J NUS 91

92 Time-dependent equations Current-driven Dynamics from vortex state Ψ 1 t + i Φ0Ψ = ( ia0) vortex-free state Critical applied current Ψ + ε vortex state (1 ) Ψ NUS 9 J Ψ normal state phase diagram near Hc1 Du: Contemp Math., 003

93 Time-dependent equations Ψ 1 t + i Φ0Ψ = ( ia0) Ψ + ε (1 Existence of time periodic solution? strong numerical evidence Ψ ) Ψ Φ curla 0 0 = J = H Analytical theory? Relation between period and J? NUS 93

94 Vortex pinning vortex motion induces electrical resistance, i.e., a loss of superconductivity! vortex pinning is needed! What features provide pinning effect? inhomogeneity, anisotropy, twin-boundary NUS 94

95 Vortex Pinning Mathematical Studies Pinning due to normal inclusion Chapman,Q.Du,Gunzburger,Peterson, 94 Aftalion,Serfarty,Sandier,Ding,Liu, Berger,Rubinstein,Sternberg, Bauman,Phillips Pinning due to anisotropy Q.Du,Gunzburger,Peterson 95, Deang,Du,Gunzburger,Peterson 97 Random pinning & de-pinning E 95, Deang,Du,Gunzburger NUS 95

96 Pinning due to variable thickness Variable thickness thin film model Chapman-Du-Gunzburger, 95, Physica D curl A 0 = H G a ( Ψ) = D a{ ( -ia 0 ) Ψ + 1 ε (1 Ψ ) }dd NUS 96

97 Vortex Pinning The variable thickness thin film model numerical simulation Chapman-Du-Gunzburger 95, Peeters et.al. 98,99 Deang-Du-Gunzburger-Peterson, 96, asymptotic limit Ding,Liu 97-98, Lin-Du 97, Jiang lower critical field Ding-Du 000, 00, time-dependent case Chapman-Richardson NUS 97

98 GINZBURG-LANDAU MODEL (stationary and time dependent) Qualitative behavior of solutions Effect of applied magnetic field (flux penetration) Effect of applied current (resistance) Effect of sample properties (parameters, temperature, anisotropy, inhomogeneity, geometry, topology) NUS 98

99 Effect of sample properties Samples with normal inclusion (S-N-S junctions) Chapman-Du-Gunzburger 95, Du,Remski, 99,00 S N S Ω Ω s n { { ( - κ ( - κ i i A ) Ψ A ) Ψ (1 (1+ Ψ Ψ ) ) + + (curl (curl A A H H ) ) } } DeGennes Proximity boundary condition, Josephson relation for the tunneling current NUS 99

100 Effect of sample properties spatial inhomogeneity produces pinning sites Ω { ( -i A) Ψ + 1 ε (α (x) Ψ ) + ( curla H) } High k limit Ω { ( - ia 0 ) Ψ + 1 ξ (α(x) Ψ ) } Deang-Du Du-Gunzburger 97 Aftalion-Serfaty Serfaty-SandierSandier NUS 100

101 Effect of sample properties Important for many applications of superconductivity Josephson Junctions dc & ac effects Superconducting weak links geometry/topology effects i h i t i h i t i t NUS 101

102 Effect of geometry/topology Rubinstein- Schatzman- Sternberg, Berger-Sternberg, 3D 1D Peeters et al NUS 10

103 Effect of geometry/topology Thin hollow sphere of constant thickness Du-Ju JCP 004, Math Comp 004 E( Ψ) = Ω { ( s ia ) Ψ + κ 0 (1 Ψ ) } d Ω A 0 = H ( y, x,0) T Parameters κ, H NUS 103

104 Vortex Nucleation with applied field Vortex nucleation on a thin spherical shell: (density plot of ψ on the upper hemi-sphere, as time increases, the color coding varies) NUS 104

105 Vortex configurations with increasing field Vortex state on the spherical shell: for different field (density plot of ψ on the upper hemi-sphere) NUS 105

106 Nucleation of vortex pairs Nucleation of vortex pairs of opposite signs near the equator (you can see a vortex and its mirror image), splitting and merging into the vortex lattice The first such observation in simulation NUS 106

107 Effect of thermal fluctuation Time-dependent stochastic G-L equations additive noises vs. multiplicative noises Dorsey, E, Bishop, Deang,Du,Gunzburger PRB 00, JCP 01 Deterministic Stochastic (multiplicative noise with a small constant variance) NUS 107

108 Effect of thermal fluctuation 3-D simulations using additive noises average steady states associated with a set of increasing values of variance Deang, Du, Gunzburger NUS 108

109 Mathematical models: a brief introduction Lecture 4: Summary Time dependent Ginzburg-Landau models Dynamics of quantized vortices Vortex pinning NUS 109

110 Mathematical models: a brief introduction Lecture 5: Variants of G-L models and Numerical methods NUS 110

111 High Temperature Superconductors Copper Oxygen Planes Other Layers Layered structure quasi-d system pancake vortices (Klemm, Vinokur, ) Lwrence-Doniach model: Chapman-Du-Gunzburger, Du-Gray, Phillips-Shin, Alama-Berlinsky-Bronsard, Alama-Bronsard-Sandier NUS 111

112 Extension to HTS d-wave G-L models Du 1999, Lin-Lin 001 Ω { ( -ia 0 ) Ψ + 1 ξ (1 Ψ ) + δ Π Ψ } p-wave G-L models Lin-Lin 003 SO(5) G-L models Alama-Bronsard ferromagnetism-superconductivity Stampli-Rice NUS 11

113 Extension to HTS d-wave G-L model - two order parameters with the s-wave & the d-wave components 4-fold symmetry DU SIAM Applied Math 1999 merging of vortices NUS 113

114 NUMERICAL METHODS Finite difference methods standard, gauge invariant Finite element methods conforming, nonconforming, multi-level implicit time marching, alternating steps Finite volume methods gauge invariant, unstructured grid Vortex/particle/spectral methods Ullah-Dorsey,Liu-Goldenfeld,Enomoto-Katsumi, Du, Chen-Hoffmann,Machida,Kaburaki,Wright et.al,crabtree et.al, Kaper et.al,mu,hoffmann-zou, Deang-Du-Gunzburger-Peterson, Du-Nicolaide-Wu, Wang-Wang, Mu-Huang, Chen-Dai, Du-Ju NUS 114

115 Finite element Du,Gunzburger,Peterson SIAM Rev 9 Min G ( Ψ, A) gauge invariance Min G A H ( Ψ,A) 1 n ( Ω,div) No divergencefree constraint! (ideal in computation) F ( Ψ, A) A H 1 ( Ω ) Min n F ( Ψ, A) = G ( Ψ, A) + div A NUS 115

116 CO-VOLUME METHODS finite volume primal/dual mesh staggered grid Based on physical principles conservation laws Green's theorem Stoke's theorem divergence theorem NUS 116

117 Differential calculus for discrete field A discrete tangential field ij τ A ij (curla) τ τ A ij h ij τ curl = A τ h edge length ij τ triangle area A r t NUS 117

118 Differential calculus for discrete field on a dual grid formed by joining circum-centers tangential field for triangle edges = normal field for polygon edges divergence in polygons NUS 118

119 Differential calculus for discrete field Gauge invariant derivatives r ( ia ) Ψ t ( Ψ Ψ e ij j i i A ij h ij ) / h ij ψ j τ A ij ψ i vector field for triangles tangential field for edges r v τ = i j cot θ ij h ij v r t ij NUS 119

120 Gauge invariant approximation of G-L ψ ij j { τ k A ij ψ + A A + (f - ij i j Ψ j + Ψ ( Ω ( NUS 10 i e j A i A ij h ij h ij ij ) Ψ cot j τ k ) θ Discrete gauge invariance f i ) / h ij Ψ j j Ψ e ij } / iκf j

121 Gauge invariant approximation of TDGL η ( Ψ n j + 1 Ψ = n j e i Φ n+ 1 j τ n ) / τ δ G ( Ψ n+ 1, A n+ 1 δψ n ) σ ((A n+ 1 jk - A n jk )/ τ n + ( Φ = n+ 1 k δ δa - Φ G ( Ψ n+ 1 j n+ 1 )/h, A ) jk n+ 1 ) Discrete gauge invariant scheme NUS 11

122 NUS 1 Gauge invariant approximation of G-L Gauge invariance & implicit enforcement Discrete maximum principle ψ 1 Discrete energy dissipation law: min of Φ Φ Φ + + Ψ Ψ Ω =, Ψ + n jk n j n k n n jk n jk jk n i n j n j j n n h A A h e n n j τ τ τ τ / ) / ( / ) A ( G 0 for ) A ( ) A ( 1 1 =, Ψ, Ψ + + J n n n n G G

123 Gauge invariant approximation of G-L Convergence under min. regularity First order in time accuracy Du, Math Comp 97 Du,Nicolaides,Wu, SIAM Num Anal 97 First order in discrete H 1 norm Second order in L norm with CVT grid Du Ju Math Comp NUS 13

124 Simulation challenge For superconducting samples of interests in device design, (in particular, HTS), there may be thousands of vortices present Scale of simulation in a square D? Given uniform grid, Ψ has n vortices n sign changes for Re{Ψ}, Im{Ψ} on bdry m grid points to resolve one change of sign NUS 14

125 Simulation challenge total mn/4 grid points on each side of D Ψ, A 4 unknowns at each grid point nonlinear systems of dimension m n /4 -d G-L m=5, n=10000, dim=6x d N-S with 51 grid, dim=5x10 8 Adaptivity? need to resolve the phase! (special feature of topological defects) Large scale 3-d simulation? alternative? NUS 15

126 Simulation challenge Parallel computation Explicit time marching (-d) Plassman et.al 96, Wright et.al 96, Crabtree et.al 97 Layer based decomposition (3-d) Du,Gray 99 Lawrence-Doniach approximation implementation on PC clusters NUS 16

127 Macroscopic vortex dynamics High Tc superconductors has large κ Lots of vortices present in real sample Models for large-scale simulation? G-L Motion law for isolated vortex Vortex density model NUS 17

128 d Vortex dynamics in the ξ=0 limit X j : isolated vortex locations with proper scaling, to leading order M X& j = H (X,..., X H : renormalized energy 1 n ) M = M(α) orthogonal H (X,..., ) = 1 X c n K ( Xj Xk ) + π j k NUS 18

129 Macroscopic/multiscale models G-L model for order parameter Motion laws for isolated vortices as the vortex core size turns to zero Neu, Pismen-Rubenstein, Chapman et.al., W.E, F.Lin, Rubenstein-Sternberg, Jarrad-Soner, Motion laws for the vortex density ω if: number of vortices vortex the vortex spacing 0 density model Chapman et.al., E, Dorsey, Lin-Xin, Lin-Zhang, Lin-Zhang, Serfaty-Sandier, Du-Zhang, NUS 19

130 Vortex dynamics Vortex density ω (x, t) n = c δ( x X j) j ω(x, t) Stream function G = c K( x Xj ) π (x, t) n j G(x, NUS 130 t) = 1 K( x y ) ω (y, t) dy π (Neu,Chapman,E,Lin,Xin,Zhang, )

131 Vortex dynamics without field Vortex density model (σ 0 if stochastic effect is included) ω + ( uω) = t - G = = Mu σ ω ω - G Existence-uniqueness: Lin-Zhang 00 Du-Zhang 003 Special case: M = Euler/NS NUS 131

132 Mathematical models: a brief introduction Lecture 5: Summary Numerical methods for G-L models NUS 13

133 Mathematical models: a brief introduction Lecture 6: Macroscopic Models Bose Einstein Condensates NUS 133

134 Vortex density model Chapman 0 SIAM Rev Facts: Magnetic field creates vortices field gradient drives vortices number of vortices conserved Generalization: Limiting equations λ u + u = ω V= M u ω t + div (ωv) =0 three dimension limiting models Μ=Ι, λ=0 ω - 1 t (ω ) = 0 pinning potential NUS 134

135 Vortex density model: Discretization Finite volume: Du 000 SINUM u, ω: vertices V: element edges Non-conforming finite element: Chen-Du M N u: piecewise linear, ω: piecewise constant, V: Raviart-Thomas Limiting equations λ u + u = ω V= M u ω t + div (ωv) = NUS 135

136 Bose-Einstein Condensation Theory of Bose-Einstein condensation 195 Recent experimental confirmation 1995 laser cooling 10 million particles 10 5 times thinner than air millimeter in dimension Nano-kelvin temperature NUS 136

137 Our focus I: Vortices: normal core BEC in rotating trap (MIT) (ENS) NUS 137

138 Theory: Gross-Pitaevskii Ψ= Ψ e iφ : complex wave function ( Ψ : particle density, φ : velocity) Energy Ω Ψ ρ ( x) Ψ ε ρ(x): trapping potential harmonic potential = ε 1/N healing length + ρ 1 ε Ψ ω i xi NUS 138

139 Comparison with Ginzburg-Landau High-κ G-L energy with pinning curl A 0 = H Ω { ( - ia0 ) Ψ + ( α (x) Ψ ) } Chapman-Du-Gunzburger, Du-Gray G-P energy in rotation frame 1 ε curl A 0 = ω Ω { ( - ia 0 ) Ψ + 1 ε (α(x) Ψ ) } Ψ =1 Striking similarity! Aftalion-Du Phys Rev A NUS 139

140 G-P energy expansion in d Estimates depends on an asymptotic expansion of energy which may be rigorously justified D { ( - ia 1 0 ) Ψ + ( α ε (x) Ψ ε curl A = ω α 0 ε (x) = α α i xi - ε A0 total energy = energy of vortex-less solution (close to α ε (x) except a boundary layer) + energy of vortex self-energy + energy of vortex interaction (renormalized energy) ) } NUS 140

141 Numerical simulation simulating MIT experiments (Du-00) Numerical Physical Simulation Experiment NUS 141

142 Vortices beyond rotation: Recent experiments (MIT group): Laser passing v through a BEC cloud flow passing through a cylinder? theory/simulation: (normal fluid) NUS 14

143 Our focus II: BEC experiments Dissipation as laser passing through cloud: MIT NUS 143

144 Boundary layer analysis Standard scaling analysis: Fetter-Feder, PRA 98 Dalfovo-Pitaevskii-Stringari, PRA 96 Schrodinger (G-P) equation h m ihφ = Φ + ( V + Ng Φ ) Φ t blow-up near the condensate edge Computational domain: 3d box Laser: modeled by an obstacle NUS 144

145 Dimensionless form: G-P: 1 iφ = t Φ + ( ρ Φ ) Φ TF ε ρ TF = x - y -z ε = 6.1E 3, a small parameter ε small Φ close to ρ TF except near the condensate edge (and the laser) cigar shaped cloud (inhomogeneous density distribution of BEC) NUS 145

146 Dissipative flow in superfluid Existing theory applies only to the center the cloud D analysis: Frisch-Pomeau-Rica, Phys. Rev. Lett. 199 below V c, the flow is stationary and dissipationless, above V c, the flow emits vortice, producing drag Numerical simulation: Huepe-Brachet, Physica D (000) Jackson-McCann-Adams, Phys. Rev. A (000) Winiecki-Jackson-McCann-Adams, J. Phys. B (000) 3D computation is challenging due to inhomogeneity NUS 146

147 Dissipative flow in superfluid Our approach: consider two distinct regions Condensate boundary: sound velocity small near condensate edge, trapping, interaction and kinetic energy are of the same order; we derive a Painleve type boundary layer equation as an effective model Far beyond boundary: system is dilute, interaction may be ignored, we have a linear regime Our aim: understand the onset of vortex nucleation in the boundary region when laser passing through an inhomogeneous cloud NUS 147

148 Boundary layer equation z Equation in the moving frame iu t + u + iv x u + ( ρ 0 z u ) u = BC: u=0, on the boundary of cylinder and at z=0 u periodic for x, y =30, ( u / p) = 0, at z=l z 0 0 = p' ' + ( ρ z p ) p 0, p 0) = 0, p(l) = ρ L ( NUS 148

149 Boundary layer equation iu t + u + iv x u + ( ρ 0 z u ) u = Rescaled velocity: v = Rescaled drag 1/ 6 v exp ε /( ω d ) z z 0 d 1 = u xu n u xu n Note that by a change of variable u=w exp(-ivx) iw t + w + ( ρ 0 z + v w ) w = NUS 149

150 Nonlinear regime: 3-d computation Numerical simulation of the G-P equation: x = ( α + i) ut + u - iv u + ( ρ 0 z u ) u z 1. Starting with u=p. Integrating GP with a damping coefficient α=-0.3~ Integrating with reduced damping α=-0.0~-0.03 or α= NUS 150

151 Linear regime: drag Total drag on cylinder, for small v v 3 / log 8/3 v Nonzero drag not due to vortex (?) but radiation of wave field at infinity! NUS 151

152 Nonlinear regime: 3d computation Small velocity: Almost stationary, Small oscillation at bottom, Smalll drag isosurface of u NUS 15

153 Vortex handle Formation of vortex handle as velocity increases, near the bottom of the obstacle, drag remains small, top of the handles move up and down along the obstacle Surface excitation Anglin NUS 153

154 Formation of vortex tube/ring Driven by surface instability NUS 154

155 Computing the drag vs. velocity --: Morse, *: 3d numerical computation Adtalion-Du-Pomeau 003 Phy Rev Letters NUS 155

156 Some other on-going works Studies of the Schrodinger vortex dynamics Multi-band G-L (MgB thin film) Nano-scale hollow cylinder, sphere (in collaboration with physicists at PSU) Vortex instability in BEC Feshbach resonance Spinor condensates NUS 156

157 Other questions of interests Time dependent G-L with current critical current (with and without pinning) effects of current/field coupling Du, Contemp Math, 39, pp , 003 Time dependent G-L with stochastic effects vortex glass, vortex liquid, collective pinning Topological dependence Time dependent G-P, optical traps, interactions with lasers, superfluid turbulence NUS 157

158 Other questions of interests Junctions, Junction arrays Networks many problems remain to be investigated NUS 158

159 Future Multi-scale modeling & computation Coupling of Ginzburg-Landau, London, vortex density, and critical state models (for device simulation) Understanding the microscopic nature of high-tc superconductivity (top-10 question in physics) We hope in the new millennium, the mystery of superconductivity will be unraveled & superconductors will be found to work at even higher temperatures --- Paul Chu NUS 159

160 Conclusion The lectures are of introductory nature, similar tutorial has been delivered earlier at IMA Mathematical study of superconductivity remains a perfect research topic for young researchers (infinite number of problems, zero resistance) Experiments Device Development Theory Science Tech Microscopic Macroscopic Mezoscale Experiment Analysis Implementation Modeling Algorithms Application Thanks to NUS for providing this opportunity NUS 160

161 In collaboration with: A. Aftalion, CNRS & Univ. of Paris VI W. Bao National Singapore Univ. J. Chapman, Oxford University Z. Chen, Chinese Academy of Sciences J. Deang, Lockheed Martin Corp S. Ding, South China Normal Univ. P. Gray, University of N. Iowa M. Gunzburger, Florida State Univ. L. Ju, IMA, UMN and Univ. of S. Carolina C. Liu, Penn State Univ. F. Lin, Courant Institute, NYU R. Nicolaides, Carnegie Mellon Univ. J. Peterson, Florida State Univ. Y. Pomeau ENS & Univ. of Arizona J. Remski, University of Michigan-Dearborn X. Wu, Hong Kong Baptist Univ. Y. Yang, Polytechnic Univ., NY P. Zhang, Chinese Academy of Sciences NUS 161

162 Thanks to the organizers and thanks to all of you NUS 16