A variational approach to the macroscopic. Electrodynamics of hard superconductors

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1 A variational approach to the macroscopic electrodynamics of hard superconductors Dept. of Mathematics, Univ. of Rome La Sapienza Torino, July 4, 2006, joint meeting U.M.I. S.M.F. Joint work with Annalisa Malusa

2 Outline 1 Introduction

3 Outline 1 Introduction 2

4 Introduction In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911)

5 Introduction In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911) The behavior (at T < T c ) of a superconducting sample in an external magnetic field H S is characterized by the Ginzburg-Landau parameter κ of the material.

6 Introduction H S In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911) The behavior (at T < T c ) of a superconducting sample in an external magnetic field H S is characterized by the Ginzburg-Landau parameter κ of the material. type I 1/ 2 type II κ

7 Introduction H S In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911) The behavior (at T < T c ) of a superconducting sample in an external magnetic field H S is characterized by the Ginzburg-Landau parameter κ of the material. Normal Hc H c : critical field (type-i) Superconduct. type I 1/ 2 type II κ

8 Introduction H S In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911) The behavior (at T < T c ) of a superconducting sample in an external magnetic field H S is characterized by the Ginzburg-Landau parameter κ of the material. Normal Hc 2 Hc H c : critical field (type-i) Superconduct. Hc 1 H c1, H c2 : critical fields (type-ii) type I 1/ 2 type II κ

9 Introduction H S In many materials, the resistance drop to an unmeasurably small value if the sample is cooled down at a temperature below its critical temperature T c = superconductivity (H. Kamerlingh-Onnes 1911) The behavior (at T < T c ) of a superconducting sample in an external magnetic field H S is characterized by the Ginzburg-Landau parameter κ of the material. Normal Hc Superconduct. Hc 2 Mixed Hc 1 H c : critical field (type-i) H c1, H c2 : critical fields (type-ii) type I 1/ 2 type II κ

10 Introduction Penetration of external magnetic field = penetrated magnetic field Supercond. Superconducting phase: thin layer (20-50 nm); no magnetic field in the bulk of the superconductor.

11 Introduction Penetration of external magnetic field = penetrated magnetic field Supercond. Mixed Superconducting phase: thin layer (20-50 nm); no magnetic field in the bulk of the superconductor. Mixed state: partial penetration in the bulk.

12 Introduction Penetration of external magnetic field = penetrated magnetic field Supercond. Mixed Normal Superconducting phase: thin layer (20-50 nm); no magnetic field in the bulk of the superconductor. Mixed state: partial penetration in the bulk. Normal conducting phase: full penetration in the bulk.

13 Introduction Advantages of type-ii superconductors Technological advantages of type-ii superconductors in mixed state: high T c (up to 135K) superconductivity properties with large magnetic fields current flow in the bulk of the sample (not only in thin layers)

14 Models for type-ii superconductors Introduction Microscopic: BCS theory (Bardeen, Cooper, Schrieffer 1957), quantum mechanical description.

15 Models for type-ii superconductors Introduction Microscopic: BCS theory (Bardeen, Cooper, Schrieffer 1957), quantum mechanical description. Mesoscopic: Ginzburg-Landau model (1950); formation of vortices (filaments), Abrikosov (1957).

16 Models for type-ii superconductors Introduction Microscopic: BCS theory (Bardeen, Cooper, Schrieffer 1957), quantum mechanical description. Mesoscopic: Ginzburg-Landau model (1950); formation of vortices (filaments), Abrikosov (1957). Macroscopic: Bean s model (1962), critical state model for the description of macroscopic electrodynamics.

17 Bean s model Introduction Bean s model (C.P. Bean, 1962) for type-ii hard superconductors: exists a critical current J c such that: J = J c in the region penetrated by the magnetic field; J = 0 otherwise.

18 Bean s model Introduction Bean s model (C.P. Bean, 1962) for type-ii hard superconductors: exists a critical current J c such that: J = J c in the region penetrated by the magnetic field; J = 0 otherwise. Anisotropy of J c, due to Cu-O planes, structure of defects, etc: exists R 3 compact convex containing a neighborhood of 0 s.t. J, in the region penetrated by the magnetic field; J = 0 otherwise.

19 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.)

20 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.) Faraday s law: curl E = µ 0 H t

21 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.) Faraday s law: curl E = µ 0 H t Ampère s law: J = curl H

22 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.) Faraday s law: curl E = µ 0 H t Ampère s law: J = curl H (Modified) Ohm s law: E = E( J)

23 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.) Faraday s law: curl E = µ 0 H t Ampère s law: J = curl H (Modified) Ohm s law: E = E( J) Examples of material laws (Ohm s law): isotropic conductor: E( J) = r J, r = resistivity

24 Introduction PROBLEM: given a superconductor Q R 3 in an external field H S (t), find the internal magnetic field H(x, t) and the electric field E(x, t). (Boundary condition: H = H S on Q.) Faraday s law: curl E = µ 0 H t Ampère s law: J = curl H (Modified) Ohm s law: E = E( J) Examples of material laws (Ohm s law): isotropic conductor: E( J) = r J, r = resistivity anisotropic conductor: E( J) = A J, A = resistivity tensor

25 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model.

26 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. In the isotropic case, the constraint J J c can be described by a vertical E J relation: E Jc J

27 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. In the isotropic case, the constraint J J c can be described by a vertical E J relation: E Jc J...that can be approximated by a power-law relation ( E( J) = c ) p J. J c

28 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. In the isotropic case, the constraint J J c can be described by a vertical E J relation: E Jc J...that can be approximated by a power-law relation ( E( J) = c ) p J. J c The electric field is determined using the additional condition E J.

29 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model.

30 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. Start from an anisotropic power law approximation for the dissipation E J: E( J) J = c ( ρ ( p ) p J) (ρ = gauge function of ).

31 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. Start from an anisotropic power law approximation for the dissipation E J: E( J) J = c ( ρ ( p ) p J) (ρ = gauge function of ). Deduce the dependence E( J) = c ( ρ ( p p 1 J)) Dρ ( J).

32 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. Start from an anisotropic power law approximation for the dissipation E J: E( J) J = c ( ρ ( p ) p J) (ρ = gauge function of ). Deduce the dependence E( J) = c ( ρ ( p p 1 J)) Dρ ( J). In the limit as p : E( J) I ( J) {0}, if J interior of, I ( J) = {λ Dρ ( J); λ 0}, if J,, if J subdifferential of the indicator function of. = gives the constraint J.

33 Introduction Problem: dependence E = E( J) in the Bean s anisotropic model. Start from an anisotropic power law approximation for the dissipation E J: E( J) J = c ( ρ ( p ) p J) (ρ = gauge function of ). Deduce the dependence E( J) = c ( ρ ( p p 1 J)) Dρ ( J). In the limit as p : E( J) I ( J) {0}, if J interior of, I ( J) = {λ Dρ ( J); λ 0}, if J,, if J subdifferential of the indicator function of. = gives the constraint J. Rigorous giustification by Γ-convergence.

34 Cylindrical symmetry Introduction Ω H=(0,0,h) H =(0,0,h ) S S Q = Ω R, cylinder with cross-section Ω R 2, smooth; H S (t) = (0, 0, h S (t)) directed along the axis of the cylinder.

35 Cylindrical symmetry Introduction Ω H=(0,0,h) H =(0,0,h ) S S Q = Ω R, cylinder with cross-section Ω R 2, smooth; H S (t) = (0, 0, h S (t)) directed along the axis of the cylinder. = By symmetry: H(x, t) = (0, 0, h(x 1, x 2, t))

36 Cylindrical symmetry Introduction Ω H=(0,0,h) H =(0,0,h ) S S Q = Ω R, cylinder with cross-section Ω R 2, smooth; H S (t) = (0, 0, h S (t)) directed along the axis of the cylinder. = By symmetry: H(x, t) = (0, 0, h(x 1, x 2, t)) = J = curl H = ( x2 h, x1 h, 0) Remark: J Dh K, where K R 2 is the rotation of the section z = 0 of.

37 Time discretization in [0, T ]: δt = T /n, t i = iδt, H i = H(t i ), E i = E(t i ).

38 Time discretization in [0, T ]: δt = T /n, t i = iδt, H i = H(t i ), E i = E(t i ). Goal: give a variational formulation of the anisotropic Bean s model starting with a power law approximation.

39 Time discretization in [0, T ]: δt = T /n, t i = iδt, H i = H(t i ), E i = E(t i ). Goal: give a variational formulation of the anisotropic Bean s model starting with a power law approximation. Power law for dissipation: E( J) J = c ( ρ ( p ) p J)

40 Time discretization in [0, T ]: δt = T /n, t i = iδt, H i = H(t i ), E i = E(t i ). Goal: give a variational formulation of the anisotropic Bean s model starting with a power law approximation. Power law for dissipation: E( J) J = c ( ρ ( p ) p J) : curl E i+1 = µ 0 Hi+1 H i δt

41 Time discretization in [0, T ]: δt = T /n, t i = iδt, H i = H(t i ), E i = E(t i ). Goal: give a variational formulation of the anisotropic Bean s model starting with a power law approximation. Power law for dissipation: E( J) J = c ( ρ ( p ) p J) : curl E Hi+1 H i+1 = µ i 0 δt = admits the variational formulation J p (h) = Ω 1 p [ρ(dh)]p + µ 0 2cδt (h h i) 2, h h s (t i+1 )+W 1,p 0 (Ω) i.e., h i+1 is the unique minimum point of J p in h s (t i+1 ) + W 1,p 0 (Ω). ρ = ρ K = gauge function of K R 2.

42 Convergence Theorem (G.C. - A. Malusa) u p h s (t i+1 ) + W 1,p 0 (Ω): unique minimum point of J p, p 1. u h s (t i+1 ) + W 1,1 0 (Ω): unique minimum point of J(u) = I K (Du) + (u h i ) 2, u h s (t i+1 ) + W 1,1 0 (Ω). Ω Then, for every q > 1, (u p ) converges to u in weak-w 1,q.

43 Convergence Theorem (G.C. - A. Malusa) u p h s (t i+1 ) + W 1,p 0 (Ω): unique minimum point of J p, p 1. u h s (t i+1 ) + W 1,1 0 (Ω): unique minimum point of J(u) = I K (Du) + (u h i ) 2, u h s (t i+1 ) + W 1,1 0 (Ω). Ω Then, for every q > 1, (u p ) converges to u in weak-w 1,q. Conclusion: the variational formulation of Bean s law is based on functional J. Given h i, we have h i+1 = u.

44 Convergence Theorem (G.C. - A. Malusa) u p h s (t i+1 ) + W 1,p 0 (Ω): unique minimum point of J p, p 1. u h s (t i+1 ) + W 1,1 0 (Ω): unique minimum point of J(u) = I K (Du) + (u h i ) 2, u h s (t i+1 ) + W 1,1 0 (Ω). Ω Then, for every q > 1, (u p ) converges to u in weak-w 1,q. Conclusion: the variational formulation of Bean s law is based on functional J. Given h i, we have h i+1 = u. Remark: variational formulation proposed by Badía-López (2002) starting from physical considerations.

45 Necessary conditions - Electric field Theorem (Dual function) a non-negative continuous function v i such that div(v i Dρ(Dh i+1 )) = h i h i+1 in Ω.

46 Necessary conditions - Electric field Theorem (Dual function) a non-negative continuous function v i such that div(v i Dρ(Dh i+1 )) = h i h i+1 in Ω. v i has an explicit representation in terms of the anisotropic principal curvatures of Ω and the normal distance from cut locus.

47 Necessary conditions - Electric field Theorem (Dual function) a non-negative continuous function v i such that div(v i Dρ(Dh i+1 )) = h i h i+1 in Ω. v i has an explicit representation in terms of the anisotropic principal curvatures of Ω and the normal distance from cut locus. Interpretation: w i = v i /δt is the (discretized) dissipated power density, and E i = w i Dρ(Dh i+1 ) is the (discretized) electric field.

48 Necessary conditions - Electric field Theorem (Dual function) a non-negative continuous function v i such that div(v i Dρ(Dh i+1 )) = h i h i+1 in Ω. v i has an explicit representation in terms of the anisotropic principal curvatures of Ω and the normal distance from cut locus. Interpretation: w i = v i /δt is the (discretized) dissipated power density, and E i = w i Dρ(Dh i+1 ) is the (discretized) electric field. Techniques developed in G.C., Malusa: to appear in Trans. Amer. Math. Soc. Isotropic case (K = ball): Cannarsa, Cardaliaguet, G.C., Giorgieri: Calc. Var. 2005

49 Selected references Badía, López, Phys. Rev. B 2002, J. Low Temp. Phys. 2003, J. Appl. Phys. 2004: anisotropic Bean s model Barrett, Prigozhin, Nonlinear Anal. 2000, preprint 2005: isotropic Bean s model, variational inequalities Brandt et al., Phys. Rev. B 1996 and 2000: numerical and experimental data

50 Candidate solution u = h i+1 Minkowski distance w.r.t. K: d(x) = min y Ω ρ0 K (x y) (ρ 0 K = polar of the gauge function of K) = viscosity solution of ρ(du) = 1 in Ω, u = 0 on Ω.

51 Candidate solution u = h i+1 Minkowski distance w.r.t. K: d(x) = min y Ω ρ0 K (x y) (ρ 0 K = polar of the gauge function of K) = viscosity solution of ρ(du) = 1 in Ω, u = 0 on Ω. Minkowski distance w.r.t. K: d (x) = min y Ω ρ0 K (x y) = min y Ω ρ0 K (y x) = viscosity solution of ρ(du) = 1 in Ω, u = 0 on Ω.

52 Candidate solution u = h i+1 Minkowski distance w.r.t. K: d(x) = min y Ω ρ0 K (x y) (ρ 0 K = polar of the gauge function of K) = viscosity solution of ρ(du) = 1 in Ω, u = 0 on Ω. Minkowski distance w.r.t. K: d (x) = min y Ω ρ0 K (x y) = min y Ω ρ0 K (y x) = viscosity solution of ρ(du) = 1 in Ω, u = 0 on Ω. Solution of the minimum problem: d(x) + h s (t i+1 ), if x Ω + = {h i > d}, h i+1 (x) = d (x) + h s (t i+1 ), if x Ω = {h i < d }, h i (x), if x Ω 0 = Ω \ (Ω + Ω ). h i+1 (x) = [h i (x) ( d (x) + h s (t i+1 ))] (d(x) + h s (t i+1 ))

53 1D heuristics d K = [ 1, 2] J(h) = h h i 2 + I K (Dh) Ω h = 0 on Ω h i Ω d

54 1D heuristics d K = [ 1, 2] J(h) = h h i 2 + I K (Dh) Ω h = 0 on Ω h h i Ω d

55 Decomposition of Ω in transport rays Ω can be decomposed in transport rays (paths of minimal distance from the boundary): two possible decompositions, one for d and one for d.

56 Decomposition of Ω in transport rays Ω can be decomposed in transport rays (paths of minimal distance from the boundary): two possible decompositions, one for d and one for d. Example: h i (y) > 0. point on cut locus Ω y ν(y) D ρ ( ν(y)) transport ray y h i+1 =hi h i+1 =d ν(y) = inward Euclidean normal of Ω at y l(y) = length of the transport ray = on each transport ray apply the 1D-heuristics.

57 Start with h(x, 0) = h 0 (x) Lip K (Ω), h 0 = h S (0) on Ω.

58 Start with h(x, 0) = h 0 (x) Lip K (Ω), h 0 = h S (0) on Ω. h i+1 = internal magnetic field at time t i+1 = { solution of the minimization problem } µ 0 min 2 h h i 2 + δt I K (Dh); h h S (t i+1 ) + W 1,1 0 (Ω) Ω

59 Start with h(x, 0) = h 0 (x) Lip K (Ω), h 0 = h S (0) on Ω. h i+1 = internal magnetic field at time t i+1 = { solution of the minimization problem } µ 0 min Ω 2 h h i 2 + δt I K (Dh); h h S (t i+1 ) + W 1,1 0 (Ω) By the existence and uniqueness theorem, h i+1 (x) = [ h i (x) (h S (t i+1 ) d (x)) ] (h S (t i+1 ) + d(x))

60 Start with h(x, 0) = h 0 (x) Lip K (Ω), h 0 = h S (0) on Ω. h i+1 = internal magnetic field at time t i+1 = { solution of the minimization problem } µ 0 min Ω 2 h h i 2 + δt I K (Dh); h h S (t i+1 ) + W 1,1 0 (Ω) By the existence and uniqueness theorem, h i+1 (x) = [ h i (x) (h S (t i+1 ) d (x)) ] (h S (t i+1 ) + d(x)) Explicit formula for monotone external field: 1. h S monotone increasing in [0, T ]: h i (x) = h 0 (x) (h S (t i ) d (x)) 2. h S monotone decreasing in [0, T ]: h i (x) = h 0 (x) (h S (t i ) + d(x))

61 The limit δt 0 For δt = T /n, n N +, construct h i as above and define h n (x, t) = h i (x), for t [t i, t i+1 )

62 The limit δt 0 For δt = T /n, n N +, construct h i as above and define h n (x, t) = h i (x), for t [t i, t i+1 ) Assume monotone external field; as n (δt 0)

63 The limit δt 0 For δt = T /n, n N +, construct h i as above and define h n (x, t) = h i (x), for t [t i, t i+1 ) Assume monotone external field; as n (δt 0) h S increasing: h n (x, t) h(x, t) = h 0 (x) (h S (t) d (x))

64 The limit δt 0 For δt = T /n, n N +, construct h i as above and define h n (x, t) = h i (x), for t [t i, t i+1 ) Assume monotone external field; as n (δt 0) h S increasing: h n (x, t) h(x, t) = h 0 (x) (h S (t) d (x)) h S decreasing: h n (x, t) h(x, t) = h 0 (x) (h S (t) + d(x))

65 The limit δt 0 For δt = T /n, n N +, construct h i as above and define h n (x, t) = h i (x), for t [t i, t i+1 ) Assume monotone external field; as n (δt 0) h S increasing: h n (x, t) h(x, t) = h 0 (x) (h S (t) d (x)) h S decreasing: h n (x, t) h(x, t) = h 0 (x) (h S (t) + d(x)) = the internal magnetic field can be explicitly computed if h S is piecewise monotone.

66 Example 'border.gp' K The section Ω, the constraints set K; Level sets and 3D-plot of the distance d.

67 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

68 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

69 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

70 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

71 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

72 Example: plot of h h s t 1 t 2 t 3 t 4 t 5 t 6 =T t

73 Hysteresis loop 8 'hyst.gp' h s 0-2 t Hysteresis loop: magnetization M = H H S versus external field H S.

74 Example: plot of w

75 Conclusion and outlook What we have done... Strong mathematical justification of the anisotropic variational formulation of Bean s law suggested by Badía and López.

76 Conclusion and outlook What we have done... Strong mathematical justification of the anisotropic variational formulation of Bean s law suggested by Badía and López. Explicit form of both magnetic field and electric field inside the superconductor; explicit computation of the dissipated power density (very important for the stability analysis of the superconducting phase).

77 Conclusion and outlook What we have done... Strong mathematical justification of the anisotropic variational formulation of Bean s law suggested by Badía and López. Explicit form of both magnetic field and electric field inside the superconductor; explicit computation of the dissipated power density (very important for the stability analysis of the superconducting phase)....and what remains to do: Nonhomogeneous samples (general Finsler metric instead of Minkowski); connections with elasticity theory and material science (e.g., dieletric breakdown).

78 Conclusion and outlook What we have done... Strong mathematical justification of the anisotropic variational formulation of Bean s law suggested by Badía and López. Explicit form of both magnetic field and electric field inside the superconductor; explicit computation of the dissipated power density (very important for the stability analysis of the superconducting phase)....and what remains to do: Nonhomogeneous samples (general Finsler metric instead of Minkowski); connections with elasticity theory and material science (e.g., dieletric breakdown). True 3D analysis (no cylindrical symmetry).

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