The Γ-Limit of the Ginzburg-Landau energy in a Thin Domain with a Large Magnetic Field

Transcription

1 p. 1/2 The Γ-Limit of the Ginzburg-Landau energy in a Thin Domain with a Large Magnetic Field Tien-Tsan Shieh tshieh@math.arizona.edu Department of Mathematics University of Arizona Tucson, AZ 85719

2 p. 2/2 Ginzburg-Landau Model G(ψ, A) = U ( ( ia)ψ A H e 2 dx. R 3 ( ψ 2 µ 2) ) 2 dx Here ψ : U C, with U: the domain occupied by the sample, ψ 2 =density of supercond. charge carriers, A : R 3 R 3 is the induced mag. potential, T c : the critical temperature under no applied field, µ 2 T c T, : material constant H e : R 3 R 3 = given external magnetic field.

3 p. 2/2 Ginzburg-Landau Model G(ψ, A) = U ( ( ia)ψ A H e 2 dx. R 3 ( ψ 2 µ 2) ) 2 dx The Basic Thermodynamic Postulate: The state (ψ,a) of the sample minimizes the Ginzburg-Landau energy.

4 p. 3/2 Little-Parks Experiment In 1961, Little and Parks observed that the phase transition temperature in thin ring is essentially a periodic function of the axial magnetic flux through the ring with a parabolic background. T h h

5 p. 4/2 Question: Here we ask : Can one (rigorously) derive a model of a thin superconducting loop in a presence of large magnetic field starting from three-dimensional Ginzburg-Landau model? This case was first treated by Richardson and Rubinstein using formal asymptotic expansion.

6 p. 5/2 The Model of our problem - Two Assumptions in our study : 1. The sample domain U is a sequence of domains {U ε } consisting of ε-neighborhoods of a limiting simple closed curve.

7 p. 5/2 The Model of our problem - Two Assumptions in our study : 1. The sample domain U is a sequence of domains {U ε } consisting of ε-neighborhoods of a limiting simple closed curve. 2. The given applied field H e take the form H e ε = He ε where H e is a given smooth magnetic field independent of ε.

8 Description of U ε Let r : [0, L] R 3 be a simple, closed C 2 curve. The triple {t,n,b} forms a Frenet Frame for the curve. The ε-neighborhood U ε is the image of the cylinder Ω = {(y 1, y 2, y 3 ) R 3 : 0 y 1 L, y y 2 3 < 1} under the mapping T ε (y) = r(y 1 ) + εy 2 n(y 1 ) + εy 3 b(y 1 ). p. 6/2

9 p. 7/2 The Ginzburg-Landau energy E ε (ψ,a) = U ε ( ( ia)ψ ) ) 2 ( ψ 2 µ A He 2 ε dx. R 3 dx

10 p. 8/2 Main difficulty E ε (ψ,a) = ( ( ia)ψ U ε ( ψ 2 µ 2) 2 ) dx + R 3 A He ε 2 dx. Main difficulty: The phase of the minimizer ψ ε oscillates rapidly as ε 0. Without shifting the phase, we can t attain compactness of minimizers ψ ε of E ε.

11 p. 8/2 Main difficulty E ε (ψ,a) = ( ( ia)ψ U ε ( ψ 2 µ 2) 2 ) dx + R 3 A He ε 2 dx. Main difficulty: The phase of the minimizer ψ ε oscillates rapidly as ε 0. Without shifting the phase, we can t attain compactness of minimizers ψ ε of E ε. We shift the phase by the special phase functions φ ε. where A e = H e. A e ε φ ε O(1) in U ε

12 p. 9/2 Define an equivalent functional This lead us to find an equivalent energy functional to E ε. F ε (ψ,a) := 1 ε 2 E ε(ψe iφ ε,a + Ae ε ) = 1 ε 2 U ε + 1 ε 2 ( i(a + Ae ε φ ε) ) ψ 2 dx U ε ε 2 ( ψ ) 2 2 µ 2 dx R 3 A 2 dx.

13 p. 10/2 Rewrite F ε 1 We decompose A e into components A e 1, A e 2 and A e 3 lying along the tangent, normal and bi-normal to the limiting curve r.

14 p. 10/2 Rewrite F ε 1 We decompose A e into components A e 1, A e 2 and A e 3 lying along the tangent, normal and bi-normal to the limiting curve r. 2 Taylor Expansion for A e 1, A e 2 and A e 3 around the the limiting curve r.

15 p. 10/2 Rewrite F ε 1 We decompose A e into components A e 1, A e 2 and A e 3 lying along the tangent, normal and bi-normal to the limiting curve r. 2 Taylor Expansion for A e 1, A e 2 and A e 3 around the the limiting curve r. 3 We set the number k ε be the closest integer to the number ( 1 L A e 1(y 1,0,0) ) L 0 ε dy 1.

16 Rewrite F ε 1 We decompose A e into components A e 1, A e 2 and A e 3 lying along the tangent, normal and bi-normal to the limiting curve r. 2 Taylor Expansion for A e 1, A e 2 and A e 3 around the the limiting curve r. 3 We set the number k ε be the closest integer to the number ( 1 L A e 1(y 1,0,0) ) L 0 ε dy 1. 4 Define the effective magnetic flux number β ε by ( 1 L ) A 1 (t, 0, 0) β ε = dt 2π L ε L k ε. 0 p. 10/2

17 p. 11/2 The special phase function φ ε Define φ ε by φ ε (y 1, y 2, y 3 ) := y1 ( A 1(t,0,0) β ε )dt 0 ε ( + y 2 A 2 (y 1,0,0) + y 3 A 3 (y 1,0,0) + y2 2 2 y 2 A 2 (y 1,0,0) + y2 3 2 y 3 A 3 (y 1,0,0) y 2y 3 y3 A 2 (y 1,0,0) + 1 ) 2 y 2y 3 y2 A 3 (y 1,0,0)

18 p. 12/2 The equivalent functional F ε F ε (ψ,a) = y1 ψ + τy 3 y2 ψ τy 2 y3 ψ i β ε y 2 H3 e + y 3H e 2 ψ ia1 dy Ω 2 ψ + + Ω Ω 1 ε y 2 ψ + i( 1 2 y 3)H e 1ψ ia 2 ψ 2 dy 1 ε y 3 ψ i( 1 2 y 2)H e 1 ψ ia 3ψ 2 dy + Ω 1 2 ψ 2 µ 2 2 dy + 1 ε 2 R 3 A 2 dx +O(ε)

19 p. 13/2 Preliminary estimates (I) There exist constants C 1, C 2, C 3, C 4, C 5 independent of ε such that for any family (ψ ε,a ε ) H 1 per(ω) H 0 with uniformly bounded energy F ε (ψ ε,a ε ), one has Ω ψ ε 4 dy C 1, A ε 2 dx C 2 ε 2, R 3 A ε 2 dx C 3 ε 2, R 3 y1 ψ ε 2 dy C 4, Ω y2 ψ ε 2 + y3 ψ ε 2 dy C 5 ε 2. Ω

20 p. 14/2 Preliminary estimate(ii) For a family of pairs (ψ ε,a ε ) H 1 (Ω) H 0 such that F ε (ψ ε,a ε ) is uniformly bounded, we have ψ ε (y 1, y 2, y 3 ) q(ψ ε )(y 1 ) p dy = 0 lim ε 0 Ω for any p [1, 6). Here q is an integral average defined by q(ψ)(y 1 ) := 1 ψ(y 1, y 2, y 3 )dy 2 dy 3. D where D is the cross-section of Ω. D

21 Function spaces H 1 per(ω) = {ψ H 1 (Ω) : ψ(0,y 2,y 3 ) = ψ(l,y 2,y 3 ) in the trace sense (0,y 2,y 3 ) Ω 0 }. H 1 per((0,l)) = {ψ H 1 ((0,L)) : ψ(0) = ψ(l)} H = the completion of the set C 0 (R 3 ; R 3 ) with respect to the norm V H := V L 2 (R 3 ;R 3 ) H 0 = {V H : V = 0}. p. 15/2

22 p. 16/2 Definition of q-convergence Define an integral average operator q : H 1 (Ω) H 1 ((0, L)) by q(ψ)(y 1 ) := 1 ψ(y 1, y 2, y 3 )dy 2 dy 3. D where D is the cross-section of Ω. D Definition 1. We say a sequence {(ψ j,a j )} H 1 per(ω) H 0 q-converges to (ψ 0,A 0 ) H 1 per((0, L)) H 0 if the sequence q(ψ j ) converges weakly to ψ 0 in H 1 ((0, L)) and the sequence {A j } converges strongly to A 0 in H 0.

23 p. 17/2 Compactness Given a sequence (ψ ε,a ε ) with F ε (ψ ε,a ε ) uniformly bounded, there exists a subsequence {(ψ εj,a εj )} that q-converges in H 1 (Ω) H 0 as ε j 0.

24 p. 18/2 The equivalent functional F ε F ε (ψ,a) = y1 ψ + τy 3 y2 ψ τy 2 y3 ψ i β ε y 2 H3 e + y 3H e 2 ψ ia1 dy Ω 2 ψ + + Ω Ω 1 ε y 2 ψ + i( 1 2 y 3)H e 1ψ ia 2 ψ 2 dy 1 ε y 3 ψ i( 1 2 y 2)H e 1 ψ ia 3ψ 2 dy + Ω 1 2 ψ 2 µ 2 2 dy + 1 ε 2 R 3 A 2 dx +O(ε)

25 p. 19/2 The limiting energy G β We guess limiting energy G β should takes the following form: G β (ψ,a) ( y1 ψ iβψ 2 + W(y 1 ) ψ 2) D dy 1 = L 0 + L 0 1 2( ψ 2 µ 2) 2 D dy1 when A H = 0 + otherwise Here W(y 1 ) := 1 8 (He 1) (He 2) (He 3) 2.

26 p. 20/2 Definition of Γ(q)-convergence Definition 2. Suppose β εj β 0 along some subsequence {ε j } 0. We say the sequence of functionals {F εj } Γ(q)-converges to the functional G β0 if the following two conditions hold: (i) Lower semi-continuity (ii) Recovering sequence

27 p. 21/2 (i) (Lower semi-continuity) For every sequence (ψ εj,a εj ) H 1 (Ω) H 0 q-converging to (ψ 0,A 0 ) H 1 ((0, L)) H 0, one has lim inf j F ε j (ψ εj,a εj ) G β0 (ψ 0,A 0 ); (ii) (Recovery sequence) For every function (ψ 0,A 0 ) H 1 ((0, L)) H 0, there exists a sequence ( ψ εj,ãε j ) H 1 (Ω) H 0 q-converging to (ψ 0,A 0 ) such that lim F ε j ( ψ εj,ãε j ) = G β0 (ψ 0,A 0 ). j

28 p. 22/2 Γ-Limit Suppose β εj β 0 [ π L, π L ] along a subsequence {ε j } 0. Then we have Γ(q) lim j + F ε j = G β0.

29 p. 23/2 Improvement of Compactness Suppose {β εj } converges to a limit β 0 [ π L, π L ] as ε j 0. Let (ψ ε,a ε ) denote a minimizer of F ε. Then passing to a subsequence, still denoted by {ε j }, one has (ψ εj,a εj ) (ψ 0, 0) strongly H 1 per(ω) H 0, where the pair (ψ 0, 0) = (ψ 0 (y 1 ), 0) minimizes G β0 in H 1 per((0, L)) H 0.

30 p. 24/2 Onset Problem The phase transition associated with onset of superconductivity is characterized by the value µ 2 ( (T c T)) at which this normal state loses stability. The direct calculation of the second variation of G β0 about normal state leads us to consider the Rayleigh quotient µ 2 c := inf ψ L 0 y 1 ψ iβ 0 ψ 2 + W(y 1 ) ψ 2 dy 1 L. 0 ψ 2 dy 1

31 p. 25/2 Little-Parks Experiment Now we proceed to calculate the phase transition curve between normal and superconducting state for a loop of uniform cross-section whose centerline is planar, which is subjected to a uniform perpendicular magnetic field H e = he z to the plane of the loop.

32 p. 26/2 In this case, the Rayleigh quotient take the form µ 2 c(h) = = ( inf ψ L 0 ( ) y 1 iβ 0 )ψ 2 dy 1 L 0 ψ 2 dy h2 = β 0 (h) h2

33 p. 27/2 Phase Transition curve Through we have µ 2 c(h e ) = µ 2 (T) = α T, β 0 (h) h2 = α T Figure of the phase transition curve T h h

34 Thank You! p. 28/2