The influence of magnetic steps on bulk superconductivity

We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.


Introduction and Main results
Motivation. The Ginzburg-Landau functional successfully models the response of a (Type II) superconducting sample to an applied magnetic field. We focus on samples that occupy a long cylindrical domain and subjected to a magnetic field with direction parallel to the axis of the cylinder. This situation has been analyzed in many papers, see for instance the two monographs [8,19]. However, in the literature, the focus was on uniform applied magnetic fields. Recently, attention has been shifted to non-uniform smooth magnetic fields in [2,3,13,17]. Such magnetic fields may arise in the study of superconducting surfaces [7] or superconductors with applied electric currents [1].
In this paper, we consider the situation when the applied magnetic field is a step function. Such fields might occur in many situations (cf. [14]). In particular • If a sample is separated into two parts, one can apply on one part a uniform magnetic field from above, and on the other part, a uniform magnetic field from below (see Figure 2). • If a sample is not homogeneous, one can have a variable magnetic permeability. This leads to a magnetic step function (cf. [6]). Here, κ > 0 is the Ginzburg-Landau parameter, a characteristic of the superconducting material, H > 0 is the intensity of the applied magnetic field, ψ ∈ H 1 (Ω, C) and A = (A 1 , A 2 ) ∈ H 1 (Ω, R 2 ). In physics, the domain Ω is the cross section of the sample, the function B 0 is the applied magnetic field, the function ψ is the order parameter and the vector field A is the magnetic potential. The configuration (ψ, A) is interpreted as follows, |ψ| 2 measures the density of the superconducting electron pairs and curl A = ∂ x 1 A 2 −∂ x 2 A 1 measures the induced magnetic field in the sample.
Discussion of Theorem 1.4. The result in Theorem 1.4 displays the strength of the superconductivity in the bulk of Ω. We will use the following notation. Let ω ⊂ R 2 be an open set, (f κ ) be a sequence in L ∞ (ω), α ∈ C and dx be the Lebesgue measure in R 2 . By writing Now we return back to the result in Theorem 1.4. Suppose that H = bκ and b ∈ (0, ∞) is a constant. Let us start by examining the case where −1 < a < 1. We observe that: (1) If 0 < b < 1, then This means that the bulk of Ω carries superconductivity everywhere, but since 0 < |a| < 1, 0 < −g(b) < −g(b|a|) and the strength of superconductivity in Ω 1 is smaller than that in Ω 2 .
with g(b|a|) < 0. In this regime, superconductivity disappears in the bulk of Ω 1 but persists in the bulk of Ω 2 . Theorem 1.5 below will sharpen this point by establishing that |ψ| is exponentially small in the bulk of Ω 1 (see Figure 3). However, in light of the analysis in the book of Fournais-Helffer [8], the boundary of Ω 1 may carry superconductivity. This point deserves a detailed analysis.
|a| , then superconductivity disappears in the bulk of Ω 1 and Ω 2 (see Figure 4). However, one might find an interesting behavior near the critical value b ∼ 1 |a| . In the spirit of the analysis in [10], one expects to find superconductivity in the bulk of Ω 2 , but with a weak strength. This superconductivity is evenly distributed and decays as b gradually increases past the value 1 |a| . (4) [Break down of superconductivity ( [12])] If b 1 |a| , one expects that ψ = 0 and superconductivity is lost in the sample. To this end, the spectral analysis in [14] must be useful. In the spirit of the book [8], this regime is related to the analysis of the third critical field(s) where the transition to the purely normal state occurs. The interesting case a = −1 is reminiscent of the situation of a smooth and sign-changing magnetic field analyzed in the paper by Helffer-Kachmar [13]. Note that Theorem 1.4 yields that superconductivity is evenly distributed in Ω 1 and Ω 2 as long as 0 < b < 1. In the critical regime b ∼ 1, one might find that superconductivity is distributed along the curve Γ that separates Ω 1 and Ω 2 , in the same spirit of the paper [13]. This behavior is illustrated in Figure 5.
Exponential decay in regions with larger magnetic intensity. Our last result establishes a regime for the strength of the magnetic field where the order parameter is exponentially small in the bulk of Ω 1 . The relevance of this theorem is that together with Theorem 1.4, display a regime of the intensity of the applied magnetic field such that |ψ| 2 is exponentially small in the bulk of Ω 1 while it is of the order O(1) in Ω 2 .

Theorem 1.5. [Exponential decay of the order parameter]
Let λ, ε, c 2 > 0 be constants such that 0 < ε < √ λ and 1 + λ < c 2 . There exist constants Unlike similar situations in [5,11], we can not extend the result in Theorem 1.5 to critical points of the functional in (1.1). The technical reason behind this is as follows. A necessary ingredient in the proof given in [5,11] is the following estimate of the magnetic energy For critical configurations, we have the following estimate from [8, Lemma 10.
To control the L 2 -and L 4 -norms of ψ, we use Theorem 1.4. But this will give that ψ L 4 (Ω) = o(1) only for H ≥ |a| −1 κ, the condition necessary to get that g(Hκ −1 ) = g(Hκ −1 |a|) = 0. This condition does not cover all the values of H in Theorem 1.5. As a substitute, we choose to control the magnetic energy by the estimate in Corollary 1.3, which is valid for minimizing configurations only. Notation.
• The letter C denotes a positive constant whose value may change from line to line.
Unless otherwise stated, the constant C depends on the function B 0 and the domain Ω, and independent of κ, H and the minimizers (ψ, A) of the functional in (1.1).
• Given > 0 and x = (x 1 , x 2 ) ∈ R 2 , we denote by the square of side length centered at x . • Let a(κ) and b(κ) be two positive functions, we write : • Let n ∈ N, p ∈ N. We use the following Sobolev spaces : On the proofs and the organization of the paper. The results in this paper can be viewed as generalizations of those in [20] already proved for the case B 0 = 1. Theorem 1.5 is reminiscent of the exponential bounds in [5]. However, the proofs in this paper are simpler than those in [20] and contain new ingredients that we summarize below: • We took advantage of all the available information regarding the limiting function g(·) proved in [9] and [3] ; • We did not used the a priori elliptic estimates, e.g. the Cκ. However, we used the simple energy bound (∇ − iκHA)ψ 2 ≤ Cκ together with the regularity of the curl-div system (cf. Theorem 4.2). This method is already used for the three dimensional problem in [15] ; • To prove Theorem 1.5, we did not established weak decay estimates as done in [5]. The paper is divided into seven sections and two appendices. The first section is this introduction. Section 2 collects the needed properties of the limiting energy g(·). Section 3 establishes an upper bound of the ground state energy. Section 4 proves the necessary estimates on the critical points of the functional in (1.1). These estimates are used in Section 5 to establish a lower bound of the ground state energy. In Section 6, we finish the proof of Theorem 1.2, Corollary 1.3 and Theorem 1.4. Section 7 is devoted to the proof of Theorem 1.5. Finally, the appendices A and B collect standard results that are used throughout the paper.
We introduce the two ground state energies As an immediate consequence, we observe that inf (2.4) and the values of m 0 (b, R, σ) and m(b, R, σ) are independent of σ ∈ {−1, 1}. In the rest of the paper, we will denote these two values by m 0 (b, R) and m(b, R) respectively, hence We cite the following result from [3] (also see [9,20]).
is continuous, non-decreasing, concave and its range is (5) There exist constants C and R 0 such that, for all R ≥ R 0 and b ∈ [0, 1],

Energy Upper Bound
The aim of this section is to prove : Before writing the proof of Proposition 3.1, we introduce some notation. If D ⊂ Ω is an open set, we introduce the local energy of the configuration (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) in the domain D ⊂ Ω as follows In Lemma A.1, we constructed a vector field F satisfying F ∈ H 1 div (Ω) and curl F = B 0 in Ω . Step 1. (Introducing a lattice of squares) We introduce the small parameter = κ −1/2 .  where Q (z) denotes the square of center z and side-length . By Assumption 1.1, the number Step 2. (Defining a trial state.) For all z ∈ I , let ϕ z ∈ C 2 (Q (z)) be the function introduced in Lemma A.2 and The function ϕ z satisfies We define the function v ∈ H 1 0 (Ω) as follows, where and u bz,Rz,σz ∈ H 1 0 (Q R ) is a minimizer of the functional in (2.1) (with (b, R, σ) = (b z , R z , σ z )). In the sequel, we will omit the reference to (b z , R z , σ z ) in the notation u bz,Rz,σz and write simply u z = u bz,Rz,σz . (3.9) Step 3. (Energy of the trial state). We compute the energy of the configuration (v, F). We have the obvious identities (cf. (3.1) and (3.2)) (3.10) Using (3.7), we write By doing the change of variable y = R z (x − z), we get where u z is the function in (3.9) and (b z , R z , σ z ) is introduced in (3.6). By using (2.4), we get Since = κ −1/2 and H ≥ c 1 κ, R z ≥ 1 (cf. (3.6)). We use Theorem (2.1) to write We insert (3.11) into (3.10) to get where N = Card I . Now, using (3.5) and the fact that − 1 2 ≤ g(·) ≤ 0, we get To finish the proof of Proposition 3.1, we use that E g.st (κ, H) ≤ E(v, F; Ω), = κ −1/2 and that the regularity of ∂Ω together with Assumption 1.1 yields

A Priori Estimates
In the derivation of a lower bound of the energy in (1.1) various error terms arise. These terms are controlled by the estimates that we derive in this section.
Proof. The inequalities in items (1) and (2) Using the bound H ≥ c 1 κ, Proposition 4.1 and the estimate in item (2) in Theorem 4.2, we get the estimate in item (3) above. Finally, the conclusion in item (4) in Theorem 4.2 is simple a consequence of the conclusion in item (3) and the Sobolev embedding of H 2 (Ω) in C 0,α (Ω).   (3) and (4) only. In fact, Assumption 1.1 ensures that the domains Ω 1 and Ω 2 satisfy the cone condition, which in turn allows us to use the Sobolev embedding theorems (cf. e.g. the proof of Lemma B.1).

Energy Lower Bound
The aim of this section is to establish a lower bound for the ground state energy in (1.2). This will be done in two steps (cf. Lemma 5.1 and Proposition 5.2 below). As a consequence of the results in this section, we will be able to finish the proof of Theorem 1.2 and Corollary 1.3.
Recall that Q (x 0 ) denotes the square of center x 0 and side length . In the statements of Lemma 5.1, Proposition 5.2 and Theorem 5.3, we will use the functional E 0 in (3.1).
Lemma 5.1. Let α ∈ (0, 1) and 0 < c 1 < c 2 be constants. There exist positive constants C and κ 0 such that, if • (1.5) holds ; then the following inequality holds where ϕ x 0 is the function introduced in Lemma A.2 and satisfying 2) Using the gauge invariance, the Cauchy-Schwartz inequality and (5.1), we write Now, by recalling the definition of u and by using the estimates ψ ∞ ≤ 1 and h ∞ ≤ 1, we deduce the following lower bound of E 0 (hψ, A; Q (x 0 )), and define the rescaled function v(x) = u( R x + x 0 ) for all x ∈ Q R = (−R/2, R/2) 2 , where the function u is defined in (5.2). The change of variable y = R (x − x 0 ) yields Since v ∈ H 1 (Q R ), then by (2.3), (2.5) and Theorem 2.1, Inserting this into the estimate in Lemma 5.1 and taking δ = , we finish the proof of Proposition 5.2.
In the next theorem, we establish a lower bound of the local energy in an open subset D of Ω. Note that, in the particular case h = 1 and D = Ω, Theorem 5.3 yields the lower bound in Theorem 1.2.

Notice that
and Recall b z and R z defined in (3.6), Let (ψ, A) be a minimizer of (1.1). We decompose E 0 (hψ, A; D) as follows : Using (5.8), |ψ| ≤ 1, |h| ≤ 1 and Item (1) in Theorem 4.2, we get On the other hand, we have the obvious decomposition of the energy in D as follows E 0 (hψ, A; Q (z)) .
Using Proposition 5.2 and the estimate in (5.7), we get Since g(·) ≤ 0, D ⊂ D and B 0 is a step function, Consequently, we get the following lower bound Now, the choice of and α in (5.5) allows us to infer from (5.10) and (5.11) Inserting these two estimates in (5.9) finishes the proof of Theorem 5.3. Multiply both sides of (6.1) by ψ, integrate by parts over Ω and use the boundary condition in (1.4) to obtain Here E 0 is the function in (3.1). Using Theorem 5.3 with h = 1 and D = Ω, we get Now, we assume in addition that (ψ, A) is a minimizer of (1.1). We will determine a lower bound of ψ 4 matching with the upper bound in (6.3). To that end, we observe that H) . (6.4) This yields that E 0 (ψ, A; Ω) ≤ E g.st (κ, H) .
The upper bound in Theorem 3.1 gives Inserting this into (6.2), we obtain It remains to prove the estimate of the magnetic energy. In fact, we get by (6.4), Proposition 3.1 and Theorem 5.3 Proof of Theorem 1.4. The proof we give does not use a priori elliptic estimates, hence, it diverges from the one used by Sandier-Serfaty in [20]. As a substitute of the elliptic estimates, we use the a priori estimates obtained by an energy argument in Theorem 4.2, and a trick that we borrow from a paper by Helffer-Kachmar in [13].
Note that the particular case D = Ω is handled by Corollary 1.3. Now we establish Theorem 1.4 in the general case when D is an open subset of Ω with a smooth boundary. The proof is decomposed into two steps: Upper bound: For ∈ (0, 1), we define the two sets Since the boundary of D is smooth, we get However, the boundary of Ω \ D might have a finite number of cusps. But we still have, Let χ ∈ C ∞ c (R 2 ) be a cut-off function satisfying where C is a universal constant. Multiply both sides of the equation by χ 2 ψ, integrate by parts over D, then use (6.9), (6.7) and ψ ∞ ≤ 1 to get Notice that χ 4 ≤ χ 2 ≤ 1. We infer from (6.10) Again, using (6.9), (6.7) and ψ ∞ ≤ 1, we obtain Consequently, in light of (6.11), we write, In a similar fashion, by using the alternative property in (6.8), we get, By (6.13), we have Using (6.5) and (6.12), we deduce that
Proof of Theorem 1.5. Define the distance function t on Ω 1 as follows : Letχ ∈ C ∞ (R) be a function satisfying where m is a universal constant. Define the two functions χ and f on Ω 1 as follows : where ε is a positive number whose value will be fixed later.
We conclude this paper by Theorem 7.2 below, whose proof is similar to that of Theorem 1.5.
Appendix A. Gauge transformation Lemma A.1. Suppose that Ω satisfies the conditions in Assumption 1.1. Let B 0 ∈ L 2 (Ω). There exists a unique vector field F ∈ H 1 div (Ω) such that curl F = B 0 .
That where C * > 0 is a universal constant.
Proof. By the definition of F and A 0 we have for all x ∈ Q (x 0 ), curl F(x) = B 0 (x)curlA 0 (x) in Q (x 0 ) .
Since Q (x 0 ) is simply connected and B 0 is constant in Q (x 0 ), we get the existence of the function ϕ x 0 .