A branched transport limit of the Ginzburg-Landau functional
We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via -convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.
In 1911, K. Onnes discovered the phenomenon of superconductivity, manifested in the complete loss of resistivity of certain metals and alloys at very low temperature. W. Meissner discovered in 1933 that this was coupled with the expulsion of the magnetic field from the superconductor at the critical temperature. This is now called the Meissner effect. After some preliminary works of the brothers F. and H. London, V. Ginzburg and L. Landau proposed in 1950 a phenomenological model describing the state of a superconductor. In their model (see (1.1) below), which belongs to Landau’s general theory of second-order phase transitions, the state of the material is represented by the order parameter , where is the material sample. The density of superconducting electrons is then given by . A microscopic theory of superconductivity was first proposed by Bardeen-Cooper-Schrieffer (BCS) in 1957, and the Ginzburg-Landau model was derived from BCS by Gorkov in 1959 (see also [FHSS12] for a rigorous derivation).
One of the main achievements of the Ginzburg-Landau theory is the prediction and the understanding of the mixed (or intermediate) state below the critical temperature. This is a state in which, for moderate external magnetic fields, normal and superconducting regions coexist. The behavior of the material in the Ginzburg-Landau theory is characterized by two physical parameters. The first is the coherence length which measures the typical length on which varies, the second is the penetration length which gives the typical length on which the magnetic field penetrates the superconducting regions. The Ginzburg-Landau parameter is then defined as . The Ginzburg-Landau functional is given by
where is the magnetic potential (so that is the magnetic field), is the covariant derivative of and is the external magnetic field. In these units, the penetration length is normalized to . As first observed by A. Abrikosov this theory predicts two types of superconductors. On the one hand, when , there is a positive surface tension which leads to the formation of normal and superconducting regions corresponding to and respectively, separated by interfaces. These are the so-called type-I superconductors. On the other hand, when , this surface tension is negative and one expects to see the magnetic field penetrating the domain through lines of vortices. These are the so-called type-II superconductors. In this paper we are interested in better understanding the former type but we refer the interested reader to [Tin96, SS07, Ser15] for more information about the latter type. In particular, in that regime, there has been an intensive work on understanding the formation of regular patterns of vortices known as Abrikosov lattices.
In type-I superconductors, it is observed experimentally [PH09, Pro07, PGPP05] that complex patterns appear at the surface of the sample. It is believed that these patterns are a manifestation of branching patterns inside the sample. Although the observed states are highly history-dependent, it is argued in [CKO04, PGPP05] that the hysteresis is governed by low-energy configurations at vanishing external magnetic field. The scaling law of the ground-state energy was determined in [CCKO08, CKO04] for a simplified sharp interface version of the Ginzburg-Landau functional (1.1) and in [COS16] for the full energy, these results indicate the presence of a regime with branched patterns at low fields.
This paper aims at a better understanding of these branched patterns by going beyond the scaling law. Starting from the full Ginzburg-Landau functional, we prove that in the regime of vanishing external magnetic field, low energy configurations are made of nearly one-dimensional superconducting threads branching towards the boundary of the sample. In a more mathematical language, we prove convergence [Bra02, DM93] of the Ginzburg-Landau functional to a kind of branched transportation functional in this regime. We focus on the simplest geometric setting by considering the sample to be a box for some and consider periodic lateral boundary conditions. The external magnetic field is taken to be perpendicular to the sample, that is for some and where is the third vector of the canonical basis of . After making an isotropic rescaling, subtracting the bulk part of the energy and dropping lower order terms (see (3.5) and the discussion after it), minimizing (1.1) can be seen as equivalent to minimizing
where we have let ,
If , since , in the limit we obtain, at least formally, that is a gradient field in the region where and therefore the Meissner condition holds. Moreover, in the regime , from (1.2) we see that and takes almost only values in . Hence can be rewritten as
where and denotes the divergence with respect to the first two variables. Therefore, from the Benamou-Brenier formulation of optimal transportation [AGS05, Vil03] and since from the Meissner condition, , the term
in the energy (1.2) can be seen as a transportation cost. We thus expect that inside the sample (this is, in ), superconducting domains where and alternate with normal ones where and . Because of the last term in the energy (1.2), one expects outside the sample. This implies that close to the boundary the normal domains have to refine. The interaction between the surface energy, the transportation cost and the penalization of an norm leads to the formation of complex patterns (see Figure 1).
It has been proven in [COS16] that in the regime , and ,
The scaling (relevant for ) corresponds to uniform branching patterns whereas the scaling corresponds to non-uniform branching ones. We focus here for definiteness on the regime , although we believe that our proof can be extended to the other one. Based on the construction giving the upper bounds in (1.3), we expect that in the first regime there are multiple scales appearing (see Figure 1):
which amounts in our parameters to
In order to better describe the minimizers we focus on the extreme region of the phase diagram , with for some fixed . In this regime, we have in particular so that the separation of scales (1.4) holds. We introduce an anisotropic rescaling (see Section 3) which leads to the functional
Our main result is a convergence result of the functional towards a functional defined on measures living on one-dimensional trees. These trees correspond to the normal regions in which and where the magnetic field penetrates the sample. Roughly speaking, if for a.e. the slice of has the form where the sum is at most countable, then we let (see Section 5 for a precise definition)
where and denotes the derivative (with respect to ) of . The ’s represent the graphs of each branch of the tree (parametrized by height) and the ’s represent the flux carried by the branch. We can now state our main result
Let with , , then:
For every sequence with , up to subsequence , weakly converges to a measure of the form with for a.e. , (where denotes the two dimensional Lebesgue measure on ) when and such that
If in addition , where , then for every measure such that and as , there exists such that and
Let us stress once again that our result could have been equivalently stated for the full Ginzburg-Landau energy (1.1) instead of (see Section 3).
Within our periodic setting, the quantization condition for the flux is a consequence of the fact that the phase circulation of the complex-valued function in the original problem is naturally quantized. It is necessary in order to make our construction but we believe that it is also a necessary condition for having sequences of bounded energy (see the discussion in Section 3 and the construction in Section 7.3). We remark that scaling back to the original variables this condition is the physically natural one .
Before going into the proof of Theorem 1.1 we address the limiting functional , which has many similarities with irrigation (or branched transportation) models that have recently attracted a lot of attention (see [BCM09] and more detailed comments in Section 5.4 or the recent papers [BW15, BRW16] where the connection is also made to some urban planning models). In Section 5, we first prove that the variational problem for this limiting functional is well-posed (Proposition 5.5) and show a scaling law for it (Proposition 5.2 and Proposition 5.3). In Proposition 5.7, we define the notion of subsystems which allows us to remove part of the mass carried by the branching measure. This notion is at the basis of Lemma 5.8 and Proposition 5.11 which show that minimizers contain no loops and that far from the boundary, they are made of a finite number of branches. From the no-loop property, we easily deduce Proposition 5.10 which is a regularity result for minimizers of . The main result of Section 5 is Theorem 5.15 which proves the density of “regular” measures in the topology given by the energy . As in nearly every convergence result, such a property is crucial in order to implement the construction for the upper bound (ii).
We now comment on the proof of Theorem 1.1. Let us first point out that if the Meissner condition were to hold, and could be written as a gradient field in the set , then and we would have
This is a Modica-Mortola [Mod87] type of functional with a degenerate double-well potential given by . Thanks to Lemma 6.2, one can control how far we are from satisfying the Meissner condition. From this, we deduce that (1.7) almost holds (see Lemma 6.5). This implies that the Ginzburg-Landau energy gives a control over the perimeter of the superconducting region . In addition, imposes a small cross-area fraction for . Using then isoperimetric effects to get convergence to one-dimensional objects (see Lemma 6.6), we may use Proposition 6.1 to conclude the proof of (i).
In order to prove (ii), thanks to the density result in Theorem 5.15, it is enough to consider regular measures. Given such a measure , we first approximate it with quantized measures (Lemma 5.18). Far from branching points the construction is easy (see Lemma 7.3). At a branching point, we need to pass from one disk to two (or vice-versa); this is done passing through rectangles (see Lemma 7.6 and Figure 3). Close to the boundary we use instead the construction from [COS16], which explicitly generates a specific branching pattern with the optimal energy scaling; since the height over which this is done is small the prefactor is not relevant here (Proposition 7.7). The last step is to define a phase and a magnetic potential to get back to the full Ginzburg-Landau functional. This is possible since we made the construction with the Meissner condition and quantized fluxes enforced, see Proposition 7.8.
From (1.2) and the discussion around (1.7), for type-I superconductors, the Ginzburg-Landau functional can be seen as a non-convex, non-local (in ) functional favoring oscillations, regularized by a surface term which selects the lengthscales of the microstructures. The appearance of branched structures for this type of problem is shared by many other functionals appearing in material sciences such as shape memory alloys [KM92, KM94, Con00, KKO13, BG15, Zwi14, CC15, CZ16], uniaxial ferromagnets [CKO99, OV10, KM11] and blistered thin films [BCDM00, JS01, BCDM02]. Most of the previously cited results on branching patterns (including [CCKO08, CKO04, COS16] for type-I superconductors) focus on scaling laws. Here, as in [OV10, CDZ17], we go one step further and prove that, after a suitable anisotropic rescaling, configurations of low energy converge to branched patterns. The two main difficulties in our model with respect to the one studied in [OV10] are the presence of an additional lengthscale (the penetration length) and its sharp limit counterpart, the Meissner condition which gives a nonlinear coupling between and . Let us point out that for the Kohn-Müller model [KM92, KM94], a much stronger result is known, namely that minimizers are asymptotically self-similar [Con00] (see also [Vie09, ACO09] for related results). In [Gol17], the optimal microstructures for a two-dimensional analogue of are exactly computed.
The paper is organized as follows. In section 2, we set some notation and recall some notions from optimal transport theory. In Section 3, we recall the definition of the Ginzburg-Landau functional together with various important quantities such as the superconducting current. We also introduce there the anisotropic rescaling leading to the functional . In Section 4, we introduce for the sake of clarity intermediate functionals corresponding to the different scales of the problem. Let us stress that we will not use them in the rest of the paper but strongly believe that they help understanding the structure of the problem. In Section 5, we carefuly define the limiting functional and study its properties. In particular we recover a scaling law for the minimization problem and prove regularity of the minimizers. We then prove the density in energy of ’regular’ measures. This is a crucial result for the main convergence result which is proven in the last two sections. As customary, we first prove the lower bound in Section 6 and then make the upper bound construction in Section 7.
2 Notation and preliminary results
In the paper we will use the following notation. The symbols , , indicate estimates that hold up to a global constant. For instance, denotes the existence of a constant such that , means and . In heuristic arguments we use to indicate that is close (in a not precisely specified sense) to . We use a prime to indicate the first two components of a vector in , and identify with . Precisely, for we write ; given two vectors we write . We denote by the canonical basis of . For and , and . For a function defined on , we denote the function and we analogously define for , the set . For and we let be the ball of radius centered at (in ) and be the analogue two-dimensional ball centered at . Unless specified otherwise, all the functions and measures we will consider are periodic in the variable, i.e., we identify with the torus . In particular, for , denotes the distance for the metric of the torus, i.e., . We denote by the dimensional Hausdorff measure. We let be the space of probability measures on and be the space of finite Radon measures on , and similarly . Analogously, we define and as the spaces of finite Radon measures which are also positive. For a measure and a function , we denote by the push-forward of by .
We recall the definition of the (homogeneous) norm of a function with ,
which can be alternatively given in term of the two-dimensional Fourier series as (see, e.g., [CKO04])
We shall write for .
The -Wasserstein distance between two measures and with is given by
where the minimum is taken over measures on and and are the first and second marginal of 555 for , we analogously define ., respectively. For measures , the -Wasserstein distance is correspondingly defined. We now introduce some notions from metric analysis, see [AGS05, Vil03] for more detail. A curve , belongs to (where stands for absolutely continuous) if there exists such that
For any such curve, the speed
exists for a.e. and for -a.e. for every admissible in (2.2). Further, there exists a Borel vector field such that
and the continuity equation
holds in the sense of distributions [AGS05, Th. 8.3.1]. Conversely, if a weakly continuous curve satisfies the continuity equation (2.4) for some Borel vector field with then and for -a.e. . In particular, we have
where by scaling the right-hand side does not depend on .
For a (signed) measure , we define the Bounded-Lipschitz norm of as
where for a periodic and Lipschitz continuous function , . By the Kantorovich-Rubinstein Theorem [Vil03, Th. 1.14], the Wasserstein and the Bounded-Lipschitz norm are equivalent.
3 The Ginzburg-Landau functional
In this section we recall some background material concerning the Ginzburg-Landau functional and introduce the anisotropic rescaling leading to .
For a (non necessarily periodic) function , called the order parameter, and a vector potential (also not necessarily periodic), we define the covariant derivative
the magnetic field
and the superconducting current
Let us first notice that and the observable quantities , and are invariant under change of gauge. That is, if we replace by and by for any function , they remain unchanged. We also point out that if is written in polar coordinates as , then
For any admissible pair , that is such that , and are -periodic, we define the Ginzburg-Landau functional as
We remark that and need not be (and, if , cannot be) periodic. See [COS16] for more details on the functional spaces we are using. Here is the external magnetic field and is a material constant, called the Ginzburg-Landau parameter. From periodicity and it follows that does not depend on and therefore, if the energy is finite, necessarily
We first remove the bulk part from the energy . In order to do so, we introduce the quantity
and, more generally,
where components are understood cyclically (i.e., ). The operator (which corresponds to a creation operator for a magnetic Laplacian) was used by Bogomol’nyi in the proof of the self-duality of the Ginzburg-Landau functional at (cf. e.g. [JT80]). His proof relied on identities similar to the next ones, which will be crucial in enabling us to separate the leading order part of the energy.
Expanding the squares, one sees (for details see [COS16, Lem. 2.1]) that (recall that )
and, for any ,
The last term integrates to zero by the periodicity of . Therefore, for each fixed , using (3.2), we have
We substitute and obtain, using and completing squares,
In particular, the bulk energy is . Since we are interested in the regime and since , the contribution of the last term in (3.5) to the energy is (asymptotically) negligible with respect to the first term in , and therefore it can be ignored in the following.
Applying (2.1) to and minimizing outside if necessary, the last two terms in can be replaced by
so that becomes
Let us notice that the normal solution , (for which we can take ) is always admissible but has energy equal to
in the regime that we consider here.
The following scaling law is established in [COS16].
For , , , sufficiently large, if the quantization condition
then the normal phase is the minority phase (typically disconnected on every slice) and there exist and (periodic) curves and such that
with . If this holds then using
Stokes Theorem on large domains the boundary of which is made of concatenations of the curves , it is possible to prove that (3.7) must hold.
As in [COS16], we will need to assume (3.7) in order to build the recovery sequence in Section 7.3.
The first regime in (3.8) corresponds to uniform branching patterns while the second corresponds to well separated branching trees (see [CCKO08, CKO04, COS16]). We focus here on the first regime, that is , and replace and by the variables , , defined according to
and then rescale
so that in particular and . Changing variables and removing the hats yields
as was anticipated in (1.2). In these new variables, the scaling law (3.8) becomes and the uniform branching regime corresponds to which amounts to , see also (1.3). Constructions (leading to the upper bounds in [COS16, CKO04, CCKO08]), suggest that in this regime, typically, the penetration length of the magnetic field inside the superconducting regions is of the order of , the coherence length (or domain walls) is of the order of , the width of the normal domains in the bulk is of the order of and their separation of order . These various lengthscales motivate the anisotropic rescalings that we will introduce in Section 4.
In closing this section we present the anisotropic rescaling that will lead to the functional defined in (1.5), postponing to the next section a detailed explanation of its motivation. We set for ,
to get , inside the sample. Outside the sample, i.e. for , we make the isotropic rescaling to get . A straightforward computation leads to , where