ELSEVIER PhysicaA236 (1997) 114-122
J.O. I n d e k e u a'*, J . M . J . v a n L e e u w e n b a Laboratorium voor Vaste Stof-Fysica en Magnetisme, Katholieke Universiteit Leuven,
Celestijnenlaan 200 D, B-3001 Leuven, Belgium b lnstituut-Lorentz, Rijksuniversiteit Leiden, P.O. Box 9506, 2300 RA Leiden, Netherlands
Abstract
We predict on the basis of the Ginzburg-Landau theory that a type-I superconductor can exhibit an interface deloealization or "wetting" transition, in which a rnacroscopically thick superconducting layer intrudes from the surface into the bulk normal phase. The condition for this transition to occur is that the superconducting order parameter I~k[ 2 is enhanced at the surface.
Surprisingly, the order of the wetting transition is controlled by a bulk material constant, the
Ginzburg-Landau parameter x. First-order wetting is predicted for 0 ~< x < 0.374, and critical wetting for 0.374 < ~ < I/x/2. The possibility of critical wetting is especially interesting because this phenomenon has largely eluded experimental verification in any system until now.
PACS: 74.55.+h; 68.45.Gd; 68.10.Cr; 82.65.Dp
Keywords: Wetting phase transitions; Type-I superconductivity; Interfaces; Critical phenomena
The purpose of this presentation is to demonstrate that type-I superconductors consti- tute an interesting new playground for wetting phenomena. Wetting phase transitions, familiar in the context of adsorbed fluids and binary liquid mixtures, also occur in the Ginzburg-Landau theory of superconductivity, contrary to previous expectations.
Moreover, the entire variety of phenomena, such as first-order wetting, prewetting, and especially critical wetting, are present. Furthermore, since thermal fluctuation effects are negligible in the superconductors under consideration, the "mean-field" results are directly relevant to experiment. Here, we give an introduction and tutorial summary of recently published research [1].
Fig. l(a) presents the situation, very familiar to the wetting community, in which a liquid drop sits on a solid substrate (S). The adsorbate consists o f two coexisting phases, liquid (L) and gas (G), and the (wedge-shaped) drop makes a contact angle
* Corresponding author. E-mail: Joseph.indekeu@fys.kuleuven.ac.be
0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved
P H S 0 3 7 8 - 4 3 7 1 ( 9 6 ) 0 0 4 0 5 - 0

J.O. Indekeu, J.M.J. van Leeuwenl Physica A 236 (1997) 114-122 115
6AS
(a)
~U. ~STP, AT I: m m m m m m m m m m m m 5 m m m m m m m m m w ~ m m m m m m m m m m r ~ m m
. m n m m m m m r w m m m l , . ~ i m u m m m r ~ m m m m u h J m m m r ~ m m m m m m m m m m r A u r z l : m m m m r ~ m m ~ i m m m m r A m m m ~ , . L - m m m m r ~ I r ~ m m m m m m | m m m r A w ~ m m m m m m m
WALL
(b)
Fig. I. (a) Liquid/gas interface making a contact angle 0 with a solid substrate. (b) Superconductor/normal interface making a contact angle 0 with the surface o f the material.
0 with the substrate or "wall". This contact angle obeys Young's law of mechanical equilibrium,
7SG = 7SL -[- 7LG COS 0,
(1) where 7IJ denotes the surface excess free energy associated with the surface or interface between phases I and J.
Fig. l(b), on the other hand, shows a portion of a superconducting material, near its outer surface or "wall" (W). We have in mind the classical superconductors, such as
Hg, A1, Sn, Pb, etc., with a surface that has been modified mechanically or chemically as we shall discuss. Needless to say the material is not a liquid but a metallic solid, and Fig. l(b) shows a possible interior struc~re, consisting of superconducting (SC) and normal (N) phases, which can coexist as thermodynamic states at the so-called temperature-dependent critical field Hc(T), in superconductors of type I. In the figure we have assumed the presence of a superconducting wedge-shaped region or "drop", and we now proceed to write the analog of Young's law
7W,N = 7W, SC + 7SC,N COS 0. (2)
We now ask a number of questions. What do we know about the various surface excess free energies featuring in (2)? What is their mutual relationship, and, in particular, are there situations in which Young's contact angle 0 vanishes? If yes, what is the character of the associated "wetting" transition? Finally, can we calculate the shape

116 J.O. lndekeu, J.M.J. van Leeuwen I Physica A 236 (1997) 114-122
P
I
GAS
T
(a)
/t
SC
0
T
Fig. 2, (a) Schematic P-T phase diagram for an ordinary compound. (b) Schematic H - T phase diagram for a type-I superconductor. of the superconducting "drop" in the vicinity of the surface? (We will not review the answer to the latter question here, but it is to be found in [2].)
In Fig. 2(a) the familiar P - T phase diagram for a simple compound is shown. In the context of wetting, it is usually the bulk first-order transition between liquid and gas that serves as the playground. If a spectator phase or "wall" is introduced, the pertinent question is whether or not the liquid phase wets the wall. More specifically, the interesting question is whether or not there exists a wetting transition point on this liquid-gas phase boundary, and if it does, what is the character of the wetting transition? In contrast, but also in analogy, Fig. 2(b) shows the H - T phase diagram
( H being the external magnetic field) of a type-I superconductor. The phase boundary between the superconducting Meissner phase and the normal phase is likewise o f first order, and terminates at a critical point in zero field. Similarly, as in Fig. 2(a), one may ask (and this question was in fact raised long ago, as we shall review) whether the superconducting phase "wets" the sample surface. Consequently, the same "wetting"- type questions as for adsorbed fluids can now be asked in the context o f the first-order
SC/N transition, taking the outer surface as "substrate" to be wetted, or, in more spirituel wording, as "wettee" [3].
Fig. 3 shows the H - T phase diagram of a type-I superconductor once more, but now focusing on the vicinity o f the bulk critical point To, where the Ginzburg-Landau theory is valid. Also, in addition to the bulk transition, at H c ( T ) , we have indicated

ZO. Indekeu, J.M.J. van Leeuwen/Physica A 236 (1997) 114-122
H
117
T
Fig. 3. Detail, near To, of the H - T phase diagram of a type-I superconductor, including the surface critical field Hc3.
X
$ sc t~ WALL
Fig. 4. Surface superconducting sheath of thickness l, between the external surface (wall) and the bulk normal phase. the possible presence of a surface phase transition, at Hc3(T), which corresponds to the critical nucleation of a surface superconducting sheath - as discovered by Saint-
James and de Gennes in 1963 [4]. Thus, in the region between Hc(T) and Hc3(T) the equilibrium state consists of the presence of a thin layer of superconducting phase, intruding between the surface and the bulk normal phase. This is sketched in Fig. 4.
Note that this layer or "sheath" cannot be of macroscopic thickness, since in that region of H and T the superconducting phase is not stable in bulk.
At this point it is useful to mention the relevant length scales, for the bulk as well as the surface properties. The thickness l of the superconducting surface sheath is typically of the order of the bulk coherence length ~, which is itself of the order of the wavelength of visible light (several 103,~) in these classical superconductors.
In addition to ~ there is the penetration depth 2 of the magnetic field, and the ratio
~c = )~/~ is the well-known Ginzburg-Landau parameter. In the Ginzburg-Landau limit
(i.e., close to To) x is a bulk material constant. For example, x ~ 0.01 for A1, 0.13 for Sn, and 0.6 for Pb. For type-I superconductors 0 ~< 1< ~< l/x/2, and for type-II,
> 1/v~.
For our "wetting" purposes it is essential to consider type-I superconductors, because only for those systems is the bulk SC/N transition of first order, so that the two phases can actually coexist, spatially separated by an interface with positive interracial tension

118 J.O. Indekeu, J.M.J. van Leeuwen/Physica A 236 (1997) 114-122
7sc,N. Indeed, one main distinction between type I and type II is that for the latter the presumed interfacial tension is negative, implying that phase separation is not stable.
Instead, like what would happen in miscible fluids, in type-II superconductors the SC and N phases mix completely, at least down to the smallest possible length scales, which is now not the molecular scale, but the quantum limit of the unit magnetic flux line. This gives rise to the well-known vortex phase in type-II materials.
With respect to the phase diagram of Fig. 3, one may now ask what happens to the sheath thickness l in the limit that bulk two-phase coexistence is approached, either by lowering H or by lowering T, or both. Does l remain finite, or does it diverge (to a macroscopic value)? Furthermore, as we monitor this limiting value of l along Hc(T), as a function of T, do we encounter a wetting point where l diverges? A preliminary an- swer to these questions was given fairly recently by Speth [5, 6], who showed that l re- mains of the order of 3, implying partial wetting and the absence of a wetting transition for all x < l/x/2. (In the limit x ~ l/x/2, however, Speth found that l does diverge, but it is due to the divergence of ~ upon approach of the bulk multicritical point.)
The conclusion that the superconducting phase does not "wet" the surface of the sample did not come as a surprise. Indeed, in one of the main textbooks on super- conductivity at the end of the 1960s, Livingston and DeSorbo remark [7]: "Another factor ignored in the simple models is the possibility that either the superconducting or normal phase "wets" the specimen surface, because of a difference between the surface energies of the external surface of normal and superconducting phases. Pippard, how- ever, has shown that this difference must be small". It is remarkable, first of all, that the term "wets" was employed that early and within the superconductivity community, and secondly, that reference is made to an even earlier work in which the relevant difference 7W, N --~W, SC is compared with a term which contributes to Ysc,N. Indeed, in this earlier paper (1951) [8] Pippard starts with the remark that Ginzburg presented evidence that 7w,N -Yw,sc could be a significant amount, comparable to a contribution to 7sc,N, but Pippard then goes on to argue that the former difference is almost certainly considerably smaller than Ginzburg's estimate.
All these previous assertions and works imply that, for the systems examined thus far, the inequality 7W,N -- 7W, SC < YSC,N does not become an equality, and therefore, there is no wetting transition. (For completeness, we mention that a "drying" transition, i.e.,
"wetting" by phase N, also does not occur.) The crucial point now is that the implicit assumption in all of the foregoing was that the surface of the specimen is against vacuum or an insulating material. This turns out to be a severe restriction as far as wetting is concerned, and we show that wetting transitions can occur for other types of surfaces.
The essential parameter in the usual description of wetting phenomena is the surface field which is a measure of the preferential adsorption of one of the coexisting bulk phases at a surface. For adsorbed fluids it is the chemical potential increment A#I. For magnets as described by, e.g., the Ising model, it is the surface magnetic field//1. We show that the corresponding quantity for superconductors is the surface extrapolation length b. It measures the distance from the surface where the superconducting order

J.O. Indekeu, J.M.J. van Leeuwen/Physica A 236 (1997) 114-122
119
\ \ f
( g0o)
0
) < ~-
Fig. 5. Superconducting wave function profiles near an external surface. The surface extrapolation length b is indicated for three different cases. parameter ~b would vanish upon a linear extrapolation of ~ away from the surface, as indicated schematically in Fig. 5. In fact, the equation defining b is the boundary condi- tion for if, which for the simple geometry that we will deal with, takes the simple form d e x=0
= b
0Ix:0. (3)
For the vacuum/SC surface considered in previous works, b = oe, which means that the gradient of ~, normal to the surface vanishes. Another fairly common case is b > 0, which corresponds to a metal/SC surface and means that the order parameter is sup- pressed at the surface. The extreme limit b---* 0 arises when one considers a surface between a superconductor (SC) and a ferromagnet which causes ff to vanish at the surface. The interesting point now is that negative values of b lead, of course, to en- hancement of ¢ at the surface, and thus would promote wetting by the SC phase. This is also what our first calculations indicated, but the question is: can b be negative for real surfaces?
The answer to that question is positive and has been known for quite some time.
In 1969 Fink and Joiner pointed out that b < 0 occurs for instance when the surface of a superconductor is treated mechanically - in their example by cold working [9].
They observed experimentally that Hc3 is increased and that this increase could be reproduced, in the Ginzburg-Landau theory, by assuming b < 0. A more detailed and quantitative study involving b < 0 was performed in the 1980s in the context of twin- ning plane superconductivity (TPS) by Khlyustikov (experiment) and Buzdin (theory), as described in a very interesting review article [10]. In the following we will not in the first place be concerned with TPS, but with enhancement of superconductivity at the outer surface of the specimen. We have in mind possible experimental realizations in which a thin layer (i.e., of thickness small compared with 2) of a material with a higher Tc (in bulk) is deposited on the surface, for the purpose of inducing b < 0.
Efforts along this line are in progress [11].

120 J.O. Indekeu, J.M.J. van Leeuwen/ Physica A 236 (1997) 114-122
The calculations we carried out, using the Ginzburg-Landau theory, have been out- lined in detail elsewhere [1], and we restrict ourselves to recalling the main ingredient here, which is the Ginzburg-Landau surface free energy functional
1 2
+ x~>O
[ V x A - p.oH] 2 + _ _ l O 1 2 ~ ( x )
}
2#0 p
(4)
The superconducting state is characterized by the wave function ~k(r) and the electro- magnetic field by the vector potential A(r). The superconductor is placed in an external constant magnetic field H . The geometry that we consider is the simplest conceivable: a half space x > 0 with superconducting material bordering another half space x < 0 with another material. For x < 0 we have a state with ~ = 0 and a magnetic field H parallel to the surface at x = 0. Let H - Hez.
As usual ~ is a measure for the temperature difference from the transition tempera- ture Tc, ~ c~ ( T - Tc), and fl is a stabilizing parameter (fl > 0). The third and fourth term between the curly brackets represent the gauge-invariant kinetic energy of the superconductor (including the coupling to the electromagnetic field) and the electro- magnetic energy density. The last term is the important local contribution from the surface at x = 0, containing the extrapolation length. Its justification is that (3) follows upon minimalization o f F with respect to fflx=O.
The Ginzburg-Landau equations for ~b and A follow from optimization of F with respect to ~k and A. Solving these equations, together with the appropriate boundary conditions, and calculating the surface free energies of the various solutions, leads eventually to the main wetting phase diagram, which we will now discuss in brief. A more detailed exposition can be found in [1].
Fig. 6 shows the wetting phase diagram at coexistence as a function of the parameters x and ~/b. We distinguish three regions according to the characteristics of the stable sheath solutions. In the lower half of the diagram we have a region l = 0 where no stable sheath solution exists. In the upper half we have on the high-~/b side a sheath of finite (microscopic) width l > 0 and in the low-~/b regime an infinitely thick (or macroscopic) sheath l = o o . The various regions are separated by phase transition lines.
For a given bulk material, i.e., for fixed x, varying the temperature corresponds to traveling horizontally in the phase diagram. Moving along bulk two-phase coexistence towards Tc corresponds to increasing
towards oc, since ~ cx IT - Tc[ -1/2, and b is essentially a surface material constant.
The line CN (continuous nucleation) ending in the tricritical point TCP, separates the region l > 0 from l = 0. The regions l > 0 and l = c~ are separated by the line
CD (critical delocalization) which ends in the critical endpoint CEP. The line CD corresponds to "critical wetting". The regions 1 = cx~ and l = 0 are separated by the line FD (first-order delocalization). This corresponds to "first-order wetting". The short stretch FN (first-order nucleation) between CEP and TCP marks the transition from no

ZO. Indekeu, Z M . Z van Leeuwen/Phys~a A 236 (1997) 114 122 121
K o., t
0.6 I > 0
0.4°s I=o¢ c \ -
0.3 ~
0.2 v FD
Ol
0.0
-0.3
,
-0.2
I = 0
FN
,
-0.1 0.0
{/b
0.1
Fig. 6. Wetting phase diagram for type-I superconductors at bulk two-phase coexistence, in the ~ versus ¢/b plane. The various regions and lines are explained in the text. sheath to a finite sheath. Finally, the line RA corresponds to reversal of preferential adsorption. To the left of that line the superconducting phase is preferred by the surface, so that 7w,sc < 7W,N. This inequality is reversed to the right of RA. It is only to the left of RA that the question of wetting by the SC phase is relevant. The alternate problem of wetting by the N phase (which we do not discuss here) belongs to the region to the right of RA.
Within the frame-work of this short presentation we omit a discussion of the global
H - T phase diagrams associated with first-order and critical wetting, respectively. This includes the phenomenon of prewetting (for first-order wetting transitions). These as- pects are thoroughly treated in [1]. We now proceed to draw the main conclusions. It is clear from the phase diagram (Fig. 6) that all type-I superconductors with a surface that is prepared such that b < 0, are bound to undergo a wetting transition at some
Tw < To, when T is varied along bulk coexistence, towards To. This is the main news with respect to previous studies, and it invalidates the general belief that the thickness of the surface sheath is always of the order of the bulk coherence length ~. Our result is in fact another realization of Cahn's general concept of critical-point wetting [12].
Note, however, that critical-point wetting does not occur for the case b = cx~ (as was already found by Speth).
The phase diagram furthermore shows that the order of the wetting transition is de- termined by a bulk material parameter, •. First-order wetting occurs for 0 < ~ < 0.374 and critical wetting for 0.374 < x < 1/x/~. This is interesting and unusual. In other systems (adsorbed fluids, Ising magnets, etc.) the order of the wetting transition is determined by surface parameters that are difficult to estimate and control. The case of a superconductor is quite different in this respect and the experimental consequences of this are important, since it is now possible to focus on studying critical wetting, for example, by choosing properly the bulk material. In particular, Pb is a good candidate, since ~c = 0.6 > 0.374 for that material.

122 J.O. lndekeu, J.M.J. van Leeuwen/Physica A 236 (1997) 114-122
The properties o f the surface are represented by a surface extrapolation length b. The interesting phenomena occur for b < 0, where the complete phenomenology o f wetting is realized. For b < 0, the surface becomes superconducting at a higher temperature than the bulk, even in zero field [1]. That is, Tcs > Tc, where Tc~ is the surface critical temperature for H = 0 . Measurement o f T~s can give an estimate o f b through b = - ~ ( T c ~ )
[1], so that both x and b can be determined for a given system.
We have presented an analysis o f the wetting and surface phase transitions in type-I superconductors on the basis o f the Ginzburg-Landau theory. Unlike for other systems
(fluids, magnets, liquid crystals, etc.), where the corresponding mean-field theory is at best qualitative and often only a poor approximation, the GL theory is quantitatively correct for classical superconductors. Therefore the results, and in particular the wet- ting phase diagram (Fig. 6), should provide a direct and quantitative guide for the experimentalist.
In closing we note that in the meantime the wetting phase diagram for twinning-plane superconductivity has also been determined. In that setting, the boundary condition for the vector potential is different from what it is for the case o f a "wall". This has impor- tant consequences for the wetting transitions, which are to be discussed elsewhere [13].
One o f us (J.O.I.) would like to thank Alberto Robledo for the invitation to present this research, and for the hospitality in Oaxaca and Mexico City. J.O.I. is Research
Director o f the Belgian National Fund for Scientific Research. This work is supported by the Belgian Inter-University Attraction Poles ( I U A P ) and the Concerted Action
Research Programme (GOA).
[1] J.O. Indekeu and J.M.J. van Leeuwen, Phys. Rev. Lett. 75 (1995) 1618; Physica C 251 (1995) 290.
[2] R. Blossey and J.O. Indekeu, Phys. Rev. B 53 (1996) 8599.
[3] The term "wettee" was coined by Ben Widom in the course of a conversation with one of us (J.O.I.).
[4] D. Saint-James and P.-G. de Gennes, Phys. Lett. 7 (1963) 306; see also P.-G. de Gennes,
Superconductivity of Metals and Alloys (Addison-Wesley, Reading. MA, 1966).
[5] W. Speth, Ph.D. Thesis, A summary of Speth's contribution is contained in Dietrich's review (Ref.
[6]), Miinchen (1986).
[6] For a review on wetting, see S. Dietrich, in: Phase Transitions and Critical Phenomena, eds. C. Domb and J.L. Lebowitz, Vol. 12 (Academic Press, London, 1988) p. 1.
[7] J.D. Livingston and W. DeSorbo, in: Superconductivity, ed. R.D. Parks, Vol.2 (Marcel Dekker, New
York, 1969)p. 1235.
[8] A.B. Pippard, Proe. Cambridge Philos. Soc. 47 (1951) 617.
[9] H.J. Fink and W.C.H. Joiner, Phys. Rev. Lett. 23 (1969) 120.
[10] For a review, see, I.N. Khlyustikov and A.I. Buzdin, Adv. Phys. 36 (1987) 271.
[11] V. Metlushko, C. Strunk and V.V. Moshchalkov, unpublished.
[12] J.W. Cahn, J. Chem. Phys. 66 (1977) 3667.
[13] F. Clarysse and J.O. Indekeu, unpublished.