ELSEVIER Physica A 244 (1997) 426-439

PHYSICA

Interface potential for nucleation of a superconducting layer J . M . J . v a n L e e u w e n a,*, J .O . I n d e k e u b

a Instituut-Lorentz, Universiteit Leiden, Nieuwsteeg 18, Leiden, Netherlands b Laboratorium voor Vaste-Stoffysica en Magnet&me, Katholieke Universiteit Leuven,

B-3001 Leuven, Belgium

Abstract

The effective interface potential V for nucleation of a superconducting surface layer in su- perconductors is derived as a function of the expelled magnetic moment M. We show that the associated variational problem with integral constraint is soluble, in contrast with the case of adsorbed fluids, where a similar constraint poses serious difficulties. Minimalization of the ef- fective potential V ( M ) gives the location of the nucleation phase transition. Depending on the temperature or other parameters this transition can be of second order and can go over in a first-order transition through a tricritical point on the nucleation line, in agreement with previous numerical work on the same phenomenon.

1. Introduction

The physics o f the interface between different thermodynamic phases and the phe-

nomenon o f wetting have been a continued interest o f Ben Widom and many o f his

contributions to this field have made intricate phase diagrams transparant for the com-

munity o f statistical physicists. The same holds for his work on tricritical points, which

due to his lucid description, are not any more a horror o f complications for the physi-

cists. In this contribution we want to show that superconductors form an interesting

arena for combining the theories o f interfacial and tricritical phenomena. In the case

o f superconductivity, the substrate to be wetted or, as Ben Widom likes to call it, the

"wettee", can simply be the surface o f the material itself. The interior or bulk material

is (most ly) assumed to be o f type I, so that two-phase coexistence between the normal

phase and the Meissner phase can take place. The physics we focus on is whether

the surface can display a superconducting "wetting" layer while the bulk is in the

* E-mail: jmjvanl@rulilO.leidenuniv,nl.

0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00230-6

J.M.J. van Leeuwen, 3. O. Indekeul Physica A 244 (1997) 426--439 427

normal phase. In particular, we will restrict our attention here to the very first onset

or "nucleation" of surface superconductivity.

The novelty of our research resides in the definition and implementation of an ef- fective interface potential V for studying the nucleation of surface superconductivity. Interface potentials are a widely used tool for studying wetting phenomena in ad-

sorbed fluids and the associated Ising-like models. We start from the recent theory of

wetting transitions in superconductors [1,2] and take our inspiration from the inter-

face potential approach developed for strongly type-I superconductors [3] as well as from experimental considerations. We show that a new integral criterion is suitable

for defining an interface potential for the general case (the entire range of type-I ma-

terials, and, as far as nucleation of surface superconductivity is concerned, also type II). The integral criterion consists of constraining the expelled magnetic moment M, which can be expressed as an integral over the magnetic induction profile. Nucleation

of surface superconductivity takes place for small M, and we thus concentrate on

an analytic expansion in M. From this we derive the exact locus of critical, tricriti-

cal and first-order nucleation transitions, in agreement with the findings of our earlier work [ 1].

Let us recall some more physics before proceeding to the calculation. It is well known that a superconducting layer may nucleate at the surface of the superconduc-

tor before the bulk becomes superconducting. This can be considered as a form of

wetting of the surface by the superconducting phase while the bulk remains in the normal phase. Generally this nucleation is a continuous process, but depending on

the surface, this (surface) transition may become a first-order transition. Recently, the phase diagram for surface phenomena has been determined on the basis of a nu-

merical solution of the Ginzburg-Landau equations [1], which give a quantitatively correct description of superconductivity in the neighborhood of the transition from the

bulk normal to the superconducting state. One of the aims of this paper is to com- plement these numerical calculations by an analysis using the interface potential for

nucleation. For deriving an interface potential one considers out-of-equilibrium states which have

a prescribed value of the order parameter associated with the transition. In the case of nucleation the experimental signature of the transition is the magnetic moment M which

is expelled by the superconducting layer. In our case M = - Mdi~ > 0 with Md,,,~ the diamagnetic moment of the sample. Near the nucleation point we thus consider the free

energy of the virtual states of the system having a given value of M. The equilibrium state is the one that minimizes the free energy as a function of the magnetic moment. On one side of the transition this is the state with M - - 0 and on the other side one finds a finite value.

In the simple geometry of a flat wall and an applied magnetic field parallel to the wall one may describe the state of the system by the vector potential a(x) and the superconducting order parameter go(x), both functions of the distance x to the wall. We work with the dimensionless units where we have scaled distances with the coherence length and the other quantities in such a way that the equations contain only the

428 J.M.J. van Leeuwen, J.O. Indekeu/Physica A 244 (1997) 426-439

Ginzburg-Landau parameter ~c, which is the ratio of the magnetic penetration depth to the superconducting coherence length. The equations are the variational equations of the free energy functional a[a, ~o]

O O

/dx{(d - h) 2 q- ~b 2 - q)2 q_ K2q~4/2 q_ a2~o 2 } -k ~(,92(0) • ( l ) a[a, qg] = J

0

Here h is the dimensionless external magnetic field and r the dimensionless inverse extrapolation length, to be explained shortly hereafter. Varying a with respect to a(x)

and qg(x) gives the Ginzburg-Landau equations

and

a ( x ) = a(x)q92(x) (2)

~(X ) = -- ~p(X ) + a2 (x )~ (X ) + ,'¢2q93(x) (3)

As we consider always the bulk to be normal we have the asymptotic conditions

d(x---~cxD)=h, qg(x---~ ~ ) = 0. (4)

Variation of a with respect to the initial values a(0) and q~(0) leads to the boundary conditions

~i(0) =h , ~(0) = ~o(0). (5)

The solution of (2) and (3) satisfying both (4) and (5) is the equilibrium solution.

From the second Eq. (5) one sees the meaning of r: it is the ratio of slope to value of the order parameter at the wall.

We want to impose the value of the expelled magnetic moment M

M = / dx (h - a ) . (6) 0

The problem then becomes the optimalization of a under the constraint (6) which leads to the optimalization of

I0J ] d[a, qo] ~ a[a, q~] + ~ dx (h - a) - M , (7)

where c~ is a Lagrange multiplier. Note that the variational Eqs. (2) and (3) remain the same for the problem (7). So the virtual profile is optimal in the same way as the equilibrium profile. We also keep the asymptotic boundary conditions (4), which we refine for the first to

a(x ~ ~ ) ~- h(x + c). (8)

J.M.J. van Leeuwen, J.O. IndekeulPhysica A 244 (1997) 426-439 429

The constant c is the shift of the asymptote to a(x) for x ~ oo. Doing the integration

in (6) we find with (8)

./-- dx(h - a) = a(0) - hc. (9)

0

As the constant c will not be independent of the boundary values a(0) and 99(0), the

optimalization of 6 with respect to the boundary values will lead to more complex boundary conditions than (5) when the value of M is given.

In the next section we construct a solution of (2) (4) for a given value of M,

which contains the parameter c as a free parameter. Then in the subsequent section we minimalize 6 with respect to the free parameter c at fixed M and so obtain the

interface potential V(M). Indeed the boundary conditions (5) are only satisfied for the equilibrium solution which minimizes V(M) with respect to M. Then we use

the interface potential for the location of the nucleation line and the position of the tricritical point on this nucleation line where the nucleation transition changes from

continuous to first-order. We conclude the paper with a comparison of our interface potential with that of the wetting problem for fluids.

Before we construct the solution we make two general remarks which facilitate the

discussion substantially.

• The analogy of the Ginzburg-Landau equations with a two-dimensional mechanical problem allows the construction of a first integral (the "energy") which takes the following form (in view of the asymptotic condition)

q)2(X) q- tj2(X) -- q~2(X) + a2(x)~92(x) + tc2q94(x)/2 + h 2 . ( 10 )

• Using the second Ginzbur~Landau equation one can transform the expression for ~r into

= / dx{--h'2q~4/2 ÷ [~J -- h] 2} q- [zqg(0) - ~b(0)]qg(0). (1 l ) o-

0

2. Solution near the nucleation point

We construct solutions of the Ginzburg-Landau equations by making an expansion

in powers of M of the following type:

a(x) = Z azj(x)MJ ( 12 ) /=0

and

q)(X) = Z ~92j+I(x)MJ+I/2 " (13) j=0

430 J.M.J. van Leeuwen, J.O, Indekeu/Physica A 244 (1997) 426 439

Here we anticipate that a small value of M will correspond to a small ~0 or a weak (and mostly thin) superconducting layer. We also know from our previous analysis [1] that the first-order correction to the vector potential is of order ~p(0) 2, so that ~0(x) is of order M I/2, which justifies the lowest order in (13). Inserting these expansions into the Ginzburg-Landau equations leads to a series of equations, one for each order in M

a0(x)=0,

~l(x) = - q~l(x) + aZ(x)qgl(x),

t~z(X ) = ao(x)~p~(x),

~3(x) = - qo3(x ) + aZ(x)cp3(x) + 2ao(x)az(x)qol(x) + K2qg~(x).

(14)

(15)

(16)

(17)

As we mentioned before, the solution is facilitated by using the first integral (10). Inserting the expansion (12) and (13) into this conservation law we get again a set of equalities for each of the integer powers of M

,~2(x)=h2, (18)

2d0(x)a2(x ) + ~b~(x)= - qg~(x) ÷ a2(x)qo~(x). (19)

It is clear that these relations are first integrals of the even Eqs. (14) and (16), the latter being combined with (15). So we may replace (14) by (18) and (16) by (19).

In order that the M in the expansion correspond to the expelled magnetisation we must obey (6) which reads with (9)

M = a(O) - hc. (20)

Inserting the expansion (12) and (13) we get the set of boundary conditions

ao(O)=hc, a2(0)= 1, a 2 n ( 0 ) = 0 , n~>2. (21)

The consecutive equations have the following solution: ao(x) is given as

ao(x) = h(x + c) , (22)

where we have taken the positive root of (18) in order to match with the asymptotic boundary condition. Note that (22) also satisfies the first of the conditions (21). The constant c is arbitrary and remains the free parameter in the subsequent terms of the solution.

The solution for q~l(X) is constructed with the aid of the auxiliary function q(x) which is the solution of the Eq. [1]

q(x) + qZ(x) = -- 1 + (hx) 2, q(x ---* c~) --* - hx. (23)

In terms of this q(x) the function q~l(x) can be expressed as

X

~P1(x)= zIp(x) , p(x) = -- exp / dx'q(x' + c) . (24) t~

o ,

o

J.M.J. van Leeuwen, J.O. IndekeuIPhysica A 244 (1997) 426 439 431

Note that q~l(x) obeys the asymptotic boundary condition. The constant D, which

equals the value ~01(0), is a temporary free constant and will be fixed in the next step.

The subsequent solutions for the azj are straightforward. One can solve ~i2j from the conservation law in terms of the functions of lower order. Then we integrate

~ c

a2/(x) = - / dx'g~2j(x'). (25)

X

We have chosen the integration constant such that a2j vanishes as x---+ vc, which is necessary in order to satisfy (8) in view of (22). Now we have to inspect the other

boundary conditions (21). We do this first for a2(x) OC

2 t 2 / 1 dx'[-(p~(x') - (o~(x') + ao(X ){pL(x )]. (26} a2(x) = 2h .

Using that

~t(x) =q(x + c)qol(x) (27)

and Eq. (23) for q(x) we may write (26) as

~ C

a~(x) = ~ 2hZ2 / dxto(x' + c)pZ(x ' ) . (28)

X

For the value at the wall we introduce the integral

r(c)= - f dxq(x)exp2 f dxlq(xl). (29)

C C

Then using the second boundary condition (21) we get the announced equation fbr Z1

7.~-- 2h r(c) (30)

In general, each equation for the q02j+l(x) contains a free integration constant which

is fixed by the boundary condition for the subsequent a2j+2(0). For later use we give two expressions which follow directly from (28). The value

of the derivative of a2(x) at the wall reads

ci2(O) = X~O(c)/2h (31)

and differentiation of r(c) with respect to c, combined with (30), leads to

c3Xl c3c = XI [q(c) - q(c)/2r(c)]. (32)

The generic equation for the q02j+l is of the form

( P 2 j + l ( X ) = - - ( D 2 j + I ( X ) -~ a2(x)~o2j+l(X) -t- g 2 j + l ( X ) , (33)

432 J.M.J. van Leeuwen, J.O. Indekeu/Physica A 244 (1997) 426-439

where gzj+l is composed of functions of lower order. The solution can be put in the form (as can be verified by substitution)

/ J ~02j+l(X)= Z2j+lp(x) -- p(x) dx' p-2(x ') dx"p(xn)g2j+l(x"). (34)

0 x I

Note that qo2j+l(x) obeys the asymptotic boundary condition (4) provided that the integral over x ~ does not diverge too strongly. It converges at the upper bound when g2j+l decays sufficiently fast. This is achieved by the decay of a2/ as given by (25).

The integration constant is Z2j+l, which equals (p2j+l(O). Hereby we have constructed a solution which still has one free parameter c. For our

purpose we need not go further than a2(x) and ~03(x). The expression for ~03(x) reads

/ J ~03(X) = Z 3 p ( x ) - ~01 ( x ) d x ' p - 2 ( x ' ) dx"pZ(x ' ')

0 x ~

× [2a0(x n )a2(x n) + ~:2~0~(xn)]. (35)

The form (35) leads to the following value for the derivative of ~p3(x) at the wall (which is useful for the next section)

OG

~b3(0) = q(c)q~3(0) - )~l / dxpe(x)[2ao(x)a2(x) + tc2~o~(x)] • (36)

0

We may reorganize the integral by using (16). Multiplying (16) with a2(x) and inte- grating over all x yields

72<3 0~3

- f dx6~(x)- a2(0)c~2(0) = / dxao(x)a2(x)q)~(x). 0 0

This gives for the derivative of (p3(X) at the wall 0<3

~b3(0 ) = q ( c ) q ~ 3 ( 0 ) 4- Zlq(C)/h 4- ( I / z I ) f dx[2dfl(x) - K2fp4(x) ] .

0

(37)

(38)

3. The interface potential

In this section we first derive by optimalization of ~ the equation which gives the value of the free parameter c as a function of the prescribed expelled magnetic moment. Then we construct the interface potential by substituting this optimal solution in the expression (11 ) for the free energy.

As we mentioned for functions satisfying (2) and (3) the variation of a is only due to the initial values of a and ~o and is given by

6a = 2[h - a(0)]ra(0) + 2[z~p(0) - ~b(0)]6~p(0). (39)

J. M.J. van Leeuwen, .L O. lndekeu / Physica A 244 (1997) 426- 439 433

Since the solution that we have constructed has the correct value of M the difference

between a and d vanishes. Thus, we may write the optimalization equation for c as

ga ~ . . . . . ?~a(O) 0 = ?c~=2[~° (0) -~b(0) ] ~ . + 2 [ h - a t u ) l ~ • (40)

This equation shows that 6qo(0) and 6a(0) are not independent when M is fixed. All

the entries can be written as expansions in M with the formulae of the previous section.

So (40) can be expanded in powers of M.

0 - - {[r - q(c)][2q(c) - q/r(c)] - c)(c)}M + O(M2). (41

We may solve this equation by a similar expansion of c in powers of M

c = co + c2 M 2 - - ' • • • (42)

The value of co follows from the lowest non-vanishing order in M. We will argue in

what follows that in the nucleation point holds

r -- q ( c o ( h ) ) . (43)

Then (41) implies

q(c0(h)) = 0 . (44)

Here the value of co(h) is defined by (144). Generally [l], the function q(x) has a negative slope for large x and a positive slope for large negative x. The maximum is

reached for co(h) which depends on the external magnetic field h. Then (43) gives the value of r as function of h and it defines a (nucleation) line in the (r,h)-plane. See

also [4]. Near the nucleation line we define

~r = r q(co(h)) , 6c = c - co(h) . (45)

By expanding the function around co(h) we find from (41)

&' 2q( c°( h ) ) 6~ . (46) 4(co(h))

In the same way, by including the next term in the expansion in powers of M in (41),

we can compute the coefficient c2, etc. Now everything is ready to compute the interface potential V, defined as the optimal

value of the free energy a for given M. It amounts to the insertion of the series into the expression (11) for the free energy. In the first step we write V as

V ( M ) = Z~6rM - B M 2 + Z l [ Z ' q ) 3 ( 0 ) - - 03(0)]M 2 +/izqo3(O)z M 2 + --. , (47)

where we have included all terms up to M 2 and where B is given by "N;

B = --/" d x [ K 2 @ ( x ) / 2 - d2(x)]. (48) /

0

434 J.M.J. van Leeuwen, J.O. IndekeulPhysica A 244 (1997) 426 439

Using (38) we have to lowest order

ZI [r~03(0) - ~b3(0)] = Zl6rtp3(0) - z~q(c)/h + 2 B . (49)

Terms of the order 6zM 2 or equivalently 6cM 2 (from q(c) cx 6c) are near the transition

of minor influence and will be dropped. Then we find

V ( M ) = z 2 6 r M + B M 2 + . . . . (50)

It is interesting to note that a contribution of the boundary term overrules the contri- bution of the bulk and turns it into its opposite value. We note that r(c) , as given by

(29), is positive (since c)(x) is negative in the whole interval). Therefore, the constant

Z12 is positive as it should be. Thus (50) concludes the construction of the interface

potential.

4. The nucleation line and the tricritical point

The potential (50) gives the free energy for a given value of M. To find the equilib- rium solution we must minimize the free energy with respect to M. The minimalization

equation reads

Z26z= - 2 B M + . . . . (51)

For B > 0 the equation is soluble for 6z < 0 and the equilibrium solution has a finite value of M, which means a finite superconducting layer. For 6~>0 the minimum is

reached at the boundary M - - 0 . Thus, the nucleation line is indeed given by fit = 0 as was anticipated in (43).

For B < 0 we do not have a stable interface potential and higher-order terms have to be included for stability. So we put

V ( M ) = Z~6zM + B M 2 + CM 3 + . . . (52)

and assume C to be positive, Near B = 0 we have the standard scenario of a tricritical point. The minimalization equation now reads

Z~6r + 2 B M + 3CM 2 = 0 (53)

which has a real solution for M when

B 2 >1 3Z26zC. (54)

The onset of nucleation for positive 6z occurs at the value where V ( M ) = 0 with M the solution of (53). This happens at the value

M = - 212&/B (55)

or for the combination (using (53))

Z~ az = B2/4C (56)

J.M.J. van Leeuwen, J.O. Indekeu/Physica A 244 (1997) 426 439 435

V(M)

/ /

/ /

/

~1 / / / C /

/ / /

/ "~. j J

/ /

f // /

/ / /

FN

0.0 0.1

M

Fig. 1. Qualitative sketch of the effective interface potential V(M) for small magnetic moment M, in the nucleation regime of the superconducting surface sheath. Curve a represents the absence of surface super- conductivity. The state with M = 0 has the minimum free energy. Note that the slope of V(M) is finite and positive in this boundary minimum. Curve b represents the presence of a stable microscopically thin superconducting sheath, with M given by the minimum of V(M). Somewhere in between a and b a critical nucleation transition occurs, represented by curve CN. At this transition the derivative of V(M) vanishes at M = 0, but the curvature remains finite. If, simultaneously with the first derivative, also the curvature vanishes, the nucleation transition becomes tricritical, as shown in curve TCN. At this transition the curva- ture of V(M ) at M = 0 changes sign, and the nucleation transition changes from critical to first-order. An example of a first-order nucleation transition is displayed in curve FN. At this transition a sheath of finite M thermodynamically coexists with the null solution. Both states correspond to (equal) minima of V(M), the null solution again being a boundary minimum.

which is well within the zone (54) of a real solution. Thus (56) gives the locus of the

first-order nuclea t ion line and (55) gives the j u m p in the magnet iza t ion M. C om bin ing

(55) and (56) one sees that the j u m p disappears at the tricritical point which is located

at B - - 0 . This criterion was used in the numer ica l solution to locate the tricritical point

[1 ]. The first-order and second-order nuclea t ion lines meet tangent ia l ly to one another

at the tricritical point, since according to (56) 3r is proport ional to B 2. B has a regular

expansion around the tricritical point, so that 3r oc (T - Tt~) 2 vanishes in a parabolic

manner . Here T denotes the temperature. The same holds when the magnet ic field is

chosen: ~z ~x (h - h t c p ) 2. A qualitative representat ion of the various shapes o f V ( M )

near M -- 0 is shown for the different types of the nuclea t ion transi t ions in Fig. 1.

Go ing back to the equi l ibr ium relat ion (51) we observe that the boundary value for

~0(x) obeys

"rqg(O) -- (9(0)~-MI/2[ZI~'C + ['cq93(0 ) - ~3(O)]M + - " "]

M1/2[Z 16"~ + 2BM/z~] • (57)

So for the equi l ib r ium solut ion the boundary condi t ion (5) for ~o(x) is fulfilled at least

to order M 5/2. The boundary condi t ion for a(x) is fulfilled to order M 2 due to (31).

436 J.M.J. van Leeuwen, J.O. Indekeu/Physica A 244 (1997) 426 439

5. Conclusion and discussion

In this paper we have shown that the critical, tricritical and first-order phase tran- sitions associated with the nucleation of surface superconductivity can be studied by means of an effective interface potential. In contrast with the critical wetting regime studied recently [5], in the nucleation regime the superconducting sheath is only mi- croscopically thin, so that the appropriate order parameter is not the wetting layer thickness l, but rather a global quantity expressed as an integral over the sheath profile.

It is worthwhile to comment in more detail on the choice of the order parameter. The usual interface potential description, in terms of V(1), where 1 is a collective coordinate indicating the (average) displacement of the interface relative to the surface or wall, is not well defined when l is microscopically small, as is the case for surface nucleation. For this type of phenomenon, in which an infinitesimal (or in any case very small) surface superconducting layer becomes stable with respect to the null solution (no superconductivity at all), a different strategy is necessary. It is well known that in general, and especially so in situations where l is difficult to define, one may have recourse to an integral criterion, in which the interface potential is obtained as a function of the "adsorption" or "coverage" [6,7]. Since the adsorption remains perfectly well defined in the limit that its magnitude tends to zero, this alternative route to V is fully adequate in principle to study nucleation. Regrettably, however, in the Landau theory of adsorbed fluids the simplest and physically most appealing integral criterion is strictly speaking insoluble [6,7] and sophistications are necessary, which in our opinion are somewhat cumbersome both from a physical and a mathematical point of view. Here we have shown that in the Ginzburg-Landau theory of superconductivity this awkward complication does not arise, so that, fortunately, the simple integral criteria corresponding to the most obvious choice(s) of physical order parameter(s), lead to a perfectly soluble variational problem.

Let us make our argument explicit by means of the following simple example. For adsorbed fluids the interface separating a thick wetting layer and bulk adsorbate can be located through the collective coordinate l defined using a simple crossing criterion

[6J, e.g.,

m ( x = l ) = O , (58)

where re(x) is the density (or concentration) profile and x the distance from the wall. The (arbitrary) value m = 0 represents a density intermediate between the bulk densi- ties of the two coexisting adsorbate phases in bulk. Near the interface m varies rapidly from one bulk density to the other. The distance l thus represents the average or mid-point location of the interface. Clearly this procedure only makes sense if the interface is sufficiently far from the wall for its structure not to be distorted by the wall. For thin layers this fails and there may not even be a crossing point x = 1 in the profile.

J.M.J. van Leeuwen, J.O. IndekeuI Physica A 244 (1997) 426-439 437

The integral criterion on the other hand, which in its simplest formulation takes the

form

f f = / d z (re(z) - mbulk ) = constant (59) J

o

(with F as the adsorption), leads to an insoluble problem (in the strict sense of the

word) because its use within a Lagrange multiplier scheme gives as Euler-Lagrange equation

d2m(x) d f (m) dx 2 ~ + :t , (60)

where f ( m ) is the (mean-field) bulk free energy density which is minimal in m = mh,,/k

and c~ is the Lagrange multiplier. The awkward problem now is that this equation is incompatible with the bulk condition m ~ mb,m for x---+ vc, unless ~ = 0. The bulk

condition indeed forces all terms in (60) to approach zero asymptotically for large x.

We now show that no such problem arises in the Ginzburg-Landau theory of super- conductivity. The order parameters appropriate to type-I superconductivity are twofold:

the wave function q)(x) and the magnetic vector potential a(x). In the presence of a thin superconducting layer near the surface of the material the thickness l and a cross-

ing criterion on qo(x= l) are not meaningful for the same reason as outlined above

for fluids. It is then natural to adopt an integral criterion. Since there are two order parameters, there are two options. The first is to impose

=-- / dx (p(x)2 = constant. (61 ) Ep !

0

We assume that the bulk phase is the normal phase, so that ~Pb,,lk = 0, and we note that the presence of the second power of ~o is a consequence of the symmetry of the theory

(the variable ~p is originally a complex function, which accidentally can be taken real in our present application [1]). The Euler-Lagrange equation which comes in place of

(60) is now

d2~p d x 2 - (p(x) q- a(x)2(p(x) q- K2fp 3 -- ~qo(x) , (62)

where 7 is the Lagrange multiplier. Since this time the Lagrange multiplier is multiplied by ~0 itself, imposition of the bulk condition q)(x) -+ 0 for x ~ vc is perfectly compatible with the use of a (non-zero) 2.

The second option, which we have adopted in this paper, is to define an integral constraint associated with the magnetic order parameter. This choice is also free from mathematical difficulties and as a matter of fact the physically most natural and appeal- ing one, because the diamaynetic moment of the sample is an experimentally directly measurable quantity, and can be expressed in the simple integral form (6). Of course,

438 J.M.J. van Leeuwen, J.O. Indekeul Physica A 244 (1997) 426-439

the two options are equivalent as far as the physical consequences, such as the location of the nucleation line, are concerned.

As far as the comparison between a local and an integral constraint is concerned we

remark the following. In the limit K--+ 0 the two approaches, based on 1 or M, respec-

tively, actually become fully equivalent, since in that limit the integrand of M becomes constant in space. The interface potential in that limit was derived and employed in

earlier work for studying the first-order wetting and prewetting transition [3]. The character of the interface potential V(M) at small M is interesting and reminis-

cent of that of V(I) at small 1, found in [3]. In both cases V is linear in its argument,

so that the minimum at l=0 or M = 0 is in fact a boundary minimum. This has cer- tain consequences for studying small fluctuations around the null state, and also for the shape of the so-called critical nucleus of a superconducting "drop" on the wall, in

analogy with the case of adsorbed fluids [8]. In the earlier work [3] there is mention

of the possibility that the minimum at l=0 may soften to become a genuine minimum with dV/dl = 0 for finite to. However, our present result for V(M) cannot resolve this

question because the connection between M and l cannot be made precise at finite ~c and small 1. Only for large l one can easily see that M and 1 become proportional to

one another.

In summary, the case of the superconductor provides a nice arena for the interface potential approach, with (i) new features associated with the "quantum" nucleation

from zero to finite superconductivity, which have no analogue in classical fluids, and (ii) new life for the natural integral constraint on the "adsorption", which has been so problematic in the classical fluid case.

Acknowledgements

With this contribution, dedicated to Ben Widom on the occasion of his 70th birthday, the authors want to express their gratitude for his incomparable way of stimulating the progress of science. Both authors have benefitted enormously from his excellence and not in the least from his example as a distiguished scientist and human being. We

thank Eivind Hauge for suggesting the idea that nucleation can be discussed in terms of an interface potential, and Christoph Strunk for discussions of the experimental measurements of the expelled magnetic moment. J.O.I. is Research Director of the Fund for Scientific Research of Flanders (FWO). This research was supported in part by the Belgian Concerted Action Programme (GOA) and Inter-University Attraction Poles (IUAP).

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