Superconductivity on networks : II. The London approach

HAL Id: jpa-00209662

https://hal.archives-ouvertes.fr/jpa-00209662

Submitted on 1 Jan 1983

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Superconductivity on networks : II. The London approach

S. Alexander, E. Halevi

To cite this version:

S. Alexander, E. Halevi. Superconductivity on networks : II. The London approach. Journal de

Physique, 1983, 44 (7), pp.805-817. �10.1051/jphys:01983004407080500�. �jpa-00209662�

Superconductivity ^on networks : II. The London approach

S. Alexander

Institute for Theoretical Physics, University ^of California at Santa Barbara, ^Santa Barbara, California 93106, ^USA

and Department ^of Physics, University of California at Los Angeles, ^Los Angeles, California 90024, ^USA and The Racah Institute of Physics (*), ^{The Hebrew} University, Jerusalem, ^Israel

and E. Halevi

The Racah Institute of Physics, The Hebrew University, Jerusalem, ^Israel

(Reçu le 22 novembre 1982, accepti ^{le 23}

1983)

Résumé.

Nous avons calculé les énergies magnétiques ^{et les} champs critiques de réseaux de filaments carrés

ou

de

tamis de Sierpinski

^et les résultats sont appliqués ^{aux amas de} percolation. ^{Les résultats sont obtenus}

par ^un développement ^des équations ^de Ginzburg-Landau ^au voisinage ^de l’amplitude ^{de London. Les effets de la} quantification ^{du flux et de courants} critiques ^{locaux sont inclus} systématiquement. ^{Pour la} percolation, ^nous

faisons l’hypothèse ^{d’une structure fractale se} reproduisant ^{à courte distance avec un} recouvrement adéquat ^{et une}

distribution d’amas du type ^{Stauffer. Nous obtenons des lois d’échelle} simples pour l’énergie magnétique, pour la

longueur ^{de cohérence} supraconductrice, pour les champs critiques ^{des amas finis et de l’amas infini. Ces résultats}

sont comparés à d’autres études du même problème ^{et nous} discutons des différences entre ces modèles. Nous discutons également ^{les résultats} expérimentaux disponibles ^et proposons d’autres voies d’approches expérimen-

tales.

Abstract.

The magnetic energies and critical fields of networks of thin wires are calculated for the square net and the Sierpinski gasket ^and the results are applied ^to percolation clusters. ^The results are obtained by

sys- tematic expansion of the Landau Ginzburg equations around the constant amplitude London limit. The role of flux quantization ^and local critical currents are included in a systematic way. For percolation ^{we assume a short}

distance self similar fractal structure of the clusters with proper crossovers and inclusion of the Stauffer cluster distribution. Simple scaling ^{forms for the} magnetic energy and for the superconducting ^coherence length, ^{and for}

the critical field of finite clusters and of the infinite cluster are obtained. The results are compared with other studies of the same problem ^{and the} origin ^{of the} discrepancies is discussed. We also discuss existing experiments ^and suggest ^some

experimental approaches.

Classification Physics ^Abstracts

74.30 - 05.40

1. Introduction.

In a recent paper [1] (referred ^{to as I} below) ^{we studied} solutions of the linearized Landau Ginzburg equations

on networks of thin wires and applied the results to the determination of the upper critical field (Hc2).

This was motivated by ^{the recent} experiments ^of

Deutscher et al. [2] measuring this field near the

percolation threshold of granular systems ^and by

de Gennes’ [3, 4] suggestion ^{that one could use such}

networks of thin wires ^{as a} model to simulate the geometry. ^In applying the results to percolation systems ^we used the idea of Gefen et al. [5] ^{that one}

should have a universal self similar fractal structure

on short distance scales. As in our investigation ^of

diffusion [6] ^we then assumed that small finite clusters have a fractal structure up ^to their radius (rs çp)

and that the infinite cluster can be described by ^a generalized Skal-Shklovskii-de Gennes [7] ^{model with}

fractals replacing ^the one-dimensional links. We have

applied ^{similar ideas to the} density ^{of states} [8].

The purpose of the present paper is ^to supplement

the results of I by studying solutions of the London

equations taking proper ^account of flux quantization

and critical current effects. As an approach ^to super-

conductivity this has considerable conceptual ^advan- tages. It also allows us to calculate the magnetic

energy and the supercurrent distribution. Our approach ^{is therefore a} generalization of that sug-

gested by ^{de Gennes} [3] ^{for the} susceptibility ^and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004407080500

applied by Stephen [9] ^{to the} Sierpinski gasket. ^In applying the results to percolation ^{we use the same}

model as in I and in references [6, 8].

As in I the emphasis ^{of our} discussion is on the

geometrical properties of the networks or supercon-

ducting ^{clusters. We therefore} completely neglect

inductive effect and assume an infinite London

penetration depth. Formally ^{this is} always justified

if the wires are thin enough. ^{We also} neglect ^all

effects of superconducting fluctuations and hysteresis

effects associated with the formation of the minimum energy vortex structures. We do not claim that these

assumptions ^are necessarily valid in all relevant

experimental situations.

In I we first developed ^a convenient algorithm

for handling ^the linearized Landau Ginzburg equa- tions on networks. We then solved the equations

on a regular ^lattice (the square net) ^{and on a self}

similar fractal (the Sierpinski gasket) ^{and then} applied

the results to percolation. This allowed us to calculate the critical fields far from the percolation ^threshold.

We did however run into difficulties in the critical

region when the characteristic length ^for super-

conducting correlations on the fractal

becomes smaller than the connectivity correlation

length (C;p).

The present paper follows a similar pattern. ^In section 2 we rederive the London equations ^{for a}

net as the constant amplitude limit of the Landau

Ginzburg equations. ^These equations ^{have been used} by ^{de Gennes} [3] ^and by Stephen [9] ^to calculate the

zero field magnetic susceptibility. Our gauge invariant formulation allows us to extend these results to high

fields.

In section 3 we expand the Landau Ginzburg equations around the constant amplitude ^solutions

and obtain the leading order corrections to the

amplitudes ^{and currents. The main} result of this

expansion ^{is to} show that the amplitude ^deviations

on a net obey equations ^{with the same structure as}

those found in I and must therefore scale the same

way.

In section 4 we solve the equations ^{for a} square net. As for the bulk the magnetic energy is found to

be linear in the field above a size dependent ^threshold.

The results for the critical field agree with those found in I and with the more recent results of Rammal

et al. [11]. ^{We find two} regimes depending ^{on the ratio}

of the Landau Ginzburg ^coherence length ^{to the}

lattice spacing. When this ratio is small a vortex core can be accommodated inside the holes of the net and no normal core appears. The upper critical field is that of the wires. When the coherence length

is large ^{we find lines of normal links.}

In section 5 we discuss the triangular Sierpinski gasket [5]. The low field susceptibility ^{was calculated}

Fig. ^1.

^A Sierpinski gasket ^{with N}

3. We indicate the notation used for the fluxes (1’ n ; Yn) ^{and the} junction points An ^at which the supercurrents associated with the flux yn-1

+ yn-1

is concentrated. These

the points ^where

one

would expect disruption if the critical currents

exceeded.

by Stephen [9] ^{and found to} be dominated by ^the largest loops ^as conjectured by ^{de Gennes} [3]. ^We rely heavily ^on Stephen’s procedure but include the effects of flux quantization ^{and critical currents.}

At high fields the magnetic ^{energy is dominated} by loops enclosing ^{one flux} quantum ^{and has an ano-} malous _power law dependence. We confirm the results of I for the critical fields for small gaskets.

For large gaskets [larger ^{than A (Eq.} (1.1))] ^{the criti-}

cal field is found to be size independent ^and equal

to that of regions of size A. The gasket retains super-

conducting ^coherence up ^to this field, contrary ^to suggestions made in I. This results from the fact that vortices on the gasket can never develop ^a

.normal core and also do not exceed local critical currents. As a result the process which drives them normal is a local one and is associated with small

loops ^{on the scale of the} amplitude correlation

length ^A.

In section 6 ^we apply the results to percolation.

For finite clusters this involves replacing ^the gasket

indices by the relevant percolation ^{indices in the} scaling expressions derived. We discuss the justification ^{of this} procedure ⁱⁿ appendix ^{B. We derive} explicit expres- sions for the experimentally observable magnetic

energy by suitably integrating ^{over the Stauffer} [12]

cluster distribution. The resulting phase diagram (Fig. 2) ^is fairly complex ^{and involves several crossover}

lines between different regimes. ^{There are several}

novel features to the results leading ^{to a} fairly complex phase diagram (Fig. 2). Below Pc ^{the low field sus-} ceptibility ^{is found to} diverge ^{with the} conductivity

index (t) (rather ^{than (t}

p)). ^{Close to} Pc(çp > A)

there is a crossover to an anomalous _p independent

flux quantization regime (f(H ) ^oc l/ç: HI +t/2v). ^For

sufficiently high ^fields (H > Hc(çp)) ^only small clusters

Fig. ^2.

^Phase diagram ^for

percolation system

function of magnetic ^{field and} (p - Pc) ^{for constant} super-

conducting ^coherence length çs. The solid line is the critical field for the infinite cluster (Hc2

Eqs. (6.23) ^and (6.27)).

The dashed line describes the critical field for finite clusters of size çp [Hc(çp)

^Eq. (6. 2)] ^{and the} dashed-dotted line the

field for flux quantization (H ocØo/ç; ^{Lf (H)} çp). ^{The two vertical} dashed lines indicate the value of P - Pc for which çp

^A(oc 03BEs2/2+03B8). The different regions

in the phase diagram

^labelled

in section 6 and

discussed there.

2#

survive and f (H) oc çs-4(Hçs)2v+P-t. Above P,,, ^we

confirm the results of I for the critical field H,, ,2*

The magnetic energy is linear in the field ^at low fields and crosses over to an anomalous power law close to pr. We also derive expressions ^{for the} magnetic

energy due ^to finite clusters above Hc2.

In section 7 we briefly discuss the results and com-

pare them to available experiments [2] ^{and to other}

theoretical predictions [13].

2. The London equations ^{for a net.}

As in I and in reference [4], ^{we are} interested in solu- tions of the Landau Ginzburg equations

on a network of thin wires. s is a running ^coordinate along ^{the wire}

where A 11 is the component ^{of the vector} potential tangential ^{to the wire at} point ^s.

At the junctions ^{of the net one has the} boundary

conditions [1, 3, 4] ^{on the} amplitudes

and on the derivatives

where the summation is ^over all outgoing ^strands-

connected at i.

We assume a constant amplitude :

Then from equation (2 .1 ) ^the supercurrent through

the strand (ij ) ^is proportional ^to

and is independent of s, ^{if the cross} section is constant.

Integrating equation (2.6) along the strand gives

where Iii ^{is the length} of the strand

and Oi ^and ø j ^{are the phases at the} respective junctions.

We also note that consistency ^with equation (2 .1 ) requires

Equation (2.4) ^{now becomes}

at the junctions.

The _presence of a magnetic ^{field can be} expressed

in a gauge invariant form by the London loop ^condi-

tion

where the summation is over the circumference of all elementary plackets (l) ^{of the network. As usual}

where 0, is the flux through ^the placket, 4lo is the flux quantum, and n, ^{is an} integer.

Equations (2.10) ^and (2.11) ^{can now be solved}

for the ^« currents » rJ.u. This has been used by ^de

Gennes [3] ^and by Stephen [9] ^to calculate the zero

field susceptibility.

We can also write down the contribution of the currents to the Landau Ginzburg free energy ^to this order

where we have used equation (2.7). ^{Since the} ampli- tude do 12 ^{comes in as a trivial factor in} equa- tion (2.13) ^we omit it in most of our discussion below.

Similarily ^{we shall} frequently ^{refer to the} aij ^as

« supercurrents » ^{or « currents ». This is a convenient}

terminology. ^{It is of course} obvious that the true

expression ^{for the} supercurrent ^{in a strand is}

where S is the cross-section of the wire. This would be important ^{if we wanted to calculate} screening

effects (i.e. ^the penetration depth).

It follows from the Euler relationships ^that equa- tions (2. 10) ^and (2.11) determine the aij uniquely

for _any finite geometry. ^{Thus the} only ^{freedom one}

has in minimizing dF is in the choice of the inte- gers ^n. This determines a vortex structure.

In addition there is the effect of the aii ^{on the} amplitudes Air ^{Since a vortex on a net does not}

have a proper ^{core one} does not necessarily ^have

normal regions. It is however obvious that when the au vary, the amplitude ^{cannot be constant} every- where. This becomes important ^{when a critical}

current can be exceeded locally. ^{For a} (sufficiently thin) single strand this occurs when

but a correct description ^{on a net} requires ^{the cohe-}

rence length ^{for the} amplitudes ^{on the net. We shall}

derive the relevant equations ⁱⁿ section 3. We actually

find in section 5 that equation (2.15) ^{is not even} qualitatively ^{correct as a critical current condition}

on a fractal.

There is an interesting difference between a net and the continuum. There are two distinct situations

depending ^on the ratio of the superconducting ^cohe-

rence length ^to the mesh size of the net. When the ratio is small vortices have no normal core and the net cannot be driven normal by ^a magnetic ^field.

The state of the system is then described by ^{a suitable}

distribution of vortex cores in the holes of the net

which minimizes the free energy (Eq. (2.13)). ^When Çs ^is large ^{one has a} qualitatively ^different regime

in which the critical current is exceeded locally ^so

that some strands are driven normal corresponding

to the normal vortex cores in the continuum.

3. Amplitudes ^{and currents.}

We want to treat fluctuations ^{on a} net in the same way

one treats fluctuations in the continuum, ^i.e. by expanding around the constant amplitude ^London

solutions.

Consider first a single ^strand connecting junctions ⁱ and j. ^{We write}

and expand ^{in ð1.} Consistency with section 2 requires

for the amplitudes (Eq. (2.3))

To leading ^{order one has} (Eq. (2.9))

where the a0ij are determined from equations (2.10)

and (2.11). ^To first order in b1 one obtains from (2.1)

and (3.4)

The explicit solutions of equation (3. 5) ^are given ⁱⁿ appendix A. To obtain the network equations ^{we have}

to use these solutions in equation (2.4). ^In general ^the

ai and therefore the AP ^{will be} different for different strands. ^{We define}

where d o ^{is a} suitably ^chosen average network ampli-

tude so that the 6ij ^{and the} 6i ^{are all} small. In the

following ^we restrict ourselves to small currents and choose

This expansion ^{thus assumes} (a 03BES)2 ^{« 1} (Eq. (2.9)).

One obtains to leading order in this parameter

and

J

where

and

One notes that the right hand side of equation (3 . 8)

has the same structure as the network equations

derived for the linearized Landau Ginzburg equations

in equation (2. .11 ) ^{of I. Thus the coherence} length

must renormalize in the same way. The left hand side of this equation ^{is the} inhomogeneous driving ^{term for} amplitude changes ^{due to the} ai. ^Thus equation (3.8)

determines the amplitude deviations (bi) ^induced.

Equation (3 . 9) gives ^the corrections to the aij(a1ij) ^{to the}

same order.

The fact that equation (3.8) displays ^{the structure}

one finds for linear problems ^{on a network seems} intuitively ^{obvious. It does however have} important implications. ^{It allows us to take over} the detailed results of the discussions in I as to the role of deadends and fractal properties. As in the situations discussed there such features of a specific ^{fractal network show}

up only through ^{their effect on} the fractal dimensio-

nally d and diffusion index 0.

We only ^{note that for} simple geometries ^{one can}

somewhat extend the range of the validity ^{of the}

formalism by ^{a better choice of} Jo’ ^{This is not} possible

in the situations we want to discuss below.

4. The square ^net.

It is instructive to apply these considerations to a

square net, and compare the results ^to those obtained from the linearized equations ^{in I and more} recently

in the detailed study of Rammal et al. [11] ^{and Simonin}

et a1. [lla]. Equation (2.10) ^{at the} junction (mn)

becomes

and from (2 .11 )

where a is the lattice spacing, I ^an integer, ^and

y

2 n(Ha’/q6o) ^{is the} proportional ^to the flux per square.

If we take l = 0 a convenient solution is

This corresponds, ⁱⁿ essence, to ^a Landau _{gauge :}

It is obvious that solutions of this type only ^make

sense for finite nets and at sufficiently ^{low fields. In} particular ^{for a} strip ^{of width L} along ^x [11] the energy per site becomes, ^from equation (2.13),

Thus the magnetic energy is quadratic ^{in the} magnetic

field (y ^oc H ), ^but diverges with the square of the strip

width (L). ^A convenient _way of expressing ^this [3] ^is

in terms of an effective loop ^area (Seff), ^{i.e. the cross-}

sectional area of a simple (say circular) loop giving ^the

same energy per unit length. Equation (4.6) ^{is then} equivalent ^to

The equivalent loop would have the size of the sample.

When L is large ^it becomes favourable to introduce vortices (l = 1) ^{with an} average spacing (wa) ^where

This minimizes the magnetic energy.

When w L, ^i.e. for high fields, ^the magnetic

energy per site becomes

which is lower than (4.6). Substituting ^the explicit

form of w (Eq. (4.7)) ^{in (4.9), one obtains}

Thus the magnetic energy becomes linear in H rather than quadratic ^{as soon as}

Thus flux quantization ^{leads to a} very dramatic decrease in the magnetic energy ^as it does for the bulk.

We note that this reduction ^occurs here purely by ^flux quantization without any normal ^cores (which ^domi-

nate the magnetic energy in the bulk).

Equation (4.9) ^is equivalent ^{to a held} dependent

effective area

The maximum current flows at the boundary ^{of the}

I

1 squares where it is approximately equal ^to

Consider now the effect on the amplitudes. ^{There are}

two regimes. If,s ^{is small}

the solution we have derived does not imply any normal strands. Thus it is qualitatively ^{correct. The}

effects on the amplitudes only implies ^small amplitude

modulations with reduced amplitudes ^{near the vortex}

cores. This holds for all fields because (4.14) ^{does not} depend ^on the field. Thus the net cannot be driven normal. We encountered this phenomenon ^{also in our}

discussion of the square ^net in reference [1].

These solutions are analogous ^to the continuum solutions except for the fact that the normal cores do not show up. To the order ^we have considered ^one is not sensitive to the detailed vortex configurations ^and only ^their density (w-1) ^is determined.

One runs into difficulties when (4.14) ^{does not} hold, ^{i.e. when}

Clearly, ^{one cannot have a} superconducting square with 1

1 if the a cannot exceed some small value

1/ ’s. ^{Flux must be} accomodated in some other way.

A new length shows up in the problem

One cannot traverse a closed path ^{with a linear dimen-}

sion larger ^{than s without} encountering ^{a normal link.}

The _presence of these normal regions allows the net

to accomodate flux continuously. For the solutions

we have considered they would be situated on strips (of ^width S) along y. The magnetic energy of the supercurrents ^becomes

when one replaces ^w by ^{s in} equation (4.9).

Equation (4.17a) neglects the contributions of the normal regions and their interaction to the _energy.

Each such region ^involves (ja links and their density

is s-1. Thus the contribution of J ^is

This is much larger ^{than the dominant field} dependent

term in (4.17a) (ya/2 çs) ^{and is} comparable ^{to the}

linear term in (4.10). Our estimates are clearly ^not good enough ^to give the coefficients correctly. ^{We note}

that equation (4.10) still differs from ^our large çs/a

result (11 ⁺ 12) in ^the value of the constant term. The critical field is now reached when

i.e.

in agreement with the results of I and with the conti-

nuum result.

A curious feature of this calculation is the fact that flux quantization ^{does not show up} explicitly. ^The

detailed form in which the flux is accomodated plays

no role to this order. The dominant field dependent

term in the energy j 2 (Eq. (4 .17b)) ^is nevertheless identical to that one would obtain by giving ^{each of}

the vortices (of density w-1 ) a ^{normal core of area} oc çs2. ^{One could} try ^to represent this situation by ^a

field dependent ^{effective area similar to} equation (4.12).

We do not see the point ⁱⁿ expressing ^{a vortex core}

energy ^{as an} effective loop ^area.

The lattice spacing (a) ^{is the} only parameter ^which shows up in ^our derivation. Actually ^it plays ^a triple

role here. First it determines the mesh size through

Its second role is ^{as an} effective bond « resistance » as

in equation (4.2)

where we use Stephen’s terminology [9] emphasizing

the equivalence of the London equations ^{to the Kir-}

choff equations ^with equivalent ^EMFs. Finally ⁱⁿ evaluating ^{critical currents we need the mass of} superconducting ^material being driven normal which is again proportional ^{to a. In} principle ^{these are three}

different quantities when the links have structure. We shall use this in applying the results of this section to the (Skal-Shklovskii-de ^Gennes [7]) ^{infinite cluster}

in section 6.

We have used a square ^net in this section. It is fairly

easy ^{to see} that the specific lattice is not important ^and

the results apply ^to general (d-dimensional) ^{nets with a}

basic Cartesian geometry.

5. The Sierpinski gasket.

This structure, introduced ^{to the} percolation problem by Gefen et al. [5] has proven instructive in getting ^some insight into the behaviour of amorphous systems because it is a fractal with a hierarchy ^of loops.

Stephen [9] has studied the low field magnetic suscep-

tibility. ^{We have studied} [1] ^{the structure of the} equa- tions and the _upper critical field using ^{the linearized}

LG equations. ^More recently, ^the complete eigenvalue spectrum ^{and the} eigenfunctions ^{in zero field were}

derived [14] and Rammal and Toulouse [15] ^obtained

some very interesting ^{results on the} eigenvalue spec- trum in magnetic ^fields.

We want to study ^the magnetic energy and the appearance of normal regions.

5.1 THE MAGNETIC ENERGY IN THE LONDON ^LIMIT.

We follow the procedure ^of Stephen [9] ^{which is}

based ^on the equivalence of equations (2. 10) ^and (2 .11 )

to the Kirchoff equations. ^Consider the effect of combi-

ning ^{three N - 1} stage gaskets ^{into an N} stage gasket.

As shown by Stephen (Eq. ^{(5) of Ref.} 9), ^{one has a}

recursion relation for the magnetic ^energy

where YN - , refers ^to each of the three N - 1 gaskets

and YN -1 ^{to the} empty ^central triangle (see Fig. 1).

This is a slight generalization. ^{When one considers}

flux quantization YN and yN ^{need not be equal. One}

has the iterative relationship

Since the only quantities which show up in equa- tion (5.1) ^{are the} combinations _y. + YM ^{we introduce}

flux quantization by writing :

where the Im ^are independent integers. The resistance

Rm ^is given by [5]

It is convenient to introduce the notation

where L. is the size of an m stage gasket ^and So

( _ ( 3/4) a2) ^{is the area of an} elementary triangle.

ho is the field for which the flux through ^an elementary triangle is 2 f/Jo. This ^is obviously ^the highest ^{field one}

has to consider. The indices introduced in 5.4 are the

fractal dimensionality [5]

and the diffusion index [1, 6, 8, 10]

The obvious generalization ^of Stephen’s [9] equa- tion (6) ^{is now}

where JN ^{is the} magnetic energy per link on an N stage gasket.

This reduces to Stephen’s ^result

when one chooses lm

^0. This is the best choice only

at very low fields. When H becomes larger ^than

it becomes favourable to choose 1m :F ^0. Thus f N ^is

the correct expression for fN only ^as long ^{as H is}

smaller than hN - 1. ^At higher ^{fields we can write}

where m is the smallest integer ^for which hm ^is larger

than H. One has lm

^{0 for all m m and} the lm ^can always ^{be chosen so that}

so that (from Eq. (5.8))

Thus, using (5.9) ^and (5.13) and the definition of m in

equation (5.11) ^one finds, ^to leading ^{order :}

where C is a constant (= 1). ^{This is} analogous ^{to the}

linear field dependence for the square ^net (Eq. (4.10))

where indeed 0=0. Equation (5.14) (or (5.11)) ^also

means that the magnetic energy ^at high ^{fields is not}

dominated by ^the largest loops. The dominant contri- butions came from loops ^{of size}

Thus, ^the gasket ^{behaves as} though ^one had smaller

independent gaskets ^{of size} L(H).

This is equivalent ^{to an effective area at low fields :}

and to a field dependent ^area

at high ^fields.

5.2 CRITICAL FIELDS AND CRITICAL CURRENTS.

We

can now determine the critical fields by comparing

the magnetic energies (Eqs. (5.9) ^or (5.14)) ^{to the}

difference between the normal and the superconduct- ing energy. Consider first the low field (H 4>0/L2)

situation (Eq. (5.9)). ^{One finds :}

which is identical to the result found in I (Eq. (6.38)).

We have seen however that this expression ^{for the}

energy is only ^{valid at} low fields (Eq. (5.10))

and therefore

We notice that A (Eq. (1.1)) ^{is the} renormalized correlation length ^on large gaskets introduced in I

(Eq. (6.41)). ^It arises here from a different argument.

So far our results have reproduced those found in I.

Consider ^now larger gaskets ^L > A. The magnetic

energy is given by equation (5.14) ^and

Thus, the critical field decreases with L (L ^Å., Eq. (5.17)) and then remains constant for all larger gaskets.

In I we noticed that the critical field given by equation (5.20) (Eq. ^{(6.46) of} I) ^{was an upper bound.}

It was there suggested that there might ^{be a lower}

field at which a large gasket ^{would break} up into smaller pieces by the appearance of normal regions ^at

the junction points (points An

Fig. 1). ^{We want to}

check if this can actually occur, i.e. if the loops ^can develop supercurrents ^{which are} sufficiently large ^to

exceed the critical current locally. ^{Since the current}

distribution on the gasket ^is highly non-uniform, ^we

need some information on the distribution of the air An explicit calculation [16] is feasible but cumber-

some. We again ^use Stephen’s electric circuit ana-

logy [9].

For a loop ^{of size} L, ^{the total} supercurrent ^is

This is obviously ^an upper bound ^on the contribution of the loop ^{to all} _(Xij. ^{At the} junction of the m - 1 stage gaskets (points Am - Fig. 1) ^{the current passes} through ^two specific ^{strands so that the} (Xij ^{for these} strands have to be comparable ^{to (Xm} (see Fig. 1).

This reflects the fact that the ramification number for the gasket ^{is constant} (independent ^{of L).}

We have to compare (Xm ^to the critical current

required ^to produce ^{a normal} region (at Am). Naively

one would be inclined to compare a"’ ^to the critical

current for a single ^strand (ljC;s) ^{as we} did for the

square lattice. This ^is however misleading ^{for a fractal.}

We have shown in section 3 that the amplitude ^fluc-

tuations obey equations, ^{on the net,} which have the

same structure as those one obtains from the linearized

equations. Thus the coherence length ^{for the} ampli-

tude is A ( oc 212+0

Eq. (1.1)) following ^{the same}

renormalization arguments ^as in section 6c of I.

This must be the size of the normal regions. ^{We there-}

fore need the critical current for a gasket of size À..

We write

and therefore

We notice that this result depends ^on the fractal nature both through ^the resistivity ^and through ^the

ramification number. For a Cartesian lattice one

would have R oc À.2-d and a current through ^a region

of size A - I oc CXÀ.d-l leading ^to ac ^oc II’s for all ^d.

Comparing equations (5.21) ^and (5.23) ^{one finds}

that _CXc is, ⁱⁿ essence, the maximum current (a.) ^{for a} gasket of size A. All larger loops have smaller values of am. Thus the critical ^current is never reached for a

large loop. ^A large gasket (L >> A) ^will exhibit the

magnetic energy of equation (5.14) up ^to the size

independent critical field H,(A) (Eq. (5.20)) where it is driven normal.

Contrary ^to the Cartesian case (section 4) ^{this is a} purely local mechanism. Large ^{vortices never have a}

normal core and the gasket is driven normal uniformly by ^the supercurrent, induced in loops ^{of size A.}

6. Percolation clusters.

We want to apply these results to the behaviour of

percolation clusters in magnetic fields. As noted in the introduction the underlying ^{idea is} [5, 6] ^that percola-

tion clusters should have a universal self similar structure on length scales small compared ^{to the} connectivity length (çp). ^{The behaviour} of the fractal

regime ^{is then assumed to} be similar to that of the

gasket. ^A straightforward scaling argument gives ^the

fractal dimensionality [5]

and the diffusion index [1, 6, 8, 10]

where t is the conductivity ^index ( Q ^oc ( p - p,y).

The loop ^{structure of a} gasket ^{seems rather} special.

We show in appendix ^{B that one is} justified ⁱⁿ applying

the gasket ^results with very plausible ^self similarity assumptions ^{on the structure of} percolation ^clusters.

For a single cluster of size rs ^the magnetic energy

becomes, ⁱⁿ analogy ^to equations (5.9) ^and (5.14) :

and

where we have added the factor lie;; ^(oc Ho ) ^{to indi-}

cate the temperature dependence. The critical field

Hc(rs) ^is given by (Eqs. (5.16) ^and (5.19))

where A is the correlation length ^for superconductivity

introduced in I

To calculate the magnetic energy below the perco- lation threshold (p Pc) ^{we have to average over the}

Stauffer cluster distribution [12]. ^As for the diffusion

problem [6] ^this changes the indices. It is also impor-

tant at high fields when only the small clusters survive.

There are three length scales in the problem, ^the per- colation coherence length (çp), the coherence length

for superconducting correlations (A - 212 ’ 0), ^{and a}

field dependent length ^scale L(H ) determined either

by ^flux quantization ^or by ^{critical current effects.}

This results in a rather complex diagram. ^We depict

this in figure ^{2 as a} function of P - Pc and I H I ^for

constant A (i.e. ^constant C;s).

Forp p, there are only finite clusters. The _magne- tic field determines two length scales. From flux

quantization ^{one has}

and from the critical field of small finite clusters

(Eq. (6. 4a))

The lines Lf(H) = çp (> ^{A) and} Lc(H) = p ( ^A)

are shown in figure ^2.

For sufficiently ^{low helds one has}

All clusters then have the Stephen magnetic energy

(Eq. (6. 3a)) ^and integrating ^over the cluster distri- bution [12] ^the magnetic energy per unit ^mass (or volume) ^{becomes :}

This holds in region ^{I of} figure ^2.

Close to p, there is ^a flux quantization regime ^{II :}

The magnetic energy for the larger ^clusters

(rs > Lf(H )) is determined by ^flux quantization ^and

has the form of equation (6. 3b). ^The magnetic energy becomes

Finally ⁱⁿ region ^III

and only clusters smaller than Lc(H) remain super-

conducting. ^The crossover occurs on the lines Lc(H) = 03BEp > A and Lf(H) = A 03BEp ^{i.e. always at the critical}

field for clusters of size 03BEp. ^{The magnetic energy is}

where we have added the normalization ^constants

(ho a) ^to emphasize ^{that the} energy is still lower than that of the normal state for all clusters (1/ ç4s ^{in these}

units).

We shall see below that regions ^{II and III also}

extend above PC.

For _p > Pc ^we invoke the results of section 4. We

replace the lattice spacing by çp ^{(y oc} Hç) ^{the bond}

resistance by R(çp) ^(oc ç;+O-d) and the bond mass by

p ^{In section 4 these were all} proportional ^{to the}

lattice spacing (a). ^{In the} London limit the energy

(per ^{unit mass} of the infinite cluster) becomes, ⁱⁿ analogy ^to equation (4.6),

We note that this is normalized to the mass of the infinite cluster (p - p,,:r. ^{If we want to normalize}

to the total mass or to unit volume of the sample ^we

would have an additional factor (p - Pc)-P ^i.e.

For a finite sample ^{one thus} predicts (m

L/çp)

for sufficiently ^low fields and çp ^{L. This size depen-}

dent regime ^{is not shown in} figure ^2.

From flux quantization ^{we have one} magnetic length ^scale (analogous ^{to w -} Eq. (4.8))

The area L}.C;p ^{encloses one flux quantum.}

The critical current for a fractal link of dimension

03BEp ( ^{À) (computed as} in 1 Eq. (5.22)) ^{is :}

For larger fractals the critical current is always (Eq. (5.23))

where we have used the result of reference [8] ^{that the}

fracton dimensionality ^is d = 2 d/2 + 9 = 4/3 ^for percolation. ^{However such} large ^{fractals are not}

driven normal uniformly ^at ac. Instead they ^break

up by the appearance of normal regions ^{of size A on}

the fractal at weak points ^of type ^{A in} figure ^{1. This is}

however irrelevant to our considerations because

a > cxc(À.) always implies ^a nonequilibrium ^distribu-

tion of vortices. For çp ^{A the critical current} (Eq. (6.18a)) ^{defines a second length}

One cannot have a continuous superconducting loop

with a linear dimension exceeding Lic(H ).

Consider first the situation at low fields close to Pc (region ^IVa

Fig. 2)

This is the London regime. ^{There are no normal} regions ^{and from} (6.15) ^{and in} analogy ^to (4.10)

On the line Lf(H) _ çp ^this crosses over to

region IIb. On this line one also has L f(H )

Lf(H).

One has

This is again ^{a London} regime ^{but the} loop ^struc-

ture is that of the fractal regime ^because Lf (H) C;p.

We also note that one has loops ^of this size also on

finite clusters which also contribute to the energy.

The magnetic energy has the same form as in region ^IIa (Eq. (6. 11)) :

but the whole infinite cluster is superconducting.

On the line Lf (H) = A the infinite cluster and all finite clusters larger than A, become normal. Thus

At higher ^fields only ^small superconducting ^clusters (rs L,(A)) survive and one enters region ^{III. The} magnetic energy is given by equation (6.13).

Further from Pc(çp ^{A) the} phase diagram ^looks

a little different.

At low fields one has region ^{IVb with a} finite density (L’(H) 03BEp)-1 of normal links. This regime ^{is best}

described in terms of a renormalized coherence length

already introduced in I and in reference [2]. ^{One has}

The energy of the magnetic ^{currents is} (Eq. (4.16))

and that of the normal regions

Thus since (Çpl À) ¹ the dominant field dependent energy f2 ^{has the same form as in} region ^IVa (Eq.

(6.21)). ^The only difference between these regimes ^is

the presence of normal lines (i.e. ^{normal vortex cores}

of radius 03BEs) ^{in IVb.}

At the field

the infinite cluster becomes normal as found in I and

suggested in reference [2].

One notes however that the field Hc2 is ^smaller

than the critical field for finite clusters of size C;p (Hc(çp) - ^Eq. (b . 4a)). One therefore crosses into

region ^V

In this region ^{one has} only finite clusters and the energy has the ^same form as in region ^{I and is} given by equation (6.9).

On the line Lc(H) = p- ^H

Hc(C;p) one crosses over to region ^III.

In I it was suggested ^{that a} large cluster would break up into smaller ^{ones at} high fields because normal regions would appear ^{at some} special points.

Our analysis ^{here does not} confirm this suggestion.

We found normal regions ^{on a} lattice in section 4 and for the infinite cluster in region ^IIb (C;p ^{A) but they}

never resulted in disrupting the fractals. The effects of flux quantization always dominate. It is ^never- theless of interest to check if this effect can ever occur

for fractals of the type ^we have considered (see Appen-

dix B). ^{We have seen that a} loop ^{of size L can} generate

a maximum supercurrent (Eq. (5.21)) :

Normal regions ^show up if am(L) ^{can become} larger

than (Eq. (5.23))

for L larger than A. The condition for this is

This does not occur for percolation (d z 4/3) ^{or for}

the Sierpinski gasket [8] (d z ^{2 In} 3/ln 5). ^{It would}

be interesting ^{to know if a fractal} obeying ^the inequa- lity (6 . 31 ) and the conditions of appendix ^{B is} possible.

7. Discussion.

We have tried to perform ^an explicit calculation for the behaviour of a percolating system ^{in a} magnetic

field. The results, ^as derived in section 6, ^{are rather} complex ^and may ^seem confusing. It, therefore, ^seems useful to summarize them here. We have assumed that a finite percolation ^cluster (rs C;p) ^{can be}

regarded ^{as a} fractal with a self similar hierarchy ^of loops. ^For low fields the magnetic energy is then

always quadratic ^{in the field} (H ) and in the amplitude

of the order parameter (oc 1/03BEs) and has the Ste-

phen [9] power law dependence on rs (Eq. (6.3a)). ^In

this regime the energy is dominated by ^the largest loop in the cluster (of ^size rs). For small clusters

(rs A) this holds until the cluster is driven normal at a size dependent critical field Hc(rs) (Eq. (b.4a)). ^For larger clusters there is an intermediate regime ⁱⁿ

which there are quantized vortices; the energy is dominated by ^smaller loops and becomes independent

of the cluster size (Eq. (6.3b)). The critical field at which the cluster _goes normal (Eq. (6.4b)) ^{is also} independent of the cluster size. The left hand side of

figure ^{2 is then} obtained when one takes into account the fact that one has a continuous distribution of cluster sizes up ^to C;p ^and integrates ^over the Stauffer distribution [12].

The new feature above pc is the presence of the infinite cluster. In the spirit ^{of the} Skal-Shklovskii- de Gennes model [7] ^we treated this as a regular (i.e.

not fractal) ^net with mesh size C;p. ^We treated the links of the net as fractals of size C;p ^{with a} hierarchy ^of subsidiary ^small loops. The low field magnetic energy is then linear in the field (above ^{a size} dependent threshold) ^{with a} coefficient which depends ^on C;p (Eqs. (6.21) ^and (6.26)). ^When C;p ^{is small} (C;p ^{A) a}

continuum picture ^is adequate. ^The supercurrents induced by ^{the field} give ^{rise to normal} regions (i.e.

the vortices have normal cores) and the critical field arises in the usual _way when the vortex density

becomes too high. Thus the infinite cluster behaves like a continuum with a renormalized coherence

length OC C;;fJ/2 C;s

^{Eq. (6. 24)). When} çp ^{is large} (çp > A) ^{the continuum} picture ^is misleading. ^The

mesh size is too large and the vortices ^can accomodate their cores in the holes of the net. Thus the vortices associated with the continuum (or regular net) ^{have no}

normal cores. Their contribution to the magnetic

energy only gives ^{rise to} Parks Little [18] oscillations

(see section 4 here and section 5 of I). ^{Once we have}

accomodated a flux quantum ^{in each hole} (H ^>

00/03BE2p) ^{the magnetic} energy is dominated by ^the

smaller loops ^{on the links} (region IIb). ^{The critical}

field (H,2) is the field at which the fractal links are

driven normal and becomes independent ^of çp. ^The

rest of the diagram ^{on the} right hand side of figure ²

results from the fact that we have included the finite clusters present.

Rammal et al. [13] ^have recently ^{tried to} apply scaling considerations to the present problem. They

obtain _very different results. In essence our approach

is also a scaling approach. ^It differs from a straight-

forward scaling ^{ansatz because we} explicitly ^attri-

bute the scaling (i.e. fractal) properties ^{to the short}

distance properties. ^{A similar} approach ^{has been}

used by ^{Gefen et ad.} [6] ⁱⁿ treating the diffusion pro- blem. The straightforward scaling expressions (e.g.

Eqs. (6.3) ^and (6.4)) ^{are then modified} by taking

into account the cluster distribution and the mass of the infinite cluster. While this reduces the symmetry ^of the expressions ^{one obtains we} believe it is essential.

Another difference between our results and those of reference [13] results from the fact that they ^{use the}

bare C;s, rather than A ^{as a} length ^{scale for} superconduc- tivity. ^{We believe our} results in I and here show clearly

that this is not justified. ^{We note that a} simple ^cross-

over argument for the coherence length ^at C;p (03BEs

p e2 S ^{= A =} C;:; C;p

^{A) gives the same result}

(Å. oc C;;/2+6). ^Finally there is the question of the role of inductive effects, ^i.e. of the fields due to the super- current distribution. These ^are explicitly ^excluded

in the calculation we have done. Thus the London

penetration depth is assumed infinite. The results of reference [13] suggest that somehow inductive effects

are implicitly ^included by ^the scaling considerations.

One notes that a purely geometric magnetic suscep-

tibility ^which depends ^on C;p ^{but not on C;s,} cannot

arise otherwise.

We note that the backbone mass index (P’) ^{ne r}

shows up in ^our calculation. This confirms the resul s of I and contradicts the speculations ^{on this} poi t

in references [2] ^and [9].

The most crucial experimental ^{test for our} predic-

tions would certainly ^{be a direct} measurement of the field dependence ^{of the} magnetization ^{which would} directly probe ^our predictions ^{for the} magneticl

energy. We would like ^to point ^{out that} hysteresis

effects could be important ⁱⁿ regions II and IV where

we have assumed ^an equilibrium ^vortex distribution.

The only specific experimental results with which

one can compare ^our prediction (or those of Ref. [12])

are those of Deutscher et al. [2]. There is clear qualita-

tive disagreement ^{in the} interesting regime ^{close to} pc.

We predict ^that H,2 ^{should be} independent of p - p,

in this regime (çp > A) ^{while the} experiments ^{show a}

critical divergence. ^{We note} that in reference [13]

Hc2 also becomes independent ^of p(çp > çs) ^{in the}

relevant regime. ^We believe this shows that somehow the model we have used does not describe the experi-

mental situation adequately.

Two obvious possibilities ^{seem to be ruled out.}

First one could have non-equilibrium ^{vortex distribu-}

tions (the ^{fields were} applied ^to superconducting samples). This could be important ⁱⁿ measurements of the magnetic ^{energy but could not} possibly ^{lead to a}

measured Hc2 higher than the real one.

A second possibility ^{is that} superconducting ^fluc-

tuations become important. ^{Close to} pc(03BEp >> ^A)

superconducting vortices have no normal cores.

In other situations the superconducting ^core energies

are very high and this tends to suppress fluctuations.

Here they ^should presumably be observable and it would be interesting ^{to look for a} Kosterlitz Thoulless

regime ^close to pc(03BEp > ^{A). We do not} believe that this

played any role in the experiments of Deutscher

et al. [2]. They used the standard ^mean field result

and the plots ^of H c2 (T) ^{versus T show no} signs ^of

deviations from mean field behaviour. We therefore have no explanation ^{for the} discrepancies.

Acknowledgments.

The authors would like to thank R. Rammal, ^{T. C.}

Lubensky, and G. Toulouse for making preprints ^of

their work available and for their interest in and

skepticism ^{as to} the results of I which led them to try the alternative approach described here. We also thank M. Stephen ^{for a} preprint of reference [9]. ^One

of us (S.A.) would also like to thank R. Rammal and G. Toulouse for discussions which clarified the

relationship between reference [13] ^{and the} present work. Discussions with G. Deutscher, ^A. Aharony,

Y. Gefen, ^R. Orbach, ^P. Chaikin, ^{and F. Zawadowski}

are gratefully acknowledged.

This work was supported ⁱⁿ part by ^{the U.S. Natio-}

nal Science Foundation, ^contract number 4-444024- 21944 and grant ^number PHY77-27084.

Appendix ^A.

We write

It is convenient to define

and