Superconducting diamagnetism near the percolation threshold

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Superconducting diamagnetism near the percolation

threshold

R. Rammal, T.C. Lubensky, G. Toulouse

To cite this version:

R. Rammal, T.C. Lubensky, G. Toulouse.

Superconducting diamagnetism near the

per-colation threshold.

Journal de Physique Lettres, Edp sciences, 1983, 44 (2), pp.65-71.

�10.1051/jphyslet:0198300440206500�. �jpa-00232144�

L-65

Superconducting diamagnetism

near

the

percolation

threshold

R. Rammal

_(*),

T. C.

_{Lubensky (**)}

and G. Toulouse

Laboratoire de

_Physique

de l’ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France

(Re!Vu le 23 aoilt 1982, uccepte le 1 er decembre ₁₉₈₂₎

Résumé. 2014 On

présente des lois d’échelle pour la

susceptibilité diamagnétique

~,

et _{pour le champ}

critique supérieur.

Hc2, de _mélanges aléatoires de _composants

_{supraconducteurs}

et isolants. En dimension

_{deux, l’exposant critique qui apparaît}

dans cette théorie est relié à d’autres _exposants de

percolation. Les

_{prédictions qui}

en résultent pour

_Hc2

sont confrontées à des données

_{expérimentales}

sur des films InGe.

Abstract. 2014 A

scaling

theory

for the

_{diamagnetic susceptibility,}

_~,

_{and the upper critical} field,

Hc2,

of random mixtures of

_{superconducting}

and

_insulating

elements is

_presented.

In two dimensions,

the new critical _exponent in this _theory is related to other

_percolation

_exponents. The

_resulting

predictions

for

_Hc2

are

_compared

with

_experiments

on InGe films. J.

_Physique

-

LETTRES 44 _{(1983) L-65 -} L-71 15 JANVIER _1983,

Classification

Physics Abstracts

74.10 - 74.70 - 05.90

Consider a random

_{superconductor-insulator}

mixture. The

diamagnetic properties

of such a

mixture and their variation across the

_percolation

threshold

_(for

the

_{superconducting component)}

raise an

_interesting

_problem.

Far on the metallic side

[1]

(p >

p~),

the zero field

_{diamagnetic susceptibility}

has a

jump

at the

superconducting

transition

_temperature

_{To. (At}

this _stage, and

_later,

we

_ignore

_any fluctuation

effects : the

_{superconducting}

transition is treated within mean-field

_theory,

the

_{emphasis being}

on the

_geometrical

effects associated with

_disorder).

On the

_insulating

side

_(p

_p~),

the

suscepti-bility

increases

_linearly

with

_temperature

below

_{To (Fig. 1).}

One

_question

is : how does the

_system

evolve from one behaviour to the other ? Otherwise

_stated,

is the infinite

_percolation

cluster at

threshold

_(p

=

pc)

dense

enough

to

give

rise to _a Meissner effect ?

Questions

of this kind _were raised

by

de Gennes

_[2,

_3]

in connection with

_experiments

done in Tel-Aviv

_{[4], measuring}

the behaviour of the critical

_field,

_Hc2’

of InGe films in the

_vicinity

of

_To.

As is the case for the

sus-ceptibility,

there is a

problem

of crossover for the critical

field,

which is

_expected

to rise as a

function of

_{temperature, T,}

with finite

_slope

on the metallic side and with infinite

_slope

on the

insulating

side

_{(Fig. 2).}

The

_{slope, dHc2/dT,}

measured on the metallic

side,

was found to

_diverge,

for p -~ p~, as :

with k = 0.6 ± 0.05. Thus the

question

of the theoretical

origin

of this

exponent

arises.

(*) Permanent address : C.R.T.B.T., B.P. 166, 38042 _Grenoble, France.

(**) Permanent address : _Department of _{Physics, University} of _{Pennsylvania,}

_{Philadelphia,}

PA 19104,

U.S.A.

L-66 JOURNAL DE PYIYSIQUE - LETTRES

Fig.

1. -

Diamagnetic

susceptibility x

as a function of temperature T, for a very

large

sample : a) p pc

(finite clusters); b) p =

/?~ c) p > _{pc (bulk behaviour).}

Fig.

2. -

Upper critical field

_{H c2, in}

the same three _regimes as in

figure

1.

This paper proposes a theoretical

description

which

_attempts

to answer these various

_questions

in a

_{simple unifying}

_way. We

_present

predictions

for the two-dimensional

problem

which differ

rather

_sharply

from those made

_by

_previous

workers

_[4,

_5].

This is a rather nascent field

which,

it is fair to _say, calls for further

_experimental

and theoretical

_study

_(in

_{particular, concerning}

the three-dimensional

case).

1. The model. - We consider

a two-dimensional mixture of

_{superconducting}

and

_insulating

grains

in a

perpendicular magnetic

field.

Following

de Gennes

_[2],

we describe the

_{superconducting}

clusters as ramified networks of

wires with

_{elementary grain}

sizes

(which

determine the transverse width of the

_wires)

that are

smaller than the London

_{penetration length}

~. The

_{diamagnetic susceptibility}

_{per unit volume}

of

_{superconducting}

matter,

7,

of such a cluster can then be

_{expressed [2, 5]}

as :

where

seff,

the effective area, is a

_{geometrical quantity}

which contains information about the

topology

of the cluster.

Since

in the

_vicinity

of the transition

_temperature

_To,

varies as :

the

_{diamagnetic susceptibility}

of a finite cluster is

predicted

to increase

_linearly

with T below

_To.

If one considers now a collection of such finite

_clusters,

as exists on the

_insulating

side of a

percolation

threshold,

p p~, the total

diamagnetic susceptibility

may be taken as the sum of

the contributions of each

cluster,

_provided

one restricts one’s attention to

_temperatures

close

enough

to

_{T 0}

that the effects of

_dipolar

interactions between clusters remain

_neglibible.

Thus,

L-67 SUPERCONDUCTING MAGNETISM AT PERCOLATION

vary

linearly

in AT. The

question

is whether its

slope,

which is a function

_{of (Pc -}

_{p), diverges}

when _{p -~ pc’}

On the metallic side

_(p

>

_pc)

in finite

samples (linear length

_L),

the same

_argument

indicates

that :

the coefficient

being

a function of _p.

2.

_Scaling

laws. - We

can assemble all available information about this

problem by writing

a

_{scaling expression}

for

_x.

For this _purpose, we first

recognize

that there are at least three

_lengths

of

_importance

in this

_{problem :}

i)

the

_length

of the

_sample,

_L,

ii)

the

_{superconducting}

coherence and

_{penetration lengths, ~S}

and

₄

which both

_diverge

in the same manner

_(3),

as the

superconducting

transition

_temperature

is

approached,

iii)

the

_percolation

correlation

_length,

_~p,

which

_diverges

as the

percolation

threshold is

approached :

There is another

_{(microscopic) length, namely}

the

_grain

size a ; but in the

region

of the

phase

diagram

which is of interest here

_(vicinity

of the transition

_temperature

around

_percolation

threshold), a

is much smaller than the three

_{previously quoted lengths}

and _may be

_ignored.

With this in

_mind,

we

_hypothesize

that the

_{diamagnetic susceptibility,}

_{per unit volume of}

superconducting

matter,

X, obeys

a

_scaling

law of the form :

where 0

is some

_exponent.

Formula

₍₆₎

can

_equivalently

be written as :

where 0

=

v5.

This

scaling

law _assumes

simple

forms in the five

regions

of the

p-T

phase

diagram,

defined on

figure

3,

which are

separated by

crossover

_boundaries,

_{corresponding}

to

_equalities

of two out of the three basic

_{lengths. Boundary}

1

_{corresponds to ~ ~}

_C;p,

_boundary

2 to L -

_~S,

boundary

3 to L -

_~p.

Fig. 3. -

p-T

phase diagram

around the transition _{temperature (T} =

_To)

and the

_percolation

threshold

(p =

_pj

for a random

_{superconductor-insulator}

mixture of finite size L. Definitions for the

_regions

I to V and for the boundaries 1 to 3 are _given in the text.

L-68 JOURNAL DE _{PI-IYSIQUE -} LETTRES

In

_region

_I,

_{~p ~}

_(çs’

_L)

and

_X

should be

_independent

of L and linear in 4T because the _system is an

_assembly

of small clusters :

In

_region

IV,

L ~

_(~,

_~p)

and

_x

should be

_independent

_{of ~p}

and linear in AT :

In

_{region V, ~ ~>}

L >

_~p

and the behaviour

_{corresponding}

to

_{equation (4)}

holds :

In

_regions

II and

_III,

a bulk

regime

is reached and the L

dependence disappears. Region

II

corresponds

to ~ ~

(L,

_~p),

and x

is a function

_{of Çs}

_{(or AT) only :}

Finally, region

III

_corresponds

to

_{~p ~ ~}

L and

_x

is a function

_{of çp}

_(or

_{Ap) only :}

Our

_{scaling hypothesis}

for

_{X, equation}

_(7),

_plus

_{previously given}

_{(section 1)}

information for the

susceptibility

of finite

clusters,

has thus allowed us to reach a rather detailed set of

_predictions

for the behaviour

_{of x.}

All

_exponents

are

_expressed

in terms of one unknown

8.

The

_exponent

v is the

two-dimensional correlation

length

exponent,

which is now

_widely

believed to be

_exactly

_4/3

_[10].

The

_question

raised

_by

de Gennes

_[2]

_concerning

the

_divergence

of the

_{slope of x on}

the

insulat-ing

side

_~8)

amounts to

_asking

whether ~ is smaller or

_larger

than 2. In order to determine the value of

5,

we shall recall some recent results

_{[5-7] bearing}

on the size

dependences

of the

diama-gnetic

susceptibility, X,

and of the

resistance, R,

for a self-similar structure called the

Sierpinski

gasket.

It has been found that :

and

where

d

is the fractal

_{dimensionality}

of the

_gasket

and 6

_represents

the same number in

₍₁₃₎

and

_(14).

We now assume that relations

₍₁₃₎

and

_{(14), rigorously}

_proven

_only

for a

_{special family}

of

fractal

_objects,

have a more

_{general validity}

and still

hold,

in

_particular,

for the

_percolation

cluster at threshold which is also a fractal

_object.

This

_{generalization,}

which is not

_{unreasonable,}

and which has

_already

been assumed

_by

S. Alexander and R. Orbach

_[8]

in a different _context,

leads to

_predictions

which have the merit of

_being

testable

_{experimentally}

and

_numerically,

as

discussed later. In the context of

_percolation,

where,

here,

d = 2

and p

is the

percolating

cluster

density

_exponent.

Equation (14)

leads then to the relation :

L-69 SUPERCONDUCTING MAGNETISM AT PERCOLATION

Since p

and t are

_presently

determined with rather

_high

_accuracy

_[7,

_{9, 10] :}

this leads to the

_prediction

_(see

also Ref.

_{[8]) :}

Consequently,

the

_{slope of x}

on the

_insulating

_side,

_equation

₍₈₁

is

_predicted

to

_diverge

at the

percolation

threshold as

with 2 v - t +

_~3

~ 1.53. This is in

_sharp

contrast with the

_analysis

of reference

_{[5], concluding}

that this

_slope

would not

_diverge.

3.

_Comparison

with numerical or

_experimental

results. - As formulated

above,

the

_scaling

law for

_X

as a function of

_~p,

_Çs

and L can be tested

_numerically.

A

_quantity

which is

_directly

computable

is the effective area, S

eff,

in situations with finite size clusters or

_sample.

This

cor-responds

to

_regions

_{I, IV,}

V for which our

predictions

can be resummarized as follows :

i)

at p =

p~,

seff(pc)

reduces to a _power law function of the

_{sample length}

L :

ii)

in the

_region

p > _p~, and L >

_~p,

one has

iii)

below _p~,

_Seffrp)

is

_predicted

to show the

_following

behaviour :

Recent numerical

_{experiments [11], using}

de Gennes

_[2]

formula for

_Serf,

lend _strong

_support

to the above

_{predicted scaling}

laws. Monte Carlo

_{calculations,}

on finite _square lattice

_samples,

also

_permit

the value of

5

to be calculated. The numerical value so

obtained,

5

= 0.79 ±

0.02,

is in

_satisfactory

_agreement

with the

_predicted

value

_equation

_(18).

In

addition,

one observes

that the finite size effects close to _p~ are in

_good

accord with the

_scaling

laws.

From the behaviour of the

_{diamagnetic susceptibility,}

described in section

2,

the behaviour of

Hc2

can be derived. This leads to a second series of

_predictions

which can be

_compared

with

experimental

data,

some of which are

_already

available

_[4].

The

_{correspondence}

_{between x}

and

_{H c2}

is based on the relation :

which is a direct _consequence of a mean-field treatment of the

_{superconducting}

transition

_[1].

Such a relation is

_clearly

satisfied on both sides of the

_percolation

threshold

_(curves

a and c of

Figs. 1

and

_2).

It allows us to derive

_expressions

for

_{H c2}

in

_regions

_{I, II,}

III. We shall leave aside finite size effects since we want to make a

_comparison

with

_experiments,

which are

performed

on

large

size two-dimensional films.

L-70 JOURNAL DE PFIYSIQUE - LETTRES In

_region

_I,

one finds :

with an

_{Hc2 increasing}

with vertical

_slope,

for p p~. In

_region

II,

Hc2

is

_independent

of

_{Ap :}

In

_region

_III,

_{H c2}

has a bulk behaviour with a linear variation in AT :

The latter law

_provides

a

_prediction

for the

_exponent

k,

defined in the introduction :

Using

the values

_given

in section 2

for 3

and v,

this

_yields

a

prediction :

k ~ 0.57

₍₂₅₎

in

_satisfactory

_agreement

with the

_experimental

data of the Tel-Aviv group

[4].

In reference

_[4],

an estimate of k was

_given

_{whereby :}

where

is the backbone

_density

_exponent

_{[12]. Using}

the

_{presently accepted}

values

_{for t,}

_~8

and

/3’ (/3’

~

0.55),

this

yields k ~

0.87,

which is in less

good

accord with the

_existing

data

_[13].

It is

worth

_noting

that our

_theory

does not make

_explicit

use of the notion of the backbone of the

infinite

cluster,

in contrast with some of the other theories

_{[4, 5]}

of

_{magnetic superconducting}

properties.

Of course, it may be that there exists a relation between

exponent

and

_exponent

t, which would make the various theories more

_{directly comparable.}

4. Conclusion. - We have

presented

a

_{scaling theory}

for the

_magnetic

_properties

of random

mixtures of

_{superconducting}

_{and insulating}

elements. This

theory

involves the three

lengths

~

~p

and

L,

and one new

_exponent

3.

In two

_dimensions,

we _argue that

5

can be

_expressed

in terms

of the

_conductivity

and

_percolating

cluster

_{exponents, t}

and

_j8,

via

_{equation (16).}

The

_resulting

prediction

for the behaviour

_{ofd~2/~ ~}

the

_{superconducting}

side is in

_good

_agreement

with

experiments

on InGe films. Our

_{scaling hypothesis}

leads to other

_predictions

which varrant

experimental investigation.

In

_particular,

it would be desirable to have any data in three

dimen-sions.

_Furthermore,

_{investigations}

of the anomalous

_temperature

_dependence

of

_Hc2

near the

percolation

threshold in both two and three dimensions would be of interest. On the theoretical

side,

though

the

_{existence of}

a

scaling

law in three dimensions appears

_plausible,

it is not clear at

all whether the

_{exponent ~}

in this case is

_simply

related to other standard

_percolation

_exponents.

Our

_{analysis, despite}

its

_good

_agreement

with the

_existing

_data,

is

_{perhaps oversimplified.}

It may well be that further

experimental

and numerical

activity

will

require

a

_theory

of a more refined

level than appears useful

today.

Acknowledgments.

- We thank R. Orbach for

numerous

stimulating

discussions and for

making

available reference

_{[8] prior}

to

_publication.

We are also

_grateful

to J. C.

_Angles

_d’Auriac,

S. Alexander and J. Vannimenus for

_helpful

comments. One of us

(T.C.L.) acknowledges

financial

support from the John Simon

_Guggenheim

Foundation and from O.N.R. under contract N00014-0106 and N.S.F. under contract DMR 79-1013.

L-71 SUPERCONDUCTING MAGNETISM AT PERCOLATION

References

[1]

TINKHAM, M., Introduction to

_{Superconductivity (McGraw-Hill)}

1975.

[2]

DE GENNES, P. _G., C. R. Hebd. Séan. Acad. Sci., Série II 292 ₍₁₉₈₁₎ 9.

[3]

DE _GENNES, P. _G., C. R. Hebd. Séan. Acad. Sci., Série II 292 ₍₁₉₈₁₎ 279.

[4]

DEUTSCHER, G., GRAVE, I. and _ALEXANDER, S.,

Phys.

Rev. Lett. 48 ₍₁₉₈₂₎ 1497 ; DEUTSCHER, G.,

Lecture Notes in _Physics vol. 149

_(Springer)

1981, p. 26.

[5]

STEPHEN, M. J., Phys. Lett. 87A (1981) 67.

[6]

ALEXANDER, S., Jerusalem

_preprint

_(1981), submitted to J. Low _{Temp. Phys.}

[7]

GEFEN, Y., AHARONY, A., MANDELBROT, B. B. and _{KIRKPATRICK, S., Phys.} Rev. Lett. 47 ₍₁₉₈₁₎ 1771.

[8]

ALEXANDER, S. and _{ORBACH, R.,} J. _{Physique-Lett.} 43 ₍₁₉₈₂₎ L-625.

[9]

STAUFFER, D.,

Phys.

Rep. 54 ₍₁₉₇₉₎ 1.

[10]

DERRIDA, B. and VANNIMENUS, J., J.

_Phys.

A 15 (1982) L-557.

[11]

RAMMAL, R. and ANGLES D’AURIAC, J. _C., C.R.T.B.T. _{preprint (1982).}

[12]

KIRKPATRICK, S., in Ill-condensed Matter, Eds. R. Balian, R. _Maynard, G. Toulouse _{(North-Holland)} 1979.

[13]

In reference

[4],

a crossover in the _{behaviour of the critical field has been observed far from the} per-colation threshold, on the metallic side. This crossover lies _clearly outside the range of _validity

of our _{theory which,} in contrast with that of reference _[4], aims _only at

_describing

the _vicinity of the

percolation

threshold.