high tc ceramics: model of granular superconductors

Applied Superconductivity Vol.1. NOS 7-9, pp. 985 - 994, 1993 Printed in Great Britain. All rights reserved

0964-1807/93 $6.00 + 0.00

Copynght @ 1993 Pergamon Press Ltd

HIGH T, CERAMICS: MODEL OF GRANULAR SUPERCONDUCTORS

A. Gerber

Van der Waals - Zeeman Laboratorium,

Universiteit van Amsterdam,

Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

Behaviour of weakly coupled high T, ceramics is compared with that of conventional granular

superconductors in the vicinity of metal-insulator transition. Copper-oxide materials present

a number of distinctive features that make them especially attractive for future study of

granularity.

A granular superconductor is usually’ characterized by two parameters. The first is

the grain’s size, on which the single grain properties depend and the second is the

barrier between the grains. Both parameters, the grain’s size and the barrier, determine

the Josephson coupling Ej. In the simplest approach, the macroscopic superconductivity

is achieved through the intergranular coupling when Ej exceeds the thermal energy of

the order of k,T. Since the coupling energy depends on the temperature, more and

more grains are coupled together as the temperature is lowered. Such mechanism is a

percolation process, at certain temperature an infinite cluster of coupled

superconducting grains is formed. Then there exists a superconducting path throughout

the sample and macroscopic superconductivity is established (one should mention that

at the percolation threshold, the superconducting volume included in the infinite cluster

is vanishingly small. The offset of the electrical transition corresponds roughly to the

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986 World Congress on Superconductivity

onset of the transition measured by ac susceptibility’ or specific heat’). When the

coupling between grains is strong (the sample’s normal state resistivity is small), the

grains become coupled immediately after the appearance of superconductivity in the

grains. The ~ndomness in the coupling energy is brought about through the

dependence on the grain size of the temperature at which the grains become

superconducting. The problem to be discussed is, therefore, a site-percolation.

When the coupling between grains is weak (the normal state resistivity higl$ one can

ignore the differences in the sizes and consider only the dist~bution of junctions

resistances, which is a bond-~rcolation problem’. The assumption that a junction

becomes connected once Ej exceeds k,T means that for any value of the sample’s

normal state resisitance there would always be a temperature below which macroscopic

su~rconductivity appears. However, when the metal consentration is low, the grains

are small and the charging energy increases. Once the latter exceeds about 1OEj

macroscopic superconductivity disappears. Instead, one finds a system of individual

superconducting grams with fully suppressed Josephson coupling, when an intergranular

charge transfer is governed by quasiparticle tunneling. Tntragranular energy gap plays

a role similar to that of dielectric one in semiconductors and macroscopic resistivity of

the sample can be idealized as thermally activated semiconductor-like: R cc

exp(A(O)/kT). Similar situation is met in random films below the percolation threshold.

Two types of intergranular charge transfer are qualitatively different. When

interg~ul~ charge transfer is governed by Josephson coupling the sample’s

superconducting - to normal state transition is characterized by positive derivative

values of dR/dT > 0 and dR/dH > 0. On the other hand, supression of the

intragranular energy gap by an applied magnetic field gives rise to negative

magnetoresistance in the case of the quasiparticle tunneling. Transition to the normal

state is then character&d by negative derivative values dRfdT < 0; dR/dH < 0. (In

fact, I-V characteristics in this case are strongly non-linear and dynamic resisitvity is

voltage dependent’). An interplay between Josephson and quasiparticle tunneling in

weakly coupled granular systems can explain the so-called quasireentrant transitiot?,

in which the resistivity decreases to a non-zero minimum and then rapidly increases at

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lower tem~ratures. The resistivity decrease is connected to the creation of a finite

cluster of Josephson couplings across closely separated grains. Since this cluster can

not short the entire system even at T=O, the minimum in R(T) can be due to an

increase in the resistance of unshorted part of the system with temperature activated

conductivity. When magnetic field is applied, the sample’s resistivity increases in the

temperature range where conductance has been dominated by Josephson tunneling and

decreases at lower temperatures, where conductance has been dominated by

quasiparticle tunneling5. This behaviour is illustrated in Fig.1.

3 I I I

0

+ + 0

++0 ++

e +

0 $p-+ %b,

@In 0

e+ 8

I I I I 6 7 8 9

-UK)

Fig. 1. Resistance of a weakly coupled Pb film as a function of temperature measured at zero

(0) and high ( f ) field. Reentrant transition is observed at zero applied field.

In granular systems these spectacular effects are observed in the vicinity of metal -

insulator transition, when metallic grams are usually small (of the order of 100 A and

less), the spacings between them are large and an intragranular contribution to the total

sample’s resistance is negligible. Variation of resistivity during the superconducting

transition indicates the state of intergranular coupling and provides only an indirect


information on the intragranular superconductivity.

Description of high temperature superconducting ceramics can be based on the same

principles as the conventional superconductors but with a number of important

modifications. The main differences are originated from a very short intrinsic

coherence length7, topological structure and chemical inhomogeneity.

Short coherence length is the main factor limiting the intergranular coupling and is

widely discussed in literature. Josephson coupling is easily suppressed by applied

magnetic field or passing current.

Typical topology of high Tc ceramics is quite different from conventional granular

systems in the vicinity of metal - insulator transition. The grains are usually large (in

the range 1 - 10 pm) tightly pressed one to another. The surfaces of the neighbouring

grains are smoothly adjusted, and the material fills almost all the volume. The

intergranular spacings are narrow and of approximately constant width, which enables

the creation of multiple parallel links between the neighbouring grains. This defines

two important properties: a) the normal resistance of the grains themselves can be of

the same order of magnitude as the intergranular or total sample’s resistance; b)

charging effects of insulated large grains are minimized.

Randomness of intragranular properties in conventional superconductors is connected

to their small dimensions. In high Tc superconductors the grains’ properties depend on

their chemical composition. The desired chemical homogeneity is, however, difficult

to be reached. As a result different grains can have different critical temperatures, their

distribution depending on chemical treatment.

Combination of these parameters gives rise to unique nonmonotonic double-peak

resistivity transitions’ as shown in Fig.2. The nonmonotonic transition is developed

under moderate magnetic fields when with temperatura reduction, the sample’s

resistance first increases, then drops down to a nonzero minimum, increases again,

and at last falls down at lower temperatures. We divide the curve into two temperature

intervals: above and below Tmi,, temperature at which the local minimum resistance

R,,,i, is reached We argue that the behaviour above this temperature, including the

enhancement of resistance below Tcp and its drop to R,i”, which was found to be


weakly de~ndent on v~ation of m~su~ng current and low applied field, is

determined by the intragranular superconducting transitions mainly and does not involve

the creation of intergranular Josephson couplings. The low-temperature part of the

transition (below T,& which is highly sensitive to the measuring conditions, is

governed by the interplay between Josephson and quasipa~icle tunneling between the

superconducting grains. Assuming an inhomogen~us chemical treatment and,

therefore, distribution of critical temperatures within different grains, the first grains

become superconducting at temperature T,,.

R WX I I , Sm-Cc cu.0 yzo.03

. ..-. _.,. : ‘..

C ‘...

._

‘.. Twit ._ T,e .,I.. __ I,..---._ 1 ‘..,_ ‘?.__

:,‘)-‘... -.:_, .L

‘. . . . . ‘x. Tc, : .,, “..,,:.,..

: ,., : ‘...Z!!::. I

‘,..’ ““W . . . . .._..... __

t TillllI

2000

a - H-O

b - H=i.B kOe

c - H-60 kOe

Fig.2. Resistance of SmCeCuO sample as a function of temperature under different applied

magnetic fields. Double peak transition is pronounced under t-i = 1.8kOe.

The appearance of su~rconducting grains between the normal ones can give two

opposite contributions to the total sample’s resistance. The total intragranular resistance

decreases, however, the intergranular resistance between the neighbouring

superconducting and normal grains is now governed by quasiparticle tunneling and,


therefore, increases. The pr~ominance of this m~hanism can explain the enh~~ment

of the resistance below TCg and the strong negative magnetoresistance in the temperature

range T, < T < T,. When the main part of the grains become superconducting, the

relative balance between the intra- and intergranular contributions can change, and

resistance drops at an effective onset temperature. At zero applied magnetic field the

su~rconducting transition in the grains is associated with the creation of percolating

chain of Josephson intergranular couplings, and zero resistance is achieved. Application

of a low magnetic field (or increase in the measuring current) does not have much

effect on the intragranular transition; however, it strongly reduces the Josephson

coupling tem~~tures. The resistance drop in this case, reflects the su~rconducting

transition of the grains alone. At temperatures below T,,,i the charge transfer is again

governed by quasiparticle tunneling between superconducting grains. The total

resistance of the sample then steeply increases with temperature reduction. In the

absence of charging effects the junction is expected to become Josephson coupled,

when its coupling energy Ej(l3) exceeds thermal energy level k,T. In the discussed

system this transition from quasiparticle - to - Josephson coupling is realized, and the

sample becomes macroscopically coupled (resistance drop at low temperatures) when

the temperature is reduced below an effective Josephson coupling temperature value

Ircjo-I>*

Similar nonmonotonic or double - step transitions can be observed in

magnetoresistance and I - V measurements. Two separated well defined transitions are

observed in each case, which enables precise definition of intra- and intergranular

critical fields and critical currents.

A number of important conclusions can be derived:

1) Anomalous nonmonotonic resistive transition observed in high temperture

superconducting ceramics can be qualitatively described in terms of intragranular and

intergr~ul~ transitions using Josephson and quasipa~icle tunneling m~hanisms. This

implies an existence of superconducting energy gap in high Tc superconductors, the

fact remaining controversial at the moment.

2) Intragranular and intergranular transitions can be separated and identified, and


intrinsic ~nt~g~nul~ ch~acte~stics of the rnate~a~ can be extracted.

3) Intragranular and intergranular normal state resistivities of high Tc ceramics are

comparable, however, their superconducting properties, like J,(H) or T,(H) are very

different.

This demands a special attention. In contrast to conventional su~rconductors in the

vicinity of metal-insulator transition Ir contributions of both intra- and intergranular

components have to be described simultaneously. We propose’, therefore, an H-T phase

diagram of granular superconductors based on the separate analyses of the intragranular

critical fields and of an assembly of Josephson junctions.

T2 Ti Tcj Tcg

Fig.3. Schematic H-T phase diagram of granular superconductor (solid lines). Dashed lines

correspond to the effective ~ntergranuiar critical field as a function of an applied magnetic

field.

For the bulk type-11 superconductor the H-T phase diagram consists of two

characteristic curves H,,(T) and H,,(T) emerging from zero at the same critical

temperature T,,. An effective critical field of an assembry of Josephson junctions can


be defined by assuming that every junction becomes superconducting when its coupling

energy Ej exceeds kT and remains proportional to the junction’s critical current under

applied magnetic field. The resulting H-T phase diagram of the granular superconductor

is schematically shown in Fig.3.

The basic difference between the temperature variation of the intragranular critical

fields and that of the assembly of the junctions is that the first two vary with a negative

second temperature derivative (d2H,1.2/dT2 < 0) with the finite values at zero

temperature H,,,,(O), whereas the second varies with d2H,/dT2 > 0 and asymptotically

increases when T -+ 0. This assures an interaction of the HEj~) curve with H&T) and

HC2(T) at temperatures T, and T2 respectively. The relative ratio between the inter- and

intragranular critical fields varies with temperature reduction, that leads to a different

response of the system to the applied fieId in three temperature ranges.

1) T, S T I Tcj : Hcj(T) < H,,(T). The intergranular Josephson-type tunneling is

suppressed before the first vortice enters inside the grains.

2) T2 S T I T, : Vortices start to enter inside the grains before the Josephson

tunneling between the grains is suppressed. For an assembly of junctions with a

distribution of I+ only a part of the Josephson junctions are destroyed when the field

has reached H,,.

3) T S T2. The sample is perfectly coupled.

The phase diagram presented in Fig.3 by the solid lines is based on the assumption

of a homogeneous distribution of the magnetic flux in all the volume of the sample.

This assumption is not met in granular su~rconductors in a low field limit. When

magnetic field is applied all the flux can be considered to be concentrated in the

intergranular regions and the effective local field there is much higher than the applied.

This intergranular flux accomodation breaks when local field exceeds H,,. The

additional vortices will accomodate themselves almost homogeneously all over the

sample’s volume. The physical state of the junctions will be influenced very little now

by enhancement of an applied field. As a result the effective critical temperature of the

assembly of intergranular junctions becomes a two-step function of the applied

magnetic field and is shown by a dashed line in Fig.3.

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Based on the diagram we can make several general predictions.

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1) The measured upper critical field is usually defined at the points where a certain

fraction (10% or 50%) of the normal-state resistance is recovered. For a sample

consisting of the mixture of strong and weak superconducting materials with

comparable values of the normal state resistances these definitions characterize mainly

the suppression of the weakest component. In the temperature range T2 < T 5 T,,

this relates mainly to the intergranular critical field H,j(T). A strong enhancement of

the measured critical field with a temperature reduction is therefore expected down to

the temperature T,.

2) The width of the R(T) transition under an applied magnetic field is equal to the

difference between the intragranular T,,(H) and intergranular Ti(H) critical

temperatures and is a nonmonotonic function of H. It has a minimum at H=O,

broadens with an increase of the applied field until some maximum value and reduces

to zero when H,,(T) curve intersects the H,j(T) one. The same arguments can be

applied to a magnetoresistance transition R(H) at different temperatures.

3) The critical current density J,(H) can be imagined as an analog of T,j(H) and

therefore is expected to be a two-step function of an applied magnetic field.

Commonly observed in high Tc superconducting ceramics enhancement of H,,(T)

with a temperature reduction, broadening of R(T) transition under applied field and a

double step critical current variation” can be qualitatively understood in this

framework.

To conclude, using the proposed phase diagram, a number of anomalous phenomena

observed in high temperature superconductors can be qualitatively described by the

interplay between intragranular and intergranular properties. New aspects of granularity

are found, including a transition from temperature activated quasiparticle tunneling to

macroscopic superconductivity with temperature reduction.

Experiments with e-doped ceramics and their interpretation have been worked out

in Laboratoire Louis Neel, CNRS Grenoble in collaboration with T.Grenet, M.Cyrot

and J.Beille to whom I am deeply indepted.


REFERENCES.

1. G.Deutscher, O.Entin-Wohlman, S.Fishman and Y.Shapira, Phys.Rev.B 21, 5041

(1980).

2. A.Gerber and G.Deutscher, Phys.Rev.B 35, 3214 (1987).

3. W.L.McLean and M.J.Stephen, Phys.Rev.B 19, 5925 (1979).

4. A.Gerber and G.Deutscher, Phys.Rev.Lett. 63, 1184 (1989).

5. A.Gerber, J.Phys.:Condens.Matter 2, 8161 (1990).

6. H.M.Jaeger, D.B.Haviland, A.M.Goldman and B.G.Orr, Phys.Rev.B 34, 4920

(1986).

7. G.Deutscher, in Early and Recent Aspects of Superconductivity, edited by

J.G.Bednorz and K.A.Muller (Springer-Verlag, Berlin, 1989), p. 174.

8. A.Gerber, T.Grenet, M.Cyrot and JBeille, Phys.Rev.Lett. 65, 3201 (1990).

9. A.Gerber, T.Grenet, M.Cyrot and J.Beille, Phys.Rev.B 45, 5099 (1992).

10. J.W.Ekin, T.M.Larson, A.M.Hermann, Z.Z.Sheng, K.Togano and H.Kumakura,

Physica C160, 489 (1989).

high tc ceramics: model of granular superconductors

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