Superconducting glass-phase diagram for ceramic YBa2Cu 3O7-δ

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Submitted on 1 Jan 1989

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Superconducting glass-phase diagram for ceramic

YBa2Cu 3O7-δ

Z. Koziol, J. Piechota, H. Szymczak

To cite this version:

Superconducting glass-phase diagram

for ceramic

YBa2Cu3O7-03B4

Z.

Kozio0142,

J. Piechota and H.

_Szymczak

Institute of

Physics,

Polish

_Academy

of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland

(Reçu

le 30

janvier

1989, révisé le 19 mai 1989,

accepté

le 19

_juillet

₁₉₈₉₎

Résumé. 2014

Nous avons mesuré la

_{susceptibility complexe}

de

_ceramiques

_{YBa2Cu3O7-03B4}

_près

de la transition

_{supraconductrice}

en fonction des

_amplitudes

du

_champ

alternatif et d’un

champ

statique

superposé.

Du _comportement du maximum de

X",

nous avons déduit une loi d’échelle universelle pour la

dependance

en fonction du

_champ

_statique

ainsi que la

_dépendance

en

température

du courant

_Josephson

_intergrains

et les

_champs

_critiques

_H0

_{(où le}

champ

est

_piégé

par les

jonctions)

et

_{HJ (où}

_apparait

la

_{supraconductivité}

en

_volume).

Ces matériaux ont des

propriétés

qui

rappellent

celles des verres de

_spins

et sont en accord

_qualitatif

avec les

_predictions

de modèles pour les

supraconducteurs

désordonnés. Abstract. 2014 AC

complex

magnetic

susceptibility

of ceramic

_{YBa2Cu3O7-03B4}

was measured close to

the

_{superconducting}

transition _temperature, as a function of the

_amplitude

of the

_alternating

magnetic

field and of a

_superimposed

dc field. From the maxima of the

_imaginary

part of

susceptibility,

the universal

_scaling

of the dc field and the _temperature

_dependence

of the

intergrain Josephson

current were _obtained,

_together

with the critical fields :

_H0,

for

_trapping

the flux

_by

the

_junctions,

and

_HJ

for the occurrence of bulk

_{superconductivity.}

The results _suggest that these materials have

_{spin-glass-like properties,}

and are

_{qualitatively}

consistent with the

predictions

of models of disordered

superconductors.

Classification

Physics

Abstracts 74.50 - 75.10N

1. Introduction.

There is a

_large

amount of work devoted to

_applying

the ideas of

_{spin-glass theory}

to

_granular

classical

_{(low Tc)}

_{superconductors}

_{[1, 2]}

and to

_{high Tc superconductors}

_[3].

We have

investigated

the

_properties

of ceramic materials. The

_important

feature of these ceramics is disorder.

_{Superconducting}

_grains

of dimensions of the order of microns are

_{separated by}

nonsuperconducting

barriers. In the

_grains,

the critical

_supercurrent

can be as

_high

as

108

A/cM2 ,

and the

_slope

of the critical field

_HC2

is of the order of a Tesla per Kelvin near the transition

_temperature

_Tc.

There

is, however,

another transition

_temperature

_Tj

connected with the occurrence of bulk

_{superconductivity}

in the

_sample,

due to the interactions between

3124

the

_{grains through Josephson}

links. At low

_{temperatures,}

the interaction Hamiltonian is written in the form

_{[2] :}

where 0

i is the

phase

of the order

parameter

of the i-th

grain

with

and

where

_Iij

is the maximum

_Josephson

supercurrent

between the i-th and

_{j-th grains,}

d (T )

is the

_temperature

_dependent

_{energy gap,} p . a is the normal state resistance of the

junction

and a is the distance between the

_grains

_[4].

The

_{Ambegaokar-Baratoff}

formula

₍₂₎

can be

_applied

to the case of

superconductor-insulator-superconductor

(S-I-S)

structures

_[5].

_Then,

close to

_Tc,

one obtains a linear

temperature

dependence

of the critical

_{supercurrent,}

_IC:

_Ic

oc

_{(Tc - T).}

In the case of

_{granular superconductivity,}

a crossover to the

_{Ginzburg-Landau dependence,}

Ic

oc

_{(Tc -}

T)3/2,

should be observed in the

_vicinity

of

_Tc,

due to the

_suppression

of the order

parameter

through

the interaction between the

_grains

_[4].

The

_{Ginzburg-Landau theory}

should, however,

apply

away from the fluctuation

regime,

which is believed to be wide in the

high Tc

oxide

_{superconductors}

_[6].

The

_goal

of our work is to

_investigate

the critical

_region

properties measuring

the field and

_temperature

_dependence

of the critical

_Josephson

supercurrent.

Below

_Tj,

_{the process of}

_freezing

of the

_quantum

_phases

of the order

_parameter

of the

_{grains begins.}

The Hamiltonian

₍₁₎

is then not

_adequate

to describe the critical

_region

properties,

but some features described

_by

₍₁₎

are still valid. A

_magnetic

field introduces frustration into the

_system.

This means that below the

_{phase-locking}

temperature

Tj

the

_magnetic

flux can be frozen in the

_sample.

Frustration is the mechanism

_leading

to the

occurrence of

_{pinning energies. Experimentally,}

a remanence of the

_{magnetisation,}

time- and

history-dependent

effects,

are observed in these

_{superconductors}

for fields well below the first critical field for the

_grains

_[7].

Due to

_this,

we can

_speak

about the existence of a critical state

in low

_fields,

and at

_temperatures

below

_TJ.

Further,

we

_apply

Bean’s critical state

_theory,

in the version

_{developed by}

Goldfarb

_[8],

to obtain the critical

_intergrain

_supercurrent

from results on ac

_{susceptibility.}

This method was used

_by

other authors

_[9],

and in our earlier work

[10].

In the case of a

_regular

network of

_exactly

identical

_grains,

the Hamiltonian of the

system

is

_periodic

in the

field,

and all measurable

_properties,

such as

_Tj,

should be also

periodic

in the field

_[2].

Because of the frustration caused

_by

disorder and the

_magnetic

field,

this

_periodicity

is

_destroyed,

and

_Tj

becomes

_{smoothly decreasing}

with the

_applied

field

_[2,

11].

As

_large

fluctuations of the order

_parameter

exist above

_Tj,

critical

_{supercurrents}

cannot

be measured at these

_{temperatures.}

_{TJ (H)}

is the border line between the normal state and bulk

_{superconductivity,}

and can be determined from the occurrence of a small

_supercurrent

in the

_sample.

Results on

Ic(T, H)

from the ac method are

_compared

with a Fraunhofer diffraction-like

_{dependence, giving}

values of the characteristic field for flux

_{trapping by}

the

intergrain junctions,

Ho(T),

and the Peterson and Ekin

_[12]

_{scaling dependence}

of

2.

_{Expérimental}

details.

Ceramic

_samples

3 mm in diameter and 5 mm

_long

were obtained

_by

the standard solid-state reaction method

_[13].

An ac

_bridge

with two identical coils and AT resistors

_(with

a _{very small}

temperature

coefficient of their

_resistances)

was immersed in a

_glass

_cryostat,

with the

sample

placed

in the middle of one of the coils. We were able to cause slow

_temperature

runs

_(about

1 K _per

_min)

of the

_sample.

The

_temperature

was monitored with a relative accuracy of about 0.1

K,

using

a calibrated carbon resistor. The

_complex

magnetic susceptibility

was measured

at 7 kHz

_using

two lock-ins. The

_experimental

data were collected

_by

a

_{microcomputer}

connected

_through

an

interface,

_working

with voltmeters in IEC-625 standard. It would not

be

_possible

in

_practice

to obtain the results

_presented

here without such automation because of the

_large

number

_{of X}

_{( T )}

traces needed for the

_analysis.

The

_{background signal}

from the

bridge

was

_compensated

a few

_degrees

above the transition

temperature,

and the

_{phase angle}

of the

_{susceptibility}

was

adjusted

at a low

_temperature

in the case of a zero de field and at

small

_amplitude

of the ac field. The measurements were done with the ac and de fields

_parallel

to the axis of the

_sample.

3. Results.

3.1 CHARACTERISTIC FEATURES OF THE AC SUSCEPTIBILITY. - The

_temperature

depend-ences of

_{susceptibility}

X’(T) (real part)

and

X" ( T ) (imaginary part)

for different

_amplitudes

and different de fields have

_already

been

_published

_many

times,

and we shall

_only

summarise the main

_typical

features

_{of X}

_{(T ;}

_{Hac ;}

_{Hdc) [7] :}

i)

In

_{good samples}

at low

T,

x ’ (T) = -

1/4 ir and does not

_depend

on the

_temperature,

while

_{X"(T) = 0 .}

ii)

With

_increasing

_T,

_X"(T)

_goes

_through

a maximum at T =

T *, and,

under certain

conditions,

a second maximum at a

_higher

_temperature,

connected with the onset of

intragrain superconductivity.

For T = T*

(Hac, Hdc)’

the ac field

_penetrates

_exactly

to the middle of the

_sample

and,

for

cylindrical infinitely long samples,

X" (T*) =

0.212/4 11’, and for

T *,

the critical

supercurrent

density

can be calculated from the relation

_{[8] :}

(Hac in

Oe,

Ic in

A/cm2,

r in

cm).

The

_validity

of this formula is based on the

_assumption

of the existence of the critical state, and on

_{Ic being independent}

of the field.

iii)

Below a certain value of

_Hac,

of the order of 0.05 Oe in our

_{experiments, independent}

of the de

_field,

one is not able to see _{any difference between the} traces

_{of X}

_(T)

for different

Hac.

We will call

_Tj

the

_temperature

of the

_{X "(T)}

maximum for such low

_Hac.

Below this

temperature,

a small

_macroscopic

_supercurrent

_begins

to flow in the bulk

_sample.

iv)

One does not measure the derivative of de

_{magnetisation}

in ac

susceptibility

measurements. _{It is easy} to obtain ac

_{susceptibility}

close to - 1/4 7r in

superimposed

de

fields,

while the

_slope

_dM/dH

in the same fields would

_give

much lower values of X. This means that the response of the

sample

to a small disturbance

(field)

variable in time is the

_{nonequilibrium}

one that has been discussed in models of

_{superconducting glasses}

[2].

3126

Fig.

1. -

Temperatures

of the maxima in the

_imaginary

_part of the ac

_{susceptibility}

for fields

Hac

(left

vertical

_scale)

measured in zero external dc field.

_According

to

_equation

₍₃₎

this curve

corresponds

to the _temperature

_dependence

of the critical supercurrent of the

_sample

_(right

vertical

scale).

The critical _temperatures

_Tc

and

_Tj

are marked

by

arrows.

Fig.

2. -

Temperatures

of the maxima in the ac

_{susceptibility}

as a function of the dc

_applied

field

(vertical scale),

obtained for different

_amplitudes

of the ac field. In the

figure,

a)

to

_p)

_correspond

to :

0.025,

0.25,

0.5, 0.75, 1.0, 1.75, 2.5, 5, 7.5, 10, 12.5, 15, 17.5, 20, and 22.5,

respectively,

in Oe. The

curve for

_Hac

= 0.025 Oe is the

HJ(T) dependence.

fields. On the basis of this

_assumption,

we will

_perform

measurements of

_T*(Hdc)

for

the field

_dependence

of the critical

_Josephson

_{supercurrent.}

Since

_X(T)

_depends

on the

history

of the

_sample

_[7]

all the data

_presented

were obtained under the same

_{field-cooling}

conditions. The reason for this is also the fact

that,

during cooling

in the

_field,

the

magnetisation

of the

_sample

is much lower than in the case of zero-field

_cooling,

which allows us to assume with an _{accuracy sufficient for} our later discussion that the internal de field is

equal

to the

_applied

de field.

3.2 BULK CRITICAL JOSEPHSON SUPERCURRENTS. -

_Figure

1

_presents

the

_amplitude

of the ac field

_(Hdc

=

0)

_{as a}

function

of the

_temperatures

T* of the

X"(T)

maxima, i. e. ,

it is

IC(T)

according

to

_(3).

The characteristic

_temperatures

of the

_sample,

_Tc

and

_Tj,

are shown

by

arrows. These results were discussed

[10],

_together

with the results for a series of

YBa2Cu3 _xFex07 -

5

samples

with different iron contents. Measurements of the

temperatures

of the

_X"(T)

maxima were

_performed

for de fields of up to

_{100 Oe,}

and the results for

temperatures

about 84 K are

_presented

in

_figure

2 on the H- T

plane.

The lines connect the

points

obtained for fixed

_amplitudes

of the ac

fields,

for fields from 0.025 Oe to 22.5

Oe,

and are drawn

_only

to

_guide

_{the eye. The} curve obtained for 0.025 Oe can be assumed to be the border curve on the H T

_plane

for the occurrence of bulk

_{superconductivity,}

and we will call it the curve of critical field for the

_Josephson

supercurrent,

HJ (T ).

As was

_already

_said,

one would obtain the same curve for

_Hac

lower than about 0.05 Oe.

4. Discussion.

4.1 CRITICAL FIELDS. - Measurements of

transport

critical currents as a function of the de field

_{[14, 15]}

have been discussed on the basis of a Fraunhofer diffraction-like

_{dependence :}

Because of

_large

disorder,

this

_equation

should be

_averaged

over the

_{junction lengths}

and

orientations,

and such

_averaging

will smear out the structure in

_{IC(T; H)}

_[12].

In

_(4),

Ho plays

the role of the critical field of the

_Josephson

_junction

for

_trapping

the flux

_by

the

junction

itself. For some fixed

_{temperatures,}

we

_{plotted straight}

vertical lines in

figure

2, and,

in this way, obtained a set of field

_dependences

of

_Ic

for these

_temperatures

_{(Fig. 3a).}

We see a

_strong

_suppression

of the current in fields of the order of a few Oe.

Now,

for each

temperature

in

_figure

_3a,

we draw a

_straight

line with a

_{slope corresponding}

to the maximum

slope

of the

_Ic(H)

_dependence,

as illustrated

_by

the broken curve for T = 84.5 K. The

dependence

of

_Ho

on

_temperature

is

_surprisingly

close to linear

_{(Fig. 3b),}

_although

the zero value of

_Ho

is obtained

_{by extrapolation}

at a

_temperature

a little

_higher

than

_{TJ (o ).}

Kwak

et al.

_[14]

_report,

from

_transport

measurements, a small

_negative

curvature of

_Ho

with

decreasing

temperature

(measurements

of Ref.

_[14]

were done in a wider

_temperature

range).

Our values of

_Ho

are consistent with these

_reports,

but are in

_disagreement

with our de

measurements on a Foner

_magnetometer

(the

full circles in

Fig. 3b).

For different

temperatures,

we have

_plotted

small

hysteresis loops

of

magnetisation

(inset

of

Fig.

3b)

and,

for all these

_loops,

we

_applied

the same

_procedure

to obtain the lower characteristic de field

(it

is not obvious where is the critical field

_Ho

in such a

_{loop ;}

in the inset in

_Fig.

3b,

this field is shown

_by

the

_arrow).

The fields obtained are _{very close} to

_being

linear with T in the

_vicinity

of

Tc, although they

are about five times smaller than those obtained from the ac

_{susceptibility}

procedure

-

we believe this to be connected

_mainly

with the time

_dependence

of the

3128

Fig.

3. -

a)

dc field

_dependences

of the critical _{supercurrents}

_(in

units of

_{Hac amplitude, Eq.}

₍₃₎₎

for temperatures from 84.5 to 89.5 K

_(matched

in the

_{Fig. ),}

obtained

by

the method described in the text.

From

_{extrapolations}

of the _steepest

_portions

of the data to zero value of

_Hac,

as illustrated

_by

the broken line for T = 84.5 K, values of the

junction

critical field

Ho(T)

_were obtained. These fields _are

represented by

open circles in

figure

b, and the fit to them,

represented by

the solid line, is discussed in the text. The inset in b shows a small

_{hysteresis loop}

of the dc

_{magnetisation}

at T = 4.2 _K, and the

method of

_obtaining

the characteristic dc field for the

_loop

marked

_by

the arrow. These fields

_(the

full circles in

_Fig.

_b)

are

_compared

with the

_Ho

measured

by

the ac method.

The field

_{Ho depends}

on the

_geometrical

dimensions of the

_junctions

and,

for

_large

junctions

(width

L >

A J),

Ho

is

given by :

where 03A60

is the flux

_{quantum, s}

= a + 2 À

g,

with À g

being

the field

penetration depth

into the

grains

and

_Àj

the

_{penetration depth}

into the

_{junction. À J depends}

on the

_intergrain

critical

current :

Since

_{À g}

is

_high

in these materials and the

_separation

between the

_grains

is

_small,

we can assume that s = 2

The

_key

to

_{understanding}

the

_temperature

_dependence

of

_1c

and

_{Ha is}

the observation that the ac method measures the

_{nonequilibrium}

_properties

of the

_system.

In each

_experiment,

the

measurements

_give

an « _average over the time » of some

_{quantum-mechanical}

«

observ-ables ». The classical

_equation

for the

_tunnelling

current

_density

in a

_{single junction}

is : I =

_le

sin

_8,

where 8 is the

_quantum

_phase

difference across the

junction.

In our case, above

Tj,

fluctuations lead to 8 = 0

and,

consequently,

to I = 0.

Below,

Tj,

some _average over time

8 :F

0 is measured

_by

determination of a finite

_{supercurrent.}

It _{is easy} to see that

Tj

should be

_{frequency dependent,}

as has been obtained in

_[16].

This leads us to the conclusion that

_Ho

should be

_dependent

in a non-standard _way on the

_temperature

_through

the

_divergence

of

_{À J}

at

_Tj

rather than at

_Tc:

where we have

_assumed,

_quite

_arbitrarily,

that

_{Ic cc}

(Tj -

T)«.

In

_figure

3a,

the solid line was drawn

_using

equation (7)

with the

_following

_parameter

values :

_Tc

= 93.7

K,

Tj

= 90.2 K and a =1.25. If the BCS

_dependence

of

Ag(T)

is

taken,

then a

_good

_agreement

with the measured values of

le is

obtained with

Àg(O)

of about 0.15 itm, a value close to those _{obtained in other ways}

_[17].

_kj

is then about 5 03BCm at

T = 84.5 K.

Fig.

4. -

Universal

_scaling

of the field

_dependence

of the

_Josephson

_supercurrent obtained from the data of

_figure

2

_(see

the

_text).

The

_symbols

from A to P refer to the values of the

_{Hac amplitudes}

in the order of

_figure

2. For

_comparison,

a Fraunhofer diffraction-like

_dependence

is drawn

_by

the solid line,

and the

_dependence

of Peterson and Ekin

[12],

calculated after

_applying

the

_{specific averaging}

3130

4.2 UNIVERSAL SCALING OF

_{Ic(T, H).}

- As

a result of our determination of

_{Ic(T, 0)}

and

Ho(T),

we can

_try

to

_plot

a universal

_temperature

independent scaling

of

Ic(H) (Fig. 4) :

where we are

_{using only}

the

_{experimental points}

of

_figure

2. For

_comparison,

the solid line shows the Fraunhofer diffraction-like

_dependence

of the

_Josephson

_supercurrent

_{(Eq. (4)).}

Quantitative agreement

between our field

_dependence

of

_1c

and the measurements and calculations of Petersen and Ekin

_{[12] (represented}

_by

the broken curve in

_Fig.

4)

is obtained. It was one of our aims to show this

_good

accordance between

transport

critical current measurements and the data derived from

_{susceptibility}

measurements,

bearing

in mind all the

approximations

involved

_{by using}

the latter method.

4.3 CRITICAL STATE OF THE JOSEPHSON SUPERCONDUCTIVITY AND THE GLASS TRANSITION.

- The critical lines

_TJ(H)

for the occurrence of bulk

_{superconductivity}

was shown in the paper of Barbara et al.

[18],

where

susceptibility

measurements were

_reported

for ceramic and

single-crystal samples.

These authors

_suggested

a

_{spin-glass-like origin}

of this critical line. There also exists a marked

_similarity

of

_transport

results

_[19]

for the field

_dependence

of the critical

_temperature

_{for appearance of} a _{very small critical} current. The

_frequency

dependence

of

_{TJ (H)}

has been shown to scale in the same _way as in the

_spin-glass

transition

[16].

The border line between localised and delocalised

_{superconductivity}

obtained

_by

us is

qualitatively

similar to that calculated

_by

Aksenov and

_Sergeenkov

_[11]

for a

_{spin glass}

transition in disordered

_{superconductors.}

Lastly,

Muller et al.

_[20]

_reported

measurements of flux

_penetration

into ceramic

_samples

in low

_fields,

and

_{successfully applied}

the critical state model of

_{Josephson superconductivity.}

Their work

_justifies

our

findings

of

_Ic

from the measurements of losses. We are thus

proposing

an alternative view of the

origin

of the

pinning

forces in these materials. We

suggest

that

_spin-glass

like

mechanism,

caused

_by

the frustration induced

_by

disorder and the

magnetic

field,

is

_leading

to the occurrence of frozen flux of the

_magnetic

field. Then there exist metastable states _{and energy barriers} to overcome for flux motion. These barriers can be

higher

than those for the mixed state of

_{Josephson superconductivity}

without frustration. The

interplay

between these two mechanisms of the

_bounding

in critical current measurements is

not clear to us.

-5. Conclusions.

We have used ac

_complex

_{magnetic susceptibility}

measurements to determine the

_dependence

of the critical

_{Josephson intergrain}

_supercurrent

on the

_temperature,

close to _the

supercon-ducting

transition

_temperature,

from the losses

_X"(T).

We made measurements of critical

currents in different de fields. The results are consistent with

_predictions

of

IC(H)

for

Josephson junctions,

smeared out

_{by large}

disorder in the

_sample.

We obtained the

temperature

dependence

of the critical field for

_trapping

the flux

_by

the

_junctions,

and the

temperature

dependence

of the critical field for occurrence of bulk

_Josephson

superconduc-tivity.

All the results

_presented

here

_strongly

_support

the existence of a

_{field-dependent}

superconducting

glass

state in the

_HTCS,

connected with

_intergrain

weak links. It was

suggested

that this

_{spin-glass-like}

mechanism is

_responsible

for flux

_pinning

and the similarities to classical critical state behavior. The

_properties

measured are

_frequency

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