Time dependent critical field transition line in an insulating spin-glass

HAL Id: jpa-00232141

https://hal.archives-ouvertes.fr/jpa-00232141

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.

Time dependent critical field transition line in an

insulating spin-glass

N. Bontemps, J. Rajchenbach, R. Orbach

To cite this version:

N. Bontemps, J. Rajchenbach, R. Orbach.

Time dependent critical field transition line in

an insulating spin-glass.

Journal de Physique Lettres, Edp sciences, 1983, 44 (1), pp.47-52.

�10.1051/jphyslet:0198300440104700�. �jpa-00232141�

Time

_dependent

critical field transition

line

in

an

_{insulating spin-glass}

N.

_Bontemps,

J.

_Rajchenbach

and R. Orbach

(*)

Laboratoire

_{d’Optique Physique,}

E.S.P.C.I., 10, rue

_Vauquelin,

75231 Paris Cedex 05, France

(Re~u le 13 _septembre 1982, revise le 25 octobre, accepte le 8 novembre 1982)

Résumé. 2014 Nous _avons

analysé

le

_{retour à l’équilibre}

de l’aimantation dans

_Eu0,4Sr0,6S

_pour diffé-rents _temps de mesure ₍₁ ms-1 s)

après

coupure

d’un

_petit

_champ

_{magnétique superposé}

à un

_grand

champ statique. Nous déduisons, pour les diverses échelles de temps, différentes

_lignes

de transition

champ-température,

qui

chacune suit la loi de De Almeida-Thouless. L’échelle de

_champ

_dépend

(ne

dépend pas)

du _{temps pour des temps} de mesure

_supérieurs

_{(inférieurs)} à 10 ms. Nous proposons

une

_{interprétation}

des résultats au vu de récentes théories.

Abstract 2014 The

change

in

_{magnetization}

of the _insulating

_{spin-glass Eu0.4Sr0.6S}

has been measured

as a function of time (1 ms-1 s) for small

_magnetic

field _changes

_superposed

on a _larger constant field. A set of

_{field-temperature}

transition lines has been obtained for different time scales.

Each

appears

to be of the De Almeida-Thouless form but with a field scale which is

_dependent

_{(independent)}

of the time

_scale

for times > () 10 ms. The results are

_analysed

in terms of recent theories.

Physique

(19’(,3) 1983,

Classification

Physics

Abstracts

75.40 - 64.60 - 78.20L

The time scale for a continuous distribution of

_spin-glass

order

_parameters

_q(x)

has

_recently

been related to the Parisi

_[1]

_{parameter x}

_{by Sompolinsky [2].}

The short time

spin-glass

behaviour

obeys

linear response

(the dynamic

region)

while the

_long

time behaviour exhibits the

thermo-dynamic

limit

_(the

static

_region).

We

_report

here measurements _{which may relate} to these

theories,

in that

_they

show

_explicitly

a time

_dependence

of the

_{field-temperature (H-T) line.}

We

have measured the

_spin-glass

critical field as a function of

_temperature

at different

experi-mental time scales in the system

Euo.4Sro.6S

using

the

_Faraday

rotation

_technique.

The critical field was defined as that

field,

_7~

which was

_required

for the

_{magnetization}

to _{appear reversible}

with reference to the

_paramagnetic

_response on the time

_{scale tm}

of our

_experiments.

A

_family

of curves for

_Hc

versus T for different _tm can be drawn

_through

the

experimental points according

to a relation of the form :

with a = 0.63 ± 0.07.

T ~(tm)

is the H = 0

spin-glass

transition

_temperature

also determined in

our

_experiments;

_{f (tm) ~}

13 for _tm =1 or 10 ms and increases for

_longer

time

_~’(tm

=1

_s)

’"

27].

(*) Permanent address : Department of _Physics,

_University

of California, Los

_Angeles

AC, 90024, U.S.A.

L-48 JOURNAL DE _{PHYSIQUE -} LETTRES

The form of

_equation

₍₁₎

_(except

for the time

_dependent

_{prefactor f(tm))}

is that first

_{proposed by}

De Almeida and Thouless

[3]

for

_Ising

_spins,

and extended to

_Heisenberg

_spins

for the

_longitudinal

susceptibility by Gabay

and Toulouse

_[4]

_(hence

the

_prefactor

_(4/5)~).

This « transition » line is now

thought

to _represent a cross-over from transverse to

_{longitudinal spin}

_freezing

_[5].

Never-theless our results appear to exhibit a

_change

in behaviour of the

_longitudinal

_{magnetization}

across such a line. That

[1 -

TIT’ (t.)]

should scale as

[g~iH~/kT~(tm)~2/3,

_regardless

of the time scale of the

_experiment,

was first

suggested by

Chamberlin et al.

_[6].

The

_sample

was a

_{single crystal platelet}

of

_Euo.4Sro.6S

₍₄

x 3 x 0.7

_mm3)

oriented

perpendi-cular to the

_magnetic

field. At a

_given

_{temperature T,}

an external field was

_applied

₍₀

to 500

_Oe)

and a small field

_{(AH -}

2

_Oe)

_superposed

the latter switched off and on. Because of the

_sample

geo-metry,

the

_{demagnetizing}

field is _very

_{large. Assuming}

the

_sample

to be a

_{quasi-infinite platelet,}

we estimate that it

approximates

to 85

_%

of the

_applied

_{field in the range where} M and H remain

proportional.

As this is the case in the

_regime

of fields we have

used,

_only

the absolute value of the field is affected. We take this « _correction ~> _into account in our

_{subsequent analysis.}

The

_change

in

_{magnetization AM(H,}

_T,

_tm)

was measured

_by

means of

_Faraday

rotation at a

_{time tm}

after the

_change

in the small field AH. The time scale _tm was varied from 1 ms to 1 s, the

_repetition

rate

for the field off-field on sequence was - 20 _tm. The

_temperature

_range was 1.3-1.8 K.

The

_principle

of a sensitive device for

_measuring

the

_Faraday

rotation has been described in reference

_[7].

This device has been

_adapted

as described in references

[8]

in order to

_perform

measu-rements over a

_large

_range of times. The

_signal

associated with the

_change

AH was fed into a multichannel

_analyser

where it was

averaged

to reach the desired

sensitivity.

The smallest rotation

angle

that we were then able to

detect,

with reference to our noise

_level,

was

10 - 3

_{degree (this}

would

correspond,

for the

_{Euo.4Sro.6S sample,}

to an

applied magnetic

field of 2 x

10-3

_Oe).

The

magnitude

of AH was

_kept

constant so _{that any transient circuit non-linearities would remain the} same as H is

_changed.

We refered our

_{magnetization}

_(rotation)

measurements as a function of time to

_high

_temperature

measurements

_(at

the same field H and time scale

_tm)

in order to establish a

_comparison

with the

paramagnetic

response. For a fixed time scale _tm at

_{temperature T,}

we found that for all H >

Hc,

the

_change

in

_{magnetization}

caused

by

AH

to within our

_experimental

_accuracy, thus

_{defining H~(T).}

In this _way we were able to _map out the H =

H~(T, tm)

line.

Figure

1 exhibits an

_example

of our

_experimental

curves for the

_Faraday

rotation

angle

A0

_{(proportional}

to

_AM)

for a

_given

time scale

_{(1 ms)}

and

_temperature

_{(1.576 K)}

with

_varying

_applied

static fields. The manner in which we extract the critical field from such data

Fig. 1. -

Experimental

decay of the _Faraday rotation angle AO as a _{function of time in the range 1 ms-10} ms,

for various

_applied

fields. The insert shows the

_plot

of ~9 (tm = 1 ms) _versus H; the hatched _part indicates

is illustrated in the insert. ~8 is measured at t =

tm, with reference to the

_equilibrium

_value,

and

plotted

versus H.

_He

is defined

_by

5 x

10- 3

_deg.

_{09(H~, tm)}

10- 3

_deg.

The

_uncertainty

on

_He

introduced

_by

this

_procedure

does not exceed 2 Oe. This

_temperature

is stabilized within ±

10 - 3

K. The raw

_{experimental points (He}

vs.

_{T e)}

are

_reported

on

_figure

2 for four different

_delay

times _tm. Our curves

_extrapolate

to various zero field

freezing

_temperature

Tg(tm)

for different _tm. With our

procedure,

we cannot

_{experimentally approach}

T

_g(tm)

closer than a few hundredths of a

_degree

K. The choice of

_{To (tm)}

will be of

_importance

later when we

_plot

our data in reduced

units,

so that the method of its determination deserves some attention.

_{Unfortunately}

we could not

_rely

on

pre-vious

_{susceptibility}

measurements

_[9]

to locate

_T~(~)

_{accurately. Though}

the _strong

_frequency

dependence

of the a.c.

_{susceptibility}

maximum as a function of

_temperature

is

_fully

consistent with

our

_results,

it is not obvious that

_TG(tm)

is to be associated with the maximum of the a.c.

suscepti-bility

for the

_{corresponding frequency.}

We thus used a different

_procedure

to determine

_T~(~).

Actually,

the

_{To (tm)}

values that we shall estimate as described further are _very close

(2

x

10-2

_K)

to those

_previously

determined

_[9].

Fig.

2. -

Experimental

values

_{of He}

versus _temperature T for _tm = 1 ms _{(full circles),10} ms _{(open circles),}

60 ms _(triangles), and 1 s _(crosses).

We found that the

_experimental

critical field

_points

fit a _power law

_{H~(tm) ~}

1 -

T/T~(~)

where

_T~(tm)

lies within an

experimental uncertainty

_range.

Choosing

Log-Log coordinates,

we

calculated a

_by

linear

_regression

of the

_{experimental points,}

and for various

T~(tm).

We then searched for the value of

_T~(tm)

which maximized the linear

regression

coefficient r

_{(1 ~ r > 0.98).}

Values of r, a and

T~(tm)

are

_reported

in table I for the various

_experimental

time scales _tm.

Remarkably,

a is found to be constant to within our

_experimental

error and close to the De Almeida-Thouless value of 0.67 whatever tm.

(This

is not

_quite

true for _tm = 1

s, but note that there

are

only

6

experimental points

so that the fit _may be somewhat less

_reliable.)

Once

_having

deter-mined

_T~(~),

we _may

plot

our measured

_He

vs.

_7~

in reduced units h =

gPHe/kT&{tm)

and

TYT~(~) :

these are exhibited in

figure

3. We

point

out that the reduced units remove the time

dependent

effects due to the time

_dependence

of

_TG,

and thus _may exhibit a relation between time

and field. In the _{range tm} > 10 ms, the h vs.

T7T~(~)

curves are

_{significantly}

_different

_showing

unambiguously

that the critical field h

_depends

on the time scale of the

_experiment

in _{this range} of time.

However,

for the 1 ms and 10 ms time

scales,

the curves _appear to coalesce. This _suggests that a minimum time scale exists for

Euo.4Sro.6S (tm

10

ms),

so that

f (tm)

is

independent

of time

_{for tm}

10 ms.

One could

_object

to our use of a small field increment

_AH,

on

_top

of a

_large

static field

_H,

to

search for the

_spin-glass

« transition ». That

is,

one could _argue that we

_only

_sample

the response around a local minimum in

configuration

space, with different

physical properties

than the

tran-L-50 JOURNAL DE _{PHYSIQUE -} LETTRES

Table I. -

Experimental

values

_of

the zero

_{field freezing}

_temperature

T’G(t,,,),

_of

the _{exponent a,} and

_of

the linear

regression

coefficient

r

for

the various

experimental

time scales _{tm. The values}

_of

T ~(tm)

and a

_correspond

to the maximum

_of

r, and the

uncertainty

range to the admitted

_{range for}

r

(column 4).

An estimate

_of

the

_{subsequent uncertainty for}

h =

g~H/kT G(tm)

and

T /T G(tm)

is indi-cated on

figure

3.

Fig.

3. - Reduced critical field h =

g#H,,Ik T’ G (t.)

_versus reduced _temperature

T/ To G t.).

The symbols _are

the same as in

_figure

2. The bars indicate _{the range} of variation of the reduced data

_{corresponding}

to the

uncertainty

range of

T ~(tm).

The dashed lines are fitted to the

_{experimental points}

with the

_f(tm)

factors listed in table I and ot = 0.67.

sition to the true

_equilibrium

state.

However,

as

already argued previously [6],

such a behaviour would

_imply

the existence of a minimum threshold field below which no irreversible

_phenomena

occur. Such a field has never been identified. We believe therefore that we are

probing

the

change

of irreversible behaviour as a whole.

Nevertheless,

it would be useful to

_perform

the same

experi-ments but

_{change H}

from 0 to

_H~.

Transient

_problems

have so far

_prevented

us from

_doing

so,

though

we do

_point

out that the behaviour at low field

_{(H ~}

10 Oe ~ 5

_AH)

does appear to fall on the same curve as for

_large

H

_{(H ~}

300

Os ~>

_{AH) (See}

_Figs.

2 and

3 ).

We now turn to a compa-rison with such theoretical results as are available.

The

_Ising

model

_analysis

of Thouless,

De Almeida and Kosterlitz

_{[10], using}

a Parisi

_approach

in a finite

_field,

finds an

_instability

in the

_high

_temperature

infinite _range

Sherrington-Kirkpatrick

(SK)

model when

_{q(0) =}

_{(1/~/2) (A/T~)~}

_{equals ~(1)}

= 1 -

(T/J),

where h is the

field,

To

the

freezing

temperature,

and

_{J 2/N}

is the variance of the

_strength

of interaction between each

_pair

of

_spins

_(for

_appropriate

normalization

_T~

=

J).

The Parisi

vector

_spin-glass

in a field

recently

by

Elderfield and

Sherrington [11].

They

find the

longitudinal

order _parameter in the

_region

1 -

(T /TG) >

(h/T G)zi3

to be

_{effectively pinned}

at

q(0) =

[(m

+

2)/2~/3]~~

_{[hlto ]213}

.

Noting

that

q(1) ~

1 -

(T/T~),

one sees that a « cross-over » condition can occur when the two

_limiting

order

_parameters

are

_equal.

Thus,

the De Almei-da-Thouless

_relationship

between

_He

and

_Tc

for the

_Ising

_{model may be}

_preserved

in the m vector

spin-glass

with

_perhaps

_only

a

_change

in

_prefactor.

It we now make the

_{Sompolinsky interpretation}

of the Parisi _parameter x, we are then led to the

following

rationale for the time

_{dependent prefactor f(t)}

in

_{equation (1). Sompolinsky}

_suggests that x = 1 be associated with short times where linear response is

obeyed,

and x = 0 with

long

times

_appropriate

to the

_{thermodynamic}

limit. The

_shape

of

_q(x)

exhibits « a

_{plateau »}

at and below x = 1

(short

times),

with _a decrease _to

q(O)

_{at x -+} 0

(long times).

We

suggest

that at fixed time

_(given

_x)

cross-over takes

_place

when

_{q(O) = q(x) ;}

that

is,

when

_{~(0) ~}

_(A/7~)~

_equals

q(x)

oc

_~(1).

In other

words,

we _argue

_that,

for h

0,

q(x) depends only

on

T/7~,

and when h increases from zero, the cross-over occurs when

(h/Tg)2/3

becomes

_equal

to

_q(x)

at that value

_{of tm}

which

_specifies

x

_(the

_Sompolinsky

_{association).}

This leads

_directly

to a

preservation

of the De Almeida-Thouless form for the relation between

_He

and

_7~,

_provided

one takes a time

_dependent

prefactor

for 1 -

(T/T~).

Said another way,

~(:c) -~ q(t)

=

!(t)[I -

(r/7~)],

where

_f(t)

is a

decreasing

function of time. The cross-over condition

_{~(0) ~ q(t)}

then

_gives

the

_relationship

between

_HG

and

_Te

exhibited in

_equation

_(1).

_Figure

3

_suggests

that

_q(x)

oc

_f(t)

is

_essentially

constant at _{the linear response value for time less than 10} ms, and decreases

significantly

for

longer

time scales

_{(smaller x).}

This is consistent with the

_shape

used for

_q(x)

_by

Parisi

_{[ 1 ],}

_Thouless,

De Almeida and Kosterlitz

_{[ 10]}

and Elderfield and

_{Sherrington [ 11].}

This

_{interpretation}

is not _{consistent with the infinite range SK}

_exchange

model. Mackenzie and

Young

have

_recently

shown

_[12]

that all relaxation times should

_diverge

in the

_{thermodynamic}

limit

_along

and below the De Almeida-Thouless line.

_However,

_Euo.4Sro,6S

is a short _range

exchange

system.

Our results exhibit a transition line of the same form as the infinite range

model,

leading

one to

_suggest

that the De Almeida-Thouless relation may continue to

_apply

to the short range case, but with a

_{time-dependent scaling}

coefficient

_{(our f(t)),}

and that a critical time scale exists below which this

_prefactor

remains constant.

Acknowledgments.

- We wish

to

_acknowledge

_very

_{helpful correspondence}

with D.

Sherring-ton and valuable discussions with

him,

and H.

_Alloul,

H.

_Bouchiat,

J.

_Ferre,

M.

_Gabay,

P.

Monod,

R.

Rammal,

J. C.

_Rivoal,

G. Toulouse and A. P.

_Young.

We also wish to

_acknowledge

H. Maletta who

_{kindly provided}

us with the

Euo.4Sro.6S sample

and J. L. Tholence who has informed us on similar measurements on CuMn and

Fe7oNi10P2o.

This research was

_supported

in

_part

_by

the U.S. National Science Foundation and the Office of Naval Research.

References

[1] PARISI, G., Phys. Rev. Lett. 23

₍₁₉₇₉₎

1754; and J. _Phys. A 13

₍₁₉₈₀₎

L115, L1887.

[2] SOMPOLINSKY, H., Phys. Rev. Lett. 47 ₍₁₉₈₁₎ 935.

[3] DE ALMEIDA, J. R. L. and THOULESS, D. _J., J. _Phys. A 11 ₍₁₉₇₈₎ 983.

[4] TOULOUSE, G. and _{GABAY, M., Phys.} Rev. Lett. 47 ₍₁₉₈₁₎ 201.

[5] CRAGG, D. M., SHERRINGTON, D. and GABAY, M.,

Phys.

Rev. Lett. 49 ₍₁₉₈₂₎ 158.

[6] CHAMBERLIN, R. V., HARDIMAN, M., TURKEVICH, L. A. and ORBACH, R., Phys. Rev. B 25

₍₁₉₈₂₎

6720.

[7] BADOZ, J., BILLARDON, M., CANIT, J. C. and RUSSEL, M. F., J. _Optics 8

₍₁₉₇₇₎

373.

[8] BONTEMPS, N., RIVOAL, J. C., BILLARDON, M., RAJCHENBACH, J. and FERRÉ, J., J.

_Appl.

_Phys. 52 ₍₃₎

L-52 JOURNAL DE _{PHYSIQUE -} LETTRES

[9] FERRÉ, J., RAJCHENBACH, J. and _{MALETTA, H.,} J. _{Appl. Phys.} 52 _{(3) (1981)} 1697.

RAJCHENBACH, J., Thèse de 3e

_cycle,

Univ. Paris-Sud (1980).

[10] THOULESS, D. _J., DE _ALMEIDA, J. R. L. and KOSTERLITZ, J. M., J. _Phys. C 13 (1980) 3271.

[11] ELDERFIELD, D. and SHERRINGTON, D., submitted for

_publication

(1982). [12] MACKENZIE, N. D. and YOUNG, A. P., Phys. Rev. Lett. 49 ₍₁₉₈₂₎ 301.