Non Ergodic Aging in Lithium-Potassium Tantalate Crystals

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Non Ergodic Aging in Lithium-Potassium Tantalate Crystals

F. Alberici, P. Doussineau, A. Levelut

To cite this version:

F. Alberici, P. Doussineau, A. Levelut. Non Ergodic Aging in Lithium-Potassium Tantalate Crystals.

Journal de Physique I, EDP Sciences, 1997, 7 (2), pp.329-348. �10.1051/jp1:1997148�. �jpa-00247331�

J.

Phys.

I £Fance 7

(1997)

329-348 _FEBRUARY

1997,

PAGE 329

Non Ergodic Aging in Lithium-Potassium Tantalate Crystals

F.

Alberici,

P.

Doussineau

and A.

Levelut (*)

Laboratoire

d'Acoustique

et

Optique

de la MatiAre Condensde (**

),

Universitd P. _et M.

Curie,

Case 78, 75252 Paris Cedex 05, France

(Received

i

July1996,

revised 12

September1996,

accepted 25 October

1996)

PACS.77.22,Gm Dielectric loss and relaxation

PACS.78.30.Ly

Disordered solids

Abstract. Isothermal kinetics of the orientational

glasses

Ki-~Li~Ta03

(0.001

< z <

0.05)

is studied after different

preparation procedures:

the

sample

is cooled from _a temperature well above the transition to the

experiInental

temperature,

generally equal

to 4,2 K and the

thermal

history

is

changed

in either

varying

the

cooling

rate or the

quenching

temperature.

Then, the dielectric _constant e' is measured from I kHz to MHz. The power-law e'

=

EL

+

/he'((t

+

to)/to)~°

fits _very well the evolution. The essential result is that, in this

equation,

all the parameters

depend

_on the

history

of the

sample. Therefore, aging

is found,

leading

to true

ergodicity breaking.

This shows that the

phase-space

can be

pictured

as a com-

plicated landscape

of

mutually

inaccessible

valleys

separated

by

_very

high

barriers, A further

insight

is provided

by

another set of experiments where temperature

cycles

_are

performed: they

are explained

by

_a temperature

dependent

hierarchical

organization

of the

phase-space-

R4sum4. La

cin4tique

isotherme de _verres orientationnels Ki-~Li~Ta03

(0,001

< _z <

0,05)

est 4tud14e

aprAs

diffArentes

proc4dures exp4rimentales

de

pr4paration

_: l'4chantillon _est refroidi

depuis

_une temp4rature ^bien

sup4rieure

h la transition

jusqu'h

la temp4rature de

l'exp4rience, g4n4ralement 4gale

h 4,2 K et son histoire thermique est

chang4e

_en faisant

varier,

soit la vitesse de

refroidissement,

soit la temp4rature de trempe. L'4volution de la constante d141ectrique e' est

alors mesur4e entre I kHz et I MHz. La loi de

puissance

^e' ₌

EL

+

/he'((t

+

to)/to)~"

rend trAs bien compte de l'4volfition observ4e. Le r4sultat essentiel est que, dans cette

4quation,

tous les

pararrbtres d4pendent

de l'histoire. Il y a donc

vieillissement,

conduisant h la brisure vraie de

l'ergodicit4,

Ceci montre que

l'espace

des phases est

repr4sentable

_{par un paysage}

compliqu4

oh des val14es mutuellement inaccessibles _sont

s4par4es

_par de trAs hautes barriAres. Des

exp4riences

de

cycles

_en

temp4rature

en apportent une vision

plus

d4tail14e _: elles

s'expliquent

en effet dons le cadre d'une

organisation h14rarchique d4pendaut

de la

temp4rature.

1. Introduction

In statistical

mechanics, systems

_are

generally

assumed _to be

ergodic:

their

equilibrium

state does not

depend

_on the initial

conditions,

_or in other

words, they

visit all allowable

points, specified by macroscopic constraints,

of their

phase-space

after _a

sufficiently long

time. These constraints _are

easily

taken into account

by

_means of the relevant

thermodynamic potential,

(* ^Author for

correspondence (e-mail: alflccr.jussieu.fr)

(** ^Associated with the Centre National de la Recherche

Scientifique:

URA 800

Les

#ditions

_de

_Physique

₁₉₉₇

a function defined in the

phase-space,

^which

presents

^minima

(stable

_or metastable

states)

and maxima

(barriers).

However,

_some

phenomena

encountered in solid state

physics

_are known to show broken

ergodicity [ii. Then,

the

phase-space

^is

^split

into

regions

or

sub-spaces separated by

barriers which _cannot be _overcome. Phase transitions _are well known

examples,

where broken

ergodicity

is

tightly

associated with broken symmetry. For

instance,

at the

paramagnetic- ferromagnetic phase

transition of _an

Ising magnet,

^{the whole}

phase-space splits

into two

symmetric halves,

each of them with _an

equilibrium

_state. Because of the

symmetry relationship

between these two

halves,

the

properties

^{of the}

equilibrium

states

present strong similarities,

such _as

opposite magnetizations

and

equal magnetic susceptibilities.

~vhen such _a material

undergoes

_a

phase

transition _on

cooling,

the choice of the final

sub-space

is done at random _{or may} be forced

by

a weak

biasing magnetic

field.

In disordered

systems, things

_are

expected

to be less

simple

since _no

symmetry operation

exists. The

accessibility

of the different

sub-spaces

of the

phase-space

of _a disordered system is

represented by

_a

complicated

and

irregular landscape

made _up of

valleys (metastable states)

and mountains

(barriers)

between them. If the barriers _are too

high

the

system

may be

trapped

in _a

particular sub-space

with _no

possibility

of _escape. If the different

sub-spaces

of the

phase-space

have different

properties (absence

of

self-similarity),

then most measured

physical quantities depend

on the

sub-space

to which the

system

is confined.

Several _non

equivalent

definitions of

ergodicity (and consequently,

of

non-ergodicity)

exist.

Hence,

_we must

clearly

indicate which is meant. A criterion

allowing

an easy

comparison

with

experiments

is convenient.

Thus,

we

adopt

the

point

of view of Classical Mechanics: _a system is

ergodic

if it evolves towards the _same

equilibrium, independently

of its initial conditions. -~s

a

result,

_we claim that _a

system

is _non

ergodic

if _we do observe that

during

^different _runs. all driven with the _same values of the control

parameters (temperature,

_pressure,

field, .),

the evolution from different initial conditions

(after

different thermal histories of the

sample,

^for

example)

leads to different

asymptotic

^values of _some characteristic

quantities (for instance, susceptibilities).

This is called trite

ergodicity breaking.

It should be noticed that this definition is

purely

static: it

only

concerns the

system

^when its evolution is terminated.

Actually,

the time scale

plays

_an

important

role here and _{a more} careful

analysis

is needed.

In

particular,

~rile

ergodici~y breaking

has to be

distinguished

from weak

ergodici~y breaking,

a term coined

by

Bouchaud [2] ^to describe _a

system

^which tends to1N.ards an

equilibrium

state but needs _an infinite time to reach it because its evolution in

phase-space

is hindered

by traps.

The model is intended to account for the slow

dynamics

of

spin-glasses.

Indeed,

the slow

dynamics

of

spin-glasses

and many other systems

depends

_on the

history

of the system. If this

dependence

is not trivial and leads _to the loss of

stationaritj< (for instance,

if _some kinetic characteristics such _as the correlation function _or the time

susceptibility

is _a

two-time

function),

then it is called

aging.

Orientational

glasses (OG)

_are

materials, generally crystals, bearing

at random

sites,

^electric and elastic moments with orientational

degrees

of freedom

[3,4].

In those

glasses,

each

dipole

and

quadrupole

is submitted to

possibly conflicting

interactions from the

crystal

field

generated by neighbouring

atoms, ^and

by

other moments located at random and

producing

_a random

field. Since the bearers of the moments _are not allowed to _move from site to

site,

the disorder of their mutual interactions is frozen in. Such interactions lead to frustration: the

dipoles

are not free _to rotate their moment locked

by high potential

barriers _so

they

_are not able to

minimize the total energy of the system. Frustration and frozen in disorder have

important

consequences on the

phase-space topology

and

ergodicity. Consequently,

these

materials,

_as

well _as

spin-glasses (SG), polymers

and _some other disordered

compounds,

_are

good

^candidates

for

experimental

evidence of

ergodicity breaking.

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 331

In the

present

_paper we

report

on measurements

(mainly dielectric) performed

_on six

potas-

sium-lithium tantalate

single crystals.

^{We have studied the kinetics}

(time dependence)

^{of their}

evolution when

they

have been

placed

out of

equilibrium.

The

dependence

_on thermal

history

of the limits deduced from numerical

extrapolation

of _very

long

time measurements allows

to assert the

non-ergodicity

of these

crystals

at low

temperatures.

^{Some of} the results _were

already published _[5j,

but others _{are new: we} have added data obtained _{on new}

samples

and data recorded with increased _accuracy. A

comparison

^is done with other

materials,

such _as

polymers

and

spin-glasses,

which _are known to also

present aging phenomena. Finally,

_some

models for the kinetics of disordered

systems

are

briefly

reviewed.

2.

Experiments

2.1. THE MATERIAL. We studied _a series of mixed

single crystals

of formula

Ki-~Li~Ta03 (KLT).

These materials result from random substitution of

Li~

_{ions to}

K~

ions in _a

KTa03

(KT)

cubic

crystal.

KT does not become ferroelectric at low

temperatures

but its dielectric

constant _e rises _as the temperature ^tends towards _zero. This is due to an increase of the

correlations between the

displacements

of the

Ta~~

_ions.

However,

the transition is

prevented by

quantum fluctuations.

Hence,

KT is called _an

incipient

ferroelectric. When the K~ ions _are

randomly

substituted

by Li~ ions,

the trend towards the

ordering

of the

Ta~~

_{ions is reduced.}

Owing

to their small

size,

the

Li~

_{ions take}

positions

^that are off-centre

refering

to the

centro-symmetric

site. This creates electric

dipoles

at random sites that _are

aligned along

one of the six

[100]

directions. Each

dipole

interacts with the others and with the

polarizable surrounding

medium. At

high temperatures,

the

dipoles constantly

reorient between their six

possible positions

and the

crystal

is in _a

paraelectric phase.

At

large

lithium concentrations

a ferroelectric

phase

_appears below'a transition

temperature Tt

at which there is

cooperative ordering (with

_some

disorder)

of the

dipoles.

At lower concentrations the

dipole

reorientations

gradually

slow down _as

temperature

is

lowered;

_a frozen

configuration

with _no

long-range

order and _no static

polarization

is created below _some temperature

Tt

and KLT becomes _an orientational _or

dipolar glass.

2.2. THE SAMPLES. We studied six different KLT

samples.

Five of them were grown

by

S. Ziolkiewicz in _{our own}

laboratory;

the other

(sample IX) by

L-A- Boatner

(Oak Ridge).

They

_are cut in

parallelepiped shape,

with

tj"pical

dimensions 4 x 4 _x 6

mm3.

All the faces of the

samples

_are oriented

perpendicular

to _one of the

[100]

^{directions of the cubic}

crystal.

The

opposite

faces _were

polished

flat and

parallel

and the two

larger

_{ones were} covered with a

thin chromium

layer creating

_a

capacitor. Although

the

shape

of the

samples

is _not best for dielectric

measurements,

we are

helped by

the

high

dielectric constant.

Moreover,

this allows to

perform

acoustic

experiments

on the _same

samples.

The lithium concentration _z of _a

sample

is deduced either

according

to the

empirical

law

proposed by

_van der Klink e~ al.

Tt(K)

=

535x~/~ relating

_x to the transition

temperature Tt

_[6] determined

by

dielectric _or acoustic

experiments,

or

according

to another

empirical

law

relating

x to the anharmonic elastic

properties [7].

^Some concentrations _were checked

by Secondary

Ion ~Iass

Spectroscopy (SIMS).

Since most of the _same

samples

_were

already used,

we

keep

the

labelling (in

_roman

numbers) adopted

in

previous publications [5, 7].

^{Table I}

gives

the concentration and the transition temperature of the

samples. Generally,

we estimate the

error on the lithium concentration _as Ax

= +0.001. The

nominally

_pure

sample

VII contains

only

_non intentional

impurities (ferric

and lithium

ions, ).

Table I. Transition

~empera~ilres

and concen~ra~ions

of

~he

samples.

sample Tt (K)

_x

VII

s

o.ooi

VI 27 0.011

V 35 0.017

VIII 46 0.025

IX 52 0.030

III 73 0.050

2.3. THE MEASUREMENT METHODS. The real and

imaginary

parts of the dielectric _con-

stant

e(u~)

=

e'(u~)

^is" _(u~) (uJ ^{is the circular}

frequency

related to the

frequency f by

_u~

= 2~

f)

were measured i<ith _a Hewlett-Packard 4192A

impedance analyser

at _seven

frequencies ranging

from 1 kHz to I MHz. Some elastic constant measurements were also

performed,

in the loo MHz range.

They provide

the real and

imaginary

_parts of the elastic constant c(u~) =

~_(u~)

ic"(u~).

For the two methods the measurements of the

imaginary part,

^while

showing

the _same be-

haviour,

are much less accurate than those of the real part. ^It was checked that the

oscillating

field of about 250 V

m~~

_{in the dielectric}

experiments

is weak

enough

that the linear response is measured. In both _cases, the isothermal evolution is recorded

during typically

20 hours

(exceptionally

40

hours).

We

emphasize

that all the measurements

reported

here _were per- formed without any

applied

static electric field _or mechanical stress, I._e., without _any

symmetry purposely

broken.

In what follows _we

report only

_our dielectric measurements, limited to

e', mostly

obtained at the

extensively

studied temperature

Texp

= 4.2 K.

2.4. THE DIFFERENT TVPES OF EXPERIMENTS. Our

experiments

_can be classified in

different

types according

to the

procedure

used to attain the

experiment temperature

Texp_

They

_are sketched in

Figure

1.

(I)

Vaxiable

Cooling Ramp.

The

sample

is cooled from _a

high temperature (about

40 K

above the transition

temperature Tt)

down to

Texp,

^at a constant rate R

=

dT/d~

chosen between R

= -0.004 K

Is

and R

= -0.3 K

Is (Fig. la).

In order _to eliminate _any

systematic effect,

the different values of R _are taken in random order.

(ii ) Interrupted Cooling Ramp.

The

sample

is cooled at _a constant rate R ₌ -0.01

Ills

down to _a

quenching

temperature

Tq

^below

Tt,

from where it is very

rapidly

cooled down at a rate of about -0.15

Ills

to

Texp (Fig. lb).

The _same R is used for _{a series} of

quenching temperatures

with

Texp

<

Tq

<

Tt.

Here too, ⁱⁿ order to eliminate any

systematic effect,

the different values of

Tq

are chosen in random order.

(iii ) Temperature Cycles.

Two

experiment temperatures Ti

and

T2

are used in three steps

(Figs.

1c and

1d).

After the

sample

_was cooled down to

Ti following

the

procedure a),

it is

kept

there for _a duration ti

Then,

the

sample temperature

is

suddenly changed (at

about -0.15

Ills)

to

T2

and

kept

_constant for t2.

Finally,

the

sample

is

suddenly

driven back to

Ti.

Measurements _are

continuously

taken all

along

the three

periods.

The temperature T2 may be smaller _or

larger

than the temperature

Tii

the two kinds of

cycle provide

different

informations [5].

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 333

@

T

)

Tt _T~

~~~

li~P li>p

t t

j @

Ti Tt

~~

~

t t

Fig,

I. Different types of

experiments reported

in this _paper:

a)

constant

cooling

rate;

b) cooling

ramp

interrupted

by a quench;

c)

temperature

cycle

with T2 >

Ti; d)

temperature cycle with Ti > T2.

(iv)

A

Tj~pical Experiment. Figure

2 shows _a

typical (although

_very

long) experiment.

After

a constant rate

cooling,

_as

explained above,

the dielectric _constant e' at _a

given frequency

(10 kHz)

and at fixed

temperature (4.2 K)

is recorded for

sample

V. This

figure

contains most features that _are

reported

at

length

in this paper.

Roughly,

_1N.e observe _a function of time that

apparently

tends towards _an

asymptotic

limit _as time goes ^to

infinity.

We

investigate

two

questions: 1)

^{what is the time variation} and

2)

on which parameters does the

asymptote depend?

The next section _answers the first _one; the

following

section examines

mainly

the second.

3. Mathematical

Analysis

of the Data

Our first

step

in the

study

of

aging

is to find

a mathematical

description

of the

data, limiting

the trials to the real

part

of the

complex

dielectric _constant e'. For that purpose, we compare

them to several functions. All _are the _sum of _{a constant}

plus

_a

time-dependent

function

tending

towards _{zero as} time goes ^to

infinity.

Before

giving

the form of these

functions,

_we remark that the

beginning

of the

experiments

is not

perfectly

defined. Thus _a time to > 0 _can be introduced _{as an}

adjustable

parameter in

order to take _a

possible delay

into account.

However,

it appears that the values of to

given by

the fits _are much

larger

that could

reasonably

be

guessed.

Thus _we consider to rather _{as an} additional

parameter

which determines the

shape

of the _curves.

We test the

adequacy

of the four functions

given

below.

They

all have the form e'

=

A+B f(t)

where

f(t)

is _a function

varying

from 1 for _t

= 0 to zero as t tends towards

infinity. They

_are

defined

only

for times t > 0. The coefficient A is the

asymptotic

value while the coefficient B is the

magnitude

of the time

dependent

_part;

they

_are both dimensionless if e' is the relative

dielectric constant.

I) Bi-exponential

e'

= A ₊ B

~~~~

_~~~~

~~

_~

_~~~~ ~~~~

(l)

+ _r

1620 loo

T,=35K

(x=0.017)

1600 ^~~~ " 4.2 K

~,_ ~, d

1580

1560

1540 10

o 50000 100000 150000 1000 10000 100000

j~) t+t~ (S)

a) b)

o.ooi

na

°

j@il llflifli%J@/Wii§@W%?

~

~

-0.001

0 100000 200000

c)

^t

(s)

Fig.

2.

a)

Time evolution of the real part e' of the dielectric constant measured

during 150,000

_s

at

f

₌ 10 kHz and Te~p ₌ 4.2 K for the _x

= 0.017

sample; b)

The _same data _as in

a)

represented as a

plot

of

(e' EL)

_versus

It

+

to)

on

logarithmic scales; c)

^Scaled

asymptotic

deviation d

(see text)

_{as a}

function of the time t.

We

try

^{this function because it}

easily

describes _a

two-step

_process

(for instance,

birth and

growth

of

domains).

It contains five free

parameters.

A time translation to amounts to

chang- ing

the coefficient B.

it Stretched

exponential

~ " ~ ~ ~

~~ilp ^~~ _)o)1)~~~

~~~

This time-honored function contains five free parameters. It is often _{used to} fit the relaxation observed in various materials.

iii)

^Inverse

logarithmic

law

~'

~ _~ ~

ln(i~llllT)

~~~

This function contains four free

parameters.

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 335

iv)

Power law

e'

= A +

b((t

+ to

)/T)~° (4)

This function _{seems to}

depend

_on five parameters.

Ho1N.ever,

these _{are not}

independent

_as

the time _{T can} be absorbed into _{a new}

magnitude

coefficient

equal

to bT°.

Consequently,

the

power law function is used in the _more convenient

following

form

e'

= A ₊ B ^~ ~°

to (5

~

which contains

only

four

independent

parameters.

For the four

functions,

the _term A is defined _as

independent

of time since it is the

asymptotic

limit at infinite time. Its value

depends

_on the

frequency f

and _on the

cooling

_rate R. The factor B _measures the

magnitude

of the

time-dependent

_part of

e';

this fonctioii of t

depends

on R and

possibly

_on

frequency.

The

comparison

of the functions _to the data is

performed by

means of _a

generalized

least _square

method,

called the

Levenberg-Marquardt

method [8]. ^The

criterium is the least value of the merit factor

Q

defined _as

=

(j ^((Fi _Jzii)2 / ^2j

^~~~

_(6a)

1-1

1N.itli

N N 2

A~

=

~=i~ ~j _fill ~>=i ~jfifi ~ _(fib)

Here the N numbers

AL

_are the measurement data while the N numbers

Fi

_are the values of the fit function calculated _at the _same

time;

the

quantity

^A is _a normalization constant which

gives

_a dimensionless merit factor

Q.

The results of the calculations _{are as} follows:

a)

^the

bi-exponential always gives

_a much poorer fit than the three other

functions;

b)

the stretched

exponential

has often _a merit factor

nearly

_as

good

_as the _po1N.er law. However this

corresponds

to unreasonable values for _some

parameters (1

ms for the characteristic time

T and _{~f <} 0.01 for the

exponent)

c)

the

logarithmic

function

provides

_a merit factor _as

good

as the power law when its

exponent

o is smaller than 0.15.

Otherwise,

the

quality

of the fit is rather

bad;

d)

The power law

gives

in _any circumstance _a very

good

fit with

only

four free

parameters.

Moreover,

_we have done the

following

test. For _a few

long lasting experiments (40 hours),

_we

have

applied

the least _square method to the entire set of data and to its first half. The calcu- lated

parameters

are the _same. This shows the

adequacy

of the _power law to the

description

of _{our curves} and

gives

_us

great

^confidence in the

asymptotic

value thus obtained.

In order to

visually

check the

quality

of the fit to the data of

Figure 2a,

_we

present

in

Figure

2b _a

plot

of

(e' e[)

_vs.

(t

₊

to)

_on

logarithmic

scales. We obtain _a

straight

line with _a

good precision. Consequently,

_we

adopt

t-he power law _as the best fit to _our data.

The

long

time behaviour is of central

importance

for

ergodicity breaking. Therefore,

_we have

accurately

verified that the form e' _m

Ki

+

K2~~°

is valid for t » to- This is done

through

the dimensionless

asymptotic

deviation d defined _as

d =

[e' Ki K2~~~) /Ki ⁽⁷⁾

We have calculated d for the

long lasting experiment (150,000 s)

shown in

Figure

2a from which the data

points

for 0 _< <

20,

000 _{s are} discarded. The result, sketched in

Figure 2c, clearly puts

^{in evidence} two features: I) the _accuracy is limited

by

the discreteness of the

capacity

measurements

(0.01 pF steps) _it)

the

proposed asymptotic

form is well followed

(the slope

of

the middle line is less than 2 _x

10~~

over

100,000 s)

_as soon as ~o becomes

effectively negligible

in front of ~.

A mathematical

identity

allows to write

[he power-law

_as

e'

= A ₊ B ^~

~

= A ₊ B

dttD(i1)e~"~ (8)

to

~

~~

~~~~~

tj

~-i -~io

(9)

n(~L)

=

Y(~)jiL

e

is _a normalized _gamma distribution of _{rates i1.} The

corresponding

normalized distribution of times 6 ₌

1/tt

is obtained for

D(6)

=

D(i1) )dtt/d6).

It is

Y(6) to

^°

~~~~

%

r(a)6

6 ^~

~ ~~~~

In the last _two

equations, Y(x)

is the Heaviside

step

^{function and}

r(o)

is the Euler gamma function.

D(6)

is _a

L#vy

distribution _[9] which behaves _as

6~(~~°)

_{for 6 ~}

cxJ. Its most

probable

value is

6mp

= to

Ill

+

o)

while its average is infinite. This _means that there is _a

large

number of

long (longer

than to times 6.

This result must not be

misleading.

It _means that the

in-phase

_response to the

periodically varying

electric field

E(t)

₌

Eo cos(u~t)

is the electric induction

~j C©

~ ~~~~

~

~"'~~~ ~°

~ ~ ~

no)

o

~'~'~°

~ ^{~ °}

~°~~~~~'

_~~~~

This

equation,

valid at least if the

frequency

is between 1 kHz and 1

MHz,

is not within the frame of standard linear response

theory

since it is neither in the time

representation

nor in the

frequency representation,

but in both.

However,

this form _agrees with the

linearity

of the

system

(which

has been checked

experimentally)

and reflects its lack of

stationarity.

4.

Physical Analysis

of the Data

Now,

_we present with _some details _our

data, examining successively

the role ofthe measurement

frequency,

of the

cooling

_rate, of the lithium

concentration,

of the

quenching

temperature and of the temperature

cycles.

4.I.

FREQUENCY

DEPENDENCE. The first noticeable feature is

that,

for _any

experiment temperature Texp,

^the

asymptotic

value

depends

_on the measurement

frequency,

in _{a manner} which

depends

_on the

sample.

The second

point

is that the kinetic

part

^does not

depend

_on the

frequency

when

Texp

= 4.2 K

(see Fig. 3a).

In this _{case, a}

simple

translation allows _to

exactly

_superpose the _curves obtained for different

frequencies:

the low

temperature aging

is not

dispersive. Therefore,

_no

frequency

dependence

_appears in the time

dependent part

of the _power law which describes these _curves.

When

Texp increases, aging

first remains

dispersionless

but becomes

dispersive

if

Texp

^is

brought

near

Tt (see Fig. 3b).

N°2 NON ERGODIC AGING IN Ki-zLi~Ta03 337

~ f=3.16 kHz

o f=31.6 kHz

o f=316 kHz

2400

,

T ₌ 27 K (x

= 0.011)

~ T ₌ 4.2 K

2350

2300

0 20000 40000 60000 80000

a)

t

(s)

a epsilon f=100 kHz T ₌ 27 K (x ₌ ^0.01 1)

. epsilon f=I MHz ^'

~ ~~ ~

1850

E'

1840

AA,

...

1830 AA,

"A,,~

,,,

'AAA'A,,,,,, ',,,,,,,,,,,,,,,,,

1820

0 10000 20000 30000 40000

bj

t

jsj

Fig.

3. Time evolution of the real part e' of the dielectric constant measured at several

frequencies

on the _{z =} 0.011

sample: a)

at Te~p ₌ 4.2 K for

f

₌ 3.16 kHz

(open triangles), f

= 31.6 kHz

(open circles)

and

f

= 316 kHz

(open diamonds); b)

at Texp ₌ 23 K for

f

= 100 kHz

(full triangles)

and

f

= 1 MHz

(full circles).

In both cases, the

high frequency

sets _are shifted in order that the first

points

of the _curves coincide. At Te~p = 4.2 K this translation allows _{to superpose} the _curves while at

Te~p = 23 K it does not.

In order _to underline the unusual character of the

dispersionless

evolution measured at 4.2

K,

we have

reported

in

Figure

4 the data obtained at different times after the

beginning

of the isothermal

evolution, displayed

_{as a} function of

frequency.

The isochronous values lie _on

parallel

lines. What is

surprising

is the

perfectly dispersionless

evolution towards the

dispersive

limit

e).

This result is

important

^{since it} allows to rule _out weak

ergodicity breaking

which would

imply

a (u~t)

scaling

for the

susceptibility _[2].

In all what

follows, except

^for the

temperature cycles,

we focus _our attention _on data recorded at

Texp

= 4.2 K. In consequence, the

frequency f

of the

measuring

^field becomes _an irrelevant

parameter

^{for the} kinetic

part.

4.2. COOLING RATE DEPENDENCE.

Figure

5 shows the time evolution of e' for

sample

VIII at different

cooling

rates R. The most

striking

result is that the asymptote

strongly depends

=

~000s

~ 27 K (x ₌ °.01'~

~ t=30000s

~ _~~K

o t=°°

, ,

E

,

o ,

2350

.

° o

' ,

. _o

,

o . _o '

o . o

~

a .

o .

2250 .

°

~ ~

2150

2.5 3.5 4.5 5.5 6.5

log

lo

Fig.

4. Real part e' of the dielectric constant _{as a} function of

frequency (on

_a

logarithmic scale)

measured at Texp = 4.2 K for different times: t

= 0 _s

(full triangles),

t ₌ 5000 _s

(open diamonds),

t = 30000 _s

(full squares)

and t ~ _cxJ

(open circles).

The isochronous _{curves are}

parallel.

150

,

T ₌ 46 K (x ₌ 0.025 )

~

T~~~ ⁼ 4.2 K

1050 ^R " 0,15

R ₌ 0.02 ©s

950

R=-0,004

850

0 20000 40000 60000

t

(s)

Fig.

5. Time evolution of the real part e' of the dielectric constant recorded at texp ₌ 4.2 K and

f

₌ 100 kHz _on the _z

= 0.025

sample

for several

cooling

rates: R

= -0,15

K/s (upper curve),

R ₌ -0.02

K/s (middle curve)

and R ₌ -0.004

K/s (lower curve).

on R _or, in other

words,

^the

equilibrium

state, ^when temperature, pressure and electric field

are

fixed, depends

_on the initial conditions. This is the definition

generally

used in Classical Mechanics for true

ergodicity breaking.

The

dependence

of the

asymptotic

value

e)

_on the

cooling

rate is

particularly

well

represented

in _a

graph

with the abscissa

equal

to

logic(-R),

since in this _{case a}

straight

line passes close to all

points. Moreover,

in order to compare all

our

data,

_{we use} for each

sample

_a reduced ordinate

e~ _le[~~,

where e[~~ ^is

conventionally

the

asymptotic value, depending

_on lithium

concentration,

obtained with the fastest

cooling (see

Tab.

II). Consequently,

_on all these

plots

the

straight

lines _{converge to a common}

point

located at the abscissa -0.52

(which corresponds

to R

= -0.3

K/s)

at the ordinate 1. This is shown in

Figure

6 where it _can be _seen that the

steepest positive slope belongs

to the

sample

with

x =

0.025;

the

slope

is smaller for the _z

= 0.017

sample

and much smaller for the _x

= 0.011

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 339

Table II.

Asymptotic

va1iles at

f

= 100 kHz obtained at the

reference cooling

rate R

=

-0.3

K/s for

the

different samples.

sample

_e[~~

VII 4585

VI 2250

v 1560

VIII rosa

IX 930

III 680

x < 0.001

- x = 0.011

+ x = 0.017

-~ x = 0.025 1.1

e :

~uJ

~ w-

---- p

~ -a~ ,,--

uJ e- -,-

_«-?

,o--

o_9 _-~--

-r-'~'~

o_---"~

i T =4.2K

0.8 ^"~

-2.5 -2 1.5 0.5

log (-R)

Fig.

6. Reduced

asymptotic

value

EL le[~i

of the real part ^{of the} dielectric constant

plotted

_{as a} function of the decimal

logarithm

of )R) ^{for the}

nominally

_pure

sample (full triangles),

the _{z =} 0.011

sample (open

_squares

),

the _{z =} 0.017

sample (open circles)

and the _z

= 0.025

sample (open diamonds).

The measurements were

performed

at Texp = 4.2 K and

f

= 100 kHz.

sample,

but still

positive;

it becomes

slightly negative

for the

nominally

_pure

sample.

The ratio _can be written _as

e)le[~~

=

alogio(-R)

+ b where _a is _an

increasing

function of the concentration _z in this range.

Other features _are well

displayed

_on the _same scale which

gives

them _a linear variation. For

instance,

the reduced initial value

e[ along

with

e)

_are

given

in

Figure

7 for

sample

V. It is

interesting

to note that the

amplitude

he'

=

e[ e)

of the kinetic effect also increases with

)R). Indeed,

the variations of

e[

and he' _seem to be correlated. The two

parameters

to ^and a

which describe the

shape

of the relaxation _curve also vary with R. For all the

samples

the time to

gets

shorter _as the

cooling

rate

gets

^faster. This parameter ^is

displayed

in

Figure 8,

with the most

probable

time

6mp

=

to/(1+ a).

The

exponent

a

slowly

increases with the

cooling

rate

(Fig. 9).

~Ve notice that this variation is inverted for the pure

sample

which

already

had

a

singular

behaviour for

e[.

4.3. LITHIUM CONCENTRATION DEPENDENCE. We

gathered

the data obtained _on all six

samples

in

Figure 10,

where the

slope die) _le[~~) Id log~o(-R)

is

given

as a fonction of the

a epsilon t _{= oo} T

= 35 K (x

= 0.017)

o epsilon t = 0 ^'

~ ~~ ~

1650

£~ o

~

o

1550

a ,

A

~ o

,

o ,

a

1450

2.5 2 1.5 -0,5

log (-R)

Fig.

7. Some characteristics of the real part e' of the dielectric constant at Texp = 4.2 K and

f

₌ 100 kHz

displayed

_{as a} function of the decimal

logarithm

of )R) ^{for the} z = 0.017

sample:

initial

(open circles)

and

asymptotic (full triangles)

values

. to (x=0.017) T, = 35 K (x

= 0.017)

a theta _mp (x=0.017) _~

~ ~ ~ ~

2500 ^~~~

2000

/s ,

~1500

^'

E ^~

~1000

A

" .

500 ^{~ ~}

o

2.5 2 1.5 0.5

log (-R)

Fig.

8. Times to

(squares)

and 9mp

(triangles)

_{as a} function of the decimal

logarithm

of )R)

plotted

at Texp = 4.2 K and

f

= 100 kHz for the _{z =} 0.017

sample.

lithium concentration _z. Two

regimes

_are

clearly

_seen: for _z < 0.025 the

slope,

I.e. the effi-

ciency

of the

cooling rate,

^increases with _z,

becoming strong

^for z = 0.025

(this already

could be _seen in

Fig. 6);

for _~ > 0.03 the effect is much smaller and it

only weakly depends

on x. It is

important

to notice that the

change

of

regime

coincides with the

change

in the nature of the

phase

transition. The dielectric constant recorded _on

heating

the two

highest

concentration

samples

shows _a

discontinuity occuring

at the _same temperature

Tt

at all

frequencies;

_more-

over, acoustic measurements

performed

_on

sample

III show the _same behaviour at the _same

temperature

[10].

^{This is the}

signature

of _a first order

phase

transition. On the

contrary,

^the dielectric constant of the four other

samples

varies

continuously

with

temperature. Figure

10

puts

ⁱⁿ

evidence,

not

only

the role of the

Li~ ions,

but

mainly

the

importance

of the state in which

they

_are frozen in at low

temperatures (ferroelectric phase

_or

dipolar glass phase).

Kleemann et al.

[11]

^have

predicted

_a

change

in the nature of the low

temperature phase

for _a

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 341

T

= 35 K (x ₌ 0.017)

T = 4.2 K

0.4 ^"~

t1 0.3

, ,

a

0.2

, ,

, ,

o-i

o

2.5 2 1,5 0.5

log (-R)

Fig,

9. Exponent _a, ^measured at Te~p ₌ 4,2 K and

f

= 100

kHz,

sketched _{as a} function of the decimal

logarithm

of )R) ^{for the} x = 0.017

sample.

2

~&~ 0.I

to .

,(

_0.05

G

.

~

g o ^{. O} .

w~

)-0,05

U 0 0.01 0.02 0.03 0.04 0.05 0.06

Concen~ation _x

Fig.

10. Slope of the reduced

asymptotic

value

ecole[~i

of the real part ^{of the} dielectric constant

versus the decimal

logarithm

of )R)

(as

it is

displayed

in

Fig. 6)

sketched _{as a} function of the lithium

concentration _z for the six KLT

samples.

critical concentration _{xc near} 0.022

(indeed

between 0.016 and

0.026).

This is

compatible

with

our results

(0.025

< xc <

0.030).

4.4.

QUENCHING

TEMPERATURE DEPENDENCE.

Figure

it shows both the

asymptotic

value

e)

and the initial value

e[,

for

sample

VIII _{as a} function of the

quenching

temperature

Tq

^which

interrupts

different

cooling

_processes, all driven with the _same rate R

= -0.009 K

Is.

The

quenching

at

Tq

^has a clear influence _on

e)

when it _occurs not far from the transition temperature

Tt,

but it

plays

_a minor role _at lower temperatures.

Anyway,

this shows that the

equilibrium

state

depends

on the

history.

This is _true

ergodicity breaking, according

to the

Classical Mechanics definition.

Here

again,

a correlation is observed between

e)

and the

aging amplitude

he':

they

simul-

taneously

decrease when the

system

^is

quenched

farther from the transition.

Perhaps

_more

surprising

_are the behaviour of the characteristic time to and of the most

probable

time

6mp

sketched in

Figure12

and of the

exponent

_a

plotted

in

Figure

13:

they steadily

_vary

(to

^and

6mp

^{decrease while} a

increases)

and seem to tend towards limit values as

Tq

^{increases from}

, epsilon t _{= oo} T ₌ 46 K (x = 0.025)

o epsilon t ₌ 0 '

~ ~ ~ ~

990 ^"~

E~ ^~

o ~

~ a

940

o o

°,

,

o o

g ^O O ~ '

, , , a , ^{' '}

1

890

~ ~~ ~~

T~

lKj

^{~~ ~~}

Fig.

11. Some characteristics of the real part e' of the dielectric constant at Texp ₌ 4.2 K and

f

= 100 kHz

displayed

_{as a} function of the

quenching

temperature ^for the _{z =} 0.025

sample:

initial

(open circles)

and

asymptotic (full triangles)

values.

. to

a the~mp 2000

. T =46 K ix=0.025)

1500

.

(,p"~'~

^~

a .

45/s

'

4J1000 .

E ,

n '

.

~ ~

500 '

' fl11

_'

'

o

0 10 20 30 40

T

(K)

Fig.

12. Times to

(full squares)

and 9mp

(full triangles)

_{as a} function of the

quenching

temperature plotted at Texp = 4.2 K and

f

= 100 kHz for the _z

= 0.025 sample.

Texp

= 4.2 K towards

Tt

" 46 K. There is _a

strong

contrast between the times to and

6mp,

and the

exponent

_{a, on one} hand and the limit

e)

_on the other hand: below about 20 K the formers

strongly

_vary while the latter is constant.

4.5. TEMPER.~TURE CYCLES.

Figure

14a shows _a

cycle experiment

with T2 >

Ti

where

the

temperatures

are

Ti

= 4.2 K and

T2

_" 29.5 K. For each of the three

parts,

relaxations

are observed. The best fits for the first and the third

parts (both

at

Ti) _give

the

asymptotic

values

e)

= 980 and

e)

=

960, respectively.

The second

asymptote

is lower than the first:

the

sojourn

at

T2

>

Ti

has acted _{as an}

annealing.

Figure

14b shows _a

cycle experiment

with

Ti

>

T2

where the

temperatures

are

Ti

₌ 29.5 K and

T2

= 4.2 K. In each of the three parts, relaxations _are observed too. But _a fundamental difference with the

previous experiment

is that the best fits for the first and the third

parts

(both

at

Ti) give equal asymptotic

values.

Moreover,

a well chosen horizontal shift of the

N°2 NON ERGODIC AGING IN Ki-xLi~Ta03 343

0.4

T=46K ix=0025)

T =42K

0.3 "P

,, , , ,

"

t ^'

0.2

, ' ,

,

, a

U-I

o

0 10 20 30 40

T

(K)

Fig.

13.

Exponent

_a, measured at Texp ₌ 4.2 K and

f

₌ 100

kHz,

sketched _{as a} function of the

quenching

temperature for the _x

= 0.025

sample.

third part towards shorter times puts the two branches onto _a

unique power-law

relaxation

curve. More

precisely, larger

the difference between

Ti

and

T2, larger

the time shift. For the

experiment displayed

in

Figure

14b the duration t2 of the second

part

in

completely annihilated,

even if _a

strong

^relaxation

obviously

acts

during

that time.

According

to the

point

of

view,

the duration t2 exists

(at T2)

or does not exist

(at Ti ).

^Such a

paradoxical

behaviour _was

already

observed for SG

[12].

5.

Comparison

with other

Experimental

Results

We have

just

^{described the}

peculiar aging

behaviour of _some OG which _are frustrated and disordered materials. One _can expect other such materials to

present comparable

behaviours and in

particular

broken

ergodicity. Obviously,

_a

special

interest must be

given

to SG.

However,

^before we consider the _case of different

materials,

_we recall that an OG

(actually

KLT

itself)

has

already

been examined from the

point

of view of

ergodicity breaking _[13].

More

precisely, cooling

rate influence has been studied _on the field induced

polarization

above and below the de Almeida-Thouless line

[14j, supposed

to be _a limit of

ergodicity

in the

temperature-electric

field

plane. However, owing

to the

experimental

method used in this _case,

symmetry (and therefore, ergodicity)

is

externally

broken

by

the

applied

field.

Therefore,

these

experiments

_are not

directly comparable

to _ours.

To _our

knowledge,

the most

complete

_{report on}

physical aging

ⁱⁿ

amorphous polymers

is the book of Struik

[15]

^where

long aging experiments

_are

reported

on a

great

^{number of}

polymers.

Those

experiments

have allowed to deduce useful

scaling

laws

but, unfortunately,

the characteristic

aging

times of those

systems

_were so

long

that

experiments

did not reach _any

asymptote.

It is not

possible

then to know if these

polymeric systems

present a true

ergodicity breaking

_or not I.e. if the

equilibrium

state

they

would reach after _a

sufficiently long

time would

depend

_or not on the initial conditions.

Anyway,

these

aging experiments

have

inspired others, particularly

on SG.

Accordingly,

numerous

experiments

have been

performed.

It _was found that the field cooled

magnetization

was not

constant,

but

slowly

varied with time

[16-19]. Moreover,

the role of _two time scales

was also

put

ⁱⁿ evidence: the

waiting

time t~

(when

_a

biasing magnetic

field is

applied)

and the observation time t

(starting

when the field is switched

off). Indeed,

it has been shown that the

important quantity

for

aging is,

at least in _a first

approximation,

the dimensionless

variable

t/t~.

Since _we

performed

all our

experiments

without

applied

electric

field,

there is