Critical behavior of acoustical waves in ferroelectric-ferroelastic phase of Tb2(MoO4)3

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Critical behavior of acoustical waves in ferroelectric-ferroelastic phase of Tb2(MoO4)3

J.M. Courdille, R. Deroche, J. Dumas

To cite this version:

J.M. Courdille, R. Deroche, J. Dumas. Critical behavior of acoustical waves in ferroelectric- ferroelastic phase of Tb2(MoO4)3. Journal de Physique, 1975, 36 (9), pp.891-895.

�10.1051/jphys:01975003609089100�. �jpa-00208327�

CRITICAL BEHAVIOR OF ACOUSTICAL WAVES IN

FERROELECTRIC-FERROELASTIC PHASE OF Tb2(MoO4)3

J. M.

COURDILLE,

^{R. DEROCHE and J. DUMAS}

Laboratoire d’Ultrasons

(*),

Université Paris

VI,

Tour

13, 4, place Jussieu,

75230 Paris Cedex

05,

^France

(Reçu

^{le 17 mars}

1975, accepté

^{le 22 avril}

1975)

Résumé. ²⁰¹⁴ Des résultats de mesures ultrasonores sont

présentés

pour la

phase

^basse tempéra-

ture d’un monodomaine de TMO à

l’approche

^de

To ~

160 °C, ^les constantes élastiques

C11

et C22

varient avec un exposant critique (0,50 ^± 0,03). Les atténuations 03B111 ^et 03B122 ^sont égales ^{et leur} exposant

critique

^est (1,45 ± 0,05). Ces résultats sont

interprétés

^sur la base des

hypothèses

suivantes :

isotropie

de la courbe de

dispersion

critique ^{au centre de la} zone,

couplage

^{linéaire en} déformation et quadratique ^en paramètre ^d’ordre.

Abstract. ²⁰¹⁴ Ultrasonic measurements are analysed ^{in the}

low-temperature phase

^{of TMO}

monodomain. A 0.50 (± 0.03) ^{critical index for elastic constants}

C11

and

C22,

^{and a 1.45} (± 0.05)

critical index for 03B111 and 03B122 attenuations of

longitudinal

^waves along ^03B1 and b axis were found.

These results can be

explained

^{on the}

hypothesis

^{of an}

isotropic

^softmode

dispersion

^curve by ^a

coupling

^{linear in} strain, ^and quadratic ^{in order} parameter. ^The equality 03B111 ⁼ 03B122 which is found is

explained

by symmetry.

Classification Physics ^Abstracts 7.270 -7.488 - 8.780

1. ^- Gadolinium

molybdate Gd2(MO04). (GMO)

and terbium

molybdate Tb2(Mo04)3 (TMO)

^under-

goes ^a

,structural phase

transition

4

2 m H m m 2 at about

160 °C,

which is the consequence of a soft mode

instability

^{in the}

high temperature phase

^at

the

( 110)

^corner

point

of the first Brillouin ^zone

[1].

At the

macroscopic ^level,

this results in the appea-

rance of a

spontaneous polarization P3 [2]

^{and a}

spontaneous

^strain u12

[2]

^both

originating

^{in the}

same atomic

displacement [3]

^{which is the true order}

parameter.

^{The critical}

mode,

^{which is}

overdamped,

has been well observed in the

high temperature phase

where its

frequency

square decreases

linearly

^with

T -

To,

^where

Tô

^{is some Curie}

temperature [1].

In the low

temperature phase,

^{the soft mode is Raman}

active in the center of the Brillouin zone, but informa- tion about it is rather uncertain

[4].

^One

fact, however,

is

unquestionable :

the mode is

overdamped

^{in the}

100°-160 °C

region [1].

Ultrasonic measurements

[5, 6]

and Brillouin diffusion

experiments [7, 8]

^have

been carried out on GMO to

study

^{elastic constant}

variations. Some data from

Brillouin

linewidths have been

published [8, 9],

^but

its

would seem difficult to

obtain a

high degree

^of accuracy

concerning

^their

temperature dependence.

New direct measurements of ultrasonic attenuation have

been given recently [ 10]

(*) Associated with the Centre National de la Recherche Scienti-

fique.

for a and ^c directions in a GMO monodomain :

a

coupling

linear in strain and

quadratic

^{in order}

parameter

^{has been} found in the

low-temperature phase,

and the critical

exponent

of the attenuation has been found in

good agreement

^{with a soft-mode}

isotropic

ⁱⁿ wave-vector space.

2. ^- In this article some ultrasonic measurements

conceming

both elastic constants and

attenuation,

obtained from a

good quality

^{TMO monodomain}

crystal,

^are

presented

^and

compared

with the GMO data mentioned above

[10].

In order to

verify

^and

complete

^{the data}

published

in a

previous

paper

[10] temperature

variations of elastic constants and attenuation have been measured

simultaneously.

^{We aim to} show that the ultrasonic attenuation is determined

by

^the

dynamic properties

of the

underdamped

soft mode in the

temperature

range in which ^we worked.

Temperature

variations of elastic constants

Cl,

and

C22 (Fig. 1)

and attenuations _aii and _a22

(Fig. 2)

of

longitudinal

^{elastic waves}

propagating

^{in a and b}

directions were

measured, using

^the

pulse-echo

^method

at 540 MHz. The TMO

crystal, carefully

oriented and

polished,

^{was set between two}

quartz

rod transducers

producing

^and

detecting longitudinal

^{acoustic waves.}

The

signal

^{was observed}

by

transmission.

Only

^the

relative variation of the ultrasonic attenuation is obtained

by recording

the first transmitted

pulse level,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003609089100

892

FIG. 1. ^- Attenuation of longitudinal ^{elastic waves versus} tempe-

rature in TMO monodomain. _{oeii -} propagation along ^a-ortho-

rhombic axis ; _a22 ^- propagation along b-orthorhombic axis ;

a°° is the background attenuation from the bond and no critical

properties.

Fic,. 2. ^- Elastic constants Cl, and C22 ^versus température ^{in TMO} monodomain. Ci 1 and C22 ^are asymptotic values determined

by log-log analysis ^of figure ^3.

as a function of

temperature.

Two electrodes were

deposited perpendicular

^to the c-axis in order to

apply

^an electric field to the

crystal

and make it a

monodomain in the

low-temperature phase.

^The

TMO ferroelectric and ferroelastic

properties

^show

that inversion of the a

and

^{b axes is}

possible simply by inverting

the electric field

polarity applied along

the c-axis

[11]. By

^this

method,

we can measure atte- nuation variations for

twg

^{axes of}

propagation

without _any

change

^{in the}

experimental set-up.

In

particular,

the bond attenuation is

exactly

^the

same for the two measurements and this

permits

exact

comparison

^of all and _(X22. It has been shown that even without an

applied

electric field

during

^the

experiment,

^the

crystal

stays ^{in a} monodomain state

throughout

the entire

temperature» range,

and all the measurements

given

^{here were} made with a zero

electric field

applied

^{to the}

crystal.

^{Elastic constants}

were measured

by

^an interferometric method and their

temperature

variations recorded with a

precision

^r

of about

10-’.

The absolute values of the elastic

constants were measured at 25 OC and 230°C

by

the same

method,

^but

they

^were obtained with a

precision

^not better than

10-2.

Table 1

presents experimental

values of critical coefficients n with the

corresponding temperature

range for which

they

are valid.

TABLE 1

Measured critical indices n

for

^the

quantities Cl,, C22,

all and

(;(22 in

^{the low} temperature

phase of

TMO. The error

given

^{here is}

only

^a

graphically

determined order

of magnitude.

3. ^- An

expression

^{for free} energy has been obtain- ed

by

^Dvorak

[12] using

the Landau method

approach, considering

the transition to be second-order. Dvorak

supposed

that the transition is

govemed by P3,

U115 u22, U33, u12,

’11 ), 112 > only

^and gave the free _energy as a function of these parameters :

expression

^{in which}

Fo

is the non-critical

part

^{of the} free _energy,

f i

^is

only

^a function of critical modes

il, >

^and

n2 ), f2

is elastic _energy,

f3

^{terms involve}

coupling

^between

polarization P3

^{and soft}

modes,

and the last four terms

explicitly

govem the

coupling

between soft modes and elastic strains u.

n1 >

and

n2)

^{are mean} values of

amplitudes

^{of the}

modes which appear in the ^center of the Brillouin

zone for T

Tc.

The soft mode in the low

temperature phase

is unknown. _{It may}

be n1 )

^or

n2)

^{or a}

linear combination of the two, ^{so it is}

simply

^written

as

n >.

^{In the}

expression of F,

^{the term}

. _

(which

^{is now written}

by

^{us as}

c)2(Ull

⁺

U22) 11 )2)

contains the order

parameter

and the strains _{u 1} and _U22 which are involved in

longitudinal

^elastic

wave

propagation along a

^{and b axes. In the}

vicinity

of the

phase transition,

^order

_parameter

fluctuations il ^must be taken into consideration. This is written

c)2(U11

⁺

U22) « Il )2

⁺

2 il >

^{i +}

n2).

^{If this term}

governs the ultrasonic

attenuation,

its contribution to the ^{two axes a} and b must be

equal :

this is well born out

experimentally.

As the first term is not

fluctuating

^{it cannot contri-}

bute to ultrasonic attenuation.

However,

^{it does} influence elastic constant variations and this effect

’

has been calculated

by

^Dvorak

[12] :

^{no variation}

FiG. 3. ^- (x°° - a(T)) plotted ^versus (Ta - T) ^{in a} logarithmic

scale. Determination of attenuation critical indice.

is

expected

^{in the}

high-temperature phase,

^{but the}

complicated expression

^{for the}

low-temperature phase

would

suggest

^{an elastic constant} modification. It is

impossible

^to

give

^{an order of}

magnitude

^{for the}

effect as few of the coefficients of the free _energy

expansion

^are known. We suppose then that this term is

unimportant

^{for both}

phases

and that the variations observed can be

explained only by fluc- tuations il

^{of the order} parameter.

(This hypothesis

is exact in the

high-temperature phase,

^where

ti >

⁼

0.)

Let us now examine

successively

^linear uil and

quadratic Ur¡2

^{terms to evaluate}

temperature

^varia- tions of elastic constants and attenuation in the

vicinity

^{of the}

phase

transition. The

frequency

^of

ultrasonic ^waves

propagating

^{in the}

crystal

^is

changed by

^the

coupling.

^{The real}

part

of the variation

gives

^an

ultrasonic

velocity and,

from

this,

^{elastic constant}

variations. The

imaginary part gives

the attenuation

produced by

^the

coupling.

The calculations which follow are based ^on the

Pytte theory

of ultrasonic anomalies

[13]

^{which can}

be

applied

^without modification to TMO.

3. 1 LINEAR COUPLING

2 Ô2 il >

nu. ^- The calcu- lation considers

coupling

^{between two}

damped

^har-

monic oscillators : the order

parameter fluctuations il having

^{as a}

preponderant spectrum frequency

^the

soft mode

frequency Sl,

and the ultrasonic

strain,

with

frequency

^{m. When w}

Q 2/r

^where

^{T -1}

^is

the soft mode

lifetime,

^linear

coupling produces

^a

step discontinuity

in elastic constants at the transi- tion

temperature,

^{and a}

l/rD2

variation in ultra- sonic attenuation. The condition ^Co

Q 2/r

^{is well}

verified over the whole

temperature

range examined.

(ro ~

^{34 x}

¹⁰¹ rad. s-1,

^{and at 140°C}

and

Using

^{the classical}

hypothesis [13],

^{r = constant}

and

(22

⁼

a(To - T),

^the

following temperature

variations were derived :

Evidently

^linear

coupling

un does not

explain

the variations observed in measurements of

Ac _(T)

c

and

a(T) in

TMO.

3.2

QUADRATIC

^COUPLING

bUt,2.

^- The theoretical results are rather more difficult to obtain in this case

which involves a third order anharmonic interaction between the critical

optical phonons

and the acoustical

phonons :

^the

Ô2

coefficient

gives

^a measurement of the

coupling strength.

The method of calculation which has been well

developed by Pytte

^{in a} very similar case of

perovskite

structural transition

[13]

gives

the results in our own notation :

where

X21J2

^and

X;21J2

^are the real and

imaginary parts, respectively,

^{of a}

susceptibility

^{x relative to}

,,2,

^the

FIG. 4. ^- C°° - C(T) plotted ^versus (T, - T) ^{in a} logarithmic

scale. Determination of elastic constant critical indices.

894

square of the order

parameter,

p the

density

^{and v}

the ultrasonic

velocity

^without

coupling.

A suscep-

tibility

^{of this} type ^is

generally

^{difficult to}

calculate,

but if we suppose the statistical

independence

^of

fluctuations of different wave vectors,

X,,2,,2(k)

^can

be

expressed

^{as a} function of xnn, ^the

susceptibility

relative to the order

parameter.

_x ^is

represented

with a

good approximation by

the harmonie oscillator

susceptibility :

where ro is the ultrasonic

frequency, Qk

^the

frequency

of a k-mode of the soft mode

dispersion

curve, and

Tk 1

^the lifetime of this k-mode. The

expression

^of

X" 2,,2 ^obtained

by

^this

decoupling

method is valid far

enough

from the transition

temperature

^{and we} have used it to

interpret

^our

experimental

^results.

Different cases must be

distinguished : overdamped (Qk « r k

^and

underdamped (Qk » rk)

soft modes.

This distinction is

important only

for the calculation of the attenuation which is

essentially

^connected

with the

dynamic

behavior of critical modes. The result for

Aclc

^{will not} be affected

by

the distinction of these two cases

The formulas which

give

^{a are in the two cases}

overdamped

^mode

underdamped

^{mode .}

The second

point

^to be examined is the variation 0

Qk

^{as a} function of

kx ky,

^and

kZ,

necessary for the

c lculation

^of

_.

^For

^this,

^the

anisotropy

^{of the}

critical

dispersion

surface in the Brillouin zone must be taken into account. Two different variations

are examined :

which is called the

3-dimensional

^{case and}

with _c b which is called the 2-dimensional case.

a,

b,

^{c are constants}

independent

^{of the}

temperature.

The calculation of

(3)

shows that the critical character is more

pronounced

^for the 2-dimensional ^case than for the 3-dimensional _one,

resulting

^{in a critical}

index

higher

in the former case. The results of this discussion which are

developed

ⁱⁿ

Pytte’s

paper

[13]

are

reported

^{in table II in which the} critical indices for elastic constants and attenuation in the different

cases are

given.

^A

comparison

of the table 1

(experi-

mental

results)

and table II

(theoretical results)

shows that in the case of

TMO, only

^one

possibility

exists : the

overdamped

soft-mode is three-dimen- sional.

TABLE II

Theoretical values

for

^{critical indices}

of

^elastic

constant and attenuation

of longitudinal

^{waves in}

function of

characteristics

of

^the

soft

^{mode in the}

case of a coupling Ur¡2

^{linear in strain and}

^quadratic

in order parameter,

from reference [13].

Using ac/c

^{and a to determine an order of}

magni-

tude for

D2/T,

^{it is found}

^that,

^at

^{140 °C,}

^{the ratio}

is ^~ 1.5 ^x

1011 rad. s-1.

Thus we have Co «

Q 2/r.

This case is

important

^because rok ⁼

Sl2/T

is the limit-

ing

condition for

validity

^{of the}

previous

calculation of initial

exponents. Furthermore,

^{in the}

temperature

range where Ü)k _

D2/T,

the influence of the central mode on ultrasonic

propagation

^must be taken into consideration

using

^{the Schwabl}

theory

^{for exam-}

ple [14].

Our value found shows that at 140,DC we are not in this

region

which is closer to the transition

temperature.

The

present

results and those of reference

[10]

together

^{form a set} of ultrasonic measurements which

give

^{us some} information on the

coupling

^between

the strain and the soft-mode

vibration,

^{and on the}

shape

of the soft-mode

dispersion

^curve.

4.

Acknowledgments.

^{- We}

gratefully acknowledge

Prof. J. P.

Chapelle (Laboratoire

^de

Physique

^Cristal-

line, Orsay, France)

^who

provided

^{us with}

high quality

^TMO

single crystals.

References

[1] DORNER, B., AXE, ^J. D., SHIRANE, G., Phys. ^{Rev. 6B} (1972)

1950.

[2] Throughout ^{this article,} the indice 3 corresponds ^to the z-axis.

Indices I and 2 correspond ^to a-axis and b-axis in the orthorhombic phase. This convention is the same for the two phases.

[3] KEVE, E. T., ABRAHAMS, ^S. C., BERNSTEIN, J. L., ^J. Chem. Phys.

54 (1971) 3185.

[4] ULLMAN, ^F. G., HOLDEN, ^B. J., GANGULY, B. N. and HARDY, J. K., Phys. ^{Rev. 8B} (1973) ^2991.

[5] HÖCHLI, U. T., Phys. ^{Rev. 6B} (1972) ^1814.

[6] EPSTEIN, ^D. J., HERRICK, ^W. V., TURCK, R. F., Solid State Commun. 8 (1970) ^1491.

[7] ITOH, S., NAKAMURA, T., Phys. ^{Lett. 44A} (1973) ^461.

[8] LUPIN, Y., HAURET, G., ^J. Physique ^{Lett. 35} (1974) ^L-193.

[9] ITOH, S., NAKAMURA, T., Solid State Commun. 15 (1974) ^195.

[10] COURDILLE, ^J. M., DUMAS, J., J. Physique ^{Lett. 36} (1975) ^L-5.

[11] CUMMINS, S. E., Ferroelectrics 1 (1970) ^11.

[12] DVORAK, V., Phys. Stat. Sol. 46b (1971) ^763.

[13] PYTTE, E., in Structural Phase Transitions and Soft Modes, (ed. by ^{E. J.} Samuelson, ^E. Anderson, J. Feder, ^Universi- tets-forlaget, Oslo) 1971, p. 151.

[14] SCHWABL, ^F. Phys. Rev. B7 (1973) ^2038.