PHYSICSBUDAPEST INSTITUTE FOR RESEARCH CENTRAL T, SIKLÓS T DANY

T U - ^ $o C t

KFKI-1981-5^4

T, SIKLÓS

MODEL DESCRIPTIONS OF

FERROELECTRIC PHASE TRANSITIONS

H ungarian ‘Academy o f ‘ Sciences CENTRAL

RESEARCH

INSTITUTE FOR PHYSICS

B U D A P E S T

—

2017

MODEL DESCRIPTIONS OF FERROELECTRIC PHASE TRANSITIONS

T. Siklós

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

Submitted to Bulletin de la Sooiete dee Sciences et des Lettree de Lódz

HU ISSN 0368 5330 ISBN 963 371 836 8

ABSTRACT

Models describing both the "order-disroder" and "displacive" ferroelec­

tric phase transitions are reviewed. By introducing a model pseudospin-phonon Hamiltonian the tunnelling motion of the atoms is also taken into account.

On the basis of the self-consistent phonon-field and molecular-field approxi­

mations a complete system of self-consistent equations for the two order parameters (average atomic displacement and average population of equilibrium positions) is obtained. The analysis of this system of equations shows that the ferroelectric phase transition (first or second order) can be either the order-disorder, displacive or mixed type, depending on the dimensionless coupling energy of atoms (or their zero-point energy).

АННОТАЦИЯ

Предложены модели для описания сегнетоэлектрических фазовых переходов как типа "порядок-беспорядок", так и типа "смещения". В более общем варианте модели учитываются также эффекты, связанные с туннелированием. Но основе при­

ближений самосогласованного фононного поля и молекулярного поля получена са­

мосогласованная система уравнений для двух параметров порядка /среднего смещения активных атомов и средней заселенности их равновесных положений/.

Качественный анализ и также численное решение уравнений показывают, что сег- нетоэлектрический фазовый переход /первого или второго рода/ может быть как типа порядок-беспорядок, типа смещения, так и смешанного типа, в зависимости от величины безразмерной энергии связи атомов /или их энергии нулевых коле­

баний/ .

KIVONAT

A ferroelektromos fázisátalakulások tárgyalására modelleket javaslunk, amelyek egyaránt jól leirják a rend-rendezetlen tipusu és a rácstorzulással járó fázisátalakulásokat. A tanulmány második részében az állapotok közötti alagutazással kapcsolatos effektusokat is figyelembe vesszük. A self-consist- ent fonon-tér és molekuláris tér közelitéseket alkalmazva a két rend-paramé­

ter - az aktiv atomok átlagos elmozdulása és egyensúlyi helyzeteik átlagos betöltöttsége - meghatározására egy self-consistent egyenletrendszert nyerünk.

Az egyenletrendszer kvalitatív vizsgálatából, éppúgy mint a numerikus számí­

tások eredményeiből láthatjuk, hogy az atomok redukált kötési energiájától /illetve a null-ponti rezgések energiájától/ függően lehet /az első-, illet­

ve másodrendű/ ferroelektromos fázisátalakulás rend-rendezetlen tipusu, rács­

torzulással járó vagy kevert tipusu.

It is generally assumed t h a t there a r e two b a s i c kinds of phase t r a nsitions (PT) in f e r r o e l e c t r i c s , one b e i n g the o r d e r - d i s o r d e r type a n d the o t h e r being t h e d i s p lacive type

(see, for instance, [1], [2]). In the f o r m e r case t h e PT results f r o m a statistical d i s o r d e r of a t o m s among several

(in the simp l e s t case between two) e q u i l i b r i u m positions.

In the l a t t e r case t h e PT is c a u s e d by l a t t i c e i n s tability against a critical vibrational m o d e (soft mode) .

Nevert h e l e s s it has been s h o w n in t h e last y e a r s that

bot h types of ferroelectric PT c a n be d e s c r i b e d w i t h i n a s ingle model and t h e r e are n o essential d i f f e r e n c e s between them (see, for instance, [3]). In the s i m p l e s t case this model is d e s c r i b e d by the H a m i l t o n i a n w h i c h is e x p r e s s e d as a sum of s i n g l e - s i t e energies, as d e t e r m i n e d by d o u b l e - m i n i m u m potential wells, a n d the h a r m o n i c c o u p l i n g s b e t w e e n atoms in different cells.

T h e nature of the PT described b y such m o d e l s has b e e n exam i n e d by applying both the C u r i e - W e i s s (or molecular-field) and the self-c o n s i s t e n t phonon-field a p p r o x i m a t i o n s . It has b e e n shown [4] by c o m p a r i n g the results of bot h appr o x i m a t i o n s that for a weak lat t i c e c o u p l i n g the c h a r a c t e r of t h e PT is o f the o r d e r - -disorder type, w h i c h is more c o n s i s t e n t l y described by the m o l e c u l a r - f i e l d approximation; f o r a s t r o n g lattice coupling the PT has to be r e l a t e d to the d i s p l a c i v e type, w h i c h can be reasonably d e s c ribed b y the s e l f - c o n s i s t e n t phonon field

2

approximation. S u c h a co n s i s t e n t d e s c r i p t i o n can b e understood under the circums t a n c e s that in the o r d e r - d i s o r d e r transition statistical fluctuations of a t oms on t o their eq u i v a l e n t

e q u i l i b r i u m p o s i t i o n play t h e main role, which is acc u r a t e l y enough d e s c ribed b y the pseud o - s p i n m odel, while in the di s - placive tra n s i t i o n the d y n a m i c a l correlations of atomic d i s ­ placements turn o u t to be m o r e essential, so the self- c o n s i s t e n t phonon - f i e l d a p p r o x imation is more efficient.

However, for a complete d e s c r i p t i o n of ferroelectric PT one has to take i n t o account b o t h m e c h a n i s m s s i m u l t a n e o u s l y

in the frame of a universal model. A unified a p p r o a c h of this type has been p r o p o s e d in [5] and we b r i e f l y c o n s i d e r it in Section 2.

However, the s i n g l e - p a r t i c l e t u n n e l l i n g m o t i o n of a t oms has not bee n taken into a c c o u n t in [5]. The i n corporation of the t u n n elling m o t i o n as an add i t i o n a l degree of freedom l e ads to c o l l ective e x c itations w h i c h may h a v e a soft m o d e c h a r a c t e r [2] or c a u s e the appearence o f a c e n t r a l peak [6],[7]. S i n c e the t u n n elling e n e r g i e s (of t h e order o f the g r o u n d state q uantum splitting) are usu a l l y much s m a l l e r than t h e c h a r a c t e ­ ristic p h o n o n energies, the r o l e of s u c h e x c itations is p r e ­ dominant at low t e mperatures № ~ k T) . O n the o t h e r hand, in

В

addition to a r e n o r m a l i z a t i o n of the p s e u d o s p i n - e n e r g y p a r a ­ meters of the De G e n n e s type [8], the h i g h e r p h o n o n excitations can lead to the structural P T of the d i s p lacive t y p e (against a certain v i b r a t i o n a l mode) at higher temperatures. In [9] ,

[10] the excitations of b o t h types w e r e taken into account s e l f - c o n s i s t e n t l y (within the v a r iational a p p r o a c h of Bogo l i u - b o v ) . This has b e e n a c h i e v e d by r e p r e s e n t i n g t h e cooperative atomic motion as a slow tunnelling process a m o n g several (in the simplest case, among two) e q u i l i b r i u m p o s i t i o n s an addi t i o n to familiar phono n - l i k e oscillations. This m o r e general typ e of ferroelectric P T is b r i e f l y reviewed in S e c t i o n 3. Some c o n ­ clusi o n s are p r e s e n t e d in the last Section.

2. UNIFIED MODEL OF FERROELECTRIC PHASE TRANSITIONS

In a model d e s c r i p t i o n of the structural PT dynamics it is convenient to use the concept of local n o r m a l c o o rdinates

[11], [12], [13] involving all a c t i v e atoms in the given criti c a l v i b rational mode. By using this repr e s e n t a t i o n a s i m p lified m o d e l H a m i l t o n i a n can be wri t t e n in the form:

P2

H = I { 2^ + U (S±) } + j I V(S., S ) . (2.1)

Here m is the c o r r e s p o n d i n g effective mass of the critical mode,

, ' # *->■ -k

the single-site U ( S i) and the pair i n t e r a c t i o n V ( S i , S^) p o t e n ­ tials define the critical dynamics o f the m odel. The l o cal normal c o o r dinate S. d e s c r i b e s a d i s t o r t i o n of the w h o l e unit

l

-f cell l. P i is t h e canonical conjugate m o m e n t u m to

It is a s s u m e d further, that the sing l e - s i t e potential U Í S J has two minima c o r r e s p o n d i n g to two e q u i l i b r i u m atomic configu-

4

rations (a=±l) in the u n i t cell. Therefore the local normal co o r d i n a t e S . can be w r i t t e n as

l

S.l - I

a = ± 1

(2.2)

H e r e о I = 1 (o) and ck =0(1) a c c o r d i n g to w hether t h e atomic c o n f i g u r a t i o n c o r responds to the state a = +1(-1) respectively.

T h e p r o j e c t i o n operator o“ i t s e l f can be e x p r e s s e d b y the p s e u d o s p i n operator

a.a l

1

2 ( 1 + a aA ) ; (a = ± 1 ) , (2.3) w h i c h is i n t roduced as a n inde p e n d e n t variable, c o m m u t i n g with

+ a -+a

the c o o r d i n a t e S, a n d the m o m e n t u m P. operators.

The co o r d i n a t e in t h e state a, s“ can be w r i t t e n as a sum of a static d i s p l a c e m e n t b? a a n d a thermal f l u c t u a t i o n u~

).l

►a li Sa = ba + ua

l l l ba = < Sa > = b

1 1 a (2.4)

w h e r e the s ymbol < ... > stands for a statistical a v e r a g e with the H a m i l t o n i a n (2.1).

T h e r e f o r e this r e p r e s e n t a t i o n of dist o r t i o n s as g i ven by E q s . (2.2), (2.4) en a b l e s one to tak e into account, a t first, the atomic r a n d o m d i s t r i b u t i o n o v e r two e q u i l i b r i u m positions in the cell, using the o p e r a t o r o “ and secondly, t h e thermal fluc t u a t i o n u? in the n e i g h b o u r h o o d of a g i v e n equi l i b r i u m position. In describing o r d e r - d i s o r d e r PT t h e latter variables are usually neglected, wh e r e a s in displacive PT it is assumed

that t h e r e is on l y one e q u i l i b r i u m p o s i t i o n in t h e cells (ct=+l or a = - l ); thus the operator a “ takes t h e same v a l u e at each lattice site i. In this g e n e r a l i z e d m o d e l we w i l l be able to study b o t h types o f PT using the full r e p r e s e n t a t i o n (2.2), (2.4).

Such a physical p i c t u r e is i n agreement with r e c e n t computer simulation and it is also a p p e a l i n g for the r e a s o n of u n i v e r s a ­ lity [6], [14]. It should be pointed o u t that t h e r e p r e s entation

(2.2) f o r atomic coor d i n a t e s as a sum of p s e u d o s p i n and p h o n o n variables was p r o p o s e d by V a k s and L a r k i n [15] in their di s c u s s i o n of o r d e r - d i s o r d e r type s t r u ctural PT (see also [1], § 6). W e g e n e r a l i z e their r e p r e s e n t a t i o n to c o n s i d e r the d i s p l a c i v e type PT as w e ll.

H a v i n g inser t e d the d e f i n i t i o n (2.2) in the H a m i l t o n i a n (2.1), it can be wr i t t e n in t h e form:

i ct=±l

(2.5)

+

V(S“ , S^) }

i,j a,0=±l 1 3 1 3

T h e e q u i l i b r i u m p o s i tions of l a t t i c e atoms Ь ^ = < S i > are d e t e r m i n e d using t h e e q u i l i b r i u m c o n d i t i o n in t h e form

i(a/3t) <P“ (t) > = < [ P“ , H ] >

which l e a d s to t h e equation

< Щ mSl)>+ l v(Si - ??> >=0

(2.6)

(2.7)

T h e phonon spectrum a n d the a v e r a g e values of the a tomic

б

d i s p l a c e m e n t c o r r e l a t i o n functions of the m o d e l can be c o n s idered w i t h i n the framework of the self-consistent p h o n o n field theory

[16] using the t h e r m o dynamical G r e e n ' s functions

D _ (t - t') = « u± (t) ; u. (t') » =

(2.8)

>( —

da) -io)(t-t')

2t\ 6 Di .(ш) ,

w h e r e o r d i n a r y notations are used [17].

Dealing w i t h the p s e u d o - s p i n subsystem a n effective H a m i l t o n i a n c a n be introduced:

H

s у

2 ijj 4 1 3

(2.9)

H e r e the e f f e c t i v e s ingle particle "field" h i and the effective

"exchange energy" J c a n be w r i t t e n on the b a s i s of vari a t i o n a l a p p r o a c h [18] in the form:

h = I 2- < (Р?Г + U(S?) > +

x “ , 2 2m i i' О

a=±l

+ l l j < V(^a ' s3) >

a,ß=±l j 4 1 3 о

i r í I , T

< v<?“ ' S?>

(2.10)

w h e r e the statistical a v e r a g i n g < •• • >q is performed o v e r p h o n o n v a r i a b l e s .

The general model has been c o n s idered in [5] for t h e one p a r t i c l e p o t e n t i a l h a v i n g a d o u b l e - m i n i m u m f o r m

u(s“) = - f (s“)2 + I (s“)4 f (2.1 1)

and for the pair interaction in the harm o n i c approximation:

V(Sal s b = i

D 2 i: (sa - S0)'

1 D ⁽2.1 2)

where the par a m e t e r s A and В d efine, respectively, t h e height of the p o t e n t i a l b a r r i e r Uq = (A /4B)and t h e distance between2

j.

the two m i n i m a 2S = 2 (A/B) . It has b e e n also a s s u m e d that

О

the critical vibrations can be d e s c ribed b y a one c o m p o n e n t local n o rmal c o o rdinata Sa = (S?, 0, 0) t h a t corre s p o n d s to a

1 X

o n e c o m p o n e n t order parameter r>a ~ < s “> . T h e i n t e r a c t i o n (2.12) acts i s otropically bet w e e n the atoms in t h e 3-dimensional

lattice. E m p l o y i n g the r e n o r m a l i z e d h a r m o n i c a p p r o x imation [16]

« (ua )3 í u. » 53 3< (ua )2 > « ua ; u, »

* J * * 3 (2.13)

for the p h o n o n s u b s ystem and t h e m o l e c u l a r field a p p r o x i m a t i o n for the p s e u d o - s p i n subsystem w i t h the e f f e c t i v e H a m i l t o n i a n

(2.9), a c l o s e d s ystem of e q u a t i o n s has b e e n o b t a i n e d for the e q u i l i b r i u m di s p l a c e m e n t s n :

Па " (1 " 3ya} + (\ + n- } fo °-a = ° » (2.14) for the a v e r a g e thermal f l uctuations y^ o f an atom in state a :

у = ~ < (ua)2 >= ?" / dwcoth

a A l A ' [ - i l m « 5 ? | u . » ] (2.15)

2k T ТГ l 1 l w+ie I

a n d for the order p a r a meter a = < a? > o r a = < a . > = 2o+ -

a l l 1

8

ст th (Jo-h)

V

In (2.14) the d i m e n s ionless d i s p l a c e m e n t na= (в/А) b a and

coupling constant f = (1/A) £ <p! w e r e introduced. The e f f e ctive j 13

exchange energy i t s e l f is d e f i n e d by t h e e q u i l i b r i u m displacements:

J ij = (A//4B) “’ij (n+ + n_)2 * (2.17)

so that above the structural PT, when n + =o , J „ becomes zero, leading to the u n i q u e solution оно .

In addition, w e quote the e x p r e s s i o n for the s p o ntaneous polarization, w h i c h is d e p e n d e n t in the present m o d e l both o n the atomic order a n d on the a t omic e q u i l i b r i u m positions, i.e., it is d e t ermined b y two order par a m e t e r s n+ and cr . In d i m e n s i o n ­ less quantities, t h e spontaneous p o l a r i z a t i o n is g i v e n by

P = s

1^

N .

1

<!>*

^s.

= 2 4 " ^{П ) +} 1

2 ° (T1+ + П )

(2.18)

By analysing o n l y the e q u i l i b r i u m conditions (2.14) o n e finds that in a d d i t i o n to the solutions n + = n = О c o r r e s p o n d i n g to the parael e c t r i c phase (with J ij= o) , nonzero solutions nа / о are also possible. In the c a s e of w e a k coupling, fQ « 1 an o r d e r - d i s o r d e r PT c a n occur s i n c e there a r e two e q u i l i b r i u m d i s ­ p l a cements n and n in the unit cell (n -n » o f « 1 ), a n d

+ - + - о

there also exists t h e solution a=o , c o r r e s p o n d i n g to the

complete disorder for an order-disorder P T . While f o r suffi­

ciently s t r o n g coupling, f — 0 . 2 5 only o n e nonzero solution m a y exist, at all temperatures e.g. n ^ o , (for c o m p l e t e atomic order o=+l) and t h e r efore only t h e d i s p l a c i v e PT is possible.

The s y s t e m of s e l f - c o n s i s t e n t e q u a tions (2.14)-(2.16) obtained for the o r d e r parameters n+ and a has been solved nume r i c a l l y in the classical l i m i t of h i g h temper a t u r e s [5].

2 T h e numerical results for cr(x) and n (x) (where т = к Т/(А /В)

i В

is the r e d u c e d temperature) are p r e s e n t e d in Figs. 1-3 for c e r t a i n values of d i m e n s i o n l e s s coupling p a r a m e t e r f . It c a n be o b s e r v e d

О

that for f <_ 0.1 an orde r - d i s o r d e r PT t a kes Diace, for f > 0 . 1 5

о о —

o n l y a d i s p l a c i v e PT is possible, while in a narrow r egion

f B 0.12 the PT is a mixed type one, d e s c r i b e d by a l l the t h r e e order pa r a m e t e r s n+ (x) , n (x) and a(x) . The temp e r a t u r e

dependence of the spontaneous p o l a r i z a t i o n (2.18) for different f Q is s h own in Fig. 4. Note t h a t in the region of t h e order- -disorder PT (f < 0.1), as c o m p a r e d to t h e ordinary Ising model, the spontaneous p o larization is d e c r e a s e d mor e r a p i d l y as the temperature is increased due to the temp e r a t u r e d e p e n d e n c e of the effective exchange energy (2.17).

We note, that t h e s e features are a l s o obtained b y a more sop h i s t i c a t e d c a l c u l a t i o n [19] b a s e d on t h e coherent potential a p p r o x i m a t i o n for the disordered lattice.

We e m p h a s i z e an important a d v a ntage of the p r e s e n t d e s c r i p ­ tion for f e r r o e l e c t r i c PT by t w o order parameters. It enables o n e in o b t a i n i n g the system of s e l f - c o n s i s t e n t e q u a t i o n s for the order parameters to choose var i o u s approx i m a t i o n s : the m o l e -

10

JTig.l. Temperature dependence of the order parameters n+ - average displacement and a~ - average pseudospin v alue, for the dimensionless coupling p a r a ­ meter f =0.10.

о

Fig.3. Same as Fig.l.3 for fo = 0.15.

Fig. 2. Same as Fig.l.3 for fQ = 0.12 .

Fig. 4. Temperature dependence

of the reduced polarization Pg for several values of the coupling parameter f .

and the self-consistent p h o n o n - f i e l d app r o x i m a t i o n for na and Уа in Eq. (2.15), which offers a s a t i s f a c t o r y d e s c r i p t i o n of the PT, both in the case f « 1 (order-disorder t r a n sition of second order) a n d in the c a s e fQ » 1 (the di s p l a c i v e PT for the parameter n ( T ) , of f i r s t order, close to second order), respectively. It is n e c e s s a r y to p o i n t out that the true order of t h e PT (first or second) cannot b e predicted in the m ean- -field approximation (see, for instance, Ref. [1]), so w e will not discuss th i s question here.

3. ORDER-DISORDER, TUNNELLING AND PHONONS IN FERROELECTRIC PHASE TRANSITIONS

In the q u a n t u m limit, T=OK, as follows f r o m the Eq. (2.16), a u n i q u e solution, o hI app e a r s (if > О a n d h >. О ) . The effect of t u n n elling be t w e e n s t a t e s a = + 1 and a = - 1

suggested in [20], makes it possible to generalize the Hamil t o n i a n (2.1) and to introduce in (2.9) the t r a nsverse field ft £ a? ,

i

w h i c h in turn m a y lead to the solution a -*■ о in the case T=OK . In t h e quantum limit the d i s p lacive P T is d e t e r m i n e d by the zero- -point vibration energy. T h e limiting values for it has bee n o b t a i n e d in b o t h cases of t h e ord e r e d (o=l) and dis o r d e r e d (a=o) lattices [21], assuming t h a t the r i g h t choice of the t r a n sverse field ft can e n s u r e the t r a n s l a t i o n f r o m 0=1 to 0=0 in the case of zer o temperature (T=OK) .

12

H owever in the model d e s c r i p t i o n in o r der to t a k e into account the tunnelling motion simul t a n e o u s l y with the statisti­

cal o r d e r - disorder and the p h o n o n o s cillations it is convenient to introduce the time - d e p e n d e n t local n o r m a l coo r d i n a t e

par t i c i p a t i n g in the given PT as de c o m p o s e d into a slow

tunnelling-like coordinate r ± w i t h a c h a r a c t e r i s t i c frequency П and a c o m p a r a t i v e l y fast d i s p l a c e m e n t of the p h o n o n type u i , w i t h c haracteristic frequency ^ü>^q

Such a r e p r e s e n t a t i o n holds u n d e r the "adiabatic" condition:

Í2 « w q . Since the tunnelling energies -flto are of t h e order of the g round state qua n t u m splitting, the adiabatic condition means that the latter ones m u s t be much smaller tha n the

c h a r a c teristic p h onon energies 'ГГшо .

H aving inserted the repr e s e n t a t i o n (3.1) into t h e general Hami l t o n i a n (2.1) w e obtain a H a m i l t o n i a n depending o n the variables r i and u . In o r d e r to separate these variables the variational appr o a c h can be used, a s s u m i n g that the system can be described by a trial H a m i l t o n i a n in the form:

S. = r. + u.

i l l < u. > = О .

l (3.1)

(3.2)

where

2m + 2 ⁹ (3.3)

*

н ({г }) =

S 1

I

i < 2 ^ + Ш г . ) }

I

ID 1

V

Неге ф _ , ^ and U ( r i ) being the variational parameters, P i are the c a n o nical c o n j u g a t e m o m e n t a to u^ a n d r^ , r e s p e c ­ tively.

For the strongly anh a r m o n i c m o t i o n d e s c ribed by Eq. (3.4) it is convenient to introduce the p s e udospin r e p r e s e n t a t i o n with r espect to the ground d o u b l e t symmetric (Yg) a n d antisymmetric

(Y ) single-particle states:

EL

p2

{ — + U(r.) } Y (r.) = e

2m l s,a l s,a Y (r.)

s,a l (3.5)

so th a t the Hami l t o n i a n (3.4) is c a s t in the w e l l known form of De Gennes [8]

+ Eо (3.6)

w h e r e the e n e r g y parameters fi, J a n d Eq are some functions of E , C.. a n d the m a t r i x elements r = < a Ir.l ß > and

a 13 aß 1 i1

2 2

r < a Ir.l a > (a,ß = s,a) , cal c u l a t e d with the w a v e functions aa 11 1

in Eq. (3.5) .

The v a r iational p a r a meters , C „ and ü ( r i ) are d e t e r ­ m i n e d from the Bogolyubov v a r i a t i o n a l approach, namely f r o m the condition of s t ationarity o f the free energy,

- w

F = Fq

+

H

>o = J^T msp {

} +

(F -H ) /к T

+ S p { e ° ° B (Н - H ) } »

О

(3.7)

14

w i t h respect to variations over these parameters or, equivalently, o v e r the c o r r e s ponding c o r relation functions. By this approach a closed system of s e l f - c o n s i s t e n t equat i o n s for all parameters ente r i n g in Eqs. (3.3), (3.4) and (3.6) c a n be obtained, which determines the phase transitions of the m o d e l and d e s c r i b e s the m u tual influence of p h o n o n and p s e u d o s p i n subsystems.

Having chosen the single-site doub l e - w e l l potential I H S ^ in the form (2.11) and the p a ir-potential V ( S i# in the h a r ­ m o n i c approximation (2.12) the m o del for the ferroelectric p hase transitions was investigated in detail in [9] , [10].

The spontaneous p o l a r i z a t i o n of the s y s t e m in th i s model is simply expressed b y the "order-disorder" (o ) and t h e displa-

Z

ci v e (n) o r der parameters as

Ps N V Ä < Г1 > - ( Ä > Гза о =

z П a (3.8)

w h e r e a z is the ave r a g e occ u p a t i o n number. The a v e r a g e "slow"

d i s p l a c e m e n t can be w r i t t e n in the form

n (T) = { .)** . (3.9)

1-P

Her e у = (B/A) < u^ > is the r e d u c e d a v e r a g e quadratic "fast"

displacement, p is the overlap of the g r o u n d state w a v e functions of the "left" and "right" u n p erturbed h a r m o n i c oscillators, the

linear combination of w h i c h was choosed in [22] for t h e trial w a v e functions.

The o rder p a r a meters n and о w e r e obtained o n the basis

Z

of the self-consistent phonon-field and molecular field

approximation sol v i n g the s y s t e m of s elf-consistent equations numericaly [22], for certain values of the dimensionless

coupling parameter fQ = (1/A) £ ф! ! = [ 1/(A - 3B < u2 > ) ] £ <p!!

j 13 1 j

and the temperature independent reduced q u a n t u m p a r a m e t e r

L 2

Aq = (А/m) /(А /В) , c haracterizing the zero-point vibrations.

The results of the numerical c a l c u l a t i o n s for о (T) , n(T)

z

and P g (T) are p r e s e n t e d in Figs. 5-8 for Aq = 0.1 a n d fQ = 0.1, 0.2 and 0.6 respectively. The c o r r e s p o n d i n g curves in Figs. 5-8 are in agreement w i t h our prev i o u s ones: Fig. 1-4. In the w e ak- -coupling limit: f « 1 , the t u n n e l l i n g effects a r e properly accounted for, c o n s i s t e n t l y to the re s u l t s by Gillis [3]. The appearance of i m a g i n a r y solutions in Figs. 5-8 is a s s i g n e d by open circles. A d i s c o n t i n u i t y itself in the d i s placive PT in the case of fQ « 1 is a ve r y well k n o w n char a c t e r i s t i c feature of the s e l f - c o n s i s t e n t phonon a p p r o x i m a t i o n [3]. In the strong- -coupling limit the displacive PT does e xist and it is properly described as p r e v i o u s l y in Sec t i o n 2, since the tunnelling effect c a n be disregarded. As it can be observed in Fig. 7.

the case Aq = 0 . 1 and fQ = 0.6 corre s p o n d s to a m i x e d type PT.

In [22] e x act self-consistent n u m e r i c a l solution of the Schrödinger e q u a t i o n (3.5) w a s also performed. The temp e r a t u r e dependence of all relevant p a r ameters for various f and A

о о

agree fairly well w i t h the results c i ted above (based on the trial w a v e s functions) for p £ 0 . 5 .

T a k i n g into account the tun n e l l i n g effects the results for a system with w e a k coupling in the low temperature limit are

16

Fig . 5. Temperature dependence of the order paramétere: n - average displacement and a - average localization, in the case of A =0.1, for the dimension-

lessu coupling parameter f =0.1 [T = k ß T / ( Ä 2/ B ) ].

Fig. 7. Same ав Fig.5.3 for to ^■ ° - 6 ■

Fig.8. Temperature dependence of the reduced polarization P for several values of the coupling parameter /0 [ T=k B T / (A V B) ] ,(Aq= 0.1) . F i g . 6. Same ав F i g . 5., for

to • ° - S ■

q u i t e different from the ones, o b t a i n e d in S e c t i o n 2. In the q u a n t u m limit (T=OK) the o r der p a r a m e t e r о in the m o l e c u l a r f i e l d a p proximation a c c o r d i n g Eq. (3.6) reads

°z ■ f- J - <>* ' % - Г- ' (ЗЛ0)

о о

w h e r e Jq = r sa fQ , and c o n s e q u e n t l y it can c h a n g e b e t w e e n the2 l i m i t s О £ a z £ 1 . In s u c h a way the zero-point v i b r a t i o n s can d e s t r o y the g r o u n d state at T=OK b o t h in the p s e u d o s p i n a n d the p h o n o n subsystems. Then a c c o r d i n g Eq. (3.8), o n e can e x p e c t that the spontaneous p o l a r i z a t i o n P v a n i s h e s e i t h e r in a o r in ц ,

s z

d e p e n d i n g on the mutual c o m p e t i o n b e t w e e n A^q g e l/fcn(3 / 4 fq ) and A . » 2 /f , i.e. d e p e n d i n g on w h i c h is the smaller of the two.

о ,p h о 3

- 17 -

Table 1.

f о 0,05 0,10 0,20 0,30 0,40 0,50

A о , s 0.37 0,50 0,76 1,095 1,587 2,50

A

о ,ph 0,44 0,62 0,89 1,099 1,26 1,40

T a b l e 1. shows, that in t h e qua n t u m limit for f < 0,3 a n order-

О

- d i s o r d e r PT t a kes p l a c e , f o r f > 0,3 w e can o b s e r v e a d i s placive PT and for fQ - 0,3 t h e PT has a m ixed character.

4. CONCLUDING REMARKS

A novel a p p r o x i m a t i o n scheme h a s been introduced, w h i c h takes into account s i m u l t a n e o u s l y all the int r i g u i n g features of the

18

structural PT, i.e. the statistical o r d e r -disorder, the tunnelling and p h o n o n osci l l a t i o n in the frame of only o n e

universal model. Our model d e s c r i p t i o n is b a s e d on the a s s u mption that the local n o r m a l c o o rdinate can be d e c o m p o s e d into a slow

"tunnelling (hopping) displacement" a n d a p h o n o n like one.

As a consequence, the e n e r g y spectrum of the cou p l e d q u a r t i c o s c illators is r e p r e s e n t e d as l o w - l y i n g strong a n h a rmonic excitations (due to the t u n n elling - in d i s t i n c t i o n fro m Ref.

[6], w h e r e the p s e u d o s p i n - f l i p - t y p e m o t i o n is a s s o c i a t e d w i t h the classical tran s f e r a c r o s s the barrier) and higher p h o n o n - -like excitations, a rather weak anh a r m o n i c i n t e r a c t i o n of which is d e s c r i b e d in the p seudoharmonic appro x i m a t i o n . However, since the e n e r g y s p e c t r u m of a particle in a local d o u b l e well

p o t e ntial has q u i t e a c o m p l e x struc t u r e (see e.g. [6]), s u c h a separation has m e r e l y an i n t e r p o l a t o r y character, i.e. it is p h y s i c a l l y i n applicable for a tempe r a t u r e r e g i o n к T-fin-liw

в

о

In particular, o n e could e xpect a m o r e c omplex r e n o r m a l i z a t i o n of the p s e u dospin p a r a m e t e r s in o r d e r - d i s o r d e r compounds,

esp e c i a l l y when the e x c i t e d atomic s t ates lie in the c r i t i c a l t e m perature region: к T ~J

в

с о о

Besides the t h e o r e t i c a l and n u m e r i c a l a n a l y s i s presented, it s hould be p o i n t e d out tha t our m o d e l reveals sati s f a c t o r i l y the essential features b o t h of the o r d e r - d i s o r d e r and d i s p l a c i v e type PT at finite and zero temperatures.

Co n c l u d i n g this r e v i e w it should be p o i n t e d out that a further deve l o p m e n t of this unified a p p r o a c h to the t h e o r y of

structural phase t r a nsitions is given in [23], w h e r e n o n l i n e a r effects for the o r d e r p a r a m e t e r n are taken into account in the s pirit of t h e central p e a k dynamics [7], [14] and soli t a r y waves [24] that p ermits one to go b e y o n d the m e a n field type approximations d i s c u s s e d in this paper.

ACKNOWLEDGEMENT

T h e author is indebted to Dr. N.M. Plakida (JINR-Dubna) and S. S t a menkovié (IBK-Vinda) for c r i t i c a l r e a d i n g the m a n u ­ script a n d wishes to express his g r a t i t u d e to the Institute for Physics, U n i v e r s i t y of Lodz, for hospitality.

20

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(in R u s s i a n ) .

[2] R. Blinc, B. 2ekS, Soft M o d e s in F e r r o e l e c t r i c s and A n t i f e r r o e l e c t r i e s ,

N o r t h - H o l l a n d , Amsterdam, 1974.

[3] N.S. Gillis, Lattice D y n a m i c s of F e r r o e l e c t r i c i t y , in "Dynamical Pro p e r t i e s of Solids"

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N o r t h - Holland, Amsterdam, 1975.

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[5] S. Stamenkovié, N.M. Plakida, V.L. Aksienov, T. Siklós, Phys. Rev. B14, 5080, 1976.

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[7] R. Blinc, B. 2eks, R.A. T a h i r - Kheli, Phys. Rev. B 1 8 , 338, 1978.

[8] P.G. De Gennes, Solid S t a t e Comm. 1, 132, 1963.

[9] S. Stamenkovié, N.M. Plakida, V.L. Aksienov, T. Siklós, F i z i k a (Zagreb) 10, S u p p l e m e n t 2, 122, 1978.

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[17]

*

D.N. Zubarev, Sov. Phys. Uspekhi, 3, 320, 1960.

N o n - e q u i l i b r i u m S t a t i s t i c a l Mechanics,

§ 16, P l e n u m Press, New-York, 1974.

[18] N.M. Plakida, Phys. Lett. 32A, 134, 1970.

[19] V.L. Aksienov, H. Breter, J.M. Kovalski, N.M. Plakida, V . B . Priezzhev, Fiz. tverd. tela 18,

2920, 1976.

[20] S. Stamenkovié, N.M. Plakida, V.L. Aksienov, T. Siklós, Acta Phys. Hung. 42, 265, 1977.

[21] S. Stamenkovié, N.M. Plakida, V.L. Aksienov, T. Siklós, Acta Phys. Hung. 4J3, 99, 1977.

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V.A. Zagrebnov, F i z i k a (Zagreb) 12, Suppl. 1 332, 1980.

[24] J.A. Krumhansl a n d J.R. Schrieffer, Phys. Rev. B 7 , 3535, 1975.

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Kroó Norbert

Szakmai lektor: Dr. Menyhárd Nóra Nyelvi lektor: Dr. Menyhárd Nóra Példányszám: 490 Törzsszám: 81-391 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. junius hó