On the theory of ferroelectric transition in the crystal LiNbO3

10 May 1999 PHYSICS LETTERS A

Physics Letters A 255 (1999) 191-200

On the theory of ferroelectric transition in the crystal LiNb03 F.P. Safaryan Department of Theoretical Physics of Armenian State Pedagogical Institute, 13 Khanjyan St., Yerevan 375010, Republic of Armenia

Received27 December 1998; acceptedfor publication 20 January 1999 Communicated by V.M. Agranovich

Abstract

A simple model for ferroelectric transition in the crystal LiNbO3 is proposed. It is considered that one of two optical branches of vibrations of the system of crystal planes in the direction of trigonal c-axis (on which the ions Li’, Nb5+ and O*- are distributed) plays the role of the “soft mode” at the ferroelectric transition in LiNbOs. The dynamic problem of these plane vibrations is solved. For frequencies of optical modes the analytical expressions depending on coordinates, charges and weights of ions are obtained. It is shown, that the frequency of one of these two optical modes, under a certain condition, tends to zero (soft mode). This condition enables one to find with high accuracy the dispositions of ions in both ferroelectric and paraelectric phases and gives an opportunity to follow the changes of these dispositions at the phase transition. Then, it is assumed that the reason of a ferroelectric transition is the thermal expansion of crystal. On the basis of this assumption the thermal shifts of ions and the Curie-Weiss law for the square of the frequency of SM are obtained. @ 1999 Elsevier Science B.V.

1. Introduction

The crystal LiNbOs (Lithium Niobate - LN) which has recently found wide applications in non-linearoptical, quantum-optical and other devices, interests us from the point of view of the ferroelectric phase transition that takes place in it when at temperatures lower than a certain degree (the Curie temperature) the paraelectric nonpolar phase turns into a polar phase (has rather large spontaneous polarization). The crystal LN is one of the experimentally well-investigated materials (the transition temperature is precisely known, the crystal structure in ferroelectric phase is well investigated, the frequency of the “soft mode” (SM) is measured at low temperatures etc.) (see Refs. [ 1,2] and references therein). Thus the LN is rather a proper material on which various theoretical constructions (both of phenomeno03759601/99/$

logical and microscopic character) can be applied to find an explanation of the nature of ferroelectric transitions occurring in the crystal substances. In this paper we proceeded from a general conception concerning the determination of the role of “soft” condensed modes of normal vibrations of the crystal lattice in ferroelectric phase transitions [ 3,4]. However in the case of the crystal LN we assumed that one of two optical branches of vibrations of the crystal planes (on which ions Lif , Nb5+ and 02- are distributed) plays in the role of SM. Such assumption is based on the well-known experimental fact that the spontaneous polarization in LN is induced in the direction of the trigonal c-axis and occurs as a result of the displacements of positive and negative sublattices in the direction of the c-axis. For the solution of the vibration problem of crystal planes it is necessary to find preliminary the part of the

- see front matter @ 1999 Elsevier Science B.V. All rights reserved.

PII SO375-9601(99)00090-O

192

EP Safatyan /Physics

potential energy of the interaction between the electrically charged planes that caused the returning force at relative movement of the planes. We have solved this problem approximately within the framework of the theory of electrostatic interaction of the crystal planes, the electrical charges on which were assumed to be distributed homogeneously and continuously. As a result, the expressions for two fundamental frequencies are obtained which depend on positions, charges and weights of ions. It is shown that the frequency of one of these optical modes under a certain condition connecting coordinates with the charges of ions, tends to zero (SM). This condition is a rather simple formula which allows to find the arrangements of ions in a unit cell and thus to determine whether the crystal can be found in a steady (when the frequency SM gets real values) or non steady phase (frequency SM is imaginary). It allows also to follow the process of phase transition because it connects the coordinates of ions in both ferroelectric and paraelectric phases. Thus, for example, it is shown that Li and Nb ions which in ferroelectric phase occupy non symmetrical positions inside surrounding octahedrons, in paraelectric phase occupy such centrosymmetrical positions when Nb ions are disposed in medium positions between two oxygen planes, while Li ions are disposed not on the oxygen plane, as it seems sometimes, but occupy equidistant positions from oxygen planes on its both sides with equal probability. Thus the calculations in this paper confirm that ferroelectric transition in LN crystal has not a character of pure transition of “displacement” type, as many authors have considered (see for example Refs. [ 5,10,11] ), and not of “order-disorder” type (see Refs. [7,8,12-15]), but most likely it is a transition of a mixed type: in terms of behaviour of Nb ions it is of “displacement” type, and in terms of behaviour of Li ions it is “order-disorder” type (the similar version is confirmed in Refs. [ 16-181). In Refs. [ 191 it is considered that ferroelectric transition in LN has a character of “order-disorder” type, but with respect to Nb ions (and not Li ions). Then, it is assumed that the reason for ferroelectric transition can be the thermal expansion of the crystal. The theoretical construction is developed which enables to find the dependence of thermal shifts of ions and the squares of the frequency of the “soft mode” on temperature within the large range of temperature change

Letters A 255 (1999) 191-200

0

2-

______:________ ________:_________I_____. ______~_________~________

4

I

I

0 ________i______

+._________I Li I+ r-------- ’ __________ ?,________c_________:_________T_____. ? -_____ --___

_--____

I

T

I

+ ________;_____. _____ _______~__________:___ 5+ 9 I

______:_________:________

Nb

______ 1 &_______\________&____ 1-------______,_________+_______ 1 ~________-I_______ I

#

I

I

I

I

I

I

I

,

I

Fig. 1. Dispositions

of crystal planes id unit cell of crystal LN.

(above and below the Curie temperature).

2. Crystal planes in LN and interaction them

between

The projection of an elementary cell of a crystal LiNbOs (which contains 6 formula units) on a plane, where the polar axis c lies, is given on Fig. 1. One can see that between two oxygen planes, which are at a distance a = 2.3 18, from each other, two planes are located: Li and Nb. The distances between close-located planes (at T = 0 K) are equal: RLt_o s Rx0 = 0.68 A, RLi-Nb E Rt2 = 0.747 A, RNb-0 E RIO = 0.883 A [ 61. The Li and Nb ions are surrounded by deformed octahedrons the bases of which (equilateral triangles of different size) lie on oxygen planes. The ions Li and Nb are displaced from centrosymmetrical positions in their octahedrons towards different sides of the c-axis by 0.475 and 0.272 8, respectively. The Li and Nb ions form a rhombic network on their planes. The distance between ions (the side of a rhombus) is equal to the constant of the lattice b (b = 5.148 A) in the perpendicular direction of the c-axis. The arrangement of ions in the direction c is also given in Fig. 1. One can see that in the direction of the c-axis the ions (with their octahedric surrounding) are disposed in the following sequence: Li, Nb, vacancy, Li, Nb, vacancy, etc. Nearest distance between the ions of Li and Nb

193

Et? Safaryan/Physics L,etters A 255 (1999) 191-200

in the direction of the c-axis is 3.075 A, while the distance between the Li and Nb planes is only 0.747 A. It means that on closely disposed planes Li and Nb ions are displaced from the opposite positions. Here we proceeded from a general assumption that the reason of ferroelectric transition is the freezing of the optical branches of vibrations of the lattice. However in the case of LN crystal we think that one of two optical modes of crystal planes’ (on which Li+, Nb5+ and 02- ions are distributed) vibrations in the direction of the trigonal c-axis plays the role of SM. In order to obtain the system of equations of motion of these planes, it is necessary to find the energy of their interaction. The electrostatic part of this energy, calculated for infinitely large planes (R/Z = 0, where I is the geometrical size of planes, R is the distance between them) is well known. It is equal to W = 2rrata2R12 ((+ is the surface density of charges). However this result is not applicable because it cannot cause a returning force. A suitable particular solution of this problem is available, for example, in the form of a logarithmic function (In Z/R) when so-called “boundary” effects are taken into account (R/l # 0) [ 91. Here we bring a more simple way of finding the necessary term in the interaction energy: first, we find it for interacting lines (the problem is precisely solved), then we generalize the obtained result in the case of interacting planes. Thus the energy of electrostatic interaction of two charged long lines is obviously equal to

where 7; = Niqi/l (qi is the charge of ions located on lines, Ni is their number) is the line density of charges. Calculation of integral ( 1) gives l-t@5 W = e2TiTj i In $-[ --I+ R$+12

Jq+Rij]

. (2)

In formula (2) we leave only the logarithmic term and apply it for interacting planes, taking into account the fact that on each plane N = l/b lines are disposed. Then for the energy of interaction of planes in the case of Rij < 1 after expanding the first term of

formula (2) in terms of a small parameter Rij/Z we obtain W = e26i8jb2 In g

,

(3)

‘I

where S; = Niqi/Zb is the surface density of charges. Returning in formula (3) to variables qi, for the energy of interaction (per pair of interacting ions) we obtain u.,

= ‘J

e2qi% In 21 b

(4)

Rij’

It is necessary to add to this energy also the energy of repulsion of planes, which has not an electrostatic nature and usually is represented as Bij/RG. Then we receive for the full energy of interaction of planes: Uij

=

*ln g++ lJ

(5)

‘J

Differentiating (5) for Rij we find the interaction force between planes. From the equilibrium condition ( U;j = 0) we find the coefficient Bij,

where R,(P) is the equilibrium distance between planes. Substituting (6) in (5)) we can then expand the potential energy in terms of plane displacements from equilibrium positions Xij = Rij - RI;O'. For Uij we receive up to the second order,

By differentiating (7) for Xii we obtain Hooke’s law with the stiffened elastic constant: Cij =

e2W?j b(R
Formula (8) is correct for interacting planes of Li and 0 (as well as for Nb and 0) because when these planes move towards each other, the ions Li (Nb) make contact with oxygen triangles and from this contact arises a repulsion force. But in the case of relative motion of Li and Nb planes there is no contact point and at first glance it seems that between these

194

El? Safaryan/Physics Letters A 255 (1999) 191-200

no compensating forces of electrostatic character It is difficult, howto notice the role such forces of atcharacter) may short-distance forces repulsion which in the Li-0 and bonds. So, total energy interaction between and Nb we can

e2F9i 21 2 . ln

Uij

=

R!?)

(9)

,,

!I

For our case of interacting planes in LN, on the basis of formulas (8) and (9) we finally have CLi_0 E C*o =

-

%7092e2

-n

,

%O

CNb_0=

c,o = --n39091

e* ,

bf?O

Ctj-Nb E C** = 9

.

(10)

21

Having the energy of interaction of planes per ion (5)) we actually reduced the dynamic problem of vibration of the system of charged planes to a problem of vibration of linear lattice, the unit cell of which (with parameter a) has three particles: Li’+, Nb5+ and 30*-. Let us denote the displacements of these particles in the sth unit cell through us, us and rs correspondingly (see Fig. 2). For the equations of motion (at the approximation of interactions between close neighbours only) we can obtain the following system of differential equations:

.-

M*fi,

= G1

(us

MO&

= C*o(vs

us)

+

c21 (v,

-

v,>

+ Czo(Ss

-

5,)

+

ClO(Vs-1

-

ys) -

u,)

0 -

Nb*

A-

Li ‘+

A

302-

30 correspondingly. The solutions of the system of Eqs. ( 11) we search in the form of travelling waves extending in chains in the direction c: u, = ueiwreiask (and two similar expressions for displacements Y, and ts). Substituting these expressions in the system ( 11) for amplitudes U, V, 5 we obtain a system of linear algebraic homogeneous equations,

-M*W2

= C2I (u =

-

iok

v>

-~>+c21(~-~), + C20(5

-

v>

,

C2o(Y - 5) + Cre( ue- iok _ 5) .

(12)

It is known that a system of linear homogeneous equations has a nontrivial solution when its determinant turns to zero. One can notice that in this case the determinant equation (A = 0), which allows to find frequencies of three vibrations (one acoustic and two optical), is a cubic one in w2. However, because the role of the soft mode may play optical vibrations, then in the determinant equation (A = 0) we can substitute k = 0. Then the frequency of acoustic vibration automatically turns to zero, and the cubic equation turns in to a square one with respect to w* which allows to find the fundamental frequencies of optical branches. Carrying out this procedure, we obtain for the determinant equation of the system ( 12) u4-BBW*+D=O,

(13)

where the following denotions are made: -C21($,+-3

9

1 5s)

n

w

B=C,o(&+&) -

Es

Fig. 2. Linear lattice of ions Li ‘+, Nb5+ and triangle of ions 02replacing the system of crystal planes.

-A40w*~

3. Vibrations of planes and soft mode

C10(&+1

US

VS

e_*

A

u

-M~w*u = Cie([e

Here it is assumed that the triangle of oxygen ions with the charge 390 and mass Me = ~YQJ(go and mc are the charge and the mass of an oxygen ion) moves as a whole in the direction of the c-axis.

Ml& =

4---a-

(14)

+c20($2+&)* 7

(11) D=

where coefficients C are defined by expressions ( 10) ; Ml, Mz, MO are the masses of particles Nb, Li and

MI

+M2+Mo (CIOGO

-

ClOC2l

-

C21C20)

.

MI M2Mo (15)

EP SafaryanIPhysics

For frequencies

of two optical modes we obtain

2 o,,=hBzt&B2-D.

2 w:, w2:

0; = B

ne2 Ml + M2 + M,-J PI b -

pz’

Ml M2Mo

w;,

(17) (18)

where the following PI = %oR:,

Table 1 The extreme values of interval of stability calculated for different defeet compositions of LN

(16)

Formula ( 16) can be given a more clear form by expanding the square root in terms of small parameter 4D/B2, then we obtain for the squares of frequencies

2 D WI=--=B

195

Letters A 255 (1999) 191-200

- q,R:,

denotions

are introduced:

- q2R:,, 3

(19)

Composition

v

ST CG HN ST

0.498 0.485 0.470 0.498

0.883 0.865 0.893 0.878

CC

0.485

0.884

R(l)

R’2’

A

A

0.68 0.695 0.675 0.675

0.696 0.706 0.69 0.70

0.258 0.418 0.144 0.314

0.677

0.73

0.278

RIO (A)

Ref.

[61 161

161 151 151

V = NL,/(NLj + NNb); ST: stoichiometric; CG: COngment; HN: nonstoichiometric; NLi: number of ions Li; NNh: number of ions Nb.

The solutions of the inequality (21) for R2a at a certain value of RIO are within the interval [ R!$ , RiA) ] where the extremal values R$) and R$ represent the roots of corresponding square Eq. (2 1) :

p 2

R(‘92) 20 = 6( a - RIO) i

-

(20)

The factor P2 is always positive at all reasonable values of parameters contained in it. As regards the parameter PI, it can change sign. Thus, the vibration with the frequency 0~2at a certain condition (D = 0 or PI = 0) which is connected with the arrangement of ions in an elementary cell and values of their charges, can be frozen in the crystal (SM) . The crystal response to this phenomenon is that for the recovery of the conservated vibrations the ions are regrouping around the new equilibrium positions (a phase transition in the crystal takes place).

4. Results and discussion 4.1. Ferroelectric

phase

Substituting in the expression ( 19) for PI the values of charges of ions qo = 2 a.u., q1 = 5, q2 = 1, we obtain for the condition of stability of the critical vibration 6Rf, - 5R;, - Rye 2 0, where R11 = a - RI - R2.

(21)

31 RT, - 60aR,o + 30a2. (22)

Thus the values of R20 (at a certain value of RIO) at which the crystal mode is steady (the frequency is real) are located in the interval [ R$ , R:A) ] , in the extreme points of this interval the frequency w becomes zero, and outside of this interval the frequency accepts an imaginary value. Extremal values of the interval of stability [ R$A’, R$ ] which are calculated on the basis of formula (22) are shown in Table 1 (columns 5 and 6) for various values of the parameter RIO found by different authors for different ionic compositions of a crystal LN. In the 4th column of the table the experimental values for distances R2e are presented. The good agreement of calculated and experimental values is evident. One can see that in a unit cell of the crystal LN the Li-plane is disposed within the interval of stability, as close as possible to the upper borders of the interval [ R$‘, R$) 1. The position close to the

lower border R$) at low temperatures, apparently, is not realized because the packing of ions in the crystal is quite compact and ions cannot get closer (the size of the ions themselves prevents this). The fact that the Li-plane prefers to be disposed close to the borders of the zone of stability, has a simple explanation. Such arrangement is energetically more favourable because the frequency of the critical mode (which tends to zero on phase transition) is small. One can note that the

196

R?? SafaryadPhysics

distance AR20 = Rexp as follows from the data 20 -R;;‘, represented in the table, is so little that the thermal expansion of a crystal can throw ions from the zone of stability to a zone where the frequency of the critical mode becomes imaginary, i.e. the thermal expansion of a crystal can, for example, become the reason for phase transition. For the spontaneous polarization which appears in the ferroelectric phase we can, apparently, write (23) where N is the number of Li or Nb ions in the elementary cell of the crystal, & is the volume of the elementary cell. Substituting in formula (23) N = 6, ~JJ = 3 18 . 1O-24 cm3 we obtain for the spontaneous polarization PO = 62mcC/cm2. Calculations for the spontaneous polarization of the crystal LiTaOs (taking into account that R2a = 0.601 A, RIO = 0.954 A [ 11) leads to the value PO = 48mcC/cm2. Experimentally observed values are 70mc C/cm2 and 50mc C/cm2 respectively [ 11. The frequency of the soft mode w2 may be calculated according to formula ( 16) or ( 17). Substituting the following values of parameters q1 = 5, q2 = 1, Sqo = 6, Rzo = 0.68 A, Rio = 0.883, R2, = 0.747, Ml = 92.9 a.u., MO = 48, M2 = 6.94, b = 5.15, we obtain wz M 4.9 . 10i3J7i = 256&i cm-‘. A coincidence with the experimentally observed value (~250 cm-’ [ lo] ) is achieved at n x 1. It should be noted that the index of non electrostatic repulsion II (from formulas (5)) (9) ) for interacting ions accepts values within the interval [5-131, but for interacting planes its value may be significantly lower since (i) at counter motion of planes not the rigid ions are encountered but the ions of Li (or Nb) are encountered with a more elastic formation - triangles of oxygen ions; (ii) slowly changing function of repulsion must correspond to the slowly changing (by a logarithmic law) function of electrostatic attraction. The frequency of the “soft” mode might be found also by another method - by expanding the full potential energy of interaction in terms of displacements of planes from their equilibrium positions. It is known that the coefficient of the quadratic term of such expansion is the energy of phonons. Thus, making such a expansion of the type (7) (see Appendix) we can

Letters A 255 (1999) 191-200

write (24) where nio = u - 5, x20 = 8 - U, x21 = u - V. The solution of the system of Eqs. ( 12) for the soft mode we receive the following values for the amplitudes of vibrations: Y/[ M 6.3, u/t M -0.99, Y/U = -6.4. Substituting into Eq. (24) these values presented through 5 and passing from normal coordinates of lattice for the frequency of SM we find 2 w=enS

(25)

c a h+vaL 2Ma

2b

where 6= 30)5-uj2

+ 6)~--5)~

R:

R;

~)u-Y)~ -

R;,

’

(26)

In the ferroelectric phase (RI = 0.883 A, R2 = 0.68 A), S = 0.41 . lOi* cme2, ya = (ehoJkT - 1) -i is the number of phonons of the lattice of type LY. In formula (25) from the sums with respect to the phonon states passing to a corresponding integral calculated on the basis of Debye approximation we obtain (27) where ou is the Debye frequency (ho = kTD, TD is the Debye temperature of the crystal), p is the density of the crystal, va is the velocity of propagation of acoustic waves in the crystal. The integral contained in formula (27) is easy to calculate for extremely low and high temperatures. Thus, for T --+ 0 the value of this integral equals ok/2 and the frequency of SM would be 3 e2n80~ -. 16 b+pv,3

f%(O) = -

(28)

Substituting in formula (28) the following typical values of parameters (measured in CGS units) @o = 5. 1013,p = 4.64, vc = 7.5. 105, n = 1, then for CT,we obtain a value which is close to the experimental one 6 x 4.5. 10’3 s-l.

EI? Safaryan/Physics Letters A 255 (1999) 191-200

4.2. Paraelectric phase

Substituting in formula (22) Rio = a/2 = 1.155 8, (the coordinate of the Nb ion in a high-temperature phase), for extreme values of the distance Rza in this case (and only in this case) the values equal by module R$) = 0.499 A, R$,’ = 13.361 - 6a = -0.499 8, are obtained. It testifies that in a high-temperature phase the ions Li’+ can be arranged with equal probability on identical distances on both sides from the oxygen planes. About such an arrangement of ions on a distance - f0.51 A is reported for example in Refs. [ 7,201. Thus our calculations confirm the version that in the paraelectric phase the Li ions can be arranged on identical distances on both sides from the oxygen plane with equal probability. 4.3. Dynamics of the phase transition If we proceed from the real structure of the crystal where the ions are not point charges but have a finite size (the ionic radius) then it becomes clear that in the crystal LiNbOs or LiTaOs the ions form a quite compact packing which forces them to be displaced at comparatively small distances. In the ferroelectric phase the ions Nb5+ and Li+ are shifted from the centers of surrounding octahedrons to different sides at values that enable their ionic sizes. Thus, for example, the distance between ions Li (Nb) and the nearby 02ion which are on the vertex of the triangle-base of the octahedron surrounding Li’+ ion is equal to 2.056 A ( 1.878 A) while the sum of ionic radii of Li and 0 (Nb and 0) is equal approximately to 2 A. This fact testifies that between Li’+ and Nb5+ ions there exists a strong repulsing force that forces the ions to be shifted to different directions. Let us note also that the sizes of triangle bases nearby Li and Nb ions are different: thus, for example, the distances from the centers of those triangles to their vertexes are equal correspondingly to 1.94 A and 1.657 A. At low temperatures the free space (“opening” or “bottleneck” [ 71) between the oxygen ions on those triangles is so small that the Li ions cannot easily pass through them from one side of the oxygen plane to the other. However, with rising of temperature, the sizes of “openings” gradually increase and Li ions are more and more drawn into the free space between the oxygen ions moving across the c-axis and dragging after the Nb ions in the same di-

197

rection. At the Curie temperature the “opening” opens enough for Li ions to pass through it. As a result, as it is shown above, the Li ions in the paraelectric phase are arranged to both sides of the oxygen plane with equal probability at a distance of 0.5 A from it. At the same time the Nb5+ ions occupy centrosymmetrical positions (RIO = 1.155 A) between nearby oxygen planes. Let us note also that besides the abovementioned temperature-shift mechanism of ions which is connected with the thermal expansion of the crystal along the perpendicular to the c-axis direction, the Li and Nb ions may shift also due to the thermal expansion of the crystal immediately along the c direction. Though the coefficient of such expansion is an order of magnitude smaller than the coefficient CYof expansion of the crystal along the perpendicular to the caxis direction (fl M 2. 10B6/K, (Y= 17.6 + 10W6/K) however in the appendix we shall show that the coefficient p of Li ions turns out to be much greater, approximately equal to p2 M 10.5 . lO+j/K than the corresponding coefficient for the oxygen frame. Thus it is easy to see that the thermal change of the distance of the Li ion from the nearby oxygen plane may be expressed according to (see Fig. 3) R20 = J2.0562 - 1.942[ 1 + (cu2 - p2)T]* M 0.68 1 - l6.26( (~2- &)T.

(29)

One can see that the distance Rza reaches its extreme value (N 0.5 A) at the Curie temperature (To x 1484 K) at the value of LYE - /I2 = 19. 10e6/K, a quite realistic value when taking into account that the commonly accepted value for (~2 is 17.6 . 10e6/K which is obtained at T = 500 K [22] with an apparent trend of its growth with the increase of T. The change of distances RIO and R2i = a - RI - R2 then could be presented in the form RIO= 0.883

1 + 16.26a,T,

(Yl= 29.5. 10-6, (30)

RZI = 0.747dm,

LY= 1.56. 10-4.

(31)

Substituting the expressions (29) -( 3 1) into formula ( 19), we obtain for the multiplier contained in the frequency of SM: PI = PIO( 1 - rT) , where Pi0 = 0.256 1 10-16, y = 1/TO = 6.72 +10m4 (the weak thermal dependence of the multiplier P2 in ( 18) is ig-

198

ER SafaryadPhysics

Letters A 255 (1999) 191-200

a b Fig. 3. Arrangement of ions Lilt and nearby triangle-base of oxygen ions. (a) Projection perpendicular to the c-axis plane; (b) projection pamile to the c-axis plane. nored), which for the temperature dependence of the square of frequency leads to (Curie-Weiss law)

(from below), A = &(l - rt), where 4 = 2.34 . 1O36cmm4 is the value of A at T = 0. The full Curie-Weiss law in this case will be w; = (coo + AAoT)2y,

where 00 = 4.67 . lOI s-t = 250 cm-’ is the frequency of SM at T = 0. But for finding the Curie-Weiss law in the full form and for high temperatures (T 2 TO) it is necessary to find the contribution of the anharmonic interaction of phonons in the frequency of the critical mode. For this anharmonic correction the following expression (see the appendix) is obtained: Aw, = AAT,

(32)

where A_

a e2kw;n(n2 + 6n + lr) 256 bdp2v; ’

(33) (34)

One can see that at T = TO (RI = 1.155 A, R-J = 0.5 A), A = 0, i.e. at the transition temperature not only the frequency of SM but also its anharmonic correction becomes zero (the whole frequency of SM turns to zero), henceforth we can write that at T -+ TO

T
In the paraelectric phase when the “opening” inside the oxygen triangle is more than the ionic radius of Li’+, the first mechanism, i.e. the temperature shift of Li and Nb ions, stops acting, and only the second mechanism, i.e. the ionic shift, remains, which displaces the Li and Nb ions in the opposite direction (with respect to the first mechanism). In this case as it is easy to see the frequency of SM is obtained to be complex and for its anharmonic correction we obtain Aw = 19AP( T - TO) (in formula (32) it is necessary to substitute the expansion of (34) in terms of p), so that for the law of Curie-Weiss (from above) we can write o=19PA(T-To),

T>To.

(36)

Substituting /3 = 10.5 . 10w6/K, A x 1.3 . 10-25, for the interval close to the point of Curie temperatures, we obtain 6 M 2.6.10” s-‘, which agrees with observed data. Within the frames of the model adduced here it is possible to explain also the fact of existence of a great

EI? SafaryanIPhysics Letters A 255 (1999) 191-200

difference between the values of the Curie temperature of the crystals LiNbOs and LiTaOs, the structures of which differ slightly from each other. The reasons for these differences are: (i) different position of the Li ion in the elementary cells (in the crystal LT the ions Li’+ are arranged closer to the critical point R2a = 0.5 8, (R20 = 0.601 A) than in the crystal LN (R:! = 0.68 A) ) , therefore at the increase of the temperature the Lit+ ions reach the critical point in LT earlier than in LN; (ii) different thermal-expansion coefficients. Thus denoting the Curie temperature for the crystal LT by TL we can write

0.601

1 - 16.26(a;

- &)T;

= 0.5,

where for Td we obtain Ti = 0.0189/( cr; - pi). Substituting czi - /3; = 19. 10e6 which is true for the LN crystal, we obtain TA = 995 K while the Curie temperature for the LN crystal is TO = 1483 K. However the value of TOapproaches its experimentally observed value (N 920 K) if one substitutes a slightly greater value (M 1.079( ai - pi) ) in formula (36) for the coefficient of thermal expansion.

5. Conclusion Here a simple model of ferroelectric transition in the crystal LiNbOs is proposed. It is assumed that on ferroelectric transition in LiNbOs one of two optical branches of the crystal plane’s vibration plays the role of “soft mode”. The corresponding theory of ferroelectric transitions, based of the calculation of these frequencies is developed. A formula is obtained which connects coordinates, charges and masses of ions, and gives an opportunity to find these intervals of coordinates of ions under which the crystal may be in a stable or nonstable phase. The calculated numerical values of coordinates of ions in both ferroelectric and paraelectric phases, and for displacements of ions in the ferroelectric phase agree with experimental data with high accuracy. The numerical values of frequencies of optical modes of crystal plane vibrations (including SM) and the numerical values of spontaneous polarization are calculated also. The results of these calculations are in satisfactory agreement with experimental data.

199

It is shown that in the paraelectric phase the Li ions with equal probability are disposed of equidistant positions on both sides of nearest oxygen planes, while the Nb ions occupy the centrosymmetrical positions between two neighbouring oxygen planes. This means that a ferroelectric transition in LiNbOs has the nature of a mixed transition: with respect to the behaviour of Li ions it is of the “order-disorder” type and with respect to behaviour of Nb ions it is of the “displacement” type. The mechanism of phase transition due to thermal expansion of crystal is proposed. Analytical expressions for the temperature dependence of ion-shifts and frequency of SM (Curie-Weiss law) for high (with respect to Curie temperature) and low temperatures are obtained. The considerable difference between the Curie temperatures of the crystal LiNbOs and LiTaOs having similar crystal structures is explained also.

Acknowledgement The author is grateful to A.M. Petrosyan and R.A. Tamazian for valuable discussions. The research described in this publication has been made possible in part by the Awards No. AP- I- 101 of the US Civilian and Development Foundation for the Independent States of the Former Soviet Union (CRDF) .

Appendix A It is known that the reason for thermal expansion of the crystal and adiabatic change of energy of lattice vibrations is the anharmonic interactions between phonons. Corresponding formulas are well known. For example, the average displacement of ionic pairs (ij) at the thermal expansion of the crystal is [ 2 I] 3B!?) kT (X)ij =

4(;p)*

’

(A.1)

iJ

where RF) is the coefficient of nth order in the expansion of the potential function of interaction in terms of displacements xij from equilibrium positions. Making this expansion for the potential function of interaction of planes we obtain

200

EI? Safaryan/Physics Letters A 255 (1999) 191-200

(A.2)

where

where

which for high temperatures (T >> TD) and, after averaging of the value of ( 1/Mama) over the phonon states leads to formula (32). B!?) = &n(n2 ‘J

+ 6n + 11).

(A.3)

IJ

Substituting the values of Bz) and B,:!’ into formula (A. 1) , for the temperature change of the distance R20 between the Li and 0 ions we obtain

References [I]

121

(X)Li-o = --

bKT

12e2

(n + -R20,

3)

(A.4)

n

which for the coefficient of thermal expansion yields a value /?2 = flLi-0 = ( bk/12e2) (n + 3)/n M - 10.3. 10-6/K(o). The minus sign means that the Li and 0 ions are displaced in opposite directions. For the expansion-coefficient (/3)~r,_o and (p)o_o we obtain respectively pl

= (p)Nb-0

= &(p)Li-07

#O-O

131 141 [5] [6]

[7] [8] [9]

= -i(P)U-0.

Thus the Li ions at thermal expansion of the LN crystal are displaced by a much greater distance than the ions Nb and 0. For the anharmonic correction of energy of lattice phonons we obtain [ 231

[IO]

[ I1 ] [ 121 [ 131

(A.51

[ 151 [ 161

where v(4) anPP

-

ri4 16M“a &I4 o4 M4P Pa

11.2 B!?) c

Y

.

(A.61

= 3e2n(n2+6n+11).A (I

[ 171

ij

Substituting (A.6) into (AS) and passing from sums over the index LYto the corresponding integral (on the basis of Debye approximation) we obtain dw

[ 141

24 - 8b9r2pu,

[ 181 [ 191 [20] [21] [22] [23]

(A.7)

M.E. Lines, A.M. Glass, Principles and application of ferroelectrics and related materials (Clarendon Press, Oxford, 1977). IS. Kuzminov, Electra-optical and nonlinear-optical crystals LN (Nauka, Moscow, 1987) (in Russian). R. Blinc. B. Zeks, Soft modes in ferroelectrics and antiferroelectrics (?, NY, 1974). H.E. Stanley, Introduction to phase transitions and critical phenomena ( ?, Oxford, 1971). SC. Abrahams, P. Marsh, Acta Cryst. B 42 (1986) 61-68; J. Phys. Sol. 34 (1973) 521. N. lyi, K. Kitavura, F. Izumi, LK. Yamamoto, T. Hayashi, H. Asano, S. Kimura, J. Sol. State Chem. 101 (1992) 340352. H. Boysen, F. Altorfer, Acta Cryst. B 50 (1994) 405414. D.R. Bimie, III, J. Am. Ceram. Sot. 74 (1991) 988-93. L.D. Landau, E.M. Lifshiz, Electrodynamics of condensed matter (Nauka, Moscow, 1982) (in Russian). W.D. Johnston, I.P. Kaminov, Phys. Rev. 168 (1968) 1045. J.L. Servoin. F. Gervais, Sol. State Commun. 31 (1979) 387-91; Ferroelectrics 25 (1980) 609-12. S.V. Ivanova, VS. Gorelik, B.A. Strukov, Ferroelectrics 21 (1978) 563-64. A.J.A. Jayataman, A.A. Balman, J. Appl. Phys. 60 (1986) 1208-10. A.F. Penna, A. Chaves, S.P.S. Porto, Sol. State Commun. 19 (1976) 491-94. C. Raptis, Phys. Rev. B 38 (1988) 10007-19. P. Prieto, A. Conzalo, Phys. State Commun. 61 (1987) 43739. I. Tonoeno, S. Marsumura, J. Phys. Sot. Jpn. 56 (1987) 163-77. M. Zang, I.F. Scott, Phys. Rev. B 34 (1986) 1880-83. J.C. Vyas, G.P. Kothial, B. Ghosh, M.K. Gupta, Cryst. Res. Technol. 30 (1995) 217-22. A.S. Barker, R. Ludon, Phys. Rev. 158 (1967) 433. C. Kittel, Introduction to Solid State Physics (?, NJ, 1962). V.V. Dzdanova, V.P. Cluev, B.B. Lemanov et al., Fizika tvjordova tela 10 (I 168) 1725 (in Russian). A.A. Maradudin, Sol. State Phys. 19 (1966) I-134.