Rhodes-Wohlfarth ratio and anomalous specific heat in displacive type ferroelectrics

ELSEVIER

Physica B 219&220 (1996) 587-589

Rhodes-Wohlfarth ratio and anomalous specific heat in displacive type ferroelectrics Masaharu Tokunaga Division of Physics, Graduate School of Science, Hokkaido University, 060 Sapporo, Japan

Abstract The R h o d e ~ W o h l f a r t h plot for ferroelectrics introduced and theoretically discussed in terms of the self-consistent p h o n o n (SCP) theory by T o k u n a g a is briefly reviewed. Specific heat is known to show no anomaly in the displacive type phase transition as far as definition is concerned. The larger Rhodes-Wohlfarth ratio is shown to result in the smaller anomaly of specific heat within the framework of the SCP theory.

1. Introduction

2. Self-consistent phonon theory

The ratio of a paraelectric m o m e n t Pc to a saturated moment Ps, Rhodes-Wohlfarth ratio r = Pc/Ps, was introduced into the field of ferroelectricity by Tokunaga [1]. The moment Pc is evaluated from the C u r i ~ W e i s s constant C and ps from the saturated spontaneous polarization P~ of a ferroelectric substance. As easily noticed, r must be unity in order~lisorder type ferroelectrics, but takes a value larger than unity in displacive type ones, which was explained in terms of the self-consistent p h o n o n (SCP) theory [1]. As far as definition is concerned, specific heat displays no anomaly at the displacive type transition temperature Tc [2]. In this paper, the anomaly of specific heat is discussed as the j u m p of specific heat ACp at Tc or the transition entropy AS. It will be easily noticed that some relation holds between r and ACp (or AS). In this paper, a brief review is given on the SCP theory. Then, this relation is shown to be derived from the SCP theory by the use of the key parameter to, which was shown to describe the displacive type mechanism in the discussion on r [1].

Let the displacement Q(R) be the local motion of atoms in the Rth unit cell associated with a soft mode. The potential energy U(R) for Q(R) is written in the form U(R) = (½)M~ 2 (s) (Q (R)) 2 + (¼)B (Q (R))4,

(1)

where M(o2(s) and B are constants depending on substances. The interaction between the dipoles eQ(R) and eQ(R') in the R and R' cells is denoted as - J(R -- R') eQ(R) eQ(R'), where e is an effective charge for these local motions. These interactions among N cells bring about the ferroelectric transition. It is convenient to write Q(R) and J(R - R') in Fourier components Q(q) and J(q) with wave vectors q's, and the starting Hamiltonian in the form H = ~(½){ - A + e 2 (J(0) - J(q))} [Q(q)l2 q

+ (¼N)B~Q(ql)Q(q2)Q(q3)Q( - ql - q2 - q3). (2)

The summation in the second terms is over ql, q2 and q~. The mode with q = 0 is considered to be unstable

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M. Tokunaga / Physica B 219&220 (1996) 587-589 3. Rhodes Wohlfarth ratio in displacive type ferroelectries

at T = 0 , - A = Me92 (s) - ezJ(0),

(3)

but believed to be stabilized by interactions with other modes through the anharmonic terms above To, Mo92(q) = - A + e2(j(0) - J(q)) + 3 B K > 0

(4)

with

K = ( 1 / N ) ~ (IQ(q)l 2 ).

(5)

The ratio r versus T~ is shown in Fig. 1 only for displacive type ferroelectrics based on the literature [4]. Although the number of available experimental data included in Fig. 1 is small even now, one of the significant conclusions is the increase of r with the decrease of To. This fact is briefly explained from the definitions of Tc and to in Eqs. (12) and (13). In the mean field approximation (MFA) [-1], I(y) becomes

q

I(y) = (1 + y ) - i

(14)

The Hamiltonian for Q(q) is written as H = ({)M~o~(q)IQ (q)l 2 + (k)(1/N) B IQ(q)l4,

(6)

Substituting Eq. (14) into Eq. (9) and expanding up to the linear term of y, we can derive the dependence of y on t,

and the mean square ([ Q(q)I z ) is assumed to be evaluated by the use only of the quadratic term in Eq. (6) in the SCP theory,

y-i = (t-

([ Q(q)[2 ) = k , T / { M ~ 2 ( q ) } .

r = Pc/Ps = (l + to) 1/2 = (1 + e2j(O)/A) 1/2

(7)

1)(1 + to),

(15)

and the ratio r [1], (16)

An SC equation for the soft mode frequency ~os(0) is obtained from Eqs. (4)-(7), 14.

i

i

i

~

i

M~o2(O) = - A + 3 B ( 1 / N ) ~ k . T / { M ~ o ~ ( O ) q

+ e2(J(O) -- J(q))}.

12

(8)

It is convenient to rewrite Eq. (8) into a simple reduced form [1], toy = t(I(y)/I(O)) - 1,

10

(9) 8

I(y) = ( 1 / N ) ~ [ y + (1 - J(q)/J(O))}

1,

(10)

x-.

°2

q

where the following abbreviations are used:

3°0

y = Mo~2(O)/{eZJ(O)},

(11)

k , Tc = e 2 {A/(3B)} J(O)/I(O),

(12)

4

o 5

7

and

6 to = e 2 j ( O ) / A

jo

9

11 I

(13)

with the reduced temperature t = T / T c . If to is zero, Eq. (9) is attributed to the sum rule in the spherical model of an Ising spin system. The introduction of the SC relation through the dependence of I(y) on y means the operation of order-disorder type mechanism even in a displacive type phase transition [3]. Without the SC relation given by l(y), Eq. (9) is reduced to that to determine the frequency in the quasi-harmonic approximation. The parameter to represents the displacive type mechanism of the decreasing frequency with decreasing temperature in addition to the order-disorder type one.

I

100

1

200

i

300

P

400

i

500

600

700

800

Tc Fig. 1. Rhodes Wohlfarth plot, r versus To, in displacive type ferroelectrics: 1 - LiT1C4H,,O6"H20, 2 BiSI, 3 - SbSBr, 4 BiSBr, 5 (CHsNHs)A1C14*, 6 - Thiourea, 7 CdzNbzOT, 8 LizGeTO15*, 9 - SbSI, 10 BaTiO3, 11 KNbO3, 12 - PbTiO3. Data are referred from Ref. [4]. The largest value is r = 36.0 (To = 34.4) in RbFeBr3* from Ref. [8]. The substances marked with * have not been confirmed to be of displacive type, but are predicted to be so from their large r values.

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M. l'okunaga / Physica B 219&220 (1996) 587-589

Magnitudes of J(0) may be different but of the same order among these substances. The dependencies of Eqs. (13) and (16) on A explain qualitatively the relation between r and Tc in Fig. 1: in the small A limit, T¢ tends to 0 and r to infinity. This is called the displacive type limit in Ref. [5]. Another conclusion from Fig. 1 is that r ~ 1 in displacive type ferroelectrics with high T~. Since to becomes 0 in the large A limit, the order-disorder type mechanism plays a significant role in the Curie-Weiss behavior in this case [3]. The substances with high T~ are known to show order-disorder type behaviors in experiments. These coincidences reveal that the Rhodes Wohlfarth ratio provides a useful criterion to discuss the mechanism of phase transition in a displacive type ferroelectric.

4. Conclusion relation between Rhodes-Wohlfarth ratio and anomalous specific heat It is easy to derive from the L a n d a u theory the relation between r and ACp or AS: r2AC, = (½)Nku.

(17)

Onodera and Sawashima [6] evaluated from experimental data the relation between r and AS on several ferroelectrics. Although the n u m b e r of data is rather small to derive an empirical relation especially for displacive ones with low To, this relation is believed to hold from the literature [4]. This relation can be shown to be derived from a statistical mechanical point of view in terms of the concept of chain-dipole introduced by Lambert and Commes [6]. Let n be the average n u m b e r of unit cells contained in a chain dipole and np the electric moment of a chain dipole. In a system with N unit cells, the n u m b e r of chain dipoles becomes m = N/n. The n u m b e r n is assumed not to vary even near T~. From the MFA, kBC = m(np) 2 and AS = mku In 2. Since P~ = m(np), we have r = n 1/2,

(18)

and the relation r2 AS = N k B In 2,

(19)

in parallel to Eq. (17). In our formulation of the SCP theory, to is a characteristic parameter to describe the displacive type mechanism. It is interesting to calculate the specific heat within the same framework. It is well known in the SCP theory that the discussion below Tc cannot be made in parallel to that above To [5]. We use the j u m p zXCp at Tc from the high temperature side to discuss the anomaly. F r o m

Eq. (6), the specific heat Cp above T~ is calculated in the reduced form: C r = ½UkB {1 - (1 + to3') dy/dt}.

(20)

The first term gives the usual equi-partition law for harmonic terms of the potential energy. The second term corresponds to the anomalous specific heat originating from the anharmonic terms. The j u m p Cp is defined as the difference from the equi-partition law: ACp = ½Nk~(dy/dt)

(at T = To).

(21)

It is easy to confirm that Eq. (21) is reduced to that of the spherical approximation for an Ising spin system in the limit to = 0. The gradient dy/dt gives the inverse of the reduced C, and ACp becomes small in the displacive, that is, the large to, limit. In order to discuss the relation between r and ACp in a more apparent form, we use the M F A in Eq. (15), and ACr becomes ACp = (½)Nkm/(to + 1).

(22)

With reference to Eq. (16), the same relation with Eq. (17) is derived also from the SCP theory. It is the merit of the SCP theory that both the quantities are given by the use of to introduced to represent the displacive type mechanism. In comparison between Eqs. (16) and (18), the longer chain dipole is related to the larger to in the SCP theory. This means that the very local motions become unstable simultaneously near T = Tc -~ 0 without the anomalous specific heat in the displacive limit where rn = 1 or to becomes infinite. In the SCP theory, the dependence of y and, then, that of Eq. (20) on t is evaluated from numerical computation, which is in progress varying to as a parameter.

References [1] M. Tokunaga, J. Phys. Soc. Japan 56 (1987) 1653; 57 (1988) 4275. 1-2] L. Tisza, Phase Transformations in Solids, eds. Smoluchowski, Mayer and Weyl (Wiley, New York, 1951) ch. 1. [3] M. Tokunaga, J. Korean Phys. Soc. 27 (1994) $30. [4] Landott-Bornstein, New Series, Group lII/28a and b, Ferroelectrics and Related Substances, eds. T. Mitsui and E. Nakamura (Springer, Berlin, 1990). 1-5] A.D. Bruce, Adv. Phys. 29 (1980) 111. [6] M. Lambert and R. Comes, Solid State Commun. 7 (1969) 4487. [7] Y. Onodera and N. Sawashima, J. Phys. Soc. Japan 60 (1991) 1247. [8] T. Mitsui, K. Machida, T. Kato and K. Iio, J. Phys. Soc. Japan 63 (1994) 839.