Tricritical phase transitions in ferroelectrics

Physica 122B (1983) 321-332 North-Holland Publishing Company

TRICRITICAL PHASE TRANSITIONS IN FERROELECTRICS

A. GORDON Department of Physics, Technion - Israel Institute of Technology Haifa, 3.2000, Israel

Received 3 March 1983

An one-dimensional model is suggested to simulate the tricritical phase transitions. The nonlinear excitations (solitary

waves and kinks) which are the cluster walls are obtained. In the weak anharmonicity case the soft-mode behaviour takes place. It is shown that in the arbitrary anharmonicity case the soft-mode behaviour with full softening is only reached in the

order-disorder regime. The domain wall energy and its thickness are calculated taking into account the nonuniform

distribution of the polarization in a transition layer between the two domains with the directions of opposite signs of

spontaneous polarization. The estimates of the energy and of the thickness are made for KHzP04 crystals near the

tricritical point. The results are compared with data which are available in literature.

1. Introduction

There is considerable interest in the question concerning the existence of tricritical points, where the phase transition changes from first to second order. The critical phenomena at these points (tricritical phase transitions [l]) are quantitatively different from those at an ordinary critical point or second-order

phase transition [2]. Theoretical examples of such changeovers were found in a number of models of

magnetic systems, and some of the properties of these points have been worked out [3-51. Experimental examples of this changeover, such as in superconducting colloids [6], the behaviour of the Ising antiferromagnet dysprosium garnet in an external magnetic field [7], phase separation in mixtures of He3 and He4 [S], the structural phase transition in ammonium halides (NH&l) [9], the phase transition in metamagnets (FeCQ have been found [lo].

Experimental results near the tricritical point in the following ferroelectrics have been published:

SbSI [ll], BaTi03 [12], Ag3AsS3 [13], KH2P04 [14], NHdHSO, [15], Ca2Sr(C2H5C00)6 and Ca2Pb(C2H5C00)6 [16], PbZr,Ti1_,03 [17]. The tricritical exponents, as derived in the mean-field approximation, (Y = 1, /3 = i, y = 1, 6 = 5 have been found in the measurements on the ferroeelectrics [14]. It has been discovered that the tricritical exponents transform into the usual critical exponents far from the tricritical point, but close to the second-order phase transition temperature, e.g., in KH2P04 [18]. The ratio of the measured Curie-Weiss law constants above and below the transition point equals 4 near the tricritical point and changes from 4 to 2 as one recedes from this point in accordance with the Landau theory (e.g., in SbSI [ll], BaTi03 [12] and KH2P04 [14]). In particular, KH2P04 crystals obey the Landau phenomenological theory to within at least 0.1 K of the phase transition temperature (see [lS] and references therein). Consequently the soft-mode picture [19,20] may be used to explain the dynamic properties of the ferroelectric crystals. Thus the 75As nuclear spin-lattice relaxation near the tricritical point in Ag3AsS3 is caused by the soft mode and has been investigated in [21]. According to the soft-mode approach the phase transition is considered as an instability of the crystal against a particular normal mode, the frequency of which decreases to zero at the phase transition temperature.

However, in recent years the soft-mode picture of structural phase transitions has been replaced to some extent by the cluster picture [22-271. In accord with the cluster picture a phase transition of the displacive type is a result of incomplete softening of a phonon mode in the temperature region T > TI, where TI is a temperature at which the clusters of the precursor order appear. In this displacive regime

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322 A. Gordon / Tricrirical phase transitions in ferroelecfrics

the atoms vibrate about their high-symmetry phase sites. The excitation spectrum is then dominated by

a softening high-temperature phonon. At the temperature 7-i the crossover from the displacive phase transition behaviour to the order-disorder regime takes place. In this order-disorder region Tt > T > T, (where T, is the phase transition temperature) the atoms vibrate around positions displaced from the high-symmetry sites to form clusters of dynamic precursor order. Within the cluster the average displacement (cluster order parameter) is non-zero.

The critical slowing down is associated with the appearance of a central peak, rather than with a complete softening of the high-temperature phase phonon. The order-disorder regime is characterized by two time scales [27]: a relatively short time scale, associated with small-amplitude harmonic vibration of the “semi-soft” mode and a long time scale, associated with the large-amplitude anharmonic motion between the two potential wells. This collective inter-well motion is assumed to be in some cases the origin of the central peak phenomena [27,28].

The experimental support for the cluster picture is provided by the observations of the central peak in inelastic neutron scattering [29] and magnetic resonance studies [30-341. The 75As nuclear magnetic resonance in KHzAsO, [31] shows up to 60 K above the phase transition temperature a spectrum, with the symmetry appropriate to the low-temperature phase indicating the presence of precursor order clusters [32]. The 75As nuclear quadrupole resonance spectrum in Ag3AsS3 exhibits some sharp “ferroelectric” type lines up to 30 K above the phase transition temperature which appear to be associated with precursor order clusters [33] rather than with the phase transition at T = 56 K [34].

The short-range and long-lived cluster excitations are apparently spatially-limited, particle-like nonlinear or soliton modes (solitary waves and kinks), which exist in addition to extended phonon modes [22-261. These cluster walls have been obtained as a result of the exact solutions of the nonlinear equations of motion in the one-dimensional T = 0 models. Within the framework of the c$” field theory in (1 + 1) dimensions, the kink solutions have been identified with cluster or microdomain walls. Further evidence in support of this connection between the microdomain walls or clusters and the central peak emerged from the molecular dynamic calculations [26]. In [35] the distribution function for two and three dimensions has been calculated using renormalization group techniques, which confirm the existence of the cluster walls in systems undergoing structural phase transitions.

The cluster wall type excitations have been obtained for the second-order phase transitions. However it is known that the ferroelectric phase transitions are first-order, close to the tricritical point

[36]. For instance, in the KH2P04 family the distance between the phase transition temperature and the point of the phase stability loss is less than 0.01 K [36]. The tricritical point may be reached in KHzP04 type crystals in comparatively small electric fields of 200-300 V/cm [37] as used in measurements of dielectric constants and of domain walls mobility.

In this paper we present an one-dimensional model which simulates the tricritical phase transition (section 2). We obtain solutions of the resulting equation of motion for the displacement field which are solitary waves and kinks (section 2). In accord with [22-261 we show that these nonlinear excitations are the cluster walls. In section 3 we consider the case of arbitrary strength of anharmonicity for the tricritical phase transitions. We obtain the expression for the soft-mode frequency as a limiting case of the weak anharmonicity. We have soliton type solutions for the strong anharmonicity. In section 4 we calculate the domain walls thickness and its energy by minimization of the free energy with respect to spontaneous polarization using the nonlinear equation for tricritical phase transitions. The results are compared with those obtained within the microscopic model [38] and with the experiment on the neutron scattering in

KD2P04 [39].

2. A microscopic model for tricritical phase transitions

This model is the development of the one-dimensional model [22,24,27] for the tricritical phase transitions. We start from the classical Hamiltonian

A. Gordon / Tricritical phase transitions in ferroelectrics

with

T = 1 c f?(Z) 1

323

(1)

(2)

and %“.

v = c VS(U(l)) + 5 c (U(l) - U(1’))2 . (3) I /,I’

Here the coordinates u(l) and p(l) describe respectively the displacements and momenta of a set of atoms of unit mass, whose high-symmetry phase equilibrium positions consists of a lattice of N sites with period a. The atoms interact through a harmonic nearest-neighbour (n.n) coupling. Vs(u) is a local double-well, short-range potential of the type (see fig. 1):

(4)

where (Y < 0, p > 0. This potential simulates the tricritical phase transition [l] at (Y = 0. This potential may appear either in uniaxial ferroelectrics or in quasi-one-dimensional ferroelectrics with hydrogen- bond chains [40], e.g. CsH2P04, CsD2P04, PbHP04 and PbDP04. In this model the phase transition occurs in a homogeneously-ordered phase, with order parameter

{u(Z))= kg. One writes the Hamiltonian in the alternative formulation [l], similar to that of the Ising model

Fig. 1. The plot of the potential V(u) = (42)~~ + (p/6) u6, where a < 0, fi > 0 as a function of the displacement u.

324 A. Gordon / Tricritical phase transitions in ferroelectrics

“.“.

v = c 4(u(l)) - c c u(l)u(l’) 1 1.1’

(6)

with the single-particle potential Vs given by

V,(u(l)) = V,(u) + 2acu*. (7)

According to [22,24,41] one may differentiate between order-disorder and displacive systems in the following manner: if U,/&, V,/K,T, 2 1, one obtains the order-disorder type, while for U,/&, V0/Ka7’, 4 1 - the displacive type where U, describes the single-particle well energy required to move an atom from the bottom to the top of one of the wells in the potential VS(u), V, is the local potential well depth which is equal to (a/3)d//P ( see fig. l), U, is a representative bond energy I?&, = 2ac(~.4(1))~ =

2acd~r/P. The order-disorder limit describes systems of the NaNOz and Ag3SbS3 type [36, 421. This is the situation which takes place in the Ising model. Indeed taking cy + *, p + CC with a/P + 1 we obtain the Ising model. This justifies the pseudo-spin formalism [43,36], in the order-disorder ferroelectrics.

In the continuum limit the Hamiltonian of the model is written as a functional of fields u(x) and k(x) (the first time derivative of u(x)).

.

Eq. (8) may be regarded as the energy functional of a classical scalar field u = u(x, t) with Lagrange density

(9)

The Euler-Lagrange equation of motion for the field u = u(x, t) which follows from eq. (8) is given by

d2u d*u c;---=(YLl+~u5.

dx2 dt2 (10)

where cg = 2ca2 is a characteristic velocity. Eq. (10) displays the two classes of solutions. Linearizing (10) under periodic boundary conditions, we

obtain the periodic, spatially extended solutions describing small-amplitude oscillations about the potential well minima-phonons

u(x, t) = -+ d 4 ; + &4(x, t) (11)

with

&4(x, t) = Su exp[i(qx - w(q)t] , (12)

d(q) = 2a + c;q2 ) (13)

where q is a wave vector. If we assume that s = x - vt, eq. (10) becomes

(14)

A. Gordon / Tricritical phase transitions in ferroelectrics 325

After the first integration of (14) we obtain the energy conservation law

where E is an integration constant (the total energy) determined by the boundary conditions. After the second integration we obtain

IfE=O,a>O,p<Oonefindsthat

(16)

(17)

Eq. (17) represents the solitary wave solution under the following boundary conditions: du/ds + 0, when s+ +m (see fig. 2).

Let us consider the solution of eq. (16) under the same boundary conditions, when E > 0, a ~0, p > 0. Taking a = -1~~1 and putting l/u2 = y, we derive the following equation

l/u2

s= kCn$-;/Ct(-; j- dy/,/Ey3-!$y2++).

eq. (18) may be written in the form

U

(18)

(19)

Fig. 2. The solitary wave solution (17).


where

(20)

Using the properties of the Weierstrass elliptic functions [44], one obtains

(21)

where zl, 22, 23 are the roots of the equation 4z3 - gzz - g, = 0. The solution (21) is obtained if A = gi - 27g$ > 0 and z, > z2 > z3. k is the module of the Jakobi elliptic functions, which is a measure of the degree of anharmonicity of the oscillation; k2 = (~2 - Z#(ZI - ZS), 0 ( k < 1, sn is the Jakobi elliptic sine. Using the expressions for z,, z2, z3.

z,=gcos;. z2 = -g cos (T) , 23 = -g cos (y) )

we find

with

(22)

(23)

Fig. 3. The kink solution (25).

A. Gordon / Tricritical phase transitions in ferroelectrics

y = arcos (1 - F) , k2 = cos Fzi!3T,6)

When A = 0, k = 1, (Y = rr, E2 = ICI/~/~/?, and we obtain the kink solutions (see fig. 3).

327

(24)

(25)

which correspond to the transfer of a particle from one minimum of the potential (4) to the other as s goes from --cc) to +m. The solutions (25) represent cluster walls because they separate regions where

u = +cm from regions where u = -qm.

3. Soft mode and nonlinear effects

The general solution of eq. (23) in case, may be represented in the form

or

with

R= p&os(~-~) .

(27)

(28)

Eq. (27) describes a periodic oscillation with the fundamental frequency [41]

o = 2&/4K(k), (29)

where m = cog/l - v’/c& 4K(k) is the period of the sn-function, K(k) being the normal elliptic integral of the first kind given by

742

K(k) = j de/d/l - k2 sin2 0 . (30) ”

If k = 0 (the harmonic case) or (El Vo) -+ 1 (the displacive case), y = 0, K(k) = 7r/2 and

w = .O = d/lal/m . (31)

Supposing J(YI - IT - T,j, one obtains the weak anharmonicity case, i.e. the displacive phase transition,


0 0.5 1.0

E/V,,

Fig. 4. The frequency o in units of VI I/ Q m as a function of the EIVO ratio (33).

with the soft-mode behaviour: w2 - IT - T,(. We also arrive at the expression for cc(t), derived from eq.

(27) for k = 0, viz.

In fig. 4 the frequency o is shown as a function of E/V, obtained from the equation

(32)

(33)

It is seen that the frequency in units w/~/)a\lm d ecreases, when E/V0 increases. The diminution is

smooth until the ratio E/V, reaches the value E/V0 = 1, where K(k) = m. Hence only in the order- disorder (deep-well) regime (E/V, = 1) the softening is full. This fact is in agreement with the cluster

picture [27] (see section 1).

4. Domain walls

Domain walls in ferroelectrics were discussed in [45] on a basis of the Landau-Ginzburg phenomenology for second-order phase transitions. However it is shown that ferroelectric phase

transitions are mostly of first order, close to the tricritical point. In some measurements of the ferroelectric domain walls the applied electric fields induce tricritical points (see section 1). Hence the domain walls in the tricritical phase transitions should be taken into consideration.

We start from expansion of a free energy, adding a term responsible for the nonuniform distribution of the polarization P: 1$(VP)2 [20]. This one writes down

F = F. + :AP2 + :BP4 + ;CPh + +f(VP)2 . (34)


The elastic strain energy terms may be taken into account by the renormalization of the expansion coefficients [45], because they are proportional to P2. Hence the problem is reduced to minimization of free energy F with respect to P. The variation of polarization in the transition layer between two domains polarized in two opposite directions is determined by thermodynamic equilibrium consideration. For the second-order phase transitions one obtains

d2P f==AP+BP’, (35)

where f, A, B are the renormalized coefficients of expansion (34) taking into account electrostrictive coupling. The domain wall solution is given by [44]

P=?d$tanh(a).

Here A = d/f/A is the domain wall thickness. For the tricritical point we find

f$=AP+CP5.

We solve eq. (37) under the domain walls boundary conditions, i.e.

dP dP dx .=_,=dx x=+z

=o.

One obtains

1

C q/5 coth2 (x/A ) - 1’

(36)

(37)

(38)

where the thickness of a domain wall A = df/A. The domain wall energy is a surface energy. Hence the domain wall energy we calculate as the

coefficient of surface tension of domain walls (see, for instance, [22]). Let us find the domain wall energy %:

which is equal to

(39)

(40)

where PO = $A/c is the polarization at the domain centre [45]. As a result of the integration we obtain

(41)

330 A. Gordon / Tricritical phase fransitions in ferroelectrics

It is seen here that 8 - (Tc - T) w ereas for the second-order phase transitions 8 - (Tc - T)3’2 [45]. h There are some experimental results near the ferroelectric tricritical points (see section 1). We use

the data on KH2P04 tricritical point studies [IS], to compare the domain wall thickness and its energy calculated here with those obtained from analysis of the experimental data [38], because the KH2POJ data are the most complete available. The KH,PO, crystals are uniaxial and hence only the 180”-domain walls must be considered.

We estimate the value of the coefficient of the Aerm with the energy of nonuniform polarization f according to [20]: f= (rr/lS)&, where d = 7.453 A [46]. One finds f = 1.16X lo-l5 cm*. When the coefficient B in the term P4 in the expansion (34) is negligible, one obtains A = 221 8, or d = 30d with A’ = 3.94 X lo-’ esu and T, - T = 6.02 X lo-* K [18]. It is 10 times greater than the value estimated from [38], based on experiments on thin samples, where the wall thickness is of the order of 2 to 3 unit cells. The domain wall energy (41) at the same temperature is O.O29erg/cm* using C = 4.2 x 10m’9esu [la], while the domain wall energy in [38] is of the order of 40 erg/cm’. The results in [38] have been obtained within the framework of the simple microscopic model of a body-centered cubic lattice occupied by point dipoles pointing along [OOl] at the absolute zero of temperature. The domain wall was assumed to be parallel to (100) plane and its energy calculated from the difference between the interaction energies of the two states with and without the domain walls. The dipolar wall energy was found to be

gdjp = 0.88P2d, where P is the value of the polarization inside the domains. The above results [38] have been obtained for zero temperature. Taking into account the temperature dependence of the spontaneous polarization according to [IX] we obtain % = 0.84 erg/cm’ at the temperature T,- T = 4.73 X 1Ol’ K, close to one used above near the trictritical point. At the same temperature we obtain here A = 241 A. The domain wall energy and thickness calculated here near the tricritical point are close to thos$ obtained in [45] near the second-order phase transition (Tc- T = 2.1 K) in the Rochelle

salt:A = 220 A and E = 0.012 erg/cm*. On approaching the tricritical point (or the phase transition temperature) the thickness of domain

wall increases. At the tricritical point the domains must vanish for energetical considerations. This approach is a macroscopic one. It can reasonably be used so long as the thickness of the transition layer is considerably greater than the lattice constant: A + d. This is the case only near the tricritical point or near the phase transition temperature. In [45] this approach was used to calculate A even at temperatures far from T,. Neutron scattering experiments in KD2P04 [39] at temperature far from T, (Tc - T = 122 K) gave the width of domain walls of around of two unit cells. As this is the only available direct measurement we shall compare it with our estimate. The experiment was carried out at temperature far from T,, where our estimates are only very approximate. Taking into account the isotopic effect of the Curie-Weiss constant [47] we obtain A = d. This result is in agreement with the

experiment (39).

5. Summary

In this paper we have considered a model for the tricritical phase transition. It was shown that there exist exact solutions of the equation of motion in this one-dimensional model. We have obtained the solitary wave and the kink solutions which exist in addition to the phonon-type solutions. The nonlinear solutions are apparently the precursor clusters of the tricritical phase transitions,

The generalization to the case of an arbitrary anharmonicity has been obtained. In the limiting case of weak anharmonicity one obtains the soft-mode behaviour. It was shown that in the case of arbitrary anharmonicity the full softening of an oscillation mode occurs only in the order-disorder regime. In the displacive regime the soft-mode frequency is not equal to zero. This is in agreement with the cluster picture [27].


The phenomenological approach was used to calculate the energy and the thickness of a domain wall near the tricritical point. These calculations were carried out for the KH,PO,-type materials. The results

were compared with experiments [39] and with the calculations based on the microscopic model [38]. It turns out that the great difference between our calculations of energy of domain walls and that of [38] decreases on taking into account the temperature dependence of spontaneous polarization. It was shown that this approach is justified near the singularity point, where the width of a transition layer is considerably larger that the lattice constant. The rough estimate of the thickness of a domain wall in KD2P04 shows agreement with the experiment [39] made far from the phase transition temperature. It was shown that the energy of a domain wall near the tricritical point ‘2Z - (TC- T) while near the second-order phase transition this energy ‘Z - (T, - T)3’2.

Acknowledgements

The author is happy to acknowledge numerous illuminating discussions with Professor J. Genossar. The author would like to thank Drs. J. Adler and L. Benguigui for the critical reading of the manuscript. The support of this research by the Wolf Foundation is gratefully acknowledged.

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Tricritical phase transitions in ferroelectrics

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