Phase transitions and relaxor properties of doped quantum paraelectrics

Journal of Physics and Chemistry of Solids 61 (2000) 167–176 www.elsevier.nl/locate/jpcs

Phase transitions and relaxor properties of doped quantum paraelectrics W. Kleemann a,*, J. Dec a,1, Y.G. Wang a,2, P. Lehnen a, S.A. Prosandeev b a

Laboratorium fu¨r Angewandte Physik, Gerhard Mercator Universita¨t Duisburg, D-47048 Duisburg, Germany b Physics Department, Rostov State University, 5 Zorge St., 344090 Rostov-on-Don, Russian Federation Received 22 February 1999; accepted 28 July 1999

Abstract Scaling properties of the paraelectric susceptibility with respect to temperature T and electric field E have been evidenced on solid solutions Sr12xCaxTiO3, x ! 1; and are explained within the framework of both the transverse Ising model (TIM) and an anharmonic oscillator model (AOM). The impurity-induced ferroelectric phase transitions occurring at x . xc < 0:002 are modeled using a heterogeneous TIM, which is solved numerically. Alternatively, we present a coupled mode model involving a TIM and an AOM for the impurity and the host lattice subsystems, respectively. Relaxor properties such as dielectric polydispersivity, quasi-first-order Raman scattering, bilinear coupling of optic and acoustic phonon modes and hyper-Rayleigh light scattering in the paraelectric regime are attributed to polar nanodomains due to fluctuations of defect-induced quenched random fields (RFs). Some of these features are modeled within the framework of the heterogeneous TIM including RF interactions. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Quantum paraelectrics

1. Introduction Doped quantum paraelectrics, such as Sr12xCaxTiO3 (SCT), K12xLixTaO3 (KTL) and KTa12xNbxO3 (KTN), x ! 1; have been studied extensively owing to their interesting properties at low temperature [1–4]. It has been found that these systems remain paraelectric when the impurity concentration x is low, while they undergo a paraelectric– ferroelectric phase transition if x exceeds a critical value xc. With increasing x the Curie temperature TC is proportional to …x 2 xc †1=2 ; when x is close to xc. It has been argued [3] that the dopants introduced into the above systems occupy off-center lattice positions because of * Corresponding author. Tel.: 149-203-379-2809; fax: 149-203379-1965. E-mail address: [email protected] (W. Kleemann) 1 On leave from Institute of Physics, University of Silesia, PL-40007 Katowice, Poland. 2 On leave from Department of Physics, Shandong University, Jinan 250100, People’s Republic of China.

their comparatively small ionic radii. When placed within the strongly polarizable host lattice of SrTiO3 or KTaO3, sizeable coupling between the dipolar impurity dynamics and the polar optic TO1 soft-mode is expected. In SCT, which has tetragonal D4h symmetry at low temperature, coupling is expected between the soft Eu component of the TO1 mode and the Ca 21 dipolar dynamics within the easy (001) plane [5]. Thus, extended polar clusters around the impurities arise, whose polar structure might be quite complicated [6]. Their size is in the order of the polar correlation length of the host lattice, j (T), and they interact ferroelectrically with distance dependent strength, kJij l / exp…2r=j† [7]. At sufficiently low temperature the overlap of the clusters eventually becomes large enough to give rise to ferroelectric instability. The phase transition thus induced is, hence, of percolative origin and implies a critical concentration, xc. In SCT a value of xc ˆ 0:002 has been reported [5]. Recently the well-known [8] transverse Ising model (TIM) has been employed to explain the phase transitional properties of doped quantum paraelectrics from a microscopic point of view [9]. The coupling of the ionic displacements is

0022-3697/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00278-4

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ourselves to showing that random fields, indeed, modify the phase transition by smearing effects, which are expected quite generally in systems with nearly degenerate order parameter [17]. We incorporate random fields into the TIM and examine some of the phase transition properties of quantum paraelectrics. They are related to precursor effects like dielectric polydispersivity [18], quasi-firstorder Raman scattering [19], bilinear coupling of optic and acoustic phonon modes [11], and hyper-Rayleigh light scattering [20]. They will be reviewed and compared to the computational results.

2. Scaling properties of Sr12xCaxTiO3 2.1. Experimental results

Fig. 1. Real part of the ac susceptibility, x 0 , vs. temperature at various bias fields E for SCT with (a) x ˆ 0, (b) 0.002 and (c) 0.007, and dependencies of x 0E =x 00 on E…x 00 †a for the same samples (d)–(f). Best data collapsing is achieved with D=g ˆ 2:03: The inset in (e) displays the field dependencies of x 0E for x ˆ 0:002 at representative temperatures.

replaced by exchange coupling between host lattice and impurity pseudo-spins. In this approximation, which will be reviewed later, the rather complicated coupling scheme can be treated self-consistently by numerical simulation within local mean-field approximation. In our previous papers [9–12] we calculated the concentration dependence of spontaneous polarization, Curie temperature and dielectric susceptibility, which fit the experimental data of KTL, KTN and SCT well. Moreover, we will show that the TIM is also able to explain scaling properties of SCT which have been discovered in recent experiments [13]. It has been evidenced by a number of experimental techniques that doped quantum paraelectrics bear many precursor characteristics, e.g. the smearing of phase transitions and extreme polydispersivity, which are similar to those of the prototypical relaxor system PbMg1/3Nb2/3O3 (PMN) [14]. The relaxor-like phenomena observed in the paraelectric regime are believed to result primarily from polar nanodomains, which are very probably due to fluctuations of quenched random fields [4,15]. In order to understand the transition into the non-ergodic glass-like low-temperature state, a spherical random bond–random field model seems to be most adequate [16]. In this paper we shall restrict

Fig. 1(a)–(c) shows the temperature dependence of the dielectric susceptibility x 0 (T, E ˆ const), of SCT with x ˆ 0; 0:002 and 0.007, measured at a frequency f ˆ 1 kHz under the influence of various axial electric fields, 0 # E # 500 kV=m: As is well known from the behavior of pure STO [21–23] the susceptibility of SCT with x , xc gradually decreases with increasing E. Finally, flat peaks emerge beyond a certain threshold field, but they do not indicate the occurrence of phase transitions contrary to previous assertions [23]. Only for samples with x . xc (e.g. Tc ˆ 18 K forx ˆ 0:007, see arrow in Fig. 1(c)) the impurityinduced relatively sharp peak occurring in E ˆ 0 indicates a phase transition. It flattens and shifts to higher T under the influence of increasing E. When recording the isotherms x 0 (T ˆ const, E) at different temperatures one obtains monotonically decaying curves. They reflect the considerable non-linearity of the susceptibility as observed previously [24]. Fig. 1(e) (inset) shows some of these curves for x ˆ 0:002; which have recently been shown [13] to follow the scaling relation

x 0 …T; E†=x 0 …T; 0† ˆ f {E·‰x 0 …T; 0†Ša }:

…1†

Eq. (1) has originally been derived for ferroelectric systems in the vicinity of their critical point [25]. Very remarkably, it contains only one fitting parameter a and does not require a priori knowledge of the critical temperature Tc. The exponent a ˆ d=…d 2 1† ˆ D=g is related to the well-known exponents d (critical isotherm), g (zero-field susceptibility) and to the gap exponent D . Fig. 1(d)–(f) show that the SCT data at all concentrations, x ˆ 0; 0:002 and 0.007, collapse onto unique master curves with very similar fitting parameters, a ˆ 2:00–2:05: It is clear that the observed scaling behavior does not indicate criticality, since the data points represented belong either to systems without any phase transition (x ˆ 0 and 0.002) or to temperature ranges far from the conventional critical region …x ˆ 0:007†: Obviously the susceptibility of SCT is inherently described by a function, which has scaling properties similar as the Langevin-type equation describing the susceptibility

W. Kleemann et al. / Journal of Physics and Chemistry of Solids 61 (2000) 167–176

of non-interacting uniaxial electric dipoles

x 0 …T; E† ˆ

nm2 E cosh22 : 2kB T 2e0 kB T

…2†

Indeed, it can be shown within the framework of two different approaches that the susceptibility of pure quantum paraelectrics may be derived from Landau–Devonshire type free energy functions F…T; E†; which are homogeneous with respect to the fields T and E and thus guarantee the desired scaling properties. An expression of the form   1 T 2 TC 2 1 F ˆ F0 …T† 1 A …3† P 1 BP4 2 EP 2 4 Tc yields Eq. (1) with g ˆ 1 and D ˆ 3=2; hence, a ˆ 3=2 [26]. Eq. (3) applies to conventional ferroelectrics, where TC is the Curie temperature and A and B are constants. P is the polarization, which is related to the susceptibility by x 0 ˆ …1=e0 †…2P=2E†T . 2.2. Transverse Ising model [27] When describing the quantum paraelectric by a system of interacting tunneling ions we start with the Hamiltonian [8,9,22,23] X X 1X H ˆ 2V Sxi 2 Jij Szi Szj 2 2m ESzi ; …4† 2 i ij i P where V is the tunneling integral, J0 ˆ j Jij is the interaction parameter, m is the dipole moment, and Sxi ; Szi are the pseudo-pin operator components. In mean field approximation the molecular field is a vector, Hi ˆ …V; 0; J0 kSzi l 1 2mE†, and the expectation value kSz l is obtained from thepequation kSzi l ˆ …Hiz =2H† tanh…H=2kB T†;  2 where H ˆ uHu ˆ V 1 …J0 kSzi l 1 2mE†2 : The inverse derivatives E 0 ; E 00 and E 000 of the field E with respect to kSz l provide the coefficients of F expanded in powers of the polarization P. After letting E ! 0 and kSz l ! 0 one obtains   1 V E0 ˆ 2 J0 ; 2V coth 2m 2kB T E 00 ˆ 0 and

…5†

  12V V V 1 E 000 ˆ coth3 12 m 2kB T kB T sinh V=kB T such that the free energy yields the standard form   TQ 2 T0 2 1 1 P 1 BP4 2 EP: F ˆ F0 …T† 1 A 2 4 T0

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although the general mathematical form of the free energy is the same as for the ordinary Landau expansion near a phase transition. Since the temperature is eliminated in the scaling law (1), Eq. (6) yields the same susceptibility scaling relation as the free energy (3) and is valid throughout the quantum paraelectric regime. Note that the fourth-order coefficient is nearly constant at low T and in the limit T , Ts ; B ˆ V=4n3 m4 : 2.3. Anharmonic coupled oscillator model [27] Following the theory of Salje et al. [28], which describes the saturation of order parameter in the ferroelectric phase at low temperature, we deduce the susceptibility in the quantum paraelectric range by applying an external electric field in order to have a non-vanishing order parameter. SrTiO3 is thus viewed as a displacive quantum system and described by the Hamiltonian  X 1 2 1 1 Hˆ Pl 1 M v20 Q2l 1 uQ4l 2M 2 4 l 2

X 1 X y 0 Q Q 0 2 E ql Ql 2 l±l 0 ll l l l

…7†

where M is the ionic mass, Pl ; Ql and ql are the momentum, displacement and charge of the lth P ion, respectively, v 0 is the harmonic frequency, u and y ˆ yll 0 are constant parameters, and E is the external field. We start with the harmonic Hamiltonian H0 and its partition function Z0,  X 1 2 1 H0 ˆ Pl 1 M V2l …Ql 2 Q l †2 ; 2M 2 l !   21 …8† ∞ X "V…n 1 12 † "V Z0 ˆ ˆ 2 sinh exp 2 : kB T 2kB T nˆ0 By inserting the moments kQ2 l ˆ s 1 Q 2 and kQ4 l ˆ Q 4 1 6sQ 2 1 3s2 ; where s ˆ k…Ql 2 Q l †2 l ˆ …"=2M V† coth…"V=2kB T†; first-order perturbation theory yields F ˆ 2kB T ln Z 1 kH 2 H0 l ( ) 3u 1 4 2 ˆ F0 …T† 1 N …s 2 sc †Q 1 uQ 2 EqQ : 2 0 4

…9†

Again, this free energy has the required form (3) and the zero-field susceptibility has the well-known Barrett form [30] …6†

TQ ˆ Ts coth…Ts =T† is the quantum temperature [28,29], which strongly depends on the saturation temperature Ts ˆ V=2kB : Inserting the polarization P ˆ 2nmkSzi l; where n is the ionic density, we easily find the linear susceptibility function x 0 ˆ …nm2 =e0 †…1=…V=2† coth…V=2kB T† 2 …J0 =4††; which has the well-known form described by Barrett [30]. One should stress that a phase transition never occurs, if T0 , Ts ;

x…0† ˆ

q2 1 C ˆ Ts coth…Ts =T† 2 T0 3e0 u s0 2 sc

…10†

with Ts ˆ "V0 =2kB and T0 ˆ ……y 2 M v20 †=3u†…M V20 =kB †; where "=2M V0 . s ˆ …y 2 M v20 †=3u . 0 in the quantum paraelectric limit, T0 , T s : The perturbed frequency is p V0 ˆ v20 1 3us0 =M with s0 ˆ s…V0 †: The preceding considerations show that the observed susceptibility scaling properties of STO (Fig. 1(d)) can be

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Fig. 2. Dielectric susceptibility x 0 as a function of temperature for doped quantum paraelectrics with different impurity concentrations (labeled on the curves): (a) simulated with (dashed lines) and without (solid lines) random fields; (b) measured on SCT with x ˆ 0; 0.002 and 0.007. The solid lines are fits to the generalized quantum Curie–Weiss law, Eq. (11), with exponents g as indicated.

explained irrespective of the microscopic ordering mechanism, displacive or order–disorder type. Below we shall try to decide on this question when considering the doped SCT system. In particular the observed deviation from the meanfield exponent, a ˆ 1:5; also has to be reconsidered. It should also be noted that the zero-field susceptibility of pure STO does not precisely fit with the Barrett law [31]. A substantially better fit is obtained with a generalized quantum Curie–Weiss law [29],

x0 ˆ

C ; …Ts coth…Ts =T† 2 T0 †g

…11†

where g ˆ 2:6 fits best to the STO data for E ˆ 0 (Fig. 2(b)). 3. Impurity-induced phase transition in Sr12xCaxTiO3 [9,10] 3.1. Heterogeneous transverse Ising model In order to mimic the heterogeneity of the doped SCT systems, two kinds of pseudo-spins have to be accounted for within the TIM approach. One kind of pseudo-spins represents the host crystal unit cells, whereas the other refers to those occupied by impurities. The Hamiltonian of the system is X X X x 1X H ˆ 2V Sxi 2 Jij Szi Szj 2 2m Ei Szi 2 V 0 sm 2 m i ij i 2

X im

J 0im Szi szm 2 2m 0

X

Em szm

respectively. Em and Ei are the electric fields at impurity and host pseudo-spin sites, respectively. In general they consist of an external electric field, Eext, and site-dependent internal random electric fields, Erand. J and J 0 can be rewritten in the form of J0 1 J rand ; if randomness of interactions is taken into account, where J0 is the average and Jrand describes the randomness. Let us consider the non-random case, Erand ˆ 0 and Jrand ˆ 0: The simulations are carried out with the parameters V ˆ 3:05; J ˆ J 0 ˆ 1:0 and V 0 ˆ 2:0 which warrant the conditions 2V . ZJ…ˆ 6:0† and 2V 0 , ZJ: The former inequality keeps the host lattice to be quantum paraelectric and by the latter the impurities tend to induce a ferroelectric instability. The thermal average of the pseudospins is obtained in mean-field approximation [8,9]. The spontaneous polarization P is proportional to the average z-component of pseudo-spins. Their thermal expectation values kSzi l ˆ kszm l

~ iu Fiz uF tanh ~ iu 2kB T 2uF

are calculated self-consistently within a supercell approach employing periodic boundary conditions and site-dependent local mean-fields X X 0 ~ i ˆ …2Vi ; 0; 2 Jij kSzj l 2 F J im kszm l† and m

j

where S and s are pseudo-spins of host and impurity lattices, respectively. Summations over i, j are for host and m for impurity pseudo-spins. Parameters without and with primes refer to those of host and impurity pseudo-spins,

…13†

~ mu Fmz uF tanh ˆ ~ 2k 2uF m u BT

m

…12†

and

~ m ˆ …2V 0m ; 0; 2 F

X

J 0mj kSzj l†:

…14†

j

It is found that spontaneous polarization at T ˆ 0 occurs only if the impurity concentration exceeds a threshold value, x . xc ; where xc < 0:01 for the above choice of

W. Kleemann et al. / Journal of Physics and Chemistry of Solids 61 (2000) 167–176

Fig. 3. Susceptibility profile on the (001) plane of an SCT-like crystal at T ˆ 0 (see text), where the units in x and y directions are lattice constants.

parameters [9]. At x . xc the Curie temperature is first rising parabolically, TC …x† / …x 2 xc †1=2 ; whereas it increases linearly at x . 0:04 as observed on KTL [32,33] and KTN [34]. This is at variance with the behavior of SCT, where the initial …x 2 xc †1=2 dependence …xc ˆ 0:002† bends into a nearly horizontal curve, Tc …x† < 40 K; at x . 0:02 [5]. Very probably this peculiar behavior is due to competing effects caused by the admixture of non-ferroelectric CaTiO3 into the quantum paraelectric SrTiO3. On the one hand, at small concentrations the impurity-induced reduction of the average tunneling probability suppresses quantum fluctuations and thus favors ferroelectricity (see earlier). On the other hand, at higher concentrations the unit cell volume is reduced by virtue of the small Ca 21 ion. This cell volume pressure effect favors repulsive forces and thus depresses ferroelectricity similarly as observed in solid solutions of Ba12xSrxTiO3 at increasing x [35]. Both effects superimposed seem to be responsible for the quasistationarity of Tc vs. x at x @ xc in SCT.

171

The solutions of Eq. (13) yield a heterogeneous distribution of local electric moments within the supercells, where sharply peaking values are found at the positions of the central impurities [9]. A similar heterogeneous distribution applies to the local response functions to an external electric field, x 0 ˆ …2nm=e0 †…2kSzi l=2E† and …2nm 0 =e0 †…2kszm l=2E†; respectively. Fig. 3 shows the distribution of the dielectric susceptibility as calculated for x ˆ 0:003 within a central layer of the corresponding supercell. Clearly, the major contribution to the average response originates from the impurity, whereas the host lattice contributes little. Fig. 2(a) shows the TIM results for the averaged susceptibility, x 0 vs. T as calculated for x ˆ 0:003, 0.005 and 0.016 in conjunction with generalized Curie–Weiss power law fits, Eq. (11). For x ˆ 0 the Barrett solution is reproduced with an exponent g ˆ 1 as expected. With increasing x the susceptibility first rises …x ˆ 0:003†; and then develops a divergence at finite temperatures, Tc …x ˆ 0:005† ˆ 0:21: Remarkably, even for these heterogeneous samples the exponent g ˆ 1:1 stays close to the mean-field value of the homogeneous system. The experimental curves, obtained for x ˆ 0; 0:002 and 0.007 and displayed in Fig. 2b look very similar to those from the TIM calculations. However, fits to the quantum power law, Eq. (11), yield quite different exponents, g ˆ 2:6; 1.7 and 1.2, respectively. In addition, it is noticed for the x ˆ 0:007 curve, that its impurity-induced phase transition at Tc ˆ 18 K appears rounded. As will be outlined below, this is a consequence of the quenched random fields, which are typical of the SCT system (see Section 4). One unmistakable merit of the heterogeneous TIM is its ability to model the experimentally found scaling behavior of the dielectric susceptibility as shown in Fig. 4(a) and (b) for x ˆ 0 and 0.003, respectively. In both cases the ratio xE =x0 collapses onto a master function vs. E(x 0) a irrespective of the temperature, 0 # T # 0:6: As expected, for x ˆ 0 the mean-field exponent a ˆ D=g ˆ 1:5 emerges. However, for x ˆ 0:003 the best-fitted value a ˆ 2 comes close to the observed ones, a ˆ 2:03–2:05: These non-classic values are probably due to the spatial heterogeneity of these systems, as will be reconsidered below within the context of a coupled mode approach. Here we remark that also the nominally pure STO system seems to behave like an inhomogeneous system with a ˆ 2:00 (Fig. 1(d)). 3.2. Coupled mode model [36]

Fig. 4. Scaled plots of xE =x0 vs. E…x0 †a for SCT-like quantum paraelectrics with impurity concentrations x ˆ 0 (a) and x ˆ 0:003 (b) simulated for various temperatures within 0 # T # 0:56; choosing a ˆ 1:5 (a) and a ˆ 2 (b), respectively.

It is well known that an optic TO1 soft mode is at the origin of the near-instability of the paraelectric phase of SrTiO3 [37,38]. It is, hence, natural to treat the host material involved in SCT as a displacive system, whereas the impurity subsystem of the Ca 21 off-center ions rather behaves as a tunneling system. Instead of the heterogeneous TIM (Section 3.1) a more physical picture of SCT might, hence be given by a coupled displacive order–disorder quantum system. A similar idea was recently pursued in a novel

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research on squaric acid [39], whose well-known ferroelectric phase transition is controlled by both a softening displacive lattice mode and the kinetics of tunneling protons [40]. Here we propose the following Hamiltonian for SCT: Hˆ

1 2 1 1 1 X y 0Q Q 0 P 1 M v20 Q2l 1 uQ4l 2 2M l 2 4 2 l±l 0 ll l l

l

2E

X

ql Ql 2 V

X

X

X 1X Jij Szi Szj 2 2m ESzi 2 i±j i

Sxi 2

i

l

llj Ql Szj

…15†

lj

It contains the essential ingredients of the TIM and the AOM Hamiltonians, Eqs. (4) and (7). They refer to different subsystems, which are counted by different indices, (l,l 0 ) and (i,j), respectively. In addition bilinear coupling between the displacements Qi and the pseudo-spin components Szj is introduced via coupling constants llj . Our perturbation approach starts with harmonically vibrating ions and tunneling non-interacting pseudo-spins described by the Hamiltonians H1 ˆ

X l

H2 ˆ 2V

1 2 1 P 1 M v2 Q2l 2M l 2 X

Sxi

2 2m

i

X

!

"

…16†

eSzi :

Z1 ˆ 2 sinh

Z2 ˆ 2 cosh

!#21

q ! V2 1 4m2 e2 : 2kB T

and …17†

Following a similar procedure as pursued in the above limiting cases, one calculates the susceptibility x ˆ …n=e0 †  …2mxkSz l 1 7…1 2 x†qQ 0 †; which contains two contributions referring to the two order parameters involved, the softmode susceptibility 0

ˆ

1 1 4 lq m1 2 3us 1 M v20 2 v

!

1 2 V V J 4l x coth 2 2 4 2 2kB T …3us 1 M v20 2 v†…1 2 x†

!;

…19† where s ˆ …"=2M v† coth…"v=2kB T† and v ˆ 1 3us=M: In these expressions the concentration of impurity pseudospins, x, plays a crucial role. Although details are still under investigation we argue that the tunneling-type contribution to x 0 bears much more the characteristics of a pure Barrett system than the displacive-type contribution. It differs considerably from the function (10) by virtue of a strongly temperature dependent numerator. Probably this difference is at the origin of the x dependence of the g exponents encountered in Fig. 2(b). Whereas our nominally pure STO has a typical impurity concentration of only x < 0:0001 (14 ppm Ca 21, 19 ppm Ba 21 and 44 ppm Al 31 [41]), the tunneling Ca 21 ions are increasingly dominant at x ˆ 0:002 and 0.007. This might explain the observed decrease of g from 2.6 to 1.2. 2

v20

4. Relaxor properties of Sr12xCaxTiO3 4.1. Random-field induced precursor phenomena

i

"v 2kB T

ˆ

2kSz l 2E

and

To first order the free energy is F ˆ 2kB T ln Z1 kH 2 H1 2 H2 l, where Z ˆ Z1…12x†N Z2xN with

Q

kS 0z l ˆ

!

X

2

and the tunnel-mode susceptibility

2Q 2E "

! # V V J mlx 1 2 coth 4kB 2kB 2kB 2kB T #; ! ˆ " V V J xl2 coth 2 …3us 1 M v20 2 v† 2 4kB 2kB 2kB T 4kB …1 2 x† q

…18†

In doped quantum paraelectrics such as SCT, KTL and KTN a number of precursor characteristics have been evidenced such as the smearing of phase transitions and extreme polydispersivity, which are similar to those of typical relaxor systems like PbMg1/3Nb2/3O3 (PMN) [14]. Fig. 5 shows the real and the imaginary parts of the dielectric permittivity, e ˆ e 0 2 ie 00 ; of SCT with x ˆ 0:002 for temperatures 1:5 , T , 15 K and frequencies 1023 , f , 104 Hz: Obviously the real part, e 0 vs. T, can be described by a Barrett-type background function (dashed line) for high frequencies, f @ 104 Hz: At low frequencies a weak dispersion step is superimposed, which gives rise to polydispersive loss peaks, e 00 vs. T. Note that the peak heights of e 00 and e 0 differ by more than one order of magnitude. SCT might hence be called a soft relaxor in contrast to PMN, where polydispersivity dominates the entire dielectric function [14]. The relaxor-like phenomena are believed to result from polar nanodomains, which exist in the temperature range between the high temperature para- and the low-temperature ferroelectric regime [4,15]. As to the origins of these nanodomains, random field [4,5], dipole glass [2], superparaelectric [42], and, most recently, random field–random bond models [16] have been proposed. Very probably, the formation of locally ordered nanodomains in SCT is due to the

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173

Fig. 7. Temperature dependence of the Brillouin shift Dn (left-hand scale) due to a TA phonon propagating along [001]c and the corresponding elastic constant c44 (right-hand scale) measured on SCT with x ˆ 0:007 (open circles). Best-fits to a linear function (curve 1a), linear–quadratic coupling theory (curve 1b, [47]) and Eq. (20) (curve 2), respectively, are plotted as solid lines (see text). The phase transition temperatures T0 ˆ 125 K and Tc ˆ 18 K are indicated by arrows. Fig. 5. e 0 and e 00 vs. T of SCT with x ˆ 0:002 measured within 1:5 # T # 15 K at frequencies 1023 # f # 104 Hz as indicated. The asymptotic Barrett law for high frequencies is indicated by a broken line.

distribution of quenched random fields rather than the distribution of random bonds. Whereas the former are ordering by virtue of their spatial fluctuations [17], the latter are disordering. While in a hard relaxor like PMN the local charge disorder is intrinsic, in SCT the origin of random fields is believed to be due to extrinsic dipolar defects like

Fig. 6. Temperature dependencies of the SHG intensity Syy measured in x…yy†x and z…yy†z geometries on SCT with x ˆ 0:002 and compared with Raman shifts, vTO1 and vTO1 24 ; measured [19] in z…yy†x geometry (circles and eye-guiding lines).

Ca 21 –V0 centers with comparatively low concentration, n < 1018 cm23 : Nevertheless, an appreciably broad size distribution of nanodomains emerges from the observed polydispersivity (Fig. 5, [18]) within Chamberlin’s model of dynamically coupled domains [43,44]. Another very convincing piece of evidence for polar nanodomains is the observation of second harmonic generation of light throughout the precursor regime as shown in Fig. 6 for SCT with x ˆ 0:002 [11,20]. Two scattering geometries, z…yy†z and x…yy†x; yield essentially the same temperature dependence, which is readily described within the framework of current theories of hyper Rayleigh scattering at precursor clusters of the polar order parameter [45,46]. The predicted [46] proportionality of the SHG signal with the inverse fourth power of the soft mode frequency is confirmed by comparison with Raman data obtained in z…yy†z scattering geometry [19]. Brillouin scattering due to the transverse c44 shear mode in z…xx†z scattering geometry lends further support to the occurrence of locally broken inversion symmetry in the precursor regime of SCT with x ˆ 0:007: As shown in Fig. 7 the continuous drop by more than 10% of the frequency shift, Dn , above the phase transition from D4h to C2v point symmetry at Tc ˆ 18 K; is compatible with bilinear coupling of the e4 shear strain with the soft Eu component of the optic TO1 mode. An appropriate description of the elastic shear constant is given by [47] q c44 …T† ˆ …ct44 †2 2 d=v2Eu …T†; …20† where ct44 refers to the tetragonal phase at high temperatures and d is a constant (best-fitted line denoted as 2). Alternatively, Hehlen et al. [48] attribute the anomalous low-T decrease of the sound velocity in STO to coupling of the strain to gradients of the electric polarization fluctuations.

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Fig. 8. Raman shifts of various soft modes measured vs. temperature on SCT with x ˆ 0:007 (a) and 0.002 (b) in various scattering geometries as indicated and connected by eye-guiding solid lines. The broken line denotes the tentative energy of the A2u phonon in the D4h phase.

The abrupt drop of Dn at T0 ˆ 125 K is due to the linear– quadratic coupling of the e4 shear strain to the antiferrodistortive order parameter at the phase transition from Oh to D4h point symmetry [49]. The function denoted as 1b is best-fitted at T , T0 to theory [50,51], whereas at T . T0 a linear dependence of c44 with the lattice expansion is assumed (curve 1a). Finally, the observation of quasi-first-order Raman scattering at the optic soft mode components Eu and A2u also strongly hints at locally broken inversion symmetry due to polar nanodomains [19]. Fig. 8 shows the temperature dependence of the Raman shifts due to low-lying optic modes in SCT with x ˆ 0:002 and 0.007. The modes designated as FMR ( ˆ ferroelectric “microregions”)-TO1 are due to coupling between very low-f relaxation modes of the nanodomains and the optic soft mode. The splitting of TO1 into A2u and Eu in D4h symmetry is ascertained by first-order scattering of the ferroelectric x ˆ 0:007 compound below Tc ˆ 18 K (Fig. 8(a), arrow). It should be mentioned that accidental degeneracies occur at T q < 40 K; where the softening optic Eu mode crosses the hardening Eg mode originating from the antiferrodistortive phase transition at T0. Very probably [52,53] this is the reason for the appearance of various anomalies reported in the literature, which were previously [54,55] attributed to a coherent low-temperature quantum state or to the onset of second sound in STO at Tq [56]. Recently [11], we observed pronounced hyper-Raman scattering peaks at Tq in SCT with x ˆ 0:014; which strongly supports the mode crossing conjecture [52,53]. 4.2. TIM calculations including random fields [12] Let us recall that the general TIM Hamiltonian, Eq. (12), allows to include randomness by introducing both random fields and random bonds. Hitherto we have done calculations

including bimodal distributions of random fields [12], Erand ˆ ^0:001; at the impurity sites, i.e. Em ˆ E 1 Erand ; but letting Ei ˆ E at the host lattice sites. Hereby the interaction of the impurity off-center dipoles with local random dipolar fields is mimicked. The calculations were ensemble averaged over enlarged supercells containing up to 27 regularly distributed impurity pseudo-spins. Fig. 9(a) shows kP2 l as a function of temperature T at different impurity concentrations. In the presence of the random fields a precursor tail is found above Tc. It should be noted that the average value of spontaneous polarization kPl goes to zero at Tc, although kP2 l survives up to temperatures far above Tc. The temperature range of the precursor increases with increasing random fields as expected. At the same magnitude of the random fields the tail is larger for the system with a lower impurity concentration. Fig. 9(b) shows the results of SHG experiments on SCT with x ˆ 0:002 (see also Fig. 6), 0.007 and 0.014, respectively. Obviously the intensity, ISHG vs. T, closely follows the simulations in Fig. 9(a). These are also in good accordance with the temperature dependence of linear birefringence measured on SCT [57] and KTN [58], respectively. Since random fields destroy phase transitions with continuous order parameters [17], they are most sensitively detected by divergent quantities. In Fig. 2(a) we compare the temperature dependences of the dielectric susceptibility x 0 calculated at different impurity concentrations with (dashed lines, Erand ˆ ^0:001) and without (solid lines) random fields, respectively. It is seen that random fields reduce the saturated dielectric susceptibility when x ˆ 0:003 , xc : For systems with x . xc ; the dielectric susceptibility no longer diverges at Tc as in the absence of random fields. Broadening of the dielectric peaks is encountered, being most pronounced when x is close to xc as seen by comparing the curves referring to x ˆ 0:005 and 0.016. The results are in accordance with the temperature

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Fig. 9. (a) Temperature dependence of kP 2l for doped quantum paraelectrics with different impurity concentrations (labeled on the curves) when random fields are taken into account (see text), and (b) temperature dependence of SHG intensity of SCT crystals with three different Ca 21 concentrations as indicated.

dependence of kP 2l shown in Fig. 9(a). Both quantities indicate that phase transitions are smeared due to the presence of random fields. In comparison with the TIM results, Fig. 2(b) shows the experimental dielectric susceptibilities of SCT with x ˆ 0; 0.002 and 0.007. Obviously these results fit better with the simulations when including random fields. In addition, it is found that the exponents g describing the temperature dependence of x 0 above Tc using Eq. (11) increase from g ˆ 1:18 and 1.12 to g ˆ 1:25 and 1.16 for x ˆ 0:003 and 0.005, respectively, when random fields are taken into account. This is in accordance with the higher values found experimentally, g ˆ 1:7 and 1.2 for x ˆ 0:002 and 0.007, respectively. We claim that the above calculations are a first promising step towards modeling of the random-field impurity system SCT. Surely, much larger supercells have to be evaluated in order to verify the nature of the nanodomains, which reflect mesoscopic fluctuations of the random field distribution.

5. Conclusion The use of quantum mechanical mean field models in order to describe the susceptibility behavior of pure and doped SrTiO3 has proven to reveal a deeper insight into the crossover from incipient to real ferroelectricity in these compounds. On the one hand, their peculiar noncritical scaling behavior has been elucidated. Generalized homogeneity of the free energy emerges from two very different theoretical approaches, the order–disorder type TIM and the displacive type AOM, in both of which the quantum temperature TQ plays a decisive role. On the other hand, by considering either a heterogeneous TIM or an appropriate mode coupling theory also the influence of the impurities has been described satisfactorily. The heterogeneous TIM turns out to be

very powerful in order to understand the doped system on a qualitative mean field level. It illustrates the heterogeneous distribution of electric moments in a very direct way. When introducing spatial randomness also clustering of impurity moments can be modeled. Very probably, however, the newly presented mode coupling concept is more adequate with respect to the underlying physics and promises to yield meaningful fitting parameters. Work is under way in order to reach this aim in near future. Understanding the relaxor nature of doped quantum paraelectrics on the ground of our microscopic models is, however, still a very ambitious task. Although we have seen some smearing effects at the phase transition, we are still far from describing correctly the mesoscopic physics, which governs the precursor regime of these materials. Very probably the dipolar interactions of cluster-like polar aggregates (“nanodomains”) have to be taken into account in addition to the short range forces appearing in the microscopic Hamiltonians. Presently, alternative theories like the local random field approach of Vugmeister and Rabitz [59] are more effective in order to describe, e.g. the relaxor-like dynamics. It starts by assuming a broad distribution of polar nanodomains. Alternatively, Potts model simulations like those of Qian and Bursill [60] might be envisaged when trying to mimic the formation of polar nanodomains in the presence of random fields and their polydispersive behavior. The vision of a complete theory taking into account all of these facets is a challenge for the future.

Acknowledgements Thanks are due to the Deutsche Forschungsgemeinschaft and to the Alexander von Humboldt-Stiftung for providing grants to J.D. and Y.G.W., respectively.

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