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1 This article was downloaded by:[tib-lizenzen - TIB Licence Affairs] On: 12 February 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Ferroelectrics Publication details, including instructions for authors and subscription information: Polarization fluctuations in pure SrTio 3 S. A. Prosandeev a ; A. E. Maslennikov a ; W. Kleemann b ; J. Dec c a Physics Dept., Rostov State University, Rostov-on-Don, Russia b Laboratorium für Angewandte Physik, Gerhard-Mercator-Universität Duisburg, Duisburg, Germany c Institute of Physics, University of Silesia, Katowice, Poland First Published on: 01 February 2000 To cite this Article: Prosandeev, S. A., Maslennikov, A. E., Kleemann, W. and Dec, J. (2000) 'Polarization fluctuations in pure SrTio 3', Ferroelectrics, 238:1, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Fermelecrrics, 2000, Vol. 238, pp Reprints available directly from the publisher Photocopying permitted by license only OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. Printed in Malaysia Polarization Fluctuations in Pure SrTi03 S. A. PROSANDEEVa, A. E. MASLENNIKOVa, W. IUEEMANNb and J. DECC aplzysics Dept., Rostov State University, 5 Zorge St., , Rostov-on-Don, Russia, bluboratorium fur Angewandte Physik, Gerhard-Mercator- Universitat Duisburg, Duisburg, Germany and 'Institute of Physics, University of Silesia, Uniwersytecka 4, Katowice, Poland (Received July 12, 1999) The dielectric susceptibility of pure strontium titanate (ST) was measured as a function of temperature in the range from 4 to 140 K. The analysis of the data obtained has shown that the Barrett formula cannot well reproduce the experimental data. In order to elucidate the nature of this discrepancy we have computed the electric zero-field susceptibility in the framework of the interacting anharmonic quantum oscillator model. A perturbation theory technique was employed. We show that the susceptibility obtained in the fourth order of the theory can nicely reproduce the experimental data. Hence, a reason for the unusual low-t behavior of the electric susceptibility in ST is the existence of large quantum fluctuations at low 7: Keywords: Paraelectrics; dielectric susceptibility; fluctuations

3 172/[7361 S. A. PROSANDEEV et al. INTRODUCTION It is well known that the dielectric susceptibility of quantum paraelectrics, such as SrTiO3 (S7) and KTa03, deviates from the Curie-Weiss law at low T. The main feature of this deviation is the saturation of the susceptibility at low T instead of a divergence. Barrett"' first described this peculiarity theoretically by taking into account the zero-point quantum vibrations of the ions. The long standing Barrett formula reduces to the simple Curie-Weiss law at high temperatures while at low ones it saturates. Despite the general quantitative similarity between the Barrett formula and experimental data, in the particular case of ST there is some deviation. Muller el QZ.[~~' first stressed this inconsistency. They showed that, when adjusted at high temperatures, the susceptibility calculated with the Barrett formula turns out to be lower by about 30 YO than the saturated experimental values. Additional information about the problem can be found from experiments on solid solutions of SrTiO3:Ca. It was shown earlier['] that even a small concentration of Ca (of about 0.02 YO) triggers a ferroeiectric-type phase transition. The dielectric susceptibility at this phase transition diverges. Nevertheless the critical parameter y describing this divergence was shown to be larger than unity@' in accordance with the theory developed by Schneider el al."' All these data strongly indicate a possible large influence of quantum fluctuations on the susceptibility. In the present work we will reconsider both the experimental and theoretical data on the low T behavior of the dielectric

4 POLARIZATION FLUCTUATIONS IN PURE SrTiO, [737]/173 susceptibility in ST. It will be shown that the discrepancy stressed by Muller et uz.[*-~~ is also confirmed by our experiment. In order to find an explanation for the inconsistency we computed the dielectric susceptibility in the framework of an interacting anharmonic quantum oscillator model. We show that the results obtained within the mean field approach just coincide with the Barrett formula, which, however, cannot reproduce the experimental data. Obviously this implies that one should go beyond this approach. As a consequence we developed a perturbation theory and computed the first few contributions. It turns out that the novel theory reproduces the experimental data rather well with reasonable parameters. EXPERIMENT Nominally pure samples of ST were obtained commercially from Crystal GmbH, Berlin. The measurements were performed on a (1 10)-oriented plate sample with the long edges parallel to [OOl],. This geometry favours an oriented state with the tetragonal c axis parallel the long edges, i.e., parallel to [OOl],. In order to make electric contacts, copper electrodes were vacuum evaporated to cover the major ac faces. The complex susceptibility, x = x' - zx", was measured with a Hewlett-Packard 4 192A im- pedance analyser at temperatures 4 < T < 140 K and frequency,f= 1 khz, with low ac probing fields, Eo = 30 V/m, in a he- lium-gas-flow cryostat.

5 174/[738] S. A. PROSANDEEV et al. Downloaded By: [TIB-Lizenzen - TIB Licence Affairs] At: 15:25 12 February 2008 PERTURBATION THEORY We start with the most general expression of the Gibbs free energy: F = -k,tin~e-':-'"'. Here En is the energy of the n-th microscopic state. We use the Salje-Wruck-Thomas Hamiltonian describing a set of interacting anharmonic quantum oscillators,f8i where PI and Q, are the operators of the momentum and coordinate of an oscillator, respectively. The other quantities have the usual meaning. The trial Hamiltonian describes the non-interacting harmonic oscillators with the effective frequency w and the average displacement Q,.[*I The difference between these two Hamiltonians is the perturbation, V. Next we use the linearization technique representing the energy in the form En = E,,""' + E,,") +..., as in the ordinary quantum perturbation theory,['] where each of the contributions has the corresponding order of smallness. Having found the matrix elements of the perturbation it is possible to represent further the result as an expansion in powers of the order parameter, Ef" = af" + p!')~q + y:'),q2 + A:)Q' + 6i')a' As the energy En can also be represented in a similar form,

6 POLARIZATION FLUCTUATIONS IN PURE SrTiO, [739]/175 EN =av + p s + y s' +,I,$3 + S s4 +..., then the coefficients can also be linearized, for instance, a,, = a!') +a:) +... Now it is possible to find the variable parameters of the model from the equilibrium conditions: af /aq = 0, dl7 /a@ = 0. Further we will be interested only in the zero-field susceptibility of a paraelectric. The zero-field dielectric susceptibility is defined asxo = (nq/so)qk(e = 0), where q is an effective charge, n is the concentration of the oscillators and Q is the permittivity of the vacuum. One can find the derivative of the order parameter by differentiating the equilibrium condition with respect to E and by considering vanishing E and Q thereafter. Note that the coefficients,in and 8, do not influence the zero-field susceptibility. From symmetry considerations it also follows that (a,)'e and p,, vanish for paraelectrics at E = 0. Thus the final result for the dielectric susceptibility is of the form The formula obtained solves the problem of deriving an initial expression for making a perturbation series. We considered the series xi'), #). xg) and so on, where each of the terms X&')contributes to the coefficients a,,, p,,, y, up to the I-th order of smallness.

7 176/[740] S. A. PROSANDEEV et a[. Y -* Y Curie-Weiss la I T [KI FIGURE 1 Experimental data and best fits to various theoretical approaches for the dielectric susceptibility in ST. RESULTS When considering only the first order terms the resulting sums over the quantum states of the oscillators can be taken analytically and the final result is of the Barrett type form. Above the first order, the susceptibility data were obtained numerically up to the

8 POLARIZATION FLUCTUATIONS IN PURE SrTiO, [7411/177 fourth order. This limitation could be justified by that we are con- sidering the case of a quantum paraelectric state, where the diver- gence of the susceptibility is absent. Figure 1 shows the results of fitting the experimental data to the theory in use. It is seen that the susceptibility can be well represented by the theory. The employed parameters are Aul M2ui = , C~vll /Mu; = and fi00 lkb = 53K. DISCUSSION In accordance with earlier studies'241 the measured temperature dependence of the dielectric susceptibility of ST has shown to reveal large discrepancies with both the Curie-Weiss law and the Barrett formula (Figure 1). The main result of the computation performed in this paper is that the dielectric susceptibility of ST may be well represented by the theory which takes into account fluctuations and treats the Hamiltonian (1) beyond a mean field approach. Similar results have been obtained within a mean field approach by taking into account mode coupling, however, those trials gave less accurate fitting fl01. As it was shown by Dec and Kleemann['I that the experimental data on the dielectric susceptibility of the solid solution SrTi03:Ca can be nicely reproduced by a generalized Barrett formula : xo = CI(Tp where T~=T,coth(T,/7') is the quantum temperature. Although y has the meaning of a critical

9 178/[742] S. A. PROSANDEEV et a1 exponent the generalized Barrett formula was found to hold also for samples in which the phase transition is absent at all. This implies that large quantum fluctuations exist also in the quantum paraelectric phase. The present work confirms previous suppositions. Owing to the large quantum fluctuations in pure ST at low T the quasicritical exponent y was found to be larger than 1 and close to 2. Acknowledgements S. A. P. appreciates the Kasa Mianowskiego for supporting his stay at the Silesian University, Katowice, Poland. J. D. and W. K gratefully acknowledge financial support by the Deutsche Forschungsgemeinschafr. References [l] J. H. Barrett, Phys. Rev. 86, 118 (1952). [2] K. A. Muller and W. Berlinger, Phys.Rev. Lett. 26, 13 (1974). [3] K. A. Miiller and H. Burkard, PhysRev. B 19,3593 (1979). [4] K. A. Miiller, Jpn. J. Appl. Phys. 24, Suppl., 89 (1985). [51 J. G. Bednorz and K. A. Muller, Phys.Rev. Lett. 52,2289 (1984). [61 J. Dec and W. Kleemann, Solid State Commun. 106,695 (1998). [71 T. Schneider, H. Beck and E. Stoll, Phys. Rev. B 13, 1123 (1976). 181 E. K. H. Salje, B. Wruck and H. Thomas, 2. Phys. B - Condens. Matter 82,399 (1991) [9] L. Landau and E. Lifshitz, Theoretical Physics, Vo1.5 (Nauka, Moscow 1976). [lo] R. Migoni, H. Bilz and D. Bauerle. Phys. Rev. Lett. 37, 1155 (1976).