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1 This article was downloaded by: [Uniwersytet Slaski] On: 14 October 2008 Access details: 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: Dielectric Susceptibility of Strontium Titanate Doped with Ca S. A. Prosandeev a ; W. Kleemann b ; J. Dec c a Physics Department Rostov State University 5 Zorge St., Rostov on Don, Russia b Laboratorium für Angewandte Physik Gerhard-Mercator-Universität Duisburg, Duisburg, Germany c Institute of Physics University of Silesia Uniwersytecka 4, Katowice, Poland Online Publication Date: 01 January 2004 To cite this Article Prosandeev, S. A., Kleemann, W. and Dec, J.(2004)'Dielectric Susceptibility of Strontium Titanate Doped with Ca',Ferroelectrics,299:1,83 87 To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be 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 Ferroelectrics, 299: 83 87, 2004 Copyright C Taylor & Francis Inc. ISSN: print / online DOI: / Dielectric Susceptibility of Strontium Titanate Doped with Ca S. A. PROSANDEEV Physics Department Rostov State University 5 Zorge St., Rostov on Don, Russia W. KLEEMANN Laboratorium für Angewandte Physik Gerhard-Mercator-Universität Duisburg D Duisburg, Germany J. DEC Institute of Physics University of Silesia Uniwersytecka 4, PL , Katowice, Poland Received in final form February 10, 2003 A theory of dielectric response of quantum paraelectric strontium titanate doped with Ca has been developed. The impurities are thought to be in two possible states, dipole and s-type. The s-type impurities soften the critical ferroelectric lattice mode due to lower spring constants Ca-O with respect to Sr-O. The dipole type Ca impurities give rise to the softening of lattice modes through a local field effect, which results in coupling of the dipole coordinates to the soft mode displacement. The theory is successfully compared with experimental data for Sr 1-x Ca x TiO 3. Keywords Dielectric properties; strontium titanate; point defects Measurements of dielectric properties of dilute single crystals of Sr 1 x Ca x TiO 3, x 1 (SCT for short) [1, 2] have revealed a strong influence of the Ca impurities on the dielectric susceptibility. E.g., Ca impurities at a concentration as low as x = enlarge the susceptibility of pure SrTiO 3 several times, although the dilute SrTiO 3 still remains in the paraelectric phase. At a concentration of x = the dielectric susceptibility shows a peak, which is attributed to a ferroelectric phase transition [1]. The drastic influence of small impurity additions on the host lattice properties is considered in the present paper theoretically. One fraction, n 1, of the Ca-impurity concentration referring to x = is assumed to contribute linearly to the free energy, F = F (α(t ) (n 1 + n 10 ))P bp4 0 EP 0. (1) The concentration n 10 takes into account the fact that some impurities exist already in nominally pure samples. α and b are the usual expansion parameters, while is a coupling constant. The linear approximation corresponds to the case when the Ca impurities can 83

3 84 S. A. Prosandeev et al. be considered as soft oscillators giving rise to softening of the host lattice dynamics. A phenomenological description of this situation was presented previously [3]. Another fraction of the Ca ions with concentration n 2 is considered to have a dipole moment and to behave like a set of tunneling ions. They enter the Hamiltonian as proposed previously [4]: H = 1 ωk 2 2 y ky k + 1 V (k 1, k 2, k 3, k 4 )y k1 y k2 y k3 y k4 4 k k 1,k 2,k 3,k 4 E z i x i n 2 Si x 1 2 n 2 j ij Si z Sz j n 2 λ ij x i S z j i i i j ij 2µn 2 E i S z i (2) In this expression the impurities are described within the framework of the transverse Ising model [5] by using pseudo-spin components S x,z i, the spin-spin interaction J 0 = j J ij and the tunneling integral. Bilinear coupling with strength λ = j λ ij between the ionic displacements x i of the charges z j and the pseudo-spin components S z j is assumed. From equilibrium conditions one obtains [6] ( α(t ) (n1 + n 10 ) + bp 2 0) P0 λ 2µ P 1 = E, (3) S z = W z 2W tan h W 2k B T. (4) Thus we consider the dipoles being in a local electric field produced by the lattice, while the host lattice ions are in a local field produced by the impurity dipoles. Here P 0 is the part of the polarization referring to the soft mode, while P 1 = 2µ(n 2 + n 20 ) S z is that due to the tunneling ions. Further, W = 2 + W 2 z, W z = 2µE + J 0 S z +λp 0. The concentration n 20 describes the dipole type impurities in the nominally pure sample. The dielectric susceptibility as obtained from the derivatives with respect to E yields ε 0 χ = dp 0 + dp 1 de de = 1 + 4λ(n 2 + n 20 )µg(e) + 4µ 2 (n 2 + n 20 )G(E)(α(T ) (n 1 + n 10 ) + 3bP 2 0) α(t ) (n 1 + n 10 ) + 3bP 2 0 λ2 (n 2 + n 20 )G(E) where G(E) = f w z /(1 J 0 f w z ) f w z = 2 tan h W 2W 3 2k B T + 1 k B [1 tan h 2 T 4W 2 2k B ]. Equation (5) does not take into account possible differences of the dipole moments and coupling T constants for different impurities. When lifting this constraint, however, a general expression can be obtained in the same way. Equation (5) contains four fitting parameters: n 2 + n 20, b, χ 0 = 1/[ε 0 (α(t ) (n 1 + n 10 ))] and µ. The value of the coupling constant is found from the expression λ = γµ/3ε 0, where γ determines the correction for the local field in the perovskite-type lattice. Adopting recent results for the Li-site in KTaO 3 [7] we assume λ = 0.2 also for the Ca site in SrTiO 3. The results of fitting the obtained expression to experimental data [8] for x = and T = 20 K are shown in Fig. 1. It is seen that this fit is rather good. The values of the parameters obtained for a few other temperatures and concentrations are represented in Table 1. Figure 2 shows the temperature dependence of the reverse χ 0 for different Ca concentrations. W 2 z W (5)

4 Dielectric Susceptibility of Strontium Titanate 85 TABLE 1 Results of Fitting Eq. (5) to the Experimental Data [8, 11] for Different Temperatures x = 0 T, K (n 2 + n 20 ), 10 5 m χ B, Jm 5 /C µ, mc χ x = T, K (n 2 + n 20 ), 10 5 m χ B, Jm 5 /C µ, mc χ x = T, K (n 2 + n 20 ), 10 5 m χ B, Jm 5 /C µ, mc χ It is seen that coupling the soft mode to the dipole impurities can explain the peculiar behavior of the dielectric susceptibility in SCT. Experimental data from EPR [9] and computations of the local fields in solid solutions of KTaO 3 and SrTiO 3 [7, 10] support the idea that the soft mode is coupled to dipolar impurities. From the data obtained it is seen that the value of the parameter n 2 + n 20 is very small in comparison with the concentration of the Ca ions, x. For the nominally pure SrTiO 3 this concentration proves to be even larger than for SCT while the value of the FIGURE 1 Comparison of experimental [8] (circles) and theoretical (solid line) data obtained for SCT with x =

5 86 S. A. Prosandeev et al. FIGURE 2 Dependence of 1/χ 0 on temperature for different concentrations of Ca in SCT (x = and 0.007) and for nominally pure SrTiO 3 (x = 0). average dipole moment is lower. It implies that in the nominally pure SrTiO 3 there exists a valuable concentration of impurities with small dipole moments while in SCT there appear comparatively larger dipole moments while being in a smaller concentration. The value of the dipole moment in SCT seems to be extremely large. Indeed, if the charge of the Ca ion is 2e then the displacement should be 0.05 nm. Nevertheless they are larger. One may suppose that groups of Ca ions are strongly correlated and influence the lattice dynamics together. The existence of such dipolar Ca impurities can be evidenced on the basis of percolation computations. Due to strong dipole-dipole interaction inside the clusters the monopole Ca impurities can loose inversion symmetry and a cluster dipole moment can appear. This dipole moment represents the individual dipole moments of all Ca ions involved in the cluster. Owing to the strong correlation of the dipoles we expect a multiwell potential acting on the total dipole moment rather than on the individual ones. This point of view should be checked by further experiments and computations. From the comparison of the obtained results for SCT with those for nominally pure strontium titanate (x = 0) it is seen that dipolar impurities already exist in the nominally pure sample. However, the value of the dipole moment in the pure sample is lower than in SCT at larger impurity concentration. Obviously the dipolar clusters are growing as x increases as it was already mentioned above. It follows from Fig. 2 that the reverse dielectric susceptibility χ 1 0 decreases with increasing Ca concentration because of the influence of the s-type Ca impurities. Obviously this decrease is not linear with increasing Ca concentration. This evidences that an increasing part of the Ca ions in SCT is in the dipole state. In conclusion we like to stress that a standard Landau-Devonshire expansion of the free energy cannot be used in the considered case in order to describe the behavior of the nonlinear susceptibility at low electric fields because of the very steep electric field dependence of the dipole contribution to the dielectric susceptibility. Only in the large E limit G(E) vanishes and merely the lattice is responsible for the nonlinear susceptibility. Thanks are due to the Deutsche Forschungsgemeinschaft for research support to S.A.P. and J.D. References 1. J. G. Bednorz and K. A. Müller, Phys. Rev. Lett. 52, 2289 (1984). 2. J. Dec, W. Kleemann, U. Bianchi, and J. G. Bednorz, Europhys. Lett. 29, 31 (1995).

6 Dielectric Susceptibility of Strontium Titanate S. A. Hayward and E. K. H. Salje, J. Phys.: Condens. Matter 10, 1421 (1998). 4. V. G. Vaks, Vvedenie v mikroskopicheskuju teoriju segnetoelektrikov, Nauka, Moscow, 1973 (in Russian). 5. R. Blinc and B. Zeks, Soft modes in ferroelectrics and antiferroelectrics, North-Holland Publishing Company, Amsterdam, W. Kleemann, J. Dec, P. Lehnen, Y. G. Wang, and S. A. Prosandeev, J. Phys. Chem Solids 61, 167 (2000). 7. S. A. Prosandeev and A. I. Riabchinskii, J. Phys.: Condens. Matter 8, 505 (1996). 8. U. Bianchi, J. Dec, W. Kleemann, and J. G. Bednorz, Phys. Rev. B 51, 8737 (1995). 9. B. E. Vugmeister and M. D. Glinchuk, Rev. Mod. Phys. 62, 993 (1990). 10. A. V. Turik, Izv. AN SSSR, ser. fiz. 57, 35 (1993) (in Russian). 11. J. Dec, W. Kleemann, and B. Westwanski, J. Phys.: Condens. Matter 11, L379 (1999).