Interpretation of elastic anomalies in SrTiO 3 at 37 K

Transcription

1 Z. Phys. B 104, (1997) ZEITSCHRIFT FÜR PHYSIK B c Springer-Verlag 1997 Interpretation of elastic anomalies in SrTiO 3 at 37 K J.F. Scott 1,2, H. Ledbetter 3 1 Max Planck Institute for Microstructural Physics, D Halle/Saale Germany 2 School of Physics, University of New South Wales, Sydney NSW 2052, Australia 3 Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Boulder, CO 80303, USA Abstract. We relate the measurements by Ming Lei and Hassel Ledbetter (1991) of elastic moduli in strontium titanate near 37 K to a dynamical anharmonic mode coupling model. It is suggested that the effects are purely dynamic and not indicative of a true structural phase transition and that neither macroscopic coherence nor second sound is involved. PACS: a for any value of S fits Φ(T ) before invoking non-mean-field exponents. 2. Hypothesis of Courtens et al.: Extra acoustic phonons Courtens et al. have reported [4,5] near 37 K an additional feature in their Brillouin spectra of strontium titanate. There is a strong decrease in the TA sound velocity and a splitting of the acoustic peak. They suggested that this may be due to second sound, or to polarization clusters. I. Introduction 1. Order parameter temperature dependence: the hypothesis of Muller et al. Muller s suggestion [1] that a macroscopically coherent quantum state might exist in strontium titanate was based on the failure of the order parameter Φ(T ) [oxygen octahedron rotation angle] to fit below 37 K a particular mean-field theory [1b]. However, it had been shown earlier by one of us [2] that this discrepancy below 37 K is eliminated if an S =1 Brillouin function is used instead of a mean-field model. Such a pseudospin model is not unreasonable in SrTiO 3 because near T c the oxygen octahedra may rotate into more than two off-center positions (around [100] or [111] directions), but it is contrary to conventional wisdom to treat this material with pseudospin models. Parenthetically we note that this revised fit also makes the need [3] for non-meanfield exponents near T 0 = 105 K much less clear or obvious. Of course a wide range of mean field fits to Φ(T ) is available, ranging from tanh τ to Langevin functions (S = ). Of course all these satisfy β =1/2, where β is the critical exponent characterizing the coexistence curve. In other ferroelectrics pseudo-spin models with S>1/2 give best fits [2], e.g. S =3/2 in TSCC. For a hydrogen-bonded ferroelectric, employing ice rules gives more than two low-lying levels important for the phase transition; hence an effective spin of 3/2 with four levels is reasonable. Similarly, S =1 or 3/2 for SrTiO 3 is not unreasonable. As a procedural protocol, one should probably show that no Brillouin function 3. Ultrasonic techniques Ultrasonic techniques, especially ultrasonic attenuation, are probably the most sensitive techniques known for revealing subtle structural phase transitions. However, peaks in ultrasonic attenuation sometimes occur for dynamic reasons that do not involve phase transitions. And even when they do signal a phase transition, its nature is not revealed in any direct way. A rather good example in ferroelectrics is the work of Spencer and Kominiak [6] on the antiferroelectric incommensurate transition in BaMnF 4 [7, 8]. Their ultrasonic data were for some years the only evidence of a phase transition, and such skepticism surrounded these data that they were not cited in subsequent work on BaMnF 4 from the same laboratory. For many years various authors have concluded that strontium titanate exhibits a structural phase transition near 37 K. A transition was reported by Sorge and Hegenbarth [9] in 1969 but not confirmed by Rehwald s precise study [10] the following year. Very recently Balashkova et al. [11] have used ultrasonic techniques on strontium titanate below 40 K and suggested that the anomalies observed may be purely dynamical and arise from domain walls. In 1991 Ming Lei and Ledbetter reported [12, 13] precise elastic measurements on single-crystal strontium titanate, which exhibited large anomalies at 65 K and 40 K. These are the temperatures at which Fleury et al. reported [14, 15] accidental crossings in the frequency of the soft optical modes responsible for the antiferrodistortive phase transition with the frequencies of those soft ferroelectric modes which

2 636 Fig. 1. (top to bottom) Poisson s ratio (ν), bulk modulus (B), Young s modulus (E), and shear modulus (G) in strontium titanate versus temperature, each normalized to unity at 295 K, showing anomalies near 37 K, 65 K (marked with vertical dashed lines) and probably 25 K. Experimental data from Ming Lei and H. Ledbetter. This specimen had an observed transition temperature at K are incipient (pure strontium titanate is not ferroelectric at any finite temperature, but would yield an extrapolated Curie temperature of ca. 30 K). Hence there is the suggestion that the anomalies might be purely dynamic and not evidence of static structural distortion(s), i.e., not real phase transitions. A simple dynamical model of optic-acoustic mode accidental crossings in frequency and the effects such crossings would generate was initially provided by Cowley [16] in 1964, but unfortunately for a true static phase transition near 105 K for which it was an inapplicable model. In the present paper we attempt to explain the ultrasonic data of Lei and Ledbetter in terms not of Cowley s harmonic mode coupling but in terms of anharmonic coupling. This is an alternative to models in which the 37 K phenomena are viewed as due to second sound or some super-phase characteristics. Fig. 2. S 11 compliance coefficient for strontium titanate versus temperature, normalized to unity at 295 K, showing a minimum at 37 K and a maximum at 65 K. This implies a dilatational anomaly at 65 K and a shear anomaly at 37 K. Data from Ming Lei and Ledbetter II. Experiment (Lei and Ledbetter) Lei s thesis [12] and subsequent report [13] provide a wealth of accurate data reproduced in Figs Figure 1 shows Young s modulus E, the shear modulus G, the bulk modulus B, and Poisson s ratio ν versus temperature T from K, normalized to the 295 K values. In each case there are two clear anomalies at ca. 65 K and 40 K. Figure 2 shows a sharp λ-shaped dip in the longitudinal compliance of strontium titanate at about 40 K. Figure 3 shows peaks near 30 K and 70 K in the C 11 elastic coefficient. Figure 4 illustrates peaks in the longitudinal elastic coefficient C L =(1/2)(C 11 + C C 44 ). Figure 5 shows C 44 alone, with a small dip at ca. 30 K and an abrupt plateau at 50 K. Figure 6 is the Debye temperature versus T, with anomalies near 25 K and 45 K. None of these anomalies exhibit any thermal hysteresis for heating/cooling. This implies that they are either perfectly Fig. 3. C 11 elastic constant for strontium titanate versus temperature, normalized to unity at 295 K. Maxima near 37 K and 70 K are observed on both heating and cooling, together with minima at K and 55 K. Note by contrast that the thermal hysteresis near 106 K is very distinct (ca. 15 K), although this transition is known to be very nearly or exactly second-order. This sugoests that the anomalies near 37 and 70 K are not static distortions second-order phase transitions or not phase transitions at all (but instead something purely dynamic). III. Basic theory The basic idea of Cowley was that linear, harmonic, mode coupling between the transverse optic (TO) soft phonon mode at wave vector q and the longitudinal acoustic (LA)

3 637 Fig. 4. Longitudinal elastic constant C L in strontium titanate versus temperature, normalized to unity at 295 K. Maxima are observed at the AFE transition at ca K and also near 37 K and 70 K, as discussed in the text. An additional peak near 20 K is not explained (Ming Lei and H. Ledbetter) Fig. 6. Debye temperature for strontium titanate calculated versus temperature from acoustic phonon velocities and normalized to ambient temperature. A 2% dip is observed between K, and a weak minimum at 65 K quires that the optic modes at ca. 15 cm 1 be thermally populated, which is true at 37 K. The same kind of process, namely two-phonon difference scattering of ω(e g ) ω(e u ), will give a line-like scattering feature near zero cm 1 whose energy is linear in wave vector q; that is, an extra feature in the Brillouin spectra for shearwave geometries. We interpret the extra acoustic like branch as two-phonon difference scattering, which will give an extra line-like feature at temperatures near where optic phonons at q = 0 cross. For a related model of two-phonon scattering in strontium titanate, see Lyons and Fleury [17] or the more recent work by Bussman-Holder [18]. IV. Present analysis 1. Qualitative arguments Fig. 5. C 44 (T ) shear coefficient in strontium titanate (Ming Lei and H. Ledbetter). In these data all the anomalies have onset at about 10 K below the expected temperatures (e.g., K for the AFE transition), possibly due to an instrumental thermal lag. A sharp peak is observed at 30 K, followed by a dip at 35 K. A plateau edge is observed at 55 K mode at the same q would depress the entire acoustic phonon branch, producing a decrease in LA sound velocity. Here we extend this basic concept to consider instead an anharmonic mode coupling between transverse acoustic modes at q = 0 [ω TA (0)] and two different soft optic modes at +q and q (q small): ω AFE (E g ; q) ω FE (E u ; q). Such an anharmonic renormalization (or decay channel) will lower the acoustic mode frequency near 37 K and hence alter Poisson s ratio and the elastic moduli. This three-phonon anharmonicity re- Firstly let us consider qualitative arguments and expectations: Near 35 K the mode crossing involves E g modes of very long wavelength (q = 0) and frequency about 10 cm 1 (Fig. 7). These transverse modes should mimic, dynamically, a distortion that softens shear modes. That is, fluctuations in transverse ionic displacements should increase near 35 K. Poisson s ratio should have a maximum near 35 K. Near 65 K the mode crossing involves only totally symmetric A 1g longitudinal modes; hence it will mimic a dilatational transition dynamically. Both predictions are in agreement with observations. What other dynamic effects should occur? At 35 K and at 65 K there will suddenly be a great increase in the number of allowed transitions at q = 0 of near zero frequency (these are just the transitions between the crossing optical modes in

4 638 be of opposite parity. To be more precise, the outer product of the two irreducible representations must include the representation of the acoustic phonon at q =0(A 2u for longitudinal acoustic LA in tetragonal strontium titanate; E u for transverse acoustic TA). Thus a crossing of a soft mode with another of the same symmetry (e.g., A 1g A 1g ) gives no effect. The opposite-parity criterion is necessary but not sufficient; thus whereas A 1g A 2u = A 2u gives and acoustic anomaly, A 1g A 1u does not (there is one silent A 1u mode at q = 0 in SrTiO 3 at ca. 173 cm 1 ). Another way of saying this is that for renormalization of the acoustic mode of irreducible representation Γ k, Γ i Γ j Γ k Γ 1, (1) where Γ i,j are the representations of the two crossing optical modes, and Γ 1 is the totally symmetric representation. The product of the three modes (acoustic, gerade optic, and ungerade optic) must of course be a scalar. It is important to note that although both the 37 K and 65 K mode crossings can affect transverse (TA E u ) and longitudinal (LA A 2u ) acoustic phonon modes, the crossing at 65 K produces only one TA (E u ) channel: Fig. 7. Soft optical phonon frequencies in strontium titanate versus temperature (Worlock et al. [15]). The antiferrodistortive modes consist of an A 1g nondegenerate branch and an E g doubly degenerate branch; these collapse to zero energy at the phase transition near 106 K. The ferroelectric modes consist of a nondegenerate A 2u branch and a doubly degenerate E u branch that are very close together in frequency. These ferroelectric modes approach zero frequency as temperature is decreased to zero Kelvin, but they remain finite since pure strontium titanate remains paraelectric at all temperatures (its extrapolated Curie temperature is below zero). The modes have accidental crossings near 37 K and 65 K. The argument in the present paper is that these crossings explain the anomalies in bulk, shear, and Young s modulus, and in Poisson s ratio A 1g E u = E u, (2) whereas that at 37 K produces two TA channels: A 2u E g = E u, E u E g = A 1u + A 2u + E u. (3a) (3b) So we will assume, in the absence of quantitative matrix elements, that the dynamics at 37 K are more shear-like than at 65 K. This will be shown to be in agreement with experiment. Fig. 7). A large increase in the density of such modes into which low frequency acoustic modes can decay [viz. ω ac goes to ω(e g ) ω(e u )] will decrease the acoustic mode frequency via three-phonon anharmonicity (and increase its linewidth). Since the Debye temperature is a linear extrapolation based on acoustic phonon frequencies, it should have minima exactly at both mode crossing temperatures (37 and 65 K). Figure 1 shows that this is true in each case. The bulk modulus is a sum of C 11 and 2C 12, so it should have minima at 35 K and/or 65 K, depending upon whether the temperature change is greater in C 11 or C 12. Only the predicted one at ca. 40 K is observed, which suggests that the change in C 12 is larger. These simple arguments therefore account for several qualitative features: That the 37 K anomalies are shear, and those at 65 K dilatational; that Poisson s ratio should have minima where Young s modulus and the shear modulus have maxima, and vice versa. Poisson s ratio will vary as the bulk modulus does only if C 12 dominates the T -dependence compared with C 11, which is apparently the situation in the present case. 2. Group theory Group theoretically, for this effect to occur, a necessary but insufficient condition is that the crossing optical modes must 3. Wave vector dependence Except in nonsymorphic crystals such as quartz, the dependence of optical mode frequency upon wave vector q is quadratic at q = 0. So the frequency of both the oddparity FE soft mode and the gerade AFE soft mode will vary as bq 2, but with different coefficients b 1 and b 2. However, if you do Brillouin scattering from such a two-phonon difference mode, the wave vector transfer from the incident and scattered laser light is q (ca cm 1 ). Thus to conserve momentum, ω 1 = ω1 0 + b 1 (q q ) 2, (4) ω 2 = ω2 0 + b 2 q 2. (5) At 37 K the two-phonon frequency will be given by ω = ω 1 ω 2 =(ω 0 1 ω 0 2)+2b 1 q +(b 1 b 2 )q 2 +b 1 (q ) 2,(6) which at the exact level crossing is ω = (2b 1 q)q, (7) which is the linear dispersion of an acoustic phonon ω = vq. Hence although each optical phonon displays quadratic dependence upon wave vector q, the wave vector dependence of the two-phonon difference mode will be linear.

5 Fermi resonance To calculate the correct quantitative intensity and linewidth of the extra acoustic branch, it will probably be necessary to include Fermi hybridization between the one- and two-phonon modes (optical difference modes and ordinary TA branch). Such Fermi resonance between one- and twophonon states is well known in SiO 2 [19] and AlPO 4 [20,21]. V. Conclusions In summary, the elastic moduli E, B, G (Young s, bulk, and shear) and Poisson s ratio all exhibit anomalies near 37 K or 65 K [22] compatible with dynamic effects caused by the accidental crossing in frequency of the antiferrodistortive soft optic modes with the ferroelectric soft optic modes [23]. It is not necessary to assume the onset of second sound [24] or of any coherent super phase characteristics [25] to account for either these data or the Brillouin data near those temperatures. Moreover, the onset of second sound would not in itself explain the changes in moduli observed near 37 K. Hence, second sound is neither necessary nor sufficient to explain the observed data. Although second sound might be a plausible explanation for the initial reports of an extra acoustic-like branch in the Brillouin spectra, it is not apparent that it can explain why the bulk, shear, and Young s modulus (or Poisson s ratio) should also exhibit anomalies of several percent at that temperature. We would expect a priori that the onset of second sound has no effect on such elastic moduli! Predictions: T 0 shifts downward rapidly from 105 K to ca. 50 K with increasing oxygen vacancy concentration. Therefore the anomalies reported near 37 K should shift, together with the mode crossing temperature, to ca K for SrTiO 3 x with x of order 0.3. We thank Prof. H. Thomas for discussions. References 1. Muller, K.A., Berlinger, W., Tosatti, E.: Z. Phys. B 84, 277 (1991); Salje, E.K.H., Wruck, B., Thomas, H.: Z. Phys. B 82, 399 (1991) 2. Kozlov, G.V., Volkov, A., Petzelt, J., Feldkamp, G.E., Scott, J.F.: Phys. Rev. B 28, 251 (1983) 3. Muller, K.A., Berlinger, W.: Phys. Rev. Lett. 26, 13 (1971) 4. Courtens, E., Coddens, G., Hennion, R., Hehlen, B., Pelous, J., Vacher, R.: Phys. Scripta T49, 430 (1993) 5. Courtens, E., Hehlen, B., Coddens, G., Hennion, B.: Physica B 219, 577 (1997) 6. Spencer, E.G., Guggenheim, H.J., Kominiak, G.J.: Appl. Phys. Lett. 17, 300 (1970) 7. Ryan, J.F., Scott, J.F.: Solid State Commun. 14, 5 (1974) 8. Cox, D.E., Shapiro, S.M., Cowley, R.A., Eibschutz, M., Guggenheim, H.J.: Phys. Rev. B 9, 5754 (1979) 9. Sorge, G., Hegenbarth, E.: Phys. Stat. Sol. 33, K79 (1969); Sorge, G., Hegenbarth, E., Schmidt, G.: Phys. Stat. Sol. 37, 599 (1970) 10. Rehwald, W.: Solid State Commun. 8, 1483 (1970) 11. Balashkova, E.V., Lemanov, V.V., Kunze, R., Martin, G., Weihnacht, M.: (in press, 1997) 12. Lei M.: PhD thesis, Univ. Colorado (1991); available from University Microfilms, Ann Arbor, Mich.; Lei M. and Ledbetter, H.: NIST report NISTIR 3980, Oxide and Oxide Superconductors: Elastic and Related Properties, 1991, US Dept. Commerce 13. Ledbetter, H.: First International SAMPE Symposium, Nov 28, 1989, Chiba, Japan 14. Fleury, P.A., Scott, J.F., Worlock, J.M.: Phys. Rev. Lett. 21, 16 (1968) 15. Worlock, J.M., Scott, J.F., Fleury, P.A.: Scattering Spectra of Solids, ed. by Wright, G.B. Berlin Heidelberg New York: Springer 1969, p Cowley, R.A.: Phys. Rev. 134, A981 (1964) 17. Lyons, K.B., Fleury, P.A.: Phys. Rev. Lett. 37, 161 (1976) 18. Bussman-Holder, A.: Workshop on Fundamental Problems in Ferroelectricity, Williamsburg, Va, Feb 1997 (Ferroelectrics, in press) 19. Scott, J.F.: Phys. Rev. Lett. 21, 907 (1968) 20. Scott, J.F.: Phys. Rev. Lett. 24, 1107 (1970) 21. Zawadowski, A., Ruvalds, J.: Phys. Rev. Lett. 24, 1111 (1970) 22. Ledbetter, H., Lei, M., Kim, S.: Phase Transitions 23, 61 (1990) 23. Scott, J.F.: Ferroelectrics Letters 20, 85 (1995) 24. Courtens, E., Hehlen, B.: Ferroelectrics 183, 25 (1996); Gurevich, V.L., Tagantsev, A.K.: Sov. Phys. JETP 67, 206 (1988) 25. Vacher, R., Pelous, J., Hennion, B., Coddens, G., Courtens, E., Mueller, K.A.: Europhys. Lett. 17, 45 (1992)