Inelastic neutron scattering study of the perovskite relaxor ferroelectric
PZN:8%PT elucidates the origin of the previously reported unusual kink on the
low frequency transverse phonon dispersion curve (known as "waterfall" effect).
We show that its position depends on the choice of the Brillouin zone and that
the relation of its position to the size of the polar nanoregions is highly
improbable. The observation is explained in the framework of a simple model of
coupled damped harmonic oscillators representing the acoustic and optic phonon
branches.Comment: 4 pages, 6 figures, LaTe

Hlinka, J.

Kamba, S.

Kulda, J.

Petzelt, J.

Randall, C. A.

Zhang, S. J.

English

arXiv

We have undertaken an inelastic neutron scattering study of the perovskite relaxor ferroelectric 

P

b

(

Z

n

1

/

3

N

b

2

/

3

)

O

3

 with 8% 

P

b

T

i

O

3

 (PZN-8%PT) in order to elucidate the origin of the previously reported unusual kink on the low frequency transverse phonon dispersion curve (known as the “waterfall effect”). We show that its position (

q

w

f

) depends on the choice of the Brillouin zone and that the relation of 

q

w

f

 to the size of the polar nanoregions is highly improbable. The waterfall phenomenon is explained in the framework of a simple model of coupled damped harmonic oscillators representing the acoustic and optic phonon branches

Randall, C.A.

Zhang, S.J.

White Rose Research Online

This is a repository copy of Origin of the "Waterfall" effect in phonon dispersion of relaxor perovskites.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/120246/Version: Submitted VersionArticle:Hlinka, J., Kamba, S., Petzelt, J. et al. (3 more authors) (2003) Origin of the "Waterfall" effect in phonon dispersion of relaxor perovskites. Physical Review Letters, 91 (10). 107602. ISSN 0031-9007 https://doi.org/10.1103/PhysRevLett.91.107602eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:cond-mat/0306692v1  [cond-mat.mtrl-sci]  27 Jun 2003Origin of the ”waterfall” effect in phonon dispersion of relaxor perovskitesJ. Hlinka, S. Kamba and J. PetzeltInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech RepublicJ. KuldaInstitute Laue-Langevin, Avenue des Martyrs, 38640 Grenoble, FranceC. A. Randall and S. J. ZhangMaterials Research Institute, Pennsylvania State University, University Park, PA 16802, U.S.A.(Dated: August 5, 2017)We have undertaken an inelastic neutron scattering study of the perovskite relaxor ferroelectricPb(Zn1/3Nb2/3)O3 with 8% PbTiO3 (PZN-8%PT) in order to elucidate the origin of the previouslyreported unusual kink on the low frequency transverse phonon dispersion curve (known as ”waterfall”effect). We show that its position (qwf) depends on the choice of the Brillouin zone and that therelation of qwf to the size of the polar nanoregions is highly improbable. The waterfall phenomenonis explained in the framework of a simple model of coupled damped harmonic oscillators representingthe acoustic and optic phonon branches.I. INTRODUCTIONFerroelectric perovskites Pb(Zn1/3Nb2/3)O3 (PZN)and Pb(Mg1/3Nb2/3)O3(PMN) and related materialshave recently attracted a great attention, among oth-ers, due to the excellent piezoelectric properties1 of theirsolid solutions with PbTiO3. Their average ABO3 per-ovskite structure is perturbed by a short-range occupa-tional ordering on B-site positions.2,3 It is known thatthis nanoscopic inhomogeneity is responsible for relaxorproperties of these materials, such as smearing and fre-quency dependence of the dielectric anomaly associatedwith the ferroelectric ordering4, and for other phenom-ena reflecting the presence of polar nanoregions (PNR)persisting hundreds of Kelvins above the temperature ofdielectric permitivity maximum5. Unfortunately, it isnot easy to deal with these inhomogeneities in a sim-ple enough model, and the understanding of the relaxorproperties is not yet satisfactory.Since these materials are available as large single crys-tals, inelastic neutron scattering can be used to study thephonon dispersion curves. The spectrum of the lowestfrequency transverse excitations usually consists of in-teracting and mutually repelling transverse acoustic andtransverse optic phonon branches. It is known that uponapproaching a displacive ferroelectric phase transition,the lowest zone center optic mode (soft mode) tends tozero frequency. Instead of that, it was repeatedly foundthat near the ”smeared” phase transition of relaxor ferro-electrics, the upper branch appears to drop precipitouslyinto the lower branch at a finite value of momentumtransfer of the order of qwf = 0.2 Å−1. For explanation ofthis ”waterfall” effect, Gehring, Park and Shirane6 pro-posed that the characteristic wave vector qwf correspondsto the size of the PNR’s, and that due to these PNR’s thesoft branch phonons with q < qwf cannot effectively prop-agate in the crystal. The idea was subsequently specifiedwithin a mode-coupling model7,8 assuming a sharp in-crease of transverse optic branch damping (inverse life-time) for q < qwf .FIG. 1: Low frequency phonon modes in PZN-8%PT mea-sured by constant energy scans at 500K. Measurements wereperformed in (a) 020 Brillouin zone (Q = (q20) scans) and in(b) 030 Brillouin zone (Q = (q30) scans).Within the past three years, it was establishedthat the ”waterfall” effect is common to a numberof relaxor ferroelectrics, including PZN7,9, PMN10,11,12,Pb(Zn1/3Nb2/3)O3 with 8% PbTiO3 (PZN-8%PT)6,PZN-15%PT13, and PMN-20%PT14. However, we havenoticed that the sharp drop (near q ≈ 0.2 Å−1) of the dis-persion curves of the single-domain tetragonal BaTiO32drawn in Fig. 6 of Ref. 15 also strikingly closely remindsthose of relaxors (compare, for example, with Fig. 2 ofRef. 6). Obviously, the explanation based on the charac-teristic size of PNR’s cannot be valid for an experimentperformed on a classic single-domain ferroelectric crystal.This letter brings new inelastic neutron scattering datashowing that such explanation does not hold for relaxorseither. We demonstrate that the ”waterfall” effect canbe ascribed to an entirely classic interference of lineshapeanomalies due to the coupled acoustic and optic brancheswithout any ad-hoc assumption of an anomalous increaseof the bare optic branch damping below qwf .For the experiment we have chosen a 3.7 g single crys-tal of PZN-8%PT, used already in our previous neutronstudy16. This as-grown, optically transparent, yellowishsingle crystal was produced by the high temperature fluxtechnique at Materials Research Institute, PennsylvaniaState University. This promising PZN-8%PT material,showing both relaxor and giant piezoelectric properties,has been studied recently by a number of techniques, andit is the one for which the possible relation of qwf to thesize of PNR’s was firstly invoked.6The present experiment was carried out on the re-cently upgraded IN8 thermal neutron spectrometer atthe ILL high flux reactor. The instrument was oper-ated with a fixed wave-number of scattered neutronskf = 3 Å−1, using horizontally focussing (vertically flat)Si crystals as monochromator and analyzer. The sam-ple was wrapped in a thin Nb foil and mounted in avacuum furnace allowing to reach temperatures up to1200K with a stability better than 1K. The sample wasoriented with the cubic [001] axis vertical. This geometryallowed us to explore the (hk0) scattering plane. Bothtransverse and longitudinal momentum resolution widthsmeasured on the (030) Bragg reflection were better than0.07 and 0.06 of reciprocal lattice unit (rlu, where 1 rlu is2π/a ≈ 1.55 Å−1), respectively. This is about equivalentto resolution widths in a setup with flat PG (pyroliticgraphite) crystals and 40’ Soller collimators used in theprevious studies by other authors. The present setupusing elastically bent Si crystals17 permits to avoid spu-rious effects from tails of reflection curves inherent to PGmosaic crystals and provides an improved energy resolu-tion of 0.8meV (FWHM), as given by the energy profileof elastic incoherent scattering from our crystal and con-firmed by an independent measurement on a vanadiumreference sample.The typical constant energy transverse scans in the020 and 030 Brillouin zones at 500K are displayed inFig. 1, and the positions of their maxima are plotted assolid symbols in Fig. 2. The apparent dispersion curveobtained in this way from the 020 Brillouin zone data hasan almost vertical rise up near qwf = 0.13 rlu. This valuecorresponds exactly to the qwf obtained from the 220 zonedata at the same temperature and same crystallographicdirection for this material in Ref. 6.On the other hand, the position of the upward riseobserved by us in the 030 Brillouin zone is at a signifi-FIG. 2: Position of maxima in constant energy transversescans performed in 0 2 0 (solid squares) and 0 3 0 (solid di-amonds) Brillouin zones (as in Fig. 1). Maxima are formingtwo distinct ”dispersion curves”. The inflection points nearqwf = 0.13 rlu and 0.06 rlu, respectively, are discussed in thetext. Open symbols show maximum intensity positions inconstant-q scans. Lines are only guides for eyes.cantly smaller value of qwf ≈ 0.06 rlu. As the same [010]-polarized modes with wave vectors along [100] are probedin both Brillouin zones, the value of qwf can hardly pro-vide a measure of the size of PNR’s6. On the contrary,the remarkable dependence of qwf on the chosen Bril-louin zone proves that the observed dispersion is merelyan apparent dispersion. In fact, it is known that maxi-mum intensity positions for response function of coupleddamped harmonic phonons can be sensitive to the dy-namical structure factors (chosen Brillouin zone).18 Fur-thermore, the striking difference between the 030 zonevalue of qwf and almost the same values of qwf in 020 and220 zones indicates that the waterfall effect is related todynamical structure factor of bare acoustic branch, whichis known to be very small in the 030 zone.11,19As a matter of fact, it turns out that the ”waterfall ef-fect” can be reproduced in a quite simple model. Letus consider the standard model20,21,22 of two coupleddamped harmonic oscillators, defined by a 2× 2 dynam-ical and damping matrices Dq and ΓqDq =(ω2TA(q) ∆(q)∆(q)∗ ω2TO(q)), (1)Γq =(ΓTA(q) ΓAO(q)ΓAO(q) ΓTO(q)), (2)where ωTA(q) and ωTO(q) describes dispersion of bareacoustic and optic branches, ∆(q) describes their mutualbilinear interaction, ΓTA(q) and ΓTO(q) stands for baremode frequency independent damping, ΓAO(q) stands forviscous interaction (bilinear in time derivatives of baremode coordinates) and q is the reduced phonon wavevector in a chosen direction (in this letter, along the cu-bic [100]). In the high temperature limit (hω << kT ),the corresponding inelastic neutron scattering intensity3is proportional to18,21I(ω, q) = kT ω−1 f(q)∗ . Im [G(ω, q)] . f(q) , (3)G(ω, q) = (Dq − i.ωΓq − ω2E)−1 , (4)where E is a 2×2 unit matrix and f(q) is a 2-componentvector composed of dynamical structure factors of bareacoustic and optic branches, respectively. At small fre-quencies, it is expected that ω.|ΓAO(q)| << |∆q| so that,as e.g. in the Ref. 18, ΓAO(q) = 0 will assumed in the fol-lowing. The wave vector dependence of dynamical matrixDq can be conveniently cast in the form which appearsin the simplest nearest-neighbor interaction models withtwo degrees of freedom per unit cell23,24,25:ω2TA(q) = A sin2(πq/2) , (5)ω2TO(q) = c+B sin2(πq/2) , (6)∆q = d sin2(πq/2) , (7)where the real parameters A,B, c and d define thestrength of dispersion of bare acoustic and opticbranches, the square of soft mode frequency and thestrength of mode mixing at zone boundary, respectively.Let us note that the dynamical matrix parametrized inthis way shows the same asymptotic behavior for q → 0as the standard model of Ref. 26, and at the same timeprovides the correct dispersion behavior also at the zoneboundary. In the same spirit, we introduceΓTA(q) = g sin2(πq/2) , (8)ΓTO(q) = h, (9)where the real parameters g and h define the dampingof the bare acoustic branch (with required asymptoticq2 dependence) and the q- independent damping of bareoptic mode, respectively.Numerical calculations were performed for a set of real-istic values (A = 100meV2, B = 150meV2, c = 15meV2,d = 100meV2, g = 6meV, h = 6meV). Constant energyprofiles calculated for the case of equal bare acoustic andoptic structure factors (f = (0.5, 0.5)) are shown in Fig. 3,while phonon dispersion curves obtained by diagonaliza-tion of the dynamical matrix Dq are shown in Fig. 4.Fig. 4 also shows that the positions of the maximum in-tensity in constant energy profiles shown in Fig. 3 formsan apparent dispersion branch which follows the acousticbranch for small q, but then it sharply rises towards theupper phonon branch near 0.1 rlu. The shape of this ap-parent dispersion curve indeed corresponds well with theexperimental findings shown for example in Fig. 2.As expected, the ”waterfall” region of the apparentdispersion curve depends noticeably on the dynamicalstructure factor. For example, a decrease of the weightof the bare acoustic structure factor shifts the ”water-fall” position towards the zone center, as it follows fromFig. 4. A similar tendency is found in the experimentaldata taken in the ”predominantly optic” 030 BrillouinFIG. 3: Constant energy scans at 1-9 meV simulated fromthe model described in the text, for structure factors f =(0.5, 0.5).FIG. 4: Dispersion curves calculated from the model de-scribed in the text. Full and dashed lines correspond to fre-quencies of repelled and bare phonon branches obtained assquare roots of eigenvalues and of diagonal elements of thedynamical matrix Dq, respectively. Full squares (on the rightside) stands for positions of maximum intensity in constantenergy profiles shown in Fig. 3, calculated for balanced struc-ture factors f = (0.5, 0.5). Full diamonds (left side of dia-gram) are calculated in a same way for the case of a smallerbare acoustic structure factor (f = (0.25, 0.75)).zone (Fig 2). In the limit of zero bare acoustic structurefactor (f = (0, 1)), the waterfall phenomenon completelydisappears (Fig. 5). On the other hand, with a slightchange of parameters, the apparent waterfall dispersioncurve can attain even a negative slope. These resultsshow that constant energy scan technique for investiga-tion of coupled acoustic-optic branches should be usedwith certain caution. More detailed discussion of themodel properties and quantitative analysis of the exper-imental data is beyond the scope of this letter and willbe published elsewhere.In conclusion, we argue that the ”waterfall” kink seen4-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.50246810121416  Energy (meV)q  (r.l.u.)FIG. 5: Dispersion curves calculated from the model de-scribed in the text. Symbols have the same meaning as in Fig4.; full squares are calculated from constant energy profilescalculated for the case of zero bare acoustic structure factor(f = (0, 1)).on the dispersion curves of several relaxor perovskites canbe explained in the framework of coupled damped har-monic oscillator models describing the interaction of theacoustic and optic branches. This kink only appears onan apparent dispersion curve, obtained from maxima ofconstant energy sections of scattered intensity, while nosuch kink is present in the dispersion of bare or coupledbranches themselves. In the light of these results, there isno reason for the position qwf of the waterfall kink to berelated to the size of the polar nanoregions. First of all,there is no need to assume the unusual increase of bareoptic branch damping below the waterfall wave vector qwfin the present model. Secondly, the ”waterfall” positionis very sensitive to the dynamical structure factor bothin our model and in our experimental data, where it be-comes reduced by as much as a factor of 2 when passingfrom the 020 to 030 Brillouin zone. It also follows that the”waterfall” anomaly in the low-energy phonon dispersionis not specific to relaxor ferroelectrics, but it can be en-countered in other systems with interacting acoustic andsoft optic branches, such as the ferroelectric perovskiteBaTiO3.AcknowledgmentsThe work has been supported by Czech grants(projects A1010213, 202/01/0612, MŠMT LA043 andAVOZ1-010-914).1 S. E. Park and T. R. Shrout, J. Appl. Phys. 82, 1804 (1997)2 A. D. Hilton, D. J. Barber, C. A. Randall, and T. R.Shrout, J. Mat. Sci. 25, 3461 (1990)3 C. A. Randall and A. S. Bhala, Jpn. J. Appl. Phys. 29,327 (1990)4 N. Setter and L. E. Cross, J. Appl. Phys. 51, 4356 (1980);F. Chu, N. Setter and A. K. Tagantsev, J. Appl. Phys. 74,5129 (1993)5 G. Burns and F. H. Dacol, Ferroelectrics 52, 103 (1983);Sol. State Comm. 48, 853 (1983)6 P. M. Gehring, S. E. Park, and G. Shirane, Phys. Rev.Lett. 84, 5216 (2000)7 P. M. Gehring, S. E. Park, and G. Shirane, Phys. Rev. B63, 224109 (2001)8 G. Shirane and P. M. Gehring, J. Phys. Soc. Jpn. 70, 227(2001)9 I. Tomeno, S. Shimanuki, Y. Tsunoda, and Y. Y. Ishii, J.Phys. Soc. Jpn. 70, 1444 (2001)10 P. M. Gehring, S. B. Vakhrushev, and G. Shirane: Fun-damental Physics of Ferroelectrics 2000: Aspen Center forPhysics Winter Workshop, editor R. E. Cohen, AIP Con-ference Proceedings, Melville, New York 535, 314 (2000)11 S. Wakimoto, C. Stock, Z.-G Ye, W. Chen, P. M. Gehring,and G. Shirane, Phys. Rev. B 66, 224102 (2002)12 P. M. Gehring, S. Wakimoto, Z.-G Ye, and G. Shirane,Phys. Rev. Lett. 87, 277601 (2001)13 D. La-Orauttapong, B. Noheda, Z.-G. Ye, P. M. Gehring,J. Toulouse, D. E. Cox, and G. Shirane, Phys. Rev. B 65,144101 (2002)14 T. Y. Koo, P. M. Gehring, G. Shirane, V. Kiryukhin, S. G.Lee, and S. W. Cheong, Phys. Rev. B 65, 144113 (2002)15 G. Shirane, J. D. Axe, and J. Harada, Phys. Rev. B 2,3651 (1970)16 J. Hlinka, S. Kamba, J. Petzelt, J. Kulda, C. A. Randall,and S. J. Zhang, J. Phys. Cond. Mat. 15, 4249 (2003)17 J. Kulda and J. Šaroun, Nucl. Inst. Meth. A 379, 155(1996)18 J. Harada, J. D. Axe, and G. Shirane, Phys. Rev. B 4, 155(1971)19 S. B. Vakhrushev and S. M. Shapiro, Phys. Rev. B 66,214101 (2002)20 A. S. Barker and J. J. Hopfield, Phys. Rev. 135, A1732(1964)21 R. Currat, H. Buhay, C. H. Perry and A. M. Quittet, Phys.Rev. B 40, 10741 (1989)22 J. Hlinka, I. Gregora and V. Vorĺıček. Phys. Rev B 56,13855 (1997)23 I. Etxebarria, M. Quilichini, J. M. Perez-Mato, P.Boutrouille, F. J. Zuniga, and T. Breczewski, J. Phys.Cond. Mat. 4, 8551 (1992)24 J. Hlinka, M. Quilichini, R. Currat, and J. F. Legrand, J.Phys.: Condens. Matter 8, 8221 (1996)25 Z. Y. Chen and M. B. Walker, Phys. Rev. B 43, 5634(1991)26 J. D. Axe, J. Harada, and G. Shirane, Phys. Rev. B 1,1227 (1969)

Origin of the "Waterfall" effect in phonon dispersion of relaxor perovskites

https://eprints.whiterose.ac.uk/120246/1/0306692v1.pdf

Origin of the "Waterfall" Effect in Phonon Dispersion of Relaxor Perovskites

Abstract

Similar works

Full text

Available Versions

White Rose Research Online