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Phonon dispersion and anharmonicity in cubic KNbO3
PII: SOO22-3697t%WOOl4-5 . ,
Pergamon
PHONON DISPERSION
J. Phys. Chem Solids Vol57, No. IO, pp. 146-1471, 1996 Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All riahts reserved 00X-3697/% Si5.00+ 0.00
AND ANHARMONICITY
M. HOLMAt
and HAYDN
IN CUBIC KNb03
CHENS
TDepartment of Physics; $Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received 27 March 1995; accepted 6 July 1995)
Abstract--Inelastic neutron scattering measurements of cubic KNb03 have been performed to study the temperature dependence and anharmonicity of the phonon branches believed to be associated with the subsequent phase transitions. Our results show that the cubic phase displays an anomalously flat and low energy transverse acoustic mode propagating in the (100)directions, absent from the previous study on the cubic phase. The transverse optic mode along the same directions shows large damping which is consistent with the previous report. A strong temperature dependent coupling occurs between these two modes, similar to the behavior reported in BaTiOa. The behavior of the other phonon branches, in contrast, show normal dispersion behavior and the phonon groups are well behaved. Computer simulations of lattice dynamics demonstrate that the phonon branches have largely harmonic behaviour except for the two anomalous modes. The transverse acoustic and lowest energy optic mode in the (100) directions can be successfully modeled with an anharmonic, non-linear polarizability model for a reduced system. This model is intrinsically anisotropic and indicates that the origin of the anomalous behavior arises from anisotropic
oxygen polarizability. This anisotropy also manifests itself in the anomalous thermal diffuseX-ray scattering that has been shown to display a planar distribution in the reciprocal space. Keywords:A. KNtQ, B. neutron scattering, D. phonons, D. anharmonicity, D. lattice dynamics.
INTRODUCEION Neutron scattering studies are an indispensable experimental technique for studying the dynamical behavior of materials and their associated phase transitions. Previous inelastic neutron scattering experiments have provided a deep understanding of the physics of structural phase transitions [l-3]. In particular the physics of the structural phase transition in strontium titanate, with the relevant coordinate being the rotation of the oxygen octahedra, has become the prototype of soft-mode materials. A structural phase transition occurs at 110 K in which an optic mode at the R (l/2, l/2, l/2) point in the Brillouin zone softens completely to zero energy as the temperature approaches T, [4]. The displacements of this optic mode are directly related to the crystal structure change. Although this theory has found notable success with some other materials [5] in addition to strontium titanate, for most materials the observed behavior does not correlate with this theory. In particular, in most materials some softening does occur but the phase transition intervenes before complete softening [6]. Inelastic neutron scattering has been previously performed on three phases of potassium niobate: the cubic [7], tetragonal[8] and orthorhombic [9, lo] phases. Of these experiments the orthorhombic phase [l l- 131 has been the most extensively studied
and cubic phase the least extensively studied. In addition there have been Raman scattering studies [14] in the tetragonal [ 151, orthorhombic [ 16, 171 and rhombohedral [17] phases but not the cubic phase. However, an IR spectroscopy [ 181study included the cubic phase which provided information on the zone center optic modes in this phase. Of other perovskite ferroelectrics, only barium titanate [ 19,201 and KTN (potassium tantalum:niobium oxygen) [21-231 show similar phase transitions and physical behavior as in potassium niobate. Potassium tantalate [24] shows similar behavior but does not undergo any phase transitions and is thus an incipient ferroelectric. The inelastic neutron scattering results of the tetragonal and orthorhombic phases correlate well with the results presented in this study. Our results however contradict the previous study [7] on the cubic phase.
EXPERIMENTAL The inelastic neutron scattering experiments were carried out at the Ames Laboratory-ORNL beam line HB-IA, at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Lab (ORNL) in collaboration with Dr R. Nicklow. The beam line HB-IA uses a double monochromating system at a fixed scattering angle to obtain an fixed initial energy 1465
M. HOLMA and H. CHEN
1466
of 14.8 meV (3584THz). the monochromators are made of pyrolytic graphite using the (002) Bragg peak for energy selection. The monochromatic beam is filtered to remove X/2 contamination, then collimated and sized before reaching the sample table. The collimation used in the experiments was 30 min before and after the sample. Two different single crystal KNbOs samples made by the top seeded technique [25] were used for the experiments. Both crystals had an approximate volume of 0.9cm3. Rocking curves determined the mosaic spread of the crystal to be less 0.2”. The samples were mounted with the [OOl]r_r~ud~ubicaxis perpendicular to the scattering plane. The phonon dispersion has been investigated at temperatures above the cubic to tetragonal phase transition temperature in the high symmetry directions [OOl]and [Ol 11.The temperatures measured were 605,525,471,445 and 429°C. For the samples used in the neutron experiments the cubic to tetragonal phase transition temperature was observed at 429°C in situ using the neutron beam. The sample was slowly raised to the highest temperature and subsequently lowered for the data collection. RESULTS AND DISCUSSION Figures 1 and 2 show the lowest energy transverse branches in the [lOO] direction (no difference was observed between in the phonon groups in the [OlO] direction) at 605 and 525°C respectively. The significant features of these data are the anomalous low energy acoustic mode and the essentially flat region for reduced wave vectors q N 0.2 extending out to the Brillouin zone boundary. Above in energy is the lowest energy transverse optic mode. Near the reduced wave vector q N 0.2 these two modes can not be resolved as separate peaks. Below q - 0.2 the transverse optic mode was not detected. This mode was also measured from the (1 00), (110) and (130) Bragg peaks but no peak corresponding to this mode was observed ,
(
,
,
01 0
,
,
,
’
0.1
’
’
0.1;&~,.35
’
’
’
’
0.4
’
0.45
’
0.5
Fig. 2. Transverse phonon modes in the [lOO]direction at T = 525°C.
below 3 THz. The reduced structure factor for modes propagating in the [ 1001direction was calculated (Fig. 3) for potassium niobate that gives a very probable interpretation of why this mode was not observed. Based on the shell mode1 of the lattice dynamics and assuming equivalent Debye-Wailer factors [26-281 for each atom the inelastic structure factor is a maximum at the zone center and smoothly decrease to a minimum at the zone boundary for this transverse acoustic mode. The inelastic structure factor for the transverse optic mode has a maximum midway between the zone center and zone boundary. However, the shell model does not give an adequate fit to the phonon dispersion for this mode. A second anomalous feature of these two modes is the behaviour as represented in the neutron groups. The transverse acoustic mode was unusually difficult to measure. Not only is the peak intensity extremely low but the background is unusually high for wave vectors q 2 0.3. The behavior of the inelastic structure factor accounts for the decrease in peak height but not the large background (see raw data in Fig. 4). For wave vectors q N 0.3 this mode is essentially flat which was observed from the defocused shape of the phonon peaks measured from different zones. 2.5
(
’
0.06
(
(
,
(
,
(
(
,
,
,
T-SOSC
+
l
2-
I 1.5
TA +
TO1
+
1
+
l0
0
0
Fig. 1. Transverse phonon modes in the [lOO]direction at T = 605°C.
0
0 0
0.5 -
TO
““““‘I
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 q.mhadwawwcbr
Fig. 3. Reduced structure factor for the transverse acoustic and optic modes.
Phonon dispersion and anharmonicity in cubic KNbOj For the lowest energy transverse optic mode the neutron group is seen to broaden substantially as the wave vector increases to the Brillouin zone boundary (see raw data in Fig. 5). This peak broadening is apparently symmetric in shape and thus can not be due to interference effects between one-phonon and multi-phonon processes. The width is an order of magnitude larger than the instrumental resolution and can be physically ascribed to anharmonic effects. Anharmonic phonon theory leads to a calculation for the damping constant of 4.2 THz at the zone boundary. This mode could not be detected for wave vectors
1467
near the zone center, it is believed to be completely overdamped in this region. The last branch measured is a transverse optic mode that is flat across the entire Brillouin zone at nearly 6THz. No branches were measured above 8THz. Figures 6 and 7 show the longitudinal scans in the [loo] direction at two temperatures. In contrast to the transverse scans the longitudinal scans showed extremely well defined neutron groups and the dispersion behaviour is normal; the phonon energy increases with increasing wave vector. The previous studies of inelastic neutron scattering, in particular in the orthorhombic and
250(a) MO.3 $&
200-
P P
150 -
100 _____--*’
______I
0
+&o____--------mm____ 0, 1.5 2 2.5 t
I
0.5
1
/Q
0 Q
,‘O
I
,O
I’
/
0
“P *s, ‘% O’--_____________________-
1
I
3
150
‘A
‘*
0
&
-9
-----co6 Q
1
$
100
‘b%*&,___._____ 50
0
0.5
2
I.5
1
2.5
3
Fig. 4. Peak width of the transverse acoustic mode for reduced wave vectors (a) q = 0.1, (b) q = 0.25 and
(c)q
=
0.35.
Fig. 5. Peak width of the transverse optic mode for reduced wave vectors (a) q = 0.3, (b) q = 0.4 and (c) q = 0.5.
1468
M. HOLMA and H. CHEN 6
,
,
,
,
1
,
7-
,
LO1 +
,
)
T&EC + +
6-
0
a. 0 -
0 LAO 5-
0
4-
0
32-
0
l0
0
’ 0.05
’ 0.1
’ ’ 0.15 0.2 RdJmd
’ ’ ’ 0.25 0.3 0.35 mmwador* 9
’ 0.4
’ 0.45
0.5
Fig. 6. Longitudinal phonon modes in the [lOO]direction at T = 605°C tetragonal phases, show similar types of anomalous behavior. The tetragonal phase shows the same transverse acoustic mode behavior, i.e. low in energy and flat for reduced wave vectors greater than 0.2. For phonon branches propagating in the [loo] direction and polarized along the [OlO]the flat region is similar in energy (1.8 THz as compared with 1.5 THz measured in the present study). There is a similar mode propagating in the [OOl] direction and polarized in either the [lOO] or [OlO] direction. These modes are purely transverse. In addition, the tetragonal phase has the flat transverse optic mode at nearly 6 THz that appears in the cubic phase. The tetragonal modes compatible with the longitudinal modes observed in the cubic phase show no difference. The only compatible mode that has distinctly changed is the pseudotransverse mode propagating in the [loo] direction and polarized close to [OOl]direction (the direction of the ferroelectric polarization in the tetragonal phase). This mode has sharply stiffened and displays normal wave vector dependence. The authors [8] also noted that the flat transverse acoustic and lowest energy transverse optic mode showed ill defined phonon groups and that anti-crossing behavior was observed. The orthorhombic phase shows a similar transverse acoustic mode that propagates in the plane
5
containing the ferroelectric polarization vector (in the orthorhombic phase the polarization is along the [loll direction with respect to the pseudo-cubic axes) and whose eigenvector is polarized strictly transverse to the ferroelectric polarization vector. The authors [9] expressed similar sentiments regarding the measurement of these neutron groups: they found unexpected difficulty measuring these groups because of a lowered intensity and a systematically higher background. In contrast with the previous results [7] on the cubic phase there is marked contradiction with those presented here. The low energy transverse acoustic mode is totally absent in that study and the optic mode is assumed to be completely overdamped throughout the entire zone. The only other phonon branch measured was the longitudinal acoustic mode in the [loo] direction and it is similar to the mode observed in the present study. Given the improved resolution of this study and the inherent difficulty in detecting the transverse acoustic mode, it is probable that it could not be observed in the previous study. Two phonon branches were measured in the [l lo] direction. These are shown in Fig. 8. The transverse optic mode was not observed in the [l lo] direction, however diffuse X-ray scattering measurements suggest that this mode is overdamped near the zone center and that the largest anisotropy in the phonon dispersion exists for the transverse optic mode in the [l lo] direction [29]. These agree with the previous study [7] and show no anomalous behavior. The significant temperature dependent behavior occurs only in the lowest energy transverse optic and transverse acoustic modes in the [lOO] direction, the branches that were difficult to measure. All other branches show no appreciable temperature dependence; including the compatibility relations this can be extended to the other phases as well. Figure 9 shows the temperature dependent behavior of the two modes. The observed temperature dependent behavior occurs only near q - 0.2 where the transverse acoustic and transverse optic modes have similar
TA
Fig. 7. Longitudinal phonon modes in the [lOO]direction at T = 525°C.
Fig. 8. Acoustic modes in the [l lo] direction at T = 525°C.
Phonon dispersion and anharrnonicity in cubic KNbOs
z$ 0
” , , , , , 0.05
o.,
1469
,;s%,1
0.15&~,,."
0.4
0.45
OU’ 0 0.05
0.5
0.1
”
”
0.1; ~.25_0&35
”
0.4
”
0.45
0.5
Fig. 9. Temperature dependence of the lowest energy phonon modes.
Fig. 11. Shell model fitted dispersion curves for longitudinal modes in the [loo] direction.
energies. The data as shown are single peak fits to the measured phonon group intensities. These data were reconsidered and fitted with a mode coupling scheme, the temperature dependent effect is an increased coupling between the transverse acoustic and optic modes near q - 0.2.
mation. The ionic charges are not equivalent to their free ion values (+l for K and +5 for Nb) and this feature is also characteristic of the oxygen perovskites. The eigenvectors for the shell model are shown in Figs 12 and 13 for the lowest energy transverse optic and transverse acoustic modes in the [loo] direction. In spite of the failure of the shell model to adequately fit the dispersion curves the eigenvectors are similar to those found in the orthorhombic phase neutron scattering results. In the transverse optic mode the oxygen ions are the predominant ones which are moving, while the niobium ions are nearly at rest. In contrast for the transverse acoustic mode all the oxygen ions have zero or nearly zero displacements while the niobium ionic displacements are quite large. This agrees well with the experimentally determined results for the compatible mode in the orthorhombic phase. The eigenvectors were not measured experimentally in this work due to the extreme difficulty in even measuring the phonon branches from one Bragg peak.
LATTICE DYNAMICAL
MODELS
The rigid shell model based on the same model used for strontium titanate [30] by Stirling was applied to the results of the inelastic neutron scattering data to determine the parameters of the lattice dynamics. This model was fitted to the measured phonon dispersion curves and also the zone center points available from infrared spectroscopy on cubic potassium niobate. The best fit dispersion curves are shown in Figs 10 and 11, which include the zone center points mentioned above. Some particular features of the model are observed in the Table 1. The transverse short range force constant between niobium and oxygen is an order of magnitude larger than all other force constants. This is also observed in the other oxygen perovskites and reflects the rigidity of the oxygen octahedra to defor-
Table 1. Shell model parameters Force constants (e’/V Transverse K-O Nb-0 o-o Longitudinal K-O Nb-0 o-o Z(K) Z(Nb) (10” cm) o(K) Wb) o(0)
b ippj 0
0.05
0.1
0.15 0.2 q,-m-r
0.25
0.3
0.55
0.4
0.45
KNbOs
KTaOs
SrTi03
13.0 355.0 -1.00
14.65 359.0 3.22
26 285 1.9
-1.01 -85 1.oo
-1.01 -68 1.085
-1.01 -43 0.74
0.8 5.125
0.82 4.74
1.52 3.3
0.0009 0.001 0.0275
0.0001
0.0035 0.0007 0.0059
0.021 0.0117
0.5
Fig. IO. Shell model fitted dispersion curves for transverse modes in the [IOO]direction.
$)K) d(Nb) d(O)
0.001
0.01 0.801
0.0052 0.797 -0.186
-0.54 0.0967 -0.0384
1470
M. HOLMA and H. CHEN
0
0
x
‘Wore shell
92’
A
M2
-0.5 0
0
Fig. 14. One dimensional anisotropic model.
-1 /:
II
0
1
0.05
0.1
II 11 0.1: ~.2&0&35
‘I
11 0.4
0.45
0.5
Fig. 12. Eigenvectors from the shell model for the transverse optic mode in the [loo] direction.
Generally the phonon branches must be measured from five to six Bragg peaks to determine the eigenvectors. The observation that the temperature dependent behavior occurs in the two lowest transverse modes and that the harmonic model is inadequate to fit these modes suggests the use of the polarizability model [3 1, 321 (Fig. 14). The fitted dispersion curves for the polarizability model are shown in Fig. 15. A pseudoone-dimensional model is sufficient to describe the lattice dynamics of the lowest energy phonon modes. The reduction to a one-dimensional system is done by grouping together the transition-metal ion and the three oxygen ions (Nb03) as a single polarizable mass, with a non-linear anisotropic polarizability. The hamiltonian is then written for the reduced system in terms of the displacements of the cores and shells of atoms 1 (the grouped Nb03) and 2 (the ion K). The hamiltonian is:
(1)
H=T+V
where
+
&r($l’- up)4]
(3)
and r#), rn?) are the core and shell masses of the ith atom, u@)and $) are the displacements of the core and shell of the ith atom and f, f(‘)‘, and &’ are the harmonic force constants for coupling between shell-shell, core-core and internal core-shell, respectively (see Fig. 14), whereas g4 is the only anharmonic term introduced in the model. The terms are summed over the two constituents over all atoms in the system, N, and the phonon energies are obtained using the Euler-Lagrange equations and the adiabatic wnditions. Details can be found in the references. An extremely good fit was obtained for these two modes. The parameters for these models are shown in Table 2. A direct comparison with the force constants of the harmonic rigid shell model is not possible but they can be compared with results presented for other oxygen perovskites [31, 321 (in the above table, for SrTi03 and KTa03). The reduced masses are nearly equal to those obtained in the fitting of potassium tantalate, but the coupling constants are significantly larger in strontium titanate. The smaller values in potassium niobate increase the net effect of anharmonit forces, supporting the view of near cancellation of forces near the phase transition. This is to be expected 4
I
2.5-
I
I
I
I
I
p
Tdo5C ,/
32.5,’
2-
I
,*
,’
,’
I’
I’
/-p-------‘
,a”
_/ ______-------
0
0.05
0.1
0.1; ~2&0.3&35
0.4
0.45
0.5
Fig. 13. Eigenvectors from the shell model for the transverse acoustic mode in the [loo] direction.
0
02
0.4
0.6 bNvs0f.q’
1.2
1.4
Fig. 15. Fitted phonon dispersion curves for T = 605°C in the [ 1001direction.
Phonon dispersion and anharmonicity in cubic KNbO, Table 2. Anisotropic oxygen polar&ability model parameters KNbOs m, (lo-** g) mz (lo-** 5)
f (lo4g~-1 r; (10; q jz\‘;J “,J) g!’ (lo4 gs-2) g4 w4
W2)
KTaOs
SrTiO,
2.9 0.65
2.88 0.649
1.549 1.461
4.05 0.55 0.0 0.035 -0.65 0.15
4.067 0.581
14.40 1.268
-0.828 0.255
-2.629 0.904
from the inherent behavior of the phonon groups. Potassium tantalate does not show the anomalous behavior of the phonon groups in the two lowest transverse modes.
CONCLUSIONS
1471
2. Shirane G., in Structural Phase Transitions and Soft Modes (Edited by E. J. Samuelserl, E. Anderson, and J. Feder). Universitetsforlaget, Oslo (1971). 3. Scott J.. Rev. mod. Phvs. 46. 83 (1974). 4. Yamada Y., Shirane d. and'Li& A., Phys. Rev. 177,848 (1969). 5. Krumhansl J., in Computing Interactions and Microstructures: Statics and Dvnamics (Edited by R. LeSar, A. Bishop, and R. Heffndr). Berlin (1987). _ 6. Shapiro S. M., in Competing Interactions and Microstructures: Statics and Dynamics (Edited by R. LeSar, A. Bishop, and R. Heffner). Springer, Berlin (1987). 7. Nunes A. C., Axe J. D. and Shirane G. Ferroelectrics 2, 291 (1971). 8. Fontana M. D., Dolling G., Kugel G. and Carabatos C., Phys. Rev. B20,3850 (1979). 9. Currat R., Comes R., Domer B. and Weisendanger E., J. Phys. C7,2521 (1974).
10. Perry C. H., Buhay H., Quittet A. and Currat R., in Lattice Dynamics (Edited by M. Balkanski). Flammarion Sciences, Paris (1977). 11. Solokoff J., Chase L. and Rytz D., Phys. Rev. B38, 597 (1988).
These neutron scattering experiments have demonstrated the similarity of behavior between the different phases of potassium niobate. The cubic phase displays an anomalously flat and low energy transverse acoustic mode propagating in the (100) directions, which is absent from the previous study on the cubic phase. The transverse optic mode along the same direction shows large damping which is consistent with the previous report. A strong temperature dependent coupling occurs between the two anomalous modes and anti-crossing cannot be ruled out. This coupling [33, 341 is different than that observed in potassium tantalate [35] but is similar to the behavior of barium titanate [36-391. The behavior of the other modes, in contrast, show normal dispersion behavior and the phonon groups are well behaved. The computer simulations of lattice dynamical models demonstrate that the phonon branches have largely harmonic behavior except for the two anomalous modes. The transverse acoustic and lowest energy transverse optic mode in the (100) directions can be modeled successfully with an anharmonic, non-linear polarizability model for a reduced system. This model is intrinsically anisotropic and indicates that the origin of the anomalous behavior arises from the oxygen anisotropic polarizability. Acknowledgements-This research was supported by the U.S. Department of Energy through the Materials Research Laboratory of the University of Illinois at UrbanaChampaign, under contract number DEFGO2-9lER45439. The authors would like to acknowledge Dr R. Nicklow and Dr J. Zaretsky for their assistance in the neutron scattering measurements at Oak Ridge National Laboratory.
REFERENCES 1. Bruce A. and Cowley R. A., Structural Phase Transitions. Taylor and Francis, London (198 1).
12. Solokoff J., Chase L. and Rytz D., Phys. Rev. B40,788 (1989).
13. Fontana M., Kugel G. and Carabatos-Nedelec C., Phys. Rev. B40,786
(1989).
14. Luspin Y., Servoin J. and Gervais F., J. Phys. C 13,376l (1980). 15. Fontana M. D., Metrat G., Kugel G. E. and Carabatos C. Phys. Stat. Solidib 103,211 (1981). 16. Quittet A., Bell M. I., Krauzman M. and Raccah P., Phys. Rev. B14, 14 (1976). 17. Quittet A. M., Fontana M., Lambert M. and Weisendanger E., Ferroelectrics 8,585 (1974). 18. Fontana M. D., Metrat G., Servoin J. L. and Gervais F., J. Phys. C 16,483 (1984). 19. Axe J. D., Harada J. and Shirane G., Phys. Rev. Bl, 3651 (1970). 20. Shirane G., Axe J. D.. Harada J. and Remeika J. P., Phys. Rev. B2, 155 (1970). 21. Fontana M., Kugel G., Foussadier L., Kress W. and Rytz D., Europhys. Lett. 23,427 (1993). 22. Rytz D., Hochli U. and Bilz H. Phys. Rev. B22, 359
(1980). 23. Fontana M., Kugel G. and Kress W., Phys. Rev. B35, 813 (1987).
24. Axe J. D., Harada J. and Shirane G., Phvs. Rev. Bl, 1227
(1970). 25. 26. 27. 28. 29.
Hurst J. and Linz A., Mater. Res. Bull. 6, 163 (1971). Hewat A., J. Phys. C6, 1074 (1973). Zaccai G. and Hewat A. W., J. Phys. C 7, 15 (1974). Hewat A. W.. J. Phvs. C 6.2559 (1973). Holma M., Takesue-N. and Chen ‘H., Ferroeiectrics 164, 237 (1995).
30. Stirling W. G., J. Phys. C5, 2711 (1972). 31. Bilz H., Benedek G. and Bussmann-Holder A., Phys. Rev. B35,4840
(1987).
32. Bussmann-Holder A., Bilz H. and Benedek G., Phys. Rev. B39,9214 (1989). 33. Mante A., Sol. State Commun. 8, 1415 (1970). 34. Migoni R., Bilz H. and Bauerle D., in Lattice Dynamics (Edited by M. Balkanski) Flammarion Sciences, Paris (1977). 35. Salce B. and Boatner L., ORNL Report (1985). 36. Comes R. and Shirane G., Phys. Rev. B5, 1886 (1972). 37. Shirane G. and Yamada Y., Phys. Rev. 177,858 (1969). 38. Comes R. and Shirane G., Phys. Rev. B5, 1886 (1972). 39. Shirane G., Frazer B. C., Minkiewicz V. J.. Leake J. and Linz A., Phys. Rev. Lett. 19,234 (1967).
Pergamon
PHONON DISPERSION
J. Phys. Chem Solids Vol57, No. IO, pp. 146-1471, 1996 Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All riahts reserved 00X-3697/% Si5.00+ 0.00
AND ANHARMONICITY
M. HOLMAt
and HAYDN
IN CUBIC KNb03
CHENS
TDepartment of Physics; $Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received 27 March 1995; accepted 6 July 1995)
Abstract--Inelastic neutron scattering measurements of cubic KNb03 have been performed to study the temperature dependence and anharmonicity of the phonon branches believed to be associated with the subsequent phase transitions. Our results show that the cubic phase displays an anomalously flat and low energy transverse acoustic mode propagating in the (100)directions, absent from the previous study on the cubic phase. The transverse optic mode along the same directions shows large damping which is consistent with the previous report. A strong temperature dependent coupling occurs between these two modes, similar to the behavior reported in BaTiOa. The behavior of the other phonon branches, in contrast, show normal dispersion behavior and the phonon groups are well behaved. Computer simulations of lattice dynamics demonstrate that the phonon branches have largely harmonic behaviour except for the two anomalous modes. The transverse acoustic and lowest energy optic mode in the (100) directions can be successfully modeled with an anharmonic, non-linear polarizability model for a reduced system. This model is intrinsically anisotropic and indicates that the origin of the anomalous behavior arises from anisotropic
oxygen polarizability. This anisotropy also manifests itself in the anomalous thermal diffuseX-ray scattering that has been shown to display a planar distribution in the reciprocal space. Keywords:A. KNtQ, B. neutron scattering, D. phonons, D. anharmonicity, D. lattice dynamics.
INTRODUCEION Neutron scattering studies are an indispensable experimental technique for studying the dynamical behavior of materials and their associated phase transitions. Previous inelastic neutron scattering experiments have provided a deep understanding of the physics of structural phase transitions [l-3]. In particular the physics of the structural phase transition in strontium titanate, with the relevant coordinate being the rotation of the oxygen octahedra, has become the prototype of soft-mode materials. A structural phase transition occurs at 110 K in which an optic mode at the R (l/2, l/2, l/2) point in the Brillouin zone softens completely to zero energy as the temperature approaches T, [4]. The displacements of this optic mode are directly related to the crystal structure change. Although this theory has found notable success with some other materials [5] in addition to strontium titanate, for most materials the observed behavior does not correlate with this theory. In particular, in most materials some softening does occur but the phase transition intervenes before complete softening [6]. Inelastic neutron scattering has been previously performed on three phases of potassium niobate: the cubic [7], tetragonal[8] and orthorhombic [9, lo] phases. Of these experiments the orthorhombic phase [l l- 131 has been the most extensively studied
and cubic phase the least extensively studied. In addition there have been Raman scattering studies [14] in the tetragonal [ 151, orthorhombic [ 16, 171 and rhombohedral [17] phases but not the cubic phase. However, an IR spectroscopy [ 181study included the cubic phase which provided information on the zone center optic modes in this phase. Of other perovskite ferroelectrics, only barium titanate [ 19,201 and KTN (potassium tantalum:niobium oxygen) [21-231 show similar phase transitions and physical behavior as in potassium niobate. Potassium tantalate [24] shows similar behavior but does not undergo any phase transitions and is thus an incipient ferroelectric. The inelastic neutron scattering results of the tetragonal and orthorhombic phases correlate well with the results presented in this study. Our results however contradict the previous study [7] on the cubic phase.
EXPERIMENTAL The inelastic neutron scattering experiments were carried out at the Ames Laboratory-ORNL beam line HB-IA, at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Lab (ORNL) in collaboration with Dr R. Nicklow. The beam line HB-IA uses a double monochromating system at a fixed scattering angle to obtain an fixed initial energy 1465
M. HOLMA and H. CHEN
1466
of 14.8 meV (3584THz). the monochromators are made of pyrolytic graphite using the (002) Bragg peak for energy selection. The monochromatic beam is filtered to remove X/2 contamination, then collimated and sized before reaching the sample table. The collimation used in the experiments was 30 min before and after the sample. Two different single crystal KNbOs samples made by the top seeded technique [25] were used for the experiments. Both crystals had an approximate volume of 0.9cm3. Rocking curves determined the mosaic spread of the crystal to be less 0.2”. The samples were mounted with the [OOl]r_r~ud~ubicaxis perpendicular to the scattering plane. The phonon dispersion has been investigated at temperatures above the cubic to tetragonal phase transition temperature in the high symmetry directions [OOl]and [Ol 11.The temperatures measured were 605,525,471,445 and 429°C. For the samples used in the neutron experiments the cubic to tetragonal phase transition temperature was observed at 429°C in situ using the neutron beam. The sample was slowly raised to the highest temperature and subsequently lowered for the data collection. RESULTS AND DISCUSSION Figures 1 and 2 show the lowest energy transverse branches in the [lOO] direction (no difference was observed between in the phonon groups in the [OlO] direction) at 605 and 525°C respectively. The significant features of these data are the anomalous low energy acoustic mode and the essentially flat region for reduced wave vectors q N 0.2 extending out to the Brillouin zone boundary. Above in energy is the lowest energy transverse optic mode. Near the reduced wave vector q N 0.2 these two modes can not be resolved as separate peaks. Below q - 0.2 the transverse optic mode was not detected. This mode was also measured from the (1 00), (110) and (130) Bragg peaks but no peak corresponding to this mode was observed ,
(
,
,
01 0
,
,
,
’
0.1
’
’
0.1;&~,.35
’
’
’
’
0.4
’
0.45
’
0.5
Fig. 2. Transverse phonon modes in the [lOO]direction at T = 525°C.
below 3 THz. The reduced structure factor for modes propagating in the [ 1001direction was calculated (Fig. 3) for potassium niobate that gives a very probable interpretation of why this mode was not observed. Based on the shell mode1 of the lattice dynamics and assuming equivalent Debye-Wailer factors [26-281 for each atom the inelastic structure factor is a maximum at the zone center and smoothly decrease to a minimum at the zone boundary for this transverse acoustic mode. The inelastic structure factor for the transverse optic mode has a maximum midway between the zone center and zone boundary. However, the shell model does not give an adequate fit to the phonon dispersion for this mode. A second anomalous feature of these two modes is the behaviour as represented in the neutron groups. The transverse acoustic mode was unusually difficult to measure. Not only is the peak intensity extremely low but the background is unusually high for wave vectors q 2 0.3. The behavior of the inelastic structure factor accounts for the decrease in peak height but not the large background (see raw data in Fig. 4). For wave vectors q N 0.3 this mode is essentially flat which was observed from the defocused shape of the phonon peaks measured from different zones. 2.5
(
’
0.06
(
(
,
(
,
(
(
,
,
,
T-SOSC
+
l
2-
I 1.5
TA +
TO1
+
1
+
l0
0
0
Fig. 1. Transverse phonon modes in the [lOO]direction at T = 605°C.
0
0 0
0.5 -
TO
““““‘I
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 q.mhadwawwcbr
Fig. 3. Reduced structure factor for the transverse acoustic and optic modes.
Phonon dispersion and anharmonicity in cubic KNbOj For the lowest energy transverse optic mode the neutron group is seen to broaden substantially as the wave vector increases to the Brillouin zone boundary (see raw data in Fig. 5). This peak broadening is apparently symmetric in shape and thus can not be due to interference effects between one-phonon and multi-phonon processes. The width is an order of magnitude larger than the instrumental resolution and can be physically ascribed to anharmonic effects. Anharmonic phonon theory leads to a calculation for the damping constant of 4.2 THz at the zone boundary. This mode could not be detected for wave vectors
1467
near the zone center, it is believed to be completely overdamped in this region. The last branch measured is a transverse optic mode that is flat across the entire Brillouin zone at nearly 6THz. No branches were measured above 8THz. Figures 6 and 7 show the longitudinal scans in the [loo] direction at two temperatures. In contrast to the transverse scans the longitudinal scans showed extremely well defined neutron groups and the dispersion behaviour is normal; the phonon energy increases with increasing wave vector. The previous studies of inelastic neutron scattering, in particular in the orthorhombic and
250(a) MO.3 $&
200-
P P
150 -
100 _____--*’
______I
0
+&o____--------mm____ 0, 1.5 2 2.5 t
I
0.5
1
/Q
0 Q
,‘O
I
,O
I’
/
0
“P *s, ‘% O’--_____________________-
1
I
3
150
‘A
‘*
0
&
-9
-----co6 Q
1
$
100
‘b%*&,___._____ 50
0
0.5
2
I.5
1
2.5
3
Fig. 4. Peak width of the transverse acoustic mode for reduced wave vectors (a) q = 0.1, (b) q = 0.25 and
(c)q
=
0.35.
Fig. 5. Peak width of the transverse optic mode for reduced wave vectors (a) q = 0.3, (b) q = 0.4 and (c) q = 0.5.
1468
M. HOLMA and H. CHEN 6
,
,
,
,
1
,
7-
,
LO1 +
,
)
T&EC + +
6-
0
a. 0 -
0 LAO 5-
0
4-
0
32-
0
l0
0
’ 0.05
’ 0.1
’ ’ 0.15 0.2 RdJmd
’ ’ ’ 0.25 0.3 0.35 mmwador* 9
’ 0.4
’ 0.45
0.5
Fig. 6. Longitudinal phonon modes in the [lOO]direction at T = 605°C tetragonal phases, show similar types of anomalous behavior. The tetragonal phase shows the same transverse acoustic mode behavior, i.e. low in energy and flat for reduced wave vectors greater than 0.2. For phonon branches propagating in the [loo] direction and polarized along the [OlO]the flat region is similar in energy (1.8 THz as compared with 1.5 THz measured in the present study). There is a similar mode propagating in the [OOl] direction and polarized in either the [lOO] or [OlO] direction. These modes are purely transverse. In addition, the tetragonal phase has the flat transverse optic mode at nearly 6 THz that appears in the cubic phase. The tetragonal modes compatible with the longitudinal modes observed in the cubic phase show no difference. The only compatible mode that has distinctly changed is the pseudotransverse mode propagating in the [loo] direction and polarized close to [OOl]direction (the direction of the ferroelectric polarization in the tetragonal phase). This mode has sharply stiffened and displays normal wave vector dependence. The authors [8] also noted that the flat transverse acoustic and lowest energy transverse optic mode showed ill defined phonon groups and that anti-crossing behavior was observed. The orthorhombic phase shows a similar transverse acoustic mode that propagates in the plane
5
containing the ferroelectric polarization vector (in the orthorhombic phase the polarization is along the [loll direction with respect to the pseudo-cubic axes) and whose eigenvector is polarized strictly transverse to the ferroelectric polarization vector. The authors [9] expressed similar sentiments regarding the measurement of these neutron groups: they found unexpected difficulty measuring these groups because of a lowered intensity and a systematically higher background. In contrast with the previous results [7] on the cubic phase there is marked contradiction with those presented here. The low energy transverse acoustic mode is totally absent in that study and the optic mode is assumed to be completely overdamped throughout the entire zone. The only other phonon branch measured was the longitudinal acoustic mode in the [loo] direction and it is similar to the mode observed in the present study. Given the improved resolution of this study and the inherent difficulty in detecting the transverse acoustic mode, it is probable that it could not be observed in the previous study. Two phonon branches were measured in the [l lo] direction. These are shown in Fig. 8. The transverse optic mode was not observed in the [l lo] direction, however diffuse X-ray scattering measurements suggest that this mode is overdamped near the zone center and that the largest anisotropy in the phonon dispersion exists for the transverse optic mode in the [l lo] direction [29]. These agree with the previous study [7] and show no anomalous behavior. The significant temperature dependent behavior occurs only in the lowest energy transverse optic and transverse acoustic modes in the [lOO] direction, the branches that were difficult to measure. All other branches show no appreciable temperature dependence; including the compatibility relations this can be extended to the other phases as well. Figure 9 shows the temperature dependent behavior of the two modes. The observed temperature dependent behavior occurs only near q - 0.2 where the transverse acoustic and transverse optic modes have similar
TA
Fig. 7. Longitudinal phonon modes in the [lOO]direction at T = 525°C.
Fig. 8. Acoustic modes in the [l lo] direction at T = 525°C.
Phonon dispersion and anharrnonicity in cubic KNbOs
z$ 0
” , , , , , 0.05
o.,
1469
,;s%,1
0.15&~,,."
0.4
0.45
OU’ 0 0.05
0.5
0.1
”
”
0.1; ~.25_0&35
”
0.4
”
0.45
0.5
Fig. 9. Temperature dependence of the lowest energy phonon modes.
Fig. 11. Shell model fitted dispersion curves for longitudinal modes in the [loo] direction.
energies. The data as shown are single peak fits to the measured phonon group intensities. These data were reconsidered and fitted with a mode coupling scheme, the temperature dependent effect is an increased coupling between the transverse acoustic and optic modes near q - 0.2.
mation. The ionic charges are not equivalent to their free ion values (+l for K and +5 for Nb) and this feature is also characteristic of the oxygen perovskites. The eigenvectors for the shell model are shown in Figs 12 and 13 for the lowest energy transverse optic and transverse acoustic modes in the [loo] direction. In spite of the failure of the shell model to adequately fit the dispersion curves the eigenvectors are similar to those found in the orthorhombic phase neutron scattering results. In the transverse optic mode the oxygen ions are the predominant ones which are moving, while the niobium ions are nearly at rest. In contrast for the transverse acoustic mode all the oxygen ions have zero or nearly zero displacements while the niobium ionic displacements are quite large. This agrees well with the experimentally determined results for the compatible mode in the orthorhombic phase. The eigenvectors were not measured experimentally in this work due to the extreme difficulty in even measuring the phonon branches from one Bragg peak.
LATTICE DYNAMICAL
MODELS
The rigid shell model based on the same model used for strontium titanate [30] by Stirling was applied to the results of the inelastic neutron scattering data to determine the parameters of the lattice dynamics. This model was fitted to the measured phonon dispersion curves and also the zone center points available from infrared spectroscopy on cubic potassium niobate. The best fit dispersion curves are shown in Figs 10 and 11, which include the zone center points mentioned above. Some particular features of the model are observed in the Table 1. The transverse short range force constant between niobium and oxygen is an order of magnitude larger than all other force constants. This is also observed in the other oxygen perovskites and reflects the rigidity of the oxygen octahedra to defor-
Table 1. Shell model parameters Force constants (e’/V Transverse K-O Nb-0 o-o Longitudinal K-O Nb-0 o-o Z(K) Z(Nb) (10” cm) o(K) Wb) o(0)
b ippj 0
0.05
0.1
0.15 0.2 q,-m-r
0.25
0.3
0.55
0.4
0.45
KNbOs
KTaOs
SrTi03
13.0 355.0 -1.00
14.65 359.0 3.22
26 285 1.9
-1.01 -85 1.oo
-1.01 -68 1.085
-1.01 -43 0.74
0.8 5.125
0.82 4.74
1.52 3.3
0.0009 0.001 0.0275
0.0001
0.0035 0.0007 0.0059
0.021 0.0117
0.5
Fig. IO. Shell model fitted dispersion curves for transverse modes in the [IOO]direction.
$)K) d(Nb) d(O)
0.001
0.01 0.801
0.0052 0.797 -0.186
-0.54 0.0967 -0.0384
1470
M. HOLMA and H. CHEN
0
0
x
‘Wore shell
92’
A
M2
-0.5 0
0
Fig. 14. One dimensional anisotropic model.
-1 /:
II
0
1
0.05
0.1
II 11 0.1: ~.2&0&35
‘I
11 0.4
0.45
0.5
Fig. 12. Eigenvectors from the shell model for the transverse optic mode in the [loo] direction.
Generally the phonon branches must be measured from five to six Bragg peaks to determine the eigenvectors. The observation that the temperature dependent behavior occurs in the two lowest transverse modes and that the harmonic model is inadequate to fit these modes suggests the use of the polarizability model [3 1, 321 (Fig. 14). The fitted dispersion curves for the polarizability model are shown in Fig. 15. A pseudoone-dimensional model is sufficient to describe the lattice dynamics of the lowest energy phonon modes. The reduction to a one-dimensional system is done by grouping together the transition-metal ion and the three oxygen ions (Nb03) as a single polarizable mass, with a non-linear anisotropic polarizability. The hamiltonian is then written for the reduced system in terms of the displacements of the cores and shells of atoms 1 (the grouped Nb03) and 2 (the ion K). The hamiltonian is:
(1)
H=T+V
where
+
&r($l’- up)4]
(3)
and r#), rn?) are the core and shell masses of the ith atom, u@)and $) are the displacements of the core and shell of the ith atom and f, f(‘)‘, and &’ are the harmonic force constants for coupling between shell-shell, core-core and internal core-shell, respectively (see Fig. 14), whereas g4 is the only anharmonic term introduced in the model. The terms are summed over the two constituents over all atoms in the system, N, and the phonon energies are obtained using the Euler-Lagrange equations and the adiabatic wnditions. Details can be found in the references. An extremely good fit was obtained for these two modes. The parameters for these models are shown in Table 2. A direct comparison with the force constants of the harmonic rigid shell model is not possible but they can be compared with results presented for other oxygen perovskites [31, 321 (in the above table, for SrTi03 and KTa03). The reduced masses are nearly equal to those obtained in the fitting of potassium tantalate, but the coupling constants are significantly larger in strontium titanate. The smaller values in potassium niobate increase the net effect of anharmonit forces, supporting the view of near cancellation of forces near the phase transition. This is to be expected 4
I
2.5-
I
I
I
I
I
p
Tdo5C ,/
32.5,’
2-
I
,*
,’
,’
I’
I’
/-p-------‘
,a”
_/ ______-------
0
0.05
0.1
0.1; ~2&0.3&35
0.4
0.45
0.5
Fig. 13. Eigenvectors from the shell model for the transverse acoustic mode in the [loo] direction.
0
02
0.4
0.6 bNvs0f.q’
1.2
1.4
Fig. 15. Fitted phonon dispersion curves for T = 605°C in the [ 1001direction.
Phonon dispersion and anharmonicity in cubic KNbO, Table 2. Anisotropic oxygen polar&ability model parameters KNbOs m, (lo-** g) mz (lo-** 5)
f (lo4g~-1 r; (10; q jz\‘;J “,J) g!’ (lo4 gs-2) g4 w4
W2)
KTaOs
SrTiO,
2.9 0.65
2.88 0.649
1.549 1.461
4.05 0.55 0.0 0.035 -0.65 0.15
4.067 0.581
14.40 1.268
-0.828 0.255
-2.629 0.904
from the inherent behavior of the phonon groups. Potassium tantalate does not show the anomalous behavior of the phonon groups in the two lowest transverse modes.
CONCLUSIONS
1471
2. Shirane G., in Structural Phase Transitions and Soft Modes (Edited by E. J. Samuelserl, E. Anderson, and J. Feder). Universitetsforlaget, Oslo (1971). 3. Scott J.. Rev. mod. Phvs. 46. 83 (1974). 4. Yamada Y., Shirane d. and'Li& A., Phys. Rev. 177,848 (1969). 5. Krumhansl J., in Computing Interactions and Microstructures: Statics and Dvnamics (Edited by R. LeSar, A. Bishop, and R. Heffndr). Berlin (1987). _ 6. Shapiro S. M., in Competing Interactions and Microstructures: Statics and Dynamics (Edited by R. LeSar, A. Bishop, and R. Heffner). Springer, Berlin (1987). 7. Nunes A. C., Axe J. D. and Shirane G. Ferroelectrics 2, 291 (1971). 8. Fontana M. D., Dolling G., Kugel G. and Carabatos C., Phys. Rev. B20,3850 (1979). 9. Currat R., Comes R., Domer B. and Weisendanger E., J. Phys. C7,2521 (1974).
10. Perry C. H., Buhay H., Quittet A. and Currat R., in Lattice Dynamics (Edited by M. Balkanski). Flammarion Sciences, Paris (1977). 11. Solokoff J., Chase L. and Rytz D., Phys. Rev. B38, 597 (1988).
These neutron scattering experiments have demonstrated the similarity of behavior between the different phases of potassium niobate. The cubic phase displays an anomalously flat and low energy transverse acoustic mode propagating in the (100) directions, which is absent from the previous study on the cubic phase. The transverse optic mode along the same direction shows large damping which is consistent with the previous report. A strong temperature dependent coupling occurs between the two anomalous modes and anti-crossing cannot be ruled out. This coupling [33, 341 is different than that observed in potassium tantalate [35] but is similar to the behavior of barium titanate [36-391. The behavior of the other modes, in contrast, show normal dispersion behavior and the phonon groups are well behaved. The computer simulations of lattice dynamical models demonstrate that the phonon branches have largely harmonic behavior except for the two anomalous modes. The transverse acoustic and lowest energy transverse optic mode in the (100) directions can be modeled successfully with an anharmonic, non-linear polarizability model for a reduced system. This model is intrinsically anisotropic and indicates that the origin of the anomalous behavior arises from the oxygen anisotropic polarizability. Acknowledgements-This research was supported by the U.S. Department of Energy through the Materials Research Laboratory of the University of Illinois at UrbanaChampaign, under contract number DEFGO2-9lER45439. The authors would like to acknowledge Dr R. Nicklow and Dr J. Zaretsky for their assistance in the neutron scattering measurements at Oak Ridge National Laboratory.
REFERENCES 1. Bruce A. and Cowley R. A., Structural Phase Transitions. Taylor and Francis, London (198 1).
12. Solokoff J., Chase L. and Rytz D., Phys. Rev. B40,788 (1989).
13. Fontana M., Kugel G. and Carabatos-Nedelec C., Phys. Rev. B40,786
(1989).
14. Luspin Y., Servoin J. and Gervais F., J. Phys. C 13,376l (1980). 15. Fontana M. D., Metrat G., Kugel G. E. and Carabatos C. Phys. Stat. Solidib 103,211 (1981). 16. Quittet A., Bell M. I., Krauzman M. and Raccah P., Phys. Rev. B14, 14 (1976). 17. Quittet A. M., Fontana M., Lambert M. and Weisendanger E., Ferroelectrics 8,585 (1974). 18. Fontana M. D., Metrat G., Servoin J. L. and Gervais F., J. Phys. C 16,483 (1984). 19. Axe J. D., Harada J. and Shirane G., Phys. Rev. Bl, 3651 (1970). 20. Shirane G., Axe J. D.. Harada J. and Remeika J. P., Phys. Rev. B2, 155 (1970). 21. Fontana M., Kugel G., Foussadier L., Kress W. and Rytz D., Europhys. Lett. 23,427 (1993). 22. Rytz D., Hochli U. and Bilz H. Phys. Rev. B22, 359
(1980). 23. Fontana M., Kugel G. and Kress W., Phys. Rev. B35, 813 (1987).
24. Axe J. D., Harada J. and Shirane G., Phvs. Rev. Bl, 1227
(1970). 25. 26. 27. 28. 29.
Hurst J. and Linz A., Mater. Res. Bull. 6, 163 (1971). Hewat A., J. Phys. C6, 1074 (1973). Zaccai G. and Hewat A. W., J. Phys. C 7, 15 (1974). Hewat A. W.. J. Phvs. C 6.2559 (1973). Holma M., Takesue-N. and Chen ‘H., Ferroeiectrics 164, 237 (1995).
30. Stirling W. G., J. Phys. C5, 2711 (1972). 31. Bilz H., Benedek G. and Bussmann-Holder A., Phys. Rev. B35,4840
(1987).
32. Bussmann-Holder A., Bilz H. and Benedek G., Phys. Rev. B39,9214 (1989). 33. Mante A., Sol. State Commun. 8, 1415 (1970). 34. Migoni R., Bilz H. and Bauerle D., in Lattice Dynamics (Edited by M. Balkanski) Flammarion Sciences, Paris (1977). 35. Salce B. and Boatner L., ORNL Report (1985). 36. Comes R. and Shirane G., Phys. Rev. B5, 1886 (1972). 37. Shirane G. and Yamada Y., Phys. Rev. 177,858 (1969). 38. Comes R. and Shirane G., Phys. Rev. B5, 1886 (1972). 39. Shirane G., Frazer B. C., Minkiewicz V. J.. Leake J. and Linz A., Phys. Rev. Lett. 19,234 (1967).