Research Article  Open Access
Temperature Dependent Variations of Phonon Interactions in Nanocrystalline Cerium Oxide
Abstract
The temperature dependent anharmonic behavior of the phonon modes of nanocrystalline CeO_{2} was investigated in the temperature range of 80–440 K. The anharmonic constants have been derived from the shift in phonon modes fitted to account for the anharmonic contributions as well as the thermal expansion contribution using the high pressure parameters derived from our own high pressure experimental data reported previously. The total anharmonicity has also been estimated from the true anharmonicity as well as quasiharmonic component. In the linewidth variation analysis, the cubic anharmonic term was found to dominate the quartic term. Finally, the phonon lifetime also reflected the trend so observed.
1. Introduction
CeO_{2} is a strategic material which has been extensively investigated and has recently been found to be of use in intermediate temperature oxide fuel cells as it easily absorbs and releases oxygen [1]. It also shows enhanced ionic conductivity with temperature [2]. Hence, the investigation of the temperature dependent behavior of CeO_{2} has gained immense interest. Under ambient conditions, it usually crystallizes in the cubic fluorite structure with space group Fm3m [3]. Nanocrystalline cerium oxide has applications in the coating technology, for example, woodcoating industry [4].
It is well known that an increase in temperature in any material introduces perturbations in the harmonic potential term, which changes the vibrational properties. Temperature affects the population of the different levels for each normal mode instigating the change of the lattice parameters. Any change of the lattice parameters with temperature is attributed to the anharmonicity of the lattice potential. As an effect of the anharmonic interaction, a nonequilibrium phonon population decays into phonons of lower energy or is scattered by thermal phonons into modes of frequencies shifted from their original positions [5]. These anharmonic interactions occur primarily due to the cubic or quartic anharmonicities, resulting in the decaying of an optical phonon into two or three acoustic phonons, respectively, which also decreases their lifetimes. As a consequence, the phonon line position and bandwidth in the Raman spectra of various amorphous or crystalline materials exhibit significant changes with temperature [6–8]. Further, the finitesize effects in nanocrystals are expected to modify the anharmonicity and the phonon decay times. It has also been reported that the anharmonic constants related to the peak widths and peak positions are higher in nanocrystals than in the bulk, implying a greater degree of anharmonicity in nanocrystals [9]. Very few experiments have been carried out to investigate the temperature dependence of Raman mode frequencies and linewidths in nanocrystals [5, 10–12].
Still fewer studies have been carried out on the anharmonic behavior of the CeO_{2} phonon modes [13–15]. Popović et al. [13] reported that three contributions are involved in the temperatureinduced Raman mode frequency shifts of nanoCeO_{2−y}: the thermal expansion, the anharmonic effect, and the phonon frequency change due to the change of nanoparticle size with temperature (confinement effect). It is natural to expect that in nanomaterials the contribution of fourphonon processes should be larger compared to that of the threephonon processes [11]. However, Spanier et al. [14] reported that phonon coupling is not stronger in the CeO_{2−y} nanoparticle system than that in the bulk counterpart. Consequently, the aim of this work is to examine the temperatureinduced changes in the vibrational properties of CeO_{2}, to identify the contribution of volume expansion and anharmonic intermode coupling to the Raman frequency shift, linewidth, and the phonon lifetime in CeO_{2}.
In this context, we report the temperature dependent Raman behavior of nanocrystalline CeO_{2} and its anharmonic constants derived using our previously published high pressure data [16]. We have analyzed the softening of the most dominant mode () in light of the theory of anharmonic phononphonon interaction and thermal expansion of the lattice.
2. Characterization at Ambient Conditions
The CeO_{2} powder used in the present study is a commercially prepared high purity powder from Johnson Matthey Rare Earth Products, UK. The powder was used in the present study in asreceived state and was not subjected to any pretreatment and, as per the manufacturer’s claims, the common metal impurities are at a ppm level. The characterization of this CeO_{2} nanopowder under ambient conditions via Xray diffraction, atomic force microscopy, and Raman spectroscopy has been discussed in detail elsewhere [17]. In brief, the crystalline phase at ambient was found to be cubic. The crystallite size of this cubic CeO_{2} was found to be 36 nm calculated using DebyeScherrer relation and about 32 nm from atomic force microscopy. The predominant Raman peak of CeO_{2} was obtained at 465 cm^{−1}, assigned as mode [17]. In Ceria, the first order Raman active mode is the triply degenerate symmetrical stretching vibration of the CeO_{8} vibrational unit. In bulk CeO_{2}, there is a single allowed Raman active triply degenerate mode centered at about 465 cm^{−1}. This mode can be viewed as a symmetric breathing vibrational mode of the O ions around each cation [18, 19]. The CeO_{2} belongs to space group Fm3m (225). This structure has two BCC unit cells and has two acoustic modes and 1 optical mode.
3. Temperature Dependent Raman Studies
The Raman spectra were recorded in the temperature range 80–440 K. The spectra were acquired using Jobin Yvon T64000 triple Raman spectrometer with the 514.5 nm line of the Ar ion laser as the source radiation. The sample temperature was varied using a continuous flow liquid nitrogen cryostat from Janis, model ST500.
The temperature dependent Raman spectra of CeO_{2} are given in Figure 1. The spectra show changes in line position, linewidth, and intensity with an increase in temperature from 80 K to 440 K. We have focused on temperature dependence of Raman frequency and linewidth of the predominant optical phonon in the studied temperature range. With temperature increase, the mode shifts to lower frequencies and the linewidth continuously increases. A vertical line on the mode in Figure 1 facilitates the visualization of the extent of shift in said phonon mode.
Hence, the observed results show temperature dependent phonon softening of mode, that is, red shift with progression of temperature. The possible reasons for red shift can be heating and phonon interactions and so forth. Heat causes expansion of the material which leads to frequency decrease. The frequency softening is significant up to a temperature of ~380 K. Above this temperature, the phonon softening is seen to slow down. It may be mentioned here that, as per the reported Debye temperature of 355 K [20, 21], the phonon softening may be said to be slowing down above the Debye temperature. This may be owing to the fact that, above the Debye temperature, the change in specific heat capacity becomes negligible. Figure 2 shows the peak positions observed at 80 K and 440 K which are centered at 468.2 cm^{−1} and 463.3 cm^{−1}, respectively. The frequency of this mode can be clearly seen to shift to lower energy side and the linewidth increases with temperature. In the studied temperature range, the total quantitative softening of the mode amounts to about 5 cm^{−1}. These shifts as well as the linewidths estimated from the “peakfit” software are graphically depicted clearly in Figure 3. It may be interesting to note here that similar to the behavior shown by mode softening the linewidth increase also shows a lower slope above about 380 K.
The observed phonon softening of the mode is examined from the theory of anharmonic phononphonon interaction and thermal expansion of the lattice. As mentioned, the variation of the frequency of the normal modes with temperature at constant pressure arises from two contributions: pure volume contribution which results from thermal expansion and puretemperature contribution which results from cubic and quartic anharmonicities (selfenergy shift). The cubic and quartic anharmonicities are described by factors which include contributions from three and fourphonon anharmonic processes [11]. The quasiharmonic term arises from the thermal expansion, while puretemperature contribution due to cubic and quartic anharmonicities is also termed as intrinsic (true) anharmonicity [9, 12]. Hence,
The quasiharmonic or thermal expansion contribution can be written aswhere (=470.1 cm^{−1}) is the harmonic frequency of mode which is obtained by extrapolating the experimental data down to 0 K. The Grüneisen parameter (=1.28) of the corresponding mode is estimated from our high pressure Raman scattering data reported elsewhere [16] and is the linear thermal expansion coefficient.
The anharmonic contribution which arises from the phononphonon interaction due to the cubic and quartic terms in the interatomic potential can be written as
The first term estimates the coupling of an optical phonon to two lowenergy phonons (threephonon process); the second term is due to the coupling of three phonons (fourphonon process), and and are the fitting parameters. We have also calculated the total anharmonicity of mode as a function of temperature using the following equation [22]:
The first term of the right hand side describes the true anharmonic contribution and the second quasiharmonic term. As mentioned, the resultant frequency shift may be either positive or negative depending on the relative magnitudes of the anharmonic terms in the interatomic potential. At lower temperatures, the main contribution to the softening is due to the positive quartic anharmonicity.
In Figure 4, we have shown the fitted experimental data points (solid circles) by using (1). The fitting parameters obtained for the present case using the above equations are given in Table 1. From the fitting parameters given in the table, it is clear that coefficients and both are negative thus explaining the phonon softening. However, coefficient dominates over indicating that threephonon processes or cubic anharmonicity is contributing significantly to the overall softening observed. It is also seen that the true anharmonic term dominates, albeit slightly, over the quasiharmonic term.

A comparison of these coefficients with those obtained by DohcevicMitrovic et al. [15] and Popović et al. [13] reveals that, in CeO_{2−y} nanocrystals, although a dominance of fourphonon process was observed in the high temperature range by these researchers, however, we have found significant contribution from threephonon process to the anharmonicity. Our results differ as we have observed a behavior in nanocrystalline CeO_{2} which has been reported for polycrystalline CeO_{2} [13]. It is understood that, in small particles of pure and doped ceria nanocrystals, when size effects have minor impact on Raman modes, fourphonon anharmonic processes prevail [9, 15]. However, when nanopowdered particles are grown enough, the size effects provoke changes of the anharmonic interactions whereby threephonon coupling prevails over the fourphonon anharmonic processes.
The linewidth of the mode was also studied as a function of temperature in the 80–440 K range, which, as expected, increased with temperature as shown in Figure 3. The FWHM of the mode increases from 4.85 cm^{−1} at 80 K to 11.43 cm^{−1} at 440 K. Hence, the linewidth increases with temperature by about 6.58 cm^{−1}.
The experimental peaks of real materials exhibit an intrinsic width which has contributions from factors such as the finite resolution of the spectrometer and impurities/defects which disturb the translation symmetry of the harmonic crystal [22, 23] apart from the anharmonic decay of the phonons. The thermal expansion is a manifestation of the lattice anharmonicity that has its own shifts in the frequency, but not on the linewidth [23]. Hence, considering that other effects due to impurities, defects, and spectrometer are constants for a particular sample, the linewidth variation would reflect the anharmonic decay of phonons. The temperature dependent phonon linewidth is then written as [24]where is the harmonic linewidth independent of temperature and it is caused by defects, including isotopic mixture; the second part of the equation is the damping part induced due to the three and fourphonon process cubic and quartic anharmonicities, respectively.
Figure 5 shows the fitting to the experimental points using (5). A good agreement can be seen between the calculated curve, given by the solid line, and the experimental points. For the present case, the parameters in (5) are cm^{−1} and and are anharmonic constants where and . It may be mentioned here that coefficients and are complementary to coefficients and which were used to estimate the contribution of anharmonicity to the phonon frequency shift. The positive coefficients describe the increase in linewidth with an increase in temperature. The factors calculated using (5) are listed in Table 2.

In this context, it may be pertinent to mention that the Klemens [25] model for the linewidth assumed that the contribution to the linewidth arises only from the decay of the optical phonon into two acoustic phonons of the same frequency and opposite momentum [3]. Balkanski et al. [11] calculated the change in linewidth and line center for silicon using cubic and quartic anharmonicity and found good agreement between theory and experiment. However, it has also been reported that [18] the temperature dependence of the linewidth of crystalline CeO_{2} can be explained completely with consideration of only the cubic anharmonic term of the crystal potential energy and is governed by the phonon occupation number. In our case, although the cubic anharmonic term is predominant, the contribution from fourphonon process is not negligible.
In addition, changes in the linewidth of the phonon in Raman and infrared spectra can be used as an indirect measurement for estimating the lifetime of strongly interacting optical phonons. The linewidth of a Raman mode is inversely proportional to its lifetime, and anharmonic linewidths (lifetimes) increase (decrease) with increase in temperatures. The estimated lifetime can be calculated by , where is the velocity of light and is the FWHM of the mode [23]. Phonon lifetime is 2.1 ps at 80 K and 0.92 ps at 440 K for the phonon of CeO_{2}. The variation of the decay time of the phonon with increase in temperature has thus been estimated and is shown in Figure 6. The discontinuity in slope observed at about 380 K is again detectable in the figure.
4. Conclusions
The behavior of the optical phonon of nanocrystalline cubic CeO_{2} as a function of temperature is investigated via Raman spectroscopy. The optical mode of CeO_{2} shows phonon softening with increasing temperature up to 440 K. This phenomenon arises from dominating cubic phononphonon interaction and linewidth broadening in the temperature range 80 to 440 K. The temperature dependent data analysis reveals that this mode is a combination of quasiharmonic and true anharmonic terms wherein the latter has a higher contribution to the total anharmonicity. The threephonon process dominated the fourphonon process. The lifetime of the firstorder optical mode is found to decrease with increasing anharmonicity. An observable discontinuity in the slopes of the phonon softening and the linewidth curves is seen at around 380 K.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to Director, NPL, for his constant encouragement and are thankful to CSIR for grant under Network Project NWP45.
References
 S. P. S. Badwal and F. T. Ciacchi, “Oxygenion conducting electrolyte materials for solid oxide fuel cells,” Ionics, vol. 6, no. 12, pp. 1–21, 2000. View at: Publisher Site  Google Scholar
 B. C. H. Steele, “Appraisal of Ce_{1y}Gd_{y}O_{2y/2} electrolytes for ITSOFC operation at 500°C,” Solid State Ionics, vol. 129, no. 1, pp. 95–110, 2000. View at: Publisher Site  Google Scholar
 B. C. H. Steele, “Mass transport in materials incorporated in electrochemical energy conversion systems,” Solid State Ionics, vol. 12, pp. 391–406, 1984. View at: Publisher Site  Google Scholar
 P. Wachter, “The magnetic susceptibility of CEO@ has shown to be paramagnetic,” in Valence Instabilities, P. Wachter and H. Boppart, Eds., p. 145, NorthHolland, Amsterdam, The Netherlands, 1982. View at: Google Scholar
 P. Mishra and K. P. Jain, “Temperaturedependent Raman scattering studies in nanocrystalline silicon and finitesize effects,” Physical Review B—Condensed Matter and Materials Physics, vol. 62, no. 22, Article ID 14790, 2000. View at: Publisher Site  Google Scholar
 G. Morell, W. Pérez, E. ChingPrado, and R. S. Katiyar, “Anharmonic interactions in beryllium oxide,” Physical Review B, vol. 53, no. 9, pp. 5388–5395, 1996. View at: Publisher Site  Google Scholar
 N. S. Yuksek, N. M. Gasanly, and A. Aydinli, “Anharmonic line shift and linewidth of the Raman modes in TlInS_{2} layered crystals,” Journal of Raman Spectroscopy, vol. 35, no. 1, pp. 55–60, 2004. View at: Publisher Site  Google Scholar
 P. Verma, S. C. Abbi, and K. P. Jain, “Ramanscattering probe of anharmonic effects in GaAs,” Physical Review B, vol. 51, no. 23, Article ID 16660, 1995. View at: Publisher Site  Google Scholar
 Z. DohcevicMitrovic, Z. V. Popović, and M. Šćepanović, “Anharmonicity effects in nanocrystals studied by Raman scattering spectroscopy,” Acta Physica Polonica A, vol. 116, no. 1, pp. 36–41, 2009. View at: Google Scholar
 M. J. Konstantinović, S. Bersier, X. Wang et al., “Raman scattering in clusterdeposited nanogranular silicon films,” Physical Review B—Condensed Matter and Materials Physics, vol. 66, Article ID 161311, 2002. View at: Google Scholar
 M. Balkanski, R. F. Wallis, and E. Haro, “Anharmonic effects in light scattering due to optical phonons in silicon,” Physical Review B, vol. 28, no. 4, pp. 1928–1934, 1983. View at: Publisher Site  Google Scholar
 S. D. Pandey, K. Samanta, J. Singh, N. D. Sharma, and A. K. Bandyopadhyay, “Anharmonic behavior and structural phase transition in Yb_{2}O_{3},” AIP Advances, vol. 3, Article ID 122123, 2013. View at: Publisher Site  Google Scholar
 Z. V. Popović, Z. DohčevićMitrović, A. Cros, and A. Cantarero, “Raman scattering study of the anharmonic effects in CeO_{2y} nanocrystals,” Journal of Physics Condensed Matter, vol. 19, no. 49, Article ID 496209, 2007. View at: Publisher Site  Google Scholar
 J. E. Spanier, R. D. Robinson, F. Zhang, S.W. Chan, and I. P. Herman, “Sizedependent properties of CeO_{2−y} nanoparticles as studied by Raman scattering,” Physical Review B, vol. 64, no. 24, Article ID 245407, pp. 1–8, 2001. View at: Google Scholar
 Z. D. DohcevicMitrovic, M. Radović, M. Šćepanović et al., “Temperaturedependent Raman study of Ce_{0.75}Nd_{0.25}O_{2_δ} nanocrystals,” Applied Physics Letters, vol. 91, no. 20, Article ID 203118, 2007. View at: Publisher Site  Google Scholar
 S. Dogra, J. Singh, N. Dilawar Sharma et al., “Phase progression via phonon modes in lanthanide dioxides under pressure,” Vibrational Spectroscopy, vol. 70, pp. 193–199, 2014. View at: Publisher Site  Google Scholar
 S. Dogra, N. D. Sharma, J. Singh, H. K. Poswal, S. M. Sharma, and A. K. Bandyopadhyay, “High pressure behavior of nanocrystalline CeO_{2} up to 35 GPa: a Raman investigation,” High Pressure Research, vol. 31, no. 2, pp. 292–303, 2011. View at: Publisher Site  Google Scholar
 T. Sato and S. Tateyama, “Temperature dependence of the linewidth of the firstorder Raman spectrum for crystalline CeO_{2},” Physical Review B, vol. 26, no. 4, pp. 2257–2260, 1982. View at: Publisher Site  Google Scholar
 W. H. Weber, K. C. Hass, and J. R. McBride, “Raman study of CeO_{2}: secondorder scattering, lattice dynamics, and particlesize effects,” Physical Review B, vol. 48, no. 1, pp. 178–185, 1993. View at: Publisher Site  Google Scholar
 V. Kanchana, G. Vaitheeswaran, A. Svane, and A. Delin, “Firstprinciples study of elastic properties of CeO_{2}, ThO_{2} and PoO_{2},” Journal of Physics Condensed Matter, vol. 18, no. 42, pp. 9615–9624, 2006. View at: Publisher Site  Google Scholar
 L. Gerward, J. Staun Olsen, L. Petit, G. Vaitheeswaran, V. Kanchana, and A. Svane, “Bulk modulus of CeO_{2} and PrO_{2}—an experimental and theoretical study,” Journal of Alloys and Compounds, vol. 400, no. 12, pp. 56–61, 2005. View at: Publisher Site  Google Scholar
 M. Millot, R. T. Zaera, V. M. Sanjose, B. J. Marc, and J. Gonzalez, “Anharmonic effects in ZnO optical phonons probed by Raman spectroscopy,” Applied Physics Letters, vol. 96, no. 15, Article ID 152103, 2010. View at: Publisher Site  Google Scholar
 K. Samanta, P. Bhattacharya, and R. S. Katiyar, “Temperature dependent E2 Raman modes in the ZnCoO ternary alloy,” Physical Review B, vol. 75, no. 3, Article ID 035208, 2007. View at: Publisher Site  Google Scholar
 K. Kamali, T. R. Ravindran, C. Ravi, Y. Sorb, N. Subramanian, and A. K. Arora, “Anharmonic phonons of NaZr_{2}(PO_{4})_{3} studied by Raman spectroscopy, firstprinciples calculations, and xray diffraction,” Physical Review B, vol. 86, Article ID 144301, 8 pages, 2012. View at: Google Scholar
 P. G. Klemens, “Anharmonic decay of optical phonons,” Physical Review, vol. 148, no. 2, pp. 845–848, 1966. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2015 Sugandha Dogra Pandey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.