The Open Condensed Matter Physics Journal


ISSN: 1874-1924 ― Volume 4, 2013

Effective Linewidth in Raman Spectra of Titanium Dioxide Nanocrystals

M Salis1, P.C Ricci*, 1, A Anedda1, 2
1 Department of Physics, University of Cagliari, s.p. 8 Km 0.7, 09042 - Monserrato, Cagliari, Italy
2 LIMINA Laboratory, University of Cagliari, s.p. 8 Km 0.7, 09042 - Monserrato, Cagliari, Italy


Raman spectra of nanocrystals titanium dioxide are discussed and the correlation between the band shape of the allowed A1g Raman mode and the crystals dimensions is discussed. Data on Raman spectra are reconsidered in the framework of a modified "hard confinement" model (MHC). The proposed model is based on the idea of using an effective linewidth parameter, which is a function of the effective dimension of the nanostructure, in spite of the intrinsic Raman band linewidth.

The comparison with standard hard confinement model reveals better agreement with the experimental results for the MHC model up to 6 nm. Moreover, the analysis permits to improve the knowledge of the phonon dispersion curve as well as the intrinsic Raman bulk parameters. An analytical form of the size-dependent peak-position in nanocrystals, useful for an approximated size estimation, has been explicated. The general structure of the model permits to extended the MHC to other nano-sized materials.

Article Information

Identifiers and Pagination:

Year: 2009
Volume: 2
First Page: 15
Last Page: 18
Publisher Id: TOCMPJ-2-15
DOI: 10.2174/1874186X00902010015

Article History:

Received Date: 10/10/2008
Revision Received Date: 06/4/2009
Acceptance Date: 6/4/2009
Electronic publication date: 28/4/2009
Collection year: 2010

© Salis et al.; Licensee Bentham Open.

open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.

* Address correspondence to this author at the Department of Physics, University of Cagliari, s.p. 8 Km 0.7, 09042 - Monserrato, Cagliari, Italy; E-mail:


Agglomerates of nanosized TiO2crystals are promising materials for electrochemical, photochemical and photovoltaic applications [1-4]. Among the several techniques used to characterize these materials, Raman spectroscopy plays a special role in the investigation of size effects on crystal vibrational properties in the nanometer scale. Depending on the crystal size nanocrystals undergo strain or non-stoichiometry which can significantly affect the Raman spectra [3, 4]. Several reports on titanium dioxide nanocrystals show that Raman shift in this material is mainly due to phonon confinement (PC) effect [1-3]. Different confinement models have been used to fit experiments with alternate successes [1-4], where the main difficulties in matching experiments can be ascribed to model oversimplifications. In this connection, it has been shown that by using the proper model parameters it is possible to greatly improve fits of theoretic Raman spectra to experimental peak-shapes [5]. These results have been obtained by modifying the "hard confinement" (HC) model, where the intrinsic Raman linewidth is replaced by a proper effective one. The latter may be affected by several unpredictable effects such as interactions between defects and band modes [6], changes of the anharmonic coupling factors [7], anisotropy [8, 9] and/or phase mixing [3]. Besides, although Raman spectra are surely subject to PC effects, they can show structures significantly scattered from the predictions based on the ideal PC problem. This difference may raise from the proper figures of the effective linewidth parameter utilized in PC models.

Early and recent data are investigated in this paper. Our goal is to show that for TiO2 nanocrystals the variations in Raman-peak positions due to the occurrence of linewidth broadening should be accounted for by the modified HC (MHC) model. In Section 2 a curve model for the size-dependent effective linewidth will be derived on from the averaged anisotropy function [8, 9]. This allows an approximated analytical form of the peak-position dependence on particle size that features all the relevant parameters. Deviations of the experimental figures from model predictions are further considered in Section 3.


In a perfect infinite crystal only phonons close to the center of the Brillouin zone (BZ) contribute to inelastic scatterings of incident radiations. When crystal sizes range in the nanometer scale, a larger portion of the BZ is allowed to effectively participate in scattering processes due to weakening of the rule q0≈0 [10, 11]. The extension of such a region in the quasi-momentum space is determined by the confinement of the q0=0 phonon wave function [11, 12]. Formally, this can be explained by considering the modulation of the phonon wave function in the infinite crystal with a suitable "weighting" function [11]. It is possible to calculate the first-order Raman spectrum of nanosized crystals on expansion by Fourier integrals (except for a constant factor) from

where C(q) stands for Fourier coefficients of the q0=0 phonon wave function,ω for the phonon dispersion curve and Г0for the intrinsic Raman band linewidth [11]. Some simplifications are required to handle eq. (1). A drastic but commonly accepted assumption considers an isotropic dispersion in a spherical BZ [11] and the function ω(q) represents an averaged dispersion curve [13]. As for phonon confinement (PC), the model which received the most credit assumes a Gaussian weighting function (hard confinement, HC) for spherical nanocrystals [1, 2, 8, 9, 13].

Accordingly, eq. (1) can be rewritten as

where L is the correlation length in the crystal, which can be assumed as the diameter of the nanoparticles, and “a” stands for the lattice parameter.

From the numerical inspection of eq. (2) it can be found that, as long as the quadratic approximation holds (A stands for a suitable curve parameter), the Raman peak-position can be calculated with a good approximation by means of [14].

where . It is possible to appreciate the effect of the effective linewidth on the peak position by replacing Г0 with Г so that

where sgnA stands for the sign of A. The latter equation clearly shows that linewidth can become important in determining the peak position when the size of nanocrystals ranges within a few nanometers (for example, if L=12axthen ).

Previous works propose to replace the effective linewidth Г0 with a function , where was referred to as the anisotropy correction which accounts for the spreading of the phonon band over the BZ to explain the anomalous broadening of Raman peak-width. This correction is defined as twice the root mean square deviations of the actual from the average [8, 9]. However, there may be other reasons for dealing with a function Г(q). For example, in bulk materials quasi-stationary modes originated by interactions between defect and band modes enlarge the frequency distribution around unperturbed modes with resonance width depending on quasi-momentum, thus affecting relaxation time in phonon-photon (or -particle) interactions [6]. The large defect disorder, which is possible in high energy state crystals, may smear the broadening over all the band modes. However, inspection of the Raman peak shapes suggests that Гo can actually be replaced by a parameter Гwhich partially depends on crystal size [5]. Thus, rather than a function Г(q), we may consider its average over the phonon wave function.

In order to account for quasi-momentum dependence in Г(q) we propose to consider the function as

which presents the same analytical structure as until phonon curves in the anisotropic BZ are considered in the quadratic approximation.

The working hypothesis is

where the dependence on crystal size of Fourier coefficients has been explicated.

When crystal size is sufficiently larger than crystal parameter a, it is possible to calculate the integrals in eq. (6) extending the upper limit to infinity:

where is to be estimated from comparison with experiments and itself integrates all the effects contributing to the enlargement of the effective linewidth. It is worth to note that eq. (7) itself represents an assumption that could only roughly estimate the function dealt with.


In Fig. (1) data on peak-widths are reported from refs. [1, 2, 5]. The peak-widths calculated from the MHC model by using eq. (7) in eq(2) are reported as well (solid curve). According to refs [1, 2, 5], the dispersion curve used for the calculations in case of Anatase phase is:

with , and ; for the curve Г(L) we used [1, 2, 5] and . It is to be pointed out that parameter λis established by fitting peak-width predictions to experimental data in the rangeL16nm. With this choice, large departures from experiments in the upper size range can be observed, but here is small and the peak-position is weakly affected by peak-width.

Peak positions were calculated by using the “anatase” values”: and the results are shown in Fig. (2, black solid curve obtained from eq. 2) in which Гo is replaced by Г Г(L)where, by comparison, the curve predicted by the non-modified HC model is reported as well (blue solid curve obtained from eq. 2).

The curves obtained from the two model are very similar at large nanoparticle dimensions (about 12 nm), while sensible differences appear where phonon confinement effect plays a major role. In this range the MHC model fits the experimental result much better than the HC model (Fig. 2). However, it is worth pointing out that both the models do not fit adequately the experimental data for particles of large dimensions, where the choice of the intrinsic Raman bulk parameters and the phonon dispersion curve are critical. Therefore the fit has been performed with small variation in the phonon dispersion curve parameters (eq. 8); the best result has been obtained for:, with and the parameterλ of eq. 7 becomes(red solid curve). The obtained value for the intrinsic Raman frequency of the A1g Raman mode is still under the spectral resolution of the experimental data. With respect to the previous case we obtained a better fit in the large size range also for the HC model, but we also observed that it failed in the low nanoparticle diameter range. The peak-widths calculated from the MHC model are not affected by these variation (Fig. 1).

Fig. (1)

Experimental and theoretical peak-widths. The latter were calculated by replacing the intrinsic linewidth with the effective one defined in eq. (7).

From comparison of data shown in Fig. (2) we note the presence of some discrepancies between model predictions and experiments that cannot be explained by linewidth broadening. Thus, to go deeper into this analysis, it is convenient to make an estimation, even though rough, of the behaviour of A (to avoid confusion, it will be referred to as AR) as a function of the nanoparticle dimension. This can be made by taking into account that when peak-width is large the peak shape features an almost Lorentz-shaped. Thus, we retrieved the searched-for parameters by fitting the MHC model to the Lorentz-like curves having the actual Raman peak-widths and -positions. This was done by using a standard Levenberg-Marquardt procedure [5, 15]. As for data in ref [5] we are able to report the fit results of the actual Raman curves. To simplify the discussion all the retrieved parameters are obtained only for the case where the fitting curve for both models gives the best results. Table 1 reports the parameters calculated for different nanoparticle dimensions: stands for the actual peak position, for the actual peak width, Г and ARfor the retrieved effective linewidth and phonon curve parameter (eq. 8 is used in eq. 2), respectively.

Fig. (2)

Experimental and theoretical peak-positions. The latter were calculated by replacing the intrinsic linewidth with the effective one defined in eq. (7). Circles show the curve calculated by means of the analytical form (3) and eq. (7).

Table 1

Actual Crystal Size L, (Mean) Peak-Position ω and -Width Г Considered in Figs. (1, 2).Гand AR Stand for the Retrieved Effective Linewidth and Phonon Curve Parameters Obtained by Using a Standard Levenberg-Marquardt Fitting Procedure

It is worth pointing out that in spite of the effective linewidths, which appear to increase almost monotonically as crystal size decreases, parameter AR shows a non-regular behaviour dependent on the actual sample. However, its value oscillates, sample by sample, around which is the expected value from the chosen . The results suggest that the mechanism determining the Г broadening and AR deviations are poorly correlated, being the former strongly connected to the effective nanoparticle dimension, while the latter ( AR) to the phonon dispersion curve. Finally, on the basis of the previous results we are able to give an analytical form of the peak-position dependence on particle size. Indeed, from eq. (3) with, and with we obtain a curve (circles) very close to the one predicted by MHC model. The choice of giving an analytical form of ωp(L)to titanium dioxide nanocrystals may be useful in characterizations of samples grown by means of properly standardized preparation methods. In particular, in the perspective of industrial applications, it could allow an approximated size estimation based on few and easily obtained experimental data.

Besides, the results show light deviation from sample to sample, and this work points out that an effective linewidth parameter, which is function of the dimension of the nanostructure, should be considered for a correct interpretation of nanoparticle Raman spectra. The model has been applied only on titanium dioxide nanocrystals but can be extended to other nano-sized materials. Further analysis on other materials are a mandatory task.


A modified hard confinement model has been proposed. We have shown that for a better matching of the HC model with Raman spectra of nanocrystals a suitable definition of an effective linewidth is required. The model is focused on the replacement of the intrinsic Raman linewidth in the HC model with a function of the actual dimension of the nanostructure. The model has been successfully applied to titanium dioxide nanocrystals and further information about the phonon dispersion curve has been obtained. An analytical form of the developed model, useful for a approximated size estimation, has been explicated.


Track Your Manuscript:


"Open access will revolutionize 21st century knowledge work and accelerate the diffusion of ideas and evidence that support just in time learning and the evolution of thinking in a number of disciplines."

Daniel Pesut
(Indiana University School of Nursing, USA)

"It is important that students and researchers from all over the world can have easy access to relevant, high-standard and timely scientific information. This is exactly what Open Access Journals provide and this is the reason why I support this endeavor."

Jacques Descotes
(Centre Antipoison-Centre de Pharmacovigilance, France)

"Publishing research articles is the key for future scientific progress. Open Access publishing is therefore of utmost importance for wider dissemination of information, and will help serving the best interest of the scientific community."

Patrice Talaga
(UCB S.A., Belgium)

"Open access journals are a novel concept in the medical literature. They offer accessible information to a wide variety of individuals, including physicians, medical students, clinical investigators, and the general public. They are an outstanding source of medical and scientific information."

Jeffrey M. Weinberg
(St. Luke's-Roosevelt Hospital Center, USA)

"Open access journals are extremely useful for graduate students, investigators and all other interested persons to read important scientific articles and subscribe scientific journals. Indeed, the research articles span a wide range of area and of high quality. This is specially a must for researchers belonging to institutions with limited library facility and funding to subscribe scientific journals."

Debomoy K. Lahiri
(Indiana University School of Medicine, USA)

"Open access journals represent a major break-through in publishing. They provide easy access to the latest research on a wide variety of issues. Relevant and timely articles are made available in a fraction of the time taken by more conventional publishers. Articles are of uniformly high quality and written by the world's leading authorities."

Robert Looney
(Naval Postgraduate School, USA)

"Open access journals have transformed the way scientific data is published and disseminated: particularly, whilst ensuring a high quality standard and transparency in the editorial process, they have increased the access to the scientific literature by those researchers that have limited library support or that are working on small budgets."

Richard Reithinger
(Westat, USA)

"Not only do open access journals greatly improve the access to high quality information for scientists in the developing world, it also provides extra exposure for our papers."

J. Ferwerda
(University of Oxford, UK)

"Open Access 'Chemistry' Journals allow the dissemination of knowledge at your finger tips without paying for the scientific content."

Sean L. Kitson
(Almac Sciences, Northern Ireland)

"In principle, all scientific journals should have open access, as should be science itself. Open access journals are very helpful for students, researchers and the general public including people from institutions which do not have library or cannot afford to subscribe scientific journals. The articles are high standard and cover a wide area."

Hubert Wolterbeek
(Delft University of Technology, The Netherlands)

"The widest possible diffusion of information is critical for the advancement of science. In this perspective, open access journals are instrumental in fostering researches and achievements."

Alessandro Laviano
(Sapienza - University of Rome, Italy)

"Open access journals are very useful for all scientists as they can have quick information in the different fields of science."

Philippe Hernigou
(Paris University, France)

"There are many scientists who can not afford the rather expensive subscriptions to scientific journals. Open access journals offer a good alternative for free access to good quality scientific information."

Fidel Toldrá
(Instituto de Agroquimica y Tecnologia de Alimentos, Spain)

"Open access journals have become a fundamental tool for students, researchers, patients and the general public. Many people from institutions which do not have library or cannot afford to subscribe scientific journals benefit of them on a daily basis. The articles are among the best and cover most scientific areas."

M. Bendandi
(University Clinic of Navarre, Spain)

"These journals provide researchers with a platform for rapid, open access scientific communication. The articles are of high quality and broad scope."

Peter Chiba
(University of Vienna, Austria)

"Open access journals are probably one of the most important contributions to promote and diffuse science worldwide."

Jaime Sampaio
(University of Trás-os-Montes e Alto Douro, Portugal)

"Open access journals make up a new and rather revolutionary way to scientific publication. This option opens several quite interesting possibilities to disseminate openly and freely new knowledge and even to facilitate interpersonal communication among scientists."

Eduardo A. Castro
(INIFTA, Argentina)

"Open access journals are freely available online throughout the world, for you to read, download, copy, distribute, and use. The articles published in the open access journals are high quality and cover a wide range of fields."

Kenji Hashimoto
(Chiba University, Japan)

"Open Access journals offer an innovative and efficient way of publication for academics and professionals in a wide range of disciplines. The papers published are of high quality after rigorous peer review and they are Indexed in: major international databases. I read Open Access journals to keep abreast of the recent development in my field of study."

Daniel Shek
(Chinese University of Hong Kong, Hong Kong)

"It is a modern trend for publishers to establish open access journals. Researchers, faculty members, and students will be greatly benefited by the new journals of Bentham Science Publishers Ltd. in this category."

Jih Ru Hwu
(National Central University, Taiwan)

Browse Contents

Webmaster Contact:
Copyright © 2021 Bentham Open