Mechanism of low-frequency Raman scattering from the acoustic vibrations of dielectric nanoparticles M. Mattarelli,* M. Montagna, and F. Rossi Dipartimento di Fisica, CSMFO group, Università di Trento, Via Sommarive 14, I-38050 Trento, Italy A. Chiasera and M. Ferrari IFN-CNR, Istituto di Fotonica e Nanotecnologie, CSMFO group, Via Sommarive 14, I-38050 Trento, Italy Received 5 September 2006; published 18 October 2006 The depolarization ratio of the quadrupolar vibrations and the relative intensity of the symmetric l =0 and quadrupolar l = 2 acoustic vibrations in the Raman spectra of some dielectric nanocrystals has been calculated. A dipole-induced-dipole model can account for the depolarized spectra from quadrupolar vibrations, but cannot be at the origin of the polarized peak from the symmetric vibration. Bond polarizability seems to be the main physical mechanism at the origin of Raman scattering from these modes. The study indicates that the quadru- polar modes or symmetric modes dominate the spectra when the dipole induced dipole or bond polarizability are more important, respectively. This result explains why semiconductor nanoparticles with covalent bonds show intense symmetric scattering, and fluoride crystals with ionic bond show Raman scattering from quadru- polar modes, and why in oxide crystals the two modes show comparable Raman activity. A comparison of the spectra of titania, zirconia, and hafnia nanocrystals offers further support to the model. DOI: 10.1103/PhysRevB.74.153412 PACS numbers: 78.30.-j, 63.22.+m, 77.84.-s Low-frequency Raman scattering is a widely used experi- mental technique for the study of the vibrational dynamics of metallic, semiconductor or dielectric nanoclusters, usually embedded in a glass. 1–9 Most theoretical approaches for the calculation of the acoustic vibrational dynamics of spheroi- dal clusters are based on the work of Lamb, which found the vibrations of a free homogeneous sphere. 10 The modes are classified in torsional and spheroidal, both labeled by three indices lmn, which describe the angular lm and radial n dependence of the displacements. As shown by Duval on the basis of simple symmetry arguments, only the spheroidal symmetric l =0 and quadrupolar l =2 spheroidal modes are Raman active. 11 Furthermore, the l =0 modes give a po- larized Raman spectrum, whereas the l = 2 modes give depo- larized spectra, allowing to distinguish the nature of the vi- brations by a comparison of the VV and HV spectra. Recently, a paper appeared with the claim that the l =0 and l = 2 spheroidal modes are not Raman active because of an odd displacement field. 12 This wrong criterium does not con- sider that even modes have usually odd displacements, as for example, the vibration of the oxygen molecule or the sym- metric stretching of the CO 2 molecule, which are Raman active even modes, having odd displacements. In any case, the explicit calculation of the average strain starting from the potential, deriving the displacement and again deriving the strain components, shows that only the l =0 and l =2 sphe- roidal modes are Raman active. 13 There are no general rules that indicate both the relative intensity of the symmetric and quadrupolar Raman peaks, appearing in the VV spectrum, or the depolarization ratio DR 2 = I HV / I VV for the quadrupolar modes. In fact, in some systems as silver, gold and PbF 2 , the quadrupolar vibrations dominate the Raman spectrum, in other systems, as CdS, Si, Ga 2 O 3 , and HfO 2 , the symmetric vibrations dominate. 3–6,8,9 In TiO 2 nanocrystals both modes are observed with similar intensities. 7 Different depolarization ratios for the quadrupo- lar vibration have been measured, ranging from about 0.3 for silver to about 0.7 for TiO 2 . 3,7 In the case of metal particles, the resonance with the sur- face plasmon excitations produces intense low-frequency de- polarized Raman scattering. 3,14,15 Here, we will limit our study to dielectric nanoparticles having electronic transition far from the excitation frequencies used in nonresonant Ra- man spectroscopy. The space-time changes of the polariza- tion are usually separated in two contributes. 16 The first one is related to the density fluctuations, which cause inelastic neutron scattering and usually most of the VV Brillouin scat- tering, due to longitudinal acoustic phonons. The second contribution is due to changes of the dipole induced dipole DID effects, caused by the motion, and to changes of the bond polarizability BP with the change of the atomic dis- tances. The induced effect contribute to the polarized Bril- louin peak due to longitudinal phonons and cause the depo- larized Brillouin peak, due to transversal phonons, and the disorder induced low-frequency Raman scattering in glasses or disordered crystals. The scattering mechanism in the low- frequency Raman scattering from the acoustic vibrations of nanoparticles is something in between to those of Raman and Brillouin scattering in bulk systems. If the particle is much smaller than the wavelength of the exciting light source, the mechanism of scattering due to density fluctuations is not active. All polarization elements are excited in phase and the particle behaves as a molecule, which can be described by an effective polarizability and by its derivatives with respect to the coordinates of the normal modes. However, the particle is sufficiently big to support acoustic modes with wavelengths much higher than the atomic sizes. Therefore, the effective polarizability tensor and the intensity of the Raman band of the active vibrations can be calculated with the method used for the calculation of the intensity of the Brillouin scattering. The important difference is that no q dependence is present, so that isotropic scattering is observed instead of the Bragg- PHYSICAL REVIEW B 74, 153412 2006 1098-0121/2006/7415/1534124 ©2006 The American Physical Society 153412-1