Updated: January 23, 2008
What's shaking in low frequency Raman
scattering off of silicon nanoparticles in SiO2?

The thermal motion of the surface of a spherical Si nanoparticle embedded in a SiO2 matrix is calculated, and compared to experimental measurements of low frequency Raman scattering. Good agreement is obtained with model calculations. It is suggested that vibrations of the nanoparticle itself are not the cause of the Raman peaks. Rather, it is vibrations of the SiO2 matrix that are being seen.

   Features of the low frequency Raman spectrum of metal (Ag, Au) and semiconductor (Si, CdS, CdSe) nanoparticles embedded in a solid glass matrix are qualitatively explained in terms of the free vibration spectrum of an isolated elastic sphere. In particular, the frequencies vary as (1/d) where d is the nanoparticle diameter. However, a number of experiments give Raman peak locations that are lower than any of the predicted free vibration frequencies. Some earlier calculations have incorporated the elastic properties of the matrix, but these have usually calculated the frequency and the damping, but have not provided the shape of the Raman spectrum.
   Details of the calculation are reported elsewhere. Here is a summary: The system under consideration is a single nanosphere of Si embedded in an infinite SiO2 glass matrix. Raman scattering detects fluctuations in the polarizability tensor. It is assumed here that these fluctuations are as a result of bulk thermal vibrations of the system. Ignoring quantum corrections, every phonon mode should have the same average energy (kBT). Assuming a temperature of 300 K, this energy is kBT = 4×10-21 J. The quantum spacing between levels of a phonon mode is hf, where h is Planck's constant and f is the frequency of the mode. The average occupation number of a phonon mode is approximately kBT / (hf). For a phonon of frequency 20 cm-1, this is 10, which is large enough for the mode to be treated classically.
   The incoming amplitude of the wave, A, at frequency ω has been calculated based on this (md30.htm).
   The boundary value problem is then solved (using Borland Turbo C++ programs scp78.c, scp79.c, scp80.c) to get the amplitude of the vibration inside the sphere. There are two amplitudes inside the sphere for the case of spheroidal vibrations: B and C. Otherwise there is a single amplitude inside the sphere, B. These amplitudes are complex numbers. What is plotted is the squared displacement of the surface of the nanoparticle. The reason for squaring is because Raman scattering intensity would vary as the square of the displacement. This displacement has r, θ and φ components, plotted in different colors in the figures below. The angular dependence of the displacement is not shown.
   This same approach has also been applied to the cases of CdS nanoparticles in a GeO2 matrix and Cd(S,Se) nanoparticles in a glass matrix.
   A separate problem which I do not address here is how to calculate the actual Raman spectrum. This is not straightforward since the polarizability of the nanoparticle will change as the nanoparticle vibrates, but it is not clear quantitatively how much.
   The same vertical scale is used in all of these figures.

Figure 1.   l = 2 spheroidal mode
The white circles are Raman spectrum data points from FIG. 2(a) in Fujii et al. Phys. Rev. B 54, (1996) 8373-8376, vertically rescaled for best fit with the red line.
(scp78b2.c, 36c.gif)

Figure 2.   l = 2 spheroidal mode
The white circles are Raman spectrum data points from Fig. 7 (a) in M. Pauthe, E. Bernstein, J. Dumas, L. Saviot, A. Pradel and M. Ribes, J. Mater. Chem. 9 (1999) 187-191, (www.rsc.org ::) vertically rescaled for best fit with the red line.
(scp78f.c, 36a.gif)

Figure 3.   l = 2 spheroidal mode
The white circles are Raman spectrum data points from Fig. 7 (b) in M. Pauthe, E. Bernstein, J. Dumas, L. Saviot, A. Pradel and M. Ribes, J. Mater. Chem. 9 (1999) 187-191, (www.rsc.org ::) vertically rescaled for best fit with the red line.
(scp78g.c, 36b.gif)

   The widths of the theoretical peaks and to some extent even the shapes are similar to the experimental data. It should be noted that the experiment is done with a sample with a distribution of particle sizes. The theoretical results suggest that the Raman spectrum would be essentially the same even if an experimental sample could be made with all particles the same size.
   The assumed density and longitudinal and transverse speeds of sound for the Si and the SiO2 are as shown in the figures. In these calculations, the Si is assumed to have isotropic elasticity. But in real silicon, the speed of sound depends on the direction of propagation. The speeds along the [100] crystal direction have been used here.
   Some insight into the nature of the physical motions associated with the peak can be indirectly obtained by changing the mechanical properties of the nanoparticle.

Figure 4.   Effect of Nanoparticle Mechanical Properties
The white circles are Raman spectrum data points from Fig. 7 (a) in M. Pauthe, E. Bernstein, J. Dumas, L. Saviot, A. Pradel and M. Ribes, J. Mater. Chem. 9 (1999) 187-191, (www.rsc.org ::)
(a) Nanoparticle is harder than silicon (b) Nanoparticle is heavier than silicon (c) Nanoparticle is harder and heavier than silicon

   The above figures demonstrate that the density or the hardness of the nanoparticle can be increased significantly without shifting the position of the maximum very much. From this, it can be concluded that the nanoparticle itself is not greatly involved in the motion leading to this peak. Thus, low frequency Raman scattering experiments on silicon nanoparticles in a SiO2 matrix may actually be probing the physical properties of the SiO2. The effect is completely unrelated to the free vibrations of a Si nanosphere.
   Another way of predicting the frequency is by using the decaying complex frequency approach [L. Saviot]. In the presence of the matrix, extra modes appear that are not present in the free sphere case. In particular, the lowest frequency l = 2 spheroidal mode has a frequency that depends mostly on the elastic properties of the matrix, and only weakly on the properties of the nanoparticle.
   Having said that, there are lots of examples of low frequency Raman scattering from nanoparticles that are accurately predicated and correctly explained in terms of vibrations of the particle, and well approximated by the free sphere situation.

References:

M. Fujii, Y. Kanzawa, S. Hayashi and K. Yamamoto, "Raman scattering from acoustic phonons in Si nanocrystals" Phys. Rev. B vol. 54, pages 8373-8376 (1996).

M. Pauthe, E. Bernstein, J. Dumas, L. Saviot, A. Pradel and M. Ribes "Preparation and characterisation of Si nanocrystallites embedded in a silica matrix" Journal of Materials Chemistry (J. Mater. Chem.) volume 9 (1999) pages 187-191. (pdf file on-line: www.rsc.org ::)

Lucien Saviot "Si nanoparticles vibration eigenmodes" noee.u-bourgogne.fr

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

For a list of related articles click here.

"Microscopic calculations on Raman scattering from acoustic phonons confined in Si nanocrystals" Jian Zi, Kaiming Zhang, Xide Xie Phys. Rev. B, Aug. 15, 1998 arXiv:cond-mat/9807293 v1 22 Jul 1998 arxiv.org/pdf/cond-mat/9807293 - c:\cofrest\9807293.pdf [email protected] http://www.fudan.edu.cn/index_ch.php
Ding Haojiang and Chen Weiqiu, "Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium"
Int. J. Solids Structures Vol. 33, No. 18, pp. 2575-2590, 1996 ::
M. Montagna and R. Dusi, "Raman scattering from small spherical particles" Phys. Rev. B 52, 10080 (1995) - about matrix effect, small influences.
"Raman scattering from fractals. Simulation on large structures by the method of moments" G.Viliani, R.Dell'Anna, O.Pilla, M.Montagna, G.Ruoco, G.Signorelli, V.Mazzacurati http://arxiv.org/pdf/cond-mat/9504039 ::
Transverse acoustic nature of the excess of vibrational states in vitreous silica http://arxiv.org/pdf/cond-mat/0209519 ::
Lingjun Wang, Guanghong Wei, Jian Zi "A planar force-constant model for phonons in wurtzite GaN and AlN: Application to hexagonal GaN/AlN superlattices" http://arxiv.org/pdf/cond-mat/9812293 ::
Raman scattering by electron-hole excitations in silver nanocrystals Authors: H. Portales, E. Duval, L. Saviot, M. Fujii, M. Sumitomo, S. Hayashi To be pub.in PRB cond-mat/0101471 http://arxiv.org/pdf/cond-mat/0101471 :: "Phonons in a nanoparticle mechanically connected to a substrate" K. Patton and M. Geller (Dec. 30, 2002 version) :: V. L. Gurevich and H. R. Schober, Phys. Rev. B 57, 11295 (1998).
Sergey Bastrukov and Pik-Yin Lai, "Oscillatory behavior of a flexible cavity in an elastic medium" J. Moscow Phys. Soc. Vol. 9 (1999) pages 71-76 ::
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