Updated: January 21, 2008
Vibrational Frequencies of Silicon Nanoparticles
The lowest few vibrational frequencies of a sphere of the anisotropic cubic crystal silicon are calculated from its bulk elastic constants.

   Solid silicon (Si) is a diamond structure crystal with cubic symmetry and density 2.329 g/cc. Its elastic constants at room temperature and pressure are C11 = 165.64 GPa, C12 = 63.94 GPa and C44 = 79.51 GPa [virginiasemi.com] [ioffe.rssi.ru] [Hall 1967]. The Zener anisotropy factor of Si is

A = 2 C44 / (C11 - C12) = 1.564

A C++ computer program cc3bmod.cpp or cc4g15.c is available [Murray 2002] to accurately compute the vibrational frequencies of an elastic sphere to within the order of 1% provided that it consists of a elastic material with cubic symmetry and Zener anisotropy factor of 2 or less.
   For comparison, if Si is assumed to be an isotropic elastic material with longitudinal sound speed sqrt(C11/ρ) = 8433 m/s and shear sound speed sqrt(C44/ρ) = 5843 m/s, the lowest frequency vibrational mode (TOR,l=2) of a sphere of radius R [Murray 2002] has ω R = 14640. For a 10 nm diameter sphere, the frequency is 15.52 cm-1.

Figure 1. Fourier transforms of lattice simulations of silicon spheres are shown. The initial disturbance of the lattice is random. For each plot, the horizontal axis is η = ω R/CS100. The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right.
(a) siran13.gif cc3bmod.cpp
(b) N=210000 cc4g13.c width=0.03 ET=2 random MT=5 Si YROWS=160 siranc1.gif
(c) siran35.gif cc3bmod.cpp
(d) 15f.gif cc4g16.c N 210000
2.9 5.1 0.03

Figure 2. Fourier transforms of lattice simulations of Silicon spheres are shown. The initial disturbance of the lattice is a velocity field corresponding to a vibrational mode of an isotropic elastic sphere. These modes can be spheroidal or torsional. For each plot, the horizontal axis is η = ω R/CS100. The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right. (C++ program cc3bmod.cpp )
(a) sitor2.gif
(b) sisph2.gif
(c) sisph0.gif
(d) sisph1.gif
(e) sisph3.gif
(f) sitor3.gif
(g) sitor1.gif
(h) 15a.gif
(i) 15b.gif
(j) N=210000 cc4g15.c width=0.05 YROWS=160 15e.gif
(k) cc4b16.cpp 15i.gif
(l) cc4b16.cpp 15h.gif

Table I. Lowest vibrational frequencies of a silicon sphere, (at room temp. and press.) from lattice simulation
type l η ω R ν (cm-1)
(2R=10 nm)		notes
tor 2 2.125 12420 13.18N=210000 cc4g13.c width=0.03 siranc1.gif
sph 2 2.130 12450 13.21 
tor 2 2.480 14490 15.39 
sph 2 2.61 15250 16.19 
sph 1 2.96 17300 18.36 
sph 0 3.825 22350 23.73N=210000 cc4g14.c width=0.03 15b.gif
tor 1 5.48 32020 34.00N=210000 cc4g15.c width=0.05 15e.gif

References:


Daniel B. Murray, "Vibrational Frequencies of an Elastic Sphere" 2002 (link to article)

D. B. Murray "Eight Point Force Molecular Dynamical Estimates of Vibrational Frequencies of an Isotropic Elastic Sphere" 2002 link to article

On-line table of mechanical properties of silicon:
www.ioffe.rssi.ru/SVA/NSM/Semicond/Si/mechanic.html
www.virginiasemi.com/pdf/Basic Mechanical and Thermal Properties of Silicon.pdf (also has variation of elastic constants with temperature)  

Original paper reporting elastic constants of silicon:
J. J. Hall, Phys. Rev. volume 161 (1967) page 756

K R Patton & M R Geller "Phonon Spectrum in a nanoparticle mechanically coupled to a substrate" Journal of Luminescence volume 94-95 (2001) pages 747-570 pdf :: ::
Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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