Vibrational Frequencies of a Silicon Sphere Struck on a Hard Substrate

Updated: January 23, 2008
Vibrational Frequencies of a Silicon Sphere Stuck on a Hard Substrate

   All of these simulations are for a sphere with the anisotropic elastic properties of crystalline silicon: C₁₁ = 166 GPa, C₁₂ = 64 GPa, C₄₄ = 80 GPa and ρ = 2.33 g/cc. The vibrational frequencies for a free silicon sphere were calculated previously md15.htm. Here, the silicon sphere of radius R is fixed to a hard substrate. The substrate surface corresponds to the x y plane. Either the [001] (Figure 1), [110] (Figure 2) or [111] (Figure 3) crystal axis of the silicon is parallel to the z axis. The center of the sphere is located at x = 0, y = 0, z = 0.5 R. All of the atoms in the crystal for which z<0 are frozen in place. That is, their positions never change and their velocities are always zero. The other atoms are free to move. This is an extremely simplified model of the adhesion of a silicon nanoparticle to a silicon substrate. More detailed adhesion models could easily be incorporated into the calculation.
   The method to calculate the vibrational frequencies of this sphere is to give it an initial disturbance and then Fourier transform the subsequent motion.
   The "atoms" in the crystal (point particles) are located on a simple cubic lattice. The unit cells are 1 � 1 � 1. The radius, R, of the sphere is made as large as possible. The number of atoms is approximately N = (4/3)πR³. Computer memory limits the number of atoms within the sphere to 9000 or less. In order to extrapolate the vibrational frequencies in the limit of many atoms, simulations are carried for R varying from 3 to over 12.
   The dimensionless frequency η = ω R / C_tran100 is used. Here, "R" is the radius of the sphere in metres. C_tran100 is the transverse speed of sound along the [100] axis. It is equal to sqrt(C₄₄/ρ) = 5859 m/s.

Figure 1. [100] orientation (C++ computer program used: cc3bmod2.cpp)

(a) In this case, the sphere is given random initial disturbances, so that all vibrational modes are excited. The lowest mode has a frequency of η = 0.77.
(cc3bmod2.cpp 39g.gif)

The next figure shows the case of a silicon nanosphere where the [110] crystal axis is normal to the surface.

Figure 2. [110] orientation (C++ computer program used: cc3bmod2.cpp)

(a) In this case, the sphere is given random initial disturbances, so that all vibrational modes are excited. The lowest mode has a frequency of η = 0.77. Note: The orientation really is [110] and not [100]

(b) The sphere is stretched along the z axis and then released. The lowest frequency mode thus excited has η = 1.75

(d) The sphere is twisted around the z axis and then released. The lowest frequency mode thus excited has η = 1.15

(e) The sphere is pushed to one side (in the x direction) and then released. The lowest frequency mode thus excited has η = 0.75

The next figure shows the case of a silicon nanosphere where the [111] crystal axis is normal to the surface.

Figure 3. [111] orientation (C++ computer program used: cc3bmod3.cpp)

(a) In this case, the sphere is given random initial disturbances, so that all vibrational modes are excited. The lowest mode has a frequency of η = 0.77.
(cc3bmod2.cpp 39h.gif)

(b) In this case, the sphere is given an initial stretch along the x-axis. The lowest mode has a frequency of η = 2.55.
(cc3bmod3.cpp 39i.gif)

(c) In this case, the sphere is bent in the x direction. The lowest mode has a frequency of η = 0.75.
(cc3bmod3.cpp 39j.gif)

   In summary, all of the low lying modes of this sphere are identifiable. The lowest frequency mode of vibration corresponds to the nanosphere shaking back and forth along the x-axis. Presumably, the actual frequency of this mode would be highly sensitive to the details of the adhesion between the silicon sphere and the substrate.
   The next highest mode, at η = 1.15, corresponds to twisting motion of the sphere, and seems unlikely to be responsive to either Raman or infrared scattering.
   The mode at η = 2.33 ([100],[110]) or 2.55 ([111]) seems to be a good candidate for observation through Raman scattering. This mode corresponds to stretching along an axis lying in the x-y plane. It is similar to the l = 2 spheroidal mode for a free sphere. (Note that the l = 2 spheroidal mode of a free silicon sphere has frequency η = 2.12 or 2.62 -- see Table I in md15.htm) Furthermore, this mode does not have any other modes lying close to it. In fact, it is the easiest mode to see in figure 1(a).
   These simulations do not address the issue of damping of these modes through connection to the substrate. However, it is apparent that the mode at η = 2.33 and 2.55 would have a relatively weak coupling to the substrate. So it should be lightly damped and have a narrow linewidth. Furthermore, its frequency should not be sensitive to the details of the adhesion of the nanosphere to the substrate.

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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