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


Molecular dynamics simulations are performed for elastic objects with the anisotropic elasticity of crystalline silicon. Simulations of a sphere fixed on a planar substrate allow vibrational frequencies to be found. Broadening of some of the vibrational modes is minimal even when the sphere is quite strongly coupled to the substrate.

   All of these simulations are for objects 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). The case of a silicon sphere attached to a rigid substrate (md39.htm) yields a spectrum of frequencies, but since there is no energy dissipation mechanism, this calculation yielded no information about the damping (i.e. lifetime) of the modes.
Figure 1. 7468 "atoms" are used to construct a model of a silicon nanosphere on a hemispherical silicon substrate.
Hemisphere radius = 14.2
Nanosphere radius = 7.1
The center of the nanosphere is 6.4 above the planar surface. The planar surface is normal to the [111] crystal axis.
(cc3bdmp3.cpp 40m.gif)
Click to see animation.
   Here, a silicon nanosphere of radius Rp is attached to a planar substrate slab also made of silicon. Ideally, the substrate is infinitely large, but in this calculation the substrate is a hemisphere with its center at (0,0,0) and radius RHS. The flat surface of the hemisphere lies in the x y plane. The geometrical center of the nanosphere is located at x = 0, y = 0, zpc = (PCZ)(RHS). Both the nanosphere and the hemisphere share the same crystal lattice. In particular, their crystal directions match. In the simulations, the normal to the flat hemisphere surface corresponded to either to [100] direction or the [111] direction.
   Energy absorbing boundary conditions are applied at the curved surface of the hemisphere so as to approximate the presence of an infinitely large substrate. To do this in a simple though inexact way, atoms on the boundary of the hemisphere are subjected to a linear velocity dependent damping force F = -b v where b was chosen to maximize the rate of energy dissipation in the system. An alternate method for selecting b is to approximate the wavelength as much smaller than the hemisphere radius. In that case, plane waves are normally incident on the hemisphere surface. Perfect absorption occurs when the damping force per unit area matches the specific acoustic impedance, z = ρ c, where c is the speed of sound, either longitudinal or transverse. Both approaches give approximately the same value of b.
   The method to calculate the vibrational frequencies of this sphere is to give it an initial disturbance and then Fourier transform the subsequent motion. Because of the possibility of damping, the Fourier peaks can be broadened.
   The "atoms" in the crystal (point particles) are located on a simple cubic lattice. The unit cells are 1 � 1 � 1. The radius, RHS, of the hemisphere is made as large as possible. The number of atoms, N , is slightly more than (2/3)π(RHS)3. Computer memory limits the total number of atoms used to construct the hemisphere and nanoparticle to 9000 or less. In order to extrapolate the vibrational frequencies in the limit of many atoms, simulations are carried out for RHS varying from 3 to 14.
   The dimensionless frequency η = ω RHS / Ctran100 is used. Here, "RHS" is the radius of the hemisphere in metres. Ctran100 is the transverse speed of sound along the [100] axis. It is equal to sqrt(C44/ρ) = 5859 m/s.
   The parameter PCZ (unfortunately called "cmz" in the figures) is varied in Figure 2 to demonstrate the broadening of the mode frequencies as the coupling to the substrate is increased.

Figure 2.     (C++ computer program used: cc3bdmp2.cpp)
(a) Rp = 0.5 RHS
zpc = 0.4 RHS
(cc3bdmp2.cpp 40a.gif)
(b)
(cc3bdmp2.cpp 40b.gif)
(c) Note the great broadening which is due to strong damping of the vibration.
(cc3bdmp2.cpp 40c.gif)
(d) Even greater damping.
(cc3bdmp2.cpp 40v.gif)		 no diagram
available
(e) In this case there is merely a substrate.
(cc3bdmp2.cpp 40w.gif)		 no diagram
available

   The frequencies of various vibrational modes are found in Figure 3.

Figure 3.     (C++ computer program used: cc3bdmp2.cpp)
(a)
(cc3bdmp2.cpp 40d.gif)

Click here to see animation of this case (360k bytes).
(b)
(cc3bdmp2.cpp 40e.gif)
(c)
(cc3bdmp2.cpp 40f.gif)
(d)
(cc3bdmp2.cpp 40h.gif)
(e)
(cc3bdmp2.cpp 40i.gif)
(f) The initial excitation is a stretching of the nanosphere along the z-axis.
(cc3bdmp2.cpp 40j.gif)
(g) The initial excitation is side to side waving of the nanosphere.
(cc3bdmp2.cpp 40k.gif)
Click here to see animation of this case (348k bytes).
(h) The initial excitation is a twist of the nanosphere around the z-axis.
(cc3bdmp2.cpp 40l.gif)

   The vibrational mode frequencies observed in Figure 3 above are summarized in Table I.

Table I.    Vibrational frequencies of a 10 nm diameter silicon nanosphere
with its geometric center 4.5 nm above the substrate surface.
substrate surface
crystal axis		 η ν (cm-1) mode type Figure reference
[100] 4.40 13.7 football
Fig. 3(a)   animation
[100] 4.7 14.6 football Fig. 3(c)
[111] 4.28 13.3 ? Fig. 3(g)
[111] 4.52 14.1 football Fig. 3(e)
[111] 5.38 16.8 football Fig. 3(e)
[111] 5.6 17.4 football Fig. 3(f)
[111] 0.8 2.5 z axis twist Fig. 3(h)
[111] 0.5 1.6 side to side
Fig. 3(g)   animation

   For comparison, the l = 2 spheroidal vibrational frequencies of a free 10 nm diameter silicon nanosphere are 13.19 cm-1 and 16.30 cm-1 (Table I in md15.htm). The frequencies 13.7, 14.6, 13.3, and 14.1 in Table I above are all within 11% of the lower free sphere l = 2 spheroidal mode. The frequencies 16.8 and 17.4 in Table I above are both within 7% of the upper free sphere l = 2 spheroidal mode. Therefore, the effect of attaching the nanosphere rather firmly to the silicon substrate is quite mild, being merely to shift the frequencies up a few percent. The broadening is extremely slight. The only case where noticeable broadening is seen is in Figure 3(f). In this motion, the "football" is along the z-axis, and therefore, stronger coupling of the mode with the substrate is to be expected. This increases the broadening.
   Ideally, the limit of large RHS should be taken. In Figure 4 a larger substrate is simulated as a convergence check. In this case only a smaller nanosphere can be simulated. The resulting η is 6.7. This is to be compared to Figure 3(a) where η = 4.40. The ratio is 1.52. Ideally, the ratio would be expected to be 3/2. Therefore, there is no indication of finite-size effects due to the size of the substrate.

Figure 4.     (C++ computer program used: cc3bdmp2.cpp)
(a)
(cc3bdmp2.cpp 40g.gif)

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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