Vibrational Mode Lifetime of a Silicon...

Updated: January 23, 2008
Vibrational Mode Lifetime of a Silicon Nanoparticle Weakly Coupled to a Substrate

Consider a spherical nanoparticle of diameter d_p. Its elastic properties are those of crystalline silicon at room temperature: C₁₁ = 166 GPa, C₁₂ = 64 GPa, C₄₄ = 80 GPa and ρ = 2.329 g/cc. The nanoparticle is connected by a cylindrical "neck" to a semiinfinite substrate also made of silicon. The neck is of length l_c and diameter d_c. The normal to the flat surface of the substrate is the [010] crystal direction. The vibrational mode being excited corresponds to dilation along the y-axis. This mode has dimensionless frequency η approximately 2.05, where ω = η C_tran100 / R, where R is the radius of the nanoparticle. C_tran100 is 5840 m/s for silicon, and is the transverse speed of sound along the [100] crystal direction. To determine the damping of this mode, it is first harmonically excited at its natural frequency. The excitation is then stopped and the velocity of one point on the sphere is plotted versus time in Figure 1 below.

Figure 1. Using Borland Turbo C++ program cc3bdm10.cpp
(a) d_c/d_p = 0.10
(b) d_c/d_p = 0.15
(c) d_c/d_p = 0.20
(d) d_c/d_p = 0.25
(e) d_c/d_p = 0.30
(g) d_c/d_p = 0.36
(g) d_c/d_p = 0.4
(h) d_c/d_p = 0.46
(i) d_c/d_p = 0.5
(j) d_c/d_p = 0.56

Table I
d_c/d_p	Q
0.10	1360
0.15	200
0.20	100
0.25	34
0.30	16
0.36	12
0.40	7
0.46	4
0.50	3
0.56	2.5

Over this entire range, the data is well fitted by

Q = 0.247 (d_c/d_p)^-3.66

Extrapolating to the case of d_p = 10 nm and d_c = 0.5 nm, Q would be 14300.
This calculation follows an analytical calculation "Phonons in a nanoparticle mechanically coupled to a substrate" by Kelly R. Patton and Michael R. Geller (preprint 2002). Their preprint is on line at: arxiv.org/PS_cache/cond-mat/0202325

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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