Page 78 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 65

ε
where s = ε − 1+ p , () =1 + 2 ³ ∞ εω ′ ′ () ξ i d ω is the dielectric
2
ξ i
π 0 ω 2 + ξ 2
constant of the metal, ε ′′ is the imaginary component of ε , and ξ is the
imaginary frequency given by ω = ξ i .
Corrections due to nonzero temperature yield [77],

ª 720 º
ζ
F T () = Fz 0 () +z 1 f () , (56)
Cas Cas « 2 »
¬ π ¼
where ζ = k Tz c = , k is Boltzmann constant, T is the absolute
B B
temperature, and

( ­ ζ 3 2π ) () ( πζϑ 3 − 4 2 45 ), for ζ ≤ 2
1
°
ζ
f ()≈ ® 2 , (57)
1
° (ζ 8π ) ( ) (πϑ 3 − 720 ), for ζ > 2
¯
ϑ
with () 13 = . 202 ... .
Roy and Mohideen [90] included originally the effects of surface
roughness, which changes the surface separation, by replacing the flat plate
with a spatial sinusoidal modulation of period λ , and the energy averaged
over the size of the plates, L, to obtain,

§ 2π · x π 2 = c 1 § A · m
<U ¨ + Az sin ¸ >= − ¦ C ¨
Casimir 3 m ¸ , (58)
© λ ¹ 720 z m © z ¹

where A is the corrugation amplitude. The corresponding Casimir force is
then given by the so-called, Force Proximity Theorem [99] relating the
parallel plate geometry and the sphere-plate geometry, namely,

F = 2π <U > (59)
R
Cas _ Roughness Cas _ Rouchness
For << L and z + z > A, where z is the average surface separation
λ
0 0
after contact due to stochastic roughness of the metal coating, they
recommend the following coefficients in (58): C = 1, C = 3 , C = 45 8,
0 2 4
C = 35 4 . A more accurate and general model for stochastic surface
6
roughness, advanced by Harris, Chen, and Mohideen [88], includes the