Page 72 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 59

They described its behavior as follows. As the actuation voltage, V ,
G
applied between the CNT and the bottom electrode increases, the beam
bends downwards causing the applied electrostatic energy to be converted
into elastic deformation energy, given by,

()]=
U [ x ³ L dx ­ EI ′ ′ z 2 + T ˆ ′ z 2 ½
z
¾ , (41)
®
Elastic 0
¯ 2 2 ¿
where E and I = r π 4 4 are the CNT Young’s modulus and moment of
inertia, respectively, and T = T + T is total stress, comprised of the
ˆ
0
residual stress, T , and the stress induced by V , which is given by,
0 G
T = ES ³ L z′ 2 dx , S = r π 2 . (42)
2L 0

Since, ignoring residual stress, the beam elastic energy must correspond
to the electrostatic energy that induced it, the total energy the state of
deformed the beam arrives at is that at which the sum of elastic and
electrostatic energies is a minimum. In the Coulomb blockade regime,
however, as the bias voltage V is raised, a discrete number of charges, nq,
populates the suspended CNT. Thus, the electrostatic energy must include
this contribution, in addition to the actuation voltage (V )-induced
G
deformation. Taking both electrostatic energy sources, into account, Sapmaz,
et al. [73] approximated the total electrostatic energy by,

2 R
2
( ) 2 ( ) ln ( ) 2 L
nq
nq
nq
U (z ())=x − nqV ≈ L − ³ z ()dxx (43)
Electrosta tic G 2
2 C () z L L R 0
G
then, minimizing the total energy with respect to z, the following equation
for the CNT bending was obtained,
( ) 2
nq
z −
IE ′′′ z T ′′ = F = . (44)
0 2
L R
where F is the electrostatic for per unit length. The bending of the doubly-
0
anchored CNT, with the boundary conditions
) 0 ( z = z (L ) = ) 0 ( ′ z = z ( ′ L ) = 0 was given as,