Page 73 - Principles and Applications of NanoMEMS Physics

60 Chapter 2

ª sinh L ξ ξ ) − º
«
F L cosh ξ L −1 (cosh L −1 » T
z () x = 0 « », ξ = . (45)
ξ
2 T « x 2 » EI
« sinh ξ x + ξ x − ξ »
¬ L ¼

Finally, the effects of charge discreteness are manifest upon examining
the maximum displacement as a function of actuation voltage, and given by
(45) and (46).

2 2 EI § Er 5 g ·
( ) Lnq
z = . 0 013 , T << ¨ <<n 0 ; ¸ (46a)
max 4 2 ¨ 2 2 ¸
Er g 0 L © q L ¹
( ) 3 / 2 L 3 / 2 EI § Er 5 g ·
nq
n
z = 24.0 , T >> ¨ >> 0 ¸ (46b)
max 3 / 1 2 3 / 1 2 ¨ 2 2 ¸
E r g L © q L ¹
0
§ V L 1 ·
n = Int ¨ G + + ¸ n δ . (47)
¨ ¸
© 2 r ln ( g /2 0 ) r 2 ¹
For a given applied voltage, (47) gives the value of n that minimizes the total
energy, where n δ is a small correction. Clearly, (45)-(47) reveal that the
beam displacement is quantized, i.e., its position changes in discrete steps
every time an electron tunnels into it.

2.3 Manifestation of Quantum Electrodynamical Forces

When the proximity between material objects becomes of the order of
several nanometers, a regime is entered in which forces that are quantum
mechanical in nature [74-76], namely, van der Waals and Casimir forces,
become operative. These forces supplement, for instance, the electrostatic
force in countering Hooke’s law to determine the beam actuation behavior.
They also may be responsible for stiction [77], i.e., causing close by
elements to adhere together and, thus, may profoundly change actuation
dynamics.

2.3.1 van der Waals Force

van der Waals forces, of electromagnetic and quantum mechanical origin,
are responsible for intermolecular attraction and repulsion. When adjacent