Introduction to Nanotechnology

13.2. NANOELECTROMECHANICAL SYSTEMS (NEMSs) 341
have a resonant frequency 1 O5 times greater, of the order of 20-30 GHz (2-3 x 10"
cycles per second). As the frequency increases, the amplitude of vibration decreases,
and in this range of frequencies the displacements of the beam can range from a
picometer (10-l~ m) to a femtometer (10-l~~).
These high frequencies and small displacements are very difficult, if not
impossible, to detect. Optical reflection methods such as those used in the micro-
meter range on the cantilever tips of scanning tunneling microscopes are not
applicable because of the diffraction limit. This occurs when the size of the object
from which light is reflected becomes smaller than the wavelength of the light.
Transducers are generally used in MEMS devices to detect motion. The MEMs
accelerometer shown in Fig. 13.1 is an example of the detection of motion using a
transducer. In the accelerometer mechanical motion is detected by a change in
capacitance, which can be measured by an electrical circuit. It is not clear that such a
transducer sensor can be built that can detect displacements as small as to
m, and do so at frequencies up to 30 GHz. These issues present significant
obstacles to the development of NEMS devices.
There are, however, some noteworthy advantages of NEMS devices that make it
worthwhile to pursue their development. The small effective mass of a nanometer-
sized beam renders its resonant frequency extremely sensitive to slight changes in its
mass. It has been shown, for example, that the frequency can be affected by
adsorption of a small number of atoms on the surface, which could be the basis for a
variety of very high-sensitivity sensors.
A weight on a spring would oscillate indefinitely with the same amplitude if there
were no friction. However, because of air resistance, and the internal spring friction,
this does not happen. Generally the frictional or damping force is proportional to the
velocity &/dt of the oscillating mass M. The equation of motion of the spring is
d2x dx
M-+b-+fi=O
dt2 dt
where K is the spring of constant.
The solution X(o) to this equation for a small damping factor b is
with the frequency o given by
(2)
X(w) = A exp - cos(ot + 6)
0 = [(E)
- ($)2]-lf2
(13.2)
(13.3)
(13.4)
Equation (13.3) describes a system oscillating at a fixed frequency o with an
amplitude exponentially decreasing in time. The displacement as a fimction of time
is plotted in Fig. 13.7a. For a clamped vibrating millimeter-sized beam, a major