Nanotechnology-Enabled Sensors

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Chapter 6: Inorganic Nanotechnology Enabled Sensors
dimensions of nanostructures. The dynamics of phonons confined in quantum
dimensions differ from their non-confined counterparts. Phonons are
responsible for transferring mechanical energy. Mechanical sensors, thermoelectric
devices and many optical measurement systems are based on
the function of phonons. Phonon vibrations affect atomic structures and
electrons. Interactions of phonons with photons, atomic structures and
electrons can be used in sensing applications and also they can be effectively
used as sensing elements. For instance, a direct application of
phonons in sensing technologies appears in acoustic phonon pulse spectroscopy
and Raman spectroscopy. However, the effects of phonons can be
unfavourable for in dimensional structures as they can broaden the spectrum
of electromagnetic measurements and add electronic noise.
6.4.1 Phonons in One-Dimensional Structures
The simplest one-dimensional structure is made of similar atoms
(monoatomic) connected in series as shown in Fig. 6.33. Despite the simplicity,
such a structure is a good example to understand the behavior of
phonons, propagation of waves and dispersion relationships. This understanding
can then be expanded to visualize more complex structures.
Let’s assume a one-dimensional lattice with the lattice constant equal to
a. When a force is exerted on the lattice the whole system moves with the
same frequency. Now, let’s consider that the system consists of N atoms
and the force applied to the n th atom is only due to neighboring atoms. If
harmonic approximation (it assumes that force is proportional to relative
displacement), Hooke’s law and the inter-atomic force constant α are used
we will obtain: 2
2
d un
M = −α
( 2u
− 1 1)
2
n un+
− un−
, (6.58)
dt
where un is the displacement of the n th particle, M is the mass of the atom
and t is time. To solve the Eq. (6.64), n coupled equations for the N atoms
in the system would have to be solved simultaneously. With a solution in
i( kxn
−ωt
)
the form of Ae (xn the location of the n th atom) the result can be
written as: 2
1/
2
4
sin( ka / 2)
M ⎟ ⎛ ⎞
= ⎜
⎝ ⎠
α
ω . (6.59)