Nanotechnology-Enabled Sensors

336
relative positions of oppositely charged ions which produces electric polarization
and therefore long-range electric fields which affect the electron
energies (this perturbation is known as piezoelectricity) . The deformation
and piezoelectric potentials take the following forms:
DP
DP 1/
2 −iq.
r
V+
q ( r)
≈ C k e . (6.66)
PE
PE −1/
2 −iq.
r
V+
( r)
≈ C k e . (6.67)
q
where C DP and C DP are the coupling strength constants for deformation and
piezoelectric potentials, respectively, and q is the phonon wavenumber.
For a quantum well with the thickness of d the matrix element is: 69
⎛ + ⎞
ψ V ˆ ( q)
ψ
δ ±
k′
±
k
DP
2
PE
2
−1
2 ⎜ C q C q ⎟
⎝
⎠ 2
≈ × sin ( q d / 2)
δ
,
2
q y
y
z k x , k′
x ± qx
k y k′
y q
2
y
2
2
[ ( 2π
/ w)
− q ]
,
(6.68)
where q is the phonon wave vector, k and k′ are the electron wavenumbers
before and after scattering, respectively. For a quantum wire element the
matrix element is: 69
ψ
Chapter 6: Inorganic Nanotechnology Enabled Sensors
⎛ +
DP
2
PE
2
−1
2 ⎜ C q C q ⎟
k V ˆ ⎝
⎠
2
2
′ ± ( q)
ψ k ≈ × sin ( q w / 2)
sin ( q d / 2)
δ , ±
2
q y
y z
z
⎞
x
[ ( ) ] [ ( ) ] q k k
z
y
′
2 2
2
2
2 2
2
2π
/ w − q q 2π
/ d − q
, (6.69)
where d and w are the dimensions of the base of the wire. From Eqs. (6.68)
and (6.69) it can be seen that momentum components parallel to the quantum
well plane and wire length are conserved.
The research on the behavior of phonons in 1D structures is becoming
increasingly popular in recent years. 70,71 In such structures the effects of
phonons have been studied in thermal transport, Raman scattering, and
electrical transport. 71 For example, it has been shown that the electron-
acoustic phonon scattering in metallic single-walled carbon nanotubes
(SWCNTs) contributes to the resistance at room temperature. It is also
shown that at low bias voltages, the scattering is weak, resulting in long
mean-free paths at room temperature. In this case, both measurements and
calculations confirm that the mean-free path is in the range of a few hundred
nanometers to several micrometers. At high bias voltages, electrons
gain enough energy to emit optical phonons. Yao et al showed that this
scattering leads to a saturation of the current at ~20 μA for such high bias
voltages. 72 Park et al investigated the scaling of the resistance in such
SWCNTs for lengths (L) ranging from 50 nm to 10 μm using the tip of an
atomic force microscope (AFM) as a movable electrode. 70 For low bias
voltages, they find a length-independent resistance for L < 200 nm, indicat-