Page 57 - Principles and Applications of NanoMEMS Physics

44 Chapter 2

= 2 ª § n π · 2 § n π y · 2 º
E ~ «¨ x ¸ + ¨ ¸ + k 2 »
¨
2m * « t ¸ ¨ w ¸ z » (1)
¹
©
©
¹
¬ s ¼
but also, that their discrete nature will be manifest. This latter feature
becomes operative when the system size along a transport dimension
becomes of the order of the carrier inelastic coherence length, and it implies
that, in addition to the quantum mechanical energy of confinement of Eq.
(1), the Coulomb charging energy required for adding or removing an
electron, E = q 2 L where L is a characteristic length in direction i,
c i i
must be taken into account [58-62]. One must then turn to quantum
mechanics to properly describe the TL behavior.
The observation [61]-[63], that the charge q in successive cells, and the
total energy, obey equations (2) and (3),

d 2 q 1
L i = (q i 1 + q i 1 − 2 q i ) (2)
−
+
dt 2 C
§ 1 § dq · 2 1 ·
H = ¦ ¨ ¨ ¨L i ¸ + (q + i 1 − q i ) 2 ¸ ¸ (3)
i © 2 L © dt ¹ 2 C ¹

whose forms are identical to the equations describing the longitudinal
vibration modes in a monatomic linear chain (MLC) [64] (see Appendix A),
Figure 2(b), motivated the application of the quantum mechanical
description of the latter to the TL. In particular, in (3), the first and second
terms account for the magnetic and electric energies in the TL inductors and

capacitors, respectively, and p = L dq and q play the roles of “momentum”
dt
and “coordinate,” respectively. Notice, however, that since q is charge, p
represents electric current.
The above TL quantization assumed the electric charge q to be a
continuous variable. As has been observed [59], however, under appropriate
circumstances, e.g., system size close to the inelastic coherence length, the
particle (or discrete) nature of electrons becomes evident. Li [61] considered
the consequences of this possibility and, accordingly, advanced a theory for
TL quantization assuming q to be discrete.
The possibility of having the charge adopt exclusively discrete values,
was introduced [61] by imposing the condition that the eigenvalues of the
charge operator q ˆ be discrete, i.e.,