Page 63 - Principles and Applications of NanoMEMS Physics

50 Chapter 2

2.2.1.2 Capacitive Transmission Line Behavior
In this design the TL is capacitive (low-impedance) and the first
bracketed term in (21) is neglected and the Schrödinger equation is given by,

− = 2 ­ q ˆ 2 + q ˆ ½ ψ = εψ , (30)
V
¾
®
2 q 2 L 2 C
e ¯ ¿
In this case, the Hamiltonian operator commutes with the charge operator
q ˆ , and consequently [60], they have simultaneous eigenstates. In particular,
the energy of the state n > is given by [67],

ε = 1 (nq − CV ) − C V , (31)
2
2
e
2C 2
where n is the number of elemental charges describing the TL state. Thus,
(31) implies that when the discrete nature of charge is at play in a low-
impedance line, the TL energy is a quadratic function of the state n of
charges.
An interesting phenomena is predicted for the current flow. In particular,
as the applied voltage increases, the TL charge can only increase in discrete
steps which are a multiple of q . Since the voltage required to cause this
e
charge to be injected into the TL is q C , it can be said that the voltage
e
axis is quantized in units of q C . Thus, the total charge of a line in the
e
ground state is given by [67],

∞ ­ ª § 1 · q º ª § 1 · q º ½
q = ¦ u ® « V − k ¨ + ¸ e » − u − V − k ¨ + ¸ e q ¾ » e , (32)
«
k= ¯ ¬ © 2 ¹ C ¼ ¬ © 2 ¹ C ¿ ¼
0
where u(z) is the unit step function. Consequently, by taking the time
derivative of (32), one obtains the corresponding current as,

∞
­ ª
I = dq = ¦ q δ V − § k ¨ + 1 · ¸ q º + δ ª V + § k ¨ + 1 · q ½ º ¾ » dV . (33)
e
e
¸
»
«
e ® «
dt k=0 ¯ ¬ © 2 ¹ C ¼ ¬ © 2 ¹ C ¿ ¼ dt
Eqn. (33) indicates that the current exhibits a series of delta-function
impulses with periodicity q C , consistent with every time a single electron
e