Page 68 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 55

A voltage source V G is connected through a small capacitor C 0, to a small
metallic island that rests over a tunnel barrier which, in turn, is in contact
with an electron reservoir. The capacitance of the tunnel barrier is denoted
C J, and the distance between the gate electrode and the small island, defining
C 0, is such that tunneling is suppressed [69]. With V G=0, the system is
neutral; the small island containing n positive charges q, which are
neutralized by an equal amount of negative charges -nq, Figure 2-24(a).
When the gate voltage increases, the number of electrons in the small island
may change by amounts q = C V , Figure 2-4(b). In particular, the field
e 0 G
induced by the gate causes an uncompensated charge nq to appear on the
island. The capacitance “seen” by the island is C 0+C J. Therefore, the
charging energy accompanying the injection of a charge q = C V is,
e 0 G
(nq − q ) 2
E = e , (36)
C
2 (C + C )
0 J
It is noticed that, while the external charge q is continuous, the island
e
charge may only increase in discrete steps of value q. Therefore, the island
charge is a step-like function of the gate voltage. As a function of
temperature, the average number of electrons in the island is given by [68]
(37), Figure 2-5.

∞
¦ ne − E C k B T

n = − ∞ (37)
∞
¦ e −E C k B T
− ∞

2 2 T=0K
Average Charge in Island (Electrons) Average Charge in Island (Electrons) -2 -2 -1 -1 1 1 0 0 -1 -1 1 1 T>0K 2 2
T=0K

T>0K

-2 -2
Injected External Charge, q e =C 0 V G (Electrons)
Injected External Charge, q e =C 0 V G (Electrons)
Figure 2-6. Average island charge versus injected charge. (After [69].)