Page 123 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 111

This fact has been utilized by Burke [148] to propose an AC circuit model
for CNTs, including electron-electron interaction, see Fig. 3.18.

Spin 1↑1 ↑ L L L K K K
Spin
Spin 1↓1 ↓ L L L K K K C C C Q Q Q
Spin
C C C Q Q Q
Spin 2↑2 ↑ L L L K K K
Spin
C C C Q Q Q
Spin 2↓2 ↓ L L L K K K
Spin
C C C
Q Q Q
C C C ES C C C ES C C C ES C C C ES
ES
ES
ES
ES
ES
ES
ES
ES
Figure 3-18. AC circuit model for interacting electrons in CNT. The four-fold degeneracy is
captured by four channels. (After [148].)
The circuit model is interpreted by Burke [148] as follows. The circuit
captures the existence of four modes, namely, three spin modes, which
corresponds to a differential excitation, and one charge mode, which
corresponds to common mode excitation. In the latter case (charge mode), all
four transmission lines appear in “parallel”, and they are characterized by an
effective line possessing a charge-mode propagation velocity and
characteristic impedance given by [148],
1 § 1 4 · 4 C v F , (85)
v p = ¨ + ¸ = v F 1 + Q ≡
L K © ¨ C Q C ES ¹ ¸ C ES g

and,

Z = 4 L K + L K = 1 h , (86)
, c CM 2
C C g 2 e
ES Q
where, L = h e 2 2 v (h is Planck’s constant) is the kinetic inductance per
K F
unit length, C = e 2 2 hv is the quantum capacitance,and
Q F
C ES = 2πε cosh − 1 ( h2 ) d (h here is the CNT-to-ground distance) is the
electrostatic capacitance (the CNT-to-ground capacitance). Typical values
for these parameters are: L = 16 nH / µ m , C = 50 aF / µ m , and
K ES
C = 100 aF / µ m . The characteristic impedance for the three spin modes is
Q
given by,