Page 101 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 89

G G
∆S = e § ¨ dt − A ⋅ l G · ¸ − e § ¨ dt − A ⋅ ld G · ¸ = − e A G ⋅ ld G , (26)
ϕ
ϕ
d
³ ¨
³ ¨
= = ABF © c ¸ ¹ = ACF © c ¸ ¹ c= ³
G
where ϕ is the scalar potential and A is the vector potential, which is
related to the magnetic field inside the solenoid by (27).
G G G G
φ = A ⋅ l = H ⋅ s , (27)
³
³
d
d
0
The remarkable aspect of this effect is that, because of (27), it predicts, and
has been confirmed, that a vector potential exists even where no magnetic
field is existent, namely, outside the solenoid in this case, and this vector
potential endows the wave functions with a phase shift difference which
establishes that the electrons may exhibit interference. In particular, the
phase shift may be expressed as,

∆ χ = e φ , (28)
= c 0

so that when ∆ χ = 2 n π there is constructive interference, and when
∆χ = 2π ( +n 1 ) 2 there is destructive interference.

3.1.4 Quantum Transport Theory
The wave nature of electrons is responsible for a number of phenomena,
such as quantized electrical conductance, resonant tunneling, and quantum
interference, which find their genesis in the quantum nature of electrons.
Since, in fact, at dimensions approaching 100nm feature sizes, these effects
are already beginning to dominate the characteristics of practical devices, the
question of how to simulate the behavior of these quantum devices has
received much attention. In this section, we focus on the principles of typical
theoretical approaches to the quantum transport of heat and electrons.

3.1.4.1 Quantized Heat Flow

In bulk devices, the rate of heat conduction per unit area is proportional to
∇

=
the temperature gradient, i.e., Fourier’s law, Q A −κ T , where κ is the
bulk coefficient of thermal conductivity. This expression assumes
κ = γ Cvl [126], where γ is a numerical factor, C is the specific heat per
p