Page 93 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 81

particular, since in the momentum interval dp, this number equals dp 2 = π ,
the corresponding current is dJ = evdp 2 = π . Therefore, the total left-right
+
current, assuming a single band, and taking into account two spins, is given
by:

∞ e E F 1 eE
e
J + = 2 ⋅ ³ vdp = ³ dE = F 1 . (1)
2 π = π = π =
0 − ∞
A similar result is obtained for the right-left current,
∞ e µ 2 eE
e
J − = 2 ⋅ ³ vdp = ³ dE = F 2 , (2)
2 π = π = π =
0 − ∞
so, the net current from left to right is:

e
J = J − J + = π = (E F 1 − E F 2 ) . (3)
−

Then, width the substitution E − E = eV , we obtain,
F1 F 2

e 2
J = V . (4)
π =
The proportionality factor between current and voltage is the quantized
conductance for a single band:

g = e 2 . (5)
0
π =
Assuming transport is occurring in N bands (channels) under the Fermi
level, the total conductance is,
g = N g ⋅ . (6)
0

s
This expression clearly reveals that the conductance is quantized in unit of
g . In reality, there is a finite probability that in going from the reservoir
0
into the wire, and vice versa, some electrons may be backscattered, in which
case the number of bands through which transport is operative is less than N.
In that case the effective value for N is conductance is given by: