Manipulating the uncertainty principle (Δx. Δp x % ~h) again, we have finally for
the conductance
G ¼
ev f
À Á 2
μα
ð6:4Þ
where α ¼ ΔN/Δm. This is a novel expression to get a measure of the conductance in
case of ballistic transport in nanowires. As one can see, in the following relation
α ¼ ΔN=Δm
one can consider the ΔN to be the uncertainty in the concentration of transmitted
electrons and Δm as defined before. Interestingly, this parameter, α can be looked
upon as the effective value of
N=m
which appears in the Drude formula for conductivity of the classical Drude model. It
is to be noted that α can be very significant in the derivation of the conductance of a
particular nanowire sample.
Now, we know that according to the Landauer–Buttiker formalism, the current
through the medium is given by
I ¼
Z þ1
À1
f
0 E
ð ÞM E
ð ÞT E
ð ÞdE
ð6:5Þ
where M(E) represents the total number of modes, f
0
(E) is the deviation from the
original electron distribution function and T(E) represents the transmission probability. This leads us to an alternative definition of the conductance, also known as the
Landauer formula (Landauer 1957) which is given by
G ¼ G 0 À MT
ð6:6Þ
where G 0 ¼ 2e
2 /h is the quantum of conductance (Taylor and Mohr 2014). Now,
taking into account the relations (6.4) and (6.6) we have
G 0 À MT ¼
ev f
À Á 2
μα
:
Using the expression for G 0 we have from here
64
B. G. Sidharth and A. Das