6.2 Conductance and Transmission
Now, let us consider the case of a carbon nanotube. Here, electrons can travel freely
through the axis of the nanowire without collision for a long time, i.e., the mean free
path (l f ¼ v f τ, with Fermi velocity v f ) for the electron is very large. This type of
transport is also known as ballistic transport (Datta 1997). Considering 1D
nanowires, the conductance is generally given by
G ¼ I=V ¼ 1=R
ð6:1Þ
where the symbols have their usual meanings. Since the electrons travel along the
axis without any collision, there is apparently no loss during the transmission. This is
a key feature of ballistic transport. Now, if any signal is transformed into an electric
signal and thereby the electrons act as carriers and are transmitted through the
nanowire with transmission probability nearly equal to unity, then we have a new
means for telecommunications.
Now, we know that the diameter of a nanotube can be written as
D ¼
a
π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
p 2 þ pq þ q 2
p
where a ¼ 0.246 (in nm), and ‘n’, ‘m’ are the integers that denote the unit vectors
along the two directions in the honeycomb lattice. Adjusting the indices ‘p’ and ‘q’
such that
p ¼ q
one can fabricate an armchair nanotube with a feasible diameter (D) which makes it
metallic along the tubular axis. We would like to consider the metallic case specifically. Now, when the mean free path (l f ) of an electron is greater than the diameter
(D) of the constriction or the quantum point contact, it necessitates us to consider the
conductance of the medium. Now, suppose an electric pulse has to be transmitted
through the nanotube medium. In that scenario, an electric field develops from which
a magnetic field accumulates. This will be useful in the explanations given later.
Now, let us find an alternative expression of the conductance (G) other than that
given in Eq. (6.1). We commence with the uncertainty principle, in terms of energy
and time. This is known as
ΔE:Δt % h
where ‘h’ is the Planck’s constant. Now, the current ‘I’ in a quantum channel can be
written as e/t, where ‘t’ is transit time and the ‘e’ is electron charge. Applying a
voltage V we get the energy as
62
B. G. Sidharth and A. Das
Now, let us consider the case of a carbon nanotube. Here, electrons can travel freely
through the axis of the nanowire without collision for a long time, i.e., the mean free
path (l f ¼ v f τ, with Fermi velocity v f ) for the electron is very large. This type of
transport is also known as ballistic transport (Datta 1997). Considering 1D
nanowires, the conductance is generally given by
G ¼ I=V ¼ 1=R
ð6:1Þ
where the symbols have their usual meanings. Since the electrons travel along the
axis without any collision, there is apparently no loss during the transmission. This is
a key feature of ballistic transport. Now, if any signal is transformed into an electric
signal and thereby the electrons act as carriers and are transmitted through the
nanowire with transmission probability nearly equal to unity, then we have a new
means for telecommunications.
Now, we know that the diameter of a nanotube can be written as
D ¼
a
π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
p 2 þ pq þ q 2
p
where a ¼ 0.246 (in nm), and ‘n’, ‘m’ are the integers that denote the unit vectors
along the two directions in the honeycomb lattice. Adjusting the indices ‘p’ and ‘q’
such that
p ¼ q
one can fabricate an armchair nanotube with a feasible diameter (D) which makes it
metallic along the tubular axis. We would like to consider the metallic case specifically. Now, when the mean free path (l f ) of an electron is greater than the diameter
(D) of the constriction or the quantum point contact, it necessitates us to consider the
conductance of the medium. Now, suppose an electric pulse has to be transmitted
through the nanotube medium. In that scenario, an electric field develops from which
a magnetic field accumulates. This will be useful in the explanations given later.
Now, let us find an alternative expression of the conductance (G) other than that
given in Eq. (6.1). We commence with the uncertainty principle, in terms of energy
and time. This is known as
ΔE:Δt % h
where ‘h’ is the Planck’s constant. Now, the current ‘I’ in a quantum channel can be
written as e/t, where ‘t’ is transit time and the ‘e’ is electron charge. Applying a
voltage V we get the energy as
62
B. G. Sidharth and A. Das