Landauer Formula is really useful in order to compute current
in nanoscale devices. Actually this formula has been included in the opensource code
which compute transport in Carbon Nanotubes and Silicon Nanowire Transistors.
Let us consider the following system, composed by a one-dimensional channel (a quantum wire) with length , and two metallic reservoirs with electrochemical potential and ( ) (Fig. 2).
We suppose that only the first one-dimensional subband is occupied,
that the electrons in the channel does not
suffer any scattering mechanism,
i.e. transport in the channel is ballistic,
and that the electrons entering the reservoirs
contacts are instantaneously in equilibrium with them.
For the moment, let the temperature be equal to zero (T=0 K)
and the contacts reflectionless, that means that
the transmission probability from the contact to contact is unitary.
Being a two-dimensional confined system, the dispersion relation
is equal to (32), where
is also referred as the cut-off energy of -th 1D subband or
The positive current, carried by states in the -subband,
Working at , the Fermi-Dirac function is a step
function, and considering constant and equal to an integer
between the energy range , the total current reads
We note that (49) can be reduced to the form
If we now relax the hypothesis of reflectionless contacts, we have to consider in (48) , the probability that an electron originated by one electrode reaches the other reservoir propagating through the channel.
If is the influx of electrons from the left reservoir,
is the flux of back-scattered electrons in the left reservoir,
and is the flux of electrons that has reached the
right reservoir (Fig. 3),
if we consider constant and equal to
between the energy range ,
we deal with the following relations
We can now study the case in which the temperature is not equal to zero.
The total current now reads
and by means of the Taylor expansion
we can express (58) as
that can be reduced to the form expressed in (50), if we define
As a numerical example, we can consider the case of a QPC with reflectionless
contacts in which the transversal potential is a harmonic
The transversal modes are equal to