By expanding the GREEN's function in terms of the basis functions the
continuity equation (3.87) can be derived as
represents the current passing through a point between and .
Note that has a unit of
due to the
one-dimensional nature of CNTs. The time derivative of the GREEN's functions
can be replaced by the relation (3.62)
where the term inside the integral is zero due to the condition stated
The next step is separating
decomposing equation (4.52). CAROLI proposed the following ansatz
in . The current is the difference between the flow of particles from left to right
and from right to left. This leads to the following expression for
It is straightforward to show that (4.54) along with an expression for
Under steady-state condition one can transform the time difference
coordinate to energy to obtain
Based on the nearest neighbor tight-binding method in mode-space (see Section 4.4)
equation (4.55) can be simplified to
where the summation runs over all the subbands contributing to transport.
The factor in (4.56) is due to double spin and double subband degeneracy.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors