where represents the current passing through a point between and . Note that has a unit of rather than 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 in (3.90).

The next step is separating from by decomposing equation (4.52). CAROLI proposed the following ansatz in [96]. 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 [96]

It is straightforward to show that (4.54) along with an expression for satisfies (4.53).

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.