4.7.2 Current

By expanding the GREEN's function in terms of the basis functions the continuity equation (3.87) can be derived as

$$\begin{array}{l}\displaystyle \underbrace{ -\frac{i\hbar\ensu... ...suremath{{\mathbf{\nabla}}}\cdot {\bf J}} \ = 0 \ , \end{array}$$ (4.52)

where $J_{_{i+1/2}}$ represents the current passing through a point between $i+1$ and $i$. Note that $J$ has a unit of $\mathrm{A}$ rather than $\mathrm{A/m^2}$ due to the one-dimensional nature of CNTs. The time derivative of the GREEN's functions can be replaced by the relation (3.62)

$$\begin{array}{lll}\displaystyle \partial_t \varrho_i \ & = &\... ...J_{_{i+1/2}}(t)\ - \ J_{_{i-1/2}}(t)}{\Delta z} \ , \end{array}$$ (4.53)

where the term inside the integral is zero due to the condition stated in (3.90).

The next step is separating $J_{_{i+1/2}}$ from $J_{_{i-1/2}}$ by decomposing equation (4.52). CAROLI proposed the following ansatz in [96]. The current $J$ is the difference between the flow of particles from left to right and from right to left. This leads to the following expression for $J_{_i}$ [96]

$$\begin{array}{l}\displaystyle J_{_{i+1/2}}(t) \ = \ -\ensurem... ...(t,t) \ - \ G^<_{_{j,k}}(t,t)H_{_{k,j}} \right) \ . \end{array}$$ (4.54)

It is straightforward to show that (4.54) along with an expression for $J_{_{i-1}}$ satisfies (4.53).

Under steady-state condition one can transform the time difference coordinate to energy to obtain

$$\begin{array}{ll}\displaystyle J_{_{i+1/2}} \ &\displaystyle ... ...\ 2\ \Re \mathrm{e}[ H_{_{j,k}}G^<_{_{k,j}}(E)] \ , \end{array}$$ (4.55)

Based on the nearest neighbor tight-binding method in mode-space (see Section 4.4) equation (4.55) can be simplified to

$$\begin{array}{l}\displaystyle J^{^\nu}_{_{i+1/2}} \ = \ \frac... ... {t}^{^\nu}_{_{i+1,i}} {G}^{<^\nu}_{_{i,i+1}}]\ \ , \end{array}$$ (4.56)

where the summation runs over all the subbands contributing to transport. The factor $4$ in (4.56) is due to double spin and double subband degeneracy. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors