G.3 Surface GREEN's Function

The main information needed to solve (G.14) are the surface GREEN's functions of $\ensuremath{{\underline{g}}}^\mathrm{r}_L$ and $\ensuremath{{\underline{g}}}^\mathrm{r}_R$. Using the recursive relation (H.10), equation (G.12) can be written as

$$\begin{array}{ll} [\ensuremath{{\underline{A}}}_{L_{i}} \ - \... ..._{R_{i,i}} \ & = \ \ensuremath{{\underline{I}}} \ . \end{array}$$ (G.22)

If the potential does not vary in the left and right contacts and if the coupling between different layers are equal, then $\ensuremath{{\underline{A}}}_{LL}$ and $\ensuremath{{\underline{A}}}_{RR}$ become semi-infinite periodic matrices with

$$\begin{array}{lllll} \ensuremath{{\underline{A}}}_{L_{1}} \ &... ... \ldots \ &= \ {\ensuremath{{\underline{t}}}_R} \ . \end{array}$$ (G.23)

Under this condition one obtains

$$\begin{array}{lllllll} \ensuremath{{\underline{g}}}^\mathrm{r... ... &=&{\ensuremath{{\underline{g}}}^\mathrm{r}_R} \ . \end{array}$$ (G.24)

Therefore, the surface GREEN's functions can be obtained by solving the quadratic matrix equations

$$\begin{array}{ll} [\ensuremath{{\underline{A}}}_L - \ensurema... ...thrm{r}_R} \ & = \ \ensuremath{{\underline{I}}} \ . \end{array}$$ (G.25)

These equations can be solved iteratively by

$$\begin{array}{ll} [\ensuremath{{\underline{A}}}_L - \ensurema... ...e m\rangle}\ & = \ \ensuremath{{\underline{I}}} \ , \end{array}$$ (G.26)

where $m$ represents the iteration number. It should be noted that the solution to (G.25) is analytic if the dimension of $\ensuremath{{\underline{A}}}_R$ is one. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors