G.1 Matrix Truncation

By defining

$$\ensuremath{{\underline{A}}} \ = \ [E \ensuremath{{\underline{I}}... ...uremath{{\underline{H}}} - \ensuremath{{\underline{\Sigma}}}_\mathrm{Scat}] \ ,$$ (G.1)

the equation (4.11) ( $\ensuremath{{\underline{A}}}\ensuremath{{\underline{G}}}^\mathrm{r}=\ensuremath{{\underline{I}}}$) can be written as,

$$\left[ \begin{array}{ccc} \ensuremath{{\underline{A}}}_{LL} & \en... ...{{\underline{I}}} & \\ & & \ensuremath{{\underline{I}}} \end{array} \right] \ ,$$ (G.2)

where

$$\ensuremath{{\underline{A}}}_{LL} \ = \ \left[ \begin{array}{cccc... ...dagger_{L_{2,1}} & \ensuremath{{\underline{A}}}_{L_{1}} \end{array} \right] \ ,$$ (G.3)

corresponds to the left semi-infinite contact,

$$\ensuremath{{\underline{A}}}_{RR} \ = \ \left[ \begin{array}{cccc... ...llet &\bullet &\bullet& \\ & & &\bullet &\bullet&\bullet \end{array} \right]\ ,$$ (G.4)

corresponds to the right semi-infinite contact, and

$$\ensuremath{{\underline{A}}}_{DD}\ =\ \left[ \begin{array}{cccccc... ...ne{t}}}_{N-1,N}^\dagger &\ensuremath{{\underline{A}}}_N \end{array} \right] \ ,$$ (G.5)

corresponds to the device region.

The coupling between the left and right contacts and device are respectively given by

$$\ensuremath{{\underline{A}}}_{LD}\ = \ \left[ \begin{array}{ccccc... ...th{{\underline{t}}}_{LD}& 0 & \bullet & \bullet & 0 & 0 \end{array} \right] \ ,$$ (G.6)

and

$$\ensuremath{{\underline{A}}}_{RD} \ = \ \left[ \begin{array}{cccc... ...t & \bullet & 0 & 0\\ 0 & 0 & \bullet & \bullet & 0 & 0 \end{array} \right] \ .$$ (G.7)

It should be noted that $\ensuremath{{\underline{A}}}_{DL}=\ensuremath{{\underline{A}}}^\dagger_{LD}$, $\ensuremath{{\underline{A}}}_{DR}=\ensuremath{{\underline{A}}}^\dagger_{RD}$, and $\ensuremath{{\underline{A}}}_{LD}$ and $\ensuremath{{\underline{A}}}_{DL}$ ( $\ensuremath{{\underline{A}}}_{RD}$, and $\ensuremath{{\underline{A}}}_{DR}$) are sparse matrices. Their only non-zero entry represents the coupling of the left (right) contact and device. From (G.2), one obtains

$$\begin{array}{l} \ensuremath{{\underline{G}}}^\mathrm{r}_{LD}... ...D}\ \ensuremath{{\underline{G}}}^\mathrm{r}_{DD}\ , \end{array}$$ (G.8)

$$\begin{array}{l} \ensuremath{{\underline{G}}}^\mathrm{r}_{RD}... ...D}\ \ensuremath{{\underline{G}}}^\mathrm{r}_{DD}\ , \end{array}$$ (G.9)

$$\begin{array}{l} \ensuremath{{\underline{A}}}_{DL} \ \ensurem... ...athrm{r}_{RD}\ = \ \ensuremath{{\underline{I}}} \ . \end{array}$$ (G.10)

Substituting (G.8) and (G.9) in (G.10), one obtains a matrix equation with a dimension corresponding to the total number of grid points in device layers,

$$[\ensuremath{{\underline{A}}}_{DD} \ - \ \ensuremath{{\underline{... ...nsuremath{{\underline{G}}}^\mathrm{r}_{DD}\ =\ \ensuremath{{\underline{I}}} \ .$$ (G.11)

The second and third terms of (G.11) are self-energies due to coupling of the device region to left and right contacts, respectively.

The GREEN's functions of the isolated semi-infinite contacts are defined as

$$\begin{array}{ll} \ensuremath{{\underline{A}}}_{LL}\ \ensurem... ...hrm{r}_{R} \ & = \ \ensuremath{{\underline{I}}} \ . \end{array}$$ (G.12)

The surface GREEN's function of the left and right contacts are the GREEN's function elements corresponding to the first edge layer of the respective contact

$$\begin{array}{ll} \ensuremath{{\underline{g}}}^\mathrm{r}_{L_... ... \ \ensuremath{{\underline{A}}}^{-1}_{RR_{1,1}} \ . \end{array}$$ (G.13)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors