H.1 Recursive Algorithm to Calculate $G^\mathrm{r}$

The DYSON equation for the retarded GREEN's function and the left-connected GREEN's function [116] are employed to calculate the diagonal blocks of the full GREEN's function recursively. The solution to the matrix equation

$$\left[ \begin{array}{cc} \ensuremath{{\underline{A}}}_{Z,Z} & \en... ...\underline{I}}} & 0 \\ 0 & \ensuremath{{\underline{I}}} \end{array} \right] \ ,$$ (H.4)

is

$$\begin{array}{lll} \ensuremath{{\underline{G}}}^\mathrm{r} &=... ...ne{U}}}\ \ensuremath{{\underline{G}}}^\mathrm{r0}\ ,\end{array}$$ (H.5)

where,

$$\ensuremath{{\underline{G}}}^\mathrm{r}\ = \ \left[ \begin{array}... ...uremath{{\underline{G}}}^\mathrm{r}_{Z^\prime,Z^\prime} \end{array} \right] \ ,$$ (H.6)

$$\ensuremath{{\underline{G}}}^\mathrm{r0}= \ \left[ \begin{array}{... ... & \ensuremath{{\underline{A}}}_{Z^\prime,Z^\prime}^{-1} \end{array}\right] \ ,$$ (H.7)

$$\ensuremath{{\underline{U}}}\ \ = \ \left[ \begin{array}{cc} 0 & ... ...ime} \\ - \ensuremath{{\underline{A}}}_{Z^\prime,Z} & 0 \end{array} \right] \ .$$ (H.8)

The left-connected retarded GREEN's function $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{q}}$ is defined by the first $q$ blocks of (H.1) by

$$\begin{array}{l}\displaystyle \ensuremath{{\underline{A}}}_{_... ...q}} = \ensuremath{{\underline{I}}}_{_{1:q,1:q}} \ . \end{array}$$ (H.9)

$\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{q+1}}$ is defined in a manner identical to $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{q}}$ except that the left-connected system is comprised of the first $q+1$ blocks of (H.1). In terms of (H.4), the equation governing $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{q+1}}$ follows by setting $Z=1:q$ and $Z^\prime=q+1$. Using the DYSON equation [(H.5)], one obtains

$$\begin{array}{l}\displaystyle \ensuremath{{\underline{g}}}^\m... ...uremath{{\underline{A}}}_{_{q,q+1}} \right)^{-1} \ .\end{array}$$ (H.10)

It should be noted that the last block $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{N,N}}$ is equal to the fully connected GREEN's function $\ensuremath{{\underline{G}}}^\mathrm{r}_{L_{N,N}}$, which is the solution to (H.1). The full GREEN's function can be expressed in terms of the left-connected GREEN's function by considering (H.4) such that $\ensuremath{{\underline{A}}}_{Z,Z}=\ensuremath{{\underline{A}}}_{_{1:q,1:q}}$, $\ensuremath{{\underline{A}}}_{Z^\prime,Z^\prime}=\ensuremath{{\underline{A}}}_{_{q+1:N,q+1:N}}$ and $\ensuremath{{\underline{A}}}_{Z,Z^\prime}=\ensuremath{{\underline{A}}}_{_{1:q,q+1:N}}$. By noting that the only non-zero block of $\ensuremath{{\underline{A}}}_{_{1:q,q+1:N}}$ is $\ensuremath{{\underline{A}}}_{_{q,q+1}}$ and using (H.5), one obtains

$$\begin{array}{lll} \ensuremath{{\underline{G}}}^\mathrm{r}_{_... ...suremath{{\underline{G}}}^\mathrm{r}_{_{q+1,q}} \ , \end{array}$$ (H.11)

Both $G^\mathrm{r}_{_{q,q}}$ and $G^\mathrm{r}_{_{q+1,q}}$ are used for the calculation of the electron density, and so storing both sets of matrices will be useful. In view of the above equations, the algorithm to compute the diagonal blocks $G^\mathrm{r}_{_{q,q}}$ is given by the following steps

• $g^\mathrm{r}_{L_{1,1}}~=~\ensuremath{{\underline{A}}}_1^{-1}$,
• For $q~=~1,~2,~\ldots ,~N-1$, (H.10) is computed,
• For $q~=~N-1,~N-2,~\ldots,~1$, (H.11) is computed.

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