G.2 Contact Self-Energies

The surface GREEN's functions defined in (G.13) enable us to rewrite (G.11) in a form very similar to (4.11),

$${[E \ensuremath{{\underline{I}}} - \ensuremath{{\underline{H}}} -... ...uremath{{\underline{G}}}^\mathrm{r}_{DD} \ = \ \ensuremath{{\underline{I}}} \ ,$$ (G.14)

where,

$$\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr... ... \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_R \ . \end{array}$$ (G.15)

All other elements of $\ensuremath{{\underline{\Sigma}}}^\mathrm{r}_\mathrm{C}$ are zero. $\ensuremath{{\underline{\Sigma}}}^\mathrm{r}_L$ and $\ensuremath{{\underline{\Sigma}}}^\mathrm{r}_R$ are self-energies due to the left and right contacts, respectively, and $\ensuremath{{\underline{t}}}_{DL}=\ensuremath{{\underline{t}}}_{LD}^\dagger$ and $\ensuremath{{\underline{t}}}_{DR}=\ensuremath{{\underline{t}}}_{RD}^\dagger$. By following the same procedure one obtains the equation of motion for the lesser and greater GREEN's functions as [116]

$$\ensuremath{{\underline{G}}}^\mathrm{\gtrless}_{DD} \ = \ \ensure... ...mathrm{r}_\mathrm{C} \right] \ \ensuremath{{\underline{G}}}^\mathrm{a}_{DD} \ ,$$ (G.16)

where,

$$\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr... ...emath{{\underline{\Sigma}}}^\mathrm{\gtrless}_R \ . \end{array}$$ (G.17)

Since the contacts are by definition in equilibrium, one obtains (Appendix D.1)

$$\begin{array}{lll} \ensuremath{{\underline{g}}}^\mathrm{<}_{_... ... \ \ensuremath{{\underline{a}}}_{_{1,1}}\ f_{R} \ , \end{array}$$ (G.18)

where $\ensuremath{{\underline{a}}}=i(\ensuremath{{\underline{g}}}^\mathrm{r}-\ensure... ...line{g}}}^\mathrm{a})=-2\Im\mathrm{m} [\ensuremath{{\underline{g}}}^\mathrm{r}]$ is the spectral function and $f_{L(R)}$ is the FERMI factor of the left (right) contact. By defining the broadening function as

$$\begin{array}{lllllll} \ensuremath{{\underline{\Gamma}}}_{C_{... ...D} & = & \ \ensuremath{{\underline{\Gamma}}}_{R}\ , \end{array}$$ (G.19)

equation (G.17) can be rewritten as

$$\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr... ...+i\ \ensuremath{{\underline{\Gamma}}}_R \ f_{R} \ . \end{array}$$ (G.20)

In a similar manner one can show that

$$\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr... ...\ensuremath{{\underline{\Gamma}}}_{R}\ (1-f_{R})\ . \end{array}$$ (G.21)

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