4.5.1 Semi-Infinite CNT Contacts

Figure 4.8 shows carbon rings of A or B-type coupled to a semi-infinite CNT acting as a contact. Each circle (rectangle) represents a carbon ring consisting of A or B-type carbon atoms. The carbon ring couples to the nearest ring, with a coupling matrix of $\ensuremath{{\underline{t}}}_1$ or $\ensuremath{{\underline{t}}}_2$, and $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{i,i}}$ is the surface GREEN's function for the $i$th ring in the left extension, ordered from the channel-contact interface. The recursive relation (G.22) can be applied to the CNT in Fig. 4.8 and gives

$$\begin{array}{ll} [\ensuremath{{\underline{A}}}_{L_{1}} \ - \... ..._{L_{2,2}} \ & = \ \ensuremath{{\underline{I}}} \ , \end{array}$$ (4.31)

where $\ensuremath{{\underline{A}}}_{L_{i}}=E\ensuremath{{\underline{I}}}_i-\ensurema... ...}}}_{L_{i}}- \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_{\mathrm{Scat}_{i,i}}$ (see Appendix G.1), and $\ensuremath{{\underline{t}}}_1$ and $\ensuremath{{\underline{t}}}_2$ are given by (4.17) and (4.18), respectively. Since the potential is invariant inside the contact, $\ensuremath{{\underline{A}}}_{L_{1}}=\ensuremath{{\underline{A}}}_{L_{2}}$. Furthermore, $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{3,3}}=\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{1,1}}$ due to the periodicity of the CNT lattice. Using these relations, (4.31) represent two coupled matrix equations with two unknowns, $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{1,1}}$ and $\ensuremath{{\underline{g}}}^\mathrm{r}_{L_{2,2}}$, which can be solved by iteration. However, in mode-space representation matrices $\ensuremath{{\underline{t}}}_1$ and $\ensuremath{{\underline{t}}}_2$ are replaced by the numbers $t_1=t$ and $t^\nu_2$, respectively. As a result, the surface GREEN's function for each mode can be calculated analytically by solving a quadratic equation

$$\begin{array}{l}\displaystyle g^\mathrm{r^\nu}_{L_{1,1}} \ = ... ... - \ 4A^2_{L_{1}}\ t_1^2}} {2{A_{L_1}}\ t_1^2 } \ . \end{array}$$ (4.32)

The self-energy of the left contact for the $\nu$th mode is therefore given by

$$\begin{array}{l}\displaystyle {\Sigma^\mathrm{r^\nu}_L} \ = \ t_1^2 \ g^\mathrm{r^\nu}_{L_{1,1}} \end{array}$$ (4.33)

A similar relation holds for the right contact self-energy.

Figure 4.8: Computing the surface GREEN's function for the left contact. The surface GREEN's function for the $i$th ring inside the contact is $g_i$.

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