4.1.3 Interface Conditions

To account for interface conditions, grid points located at the boundary of the segments (see Fig. 4.4-a) are assigned three values, one for each segment (see Fig. 4.4-b) and a third value assigned directly to the interface, which can be used to formulate more complicated interface conditions such as, for example, interface charges.

As discussed in Section 4.1, charges on the CNT are approximated as sheet charges at the CNT-insulator interface. The boundary flux due to interface charges is simply added to the segment fluxes given by (4.5)

$$\begin{array}{ll} F_{i} \ + \ F_{i$$ (4.8)

with

$$\begin{array}{ll} F_{i$$ (4.9)

where $A_{i}=A_{i'}$ and is the interface charge density at some point along the insulator-CNT interface given by (4.2). This method satisfies the condition of the discontinuity of the electric displacement in the presence of interface charges. For the CNT the free-space relative permittivity, $\varepsilon_\mathrm{CNT}=1$, is assumed [241].

Figure 4.4: Interface points as given in a) are split into three different points having the same geometrical coordinates b) [240].

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