2.7.2 Diffusive Transport

In case of incoherent transport, the electron wave-functions can not be described by a single phase over the entire sample. If one considers multiple reflections, for example between scattering centers $ i=1$ and $ i=2$, the overall transmission probability $ \mathcal{T}_{12}$ between these two scattering centers is given by

\begin{displaymath}\begin{array}{l}\displaystyle \mathcal{T}_{12} \ = \ \mathcal...
...}_1\mathcal{T}_2}{1-\mathcal{R}_1\mathcal{R}_2} \ , \end{array}\end{displaymath} (2.26)

where the effect of interference during a scattering event is neglected. One can rewrite (2.26) as

\begin{displaymath}\begin{array}{l}\displaystyle \frac{1-\mathcal{T}_{12}}{\math...
...\ + \ \frac{1-\mathcal{T}_{2}}{\mathcal{T}_{2}} \ . \end{array}\end{displaymath} (2.27)

Using the formula for the resistance given by (2.25) and (2.27), one can see that the resistance of the wire is additive, $ R_{12}=R_{1}+R_{2}$. Applying this result to the case of $ N$ scatterers yields

\begin{displaymath}\begin{array}{l}\displaystyle R \ - \ R_\mathrm{c} \ = \ R_{0...
...m N \ R_{0} \ \frac{1-\mathcal{T}}{\mathcal{T}} \ . \end{array}\end{displaymath} (2.28)

where $ \mathcal{T}$ is an average transmission probability for an individual scattering event over a mean free path. Thus the total resistance is given by a series connection of microscopic resistances. This is nothing but OHM's law, according to which the microscopic resistance is proportional to $ L$. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors