Conductance and shot noise of pseudodiffusive and diffusive electron transport through graphene nanoribbons

Using KWANT or your own Python code, setup a two-terminal graphene nanoribbon with zigzag edges (ZGNR) where semi-infinite ZGNR leads of width $W$ are attached to central region of length $L$ and same width (in other words, if the central region is clean your two-terminal device is just an infinite homogeneous ZGNR). In the calculations below, you can fix the width at some value that would allow you to perform calculations on available computational resources while length will be varied.

Compute the zero-temperature conductance $G={\frac {2e^{2}}{h}}\sum _{n}T_{n}$, and the Fano factor, $F={\frac {\sum _{n}T_{n}(1-T_{n})}{\sum _{n}T_{n}}}$, quantifying the shot noise power, $S(0)=2F\langle I\rangle$, as a function of the length of the central region for two different device setups discussed below. Here $T_{n}$ is the transmission eigenvalue which is obtained by diagonalizing Hermitian matrix $\mathbf {t} \mathbf {t} ^{\dagger }$, where $\mathbf {t}$ is the transmission submatrix of the scattering matrix of the device.

Pseudodiffusive transport in clean ZGNRs

For this part of the Project, the central region in your two-terminal ZGNR device should have geometry $W>L$ and be covered by a gate electrode whose voltage is modeled as an on-site potential. The parameters like the gate voltage and Fermi energy can be set similarly to those in Fig. 5 of Phys. Rev. B 76, 205433 (2007).

Diffusive transport and localization in ZGNRs with vacancies

For this part of the Project, introduce vacancies [i.e., remove carbon atoms, see Phys. Rev. Lett. 115, 106601 (2015)] into the central region of your two-terminal ZGNR device of concentration $n_{\mathrm {ad} }=10\%$ and set the Fermi energy very close to the Dirac (or charge neutral) point $E=0$. Due to random distribution of vacancies, proper results for $G$ and $F$ should be obtained by disorder averaging them over at least 10 different vacancy configurations. An alternative is to use only one vacancy configuration and average $G$ and $F$ around some small interval (which also acts like simplistic dephasing), such as $[-0.01\gamma _{0},0.01\gamma _{0}]$, where $\gamma _{0}$ is the nearest-neighbor hopping in the tight-binding model of ZGNR. Note that wire of length $L$ will enter the diffusive transport regime, defined by electronic mean free path $\ell$ being smaller than $L$, once the Fano factor reaches $F\simeq 1/3$, see Phys. Rev. B 77, 081410(R) (2008). This should allow you to estimate the mean free path $\ell$. Examine localization properties at the Dirac point and away from it by plotting $G$ vs. length or $G$ vs. vacancy concentration at fixed length, akin to Figure 4 in Phys. Rev. Lett. 115, 106601 (2015) (where conductivity of bulk samples rather than conductance of nanowires was studied).