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 the second part of the Project, introduce vacancies by removing carbon atoms from the central region of your two-terminal ZGNR device (where you do not use anymore gate potential as in the first part of the Project). You can start by using some large concentration $n_{\mathrm {ad} }=10\%$ of vacancies, and later also explore dependence on this number akin to analysis in , as in Phys. Rev. Lett. 115, 106601 (2015) for bulk graphene. Use the Fermi energy very close to the Dirac point at $E_{F}=0$, as well as exactly at this energy, and compute $G$ and $F$ as a function of lenght $L$ of the central region, as well as $n_{\mathrm {ad} }=10\%$ at some fixed length. Due to random distribution of vacancies, results for $G$ and $F$ should be obtained by disorder averaging them over at least 10 different vacancy configurations. Due to non-Gaussian distribution of these quantities over an ensemble of impurity configurations, the best is to plot geometric instead of arithmetic average, where geometric average is defined by:

${\overline {G}}={\sqrt[{N}]{G_{1}{\dot {G}}_{2}\cdots G_{N}}}$

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).