# Computational Atomistic Modeling in Carbon Flatland and Other 2D Nanomaterials
## 2 Electronic Transport in Polycrystalline Graphene
In this framework, the zero-temperature linear conductance is given by the Landauer formula:  
\[G=\frac{e^{2}W}{\pi h}\int_{BZ}\mathcal{T}(E,k_{y})dk_{y}, \tag{1}\]  
where \(W\) is the transverse width of the system, \(\mathcal{T}\) is the transmission probability across the system, a function of carrier energy \(E\) and transverse momentum \(k_{y}\). The function \(\mathcal{T}(E,k_{y})\) is calculated through a fully quantum-mechanical approach:  
\[\mathcal{T}(E,k_{y})=Tr\left[\Gamma_{L}\mathcal{G}\Gamma_{R}\mathcal{G}^{ \dagger}\right], \tag{2}\]  
where the retarded Green's function \(\mathcal{G}\) is determined in terms of the device Hamiltonian \(H_{D}\) as:  
\[\mathcal{G}(E,k_{y})=\left[E+i0^{+}-H_{D}(k_{y})-\Sigma_{L}-\Sigma_{R}\right] ^{-1}. \tag{3}\]  
In Equation (3), \(\Sigma_{L(R)}\) represents the left (right) self-energy, describing the left (right) lead-to-device coupling. \(\Gamma_{L(R)}\) in Equation (2) is the transfer rate at the left (right) contact determined as \(\Gamma_{L(R)}=i(\Sigma_{L(R)}-\Sigma_{L(R)}^{\dagger})\). To investigate quantum transport in PG, GBs are modeled as periodic lines of defects with a periodicity that is expressed in terms of two translation vectors \(\vec{d}_{L}=(n_{L},m_{L})\) and \(\vec{d}_{R}=(n_{R},m_{R})\) in the left and right domains, respectively. [PERSON] et al. [41] demonstrated that GBs exhibit either metallic or semi-conducting transport regime depending on the interface nature. If \(n_{L}-m_{L}=3q\) but \(n_{R}-m_{R}\
eq 3q^{\prime}\) (or vice versa; \(q,q^{\prime}\in\mathbb{Z}\)), a transport gap appears as the allowed momentum-energy manifolds of the two crystalline domains are significantly misaligned. In the case of massless Dirac fermions, this transport gap is given by:  
\[E_{g}=\hbar v_{F}\frac{2\pi}{3|\vec{d}|}\approx\frac{1.38}{|\vec{d}|\text{[ nm]}}\text{[eV]}, \tag{4}\]  
where \(v_{F}\) is the Fermi velocity, and \(|\vec{d}|\) the common GB period (after applying possibly strain to obtain matching of both domains). Interestingly, the transport gap depends only on the GB periodicity. GBs are subsequently classified as either transparent (class-I) or reflective (class-II) to electronic transport. First, a symmetric GB with \(\vec{d}=(2,1)\) and \(|\vec{d}|=6.5\,\text{\AA}\) is investigated as illustrated in Figure 1. The system is transparent, as expected from its symmetry, with a conductance comparable to pristine graphene despite the large concentration of pentagon-heptagon pairs (see Figure 1a,c). In the asymmetric case (see Figure 1b,d), a transport gap of \(0.92\,\text{eV}\) appears when \(\vec{d}_{L}=(5,0)\), \(\vec{d}_{R}=(3,3)\) and \(|\vec{d}|=12.6\,\text{\AA}\), in relative accordance with Equation (4). Outside the transport gap, transmissions are considerably smaller compared to pristine graphene. However, systems considered in Figure 1a,b have short periodicities and are not representative of real GBs which exhibit a more complicated atomic structure. Consequently, more realistic GBs constructed from more complicated arrangements of polygons are studied hereafter. First, an asymmetric class-I GB is examined. This system is defined by \(\vec{d}_{L}=(11,3)\), \(\vec{d}_{R}=(10,5)\) and \(|\vec{d}|=32.0\,\)A, as illustrated in Figure 1e. The corresponding transmissions are significantly lower than pristine graphene, indicating an increased scattering due to the presence of dangling bonds. An asymmetric class-II GB is then considered, with \(\vec{d}_{L}=(24,0)\), \(\vec{d}_{R}=(16,12)\) and \(|\vec{d}|=59.6\,\)A (see Figure 1f). The interface was constructed by creating a vacancy after repeating \(n\) times an initial pattern (here \(n=4\)), which corresponds to applying a perturbation approach to the periodicity assumption. The related transmissions depicted in Figure 1d show a transport gap of 0.76 eV. However, a logarithmic scale plot analysis reveals non-zero conductance as large as \(10^{-2}\) G\({}_{0}\) nm\({}^{-1}\) in the transport-forbidden energy range.