# Computational Atomistic Modeling in Carbon Flatland and Other 2D Nanomaterials
## 3 Optical Absorption in Graphene and Borophene  
Owing to their extreme thinness, 2D materials are expected to absorb only a small amount of light. Therefore, even metallic 2D layers can be almost transparent in particular in the visible range. Such transparent conducting layers are of broad interest for various applications, e.g., as front current collector in solar panels. To assess the optical properties of 2D materials, a key quantity is the optical conductivity \(\sigma\), whose real part is related to the imaginary part of the dielectric function \(\varepsilon\) (Equation (5)). Based on vertical interband electronic transitions, \(\sigma\) can be routinely computed with both TB and DFT approaches using the Kubo formula (Equation (6)). \[\Re(\sigma_{\alpha\beta}(\hbar\omega)) = \omega\varepsilon_{0}\Im(\varepsilon_{\alpha\beta}(\hbar\omega)), \tag{5}\] \[\Re(\sigma_{\alpha\beta}(\hbar\omega)) = \frac{2e^{2}\hbar}{iS}\sum_{\mathbf{k}\in\mathbb{BZ}}\sum_{n,m} \frac{f\left(E_{n}\right)-f\left(E_{m}\right)}{E_{n}\left(\mathbf{k}\right)-E _{m}\left(\mathbf{k}\right)}\] (6) \[\times \frac{\left\langle n\right|\phi_{\alpha}\left|m\right\rangle \left\langle m\right|\phi_{\beta}\left|n\right\rangle}{\hbar\omega+E_{n}\left( \mathbf{k}\right)-E_{m}\left(\mathbf{k}\right)+i\eta},\]  
where \(\alpha,\beta\) are the principal in-plane directions (\(x\) and \(y\)), \(S\) is the cell surface, \(f(E)\) is the Fermi-Dirac distribution function, \(E_{n,m}\left(\mathbf{k}\right)\), \(\left|n\right\rangle\) and \(\left|m\right\rangle\) are the eigenenergies and eigenstates, \(\widehat{\upsilon}_{\alpha/\beta}\) is the velocity operator along a given direction, and \(\eta\) is a small broadening. The longitudinal optical conductivity components are \(\sigma_{xx}\) and \(\sigma_{yy}\) while \(\sigma_{xy}\) is the optical Hall conductivity. Both are given in units of the universal value of \(\sigma_{0}=\frac{e^{2}}{4\hbar}\) (\(\sigma_{0}=\alpha\pi\varepsilon_{0}c\) with \(\alpha\), \(\varepsilon_{0}\) and \(c\) the fine-structure, the vacuum electric permittivity and the speed of light constants, respectively). As a consequence of intraband transitions  
Figure 3: Aharonov–Bohm interferences in polycrystalline graphene in the quantum Hall regime: (**a**) Schematic of considered systems, (**b**) computed conductance as a function of the magnetic field, and (**c**,**d**) images of the local density of left-injected electronic states at magnetic fields (**c**) \(B_{1}\) (conductance peak), marked in red in (**b**,**d**) \(B_{2}\) (conductance valley) marked in blue in (**b**). Reprinted and adapted with permission from reference [46]. in metallic systems, Drude corrections are added to \(\varepsilon\) (The real part of \(\varepsilon\) can also be corrected by the Drude term with \(\Re(\varepsilon^{\text{[PERSON]}}(\hbar\omega))=-\frac{\omega_{p}^{2}}{\omega^{2}+ \gamma^{2}}\))  
\[\Im(\varepsilon^{\text{[PERSON]}}(\hbar\omega)) = \frac{\gamma}{\omega}\frac{\omega_{p}^{2}}{\omega^{2}+\gamma^{2}}, \tag{7}\]  
where \(\omega_{p}\) (\(\omega_{p}=\sqrt{\frac{\hbar\omega^{2}}{m\varepsilon_{0}}}\) with \(n\) the electron density and \(m\) an effective mass. In practise, \(\omega_{p}\) is computed through Kramers-Kronig relation) and \(\gamma\)=\(\tau^{-1}\) are the plasmon frequency and the inverse scattering time of the Drude model, respectively. Light propagation is considered orthogonal to the 2D layers and the associated linearly oscillating electric field is thus in-plane along the \(\alpha\) direction.