# Computational Atomistic Modeling in Carbon Flatland and Other 2D Nanomaterials
## 3 Optical Absorption in Graphene and Borophene
Photon absorption is therefore by essence a 3D phenomenon and to account for the 2D nature of the atomically-thin structure within DFT calculation, the computed dielectric function \(\varepsilon\) has to be renormalized [58; 59] (including both interband transitions and Drude terms), as \(\varepsilon^{\text{2D}}=\varepsilon^{\text{3D}}\frac{H}{\hbar}\) with \(H\) the total height of the DFT 3D periodic cell which includes the vacuum distance introduced to separate the periodic images of the layer, and \(h\) the polarizable electronic thickness which can be estimated e.g., from van der Waals atomic radii [59]. To compute the transmittance \(T\), general Fresnel equations should be used or at least their limit for (ultra-)thin film which include secondary reflections at top and back side interfaces with external media [59; 60; 61], here vacuum:  
\[T(\hbar\omega)=\frac{1}{|1+\frac{\omega n\hbar\Im(n)}{c}|^{2}}, \tag{8}\]  
where \(n=\sqrt{\varepsilon}\) is the complex refractive index. The synthesis of monolayers of boron atoms, so-called borophene, has been reported in 2015-2016 [62; 63] showing a rich structural polytype. In particular three important structures have been reported, including a buckled triangular lattice and two flat layers named \(\beta_{12}\) and \(\chi_{3}\) (Figure 4a). These three borophene layers are (semi-)metallic as clearly shown by their electronic bandstructure and density of states (Figure 4b). Nevertheless and interestingly, the buckled triangular borophene layer displays a zero optical conductivity up to 3 eV (Figure 4c) and, hence, absorbs light only in the UV range or for higher photon energy (with a strong anisotropic character between \(\sigma_{xx}\) and \(\sigma_{yy}\)) [59]. In contrast, \(\beta_{12}\) and \(\chi_{3}\) borophene layers absorb light in the visible range starting from \(\sim\)0.5 and \(\sim\)1.0 eV, respectively, with a value varying between 0.5 \(\sigma_{0}\) and 1.5 \(\sigma_{0}\). For comparison, graphene exhibits a value slowly increasing from 1.0 \(\sigma_{0}\) up to roughly 2.0 \(\sigma_{0}\) at the end of the visible range. This value of 1.0 \(\sigma_{0}\) for graphene corresponds to the well-known absorbance of \(\pi\alpha\sim\) 2.3% of light, which gives, with reflectance being nearly zero (\(\sim\)0.05%), the high transmittance displayed in Figure 5. Again, buckled triangular borophene is fully transparent in this visible region neglecting the Drude corrections which, depending on the scattering time \(\tau\), can give rise, at most, to a decrease of transmittance down to 96% in the visible range, i.e., comparable to graphene. Linearly polarized light can be rotated when going through a material in the presence of a magnetic field. This phenomenon is known as Faraday rotation and originates from the optical Hall effect (OHE) for which \(\sigma_{xy}(\omega)\
eq 0\). The OHE has been investigated in thin films and more recently in 2D materials. Strain engineering has been proposed as an alternative to magnetic fields to induce finite optical Hall conductivity [64]. Indeed, without any symmetry breaking mechanisms, \(\sigma_{xy}\) is normally zero after integrating over the whole Brillouin zone (BZ). Uniaxial strain in graphene, but also shear strain, allows to split in energy the van Hove singularities and the related optical absorption peaks (see \(\sigma_{xx}\) in Figure 6). Because of this strain-induced asymmetry, the BZ integration at a given energy yields a positive or negative \(\sigma_{xy}\) (see \(\sigma_{xy}\) in Figure 6 for light linearly polarized along oblique direction \(\phi=45^{\circ}\) and for two strain directions \(\theta=0^{\circ}\) and \(90^{\circ}\) corresponding to armchair and zigzag respectively) which are responsible for clockwise or anti-clockwise Faraday rotation (Figure 6(inset)). The Faraday rotation angle is however limited to a few degrees for a strained monolayer and its exact value depends on the strain amplitude and the orientation with respect to light polarization.