Electrodynamics of two-dimensional materials: Role of anisotropy

Two-dimensional (2D) materials are intrinsically anisotropic, and an accurate description of their out-of-plane response to an electromagnetic field is more and more important as new materials with diverse properties are proposed. Their electromagnetic properties are often modeled using a single sheet with a surface susceptibility or conductivity or by means of a thin film of finite thickness with an effective bulk permittivity. The discordances between these two approaches lead to two irreconcilable interpretations of the optical characterizations and uncertain predictions of electromagnetic responses. Here, we fully account for the particular anisotropy of 2D materials and reconcile both approaches. We propose a unified description for the electromagnetic properties that applies to 2D heterostructures for all polarizations and at all angles of incidence. In particular, we determine the class of materials for which both models can be used indifferently and when particular care should be taken to select the thickness and the tensorial response of the effective thin film. We illustrate our conclusions on extensively studied experimental quantities such as transmittance and ellipsometric data of graphene and metal dichalcogenides. We discuss similarities and discrepancies reported in the literature when single-sheet or thin-film models are used.

Figure 1

Schematic representation of the two configurations. Left: current sheet model. Right: thin film with an effective material $f$ extending over a distance $d_a$ ($d_b$) on the $a$ ($b$) side. The wave vectors of the forward $F$ and backward $B$ fields in media $a$ and $b$ are denoted by ${\vec{k}}_{F,B}^{a,b}$. The reference frame is $Oxyz$.

Figure 2

Relative difference $\mathrm{\Delta }T$ between the transmittance computed in the single sheet and in the anisotropic thin-film models, with respect to the real and imaginary parts of the single-sheet susceptibility, related to the conductivity by $k_0{\chi }_x^s=\mathrm{i}{\sigma }_x^s/\left({\varepsilon }_{0}c\right)$. The system considered is air/2D material/glass. The refractive index of glass is taken to be 1.5. (a) Infrared EM radiation ($\lambda =1550\mathrm{nm}$). (b) Visible light ($\lambda =700\mathrm{nm}$).

Figure 3

Same as Fig. 2, but for the isotropic thin-film model. The circled areas indicate graphene's adimensional parameter $k_0{\chi }^s$. To determine it we use the Kubo formula at a fixed wavelength of (a) 700 nm or (b) 1550 nm over a range of Fermi levels from 0.05 to $1\mathrm{eV}$ and a range of relaxation times from 10 to $200\mathrm{fs}$. Within this range of parameters, an artificial plasmonic resonance appears in the left panel.

Figure 4

Difference of transmittance $\mathrm{\Delta }T$ between the isotropic and anisotropic thin films of graphene for a system of air/graphene/glass. The refractive index of glass is taken to be 1.5. Graphene is modeled using the Kubo formula with $E_F=0.4\mathrm{eV}$, $\tau =100\mathrm{fs}$. The red dashed line represents a thickness equal to $1/1000$ of the wavelength.

Figure 5

Reduced transmittance [brackets in (17) and (18)] for an air/graphene/glass structure: $n_a=1$, $n_b=1.5$, ${\chi }_x^s=(1.50+2.29i)\mathrm{nm}$ at a wavelength of $634\mathrm{nm}$ [15]. Curves are for the anisotropic single sheet with no out-of-plane response (${\chi }_z^s=0$), the isotropic thin-film model (${\varepsilon }_z^f={\varepsilon }_x^f$) with $d_f=0.34\mathrm{nm}$ and $d_f=5\mathrm{nm}$, and isotropic current sheet model (${\chi }_z^s={\chi }_x^s$).

Figure 6

Values of $\mathrm{Re}\left[{\chi }_x^s\right]$ for a ${\mathrm{MoS}}_2$ single layer retrieved with the isotropic thin-film model (${\left[{\chi }_x^s\right]}_i$, blue curve and crosses) and the anisotropic current sheet model (${\left[{\chi }_x^s\right]}_a$, black curve and squares). Data are from Ref. [20]. The red curve (diamonds) is calculated from ${\left[{\chi }_x^s\right]}_i$ shifted by $-{\varepsilon }_x^b{d}_f$ and is almost superimposed on the black curve. The inset displays the difference ${\left[{\chi }_x^s\right]}_i-{\left[{\chi }_x^s\right]}_a$ (green curve with pluses) and $-{\varepsilon }_x^b{d}_f$ (cyan curve with circles). The dielectric function of N-BK7 glass (substrate used in Ref. [20]) is taken from Sellmeier's equation provided by Schott [35].