Optical modeling of single and multilayer two-dimensional materials and heterostructures

Bidimensional materials are ideally viewed as having no thickness, as their name suggests. Their optical response have been previously modelled by a purely bidimensional surface current or by a very thin film with some contradictory results. The advent of multilayer stacks of bidimensional materials and combinations of different materials in vertical van der Waals heterostructures highlights, however, that these materials have a finite thickness. In this article, we propose a new model that reconciles both approaches and we show how volume properties of stacked bidimensional layers can be calculated from the bidimensional response of each individual layer, and conversely. In our approach, each bidimensional layers is surrounded by vacuum and described as a kind of transfer matrix with intrinsic parameters that do not depend on the external medium. This provides a link between continuous thin films and discrete layers. We show how to model heterostructures of bidimensional materials and identify the parameters of the current sheet that represents the bidimensional material in the zero-thickness limit, namely the in-plane surface susceptibility and the out-of-plane displacement susceptibility. We show that our unified model is perfectly compatible with existing ellipsometric data with the same reliability as the existing interface model but with different values of the surface susceptibility or bulk dielectric function. We discuss in detail the origin of the discrepancies and show that our approach allows to determine intrinsic properties of the bidimensional materials with the advantage that multilayer and monolayer systems are described in a same framework.

Figure 1

Geometries considered in different models. (I) surface current used in AnisInt model and MicroDip model; (II) thin film used in IsoFilm model; (III) and (IV) continuous and discrete layer geometries used in our new AnisLay model, and compatible with multilayer systems. Surrounding materials are labeled $a$ and $b$. The interface (I) is described with a surface susceptibility ${\chi }^s$ (red line). The thin film (II) is described by a permittivity ${\varepsilon }_m$ and a thickness $d$. The discrete layer (III and IV) is modeled using either a bulk permittivity ${\varepsilon }_m$ or a current sheet with surface susceptibility ${\chi }^s$ (red line). Each layer is surrounded with vacuum ($v$) and has a total thickness $d$.

Figure 2

Adding a 2D material on stacked layers. Illustration of volume properties of a continuous medium on the left, with the transfer matrix ${\mathcal{V}}_m$, and of a discrete medium on the right, with the single layer matrix ${\mathcal{L}}_m$. The red line indicates surface currents associated to one layer.

Figure 3

Relations between the different models appearing in Table 1. Red arrows indicate loss of information. The green arrow requests additional information. Unnamed models are intermediate steps to link our model to the existing ones. Transfer matrices ${\mathcal{I}}_{pq}, {\mathcal{S}}_{vv}$ and ${\mathcal{L}}_m$ are defined in the text. The parameters appearing at the bottom of each box denote the quantities used in the transfer matrices and/or the TE and TM reflection coefficients, and hence in the ellipsometric parameters.

Figure 4

Hybrid structure made of alternated layers of type 1 and 2. The structure is modeled as a set of hybrid layers ($h$).

Figure 5

Graphene reflectance and ellipsometric curves at a wavelength of 633 nm. Our new model is compared to the interface response of the existing models. No experimental data are plotted here. Acronyms corresponds to those of Table 1. MicroDip model and AnisInt model (blue line) describe the interface and provide the same analytical expressions if (59) is applied. The AnisLay model plotted with the parameters of the AnisInt model differs strongly from it (red dashed). An optimal tuning of the AnisLay model parameters provides almost perfectly matching curves (green diamonds).

Figure 6

${\mathrm{MoS}}_2$ reflectance and ellipsometric curves at a wavelength of 633 nm. Our new model is compared to the interface response of the existing model. No experimental data are used here. Acronyms correspond to those of Table 1. MicroDip model and AnisInt model (blue line) describe the interface and provide the same analytical expressions if (59) is applied. The AnisLay model plotted with the parameters of the AnisInt model differs strongly from it (red dashed). An optimal tuning of the AnisLay model parameters provides almost perfectly matching curves (green diamonds).

Figure 7

Multilayer system. Schematic representation of the different models. $a$ (green) and $b$ (blue) denote surrounding media, $v$ (white) represents vacuum. Red lines correspond to surface currents. The interface configurations (MicroDip model and AnisInt model) do not contain vacuum between the surface current and the surrounding material. In the layer model (AnisLay model) surface currents are separated by vacuum. In other representations, a single current sheet is considered, with vacuum on one or two sides to compensate for the physical thickness.

Figure 8

Reflectance of multilayer systems of graphene sheets immersed in a polymer with refractive index 1.4233. The five geometries of Fig. 7 are considered. Left: 10 layers. Right: 100 layers. Top, TE reflectance. Bottom, TM reflectance.

Figure 9

Reflectance of multilayer systems of ${\mathrm{MoS}}_2$ sheets immersed in a polymer with refractive index 1.4233. The five geometries of Fig. 7 are considered. Left: 10 stacked layer. Right: 100 layers. Top, TE reflectance. Bottom, TM reflectance.