Optical modelling of 2D materials and multilayer systems: a complete picture

Bidimensional materials are, as their name suggests, ideally viewed as having no thickness. The advent of multilayer stacks of 2D 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 show how volume properties of stacked 2D layers can be calculated from boundary conditions and conversely. We introduce the layer, surrounded by vacuum, as a kind of transfer matrix with intrinsic parameters. This provides a link between continuous and discrete media, and a connection with the interface reflection and transmission coefficients calculated from microscopic models. We show how to model hybrid systems and identify in the zero-thickness limit the intrinsic parameters of the current sheet that represents the 2D material, namely the in-plane surface susceptibility $\chi^s$ and the out-of-plane parameter $\xi^s$ that corresponds to a displacement susceptibility. By considering anisotropic layers under the assumption that TE and TM modes are not coupled, we provide a unified vision of the different models used in optical characterization of 2D materials, including ellipsometry. We build on this to model 3D structures layer per layer and identify their effective permittivity and propagation constants. We show that our model fits existing ellipsometric data with the same reliability as the existing interface models, but with the advantage that multilayer and monolayer systems are described in a same way

Anisotropy and effective medium approach in the optical response of two-dimensional material heterostructures

Two-dimensional (2D) materials offer a large variety of optical properties, from transparency to plasmonic excitation. They can be structured and combined to form heterostructures that expand the realm of possibility to manipulate light interactions at the nanoscale. Appropriate and numerically efficient models accounting for the high intrinsic anisotropy of 2D materials and heterostructures are needed. In this article, we retrieve the relevant intrinsic parameters that describe the optical response of a homogeneous 2D material from a microscopic approach. Well-known effective models for vertical heterostructure (stacking of different layers) are retrieved. We found that the effective optical response model of horizontal heterostructures (alternating nanoribbons) depends on the thickness. In the thin layer model, well adapted for 2D materials, a counterintuitive in-plane isotropic behavior is predicted. We confront the effective model formulation with exact reference calculations such as ab initio calculations for graphene, hexagonal boron nitride (hBN), as well as corrugated graphene with larger thickness but also with classical electrodynamics calculations that exactly account for the lateral structuration.</p

Anisotropy and effective medium approach in the optical response of 2D material heterostructures

2D materials offer a large variety of optical properties, from transparency to plasmonic excitation. They can be structured and combined to form heterostructures that expand the realm of possibility to manipulate light interactions at the nanoscale. Appropriate and numerically efficient models accounting for the high intrinsic anisotropy of 2D materials and heterostructures are needed. In this article, we retrieve the relevant intrinsic parameters that describe the optical response of a homogeneous 2D material from a microscopic approach. Well-known effective models for vertical heterostructure (stacking of different layers) are retrieved. We found that the effective optical response model of horizontal heterostructures (alternating nano-ribbons) depends of the thickness. In the thin layer model, well adapted for 2D materials, a counter-intuitive in-plane isotropic behavior is predicted. We confront the effective model formulation with exact reference calculations such as ab-initio calculations for graphene, hexagonal boron nitride (hBN), as well as corrugated graphene with larger thickness but also with classical electrodynamics calculations that exactly account for the lateral structuration

Electrodynamic models of 2D materials: can we match thin film and single sheet approaches?

The electromagnetic properties of 2D materials are modeled either as single sheets with a surface susceptibility or conductivity, or as thin films of finite thickness with an effective permittivity. Their intrinsic anisotropy, however, has to be fully described to reliably predict the optical response of systems based on 2D materials or to unambiguously interpret experimental data. In the present work, we compare the two approaches within the transfer matrix formalism and provide analytical relations between them. We strongly emphasize the consequences of the anisotropy. In particular, we demonstrate the crucial role of the choice of the thin film's effective thickness compared with the parameters of the single sheet approach and therefore the computed properties of the 2D material under study. Indeed, if the isotropic thin film model with very low thickness is similar to an anisotropic single sheet with no out-of-plane response, with larger thickness it matches with a single sheet with isotropic susceptibility, in the reasonable small phase condition. We illustrate our conclusions on extensively studied experimental quantities such as transmittance, ellipsometry and optical contrast, and we discuss similarities and discrepancies reported in the literature when using single sheet or thin film models

Anisotropy and effective medium approach in the optical response of 2D material heterostructures

2D materials offer a large variety of optical properties, from transparency to plasmonic excitation. They can be structured and combined to form heterostructures that expand the realm of possibility to manipulate light interactions at the nanoscale. Appropriate and numerically efficient models accounting for the high intrinsic anisotropy of 2D materials and heterostructures are needed. In this article, we retrieve the relevant intrinsic parameters that describe the optical response of a homogeneous 2D material from a microscopic approach. Well-known effective models for vertical heterostructure (stacking of different layers) are retrieved. We found that the effective optical response model of horizontal heterostructures (alternating nano-ribbons) depends of the thickness. In the thin layer model, well adapted for 2D materials, a counter-intuitive in-plane isotropic behavior is predicted. We confront the effective model formulation with exact reference calculations such as ab-initio calculations for graphene, hexagonal boron nitride (hBN), as well as corrugated graphene with larger thickness but also with classical electrodynamics calculations that exactly account for the lateral structuration

Sociale ongelijkheid in sterfte in België. Vele dimensies, vele oorzaken

Analyses de l'évolution des inégalités sociales de mortalité en Belgique et de ses déterminants (emploi, logement, niveau d'instruction, état-civil

Inégalités sociales de mortalité en Belgique. De multiples dimensions, de multiples causes

Analyse de l'évolution des inégalités sociales de mortalité en Belgique et de ses déterminants (état-civil, emploi, niveau d'instruction et conditions de logement