Theoretical investigation of patterned two-dimensional semiconductors for tailored light--matter interactions

TL;DR

The paper develops a tutorial-style theoretical framework for tailoring light–matter interactions in patterned 2D semiconductors, using a 2D optical conductivity σ(ω) at interfaces. It presents two ribbon-patterning approaches (edge-condition and Fourier-series) to compute optical responses and two spherical geometries (hollow shells and coated cores) via a modified Mie theory that incorporates a surface coating g(ω) = i σ(ω)/(ε0 ω R). Applying these methods to hBN in the UV and WS2 in the visible reveals exciton-polariton–related resonances in ribbons and strong pattern-dependent hybridization in spheres, with hollow spheres showing particularly rich tunability. The framework generalizes to any polaritonic 2D material, providing a versatile toolset for designing nanoscale patterns that control optical response.

Abstract

We introduce theoretical methods for describing the optical response of two-dimensional (2D) materials patterned at the nanoscale into both arrays of ribbons along a planar surface and spherical particles. Fourier-Floquet decompositions of the electromagnetic fields are used in order to obtain the reflectance, transmittance and absorbance of the nanoribbon array. The spherical particles consist of a vacuum or dielectric core, coated by single 2D material layers. A Mie theory, with boundary conditions modified to accommodate a 2D material at the interface, is applied to theoretically examine these spherical particles. As examples of 2D materials, we consider the excitonic response of hexagonal boron nitride in the ultraviolet, and of the transition-metal dichalcogenide WS2 in the visible. The most important steps and equations for implementing the various methods are provided as a means to an easy introduction to the theory of patterned 2D materials. This renders the article a toolset for investigating the patterning of any 2D material with the intention to tune their optical response and/or introduce hybridization schemes with their excitons. The methods are not restricted to exciton polaritons in 2D semiconductors, but can be applied, by simple replacement of the optical conductivity, to 2D materials exhibiting any polaritonic response.

Figures (7)

• Figure 1: Optical conductivity of monolayer (a) hBN, obtained as discussed in Sec. \ref{['sec:hBN']}, and (b) WS2, obtained as discussed in Sec. \ref{['sec:WS2']}. Both optical conductivities are normalized to $\sigma_0 ={e^2}/{4\hbar }$ and given in terms of the real (imaginary) part as the blue (red) curve. The vertical dashed lines denote the main excitons in the two materials.
• Figure 2: Schematic (upper) and illustration (lower) of the system described in Sec. \ref{['sec4']} consisting of nanoribbons of a 2D semiconductor of optical conductivity $\sigma(\omega)$ that are infinitely long in the $y$ direction, with a finite width $w$ and a periodicity $L$ along $x$. The interface of nanoribbons is placed between dielectrics $\varepsilon_1$ and $\varepsilon_2$, and it is illuminated by plane waves of angular frequency $\omega$ incident at an angle $\theta$ onto the interface.
• Figure 3: Absorbance of nanoribbons of a 2D semiconductor encapsulated by a dielectric $\varepsilon_1 = 3$ above, and a dielectric $\varepsilon_2 = 4$ below, with radiation at normal incidence $\theta=0$. (a) and (e) show contour maps of the absorbance for hBN nanoribbons computed using the edge condition method and the Fourier series method, respectively. The excitonic resonances of hBN are given as the horizontal dashed white lines, and the vertical solid white line marks a ribbon width of $w=6\ \mathrm{nm}$, where the absorbance of hBN nanoribbons at this width is given in (b) and (f) for three periodicities $L$ of the nanoribbon arrays computed using the edge condition method and the Fourier series method, respectively. (c) and (g) show contour maps of the absorbance for WS2 nanoribbons computed using the edge condition method and the Fourier series method, respectively. The $A$ and $B$ peaks of WS2 are given as the horizontal dashed white lines and the vertical solid white line marks a ribbon width of $w=10\ \mathrm{nm}$, where the absorbance of WS2 nanoribbons at this width is given in (d) and (h) for three periodicities $L$ of the nanoribbon arrays computed using the edge condition method and the Fourier series method, respectively.
• Figure 4: Comparison of the edge condition and Fourier series method when applied to hBN nanoribbons of varying width $w$ with a periodicity of $L=2w$ in an environment $\varepsilon_1=3$ and substrate $\varepsilon_2=4$ for radiation of normal incidence $\theta=0$. (a) shows the spectral position of the two main peaks and (b) shows normalized peak values in absorbance for the edge condition (Fourier series) method as the blue (red) curves.
• Figure 5: Illustration (left) and schematic (right) of the system described in Sec. \ref{['sec3']} consisting of a spherical particle of radius $R$ and a relative permittivity $\varepsilon_1$ coated by a 2D material of conductivity $\sigma(\omega)$. The particle is placed in a medium of relative permittivity $\varepsilon_2$ and is illuminated by a plane wave of angular frequency $\omega$.