Routing Light Emission from Monolayer MoS$_2$ by Mie Resonances of Crystalline Silicon Nanospheres

A dielectric Mie-resonant nanoantenna is capable of controlling the directionality of the emission from nearby quantum emitters through the excitation of multiple degenerate Mie resonances. A crystalline silicon nanosphere (Si NS) is a promising candidate for a dielectric nanoantenna because crystalline Si has a large refractive index (3.8 at 650 nm) and the small imaginary part of a complex refractive index (0.015 at 650 nm) as an optical material. In this work, we control the emission directionality of excitons supported by monolayer transition metal dichalcogenides (1L-TMDCs) using a Si NS. We first discuss the condition to extract the emission preferentially towards the Si NS side from the analytical calculations. We then study the photoluminescence (PL) of 1L-TMDCs on which differently sized single Si NSs are placed. We show that the PL spectral shape strongly depends on the emission direction, and that the emission toward the Si NS side (top) with respect to the opposite side (bottom) is the largest at wavelengths between the magnetic dipole and electric dipole Mie resonances of a Si NS. Finally, we quantitatively discuss the spectral shape of the top-to-bottom ratio from numerical simulations.