Dielectric permittivity extraction of MoS$_2$ nanoribbons using THz nanoscopy

The nanoscale optical properties of high-quality MoS$_2$ nanoribbons are investigated using THz nanoscopy based on a scattering-type scanning probe. The nanoribbons comprise a multi-layer core, surrounded by monolayer edges. A featureless complex permittivity spectrum covering the range 0.6-1.6 THz is extracted from experimental time-domain measurements through a minimization procedure, adopting an extended finite-dipole model of the probe-sample interaction. Real-space mapping of the nanoribbon reveals variations in the local permittivity down to the instrument-limited resolution, on the order of 30 nm. Clustering analysis statistically identifies regions of lower apparent permittivity that we attribute to a high curvature at the edges of the nanoribbon causing an increase in local material strain or cross-talk in the measured signal with topography-induced measurement artifacts. The core of the nanoribbon contains two regions that follow tightly distributed, but slightly shifted Gaussian statistics in complex permittivity space, with the real part mean of both distributions lying around 5.4 and compatible with literature values of the static permittivity of thin-film MoS$_2$ reported previously. Our results show that the nanoribbons exhibit a modest degree of dielectric variation at the nanoscale that could be explained by heterogeneous doping or variations in the local defect density. We believe that our approach could be useful for the direct real-space measurement of dielectric disorder in other low-dimensional semiconducting material systems.