Nano-spectroscopy of excitons in atomically thin transition metal dichalcogenides

Excitons play a dominant role in the optoelectronic properties of atomically thin van der Waals (vdW) semiconductors. These excitons are amenable to on-demand engineering with diverse control knobs, including dielectric screening, interlayer hybridization, and moiré potentials. However, external stimuli frequently yield heterogeneous excitonic responses at the nano- and meso-scales, making their spatial characterization with conventional diffraction-limited optics a formidable task. Here, we use a scattering-type scanning near-field optical microscope (s-SNOM) to acquire exciton spectra in atomically thin transition metal dichalcogenide microcrystals with previously unattainable 20 nm resolution. Our nano-optical data revealed material- and stacking-dependent exciton spectra of MoSe2, WSe2, and their heterostructures. Furthermore, we extracted the complex dielectric function of these prototypical vdW semiconductors. s-SNOM hyperspectral images uncovered how the dielectric screening modifies excitons at length scales as short as few nanometers. This work paves the way towards understanding and manipulation of excitons in atomically thin layers at the nanoscale.


Supplementary Note 1: The analytical relation between the near-field signal and the permittivity
In this section, we derive a simplified, yet intuitive relation between the scattering response and the permittivity of the TMDs before performing the numerical simulation in the next section. It should be noted that all the data analyses in the manuscript are based on rigorous numerical simulation discussed in the next section. The sample structure includes three regions: air (denoted by subscript "a") at the top, TMDs with thickness in the middle, and semi-infinite substrate (denoted by subscript "s").
The Fresnel reflection coefficient for p-polarized electric field servers as an estimate for the scattering response in this section and is given by Here, = − 0 ( − 1) is the sheet conductivity of van der Waals TMDs, where 0 is vacuum permittivity, is the relative permittivity of TMDs. For atomically thin TMDS, only the in-plane conductivity contributes to the reflectivity 1 . For other materials, to take the anisotropy of materials into consideration, we use both in-plane and out-of-plane permittivity. , are the in-plane permittivities of the substrate and air, respectively.
, are the out-of-plane permittivities of the substrate BN and air, respectively. = √ • . For air, = = ; for the h-BN, the in-plane and our-of-plane permittivity do not have much difference in the spectral range we investigated, so we assume = = 3 .
( corresponds to air or substrate) is the wave vector along out-of-plane direction, which is given by , where is the in-plane wave vector.
At large , which is determined by the tip radius, we can make the approximation, From these definition, we can get, We denote Δ as the difference in reflection coefficients for structures including and excluding the TMDs.
Here, the term encodes the properties of the TMDs and determines the near-field signal.
This is the key result of this part.
At the far-field limit and in the near-infrared spectral range, the large damping at the room temperature makes that the real dielectric function remains positive and the imaginary dielectric function has a finite positive value. With a layer thickness of 0.7 nm, this yields ( − 1) ≪ 1.
At the near-field limit, is determined by the tip apex. The dominant momentum is about 1/ , where is the tip radius (~25 nm). When the thickness of the material is smaller than the tip radius, ( − )( − 1) remains marginal compared to ( − ) ( + ). We can evaluate the two terms in the denominator using = 3.8, = 1, the terms ( − )( + ) is about 16.24, while the second term Here we take = 0.7 nm, = 4 × 10 5 −1 , and =

10.
From the above discussion, we find that in the few-layer TMD case, Here, 0 is the complex reflection without TMDs. From the formula, we can clearly see that Δ is proportional to TMD's permittivity.
The near-field signal can be denoted as 2,3 , where the weighting function ( ) denotes the momentum distribution and is determined by the tip geometry. Here all the data can be normalized to the substrate (h-BN). In weak resonance limit, of the substrate and the TMDs do not have much dependence on the .
Therefore, the normalized near-field signal is approximately proportional to Δ .
The key results derived from this formula can be summarized as follows: 1) When the excitation energy is far away from the excitonic resonance energies, the imaginary part of permittivity is zero. So the phase signal for the sample with and without TMDs are the same. Generally, the phase signal peaks around the exciton resonance energy, where the imaginary part of the permittivity is maximum.
2) The near field signal is proportional to the layer thickness if the change of the permittivity is negligible. So the near field amplitude linearly increases as a function of the layer numbers (see Supplementary Figure 7).
3) The reflectivity is proportional to the momentum q. Therefore, the large momentum given by the sharp tip in the near-field experiment boosts the light-TMDs interaction. This is one of the factors that give raise to strong near-field interaction at the nanometer scale.
We note that in this section we focus on the case that the dielectric function has large damping. When damping is reduced, such as through low-temperature experiments, the real dielectric function will go across zero and will strongly depend on the momentum q. These consequences give rise to different relation between the near-field signal and the dielectric function. TMDs around the exciton resonance frequency can be described by a Lorentz oscillator (eq. 1 in the main text). Once the dielectric function of each layer is known, ( , ) can be calculated following the transfer matrix formalism and can be evaluated using equation (8).
In our case, we aim to solve the reverse problem where is experimentally measured and the TMD dielectric function is to be solved. This problem can be approached using various techniques [8][9][10][11] . Here we employ a fitting procedure: the parameters in equation (1) in the main text are used as free parameters. Staring with estimated initial values for these fitting parameters, iterative optimization algorithms such as the BFGS or the Powell methods are then used to minimize the error function, which is defined as the sum of squared differences, in the parameter space. The parameters that give the best fit are considered as the extracted value and the corresponding ε( ) for the TMD is calculated using equation 1 in main text.
of the realistic tip geometry, its validity has been demonstrated in numerous studies 3,12,13 .
Recent simulation study has also demonstrated that the radiation pattern from a realistic tip geometry is very reminiscent of the point-dipole located at the tip apex 5