Unravelling strong electronic interlayer and intralayer correlations in a transition metal dichalcogenide

Electronic correlations play important roles in driving exotic phenomena in condensed matter physics. They determine low-energy properties through high-energy bands well-beyond optics. Great effort has been made to understand low-energy excitations such as low-energy excitons in transition metal dichalcogenides (TMDCs), however their high-energy bands and interlayer correlation remain mysteries. Herewith, by measuring temperature- and polarization-dependent complex dielectric and loss functions of bulk molybdenum disulphide from near-infrared to soft X-ray, supported with theoretical calculations, we discover unconventional soft X-ray correlated-plasmons with low-loss, and electronic transitions that reduce dimensionality and increase correlations, accompanied with significantly modified low-energy excitons. At room temperature, interlayer electronic correlations, together with the intralayer correlations in the c-axis, are surprisingly strong, yielding a three-dimensional-like system. Upon cooling, wide-range spectral-weight transfer occurs across a few tens of eV and in-plane p–d hybridizations become enhanced, revealing strong Coulomb correlations and electronic anisotropy, yielding a two-dimensional-like system. Our result shows the importance of strong electronic, interlayer and intralayer correlations in determining electronic structure and opens up applications of utilizing TMDCs on plasmonic nanolithrography.

Using the W-VASE analysis program, a model of the sample is created which includes the MoS2 bulk material and surface effects (e.g. roughness, oxidation etc.). As the sample thickness is very large, the complex dielectric function of the MoS2 can be determined through the best fits to the output data  and . Single crystal MoS2 is known to be anisotropic so the spectroscopic ellipsometer was used in the Mueller-Matrix mode with a rotational sample stage. 1 parts of the complex dielectric function, respectively, on a linear scale. The reflectivity data taken from the SUV beamline are selected in the range 3.5eV -45eV, whilst the ellipsometry results cover the range 0.6eV -5. 5eV, providing ~3.0eV overlap for normalization.
Spectroscopic ellipsometry allows us to determine the complex dielectric function of a material from which the self-normalised reflectivity can be calculated using the following: where n and k are the refractive index and absorption coefficient respectively and can be calculated from the real ( 1 ) and ( 2 ) imaginary parts of the complex dielectric function using: The soft X-ray reflectivity is then normalised using spectroscopic ellipsometry at the lower energy range, 3 and X-ray data is tabulated at the high energy range from 30 eV onward. 4 In order to determine the complex dielectric function of the MoS2 across the whole energy range, we use a model based on Kramers-Kronig-transformable Drude-Lorentz oscillators of the form: where ∞ is the high-frequency dielectric constant and , , 0, and Γ are the plasma frequency, the transverse frequency (eigenfrequency), and the line width (scattering rate) of the k-th oscillator, respectively. Using the new methodology introduced here, our model is constrained by the complex dielectric function measured with spectroscopic ellipsometry.
Therefore, we are able to obtain reflectivity with very high accuracy and resolution as a function of temperature in such a broad energy range.
Supplementary Within a single measurement, spectroscopic ellipsometry measures the change in phase and amplitude of polarised light after being transmitted or reflected from a material, from these we can simultaneously determine the complex dielectric function, loss function and reflectivity. For data that cover a limited spectral range, the derived complex dielectric function towards these limits is not accurate. Supplementary Fig. 3 shows the complex dielectric function determined from a small energy range (SE data) compared to a large energy range (Combined SE and high-energy reflectivity data). It is clear that the complex dielectric function is severely underestimated above 2.5eV using data from only a limited energy range. Therefore, low energy measurements, especially those trying to determine the complex dielectric function and spectral weight transfer, are not always valid. Higher energy band measurements must also be taken into consideration.
Supplementary Figure  calculations. 5 More numbers of valence and conduction bands mean that there are more higher energy bands involved and a higher number of valence and conduction bands is needed to properly calculate the spectrum in the energy range of interest. We utilize Nv = Nb = 7 to give a converging result.
From Supplementary Fig. 7, we see that the single-band transition Nv = Nc = 1 can already form the excitonic peaks A and C but the excitonic peak B is missing. By increasing the number of conduction bands at high energies, we can see that low-energy excitations are enhanced, including these excitons. This further supports that the low-energy excitations are determined by high-energy bands. We note that our calculated values for the three excitonic peaks are in average overestimated by ~0.57 eV, which is due to the limited of number of kpoints in the GW calculations.