Giant optical anisotropy in transition metal dichalcogenides for next-generation photonics

Large optical anisotropy observed in a broad spectral range is of paramount importance for efficient light manipulation in countless devices. Although a giant anisotropy has been recently observed in the mid-infrared wavelength range, for visible and near-infrared spectral intervals, the problem remains acute with the highest reported birefringence values of 0.8 in BaTiS3 and h-BN crystals. This issue inspired an intensive search for giant optical anisotropy among natural and artificial materials. Here, we demonstrate that layered transition metal dichalcogenides (TMDCs) provide an answer to this quest owing to their fundamental differences between intralayer strong covalent bonding and weak interlayer van der Waals interaction. To do this, we made correlative far- and near-field characterizations validated by first-principle calculations that reveal a huge birefringence of 1.5 in the infrared and 3 in the visible light for MoS2. Our findings demonstrate that this remarkable anisotropy allows for tackling the diffraction limit enabling an avenue for on-chip next-generation photonics.


INTRODUCTION
Optical anisotropy plays a crucial role in light manipulation owing to birefringence phenomena, namely, doubling the incoming light into two different rays (called ordinary and extraordinary for uniaxial optical materials), which results in spatial and polarization separation, 1 through versatile optical components, [2][3][4] including polarizers, wave plates, multilayer mirrors, and phase-matching elements.Their performance primarily depends on the phase retardance (φ) between ordinary and extraordinary rays, which is proportional to the thickness (d) of the device and the birefringence (Δn) of the constituting materials.Thus, a large birefringence is highly favorable and beneficial since it leads to more compact and efficient devices.][7][8][9][10] Even the record-holders quasi-one-dimensional BaTiS3 and layered h-BN crystals improve this result by less than twofold (Δn ~ 0.8). 11,12The problem is partially solved in the mid-infrared range by large anisotropy in the biaxial van der Waals (vdW) crystals α-MoO3 and α-V2O5. 13,14Still, these materials become mostly isotropic in the visible and near-infrared light. 15Meanwhile, artificial design can offer large birefringence in metamaterials and metasurfaces. 16,17However, its widespread usage is impeded by optical losses and fabrication challenges. 18s a result, natural materials with giant anisotropy (Δn > 1) are in high demand both for scientific and industrial purposes.In this regard, transition-metal dichalcogenides (TMDCs) in a bulk configuration are promising candidates because of their strongly anisotropic vdW structure, which naturally leads to a large intrinsic birefringence.In particular, while MoS2 solids adopt an in-plane crystalline layered structure through strong ionic/covalent bonds between molybdenum (Mo) and sulfur (S) atoms, the out-of-plane link of these layers occurs via weak vdW forces in trigonal prismatic configuration 19 , as illustrated in Figure 1a.
As a consequence, a strong optical anisotropy emerges in TMDCs.A diagonal permittivity tensor can describe it with two optical constants corresponding to the crystallographic ab-plane and the c-axis. 20Interestingly, these anisotropic properties of TMDCs were qualitatively demonstrated back in 1967 by Liang et al., 21 but only currently attracted significant importance in experiments dealing with novel regimes of light-matter interactions 22,23 comprising excitonpolariton transport, 24 Zenneck surface waves, 25 tunable modal birefringence, 26 and anapoleexciton polaritons. 27Although a recent pioneering work by Hu and co-workers 20 reported a birefringence value of Δn = 1.4 for MoS2 at λ = 1530 nm, the values of asymmetric dielectric responses of MoS2 in a wide wavelength interval have so far remained unknown.Most likely, it stems from inherent experimental difficulties while measuring a high refractive index of anisotropic materials, which we overcome here by joining together far and near-field characterization techniques.The method allows us to obtain the full dielectric tensor in a wide wavelength range (360 -1700 nm) and reveals giant birefringence for MoS2 as high as Δn ~ 1.5 in the infrared and Δn ~ 3 in the visible spectra.This outstandingly large optical anisotropy accompanied by a high refractive index n ~ 4 paves the way for highly efficient optics and light manipulation.

RESULTS
The measurement of the anisotropic optical response of TMDCs is a challenging task because of multiple experimental obstacles.One of the hardest part is the implementation of traditional spectroscopic diffraction-limited characterization techniques, including transmittance, reflectance, and ellipsometry measurements, since bulk TMDCs are usually prepared by exfoliation method and, as a result, the samples obtained have lateral dimensions of tens micrometers.A second difficulty is related to the measured signal's low-sensitivity to the out-of-plane components attributed to a large in-plane refractive index n ~ 4. For instance, in ellipsometry configuration, incident light at 80° gives the refraction angle of only 14.3° according to the Snell's law, which implies that the probed electric field of the refracted light mainly lies along the layers and thus, is almost insensitive to the out-of-plane dielectric component.
To overcome the latter, we prepared an exfoliated thin MoS2 films on 285 nm SiO2/Si substrate and verified their 2H semiconducting configuration by the resonant Raman 28 demonstrated in the inset of Figure 1a since MoS2 exists in nature in three phase modifications: semiconducting (2H and 3R) and metallic (1T). 29The thick layer of silicon oxide will produce an interference-like pattern for ordinary and extraordinary beams, which can readily be detected employing phasesensitive techniques such as spectroscopic ellipsometry (SE).For this reason, we performed imaging spectroscopic ellipsometry (ISE) measurements in the 360 -1700 nm wavelength range, given that it allows measuring samples down to several micrometers since it is a hybrid of ellipsometer and microscope (Methods). 30It allowed us to record the ellipsometry signal Ψ and Δ (Methods) from several regions of interest (ROI) of the flakes within the selected field of view, as indicated in Figure 1b.As a result, multiple sample analysis was implemented to increase data reliability (see Supporting Information).The resulting ellipsometry spectra in Figure 1c indeed shows a pronounced asymmetrical interference-like peak at around 900 nm induced by a large phase difference between ordinary and extraordinary beams, indicating (without any modeling of the experimental curves) a large birefringence stemming from a strong anisotropy between the caxis and the ab-plane.Notwithstanding the noticeable anisotropic feature at 900 nm in the measured spectra, to accurately retrieve the complete dielectric tensor of MoS2 and enable predictive capabilities for future advanced optical devices using this material, it is imperative to develop an accurate dielectric function model.In that case, the best route towards a dielectric description is to utilize the crystallographic features of MoS2.Briefly, in its 2H structure consecutive layers are linked by weak vdW forces and rotated by 180° with respect to each other leading to a strong suppression of interlayer hopping for both electrons and holes, and, thus, preventing the formation of tightly bound interlayer electron-hole pair upon light illumination, the so-called excitons. 31,32Therefore, along the c-axis, the material is transparent, and its dielectric response is described by a Cauchy model, which is an evident consequence of Kramers-Kronig relation between real (n) and imaginary (k) parts of the refractive index and material transparency. 33In contrast, the confinement of electron and holes within the layer results in enormous binding energy (~ 50 meV) for intralayer A-and B-excitons at the visible range similar to its monolayer counterpart. 34At the same time, at ultraviolet wavelengths, it supports C and C' exciton complexes due to the nest banding effects and complex atomic orbital contributions. 35This excitonic behavior for ab-plane is best described by the Tauc-Lorentz oscillator model (Methods) because it captures two the most essential physical features: 36 (i) at low photon energies, excitons cannot be excited, as a consequence, absorption, or equivalently the imaginary part of refractive index (k), is equal to zero in this wavelength range and (ii) excitonic peaks exhibit an asymmetric shape due to phonon coupling of bright (excited by light) and dark (not excited by light) excitons.
Using these models for describing the optical properties of MoS2, we fitted the experimentally measured ellipsometric parameters Ψ and Δ for both ROIs at the same time (see Supporting Information).The resulting ordinary (along the layers) and extraordinary (perpendicular to the layers) optical constants, and birefringence are displayed in Figure 2. As expected, the material along the c-axis is transparent, even at ultraviolet and visible wavelengths.It confirms that excitons are formed in the layers and provide a dichroic window from ~ 900 nm where the absorption of both ordinary and extraordinary light becomes negligible.Of immediate interest is the giant birefringence of Δn ~ 1.5 in the infrared and Δn ~ 3 in the visible ranges, which can serve as a platform for optical engineering in the creation of novel devices for the photonic application.
Furthermore, as we have not specified particular properties of MoS2 other than its in-plane excitonic nature and the out-of-plane transparency, our conclusions are quite general, applying equally well to other semiconductor members of layered TMDCs with hexagonal, tetragonal, and trigonal atomic structure, 1 including MoSe2, WS2, and WSe2.Due to the lack of absorption mechanisms in the mid-and long-wave infrared regions, we believe that this anisotropy persists up to frequencies of the phonon modes in MoS2 at about 450 cm -1 (22 µm) far beyond the measured edge of 1700 nm, 37 where one could observe extremely strong and anisotropic reststrahlen bands as in the case of α-MoO3 and α-V2O5. 14,38As compared in Figure 2c, the birefringence obtained for MoS2 in the visible and near-infrared spectral intervals is several times larger than for previous record-holders BaTiS3 and h-BN, 11,12 and an order of magnitude exceeding the values of currently used birefringent materials.
Particular attention should be given to the absolute values of the refractive indices in Figure 2b, specifically to their in-plane component.The high value of ~ 4.1 is comparable with traditionally used isotropic high-refractive-index semiconductors, 39 including Si (~ 3.6), 40 Ge (~ 4.3), 41 and GaSb (~ 3.9) 42 as illustrated in Figure 2b.Such a large refractive index for MoS2 opens the door for lossless subwavelength photonics with the resolution of ~ 100 nm, which can easily rival with plasmonics platform, yet does not suffer from losses as well as nonlinear nanophotonics 43 because the magnetic dipole (MD) Mie-resonance in MoS2 nanoparticles is strongly affected by the refractive index values. 44For instance, the spectral position of MD-resonance for a spherical particle is approximately defined by λMD ≈ nD, with D being the sphere's diameter. 45Besides, its anisotropic behavior allows its use as nanoresonators even for photon energies higher than the electronic bandgap thanks to the absence of absorption along the c-axis, while for conventional isotropic materials (c-Si, Ge, GaAs, and GaSb) this option is closed.Therefore, TMDCs provide device miniaturization, and their birefringence enables fine-tuning the resonance position in a wide spectral range by altering the light polarization, which is roughly nabD -ncD ≈ 450 nm for a typical diameter of 300 nm.8][9][10][11][12] For an unambiguous validation of the extracted dielectric function, we analyzed the planar transverse magnetic (TM) waveguide modes propagating in MoS2 flakes employing a scatteringtype scanning near-field optical microscope (s-SNOM, Methods).By recording the scattered radiation, nanoscale images corresponding to the field distribution associated with the guided mode is obtained (Figure 3a-b).The effective TM-waveguide mode index (neff,TM) strongly depends on the material anisotropy allowing to probe anisotropic response, in-plane (nab) and outof-plane (nc) refractive indices, and determined by: 20 where d is the thickness of the MoS2 flake, λ is the incident wavelength, nAir = 1 and nSiO2 = 1.45 are air and SiO2 refractive indices, and m is the mode order.We used incident wavelengths in the range 1470 -1570 nm and 632.8 nm to excite guiding modes by focusing light into the apex of the s-SNOM tip, which allows for momentum matching conditions.The excited mode propagates in the MoS2 nanoflake as cylindrical waves, which interfere with the illuminating plane wave giving rise to interferometric patterns of the near-field, 24 as clearly seen in Figure 3c-d.It is worth mention that while most of the previous works with s-SNOM for TMDCs focus only on the nearfield amplitude, 20,24 the most accurate results are obtained by analyzing the phase as well 47 (see Supporting Information for comparison).To retrieve the effective waveguide mode index, in Figure 3e, we analyzed the Fourier transform (FT) of individual line scans from Figure 3d.The resulting FT has two pronounced peaks: one around zero due to background originating mostly from a strong tip-sample coupling, and the second one associated with the planar TM-waveguide mode of interest.Note that there are no peaks in the left part of Figure 3e (for negative values of q), indicating that no modes propagate in the backward direction (from the edge to the tip).The latter implies that mode scattering by the edge is far more efficient than mode edge reflection or launching.Otherwise, we would observe standing waves with a cosine form, whose FT would be symmetrical and which are predominantly observed in nano-infrared imaging of graphene plasmons 48,49 and hexagonal boron nitride (hBN) polaritons. 50The primary reason for the observed tip-launching and edge-scattering mechanisms is the relatively small momenta of the modes 24 since it is much closer to the free-space photon wavevector (k0) than in the studies of graphene 48,49 or hBN. 50For small momenta, the effective mode index is connected with that determined from the FT (ns-SNOM,FT) by momentum conservation along the edge direction: 24  eff,TM =  s−SNOM,FT + cos() • sin() where in our case  = 45° is the angle between the illumination wavevector and its projection  ∥ on the sample surface plane and  = 80° is the angle between  ∥ and the sample edge.Based on the extracted  eff,TM , we constructed the energy (E = h•c/λ)-momentum (q = 1/λ) dispersion relation of the waveguide mode.The obtained experimental (q, E) data points (green triangles) are overlaid on top of the calculated dispersion color map in Figure 4a by using constants from Figure 2a.For reference, we also added the dispersion for an isotropic model, assuming the optical constants to be the same for all crystallographic axes and equal one from ab-plane.Notably, for the visible spectral range, where excitons start playing a role, the isotropic model (Figure 4b) predicts the absence of guided modes owing to high material absorption.Conversely, our anisotropic results and near-field measurements tell us that even for this spectral interval, guided modes exist, which explains the recently discovered excitons polaritons in TMDs. 24Therefore, the excellent agreement between the experiment and theory validates our dielectric permittivity of MoS2 allowing for predicting capabilities in future photonic devices, including polarizationmaintaining fibers 4 and polarization tunable Mie-nanoresonators for nonlinear photonics. 43

DISCUSSION
Optical anisotropy lies behind the functionality of many optical devices such as polarizers and wave plates, to name a few.The outstanding importance of this phenomenon leads to an active investigation of anisotropic materials and expansion of their application scope; for instance, birefringence allows the control of boundary states in the continuum. 51However, the apparatus efficiency and compactness mostly depend on absolute birefringence values, which in most cases are moderate with the best result (Δn < 0.8), reported in h-BN and BaTiS3 crystals in the visible and near-infrared ranges.We believe that these limitations can be outperformed by the family of TMDCs materials, whose inherent intralayer excitonic behavior results in large anisotropy along and perpendicular to the layers.To validate the concept, we have shown a giant (Δn > 1.5) broadband anisotropy for MoS2 employing far and near-field techniques.From a wider perspective, our result establishes new avenues for next-generation nanophotonics based on TMDCs, for example, in tunable Mie-nanoresonators, and exciton-polariton physics.

Figure 1 .
Figure 1.Anisotropy in MoS2. a Schematic illustration of the MoS2 layered structure: the giant

Figure 2 .
Figure 2. Optical anisotropy of MoS2. a Real (n) and imaginary (k) parts of the dielectric function

Figure 3 .
Figure 3. Waveguide modes imaged by s-SNOM.a Topography image of the analyzed MoS2

Figure 4 .
Figure 4. Dispersion of a planar MoS2 waveguide.a-b Transfer matrix calculations for the