## Introduction

Layered materials with weak out-of-plane van der Waals bonds attract great research interest. These materials are studied in natural1,2 and artificial formations3,4,5,6. Based on the stoichiometric composition and crystal structure, one can classify the natural van der Waals materials into graphene, transition metal dichalcogenide or mono-chalcogenide, sulfosalt, oxide, neo-, phyllosilicate and phosphate families. Artificial heterostructures are assembled by stacking atomic layers of van der Waals materials on top of each other3,7. Alternatively, one can categorise layered materials based on the feasibility of their exfoliation into separate atomic sheets8, with its quantitative measure being an energy required for the isolation or separation of a singular atomic layer from the bulk9,10. In first principles calculations, the isolation energy can be estimated by evaluation of the interlayer binding energy per unit area. Within this approach, it is reasonable to distinguish naturally, potentially cleavable, and robust materials with threshold exfoliation energies of the order of8 10 meVÅ−2, 100 meVÅ−2, and 1000 meVÅ−2, respectively. Apart from natural van der Waals formations, some of the cleavable materials have no van der Waals bonds in their structure. They form interlayer bonds of a different nature, e.g., covalent, but of comparable strengths and bear the name of non-van der Waals materials11,12,13,14,15,16,17,18. Vivid examples are atomic sheets of Fe2O315, AgCrS216 and FeS219 obtained via sonication-assisted and cation-intercalation exfoliation methods, respectively.

In this article, we explore the structural and optical properties of the hexagonal phase of InGaS3 thereby expanding the list of non-van der Waals materials. Contrary to the previous theoretical work, which predicts that InGaS3 has an orthorhombic crystal structure with well-defined atomic layers connected by van der Waals bonds20, we find that our samples have a hexagonal crystal structure with no van der Waals bonds in the unit cell. Surprisingly, those can be cleaved down to atomically thin sheets owing to the weak out-of-plane non-van der Waals bonds. We confirm the crystal structure by X-ray crystallography and discover the weak bonds by first-principles calculations. In addition, we study the optical properties both experimentally by imaging spectroscopic ellipsometry and theoretically by first principles calculations. Ab initio calculations are performed for orthorhombic and hexagonal phases, with only the latter agreeing with the experiment. As a result, we conclude that the hexagonal InGaS3 is a wide bandgap (2.73 eV) cleavable semiconductor with high-refractive index (>2.5) and negligible optical losses in the visible and near-infrared spectral ranges, which makes it a unique material for the visible range nanophotonics and Mie-tronics.

## Results

### Structure and morphology of non-van der Waals InGaS3

Typical InGaS3 crystals appear with yellow-to-lustrous grey shades as displayed in Fig. 1 (a). Such colours are determined by the 1:1:3 ratio distribution of Indium: Gallium: Sulphur atoms (see EDX and XPS spectroscopy patterns in Supplementary Figure 1). The crystals exhibit a hexagonal arrangement of III-III-VI group elements in the P65 space group with lattice parameters of a = b = 6.6 Å and c = 17.9 Å as presented in the inset of Fig. 1 (a). Our crystallographic imaging results confirm its hexagonal structure. It was refined in a chiral P65 space group giving an appropriate value of Flack’s parameter of 0.06(2) being close to zero. Figure 1 (b, c) demonstrates the selected X-ray diffraction patterns across a*b* and c*b* crystallographic planes. Notably, a similar hexagonal structure of InGaS3 was previously reported with a high R-value, 8.1 %, in the chiral P61 space group21. However, testing the space group for the dataset collected from our samples resulted in an unsatisfactory Flack’s parameter of 0.94(3) which is in support of a swap within an enantiomorphic pair, P61 - > P6522. Further details on data collection, refinement parameters and interatomic distances are provided in Methods and Supplementary Tables 1 and 2. The hexagonal structure of the studied material was alternatively verified through transmission electron microscopy (see Supplementary Figure 2 for diffraction pattern across a*b* crystallographic plane reflex interpretations).

The hexagonal InGaS3 contains various structural bonds, whose strengths were estimated by first-principles calculations based on the density functional theory (DFT). To obtain isolation energy of individual atomic sheets, we estimated the difference between the ground-state energy of the relaxed structure and all of its unrelaxed substructures. Afterwards, we looked for the planes with the minimal binding energies to determine potentially breakable, or, in our case, cleavable directions. Excluding the relaxation energy of substructures, we found the exfoliation energy of Eexf ≈ 53 meVÅ−2 for the plane shown in Fig. 1d. Notably, the exfoliation energy decreased when we added more atomic layers into our model. In the case of five layers, we obtained Eexf = 47 meV/Å−2. After accounting for the further relaxation of the isolated layers23, we obtained an even smaller value of E*exf ≈ 21 meVÅ−2. It is important to note, that some alternative and DFT accuracy approaches, such as MLIP provide computationally faster studies of mechanical properties of materials under the external deformations including exfoliation effects24,25. Figure 1e compares our results with the evaluated exfoliation energies of conventional van der Waals (and non-van der Waals) materials. Notably, the energies of conventional van der Waals materials cover the range of 20 to 60 meVÅ−2, whereas non-van der Waals materials fall into a broader range from 25 to 180 meVÅ−2. Among non-van der Waals materials, InGaS3 has a relatively low exfoliation energy, which is even lower than that of some van der Waals materials, such as PdS2 and PdSe2.

Slightly elevated temperature treatment within the standard exfoliation procedure (see Methods)7 allows for the cleavage of InGaS3 crystals into individual nanosheets with atomically smooth surfaces (see Fig. 2a). However, ultrathin flakes are often highly inhomogeneous or cracked (see Supplementary Figure 3 for the atomic-force micrographs of inhomogeneous atomic sheets). This can be attributed to the lack of strength of some in-plane bonds within the material, leading to their random breakage during exfoliation. Figure 2b–d show AFM scans of pristine atomic sheets with nearly atomically flat surfaces (root-mean-square roughness of 0.3 nm). Thicknesses of these sheets correspond to one (1 L), two (2 L), three (3 L), four (4 L) and five (5 L) monolayers.

Figure 2f shows Raman spectra of the hexagonal InGaS3 nanosheets taken at room and liquid nitrogen temperatures. Obtained spectral profiles have a complex form due to the overlapping bands, although peaks are slightly sharper at lower temperatures. The profiles display depths at wavenumber ranges of 180–200 cm–1 and 270–290 cm−1, which is in good agreement with our lattice dynamics calculations (see vertical bars on the inset of Fig. 2f). Those predict an absence of vibrational modes within these wavenumber ranges (see Methods for calculation details). For hexagonal InGaS3 in the P65 space group, we found 90 normal vibrational modes in the centre of the first Brillouin zone. Those can be represented by the expression Γv = 15 A + 15B + 15E1 + 15E2. Here, A + E1 are acoustical translational modes, B are the silent modes, 14 A + 14E1 are infrared-active modes, while 14 A + 14E1 + 15E2 are Raman-active modes. Their wavenumbers at zero pressure are presented in Supplementary Table 3. The contribution of individual atoms to the vibrational modes, i.e., the partial phonon density of states (pPDoS), is depicted in Supplementary Figure 4. According to our pPDoS evaluations, the strong peak at 125 cm−1 is a mixed vibration of Gallium and Indium atoms, while the weak shoulder at the low-wavenumber region (157 cm–1) is related to Gallium translations. The middle range of the spectra (from 200 to 270 cm−1) associates with Sulphur vibrations. The high-wavenumber spectral part, above 290 cm–1, mostly associates with Ga-S vibrations. An example of Gallium-Sulphur stretching-like mode is shown in Fig. 2 (f, right inset). Tetrahedral GaS4 complexes combined in chains connected by common Sulphur atoms are capable of specific rotational vibrations, such as those demonstrated in Fig. 2 (f, mid inset). In this case, the rotation of triangle-like GaS3 units is observed. Raman spectra of thinner, up to bilayer, sheets are shown in Supplementary Figure 5. They exhibit no apparent shift in the peak positions and an expected decline in the peak amplitudes.

### Anisotropic optical properties and crystal structure reckoning of non-van der Waals InGaS3

We investigated the anisotropic dielectric tensor (Fig. 3a) of hexagonal InGaS3 through imaging spectroscopic ellipsometry in the tiny region of interest of 10 μm2 within the area of exfoliated sheets. The ability to focus on the homogeneous high-quality region is a key benefit of our imaging technology26. Furthermore, we simultaneously recorded and analysed optical response from four InGaS3 sheets to ensure a great precision and reproducibility of our results (see Methods and Supplementary Information for details). Тhe dielectric function of hexagonal InGaS3, similarly to transition metal dichalcogenides, is best represented by the Tauc-Lorentz oscillators27,28 and the Cauchy model29 across the crystallographic ab-plane and c-axis, respectively. Despite this, owing to the high thickness of monolayers and the non-van der Waals interaction in InGaS3, the measured birefringence is relatively weak with Δn ~ 0.1 (the inset in Fig. 3a), contrasting the huge anisotropy of Δn ~1.5 in transition metal dichalcogenides with natural van der Waals bonds26. To explain this relatively weak anisotropy, we have performed additional DFT calculations (see Fig. 3b). The comparison of plots in Fig. 3a, b reveals a good agreement between experimental and theoretical results. Furthermore, our evaluations independently confirm the crystal structure of InGaS3 (see inset of Fig. 1a) since the dielectric response is a fingerprint of the material’s electronic bandstructure (see Supplementary Information for details). We also evaluated optical constants for the orthorhombic phase of InGaS3 predicted theoretically in recent work20. The obtained constants, presented in Supplementary Figure 7, clearly disagree with our experimental results, which indirectly indicates the hexagonal crystal structure of our sample.

Additionally, a good agreement between experimental and theoretical optical responses (see Fig. 3) implicitly shows a small influence of spin-orbit coupling (SOC) in contrast to transition metal dichalcogenides,24 where SOC leads to noticeable mismatch between DFT and experimental results. Hence, our technique, which combines a spectroscopic ellipsometry with density functional calculations unambiguously confirms the hexagonal structure of InGaS3 in addition to the X-ray analysis and can be potentially used for the identification of crystal structures of other materials.

Apart from optical constants, we also evaluated the electronic bandstructure of hexagonal InGaS3 (Fig. ) and individual orbital-resolved density of electronic states (see Supplementary Figure 6) using the Heyd-Scuseria-Ernzerhof hybrid functional (see Methods)30. For the latter, the main contributions to the bottom of conduction bands come from s-states of Gallium, Indium, and p-states of Sulphur atoms. From Fig. , we conclude that InGaS3 is an indirect bandgap semiconductor with Eg = 3.0 eV, which is close to bandgap (Eg = 2.73 eV) obtained experimentally from ellipsometry analysis. Moreover, we observed an excitonic peak at Eexc = 2.67 eV (464 nm) in optical response of the material, which is shown in Fig. 3a.

Its excitonic binding energy is Eb = Eg − Eexc = 60 meV. Interestingly, these values (Eg = 2.73 eV and Eb = 60 meV) are in line with bandgaps and excitonic binding energies of traditional semiconductors but contrast with transition metal dichalcogenides, such as MoS2 (Fig. 3d). We believe it results from two main reasons: (i) the unit cell along the crystallographic c-axis of InGaS3 is about 3 times larger than for MoS2; (ii) InGaS3 refractive index is about 2 times smaller compared to MoS2. Hence, we estimate that InGaS3 should have much weaker quantum confinement making it closer to 3D rather than 2D materials. Therefore, InGaS3 does not exhibit extraordinary excitonic effects and birefringence as MoS2. This indicates that the exciton in InGaS3 has the same physical origin as in traditional semiconductors, and thus, can be described by the Wannier-Mott model31.

Finally, we would like to highlight the strong dielectric response of InGaS3 (Fig. 4a). Its refractive index is comparable to indices of classical high-index materials, such as silicon (Si)32 and titanium oxide (TiO2)33,34, as well as emerging high-index materials, such as Gallium Phosphide (GaP)35 and tin(IV) Sulphide (SnS2)36, as shown in Fig. 4a. More importantly, InGaS3 has zero optical losses up to 2.57 eV (485 nm), as can be seen in Fig. 4b. This value lies between optical bandgaps of SnS2 (Eg = 2.18 eV) and TiO2 (Eg = 3.35 eV). Hence, InGaS3 covers an important gap in visible spectral range for high-refractive index materials (see inset in Fig. 4b). As a result, we conclude that it is an outstanding material for the novel research direction of all-dielectric meta-optics in the visible range35,37. To demonstrate that InGaS3 outperforms other transparent materials in the visible range, we have studied planar waveguides on a quartz substrate. Setting the operating free space wavelength to 500 nm, we find (Fig. 4c) that InGaS3 allows for a much stronger field confinement of λ/9 than TiO2 (λ/5.6) and with the much thinner core layer (32 nm vs 57 nm). These results are surprising in light of a relatively small refractive index difference of about 18% between InGaS3 and TiO2 at 500 nm. Although InGaS3 and TiO2 waveguides are both very small, mode length evaluated here governs the Purcell enhancement of spontaneous emission38. As a result, single-photon sources and nanoLEDs coupled to InGaS3 waveguides would have higher internal quantum efficiency and their radiation would be better collected by the waveguide.

## Discussion

To sum up, we have demonstrated that hexagonal InGaS3 can be cleaved down to thicknesses of individual monolayers even though it is not composed of atomic layers connected by van der Waals bonds within its unit cell. It is made possible due to a specific arrangement of atoms across the crystallographic c-axis, where the exfoliation energy diminishes to 53 meVÅ−2 for a selected atomic plane. This exfoliation energy value is comparable to conventional van der Waals materials. Furthermore, the hexagonal InGaS3 is well suited for next-generation nanophotonics, Mie-tronics39, and expands high-refractive index pallette40. In particular, it meets two major requirements of Mie-tronics: high-refractive index (n > 2.5) and broadband transparency (k ≈ 0 above 465 nm). Although, its refractive index is only about ~18 % higher than of traditional TiO241, we show that InGaS3 planar waveguides have by 55 % stronger field confinement achieved at 40 % lower thickness than TiO2 waveguides. Hence, we expect that all-dielectric nanostructures, such as waveguides42, Mie-resonance nanoparticles43, and subwavelength metasurfaces44 based on InGaS3 would benefit greatly from its higher refractive index. Additionally, it exhibits an out-of-plane optical anisotropy (Δn ~ 0.1), in contrast to conventional high-refractive index materials, which may greatly extend its scope of applications. Therefore, InGaS3 possesses a unique combination of optical properties, such as high-refractive index, zero optical losses and out-of-plane anisotropy also offering convenient thickness control due to its structural properties.

## Methods

### Sample preparation

Bulk InGaS3 crystals were purchased from 2D Semiconductors (Scottsdale, USA) and exfoliated on top of required substrates: Si/SiO2 and quartz. Prior to exfoliation, the corresponding substrates were subsequently cleaned in acetone, isopropanol alcohol, and deionized water. Then, those were subjected to oxygen plasma (O2) to remove the ambient adsorbates. Following the plasma cleaning, substrates were put on a hot plate at the temperatures of 120–140 °C for 2 min. After the heating, the scotch-tape from Nitto Denko Corporation (Osaka, Japan) with loaded crustal InGaS3 was brought in contact with substrates. Afterwards, the tape was removed completing the exfoliation procedure.

### X-ray crystallography

X-ray diffraction analysis of InGaS3 single crystals was performed on Bruker D8 QUEST diffractometer with Photon III CMOS detector using Mo Ka radiation (l = 0.71073 Å) focused by multilayer Montel mirror. The full dataset was collected at the temperature of 100 K within ω-scans indexed with cell-now. It was integrated through SAINT from the SHELXTL PLUS package45. Absorption correction was completed by a multiscan approach implemented from SADABS46. The crystal structure was solved by direct methods and refined anisotropically with full-matrix F2 least-squares technique using SHELXTL PLUS package. The structure was refined in a chiral P65 space group with the resulting Flack parameter of 0.06 (2) being close to zero47. The second virtual inversion twin-component was added to refinement by TWIN/BASF instruction to account for the absolute structural parameter. This led to a slight reduction of R-value from 2.50 to 2.49%.

### Density functional theory

Calculations of electronic bandstructure and optical constants were performed using the Vienna ab-initio simulation package VASP48,49 within generalized gradient approximation (GGA)50. The Heyd-Scuseria-Ernzerhof (HSE) hybrid functional51 in combination with Perdew-Burke-Ernzerhof (PBE)52 potentials was applied for accurate description of the electronic bandstructure. Electron-ion interactions were described by projector-augmented wave (PAW) method52,53. The cutoff energy for the plane-wave basis was set to 400 eV. The first Brillouin zone of the supercells was sampled with a 6 × 6 × 3 Monkhorst-Pack mesh54 of k-points for bulk supercell and 6 × 6 × 1 for monolayer case. The maximal force convergence tolerance settings for geometry optimization were set to 0.01 eVÅ−1.

### Lattice dynamics calculations

First-principle lattice dynamics calculations were carried out with the CASTEP code package55. The crystal structure was fully optimized using LD approximation with CA-PZ exchange-correlation functional based on Ceperley and Alder numerical representation56 parameterized by Perdew and Zunger57. Maximal force and stress tensor convergence tolerance settings for geometry optimization were set to 0.01 eVÅ−1 and 0.02 GPa, respectively. 5s2 5p1, 3d10 4s2 4p1, and 3s2 3p4 orbitals were considered occupied with valence electrons for Indium, Gallium and Sulphur, respectively. Plane-wave cutoff energy was set to 880 eV for 4 × 4 × 2 sampling of the first Brillouin zone by the Monkhorst-Pack method.

### Atomic Force Microscopy

The thickness and surface morphology of InGaS3 sheets were accurately characterized by an atomic force microscope (NT-MDT Ntegra II) operated in a HybriD mode at ambient conditions. AFM images were acquired using silicon tips from TipsNano (GOLDEN, NSG 10) with spring constant of 11.8 N/m, head curvature radius <10 nm, and resonant frequency of 240 kHz. The measurements were performed at a scan rate of 0.3 Hz and 512-pixel resolution. The obtained datasets were analysed by the Gwyddion software.

### Imaging spectroscopic ellipsometry

To analyze anisotropic optical response of InGaS3, we used commercial imaging spectroscopic ellipsometer Accurion nanofilm_ep4 in the nulling mode. Ellipsometry spectra were recorded for four sheets with various thicknesses (h = 3.6 nm, 68.5 nm, 103.0 nm, and 277.4 nm) in the spectral range from ultraviolet (360 nm) to near infrared (1700 nm). For ellipsometry analysis, we followed the algorithm described in Supplementary Note 2 of our recent work26.

### Eigenmode simulations

To study planar photonic waveguides with anisotropic InGaS3 and isotropic TiO2 cores, we employed a finite element method solver implemented in COMSOL Multiphysics software. The mode length was evaluated as:

$$l_{mode} = \frac{{{\int}_{ - \infty }^\infty {W_{{{{\mathrm{em}}}}}dy} }}{{max\left( {W_{em}} \right)}}$$
(1)

where

$$W_{em} = \frac{1}{{16\pi }}\left[ {\frac{{\partial \left( {\omega \varepsilon } \right)}}{{\partial \omega }}\left| E \right|^2 + \left| H \right|^2} \right]$$
(2)

is the electromagnetic energy density in dispersive non-magnetic media.