Topological phase singularities in atomically thin high-refractive-index materials

Atomically thin transition metal dichalcogenides (TMDCs) present a promising platform for numerous photonic applications due to excitonic spectral features, possibility to tune their constants by external gating, doping, or light, and mechanical stability. Utilization of such materials for sensing or optical modulation purposes would require a clever optical design, as by itself the 2D materials can offer only a small optical phase delay – consequence of the atomic thickness. To address this issue, we combine films of 2D semiconductors which exhibit excitonic lines with the Fabry-Perot resonators of the standard commercial SiO2/Si substrate, in order to realize topological phase singularities in reflection. Around these singularities, reflection spectra demonstrate rapid phase changes while the structure behaves as a perfect absorber. Furthermore, we demonstrate that such topological phase singularities are ubiquitous for the entire class of atomically thin TMDCs and other high-refractive-index materials, making it a powerful tool for phase engineering in flat optics. As a practical demonstration, we employ PdSe2 topological phase singularities for a refractive index sensor and demonstrate its superior phase sensitivity compared to typical surface plasmon resonance sensors.

O ptical waves carry energy and information encoded in their electric field amplitude, phase, and polarization. While light field amplitude is still used the most in various applications, optical phase manipulation could lie in the core of next-generation information technologies [1][2][3] . Generally, the phase acquired by light upon reflection, scattering, or transmission varies rapidly when the amplitude of the light (reflected, scattered, or transmitted, respectively) goes to zero [4][5][6][7] . Reflection zeros can be encountered in the effect of plasmonic blackbody 8 , perfect absorption 9,10 , coherent perfect absorption 11 , Brewster angle 12,13 or more sophisticated examples of zero-reflection modes 5,14 . Such zeros of response function always reveal phase singularities 4,5,15,16 which are accompanied by a non-trivial topological charge C ¼ 1 2π H γ gradðφÞds, where φ is the response function phase while the integration is performed along a path γ enclosing the singular point (zero response point) in twodimensional parameter space 5,15,16 .
To demonstrate how phase singularities associated with zeroreflection can be topologically protected, let us consider light reflection from a planar structure shown in Fig. 1a, where an atomically thin layer of a high-refractive index material (HRIM) is placed on top of SiO 2 /Si substrate. For a given thickness of HRIM film, an angle of incidence (ϕ), the photon energy (E), and polarization, reflection from the structure can be made exactly zero by calculating appropriate values for the dielectric permittivity of the HRIM film (Re(ε) and Im(ε)), which can be always found by the nature of Fresnel coefficients for the structure 17 . When we change an angle of incidence and a photon energy in some range, a zero-reflection surface will appear, see the blue surface in Fig. 1b. Any point on the zero-reflection surface corresponds to zero-reflection (at some angle of incidence and light wavelength) for the studied structure. Now we plot an actual dependence of the dielectric constants HRIM on the photon energy on the same graph (the so-called material dispersion curve), see the red line in Fig. 1b. In the presence of a reasonably large resonance feature, this curve would look like a spiral in the space of (E, Re(ε) and Im(ε)) and hence will inevitably cross the surface of zero-reflection as shown in Fig. 1b. For a Lorentz resonance feature 17 , there will be two intersection points between the material dispersion curve and the zero-reflection point resulting in two zeros of reflection from the structure in Fig. 1a. It is necessary to stress that these intersection points are protected by the Jordan theorem 5,6 . Indeed, minor variations of the material dispersion curve caused by material imperfections will not change the relative alignment of the curve and the zero-reflection surface and cannot lead to the disappearance of the zero-reflection points which lead to the idea of topological darkness 5,6 . Our structure relies on the atomic flatness of the interfaces and on the sharp changes of the refractive indexes, which offer a non-trivial possibility to realize zero-reflection and topological singularities for atomically thin layers in this structure at visible light, which was never achieved before.
Zero-reflection implies perfect absorption of the light that falls onto the discussed structure (as transmission through the silicon substrate should be zero). Zero-reflection entails phase singularity due to the singular nature of light phase at zero light amplitude a φ φ φ Fig. 1 Topology of the reflection phase near the singular point. a Schematics of generalized structure for observation of phase singularities, arising from interaction of Fabry-Perot resonator's (280 nm SiO 2 /Si) modes with ultrathin films of HRIM. b Phase singularity point arises when zero-reflection surface of the system HRIM/SiO 2 /Si intersects with the material dispersion curve of HRIM. c In close vicinity of zero-reflection points, phase becomes singular and acquires topological charge C ¼ À1 or þ1, corresponding to -2π or +2π phase round-trip accumulation. d Phase has opposite π-gradient for angles slightly above and below singular point (dashed lines in panel (c)) giving rise to topological charge with 2π round-trip around zero-reflection point. The inset is a dielectric permittivity of the model HRIM used for calculation of (b) and (c).
(where the phase of light is not defined). Figure 1c represents the map of the p-polarized wave reflection phase in space of E and ϕ, which contains two phase singularities corresponding to the zeroreflection points with topological charges equal to À1 and +1 (the topological points of Fig. 1b). Of immediate interest is a bifurcation behavior of optical phase in the vicinity of topological point (Fig. 1d). In other words, phase reveals abrupt ±π-jumps near a zero-reflection when plotted as a function of wavelength for a fixed incidence angle close to a phase singularity. It gives an indispensable degree of freedom for efficient phase manipulation.
The most exciting consequence of such optical phase control is the realization of "flat optics" paradigm-flexible manipulation of the optical wavefront by an arrangement of subwavelength planar objects to shape the desired phase pattern 1,18-20 . Such "flat optics" paradigm enables miniaturized metalenses [21][22][23] and metaholograms [24][25][26] , and two-dimensional (2D) materials 27 such as graphene 28 , transition metal dichalcogenides 29 , and organic semiconductors 30 provide an excellent platform for the implementation of these components. Although these works constitute an important step towards truly flat optics with electrically controlled properties, the efficiency of current devices is limited by the fundamental constraints: the majority of optical devices require a sufficient phase accumulation, while for atomically thin layers, this value is naturally low 31 . Indeed, the typical wavefront manipulation applications require that the phase accumulated by a light wave upon interaction with such a device can be tuned at least within the range of π. However, in the monolayer limit (t ∼ 0.65 nm and n ∼ 4) for visible light (λ ∼ 60 nm), the resulting phase delay, which is approximately determined by the optical thickness of the 2D material layer, is only about 0.01π. Consequently, finding new ways to induce strong optical phase variations in atomically thin structures is vital for flat optics.
Here, we demonstrate a platform for efficient optical phase manipulation presented by atomically thin high-refractive-index materials (HRIMs) that often possess excitonic resonances. We experimentally observe zero-reflection, phase singularities, and rapid phase variation of reflected light in extremely thin layers (down to single monolayer!) of PdSe 2 , graphene, MoS 2 , and WS 2 films placed on SiO 2 /Si substrate. Combined theoretical and experimental analysis indicates that the zero-reflection points are accompanied by a non-trivial topological charge. We derive an analytical condition for such points to occur in layered structures containing optically thin films and predict the occurrence of these points in structures containing a broader family of atomically thin HRIMs and substrates. The observed effect is highly robust and does not require complicated fabrication steps guaranteeing its reproducibility and reliability. In contrast to optical darkness observed for dielectric materials and multilayers which disappear with layer irregularities (e.g., at Brewster angle conditions) the effect is topologically protected. It can be used in numerous applications, including label-free bio-and chemical sensing, photo-detection and photo-harvesting, perfect light absorption in 2D monolayers, quantum communication, and security. As a practical application of this platform, we demonstrate a refractive index sensor that can rival modern plasmon resonance-based counterparts 32 . Therefore, our phase engineering approach as a whole offers an advanced tool for current and next-generation 2D flat optics.

Results
Phase singularities in reflection. To examine the topological properties of the light reflectance from thin HRIM films placed on SiO 2 /Si substrates, we utilized spectroscopic ellipsometry (Methods). The unique advantage of this technique is the simultaneous determination of reflection amplitude and phase in terms of the ellipsometric parameters Ψ and Δ, which are defined through the complex reflection ratio ρ: 33 where r p and r s are the amplitude reflection coefficients of p-and s-polarized plane waves. Therefore, ellipsometry provides us with the information not only about the reflected light amplitude, but also about the light phase. Unexpectedly, we found that light reflection measured from 5.1 nm thick PdSe 2 as well as for monolayers of graphene, MoS 2 , and WS 2 on SiO 2 /Si substrate showed a number of zero-reflection points as explained in Fig. 2. Remarkably, the measured amplitude parameter Ψ in Fig. 2a is in excellent agreement with the simulated spectrum in Fig. 2b calculated using the transfermatrix method 34 . However, the spectra of Ψ alone do not definitively indicate if an exact zero was attained in reflection at the position of any of Ψ dips. This can be deduced from the behavior of relevant phase which was measured using spectroscopic ellipsometry (phase Δ). The angle-dependent spectrum of the measured ellipsometric phase Δ in Fig. 2c, d clearly indicates that the reflected phase is undefined in the vicinity of certain points in the energy-incidence angle parameter space and possesses non-trivial topological charge C. Therefore, for the first time we experimentally proved that these points are phase singularities, which can occur if and only if the response function (reflection in our case) takes zero magnitude at that point. Figure 2 reveals three such phase singularities for PdSe 2 , two for graphene, and one for MoS 2 and WS 2 monolayers. Interestingly, PdSe 2 has the largest number of topological phase singularities ( Fig. 2a and Supplementary Note 1): while two of them are associated with substrate phase singularities of the substrate 35 , three of them are a direct consequence of a unique material properties of PdSe 2 (see the section Morphological and optical study of PdSe 2 ). It is worth mention that these phase singularities do not originate from Brewster or interference effect [36][37][38] . Indeed, the Brewster effect requires a lossless dielectric while the interference effect needs a substantial phase accumulation, which is negligible for atomically thin PdSe 2 . Hence, our concept of topological phase singularities ( Fig. 1) generalizes the phase effects of zero-reflection points. Most of these phase singularities are associated with Ψ ¼ 0 (equivalently ρ ¼ 0 and, hence, r p ¼ 0), whereas one for graphene has Ψ ¼ 90 (equivalently ρ ¼ 1 or r s ¼ 0). Simulated angle-dependent spectrum of Δ in Fig. 2 again demonstrates remarkable agreement with the experimental data, correctly predicting spectral positions of all phase singularities. Close examination of Fig. 2a reveals that although the exact zero-reflection around 4 eV and 30°is not attained owing to experimental non-idealities (roughness, finite coherence of light source, etc.), the phase singularity is almost unchanged compared to theoretical predictions (Fig. 2c, d) thanks to topological protection (Supplementary Note 2).
Previous works 4-6 realized these topological phase singularities only in metallic nanostructures through careful engineering of optical properties of nanostructured materials. Later, singular phase behavior was achieved in simple plasmonic heterostructures 39 where thin layers of metals (~20 nm) and dielectric were used to generate zero reflection and phase singularities. Figure 2 proves that the heterostructure approach is quite general and could be realized even for 2D materials with ultimate atomic thickness in the simplest structure-2D material/SiO 2 /Si. The reason for this counterintuitive result is a rapid dielectric function variation (for example, due to excitons in TMDCs) in atomically thin HRIM. This rapid variation guarantees intersection with the zero-reflection surface (Fig. 1b). Note that a thick layer is unsuitable for this purpose because absorption in that layer will prohibit interaction with the substrate's Fabry-Perot resonances.
To predict the position zero-reflection points for p-and s-polarized reflection in our structure shown in Fig. 1a, we derived analytical expressions for the permittivities ε p and ε s of a thin film placed on a dielectric substrate which would result in the absence of the reflection for p-and s-polarized light, respectively (Supplementary Note 3): where t and d are thicknesses of the high-refractive-index material and dielectric layer respectively; ε 1 , ε 2 , and ε 3 are the dielectric permittivity of top halfspace, dielectric layer and bottom halfspace, in our case, that is air, SiO 2 , and Si, respectively; is the normalized z-component (perpendicular to layers) of the wavevector in medium number i, k 0 ¼ ω=c, ω is the frequency, c is the speed of light, and ϕ is the angle of incidence. If the bottom halfspace is filled with a perfect electric conductor, the expressions can be simplified greatly to: where ε 0 p;s and ε 00 p;s are the real and imaginary parts of dielectric permittivity ε p;s .
Equations (2) and (3) define a zero-reflection surface in the parameter space of wavelength, real and imaginary parts of the permittivity (λ, Re[ε] and Im[ε], respectively). Intersections of the material dispersion curve in this parameter space with the zero-reflection surface of the system (thin film of HRIM/SiO 2 /Si) define zero-reflection points for the particular material of the film, as shown in Fig. 1b. The topology of mutual arrangement of the curve and the surface underlies the robustness of the zeroreflection effect to external perturbations (roughness, temperature change, etc.). If a perturbation is introduced to the thin film, a displacement of the material dispersion curve and/or the zeroreflection surface will only result in a shift of the zero-reflection point in the parameter space, but will not lead to its disappearance (Fig. 1b). This argument is in line with nontrivial topological charges of the observed phase singularities: small changes in the parameters of the system cannot lead to a change in the phase round-trip around a point, since it is an integer of 2π, thus making the zero-reflection point topologically protected 40 .
Spectral positions of topological phase singularities can be controlled by either the thickness or the dielectric permittivity of the material. The derived analytical expressions allow us to generalize the effect of phase singularities to other highrefractive-index materials (Supplementary Note 1). Hence, the effect of rapid phase change is universal for all atomically thin materials and substrates. It allows us hereafter to focus on PdSe 2 , which demonstrates rich Ψ and Δ spectra with a series of peaks and dips in Fig. 2a, b. We begin with a detailed characterization of PdSe 2 film, and then switch to unique applications and features of topological phase gradient.
Morphological and optical study of PdSe 2 . PdSe 2 thin films were prepared through chemical vapor deposition (CVD) 41 resulted in a uniform sample as confirmed by representative optical and The insets in panels (c) and (d) are phase behavior for incidence angles slightly above (ϕ = 30.5°) and below (ϕ = 30°) topological zero. e-g, k-m Experimental and h-j, n-p simulated Ψ and Δ for graphene, MoS 2 , and WS 2 on SiO 2 /Si. In close vicinity of topological points, phase becomes singular and acquires topological charge C ¼ À1 or þ1. Optical constants of PdSe 2 for simulations are taken from Fig. 3k. Meanwhile optical constants for graphene, MoS 2 , and WS 2 were adopted from several reports 62,63 . scanning electron microscopy (SEM) images in Fig. 3b, c. X-ray diffraction (XRD) spectrum showed pronounced peak in Fig. 3b validating the high crystallinity of the film 42 . Next, we validated the material's purity by X-ray photoemission spectroscopy (XPS) in Fig. 3g-h. It shows that Se:Pd atomic concentration ratio equals 1.92, close to the expected value of 2. Additionally, Raman spectra in Fig. 3e-f have characteristic phonon modes A 1 g , A 2 g , B 2 1g , and A 3 g inherent to PdSe 2 with puckered pentagonal crystal structure presented in Fig. 3a 43 . This crystal configuration naturally has high geometrical and, therefore, high optical anisotropy (see Supplementary Note 4) 44,45 .
To investigate anisotropic optical response, we measured Mueller matrices (Methods), which nonzero off-diagonal elements relate to sample anisotropy (see Supplementary Note 4). Interestingly, Mueller matrix' elements vary from point to point, as seen from Fig. 3i, j. Conceivably, it comes from the random growth during the CVD synthesis since Mueller matrices' values in Fig. 3i follow Gaussian law for random numbers. A similar random local anisotropic response is observed by polarized optical microscopy and Raman spectroscopy (Supplementary Note 4). Hence, at a macroscopic scale, our PdSe 2 layer exhibits an isotropic dielectric response. It allowed us to investigate optical constants by classical ellipsometric and reflectance measurements (see Supplementary Note 4) using the isotropic model for PdSe 2 with 5.1 nm thickness obtained by atomic force microscopy (AFM) in Fig. 3d. The resulting broadband dielectric function is presented in Fig. 3k. As expected, PdSe 2 has pronounced excitonic peaks 46 and a metallic Drude response caused by p-doping revealed by XPS (see Supplementary Note 5). Note that excitonic peaks of PdSe 2 align with Fabry-Perot resonances of the standard SiO 2 (280 nm)/Si substrate, making system PdSe 2 /SiO 2 (280 nm)/Si promising for enhancement of the PdSe 2 optical response.
Applications of topological zeros: sensing. Simple planar structures studied here are easy to incorporate and leverage in industrial and scientific devices where the optical phase plays a critical role 5,24,25,32,47 . The most prominent practical examples are holography 24,25,47 , image processing 48,49 , label-free bio-or chemical sensing 50-53 , gas sensors 35 and quantum key distribution 54,55 . To validate the concept, we demonstrated that the liquid/PdSe 2 /SiO 2 /Si system is already an ultrahigh sensitive sensor owing to rapid phase change around the topological point. Note that for sensing measurements we used ellipsometer in the most accurate nulling mode (Methods) and 7.1 nm PdSe 2 thin film to have topological zero in the operation range of our device since liquid changes zero's spectral and angle position according to Eqs. (2)(3).
For demonstration, we used water with 0, 2.5, 5, 15, and 20% volume concentration of isopropanol. Notably, the measured Ψ and Δ in water are in agreement with the predicted values (see Supplementary Note 6) whereby confirming the water stability of PdSe 2 in addition to its recently shown air stability 43 . Then, to alter the refractive index (RI), we injected isopropanol into the solution and recorded Ψ and Δ (Fig. 4a, b) for each water solution. As predicted, the change in amplitude response, Ψ, (Fig. 4a, c) is relatively small due to the resonance's topological nature, whereas Δ (Fig. 4b, d) shows a dramatic dependence on RI of liquid. Noteworthy, the phase sensitivity in our device of 7.5 × 10 4 degrees   32 . Therefore, our biosensor design can greatly help in dealing with the current coronavirus situation since the investigated system PdSe 2 /SiO 2 /Si is already a ready-to-use scalable, cheap, easy-to-use and fabricate, robust device with outstanding performance thanks to the pronounced phase effect in topological points.
Evolution of phase singularities. So far we have considered and observed reflection phase singularities with unitary topological charge, wherein the argument makes a ± 2π round-trip around the singularity. These points, however, are not stationary and evolve with the material parameters, as Eqs. (2)(3) suggest. When two points with opposite charges (+1 and −1) meet in the parameter space, they annihilate leaving no phase singularity. We theoretically observe such annihilation with variation of the PdSe 2 film thickness, t, when two phase singularities with opposite topological charges meet at around t % 3.2 nm (Fig. 5a). This case happens when the material dispersion curve in Fig. 5b becomes tangent to zero-reflection surface. The corresponding angleresolved spectrum of Δ shown in Fig. 5b plotted for t ¼ 3.2 nm reveals the absence of any phase singularities. Therefore, by varying the thickness one is able to control the position and amount of phase singularities, which appear or disappear in pairs, so that the total topological charge preserves.
Next, we examine the possibility of phase singularities with higher topological charges, jCj> 1. One potential opportunity for the emergence of a non-unitary topological charge is when a phase singularity either in r p or r s exhibits a non-unitary charge. Unfortunately, for non-magnetic materials in a planar system, zeros in r p or r s are essentially non-degenerate (Supplementary Note 8).
However, since ρ is the ratio of two reflection coefficients, another possibility is when a C ¼ ± 1 phase singularity of r s coincides in the parameter space with a C ¼ Ç1 phase singularity of r p . This gives rise to a C ¼ ± 2 phase singularity of ρ, which can be detected by spectroscopic ellipsometry. Equating the righthand sides of Eqs. (2) and (3), we obtain an equation that determines, for a given substrate, the position of the point at which the zero-reflection conditions for both polarizations coincide; then we can immediately calculate the dielectric constant of the film that satisfies this condition. The charge of a point is determined by the derivative of the material dispersion curve, so the last step is to choose the direction of the material dispersion curve near the zero-reflection point. By using the set of parameters satisfying these conditions (Supplementary Note 9), we observe a C ¼ þ2 phase singularity in the spectrum of ρ,   s-polarization only by the transverse one. In that case, it is possible to select zero-reflection conditions for different polarizations entirely independently. Therefore, in contrast to isotropic materials, for which the position of points with a topological double charge depends on the properties of the substrate, in the case of in-plane anisotropic materials, it is potentially possible to obtain a point with C ¼ ± 2 at any point in the space of angles and frequencies (see Supplementary Note 9). One of such cases is shown in Fig. 5d.
For C ¼ ± 2 singularities demonstrated in Fig. 5c, d, the phase makes a ± 4π round-trip around such a singularity, thus increasing the local phase gradient approximately by a factor of two, which could significantly improve phase applications. For instance, the sensitivity of the corresponding refractive index sensor may increase approximately two-fold. However, similar to high-order optical vortices 56,57 , high-order topological charges are structurally unstable as a small perturbation can deform its singular region, reducing them to a collection of stable unitary singularities.
Noteworthy, in our study, the topological points appear in the dependence of the reflection amplitude on the angle of incidence and the wavelength, in contrast to topology in the spatial distribution of electric field [58][59][60] . Quite surprisingly, the properties of such different topological charges and experimental manifestation are very similar. Those include the conservation of the topological charge under perturbations, birth and annihilation in pairs of opposite charges. The analogy of topological properties between our work and spatial singularities gives an opportunity for the future development of zero-reflection points with a variety of topological effects such as fractional topological charges 56,57 .
The scenario of non-unitary phase singularities is somewhat reminiscent of exceptional points in non-Hermitian optical systems 61 . Such points, which correspond to coalescent eigenstates of a non-Hermitian Hamiltonian, feature a strong ðϵ À ϵ 0 Þ 1=ðNþ1Þ dependence of the eigenenergies of the optical system on a perturbation parameter ϵ in the vicinity of the exceptional point ϵ 0 (with N being the order of the exceptional point), which has been used for boost the sensitivity.
Additionally, upon appropriate engineering of the system the same concept of topological phase manipulation could also be applied in the transmission regime, thus bridging our phase engineering approach with metasurfaces.

Discussion
Flat optics enable the design of optical components into thin, planar, and CMOS-compatible structures. Coupled with 2D materials, it evolves into 2D flat optics with ultracompact and tunable devices. Nevertheless, atomically thin optical elements suffer from low efficiency of phase manipulation. To lift this limitation and achieve phase control with 2D materials, we utilized topologically protected zeros of a simple heterostructure. We showed both experimentally and theoretically that a whole set of high-index 2D materials could provide rapid phase variations revealed by spectroscopic ellipsometry. In addition, we demonstrate that topological approach leads to high-performance devices on the sensing example and propose the future direction of topological phase effects such as annihilation and high-order charges. From a broader perspective, our results open new avenues for effective application of atomically thin highrefractive-index materials as phase materials in photonics.   (1 -5 eV) in steps of 1 nm and multiple angles of incidence in the range 15°-80°with 0.5°s tep. Ellipsometer has a single chamber monochromator with two gratings: 1200 g/ mm for visible light (248 -1040 nm) with 4.6 nm bandwidth and 600 g/mm for near-infrared interval (1040-1240 nm) with 9.2 nm bandwidth. For accurate refractive index sensing, we performed the nulling ellipsometry with Accurion nanofilm_ep4 ellipsometer. During the measurement light was initially directed through the polarizer then through the compensator, whose settings were adjusted until the reflection from the sample became linearly polarized. Afterward, the analyzer was set to achieve the minimum in the signal at the photodetector. The final positions of polarizer and analyzer at signal's minimum uniquely define ellipsometric parameters Ψ and Δ. Measurements were done over a wavelength range from 300 to 360 nm in steps of 0.1 nm with 4 nm bandwidth and at 49.4°incident angle corresponding to the singular point of PdSe 2 in water.
To probe anisotropic response, we also measured 11 elements of Mueller Matrix (m 12 , m 13 , m 21 , m 22 , m 23 , m 24 , m 31 , m 32 , m 33 , m 34 ) on Accurion nanofilm_ep4 ellipsometer over 400-1000 nm wavelength range in 5 nm step with 4 nm bandwidth at 50°incident angle in a rotation compensator mode to get access to Stokes parameters in the input branch of ellipsometer.
Optical visualization. The surface images (2400 × 2400 pixels) of PdSe 2 were captured by an optical microscope (Nikon LV150L) with a digital camera DS-Fi3.
Atomic force microscopy. The thickness and surface morphology of the PdSe 2 film were accurately characterized by an atomic force microscope (AFM, NT-MDT Spectrum Instruments "Ntegra") using AFM in a peak-force mode at ambient conditions. AFM measurements were carried out using cantilever tips from NT-MDT (ETALON, HA_NC) with a spring constant of 3.5 N/m, a tip radius < 10 nm and a resonant frequency of 140 kHz. Images of the PdSe 2 surface were taken with 512 × 512 pixels and scan rate of 0.5 Hz, after that data were analyzed by Gwyddion software.
Scanning electron microscopy. Scanning electron microscopy (SEM, JEOL JSM-7001F) with a Schottky emitter in secondary electron imaging mode with a voltage of 30 kV and current of 67 µA, and a working distance of~6.3 mm was applied to study in detail surface features and homogeneity of the PdSe 2 film surface within different areas using 1960 × 1280 pixel scan.
XPS characterization. The chemical states of the elements in the film, as well as the valence band were analyzed by X-ray photoelectron spectroscopy (XPS) in Theta Probe tool (Thermo Scientific) under ultrahigh vacuum conditions (base pressure < 10 −9 mBar) with a monochromatic Al-K α X-ray source (1486.6 eV). Photoelectron spectra were acquired using fixed analyzer transmission (FAT) mode with 50 eV pass energy. The spectrometer energy scale was calibrated on the Au 4f 7/2 line (84.0 eV).
X-ray diffraction. X-ray powder diffractometrer (XRD, Thermo ARL X'TRA) equipped with Cu Kα radiation λ = 0.154 nm was used to characterize the crystalline structure and phase of PdSe 2 film. The XRD pattern was taken at ambient conditions by 2θ-scan over the range of 20°-75°with a step of 0.05°and accumulation time of 2 s.
Reflectance measurements. Fourier-transform spectrometer Bruker Vertex 80 v has been used to measure the reflection coefficient at the normal incident at room temperature in the frequency range from 1000 to 24,000 cm −1 , as the reference was used the reflection from the gold mirror.
Raman characterization. The experimental setup used for Raman measurements was a confocal scanning Raman microscope Horiba LabRAM HR Evolution (HORIBA Ltd., Kyoto, Japan). All measurements were carried out using linearly polarized excitation at wavelengths 532 and 632.8 nm,1800 lines/mm diffraction grating, and ×100 objective (N.A. = 0.90), whereas we used un-polarized detection to have a significant signal-to-noise ratio. The spot size was approximately 0.43 μm. The Raman images were recorded by mapping the spatial dependence of Raman intensity integrated at the main Raman peaks within the shift range 136-156 cm −1 , for each of the 45 × 45 points in the scan, with an integration time of 500 ms at each point and incident power P = 1.7 mW.