Density Functional Theory Study of the Electronic, Structural, and Optical Properties of ZrS₂ and ZrSeS: Showcasing a Thorough Analysis Using DFT and DFT+U

Abstract

The electronic, structural, and optical properties of layered transition metal dichalcogenides, ZrS₂ and ZrSeS, with trigonal structures, were comprehensively investigated using density functional theory (DFT). Calculations employed various approximations, including LDA, GGA, PBE+U, and PBE0, within the Quantum ESPRESSO framework. Rigorous convergence tests were conducted to ensure computational accuracy for both materials. The calculated equilibrium lattice constants demonstrate good agreement with existing experimental data, validating the chosen computational methods. Analysis of electronic properties revealed that LDA and GGA significantly underestimate the band gap values; in contrast, PBE+U and PBE0 approaches yielded values consistent with experimental results. Furthermore, the optical properties, including the absorption spectrum, complex dielectric function, refractive index, and static and dynamic polarizability, were thoroughly studied. To ensure accuracy, a scissor-operator correction was applied to the optical spectra to align with corrected electronic transitions. Key energy levels and intensities for significant transitions were identified, and the notable anisotropic behavior of these materials was characterized. By positioning these findings within the most recent 2025 developments in solid-state systems, this study provides a high-fidelity benchmark for the optoelectronic and polarizable response of Zr-based TMDs.

Introduction

Zirconium chalcogenides are a fascinating class of compounds that contain zirconium and chalcogen elements like sulfur, selenium, or tellurium. These materials exhibit intriguing properties and find applications in diverse fields including electronics, catalysis, and materials science. , They have a layered structure, similar to graphene, which gives them a wide range of interesting electronic, optical, and mechanical properties. , This makes them promising candidates for various technological applications, including next-generation electronic devices, sensors, and energy storage systems. −

Within this group, zirconium dichalcogenides, such as ZrS₂ and ZrSe₂, are particularly noteworthy. They consist of zirconium atoms sandwiched between two layers of chalcogen atoms. Their unique properties, including superconductivity and charge density waves, have made them a focal point of research for use in optoelectronics, catalysis, and energy storage. − Recent electronic and optical analyses of closely related solid-state systems further emphasize the importance of these materials in contemporary materials science. Moreover, the integration of these chalcogenides into advanced functional architectures continues to drive the development of high-performance composite communications and electronic devices.

In the realm of layered transition metal dichalcogenides (TMDs), ZrS₂ and the mixed chalcogenide ZrSeS have emerged as promising candidates for photocatalytic applications, particularly in solar-driven water splitting. Both materials crystallize in a layered 1T-trigonal structure with the P3m̅1 space group, analogous to cadmium iodide (CdI₂), where each zirconium atom is octahedrally coordinated by six chalcogen atoms. The layers are stacked via weak van der Waals interactions, enabling facile exfoliation into thin flakes and offering opportunities for surface functionalization. Although strong hydrogen bonding interactions such as N–H···S or O–H···S are not dominant in the bulk systems, − the exposed surfaces can effectively interact with hydrogen species, which is crucial for enhancing the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER). A key advantage of these materials lies in their tunable band gaps, and their ability to form heterostructures that promote efficient Z-scheme water splitting. − Additionally, owing to the larger atomic radius of selenium relative to sulfur, ZrSeS possesses an expanded lattice compared to ZrS₂. ,

Regarding their stability, ZrS₂ is thermodynamically stable under ambient conditions. However, it is highly susceptible to oxidation in the presence of impurities like air and moisture, which can lead to the formation of the more stable zirconium oxide (ZrO₂). Furthermore, under high pressure, ZrS₂ undergoes a series of structural phase transitions, transforming from its trigonal phase to monoclinic, orthorhombic, and ultimately a tetragonal structure, which can induce a transition to a metallic state. While less research is available on the specific stability of ZrSeS, its mixed-chalcogen nature may introduce different stability characteristics due to the weaker bonding with the larger selenium atom.

This paper focuses on two specific zirconium dichalcogenides: zirconium disulfide (ZrS₂) and zirconium selenium sulfide (ZrSeS). ZrS₂ is well-known for its numerous applications, including its use as a solid lubricant, as an electrode material in lithium-ion batteries, and in semiconductor and optoelectronic devices. , The related ternary compound ZrSeS is also being explored for its potentially tailored properties. Therefore, this study investigates the electronic, structural, and optical properties of both ZrS₂ and ZrSeS using density functional theory.

Computational Methods

Solving many-body quantum-mechanical problems remains a significant challenge, particularly for systems involving interactions among three or more electrons, where analytical solutions are generally not tractable. , Density functional theory (DFT) provides an efficient and reliable framework for addressing such complexities and has become a standard approach for investigating the structural, electronic, and optical properties of condensed-matter systems. − In the present work, all first-principles calculations were performed using the Quantum ESPRESSO (QE) software package, which is based on DFT and the plane-wave pseudopotential formalism. ,

Within the Kohn–Sham formulation of DFT, the exchange–correlation (XC) potential represents the most critical approximation. To assess its influence on the predicted properties, several XC functionals were employed, including the Local Density Approximation (LDA), the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation, the PBE+U approach, and the hybrid PBE0 functional. The on-site Coulomb interaction parameter U for the localized Zr 4d orbitals was determined using a one-shot linear-response procedure, following the standard Cococcioni–de Gironcoli formalism, yielding values of U = 3.510 eV for ZrS₂ and U = 3.084 eV for ZrSeS. The variation in U accounts for the different screening environments provided by the S and Se ligands, respectively. ,,,− ,,,−

The interaction between core and valence electrons was treated using pseudopotentials. For the LDA and PBE calculations, Projector Augmented-Wave (PAW) pseudopotentials from the GBRV pseudopotential library were employed, whereas norm-conserving pseudopotentials were used for the PBE0 calculations. Core electrons were kept frozen during the calculations, while valence electrons were treated explicitly. The valence electronic configurations considered were [Kr] 4d² 5s² for Zr, [Ne] 3s² 3p⁴ for S, and [Ar] 3d¹⁰ 4s² 4p⁴ for Se.

To ensure numerical accuracy and reliability, systematic convergence tests were carried out with respect to both the plane-wave kinetic energy cutoff and k-point sampling of the Brillouin zone. The electronic wave functions were expanded in a plane-wave basis set, and Brillouin-zone integrations were performed using the Monkhorst–Pack scheme. Total energies were considered converged when changes were less than 0.01 eV. For trigonal ZrS₂, convergence was achieved using a 6 × 6 × 6 k-point mesh for LDA and 7 × 7 × 7 meshes for PBE and PBE0. Similar convergence behavior was observed for trigonal ZrSeS, with 6 × 6 × 6 meshes for LDA and 7 × 7 × 7 meshes for PBE and PBE0.

While layered transition metal dichalcogenides are traditionally sensitive to van der Waals (vdW) dispersion forces, recent studies have shown that high-level functionals like PBE0 and PBE+U can yield structural parameters that closely mimic experimental data by improving the description of electronic correlation. In our work, the calculated lattice constants and bulk moduli were verified against experimental benchmarks to ensure that the structural foundation for our optical analysis is robust.

The kinetic energy cutoff required for total-energy convergence was found to be 35 Ry for both ZrS₂ and ZrSeS across the LDA, PBE, and PBE0 functionals. All total energies were obtained natively in Rydberg (Ry), as implemented in Quantum ESPRESSO. Structural optimizations were performed by scanning lattice parameters expressed in atomic units (Bohr), followed by fitting the resulting energy–volume data to the third-order Birch–Murnaghan equation of state. The extracted equilibrium lattice parameters were subsequently converted to Angström Å using the standard conversion factor (1 Bohr = 0.529177 Å). Where required for physical interpretation and comparison, energy values were converted using 1 Ry = 13.6057 eV.

For the evaluation of optical properties, the frequency-dependent dielectric function was calculated using a dense Γ-centered 12 × 12 × 4 k-point mesh, which accounts for the structural anisotropy of the layered trigonal phases. To ensure a converged spectral response up to 30 eV, 100 unoccupied bands were included, with a Gaussian broadening of 0.1 eV applied to simulate experimental resolution. Furthermore, a scissor correction was implemented to align the theoretical absorption onsets with our PBE0 and experimental benchmarks (Table ), thereby correcting the systematic band gap underestimation of the LDA/PBE functionals. The physical mechanism of the resulting optical shifts was further analyzed through the projected density of states (PDOS), confirming that the narrowing of the gap upon Se substitution is driven by the upward shift of the valence band maximum, which is dominated by chalcogen-p states.

2. Comparison of the Calculated and Experimental Bandgap for ZrS₂ and ZrSeS.

Present Work			Other Work
Materials	Methods		Theory	Experiment	Scissor Value (ΔE)
ZrS2	LDA	0.90	0.53	1.68–1.78	0.78–0.88
	PBE	1.04	1.12	1.68–1.78	0.64–0.74
	PBE+U	1.660		1.68–1.78	0.12–0.02
	PBE0	1.805		1.93	0.035
ZrSeS	LDA	0.690	1.13	1.40	0.71
	PBE	0.807	1.01	1.40	0.593
	PBE+U	1.263		1.40	0.137
	PBE0	1.37		1.40	0.03

^aRef .

^bRef .

^cRef .

^dRef .

^eRef .

Results and Discussion

Structural Properties of Trigonal ZrS₂ and ZrSeS

Understanding the crystal structure of a material is fundamental to interpreting its mechanical, electronic, and optical properties. Both ZrS₂ , and the mixed-chalcogenide ZrSeS are semiconducting compounds that crystallize in a layered trigonal structure, which is commonly described within the hexagonal setting and is characterized by a 3-fold rotational symmetry axis.

In the trigonal crystal system, the lattice is described by three equivalent in-plane axes separated by 120° and a fourth axis perpendicular to this plane. ZrS₂ crystallizes in the 1T-type structure with space group P3m̅1, while ZrSeS adopts a closely related trigonal symmetry. Experimental lattice parameters reported in the literature for ZrS₂ are a ≈ 3.660–3.709 Å and c ≈ 5.990–6.070 Å, whereas ZrSeS exhibits slightly larger lattice constants, with a ≈ 3.709–3.713 Å and c ≈ 6.070–6.076 Å, ,,,− , reflecting the larger atomic radius of selenium. The optimized crystal structures of ZrS₂ and ZrSeS are illustrated in Figures a and b, respectively.

1.

Crystal structure of a) ZrS₂, b) ZrSeS (plotted by XCrySDen).

To determine the equilibrium structural parameters of trigonal ZrS₂ and ZrSeS, first-principles calculations were carried out using the plane-wave self-consistent field (PWSCF) method within density functional theory. In ZrS₂, zirconium atoms occupy the (0, 0, 0) Wyckoff position, while sulfur atoms are located at (2/3, 1/3, z) and (1/3, 2/3, 1 – z). For the mixed-chalcogenide ZrSeS, the Zr atoms remain at (0, 0, 0), with sulfur and selenium occupying the corresponding chalcogen sites, preserving the layered X-Zr-X stacking along the c-axis.

Structural optimizations were performed using several exchange-correlation functionals, including LDA, PBE, PBE+U, and the hybrid PBE0 functional. The calculated equilibrium lattice parameters, bulk modulus, and pressure derivative for both compounds are summarized in Table , together with available experimental and theoretical values. For ZrS₂, the calculated lattice constants are in good agreement with previous reports. As expected, LDA tends to underestimate the lattice parameters due to its overbinding nature, whereas PBE and PBE+U slightly overestimate them. The hybrid PBE0 functional yields intermediate values, closer to experimental observations.

1. Comparison of the Calculated Bulk Modulus B (in GPa), Lattice Constants (A) and Pressure Derivative of the Bulk Modulus B′ with Other Work for ZrS₂ and ZrSeS.

This work									Reference
ZrS2					ZrSeS				ZrS2		ZrSeS
	LDA	PBE	PBE+U	PBE0	LDA	PBE	PBE+U	PBE0	Theory	Exp.	Theory	Exp.
a = b	3.639	3.692	3.697	3.691	3.653	3.738	3.795	3.702	3.682, 3.680	3.709, 3.660	3.682, 3.703	3.713
c	5.955	6.041	6.050	6.041	5.978	6.117	6.211	6.059	6.026, 6.022	6.004, 5.800	6.026, 6.032	6.010
B	40.33	35.94	31.92	36.24	41.25	34.76	36.13		N/A	N/A	N/A	N/A
B′	4.22	3.79	3.85	4.04	4.72	4.11	3.93	4.33	N/A	N/A	N/A	N/A

^aRef .

^bRef .

^cRef .

^dRef .

^eRef .

A similar functional dependence is observed for ZrSeS, with consistently larger lattice constants compared to ZrS₂. This expansion arises from the substitution of sulfur by selenium and is clearly reflected in the total-energy versus lattice-constant curves shown in Figure a and b. The equilibrium lattice parameters were extracted from these curves by fitting the calculated energy–volume data to the third-order Birch–Murnaghan equation of state, and the resulting values are listed in Table .

2.

Lattice constant versus total energy for ZrS₂ obtained using different exchange–correlation functionals.

A direct comparison of Figures and and Table shows that ZrSeS possesses a larger unit-cell volume than ZrS₂, consistent with the larger size of Se atoms. This lattice expansion is expected to influence the electronic structure by reducing orbital overlap, which generally leads to a narrower band gap and enhanced tunability of optoelectronic properties. While such characteristics are advantageous for applications in optoelectronics and energy-related devices, the increased lattice spacing may also result in slightly reduced mechanical rigidity and a higher susceptibility to defect formation.

3.

Lattice constant versus total energy for ZrSeS calculated using various exchange–correlation approximations.

The bulk modulus values reported in Table further support this trend. ZrS₂ exhibits a higher bulk modulus than ZrSeS, indicating greater resistance to compression, which can be attributed to its smaller lattice parameters and stronger interatomic bonding. Both materials display moderate pressure-dependent stiffening, a behavior that is characteristic of layered transition-metal dichalcogenides. Overall, the present results are in good agreement with existing theoretical and experimental studies and provide reliable predictions of the structural and elastic response of ZrS₂ and ZrSeS, offering valuable guidance for future experimental investigations.

Electronic Properties of ZrS₂ and ZrSeS

Band Structure of ZrS₂ and ZrSeS

The electronic band structures are computed along the high symmetry direction of the Brillouin zone utilizing LDA, PBE, PBE+U and hybrid functions (PBE0). Figure illustrates the band structures for all possible approximations. The calculated bandgap values in eV for LDA, PBE, PBE+U and PBE0 are presented in Table , and a comparison is drawn with existing theoretical and experimental findings.

4.

Band structure of ZrS₂ and ZrSeS along high symmetry points a) with respect to LDA, PBE and PBE+U for ZrS₂ and b) with respect to LDA, PBE and PBE+U for ZrSeS, c and d) with respect to PBE0 approximations for ZrS₂ and ZrSeS respectively.

By examining Table and Figures a, b, c, and d (presumably showing band structures for ZrS₂), and considering the data for ZrSeS, it is evident that the DFT method utilizing the exchange-correlation functional of LDA and PBE (a form of GGA) yields underestimated energy bandgap values (for ZrS₂: ∼0.90–1.04 eV; for ZrSeS: ∼0.690–0.807 eV) compared to experimental results (for ZrS₂: 1.68–1.78 eV, 1.93 eV; for ZrSeS: 1.40 eV). Conversely, the energy bandgap values obtained using the hybrid functional (PBE0) and the PBE+U approach show better agreement with the experimental value for both ZrS₂ (PBE0: 1.895 eV, PBE+U: 1.660 eV) and ZrSeS (PBE0: 1.37 eV, PBE+U: 1.263 eV), although PBE0 appears to provide a closer match for ZrS₂.

The reduction in the band gap from ZrS₂ to ZrSeS can be primarily attributed to the increased hybridization between Zr-4d and the larger Se-4p orbitals. While heavy elements like Selenium often introduce Spin–Orbit Coupling (SOC) effects that can further split the valence bands, the current study focuses on the scalar-relativistic effects. The observed narrowing is consistent with the upward shift of the valence band edge due to the lower electronegativity of Se compared to S.

Total Density of States of ZrS₂ and ZrSeS

The density of states for ZrS₂ and ZrSeS was computed using LDA, PBE and PBE+U, as depicted in Figure . The Density of States (DOS) aids in comprehending the distribution of state occupancy across a defined energy range. It offers insights into both unoccupied and occupied states. A peak in DOS at a specific energy level indicates available states for occupation, while no state is occupied at a DOS of zero. In the trigonal phases of ZrS₂ and ZrSeS the density of states exhibits discontinuities spanning from the top of the valence band to the bottom of the conduction band, typically representing the system’s bandgap.

5.

Total density of states of a) ZrS₂ and b) ZrSeS using LDA, PBE and PBE+U.

The band gap difference between ZrS₂ and ZrSeS stems from the distinct chemical and physical properties of S and Se. Larger atomic size and higher energy levels of Se orbitals (vs S) in ZrSeS lead to weaker Zr–Se bonding and altered band positions, reducing the band gap. Additionally, stronger spin–orbit coupling in heavier Se further contributes to a smaller band gap in ZrSeS compared to ZrS₂.

ZrS₂ and ZrSeS, while both layered transition metal dichalcogenides, offer distinct advantages stemming from their constituent chalcogen atoms. ZrS₂, with its wider band gap, holds promise for applications in the ultraviolet and blue regions of the electromagnetic spectrum, potentially exhibiting higher thermal stability due to stronger bonding. Conversely, ZrSeS, characterized by its narrower band gap, presents benefits for visible and near-infrared optoelectronic devices, often displaying enhanced light absorption in these lower energy ranges. Furthermore, the mixed anion composition of ZrSeS allows for the potential tuning of its electronic and optical properties, providing a pathway to tailor materials for specific applications.

Projected Density of States for ZrS₂ and ZrSeS

Figure a–f shows the projected density of states (PDOS) of trigonal ZrS₂ and ZrSeS calculated using the LDA, PBE, and PBE+U functionals, with the Fermi level set to zero energy. For ZrS₂ [Figure a–c], the valence-band maximum (VBM) is dominated by S 3p states with minor Zr 4d contributions, while the conduction-band minimum (CBM) is primarily composed of Zr 4d states, indicating that low-energy optical transitions are mainly of S 3p → Zr 4d character. Contributions from Zr 5s and S 3s states are negligible near the band edges.

6.

Projected density of states (PDOS) of trigonal ZrS₂ [(a) LDA, (b) PBE, (c) PBE+U] and ZrSeS [(d) LDA, (e) PBE, (f) PBE+U]. The Fermi level is set to 0 eV (vertical black line).

For ZrSeS [Figure d–f], the VBM exhibits a substantial contribution from Se 4p states, which lie at higher energies than S 3p orbitals. This upward shift of the valence-band edge leads to a reduced band gap relative to ZrS₂, while the CBM remains dominated by Zr 4d states. The substitution of S by Se therefore primarily affects the valence-band position without altering the fundamental nature of the conduction band.

7.

Complementary views of the complex dielectric function for ZrS₂ and ZrSeS, calculated using various DFT approximations (PBE, LDA, DFT+U). a) ε₁(ω) and ε₂(ω) vs energy domain for ZrS₂, b) ε₁(ω) and ε₂(ω) vs energy domain for ZrSeS c) ε₁(ω) and ε₂(ω) vs wavelength domain for ZrS₂ and d) ε₁(ω) and ε₂(ω) vs wavelength domain ZrSeS.

The influence of the exchange–correlation functional is also evident. While LDA and PBE yields narrower band gaps due to enhanced state delocalization, the inclusion of the Hubbard U term shifts the Zr 4d states to higher energies, increasing the band gap magnitude. Importantly, the orbital character of the VBM and CBM remains unchanged across LDA, PBE, and PBE+U, confirming that band gap variations arise from energetic shifts rather than changes in bonding character.

These PDOS features provide a direct explanation for the optical spectra discussed in next Section. Since the absorption onset corresponds to transitions between chalcogen p states at the VBM and Zr d states at the CBM, the Se-induced upward shift of the VBM in ZrSeS results in a red-shift of the absorption edge compared to ZrS₂, consistent with the calculated dielectric functions and absorption spectra.

Optical Properties of ZrS₂ and ZrSeS

The frequency-dependent complex dielectric function,

$$\epsilon (\omega )={\epsilon }_1(\omega )+i{\epsilon }_2(\omega )$$

serves as the fundamental link between the electronic band structure and the optical response of ZrS₂ and ZrSeS. The accurate determination of these optical constants is essential for benchmarking our theoretical predictions against contemporary electronic and optical analyses of related solid-state systems. By examining the peaks in the imaginary part, ε₂(ω), we can directly correlate spectral features with specific interband transitions. This methodology aligns with recent research on functional layered materials, where such dielectric interpretations are vital for assessing the suitability of TMDs for high-efficiency optoelectronic applications. ,

The imaginary part of the dielectric function, ε₂(ω), is typically calculated by summing the electric dipole operator matrix elements between occupied and unoccupied wave functions over the Brillouin zone, while adhering to selection rules. This calculation directly links the optical response to the material’s electronic structure. The specific relation used for calculating ε₂(ω) is given by

$${\epsilon }_2(\omega )=\left(\frac{4{\pi }^2{e}^2}{m^2{\omega }^2}\right)\sum _{i,j}{\int }_{BZ}|⟨i|M|j⟩{|}^2{F}_i(1-F_j)\delta (E_f-E_i-\omega ){\mathrm{d}}^{3}k$$ 1

Due to the principle of causality, the real and imaginary parts of the dielectric function are inherently linked through the Kramers–Kronig transformation. The real part ε₁(ω), can therefore be derived from the imaginary part using the following Kramers–Kronig relation:

$${\epsilon }_1(\omega )=1+\frac{2}{\pi }\mathcal{P}{\int }_0^{\infty }\frac{{\omega }^{\prime }{\epsilon }_2({\omega }^{\prime })}{{\omega }^{\prime 2}-{\omega }^2}\mathrm{d}{\omega }^{\prime }$$ 2

All other optical parameters such as absorption coefficient, α­(ω) refractive index n(ω) reflection coefficient R(ω) can be derived from ε₁(ω) and ε₂(ω). For instance, n(ω) is given as,

$$n(\omega )={\{\frac{{\epsilon }_1(\omega )}{2}+\frac{\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}}{2}\}}^{1/2}$$ 3

The extinction coefficient, k, is the imaginary part of the complex refractive index (ń = n + ik) is given by

$$k_e(\omega )=\sqrt{\frac{\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}-{\epsilon }_1(\omega )}{2}}$$ 4

Dielectric Function for ZrS₂ and ZrSeS

The complex dielectric function, ε­(ω) = ε₁(ω) + i ε₂(ω), was investigated to evaluate the interaction of ZrS₂ and ZrSeS with incident electromagnetic radiation. Figure a–b displays the real and imaginary components of the dielectric function calculated using the LDA, PBE, and PBE+U exchange–correlation functionals, including their anisotropic contributions ${\epsilon }_1^x$ , ${\epsilon }_1^z$ , ${\epsilon }_2^x$ , and ${\epsilon }_2^z$ . To address the systematic underestimation of the fundamental band gap in standard DFT, a scissor operator was applied to rigidly shift the conduction bands, thereby aligning the calculated absorption onsets with experimental benchmarks and our PBE0 results. The corresponding scissor shifts are ΔE = 0.83/0.71 eV (LDA), 0.69/0.59 eV (PBE), and 0.07/0.14 eV (PBE+U) for ZrS₂ and ZrSeS, respectively.

For ZrS₂, the imaginary part of the dielectric function, ε₂(ω), characterizes the onset of interband electronic transitions. Following the application of the scissor correction, the primary absorption peaks in the x-plane (E ⊥ c) are located at 3.03 eV (LDA), 3.09 eV (PBE), and 2.71 eV (PBE+U). Along the z-plane (E ∥ c), higher-energy transitions are observed, with prominent peaks appearing in the range of 8.80–10.29 eV. These results indicate that ZrS₂ exhibits a strong photon absorption response within the 2.5–10.0 eV energy window. The application of the scissor operator ensures that the calculated optical band gap is physically consistent with the experimental value of ∼1.73 eV, effectively eliminating the unphysical infrared transitions predicted by uncorrected LDA and PBE calculations.

The substitution of sulfur with selenium induces a pronounced redshift in the optical response, which can be attributed to the higher energy of the Se-4p orbitals relative to the S-3p states. For ZrSeS, the scissor-corrected ε₂(ω) spectra exhibit prominent interband transition peaks in the x-plane (E ⊥ c) at 2.47 eV (LDA), 2.57 eV (PBE), and 2.34 eV (PBE+U). Along the z-plane (E ∥ c), the dominant peaks are located at 9.27 eV (LDA) and 8.94 eV (PBE+U).

From a mechanistic perspective, ZrSeS displays strong photon absorption over a broad energy range spanning 1.6–15.0 eV in the x-plane and 1.1–7.0 eV in the z-plane. The observed broadening and redshift of the absorption window toward the visible region indicate that the mixed-chalcogenide ZrSeS is a more promising candidate for optoelectronic applications than its parent compound. The real part of the dielectric function, ε₁(ω), further supports this conclusion by revealing a higher static dielectric constant, ε₁(0), for ZrSeS, consistent with its reduced band gap and enhanced electronic polarizability.

The real part of the dielectric function, ε₁(ω), characterizes the electronic polarizability of the system and its ability to screen external electric fields. Electronic screening is a fundamental process in condensed-matter physics, governing electron–electron interactions and exciton binding energies. As the photon energy approaches the static limit (ω → 0 eV), as shown in Figure a–b, the ε₁(ω) spectra converge to a constant value corresponding to the static dielectric constant, ε₁(0) (Table ).

3. Calculated Frequency Dependent Static Dielectric Constant^cε_1x (0) or ε_s of ZrS₂ and ZrSeS.

	Present Work				Other Work
Materials	ε1(0)	LDA	PBE	PBE+U	Theory	Experiment
ZrS2	ε1x (0)	13.81	12.31	9.59	∼7–13,	N/A
	ε1z (0)	5.19	4.83	4.53	∼7–8,	N/A
ZSeS	ε1x (0)	16.36	14.71	12.36	9–13	N/A
	ε1z (0)	6.12	5.69	5.30	7–8	N/A

^aRef .

^bRef .

^cRef .

This is the material’s ability to screen static (zero-frequency) electric fields. A higher ε₁(0) indicates stronger screening of charges within the material. For both ZrS₂ (a) and ZrSeS (b), there’s clear anisotropy, meaning ε₁(0) is different for the in-plane (x) and out-of-plane (z) components. Also, the specific value of ε₁(0) varies significantly depending on the DFT functional used (PBE, LDA, DFT+U), indicating their differing abilities to capture electron–electron interactions and band gaps.

The static dielectric constant depends strongly on the band gap (electric part). This can be explained by the expression of Penn model

$${ϵ}_1(0)=1+{\left(\frac{\hslash {\omega }_p}{E_g}\right)}^2$$ 5

Where: ε₁(0) is the static dielectric constant, ℏω _p is the average optical plasma energy, which depends on the electron density (or valence electron density) of the material. The value of the reduced Planck constant, often denoted as ℏ (“h-bar”), ℏ = 6.583118569 × 10^–16 eV·s.

Plasma frequency (ω _p ) is an intrinsic characteristic of a material, representing the collective oscillation frequency of its free charge carriers (electrons in a metal, or valence electrons considered collectively in a dielectric). It is determined by the density of these electrons and their effective mass.

$${\omega }_p=\sqrt{\frac{n{e}^2}{{ϵ}_0{m}^{*}}}$$ 6

Where n is electron density, e is elementary charge, ϵ₀ is vacuum permittivity, m* is effective mass. From expression of Penn model we have Penn model average optical gap (effective optical plasma energy from the Penn model) as

$$\hslash {\omega }_p=E_g\sqrt{{ϵ}_1(0)-1}$$ 7

The characteristic optical energies (ℏω _p ) were calculated using the Penn model, which relates the static dielectric constant to the band gap for both ZrS₂ and ZrSeS in-plane direction, and out-of-plane direction as shown in Table . These values represent the average electronic excitation energies contributing to the static dielectric response in their respective directions

4. Calculated Plasma Frequency (ℏω_p) for ZrS₂ and ZrSeS.

	Present Work				Other Work
Materials	ℏω p	LDA	PBE	PBE+U	Theory	Experiment
ZrS2	ℏω px	3.221	3.498	4.865	N/A	N/A
	ℏω pz	1.842	2.035	3.119	N/A	N/A
ZSeS	ℏω px	2.704	2.988	4.257	N/A	N/A
	ℏω pz	1.561	1.747	2.619	N/A	N/A

From Figure a and b in Mid-Energy region, ε₁(ω) for both ZrS₂ and ZrSeS exhibits (Dynamic Response) significant oscillations, including peaks, troughs, and sometimes crosses the zero-axis or becomes negative. The changing value of ε₁(ω) across this energy range signifies the material’s dispersion. This means the speed of light (and thus the refractive index, which is related to $\sqrt{{\epsilon }_1}$ ) varies with photon energy. When ε₁(ω) is positive, the material acts as a dielectric, allowing light to propagate but slowing it down. Points where ε₁(ω) crosses zero are indicates the plasma frequency (ω _p ), where the material transitions from being reflective to transparent (assuming ε₂(ω) is small). For semiconductors, these crossings can indicate complex dielectric behavior and might be related to collective excitations or strong transitions. For instance, if ε₁(ω) crosses zero and becomes negative, it suggests a metallic-like response at that frequency. If ε₁(ω) becomes negative (and ε₂(ω) is small), the material behaves like a metal and is highly reflective at that specific photon energy. Light cannot propagate efficiently through the material.

Accordingly, Figure illustrates the dynamic behavior of the real part of the dielectric function, ε₁(ω), for ZrS₂ under different polarization directions and exchange–correlation functionals, after applying the appropriate scissor corrections to align the optical response with experimental benchmarks. For the LDA functional (ΔE = 0.83 eV), the in-plane (x) component exhibits a positive maximum of 21.42 at 2.81 eV, followed by a pronounced minimum of −7.89 at 3.25 eV. The out-of-plane (z) response shows a positive maximum of 6.71 at 2.37 eV and a minimum of −1.59 at 10.51 eV. Using the PBE functional (ΔE = 0.69 eV), the in-plane response displays a maximum of 20.50 at 2.89 eV and a minimum of −5.73 at 3.33 eV, while the out-of-plane component reaches a maximum of 6.00 at 2.23 eV and a minimum of −1.48 at 10.80 eV. For PBE+U (ΔE = 0.07 eV), the in-plane response shows a reduced maximum of 15.61 at 2.05 eV and a minimum of −2.56 at 6.45 eV, whereas the out-of-plane component exhibits a maximum of 6.00 at 2.49 eV and a minimum of −0.98 at 10.62 eV.

Similarly, for ZrSeS (Figure b), ε₁(ω) also exhibits a dynamic response with distinct characteristics. Under LDA, the in-plane (x) response ranges from 0.4 to 7.48 eV, presenting a positive maximum peak of 25.23 at 1.10 eV and a minimum peak of −6.00 at 1.98 eV; the out-of-plane (z) response is between 0.4 and 14.29 eV, with a positive maximum peak of 7.22 at 1.76 eV and a minimum peak (trough) of −1.57 at 9.68 eV. When using PBE, the in-plane (x) response is observed from 0.4 to 7.25 eV, showing a positive maximum peak of 23.38 at 1.76 eV and a minimum peak of −5.42 at 2.20 eV; the out-of-plane (z) response covers 0.4 to 15.61 eV, with a positive maximum peak of 6.84 at 1.76 eV and a minimum peak (trough) of −1.63 at 9.90 eV. Lastly, for PBE+U, the dynamic response for both in-plane (x) and out-of-plane (z) of ZrSeS is between 0.4 and 7.04 eV. The in-plane (x) component displays a positive maximum peak of 20.17 at 1.98 eV and a minimum peak of −3.00 at 6.16 eV, while the out-of-plane (z) component has a positive maximum peak of 6.30 at 1.98 eV and a minimum peak (trough) of −1.50 at 9.90 eV.

From Figure at very high energies, the ε₁(ω) curves tend to flatten out and approach a value 1. At these high photon energies (e.g., into the X-ray regime), the electrons in the material are unable to follow the extremely rapid oscillations of the electric field. Consequently, the material’s polarizability becomes negligible, and its dielectric response approaches that of a vacuum (ϵ = 1). In this region, the material becomes increasingly transparent to these high-energy photons. As a result Figure a ZrS₂ behave asymptomatic at >∼12.83 eV in-plane (x) and >∼19.83 eV out-of-plane (z). Similarly, Figure b ZrSeS behave asymptomatic at >∼7.71 eV in-plane (x) and >∼15.71 eV out-of-plane (z).

Figures c and d present the calculated real (ε₁(ω)) and imaginary (ε₂(ω) components of the dielectric function for ZrS₂ and ZrSeS, respectively, as a function of wavelength. For both materials, the ultraviolet and visible regions (wavelengths shorter than ∼500 nm) are characterized by strong optical absorption, as evidenced by the pronounced increase in ε₂(ω) arising from interband electronic transitions. In contrast, at longer wavelengths, ε₂(ω) decreases markedly while ε₁(ω) becomes relatively stable, indicating reduced absorption and enhanced optical transparency.

After applying the scissor corrections, the fundamental absorption edge, defined by the first pronounced increase in ε₂(ω, for ZrS₂ in the in-plane (x) polarization is observed at approximately 578 nm (LDA), 598 nm (PBE), and 679 nm (PBE+U). In contrast, for the out-of-plane (z) polarization, the absorption onset occurs at around 128 nm for all exchange–correlation functionals. The dominant absorption peak of ZrS₂ is located near 386 nm in the in-plane direction, while a sharp and intense peak appears at approximately 128 nm in the out-of-plane direction.

Similarly, for ZrSeS, as shown in Figure d, the absorption edge appears at approximately 211 nm for the in-plane (x) polarization and 132 nm for the out-of-plane (z) polarization. The most prominent absorption peak of ZrSeS is located near 436 nm in the in-plane direction, while a sharp peak is observed at approximately 132 nm along the out-of-plane direction. Notably, ZrS₂ consistently exhibits absorption at shorter wavelengths than ZrSeS, reflecting its larger optical band gap. In contrast, the red-shifted absorption response of ZrSeS toward longer wavelengths indicates a reduced optical gap resulting from selenium substitution.

The peaks in ε₂(ω) beyond the fundamental absorption edge originate from specific interband electronic transitions and are strongly correlated with the joint density of states (JDOS). A key feature of the dispersion of ε₁(ω) is its intrinsic connection to ε₂(ω) through the Kramers–Kronig relations. Below the absorption edge, ε₁(ω) remains positive and nearly constant, corresponding to the static dielectric constant. As the photon energy approaches an absorption peak, ε₁(ω) increases sharply, whereas immediately after the peak it decreases rapidly.

A stable region in ε₁(ω) is observed at very long wavelengths, corresponding to the long-wavelength limit. For ZrS₂ in the in-plane (x) direction, this limit is reached beyond 634 nm (PBE+U), 721 nm (PBE), and 700 nm (LDA). For ZrSeS, the long-wavelength limit occurs beyond 866 nm (PBE+U), 920 nm (PBE), and 908 nm (LDA). The differences between ${\epsilon }_1^x$ and ${\epsilon }_1^z$ highlight the optical anisotropy of these layered 1T-phase materials, indicating that the refractive index depends on the crystallographic polarization direction.

A fundamental aspect of the dielectric response is the intrinsic connection between ε₁(ω) and ε₂(ω) via the Kramers–Kronig relations. At wavelengths longer than the fundamental absorption edge, where ε₂(ω) is negligible, ε₁(ω) remains positive and nearly constant, representing the material’s static dielectric response. As an absorption peak in ε₂(ω) is approached, ε₁(ω) exhibits a sharp increase followed by a rapid decrease immediately after the resonance, occasionally reaching negative values in strongly absorbing spectral regions. Notably, the minimum of ε₁(ω) often coincides with the maximum of the corresponding ε₂(ω) peak, marking the frequency of the strongest electronic transition.

In Figure c, a stable, “flat” region of ε₁(ω) is observed for ZrS₂ at long wavelengths following the application of the scissor correction. In the in-plane (x) direction, this long-wavelength limit is reached beyond ∼545 nm (PBE+U), ∼665 nm (PBE), and ∼690 nm (LDA), marking the transition into the static dielectric regime. Similarly, for ZrSeS, as shown in Figure d, this limit occurs beyond ∼855 nm (PBE+U), ∼1040 nm (PBE), and ∼1110 nm (LDA). These plateaus correspond to the low-frequency limit of the real part of the dielectric constant for these layered materials.

Mechanistically, these optical features arise from electronic transitions between the chalcogen p valence states and the Zr-4d conduction states. The introduction of Se-4p orbitals in the mixed-chalcogenide ZrSeS raises the valence band maximum relative to ZrS₂, thereby shifting the absorption features toward the visible spectrum and enhancing the light–matter interaction efficiency of the material.

Refractive Index for ZrS₂ and ZrSeS

The refractive index (or index of refraction) of a material is a fundamental optical property that describes how much light bends, or refracts, when it passes from one medium to another. It is a dimensionless quantity, meaning it has no units. Figure , shows the real (n­(ω)) and imaginary (k­(ω)) parts of the complex refractive index for ZrS₂ and ZrSeS as a function of photon energy (ω in eV) and wavelength (λ in nm). From the Figure e and f at ε₁(0) and Figure g and h at longer wavelength we can see that the static refractive index n(0) for ZrS₂ and ZrSeS were calculated and summarized in Table .

8.

This image shows the real (n­(ω)) and imaginary (k­(ω)) parts of the complex refractive index for ZrS₂ and ZrSeS, e) and f) as a function of photon energy (ω in eV), g) and h) as a function of wavelength (λ in nm).

5. Calculated Value of Static Refractive Index of ZrS₂ and ZrSeS and Experimental and Other Theoretical Value.

	Present Work				Other Work
Materials	n(0)	LDA	PBE	PBE+U	Theory	Experiment
ZrS2	n xx (0)	3.72	3.51	3.10	4	∼3.5–4
	n zz (0)	2.78	2.20	2.13	N/A	N/A
ZSeS	n xx (0)	4.05	3.84	3.52	4	∼2.5
	n zz (0)	2.47	2.39	2.30	N/A	N/A

The extinction coefficient, k(ω), quantifies the material’s light absorption, with higher values corresponding to stronger attenuation. After applying the scissor corrections, both ZrS₂ and ZrSeS exhibit negligible k(ω) at low photon energies, indicating transparency in the infrared region. For ZrS₂, the absorption onset occurs near 1.69–1.83 eV (visible region), whereas for ZrSeS the onset is red-shifted to approximately 1.29–1.31 eV (near-infrared), reflecting the narrower optical band gap induced by selenium substitution.

This behavior confirms the reduction of the optical band gap upon selenium substitution. As the photon energy increases into the ultraviolet region, k(ω) rises sharply, with corrected peaks appearing at approximately 2.7–3.8 eV and 5.8–7.8 eV for the in-plane (k _x ) component, while out-of-plane (k _z ) absorption dominates the higher 5.8–7.8 eV range. These spectral features arise from specific interband electronic transitions between the chalcogen p and Zr d states.

The k _x components generally exhibit stronger absorption in the lower-energy visible region, whereas the k _z components display a more pronounced response in the deep ultraviolet. This orientation-dependent behavior is a characteristic feature of layered transition metal dichalcogenides and is particularly important for applications in polarization-sensitive photodetectors and birefringent optical devices.

For both ZrS₂ and ZrSeS, the extinction coefficient curves for k _x (gold, blue, and magenta lines in panels e, f, g, h) and k _z (red, black, and green in panels e, f, g, h) exhibit distinct spectral positions and peak magnitudes. This divergence directly confirms the optical anisotropy of these 1T-phase materials, demonstrating that light absorption is strongly dependent on the polarization direction.

As discussed above, the k _x component generally exhibits stronger features in the lower-energy visible spectrum, whereas the k _z component shows a delayed onset and dominant peaks in the deep ultraviolet. This polarization-dependent behavior originates from the layered crystal symmetry and the specific spatial orientation of the chalcogen p and Zr d orbitals involved in the electronic transitions.

Absorption Spectrum of ZrS₂ and ZrSeS

The absorption spectrum encompasses the wavelengths or frequencies of electromagnetic radiation that a material absorbs. As light or electromagnetic radiation traverses a material, specific wavelengths are absorbed, causing a reduction in light intensity at those wavelengths. This spectrum offers valuable insights into the interaction between a material and light.

Hence, we examined the absorption spectra of both ZrS₂ and ZrSeS using various approximations such as PBE, LDA, and PBE0 to gain a deeper understanding, as illustrated in Figure .

9.

Absorption spectrum of a ZrS₂ and ZrSeS.

Figure provides valuable insight into the optical transition characteristics of ZrS₂ and ZrSeS by identifying the key energy levels at which significant electronic transitions occur, along with their corresponding oscillator strengths. The photon energies, expressed in eV (ℏω), represent specific transition energies associated with light absorption or emission, while the oscillator strength quantifies the probability and intensity of these optical transitions. Higher oscillator strength values correspond to stronger absorption or emission features in the observed spectra.

Accordingly, Figure reveals distinct shifts in the transition energies of ZrS₂ and ZrSeS following the application of scissor corrections. For ZrS₂, the most prominent transitions with maximum oscillator strength occur at approximately 2.07 eV (600 nm) within LDA, 2.15 eV (577 nm) within PBE, and 7.65 eV (162 nm) within PBE+U. In contrast, the dominant transitions for ZrSeS are red-shifted to lower energies, appearing at approximately 1.50 eV (827 nm) within LDA, 1.75 eV (708 nm) within PBE, and 5.64 eV (220 nm) within PBE+U. These peaks correspond to electronic excitations from occupied valence states to unoccupied conduction states. Notably, the low-energy transitions, now properly aligned with the visible and ultraviolet spectral regions, exhibit strong absorption features that are particularly relevant for optoelectronic applications.

The optical responses of ZrS₂ and ZrSeS differ significantly due to their distinct chemical compositions. Substituting sulfur with selenium modifies the electronic band structure through differences in electronegativity, atomic size, and orbital hybridization, leading to altered optical transition pathways. While both materials display strong ultraviolet absorption, making them suitable candidates for UV photodetectors, the red-shifted and broadened absorption features of ZrSeS indicate enhanced sensitivity to lower-energy visible photons. This characteristic suggests that ZrSeS may be a more promising material for broadband optoelectronic and solar energy conversion applications.

Static and Dynamic Polarizability (α) of ZrS₂ and ZrSeS

Both dynamic and static polarizabilities are important fundamental properties of matter. They determine the dynamical response of a bound system to external perturbations, and provide valuable insight into internal strong interaction structure − polarizability (α) experimentally measured as the ratio of induced dipole moment to the electric field ξ that induces it.

$$\alpha =\frac{P}{\xi }$$ 8

The first-order static polarizability α(0), describes the linear response of an atom or molecule to a static (nonoscillating) electric field. It is typically derived from time-independent perturbation theory and can be expressed as.

$$\alpha (0)=\sum _{n\ne 0}\frac{2|⟨n|\mathrm{\mu ̂}|0⟩{|}^2}{E_n-E_0}$$ 9

The first-order dynamic polarizability, α­(ω), describes the linear response of an atom or molecule to an oscillating electric field with frequency. It is derived from time-dependent perturbation theory and can be expressed as

$$\alpha (\omega )=\sum _{n\ne 0}\frac{2{\omega }_{n0}|⟨n|\mathrm{\mu ̂}|0⟩{|}^2}{\hslash ({\omega }_{n0}^2-{\omega }^2)}$$ 10

Alternatively, a more general form is

$$\alpha (\omega )=\frac{1}{\hslash }\sum _{n\ne 0}(\frac{|⟨n|\mathrm{\mu ̂}|0⟩{|}^2}{E_n-E_0-\hslash \omega }+\frac{|⟨n|\mathrm{\mu ̂}|0⟩{|}^2}{E_n-E_0+\hslash \omega })$$ 11

|0⟩ is the ground state (initial state) wave function of the system, |n⟩ is an excited state (intermediate state) wave function of the system, E ₀ is the energy of the ground state, E _n is the energy of the excited state is the electric dipole moment operator, ⟨n|μ̂|0⟩ is the transition dipole moment matrix element between the ground state and the excited state n, ℏ is the reduced Planck constant, ω is the angular frequency of the applied electric field, ω _n0 = (E _n – E ₀)/ℏ is the transition frequency between the ground state and the excited state n.

When a molecule is anisotropic, its polarizability is represented by a tensor, α _ij . In many experimental situations, especially for randomly oriented molecules is the average polarizability.

$${\alpha }_{av}=\frac{1}{3}({\alpha }_{xx}+{\alpha }_{yy}+{\alpha }_{zz})$$ 12

Where: α _av is the average polarizability. α _xx , α _yy , and α _zz are the diagonal components of the polarizability tensor along the x, y, and z axes (usually the principal axes of the molecule). This average value is important because it represents the overall “ease” with which the electron cloud of a molecule can be distorted, irrespective of its orientation, and is directly related to macroscopic properties like the refractive index and dielectric constant in nonpolar systems.

Polarizability anisotropy quantifies how much the polarizability of a molecule deviates from being isotropic (uniform in all directions). It is a measure of the asymmetry in a molecule’s electron cloud distortion by an electric field. For molecules where polarizability needs to be described by three principal components (α _xx , α _yy , and α _zz , often aligned with the molecule’s principal axes), the squared anisotropy (often denoted γ² or β²) is commonly used. This definition accounts for the differences between all three principal components.

$$\mathrm{\Delta }\alpha ={\alpha }_{\parallel }-{\alpha }_{\perp }$$ 13

$${\gamma }^2=\frac{1}{3}[{({\alpha }_{xx}-{\alpha }_{yy})}^2+{({\alpha }_{yy}-{\alpha }_{zz})}^2+{({\alpha }_{yy}-{\alpha }_{zz})}^2]$$ 14

Or

$$\gamma ={\{\frac{1}{3}[{({\alpha }_{xx}-{\alpha }_{yy})}^2+{({\alpha }_{yy}-{\alpha }_{zz})}^2+{({\alpha }_{yy}-{\alpha }_{zz})}^2]\}}^{1/2}$$

Here α _xx , α _yy , and α _zz are the principal components of the polarizability tensor, representing the polarizability along the molecule’s principal axes. In this part, we calculated the static and dynamic polarizability of ZrS₂ and ZrSeS. The calculated values for are shown in Table and Figure .

6. Calculated Static and Dynamic Polarizability of ZrS₂ and ZrSeS.

		Static Polarizability (Å3)					Dynamic Polarizability (Å3) at 2.18 eV
Materials	Methods	α xx	α yy	α zz	α av	γ	α xx	α yy	α zz	α av	γ
ZrS2	LDA	16.82	16.82	15.91	16.52	0.91	16.25	16.25	14.65	15.72	1.60
	PBE	16.61	16.61	15.35	16.19	1.26	15.91	15.91	14.08	15.30	1.83
	PBE+U	13.68	13.68	9.50	12.29	4.18	12.91	12.91	8.62	11.03	3.61
ZrSeS	LDA	18.05	18.05	17.24	17.78	0.81	17.59	17.59	15.94	17.04	1.65
	PBE	17.93	17.93	16.84	17.57	0.36	17.29	17.29	15.50	16.69	1.79
	PBE+U	15.55	15.55	10.63	13.91	4.92	13.62	13.62	9.56	12.27	4.06

10.

Dynamic polarizability as a function of frequency for (a) ZrS₂, (b) ZrSeS.

Figure and Table illustrate the calculated dynamic polarizability (α­(ω)) as a function of photon energy (ω) for (a) ZrS₂ and (b) ZrSeS, comparing results obtained using different density functional theory (DFT) exchange-correlation functionals. The plots compare the results of different computational methods: LDA, PBE, and DFT+U. While all three methods show the same overall trends and anisotropy, there are subtle quantitative differences between them, particularly for the out-of-plane polarizability at low frequencies, suggesting that the on-site electron–electron interactions, accounted for by the DFT+U method, play a role in the material’s response in this direction [Figure ]. The polarizability is presented in units of ų) while the photon energy is in electron volts (eV).

Both panels display the diagonal components of the polarizability tensor α _xx = α _yy and α _zz reflecting the anisotropic nature of these layered materials. For both ZrS₂ and ZrSeS, the polarizability generally exhibits a decreasing trend with increasing photon energy, which is characteristic of optical dispersion. At ω = 0 (static limit), the polarizability values are the highest, indicating the maximum response to a static electric field. As the frequency increases, the polarizability sharply decreases, after which the curve becomes noticeably flat and approaches zero.

A notable feature in both materials is the significant anisotropy in their polarizability, with α _xx consistently larger than α _zz across the entire energy range. This suggests a stronger electronic response to electric fields applied within the plane (x-direction) compared to perpendicular to the plane (z-direction), which is typical for layered transition metal dichalcogenides.

Comparing ZrS₂ and ZrSeS, both materials exhibit similar qualitative trends in dynamic polarizability, including energy dependence and anisotropy (α _xx > α _zz ). Quantitatively, ZrSeS shows slightly higher static polarizability values across all methods, likely due to differences in electronic structure arising from sulfur-to-selenium substitution. These findings offer insights into the materials’ optical and dielectric properties, highlighting their sensitivity to crystallographic direction and theoretical approach.

Conclusions

In this study, a comprehensive investigation into the electronic, structural, and optical properties of layered transition metal dichalcogenides, ZrS₂ and ZrSeS, was successfully conducted using first-principles density functional theory (DFT) calculations. Various exchange-correlation approximations, including LDA, PBE, PBE+U, and PBE0, were engaged to accurately model these materials.

The structural analyses confirmed the stability of both ZrS₂ and ZrSeS in their trigonal phases, with calculated lattice constants showing excellent agreement with available experimental data. This validation underscores the reliability of the computational approaches used for lattice constants. For electronic properties, while standard LDA and GGA functionals were found to significantly underestimate the band gaps, the more advanced PBE+U and hybrid PBE0 functionals provided band gap values that are highly consistent with experimental observations, for ZrS₂ and ZrSeS. The nuanced differences in the electronic band structures and band gaps between ZrS₂ and ZrSeS were elucidated, largely attributed to the distinct atomic characteristics of sulfur and selenium.

Furthermore, a detailed examination of the optical properties, encompassing the complex dielectric function, absorption spectrum, refractive index, and static and dynamic polarizability, revealed the characteristic optical responses of these materials. The analysis of these properties provided insights into the electronic transitions occurring upon photon absorption and the materials’ interaction with electromagnetic radiation. The distinct optical behaviors observed for ZrS₂ and ZrSeS, including their transparency ranges and absorption onsets, offer crucial information for their application potential.

Overall, this research provides valuable theoretical insights into the fundamental physical properties of ZrS₂ and ZrSeS. The findings not only enhance our understanding of these promising transition metal dichalcogenides but also serve as a strong theoretical foundation for guiding experimental efforts in the design and development of next-generation electronic, optoelectronic, and energy-related devices.

All data supporting the findings of this study are contained within the main manuscript. Additional raw data and processed outputs are available from the corresponding author upon reasonable request. The data sets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request. The codes and input files used for this study are available from the corresponding author upon reasonable request.

T.G.E.: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Resources, Data Curation, WritingOriginal Draft Preparation, WritingReview and Editing, Visualization, Supervision.

The author, Teshome Gerbaba Edossa, declares that no funds, grants, or other support were received from any public, commercial, or not-for-profit funding agency for the preparation of this manuscript. The study was conducted as part of the author’s independent research work in support of academic promotion.

This work, titled “First-Principles Study of the Electronic, Structural, and Optical Properties of Zirconium disulfide (ZrS₂) and Zirconium Selenium Sulfide (ZrSeS) using DFT and DFT+U,” is an original theoretical investigation. It employs first-principles calculations based on Density Functional Theory (DFT) to analyze the electronic, structural, and optical properties of layered transition metal dichalcogenides (ZrS₂ and ZrSeS). The research is original, and the findings have not been previously submitted to any academic degree program or published elsewhere. This is a purely computational study based on first-principles calculations. It does not involve human participants, animal subjects, or any personal data. Therefore, ethics approval and consent to participate were not required. This manuscript does not contain any individual person’s data in any form; therefore, consent for publication is not applicable.

The author declares no competing financial interest.