Predicted behavior of a proposed novel hexa-structured layered Sb₂S₃: potential optoelectronic applications in the GHz range

Abstract

This study investigates a proposed novel layered orthorhombic phase of antimony sulfide (Sb₂S₃) using first-principles density functional theory (DFT). The theoretical analysis reveals strong anisotropy in the electronic and optical properties, with metallic character along the in-plane (x/y) directions and a small direct band gap (0.44–0.52 eV) along the out-of-plane (z) direction. A band gap value of 0.44 eV is obtained from band-structure calculations, while that of 0.52 eV is derived from Tauc plot analysis. Dielectric tensor analysis shows Drude-like behavior, with negative real permittivity in selective wavelength regions, indicating that the material functions as a natural hyperbolic metamaterial (HMM)—specifically, type-I in the 70–100 nm range and type-II in the 120–150 nm range. This property enables subwavelength wave propagation along specific crystal directions, suggesting potential applications in chip-integrated waveguides, UV-visible photonic circuits, and GHz-range optoelectronic devices. The high optical absorption, metallic conductivity, and Lorentz oscillator-fitted optical response—comparable to noble metals—position this phase of Sb₂S₃ as a promising candidate for anisotropic plasmonic devices, hyperlensing, and directional photodetection. These insights provide a pathway toward utilizing this natural HMM structure in future photonic, plasmonic, and photovoltaic technologies.

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1. Introduction

Chalcogenides are basically compounds of antimony and cadmium with oxygen group elements like sulphur (S), sellenium (Se), and tellurium (Te).¹ They are studied separately due to the vastness of the subject on oxides. In the early 1990's, chalcogenides were studied as a popular candidate for optical data storage.² However, with the decreasing popularity of compact discs (CDs), investigation along this line has gone out of vogue. In recent years, research on chalcogenides such as antimony sulphide (Sb₂S₃) has regained attention due to their potential application as photovoltaics.^3,4 Sb₂S₃ has a band-gap tunable in the range of 1.55 to 2.4 eV,⁵ near to the Shockley–Queisser limit. The Shockley–Queisser limit⁶ theoretically shows that a maximum efficiency of 33.7% can be obtained from a single-junction solar cell, assuming a band gap of 1.34 eV. Along with this, Sb₂S₃ has a high absorption coefficient in the visible region,⁷ it occurs abundantly in nature, and it is not a toxic material.

Sb₂S₃ is known to have a layered orthorhombic crystal structure and has been studied in detail.⁸ Sb₂S₃ single crystals are optically a biaxial system.⁹ For this reason, the linear dielectric tensor of the Sb₂S₃ solid solution has three independent components that are the diagonal elements of the linear dielectric tensor.¹⁰

In 1996, a new layered structure of Sb₂S₃ was reported.¹¹ Unfortunately, it did not excite interest and a detailed investigation of the electronic and optical properties of this structure was not taken up. This paper does a theoretical investigation of the band gap of this new structure and determines its electrical conductivity and optical properties. Based on the knowledge of the optical properties, comments are made on how electromagnetic radiation would interact with this structure.

2. Structural parameters

The lattice parameters of this proposed Sb₂S₃ layered orthorhombic face centred cubic structure were taken as a = 7.47, b = 14.2 and c = 11.47 Å (with α = β = γ = 90°). The structure (see Fig. 1) is layered, with each layer consisting of a Sb–S–Sb atomic chain and S–Sb–S bondings at an angle of 60°. This results in a honeycomb-like structure in the ‘x–y’ plane. Each unit cell is made up of three layers, with a displaced central layer due to the partial coulombic force between them. First principles calculations were performed to understand the behaviour of this layered Sb₂S₃ using structure parameters and details of the method discussed in the next section.

Fig. 1 (a) Structure of Sb₂S₃ in the orthorhombic chicken-net phase. The lattice parameter of this proposed Sb₂S₃ layered orthorhombic face centred cubic structure was taken as a = 7.47, b = 14.2 and c = 11.47 Å. (b) Atomic arrangement in a single layer is shown to highlight hexagonal shape formation (not to scale), which gives the name hexa-structured layered Sb₂S₃.

3. Methods

First-principles computations based on density functional theory (DFT) were performed for this proposed structure using Quantum Expresso (QE).^12,13 The norm conserving (NC)/ultrasoft pseudopotential (PP)¹⁴ can be employed from the PPs created using the Martins-Troullier method, which replaces an element's/atom's electrons with its effective potential. FHI98PP software was used to generate these potentials. The aforementioned PP is used to determine the electrical, dielectric, and thermodynamic properties of various materials. The calculations of the band structure, PDOS and dielectric constant were carried out using NCPPs and the Perdew–Burke–Ernzerhof (PBE) exchange correlation functional^15,16 with pw.x, projwfc.x and epsilon.x codes, respectively. Specifically, the PBE functional is a widely used generalized gradient approximation (GGA) exchange correlation. The band structure and dielectric calculations were performed for a grid of 12 × 12 × 12 for 200 bands. The K-path sequence was chosen based on the Brillouin zone for an orthorhombic face-centered cubic structure.¹⁷ For the computation of the dielectric constant, the energy grid was set to 0–30 eV with 30 000 points, and broadening was assumed to be the Gauss type with 0.136 eV interbond contribution and 0 eV intraband contribution.

4. Results and discussion

4.1. Electronic structure

The band structure calculation of Sb₂S₃ along the first Brillouin zone for the orthorhombic face centred structure is shown in Fig. 2. Results of Fig. 2 show metallic behavior (‘A’ and ‘B’) along the Γ–X direction and (‘C’) along the A1–Y direction. These directions are associated with the ‘x/y’ directions of the real crystal. Thus, this proposed Sb₂S₃ is essentially metallic in nature. However, in the T–Z direction (‘D’) which corresponds to the z-direction of the real crystal, a direct band-gap of 0.44 eV exists locally. The strong anisotropic nature of this proposed Sb₂S₃ structure can be appreciated. It is important to note that the calculations were performed using the PBE-GGA functional, which is known to underestimate band gaps. Incorporating GW corrections—which account for electron–electron interactions—would likely yield more accurate results, shifting the band gap to higher values. However, as this is a preliminary study, GW corrections have not been included.

Fig. 2 Structure of Sb₂S₃ in the orthorhombic chicken-net phase.

The calculated DOS and projected DOS for Sb₂S₃ with the Fermi energy level (E_F) represented by a reference line at zero energy is shown in Fig. 2.

where PDOS is the projected DOS on the atomic wavefunctions with component m (m = 0, ± 1, ± 2, …). The results of DOS reveal how the specific atomic or molecular orbital configuration contributes to the overall electronic structure and its other related properties. The main contribution in the case of Sb and S comes from the interaction between their p-orbital.

4.2. Phonon band structure

Fig. 3 shows the calculated phonon dispersion of the proposed layered orthorhombic Sb₂S₃ phase. The calculation validates the structural and electronic interpretations presented above. The absence of imaginary modes across the Brillouin zone confirms the dynamical stability of the structure, supporting its feasibility as a realizable material. The spectra have two distinct vibrational regimes: low-frequency acoustic modes (100–300 cm⁻¹) (in the x/y directions) associated with collective lattice vibrations and inter-layer interactions, and secondly the high-frequency optical modes (900–1000 cm⁻¹) (along the z-direction) attributed to Sb–S bond stretching within the honeycomb-like layers in the x/y directions. The three acoustic branches converge smoothly to zero at the G-point, while the optical modes extend up to ∼1000 cm⁻¹, consistent with the strong Sb–S bonding environment inferred from electronic and optical analyses. The flat nature of several optical branches indicates pronounced vibrational anisotropy, mirroring the electronic anisotropy that gives rise to the observed hyperbolic dielectric response. Thus, the phonon spectrum not only affirms the stability of the predicted phase but also reinforces the intrinsic hyperbolic behavior of the material, wherein in-plane metallicity contrasts with out-of-plane semiconducting characteristics.

Fig. 3 Phonon dispersion relationship of Sb₂S₃ along the high-symmetry directions of the Brillouin zone. The calculated phonon branches show good agreement with theoretical predictions detailed in the previous section.

4.3. Optical/dielectric properties

The optical properties of materials can be explored through their linear response to electromagnetic radiation. The real and imaginary parts of the frequency dependent dielectric tensor, ε(ω) = ε_r(ω) + ε_i(ω) as a function of the incident photon's energy, can be determined from the band structure calculation via an epsilon code of QE. The imaginary part of the dielectric constant can be obtained using the following equation:¹⁸where 〈f|P_α|i〉 and 〈i|P_β|f〉 are dipole matrix elements in the α and β directions with i and f as the initial and final states, respectively. W_n is the Fermi distribution function of the n^th state. The real part of the dielectric constant can be evaluated using the following Kramer–Kronig relationship:¹⁹where P is the principal value of the integral.

The dielectric constants along the ‘x’ and ‘y’ directions were found to be comparable, and hence, here, we report the average value of the dielectric constant along these two directions and report the results.

The Fig. 4(a) inset shows that the real part of the dielectric constant (ε_r) has negative values in the UV-visible region (120 < λ < 150 nm) in the ‘x/y’ direction. This is clearly indicative of the metallic/plasmonic behavior in this plane. The peaks of the dielectric constant in the ‘x/y’ direction are those of electronic transitions beyond the band edge. The values of real and imaginary ε_z (Fig. 4b) remain positive until 30 eV along the van der Waals (vdW) direction.

Fig. 4 The dielectric constants were calculated using the epsilon.x routine available in QE. (a) and (b) show the variation of ε_r(ω) and ε_i(ω) in the ‘x/y’ direction and ‘z’ direction, respectively, of Sb₂S₃ in its new orthorhombic chicken-net phase. The inset shows the variations in the UV-visible to IR region (≈38–1880 nm).

Fig. 4 suggests that the dielectric constant in all directions has metallic behavior and follows the Drude model.²⁰ The dashed line in Fig. 4 shows the best fit of the equations

Fit on ε_x/y returns values of relaxation time (τ) and ω_p, which is related to the density of conduction electrons (N) and the effective optical mass (m_o)

as 5.8 × 10⁻¹⁶ s (58 fs) and 2 × 10¹⁶ Hz, respectively. Similarly, fit to ε_z data (Fig. 4b) gives 2.6 × 10⁻¹⁶ s (26 fs) and 2 × 10¹⁶ Hz, respectively. These values are very similar to the values of silver and gold²⁰ (see Table 1).

Table 1 Comparison of the optical properties of the proposed Sb₂S₃ structure with those of noble metals like Ag and Au

	Present Sb2S3	Ag	Au
ω p	≈2 × 10^16 Hz	≈1.39 × 10^16 Hz	≈1.37 × 10^16 Hz
ε r	−ive in UV	in the vis/IR region	in the vis/IR region
	Anisotropic	Isotropic	Isotropic
	(Type I and II HMM)	(Drude)	(Drude)
τ	26 fs in the z-direction	30 fs	9 fs
	58 fs in the x/y-direction
E g	0.44 eV in the z-direction	Metallic	Metallic
	0.03 eV in the x/y-direction

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Another interesting fact noticeable from Fig. 4 is that ε_x/y is negative in the UV-visible range (120 < λ < 150 nm), while ε_z is negative in the UV-visible range (70 < λ < 100 nm). This would mean that Sb₂S₃ exhibits the behavior of type-I hyperbolic meta materials (HMM) in the range of 70 < λ < 100 nm and that of type-II HMM in the wavelength range of 120 < λ < 150 nm.²¹

The distinction between type-I and type-II HMM is based on the signs of the dielectric permittivity tensor. In type-I HMM, the in-plane permittivity is positive while the out of plane permittivity is negative. This results in a hyperbolic dispersion where high-k modes can propagate primarily within the plane of the material. In contrast, type-II HMM exhibits negative in-plane permittivity and positive out-of-plane permittivity, leading to out-of-plane propagation of high k-modes. These differences give rise to a distinction in isofrequency surfaces, i.e. hyperboloids oriented differently showing how light waves travel through the material.

The distance between two planes (d) of this structure is ≈7 nm. In normal waveguides, the guiding walls would only support waves adhering to the cut-off conditions, λ < 2d. Higher wavelengths would attenuate. However, in our natural HMM structure, the ‘xy’ planes would act as guiding walls and in the wavelength regime of type-II HMM allow waves of high wavelength to propagate without attenuation. This argues well for optoelectronic devices where waveguides in the Giga-Hertz (GHz) range can be designed on chips.

The optical mass and hence the optical density is the same in all directions (ω_p is the same in all directions). The order of relaxation time (as also the order of ω_p) is typical of metals. However, the point of interest here is the lower relaxation time along the ‘z’ axis. As was seen from Fig. 2, the flat band in the T–Z direction implied a large effective mass along the vdW direction. This makes the ‘z’-axis the slow axis (low mobility) and would suggest a lower relaxation time.

The absorption coefficient α(ω) can be obtained from the dielectric constant using the relationship:

The Fig. 5 insets show that the absorption along the ‘x/y’ directions (a) is an order of magnitude more than that along the ‘z’-axis (b). Also, the band edges in these directions are marked by regions of maximum absorption. A more precise value of the band edge is obtained from Tauc's method²² where a linear segment of (αhν)² is plot with respect to hν. Extrapolation of a fit line to the ‘x’-axis of the curve, where (αhν)² = 0, gives the band gap. The band gap along the ‘x/y’ direction and the ‘z’ direction work out to be 0.03 eV and 0.52 eV, respectively. The result of 0.52 eV is similar to the value obtained from Fig. 2 of 0.44 eV. The discrepancy between the 0.44 eV and 0.52 eV values arises not from differing DFT functionals or structural changes, but from the inherent precision limits of the respective methods. Both values fall within the margin of error and correspond to the same band edge along the z-direction.

Fig. 5 The absorption inset of (a) and (b) shows the absorption (a) in the ‘x/y’ direction and along the ‘z’ direction, respectively. The absorption in the ‘x/y’ direction is orders of magnitude stronger than that along the ‘z’ axis. The graphs also show the plot between (αhν)² and hν from which the band-gaps can be calculated by Tauc's method. The ‘x/y’ direction is metallic with a low band-gap of 0.03 eV, while the band-gap along the ‘z’ axis is around 0.52 eV.

Using the simulated complex dielectric tensor components, the refractive index of the material in different directions can be obtained using the following equation:²³

Fig. 6 shows the variation in the refractive index of our structure with wavelength. The dispersive behaviour of a material can be modeled using the Lorentz oscillator model²⁴ given by the following relationship (the real part of the refractive index):where this model assumes absorption due to resonance taking place between the valence electrons oscillating at their natural frequency (ω_o) and the incident light wave. In the case of molecules with different types of bonds and hence electrons oscillating with different natural frequencies, then as many terms will appear in eqn (6). The equation is then given as follows:The coefficient ‘B’ is the strength or amplitude of oscillation, and ‘C’ is a measure of the damping. Fig. 6 shows the variation of the refractive index in the full range of study. The dashed line in the figure shows the fit made using eqn (6) and (7).

Fig. 6 The variation in refractive index with respect to frequency as calculated using QE. The variation is in agreement with the Lorentz-oscillator model over the full frequency range.

n _x/y and n_z have similar resonant frequencies (1.5 and 1.6 × 10¹⁶ Hz, respectively), strength of oscillator and damping. However, an additional peak is found at ω_o ≈ 4.8 × 10¹⁵ Hz in n_x/y. This peak is marked with a higher strength of oscillation and a negligible damping factor. The energy associated with the resonant oscillators in the ‘z’ direction is 10.86 eV and is due to the collective oscillations of bound electrons (in valence orbitals) perpendicular to the ‘xy’ plane (i.e. along the ‘z’-axis) driven by the electric field of incident light. These oscillations are slightly blue-shifted in the ‘x/y’ directions due to the lower restoring force caused due to the Sb–S bonds. The second dominant peak seen in the ‘x/y’ direction confirms the influence of Sb–S bonding. Peaks at around 3.0–3.4 eV are typically assigned to interband transitions involving sulfur p-orbitals and antimony p-orbitals.²⁵ The values of the Lorentz oscillator model as fit to our refractive indices simulation further confirm our structure and its behavior.

5. Conclusions

In this study, a newly proposed layered orthorhombic phase of Sb₂S₃ was investigated theoretically, using density functional theory (DFT) to evaluate its structural, electronic, and optical properties. The band structure reveals strong anisotropy, with metallic behavior in the ‘x/y’ directions and a small direct band gap (0.44–0.52 eV) along the out-of-plane (‘z’) direction. This anisotropy is reflected in the optical properties as well, with a significantly higher absorption coefficient and plasmonic behavior in the x/y directions compared to the z-direction. Dielectric constant analysis confirms the metallic-like response and shows good agreement with the Drude model. Fitting of the dielectric and refractive index spectra using the Lorentz oscillator model reveals resonant peaks corresponding to interband transitions (involving sulfur p- and antimony p-orbitals) and collective oscillations of bound electrons. A secondary resonance peak in the in-plane direction, along with a red-shifted primary peak compared to the z-direction, underscores the role of Sb–S bonding in modulating the optical response. The optical parameters, such as relaxation time and plasma frequency, are comparable to noble metals like silver and gold, making the material a promising candidate for anisotropic plasmonic and optoelectronic applications in the UV-visible and near-IR ranges. This study provides valuable insight into the directional optoelectronic behavior of the proposed layered structure and its behavior as a natural 2-D hyperbolic material. The results open up pathways for its use in photonic and photovoltaic technologies.

Author contributions

Kuldeep Kumar: writing original draft, writing – review and editing, and methodology. P. Arun: writing original draft, writing – review and editing, and supervision.

Conflicts of interest

There are no conflicts to declare.

Data availability

This study was carried out using available data in the article P. Arun and A. G. Vedeshwar, J. Mater. Sci., 1996, 31, 6507–6510.

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