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Abstract
In this study, ternary aluminum-based chalcogenide materials are discussed since these are found to be very appealing for multifunction devices. Here, the structural, optoelectronic, mechanical, and vibrational properties of RbAlTe2 and CsAlTe2 are observed via density functional theory (DFT). An indirect energy band gap is noted to be increased from 1.33 eV to 1.96 eV for RbAlTe2 and 1.28 eV to 1.83 eV for CsAlTe2 by employing improved functional as modified by Trans and Blaha. The calculated formation energy appears to be decreasing, such as -4.39 and -3.83 eV for RbAlTe2 and CsAlTe2, respectively. The investigation of PDOS revealed that Rb-d, Cs-p, Al-p/s, and Te-p orbitals are located prominently and contribute mainly to boosting the conduction mechanism. The optical results declare CsAlTe2 as the strongest absorptive substance, which may be used to devise optoelectronic and photovoltaic devices. Moreover, six independent elastic constants show that these are mechanically stable materials, their brittle nature is confirmed by obeying Born’s stability requirements. According to the density functional perturbation theory (DFPT) approach used for analyzing phonon dispersion, there is no imaginary phonon frequency in both cases (RbAlTe2 and CsAlTe2). The overall results show that the studied materials are potential candidates for applications in photovoltaic and optoelectronic devices.
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1. Introduction
Recent technological developments are essential to progress, particularly in the fields of thermoelectric and optoelectronics. Many investigators from areas of physics, chemistry, and material science are interested in ternary chalcogenides because these compounds can accommodate abundant nontoxic components in their composition [1]. Such types of chalcogenides are suitable for the latest innovations due to their unique structural composition with significant chemical or physical properties [2].
Xiao et al. [3] studied the thermoelectric, optoelectronic, and photocatalytic characteristics of B2ZnSe3. Reshak et al. [4] analyzed the structural and optoelectronic characteristics of Cu2ZnSnS4 and Cu2ZnSnSe4 ternary chalcogenide materials. The magnetic influence of BaFe2Se3 material was examined by Pomjakushina et al. [5]. In 2019, Abbas et al. [6] investigated the structural, transport, and electrical characteristics of BaCu2GeX4 (X = S, Se) using the DFT approach. They also determined birefringence, which represents a crucial factor in figuring out whether the substance is suitable for thermoelectric or optoelectronic applications. The first principle’s simulation was used to observe the thermoelectric, optical, and electrical properties of the alkali chalcogenides SrCdSnM4 (M = S, Se, Te) [7]. They found that the optical findings of the examined substances exhibit polarization anisotropy, making them appropriate for use in optoelectronic devices. In 2022, Hegazy et al. [8] reported comprehensive thermoelectric properties of chalcogenide HgAl2M4 (M = S, Se) materials for photovoltaic applications. Zhou et al. [3] have examined the visible light-sensitive chalcogenides Ba2ZnSe3. Saidi et al. [9] addressed the optoelectronic characteristics of the additional chalcogenides Ag2CdSnS4 material in 2018 and found that they possess significant absorptivity (above 107/cm). The PBE-GGA and TB-mBJ functionals were employed to reveal the structural, optical, and mechanical properties of BInS2 (B = K, Rb, and Cs) by Bouchenafa et al. [10] who observed that KlnS2 material exhibits the greatest Young’s modulus and thermal conductivity. In 2022, Sujith et al. [11] predicted the lattice constants and cell volume of Ba2CdX3 (X = S, Se, Te) ternary chalcogenides using the PBE-GGA functional which have shown good coherence with experimental findings. The small band gap value and the large absorption coefficient of such compounds are appropriate for photovoltaic devices. Al-Douri et al. [12] observed that the KBTe2 (B = In, Al) are attractive candidates for nonlinear optics, optoelectronic and photovoltaic devices. In 2019, the vibrational properties of RbInSe2 were described in detail by Guc et al. [13]. Kim et al. [14] synthesized these materials and established their monoclinic system. Benmakhlouf et al. [15] carried out a theoretical analysis of the electronic, structural, optical, and elastic properties of the materials KAlM2 (M = Se and Te). In 2022, Khan et al. [16] used TB-mBJ functional within the DFT to determine the structural, electronic, optical, and thermoelectric properties of NLiM (M = S, Se and Te) where all materials are found very useful in thermoelectric devices. The optical, electronic elastic characteristics of the KBS2 (B = La, Y) and KGaQ2 (Q = S, Se, and Te) chalcogenides have also been explored using a first-principles study [17,18]. Gull et al. [19] had examined the optoelectronic, and thermoelectric behavior of MgZnO2 and MgZnS2 chalcogenides and reported these materials as wide bandgap semiconductors.
According to the literature review, the structural, optoelectronic, elastic, and vibrational properties of BAlTe2 (B = Rb, Cs) are examined for the first time by first principles approach while employing the Full Potential Linearly Augmented Plane Wave (FPL-LAPAW) technique. Where the PBE-GGA and TB-mBJ functionals are used for calculations. These theoretical assumptions are motivated by the absence of experimental and theoretical research on these materials. Our significant effort will encourage other investigators to conduct experimental investigations on these novel substances, i.e., BAlTe2 (B = Rb, Cs) for optoelectronic as well as photovoltaic applications.
2. Method of calculations
The ab initio technique is used to examine the structural, optoelectronic, mechanical, and vibrational properties of RbAlTe2 and CsAlTe2 using the FP-LAPAW approach [20] within the framework of WIEN2 K software. PBE-GGA [21,22] along with the TB-mBJ [23] functional are used to calculate the exchange-correlation potential because the TB- mBJ functional is assumed to be concise explanatory component of the exchange potential that can estimate accurate value of the electronic band gap in comparison to GGA and LDA functionals. To reduce the atomic positions, cell volume, and lattice constants of such ternary chalcogenides, we applied the conjugate gradient method [24]. To obtain optimized and exact lattice constants, the Muffin-Tin Radius (RMT) is reduced by 6.5%. The cut-off kinetic energy value is fixed at -6.0 Ry, which is deemed high enough to obtain adequate convergence. 0.001 Ry convergence energy is opted to achieve system stability during self-consistent. The calculation of the Brillouin zone (BZ) was carried out by employing a grid of 2000 k-points with RMT × Kmax = 10, in which RMT denotes the shortest muffin-tin radius while Kmax denotes the largest reciprocal lattice vector, along with the Gaussian parameter ${G_{max}} = 12\; {({a.u} )^{ - 1}}.$ The ionic radii of spheres are chosen as 2.50, 2.24 and 2.50 for B = (Rb, Cs, Fr), Al and Te, respectively, for every component of the considered substances. The Voigt-Reuss-Hill approaches [25,26] are used to compute the mechanical properties. The optical characteristics of these chalcogenides are calculated by applying Kramers-Kronig relations [27] through modified functional (TB-mBJ) with Fermi energies, i.e., 0.2198 eV for RbAlTe2 and 0.2902 eV for CsAlTe2, respectively. The smearing factor of 0.1 eV was opted during the calculations of these optical parameters including dielectric functions.
3. Results and discussions
3.1 Structural properties
Aluminum based ternary chalcogenides BAlTe2 (B = Rb, Cs) having body-centered tetragonal geometry with a $14/mcm$ space group (No. 14) are illustrated in Fig. 1 which exhibit lattice parameters $a = b \ne c$. The tetragonal structure of BAlTe2 (B = Rb, Cs) possesses 8 atoms in a unit cell comprising 2 atoms of (B = Rb, Cs), 2 atoms of Al and 4 atoms of Te. To reduce the atomic positions, cell volume and lattice constants of such ternary chalcogenides, we applied the conjugate gradient method [24]. The force as well as the c/a ratio is optimized to determine the total energy in terms of volume and fully relaxed the ionic positions to achieve the ground state energy.
The lattice parameters for RbAlTe2 and CsAlTe2 are optimized through Birch-Murnaghan’s [28] equation of state. The ground state energy versus c/a ratio optimization plots are represented in Fig. 2. The results are compiled in Table 1. The computed values of lattice parameters are $a = b = 8.7293\; Å\,\textrm{and}\; c = $ 7.0398 Å for RbAlTe2 and $a = b = 8.7431\; {Å}\; \textrm{and}\; c = $ 7.0176 Å for CsAlTe2 having ${E_o}$ -67272.8969 Ry and -86508.0853 Ry, respectively as determined by PBE-GGA functional. These values vary depending on the atomic size of B-cations. The value of ground state energy (${E_o}$) is declining (CsAlTe2 < RbAlTe2) because the number of electronic shells increases. This is due to the fact that, in RbAlTe2, the more powerful valence electrons seem to be tightly bound and take less energy to break their bonds in comparison to CsAlTe2. Furthermore, the cohesive $({\textrm{E}_{\textrm{coh}}})$ and formation $({\textrm{E}_{\textrm{for}}})$ energies for BAlTe2 (B = Rb, Cs) are calculated to determine their structure and thermodynamic stability. The quantal of energy required to break substance into its individual atoms is known as cohesive energy/binding energy. The equation below illustrates how cohesive energy is expressed [29]:
In Eq. (1), ${N_B},{N_{Al}}$ and ${N_{Te}}$ represents the numbers of B, Al and Te atoms in the $BAlT{e_2}$ $({B = Rb,\; Cs} )$ unit cell structure, $E_{tot}^{BAlT{e_2}}$ signifies the total energy of $RbAlT{e_2}$ and $CsAlT{e_2}$ materials, $E_{tot}^{B({atom} )},\; \; E_{tot}^{Al({atom} )}$ and $E_{tot}^{Te({atom} )}$ are the energies determined for single isolated $({B = Rb,\; Cs} ),\; \; Al$ and $Te$ atoms as summarized under Table 2. The calculated cohesive energy is as under $E_{tot}^{RbAlT{e_2}} ={-} 6.82\,\textrm{eV}/\textrm{atom}$ and $E_{tot}^{CsAlT{e_2}} ={-} 5.88\,\textrm{eV}/\textrm{atom}$. These compounds can be considered more stable owing to the lower negative values resulting from these thermodynamic variables. RbAlTe2 can be considered substantially more stable compound due to its cohesive energy.
Table 1. The tetragonal optimized parameters for RbAlTe2 and CsAlTe2 compounds.
Table 2. The computed cohesive energy $({\textrm{E}_{\textrm{coh}}}$/atom) for BAlTe2 (B = Rb, Cs).
Moreover, the difference between a crystal’s energy and its individual component’s energy in nmal state is known as formation energy. The following formula gives the formation energy of BAlTe2 (B = Rb, Cs) materials [30]:
Here, $E_{tot}^{B({solid} )},\; \; E_{tot}^{Al({solid} )}$ and $E_{tot}^{Te({solid} )}$ have the total energies for the aforementioned materials in the solid form as summarized under Table 3. The computed formation energy/atom for $\textrm{RbAlT}{\textrm{e}_2} ={-} 4.39\,\textrm{eV}$ and $\textrm{CsAlT}{\textrm{e}_2} ={-} 3.83\,\textrm{eV}$. . These negative values represent the stability of such materials in the tetragonal structure [30]. So far as, it is crystal evident that previously neither theoretical calculations nor experimental efforts are made to examine the compounds such as RbAlTe2, and CsAlTe2.
Table 3. The computed formation energy $({\textrm{E}_{\textrm{form}}}$/atom) for BAlTe2 (B = Rb, Cs).
3.2 Electronic properties
3.2.1 Electronic band structure
To understand the behavior of electronic band structure evaluated theoretically is an essential feature for semiconductor technological applications [31]. Band structure illustrates the nature of electron and their state in a particular material [32] while elaborating their momentum and energy of the electrons. The two functional PBE-GGA and PBE + TB-mBJ are utilized to discuss the energy band structure for the RbAlTe2 and CsAlTe2 in first BZ in tetragonal phase as illustrated in Fig. 3 and 4. It is observed from Fig. 3 and 4 that the valence band (VB) maxima lies at Г point, whereas the conduction band minima occur at H point of the BZ demonstrating that such compounds possess an indirect band gap leading to their semiconductor behavior. In case the material exhibits an indirect band gap, the photons are not released due to exciting electrons rather these electrons travel through an intermediate state by transferring their momentum to the crystal’s lattice. Therefore, the electronic structure of RbAlTe2 and CsAlTe2 with indirect bandgap may be more effective absorbers as well as light emitters [33]. The computed PBE-GGA electronic band gaps are 1.33 eV and 1.28 eV for RbAlTe2 and CsAlTe2, respectively. Nevertheless, the measured PBE + TB-mBJ band gaps are 1.96 eV and 1.83 eV for RbAlTe2 and CsAlTe2, respectively. These materials are typically suitable for optoelectronics like devices that need wide bandgap energy to work more effectively and produce the optimum results. The use of two distinct exchange potentials has contributed to fluctuations in the band gap values for BAlTe2 (B = Rb, Cs). The band gap value for RbAlTe2 and CsAlTe2 shows a considerable increase from 1.33 eV to 1.96 eV and 1.28 eV to 1.83 eV, respectively, by employing the PBE + TB-mBJ functional. As a result, it is determined that the PBE + TB-mBJ technique seems more suitable for computing the band gap values of these compounds as much closer to those determined experimentally [23]. Moreover, Koller et al. [34] has tested TB-mBJ exchange potential on various types of solids and their calculated band gaps that has shown relatively better values through TB-mBJ exchange potential as compared to those obtained through standard local density and generalized gradient approximations. These chalcogenides can be utilized in optoelectronic devices because the values of bandgap of the examined materials fall in the visible region of electromagnetic radiations.
The bandgap values for BAlTe2 (B = Rb, Cs) along with the formerly reported theoretical and experimental results about similar other materials are listed in Table 4. It is iterated that our materials are novel combinations of chalcogenides that have primarily been discussed neither theoretically nor experimentally. Comparatively bandgap values for BAlTe2 (B = Rb, Cs) are much better than the similar theoretical results [12,35], nonetheless, these are small as compared to the experimentally reported values for other combinations of chalcogenides [36,37].
Table 4. The calculated energy bandgap (eV) of BAlTe2 (B = Rb, Cs) in chalcogenides phase as compared to other experimental and theoretical results.
3.2.2 Density of states
The total density of states (TDOS) and partial density of states (PDOS) are determined in the range of energy from -6 to 7 eV to examine total behavior of materials as shown in Fig. 5 and 6. Figures 5 (a-b) for TDOS for RbAlTe2 and CsAlTe2 reveal the fact that no states crossed the Fermi level. Therefore, a significant electronic band gap appeared between the lowest conduction state and the greatest valence state. The band gap values obtained through TDOS are consistent with the forbidden gaps calculated from electronic band structure as illustrated in Figs. 3 and 4. As a result, it may be stated that the materials in discussion are semiconducting nature. As the fermi level is found near the maximum edge of the valence states, indicating that both of these materials are p-type semiconductors. It is worth to mention that PBE + TB-mBJ potential is utilized to calculate PDOS as demonstrated in Fig. 6 (a-b) which estimates relatively more effective results for electronic properties [38].
The PDOS explains that Rb-d states play a major role in the formation of the conduction band around the energy of 4 eV, while Rb-p/s states have contributed slightly into the formation of the valence band. Localized states of the cesium atom have not contributed adequately to the creation of the conduction band; nevertheless, these states have contributed sufficiently in the far region of the valence band. The major peak of cesium atom is observed around -4 eV energy. Additionally, Al-s orbitals possess a maximum contribution in the formation of the valence band and conduction band, whereas 3p states of aluminum participate minimally in both conduction and valence band formation. Furthermore, PDOS indicates that chalcogen element have completely filled s and d states, but the p states are partially filled. So, unfilled 5p orbitals of the chalcogen element (Te) are found partaking chiefly in formation of the valence band (near to the Fermi level) whereas it influences minutely in the far region of the conduction band. The maximum peak of Te-5p states is seen in the vicinity of -1.2 eV energy related to the valence regions (near the Fermi level).
3.3 Optical properties
The optical behavior of substances to incident electromagnetic radiations plays an important role in the field of photonics as well as optoelectronics. The material’s optical properties lead to an efficient and comprehensive approach to examining the electronic band structure [39]. In this section, we calculated the optical properties of BAlTe2 (B = Rb, Cs) using modified functional (TB-mBJ). The refractive index $n(\omega )$, extension coefficient $k(\omega )$, complex dielectric constant $\mathrm{\epsilon} (\omega )$, absorption α(ω), optical conductivity $\sigma (\omega )$, energy loss function $L(\omega )$ and reflectivity $R(\omega )$ of the studied chalcogenides are discussed to determining the optical response for several uses in optoelectronics. The computed optical behavior of RbAlTe2 and CsAlTe2 in the range of energy 0-14 eV is depicted in Fig. 7–10. The optical behavior of the incident rays is plotted for the electric field vector $\vec{E}$ that is polarized both in parallel and perpendicular to the tetragonal c-axis of the crystalline structure in tetragonal phase. It is observed that both materials, i.e., BAlTe2 (B = Rb, Cs) exhibit an optical anisotropy.
Fig. 7. Computed real and imaginary components of dielectric function (in both ordinary as well as extraordinary directions of polarizations) for RbAlTe2 and CsAlTe2 compounds.
Fig. 8. The computed ${n_1}(\mathrm{\omega } )$ and ${k_1}(\omega )$ (in both ordinary and extraordinary directions of polarizations) for RbAlTe2 and CsAlTe2.
Fig. 9. Computed α(ω) and $L(\omega )$ (in both ordinary and extraordinary directions of polarizations) for RbAlTe2 and CsAlTe2.
Fig. 10. Computed $\sigma (\omega )$ and $R(\omega )$ (in both ordinary and extraordinary directions of polarizations) for RbAlTe2 and CsAlTe2.
3.3.1 Complex dielectric function
The $\mathrm{\epsilon}(\omega)$ is an extensive explanation of the optical performance of solid substances. It comprises on two recognized elements $\mathrm{\epsilon}(\omega)= {\mathrm{\epsilon}_1}(\omega )+ i{\mathrm{\epsilon}_2}(\omega )$ that define the spectral response [40]. Here, ${\mathrm{\epsilon}_1}(\omega )$ is a real part of the dielectric function that describes the polarization or dispersion of the incident light, while ${\mathrm{\epsilon}_2}(\omega )$ is an imaginary function to explain the absorption property of the electromagnetic radiation. Both these components are correlated via the Kramers-Kronig expression [41,42].
Figures 7 (a-d) illustrate the computed real $\mathrm{\epsilon}_1^{ /{/} }(\omega )\; [{\mathrm{\epsilon}_1^ \bot (\omega )} ]$ as well as imaginary $\mathrm{\epsilon}_2^{ /{/} }(\omega )\; [\mathrm{\epsilon}_2^ \bot (\omega )]$ components in tetragonal phase of RbAlTe2 and CsAlTe2 for two electric polarization $\vec{E} /{/} c\; and\; \vec{E} \bot c.$ The first part ${\mathrm{\epsilon} ^{ /{/} }}(\omega )$ is referred to as extraordinary dielectric function since it is parallel to the optic axis (i.e., $\vec{E} /{/} c\; axis)$ and the ${\mathrm{\epsilon} ^ \bot }(\omega )$ is referred to as ordinary dielectric function since it is perpendicular to the optic axis (i.e., $\vec{E} \bot c\; axis)$. The computed spectrum of linear optical behavior is found along both polarization directions that are denoted as $({{\mathrm{\epsilon}_o},{\mathrm{\epsilon}_e}} ),({{n_o}\; {n_e}} )\; etc.$
As shown in Figs. 7(a-b), the static real dielectric constant $\mathrm{\epsilon}_1^{ /{/} }(0 )\; [\mathrm{\epsilon}_1^ \bot (0 )]$ as determined by static filed is noted to be 7.05 [5.90] for RbAlTe2 and 7.50 [6.33] for CsAlTe2, respectively. In these examined materials, its larger value is observed in that case where lower value of bandgap is reported according to Penn’s model [43]. The static value is correlated with the energy bandgap by ${\mathrm{\epsilon}_1}(0 )= 1 + \hbar {\omega _P}/{E_g}$. It is clear from Fig. 7 (a, b) that when the photon energy increases, the $\mathrm{\epsilon}_1^{ /{/} }(\omega )\; [\mathrm{\epsilon}_1^ \bot (\omega )]$ graph rises beyond the critical values $\mathrm{\epsilon}_1^{ /{/} }(0 )\; [\mathrm{\epsilon}_1^ \bot (0 )]$ to reach a greatest polarization at 14.03 (2.81 eV) [9.96 (2.68 eV)] for RbAlTe2 and 15.1 (2.83 eV) [10.80 (2.76 eV)] for CsAlTe2. The prominent peaks of the real component of dielectric function $\mathrm{\epsilon}_1^{ /{/} }(\omega )\; [\mathrm{\epsilon}_1^ \bot (\omega )]$ are 4.26 [7.85] for RbAlTe2 and 5.06 [9.20] CsAlTe2, respectively, with energies ranging from 4.34 eV - 4.39 eV. Therefore, above resonance to greater photon energy, $\mathrm{\epsilon}_1^{ /{/} }(\omega )\; [\mathrm{\epsilon}_1^ \bot (\omega )]$ drastically declines to a negative magnitude of -3.80 [-1.93] for RbAlTe2 and -3.55 [-2.26] for CsAlTe2. This demonstrates a plasmonic effect occurs whenever incident light interacts with free electrons in a material. This effect is strongest at the resonance frequency of the material that causes the light to disperse upon striking the material’s surface [44]. Above 6 eV declining behavior of $\mathrm{\epsilon}_1^{ /{/} }(\omega )\; [\mathrm{\epsilon}_1^ \bot (\omega )]$ to negative values indicates that light is reflecting from surface of the material where substance behaves as a metal [45,46].
The calculation of ${\mathrm{\epsilon}_2}(\omega )$ integrate all the possible electronic transitions in the material that can absorbs photon at the given frequency. This integration considers the energy levels and transition possibilities of the electron in the material. Therefore, in the present situation, the intra-band transition can’t be considered that identifies its metallic nature [47]. It can be seen from Fig. 7 (c, d) that after obtaining the threshold frequency, the magnitude of $\mathrm{\epsilon}_2^{ /{/} }(\omega )\; [{\mathrm{\epsilon}_2^ \bot (\omega )} ]$ increases sharply and reaches its highest values of 16.22 [5.81] for RbAlTe2 and 16.41 [6.82] for CsAlTe2 in the energy range of 3.60-3.63 eV. This is an obvious indication that incoming light from visible region of the incident radiation is absorbed by the material’s surface, where the light dispersion is lowest. Therefore, extraordinary ray $\mathrm{\epsilon}_2^{ /{/} }(\omega )$ shows two pronounced peaks at 5.04 eV (RbAlTe2) and 4.91 eV (CsAlTe2) as a result of inner transitions.
3.3.2 Refractive index and extension co-efficient
The refractive index is also a complex function consisting on its real and imaginary components. The relationship between these two components can be expressed as $n(\omega )= {n_1}(\mathrm{\omega } )+ i\; {k_1}(\omega )$, wherein ${n_1}(\mathrm{\omega } )$ exhibits its real part that describes the phase shift (dispersion) of the light whenever passes through the material and ${k_1}(\omega )$ indicates the imaginary/extension coefficient which describes the absorption of the light by the material [48]. Figures 8 (a-d) illustrate the calculated ${n_1}(\mathrm{\omega } )$ and ${k_1}(\omega )$ of RbAlTe2 and CsAlTe2 in tetragonal phase for both directions, i.e., ordinary ($\vec{E}$ ⊥ to c-axis) and extraordinary ($\vec{E}$ // to c-axis) where Figs. 8 (a, b) depicts the trend of $n_1^{ /{/} }(\omega )\; [{n_1^ \bot (\omega )} ]$ for BAlTe2 (B = Rb, Cs). The static values $n_1^{ /{/} }(0 )\; [{n_1^ \bot (0 )} ]$ of RbAlTe2 and CsAlTe2 materials are 2.65 [2.43] and 2.74 [2.51], respectively. Afterwards, $n_1^{ /{/} }(\omega )\; [{n_1^ \bot (\omega )} ]$ rises on increasing energy to achieve the greatest value, i.e., 3.80 [3.21] at 2.81 eV and 3.96 [3.32] at 2.89 eV photonic energy for RbAlTe2 and CsAlTe2, respectively. Thus, the ${n_1}(\mathrm{\omega } )$ has displayed a notable optical anisotropy. In comparison, CsAlTe2 has shown large value of ${n_1}(\mathrm{\omega } )$ narrating that many pseudo electrons can interact with the incoming radiation, resulting in greater polarization while decreasing the speed of light. On further growth of the photon energy, $n_1^{ /{/} }(\omega )\; [{n_1^ \bot (\omega )} ]$ steadily declines till its lowest values of 0.23 [0.19] for RbAlTe2 and 0.35 [0.32] for CsAlTe2 at 13.5 eV. It is an established fact that large bandgap semiconductors exhibit lower value of $n(\omega )$ [49]. The difference between the extraordinary $({{n_e}} )$ and ordinary $({{n_o}} )$ refractive indices can be correlated to the birefringence $({{\Delta} n} )$ variable at zero photonic energy which is also shown in Table 5. Our study unveils that RbAlTe2 and CsAlTe2 exhibits a positive birefringence. This implies that these materials can be potential candidate for 2nd harmonic generation, particularly in the far-ultraviolet region of the electromagnetic radiations [12].
Table 5. A comparison of static dielectric function ${\mathrm{\epsilon}_1}(0 )$, refractive index ${n_1}(0 )$, reflectivity $R(0 )$ and absorption coefficient $\alpha (\omega )$ in (cm)-1 along both polarization directions for BAlTe2 (B = Rb, Cs) materials and difference between the ordinary and extraordinary refractive index (${\Delta} n$) is also reported.
Figures 8 (c, d) displays $k_1^{ /{/} }(\omega )\; [{k_1^ \bot (\omega )} ]$ as a function of the light energy for RbAlTe2 and CsAlTe2, where the $k_1^{ /{/} }(\omega )\; [{k_1^ \bot (\omega )} ]$ follow the same trend of $\mathrm{\epsilon}_2^{ /{/} }(\omega )\; [{\mathrm{\epsilon}_2^ \bot (\omega )} ]$ (Figs. 7 (c, d)). After attaining threshold frequency, $k_1^{ /{/} }(\omega )\; [{k_1^ \bot (\omega )} ]$ rises with the photon energy and achieve its highest values such as 2.67 [2.22] for RbAlTe2 and 2.58 [2.36] for CaAlTe2 in the energy window between 5.4 eV and 5.7 eV. Their values vary in a greater energy range up to 13.5 eV. CsAlTe2 is shown to be a more effective absorber of incoming radiation than RbAlTe2, particularly at 13.5 eV photon energy. The relationship between these two components ${n_1}(\mathrm{\omega } )$ and ${k_1}(\omega )$ is given below:
3.3.3 Absorption co-efficient and energy loss function
The capacity of any material for absorbing incident photons is described by its absorption coefficient α(ω) which is connected with the dielectric constant by the following equation [50]:
The α(ω) is shown in Figs. 9 (a, b) for BAlTe2 (B = Rb, Cs) chalcogenides which can assist to identify the materials for their use in effective solar cell devices. After gaining threshold frequency, ${\alpha ^{ /{/} }}(\omega )\; [{{\alpha^ \bot }(\omega )} ]$ increases substantially to 3.74 eV [5.97 eV] for RbAlTe2 and 3.65 eV [5.72] for CsAlTe2, respectively. An unanticipated decline in the absorption of incoming radiation is noted for both the cases, i.e., RbAlTe2 and CsAlTe2 within the energy range of 6 eV to 11 eV. In the larger energy region, the variations in the α(ω) values were produced through various transition rates. However, it is observed that absorptivity increases above the 11.0 eV energy. Hence, the highest values of ${\alpha ^{ /{/} }}(\omega )\; [{{\alpha^ \bot }(\omega )} ]$ is found to be 1.1468 × 106 cm-1 [1.1110 × 106 cm-1] for RbAlTe2 and 1.9330 × 106 cm-1 [1.6990 × 106 cm-1] CsAlTe2 at 13.5 eV energy. It is seen from the Figs. 9 (a, b) that the examined materials exhibit optically anisotropy. Therefore, CsAlTe2 is declared as the strongest absorptive substance that can be used in optoelectronics and photovoltaic devices.
As illustrated in Figs. 9 (c, d), $L(\omega )$ is also essential parameter to study the interaction of radiations with the matter. It describes the amount of lost energy by the radiation on passing through the material’s surface due to interactions with swiftly moving electrons in the material [51]. Inner-shell excitations, photon excitations, inner-band transitions and plasmons are the primary originating components that contribute to the loss function. The $L(\omega )$ is related to the dielectric function with the following formula for $L(\omega )$:
As noted from Figs. 9 (c, d), $L(\omega )$ is approximately zero up to 2.6 eV energy, thereafter, it increases steadily with its sharp rise in each polarization direction (ordinary and extraordinary) reaching to its plasmonic peaks in the energy ranging from 8.25 and 12.97 eV. The absorption is smallest where maximal $L(\omega )$ is noticed. Therefore, a significant anisotropy is found in both polarization directions. In the ultraviolet region, the ${L^{ /{/} }}(\omega )\; [{{L^ \bot }(\omega )} ]$ computed for RbAlTe2 and CsAlTe2 at 13.56 eV are determined to be 0.68 [0.67] and 0.23 [0.29], respectively. Comparatively, CsAlTe2 has the smallest amount of $L(\omega )$ that is appropriate for absorptivity of incident radiations.
3.3.4 Optical conductivity and reflectivity
Optical conductivity is a measure of how well a material conducts light. It can directly be calculated using the following basic formula [52]:
Here, n is refractive index and c indicates speed of light. Figures 10 (a, b) shows graph of $\sigma (\omega )$ versus energy. Firstly, the value of $\sigma (\omega )$ in both polarization direction (ordinary and extraordinary) is zero till the incident photon energy approach to the bandgap. It is noticed that $\sigma (\omega )$ begins to rise if incident photons possessing a threshold frequency strike the surface of material. Therefore, after 2.40 eV, the significant peaks rise in extraordinary cases, with values of 9133.81 (Ωcm)-1 and 900.31 (Ωcm)-1 rising at different levels of energy, i.e., 5.02 eV and 4.93 eV for RbAlTe2 and CsAlTe2, respectively. High conductivity demonstrates that such substances are strong in $\sigma (\omega )$ in a greater energy region beyond the minimum frequency. Beyond 4.93 eV, $\sigma (\omega )$ gradually declines, reaching to its minimum value in each polarization direction. Comparative findings indicate that CsAlTe2 also has unique conductivity that can make it the best material for use in optoelectronic devices.
Reflectivity is also key parameter that solely determines a portion of energy reflected from surface of the solid materials. It is related with $n(\omega )$ as follows [53]:
The Figs. 10 (c, d) portrays the calculated $R(\omega )$ verses photon energy of tetragonal RbAlTe2 and CsAlTe2 for both directions; ordinary ($\vec{E}$ ⊥ c-axis) and extraordinary ($\vec{E}$ // c-axis). The zero-frequency limit ${R^{ /{/} }}(0 )\; [{{R^ \bot }(0 )} ]$ is found to be 0.20 [0.17] for RbAlTe2 and 0.21 [0.19] for CsAlTe2. However, on increasing frequency, beyond the threshold frequency, $R(\omega )$ upsurges to ~ 3.0, i.e., 1st peak is noticed at 2.87 eV energy in each polarization direction. In higher energy (~ 2.87 eV to 12.5 eV), its value fluctuates in between 0.24 to 0.5 for both polarization directions (ordinary and extraordinary). The ${R^{ /{/} }}(\omega )\; [{{R^ \bot }(\omega )} ]$ reaches its utmost value ~ 0.58 [0.61] for RbAlTe2 and ~ 0.63 [0.60] for CsAlTe2 at 13.56 eV energy. Comparatively, the high $R(\omega )$ for both polarization direction seems very small which may not affect the absorptivity of our material being significant in magnitude, i.e., 1.9330 × 106 cm-1 [1.6990 × 106 cm-1] (Fig. 9(a, b)) for CsAlTe2. The low reflection further supports our belief that the studied chalcogenides can be very effective ones for potential use in optoelectronic applications. In these materials, considerable anisotropy is found in all the optical parameters. The comparative data presented in Table 5 indicates that our considered compounds, i.e., BAlTe2 (B = Rb, Cs) have greater values of optical parameters in both polarization directions, especially the absorption coefficient is significantly better than other identical materials. So, BAlTe2 (B = Rb, Cs) are potential candidates for optoelectronic applications.
3.4 Mechanical properties
3.4.1 Elastic constants $({\mathbf{ C}_{\mathbf{ ij}}})$
The elastic constants have a significant role in determining physical properties of the materials [55]. A thorough exploration of the elastic properties can extract necessary details about its mechanical, thermodynamic, structure stability and dynamic behaviors. Any crystal structure’s macroscopic deformation has a direct connection to the elastic constants ${C_{ij}}\; ({i,\; j = 1,\; 2,\; 3,\; 4,\; 5\; and\; 6} )$ which can be used to calculate strains and elastic energy produced in solid substances as a result of thermal, internal and external stresses [56]. Typically, 6 × 6 matrix is used to define the elastic constants ${C_{ij}}$ of tetragonal crystal structure with $14/mcm$ space group. These six distinct elastic constants include $({C_{11}},\; {C_{33}},\; {C_{44}},\; {C_{66}},\; {C_{12}}$ and ${C_{13}})$. These elastic constants computed for BAlTe2 (B = Rb, Cs) are presented under Table 6 that may be used as reference for further investigation by the future researcher since these are novel results. The system’s mechanical stability is determined by elastic constants. The ${C_{11}}$ and ${C_{33}}$ define the resistance to linear compression, whereas ${C_{44}},\; {C_{66}},\; {C_{12}}$ and ${C_{13}}$ determine the resistance to elastic shear deformations [57]. The stress tensor components for small strains are determined at 0 GPa pressure and the energy is utilized according to the lattice strain to keep the volume constant [58]. The mechanical stability of tetragonal substances should meet the following Born stability requirements [59]:
Table 6. Numerical calculations of elastic constants for BAlTe2 (B = Rb, Cs) chalcogenides.
Our chalcogenides, i.e., RbAlTe2 and CsAlTe2 are perceived to be mechanically stable. ${C_{11}}$ and ${C_{33}}$ have greater values as compared to ${C_{44}},\; {C_{66}},\; {C_{12}}$ and ${C_{13}}$ indicating that elastic share deformation may occur very frequently rather to the linear compression deformation.
3.4.2 Elastic moduli
The bulk modulus $B\; ({\textrm{GPa}} )$ and shear modulus $G\; ({\textrm{GPa}} )$ that are frequently employed to describe the elasticity of polycrystalline substances [60] can be calculated via ${C_{ij}}$ (single-crystal elastic constants) through Voigt-Reuss-Hill’s approximation [61–63]. Usually, the bulk moduli measure a material’s resistance to volume change with hydrostatic pressure, whereas the shear moduli determine the resistance against shape change carried through a share force [64]. According to the Voigt approximation, the results for bulk modulus ${\textrm{B}_{\textrm{V\; }}}({\textrm{GPa}} )$ and shear modulus ${\textrm{G}_{\textrm{V\; }}}({\textrm{GPa}} )$ are computed as under [61,62]:
Whereas, according to Reuss approximation, the bulk modulus ${\textrm{B}_\textrm{R}}({\textrm{GPa}} )$ and shear modulus ${\textrm{G}_\textrm{R}}({\textrm{GPa}} )$ are calculated as under:
The bulk modulus $B\; ({\textrm{GPa}} )$ and shear modulus $G\; ({\textrm{GPa}} )$ can be determined by the arithmetic mean of Voigt and Reuss limits:
$B$ and G are used to calculate Young’s modulus $E\; ({\textrm{GPa}} )$ and Poison’s ratio $(\mathrm{\nu } )$ in the following manner:
Our findings of B, G, E, $B/G$, ν, universal anisotropy $({{A^U}} )$ index, Vicker Hardness ${H_\textrm{V}}$ and Debye temperature ${\theta _D}$ for RbAlTe2 and CsAlTe2 are presented in Table 7.
- 1) Bulk Modulus: Its value follows a pattern like CsAlTe2 > RbAlTe2. The increase of B from RbAlTe2 to CsAlTe2, indicates that RbAlTe2 material has greater compressibility $({\beta \; \sim \; 0.04\; GP{a^{ - 1}}} )$ compared to CsAlTe2 material $({\beta \; \sim \; 0.03\; GP{a^{ - 1}}} )$. Since the calculated values of B are typically low, i.e., $( < 30\; GPa)$ [65] and therefore, these substances be characterized as comparatively soft substances.
- 2) Shear Modulus: The relation among the resistance to shear distortions (deformation) is known as shear moduli [66]. As compared to B, G is better indicator of hardness [67]. Its values show a pattern of CaAlTe2 > RbAlTe2 that adopt similar trend as the bulk modulus. It is found that RbAlTe2 has the smallest shear modulus, whereas CsAlTe2 has the greatest. These chalcogenides are more resistive as they fulfil the condition that $B > G.$
- 3) Young’s Modulus: $E$ defines the stiffness and hardness of each substance [68]. It is found that E calculated for CsAlTe2 is greater than RbAlTe2, indicating that CsAlTe2 is stiffer than RbAlTe2.
- 4) Pugh’s Ratio: $B/G$ ratio can be used as a projector to distinguish the ductile/brittle behavior of materials. According to Paugh’s empirical rule [69] if $({B/G > 1.75} )$ the material is ductile in nature, while if $({B/G < 1.75} )$ the material is brittle. Our results show that the considered chalcogenides are brittle materials since Paugh’s ratio of BAlTe2 (B = Rb, Cs) is smaller than the standard value (1.75).
- 5) Poisson’s Ratio: The strength of covalent bond direction and volume variation that occur during uniaxial distortion are measured by ν. Covalent materials have Poisson’s ratio (ν ∼ 0.1), whereas ionic materials have a Poisson’s ratio (ν ∼ 0.25) [70]. In the present case, both material’s ν is greater than 0.25 as given in Table 7. So, it may be declared that ionic bonds play a considerable role during the net interatomic reactions in such tetragonal chalcogenide materials.
- 6) Universal Anisotropy (AU): The literature reveals that if the magnitude of AU is one the materials be classified as isotropic else anisotropic if it is less one. We have used the following well-known relation to compute ${A^U}$ for the considered compounds [71]:
As tabulated in the Table 7, anisotropy factor examined for both RbAlTe2 and CsAlTe2 is less than unity, i.e., 0.54 for RbAlTe2 and 0.44 for CsAlTe2 depicting the elastic anisotropic behavior of both these compounds.
- 7) Vickers Hardness: Hardness can be defined as a material’s resistance to local plastic deformation. High shear and bulk moduli are essential characteristics of a hard material [64]. Using Tian-empirical equation [72] the Vickers hardness can be determined using $\textrm{B}$ and $\textrm{G}$ as under:
The computed values of Vickers hardness (${H_\textrm{V}})$ of the considered materials are shown in Table 7. We observed that both the considered compounds have low ${H_\textrm{V}}$ as compared to formerly reported experimental results for different other materials as referred to in Refs. [73,74]. Thus, these materials have poor resistance to dents as well as scratches.
- 8) Debye Temperature: The temperature of the highest normal mode of vibration in a crystal is known as Debye Temperature and represented as ${\theta _D}$. It has strong relationship with number of solid properties, including specific heat, thermal expansion, phonons, melting temperature and thermal conductivity. Large thermal conductivity of the material is implied by large Debye Temperature [75]. The averaged sound velocity $({V_m})$ is used to calculate the Debye Temperature of crystals by the following formula [76]:
In Eq. (21), $\textrm{h}$ and ${\textrm{k}_\textrm{B}}$ represent the Plank & Boltzmann constants, $\rho $ and ${N_A}$ indicates the charge density and Avogadro's number, M and n signifies the molecular mass and number of atoms in the structure. ${\theta _D}$ value 184.09 K for RbAlTe2 and 193.18 K for CsAlTe2 is obtained from the computed average velocity $({{V_m}} ).$ ${\theta _D}$ value extends to the temperature range of 38 K for cesium to 2230 Kor diamonds [45]. As a result, the estimated value of ${\theta _D}$. is slightly low demonstrating that the BAlTe2 compounds can have a fairly low thermal conductivity.
Table 7. Computed Bulk Modulus, Shear Modulus and Young’s Modulus$,$ Pugh’s Ratio, Poisson’s Ratio, Universal Anisotropy index, Vicker’s Hardness, Debye Temperatures and average sound velocity $({{V_m}\; m/s\; } )$ of RbAlTe2 and CaAlTe2 materials.
3.5 Vibrational properties
This research provides precise predictions for certain variables that can be difficult to compute experimentally, such as silent mode frequencies and vibrational eigenvectors. Furthermore, the study of phonon dispersion is also considered to be helpful for determining structural stability as well as thermal properties. Generally, it is examined using several experimental techniques, including neutron scattering and Raman spectroscopy [77] which include the interaction of phonons with particles or electromagnetic waves. Therefore, in the current study, phonon dispersion is explored using Raman and infrared (IR) spectroscopies. For the purpose, DFPT [78] technique is regarded as a precise and efficient way to determine the vibrational characteristics of different substances. The literature reveals that DFPT is mostly employed for semiconductors where the phonon spectrum only needs some wave vectors [79]. Here, we used DFPT approach to examine the vibrational behavior $\textrm{of}$ BAlTe2 (B = Rb, Cs).
Figures 11 (a, b) display phonon dispersion curves. Fortunately, no imaginary phonons are noted throughout the BZ of the opted wave vectors demonstrating that the tetragonal phase of RbAlTe2 and CsAlTe2 is structurally and dynamically stable. There may be $({3\textrm{N} - 3} )$ optical modes for every wave vector $(\textrm{k} )$, therefore, BAlTe2 (B = Rb, Cs) have 24 vibration modes due to 8 atoms in the unit cell structures. Out of these 24, the phonon dispersion curve shows 3 acoustic modes (Fig. 11 (a, b)). In these acoustic phonon modes, atoms vibrate ‘in phase’ reason being the acoustic mode frequency approaches zero at the Г symmetry point rather to show some polarization. While, the remaining 21 modes are optical modes, where atoms vibrate out of phase. These modes appear at finite values of the frequency rather to have null value at Г symmetry point.
Fig. 11. Phonon Dispersion Curves along symmetry direction for (a) RbAlTe2 (b) CsAlTe2 ternary chalcogenides.
Tables 8 and 9 depicts some active vibrational modes for RbAlTe2 and CsAlTe2 as determined through DFPT approach (Raman and IR spectroscopy). It is important to mention that heretofore no researcher has made any effort to examine such Raman and IR modes of vibrations for these materials. Among 21 optical modes of vibration, seven are Raman active with five IR active modes as shown in Table (8, 9), however, the remaining 9 are found inactive. The significant optical modes are observed at highest frequency such as 344.15 cm-1 and 343.06 cm-1 for RbAlTe2 and CsAlTe2, respectively. The predicted modes of vibrations, as narrated above, may be useful for future spectroscopic studies if these materials are synthesized experimentally.
Table 8. Phonon modes of ternary chalcogenide RbAlTe2 at Г symmetry point by DFPT technique.
Table 9. Phonon modes of ternary chalcogenide CsAlTe2 at Г symmetry point by DFPT technique.
4. Conclusions
We have mainly emphasized to investigate the structural, optoelectronic, mechanical and vibrational behavior of BAlTe2 (B = Rb, Cs) chalcogenides in the tetragonal phase while employing WIEN2 K code. The computed values of the lattice parameters for RbAlTe2 are $a = b = 8.7293\; {Å}\,\,\mathrm{and}\; c = 7.0398{Å}$, while for CsAlTe2 these are found as $a = b = 8.7431\; {Å}\; \textrm{and}\; c = 7.0176\; {Å}$. By employing PBE + TB-mBJ functional, the values of bandgap increase considerably from 1.33 eV to 1.96 eV and 1.28 eV to 1.83 eV for RbAlTe2 and CsAlTe2, respectively. The PBE + TB-mBJ functional predicts greater bandgap compared to PBE-GGA. We also calculated TDOS and PDOS for both compounds and the outcomes acquired for bandgap energies of such compounds are in consistence with those obtained by electronic band structures. Both these compounds have shown their semiconducting behavior. The negative cohesive energy shown that considered materials are structurally stable. Additionally, the computed values of the elastic constants have depicted that RbAlTe2 and CsAlTe2 are mechanically stable materials. According to the DFPT approach used for analyzing phonon dispersion, there is no imaginary (negative) phonon frequency in both cases such as RbAlTe2 and CsAlTe2. It is anticipated that our theoretical estimations will surely motivate future researchers to synthesize these materials experimentally owing to be extremely useful for new technological applications such as optoelectronics and photovoltaics.
Disclosures
I declare that the authors have no competing interests or other interests that might be perceived to influence the results and/or discussion reported in this paper
Data availability
The data will be available on request if and when required.
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