A first principle investigation of electronic, mechanical, optical and transport properties of A2AgAlI6(A = Rb, K, Na) for energy harvesting

The structural, elastic, and optoelectronic properties of cubic double halide perovskites A2AgAlI6 (A = Na, K, Rb) were calculated using the full potential linearized augmented plane wave method. The structural stability of these materials was demonstrated using Goldsmith’s tolerance and modified tolerance. The optoelectronic properties were analyzed using the complex dielectric function and density of states. The potential application of this compound is indicated by the absorption and conduction of light in the visible spectrum. The direct bandgap values of 1.77 eV, 1.74 eV, and 1.64 eV for the compound A2AgAlI6 (A = Rb, K, Na) suggest its usefulness in solar panels. The electrical and thermal conductivities, and Seebeck coefficient of A2AgAlI6 (A = Rb, K, Na) were also determined.


Introduction
Global challenges of power shortage and environmental contamination can hinder world progress [1].Solar energy has been identified as a clean and large energy source that can address these issues [2,3].Silicon is, by far, the most common semiconductor material used in solar cells, representing approximately 95% of the modules sold today.The maximum possible room-temperature power conversion efficiency of a single junction, crystal Si solar cell under 1-sun illumination, according to the laws of thermodynamics, is 32.33%.This limit is based on the assumptions of perfect solar absorption and no losses due to non-radiative charge-carrier recombination.The energy conversion efficiency of silicon solar cells in the lab reached a record value of 25% in 1999 (the PERL cell based on p-type silicon ) which stood unsurpassed for 15 years.The record efficiency rose to25.6% in 2014 and to date (in 2017), developed by Kaneka Corporation, is able to achieve 26.7% [4,5].On the other hand the power conversion efficiency (PCE) of single-junction halide perovskite solar cell progressed from 3% to a certified value of 25.5%, the highest value obtained for thin-film photovoltaics over the past decade, making them a leading technology in photovoltaics [6,7].Hybrid (organic-inorganic) halide perovskites have rapidly improved power conversion effectiveness and are much cheaper to manufacture than traditional silicon solar cells [8].Research has focused on understanding the special characteristics of these alloys to maximize their potential in photovoltaics and optoelectronics.Recent research has seen a significant expansion in hybridperovskites, which are easy to produce and lightweight.However, the harmfulness and poor stability of some hybrid perovskites, such as CH 3 NH 3 PbX 3 (X = I, Cl, Br), limit their usefulness in solar systems [8,9].
Overall, halide perovskite solar cells have shown great promise as cheaper and more efficient alternatives to traditional silicon solar cells.However, further research is needed to address the issues of toxicity and stability of these materials [9].Double perovskite materials that are inorganic and non-toxic are being considered as alternatives to lead-based perovskite materials [10,11].Giustino and Snaith predicted the possibility of new elements in halide double perovskites [12].In this study, density functional theory (DFT) calculations were used to study aluminium based double halide perovskite.A H Slavney et al in 2016, were reported the synthesis of the 3D double perovskite Cs 2 AgBiBr 6 .The material has an indirect bandgap of 1.95 eV, which is suited for coupling with a Si absorber in a tandem solar cell [13].However, the band gaps of these compounds are indirect, which is not ideal for applications in thin film photovoltaics.G Volonakis et al in 2017, proposed a halide double perovskites, Our first-principles calculations indicate that the compounds Cs 2 InAgCl 6 exhibit direct band gaps of 2.6 eV by using HSE hybrid functional [14].In 2022, M A Mebed et al have explored the electronic, optical, and thermoelectric characteristics of Rb 2 AgBiX 6 (X_Br, I) by using modified Becke and Johnson potential through DFT approach.The indirect band gaps of 1.88 eV and 1.22 eV are computed for Rb 2 AgBiBr 6 and Rb 2 AgBiI 6 , respectively [15].The incorporation of aluminum imparts unique properties to these materials, making them attractive for optoelectronic applications.The abundance and low cost of aluminum ensure a stable and economically viable supply for large-scale production.The introduction of halides enhances the tunability of the bandgap, crucial for efficient light absorption and emission in optoelectronic devices.Aluminum-based halide perovskites exhibit strong light absorption, efficient charge transport, and excellent luminescent properties, making them suitable for applications like solar cells and display technologies.Furthermore, their low toxicity compared to other metal-based perovskite materials makes them environmentally friendly and aligned with sustainable development goals.Similarly, limited cesium reserves and its high demand in other fields, the feasibility of using cesium in photo-electronic devices is questionable [16].
To address this issue, researchers have investigated the potential of substituting Cs with other alkali metals such as A 2 AgAlI 6 (A = Rb, K, Na) and we have examined the viability of these three compounds.

Computational details
The WIEN2k simulation package was used to compute the structural, optoelectronic, and mechanical properties of these materials [17].To obtain the ground state properties, the Trans-Blaha Modified Becke-Johnson (TB-mBJ) [18] approximation method, along with the GGA-PBEsol approximation, was used to solve the 'Kohn-Sham equation' and estimate the exchange-correlation potential [18].Spherical harmonic functions were used inside non-overlapping muffin tin spheres that enclose the atomic sites, whereas a plane wave basis set was used in the interstitial region of the unit cell to expand the wave functions, charge density, and potential [19].For Rb/K/Na, Al, Ag, and I, the radii of the muffin tin spheres (RMT) were selected as 2.5, 2.26, 2.5, 2.14, 2.5, and 2.5, respectively.We used R mt K max = 7.0, where Kmax is the largest value of the reciprocal lattice vector used in the plane wave expansion, and R mt is the smallest radius of the muffin-tin sphere.In the muffin tins, we expanded the angular momentum up to l max = 10.The greatest vector value in the charge density is G max , which has a value of 12.The cutoff energy of −9.0 eV was selected to define the separation of valence and core states.To generate the charge density in each phase of self-consistency, the calculations were done using 3000 k-points for the Brillouin zone.The iterations for energy and charge convergence were taken at 0.00001 Ry and 0.0001e to attain better results.The elastic properties were calculated with the help of the Elast code [20].The electrical properties of these materials were calculated using the Boltztrap2 code [21].

Structural properties
The energy-volume optimization curve confirms the stability of a material by evaluating its total energy at different volumes and identifying the volume corresponding to the lowest energy state.A stable material exhibits a broad, flat curve, indicating a range of volumes where the material is energetically favorable and less prone to structural distortions or phase transitions.Conversely, an unstable material shows a narrow and steep curve, indicating a small volume range where it is energetically favorable and more susceptible to degradation or loss of desirable properties.Thus, analyzing the energy-volume optimization curve provides valuable insights into the stability and structural properties of materials.To assess structural stability, the material's crystal structure can be optimized using equations such as the Birch Murnaghan equation of state and Goldschmidt's rule, which consider bond length and effective ionic radius [22,23].The combination of anions and cations can determine the bond structure, structural stability, carrier transport behavior, and potential applications of the material.In the case of halide double perovskite materials in cubic structures, the space group Fm 3̅ m (225) is utilized [24,25].Table 1 provides the computed values of lattice constants, tolerance factors, modified tolerance factors, and octahedral factors of A 2 AgAlI 6 (where A = Rb, K, Na).The values of tolerance, modified tolerance, and octahedral factors fall within the ranges of 0.87 < t G < 0.95, τ < 4.18, and 0.44 < μ < 0.90, respectively [26,27].These factors are useful in determining the suitability of the material for specific applications.The energyvolume optimization curve confirms the stability of a material by evaluating its total energy at different volumes and identifying the volume corresponding to the lowest energy state as shown in figure 1.

Density of states
The density of states (DOS) is a key electronic feature for any material.Figures 2(a  5 , respectively.As shown in figures 2(a)-(f) the contribution of valence electrons to hybridization and inter-band transitions in the valence and conduction bands is illustrated [28].At the valence band edge, the d orbitals of Ag and p orbitals of iodine make a significant contribution, and electrons transition from the valence band to the conduction band as carriers gain energy.

Band structure
We utilized the PBEsol-GGA and TB-mBJ exchange-correlation functions to accurately predict the bandgaps as LDA and GGA methods often underestimate them [29].Electronic properties of a material play a significant role in determining its applications, which includes the band structure and electron dispersion within these bands.Using DFT, we calculated the electronic properties of the inorganic material under study.The band structures of the titled materials computed by applying the TB-mBJ potential are presented in figures 3(a)-(c).The calculated direct bandgaps for K 2 AgAlI 6 , Na 2 AgAlI 6 , and Rb 2 AgAlI 6 are 1.74 eV, 1.77 eV, and 1.64 eV, respectively.Among these, Rb 2 AgAlI 6 exhibits the lowest bandgap, making it more suitable for solar cells in the visible light region.The valence band maxima (VBM) and conduction band minima (CBM) are symmetrically positioned, indicating a direct bandgap.The interband transitions of the titled materials occurred directly at the Γ-symmetric points and indirectly at other symmetric points.

Mechanical properties
Highly sought-after in technology applications are materials that possess non-corrosive, lightweight, hightemperature stability, stiffness, and ductility.The elasticity, structural stability, bonding properties, stiffness, anisotropic characteristics, ductility, and brittleness of materials can be determined by computing their elastic  constants.These elastic parameters also enable the prediction of materials' response to applied stress, and are the second-order derivatives of total energy or potential.In the cubic system, there are three independent elastic constants: C 11 , C 12 , and C 44 , which can be estimated by using the appropriate distortions.The stiffness of the material's shape is represented by C 12 and C 44 .The elastic constants must have positive values and meet the Born Huang stability criterion [30], where ( ) This criterion provides information about the strain energy after applying elastic deformation.The elastic constants for the aforementioned compounds were calculated using the Elast package, and the calculated C 11 , C 12 , and C 44 are non-negative and satisfy the Born Huang stability criteria.Using these elastic constants, several mechanical parameters such as shear, bulk and Young's moduli, Poisson ratio, Pugh ratio, anisotropic factor, The anisotropic nature of our proposed material arises from the anisotropic factor deviating from unity, which is caused by a significant difference between the longitudinal and shear elastic constants.Specifically, the C 11 values of our titled compound are higher than the C 12 and C 44 values, and the bulk modulus B is also greater than the shear modulus G.This indicates that our material has a higher resistance to volumetric deformation than shape deformation.Additionally, our material has a higher Young's modulus than both the bulk and shear moduli.The values of Cauchy's pressure, Pugh's ratio, and Poisson's ratio suggest that Rb 2 AgAlI 6 and Na 2 AgAlI 6 are ductile, while K 2 AgAlI 6 exhibits brittle behavior.The relationship between elastic constants and thermodynamic properties can be explained through the Debye temperature, which is a physical phenomenon.Specifically, by calculating the Debye temperature, we can determine the elastic constants and mechanical properties of a material at low temperatures.The typical sound velocity is used to estimate the Debye temperature θ D , which can be calculated using a formula as shown below.
In the following equation, the constants k, h, n, M, and N A correspond to Boltzmann's constant, Planck's constant, the number of atoms per unit cell, molecular mass, and Avogadro's number, respectively.The average sound velocity V m is calculated using both the longitudinal and transverse components of sound velocity.The Debye temperature values for each material are presented in table 2, where K 2 AgAlI 6 has a larger value compared to the others.Since the Debye temperature and specific heat capacity are directly proportional, K 2 AgAlI 6 is more thermodynamically stable than the others.

Optical properties
The frequency-dependent complex dielectric constants function can be used to analyze a material's optical characteristics.i .

( ) ( ) ( ) e w e w e w = +
The Kramer Kronig relations acquire the real portion 1 ( ) e w whereas adding all the interband transitions from unoccupied to occupied states took the imaginary part 2 ( ) e w [32].
Where, m: Mass of electron, e: charge of an electron, : w the angular frequency of electromagnetic radiation [33,34].Optical transitions were calculated using the dipole matrix M e .
cv ck vk j  j = á   ñ The 1 ( ) e w and imaginary 2 ( ) e w dielectric functions, respectively, are used to compute all other key optical properties such as refractive index n , ( ) w absorption ( ) a w and extinction coefficients k , ( ) w conductivity ( ) s w and reflectivity R ( ) w [33,34].
where ħ and p w are the Planck constant and is plasma frequency respectively.The computed real dielectric constant 1 ( ) e w describes the dispersion of light that occurs when light traverses through the medium.The static dielectric values are found 3.5 for all titled materials.Since these values are inversely related to the corresponding bandgaps retrieved from the computed band structures, Penn's model may be demonstrated to be satisfied.The dielectric constants 1 ( ) e w real and 2 ( ) e w complex part are shown in figures 4(a) and (b).The 1 ( ) e w of A 2 AgAlI 6 (A = Na, K, Rb) attain their highest values at 4.5 eV.It is notable that after 8.1 eV, the negative values of 1 ( ) e w are found.It shows considerable reflection, indicating that photons of this energy cannot go through materials.The 2 ( ) e w spectrum of the A 2 AgAlI 6 (A = Na, K, Rb) material is connected to their light absorption behavior.With decreasing bandgap values, light absorption thresholds change to lower photon energy.The main peaks of 2 ( ) e w appeared at 7.5 eV in the energy spectrum.The highest peaks are detecting in the deep UV light region which is composed of electronic transitions from the top valence bands to the conduction bands far away from the Fermi surface.It could point to the possibilities of its

Transport properties
The bandgap and carrier concentration are related to the Seebeck coefficient and electrical conductivity, respectively.The electrical conductivity has an inverse relationship with the Seebeck coefficient.Figures 6(a) and (b) display the Seebeck coefficient and electrical conductivity, respectively.The Seebeck coefficient is positive, indicating that the majority of carriers are electrons and that the materials under study are n-type semiconductors.Its values are 205 μVK −1 , 210 μVK −1 , 215 μVK −1 at room temperature and decrease to 175 μVK −1 ,175 μVK −1 , 176 μVK −1 for Na 2 AgAlI 6 , K 2 AgAlI 6 , and Rb2AgAlI6 respectively.The Seebeck coefficient of K 2 AgAlI 6 is greatest because it has lowest electrical conductivity.The Seebeck coefficient and free charge concentration have an inverse relationship.Since the Seebeck coefficient decreases as the temperature rises like Rb 2 Ag(Ga/In)Br 6 as reported by Gourav et al in 2022, free charge contributions also increase [37].Since conductivity rises linearly with temperature, materials exhibit semiconductor behaviour.The quantity of holes or electrons that are available for conduction was measured by electrical conductivity.Because more electrons are created by bond breakage and have a high kinetic energy at high temperatures, the conductivity of Rb

= +
where k e and k ph are the thermal conductivities of electrons and phonons, respectively.Due to the limitations of the BoltzTraP code, which is based on classical theory, only the electronic component of the computations is made at this time, and the contribution of phonons is omitted.The thermal conductivity per relaxation time is shown in figure 6(c).As can be observed, these compounds' electronic thermal conductivities rose linearly as temperature rose.The power factor (P F S . . ), the product of double times Seebeck coefficient and electrical conductivity [26,[38][39][40], can be used to measure a material's efficiency.If a material has a high P.F., it will extract heat effectively.Plotted against temperature are the computed P.Fs per relaxation time for A 2 AgAlI 6 (A = Na, K, Rb) shown in the figure 6(d).Up to 700 K, the P.F. of A 2 AgAlI 6 (A = Ns, K, Rb) is observed to rise with rising temperature, but after this point, it falls as a result of a falling Seebeck coefficient.Similar trends of these properties are shown in Cs 2 InAgX 6 (X = Cl, Br, I) as reported by F Aslam et al, 2021 [41].

Phonon properties
Figures 7(a)-(c) depict the computed phonon dispersion curves of A 2 AgAlI 6 (A = Na, K, Rb).(X-L-Γ-X-W-K) has been selected as the symmetry direction.The findings show that every mode is positive, and the lack of any negative phonon modes supports the dynamical stability of the materials under investigation.As opposed to K 2 AgAlI 6 and Na 2 AgAlI 6 , it is clear that the phonon modes for Rb 2 AgAlI 6 are at lower frequencies.The fundamental cause of this change is the larger atomic weights of Rb and K than Na.The acoustic modes of Na 2 AgAlI 6 have a few kinks, and the mode separations around the 60 cm −1 frequency range are getting wider [42][43][44].

Conclusion
This study investigates the theoretical estimation of the structural stability and electronic, elastic, and optical properties of the inorganic halide double perovskite compound A 2 AgAlI 6 (A = Na, K, Rb) using PBEsol-GGA and TB_mBJ approximations.The cubic structures of A 2 AgAlI 6 (A = Na, K, Rb) exhibit stability and follow the symmetry of the Fm 3̅ m space group.To assess thermodynamic and structural stability, the tolerance factor was calculated as 0.917 for Rb 2 AgAlI 6 , and the modified tolerance factor was below the limit of 4.18.The results of the total density of states (TDOS) reveal that the iodine-p orbital bands and Ag d-orbital bands make significant contributions to the partial density of states (PDOS).The positive elastic constant results confirmed that the compound's mechanical stability, which was further supported by the ductile and anisotropic nature of the materials as indicated by the elastic constants.The band structure and density of states of the materials indicated a small direct bandgap, reflecting its semiconducting nature.Another advantage feature is that it is a lead-free double perovskite material.Additionally, the optical properties demonstrate a strong optical response in the visible and ultraviolet range, suggesting potential applications in solar cells, ultraviolet detectors and photocatalysis.Based on the calculated optoelectronic and thermoelectric properties, A 2 AgAlI 6 (A = Na, K, Rb) can be considered a good candidate for energy harvesting.