Inorganic M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives for next-generation solar cells and optoelectronics: in-depth analysis of stability, optoelectronic features, and temperature-dependent carrier mobilities

Abstract

The burgeoning realm of energy-harnessing technologies, coupled with significant advancements in optoelectronics, necessitates the exploration of compelling semiconductors that can effectively bridge existing technological divides. In this context, the present study meticulously examines M₃ACl₃ semiconductors to elucidate their multifaceted stability, encompassing structural, phase, dynamic, mechanical, and thermodynamic dimensions. Furthermore, this inquiry unveils insights into their optoelectronic properties, with a particular focus on temperature-induced carrier mobility behaviors. In this analysis, HSE06 calculations reveal the direct band gaps of the studied materials in the range of 1.7 to 2.9 eV. Specifically, Ba-containing materials possess the lowest band gaps, corresponding to the red region of the visible spectrum, while the others fall within the green and blue ranges. Moreover, these materials exhibit low effective masses, with Ba-containing variants showing the lowest effective masses and elevated relative effective mass ratios, indicating their pronounced charge carrier generation rates. In the optical domain, the absorption onset is observed in the visible range corresponding to the band gaps, displaying vis-UV light absorption characterized by high coefficients. Additionally, average mobilities range from 38.2 to 192.98 cm² V⁻¹ s⁻¹ at 300 K for a carrier concentration of 10²⁰ cm⁻³, showcasing remarkable electron and hole mobilities of 173 and 138.16 cm² V⁻¹ s⁻¹, respectively, for Ba₃PCl₃. This study reveals the robust stabilities and pertinent optoelectronic features of the studied materials, indicating their potential applications in tandem solar cells and other optoelectronic devices such as RGB light-emitting diodes, semiconductor displays, and UV emitters and sensors.

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1 Introduction

Green energy sources offer an eco-friendly, cost-effective, and sustainable power supply, which is a necessity of today.¹ Solar energy with its global availability is one of the promising green energy sources, offering a mass-scale production of clean power. Currently, solar photovoltaics shares 5.4% of the total worldwide power generation and 8.3% of total renewable source generated power.² In addition to harnessing this green source on a large scale, cutting-edge research of innovative materials is progressing. Compared to conventional solar cells with silicon of indirect band gap characteristics, metal lead-based halide perovskite materials with direct band gaps have exhibited exceptional power conversion efficiencies (PCE) surpassing 25%.^3,4 However, despite their high efficiencies, low-cost production, lightweight designs, and stable structures, perovskite-driven solar cells are still on their way to widespread acceptance.^5–7 The significant hurdles that prevent commercialization include lead-based toxicity and inferior stability at high temperatures, in moisture, and in electromagnetic environments. Lead is particularly problematic because of its bioaccumulation hazards.^8,9

The quest for lead-free perovskites and their derivatives is progressing rapidly, and recently, novel inorganic halide perovskite derivatives (A₃BX₃) have garnered the attention of researchers due to their exceptional structural attributes and direct band gap-tuned optoelectronic characteristics. Furthermore, the ability to modulate their band gaps under varying pressures and strains amplifies their versatility, facilitating the development of tunable optoelectronics.^10–12 Haque et al. provides insights into the optoelectronic characteristics of Ba₃SbX₃ (X = F, Cl) compounds, showing band gaps of 0.943 and 0.908 eV for X = F and Cl, respectively, and revealing absorption maxima within the ultraviolet range. The authors remarked that these materials are promising for a range of optoelectronic applications, including UV photodetectors, deep-UV optical filters, and photovoltaic systems. Furthermore, the pressure-induced diminution of the band gaps in these materials underscores their potential for practicality in tunable optoelectronic devices.¹³ Apurba et al. investigated the thermoelectric and optoelectronic properties of Ca₃PX₃ (for X = Cl, Br, and I) within the framework of DFT-PBE, revealing direct band gaps of 1.4909, 1.9502, and 2.2058 eV, respectively. Their findings also indicated that the band gaps experience a contraction under compressive strain, while an expansion occurs under tensile strain, highlighting the excellent potential of these materials for enhancing light manipulation in solar energy applications and energy retention technologies.¹⁴

Additionally, the A₃BX₃ class of perovskite-derivatives shows high structural stability and thermal efficiencies.¹⁵ The shift in their electronic behavior under stress, such as the transition from semiconducting to metallic, further emphasizes their potential for innovative optoelectronic solutions.¹⁶ Furthermore, solar cells based on these materials exhibit a remarkable ability to sustain elevated open-circuit voltages.¹⁷ Also, the octahedral symmetry inherent in A₃BX₃ crystals may engender a shift in the p–s band edge state, which can markedly enhance the optical performance of the cells.¹⁸ Cumulatively, these investigations elucidate the significant potential of A₃BX₃ perovskite-derivatives and underscore their prospective applications across a diverse array of optoelectronic devices functioning within the visible and ultraviolet spectra.

Simulating solar cell devices necessitates intricately linked optoelectronic and transport properties, encompassing electron affinities, band gaps, dielectric constants, valence and conduction band density of states, charge carrier concentrations, and mobilities. The efficacy of optoelectronic devices depends on the carrier dynamics of semiconductors. In particular, the carrier mobilities are critical determinants of the interplay between carrier transport and recombination processes and thus play a vital role in determining optoelectronic performance.^19–21 Numerous studies have revealed the finite limits of the intrinsic electron and hole mobilities of halide perovskites.^22,23 In a theoretical investigation, Poncé et al. reported the electron and hole mobilities of CsPbI₃ as 68 and 76 cm² V⁻¹ s⁻¹, respectively, at 327 K, 41 cm² V⁻¹ s⁻¹ for CsPbBr₃, and a range of 30 to 80 cm² V⁻¹ s⁻¹ for MAPbI₃ halide perovskite. They examined that the mobilities of these materials are ultimately limited to 80 cm² V⁻¹ s⁻¹ at ambient temperatures and are restrained due to the fluctuations in the Pb–I bond.²² Cucco et al. also computed the hole mobility of 41.6 cm² V⁻¹ s⁻¹ at room temperature for bulk CsPbBr₃.²³ However, the exact carrier mobilities for A₃BX₃ compounds are not found in the prior investigations.

Due to a lack of exact theoretical or empirical data for mobilities of A₃BX₃, researchers have resorted to employing approximated or averaged mobilities in their analyses of optimizing A₃BX₃ based solar cells. For instance, Ria et al. have adopted mean electron and hole mobilities of 100 and 20 cm² V⁻¹ s⁻¹, respectively, for perovskite-derived Ca₃NF₃, Ca₃NCl₃, Ca₃NBr₃, and Ca₃NI₃ semiconductors.²⁴ Similarly, Islam et al. and Ghosh et al. have utilized an average mobility of 50 cm² V⁻¹ s⁻¹ for Ca₃NI₃ and Ba₃NCl₃.^25,26 In a parallel vein, Rahman et al. have employed the same statistics, 50 cm² V⁻¹ s⁻¹, for the electron and hole mobilities of Ba₃NCl₃, Sr₃NCl₃, and Ca₃NCl₃ chlorides.²⁷ However, the acquisition of experimentally validated or accurately predicted mobilities of these materials remains the most coveted attribute in the advancement of optoelectronics.

Viewing the substantial scope of inorganic A₃BX₃ halide perovskite-derivatives in solar cells, this study for the first time thoroughly investigated M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) compounds that encompass the accurate prediction of their structural and dynamical stability characteristics and mechanical, thermodynamic, electronic, and optical properties employing the HSE06 functional, and most importantly, the prediction of carrier mobilities using the first-principles electron–phonon interaction physics, as implemented in Quantum Espresso coupled EPW (Electron–Phonon Wannier) code. Moreover, the impact of alkaline earth cations on the optoelectronic features of these materials is examined. Furthermore, this investigation is pioneering in its analysis of thermodynamic properties, the temperature-dependent dynamic stability derived from phonon dispersion spectra, and the temperature-dependent carrier mobilities of the studied pnictides. This study presents a foundational analysis of the optoelectronic properties of the materials studied, which are prerequisites for designing efficient solar cells. The explicit goal of this work is to establish a robust material-level understanding to inform subsequent device optimization studies.

2 Methodology

2.1 Framework for structural and electronic analysis

This investigation has conducted first-principles calculations using the Vienna Ab Initio Simulation Package (VASP).^28–30 It has harnessed projected augmented wave potentials to adeptly account for ion–electron interactions. The PBE functional has been deployed to ascertain the structural parameters. However, for the assessment of optoelectronic properties, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional,³¹ which encapsulates the Hartree–Fock exchange energy, has been adopted due to its remarkable accuracy in capturing exchange-correlation phenomena. The relaxation of crystal structures included relaxation of shape, volume, and ionic positions. Strict criteria for relaxation and energy calculations are adopted to ensure the convergence of static energies to within 1 × 10⁻⁷ eV, with maximum Hellmann forces on each atom restrained to less than 1 × 10⁻⁶ eV Å⁻¹. Also, a gamma scheme sampled 6 × 6 × 6 k-point grid and a plane-wave basis with a cutoff energy of 350 eV, determined through convergence tests (see Fig. S1 and S2 of the SI for the details), have been utilized. Gaussian smearing with a smearing width of 0.02 is used for k-point sampling. However, the density of states is computed with a finer k-point grid of 8 × 8 × 8 for all studied cases. Structural optimization and static energy computations revealed the optimized lattice parameters, lattice energies, interatomic bond lengths, and electron density distributions.

2.2 Framework for mechanical analysis

The mechanical attributes are meticulously scrutinized by using the energy-strain methodology, a preeminent technique for deducing theoretical elastic constants. This approach facilitates an exacting computation of the energy fluctuations induced by minute deformations relative to the equilibrium lattice configuration. For a given crystal system acted upon by a strain ε_i, the elastic energy (ΔE(V′, ε_i)) within the framework of harmonic approximation can be expressed using eqn (1).³²Here, E(V′, ε_i) and E(V_o) denote the energies of distorted and equilibrium crystal systems, respectively, and C_ij is the elastic stiffness tensor that is obtained through the second-order derivative of elastic energy with respect to strain. For cubic crystal systems of the investigated materials, the elastic stiffness tensor consists of three types of elastic constants, C₁₁, C₁₂, and C₄₄, which are generated from the application of tri-axial shear strain ε = (0, 0, 0, δ, δ, δ). This calculation has computed the elastic energies for the applied strain values of −0.015, −0.010, −0.005, 0.000, 0.005, 0.010, and 0.015. Following the computation of elastic constants, the mechanical stability is evaluated by examining Born's stability criteria, which include the following conditions: C₁₁ > 0, C₄₄ > 0, C₁₁ − C₁₂ > 0, and C₁₁ + 2C₁₂ > 0. Employing the Voigt bounds for the bulk modulus (B) and shear modulus (G), this analysis computed Young's moduli (E) and Poisson's ratios utilizing eqn (2) and (3). The machinability index (μ_M), Kleinman parameters (ζ), and the Zener anisotropic index (A) are subsequently calculated viaeqn (4)–(6). Additionally, Cauchy pressures, linear compressibilities, Vickers hardness, and acoustic velocities are ascertained within the framework of state-of-the-art mechanical property analysis implemented in Vaspkit.^32–35

2.3 Framework for stability analysis

Structural stability assessments entail the meticulous computation of cohesive and formation energies to evaluate phase stability, while phonon spectra are scrutinized to ascertain dynamic stability. The cohesive and formation energies are determined by employing eqn (7) and (8).

N = N_M + N_A + N_Cl (9)

Here, E_i denotes the energies per atom of isolated atoms, and E_b signifies the energies per atom of the stable bulk phases of an element or a structure.

Furthermore, to assess the dynamic stability of the materials under investigation, the phonon dispersion relations across a temperature range of 200 to 500 K are computed. This meticulous evaluation facilitates a deeper understanding of the vibrational properties and stability characteristics of the materials at elevated temperatures. This analysis utilizes the temperature-dependent volumetric data derived from the QHDM. In this investigation, the super-cell method for determining phonon dispersion is adopted. It is a robust computational technique that facilitates the assessment of the dynamical matrix through finite difference approximations.^36,37 This approach necessitates the construction of super-cells wherein atomic positions are made knowingly perturbed for a given crystal system. Then, energy calculations are performed systematically across all perturbed configurations that lead to the computation of force constants. Utilizing these force constants, phonon dispersion spectra are computed for the prescribed mesh sampling.

2.4 Framework for thermodynamic analysis

Employing the Quasi-Harmonic Debye Model (QHDM),³⁸ this study has examined the pressure and temperature effects of thermodynamic properties, including volume, bulk moduli, heat capacities, Debye temperatures, and thermal expansion coefficients. This model tackles anharmonic effects by employing a harmonic approximation for all crystalline structures, even those that exist out of equilibrium. Essentially, this provides quasi-anharmonicity by the volume dependence of phonon frequencies. The efficacy of this approach in forecasting thermodynamic properties and the phase stability of solids is well-established.^39–41 This model incorporates the anharmonic effects through vibrational Helmholtz free energy function F_vib as given by eqn (10),³⁸

F_vib = nk_BT(9ω/8 + 3 ln(1 − e^−ω) − D(ω)) (10)

Here, ω = θ_D/T, n is the number of atoms in the formula unit of a given material, and k_B is the Boltzmann constant. θ_D = θ_D(V) is the Debye temperature, and D is the Debye integral, which are given by eqn (11) and (12).^42,43Here, M is the molar mass of the formula unit whereas f(σ) is the Poisson ratio functional and B_s(V) is the static bulk modulus, and these are given by eqn (13) and (14).³⁸

Poisson ratio σ is set to 0.25, a commonly used value for Cauchy solids.³⁸

The calculation of the Debye temperature at the equilibrium volume for a given temperature and pressure leads to the computation of thermodynamic properties at that temperature and pressure, as determined from eqn (15) and (16).^38,42

C_p = C_v (1 + αγT) (17)

2.5 Framework for carrier mobility analysis

The carrier mobilities of the examined materials are computed utilizing the framework of density functional perturbation theory in conjunction with maximally localized Wannier functions, as executed in the EPW software.^44,45 This methodology adeptly addresses electron–phonon interactions and facilitates the precise calculation of electron–phonon matrix elements across ultra-dense momentum grids. Subsequently, EPW employs the ab initio linearized Boltzmann Transport Equation (BTE), which elucidates the first-order response of the carrier distribution function (∂E_βf_nk) to external perturbations, as delineated in eqn (20), to unveil the electronic transport properties.⁴⁶

In this representation, f_nk signifies the electron distribution function, wherein n refers to a Kohn–Sham band index and k denotes a wave vector. V_nkβ = ℏ⁻¹∂ε_nk/∂k_β epitomizes the intra-band velocity matrix element associated with the eigenvalue ε_nk. The symbol δ denotes the Dirac delta function. The second term on the right-hand side represents an aggregate of Brillouin zone-integrated electron–phonon interactions, elucidating the scattering of electrons from state nk to state mk + q through the phonon of branch v and wave vector q. This linear response framework incorporates thermal influences derived from the Fermi–Dirac and Bose–Einstein equilibrium distribution functions for electrons (f^o_nk) and phonons (n_qv), respectively. In this formulation, τ_nk denotes the carrier relaxation time, which can be ascertained from eqn (21),⁴⁶ delineating the scattering rates by Fermi's golden rule.

Subsequently, the conductivities (σ_αβ) and mobilities (μ_αβ) are calculated through eqn (22) and (23).⁴⁶

μ_αβ = σ_αβ(eN_c)⁻¹ (23)

Here, N_c denotes the carrier concentrations.

The prediction and effectiveness of this methodology are widely acknowledged within the materials science community.^44,46–48 S. Poncé et al. developed software to benchmark experimental results for several semiconductors, including diamond, Si, GaAs, 3C-SiC, GaP, AlP, cubic BN, AlAs, AlSb, and SrO.^44,47 Likewise, employing the same method, Lee et al. conducted a thorough benchmarking of the variation in electron and hole mobilities with temperature and carrier concentration in silicon semiconductors.⁴⁶ Additionally, S. Poncé et al. reported comparable experimental results for the electron and hole mobilities in wurtzite gallium nitride.⁴⁹ Recently, Yin et al. calculated the mobilities of CsSnBr₃ inorganic halide perovskite, over a temperature range of 300 to 500 K, and found that their results align with experimental data.⁵⁰

2.6 Framework for optical feature analysis

Utilizing the optical response theory as implemented within VASP, this study has elucidated the optical characteristics of the materials under scrutiny. The salient feature that delineates the optical response of a material to incident electromagnetic radiation is the complex dielectric function ε_ω, which exhibits dependence on the incident photon frequency ω. The subsequent eqn (24) delineates the formulation of the dielectric function.³²

ε_ω = ε^r_ω + iεⁱ_ω (24)

Here, ε^r_ω and εⁱ_ω are the real and imaginary components of the dielectric function, encapsulating the two principal responses of a material to incident radiation, which are polarizability and optical transitions. Specifically, the imaginary part bears paramount significance; it is computed initially, from which other optical properties are subsequently derived. The following eqn (25) articulates the Cartesian tensor representing the imaginary part of the dielectric function.⁵¹

In this representation, Ω signifies the volume of the primitive cell, w_k denotes the weights of the k-points ensuring the integral evaluates to 1, while e_a and e_b represent the unit vectors denoting the a and b bands, respectively. Furthermore, c_k and v_k epitomize the states of conduction and valence bands within the reciprocal k-space of the irreducible Brillouin zone, with z_ck and z_vk indicating the periodic functions of the conduction and valence band states, respectively. At any given k-point, the index c traverses the unoccupied states of the conduction band, while the index v iterates over the occupied states of the valence band.³² In the presence of cubic crystalline symmetry, the off-diagonal components of the optical tensor dissipate, leaving only the diagonal components that are equivalent, thereby transforming the tensor into a scalar representation for cubic systems.⁵²

Subsequently, by employing the Kramers–Kronig transformation as eqn (26) (ref. 51) describes, the real part of the dielectric function is computed.

In this transformation, P represents the principal value. Subsequent calculations of optical characteristics utilize both real and imaginary parts. Eqn (27)–(31) describes the absorption coefficient (α_ω), extinction coefficient (K_ω), refractive index (n_ω), reflectivity (R_ω), and the energy-loss function (L_ω), respectively.^32,51,53

3 Results and discussion

3.1 Structural properties

The studied inorganic M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives crystallize in the Pm3̄m cubic crystal structures. Their optimized crystal structures are provided in Fig. S3 of the SI in detail. The unit cell of these structures comprises seven atoms. Each M²⁺ cation is coordinated with four co-planer Cl⁻¹ ions, and two A⁻³ ions, where M–A bond lengths are equivalent to M–Cl, forming an octahedral geometry, as depicted in Fig. 1, while each A⁻³ ion is coordinated with 6 M²⁺ ions, forming an AM6 octahedral. The M²⁺, A⁻³, and Cl⁻¹ ions are located at 3d (0, 0.5, 0), 1a (0, 0, 0), and 3c (0.5, 0.5, 0) Wyckoff positions, respectively. Γ, X, M, Γ, R, and X represent the high-symmetry k-points in the irreducible Brillouin zone of these structures, which are used to examine the phononic and electronic structures.

Fig. 1 3D crystalline structure showing MA₂Cl₄ symmetry, and the k-path of IBZ of M₃ACl₃ (M = Ca, Ba, Sr A = N, P, As) perovskite-derivatives.

Inter-atomic bond lengths at 0 and 300 K, listed in Table S1 of the SI, delineate equivalent M–A and M–Cl bond lengths. These bond lengths widen as A varies from N to P to As and M varies from Ca to Sr to Ba, which is related to the atomic radii of formula elements. These computed bond lengths agree with the results of prior investigations.^12,57,58

Moreover, the plots of charge density distribution, as shown in Fig. 2, enlighten the nature of these bonds. These illustrate the contours of charge densities of studied compounds along the [200] crystallographic plane. The color bar on the right displays the scale of charge density intensity. The charge distribution illustrated in these plots reveals a pronounced overlap of electronic densities between the M and A atoms across the majority of these materials, with the notable exception of Ca₃NCl₃. This overlap diminishes the distinct characteristics of these related atoms, thereby indicating the formation of covalent bonds between the M and A species. In these structures, each A ion is encircled by six M ions, facilitating a shared charge among them. The plots further delineate varying extents of charge overlap, which correspond to the level of charges shared between the M and A atoms. Specifically, Ca₃PCl₃, Ca₃AsCl₃, and Sr₃NCl₃ exhibit a single level of charge overlap, whereas Sr₃PCl₃, Sr₃AsCl₃, Ba₃NCl₃, and Ba₃AsCl₃ demonstrate two levels of overlapping interactions. Notably, Ba₃PCl₃ is characterized by a triad of charge overlapping levels. Conversely, the absence of coincidental charge density contours between M and Cl suggests a predominantly ionic bonding character. It is also evident from the elongation edges of the charge density distribution of Cl towards the M atoms. However, the interactions between A and Cl atoms exhibit anti-bonding characteristics. Recent studies have highlighted the presence of various bonding types within these materials. For instance, Rahman et al. identified covalent bonds forming between strontium and nitrogen, alongside ionic bonds between strontium and chlorine, and anti-bonding characteristics in the compound Sr₃NCl₃.⁵⁴ Similar findings have been noted in related compounds, such as Sr₃PCl₃,¹² Sr₃AsCl₃,⁵⁹ and Ba₃NCl₃,⁶⁰ indicating a consistent pattern in the types of bonding among these materials.

Fig. 2 Charge density distributions of M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives along the [200] crystallographic plane.

Fitting of the fourth-order Birch–Murnaghan equation of state, as expressed in eqn (32),⁶¹ to the energy–volume data of all studied materials, yielded the equilibrium energies and volumes (E_o, V_o), bulk moduli (B), and their derivatives (B′). The fitted energy–volume curves are provided in Fig. S4 of the SI. Additionally, the Debye temperatures are calculated using the equilibrium volumes and bulk moduli in eqn (12). Table 1 provides a summary of the lattice constants and all these structural properties, as calculated in the PBE computational domain.

Table 1 Structural properties of the examined M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives as determined in the PBE domain

Materials	a (Å)	V st (Å)^3	V o (Å)^3	E st (eV)	E o (eV)	B o (GPa)		θ D (K)	E c (eV per atom)	E f (eV per atom)
Ca3NCl3	5.46	162.77	158.84	−32.005	−32.007	50.19	4.42	403.97	4.538	−1.769
(PBE)^27	5.42
Ca3PCl3	5.76	190.97	186.78	−30.832	−30.834	40.17	4.20	358.87	4.369	−2.024
Ca3AsCl3	5.83	198.31	193.78	−30.026	−30.029	37.59	4.19	322.80	4.254	−2.007
Sr3NCl3	5.79	194.19	191.36	−30.953	−30.955	41.41	4.56	299.92	4.386	−1.744
(NC-PBE)^54	5.74
Sr3PCl3	6.10	227.31	222.18	−30.108	−30.110	33.87	4.34	272.10	4.264	−2.046
(USP-PBE)^12	6.05		221.35			30.42	2.56	282.07
Sr3AsCl3	6.16	233.24	229.93	−29.391	−29.393	31.77	4.33	251.61	4.162	−2.041
(NC-PBE)^55	6.12		229.22			31.6				−4.40
Ba3NCl3	6.19	237.55	233.25	−31.358	−31.361	34.55	4.69	240.23	4.442	−1.684
Ba3PCl3	6.47	271.32	267.08	−30.768	−30.770	28.73	4.46	220.58	4.355	−2.022
(NC-PBE)^56	6.44					27.745
Ba3AsCl3	6.55	280.61	275.51	−30.149	−30.151	27.00	4.45	206.84	4.268	−2.032

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The bulk modulus and its derivative, which measure a material's resistance to compressive forces and the variation of stress under pressure, are observed to range from 25 to 50 GPa, with the derivative from 4.1 to 4.7. This range of bulk moduli is typically preferred for applications involving relatively low-pressure forces, such as mass-scale optoelectronic devices, including solar cells, displays, optical sensors, dimming functions, optical amplifiers, and optocouplers. Materials possessing this range of bulk moduli may prove to be more economically viable and simpler to process than those exhibiting a high bulk modulus. Additionally, the positive values of B′ suggest that these materials become approximately four times harder when subjected to pressure. Moreover, the Debye temperatures are within the 205 to 405 K range which is a typical range for solids. These exhibit that rigidity in these materials decreases as volume expands due to the increased size of M and A atoms.

Moreover, the materials under investigation exhibit both phase stability and chemical stability as evidenced by their positive cohesive (E_c) and negative formation (E_f) energies listed in Table 1, as computed from eqn (7) and (8). Also, negative formation energies indicate the experimental formability and viability of the studied compounds. The comparison to prior investigations, which have employed Norm-Conserving (NC) and Ultra-Soft Pseudopotential (USP) with the PBE functional,^12,54–56 shows that our results align with these.

3.2 Mechanical properties

This investigation conducted the computations for mechanical properties based on the strain–-energy theoretical framework. These computations provide the elastic stiffness tensor (C_ij) comprising the elastic constants that are essential mechanical features for characterizing structural integrity and mechanical strength of materials. Essentially, these parameters signify the degree to which a substance deforms under an applied load and its capacity to revert to its original configuration upon the cessation of that load.^62–64 The cubic Pm3̄m crystal structures of the studied compounds exhibit three independent elastic constants, namely C₁₁, C₁₂, and C₄₄. These facilitate the examination of the material's anisotropic characteristics, mechanical stability, stiffness, and ductility or brittleness.

By Born's criteria for mechanical stability, a material is deemed mechanically stable under zero stress if its second-order elastic stiffness tensor (C_ij) is positive-definite, indicating that all its eigenvalues are strictly positive. More precisely, the requisite conditions include C₁₁ − C₁₂ > 0, C₄₄ > 0, and C₁₁ + 2C₁₂ > 0.⁶⁵ As listed in Table 2, the computed elastic constants are positive and manifest fulfilling the mentioned conditions and thus reveal the mechanical stability of the studied compounds. Moreover, the higher C₁₁ than C₁₂ and C₄₄ also implies superior resistance towards compression along the x-axis.

Table 2 Mechanical properties: elastic constants, Born's stability criterion, Pugh ratios (B/G), Cauchy pressures (P_c), Poison ratios (ν), Kleinman parameters (ζ), machinability indices (μ_M), and melting temperatures (T_m) of studied M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) compounds

Materials	C 11 (GPa)	C 12 (GPa)	C 44 (GPa)	C 11 − C12	C 11 + 2C12	Pugh ratio	P c	ν	ζ	μ M	T m (K)
Ca3NCl3	113.08	18.80	28.22	94.28	150.69	1.45	−9.4	0.22	0.35	1.78	1221.3
Ca3PCl3	93.49	14.91	22.65	78.58	123.31	1.45	−7.7	0.22	0.34	1.81	1105.5
Ca3AsCl3	91.82	10.18	21.41	81.64	112.19	1.34	−11.2	0.20	0.28	1.75	1109.57
Ca3AsCl3 (ref. 58)	97.7	11.87	20.94					0.108
Sr3NCl3	94.27	14.74	20.47	79.53	123.75	1.54	−5.7	0.23	0.34	2.02	1110.1
Sr3PCl3	79.49	10.67	15.63	68.82	100.83	1.56	−5.0	0.24	0.31	2.15	1022.8
Sr3PCl3 (ref. 12)	76.31	7.47	16.24			1.38	−8.76	0.208	0.247	1.38	1004.02
Sr3AsCl3	75.81	9.31	15.59	66.51	94.43	1.48	−6.3	0.22	0.29	2.02	1001.1
Sr3AsCl3 (ref. 55)	75.07	10.03	15.21			1.529	−5.18	0.231	0.284	2.08
Ba3NCl3	78.09	12.57	13.59	65.52	103.22	1.76	−1.0	0.26	0.34	2.53	1014.5
Ba3NCl3 (ref. 60)	77.90	12.97	14.01			1.76		0.27
Ba3PCl3	70.37	10.34	10.97	60.03	91.04	1.82	−0.6	0.27	0.32	2.77	968.9
Ba3AsCl3	63.21	10.40	10.41	52.81	84.02	1.83	0.0	0.27	0.35	2.69	926.6

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Theoretically, a Pugh ratio lower than 1.75,⁶⁶ a negative Cauchy pressure (P_c),⁶⁷ and a Poisson ratio of less than 0.26 all serve as criteria for distinguishing between brittle and ductile behavior in materials.⁶⁸ The brittle behavior of the studied materials having Ca and Sr is established by their Pugh ratios of less than 1.75, negative Cauchy pressures, and Poisson ratios of less than 0.26. However, studied Ba-containing materials exhibit ductile behavior as manifested by their Pugh ratios higher than 1.75, Poisson ratios greater than or equal to 0.26, and approximately zero Cauchy pressures.

The graphical representation of the relationships between bulk, shear, and Young's moduli with respect to the investigated materials is depicted in Fig. 3. It illustrates a distinct correlation in which compounds containing Ca exhibit enhanced rigidity and stiffness, categorizing them as the most resilient among the studied materials. In contrast, those fortified with Sr display intermediate mechanical properties, while barium-containing compounds manifest the lowest degrees of rigidity and stiffness. Typically, pronounced rigidness and stiffness suggest a propensity for brittle behavior in materials, and ductile nature contrarily.⁶⁹ Therefore, these findings also underscore the brittle character of Ca- and Sr-containing compounds and the ductile nature of Ba-containing compounds. Furthermore, it demonstrates that the linear compressibility increases along with a diminution in Vickers hardness for the sequence of examined compounds. These trends indicate that the studied Ba-containing materials exhibit greater compressibility and lower hardness, traits that are crucial for applications requiring enhanced malleability. These graphical illustrations further elucidate an inverse correlation between linear compressibility and bulk modulus, consistent with the understanding that materials exhibiting diminished compressibility generally manifest heightened bulk moduli and hardness.³³ Additionally, the numerical data of these mechanical parameters, as described in Table S2 of the SI, are compared with the theoretical findings,^12,55,58,60 showing close agreement with them.

Fig. 3 Graphical representation of bulk moduli (B), Young's moduli (E), shear moduli (G), linear compressibilities (β), and Vickers hardness (H_v) as a function of the studied compounds.

Another essential mechanical metric is the dimensionless Kleinman parameter (ζ) that articulates the relative contribution of bond-bending and bond-stretching effects to the material's resistance against applied stress or strain, taking values from 0 to 1. A lower Kleinman parameter denotes a predominance of bond-bending contributing mechanical strength, often associated with brittle and rigid materials, while a higher value implies a primary contribution from bond-stretching, typically aligned with more ductile materials exhibiting reduced hardness.^70,71 The materials studied display Kleinman parameter values in the range of 0.28 to 0.35, indicating a dominance of the bond-bending effect and elucidating their brittle character.

Moreover, the machinability index (μ_M), indicating how easily a material can be machined or shaped, is typically expressed as B/C₄₄ for cubic crystals. A higher index suggests better machinability, meaning the material is more easily cut or molded.^72,73 The machinability indices increase going from Ca₃NCl₃ to Ba₃AsCl₃ as shown in Table 2. It indicates the increasing cutting capacity or ease in shaping in this material sequence. So, this supports the brittleness of the studied Ca- and Sr-based materials and the ductile behavior of Ba-based ones.

Fine et al. determined a least squares fit formula from the elastic constant C₁₁ and melting temperatures (T_m) for cubic elemental metals and intermetallic compounds,⁷⁴ as shown in eqn (33).

T_m = 553 + 5.91C₁₁ (33)

Leveraging this formula, the melting temperatures of the studied cubic materials are determined. A decrease in T_m values going from Ca₃NCl₃ to Ba₃AsCl₃ can be observed in Table 2, which corresponds to a decreasing trend in their hardness. These values follow a pattern according to the hardness of these materials. Utilizing this empirical relation, Hosen has also determined the melting temperatures of cubic Sr₃BCl₃ (B = As, Sb),⁵⁵ which aligns with our results.

Acoustic velocities have also been computed for the materials under investigation, highlighting the speed at which sound waves propagate through these materials. This phenomenon reflects the hardness and stiffness of the solids. Generally, higher acoustic velocities indicate a higher material's hardness and the inverse is true. Referring to the values in Table 3, it is evident that longitudinal wave velocities (LWV), also referred to as compressional wave speeds, surpass transverse wave velocities (TWV) or shear wave speeds. This disparity arises from the higher bulk moduli and lower shear moduli values that are used in computing these velocities. Furthermore, both types of wave velocities exhibit a decreasing trend from Ca₃NCl₃ to Ba₃AsCl₃, which corresponds with the observed hardness trends of these materials.

Table 3 Mechanical properties: anisotropic parameters (A, UEA) and acoustic speeds of M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives

Materials	A	UEA	LWV (km s^−1)	TWV (km s^−1)	AWV (km s^−1)
Ca3NCl3	0.60	0.32	6.19	3.71	4.11
Ca3PCl3	0.58	0.37	5.86	3.51	3.89
Ca3AsCl3	0.53	0.52	5.36	3.28	3.62
Sr3NCl3	0.52	0.55	4.81	2.84	3.14
Sr3PCl3	0.45	0.79	4.57	2.69	2.98
Sr3PCl3 (ref. 12)	0.47		4.464	2.710	2.995
Sr3AsCl3	0.47	0.72	4.32	2.57	2.85
Sr3AsCl3 (ref. 55)	0.467
Ba3NCl3	0.42	0.99	3.99	2.27	2.52
Ba3PCl3	0.37	1.32	3.89	2.19	2.44
Ba3AsCl3	0.39	1.12	3.67	2.06	2.29

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The Poisson ratio within the 0.25 to 0.5 range indicates that central force interactions predominantly govern the behavior of a crystal. In such a case, the interatomic forces are primarily directed toward the centers of the atoms, a characteristic commonly observed in cubic crystal structures. When the Poisson ratio falls outside this range, non-central force interactions become more significant as the characteristics of non-cubic or anisotropic crystals. Moreover, the Cauchy relations hold for central force crystals and deviate in the case of non-central crystals.^75,76 This calculation shows that the Poisson ratios of the studied Ba-containing materials greater than 0.25 signify that these are central force crystals that also obey Cauchy relations. However, the Poisson ratios of Ca- and Sr-containing materials are closely below 0.25, suggesting that their anisotropic character originates from non-central interatomic forces. The elastic anisotropy of the studied materials reveals the directional dependence of their mechanical properties. In this regard, the Zener anisotropic factor (A) and the Universal Elastic Anisotropy (UEA) index offer a quantitative measure of the anisotropy of compounds. Specifically, the Zener anisotropic factor accounts for the anisotropy of cubic crystals.^77,78 Factor A equals 1 indicating a state of zero anisotropy, and conversely, any deviation from 1 signifies the anisotropic behavior. In contrast, the UEA index is zero for isotropic materials and nonzero otherwise. Both parameters are detailed in Table 3, which illustrates the anisotropic behavior of the studied materials. Transitioning from Ca₃NCl₃ to Ba₃AsCl₃, the anisotropy indices gradually increase, highlighting more direction dependence of mechanical behavior. Fig. 4 presents 3D topological surface plots that depict the form of elastic anisotropy of Young's moduli, shear moduli, Poisson ratios, and linear compressibilities. Meanwhile, Fig. S5 of the SI shows the 2D contours of elastic anisotropy and highlights the minima, maxima, and the relevant anisotropy factor of these properties. These demonstrate isotropic behavior in linear compressibilities while showing directional dependence in Young's moduli, shear moduli, and Poisson ratios for all the materials studied. Additionally, although the anisotropic plots are identical for each material, the indices of anisotropy vary, and Ba₃AsCl₃ shows the highest indices of anisotropy.

Fig. 4 3D surface plots of elastic anisotropy of Young's moduli, linear compressibility, shear moduli, and Poisson ratios of the investigated materials.

Our computed mechanical properties align with previous theoretical investigations.^12,55,58,60 While no pertinent experimental work currently exists, our findings may prove instrumental in devising an experimental framework for these materials.

3.3 Dynamic stability analysis

Phonon dispersion spectra are shown in Fig. 5, emphasizing the dynamic stability of materials under investigation. This figure illustrates the dispersion spectra at high-symmetry points in the first Brillouin zone over a temperature range from 200 to 500 K, with a step size of 50 K. These spectra evidence that the N-containing materials lack dynamic stability, as indicated by the presence of negative frequency modes across the entire temperature range. Contrarily, the Sr- and Ba-based materials exhibit positive phonon frequencies, highlighting their dynamic stability. These spectra demonstrate that phonon frequencies decrease as the temperature increases, which transpires due to the rising Grüneisen parameters and thermal expansion coefficients with temperature rise of the investigated materials. Despite frequency decreases with temperature, the dynamic stability of the Sr- and Ba-based materials is not disregarded by any temperature change.

Fig. 5 Graphical representations of temperature-dependent phonon dispersions of the examined compounds.

As can be observed, optical phonon frequencies decrease with an increase in temperature, while acoustic phonon frequencies remain largely unaffected. These spectra indicate that the highest optical frequencies observed are 249.13, 217.04, 207.97, 164.12, 158.41, and 123.47 cm⁻¹ for Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, Ba₃PCl₃, and Ba₃AsCl₃, respectively. These frequencies correspond to temperatures of 358.44, 312.27, 299.22, 236.13, 227.92, and 177.65 K. Furthermore, multiplying these temperatures by 3/2 yields values of 537.66, 468.405, 448.83, 354.195, 341.88, and 266.475 K, at which point the heat capacities of these materials at constant volume approximately approach the Dulong–Petit limit of 3R. These values indicate that the examined materials reach their maximum heat capacity at relatively low temperatures. Additionally, a steeper slope in the acoustic modes suggests increased acoustic velocities within these crystals. All the dispersion spectra are alike along the k-path but with differences in frequencies. Looking into the Ba₃AsCl₃ spectra as a case, among the acoustic modes, two low-frequency transverse acoustic (TA) modes correspond to transverse waves and shear speeds in solids, while the longitudinal acoustic (LA) mode with a steeper slope signifies a longitudinal wave, indicative of a higher longitudinal speed. Likewise, the lower-frequency transverse optical (TO) modes represent transverse waves, and the higher-frequency longitudinal optical (LO) modes are associated with the most energetic longitudinal waves. Other spectra have a resemblance with differences in frequencies.

The lack of dynamic stability in studied N-based materials indicates the off-centering of the B-site cation (N-site), which is a common concern in perovskite structures, leading to distortions and the prospect of low-symmetry tilted structures. Essentially, the relatively small radius of N atoms in these structures causes substantial voids within the lattice, undermining structural compactness and leading to the emergence of imaginary frequency phonon modes. This inherent dynamic instability of M₃NCl₃ materials signifies a propensity for structural degradation, rendering them susceptible to phase transitions or structural decomposition. Furthermore, their instability can preclude electron–phonon coupling and compromise carrier mobility calculations, thereby making it futile to predict transport properties. Due to these implications, the M₃NCl₃ materials are dropped for further analysis in this work. However, the low-symmetry tetragonal and rhombohedral configurations, as potential distortions of their tilted structures, necessitate a parallel comprehensive investigation.

3.4 Thermodynamic properties

Utilizing QHDM, this study has computed the thermodynamic properties under varying pressure and temperature conditions where pressure ranges from 0 to 10 GPa, and temperature ranges from 0 to 500 K. Fig. 6 illustrates the cell volumes, isothermal bulk moduli, and Debye temperatures for the given pressure range at 300 K and for the given temperature range at a static pressure of 0 GPa. These characteristics demonstrate that the investigated structures exhibit typical contraction under the influence of applied pressure while they undergo an expansion with increasing temperature. Under compressive forces, atomic constituents are compelled into closer proximity, effectively diminishing the overall crystallographic volume and augmenting its hardness. This improvement of hardness is evident by the observable elevation in both the bulk modulus and Debye temperatures. Conversely, temperature elevation induces dilation of the lattice parameters, culminating in an overall enlargement of the crystalline volume. This thermal expansion is attributable to the enhanced vibrational motion of atoms within the lattice structure, which subsequently leads to a reduction in the material's hardness. Declining trends in both bulk modulus and Debye temperature with temperature rise also substantiate this reduction in hardness. The relationship delineated by eqn (12) elucidates the intricate dependence of the Debye temperature, θ_D, on the bulk modulus. It translates that an elevation in the bulk modulus under applied pressure leads to a heightened θ_D. This increase signifies enhanced material stiffness and elevated melting points. Conversely, an increase in temperature induces a decrement in the bulk modulus, thereby resulting in a concomitant reduction of θ_D.

Fig. 6 Thermodynamic trends of unit cell volumes, bulk moduli, and Debye temperatures of M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds.

Fig. 7 illustrates the molar heat capacities (C_p and C_v) per formula unit. It shows that the heat capacities are elevating when moving from Ca₃NCl₃ to Ba₃AsCl₃. This trend is attributed to the lowering of the frequencies of the acoustic phonons in this series of materials, as shown in Fig. 5. Additionally, the elevated heat capacities correspond to the increased number of electrons near the Fermi level that enhances the electronic contribution to heat capacities. However, the electronic contribution is always smaller than the phonon contribution. Consequently, the studied Ba-based materials are comparatively more stable structures that can absorb, withstand, and store more thermal energy.

Fig. 7 Thermodynamic behavior of molar heat capacities (C_p, C_v) of M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds.

Under elevated pressure conditions, both molar heat capacities exhibit a decrement, thereby diminishing their thermal capacity. It indicates that the material's temperature will rise quickly with minimal thermal energy input under high-pressure regimes. This phenomenon occurs due to the augmented stiffness of the studied materials, which is reflected in their enhanced bulk moduli and elevated Debye temperatures, as illustrated in Fig. 6. Furthermore, the lattice vibrational frequencies mount under pressure. These high-frequency oscillations, primarily associated with optical phonons, contribute less to the overall heat capacity in comparison to the low-frequency acoustic phonons.

Upon increasing the temperature, the molar heat capacities increase rapidly up to 100 K, primarily due to the significant contribution of low-frequency acoustic phonons. Beyond this point, both heat capacities gradually rise to saturation values where the contribution of acoustic phonon modes to heat capacity becomes less, while optical phonons contribute. The Debye temperatures, which indicate the highest degree of vibrational motion in solids, as summarized in Table 1 are 403.97, 358.87, 322.80, 299.92, 272.10, 251.61, 240.23, 220.58, and 206.84 K traversing from Ca₃NCl₃ to Ba₃AsCl₃, respectively. Theoretically, the heat capacities of these materials will reach the Dulong–Petit limit at temperatures exceeding their Debye temperatures. The Dulong–Petit limit is equal to 3NR, which for these materials amount to 174.59 J mol K⁻¹. Considering the Debye temperatures, the barium-based materials will reach their Dulong–Petit limit at lower temperatures than the others. Specifically, Ba₃AsCl₃, with the lowest Debye temperature, will saturate at low temperature, while Ca₃NCl₃, having the highest Debye temperature, will reach saturation at a higher temperature, as Fig. 7 depicts.

Fig. 8 delineates the influence of pressure and temperature on thermal expansion coefficients (α) and entropies (S) of the studied materials, revealing no discontinuities during thermal fluctuations, indicative of their thermal stability. An inverse correlation between α and compressive forces is observed, as characterized by eqn (19). Increased phonon frequencies from compression enhance material hardness, thereby reducing α, which benefits in mitigating crack propagation at thermal interfaces. In contrast, elevated temperatures induce volumetric expansion, yielding softening behavior accompanied by reduced bulk moduli and Debye temperatures, which enhances α in line with heat capacity trends. The increase in α above 100 K is attributed to anharmonic lattice vibrations prevailing at elevated temperatures, consistent with the Grüneisen relation α ∝ γC_v/3B, where γ amplifies with bond softening. However, the high α in the range of 8 to 9 × 10⁻⁵ K⁻¹ at 300 K of the studied materials exhibits mismatch of nearly Δα ≈ 7.5 × 10⁻⁵ K⁻¹ with standard substrates that are generally used in silicon or thin-film solar cells (e.g., soda lime glass with α ≈ 9.1–9.5 × 10⁻⁶ K⁻¹,⁷⁹ engendering interfacial stresses, which could lead to delamination unless mitigated by negative thermal expansion interlayers or buffer layers with intermediate coefficients.⁸⁰

Fig. 8 Thermodynamic behavior of thermal expansion coefficients (α), and entropies (S) of M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds.

Moreover, the entropies of the investigated systems decrease under heightened pressure and increase with temperature. Elevated pressure constrains molecular motion, resulting in a more ordered configuration and reduced entropy due to a decrease in accessible microstates. Conversely, increased temperature augments crystal volume and kinetic energies, boosting the number of microstates and fostering disorder, thereby elevating entropy. The pronounced entropy at higher temperatures indicates phase stability against decomposition, impeding halide migration in solar cells during operational heating.

Additionally, the thermodynamic variables of the studied materials at ambient temperature are delineated in Table S3. These reveal lower values of B and θ_D, and higher values of C_p, C_v, and α for Ba₃PCl₃ and Ba₃AsCl₃ than others, which underscores their comparatively lower hardness. This investigation found no theoretical or experimental record of the thermodynamic properties of the studied inorganic halide perovskite derivatives. Our research bridges this gap and presents for the first time a comprehensive understanding of the thermodynamic variables of the investigated materials. These thermodynamic characteristics manifest a consistent and predictable behavior devoid of any anomalous trends or discontinuities, thereby highlighting the thermal stability of the studied material. Such stability is necessary in applications where materials are subjected to fluctuating thermal environments, ensuring reliability and performance under varying conditions.

3.5 Electronic properties

Following structural optimization and QHDM calculation of lattice volumes at 300 K, the electronic properties, including band structures, the partial density of states (PDOS), carrier effective masses, and carrier mobilities, are determined. Since the accuracy and efficacy of HSE06 calculations have been agreed upon by the scientific community,^81–84 this study utilized the HSE06 exchange-correlation functional to ensure the accuracy of results.

3.5.1 Band structures. The band structures are meticulously calculated along the high-symmetry k-path Γ–X–M–Γ–R–X within the irreducible Brillouin zone of the reciprocal lattice, as depicted in Fig. 9. In these plots, 0 eV represents the Fermi level. This analysis reveals the direct band gaps positioned at Γ, exhibiting values ranging from 1.711 eV for Ba₃AsCl₃ to 2.887 eV for Ca₃PCl₃, aligning with the visible spectrum. The band gaps for Ba₃PCl₃ and Ba₃AsCl₃ belong to the red wavelengths, whereas those for Sr₃PCl₃ and Sr₃AsCl₃ correspond to the blue wavelengths, and the band gaps for Ca₃PCl₃ and Ca₃AsCl₃ belong to the violet region of the spectrum. This range of band gaps is typically favorable for solar cells and various other optoelectronic applications.^85–87 Furthermore, these direct band gap materials are highly advantageous for solar absorbers because they enable the efficient generation of charge carriers – electrons and holes – through the absorption of photons. This process occurs without energy loss caused by photon–phonon interactions, which also enhances the rate of electron–hole pair generation, leading to improved optoelectronic performance of devices.^88,89 Various studies describe that the Fermi level near the valence band maximum (VBM) is the feature of p-type semiconductors, while contrarily, the Fermi level near the conduction band minimum indicates n-type semiconducting properties.^90–92 The Fermi levels of the studied semiconductors align near the VBM, indicating their p-type characteristics. It denotes that these materials can serve as a hole-transport layer in optoelectronic devices.

Fig. 9 Graphical representations of the band structures (HSE06) along the high symmetry k-path in IBZ of the examined M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds at 300 K.

Moreover, the band gaps are observed to diminish as the atomic size of the alkaline earth metal (M) progresses from Ca to Sr to Ba in these structures. This attenuation is attributed to the augmented orbital overlap resulting from the increased atomic dimensions. Such pronounced overlap consequently narrows the disparity in energy between the valence and conduction bands.⁹⁵

Table 4 exhibits a comparative analysis of the band gaps, demonstrating that our results are in close alignment with previously calculated findings using the HSE06 methodology. Given that these materials are novel and have only recently begun to be scrutinized through theoretical investigations, no precise experimental comparisons are currently available for validation. Typically, band gaps less than or about 1.7 eV are deemed optimal for the top cells in tandem photovoltaic systems.⁹⁶ In contrast, band gaps ranging from 2.5 to 2.9 eV present an excellent choice for optoelectronic applications, including light-emitting diodes (LEDs), as well as laser diodes and ultraviolet sensors. Specifically, for the development of solid-state lighting, semiconductor displays, and multi-color red-green-blue LEDs, semiconductors with band gaps in the 1.7 to 3.1 eV range are desirable.⁹⁷ Consequently, the studied Ba-containing variants emerge as particularly advantageous for solar cell applications, whereas Ca- and Sr-containing compounds offer promising prospects for a variety of other optoelectronic devices.

Table 4 Comparison of computed band gaps of the examined materials to the available data

Materials	Electronic band gap (eV)
	Our work (HSE06)	GGA-PBE	HSE06
Ca3PCl3	2.887	2.109 (ref. 57)	2.856 (ref. 57)
Ca3AsCl3	2.873	1.742 (ref. 58)
Sr3PCl3	2.604
Sr3AsCl3	2.570	1.70,^55 1.65 (ref. 93)	2.47 (ref. 94)
Ba3PCl3	1.719	0.997 (ref. 56)	1.64 (ref. 56)
Ba3AsCl3	1.711

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3.5.2 Partial density of states. The partial density of states (PDOS), illustrated in Fig. 10, elucidates the contribution of diverse atomic orbitals to the valence and conduction bands. It becomes evident that P-p and As-p states predominantly constitute the valence band edges, whereas Ca-d, Sr-d, and Ba-d states largely contribute to the conduction band edges in the respective examined compounds. These materials manifest covalent bonding interactions between M and A atoms, as depicted in Fig. 2, wherein A atoms engage in electronic charge sharing with M atoms. The PDOS analysis shows that the covalent bonding between M and A atoms originates from interactions between the M-d and A-p states. In their parent structures, the Ca-3d, Sr-4d, and Ba-5d states are hybridized with the P-3p and As-4p states, which facilitates charge sharing. While the Ca-3d states are unoccupied in their elemental form, they attain charges during the hybridization process with the P-3p and As-4p states in Ca₃PCl₃ and Ca₃AsCl₃, respectively. Laurien et al. reported a comparable density of states in CaP₃ and CaAs₃, demonstrating that Ca-d states predominantly facilitate the conduction band edge and P-3p and As-4p orbitals principally contribute to the valence band edge, respectively, while other states are relatively diminished. They reported that Ca-3d states hybridized with P-3p and As-4p orbitals, leading to covalent bonding between Ca, and P and As, respectively. In their analysis, Ca-3p and P-3p or As-4p exhibit anti-bonding interactions, while p–d interactions predominate for which the interaction mechanism is described in reference Fig. 4.⁹⁸ Analogously, in our investigation, Ca-3d orbitals exemplify interaction with P-3p or As-4p states in a covalent bonding context at the conduction band edge for both Ca₃PCl₃ and Ca₃AsCl₃, respectively. In contrast, Apurba et al. reported the absence of Ca-3d states in the conduction band of the Ca₃PCl₃ material,¹⁴ positing covalent interactions between P–Cl and Ca–Cl, whereas our findings accentuate the covalent bond between Ca–P alongside ionic bonding between Ca–Cl and the anti-bonding dynamics of P–Cl.

Fig. 10 Graphical representation of partial density of states (HSE06) of the examined M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds at 300 K.

Like the orbital interactions in the studied Ca-containing materials, the Sr- and Ba-containing compounds exhibit p–d bonding interactions and p–p anti-bonding interactions. In the bottom line, for M₃ACl₃ compounds, the hybridized A-p and M-d states interact at the conduction band edge, constituting a covalent bond between them while the interactions among A-p and M-p states, as well as A-p and Cl-p states, exhibit anti-bonding characteristics. Other states contribute negligibly to valence and conduction band edges. Moreover, the Ba-5d states are more delocalized due to the screening effect of electrons as compared to the Ca-3d and Sr-4d states, therefore the Ba-5d states hybridize more strongly with P-3p or As-4p orbitals. This not only results in diminished band gaps of studied Ba-based studied but the stronger p–d bonding interactions also stabilize these compounds.

3.5.3 Effective masses. The effective masses of carriers in semiconductors are crucial for analyzing their carrier mobilities, transport properties, and photocatalytic performance, which relies on the generation, separation, and dispersion of photo-induced carriers. Lower effective masses of carriers indicate enhanced photocatalytic activity and contrariwise.^12,99 This study computed the effective masses of electrons and holes of the studied P- and As-containing semiconductors by utilizing eqn (34) along the Γ–M, Γ–R, and Γ–X directions in the reciprocal space over the VBM and CBM, respectively.Here, E_k represents the energy function of the k vector, illustrating the dispersion relation. Table 5 describes the electron and hole effective masses that represent essentially the multiple of m_e, the intrinsic mass of the electron.

Table 5 Carrier effective masses and the relative ratios of M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) compounds along Γ-centered directions

Materials									D
	Γ–M	Γ–R	Γ–X	Avg.	Γ–M	Γ–R	Γ–X	Avg.	Γ–M	Γ–R	Γ–X	Avg.
Ca3PCl3	0.736	0.537	2.071	1.114	0.913	0.654	0.933	0.833	0.806	0.821	2.219	1.337
Ca3AsCl3	0.745	0.540	2.056	1.113	0.866	0.614	0.886	0.788	0.860	0.879	2.321	1.412
Sr3PCl3	0.690	0.516	1.863	1.023	1.014	0.564	1.038	0.872	0.680	0.915	1.794	1.173
Sr3AsCl3	0.699	0.515	1.910	1.041	0.936	0.523	0.958	0.805	0.747	0.985	1.994	1.293
Ba3PCl3	0.680	0.502	1.978	1.053	0.893	0.432	0.921	0.748	0.761	1.162	2.148	1.407
Ba3AsCl3	0.692	0.503	2.051	1.082	0.828	0.404	0.854	0.695	0.836	1.245	2.401	1.557

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It is noted that both types of carriers – electrons and holes – show different effective masses depending on the direction. Notably, all studied materials exhibit lower electron effective masses than holes along the Γ–M and Γ–R directions that are associated with relatively high edge conduction band dispersion and a flat valence band at Γ, respectively. However, the exception is witnessed for the studied Ba-based materials, which have lower hole masses than electrons along the Γ–R direction. Contrarily, along the Γ–X direction, electrons have a larger effective mass than holes in all studied materials, which is associated with a relatively more flat conduction band than the valence band in this direction. Specifically, the effective masses of both carriers along the Γ–M and Γ–R directions are lower compared to the Γ–X direction. It indicates the higher carrier mobilities along the diagonal directions but lower mobilities along b₂-direction as depicted by the high symmetry path in Fig. 1. This behavior is contingent upon the spatial distribution of the electrostatic potential within the material. The elevated effective masses of carriers along the Γ–X directions suggest a more pronounced potential along the b₂ direction compared to alternative pathways. Consequently, carriers exhibit preferential mobility along the remaining two directions rather than along the b₂ axis.⁹⁹ Liu et al. reported similar band structures and effective masses of Ba₃MX₃ (M = As, Sb, X = Cl, Br, I). They observed higher electron effective mass than holes along the Γ–X direction, which were due to the flat conduction band dispersion.¹⁰⁰ Our computed effective masses and their trends along different directions for the Ba₃AsCl₃ exactly match Liu et al. findings for the same material.

Moreover, both electron and hole effective masses along the Γ–M direction are moderate and reasonable in all materials. But remarkably, the effective masses along the Γ–R direction are small and highly desirable for thin-film solar cells and other optoelectronic applications. On the other hand, the average electron effective masses are greater than those of holes, indicating high hole mobilities and efficient hole transport in these semiconductors. This suggests that they can function effectively as hole transport layers in optoelectronic devices, particularly in solar applications. Hosen has also reported observing higher electron effective masses and lower hole effective masses for Sr₃AsCl₃ and Sr₃SbCl₃.⁵⁵ However, both types of carrier masses are below 1.5, indicating higher mobility, lower recombination rates, and improved optoelectronic performance of the studied materials.^101–103

The relative ratio of the effective masses of photo-induced electrons and holes is noteworthy for evaluating the rates of recombination of charge carriers. This ratio is also crucial for illustrating the generation process of charge carriers and analyzing the photocatalytic performance. The deviation of D from 1 attributes the difference in electron and hole mobilities, indicating higher rates of recombination of charge carriers and poor optoelectronic efficiencies.^99,104Table 5 shows that the D values along all directions deviate from 1, which infers the lower recombination rates of electron–hole pairs in the investigated compounds. Specifically, along the Γ–X direction, the ratio deviates more than in the other two directions. It indicates lower recombination rates in this direction, resulting in a higher generation of photo-induced carriers and a longer lifespan for these carriers.¹⁰⁵ Zhang et al. reported the D values for bismuth oxyhalides (BiOX, X = Cl, Br, I) to comprehend their photocatalytic properties. They reported the D values in the range from about 0.7 to 21 along various directions of k vectors. They found that the facets with D values 1 are ill-suited for photocatalytic activity.¹⁰⁴

In summary, the electronic characteristics of the investigated compounds reveal band gaps situated within the visible spectrum, underscoring their potential utility in vis-UV-driven optoelectronic applications. Furthermore, the effective masses and D ratios signify superior carrier mobilities and diminished recombination rates of electron–hole pairs, culminating in enhanced optoelectronic efficacy. Notably, the Ba₃PCl₃ and Ba₃AsCl₃ compounds exhibit greater promise for solar cell applications, attributable to their smaller band gaps, lower effective masses, and elevated D values.

3.5.4 Carrier mobilities. This investigation computed the carrier mobilities (μ_e, μ_h) in the temperature range of 200 to 500 K for various carrier concentrations (N). The electron–phonon coupling matrix elements |g_mnv(k, q)|² demand a dense k-point (for electronic states) and q-point (for phonon wave-vectors) grid. Generally, materials with complex Fermi surfaces, such as anisotropic or multi-band systems, require a dense grid to capture an increasing number of states. However, our investigated systems are not multi-band systems. Moreover, these systems exhibit phonon stability with the absence of soft modes and van Hove singularities or Kohn anomalies. Therefore, this study utilized a sufficiently dense grid of 100 × 100 × 100 points to ensure accuracy and precision. Moreover, the long-range LO–TO splitting polarization effect in electron–phonon interactions is accounted for by the Non-Analytic Correction term included in EPW. Fig. 12 shows the variation of xx-, yy-, and zz-components of the carrier mobility tensor under varying carrier concentrations at 300 K. The results indicate that mobilities remain relatively constant at low concentrations, ranging from 10¹³ to 10¹⁷ cm⁻³. However, beyond this threshold, a peak is observed for all studied materials except for the electron mobility of Ba₃PCl₃, which decreases above 10¹⁹ cm⁻³ but shows an increase after reaching the 10²¹ cm⁻³ density level.

Fig. 11 illustrates the fundamental scattering mechanisms that influence the relationship between mobilities and carrier concentrations, using arbitrary values for illustration. In regime I, which corresponds to low-to-moderate carrier concentrations up to 10¹⁹ cm⁻³, unscreened electron–phonon scattering dominates the transport phenomena. The long-range electron–phonon interactions remain unaffected by the rise in carrier concentration, and the scattering rate shows invariance, resulting in constant mobility. Beyond this threshold, in regime II, the screening effect becomes dominant, overwhelming other scattering mechanisms, until very high carrier concentrations. The influence of carrier screening substantially reduces polar optical phonon scattering, also known as Fröhlich interactions. At high concentrations, within regime III, mobility decreases because of band non-parabolicity,¹⁰⁶ which causes the Fermi level to drop deeper into the conduction band. This shift increases the effective masses of the carriers, leading to reduced mobility. Moreover, optical phonon scattering becomes more significant, and inelastic emission of LO phonons becomes prominent. Moreover, acoustic phonon scattering, a short-range interaction, is independent of carrier concentration, as depicted. Therefore, the mobility does not depend on low-energy acoustic phonon scattering.

Fig. 11 Schematic illustration of the scattering mechanism and mobility driven by carrier concentration.

Our results align well with previous studies. Jacoboni et al. documented similar peak mobility of GaAs at a concentration of 3 × 10¹⁸ cm⁻³ due to the screening effect, followed by a drop from non-parabolicity.¹⁰⁷ Wu et al. have reported theoretical results of carrier concentration-dependent electron–hole mobilities of CsBX₃, Cs₂BX₆, and for Cs₃B₂X₉ compounds. The authors witnessed that the mobilities for each compound remain nearly invariant within the carrier concentration range of 10¹⁶ to 10¹⁹ cm⁻³.¹⁰⁸ In a couple of experimental investigations, Tanase et al. observed analogous phenomena concerning the hole transport layers of field-effect transistors and light-emitting diodes.^109,110 Their findings reveal that mobilities remain static up to a threshold of 10²² cm⁻³, after which they exhibit an ascendant trend, adhering to a power law described by the expression (μ ∼ N^{T_exp,on/T−1}), as articulated in reference eqn (3).¹⁰⁹ Here, T_exp,on signifies a quantification of the breadth of the exponential density of states. Likewise, Li et al. analytically corroborated the presence of analogous effects concerning mobility versus carrier concentrations, specifically the escalation of mobility following surpassing a threshold concentration.

Furthermore, all three components of hole mobilities are higher than electron mobilities for Ca₃PCl₃, Sr₃PCl₃, and Ba₃AsCl₃ compounds, which is due to their lower hole effective masses along the Γ–X direction of the Brillouin zone, as Table 5 expresses. Zhang et al. also reported higher hole mobilities than electron mobilities of BSb semiconductors at 300 K and elucidated that higher mobilities are due to small effective masses, high sound speeds, and high optical phonon frequencies.¹¹¹ The zz-component of electron mobility is notably higher in four structures: Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, and Ba₃PCl₃, which is due to their lower electron effective masses along Γ–X. Fig. 12 also illustrates that Ba₃AsCl₃ demonstrates isotropic behavior across all tensor components, exhibiting uniform characteristics for both electron and hole mobilities. Moreover, as delineated in Table 6, Ca₃PCl₃ exhibits the congruent maximum mobilities for holes and electrons at a concentration of 10²⁰ cm³, and Ba₃PCl₃ holds the highest electron mobility within the low to intermediate concentration range.

Fig. 12 Carrier mobilities at 300 K versus carrier concentration.

Table 6 Maximum electron and hole mobilities at 300 K

Materials	N h (cm^−3)	μ ^maxh (cm^2 V^−1 s^−1)	N e (cm^−3)	μ ^maxe (cm^2 V^−1 s^−1)
Ca3PCl3	10^20	233.56	10^20	233.56
Ca3AsCl3	10^21	31.57	10^22	78.63
Sr3PCl3	10^20	94.86	10^20	104.36
Sr3AsCl3	10^20	52.59	10^20	128.15
Ba3PCl3	10^20	158.10	10^13 to 10^17	515.27
Ba3AsCl3	10^22	108.37	10^22	67.76

---

Fig. 13 elucidates the temperature dependence of carrier mobilities across various concentrations. Conventionally, carrier mobilities exhibit a decrement with an elevation in temperature,⁴⁷ a trend that our findings corroborate. This phenomenon arises due to the weakening of electron–phonon coupling at elevated temperatures, which diminishes the scattering rates and consequently leads to reduced mobilities. However, concurrently, the density of phonons augments according to the Bose–Einstein statistics, whereby the occupancy of phonons in a specified energy state proliferates with increasing temperature.¹¹² The proliferation of phonons amplifies phonon–phonon interactions, commonly referred to as Umklapp scattering, which becomes increasingly prominent at elevated temperatures and predominantly contributes to the attenuation of carrier mobility.¹¹³

Fig. 13 Carrier mobilities of the studied materials as a function of temperature for various carrier concentrations (N).

These findings elucidate that Ba₃PCl₃ possesses the most elevated carrier mobilities among all evaluated materials, showcasing a peak zz-component of electron mobility at an extraordinary 1170 cm² V⁻¹ s⁻¹ at a concentration of 10¹⁹ cm⁻³ at 200 K. Conversely, Ca₃AsCl₃ demonstrates the lowest mobilities at the same concentration level. These results underscore a pronounced transition from superior hole mobilities to enhanced electron mobilities correlated with variations in concentration levels. At 10¹⁹ cm⁻³, Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, and Ba₃AsCl₃ manifest superior hole mobilities, a trend that undergoes inversion at 10²² cm⁻³ for Ca₃PCl₃, at 10²⁰ cm⁻³ for both Ca₃AsCl₃ and Sr₃PCl₃, and at 10²¹ cm⁻³ for Ba₃AsCl₃. Notably, Ba₃PCl₃ consistently displays higher electron mobilities across all examined concentration levels. In probing MoS₂, Poncé et al. identified superior hole mobilities compared to electron mobilities for temperatures beneath approximately 250 K.¹¹⁴ In parallel, Xia et al. carried out an experimental inquiry into the electrical mobility of defect-free single crystals of MAPbI₃, reporting μ_e = 33 cm² V⁻¹ s⁻¹ and μ_h = 50 cm² V⁻¹ s⁻¹ at 300 K. They also performed mobility calculations employing the Boltzmann Transport Equation, which also indicated higher hole mobilities than electron mobilities for temperatures spanning 0 to 400 K.¹¹⁵

Moreover, our findings demonstrate that hole mobilities exhibit a rapid decline in comparison to electron mobilities as temperature increases. This phenomenon is attributable to the pronounced increase in the effective mass of holes with rising temperatures. A comparable observation is documented by Zhou et al., who conducted a study on the mobilities of GaP, InP, GaAs, InAs, GaSb, and InSb within a temperature range of 100 to 500 K. They discovered that electron mobilities decay by one order of magnitude across this temperature range, while hole mobilities decline at a fast rate.⁴⁸ Herz cataloged the mobilities of various films, single crystals, thin films, and poly-crystalline specimens of halide perovskites, ranging from a mere 0.2 cm² V⁻¹ s⁻¹ for FAPbI₃ films to an impressive 2320 cm² V⁻¹ s⁻¹ for MASnI₃ poly-crystalline materials, while the majority of measurements fall below 50 cm² V⁻¹ s⁻¹.¹¹⁶ This compilation of data reveals significant discrepancies in mobilities, presenting a wide spectrum of mobility values for a single material. For instance, reference Table 1 (ref. 116) illustrates that for single crystalline MAPbI₃, reported mobility values fluctuate between 2.5 and 164 cm² V⁻¹ s⁻¹ for hole mobility and from 115 to cm² V⁻¹ s⁻¹ for the electron–hole sum mobility, indicative of a staggering discrepancy of nearly two orders of magnitude. Given that identical single crystals sharing the same stoichiometry and crystallographic structure should yield consistent charge-carrier mobilities, and this extensive variation can be primarily attributed to experimental uncertainties.

Table 7 lists the averaged mobilities and their corresponding Fermi levels at 300 K for a concentration level of 10²⁰ cm⁻³. In an ab initio method study, Poncé et al. documented the electron and hole mobilities of CsPbI₃ iodides to be 68 and 76 cm² V⁻¹ s⁻¹, respectively, at 327 K, and an average 41 cm² V⁻¹ s⁻¹ for CsPbBr₃ at room temperature. In the case of MAPbI₃, the electron and hole mobility averages at room temperature fluctuate within a range of 30 to 80 cm² V⁻¹ s⁻¹ at ambient temperature, attributable to the fluctuations in the Pb–I bond structure.²² Similarly, Cucco et al. computed the hole mobility of 41.6 cm² V⁻¹ s⁻¹ for bulk CsPbBr₃ at room temperature.²³ This comparison illustrates reasonable agreement between our results and findings of ABX₃ type halide perovskites. Instead, Ba₃PCl₃ exhibits more distinguished electron and hole mobilities of 173.0 and 138.16 cm² V⁻¹ s⁻¹, respectively. In this view, our findings indicate the commendable performance of the evaluated semiconductors and provide a substantial reference point for future investigations.

Table 7 Average carrier mobilities at 300 K for N = 10²⁰ cm⁻³

Materials	ε ^eF (eV)	[small mu, Greek, macron] e (cm^2 V^−1 s^−1)	ε ^hF (eV)	[small mu, Greek, macron] h (cm^2 V^−1 s^−1)
Ca3PCl3	8.039	82.23	5.247	192.98
Ca3AsCl3	8.425	38.20	5.394	25.06
Sr3PCl3	7.692	80.96	5.212	82.11
Sr3AsCl3	7.954	80.78	5.327	43.43
Ba3PCl3	7.249	173.00	5.447	138.16
Ba3AsCl3	7.399	60.32	5.455	90.33
CsPbI3 (ref. 22)		68		76
CsPbBr3 (ref. 22)		41		41
CsPbBr3 (ref. 23)				41.6

---

3.6 Optical properties

This investigation entails the dielectric functions (ε^r_ω, εⁱ_ω), absorption (α_ω) and extinction coefficients (κ_ω), along with the reflective (R_ω) and refractive (n_ω) indices, and energy-loss functions of the materials under examination. This analysis employs hybrid functional (HSE06) energy calculations leveraging the QHDM-computed lattice parameters at 300 K. It is observed that in these calculations, the off-diagonal tensor components of the dielectric function diminish, and the diagonal elements are all equal due to the cubic structural symmetry of these materials. Therefore, all optical properties manifest isotropic functions of incident photon frequency (ω). The dielectric function plays a pivotal role in elucidating the optical characteristics of materials. It delineates the material's response to incident photons or applied electric fields. Typically, the real part of the dielectric function elucidates the polarizability that occurs due to incident photons or in the presence of an electric field.^117,118 Moreover, the peaks in the real part correlate with the material's resonance frequencies, indicating electronic polarization that occurs in the ultraviolet region or ionic polarization that is generally evident in the infrared domain. These peaks signify that materials exhibit pronounced responsiveness to incident photons whose frequencies coincide with the resonance frequencies. Conversely, the imaginary component encapsulates the optical transitions between the valence and conduction bands in reaction to incident photons. The peaks in the imaginary component signify a substantial optical transition density.

Fig. 14 delineates that the ε^r_ω functions are non-zero at null frequency or under static conditions, and this value is characterized as the dielectric constant (ε_o) of the material. The computed values for Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, Ba₃PCl₃, and Ba₃AsCl₃ are 2.68, 2.38, 2.61, 2.68, 2.35, and 2.41, respectively. This parameter is critically significant in the design and optimization of solar cells and optoelectronic devices. Generally, elevated dielectric constant values indicate diminished electron–hole pair recombination rates. Also, a higher ε_o effectively mitigates the coulombic interaction between electrons and holes.^119–121 Furthermore, the dielectric constant has an inverse relation with band gaps so a higher dielectric constant is indicative of a lower band gap.¹²² Consequently, superior dielectric constants are intrinsically linked to the enhanced optoelectronic performance of materials.¹²³ The values exhibited by the studied materials are commendable and promising for optoelectronic applications. The peak values of the ε^r_ω function are 4.20, 3.89, 3.96, 4.12, 3.21, and 3.35 at frequencies of 3.19, 3.18, 3.01, 2.80, 2.10, and 2.10 eV for Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, Ba₃PCl₃, and Ba₃AsCl₃, respectively, and they are within the visible spectrum demonstrating a robust interaction with visible light that leads to augmented polarization effects. Conversely, the shallow region displaying a negative ε^r_ω function manifests a pronounced light reflection, particularly in the Ca- and Sr-containing variants within the energy range of approximately 9 to 12 eV. In contrast, the Ba-based compounds exhibit no negative ε^r_ω function, resulting in stronger interactions with incident light and minimal reflection. These trends are further elucidated by the R_ω function, which reveals minimal reflection in Ba-based materials, while prominent reflection peaks correspond to the negative region of the ε^r_ω function.

Fig. 14 Dielectric functions (ε^r_ω, εⁱ_ω), absorption coefficients (α_ω), extinction coefficients (κ_ω), refractive indices (n_ω), and reflectivity spectra (R_ω) of the examined M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) perovskite-derivatives.

The spectra of εⁱ_ω, α_ω, and κ_ω elucidate the intrinsic mechanisms underlying optical transitions and the initiation of absorption in the visible spectrum. These demonstrate the onset of optical transitions and absorption in the visible range of light. The onset frequencies are concordant with the band gap energies investigated, aligning with the red, green, and blue segments of the visible spectrum for the studied Ca-, Sr-, and Ba-based materials, respectively. The transition peaks exhibited in the εⁱ_ω functions demonstrate pronounced maxima within the visible spectrum and extend into the proximal violet region. Conversely, the more elevated peaks of absorption lie in the ultraviolet domain. This contrast is attributable to the optical transitions encompassing both rates of photon absorption and emission processes; hence, a prominent peak does not necessarily imply elevated absorption. In the instances of Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, and Sr₃AsCl₃, the absorption peaks are situated near the violet edge and progress into the ultraviolet range. However, the studied Ba-containing compounds feature an initial albeit diminutive peak of εⁱ_ω within the visible domain, specifically around the green wavelength at approximately 2.5 eV, while the subsequent peaks reside in the ultraviolet region near 5 eV and 8.5 eV. Thus, Ba-based variants demonstrate relatively adequate absorption capabilities within the visible spectrum. Although each of the investigated materials exhibits absorption within the visible range, their peak responses predominantly linger in the ultraviolet realm. This characteristic renders them promising candidates for vis-UV driven optoelectronic applications.

The refractive index (n₀) is a critical design parameter in the realm of optoelectronic devices. The utilization of semiconductors in optoelectronic devices is profoundly influenced by the intrinsic characteristics and magnitude of both the refractive index and band gap of materials, which are instrumental in evaluating device performance.¹²⁴ Various theoretical models, including the Penn model,¹²⁵ delineate an inverse correlation between the refractive index and band gap energy, suggesting that a diminished band gap is indicative of a heightened refractive index.¹²⁶ The refractive index serves as a metric for the transparency or opacity of materials to incident light. The frequency dependence of the refractive index, n_ω, exhibits trends that closely mirror those of the real component of the dielectric functions, as they are interconnected through eqn (29). At zero frequency, the refractive indices recorded are 1.64, 1.54, 1.62, 1.64, 1.53, and 1.55 for Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, Ba₃PCl₃, and Ba₃AsCl₃, respectively. It underscores the notion that these perovskite-derivatives exhibit opacity under static conditions. Moreover, their opaqueness intensifies with increasing incident photon energies within the visible spectrum, while their refractive indices decrease in the ultraviolet region, particularly for higher frequencies from 10 to 12 eV, signifying substantial reflective properties in that domain.

The energy loss function, derived from the imaginary component of the inverse of the complex dielectric function (−Im[1/ε(ω)]), elucidates the dissipation of electron energy during electronic excitation phenomena within semiconductors. This construct is paramount for a meticulous quantitative assessment of inelastic scattering during electron transport.¹²⁷ The most elevated peak of this function corresponds to the plasma frequency.^128,129 The plasma frequencies observed for the materials under examination are 11.68, 10.5, 11.28, 11.21, 9.16, and 8.87 eV for Ca₃PCl₃, Ca₃AsCl₃, Sr₃PCl₃, Sr₃AsCl₃, Ba₃PCl₃, and Ba₃AsCl₃, respectively, as illustrated in Fig. 15. These frequencies indicate that incident photons at these energies will primarily be expended in generating inelastic scattering, thereby impeding the generation of the electron–hole pairs. Consequently, one can observe a conspicuous reduction of the absorption coefficients at these corresponding frequencies. In a theoretical study, Gosh et al. recorded peaks at 9.54, 9.44, and 9 eV for Sr₃AsF₃, Sr₃AsCl₃, and Sr₃AsBr₃, respectively.⁵⁹ Our findings show alignment with these observations. Beyond these plasma frequencies, the materials exhibit transparency, which can be checked in our results by the asymptotically approaching values of refractive index to 1.

Fig. 15 Energy loss functions of the examined M₃ACl₃ (M = Ca, Sr, Ba, A = P, As) perovskite-derivatives.

The band gap range for the top cell of tandem solar cells is documented as 1.5–2.2 eV.^130,131 The calculated band gaps of Ba₃PCl₃ and Ba₃AsCl₃ fall within this spectrum. With their ideal band gaps and notable optical properties, these compounds demonstrate considerable promise for application in solar cell technologies, particularly in the top cell of tandem heterojunction cells.¹³² Their impressive absorption coefficients across the visible spectrum further indicate their favorable optical performance.

Other examined materials demonstrate band gaps spanning 2.5 to 2.9 eV, signifying their potential application in photovoltaic cells for selective UV absorption. Furthermore, their elevated UV absorption coefficients facilitate effective photon capture within the crucial short-wavelength spectrum, thereby diminishing thermalization losses.^133,134 Conversely, the relatively low refractive indices, ranging from 1.53 to 1.64, reduce reflection losses at interfaces, thereby enhancing light coupling between sub-cells. The notable peaks in UV absorption suggest their effectiveness for UV emitters, particularly in UV-based LED applications such as sterilization and purification systems,¹³⁵ where excitonic absorption can improve the efficiency of radiative recombination. The dielectric constants in the 2.35 to 2.68 range indicate compatibility with common charge-transport layers (e.g., TiO₂, ZnO), reducing interfacial defects in heterostructures. Moreover, absorption coefficients within the visible spectrum facilitate the utilization of down-conversion phosphors in converted LED technologies.^136,137 The combination of low ε_o values in the 2.3–2.68 range and moderate refractive indices between 1.53 and 1.64 makes these materials suitable for anti-reflection coatings with reduced parasitic absorption, which is critical for photodetectors.

4 Conclusions

The scrutinized M₃ACl₃ (M = Ca, Sr, Ba, A = N, P, As) perovskite-derivatives manifest covalent interactions between M and A atoms alongside the ionic bonding of M with Cl atoms and display anti-bonding characteristics between A and Cl atoms, as delineated through their charge density distributions and density of states analyses, except Ca₃NCl₃ which exhibits all ionic bonds. Within these compounds, a hybridization of A-p and M-d electron states engenders bonding interactions that constitute the valence and conduction band edges. The electronic band gaps range from 1.7 to 2.9 eV, as determined through HSE06 calculations, facilitating direct optical transitions between valence and conduction bands at the high-symmetry Γ point. The delocalized Ba-5d electrons significantly enhance p–d interactions, culminating in reduced band gaps for Ba₃PCl₃ and Ba₃AsCl₃ as compared to Ca- and Sr-containing materials.

The mechanical properties underscore the inherent brittleness of the Ca- and Sr-containing materials as evidenced by their Pugh ratios of less than 1.75, negative Cauchy pressures, and Poisson ratios of less than 0.26, contrasting with the ductile behavior exhibited by Ba-based compounds. Furthermore, these structures exhibit both structural and phase stability, as evidenced by their negative formation energies and positive cohesive energies. Mechanical stability is affirmed through adherence to Born's stability criteria, while dynamic stability is underscored by the phonon dispersion spectra observed between 200 and 500 K. Contrarily, the N-based structures examined lack dynamic stability, attributable to the negative frequencies of their phonon modes across all temperatures.

Additionally, the average carrier mobilities of the analyzed materials span from 38.2 to 192.98 cm² V⁻¹ s⁻¹ at 300 K and a carrier concentration of 10²⁰ cm⁻³. All the materials investigated demonstrate reasonably good mobilities; however, Ba₃PCl₃ stands out with exceptional electron and hole mobilities of 173 and 138.16 cm² V⁻¹ s⁻¹, respectively. These impressive mobilities highlight its superior optoelectronic capabilities, making it a noteworthy candidate in the field.

The optical spectra reveal that the initiation of optical transitions and absorption occurs within the visible spectrum, corresponding to their respective band gap energies, thus facilitating the absorption of red, green, and blue light for Ca-, Sr-, and Ba-containing compounds, respectively. Specifically, the Ba-containing variant demonstrates promising absorption coefficients of 1.616 and 1.675 × 10⁵ cm⁻¹ at 2.6 eV. However, these spectra exhibit peaks in the ultraviolet range. Consequently, the optical characteristics accentuate the applicability of these materials in devices operating across the visible to UV spectrum. Specifically, Ba-based materials show exceptional promise for photovoltaics, particularly in tandem solar cells, while the others are suitable for a spectrum of optoelectronic applications, including RGB light-emitting diodes, thin-film transistors, diode lasers, semiconductor displays, and UV emitters and sensors.

Moreover, the elucidated optoelectronic properties of the examined materials furnish essential design parameters, encompassing band gaps, carrier mobilities, effective masses of charge carriers, dielectric constants, absorption spectra, and refractive indices, among others, for the computation of PCE of optimized solar cells constructed from these materials.

Conflicts of interest

There is no conflict of interest.

Data availability

The authors confirm that the data presented in the main article, as well as in the SI, which support the findings of this study, are available upon reasonable request.

Supplementary information is available. See DOI: https://doi.org/10.1039/d5ta05368d.

Acknowledgements

The work was sponsored by the National Natural Science Foundation of China (No. 12374017) and the Innovation Program for Quantum Science and Technology (2021ZD0303303). The computing time of the Supercomputing Center of the University of Science and Technology of China is acknowledged.

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