Pressure-induced multi-functional property analysis of lead-free tin based halide perovskites ASnCl₃ (A = Ga, In, Tl) for advanced optoelectronic applications

Abstract

This work examines the structural, electrical, and optical characteristics of lead-free tin-based halide perovskites, ASnCl₃ (A = Ga, In, and Tl), as environmentally friendly substitutes for lead-based perovskites in solar applications. Hydrostatic pressures ranging from 0 to 8 GPa cause all three compounds – GaSnCl₃, InSnCl₃, and TlSnCl₃ – to display a 3D-cubic perovskite structure. When pressure is applied, lattice parameters and unit cell volumes decrease as follows. For GaSnCl₃, they decrease from 5.554 Å and 171.346 A³ to 5.161 Å and 137.515 A³; for InSnCl₃, they are from 5.568 Å and 172.623 A³ to 5.178 Å and 138.891 A³; and for TlSnCl₃, they are from 5.573 Å and 173.146 A³ to 5.184 Å and 139.352 A³. This suggests that the structure is stable under compression. These compound's formation enthalpies attest to their thermodynamic stability over the investigated pressure range. All three Sn-based compounds exhibit a direct bandgap at 0 GPa, and as pressure increases the band gap decreases, which suggests a tunable electronic structure. TlSnCl₃ demonstrates a significant change from semiconductor to metallic behavior at higher pressures. The optical absorption spectra of the materials shift towards longer wavelengths (redshift) as pressure increases, enhancing the light absorption capabilities of these compounds. The compounds exhibit enhanced mechanical stability and ductility with increasing pressure, as indicated by their bulk, shear, and Young's modulus. Poisson's ratio values for these materials are in the range of 0.372 to 0.441 for GaSCl₃, 0.355 to 0.418 for InSnCl₃, and 0.349 to 0.413 for TlSnCl₃, which highlights their ductile nature. GaSnCl₃, InSnBr₃, and TlSnCl₃ exhibit diamagnetic behavior both under normal conditions and with increased pressure. Thermal conductivity and stability are enhanced with increased pressure, making these materials suitable for high-temperature applications. The ability to tune the properties of these compounds through pressure makes them promising candidates for next-generation optoelectronic devices, energy storage, and conversion systems.

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Introduction

The first perovskite-based research began in the early 19th century with the mineral perovskite (calcium titanium oxide, CaTiO₃) first being discovered in 1839 by German mineralogist Gustav Rose in the Ural Mountains of Russia.¹ Scientists are presently dedicating substantial resources to discover materials that have practical applications, such as solar energy converters, memory devices, optical devices, sensors, and high-temperature sensors.^2–4 First-principles calculations can elucidate the physical characteristics of a wide range of materials, and their findings are corroborated by practical investigations on these materials.^5,6 Halogen-based cubic perovskites with the conventional formula ABX₃ are attracting attention due to their remarkable characteristics, such as photodegradability, di-electricity, ferroelectricity, pyroelectricity, magnetism, and superconductivity.^7–14 From a fundamental and technical perspective, these perovskites are becoming increasingly intriguing. The formula ABX₃ represents a compound where X represents a halogen anion, while A and B represent monovalent and divalent cations, respectively.¹⁵ Almost all of the lead-based studies by different researchers such as Rehman et al.,¹⁶ Ghani et al.,¹⁷ Selmani et al.,¹⁸ Sidik et al.,¹⁹ Khan et al.²⁰ illustrate that Pb-based cubic perovskites have a larger band gap ranging from 1.801 eV to 2.337 eV, 1.35 eV to 2.29 eV, 1.87 eV to 0.98 eV, 1.24 eV to 2.27 eV, and 0.87 to 1.89 eV, respectively.^21,22 However, Sn-based perovskites exhibit a smaller band gap, which is crucial for absorbing more photons, generating a higher number of charge carriers, and enhancing efficiency in energy-conversion and light-emission devices. In contrast, lead-based perovskites can sometimes suffer from phase segregation and instability under light exposure.^23–34 However, the incorporation of cations like Ga, In, and Tl promotes structural and chemical stability within the perovskite lattice. Furthermore, the ability of ASnCl₃ (where, A = Ga, In, and Tl) to respond to pressure by tuning their structural, optical, and electronic properties makes these materials highly suitable for optoelectronic and energy-related applications. The choice of studying GaSnCl₃, InSnCl₃, and TlSnCl₃ is motivated by the need for non-toxic, environmentally friendly alternatives, making them a superior choice compared to lead-based perovskites.

Earlier research on doped Cs-based cubic halide perovskites was mainly focused on the investigation of their thermal and optical properties.^35,36 Theoretical analysis of perovskite materials also reveals the potential effect of their use in optoelectronics and photocatalytic applications because of their intermediate band gap transition from an indirect band gap to direct band gap.^37–42 Materials capable of mitigating the heating effects and reducing the efficiency of optical electronics can generate phonons via the indirect band gap.^43,44 In addition, the physical properties of perovskite compounds can be modified through processes such as doping,^45,46 chemical alteration,⁴⁷ and the application of hydrostatic pressure.^{21,22,48–60} The latest study has shown that Sn-based cubic halide perovskites, such as LiXCl₃ (X = Sn, Pb)¹⁷ and ZSnCl₃ (Z = Na/K),⁶¹ have significant potential for a wide range of optoelectronic applications. However, the impact of varying hydrostatic pressure has great potential as a viable option for solar cell applications, such as NaGeX₃.⁶² The existence of Pb poses significant risks to human health and the environment, making the need for lead-free cubic halide perovskites a high priority. Sn-based lead-free cubic halide perovskites, such as ASnX₃ (A = K, Rb; X = Cl, Br, I),⁶³ RbSnX₃ (X = Cl, Br),⁶⁴ KMCl₃ (M = Ge, Sn),⁶⁵ and CsSnCl₃,⁵⁴ have gained significant attention due to their environmentally friendly nature. These materials have been extensively studied under pressure-driven conditions. A thorough investigation was carried out using first-principles density functional theory (DFT) to analyze the impact of hydrostatic pressure on the electronic structure of CsSnX₃ (X = I, Br, Cl) compounds.⁶⁶ The objective of the study carried out by Imtiaz Ahamed Apon et al.⁶⁷ was to investigate the mechanical, magnetic, elastic, electrical, and optical characteristics of the halide-based perovskites FrSnX₃ (X = Cl, Br, and I) under hydrostatic pressures ranging from 0 to 6 GPa. In 2024, Hasan et al. undertook a study to achieve the greatest bandgaps,⁶⁸ which were measured at 1.14, 0.8, and 0.645 eV, respectively. Insufficient research has been conducted on the effects of hydrostatic pressure on InSnCl₃, GaSnCl₃, and TlSnCl₃, apart from the alkali materials. This also includes the results of the initial DFT calculations for various compounds, namely TlBX₃ (B = Ge, Sn; X = Cl, Br, I),⁶⁹ TlSnX₃ (X = Cl, Br, or I),⁷⁰ TlZX₃ (Z = Ge, Sn, Be, Sr; X = Cl, Br, I),⁷¹ InSnX₃ (X = Cl, Br, I),⁷² and InACl₃ (A = Ge, Sn, Pb).²⁰

The mystery from the deep ocean to outer space is still not fully uncovered, and the characteristics of the environment are also uncertain, particularly the hydrostatic pressure that intensifies with increasing ocean depth.⁷³ In tin-based perovskites (ASnCl₃, A = Ga, In, and Tl), the choice of A-site cations is crucial for stability and analyzing material properties.⁷⁴ Ga, In, and Tl provide suitable ionic radii for structural integrity, particularly under pressure. Their electronic configurations of Ga (3d¹⁰ 4s² 4p¹), In (4d¹⁰ 5s² 5p¹), and Tl (5d¹⁰ 6s² 6p¹) introduce fine-tuning the bandgap, enhancing light absorption and charge carrier mobility for optoelectronic applications. These Ga, In, and Tl cations also improve environmental stability by resisting oxidation,^75,76 whereas tin-based perovskites like CsSnCl₃ are prone to oxidation, which degrades their performance.⁷⁷ In-based perovskites show promising behavior in thermoelectrics, the Tl heavy atom effect aids spintronics,⁷⁸ and finally Ga offers a balance of stability and performance in optoelectronic devices.

In this research, hydrostatic pressures are applied to lead-free tin-based halide perovskites (ASnCl₃, where A = Ga, In, Tl). The results show that as pressure increases, the material's lattice parameters and unit cell volumes decrease, confirming their structural stability. This pressure application leads to a reduction in the band gap, allowing the tuning of the electronic structure, which is critical for optoelectronic applications. Additionally, optical absorption spectra shift towards longer wavelengths (redshift) with increasing pressure, enhancing the material's light absorption capabilities.

Computational method

This research is based on density functional theory (DFT) computations using the CASTEP packages,⁷⁹ which also provides numerous and significant benefits for various optoelectronic application research. This research also utilizes the Vanderbilt-type ultrasoft pseudopotential⁸⁰ along with the Perdew–Burke–Ernzerhof (PBE)⁸¹ program parameterized generalized gradient approximate (GGA) exchange–correlation functional for property analysis. The rSCAN meta-GGA and GGA-PBESol functionals were also used to compare the accuracy of the GGA-PBE functional in this study. These methods are utilized in the ab initio self-consistent field linear combination of atomic orbitals (SCF-LCAO) computer program CASTEP. The objective is to produce the geometrically optimized unit cell architecture of ASnCl₃ (A = Ga, In, and Tl) perovskites. Limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) was employed as the optimized algorithm by targeting energy convergence at 2.0 × 10⁻⁶ eV per atom displacement of 0.001 Å. The k-point mesh used has dimensions of 8 × 8 × 8. It allows for distinguishing between core states and valence levels. This distinction is facilitated by the plane-wave basis sets with a cut-off energy of 700 eV with a max iteration of 500. This strategy also uses 500 SCF cycles with a maximum force of 0.025 eV Å⁻¹. The pseudopotentials used were OTFG with a Koelling–Hamon relativistic treatment. The electronic configurations for the calculation were Ga [3d¹⁰ 4s² 4p¹], Sn [4d¹⁰ 5s² 5p²], and Cl [3s² 3p⁵] for GaSnCl₃; In [4d¹⁰ 5s² 5p¹], Sn [4d¹⁰ 5s² 5p²], and Cl [3s² 3p⁵] for InSnCl₃; and Tl [4f¹⁴ 5d¹⁰ 6s² 6p¹], Sn [4d¹⁰ 5s² 5p²], and Cl [3s² 3p⁵] for TlSnCl₃.

Results and discussion

Structural properties

The optimized external stress with crystal structure and the intriguing physical properties of the perovskite materials under research are examined here in the hydrostatic pressure range of 0 to 8 GPa and the calculation was conducted on the space group Pm3m (no. 221). InSnCl₃, GaSnCl₃, and TlSnCl₃ exhibit 3D-cubic crystal structures with the structural formula ABX₃. The A atom occupies the 1a Wyckoff site at coordinates (0, 0, 0), positioned at the corners of the cube. The B atom, tin (Sn), is situated at the 1b Wyckoff site at (0.5, 0.5, and 0.5) which is located at the center within the body of the cube. The X atoms, which can be Cl occupy the 3c Wyckoff sites at (0, 0.5, 0.5), positioned at the face-centered locations of the cube (Fig. 1).

Fig. 1 The crystal structure of ASnCl₃ (A = Ga, In, and Tl).

The analysis of the materials' structural properties provides a great comprehensive understanding of how pressure influences the physical and other similar properties of GaSnCl₃, InSnCl₃, and TlSnCl₃. This understanding is very important for their potential use in advanced materials and optoelectronic devices. Table 1 illustrates the lattice parameters, unit cell volumes and the formation enthalpy of the compounds GaSnCl₃, InSnCl₃, and TlSnCl₃.

Table 1 The lattice constants, unit cell volume & formation enthalpy of the compounds ASnCl₃ (A = Ga, In, and Tl) by varying different pressures

Compound	Calculated data	Pressure (GPa)
		0	2	3	5	6	8
GaSnCl3	a (Å)	5.554	5.419	5.365	5.271	5.232	5.161
	V (Å^3)	171.346	159.181	154.458	146.457	143.232	137.515
	ΔEf (eV per atom)	−3.338	−2.928	−2.733	−2.357	−2.177	−1.827
	Band gap (eV)	0.248	0	0	0	0	0
InSnCl3	a (Å)	5.568	5.434	5.378	5.287	5.250	5.178
	V (Å^3)	172.623	160.481	155.619	147.840	144.708	138.891
	ΔEf (eV per atom)	−3.303	−2.888	−2.691	−2.313	−2.130	−1.777
	Band gap (eV)	0.690	0.115	0	0	0	0
TlSnCl3	a (Å)	5.573	5.439	5.385	5.293	5.253	5.184
	V (Å^3)	173.146	160.909	156.189	148.303	144.989	139.352
	ΔEf (eV per atom)	−3.425	−3.010	−2.812	−2.432	−2.249	−1.894
	Band gap (eV)	0.878	0.300	0.044	0	0	0

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The compounds were subjected to hydrostatic pressures ranging from 0 to 8 GPa. The lattice parameter of the GaSnCl₃ reduces from 5.554 Å at ambient pressure to 5.161 Å at 8 GPa. Similarly, the lattice parameters reduced from 5.568 Å to 5.178 Å, and from 5.573 Å to 5.184 Å for the compounds InSnBr₃ and TlSnCl₃ respectively. The crystal lattice is compressed under high pressure, and this results in a compact arrangement of atoms.⁸² The substitution of the ‘A’ atom position by different sizes eventually affects the lattice constants and volumes by demonstrating the octahedral factor effect due to the periodic pattern of the size of the atoms. The structural properties are influenced by the size of the A site ion; compounds with larger A site ions exhibit higher lattice constants and atomic volumes.⁶⁷

The volume of the unit cell also decreases after applying hydrostatic pressure up to 8 GPa. The volume of GaSnCl₃ decreases from 171.346 ų to 137.515 ų, InSnBr₃ from 172.623 ų to 138.891 ų and TlSnCl₃ from 173.146 ų to 139.352 ų at 8 GPa applied pressure. The relationship between energy and volume is depicted using the Birch–Murnaghan equation of state, demonstrating that energy decreases with decreasing unit cell volume until reaching a minimum at the ground-state energy. Murnaghan's equation of state is as follows:

This behavior is consistent with findings by Das et al. (2023), who observed that increasing hydrostatic pressure compresses the crystal lattice of cubic halide perovskites.⁶⁵ Fig. 2 shows the EOS Birch–Murnaghan relationship between the volume and applied pressure of the compounds GaSnCl₃, InSnCl₃ and TlSnCl₃.

Fig. 2 The total energy as a function of unit cell volume of the cubic ASnCl₃ (A = Ga, In, Tl).

The formation enthalpy of these compounds was calculated using the formula:

Here E_tot. is the total energy of the unit cell, and E_s are the energies of the individual atoms (A = In, Ga, Tl). The calculated values are presented in Table 1. The table shows that each compound retains negative values at each pressure which indicates the thermal stability of the compounds. The formation energy of each compound becomes less negative with increasing pressure, indicating a reduction in stability. For GaSnCl₃ the formation enthalpy increases from −3.338 eV per atom at 0 GPa to −1.827 eV per atom at 8 GPa. Similarly, for InSnBr₃ and TlSnCl₃ the formation enthalpy rises from −3.303 eV per atom to −1.777 eV per atom, and from −3.425 eV per atom to −1.894 eV per atom respectively. As the pressure increases, the formation enthalpies have less negative values which depicts that the compounds are becoming less thermodynamically stable as pressure increases.

The tolerance factor (t) of these compounds is calculated by using the formula stated below:

Here, r_A, r_B, and r_X are the ionic radii of the A-site cation, B-site cation, and halide anion, respectively, which fall within the range of 0.89 to 1.1 for a stable cubic structure.⁸³ For InSnCl₃, GaSnCl₃, and TlSnCl₃, the tolerance factors are 0.74, 0.69, and 0.76 respectively. This indicates that the structures are slightly distorted at ambient and hydrostatic pressures also suitable for research. A comparison of lattice constants, unit cell volumes and other parameters of this study with other studies is shown in Table 2. It demonstrates that the calculated lattice constants and band gaps align closely with previous work. Larger halide atoms result in increased lattice constants and decreased band gaps. The results for GaSnCl₃, InSnCl₃ and TlSnCl₃ offer several advantages when compared to existing data from other studies. The band gap of GaSnCl₃ is 0.248 eV, for InSnCl₃ it is 0.690 eV and for TlSnCl₃ it is 0.878 eV which are calculated using GGA-PBE. The PBESol functional study reveals that CsSnCl₃ exhibits a band gap of 0.55 eV larger than this finding.⁸⁴ Other studies suggest that RbSnCl₃ also exhibits a larger band gap than our studies, which demonstrates that this study is more suitable for applications with lower band gap requirements and potentially enhances their utility in specific electronic and optoelectronic applications.

Table 2 Comparison between the lattice constants (Å), unit cell volume (ų) and band gap (eV) of our compounds ASnCl₃ (where, A = Ga, In, and Tl) and others

Ref.	Compound	Band gap (eV)	Lattice constants (Å)	Volume (Å^3)	Function
61	NaSnCl3	1.36	5.598	175.447	HSE06
	KSnCl3	1.47	5.631	178.574
	NaSnCl3	0.71	5.598	175.447	PBE
	KSnCl3	0.78	5.631	178.574
	NaSnCl3	1.04	5.598	175.447	RPBE
	KSnCl3	1.10	5.631	178.574
65	KSnCl3	0.91	5.58	173.80	GGA-PBE
84	CsSnCl3	0.55	5.510	—	PBESol
		1.4	5.750	—	rPBE
		2.91	5.563	—	GW
85	CsSnCl3	2.8	5.620	—	Experimental
	CsSnBr3	2.0	5.870	—
	CsSnI3	1.3 to 1.4	6.230	—
86	CsSnCl3	3.0	5.560	—	Experimental
	CsSnBr3	1.7 to 2.1 eV	5.804	—
54	CsSnCl3	0.943	5.61	176.56	GGA-PBE
87	RbSnCl3	0.88	5.596	—	GGA-PBE
	RbSnBr3	0.61	5.891	—
	KSnCl3	0.89	5.596	—
	KSnBr3	0.57	5.873	—
88	RbSnCl3	0.878	5.581	—	GGA-PBE
	RbSnBr3	0.556	5.853	—
	RbSnI3	0.383	6.249	—
89	FrSnCl3	1.050	5.64	—	GGA-PBE
	FrSnBr3	0.670	5.90	—
	FrSnI3	0.420	6.27	—
67	FrSnCl3	1.046	5.645	179.855	GGA-PBE
	FrSnBr3	0.675	5.906	206.038
	FrSnI3	0.485	6.299	249.927
20	TlGeCl3	0.910	5.262	145.658	GGA-PBE
	TlGeBr3	0.570	5.521	168.246
	TlGeI3	0.420	5.897	205.024
	TlSnCl3	0.880	5.573	173.073
	TlSnBr3	0.520	5.826	197.788
	TlSnI3	0.300	6.203	238.711
This work	GaSnCl3	0.248	5.554	171.346	GGA-PBE
	InSnCl3	0.690	5.568	172.623
	TlSnCl3	0.878	5.573	173.146
	GaSnCl3	0.000	5.435	160.564	GGA-PBESol
	InSnCl3	0.078	5.443	161.263
	TlSnCl3	0.281	5.451	161.984
	GaSnCl3	0.000	5.486	165.120	m-GGA-RSCAN
	InSnCl3	0.455	5.501	166.512
	TlSnCl3	0.647	5.499	166.292

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GaSnCl₃ and InSnCl₃ exhibit similar or more compact parameters of lattice constants and unit cell volume compared to other compounds. The lattice constants and volumes of InSnCl₃ are smaller than those of the experimentally synthesized compounds CsSnCl₃ and CsSnBr₃.⁸⁵ This illustrates that the compounds ASnCl₃ (A = Ga, In and Tl) are more stable and less prone to expansion under hydrostatic applied pressure conditions. The compound TlSnCl₃ exhibits a band gap of 0.880 eV⁶⁹ which is closer to this study with a value of 0.878 eV. Our results provide competitive values that align well with existing data with more precise measurements that can be beneficial for practical applications.

Electronic properties

Investigating the electronic properties is crucial for understanding the overall behavior of InSnCl₃, GaSnCl₃, and TlSnCl₃. The electronic band structures and density of states (DOS) were analyzed using the PBE functional within the GGA approximation, emphasizing the high symmetrical directions of the Brillouin zone. The band gap values and DOS for these compounds under hydrostatic pressure ranging from 0 to 8 GPa are presented in Fig. 3 and 4, respectively. The range for the valence band to conduction band is −30 eV to +30 eV and 0 eV represents the Fermi level (E_F).

Fig. 3 Band structure of ASnCl₃ (A = Ga, In, and Tl) under pressure.

Fig. 4 Total density of states (TDOS) of ASnCl₃ (A = Ga, In, and Tl) under pressure.

The band gap represents the energy difference between the conduction and valence bands. The observed materials exhibit a direct bandgap because of the high symmetry of the crystal structure. As pressure increases, the band gap decreases for all InSnCl₃, GaSnCl₃, and TlSnCl₃ compounds. Larger atoms at the A sites result in smaller band gaps. Under high pressures up to 8 GPa the compounds become conductive. Fig. 3 shows that GaSnI₃ and TlSnI₃ exhibit conductive properties at 2 GPa, and InSnCl₃ requires 5 GPa to achieve similar conductivity.

At 0 GPa, the DOS of GaSnCl₃ around the Fermi level exhibits a small bandgap (no states around 0 eV), indicating a semiconducting nature at ambient pressure. The states below the Fermi level are dense from −10 eV to around −3 eV, which indicates the valence band. There is a clear separation of the conduction band, which starts slightly above 1 eV. With increasing pressure of GaSnCl₃ from 2 GPa to 8 GPa, the bandgap narrows as indicated by the states moving closer to the Fermi level, particularly at 2 GPa and 3 GPa. At higher pressures (6 GPa and 8 GPa), the gap diminishes significantly, and states near the Fermi level begin to appear, indicating a transition towards a more metallic character. The increased states around 0 eV suggest that GaSnCl₃ may become conductive at these higher pressures.

At 0 GPa, InSnCl₃ also exhibits a noticeable bandgap around the Fermi level, although smaller than GaSnCl₃, suggesting it has semiconducting properties at ambient pressure. The valence band states are concentrated between −10 eV to about −2 eV, with a well-defined gap separating the conduction band. With increasing pressure of InSnCl₃, the bandgap narrows progressively. By 3 GPa and beyond, states around the Fermi level begin to appear, particularly at 6 GPa and 8 GPa, where the DOS plot shows significant electronic states near and at the Fermi level. This behavior implies a pressure-induced semiconducting-to-metallic transition, like GaSnCl₃, but the transition occurs more gradually in InSnCl₃.

At 0 GPa, TlSnCl₃ demonstrates a bandgap around the Fermi level, showing its semiconducting nature. The valence band is more widespread, extending from −10 eV to approximately −1.5 eV. The conduction band starts above the gap, at about 1.5 eV. As pressure increases, the bandgap narrows similarly to the other two compounds. However, the transition seems more rapid in TlSnCl₃, with states appearing near the Fermi level as early as 3 GPa. At 5 GPa and 6 GPa, the DOS around the Fermi level increases significantly, suggesting that TlSnCl₃ undergoes a faster transition to a metallic state compared to GaSnCl₃ and InSnCl₃. At 8 GPa, the Fermi level intersects several states, indicating strong metallic behavior.

The partial density of states (PDOS) of InSnCl₃, GaSnCl₃, and TlSnCl₃ at 0 to 8 GPa are illustrated in Fig. 5 which range from −16 eV to +16 eV. The significant peaks are observed below the Fermi level between −5 eV and 0 eV where the maximum peak is observed by the s-orbital. In the 5 eV energy range, the p-orbital and d-orbital show significant contributions. Above the Fermi level, fewer states are available, which can be observed from the smaller peaks primarily between 0 eV and 5 eV, which are typical of non-metallic or semiconductor materials states. The contribution of f-orbitals is quite low for visualization. These observations demonstrate that the energies below the Fermi level correspond to bonding states, while there is the presence of flat PDOS. This detailed orbital analysis helps in understanding the electronic properties and potential semiconductor and non-metallic material-based applications.

Fig. 5 The partial density of states (PDOS) of ASnCl₃ (A = Ga, In, and Tl) under pressure.

Charge density

The charge density for ASnCl₃ (A = Ga, In, and Tl) shows irregular charge density lines, which indicates the bonding between atoms. The effect of hydrostatic pressure up to 8 GPa on charge distribution was investigated for the compounds GaSnCl₃, InSnCl₃ and TlSnCl₃. Fig. 6 shows the change in the spherical charge density lines and elliptical lines around the atoms Ga, In and Tl with respect to Cl and Sn atoms along the (100) plane and (200) plane respectively. The electron density is shown with a scale on the right side where blue indicates lower electron density and red indicates higher electron density.

Fig. 6 Charge density of ASnCl₃ (A = Ga, In, and Tl) for pressures of 0 GPa and 8 GPa.

The compounds ASnCl₃ (A = Ga, In, and Tl) exhibit a covalent nature between “A” site atoms and the Cl atom which signifies the hardness nature of the compounds. The local field surrounding the A (Ga, In and Tl) atoms and Cl atoms represents the hybridization among the atomic states. The atoms with the highest electronegativity charges are attracted by most surrounding charges. The stronger covalent bond is observed at Sn–Cl bonding and A–Cl bonding depicts ionic bonds which are illustrated in Fig. 6. The surrounding area of the atoms also demonstrates that the covalent bonds between Sn–Cl are weaker compared to the ionic bonds between A–Cl (A = Ga, In, and Tl). After increasing pressure up to 8 GPa, the bonding between Ga/In/Tl and Cl atoms grows within more surrounding areas and the charge shows more elliptical contours around Sn–Cl atoms in the (200) plane.

The chemical bonding characteristics of RbSnX₃ (X = Cl, Br) compounds from analyzing charge density mapping were investigated by Rashid et al. (2022).⁸⁸ This article shows that the bonding between Rb and Cl/Br atoms belongs to strong ionic bonding compared to the covalent bonding between Sn and Cl/Br atoms at 0 GPa. On increasing pressure up to 10 GPa, the study revealed that the overlapping of charge density along the (100) plane and (200) plane increased compared to the 0 GPa mapping. This analysis reveals the electronic bond nature between the atoms in a compound and the impact of different atoms on the electronic properties and charge density mapping.

Optical properties

The optical properties of the ASnCl₃ (A = Ga, In, and Tl) compounds provide a deep understanding of the electronic structure and density of states. The optical properties of materials such as conductivity, dielectric function, absorption, refractive index, and loss function also help in investigating material behaviors and reveal the interactions with the electromagnetic radiation, and are included in this section.

Absorption. The absorption coefficient (α) rate of the ASnCl₃ compounds is presented in Fig. 7. The X-axis shows the function of photon energy in the range 0 to 40 eV and the Y-axis shows the absorption coefficient. After increasing the pressure up to 8 GPa, the peak height of the absorption coefficient is slightly increased and shifted towards the higher energy range for each compound. The absorption remains low at the lowest energy and has slight fluctuation influenced by the pressure. This also demonstrates how different hydrostatic pressures can modify the optical absorption properties of ASnCl₃ by impacting both the position and magnitude of its absorption peaks.

Fig. 7 Calculated pressure-induced spectra of absorption (α) for ASnCl₃ (A = Ga, In, and Tl).

Conductivity. The conductivities of the ASnCl₃ (here A = Ga, In and Tl) compounds include both real and imaginary parts. The photon energy range was 0 to 30 eV under applied pressure up to 8 GPa as depicted in Fig. 8. For GaSnCl₃ the real part of the conductivity shows a peak of 6 S m⁻¹ between the 5 eV to 10 eV energy range at 0 GPa applied pressure. As the pressure increases to 2 GPa (magenta line), the peaks shift slightly in position and magnitude. At 3 GPa (blue line), the peaks become more pronounced, especially around 5 eV and 10 eV. At 5 GPa (cyan line), the peaks shift to higher energies with a notable increase in conductivity around 10 eV. At 6 GPa (green line), the peak pattern remains similar with minor adjustments. Finally, at 8 GPa (red line), the peaks are sharper and more distinct, particularly around 10 eV and 20 eV, indicating higher conductivity within specific energy ranges. In the imaginary conductivity graph, the values range from −4 to 4 S m⁻¹. At 0 GPa, notable fluctuations with positive and negative peaks are observed around 10 eV and 20 eV. With 2 GPa, the overall pattern remains but with slight changes in magnitude. At 3 GPa, peaks around 10 eV become more pronounced, and a trough appears around 15 eV. At 5 GPa, peaks shift to higher energies with increased positive peaks around 10 eV. At 6 GPa, the fluctuations are consistent but with minor amplitude changes. At 8 GPa, peaks and troughs become more defined, especially around 10 eV and 20 eV, indicating stronger phase shifts at higher pressures.

Fig. 8 Calculated pressure-induced imaginary and real spectra of optical conductivity (σ) for ASnCl₃ (A = Ga, In, and Tl).

The graphs provide the imaginary and real components of the conductivity (σ) for InSnCl₃ as a function of photon energy (E) in electron volts (eV) under different hydrostatic pressures ranging from 0, 2, 3, 5, 6, and 8 GPa. The observation from the imaginary conductivity graph shows oscillatory behavior with both positive and negative values which also indicates the energy storage and dispersion characteristics for increasing amplitude in the higher pressure's oscillation. On the other hand, the real conductivity graph represents generally positive values with distinct peaks at specific photon energies revealing an actual conducting behavior and trend of increasing magnitude with higher pressure. Each graph reveals that increasing pressures can affect the materials' conductivity, where higher pressures enhance both reactive and resistive properties of InSnCl₃ by reflecting a complex interplay between the materials’ increased pressure and photon energies in determining the materials’ electrical properties.

For TlSnCl₃, the real and imaginary components of the conductivity (σ) as a function of photon energy (E) are shown in Fig. 8. The real part of conductivity increased with increasing pressure up to 8 GPa. The highest peak of the real part of conductivity is observed in the ultraviolet range of this curve. Increasing pressure shifts both the real and imaginary parts, but the overall patterns remain consistent. The real conductivity values are generally higher than the imaginary ones, indicating a more significant energy dissipation component in the material's response. Both graphs together demonstrate how TlSnCl₃ conductivity, in terms of energy loss (real part) and stored energy (imaginary part), varies with photon energy and applied pressure.

Dielectric function. Dielectric function is defined as follows:

ε(ω) = ε₁(ω) + iε₂(ω) (4)

In this case, ε₁(ω) and ε₂(ω) represent the real and imaginary dielectric function components, whereas η(ω) and k(ω) represent the refractive index components.

We analyzed optical characteristics such as absorption coefficient, calculated from ε₁(ω) and ε₂(ω). ASnCl₃ (A = Ga, In, and Tl) molecules are isotropic and homogenous, according to our research.

Imaginary part of the dielectric function. The analysis of TlSnCl₃, GaSnCl₃, and InSnCl₃ under pressures from 0 to 8 GPa reveals several significant changes and benefits. As pressure increases, the main peak intensity of the dielectric function generally rises for all compounds, indicating enhanced optical absorption and dielectric response. Peaks also broaden slightly, suggesting a wider range of electronic transitions, and shift towards higher energies, reflecting changes in the electronic band structure. Higher pressures smooth out finer spectral features, particularly noticeable in the 8–15 eV range, while the dielectric function increases at lower energies (0–5 eV), potentially indicating improved conductivity or polarizability in Fig. 9.

Fig. 9 Calculated pressure induced imaginary spectra of dielectric function (ε) for ASnCl₃ (A = Ga, In, and Tl).

These pressure-induced changes offer several benefits. The ability to tune the optical and electronic properties through pressure adjustments provides valuable control over material characteristics. Increased peak intensities could enhance light absorption, which is beneficial for photovoltaic applications. The shift in peaks suggests that pressure can be used for bandgap engineering, crucial for semiconductor applications. The improved low-energy response may lead to better electrical conductivity at higher pressures. Overall, understanding these pressure effects aids in designing materials with specific optical and electronic properties for a variety of applications.

Real part of the dielectric function. The graphs (Fig. 10) illustrate the real part of the dielectric function (ε₁) and conductivity (σ) as a function of photon energy for InSnCl₃, TlSnCl₃, and GaSnCl₃ under varying pressures from 0 GPa to 8 GPa. For the dielectric function (ε₁), each material displays a prominent peak around 4–6 eV, which gradually decreases as photon energy increases. As pressure rises, these peak positions shift slightly, and the overall value of ε₁ diminishes, particularly around the initial peak. This effect is more noticeable in TlSnCl₃, where the ε₁ values decrease significantly with pressure compared to InSnCl₃ and GaSnCl₃. Regarding conductivity (σ), multiple peaks appear between 0 eV and 30 eV, representing different optical transitions in Fig. 10. With increasing pressure, these peak positions also shift slightly, and their magnitudes vary, although not as drastically as in the dielectric function.

Fig. 10 Calculated pressure induced real spectra of dielectric function (ε) for ASnCl₃ (A = Ga, In, and Tl).

These pressure-induced changes offer several benefits. The ability to tune the optical properties through pressure adjustments is advantageous for applications like photo detectors and solar cells. Understanding how pressure affects these properties enhances insights into material stability under different conditions, essential for high-pressure environment applications. Furthermore, the shift in peaks suggests potential for engineering specific electronic band structures, making these materials suitable for various electronic and photonic applications, depending on the desired optical response. Overall, the pressure dependence of the dielectric function and conductivity highlights the potential for precise control of these materials' optical properties, enabling their use in advanced technological applications.

Such a negative dielectric constant indicates the presence of metallic or plasmonic behavior, which can significantly affect the material's optoelectronic properties, making it suitable for applications that leverage these unique characteristics.⁶⁹

Islam and Hossain et al. in 2020 observed that increasing pressure on CsSnCl₃ perovskites leads to a rise in the static peak of dielectric constant which indicates a reduction in charge carrier recombination rate and enhanced efficiency of optoelectronic devices. The imaginary part of the dielectric function increases significantly with pressure that shifts to lower energy regions, suggesting an increase in absorption and transparency in high energy regions above 26 eV.³⁸

Loss function. The provided images depict graphs of the loss function as a function of photon energy (E) for materials InSnCl₃, TlSnCl₃, and GaSnCl₃ under various pressures from 0 GPa to 8 GPa in Fig. 11. Key observations reveal that the peaks in the loss function shift toward higher photon energies as pressure increases across all materials, indicating enhanced resilience in their electronic structures to energy losses under pressure. The intensity of these peaks also varies with pressure, suggesting changes in the optical and electronic properties. This trend is consistent across all three materials, showing that pressure induces similar modifications in their electronic structures.

Fig. 11 Calculated pressure-induced spectra of loss function of ASnCl₃ (A = Ga, In, and Tl).

These changes offer several benefits. The ability to tune the electronic and optical properties through pressure makes these materials suitable for applications requiring specific photon energy responses, such as in optoelectronics or sensor technologies. Understanding how these materials behave under pressure provides valuable insight into developing new materials with desirable properties for high-pressure environments.

Refractive index. The graphs display the imaginary and real part of the refractive index as a function of photon energy for TlSnCl₃, GaSnCl₃, and InSnCl₃ under pressures ranging from 0 to 8 GPa, illustrating how the optical properties of these materials evolve under different pressures in Fig. 12. As pressure increases, the peaks in the refractive index exhibit slight shifts, indicating changes in the materials' electronic structures and optical absorption characteristics. Each material responds differently to pressure, showing unique variations in their refractive index across the photon energy spectrum.

Fig. 12 Calculated pressure induced spectra of the refractive index (n₁) of ASnCl₃ (A = Ga, In, and Tl).

These observations offer several benefits. Understanding the changes in refractive index with pressure aids in designing materials for specific optical applications, such as pressure sensors and adaptive optical devices. The ability to tune the optical properties by applying pressure allows for dynamic control over material behavior, making these materials suitable for devices requiring adjustable optical properties. Insights from these graphs can inform the development of optoelectronic devices, like lasers and photodetectors, which demand precise control over light–matter interactions under varying environmental conditions. Additionally, examining the impact of pressure on these materials provides information about their structural stability, essential for long-term use in high-pressure environments. Overall, these graphs offer valuable data for optimizing materials for advanced optical and electronic systems. Mitro et al. in 2022 highlighted that increasing pressure on RbGeX₃ perovskites (X = Cl, Br) results in a higher static refractive index, indicating greater light absorption potential. This suggests enhanced suitability for optoelectronic and photovoltaic device applications, generating considerable interest among researchers.⁹⁰ Applying pressure makes certain materials better at absorbing photons in a specific energy range. Photo detectors, solar cells, and optical sensors need this property to absorb light.

Reflectivity. One of the most important factors in the photovoltaic performance of perovskite materials is their reflectivity. The low reflectance values over the visible range are shown by the perovskite compounds ASnCl₃ in Fig. 13. For the Cl-based compound, the reflectance is around 11%. The ASnCl₃ compounds show their strongest peak within the 0–30 eV range. When site A is changed, there is also a change in reflectance.

Fig. 13 Calculated pressure induced spectra of the reflectivity (ρ) of ASnCl₃ (A = Ga, In, and Tl).

Haq et al. in 2021 illustrated that increasing pressure on KCaCl₃ perovskites results in a rise in reflectivity in the low-energy region which decreases photovoltaic efficiency. While in the high-energy zone, the material's reflectivity increases suggesting better results for solar heating reduction.⁹¹ Although it stays in the infrared region and grows somewhat for all compounds, the range of the maximum peak is slightly affected by induced pressure. Due to the low reflectivity values in the low-energy range, ASnCl₃ shows promising candidate for solar cell application, even under pressure. Additionally, these compounds are recommended for use as covering materials to reduce solar heating, owing to their higher reflectivity values in the high-energy range.⁹²

Mechanical properties

Table 3 presents the elastic stiffness constants matrix C_ij for the compounds GaSnCl₃, InSnBr₃, and TlSnCl₃ calculated under different pressures (0, 2, 3, 5, 6, and 8 GPa). These constants, C₁₁, C₁₂, and C₄₄, characterize the compounds' mechanical properties and how they change with pressure.

Table 3 The elastic stiffness constants, C_ij, of the ASnCl₃ (A = Ga, In, and Tl) compounds under different pressures

Ref.	Compound	Elastic constants	Pressure (GPa)
			0	2	3	5	6	8
This work	GaSnCl3	C11	51.251	69.263	77.823	95.920	103.910	119.945
		C12	8.441	11.189	12.416	15.035	16.196	18.849
		C44	1.882	0.351	−0.303	−1.371	−1.960	−3.180
		C12–C44	6.559	10.838	12.719	16.406	18.156	22.029
	InSnCl3	C11	51.962	70.081	79.289	96.167	104.146	120.172
		C12	9.345	11.942	13.534	16.243	17.733	20.372
		C44	3.103	2.240	1.780	0.850	0.397	−0.599
		C12–C44	6.242	9.702	11.754	15.393	17.336	20.971
	TlSnCl3	C11	51.204	69.761	78.687	95.862	104.107	119.799
		C12	9.017	12.137	13.660	16.552	17.976	20.832
		C44	3.308	2.420	2.054	1.254	0.832	−0.015
		C12–C44	5.763	9.717	11.606	15.298	17.144	20.847
70	TlSnCl3	C11	51.52	—	—	—	—	—
		C12	8.81	—	—	—	—	—
		C44	3.38	—	—	—	—	—
		C12–C44	5.43	—	—	—	—	—
	TlSnBr3	C11	45.44	—	—	—	—	—
		C12	6.78	—	—	—	—	—
		C44	2.89	—	—	—	—	—
		C12–C44	3.89	—	—	—	—	—
	TlSnI3	C11	37.99	—	—	—	—	—
		C12	4.39	—	—	—	—	—
		C44	2.66	—	—	—	—	—
		C12–C44	1.73	—	—	—	—	—

---

For GaSnCl₃, the longitudinal stiffness constant C₁₁ increases significantly from 51.25 at 0 GPa to 119.95 at 8 GPa which indicates enhanced resistance to deformation with increasing pressure. The transverse stiffness constant C₁₂ rose from 8.44 to 18.85 while the shear stiffness constant C₄₄ decreased from 1.88 to −3.18 depicting that the shear stability of the compound decreased after increasing pressure. Similarly, InSnCl₃ shows increments from 51.96 to 120.17 and from 9.35 to 20.37 for the C₁₁ and C₁₂ matrix across the same applied pressure range from 0 to 8 GPa. But C₄₄ decreased from 3.10 to normal pressure and reached −0.60 at 8 GPa applied pressure which indicates the reduced shear resistance with the increase of pressure. The matrix C₁₁ and C₁₂ also grew from 51.20 to 119.80 and from 9.02 to 20.83 respectively, for the compound TlSnCl₃. On the other hand, the value of C₄₄ decreases from 3.31 to −0.02 which indicates the diminishing shear stability of this compound.

Table 4 presents the calculated mechanical properties of three compounds (GaSnCl₃, InSnCl₃, and TlSnCl₃) under varying pressures, measured in gigapascals (GPa). The properties considered are the bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), and the ratio of bulk modulus to shear modulus (B/G), though the latter is not provided in the table and would need to be calculated separately. The bulk modulus (B) rises significantly from 22.711 GPa to 52.548 GPa with increasing pressure for GaSnCl₃ which indicates the material becomes less compressible. The shear modulus (G) increases a little, then slightly decreases and finally increases to 6.389 GPa. The Young's modulus (E) increases from 17.369 to 18.421 GPa after applying pressure up to 8 GPa. The Poisson's ratio (ν) increases from 0.372 to 0.441 which indicates that the material becomes more ductile under applied pressure.

Table 4 Calculated mechanical properties of ASnCl₃ (A = Ga, In, and Tl) under different pressures

Ref.	Compound	Calculated data	Pressure (GPa)
			0	2	3	5	6	8
This work	GaSnCl3	B (GPa)	22.711	30.547	34.218	41.997	45.434	52.548
		G (GPa)	6.327	6.203	6.195	6.507	6.499	6.389
		E (GPa)	17.369	17.429	17.527	18.563	18.609	18.421
		ν	0.372	0.404	0.414	0.426	0.431	0.441
		B/G	3.589	4.924	5.523	6.453	6.990	8.224
	InSnCl3	B (GPa)	23.550	31.321	35.452	42.884	46.537	53.639
		G (GPa)	7.549	8.261	8.542	8.946	9.090	9.296
		E (GPa)	20.462	22.782	23.721	25.093	25.603	26.366
		ν	0.355	0.378	0.388	0.402	0.408	0.418
		B/G	3.119	3.791	4.150	4.793	5.114	5.769
	TlSnCl3	B (GPa)	23.079	31.345	35.336	42.989	46.686	53.821
		G (GPa)	7.707	8.399	8.761	9.331	9.548	9.879
		E (GPa)	20.805	23.131	24.278	26.105	26.816	27.929
		ν	0.349	0.377	0.385	0.398	0.404	0.413
		B/G	2.994	3.731	4.033	4.606	4.889	5.447
70	TlSnCl3	B (GPa)	22.99	—	—	—	—	—
		G (GPa)	7.81	—	—	—	—	—
		E (GPa)	20.82	—	—	—	—	—
		ν	0.33	—	—	—	—	—
		B/G	2.94	—	—	—	—	—

---

For InSnCl₃, the bulk modulus (B) increases from 23.550 GPa to 53.639 GPa and the shear modulus (G) increases from 7.549 GPa to 9.296 GPa after applying pressure up to 8 GPa. The Young's modulus (E) also increases from 20.462 GPa to 26.366 GPa, which depicts the enhanced stiffness of the compound. The Poisson's ratio (ν) rises from 0.35519 to 0.418 demonstrating that the ductility of the compound is increased after applying pressure. TlSnCl₃ exhibits that the bulk modulus (B) has increased from 23.079 GPa to 53.821 GPa and the shear modulus (G) from 7.707 GPa to 9.879 GPa. Also, the Young's modulus (E) increases from 20.805 GPa to 27.929 GPa which demonstrates improved rigidity under pressure. The Poisson's ratio (ν) also increases from 0.349 to 0.413 showing that the material becomes more ductile as pressure is applied.

Anisotropic properties

The anisotropy of the elastic moduli of ASnCl₃ (A = Ga, of, and Tl) compounds was demonstrated by three-dimensional surface constructions (3D contour plots) and their corresponding two-dimensional projections (2D contour plots), generated using the ELATE program,⁶⁹ which used calculated values of the elastic constant C_ij. In an isotropic crystal, such representations are often shown as flawless spheres in three-dimensional space. For anisotropic materials like ASnCl₃ (where A = Ga, In, and Tl), the three-dimensional surfaces diverge from a spherical configuration, signifying varying mechanical characteristics across distinct crystallographic orientations.⁹³

The subsequent formulas delineate the equations for the shear anisotropic components,

Because of the cubic symmetry,The Zener isotropic factor A can be defined asFor an isotropic material, A = A₁ = A₂ = A₃ = 1 and the variation from unity corresponds to the anisotropy of a material.⁹⁴

Fig. 14–16 illustrate the 3D anisotropy contour plots for the Young's modulus (E, in GPa), shear modulus (G, in GPa), and Poisson's ratio (ν) of the ASnCl₃ (A = Ga, In, and Tl) composites. These plots visually highlight the amount of elastic anisotropy present in these materials. Anisotropy is evident in the non-spherical morphology of the three-dimensional surface, which exhibits directional dependence of the elastic properties within the crystal lattice.

Fig. 14 Young's modulus of (a) and (a′′) GaSnCl₃, (b) and (b′′) InSnCl₃, and (c) and (c′′) TlSnCl₃.

Fig. 15 Shear modulus of (a) and (a′′) GaSnCl₃, (b) and (b′′) InSnCl₃, and (c) and (c′′) TlSnCl₃.

Fig. 16 Poisson's ratio of (a) and (a′′) GaSnCl₃, (b) and (b′′) InSnCl₃, and (c) and (c′′) TlSnCl₃.

This analysis involves the calculation and plotting of minimum and maximum values of Young's modulus, shear modulus and Poisson's ratio for ASnCl₃ (A = Ga, In, and Tl) compounds in different crystallographic orientations. These values are presented in Table 4 to understand these parameters (v > G > E). The ranking of the three properties, Poisson's ratio (ν), shear modulus (G), and Young's modulus (E), reveals that Poisson's ratio exhibits the highest level of anisotropy, followed by shear modulus, and finally Young's modulus. The 3D illustrations are significant for understanding the variations in elastic properties of the compounds ASnCl₃ (A = Ga, In, and Tl) with distinct crystal lattice orientations.

Magnetic properties

A crystal lattice demonstrates diamagnetic properties when two spins are aligned in opposite directions, thus cancelling out each other's spin. Upon examining the graph of the band structure in Fig. 17, it is evident that the ‘α’ and ‘β’ lines overlap with each other. They lie on the same line and, even after rising hydrostatic pressures, they remain in the same overlaid locations. The electron densities of the ASnCl₃ (A = Ga, In, and Tl) combination, where almost all of them span a range from −10 eV to 10 eV, are shown in Fig. 18. The ‘α’ and ‘β’ lines correspond to the electron spins, with ‘α’ representing the positive spin and ‘β’ representing the negative spin. The ‘α’ line indicates an upward spin, while the ‘β’ line indicates a downward spin. These two lines exhibit perfect symmetry and are exact reflections of one another. According to the results, the compounds exhibit diamagnetic characteristics when subjected to a magnetic field.

Fig. 17 Calculated band structure of ASnCl₃ (A = Ga, In, and Tl) with spin-up and spin down channels calculated.

Fig. 18 Calculated DOS of ASnCl₃ (A = Ga, In, and Tl) with spin-up and down channels.

In the density of states graphs, the number of electrons that may occupy a given state is shown vertically, while their energy level is shown horizontally. Positive DOS refers to states occupied by electrons, whereas negative DOS refers to states filled by holes. As the pressure varies from 0 GPa to 8 GPa, the energy levels of the compounds ASnCl₃ (A = Ga, In, and Tl) show negligible fluctuation in the highest peak, but they stay constant across the α and β states of the material. Therefore, the compounds' diamagnetic behavior is unaffected by the application of pressure. No matter how much pressure is applied, the material's magnetic properties will not change.

Thermal properties

A material's thermal properties, which include heat conduction, expansion, and phase transitions, determine temperature changes. Transverse velocity (V_t) denotes the speed at which shear waves, or transverse waves, travel through the materials, moving perpendicular to the direction of propagation. Longitudinal velocity (V_l) denotes the speed at which compressional or longitudinal waves travel, moving in the same direction as the wave's propagation. The mean sound velocity (V_m) is the average velocity at which both transverse and longitudinal waves travel, moving in the velocity at which both transverse and longitudinal sound waves pass through the materials.⁹⁵ The parameters can be expressed in meters per second (m s⁻¹), with θ_D denoting the Debye temperature, which is defined as the highest temperature at which the specific heat capacity remains invariant.⁹⁵ The material's lowest limit of heat conduction is denoted by the minimum thermal conductivity (K_min), which is mainly determined by phonon transport.⁹⁶ The phonon thermal conductivity (K_ph) refers to the measure of how efficiently heat is transferred through a material via lattice vibrations.⁷⁹ These two parameters are expressed in watts per meter per kelvin (W m⁻¹ K⁻¹).

Table 5 provides a comprehensive comparison of the thermal properties of the compounds GaSnCl₃, InSnCl₃, and TlSnCl₃ for different applied pressures. For transverse sound velocity (V_t), InSnCl₃ generally has the highest values, peaking at 48 532.53 m s⁻¹ at 3 GPa, while GaSnCl₃ V_t decreases from 47 061.72 m s⁻¹ at 0 GPa to 42 366.7 m s⁻¹ at 8 GPa. TlSnCl₃ shows a steady increase in V_t from 43 258.14 m s⁻¹ to 43 939.8 m s⁻¹ across the pressure range. The longitudinal sound velocity (V_l) increases for all compounds with pressure. TlSnCl₃ exhibits the highest V_l at higher pressures, rising from 370 646.4 m s⁻¹ at 0 GPa to 585 500.4 m s⁻¹ at 8 GPa. In comparison, GaSnCl₃ has the lowest V_l, increasing from 298 303.7 m s⁻¹ to 466 234.6 m s⁻¹ over the same pressure range.

Table 5 Thermal properties of the compounds ASnCl₃ (A = Ga, In, and Tl) under different pressures

Compound	Pressure	This work
		ρ	Vt	Vl	Vm	θD	Kmin	Kph
GaSnCl3	0	2.85685	47 061.72	298 303.7	124 42.83	114.05484	0.0162	0.968
	2	3.07518	44 912.93	345 504.8	11 878.13	111.58439	0.0163	0.653
	3	3.16914	44 213.66	366 908.9	11 694.06	110.9629	0.0164	0.577
	5	3.34227	44 125.27	411 542.6	11 671.69	112.7318	0.0169	0.528
	6	3.41752	43 608.31	429 986.7	11 535.31	112.2445	0.0170	0.489
	8	3.55960	42 366.7	466 234.6	11 207.42	110.5448	0.0169	0.414
InSnCl3	0	3.26950	48 053.36	331 528.8	12 706.88	116.1872	0.0165	1.377
	2	3.51669	48 470.21	385 862	12 819.38	120.0984	0.0175	1.205
	3	3.62657	48 532.53	412 161.4	12 836.61	121.4995	0.0179	1.129
	5	3.81740	48 410.34	457 432.1	12 805.24	123.2923	0.0184	1.015
	6	3.90020	48 277.6	478 309	12 770.46	123.8401	0.0187	0.965
	8	4.06335	47 832.31	517 998.5	12 653.18	124.3904	0.0190	0.875
TlSnCl3	0	4.1186	43 258.14	370 646.4	11 441.64	104.5130	0.0148	1.328
	2	4.43180	43 533.65	434 218.8	11 515.63	107.7903	0.0157	1.120
	3	4.56571	43 806.5	463 329	11 588.11	109.5506	0.0161	1.077
	5	4.80816	44 054.04	516 258.9	11 654.03	112.0903	0.0167	1.000
	6	4.91842	44 060.03	540 592.6	11 655.77	112.95749	0.0170	0.963
	8	5.1170	43 939.8	585 500.4	11 624.2	114.14765	0.0174	0.895

---

Mean sound velocity (V_m) generally decreases with pressure for all compounds, with GaSnCl₃ showing the most significant decrease from 12 442.83 m s⁻¹ at 0 GPa to 11 207.42 m s⁻¹ at 8 GPa. In contrast, InSnCl₃ and TlSnCl₃ exhibit smaller decreases, with InSnCl₃ maintaining the highest V_m throughout. The Debye temperature (D) increases with pressure for InSnCl₃ and TlSnCl₃, with InSnCl₃ having the highest θ_D, rising from 116.19 K at 0 GPa to 124.39 K at 8 GPa. GaSnCl₃ D slightly decreases from 114.05 K to 110.54 K, indicating a different response to pressure compared to the other compounds. Minimum thermal conductivity (K_min) increases slightly for all compounds, with InSnCl₃ showing the highest increase from 0.01655 W m⁻¹ K⁻¹ at 0 GPa to 0.01905 W m⁻¹ K⁻¹ at 8 GPa. TlSnCl₃ and GaSnCl₃ also see increases, though at lower rates. Phonon thermal conductivity (K_ph) decreases for all compounds as pressure increases, with InSnCl₃ showing the highest initial value of 1.37781 W m⁻¹ K⁻¹ at 0 GPa, which decreases to 0.87508 W m⁻¹ K⁻¹ at 8 GPa. GaSnCl₃ shows a more significant decrease, from 0.96832 W m⁻¹ K⁻¹ to 0.41429 W m⁻¹ K⁻¹, indicating a more pronounced reduction in thermal conductivity under pressure.

Phonon analysis

The phonon dispersion curves for GaSnCl₃, InSnCl₃, and TlSnCl₃ provide insights into the vibrational dynamics of these lead-free tin-based halide perovskites, as observed along high-symmetry points in the Brillouin zone (X, R, M, G, and back to R). These curves represent the variation in phonon frequencies (in cm⁻¹) across the Brillouin zone, reflecting the stability and lattice dynamics of the materials. The absence of negative frequencies across all three compounds in Fig. 19 confirms their dynamical stability, indicating that the lattice structures are mechanically stable under ambient conditions. The dispersion plots feature three acoustic branches that start from the gamma (G) point at zero frequency, consistent with translational invariance, and numerous optical branches at higher frequencies, arising from the vibrations of individual atomic sublattices.

Fig. 19 Phonon dispersion curves of GaSnCl₃, InSnCl₃, and TlSnCl₃ perovskites.

For GaSnCl₃, the dispersion curves show a distinct separation between the acoustic and optical branches, with the highest optical modes reaching approximately 8 cm⁻¹. This separation suggests a clear distinction in the vibrational modes contributed by the lighter and heavier atoms in the structure. In comparison, the dispersion curves for InSnCl₃ and TlSnCl₃ exhibit similar frequency ranges but with slightly less pronounced separation between the acoustic and optical branches. The optical branches in InSnCl₃ and TlSnCl₃ are more distributed, which could be attributed to differences in atomic masses and bonding characteristics within these compounds. The transitions between high-symmetry points, such as X to R, R to M, and M to G, reveal detailed vibrational properties, including mode softening and coupling. These characteristics influence the thermal and mechanical properties of the materials, essential for their application in optoelectronics.

Population analysis

Table 6 provides a detailed analysis of the electronic structure and charge distribution for the compounds GaSnCl₃, InSnBr₃, and TlSnCl₃. This analysis is conducted under two different pressure conditions (0 and 8 GPa) and examines parameters such as charge spilling, Mulliken atomic populations, and charges determined by both Mulliken and Hirshfeld methods.

Table 6 Mulliken and Hirshfeld charge analysis of different atoms of ASnCl₃ (A = Ga, In, and Tl)

Compound	Pressure	Charge spilling (%)	Species	Mulliken atomic populations					Mulliken charge	Hirshfeld charge
				s	p	d	f	Total
GaSnCl3	0	0.18	Ga	1.98	0.30	10.00	0.00	12.28	0.72	0.51
			Sn	1.91	1.24	10.00	0.00	13.15	0.85	0.27
			Cl	1.96	5.56	0.00	0.00	7.52	−0.52	−0.26
	8	0.24	Ga	1.99	0.38	10.00	0.00	12.37	0.63	0.43
			Sn	1.78	1.42	10.00	0.00	13.20	0.80	0.16
			Cl	1.95	5.52	0.00	0.00	7.47	−0.47	−0.20
InSnIn3	0	0.18	In	1.97	0.31	10.00	0.00	12.28	0.72	0.47
			Sn	1.91	1.24	10.00	0.00	13.15	0.85	0.28
			Cl	1.96	5.56	0.00	0.00	7.52	−0.52	−0.25
	8	0.23	In	1.97	0.40	10.00	0.00	12.37	0.63	0.38
			Sn	1.80	1.42	10.00	0.00	13.21	0.79	0.17
			Cl	1.95	5.52	0.00	0.00	7.47	−0.47	−0.19
TlSnCl3	0	0.15	Tl	3.97	6.30	10.00	0.00	20.28	0.72	0.47
			Sn	1.91	1.24	10.00	0.00	13.15	0.85	0.29
			Cl	1.96	5.56	0.00	0.00	7.52	−0.52	−0.25
	8	0.19	Tl	3.96	6.39	10.00	0.00	20.35	0.65	0.39
			Sn	1.80	1.42	10.00	0.00	13.22	0.78	0.18
			Cl	1.95	5.52	0.00	0.00	7.48	−0.48	−0.19

---

The pressure applied in the analysis affects the electronic structure of the compounds, with values provided for 0 and 8 units. Charge spilling refers to the percentage of electron charge not accounted for in Mulliken population analysis, with lower values (below 0.25%) indicating more reliable results. The species column identifies the atom in the compound, while Mulliken atomic populations describe the distribution of electrons across various atomic orbitals (s, p, d, f). The total column sums these orbital populations, measuring each atom's total electron count. The Mulliken charge is the net charge on an atom where a positive value represents electron lost and negative value represents electron gain. Also, the Hirshfeld charge shows the perspective on atomic charge distribution.

At normal ambient pressure, GaSnCl₃ shows contribution of the d orbitals with both Ga and Sn atoms, with Mulliken charges of +0.72 and +0.85 respectively. These two orbitals showed partial positive contributions while the Cl atom exhibits −0.52 Mulliken charge as this atom gained electrons. When hydrostatic pressure was applied up to 8 GPa, the Mulliken charge for Ga and Sn atoms decreased to +0.63 and +0.80 respectively, which enhanced the electronic density stability. For InSnCl₃ perovskite, the In, Sn and Cl atoms show Mulliken charges of +0.72, +0.85 and −0.52 with Hirshfeld charges of +0.47, +0.28 and −0.25 respectively at 0 GPa pressure. With 8 GPa applied pressure, the charge distribution reduced the positive Mulliken charges for In and Sn, and increased them for Cl which indicates enhanced electronic stability. Similarly, Mulliken and Hirshfeld charges were reduced for Tl and Sn atoms but increased for Cl atoms in the TlSnCl₃ perovskite after 8 GPa applied pressure. The low charge density demonstrates the accuracy of the Mulliken charges. Furthermore, increasing pressure results in covalent interaction between atoms and increased stability of the compounds.

Conclusion

This research study offers a brief and comprehensive analysis of the characteristics and attributes of ASnCl₃ (A = Ga, In, Tl) perovskites, with Cl as the substituent. The investigation employs density functional theory (DFT) calculations utilizing the CASTEP algorithm. When the hydrostatic pressure reached 8 GPa, the ASnCl₃ compound (where, A can be Ga, In, or Tl) saw a decrease in its lattice properties and unit cell volume. The inherent stability of all compounds confirms their mechanical stability. During the mechanical examination, it was shown that ASnCl₃ (where A represents Ga, In, or Tl) exhibit greater stiffness and more pronounced elastic anisotropy compared to the other compounds. Analysis of the electronic band structure at 0 GPa indicates that the compound demonstrates semiconductor properties, with a direct band gap observed along specific crystallographic orientations. As the pressure decreases progressively, the band gap increases, resulting in the compound transitioning to a conductive state at various gigapascals (GPa). Through the analysis of increasing pressure based on the analysis of Pugh's ratio and Poisson's ratio, it is shown that ASnCl₃ compounds (where, A represents Ga, In, Tl) are inclined towards increased ductility. These materials have the potential to be useful in circumstances where a high level of ductility is necessary for practical application. Upon optical investigation, it was shown that ASnCl₃ compounds (where, A represents Ga, In, or Tl) exhibited higher absorptivity, photoconductivity, and loss function in the ultraviolet (UV) region compared to the other compounds. The Mulliken atomic population and Hirshfeld charge analyses demonstrated significant alterations in atomic populations under pressure, particularly for Sn and Cl atoms. This suggests the possibility of modifiable electrical properties. The compounds exhibited diamagnetic characteristics, providing additional evidence for the suitability of ASnCl₃ (A = Ga, In, Tl) as an environmentally friendly inorganic perovskite material for high-performance solar cells and diverse optoelectronic applications, especially in devices that are responsive to ultraviolet radiation.

Author contributions

Md Ratul Hasan contributed to methodology, data curation, and formal analysis, writing the original draft; Imtiaz Ahamed Apon contributed to formal analysis, conceptualization, formal analysis, writing the original draft and reviewing and editing of the manuscript; Md. Mafidul Islam, Asab Uzzaman Azad, and Md. Solayman contributed to data curation, reviewing and editing of the manuscript; Md. Salman Haque helped with supervision and formal analysis throughout the whole research project.

Compliance with ethical standards

All procedures were performed following ethical standards.

Data availability

The data that support the findings of this study are available on request from the corresponding author.

Conflicts of interest

The authors declare that they have no competing interests.

Acknowledgements

We declare that this research is not supported or funded by any organization or individual.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ma00039d

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