Investigation of structural, opto-electronic and thermoelectric properties of titanium based chloro-perovskites XTiCl₃ (X = Rb, Cs): a first-principles calculations

Abstract

Perovskites are a significant class of materials with diverse uses in modern technology. The structural, electronic, elastic, thermoelectric, and optical properties of RbTiCl₃ and CsTiCl₃ perovskites were estimated using the FP-LAPW method within the framework of density functional theory. The exchange–correlation energy of both analyzed systems was calculated using the Generalized Gradient Approximation (GGA) functional. The structures are optimized and lattice constants of 5.08 Å and 5.13 Å are found for XTiCl₃ (X = Rb, Cs), respectively. The structural analysis reveals that they have cubic symmetry. Their half metallic nature was proved by their metallic nature in one spin channel and semiconducting nature in the opposing spin channel. Densities of states are calculated to predict the interaction of orbitals of distinct atoms in the compounds. From the results of optical response, it is found that these compounds show high optical absorption in the visible region of light. Moreover, thermoelectric properties of the studied materials are calculated as a function of chemical potential at different temperatures using the theory of semi-classical Boltzmann transport within BoltzTrap code. The thermoelectric response shows that the investigated compounds as p-type can be beneficial in overcoming the global warming issue.

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Introduction

Researchers are looking for materials that may exhibit potential applications and fulfill the need of modern technology. Literature reveals that perovskites have been investigated frequently due to their remarkable physical and chemical features such as piezoelectricity, ferro-electricity, superconductivity, thermoelectricity, catalytic activity, ferromagnetism, half-metallic and magneto-resistance, and thermoelectric properties.^1–3

Perovskites can be described in terms of the general formula ABX₃, where A & B are cations and X is an anion. Halide perovskites refers to perovskites in which X belongs to halide anions. A literature study reveals that halide perovskites have been studied for a number of device applications, including solar-cells, topological insulators, superconductors and spintronics.^4,5

Spintronic applications are based on half-metallicity. It is an emerging phenomenon that refers to compounds exhibiting a metallic nature for one spin channel and insulating or semiconducting for the opposite spin channel.⁶ Half metallic (HM) compounds were found in 1980 and have been extensively researched for spin injection both theoretically and experimentally over the last decade.^7–12 This is a widespread phenomenon that has been observed in a variety of other materials, including Heusler alloys, transition metal oxides, and dilute magnetic semiconductors^13–19 The spin dependent unique property of HM allows for an opportunity to construct novel devices such as magnetic sensors and non-volatile magnetic random access memory that operate under the combined effect of spin-polarization and microelectronics.^20,21

Among the oxide perovskites, MgMnO₃ is a semiconductor at ambient temperature and has an insulating response at low temperatures.²² Thota and his research group further investigated MgMnO₃ experimentally using the sol–gel method and discovered that MgMnO₃ is stable in cubic phase at low temperatures.²³ The same research group further investigated Manganese oxide perovskite GaMnO₃ for magnetic and thermoelectric properties.²⁴

Recently, halide perovskites have been studied to find more suitable HM materials. In the literature we found that complex alkali metals fluorides are widely explored by scientists owing to their vast applications as fluorinating agents and as catalysts in organo-fluorochemical chemistry.^25,26

The thallium based chloroperovskite TlXCl₃ (X = Ca and Cd) compounds have been reported by Shams U. Zaman et al.²⁷ They suggested that these can be used as scintillating materials. Ternary chloride perovskites have a broad range of applications among halide perovskite compounds due to wide bandgap and maximum valued optical absorption coefficients and having bright photoluminescence of narrow band, low exciton binding energies, and long-range carrier diffusion. Due to these properties, such materials are widely examined theoretically as well as experimentally.^28–31

Moreover, fluoro-perovskites have considerable applications in photoluminescence, high temperature superconductors, colossal magneto-resistivity and piezoelectricity. Motivated by the literature review, we presented a detailed simulation-based analysis on the structural, electronics, thermoelectric, and optical properties of CsTiCl₃ and RbTiCl₃ halide perovskites in this paper. To carry out our calculations, we employed Tran–Blaha modified Becke–Johnson (Tb-mbj) approximation to find the exchange–correlation energy and full-potential linearized augmented plane wave (FP-LAPW) method to treat nuclei–electrons interaction within the framework of density functional theory (DFT). It is hoped that the present study would motivate the experimentalists to conduct experiments and further explore this field.

Computational details

To perform the present study, we have employed Wien2k code under the frame work of DFT.^15,32 Wien2K works on the principle of pull potential leading to FP-LAPW method.³³ Tb-mbj approximation is selected to estimate the exchange–correlation energy.³⁴ The cut-off energy of 500 eV was employed with taking titanium (Ti) as center of the first Brillouin zone of both the compounds.

The structural parameters and volume versus energy curve are calculated with the help of Birch–Murnaghan equation. Various elastic parameters are calculated from the elastic constants which are determined by using IR-Elastic package.³⁵ Optical properties are determined from the real and imaginary part of dielectric function ε(ω) = ε₁(ω) + iε₂(ω). The real part ε₁(ω) is calculated from the momentum matrix of occupied and unoccupied energy states, while the imaginary part is determined from the real part using Kramers–Kronig relations. Thermal properties are calculated using BoltzTrap code.³⁶ To achieve the best results, the cut-off value of 1 max was set to 10, RKmax was set to 2.5, and 2000 K-points were chosen.

Structural properties

In this subsection, we investigated the structural properties of chloro-perovskites XTiCl₃ (X = Cs or Rb). Both compounds have an ideal cubic structure with the same unit formula having space group Pm3̄m (221). In its unit cell X = Cs or Rb atom lies at the corner of the unit cell with positions (0,0,0) and titanium atom occupies the center of the unit cell with the position (0.5,0.5,0.5) to which six chlorine atoms are attached at center positions (0,0.5,0.5). Fig. 1 depicts the crystallographic unit cell of these compounds. For the verification of structural stability of these perovskite, we use Goldschmidt tolerance factor having formula;³⁷

Fig. 1 The crystallographic unit cell of RbTiCl₃.

In the above formula, r_A, r_B and r_X represents the ionic radii of A, B and X atoms respectively. The literature shows that if 0.7 < τ < 1, then material will possess stable structure and the structure of material will be more stable if the tolerance factor is close to 1. The tolerance factor values for CsTiCl₃ and RbTiCl₃ are found to be 0.98 and 0.97 respectively, indicating that both crystal structures are stable, however CsTiCl₃ has a more stable crystal structure than RbTiCl_3. Moreover, the octahedral factor is also evaluated for both compounds and presented in Table 1, which indicates the stability of these compound.³⁸ The stability of this structure is also checked by plotting the changes in total energy as a function of volume has been shown in Fig. 2. The goodness of fit (R²) is also evaluated and presented in Table 1.

Table 1 The computed structural parameters for RbTiCl₃ and CsTiCl₃

Structural parameters	RbTiCl3	CsTiCl3
Equilibrium lattice constant (ao)	5.08	5.13
Bulk modulus (B0)	35.49	34.48
Ground state volume (V0)	887.21	918.56
Bulk modulus first derivative B′	5.04	4.66
Energy at ground state (E0)	−10440.18	−20057.79
Tolerance factor	0.97	0.98
Octahedral factor	0.83	0.88
Goodness of fit (R^2)	0.952	0.961

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Fig. 2 The optimized volume curve for RbTiCl₃ and CsTiCl₃ perovskites.

Electronic properties

To study the semiconducting nature of materials we calculate their electronic properties. The electronic band structure and density of states are calculated for both compounds in spin up and spin down directions using the generalized gradient approximation of Tran–Blaha modified Becke–Johnson (Tb-mbj). The electronic band structures for the investigated systems are depicted in Fig. 3. The band structure analysis shows that both understudy perovskites exhibited indirect band gap semiconductors in spin down channel. In case of spin dn channel, their estimated energy band gap values are found 5.2 eV and 5.5 eV for RbTiCl₃ and CsTiCl₃, respectively. However, in case of spin up direction, both the compounds are found metallic as their valence and conduction bands overlap; hence the materials are half metallic compounds.³⁹ To further investigate the contribution of each valence orbital of every constituent atom of the compound to the valence and conduction bands, we calculate their total density of states (TDOS) and projected density of states (PDOS). The density of states also conform the half metallic behavior of the studied compounds. The plots of calculated density of states for RbTiCl₃ and CsTiCl₃ are presented Fig. 4 respectively. The PDOS graphs show that in the case of RbTiCl3, the d-states of Ti and Cl contribute significantly to the valence band while only the Ti-d state contributes significantly to the conduction band. However, in case of CsTiCl_3, Ti-d and Cl-p sates principally form the valence band while Ti-d and Cs-d states play a major rule in the conduction band near the Fermi level. Further, it can also be noticed from the study of PDOS that hybridization between the d-states of Ti and Cl atoms and Ti-d and Cl-p sates take place in the valence bands of XTiCl₃ (X = Rb, Cs), respectively. However, in case of CsTiCl₃, there is occurrence of hybridization between the d sates of Ti and Cs atoms in conduction, while no overlapping of orbitals are observed in conduction band of RbTiCl₃.

Fig. 3 The obtained band structure for XTiCl₃ (X = Rb, Cs) in both spin up and spin down.

Fig. 4 The TDOS and PDOS of (a) CsTiCl₃ and (b) RbTiCl₃.

Optical properties

The well known dielectric function ε(ω) = ε₁(ω) + iε₂(ω) describes the optical response of the medium at all photon energies. The imaginary part explains the compound's absorptive behavior and is directly related to the electronic band structure of a material. The Imaginary part of the dielectric function ε₂(ω) is represented by:⁴⁰Here E_i indicates the energy of electron in the i-th state with crystal wave vector k, i is the initial states and j is the final states, f_i represent the ith state Fermi distribution function and M is the dipole matrix. The real part ε₁(ω) of the dielectric function can be obtained through Kramers–Kronig relation from the imaginary part. At 0.93 eV and 1 eV energies, the static dielectric constant of compound XTiCl₃ (X = Cs or Rb) is 14.2 and 14.6, suggesting that RbTiCl₃ has a larger real part. This effect is caused by the inverse proportionality between the energy band gap and the real part. Fig. 5(a) indicates that the spectra of CsTiCl₃ declines and subsequently rises to a maximum value of 6.4 at 4 eV, while RbTiCl₃ has a maximum value of 5.98 at 5 eV. The negative value of real part ε₁(ω) for both compounds indicates the optical conductivity at this energy.⁴¹ Fig. 5(b) shows that the threshold value of imagery part for CsTiCl₃ is 11.19 and for RbTiCl₃ is 11.35, then rises to a maximum value of 4.3 at 7.5 eV energy for CsTiCl₃ and 3.8 at 8 eV energy for RbTiCl₃. The system indicates a fluctuating reduction that ends at saturation values. The fluctuating nature of imaginary spectra is caused by band to band transition in the compounds.

Fig. 5 The curve of obtained (a) real part and (b) imaginary part of dielectric function for XTiCl₃ (X = Cs, Rb).

Fig. 6(a) depicts the relationship between refractive index and incident photon energy. The figure clearly shows that the value of n(0) for CsTiCl₃ is 3.94 and for RbTiCl₃ is 4.01. Figure indicates that the refractive index increases and eventually reaches a maximum value of 2.5 for CsTiCl₃ at 4 eV and 2.4 for RbTiCl₃ at 5 eV energy, after which the refractive index drops to 0.4 and 0.3 for CsTiCl₃ and RbTiCl₃, respectively. Both compounds have refractive index values larger than one, indicating that incident photons lose energy when they interact with the electrons of the compound. Fig. 6(b) shows the absorption coefficient of CsTiCl₃ and RbTiCl₃ and it is clear that CsTiCl₃ has a threshold absorption 0.001 and RbTiCl₃ has threshold absorption 0.0011 and a maximum absorption 250 at 35 eV energy in the visible region and 249.5 at 35 eV in the visible region energy for CsTiCl₃ and RbTiCl₃, respectively. Fig. 6(c) depicts the optical conductivity of CsTiCl₃ and RbTiCl₃ as a function of incident photon energy. From the figure we can see that CsTiCl₃ has high optical conductivity of 7590 in the visible region and RbTiCl₃ has a high optical conductivity of 6500 in the visible region.

Fig. 6 The curve of (a) refractive index, (b) absorption coefficient, (c) reflectivity, and (d) conductivity of XTiCl₃ (X = Cs, Rb).

Fig. 6(d) shows the reflectivity curve of both compounds. From the figure it is clear that both compounds have a transmitting nature in the energy range of 0 to 11 eV and an excellent reflectivity in the energy range of 11 to 22 eV. From figure it is also clear that CsTiCl₃ has maximum reflectivity of 0.44 at 15 eV energy and RbTiCl₃ has maximum reflectivity of 0.26 at 22 eV energy, which justifies that CsTiCl₃ is better reflector than RbTiCl₃.

Thermoelectric properties

Currently, the world is facing the issue of energy crises. It has been estimated that more than half of the energy from primary energy sources is lost as waste heat during their use. This has badly affected the GDP of the world.^42–44 Scientists across the world have focused their research to discover smart thermoelectric materials. Thermoelectricity is a phenomenon in which the waste heat can be converted into useable clean and green energy and vice versa and materials with these properties are referred to as thermoelectric materials.^45,46 The thermoelectric performance of a material can be estimated in terms of figure of merit, abbreviated as ZT. Mathematically it can be written as;⁴⁷ ZT = S²σT/κ_e where σ, S, T and κ_e represent the electrical conductivity, Seebeck coefficient, absolute temperature and thermal conductivity at which the thermoelectric properties are measured, respectively. Aiming that, we have for the first time calculated the electrical conductivity (σ), Seebeck coefficient (S), power factor (PF) and thermal conductivity (κ_e) for XTiCl₃ (X= Cs, Rb) compounds for different temperatures as a function of chemical potential (μ) and are presented in Fig. 7 and 8.

Fig. 7 (a) The Seebeck coefficient (S), (b) the power factor (S²σ), (c) the electronic thermal conductivity (κ_e) and (d) the electrical conductivity (σ), versus chemical potential (μ) at different temperatures for CsTiCl₃.

Fig. 8 (a) The Seebeck coefficient (S), (b) the power factor (S²σ), (c) the electronic thermal conductivity (κ_e) and (d) the electrical conductivity (σ), versus chemical potential (μ) at different temperatures for RbTiCl₃.

Seebeck coefficient is measured in terms of voltage induced at the cost of temperature gradient across a conductor. The sign of S refers to the type of dominant charge carriers. Positive S means p-type (holes as dominant charge carriers), while negative S refers to n-type semiconductor (electrons as main charge carriers). The room temperature S values for XTiCl₃ (X= Cs, Rb) as a function of μ for different temperatures are plotted and shown in Fig. 7(a) and 8(a).

The figures indicate that both compounds have significant Seebeck coefficient values both in the positive and negative range which shows that both type of charge carriers are present in the systems. It is also worth noting that S exhibits its maximum values at small chemical potential values. Both the materials exhibit significant values of S ∼ 160 μK V⁻¹ at 300 K. However, when temperature in increased from 300 K to 900 K no significant change have been noticed in S for both the investigated materials.

Electrical conductivity (σ) is an important thermoelectric parameter that describes how the movement of charge carriers from a higher temperature region to a lower temperature region causes an electric current.

For good thermoelectric performance, a compound should exhibit higher value of electrical conductivity. The plots of electrical conductivity as a function chemical potentials at different temperatures are displayed in Fig. 7(b) and 8(b).

It can be noticed from the plots that the value of σ is higher at positive chemical potential than in the negative μ region. This result implicates that electron doping in these XTiCl₃ (X= Cs, Rb) materials will be more beneficial for thermoelectric applications than hole doping.

At 300 K, the maximum value of σ for CsTiCl₃ and RbTiCl₃ is found to be ∼6.5 × 10²⁰ Ωms⁻¹ (4 eV) and ∼7.5 × 10²⁰ Ωms⁻¹ (4.8 eV), respectively. However, when temperature is increased from 300 K to 900 K, σ are decreased and attain their lowest values (∼5.5 × 10²⁰ Ωms⁻¹ and ∼5.5 × 10²⁰ Ωms⁻¹) for CsTiCl₃ and RbTiCl₃, respectively. The slight decrease in the electrical conductivity with temperature rising may be attributed to the increasing charge carrier concentration plus collision and scattering phenomenon as well.

The phenomenon of heat conduction in a material by moment of free electrons and lattice vibration is termed as thermal conductivity κ_e. Fig. 7(c) & 8(c) illustrate the variation of electronic thermal conductivity versus chemical potential at 300 K, 600 K and 900 K.

It can be noticed from the figures that both the compounds exhibited higher thermal conductivity for positive chemical potential and their thermal conductivity is found to be ∼22 × 10¹⁴ W Ωms⁻¹ (4 eV) at 300 K. However, in contrast to electrical conductivity, electronic thermal conductivity is increased drastically when temperature goes from 300 K to 900 K and achieve their maximum peaks (∼5.5 × 10¹⁴ W Ωms⁻¹ for CsTiCl₃∧60.0 × 10¹⁴ W Ωms⁻¹ for RbTiCl₃). It can also be noticed from the plots that both electrical and thermal conductivities show similar profile and their ratio bears a constant value, hence the results follow the Wiedemann–Franz law: k = σLT.

The efficiency of a material can be estimated in term of power factor (PF). PF can be determined from Seebeck coefficient and electrical conductivity (PF = S²σ).⁴⁸ The graphs of power factor versus chemical potential at temperatures 300 K, 600 K and 900 K are shown in Fig. 7(d) and 8(d).

Both the compounds exhibited their maximum PF (90.0 × 10¹¹ W mK⁻² s⁻¹ (2.4 eV) for CsTiCl₃ and 50.0 × 10¹¹ W mK⁻² s⁻¹ (2.9 eV) for RbTiCl₃) at 300 K. However, at elevated temperature, their Pf decrease and attain minimum values at 900 K. It is also interesting to note that the power factor for p-type region is higher of the studied systems than n-type region. The thermoelectric study of the under investigated compounds reveals that these materials can be employed for the construction of thermoelectric applications at ambient temperature.

Elastics properties

Elastic properties of material have important applications in engineering and medical fields, using the material with best elastic properties are too much important. The mechanical and dynamical characteristics of material depend upon elasticity of material and related with each other through elastic constants. Elastic constant takes a crucial role in shaping the properties of material and shows its reaction to external applied forces. Three elastic constants C₁₁, C₁₂ and C₄₄ are sufficient to completely describe the mechanical properties of cubic compounds. Elastic constant values are computed and correlated with the literature. The calculated values of all the elastic parameters are shown in Table 2.

Table 2 Calculated value of bulk modulus (B), shear modulus (G) Young's modulus (E), anisotropic factor (A), Kleinman parameter (ζ), Pugh ratio (B/G)

Compounds	C11	C12	C44	B	A	G	E	B/G
RbTiCl3	59.5	29.2	8.5	36.8	0.40	18.5	42.2	2.04
CsTiCl3	55.9	24.8	6.84	34.9	0.14	15.2	46.40	2.29

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To ensure the mechanical stability of cubic perovskites, the Born–Huang criteria C₁₁ > 0, C₄₄ > 0, (C₁₁–C₁₂) > 0 and C₁₂ < B < C₁₁ are determined for all the understudy compounds indicating their stability. Specific mechanical properties such as bulk modulus, young modulus, anisotropy factor and shear modulus can be estimated by using the following formulas:⁴⁹

The computed result listed in Table 2 shows that the value of B (bulk modulus) for RbTiCl₃ (36.8) is greater than that for CsTiCl₃ (34.9), concluding that if both compounds are compressed from all sides RbTiF₃ withstand more with respect to CsTiCl_3.

The shear modulus (G) represents the toughness of the system. The computed data calculated for shear modulus is listed in Table 2. It is clear from the table values that shear modulus (G) for RbTiCl₃ is greater than CsTiCl₃. It determines that RbTiCl₃ is harder than CsTiCl₃. So, RbTiCl₃ will show greater opposition to deformation than CsTiCl₃.

The stiffness of the material can be represented by parameter known as young modulus (E). From Table 2 the computed values for the young modulus indicate that CsTiCl₃ is stiffer than RbTiCl₃. Ductility and brittleness are of much importance in an engineering point of view. Ductility and brittleness of a compound can be indicated by using Pugh ratio (B/G). According to criteria of high Pugh ratio material will be ductile if B/G > 1.75 and brittle if B/G < 1.75. Thus B/G values for these compounds from Table 2 indicates that both compound have ductile nature.

Anisotropic factor (A) is the parameter that indicates whether or not the material structural properties remain the same in all directions. For the medium to be isotropic, value of A is equal to 1 else, anisotropic. From Table 2 the computed values indicate the anisotropic nature.

Phonopy spectra

Phonons are a crucial component of dynamic behaviors and thermal characteristics, which are the main aspects of the fundamental problems of materials science. The phonon dispersion band structure of cubic ternary RbTiCl₃ and CsTiCl₃ are investigated using the WIEN2K package as depicted in Fig. 9. It can be seen from Fig. 9 that the phonon dispersion curves for both materials are positive and there are no negative values of dispersion curves (imaginary phonon frequencies). The positive dispersion curves confirm the phonon dynamical stability of both compounds. Our result are analogy to BaMCl₃ (M = Ag, Cu).⁵⁰

Fig. 9 Calculated phonon dispersion curves of (a) CsTiCl₃ and (b) RbTiCl₃.

Conclusion

Shortly, in this research work, we have calculated the structural, electronic, elastic, optical and thermoelectric properties of XTiCl₃ (X = Rb, Cs) perovskites using FP-LAPW numerical method in Wien2K under the umbrella of DFT. The structures of both the compounds were found in cubic phase. The lattice constant were optimized and found 5.08 and 5.13 for XTiCl₃ (X = Rb, Cs), respectively. The compound shows 100% of spin polarization for electronic studies, with spin up states conducting and spin down states semi-conducting. Moreover, the optical response of these compounds like optical absorption, reflectivity and refractive index show that these materials can be used as a UV absorbers. The mechanical properties reveals that XTiCl₃ (X = Rb, Cs) satisfy the stability criteria in cubic structure. The dynamical stability of the corresponding compounds is conformed from phonon spectra. Thermoelectric properties like electrical conductivity, thermal conductivity, Seebeck coefficient, and power factor were estimated using BoltzTrape code, based on the semi-classical Boltzmann transport theory. The thermoelectric study indicates that both compounds are the promising candidates for thermoelectric applications when used as p-type semiconductor.

Conflicts of interest

The authors have no conflict of interest.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University Abha 61421, Asir, Kingdom of Saudi Arabia for funding this work through the Small Groups Project under grant number RGP.1/123/43. Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R24), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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