# The First-Principles Investigation of Structural Stability, Mechanical, Vibrational, Thermodynamic, and Optical Properties of CaHfS3 for Optoelectronic Application

Authors: Mulugetta Kassa; Nebiyu Debelo; Menberu Woldemariam
Journal: Advances in Condensed Matter Physics
 (2023)

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## Abstract

In this study, the structural, electronic, elastic, phonon vibration, thermodynamic features, and optical properties of the orthorhombic phase of (space groupPnma) CaHfS3 were examined by first-principles calculations utilizing the plane wave ultrasoft pseudopotentials in generalized gradient approximations (GGAs) and with Hubbard on-site correction (DFT + U). To improve the value of the band gap, the exchange correlation potential is also approximated with Hubbard correction (GGA + U). The equilibrium state properties such as lattice parameters, unit cell volume, bulk modulus, and its derivative were calculated and are in good agreement with the existing data. The mechanical properties such as bulk modulus, shear modulus, Young’s modulus, and elastic anisotropy were determined from the obtained elastic constants. The ratio of bulk modulus to shear modulus confirms that the orthorhombic phase of CaHfS3 is a ductile material. In addition, the longitudinal sound velocity, transverse sound velocity, and Debye temperature for CaHfS3 have been computed. The absence of negative frequencies in the phonon dispersion curve and the phonon density of states confirm that CaHfS3 in the orthorhombic phase is dynamically stable. The thermodynamic parameters such as free energy, entropy, and heat capacity were examined with variations in temperature. Finally, the absorption coefficient, dielectric constant, energy loss function, reflectivity, and refractive index are discussed in detail in the spectral range 0–1.6 Ry (21.77 eV). The polarizations along (100), (010), and (001) directions significantly show different optical responses.


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## Body

## 1. Introduction

Perovskite materials recently have attracted the interest of scientists and engineers due to their numerous potential advantages. Perovskite materials are crystals that have a chemical formulaABX3 or equivalent crystal structure, where A and B are two cations (very different sizes or different valences) and X is the anion that bonds with both [1]. These materials could have a significant impact on a variety of fields, including energy production, information storage, and solar cell manufacturing. In the past decades, due to their possible applications in many industrial and engineering domains [2], perovskite materials with different compositions and structures have attracted much attention. These perovskite material classes have been recognized as semiconductors for photovoltaic and optoelectronic applications, such as photo-detection and light-emitting devices [2], in addition to being used mainly for dielectric applications, such as piezoelectric and ferroelectrics [3]. The most commonly used perovskite materials for photovoltaics and optoelectronics are hybrid organic-inorganic metal halide-based materials as light-absorbing active layers [4]. In 1970, Weber studied the synthesis and physical properties of -CH3NH3PbX3 (X = Cl, Br, l) organometallic lead halide perovskite [5]. During this time, the organometallic lead halide perovskites have emerged as a candidate and promising for light harvesting in the solar cell as reported in 2009 [6], for which the efficiency reached 22.7% in 2018 [7], starting from 3.8% in 2009 [6]. However, this category of hybrid perovskite materials suffers from long-term instability in ambient air due to the hygroscopic amine cations as well as from the presence of lead toxicity [8], which has slowed down their industrial application and commercialization. To address the stability and toxicity issues of halide perovskites, it is promising to use the synthesized chalcogenide perovskites as an alternative family of materials to replace halide perovskites for photovoltaic and other optoelectronic applications [9]. To date, chalcogenide perovskites having similar kinds of optoelectronic properties have received attention due to their promising photovoltaic and thermoelectric properties, with initial studies conducted on oxide perovskites that have good band gaps for optical absorption [10]. Recently, because of their advantages of being abundant on Earth, having a low impact on the environment, and having good thermal and aqueous stability for applications in photovoltaic and optoelectronic applications, chalcogenides perovskites are the subject of research investigation [10].Chalcogenide perovskites assume an ABX3 configuration with A, and B are elements with a combined valence of 6 (with different valences), while X is typically S or Se [11, 12]. These materials are found to be belonging to a new class of ionic semiconductors [11]. Mostly, a family of chalcogenide and oxy-sulfide perovskite compounds including BaZrS3, BaHfS3, BaTiS3, CaZrS3, CaHfS3, SrZrS3, SrHfS3, SrTiS3, and BaZr (OxS1-x)3 have been identified and studied [1, 13–15]. Due to their predicted strong iconicity, they may display special physical characteristics that are free of deep level flaws, and advantageous for energy harvesting and other optoelectronic applications.For the past few years, theoretical calculations based on the density functional theory (DFT) have been used to reveal and predict the structural, mechanical, electronic, optical, and thermal properties of crystals. CaHfS3, which is the focus of this work, is a family of chalcogenide perovskite crystal material that has been considered theoretically for optoelectronic applications in this study. CaHfS3 has been classified as a promising and attractive chalcogenide ionic semiconductor for optoelectronics and photovoltaic applications [11, 12, 16]. On the other hand there is limited knowledge on the physical properties of this material in both theoretical and experimental aspects. To the best of our knowledge, the structural, elastic, electronic, phonon dispersion, and thermodynamic properties of CaHfS3 are not yet well investigated from computational perspective for optoelectronic application. Moreover, investigation of mechanical properties, phonon dispersion, and thermodynamic properties of CaHfS3 using the first principle computational methods remains unexplored. In this paper, the structural, elastic, electronic, and phonon dispersion relation and thermal properties of CaHfS3 are carefully examined. The electronic properties are calculated by considering GGA-PBE [17] and with the Hubbard correction GGA + U [18] for exchange correlation potential using Quantum Espresso Package (QE). In addition to this, phonon dispersion and the thermodynamic properties are studied using a 1 × 1 × 2 (in x, y, and z direction, respectively) supercell containing 40 atoms as in [19] with density functional perturbation theory (DFPT) as implemented in Quantum Espresso. Furthermore, implementing time-dependent density functional perturbation theory (TD-DFPT), optical properties are estimated. Finally with these predictable properties, these materials are particularly attractive for photovoltaic and optoelectronic applications and therefore call for experimental verification.


## 2. Computational Details

In this work, the first principle computations were carried out in the Quantum Espresso (QE) software package [20] based on the density functional theory (DFT) as implemented within the generalized gradient approximation (GGA) functional [17] and with the Hubbard correction GGA + U [18]. The effective Hubbard parameter (Ueff) was calculated iteratively for Hf-d orbitals using a linear response formalism, utilizing DFPT with the orthoatomic projection method [21, 22] (calculated value of Ueff = 2.2167). For this study, a cell with 20 atoms (4-Ca, 4-Hf, and 12-S) in orthorhombic phase for structural, elastic, and electronic property calculations were used. The ultrasoft pseudopotentials (US-PP) obtained from standard solid-state pseudopotential (SSSP) libraries [23] were used to treat the interaction of the electrons with the ion cores as in [24]. The corresponding valence electrons considered for the calculations are Ca-[Ar] 4s2, Hf-[Xe] 4f14 5d2 6s2, and S-[Ne] 3s2 3p4. The convergence of energy, force, and cell pressure with respect to the plane-wave cutoff and k-points mesh, were carefully tested. Crystal structure optimization was conducted using a plane wave cutoff energy of 55 Ry, density cutoff energy 220 Ry and the Brillouin zone with a 3 × 3 × 3 Monk horst-Packk-point grid [25] based on the convergence criteria (energy 10−4 Ry), (force 10−3 Ry/Bohr), and (cell pressure 0.5 kbar) with the help of Brayden-Fletcher-Goldfarb-Shannon (BFGS) method [26]. Using the optimized structure, the elastic properties were calculated by density functional theory (DFT) within Quantum Espresso package [20]. In addition, the thermal properties were calculated with the help of density functional perturbation theory (DFPT) within a Quantum Espresso package [19]. For the study of phonon dispersion relations and thermodynamic properties of CaHfS3, a supercell of 1 × 1 × 2 in x, y, and z directions with 40 atoms was generated and utilized in calculations. Finally, the optical properties were investigated with the help of time-dependent density functional perturbation theory (TD-DFPT) as implemented in Quantum Espresso package.


## 3. Results and Discussion

### 3.1. Crystal Structure

The crystal structure of a material describes the arrangement of the atoms or molecules in a unit cell in three dimensions. The structure is described in terms of the symmetry elements present and the atomic positions relative to each other. The symmetry elements can be translational, rotational, or reflective. The most common crystal structures are based on the cubic, hexagonal, and tetragonal crystal systems. Most of the materials’ properties are governed by their crystal structures. The equilibrium lattice constant is a fundamental part of the structural information related to crystal structures. In this study, CaHfS3 crystals in the orthorhombic crystal system of space group Pnma (62) and point group D2h (mmm) with the crystallographic structure GdFeO3-type were considered, as shown in Figure 1.Figure 1
The GdFeO3-type crystallographic structures of CaHfS3.The structural stability is determined with the help of Goldschmidt tolerance factor (τ) using the ionic radius of Ca+2, Hf+4, and S−2 [27]. The calculated value of tolerance factor (τ) is found to be 0.83 lies, which within a distorted perovskite range 0.71≤τ≤0.9, indicating that CaHfS3 stability is defined, and Pnma symmetry is observed.Here, structural optimization was performed for CaHfS3 using the cutoff energy and k-point grid size values that were estimated from the convergence test with GGA-PBE and GGA + U approximations. From this, the optimized equilibrium lattice constants were found to be a = 6.561, b = 7.015, and c = 9.592 angstrom with GGA-PBE and a = 6.581, b = 7.039, and c = 9.627 angstrom with GGA + U approximations. These values are in good agreement with the theoretical and experimental values, as summarized in Table 1. Furthermore, the static lattice potential corresponding to total energy was calculated using a series of strained lattices. These results allow for the calculation of the equilibrium unit cell volume, bulk modulus, and its pressure derivative. A sequence of volume-dependent total energy calculations can be fitted to an equation of state consistent with Murnaghan [32].(1)EV=E0+B0VB0′V0/VB0′+1−BoV0B0′−1,where B0 is an equilibrium bulk modulus that effectively measures the curvature of the energy versus volume curve about the relaxed volume V0 and B0′ is the derivative of the bulk modulus. The calculated values of lattice constant, bulk modulus, volume, and pressure derivatives of bulk modulus from the equation of states are summarized in Table 1. The values obtained with respect to GGA + U approximations results in an increment of the lattice constants and volume, but in decrement of bulk modulus and its derivative.Table 1
Calculated values of equilibrium lattice constant, unit cell volume, bulk modulus, and its derivative of CaHfS3 in comparison to existing works.
SourceLattice parametersA0Vol.A03B (GPa)B′abcCalculated value(GGA-PBE)6.5617.0159.592441.47673.43.86GGA + U6.587.049.63446.4571.83.79Others[28]6.456.969.48425.576——[29]7.0159.8526.54451.991——Expt.[30]6.9839.5376.518434.078——[31]6.526.989.54434.162——


### 3.2. Elastic Properties

Elastic properties play a vital role in the governing of a material’s properties and behavior. This is due to the fact that elasticity is one of the most important properties of a material. Elasticity is a measure of how much a material will change in size when subjected to a force. This can be important in many fields, such as engineering and physics. Due to their relationship to mechanical properties of the material, such as elastic moduli, Poisson’s ratio, and elastic anisotropy factor, crystals elastic constants provide key knowledge for the study of mechanical properties of materials. Calculations of the stresses produced by slight deformations of the equilibrium primitive cell led to the determination of the complete elastic constant tensor. Starting from the generalized form of Hooke’s law, the elastic constant tensorsCijkl are given by [33](2)Cijkl=∂σij∂εklχ=1V∂2E∂εij∂εklχ,where E stands for the Helmholtz free energy, εij and εkl are the applied stress and Eulerian strain tensors, and χ stands for the coordinates. Moreover, in orthorhombic symmetry system of Laue class D2h (mmm) group, there are nine independent elastic constants that should satisfy the well-known Born stability criteria [34];(3)C11+C22−2C12>0,C11+C33−2C13>0,C22+C33−2C23>0,C11>0,C22>0,C33>0,C44>0,C55>0,C66>0,C11+C22+C33+2C12+2C13+2C23>0,C12+C13+C23<3B<C11+C22+C33.In this study, the calculated elastic constantsCij, bulk moduli as well as the shear modulus are shown in Tables 2 and 3. From Table 2, we can find that Cij>0, C11 > C12, and 2C13 < (C11 + C33), which shows that our results satisfy the required born mechanical stability conditions for CaHfS3. Moreover, using the Voigt-Reuss-Hill (VRH) average approximation, mechanical parameters such as the Young’s modulus (E), Poisson’s ratio η, and shear modulus (G) are computed from the calculated elastic constants as [35](4)G=12GV+GR,B=12BV+BR,where BR and GR are the Reuss bulk and shear moduli, and Bv and Gv correspond to the Voigt bulk and shear moduli. The Voigt bound values for orthorhombic crystals are obtained using the elastic constant matrix Cij as [36]:(5)BV=19C11+C22+C33+2C12+C13+C23GV=115C11+C22+C33−C12+C13+C23+3C44+C55+C66.Table 2
Calculated elastic constants of CaHfS3.
C11C22C33C44C55C66C12C13C23GGA-PBE131.47164.27153.6146.8331.4455.1563.5540.4738.83GGA + U106.17149.58135.4742.3527.7748.1861.7434.7237.26Table 3
Mechanical properties calculated from elastic constants of CaHfS3.
GvGRBvBRBGB/GEηAuGGA-PBE47.2444.5881.2680.2780.7645.911.76115.780.260.31GGA + U40.8338.2973.2072.8273.039.561.85105.110.270.34Also, the Reuss bound values obtained from the Elastic compliancesSij are(6)BR=S11+S22+S33+2S12+S13+S23−1GR=154S11+S22+S33−S12−S13−S23+3S44+S55+S66−1.Moreover, the Young’s modulus (E) and Poisson’s ratio η are given by [36, 37](7)E=9BG3B+G,η=3B−2G23B+G.The elastic constants and calculated mechanical properties such as bulk modulus (B), Young’s modulus (E), Poisson’s ratio (η), and shear modulus (G) from the elastic constant using the Voigt-Reuss-Hill (VRH) average approximation [35] are shown in Table 3.The calculated bulk modulus values from elastic constants are found to be 80.76 GPa with GGA-PBE and 73.0 GPa with GGA + U. The value calculated with GGA + U approximation is in good agreement with the value calculated from the equation of state (73.4 GPa) for CaHfS3 as in Table 1. Furthermore, determining the ratio of B/G is important for the brittle and ductile behavior of materials during material fabrication. The ductility and brittleness of materials can be determined based on the value of the B/G ratio, according to Pugh [38]. The threshold value that distinguishes between brittleness and ductility tendencies in materials is 1.75. The material behaves ductile when B/G > 1.75; otherwise, it exhibits brittle characteristics. From Table 3, our calculated ratio for B/G value is 1.76 and 1.85 (>1.75) with GGA-PBE and GGA + U, respectively, for CaHfS3. This indicates that CaHfS3 in GdFeO3-type phase is a ductile.Poisson’s ratio (η) is another mechanical parameter that provides information about the feature of the bonding forces. In the evaluation of Poisson’s ratio, the values 0.25 and 0.5 are the lower and upper limits of the central force, respectively [37, 39]. From Table 3, the calculated Poisson’s ratio η for CaHfS3 is 0.26 and 0.34 (which is between 0.25 and 0.5) with GGA-PBE and GGA + U, respectively, indicating that the interatomic forces are central.The universal anisotropic indexAu is a measure to determine whether an elastic material is anisotropic or isotropic based on the contributions of its bulk and sheared moduli [40](8)Au=5GvGR+BVBR−6,where GV and BV are sheared and bulk modulus obtained from Voigt bound approximation, respectively. Similarly, GR and BR are sheared and bulk modulus acquired from Reuss bound approximation. The material is known to be isotropic if the value of Au=0; otherwise, it Au≠0 refers to the anisotropic mechanical properties. The variation of Au from zero value defines the level of elastic anisotropy. Based on this, the calculated value of Au is 0.31 and 0.34 with GGA-PBE and GGA + U, respectively, for CaHfS3, indicating that the CaHfS3 was found to be anisotropic.


### 3.3. Debye Temperature and Speed of Sound

The Debye temperature is an important physical quantity to describe phenomena of solid-state physics which are associated with lattice vibration, elastic constants, specific heat and melting point. At low temperatures the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. The Debye temperatureθD can be estimated from the averaged sound velocity, Cm, given by [41](9)θD=hkB3n4πNAρM1/3Cm,where h is Planck’s constant, kB is Boltzmann’s constant, NA is Avogadro’s number, ρ is density, M is molecular weight, and n is the number of atoms in a formula unit and average sound velocity, Cm, is(10)Cm=132Ct3+1Cl3−1/3,where Cl and Ct are the longitudinal and transverse sound velocity. These velocities are obtained from density, shear and bulk modulus of the material as [42](11)Cl=B+4/3Gρ1/2,Ct=Gρ1/2.The calculated values of longitudinal (compressional) sound velocity and transverse (shear) sound velocity and Debye temperature for CaHfS3 is given in Table 4. Unfortunately, as our knowledge, there are no previous works available on these properties for CaHfS3 phases. Future experimental work will test our calculated results.Table 4
Calculated values of sound velocities and Debye temperature of CaHfS3.
Clm/sCtm/sCmm/sθDkGGA-PBE5475.743113.763461.38365.41GGA + U5182.012906.473234.62341.99


### 3.4. Electronic Properties: Density of States (DOS) and Band Structure

In this study, the electronic band structure along the high symmetry direction of the Brillouin zone was estimated using the GGA-PBE functional and the Hubbard correction (GGA + U) for the exchange-correlation potential of CaHfS3 as shown in Figure 2. The calculated band gap value is compared with the existing theoretical band gap values in Table 5. With the help of the approximation of GGA-PBE exchange-correlation potential, the calculated band gap value is 1.526 eV for CaHfS3. Moreover, the band gap was calculated with the Hubbard correction for onsite interaction, yielding a band gap value of 1.86 eV, which is less than the value (2.5 eV) estimated by reference [28] using HSE06 hybrid functional. To our knowledge, there is no experimental data to compare it with the calculated values.Figure 2
Band structure of CaHfS3 with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Table 5
Calculated band gap value (eV) of CaHfS3 in comparison to the existing works.
SourceGGA-PBEHSE06GGA + UThe calculated value1.526—1.86Others[29]1.51——[28]—2.5—Expt.NAIn addition to this, the total and partial density of states were obtained for the equilibrium states of the phases using GGA-PBE and GGA + U (on-site Hubbard correction) approximations for exchange correlation interaction of CaHfS3 as shown in Figures 3 and 4, respectively. The result shows that for the low-lying states at the maximum valence band mostly contributed by S (2p) orbitals and minimum conduction band mainly dominated by Hf (5d) orbitals. On the other hand, the rest of the orbitals observed to have small contributions on both maximum valence and minimum conduction bands.Figure 3
Total density of states of CaHfS3 with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Figure 4
The partial density of states of CaHfS3.with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Moreover, the other important parameter calculated from the band gap is the effective masses of the electron and the hole. The effective masses can be obtained by fitting the energy of the valence band maximum (holes) and conduction band minimum (electrons) to a quadratic polynomial in the reciprocal lattice vectork based on the equation(12)meh∗=±ℏ2d2Edk2−1.The calculated electron (hole) effective masses for CaHfS3 along the Γ−Z directions are 0.67(0.47) m0 (electron rest mass).


### 3.5. Phonon Vibrations

Dynamic behaviors and thermal properties, which are central topics in the core problems of materials science, are significantly influenced by phonon vibration. One of the basic elements to consider when studying the phase stability, phase transformations, and thermodynamics of these materials is the phonon frequency of crystalline structures. The phonon density of states as a function of frequencyfω is given by [43, 44](13)fω=1N∑qjδω−ωqj,where N is the number of unit cells in a crystal. Divided by N, fω is normalized so that the integral over frequency becomes 3na, where na is a number of atoms.Moreover, considering atom specific phonon density of states projected along a unit direction vectorn^, is defined as [44](14)fkω,n^=1N∑qjδω−ωqjn^,eqjk2.From the canonical distribution in statistical mechanics for phonons under the harmonic approximation, the energyEn of the phonon system is given as(15)E=∑ℏωqj12+1expℏωqj/kBT−1,where T and ℏ are temperature, and reduced Planck constant, respectively. Here from statistical mechanics, n¯=1/expℏωqj/kBT−1 gives the mean phonon number distribution function.A supercell of 1 × 1 × 2 (inx, y, and z-direction) containing 40 atoms was created to study the phonon dispersion relation for CaHfS3, using a DFPT and DFPT + U with Quantum Espresso as implemented in [19]. It is known that a crystal constituent of 40 atoms in bulk system (three dimensions x, y, and z coordinates) has 120 degrees of freedom (3na, where na is number of atoms). The phonon dispersion relation for frequency bands and frequency density of states were calculated and displayed as shown in Figures 5 and 6, respectively. As indicated on Figure 5, it was observed that there are three (3) acoustic branches and 117 optical branches 3na−3 mode of vibrations. Here also the results showed that CaHfS3 possesses no imaginary (no negative terms) phonon frequencies mode in both DFPT and DFPT + U. Hence, it is structurally and dynamically stable which agrees with the results of the analysis of the elastic constants and Goldschmidt tolerance factor analysis.Figure 5
Phonon dispersion relation for CaHfS3 with respect to DFPT (a) and DFPT + U (b).
(a)(b)Figure 6
Phonon density of states ofCaHfS3 with respect to DFPT (a) and DFPT + U (b).
(a)(b)


### 3.6. Thermodynamic Properties

Thermodynamic properties of materials are one of the underlying principles of solid-state research and industry. The investigation of these properties is important in order to reveal their specific behavior when these materials are under high pressure and temperature. With the use of thermodynamic relations, a number of thermal properties, such as constant volume heat capacityCV, Helmholtz free energy F, and entropy S, can be computed as functions of temperature as [19, 43, 45, 46](16)F=12∑qjℏωqj+kBT∑qjln1−exp−ℏωqjkBT,S=12T∑qjℏωqjcothℏωqj2kBT−kB∑qjln2sinhℏωqj2kBT,Cv=∑qjkBℏωqjkBT2expℏωqj/kBTexpℏωqj/kBT−12.Here also, a supercell of 1 × 1 × 2 (inx, y, and z directions) containing 40 atoms was used to study the thermodynamic properties of CaHfS3, using DFPT and DFPT + U with Quantum Espresso as implemented in [19]. At a limited temperature ranging from 0K to 1000 K, thermodynamic parameters such as enthalpy, free energy, entropy, and heat capacity were computed and plotted as shown in Figure 7. In this consideration, the volume and temperature are independent variables. The overall feature of the results remains the same in both DFPT and DFPT + U approximation. In Figure 7, at 0 K, it can be seen that the entropy and heat capacity values are zero, which is in agreement with the third law of thermodynamics. While, with an increase in temperature, the free energy gradually decreases while the entropy increases quickly, following the reasonable trends shown in [43, 47, 48]. As a result, the enthalpy increases linearly with the increase in temperature. The increase in enthalpy at high temperatures leads to a decrease in free energy, which is associated with defects. It is clearly observed that for the temperature below 400 K, the heat capacity increases rapidly, whereas for the temperature above 400 K, it increases slowly (almost increasing linearly) with temperature and gradually approaches the Dulong–Petit classical limit (495.35 J/K/mol) owing to the anharmonic approximations of the Debye model as observed in [39, 43, 49]. The calculated heat capacity graph is also smooth and continuous, confirming that there is no phase change occurring in CaHfS3 up to 1000 K as in [43]. These results can be helpful for further analysis and future experimental verifications.Figure 7
Free energy, entropy, and specific heat ofCaHfS3 with respect to temperature variation.


### 3.7. Optical Properties

Optical properties are the physical properties of materials that determine how the materials respond to interaction with electromagnetic radiation (photon energy), including visible light. The optical property of a material can be determined from the complex dielectric functionεω=ε1ω+iε2ω within an independent particle formalism [50]. The imaginary part ε2ω is determined from the momentum dipole transition matrix elements between the occupied and the unoccupied electronic states along the long wave length limit [51](17)ε2ijω=4π2e2Vmω2×∑knn′σψnσuiψn′σψn′σujψnσ×fkn1−fkn′σEkn′−Ekn−ℏω.The dispersion of the real part of the dielectric function is computed from the imaginary part of the dielectric function by using Kramers–Kronig relations:(18)ε1ω=1+2πΡ∫0∞ω′ε2ωω′2−ω2dω,where P represents the principal value of the integral. With the help of the real and imaginary part of the dielectric function, other optical parameters such as the refractive index nω, the reflectivity Rω, the absorption coefficient βω, and the electron energy loss function Lω were calculated.(19)nω=ε1ω+ε1ω2+ε2ω221/2,βω=2ωε1ω2+ε2ω2−ε1ω1/2,Rω=nω−12+kω2nω+12+kω2,Lω=Im−εω=ε2ωε1ω2+ε2ω2.In this study, the optical properties ofCaHfS3 was studied with the polarization along the X-axis (100), Y-axis (010), and Z-axis (001) within time-dependent density functional perturbation theory (TD-DFPT) as implemented in Quantum Espresso package. We investigated the optical properties including dielectric function, optical absorption and conductivity, reflectivity, refractive index as well as the extinction coefficient for the considered CaHfS3 for the photon energy from 0 to 1.6 Ry (21.77 eV) and the details of the calculations are presented. The calculated real ε1ω (dispersive) and imaginary ε2ω (absorptive) parts of the dielectric functions along X-axis (100), Y-axis (010) and Z-axis (001) for CaHfS3 are shown in Figure 8. As described in Figure 8, the peak of the real of the dielectric function ε1 is 12 at 0.21 Ry, 12.83 at 0.16 Ry and 12.01 at 0.16 Ry along (100), (010), and (001) polarization direction, respectively. The static dielectric constant ε1ω⟶0 is 7.93, 8.09 and 7.90 along (100), (010), and (001) polarization direction, respectively. On the other hand, the peal of the imaginary part of the dielectric function ε2 is 10.80 at 0.24 Ry, 11.35 at 0.29 Ry, and 10.94 at 0.24 Ry along (100), (010), and (001) polarization direction, respectively. Here, we can observe that the dielectric constant along (010) polarization direction shows greater value (12.83) than other directions. Overall, the predicted high dielectric constants are the desired properties for photovoltaics applications as a dielectric constant value of 10 (and greater than 10) are good enough to produce exciton binding energy value lower than 25 meV (1.84 mRy) at room temperature as in [52], which indicates that CaHfS3 is favorable for photovoltaic devices.Figure 8
The real (a) and imaginary (b) part of dielectric function ofCaHfS3.
(a)(b)In addition to this, the reflectivity and refractive index are two important parameters necessary for photovoltaic and optoelectronic applications, are calculated and plotted as shown in Figure9. The optical reflectivity shows a greater value (39%) at about energy of 0.30 Ry along (010) polarization direction. The static reflectivity is 22.6%, 23.0%, and 22.6% along (100), (010), and (001) polarizations direction, respectively. This implies that highest reflection occurs along Y-axis (010) polarization direction and least reflection occurs on both (100) and (001) polarization directions. The peak of the refractive index is 3.63 at 0.22 Ry, 3.61 at 0.16 Ry, and 3.60 at 0.22 Ry along (100), (010), and (001) polarization directions, respectively. The estimated refractive index exists in the range of 0.52–3.63 for CaHfS3.Figure 9
The optical reflectivity and refractive index ofCaHfS3.The other two important optical parameters are the optical absorption function and the energy loss function, which are calculated and plotted in Figure10. The absorption coefficient gives important knowledge on the light harvesting ability of a material, which affects the efficiency of solar energy conversion of corresponding solar cells as a consequence. As shown in Figure 10, the energy dependence of the absorption spectrum is given. The fundamental absorption edge starts at about 0.14 Ry (1.90 eV) that corresponds to the direct Γ−Γ transition or fundamental band gap. This originates from the transition from the S (2p) states located at the top of the valence band to the empty Hf (5d) states located at the bottom of the conduction bands. The absorption spectrum peaks are mainly found in the ultraviolet region. The loss function describes the energy loss of a fast electron traversing the material [53]. The sharp peak is associated with the collective plasma oscillation [54]. The computed energy loss function has a peak value of 2.41 at 1.44 Ry, 2.30 at 1.44 Ry, and 2.11 at 1.45 Ry along (100), (010), and (001) polarizations directions, respectively. This indicates that a plasma oscillation occurs about energy of 1.44 Ry (19.59 eV), which is far away from the peak position of absorption for CaHfS3.Figure 10
The optical absorption coefficient and energy loss function forCaHfS3.


## 3.1. Crystal Structure

The crystal structure of a material describes the arrangement of the atoms or molecules in a unit cell in three dimensions. The structure is described in terms of the symmetry elements present and the atomic positions relative to each other. The symmetry elements can be translational, rotational, or reflective. The most common crystal structures are based on the cubic, hexagonal, and tetragonal crystal systems. Most of the materials’ properties are governed by their crystal structures. The equilibrium lattice constant is a fundamental part of the structural information related to crystal structures. In this study, CaHfS3 crystals in the orthorhombic crystal system of space group Pnma (62) and point group D2h (mmm) with the crystallographic structure GdFeO3-type were considered, as shown in Figure 1.Figure 1
The GdFeO3-type crystallographic structures of CaHfS3.The structural stability is determined with the help of Goldschmidt tolerance factor (τ) using the ionic radius of Ca+2, Hf+4, and S−2 [27]. The calculated value of tolerance factor (τ) is found to be 0.83 lies, which within a distorted perovskite range 0.71≤τ≤0.9, indicating that CaHfS3 stability is defined, and Pnma symmetry is observed.Here, structural optimization was performed for CaHfS3 using the cutoff energy and k-point grid size values that were estimated from the convergence test with GGA-PBE and GGA + U approximations. From this, the optimized equilibrium lattice constants were found to be a = 6.561, b = 7.015, and c = 9.592 angstrom with GGA-PBE and a = 6.581, b = 7.039, and c = 9.627 angstrom with GGA + U approximations. These values are in good agreement with the theoretical and experimental values, as summarized in Table 1. Furthermore, the static lattice potential corresponding to total energy was calculated using a series of strained lattices. These results allow for the calculation of the equilibrium unit cell volume, bulk modulus, and its pressure derivative. A sequence of volume-dependent total energy calculations can be fitted to an equation of state consistent with Murnaghan [32].(1)EV=E0+B0VB0′V0/VB0′+1−BoV0B0′−1,where B0 is an equilibrium bulk modulus that effectively measures the curvature of the energy versus volume curve about the relaxed volume V0 and B0′ is the derivative of the bulk modulus. The calculated values of lattice constant, bulk modulus, volume, and pressure derivatives of bulk modulus from the equation of states are summarized in Table 1. The values obtained with respect to GGA + U approximations results in an increment of the lattice constants and volume, but in decrement of bulk modulus and its derivative.Table 1
Calculated values of equilibrium lattice constant, unit cell volume, bulk modulus, and its derivative of CaHfS3 in comparison to existing works.
SourceLattice parametersA0Vol.A03B (GPa)B′abcCalculated value(GGA-PBE)6.5617.0159.592441.47673.43.86GGA + U6.587.049.63446.4571.83.79Others[28]6.456.969.48425.576——[29]7.0159.8526.54451.991——Expt.[30]6.9839.5376.518434.078——[31]6.526.989.54434.162——


## 3.2. Elastic Properties

Elastic properties play a vital role in the governing of a material’s properties and behavior. This is due to the fact that elasticity is one of the most important properties of a material. Elasticity is a measure of how much a material will change in size when subjected to a force. This can be important in many fields, such as engineering and physics. Due to their relationship to mechanical properties of the material, such as elastic moduli, Poisson’s ratio, and elastic anisotropy factor, crystals elastic constants provide key knowledge for the study of mechanical properties of materials. Calculations of the stresses produced by slight deformations of the equilibrium primitive cell led to the determination of the complete elastic constant tensor. Starting from the generalized form of Hooke’s law, the elastic constant tensorsCijkl are given by [33](2)Cijkl=∂σij∂εklχ=1V∂2E∂εij∂εklχ,where E stands for the Helmholtz free energy, εij and εkl are the applied stress and Eulerian strain tensors, and χ stands for the coordinates. Moreover, in orthorhombic symmetry system of Laue class D2h (mmm) group, there are nine independent elastic constants that should satisfy the well-known Born stability criteria [34];(3)C11+C22−2C12>0,C11+C33−2C13>0,C22+C33−2C23>0,C11>0,C22>0,C33>0,C44>0,C55>0,C66>0,C11+C22+C33+2C12+2C13+2C23>0,C12+C13+C23<3B<C11+C22+C33.In this study, the calculated elastic constantsCij, bulk moduli as well as the shear modulus are shown in Tables 2 and 3. From Table 2, we can find that Cij>0, C11 > C12, and 2C13 < (C11 + C33), which shows that our results satisfy the required born mechanical stability conditions for CaHfS3. Moreover, using the Voigt-Reuss-Hill (VRH) average approximation, mechanical parameters such as the Young’s modulus (E), Poisson’s ratio η, and shear modulus (G) are computed from the calculated elastic constants as [35](4)G=12GV+GR,B=12BV+BR,where BR and GR are the Reuss bulk and shear moduli, and Bv and Gv correspond to the Voigt bulk and shear moduli. The Voigt bound values for orthorhombic crystals are obtained using the elastic constant matrix Cij as [36]:(5)BV=19C11+C22+C33+2C12+C13+C23GV=115C11+C22+C33−C12+C13+C23+3C44+C55+C66.Table 2
Calculated elastic constants of CaHfS3.
C11C22C33C44C55C66C12C13C23GGA-PBE131.47164.27153.6146.8331.4455.1563.5540.4738.83GGA + U106.17149.58135.4742.3527.7748.1861.7434.7237.26Table 3
Mechanical properties calculated from elastic constants of CaHfS3.
GvGRBvBRBGB/GEηAuGGA-PBE47.2444.5881.2680.2780.7645.911.76115.780.260.31GGA + U40.8338.2973.2072.8273.039.561.85105.110.270.34Also, the Reuss bound values obtained from the Elastic compliancesSij are(6)BR=S11+S22+S33+2S12+S13+S23−1GR=154S11+S22+S33−S12−S13−S23+3S44+S55+S66−1.Moreover, the Young’s modulus (E) and Poisson’s ratio η are given by [36, 37](7)E=9BG3B+G,η=3B−2G23B+G.The elastic constants and calculated mechanical properties such as bulk modulus (B), Young’s modulus (E), Poisson’s ratio (η), and shear modulus (G) from the elastic constant using the Voigt-Reuss-Hill (VRH) average approximation [35] are shown in Table 3.The calculated bulk modulus values from elastic constants are found to be 80.76 GPa with GGA-PBE and 73.0 GPa with GGA + U. The value calculated with GGA + U approximation is in good agreement with the value calculated from the equation of state (73.4 GPa) for CaHfS3 as in Table 1. Furthermore, determining the ratio of B/G is important for the brittle and ductile behavior of materials during material fabrication. The ductility and brittleness of materials can be determined based on the value of the B/G ratio, according to Pugh [38]. The threshold value that distinguishes between brittleness and ductility tendencies in materials is 1.75. The material behaves ductile when B/G > 1.75; otherwise, it exhibits brittle characteristics. From Table 3, our calculated ratio for B/G value is 1.76 and 1.85 (>1.75) with GGA-PBE and GGA + U, respectively, for CaHfS3. This indicates that CaHfS3 in GdFeO3-type phase is a ductile.Poisson’s ratio (η) is another mechanical parameter that provides information about the feature of the bonding forces. In the evaluation of Poisson’s ratio, the values 0.25 and 0.5 are the lower and upper limits of the central force, respectively [37, 39]. From Table 3, the calculated Poisson’s ratio η for CaHfS3 is 0.26 and 0.34 (which is between 0.25 and 0.5) with GGA-PBE and GGA + U, respectively, indicating that the interatomic forces are central.The universal anisotropic indexAu is a measure to determine whether an elastic material is anisotropic or isotropic based on the contributions of its bulk and sheared moduli [40](8)Au=5GvGR+BVBR−6,where GV and BV are sheared and bulk modulus obtained from Voigt bound approximation, respectively. Similarly, GR and BR are sheared and bulk modulus acquired from Reuss bound approximation. The material is known to be isotropic if the value of Au=0; otherwise, it Au≠0 refers to the anisotropic mechanical properties. The variation of Au from zero value defines the level of elastic anisotropy. Based on this, the calculated value of Au is 0.31 and 0.34 with GGA-PBE and GGA + U, respectively, for CaHfS3, indicating that the CaHfS3 was found to be anisotropic.


## 3.3. Debye Temperature and Speed of Sound

The Debye temperature is an important physical quantity to describe phenomena of solid-state physics which are associated with lattice vibration, elastic constants, specific heat and melting point. At low temperatures the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. The Debye temperatureθD can be estimated from the averaged sound velocity, Cm, given by [41](9)θD=hkB3n4πNAρM1/3Cm,where h is Planck’s constant, kB is Boltzmann’s constant, NA is Avogadro’s number, ρ is density, M is molecular weight, and n is the number of atoms in a formula unit and average sound velocity, Cm, is(10)Cm=132Ct3+1Cl3−1/3,where Cl and Ct are the longitudinal and transverse sound velocity. These velocities are obtained from density, shear and bulk modulus of the material as [42](11)Cl=B+4/3Gρ1/2,Ct=Gρ1/2.The calculated values of longitudinal (compressional) sound velocity and transverse (shear) sound velocity and Debye temperature for CaHfS3 is given in Table 4. Unfortunately, as our knowledge, there are no previous works available on these properties for CaHfS3 phases. Future experimental work will test our calculated results.Table 4
Calculated values of sound velocities and Debye temperature of CaHfS3.
Clm/sCtm/sCmm/sθDkGGA-PBE5475.743113.763461.38365.41GGA + U5182.012906.473234.62341.99


## 3.4. Electronic Properties: Density of States (DOS) and Band Structure

In this study, the electronic band structure along the high symmetry direction of the Brillouin zone was estimated using the GGA-PBE functional and the Hubbard correction (GGA + U) for the exchange-correlation potential of CaHfS3 as shown in Figure 2. The calculated band gap value is compared with the existing theoretical band gap values in Table 5. With the help of the approximation of GGA-PBE exchange-correlation potential, the calculated band gap value is 1.526 eV for CaHfS3. Moreover, the band gap was calculated with the Hubbard correction for onsite interaction, yielding a band gap value of 1.86 eV, which is less than the value (2.5 eV) estimated by reference [28] using HSE06 hybrid functional. To our knowledge, there is no experimental data to compare it with the calculated values.Figure 2
Band structure of CaHfS3 with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Table 5
Calculated band gap value (eV) of CaHfS3 in comparison to the existing works.
SourceGGA-PBEHSE06GGA + UThe calculated value1.526—1.86Others[29]1.51——[28]—2.5—Expt.NAIn addition to this, the total and partial density of states were obtained for the equilibrium states of the phases using GGA-PBE and GGA + U (on-site Hubbard correction) approximations for exchange correlation interaction of CaHfS3 as shown in Figures 3 and 4, respectively. The result shows that for the low-lying states at the maximum valence band mostly contributed by S (2p) orbitals and minimum conduction band mainly dominated by Hf (5d) orbitals. On the other hand, the rest of the orbitals observed to have small contributions on both maximum valence and minimum conduction bands.Figure 3
Total density of states of CaHfS3 with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Figure 4
The partial density of states of CaHfS3.with respect to GGA-PBE (a) and GGA + U (b).
(a)(b)Moreover, the other important parameter calculated from the band gap is the effective masses of the electron and the hole. The effective masses can be obtained by fitting the energy of the valence band maximum (holes) and conduction band minimum (electrons) to a quadratic polynomial in the reciprocal lattice vectork based on the equation(12)meh∗=±ℏ2d2Edk2−1.The calculated electron (hole) effective masses for CaHfS3 along the Γ−Z directions are 0.67(0.47) m0 (electron rest mass).


## 3.5. Phonon Vibrations

Dynamic behaviors and thermal properties, which are central topics in the core problems of materials science, are significantly influenced by phonon vibration. One of the basic elements to consider when studying the phase stability, phase transformations, and thermodynamics of these materials is the phonon frequency of crystalline structures. The phonon density of states as a function of frequencyfω is given by [43, 44](13)fω=1N∑qjδω−ωqj,where N is the number of unit cells in a crystal. Divided by N, fω is normalized so that the integral over frequency becomes 3na, where na is a number of atoms.Moreover, considering atom specific phonon density of states projected along a unit direction vectorn^, is defined as [44](14)fkω,n^=1N∑qjδω−ωqjn^,eqjk2.From the canonical distribution in statistical mechanics for phonons under the harmonic approximation, the energyEn of the phonon system is given as(15)E=∑ℏωqj12+1expℏωqj/kBT−1,where T and ℏ are temperature, and reduced Planck constant, respectively. Here from statistical mechanics, n¯=1/expℏωqj/kBT−1 gives the mean phonon number distribution function.A supercell of 1 × 1 × 2 (inx, y, and z-direction) containing 40 atoms was created to study the phonon dispersion relation for CaHfS3, using a DFPT and DFPT + U with Quantum Espresso as implemented in [19]. It is known that a crystal constituent of 40 atoms in bulk system (three dimensions x, y, and z coordinates) has 120 degrees of freedom (3na, where na is number of atoms). The phonon dispersion relation for frequency bands and frequency density of states were calculated and displayed as shown in Figures 5 and 6, respectively. As indicated on Figure 5, it was observed that there are three (3) acoustic branches and 117 optical branches 3na−3 mode of vibrations. Here also the results showed that CaHfS3 possesses no imaginary (no negative terms) phonon frequencies mode in both DFPT and DFPT + U. Hence, it is structurally and dynamically stable which agrees with the results of the analysis of the elastic constants and Goldschmidt tolerance factor analysis.Figure 5
Phonon dispersion relation for CaHfS3 with respect to DFPT (a) and DFPT + U (b).
(a)(b)Figure 6
Phonon density of states ofCaHfS3 with respect to DFPT (a) and DFPT + U (b).
(a)(b)


## 3.6. Thermodynamic Properties

Thermodynamic properties of materials are one of the underlying principles of solid-state research and industry. The investigation of these properties is important in order to reveal their specific behavior when these materials are under high pressure and temperature. With the use of thermodynamic relations, a number of thermal properties, such as constant volume heat capacityCV, Helmholtz free energy F, and entropy S, can be computed as functions of temperature as [19, 43, 45, 46](16)F=12∑qjℏωqj+kBT∑qjln1−exp−ℏωqjkBT,S=12T∑qjℏωqjcothℏωqj2kBT−kB∑qjln2sinhℏωqj2kBT,Cv=∑qjkBℏωqjkBT2expℏωqj/kBTexpℏωqj/kBT−12.Here also, a supercell of 1 × 1 × 2 (inx, y, and z directions) containing 40 atoms was used to study the thermodynamic properties of CaHfS3, using DFPT and DFPT + U with Quantum Espresso as implemented in [19]. At a limited temperature ranging from 0K to 1000 K, thermodynamic parameters such as enthalpy, free energy, entropy, and heat capacity were computed and plotted as shown in Figure 7. In this consideration, the volume and temperature are independent variables. The overall feature of the results remains the same in both DFPT and DFPT + U approximation. In Figure 7, at 0 K, it can be seen that the entropy and heat capacity values are zero, which is in agreement with the third law of thermodynamics. While, with an increase in temperature, the free energy gradually decreases while the entropy increases quickly, following the reasonable trends shown in [43, 47, 48]. As a result, the enthalpy increases linearly with the increase in temperature. The increase in enthalpy at high temperatures leads to a decrease in free energy, which is associated with defects. It is clearly observed that for the temperature below 400 K, the heat capacity increases rapidly, whereas for the temperature above 400 K, it increases slowly (almost increasing linearly) with temperature and gradually approaches the Dulong–Petit classical limit (495.35 J/K/mol) owing to the anharmonic approximations of the Debye model as observed in [39, 43, 49]. The calculated heat capacity graph is also smooth and continuous, confirming that there is no phase change occurring in CaHfS3 up to 1000 K as in [43]. These results can be helpful for further analysis and future experimental verifications.Figure 7
Free energy, entropy, and specific heat ofCaHfS3 with respect to temperature variation.


## 3.7. Optical Properties

Optical properties are the physical properties of materials that determine how the materials respond to interaction with electromagnetic radiation (photon energy), including visible light. The optical property of a material can be determined from the complex dielectric functionεω=ε1ω+iε2ω within an independent particle formalism [50]. The imaginary part ε2ω is determined from the momentum dipole transition matrix elements between the occupied and the unoccupied electronic states along the long wave length limit [51](17)ε2ijω=4π2e2Vmω2×∑knn′σψnσuiψn′σψn′σujψnσ×fkn1−fkn′σEkn′−Ekn−ℏω.The dispersion of the real part of the dielectric function is computed from the imaginary part of the dielectric function by using Kramers–Kronig relations:(18)ε1ω=1+2πΡ∫0∞ω′ε2ωω′2−ω2dω,where P represents the principal value of the integral. With the help of the real and imaginary part of the dielectric function, other optical parameters such as the refractive index nω, the reflectivity Rω, the absorption coefficient βω, and the electron energy loss function Lω were calculated.(19)nω=ε1ω+ε1ω2+ε2ω221/2,βω=2ωε1ω2+ε2ω2−ε1ω1/2,Rω=nω−12+kω2nω+12+kω2,Lω=Im−εω=ε2ωε1ω2+ε2ω2.In this study, the optical properties ofCaHfS3 was studied with the polarization along the X-axis (100), Y-axis (010), and Z-axis (001) within time-dependent density functional perturbation theory (TD-DFPT) as implemented in Quantum Espresso package. We investigated the optical properties including dielectric function, optical absorption and conductivity, reflectivity, refractive index as well as the extinction coefficient for the considered CaHfS3 for the photon energy from 0 to 1.6 Ry (21.77 eV) and the details of the calculations are presented. The calculated real ε1ω (dispersive) and imaginary ε2ω (absorptive) parts of the dielectric functions along X-axis (100), Y-axis (010) and Z-axis (001) for CaHfS3 are shown in Figure 8. As described in Figure 8, the peak of the real of the dielectric function ε1 is 12 at 0.21 Ry, 12.83 at 0.16 Ry and 12.01 at 0.16 Ry along (100), (010), and (001) polarization direction, respectively. The static dielectric constant ε1ω⟶0 is 7.93, 8.09 and 7.90 along (100), (010), and (001) polarization direction, respectively. On the other hand, the peal of the imaginary part of the dielectric function ε2 is 10.80 at 0.24 Ry, 11.35 at 0.29 Ry, and 10.94 at 0.24 Ry along (100), (010), and (001) polarization direction, respectively. Here, we can observe that the dielectric constant along (010) polarization direction shows greater value (12.83) than other directions. Overall, the predicted high dielectric constants are the desired properties for photovoltaics applications as a dielectric constant value of 10 (and greater than 10) are good enough to produce exciton binding energy value lower than 25 meV (1.84 mRy) at room temperature as in [52], which indicates that CaHfS3 is favorable for photovoltaic devices.Figure 8
The real (a) and imaginary (b) part of dielectric function ofCaHfS3.
(a)(b)In addition to this, the reflectivity and refractive index are two important parameters necessary for photovoltaic and optoelectronic applications, are calculated and plotted as shown in Figure9. The optical reflectivity shows a greater value (39%) at about energy of 0.30 Ry along (010) polarization direction. The static reflectivity is 22.6%, 23.0%, and 22.6% along (100), (010), and (001) polarizations direction, respectively. This implies that highest reflection occurs along Y-axis (010) polarization direction and least reflection occurs on both (100) and (001) polarization directions. The peak of the refractive index is 3.63 at 0.22 Ry, 3.61 at 0.16 Ry, and 3.60 at 0.22 Ry along (100), (010), and (001) polarization directions, respectively. The estimated refractive index exists in the range of 0.52–3.63 for CaHfS3.Figure 9
The optical reflectivity and refractive index ofCaHfS3.The other two important optical parameters are the optical absorption function and the energy loss function, which are calculated and plotted in Figure10. The absorption coefficient gives important knowledge on the light harvesting ability of a material, which affects the efficiency of solar energy conversion of corresponding solar cells as a consequence. As shown in Figure 10, the energy dependence of the absorption spectrum is given. The fundamental absorption edge starts at about 0.14 Ry (1.90 eV) that corresponds to the direct Γ−Γ transition or fundamental band gap. This originates from the transition from the S (2p) states located at the top of the valence band to the empty Hf (5d) states located at the bottom of the conduction bands. The absorption spectrum peaks are mainly found in the ultraviolet region. The loss function describes the energy loss of a fast electron traversing the material [53]. The sharp peak is associated with the collective plasma oscillation [54]. The computed energy loss function has a peak value of 2.41 at 1.44 Ry, 2.30 at 1.44 Ry, and 2.11 at 1.45 Ry along (100), (010), and (001) polarizations directions, respectively. This indicates that a plasma oscillation occurs about energy of 1.44 Ry (19.59 eV), which is far away from the peak position of absorption for CaHfS3.Figure 10
The optical absorption coefficient and energy loss function forCaHfS3.


## 4. Conclusion

In this study, the first-principles calculations using the Quantum Espresso software package were performed to study the structural, electrical, elastic, phonon dispersion relation, temperature-dependent thermodynamic characteristics, and optical properties of orthorhombicCaHfS3. The elastic characteristics of the material were computed using the optimized structure. The density functional perturbation theory (DFPT) and DFPT + U implemented in Quantum Espresso software package was also used to calculate the phonon dispersion and thermal characteristics. The optical properties were also studied with the help of time-dependent density functional perturbation theory (TD-DFPT) as implemented in Quantum Espresso package. The computed lattice parameters are a = 6.561, b = 7.015, and c = 9.592 angstrom with GGA-PBE, and a = 6.581, b = 7.039, and c = 9.627 angstrom with GGA + U approximations, and comparable to the existing theoretical and experimental results. The elastic constants of CaHfS3 were obtained and used to calculate the mechanical parameters such as bulk modulus, shear modulus, Young’s modulus, and elastic anisotropy. The Poisson’s ratio is 0.26 (0.34) with GGA-PBE (GGA + U) and indicates that the interatomic forces are central. The value of B/G (1.76) confirms that CaHfS3 is ductile. The calculated band gap values were 1.526 eV and 1.86 eV with respect to GGA-PBE and GGA + U approximations, respectively. The absence of imaginary (negative frequencies in the figures) frequencies in phonon dispersion curve and the phonon density of states indicate that the structure is dynamically stable. The temperature dependence of thermodynamic parameters including enthalpy, entropy, free energy, and heat capacity is calculated and analyzed. Main optical properties such as absorption coefficient, dielectric constant, energy loss function, reflectivity, and refractive index are calculated and discussed in detail in the spectral range 0–1.6 Ry (21.77 eV). The polarizations along X-axis (100), Y-axis (010), and Z-axis (001) found exhibit significantly different optical responses confirming anisotropic behavior. Optical properties also show relatively high reflectivity and absorption in the ultraviolet region. To our knowledge there is no experimental data available for comparison for elastic, thermal and optical properties. These results will be hopeful that it can inspire researchers for the future further works.



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*Source: 9332399-2023-02-23.xml*