2.1. Computational First-Principle Approach

The conventional unit cell structures of Cs₂HfInBr₆ (each comprises 40 atoms), as shown in Figure 1a, were fully optimized to ground state energy through DFT implementations. This study delves into the detailed analysis of perovskite crystal structure, electronic band dispersions and state density, elastic modulus, and optical properties using the plane wave pseudopotentials method, specifically implemented through the Cambridge Serial Total Energy Package (CASTEP) in Materials Studio [38]. The evaluation of exchange–correlation interactions within Kohn–Sham orbitals during geometry optimization and subsequent electronic energy calculations was performed using the Perdew–Burke–Ernzerhof (PBE) function within the generalized gradient approximation (GGA) [39, 40].

A wavefunction energy cutoff of 500 eV for prolonging plane waves and an 8 × 8 × 8 Monkhorst–Pack k-point mesh centered at Γ were utilized to sample the first Brillouin zone (BZ) of the Cs₂HfInBr₆ double perovskite crystal [41]. Lattice constant, cell volume, and atom positions were optimized without any symmetry constraints until the residual force per atom is <0.006 eV/Å. The self-consistent electronic relaxation was carried out with convergence energy and force criteria set at 10⁻⁶ eV and 0.02 eV/Å, respectively, and the calculations considered spin polarizations. In this study of the Cs₂HfInBr₆ double perovskite, the electronic configurations of valence states are Cs⁺¹:5p⁶6s¹, Hf⁺⁴:4f¹⁴5d²6s², In⁺³:4d¹⁰5s²5p¹, and Br⁻¹:3d¹⁰4s²4p⁵. Each Hf and In atom is coordinated with six Br atoms, forming alternating (HfBr₆)²⁻ and (InBr₆)³⁻ octahedra, while Cs atoms occupy the voids within the lattice to maintain charge neutrality. Vanderbilt [42] ultrasoft pseudopotential in the CASTEP code was utilized for the atomic potentials. To understand the pressure influence on the optoelectronic and mechanical attributes of Cs₂HfInBr₆, double perovskite is studied within the range spanning from 0 GPa to 25 GPa.

2.2. Device Architecture Modeling and Simulation

First-principle calculations optimized the nonleaded noble Cs₂HfInBr₆ material could utilized as the light photon-absorbing material in a photovoltaic device featuring an n-i-p configuration. The schematic representation of the multilayer PSC structure is depicted in Figure 1c. The modeled device utilizes Spiro-OMeTAD as the electron transport layer (ETL) and cadmium sulfide (CdS) as the hole transport layer (HTL). It consists of In-doped tin oxide (ITO) on a transparent glass substrate as the top layer, serving as the cathode, with gold (Au) acting as the back metal anode contact. ITO, chosen for its optical transparency, substrate adhesion, and chemical inertness, is utilized as the cathode layer, while Au, selected for its enhanced electrical conductivity, stability, and compatibility with perovskite materials, serves as the back metal contact. The layer thicknesses in the structure (ITO, CdS, Cs₂HfInBr₆, Spiro-OMeTAD, and Au contact) are 100, 50, 600, 50, and 40 nm, respectively. For the necessary equations used in the computational calculations, please see the Supporting Information.

3.1. Structural Phase Stability

The CASTEP performed geometry-optimized double perovskite Cs₂HfInBr₆ which crystallizes in the cubic phase of the Fm$$m (no. 225) space group at room temperature, as represented in Figure 1b. The double perovskite crystal consists of 8 Cs atoms at the Wyckoff position 8c of the cell with coordinates (1/4 1/4 1/4), 13 Hf atoms are at the Wyckoff site 4b at (1/2 1/2 1/2) coordinates, and 14 In atoms are at Wyckoff site 4a positioned at (0 0 0), while 5 halide Br atoms occupy the Wyckoff site 24e with (x, 0, 0) positions. Within the crystal, B-site’s 13 (Hf–Br₆) and 14 (In–Br₆) atomic compositions construct an inorganic octahedral framework, as shown by the polyhedral formation in Figure 1b. Utilizing the ionic Shannon radii of individual atoms in the Cs₂HfInBr₆ compounds yields a tolerance factor of 0.89, aligning with the stable range of 0.81 < t < 1.1, thus confirming their stability in accordance with the Goldschmidt rule following equation (Supporting Information Equation (2)). Analyzing crystal stability under different physical conditions (i.e., pressure and volume), we employed Birch–Murnaghan equation (Supporting Information Equation (1)) of states to optimize energy/pressure–volume behavior, investigating optimal lattice constants for Cs₂HfInBr₆ double perovskites. The ground state structure was determined by minimizing the crystal energy with respect to the volume, as shown in the E–V fitting plot in Figure 2a, confirming the structural stability of these perovskites.

The Cs₂HfInBr₆ perovskites exhibit a consistent decrease in equilibrium lattice constants from 7.97 to 6.96 Å as hydrostatic pressure varies from 0 to 25 GPa, as exhibited in Figure 2b and listed in Table 1. This trend results from structural atomic position modifications under pressure, aiming to minimize energy and resulting in decreased interatomic distances. Volumes under pressure variations are consistent with lattice parameters, as shown in Figure 2b. The formation energy (ΔE_f) is a crucial indicator of thermodynamic stability in DHPs, reflecting the energy balance during defect creation and compound formation. Lower formation energies suggest enhanced stability and favorable synthesis conditions. Calculated formation energies with equation (Supporting Information Equation (4)) are detailed in Table 1, and trends under varying pressure are depicted in Figure 2b, with negative values indicating thermodynamic stability and exothermic behavior for the noble compound. However, with increasing pressure, this parameter value goes up from −3.64 to −2.72 eV/atom in the negative magnitude.

Since the stability of the double perovskites is important for the suitability of fabrication and commercialization, we also need to check the thermodynamic stability of Cs₂HfInBr₆ materials, which can be demonstrated by obtaining their decomposition enthalpies (ΔH). The basic and most convenient decomposition method for our considered double perovskite is breaking them into their corresponding binary compounds. We calculate the ΔH of Cs₂HfInBr₆ utilizing the GGA–PBE exchange functional from the total energies with respect to three suitable decomposition pathways. The difference between the total energy of the considered material and the total energy of the disintegrated binary compounds is used to determine the decomposition enthalpy of Cs₂HfInBr₆ double perovskite.

The minimized total energy was obtained for each compound after full relaxation was secured. A positive value of ΔH indicates the spontaneous nature and instability of the materials, while a negative enthalpy confirms the thermodynamic stability of the compounds. We have taken binary crystal systems of CsBr cubic ($$), Cs₃Hf₂Br₉ trigonal ($$), InBr₃, and HfBr₃ monoclinic (c2/m), Cs₃Hf₂Br₅ hexagonal (P63/mmc), and InBr with cubic ($$), respectively. On these pathways, this crystal showed thermodynamically stable. As far as literature goes, there is no accessible experimental data for the structural behavior of the double Cs₂HfInBr₆ halide perovskites. Nevertheless, our computed structural data and behaviors align with prior experimental and theoretical studies on other analogous compounds [46].

3.2. Dynamic Stability

Phonon dispersion describes how vibrational energy (frequency) varies with momentum (wave vector) in a crystal lattice. Figure 3 presents the calculated phonon dispersion for the Cs₂HfInBr6 compound, obtained using the CASTEP code and DFPT [38]. The plot includes the longitudinal optical/transverse optical (LO/TO) splitting effect caused by the electric field, analyzed along high-symmetry paths in the BZ under hydrostatic pressures of 0, 3, and 7 GPa. The phonon spectrum consists of 30 branches arising from the 10 constituent atoms, comprising three acoustic modes and 27 optical modes. Importantly, the absence of imaginary frequencies in the phonon spectra at all pressures confirms the compound’s kinetic stability. These results suggest that Cs₂HfInBr₆ is dynamically stable under different hydrostatic pressure range investigated, with no structural instabilities detected.

3.3. Bader Charge Analysis

The Bader charge analysis reveals pressure-dependent electronic and bonding changes in Cs₂HfInBr₆. At 0 GPa, the Bader charges for In, Hf, Cs, and Br are 0.66581, 1.791, 0.87561, and −0.701441, reflecting ionic bonding with balanced charge distribution, as tabulated in Table 2. Also in Figure 4a, the electron localization function (ELF) map at 0 GPa pressure shows localized electron density around Br atoms and moderate delocalization around Cs and In atoms, indicating ionic In─Br and Cs─Br interactions [47].

At 3 GPa pressure-enhanced Coulomb interactions increase electron localization around Hf and Cs (charges rise to 1.884825 and 1.281884), while Br becomes more electronegative (−0.71759), indicating stronger ionicity [48]. For validation, the ELF maps (Figure 4 b) show sharper electron localization around Br and more pronounced bonding regions between Br and Cs/In, suggesting stronger ionic In─Br and Cs─Br bonds under pressure. Furthermore, at 7 GPa, lattice compression and orbital hybridization reduce Hf and Cs localization (charges drop to 1.00061 and 0.791424), and Br’s charge becomes less negative (−0.54423), signifying a shift toward covalent Hf─Br bonding [47, 48]. The localization map at this pressure shows enhanced delocalization in Hf─Br bonds presented in Figure 4c, with reduced localization around Cs and In, indicating pressure-induced covalency in Hf─Br and weaker ionic interactions for Cs─Br and In─Br bonds. This highlights the potential of pressure as a tool to fine-tune the electronic and structural properties of Cs₂HfInBr₆, opening avenues for advanced applications in optoelectronics, high-pressure sensing, and energy materials.

3.4. Electronic Properties

To understand the energy distribution nature of electrons in the noble Cs₂HfInBr₆ double perovskite along with different applied pressure effects on its electronic states and consequently optical behavior, energy band profile (E–K dispersions) and electron DOS are evaluated within GGA–PBE functional. The E–K dispersions are represented in Figure 5a–g for the HfIn-based double perovskite under different pressure conditions. Figure 5 illustrates the electronic energy band structures along high-symmetry paths in the first BZ. The band edge dispersions in Cs₂HfInBr₆ exhibit a consistent trend that the conduction band minima (CBM) and valence band maxima (VBM) remain at the same K-path within the BZ. This alignment persists up to the applied pressure of 16 GPa from ambient conditions, corroborating the semiconducting nature of the material with direct band gaps. Usually, following Shockley–Queisser efficiency, a suitable absorber material for efficient photoelectric conversion should have a bandgap within 0.9–2.2 eV [49]. At ambient pressure (0 GPa), Cs₂HfInBr₆ exhibits a semiconducting nature with a band gap of 2.08 eV. However, applying pressures ranging from 1 to16 GPa leads to a reduced electronic band gap from 1.97 to 0.29 eV, indicative of a potential semiconducting to conduct transition behavior. The band edge dispersions of the conduction and valence band exhibit a similar trend under pressure variations from 0 to 16 GPa, as depicted in Figure 5a–f. The conduction band structure undergoes downward shifts toward the Fermi level with increasing pressure, while increased pressure leads to shifts down in valence band states other than the VBM toward deeper energy levels. It is worth noting that there is an inverse relationship between band gap and external pressure [50], as well as the repulsion of the maximum valence band and other valance bands [51, 52] which leads to an increase in the potential between electrons and ions, responsible for reducing lattice parameters (Table 1). Orbitals from Br-p and In-s atoms can hybridize, forming bonding and antibonding states. The highest valence band is often an antibonding state, which naturally lies higher in energy than bonding states [53]. Also, in Cs₂HfInBr₆ with lower symmetry and applied pressure, the crystal field as well the lattice can cause the splitting of degenerate states, raising one state higher in energy [54, 55]. Furthermore, at 25 GPa, there is observable overlap between both the CBM and VBM along different high symmetry k-paths; specifically, the VBM shifts to the left between the W → L k-points, while the CBM shifts to the G point from L. This implies a direct-to-indirect transition in the conduction mechanism alongside a semiconductor-to-metallic phase shift in the studied double HfIn halide perovskite under pressure [56]. Moreover, the presence of flat bands within the conduction band (circled in Figure 5b) hints a potential secondary phase may contribute to the onset of superconductivity [57]. Intriguingly, as pressure increases, the flat bands near the Fermi level realize the transition to a dispersive nature, as illustrated in Figure 5f,g.

Pressure-induced changes in effective mass enable tailored electronic behavior of the Cs₂HfInBr₆ perovskite to improve device performance in photovoltaics, transistors, sensors, and advanced nanoelectronics [58]. In semiconductor physics, effective mass, inversely related to CBM and VBM bandwidth, is determined by the curvature of the energy dispersion relation near band edges [59]. A reduced bandwidth implies a more localized wavefunction, resulting in a higher effective mass for charge carriers. The conduction band edge shows superior dispersion with a larger bandwidth compared to the valence band. Influenced by Cs s states for the bottom conduction band and p and d states of B-site cations for the top valence band, bandwidth narrowing occurs due to reduced localization. Effective masses (m^∗) for electrons and holes are determined using band-edge eigenvalues and wave vectors near the VBM and CBM through equations (Supporting Information Equations (5) and (6)) and detailed in Table 1. Figure 2c illustrates the correlation between band structures and effective masses, showcasing the impact of pressure on electronic charge-carrier dynamics and band gaps in Cs₂HfInBr₆ double perovskite. In the L k-space, the effective mass of electrons (m^∗) increases from 0.257 at 0 GPa to 0.495 at 16 GPa with increasing pressure. This behavior arises from the compression of the lattice under applied pressure, which reduces interatomic bond lengths and induces octahedral tilts. These structural changes modify the band edge profiles of the perovskites, as shown in Figure 5, and affect the movement of charge carriers by altering the electronic band structure [58, 60]. Consequently, the deformation of the crystal lattice and resulting changes in electronic band profiles influence the optical properties, suggesting pressure-induced band modulation effect under applied pressure [61]. However, at 25 GPa, the band gap becomes 0 eV, the material becomes metallic in nature, and electrons can move freely in the whole crystal lattice, which reduces its effective mass. As pressure compresses the crystal lattice, increased electronic localization within the crystal lattice results in higher effective mass for electrons, reducing electron mobility due to constrained movement in the tighter lattice. Though electron mobility reduces the effect of applied pressure for the Cs₂HfInBr₆ perovskite, m_e ^∗ values are still smaller than the standard MAPbI₃ (30% PCE) [62]. Therefore, deliberate reduction of electron mobility can enhance characteristics in switching, stability, or sensing applications, while the usual goal is to maximize mobility for better device performance and efficiency. Besides, Table 1 and Figure 1c show that under 0–25 GPa pressure, hole effective mass (m_h ^∗) consistently decreases from 0.181 to 0.096. This suggests higher hole mobility within the crystal lattice, facilitating faster charge transport and more effective charge extraction with an applied electric field. Semiconductors with lower hole effective mass would improve electrical conductivity and enhance charge carrier dynamics, making them suitable for applications like transistors, solar cells, and photodetectors, where efficient charge transport is crucial for energy conversion [63].

To understand the atomic orbital contribution to optoelectronic behavior of the Cs₂HfInBr₆ perovskite by influencing both CBM and VBM, the GGA–PBE function employed the total density of states (TDOS) and partial density of states (PDOS) which are evaluated and represented in Figure 5h,i from −10 to 10 eV energy range. Aligning with the band structure, the TDOS plot in Figure 5h, the top of the valence band (VBM) is predominantly localized with In and Br atoms, encompassing the Fermi level from −6 to 0 eV. The TDOS peaks attributed to B-site atoms are situated around 2.5 eV in the conduction band and −5 eV in the valence band. With increasing pressure, the TDOS plot indicates a shift of the conduction band edge toward the Fermi level more than the VBM, resulting in the band merging of both CBM and VBM at 25 GPa, where Br atom dominates the Fermi level. PDOS refines TDOS by identifying atomic orbitals contributing to electronic behavior, providing insights into energy-dependent properties and photon–matter interaction control at the quantum level. It also reveals bonding characteristics and hybridization effects on material properties by identifying orbitals influencing valence and conduction bands. In-5s and Br-4p orbitals dominate in shaping the valence band edges to the fermi level, with a slight contribution from Br-4s. In contrast, the conduction band is ~2.5 eV above the ground state and experiences hybridization induced by external pressure among In-5s, Br-4p, and Br-4s orbitals, lowering the CBM at the L point to the fermi energy as evident from PDOS plots in Figure 5i. As pressure increases, the dominance in the valence band near the fermi level shifts from In-5s to Br-4s (at 5 GPa), while in the conduction band, Cs-4s and 4p states increase, and Br-s orbital contributions decrease. At 25 GPa, band merging by Cs-s, Br-s orbitals’ hybridization at Fermi level results in a transition from semiconducting to metallic phase as also depicted in band structure (Figure 5g), where the dominance of Br-4s orbital is evident. The pressure-induced electronic energy analysis indicates band edge shifts attributing to the effect of defect energies by Hf and In ions, aligning with experimental observations for persistent structural deformation and desired band gap transformation in perovskite compounds [25]. The observed controllable band gap shifts and phase transitions in Cs₂HfInBr₆ sets it as a promising perovskite for advanced applications in spin-optoelectronics, transistors, etc., leveraging tailored carrier dynamics for improved performance and versatility beyond efficient photovoltaic devices.

3.5. Photon–Matter Interactions of Cs₂HfInBr₆ Perovskite

Optical analysis of Cs₂HfInBr₆ perovskites reveals the intricate interaction between photons and electronic structures, influencing photon absorption, electron excitation, and emission crucial for solar cells and optoelectronic devices. Understanding quantum-level light–matter interactions is paramount for optimizing halide perovskite device performance, aligning with electronic properties that facilitate tailored applications in transistors, detectors, lasers, sensors, and photonics. Implementing with GGA–PBE, optical properties of the Cs₂HfInBr₆ perovskite under different pressure conditions are evaluated, including dielectric functions, photon absorption spectra, refractive index and extinction, reflectivity, and corresponding optical loss in the energy range of 0–30 eV and illustrated in Figure 6. Modifications in lattice structure, band gap, and electronic properties induced by pressure can directly affect the dielectric function [64]. Transitions of electrons between valence and conduction bands induce alterations in dielectric behavior, influencing charge carrier polarization and mobility within the crystal lattice in response to an electric field, affecting the material’s energy storage and transmission efficiency [63]. Moreover, dielectric properties govern the interaction between incident light and collective electron oscillations (plasmons), shaping the plasmonic behavior of plasmons propagating, resonating, and interacting with incident light, subject to applications in light manipulation, sensing, and energy harvesting [65]. Comprised of the real part (ε₁(ω)) and the imaginary part (ε₂(ω)), the dielectric function is expressed as ε(ω) = ε₁(ω) + iε₂(ω). By utilizing momentum matrix components between occupied and unoccupied wave functions, equation (Supporting Information Equation (7)) from the Kohn–Sham equation was utilized to derive the imaginary dielectric constant (IDC), ε₂(ω). The ε₂(ω) shapes the optical absorption bandwidth in semiconducting perovskites, with peaks indicating resonant frequencies tied to excitonic effects and electron–hole pair (EHP) creation during interband electronic transitions [66].

Variations in ε₂(ω) dispersions with applied pressure for the HfIn based HDPs are represented in Figure 6b. The first peak in ε₂(ω) signifies absorption energy associated with transitions from valence to conduction bands. As illustrated in Figure 4b, the compound in all pressure conditions shows a subdued magnitude of IDC at E = 0 eV, suggesting negligible energy dissipation within crystal systems at zero photon energy. The initial sharp peak of IDC for the Cs₂HfInBr₆ perovskite at 0 GPa occurs at E = 4.3 eV, as depicted in Figure 6b, and shifts to E = 2.5 eV with increasing pressure. The dielectric function’s peak at 4.3 eV signifies strong absorption of vacuum UV light by the perovskite, which shifts to the visible spectrum under pressure, as evident in Figure 6c. The UV peak implies excitonic effects, influencing EHP creation and movement to regulate charge carrier dynamics [32]. At 0 GPa, the material may seem transparent to visible light, transitioning to a solar radiation absorber under increased pressure [24]. This tunability can be advantageous for these materials for designing advanced opto-spin-electronic applications.

In Figure 6a, the real part of the dielectric constant (RDC) ε₁ (ω) spectra for Cs₂HfInBr₆ under pressures ranging from 0 to 16 GPa, calculated using the Kramers–Kronig transform following equation (Supporting Information Equation (8)), demonstrates the material’s polarization response to an external electric field across the same energy range as ε₂(ω). The dipole momentum transfer matrix $$ between $$ and $$ in ε₁(ω) influence polarization behavior of the material in effect of external electric field induced charge displacement within crystal lattice directly govern the dispersion behavior. Thus, quantifying susceptibility with ε₁ (ω) is vital for understanding the material’s energy storage ability and dielectric response to electromagnetic waves portraying polarization behavior and charge displacement in the crystal lattice under an applied electric field by P(ω) = χ(ω)ε₀E(ω) = [ε₁(ω) − 1]ε₀E(ω). The oscillating electric field in a linear optical field induces polarization with frequency ω universally, using low to medium light intensities of electromagnetic waves. The real component of the dielectric constant ε₁(ω) underwent a significant rise at 0 eV until reaching its peak value, followed by a decline toward unity after an inflection point around 16 eV. The increase in pressure results in an increasing trend in real dielectric behavior, as depicted in Figure 6a. The peak magnitude in ε₁(ω) shifts from UV to visible range under pressure, with ε₁(ω) trends paralleling ε₂(ω) except at 5 GPa, where ε₁(ω) decreases from 0 GPa. Besides, higher values of ε₁ lead to higher refractive indices, which is comparable with Figure 7a, which shapes the light propagation behavior in the material. Higher dielectric constants in materials correspond to reduced charge–carrier recombination rates, thereby improving optoelectronic device performance, particularly influencing charge transport in devices such as transistors [30]. Figure 6a,b illustrates that increasing pressure leads to higher magnitudes in both the real and imaginary parts of the dielectric function at energies (E) for HfIn-based double perovskite, indicating Cs₂HfInBr₆ halide perovskites’ potential for practical optoelectronic devices. The trend in static dielectric ε₁(0) with respect to E_g indicates how the band gap and plasmonic effects influence the dielectric response at low frequencies following the relation (Supporting Information Equation 9). When the plasmon energy (ℏω_p) is comparable to or smaller than the band gap (E_g), ε₁(0) tends to deviate from unity. As depicted in Figure 3, a diminishing band gap implies a rising influence of plasmonic oscillations on the static dielectric constant ε₁(0); diminishing with higher energy signifies the prevalence of electronic transitions associated with the band gap. Also, a higher static dielectric constant for the Cs₂HfInBr₆ double perovskite implies a greater ability to polarize in response to the applied electric field, leading to increased energy storage capacity. At photon energies above 25 eV, the imaginary component of the complex dielectric function tends toward zero, while the real component approaches unity, suggesting that all phases of the perovskites demonstrate transparency with limited absorption in higher energy ranges, as also noted in the absorption and transparency edges depicted in Figure 6c.

Furthermore, the exciton binding energy is related to the static dielectric constant and effective mass of the electrons and holes. The low carrier effective mass and high dielectric constant could lead to low exciton binding energy. The In-based double perovskites Cs₂BB’Cl₆ (BB’ = InBi, AgIn, NaIn) exhibit considerably small values of exciton binding energy ranging from 18 to 650 meV [67]. We evaluated the exciton binding energy using the Wannier–Mott formula [68] (Supporting Information Equation (17)). Our calculated values of $$ are in the range of 5.74–34.26 meV, which are small and comparable to the exciton binding energy of the other double perovskites. The result of $$ at equilibrium is 34.26 meV, which is lessened to 16.43 and 5.74 meV at 3 and 5 GPa pressure, respectively. However, the values of $$ rose to 14.45 and 17.93 meV at 7 and 12 GPa, respectively. Finally, a slight declination has been observed with a value of 15.05 at 16 GPa. The pressure-dependent photoabsorption coefficient α(ω) in Cs₂HfInBr₆ double perovskite compounds, depicted in Figure 6c, elucidates the variation in light absorption ability with photon energy (eV). The assessment of the light-harvesting efficiency in practical devices relies on accurately estimating the absorption profile, which is determined using dielectric functions and photon energy following equations (Supporting Information Equations (10) and (11)). The absorption edge initiates at zero photon energy with a sharp rise, covering a bandwidth of ~20 eV and marked by three prominent peaks in accordance with the imaginary dielectric response displayed in Figure 6b. Under varied pressure conditions, Cs₂HfInBr₆ perovskite exhibits a substantial absorption coefficient, reaching a magnitude of 10⁵/cm (Figure 6c), making it highly advantageous for applications in optoelectronic devices geared toward efficient solar energy conversion comparable to other highly efficient perovskites [69, 70]. Almost across the whole EM spectrum, it is evident from Figure 6c that increasing pressure results in enhanced absorption behavior for the considered HfIn based halide double perovskite where 16 GPa applied compound dominated over others. Significantly, this Cs₂HfInBr₆ perovskite under all pressure shows photon absorption of very large magnitudes under infrared (IR) energy where the 5 GPa-based compound shows the highest magnitude, which opens horizons to the perovskite’s various IR applications under optimal pressure like IR detector and sensor, LED, communications and thermal imaging, IR spectroscopy, and so on [71]. In the visible range (inset for clarity), the absorption coefficient peak consistently increases with a leftward shift to lower energy under elevated pressures (0–16 GPa). This indicates an enhanced capacity to absorb solar energy radiation within the visible spectrum. This improvement is particularly beneficial for light-harvesting photovoltaic applications. The Cs₂HfInBr₆ perovskite exhibits its highest peak at 14.7 eV in the ultraviolet region, with a predominant absorption bandwidth in that range. Besides, absorption peaks in the energy spectrum denote resonant frequencies signifying electronic transitions and excitonic effects, including EHP generation. Higher absorption peaks, as observed for the Cs₂HfInBr₆ perovskite with pressure, suggest strong interactions between photons and electrons, contributing to enhanced conductivity, which can be observed in Figure 6d. Cs₂HfInBr₆ demonstrated a larger magnitude spanning from IR to UV spectrum improves the absorption coefficient and outperforms the other studied perovskites.

Optical conductivity (σ), which pertains to the photon absorption induced conductivity in a material, is influenced by photon energy-dependent charge carrier excitation and electronic transitions between valence and conduction bands. Figure 6d depicts the pressure-dependent optical conductivity (σ(ω)) of Cs₂HfInBr₆ from 0 to 16 GPa. The figure reveals that enhanced photon absorption corresponds to an increased rate of photoconductivity, due to its lowering band gap of the perovskite compound under applied pressure as observed in Figures 5 and 6c. The variations in σ(ω) closely mirror those in α(ω), signifying the conditioned absorption of incoming energies with k(ω) and ε(ω). At 0 GPa, the perovskite demonstrates low conductivity (σ) by equation (Supporting Information Equation (12)) in the visible region, which increases with rising pressure. The maximum σ aligns with negative ε₁(ω) at around 15 eV, corresponding to the highest absorption peak (Figure 6a,c). At 5 GPa, the perovskite exhibits the highest σ in the IR region while dominating the visible region at 16 GPa (Figure 6d). Like absorption behavior, the σ peak shifts to lower energy with increasing pressure.

Light propagation behavior with the phase velocity of the incident wave in the absorber medium is governed by refractive index n(ω), while the extinction coefficient k(ω) determines how strongly the material absorbs or scatters light and transmits light through the materials. Figure 7a,b represents the n(ω) profile and k(ω) profile under varying pressure conditions for the Cs₂HfInBr₆ perovskite. A higher refractive index aids in trapping light within the perovskite material through total internal reflection, while a higher extinction coefficient contributes to enhanced light absorption and reduced transmission. The real part of the refractive index is primarily influenced by the RDC, resulting in identical dispersions between n and ε₁(ω) shaped by the equation (Supporting Information Equation (14)). Illustrated in Figure 7a, the refractive index (n) reaches its highest peaks at low energies (2–3 eV), progressively decreasing in the high-energy region up to 20 eV, and then stabilizing at all pressure conditions, indicating maximal refractive index at low photon energies for these perovskites. Among all different applied pressure, highest value for n(ω) is observed at 16 GPa, gradually increasing from 0 GPa. A higher static refractive index ($$) greater than 3 suggests that the noble Cs₂HfInBr₆ perovskite has strong light–matter interactions, potentially leading to enhanced light trapping, and total internal reflection, which can offer advantages in sophisticated optical devices such as waveguides and photonic crystals [72]. The extinction coefficient k(ω) as depicted in Figure 7b signifies the material’s light absorption capability and photon wave propagation depth through the material, reflecting energy absorption and dissipation. The extinction coefficient (k) peaks at the absorption maximum before declining as material absorption decreases in Figure 6c. Notably, the square root relationship with the dielectric allows for a large negative RDC, resulting in a substantial imaginary refractive index at zero photon energy. This phenomenon enables the material to strongly weaken and reflect electromagnetic waves, even without significant absorption, as illustrated in Figures 6b and 7b,c. The reflectivity R(ω), elucidating the material’ s surface optical behavior, is a crucial optical parameter for materials in photovoltaic and other optoelectronic applications. Simultaneously, the optical loss function (OLF) L(ω) delineates the energy dissipation of electrons in photon interactions. The profiles of R(ω) and L(ω) under applied pressure for Cs₂HfInBr₆ are evaluated using the relations (Supporting Information Equations (13) and (14)) and depicted in Figure 7c,d, respectively. High reflectivity in optical materials can lead to the loss of incident photons, reducing device efficiency. Therefore, the reflectivity profile assists in designing fabrication strategies to minimize photon losses and enhance light absorption in devices. Under visible photon exposure, the Cs₂HfInBr₆ double perovskite exhibits reflectivity of ~9% at 0 GPa, maintaining around 10% reflections up to 5 GPa, and experiences an increase to 20% reflectivity at 16 GPa as shown in Figure 7c. Upon transitioning to the ultraviolet region, the perovskite phase displays its highest reflectivity peak at ~15 eV, corresponding to the real dielectric function ε₁(ω) in Figure 6a. The relatively lower reflectivity at low energy regions, both with and without pressure, suggests the potential of Cs₂HfInBr₆ perovskite for solar absorber, light harvesting applications, and as a coating material, given its higher reflectivity at high energy regions. As portrayed in Figure 7d, photon energy loss due to electronic excitations in Cs₂HfInBr₆ indicates that most photon energy losses occur in the UV region. In the low-energy regions, the loss function is consistently below 0.25 under all pressure conditions, while in the high-energy region, the loss function decreases with applied pressure.

3.6. Mechanical Properties

The mathematical representation of elastic constants bridges the mechanical and dynamic traits of materials, influencing their behavior under external forces and finding applications in diverse fields. The elastic tensor C_ij of a crystal offers a detailed understanding of its mechanical stability, structural integrity, and mechanical characteristics. A material’s capacity to deform under stress and return to its initial configuration after removing the load defines its elastic behavior. Anisotropic behavior, binding between neighboring atomic planes, and the material’s general structural stability can all be understood by looking at elastic constants. To comprehensively describe the mechanical properties of a cubic HfIn-based halide perovskite compound under varying pressures, three independent components of the elastic tensor, namely, C₁₁, C₁₂, and C₄₄ and the Cauchy pressure (C₁₂-C₄₄), are calculated using the finite strain theory within the first-principle DFT calculations. The elastic constants and pertinent mechanical parameters of cubic Cs₂HfInBr₆ double perovskite are detailed in Table 3 [73] and the material’s mechanical behavior is portrayed in Figure 8. The variations in elastic tensors indicate a consistent increase in these constants with the progressive application of pressure. This upward trend is likely attributed to the enhancement of B–X interactions within the 0–25 GPa range, in effect of strengthening lattice confinement corroborated by the observed variation in lattice constants under pressure, as depicted in Figure 2b. The computed elastic constants, in accordance with the established Born stability criteria illustrated in Figure 8a, affirm the mechanical stability of the investigated compound under applied pressure. The material’s ductile-brittle nature is primarily evident through variations in Cauchy pressure (C₁₂-C₄₄), representing the pressure-average shear stress nature. Figure 8a demonstrates a positive Cauchy pressure in each case, signifying the material’s ductility. The observed trend suggests that increasing pressure enhances the material’s ductile characteristics, making it more prone to plastic deformation without fracture.

Utilizing the determined single-crystal elastic constants, the mechanical parameters of the Cs₂HfInBr₆ halide perovskite’s polycrystalline form are tabulated in Table 3, evaluated through Voigt–Reuss approximations [74]. In accordance with Hill’s theory, bulk (B) and shear (G) modulus for the considered perovskite exhibit elevated values, indicative of higher crystal strength against both compression strain and shear deformation. Increased pressure correlates with increased magnitudes for both parameters, as illustrated in Figure 8b as the B/G–Pugh’s ratio variations for the perovskite indicate ductile characteristics, surpassing the critical threshold of 1.75. This observation suggests that the applied pressure consistently enhances the ratio, affirming the ductile nature of the material. Quantifying resistance to longitudinal tension via Young’s modulus (Y) using B and G values reveals that increasing pressure results in higher Young’s modulus, signifying greater stiffness and reduced deformation under stress, affirming the suitability of the noble Cs₂HfInBr₆ double perovskite for structural integrity and diverse applications, including electronics and structural materials.

The Poisson ratio (ʋ) in perovskites describes the material’s tendency to contract laterally when compressed longitudinally, and brittle to ductile transition occurs at 0.26. Perovskite crystals demonstrate diverse phonon behaviors on specific crystal orientations, with (112) exhibiting shear and quasi-shear phonons and (100) tending only to longitudinal phonons [75]. This influences crystal orientation and phonon generation, resulted in thermal expansion coefficients or deformation potential along crystal axes, contributing to our understanding of the Poisson ratio. Cs₂HfInBr₆ double perovskite exhibits ductility at ambient pressure (0.33) as in Figure 8b, which increases to 0.39 at 25 GPa, reflecting the crystal’s ability to deform without fracture, indicative of a favorable atomic structure for plasticity. This observed behavior suggests potential fabrication suitability in thin film optoelectronics, flexible electronics, or structural phase change materials [75, 76].

Anisotropy influences directional mechanical properties, while average sound velocity impacts electronic and photonic device efficiency, and the Debye temperature characterizes maximum atomic vibration energy, aiding thermal management in electronics. The synergistic interplay of these factors is essential in designing perovskite-based devices with optimized mechanical, electronic, and thermal properties, advancing applications in electronics, photonics, and energy conversion. The anisotropic index in perovskite crystals, impacting stress, strain, and bonding strength along crystallographic planes, is quantified by the Zener anisotropy factor (A) for the cubic Cs₂HfInBr₆ halide perovskite, as detailed in Table 3. The decreasing Zener anisotropy factor (A) from 0.22 at 0 GPa to 0.07 at 25 GPa, as portrayed in Figure 8c, indicating nonunity, suggests a transition toward anisotropic elastic behavior in the material. This implies potential mechanical strength and ductility improvements, supported by Cauchy pressure and B/G behavior, as shown in Figure 8a,b. Reduced anisotropy in Cs₂HfInBr₆ under pressure enhances its potential as an energy absorber material for high-pressure conditions, as observed in Figure 8c, with implications for pressure-sensitive devices like sensors or switches. This pressure-tunable mechanical anisotropy opens possibilities for the Cs₂HfInBr₆ in designing metamaterials, potentially including negative compressibility or cloaking. The anisotropy reduction under pressure may improve optical, thermoelectric, and energy storage device performance, ensuring uniform stress distribution and potentially enabling pressure-controlled LEDs or lasers for diverse applications like displays, optical communication, and biosensing.

The variations in calculated average velocity (v_m) and debye temperature (Θ_D) are illustrated in Figure 8c under applied pressure of 0–25 GPa for the HfIn-based cesium-bromide double perovskite. The Debye temperature data in Table 3 for Cs₂HfInBr₆ perovskite reveal a nonlinear trend, where values ascend from 118.96 K (0 GPa) to 173.89 K (25 GPa). This suggests a pressure-driven elevation in the average vibrational energy within the crystal lattice with Debye hardness, implying improved thermal conductivity and optimal device efficiency. The calculated mean velocity of sound (v_m) in Cs₂HfInBr₆ perovskite is the highest at 1727.94 m/s, exhibiting a pressure-dependent increase from 1518.80 m/s at 0 GPa to 1750.97 m/s at 12 GPa and a slight decrease at 25 GPa as in Table 3. This variation implies a positive correlation between pressure and longitudinal acoustic velocity, influencing the material’s lattice dynamics and having implications for device design relying on acoustic properties. However, Table 3 also displays the estimated melting temperatures (T_m) for the perovskite under different pressures using the C₁₁ value. The data indicate that ambient temperatures can be employed to manufacture perovskite compounds. The melting temperatures show a coherent trend, increasing from 882.19 to 1807.69 K as pressure rises. This behavior is consistent with principles of phase transitions, where higher pressure leads to a more ordered lattice structure, demanding elevated temperatures for the solid-to-liquid transition. The compression-induced stronger atomic interactions require increased thermal energy for bond breaking during melting. These insights into pressure-dependent melting temperatures are valuable for industrial applications in high-pressure environments.