The structural, magnetic, optoelectronic, and mechanical characteristics of NaGeX3 perovskites under pressure for soler-cell applications

This study examines the physical properties of germanium-based halide perovskite through Density Functional Theory (DFT) computations. The physical, optical, mechanical, and magnetic properties of NaGeX3 (X = Cl, Br, and I) were examined with the effects of hydrostatic pressure applied externally. The compounds were subjected to pressure variations ranging from 0 to 5 GPa. The results indicate a decrease in the band gap from the infrared to the visible spectrum. For NaGeCl3, NaGeBr3, and NaGeI3 the band gap decreased from 0.766 eV, 0.497 eV, and 0.400 eV to 0 eV, respectively, indicating the metallic behavior. The mechanical properties of NaGeX3 (X = Cl, Br, and I) demonstrate that for all three compounds, Bulk Modulus (B), Shear Modulus (G), Young’s Modulus (E), Poisson’s ratio (ν), and Pugh’s ratio B/G all increase with increasing pressure. It demonstrates that all these NaGeX3 (X = Cl, Br, I) compounds are ductile in nature. The compounds are determined to be diamagnetic based on their magnetic property investigation, which reveals no notable changes in behavior up to 5 GPa of rising pressure. To gain a better understanding of the properties of the material when incident light strikes its surface, researchers also looked in at optical absorption, reflectivity, dielectric constants, refractive index, conductivity, and loss functions. Pressure-induced NaGeX3 perovskite compounds, where X = Cl, Br, and I, show an increase in dielectric constant as pressure rises, suggesting a decrease in charge carrier recombination rates and a possibility for higher optoelectronic device efficiency. For all NaGeX3 compounds (where X = Cl, Br, and I), the maximum absorption coefficient peaks are located around 3 eV, indicating that increasing pressure increases optical conductivity. Additionally, they have significantly low reflectance throughout the visible spectrum and very narrow band gap, which indicates significant absorption and the possibility of effective Near-Infrared (NIR) Sensors, photodetector etc applications.


Introduction
Scientists have focused on a class of structures termed perovskites, which possess the crystal structure of ABX 3 with a wide range of chemical compositions [1] where 'A' indicates a cation, 'B' indicates another cation, and 'X' indicates an anion, usually a halogen or oxide [2].The notable chemical [3] and physical characteristics [4] of perovskites like optical, electrical, mechanical, and thermal capabilities, have drawn the interest of many researchers [5][6][7][8].The uses of perovskite materials are increasing in solar energy research [9,10], particularly in light-emitting diodes (LEDs) [11] and photovoltaics [10].Furthermore, these types of compounds are found to have applications in insulators [12], conductors [13,14], semiconductors [14] and superconductors [15].Their promising characteristics lead to the development of efficient energy technologies, which in turn play a vital role in their roles.For nearly two decades, researchers have been examining these materials to better understand how they function as the absorber layer in solar cell applications [16].
Perovskite structures exhibit a wide range of physical properties, including dielectric, photocatalytic, ferroelectric, piezoelectric, magnetic, superconducting, and ionic conductivity.Because of their versatility, these structures have a wide range of possible uses in science and technology [17][18][19][20][21][22][23][24].Using first-principles calculations, the physical and chemical properties of the compounds have been examined [25][26][27][28][29][30][31][32].In recent times, band gap tuning of semiconductors and insulators has become an interest of research where the band gap of the compounds can be reduced using hydrostatic pressure.Consequently, the compounds show better optical, electrical, and mechanical properties and change from being semiconductors to metallic structures [33,34].Ghani et al (2023) explore the band gap properties of RbXCl 3 (X = Ge, Sn and Pb) halide perovskites using the density functional theory (DFT) with the Generalized Gradient Approximation (GGA) method.The calculated band gaps for RbGeCl 3 , RbSnCl 3 , and RbPbCl 3 are reported to be 1.03 ev, 1.07 ev, and 2.13 eV, respectively, showing their potential for solar cell applications [35].In another study, Ahmad et al (2023) investigated the influence of hydrostatic pressure on RbGeX 3 (X = Cl, Br, and I) perovskites, finding a decrease in bandgap and changes in volume and lattice parameters.This study, using density functional theory, reveals a narrowing bandgap with increasing pressure, showing potential for different optoelectronic and solar-cell applications [36].The physical properties of lead-free CH 3 NH 3 SnI 3 perovskite, a promising alternative due to its eco-friendliness and wider absorption spectrum, have been explored by Paschal et al (2020).Through density functional theory calculations, they investigated the structural and electronic features using three exchangecorrelation functionals: PBE, PBEsol, and LDA.The study reveals a direct bandgap at gamma symmetry points, with bandgap values of 1.12 eV (PBE), 0.98 eV (PBEsol), and 0.46 eV (LDA).Comparing these results with experimental data, the PBE functional shows the best agreement [37].The potential of metal halide perovskites for narrowband photodetectors was highlighted by Lu et al's review where emphasis was put upon recent progress in perovskite photodetectors [38].Vashishtha et al's review explores recent advancements in nearinfrared (NIR) perovskite light-emitting diodes (LEDs) which underlines their potential for lighting applications [39].The study delves into structural frameworks and challenges in achieving high-performance NIR-LEDs which offers insights into strategies for further improvement.A study on the NaGeX 3 (X = Cl, Br, and I) compounds has already been conducted in a research article but the pressure-induced properties of these compounds have not been analyzed yet using first principle calculations [40].Therefore, this study considers the impact of pressure-induced phase changes on structural, electronic, optical, mechanical, and magnetic properties, offering insights into the development of innovative energy materials.
The manuscript is structured as follows: section 2 provides an overview of the theoretical methodologies utilized for property determination.In section 3, the results of computational analysis are presented and discussed.Finally, section 4 offers conclusions derived from the study.This research contributes valuable insights that may encourage materials scientists to explore the effects of hydrostatic pressure on halide materials, with potential implications for diverse device applications.The main objective of this work is to investigate the geometrical, electronic, magnetic, optical, and mechanical properties of the cubic halide perovskite NaGeX 3 (X = Cl, Br, and I) under different hydrostatic pressures.By using varied hydrostatic pressures, the lattice constant, compound volume, formation enthalpy, phase transitions, and the separation between cations and anions have also been analyzed.From the methodologies and findings outlined in this study, it is expected that NaGeX 3 (X = Cl, Br, and I) can be a promising candidate which may pave the way for new potential optoelectronic applications in various fields.

Computational method
The theory behind perovskites is based on density functional theory (DFT) methods, combined with the planewave pseudo-potential (PW-PP) total energy approach and the CASTEP code [41].The theory of perovskites is the basis for these computations.Using the Vanderbilt-type ultrasoft pseudo-potential [42], we investigate the valence electron-ion core interaction.The generalized gradient approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) functional are used to calculate the exchange-correlation impact [43].The cut-off energy of 700 eV has been selected for the plane-wave basis set to differentiate between core states and valence levels.The k-point mesh with dimensions of 8 × 8 × 8 is utilized to incorporate unique point sampling throughout the Brillouin zone [44].The Monkhorst-Pack approach is used to carry out this implementation.The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization method is also employed in geometry optimization to rapidly determine the structure with the lowest energy [45].We found that the ionic Hellmann-Feynman force was weaker than 0.01 eV Å, the greatest displacement was smaller than 5 × 10 −4 Å, the maximum stress was around 0.02 GPa, and the total energy difference was around 5 × 10 −6 eV/atom.There was no deviation from the expected range for any of these values.These values were considered to determine the correct convergence factors.Here, we optimize the crystal structure of the perovskite materials under investigation and explore their intriguing physical properties in the hydrostatic pressure range of 0 to 5 GPa.The goal is to obtain optimal performance.To construct the ideal crystal structure, the software VESTA is used [46] and to determine the elastic constants, the 'finite-strain' method of the CASTEP code is utilized in this calculation [47].

Structural properties
The compound illustrated in figure 1 is NaGeX 3 (where, X = Cl, Br, and I) which is a metal halide semiconductor with a cubic crystal structure.It belongs to space group Pm-3 m (221) [48].The unit cell consists of five atoms, which are illustrated as a single formula unit.Na is at the 1a Wyckoff position (0, 0, 0), Ge is at the 1b position (0.5, 0.5, 0.5) of the body-centered position, and X (Cl, Br, and I) is at the 3c Wyckoff position (0, 0.5, 0.5) [49].Initially, the atom locations and lattice parameters were optimized using normal stress.The properties were then determined using hydrostatic pressure of up to 5 GPa.
A representation of the compound's formation energy is given by equation (1).In this equation, the components Es (Na), Es (Ge), and Es (X) represent the energies of the atoms Na, Ge, and X (Cl, Br, and I), respectively.Etot (NaGeX 3 ) represents the total energy of the unit cell (NaGeX 3 ), and N represents the number of atoms that are contained within the unit cell [50].The calculation of the electrical, optical, and mechanical properties involves the utilization of all of the parameters that were presented before.Table 1 illustrates the estimated formation enthalpy, lattice constant, and unit cell volume.
The negative values of the formation enthalpy for this material indicate its thermal stability, signifying that the compound NaGeX 3 (X = Cl, Br, and I) is energetically favorable to form.As the hydrostatic pressure increases, the formation enthalpy of NaGeX 3 (X = Cl, Br, and I) decreases, as shown in table 1.This suggests that under high-pressure conditions, the stability of the compound is decreased compared to normal atmospheric condition.However the compound stay stable at 5 GPa hydrostatic pressure as the formation energy is negative at that pressure.The decrease in lattice parameters of NaGeX 3 (X = Cl, Br, and I) with increasing pressure results in a reduction in the bond length and volume of the unit cell.Consequently, the electrons within the atoms are held closer to the nuclei, leading to stronger bonds between the atoms.As a result, more energy is required to break these bonds as observed by an increase in the formation enthalpy of the compounds.This phenomenon highlights the role of pressure in influencing the stability and bonding within NaGeX 3 (X = Cl, Br, and I), contributing to its overall thermal stability.Figure 2 illustrates that the lattice constant and unit cell volume of NaGeX 3 (X = Cl, Br and I) drops quickly with increasing hydrostatic pressure.As pressure rises, there is a discernible decrease in the bond lengths of the NaGeX 3 compounds (where X is Cl, Br and I).This decrease denotes an increased interaction between the atoms Na-X and Ge-X by bringing them closer together.As a result, the unit cell volume as a whole drops, and the lattice constants contract accordingly.This occurrence demonstrates the influence of external pressure on the structural characteristics of the compounds.

Electronic properties
The investigation of the electrical characteristics, such as band structure and density of state (DOS), has helped us gain a better understanding of the phase transition that occurs in the NaGeX 3 (X = Cl, Br, and I) perovskite, also known as the transition from a semiconductor to a metal.The electronic band structure and the DOS were simulated at a variety of hydrostatic pressures, as shown in figures 3 and 4.This was accomplished by employing the generalized gradient approximation (GGA) of the Perdew-Berke-Ernzerhof (PBE) exchange-correlation theoretical framework.According to the semi-conductive theory, the band at the Fermi level is an essential component in gaining an understanding of the material's physical behavior.When no external pressure is applied at the R-point, the bandgaps of NaGeCl 3 , NaGeBr 3 , and NaGeI 3 are 0.766 eV, 0.497 eV, and 0.400 eV, respectively.The GGA-PBE functional is recommended for pressure studies on materials as it yields more accurate results than the used functional and the difference between the bandgap (Eg) and the band structure with applied pressure is different [51,52].To find the band structure of the NaGeX 3 (X = Cl, Br, and I) perovskite, we used the GGA-PBE functional.Hydrostatic pressures ranging from 0 GPa to 5 GPa were used to calculate the band structure.The VB and CB peaks start to shift towards E F at the R-point as the pressure starts to rise. Figure 3 shows that at 5 GPa, 3 GPa, and 3 GPa, respectively, the bandgap for NaGeCl 3 , NaGeBr 3 , and NaGeI 3 is zero.According to figure 3, the valence band maximum (VBM) and conduction band minimum (CBM) of all three halides reached the Fermi level at 5 GPa, which signifies a change from semiconductor to metallic behavior [53].It was at this stage that NaGeX 3 started to display metallic characteristics.showed negative correlation with our reference work and hence, we decided to stay with the GGA-PBE functional based values.
Figure 4 illuminated below demonstrates the density of states (DOS) of the compound NaGeX 3 (X = Cl, Br and I) as a function of photon energy under different hydrostatic pressures.The (DOS) graph demonstrates the energy distribution of different states within materials.From the DOS graph, the electron density of a compound can be observed, and the maximum peak of the graph demonstrates the maximum electron density.This analysis helps to understand the thermal and electronic properties of materials [54].There was no sudden change of peaks observed towards the Fermi Level with increasing pressure 0 GPa to 5 GPa which indicates the semiconductor nature of the compounds.

Band structure
However, when pressure is applied, the peak moves toward lower energy from the Fermi level for pressure 0 GPa to 5 GPa for all compounds.This means that the band gap must be reduced as pressure is applied for all compounds.The pressure also reduces the peak for all compounds, indicating that the electron density might be reduced with pressure.

Optical properties
Since perovskite material is utilized as an absorber layer in solar cells [16], its optical qualities are crucial for determining how to employ the compound in optoelectronic devices.This is because the performance of the material depends on these features.Under hydrostatic pressure, the compound NaGeX 3 (X = Cl, Br and I) is

Calculated data Compound
Other work [39] This work studied for its optical characteristics, which include reflectivity, conductivity, refractive index, absorption, dielectric function, and loss function.As the hydrostatic pressure is increased, the effects are almost the same for all compounds.

Absorption
The optical absorption spectra of NaGeX 3 (X = Cl, Br and I) perovskite, which were analyzed, are shown in figure 5 Cl, Br, and I are the elements that they represent., it is essential to have a solid understanding of the optical absorption coefficient and solar energy conversion efficiency [55].It is possible to define the optical absorption coefficient as the degree to which light of a certain wavelength and energy penetrates a substance before being absorbed by the material.The observed absorption peak values for NaGeCl 3 , NaGeBr 3 , and NaGeI 3 perovskite compounds under varying hydrostatic pressures present interesting insights into their optical characteristics.Starting with NaGeCl 3 , the absorption peak at 15 eV energy increased from 2.25 × 10 5 cm −1 at 0 GPa to 2.5 × 10 5 cm −1 at 5 GPa.This signifies a notable enhancement in the material's ability to absorb light with increasing pressure, suggesting a favourable trend for optoelectronic applications.Similarly, NaGeBr 3 exhibited a rise in absorption peak from 2 × 10 5 cm −1 at 0 GPa to 2.25 × 10 5 cm −1 at 5 GPa, indicating improved light absorption capabilities under pressure.Comparatively, NaGeI 3 displayed a rise in absorption peak from 1.9 × 10 5 cm −1 at 0 GPa to 2.2 × 10 5 cm −1 at 5 GPa, showing a slightly lower but still significant enhancement.These results suggest that all three compounds experience an increase in their ability to absorb light across the studied pressure range, with NaGeCl 3 showing the highest enhancement.This trend implies that as pressure increases, the compounds become more efficient at converting incident light into electronic excitations, which is crucial for applications in photovoltaics and optoelectronics.The enhanced absorption coefficients and tunable optical properties of NaGeX 3 (X = Cl, Br, and I) perovskite compounds under pressure offer promising avenues for solar energy harvesting, optoelectronic devices, sensors, transparent conductive films, energy-efficient coatings, and quantum dot technologies.

Conductivity
Figure 6 shows the Optical conductivity (σ) (real part) of the compound NaGeX 3 (X = Cl, Br, and I) across various hydrostatic pressures, encompassing photon energies up to 30 eV which is accountable for photoconductivity [56].The conductivity of the material is the result of the substance's ability to liberate free carriers when it absorbs energy.This resemblance exemplifies how the behavior of the material in terms of conducting electricity is comparable to its capacity to absorb energy.
The increase in the real part of conductivity peaks for NaGeX 3 (X = Cl, Br, and I) compounds, particularly from 3.5 S m −1 to 4.5 S m −1 for NaGeCl 3 , 3.6 S m −1 to 4.6 S m −1 for NaGeBr 3 , and 4 S m −1 to 5.5 S m −1 for NaGeI 3 , signifies an enhancement in their electrical conductivity under increasing hydrostatic pressure from 0 GPa up to 5 GPa.This suggests that these materials become more efficient in conducting electricity as pressure is applied.The higher conductivity values at elevated pressures indicate improved charge carrier mobility within the compounds, which is crucial for various electronic and optoelectronic applications.The rise in the imaginary part of conductivity peaks for NaGeX 3 (X = Cl, Br, and I) compounds, such as the increase from −2 S m −1 to 3 S m −1 for NaGeCl 3 , −2 S m −1 to 2.6 S m −1 for NaGeBr 3 , and −2.4 S m −1 to 2.6 S m −1 for NaGeI 3 , indicates an elevation in their absorption capability with increasing hydrostatic pressure.This suggests that these materials become more effective in absorbing incident light energy as pressure is applied.The higher values of the imaginary part of conductivity at elevated pressures imply increased absorption of photons by the materials, which is beneficial for various optoelectronic applications such as solar cells and photodetectors.

Dielectric function
When it comes to optoelectronic devices, the static peak of the dielectric function is a crucial parameter that plays a significant role in defining the charge carrier recombination rate as well as the overall efficiency [57].An increase in the dielectric constant of a material is an indication of a decrease in the rate at which charge carriers recombine.The real and imaginary components of the dielectric constant for pressure-induced NaGeX 3 (X = Cl, Br, and I) up to 30 eV of light energy are shown in this figure, respectively.The pressure range for this dielectric constant is from 0 to 5 GPa. Figure 7 demonstrates that the UV region's static peak of the dielectric constant increases in both the real and imaginary sections of NaGeX 3 (X = Cl, Br, and I) perovskite compounds when pressure increases.The static value of the dielectric constant is typically lower for materials with a larger band gap [58].Because of this, the pressure-induced NaGeX 3 (X = Cl, Br, and I) metal halide has a higher static dielectric constant.Applying pressure causes a decrease in the band gap, which is the reason for this  phenomenon [59].The imaginary component of a material's dielectric function is connected to the band structure of the material, which indicates the nature of the material's absorption.As pressure rises, the imaginary component peaks of the dielectric functions for all compounds get larger.Corresponding to the results of the absorption spectra given in figure 5, the curve shifts towards the low-energy region.For all pressure-induced NaGeX 3 (X = Cl, Br, and I) compounds, the imaginary portion of the dielectric constant approaches zero at higher energy regions (around 20 eV), while the real part approaches unity.Among its properties, the dielectric constant has this quality.Based on this finding, it follows that all pressure-induced samples are very transparent and, thus, show little absorption in the high-energy region (above 20 eV). Figure 5 shows the profiles of the absorption coefficients, and this is the connection between them.

Loss function
The loss function indicates how much energy a compound loss for optoelectronic transformation.Here the peak of the graph indicates the value of loss function.Greater the peak greater the loss function.Figure 8 indicate that the loss function increase with increasing pressure for all compound.However the increase in not same for all compounds.In case of NaGeCl 3 it start increasing gradually but it jumped suddenly when the 5 GPa pressure is applied.But in case of NaGeBr 3 in increases gradually and it increasing rate is lowest among these three compounds.The NaGeI 3 shows same nature as NaGeBr 3 but it increasing rate is higher than NaGeBr 3 .

Refractive index
The refractive index of a substance is determined by dividing its incident ray angle (θ) by its refractive ray angle after passing through a medium [60].The positive part shows how much the material can change the incident light's speed.The imaginary part is a representation of the light's energy lost through the compound.For each of these compounds, the refractive index increases in the low energy region and progressively decreases in the high energy region.The dielectric function values are computed in relation to the material's refractive index.There is a relation between the real and imaginary components of the refractive index and the dielectric function.The relationship is mentioned below: e w e w e w = +i Dielectric Function, 2 m w e w e w The portion of the refractive index (μ 1 ) that appears in figure 9 for each NaGeX 3 (X = Cl, Br, and I) compound.This value gradually decreases as the energy level increases.The increase in the real part of the refractive index peaks for NaGeX 3 (X = Cl, Br, and I) compounds, such as the rise from 2 to 2.4 for NaGeCl3, 2.2 to 2.6 for NaGeBr 3 , and 2.4 to 3 for NaGeI 3 , signifies an enhancement in their light-bending ability with increasing hydrostatic pressure.This suggests that these materials exhibit stronger light confinement and higher optical density as pressure is applied.The increasing trend in the refractive index peaks indicates a shift towards higher photon energies, implying a greater ability to interact with light in the visible and UV regions.NaGeI 3 demonstrates the highest refractive index peak at both ambient and pressurized conditions, indicating its potential for optical devices requiring high refractive indices.The imaginary part of the refractive index for NaGeX 3 (X = Cl, Br, and I) compounds, such as the rise from 1 to 1.5 for NaGeCl 3 , 1.3 to 1.4 for NaGeBr 3 , and 1.5 to 1.8 for NaGeI 3 , indicates an escalation in their light absorption capacity with increasing hydrostatic pressure.This suggests that these materials become more effective at absorbing photons in the specified energy range as pressure is applied.The higher values of the imaginary part signify increased light attenuation within the material, implying enhanced absorption and reduced transmission.This characteristic is crucial for applications requiring light absorption, such as photodetectors, solar cells, and optical sensors.NaGeI 3 exhibits the highest imaginary refractive index at both ambient and pressurized conditions among the compounds, indicating its potential for efficient light absorption in the specified energy range.

Reflectivity
Reflectivity is a crucial optical property of perovskite materials that significantly impacts their performance in photovoltaic applications.The perovskite compounds NaGeX 3 (X = Cl, Br, and I), as depicted in figure 10, exhibit a notable low reflectance across the visible spectrum, with values around 12% for Cl, 14% for Br, and 16% for I. Interestingly, the reflectance increases in the infrared region and reaches its peak in the ultraviolet range.Particularly for NaGeCl 3 , the most substantial peak is observed in the 15-20 eV range.When X is changed to Br or I, the magnitude of the reflectivity spectra shifts, yet the positions and energies of the peaks remain relatively constant, typically in the 13-15 eV and 10-12 eV ranges, respectively.The impact of pressure on the reflectivity spectra is minimal, with the greatest peak mostly remaining in the infrared region but slightly expanding.Notably, among all the compounds, NaGeI 3 exhibits the highest variability in its reflectivity peak under pressure.The low reflectance values across these compounds signify a strong absorption capability and significant transmittance, making them promising candidates for solar cell applications.Moreover, their relatively low reflectivity values in the low-energy regions, regardless of pressure, suggest potential for efficient solar energy absorption.Additionally,  their higher reflectivity in the high-energy region indicates suitability as coating materials to mitigate solar heating effects.These findings underscore the potential versatility and applicability of NaGeX 3 (X = Cl, Br, and I) perovskite compounds in solar energy harvesting and related technologies.

Mechanical properties
In this section the elastic constants are calculated and discussed from the simulated value.Because the mechanical stability depends on elastic constant.A compound is mechanically stable when it satisfy Born stability criteria [61].Which is, Table 3 demonstrate the Born stability criteria of elastic constant for all different halide of NaGeX 3 (X = Cl, Br, and I) in all applied pressure.All the calculated value of elastic constants are positive which means it satisfy the Born stability criteria.Table 3 demonstrates that current study is valid because the calculated elastic constants are corresponded with known theoretical results.
There are another useful matric which is called the Cauchy pressure (C 12 -C 44 ).By the value of the Cauchy pressure we can predict the mechanical nature of a materials.It can predict that the materials is either brittle or ductile.The positive value of the matric indicate ductility and negative value indicate brittleness of a materials [62].So, from table 3 we find that the value Cauchy pressure for all compounds (NaGeCl 3 , NaGeBr 3 , and NaGeI 3 ) in applied hydrostatic pressure is positive which indicate that all compounds are ductile in nature even after applying pressure [63].
The analysis of the mechanical properties of a materials it is important to calculate the bulk modulus B, shear modulus G, Young's modulus E, B G ratio, and Poisson's ratio ν.We use some formula to calculate those properties which is listed below, ( ) The value of those properties are calculated and listed in table 4 by applying these formula with computed value of elastic constants.In addition, table 4 illustrates that there is a substantial degree of concordance between all of the mechanical properties that were calculated and the outcomes of the theoretical calculations.It is one of the most important mechanical properties that represent the stiffness of a material and the bulk modulus of the material.There is a correlation between the rise in hydrostatic pressure and the increase in the predicted bulk modulus of the compound for all halides.This suggests that the ductility of those compounds decreases as the pressure increases.Another significant quantity is the shear modulus, which indicates the degree to which a material undergoes plastic deformation when subjected to an external force.According to the data shown in table 4, the shear modulus (G) values of all compounds increase as the hydrostatic pressure increases.This indicates that the stiffness of the compounds increases as the pressure increases.When it comes to the Young's modulus of any compound, a similar pattern may be seen.Among the many important metrics that may throw light on the bonding and flexibility of a material, Poisson's ratio (ν) is one of the most important.One can determine the ductile or brittle character of a compound by calculating its critical value, which is 0.26 [64].Based on table 1, the molecule NaGeCl 3 has a value of v that increases as pressure increases, ranging from 0.32 to 0.34.Accordingly, NaGeCl 3 is ductile by nature, and pressure causes it to become more so.The nature of NaGeBr 3 is similar to that of NaGeCl 3 , with a value of ν ranging from 0.31 to 0.32; however, it appears less ductile.NaGeI 3 has a rising value of ν, which ranges from 0.28 to 0.33.Therefore, NaGeI 3 has the same nature as the rest.NaGeCl 3 has the highest ductility in nature out of all three of these compounds, though.The mechanical properties of NaGeX 3 (X = Cl, Br, and I) are compared with other works represented in table 5. Nonetheless, we can observe that a compound becomes more ductile as pressure rises.All halides experienced this, but NaGeI 3 experienced the greatest fluctuation from 0 GPa.NaGeI 3 also shows a peculiar behaviors of V because for other halide it increase with increasing pressure but for I it decrease first then start increasing.Another parameter to forecast the mode of failure of materials is the Pugh's ratio B G , ( ) which is determined by dividing the bulk modulus by the shear modulus.Pugh's ratio's critical value in this instance is 1.75, which can be used to distinguish between a material's brittleness and ductility [71].Table 3 shows that all compounds have a value of B G larger than 1.75, indicating that all compounds are ductile by nature.The value of B G has grown in response to rising hydrostatic pressure, indicating that NaGeCl 3 and NaGeBr 3 ductility is rising as pressure rises.However, only NaGeI 3 exhibits considerable variation; the value of B G increases with increasing pressure after initially decreasing.
The ELATE program was used to create three-dimensional (3D) graphical representations showing the direction dependence of Young's Modulus, Shear Modulus, and Poisson's Ratio at pressures of 0 and 5 GPa to investigate the anisotropic nature.The graphical representations are shown in figures 11-13, respectively.Spherical 3D plots show anisotropy, whereas spherical plots show perfect isotropy.The analyzed compounds exhibit anisotropy in every direction, as seen in the 3D graphs in figures 11-13.At 5 GPa pressure, the spherical  3D plots show a greater divergence compared to 0 GPa pressure, indicating that the three compounds are more anisotropically influenced by the pressure.

Magnetic properties
The magnetic property shows the characteristics of a material in the presence of magnetic field.The magnetic properties of a material are primarily associated with the arrangement and behavior of its electrons.The calculations were performed with electrons are polarized at co-linear positions.Here half of the electrons are at upward direction (denoted as α) and the other half are at downward directions (denoted as β).After calculating the band gap and Density of states of the compounds when electrons are polarized, two different lines are observed in figure 15.From the graph of the Band structure in figures 14 and 15, it is observed that both 'α' and 'β' lines superimposed each other.They lies on the same lines and after increasing hydrostatic pressures they stays at the same superimposed positions.Also the density of states lines lies on the similar path but exact the  They are like mirror image of each other which denotes the diamagnetic behavior of the compounds [69].When pressure is increased, the highest peak of the compounds NaGeX 3 (X = Cl, Br, and I) decreased gradually.Same phenomena is denoted for changing the halide the peak is highest for Cl, it decreased for Br and lowest for I.There another phenomena is denoted that decreasing of highest peak due to applying pressure is highest in Cl.It decrease less in Br and I shows lowest decrease in peak due to pressure.

Conclusion
Using density-functional theory (DFT) simulations, the mechanical, optical, structural, electrical, and magnetic characteristics of the NaGeX 3 (X = Cl, Br, and I) compound were investigated in a cubic perovskite structure.Here, X stands for Cl, Br, or I.With increasing external pressure, the lattice properties and cell volumes gradually decrease.As a result of increasing external pressure, the band gap closes, and the material's behavior changes from semiconducting to metallic.The material's optical conductivity and absorption spectra both show a significant improvement when compressed, suggesting its potential use in solar cells.Optoelectronic devices may benefit from pressurized and non-pressurized compounds due to their elevated static dielectric constants.The band gap of NaGeX 3 perovskite compounds (where X = Cl, Br, and I) decreases dramatically as the pressure increases, causing them to change behavior from semiconductor to metallic.This pressure-induced change is accompanied by a decrease in atom-to-atom bond length, lattice constant, and cell volume, facilitating easy electron transitions between valence and conduction bands with minimal photon energy variation.The compounds also demonstrate ductile properties, as evidenced by Poisson's ratio (ν) values ranging from 0.28 to 0.34 and Pugh's ratio B G ( ) greater than 1.75.NaGeI 3 's volume initially decreases under pressure before rising, indicating its potential for varied applications.Moreover, the compounds exhibit diamagnetic behavior unaffected by external pressure.The optical properties of NaGeX 3 (X = Cl, Br, and I) perovskite compounds, including absorption, reflectivity, dielectric constants, refractive index, conductivity, and loss functions, were investigated and the fact that is to be highlighted is that these properties returned favorable output indicating narrow band gap, which indicates significant absorption and the possibility of application in the domain of Near-Infrared (NIR) Sensors and photodetectors.The increased UV-region dielectric constant peaks under pressure suggest reduced charge carrier recombination rates, enhancing optoelectronic device efficiency.NaGeCl 3 particularly shows a notable absorption coefficient peak of 2.5 × 10 5 cm −1 in the 13-15 eV UV region.Overall, the compounds exhibit low reflectance across the visible spectrum, indicating strong absorption and promising potential for further scientific applications.

Figure 2 .
Figure 2. The change in Lattice Constant (a) and Unit Cell Volume (V) of NaGeX 3 (X = Cl, Br and I) under different pressure.
(X = Cl, Br, and I) with other works.It also shows the variations of Band Gap values when other functionals like GGA-PBESol.The values obtained via use of GGA-PBE functional stays more true to the values observed in the reference works while use of GGA-PBESol
represents the real & imaginary part of the dielectric function while μ (ω) & k (ω) represents the real & imaginary part of the refractive index.

Figure 8 .
Figure 8. Calculated pressure induced spectra of Loss Function of NaGeX 3 (X = Cl, Br, and I).

Figure 14 .
Figure 14.Calculated Band Structure of NaGeX 3 (where X = Cl, Br, and I) with spin-up and down channels calculated.

Figure 15 .
Figure 15.Calculated Band Structure of NaGeX 3 (where X = Cl, Br, and I) with spin-up and down channels calculated.

Table 1 .
The computed values of lattice parameter (a), volume (V), formation enthalpy (ΔE f ), and Band Gap of NaGeX 3 (X = Cl, Br, and I) under different pressures.

Table 2
demonstrates the comparison of Band gap of NaGeX 3

Table 2 .
Comparison of the band gap values of other work with this work.

Table 3 .
The Elastic stiffness constants, C ij (in GPa) of the compounds NaGeX 3 (X = Cl, Br, and I) under different pressure.

Table 4 .
The mechanical characteristic of the compound NaGeX 3 (X = Cl, Br, and I) under different pressure.

Table 5 .
Comparison of Mechanical properties with other works.