First-principles investigations on the conducting photocatalytic behaviour in SrZrxGe1−xO3 (x = 1, 0.96, 0.92 and 0.88)

Structural properties

SrZrO₃ (Orthorhombic) is Perovskite crystal passing in the orthorhombic Pnmaspace group where the Sr²⁺ is bonded in an 8-coordinate geometry to eight O²⁻ atomsto Fig. 1. Sr–O bond distanceisrange from 2.52 to 2.90 Å. Zr⁴⁺ bonds with six O²⁻ atoms to produce corner-sharing ZrO₆ octahedra. The corner-sharing octahedral tilt angles are 24°. All Zr-O bond lengths are the 2.10 Å. There are two incompatible O²⁻ sites. In the first O²⁻ site, O²⁻ bonds in a 5-coordinate geometry to three equivalent Sr²⁺ and two equivalent Zr⁴⁺. In the second O²⁻ site, O²⁻ is linked in a 4-coordinate geometry to two equivalent Sr²⁺ and Zr⁴⁺ atoms⁵⁰. All Zr–O bond lengths are at 2.10 Å. The O²⁻ is bonded in a distorted linear geometry to four equivalent Sr²⁺ and two equivalent Zr⁴⁺ atoms. The Zr-site atoms are located at the 4a Wyckof site and in the fractional coordinates (0, 0, 0), and the Sr-site cations are at the 4c Wyckof site with the fractional coordinates (0.0069, 0.531, 1/4). There are two O-sites 4c and 8d Wyckof as anion with the fractional coordinates (0.925, 0.981, 1/4) and (0.214, 0.285, 0.460), respectively. It has the optimized lattice values of a = 5.847 Å, b = 5.912 Å, c = 8.298 Å, and angles; α = 90.00 Å, β = 90.00Å, and γ = 90.00Å respectively. In the similar situation, lattice values and condition, the SrZr_0.96Ge_0.04O_3, SrZr_0.92Ge_0.08O_3, and SrZr_0.88Ge_0.12O₃ were designed and optimized from the materials studio. All crystals have density 5.25 g/cm³ showing in Fig. 1a–d placed after the simulation by GGA with PBE, which were considered as the standard functional of DFT to calculate the electronic structures and optical properties of a crystal having heavy metals. Additionally, the optimization was done to ensure that all crystals had the identical lattice characteristics.

Electronic structures

Scandalously, the energy band structure or band theory of solid explains how a solid may serve as an insulator, semiconductor, or even a metal⁵¹. Additionally, the photocatalytic activity of several optical features, such as optical absorption, refractivity, reflectivity, and loss function, is highly correlated with the electronic band theory. The electronic band structure of semiconductors generally consists of a significantly low energy of VB and a considerably high energy of conduction band (CB), and the bandgap has considered the energy gap between CB and VB. A potential method for increasing the activity of photocatalytic processes, specifically the absorption of sunlight through various semiconductors for the breakdown of aqueous organic pollutants, is to precisely match the band gap with respect to the wavelength of UV radiation with which it comes into contact. However, semiconductor photocatalyst, SrZrO₃ has the experimental value of bandgap at 3.72 eV^49,52,53 as the reference crystal, and recoded band gap at 3.72, 3.73, 3.69, 3.64, and 3.63 eV, through the GGA with PBE, GGA with RPBE, GGA with PW91, GGA with WC, and GGA with PBEsol, respectively. Therefore, the references crystal can absorb the wave length about 332.00 nm light, which is much smaller compared to the visible light, and it is the main drawback for this crystal. For that fact, doping was performed by Ge atom with 4%, 8%, and 12% replacing Zr atom in reference crystal (SrZrO₃). The band gap and electronic band structures for SrZrO₃, SrZr_0.96Ge_0.04O_3, SrZr_0.92Ge_0.08O_3, and SrZr_0.88Ge_0.12O₃ through the five functionals of GGA method has listed in the Table 1 and the Fig. 2a–t.

From Fig. 2, it has been observed that both the minimum of conduction bands (MCB) and the maximum of valence bands (MVB) for SrZrO₃, SrZr_0.92Ge_0.08O_3, and SrZr_0.88Ge_0.12O₃crystals are obtained in the G symmetry point, whereas for SrZr_0.96Ge_0.04O₃ obtained at the middle of G and Q symmetry point. Consequently, all crystals are found in the direct bandgap in case of all functionals.

In case of 4% doping, the band gaps by five functional—GGA of PBE, RPBE, PW91, WC, and PBEsol—are at 2.43 eV, 2.48 eV, 2.42 eV, 2.39 eV, and 2.40 eV, respectively, that conveys about 512 nm wavelength of absorbed light andalso must be stayed in the visible range of light. In case of SrZr_0.92Ge_0.08O₃ or 8% doping, the band gap was more declined at 2.18 eV, 2.24 eV, 2.18 eV, 2.14 eV and 2.15 eV through five functionals of GGA method, respectively while the most accepted band gap was selected at 2.18 eV having two similarity by GGA with PBE and GGA with PW91. That band gap corresponds to the 568.73 nm wave length. Finally, for the SrZr_0.88Ge_0.12O_3, the lowest band gap was recorded at 1.20 eV, 1.20 eV, 1.16 eV, 1.14 eV and 1.15 eV, by the five functionals (PBE, RPBE, PW91, WC, and PBEsol). A band gap in this range enables the photocatalyst to absorb a wider range of visible light, from shorter wavelengths (blue light) to longer wavelengths (red light), optimizing the creation of electron–hole pairs required to drive photocatalytic reactions. Doping introduces impurity states that reduce the band gap, increasing visible light absorption while retaining enough energy levels for successful charge carrier separation and redox potential. This balance between broad light absorption and the ability to drive chemical processes is critical for obtaining peak photocatalytic efficiency, since a narrower band gap below 1.80 eV provides an excellent compromise between broadening the absorption range and maintaining appropriate photocatalytic power⁵⁴. Adding 12% Ge to SrZrO₃ improves photocatalytic activity by lowering the band gap to below 1.80 eV, allowing for better visible light absorption. Finally, the crystals were investigated with two more functionals GGA with WC and GGA with PBEsol, which are ended up with approximately similar band gap at 3.64 eV and 3.63 eV for SrZrO₃, 2.39 eV and 2.40 eV for SrZr_0.96Ge_0.04O₃, 2.14 eV and 2.15 eV for SrZr_0.92Ge_0.08O₃, and 1.14 eV and 1.15 eV for SrZr_0.88Ge_0.12O₃, respectively. Therefore, this study is concluded for these crystals that the GGA with WC and GGA with PBEsol functionals convey the close magnitude although the GGA with PBE and GGA with RPBE show to the closer value nearing to the experimental value. However, the GGA with PBE is the most acceptable method for these crystals in compared to other GGA functionals. The p-type (HOMO) semiconductor material’s Fermi energy is always closer to the VB⁵⁵. It is clear from looking at Fig. 2a–t that every band structure is virtually at the Fermi energy level of 0. As a result, it is compelling that these are materials with p-type (HOMO) characteristics.

The comparable study among the SrZrO₃, SrZr_0.96Ge_0.04O_3, SrZr_0.92Ge_0.08O_3, and SrZr_0.88Ge_0.12O₃by band gap under similar conditions from these five functionals suggest that for all methods it follows a similar trend i.e., SrZrO₃ contains the largest bandgap. The decreasing order of bandgap is obeyed with an increasing amount of doping. The lowest bandgap was obtained for 12% doping of Ge by replacing Zr for all five functionals. A similar bandgap of 3.72 eV for SrZrO₃ was obtained from both functional GGA with PBE and GGA with RPBE which is in accordance with the previous reference literature and consequently, GGA with PBE has been taken as a standard method for further calculations.

DFT + U is chosen over standard GGA for more accurate representations of electronic properties in doped systems, especially when localized states play a key role. For broad trends and basic studies of band structure, any of the GGA functionals (PBE, RPBE, PW91, WC, and PBESOL) can provide roughly comparable results, making them a good choice for computational efficiency.

All GGA functionals show a considerable decrease in band gap relative to undoped SrZrO₃, ranging from 2.39 to 2.48 eV. The 4% Ge doping reduces the band gap by approximately 1.2 to 1.3 eV. The DFT + U method has a larger band gap (3.335 eV) than any other GGA functionals, presumably due to the U correction influencing the electronic states created by Ge doping. All GGA functionals predict similar band gap values, ranging from 3.63 eV (PBESOL) to 3.73 eV (RPBE), with minor variation (± 0.1 eV). DFT + U predicts a somewhat lower band gap (3.297 eV), implying that adding the U correction narrows the gap for this material. As the Ge doping concentration rises to 8%, the band gap decreases even further. GGA predicts gaps ranging from 2.14 eV (WC) to 2.24 eV (RPBE). The DFT + U band gap (2.557 eV) is higher than predicted by the GGA functionals, but significantly lower than at 4% Ge doping. With 12% Ge doping, the band gap values from GGA calculations are further reduced, ranging from 1.14 eV (WC) to 1.20 eV (PBE and RPBE). This significant drop demonstrates the impact of extensive doping on band gap shrinking. The DFT + U band gap (2.297 eV) is significantly higher than the GGA values, showing that DFT + U captures additional electronic effects that may not be reflected in standard GGA.

The band structures obtained from DFT calculations using the DFT + U (Hubbard U) method and the GGA method can be different due to the distinct approximations and treatments of electronic correlations. The GGA method incorporates a simple exchange–correlation functional that considers the electron density and its gradient. It is a local approximation that works well for many systems but can struggle with materials with strongly correlated electrons, such as transition metals and metal oxides. The DFT + U method is designed to handle strong electron correlations in localized orbitals, commonly encountered in transition metal oxides and some rare earth materials. It introduces an additional Hubbard-like term (U) to the standard DFT functional, which helps correct the deficiencies of GGA in capturing strong electronic correlations. On the other hand, the GGA method tends to delocalize electrons and can struggle to properly describe the localization of d and f orbitals, which is important in correlated systems. In this case, the DFT + U method can better capture the localized nature of certain orbitals, leading to improved descriptions of electronic states in systems with strong correlations. GGA is computationally less expensive than DFT + U, which involves additional terms and self-consistency iterations to calculate the Hubbard U parameter. The DFT + U method is more computationally demanding due to the Hubbard U parameter calculation and self-consistency iterations. While GGA can work well for many materials, it can introduce systematic errors in the band structure, especially for strongly correlated systems. DFT + U has its limitations too, and the accuracy of the method depends on the appropriate choice of the Hubbard U parameter. If the U value is not well-tuned for a specific system, it may lead to inaccurate results. The calculative value by DFT + U is shown in Table 1.

In summary, GGA is a more straightforward and computationally efficient method that performs well for many materials, but it may not capture strong electron correlations accurately. On the other hand, DFT + U is a more sophisticated method specifically designed to handle systems with strong electronic correlations, but it requires careful tuning and can be computationally more expensive. The choice between these methods depends on the nature of the system being studied and the specific properties of interest. Researchers often use a combination of different methods to gain a more comprehensive understanding of the electronic structure of materials. In this study, GGA conveys the more accurate and overlapping than DFT + U shown in Fig. 3.

Density of states (DOS) and partial density of states (PDOS)

In order to determine the carrier concentrations and energy distributions of carriers inside a semiconductor, the DOS is defined as the total number of electronic states that are possible in a system. The DOS and PDOS are directly related to the chemical reactivity descriptors, including highest occupied molecular orbital (HOMO), lowest unoccupied molecular orbital (LUMO), and HOMO–LUMO gap, which are used to calculate ionization potential, electron gravity, hardness, softness, and electron affinity of any crystals^56,57. HOMO is equal to the VB (the negative magnitude of DOS) while LUMO conveys to the conduction band indicating the positive side of DOS. In case of a super cell model from DFT calculation, the sharp peak of DOS indicates their n and p type semiconductor. If the sharp peak of DOS is closer to the CB, it will be n-type semiconductor, and it is vice versa for p type semiconductor. From the Fig. 4a, the sharp peak for all crystals is obtained in VB, so they are p-type semiconductors. As a result, increasing the doped atom’s surface area in photocatalytic activity has an impact on hole formation.

The method of GGA with PBE was used to calculate the DOS and PDOS of Sr, Zr, Ge, and O elements for crystals SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, andSrZr_0.88Ge_0.12O₃. There isa glut of literature for calculating the DOS and PDOS from LDA to GGA method^58,59. Figure 4a illustrates the comparative study of SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O3 crystals orbital contribution which creates the DOS and PDOS. It is clear from Fig. 4 that the TDOS is higher for SrZr_0.92Ge_0.08O₃ (blue color) and SrZr_0.88Ge_0.12O₃ (light green color) than the SrZrO₃, and SrZr_0.96Ge_0.04O₃.

Partial density of states (PDOS) of constituents of elements are measured for SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃ crystals and illustrated in Fig. 4b–e, respectively. All of them suggest the highest contribution of p-orbital and d-orbitals for VB and CB, respectively. The comparative study of s, p, and d orbitals for SrZrO₃ has been depicted in Fig. 4f and g which suggested that the VB consists of the 3p of the Sr, and 4d, 4p, and 5s orbital contribution of the Zr atom whereas CB is consists of 3d of Sr, and 4p and 4d orbital contribution of Zr atom. Figure 4h–j illustrates orbital contribution from different atoms Sr, Zr, and Ge for SrZr_0.96Ge_0.04O₃ which implies that 3p of Sr, 4p, 5s of Zr, and 4p and 3d of Ge makes the major contribution to VB whereas 3d of Sr, 4d of Zr and 4p of Ge to the CB. Again, Fig. 4k–m shows the comparable orbital contribution from different atoms (Sr, Zr, and Ge) for SrZr_0.92Ge_0.08O₃ and shows that 3p of Sr, 4p, 5s of Zr and 4p and 3d of Ge makes donations to the VB while 3d of Sr, 4d of Zr and 4s, 4p of Ge to the CB, respectively. Finally, Fig. 4n–p depicts the orbital contribution from different atoms (Sr, Zr, and Ge) for SrZr_0.88Ge_0.12O_3, and represents that VB consists of Sr 3p, Zr 4p, and Ge 4p. In addition, CB consists of 3d, 5s of Sr, 4d of Zr and 4p of Ge. Doping with 4, 8, and 12% by Ge atom replacing in Zr, which contributes the overlapping the 4p and 3d orbital,is resulted to show the lowering in CB consequently reducing the bandgap.

Photocatalytic activity

From the perspective of semiconductor based photocatalysts, and its behaviors, the role of photocatalysis is to initiate or hasten specific reduction and oxidation (redox) reactions in the presence of irradiated semiconductors as catalyst and light. When the semiconductor catalyst is illuminated with photons energy in equal to or greater than their band-gap energy (E_g), firstly, an electron (e_cb^_) is promoted from the VB into the CB, leaving a hole (h_vb⁺) behind. Then the excited electrons and holes migrate to the surface where electrons (e_cb^_) and holes (h_vb⁺) can act as the reductant and oxidant to react with electron donors (D) and electron acceptors (A) adsorbed on the semiconductor surface^60,61. The D-π-A structure simultaneously expands π-electron delocalization and promotes intramolecular charge transfer, thus accelerating the photocatalytic reaction¹⁷.

Electrons in the conduction band interact with the oxygen molecules that are adsorbed on the surface to form highly reactive superoxide anion radicals (·O⁻²). However, when the holes interact with the hydroxyl groups adsorbed on the surface, they produce extremely reactive hydroxyl radicals. Furthermore, the dissociation of water molecules also generates hydroxyl radicals. These two highly reactive radicals react with the MB dye molecules adsorbed on the catalyst, leading to their degradation. The photocatalytic degradation process can be summarized by a series of reactions which are as follows⁶⁰:

The intricate dance between doped crystals and adsorbed molecules is a cornerstone of advancements in catalysis and sensor technology. Researchers employ computational techniques, notably density functional theory (DFT), to unravel these complex interactions.

A typical model features a doped crystal surface interacting with an adsorbate. The study on N and Br-doped TiO₂ (101) explores water adsorption using DFT calculations, revealing that doping enhances surface reactivity and increases adsorption energy compared to undoped TiO₂⁶². The study investigates CO₂ gas adsorption on transition metal-doped BiFeO₃ using DFT, revealing strong chemisorption on Mo-doped structures, highlighting their potential for gas sensing applications⁶³. The study of SO₂ adsorption asses on low-Miller index CoP surfaces using DFT, analyzing adsorption energy, electronic properties, and surface terminations to identify the most stable and active hydro treating catalyst⁶⁴. However, in this study, an attempt was taken to evaluate the photocatalytic activity of SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃ against methyleneblue (MB) dye by molecular adsorption tools. The study shows the higher stability for reference crystal (SrZrO₃) with lower total energy of 353.691 eV, while SrZr_0.88Ge_0.12O₃demonstrates lower stability with higher energy of 952.124 eV after molecular optimization loading with the MB dye as an absorbent (Table 2).

Increased Ge doping in SrZrO₃ leads to reduced total energy values. SrZrO₃ has a total energy of − 353.691 eV, while SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, andSrZr_0.88Ge_0.12O₃ have lower total energies of − 341.567 eV and − 935.719 eV, respectively. For SrZr_0.88Ge_0.12O₃ and O₃, the total energy reduces to − 952.124 eV. Incorporating additional Ge into the SrZrO₃ structure leads to increased energy stability.

Next, adsorption energy provides information on the strength of the interaction between the catalyst’s surface and adsorbents (in this case, methylene blue molecules). Adsorption energy drops from − 16.1986 eV for pure SrZrO₃ to − 15.996 eV with SrZr_0.96Ge_0.04O₃, and O₃. Higher doping concentrations (8% and 12%, respectively) result in a considerable reduction to − 3.292 and − 3.293 eV. This shows that increased Ge doping reduces adsorption strength, potentially impacting the material’s catalytic activity.

In case of deformation energy, it refers to the amount of energy required to deform the catalyst surface during adsorption. The readings change as the Ge content increases. The energy required for surface deformation increases from − 6.130 eV for pure SrZrO₃ to − 9.911 eV with SrZr_0.96Ge_0.04O₃, and O₃. At higher doping levels, it abruptly reduces to − 0.708 eV for SrZr_0.92Ge_0.08O₃, but subsequently increases to − 3.292 eV for SrZr_0.88Ge_0.12O₃, indicating varying surface stability and adaptation to adsorbents with different doping concentrations.

The methylene blue energy, which may indicate methylene blue-specific adsorption strength, changes somewhat as the Ge content increases. The results vary from − 5.380 eV for pure SrZrO₃ to − 5.449 eV for SrZr_0.96Ge_0.04O₃ with a drop to − 3.291 eV for greater Ge doping levels. This pattern indicates that the particular interaction with methylene blue declines as the Ge level above 4%.

The catalyst’s energy varies depending on Ge doping. Pure SrZrO₃ has a negative charge of − 2.150 eV, which decreases with doping to − 1.549 eV for SrZr_0.96Ge_0.04O₃. Further doping reduces energy to − 2.532 eV for SrZr_0.92Ge_0.08O₃, and − 3.105 eV for SrZr_0.88Ge_0.12O₃. This could indicate that differing doping levels affect the catalyst’s intrinsic stability or electrical structure. Hence, SrZr_0.88Ge_0.12O₃ shows the highest photocatalytic activity against Methyleneblue dye (Table 2). Here the crystals were as catalyst in case of dye degradation where the catalytic energy has recorded at − 2.150, − 1.549, 2.532, and − 3.105 for SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃ in which the highest catalytic energy shows for SrZr_0.88Ge_0.12O₃ against Methyleneblue dye. The main reason is because doped atoms have a huge surface area, which renders them unstable and reduces their band gap, resulting in high catalytic energy.

Stability of crystals

Any alloys exhibit thermodynamic stability (negative formation energy), dynamical stability (no imaginary phonon frequencies), and mechanical stability (satisfying Born’s criteria). Their half-metallic/metallic nature enhances robustness for spintronic applications, making them promising for magnetic and electronic devices⁶⁵. In addition, exhibits thermodynamic, mechanical, and dynamical stability, confirmed by negative formation energy, elastic constants satisfying Born’s criteria, and the absence of imaginary phonon frequencies⁶⁶. The stability of a crystal can be evaluated based on its total energy, where lower total energy values generally indicate higher stability. In this study, the reference SrZrO₃ structure exhibits the highest stability with the lowest total energy of − 353.691 eV from Table 2. As Ge doping increases, the total energy changes, suggesting a variation in structural stability. SrZr₀.₉₆Ge₀.₀₄O₃ has a total energy of − 341.567 eV, slightly higher than pure SrZrO₃, indicating a minor reduction in stability. However, for higher doping concentrations, the trend reverses. SrZr₀.₉₂Ge₀.₀₈O₃ and SrZr₀.₈₈Ge₀.₁₂O₃ exhibit significantly lower total energies of − 935.719 eV and − 952.124 eV, respectively, implying enhanced stability. This stabilization effect at higher Ge doping levels may be attributed to the structural reorganization and optimized electronic interactions within the crystal lattice. The decreasing adsorption energy with increasing Ge content further supports the idea that excessive doping reduces interaction strength, affecting the material’s catalytic properties. Overall, while lower Ge doping reduces stability, higher concentrations (8% and 12%) improve it, making SrZr₀.₈₈Ge₀.₁₂O₃ the most stable among the doped structures. The SrZrO₃ structure is more stable than the doped ones, as indicated by its more negative total and adsorption energies. Doping with Ge reduces stability, with increased doping showing lower rigid adsorption and deformation energy, suggesting decreased mechanical and adsorption strength. The catalyst energy is relatively consistent, but SrZr₀.₈₈Ge₀.₁₂O₃ has the least negative value, implying lower catalytic stability. Overall, pure SrZrO₃ is the most stable configuration, while doping reduces overall stability.

Optical properties

Photocatalysts’ conductivity, reflectivity, refractive index, and loss function depend on a variety of active sites, including light absorption, charge mobility, band gap, and electron–hole transport. Additionally, a material’s large surface area plays a crucial role in the absorption of pollutants since a greater surface area creates a larger active surface area, which in turn speeds up the degradation or oxidation process. In this study, the reflectivity, absorption, refractive index, dielectric function, optical conductivity, and loss functions for all the crystals SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃ have been investigated and illustrated in Figs. 5, 6, 7, 8, 9 and 10.

Optical reflectivity

The quantity of incoming light on a substance’s surface may be determined from the optical reflectivity statistics of that material. Additionally, the incident light on the surface and absorption are connected. Higher UV or visible light absorption resembles lower reflectivity⁶⁷. In this study, the reflectivity of SrZrO₃, SrZ_r0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O_3, and SrZr_0.88Ge_0.12O₃ have been recorded within the range of energy 0.0–5.0 eV (shown in Fig. 5) and observed to follow the similar pattern throughout the energy. The reflectivity increases with the photon energy for all the crystals with a breakdown within from 3.0 to 3.5 eV photon energy. Within the photon energy in the range of 0.0–3.5 eV, the highest reflectivity was recorded for SrZr_0.88Ge_0.12O₃, whereas within 4.0–5.0 eV of photon energy, the highest value of reflectivity was recorded for SrZrO_3.

Absorption

Key elements that pertain to the absorption end of a photocatalyst are band gap energy and absorption coefficient, which are photocatalyst light absorption properties. Furthermore, it is notable that enhanced photocatalytic activity is revealed by more spectacular absorption^48,68. The optical absorption recorded for SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃ and SrZr_0.88Ge_0.12O₃ are depicted in Fig. 6, and the obtained absorbance peaks are ascribed to the photo transition energies from the MVB to MCB under UV and visible light irradiation, which indicates that this material can absorb photons of UV and visible range (Fig. 6 revealed that Absorbance increases with increasing the photon energy). Within the photon range of 0.0–4.0 eV photon energy, the highest absorbance was recorded for the crystal with 12% Ge doping by replacing Zr i.e., for SrZr_0.88Ge_0.12O₃ as doping decreases the bandgap. After that, an increase in absorbance with respect to photon energy was seen for all crystals with the same value, revealing that doping had no influence on absorbance at higher photon energies than 4.0 eV.

Refractive index

Throughout the whole process of chemical breakdown from solution, the refractive index of a material is an essential metric for detecting photon absorption. The relationship between medium density and refractive index is well established⁶⁹. A complicated refractive index may be defined to easily contain this as well⁷⁰.

The real part \(\eta \left(\omega \right)\) is the refractive index and demonstrates the phase velocity, whereas the imaginary part \(k\left(\omega \right)\) is known as the extinction or absorption coefficient, though \(k\left(\omega \right)\) can also correspond to the mass attenuation coefficient even it indicates the amount of attenuation at the time of the electromagnetic wave passes through the material. Figure 7 displays the refractive index as a function of photon energy where the red, green, dark green and blue lines represent the real part of SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃, respectively, and the dark blue, pink, navy blue and violet lines stand for the imaginary part. It’s noteworthy that the relationship between the real and imagined parts is inverse. At the initial point of photon energy, the refractive index is higher for the real part while the imaginary part stays almost closed to zero. After that, the higher refractive index was recorded for the crystal with 12% doping with Ge by replacing Zr at the photon energy of 0.0–3.0 eV. At higher photon energy than 3.0 eV, there is a decrease in refractive energy with increasing doping.

Dielectric function

The dielectric function defines how a material reacts to an electric field by being polarized, with positive and negative charges shifting in opposite directions to form an opposing internal field. This polarization is critical for energy storage in devices such as capacitors, where dielectric materials improve charge storage by reducing the effective electric field. The function is related to the material’s permittivity, which determines how easily it can be polarized¹⁷; more permittivity allows for more energy storage. The dielectric function also varies with the frequency of the applied field, which influences material behavior in various electromagnetic environments. The equation of Kramers–Kronig relationship for solid materials represents the dielectric function, which is the primary function used to compute several optical characteristics like reflectivity and refractive index after adsorption⁷¹ which is given as:

Here, ε₁ (ω) indicates the real part of the dielectric constant, and ε₂(ω) stand for the dielectric loss factor (imaginary part). Dielectric functions are spatially related molecular properties of a material that are materially equivalent to the absolute permittivity or permittivity. The potential energy stored in the electric field is maintained by the real part of the dielectric constant and the imaginary part shows the opposite even for the electric potential. Figure 8 shows that the actual component is consistently higher than the imagined part. At the beginning of photonic energy for the real part, the undoped crystal has a lower value than the others, and 12% doped crystal was found to have a higher dielectric value at the beginning of photonic energy. Both the undoped and 12% doped crystals have the same value at 2.5 eV; however, beyond that, they begin to exhibit the opposite behavior. On the other contrary, the imaginary part comprises the same value for all crystals up to 1.0 eV. A higher value of dielectric function was recorded for the 12% doped SrZr_0.88Ge_0.12O₃ crystal at higher photon energy.

Conductivity

The optical conductivity, which is directly connected to the placement of electrons between the conduction band and the valance band while taking into consideration the induced current density and electric field, is what allows for the electronic transition. This means that these transport electrons are created due to the presence of free holes and free electrons in the crystalline material. The conductivity for real part (\(\sigma_{r}\)) and imaginary part (\(\sigma_{i}\)) for the dielectric function \(\varepsilon \left( \omega \right)\) is related through the term^72,73 given below:

According to the equation \(\sigma_{i} = \omega \varepsilon_{2} \left( \omega \right)\), conductivity is proportional to the real and imaginary parts of the dielectric constant.

The band gap of an active photocatalyst between the valance band and the conduction band must be less than 1.8 eV⁷⁴. From Fig. 9, the conductivity for doped and undoped is at about 0.0 eV for both the real and imaginary parts, which is almost overlapping for an excellent photocatalyst. For both the real and imaginary parts, the highest conductivity was obtained for the SrZr_0.88Ge_0.12O₃ crystal by 12% doping of Ge whereas 8% doped crystal SrZr_0.92Ge_0.08O₃ occupies the second position.

The graph presents the real and imaginary parts of the conductivity for three different compounds—SrZrO₃, SrZr_0.96Ge_0.04O₃, SrZr_0.92Ge_0.08O₃, and SrZr_0.88Ge_0.12O₃—plotted against photon energy (eV) from the Fig. 9. For the fact of real part of conductivity, the real part of the conductivity shows a sharp increase at lower photon energies, followed by a plateau as the energy increases for SrZr_0.96Ge_0.04O₃. Next, SrZr_0.92Ge_0.08O₃ exhibits a similar trend to SrZr_0.92Ge_0.08O₃, with a rapid rise at lower energies and a leveling off at higher energies. AlsoSrZr_0.92Ge_0.08O₃ shows a steep increase at low photon energies, followed by a steady state at higher energies. In case of imaginary part of conductivity, the imaginary part initially rises, then fluctuates with peaks at certain photon energies, and eventually decreases for SrZr_0.96Ge_0.04O₃ than SrZrO_3. Next, SrZrO₃ follows a similar pattern with an initial increase, fluctuations, and peaks at specific energies. AlsoSrZr_0.88Ge_0.12O₃ demonstrates a rise in the imaginary part, with fluctuations and a subsequent decline. All three compounds exhibit a consistent trend in both the real and imaginary parts of the conductivity as photon energy changes.