The structural formula of the osmium diiodides is of the OsI₂ type, similar to other well-known transition metal dihalides (TMDHs) [20,21]. It crystallizes in two different phases: hexagonal (2H, space group p6¯m2) and trigonal (1T, space group p3¯m1). The optimized crystal structures of the 1T phase of the pure, point-defective (Os/I vacancy), and Ru-doped OsI₂ monolayers are illustrated in Fig. 1. In the onward discussion, these systems are denoted by OsI₂:pure, OsI₂:V_Os, OsI₂:V_I, and OsI₂:Ru, Fig. 1a–d, respectively. The side view of the OsI₂ monolayer is shown in Fig. 1e. The primitive cell of the OsI₂ monolayer is shown as a red-dashed parallelogram in Fig. 1a. The Os and I atoms are shown in silver and pink, respectively. The monolayer OsI₂ possesses a trigonal symmetry and belongs to a space group number 164 (p3¯m1). In its crystal structure, each central Os atom is octahedrally coordinated to six nearest-neighbor I-type atoms, forming two sublayers on either side, Fig. 1e. A 3 × 3 × 1 supercell of 2D OsI₂ monolayer is prepared by the repetition of the single unit cell three times along the in-plane a and b directions, respectively. The lattice constant of the optimized 3 × 3 × 1 supercell containing 27 atoms (nine Os and eighteen I) of 2D OsI₂:pure monolayer is a = b = 12.17 Å. The crystal lattice parameters obtained are similar to those of transition metal 27 atoms (nine Os and eighteen I) as reported in the 2D material database [47]. Additionally, the calculated lattice constant of OsI₂ is comparable to those of other reported TMDH monolayer materials [20,21]. The calculated lattice constant of the single OsI₂ unit cell and Os-Os and Os-I distances agree with the reported values [30].

These crystal lattice parameters slightly deviate in the point-defective (Os/I vacancy) and Ru-doped OsI₂ monolayer. The created Os and I vacancies, which are a type of point defect, are shown as dashed circles in Fig. 1b,c. Alternatively, these point defects can be the missing of an atom from one of the crystal lattice sites. These point defects in the OsI₂ monolayer slightly decreased the lattice constant in the OsI₂:V_Os and OsI₂:V_I systems; see Fig. 1b,c, respectively. The lattice constants (e.g., the Os-Os distance) of OsI₂:V_Os and OsI₂:V_I are reduced to 4.04 Å and 4.02 Å, respectively. The Os vacancy in the OsI₂ monolayer gives rise to a hexagonal-type vacant region with I atoms at all corners of the hexagon, shown in Fig. 1b. In contrast, Os and I atoms are situated alternatively on the corners of the hexagon in the OsI₂ with an I vacancy, shown in Fig. 1c. The I-I distance around the vacant Os site in the OsI₂:V_Os is 3.92 Å, as depicted in Fig. 1b, whereas the Os-I distance around the vacant I site in the OsI₂:V_I is 2.65 Å, as depicted in Fig. 1c. The Ru-doped site in the OsI₂:Ru monolayer is also indicated by a dashed circle, as shown in Fig. 1d. A single Ru atom is inserted at one of the equivalent vacant Os sites by substitutional doping, which corresponds to the 3.70% doping of Ru in a 3 × 3 × 1 supercell of the OsI₂ monolayer. The Ru substitutional doping can also be considered an Os-vacancy filler, as it is clear from Fig. 1b,d. The lattice constant of the OsI₂:Ru monolayer is 4.05 Å. The I-I distance around the dopant site, shown by a hexagon in Fig. 1d, is 3.63 Å, and the Ru-I distance is 2.73 Å, which is slightly greater than the Os-I distance at the dopant site in the pure material. It may be attributed to the difference in atomic radii between the removed Os atom and the inserted Ru atom. It is important to note that the atomic radii of Os and Ru are 185 and 178 pm, respectively. The dopant Ru has a smaller atomic radius than the Os atom, leading to an increase in the Ru-I distance (2.73 Å) than the Os-I distance (2.71 Å); thereby slightly decreasing the lattice constant of OsI₂:Ru compared to that of the pure material.

The structural and energetic stability of these OsI₂ systems was verified by calculating their cohesive energies. It is the amount of energy required to separate the constituent atoms in a solid crystal into their free state of neutral atoms. The cohesive energies (EC) were calculated using the following formula (1)EC=EOsI2−EOs−EInOs+nI

In Eq. (1), EOsI2 is the total energy of the OsI₂ monolayer, EOs and EI are the total energies of the isolated Os and I atoms, respectively, whereas nOs/nI is the number of Os/I atoms. According to the mathematical expression of the cohesive energy in Eq. (1), its more negative value indicates the most energetically favorable structure. The obtained cohesive energies of the OsI₂:pure, OsI₂:V_Os, OsI₂:V_Os, and OsI₂:Ru are −4.33, −4.16, −4.31, and −4.30 eV, respectively. These are also listed in Table 2. The pure OsI₂ monolayer has a more negative value of the cohesive energy. It is the most energetically stable structure, indicating the formation of strong chemical bonds between Os and I atoms. It results in the stronger interaction between Os and I atoms in the OsI₂:pure, indicating a short bond length compared to those with a less cohesive energy of −4.30 eV in the OsI₂:Ru, Fig. 1a,d. However, the stability of other materials is slightly reduced, indicating elongation of chemical bond lengths, thereby weakening the bonds. It is generally believed that a high cohesive energy in a geometrical structure usually implies a lower total energy of the corresponding material, indicating energetically favorable crystal structures. Moreover, cohesive energies are also affected by the geometry of the crystal structure. The calculated cohesive energies suggest that these structures are stable and energetically favorable to form. Additionally, the high cohesive energy of these structures indicates that they are mechanically strong. In other words, the geometrical structures with a high cohesive energy possess a high mechanical strength. Furthermore, the phonon dispersion of the OsI₂ monolayer exhibits its dynamical stability due to the absence of negative phonon frequencies, as reported by others [30].

Here, the Kohn-Sham electronic band structure and density of states (DOS) were discussed to investigate the electronic behavior of the OsI₂ monolayers. In the Kohn-Sham approximation, the band structure is the set of energy eigenvalues as a function of wave vector. The Kohn-Sham electronic band structure was calculated along the high symmetry paths (Γ-M-K-Γ) of the first BZ. Fig. 2a,b shows the band structure and DOS of OsI₂:pure monolayer. In the valence band (VB) and conduction band (CB), there is a set of energy bands separated by a finite energy gap. For instance, there are two sets of bands in the VB and CB, spanning −4.0 to 0.0 eV and 0.5 to 4.0 eV, respectively. The subbands in the VB are separated by a gap of 0.61 eV, where the top and bottom of the lower and upper subbands lie at the K and Γ-points of the BZ. Similarly, the bottom and top of the upper and lower subbands in the CB lie along the Γ-M path and at the Γ-point, respectively. These subbands in the CB are separated by 1.08 eV. The bottom of the lower subband and top of the upper subband in the CB and VB are located between the Γ and M points, exhibiting a bandgap of about 1.20 eV. Thus, the 2D OsI₂:pure monolayer is a direct bandgap semiconductor for a range of k-values (in Bohr⁻¹) between 0.1063 and 0.1122. It exhibits a direct bandgap of about 1.27 and 1.34 eV at M- and K-points, respectively. The obtained bandgap is comparable to the reported value using the PBE functional [30]. However, the GGA-PBE underestimates [48] the bandgap value due to the involved self-interaction error in treating the exchange-correlation energy. This difficulty in determining the bandgap can be overcome by using a hybrid functional (such as HSE06) for the exchange-correlation. We employed the OpenMX DFT code to calculate the physical properties of the 2D OsI₂ monolayer, which does not include HSE06 functionality. Therefore, we reported all the results in the present study using the PBE functional. The obtained bandgap of OsI₂:pure monolayer can be verified with the total density of states (TDOS), as shown in Fig. 2b. There are zero available DOS near the Fermi level within the energy range −0.61 to +0.59 eV, indicating a bandgap of 1.20 eV between the VB and CB.

Electronic properties can be significantly altered by point defects such as vacancies (missing atoms) and by doping, both of which fall within the scope of the present discussion. In the present study, two types of vacancy were created: one by removing a single Os atom and the other by removing a single I atom from the crystal structure of the OsI₂ monolayer. These systems are referred to as the OsI₂:V_Os (Os-point defect/vacancy) and OsI₂:V_I (I-point defect/vacancy), respectively, and are shown in Fig. 1b,c, respectively. The electronic band structure of the OsI₂:V_Os monolayer is illustrated in Fig. 2c. Like the OsI₂:pure monolayer, two subbands in the VB and CB can be observed in the OsI₂:V_Os monolayer. These subbands are separated by a gap of 0.22 and 1.21 eV in the VB and CB, respectively. Now, the top of the VB (−0.02 eV) is at the M-point of the BZ, and the bottom of the CB (+1.16 eV) is still lying in the Γ-M path of the BZ. Thus, a single Os vacancy negligibly changes the band gap of the OsI₂ monolayer. Moreover, the whole electronic structure is observed to be slightly shifted compared to that of the pure material due to the shifting of the Fermi level caused by the presence of dangling bonds or unpaired electrons (due to the missing Os atom). Comparing the band structures of the pure and Os vacancy-defected OsI₂ monolayer, it is evident that the vacancy leads to more electronic bands to appear, e.g., new localized energy levels, in the VB and CB, which are solely attributed to the single vacancy defect (or the point defect/missing atom). This point defect creates localized defect energy levels close to the band edges, forming shallow donor and acceptor levels that shift the Fermi level. In particular, the Os point defect in the OsI₂ monolayer creates localized electronic energy states near the top of the VB. It shifts the Fermi level close to the VB. These changes in the electronic structure, due to electron deficiency caused by a point defect (e.g., an Os vacancy), can significantly modify a material’s optical response. The change in the band structure of the OsI₂ monolayer with an Os point defect is clear from the DOS; see Fig. 2d. The appearance of additional localized electronic DOS is evident from Fig. 2d. These localized energy states can be attributed solely to the Os point defect, i.e., the missing single Os atom from one of the crystal lattice sites of the pure host material.

An explanation similar to that for the Os point defect can also be applied to the I vacancy defect to explain the electronic properties of the OsI₂ monolayer. As with the Os point defect, two subbands are observed in VB and CB, as shown in Fig. 2e. In the I point defect, shallow and deep localized defect energy levels are observed both close to and far from the VB and CB edges, resulting in a shift in the Fermi level position due to the presence of the point defect. A nearly flat band in the electronic band structure of OsI₂:V_I is observed at the top and bottom of the VB and CB, respectively. Alternatively, these flat bands occur in the gap region of the OsI₂ monolayer, and they are solely attributed to the I point defect, a missing I atom from one of the crystal sites in the host material. Similar to the Os vacancy, the Fermi level shifted towards the VB for a single I vacant position in the OsI₂ monolayer. These changes in electronic band structure due to the I-point defect can be corroborated with the DOS, as shown in Fig. 2f. The localized electronic energy levels within the bandgap are evident due to the I-point defect in the OsI₂ monolayer. Similar to the Os point defect, the I point defect has almost a negligible effect on the bandgap of the OsI₂ monolayer.

Next, we discuss the effect of Ru-doping on the band dispersion of the OsI₂ monolayer. This Ru doping can also be considered a filler for the Os vacancy in the OsI₂:V_Os monolayer. Its band structure and DOS are shown in Fig. 2g,h. It is observed that Ru doping introduces new localized electronic states into the band structure. In the Ru-doped OsI₂ monolayer, there are two subbands in the VB and CB, as in the pristine material. The upper and lower subbands in the VB and CB appear at energies between −0.54 and −1.91 eV and +0.73 and +1.73 eV, respectively. Additionally, two energy bands below the bottom subband in the CB are evident in the energy range between +0.53 and +0.71 eV and are solely attributed to Ru doping. Unlike p- or n-type doping, Ru-doping does not introduce free electronic charge carriers because it is isoelectronic (or isovalent) with the host atom; it does not add or remove free carriers such as electrons or holes. In other words, isovalent doping is considered in the present case to recover vacancies. A slight band edge shifting in the VB and CB is observed due to the difference in the atomic radii of the host Os (185 pm) and Ru dopant (178 pm) atoms. The difference in atomic radii can also slightly distort the crystal lattice locally. Moreover, a direct band gap of approximately 1.03 eV, at the k value of 0.115 Bohr⁻¹, can be achieved due to the occurrence of the intermediate localized energy bands below the CB minimum. However, a direct bandgap of approximately 1.07 eV is observed at the M-point of the BZ. The energy difference between the VBM at the k value of 0.115 Bohr⁻¹ and at the M-point is nearly 0.03 eV, which is very small. Therefore, the Ru-doped OsI₂ can also be considered as a Quasi (semi)-direct bandgap semiconductor at the M-point of the BZ, which is similar to that observed in the 2D hexagonal SnTe monolayer [49]. The DOS corroborates the band structure of OsI2:Ru; see Fig. 2h. New electronic states are evident in the DOS plot, arising from Ru substitutional doping at an Os site in the OsI₂ crystal lattice. The doping-induced or impurity energy levels can be identified in Fig. 2h near the bottom of the CB. These impurity energy levels also appear deep in the VB and the CB, as shown in Fig. 2h.

The DOS projected on the constituent Os and I atoms is depicted in Fig. S1 of the Supplementary Material (SM). The Os atom has a dominant contribution to the total DOS in both the VB and CB. The VB below the Fermi level is mainly populated by the Os atom, Fig. S1a–d of the SM. The I atom contributes less to the total DOS. However, a slight degree of hybridization is also observed in the Ru-doped OsI₂ monolayer. The impurity electronic energy levels below the bottom of the CB are evident as sharp peaks in the DOS of OsI₂:Ru, as shown in Fig. S1d. Moreover, a few impurity levels due to Ru doping also appear in the top VB subband below the Fermi level, as shown in Fig. S1d. All the discussion above shows that point defects (vacancies) can significantly modify the electronic properties, thereby altering the dielectric and optical responses of the OsI₂ monolayer. Moreover, the isoelectronic substitution of the Ru atom at the Os vacancy contributes to defect recovery in the OsI₂:V_Os monolayer. This means that isoelectronic Ru substitutional doping at the Os site recovers almost all the electronic properties of the pure OsI₂ monolayer. A similar phenomenon has also been reported in the rhenium diselenide (ReSe₂) monolayers in the presence of various kinds of point defects [50]. Therefore, we have also systematically studied the effects of point defects and isoelectronic doping on the dielectric properties and the optical responses of the OsI₂ monolayer, as discussed in the sections below.

The defect and doping cause a redistribution of electronic charge. The calculated net charge on each atom in the OsI₂:pure configuration is shown in Fig. S2 of the SM. It can be noticed that each Os and I atom in an intrinsic OsI₂ monolayer has a net electronic charge of about −0.355 and +0.177, respectively. This means that each Os atom can accept 0.355 electrons from the two bonded I atoms, each I atom having a net charge of about 0.177. In other words, each I atom has electron donor features, while each Os atom acts as an electron acceptor. However, this electronic distribution is disrupted by point defects and substitutional doping. This means the electronic redistribution occurs in the presence of point defects created by removing an atom from a crystal lattice site. For instance, the change in the electronic distribution in the presence of the Os vacancy is shown in Fig. S3 of the SM. It can be noticed that the effect of the Os vacancy significantly affects the electronic charges on other atoms throughout the crystal lattice. All the I atoms nearest to the vacant Os site now have a reduced electron-donating capability, down to 0.054 electrons compared to 0.177 electrons in the pure OsI₂ monolayer. The next-nearest neighbor of the same type as the vacant site is also significantly affected, showing a slightly reduced electron-accepting feature. A similar effect on the electron-donating and accepting features can also be noticed in the presence of an I vacancy in the OsI₂ monolayer, as shown in Fig. S4 of the SM. Here, the electronic charge on each nearest I atom is slightly increased, indicating these nearest I atoms to the vacant I site have more electron-donating capability. At the same time, however, the electron-accepting capability of the nearest Os atom to the vacant I site is slightly reduced. The effect of isoelectronic Ru doping on the electronic distribution of the OsI₂ monolayer is shown in Fig. S5 of the SM. It is observed that the Ru doping at one of the equivalent Os sites in the OsI₂ crystal lattice significantly changes the electronic charges on each nearest bonded I atom. The electronic charge on each nearest I atom bonded to the Ru atom is now reduced to +0.137 from its value +0.177 in the pure OsI₂ monolayer, indicating a reduced electron-donating feature. However, the Ru atom can accept about 0.153 electrons from surrounding atoms. The electronic charges on the nearest Os atoms to the Ru-doping site are not significantly affected. However, the next-nearest I atoms have a slightly higher electronic charge than in the pure OsI₂ monolayer.

These changes in electronic distribution, caused by point defects and doping, lead to differences in the electron density between the considered monolayer materials, as shown in Fig. S6 of the SM. From Fig. S6a, it can be observed that most of the difference electronic density is uniformly concentrated around the Os atoms, with yellow and blue indicating electron-rich and electron-deficient regions, respectively, and the amount is negligible at each I site. This difference electron density distribution is disturbed in the presence of the point defect (Os/I vacancy); see Fig. S6b,c for the Os and I vacancy, respectively. For instance, the Os/I vacancy leads to a change in the difference electron density around the Os/I atoms; see Fig. S6b,c. However, the change in the difference electron density due to the Os vacancy in the OsI₂ monolayer can again be recovered by isovalent Ru substitution, as shown by comparing Fig. S6a,d. This indicates that isoelectronic Ru doping acts as a defect-recovery agent, suggesting the successful formation of various Os- and Ru-based dihalide/chalcogenide hybrids, similar to the RuOsSe₂ hybrid [29]. Fig. 3 shows the total spectral weights and the orbital-projected spectral weights in the materials under investigation. For instance, the green circles in Fig. 3a,d,g,j show the total spectral weights, while solid black curves show the conventional band structure obtained from the DFT calculations. On the contrary, Fig. 3b,c,e,f,h,i,k,l shows the orbital decomposed band structures (orbital projected spectral weights) of the constituent elements in the OsI₂:pure, OsI₂:V_Os, OsI₂:V_I, and OsI₂:Ru monolayers, respectively. Red is the spectral weight of the ‘s’ orbitals, green is the spectral weight from the sum of all ‘p’ orbitals, blue is the spectral weight from the sum of all ‘d’ orbitals, and magenta is the spectral weight from the sum of all ‘f’ orbitals. The radius of circles reflects the magnitude of the spectral weight. The Os-d orbital has made a predominant contribution to the top of the VB in all materials, while a slightly less contribution in forming the CBM. These observations are consistent with the atom-projected DOS shown in Fig. S1a for the pure material, as reported in the SM. However, it is the Ru-d orbital that makes a predominant contribution near the VBM and the CBM (Fig. 3k), consistent with the atom-projected DOS shown in Fig. S1d of the SM. Moreover, the hybridization between the transition metal’s ‘d’ orbital and halogen (I) ‘p’ orbital is obvious from the middle and the right columns panel.

The material’s dielectric response is an intrinsic property and varies from material to material. In other words, it demonstrates the material’s ability to be polarized due to the shifting of electric charges in the presence of an applied field. This phenomenon enables the determination of the material’s permittivity (dielectric constant). The material’s dielectric response depends on factors such as its composition, physical state, and the frequency of the applied field. A material with a higher permittivity exhibits a stronger response to light. The frequency dependence of the dielectric response makes it a complex quantity. It can be expressed as ε(ω)=ε1(ω)+iε2(ω). Here, the ε1(ω) and ε2(ω) are the frequency-dependent real and imaginary parts of the complex dielectric function. The real part indicates the material’s polarizability and its dispersion of light. It demonstrates the material’s ability to store electrical energy. In contrast, light absorption (or extinction) is governed by the imaginary part, as the light travels through the material, and it contains all the information about electronic transitions. The mathematical expression of the ε2(ω) is as follows [51,52] (2)ε2(ω)=4π2e2Vm2ω2∑nn′⟨kn|pi|kn′⟩⟨kn′|pj|kn⟩fkn(1−fkn′)δ(Ekn′−Ekn−ℏω)

In Eq. (2), e and m are the electronic charge and mass, V is the unit cell volume, p is the momentum operator, |kn⟩ is the wave function with an eigenvalue of Ekn, and fkn is the Fermi distribution function. However, the Kramer-Kronig transformation calculates the real dielectric function [53] as follows (3)ε1(ω)=1+2πP∫0∞ω′ε2(ω)dω′ω′2−ω2

The frequency-dependent optical parameters, such as absorption (α(ω)), extinction (κ(ω)), transmission (T(ω)), and reflection (R(ω)) coefficients can easily be calculated, including the refractive index, n(ω), and energy loss function, L(ω), can be calculated as follows.(4)α(ω)=ω2([ε12(ω)+ε22(ω)]12−ε1(ω))12 (5)n(ω)=12([ε12(ω)+ε22(ω)]12+ε1(ω))12 (6)κ(ω)=12([ε12(ω)+ε22(ω)]12−ε1(ω))12 (7)R(ω)=(n(ω)−1)2+k2(ω)(n(ω)+1)2−k2(ω) (8)L(ω)=ε2(ω)ε12(ω)−ε22(ω)where all other symbols have their usual meaning.

The calculated frequency-dependent imaginary dielectric function and the absorption coefficient for OsI₂ monolayers are illustrated in Fig. 4. The material’s linear response in the visible and ultraviolet (UV) regions is indicated in the blue and green double-headed arrow at the top of each plot. The material’s response to the light field is indicated by the x (E||x) and z (E||z) components in each plot. From Fig. 4, it can be noticed that the material’s in-plane responses to the incident light field are the same. This means that the material exhibits an in-plane isotropic response to the light field, e.g., the in-plane components of ε2(ω) overlap; see Fig. 4a. Therefore, only the x component of the material’s in-plane response is shown in all plots. The in-plane components of all studied physical quantities are illustrated in red and blue throughout the manuscript. From the imaginary dielectric response of all materials, along with their band structures and DOS, the different electronic transitions between the VB and CB can be identified. The ε2(ω) in the OsI₂:pure monolayer is shown in Fig. 4a. The ε2(ω) exhibits different maxima at photon energy of 1.78, 2.40, and 4.58 eV for the x-component. However, the z-component exhibits a maximum at 1.73, 4.81, 5.71, and 6.76 eV, respectively. The onset of ε2(ω), and the entire spectrum is slightly shifted towards the ultraviolet (UV) region, for the E||z than that of the E||x. However, the onset of ε2(ω) for both components belongs to the infrared (IR) region of electromagnetic radiation. The x-component of ε2(ω) exhibits a maximum at the interface of the visible and UV regions, which is entirely in the UV region for E||z. It indicates that the OsI₂:pure monolayer exhibits significant visible-light absorption for E||x; see Fig. 4b. In contrast, most of the light absorption along the z-direction occurs in the UV region, Fig. 4b. The onset of light absorption for both components is in the IR region, at 0.83 eV (E||x) and 1.30 eV (E||z); see Fig. 4b. The OsI₂:pure monolayer exhibits significant light absorption in the UV region for both components due to carrier transitions between occupied and unoccupied states in the VB and CB, Fig. 4b, as is clear from the electronic band structures.

The dielectric response of the OsI₂ monolayer in the presence of a point defect (Os and I vacancy) is shown in Fig. 4c,e, while the corresponding absorption coefficient is illustrated in Fig. 4d,f. It can be observed that the OsI₂ monolayer exhibits markedly different dielectric response and light absorption in the presence of point defects. In both cases, the ε2(ω) exhibits a peak (x-component) in the low-energy IR region, which was absent in the pure OsI₂ monolayer, e.g., the point defect shifts the peak to lower energy. This feature is solely attributable to defect states in the OsI₂ monolayer due to Os and I vacancies, which modify electronic transitions. In the presence of vacancies, the entire spectrum of the ε2(ω) for the z-component is shifted towards the higher energy compared to the x-component. However, the peaks around 5.0 eV (E||x) and 7.0 eV (E||z) are intact in all cases, which lie in the UVC region. Light absorption is significantly increased in the UV region in the presence of point defects (Os and I vacancies), as shown in Fig. 4d,f, respectively. The increased absorption in the defective OsI₂ monolayer can be explained as follows. The periodic potential is disrupted by defects, leading to localized electronic energy levels within the gap region. These defect energy levels within the bandgap allow electrons to excite themselves at lower photon energies. It leads to a redshift of the material’s absorption edges, which expands the absorption energy range; hence, a broader range of wavelengths can be absorbed by the material. In other words, the created defect sites act as additional photon-absorption centers. It increases the material’s photon capture ability, thereby enhancing light absorption in the defective OsI₂ Monolayer. Their higher absorption indicates that these are potential UV absorbers, similar to the 2D magnesium dihalides [54,55].

The effect of doping on the ε2(ω) and the absorption coefficient, α(ω), on the OsI₂ monolayer is elucidated in Fig. 4g,h, respectively. The onset of ε2(ω) and α(ω) exhibits a redshift, e.g., the low-energy response is shifted to lower energies due to a slight reduction in the bandgap caused by Ru-doping at one of the equivalent Os sites in the host OsI₂ monolayer. As mentioned earlier, Ru doping slightly distorts the host crystal lattice, thereby changing the electronic band structure; in particular, a slightly reduced bandgap is observed. It slightly changes the fundamental absorption edges, and a redshift is seen in both the ε2(ω) and α(ω) of the OsI₂:Ru. It is worth mentioning here that the host and dopant are both isovalent. An isovalent dopant can control the intrinsic point defect. However, the host Os (185 pm) and dopant Ru (178 pm) have a small atomic radius difference of 7 pm. Hence, the host crystal lattice does not deform much. It can also lead to the formation of a new composite material similar to RuOsSe₂ [29]. The change in the absorption edges is due to intermediate localized impurity electronic states induced by isovalent Ru doping in the OsI₂ monolayer. Therefore, it is seen that the Ru can easily be incorporated into the OsI₂ crystal lattice due to its isovalent nature and similar atomic radii to those of the host Os. After the Ru substitutional doping at the Os vacant site in the OsI₂:V_Os monolayer, the material exhibits properties similar to those of the pure OsI₂ monolayer. This means that isoelectronic Ru doping at one of the Os-vacant sites in the OsI₂ monolayer almost recovers the changes induced by the Os vacancy. Therefore, the isoelectronic substitution of the Ru atom at the Os vacancy contributes to defect recovery. All the considered materials exhibit significant light absorption across the visible to the UV region, with a small absorption in the IR region as well. Hence, these materials are excellent absorbers across a wide range of wavelengths in the electromagnetic spectrum, from the IR to the visible to the UV region, demonstrating their potential for optoelectronic device applications.

The extinction coefficient, κ(ω), for these materials is shown in Fig. S7 of the SM. The extinction coefficient is closely related to the absorption coefficient. It quantifies the loss of intensity due to absorption. However, the κ(ω) is the total light attenuation; it is generally the sum of absorption and scattering. The material’s higher extinction coefficient means that it absorbs light more effectively, e.g., the light photons strongly interact, which are quickly absorbed by the material. In other words, the material’s higher extinction coefficient shows higher light absorption (see Fig. S7 of the SM). The peak position in the extinction coefficient occurs nearly at the same energies as that of the absorption coefficient. Therefore, the calculated extinction coefficient is consistent with the absorption spectrum of the materials under consideration. A material’s higher extinction coefficient can have potential applications in the sensitivity detection. Additionally, both the imaginary dielectric function and the absorption coefficient behave differently for different orientations of the light field. This means that both are anisotropic in response to the photon field. However, this anisotropy in these quantities is prominent only in the IR-to-visible-to-UV region; for energies above 15.0 eV, all the considered OsI₂ monolayers exhibit isotropic behavior. This suggests that it can be useful for polarization-dependent optical instruments and devices. Energy losses are always associated with light absorption and extinction. The average energy losses are shown in Fig. S8 of the SM. It is observed that most of the energy losses are concentrated around 11.38 and 13.54 eV for pure (Ru-doped) and point-defective OsI₂ monolayers, respectively, indicating that light absorption and extinction are prominent in the UV region. The maxima in the loss spectrum corresponds to plasma resonances, and the associated frequency is called the plasma frequency.

Next, we discuss the frequency-dependent real dielectric function, ε1(ω), and the refractive index, n(ω), in the defective and doped OsI₂ monolayers, as illustrated in Fig. 5. The real dielectric function indicates the material’s ability to store the electrical energy, which shows the material’s ability to be polarized by the field. It can also be thought of as the induced internal field opposes the changes inside the material due to the external field. At the same time, the frequency-dependent refractive index quantifies the bending or slowing of light when it enters the material. The ε1(ω) is related to the phase velocity; however, the n(ω) determines the amount of light refraction or the bending of light when it enters the material. The calculated ε1(ω) in an intrinsic OsI₂ monolayer (OsI₂:pure) is shown in Fig. 5a. The most important is the static dielectric constant, ε1(0), which is the value of the real dielectric constant at zero photon energy. For OsI₂:pure, the ε1(0) is nearly 4.05 and 2.15 along the E||x- and E||z. The ε1(0) in an intrinsic OsI₂ monolayer is slightly higher than that of the standard value of the silica (3.9) [56], demonstrating its applications as a gate dielectric. It increases smoothly with increasing photon energy, reaching its peak values of 4.91 and 3.22 at 1.24 and 4.32 eV for E||x and E||z, respectively. These peak values indicate that the material is strongly polarized due to the applied field. At these energies, the frequency of the applied field matches the material’s natural oscillation frequency, e.g., the resonant frequency of the bound charges within the material, thereby increasing the amplitude of the oscillating charges and giving rise to a sharp peak in the real dielectric function, Fig. 5a. The induced polarization is strongly in phase with the electric field below the resonance. However, this phase was disrupted by further increasing the photon frequency, resulting in a decrease in ε1(ω). A sharp drop in the ε1(ω) is observed, after its peak value at 1.24 eV, with further increasing photon frequency. It is due to the phase lag in the induced polarization with the external field. It demonstrates that the material’s ability to store electrical energy decreases as photon frequency increases. Therefore, Fig. 5a shows that the intrinsic OsI₂ monolayer exhibits better performance for storing electrical energy in the IR region, with a continuous decrease in the visible and UV regions. It also shows a slight negative value at specific photon energies, demonstrating its metallic character at those energies. It is due to the opposite orientation of the induced dielectric displacement to the applied field, exhibiting an unusual material’s polarization in response to the external field. It is a useful property for creating superlenses. On the other hand, the z-component exhibits a wide energy range for storing electrical energy across the far IR-to-visible-to-low-energy-UV region. Moreover, the different behavior of the real dielectric function under different light polarization directions again shows that the material exhibits anisotropy. However, this anisotropy disappears at energies above 12.0 eV.

The real dielectric function, ε1(ω), is significantly affected in the defective OsI₂ monolayer due to the presence of a point defect (Os/I vacancy). The effect of the point defect, the Os and I vacancy, on the ε1(ω) in the pristine OsI₂ monolayer (OsI₂:pure) is depicted in Fig. 5c,e, respectively. The static dielectric constants of Os and I-defective OsI₂ monolayers are significantly increased in comparison to the intrinsic ones. The observed value of the ε1(0) in the OsI₂:V_Os, Fig. 5c, and OsI₂:V_I are approximately 9.78 and 6.03, respectively. Therefore, the Os and I vacancies in the OsI₂ monolayer not only enhance the dielectric constant but also give rise to a high-k dielectric feature. Hence, both OsI₂:V_Os and OsI₂:V_I are useful for realizing electronic devices based on high-k dielectric materials. Also, their higher dielectric constant indicates that they are more suitable than intrinsic materials for storing electrical energy and can be more strongly polarized by the light field. However, the static dielectric constant of the OsI₂ monolayer is not significantly affected by these point defects along the z-direction, which is approximately 2.53 and 2.41 for OsI₂:V_Os and OsI₂:V_I, respectively. The static dielectric constants are listed in Table 3. An induced depolarization effect is also observed in both materials, leading to a sharp drop in the ε1(ω) with increasing photon frequency, as shown in Fig. 5c,e. The Ru-doping slightly alters its static value relative to its intrinsic value; see Fig. 5g. The ε1(0) in the OsI₂:Ru is 4.15 and 2.19 for E||x and E||z, respectively. Again, the effect of Ru-doping does not deviate much from the response of the real dielectric function in comparison to that of the intrinsic response. This is due to the isoelectronic (or isovalent) nature and similar atomic radii of the dopant Ru and host Os atoms, which suggest that Ru doping can be readily incorporated into the preparation of different hybrids without significantly altering their physical properties. It reveals that Ru substitutional doping at one of the Os vacant sites in the OsI₂:V_Os monolayer contributes to the defect recovery, yielding optical responses similar to those of the pure and Ru-doped OsI₂ monolayers.

Refractive index is another important parameter of an optical material. It is the quantity that determines the bending of light when it enters a medium of a different refractive index. It is a material-dependent and intrinsic property. It also depends on the wavelength and is generally a complex function. Thus, it can be expressed as, n¯(ω)=n(ω)+iκ(ω). Here, the real part represents the refractive index, which quantifies the bending of light. The imaginary part is the extinction coefficient, which accounts for light absorption in the medium and was discussed earlier and shown in Fig. S7 of the SM. The static refractive index, n(0), is its value at the zero energy of the incident photon field. It is an important measure for a material to be applied in various optical and optoelectronic instruments and devices. It can be determined as n(0)=ε1(0). The static refractive indices of OsI₂:pure, OsI₂:V_Os, OsI₂:V_I, and OsI₂:Ru monolayers are 2.02 (1.46), 3.13 (1.59), 2.46 (1.55), and 2.04 (1.48), respectively, for x(z)-components of the light polarization. It can be noticed that the static refractive index of the OsI₂ monolayer is enhanced significantly in the presence of point defects (Os and I vacancy). It is the maximum for the OsI₂:V_Os among all the materials studied, indicating that the speed of light is affected much and slowed down when it enters the OsI₂:V_Os than that of the intrinsic material. In other words, the OsI₂:V_Os monolayer is a strongly refracting material among other studied materials, and hence, exhibits more light bending. This property of the point defect in the OsI₂ monolayer can be utilized to manipulate the light propagation through it. Moreover, a less light refraction is observed for the z-component in all materials.

The refractive index increases from its static value as the photon frequency increases. For instance, the x-component of the refractive index in OsI₂:pure increases from its static value of 2.02 to 2.23 at 1.27 eV in the IR region, Fig. 5b. In contrast, the z-component shows a maximum refractive index of 1.82 at 4.38 eV. These static values (listed in Table 4) are slightly higher than those reported for the 2D magnesium dihalides [54,55]. The refractive index of all the studied materials shows a sharp decrease after their first maximum as the photon frequency increases, Fig. 5b,d,f,h for OsI₂:pure, OsI₂:V_Os, OsI₂:V_I, and OsI₂:Ru monolayers, respectively. The reason behind a sharp decrease in refractive index is also the same as in the case of the real dielectric function, as already discussed above. The refractive index of the intrinsic and Ru-doped OsI₂ monolayer is approximately the same, again emphasizing that it is due to the isovalent doping in the host material. It also suggests that the Ru-doping contributes to defect recovery in the OsI₂:V_Os monolayer. Moreover, as mentioned earlier, the atomic radii of the host and dopant are also similar, differing by 7 pm. It leads to a less-distorted crystal lattice in the host material, resulting in behavior that is almost the same. It is another indication of the successful incorporation of Ru into the host lattice without significant changes in its properties, demonstrating the formation of various hybrids. However, a thorough experimental study is needed to verify it, which is beyond the scope of the present work. Moreover, the refractive index is polarization-dependent, indicating an anisotropic response to light fields with different polarization directions. This anisotropy, however, disappears for energies greater than 15.0 eV. Therefore, the materials studied exhibit birefringence. Hence, these materials are birefringent, exhibiting double refraction. The incident light is split into two rays: the mutually perpendicular ordinary and the extraordinary rays. They travel at different speeds due to the differences in the refractive index. The difference between the refractive indices is called the birefringence, as shown in Fig. S8 of the SM. It can be noticed that all materials exhibit negative birefringence below 5.0 eV and positive birefringence above it, which are related to the material’s optical anisotropy. The negative birefringence implies that the refractive index of the medium in the direction of propagation of an ordinary ray is greater than that of the propagation direction of an extraordinary ray. This is the reason why extraordinary light waves travel faster than ordinary light waves. However, in the case of positive birefringence, the speed of the ordinary light wave is greater than the speed of the extraordinary light wave. It is an important phenomenon with applications in liquid crystal displays for light control and image creation. Optical waveplates exploit this phenomenon to control and modify the polarization state of light (e.g., from linear to circular and vice versa). In materials science, it is used to identify and study crystal lattice structures.

The calculated percentage reflection and transmission coefficients of the materials under consideration are presented in Fig. 6 as a function of energy. It can be observed from Fig. 6a,g that the static light reflection (static reflection coefficient, R(0)) along the x(z)-direction is below 12% (4%) for the pure and Ru-doped OsI₂ monolayers, respectively. An overall reflection coefficient of less than 25% in the IR-to-visible-to-UV region indicates their potential as antireflective coating materials, as shown in Fig. 6a,g. However, the corresponding static transmission coefficients, T(0), are greater than 87% (97%) for E||x(z) in the pure and Ru-doped OsI₂ monolayers; see Fig. 6b,h, respectively. However, the obtained transmission coefficient remains above 75% over the energy range 0–20 eV. Hence, these materials may be useful for polarization-dependent light transmission filters. The static reflection coefficients in the defective OsI₂ monolayer, with a point defect of Os and an I vacancy, are slightly increased compared to those of the pure and doped monolayer. The Os and I vacancies result in static reflection coefficients of less than 27% and 18%, respectively, as shown in Fig. 6c,e, whereas the corresponding transmission coefficients are greater than 72% and 82%, respectively; see Fig. 6d,f. However, the average transmission is almost 85% or higher between 0 and 20 eV, indicating that these are also useful for antireflective materials in the optical and optoelectronic instruments and devices. Their reflection coefficient in the IR and visible regions is slightly higher than that observed in a few magnesium-based 2D halide materials [54,55,57]. Moreover, the reflection and transmission coefficients, after doping of the Ru atom at the vacant Os site in the OsI₂:V_Os monolayer, exhibit a light response similar to that of the pure material. This suggests that the isoelectronic Ru doping contributes to the vacancy recovery in the OsI₂:V_Os monolayer. Furthermore, the reflection and transmission coefficients of these materials depend on the polarization state of the light. This further emphasizes that these are strongly anisotropic optical materials with potential for anisotropic optoelectronic applications.