Band Gap Engineering and Electronic Structure of Cu₂Ni(Sn,Ge,Si)Se₄ Kesterites: A DFT Perspective on New Earth-Abundant Semiconductors for High-Performance Photovoltaics

2. MATERIALS AND METHODS

In this study, the properties of kesterite compounds from the Cu₂Ni(Sn, Ge, Si)Se₄ family were analyzed using density functional theory (DFT). All calculations were carried out using the Vienna Ab initio Simulation Package (VASP) [40], which is based on the projector augmented-wave (PAW) method [41]. Exchange-correlation interactions were treated using several functionals, including the generalized gradient approximation (GGA) with the PBE and PBEsol parametrizations, and the meta-GGA SCAN (Strongly Constrained and Appropriately Normed) functional [42]. The SCAN and HSE06 (Heyd-Scuseria-Ernzerhof) hybrid functional [43] were primarily used due to their superior accuracy in describing structural, electronic, and optical properties, particularly the band gap.

Prior to performing structural relaxations and optoelectronic property calculations, convergence tests were conducted using the PBE functional to determine optimal values for the plane-wave energy cutoff (ENCUT) and k-point mesh density (tested using both Monkhorst-Pack and Γ-centered schemes). The convergence criteria were based on identifying the minimum ENCUT and k-point density beyond which no significant change in total energy was observed, while further increases resulted in substantial computational cost without meaningful improvements in accuracy.

The optimized parameters obtained from these convergence tests were then employed in subsequent geometry optimization calculations using the SCAN functional. Electronic structure and optical property calculations were further refined using the hybrid HSE06 functional. The exchange-correlation energy was calculated as:

$$E_{xc}^{HSE}=a{E}_x^{HF,SR}\left(\omega \right)+\left(1-a\right)E_x^{PBE,SR}\left(\omega \right)+E_x^{PBE,LR}\left(\omega \right)+E_x^{PBE}$$

where $\omega$ is an adjustable parameter governing the extent of short-range interactions. Parameter α is the fraction of Hartree-Fock exchange (HF). The hybrid functional ($E_{xc}^{HSE}$) is equivalent to PBE0 [44] for $\omega =0$ and asymptotically reaches the popular PBE for $\omega \to 0$. The HSE06 hybrid functional employed a screening parameter ω = 0.2 Å^-1 and a Hartree-Fock exchange fraction α = 0.25, in line with standard practice for semiconductor systems.

Thus, the geometry of Cu₂Ni(Sn, Ge, Si)Se₄ crystals was optimized until the residual forces on each atom were less than 0.01 eV/Å. The tetragonal structure (space group I-42m), which is characteristic of kesterites, was considered. For the analysis of the electronic structure, both total and partial density of states (DOS and PDOS) were calculated. Optical properties, including the absorption coefficient, were computed based on the dielectric function obtained using the linear response method. To accurately determine the band gap, the hybrid HSE06 functional was employed, as it corrects the well-known underestimation of band gaps in the GGA approximation. Additionally, all DOS and PDOS analyses were performed using spin-polarized calculations to capture potential magnetic effects, especially in Cu₂NiSnSe₄. Effective masses were extracted via second derivative fitting of the HSE06 band edges using a parabolic approximation. All calculations employed the VASP 6.4.2 code with PAW pseudopotentials (valence: Cu: 3d¹⁰4s¹, Ni: 3d⁸4s², Sn 5s²5p², Ge: 4s² 4p², Si: 3s² 3p², Se: 4s² 4p⁴). The choice of SCAN and HSE06 functionals was motivated by their documented accuracy in reproducing structural and bandgap parameters in transition metal chalcogenides.

3. RESULTS AND DISCUSSION

To ensure high accuracy of the calculations, convergence tests were performed for the plane-wave cutoff energy (ENCUT) and the k-point mesh used to sample the Brillouin zone. In particular, the convergence of ENCUT and k-points was analyzed using both Monkhorst-Pack and Γ-centered grids, which enabled the determination of optimal cutoff energy values (Figure 1).

As shown, starting from 450 eV, the total energy stabilizes for both sampling schemes, and further increases in ENCUT do not lead to significant changes in energy. The advantage of this approach lies in the fact that, despite the differing behavior of the total energy with respect to k-point mesh density, good convergence is observed for both Monkhorst-Pack and Γ-centered schemes. This confirms the correctness of the chosen Brillouin zone discretization parameters and the reliability of the computational results.

These results demonstrate that both schemes yield comparable total energies, indicating the robustness of the applied methodology for modeling the electronic structure of the investigated kesterite compounds. Nevertheless, for the Cu₂NiGeSe₄ structure, k-point mesh optimization was performed at ENCUT = 450 eV, using a 4×4×2 Monkhorst-Pack grid (Figure 2). However, for electronic structure and optical property calculations, the cutoff energy and k-point mesh density were increased to 600 eV and 6×6×3, respectively. Similar convergence tests were also carried out for the Cu₂NiSnSe₄ and Cu₂NiSiSe₄ phases.

The Monkhorst-Pack scheme samples k-points uniformly across the Brillouin zone without necessarily including the Γ-point, which can be efficient for periodic systems with certain symmetries. In contrast, the Γ-centered grid includes the Γ-point explicitly and often provides better convergence behavior in systems with lower symmetry or when fine details around the Γ-point are critical. In our calculations, both schemes were evaluated for convergence of total energy, and the difference in total energy between them was found to be less than 1 meV per atom. This confirms that the selected k-point mesh provides sufficient precision without unnecessary computational cost.

Based on the optimal computational parameters determined through a series of convergence tests, we performed structural optimization using two widely adopted standard functionals (GGA and PBEsol), as well as the recently recommended, high-accuracy SCAN (Strongly Constrained and Appropriately Normed) meta-GGA functional. Figure 3 presents the optimized structural fragments illustrating the coordination environments of the central atoms (Sn, Ge, Si) in Cu₂NiSnSe₄, Cu₂NiGeSe₄, and Cu₂NiSiSe₄ compounds. These were obtained from SCAN-relaxed structures, where selenium (Se) atoms form tetrahedral complexes around the central X atoms (X = Sn, Ge, Si). The bond length of X–Se is observed to decrease progressively from Sn to Si. The crystal structures were visualized using the VESTA software package [45].

This gradual reduction in bond length leads to increased lattice rigidity, which in turn affects the mechanical, thermodynamic, and electronic properties of the materials. Shorter bonds, as seen in Cu₂NiSiSe₄, may contribute to a wider band gap and enhanced dielectric constant, whereas longer bonds, as in Cu₂NiSnSe₄, can provide improved charge carrier mobility, which is beneficial for thermoelectric applications [46].

This reduction in interatomic distances correlates well with the decreasing ionic radii of the central atoms (Sn > Ge > Si) and the strengthening of covalent bonding. This trend is consistent with the observed decrease in the unit cell volume (Figure 4) and explains the variations in the electronic properties of the materials. The shortening of the Si–Se bond compared to the Sn–Se bond may further enhance the hybridization between Se-4p states and the X-orbitals (X = Sn, Ge, Si), which undoubtedly influences the dielectric function and the optical absorption coefficient.

Table 1 presents a comparative analysis of the calculated lattice parameters for the investigated Cu₂Ni(Sn, Ge, Si)Se₄ systems, obtained using different exchange-correlation functionals: PBE (GGA), PBEsol, and SCAN. While this study focuses on widely accepted SCAN and HSE06 methods, more advanced approximations such as GW and Bethe-Salpeter Equation (BSE) methods are under consideration for future work to capture excitonic and quasiparticle effects. Unfortunately, no experimental data were found in the literature to directly compare the structural properties of these compounds, as their crystallographic features are reported here for the first time. Existing publications provide only limited information, mainly related to synthesis, electronic structure, and photovoltaic efficiency of Cu₂NiSnSe₄-based kesterites [47,48,49].

According to the results, a clear trend is observed for all three compounds (Cu₂NiSiSe₄, Cu₂NiGeSe₄, and Cu₂NiSnSe₄): the PBE (GGA) functional yields the largest values of the lattice parameters a, b, and c, while the PBEsol functional provides the smallest values. The SCAN functional occupies an intermediate position, showing good agreement with other computational data and with trends reported in the literature [50,51,52,53,54]. For instance, in the case of Cu₂NiSiSe₄, the lattice parameters are a = b = 5.491 Å and c = 11.044 Å with PBE; a = b = 5.373 Å and c = 10.911 Å with PBEsol; and a = b = 5.406 Å and c = 11.048 Å with SCAN. A similar trend is observed for Cu₂NiGeSe₄ and Cu₂NiSnSe₄, where the PBE functional predicts the highest lattice parameters and PBEsol the lowest.

These differences in lattice constants and unit cell volume are attributed to the varying accuracy of the exchange-correlation functionals. PBE (GGA) tends to overestimate lattice parameters due to its limited treatment of electron correlation. PBEsol, designed specifically for solids, generally yields more accurate results for cell volumes and lattice constants. The SCAN functional, as a meta-GGA approach, offers a more accurate description not only of geometry but also of the electronic structure.

All functionals predict a tetragonal structure with angles α = β = γ = 90°, which is consistent with the expected symmetry of these compounds. The unit cell volume V also exhibits a functional-dependent trend and decreases linearly from Cu₂NiSnSe₄ to Cu₂NiGeSe₄ and Cu₂NiSiSe₄ (Figure 4). This observation is in agreement with the systematic shortening of the Sn–Se, Ge–Se, and Si–Se bond lengths (Figure 5).

A clear trend of linear decrease in the unit cell volume is observed as we move from Cu₂NiSnSe₄ to Cu₂NiSiSe₄. This is due to the reduction in the ionic radius of the central atom (Sn → Ge → Si). This behavior is consistent with expectations for similar systems, where the substitution of a larger atom (Sn) with a smaller one (Si) leads to shorter interatomic distances and, consequently, a decrease in the cell volume. This trend aligns with the expected behavior for such systems. These structural variations directly correlate with electronic properties, particularly band gap evolution. The decrease in unit cell volume and X–Se bond length enhances orbital overlap between Se p-states and X s/p-orbitals, modifying the band edges and increasing the band gap. This structural-electronic interplay is clearly reflected in the band structure shifts observed in Figure 6. The results show that the choice of exchange-correlation functional significantly influences the predicted lattice parameters and unit cell volume. SCAN and PBEsol provide more accurate results compared to PBE, making them preferable for modeling such systems.

The X-ray diffraction patterns shown in Figure 5 for the kesterites of the Cu₂Ni(Sn, Ge, Si)Se₄ system display shifts of the X-ray peaks in the high-angle region, indicating a reduction in the lattice parameters. This supports the earlier discussion regarding the shortening of the Se–central atom bond lengths. The intensities of the peaks also vary, which may be attributed to differences in atomic scattering factors and the degree of structural order.

The interaction between selenium and the central atom has a partially ionic character; however, the spatial distribution of ligands influences the electron density, which is crucial for photovoltaic and thermoelectric applications. Therefore, the modification of the central atom in these structures regulates their fundamental properties, which should be considered when selecting materials for energy technologies.

To complement the structural analysis, we performed a comparative evaluation of the total energies and formation energies for the Cu₂Ni(Sn,Ge,Si)Se₄ compounds in order to assess their relative thermodynamic stability. The formation energy was computed as the difference between the total energy of each compound and the sum of the total energies of its constituent elements in their reference states. The results are summarized in Table 2.

As seen from Table 2, there is a clear trend of decreasing total and formation energies when moving from Sn to Ge and Si, indicating enhanced energetic stability for the compounds with smaller group-IV elements. This behavior is consistent with the observed reduction in lattice volume and stronger X–Se bonding, and further supports the structural integrity of Cu₂NiSiSe₄ as the most stable member of the series.

After careful relaxation of the studied materials, we performed calculations to investigate their electronic band structures. Important parameters such as the band gap, band structure, total and partial density of states, as well as the effective masses of electrons and holes, were analyzed and interpreted. Additionally, using the well-optimized structures obtained with the exchange-correlation functionals GGA, PBE-sol, and SCAN, we determined the values of the band gap using the hybrid functional HSE06 and the modified mBJ functional [55,56,57], and the results are compared in Table 3. Figure 6a-c show the energy band distributions (band structures) of the studied materials Cu₂NiSnSe₄, Cu₂NiGeSe₄, and Cu₂NiSiSe₄. The displayed band structures demonstrate the dependence of electron energy on the wavevector along the directions of the Brillouin zone (G-X-P-N-G-M-S-G). The analysis reveals key changes in the band gap width and the shape of the energy bands when substituting Sn with Ge and Si.

The analysis shows that replacing Sn atoms with Ge and Si in the crystal lattice leads to significant changes in the band gap (Eg) and the shape of the energy bands, which directly affects their applicability in optoelectronics and solar cells. Moving from Sn to Ge and Si results in an increase in the band gap, due to the reduction in atomic radius and the increase in electronegativity of the elements. This strengthens the interatomic interactions and modifies the crystal field, shifting the conduction band and valence band relative to each other. For example, Cu₂NiSnSe₄, with the smallest band gap width (Figure 6a), exhibits properties suitable for operation in the infrared range, making it promising for photodetectors and sensors sensitive to low-energy radiation. Meanwhile, Cu₂NiGeSe₄ and Cu₂NiSiSe₄, with a wider band gap (Figure 6b and 6c), are better at absorbing photons from the visible spectrum, which is critical for enhancing the efficiency of solar cells, as the majority of solar radiation is concentrated in this range.

An important aspect is also the type of band gap: the presence of direct transitions (e.g.; at the G point, where the conduction band minimum and the valence band maximum coincide in momentum) promotes more efficient generation and recombination of electron-hole pairs. This enhances luminescence and light absorption, which directly improves the efficiency of light-emitting diodes (LEDs) and lasers. For solar cells, materials with a wide and direct band gap, such as Cu₂NiSiSe₄ (Figure 6c), minimize energy losses due to heat dissipation and increase the output voltage, which is especially important for the development of high-performance photovoltaic panels. Furthermore, the ability to fine-tune the band structure by substituting elements in the crystal lattice opens pathways for designing materials with specific optical and electronic properties. For example, Ge-containing compounds could serve as a compromise between narrow and wide band gaps, adapting to the specific requirements of hybrid devices [58, 59]. Thus, the observed changes in the band structures reflect the fundamental relationship between the atomic characteristics of elements and the functional properties of materials, enabling the targeted optimization of these materials for alternative energy, optoelectronics, and microelectronics applications, reducing the cost of experimental studies and accelerating the adoption of new technologies.

Although the systems under investigation are considered for the first time, and there is a lack of direct comparison data in the literature, the calculation results show that the hybrid functional HSE06 gives the highest values for the band gap. This is consistent with the fact that standard functionals, including GGA, PBEsol, as well as SCAN, are known to significantly underestimate the band gap. It is particularly noteworthy that calculations using HSE06, based on prior structural optimization with the SCAN method, show a good correlation with the expected values of the band gap. It is important to note that the calculated values are subject to several computational uncertainties. These primarily stem from the choice of exchange-correlation functional, the k-point mesh density, and the treatment of pseudopotentials. For instance, different functionals (PBE, PBEsol, SCAN, HSE06) produce varying bandgap values due to differences in the treatment of electronic correlations. In addition, the estimation of effective masses relies on a parabolic approximation near the band edges, which may not fully capture band anisotropy or multi-valley effects. These approximations may introduce deviations on the order of ± 0.1–0.2 eV in the energy levels and ± 10–20% in carrier effective masses, which should be considered when comparing with experimental or high-level theoretical results.

Next, using the band structure curves shown in Figure 6 for the compounds Cu₂NiSnSe₄, Cu₂NiGeSe₄, and Cu₂NiSiSe₄, the Fermi levels and approximate values for the effective masses of electrons and holes near the band gap edges were computed (Table 4). The calculations were performed by numerically approximating the curvature of the energy bands near the conduction band minimum (CBM) and the valence band maximum (VBM) using a parabolic approximation [60]. The effective mass was determined according to the following expression:

$$m_{h\left(e\right)}^{\mathrm{*}}={\displaystyle \frac{{\mathrm{\hslash }}^2}{\raisebox{1ex}{$d^2{E}_{v\left(c\right)}$}\!\left/ \!\raisebox{-1ex}{$d{k}^2$}\right.}}$$

where ${\hslash }^2$ - is the reduced Planck constant, E(k) is the energy dependence on the wavevector k, and ${\displaystyle \frac{d^2{E}_{v\left(c\right)}}{d{k}^2}}$​ is the second derivative of energy with respect to k, which determines the curvature of the band.

Based on the analyzed band diagrams, it is evident that the Cu₂NiSiSe₄ system exhibits the flattest valence band in the vicinity of the valence band maximum, indicating heavier hole masses. In contrast, the Cu₂NiSnSe₄ system displays bands with greater curvature, suggesting a smaller effective carrier mass. The Fermi levels for all three systems lie within the range of 5–5.2 eV, which is typical for semiconductors with a wide band gap. A visual inspection of the band structures indicates that the electron masses range from 0.28 to 0.33 m₀, while the effective hole masses are more sensitive to composition and vary from 0.52 to 0.70 m₀. These band structure characteristics could play a crucial role in determining the transport properties and the suitability of these compounds for photovoltaic or thermoelectric applications.

Figure 7 shows the linear dependence of the effective carrier masses on the nature of the substituted IV-group element (Sn → Ge → Si) in the Cu₂Ni(Sn, Ge, Si)Se₄ structure. A clear increasing trend is observed for both electrons and holes, which can be attributed to the decrease in atomic radius and the increase in electronegativity of the element as one moves from Sn to Si. This leads to a stronger localized interaction between atoms and, consequently, to a greater band curvature, which determines a lower carrier mobility (and thus a higher effective mass).

and holes ($m_e^{\mathrm{*}}/m_0$) of semiconductors of the Cu₂Ni(Sn, Ge, Si)Se₄ system.

A clear linear trend in the effective hole mass variation is particularly evident, which may indicate a more homogeneous and predictable modification of the valence band when substituting Sn atoms with lighter Ge and Si. While the effective electron masses also show a similar increase, they exhibit slightly higher uncertainties and deviations from ideal linearity. This could be due to the more complex nature of the conduction band and the influence of the transition metal Ni, whose 3d states may contribute additional components to the conduction band, making it more sensitive to crystallographic and chemical modifications [61-67]. It is also important to note that the data were calculated using a parabolic approximation of the band curvature near the minimum/maximum point, which inherently introduces certain methodological errors. In real systems, the bands may be anisotropic, complicating the precise determination of the effective mass, especially in the case of multi-band conduction minima.

Valence band maximum (VBM) energy levels for Cu₂Ni(Sn, Ge, Si)Se₄ system compounds relative to the standard redox potentials of O₂/H₂O and H⁺/H₂, expressed in electron volts (eV) with respect to the vacuum level (Figure 9). The presented data provide an estimate of the potential of these materials for use in photovoltaic devices. These band edge alignments and bandgap values demonstrate that Cu₂Ni(Sn,Ge,Si)Se₄ compounds are suitable for integration as absorber layers in single- or multi-junction thin-film solar cells. In particular, Cu₂NiSiSe₄, with its wide and direct bandgap, is an excellent candidate for the top cell in tandem solar architectures, while Cu₂NiSnSe₄ could serve in infrared photodetectors and thermoelectric applications. The tunability of the band structure through elemental substitution offers versatile design opportunities in optoelectronics and sustainable energy technologies.

Figure 8. Band edge position evaluated for the kesterites of the Cu₂Ni(Sn, Ge, Si)Se₄ system.

Figure 8. Band edge position evaluated for the kesterites of the Cu₂Ni(Sn, Ge, Si)Se₄ system.

In the context of photovoltaic applications, such as chalcogenide or perovskite-based solar cells, the energy level diagram plays a crucial role in the design of heterojunctions. Efficient carrier transfer requires the alignment of the valence band maximum (VBM) and conduction band minimum (CBM) of the absorbing material with the energy levels of the electron transport layer (ETL) and hole transport layer (HTL). For instance, an excessively low VBM can create a barrier for hole extraction, thereby reducing the overall efficiency of the device. A comparison with well-known materials, such as CZTS (VBM ~-5.2 eV) or perovskites (VBM ~-5.4 eV), suggests that Cu₂NiXSe₄ compounds could be promising candidates for thin-film solar cells, provided their electronic structure is optimized and compatible transport layers are selected.

Figure 9 presents a three-dimensional density of states (DOS) diagram for the three chalcogenide-type compounds in the Cu₂Ni(Sn,Ge,Si)Se₄ family, providing valuable information about the electronic structure of these materials and enabling a comparative analysis of their properties. The calculated DOS profiles correlate with the band structure features (Figure 6), providing a consistent picture of the evolving electronic states across the series.

Figure 9. Comparative analysis of the density of states (DOS) of the Cu₂Ni(Sn, Ge, Si)Se₄ system.

Figure 9. Comparative analysis of the density of states (DOS) of the Cu₂Ni(Sn, Ge, Si)Se₄ system.

The primary difference between the compounds Cu₂NiSiSe₄, Cu₂NiGeSe₄, and Cu₂NiSnSe₄ lies in the changes to their electronic structure, particularly in the vicinity of the band gap. As the atomic number of the central atom increases (Si → Ge → Sn), a narrowing of the band gap is observed, which enhances the absorption of the solar spectrum and improves the efficiency of solar panels. Additionally, the density of states near the Fermi level changes, influencing the conductivity of the material and the generation of charge carriers. This effect is significant for optimizing materials for use in solar cells and electronics [68-71]. A comparative analysis of the density of states for these compounds helps evaluate their potential as active layers in solar cells and in the creation of heterostructures. To better understand the contribution of each specific element to the formation of states in the valence and conduction bands, partial density of states for Cu₂NiSnSe₄, Cu₂NiGeSe₄, and Cu₂NiSiSe₄ are presented in Figure 10 (a-c).

The partial density of states (PDOS) graphs for Cu₂NiSnSe₄ (a), Cu₂NiGeSe₄ (b), and Cu₂NiSiSe₄ (c) presented in the figure show the contributions of Cu and Ni (particularly their d-orbitals), as well as the p-orbitals of Se and elements X (Sn, Ge, Si) to the formation of the electronic structure. The analysis reveals that the valence band is primarily formed by the hybridization of the p-orbitals of Se and X, while the conduction band is determined by the d-states of Cu and Ni. The projected density of states (PDOS) results also support the observed trends in the band structure (Figure 6) and effective mass analysis (Figure 7). Specifically, the increasing contribution of Ni-3d orbitals near the conduction band minimum (CBM), particularly in Cu₂NiSiSe₄, correlates with the enhanced localization of conduction states and the observed increase in electron effective mass. Simultaneously, the narrowing of the valence band and increasing dominance of Se-4p and X-s/p (X = Ge, Si) orbitals near the valence band maximum (VBM) results in flatter valence bands and larger hole effective masses. These effects are consistent with the more pronounced curvature of the bands in Cu₂NiSnSe₄ (indicative of lighter carriers) and the flatter bands in Cu₂NiSiSe₄.

Furthermore, as the atomic number of the central atom decreases (Sn → Ge → Si), the DOS near the Fermi level becomes more asymmetric and sparser, reflecting the wider bandgap and more semiconductor-like behavior. The reduction in the density of states around the Fermi energy supports the transition from a weakly metallic (or narrow-gap) character in Cu₂NiSnSe₄ to a non-magnetic wide-bandgap semiconductor character in Cu₂NiSiSe₄. This demonstrates how the nature of the chemical bonds and orbital hybridization not only governs the energy dispersion in the band structure but also directly impacts the material’s transport and optical behavior.

The results of the density of states calculations, performed considering spin polarization (Figure 11), allow for the analysis of the distribution of electronic states across spin sublevels -spin-up (upper curves, black line) and spin-down (lower curves, red line). The Fermi level is set to zero.

The analysis of the results presented in Figure 11(a) shows that the compound Cu₂NiSnSe₄ exhibits a pronounced asymmetry between the density of states for the spin-up and spin-down states near the Fermi level. The upper and lower spins have different intensity distributions, indicating the presence of spontaneous spin polarization and, consequently, ferromagnetic behavior of the material. It is also evident that the density of states at the Fermi level remains non-zero for both spin-up and spin-down states, suggesting a metallic or semi-metallic nature of the compound. In the case of Cu₂NiGeSe₄ (Figure 11(b)), spin asymmetry is still present, although it is weaker compared to Cu₂NiSnSe₄. The upper and lower spins show more similar distributions, although the density of states at the Fermi level remains non-zero. This indicates a weakening of the magnetic order, a possible reduction in the magnetic moment, and the retention of conductive properties. In Figure 11(c), for Cu₂NiSiSe₄, the densities of states for the upper and lower spins are nearly symmetric with respect to the Fermi level. A significant reduction in the total density of states near the Fermi level is observed, especially compared to the previous compounds. This pattern indicates the disappearance of spontaneous magnetic polarization and the formation of a non-magnetic state [72-80]. Moreover, the decrease in the density of states near the Fermi level suggests the presence of a band gap and a semiconductor character for the material.

Thus, as one moves from Cu₂NiSnSe₄ to Cu₂NiSiSe₄, a clear trend is observed: the density of states at the Fermi level decreases, and the distributions of the upper and lower spins become more aligned. This indicates a gradual transition from a ferromagnetic metallic state to a non-magnetic semiconductor, highlighting the influence of the Sn, Ge, Si on the magnetic and electronic properties of Cu₂Ni(Sn, Ge, Si)Se₄ compounds.