1 Introduction

Spinel ferrite nanomaterials have attracted considerable attention due to their distinctive properties and broad range of applications, including data storage, batteries, and catalysts [1]. These magnetic substances can exhibit either semiconducting or conducting behavior [2, 3]. With their unique features, they play a key role in the development of innovative electronic devices [4, 5]. Various experimental methods can be employed to uncover their electrical, magnetic, and other characteristics [6]. Theoretical models have suggested that they demonstrate noticeable spin polarization resulting from spin-dependent behavior [7,8,9]. This is fundamental in the field of spintronics, a field focused on the significance of electron spin, which has been attractive for a considerable period [10, 11]. The spin of electrons has the potential to enable novel multifunctional devices characterized by improved speed and efficiency, while consuming less power. Such devices could have diverse applications in technology. Their behavior can be manifested through spin-dependent electronic structure and magnetic properties, which rely on the manipulation of spin [11].

Incorporating rare earth (RE) atoms into spinel ferrites has a notable influence on the electronic, chemical, and magnetic properties of a material, as shown by various studies [12,13,14,15,16,17]. The introduction of RE atoms can bring about alterations in the crystal structure of a spinel ferrite. This transformation arises from the disparity in ionic radii between the RE ions and the host metal ions in the lattice, potentially resulting in lattice distortion [18, 19]. The substitution of larger RE ions for smaller Fe3+ ions, leading to an adjustment in the pristine structure, exerts an impact on the relevant characteristics. This is attributed to spin–orbit coupling and the presence of their unpaired 4f electrons [20].

Density functional theory (DFT) has found extensive application in various scientific investigations focused on nanomaterials [21]. Through appropriate approximations, it becomes feasible to observe a spin asymmetry in the pertinent spectra elucidating the electronic structure. Utilizing the DFT, it has been demonstrated that modifying spin polarization can be achieved by introducing transition metal (TM) atoms or making alterations in the atomic configuration [22]. One drawback of DFT computations lies in their potential inability to accurately consider orbital hybridization [23]. This drawback may lead to an underestimation of the energy gap in electronic structure calculations. Nevertheless, this challenge can be addressed by introducing a Hubbard term (U) [24, 25] to factor in the influence of valence electrons. The inclusion of the U term has a notable influence on electronic structure computations, enabling adjustment in the energy band gap as required. In the present investigation, the U term is employed to comprehensively evaluate the effects of RE ions on the spin-dependent properties of Co0.5Ni0.5Fe2O4 (CN) spinel ferrites. Among these spinel ferrites, those based on Co–Ni are particularly significant due to their promise in relevant technological applications [3, 18, 26]. Extensive investigation has already been conducted on undoped nanomaterials, including CoFe2O4, NiFe2O4, and CNs [3, 27,28,29,30,31]. While existing literature has discussed the integration of either single or two distinct RE ions into the spinel ferrites [18, 32, 33], a collective RE doping has rarely been studied [13] to the best of our current knowledge. Ref. [13] was an experimental work on Pr, PrY and PrYDy doped CNs (RCNs), where magnetic properties were focused. In this context, the present study probes into the mechanisms originating from these RE atoms (Pr, Y and Dy) in order to uncover their influence on the spin-resolved characteristics of CNs. We explore three structural configurations resulting from (i) Pr, (ii) Pr + Y (PrY) and (iii) Pr + Y + Dy (PrYDy) substitutions for Fe atoms at specific sites. The spin resolved total density of states (\({\text{TDOS}}_{\uparrow \downarrow }\)), partial density of states (\({\text{PDOS}}_{\uparrow \downarrow }\)), magnetic moment and relevant optical parameters all demonstrate their impact on the properties of CNs. Notably, the pristine CN exhibits a spin-dependent band gap which diminishes upon the addition of RE atoms. Co-doping (PrY) and ternary-doping (PrYDy) play a crucial role, highlighting their potential in achieving desired characteristics of RCNs. These findings, coupled with pertinent data on optical properties, hold great promise for the advancement of RE-doped CN materials with applications in the fields of spintronics and optoelectronics. Section 2 provides an introduction to the methodology, Sect. 3 presents the numerical findings and discussions, and Sect. 4 concludes with final remarks.

2 Methodology

DFT plays a significant role in revealing various characteristics of spinel ferrites. Prior to conducting experiments, a proper modeling of desired materials with DFT calculations can exhibit their unique characteristics. In the present work, employing the DFT, numerical computations are implemented to examine how RE dopants affect the electronic, magnetic, and optical properties of CNs. Their designs and investigations were carried out by means of DFT-based QuantumATK software [34]. First, we construct the pristine CN using a unit cell consisting of 56 atoms (4 Co, 4 Ni, 16 Fe, and 32 O atoms) with a lattice parameter of 8.335 Å. Next, we build RE-doped CNs (RCNs) which are Pr-doped CN (PCN); CN co-doped with Pr and Y (PYCN); CN ternary doped with Pr, Y, and Dy (PYDCN). To model RCNs, individual RE dopants are introduced in place of the Fe atoms: PCN (with a Pr concentration of 1.8%); PYCN (with Pr and Y co-doping at a concentration of 3.6%); PYDCN (with Pr, Y, and Dy ternary doping at a concentration of 5.4%). Then, these structures are relaxed by setting a force tolerance of 0.01 eV/Å, resulting in optimized distances in the unit cell of the RCNs. As a representative system, the PYDCN structure is depicted by the unit cell shown in Fig. 1.

Fig. 1
figure 1

Unit cell representing the PYDCN structure

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Subsequent DFT calculations were performed using the spin-dependent generalized gradient approximation (SGGA + U) with the Perdew-Burke-Ernzerhof (PBE) functional (SGGAU.PBE) [35] to account for the exchange and correlation effects of the electrons. Here, U represents the on-site Coulomb potential (Hubbard term), which corrects the DFT approximation to properly describe the hybridization of RE-4f and TM-3d orbitals. Following a preliminary analysis specific to each system, the U correction term was set to 3.3, 3.5, and 5.5 eV for Co-3d, Fe-3d, and Ni-3d electrons, respectively. We employ SGGA + U as it has been demonstrated to adequately capture the influence of cation distribution in spinel ferrites [3, 36]. The SGGA + U approximation, which provides spin dependent DFT calculations, can yield the corresponding magnetic characteristics. Norm-conserving PseudoDojo [37] and Fritz-Haber-Institute [38] pseudopotentials were utilized for the ion cores of both host and RE atoms, respectively. Co, Ni, Fe, RE, and O atoms are identified by their valence electrons; Co: 4s23d7, Ni: 4s23d8, Fe: 4s23d6, Pr: 6s24f3, Y: 5s24d1, Dy: 6s24f10, O: 2s22p4. For the geometry optimization and self-consistent calculations, a mesh cutoff energy of 125 Hartree with a (4,4,4) k-point mesh within the Monkhorst–Pack scheme [39] is adopted. TDOS spectra and optical properties are revealed using (7,7,7) k-points. The electronic structure of valence electrons for the host and RE atoms was characterized using medium and double zeta polarized basis sets of local numerical orbitals, respectively.

3 Numerical results

To our current knowledge, this study marks the first theoretical exploration encompassing the collective integration of Pr, Y, and Dy into CNs. The electronic structure properties, which are contingent on the spin orientation, are exhibited in both the total density of states (TDOS) and partial density of states (PDOS). In Fig. 2, the TDOS of each system is depicted with the Fermi energy (EF) as the reference point (set to zero). In Fig. 2, the spin-up (\({\text{TDOS}}_{\uparrow }\)) and spin-down (\({\text{TDOS}}_{\downarrow }\)) states are noticeably dissimilar, leading to asymmetric spectra. The elimination of orbital degeneracy underscores the spin-dependent alterations in TDOS, highlighting magnetism primarily due to the spin-splitting of d-band electrons of TMs across all systems. This implies the emergence of spin polarization at specific energy levels, a key aspect in the field of spintronics. The introduction of RE atoms induces modifications in the TDOS, generating new states in proximity to the EF, and influences the spin resolved properties as they possess a strong spin–orbit coupling. This fundamentally transforms the spin-dependent electronic structure characteristics of the pristine CN material, as distinctly illustrated in Fig. 2b–d. These shifts in the spin-resolved spectra are governed by magnetic interactions, contingent upon both the individual RE atoms and the doping sites. The combined \({\text{DOS}}_{\uparrow \downarrow }\) spectra of CN and RCNs reveal their spin-dependent semiconducting nature, featuring an energy band gap (Eg) of 1.70 eV for the CN (Fig. 2a). The RE atoms, exhibiting semicore electronic levels, can drastically influence the electronic (and optical) properties due to formation of additional impurity states within the Eg. Each RE atom introduces an impurity band, resulting in the modification (narrowing) of the Eg based on the dopant type and concentration, as depicted in Fig. 2b–d. The reduction in Eg due to RE doping can be attributed to factors such as alterations in the unit cell (leading to changes in lattice parameters) and the presence of additional energy states around the EF, which lowers the band gap. It's worth noting that DFT calculations can yield Eg values which are different from experimental ones. Besides, numerical results are contingent on approximations chosen. For example, experimental values for Eg in NiFe2O4 were reported as 1.52 eV [29] and 1.56 eV [40], while through the GGA + U approximation, it was theoretically computed as 1.10 eV [41]. Consequently, we can assert that our Eg result for pristine CNs aligns well with anticipated theoretical values. Our findings regarding the diminishing of Eg with RE dopants concur with prior research on cobalt ferrites [42] and zinc ferrites where the band gap was decreased when a certain amount of Y atoms (replacing the Fe ions) were introduced [43]. Therefore, it is evident that the band gap can be effectively controlled and adjusted through the incorporation of RE dopants. Nevertheless, it's important to note that the specific effects may vary depending on the specific RE ion used and its concentration. Distinct RE atoms can be applied to enhance the characteristics of spinel ferrites, as demonstrated in earlier studies [17, 44]. These outcomes imply the interplay between spin-dependent electronic structure and the effects of RE dopants on the electronic properties of CNs, suggesting insights into potential applications in spintronics and materials engineering.

Fig. 2
figure 2

\({\text{TDOS}}_{\uparrow \downarrow }\) and \({\text{PDOS}}_{\uparrow \downarrow }\) for a CN, b PCN, c PYCN and d PYDCN structures in the energy range \(-11 \text{eV}\le E\le 11 \text{eV}\). Zero energy refers to EF

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Figure 2 also displays the spin-dependent partial density of states (\({\text{PDOS}}_{\uparrow \downarrow }\)) projected onto the orbitals, revealing the contributions of TM and RE atoms to the \({\text{TDOS}}_{\uparrow \downarrow }\). In Fig. 2a, the \({\text{PDOS}}_{\uparrow \downarrow }\) spectra of CN exhibit distinct peaks originating from the s, p, and d orbitals of Ni and Co atoms. These spectra clearly emphasize that, near the \({\text{E}}_{\text{F}}\), the primary influence arises from Ni-3d and Co-3d orbitals. Figure 2b–d illustrate the impact of Pr, Y, and Dy atoms along with their respective orbitals on the electronic states, as shown in the PDOS spectra. The \({\text{PDOS}}_{\uparrow \downarrow }\) of PCN (Fig. 2b) reveals that the spin-up Pr-4f orbitals contribute to the valence band just below the \({\text{E}}_{\text{F}}\). In the conduction band, the dominant contribution to \({\text{TDOS}}_{\uparrow \downarrow }\) stems from Pr-4f orbitals within the energy range of 2.5 eV to 8.8 eV, while its d orbitals are primarily influential around 5 eV. The \({\text{PDOS}}_{\uparrow \downarrow }\) spectra of PYCN (Fig. 2c) exhibit a prominent spin-up Pr-4f peak at -2.5 eV (valence band) and just above the \({\text{E}}_{\text{F}}\) (conduction band). In the conduction band, approximately 5 eV, both Pr (via d and f orbitals) and Y (via d orbitals) exert significant influence. Regarding the \({\text{PDOS}}_{\uparrow \downarrow }\) of PYDCN in Fig. 2d, it is evident that both spin-up Pr-4f and spin-down Dy-4f orbitals play a role just below the \({\text{E}}_{\text{F}}\) in the valence band. Additionally, spin-up Dy-4f orbitals around -5 eV contribute to the valence band of \({\text{TDOS}}_{\uparrow \downarrow }\). Within the conduction band, all RE atoms participate in electronic states: Pr-4f (between 2.5 and 8.8 eV) and d orbitals (at approximately 6 eV); both Y-d and Dy-d orbitals around 5 eV; with a Dy-4f peak just above the \({\text{E}}_{\text{F}}\). This demonstrates the discernible contribution of (spin down) Dy-4f orbitals in the vicinity of \({\text{E}}_{\text{F}}\), as depicted in Fig. 2d. The interaction between the RE atom(s) and TMs involves a hybridization between RE-4f and TM-3d orbitals. This coupling serves two roles: firstly, it can tune the magnetism by virtue of a fractional charge transfer. Secondly, RE atoms raise the population of delocalized electrons, thereby influencing electronic structure characteristics.

Mulliken analysis [45] is a method used to understand the extent of charge transfer between atoms in a material. Therefore, the degree of charge transfer is determined through this method. The accompanying self-consistent electron density can be visualized using the mesh cutoff, which dictates the density grid sampling for any atomic configuration [34]. This visualization helps in understanding the self-consistent electron density of the system. Figure 3 illustrates it, presenting the redistribution of electrons upon the introduction of RE atom(s). This redistribution pattern provides insights into the electronic structure changes induced by the presence of dopants. A higher charge density along the bond indicates an increased electron transfer between neighboring atoms. The accompanying magnetism induced in each material can be discerned through the magnetic moment, which is determined via Mulliken analysis, yielding the charge distribution within a system. The average magnetic moments per atom are respectively computed to be \(1.002 {\mu }_{B}\), \(1.135 {\mu }_{B}\), \(1.076 {\mu }_{B}\) and \(1.002 {\mu }_{B}\) for CN, PCN (2.071 \({\mu }_{B}\) for Pr), PYCN (1.969 \({\mu }_{B}\) and 0.095 \({\mu }_{B}\) for Pr and Y, respectively), and PYDCN (2.076 \({\mu }_{B}\), 0.093 \({\mu }_{B}\) and 5.068 \({\mu }_{B}\) for Pr, Y and Dy, respectively). Consequently, the highest magnetic moment is observed in Pr-doped CN compared to the other structures. This underscores the distinctive role of the RE atom in influencing the magnetism of RCNs. A previous work demonstrated that the introduction of different RE atoms can lead to either an increase or decrease in the magnetic moment of RE-doped CoFe2O4 materials [46]. This variation in magnetic moment with different RE atoms aligns with the findings presented in this study. Specifically, in the case of PCN, the augmentation of the magnetic moment can be attributed to the presence of unpaired 4f electrons from Pr atoms. The magnetic properties and associated characteristics of spinel ferrites primarily stem from the interactions between metal cations located at tetrahedral (A) and octahedral (B) sites. Consequently, magnetization is influenced by factors affecting these A-B interactions. This strength can be modified by introducing foreign atoms, impurities, or making morphological adjustments. For instance, alterations in the distribution of Fe ions and surface effects can lead to a reduction in the magnetic moment [47]. Substituting Fe cations with paramagnetic ions results in a non-collinear spin arrangement [48], leading to weaker A–B interactions. This study highlights that the presence of RE atoms in CN structures effectively adjusts not only magnetism but also spin-dependent spectra and band gap. This adjustment capability is significant for tailoring the properties of CNs to meet specific requirements, yielding opportunities for the fabrication of customized magnetic materials with desired functionalities.

Fig. 3
figure 3

Cut-plane representation of electron density for both pristine and RE doped structures

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Utilizing first principles calculations has demonstrated its effectiveness in scrutinizing the optical properties of nanomaterials, as exemplified in various research papers [3, 7, 8, 21]. In this context, to ensure comprehensive coverage, we also explore the optical characteristics of RCNs. These characteristics are elucidated through parameters such as absorption coefficient (\({\alpha }_{a}\)), dielectric constant (\(\upvarepsilon\)), refractive index (n), reflectivity, extinction, and energy loss function, all depicted in Fig. 4. The corresponding peak values at certain energies are listed in Table 1 for the sake of clarity. The absorption coefficient can be expressed in terms of the dielectric constant as follows:

$$\alpha _{a} = \omega \sqrt 2 \left[ {\left( {\text{Re} \left[ \varepsilon \right]^{2} + \text{Im} \left[ \varepsilon \right]^{2} } \right)^{{1/2}} - \text{Re} \left[ \varepsilon \right]} \right]^{{1/2}}$$
(1)

where \(\omega\) represents the photon frequency, and \(\text{Re}\left[\upvarepsilon \right]\) (\(\text{Im}\left[\upvarepsilon \right]\)) signifies the real (imaginary) component of the dielectric constant, which is given by \(\varepsilon =\text{Re}\left[\upvarepsilon \right]+\text{Im}[\upvarepsilon ]\). The imaginary component, \(\text{Im}\left[\upvarepsilon \right]\), is defined by the expression:

$${\text{Im}}\left[ {\upvarepsilon } \right] = \frac{{\hbar^{2} e^{2} }}{{\pi m^{2} \omega^{2} }}\sum \smallint \left| {M_{nl}^{{kk^{\prime}}} } \right|^{2} \left( {1 - f_{n}^{k} } \right)\delta \left( {\omega_{n}^{k} - \omega_{l}^{k} - \omega } \right)d^{3} k$$
(2)
Fig. 4
figure 4

a Absorption coefficient, b Imaginary component of dielectric constant, c Refractive index, d Reflectivity, e Extinction and f Energy loss function for CN and RCNs

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Table 1 Peak values of optical parameters at certain energies
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In this equation, \({M}_{nl}^{k{k}{\prime}}\) represents the momentum operator (with \(nk\) and \(lk{\prime}\) being eigenfunctions), and \({f}_{n}^{k}\) denotes the Fermi function. Furthermore, the real component, \(\text{Re}\left[\upvarepsilon \right]\), can be determined from the imaginary component, \(\text{Im}\left[\upvarepsilon \right]\), through the application of the Kramers–Kronig transformation [49]:

$${\text{Re}}\left[ \varepsilon \right] = 1 + \frac{2}{\pi }\int_{0}^{\infty } {\frac{{\omega ^{\prime } {\text{Im}}\left[ {\varepsilon \left( {\omega ^{\prime } } \right)} \right]}}{{\omega ^{{\prime 2}} - \omega ^{2} }}d\omega ^{\prime } }$$
(3)

Figure 4a depicts the relationship between the absorption coefficient and energy. This graph visually represents how materials interact with different energy levels of incident light. It reveals a notable enhancement in absorption due to the presence of RE atoms. The absorption sharply increases, reaching its peak around 0.5 eV, and then rapidly diminishes to zero shortly after reaching 4 eV for all RCNs. Among these, PYDCN demonstrates the highest absorption peak, while CN exhibits the lowest. This suggests significant absorption in the infrared region, aligning with findings from a previous study on europium-doped CoFe2O4 ferrites [50]. The present work suggests that the enhanced absorption properties make RCNs valuable for efficiently detecting and sensing light across different energy ranges. Hence, their absorption spectra hold potential applications in various fields, including sensors and photodetectors.

The representation of the imaginary component of the dielectric constant in Fig. 4b proves highly informative in evaluating the optical behavior of structures. This component indicates the energy needed for a transition from the valence to the conduction band. Across all structures, a distinct peak emerges at approximately 0.1 eV, followed by a subsequent decrease in \(\text{Im}\left[\upvarepsilon \right]\) with increasing energy. Such a peak suggests a common energy threshold for electronic transitions in a material. The trend observed in \(\text{Im}\left[\upvarepsilon \right]\) aligns with the electronic properties highlighted in spin-dependent spectra, indicating a correlation between the optical behavior and the electronic structure of the materials. The specific energy levels referring to maxima can be elucidated through an examination of the evolution of the TDOS. The introduction of RE doping leads to an enhancement of peak intensity in \(\text{Im}\left[\upvarepsilon \right]\), reaching its maximum in the case of PCN. The intensity of a peak in the imaginary part of the dielectric constant at a specific energy is indicative of the absorption frequency range. Higher intensity suggests greater absorption potential at that energy level.

Another crucial optical parameter is the refractive index, which defines the behavior of an electromagnetic wave within a medium. It can be calculated from the dielectric constant at a specific frequency (\(\omega\)) using the formula:

$$n = \frac{1}{\sqrt 2 }\left[ {\left( {{\text{Re}}\left[ {\upvarepsilon } \right]^{2} + {\text{Im}}\left[ {\upvarepsilon } \right]^{2} } \right)^{1/2} + {\text{Re}}\left[ {\upvarepsilon } \right]} \right]^{1/2}$$
(4)

Figure 4c displays the refractive index, revealing a sharp decline below 0.5 eV, a trend consistent across all materials. This data is crucial for understanding how different materials interact with light, particularly at lower energies. The reflectivity, as depicted in Fig. 4d, illustrates that the peaks observed in both pristine and RE-doped materials fall within the infrared range and are absent in the visible spectrum. Hence, they are particularly reflective at certain wavelengths within the infrared range but not in the visible range of light.

Figure 4e and f present the extinction and energy loss function, respectively. The extinction spectrum is a measure of how much light is absorbed or scattered by a material over a range of wavelengths. The extinction spectrum (Fig. 4e), following a similar pattern as the imaginary dielectric constant (Fig. 4b), exhibits peaks in the infrared region around 0.2 eV for all systems. Such a pattern typically relates to how materials absorb and emit light. Among the systems, RCNs demonstrate stronger peaks, with PCN exhibiting the highest extinction. The energy loss within the medium is characterized by the energy loss function (describing how energy is dissipated during the interaction with light), which displays peaks in the vicinity of 2 eV (Fig. 4f). Upon doping, these peaks are slightly shifted to higher energies. This indicates that doping affects the energy dissipation behavior of the CN. Notably, the energy loss function indicates the highest intensity for PYDCN. It means that the PYDCN absorbs or dissipates energy most effectively.

The observations from the optical spectra presented in Fig. 4 suggest that RE doping can exert an influence on the optical properties of spinel ferrites. Moreover, appropriate levels of RE content may further enhance the optical characteristics of CN-based materials. This has potential implications for applications in areas such as sensors and optoelectronic devices.

4 Conclusion

We conducted first principles calculations to investigate the electronic structure, magnetic properties, and optical characteristics of RCNs. Our results indicate that the substitution of Fe atoms in CNs with a peculiar RE ion or a combination of different RE dopants leads to tunable spin-dependent spectra, band gaps, magnetic behaviors, and optical parameters. These distinctive spin-dependent properties are contingent on the choice of RE atom or the introduction of multiple distinct RE atoms (co- or ternary-doping) in CNs. The optical analysis of RCNs demonstrates that RE doping can serve as a powerful tool for manufacturing optical sensors and photodetectors. Consequently, RCNs have the potential to be employed in the fabrication of specialized magnetic systems, making them valuable candidates for applications in both spintronics and optoelectronics.