Spin effect on electronic, magnetic and optical properties of spinel CoFe2O4: A DFT study

Materials Science & Engineering B 253 (2020) 114496

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Spin effect on electronic, magnetic and optical properties of spinel CoFe2O4: A DFT study

T

⁎

A. Hossain, M.S.I. Sarker , M.K.R. Khan, M.M. Rahman Department of Physics, University of Rajshahi, Rajshahi 6205, Bangladesh

A R T I C LE I N FO

A B S T R A C T

Keywords: Spin polarization Magnetic moments Elastic constants Optical properties Surface plasmon resonance

This report demonstrates the structural, electronic, magnetic, elastic, and optical properties of spinel CoFe2O4 using generalized gradient approximation (GGA). Both the spin and non-spin polarized density functional theory (DFT) have been used to study the influence of spin interactions on electronic structures, spin magnetic moments, and optical properties. The calculated magnetic moments of CoFe2O4 from spin density of states are 6.98 μB per formula unit. The Fe and Co ions prefer high spin orientations owing to the cationic polarization because of crystal field strength and intra-atomic exchange interactions, which induces large spin magnetic moments. The high values of spin magnetic moments confirm strong spin orbit coupling due to strong electron-electron interactions and can be a promising for spintronic application. Moreover, the calculated high reflectivity of CoFe2O4 material (~100%) in the Infrared-Visible-Ultraviolet region up to ~30 eV, which suggesting that the CoFe2O4 can also be a good candidate for solar reflector.

1. Introduction The spinel compounds of AB2O4 family (such as NiFe2O4, MgFe2O4 and CoFe2O4 etc.) are very important in materials science and engineering due to their wide range of aptness and outstanding properties. In stoichiometric formula of AB2O4 structure A, B and O are the divalent cations, trivalent cations and divalent anions, respectively. For inverse spinel oxide, A atom shared by octahedral sites and B atom shared evenly by both tetrahedral and octahedral sites. Spinel ferrites have great attraction due to its rich magnetic and electronic properties. Particularly, the spinel CoFe2O4 has great importance due to its unique physical and chemical properties. It exhibits high Curie temperature, low coercivity, moderate saturation magnetization, high magnetic moment, large magneto crystalline anisotropy, high magnetostrictive coefficient, excellent chemical stability, and mechanical hardness [1–8]. These properties assign CoFe2O4 as a technologically important and suitable for high density magnetic recording media [9], ferro-fluid applications, biomedicine, magnetic resonance imaging, biosensors, magnetic hyperthermia-based therapy [10], data storage, magnetic refrigerators and microwave devices [11]. The arrangement of divalent and trivalent cations in tetrahedral and octahedral voids plays a crucial role on its electronic structures as well as on physical properties of spinels. The cation distribution of CoFe2O4 can be expressed as: (Co1 − x Fex )Td [Cox Fe2 − x ]Oh , where, x is the degree of

⁎

inversion parameter. For normal spinels (x = 0), the tetrahedral (Td) and octahedral (Oh) sites are occupied by Co2+ and Fe3+ cations, respectively, while in the inverse spinels (x = 1) all the Co2+ cations occupy the octahedral sites and Fe3+ cations occupy both tetrahedral and octahedral sites. Combining a divalent cation with an inversion degree offers a huge variety of structural, electronic, and magnetic properties of spinel ferrites [12]. The magnetic properties of spinel cobalt ferrites are contributed by the super-exchange interaction between the metal ions located at the tetrahedral and octahedral sites [13]. Moreover, spinel CoFe2O4 demonstrated ferrimagnetic ground state with high spin orientations on tetrahedral to octahedral sites [14,15]. Experimental studies so far dealt with structural, magnetic and electrical properties of CoFe2O4 [4,16–20]. Many theoretical studies have been performed on inverse and normal spinel of CoFe2O4 using DFT theory through various approximations such as local spin density approximation (LSDA) [21,22], Generalized Gradient approximations (GGA) or by introducing on-site Coulomb repulsion energy (U) through the LSDA + U [23] and GGA + U approaches [24] or even by using the self-interaction corrected (SIC)-LSDA method [25]. The LSDA and GGA approaches generally describe these materials to be half-metallic or metallic, if no distortions are included. The SIC-LSDA method, which is parameter free, may provide a better description of correlations than LSDA, but requires a much heavier computing resource than LDA or GGA. All the

Corresponding author. E-mail address: [email protected] (M.S.I. Sarker).

https://doi.org/10.1016/j.mseb.2020.114496 Received 23 May 2018; Received in revised form 3 October 2019; Accepted 8 January 2020 0921-5107/ © 2020 Elsevier B.V. All rights reserved.

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computational studies discussed above, focused on magneto crystalline anisotropy energy [21], magnetic moments of different ferrites [22], the electronic structure of charge-ordering [23] and electronic structure of normal and inverse spinel ferrites [24,25]. Although above studies have proven to give improved results for spinel ferrites [26]. However, they did not consider the spin polarization effect in their calculation resulting in the suppression of density of states (DOS). In this study, we have taken the spin polarization effect into consideration for better understanding of CoFe2O4 properties. We applied the GGA approximation to study the electronic structure, magnetic and optical properties of CoFe2O4. We investigated the CoFe2O4 as highly spin polarized materials, because the highly spin polarized materials are important for increasing spin polarization of currents injected into semiconductor heterojunction for an optimal mobility of spintronic devices. For this reason, the electronic structures have been computed using both spin and non-spin polarized consideration with vivid picture of interactive spins. The electronic configuration depends on the occupancy of Fe and Co in tetrahedral and octahedral voids. The Fe and Co ions prefer high spin orientations due to relative strength of the crystal field and intra-atomic exchange field. Therefore, this study manifests the significance of electronic spin and reveals the actual picture of electronic spin interactions. Furthermore, the optical parameters (dielectric function, refractive index, loss function, absorption, and optical conductivity), as a different domain of material properties, have been studied as a function of incident photon energy up to 30 eV to explore whether this material could be used in photonic application.

(2)

ν=

3B − 2G 2(3B + G )

(3)

4π 2e 2 ε2 (ω) = ⎛ 2 2 ⎞ ⎝m ω ⎠ ⎜

⎟

∑ ∫ 〈i|Md |j〉2 fi (1 − fi ) δ (Ef

− Ei − ω) d3k

i, j

(4)

where, Md is the dipole matrix, i and j are the initial and final states respectively, fi is the Fermi distribution function for the i-th state, and Ei is the energy of electron in the i-th state. The real part ε1(ω) of dielectric function can be evaluated from ε2(ω) using Kramers-Kronig relations [35], as follows

ε1 (ω) = 1 +

2 ∞ ω′ε2 (ω′) dω′ P∫ π 0 (ω′2 − ω2)

(5)

where, P implies the principal value of the integral. The knowledge of both the real and imaginary parts of the dielectric tensor allows the calculation of important optical functions. We analyzed the reflectivity R(ω), the absorption coefficient I(ω), the optical conductivity σ(ε), the electron energy loss spectrum L(ω), as well as the refractive index n(ω) and the extinction coefficient k(ω). 3. Results and discussions

The computations were carried out using modelling and simulation software packages, Materials Studio version 7.0, designed by Accelrys, Inc. The geometry optimization and electronic structure calculation of cobalt ferrites were performed by using ab initio techniques employed in CASTEP code [27] based on density functional theory (DFT). In this study, we used the energy cut-off and number of k-points that measures how well one discrete grid has appointed the continuous integral. The optimizations were performed through plane wave energy of 500 eV and 3 × 3 × 3 Monkhorst-Pack [28] grid parameter for sampling of the Brillouin zone for CoFe2O4. The treated valence electron configurations are 2 s2 2p4 for O, 3d6 4 s2 for Fe and 3d7 4 s2 for Co, respectively. The interactions among valence electrons and ions were considered using the Vanderbilt type ultrasoft pseudopotential (UPP) formalism [29–31]. UPPs attain much softer pseudo-wave function that considerably used fewer plane waves for calculations of the same accuracy. To obtain the ground state energy configuration of CoFe2O4, we utilized the BroydenFletcher-Goldfarb-Shanno (BFGS) minimization technique [32].The GGA of the Perdew-Burke-Ernzerhof (PBE) formalism was used to evaluate the exchange-correlation energy [33]. Optimization was operated using convergence thresholds of 10-5 eV/atom for the total energy and 10-3 Å for maximum displacement. Maximum force and stress were 0.03 eV/Å and 0.05 GPa respectively for all the calculations. The elastic constants, Cij were calculated by the ‘stress-strain’ method. The elastic constants were defined as the second derivatives of the ground state energy with respect to strain component. Mathematically it can be written as,

1 ⎛ ∂ 2E ⎞ ⎜ ⎟ V ⎝ ∂εi ∂εj ⎠

9BG 3B + G

The optical properties of CoFe2O4 can be described entirely by complex dielectric function ε (ω) = ε1 (ω) + iε2 (ω) , which correlated with interactions of photons and electrons at all frequencies. The imaginary part ε2(ω) of dielectric function from the momentum matrix elements between the occupied and unoccupied wave functions, which is given by,

2. Computational modelling

Cij =

Y=

3.1. Structural analysis The crystalline structure of CoFe2O4 possesses face-centered cubic (fcc) spinel oxide in the space group Fd-3 m (2 2 7) [2,36–39]. To investigate the structural properties of ground state configuration of the CoFe2O4 spinel oxide, at first geometry optimization is performed with zero applied pressure using CASTEP code [27]. Then the optimized unit cell, lattice parameter and equilibrium volume are also obtained from this simulation. The crystallographic structural unit cell of CoFe2O4 consists of eight formula units. A unit cell of CoFe2O4 has 8 Co atoms, 16 Fe atoms and 32O atoms. The structural formula unit of spinel CoFe2O4 is shown in Fig. 1. In an fcc lattice, the O2− ions are tetrahedrally and octahedrally coordinated by divalent Co and trivalent Fe ions. According to cationic distribution as stated in introduction, 8 Co2+ and 8 Fe3+ cations occupied in octahedral sites, while the other

(1)

where, E and V are the energy and volume of the unit cell, respectively. Nye gives a full account of the symmetry of stress, strain, and elastic constants [34]. The elastic properties such as the bulk modulus, Poisson ratio, Young’s modulus and so forth were computed from the values of Cij. Further, the calculated bulk modulus B and shear modulus G allow us to obtain the Young’s modulus, Y and the Poisson’s ratio, ν as

Fig. 1. The crystallographic structure of CoFe2O4 in three dimensions. 2

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Fig. 2. Geometrically spin orientations between (a) Fe3+ (Td) and Fe3+ (Oh), (b) Fe3+ (Oh) and Co2+ (Oh), and (c) Fe3+ (Td) and Co2+ (Oh). Where, Td and Oh indicate tetrahedral and octahedral sites, respectively.

eight Fe3+ cations occupied in tetrahedral sites. Each octahedral Co2+ ion contains two unpaired electrons at the tg levels, while one unpaired electron at t2g levels, respectively. Each tetrahedral and octahedral Fe3+ ion contains two unpaired electrons at eg and three unpaired electrons at t2g levels, respectively as schematically shown in Fig. 2. The Co and Fe occupied in Wyckoff positions 16c (1/8, 1/8, 1/8) and 8b (1/ 2, 1/2, 1/2), respectively, whereas O atom occupied in 32e (0.25027, 0.25027, 0.25027) position in the lattice sites [36–38]. The size and shape of the unit cell is defined from the lattice constant. Generally, in three-dimensional lattice it has three mutually perpendicular axes referred to as a, b and c along x, y, and z directions. In cubic crystals, these values of three axes are equal which is found about 8.363 Å. The values of axial angles are also equal which is α = β = γ = 90°. The volume of the unit cell is calculated using the optimized lattice parameter. The values of structural parameters are consistent well with other theoretical and experimental data, which are listed in Table 1.

between the dxy, dyz and dzx orbitals and surrounding anion orbitals, while the order is reversed in octahedral sites (Fig. 2) due to direct repulsion of the dz2 and dx2−y2 orbitals [16]. The electronic structure is influenced by the interactions between the up-spin and down-spin states, which arises due to strength of intra-atomic exchange field and crystal field [36,42]. The lowest energy band lying in the range from −10.19 eV to −5.90 eV is mainly arising due to hybridization of O 2p states. The higher values of DOS at −3.09 eV manifests the hybridization of Fe 3d states and −1.58 eV and −0.63 eV are contributed from Co 3d hybridization, which are seen also from Fig. 3b and c, respectively. The calculated total magnetic moments are found to be 6.98 μB , that indicates the strong SOC which is a consequence of strong electronelectron interactions. The higher SOC is technologically important and suitable for spintronic applications. The partial DOS (PDOS) determine the occupied electronic states per unit energy contributing from individual atoms. The Fe 3d electrons provide PDOS and SDOS are 26.28 and 26.05 states/eV, respectively, per unit cell at Fermi level as presented in Fig. 3b. In addition, it is seen from Fig. 3c that the values of PDOS and SDOS at Fermi level are 23.96 and 17.74 states/eV, respectively per unit cell due to Co 3d electrons. The Fe3+ ion has half-filled (d5) orbital that prefer high spin orientations for both octahedral and tetrahedral sites (Fig. 2a). On the other hand, Co2+ ion has two localized down-spin d-electrons, in addition to the five up-spins (Fig. 2b-c). Thus the imbalances of spin orientations strengthen crystal field and intra-atomic exchange field which produce large spin magnetic moments. Besides that, the non-spin polarized PDOS values of Fe and Co 3d states are 17.39 and 13.13 states/eV, respectively (Fig. 3b and c). The large values of SDOS of Fe and Co atoms yield the high spin polarization in the presence of high magnetic moments of the atoms. The calculated spin magnetic moments for Fe and Co atoms are found to be 3.24 μB and 2.28 μB , respectively. Moreover, the non-spin polarized PDOS value of O atoms is 0.73 states/eV at the Fermi level which shown in Fig. 3d. It reveals that the O atoms have also a small contribution in TDOS and SDOS. It is found that the values of PDOS and SDOS of O atoms at Fermi level are 1.62 and 0.29 states/eV, respectively, per unit cell. On the other hand, the large value of DOS in the entire energy bands from −10.19 eV to −5.90 eV found to be 18.4 states/eV and 20.4 states/eV, which mainly contributed from the O 2p electronic states. The contribution of O atom in SDOS is due to polarized DOS in the presence of small magnetic moment of O atoms. The calculated magnetic moments are compared with the other theoretical and experimental data listed in Table 2. The obtained results are consistent, but small variations appeared due to cationic arrangements in octahedral and tetrahedral sites.

3.2. Electronic properties: density of states (DOS) and magnetic moments The DOS determines the number of electronic states per unit energy. The Fermi level position is taken as reference point for determining DOS. The DOS of CoFe2O4 is shown in Fig. 3a. It is seen from Fig. 3a that the total density of states (TDOS) reveals non-zero values at Fermi level, EF. These non-zero values of TDOS manifest the metallic nature of CoFe2O4 with the evidence of zero energy gap [37]. At the Fermi energy, EF the values of total non-spin polarized DOS (NSDOS) is 31.99 states/eV. On the contrary, considering the electronic spin the DOS becomes higher. In this case, the values of TDOS and spin DOS (SDOS) are 55.92 and 43.79 states/eV per unit cell, respectively. The high values of SDOS at Fermi level arise due to cationic polarizations that observe the magnetic states, which are expected for this type of spinel oxide. The spin polarizations manifest the splitting of electronic energy states due to strong spin orbit coupling (SOC). To illustrate the nature of magnetic properties of spinel CoFe2O4 spin configuration have been studied. The above investigation of the DOS suggests that the polarization shifts between up-spin and downspin states causes induced magnetic moments contributed by Co, Fe and O atoms. According to crystal field theory, the eg levels are lower than t2g levels in a tetrahedral sites owing to the direct electrostatic repulsion Table 1 The lattice constant (Å) and Volume (Å)3 for spinel CoFe2O4. Compounds

Lattice constant (Å)

Volume (Å)3

Remarks

CoFe2O4

8.363 8.384 8.408 8.379 8.403 8.378 8.270 8.393

584.91 – – 588.31 593.34 – – –

Present work Calc. [36] Expt. [37] Expt. [38] Expt. [2,39] Expt. [40] Calc. [24] Expt. [41]

3.3. Elastic properties The elastic constants described the response of a material to an applied stress. They provide a link between the mechanical and dynamic information concerning the nature of the forces operating in solids, especially for the stability and stiffness of materials [42,43]. The 3

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Fig. 3. (a) Total DOS of CoFe2O4, (b) Partial DOS of Fe (3d), (c) Partial DOS of Co (3d) and (d) Partial DOS of O (2p) atoms as a function of energy in eV. (The red solid line and blue dashed line indicate the electronic and spin DOS with spin polarized DFT. The dotted black line indicates the DOS with non-spin polarized consideration.)

Table 2 The calculated total and atomic spin magnetic moments for CoFe2O4. Compound

μtot (μB )

μFe (μB )

μCo (μB )

μO (μB )

Remarks

CoFe2O4

6.98 7 6.42 7

3.24 – 2.11 3.57

2.28 – 1.92 2.58

0.04 – 0.04 0.07

Present work Calc. [36] Calc. [37] Calc. [25]

Table 3 The calculated Elastic constants, Cij, bulk modulus B, Shear modulus G, Young’s modulus Y, Poission’s ratio σ and G/B ratio (all are in GPa) for CoFe2O4. C11

C12

C44

B

G

Y

σ

G/B

Remarks

379 387

125 126

141 133

230 233

53 49

148 125

0.39 0.41

0.23 0.21

Present work Calc. [46]

strain and is used to provide a measure of stiffness, i.e., the larger the value of Y, the material is stiffer. The calculated value of the Young’s modulus, Y of CoFe2O4 is found to be 148 GPa that shows a better performance of the resistance to shape change and against uniaxial tensions.

mechanically stable phases or macroscopic stability is dependent on the positive definiteness of stiffness matrix. For the stability of cubic lattice, the following conditions known as the Born criteria [44] must be carried out: C11 > 0, C11 – C12 > 0, C44 > 0. The calculated elastic constants completely satisfy the above conditions, indicating that the cubic phase of CoFe2O4 is mechanically stable. The simulated results of bulk modulus B, shear modulus G, Young’s modulus Y (all are in GPa), and Poisson’s ratio v at zero pressure calculated using elastic constants are 230, 53, 148 and 0.39, respectively. The ductility of a material can be roughly estimated by the ability of performing shear deformation, such as the value of shear-modulus to bulk-modulus ratios. Thus, a ductile plastic solid would show low G/B ratio (< 0.5); otherwise, the material is brittle [45]. The calculated G/B ratio is 0.23 (Table 3) for the investigated material, indicating that the compound is ductile plastic solid [46]. The Young’s modulus is defined as the ratio between stress and

3.4. Optical properties The optical properties such as dielectric function, refractive index, absorption, loss function, reflectivity and optical conductivity of CoFe2O4 are computed as a function of incident photon energies up to 30 eV for [1 0 0] polarization direction. For the metallic compounds both inter-band and intra-band transitions contribute to dielectric functions. A Drude term with unscreened plasma frequency 3 eV and damping 0.05 eV has been used to enhance the low energy part of the spectrum. Fig. 4 a shows the calculated dielectric function and refractive index 4

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Fig. 4. (a) Dielectric function and (b) refractive index of CoFe2O4 as a function of incident photon energy, eV.

0.9 eV. The loss spectrum reveals quite different picture reported than the value of [46], due to electronic spin interactions. The energy loss spectrum shows some characteristics peaks at 9.8 eV, 11.6 eV, 22.0 eV and 26.7 eV, respectively reveal surface plasmonic excitations at the metal-dielectric interface due to optical transition instead of absorbing high energy photons. These types of loss spectroscopy predicted that the compound is applicable for guiding of light below the diffraction limit (near-field optics), non-linear optics, biosensors etc [47]. Moreover, the peaks of loss spectrum indicate the large magneto-optical kerr effect (MOKE) which is expected for high density storage systems [48]. In the visible energy range, the energy loss function of CoFe2O4 is quite low. Fig. 5b shows the absorption spectra for CoFe2O4 compound. The absorption spectra begin at zero photon energy, which confirms the metallic nature of CoFe2O4 as expected. The spectra rise sharply until 1.17 eV and then decreases as an exponential function to reach a constant value in the energy range ~4–16.4 eV. The peak at 1.17 eV arises due to transition of Co/Fe electrons from p to d states. In the energy range ~16.4–19.3 eV, it is observed that the CoFe2O4 material possesses zero absorption, this indicates that the material is transparent in this region. The zero-absorption coefficient also observed in the energy range ~28.9–30 eV. The calculated optical reflectivity as a function of incident photon energy is presented in Fig. 6a. It is observed that CoFe2O4 has higher reflectivity in infrared, visible and ultraviolet regions. The maximum reflectivity (~100%) observed with almost a constant value in the energy range ~2.2–30 eV. Thus, the result shows that CoFe2O4 can be used as a best reflector and also capable to reduce solar heating [49]. The optical conductivity of CoFe2O4 compound is depicted in Fig. 6b. The optical conductivity begins with zero photon energy that indicates the materials have no energy gap. This information is evident from DOS calculation, which also explained earlier in Fig. 3. The

of CoFe2O4 as a function of photon energy. The dielectric function explains how a material response to the electromagnetic radiation, in particular to visible light. Both the real and imaginary part of dielectric function indicates the metallic characteristics of the material. The real part of dielectric function exhibits characteristic peak at 0.82 eV. The negative value of the real part of dielectric function indicates that the CoFe2O4 shows the intra-band Drude-like behavior. At high energy region, the real part of dielectric function tends to unity and the value of imaginary part possesses very close to zero. It manifests itself as almost transparent material with very low absorption in IR region and confirms that the material is optically isotropic. Therefore, the dielectric formalism is mainly arises from electronic polarizability, since in high energy region the effect of ionic and dipolar polarizability is negligible. The simulated result of the refractive index of CoFe2O4 is presented in Fig. 4b. The real part of refractive index determines the phase velocity and the imaginary part determines the amount of absorption loss when an electromagnetic wave passes through the materials. It is seen in Fig. 4b that the real part of refractive index, n sharply decreased in low energy region up to 1.6 eV and then it is almost constant in high energy region. Moreover, extinction coefficient linearly increased up to 0.53 eV and then decreases exponentially that indicates how incident energy absorbed in the materials. The computed energy loss spectra are shown in Fig. 5a. The energy loss function of materials is very important parameter in the dielectric formalism, which is useful to understand the screened excitation spectra, especially the collective excitations produced by the swift charges inside a solid. Loss function refers to the how fast electron traversing in a material. Hence, the study of loss function is most important in material study. The highest peak of the energy loss spectrum appears due to bulk plasmonic excitation at particular incident photon energy as well as the corresponding frequency known as the bulk plasma frequency. In loss spectrum, the plasma frequency is located at

Fig. 5. (a) Loss function and (b) absorption of CoFe2O4 as a function of incident photon energy, eV. 5

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Fig. 6. (a) Reflectivity and (b) conductivity of CoFe2O4 as a function of incident photon energy, eV.

conductivity sharply increases in low energy region and appears a peak at 0.50 eV. The optical conductivity as well as electrical conductivity increases in infrared region due to absorbing the incident photon energy [50].

[10] [11] [12] [13] [14]

4. Summary

[15] [16]

In summary, the spin polarized DFT analysis is performed on inverse spinel CoFe2O4 material to study the structural, electronic, mechanical, and optical properties by employing ultrasoft pseudopotential. The calculated lattice constant well agrees with the available experimental data. The computational electronic structure analysis confirms the zero energy gaps of the CoFe2O4 materials. The DOS calculation shows that the electronic properties are mainly attributed due to contribution of 3d electrons of Fe and Co atoms. The large values of DOS at Fermi level exhibit strong electron-electron interactions. The calculated total spin magnetic moments is found to be 6.98 μB , is a consequence of high spin polarization. The exchange interaction among up and down spin states in octahedral and tetrahedral sites owing to the crystal field induces high spin magnetic moments. The highly spin polarized material is required for optimal performance of spintronic devices. The mechanical investigations reveal that the material is ductile plastic solid. In addition, the optical measurements show that CoFe2O4 has high reflectivity (~100%) in the entire energy range of 0–30 eV. The optical conductivity starts with zero photon energy that shows also the zero energy gap of CoFe2O4 material. Thus, the CoFe2O4 is a promising material for spintronic applications, as well as it may be used as a good solar reflector. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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