The relaxed CdFe₂O₄ structure has been optimized in ferromagnetic (FM) and antiferromagnetic (AFM) states, and stable FM state is confirmed due to more energy released in FM state than in AFM state. While calculating the AFM, the nearest neighbor layers are flipped in alternative layer to add the spin contribution. The calculated ground state energy in AFM state for all configurations is compared with FM state energy. Stability of FM state is further confirmed due to negative sign of computed enthalpy of formation, as listed in Table 1. Hence, compound CdFe₂O₄ keeps less energy as compared to the sum of the atomic state energies of Cd, Fe, and O. The ground state parameters like lattice constant, bulk modulus and its pressure derivative calculated through Morganʼs equation of state are also listed in Table 1. The electronic structure of the studied compound which is revealed through PBEsol functional shows metallic behavior, while TB-mBJ improves the electronic structure and ferromagnetic semiconducting nature, as can be seen in Figs. 1 and 2, respectively. The spin polarization P is extracted by using , where and show electron densities at E_F in respective spin orientations. The applications of TB-mBJ suggest that spin polarization reduces to zero (in computations with PBEsol that was nonzero due to metallic nature) because in both spin channels no state crosses E_F.^[23–25]

Fig. 1.

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	Fig. 1. (color online) Spin polarized band structures of CdFe2O4 computed by applying PBEsol and PBEsol + mBJ functional.

Fig. 2.

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	Fig. 2. (color online) (a), (b) Total (TDOS) and partial density of states (PDOS) determined for CdFe2O4 for both spin configuration by applying PBEsol and PBEsol + mBJ functional.

Table 1.

Table 1.

Table 1. Various parameters calculated for CdFe2O4 in the FM state by using PBEsol and PBEsol + mBJ potentials. . Properties PBEsol PBEsol + mBJ Experimental/Others ao/Å 8.71 8.70 Exp. 8.715a, Theo. 8.69c ΔE/meV 3.4 – Hf/eV –1.77 – ε1(0) 6.64 3.52 n(0) 2.57 1.87 R(0) 0.1945 0.0929 ΔEg/eV – 1.86 Exp. 1.76b , Theo.1.80d ΔJT 1.49 0.64 Δx(d) 3.81 6.79 Δx(pd) 0.00 –0.600 ΔEC/eV 0.1476 –2.575 ΔEV/eV 0.6260 –0.706 Noα 0.0367 –0.643 Noβ 0.1565 −0.176 Mtot/µB 18.00 19.99 Theo. 19.99d MCd/µB 0.09 0.97 MO/µB 2.51 4.06 MFe/µB 14.0 11.15 MInt/µB 3.89 2.32 a Ref. [4], b Ref. [10], c Ref. [15], d Ref. [50].

Table 1. Various parameters calculated for CdFe2O4 in the FM state by using PBEsol and PBEsol + mBJ potentials. .

The band structures (BS) and density of states (DOS) computed by using TB-mBJ potential, as shown in Figs. 1(a),1(b),2(a), and 2(b), illustrate that CdFe₂O₄ is ferromagnetic semiconductor because both spin channels show direct band gap (E_g) at the Gamma point in the first Brillion zone and no state crosses E_F. The depletion of states and appearance of E_g around E_F in both spin channels are due to the exchange interactions. The presence of exchange interactions is evident because of unequal shift in energy in the spin polarized electronic structures, suggesting that the illustrated semiconducting behavior is also associated with a stable ferromagnetic state. On the other hand, computations performed through solely PBEsol show metallic response, for in both spin channels states cross E_F. the Fermi level moves, for spin up channel inside the valence band and for spin down channel inside the conduction band. Therefore, the holes for up spin channel and electrons for down spin channel are available for conduction process. To reveal the origin of ferromagnetism in detail, total and partial density of states are presented in Figs. 2(a) and 2(b). When computations are done by using TB-mBJ potential, the Fe d-states hybridize with O p-states in the energy region from 2 eV to 4 eV in the conduction band for down spin channel, and −1 eV to −5 eV in the valence band for up spin channel, while this hybridization energy range is contracted when PBEsolis is used. Basically, exchange splitting of 3d-states of Fe appears, due to octahedral environment of oxygen (FeO₆) causing a net crystal field, as schematically shown in Fig. 3. The crystal field splits the 3d-states into doublet e_g ( ) and and triplet states t_2g (d_xy, d_yz, and d_zx), with increased and decreased energy respectively.^[26–28] The degenerate orbitals can be suppressed through symmetry reductions, and this could be achieved by elongation or stretching bonds along z-axis, which are named as Z-out and Z-in distortions respectively. Therefore, orientation of the bonds and lowering symmetry make t_2g and e_g states strongly non-degenerate, thus distorting the octahedral symmetry into tetragonal one, as shown in Fig. 3. Because of that, , d_yz, and d_zx orbitals are lowered in energy but and d_xy orbitals has raised energy. This effect elongates the bond along z-direction, which is the preferred direction, and usually occurs when degeneracy in the e_g level with orbitals d⁴, d⁷, d⁸, etc. occurs. The Fe atom has seven valence electrons in the 3d-state, therefore, such elongations are preferred. To investigate the details of the (John Teller) JT distortions, the crystal field and JT energies are calculated by the relations; for up, for down spin channel, and respectively. The values are presented in Table 1. Here and represent the crystal field energy for spin up and spin down channels. The direct exchange energy ( and indirect exchange energy ( ) due to Fe-3d spin orientations are also calculated and reported in Table 1. The comparison of crystal field and John–Teller energies with the exchange energies ( depicts strong role of exchange interactions.^[29–33] The negative indirect exchange energy suggests that the dominant character of electronic spin is to form the more attractive spin down channel, as the reduced energy is more evident through the computations done by using PBEsol + mBJ potential. At the same time, computing indirect exchange energy through PBEsol potential gives out zero, which is due to the metallic nature. Similarly, using PBEsol + mBJ potential instead of PBEsol improves direct exchange energy and reduces John Teller energy. The magnetic moments of compound and individual atoms like Fe, Cd and O calculated by using PBEsol functional and TB-mBJ potential are listed in Table 1.

Fig. 3.

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	Fig. 3. (color online) The splitting of 3d-states of Fe (III) by John Teller distortion (JTE) at octahedral position in the spinel structure of CdFe2O4.

The magnetic moment with PBEsol functional is an integer while for TB-mBJ potential is a fractional, and that gives the evidence of convergence of metallic to semiconductor. The reduction of magnetic moment of Fe and its sharing to the nonmagnetic sites elucidate the strong hybridization between 3d-states of Fe and 2p-states of O. Computations through TB-mBJ illustrate that shifting of magnetic moment from magnetic to nonmagnetic site is stronger as compared to the computations through PBEsol functional. It suggests that in the studied compound, exchange mechanism is more favored in the ferromagnetic semiconducting than in the metallic compounds. The magnetic moments determined by the varying symmetries such as Fd3m, P3m1, and Pnma are 19.99 μ_B, 199 μ_B and 40 μ_B, respectively. The computed magnetic moment for P3m1 phase is higher, but being the lowest for Fd3m phase, while computed enthalpy of formations shows that CdFe₂O₄ has the highest stability in Fd3m phase, and this is the reason we have preferred to study the cubic phase of CdFe₂O₄.^[32,33]

The details of the electronic transitions, illustrating material interactions with the impinging radiations w.r.t. in various symmetry directions of the first Brillion zone are observed through inter-and intra-band transitions, which cause varying excitation and de-excitation rates. While, the inter-band transitions are sufficient for precisely revealing the optical behaviors, and can be expressed by complex dielectric constant .^[34–37] The real part, , shows dispersion, while imaginary part, , indicates light absorption, and both are related through Kramer–Kronig relation.^[38] Both and are of fundamental importance because of being used to compute many other optical parameters.^[36,37] We have applied PBEsol and PBEsol + mBJ potential to determine various optical parameters, within 0–10 eV, as illustrated in Figs. 4–5. The determined by PBEsol and PBEsol + mBJ are plotted in Fig. 4. It is evident that computed by using PBEsol potential reaches maximum at 1.76 eV. After attaining minimum at 3.18 eV, it again becomes maximum at 4.06 eV. The large intensity peaks depict sharply altering polarization to exhibit plasmonic excitations, of which appearance is due to the metallic nature. In contrast, using PBEsol + mBJ potential shows linearly enhancing up to 3.7 eV, and if it is above this energy, it decreases to the minimum without any considerable features. This is caused by the semiconducting characters. Hence, mBJ potential improves the band gap, as already shown in Fig. 1. From 5.8 eV to 8.2 eV, , when using PBEsol potential to compute, it becomes negative, revealing reflective nature to the incoming radiations. While PBEsol + mBJ potential does not expose this fact. Interestingly, static dielectric constant and E_g (computed by using PBEsol and PBEsol + mBJ potentials) have been found to satisfy Pennʼs relation ,^[39] as shown in table 1.

Fig. 4.

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	Fig. 4. (color online) (a) Real and (b) imaginary dielectric constants, (c) refractive index n(ω) and extinction co-efficient k(ω) calculated for CdFe2O4 in FM phase by employing PBEsol and PBEsol + mBJ functional.

Fig. 5.

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	Fig. 5. (color online) (a) Reflectivity, (b) absorption coefficient, and (c) optical conductivity calculated for CdFe2O4 in FM phase by employing PBEsol and PBEsol + mBJ functional.

The , showing absorptive nature for the incoming radiations, exhibits a threshold energy for the absorption, which are 1.43 eV and 2.36 eV, after calculating with PBEsol and PBEsol + mBJ respectively. The metallic response due to PBEsol potential shows sharp peaks in , induced by different resonance frequencies. This is evident in the computed band structure having states crossing E_F in both spin channels (see Fig. 1). Owing to the semiconducting character as elucidated through PBEsol + mBJ potential, both channels ignore the presence of free carries, due to which, uniformly improves with energy and shows maximum absorption at 4.7 eV, and then also uniformly deceases to minimum, as shown in Fig. 4(b).

The refractive index n(ω) and extinction coefficient k(ω), as plotted in Figs. 4(c) and 4(d), show a nature similar to that determined for and . These parameters are related according to the relations and .^[40] The refractive index n(ω) shows transparency while extinction coefficient k(ω) describes light attenuating capability of a material. The zero frequency limits for and n(ω), illustrated as and n(0) respectively, satisfying the relation. The values of and n(0) are higher for PBEsol potential than PBEsol + mBJ potential because the later potential improves the band, making the material more suitable to the optoelectronic applications.

To probe surface quality, light reflected by the material surface is observed by computing reflectivity R(ω) using PBEsol and PBEsol + mBJ potentials, as presented Fig. 5. There are large variations in the R(ω) determined by using PBEsol, as compared to that determined by using PBEsol + mBJ potentials, and it is again due to electronic structure difference that varies the carrier distribution. The R(0) determined through PBEsol + mBJ potential (0.0929) is much smaller than extracted through PBEsol potential (0.195), suggesting optoelectronic applications. Similarly, absorption coefficient α(ω) showing light absorption, as depicted in Fig. 5(b), exhibits critical values of 1.43 eV and 2.36 eV after applying PBEsol and PBEsol + mBJ potential, respectively. It agrees well to the , k(ω), and E_g values (extracted from the band structures, E_g is taken from the difference of valence band maxima and conduction minima, although the available carries for PBEsol make the studied compound metallic like system), showing the precision of the results. Moreover, according to the relation, , photons having shorter wavelength ( ) than a critical value are more susceptible to absorption, suggesting the significant role of PBEsol + mBJ potential in elucidating those critical values for suggested attractive optical applications.^[41]

The optical conductivity, σ (ω), showing optically activated free charge carriers, is computed by using both potentials, as expressed in Fig. 5(c). A part of impinging photons energy equal to E_g is consumed by the material, while remaining energy is responsible in generating free carriers (electrons and holes) to induce a forward current. As being expected, σ(ω) computed by using PBEsol potential illustrates more carriers for conduction than that computed by employing PBEsol + mBJ potential (because it showed semiconducting nature). Hence, the use of mBJ potential improves the band statures to enhance the feasibility of optical applications.

The imbalanced demand and supply chain of energy has stimulated researchers to search for the materials exhibiting high waste heat for useful energy conversion efficiencies. Such thermoelectric materials are gauged in terms of high electrical conductivity (σ/τ) and low thermal conductivity (κ/τ), to enhance the power factor and the figure of merit or the thermal efficiency , where S is Seebeck coefficient and τ is relaxation time.^[42–46] Mostly, in semiconductors, σ and S exhibit opposite trend. The thermoelectric nature of CdFe₂O₄ is investigated by computing σ, S, electronic part of the thermal conductivity κ_e and PF, within 200–800 K, and are plotted in Figs. 6(a)–6(d). The semi-classical theory based on BoltzTraP code^[47] is used for these computations. As shown in Fig. 6(a), electrical conductivity linearly increases from to by increasing temperature from 200 to 800 K respectively. This could be justified by the fact that increasing temperature enhances thermal energy of the electrons to freely form a high conducting state.

Fig. 6.

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	Fig. 6. (a) Electrical conductivity, (b) Seebeck coefficient, (c) thermal conductivity, and (d) power factor extracted for CdFe2O4 by applying BoltzTraP code.

As total thermal conductivity (κ /τ) arises due to both electrons and phonons, the Fourier law, written as (q and illustrate heat flow per unit time and the temperature gradient), can describe the thermal conductivity.^[48] The electronic thermal conductivity also linearly improves from to by increasing temperature from 200 K to 800 K (see Fig. 6(b).

It is important to notice a potential difference ( ) due to temperature gradients ( ), because the ratio of and expresses the Seebeck coefficient (S) that is useful in deciding the efficiency of a thermocouple. The S computed for CdFe₂O₄ is plotted in Fig. 6(c). It is evident in Fig. 6(c), that S initially increases with temperature and reaches at 350 K. Above that, it decreases linearly and becomes at 800 K. The decaying Seebeck coefficient may be due to the negative coefficient of temperature in typical semiconductors. Power factor (PF), which is related to the product of σ and square of S, is very useful to reveal thermoelectric application, and is also computed and presented in Fig. 6(d). As can be observed in Fig. 6(d), PF linearly increases from to with rising temperature from 200 K to 800 K. Therefore, linearly increasing PF having very high magnitude suggests the potential in CdFe₂O₄ for commercial thermoelectric applications.^[49]