The electronic, magnetic, and optical properties of double perovskite La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc)

Abstract

2D La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskites are considered promising potential materials for electronic, magnetic, and optoelectronic applications. Herein, we thoroughly investigated the ground state electronic, magnetic, and optical properties of half-metallic ferromagnetic (HM-FM) La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskites using the full-potential linearized augmented plane wave (FP-LAPW) method within the generalized gradient (GGA) and (GGA (+U)) approximations based on density functional theory (DFT). Full structural optimization was carried out by using the FP-LAPW method. The electronic band structure (BS) and the total density of states (DOS) calculations confirm that all these double perovskite compounds are half-metallic (HM). The total spin magnetic moments calculated confirm the half metallic and ferromagnetic nature of the materials. Optical properties were determined for their potential use in opto-electronic devices. Based on our results, we concluded that La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds are half metal ferromagnetic and have potential applications as ferromagnetic insulators in spintronic, magnetic and optoelectronic devices.

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1. Introduction

Perovskites are extensively studied oxides, and exhibit potential applications as insulators/dielectrics in superconductors and a range of materials from diamagnetic to colossal magneto-resistive (CMR).^1–3 BaTiO₃ and PbZrO₃ are well-known perovskites used in the piezoelectric, dielectric, optoelectronic and microelectronic industries.^4–6 Among the different classes of perovskites are ordered double perovskites (ODP), general formula, A₂B′B′′O₆, where “A” is a rare-earth or alkaline-earth element, while B′ and B′′ are different transition-metal ions. These have attracted attention due to their ability to accommodate an extensive range of rare earth elements at the A-site, significantly modifying the bond length and bond angle of the octahedra, and two different types, sizes and valence states of cations at the B′ and B′′-sites, offering the opportunity to manipulate, tune and tailor their physical properties according to industrial and technological needs.^7,8 ODP usually exhibit half-metallic (HM),^9,10 ferrimagnetic (FiM),^11,12 ferromagnetic (FM),^13,14 antiferromagnetic (AF),^15,16 half-metallic ferromagnetic (HM-FM),^17,18 half-metallic ferrimagnetic (HM-FiM),^17,19 nonmagnetic (NM),^18,20 ferroelectric (FE),^12,21 and semiconducting (SC)^12,18,21 responses near room temperature. Half-metallic materials have well-defined band gaps in one spin state and metallic bands in the other spin channel.²¹ Thus, HM materials have quantization of magnetic moment, zero spin susceptibility and 100% spin polarization at the Fermi level.^21–23 HM materials have potential application in data storage memory and magnetic recording due to their single spin charge carriers.^24,25 J. B. Philipp et al.²⁶ reported the structural, transport and optical properties and doping effects in half-metallic double perovskites (HM-DP) A₂CrWO₆ with A = Sr, Ba and Ca. The saturation magnetization and the influence of steric effects on the Curie temperature, T_C, were determined by changing the alkaline earth ion on the A site and a maximum Curie temperature (T_C = 458° K) was reported for Sr₂CrWO₆, which has an almost undistorted perovskite structure and a tolerance factor T_f ≈ 1. The researchers showed that the structural changes in Ba₂CrWO₆ and Ca₂CrWO₆ result in a reduction of the Curie temperature, suggesting that, in general for double perovskites, an optimal T_C can be achieved only for T_f ≈ 1. Furthermore, they found that both the Curie temperature and saturation magnetization M_S reduce due to electron doping in Sr₂CrWO₆ by partial substitution of Sr²⁺ by La²⁺. Room temperature colossal magneto resistance and high temperature T_C transition for Sr₂FeMoO₆ systems confirm half-metallic (HM) behavior. Half-metallic ferromagnetic (HM-FM) nature is observed for La₂CrCoO₆ (LCCO),²⁷ La₂CrNiO₆ (LCNO),²⁸ La₂ScNiO₆ (LSNO),²⁷ La₂VNiO₆ (LVNO)²⁷ and La₂VScO₆ (LVSO)²⁷ compounds, validating their potential suitability for spintronic applications such as in magnetic sensors, magneto resistive random-access memory (MRRAM) and solid fuel cells. A half-metallic antiferromagnetic (HM-AFM) response is observed in LaAVRuO₆ (A = Ca, Sr, and Ba) systems using the local spin-density approximation (LSDA).²⁹ Chen et al.³⁰ reported that La₂VMnO₆ has half-metallic antiferromagnetic (HM-AFM) and ferromagnetic (HM-FM) phases and the antiferromagnetic phase is more stable by 0.1 eV per cell than the ferromagnetic phase. The magneto resistance (MR) in ordered and disordered double perovskite oxide Sr₂FeMoO₆ was investigated by D. D. Sarma et al.³¹ They concluded that the ordered sample exhibits a sharp low-field response, followed by moderate changes at higher fields, and the disordered sample was characterized in the absence of a low-field response, confirming that the low-field response depends critically on the half-metallic ferromagnetism, while the high-field response depends on the overall magnetic nature of the sample, still in the absence of the half-metallic state. Although double perovskites are widely studied materials due to their potential applications, no detailed theoretical and experimental studies are available for La based La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskite compounds. Herein, we conclusively theoretically predict the structural, electronic, magnetic and optoelectronic properties of La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskites using first principle generalized gradient GGA^32–34 and GGA (+U)^35,36 approximation calculations, considering the ferromagnetic state in the tetragonal structure (14/mmm, No. 139). Since La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskites are strongly electron correlated systems requiring enhanced description, GGA calculations were corrected by using a strong-correlation correction GGA (+U). We revealed that the electronic magnetic and optoelectronic properties of these materials support their potential use in capacitors, photocells, data storage devices and automobile license plates.

2. Method of calculation

We performed structural, electronic, magnetic and optical property calculations for La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskite compounds using GGA^32–34 and GGA (+U)^35,36 (hybrid term) techniques within the framework of density functional theory (DFT).^37,38 Full structural optimization was carried out by using the full-potential linearized augmented plane wave (FP-LAPW) method^39,40 as implemented in WIEN2K software⁴¹ (updated UNIX versions of the original Wien2K code are called Wien93, Wien95 and Wien97). The WIEN2K software was used to calculate the lattice parameter, band structure, charge density and exchange-correlation potential (V_Exc). The charge density and potential were extended in terms of spherical harmonics inside the muffin-tin spheres in terms of the augmented plane wave (APW)^32–36 outside the muffin-tin spheres. In WIEN2K, the basis set APW+ local is used inside the atomic sphere for chemically important orbitals, while FP-LAPW is used for others, therefore it is based on the combination of both techniques.^39–41 The strong electron correlation systems, such as transition metal oxides, were described by the on-site electron correlation correction GGA (+U) method.^35,36 The main control of the convergence in co-relation with the basis size is by means of the parameter, RK_max, the product of the largest plane wave vector (K_max), the smallest radius of the sphere (R) and the cutoff value of the angular momentum (L_max). In our calculations, for excellence convergence of the charge, a plane wave cut-off value of R_MTK_max = 6 is used in the interstitial region, up to 120 K-point numbers were used for the Brillouin zone for the 14/mmm structures and L_max = 6 was used for the charge density and potential.

3. Results and discussion

3.1. Electronic band structures and density of states

The band structure (BS) and density of states (DOS) give information about the electronic properties^27,28,36 of the materials, and were investigated using GGA and GGA (+U) for the La₂BB′O₆ (B = Cr, V and B′ = Co, Ni and Sc) double perovskite systems, as illustrated in Fig. 1 and 2. We observed that the band structures are crowded in both the valence and conduction band. GGA calculations for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ systems (Fig. 1(a–j) and Table 1) reveal overlapped valence and conduction bands around the Fermi level in spin-up states with a metallic response, whereas in the spin-down channel there was no overlapping around the Fermi level and energy gaps of 0.94 eV, 1.14 eV, 1.63 eV, 1.22 eV and 3.53 eV were exhibited, confirming a non-metallic semiconducting response, validating half-metallic (HM) behavior in good agreement with previously reported work.^37–39

Fig. 1 (a–j) Band structures of La₂CrCoO₆ (LCCO), La₂CrNiO₆ (LCNO), La₂ScNiO₆ (LSNO), La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) using the GGA scheme.

Fig. 2 (a–j) Band structure of La₂CrCoO₆ (LCCO), La₂CrNiO₆ (LCNO), La₂ScNiO₆ (LSNO), La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) using the GGA (+U) scheme.

Table 1 Spin-up and spin-down state values of the bandgap of La₂CrCoO₆ (LCCO), La₂CrNiO₆ (LCNO), La₂ScNiO₆ (LSNO), La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) compounds calculated via GGA and GGA (+U)

Serial no.	Compound	Spin	Present work Eg (eV)		Other theoretical work Eg (eV)
			GGA	GGA (+U)	GGA	GGA (+U)
1	La2CrCoO6	Spin-up	Metallic	Metallic	Metallic	Metallic
		Spin-down	0.945296	1.667336	0.94 (ref. 28)	0.762 (ref. 27)
2	La2CrNiO6	Spin-up	Metallic	Metallic	Metallic	Metallic
		Spin-down	1.149897	1.211155	1.14 (ref. 28)	1.007 (ref. 27)
3	La2ScNiO6	Spin-up	Metallic	Metallic	Metallic	Metallic
		Spin-down	1.632651	3.855420	1.63 (ref. 28)	1.497 (ref. 27)
4	La2VNiO6	Spin-up	Metallic	Metallic	Metallic	Metallic
		Spin-down	1.225672	2.00800	1.22 (ref. 28)	1.116 (ref. 27)
5	La2VScO6	Spin-up	Metallic	Metallic	Metallic	Metallic
		Spin-down	3.537194	3.566456	3.53 (ref. 28)	3.238 (ref. 27)

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Fig. 2(a–j) represent the GGA (+U) calculations for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds, showing that in the spin-down channel the energy gaps for these compounds were 1.66 eV, 1.21 eV, 3.85 eV, 2.00 eV and 3.56 eV, respectively. However, overlapped valence and conduction bands are observed in the spin-up channel, confirming the metallic response. Both the values of the band gap calculated via GGA and GGA (+U) calculations listed in Table 1 and represented in Fig. 1 and 2 indicate that La₂CrCoO_6, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ are direct bandgap materials since the maxima of the valence bands and minima of the conduction bands lie at the sigma (Γ) symmetry points.

The band structure calculated by GGA and GGA (+U) shows that the width of the band gaps increases by a specific amount of energy (0.5 eV). The band gap or energy gap (E_g) calculated for La₂VScO₆ is the maximum and equal to 3.537 eV and 3.566 eV whereas La₂CrCoO₆ has the minimum bandgap of 0.945 eV (according GGA), and the La₂CrNiO₆ system has the minimum band gap of 1.211 eV (according to GGA (+U)). The calculated band gaps in terms of electron volts (eV) using GGA and GGA (+U) calculations are illustrated in Table 1 and are in good agreement with other previously reported theoretical work.

The behaviors of the band structures and electronic distributions (states) of the La₂CrNiO₆, La₂ScNiO_6, La₂VNiO₆ and La₂VScO₆ compounds were calculated by total DOS (TDOS) using the GGA and GGA (+U) schemes as shown in Fig. 3 and 4(a–e). The TDOS calculations reveal that in all the compounds a maximum contribution above the Fermi level (E_F), arises due to Tot-La in the spin-up as well as spin-down states. The valence and conduction bands overlapping in the spin-up channel validates the metallic behavior, whereas there is a clear gap between the valence and conduction bands for the spin-down channel displaying semiconducting behavior, hence confirming that all these compounds are half-metallic. The main reasons for the half metallicity are p–d hybridization and double exchange. The stronger p–d hybridization in these compounds leads to the p-state being pushed above the Fermi level (E_F) in the spin-up channel, while the band extends in the spin-down channel with E_F exhibiting the double exchange effect, referred to as half-metallicity. In the case of La₂CrNiO₆ (Fig. 3(b) and 4(b)) in both the spin-up and spin-down channels above the Fermi level, La-f has the maximum contribution, whereas below the Fermi level there is a small contribution from Cr-d and Ni-d to the total density of states (TDOS) and the peak value of the TDOS in the conduction band is nearly 48 (states per eV) at 3 eV. Similarly, we observed that in the case of the La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds (Fig. 3(c–e) and 4(c–e)) there is a small contribution of Ni-d to the total density of states (TDOS) below the Fermi level (E_F), while above the Fermi level there is a maximum contribution of La-f to the (TDOS). Furthermore, there is a negligible contribution of O-p to the total density of states (TDOS) below the Fermi level and a small contribution of Sc-d and V-d in the conduction bands, for both the spin-up and spin-down channels.

Fig. 3 (a–e) Total density of states (TDOS) of La₂CrCoO₆ (LCCO), La₂CrNiO₆ (LCNO), La₂ScNiO₆ (LSNO), La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) compounds calculated by the GGA scheme.

Fig. 4 Total density of states (TDOS) of La₂CrCoO₆ (LCCO), La₂CrNiO₆ (LCNO), La₂ScNiO₆ (LSNO), La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) compounds calculated by the GGA (+U) scheme.

The DOS is nearly the same for both the GGA and GGA (+U) calculations and the maximum value of the density of states above the Fermi level is at approximately 1.25 eV.

3.2. Magnetic properties

The magnetic behaviors of the La₂VNiO₆, La₂VScO₆, La₂ScNiO₆, La₂CrNiO₆ and La₂CrCoO₆ double perovskite compounds were calculated by spin polarized GGA and GGA (+U) approximations by calculating the interstitial region (m^Intst), individual atom (m^{S1, S2, S3, S4}) and total cell (m^Cell) magnetic moments. The detailed values of the magnetic moments calculated are listed in Tables 2 and 3. In the generalized gradient approximation calculation, spinel compound transition metal d-state electrons with oxides are not strongly correlated, which keeps them near the Fermi level (E_F). Therefore, for the transition metal d-state electrons we used GGA (+U) calculations with the U parameter changing from 1.5 eV to 3.0 eV. Choosing an appropriate value of U (ranging from 1.5 eV to 3.0 eV), splits the band structure of these compounds but the total magnetic moments remain the same and the energy gaps increase from 1.2 eV, 3.5 eV, 1.6 eV, 1.1 eV and 0.9 eV to 2.0 eV, 3.6 eV, 3.3 eV, 2.9 eV and 1.6 eV in spin-down channels. The detailed values of the magnetic moments calculated via the GGA and GGA (+U) approximations of all these compounds are illustrated in Tables 2 and 3, and confirm the half-metallic ferromagnetism.

Table 2 The magnetic moments of half-metallic ferromagnetic (HM-FM) double perovskites, along with interstitial region (m^Intst), individual atom (m^S1, m^S2, m^S3, m^S4, m^S5) and total cell (m^Cell) magnetic moments by GGA calculation

Serial no.	Materials	m^Intst	m^S1	m^S2	m^S3	m^S4	m^S5	m^Cell
1	La2CrCoO6	0.27150	0.01201	2.32582	0.18412	0.01326	0.01290	2.88359
2	La2CrNiO6	0.28269	0.01420	1.99849	1.38930	0.04885	0.05090	4.00017
3	La2ScNiO6	0.01170	−0.00220	0.04520	0.77638	0.02802	0.02834	0.99828
4	La2VNiO6	0.15326	0.01338	0.89250	1.48145	0.06907	0.06982	2.97139
5	La2VScO6	0.42797	0.07356	1.42382	0.03688	−0.00487	−0.00447	2.00820

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Table 3 The magnetic moments of half-metallic ferromagnetic (HM-FM) double perovskites, along with interstitial region (m^Intst), individual atom (m^S1, m^S2, m^S3, m^S4, m^S5) and total cell (m^Cell) magnetic moments by GGA (+U) calculation

Serial no.	Compound	m^Intst	m^S1	m^S2	m^S3	m^S4	m^S5	m^Cell
1	La2CrCoO6	0.32371	0.02128	2.38831	0.15488	0.01886	0.01511	3.00761
2	La2CrNiO6	0.26399	0.01282	1.97414	1.40242	0.05606	0.05715	4.00693
3	La2ScNiO6	−0.01631	0.00239	0.01728	1.31964	−0.05725	−0.05270	1.00007
4	La2VNiO6	0.15399	0.01325	0.89531	1.48322	0.06848	0.06922	2.97283
5	La2VScO6	0.42984	0.07343	1.42780	0.03685	−0.00532	−0.00506	2.01050

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The transition metals Cr, Co, Ni, Sc, and V have the maximum contribution to the total spin magnetic moments, whereas La has the minimum contribution. In the La₂CrCoO₆ (LCCO) and La₂CrNiO₆ (LCNO) compounds, Cr has the maximum contribution to the total spin magnetic moment, whereas in the La₂ScNiO₆ (LSNO) and La₂VNiO₆ (LVNO) compounds, Ni has the maximum contribution. However, in the case of the La₂VScO₆ (LVSO) compound, V has the maximum contribution to the total spin magnetic moment. Furthermore, in all these compounds La₂CrNiO₆ (LCNO) has the maximum total spin magnetic moment of 4.0 μ_B, whereas La₂ScNiO₆ (LSNO) has the minimum total magnetic moment of 0.9 μ_B.

3.3. Optical properties

We have investigated the optical properties of the half-metallic ferromagnetic ordered double perovskite compounds La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆, including the optical conductivity σ(ω), reflectivity R(ω), refractive index n(ω), absorption co-efficient α(ω) and real ε₁(ω) and imaginary ε₂(ω) parts of the dielectric constant for various energies (in eV), via the GGA and GGA (+U) techniques. The calculated optical conductivity (σ(ω)) is illustrated in Fig. 5(a and b). The optical conductivity (σ(ω)) is the ratio of the current density to the electric field and depends on the frequency or energy of the incident photons, and is represented by the equation:

σ(ω) = J(ω)/E(ω) (1)

Fig. 5 (a, b) Optical conductivity (σ(ω)) and (c, d) absorption co-efficient (α(ω)) of the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds calculated by the GGA and GGA (+U) processes.

The calculated values of the optical conductivity (σ(ω)) are listed in Table 4, and show that the optical conductivity (σ(ω)) varies from 11.205 to 14.927. La₂CrCoO₆ (LCCO) has the maximum value of 14.927 at 20.993 eV, whereas La₂ScNiO₆ (LSNO) has the minimum value of 11.205 at 21.565 eV. The values of the optical conductivity calculated by using the GGA scheme for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds are 14.927, 14.181, 11.205, 14.377 and 11.680 at 20.993 eV, 20.911 eV, 21.565 eV, 20.830 eV and 20.367 eV. The values of the optical conductivity using the GGA (+U) scheme for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds are 14.958, 14.317, 9.286, 14.377 and 11.680 at 20.990 eV, 20.912 eV, 22.000 eV, 20.830 eV and 20.367 eV. The results obtained for the optical conductivity are nearly the same for both the GGA and GGA (+U) techniques. Furthermore, the value of the optical conductivity (σ(ω)) for La₂VNiO₆ (LVNO) and La₂VScO₆ (LVSO) is exactly same for both schemes. The absorption coefficients α(ω) calculated for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds shown in Fig. 5(c and d) quantify the incident light absorbed by the material (instead of being reflected or refracted), giving information about the transformation of electromagnetic energy into the internal energy of the absorber.

Table 4 The optical conductivities σ(ω) and absorption co-efficients α(ω) of the half-metallic ferromagnetic (HM-FM) double perovskite compounds, calculated by the GGA and GGA (+U) schemes

Serial no.	Compound	GGA		GGA (+U)		GGA		GGA (+U)
		σ(ω)	E (eV)	σ(ω)	E (eV)	α(ω)	E (eV)	α(ω)	E (eV)
1	La2CrCoO6	14.927	20.993	14.958	20.990	401.082	21.973	401.871	21.973
2	La2CrNiO6	14.181	20.911	14.317	20.912	394.128	21.918	395.603	21.918
3	La2ScNiO6	11.205	21.565	9.286	22.000	378.128	22.136	376.096	22.204
4	La2VNiO6	14.377	20.830	14.377	20.830	390.602	21.837	390.602	21.837
5	La2VScO6	11.680	20.367	11.680	20.367	376.535	22.354	376.535	22.535

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The values of the absorption coefficient α(ω) calculated via GGA and GGA (+U) listed in Table 4 indicate that of all five of these compounds, La₂CrCoO₆ (LCCO) has the maximum value of 401.082 at 21.973 eV, showing that this compound has more potential to absorb incident photons and can be used for absorption in optical devices.

The absorption power for incident photons in the La₂VScO₆ compound is the minimum (376.535 at 22.354 eV), indicating that this compound may be less useful for optical and optoelectronic applications than the other four compounds. The value of the energy observed for the maximum absorption coefficient (α(ω)) is in the range of 21 eV to 22.5 eV. Interestingly, the absorption coefficient (α(ω)) calculated for La₂VNiO₆ is α = 390.602, exactly the same for both schemes, and both schemes exhibit the same value of the energy (E = 21.837 eV). The values of the optical absorption coefficients α(ω) are specific for each compound, validating the potential application in optics where a specific ultraviolet radiation (UV) wavelength of absorption is required, such as in sunglasses, colored filters, dyes etc.

Fig. 6 represents the refractive indices (n(ω)) of all five compounds. The refractive index n(ω) is a critical feature of semiconductors, revealing the amount of light bent or refracted. The value of n(ω) increases with increasing energy (eV). The GGA and GGA (+U) calculations report higher values of the refractive index n(ω) for La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds of 6.005 (at 0.013 eV), 9.602 (at 0.038 eV), 4.928 (at 0.122 eV), 3.943 (at 0.013 eV) and 3.979 (at 0.013 eV), and 6.322 (at 0.013 eV), 8.732 (at 0.013 eV), 4.614 (at 0.149 eV), 3.820 (at 0.019 eV) and 3.979 (at 0.013 eV), indicating their potential use in optoelectronic applications, primarily in electronic displays, such as LCDs, OLEDs, and quantum dot (QDLED) televisions. The refractive index (n(ω)) calculated for La₂CrNiO₆ has the maximum value of 9.602 at 0.038 eV, whereas a minimum value of 3.943 at 0.013 eV is reported for the La₂VNiO₆ compound, as compared to the other compounds. There is no large difference between the refractive indices (n(ω)) calculated by GGA and GGA (+U). The refractive indices (n(ω)) measured for the La₂VNiO₆ and La₂VScO₆ compounds are nearly the same at 0.013 eV in the GGA scheme. The refractive indices (n(ω)) of La₂CrCoO₆ and La₂ScNiO₆ are equal to 6.0 and 4.9 at 0.013 eV and 0.122 eV, respectively. The n(ω) values observed for all these compounds are non-linear. Initially there is a sharp increase and then a decrease with increasing energy (eV), etc., which is in good agreement with previously reported theoretical and experimental results related to 2D double perovskites.^40–44

Fig. 6 (a, b) Refractive index (n(ω)) and (c, d) reflectivity R(ω) of the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds calculated by the GGA and GGA (+U) processes.

The optical reflectivity R(ω) values in relation to the incident energy calculated via GGA and GGA (+U) shown Fig. 6(c and d) confirm the reflective nature of these compounds. The optical reflectivity R(ω) is the ratio of the energy of a wave reflected from a surface to the energy possessed by the wave striking the surface. The GGA and GGA (+U) values of the reflectivity R(ω) for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds of 0.534, 0.696, 0.475, 0.389 and 0.369, and 0.558, 0.662, 0.454, 0.389 and 0.369 indicate that the surfaces of these materials are smooth and have an ordered crystalline structure. Among all five of these compounds, La₂CrNiO₆ has the maximum value of 0.696 at 0.04 eV, whereas La₂VScO₆ has the lowest value of 0.369 at 0.013 eV in both the GGA and GGA (+U) schemes.

The calculated real ε₁(ω) and imaginary ε₂(ω) parts of the dielectric constant as a function of energy (eV) calculated via both the GGA and GGA (+U) schemes are shown in Fig. 7. The dielectric constant, known as the relativity permittivity or specific inductive capacity, is a dimensionless number expressing a material's ability to support an electrostatic field while dissipating a minimum amount of energy in the form of heat. The calculated values of the static dielectric constants for the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds confirm that all these compounds display a dielectric response, and according to GGA approximations the values of the real parts of the dielectric constant ε₁(ω) are in the range of 14.172 to 80.617, and the values of the imaginary parts are in the range of 7.867 to 78.097. Furthermore, according to the GGA (+U) scheme, the values of the real parts of the dielectric constant ε₁(ω) are in the range of 14.172 to 68.613, and the values of the imaginary parts ε₂(ω) are in the range of 7.867 to 58.682.

Fig. 7 Real ε₁(ω) and imaginary ε₂(ω) parts of the dielectric constants of the La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ compounds calculated by the GGA and GGA (+U) processes.

It is worth noting that the value of the ε₂(ω) is zero until absorption starts after the photon energy reaches the band gap energy, which determines the threshold for a direct optical transition between the VBM and the CBM. The dielectric constant values describe the effect of the electric field on a material and the higher the dielectric constant, the more the material tends to reduce any field set up in it. In both the GGA and GGA (+U) processes, the calculated values of the real ε₁(ω) and imaginary ε₂(ω) parts of the dielectric constant for La₂CrNiO₆ (LCNO) are the highest, which indicates that this material is more dielectric than the others. La₂VNiO₆ (LVNO) is the least dielectric material, indicated by the obtained values of the dielectric constant. The materials with high dielectric constants are useful in the manufacture of high-value capacitors.

4. Conclusions

In this study, we investigated the influence of 3d transition metal incorporation at the B (Cr and V) and B′ (Co, Ni and Sc) sites of the 2D La₂BB′O₆ double perovskites on the electronic, magnetic, and optoelectronic properties. Full structural optimization was carried out by using the full-potential linearized augmented plane wave technique as implemented in WIEN2K software. The results are in good agreement with previously reported work. The electronic band structures in the spin-up and spin-down states reveal that all five of these compounds are half-metallic. In the case of the spin-up channel, La₂CrCoO₆, La₂CrNiO₆, La₂ScNiO₆, La₂VNiO₆ and La₂VScO₆ display metallic features, while they display non-metallic features in the spin-down channel. The calculated density of states gives a detailed explanation of the band structure, and the electronic distributions of these compounds drawn in terms of the total density of states verify the half-metallic nature. The pd-hybridizations, double exchange correlations (strong-correlation correction (GGA (+U))), ground-state energies and calculated total magnetic moments (2.9 μ_B, 2.0 μ_B, 0.9 μ_B, 4.0 μ_B and 2.8 μ_B) of these compounds confirm the ferromagnetic behavior. Therefore, the overall nature of these materials is half-metallic ferromagnetic. We have calculated the optical conductivities σ(ω) and absorption co-efficients α(ω), and the maximum values are 14.927 and 401.08 at 21.97 eV for the La₂CrCoO₆ compound. The refractive index (n(ω)) and optical reflectivity (R(ω)) calculations reveal that La₂CrNiO₆ has the highest values of 9.602 at 0.038 eV and 0.696 at 0.04 eV, whereas La₂VNiO₆ has the lowest values of 3.943 at 0.013 eV and 0.389 at 0.013 eV, compared to all the other compounds. The real and imaginary parts of the dielectric constant show that La₂CrNiO₆ has the maximum value, which means that this compound is more dielectric than the others. The fully spin-polarized strong magnetic element induces the weak magnetic element, and slightly increases the positive (or negative) magnetic moment, therefore it is regarded as the main cause of the half-metallic properties. In summary, based on our calculations, we revealed that 2D La₂BB′O₆ double perovskites with 3d transition metals incorporated at the B (Cr and V) and B′ (Co, Ni and Sc) sites have potential applications in spintronic, multifunctional, and optoelectronic devices.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. The data may be provided on request.

Conflicts of interest

I hereby confirm that the work reported in this manuscript is novel and has no conflict of interest.

Acknowledgements

All my colleagues and fellows from the Material Modelling Lab are acknowledged for valuable suggestions and comments in this research work.

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