Half-metallicity of novel halide double perovskites K₂CuVCl₆ and Rb₂CuVCl₆: application in next-generation spintronic devices

Abstract

This work reports the determination of structural, electronic, half-metallic and magnetic properties of new double perovskites K₂CuVCl₆ and Rb₂CuVCl₆ using the full-potential linearized augmented plane wave plus local orbitals method incorporated in the WIEN2k code. The calculations performed for this prediction were framed using the density functional theory, and the exchange and correlation potential were described using the generalized gradient approximation of TB-mBJ (Tran–Blaha modified Becke–Johnson). The structural properties confirmed the stable ferromagnetic ground state of the two studied compounds. The equilibrium structural parameters, such as lattice constant (a₀), bulk modulus (B₀), their first pressure derivative (B′) and minimum of the total energy (E₀), were determined for both the compounds. The electronic properties showed that the studied perovskite compounds were completely half-metallic materials. The half-metallic gap (E_HM) values for the compounds were 1.119 eV (for K₂CuVCl₆) and 1.088 eV (for Rb₂CuVCl₆). The exchange-splitting energy (Δ(d)) was found to be large for both the compounds (Δ(d) = 3.482 eV for K₂CuVCl₆ and Δ(d) = 3.380 eV for Rb₂CuVCl₆). The calculated total magnetic moments of the two studied materials indicated major contributions from V atoms and minor contributions from Cu atoms. Owing to p–d hybridization, feeble magnetic moments were exhibited by the non-magnetic K, Rb, Cu and Cl sites, while the atomic magnetic moment of V atoms decreased from its free space charge of 3.00 μ_B.

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1. Introduction

Half-metallic materials play vital roles in the development of new devices based on spintronic and optoelectronic technologies. In the last decade, half-metallic materials have attracted increasing attention of researchers owing to their excellent electronic, magnetic and optical properties. Generally, half-metallic compounds exhibit two different electronic behaviors depending on the two spin directions: a semiconducting character in one spin direction and a metallic nature in the other spin direction.¹ Research in this domain was initiated by the prediction of the electronic structure of half-Heuslers NiMnS and PtMnS by Groot et al.² The most prominent works after this prediction were on perovskite materials, such as La_0.70Sr_0.30MnO₃ and Sr₂FeMoO₆;^3,4 Heusler alloys, such as Co₂MnSi and Co₂FeSi;^5,6 and TM-doped chalcogenide and pnictide semiconductors.⁷

In the present time, the half-metallicity has appeared in several types of materials, such as: TM (transition metal) doped semiconductors, exploring half-metallicity in NiO via TM and NTM doping: Insights from LDA and LDA-SIC approaches,⁸ investigating half-metallic behavior of MnO doped with TM and NTM: LDA, LDA-SIC and LDA + U analysis;⁹ half-Heusler alloys, study on the phase stability, mechanical and half-metallic properties of half-Heusler alloys FeMnZ (Z = Si, Ge and Sn),¹⁰ half-metallicity and thermoelectric performance: A multifaceted investigation of Zr-based half-Heusler alloys;¹¹ full-Heusler alloys, a spin-polarized analysis of the half-metallicity, mechanical, structural and optoelectronic attributes of full-Heusler XVCo₂ (X = B and P) alloys,¹² revealing half-metallicity: predicting large band-gaps in halogen-based full-Heusler alloys;¹³ quaternary Heusler alloys, systematic study of structural, elastic, electronic, magnetism and half-metallic properties for the quaternary alloys: Heusler type VZrReZ (Z = Si, Ge and Sn),¹⁴ ab initio study of quaternary Heusler alloys LiAEFeSb (AE = Be, Mg, Ca, Sr or Ba) and prediction of half-metallicity in LiSrFeSb and LiBaFeSb;¹⁵ perovskites materials, a DFT theoretical prediction of new half-metallic ferromagnetism, mechanical stability, optoelectronic and thermoelectric properties of ZnCrO₃ perovskites for spintronic applications,¹⁶ half-metallic ferromagnetic and optical properties of YScO₃ (Y = Ni, Pd, and Pt) perovskite: a first principles study;¹⁷ and double perovskite materials, harnessing the half-metallicity and thermoelectric insights in Cs₂AgMBr₆ (M = V, Mn, Ni) double halide perovskites: a DFT study,¹⁸ tunability of half metallicity and thermoelectric indicators in Na₂TaX₆ (X = Cl, Br) vacancy ordered double perovskites.¹⁹ Moreover, recent studies have highlighted the growing interest in simple and double perovskites owing to their tunable properties and potential in optoelectronics, catalysis, and energy applications.^20–33

In this study, we computed the structural, electronic and magnetic features of the new double perovskites K₂CuVCl₆ and Rb₂CuVCl₆ in order to prove their complete half-metallic property, and the calculations were carried out using the full-potential linearized augmented plane waves with local orbitals (FP-LAPW + lo) method embedded in the WIEN2k package. This study was conducted in order to determine the large half-metallic energy gap (E_HM) and high Curie temperature (T_C) corresponding to the new double perovskites K₂CuVCl₆ and Rb₂CuVCl₆ using the TB-mBJ–GGA approximation. This study is crucial for the development of spintronics, enabling low-energy storage and computing devices. It also highlights the prospects of these materials in optoelectronics, particularly for advanced LEDs and solar cells. Finally, these findings can be utilized in thermoelectricity to enhance the energy efficiency of industrial systems.

The rest of this article is given as follows: simulation method and calculation details are mentioned in Section 2. Obtained results are discussed in Section 3. The conclusions of this prediction are listed in Section 4.

2. Computational details

In this work, the DFT (density functional theory)³⁴ calculations were carried out by employing the full-potential linearized augmented plane waves plus local orbitals (FP-LAPW + lo) method^35,36 implemented in the WIEN2k code.³⁷ The potential of exchange and correlation is described using both Perdew–Burke–Ernzerhof generalized gradient approximation (PBE–GGA)³⁸ and Tran–Blaha modified Becke–Johnson potential of the generalized gradient approximation (TB-mBJ–GGA).^39,40 The computational parameters employed in this prediction approach are as follows: the matrix size R_MT × K_max parameter, which represents the extension of the plane waves and the convergence of the energy eigenvalues, is taken as 8, where K_max denotes the large modulus of the reciprocal lattice vector, and R_MT denotes the small radius of muffin-tin spheres. The cut-off energy (E_cut-off), which evaluates the number of plane waves used in the basic function, was chosen as −6 eV. The Fourier expansion factor (G_Max) that describes charge density translation and wave function expansion was set to 12. The irreducible Brillouin zone integrations were founded on the 11 × 11 × 11 mesh, in which 56 k-points were produced. The maximum norm of the angular momentum was equal to l_max = 10. The computed radii of the K, Rb, Cu, V, and Cl atoms reported in this study were 2.50, 2.50, 2.35, 2.30 and 2.02 a.u., respectively. The K (4s¹), Rb (5s¹), Cu (3d¹⁰4s¹), V (3d³4s²) and Cl (3s²3p⁵) states were treated as valence electrons. The anti-ferromagnetic calculations were carried out using the super-cell size of 1 × 1 × 2. Furthermore, the self-consistent process was set to stop iterations when the convergence energy becomes greater than 10⁻⁴ Ry.

3. Results and discussion

3.1. Structural properties

3.1.1. Crystal structure. The current double perovskites X₂CuVCl₆ (X = K and Rb) crystallize in the cubic system with the space group Fm3̄m (no. 225).⁴¹ The atomic positions are given as follows: the X atoms occupy the 8c (1/4, 1/4, 1/4) positions, the Cu and V atoms occupy the 4a (0, 0, 0) and 4b (1/2, 0, 0) positions, respectively, and Cl atoms are located at the 24e (0.2504, 0, 0) positions.⁴¹ The crystal structure of the double perovskite compound is depicted in Fig. 1.

Fig. 1 Crystal structure of (a) ABX₃ simple perovskite and (b) A₂BB′X₆ double perovskite compounds (A = K and Rb, B = Cu, B′ = V and X = Cl).

3.1.2. The equilibrium lattice parameters of the studied double perovskites. The optimized structural parameters were fitted using the empirical Birch–Murnaghan equation of state (EOS).^42,43 The fitted E–V curves of the K₂CuVCl₆ and Rb₂CuVCl₆ double perovskites obtained using the framework of the Birch–Murnaghan equation in the paramagnetic (PM), ferromagnetic (FM) and anti-ferromagnetic (AFM) phases are illustrated in Fig. 2. It can be observed that the ferromagnetic state is the stable ground phase for all the studied compounds. The optimized structural parameters, such as the equilibrium lattice constant (a₀), bulk modulus (B₀), its first pressure derivative (B′) and the minimum of the total energy (E₀), of both K₂CuVCl₆ and Rb₂CuVCl₆ are summarized in Table 1. No experimental data or theoretical values have been reported in the literature for these double perovskite materials, so our structural results of these K₂CuVCl₆ and Rb₂CuVCl₆ compounds are more and more useful for further experimental and predicted works in the spintronic and optoelectronic areas.

Fig. 2 Plot of total energy fitting versus cell-volume of (a) K₆CuVCl₆ and (b) Rb₆CuVCl₆ double perovskite compounds.

Table 1 Equilibrium lattice parameters of X₆CuVCl₆ (X = K and Rb) double perovskite materials in PM, FM and AFM phases calculated using the GGA approximation. The results are compared with those of the K₂ScCuCl₆ and Rb₂ScCuCl₆ double perovskite compounds

Material	Configuration	Lattice constant a0 (Å)	Bulk modulus B0 (GPa)	B′	Minimum of the total energy E0 (Ry)
K2CuVCl6	PM phase	9.8142	40.7357	4.8572	−13156.1424
	FM phase	9.8513	39.0165	5.1508	−13156.2112
	AFM phase	9.8417	39.0712	4.9912	−13156.1957
Rb2CuVCl6	PM phase	9.8978	39.6915	4.9354	−22673.5721
	FM phase	9.9357	38.3989	4.9421	−22673.6420
	AFM phase	9.9214	38.4157	4.9321	−22673.6249
K2ScCuCl6 (ref. 44)	PM phase	9.98	35.76	5.00	−12786.28
Rb2ScCuCl6 (ref. 44)	PM phas	10.08	49.09	8.47	−22303.72

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3.1.3. Cohesive energy and formation energy.
3.1.3.1. Cohesive energy. In a solid, the atoms are bound together by cohesive energy (E_Coh). The E_Coh energy value is estimated as the difference between the total energy of the material and the sum of different atomic energies of the constituents at infinite separation.⁴⁵ The E_Coh energy of the X₂CuVCl₆ (X = K and Rb) materials was calculated using the following formula:

E_Coh(X₂CuVCl₆) = (2E^X + E^Cu + E^V + 6E^Cl) − E₀ (1)

where E^X, E^Cu, E^Vand E^Cl represent the constituent energies of X, Cu, V and Cl atoms, respectively; and E₀ indicates the total energy of the solid. The calculated E_Coh energies of the current compounds at their ferromagnetic stable state are listed in Table 2. The E_Coh energies of both compounds have a positive sign, which confirms that the compounds are stable in the ferromagnetic phase.

Table 2 Computed cohesive energy E_Coh (in Ry) of the equilibrium X₆CuVCl₆ (X = K and Rb) double perovskite materials and energies of individual X, Cu, V and Cl atoms

Material	E0	E^X	E^Cu	E^V	E^Cl	ECoh
K2CuVCl6	−13156.211232	−1204.191254	−3309.685742	−1898.056927	−922.888706	2.753819
Rb2CuVCl6	−22673.642044	−5962.915605	−3309.685748	−1898.056928	−922.888707	2.735916

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3.1.3.2. Formation energy. Formation energy (E_f) is the energy required for maintaining material stability in its favorable crystalline structure under the theoretical temperature of 0 K; the negative value of E_f denotes the stable structure of the material and confirms the strong bonding between the atoms in their favorable crystal structure.⁴⁶

The E_f energy of the studied X₂CuVCl₆ (X = K and Rb) compounds was calculated according to the following relationship:

E_f(X₂CuVCl₆) = E₀ − (2E^X + E^Cu + E^V + 6E^Cl) (2)

where E^X, E^Cu, E^V and E^Cl are the corresponding atomic energies in their crystal structure, while E₀ is the total energy of the material; the obtained values of E_F energy are tabulated in Table 3. The negative E_F energy values obtained for all compounds (Table 3) strongly confirm that these materials are stable in their perovskite crystal structure.

Table 3 Computed formation energy E_f (in Ry) of the equilibrium X₆CuVCl₆ (X = K and Rb) double perovskite materials and atomic energies of the X, Cu, V and Cl atoms in their stable ferromagnetic ground phase. The results are compared with those of the K₂ScCuCl₆ and Rb₂ScCuCl₆ double perovskite compounds

Material	E0	E^X	E^Cu	E^V	E^Cl	Ef
K2CuVCl6	−13156.211232	−1204.200105	−3309.691002	−1898.059868	−922.890102	−2.719540
Rb2CuVCl6	−22673.642044	−5962.920203	−3309.691002	−1898.059868	−922.890102	−2.710156
K2ScCuCl6 (ref. 44)	−12786.28	—	—	—	—	−2.75
Rb2ScCuCl6 (ref. 44)	−22303.72	—	—	—	—	−2.27

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3.1.4. Dynamic stability. The computed phonon dispersion curves of the studied X₂CuVCl₆ (X = K and Rb) materials are depicted in Fig. 3 along the R–Γ–X–Z–M–Γ path in the three-dimensional (3D) Brillouin zone. It can be seen that the imaginary frequency modes (frequencies with a negative sign) are absent in the phonon dispersion curves of both compounds, which confirms their dynamic stability.

Fig. 3 Phonon dispersion of (a) K₂CuVCl₆ and (b) Rb₂CuVCl₆ double perovskite materials along the R–Γ–X–Z–M–Γ path.

3.2. Elastic properties

The elasticity tensor of a cubic system is defined by three independent elastic constants, namely C₁₁, C₁₂ and C₄₄. The elastic constants (C₁₁, C₁₂ and C₄₄) of the X₂CuVCl₆ (X = K and Rb) materials were computed using the IRelast program⁴⁷ implemented in the WIEN2k package,²³ and the values obtained by the GGA approximation are shown in Table 4. The Born's criteria^48,49 for assessing the mechanical stability of materials that crystallize in the cubic system are as follows:^48,49

Table 4 Elastic constants C_ij (in GPa), bulk modulus B₀ (in GPa), shear modulus G and Pugh's ratio B₀/G of the equilibrium X₆CuVCl₆ (X = K and Rb) double perovskite materials in their stable ferromagnetic ground state calculated using the GGA approximation. The results are compared with those of the K₂ScCuCl₆ and Rb₂ScCuCl₆ double perovskite compounds

Material	C11	C12	C44	B0	G	B0/G
K2CuVCl6	69.512	23.012	7.561	38.512	12.096	3.184
Rb2CuVCl6	91.473	11.110	11.242	37.898	19.304	1.963
K2ScCuCl6 (ref. 44)	68.63	16.76	6.83	34.15	11.75	2.91
Rb2ScCuCl6 (ref. 44)	89.74	28.43	10.27	48.87	16.01	3.05

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Therefore, the GGA results confirm that the studied X₂CuVCl₆ (X = K and Rb) compounds are mechanically stable.

The shear modulus (G) is defined as a parameter for evaluating the resistance of a material to plastic strain; it is calculated as the average of the Voigt⁵⁰ and Reuss⁵¹ approaches, and it is reformulated according to the Hill's approach:⁵²

where

Bulk modulus (B₀) is the factor that determines the resistance to fracture.⁵³ Moreover, Pugh's ratio (B₀/G) is another parameter that identifies the brittleness and ductility of materials.^54,55 The material becomes brittle when the B₀/G parameter is less than 1.75; when the value is higher (B₀/G > 1.75), the material will have a ductile nature. According to the Pugh ratios reported in Table 4, the present materials are ductile materials. Therefore, the two studied materials are held in their form by metallic bonding.

3.3. Electronic properties

In this work, the electronic band structures and density of states of the double perovskite materials K₂CuVCl₆ and Rb₂CuVCl₆ were predicted using both GGA and TB-mBJ–GGA functionals, aiming to explore their electronic properties and assess their potential for practical applications.

3.3.1. Spin-polarized electronic band structures. The spin-polarized electronic band structures of the equilibrium K₂CuVCl₆ and Rb₂CuVCl₆ compounds were evaluated along the high-symmetry directions and are presented in Fig. 4 and 5, respectively. The differences observed between the GGA and TB-mBJ–GGA results highlight a pronounced exchange splitting between the spin-up and spin-down channels. Specifically, the spin-up band structures of both compounds exhibit metallic behavior, as evidenced by the overlap between the valence and conduction bands. In contrast, the spin-down band structures display semiconducting characteristics, with the Fermi level positioned within a distinct energy gap. Therefore, the studied double perovskite materials strongly exhibit a complete half-metallic nature in the cubic structure. Moreover, the half-metallic gap (E_HM) of each equilibrium material was evaluated in order to determine the Curie temperature. First, the E_HM gap energy is defined as the minimum between the lowest energies of the minority-spin and majority-spin conduction bands with respect to the Fermi level, and the absolute values of the highest energies of minority-spin and majority-spin valence bands.^56,57 The computed E_HM gap energies of both K₂CuVCl₆ and Rb₂CuVCl₆ materials are listed in Table 5. The E_HM energies obtained using the TB-mBJ–GGA approximation are increasingly higher. Therefore, the TB-mBJ–GGA results confirm the high Curie temperatures of the studied double perovskite materials, making them suitable for application in spintronic and optoelectronic devices.

Fig. 4 Spin-polarized electronic structure of the equilibrium double perovskite K₂CuVCl₆ calculated using (a) PBE–GGA and (b) TB-mBJ–GGA functionals.

Fig. 5 Spin-polarized electronic structure of the equilibrium double perovskite Rb₂CuVCl₆ calculated using (a) PBE–GGA and (b) TB-mBJ–GGA functionals.

Table 5 Computed spin-down band gap energy E_g (in eV), half-metallic gap energy E_HM (in eV), and 3d-V exchange-splitting Δ(d) (in eV) of the equilibrium X₆CuVCl₆ (X = K and Rb) double perovskite materials; the results obtained using the TB-mBJ–GGA scheme are shown in parentheses

Material	Eg	EHM	Δ(d)
K2CuVCl6	1.7143 (3.8368)	0.0997 (1.1188)	2.0228 (3.4822)
Rb2CuVCl6	1.6055 (3.7552)	0.0333 (1.0880)	1.9438 (3.3804)
K2ScCuCl6 (ref. 44)	— (1.55)	—	—
Rb2ScCuCl6 (ref. 44)	— (1.43)	—	—

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3.3.2. Electronic density of states. It is essential to investigate the electron density of states (DOS) projections for each studied material in order to know the electronic details of their electronic structures. The partial density of state (PDOS) calculations were performed by employing both GGA and TB-mBJ–GGA approximations; the PDOS curves of the equilibrium K₂CuVCl₆ and Rb₂CuVCl₆ compounds are shown in Fig. 6 and 7, respectively.

Fig. 6 Spin-polarized partial density of states (PDOS) of the equilibrium double perovskite K₂CuVCl₆ calculated using (a) PBE–GGA and (b) TB-mBJ–GGA approximations.

Fig. 7 Spin-polarized partial density of states (PDOS) of the equilibrium double perovskite Rb₂CuVCl₆ calculated using (a) PBE–GGA and (b) TB-mBJ–GGA approximations.

For the K₂CuVCl₆ material (Fig. 6), the PDOS curves determined by the TB-mBJ–GGA scheme reveal the following: the 4s-Cu electrons in the spin-up and spin-down states appear in the energy spectra from −6 to −5 eV and from 4.5 to 8 eV, respectively; energy bands of the 4p-Cu spin-up and spin-down electrons are spotted in the energy range from −4.50 to −4 eV; the spin-up 3d-V electrons are located in the energy zones between −4.50 and −2 eV, around the Fermi level (E_F), and between 2.75 and 3.80 eV, while the spin-down 3d-V electrons are spotted in the energy regions between −4.50 and −2 eV and between 2.80 and 3.75 eV; moreover, the energy bands of the 3s-Cl spin-down electrons are located in the energy spectrum between −3.40 and −3 eV. Therefore, the PDOS of the K₂CuVCl₆ compound is mainly contributed by the 3d electrons of V atoms, where they are at the origin of the half-metallic aspect of the material. In addition, strong hybridization is also observed between the 3d-V states and the 4p-Cu states in the energy spectrum between −4.50 and −4 eV.

For the Rb₂CuVCl₆ material, Fig. 7 depicts that the bands in the energy ranges from −5.90 to −5 eV and 4.50 to 7 eV belong to the spin-up and spin-down 4s-Cu states, respectively; the energy bands between −4.50 and −3.75 eV and between 7 and 11 eV arise from the 4p-Cu spin-up and spin-down components; the spin-up and spin-down energy bands observed at around −10 eV and from 7 to 11 eV belong to the 3p states of the Cl atoms; on the other hand, the 3d-V states occupy the spin-up energy bands located in the energy regions from −4.25 to −2 eV, around the Fermi level (E_F), and from 2.30 to 3.50 eV; whereas, the spin-down 3d-V states lead to bands in the energy regions from −4.25 to −2 eV and 2.40 to 3.55 eV. We can conclude that the 3d-V states are responsible for the half-metallicity of the equilibrium Rb₂CuVCl₆ double perovskite as they produce an energy gap in the spin-down direction and metallic behavior in the spin-up direction around the Fermi level. Furthermore, the hybridization of the 3d-V states and the 4p-Cu states is observed in the energy range of −4.50 to −3.75 eV.

In the X₂CuVCl₆ (X = K and Rb) halide double perovskites, the 3d-V states influence the electronic structure by introducing an exchange and splitting process (Δ(d)), where Δ(d) energy is defined as the energy difference between the highest d-orbital peaks of the corresponding spin-down and spin-up states.⁵⁸ It is expressed as follows:

Δ(d) = E_d(↓) − E_d(↑) (6)

The computed Δ(d) energy values of the studied K₂CuVCl₆ and Rb₂CuVCl₆ compounds are reported in Table 5. The TB-mBJ–GGA results are considered relatively improved owing to the high Δ(d) energy observed for the studied compounds (Table 5).

3.4. Magnetic properties

3.4.1. Computed magnetic moments. The total, interstitial and atomic moments of the equilibrium X₂CuVCl₆ (X = K and Rb) halide double perovskites were calculated by applying the FP-L/APW + lo method, and the results obtained using GGA and TB-mBJ–GGA approximations are reported in Table 6. Notably, the calculated total magnetic moments (M_Tot) of both materials are mainly contributed by the atomic magnetic moment of V atoms, whereas feeble contributions arise from the interstitial zone. The M_Tot value was found to be an integer value of 2 μ_B for both K₂CuVCl₆ and Rb₂CuVCl₆, confirming the half-metallic property of these materials. The atomic magnetic moment of the V atom is decreased compared to its free space charge of 3 μ_B, and small magnetic moments are brought into the nonmagnetic X, Cu, and Cl sites; this phenomenon is mostly due to strong p–d hybridization between the 3d-V and 4p-Cu states during the exchange and splitting process. Notably, an opposition of signs between the magnetic moments of V and Cu atoms can be seen; this means that the valence electrons of V and Cu atoms interact in an antiferromagnetic manner. Moreover, the results obtained using the framework of TB-mBJ–GGA approximation are increasingly improved.

Table 6 Magnetic moments M (in μ_B) and Curie temperature T_C (in K) of the equilibrium X₆CuVCl₆ (X = K and Rb) halide double perovskites computed using the PBE–GGA and TB-mBJ–GGA functionals; the TB-mBJ–GGA-estimated values of magnetic moments are shown in parentheses

Material	Magnetic moment (μB)						Curie temperature (K)
	MTot	M (interstitial)	M (X)	M (Cu)	M (V)	M (Cl)	TC
K2CuVCl6	2.1082 (2.0000)	0.2402 (0.2454)	0.0006 (0.0002)	−0.0133 (−0.4792)	1.9546 (2.3109)	−0.0124 (−0.0129)	814.0870
Rb2CuVCl6	1.9748 (2.0004)	0.2353 (0.2321)	0.0007 (0.0002)	−0.1337 (−0.4792)	1.9543 (2.3223)	−0.0137 (−0.0125)	898.1217

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3.4.2. Computed curie temperature. The Curie temperature (T_C) is defined in magnetic materials as the point of separation between paramagnetic and ferromagnetic configurations. We evaluated the T_C temperature according to the following equation based on the Heisenberg model:⁵⁹where J_ij is the exchange interaction parameter estimated according to the following formula:

The calculated T_C temperatures of the equilibrium X₂CuVCl₆ (X = K and Rb) halide double perovskites using the TB-mBJ–GGA scheme are listed in Table 6. Notably, the T_C values of both compounds are high.

4. Conclusions

In summary, the FP-LAPW + lo method was applied in this work to evaluate the structural, elastic, electronic and magnetic properties of X₂CuVCl₆ (X = K and Rb) halide double perovskite compounds at equilibrium. The main conclusions of this prediction model are summarized below:

(i) Analyses of the structural properties of both compounds show that the ferromagnetic state is the stable ground phase.

(ii) The optimized structural parameters (a₀, B₀, B′ and E₀) of the compounds are reported in this study.

(iii) The calculated elastic constants of both materials confirm their mechanical stability and ductility.

(iv) The calculated electronic properties of the compounds using both GGA and TB-mBJ–GGA parameterizations prove that they are entirely half-metals with a large half-metallic gap.

(v) The half-metallic gaps corresponding to the two halide double perovskites are calculated using GGA and TB-mBJ–GGA functionals.

(vi) The 3d-V exchange-splitting energy (Δ(d)) calculated using this approach was found to be large for both studied compounds.

(vii) The estimated magnetic properties of the two compounds reveal that:

(1) The atomic magnetic moment of the V atom contributes mainly to the total magnetic moment.

(2) The opposite signs of the atomic magnetic moments of V and Cu atoms reveal that the valence electrons of V and Cu atoms interact in an antiparallel manner during the exchange and splitting process.

(3) The hybridization between 3d-V and 4p-Cu electrons reduces the magnetic moment of the V atom from its free space charge and produces weak magnetic moments at the non-magnetic X, Cu and Cl sites.

Data availability

Data supporting the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author (E-mail: fatmimessaud@yahoo.fr) upon reasonable request.

Conflicts of interest

The authors declare that they have no conflict of interest.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number NBU-FFR-2025-310-11.

References

1. W. E. Pickett and J. S. Moodera, Phys. Today, 2001, 54, 39–44.
2. R. A. de Groot, F. M. Mueller, P. G. van Engen and K. H. J. Buschow, Phys. Rev. Lett., 1983, 50, 2024–2027.
3. R. J. Soulen Jr, J. M. Byers, M. S. Osofsky, B. Nadgorny, T. Ambrose and A. Barry, Science, 1998, 282, 85–88.
4. K. L. Kobayashi, T. Kimura, H. Sawada, K. Terakura and Y. Tokura, Nature, 1998, 395, 677–680.
5. I. Galanakis, Phys. Rev. B:Condens. Matter Mater. Phys., 2005, 71, 012413.
6. S. Wurmehl, G. H. Fecher, H. C. Kandpal, V. Ksenofontov, C. Felser and H.-J. Lin, Appl. Phys. Lett., 2006, 88, 032503.
7. M. Nakao, Phys. Rev. B:Condens. Matter Mater. Phys., 2004, 69, 214429.
8. A. Nfissi, R. El Fdil, A. Samih, D. Kabouchi, Z. Fadil, C. J. Raorane, A. A. Ifseisi, M. E. Assal and E. Salmani, J. Magn. Magn. Mater., 2024, 610, 172563.
9. A. Nfissi, D. Kabouchi, R. El Fdil, A. Samih, Z. Fadil, C. J. Raorane, K. H. Mahmoud, A. S. A. Alsubaie and E. Salmani, Solid State Commun., 2024, 389, 115596.
10. G. Li, Z. Li and Z. Liu, J. Magn. Magn. Mater., 2025, 614, 172742.
11. B. Gurunani and D. C. Gupta, Mater. Sci. Eng., B, 2025, 311, 117783.
12. F. Firdous, Q. Ain, S. A. M. Issa, H. M. H. Zakaly and J. Munir, RSC Adv., 2024, 14, 34679–34689.
13. I. Muhammad, S. Ahmed, N. Ullah, M. Mushtaq, M. Liaqat, X. Tian and J.-M. Zhang, Results Phys., 2024, 57, 107419.
14. F. Faid, H. Mebarki, K. Mokadem, F. M. Abdalilah, A. Benmakhlouf, M. Khatiri and T. Helaimia, J. Magn. Magn. Mater., 2024, 605, 172345.
15. J. K. Sharma, A. Dhamija, A. Pal and J. Kumar, Comput. Mater. Sci., 2024, 232, 112679.
16. M. Jamil, M. Manzoor, A. Kumar, A. Agrawal, R. Almeer, Y. A. Kumar and R. Sharma, Solid State Commun., 2024, 394, 115702.
17. H. Kerrai, A. Zaim and M. Kerouad, Vacuum, 2024, 226, 113341.
18. M. Y. Sofi, M. S. Khan and M. A. Khan, Mater. Sci. Semicond. Process., 2025, 186, 109023.
19. M. A. Yasir, G. M. Mustafa, M. A. Ameer, N. A. Noor, S. Mumtaz and I. M. Moussa, Mater. Sci. Eng., B, 2025, 311, 117830.
20. M. Husain, H. Albalawi, M. Al Huwayz, N. Al Saqri, R. Khan and N. Rahman, Phys. Scr., 2023, 98, 105935.
21. N. Rahman, A. Rauf, M. Husain, N. Sfina, V. Tirth, M. Sohail, R. Khan, A. Azzouz-Rached, G. Murtaza, A. A. Khan and S. A. Khattak, RSC Adv., 2023, 13, 15457–15466.
22. M. Husain, N. Rahman, H. Albalawi, S. Ezzine, M. Amami, T. Zaman, A. U. Rehman, M. Sohail, R. Khan, A. A. Khan, Tahir and A. Khan, RSC Adv., 2022, 12, 32338–32344.
23. S. A. Shah, M. Husain, N. Rahman, M. Sohail, R. Khan, A. Alataway, A. Z. Dewidar, H. O. Elansary, L. Abu El Maati, K. Yessoufou, A. Ullah and A. Khan, Materials, 2022, 15, 5684.
24. S. A. Shah, M. Husain, N. Rahman, M. Sohail, R. Khan, A. A. Khan, A. Ullah, S. A. M. Abdelmohsen, A. M. M. Abdelbacki, A. M. El-Sabrout, H. O. Elansary and A. Khan, RSC Adv., 2022, 12, 8172–8177.
25. N. Rahman, M. Husain, V. Tirth, A. Algahtani, H. Alqahtani, T. Al-Mughanam, A. H. Alghtani, R. Khan, M. Sohail, A. A. Khan, A. Azzouz-Rached and A. Khan, RSC Adv., 2023, 13, 18934–18945.
26. A. A. Pasha, H. Khan, M. Sohail, N. Rahman, R. Khan, O. H. Alsalmi, D. Abduvalieva, K. M. Abualnaja, A. El Jery and M. Adrdery, Mater. Adv., 2023, 4, 6645–6654.
27. W. Ullah, M. Husain, N. Rahman, N. Sfina, M. Elhadi, R. Khan, M. Sohail, A. Azzouz-Rached, M. Uzair, A. A. Khan and A. Khan, Silicon, 2024, 16, 3021–3032.
28. U. Hameed, H. Ullah, S. Z. Abbas, K. Safeen, K. M. Alotaibi, A. Safeen, S. Yasin, G. Murtaza, F. Khalil, S. Ali, G. Asghar and R. Khan, J. Phys. Chem. Solids, 2025, 199, 112519.
29. M. S. Khan, M. Sohail, N. Rahman, R. Khan, H. E. Ali, J. Jamil, B. S. Abdullaeva, K. M. Abualnaja, G. Alosaimi, Y. Khairy and H. Algarni, Chem. Pap., 2025, 79, 107.
30. N. S. Alsaiari, I. Ahmed, S. Hanf, M. M. Rekha, M. Kundlas, M. Ouladsmane, K. Muhammad and J. Rehman, J. Inorg. Organomet. Polym., 2025 DOI:10.1007/s10904-025-03746-z.
31. N. Rahman, K. M. Abualnaja, S. Belhachi, N. Sfina, M. Husain, B. M. Al-Khamiseh, A. Azzouz-Rached, H. A. Althobaiti, S. Ullah, R. Khan and M. Asghar, J. Inorg. Organomet. Polym., 2025, 35, 1439–1452.
32. N. Sfina, N. Rahman, S. Belhachi, M. Husain, B. M. Al-Khamiseh, K. M. Abualnaja, G. Alosaimi, A. Azzouz-Rached, S. Ullah, A. U. Rashid and R. Khan, J. Inorg. Organomet. Polym., 2024, 34, 6102–6113.
33. M. Husain, N. Rahman, A. Azzouz-Rached, N. Sfina, K. M. Abualnaja, G. Alosaimi, M. Asad, R.-D. Hamza, V. Tirth, N. U. Khan, Q. Humayun, R. Khan, R. Ahmad, A. Samreen, A. A. Khan and J. Lu, Inorg. Chem. Commun., 2024, 164, 112424.
34. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864.
35. K. M. Wong, S. M. Alay-e-Abbas, A. Shaukat, Y. Fang and Y. Lei, J. Appl. Phys., 2013, 113, 014304.
36. K. M. Wong, S. M. Alay-e-Abbas, Y. Fang, A. Shaukat and Y. Lei, J. Appl. Phys., 2013, 114, 034901.
37. P. Blaha, K. Schwarz, P. Sorantin and S. K. Trickey, Comput. Phys. Commun., 1990, 59, 339.
38. J. P. Perdew, S. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
39. F. Tran and P. Blaha, Phys. Rev. Lett., 2009, 102, 226401.
40. D. Koller, F. Tran and P. Blaha, Phys. Rev. B:Condens. Matter Mater. Phys., 2011, 83, 195134.
41. G. M. Mustafa, R. Waqar, S. Saba, N. A. Noor, Z. Farooq, M. Imran, R. B. Behram and Y. M. Alanazi, Phys. Scr., 2023, 98, 065703.
42. F. D. Murnaghan, Proc. Natl. Acad. Sci. U. S. A., 1944, 30, 244–247.
43. S. L. Shang, Y. Wang, D. Kim and Z.-K. Liu, Comput. Mater. Sci., 2010, 47, 1040–1048.
44. A. Ayyaz, G. Murtaza, A. Usman, N. Sfina, A. S. Alshomrany, S. Younus, S. Saleem and Urwa-tul-Aysha, J. Inorg. Organomet. Polym., 2024, 34, 3560–3575.
45. J. J. Kim, H. W. Seong and H. J. Ryu, J. Nucl. Mater., 2022, 566, 153780.
46. A. Yakoubi, O. Baraka and B. Bouhafs, Results Phys., 2012, 2, 58–65.
47. J. B. Kramers-Kronig, Am. J. Phys., 2011, 79, 1053–1059.
48. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1956.
49. J. Wang and S. Yip, Phys. Rev. Lett., 1993, 71, 4182.
50. W. Voigt, Lehrbuch der Kristallphysik, Taubner, Leipzig, 1928.
51. A. Reuss, Z. Angew. Math. Mech., 1929, 9, 55.
52. R. Hill, Proc. Phys. Soc., London, 1953, 65, 909.
53. G. Vaitheeswaran, V. Kanchana, R. S. Kumar, A. L. Cornelius, M. F. Nicol, A. Svane, A. Delin and B. Johansson, Phys. Rev. B:Condens. Matter Mater. Phys., 2007, 76, 014107.
54. S. F. Pugh, Philos. Mag., 1954, 45, 823.
55. G. Simmon and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: a Handbook, M.I.T Press, Cambridge, 1971.
56. K. L. Yao, G. Y. Gao, Z. L. Liu and L. Zhu, Solid State Commun., 2005, 133, 301–304.
57. G. Y. Gao, K. L. Yao, E. Sasioglu, L. M. Sandratskii, Z. L. Liu and J. L. Jiang, Phys. Rev. B:Condens. Matter Mater. Phys., 2007, 75, 174442.
58. A. Zunger, Solid State Phys.: Condens. Matter, 1986, 39, 275.
59. J. Kubler, G. H. Fecher and C. Felser, Phys. Rev. B:Condens. Matter Mater. Phys., 2007, 76, 024414.

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