Stable antiferromagnetism and semiconducting-to-metal transition in ALaCuOsO₆ (A = Ba and Sr): strain modulations

Abstract

Double perovskite oxides (DPO) with antiferromagnetic ground state have received much consideration as they exhibit small stray-field and ultra-fast spin dynamics, which is extremely convenient for high-density and high-frequency data storage devices. It is a well-established fact that strain can easily tune the physical properties of the materials; therefore, the electronic and magnetic properties of recently synthesized ordered ALaCuOsO₆ (A = Ba and Sr) DPO under biaxial ([110]) strain are investigated using ab initio calculations. Our results revealed that the unstrained systems exhibit semiconducting states having energy band gaps (E_g) of 0.28 and 0.39 eV for A = Ba and Sr, respectively. Along with this, both structures exhibit AFM ground state due to a strong AFM coupling between partially filled high-energy Cu⁺ e¹_g↑ and low-energy empty Os⁺⁵ t⁰_2g↓ orbitals. The calculated partial spin moments of Cu and Os ions are 0.65/0.66 and 1.58/1.60μ_B in a Ba-/Sr-doped system having electronic configurations of 3d⁹ (t³_2g↑t³_2g↓e²_g↑e¹_g↓) with S = 0.5 and 5d³ (t³_2g↑) with S = 1.5, respectively. The robustness of AFM spin ordering is affirmed under the strain effects. The most striking feature of the present study is that Ba- and Sr-doped systems demonstrate an electronic transition from semiconductor to metal at critical tensile strains of +4% and +5% along with improved magnetism as well as Néel temperature, respectively. However, the magnetic ground state remains robust against applied strains in both cases. Hence, the present study shows that strain engineering could be a practical tool to modulate the electronic and magnetic properties of DPO to further enhance their technological applications in spintronics.

---

1 Introduction

Double perovskite oxides (DPO) with the general formula A₂BB′O₆ based on B = 3d and B′ = 4/5d heavy transition metals often exhibit exotic electronic and magnetic characteristics due to the strong relationship among electronic correlation, spin–orbit coupling (SOC), and crystal field effects.^1–6 As an octahedral distortion of the magnetic ions results in an exceptionally prosperous physics. Such as osmium ion having 5d⁵ electronic configuration has got much attention due to SOC-associated Mott-insulating (MI) antiferromagnetic (AFM) ground states that offer promising applications for high-density storage data devices with high frequency and ultrafast spin motions.^7–9 In this respect, ordered DPO has attracted much attention as it is a well-known fact that magnetism in DPO substantially depends on B-site ordering. In most cases, AFM magnetic ordering is dominant due to the magnetic B-site cation, which leads the exchange coupling to the next nearest neighbor (NNN) ions.^10–13 On the other hand, in the presence of both magnetic B and B′ cations, the magnetic interactions are governed by the nearest neighbor and NNN coupling, which leads the systems to a ferromagnetic (FM) or ferrimagnetic (FiM) spin state.

Previously, it has been affirmed that DPO particularly having Cu⁺² d⁹ and d⁰ cations at B and B′ sites, respectively, results in interesting low-dimensional magnetism^14–16 due to John–Teller distortion. This is associated with elongation and shortening of Cu–O bond lengths along the c-axis and in the ab-plane, respectively. Moreover, it is also found that Cu in the a+2 state contains partially filled e_g orbitals, which substantially improves the magnetic anisotropy of the system along the ab-plane. Very recently, Thakur et al.¹⁷ synthesised Cu⁺² based DPO ALaCuOsO₆ (A = Ba and Sr) which shows complete A- and B-site order having a tetragonal structure with space group no. 82 (I4̄). The authors observed that the A = Ba/Sr system displays AFM spin ordering as a magnetic ground state at low Néel temperature (T_N) of 25 and 50 K, respectively. Furthermore, both Ba- and Sr-doped systems demonstrate semiconducting (SC) characteristics with an estimated energy band gap (E_g) of 0.26 and 0.34 eV, respectively.

To further improve the functionalities of DPOs, it is found that the strain strategy is one of the most common techniques. It is experimentally observed that Sr₂FeMoO₆ films deposited on the SrTiO₃ substrate exhibit metallic behavior for thicknesses below 44 nm and become half-metal (HM) above this thickness.¹⁸ Similarly, Bark et al.¹⁹ found that when a biaxial compressive strain along the [110]-direction is applied on the LaAlO₃/SrTiO₃ structure, an electronic transition from insulator to metal occurs. In a similar fashion, Sr₂CoNbO₆ DPO demonstrates an increasing and decreasing trend in the electrical resistivity as well as in E_g values under the influence of tensile and compressive strains, respectively.²⁰ In addition, an induced strain of 0.05% leads the Sr₂CrReO₆ epitaxial HM film to the SC state with an E_g of 0.21 eV at a very low temperature of 2 K when grown on the SrTiO₃ substrate.²¹ Similar to experimental investigations, Yang et al., all-electron calculations show that application of a feasible strain value to an ordinary band insulator having heavy elements leads the system to a topological insulator.²² Along with this, a FM SC to metal and HM transition is confirmed in La₂FeMnO₆ for −8% compressive and +1% to +5% tensile strains, respectively.²³ Similarly, a FiM SC to HM transition is observed in Lu₂NiIrO₆ DPO for −8% and −6% biaxial and hydrostatic strains, respectively.²⁴ Equivalently, a FiM to FM magnetic transition occurs in the Sr₂CrReO₆ motif at +2% biaxial ([110]) and hydrostatic ([111]) strains.²⁵ Very recently, Rubab et al. found an insulator-to-metal transition in a monoclinic LaSr_1−xCa_xNiReO₆ DPO for x = 0.0 and 0.5 at −3% biaxial compressive strain. It is also confirmed that the combined effect of strong correlation and SOC is necessary to get the correct ground states (both electronic and magnetic) of the systems.²⁶

Motivated by the above discussions, we investigate the impact of biaxial ([110]) strain on the electronic and magnetic properties of recently synthesized ALaCuOsO₆ (A = Ba and Sr) DPO through density functional theory calculations. Our results indicate that both unstrained systems exhibit AFM SC state, which is in good agreement with the experiment. Interestingly, it is established that Ba- and Sr-doped systems display an electronic transition from SC to metal at a critical tensile strain of +4% and +5%, respectively. The robustness of the magnetic ground state is also affirmed under the strain effects. Furthermore, the computed moments of Cu and Os ions confirmed that they have S = 0.5 and S = 1.5 with electronic configurations of 3d⁹ (t³_2g↑t³_2g↓e²_g↑e¹_g↓) and 5d³ (t³_2g↑), respectively. Moreover, the structural stability of the unstrained and strained systems is taken into account by computing the enthalpies of formation (ΔH_f). Finally, the variation in the T_N magnitude is studied as a function of strain.

2 Computational and structural details

The present calculations were performed using the full-potential linearized augmented plane wave method (FP-LAPW) as implemented in the WIEN2K code.²⁷ The generalized gradient approximation (GGA) is used on the exchange–correlation functional within the Wu-Cohen framework²⁸ along with the inclusion of Hubbard parameter U of 7.5, 4.0, and 2.6 eV on the La 4f, Cu 3d, and Os 5d orbitals,²⁹ respectively. Moreover, SOC effects are also attributed to the scalar relativistic form. A maximum value of l_max = 12 is chosen for the wave function expansion inside the atomic spheres, while the plane-wave cutoff is set to R_mt × K_max = 7 with G_max = 24. A 9 × 9 × 6 k-mesh having 84-points within the irreducible wedge of the Brillouin zone is found to be well converged. Along with this, the atomic positions of all doped systems are fully relaxed by minimizing the atomic forces up to 5 mRy au⁻¹. Self-consistency is taken into account for a total charge/energy convergence of less than 10⁻⁵ C/10⁻⁵ Ry.

Pristine ALaCuOsO₆ (A = Ba and Sr) DPO crystallizes in a tetragonal cell of space group no. 82 (I4̄).¹⁷ The experimentally and theoretically calculated lattice parameters as well as atomic positions of both structures are listed in Table 1, which are a good match. The schematic representation of the ALaCuOsO₆ DPO in three different magnetic states such as FM, FiM, and AFM spin ordering is shown in Fig. 1(a)–(c), respectively.

Table 1 Experimentally (exp.) and theoretically (theo.) calculated lattice parameters (a and c) in Å, atomic positions, Cu–O/Os–O band lengths in Å, and enthalpies of formation (ΔH_f) in eV of ALaCuOsO₆ (A = Ba and Sr) double perovskite oxides in a tetragonal cell of space group no. 82 (I4̄)

	BaLaCuOsO6 (Ba-dop.)		SrLaCuOsO6 (Sr-dop.)
	Exp.^17	Theo.	Exp.^17	Theo.
A	5.5230	5.5209	5.4728	5.4697
c	8.3773	8.3768	8.2905	8.2893
A(x, y, z)	(0, 0.5, 0.25)	(0, 0.5, 0.25)	(0, 0.5, 0.25)	(0, 0.5, 0.25)
La(x, y, z)	(0, 0.5, 0.75)	(0, 0.5, 0.75)	(0, 0.5, 0.75)	(0, 0.5, 0.75)
Cu(x, y, z)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)
Os(x, y, z)	(0, 0, 0.5)	(0, 0, 0.5)	(0, 0, 0.5)	(0, 0, 0.5)
O1(x, y, z)	(0, 0, 0.265)	(0, 0, 0.263)	(0, 0, 0.265)	(0, 0, 0.262)
O1(x, y, z)	(0.281, 0.206, 0.029)	(0.276, 0.203, 0.027)	(0.281, 0.206, 0.029)	(0.279, 0.210, 0.028)
Cu–O1	2.217	2.220	2.156	2.149
Cu–O2	1.938	1.951	1.898	1.901
Os–O1	1.967	1.960	1.989	1.955
Os–O2	2.036	1.977	2.013	1.977
ΔHf	—	−21.6	—	−20.4

---

Fig. 1 The crystal structures of ALaCuOsO₆ (A = Ba and Sr) double perovskite oxide in (a) ferromagnetic (FM), (b) ferrimagnetic (FiM), and (c) antiferromagnetic (AFM) spin ordering. The oxygen ions are surrounded by CuO₆ (blue) and OsO₆ (camel) octahedra, respectively. The a, b, and c-axes are the physical dimensions along the x, y, and z-directions, respectively.

3 Results and discussion

3.1 Unstrained systems

First, the structural stability of the systems is estimated by computing the ΔH_f of ALaCuOsO₆ (A = Ba and Sr) DPO aswhere is the ground state total energy (E_tot.) of the tetragonal phase of ALaCuOsO₆. Similarly, E^{A-bcc/trigonal}_t, E^La-hcp_t, E^Cu-hcp_tot., E^Os-hcp_tot., and are the ground state E_tot. of the fully relaxed crystal structures of Ba/Sr, La, Cu, Os atoms, and oxygen molecule having space group no. 229 (Im3̄m)/166 (R3̄m), 194 (P6₃/mmc), 225 (Fm3̄m), 194 (P6₃/mmc), and 12 (C2/m), respectively. The calculated ΔH_f for BaLaCuOsO₆ and SrLaCuOsO₆ systems are −21.6 and −20.4 eV (see Table 1), respectively. The “−” values of ΔH_f support the exothermic reaction and confirm the structural stability of both motifs.

Next, we analyze the magnetic ground state of the ALaCuOsO₆ (A = Ba and Sr) DPO by supposing three types of magnetic structures such as FM (↑↑↑↑), FiM (↑↑↓↓), and AFM (↑↓↑↓) as shown in Fig. 1(a)–(c), respectively. Here, it is to be noted that small and large arrowhead lengths represent the Cu and Os spin moment amplitudes, respectively. The magnetic ground state of Ba- and Sr-doped La₂CuOsO₆ motifs is computed by comparing the E_tot. of FM, FiM, and AFM spin ordering as ΔE₁/ΔE₂ = E_AFM − E_FM/E_AFM − E_FiM. The calculated values of ΔE₁/ΔE₂ are −112/−52 meV and −163/−48 meV for BaLaCuOsO₆ and SrLaCuOsO₆ motifs, respectively. The “−” sign of ΔE indicates that AFM spin ordering is more energetically stable than FM and FiM ordering in each case, which is in good agreement with the experimental observations.¹⁷ This means that a strong AFM coupling is dominant between Cu and Os ions in both in-plane and out-of-plane directions.

Regarding the electronic properties of unstrained ALaCuOsO₆ DPO in a stable AFM spin ordering, we plotted the computed spin-polarized total density of states (TDOS) for Ba- and Sr-doped systems in Fig. 2(a) and (b), respectively. One can see that the spin-majority (N↑) and spin-minority (N↓) channels are degenerate and not crossing the Fermi level (E_F) in both systems, which confirms the AFM SC state of Ba- and Sr-doped structures having E_g of 0.28 and 0.39 eV, which are in excellent agreement with the recent experimentally observed values of ∼0.26 and 0.34 eV,¹⁷ respectively. For a more deep understanding of the contribution of the electronic states near the Fermi level (E_F), we plotted the spin-polarized spin-up (↑-blue color)/spin-down (↓-red color) Cu 3d and Os 5d states in Fig. 3(a, a′) and (b, b′) for ALaCuOsO₆ (A = Ba (left column) and Sr (right column)) DPO. From Fig. 3, one can see that N↑ of Cu↑/Os↑ 3d/5d states (see the blue color) is equivalent to that of N↓ (see the red color) in the case of Cu↓/Os↓ 3d/5d states. Similarly, N↓ of Cu↑/Os↑ 3d/5d states (see the blue color) is identical to that of N↑ of Cu↓/Os↓ 3d/5d states (see the red color). This leads the systems into a perfect AFM state. Furthermore, it is found that the conduction band edges are mainly comprised of the Cu 3d states, while Os 5d orbitals are dominant at the valence band edges in both cases (see Fig. 3). In addition, the spin-polarized band structures within the GGA+U+SOC method are also plotted in Fig. 2(a′) and (b′) for further clear insight into the SC states in Ba- and Sr-doped motifs, respectively. Hence, the calculated band structures establish the fact that both systems display a well-defined E_g, which affirms the computed TDOS in Fig. 2.

Fig. 2 GGA+U+SOC calculated spin-polarized total density of states (TDOS) along with band structures in the most stable antiferromagnetic spin ordering of (a and a′) Ba- and (b and b′) Sr-doped La₂CuOsO₆ double perovskite oxides. For clarity, the vertical and horizontal dotted lines represent the Fermi level in each density of state and band structure case.

Fig. 3 GGA+U+SOC calculated spin-polarized partial density of states of (a/a′) Cu 3d↑ (3d↓) and (b/b′) Os 5d↑ (5d↓) in Ba-/Sr-doped La₂CuOsO₆ double perovskite oxides.

Now, we discuss the magnetism in these structures by analyzing the partial spin magnetic moments (m_s) of the magnetic ions. The calculated m_s of the Cu1/Cu2 and Os1/Os2 ions are +0.65/−0.65μ_B and +1.58/−1.58μ_B in the Ba-doped system, respectively. Similarly, the computed m_s of the Cu1/Cu2 and Os1/Os2 ions are +0.66/−0.66μ_B and +1.60/−1.60μ_B in the Sr-doped system, respectively. Here, it is worth mentioning that the “−” and “+” signs of m_s between Cu1 and Cu2 (Os1 and Os2) ions along with the same magnitudes confirm the AFM ordering between them in both doped systems. Cu and Os ions are in +2 and +5 states having electronic configurations of 3d⁹ (t³_2g↑t³_2g↓ e²_g↑ e¹_g↓) with S = 0.5 and 5d³ (t³_2g↑t⁰_2g↓ e⁰_g↑ e⁰_g↓) with S = 1.5 in these motifs, respectively. Therefore, the only possible exchange coupling between the partially filled e_g of Cu⁺² and t_2g of Os⁺⁵ ions via oxygen in ALaCuOsO₆ DPO is shown in Fig. 4 as described in ref. 30. On the other hand, the higher energy Os e⁰_g orbitals are empty and do not contribute to this exchange process. Similarly, t⁰_2g orbitals of Cu are occupied and also do not take part in interatomic exchange coupling. Hence, the electron hopping only occurs between partially filled high energy Cu e¹_g↑ and low energy t⁰_2g↓ states of Os ions via oxygen 2p² orbitals as displayed in Fig. 4. Next, we plotted the three-dimensional spin magnetization density iso-surface plot having an iso-value of ±0.06 e Å⁻³ for the Ba-doped system in Fig. 5 for the direct observation of magnetism. The large densities around Os ions than that of Cu ions affirm that Os is the main contributor to magnetism along with a substantial contribution from Cu ions, which further confirms the calculated m_s of these ions. Furthermore, Cu 3d⁹ is in a +2 state having the electronic configuration of t³_2g↑t³_2g↓e²_g↑e¹_g↓; therefore, the e_g orbital characteristics are manifested in the isosurfaces of both Cu ions. Similarly, the Os 5d³ ion remains in a +5 state with t³_2g↑t⁰_2g↓e⁰_g↑e⁰_g↓ electronic configuration; thus, only t_2g orbital characteristics are evident in the isosurfaces of Os ions. Here, it is also worth noting that Cu1 and Cu2 ions have the same spin-density magnitudes but with different colors (see Fig. 5). This means that Cu spins are anti-aligned with each other, and a similar trend is also established for Os ions. Hence, the spin-magnetization density plot again verifies the AFM spin ordering in the Ba-doped motif as well.

Fig. 4 Superexchange mechanism is displayed between Cu 3d and Os 5d orbitals via oxygen 2p state by meditating the feasible Cu⁺²–O⁻²–Os⁺⁵ (e¹_g↑-2p²-t⁰_2g↓) coupling.

Fig. 5 Calculated spin-magnetization density isosurface plot having an iso-value of ±0.06 e Å⁻³ in the most stable antiferromagnetic (AFM) spin ordering of Ba-doped La₂CuOsO₆ double perovskite oxide. For clarity, La, Ba, and oxygen ions are omitted. The different colors of Cu1/Os1 (blue) and Cu2/Os2 (red) represent that these ions are anti-aligned with each other, which further confirms the AFM spin state in this system.

Finally, we determined the T_N of the Ba/Sr-doped La₂CuOsO₆ systems using the classical Heisenberg Hamiltonian:³¹

where S_i = (S_i(S_i + 1))^1/2 is the amplitude of spin for the ith ion and J_ij is the spin exchange interaction term between i and j ions. The ionic coupling is considered up to the first (J₁) and the second (J₂) nearest neighbors. Hence, E_tot. per magnetic ion in various magnetic spin configurations is estimated as

E_tot.(FM) = E₀ + (S₁(S₁ + 1))^1/2(S₂(S₂ + 1)^1/2)(2J₁ + 4J₂) (3)

E_tot.(FiM) = E₀ + (S₁(S₁ + 1))^1/2(S₂(S₂ + 1)^1/2)(−2J₁ + 4J₂) (4)

E_tot.(AFM) = E₀ + (S₁(S₁ + 1))^1/2(S₂(S₂ + 1)^1/2)(−2J₁ − 4J₂) (5)

Here E₀ describes the energy of the paramagnetic system. Using the above equations, the computed J₁ and J₂ for Ba-/Sr-doped systems are 0.47/0.93 and 0.34/0.69 meV, respectively. Moreover, S₁ = 1/2 and S₂ = 3/2 are used for the Cu⁺² and Os⁺⁵ spin states, respectively. As both J have positive values, the AFM coupling between Cu and Os via oxygen is confirmed as explained in ref. 32. Hence, the T_N of doped motifs has been examined using the mean-field approximation as³³Thus, the calculated T_N is 30 and 60 K for Ba- and Sr-doped structures, which are close to the experimentally observed values of 25 and 50 K, respectively.¹⁷

3.2 Strained systems

Here, we study the impact of biaxial (i.e., along the [110]-direction) strain on the physical properties of ALaCuOsO₆ (A = Ba and Sr) DPO. The strain effects are realized by changing the lattice constants along the ab-plane, where the experimental lattice parameters are taken as a reference point (0% strain). The strain is constrained in the range of −5% to +5% as it is a well-known fact that a small lattice mismatch between the grown material and the substrate sufficiently reduces the possibility of the defect at the contact point.³⁴ The “−” and “+” signs indicate the compressive and tensile strains, respectively. First, to confirm the dynamical stability of the ALaCuOsO₆ (A = Ba and Sr) structures under biaxial ([110]) strain, the phonon band dispersion calculations have been performed at two extreme strain values of −5% (compressive) and +5% (tensile) strains. Therefore, the computed phonon band structures along the high symmetry directions in the BaLaCuOsO₆ and SrLaCuOsO₆ systems for −5%/+5% biaxial ([110]) strain are shown in Fig. S1 of the (ESI†), respectively. One can see that the phonon branches are free from any imaginary frequencies, which affirms the dynamical stability of both structures. Next, we investigate the stability of the magnetic ground state of the ALaCuOsO₆ system against strain. To do this, we plotted the ΔE₁ = E_AFM − E_FM and ΔE₂ = E_AFM − E_FiM as a function of biaxial strains for Ba-/Sr-doped La₂CuOsO₆ structures in Fig. 6(a) and (b), respectively. The negative values of ΔE₁ and ΔE₂ in both cases affirm that the AFM structures emerge as the lowest energy state against applied strains as predicted in the case of unstrained ones. This confirms the robustness of AFM spin ordering in the Ba-/Sr-doped motif against considerable biaxial compressive and tensile strain. Now, the structural stability of the systems under biaxial strain is studied. Thus, we plotted the calculated values of ΔH_f with respect to strain in Fig. 7 for Ba- and Sr-doped structures. Our results clearly established that all the motifs maintained negative values of ΔH_f for both doped systems against each strain magnitude, which confirms their structural stability as well. However, systems show less stability when compressive strain is applied from −1% to −5% due to a lower negative amplitude of ΔH_f than that of unstrained ones. On the other hand, tensile strained structures displayed more strength as they have higher negative ΔH_f values as compared to unstrained motifs (see Fig. 7).

Fig. 6 GGA+U+SOC calculated (a) ΔE₁ = E_AFM − E_FM (energy difference between antiferromagnetic (AFM) and ferromagnetic (FM) spin ordering) and (b) ΔE₂ = E_AFM − E_FiM (energy difference between AFM and ferrimagnetic (FiM) spin ordering) as a function of biaxial strains along the ab-plane (i.e., along the [110]-direction) ranging from −5% to +5% in Ba-/Sr-doped La₂CuOsO₆ double perovskite oxide. The “−” and “+” signs indicate the compressive and tensile strains, respectively. Moreover, the “−” sign in ΔE represents that the AFM spin state is the lowest energy state in both cases.

Fig. 7 Calculated enthalpies of formation (ΔH_f) as a function of biaxial strains ranging from −5% to +5% along the [110]-direction in the most stable antiferromagnetic spin ordering of Ba- and Sr-doped La₂CuOsO₆ double perovskite oxides.

Now, we study the strain-induced variations in the electronic properties of ALaCuOsO₆ (A = Ba and Sr) DPO. Hence, first, we plotted the calculated E_g as a function of biaxial ([110]) strains ranging from −5% to +5% in the most stable AFM spin ordering of the Ba-/Sr-doped motif in Fig. 8. It is evident that E_g increases linearly for the compressively strained structures in each case. In contrast, the E_g magnitude continuously decreases for tensile strained systems and eventually becomes zero at +4/+5% for the Ba-/Sr-doped motif as displayed in Fig. 8. This means that an electronic transition from SC-to-metallic state occurs (see the arrowheads in Fig. 8) at these critical tensile strains. The increase/decrease in the E_g magnitude as a function of biaxial [110] compressive/tensile strain can be explained by considering the metal–anion bonding interactions using the Jaffe and Zunger ligand-field type model.^35,36 From Fig. 3, one can see that the valence band edges mainly come from Os 5d states with a small contribution from Cu 3d states. On the other hand, the conduction band edges are mainly comprised of the Cu 3d states in both systems because they lie at high energy levels as compared to the other cations. Thus, Cu–O/Os–O bond length coupling results in bonding and anti-bonding states in the valence and conduction band edges, respectively, as discussed in ref. 37–39. Usually, it is well established that a strong bond length bonding increases the E_g between the bonding and anti-bonding states. Hence, the Cu–O/Os–O bond lengths become shortened as the compressive strain magnitude increases. This moves the anti-bonding states in the conduction band towards higher energies, which results in the increase in E_g. On the other hand, tensile strain lengthens the Cu–O/Os–O bond lengths, which shifts the anti-bonding states toward lower energies and reduces the E_g. So this is surely the valid reason for the variation of E_g with respect to metal–anion bonding.

Fig. 8 GGA+U+SOC calculated energy band gap (E_g) as a function of biaxial strain ranging from −5% to +5% along the [110]-direction in the most stable antiferromagnetic spin ordering of Ba- and Sr-doped La₂CuOsO₆ double perovskite oxides. For clarity, the semiconducting (SC)-to-metal transitions (MT) are highlighted by the corresponding color arrowheads in each case.

For quantitative analysis, we plotted the GGA+U+SOC calculated spin-polarized TDOS of ALaCuOsO₆ (A = Ba and Sr) DPO in their most stable AFM spin ordering as a function of biaxial ([110]) compressive strain ranging from −1% to −5% in Fig. SS of the ESI.† One can see that conduction band edges of TDOS solely shift toward higher energies as the compressive strain increases from −1% to −5% (see Fig. S2(a–e) and 2S(a′–e′) (ESI†) for Ba- (left column) and Sr- (right column) doped motifs, respectively). This means that E_g increases in both cases and confirms the plotted data in Fig. 8. Similarly, the GGA+U+SOC computed spin-polarized TDOS of Ba- and Sr-doped systems in their most stable AFM spin ordering as a function of biaxial ([110]) tensile strains ranging from +3% to +5% is plotted in Fig. 9 (left column) and (right column), respectively. Our results revealed that as the tensile strain increases, the conduction band edges of TDOS substantially move towards lower energies (i.e., towards E_F) in both cases. A small E_g exists for + 3% strain in the Ba-doped system as displayed in Fig. 9(a). Surprisingly, the E_g becomes zero at a critical strain of +4% (see Fig. 9(b), left column) in the Ba-doped system and a similar metallic behavior is also observed for +5% tensile strain (see Fig. 9(c)). In the case of the Sr-doped motif, both +3% and +4% tensile strained structures exhibit SC behavior (i.e., a small E_g exists between the conduction and valence band) as shown in Fig. 9(a′) and (b′) in the right column, respectively. Interestingly, an electronic phase transition from SC to metal occurs (i.e., E_g becomes zero) at +5% strain (see Fig. 9(c′), right column).

Fig. 9 GGA+U+SOC calculated spin-polarized total density of states for (a and a′) +3%, (b and b′) + 4%, and (c and c′) + 5% biaxial tensile strains along the [110]-direction in the most stable antiferromagnetic spin ordering of Ba- (left column) and Sr- (right column) doped La₂CuOsO₆ double perovskite oxides.

For more clarification of the electronic transitions, we plotted the GGA+U+SOC calculated spin-polarized band structures along with the high symmetry directions of the tetragonal Brillouin zone in Ba-doped La₂CuOsO₆ DPO for +3%, +4%, and +5% tensile strains in Fig. 10(a)–(c), respectively. It is evident that the system under +3% strain displays a SC behavior because a small E_g lies between the valence and conduction bands (see Fig. 10(a)) and confirms the calculated TDOS in Fig. 9(a). Interestingly, a few bands (green ones) emerge at the E_F and E_g becomes zero for a +4% strain as displayed in Fig. 10(b), which further strengthens the calculated TDOS in Fig. 9(b). Moreover, a similar metallic nature is also predicted for the +5% strain-induced band structure (see Fig. 10(c)) as TDOS does in Fig. 9(c). Similarly, the calculated GGA+U+SOC spin-polarized band structures for the Sr-doped system for +3%, +4%, and +5% tensile strains are plotted in Fig. S3(a)–(c) of the ESI,† respectively. A SC nature is visible for +3% (see Fig. S3(a) of the ESI†) and +4% (see Fig. S3(b) of the ESI†) modulated band structures as TDOS does in Fig. 9(a′) and (b′), respectively. Interestingly, a SC-to-metal transition occurs (a few green bands appear at E_F in Fig. S3(c) of the ESI†) in the +5% tensile strained band structure for the Sr-doped motif, which endorses the calculated TDOS in Fig. 9(c′).

Fig. 10 GGA+U+SOC calculated spin-polarized band structures for (a) +3%, (b) +4%, and (c) +5% biaxial tensile strains along the [110]-direction in the most stable antiferromagnetic spin ordering of Ba-doped La₂CuOsO₆ double perovskite oxide. For clarity, a few bands are highlighted with green color which are near/at the Fermi level in each case.

After that, we studied the impact of strain on the system magnetism. To do this, the calculated m_s of Cu↑/Cu↓ and Os↑/Os↓ are plotted in Fig. 11(a) and (b) as a function of biaxial strain ranging from −5% to +5% along the [110]-direction in the most stable AFM spin ordering of Ba-doped La₂CuOsO₆ double perovskite oxide, respectively. One can see that the m_s of Cu↑/Cu↓ and Os↑/Os↓ m_s slightly decreases from 0.65 to 0.53μ_B and 1.58 to 1.38μ_B when the biaxial compressive strain is applied from 0% to −5% (see Fig. 11(a) and (b)), respectively. In contrast, the m_s value increases from 0.65 to 0.81μ_B and 1.58 to 1.82μ_B when the biaxial tensile strain is applied from 0% to +5%, respectively. Here, it is very important to note that m_s of both Cu ions have the same magnitude but are opposite in direction at each strain level, which results in a perfect AFM state. A similar trend of m_s is also found versus strain for Os ions. Moreover, comparable variations in m_s magnitudes with respect to strain are observed in Cu/Os ions for the Sr-doped system (which are not shown here). The decrease/increase in the m_s amplitude due to compressive/tensile strain can be understood by taking the hybridization effects between ions. As the bond length between ions becomes shorter due to compressive strain, strong hybridization occurs among the neighboring ions. This leads to more charge transfer and the local m_s value of each ion is decreased. In contrast, as the bond length between ions increases due to tensile strain, the ions are less hybridized. This restricts the charge transfer and results in the increase of m_s of each ion.

Fig. 11 GGA+U+SOC calculated partial spin moments in spin up(↑)/spin down(↓) states for (a) Cu and (b) Os ions as a function of biaxial strains ranging from −5% to + 5% along the [110]-direction in the most stable antiferromagnetic spin ordering of Ba-doped La₂CuOsO₆ double perovskite oxide.

Finally, we plotted the calculated T_N in the ALaCuOsO₆ (A = Ba and Sr) structure as a function of the biaxial ([110]) strain in Fig. S4 of the ESI.† One can see that as the magnitude of compressive and tensile strains increases, T_N values are also enhanced in both cases (see Fig. S4 of the ESI†). This is due to the fact that the compression and elongation of the lattice constants along the [110]-direction (i.e., along the ab-plane) distorted the polyhedra in the c-out and c-in directions, respectively. Hence, distorted DPOs show higher T_N amplitude than that of undistorted ones as discussed in ref. 40–42.

4 Conclusion

In summary, GGA+U+SOC spin-polarized calculations were performed to investigate the electronic and magnetic characteristics of ALaCuOsO₆ (A = Ba and Sr) double perovskite oxide under the influence of biaxial ([110]) strain. The negative values of the formation of enthalpy confirm the structural stability of the motifs. Our results revealed that both unstrained systems displayed an antiferromagnetic (AFM) semiconducting state, which is attributed to a strong AFM coupling between partially filled high-energy Cu⁺² e¹_g↑ and low-energy empty Os⁺⁵ t⁰_2g↓ orbitals. The computed partial spin moments of Cu and Os ions are 0.65/0.66 and 1.58/1.60μ_B in Ba-/Sr-doped systems having electronic configurations of 3d⁹ (t³_2g↑t³_2g↓e²_g↑e¹_g↓) with S = 0.5 and 5d³ (t³_2g↑t⁰_2g↓e⁰_g↑e⁰_g↓) with S = 1.5, respectively. It is found that AFM spin ordering in both systems is a stable magnetic ground state under the influence of biaxial strain. Strikingly, Ba- and Sr-doped structures demonstrate an electronic transition from semiconductor to metal at a critical tensile strain of +4% and +5% along with improved magnetism as well as Néel temperature, respectively.

Author contributions

S. Nazir: calculation, writing-original draft, editing, supervision, and validation. Yingchun Cheng: analysis and review.

Conflicts of interest

We have no any conflict of interest.

Acknowledgements

The computational work was supported by the University of Sargodha, Sargodha, Pakistan.

Notes and references

1. W. W. Krempa, G. Chen, B. Y. Kim and L. Balents, Annu. Rev. Condens. Matter Phys., 2014, 5, 57–82.
2. Y. Du and X. Wan, Comput. Mater. Sci., 2016, 112, 416–427.
3. J. G. Rau, E. K.-H. Lee and H.-Y. Kee, Annu. Rev. Condens. Matter Phys., 2016, 7, 195–221.
4. R. Schaffer, E. K.-H. Lee, B.-J. Yang and Y. B. Kim, Rep. Prog. Phys., 2016, 79, 094504.
5. C. Martins, M. Aichhorn and S. Biermann, J. Phys.: Condens. Matter, 2016, 29, 263001.
6. G. Cao and P. Schlottmann, Rep. Prog. Phys., 2018, 81, 042502.
7. T. Jungwirth, X. Marti, P. Wadley and J. Wunderlich, Nat. Nanotechnol., 2016, 11, 231.
8. V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono and Y. Tserkovnyak, Rev. Mod. Phys., 2018, 90, 015005.
9. H. Bai, X. Zhou, Y. Zhou, X. Chen, Y. You, F. Pan and C. Song, J. Appl. Phys., 2020, 128, 210901.
10. A. K. Paul, M. Jansen, B. Yan, C. Felser, M. Reehuis and P. M. Abdala, Inorg. Chem., 2013, 52, 6713–6719.
11. A. K. Paul, M. Reehuis, V. Ksenofontov, B. Yan, A. Hoser, D. M. Tobbens, P. M. Abdala, P. Adler, M. Jansen and C. Felser, Phys. Rev. Lett., 2013, 111, 167205.
12. R. Morrow, J. R. Soliz, A. J. Hauser, J. C. Gallagher, M. A. Susner, M. D. Sumption, A. A. Aczel, J. Yan, F. Yang and P. M. Woodward, J. Solid State Chem., 2016, 238, 46–52.
13. O. N. Meetei, O. Erten, M. Randeria, N. Trivedi and P. Woodward, Phys. Rev. Lett., 2013, 110, 087203.
14. D. Iwanaga, Y. Inaguma and M. Itoh, J. Solid State Chem., 1999, 147, 291–295.
15. S. Vasala, J. G. Cheng, H. Yamauchi, J. B. Goodenough and M. Karppinen, Chem. Mater., 2012, 24, 2764–2774.
16. V. Sami, S. Hassan, M. Elvezio, C. Omar, C. Ting-Shan, C. Jin-Ming, H. Ying-Ya, Y. Hisao and K. Maarit, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 134419.
17. G. S. Thakur, H. L. Feng, W. Schnelle, C. Felser and M. Jansen, J. Solid State Chem., 2021, 293, 121784.
18. T. Fix, D. Stoeffler, S. Colis, C. Ulhaq, G. Versini, J. P. Vola, F. Huber and A. Dinia, J. Appl. Phys., 2005, 98, 023712.
19. C. W. Bark, D. A. Felker, Y. Wang, Y. Zhang, H. W. Jang, C. M. Folkman, J. W. Park, S. H. Baek, H. Zhou, D. D. Fong, X. Q. Pan, E. Y. Tsymbal, M. S. Rzchowski and C. B. Eom, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 4720–4724.
20. K. Ajay, S. Rishabh, P. Akhilesh, D. Sandeep, M. Muralidhar, U. Kazuyuki, M. Masahiko and D. D. Rajendra, J. Appl. Phys., 2020, 128, 025303.
21. A. J. Hauser, J. R. Soliz, M. Dixit, R. E. A. Williams, M. A. Susner, B. Peters, L. M. Mier, T. L. Gustafson, M. D. Sumption, H. L. Fraser, P. M. Woodward and F. Y. Yang, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 161201.
22. K. Yang, W. Setyawan, S. Wang, M. B. Nardelli and S. A. Curtarolo, Nat. Mater., 2012, 11, 614.
23. Q. Yan, W. Haiping, K. ErJun, L. Jian, L. Ruifeng, L. Yuzhen, T. Weishi, X. Chuanyun and D. Kaiming, J. Appl. Phys., 2013, 114, 063713.
24. S. Nazir, Phys. Chem. Chem. Phys., 2020, 22, 17969–17977.
25. Q. U. Ain, S. Naseem and S. Nazir, Sci. Rep., 2020, 10, 13778.
26. F. Rubab and S. Nazir, Phys. Chem. Chem. Phys., 2022, 24, 17174–17184.
27. P. Blaha, K. Schwarz, T. Fabien, R. Laskowski, G. K. H. Madsen and L. D. Marks, J. Chem. Phys., 2020, 152, 074101.
28. P. John, B. Kieron and E. Matthias, Phys. Rev. Lett., 1996, 77, 3865–3868.
29. E. C. Calderon, J. J. Plata, C. Toher, C. Oses, O. Levy, M. Fornari, A. Natan, J. M. Mehl, G. Hart, B. M. Nardelli and S. Curtarolo, Comput. Mater. Sci., 2015, 108, 233–238.
30. S. Jana, P. Aich, A. P. Kumar, K. O. Forslund, E. Nocerino, V. Pomjakushin, M. Mansson, Y. Sassa, P. Svedlindh, O. Karis, V. Siruguri and S. Ray, Sci. Rep., 2019, 9, 18296.
31. M. Pajda, J. Kudrnovsky, I. Turek, V. Drchal and P. Bruno, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 64, 174402.
32. M. D. I. Bhuyan, R. Hossain, F. Ara and M. A. Basith, Phys. Chem. Chem. Phys., 2022, 24, 1569–1579.
33. W. Chen, J. George, G. B. Varley, G. M. Rignanese and G. Hautier, npj Comput. Mater., 2019, 5, 72.
34. K. Yang, S. Nazir, M. Behtash and J. Cheng, Sci. Rep., 2016, 6, 34667.
35. J. E. Jaffe and A. Zunger, Phys. Rev. B: Condens. Matter Mater. Phys., 1983, 27, 5176–5179.
36. J. E. Jaffe and A. Zunger, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 29, 1882–1906.
37. K. Yoodee, J. C. Woolley and V. Sa-yakanit, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 30, 5904–5915.
38. S.-H. Wei and A. Zunger, Phys. Rev. Lett., 1987, 59, 144–147.
39. P. Sainctavit, J. Petiau, A. M. Flank, J. Ringeissen and S. Lewonczuk, Physica B, 1989, 158, 623–624.
40. A. V. Powell, J. G. Gore and P. D. Battle, J. Alloys Compd., 1993, 201, 73–84.
41. R. C. Currie, J. F. Vente, E. Frikkee and D. J. W. IJdo, J. Solid State Chem., 1995, 116, 199–204.
42. T. Ferreira, G. Morrison, J. Yeon and H.-C. zur Loye, Cryst. Growth Des., 2016, 16, 2795–2803.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp04548f

---