1 Introduction

A few multiferroic materials that possess both ferroelectricity and magnetism [1] are currently intensively investigated due to their potential technological applications in a variety of fields [2]. In multifunctional material technology, it is advantageous to have a high critical temperature and a high value of spontaneous polarization simultaneously. For this purpose, perovskite materials with general formula AB O3 are extensively studied. The majority of known magnetic-ferroelectric perovskites contain Bi or Pb atoms at A-site which have a lone pair effect and a transition metal atom having unfilled d orbital at B-site [3]. A-cation leads toferroelectricity while B-cation leads to magnetism [4]. Along the same line of thought, compounds like BiFeO3 [5] and PbVO3 [6] have been synthesized and they display a large spontaneous polarization as well as large critical temperatures simultaneously with antiferromagnetic spin ordering at the B-site [4]. Following this idea, multiferroic BiCoO3 (BCO) has been carried out; BCO adopts a tetragonal PbTiO3 structure type where the cobalt ion is coordinated by five oxygen ions which make a pyramidal polyhedron [7]. The pyramidal coordination for Co ion arises from the noncentrosymmetric position of Bi3+ atom [8]. The enclosed d6 shell of Co3+ ion possesses three possibilities of spin configuration: low-spin (LS), intermediate-spin (IS), or high-spin (HS) configurations. BiCoO3 is an insulator antiferromagnetic (AFM) below TN= 470 K with antiferromagnetic in the ab-plane stacked ferromagnetically in the c-direction (AFM-C)-type spin order, where the spin magnetic moments of cobalt ions are antiferromagnetically aligned in xy-plane and are ferromagnetically stacked along the z-axis [8, 9].

At ambient conditions, BiCoO3 has a tetragonal structure with a high tetragonality (1.27) associated with a large spontaneous polarization (170 μ C/cm2) [10] and a HS state-AFM-C spin ordering of cobalt ion [4]. Decreasing the temperature, it is found that the spin electron of Co3+ retains its HS state up to low temperature which prevents the spin-state transition as in the case of LaCoO3 [11].

Bi3+ and La3+ have a similar ionic radius; consequently, the local Co–O bond lengths in BiCoO3 at high pressure-nonmagnetic (LS) phase is identical to those of LaCoO3 in its ambient condition phase (LS state: Co–O = 1.925 Å at 5 K) [12, 13].

The A-site chemical modification of the AB O3 structure is expected to affect the ferroelectric (FE) properties. Kan et al. [14] have shown that an enhanced dielectric response is obtained in BiFeO3 compounds in which Bi is partially substituted by a rare-earth RE ions (RE = La, Sm, Gd, Dy, etc.).

In this article, we show the influence of cation size on the structural, electronic, and magnetic properties of M CoO3 (M= Bi, La) in tetragonal and rhombohedral structures using local spin-density approximation (LSDA)+ U calculations. We also discuss the bonding mechanism effects on magnetism and ferroelectricity phenomena in these systems.

2 Computational Methods

The calculations have been performed within density functional theory (DFT) implemented in the Wien2k package [15] based on the hybrid full-potential L/APW + lo method [16]. In this formalism, the unit cell is divided into two regions, nonoverlapping muffin-tin (MT) spheres, inside of which the basis functions are expanded in spherical harmonics and the basis functions in the interstitial region, outside the MT spheres, are plane waves. For the exchange-correlation potential, we employed a local-density approximation (LDA) built from the Perdew-Wang parametrization [17]. In order to describe the behavior of the localized Co d electrons, we have included the orbital-dependent, on-site Coulomb potential (Hubbard U) in the calculation within the so-called LSDA+ U method [18]. Energy convergence in terms of the number of k-points has also been achieved. The bulk moduli and the equilibrium lattice parameters were evaluated using the Murnaghan equation of state [19] to fit the volume-energy curves. The electronic states of atoms in the crystal were chosen with the valence configurations of Bi: 5d10 6s2 6p3, La: 5s2 5p6 6s2 5d1 4f0, Co: 3p6 3d7 4s2, and O: 2s2 2p4, and we have adopted the values of 2.5 Bohr for La/Bi elements, 1.85 Bohr for cobalt, and 1.6 Bohr for oxygen, as MT radii. The number of plane-wave energy cutoffs RMT*KMAX was chosen to be 8.0 for all calculations. A Monkhorst-Pack k-mesh of 12 × 12 × 9 and 10 × 10 × 10 were used for tetragonal and rhombohedric structures, respectively [20]. The convergence of the self-consistent cycles (SCF) was assumed when the energy difference between them was less than 10−4 Ry.

3 Results and Discussion

3.1 Magnetic-Phase Stability

Bismuth cobaltite based oxide BiCoO3 crystallize in tetragonal structure with P4mm space group No. 99 (Fig. 1) In order to explore the magneticphase stability, we compare the total energy of BiCoO3 in different magnetic phases relative to the paramagnetic (PM) state. Calculations are done for an artificial nonspin-polarized case, as well as ferromagnetic (FM), AFM-C, antiferromagnetic in three directions (AFM-G), and ferromagnetic ab planes stacked antiferromagnetically along the c axis (AFM-A). The calculated results are shown in Fig. 1. We find that the total energy of the AFM-C phase is the lowest, which is consistent with experimental observations [21, 22] and other theoretical results [23,24,25].

Fig. 1
figure 1

Magnetic stability: a energies of various spin configurations of BiCoO3 relative to the nonmagnetic case; b tetragonal AFM-C crystalstructure; and c variation of energies in various spin configurations of LaCoO3 using LSDA+ U (U = 6 eV)

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According to Fig. 1a, we can see that AFM-C ordering is favored, and the superexchange coupling is dominated in ab-plane which confirms the gain energy of AFM-G and AFM-C compared with FM and AFM-A (the ab-plane stacked ferromagnetically). The spinordering priority of AFM-C compared with AFM-G is due to the charge transfer between neighboring oxygen atoms This energy gain make AFM-C magnetic structure more favorable compared with AFM-G.

The groundstate determination, lattice parameters, electronic structure, and magnetic properties of BiCoO3 are calculated by employing the LSDA+ U schemes, which give the accurate magnetic structure. The value of U does not affect the magnetic structure result.

Next, the magnetic stability of the hypothetical compound tetragonal LaCoO3 was investigated with the selected Hubbard parameter of BiCoO3 tetragonal (U= 6 eV) and presented in Fig. 1c. It is clear that the same spin ordering appears in hypothetical LaCoO3 tetragonal, which is due to the similar atomic size of both La and Bi atoms.

The prototypical LaCoO3 has a rhombohedrally distorted pseudo-cubic perovskite structure with space group R3c. Compared with the cubic structure, the corner-shared CoO6 octahedra are tilted and the point symmetry of the central Co ion is reduced from the cubic Oh to the trigonal C3i one. All the Co–O bond lengths lCo−O are kept the same. The ground state of LaCoO3 is in diamagnetic phase in the LS [26]. To confirm this result, we have done two different calculations of LaCoO3, no spin-polarized calculation and spin-polarized one.

There are three possible configurations of spin in Co+ 3 ions, the nonmagnetic state is derived from LS and the magnetic state is derived from IS or HS.

To our knowledge, BiCoO3 rhombohedral is a hypothetical compound. Figure 2 shows the nonspin-polarized and spin-polarized states for La/BiCoO3 and it is clear that both materials are nonmagnetic for all values of Hubbard parameter Ueff (see Fig. 2a, b). The determination of spin state in this rhombohedral material is an important step to understand the magnetic properties of these compounds. The nonmagnetic state in both compounds explains the LS (S = 0). Since LaCoO3 is a strong correlated material, certainly, the effect of Hubbard parameter Ueff is clear. For this, we performed the so-called fixed spin-moment (FSM) calculations [27] within the LSDA+ U formalism using Wien2k-code with the variation of Ueff (see Fig. 2d). For both cases, the variation energy with total magnetic moment predicts the lowspin state for all U values (U= 0 to 6) and beyond U= 6 there is change of spin state from LS to IS. So, the total magnetic moment that corresponds to the IS state is around 4 μ B which gives to the Co ion a local magnetic moment (MM) of 2 μ B. So, we can conclude that the Hubbard parameter Ueff affects the result and we should select the best value carefully. Finally we must use Ueff value equal or less than 6 eV.

Fig. 2
figure 2

FSM curves for LaCoO3 (a) and BiCoO3 (b) (Ueff = 0 to 7 eV) obtained by LSDA+ U calculation c The rhombohedral unit cell d Variation of MM of Co-ion versus Ueff for LaCoO3 and BiCoO3

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3.2 Structural Properties of M CoO3 (M= Bi, La) Compounds

In this section, we have calculated the total energies of Bi/LaCoO3 materials in tetragonal and rhombohedral structures, with and without Hubbard U parameter application. Calculations were performed using LSDA+ U for U= 0, 3, 4, 5, 6 and 7. In Fig. 3a–d, we have presented only the results for LSDA+ U(U= and 6 eV).

Fig. 3
figure 3

Variation of total energies with LSDA+ U (U = 0 and 6 eV)

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The effect of U effective on the structural stability is clear in BiCoO3 (BCO) than LaCoO3 (LCO). From Fig. 3b, d, it is clear that the LCO adopts a rhombohedric structure which is experimentally confirmed [26] and U parameter does not affect the structural stability. However, the experimental result shows a tetragonal groundstate structure for BiCoO3 [23,24,25] which does not appear in our LSDA calculation (U= eV) (see Fig. 3a). This failure of LSDA was corrected by application of LSDA+ U with a large value of Ueff.

Theoretical and experimental values of lattice parameters (a, c/a) and internal parameter z are represented in Table 1.

All past works suggest that BiCoO3 should rather be categorized as a Mott insulator, which requires DFT+ U calculation, but unfortunately, the application of U parameter in LSDA increases the volume and tetragonality (c/a). However, this exchange-correlation approximation gives good results in electronic properties. Figure 4a, b and Table 1 show the effect of Hubbard U parameter on lattice parameters and atomic positions of BiCoO3 compound; Fig. 4c shows also the effect of Ueff on the volume and c/a of BiCoO3, which confirms that Ueff= 6 eV is a best value for giving a good structural parameters of ground state.

Table 1 Optimized lattice constants and atomic positions of tetragonal BiCoO3 in AFM-C ordering compared with theoretical and experimental data (values set in italics are in good agreement with experimental data)
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Fig. 4
figure 4

a Variation of zCo, zO1, and zO2 versus Ueffb Variation of volume and c/a versus Ueffc Variation of a and c versus Ueff for BiCoO3

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We have studied with success the ground state of our systems compared with experimental and theoretical investigations. BiCoO3 adopts a tetragonal structure because the existence of Bi ion and especially s orbital in valence states that promotes a shift of cobalt position from his centrosymmetric site (lone pair mechanism). However, in LaCoO3, there is no displacement of Co atom and no lone pair effect.

We present in Table 2 all structural data of rhombohedral LaCoO3 compound and Fig. 5 shows the Hubbard potential effect on the structural parameters.

Table 2 Optimized lattice constants and atomic positions of rhombohedral LaCoO3 in NM state compared with theoretical and experimental data (values set in italics are in good agreement with experimental data)
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Fig. 5
figure 5

a Variation of a and α versus Ueff value for LaCoO3b Variation of internal parameter x versus Ueff value for LaCoO3

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3.3 Band Structures

First, we will discuss the main features of the Bi/LaCoO3 band structures E(k) and the effect of U parameter in the standard LSDA calculations. Figure 6 shows the band structures E(k) along the high-symmetry lines in an energy range close to the Fermi level of BiCoO3 and LaCoO3, respectively.

Fig. 6
figure 6

(a) Variation of band gap and spin magnetic moment of BiCoO3 and LaCoO3 in tetragonal structure, with different Ueff values comparing with experimental data; (b) Variation of band gap of LaCoO3 and BiCoO3 in rhombohedric structure, with different Ueff values comparing with experimental data

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There is a clear effect of Ueff on band structures in both compounds through an important correction of states from metallic to insulator nature.

Table 3 presents a different values of gap energies for Bi/LaCoO3 with LSDA+ U(U=− 7 eV) compared with experimental and theoretical data. Our results shown in Fig. 6 confirm the good choice of a large Ueff (U = 6 eV).

Table 3 Calculated results of gap size (Eg) and magnetic moment per Co for BCO and LCO materials (values set in italics are in good agreement with experimental data)
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For both compounds, an indirect band gap is localized (the valence-band maximum at M and the conduction-band minimum at Z for BCO, the valence-band maximum at Γ and the conduction-band minimum at F for LCO). Using LAPW-LSDA+ U, we have predicted the band gap and magnetic moment with a good accuracy. Since the ionic radius of La and Bi are nearly equal, and cobalt environment are the same in both structures, we have found equivalent band gap energies in both materials.

3.4 Density of States and Magnetic Moment

Total and partial DOS of BiCoO3 and LaCoO3 in the AFM-C tetragonal structure obtained by LSDA+ U (U= 6 eV) are shown in Figs. 7 and 8. TDOS of LCO presents a metallic character using LSDA and a semiconductor character by LSDA+ U. For BCO, the insulator character is present, and the band gap energy increases with the application of Ueff. From Fig. 7, we notice that there are unoccupied Co 3d states (hybridized with O 2p orbital) extending from 2 to 2.8 eV and the width of this unoccupied band is about 0.8 eV. According to partial DOS of Bi, Co, and O, there are occurrences of strong hybridization effects O 2p-Co 3d and Bi (6s,6p)-O 2p states. It is accepted generally that the ferroelectricphase transition arises from a delicate balance between long-range Coulomb forces (which favors the ferroelectric state) and short-range repulsions (which favors the nonpolar structure) [25]. The hybridization between Co 3d and O 2p leads to the charge transfer from O to Co; thus, O (2.535) and Co (4.5884↑ and 1.6357↓) are not fully ionized. The covalent Co–O bonds which extended from − 7.5 eV to EF, will weaken the short-range repulsions and reduce the total energy, which is favorable for the ferroelectricphase transition. Moreover, the Bi s state is fully occupied and acts as a lone pair state causing the off centering of Co ion [25]. The covalence of Bi–O (− 10 to − 9 eV) causes a partially charge transfer from O to both Bi, which reduces the total energy and enhances the stability of the ferroelectric structure of BiCoO3. On the other hand, the hybridizations of Co–O cause the large decrease of spin magnetic moments of Co ions 2.97 μ B in comparison with the d6 configuration 4 μ B. Moreover, Fig. 7 shows a hybridization between Co dxz, dyz, and O pz, which is missing between dxy and pz. This remark confirms the localization character of dxy orbital. From Fig. 7, we can observe the occupancy of Co d orbital as follows: dxy↑, dxz↑, dyz↑, \(d_{z}^{2} \uparrow \), and \(d_{\mathrm {x}}^{2}\)−y2↑↓. This result confirms that the t2gorbital is higher than egorbital, where the down electron prefers egorbital. The last result shows the HS state in BCO-T with magnetic moment of 2.97 μ B reduced by Co–O covalent bond.

Fig. 7
figure 7

Band structures of La/BiCoO3 using Ueff = 0 and 6 eV

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Fig. 8
figure 8figure 8

Total and partial DOS of BiCoO3 and LaCoO3 in the NM rhombohedral structure obtained by LSDA+ U (U = 6 eV)

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The crystal field in BCO is present due to the octahedral environment of Co ion. From our BCO-DOS, there is a splitting of partial d-orbitals of cobalt due to the Jahn-Teller effect that arises because regular octahedra deformed into a tetragonal distortion to lift the cubic symmetry. Nevertheless, in this type of transition metal oxides, the C-type or G-type antiferromagnetic spin state results from the superexchange mechanism. In the case of BiCoO3, the C-type AFM is favored because of the existence of charge transfer between nearest-neighbor oxygen, where the transport properties follow Co–O–O–Co path. O–O charge transfer reduces the total energy of system and makes the AFM-C state more stable than AFM-G. In the hypothetical tetragonal AFM-C LaCoO3, we remark the same hybridization and covalent bond between Co and O atoms (see Fig. 7). However, the absence of hybridization between La s and p orbitals and O p orbital makes the lone pair effect absent, which disadvantage the Co displacement and ferroelectric phase.

Figure 8 shows the calculated density of states in nonmagnetic configuration (LS: \(t^{6}_{\mathrm {2g}}e_{\mathrm {g}})\) for the rhombohedric LaCoO3, where t2g is fully occupied with triply degenerated dxy, dxz, and dyz orbitals (same energies in DOS). The valence band is formed by a mixture of Co states and O 2p orbitals while the doubly degenerate d x2−y2 and \(d_{\mathrm {z}}^{2}\) orbitals, which represent the eg manifold, form the conduction band and the valence band, which is characterized by two peaks at − 0.5 and − 4.5 eV below the Fermi energy, representing strong hybridization of electrons of O and Co atoms with some small contribution from La atom. In the hypothetical rhombohedric BiCoO3, there is a small hybridization between Bi s and O p orbitals, which can cause a displacement of Co atom and consequently the tilt of the octahedral.

4 Conclusion

Using the first-principle DFT method and LSDA+ U calculations, we have performed a detailed investigation on the structural and magnetic ground states of strongly correlated perovskites, BiCoO3 and LaCoO3. We have found a critical value of Ueff= 6 eV that induces a spinstate transition in the bulk rhombohedral LaCoO3 structure. The variation of U induces a structural transition from rhombohedric to tetragonal in bulk BiCoO3. Our result shows that the correct tetragonal AFM-C structure is stable for Ueff values more than 4.0 eV. According to these two critical values of Ueff (between 4.0 and 6.0 eV), BiCoO3 adopts the antiferromagnetic state with C-type spin ordering and LaCoO3 was found nonmagnetic with lowspin configuration of Co d electrons. Structural parameters of BiCoO3 compound in tetragonal-AFM-C structure and of LaCoO3 in rhombohedric-NM structure using Ueff= 6 eV, are found in good agreement with other experimental and theoretical works.

The band structures of these compounds confirm the insulator character for both materials with a large band gap. We have also examined the density of states for the synthesized T-BiCoO3-AFM-C and R-LaCoO3-NM compounds and the hypothetical T-LaCoO3-AFM-C and R-BiCoO3-NM compounds. There are occurrences of strong hybridization effects O 2p-Co 3d and Bi (6s, 6p)-O 2p states. The covalent Co–O bonds will weaken the short-range repulsions and reduces the total energy, which is favorable for the ferroelectricphase transition. Moreover, the Bi s state is fully occupied and acts as a lone pair state causing the off centering of Co ion. On the other hand, the hybridizations of Co–O cause the large decrease of spin magnetic moments of Co ions at 2.97 μB in comparison with the d6 configuration 4 μ B. Our results show the HS state in BCO-T with a magnetic moment of 2.97 μB reduced by Co–O covalent bond. There is a splitting of partial d-orbitals of cobalt due to the Jahn-Teller effect that arises by the octahedral deformation to lift the cubic symmetry. From our calculations, there are clear similarities between BCO-R and LCO-R. However, the absence of hybridization between La s and p orbitals and O p orbital does not induce the lone pair effect and consequently the ferroelectric phase is not favored.