Magnetic Properties and Spontaneous Polarization of La-, Mn- and N-Doped Tetragonal BiFeO3: A First-Principles Study

Multiferroic materials have been receiving attention for their potential applications in multifunctional devices. Chemical substitution is an effective method for improving the physical properties of BiFeO3 (BFO). However, different experimental results have been reported for Lanthanum- (La-) and Manganese (Mn) -doped BFO ceramics. Here, we systematically studied the magnetic properties and spontaneous polarization of La-, Mn-, and Nitrogen (N) -doped tetragonal BiFeO3 using density functional theory with the generalized gradient approximation and U-value method. The calculated results demonstrated that the systems show ferromagnetism with Mn and N doping, whereas no magnetization was found with La doping in G- and C-type antiferromagnetic orderings. Our research further revealed that the ferromagnetism is attributed to the p-d orbital hybridization. Berry-phase polarization calculations predicted a large polarization of 149.2 µC/cm2 along the [001] direction of pure tetragonal BFO. We found that La and N substitution had little influence on the spontaneous polarization, whereas Mn substitution reduced the spontaneous polarization. The reduced energy barrier heights of the doped systems indicate the reduced stability of the off-centering ferroelectricity against the thermal agitation. These findings provide greater understanding for controlling and tuning the multiferroic properties of BFO.


Introduction
Multiferroic materials are receiving considerable attention for their potential applications in multifunctional devices [1,2]. BiFeO 3 (BFO) is the most promising candidate for practical applications among ABO 3 -type perovskites due to its multiferroic properties at room temperature (T C~1 100 K and T N~6 40 K). However, some serious shortcomings of BFO limit its practical applications in devices, such as high leakage current and weak net magnetization. Therefore, numerous studies have been undertaken to solve these problems. Chemical substitution is an effective method to improve the performance of BFO and many experimental studies have been reported on the A-and B-site substitution of BFO . Rare earth and transition metals are employed to enhance the multiferroic properties of BFO with their substitution at the A-and B-site, respectively. Among them, lanthanum (La) and manganese (Mn) elements doped at the A-site and B-site of BFO, respectively, have been extensively investigated. However, different experimental results were reported in La-and Mn-doped BFO thin films. Some experimental investigations reported that La-doped BFO films enhanced ferroelectric remnant polarization and improved ferromagnetic properties [8,[17][18][19]. However, You et al. [20] reported that La doping destabilized the in-plane polarization component, and Lazenka et al. [21] reported that La-doped BFO film showed weak ferromagnetism. Besides, some groups [22,23] reported that the Mn-doped BFO system showed intrinsic weak magnetism, whereas other groups [24,25] observed enhanced magnetism in Mn-doped BFO thin films. Therefore, a systematic theoretical investigation was necessary to clarify these issues.
High-strain BFO thin film has a tetragonal structure, which was reported in many experimental and theoretical studies [26][27][28][29][30]. The tetragonal structure is not only suitable for improving the application of device miniaturization but may also enhance the multiferroic properties. Therefore, in this work, the multiferroic properties of La and Mn substitution at the A-site and B-site of tetragonal BFO were investigated by using first-principles. Additionally, the multiferroic properties of nitrogen (N) substitution at the O-site of tetragonal BFO was also investigated as a new method of improving the multiferroic properties of BFO because room temperature ferromagnetism was reported in N-doped BaTiO 3 and SrTiO 3 ferroelectric materials [31,32]. Our work highlights the explicit physical mechanism of the multiferroic properties of La-, Mn-, and N-doped BFO films and provides useful guidance for improving the multiferroic properties of BFO.
In the calculation, we used a kinetic energy cutoff of 30 Ry; Monkhorst-Pack k-point grids [41,42] of 3×3×3 were used for structure optimizations and then the denser 4×4×3 k-points were adopted for the total energy calculations. The Methfessel and Paxton Fermi smearing method with a smearing width of 0.01 Ry was used [43]. The calculation conditions were checked to obtain converged results. The ions were relaxed until the interatomic forces on them were less than 10 −3 Ry/bohr. Both G-type and C-type antiferromagnetic (AFM) spin orderings in tetragonal BFO were investigated in our calculation since many first-principles and experimental works reported that the two magnetic structures nearly degenerate [29,[44][45][46]. The optimized lattice constants of tetragonal BFO were a = 3.77 Å and c/a = 1.30 Å, which agreed well with the experimental results [30]. A 2 × 2 × 2 supercell containing eight BFO formula cells for tetragonal structure was built based on the optimized lattice constants, as seen in Figure 1a,b.

Results and Discussion
We first investigated the magnetism of La, Mn, and N substitution at the A-, B-, and O-site of BFO, respectively. The calculated magnetic moments were both one µ B for Mn-and N-doped in BFO with G-AFM and C-AFM orderings. However, La dopant did not affect magnetization because no electron was added or removed for La 3+ replacing Bi 3+ . Thus, the reported apparent improvement inmagnetism in La-doped BFO is not caused by La substitution but originates from some other intrinsic defects [47]. N dopant caused the redistribution of charge among Fe, O, and N ions. However, N is not an ideal trivalent in our calculation, causing the N ion and neighboring O ions to have magnetic moments (0.12 µ B for one N and 0.45 µ B for all O ions) and the antiparallel two Fe ions to have nonequilibrium magnetic moments (3.96 µ B and −3.53 µ B , respectively). To investigate the magnetic coupling between magnetic ions, two Fe (or O) ions were substituted by Mn (N) atoms in the supercell. In our earlier work [48], we investigated the magnetic coupling between two Mn ions in tetragonal BFO. Results indicated that Mn atoms prefer to assemble and exhibit a parallel spin alignment with a total moment of 8 µ B . The robust ferromagnetism was attributed to the stronger p-d hybridization between Mn-3d, Mn-4p, and neighboring O-2p orbitals. For N substitution at the O-site, there are two nonequivalent O ions (i.e., the apical and basal plane atoms of the inverted pyramid structure) in BFO; thus, we discussed two cases when two N replace two O ions in the G-AFM and C-AFM magnetic lattices. The first case contained five nonequivalent possible positions (i.e., C f1 -C f5 configurations) and the second case contained seven nonequivalent possible positions (i.e., C S1 -C S6 and C 34 configurations), as seen in Figure 1c,d. The calculated results are listed in Tables 1 and 2. From the table, we can see that the magnetic properties of N-doped BFO are related to the distribution of N atoms. We found that the C S3 configuration is favorable in terms of energy and shows robust ferromagnetic coupling with a total magnetic moment of 2 µ B . Table 1. Values of N···N distance (d N-N ) (Å), magnetic ordering of ground state, relative energy, and total magnetic moment (M tot ) calculated for each configuration of two nitrogen (N) atoms doped in BiFeO 3 (BFO) with G-antiferromagnetic (AFM) ordering. The energy difference between ground state of each configuration and C S3 configuration is the relative energy. The negative value of magnetic moment denotes its direction is downward. We also investigated the origin of ferromagnetism in terms of electronic structure. Figure 2a,b display the spin charge density of the C S3 configuration in G-AFM and C-AFM orderings, respectively. The parallel spin alignment of the two N ions is mediated by the neighboring Fe ion, which is in antiparallel spin alignment with the N ions. Figure 2c,d show the partial density of states (DOS) of Fe and N ions corresponding to Figure 2a,b, respectively. As can be seen in Figure 2c,d, there is an overlap between Fe 3d states and N 2p states at the Fermi level, which indicates the stronger interaction between Fe 3d electrons and N 2p electrons. This further reveals that the ferromagnetic coupling originates from the p-d hybridization. The C S1 and C S2 configurations in which the two N ions substitution at the equivalent positions and link to the same Fe ion own a net magnetic moment of 0 µ B . Further investigation found that the two N ions are not spin polarized and thus show nonmagnetic. The N-N pairing interaction leads to a nonmagnetic state that is attributed to the ppσ bonding state and ppπ states, which accommodate all 10 electrons and caused the ppσ antibonding state to be empty [49]. We then calculated the spontaneous polarization of La, Mn, and N-doped tetragonal BFO respectively based on the Berry-phase theory of polarization [50,51]:

Configurations
where Z i is the atomic number of the ith atom, u λ v are the Bloch wave functions, Ω is the volume of the unit cell, and the integral is performed over the first Brillouin zone. For comparison, the spontaneous polarization of pure tetragonal BFO was also calculated. In order to obtain the spontaneous polarization of the ferroelectric phase of tetragonal BFO, the non-polar reference state (centrosymmetric structure, λ = 0, labeled with 0% distortion) and ferroelectric state (λ = 1, labeled with 100% distortion) were selected as the end points. We inserted several intermediate structures along an idealized "switching path" between the non-polar reference state and the ferroelectric state. Considering the polarization quantum eR/Ω, which was 113 µC/cm 2 along the [001] direction in BFO unit cell, to jump between branches of the polarization lattice, e is the electronic charge, R is the lattice vector in the direction of polarization, and Ω is the volume of the unit cell [50,51]. The spontaneous polarization is the difference between the two values of end points on the same branch of polarization lattice. Figure 3 shows the calculated polarization value for pure tetragonal BFO. The red line denotes one real evolution of polarization with distortion, and the spontaneous polarization was calculated by 156.1−6.9 = 149.2 µC/cm 2 , which agrees well with other theoretical calculated results [26] and experimental results [52,53]. Therefore, this method can be credibly used for spontaneous polarization predictions of La-, Mn-, and N-doped BFO with G-AFM and C-AFM orderings. Figure 4 shows the calculated total energy as a function of percentage distortion from the centrosymmetric structure to the ferroelectric state (+P4mm structure) for BFO. Energy varied continuously from the maximum (centrosymmetric structure) to the minimum (+P4mm structure). Therefore, we speculate that the energy curve will be a symmetrical as the distortion from centrosymmetric structure to the −P4mm structure with opposite polarization will form a double-well potential. The barrier height of the double-well potential was about 2.335 eV, which is in good agreement with the experimental results reported in BFO films grown on La-doped SrTiO 3 substrates [53].   Table 3 shows the calculated results for one La-, Mn-, and N-doped BFO at the A-, B-and O-site, which correspond to doping concentrations of 12.5%, 12.5% and 4.17%, respectively. From Table 3, the values for spontaneous polarization for La-and N-doped BFO almost did not changes in C-AFM ordering and only changed minimally in G-AFM ordering compared with pure BFO. For Mn dopant, the value of spontaneous polarization tended to decrease, especially in C-AFM ordering, where the value reduced to 123.8 µC/cm 2 . Notably, the barrier heights of the double-well potential all reduced for La, Mn and N doping, which reduced the stability of ferroelectricity against thermal agitation. Thus, the reported apparent improvement in ferroelectric polarization by La doping is not caused by the enhanced off-centering ferroelectric polarization but originates from the P4mm tetragonal structure, which is transformed from rhombohedral structure with increasing La doping concentration [54][55][56][57]. We next used G-AFM BFO as an example to investigate the origin of spontaneous polarization for La-, Mn-, and N-doped BFO from charge density distribution. Figure 5a-c show the charge density distribution of La-, Mn-, and N-doped BFO, respectively. The ionic interaction between La and O is stronger than that of Bi and O (Figure 5a), the Mn-O and Fe-N covalent interaction is stronger than that of Fe-O at the equivalent position (Figure 5b,c). The difference charge density analysis further demonstrated the charge transfer during the bonding process for La-, Mn-, and N-doped BFO (Figure 5d-f, respectively). Here the difference charge density is got by the difference charge density of La, Mn and N-doped BFO minus that of pure BFO, respectively. From which we can obtain the charge-transfer distribution when La, Mn and N substituted at A-, B-and O-site of BFO, respectively. For La-and Mn-doped BFO, the distribution in charge transfer was asymmetric along the [001] direction, which was the cause of a small change in spontaneous polarization. For N-doped BFO, however, the distribution in charge transfer was symmetrical and therefore showed no change in spontaneous polarization. Therefore, the charge transfer is the main cause of polarization change for La-, Mn-, and N-doped tetragonal BFO.

Conclusions
In conclusion, we systematically investigated the magnetic properties and spontaneous polarization for La-, Mn-, and N-doped tetragonal BFO using first-principles. In terms of magnetic properties, Mn-and N-doped tetragonal BFO showed enhanced ferromagnetism that was mediated by p-d hybridization. However, La-doped tetragonal BFO was nonmagnetic. In terms of spontaneous polarization, La and N substitution had little influence on the spontaneous polarization, whereas Mn substitution had a tendency to reduce the spontaneous polarization. However, the spontaneous polarization values remained substantial compared with prototypical ferroelectric materials (BaTiO 3 and KNbO 3 ). Consequently, the enhanced magnetic properties and the large spontaneous polarization values for Mn-and N-doped tetragonal BFO propose a new route for improving the multiferroic properties of BFO film.
Author Contributions: Q.W. conceived the work; Q.T. and Q.W. performed the calculations and wrote this paper; and Y.L. made valuable comments on this manuscript.
Funding: This work was supported by the National Natural Science Foundation of China (Grant Nos. 61564010, 11304273 and 11764046).

Conflicts of Interest:
The authors declare no conflicts of interest.