Calculation of exchange integrals and Curie temperature for La-substituted barium hexaferrites

As the macro behavior of the strength of exchange interaction, state of the art of Curie temperature Tc, which is directly proportional to the exchange integrals, makes sense to the high-frequency and high-reliability microwave devices. Challenge remains as finding a quantitative way to reveal the relationship between the Curie temperature and the exchange integrals for doped barium hexaferrites. Here in this report, for La-substituted barium hexaferrites, the electronic structure has been determined by the density functional theory (DFT) and generalized gradient approximation (GGA). By means of the comparison between the ground and relative state, thirteen exchange integrals have been calculated as a function of the effective value Ueff. Furthermore, based on the Heisenberg model, the molecular field approximation (MFA) and random phase approximation (RPA), which provide an upper and lower bound of the Curie temperature Tc, have been adopted to deduce the Curie temperature Tc. In addition, the Curie temperature Tc derived from the MFA are coincided well with the experimental data. Finally, the strength of superexchange interaction mainly depends on 2b-4f1, 4f2-12k, 2a-4f1, and 4f1-12k interactions.

Owing to the large magnetocrystalline anisotropy, high Curie temperature T c and saturation magnetization M s , barium hexaferrites [BaFe 12 O 19 , BaM] are of great interest for magnetic recording, microwave magnetic devices, and permanent magnets 1-3 . Many schemes [3][4][5][6][7] have been attempted in the past few decades to improve the intrinsic magnetic properties of barium hexaferrites. The typical representative is La-based substitutions [8][9][10] . La-substitutions, which are benefit to improve the saturation magnetization and magnetocrystalline anisotropy, however, are detrimental to enhance the Curie temperature [8][9][10]11 . With the requirements of wide operating temperature range of microwave devices and components, it is very important to quantitatively explore the relationship between the Curie temperature T c and exchange integrals.
The Heisenberg model provides an opportunity to realize account for a large amount of the basic physical laws of ferrites from a phenomenological description. Especially, for a wide range of spinel ferrites 12,13 , the exchange interactions have been investigated based on the molecular field theory employing the generalized gradient approximation (GGA) and local spin density approximation (LSDA) methods. So far, the work of the relationship between the Curie temperature T c and exchange integrals for barium hexaferrites has been mainly concentrated on the undoped samples using the nonlinear fitting methods: Isalgué et al. 14 suggested that 12k sublattice of barium hexaferrites is subject to a strong exchange interaction for the sake of the link between R (R*) and S (S*) blocks, and Grill et al. 15  triads demonstrate the comparatively strong exchange coupling. Unfortunately, the above calculations neglect the intra-sublattice interactions and strongly correlated 3d electrons of Fe, and then overestimate the exchange integrals and Curie temperature T c . In short, in terms of La-substituted barium hexaferrites, there are seldom researches that quantitatively explore the relationship between the Curie temperature T c and exchange integrals. And the exchange integrals as a function of the effective value U eff have not been also investigated completely within the framework of the density functional theory. Furthermore, the Curie temperature T c has not been realized based on the exchange Scientific RepoRts | 6:36200 | DOI: 10.1038/srep36200 integrals by the molecular field approximation (MFA) and random phase approximation (RPA) methods. So this paper would focus on solving these issues.

Results and Discussion
Crystal and Magnetic structure. The crystal structure of M-type hexaferrites could be described as is a spinel block with only two layers, and R = (Ba 2+ Fe 6 3+ O 2− 11 ) 2− is a barium containing hexagonal block with three oxygen layers: S* and R* are obtained from S and R blocks, respectively, by a rotation of 180° about c axis 16 . As shown in Fig. 1, 24 Fe 3+ ions of M-type hexaferrites are distributed in five different sublattices: 3 parallel sites (12k, 2a and 2b) and 2 antiparallel sites (4f 1 and 4f 2 ) 17 . La-based substitutions could contribute to some Fe 3+ transforming into Fe 2+ at the 2a and 4f 2 sites 18 . The X-ray diffraction (XRD) patterns indicated that the compounds are crystallized in a magnetoplumbite hexagonal structure (see Fig. 2). According to the relationship between the cation distribution and the magnetic moments at 0 K 19,20 , the cation distribution of La-substituted barium hexaferrites is summarized in Table 1. In order to conform the results, the  photoelectron counting area of Fe 2+ and Fe 3+ ions is presented in Fig. 3, and the molar ratio of Fe 2+ /(Fe 2+ + Fe 3+ ) for LaFe 12 O 19 sample is approximately 9%.

Ab initio calculation of exchange interactions.
For the exchange interaction between two spins in the isotropic Heisenberg mode, the ferrimagnetic spin configurations (up or down) are considered. The exchange energy per unit cell in the complex system with N magnetic sublattices could be then written as where n i is the number of ith sublattice, z ij is the number of neighboring sites in jth sublattice to ith sublattice, J ij is the exchange integrals, S i and S j represent the spins in the ith and jth sublattices, σ α are equal to ± 1, and the index α is the spin arrangement of the sublattices. We denote α 0 as the ground state, and α n (n ≠ 0) as the relative state. The neighboring z ij and corresponding distance r ij for five sublattices are given in Table 2. The difference between the exchange energy of α n and α 0 is Note that n i z ij = n j z ji . When the spin of a single sublattice is inverted relative to the ground state, we get When the spins of two sublattices inverted relative to the ground state are considered, we then obtain.
Thereby the exchange integral could be given by As mentioned above, the Fe 3+ and Fe 2+ ions in the five sublattices are anti-ferromagnetically coupled with each other. The orbit angular momentum is frozen and hence the magnetism of the Fe 3+ and Fe 2+ ions mainly results from the spin angular momentum S = 2.5 and S = 2.0, respectively 20 . For the mixed valence Ba/La hexaferrites, the spin angular momentum S i in the ith sublattice could be assumed to be.
Calculations of Curie temperature. In the following one calculated the Curie temperature T c employing the Heisenberg Hamiltonian. The common calculations of Curie temperature T c derived from the Heisenberg model contain the mean-filed approximation (MFA) and random-phase approximation (RPA) methods 23 . The mean-field approximation is based on the notion of single-spin excitations, and the Hamiltonian is 24 is the Brillouin function, k B is the Boltzmann constant, and T is the temperature in K. When T is very high, such as and < S iz > are rewritten as  In fifth order systems, there are five solutions in Eq. (13). The highest positive T is the desired Curie temperature T c .
In the random-phase approximation, it is assumed that the thermal disordering is achieved by the excitation of independent spin waves 25 . The equation of motion for the Green function (analogically to Callen) 26 is given that where α, δ(τ ), Φ (τ ) and ∧ s i A z , are an auxiliary, unit-impulse function, unit-step function, and spin operators operating in the unit cell i at the basis site.
is a commutator, = ± , , , 27 , the double commutator could be simplified by applying the RPA decoupling. For convenience, the matrix N(q) could be defined as where T c is the Curie temperature, k B is the Boltzmann constant, and S A is the spin quantum numbers. By the iterative methods to solve this self-consistent set of < > ∧ s A z , the Curie temperature T c could be then obtained. Table 4 shows the experimental and calculating values of Curie temperature. It is concluded that the experimental Curie temperature is reproduced by calculations for the effective value U eff ≈ 6.7 eV, and the MFA and RPA estimations provide an upper and lower bound of the Curie temperature. In the case of La-substituted barium hexaferrites, the T c determined by the RPA is in good agreement with the experimental T c . This could be explicated: the fluctuations of spin wave in MFA (i.e., fluctuations in the magnitudes of the atomic moments) are generally neglected, and hence the arithmetic average takes all the magnon values with the equal weight. While in RPA, this is the harmonic average, and the weight decreases with the increasing spin-wave energy 25,28 .

Conclusions
The composition of Ba 1−x La x Fe 12 O 19 (x = 0.0, 0.5 and 1.0) were prepared by a conventional ceramic method. Thirteen exchange interactions were calculated by the DFT and GGA + U. With the increase of La contents, the 2a-4f 1 , 2a-12k, 2b-4f 1 , and 2b-12k interactions increase, the 4f 1 -4f 2 , 4f 1 -12k, 4f 2 -12k, 2f 1 -2f 1 , 2f 2 -2f 2 , and 4k-8k interactions decrease, while the 2b-4f 2 interaction has a slight change. The Curie temperature was then calculated using the MFA, and RPA estimations. The RPA is more coincident with the experiments for the effective value U eff ≈ 6.7 eV.   were calcined at 800 °C for 2 h and then second-milled with 3.0 wt% Bi 2 O 3 for 8 h. After being further dried at 90 °C, the powders were granulated, pressed and sintered at 1050 °C for 2 h in air. Essential for preventing decomposition into Fe 2 O 3 and LaFeO 3 /BaFe 2 O 4 is rapid cooling. The X-ray diffraction (XRD) patterns were identified on Bruker D8 Advance X-ray diffractometer with Cu-Kα radiation. The binding energy of iron ions was acquired by X-ray photoelectron spectroscopy (XPS) XSAM800. The hysteresis loops of the samples at 1.8 K (approaching 0K) were measured by Quantum Design SQUID VSM under the applied static magnetic fields up to 6T. The experimental values (E.V.) of Curie temperature for La-substituted barium hexaferrites were measured by the Thermal Gravimetric Analyzer (TGA) under N 2 atmosphere using a TA-Q50 series analyzer system.

Methods
Computational details. The total energies and forces were calculated using the density-functional theory (DFT) with the Generalized Gradient Approximation (GGA) as parameterized by the Perdew-Burke-Ernzerhof (PBE) in VASP 29,30 . In structure optimization, we adopted the Conjugate Gradient (CG) method to optimize the lattice parameters and the position of ions until the force on each ion was less than 0.03 eV/Å. The plane-wave cutoff energy and convergence criteria were 500 eV and 10 −7 eV, respectively. The reciprocal space was sampled with an 11 × 11 × 1 Monkhorst-Pack mesh 31 . All the calculations were spin polarized according to the Gorter's ferrimagnetic ordering of the magnetic moments 32 . For improved description of 3d electrons in iron ions, the generalized gradient approximation with Coulomb and exchange interaction effects (GGA+ U) were employed, where an on-site potential is added to introduce intra-atomic interactions between the strongly correlated electrons 33 . We employed an effective value (U eff ) equal to the difference between the Hubbard parameter U and the exchange parameter J 34 . To study how the results depend on U eff , three values (3.4, 6.7, and 10.4 eV) on Fe atoms were adopted on the basis of many rigorous calculations of barium hexaferrites 31,35,36 .