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 Tc and saturation magnetization Ms, barium hexaferrites [BaFe12O19, BaM] are of great interest for magnetic recording, microwave magnetic devices, and permanent magnets123. Many schemes34567 have been attempted in the past few decades to improve the intrinsic magnetic properties of barium hexaferrites. The typical representative is La-based substitutions8910. La-substitutions, which are benefit to improve the saturation magnetization and magnetocrystalline anisotropy, however, are detrimental to enhance the Curie temperature891011. 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 Tc 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 ferrites1213, 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 Tc 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 confirmed that Fe3+2-O-Fe3+4, Fe3+2-O-Fe3+4, Fe3+4-O-Fe3+12, and Fe3+4-O-Fe3+12 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 Tc.

In short, in terms of La-substituted barium hexaferrites, there are seldom researches that quantitatively explore the relationship between the Curie temperature Tc and exchange integrals. And the exchange integrals as a function of the effective value Ueff have not been also investigated completely within the framework of the density functional theory. Furthermore, the Curie temperature Tc has not been realized based on the exchange 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 SRS*R*, where S = (Fe63+O2−8)2+ is a spinel block with only two layers, and R = (Ba2+Fe63+O2−11)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 axis16. As shown in Fig. 1, 24 Fe3+ ions of M-type hexaferrites are distributed in five different sublattices: 3 parallel sites (12k, 2a and 2b) and 2 antiparallel sites (4f1 and 4f2)17. La-based substitutions could contribute to some Fe3+ transforming into Fe2+ at the 2a and 4f2 sites18. 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 K1920, the cation distribution of La-substituted barium hexaferrites is summarized in Table 1. In order to conform the results, the photoelectron counting area of Fe2+ and Fe3+ ions is presented in Fig. 3, and the molar ratio of Fe2+/(Fe2++Fe3+) for LaFe12O19 sample is approximately 9%.

Figure 1

The unit cell and spin configurations for Ba1−LaFe12O19: Purple, red, and gold spheres denote Ba/La, O and Fe atoms.

The arrows represent the local magnetic moment at each atom site.

Figure 2

The X-ray diffraction patterns for Ba1−LaFe12O19 samples.

(a) x = 0.0, (b) x = 0.5, (c) x = 1.0. And the main (hkl) peaks from JCPDF Card No. 43–0002 for BaFe12O19 are also plotted.

Table 1

The cation distribution for Ba1−La Fe12O19 samples.

Samples	σs (emu/g)	Cation distribution
x = 0.0	100.50	(Ba2+Fe23+[Fe3+]O32− Fe23+[(Fe3+)(Fe63+)]O162−)2 24
x = 0.5	100.31	(Ba0.52+La0.53+Fe1.7623+Fe0.2382+[Fe3+]O32− Fe23+[(Fe0.7383+Fe0.2622+)(Fe63+)]O162−)2
x = 1.0	100.28	(La3+Fe1.5083+Fe0.4922+[Fe3+]O32− Fe23+[(Fe0.4923+Fe0.5082+)(Fe63+)]O162−)2

Figure 3

The representative XPS spectra of iron ions for LaFe12O19 sample.

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 ni is the number of ith sublattice, zij is the number of neighboring sites in jth sublattice to ith sublattice, Jij is the exchange integrals, Si and Sj represent the spins in the ith and jth sublattices, and are equal to ±1, and the index α is the spin arrangement of the sublattices. We denote α0 as the ground state, and αn (n ≠ 0) as the relative state. The neighboring zij and corresponding distance rij for five sublattices are given in Table 2. The difference between the exchange energy of αn and α0 is

Table 2

The number of nearest Fe neighbors and corresponding distance for five sublattices in the BaFe12O19.

	2a		2b		4f1		4f2		12k
	zij	zij	zij	zij	zij	rij	zij	rij	zij	rij
2a	6	0.589	2	0.580	6	0.346	6	0.557	6	0.305
2b	3	0.580	6	0.589	6	0.619	6	0.367	6	0.371
4f1	2	0.346	3	0.619	3	0.363	1	0.379	6	0.350
									3	0.356
4f2	3	0.557	3	0.367	1	0.379	1	0.277	6	0.351
12k	1	0.305	1	0.371	2	0.350	2	0.351	2	0.291
					1	0.356			2	0.298

Note that nz = nz. 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 Fe3+ and Fe2+ ions in the five sublattices are anti-ferromagnetically coupled with each other. The orbit angular momentum is frozen and hence the magnetism of the Fe3+ and Fe2+ ions mainly results from the spin angular momentum S = 2.5 and S = 2.0, respectively20. For the mixed valence Ba/La hexaferrites, the spin angular momentum S in the ith sublattice could be assumed to be.

where t1 and t2 are the Fe3+ and Fe2+ numbers in the ith sublattice, respectively. Since the exchange interactions between two magnetic ions decrease with the increase of the corresponding distance, the distance more than 4 Å could be neglected21. As shown in Table 2, the distance of neighboring 4f1, 4f2, and 12k sublattices is smaller than 4 Å, and in fact these sublattices have also the nearest Fe neighbors in the same sublattices. Ten inter-sublattice interactions (2a-2b, 2a-4f1, 2a-4f2, 2a-12k, 2b-4f1, 2b-4f2, 2b-12k, 4f1-4f2, 4f1-12k, and 4f2-12k) and three intra-sublattice interactions (4f1-4f1, 4f2-4f2, and 12k-12k) were thus considered. For intra-sublattice interaction calculations, in order to preserve the highest symmetry, one divided the 4f1, 4f2, and 12k sublattices into (2f1, 2f1), (2f2, 2f2), and (4k, 8k), respectively. The representative calculations (x = 0.5) of energy difference per unit cell for each relative state are given in Table 3. According to the Eq. (5), thirteen exchange integrals as a function of Ueff (see Fig. 4) could be obtained. The 2a-2b and 2a-4f2 interactions are very small with |J2| and |J2| < 0.05, which is ascribed to the Fe2-O-Fe2, nd Fe2-O-Fe4ngles approach 90°14. The different ionic radii of La3+ (1.22 Å), and Ba2+ (1.47 Å)11 causes some changes in exchange interactions of Ba/La hexaferrites: The 2a-4f1, 2a-12k, 2b-4f1, and 2b-12k interactions increase, and whilst the 4f1-4f2, 4f1-12k, 4f2-12k, 2f1-2f1, 2f2-2f2, and 4k-8k interactions decrease with the increase of La concentrations. It is emphasized, however, 2b-4f2 interaction has a slight change (about 0.1 eV). This is associated with the strong effects of barium or lanthanum ions on the nearest neighboring iron ions (2b and 4f2 sublattices)22.

Table 3

The energy difference △(U eff) in eV between the ground state and excited state.

S1	S2	△(3.4)	△(6.7)	△(10.4)
2a	—	0.853	0.607	0.565
2b	—	0.802	0.476	0.285
4f1	—	3.218	1.647	0.951
4f2	—	3.349	1.814	1.268
12k	—	3.986	1.972	1.244
2a	2b	1.649	1.082	0.853
2a	4f1	1.882	0.861	0.762
2a	4f2	4.201	2.420	1.831
2a	12k	5.128	2.598	1.800
2b	4f1	3.921	2.086	1.260
2b	4f2	1.694	0.835	0.741
2b	12k	5.707	2.918	1.795
4f1	4f2	6.738	3.545	2.252
4f1	12k	2.548	0.758	0.341
4f2	12k	2.718	0.842	0.673
2f1	—	1.556	0.809	0.470
2f2	—	1.645	0.891	0.633
4k	—	0.758	0.453	0.267
8k	—	2.013	0.963	0.781

Figure 4

The exchange integrals as a function of Ueff for Ba1−LaFe12O19 samples.

(a) x = 0.0, (b) x = 0.5, (c) x = 1.0.

Calculations of Curie temperature

In the following one calculated the Curie temperature Tc employing the Heisenberg Hamiltonian. The common calculations of Curie temperature Tc derived from the Heisenberg model contain the mean-filed approximation (MFA) and random-phase approximation (RPA) methods23.

The mean-field approximation is based on the notion of single-spin excitations, and the Hamiltonian is24

where g and μB are the Lande factor and Bohr magneton, respectively. The molecular field could be thus defined as

with

where is the Brillouin function, k is the Boltzmann constant, and T is the temperature in K. When T is very high, such as kT  ≫ μBHi, the and are rewritten as

Introducing the exchange integral Jij, we get

i.e.,

which has nonzero solution only if the determinant

In fifth order systems, there are five solutions in Eq. (13). The highest positive T is the desired Curie temperature Tc.

In the random-phase approximation, it is assumed that the thermal disordering is achieved by the excitation of independent spin waves25. The equation of motion for the Green function (analogically to Callen)26 is given that

where α, δ(τ), Φ(τ) and are an auxiliary, unit-impulse function, unit-step function, and spin operators operating in the unit cell i at the basis site. Here is a commutator, , and the mean value with being the Heisenberg Hamiltonian and β = 1/(kBT). According to Tyablikov 27, the double commutator could be simplified by applying the RPA decoupling. For convenience, the matrix N(q) could be defined as

After performing time and lattice Fourier transformations, the Green’s function could be expressed by the following form

where I is unity matrix. Based on the solutions of Green’s function, as described in Ref. 28, the self-consistent equation for the mean values could be written as

where Tc is the Curie temperature, k is the Boltzmann constant, and S is the spin quantum numbers. By the iterative methods to solve this self-consistent set of , the Curie temperature Tc 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 Ueff ≈ 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 Tc determined by the RPA is in good agreement with the experimental Tc. 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 energy2528.

Table 4

The experimental and calculating Curie temperature as a function of U eff for Ba1−La Fe12O19 samples.

Ueff (eV)	x = 0.0			x = 0.5			x = 1.0
	3.4	6.7	10.4	3.4	6.7	10.4	3.4	6.7	10.4
E.V. (K)	723	723	723	705	705	705	695	695	695
MFA (K)	1514	980	629	1450	951	581	1419	928	579
RPA (K)	1009	653	419	967	634	387	946	619	386

Conclusions

The composition of Ba1−LaFe12O19 (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-4f1, 2a-12k, 2b-4f1, and 2b-12k interactions increase, the 4f1-4f2, 4f1-12k, 4f2-12k, 2f1-2f1, 2f2-2f2, and 4k-8k interactions decrease, while the 2b-4f2 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 Ueff ≈ 6.7 eV.

Methods

Experimental procedures

The compositions of Ba1−LaFe12O19 (x = 0.0, 0.5 and 1.0) were fabricated by a conventional ceramic method. The analytical-grade raw materials, BaCO3, La2O3, and Fe2O3 were weighed in stoichiometric proportion and mixed homogeneously in zirconia ball mills for 12 h. The slurries, after being dried, were calcined at 800 °C for 2 h and then second-milled with 3.0 wt% Bi2O3 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 Fe2O3 and LaFeO3/BaFe2O4 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 N2 atmosphere using a TA-Q50 series analyzer system.

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 VASP2930. 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 mesh31. All the calculations were spin polarized according to the Gorter’s ferrimagnetic ordering of the magnetic moments32. 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 electrons33. We employed an effective value (Ueff) equal to the difference between the Hubbard parameter U and the exchange parameter J34. To study how the results depend on Ueff, three values (3.4, 6.7, and 10.4 eV) on Fe atoms were adopted on the basis of many rigorous calculations of barium hexaferrites313536.

Additional Information

How to cite this article: Wu, C. et al. Calculation of exchange integrals and Curie temperature for La-substituted barium hexaferrites. Sci. Rep. 6, 36200; doi: 10.1038/srep36200 (2016).

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