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

Abstract

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.

An ab initio study of the magnetic properties of strontium hexaferrite

“…Figure 4 presents the dependence of the most relevant exchange constants on U for the ES case (the same trend is found for the other three cases). A strong decay in the strength of almost all interactions is apparent as U increases, which is in line with the results of Wu et al 19 and Novak et al 12 for Ba hexaferrite and derives from the fact that the Hubbard term tends to localize the d -states at the Fe sites and hence, their MMs become less influenced by the precise magnetic state of the neighboring ions. Furthermore, all J s remain positive indicating a robust anti-ferromagnetic character.…”

“…It turns out that, by far, the most relevant parameter is U, for which a moderate value of 2–3 eV provides excellent agreement with all the available data. Our results are at contrast with similar studies on Ba-hexaferrite 12 , 19 where anomalously large U values of 6–10 eV were required to obtain, via the random phase approximation (RPA) or a mean field approach, temperatures close to the experimental ones.

Figure 1 Atomic structure of the SFO.

…”

“…Exchange constants, , between sublattices i and j , have been calculated from the energy differences between different spin-collinear configurations. To this end, and starting from the usual Heisenberg Hamiltonian, the exchange energy for a given spin configuration , is expressed as 19 , 42 : where is the number of atoms in the i -th sublattice, the number of nearest neighbors to an i -th ion that belong to sublattice j , and are the spin vectors of the ions in the i / j sublattice.…”

“…Based on Eq. ( 1 ), and assuming that the modulus of the spin vector of the i -th ions does not change between different spin configurations (that is, independent of ), it is straigthforward to show that the exchange constants between sublattices i and j can be directly determined from the expression 19 : where corresponds to the DFT total energy of the ground state spin configuration , to that of the spin configuration where the spin of sublattice i / j has been inverted with respect to 0, and to that where the spins of both lattices i and j have been simultaneously flipped.…”

“…From the theoretical side, and since the early work of Fang et al 9 where Gorter’s prediction 10 on the ferrimagnetic arrangement of the Fe sublattices was confirmed via ab initio calculations, M-type hexaferrites (M = Ba,Sr) have been extensively studied 11 – 18 A number of properties have been derived in these works including, among others, the electronic gap, the saturation magnetization ( ) and the MMs on the Fe ions together with their associated spin-resolved density of states (DOS), the MCA 13 , 15 , 16 as well as exchange couplings ( Js ) subsequently employed in the estimation of 12 , 19 or the evolution of with temperature 14 . In addition, a large effort has been put in exploring substitutional elements either for the M-ion (La 11 , 15 , 19 , Pb 17 , Pr 15 , Nd 15 ) or the Fe ions (Al 16 , Zn–Sn 13 ) in order to improve the material’s magnetic performance. Most of these studies have relied on the DFT + U framework, paying special attention to the precise value of the Hubbard term U, for which values in the 3–10 eV range have been considered.…”