Ferrites, a class of ceramic-like materials with magnetic properties, are useful in many types of electronic and magnetic devices. The crystal structures of ferrites are classified into 4 types, including spinel ferrites, garnet structure, ortho ferrites (perovskite), and hexagonal ferrites.^[1]

Hard ferrites are important in many industrial applications such as permanent magnetic materials, electric motors,^[2] microwave devices,^[3] and data storage for the telecommunication,^[4,5] medical,^[6] and environmental uses.^[7] Magneto-plumbite type hexaferrites (M type) of AFe₁₂O₁₉ (A = Sr, Ba, Pb) are high-performance hard magnetic materials with high Curie temperatures, high magnetism, and high magneto-crystalline anisotropy, great chemical stability, and high resistance against corrosion; these compounds are synthesized by the sol-gel method as many other hard magnetic materials.^[8–13]

Structurally, the oxygen ions are positioned in the polyhedron’s corners. The space group of AFe₁₂O₁₉ is P63/mmc with five non-equivalent crystalline positions for Fe (namely 2a, 2b, 4f₁, 4f₂, and 12k). There are six coordination numbers of Fe for the positions 2a, 4f₂, and 12k, five coordination numbers for 2b, and four coordination numbers for 4f₁. The magnetic structure of the hexagonal ferrite is ferrimagnetic, in which the magnetic moments of Fe³⁺ are parallel at positions 2a, 2b, and 12k, while anti-parallel at the positions 4f₁ and 4f₂ along c-axis.^[14]

Studies in recent years have shown that pure barium hexaferrite as one of the most famous ones among hard magnetic materials with high crystalline magnetic anisotropy along c axis can be transformed into a soft magnetic material with planar crystalline magnetic anisotropy by replacing its ions with different cations such as Fe³⁺, Ti⁴⁺, Co²⁺, Li³⁺, and Ce⁴⁺, so that this pattern has similar results for other magneto-plumbite structures.^[15] Recently, studies of the Fe³⁺ and Sr²⁺ substitution with metal and rare alkali ions have been done to improve different properties. Doping elements such as Ce, Zn, Mn, Sm, Co, Yb, and Gd has been carried out, in each of which the effects of the doped element on the magnetic behavior and electrical properties of the structure have been studied.^[16–18]

Studies on garnet and perovskites and other magnetic nano-particles have also shown that doping the above elements has a significant effect on their thermal stability, crystallization, physical, magnetic, and optical properties.^[19–22] The carrier doping positive effects on the magnetic properties of defective graphene have been detected.^[23] Using first-principle calculations, researchers have investigated the mechanical, structural, and electronic properties and formation energy of 25 kinds of III–V binary monolayers in detail.^[24] Different magnetic and dynamic properties studies on iron nanowire by Monte Carlo simulation and coercivity mechanisms in nanostructured permanent magnets, as well as the density functional theory (DFT) study on Dirac cones in bilayered perovskites, have been conducted in different research studies.^[25–27]

In addition to investigating the different methods of experimental synthesis of these ferrites and their application in industry, there have been numerous studies on the relating computational methods using the density functional theory,^[28,29] such as the calculation of exchange integrals in the strontium hexaferrite structure in order to study the antiferromagnetic exchange interactions,^[30] calculation of the electronic band structure, splitting parameters investigation of the electronic properties in the doped structure, charge localization, and crystalline magnetic anisotropy in the doped structures.^[31]

One of the most commonly used physical states of magnetic materials is the half-metallic state. Half-metals are a class of materials that are well-known candidates in the spintronics industry for enhancing magnetic memory, and application in tunnel magnetoresistances (TMRs) and giant magnetoresistances (GMRs).^[32,33] These materials are semiconductors in one spin channel and metals in the other.^[34–38] One of the problems with these materials is that the half-metallic property is often disturbed or severely decreased at the surface of the material. Thus, researchers have always been looking for a way to solve this problem, and the best way up to now, is engineering the half-metal to obtain a spin polarization state of 100% and an integer magnetic moment.^[39–44]

In this work, the Gd atom with the atomic number 64 from the lanthanide family is selected for doping in strontium hexa-ferrite. This element has a single magnetic moment of 8μ_B and is very capable for changing the physical properties of the material. After the introduction, the details and methods of calculation are presented in the second section. In the third section, after optimizing the input parameters, we study the structural properties and thermodynamic stability of the pure and doped cases. Finally, the fourth section provides the electronic and magnetic properties, followed by the conclusion in the fifth section.

We performed self-consistent calculations using the WIEN2k computational code based on the density functional theory with full potential. To solve the Kohn–Sham equations governing the problem, we used the augmented plane wave method with local orbital for the valence and semi-core electrons (APW+lo). We applied the generalized gradient approximation (pbe-GGA) and GGA plus Hubbard U parameter (GGA+U) to calculate the exchange correlation potential,^[39,42,43] and the space group p63/mm3 (194) was employed to construct the graphene flat structure, which belongs to the hexagonal space group family. The radii of proper muffin-tin spheres without overlap for Sr, Fe, O, and Gd atoms were considered 2.45 Å, 1.74 Å, 1.5 Å, and 2.21 Å respectively. We chose the cut-off radius of the wave functions in the inter-position zone regions R_kmax and G_max equal to 8 and 14 respectively. In the first Brillouin zone, we chose 500 points which were reduced to 42 points taking advantage of symmetry.

One of the hexagonal ferrites’ family is strontium hexa-ferrite (SrFe₁₁O₁₉). In the unit cell of this structure, 64 atoms are assumed to have 24 symmetries in the crystal by applying the p63/mm3 (194) space group, and these 64 atomic positions are reduced to 11 atomic positions. The first position consists of two Sr atoms, the positions 2–6 consist of 24 Fe atoms, and the atomic positions 7–11 contain 38 O atoms (Fig. 1(a)).

Fig. 1.

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	Fig. 1. (a) The SrFe12O19 structure, (b) the SrGdFe11O19-I structure, and (c) SrGdFe11O19-II structure.

In order to change the physical properties of the material, we substitute the magnetic metal Gd instead of the Fe atoms in the two positions 2 and 3, represented by SrGdFe₁₁O₁₉-I and SrGdFe₁₁O₁₉-II, respectively, as shown in Figs. 1(b) and 1(c). By introducing the Gd atoms with a larger atomic radius than Fe, the structural order of the pure crystals is slightly disrupted, and after optimizing, the bond lengths are calculated before and after substituting the impurity in all the three crystals as shown in Table 1.

Table 1.

Table 1.

Table 1. The bond lengths in the crystals SrFe11O19, SrGdFe11O19-I, and SrGdFe11O19-II. . Bond lengths SrFe12O19 SrGdFe11O19-I SrGdFe11O19-II Fe–O/Å 1.8169 1.8170 1.9445 Sr–O/Å 2.9173 2.7481 2.9309 Fe–Sr/Å 3.7002 3.5483 3.7403 Gd–O/Å – 2.2531 2.0479

Table 1. The bond lengths in the crystals SrFe11O19, SrGdFe11O19-I, and SrGdFe11O19-II. .

As can be seen, the Gd impurity has changed the bond lengths. Changes in the crystals’ structures also change their physical properties. To study the thermodynamic stability and structural properties of a crystal in computational physics, it is necessary to study the total energy of the system and its changes versus volume (E–V). For this purpose, the total energy of the crystal is calculated for different volumes and then, using a proper equation of state, namely, the Birch–Murnaghan equation, the energy as a function of volume is fitted for the pure SrFe₁₁O₁₉ and the doped SrGdFe₁₁O₁₉-II as shown in Fig. 2. Using the minimum of the E–V diagram, the equilibrium volume is calculated, then using this equilibrium volume, the appropriate lattice constants for the crystal are obtained. It is noteworthy that the existence of this minimum energy attests to the thermodynamic stability of these structures.

Fig. 2.

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	Fig. 2. The E–V diagram of (a) the pure compound SrFe11O19 and (b) the doped SeGeFe11O19-II.

To study a system theoretically, one should be able to realize it at the first place. This is confirmed using some measures defining the feasibility of a system, one of which is connected to its thermodynamic stability, namely, the cohesive energy E_C (in units of eV)

In condensed matter physics, the most important quantities for studying the electronic and magnetic properties of the crystal are the density of states (DOS) and the band structure, which give us important information about the physical properties of the material under investigation. Figure 3 shows the total DOS diagram for the three crystals SrFe₁₂O₁₉, SrGdFe₁₁O₁₉-I, and SrGdFe₁₁O₁₉-II in the two spin up and down modes using the PBE-GGA functional.

Fig. 3.

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	Fig. 3. Total DOS contribution of the spins up and down to (a) SrFe12O19, (b) SrGdFe11O19-I, and (c) SrGdFe11O19-II using PBE-GGA method.

In the pure SrFe₁₁O₁₉ crystal in spin down, the DOS diagram crosses the Fermi surface and has metallic property in this channel; but in the spin up, we see a splitting at the Fermi surface so that the compound in this channel is a semiconductor with 0.5 eV spin flip gap. This means that, overall, the crystal is a half metal. In the SrGdFe₁₁O₁₉-I crystal, the DOS diagram intersects the Fermi surface at both the spins up and down, representing the metallic property of this compound. However, in the SrGdFe₁₁O-II crystal, the material behaves like a semiconductor with a band gap of about 0.4 eV in the spin-up, which is lower than that of the pure state. At the spin down channel, on the other hand, as in the two other structures, the Fermi surface is cut by the DOS diagram, showing the metallic property, so this crystal is also a half-metal.

GGA does not correctly calculate the correlation between the d and f electrons, hence, in the calculations using the computational approach, the Columbian repulsion energy (U) is calculated and used for the d and f orbitals of Fe and Gd, where U = U_eff and J = 0. To study the dependence of the results on U, we apply U = 5.3 eV for Fe atoms and U = 8.3 eV for Gd, following the evaluation performed for SrFe₁₂O₁₉ and SrGdFe₁₁O₁₉-I and SrGdFe₁₁O₁₉-II structures. The total DOS diagram for these structures is shown in Figs. 4(a)–4(c).

Fig. 4.

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	Fig. 4. The total DOS contribution of the spins up and down to (a) SrFe12O19, (b) SrGdFe11O19-I, and (c) SrGdFe11O19-II using GGA+U method.

Under the effect of the U parameter and the dependence of the exchange integrals on it, SrFe₁₂O₁₉ turns to a magnetic semiconductor and is changed significantly from the case calculated with the PBE-GGA functional. In spin up, the energy gap is increased from 0.5 eV to 3.7 eV, and in the spin down, an energy band gap of 2.27 eV is observed with a spin flip gap of about 1.0 eV. SrGdFe₁₁O₁₉-I and SrGdFe₁₁O₁₉-II are also converted to magnetic semiconductors, their electronic states change and a decrease in the spin down energy gap is observed, which are obtained as 2.12 eV and 1.80 eV, respectively.

The band structure is created by the overlap of atomic wave functions in the k space. Figure 4 shows the band structure calculated for the pure SrFe₁₂O₁₉ compound and in the presence of the doped Gd atom at positions 2 and 3 for the two spin states up and down with the PBE-GGA functional.

In the pure structure of SrFe₁₁O₁₉ and at the spin up, a spin flip gap of 0.5 eV is observed at point M, which is a direct energy gap. But in the spin down, many of energy bands cut the Fermi surface and this spin plays the major role in electron transport. In the presence of the impurity Gd at position 2 in both spins up and down, the energy bands cut the Fermi energy, resulting a metallic nature; due to high density of the bands in spin down, again, the electron transport is expected in this spin channel. In the presence of the impurity Gd at position 3, i.e., in SrGdFe₁₁O₁₉-II at spin up, an indirect energy band gap of about 4 eV is observed in the band structure, which is less than that of the pure case. In spin down, the density of the bands is also extremely high which is clearly proportional to the DOS diagram at the Fermi level.

The most important aspect of ferrites is their high magnetic properties, making them highly useful in the electronics industry. The magnetic property of a material depends on the magnitude of its magnetic moments of the atoms forming that material and the interaction of the moments. Fully filled orbitals have no contribution to the magnetic moment, while the filling orbitals have a major role. Most of the magnetic moment relates to free atoms and ions, which causes permanent magnetic moment so the material becomes ferrimagnetic. The Gd atom from the lanthanide elements group has the electron arrangement of [Xe]4f⁷5d¹6s², which in isolation has a high magnetic moment of about 8μ_B and we expect to find more interesting properties by injecting this element into SrFe₁₂O₁₉. By comparing the total DOS diagram in the spins up and down, one can guess the magnetic or non-magnetic state of the crystal, and the more asymmetric both the up and down diagrams are, the stronger the magnetic properties of the material are. According to the DOS diagram of the all crystals in Figs. 3 and 4, and the band structure in Fig. 5, it is clear that the spins up and down diagrams in all three structures are completely asymmetric with respect to the horizontal line, so that all the three crystals have magnetic behavior.

Fig. 5.

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	Fig. 5. Band structure for (a) pure SrFe12O19, (b) SrGdFe11O19-I, and (c) SrGdFe11O19-II.

One of the problems with magnetic materials, especially the Heuslers, is that their half-metallic properties are often disrupted or severely reduced.^[45–47] One way to solve this problem, and the best one up to now, is to look for a polarization state of 100%, or in other words, the magnetic moment of the whole crystal being an integer. In Table 2, we summarize the total and partial magnetic moments of the constituent atoms for all the three structures mentioned above using the PBE-GGA functional. In the pure crystal SrFe₁₂O₁₉, the total magnetic moment is 64μ_B, which is a large integer number. Therefore, strontium hexaferrite is a hard magnetic material with a high magnetization intensity. Observing the partial magnetic moment of the atoms, it is expected that the Fe atom has the greatest influence on the crystal magnetization, and this is due to the half-filled d orbitals of this element, while the contribution of Sr and O is nearly negligible. The atoms at the positions 2 and 6 have less magnetic moments than the three Fe atoms. We therefore perform the substitution action of the Gd atom once for the Fe atoms in position 2 and once for position 3. In the SrGdFe₁₁O₁₉-I crystal, according to Table 2, we obtain a total magnetic moment of 71.68μ_B which is higher than the pure state, i.e., the magnetic moment in this crystal is increased relative to the pure state. The results indicate that the Gd atom with 6.86μ_B has the major effect on intensifying the magnetization of the entire crystal. However, the moment of Gd atom is less than that of its isolated state, which is because of hybridization between the orbitals of adjacent atoms when Gd is placed in the crystal. The remarkable point about this structure is that the total magnetic moment is not an integer, which can affect the magnetic stability of this material and may decrease its intensity over time.

By substituting the Gd atom in the atomic position 3 with two symmetric Fe atoms, the structure of SrGdFe₁₁O₁₉-II is formed. As can be seen in Table 2, the total magnetic moment obtained for this structure is 68μ_B which is higher than the pure state but lower than the SrGdFe₁₁O₁₉-I. The remarkable point about this structure is that the total magnetic moment is an integer which can be very useful, because the doped crystal is a hard magnetic material and its magnetic intensity has also increased. We should note that, the magnetic properties of these structures change drastically by applying the Hubbard parameter. In SrFe₁₂O₁₉, the magnetic moment is increased steadily from 64.00μ_B to 120.0μ_B, which is in good agreement with a previous report.^[48] However, similar to the pure state from a magnetic point of view, by introducing the Gd atom, the magnetic intensity increases to 124.0μ_B for both SrGdFe₁₁O₁₉-I and SrGdFe₁₁O₁₉-II compounds.

For a closer look at the electronic and magnetic properties of these structures, the electronic density of states graph of the atoms forming these crystals, which specifies the bond type and the contribution of the atoms to the total DOS diagram, can be very beneficial. Due to the more interesting and practical magnetic properties of SrGdFe₁₁O₁₉-II than the other structure, the DOS diagrams of the atoms of this crystal are shown in Fig. 6 for the two spins up and down using the PBE-GGA functional.

Fig. 6.

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	Fig. 6. PDOS of the SrGdFe11O19-II structure.

Table 2.

Table 2.

Table 2. Atomic magnetic moments (AMMs) of the atoms, together with the total magnetic moment (TMM) of SrFe12O19, SrGdFe11O19-I, and SrGdFe11O19-II. . SrFe11O19/μB SrGdFe11O19-I/μB SrGdFe11O19-II/μB TMM 64.00 71.68 68.00 AMM-atom 1 Sr 0.00250 Sr 0.00677 Sr 0.01312 AMM-atom 2 Fe 1.33087 Gd 6.86616 Fe 1.26978 AMM-atom 3 Fe 3.65864 Fe 3.19686 Gd 6.82858 AMM-atom 4 Fe 3.37430 Fe 3.27279 Fe 3.29875 AMM-atom 5 Fe 3.57487 Fe 3.10673 Fe 3.44749 AMM-atom 6 Fe 1.35474 Fe 1.33457 Fe 1.32777 AMM-atom 7 O 0.05646 O 0.02623 O 0.01662 AMM-atom 8 O 0.11468 O 0.12319 O 0.10521 AMM-atom 9 O 0.32360 O 0.19861 O 0.16119 AMM-atom 10 O 0.09990 O 0.12767 O 0.09267 AMM-atom 11 O 0.11669 O 0.10424 O 0.09867

Table 2. Atomic magnetic moments (AMMs) of the atoms, together with the total magnetic moment (TMM) of SrFe12O19, SrGdFe11O19-I, and SrGdFe11O19-II. .

As in the figure, the major effect of the Gd atom is in the valence region with the energy of −2.7 eV to −4.2 eV. Around the Fermi surface, the iron atom contribution is higher than the other elements. Also, due to the strong differences in the peaks of the density of states and the non-complete overlap, the bond type in this material is ionic. Unlike the spin up state, the maximum impact of the Gd atom in the spin down channel is in the conduction region with the energy range of 1.5 eV to 2.7 eV; and this has intensified the magnetic state of the crystal. However, at the Fermi level, the dominant state still belongs to the Fe atoms.

The more asymmetric diagram of an atom at the spins up and down corresponds to more effects on the magnetization of that material. As can be seen, the most asymmetric cases are Gd and Fe, so they have the most influence on the total crystal magnetization. The two DOS diagrams of the two atoms Sr and O are more symmetrical, and as observed in the magnetic moment of these atoms, their effect on the magnetization of the crystal is much less.

The structural properties, thermodynamic stability, and electronic and magnetic properties of SrFe₁₂O₁₉ hexa-ferrite are first studied at pure state using the PBE-GGA and GGA+U functionals. The E–V diagram shows that this structure is thermodynamically quite stable. The examination of the DOS and the band structure diagrams with PBE-GGA reveals that the crystal is a magnetic half metal with a total magnetic moment of 64μ_B. The partial magnetic moments of the crystal atoms show that the Fe atoms have the most influence on the crystal magnetization. The Fe atoms demonstrate different magnetization and are divided into two groups from the point of view of magnetization intensity, which we have labeled as the positions 2 and 3, with the position 3 having a much higher magnetization. In the next step in the two separate structures, instead of Fe atoms, we have replaced Gd from the lanthanide family into the positions 2 and 3. In the SrGdFe₁₁O₁₉-I structure where the substituted atom is in position 2, the magnetic moment is significantly increased but it is non-integer. From the electronic point of view, the material becomes a magnetic metal. However, in the SrGdFe₁₁O-II crystal, where the Gd atom is in the position 3, an integer magnetic moment of 68μ_B is obtained which is larger than the pure state. Hence, this structure can be very applicable in the magnetic memory industry, since, beside magnetic stability, it has a higher magnetic moment than the pure state. By investigating the electronic structure, SrGdFe₁₁O₁₉-II is found to be a half metal, with conducting and semiconducting behavior at spins down and up, respectively, albeit with a lower energy band gap compared to the pure case. By obtaining the magnetic moment as well as the DOS diagram of the atoms constituting the structure of SrGdFe₁₁O₁₉-II, it is found that the major effect on the magnetization intensity is related to the Gd and Fe atoms respectively. Utilizing the Hubbard parameter in these calculations, all the three compounds become magnetic semiconductors. However, the spin down gap is reduced by doping the Gd atom. The magnetic moment sharply increases and is obtained as 120μ_B and 124μ_B for the pure state and the other two structures respectively. Thus, the increase of the magnetic moment with doping the Gd atom is confirmed using both PBE-GGA and GGA+U functionals.