We use first-principles total-energy calculations based on density functional theory to study the site occupancy and magnetic properties of Al-substituted M-type strontium hexaferrite SrFe12−xAlxO19 with x = 0.5 and x = 1.0. We find that the non-magnetic Al3+ ions preferentially replace Fe3+ ions at two of the majority spin sites, 2a and 12k, eliminating their positive contribution to the total magnetization causing the saturation magnetization Ms to be reduced as Al concentration x is increased. Our formation probability analysis further provides the explanation for increased magnetic anisotropy field when the fraction of Al is increased. Although Al3+ ions preferentially occupy the 2a sites at a low temperature, the occupation probability of the 12k site increases with the rise of the temperature. At a typical annealing temperature (>700 °C) Al3+ ions are much more likely to occupy the 12k site than the 2a site. Although this causes the magnetocrystalline anisotropy K1 to be reduced slightly, the reduction in Ms is much more significant. Their combined effect causes the anisotropy field Ha to increase as the fraction of Al is increased, consistent with recent experimental measurements.

Strontium hexaferrite, SrFe12O19 (SFO), is one of the most commonly used materials for permanent magnets, magnetic recording and data storage, and components in electrical devices operating at microwave/GHz frequencies, due to its high Curie temperature, large saturation magnetization, excellent chemical stability, and low manufacturing cost.1–5 However, in comparison with Nd-Fe-B and magnet, the coercivity of the SFO is low and presents a significant limitation in its application. Therefore, enhancement of the coercivity is an important research topic for the strontium hexaferrite.

In order to tailor the magnetic properties such as magnetization and coercivity, various cation substitutions in the M-type hexaferrites have been investigated. For example, the substitution of La,6,7 Sm,8 Pr,9 and Nd10 in the SFO increased coercivity moderately while the substitution of Zn-Nb,11 Zn-Sn,12–14 and Sn-Mg4 decreased coercivity. However, the coercivity of the M-type hexaferrites is not increased significantly by these cation substitutions and is still much smaller than that of Nd-Fe-B magnet.15 

Al substitution in the M-type hexaferrite has been more effective in enhancing coercivity.16–20 Particularly, Wang et al. synthesized Al-doped SFO SrFe12−xAlxO19 (SFAO) with Al content of x = 0–4 using glycin-nitrate method and subsequent annealing in a temperature over 700 °C obtaining the largest coercivity of 17.570 kOe, which is much larger than that of SFO (5.356 kOe) and exceeds even the coercivity of the Nd2Fe17B (15.072 kOe).1 Wang and co-workers also observed that the coercivity of the SFAO increases with increasing Al concentration at a fixed annealing temperature. These results call for a systematic understanding, from first principles, of why certain combinations of dopants lead to particular results. This theoretical understanding will be essential in systematically tailoring the properties of SFO.

There have been several previous first-principles investigations of SFO. Fang et al. investigated the electronic structure of SFO using density-functional theory (DFT).21 Park et al. have calculated the exchange interaction of SFO from the differences of the total energy of different collinear spin configurations.22 In spite of the importance of substituted SFO, only a few theoretical investigations have been done. Magnetism in La substituted SFO has been studied using DFT.23,24 The site occupancy and magnetic properties of Zn-Sn substituted SFO has been investigated.14 

In this work, we use first-principles total-energy calculations to study the site occupation and magnetic properties of Al substituted M-type strontium hexaferrite SrFe12−xAlxO19 with x = 0.5 and x = 1.0. Based on DFT calculations, we determine the structure of various configurations of SFAO with different Al concentrations and compute the occupation probabilities for different substitution sites at elevated temperatures. We show that our model predicts a decrease of saturation magnetization as well as a decrease in magnetocrystalline anisotropy K1, and the increase of the anisotropy field Ha as the fraction of Al is increased, consistent with recent experimental measurements.

SFO has a hexagonal magnetoplumbite crystal structure that belongs to P63/mmc space group. Fig. 1 shows a unit cell of SFO used in the present work that contains 64 atoms of two formula units. Magnetism in SFO arises from Fe3+ ions occupying five crystallographically inequivalent sites in the unit cell, three octahedral sites (2a, 12k, and 4f2), one tetrahedral site (4f1), and one trigonal bipyramidal site (2b) as represented by the polyhedra in Fig. 1(a). SFO is also a ferrimagnetic material that has 16 Fe3+ ions with spins in the majority direction (2a, 2b, and 12k sites) and 8 Fe3+ ions with spins in the minority direction (4f1 and 4f2 sites) as indicated by the arrows in Fig. 1(b).

FIG. 1.

(a) A unit cell of SFO containing two formula units. Two large gold spheres are Sr atoms and small gray spheres are O atoms. Colored spheres enclosed by polyhedra formed by O atoms represent Fe3+ ions in different inequivalent sites: 2a (blue), 2b (cyan), 12k (purple), 4f1 (green), and 4f2 (red). (b) A schematic diagram of the lowest-energy spin configuration of Fe3+ ions of SFO. The arrows represent the local magnetic moment at each atomic site. (For interpretation of the references to color in this figure caption, the reader is referred to the online version of this paper.)

FIG. 1.

(a) A unit cell of SFO containing two formula units. Two large gold spheres are Sr atoms and small gray spheres are O atoms. Colored spheres enclosed by polyhedra formed by O atoms represent Fe3+ ions in different inequivalent sites: 2a (blue), 2b (cyan), 12k (purple), 4f1 (green), and 4f2 (red). (b) A schematic diagram of the lowest-energy spin configuration of Fe3+ ions of SFO. The arrows represent the local magnetic moment at each atomic site. (For interpretation of the references to color in this figure caption, the reader is referred to the online version of this paper.)

Close modal

Total energies and forces were calculated using DFT with projector augmented wave (PAW) potentials as implemented in VASP.25,26 All calculations were spin polarized according to the ferrimagnetic ordering of Fe spins as first proposed by Gorter.21,27 A plane-wave energy cutoff of 520 eV was used both for pure SFO and Al-substituted SFO. Reciprocal space was sampled with a 7 × 7 × 1 Monkhorst-Pack mesh.28 with a Fermi-level smearing of 0.2 eV applied through the Methfessel-Paxton method.29 We performed relaxation of the electronic degrees of freedom until the change in free energy and the band structure energy was less than 10−7 eV. We performed geometric optimization to relax the positions of ions and cell shape until the change in total energy between two ionic step was less than 10−4 eV. Electron exchange and correlation was treated with the generalized gradient approximation (GGA) as parameterized by the Perdew-Burke-Ernzerhof (PBE) scheme.30 To improve the description of localized Fe 3d electrons, we employed the GGA + U method in the simplified rotationally invariant approach described by Dudarev et al.31 This method requires an effective U value (Ueff) equal to the difference between the Hubbard parameter U and the exchange parameter J. We chose Ueff equal to 3.7 eV for Fe based on the previous result.14 

The substitution of Fe3+ ions by Al3+ ions considerably affects the unit cell parameters. We have calculated the lattice parameters of pure and Al-substituted SFO by relaxing ionic positions as well as the volume and shape of the unit cell. In all cases, the final unit cell was found to remain hexagonal. In the case of pure SFO, the lattice parameters a and c were found to be 5.93 Å and 23.21 Å in good agreement with the experimental values of a = 5.88 Å and c = 23.04 Å, respectively;19,32 the deviation between the experimental and the theoretical values is less than 1%. In the case of x = 0.5 in SrFe12−xAlxO19, the lattice parameters a and c were calculated to be 5.92 Å and 23.16 Å, respectively, while the volume of the unit cell was reduced by 0.61%. For x = 1.0, a = 5.91 Å and c = 23.04 Å were found, and reduction in the unit cell volume was 2.51%. Fig. 2 shows that the reduction of unit cell volume predicted by our DFT calculation is consistent with the experimental results.1,19

FIG. 2.

Comparison of calculated and experimental (Ref. 19) volume of the unit cell of SrFe12−xAlxO19 as a function of the fraction of Al x.

FIG. 2.

Comparison of calculated and experimental (Ref. 19) volume of the unit cell of SrFe12−xAlxO19 as a function of the fraction of Al x.

Close modal

We investigated the site preference of Al substituting Fe in SrFe12−xAlxO19 for (i) x = 0.5 and (ii) x = 1.0. The x = 0.5 case corresponds to the condition where one Al atom is substituted in the unit cell, while two Al atoms were substituted in the case of x = 1.0, as shown in Fig. 3. To determine the site preference of the substituted Al atoms, the substitution energy of configuration i was calculated using the following expression:

Esub(i)=E(SFAO(i))E(SFO)αnαϵ(α),
(1)

where E(SFAO(i)) is the total energy per unit cell at 0 K for SFAO in configuration i while E(SFO) is the total energy per unit cell at 0 K for SFO. ϵ(α) is the total energy per atom for element α (α = Al, Fe) at 0 K in its most stable crystal structure. nα is the number of atoms of type α added or removed: if two atoms are added, then nα = +2 while nα = −1 when one atom is removed. The configuration with the lowest Esub is concluded to be the ground state configuration, and the corresponding substitution site is the preferred site of Al atoms at 0 K.

FIG. 3.

The structures of SrFe12−xAlxO19 with spins oriented in the easy axis (001): (a) configuration [2a] for x = 0.5 and (b) configuration [2a, 12k].1 for x = 1.0. Al atoms are labeled and other atoms are colored as in Fig. 1.

FIG. 3.

The structures of SrFe12−xAlxO19 with spins oriented in the easy axis (001): (a) configuration [2a] for x = 0.5 and (b) configuration [2a, 12k].1 for x = 1.0. Al atoms are labeled and other atoms are colored as in Fig. 1.

Close modal

To understand the site preference of the substituted Al3+ ions at higher temperatures, we compute the formation probability of configuration i using the Maxwell-Boltzmann statistical distribution33 

Pi=giexp(ΔGi/kBT)jgjexp(ΔGj/kBT),
(2)

where gi is the multiplicity of configuration i (number of equivalent configurations) and

ΔGi=ΔEsub(i)+PΔViTΔSi
(3)

is the change of the free energy of configuration i relative to that of the ground state configuration; ΔEsub (i), ΔVi, and ΔSi are the substitution energy change, volume change, entropy change for configuration i; P and kB are the pressure and Boltzmann constant.

For the x = 0.5 concentration, one Al atom is substituted at one of the 24 Fe sites of the unit cell, as shown in Fig. 3(a). The application of crystallographic symmetry operations shows that many of these Fe sites are equivalent and leaves only five inequivalent structures. We label these inequivalent configurations using the crystallographic name of the Fe site: [2a], [2b], [4f1], [4f2], and [12k]. These structures were created by substituting one Al atom to the respective Fe site of a SFO unit cell and performing full optimization of the unit cell shape and volume, and ionic positions.

Table I lists the results of our calculation for all five inequivalent configurations in the order of increasing substitution energy. The lowest Esub is found for configuration [2a] shown in Fig. 1(a). We can conclude that at 0 K, the most preferred site for the substituted Al atom is the 2a site. We used Eq. (2) to compute the probability to form each configuration as a function of temperature. Since the volume change among different configurations is very small (less than 0.1 Å3), we can safely regard PΔV term to be negligible (in the order of 10−7 eV at the standard pressure of 1 atm) compared to the ΔEsub (i) term in Eq. (3). The entropy change ΔS has a configurational part, ΔSc, and a vibrational part, ΔSvib.34 For binary substitutional alloys such as the present system, ΔSvib is around 0.1–0.2 kB/atom, and ΔSc is 0.1732 kB/atom.33 Therefore, we set ΔS = 0.3732 kB/atom.

TABLE I.

Five inequivalent configurations of SrFe12−xAlxO19 with x = 0.5. g is the multiplicity of the configuration. Esub is the substitution energy of the SFAO. The total magnetic moment (mtot) and its change with respect to SFO (Δmtot) are also given. All values are for a double formula unit cell containing 64 atoms. Energies are in eV while magnetic moments are in μB.

ConfigurationsgEsubmtotΔmtot
[2a−6.04 35 −5 
[12k12 −6.00 35 −5 
[4f2−5.63 45 +5 
[2b−5.60 35 −5 
[4f1−5.57 45 +5 
ConfigurationsgEsubmtotΔmtot
[2a−6.04 35 −5 
[12k12 −6.00 35 −5 
[4f2−5.63 45 +5 
[2b−5.60 35 −5 
[4f1−5.57 45 +5 

Fig. 4 displays the temperature dependence of the formation probability of different configurations of SrFe12−xAlxO19 with x = 0.5. The doped Al3+ ions mainly replace Fe3+ ions from the 2a and the 12k sites. The formation probabilities of [2b], [4f1], and [4f2] are negligible and not shown in Fig. 4. The probability that the doped Al3+ ion replaces Fe3+ ion from the 2a site is maximum at 0 K and it falls as temperature increases, while the occupancy of Al3+ at the 12 k site rises with temperature. The two curves cross at T ∼ 220 K. At a typical annealing temperature of 1000 K for SFAO,1 the site occupation probability of the site 2a and 12k is 0.196 and 0.798, respectively. Thus, during the annealing process of the synthesis of the SFAO, the doped Al3+ ions are more likely to replace Fe3+ ions from the 12k site than the 2a site despite higher substitution energy.

FIG. 4.

Temperature dependence of the formation probability of different configurations of SrFe12−xAlxO19 with x = 0.5. The configurations with negligible probability are not shown.

FIG. 4.

Temperature dependence of the formation probability of different configurations of SrFe12−xAlxO19 with x = 0.5. The configurations with negligible probability are not shown.

Close modal

For the x = 1.0 concentration, two Al atoms are substituted at two of the 24 Fe sites of the unit cell, as shown in Fig. 3(b). These Fe sites have more than one equivalent site. Substitution of Al atoms breaks the symmetry of the equivalent sites of pure SFO. Out of all C(24, 2) = 276 possible structures, many of the structures are crystallographically equivalent. On applying crystallographic symmetry operations, the number of inequivalent structures reduces to 40. We label these inequivalent configurations using the convention of [(site for the first Al), (site for the 2nd Al)] (unique index). For example, when two Al atoms are substituted at the 2a and 12k sites, there are 2 inequivalent configurations, which are labeled as [2a, 12k].1 and [2a, 12k].2. These structures are fully optimized, and their substitution energies are calculated using Eq. (1). When there is more than one inequivalent configuration, we assign the unique index in the order of increasing Esub.

Table II lists the ten lowest energy configurations of SrFe12−xAlxO19 with x = 1.0. The configuration [2a, 2a] where two Al3+ ions replace Fe3+ ions from two 2a sites has the lowest Esub, and it is the most energetically favorable configuration at 0 K. To investigate the site occupation at nonzero temperatures, we compute the formation probability of each configuration using Eq. (2). Similar to the previous case, the volume change among different configurations is very small (less than 0.7 Å3) and we can safely ignore the PΔV term. The entropy term is calculated in the same way as the x = 0.5 case. Fig. 5 shows the variation of the formation probability of different configurations with temperature. We note that due to low multiplicity of the configuration [2a, 2a], its formation probability falls rapidly as temperature increases. On the other hand, the formation probability of the configuration [2a, 12k] (sum of the formation probabilities for all [2a, 12k].n configurations) increases steeply and reaches a maximum value at 50 K and then falls with temperature. Fig. 5 shows that the formation probability of the [2a, 12k] configuration becomes larger than that of [2a, 2a] beyond T ∼ 10 K, which is a much lower transition temperature than in the x = 0.5 case.

TABLE II.

Ten lowest energy inequivalent configurations of SrFe12−xAlxO19 with x = 1.0. g is the multiplicity of the configuration. Esub is the substitution energy per Al atom. The total magnetic moment (mtot) and its change with respect to SFO (Δmtot) are also given. All values are for a double formula unit cell containing 64 atoms. Energies are in eV while moments are in μB.

ConfigurationsgEsubmtotΔmtot
[2a, 2a−6.056 30 −10 
[2a, 12k].1 12 −6.054 30 −10 
[2a, 12k].2 12 −6.041 30 −10 
[12k, 12k].1 −6.025 30 −10 
[12k, 12k].2 12 −6.025 30 −10 
[12k, 12k].3 12 −6.027 30 −10 
[12k, 12k].4 12 −6.025 30 −10 
[12k, 12k].5 −6.023 30 −10 
[12k, 12k].6 −6.017 30 −10 
[12k, 12k].7 12 −6.014 30 −10 
ConfigurationsgEsubmtotΔmtot
[2a, 2a−6.056 30 −10 
[2a, 12k].1 12 −6.054 30 −10 
[2a, 12k].2 12 −6.041 30 −10 
[12k, 12k].1 −6.025 30 −10 
[12k, 12k].2 12 −6.025 30 −10 
[12k, 12k].3 12 −6.027 30 −10 
[12k, 12k].4 12 −6.025 30 −10 
[12k, 12k].5 −6.023 30 −10 
[12k, 12k].6 −6.017 30 −10 
[12k, 12k].7 12 −6.014 30 −10 
FIG. 5.

Temperature dependence of the formation probability of different configurations of SrFe12−xAlxO19 with x = 1.0. For clarity only the configurations with significant formation probability are labeled.

FIG. 5.

Temperature dependence of the formation probability of different configurations of SrFe12−xAlxO19 with x = 1.0. For clarity only the configurations with significant formation probability are labeled.

Close modal

We can calculate the occupation probability of Al at nonzero temperatures for a given site by adding all formation probabilities of the configurations where at least one Al3+ ion is substituted in that site. At the annealing temperature of 1000 K, the occupation probability of Al for 12k site is 79.8% for x = 0.5 as given in Table IV. The same probability is increased to 97.7% for x = 1.0 as calculated by adding the P1000's for all configurations that contain the 12k site. This means that the fraction of Al3 + ions occupying the 12k site increases when the fraction of Al is increased from x = 0.5 to x = 1.0. This conclusion is in agreement with the previously reported measurements.1,16,35

TABLE IV.

The saturation magnetization (Ms), magnetocrystalline anisotropy energy (MAE), magnetocrystalline anisotropy constant (K1), and anisotropy field (Ha) for SFO and SFAO for a temperature of 0 K. x is the Al fraction in SrFe12−xAlxO19 and V is the volume of the unit cell in Å3. P1000 is the formation probability at 1000 K. The averaged quantities are weighted by P1000. Ms is in emu/g, MAE in meV, Ha in kOe, and K1 in kJ m−3.

xConfigurationsMsMAEVK1HaP1000MsK1Ha
0.0 SFO 110.19 0.85 707.29 193 7.35 1.000 110.19 193 7.35 
0.5 [2a93.33 0.95 703.29 216 9.38 0.196 93.29 189 8.18 
 [12k93.33 0.80 703.19 182 7.90 0.798    
 [2b93.33 0.67 702.82 152 6.62 0.003    
 [4f1119.99 0.86 704.22 196 6.61 0.001    
 [4f2119.99 0.83 702.58 189 6.38 0.001    
1.0 [2a, 2a81.11 0.99 698.94 227 11.41 0.019 81.49 184 9.23 
 [2a, 12k81.11 0.88 699.08 202 10.13 0.379    
 [12k, 12k81.11 0.75 698.66 172 8.64 0.585    
 [12k, 4f2108.15 0.78 690.64 181 6.74 0.007    
 [12k, 4f1108.15 0.80 700.29 183 6.92 0.004    
 [12k, 2b81.11 0.62 698.98 142 7.14 0.002    
 [4f2, 4f2135.19 0.80 697.96 184 5.53 0.000    
 [4f2, 4f1135.19 0.83 699.62 191 5.74 0.000    
 [4f2, 2b108.15 0.60 698.82 138 5.19 0.000    
 [4f2, 2a108.15 0.90 698.53 206 7.78 0.002    
 [4f1, 4f1135.19 0.86 701.38 196 5.95 0.000    
 [4f1, 2b108.15 0.65 699.95 149 5.62 0.000    
 [4f1, 2a108.15 0.91 700.11 208 7.87 0.001    
 [2b, 2b81.11 0.45 698.86 103 5.19 0.000    
 [2b, 2a81.11 0.74 698.82 170 8.53 0.001    
xConfigurationsMsMAEVK1HaP1000MsK1Ha
0.0 SFO 110.19 0.85 707.29 193 7.35 1.000 110.19 193 7.35 
0.5 [2a93.33 0.95 703.29 216 9.38 0.196 93.29 189 8.18 
 [12k93.33 0.80 703.19 182 7.90 0.798    
 [2b93.33 0.67 702.82 152 6.62 0.003    
 [4f1119.99 0.86 704.22 196 6.61 0.001    
 [4f2119.99 0.83 702.58 189 6.38 0.001    
1.0 [2a, 2a81.11 0.99 698.94 227 11.41 0.019 81.49 184 9.23 
 [2a, 12k81.11 0.88 699.08 202 10.13 0.379    
 [12k, 12k81.11 0.75 698.66 172 8.64 0.585    
 [12k, 4f2108.15 0.78 690.64 181 6.74 0.007    
 [12k, 4f1108.15 0.80 700.29 183 6.92 0.004    
 [12k, 2b81.11 0.62 698.98 142 7.14 0.002    
 [4f2, 4f2135.19 0.80 697.96 184 5.53 0.000    
 [4f2, 4f1135.19 0.83 699.62 191 5.74 0.000    
 [4f2, 2b108.15 0.60 698.82 138 5.19 0.000    
 [4f2, 2a108.15 0.90 698.53 206 7.78 0.002    
 [4f1, 4f1135.19 0.86 701.38 196 5.95 0.000    
 [4f1, 2b108.15 0.65 699.95 149 5.62 0.000    
 [4f1, 2a108.15 0.91 700.11 208 7.87 0.001    
 [2b, 2b81.11 0.45 698.86 103 5.19 0.000    
 [2b, 2a81.11 0.74 698.82 170 8.53 0.001    

In Table III, we compare the contribution of different sublattices to the total magnetic moment in Al-substituted SFO. To see the effect of Al3+ ions in different substitution sites, we split the entries of sublattices containing these ions (2a and 12k). As expected, Al3+ ions carry negligible magnetic moment regardless of their substitution sites. Consequently, when they replace Fe3+ ions in the minority spin sites (4f1 and 4f2), they eliminate a negative contribution and hence increase the total magnetic moment. On the other hand, when they replace Fe3+ ions in the majority spin sites (12k, 2a, and 2b), they eliminate a positive contribution and hence reduce the total magnetic moment. For the x = 0.5 case, the most probable sites are 12k and 2a (majority sites) and the net magnetic moment of the unit cell is reduced by 5 μB. For the configuration [2a, 12k].1 of the x = 1.0 case, two Al atoms are substituted in the 2a and 12k sites, there is a reduction of 10 μB in the total magnetic moment per unit cell.

TABLE III.

Contribution of atoms in each sublattice to the total magnetic moment of Al-substituted SFO structures [12k], [2a], and [2a, 12k].1 compared with pure SFO. All magnetic moments are in μB. Δm is measured relative to the values for the pure SFO. Note that the total magnetic moment of the unit cell (mtot) is slightly different than the sum of local magnetic moments due to the contribution from the interstitial region.

siteSFO[12k][2a][2a, 12k].1
AtomsMatomsMΔmatomsmΔmatomsmΔm
2d 2 Sr −0.006 2 Sr −0.006 0.000 2 Sr −0.006 0.000 2 Sr −0.006 0.000 
2a 1 Fe 4.156 1 Fe 4.155 −0.001 1 Al −0.010 −4.166 1 Al −0.010 −4.166 
 1 Fe 4.156 1 Fe 4.156 0.000 1 Fe 4.156 0.000 1 Fe 4.156 0.000 
2b 2 Fe 8.098 2 Fe 8.086 −0.012 2 Fe 8.100 0.001 2 Fe 8.090 −0.008 
4f1 4 Fe −16.152 4 Fe −16.189 −0.037 4 Fe −16.268 −0.116 4 Fe −16.304 −0.152 
4f2 4 Fe −16.384 4 Fe −16.420 −0.036 4 Fe −16.382 0.002 4 Fe −16.418 −0.034 
12k 1 Fe 4.172 1 Al 0.000 −4.172 1 Fe 4.170 −0.002 1 Al −0.001 −4.173 
 11 Fe 45.884 11 Fe 45.861 −0.023 11 Fe 45.872 −0.012 11 Fe 45.846 −0.038 
4e 4 O 1.416 4 O 1.304 −0.112 4 O 1.424 0.008 4 O 1.316 −0.100 
4f 4 O 0.360 4 O 0.281 −0.079 4 O 0.310 −0.050 4 O 0.230 −0.129 
6h 6 O 0.124 6 O 0.115 0.009 6 O 0.134 0.010 6 O 0.117 −0.007 
12k 12 O 1.016 12 O 0.877 −0.129 12 O 0.548 −0.468 12 O 0.404 −0.612 
12k 12 O 2.140 12 O 1.895 −0.245 12 O 2.088 −0.052 12 O 1.839 −0.301 
m  38.980  34.114 −4.837  34.136 −4.845  29.259 −9.720 
mtot  40  35 −5  35 −5  30 −10 
siteSFO[12k][2a][2a, 12k].1
AtomsMatomsMΔmatomsmΔmatomsmΔm
2d 2 Sr −0.006 2 Sr −0.006 0.000 2 Sr −0.006 0.000 2 Sr −0.006 0.000 
2a 1 Fe 4.156 1 Fe 4.155 −0.001 1 Al −0.010 −4.166 1 Al −0.010 −4.166 
 1 Fe 4.156 1 Fe 4.156 0.000 1 Fe 4.156 0.000 1 Fe 4.156 0.000 
2b 2 Fe 8.098 2 Fe 8.086 −0.012 2 Fe 8.100 0.001 2 Fe 8.090 −0.008 
4f1 4 Fe −16.152 4 Fe −16.189 −0.037 4 Fe −16.268 −0.116 4 Fe −16.304 −0.152 
4f2 4 Fe −16.384 4 Fe −16.420 −0.036 4 Fe −16.382 0.002 4 Fe −16.418 −0.034 
12k 1 Fe 4.172 1 Al 0.000 −4.172 1 Fe 4.170 −0.002 1 Al −0.001 −4.173 
 11 Fe 45.884 11 Fe 45.861 −0.023 11 Fe 45.872 −0.012 11 Fe 45.846 −0.038 
4e 4 O 1.416 4 O 1.304 −0.112 4 O 1.424 0.008 4 O 1.316 −0.100 
4f 4 O 0.360 4 O 0.281 −0.079 4 O 0.310 −0.050 4 O 0.230 −0.129 
6h 6 O 0.124 6 O 0.115 0.009 6 O 0.134 0.010 6 O 0.117 −0.007 
12k 12 O 1.016 12 O 0.877 −0.129 12 O 0.548 −0.468 12 O 0.404 −0.612 
12k 12 O 2.140 12 O 1.895 −0.245 12 O 2.088 −0.052 12 O 1.839 −0.301 
m  38.980  34.114 −4.837  34.136 −4.845  29.259 −9.720 
mtot  40  35 −5  35 −5  30 −10 

Magnetic anisotropy determines the capacity of a magnet to withstand external magnetic and electric fields. This property is of considerable practical interest, because anisotropy is exploited in the design of the most magnetic materials of commercial importance. The magnetocrystalline anisotropy energy (MAE) is one of the main factors that determine the total magnetic anisotropy of the material. To investigate the effect of Al substitution on the magnetic anisotropy of SFO, we have computed the MAE and the magnetic anisotropy constant of SrFe12−xAlxO19 for x = 0, 0.5, and 1. The MAE, in the present case, is defined as the difference between the two total energies where electron spins are aligned along two different directions37 

EMAE=E(100)E(001),
(4)

where E(100) is the total energy with spin quantization axis in the magnetically hard plane and E(001) is the total energy with spin quantization axis in the magnetically easy axis. Using the MAE, the uniaxial magnetic anisotropy constant K1 can be computed38,39

K1=EMAEVsin2θ,
(5)

where V is the equilibrium volume of the unit cell and θ is the angle between the two spin quantization axis orientations (90° in the present case). The anisotropy field Ha can be expressed as40 

Ha=2K1Ms,
(6)

where K1 is a magnetocrystalline anisotropy constant and Ms is saturation magnetization.

The results for the MAE, the magnetocrystalline anisotropy constant K1, and anisotropy field Ha for SFAO with different Al concentration are presented in Table IV. To compare with experimental results, we also compute the weighted average of K1 and Ha using the formation probability P1000 at a typical annealing temperature of 1000 K.1 We note that SFAO considered in the present work loses most of its magnetic properties at typical annealing temperatures (1000 K or higher) that are near or above its Curie temperature. The magnetic properties listed in Table IV refer to their ground state properties at the temperature T = 0 K. We use the formation probability at 1000 K to compute the weighted averages as the crystalline configurations of SFAO will be distributed according to this value during the annealing process.

Table IV shows that Ms decreases as the concentration of Al x is increased from 0 to 0.5 to 1.0, consistent with the previous experimental results.1,20,36,41 Our calculation also shows that K1 decreases as the concentration of Al x is increased from 0 to 0.5 to 1.0. At a low temperature Al atoms prefer to occupy the 2a sites, which would have increased K1 (see K1 values for [2a] and [2a, 2a] in Table IV). However, the formation probability of the configurations involving 12k site (such as [12k], [2a, 12k], and [12k, 12k]) increases significantly as the temperature rises due to the entropy contribution of the free energy. At the annealing temperature, Al3+ ions are much more likely to occupy the 12k site than the 2a site. This causes the magnetocrystalline anisotropy constant K1 of Al-substituted SFO to be reduced with the increase of Al fraction x, consistent with the experimental measurement reported by Albanese.36 Despite this, Ms is reduced more significantly than K1 and this causes the anisotropy field Ha in Eq. (6) to increase as the concentration of Al x is increased from 0 to 0.5 to 1.0, as shown in Table IV. We compare the calculated values of Ms, K1, and Ha for SFO and SFAO with the experimental values19,20,36 in Table V. The changes in these three magnetic properties as the Al fraction x is increased are consistent with those of the measured values. We note that the calculated values are for 0 K while the experimental values are measured at the room temperature. The result is also consistent with several other experimental results.1,41

TABLE V.

Comparison of calculated and experimental magnetic properties of SFO and SFAO. The calculated values are for 0 K while the experimental values are measured at the room temperature. The saturation magnetization (Ms), magnetocrystalline anisotropy constant (K1), and anisotropy field (Ha) for different Al fraction x in SrFe12−xAlxO19 are given. Relative difference w.r.t. the pure SFO values are also given in the parentheses.

xMs (emu/g)K1 (kJ m−3)Ha (kOe)
CalculatedExpt.aCalculatedExpt.bCalculatedExpt.c
0.0 110.19  59.33  193  450  7.35  16.9  
0.5 93.30 (−15.3%) 46.68 (−21.3%) 189 (−2.1%) 445 (−1.2%) 8.18 (+11.3%) 19.0 (+12.4%) 
1.0 81.49 (−26.0%) 43.49 (−26.7%) 184 (−4.7%) 436 (−3.2%) 9.23 (+25.6%) 21.8 (+28.7%) 
xMs (emu/g)K1 (kJ m−3)Ha (kOe)
CalculatedExpt.aCalculatedExpt.bCalculatedExpt.c
0.0 110.19  59.33  193  450  7.35  16.9  
0.5 93.30 (−15.3%) 46.68 (−21.3%) 189 (−2.1%) 445 (−1.2%) 8.18 (+11.3%) 19.0 (+12.4%) 
1.0 81.49 (−26.0%) 43.49 (−26.7%) 184 (−4.7%) 436 (−3.2%) 9.23 (+25.6%) 21.8 (+28.7%) 
a

Reference 19.

b

Values taken from Fig. 12 of Ref. 36.

c

Values taken from Fig. 14 of Ref. 20.

Using the first-principles total energy calculations based on density functional theory, we obtained the ground state structures and associated formation probabilities at finite temperatures for Al-substituted SFO, SrFe12−xAlxO19 with x = 0.5 and 1.0. The structures derived from our calculations show that the total magnetic moment of the SFO unit cell is reduced as the fraction of Al atoms increases. This reduction of magnetization is explained by the fact that the non-magnetic Al atoms prefer to replace Fe3+ ions at two of the majority spin sites, 2a and 12k, eliminating their positive contribution to the total magnetization. Our model also explains the increase of the observed anisotropy field when the fraction of Al in SFO is increased. At the annealing temperature, Al3+ ions are much more likely to occupy the 12k site than the 2a site. Although this causes the magnetocrystalline anisotropy to decrease slightly, the reduction in the saturation magnetization is larger and their combined effect causes the magnetic anisotropy field of Al-substituted SFO to be increased with increase of Al fraction x. Our results are consistent with the available experimental measurement on Al-substituted SFO.

This work was supported in part by the U.S. Department of Energy ARPA-E REACT program under Award No. DE-AR0000189 and the Center for Computational Science (CCS) at Mississippi State University. Computer time allocation has been provided by the High Performance Computing Collaboratory (HPC2) at Mississippi State University.

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