Maximum energy product at elevated temperatures for hexagonal strontium ferrite (SrFe12O19) magnet

Journal of Magnetism and Magnetic Materials 355 (2014) 1–6

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Maximum energy product at elevated temperatures for hexagonal strontium ferrite (SrFe12O19) magnet Jihoon Park a, Yang-Ki Hong a,n, Seong-Gon Kim b, Sungho Kim b, Laalitha S.I. Liyanage b, Jaejin Lee a, Woncheol Lee a, Gavin S. Abo a, Kang-Heon Hur c, Sung-Yong An c a

Department of Electrical and Computer Engineering and MINT Center, The University of Alabama, Tuscaloosa, AL 35487, USA Department of Physics & Astronomy and Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39792, USA c Corporate R&D Institute, Samsung Electro-Mechanics Co., Ltd., Suwon-si, Gyeonggi-Do 443-743, South Korea b

art ic l e i nf o

a b s t r a c t

Article history: Received 3 August 2013 Received in revised form 11 November 2013 Available online 22 November 2013

The electronic structure of hexagonal strontium ferrite (SrFe12O19) was calculated based on the density functional theory (DFT) and generalized gradient approximation (GGA). The GGAþU method was used to improve the description of localized Fe 3d electrons. Three different effective U (Ueff) values of 3.7, 7.0, and 10.3 eV were used to calculate three sets of exchange integrals for 21 excited states. We then calculated the temperature dependence of magnetic moments m(T) for the five sublattices (2a, 2b, 12k, 4f1, and 4f2) using the exchange integrals. The m(T) of the five sublattices are inter-related to the nearest neighbors, where the spins are mostly anti-ferromagnetically coupled. The five sublattice m(T) were used to obtain the saturation magnetization Ms(T) of SrFe12O19, which is in good agreement with the experimental values. The temperature dependence of maximum energy product ((BH)max(T)) was calculated using the calculated Ms(T). & 2013 Elsevier B.V. All rights reserved.

Keywords: Strontium ferrite Exchange energy Exchange integral Temperature dependence Sublattice magnetization Maximum energy product

1. Introduction Magnetization of 74 emu/g (377 emu/cm3) and effective anisotropy constant (Keff) of 3.57  106 erg/cm3 at 300 K, and Néel temperature (TN) of 750 K were reported for single crystal hexagonal strontium ferrite (SrFe12O19: SrM) [1]. There have been many attempts to improve the magnetic properties of SrM by substituting foreign cation(s) for its Fe or Sr cation. The substituted cations include: La– Co [2–12], Al [13], Cu–La [14], La–Zn [15,16], Mn–Co–Sn [17], Mn–Co– Ti [18], Mn–Co–Zr [19], Nd–Sm [20], Pb [21], Sn–Zn [22], Zn–Nd [23], Sn–Mg [24], and Er–Ni [25]. The hexagonal strontium ferrite has 24 magnetic Fe3þ ions distributed among five crystallographically distinct magnetic sublattices, namely 2a, 2b, 12k, 4f1, and 4f2. Magnetic moment and magnetocrystalline anisotropy of SrM originate from electronic structure of Fe3 þ ions at the five distinct magnetic sublattices. Therefore, there are 15 possible exchange integrals. Since each magnetic site magnetically behaves differently at a given temperature, temperature dependent magnetic hyperfine field and magnetic moment of each magnetic site have been studied [26,27] to investigate the magnetization of SrM at elevated temperatures. There are two different approaches to determine temperature dependence of magnetization for each sublattice of hexaferrite Sr(or

n

Corresponding author. Tel.: þ 1 205 348 7268. E-mail address: [email protected] (Y.-K. Hong).

0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.11.032

Ba)Fe12O19. One approach is to calculate five sublattice magnetizations from five sublattice magnetic hyperfine fields (MHF), which are determined by Mössbauer spectra [26]. It is shown that 4f1 and 4f2 sublattices have almost the same temperature dependent magnetization behavior. Therefore, the other approach assumes only two sublattices, namely spin-up parallel (2a, 2b, and 12k sites) and spin-down anti-parallel (4f1 and 4f2) lattices, instead of five distinct sublattices [27]. Also, all exchange integral (Jij) were assumed to be identical except for J4f2–12k to calculate sublattice magnetizations using molecular field theory. In this paper, we present the calculation of fifteen exchange integrals for SrFe12O19, based on the density functional theory (DFT), to obtain the temperature dependence of magnetic moment mi(T) of the five distinct sublattices. Then, the five mi(T) were used to estimate temperature dependent maximum energy product, (BH)max(T), at elevated temperatures. Electron exchange and correlation were treated with the generalized gradient approximation (GGA), and GGA þU method was used to improve the description of localized Fe 3d electrons. 2. Magnetic structure of strontium ferrite SrM is a hexagonal ferrimagnetic material with five magnetic sublattices (2a, 2b, 12k, 4f1, and 4f2). There are two formula units of SrFe12O19 in one unit cell. The magnetic moment of SrM originates

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J. Park et al. / Journal of Magnetism and Magnetic Materials 355 (2014) 1–6

Fig. 1. The unit cell (left), spin configurations of Fe3 þ at each layer (middle), and lattice constants and number of Fe3 þ at each sublattice (right) for hexagonal strontium ferrite (SrFe12O19) [28]. The spins at the nearest neighbors are anti-ferromagnetically coupled as shown (middle).

from Fe3 þ ions at each magnetic sublattice. As shown in Fig. 1, the Fe3 þ ions at the five sublattices are anti-ferromagnetically coupled with each other. It appears that the magnetism of the ground state Fe3 þ ion is mainly due to spin moment S (¼5/2), with no orbital contribution L (¼ 0). Therefore, the total angular momentum J (¼ SþL) is 5/2 for the Fe3 þ ion. The magnetic moment of the Fe3 þ ion, therefore, is 5 μB in the ideal case. Since the spins in the 2a, 2b, and 12k sublattices are parallel to each other, but anti-parallel to those of the 4f1 and 4f2 sublattices, the net magnetic moment of one SrFe12O19 unit cell (two formula units) is 40 μB. These interrelated magnetic sublattices to the nearest neighbors lead to strong mutual interdependence of the five sublattices.

Table 1 The five magnetic sublattices (2a, 2b, 12k, 4f1, and 4f2) in the unit cell of SrFe12O19. ni is the number of Fe3 þ ions at ith sublattice in the unit cell, and s(0)i ¼ þ 1 or  1 represents the ground state spin configuration at ith sublattice. Fe sublattice

ni (# of Fe)

s(0)i (↑ or ↓)

2a 2b 12k 4f1 4f2

2 2 12 4 4

þ1 þ1 þ1 1 1

Therefore, the exchange energy can be rewritten as Eex ¼

3. Method of calculations In order to derive temperature dependence of saturation magnetization Ms(T), it is first needed to calculate the exchange integrals. The exchange integrals are obtained by the difference between exchange energies of ground state and all excited states of sublattices. The exchange energy per unit cell in a system with N magnetic sublattices can be defined as Eex ¼

N !! 1 N ∑ ∑ ni zij J ij ð S i S j Þ; 2 i ¼ 1 jai ¼ 1

ð1Þ

where ni is the number of Fe3 þ ions at ith sublattice, zij is the number of neighboring Fe3 þ ions at jth sublattice to ith sublattice, ! Jij is the exchange integrals, and S i is the spin of the Fe3 þ ions at ith sublattice. When the ferrimagnetic spin configurations, i.e. only ! ! up and down, are considered, the S i and S j can be written as !! ðαÞ S i S j ¼ Si Sj sðαÞ i sj ;

ð2Þ

where sðαÞ i ¼ 7 1, and index α is the sublattice spin arrangement.

N 1 N ∑ ∑ n z J S S sðαÞ sðαÞ : 2 i ¼ 1 j a i ¼ 1 i ij ij i j i j

ð3Þ

Now, the exchange integral [29] is written as ð0Þ J ij ¼ ðΔij  Δi  Δj Þ=ð4Si Sj ni zij sð0Þ i sj Þ;

ð4Þ

where Δij is the difference of exchange energies between the excited state at both i and j sublattices and the ground state, Δi is between the excited state at i sublattice and the ground state, and Δj is between the excited state at j sublattice and the ground state. The ni and s(0)i for the five sublattices are given in Table 1, and the number of the nearest neighboring Fe3 þ ions at jth sublattice to ith sublattice and corresponding distance rij are listed in Table 2. In order to calculate the energy difference (Δi , Δj , and Δij ), the first principles calculations were performed using VASP [30] with the relaxed structure of SrFe12O19. The VASP is a package for performing ab initio quantum mechanical molecular dynamics (MD), mainly based on density functional theory (DFT) [31,32]. For improved description of 3d electrons in Fe3 þ ion, generalized gradient approximation with Coulomb and exchange interaction effects (GGAþU) were employed, in which an on-site potential is

J. Park et al. / Journal of Magnetism and Magnetic Materials 355 (2014) 1–6

2a

2b

4f1

4f2

12k

zij

rij

zij

rij

zij

rij

zij

rij

zij

rij

2a 2b 4f1

6 2 3

0.593 0.580 0.348

2 6 3

0.580 0.593 0.621

6 6 3

0.348 0.621 0.365

6 6 1

0.560 0.369 0.380

4f2 12k

3 1

0.560 0.307

3 1

0.369 0.370

1 2 1

0.380 0.353 0.358

1 2

0.275 0.352

6 6 6 3 6 2 2

0.307 0.370 0.353 0.358 0.352 0.293 0.300

added to introduce intra-atomic interactions between the strongly correlated electrons. The energy of the U term [33] is defined as E¼

ðU JÞ ∑ðnm;s  n2m;s Þ; 2 s

ð5Þ

where U is the Coulomb interaction parameter (energy necessary to add an electron to an already occupied localized orbital in an atom), J is the exchange interaction parameter (exchange between doubly occupied sites), and n is the occupation number of a d-orbital number m with spin s. As can be seen from Eq. (5), the total energy depends on the difference between U and J, which is defined as the effective U, Ueff. The Ueff affects the total energy difference between excited and ground states, thereby Jij. The Jij were applied to the Brillouin function to derive m(T) for the five sublattices (2a, 2b, 12k, 4f1, and 4f2), consequently (BH)max.

4. Results 4.1. Exchange integrals (interactions between the five magnetic sublattices) The relaxation for the SrM system gives the lattice constants of a and c of 5.932 and 23.206 Å, respectively. These constants are in good agreement with experimental ones of 5.88 and 23.04 Å [34]. The energy difference Δ between the ground state and excited states was calculated with three different values of Ueff ( ¼3.7, 7.0, and 10.3 eV). All possible excited spin configurations, 21 sets at the five different magnetic sublattices, were considered. The calculated energy difference Δ and magnetic moments per unit cell for each excited state are given in Table 3. It can be seen that the energy difference Δ is smaller for larger Ueff, while the magnetic moment stays almost constant. However, Ueff values larger than 10.3 eV cause unreliable energy and magnetic moment for SrM. The calculated energy difference Δ was then used to obtain exchange integrals using Eq. (4). The input parameters ni, zij, and s(0)i for each excited state were taken from Tables 1 and 2. Fig. 2 shows exchange integrals for all the possible excited states as a function of Ueff. The exchange integrals become smaller as larger Ueff is used. 4.2. Temperature dependence of saturation magnetization Ms(T) and maximum energy product (BH)max(T) The exchange integrals and magnetic moments calculated with Ueff ¼7.0 eV were used to obtain the m(T) for the five sublattices (2a, 2b, 12k, 4f1, and 4f2). The magnetic moments at 0 K (m0,i) for 2a, 2b, 12k, 4f1, and 4f2 sublattices are 4.427, 4.365, 4.432, 4.362, and 4.400 μB, respectively. These moments were calculated by

Table 3 The energy difference Δ(Ueff) in eV between the ground state and excited states. The magnetic moment m in μB are also listed for the excited spin configurations. The s1 and s2 denote the excited sublattices. s1

s2

Δ(3.7)

Δ(7.0)

Δ(10.3)

m(3.7)

m(7.0)

m(10.3)

2a 2b 4f1 4f2 12k 2a 2a 2a 2a 2b 2b 2b 4f1 4f1 4f2 2f1–1 2f1–2 2f2–1 2f2–2 8k 4k

– – – – – 2b 4f1 4f2 12k 4f1 4f2 12k 4f2 12k 12k – – – – – –

0.914 0.815 3.130 3.164 3.818 1.727 2.012 4.080 4.926 3.874 1.692 5.362 6.477 2.609 2.766 1.523 1.523 1.569 1.569 0.782 2.060

0.519 0.456 1.663 1.691 2.091 0.975 1.112 2.215 2.630 2.089 0.931 2.885 3.432 1.455 1.539 0.816 0.816 0.843 0.843 0.496 1.196

0.163 0.153 0.516 0.552 0.660 0.316 0.346 0.716 0.829 0.660 0.306 0.916 1.091 0.470 0.490 0.255 0.255 0.271 0.271 0.144 0.365

20.00 20.00 79.99 80.00  80.00 0.00 60.01 60.00  100.00 59.99 60.01  99.99 120.00  39.99  40.00 60.00 60.00 59.99 60.00 0.00  39.99

19.99 19.99 80.00 79.99  79.99 0.00 59.99 60.00  99.84 60.00 59.99  100.00 120.00  40.00  40.00 60.00 59.99 60.00 60.00 0.00  40.00

20.00 20.00 80.00 79.99  79.99 0.00 59.99 59.99  99.99 60.00 59.99  100.00 120.32  39.99  40.00 59.99 59.99 59.99 59.99 0.00  39.99

9 8 7 6

Jij (meV)

Table 2 The number of nearest neighbors and corresponding distance for the five magnetic sublattices (2a, 2b, 12k, 4f1, and 4f2) in the unit cell of SrFe12O19. zij is the number of neighbors (Fe3 þ ions) at jth sublattice to ith sublattice, and rij is the corresponding distances in nm. The second-nearest neighbors are also given if rij is shorter than 0.4 nm.

3

5 4 3 2 1 0 3

4

5

6

7

8

9

10

11

Ueff (eV) Fig. 2. The sublattice exchange integrals as a function of Ueff (3.7, 7.0, and 10.3 eV).

first principles calculations. In the case of a multiple sublattice system, the number of Brillouin functions is the same as the number of sublattices. Therefore, there are five different Brillouin functions for SrM. The calculated exchange integrals for 2a  4f1, 2b  4f2, 2b  12k, 12k  4f1, and 12k  4f2 in Fig. 2 were considered for the derivation of the five Brillouin functions, since those exchange integrals are dominant. It is noted that the spins at each sublattice are mostly anti-ferromagnetically coupled with the spins at the nearest sublattices. By considering the five exchange integrals, the m(T) for the five sublattices in SrM can be written as ma =m0;a ¼ B½ðs2 =kB TÞj2N af 1 J af 1 ðmf 1 =m0;f 1 Þj

ð6aÞ

mb =m0;b ¼ B½ðs2 =kB TÞj2Nbf 2 J bf 2 ðmf 2 =m0;f 2 Þ þ2N bk J bk ðmk =m0;k Þj

ð6bÞ

mk =m0;k ¼ B½ðs2 =kB TÞj2N kf 1 J kf 1 ðmf 1 =m0;f 1 Þþ 2N kf 2 J kf 2 ðmf 2 =m0;f 2 Þ þ 2N kb J kb ðmb =m0;b Þj

ð6cÞ mf 1 =m0;f 1 ¼ B½ðs2 =kB TÞj2N f 1 a J f 1 a ðma =m0;a Þ þ 2N f 1 k J f 1 k ðmk =m0;k Þj ð6dÞ

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J. Park et al. / Journal of Magnetism and Magnetic Materials 355 (2014) 1–6

mf 2 =m0;f 2 ¼ B½ðs2 =kB TÞj2N f 2 b J f 2 b ðmb =m0;b Þ þ 2N f 2 k J f 2 k ðmk =m0;k Þj: ð6eÞ The numbers of nearest neighbors Nij in Eq. (6) were taken from Table 2 as

ηa ¼ 0:5175 ηb ¼ 0:3565 ηk ¼ 0:7332 ηf 1 ¼ 0:2451 ηf 2 ¼ 0:5640

N af 1 ¼ 6 N bf 2 ¼ 6 and N bk ¼ 6

Also, the exchange integrals in Eq. (8) can be rewritten as

N kf 1 ¼ 3; Nkf 2 ¼ 2; and N kb ¼ 1

J af 1 TN ¼ kB 4sðs þ 1Þηa

ð9aÞ

J bf 2 TN ¼ kB 4sðs þ 1Þð1 þ bÞηb

ð9bÞ

N f 1 a ¼ 3 and N f 1 k ¼ 9 N f 2 b ¼ 3 and N f 2 k ¼ 6: Therefore, after substituting the above numbers for Nij, Eq. (6) becomes Eq. (7). ma =m0;a ¼ B½ðs2 =kB TÞj12J af 1 ðmf 1 =m0;f 1 Þj

ð7aÞ

J kf 1 3T N ¼ kB 2sðs þ1Þð3 þ 2c1 þ c2 Þηk

ð9cÞ

mb =m0;b ¼ B½ðs2 =kB TÞj12J bf 2 ðmf 2 =m0;f 2 Þ þ 12J bk ðmk =m0;k Þj

ð7bÞ

Jf 1 k TN ¼ kB 2sðs þ1Þðd þ 3Þηf 1

ð9dÞ

Jf 2 k TN ¼ : kB 2sðs þ1Þðeþ 2Þηf 2

ð9eÞ

mk =m0;k ¼ B½ðs2 =kB TÞj6J kf 1 ðmf 1 =m0;f 1 Þ þ 4J kf 2 ðmf 2 =m0;f 2 Þ þ 2J kb ðmb =m0;b Þj

ð7cÞ mf 1 =m0;f 1 ¼ B½ðs2 =kB TÞj6J f 1 a ðma =m0;a Þ þ 18J f 1 k ðmk =m0;k Þj

ð7dÞ

mf 2 =m0;f 2 ¼ B½ðs2 =kB TÞj6J f 2 b ðmb =m0;b Þ þ 12J f 2 k ðmk =m0;k Þj

ð7eÞ

In order to calculate Jij as a function of the TN, limits were taken for the case that T approaches to TN. For example, Eq. (7a) can be
  written as ma =m0;a ¼ 12J af 1 s2 =kB T Bðmf 1 =m0;f 1 Þ . After simple




re-arrangement of the above equation, T becomes T ¼ 12J af 1 s2 =kB
    Bðmf 1 =m0;f 1 Þ= ma =m0;a . Therefore, T N ¼ 12J af 1 s2 =kB Bðmf 1 =m0;f 1 Þ=   ma =m0;a Þ when T approaches TN. Now, a conversion ratio parameter ηa is introduced to obtain Bðmf 1 =m0;f 1 Þ ¼ ηa Bðma =m0;a Þ. Then,
  the equation can be rewritten as T N ¼ 12J af 1 s2 =kB ηa Bðma =m0;a Þ=       ma =m0;a Þ. Since Bðma =m0;a Þ = ma =m0;a ¼ ðs þ 1Þ=3s for the case of T ETN, five TN equations for the five sublattices are obtained as TN ¼

4sðs þ 1ÞJ af 1 ηa kB

4sJ bf 2 ð1 þ bÞηb ðs þ 1Þ TN ¼ kB

After inserting the above exchange integral expressions of Eq. (9) and the calculated values of M0,i into Eq. (7), m(T) for five sublattices becomes     mf =m0;f 1 mf =m0;f 1 ma  0:2 coth 0:2  2:025 1 ¼ 1:2 coth 1:2  2:025 1 m0;a T=T N T=T N

ð10aÞ    mf 2 =m0;f 2 mk =m0;k mb ¼ 1:2 coth 1:2 2:3 þ 0:6383 m0;b T=T N T=T N    mf 2 =m0;f 2 mk =m0;k þ 0:6383  0:2 coth 0:2 2:3 T=T N T=T N " mk m0;k

"

TN ¼

2sJ kf 1 ðs þ 1Þ ð3 þ 2c1 þ c2 Þηk 3 kB

ð8cÞ

TN ¼

2sJ f 1 k ðd þ3Þηf 1 ðs þ1Þ kB

ð8dÞ

TN ¼

2sJ f 2 k ðe þ 2Þηf 2 ðs þ 1Þ; kB

ð8eÞ

where b, c1, c2, d, and e are b ¼ J bk =J bf 2 c1 ¼ J kf 2 =J kf 1 c2 ¼ J kb =J kf 1 d ¼ J f 1 a =J f 1 k e ¼ J f 2 b =J f 2 k : The conversion ratio parameters ηi in Eq. (8) can be calculated using the experimental TN of 750 K [1] and the exchange integrals at Ueff ¼ 7 eV in Fig. 2. The following conversion ratio parameters were obtained.

mf 1 mf 2 m þ 0:2356 þ 0:1064 b m0;f 1 m0;f 2 m0;b

 0:2 coth 0:2 1:0869

#   = T=T N

!

mf 1 mf 2 m þ 0:2356 þ 0:1064 b m0;f 1 m0;f 2 m0;b

!

  = T=T N

#

ð10cÞ

ð8aÞ

ð8bÞ

¼ 1:2 coth 1:2 1:0869

ð10bÞ

mf 1 m0;f 1

mf 2 m0;f 2

     ma m ¼ 1:2 coth 1:2 1:0127 þ 1:8665 k = T=T N m0;a m0;k      ma m  0:2 coth 0:2 1:0127 þ 1:8665 k = T=T N m0;a m0;k      m m ¼ 1:2 coth 1:2 1:1505 b þ 1:9571 k = T=T N m0;b m0;k      mb m  0:2 coth 0:2 1:1505 þ 1:9571 k = T=T N m0;b m0;k

ð10dÞ

ð10eÞ

Since there is strong mutual interdependence among the above five equations, numerical analysis was performed, and the plotted temperature dependence of magnetic moment m(T) for the five sublattices is given in Fig. 3. The temperature behavior of magnetic moment is the same behavior as well-known, typical ferrimagnetic behavior of ferrite. The magnetic moment at 12k sublattice decreases more rapidly than the other sublattices as the temperature increases, even though the TN for all the sublattices is the same. After having converted magnetic moment to magnetization, the calculated and experimental [1] Ms(T) are compared in Fig. 4. The number of Fe3 þ ions at 12k sublattice per unit cell is 12, which is the same number as all the other Fe3 þ ions summed, and the spin directions for the 4f1 and 4f2 sublattices are opposite to the 2a, 2b and 12k sublattices. Therefore, the total Ms(T) for SrM is very likely to follow m(T) for the 12k sublattice.

J. Park et al. / Journal of Magnetism and Magnetic Materials 355 (2014) 1–6

5

5

14

From calculated M(T ) Experiment - Ref. [36] Experiment - Ref. [37]

4 12

2a 2b 12k

2 1

(BH )max (MGOe)

Magnetic Moment ( µ B)

3

0 -1

4f 1

-2

8 6 4 2

4f 2

-3

10

0

-4

0

-5

0

100

200

300

400

500

600

700

100

Temperature (K) Fig. 3. The calculated sublattice magnetic moments as a function of temperature m (T). The exchange integrals used to calculate the sublattice magnetic moments are from the exchange energies with Ueff of 7.0 eV.

200

300

400

500

600

700

800

Temperature (K)

800

Fig. 6. The calculated maximum energy product as a function of temperature (BH)max(T). The open circles are the calculated data from the calculated temperature dependences of saturation magnetization Ms(T). Red and blue squares indicate experimental room temperature (BH)max (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

600

of 100–700 K is larger than half of the remnant magnetization Br/2 for SrM as shown in Fig. 5. By neglecting pinning of the nucleation and assuming only coherent rotation with a rectangular magnetic hysteresis loop, Br(T) can be taken the same as the temperature dependence of the saturation magnetization Ms(T), and nucleation field (HN) is also the same as Hci. Therefore, (BH)max(T) can be calculated by the following equation [35].

Magnetization (emu/cm3)

500 400 300

B2r ðTÞ 4

200

ðBHÞmax ðTÞ ¼

100

Fig. 6 shows the calculated (BH)max(T). It reproduces the experimental (BH)max values at room temperature (4.8 [36] and 4.0 [37] MGOe) reasonably well. The difference may be attributed to imperfect magnetic orientation and existence of secondary phases, including hematite.

0 0

100

200

300

400

500

600

700

800

ð11Þ

Temperature (K) Fig. 4. The calculated and experimental saturation magnetizations as a function of temperature Ms(T). The open circles are the Ms(T) from the calculated sublattice m (T), and full circles are experimental Ms(T).

11

Semi-empirical Hci(T ) - Ref. [1]

10

Calculated Br/2(T )

Br/2 (kG) or Hci (kOe)

9 8 7 6 5 4 3 2 1 0 0

100

200

300

400

500

600

700

800

Temperature (K)

5. Conclusion The exchange integrals of hexagonal strontium ferrite (SrFe12O19) were calculated in order to obtain the temperature dependences of magnetic moments m(T) for the five magnetic sublattices (2a, 2b, 12k, 4f1, and 4f2). The density functional theory (DFT) and generalized gradient approximation with Coulomb and exchange interaction effects (GGAþU) were used for all calculations. The exchange integrals with Ueff of 7.0 eV were then employed to the five interrelated Brillouin functions for the sublattice m(T) calculations. The saturation magnetization Ms(T) for SrFe12O19 was obtained by summation of the sublattice m(T) and compared with an experimental Ms(T). It was found that the calculated Ms(T) was in good agreement with the experimental values. The temperature dependence of maximum energy product (BH)max(T) was also calculated from the Ms(T). The calculated (BH)max(T) shows that the value at 300 K is 5.9 MGOe, which is close to the experimental values of 4.8 MGOe [36] and 4.0 MGOe [37].

Fig. 5. The temperature dependence of half-remnant magnetization Br/2(T) and intrinsic coercivity Hci(T). The open circles are the calculated Br/2(T), and full squares are semi-empirical Hci(T).

Acknowledgments

The figure of merit for a permanent magnet is the maximum energy product (BH)max. Accordingly, (BH)max(T) is calculated based on the calculated Ms(T) and semi-empirical intrinsic coercivity (Hci(T)) [1]. The intrinsic coercivity (Hci) over the temperature range

This work was supported in part by the U.S. Department of Energy ARPA-E REACT Program under Award number DE-AR0000189 and the E.A. “Larry” Drummond Endowment at the University of Alabama. The authors thank Xingang Xia at the Electrical and

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