Density functional calculations of the Mössbauer parameters in hexagonal ferrite SrFe12O19

Author’s Accepted Manuscript Density Functional Calculations of the Mössbauer Parameters in Hexagonal Ferrite SrFe12O19 Hidekazu Ikeno

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To appear in: Physica B: Physics of Condensed Matter Received date: 18 December 2016 Accepted date: 26 January 2017 Cite this article as: Hidekazu Ikeno, Density Functional Calculations of the Mössbauer Parameters in Hexagonal Ferrite SrFe12O19, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.01.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Density Functional Calculations of the Mössbauer Parameters in Hexagonal Ferrite SrFe12O19 Hidekazu Ikeno Nanoscience and Nanotechnology Research Center, Research Organization for the 21st Century, Osaka Prefecture University, 1-2 Gakuen-cho, Naka-ku, Sakai Osaka 599-8570, Japan e-mail: [email protected] Abstract Mössbauer parameters in a magnetoplumbite-type hexagonal ferrite, SrFe12O19, are computed using the all-electron band structure calculation based on the density functional theory. The theoretical isomer shift and quadrupole splitting are consistent with experimentally obtained values. The absolute values of hyperfine splitting parameters are found to be underestimated, but the relative scale can be reproduced. The present results validate the site-dependence of Mössbauer parameters obtained by analyzing experimental spectra of hexagonal ferrites. The results also show the usefulness of theoretical calculations for increasing the reliability of interpretation of the Mössbauer spectra. Keywords: APW+lo method; GGA+U; Isomer shift; Quadrupole splitting; Hyperfine splitting; 1. Introduction Hard ferrites are important materials for many industrial applications, such as electric motors, microwave devices, and data storage. The magnetoplumbite-type (M-type) hexagonal ferrites AFe12O19 (A=Sr, Ba, Pb), also known as hexaferrites, are the most commonly used hard magnetic material, because of their chemical stability and cost efficiency [1]. Fig. 1 shows the crystalline structure of M-type AFe12O19. The oxygen ions are located at the corners of the polyhedra. The space group of AFe12O19 is P63/mmc (No. 194). There are five crystallographically inequivalent Fe sites, (referred as 2a, 2b, 4f1, 4f2, and 12k) which are represented by different colors in Fig. 1. The coordination numbers of Fe are six for 2a, 4f2 and 12k sites, five for the 2b site, and four for the 4f1 site. The magnetic structure of the hexagonal ferrite is ferrimagnetic, where magnetic moments of Fe3+ at 2a, 2b, and 12k sites are parallel while those at 4f1 and 4f2 sites are anti-parallel along the c-axis, as shown in the

arrows in Fig. 1 [2][3]. Compared to spinel ferrites, hexagonal ferrites have higher magnetic coercivity and residual magnetization. Replacing Fe by a small amount of Co enhances the properties of materials including the saturation magnetization and the magnetic anisotropy. However, the role of Co atoms on the magnetic properties of hexagonal ferrites is not fully understood. The location of Co substitution sites is still controversial despite a considerable number of experiments using various spectroscopic techniques including x-ray absorption spectroscopy (XAS) and its magnetic circular dichroism (XMCD), nuclear magnetic resonance (NMR), and Mössbauer spectroscopy [4]-[9]. In order to fully understand the local atomic structures as well as the electronic structures of hexagonal ferrites, the interpretation of experimental spectra by means of theoretical calculations is highly desirable. Mössbauer spectroscopy is a powerful technique for the characterization of atomic species in materials. It monitors transitions of nuclear quantum states due to the recoil-free resonant absorption and emission of 𝛾 -rays. The observed fine structures reflect the interaction between nuclei and electron charge distributions, and hence are sensitive to the oxidation states, local atomic coordination, and symmetries. This technique can also be applied to non-crystalline solids and can detect signals from dilute atomic species in materials. On the other hand, the interpretation of the obtained spectra is usually made in an empirical manner, i.e., by comparing the experimental spectra of other reference compounds. In modern science, the chemical composition of the target materials becomes complicated. Sometimes impurity/dopant atoms and those located at surfaces, interfaces, and grain boundaries play crucial role in material properties. The local atomic environment of those atoms is usually different from those in the bulk, and hence the peak assignment of the signals from such atoms would be a difficult task. The interpretation of experimental spectra by means of theoretical calculations is highly desirable. All Mössbauer parameters, including isomer shift, quadrupole splitting, and hyperfine splitting, are directly related to the electron charge densities near nuclear positions, which can be obtained by electronic structure calculations. In this work, the Mössbauer parameters in the hexagonal ferrite, SrFe12O19, are computed from the electron charge densities obtained by all-electron band structure calculations based on the density functional theory (DFT). The theoretical hyperfine parameters are consistent with those obtained experimentally, which contributes to improve the reliability of the interpretation of Mössbauer spectra and site identification in hexagonal ferrites.

Fig. 1 Crystalline structure of magnetoplumbite type (M-type) hexagonal ferrite, SrFe12O19. Sr and five inequivalent Fe sites are represented as balls with different colors. Oxygen ions are located on the corner of polyhedra that are not shown explicitly for clarity. The arrows represent the direction of magnetic moments.

2. Computational Procedure for Mössbauer Parameters Isomer shifts, quadrupole splitting, and hyperfine splitting strongly depend on the electron charge distribution at nuclear positions which can be computed using appropriate first-principles calculations. The background theory and detailed procedure for calculating Mössbauer parameters can be found in the literature [10]-[12]. Here, I briefly review how to compute those parameters from electron charge distributions.

Isomer shift in Mössbauer spectroscopy originates in the electric monopole interaction between the nucleus and electron charge distribution. A real nucleus has finite size and the size of nucleus changes before and after 𝛾 radiation. The isomer shift reflects the difference of nuclear-electron interactions between the ground and excited states due to the change of the nucleus size. Hence, the isomer shift 𝛿 is directly related to the electron charge distribution at the nuclear position and can be expressed as sample

𝛿 = 𝛼(𝜌0

− 𝜌0reference ), (1)

where 𝛼 is a calibration constant whose value has been reported in several references [10][11][12]. The value 𝛼 = −0.291 (Bohr3 · mm/s) obtained from the DFT calculations by sample

Wdowik et al. was adopted in the present work [10]. 𝜌0

and 𝜌0reference are the charge

densities at the nuclear positions of the sample (absorber) and reference (source) compound. The reference compound is usually chosen as bcc-Fe. Quadrupole splitting, 𝛥, that reflects the interaction between the nuclear quadrupole moment and electric field gradient (EFG), can be expressed as 𝛥=

𝑐 𝜂2 𝑒𝑄|𝑉𝑧𝑧 |√1 + . 2ℎ𝜈 3 (2)

Here, Q is the nuclear electric quadrupole moment, 𝜂 = (𝑉𝑥𝑥 − 𝑉𝑦𝑦 )/𝑉𝑧𝑧 is the asymmetry parameter, and Vxx, Vyy, and Vzz denote the principal components of the electric field gradient tensor on the nucleus which satisfies 𝑉𝑥𝑥 + 𝑉𝑦𝑦 + 𝑉𝑧𝑧 = 0. The value of the nuclear electric quadrupole moment for 57Fe, Q=0.16 b (barn; 1b = 10-28 m2), was used in the present work as determined by Dufek et al., by comparing experimental and theoretical data for a many numbers of Fe compounds [14]. The principal component of EFG can be computed from the electronic charge distribution 𝜌(𝑟) by 𝑉𝑧𝑧 = ∫ 𝜌(𝑟)

2𝑃2 (cos 𝜃) 𝑑𝑟 , 𝑟3 (3)

where the 𝑃2 (𝑥) is the second-order Legendre polynomial which projects the 𝑙 = 2 component of the charge density [15]. Hyperfine splitting is the Zeeman splitting of the nuclear spin states due to the internal magnetic field. In other words, it reflects the interaction between nucleus and internal magnetic field. The hyperfine splitting 𝐵tot can be decomposed theoretically into several

contributions as 𝐵tot = 𝐵c + 𝐵dip + 𝐵orb . (4) where, 𝐵c is the Fermi contact term, 𝐵dip is the dipolar field from the on-site spin, and 𝐵orb is the term associated with the orbital moment. The formulae to compute each term of hyperfine splitting can be found in Refs. [16][17]. Note that the 𝐵orb does not appear without including the spin-orbit coupling in the valence states. 3. Results Mössbauer parameters in magnetoplumbite type (M-type) hexagonal ferrite SrFe12O19 were computed. The lattice constants (a = 5.8836Å and c = 23.0376Å) and atomic positions were obtained from the experimental literature [18]. The ferrimagnetic coupling as shown in Fig. 1 was considered as the initial magnetic ordering. All electron calculations were performed under the periodic boundary condition using the augmented plane wave plus local orbital (APW+lo) method implemented in the WIEN2k code [19]. The exchange-correlation interaction was considered within the framework of the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional [20]. The strong on-site Coulomb interaction on the localized Fe-3d electrons were treated with the GGA+U approach following the formalism introduced by Anisimov et al. [21]. The muffin-tin radii, RMT, for Sr, Fe, and O atoms were chosen to be 2.50, 1.85, and 1.64 (Bohr), respectively. The product of the minimum muffin-tin radius and the maximum reciprocal space vector, Kmax, i.e., the plane-wave cutoff, RMTKmax, was fixed at 8.0 (Bohr · Ry1/2) for all calculations. The relativistic effects were fully taken into account for core states by solving the Dirac equation, while the valence electrons were treated within the scalar relativistic approximation plus spin-orbit coupling. The k-points were sampled in accordance with the Monkhorst-Pack scheme [22] with 4×4×1 mesh. Density functional calculation for the ferromagnetic bcc-Fe, which is the reference system to determine the isomer shift, was also performed with the same RMTKmax and the same functional. The GGA+U method might yield an incorrect electronic structure for itinerant metal. Thus, the normal GGA calculations were performed for bcc-Fe with a 30×30×30 k-mesh in the Brillouin zone. sample

Tables 1 and 2 summarize the difference of charge densities (𝜌0

− 𝜌0reference )

between SrFe12O19 and reference bcc-Fe, and the EFG with the asymmetry parameter (𝜂), on each Fe site obtained by GGA (Ueff = 0) and GGA+U (Ueff = 4-8eV) calculations. Clear site saple

dependence of the values can be found in these tables. The magnitude of |𝜌0

− 𝜌0reference |

(Table 1) decreases with decreasing the coordination number as 4f2 > 12k ≈ 2a > 2b > 4f1. This suggests the isomer shift also decreases in this order. Significant site dependence can be found for the EFG (Table 2). The magnitude of the EFG on the bipyramidal 2b site, which is a very unique coordination geometry appearing in hexagonal ferrite, is one order larger than those on other octahedral and tetrahedral sites. The theoretical hyperfine splitting on each Fe site is summarized in Table 3. The contributions of the Fermi contact term (𝐵c ), the dipolar field (𝐵dip ), and orbital term (𝐵orb ) are also shown. As can be seen, the Fermi contact term dominates the hyperfine splitting. The Fermi contact term is proportional to the spin-density at the nucleus, 𝜌↑ (0) − 𝜌↓ (0), which accounts for the polarization of core-electrons due to the finite 3d moments [13]. Figure 2 shows the theoretical isomer shift (𝛿), quadrupole splitting (𝛥), and hyperfine splitting (𝐵𝑡𝑜𝑡 ) obtained by the GGA+U (Ueff = 5eV) calculation. The experimental values reported by Evans et al., are also drawn for comparison [23]. The calibration constant 𝛼 = −0.291 (Bohr3 · mm/s) and the nuclear quadrupole moment 𝑄 = −0.16 (bahn) were used for calculating the isomer shift and quadrupole splitting, respectively. The sign of hyperfine splitting is distinguished by the open and filled circles in Fig. 2 (c), which cannot be determined experimentally. The theoretical isomer shift and quadrupole splitting are consistent with experimentally obtained values. The theoretical hyperfine splitting is significantly underestimated. This underestimation can mainly be ascribed to the error in the exchange-correlation functional. However, the errors between theoretical and experimental values are systematic, and the relative scale of hyperfine splitting on each Fe site can be reproduced. sample

Table 1. Difference of charge densities at nuclear position, 𝜌0

− 𝜌0reference , for

SrFe12O19 and bcc-Fe (Bohr -3) Ueff (eV)

0

4

5

6

7

8

Fe (2a)

-1.506

-1.164

-1.093

-1.019

-0.939

-0.874

Fe (2b)

-1.494

-1.006

-0.897

-0.790

-0.688

-0.588

Fe (4f1)

-1.359

-0.885

-0.772

-0.661

-0.551

-0.451

Fe (4f2)

-1.584

-1.273

-1.197

-1.125

-1.051

-0.981

Fe (12k)

-1.598

-1.171

-1.071

-0.976

-0.884

-0.797

Table 2. Electric field gradient (EFG) at Fe sites in SrFe12O19 (1021 V/m2). The asymmetry parameter 𝜂 for the 12k site is shown in the parenthesis. The values of 𝜂 for other sites are ideally zero and are not shown explicitly.

Ueff (eV)

0

4

5

6

7

8

Fe (2a)

-0.146

-0.032

-0.065

-0.100

-0.133

-0.165

Fe (2b)

9.420

13.271

13.623

13.870

14.040

14.150

Fe (4f1)

0.351

0.848

0.886

0.909

0.922

0.928

Fe (4f2)

1.638

1.459

1.464

1.466

1.467

1.466

1.987

2.329

2.379

2.426

2.469

2.510

(0.224)

(0.613)

(0.623)

(0.620)

(0.611)

(0.598)

Fe (12k)

Table 3. Theoretical hyperfine splitting B (T) at Fe sites in SrFe12O19 Ueff (eV)

Fe (2a)

Fe (2b)

Fe (4f1)

Fe (4f2)

Fe (12k)

0

4

5

6

7

8

Bc

-35.40

-38.39

-38.56

-38.54

-38.50

-38.55

Bdip

-0.54

-0.09

-0.06

-0.04

-0.03

-0.03

Borb

0.77

0.36

0.29

0.23

0.17

0.11

Btot

-35.17

-38.12

-38.33

-38.35

-38.36

-38.47

Bc

-22.14

-27.02

-27.57

-28.14

-28.26

-28.59

Bdip

2.56

2.29

2.19

2.07

1.93

1.78

Borb

0.64

0.59

0.54

0.48

0.42

0.35

Btot

-18.94

-24.14

-24.84

-25.59

-25.91

-26.46

Bc

34.76

38.52

38.70

38.54

38.29

38.04

Bdip

0.07

-0.03

-0.04

-0.05

-0.05

-0.05

Borb

-0.93

-0.51

-0.42

-0.34

-0.26

-0.18

Btot

-33.90

37.98

38.24

38.15

37.98

37.75

Bc

34.87

41.15

41.33

41.30

41.50

41.44

Bdip

-1.21

-0.01

0.02

0.05

0.06

0.07

Borb

-0.87

-0.37

-0.29

-0.22

-0.16

-0.10

Btot

32.79

40.77

41.06

41.13

41.40

41.41

Bc

-28.97

-32.94

-33.30

-33.38

-33.48

-33.51

Bdip

0.60

0.28

0.24

0.21

0.19

0.17

Borb

0.86

0.40

0.33

0.26

0.20

0.14

Btot

-27.51

-32.26

-32.73

-32.91

-33.09

-33.20

Isomer Shift (mm/s)

0.4

(a) 0.3

0.2

0.1

Calc. Exp.*

0

2a

2b

4f1

4f2

12k

Quadrupole Splitting (mm/s)

Fe site 2.5

(b) Calc.

2.0

Exp.*

1.5 1.0 0.5 0 2a

2b

4f1

4f2

12k

Fe site

Hyperfine Field (T)

50

(c)

40 30 20 Calc.

10

(+)

(-)

Exp.*

0 2a

2b

4f1

4f2

12k

Fe site Fig. 2 Theoretical values of (a) the isomer shift, (b) quadrupole splitting, and (c) hyperfine field parameters obtained by GGA+U calculations (Ueff = 5eV). They are compared with the experimentally obtained values reported in Ref. [23]. The filled and open circles drawn in (c) denote the difference of the sign of hyperfine field parameters, which cannot be determined experimentally.

4. Conclusions Mössbauer parameters at the five Fe sites in the M-type hexagonal ferrite, SrFe12O19, were calculated. The all-electron APW+lo method based on the DFT was used, which enables us to compute electron charge density near the nucleus with high accuracy. The theoretical isomer shift and quadrupole splitting in SrFe12O19 showed good agreement with the values obtained experimentally. The significant underestimation of hyperfine splitting parameters was found, which can be ascribed to the error in the exchange-correlation functional. However, the relative scale of hyperfine splitting was consistent with experimental results. The present results demonstrate that the present DFT calculation enable us to quantitatively predict the Mössbauer parameters in various transition metal complex oxides. The theoretical calculations of Mössbauer parameters are also useful for refining the hyperfine parameters, and for increasing the reliability of the interpretation of Mössbauer spectra. Acknowledgments This work was supported by the Grant-in-Aid for Scientific Research on Innovative Areas “Exploration of nanostructure-property relationships for materials innovation,” Grant No. 26106518, and Grant-in-Aid for Scientific Research (A) No. 26249092 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

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