First-principle study of the electronic structures and optical properties of six typical hexaferrites

Computational Materials Science 105 (2015) 27–31

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First-principle study of the electronic structures and optical properties of six typical hexaferrites Wenming Sun a,⇑, Liang Zhang b, Jing Liu a, Hong Wang a,⇑, Yan Zuo a, Yuxiang Bu c a

State Key Laboratory of Green Building Materials, China Building Materials Academy, 100024 Beijing, China NeoTrident Technology Ltd., 201204 Shanghai, China c Department of Chemistry, Shandong University, 250100 Jinan, China b

a r t i c l e

i n f o

Article history: Received 23 January 2015 Received in revised form 13 April 2015 Accepted 16 April 2015 Available online 16 May 2015 Keywords: Density functional theory Hexaferrite Electronic structure Static dielectric constant

a b s t r a c t The structural, electronic and optical properties of hexagonal ferrites MFe12O19 (M = Sr, Ba, Pb, Sr0.5Ba0.5, Sr0.5Pb0.5 and Ba0.5Pb0.5) are calculated by plane-wave pseudopotential density functional theory with general gradient approximation (GGA) and GGA+U. The calculated lattice constants and band gaps are in good agreement with the available experimental and theoretical values. Lattice constants change corresponding to the cation radii at M-sites. The electronic structure shows that all the six hexaferrites are narrow gap semiconductors and Sr2+ and Ba2+ at M-sites have little contribution to the DOS at the vicinity of Fermi level due to the ionic bond interaction nature between M2+ and O2. It should be noted that for Pb2+, comparing with the narrow localized s-states of Sr2+ and Ba2+, there is a significant broadening of its s-states from 7 eV to the Fermi level, indicating its minority donation to the valence band near Ef. The six MFOs (FO refers Fe12O19) could be classified into two categories: Pb-containing hexaferrites (PFO, SPFO and BPFO) and others (SFO, BFO and SBFO). The former has larger static dielectric constants. This study will serve as the base for the investigation of the correlation among factors such as site preferences, properties and substitution strategies for MFOs. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Hexaferrites MFe12O19 (acronym MFO, with FO = Fe12O19 and M = Sr, Pb, Ba) have been a subject of intensive study for several decades, owing to their intriguing physical and chemical properties such as large saturation magnetization, weak temperature dependent coercivity, excellent chemical stability and low cost [1,2]. Much attention is devoted to their applications in some commercial and military fields, such as electromagnetic wave absorbers, high-density magnetic recording media and microwave devices [3,4]. Attempts have been made to modify the electronic and magnetic properties by introducing impurities to substitute either M2+ or Fe3+ sites or both due to their complex magnetoplumbite (M-type) structure with mono-substitution and dual-substitution strategies. For mono-substitution, Fe3+ is usually replaced by Al3+ in order to obtain larger coercivity [5]. In the dual-substitution strategy, Fe–Fe pairs substituted by Zr–Cd [6], Er–Ni [7], Zn–Nb [8], Zn–Sn [9–11], Sn–Mg [12] Ti–Mg [13] and Ti–Co [13] pairs are investigated to tune saturation magnetization and coercivity ⇑ Corresponding authors. E-mail addresses: [email protected] @cbmamail.com.cn (H. Wang). http://dx.doi.org/10.1016/j.commatsci.2015.04.021 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

(W.

Sun),

hongwang2

of hexaferrites. La–Co pair replacement of Sr–Fe pair is also reported [14–17]. Studies on M-type hexaferrites were mainly focused on experimental approaches to the material preparation, microstructure characterization and physical properties measurement. There are only limited publications of first principle study on the intrinsic electronic, optical and magnetic properties of hexaferrites. In an early work, Fang et al investigated the electronic and magnetic structure of SFO using density functional theory (DFT) method [18] and confirmed Gorter’s prediction [19], that the most stable form of this hexaferrite is a ferrimagnet with Fe3+ in 4f1 and 4f2 sites having the spin polarization anti-parallel to the other Fe3+ cations. In 2005, Novak and coworkers group calculated the exchange interactions in BFO [20]. Recently, Feng et al investigated the magnetic anisotropy change of Co–Ti pair substituted SFO [21]. The site preferences for impurity cations in SFO and BFO were also studied using DFT method [11,22]. To our best knowledge, systematic study on electronic properties variation caused by the cations change at M-sites has not been reported, and theoretical analysis of how cations affect electronic, optical and magnetic properties in MFOs is still need. In this study, six typical hexaferrites MFOs (M = Sr, Pb, Ba, Sr0.5Ba0.5, Sr0.5Pb0.5 and Ba0.5Pb0.5) are selected as versatile precursors to explore the influence of geometrical, electronic and optical properties with the

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W. Sun et al. / Computational Materials Science 105 (2015) 27–31

cations variation at M-sites. This work will be a basis for establishing the correlation between factors such as site occupancy preferences, properties and substitution strategies for MFOs, which paves the way toward the hexaferrites design for special applications.

2. Method The calculations were performed within the framework of DFT, using the plane-wave basis-set and Vanderbilt-type ultrasoft pseudopotential [23]. The valence shells for Sr, Ba, Pb, Fe and O are 4s4p5s, 5s5p6s, 5d6s6p, 3d4s and 2s2p, respectively. The Perdew–Burke–Ernzerhof (PBE) parameterization utilizing generalized-gradient approximation (GGA) scheme was adopted to deal with the exchange–correlation interactions [24]. Since local approach (LDA) or GGA for exchange and correlation failed to correctly describe the band gap In order to obtain an accurate electronic structure, a Coulomb correction of U = 3.7 eV in GGA+U was implemented to overcome this well-known shortcoming. The parameter U was tested and we chose a value of convergence for the local magnetic moments in SFO, as explained in Section 3.2. A plane wave energy cutoff of 500 eV was applied, and Brillouin zone integration was performed over 5  5  1 grid points using the Monkhorst–Pack scheme [25]. The convergence with respect to k-point density has been thoroughly checked. The lattice constant of SFO does no change with denser k points samplings such as 7  7  2 and 10  10  2. A denser k-point grid (10  10  2) decreased the total energy only 2.5 meV. Geometry optimizations were performed to fully relax the atomic internal coordinates (within the BFGS minimization algorithm [26]) and the lattice parameters till the total energy convergence of 1.0  106 eV per atom and residual force to 0.03 eV/Å. Using the electronic structure, we can get the linear optical properties, which are described by the complex dielectric function defined by

e ¼ e1 þ ie2

3. Results and discussion 3.1. The structures of MFO (M = Sr, Ba, Pb, Sr0.5Ba0.5, Sr0.5Pb0.5 and Ba0.5Pb0.5) The crystal structure of hexaferrites belongs to the space group P63/m. The optimized geometry of SFO is shown in Fig. 1. The lattice parameters for SFO obtained in our DFT calculation are a = 5.864 and c = 23.105 Å which agrees fairly well with parameters measured experimentally (a = 5.883 and c = 23.038 Å) [29]. All the lattice parameters for the six optimized hexaferrites are listed in Table 1. The results show that BFO has the largest c, followed by PFO and SFO. It correlates with the effective radii of cations in Msites (Ba2+ (135 pm) > Pb2+ (119 pm) > Sr2+ (118 pm)) [30]. Lattice constants from our DFT optimized BFO are also in good agreement with the experimental values [31], indicating our calculation method is reasonable and valid. The primitive cell has 11 inequivalent sites: one Sr site of multiplicity 2, five oxygen sites of multiplicity 4, 4, 6, 12 and 12 and five iron sites of multiplicity 2, 2, 4, 4 and 12, respectively. Based on their coordination with oxygen, the Fe3+ cations are positioned at five crystallographically different sites as shown in Fig. 1, namely 2a, 2b, 4f1, 4f2 and 12 k. It is worth noting that in our calculation, we adopt the most stable spin configuration following Gorter’s prediction [19] and Fang’s calculation [18], that is, spin polarization of Fe3+ in 4f1 and 4f2 sites are opposite to that of the other Fe3+ cations. According to the previous reports, all the other spin configurations would cost at least 0.84 eV for BFO [22] and 1.44 eV for SFO [18], respectively. 3.2. The electronic properties of MFO (M = Sr, Ba, Pb, Sr0.5Ba0.5, Sr0.5Pb0.5 and Ba0.5Pb0.5) As mentioned above, traditional DFT methods fail to obtain the accurate band gap in solid state physical calculations. Nowadays,

ð1Þ

where e1 and e2 are the real and imaginary components of the dielectric constant, respectively. The imaginary part is directly related to the electronic band structure and optical matrix elements, which can describe the absorptive behavior. By summing all possible transitions from occupied to unoccupied states, the interband contribution to the imaginary part can be calculated. In the simple dipole approximation, the imaginary part has the following expression:

e2 ðxÞ ¼

 2 2 X Z 4p e 3 hijMjji2 f i ð1  f i Þ  dðEj;k  Ei;k  xÞd k m2 x2 i;j k

ð2Þ

where e is the charge of free electrons, m is the mass of free electrons, x is the frequency of incident photons, i and j are the initial and final states, respectively, M is the dipole matrix, fi is the Fermi distribution on the i-th states and Ei,k is the electronic energy on i-th state with reciprocal crystal vector k. Summing over k is actually samples the whole region of Brillouin zone in the k space. Within the long wavelength limit, once e2 is calculated as a function of frequency, the real part e1 is derived out via Kramer–Kronig relationship [27]. Formula 1 and 2 show the e1 and e2 are the response of the incident light. In industrial applications, low-cost and reliable polycrystalline ferrites are used, hence in our calculation, only ploycrystalline polarization is set for simplicity, where optical properties are averaged over all polarization directions as a fully isotropic average, thereby imitating an experiment on a polycrystalline sample. All the total energy calculations were performed in the reciprocal space using the Cambridge serial total energy package (CASTEP) code [28].

Fig. 1. Crystal structure of M-type hexaferrite. Red, brown and green colored spheres represent oxygen, iron and M atoms (M = Sr, Ba, and Pb). In the right graph, the spin directions for different Fe3+ at 2a, 2b, 12 k, 4f1and 4f2 are labeled with different colors (black and red). In order to make it clear, oxygen atoms are not displayed inside. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

W. Sun et al. / Computational Materials Science 105 (2015) 27–31 Table 1 Lattice constants and band gaps for the six hexaferrites. The values in the parentheses are from the corresponding references.

SFO BFO PFO SBFO SPFO BPFO

a (Å)

c (Å)

Band gap (eV)

5.864 (5.88) [29] 5.876 (5.90) [31] 5.868 5.870 5.865 5.877

23.105 (23.04) [29] 23.342 (23.24) [31] 23.132 23.183 23.107 23.187

0.876 (0.93) [11] 0.866 1.006 0.858 0.862 0.907

many strategies have been implemented to overcome this limitation. For instance, the newly developed nonlocal exchange-correlation functionals [32–35] result from the generalized Kohn-Sham procedure and are intended to improve on the description of band gaps in insulators and semiconductors compared with LDA and GGA. Meanwhile GW approximation scheme [36,37], in which many body effects are considered, has been confirmed valid in band gap calculations. Unfortunately, the additional accuracy comes at the price of much more time consuming calculations. Considering the size of our system and their derivatives after flexible substitutions, which we are about to carry through in the next step, the GGA+U method was employed in our spin-polarized systems to deal with the ‘strongly correlated’ Fe-3d electrons. An effective value-Ueff is required in this method, which equals to the difference between the Hubbard parameter U and the exchange parameter J. Since there is no measured band gap value of SFO, the Ueff value (simply U from now on) was determined by investigating the dependency of local magnetic moments of Fe3+ on parameter U as stated by Liyanage et al. [11]. With the increasing of U, the absolute values of local magnetic moments increase monotonically for Fe3+ cations in SFO. This is consist with Liyanage’s report [11]. By adjusting the U, they reproduce the local magnetic moments of Fe3+ which are obtained by HSE calculation. To be comparable with the previous report, the same value (U = 3.7 eV) is adopted in our band gap calculation for SFO. The variation of band gaps of SFO with respect to the U parameter is shown in the inset of Fig. 2. Without U, SFO’s band gap vanishs, accordingly SFO shows a metallic behavior.

29

Band gap for SFO is 0.876 eV, which is in good agreement with Liyanage’s GGA+U and HSE results (0.93 and 1.19 eV) [11]. The band gaps for the six hexaferrites are calculated using GGA+U (U = 3.7 eV) method as given in Table 1. All the six MFOs show narrow gap semiconductor behavior. PFO has the largest band gap (1.006 eV), while SBFO has the lowest one (0.858 eV). The difference between them is only 0.14 eV. To quantitatively exhibit the effect due to the variation in Msites, we listed the results of Mulliken population analysis in Table 2. The charges in 2b for SFO and PFO are similar (0.96 and 0.92e), and both are smaller than that for BFO (1.23e). The population on the 2b sites changes more significant than other sites for iron cations, which should be mainly attributed to the short distance between 2b and M-site (3.38 Å). For 4f2 and 12k, the distances between them and M-sites are about 3.6 Å. While 2a and 4f1 are further away from M-sites (more than 6.1 Å). As a result, the inequivalent sites 2b, 4f2 and 12k are more sensitive to the variation of M-sites compared with 2a and 4f1. This influence is distance-dependent. This can also be verified from charge population for SBFO, BPFO and SPFO. When M-sites are occupied by different cations, the symmetry of MFOs reduces to P3m1, The charge population near specific M sites only depends on cation type of M. For example, charge of 2b nearby Sr in SBFO is the same as in SFO, so is the Ba in BFO. For the cations in M-sties, Pb2+ cation has the largest charge (1.9e) compared with the other two cations, indicating that Pb–O interaction is stronger than Sr–O and Ba–O. This correlates with the bond lengths between M and its nearest oxygen atom. It should be pointed out that the variation in M-sites does not change the spin density (Fig. 2) and total magnetic moments of MFOs and has only slightly influence on the local magnetic moment of Fe3+ cations. In all systems we investigated here, the total magnetic moment of each system is 40lB/unit-cell. The total density of states (DOS) for SFO, BFO and PFO are shown in Fig. 3a. The overall profile of them are similar. After analyzing the partial DOS, we found Sr2+ has little contribution to the valence band and conduct band near Fermi level. The valence band is mainly composed by 2p of oxygen and 3d of iron from -9 eV to Fermi level. The significantly hybridization between Fe-3d and O2p reveal the covalent nature between them (Fig. 3b). The

Fig. 2. Variation of local magnetic moments of the five inequivalent Fe sites (2a–black, 2b–red, 12k–blue, 4f1–cyan, 4f2–magenta) of SFO with respect to the U parameter. The inset in the left graph shows the variation of band gaps for SFO with respect to the U parameter. The right graph represents the spin density contour map of BPFO (U = 3.7 eV). The slice crosses both Ba and Pb cations, and is parallel with z-axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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W. Sun et al. / Computational Materials Science 105 (2015) 27–31

Fig. 3. Total DOS (a) of SFO, BFO and PFO in their optimized geometries (U = 3.7 eV). The DOS with positive and negative sign represent the density of spin-up and spin-down electrons in this system, respectively. Graph (b) represents the PDOS for Fe-d (black line) and O-p (red line) in SFO. Graph (c) represents the PDOS (s-states and p-states) for M2+ cations in SFO, BFO and PFO. DOS of SFO, BFO and PFO are presented by black, blue and red solid lines, respectively. In all graphs, the Fermi level is set at the origin of the energy scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2 Calculated Mulliken population (in e) and local magnetic moment r (in lB) for cations in MFO. For SBFO, BPFO and SPFO, since the symmetry is reduced due to the different cations in M-sites, the charge and localized magnetic moment are influenced, and the change could be evaluated by comparing with the values in parentheses.

Fe2a

SFO

BFO

PFO

SBFO

BPFO

SPFO

Charge

1.12 4.04 0.96 3.92 1.13 3.96 1.14 3.94 1.06 4.04 1.39 0.02

1.13 4.04 1.23 3.94 1.14 3.96 1.2 3.98 1.07 4.04

1.09 4.04 0.92 3.98 1.1 3.96 1.08 3.96 1.01 4.04

1.12 4.04 0.96 (1.23) 3.94 1.13 3.96 1.14 (1.2) 3.96 1.06 (1.07) 4.04 1.39 0.02 1.57 0.02

1.11 4.04 0.92 (1.23) 3.98 (3.94) 1.13 (1.11) 3.96 1.2 (1.08) 3.96 (3.94) 1.01 (1.07) 4.04

r Fe2b

Charge

r Fe4f1

Charge

r Fe4f2

Charge

r Fe12k

Charge

r Sr

Charge

r Ba

Charge

r Pb

Charge

r

1.57 0.02 1.90 0.04

interaction between O2 and Sr2+ is not covalent due to the long distance between them and no significant electron density coupling. For Sr2+ and Ba2+, they belong to the same group as alkaline earth metals, s and p states lie deeply below the Fermi level at least 10 eV (Fig. 3c). The variation of Sr/Ba in M-site does not have significant contribution to the total electronic properties, such as the composition of valence states top and band gap due to the ionic nature of M–O interaction. The change in M-sites only influence electronic structures near Fermi level via disturbing the Fe3+ and O2 nearby indirectly. For Pb2+, its d state lie deeply below the Fermi level about 15 eV, while the p band mainly lies above Fermi level about 5 eV with a sharp peak there as shown in Fig. 3c. The Pb-s state broadens widely and significantly from 7 eV to Fermi level with three significant peaks compared with

1.57 0.02 1.90 0.04

1.1 4.04 0.92 3.98 1.1 3.96 1.08 3.94 1.01 4.04 1.39 0.02

(0.96) (3.92)

(1.14) (3.96) (1.06)

1.91 0.04

the localized s state in Sr and Ba, revealing a stronger hybridization in valence band. Pb is involved most in the Fe–O hybridization near the Fermi level due to its shortest Pb–O distance compared with Sr–O and Ba–O. 3.3. The optical properties of MFO (M = Sr, Ba, Pb, Sr0.5Ba0.5, Sr0.5Pb0.5 and Ba0.5Pb0.5) The dielectric function of all the six system is depicted in Fig. 4. The trend of static dielectric constants of the six system is PFO (9.12) > SPFO (8.94) > BPFO (8.78) > SFO (8.68) > SBFO (8.57) > BFO (8.37). From the inset of Fig. 4, it is noted that the peaks of e1 for all the species containing Pb (i.e. PFO, SPFO and BPFO) are significantly higher than those without Pb. Apparently, the static

W. Sun et al. / Computational Materials Science 105 (2015) 27–31

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Fig. 4. Dielectric function for SFO (black solid line), BFO (blue solid line), PFO (red solid line), SBFO (green solid line), BPFO (yellow solid line) and SPFO (cyan solid line). The left and right graph represent the real (e1) and imaginary (e2) part, respectively. The insets represent the scaled fist peak and trough in the left graph and the first peak in the right graph. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dielectric constants will increase once Pb as a participator, however it decreases if Ba is in. There are two valleys for the ranges about 4.6–7.8 eV and 16–18 eV for MFOs. This indicates that the hexaferreties show a metallic behavior in these frequency zones. The e2 shape for all the systems is similar. There is a significant peak for the imaginary part near 3.5 eV, which is mainly due to the interband transitions from the O 2p states in valence band to the Fe 3d states in the conduction band. During the variation of M-site, the peaks does not change significantly in the low energy scale. 4. Conclusion In summary, the structural, electronic, and optical properties of MFOs were calculated by performing DFT with GGA and GGA+U. Lattice constants show that BFO has the largest c, followed by PFO and SFO. This correlates with the effective cation radii (Ba2+ (135 pm) > Pb2+ (119 pm) > Sr2+ (118 pm)). The electronic structures indicate that the Sr2+ and Ba2+ at M-sites has little contribution to the valence band near Fermi level due to the ionic nature of M–O. For PBO, due to the shorter Pb–O distances than Sr–O and Ba–O, there is a significant broadening for its s-states from 7 eV to the Fermi level, indicating its donation to the valence band near Ef. Based on the static dielectric constant and the peaks for dielectric function, the six MFOs could be grouped into two categories: Pb-containing MFOs (PFO, SPFO and BPFO) and others (SFO, BFO and SBFO). The former has larger static dielectric constants. This study will pave the way for the investigation of the correlation among factors such as site occupancy preferences, electronic, optical and magnetic properties and substitution strategies of MFOs in the next step of our blueprint.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

Acknowledgments This research was sponsored by National High Technology Research and Development Plan of China (Grant No. 2015AA034204), National Natural Science Foundation of China (Grant No. 51272245, 21373123), National Science & Technology Pillar Program during the 12th Five-year Plan Period of China (Grant No. 2013BAE12B01) and NSF of China (Grant No. ZR2013BM027).

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