# Site-selective cobalt substitution in La–Co co-substituted magnetoplumbite-
type ferrites: 59Co-NMR and DFT calculation study

Hiroyuki Nakamura1, Hiroto Ohta2, Ryuya Kobayashi1, Takeshi Waki1, Yoshikazu
Tabata1, Hidekazu Ikeno3 and Christian Mény4 1 Department of Materials Science
and Engineering, Kyoto University, Kyoto 606-8501, Japan  2 Department of
Molecular Chemistry and Biochemistry, Faculty of Science and Engineering,
Doshisha University, Kyotanabe 610-0321, Japan  3 Department of Materials
Science, Graduate School of Engineering, Osaka Metropolitan University, Sakai,
Osaka, 599-8570, Japan  4 Institut de Physique et Chimie des Matériaux de
Strasbourg (IPCMS), UMR7504, CNRS-Université de Strasbourg, 23, rue du Loess,
F-67034 Strasbourg Cedex 02, France<EMAIL_ADDRESS>
###### Abstract

The La–Co co-substituted magnetoplumbite-type (M-type) ferrites $A$Fe12O19
($A$ = Ca, Sr and Ba, ion sizes Ca2+ $<$ Sr2+ $<$ Ba2+) with Co compositions
around 0.2 have been subjected to 59Co-NMR. The results show that Co occupies
the 4f1, 2a and 12k sites, and that the smaller the $A$ ion, the more Co tends
to occupy the 4f1 minority spin site, which is effective in enhancing both
uniaxial anisotropy and magnetisation. First-principles total energy
calculations based on density functional theory (DFT) of undoped $A$Fe12O19
and a supercell ($2\times 2\times 1$ of the unit cell) in which 1/96 of Fe3+
is replaced by Co2+ were performed to predict the stable structure and Co
occupancy sites. The results show that regardless of $A$, Co is most stable
when it occupies the 4f1 site, followed by the 2a and 12k sites with energy
differences on the order of 100 meV, and Co practically does not occupy the 2b
and 4f2 sites. As the $A$ ion becomes smaller, the energy difference when Co
occupies each Fe site tends to increase, and the Co occupancy of the 4f1 site
also increases. The site selectivity of Co can be roughly explained as a
result of the difference in uniaxial strain along the $c$-axis associated with
the difference in $A$. However, the influence of the $A$ ion differs between
the R and S blocks and the local strain also has a secondary effect on the Co
distribution. Based on these results, the guidelines for improving the
performance (anisotropy and magnetisation) of La–Co co-substituted M-type
ferrite magnets with a limited amount of Co can be summarised as follows: It
is effective to select as small $A$ ions as possible and to post-anneal at low
temperature or cool slowly to concentrate Co at the 4f1 site in tetrahedral
coordination.

Keywords: ferrite magnet, anisotropy, NMR, DFT calculation, magnetoplumbite-
type ferrite

Figure 1:  Crystal structure of the magnetoplumbite-type (M-type) $A$Fe12O19
drawn by VESTA [4]. The space group is $P6_{3}/mmc$. The unit cell is equal to
two formula units. The $A$ ion (black circle) occupies the 2d site surrounded
by 12 oxygen ions. Iron (red circles) has five crystallographically
inequivalent sites (2a, 2b, 4f1, 4f2 and 12k) and its oxygen coordination
polyhedra are represented by different colours. The 2a, 4f2 and 12k sites are
6-coordinated (octahedral), the 4f1 site is 4-coordinated (tetrahedral) and
the 2b site is 5-coordinated (bipyramidal). The 2a, 2b, 4f1 and 4f2 sites have
trigonal rotational symmetry around the $c$-axis, but the 12k site is not
trigonally symmetric, with the local principal axis tilted from the $c$-axis
resulting in three types of 12k-Fe. Oxygen (blue circles) has five
crystallographically inequivalent sites (4e, 4f, 6h, 12k1, 12k2). All Fe is
trivalent and is expected to have a spin moment of 5 $\mu_{\mathrm{B}}$/Fe.
$A$Fe12O19 is ferrimagnetic; the Fe at the 2a, 2b and 12k sites has the
majority spin and the Fe at the 4f1 and 4f2 sites has the minority spin. S and
R denote blocks and the M-type structure has SRS∗R∗ stacking, where * denotes
a 180∘ rotation around the $c$-axis.

##  1 Introduction

Ferrite magnets are used in a wide range of equipment because they are
extremely cost effective compared with rare earth magnets [1]. Higher
performance and lower cost of ferrite magnets are the demands of society. Many
attempts have been made to improve the performance of ferrite magnets [2].
Many of these are elemental substitutions, and the substitution of Co for Fe
is a successful example [3]. However, much of the development is trial and
error and not necessarily based on clear guiding principles. Against this
background, the aim of this study was to propose guidelines beyond trial and
error for the development of the next generation of high performance magnets
and the effective use of resources. To improve the performance of
magnetoplumbite-type (M-type) ferrites, improvements in magnetic anisotropy
(coercivity), magnetisation (remanence), or both, are required. This paper
focuses on methods to improve magnetic anisotropy without degrading or even
improving magnetisation.

The base material of ferrite magnets is the hexagonal M-type ferrite
$A$Fe12O19 ($A$ = Ba, Sr, Pb…) (figure 1). The space group of the M-type
structure is $P6_{3}/mmc$, and the unit cell contains two formula units. All
Fe atoms are located in the gap between O and $A$. There are five
crystallographically inequivalent Fe sites in the Wyckoff symbols: 2a
(octahedral coordination), 2b (bipyramidal), 4f1 (tetrahedral), 4f2
(octahedral) and 12k (octahedral). The M-type structure can be divided into
the S block (Fe6O8)2+ including 2a and 4f1 sites and the R block ($A$Fe6O11)2-
including 2b and 4f2 sites, with the 12k site in the plane perpendicular to
the $c$-axis at the boundary between the S and R blocks. The stacking is
described as SRS∗R∗, where * represents a 180∘ rotation around the $c$-axis.
All Fe ions in $A$Fe12O19 are trivalent (3d5) and have a spin magnetic moment
of 5 $\mu_{\mathrm{B}}$ ($S=\frac{5}{2}$). The M-type ferrites are
ferrimagnets in which the magnetic moment of Fe3+ is ordered by superexchange
interactions through O2-. The 2a, 2b and 12k sites are the majority spin
sites, and the 4f1 and 4f2 sites are the minority spin sites, resulting in a
net magnetic moment of $20~{}\mu_{\mathrm{B}}$ per formula unit. Isolated Fe3+
with no orbital component in the magnetic moment do not normally cause
magnetic anisotropy. The presence of the asymmetric 2b site, which exhibits
strong single-ion anisotropy [5] via interaction with the coordinating oxygen
ions, appears to be essential for uniaxial anisotropy in M-type ferrites.
However, contributions other than the 2b site are also important for a
quantitative explanation of the anisotropy, as has been better understood by
recent theoretical work [6, 7, 8].

Various element substitutions have been attempted to improve the performance
of M-type ferrite magnets. Among these, La–Co co-substituted M-type SrFe12O19,
where some of the Fe3+ (approximately 2–3%) in SrFe12O19 is replaced by Co2+
and some of the Sr2+ by La3+ as charge compensation, has been reported to
improve the coercivity by approximately 30% and the residual magnetisation by
a few percent compared to the undoped material [3]. The residual magnetisation
is slightly improved because Co2+, which has a smaller magnetic moment than
Fe3+, mainly occupies the minority spin sites, thereby increasing the net
magnetisation. It is easy to imagine that the increase in coercivity is due to
the enhanced uniaxial magnetic anisotropy caused by the residual (or induced)
orbital magnetic moment of Co2+. However, the mechanism of enhancement has
long been unclear due to the ambiguity of the Co occupancy sites. However,
recent studies have shown that Co2+ mainly occupies the 4f1 site and partially
occupies the 2a and 12k sites. The fact that Co occupies three sites at 2a,
4f1 and 12k was first proposed by a combined Rietveld analysis of neutron
diffraction, extended X-ray absorption fine structure (EXAFS) and X-ray
magnetic circular dichroism (XMCD) [9]. This result was supported by the atom
location determined using the channeling-enhanced microanalysis (ALCHEMI)
method [10]. Recent NMR [11] and Mössbauer experiments [12, 13] are consistent
with the site occupancy of Co.

La–Co co-substituted M-type ferrite magnets with $A$ ions other than Sr have
also been developed. It is known that the case of $A$ = Ca has a higher
coercivity than $A$ = Sr, even with the same amount of Co substitution [14].
In contrast, the coercivity for $A$ = Ba is lower than for $A$ = Sr [15]. This
suggests that even with the same amount of Co, the Co distribution is not the
same and depends on the $A$ ion. The Co distribution in La–Co co-substituted
CaFe12O19 has also been studied by a combined analysis of Rietveld refinement
of neutron diffraction, EXAFS and XMCD [16, 17, 18]. The occupancy sites of Co
are the same as in SrFe12O19, but the 4f1-site occupancy of Co has been
reported to be greater in the Ca–La ferrite.

It should also be noted that physical properties such as coercivity can vary
greatly depending on the synthesis process and heat treatment conditions, even
for materials of the same composition. Such cases arise when dopants occupy
several non-equivalent sites and their stability is antagonistic at relatively
low energy scales. This is because the cation distribution at room temperature
is not necessarily the ideal thermal equilibrium state at room temperature,
but is more or less a frozen state of the high temperature distribution. In
other words, the final physical properties depend on the temperature at which
the cation distribution is frozen, that is, the heat treatment applied to the
sample. Spinel compounds have long been the focus of studies of cation
distributions at finite temperatures [19]. Some simple models commonly discuss
the temperature dependence of the cation distribution from the probability of
occupancy via the Gibbs free energy [19]. In recent years, many examples have
been reported where density functional theory (DFT) has been used to calculate
the energy change of a state in which a particular site is replaced by a
particular element, which is then used to calculate the probability of
occupancy at a finite temperature [20, 21, 22]. The application of this method
to M-type ferrites is growing rapidly [21, 22], and similar discussions can be
made in this study.

59Co ferromagnetic nuclear resonance experiments are a convenient way to
obtain information on the site-to-site distribution of Co in ferromagnets and
microscopic magnetism, provided that the occupancy sites can be assigned to
the resonance signals. We performed 59Co-NMR experiments on samples of La–Co
co-substituted SrFe12O19 with widely varying Co compositions (from 0.15 to
0.93 Co) to investigate the Co distribution over multiple Fe sites and the
role of Co occupying each site [23]. The results showed that Co was commonly
distributed over three Fe sites (4f1, 2a and 12k), with the 4f1 site being the
most preferentially occupied. We evaluated the Co content dependence of the
local anisotropy at each site by measuring the signal enhancement factor, and
found that the macroscopic anisotropy was positively correlated with the local
anisotropy of Co at the 4f1 site and inversely correlated with those at the 2a
and 12k sites: see the second paragraph of section 3.2 of [23] for details,
and [24, 25] for this methodology. Based on these results, we proposed for the
first time that only Co at the 4f1 site contributes to the enhancement of
uniaxial anisotropy. Although no orbital degeneracy of Co2+ at the
tetrahedrally coordinated site would be expected if the coordination
polyhedron were a regular tetrahedron, we concluded that the Co at the 4f1
site, which is trigonally symmetric, is the main cause of the enhancement of
uniaxial anisotropy. From these results, we proposed that the magnetic
anisotropy and saturation magnetisation of ferrite magnets can be improved by
concentrating Co at the 4f1 site, and the production cost can also be reduced
by the effective use of Co. However, the actual method for concentrating Co at
the 4f1 site remains debatable. In La–Co co-substituted SrFe12O19, the
4f1-site occupation ratio of Co was unexpectedly increased with increasing
La–Co substitution. It was suggested that the 4f1-site selectivity of Co could
be enhanced by a decrease in $c/a$, that is, compressive strain along the
$c$-axis, associated with the substitution of La3+, which is smaller in size
than Sr2+. However, as a series of samples with very different Co
concentrations were used for the measurements, we could not rule out the
possibility that the Co distribution changes due to changes in the Co
concentration itself and could not clearly conclude that uniaxial strain
affects Co site selectivity.

In this study, we focused on the change in Co distribution in La–Co co-
substituted $A$Fe12O19 due to the difference in $A$-ion size. In other words,
similar to the previous study [23], 59Co-NMR experiments were performed on
La–Co co-substituted $A$Fe12O19 ($A$ = Ca, Sr and Ba) with a common Co
composition to compare the Co site selectivity and investigate the effect of
$A$-ion size. We also evaluated the stability of Co occupying each Fe site and
the crystal structure parameters of structure-optimised $A$Fe12O19 and Co
substituted $A$Fe12O19 by first-principles total energy calculations based on
DFT and discussed the agreement with experiments, as well as the factors
controlling the Co site selectivity. As mentioned above, to further improve
the coercivity and magnetisation of La–Co co-substituted M-type ferrite
magnets and to make effective use of Co, it is necessary to concentrate Co2+
more efficiently at the 4f1 site. Establishing a simple guiding principle for
this will accelerate the development of next-generation high performance
ferrite magnets. In this study, we show that the selection of $A$ ions that
reduce the $c$-axis length is effective for this purpose. We also discuss the
secondary effect of the local strain on Co site selectivity. From the results
of DFT calculations, we show that the Co site-to-site distribution is
temperature sensitive in the temperature range where the M phase is stable,
and that post-annealing at low temperature and/or slow cooling is effective in
concentrating Co at the 4f1 site.

##  2 Methods

###  2.1 Our strategy

Since the development of La–Co co-substituted high-performance ferrite magnets
[3], the mechanism of coercivity (uniaxial magnetic anisotropy) enhancement
has been extensively studied, but the debate has long remained confused. The
main reason for this confusion was that the Co occupancy sites were
experimentally ambiguous [26]. The amount of Co substituted for Fe is at most
a few percent of Fe, and Co is distributed over several sites in five
crystallographically inequivalent Fe sites. In other words, the identification
of Co occupancy sites in La–Co co-substituted M-type ferrite is an ultra-trace
element analysis in a multi-site compound and requires extremely high
experimental precision. Despite these difficulties, early studies paid
insufficient attention to sample homogeneity, compositional reliability,
experimental precision and analytical validity.

####  2.1.1 Sample

In many previous studies of M-type ferrites, the reagents were mixed to a
certain composition and then calcined, and the initial composition was taken
as the nominal composition. Compositional analysis after synthesis was rarely
performed. This is because the main objective of these studies was to achieve
high performance. In such samples, the actual composition rarely matches the
initial composition due to the formation of secondary phases, precipitation of
different phases at grain boundaries and segregation of dopants on grain
surfaces. Permanent magnets are composite materials that skilfully exploit the
different functions of crystal grains and grain boundaries. The direct use of
such materials is inappropriate for evaluating the intrinsic properties of the
matrix phase. In fact, 59Co-NMR signals of various secondary phases have been
observed in NMR experiments on La–Co co-substituted M-type ferrite magnets,
making identification of the essential signal difficult. It is also known that
Co segregates on the grain surface depending on the heat treatment conditions
and plays a different role from that in the grain [27]. More importantly, the
valence of transition metals in M-type ferrite is generally unstable. La–Co
co-substituted M-type ferrites were originally designed to have equal amounts
of La and Co, with all Co being divalent, but in reality the divalent (3d6)
and trivalent (3d5) states of Fe are energetically antagonistic and generally
coexist [28, 29]. The abundance of Fe2+ and Fe3+ is dependent on the oxygen
potential during synthesis and requires a sufficiently high oxygen potential
to maintain all Fe in the trivalent state [30, 31]. In other words, if a
sample is synthesised at a low oxygen potential, the formation of Fe2+ is
inevitable. This results in a discrepancy between the design (initial)
composition and the actual composition.

The study of the intrinsic properties of materials requires the use of quality
controlled samples, whose phases and compositions have been identified from a
solid state chemistry point of view. For this purpose, it is essential to
determine the actual composition of the sample after synthesis by
compositional analysis, even if the sample is single phase. As mentioned in
section 1 and discussed in more detail in section 4.2, the dopant distribution
in M-type ferrite can be strongly dependent on the cooling conditions or post-
annealing of the sample, even if the compositions of the samples are the same.
This means that the heat treatment conditions should be carefully controlled
so that they are the same when comparing the properties of different samples.
The necessary conditions for the sample used in this study were as follows:
(i) a state close to equilibrium was achieved by properly controlling the heat
treatment conditions, (ii) it was single phase (no secondary phases), (iii)
the composition was uniform throughout the sample, and (iv) the actual
composition was determined. To satisfy these conditions, we used single
crystal samples grown by the flux method using a platinum crucible, which is a
popular growth method for M-phase single crystals [32]. To satisfy (i), the
sample was slowly cooled at a predetermined cooling rate. With natural
cooling, it was difficult to standardise the cooling conditions for each
sample. The samples were grown in air (oxygen partial pressure 0.2 atm) or in
oxygen gas flow (1 atm); these atmospheres are easy to handle and allow stable
crystal growth, facilitating (i)–(iii). However, Fe2+ contamination is
inevitable and care must be taken to ensure that the effect of Fe2+ does not
vary significantly from sample to sample. In the following NMR experiments,
powdered crystals were used; the use of single crystals as raw material
guarantees (ii). Although other crystal growth methods, such as the travelling
solvent floating zone (TS-FZ) method, are known to grow large single crystals
[33], crystals grown in a crucible were used in this study. This is because,
while the FZ method facilitates compositional control by growing crystals from
the melt in a closed space isolated in air, it limits the freedom of atomic
flow in the melt, and the sample is generally cooled rapidly, which may cause
the synthesised crystals to deviate from their equilibrium state. In the
crucible flux method, the crystals grow gently in the melt without significant
perturbation, resulting in the growth of crystals in a state close to
equilibrium (note that this is not necessarily the ideal equilibrium phase, as
described in section 1), and (i) and (iii) are satisfied. Indeed, 59Co-NMR
spectra of La–Co co-substituted SrFe12O19 show slightly different
characteristics for crystals synthesised by the TS-FZ method and crystals
synthesised by the flux method using a crucible [11].

Wavelength dispersive X-ray spectrometry (WDX) with a scanning electron
microscope (SEM) was used to satisfy condition (iv). Unlike energy dispersive
X-ray analysis (EDX), WDX can completely separate the Fe and Co spectra,
providing a highly accurate analysis of Co composition. X-ray analysis, which
only observes the sample surface, has been criticised for its low accuracy.
However, comparison with inductively coupled plasma atomic emission
spectrometry (ICP-AES), which gives the average value for the bulk sample,
shows that the two analyses are in good agreement [33, 34]. The high
reliability of WDX has been demonstrated in many studies [28, 31, 34, 35, 36,
37, 38, 39, 40, 41]. In addition, WDX, which analyses a narrow spot where
X-rays are irradiated, guarantees (iii) by sampling different parts of the
crystal.

We have applied this sample strategy to systematic studies of La substituted
and La–Co co-substituted SrFe12O19 and CaFe12O19. Their structural and
magnetic properties have been reported in detail in [28, 34, 35]. In this
study, among the single crystals synthesised in these series of studies, we
selected those with a Co composition close to 0.2 and in sufficient quantity
(see table 1 and section 2.2 for details). Not all samples had a Co
composition of exactly 0.2. This is because the final composition of flux-
grown M-phase crystals is difficult to deliberately control; the M-phase melts
incongruently [42], and the elemental fraction of the reagent before melting
and the composition of the grown crystal are generally different.
Nevertheless, the compositions determined by WDX were sufficiently accurate
and the samples used in this study satisfy conditions (i)–(iv) above. It
should be noted that the information used in this study is the ratio of Co
occupancy among sites, not the absolute amount, so that differences in Co
composition within reasonable limits will not affect the outcome of the
discussion. As the samples used in this study were synthesised at relatively
low oxygen potentials, the formation of Fe2+ was inevitable. However, since
the design concept of La–Co co-substitution is to prevent the formation of
Fe2+, and the purpose of this study was to investigate the site-to-site
distribution of divalent Co, we assumed that the effect of Fe2+ in the samples
used in this study could be neglected. In fact, the effect of Fe2+ on the
magnetic properties is not significant in the present samples; the presence of
Fe2+ reduces the uniaxial magnetic anisotropy in regions of low Fe2+
concentration [33, 43]. The physics of Fe2+ in M-type ferrite is certainly
important [44, 45], but is not considered in this study; the discussion is
limited to outside its influence. In La substituted CaFe12O19, the Fe/(Ca +
La) content ratio is known to be less than 12 [46]. Some Ca may occupy Fe
sites, but the origin of this anomaly has not been fully understood [18, 35,
47, 48]. We will not address this point in detail in this study.

####  2.1.2 Evaluation method for site-to-site distribution of Co

What we need in this study is a rapid method for assessing the site-to-site
distribution of Co. The current understanding of the Co occupancy sites in
La–Co co-substituted SrFe12O19 and CaFe12O19 was first achieved by combining
synchrotron radiation with neutron experiments [9, 16, 17, 18]. Experiments
requiring large-scale facilities are certainly useful, but lack immediacy and
availability. Rapid on-site evaluation of synthesised samples is needed to
accelerate research.

Nuclear methods, such as Mössbauer spectroscopy and NMR, have been widely used
to assess cation distribution with high immediacy and availability. However,
their application has historically caused confusion in the identification of
Co sites in La–Co co-substituted M-type SrFe12O19 [26]. Recently, a series of
studies have been carried out to investigate the validity and interpretation
of 57Fe Mössbauer spectroscopy and 57Fe, 59Co-NMR in this research area [11,
12, 13, 49]. As a result, we have confirmed that although the 57Fe probe
methods are useful for discussing microscopic magnetism, for example through
experiments under a magnetic field, they are not ideal for quantifying the Co
distribution. In the case of M-type ferrite, quantitative analysis of the Co
distribution requires the detection of trivial intensity changes in the 57Fe
spectrum, which is composed of overlapping multi-components. Information on Co
is obtained from changes in the number of Fe atoms, but the effect of Co
substitution is attenuated because Co is distributed over multiple sites. The
intensity of the 57Fe signal is more sensitive to the substitution modified
environment than to the number of Fe atoms occupying the site [11, 12]. In
addition, the lattice anharmonicity of M-type ferrites [50, 51, 52, 53] makes
it difficult to evaluate the true signal intensity and challenges the
assumption that the signal intensity is proportional to the abundance of the
corresponding nuclei. Thus, the shortcomings of the 57Fe probe methods are
more pronounced in the quantitative evaluation of the Co distribution. This is
in contrast to the fact that the 57Fe probe experiment works well for
discussing more dramatic changes such as the crystal structure transition in
M-type ferrites [43, 45].

In 59Co-NMR, on the other hand, the integrated intensity of the signal
corresponds directly to the number of Co atoms occupying a particular site. In
principle, therefore, 59Co-NMR provides simple and direct information on the
Co occupancy sites and substitution amounts, provided that the site assignment
of the signal is complete. The NMR signal intensity of 59Co nuclei is
generally extremely strong, and especially in ferromagnetic materials the
signal intensity is greatly enhanced; thus, significant information can be
extracted, even with a small amount of Co. The problem in previous studies of
59Co-NMR on La–Co co-substituted M-type ferrites [49, 54, 55] was the site
assignment of the signal and possibly the sample used. The site assignment of
the 59Co-NMR signal of La–Co co-substituted SrFe12O19 has already been
completed with respect to the distinction between tetrahedrally and
octahedrally coordinated sites (see section 2.1 in [23]). We believe that this
conclusion is also valid for other La–Co co-substituted M-type ferrites.

In this study, the NMR spectra were recorded on a home-built state-of-the-art
broadband spectrometer at the Institut de Physique et Chimie des Matériaux de
Strasbourg (IPCMS) using an untuned (high-pass design) probe [24, 25], which
ensures good reproducibility of the frequency spectrum. This system is
particularly suitable for the measurement of ferromagnetic (and ferrimagnetic)
materials, whose resonance signals are distributed over a wide frequency
range. When there are multiple magnetic sites in different local environments,
as in M-type ferrites, the local anisotropy of each site may be different. In
such cases, the signal enhancement factor will be different at each site, and
it is necessary to correct for the frequency dependence of the enhancement
factor in order to evaluate the distribution of nuclei corresponding to the
resonance frequency. The enhancement factor at each frequency point was
evaluated by measuring the signal intensity at each frequency using multiple
radiofrequency (RF) field powers. The RF power dependence of the signal
intensity was fitted with a suitable function, the width of which was also
taken into account. From the above series of procedures, the enhancement
factor was used to correct the signal intensity. Monitoring the enhancement
factor also facilitates the differentiation of NMR signals from the magnetic
domains and walls. Intensities were corrected by dividing by the square of the
frequency, which is standard in NMR methods. The NMR spectrum corrected by the
above process corresponds to the distribution of nuclei corresponding to the
resonance frequencies and is independent of the value and distribution of the
enhancement factor. Furthermore, it is easy to calculate the local magnetic
anisotropy of each environment from the signal enhancement factor, providing a
unique insight into the magnetic properties of the sample, which has been
described in detail in [24, 25].

Note that the signal intensity is corrected by dividing it by the square of
the frequency and that the resonance frequency for our materials is
significantly spread over the frequency domain (the lower and upper frequency
limits differ by an order of magnitude, see section 3.2). This means that the
correction factor differs by a factor of $10^{2}$ between the lowest and
highest frequencies. Thus, if the correction factors contain potential errors,
however small, these errors will be amplified in the final intensities after
correction. Therefore, the final quantitative value of the Co distribution may
contain an error of the order of 10%. Nevertheless, the quantitative nature of
the system at IPCMS is exceptional for this type of experiment. Moreover, the
errors contained in the correction coefficients are expected to vary
systematically with frequency, so that the qualitative trend of the intensity
distribution cannot be seriously affected. Furthermore, it should be noted
that even sophisticated EXAFS and XMCD measurements show overlapping spectra
of several components derived from Co occupying several sites. In order to
quantitatively assess the amount of Co at each site, the intensity of each
component must be decomposed by fitting. In contrast, in the 59Co-NMR
experiment, the signal derived from Co occupying each site was observed
without overlap (see figure 2). In short, the direct quantification, speed and
simplicity of the 59Co-NMR method are outstanding among experimental
techniques for the same purpose. It is therefore undeniable that the 59Co-NMR
is the most appropriate method for this study. This study follows a series of
studies in which 59Co-NMR experiments have been applied to La–Co co-
substituted M-type ferrites from the above perspective [23, 49].

In this study single crystals were powdered for NMR measurements. As explained
in section 2.1.1, the reason for using single crystals is to ensure that the
sample is single phase. As the main objective of this study was to assess the
Co occupancy between sites, single crystals were powdered. Sakai et al [11]
reported a difference in NMR results between powdered and single crystal
samples for the same sample type; there is anisotropy in the excitation of Co
nuclei in this system, and this anisotropy is site dependent [23]. This
complicates the process of evaluating Co occupancy ratios from single crystal
NMR spectra. In particular, the fact that the 12k site has low local symmetry
and multiple local symmetry axes further complicates the analysis. In this
study, we have avoided this problem by powdering the crystals and measuring
the average intensity.

####  2.1.3 Prediction of Co site preference

First-principles total energy calculations based on DFT have been used to help
interpret the experimental results. The subject of discussion is the site-to-
site distribution of Co, that is, atomic-level issues. One of the objectives
is also to perform structural optimisation calculations to discuss the stable
structure of the crystal. Therefore, the Vienna Ab initio Simulation Package
(VASP) [56, 57, 58, 59] with a plane-wave basis and the projector-augmented
wave (PAW) method [60, 61], which is suitable for total energy calculations
with fast structure optimisation, was chosen as the calculation code. The use
of the PAW method reduces the computational load. However, the PAW method does
not explicitly treat the core orbitals. Therefore, VASP is less effective in
discussing phenomena where the contribution of the core electrons is dominant,
such as hyperfine interactions at transition metal sites. Therefore, this DFT
calculation is only used to predict the Co site preference. The physics of the
anharmonic lattice vibrations of Fe at the 2b site [50, 51, 52, 53] was also
not investigated in depth in this study. Anharmonic fluctuations at the 2b
site are likely to be intrinsically important for the multiferroic nature of
hexagonal ferrites [53, 62]. However, we assume that the effect of Co
substitution on the anharmonicity of the 2b site is not pronounced, and the
possibility of Co substitution at the 2b site has been experimentally ruled
out [9, 16, 17, 18].

Previous DFT studies have attempted to predict Co occupancy sites in La–Co co-
substituted SrFe12O19 [10, 63]. Hui et al [63] predicted that the priority
occupancy site of Co is the 2a site in their calculations using generalised
gradient approximations (GGA) [64, 65]. This differs from the current view.
Ohtsuka et al [10] performed calculations using GGA with Hubbard-type orbital
interaction (GGA+$U$), but failed to reproduce the experimental results of the
Co occupancy site. Calculations using GGA+$U$ are relatively expensive, but
require the provision of appropriate Hubbard $U$ values for the transition
metal atoms. However, in addition to the fact that the appropriate $U$ is not
clear a priori, it has already been pointed out that DFT calculation results
for M-type ferrites are sensitive to $U$ [45, 66, 67]. Our preliminary
calculations using GGA+$U$ showed that even if a particular $U$ could explain
the properties of a particular substance, the same $U$ could not uniformly
explain the overall trend among different substances. Therefore, we will
attempt a more accurate calculation that is feasible on realistic time scales
and does not include arbitrary parameters, such as $U$. For this purpose, we
have chosen the Heid-Scuseria-Ernzerhof hybrid exchange-correlation functional
(HSE06) [68]. To facilitate the convergence of the calculations, we will first
perform a structure optimisation with the GGA+$U$ calculation using the
appropriate $U$ and use the resulting structure parameters as initial values
for the HSE06 calculation.

Calculations using HSE06 are relatively computationally intensive, and in
order to perform the calculations in a realistic time, the structural model
must be suitably simplified. The model for the Co substituent of $A$Fe12O19 is
described below. First, the interaction between different Co atoms was
neglected, assuming that the Co atoms were sufficiently dilute. This is
because considering the interactions between Co atoms requires considering
many cases with different arrangements of neighbouring Co atoms. We consider a
supercell of size $2\times 2\times 1$ times the unit cell as the smallest
supercell that satisfies the condition that Co is sufficiently dilute and
replaces one of the $24\times 4=96$ Fe atoms with a Co atom. This corresponds
to a substitution of 0.125 Co for a Fe+Co composition of 12. This Co
composition is lower than that of the samples used in the NMR experiment
($\sim$0.2). However, it was assumed that the condition where Co is
sufficiently diluted is also fulfilled in the experiment, and a comparison was
made between the calculations and the experiment. Supercells of the five
types, with Co substituted at each Fe site, were studied.

Second, the effect of La substitution was assumed to be replaced by the effect
of one additional electron per supercell. In real materials, La3+ is replaced
by $A^{2+}$ at the same time as the Co substitution for charge compensation.
This means that Fe3+ in $A$Fe12O19 is replaced by Co2+ instead of Co3+.
However, in models with the co-substitution of La and Co, there are multiple
relative positions of Co and La, and the number of such cases is large. This
assumption is equivalent to neglecting the interaction between La and Co, the
model seems reasonable as a first approximation. The following calculations
confirm that the value of the Co bond valence sum is in the range of +1.9 to
+2.3 in all cases (except for 4f2-Co for $A$ = Ca, see section 3.3.3). The
magnitude of the magnetic moment at this time was approximately 2.6
$\mu_{\mathrm{B}}$/Co. In other words, Co was converging to a high spin state,
close to the divalent state. This suggests that the structural model is
working to some extent. In fact, it is quite possible that the partial
substitution of the $A$ site by La affects the electronic state of the system
and the priority occupancy site of Co. To confirm this, it is necessary to
perform structural optimisation calculations for systems with both La and Co
substitutions; this will be reserved for future research.

Table 1: List of samples used for 59Co-NMR measurements, their lattice
constants at room temperature and their anisotropy fields at 5 K.

Label | Chemical formula |  $r(A^{2+})$† | $a$ | $c$ | $c/a$ | $H_{\mathrm{A}}$ | Ref.
---|---|---|---|---|---|---|---
|  | (Å) | (Å) | (Å) |  | (T) |
Ca17 |  Ca0.77La0.37Fe11.70Co0.17O19 | 1.34 | 5.8890 | 22.919 | 3.8918 | 2.689 | [35]
Sr28 |  Sr0.68La0.32Fe11.72Co0.28O19 | 1.44 | 5.8839 | 23.005 | 3.9098 | 2.540 | [23]
Ba20 |  Ba0.70La0.30Fe11.80Co0.20O19 | 1.61 | 5.8910 | 23.141 | 3.9282 | 2.083 | –

† 12-coordinated ionic radius [69].

###  2.2 Sample preparation and characterization

The samples used for the 59Co-NMR experiments were selected from our stock of
crystals, one each for $A$ = Ca, Sr and Ba in La–Co co-substituted $A$Fe12O19,
following the strategy described in section 2.1.1. Samples with Co
compositions around 0.2 were selected (see table 1). These three samples are
referred to as Ca17, Sr28 and Ba20, respectively. For comparison of the
$A$-ion size, the ionic radius of the 12-coordination by Shannon [69] is added
to table 1.

Crystal growth techniques for Ca17 and Sr28 have been reported in [35, 23].
Systematic studies of flux-grown single crystals of La–Co co-substituted
SrFe12O19 and CaFe12O19 have been reported in detail [28, 35]. The Ba20
crystals were grown under the same conditions as Ca17. All crystals were
cooled at $-5$ ∘C/h above 1000 ∘C and $-200$ ∘C/h below 1000 ∘C to ensure the
same cooling conditions. All compositions shown in table 1 are actual
compositions determined for the grown crystals using SEM-WDX (INCA wave 500,
Oxford Instruments attached to SEM S-3500H, Hitachi). For Sr28 and Ba20, the
(Fe+Co)/($A$+La) composition ratio was 12, whereas for Ca17, the
(Fe+Co)/(Ca+La) ratio was greater than 12, which is characteristic of
CaFe12O19 as mentioned in section 2.1.1. The room temperature lattice
constants given in table 1 [23, 35] were estimated by X-ray diffraction (XRD)
using a PANalytical X’Pert PRO Alpha-1 for partially powdered single crystals.

The anisotropy field $H_{\mathrm{A}}$ at 5 K shown in table 1 [23, 35] was
obtained from the analysis of single crystal magnetisation curves, which were
measured with a SQUID magnetometer (Quantum Design, MPMS). The anisotropy
energy $E_{\mathrm{A}}$ was evaluated from the region bounded by the easy and
hard axis magnetisation curves. Using the values of $E_{\mathrm{A}}$ and the
saturation magnetisation $M_{\mathrm{s}}$, $H_{\mathrm{A}}$ was obtained from
the relation $H_{\mathrm{A}}=2E_{\mathrm{A}}/M_{\mathrm{s}}$. The results of
our systematic magnetic measurements of La–Co co-substituted CaFe12O19 and
SrFe12O19 have been reported previously [28, 35]. The magnetisation curve of
single crystalline BaFe12O19 [70] is qualitatively the same as that of
SrFe12O19, except that the magnetic field at which the hard axis magnetisation
reaches saturation (i.e., $H_{\mathrm{A}}$) is slightly lower. In the B20
material used in this study, the effect of La–Co co-substitution on BaFe12O19
is qualitatively the same as that on SrFe12O19. Although the $M_{\mathrm{s}}$
values of the three samples were almost the same, the absolute value of
$M_{\mathrm{s}}$ is not discussed in this paper for the following reasons.
Taking into account the Co distribution revealed in section 3.2, the change in
magnetisation with Co substitution in this system is expected to be at most
0.5%. Co substitution makes the magnetisation curve convex, and suppresses the
saturation of magnetisation even under a magnetic field of 7 T [28, 35]. In
addition, when measuring the hard axis magnetisation of single crystals, the
sample may tilt slightly during the measurement due to the large torque
exerted on the sample by the magnetic field. Therefore, it is technically
difficult to experimentally detect the expected change in $M_{\mathrm{s}}$ due
to Co substitution, and even if such a change is observed, it is not
appropriate to simply discuss its origin.

###  2.3 NMR

The 59Co-NMR experiments were performed at IPCMS, following the strategy
described in section 2.1.2. The nuclear quantities for 59Co are the nuclear
spin $I=7/2$, the gyromagnetic ratio $\gamma/2\pi=10.05$ MHz T-1, the
quadrupole moment $Q=42.0\times 10^{-30}$ m2 and a natural abundance of 100%.
For the reasons described in section 2.1.2, the single crystals were powdered
for NMR measurements. Measurements were performed at 2 K under zero external
field in the frequency range 40–600 MHz using 1.28 MHz steps. A classical
spin-echo sequence of RF pulses was used. To observe magnetic domain
excitations, we used wide pulses (12.8 $\mu$s) separated by a 5 $\mu$s delay.
The maximum RF power range of our setup was used ($\leq$ 32–40 dBm). Each
spectrum results in 5,000 to 10,000 frequency scans, depending on the mass of
the sample (100 to 200 mg). To evaluate the signal enhancement factor, the
signal intensity was measured as a function of RF power (four or more points)
at each frequency. The NMR data for Sr28 have already been reported in [23]
and some of the results of this study are quoted from them.

###  2.4 DFT calculations

The stable structures and total energies of the undoped materials and Co
substituents of $A$Fe12O19 ($A$ = Ca, Sr and Ba) were evaluated by structure-
optimisation calculations using VASP. The convergence condition was set to
$1\times 10^{-5}$ eV ($1\times 10^{-4}$ eV for GGA+$U$ preliminary
calculations) for the electronic self-consistent loop and 0.01 eV/Å for the
structure optimisation. The $k$-mesh was $2\times 2\times 1$ and the energy
cutoff was 500 eV. The experimental values reported in [52, 71] were used as
the initial values of the structural parameters in the GGA+$U$ calculations.
However, since CaFe12O19 does not exist in the equilibrium phase, the value of
SrFe12O19 was used for the calculations when $A$ = Ca.

Calculations for undoped $A$Fe12O19 were carried out using the following
procedure, assuming that the Fe sublattice has a conventional ferrimagnetic
structure collinear along the $c$-axis. (i) Preliminary GGA+$U$ calculations
using arbitrary values in the range 4 to 6 eV as $U$ for Fe. (ii) HSE06
calculations with the structural parameters optimised in (i) as initial values
(the results are treated as final). (iii) Optimal GGA+$U$ calculations using
the $U$ value ($U$(Fe) = 4.1 eV) obtained by the method described at the end
of this section (the results are treated as reference data).

For the Co substituents, as described in section 2.1.3, the $2\times 2\times
1$ supercell of the $A$Fe12O19 unit cell was considered, in which one of the
Fe ions was replaced by Co2+ (valence initially set to 2), and then an
electron was added. Supercells of the five types, with Co substituted at each
Fe site, were studied. The magnetic moments $m$ of the host Fe and the doped
Co were assumed to be collinear along the $c$-axis. In this case, $m$(Co) can
be parallel or antiparallel to $m$(Fe). We calculated both cases for $A$ = Sr,
but only the parallel case for $A$ = Ca and Ba, considering the results for
$A$ = Sr (section 3.3.2). The calculations were carried out as follows: (i)
Preliminary GGA+$U$ calculations with $U$(Fe) = 6 eV and $U$(Co) = 5 eV. (ii)
HSE06 calculations with the structural parameters obtained in (i) as initial
values (the results are treated as final).

The optimal $U$ values in the GGA+$U$ calculations were sought to reproduce
the total energy difference given by the HSE06 calculations for the Co
substituents of SrFe12O19. Under the constraint $U$(Fe)/$U$(Co) = 1.2, the
HSE06 results are best reproduced when $U$(Fe) = 4.1 eV and $U$(Co) = 3.4 eV.
Based on this result, the structural parameters of undoped $A$Fe12O19 were
again optimised by GGA+$U$ calculations assuming $U$(Fe) = 4.1 eV (reference
data in tables 3 and 5). However, the GGA+$U$ calculations for the Co
substituents of CaFe12O19 using the same $U$ values failed to reproduce the
experimental tendency of the Co distribution (section 3.2); the calculations
predicted that 12k-Co and 2a-Co are more stable than 4f1-Co in CaFe12O19.

##  3 Results

###  3.1 $A$-ion dependence of structural and magnetic parameters

The lattice constants of the three samples measured at room temperature and
the anisotropy field, $H_{\mathrm{A}}$, at 5 K are listed in table 1. The
values for Ca17 and Sr28 are taken from [28, 35]. The parameter $a$ changes
little and the difference in $a$ is at most 0.1%. In contrast, $c$ is highest
for $A$ = Ba and lowest for $A$ = Ca with a difference of approximately 1%.
This means that the effect of the $A$-ion size is anisotropic. This may be
because $a$ is determined by the skeleton of the oxygen ions, whereas $c$
(particularly the R-block length) directly reflects the effect of the $A$-ion
size. Thus, the change in $A$ ions acts as a uniaxial strain. In the
following, $c/a$ is used as a parameter to characterise the structure of the
three samples.

As shown in table 1, the anisotropic field $H_{\mathrm{A}}$ (at 5 K) is
negatively correlated with the $A$-ion size and the value of $c/a$.

###  3.2 NMR

Figure 2(a) shows the 59Co-NMR spectra of the three samples measured at zero
field and 2 K. Three types of signals, S1, S2 and S3, were observed in all
samples as in the previous studies [23, 49]. An additional signal (S4) was
observed at $\sim$530 MHz in Sr28 [23]. The origin of S4 has been attributed
to high spin Co3+, which may be present in very small amounts due to an
unexpected charge imbalance. However, its intensity is less than 0.2% of the
total Co signal intensity and is therefore ignored in this study. No S4
signals of significant intensity were observed for Ca17 and Ba20. Although the
signal intensity ratios vary from sample to sample, it is noteworthy that
three similarly shaped signals were observed in almost all common frequency
ranges. This indirectly suggests that various parameters other than the
differences in $A$ ions remain nearly constant during crystal growth. The
intensity of S1 was the highest for all samples. In figure 2(a) the signal
intensity is normalised to the intensity of S1. The centre frequencies of S1,
S2 and S3, evaluated by fitting to a Gaussian function, are listed in table 2.
They differed slightly from sample to sample. This frequency shift is
discussed in the last part of section 4.3.

Figure 2(a) shows that the intensities of S2 and S3 tend to decrease with
decreasing $A$-ion size ($A$ = Ba $\rightarrow$ Sr $\rightarrow$ Ca). The
proportions $P(j)$ ($j$ = S1, S2, S3) of the integrated intensities of S1, S2
and S3 for each sample are shown in figure 2(b). As the $A$-ion size
decreased, $P(\mathrm{S1})$ increased. $P(\mathrm{S3})$ decreased slightly and
the change was small. In contrast, $P(\mathrm{S2})$ decreased significantly
and at $A$ = Ca it decreased to approximately 1.5% of the total. That is,
although S2 and S3 were observed in similar frequency ranges, the $A$-ion
dependence of their intensities was large for $P(\mathrm{S2})$ and small for
$P(\mathrm{S3})$.

The site assignments of the S1, S2 and S3 resonances when $A$ = Sr have been
described in detail in section 3.1 of [23]. In this study, we apply this
conclusion to the other $A$-ion cases as well; the S1 resonance is assigned to
4f1-Co and the remaining S2/S3 are assigned to 2a/12k-Co. Kobayashi et al [16,
17, 18] reported that Co occupies three sites: 2a, 4f1 and 12k, with 4f1
occupying the largest fraction, also for La–Co co-substituted CaFe12O19,
according to studies using synchrotron radiation and neutrons. This result
supports our site assignment. In [23] we could not conclude whether S2 or S3
was assigned to 2a-Co or 12k-Co. However, considering the cation diffusion
processes during the synthesis procedure, the occupancy of 12k-Co is expected
to be sensitive to various conditions (including $A$-ion differences), whereas
that of 2a-Co is relatively invariant (see section 4.2 for details). The
$A$-ion dependence of the signal intensity was large in S2 and small in S3
(figure 2(b)), qualitatively suggesting an S2 $\rightarrow$ 12k and S3
$\rightarrow$ 2a assignment.

Figure 2:  (a) 59Co-NMR spectra for La–Co co-substituted $A$Fe12O19 measured
at zero applied field and at 2 K. The intensity was normalised so that the
maximum intensity of S1 is the same. (b) Intensity ratios of S1, S2 and S3 for
the three samples.  Figure 3:  (a) Dependence of the anisotropy field on the
4f1 site selectivity of Co. (b) $c/a$ dependence of the 4f1 site selectivity
of Co in La–Co co-substituted $A$Fe12O19 ($A$ = Ca, Sr and Ba) evaluated from
the 59Co-NMR signal intensity. The straight lines in (a) and (b) are visual
aids.  Table 2:  59Co ferromagnetic nuclear resonance frequencies of La–Co co-
substituted $A$Fe12O19 at zero field and at 2 K, corresponding internal fields
and their site assignments. Each 59Co-NMR line was fitted with a Gaussian
function and the median was taken as the resonance frequency. The direction of
the local magnetic moment at each site and the direction of the internal field
are defined as positive if they are parallel and negative if they are
antiparallel.

  * Signal |  $\nu_{\mathrm{res}}$ (MHz) |  |  $H_{\mathrm{int}}$ (T) | Site
---|---|---|---|---
| Ca17 | Sr28 | Ba20 |  | Ca17 | Sr28 | Ba20 |
S1 | 82.4(4) | 83.6(4) | 89.4(2) |  |  $-8.20$(4) |  $-8.32$(4) |  $-8.90$(2) | 4f1
S2 | 327.0(29) | 308.3(6) | 301.3(4) |  | 32.54(29) | 30.68(6) | 29.98(4) | 12k
S3 | 384.5(15) | 389.1(5) | 391.8(2) |  | 38.26(15) | 38.72(5) | 38.99(2) | 2a

Based on the site assignment above, the Co 4f1 site occupancy $P$(4f1) was
highest at $A$ = Ca and lowest at $A$ = Ba. Figure 3(a) shows the anisotropy
field $H_{\mathrm{A}}$ of each sample at 5 K plotted against $P$(4f1) (=
$P$(S1)). These values are almost proportional to each other, indicating that
the 4f1 site selectivity of Co does indeed dominate the magnetic anisotropy.

As mentioned in section 3.1, the difference in $A$ ions appears as a uniaxial
strain. Figure 3(b) shows the $c/a$ dependence of $P$(4f1) for each sample.
$P$(4f1) increased with decreasing $c/a$. It is therefore hypothesised that
the uniaxial compressive strain of the crystal improves the 4f1 site
selectivity of Co and consequently the uniaxial magnetic anisotropy. As the
4f1 site is a minority spin site, the magnetisation is simultaneously
increased. To quantitatively evaluate the effect of different Co compositions
of the samples used in the experiments, one can refer to a previous study on
La–Co co-substituted SrFe12O19 [23]. In this system, the change in $c/a$ was
approximately 0.08% when the Co concentration was changed by approximately
0.1. Assuming that the Co occupancy ratio depends only on the value of $c/a$,
the expected change in $P$(4f1) does not exceed 0.02 (see figure 3(b)), which
is sufficiently small compared with the change when the $A$ ion is replaced.
When $A$ = Ca, the Ca composition is greater than its stoichiometric
composition. If excess Ca occupies the 4f1 site, as proposed by Kobayashi et
al [18], the number of 4f1 sites available for Co to occupy would be reduced,
and hence $P$(4f1) in Ca17 would be expected to be suppressed compared to Sr28
and Ba20. However, this was actually increased. This suggests that even if Ca
partially occupies the 4f1 site, the effect is insignificant and does not
affect the discussion here.

###  3.3 DFT calculations

Table 3:  Lattice parameters, $c$-axis lengths of S and R blocks, and unit
cell volumes of $A$Fe12O19 ($A$ = Ca, Sr and Ba) obtained by structure-
optimisation DFT calculations with HSE06 and GGA+$U$ ($U$(Fe) = 4.1 eV).
Available experimental values at room temperature are also given. No
experimental values are available for $A$ = Ca because the undoped CaFe12O19
is not stable.

  * $A$ | Method |  $a$ (Å) |  $c$ (Å) | $c/a$ |  $c$(S) (Å) |  $c$(R) (Å) |  $V$ (Å3) | Ref.
---|---|---|---|---|---|---|---|---
Ca | HSE06 | 5.880 | 22.892 | 3.893 | 5.031 | 6.415 | 685.4 |
| GGA+$U$ | 5.917 | 22.963 | 3.881 | 5.041 | 6.441 | 696.2 |
| Experiment | – | – | – | – | – |  |
Sr | HSE06 | 5.885 | 23.017 | 3.912 | 5.025 | 6.483 | 690.4 |
| GGA+$U$ | 5.931 | 23.158 | 3.905 | 5.052 | 6.527 | 705.5 |
| Experiment | 5.884 | 23.050 | 3.917 | 5.039 | 6.486 | 691.1 | [52]
Ba | HSE06 | 5.897 | 23.162 | 3.928 | 5.013 | 6.568 | 697.5 |
| GGA+$U$ | 5.943 | 23.307 | 3.921 | 5.038 | 6.615 | 713.0 |
| Experiment | 5.892 | 23.183 | 3.935 | 5.019 | 6.573 | 697.0 | [71]

Table 4:  Atomic coordinates of $A$Fe12O19 ($A$ = Ca, Sr and Ba) obtained by
structure-optimisation DFT calculations with HSE06. Experimental data for
SrFe12O19 and BaFe12O19 were taken from [52] and [71], respectively.

  * Element/Site | HSE06 | GGA+$U$ | Experiment
---|---|---|---
| $x$ | $y$ | $z$ | $x$ | $y$ | $z$ | $x$ | $y$ | $z$
CaFe12O19 |  |  |  |  |  |  |  |  |
Ca/2d | 0.6667 | 0.3333 | 0.2500 | 0.6667 | 0.3333 | 0.2500 | – | – | –
Fe/2a | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | – | – | –
Fe/2b | 0.0000 | 0.0000 | 0.2500 | 0.0000 | 0.0000 | 0.2500 | – | – | –
Fe/4f1 | 0.3333 | 0.6667 | 0.0270 | 0.3333 | 0.6667 | 0.0275 | – | – | –
Fe/4f2 | 0.3333 | 0.6667 | 0.1913 | 0.3333 | 0.6667 | 0.1915 | – | – | –
Fe/12k | 0.1691 | $0.3382$ | 0.8902 | 0.1688 | 0.3375 | 0.8902 | – | – | –
O/4e | 0.0000 | 0.0000 | 0.1521 | 0.0000 | 0.0000 | 0.1525 | – | – | –
O/4f | 0.6667 | 0.3333 | 0.0556 | 0.6667 | 0.3333 | 0.0557 | – | – | –
O/6h | 0.1817 | $0.3633$ | 0.2500 | 0.1821 | 0.3643 | 0.2500 | – | – | –
O/12k1 | 0.1572 | $0.3144$ | 0.0527 | 0.1563 | 0.3125 | 0.0530 | – | – | –
O/12k2 | 0.5056 | $0.0112$ | 0.1525 | 0.5061 | 0.0123 | 0.1523 | – | – | –
SrFe12O19 |  |  |  |  |  |  |  |  |
Sr/2d | 0.6667 | 0.3333 | 0.2500 | 0.6667 | 0.3333 | 0.2500 | $\frac{1}{3}$ | $\frac{2}{3}$ | $\frac{1}{4}$
Fe/2a | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0 | 0 | 0
Fe/2b | 0.0000 | 0.0000 | 0.2500 | 0.0000 | 0.0000 | 0.2500 | 0 | 0 | 0.2542
Fe/4f1 | 0.3333 | 0.6667 | 0.0270 | 0.3333 | 0.6667 | 0.0271 | $\frac{1}{3}$ | $\frac{2}{3}$ | 0.0272
Fe/4f2 | 0.3333 | 0.6667 | 0.1909 | 0.3333 | 0.6667 | 0.1910 | $\frac{1}{3}$ | $\frac{2}{3}$ | 0.1909
Fe/12k | 0.1689 | $0.3378$ | 0.8908 | 0.1688 | 0.3376 | 0.8909 | 0.1689 | $2x$ | 0.8907
O/4e | 0.0000 | 0.0000 | 0.1516 | 0.0000 | 0.0000 | 0.1520 | 0 | 0 | 0.1516
O/4f | 0.6667 | 0.3333 | 0.0552 | 0.6667 | 0.3333 | 0.0554 | $\frac{2}{3}$ | $\frac{1}{3}$ | 0.0552
O/6h | 0.1817 | $0.3634$ | 0.2500 | 0.1819 | 0.3638 | 0.2500 | 0.1817 | $2x$ | $\frac{1}{4}$
O/12k1 | 0.1570 | $0.3141$ | 0.0524 | 0.1568 | 0.3136 | 0.0525 | 0.1565 | $2x$ | 0.0527
O/12k2 | 0.5041 | $0.0083$ | 0.1511 | 0.5044 | 0.0088 | 0.1512 | 0.5047 | $2x$ | 0.1508
BaFe12O19 |  |  |  |  |  |  |  |  |
Ba/2d | 0.6667 | 0.3333 | 0.2500 | 0.6667 | 0.3333 | 0.2500 | $\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{4}$
Fe/2a | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0 | 0 | 0
Fe/2b | 0.0000 | 0.0000 | 0.2500 | 0.0000 | 0.0000 | 0.2500 | 0 | 0 | $\frac{1}{4}$
Fe/4f1 | 0.3333 | 0.6667 | 0.0272 | 0.3333 | 0.6667 | 0.0272 | $\frac{1}{3}$ | $\frac{2}{3}$ | 0.0273
Fe/4f2 | 0.3333 | 0.6667 | 0.1903 | 0.3333 | 0.6667 | 0.1903 | $\frac{1}{3}$ | $\frac{2}{3}$ | 0.1904
Fe/12k | 0.1687 | $0.3374$ | 0.8918 | 0.1686 | 0.3372 | 0.8919 | 0.1687 | $2x$ | 0.8917
O/4e | 0.0000 | 0.0000 | 0.1508 | 0.0000 | 0.0000 | 0.1511 | 0 | 0 | 0.1504
O/4f | 0.6667 | 0.3333 | 0.0555 | 0.6667 | 0.3333 | 0.0549 | $\frac{2}{3}$ | $\frac{1}{3}$ | 0.0548
O/6h | 0.1818 | $0.3635$ | 0.2500 | 0.1820 | 0.3640 | 0.2500 | 0.1821 | $2x$ | $\frac{1}{4}$
O/12k1 | 0.1568 | $0.3136$ | 0.0519 | 0.1565 | 0.3130 | 0.0520 | 0.1563 | $2x$ | 0.0521
O/12k2 | 0.5021 | $0.0042$ | 0.1491 | 0.5023 | 0.0046 | 0.1492 | 0.5024 | $2x$ | 0.1493

Table 5:  Volume of the Fe–O coordination polyhedron of each Fe site in
undoped $A$Fe12O19 ($A$ = Ca, Sr and Ba) calculated from atomic coordinates
obtained by structure-optimisation DFT calculations with HSE06 and GGA+$U$
($U$(Fe) = 4.1 eV) (table 4). Experimental values calculated from the
available atomic coordinates at room temperature are also given. No
experimental values are available for $A$ = Ca, as the undoped material
CaFe12O19 is not stable.

  * $A$ | Method |  $V$ (Å3) | Ref.
---|---|---|---
|  | 2a | 2b | 4f1 | 4f2 | 12k |
Ca | HSE06 | 10.714 | 6.650 | 3.454 | 10.527 | 10.926 |
| GGA+$U$ | 10.805 | 6.758 | 3.557 | 10.711 | 11.039 |
| Experiment | – | – | – | – | – |
Sr | HSE06 | 10.693 | 6.730 | 3.461 | 10.646 | 10.946 |
| GGA+$U$ | 10.922 | 6.863 | 3.559 | 10.871 | 11.187 |
| Experiment | 10.706 | 6.737 | 3.498 | 10.730 | 10.893 | [52]
Ba | HSE06 | 10.671 | 6.860 | 3.475 | 10.829 | 10.983 |
| GGA+$U$ | 10.897 | 7.002 | 3.572 | 11.062 | 11.223 |
| Experiment | 10.671 | 6.871 | 3.482 | 10.783 | 10.999 | [71]

####  3.3.1 Undoped $A$Fe12O19

We have carried out structure-optimisation DFT calculations on $A$Fe12O19 to
obtain information on the stable lattice constants and local structure of
$A$Fe12O19 in the undoped form. The lattice constants obtained from the
calculations are given in table 3, the atomic coordinates in table 4 and the
volume $V$ of the Fe–O coordination polyhedron of each Fe site calculated from
the atomic coordinates in table 5. The values obtained using GGA+$U$ are also
listed for reference, together with calculations using HSE06. The experimental
values at room temperature [52, 71] are also given for Sr and Ba ferrites. The
S-block length $c$(S) and R-block length $c$(R) calculated for $A$Fe12O19 are
also given in table 3, together with the $c$-axis length. The sum of $c$(S)
and $c$(R) is equal to $c/2$. As shown in figure 1, the 2b and 4f2 coordinated
polyhedra belong to the R block, while the 2a and 4f1 coordinated polyhedra
belong to the S block. The 12k site is located at the boundary between the R
and S blocks. As a general trend, the experimental values in tables 3, 4 and 5
are closer to the values calculated using HSE06 than those calculated using
GGA+$U$. This indicates that the calculations using HSE06 are more accurate
than those using GGA+$U$. In the following discussion only the values
calculated using HSE06 are used.

Table 6:  Energy change evaluated by DFT calculations with HSE06 when one of
the Fe atoms in the $A$Fe12O19 supercell is replaced by a Co atom, based on
the energy when Co occupies the 4f1 site and has a moment in the same
direction as the host Fe.

  * $A$ | Momenta |  $\Delta E$ (meV)
---|---|---
|  | 2a | 2b | 4f1 | 4f2 | 12k
Cab | $\uparrow\uparrow$ | 43 | 1628 | 0 | – | 174
Sr | $\uparrow\uparrow$ | 30 | 840 | 0 | 818 | 139
Ba | $\uparrow\uparrow$ | 66 | 796 | 0 | 797 | 109
Sr | $\uparrow\downarrow$ | 334 | 954 | 260 | 1222 | 362

a Relative spin direction of substituted Co and host Fe. $\uparrow\uparrow$
and $\uparrow\downarrow$ denote cases where the moment of the substituted Co
is in the same or opposite direction to the host Fe, respectively.
b No stable solution with Co2+ occupying the 4f2 site was obtained in the case
of $A$ = Ca (see text).

Figure 4:  Energy change evaluated by DFT calculations with HSE06 when each Fe
site in SrFe12O19 is replaced by Co. Both cases where the Co moment is in the
same direction as and opposite to the magnetic moment of the host Fe are
shown. The energy of the case where Co occupies the 4f1 site and the moment of
Co is in the same direction as the magnetic moment of the host Fe is taken as
the origin.

####  3.3.2 Co substituted SrFe12O19

The total energy $E(i)$ of the supercell, in which some Fe3+ in SrFe12O19 was
replaced by Co2+, was evaluated by DFT calculations with HSE06, where $i$
denotes the Fe sites (2a, 2b, 4f1, 4f2 and 12k) occupied by Co. In all cases,
Fe and Co converged to trivalent and divalent high spin states, respectively.
Calculations were carried out for both cases where the moment of substituted
Co is in the same direction ($\uparrow\uparrow$) and opposite direction
($\uparrow\downarrow$) to the magnetic moment of Fe in the host (Fe site where
Co is substituted). When Co occupies the majority spin sites (12k, 2a, 2b),
$\uparrow\uparrow$ indicates that the Co moment is parallel to the net
magnetisation, $\uparrow\downarrow$ indicates that the Co moment is
antiparallel to the net magnetisation, and when Co occupies the minority spin
sites (4f1, 4f2), note that $\uparrow\uparrow$ indicates that the Co moment is
antiparallel to the net magnetisation and $\uparrow\downarrow$ indicates that
the Co moment is parallel to the net magnetisation.

The total energy was minimised when Co occupied the 4f1 site with
$\uparrow\uparrow$. The energy difference $\Delta E(i)$ was then defined with
respect to $E(\textrm{4f}_{1})$ for $\uparrow\uparrow$. These values are shown
in table 6 and figure 4. First, regardless of the site occupied by Co, the
$\uparrow\uparrow$ case has an energy of 100–400 meV lower than the
$\uparrow\downarrow$ case. Therefore, the magnetic moment of Co in the ground
state is always stable in the same direction as that of the host Fe. Although
there is no experimental evidence that $A$Fe12O19 is $\uparrow\uparrow$ in
La–Co co-substituted $A$Fe12O19, the above results provide the first evidence
for the assumption of $\uparrow\uparrow$ in this system. In the following,
only the case of $\uparrow\uparrow$ is considered.

The total energy $E(i)$ of Co substituted SrFe12O19 is
$E(\textrm{4f}_{1})<E(\textrm{2a})<E(\textrm{12k})\ll
E(\textrm{4f}_{2})<E(\textrm{2b})$. Thus, Co occupies the 4f1 site in the
ground state. However, as discussed in section 4.2, the cation distribution in
a real material freezes at a certain finite temperature. $\Delta
E(\textrm{2b})$ and $\Delta E(\textrm{4f}_{2})$ are large of $\sim$1 eV,
whereas the synthesis temperature of the M phase never exceeds 2000 K;
therefore, Co practically does not occupy 2b or 4f2 sites. On the other hand,
$\Delta E(\textrm{2a})$ and $\Delta E(\textrm{12k})$ are $\sim$100 meV;
therefore, if we assume that the cation distribution freezes at approximately
1000 K, Co is distributed in three sites, 4f1, 2a and 12k. In principle, it is
impossible to selectively replace only one specific site with Co that is more
or less distributed among the three sites. The above calculations are in
agreement with the experimental results (figure 2), in which Co was
distributed over three sites and mainly occupied the 4f1 site.

Figure 5:  Energy change of $A$Fe12O19 ($A$ = Ca, Sr and Ba) evaluated by DFT
calculations with HSE06 when each Fe site is replaced by Co. The origin is the
energy when Co occupies the 4f1 site. In the case of $A$ = Ca, a state where
Co2+ occupies the 4f2 site was not obtained as a stable solution (see text).
The right figure is an enlarged view of the low energy part.

####  3.3.3 Co-substituted CaFe12O19 and BaFe12O19

To test the hypothesis that the Co distribution correlates with the uniaxial
strain of the crystal, we performed the same calculations as in the previous
section for CaFe12O19 and BaFe12O19 with different $A$-ion radii, and compared
the results. We assumed that $\uparrow\uparrow$ is generally more stable than
$\uparrow\downarrow$ for La–Co co-substituted $A$Fe12O19, and for $A$ = Ca and
Ba we only calculated in the $\uparrow\uparrow$ case.

Since $E(\textrm{4f}_{1})$ was minimal for all $A$ = Ca, Sr and Ba, we defined
$\Delta E(i)$ with $E(\textrm{4f}_{1}$) as the origin, as in the previous
section. The $\Delta E(i)$ values for each system are shown in figure 5 and
table 6. In the case of $A$ = Ca, calculations using HSE06 for the supercell
with Co substituted for Fe at the 4f2 site show that the Co2+ state is
unstable. In other words, the Co valence converged to a significantly higher
state ($\sim$+2.5) even when the Co valence value was initially set to +2, and
the Fe valence at the 2a site decreased accordingly. As a result, the
magnetisation per supercell did not change from the undoped state. Therefore
the total energy of the 4f2-Co case for $A$ = Ca cannot be compared with the
other cases. The relationship
$E(\textrm{4f}_{1})<E(\textrm{2a})<E(\textrm{12k})\ll
E(\textrm{4f}_{2})<E(\textrm{2b})$ holds in all cases (except the Ca 4f2-Co
case), but the overall trend is that the distribution of $\Delta E(i)$ is
largest for $A$ = Ca and smallest for $A$ = Ba. In all cases, $\Delta
E(\textrm{2a})$ and $\Delta E(\textrm{12k})$ are of the order of 100 meV,
whereas $\Delta E(\textrm{4f}_{2})$ and $\Delta E(\textrm{2b})$ are of the
order of 1 eV. This implies that at finite temperatures of $\sim$1000 K, Co
tends to aggregate at the most stable 4f1 site when $A$ = Ca, whereas it tends
to be evenly distributed among the three sites when $A$ = Ba. This trend is in
qualitative agreement with the experimental results (figure 2), confirming
that the Co distribution is correlated with the uniaxial strain of the
crystal. The quantitative trends are discussed in section 4.2.

##  4 Discussion

###  4.1 Orbital contribution to hyperfine fields

The internal magnetic field at the Co nucleus is given by
$H_{\mathrm{int}}=A_{\mathrm{spin}}m_{\mathrm{spin}}+A_{\mathrm{orb}}m_{\mathrm{orb}}$,
where $A_{\mathrm{spin}}$ and $A_{\mathrm{orb}}$ are the spin and orbital
components of the hyperfine coupling constants, respectively, and
$m_{\mathrm{spin}}$ and $m_{\mathrm{orb}}$ are the spin and orbital moments,
respectively. Since $|A_{\mathrm{spin}}|\ll|A_{\mathrm{orb}}|$ in general,
when the magnetic moment of Co has an orbital component $m_{\mathrm{orb}}$,
the resonance frequency depends significantly on the magnitude of
$m_{\mathrm{orb}}$ and cannot be predicted in advance. Conversely, the
frequencies of S1, S2 and S3 differ significantly due to the different
magnitudes of $m_{\mathrm{orb}}$. The S1 resonance shifts to the high
frequency side when an external magnetic field is applied [11]. In other
words, an external magnetic field is added to the internal field at the 4f1
site. Since the 4f1 site is a minority spin site, the magnetic moment of Fe at
the 4f1 site is antiparallel to the external field when the sample particles
are oriented in the magnetic field. Furthermore, we assumed that the magnetic
moment of the substituted Co is in the same direction as the magnetic moment
of the host Fe, considering the results of the DFT calculations (section
3.3.2). In this case, the above observation implies that the sign of the
internal field at Co occupying the 4f1 site is negative. However, since the 2a
and 12k sites are in octahedral coordination, there may be degeneracy in the
ground state orbitals and residual orbital moments. In general, since
$A_{\mathrm{orb}}$ has a large positive value, the orbital moment produces a
relatively large field. Therefore, the internal fields of S2 and S3 were
assumed to be positive, although their sign was experimentally unknown. The
magnitude of the internal field at each site, evaluated on the basis of the
above assumptions, is shown in table 2.

Assuming $A_{\mathrm{spin}}=-12$ T/$\mu_{\mathrm{B}}$ [72] and
$A_{\mathrm{orb}}=+65$ T/$\mu_{\mathrm{B}}$ [73], and fixing
$m_{\mathrm{spin}}$ at 3 $\mu_{\mathrm{B}}$/Co for convenience, the Co orbital
moments $m_{\mathrm{orb}}$ at 4f1 (S1), 12k (S2) and 2a (S3) for the sample
with $A$ = Sr are estimated to be approximately 0.4, 1.0 and 1.1
$\mu_{\mathrm{B}}$/Co. Although these may be overestimated (if
$|A_{\mathrm{spin}}|$ and $m_{\mathrm{spin}}$ are overestimated), the relative
magnitude appears to be reliable. These results suggest that relatively large
$m_{\mathrm{orb}}$’s remain in Co occupying the 2a and 12k sites in octahedral
coordination, while a small $m_{\mathrm{orb}}$ is induced in Co at the 4f1
site in tetrahedral coordination through spin-orbit interactions and
interactions with ligands. The orbital field at the 4f1 site is not large
enough to cancel the spin component of the internal field, resulting in a
relatively small negative internal field. At the 2a and 12k sites, the orbital
field was approximately twice as large as the spin field, resulting in a large
positive magnetic field.

According to [23], only Co at the 4f1 site was effective in enhancing the
macroscopic anisotropy. At first glance, it may seem strange that despite the
large orbital moments of Co at the 2a and 12k sites, they do not contribute to
the anisotropy enhancement. However, there is no contradiction if only the Co
at the 4f1 site is uniaxially anisotropic and the Co at the 2a and 12k sites
is in-plane anisotropic. Cluster calculations taking into account interactions
with ligands show that the sign of the magnetic anisotropy of each site is
sensitive to the local strain and can be positive or negative [6, 7].
Therefore, it is not unnatural for the above situation to be realised in
M-type ferrites. It is concluded that the presence of Co in octahedral
coordination is undesirable for enhancing macroscopic uniaxial anisotropy, and
that Co at the 2a and 12k sites should be reduced as much as possible to
improve performance as a hard magnet.

Figure 6:  (a) Temperature dependence of the probability of Co occupancy at
each Fe site evaluated based on the energy difference obtained from DFT
calculations with HSE06 for $A$Fe12O19 ($A$ = Ca, Sr and Ba). (b) Schematic of
Co distribution of $A$Fe12O19 expected at specific temperatures (400, 800 and
1000 K).

###  4.2 Temperature dependence of Co distribution

The results of the DFT calculations provided information on the ground state.
In real materials, the cation distribution freezes at a finite temperature.
Therefore, we considered the cation distributions at finite temperatures using
the Gibbs free energy. The Gibbs free energy change when Co occupies site $i$
is given by $\Delta G(i)=\Delta E(i)+p\Delta V(i)-T\Delta S(i)$, where $\Delta
E(i)$ is the energy difference in the ground state when Co occupies site $i$,
as defined in section 3.3.2. $\Delta V(i)$ and $\Delta S(i)$ are the
corresponding volume and entropy changes, respectively, $p$ is the pressure,
and $k_{\mathrm{B}}$ is the Boltzmann constant. The $pV$ term can affect the
results, mainly through the thermal expansion of the material. However, the
volume change of SrFe12O19 between 0 K and 1000 K is approximately 3% [74],
and the change in $pV$ at this time can be roughly estimated to be on the
order of 1 neV. This is sufficiently small compared with the $\Delta E$ we are
discussing (on the order of 10–100 meV) to be negligible. The entropy change
$\Delta S(i)$ consists of the contributions of the configuration and the
vibrations (phonons, magnons, etc). If the number of sites to be exchanged
does not change with temperature, the contribution from the configuration
entropy does not affect the statistical average of the physical quantity. The
vibrational contribution is estimated to be approximately 0.1–0.2
$k_{\mathrm{B}}$/atom for binary substitutional alloys [75]. The corresponding
$T\Delta S(i)$ is of the order of 10 meV at $T=1000$ K, and the possibility of
affecting the total Gibbs energy cannot be ruled out. However, the site
dependence of $T\Delta S(i)$ due to vibrations is non-trivial and is ignored
for simplicity in this analysis. In this case, the probability $P(i)$ of Co
occupying site $i$ at temperature $T$ is determined by $\Delta E(i)$. Using
classical Boltzmann statistics, $P(i)$ is given by $P(i)=g_{i}\exp(-\Delta
E(i)/k_{\mathrm{B}}T)/\sum_{j}g_{j}\exp(-\Delta E(j)/k_{\mathrm{B}}T)$, where
$g_{i}$ is the multiplicity in the unit cell of site $i$.

Figure 6(a) shows the temperature dependence of $P(i)$ for each sample. In the
equilibrium state at 0 K, Co should only occupy the 4f1 site. However, as the
freezing temperature of the cation increases, the probability of occupancy of
the 2a site in the first excited state $P(\mathrm{2a})$ increases.
Subsequently, the probability of occupancy of the 12k site in the second
excited state $P(\mathrm{12k})$ increases. At relatively high temperatures,
$P(\mathrm{2a})$ and $P(\mathrm{12k})$ are reversed . At temperatures above
1000 K, $P(\mathrm{2a})$ saturates or slowly decreases. This is because
$P(\mathrm{2a})$ approaches the upper limit with a small multiplicity of
sites, whereas the 12k sites have a sufficient number of sites. Close to the
upper temperature limit where the M phase is stable, $P(\mathrm{4f}_{1})$ and
$P(\mathrm{12k})$ are comparable.

As mentioned in section 2.1.1, the samples used in this study may have
contained Fe2+. From the charge balance, the amount of Fe2+ is approximately
0.04–0.1 relative to 12 Fe. It is known that Fe2+ prefers to occupy the 2a
site [44]. Therefore, in the equilibrium state at 0 K, Fe2+ is expected to
occupy at most 10% of the 2a site. However, the extra electron associated with
Fe2+ is expected to hop from site to site as the temperature increases [45].
The effect is averaged over all Fe sites and should not significantly affect
the discussion here.

To observe the $A$-ion dependence in detail, figure 6(b) shows the Co
distribution for each system at specific temperatures (400, 800 and 1000 K).
As a qualitative trend, $P(\textrm{4f}_{1})$ is largest for $A$ = Ca and
smallest for $A$ = Ba at relatively high temperatures, but this relationship
does not necessarily hold at temperatures below 500 K. That is,
$P(\textrm{4f}_{1})$ of BaFe12O19 increases relatively rapidly at low
temperatures.

In actual material synthesis, the cation distribution at a given temperature
is not necessarily maintained at room temperature, even if heat treatment is
carried out at that temperature. For example, in the spinel compound MgAl2O4,
the time required to equilibrate the ion distribution is a few seconds above
1400 K, but more than 10,000 min below 900 K [19]. Assuming that our compounds
require a similar duration to equilibrate, at an experimental cooling rate,
e.g. $-200$ ∘C/h, the cation distribution above 1200 K is transformed to a low
temperature equilibrium state, whereas at a lower temperature below 900 K the
temperature drops before the cation distribution reaches equilibrium.
Therefore, under normal sample preparation conditions, a cation distribution
at $\sim$1000 K is maintained at room temperature. The Co distribution at 1000
K in figure 6(b) shows that $P$(4f1) is the highest for $A$ = Ca and the
lowest for $A$ = Ba. This trend is qualitatively consistent with the trends
observed experimentally (figure 2). Quantitatively, $P$(4f1) tends to be
overestimated in the 59-Co NMR experiments. As mentioned in section 3.2, the
experimental error in $P(j)$ seems to be too large to make quantitative
comparisons. Nevertheless, the calculations confirmed that uniaxial
compressive strain improves the 4f1 site selectivity of Co.

As mentioned above, $P(\textrm{4f}_{1})$ increases as a lower temperature
cation distribution is achieved. Since the time to reach equilibrium is longer
at lower temperatures, it is expected that $P(\textrm{4f}_{1})$ will increase
by post-annealing at lower temperatures or by slowing the cooling rate. In
other words, it is concluded that post-annealing and slow cooling at low
temperatures are effective in increasing the uniaxial anisotropy and improving
the coercivity and magnetisation of Co-substituted ferrite magnets. When the
temperature variation of $P(\mathrm{4f}_{1})$ is steeper in BaFe12O19 than in
SrFe12O19 and CaFe12O19, as shown in figure 6(a), the post-annealing effect is
most pronounced in the BaFe12O19 system.

The site assignment of the S2 and S3 Co-NMR signals is mentioned at the end of
this section. As mentioned above, when a sample is prepared by the usual solid
state reaction method, an equilibrium cation distribution of approximately
800–1500 K is frozen, regardless of the cooling rate or post-annealing, taking
into account the diffusion rate of the atoms. Therefore, if we look more
closely at the variation of $P(i)$ in the 800–1500 K range in figure 6(a), the
following general trends can be expected. (i) The temperature variation of
$P$(2a) is relatively small and almost independent of the freezing
temperature. (ii) Although the freezing temperature dependence of $P$(4f1) and
$P$(12k) is large, the signs of their temperature coefficients are opposite,
and the sum of $P$(4f1) and $P$(12k) tends to be constant. (iii) The $A$-ion
dependence is largest at $P$(12k). These facts suggest that $P$(2a) is
relatively insensitive to perturbations such as $A$-ion exchange, freezing
temperature variation, and concentration changes, whereas $P$(12k) is expected
to be sensitive to these perturbations. This may be mainly due to the higher
multiplicity of sites at the 12k site compared with the 2a site. Figure 2(b)
shows that the experimental $A$-ion dependence was large for S2 and small for
S3. Furthermore, according to the 59Co-NMR spectrum of La–Co co-substituted
SrFe12O19 reported in [23], the intensity of S3 hardly changes with the Co
concentration, whereas that of S2 decreases significantly with increasing Co
concentration (with compression along the $c$-axis). In the light of the above
expected trends and experimental results, it seems reasonable to assign S2
$\rightarrow$ 12k and S3 $\rightarrow$ 2a. It should be noted that this
argument is only an inference based on qualitative trends. However, this site
identification, whether correct or incorrect, does not affect the discussion
of the 4f1 site selectivity of Co in this study.

Figure 7:  $A$-ion dependence of the relative change in structural parameters
of undoped $A$Fe12O19 with respect to CaFe12O19 obtained by DFT calculations
with HSE06. (a) R-block length, S-block length and $c$. (b) Volume of the
coordination polyhedron of each Fe site and $c/a$

###  4.3 Effect of local structural distortion on Co distribution

In section 3.3.3, we noted that Co distribution correlates with the uniaxial
strain of the crystal. If the energy change associated with the Co
substitution (figure 5) is determined only by the uniform uniaxial lattice
strain, then $\Delta E(i)$ is expected to vary monotonically with the $A$-ion
size, and indeed $\Delta E(i)$ decreases monotonically with increasing the
$A$-ion size ($c/a$ decreases) at most sites. However, the 2a site does not
follow this trend and $\Delta E(2a)$ varies non-monotonically. That is,
$\Delta E(2a)$ decreases for $A$ = Ca $\rightarrow$ Sr, but conversely
increases for $A$ = Sr $\rightarrow$ Ba. To investigate the cause, we discuss
the effect of local deformation in the lattice based on the atomic coordinates
of undoped $A$Fe12O19 evaluated by DFT calculations (table 4).

The relative changes in $c$(S) and $c$(R) (with respect to CaFe12O19) are
shown in figure 7(a). As the $A$-ion size increased, $c$(R) increased in
response to $c$, but the amount of change was approximately three times that
of $c$. In contrast, $c$(S) showed a negative dependence on the $A$-ion size,
but the change was small. This means that the effect of $A$-ion expansion is
not uniform within the crystal and the S block absorbs some of the expansion
of the R block. This may be because the R block to which the $A$ ion belongs
is directly affected by the $A$ ion, whereas the S block is not.

Similar non-uniform changes can also be observed in the volume $V(i)$ (table
5) of the coordination polyhedron at each Fe site. Figure 7(b) shows the
relative change $\Delta V(i)$ of $V(i)$ (with respect to CaFe12O19). The
changes in $c/a$ are included in the same figure for comparison. The $\Delta
V$(2b) and $\Delta V$(4f2) at the R-block sites increased significantly with
the expansion of the $A$ ion (increase in $c/a$). The $\Delta V$(12k) at the
R/S block boundary increased only slightly with the expansion of the $A$ ion,
which may be due to the combined effect of the large expansion of the R block
and the small contraction of the S block, as well as the greater degree of
freedom in the position of the oxygen ions forming the 12k coordination
polyhedron than at other sites. In contrast, $\Delta V$(2a) decreased slightly
with increasing $A$-ion size. This is because the height of the 2a
coordination polyhedron along the $c$-axis is strongly correlated with the
S-block length, as can be seen from the similarity with the relative change in
the S-block length $c$(S) in figure 7(a).

Table 7:  Coordinates of oxygen ions forming 2a and 4f1 coordination
polyhedra, and aspect ratios of 2a and 4f1 coordinated polyhedra in undoped
$A$Fe12O19 ($A$ = Ca, Sr and Ba) obtained by DFT calculations with HSE06.

  * $A$ | Oxygen coordinatea | Aspect ratiob
---|---|---
| $x$ | $z$ | $z^{\prime}$ |  $r$(2a) |  $r$(4f1) |  $r$(12k)
Ca | 0.1572 | 0.0527 | 0.0556 | 1.0445 | 0.9848 | 0.9707
Sr | 0.1571 | 0.0523 | 0.0553 | 1.0429 | 0.9823 | 0.9677
Ba | 0.1568 | 0.0519 | 0.0547 | 1.0407 | 0.9790 | 0.9625

a Coordinate in O-12k1 $(x,2x,z)$ and O-4f
$(\frac{2}{3},\frac{1}{3},z^{\prime})$.
b Ratio of the oblique (side length of side triangle) $h$ to the base (side
length of base triangle) $b$, $r=h/b$. $r$(12k) is the average of the
different $r$ values in the 12k octahedron.

Figure 8:  (a) 2a and 4f1 coordination polyhedra viewed along the $c$-axis.
Only the oxygen ions are shown. The atomic coordinates of the ligand oxygen
ions (closed circles, O-12k1 site) in the 2a octahedron (yellow) are given by
$(x,2x,z)$. In this case, the length of the base is $3x\times a$ and the
height along the $c$-axis is $2z\times c$. The vertices of the 2a octahedron
are also those of the base triangle of the 4f1 tetrahedron (red). The 4f1
tetrahedron has a base length $(1-3x)\times a$. The sum of the base length of
the 4f1 tetrahedron and the base length of the 2a octahedron is equal to the
lattice constant $a$. The atomic coordinates of the remaining vertex of the
4f1 tetrahedron (open circle, O-4f site) are given by
$(\frac{2}{3},\frac{1}{3},z^{\prime})$ and have $c$-axis degrees of freedom
independent of the 2a octahedron. The $c$-axis height of the 4f1 tetrahedron
is $(z+z^{\prime})\times c$. If the $c$-axis displacements of O-12k1 and O-4f
are comparable, the volumes of the 4f1 tetrahedron and the 2a octahedron are
determined by the $c$-plane displacement of O-12k1 and show a trade-off
tendency if $a$ is invariant. (b), (c) Relative changes in the coordinates of
the oxygen atoms constituting the 2a and 4f1 coordination polyhedra ($x$ and
$z$ in O-12k $(x,2x,z)$ and $z^{\prime}$ in O-4f
$(\frac{2}{3},\frac{1}{3},z^{\prime}))$ and lattice constants ($a$ and S-block
length $c$(S)) in undoped $A$Fe12O19 ($A$ = Ca, Sr and Ba) (CaFe12O19 basis)
obtained by DFT calculations with HSE06.  Figure 9:  (a) The $A$-ion
dependence of the ratio of the oblique side to the base of the 2a, 4f1 and 12k
coordinated polyhedra in $A$Fe12O19 ($A$ = Ca, Sr and Ba) obtained by DFT
calculations with HSE06. (b) Internal magnetic fields at nuclear sites of Co
occupying 2a, 4f1 and 12k sites in La–Co co-substituted $A$Fe12O19.

It is interesting to note the $A$-ion dependence of the volume of the 4f1
coordination polyhedron belonging to the same S block. The $A$-ion dependence
of $\Delta V$(4f1) (figure 7(b)) shows a trend that contradicts the change in
the length of the S block (figure 7(a)) to which it belongs. This is due to
the fact that $\Delta V(\textrm{2a})$ and $\Delta V(\textrm{4f}_{1})$ are not
independent. Figure 8(a) schematically shows the projection of the S block
onto the $c$-plane. The 4f1 tetrahedron shares the vertices of its base
triangle with the 2a octahedron, and the sum of the lengths of the base of the
4f1 tetrahedron and the base of the 2a octahedron is equal to the lattice
constant $a$. The Wyckoff positions of the coordinating oxygens of the 2a
octahedron are all 12k1 $(x,2x,z)$, the length of the base is $3x\times a$ and
the height along the $c$-axis is $2z\times c$. The 4f1 tetrahedron shares the
vertices of its base triangle with a 2a octahedron and has a base length of
$(1-3x)\times a$. The remaining vertex of the 4f1 tetrahedron is 4f
$(\frac{2}{3},\frac{1}{3},z^{\prime})$ and has $c$-axis degrees of freedom
independent of the 2a octahedron. In this case, the height of the 4f1
tetrahedron along the $c$-axis is $(z+z^{\prime})\times c$. Thus, if the
$c$-axis displacements of the oxygen ions at the 12k1 and 4f positions are
comparable, $V$(4f1) and $V$(2a) are determined by the in-plane displacement
of the oxygen ion at the 12k1 position. If $a$ is invariant, $\Delta V$(4f1)
and $\Delta V$(2a) show a trade-off tendency.

The coordinates of the oxygen positions 12k1 $(x,2x,z)$ and 4f
$(\frac{2}{3},\frac{1}{3},z^{\prime})$ in the 2a and 4f1 coordination
polyhedra in undoped $A$Fe12O19 ($A$ = Ca, Sr and Ba) calculated by DFT
calculations are given in table 7. The relative changes with respect to
CaFe12O19 are shown in figures 8(b) and (c). The $z$ and $z^{\prime}$, which
correspond to the heights of the coordination polyhedra along the $c$-axis,
vary almost similarly to $A$, and the displacement of the oxygen ions along
the $c$-axis in the S block is almost uniform. However, $x$, which corresponds
to the length of the sides of the triangle in the $c$-plane, decreases with
increasing $A$-ion size (slightly increasing for $a$). This means that the
base of the 4f1 tetrahedron is enlarged and the base of the 2a octahedron is
reduced. This explains why the $A$-ion dependence of $\Delta V(\textrm{2a})$
and $\Delta V(\textrm{4f}_{1})$ show opposite trends, but it is not clear
whether the 2a octahedron or the 4f1 tetrahedron is expanding or contracting.

Next we look at the changes in the 2a-octahedron and the 4f1-tetrahedron from
a symmetry point of view. Table 7 and figure 9(a) show the ratio $r=h/b$ of
the length $b$ of the base (side of a triangle parallel to the $c$-plane) and
$h$ of the hypotenuse (side of a side triangle) of each polyhedron, calculated
using DFT calculations, where $r=1$ corresponds to a regular polyhedron. In
the 2a octahedron, $r>1$, the octahedron is elongated along the $c$-axis, but
the stresses due to the $A$-ion expansion cause $r$ to decrease and the 2a
octahedron approaches the regular octahedron. On the other hand, in the 4f1
tetrahedron, $r<1$, the tetrahedron contracts along the $c$-axis, but the
stresses due to $A$-ion expansion reduce $r$ and further increase the
distortion. That is, the 2a octahedron (containing isotropic Fe3+) has room to
stabilise against $c$-axis stress by increasing symmetry, whereas the 4f1
tetrahedron has no such room. As a result, the 4f1 tetrahedron expands, and
the 2a octahedron, which has a high symmetry and spatial margin, absorbs the
distortion, thereby stabilising the whole system. It can be said that the 2a
octahedron plays a buffering role in absorbing strain in the crystal.

Comparing figure 5 and figure 7(b), the change in $\Delta E(i)$ is
approximately inversely correlated with the change in $\Delta V(i)$. In other
words, a large positive volume change in the coordination polyhedron tends to
increase the probability of Co occupancy. This may be due to the decrease in
elastic energy loss when Co2+, which has a larger ionic radius than Fe3+,
occupies the Fe sites. Finally, the non-monotonic $A$-ion dependence of
$\Delta E(\textrm{2a})$ shown in figure 5 can be interpreted as the 2a
coordination polyhedron contracting with the expansion of $A$, unlike the
other sites. The fact that there is a trade-off between $V$(4f1) and $V$(2a)
may become apparent during the final optimisation process for the performance
of the M-type ferrite magnet. In conclusion, lattice distortion is not
necessarily uniform and differences in local distortion also have a secondary
effect on the distribution of Co sites.

We mentioned in section 3.2 that the resonance frequencies of the S1, S2 and
S3 resonances (table 2) shift slightly depending on the $A$ ion. The resonance
frequencies were converted to internal fields based on the site assignment
described above, and are shown in figure 9(b). The internal field shifted
positively at the 2a site and negatively at the 4f1 and 12k sites with
increasing $A$-ion size. The aspect ratio $r$ in figure 9(a) corresponds to
the deviation from the cubic symmetry. The $r$ of the 12k coordination
octahedron, which is the average of the base and oblique sides at the 12k
site, has also been added to the figure. Comparing (a) and (b) in figure 9 we
can see that as $r$ approaches 1, that is, as the coordination polyhedron
approaches cubic symmetry, the internal field shifts in the positive
direction. Assuming that $m_{\mathrm{spin}}$ is invariant, a positive shift in
the internal field corresponds to an increase in $m_{\mathrm{orb}}$. Thus, it
can be seen that $m_{\mathrm{orb}}$ tends to increase as the symmetry
approaches cubic symmetry.

##  5 Summary

In the La–Co co-substituted M-type ferrite $A$Fe12O19, Co mainly occupies the
4f1 site (minority spin site) in tetrahedral coordination, and some Co
occupies the 2a and 12k sites (majority spin sites) in octahedral
coordination. However, as proposed in [23], only Co occupying the 4f1 site was
found to be effective in enhancing uniaxial anisotropy. To improve the
performance (anisotropy and magnetisation) of M-type ferrite magnets with a
limited amount of Co, it is important to concentrate Co at the 4f1 site.
However, their magnetic properties depend on the type and concentration of the
$A$ ions, even if the Co content is the same. Previous studies have shown that
the decrease in $c$-axis length due to contraction of the $A$ ion, that is,
uniaxial compressive strain, increases the 4f1 site selectivity of Co. In this
study, we performed 59Co-NMR on La–Co co-substituted M-type ferrites with
different $A$ ions, that is, $A$ = Ca, Sr and Ba (ion sizes are Ca2+ $<$ Sr2+
$<$ Ba2+) with Co compositions close to 0.2, and experimentally confirmed the
site distribution of Co. At the same time, DFT calculations were performed for
undoped $A$Fe12O19 and ($A$Fe12O19)8 with Co2+ replacing one Fe atom,
[($A$Fe11.875Co0.125O19)8]-, and the stable structure and preferential site
occupancy of Co were predicted using first-principles calculations. The
results were as follows.

  1. 1.

In La–Co co-substituted $A$Fe12O19, it was experimentally confirmed that Co
concentrates at the 4f1 site as $A$ ions become smaller.

  2. 2.

The resonance frequency (resonance field) of Co occupying each Fe site was
distributed over a wide frequency (field) range, which was mainly attributed
to the difference in the orbital components of the magnetic moment. The
increase in local symmetry at each site correlates with an increase in the
orbital field.

  3. 3.

DFT calculations confirmed that when Co occupies the Fe site, its magnetic
moment is stable in the same direction as the moment of the host Fe.

  4. 4.

According to DFT calculations, the most stable case for any $A$ ion is when Co
occupies the 4f1 site, followed by the 2a and 12k sites with energy
differences of the order of 100 meV. Co at the 2b and 4f2 sites is unstable
(the energy difference is of the order of 1 eV) and Co practically does not
occupy these sites. Thus, Co is expected to occupy mainly the 4f1 site, with
some Co occupying the 2a and 12k sites, which is in agreement with the
experimental results.

  5. 5.

DFT calculations predicted that the energy difference of Co occupying
different Fe sites tends to be larger for smaller $A$ ions. This means that
the smaller the $A$ ion, the higher the 4f1 site selectivity for Co, which is
in agreement with the experimental results.

  6. 6.

According to the DFT calculations, the strain due to the contraction of the
$A$ ions is not uniform throughout the crystal, but has a positive effect
(contraction) in the R block and a negative effect (expansion) in the S block.
Therefore, the Co occupancy of each Fe site is not necessarily determined by
uniform strain, but is also influenced by differences in the local strain. In
particular, the order of Co occupancy at the metastable 2a and 12k sites may
depend on the type of $A$ ion.

  7. 7.

Because the ionic size of Co2+ is larger than that of Fe3+, the probability of
Co2+ occupancy generally tends to increase with increasing volume of the
oxygen coordination polyhedron at each Fe site. However, the crystal
structure, particularly the S block, is characterised by a trade-off between
the volume of the 4f1 coordination polyhedron and that of the 2a coordination
polyhedron, which may be important in the final stage of performance
optimisation.

  8. 8.

As the cation distribution is frozen at a finite temperature in the actual
sample preparation, the 4f1 site occupancy of Co increases as the cation
distribution is realised at a lower temperature, taking into account the
effect of thermal excitation. Therefore, post-annealing at low temperature and
slow cooling is expected to be effective in improving performance.

The above results are briefly summarised in terms of guidelines for improving
performance (anisotropy and magnetisation) of La–Co co-substituted M-type
ferrite magnets with a limited amount of Co. It is effective to select $A$
ions with the smallest possible size and to post-anneal at a low temperature
or cool slowly to concentrate Co at the 4f1 site in tetrahedral coordination.

This work was supported by the Elements Strategy Initiative Center for
Magnetic Materials (ESICMM), Grant Number JPMXP0112101004, through the
Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan,
and JSPS KAKENHI 17K06793, 19K05002 and 22H00263. The authors would like to
thank H. Nishida for his help in preparing Ba-based crystals.

## Conflict of interest

The authors declare no competing interests.

## ORCID iDs

Hiroyuki Nakamura https://orcid.org/0000-0001-7085-4800
Hiroto Ohta https://orcid.org/0000-0002-0268-9468
Takeshi Waki https://orcid.org/0000-0002-3294-9814
Yoshikazu Tabata https://orcid.org/0000-0002-4447-2160
Hidekazu Ikeno https://orcid.org/0000-0002-3840-4049
Christian Mény https://orcid.org/0000-0003-0718-161X

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