# Giant magnetic and optical anisotropy in cerium-substituted M-type strontium
hexaferrite driven by 4$f$ electrons

Churna Bhandari1<EMAIL_ADDRESS>(corresponding author) Durga Paudyal1,2 1The
Critical Material Institute, Ames National Laboratory, Iowa State University,
Ames, IA 50011, USA
2Department of Electrical and Computer Engineering, Iowa State University,
Ames, Iowa 50011, USA

###### Abstract

By performing density functional calculations, we find a giant
magnetocrystalline anisotropy (MCA) constant in abundant element cerium (Ce)
substituted M-type hexaferrite, in the energetically favorable strontium site,
assisted by a quantum confined electron transfer from Ce to specific iron (2a)
site. Remarkably, the calculated electronic structure shows that the electron
transfer leads to the formation of Ce3+ and Fe2+ at the $2a$ site producing an
occupied Ce($4f^{1}$) state below the Fermi level that adds a significant
contribution to MCA and magnetic moment. A half Ce-substitution forms a
metallic state, while a full substitution retains the semiconducting state of
the strontium-hexaferrite (host). In the latter, the band gap is reduced due
to the formation of charge transferred states in the gap region of the host.
The optical absorption coefficient shows an enhanced anisotropy between light
polarization in parallel and perpendicular directions. Calculated formation
energies, including the analysis of probable competing phases, and elastic
constants confirm that both compositions are chemically and mechanically
stable. With successful synthesis, the Ce-hexaferrite can be a new high-
performing critical-element-free permanent magnet material adapted for use in
devices such as automotive traction drive motors.

##  I Introduction

The increasing demand for high-performance permanent magnets for higher energy
efficiency miniaturized devices with the limited global supply of critical
rare-earth elements has urged the magnetism community to discover an
alternative to critical rare-earth-based magnets. Although the hexaferrites
are useful critical rare-element-free permanent magnets, the low  value of
maximum energy product (BH${}_{\text{max}})$[1] limits their usage for high-
performance magnetic devices. Continuous theoretical and experimental efforts
have been made to improve their intrinsic magnetic properties. In particular,
an electron doping via $5d$-element La substitution is a promising approach in
uplifting the magnetocrystalline anisotropy (MCA) constant ($K_{1}$) of ${\rm
SrFe}_{12}{\rm O}_{19}$[2, 3], albeit it reduces the net magnetic moment. The
magnetic rare-earth elements are desirable over La because a large spin-orbit
coupling (SOC) in $4f$-states changes the MCA drastically, as seen in other
rare-earth-based permanent magnets[4, 5, 6, 7, 8, 9] including Sm-substituted
hexaferrites[10]. Ce substitution is preferred in hexaferrite as Ce is
naturally abundant and is a low-cost $4f$-element. Although a few experiments
have been performed with the Ce-substitution [4, 5, 11, 6], its site
preference and effect on stability and the electronic, magnetic, and optical
properties remain elusive.

Experimentally, Ce-substitution has been made in the Ba-hexaferrites, e.g.,
20% in Ba-site and $\sim 5\%$ in Fe-site. Although these experiments adopt a
sol-gel approach, sample preparations differ. Mosleh et al.[6] used Fe(NO3)3,
Ba(NO3)2, CeO2, ethylenediamine tetraacetic acid (EDTA(C10H16N2O8), citric
acid (C6H8O7), and ammonia solution as precursor to make Ba1-xCexFe12O19
$(0\leq x\leq 0.2)$, while Pawar et al.[11] used Ba(NO3)2, Ce(NO3)36H2O and
Fe(NO3)39H2O, and C6H8O7 in forming BaCexFe12-xO19 $(0\leq x\leq 0.3)$. Chang
et al. [5] used slightly different precursor to synthesize BaCe0.05Fe11.95O19.
These reports point to two contrasting site preferences, for instance, the Sr-
site in Ref. [6] and the Fe-site in Ref. [11, 5].

Mosleh et al.[6] found no specific trend in the magnetization and coercivity
by increasing Ce concentration; for instance, they found an increase of
magnetization for $x=0.1$ and then a decrease for $x=0.2$. Pawar et al.[11]
found a decrease in the magnetization and Curie temperature (T${}_{\text{C}}$)
as well as an increase of coercivity with an increase of Ce where the site
occupancy is different. Chang et al.[5] report that  the tendency increases in
anisotropy field with Ce doping. In a separate experiment, Banihashemi et al.
[12] also synthesized Sr1-xCexFe12O19 (x$=$0.0, 0.05, 0.10) utilizing
hydrothermal process and confirmed the formation of the nano-crystalline phase
of the M-type hexaferrite from x-ray diffraction (XRD) and Fourier transform
infrared spectroscopy (FTIR). They identified the band gap dependency
(1.37–1.51 eV) on the Ce content. However, to the best of our knowledge, no
theoretical studies are available in the literature on Ce-substituted
hexaferrite.

In this paper, we predict the stability and electronic, magnetic, and optical
properties of Ce-substituted strontium hexaferrites by performing ab initio
calculations with density functional theory (DFT). We find the following key
results. i) Both full and half Ce-substituted compositions have negative
formation energies and are mechanically stable. ii) Ce donates an electron to
Fe($2a$), leading to significant changes in the electronic structure resulting
in a metal for $x=0.5$ and a semiconductor for $x=1$. iii) The net magnetic
moment and $K_{1}$ are improved with Ce, especially $K_{1}$ is enhanced by one
order of magnitude. iv) The computed optical spectra for $x=1$ show
significant anisotropy between light polarization in parallel and
perpendicular directions.

##  II Methodology and Crystal Structure

We used Vienna Simulation Package (VASP)[13, 14] to solve Kohn-Sham eigenvalue
equations within the projector augmented wave (PAW) method and Perdew-Burke-
Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation
functional including the on-site electron-correlation[15, 16]. Fully self-
consistent charge density was obtained using plane wave total energy cut-off
500 eV for all elements and $7\times 7\times 1$ k-mesh for the Brillouin zone
integration. For GGA + $U$ calculations, including structural relaxation, we
used Dudarev’s[17] $U_{eff}=U-J=3$ and $4.5$ eV for Ce($4f)$ and Fe($3d$)
orbitals following the previous theoretical reports[18, 19], and
Heyd–Scuseria–Ernzerhof (HSE06) calculations. We employed HSE06 with Hartree-
Fock (HF) mixing parameter, $\alpha=0.25$ and the screening parameter, $w=0.2$
Å-1 in the electronic and optical properties calculations[20, 21]. For the
magnetic anisotropy constant, we used fully self-consistent noncollinear GGA +
$U$ \+ SOC calculations  in standard 64 atom hexaferrite unit cell. The unit
cell consists of Wyckoff’s positions: Sr at $2d$, Fe at $2a$, $2b$, $12k$,
$4f_{1}$, and $4f_{2}$, and O at $4f$, $4e$, $6h$, and two $12k$ [19]. The
full Ce-substitution at Sr $2d$-site retains the original crystal symmetry to
$P6_{3}/mmc$, space group no. 194, while a half-Ce slightly lowers the crystal
symmetry $P6m2$, space group no. 187 (hexagonal),  including the atomic site-
symmetries. The chosen k-mesh produces well converged MCA in M-type
hexaferrite structures[22, 19]

The optimized structural parameters are given in Table 1. The GGA + $U$
overestimates volume (Table 1) for experimentally known ${\rm SrFe}_{12}{\rm
O}_{19}$[23], and a similar trend is seen in the full and partial Ce-
substitutions. The Ce-substitution leads to a reduction in volume as the
atomic size of Ce3+ (1.01 Å) is slightly smaller than Sr2+ (1.12 Å).
Specifically, the reduction is due to a slight decrease in the out-of-plane
lattice constant ($c$) (Table 1).

Table 1: DFT optimized lattice constants in Å and volume per unit cell, V, in Å3 of Sr1-xCexFe12O19 with GGA + $U$ calculations. $x$ | a | b | c | c/a | V
---|---|---|---|---|---
$0$ | 5.927 | 5.927 | 23.17 | 3.909 | 704.91
$0.5$ | 5.933 | 5.933 | 23.016 | 3.879 | 701.69
$1$ | 5.927 | 5.927 | 22.919 | 3.867 | 697.45

##  III Formation energy and Mechanical Properties

###  III.0.1 Formation energy

Now we discuss the formation energy of ${\rm CeFe}_{12}{\rm O}_{19}$ ($x=1$)
to examine chemical stability. The magnetic order of Fe sublattices remains
the same as the original Sr-hexaferrite. This is confirmed by total energy
calculations of all possible spin configurations in all Ce-doped compositions.
All the calculations hereafter correspond to the Sr-hexaferrite type magnetic
state unless explicitly stated. It is well-known that CeO and Fe2O3 both exist
in crystalline form[24, 25]. We compute the formation energy of ${\rm
CeFe}_{12}{\rm O}_{19}$, using the relation

$E_{f}({{\rm CeFe}_{12}{\rm O}_{19}})=E({{\rm CeFe}_{12}{\rm
O}_{19}})-E({\text{CeO}})-6E({\text{Fe}}_{2}{\text{O}}_{3}),$ (1)

where $E$(${\rm CeFe}_{12}{\rm O}_{19}$), $E$(CeO), and $E$(Fe2O3) are total
energies of ${\rm CeFe}_{12}{\rm O}_{19}$, CeO, and Fe2O3 per formula unit
(f.u.), which yields $E_{f}=-3.31$ eV per f.u. Alternatively, we compute the
formation energy with respect to Ce2O3, Fe2O3, and FeO using the expression

$\displaystyle E_{f}(\text{CeFe}_{12}{\rm O}_{19})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}[2E({\rm CeFe}_{12}{\rm
O}_{19})-E(\text{Ce}_{2}\text{O}_{3})$ (2)
$\displaystyle-11E(\text{Fe}_{2}\text{O}_{3})-2E(\text{FeO})],$

and find -2.08 eV per f.u. In both cases, ${\rm CeFe}_{12}{\rm O}_{19}$ is
chemically stable. Further, we compute $E_{f}$ relative to CeO2 phase as

$\displaystyle E_{f}$ $\displaystyle=$ $\displaystyle
2E(\text{Ce}\text{Fe}_{12}\text{O}_{19})-E(\text{CeO}_{2})$ (3)
$\displaystyle-12E(\text{Fe}_{2}\text{O}_{3})-E(\text{Ce}),$

which is -3.88 eV /f.u.
We also examine the formation energy for $x=0.5$ with respect to SrO, CeO, and
Fe2O3 as

$\displaystyle
E_{f}(\text{Sr}_{0.5}\text{Ce}_{0.5}\text{Fe}_{12}\text{O}_{19})$
$\displaystyle=$ $\displaystyle
E(\text{Sr}_{0.5}\text{Ce}_{0.5}\text{Fe}_{12}\text{O}_{19})$ (4)
$\displaystyle-0.5E(\text{CeO})-0.5E(\text{SrO})-6E(\text{Fe}_{2}\text{O}_{3})$

and find $E_{f}=-2.94$ eV /f.u. It is higher than for $x=1$ obtained with a
similar expression, meaning a half Ce-substituted composition is chemically
less stable.

To clarify Ce site-preference in Fe-sites, we compute the self-consistent
total energy for  (Sr2Ce1Fe23O38, $x\sim 0.04$) one Ce substituted for one Fe
atom in $4f_{2}$, $4f_{1}$, $12k$, $2b$, and $2a$ sites in GGA + $U$, in fully
relaxed geometries. The computed energies are 0, 0.51, 1.03, 1.06, and 1.66 in
eV relative to the $4f_{2}$ site, which indicates $4f_{2}$ as the most
favorable Fe site. Interestingly Ce enhances the unit cell volume by $\sim
2\%$ as compared to Sr-hexaferrite, presumably due to its larger ionic radius.
In contrast, the volume shrinks by $\sim 1\%$ if Ce replaces Sr. As noted
earlier, the Sr has a larger ionic size, in which Ce fits relatively easily,
with minimal distortion of crystal structure, while in Fe-sites, Ce creates
more spacing that leads to lattice expansion.

To further scrutinize the site preference, we compute the formation energy of
one Ce substituted Sr-hexaferrite in the lowest energy Fe site, i.e.,
$4f_{2}$, by using the relation

$\displaystyle E_{f}(\text{Sr}_{2}\text{Ce}\text{Fe}_{23}{\text{O}}_{38})$
$\displaystyle=$ $\displaystyle
E(\text{Sr}_{2}\text{Ce}\text{Fe}_{23}{\text{O}}_{38})-E(\text{Ce}\text{O}_{2})$
(5)
$\displaystyle-2E(\text{SrO})-11E(\text{Fe}_{2}\text{O}_{3})-E(\text{FeO}).$

It gives -1.88 eV/f.u., significantly larger than Ce at Sr-site. The formation
energy of other compositions for Ce at other remaining Fe-sites is even larger
as the total energy is larger than for Ce at the 4$f_{2}$ site. The formation
energy results, therefore, confirm that Ce has a strong Sr-site preference
over the Fe-sites.

Experimentally Ce is doped in Ba-hexaferrites where the site preference is
much debated. The experimental reports show that site occupancy depends on how
the sample is synthesized. Indeed, there are some differences in sample
preparation. For example, Mosleh et al.[6] used slightly different
experimental precursor than Pawar et al.[11]. Both experiments report lattice
parameters obtained with x-ray diffraction (XRD), which behave oppositely to
the pristine system. Ce at the Ba site is shown to reduce the cell volume in
Ref. [6]. In contrast, in Refs. [11] and 5], the authors show an increase in
volume with Ce at the Fe site. These trends are consistent with our results.

###  III.0.2 Mechanical and dynamical stability

Mechanical stability is described by calculating the stiffness tensor using
the energy-strain relationship, which, in Voigt notation, is

$E=\dfrac{V_{0}}{2}\sum_{i=1}^{6}\sum_{j=1}^{6}C_{ij}\epsilon_{i}\epsilon_{j},$
(6)

where $V_{0}$, $C_{ij}$, and $\epsilon_{i}$ are the volume of unit cell, the
elastic constants, and the components of strain tensor, respectively. As the
crystal structure of Ce-substituted hexaferrites is hexagonal (space group
$P6_{3}/mmc$), its stability can be described by five independent non-zero
elastic constants $C_{11},C_{12},C_{13},C_{33}$, and $C_{44}$.

Table 2: Elastic constants ($C_{ij}$), bulk modulus (K), and shear modulus (G) in Voigt notation (in units of GPa) of Sr1-xCexFe12O19 obtained with GGA $+U$ calculations. $C_{ij}$ | $x=1$ | $x=0.5$ | $x=0$
---|---|---|---
$C_{11}$ | 297.932 | 226.311 | 283.214
$C_{12}$ | 130.157 | 109.500 | 143.012
$C_{13}$ | 110.131 | 84.067 | 102.011
$C_{33}$ | 267.509 | 203.541 | 248.651
$C_{44}$ | 71.192 | 43.796 | 65.769
K | 173.801 | 134.603 | 167.461
G | 79.422 | 54.435 | 71.698

The elastic stiffness tensor (not given here) is the definitive positive and
has all six eigenvalues positive for all compositions. All three elastic
stability criteria for hexagonal crystal [26], e.g., $C_{11}>|C_{12}|$,
$2C_{13}^{2}<C_{33}(C_{11}+C_{12})$, and $C_{44}>0$ are satisfied by the
computed elastic constants (Table 2) for substituted as well as pure
compositions confirming their mechanical stability. Interestingly, bulk
modulus (K) and shear modulus (G) are similar for $x=0$ and $x=1$ compositions
while smaller for $x=0.5$. These results suggest that a half-Ce-substitution
makes it less stiff, likely due to the peak of the density of states at the
Fermi level, which we discuss later. The zone-center ($\Gamma$-point) phonon
frequencies are all positive, indicating that these are likely dynamically
stable. Currently, we succeed in the full Brillouin zone phonon dispersion
calculations, as shown in Fig. 1, for pure strontium hexaferrite, and we find
no soft-mode frequencies that confirm the dynamical stability. We believe Ce
and La substituted-hexaferrite,  where an extra electron is added to the
system, presumably exhibit similar phonon behaviors at finite k-points.
However, a thorough study of phonon behaviors  and thermodynamic stability
with element substitutions in hexaferrites requires computationally time-
consuming calculations, which is worth investigating in the future.

Figure 1: Phonon dispersion in Sr-hexaferrites computed along the high
symmetry k-points with GGA + $U$.

Figure 2: GGA + $U$ \+ SOC computed band structure of Sr1-xCexFe12O19 for (a)
$x=1$ and (b) $x=0.5$. (c) Comparison of partial density of states (PDOS) of
individual atoms for $x=0.5$ (in black) and $x=1$ (in red)  obtained with GGA
+ $U$. (d) Electron charge density contour occupying the valence bands near
the Fermi level (in e/Å3) for $x=1$. The occupied region of PDOS corresponds
to the net charge (1 e) occupying a single $4f$-orbital of Ce, confirming the
formation of Ce3+. The Fe($2a$)-PDOS occupied by transferred electrons is
shown in bottom panel (c). Consistent with the electronic structure of $x=1$,
the charge contours indicate the localization of Ce-substituted extra
electrons near the Fe($2a$) and the localization of one electron left in
Ce($4f$)-orbitals. The shape of the charge contour of electrons at the Ce atom
is polarized along the $c$-axis, which is responsible for enhancing MCA. For
$x=0.5$, the same is true except Fe($2a$)-$3d_{z}^{2}$ is a half-filled that
leads to a metal (b). The $3d_{z}^{2}$ bands split into two manifolds as the
net charge transfer slightly differs between two Fe($2a$) due to the proximity
effect.

##  IV Electronic structure

The band structure for $x=1.0$ is shown in Fig. 2(a). The GGA + $U$ \+ SOC
predicts an indirect band gap of 0.5 eV analogous to La-substitution[19]. The
reduction in the band gap is due to the appearance of a localized band in the
gap region of the parent compound. The narrow band corresponds to the
$3d_{z}^{2}$ orbital of Fe($2a$) as shown in Fig. 2 (c) (in red). Besides,
there are extra new features in band structure with varying Ce. First, four
occupied $3d$ bands show up slightly above the highly localized Ce($4f$) band.
Interestingly, these $3d$ bands are pushed down to the $4f$ band for $x=0.5$.

Figure 3: Partial density of states (PDOS) calculated using HSE06 for $x=1$
showing a band gap and electron localization.

Second, the band structure [Fig. 2(b)] shows a metallic state for $x=0.5$ as
the Fermi level lies in the middle of the spin-polarized (spin-$\downarrow$)
band and PDOS (c) in black. The half-filled bands are split into two, likely
due to a disproportionate amount of charge transfer from Ce, as the two
Fe($2a$) are not at the equal distance from Ce. Interestingly, the half-filled
bands are more dispersive along the planar distance,  indicating asymmetry in
the electron’s effective mass, which may lead to interesting transport
properties.

To better understand electron-electron correlation in $3d$ and $4f$ states, we
perform more advanced HSE06 calculations for $x=1$. We show HSE06 calculated
total and partial DOS of Ce and Fe($2a$) in Fig. 3. For completeness, we
compute the band gap of the pure system: HSE06 yields 2.4 eV, which is
slightly higher than the experimental (1.9 eV) and GGA + $U$ (1.74 eV) values.
We note that $U$ for Fe($3d$) is well established in the hexaferrites[22, 19],
while for Ce$(4f)$, it is unknown priori. With the help of HSE06 results, we
repeat the GGA + $U$ calculations by fine-tuning $U_{eff}$ values for both
$4f$ and $3d$ orbitals. The GGA + $U$ results correspond to the optimal
$U_{eff}$ value, as stated in the method section (II). Qualitatively, both the
GGA + $U$ and the HSE06 yield a similar band gap.  Not surprisingly, the
theoretical indirect band gap is smaller than the experiment. The experimental
band gap of (1.31-1.51 eV) is a direct optical band gap, as it was obtained
from a linear fit of $\alpha^{2}(h\nu)$ vs. $h\nu$. The calculated GGA + $U$
\+ SOC direct band gap of $~{}\sim 0.8$ eV at $\Gamma$ is also underestimated
from the experiment, as discussed in Ref. [12], probably due to the presence
of impurities in the low Ce-doped sample.

To get a more quantitative picture of the localization of transferred
electrons, we compute the net electrons in individual atoms by integrating the
occupied PDOS in GGA + $U$. First, we analyze the valence state of Ce, which
tends to fluctuate between Ce3+ and Ce4+[27, 28, 29]. Ce-atom consists of four
valence electrons with two $6s$, one $5d$, and one $4f$, meaning it can donate
two more electrons than Sr: $5s^{2}$. We find the net charge $\sim 1$e
residing in Ce confirming Ce3+ which is in agreement with Mosleh et al.’s [6]
report with Fourier-transform infrared spectroscopy (FTIR) data in Ce-
substituted hexaferrites. Next, we analyze the net charge transfer to the
remaining atoms. The extra electron occupies symmetry allowed[19] Fe at
$2a$-site as given in Table 3. For $x=0.5$, the Fe($2a$)-atoms share an
electron leading to a half-filled $3d_{z}^{2}$-orbital. On the other hand, the
amount of net electron transfer in the other Fe-sites is negligible. The sharp
localization of electron at $2a$-site is also confirmed by a charge density
contour [Fig.2(c)] computed for $x=1$ within the 1 eV energy below the Fermi
level.

Table 3: Spin and orbital magnetic moment (in $\mu_{B}$) of individual atom in Sr1-xCexFe12O19 ($x=0.5,1$), the total magnetic moment per unit cell including the interstitial contribution, and the net amount of electron transfer ($q_{t}$) (in e) to different atomic sites in units of electrons. For oxygen, only the average value is given. | $x=1$ |  |  | $x=0.5$ |  |
---|---|---|---|---|---|---
Atom | $\mu_{s}$ | $\mu_{l}$ | qt | $\mu_{s}$ | $\mu_{l}$ | qt
Fe($2a$) | 3.64 | 0.01 | 1.679 | 4.02 | 0.05 | 0.800
Fe($2b$) | 4.14 | 0.01 | 0.000 | 4.12 | 0.03 | 0.000
Fe($12k$) | 4.24 | 0.02 | 0.031 | 4.25 | 0.02 | 0.020
Fe($4f_{1}$) | -4.19 | -0.02 | 0.004 | -4.19 | -0.02 | 0.002
Fe($4f_{2}$) | -4.11 | -0.02 | 0.059 | -4.13 | -0.02 | 0.020
O | 0.10 | 0.0 | 0.227 | 0.11 | 0.00 | 0.100
Ce($2d$) | -0.27 | 2.17 |  | -0.92 | 0.85 | -
Total | 37.41 | 4.48 | 2.0 | 38.01 | 1.09 | 1.0

##  V Magnetic properties

Now, we discuss the magnetic moments as presented in Table 3. The spin moments
of all Fe-atoms are, to some extent, the same as in ${\rm SrFe}_{12}{\rm
O}_{19}$ except for Fe($2a$) in which the moment is reduced by about 0.5
$\mu_{B}$ for Fe2+ ($S=2$). As the $4f$-states are less than half-filled, the
signs of orbital and spin magnetic moments follow Hund’s rule. For $x=0.5$,
the Ce orbital moment is significantly quenched, likely due to a stronger
effect of the crystalline electric field in the metallic state. Interestingly,
the spin and orbital moments nearly cancel, leading to no gain in the net
magnetic moment. Indeed, experimentally, the magnetic moment is also shown to
decrease for $x=0.2$[6], which agrees with our results for $x=0.5$. On the
other hand, for $x=1$, Ce orbital moment (2.17 $\mu_{B}$) is much larger than
the spin moment (-0.27 $\mu_{B}$), which leads to an increase of the net
magnetic moment about 1 $\mu_{B}$ relative to ${\rm SrFe}_{12}{\rm O}_{19}$
(20 $\mu_{B}$). These results suggest that Ce controls the net magnetic moment
of hexaferrites.

Next, we compute T${}_{\text{C}}$ by taking the energy difference between the
ferromagnetic and ferrimagnetic structure of the nearest neighbor Fe-
sublattices, as T${}_{\text{C}}=S^{2}\bar{J}_{ij}/k_{B}$ where $S=5/2$ is spin
of each Fe and $\bar{J}_{ij}$ is average nearest neighbor exchange
interaction. The calculated T${}_{\text{C}}$ are 825.53 K and 777.74 K for
$x=1$ and 0.5, which are slightly smaller than for ${\rm SrFe}_{12}{\rm
O}_{19}$ (893 K). The predicted T${}_{\text{C}}$ for CeFe12O19 (CeM) is
slightly smaller than for ${\rm LaFe}_{12}{\rm O}_{19}$ (851 K), possibly due
to the neglect of the exchange interaction between Ce3+ and Fe3+ in  our
oversimplified mean-field approximation calculations.

We further calculated the T${}_{\text{C}}$ using the static linear-response
approach for CeM and LaFe12O19(LaM), as implemented in a Green’s function
(GF), the linearized muffin-tin orbital (LMTO) method[30, 31] in conjunction
with the atomic sphere approximation (ASA) and the local density approximation
(LDA) and $U$ ($U_{eff}=3$ eV for Ce and 4.5 eV for Fe). The effective
exchange interaction $J_{0}(i)$ for magnetic ion at site $i$ is obtained by
using the relation $J_{0}(i)=\sum_{j}J_{ij}$, where $j$ is the number of the
nearest neighbor atoms and $J_{ij}$ is the pairwise exchange coupling for
Heisenberg Hamiltonian $H=-\sum_{i,j}J_{ij}\hat{\bf e_{i}}.\hat{\bf e_{j}}$,
$\hat{\bf e_{i}}$ and $\hat{\bf e_{j}}$ are unit vectors along the direction
of local magnetic moments. Then, T${}_{\text{C}}$ is computed using the mean-
field approximation, i.e., T${}_{C}=\dfrac{2}{3}\sum_{i}J_{0}(i)/k_{B}$. The
computed nearest neighbor $J_{ij}$ are given in Table 4, which correctly show
the ordering of magnetic ions, i.e., positive and negative values for
antiferromagnetically and ferromagnetically aligned spins. As expected, the
leading magnetic exchange-couplings are found between the nearest neighbor
Fe3+ ions at $2a-4f_{1}$, $2a-12k$, $2b-4f_{2}$, $2b-12k$, $12k-4f_{1}$,
$12k-4f_{2}$, and $12k-12k$ sites, while the exchange couplings with Ce3+ are
negligible.

Table 4: Exchange coupling (in meV) between the nearest neighbor magnetic ions in Ce-hexaferrites computed from GF-ASA in the LDA + $U$. Fe atoms in different sites are labeled by corresponding Wyckoff positions. | Ce | $2a$ | $2b$ | $12k$ | $4f_{1}$ | $4f_{2}$
---|---|---|---|---|---|---
Ce | 0.00 | 0.01 | 0.97 | 0.36 | 0.29 | 0.00
$2a$ | - | -0.12 | 0.00 | -3.50 | 19.43 | 0.00
$2b$ | - | - | -0.98 | -9.36 | 0.80 | 24.08
$12k$ | - | - | - | -10.34 | 18.23 | 15.24
$4f_{1}$ | - | - | - | - | -2.91 | -0.57
$4f_{2}$ | - | - | - | - | - | -0.52

We found T${}_{\text{C}}=905$ K for LaM and 914 K for CeM, which agree
qualitatively with respective values obtained with an oversimplified approach.
For CeM, T${}_{\text{C}}$ obtained by including the exchange interactions of
Ce3+ with Fe3+ is slightly larger. Overall there are no significant
differences in T${}_{\text{C}}$ with La and Ce substitutions.

##  VI Magnetocrystalline anisotropy

Table 5: GGA + $U$ \+ SOC calculated magnetic anisotropy constant, $K_{1}$, in Sr1-xCexFe12O19 ($x$ is a variable $0$ to $1$) and its comparison with experiment and previous theory for Sr/LaFe12O19. The values are presented in units of MJ/m3 for $U_{eff}=2-4$ for Ce and $4.5$ eV for Fe. $K_{1}$ | $x=0$ | $x=1$ | $x=0.5$ | ${\rm LaFe}_{12}{\rm O}_{19}$
---|---|---|---|---
This work111$U_{eff}=4$ eV for Ce | 0.35 | 10.31 | 2.70 | 0.66
This work222$U_{eff}=3$ eV for Ce | 0.35 | 8.14 | 1.71 | 0.66
This work333$U_{eff}=2$ eV for Ce | 0.35 | 6.34 | 1.14 | 0.66
Expt.[32, 33] | $0.35-0.36$ | - | - | $0.5-0.8$

Now we turn our discussion to magnetic anisotropy. To predict MCA, we employ a
fully self-consistent relativistic non-collinear scheme to compute the total
energy in two different crystalline directions, viz., $E_{001}$
(magnetocrystalline easy) and $E_{100}$ (magnetocrystalline hard) along the
crystalline $c,~{}i.e.,[0,0,1]$ and $a,~{}i.e,[1,0,0]$ directions, including
SOC in GGA + $U$. The magnetic anisotropy energy ($MAE$) is then obtained as
$MAE=E_{100}-E_{001}$ from which we evaluate $K_{1}$ as $K_{1}=MAE/V$, where
$V$ is the unit cell volume.

Figure 4: On-site anisotropy energy ($E_{aniso}$) contributions. The Ce has
the highest $E_{aniso}$ contribution. Fe atoms are labeled by their
corresponding Wyckoff position. The onsite Fe contribution is positive at
$12k$ and $2b$ and negative at $2a$. The contribution is negligible at
$4f_{1}$ and $4f_{2}$-sites.

$K_{1}$ is computed for different values of $U_{eff}$ for Ce(4$f$) [Table 5].
For $U_{eff}\sim 3$ eV, the computed value of $K_{1}$ in 100% Ce-doped
hexaferrite is nearly 8 times larger than in strontium-hexaferrite. These
results clearly show a big influence of $4f$-electrons. Qualitatively, the
effect of $4f$-electrons can be understood by the charge density contours of
occupying states near the Fermi level. As seen from Ce’s prolate type of
charge density contour, it costs huge energy to rotate it in other directions.
The most direct quantitative evidence for the site-contribution to MAE can be
obtained from the energy difference between easy and hard axes (anisotropy
energy) viz., $E_{aniso}=E_{soc}[100]-E_{soc}[001]$, where $E_{soc}[001]$ and
$E_{soc}[100]$ are atomic SOC energies for magnetization directions $[0,0,1]$
and $[1,0,0]$. The net MAE is obtained from the expression[34]
$\dfrac{1}{2}\sum_{i}E_{aniso}^{i}$, where $i$ denotes the atomic site. Figure
4 shows the $E_{aniso}$ to MAE for different Ce substitutions. For $x=0$, the
MAE is primarily governed by Fe($2b$) and Fe($12k$) with some contribution
from Fe at $4f_{1}$ and $4f_{2}$. Although small, Fe($2a$) gives a planar
contribution. Interestingly, electron doping to Fe($2a$) results in uniaxial
or planar MAE depending on the substituted elements. For instance, it produces
a uniaxial contribution and is a leading term in La substituted hexaferrite.
In CeM, the shape of the $3d_{z}^{2}$ orbital is significantly distorted [Fig.
1(d)] from the $c$-direction, unlike in LaM, probably due to the $4f$
electrons. The Fe($2b$) site contribution is uniaxial and remains unchanged
after substitutions. The $4f_{1}$ or $4f_{2}$ contribution changes sign;
however, it does not affect the MAE since it is much smaller. Of note, as
expected in $4f$-electron systems, the Ce-site contribution is larger by
orders of magnitude than other sites.

A similar $4f$-electrons effect in MAE is found for $x=0.5$, while the
increase is not proportional to Ce. In the metallic system, the nonlinear
trend in MAE is expected; because the itinerant electrons can perturb the $4f$
charge distribution and affect the crystalline electric field. Indeed, the Ce-
site contribution to MAE [Fig. 4] is reduced, consistent with the reduced Ce-
orbital magnetic moment. Lastly, the Fe-sites contribution to MAE is less
significant.

Table 6: Calculated magnetic properties and comparison with available experiments (for $x=0$) at room temperature otherwise stated for Ce-substituted compositions (Sr1-xCexFe12O19) with  $U_{eff}=3$ eV for Ce and $4.5$ eV for Fe viz., magnetic anisotropy constant ($K_{1}$), saturation magnetization ($M_{S}$), anisotropy field ($H_{a}$), BHmax, and magnetic hardness parameter ($\kappa$) (units are given inside the parenthesis). parameter | $x=0$ | $x=1$ | $x=0.5$ | Expt[35]
---|---|---|---|---
$K_{1}$ (MJ/m3) | 0.35 | 8.14 | 1.71 | 0.35
$M_{S}$ (T) | 0.60 | 0.64 | 0.59 | 0.31444Experiment at 1100 ∘ C for $x=0.1$ Ref.6, 0.48 - 0.59555 Experiment at 5 K for $x=0$ Ref. [3]
$H_{a}$ (KOe) | 14.58 | 320.04 | 72.56 |  $14.88^{a}$-20.01
BHmax (kJ/m3) | 72.42 | 81.12 | 69.82 | 45-$69.53^{a}$
$\kappa$ | $1.13$ | 5.01 | 2.47 | 1.35

Table 6 summarizes the computed magnetic properties of Ce-substituted
compositions and compares them with experimental values of pure Sr-
hexaferrite. The saturation magnetization ($M_{S}$) is enhanced in full Ce-
hexaferrite. However, 50% Ce substitution does not increase the $M_{S}$.  Our
calculated $M_{S}$ lies within the range of experimental values, which vary
from 0.31 to 0.59 T depending on the temperature, sample preparation
techniques, and micromagnetic structures. Magnetic anisotropy field ($H_{a}$)
can be estimated using the relation

$H_{a}=\dfrac{2K_{1}}{M_{S}}.$ (7)

For Ce-substituted compositions, $H_{a}$ enhances by about a factor of 10,
consistent with the predicted value of $K_{1}$.

BHmax which is estimated from $M_{S}$ as

$\text{BH}_{max}=\dfrac{\mu_{0}M_{S}^{2}}{4}.$ (8)

BHmax is the largest for full Ce. Magnetic hardness parameter $\kappa$ is
given by

$\kappa=\sqrt{K_{1}/\mu_{0}M_{S}^{2}},$ (9)

and it satisfies the criterion[35] ($\kappa>1$) of hard magnet for all
compositions. The full Ce substitution of Sr produces the hardest permanent
magnet with the largest value of $\kappa$.

##  VII  Optical anisotropy

To understand the effect of Ce($4f$)-states in optical properties of
CeFe12O19, we compute the optical absorption coefficient

Figure 5: Optical absorption coefficient, $\alpha(\omega)$, of full Ce-
substituted hexaferrite ($x=1$), calculated with GGA + $U$ \+ SOC (a) and
HSE06 + SOC (b). The strength of dipole-transition between $4f$ states is
rather strong compared to $3d$-states, resulting in two peaks $\beta$ and
$\gamma$ at $\sim 2$ and $\sim 4$ eV in GGA + $U$ \+ SOC. These peaks shift by
$\sim 1$ eV with HSE06. In the HSE06 results, an additional peak appears in
the low energy region around 1.3 eV for $E\parallel c$.

$\alpha(\omega)=\dfrac{\omega\varepsilon_{2}(\omega)}{cn(\omega)},$ where $c$,
$\omega$, and
$n(\omega)=\dfrac{1}{\sqrt{2}}[\varepsilon_{1}(\omega)+\sqrt{\varepsilon_{1}(\omega)^{2}+\varepsilon_{2}(\omega)^{2}}]^{\frac{1}{2}}$,
and $\varepsilon_{2}(\omega)$ are speed of light, photon energy, refractive
index, and the imaginary part of the macroscopic dielectric constant from
which the real part of the macroscopic dielectric constant
$\varepsilon_{1}(\omega)$ is obtained using Kramers-Kronig transformation. The
$\varepsilon_{2}(\omega)$ is calculated from the expression

$\varepsilon_{2}(\omega)=\frac{4\pi^{2}e^{2}}{m^{2}\omega^{2}}\sum_{v,c}\int_{BZ}\frac{d^{3}k}{(2\pi)^{3}}|\hat{e}\cdot{\bf
M}_{cv}({\bf k})|^{2}\delta(\hbar\omega-\epsilon_{c{{\bf k}}}+\epsilon_{v{{\bf
k}}}),$ (10)

where $\hat{e}$ and ${{\bf M}}_{cv}({{\bf k}})=\langle\psi_{c{{\bf k}}}|{\bf
p}|\psi_{v{\bf k}}\rangle$ are the light polarization vector and the momentum
matrix element between the conduction and the valence states. We find
significant changes in the $\alpha(\omega)$ as compared with ${\rm
LaFe}_{12}{\rm O}_{19}$, where vertical transitions occur between Fe($3d$)
states as discussed in our previous paper[19]. Overall the intensity of
$\alpha(\omega)$ is enhanced considerably with Ce, including the appearance of
two main peaks labeled by $\beta$ and $\gamma$ in Fig. 5. It is expected as
the $4f$-states are located near the Fermi-level, for which the strength of
matrix elements of dipole moments corresponding to the transition from the
occupied $4f$ to unoccupied $4f$-states is much stronger. The unoccupied
$4f$-states are split into two manifolds because of the combined effects of
crystal-field, exchange, and SOC. $\beta$-peak corresponds to the transition
from the occupied $4f$ to lower unoccupied at $\sim 2$ eV, and $\gamma$
corresponds to upper unoccupied $4f$-states at $\sim 4$ eV. The optical
absorption is spatially localized to Ce-atom, which could be interesting to
examine experimentally. The other interesting result is enhanced anisotropy in
optical absorption. Indeed, the real parts of dielectric constant
$\varepsilon_{1}$ are 6.94 for $E\parallel a$ and 6.18 for $E\parallel c$
consistent with $\alpha(\omega)$.

HSE06 calculations reveal qualitatively similar optical spectra, except now
that the absorption peaks are shifted. In particular, $\beta$-peak is shifted
by about 1 eV, while $\gamma$-peak shifts by about 0.5 eV. These are
consistent with the alignment of Ce($4f$)-bands relative to the Fermi level.
The height of $\beta$-peak is similar, while $\gamma$-peak is reduced by about
2. Interestingly, HSE06 + SOC produces another peak of about 1.3 eV
corresponding to the interband transition between the Fe($3d$)-states for
$E\parallel c$, which is much weaker with the GGA + $U$ \+ SOC calculations.
The magnitudes of $\varepsilon_{1}$ are 3.94 for $E\parallel a$ and 3.65
$E\parallel c$. The difference in the dielectric constants between the two
approaches is expected, partly due to the use of different numbers of virtual
states in the calculations. Unfortunately, the convergence of $\varepsilon$ in
HSE06 calculations is always problematic due to the huge computational cost.
However, both approaches yield a similar trend in the anisotropy.

##  VIII Metastable sites and magnetism

By performing the GGA + $U$ \+ SOC calculations, we examine the effect of Fe-
site substitutions by Ce in magnetic properties and local spin quantization.
Depending on the Ce-site occupancy, the net magnetic moment changes from 34
$\mu_{B}$ to 44 $\mu_{B}$, as given in Table 7. To illustrate it, we discuss
the PDOS of Ce and Fe (nearest to Ce) while replacing one Fe at $4f_{2}$ (the
most favorable Fe site) in Fig. 6. The $4f$-states are delocalized below the
Fermi level. Quite interestingly, one electron of Ce transfers only to the
nearest Fe($2a$), forming a localized $3dz^{2}$ state. It reduces Fe3+ to
Fe2+, as found in Ce at Sr-site. Ce remains to be Ce3+, which helps to
increase the net magnetic moment to 44 $\mu_{B}$ for the following reasons:
(i) the negative spin magnetic moment contribution of about 5 $\mu_{B}$ at
$4f_{2}$ of one Fe-atom vanishes and (ii) the magnetic moment of Ce itself is
much smaller $\sim$ 1 $\mu_{B}$. A similar situation holds in $4f_{1}$-site.
At the $2b$ site, Ce gives an extra electron to Fe($2a$), producing Fe2.5+ and
further reducing the net moment to 34 $\mu_{B}$. Interestingly, at the $12k$
site, our calculations show that the Ce donates an extra electron to the
nearest Fe($12k$), resulting in a similar net moment.

Figure 6: PDOS of Ce($4f$) and Fe($3d$) at $2a$ (the nearest to the Ce in the
unit cell) computed with GGA + $U$ \+ SOC for Ce-substituted at Fe-$4f_{2}$
site. The extra Ce electron transfers to the Fe($3d$) states localized below
the Fermi level.

Next, we  calculate total energies by choosing the spin quantization
directions along [0,0,1], [1,0,0], and [1,1,1] using self-consistent GGA + $U$
\+ SOC method and investigate the Ce-induced spin-canting by analyzing the
$E_{aniso}$ and $MAE$.  Fortunately, we did not have to constrain the spins
because they remained aligned with the initialized directions even after
completing the full self-consistent calculations. The computed $E_{aniso}$
(not shown here) suggest that the local spin magnetic moment directions of
Ce($4f_{2}$) and Fe($2a$) are not along the [0,0,1] (Gorter’s type[36]),
rather they are along the [1,1,1], while the directions of other remaining Fe-
spin magnetic moments remain the same. As Ce($4f$) with a large $E_{aniso}$
primarily controls the magnetic anisotropy, the whole spin-system becomes uni-
axial along the crystalline [1,1,1] direction with $MAE=E_{001}-E_{111}=0.45$
meV. This local spin-canting could arise due to the following reasons: (i)  a
complex spin structure around the defect and (ii) a significant change in the
crystalline electric field. Experimentally Sm is already known to replace
Sr[10]. In a separate experimental work[7], authors report that Sm causes a
spin-canting at Fe-sites. Pawar et al. also discussed the possibility of spin
canting if Ce is doped at Fe-sites[11]. Although our DFT calculations suggest
Fe-spins canting near the impurity, it does not dramatically change the net
magnetic moment (see Table VII) as reported in experiment[7], in which $M_{S}$
is shown to decrease significantly by increasing the Ce concentration.

Table 7: GGA + $U$ calculated the net magnetic moments ($m$) (in $\mu_{B}$) for SrCeFe23O19 by replacing a Fe atom with Ce in different sites and the optimized lattice constants $a$ and $c$ (in Å).  parameter | $2a$ | $2b$ | $12k$ | $4f_{1}$ | $4f_{2}$
---|---|---|---|---|---
a | 5.95 | 5.99 | 5.98 | 5.96 | 5.96
c | 23.48 | 23.15 | 23.36 | 23.41 | 23.68
$m$ | 36 | 34 | 34 | 44 | 44

##  IX Conclusion

In summary, by performing DFT, including hybrid functional calculations and
analyzing formation energies and elastic constants, we uncover the lattice
stability of Ce-substituted M-type strontium hexaferrites. Both full- and
half-Ce-substituted compositions are chemically and mechanically stable. The
full Ce results in fully occupied narrow $3d_{z}^{2}$-derived bands of
Fe($2a$), yielding a reduced band gap. On the other hand, the half-Ce leads to
a half-filled $3d_{z}^{2}$-band producing a metallic state. These results,
including the computed net amount of 4f electron in Ce, confirm the formation
of Ce3+. The Ce increases the MCA of the parent compound by at least an order
of magnitude, which is a direct consequence of Ce(4f)-electrons. The net
magnetic moment also increases with the full Ce as the net Ce-contribution
results from the orbital moment following Hund’s rule. In addition, due to
Ce($4f$), the computed optical spectra for CeFe12O19 show significant
anisotropy between light polarization in parallel and perpendicular directions
with enhanced transitions between Ce($4f$) and conduction bands. If grown
successfully, we believe the Ce-substituted M-type hexaferrite can be a
potential candidate material for a high-performance permanent magnet.

###### Acknowledgements.

This work is supported by the Critical Materials Institute, an Energy
Innovation Hub funded by the U.S. Department of Energy, Office of Energy
Efficiency and Renewable Energy, Advanced Manufacturing Office. The Ames
National Laboratory is operated for the U.S. Department of Energy by Iowa
State University of Science and Technology under Contract No. DE-
AC02-07CH11358.

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