# Origin of Perpendicular Magnetic Anisotropy in CoxFe3-xO4+δ Thin Films
Studied by X-ray Magnetic Circular and Linear Dichroisms

Jun Okabayashi1,∗, Masaaki A. Tanaka2, Masaya Morishita2, Hideto Yanagihara3,
and Ko Mibu2 1Research Center for Spectrochemistry, The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan
2Graduate School of Engineering, Nagoya Institute of Technology, Nagoya
466-8555, Japan
3Institute of Applied Physics, University of Tsukuba, Tsukuba 305-8573, Japan

###### Abstract

We investigate the element-specific spin and orbital states and their roles on
magnetic anisotropy in the Co-ferrite (CoxFe3-xO4+δ (001)) thin films which
exhibit perpendicular magnetic anisotropy (PMA). The origin of PMA in the low
$x$ region ($x$ $<$ 1) can be mainly explained by the large perpendicular
orbital magnetic moments in the Co2+ (3$d^{7}$) states detected by X-ray
magnetic circular and linear dichroisms (XMCD/XMLD). The XMLD for a PMA film
($x=0.2$) with square hysteresis curve shows the oblate charge distribution in
the Co2+ site, which is consistent with the change in local nearest neighbor
distance in Co detected by extended X-ray absorption fine structure analysis.
Our finding reveals that the microscopic origin of PMA in Co-ferrite comes
from the enhanced orbital magnetic moments along out-of-plane [001] direction
through in-plane charge distribution by tensile strain, which adds the
material functionalities in spinel ferrite thin films from the viewpoint of
strain and orbital magnetic moments.

Perpendicular magnetic anisotropy (PMA) is one of the crucial issues in the
spintronics research field from the viewpoints of thermal stability
enhancement and high-density storage technology with the low current
magnetization switching. Until now, tremendous efforts have been devoted to
design the materials with large PMA to add the novel functionalities and
external field controlling of magnetic properties ref1 ; ref2 ; ref3 ; ref4 ;
ref5 . Among them, the magnetic oxides possess a number of attractive
advantages for the spintronics devices because of their semiconducting or
insulating behavior with high spin polarization. These magnetic oxides can be
utilized as spin-filtering effects, resulting in high tunnel magnetoresistance
with multiferroic properties by controlling the tensile or compressive strains
in the thin films ref6 ; ref7 ; ref8 .

Most famous spinel-type compounds for spintronics application might be Co-
ferrites (CoFe2O4: CFO) epitaxial thin films because of their insulating
advantages as spin-filtering, high-frequency performance, and magneto-
strictive properties ref9 ; ref10 ; ref11 ; ref12 ; ref13 ; ref14 ; ref15 ;
ref16 . Further, the interfacial proximity effect, emerging on films with non-
magnetic heavy-metal elements which possess a large atomic spin-orbit
interaction ref17 ; ref18 ; ref19 , also develops as the spin-orbitronics
researches through the interfacial spin-orbit coupling. Recent development
appends further functionality of large PMA in CFO in the order of 106 J/m3 by
tuning the Co compositions ref20 ; ref21 ; ref22 ; ref23 . Since CFO exhibits
high magnetostriction coefficient ($\lambda_{100}=–590\times 10^{-6}$), the
strain from the substrates propagates into the CFO layer, which brings the
local distortion around the transition-metal (TM) cation sites. In fact, the
lattice mismatching between MgO (4.21 $\mathrm{\AA}$) and CFO (8.38
$\mathrm{\AA}$) forming almost the double period brings the epitaxial
crystalline growth with a few percentages of tensile strain into CFO thin
layer. Depending on the Co compositions, the lattice constants of CFO can be
tuned systematically. Structural and magnetic properties in the CFO thin films
can be controlled by the cation vacancies and anti-phase boundaries which are
generated during the crystalline growth and suppress the saturation
magnetization values through the antiparallel spin coupling ref24 ; ref25 ;
ref26 ; ref27 . Historically, magnetic anisotropy in bulk CFO has been
understood as an effect that the average of $\textless 111\textgreater$ easy
axes direction produces cubic anisotropy in Co2+ ref28 ; ref29 ; ref30 .
Recent theoretical and experimental investigations for CFO thin films suggest
that the tensile strain into CFO layer produces the PMA ref31 ; ref32 ; ref33
; ref34 ; ref35 ; ref36 ; ref37 ; ref38 . Squareness of out-of-plane
hysteresis curves depends on the growth mode, such as molecular-beam epitaxy,
pulsed-laser deposition (PLD), or sputter deposition. It is also reported that
the PMA increases in off-stoichiometric CFO films with low Co compositions
ref22 ; ref23 . Although the PMA in CFO is strongly demanded for spinel-type
oxide spintronics and spin-orbitronics by controlling the orbital states ref39
, the microscopic origin of the PMA appearing in the low Co compositions of
CFO has not been clarified yet.

In order to understand the PMA in CFO, the element-specific orbital magnetic
moments ($m_{\mathrm{orb}}$) and their orbital anisotropy have to be examined
explicitly. In particular, $m_{\mathrm{orb}}$ is expected to be sensitive to
the strain and local symmetry from the ligand fields around the TM ions. X-ray
magnetic circular and linear dichroisms (XMCD / XMLD) with their magneto-
optical sum rules are powerful tools to detect the element-specific spin and
orbital states in each site with the charge asphericity ref40 . Especially,
XMLD sum rules can probe the orbital anisotropy ($Q=3L_{z}^{2}-L^{2}$) in the
notation of orbital angular momentum ($L$) and electric quadrupoles ($Q$).
Furthermore, since the Fe $L$-edge spectra consist of the signals of three
kinds of Fe2+ or Fe3+ states with octahedral $O_{h}$ and tetrahedral $T_{d}$
symmetries, the XMCD and XMLD magneto-optical sum rules for the integrals of
Fe $L$-edge spectra cannot be adopted without separating each contribution.
Site- and state-resolved analyses become a crucial role for understanding the
origin of PMA. For the analysis of XMCD and XMLD spectra, the ligand-field
multiplet (LFM) cluster-model simulations including the configuration
interaction (CI) approach are used to determine the site-specific electronic
structure parameters accompanied by spin and orbital magnetic moments and
their anisotropies ref41 . Until now, the XMCD and XMLD for CFO with in-plane
anisotropy have been extensively studied with angular dependence between
incident polarized beam, magnetic field, and sample surface normal ref42 ;
ref43 ; ref44 ; ref45 . However, the X-ray magnetic spectroscopy
investigations for the PMA films with square magnetization–magnetic field
($M$-$H$) hysteresis shape in low Co compositions have not been pursued. We
have proceeded the XMLD measurements for probing the orbital anisotropy in
several PMA systems by using remanent magnetization states in the polar Kerr
geometry ref46 ; ref47 . On the other hand, recent operando-XMCD technique
enables manipulation of out-of-plane and in-plane $m_{\mathrm{orb}}$ by
controlling strain in the magnetic multilayers, which is a microscopic origin
of magneto-striction; that leads to the observation of an effect proposed as
”orbital-striction effect” ref48 . The PMA in strained CFO can be also
categorized as a novel orbital-controlled material system by using the orbital
degeneracy in Co2+ 3$d^{7}$ system in $O_{h}$ ligand field. Furthermore,
extended x-ray absorption fine structure (EXAFS) analysis through the
multiple-scattering process is also employed to determine the element-specific
structural characterization. The precise determinations of spin and orbital
states become a key solution to develop a strain-induced orbital physics and
their applications using CFO.

In this study, considering above research motivations, we aim to investigate
the origin of PMA in CFO from the viewpoint of element-specific spin and
orbital magnetic moments and their anisotropy using XMCD and XMLD with LFM
calculations and local structure detection by EXAFS analysis.

The CoxFe3-xO4+δ (001) samples ($x$=0.2 and 0.6) were prepared by PLD on the
MgO (001) substrates with the similar recipe of reference ref23 . The PLD was
performed under the conditions of the substrate temperature of 300∘C, the
background oxygen pressure of 6.0 Pa, and the deposition rate of 0.03 nm/s. A
neodymium-doped yttrium-aluminum-garnet laser at the double frequency (532 nm)
with the pulse width of 6 ns and the repetition rate of 30 Hz was used. The
energy density of the laser beam was controlled to 1 J/cm2 by an optical lens.
The 13-nm-thick CFO (001) layer with 1-nm-thick Cu capping layer was deposited
for various Co compositions ($x$). Excess oxygens accompanied by possible
cation vacancies are described as $\delta$. The case of $x$=0.2 and 0.6
exhibits the PMA and in-plane anisotropy, respectively. The in-plane
anisotropy in Co0.6Fe2.4O4+δ is produced by the post annealing at 400∘C after
the growth by releasing the strain. The results of sample characterization by
X-ray diffraction (XRD) and magnetization measurements are shown in
Supplemental Material (Fig. S1) ref49SM .

The XMCD and XMLD were performed at BL-7A and 16A in the Photon Factory at the
High-Energy Accelerator Research Organization (KEK-PF). For the XMCD
measurements, the photon helicity was fixed, and a magnetic field of $\pm$1.2
T was applied parallel to the incident polarized soft x-ray beam, to obtain
signals defined as $\mu$\+ and ${\mu}-$ spectra. The total electron yield mode
was adopted, and all measurements were performed at room temperature. The
X-ray absorption spectroscopy (XAS) and XMCD measurement geometries were set
to normal incidence, so that the directions of photon helicity axis and the
magnetic field were parallel and normal to the surface, enabling measurement
of the absorption processes involving the normal components of the spin and
orbital magnetic moments. In the XMLD measurements, the remanent states
magnetized out-of-plane were adopted. For grazing incident measurements in
XMLD, the tilting angle between incident beam and sample surface normal was
kept at 60∘. The direction of the electric field ($\bf{E}$) of the incident
linearly polarized synchrotron beam was tuned horizontally and vertically by
undulator. We define the sign of XMLD by the subtraction of the
($\bf{M}{\parallel}{\bf{E}}$)$-$(${\bf{M}}{\perp}{\bf{E}}$) spectra with
respect to the magnetization $\bf{M}$. The EXAFS measurements at Co $K$-edge
were performed at BL-12C in KEK-PF using the florescence yield mode with a
19-element solid-state detector at room temperature.

Figure 1 shows the XAS and XMCD of Co0.2Fe2.8O4+δ for Fe and Co $L$-edges at
the normal incidence setup. Spectra are normalized at each absorption edge.
Because of the composition ratio of Co:Fe=1:14, the XAS intensities of Co are
suppressed. XMCD intensity ratio to XAS is estimated to be 5$\%$ and 47$\%$
for Fe and Co $L_{3}$ edge, respectively. The raw data are displayed at the
Supplemental Material (Fig. S2). XAS and XMCD line shapes for Fe $L$-edges
show distinctive features due to the three kinds of Fe states (Fe3+ in
$O_{h}$, Fe3+ in $T_{d}$, and Fe2+ in $O_{h}$). For the Fe $L$-edges, although
the difference in XAS is small, clear differential XMCD line shapes are
detected. The Fe3+ state with $T_{d}$ symmetry exhibits opposite sign, which
is common for the spinel ferrite compounds. On the other hand, in the case of
Fe3O4, the Fe2+ component is more enhanced ref49 . Large XMCD signals in Co
$L$-edge correspond to the saturated magnetized states. Within the orbital sum
rule, the large $m_{\mathrm{orb}}$ gives rise to the asymmetric XMCD line
shapes. Since the Co site is almost identical as Co${}^{2+}(O_{h})$ symmetry,
the sum rules can be applicable for the Co XMCD spectra. The spin and orbital
magnetic moments for Co2+ sites are estimated as 1.32$\pm$0.20 and
0.63$\pm$0.09 $\mu_{\mathrm{B}}$, respectively, using an electron number of
7.1. The error bars mainly originate from the estimation of background in XAS
which is used for the sum rule analysis. Large $m_{\mathrm{orb}}$ originates
from the orbital degeneracy in $d^{7}$ electron system in the strained Co
site. The multiplet structures are almost similar to the previous report by
the LFM calculations along the easy-axis direction ref44 ; ref45 . In order to
confirm the PMA, the element-specific magnetic hysteresis curves are measured
at each peak in Fe and Co $L_{3}$ edge. As shown in Fig. 1(c), clear square
shapes are detected at each photon energies due to the out-of-plane easy axis,
which is almost consistent with the magnetization measurement shown in Fig. S1
ref49SM . The square $M$-$H$ curves for all energies suggest the strong
exchange coupling among each site. The opposite sign in the $M$-$H$ curves at
Fe3+ $T_{d}$ site and Fe2+/Co2+ $O_{h}$ sites originates from the
antiferromagnetic super-exchange interaction among these sites. These suggest
that the small amounts of anisotropic Co sites with large $m_{\mathrm{orb}}$
govern the easy axis direction of Fe sites and stabilize the PMA.

Figure 2 shows the $\bf{E}$ vector polarization dependent XAS, where the
electric field $\bf{E}$ is parallel and perpendicular to the out-of-plane
magnetization direction. The differential line shapes were similar to those of
previously reported spectra ref44 . Because of small difference in XAS by the
horizontal (parallel) and vertical (perpendicular) beams as shown in the inset
of Fig. 2, XMLD intensities are also suppressed and displayed in the different
scales. For Fe $L$-edge XMLD, the overlapping of three kinds of components
brings complex differential line shape. We note that the integrals of the XMLD
line shapes are proportional to the charge asphericity within the XMLD sum
rule ref50 . For Co $L$-edge XMLD of the Co2+ $O_{h}$ site, we confirmed that
the integral of XMLD converges to a negative value, deducing that the sign of
$Q_{\mathrm{zz}}$, where $Q_{\mathrm{zz}}$ is the diagonal tensor component of
electric quadrupole moment describing the charge distribution asphericity as
$Q_{\mathrm{xx}}+Q_{\mathrm{yy}}+Q_{\mathrm{zz}}=0$, is negative with the
order of 10-1. The result means the in-plane orbital states are strongly
coupled with $\bf{E}$. This value can be also estimated from the XMCD spin sum
rule considering the magnetic dipole $m_{T_{z}}$ term. Since the measurement
geometry of 60∘ tilted from the sample surface normal cannot detect precise
orthogonal direction, the oblique angle suppresses the linear dichroism
intensity to $\sqrt{3}/2$. The orbital polarization of Co 3$d$ states means
the pancake-type oblate charge distribution, which enhances out-of-plane
component of $m_{\mathrm{orb}}$. Therefore, with both XMCD and XMLD combined,
it can be concluded the in-plane tensile strain triggers the changes of charge
distribution along in-plane direction, resulting in the large out-of-plane
$m_{\mathrm{orb}}$ and the PMA.

For the analysis of XMCD and XMLD spectra, we employed cluster-model
calculations including the CI for Co2+ and Fe sites in Co0.2Fe2.8O4+δ as
tetrahedral ($T_{d}$) TMO4 and octahedral ($O_{h}$) TMO6 clusters, modeled as
a fragment of the spinel-type structures. The Hamiltonian included the
electronic structure parameters of full on-site TM 3$d$–3$d$ (valence–valence)
and 2$p$–3$d$ (core–valence) Coulomb interactions ($U$) and the $T_{d}$ or
$O_{h}$ crystal fields (10$Dq$) in the TMs, along with the hybridization
between the TM 3$d$ and O 2$p$ wave functions. The charge-transfer energy was
defined as $\Delta$=$E(3d^{n+1}\underline{L})-E(3d^{n})$, where
$\underline{L}$ denotes a hole in a ligand $p$ orbital. The hybridization
between the TM 3$d$ and O 2$p$ states was also parameterized in terms of the
Slater–Koster parameters ($pd\sigma$) and ($pd\pi$), where the relation
$(pd\sigma)=–1/2(pd\pi)$ was used ref51 . The parameters ($pp\sigma$) and
($pp\pi$) were always set to zero. In all cases, a Gaussian broadening was
used to simulate spectral broadening.

As shown in Fig. 3, spectral line shapes of XMCD and XMLD can be reproduced by
the LFM calculations qualitatively, at least the peak positions, with three Fe
states and uniformed Co state. The fitting parameters are listed in Table I.
We emphasize the same adjusted parameters of $\Delta$, $U$ and $pd\sigma$ and
intensity ratios of each component for XMCD and XMLD are applicable for the
fitting. In comparison with the previous first-principles calculations of CFO,
the values of $U=5$ eV are also plausible from the viewpoints of LFM
calculations ref52 ; ref53 ; ref54 ; ref55 . The multiplet parameter values
are almost similar to the previous report ref56 . For the Co2+ 3$d^{7}$ case,
the $t_{2g}$ states are split into two levels by the tetragonal distortion,
and the lowest $xy$ states are occupied by one of the down spin electrons. The
other electron occupies the degenerated $yz$ or $zx$ states ref35 . We
determined the tetragonal distortion for the Co 3$d$ states
($D_{\mathrm{tet}}$) to be 0.02 eV in order to reproduce the XMCD and XMLD
spectral line shapes qualitatively. However, the best fitted parameter sets
shown in Table I are not sensitive to the parameters caused by the strain. The
intensity ratio of Fe3+ $T_{d}$, Fe3+ $O_{h}$, and Fe2+ $O_{h}$ is
deconvoluted to be 1:1.3:0.25 in the fitting of Fe $L_{3}$-edge XMCD and XMLD
affecting the decrease of Fe2+ states in the inverse spinel structure. The Co
$L$-edge line shapes are reproduced with only a single Co2+ site, except the
peak at 779.5 eV in XMLD.

In order to confirm the local environment around the Co atoms, XAFS
measurements with EXAFS analysis in the Co $K$-edge were performed to deduce
the nearest neighbor distance through the Fourier transform. As shown in Fig.
4, the Co $K$-edge absorption spectra with EXAFS oscillatory behaviors are
observed for $x=0.2$ (PMA) and $x$=0.6 (in-plane anisotropy). We note that the
Fe $K$-edge XAFS cannot be separated in the Fourier transform without some
assumption because of the contributions of three kinds of states. Clear XAFS
oscillation is detected and plotted in the wave number $k$ in the inset of
Fig. 4(a). XAFS oscillation functon $k^{3}{\chi}(k)$ can be fitted by FEFF8
program ref57pre  and the Fourier transformed EXAFS profile is also displayed
in Fig. 4(b) considering the information up to second nearest neighbor from
the Co site. By using the fitting procedure, the nearest Co-O bond length in
the Co sites is 2.09 and 2.08 Åfor PMA and in-plane cases, respectively, which
is almost identical with the previous study ref57 . The expansion of nearest
neighbor in the PMA film is qualitatively consistent with the XRD shown in
Fig. S1 ref49SM . Because of the effect of phase shift, the peak position in
EXAFS is not related to the bond length directly. Coordination number also
becomes large in the case of PMA because of the distorted local environment
around Co site, which corresponds to the intensity of EXAFS profile in Fig.
4(b). These suggest the tensile local distortion triggers the PMA in the Co
site of strained CFO.

Considering the above results, we discuss the electronic structures of CFO.
First, we discuss the suppression of Fe2+ with increasing the Co composition.
The previous report of Mössbauer spectra for related CFO with PMA suggests the
suppression of Fe2+ ref23 , where $\sim$100 nm from the interface was probed
using incident $\gamma$-rays and emitted conversion electrons. Since the
probing depth region of XMCD is beneath 3 nm from the surface, finite Fe2+
states in XMCD is located at the surface region. The previous report suggests
that the Fe2+ state is located at the surface region with inevitable lateral
inhomogeneity ref57re .

Second, we discuss the occupation of cation sites by the charge neutrality and
the number of cation elements. Using the Co concentration and charge
neutrality in the inverse spinel structure of the formation of spinel
structure of $AB_{2}$O4 formula unit;
Fe${}^{3+(T_{d})}$(Co${}^{2+(O_{h})}$Fe${}^{2+(O_{h})}$Fe${}^{3+(O_{h})}$)O4,
the following two equations are deduced for Co0.2Fe2.8O4+δ;
14$n_{\mathrm{Co}^{2+}}$=$n_{\mathrm{Fe}^{2+}}$+$n_{\mathrm{Fe}^{3+}}$, and
$2n_{\mathrm{Co}^{2+}}$+2$n_{\mathrm{Fe}^{2+}}$+3$n_{\mathrm{Fe}^{3+}}$=8,
where $n$ is the number of each element and the cation vacancy is implicitly
incorporated as $\delta$. Since the ratio of $n_{\mathrm{Fe}^{3+}(T_{d})}$,
$n_{\mathrm{Fe}^{2+}(Oh)}$, and $n_{\mathrm{Fe}^{3+}(O_{h})}$ is estimated
from the XMCD and XMLD analyses, these four parameters are calculated to be
$n_{\mathrm{Fe}^{3+}(O_{h})}$=1.34, $n_{\mathrm{Fe}^{3+}(T_{d})}$=1.03,
$n_{\mathrm{Fe}^{2+}(O_{h})}$=0.26, and $n_{\mathrm{Co}^{2+}(O_{h})}$=0.19.
These suggest that the existence of excess cation vacancies ($n_{\mathrm{v}}$)
can be estimated to be 0.185 from the equation for the cation site,
$n_{\mathrm{v}}=3-n_{\mathrm{Fe}^{3+}(O_{h})}-n_{\mathrm{Fe}^{3+}(T_{d})}-n_{\mathrm{Fe}^{2+}(O_{h})}-n_{\mathrm{Co}^{2+}(O_{h})}$.
We emphasize that the cation vacancy can be estimated quantitatively from the
analysis of XMCD.

Third, we discuss the relationship between strain and orbital magnetic
moments. For the Co2+ 3$d^{7}$ system, the ground state ${}^{4}F$ multiplet
energy split by the cubic crystal field to $\Gamma_{4}$ state, and further
effects by trigonal or tetragonal field and exchange coupling stabilize the
lowest 3$d^{7}$ states. The splitting of $t_{2g}$ state promotes the
degeneracy for spins as analogous to the 3$d^{2}$ system of spinel-type V3+
compounds ref58 ; ref59 . Large $m_{\mathrm{orb}}$ is recovered by the spin-
orbit coupling $\xi_{\mathrm{Co}}$ of 70 meV. Since the uniaxial anisotropy in
spinel-type Co ferrite is inevitably formed from the structural and electronic
configuration, the out-of-plane easy axis can originate from the tensile
strain effect. In the theoretical model calculation, the PMA energy ($K$)
deduced from the energy difference along the axis direction can be expressed
as $K=E(100)-E(001)=-B_{1}\chi$, where $B_{1}$ and $\chi$ is the magneto-
elastic coefficient and strain, respectively ref34 . Phenomenological $B_{1}$
of 1.4$\times$108 J/m3 consists of the elastic constant and magneto-striction
coefficient at room temperature for CoFe2O4. In the case of $x$=0.2, $B_{1}x$
is applied to estimate $K$. Considering the lattice constants of
$a_{\mathrm{\parallel}}=8.42\mathrm{\AA}$ and
$a_{\mathrm{\perp}}=8.29\mathrm{\AA}$ estimated from the XRD, the strain of
($a_{\mathrm{\parallel}}-a_{\mathrm{\perp}}$)/$a_{0}{\sim}$1.5 $\%$ was
deduced using the bulk CFO value $a_{0}=8.38\mathrm{\AA}$ ref32 . Therefore,
the PMA energy is roughly estimated to $0.51\times 10^{6}$ J/m3. On the other
hand, microscopic estimation of the PMA energy can be performed using
anisotropic $m_{\mathrm{orb}}$ as $K=1/4{\xi}{\Delta}m_{\mathrm{orb}}$, where
spin-orbit coupling constant is $\xi$ and ${\Delta}m_{\mathrm{orb}}$ is the
difference between out-of-plane and in-plane $m_{\mathrm{orb}}$. Although the
in-plane $m_{\mathrm{orb}}$ cannot be detected unless applying high magnetic
field to saturate the magnetization along the hard axis, we used the value in
the reference with extrapolation in low Co concentration ref44 . Assuming the
value of ${\Delta}m_{\mathrm{orb}}$ to be 0.2 $\mu_{\mathrm{B}}$, the PMA
energy of Co can be estimated to be $1.6\times 10^{6}$ J/m3 with the
assumption of the Co composition $x$=0.2 and strained CFO volume. The value of
$K$ from XMCD is overestimated compared with that from magnetization
measurements, in general, by a pre-factor of 0.1 ref60 . These estimations
suggest that the PMA energy can be explained mainly by the contribution of
small amount of anisotropic Co sites. Furthermore, the macroscopic
magnetostriction effect, which the relation between strain and $K$, can be
recognized including the $m_{\mathrm{orb}}$ from the viewpoint of electron
theory as orbital-elastic effect. The contributions from the Fe3+ (3$d^{5}$)
sites are negligible because of half-filled quenched orbital magnetic moments.
The contribution from the Fe2+ (3$d^{6}$) sites is also quenched since the
strained undegenerated level is occupied. Therefore, the origin of PMA in
Co0.2Fe2.8O4+δ can be predominantly explained by the enhanced
$m_{\mathrm{orb}}$ of 1/14 Co compositions through the strained charge
distributions, which can be recognized as diluted doped magnetic oxide
systems.

Finally, we discuss the origin of PMA in the spinel-type oxides from the
viewpoint of tensile or compressive strain and $m_{\mathrm{orb}}$. The strain
in CFO is derived from the tensile strain in the film, which is detected by
the EXAFS analysis and confirmed by the sign of XMLD. Recent report shows that
the PMA in spinel-type NiCo2O4 originates from the compressive strain in the
film ref61 ; ref62  and that the enhancement of $m_{\mathrm{orb}}$ in XMCD
measurements is not distinctive with the 2+ and 3+ mixed valence states of
both Co and Ni sites ref63 . Therefore, the PMA in spinel-type oxides can be
categorized by the tensile or compressive strain. Because of tensile strain,
oblate-type charge distribution induces $m_{\mathrm{orb}}$ along out-of-plane
direction, which is similar to the case of enhanced $m_{\mathrm{orb}}$ in
Co/Pd multilayer ref66 . On the other hand, in-plane compression or out-of-
plane elongation modulates the prorate-type charge distributions, which
induces the quadrupole moments without enhancing $m_{\mathrm{orb}}$ as
discussed in strained Mn3-xGa system ref47 .

In summary, using XMCD and XMLD, we investigated the element-specific orbital
magnetic moments and their anisotropy in Co-ferrite thin films with low Co
concentration which exhibit the PMA. The origin of PMA in CFO is explained by
the large $m_{\mathrm{orb}}$ in the Co2+ (3$d^{7}$) states. The tensile
lattice strain, which is deduced from EXAFS, induces the out-of-plane
$m_{\mathrm{orb}}$ through the anisotropic in-plane charge distribution.
Furthermore, a novel method to estimate the number of cation vacancy from XMCD
line shapes was also proposed. Our finding clearly reveals that the
controlling strain modulates $m_{\mathrm{orb}}$, which opens up the material
functionalities in the oxide spin-orbitronics by strain engineering,
especially in the TM $d^{7}$ systems.

###### Acknowledgements.

This work was partially supported by JSPS KAKENHI (Grant No. 16H06332), the
Kato Science and Technology Foundation, the Izumi Science and Technology
Foundation, and the Telecommunications Advancement Foundation. Parts of the
synchrotron radiation experiments were performed under the approval of the
Photon Factory Program Advisory Committee, KEK (Nos. 2019G028 and 2021G069).

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Table 1: The Electronic structure parameters used in the LFM cluster calculation. The units of each value is in eV.  |  $\Delta$ |  $U$ |  $(pd\sigma)$ |  10$Dq$
---|---|---|---|---
Fe2+ ($O_{h}$) | 6.5 | 6.0 | 1.2 | 0.9
Fe3+ ($O_{h}$) | 0.5 | 6.0 | 1.2 | 0.9
Fe3+ ($T_{d}$) | 4.5 | 6.0 | 2.0 | -0.5
Co2+ ($O_{h}$) | 6.5 | 6.0 | 1.3 | 0.5
Figure 1: XAS and XMCD of Co0.2Fe2.8O4+δ film. $\mu$\+ and $\mu-$ in XAS
denote the magnetic field direction along the incident photon beams at normal
incident geometry. Difference of ${\mu^{+}}-{\mu^{-}}$ is defined as XMCD
spectra for (a) Fe $L$-edge and (b) Co $L$-edge. Both XAS and XMCD intensities
are normalized by photon flux. (c) Magnetic-field dependence at each fixed
photon energy; 708, 709, and 778 eV corresponding to Fe2+ $(O_{h})$, Fe3+
$(T_{d})$, and Co2+ $(O_{h})$ peaks, respectively. Figure 2: XAS and XMLD of
Co0.2Fe2.8O4+δ film of (a) Fe and (b) Co $L$-edges. Spectra were taken at the
grazing incident setup where $\bf{E}$ of the incident beam and direction of
magnetization $\bf{M}$ were parallel (horizontal) and perpendicular
(vertical), respectively. The inset displays an illustration of the XMLD
measurement geometry.  Figure 3: Calculated XMCD and XMLD spectra for (a,b) Fe
and (c,d) Co $L$-edges. For Fe $L$-edge, three components of Fe2+ $(O_{h})$,
Fe3+ $(O_{h})$, and Fe3+ $(T_{d})$ sites are also shown. Total spectra are
shown in black solid line. Open dotted curves are experimental XMCD and XMLD
shown in Figs. 1 and 2.

Figure 4: XAFS and EXAFS of Co0.2Fe2.8O4+δ (PMA) and Co0.6Fe2.4O4+δ (in-plane
anisotropy). (a) Co $K$-edge XAFS. Inset shows corresponding $k$ oscillation
in $k^{3}\chi(k)$ for $x=0.2$. Dot curve shows the fitting result. (b) EXAFS
profile after the Fourier transform. Inset shows the fitting result in
$x=0.2$.


# Permanence based Hidden Community and Graph Recovery in Social Networks
††thanks: ∗ Corresponding author. School of Computing, Gachon University,
Republic of Korea. (email<EMAIL_ADDRESS>
Jaeyoung Choi∗ and Wooseok Sim

###### Abstract

Due to the recent development of data analysis techniques, technologies for
detecting communities through information expressed in social networks have
been developed. Although it has several advantages, including the ability to
effectively share recommended items through an estimated community, it may
cause personal privacy issues. Therefore, recently, the problem of hiding the
real community well is being studied at the same time. As an example, Mittal
et al., proposed an algorithm called NEURAL that can hide this community well
based on a metric called permanence. Based on this, in this study, we propose
a Reverse NEURAL (R-NEURAL) algorithm that restores the community as well as
the original graph structure using permanence. The proposed algorithm includes
a method for well restoring not only the community hidden by the NEURAL
algorithm but also the original graph structure modified by recovered edges.
We conduct experiments on real-world graphs and found that the proposed
algorithm recovers well the hidden community as well as the graph structure.

##  I Introduction

The community detection (CD) problem is a problem that infers which nodes are
included in which groups in a network or graph through graph connectivity
information. In social networks, it is regarded as an important problem since
many users are organized into nodes and each user has complex connections [1,
2]. In this CD problem, the community is inferred by classifying cases in
which the connectivity between nodes in a specific group is greater than the
connectivity with other groups. Recently, many algorithms have been proposed
to solve this CD based on the increase of data in social networks and the
development of many machine-learning techniques [3, 4]. Many of the proposed
CD algorithms are made based on the score function for the community structure
in the graph. Examples include modularity [5], conductance [6], coverage, or
partition quality [2]. Modularity is the most used metric which is the ratio
between edges in the community and the edges distributed over the graph. This
indicates how well it is structured when compared to uniform connections. The
conductance is a fraction of edges that have the outside direction from the
community and the total number of edges in the community. Coverage is the
coverage of all edges in the graph and the edges of the intracommunity.
Partition quality is the ratio of the number of two pairs in the same
community. In addition, a new score function called permanence has recently
been proposed, which is a score function based on the vertices of the graph
[7]. This metric is based on the ratio of connections between inter-community
and intra-community. Unlike the community detection approaches so far, the
authors in [8] conducted a study that could well hide the community structure
in the graph. They proposed a community deception algorithm named NEURAL,
which is a method of selecting a specific edge in the graph based on this
permanence score and adding or deleting it to intracommunity or inter-
community. Motivated by the algorithm they proposed, we suggest a method
called Reverse NEURAL (R-NEURAL) that can simultaneously restore the original
community structure and graph structure after NEURAL (a deception algorithm)
is applied as shown in Figure 1. In R-NEURAL, the main approach is to recover
edges from hidden graphs based on permanence.

Figure 1: Example of community and graph recovery process after the deception
of the target community using edge updates. (CS: Community Structure).

##  II Model and Algorithms

###  II-A Preliminaries

We consider an undirected connected graph $G=(V,E)$, where $V$ is a set of
nodes and $E$ is the set of edges of the form $(i,j)$ for $i,j\in V$. After
applying a CD algorithm on $G,$ we assume that there are $k\geq 1$ detected
communities and we denote a set of detected communities by
$CS=\\{C_{1},...,C_{k}\\}.$ We also assume that there are no overlapping
communities, i.e., $C_{i}\cap C_{j}=\emptyset$ for any $i\neq j.$ For each
$C_{i}$, if an edge $(u,v)$ is in $C_{i},$ it is denoted by intracommunity
edge whereas for an edge $(u,v)$, where $u\in C_{i}$ and $v\in C_{j}$ with
$C_{i}\cap C_{j}=\emptyset$, we denote it as a intercommunity edge.

###  II-B Permanence

As a measure of community structure in the graph, we use the permanence
proposed by Chakraborty et al.[7]. This is a vertex-centric score function
that quantifies how well node $v$ is affiliated with community $C$. To see
this, let $I(v)$ be the internal connections of a node $v$ within its own
community and let $E_{max}(v)$ be the maximum connections of $v$ to its
neighboring communities. Next, we let $C_{in}(v)$ be the fraction of actual
and possible number of edges among the internal neighbors of $v$. Then, the
permanence of $v$ in $G$ is given by:

$\displaystyle
Perm(v,G)=\frac{I(v)}{E_{max}(v)}\times\frac{1}{deg(v)}-(1-C_{in}(v)),$ (1)

where $deg(v)$ is the node degree of $v.$ This score function indicates that
if the internal connectivity $I(v)$ of vertex $v$ is greater than the external
connectivity $E_{max}(v)$, the corresponding value increases and forming a
close clique and is divided by the degree of $v$ to normalize it [7]. As an
extension, the permanence of the graph $G$ is defined by $Perm(G)=\sum_{v\in
G}Perm(v,G)/|V|,$ where $|V|$ is the number of nodes in $G$. Then the graph
permanence is an average of node permanence over the graph.

###  II-C Community Deception Algorithm - NEURAL [8]

1

Input: Graph $G,$ Target community $C,$ Budget $B>0$

Output: Updated graph $G^{\prime}$

2

3

4$\mathcal{P}_{l,ad}=0$ and $\mathcal{P}_{l,de}=0$;

5while  _$B >0$_ do

6 Step1: For $u\in C$ and $v\in C^{\prime}$, where $C^{\prime}$ is the
community of maximum external pull for $u,$ set $G^{\prime}$ to be the graph
with added edge $(u,v)$. Compute $Perm(u,G)$ and $Perm(u,G^{\prime})$ and
choose the best edge satisfying:

7 $(u^{*},v^{*})=\arg\max_{u\in C,v\in
C^{\prime}}Perm(u,G)-Perm(u,G^{\prime})$;

8 Set $\mathcal{P}_{l,ad}=\mathcal{P}(G)-\mathcal{P}(G^{\prime})$ for
$(u^{*},v^{*})$;

9 Step2: For $w,z\in C$, set $G^{\prime}$ to be the graph with deleted edge
$(w,z)$. Compute $Perm(w,G)$ and $Perm(w,G^{\prime})$ and choose the best edge
satisfying:

10 $(w^{*},z^{*})=\arg\max_{w,z\in C}Perm(w,G)-Perm(w,G^{\prime})$;

11 Set $\mathcal{P}_{l,de}=\mathcal{P}(G)-\mathcal{P}(G^{\prime})$ for
$(w^{*},z^{*})$;

12 if  _$\mathcal{P}_{l,ad}\geq\mathcal{P}_{l,de}$ and $\mathcal{P}_{l,ad}>0$_
then

13 $G\leftarrow(V,E\cup(u^{*},v^{*}))$

14 else if  _ $\mathcal{P}_{l,de}>0$_ then

15 $G\leftarrow(V,E\setminus(w^{*},z^{*}))$

16 $B\leftarrow B-1$;

17$G^{\prime}\leftarrow G$;

18Return $G^{\prime}$;

Algorithm 1 NEURAL [8]

For community deception, the authors [8] proposed Network Deception Using
Permanence Loss (NEURAL) algorithm in their work. The main idea of the
algorithm is that some intracommunity or inter-community edges are updated
(added or deleted) based on the permanence difference between before the
update and after the edge update. To see this more precisely, we first define
a permanence loss [8] of two graphs $G$ and $G^{\prime}$, where $G^{\prime}$
is the graph result from some edge update on $G,$ as follows:

$\displaystyle\mathcal{P}_{l}=Perm(G)-Perm(G^{\prime}).$ (2)

Then, the goal of designing the NEURAL is to maximize the $\mathcal{P}_{l}$.
For this, it is necessary to minimize the right term $Perm(G^{\prime})$ since
the permanence of $G$, $Perm(G)$ is fixed if the graph is given. The NEURAL
algorithm includes two main steps as depicted in Algorithm 1. Step 1 describes
the process of finding an edge to add to another community in the target
community. That is, for given original graph $G$, Target community $C$ and
budget $B>0$, which will be used for the edge update, the NEURAL first select
nodes pair $(u,v)$, for $u\in C$ and $v\in C^{\prime}$, where $C^{\prime}$ is
the community of maximum external pull of $u.$ Then, it computes the
permanence $Perm(u,G)$ for the original graph $G$ and $Perm(u,G^{\prime})$ for
the updated graph $G^{\prime}$ with adding the edge $(u,v)$. Next, it computes
the difference between them for all $u\in C$ and $v\in C^{\prime}$ and finds
the maximum $(u^{*},v^{*})$ of the difference (lines 3-4). Then, it calculated
the difference of permanences between the original graph $G$ and the updated
graph $G^{\prime}$ as $\mathcal{P}_{l,ad}$. On the other hand, Step 2 includes
the process of finding an edge to delete within the same community. In this
process, similar to step 1, calculate the permanence of the selected edge to
find the edge $(w^{*},z^{*})$ that maximizes the difference and put it in
$\mathcal{P}_{l,de}$ (lines 6-8). 111The reason why it is possible to find
edges greedily as in Step 1 and Step 2 is that all permanences satisfy
submodularity as shown in [8]. Adding an edge in the inter-community and
deleting an edge in the intra-community is an operation that increases the
permanence gain as proven in [8], so only these two have been considered.
Finally, the algorithm compares the values of $\mathcal{P}_{l,ad}$ and
$\mathcal{P}_{l,de}$ obtained in this way and performs an edge update that
satisfies a larger permanence gain, one budget is used (lines 9-13). This
process is repeated until the total budget $B$ is exhausted, and the last
updated graph is output as $G^{\prime}$.

###  II-D Proposed Algorithm - Reverse NEURAL (R-NEURAL)

As a simple approach to the recovery of hidden communities that applied the
NEURAL algorithm, we propose a Reverse-NEURAL (R-NEURAL) in this paper. This
includes the reverse application of the two cases below based on NEURAL.
Before describing that, we let $G^{\prime}$ be the output graph after applying
the NEURAL and let $G^{\prime\prime}$ be the graph after updating edge
deleting, or adding on $G^{\prime}$ to recover the original graph $G$. Then,
we introduce permanence gain, $\mathcal{P}_{g}$, as follows:

$\displaystyle\mathcal{P}_{g}=Perm(G^{\prime\prime})-Perm(G^{\prime}).$ (3)

The above permanence gain seems quite similar to the permanence loss in (2).
However, the goal of maximizing the permanence loss is to find $G^{\prime}$
that minimizes the permanence of the new graph $G^{\prime}$ whereas the goal
of permanence gain $\mathcal{P}_{g}$ is to maximize the permanence of the
graph $G^{\prime\prime}$ recovered from $G^{\prime}$. To do this, we reverse
the following two in Algorithm 1, which is described in Algorithm 2.

  * $(1)$

In Step 1, since inter-community edges are newly connected in Step 1 of the
NEURAL algorithm, we work to find and remove edges that exist between
intercommunity. Due to the lack of knowledge of which edges have been added,
R-NEURAL uses also a similar measure, permanence gain $\mathcal{P}_{g,de}$, to
select edges to be deleted.

  * $(2)$

In Step 2, based on the method of finding the permanence loss while removing
the intracommunity edge in NEURAL, we consider the addition of an
intracommunity edge to $G^{\prime}$ and raise the permanence of
$G^{\prime\prime}$ to $\mathcal{P}_{g,ad}$.

Finally, we compare $\mathcal{P}_{g,de}$ and $\mathcal{P}_{g,ad}$ and proceed
with the operation for the larger one and consume one budget. We examine how
well the community structure hidden by NEURAL can be recovered through this
simple opposite approach.

1

Input: Output Graph $G^{\prime},$ Target community $C,$ Budget $B>0$

Output: Updated graph $G^{\prime\prime}$

2

3

4$\mathcal{P}_{l,ad}=0$ and $\mathcal{P}_{l,de}=0$;

5while  _$B >0$_ do

6 Step1: For $u\in C$ and $v\in C^{\prime}$, where $C^{\prime}$ is the
community of maximum external pull for $u,$ set $G^{\prime\prime}$ to be the
graph with deleted edge $(u,v)$. Compute $Perm(u,G)$ and $Perm(u,G^{\prime})$
and choose the best edge $(u^{*},v^{*})$ with

7 $\arg\max_{u\in C,v\in
C^{\prime}}Perm(u,G^{\prime\prime})-Perm(u,G^{\prime})$;

8 Set
$\mathcal{P}_{g,de}=\mathcal{P}(G^{\prime\prime})-\mathcal{P}(G^{\prime})$ for
$(u^{*},v^{*})$;

9 Step2: For $w,z\in C$, set $G^{\prime\prime}$ to be the graph with added
edge $(w,z)$. Compute $Perm(w,G)$ and $Perm(w,G^{\prime})$ and choose the best
edge satisfying:

10 $(w^{*},z^{*})=\arg\max_{w,z\in
C}Perm(w,G^{\prime\prime})-Perm(w,G^{\prime})$;

11 Set
$\mathcal{P}_{g,ad}=\mathcal{P}(G^{\prime\prime})-\mathcal{P}(G^{\prime})$ for
$(w^{*},z^{*})$;

12 if  _$\mathcal{P}_{g,de}\geq\mathcal{P}_{g,ad}$ and $\mathcal{P}_{g,de}>0$_
then

13 $G^{\prime}\leftarrow(V,E\setminus(u^{*},v^{*}))$

14 else if  _ $\mathcal{P}_{g,ad}>0$_ then

15 $G^{\prime}\leftarrow(V,E\cup(w^{*},z^{*}))$

16 $B\leftarrow B-1$;

17$G^{\prime\prime}\leftarrow G^{\prime}$;

18Return $G^{\prime\prime}$;

Algorithm 2 Reverse NEURAL (R-NEURAL)

##  III Numerical and Simulation Results

In the simulation, we apply several different CD algorithms for $(i)$ original
graph $G$, $(ii)$ updated graph $G^{\prime}$ after NEURAL, and $(iii)$
recovered graph $G^{\prime\prime}$ after R-NEURAL, respectively. As an
original graph, we use three different real-world graphs: Dolphin social
network (Dol, 62 nodes) [9], and word adjacencies (Adjn, 112 nodes) [9], and
Polbook (105 nodes) [9], respectively. As evaluation metrics, we consider
three measures: modularity (M), coverage (C), and partition quality (PQ). In
the experiments, we set $B=0.3|V_{C}|$, i.e., 30% of the size of the target
community $C$ as in [8]. As result, in Table I, we obtain three evaluation
results for two real-world graphs, we see that there is an increase in most
evaluation metrics for each graph $G^{\prime\prime}$ compared to those of
$G^{\prime}$. Among them, both graphs show that the partition quality score
improves the most, which is due to the process of community recovery based on
permanence. Next, we use two CD algorithms: Louvain [10] and InfoMap [11] in
Table II. We obtain the three evaluation results with respect to the applied
CD algorithms for one real-world graph (Dol). In this case, we also check that
the detection performance is improved in the recovered graph
$G^{\prime\prime}$ in most cases. Finally, to check how similar the recovered
graph was to the original graph, we obtained the similarity measure between
the two graphs using the well-known eigenvector similarity. To do this, we
first measured the similarity between $G$ and the recovered graph
$G^{\prime\prime}$ (sim($G$,$G^{\prime\prime}$)) together with the similarity
between the original graph $G$ and the updated graph $G^{\prime}$
(sim($G$,$G^{\prime}$)) for deception in Figure 2. We compared the similarity
of Dol, Adjn, and Polbook graphs. As a result, all graph $G^{\prime\prime}$ in
the three real-world graphs can recover the original graph from the updated
graph $G^{\prime}$.

TABLE I: Two real-world graphs: Dol (left) and Adjn (right). Metrics |  $G$ (Dol) | $G^{\prime}$ | $G^{\prime\prime}$ |  $G$ (Adjn) | $G^{\prime}$ | $G^{\prime\prime}$
---|---|---|---|---|---|---
M | 0.5202 | 0.4822 | 0.4866 | 0.2941 | 0.2911 | 0.2920
C | 0.7447 | 0.7119 | 0.7123 | 0.4682 | 0.4479 | 0.4448
PQ | 0.8381 | 0.8239 | 0.8324 | 0.8394 | 0.8323 | 0.8351
TABLE II: Two CD algorithms: Louvain (left) and Infomap (right). Metrics |  $G$ (Dol) | $G^{\prime}$ | $G^{\prime\prime}$ |  $G$ (Dol) | $G^{\prime}$ | $G^{\prime\prime}$
---|---|---|---|---|---|---
M | 0.5205 | 0.4796 | 0.4854 | 0.5219 | 0.4198 | 0.4570
C | 0.7418 | 0.7068 | 0.7063 | 0.7494 | 0.8400 | 0.7198
PQ | 0.8407 | 0.8285 | 0.8525 | 0.8358 | 0.7298 | 0.8371
Figure 2: Eigenvector similarity from the original graph $G$. The orange line
shows how close the graph $G^{\prime\prime}$ is recovered to $G$ compared to
$G^{\prime}$.

##  IV Conclusion

In this study, we proposed a reverse NEURAL algorithm that restores the
community and graph structure using permanence. From the simulation on real-
world graphs, we have checked that our proposed algorithm recovered not only
the target community hidden by the NEURAL algorithm but also the original
graph structure modified by edges. As future work, we will consider an
additional query process for the selected edges in the algorithm to check
whether it was in the original graph or not to improve the recovery
performance.

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