Present address: ]Institute of Materials Structure Science, High Energy
Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan

#  Intrinsic coercivity induced by valence fluctuations in $4f$-$3d$
intermetallic magnets

Hiroaki Shishido1<EMAIL_ADDRESS>Tetsuro Ueno2 Kotaro Saito3
Masahiro Sawada4 Munehisa Matsumoto5 [  1Department of Physics and
Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai,
Osaka 599-8531, Japan
2Quantum Beam Science Research Directorate, National Institutes for Quantum
and Radiological Science and Technology, 1-1-1 Kouto, Sayo, Hyogo 679-5148,
Japan
3Institute of Materials Structure Science, High Energy Accelerator Research
Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
4Hiroshima Synchrotron Radiation Center, Hiroshima University, Higashi-
Hiroshima, Hiroshima 739-0046, Japan
5Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa
277-8581, Japan

###### Abstract

Temperature dependence of magnetization curves of well homogenized samples of
Ce(Co1-xCux)5 ($0\leq x\leq 0.7$), a family of representative $4f$-$3d$
intermetallic magnets found in rare-earth permanent magnets, is measured. A
remarkable enhancement of intrinsic coercivity is observed with $x=0.3$ and
$x=0.4$, persisting to higher temperatures. This experimental observation is
theoretically attributed to an effect of electronic correlation among
$4f$-electrons. That is, an intrinsic pinning happens originating in an
anomalously enhanced magnetic anisotropy energy contributed by an order of
magnitude stronger charge-transfer process between $4f$-electrons and
$3d$-electrons, than the conventional crystal field effects. It is
demonstrated that the $4f$-$3d$ charge-transfer process depends on the
direction of magnetization in the middle of a crossover of the valence state
of Ce between CeCu5 with robust Ce3+ and CeCo5 with the mixed valence state.

###### pacs:

75.50.Ww, 75.10Lp, 75.30.Gw, 71.15.Rf

## Introduction

Rare earth permanent magnets (REPMs) make a particular research target where
the forefronts of fundamental physics meet the critical demands from society
and industry [1]. Among the intrinsic magnetism required for permanent
magnets, magnetization and Curie temperature carried mostly by $3d$-electrons
of Fe-group elements involve the long-standing problem of metallic magnetism
[2], and magnetic anisotropy is realized dominantly by $4f$-electrons,
typically as an outcome from combined effects of crystal fields and spin-orbit
interaction (SOI). SOI brings about novel topologically nontrivial electronic
states with which novel functionalities are extensively investigated in the
past decade [3], and has been one of the central topics in recent condensed
matter physics. Also, the multiple-scale problem to bridge over the gap
between the microscopic magnetism and the macroscopic properties of bulk
materials is not yet fully solved [4, 5]. Given the recent shifts to electric
vehicles and introduction of robotics in various aspects of daily lives and
industry, permanent magnets will be one of the most critically important
materials in the upcoming decades. It is an important problem in many respects
to construct the fundamental understanding of permanent magnets.

In this work, we point to a potential utility of Ce, which is relatively
abundant among all of the rare earth elements [6], exploiting the valence
fluctuations coming from the $4f$-electrons that reside on the border of
localization and delocalization [7]. While the typical magnetic anisotropy
energy is understood to be driven by potential energy of well localized
$4f$-electrons in the crystal fields, we argue that valence-fluctuations
combined with SOI bring about an effective magnetic anisotropy driven by
kinetic energy in the hybridization of delocalized $4f$-electrons and other
conduction electrons. This kinetic-energy-driven anisotropy originating in
charge fluctuations has an order of magnitude stronger energy scale than the
magnetic anisotropy driven by potential energy. Also the relatively strong
coupling between $4f$-$3d$ electrons helps to render such kinetic-energy
driven anisotropy more resistant against high temperature fluctuations [8].

We present a case study with Ce(Co,Cu)5. The prototype, CaCu5-type structure,
(For a picture, see the supplementary material [9].) makes one of the most
fundamental building blocks among typical compounds in REPMs such as R2T17,
RT12, and R2T14B (R=rare earth and T=transition metals ) [11, 10, 12]. Thus,
the physics revealed with RT5 should lead to a versatile guideline of material
design for REPMs. In Ce(Co,Cu)5, there is a crossover in the valence state of
Ce between CeCo5 with mixed valence state of Ce [13] and CeCu5 with trivalent
Ce [14, 15]. Between $3d$-electron ferromagnetism of CeCo5 and $4f$-electron
antiferromagnetism of CeCu5 [14], we observe the enhanced coercivity and
attribute it to the effects of valence fluctuations combined with SOI. The
whole composition range was investigated in Refs. [16, 17] and magnetism
trends in the Cu-rich compounds were investigated [18], while positive
contribution of valence fluctuations to the intrinsic magnetism has not been
remarked so far [19].

The materials RCo5 (R=rare earth) with Co partly substituted by Cu belonged to
the earliest REPMs [20, 21, 22] developed in 1960’s. This eventually lead to
the Sm-Co-based magnets that were the champion magnet in 1970’s before the
advent of Nd-Fe-B magnets [23, 24]. While the main-stream magnets attracted
attention both technologically and fundamentally since 1980’s [10], the
understanding of the early REPMs was not completely established. The
fundamental understanding of the Nd-Fe-B magnets are still being constructed
[25, 4]. In these respects, all possible roles of $4f$-electron physics in the
$3d$-electron ferromagnetism are worth investigating comprehensively. Valence
fluctuations characterize some of the important rare earth elements such as Ce
and Sm.

## Experimental setup

Poly-crystalline samples of Ce(Co1-xCux)5 for $x=0$, $0.3$, $0.4$, $0.5$,
$0.6$, and $0.7$ were grown by arc-melting at Ar atmosphere from elementary
materials of 3N(99.9%)-Ce, 3N-Co and 4N-Cu. They were wrapped in Ta-foil,
sealed in a quart tube under vacuum and annealed at 850-1050∘C for 5-24 days
according to each concentration. The samples were characterized by X-ray
diffraction (XRD), scanning electron microscope (SEM) and energy dispersive
X-ray spectrometry (EDX). XRD measurements were performed by a SmartLab
(Rigaku) with Cu-K$\alpha$ radiation. SEM images and EDX mapping were
collected with a SU8010 (Hitachi) and XMax50 EDX (Oxford instruments).
Magnetization measurements were carried out with a MPMS3 (Quantum design) for
the magnetic field range of $-70$ kOe to $70$ kOe in the temperature range of
$1.9$ K to $300$ K. X-ray absorption spectroscopy (XAS) experiment was
performed at the HiSOR BL-14 at the Hiroshima Synchrotron Radiation Center
(HSRC), Hiroshima University, Japan [26]. The details of the experimental
station are described in Refs. 27 and 28. Sample was filed by a diamond file
in the ultra high vacuum (UHV) to obtain clean surface. Ce $M_{4,5}$ XAS
spectra were measured using the total electron yield method by measuring the
sample drain current. The sample temperature was maintained during the
measurements using a closed-cycle He refrigerator.

## Characterization of the samples

Main panel of Fig. 1(a) shows the powder XRD pattern for Ce(Co0.7Cu0.3)5. All
peaks are well indexed to belong to the CaCu5-type phase (Space group:
$P6/mmm$ #191, and no contamination was detected within the experimental
errors. Rietveld XRD refinements were carried out by RIETAN-FP [29] for the
samples. Obtained lattice constants $a$ and $c$ follow the Vegard’s law as
shown in the inset of Fig. 1(a). It indicates no significant phase separation
occurs even in intermediate Cu concentrations.

Figure 1: (Color online) (a) The powder diffraction pattern for
Ce(Co0.7Cu0.3)5. Red dots, black and blue lines denote the experimental data,
fitting results and their difference. SEM-EDX chemical mapping of
Ce(Co0.7Cu0.3)5 for (b) Ce, (c) Co, (d) Cu, and (e) overlay of them on the SEM
image. (f) SEM image corresponding to the SEM-EDX mapping.

Figures 1(b)–(f) show a typical SEM image (Fig. 1(f)) and SEM-EDX mapping
(Figs. 1(b)–(e)) of Ce(Co0.7Cu0.3)5. SEM images of Ce(Co1-xCux)5 shows no
particular microstructure that may work as an extrinsic pinning center.
Moreover, chemical distributions of Ce(Co1-xCux)5 are homogeneous as shown in
Figs. 1 (b)–(e). Considering XRD and SEM-EDX results, we can conclude that the
samples are single phase and homogeneous. It implies that the observed bulk
properties presented below can be attributed to the intrinsic properties.

Figure 2: (Color online) (a) Magnetization curves of Ce(Co0.7Cu0.3)5 at
various temperatures and (b) Temperature dependence of coercive force $H_{\rm
c}$ for various substitutions of Ce(Co1-xCux)5.

## Results: measurement of bulk magnetization curves

Reflecting a ferromagnetic ordering, magnetization curves of Ce(Co0.7Cu0.3)5
exhibit hysteresis loops, as shown in Fig. 2(a). Hysteresis loops show abrupt
drop across zero magnetic field. This anomaly becomes more pronounced at
$x=0.4$, and hysteresis loops show snake-like form at $x=0.5$ and $0.6$ (shown
in the Supplementary Material [9]). The origin of this anomalous behavior is
not clear yet, but finite concentration distributions could provide negligible
difference on coercive force $H_{\rm c}$ and saturated magnetization ($M_{\rm
s}$), and the combination of different hysteresis loops gives these anomalous
loops at intermediate concentration region. For Ce(Co0.7Cu0.3)5, $H_{\rm c}$
of 9 kOe increases with decreasing temperature, and reaches 24.6 kOe at 1.9 K.
Temperature dependence of $H_{\rm c}$ for each substitution are summarized in
Fig. 2 (b). The measured $H_{\rm c}$ for CeCo5 is almost temperature
independent and weak, indicating that CeCo5 is effectively a soft magnetic
material [30].

The measured $H_{\rm c}$ at 1.9 K has a maximum at $x=0.3$, decreases almost
linearly with increasing $x$. More interestingly, $H_{\rm c}(T)$ for $x=0.5$
and $0.6$ exhibit a downward convex temperature dependence, whereas that for
$x=0.4$ shows a slightly upward convex temperature dependence. The temperature
dependence of $H_{\rm c}(T)$ with $x=0.3$ is almost linear. Such significant
enhancement of the temperature resistance in $H_{\rm c}(T)$ around $x=0.3$ and
$x=0.4$ should come from the direct hybridization between $4f$-electron state
and the conduction electron states that include the ferromagnetic
$3d$-electron state of Co, in contrast to the weak temperature resistance in
the materials with the indirect exchange coupling between well localized
$4f$-electrons in Nd and $3d$-electrons [8].

## Microscopic probing of the Ce valence state

Figure 3: (Color online) Soft X-ray absorption spectra of Ce(Co1-xCux)5
($x=0.3$, $0.4$, and $0.6$) at 20 K. The arrows indicate the presence of the
contribution from the $(4f)^{0}$ state of Ce4+ in the target materials as the
initial state of the core-electron excitation [31], which is gradually
suppressed from $x=0.3$ to $x=0.6$.

The measured XAS results at $20~{}\mbox{K}$ for various $x$ are shown in Fig.
3. The absorption peak corresponding to the $(4f)^{0}$ state of Ce4+ in the
target materials is located at around 5 eV higher from the main peak
originating in the $(4f)^{1}$ state of Ce3+. Here we refer to Ref. 31 for the
assignment of the peaks. We focus on the relative trends among the samples
concerning the contribution from the Ce4+ state and we see that the measured
data clearly shows the trend of the valence fluctuations being suppressed as
the concentration of Cu increases. Here we note that $T$-dependence of XAS was
relatively minor as compared to the $x$-dependence [9].

## A supporting data set of electronic structure calculations

The crossover in the valence state of Ce by Cu substitution in CeCo5 can be
simulated by imposing Coulomb $U_{4f}$ among $4f$-electrons. The realistic
range of the parameter $U_{4f}$ is upper-bounded by the localized limit at
$U_{4f}=5$ eV. Between such localized limit and the mixed valence region with
$U_{4f}\ll 5$ eV to which the experimental pristine CeCo5 seems to correspond,
the crossover in the valence state of Ce can be numerically observed as the
parameter $U_{4f}$ is varied starting from $U_{4f}=0$. In this way, we below
present a set of electronic structure calculations based on the density
functional theory (DFT) [32] combined with the $U_{4f}$ parameter for CeCo5,
so called the DFT+U method [33], with which we describe the trend in the
valence state of Ce in Ce(Co1-xCux)5 from $x=0$ to $x$ closer to $1$.

We use the DFT code, OpenMX [34, 35, 36, 37, 38], based on a linear
combination of pseudoatomic orbitals. The DFT+U method and the SOI works on
the $j$-dependent pseudopotential [39, 40] for the description of the non-
collinear magnetism within DFT (Here $j$ refers to the total angular momentum
incorporating both of spin and orbital). We proceed with the following two
steps: a) starting with the experimental lattice constants, the crystal
structure is computationally optimized, and then b) the intrinsic magnetic
anisotropy is estimated by the calculation of the energy while putting a
constraint on the direction of magnetization. Here the SOI is turned on for
b). Calculated energy is plotted with respect to the angle $\theta$ of the
direction of magnetization as measured from the $c$-axis. The particular DFT+U
scheme we use here is based on the formulation in Ref. 41 as implemented in
OpenMX. We set another parameter $U_{3d}=2.0$ eV for Co in order to describe a
set of realistic scales in the $3d$-electron part [9].

Figure 4:  (Color online) (a) Calculated energy as a function of $\theta$, the
angle between the magnetization and the c-axis, for CeCo5 within GGA+U. The
parameters in the calculations are $0\leq U_{4f}\leq 3.5~{}\mbox{eV}$ and
$U_{3d}=2.0~{}\mbox{eV}$. (b) Visualized difference of the calculated spin
density between $\theta=10^{\circ}$ and $\theta=90^{\circ}$ with
$U_{4f}=3.5~{}\mbox{eV}$ and $U_{3d}=2.0~{}\mbox{eV}$. The red (green) region
corresponds to the area where the spin density has increased (decreased),
respectively. The path of a charge transfer process from the Ce atom to the
Co($2c$) atom can be seen.

Calculated energy $U_{\rm total}$ as a function of $\theta$ is shown in Fig.
4(a). We observe the enhancement of the effective magnetic anisotropy energy
from $U_{4f}=2$ eV to $U_{4f}=3.5$ eV, gradually showing a larger jump in the
calculated energy. We observed that this type of jump at $\theta_{\rm jump}$
is encountered irrespective of the ways the particular angle $\theta_{\rm
jump}$ is approached. The overall calculations proceed as close as possible to
the conventional calculations of magnetic anisotriopy energy, always starting
from a converged electronic state in the crystal structure optimization in a)
above and then imposing the constraint on the direction of magnetization in
b). When the convergence is not reached in b) at some $\theta$, other
converged electronic state with a neighboring value of $\theta$ is taken as a
starting point. In this way, the jump in the calculated energy as a function
of $\theta$ is robustly observed. Accordingly enhanced effective magnetic
anisotropy energy does not seem to be an artifact of our numerics.

Furthermore, we can attribute the observed jump in $U_{\rm total}(\theta)$ to
the valence fluctuation and the associated charge transfer between Ce and Co,
using a visualization of the electronic state in a convincing way as follows.
For the calculated electronic structure with $U_{4f}=3.5$ eV, difference of
the calculated spin density is taken between $\theta=10$∘ and $\theta=90$∘ and
its real-space picture is shown in Fig. 4(b). A charge transfer path between
Ce and Co($2c$) is observed. This indicates that the jump in the calculated
energy originates in the delocalization of $4f$-electron in Ce, which hops to
the neighboring Co($2c$) atoms, and the magnetic anisotropy has been enhanced
by the charge transfer energy. This particular direction-dependent $4f$-$3d$
charge transfer process combined with spin-orbit interaction contributes to
the anomalous enhancement of the intrinsic magnetic anisotropy. Since the
energy scale of $4f$-$3d$ hybridization in Ce is $O(0.1)$ eV [31] and the
scales of the magnetic anisotropy energy originating in the crystal fields is
of the order of $O(10)$ meV, the effective magnetic anisotropy can be enhanced
by an order of magnitude by the involvement of the valence fluctuations. With
such enhancement of the effective uni-axial anisotropy $K$ and the reduction
of magnetization as realized by the dilution effect caused by Cu into
ferromagnetic Co, the domain wall width spanning the length scale of
$\pi\sqrt{A/K}$ (here $A$ is the exchange stiffness involving magnetization)
[42], which is in the order of $O(1)$ nm for pristine CeCo5, can narrow down
to the scale of $O(1)$ Å. Then an atomic-scale pinning of the domain walls can
be at work. With the measurements of our homogenized samples and the
supporting calculations, this mechanism of an intrinsic coercivity seems to be
plausible.

## Discussions: implications on Sm compounds

The relevance of valence fluctuations for the anomalous magnetic anisotropy
may hold for other rare earth elements with valence fluctuations, such as Sm,
Eu, Tm, Yb, and Pr under pressure, and for early actinides such as U, Np, and
Pu, where the $5f$-electrons located on the border of localization and
delocalization [7]. A possible close example can happen with Sm-based
compounds. For another representative $4f$-$3d$ intermetallic ferromagnet for
REPMs, Sm2Fe17N3, an electronic structure calculation pointed to a divalent
state of Sm in the ground state [43]. In contrast, the XAS data of the same
compound and related ones at finite temperatures indicate that Sm is in the
trivalent state [44]. Even though some sample dependence can be suspected
[45], these past claims point to the possibility of a valence transition or a
crossover for Sm at very low temperatures. Common physics with Ce(Co,Cu)5 can
enhance the magnetic anisotropy of Sm-based Fe-group intermetallics, which may
be at work in Sm(Co,Cu)5 in the early days of REPM [21], where an enhancement
of the coercivity was reported for the intermediate region in the
concentration of Cu.

## Discussions: materials design for Fe-rich cases

Since CeFe5 does not exist as a bulk stable material and it often happens that
Cu does not mix with Fe, we have not explicitly addressed the Fe-rich cases.
However, the physics with the delocalized $4f$-electrons of Ce in the
ferromagnetically polarized conduction bands should generically hold: the Fe-
rich Ce intermetallics and their less ferromagnetic counterparts that may be
realized by mixing Fe with Al or Ga, instead of Cu, should show an analogously
enhanced anisotropy induced by valence fluctuations. This may be exploited in
order to implement an intrinsic coercivity in the possible fabrication of a
rare-earth permanent magnet based on Fe-rich $4f$-$3d$ intermetallics.

## Conclusions and outlook

We have described how the valence fluctuations of Ce can be relevant for the
enhanced magnetic anisotropy and its associated intrinsic coercivity with the
case of Ce(Co1-xCux)5, where the intrinsic coercivity is maximized around
$x=0.3$. This particular coercivity is characterized by its relatively weak
temperature dependence due to the direct hybridization mechanism of the
delocalized $4f$-electrons and conduction electrons. The temperature
resistance can pave the way toward the possible technological applications at
elevated temperatures.

Apparently, the intrinsic coercivity associated with valence fluctuations has
not been discussed so far. This points to a possibility that there are
potentially several mechanisms of the coercivity of permanent magnets. There
can be some more coercivity mechanisms that are not well explored but are
relevant both for fundamental understanding and industrial applications. In
order to construct a versatile understanding of complicated materials such as
REPMs, spanning from the Nd-Fe-B magnets [23, 24, 10] to Sm-Co magnets and
possible new ones [46], it is desirable to implement a more comprehensive way
of material characterization utilizing multiple probes and to fairly
investigate a range of the possible mechanisms based on the fundamentals of
statistical physics and condensed-matter physics.

###### Acknowledgements.

Inspiring discussions with Y. Kuramoto and introduction to the present problem
given by T. Akiya are gratefully acknowledged. The present project was partly
supported by JSPS KAKENHI Grant No. 15K13525 and a Hitachi Metals-Materials
Science Foundation Research Fellowship. The XAS experiments were performed at
HSRC with the approval of the Proposal Assessing Committee (proposal Nos.
16AG045 and 17BG017). Numerical calculations for the present work were done in
the facilities in ISSP Supercomputer Center, University of Tokyo.

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  * [46] For recent prospects, see e.g. G. C. Hadjipanayis, A. M. Gabay, A. M. Schönhöbel, A. Martín-Cid, J. M. Barandiaran, D. Niarchos, Engineering 6, 141 (2020).