#  Maximizing intrinsic anomalous Hall effect by controlling
the Fermi level in simple Weyl semimetal films

Mizuki Ohno  Department of Applied Physics, University of Tokyo, Tokyo
113-8656, Japan  Quantum-Phase Electronics Center (QPEC), University of Tokyo,
Tokyo 113-8656, Japan  Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8550, Japan  Susumu Minami  Department of Physics, University of
Tokyo, Tokyo 113-8656, Japan  Yusuke Nakazawa  Department of Applied Physics,
University of Tokyo, Tokyo 113-8656, Japan  Quantum-Phase Electronics Center
(QPEC), University of Tokyo, Tokyo 113-8656, Japan  Shin Sato  Department of
Applied Physics, University of Tokyo, Tokyo 113-8656, Japan  Quantum-Phase
Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan  Markus
Kriener  RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
Ryotaro Arita  Department of Applied Physics, University of Tokyo, Tokyo
113-8656, Japan  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan  Masashi Kawasaki  Department of Applied Physics, University
of Tokyo, Tokyo 113-8656, Japan  Quantum-Phase Electronics Center (QPEC),
University of Tokyo, Tokyo 113-8656, Japan  RIKEN Center for Emergent Matter
Science (CEMS), Wako 351-0198, Japan  Masaki Uchida
<EMAIL_ADDRESS>Department of Physics, Tokyo Institute of
Technology, Tokyo 152-8550, Japan  PRESTO, Japan Science and Technology Agency
(JST), Tokyo 102-0076, Japan

###### Abstract

Large intrinsic anomalous Hall effect (AHE) originating in the Berry curvature
has attracted growing attention for potential applications. Recently proposed
magnetic Weyl semimetal EuCd2Sb2 provides an excellent platform for
controlling the intrinsic AHE because it only hosts a Weyl-points related band
structure near the Fermi energy. Here we report the fabrication of EuCd2Sb2
single-crystalline films and control of their anomalous Hall effect by film
technique. As also analyzed by first-principles calculations of energy-
dependent intrinsic anomalous Hall conductivity, the obtained anomalous Hall
effect shows a sharp peak as a function of carrier density, demonstrating
clear energy dependence of the intrinsic AHE.

Topological semimetals, characterized by massless Dirac band dispersion in the
low energy excitation, exhibit unique magnetotransport due to their nontrivial
band topology in the momentum space Burkov et al. (2011); Wang et al. (2017);
Armitage et al. (2018); Lv et al. (2021). Recently reported magnetic
topological semimetals further enrich the understanding of magnetotransport,
where entanglement between magnetism and nontrivial topology can lead to
exotic magnetotransport such as a large anomalous Hall effect (AHE) Miyasato
et al. (2007); Nagaosa et al. (2010); Xiao et al. (2010). Integration of the
Berry curvature for all occupied states gives rise to a large intrinsic
contribution to anomalous Hall conductivity. Namely, anomalous Hall
conductivity is closely related to the electron occupation determined by the
position of Fermi level Thouless et al. (1982); Xiao et al. (2010).

Paired Weyl points (WPs) in magnetic Weyl semimetals can be viewed as
monopoles with opposite chirality corresponding to the sources (W$+$) and
sinks (W$-$) of the Berry curvature (Fig. 1(a)). Due to the Berry flux of
paired WPs, anomalous Hall conductivity $\sigma_{\mathrm{xy,AHE}}$ is expected
to exhibit a large peak at the energy position of WPs $(E_{\mathrm{W}})$
Burkov and Balents (2011); Lu et al. (2015) and be strongly dependent on the
Fermi level. Specifically, $\sigma_{\mathrm{xy,AHE}}$ can be maximized if the
Fermi level can be adjusted to $E_{\mathrm{W}}$ Noky2018 ; Muechler2020 .
However, such an energy dependence of $\sigma_{\mathrm{xy,AHE}}$ has not been
demonstrated yet due to high carrier densities ($\sim 10^{20}$–$10^{22}$ cm-3)
and complicated band structures of conventional magnetic Weyl semimetals,
while the modulation of $\sigma_{\mathrm{xy,AHE}}$ has been attempted by
chemical doping and pressure application for css Shen et al. (2020a, b); Zhou
et al. (2020); Thakur et al. (2020), Mn3Sn Kuroda et al. (2017); Khadka et al.
(2020); Chen et al. (2021a, b), and Mn3Ge Kiyohara et al. (2016).

In this context, EuCd2As2 and isostructural EuCd2Sb2, where one of three Cd
atoms is substituted with Eu in the parent Dirac semimetal Cd3As2, provide an
ideal platform for experimentally demonstrating the peculiar energy dependence
of intrinsic anomalous Hall conductivity. This is because they have a simple
band structure near the Fermi energy ($E_{\mathrm{F}}$) and much lower carrier
densities ($\sim 10^{18}$–$10^{19}$ cm-3) Soh et al. (2019); Su et al. (2020).
On the other hand, difficulties in the growth of thin films have prevented the
study of carrier density dependence with taking these advantages.

Fig. 1:  (a) Energy dispersion of paired Weyl points (WPs) and energy-
dependent anomalous Hall conductivity $\sigma_{xy,{\mathrm{AHE}}}$. Weyl
points W$+$/W$-$ are the source/sink of the Berry curvature. (b) Crystal
structure of EuCd2Sb2 Artmann et al. (1996) drawn using VESTA Momma2011 . (c)
Energy dispersion calculated for EuCd2Sb2 films, showing four pairs of WPs
near the Fermi energy. They emerge along the $\Gamma-$Z direction in the
Brillouin zone when magnetic moments are forcedly aligned along the $c$ axis.
Blue/green circles represent WPs (W1-W4) with plus/minus chirality. (d) Cross-
sectional schematic illustration and (e) top-view photograph of the
electrostatic gate device fabricated from a EuCd2Sb2 film, where holes are
depleted (accumulated) when $V_{\mathrm{G}}$ is positively (negatively)
biased.

EuCd2Sb2 has a trigonal crystal structure with space group $P$-$3m1$ Artmann
et al. (1996); Goryunov et al. (2012); Soh et al. (2018); Su et al. (2020). It
consists of an alternate stacking of triangular Eu layers and Cd2Sb2 layers
(Fig. 1(b)). Eu atoms contribute to magnetism in this compound, forming
$A$-type antiferromagnetic structure below the Néel temperature
$T_{\mathrm{N}}=7.4$ K with Eu2+ magnetic moments lying in the $ab$ plane
Goryunov et al. (2012); Soh et al. (2018). As shown in Fig. 1(c), four pairs
of WPs (W1-W4) near $E_{\mathrm{F}}$ along the $k_{z}$ direction are protected
by $C_{3z}$ symmetry when the magnetic Eu moments are ferromagnetically
polarized along the $c$ axis under the out-of-plane magnetic field Su et al.
(2020). There are no other bands near $E_{\mathrm{F}}$ and all the hole
carriers occupy the WPs-related valence bands. Therefore, it is expected that
the Fermi level dependence of intrinsic anomalous Hall conductivity can be
demonstrated by modulating hole carrier densities in EuCd2Sb2 with film
techniques.

Here we report the fabrication of single-crystalline EuCd2Sb2 films by
molecular beam epitaxy and investigation of their anomalous Hall effect using
film techniques. By adjusting the Fermi level with controlling Sb flux
conditions and also performing electrostatic gating experiments, we find that
their anomalous Hall effect is largely enhanced compared to bulks and strongly
dependent on the carrier density with a sharp peak, as also demonstrated by
first-principles calculations.

EuCd2Sb2 films were grown in an Epiquest RC1100 chamber Nakazawa et al.
(2019); Ohno et al. (2021a, b); Uchida et al. (2021) on single-crystalline
(111)A CdTe substrates. In-plane lattice mismatch between EuCd2Sb2 and CdTe is
2.4% as shown in Fig. 2(b). CdTe substrates were etched with 0.01%
Br2-methanol before loading it into the chamber and then heated to 750∘C with
Cd flux supplied to obtain an atomically smooth surface Nakazawa et al.
(2019). After annealing the substrate, it was cooled to the growth temperature
of 360∘C. The beam equivalent pressures were measured by an ionization gauge,
and were set to $1.2{\times}10{{}^{-5}}$ Pa for Eu, 5.0$\times 10^{-4}$ Pa for
Cd, and $8.5\times 10^{-6}$ Pa for Sb during the co-deposition, which is the
same growth condition as EuCdSb2 films on Al2O3 (0001) substrates Ohno et al.
(2021b). This Cd-rich growth condition offers advantages in reducing carrier
densities in EuCd2Sb2 films, because Cd deficiency is a major origin of the
hole carriers. The films were then annealed $in$-$situ$ at 500∘C for 5 minutes
under exposure of Cd flux. The film thickness was set at 50 nm, and the growth
rate was about 0.07 Å/s.

The films were patterned to Hall bars with a channel width of 10 $\mu$m
through conventional photolithography, Ar ion milling, and chemical wet
etching processes: 10 nm thick Ni and 50 nm thick Au electrodes were deposited
for ohmic contact by electron beam evaporation, and then 30 nm thick Al2O3 was
deposited as gate dielectric by atomic layer deposition, as shown in Figs.
1(d) and 1(e). Low-temperature magnetotransport was measured using a Quantum
Design Physical Property Measurement System cryostat equipped with a 9 T
superconducting magnet. Hall measurements were performed using a lock-in
technique. The excitation current was kept constant at 0.5 mA with a frequency
of 13 Hz. For gating experiments, DC bias ($V_{\mathrm{G}}=-8\sim+8$ V)
voltage was applied to the gate electrode.

First-principles calculations of the band structure with spin-orbit coupling
were performed using the VASP package Kresse and Furthmüller (1996a, b);
Kresse and Joubert (1999) for experimentally determined film lattice
parameters. The generalized gradient approximation of Perdew-Burke-Ernzerhof
was adopted for the exchange-correlation functional Perdew et al. (1996). The
correlation effect was considered by a Hubbard $U$ correction of $U=4.5$ eV
for the Eu-$f$ orbitals Dudarev and Botton (1998), which was determined by
comparing the band structures revealed by angle-resolved photoemission
spectroscopy and first-principles calculations Su et al. (2020). $16\times
16\times 10$ $k$-point mesh with Monkhorst-Pack scheme Monkhorst and Pack
(1976) was used for the Brillouin zone sampling of the primitive cell and
Gaussian smearing with a width of $0.02$ eV was applied. From the Bloch states
obtained in the DFT calculation, a Wannier basis set is constructed by using
the Wannier90 code Pizzi et al. (2020). The basis consists of $s$,
$f$-character orbitals localized at the Eu site, $s$, $d$-character orbitals
at the Cd, and $s$, $p$-character orbitals at the Sb. The intrinsic anomalous
Hall conductivity is computed using a $k$-point mesh of $300\times 300\times
300$ Wu et al. (2018).

Fig. 2:  (a) XRD $\theta$-2$\theta$ scan of a EuCd2Sb2 film grown on a CdTe
substrate. Substrate peaks are marked with an asterisk. (b) Crystal structures
of EuCd2Sb2 film and CdTe substrate, viewed along the out-of-plane direction.
(c) Cross-sectional image of EuCd2Sb2 film, taken by high-angle annular dark-
field scanning transmission electron microscopy. (d) Atomic arrangement of
EuCd2Sb2 viewed along [110] direction. Energy dispersive x-ray spectrometry
maps taken for (e) Eu $L$, (f) Cd $L$, and (g) Sb $L$ edge in the boxed region
in (c).

Figure 2 shows structural characterization of EuCd2Sb2 film. As shown in the
x-ray diffraction (XRD) $\theta$-2$\theta$ scan in Fig. 2(a), reflections from
the (001) EuCd2Sb2 lattice planes are observed without any impurity phases.
Its out-of-plane lattice constant along the $c$-axis is calculated to be
$7.73$ Å. Figure 2(c) shows a cross-section image of the EuCd2Sb2, taken by
high-angle annular dark-field scanning transmission electron microscopy. The
periodic atomic arrangement corresponding to the EuCd2Sb2 crystal structure in
Fig. 2(d) is clearly confirmed. Elemental maps are also taken by energy
dispersive x-ray spectrometry, as shown in Figs. 2(e)-2(g). The alternate
stacking of Eu layers and Cd2Sb2 layers is clearly resolved. Further
structural characterization was performed for examining the in-plane epitaxial
relation (See Supplementary Materials for details SM ).

Fig. 3:  Magnetotransport measured for a EuCd2Sb2 film with a carrier density
$p=0.8\times 10^{19}$ cm-3. (a) Temperature dependence of longitudinal
resistivity $\rho_{\mathrm{xx}}$, with a clear kink at the Néel temperature
$T_{\mathrm{N}}=7.4$ K. The inset shows measurement configuration. (b)
Magnetoresistance $(\rho_{\mathrm{xx}}(B)-\rho_{\mathrm{xx}}(0$
$\mathrm{T}))/\rho_{\mathrm{xx}}(0$ $\mathrm{T})$ taken with sweeping the out-
of-plane magnetic field at various temperatures. The saturation field is
estimated $B_{\mathrm{sat}}=3.4$ T at $T=2.0$ K. (c) Hall resistivity
$\rho_{\mathrm{yx}}$ taken at various temperatures. (d) Anomalous Hall
resistivity $\rho_{\mathrm{yx,AHE}}$, obtained by subtracting the ordinary
term at various temperatures. Anomalous Hall resistivity
$\rho_{\mathrm{yx,AHE}}$ of EuCd2Sb2 films with (e) different carrier
densities and (f) electrostatic gating at 2.0 K.

Figure 3 summarizes fundamental magnetotransport of the EuCd2Sb2 film. As
shown in Fig. 3(a), longitudinal resistivity $\rho_{\mathrm{xx}}$ exhibits a
metallic behavior in the whole temperature regime. Below 50 K,
$\rho_{\mathrm{xx}}$ gradually increases upon cooling until it shows a clear
kink at 7.4 K. Then it exhibits a sudden drop upon further cooling. This kink
observed at 7.4 K corresponds to Néel temperature $T_{\mathrm{N}}$ Goryunov et
al. (2012); Soh et al. (2018); Su et al. (2020). The kink disappears at the
out-of-plane magnetic field $B=9$ T, where the Eu2+ magnetic moments are
completely aligned along the $c$-axis Soh et al. (2018); Su et al. (2020).
This low temperature behavior can be understood considering the scattering of
conduction electrons by the localized Eu2+ spins. Figure 3(b) presents
magnetoresistance (MR) taken by sweeping the out-of-plane magnetic field at
various temperatures. At $T=2.0$ K, a peak ends at the saturation field of
$B_{\mathrm{sat}}=3.4$ T, and which shifts to lower fields upon increasing
temperature and then disappears above ${T_{\mathrm{N}}}$. This peak below
$B_{\mathrm{sat}}$ is ascribed to an increase of magnetic scattering in the
canted spin structure. At high-fields above $B_{\mathrm{sat}}$, the Eu2+ spins
are fully polarized along the $c$-axis, resulting in reduced magnetic
scattering. At higher temperatures above $T_{\mathrm{N}}$, the peak becomes
smaller and shifts to higher magnetic fields. Finally, the peak disappears and
conventional positive MR appears.

Figure 3(c) shows Hall resistivity $\rho_{\mathrm{yx}}$ measured at various
temperatures. At temperatures above 50 K, Hall resistivity depends almost
linearly on the magnetic field. In contrast, it deviates from the linear
behavior upon cooling below 50 K. Here $\rho_{\mathrm{yx}}$ is expressed as
$\rho_{\mathrm{yx}}=R_{\mathrm{H}}B+\rho_{\mathrm{yx,AHE}}$ with the ordinary
Hall coefficient $R_{\mathrm{H}}$ and the anomalous Hall resistivity
$\rho_{\mathrm{yx,AHE}}$. Deviation from the linear behavior disappears above
$B_{\mathrm{sat}}$, showing clear contribution of the anomalous Hall term (See
Supplementary Materials [39]). By subtracting the ordinary term with a linear
fit, the anomalous term is obtained as shown in Fig. 3(d).
$\rho_{\mathrm{yx,AHE}}$ taken at $T=2.0$ K exhibits a kink at
$B_{\mathrm{sat}}=3.4$ T. With increasing temperature, $B_{\mathrm{sat}}$
shifts to lower fields below $T_{\mathrm{N}}$ and then disappears above
$T_{\mathrm{N}}$, as confirmed in MR. Besides, a broad but pronounced peak
appears around $B_{\mathrm{sat}}$ below $T_{\mathrm{N}}$, showing deviation
from the magnetization behavior. While this peak suggests the existence of the
so-called topological Hall term, it is unlikely that this is caused by the
real-space spin Berry phase on noncoplanar spin configuration during simple
spin polarization process. Rather, this may be understood by intrinsic AHE
corresponding to band structure changes during the polarization process, as
also reported for other Eu compounds Cao et al. (2021); Takahashi et al.
(2018); Mayo2021 ; Shen et al. (2021).

Here we define the saturation value of $\rho_{\mathrm{yx,AHE}}$ at 9 T as
$\rho_{\mathrm{yx,AHE}}^{\mathrm{sat}}$. It is expected that
$\rho_{\mathrm{yx,AHE}}$ is dominated by intrinsic AHE in the forced
ferromagnetic phase. In addition, $\sigma_{\mathrm{xx}}$ and
$\sigma_{\mathrm{xy,AHE}}$ are calculated by

$\displaystyle\sigma_{\mathrm{xx}}$
$\displaystyle=\frac{\rho_{\mathrm{xx}}(B=0\mathrm{~{}T})}{\left(\rho_{\mathrm{yx,AHE}}^{\mathrm{sat}}\right)^{2}+\rho_{\mathrm{xx}}(B=0\mathrm{~{}T})^{2}},$
(1)

and

$\displaystyle\sigma_{\mathrm{xy,AHE}}$
$\displaystyle=\frac{\rho_{\mathrm{yx,AHE}}^{\mathrm{sat}}}{\left(\rho_{\mathrm{yx,AHE}}^{\mathrm{sat}}\right)^{2}+\rho_{\mathrm{xx}}(B=0\mathrm{~{}T})^{2}}.$
(2)

AHE has been roughly categorized into three regimes based on the range of
$\sigma_{\mathrm{xx}}$ Nagaosa et al. (2010); Onoda et al. (2006). For
EuCd2Sb2 films in this study, $\sigma_{\mathrm{xx}}$ is below $1\times 10^{4}$
$\Omega^{-1}$cm-1, and the carrier density is located between $0.3\times
10^{19}$ and $4.8\times 10^{19}$ cm-3. This suggests that the anomalous Hall
velocity induced by the Berry curvature is suppressed with following the
scaling relation $\sigma_{\mathrm{xy,AHE}}\propto\sigma_{\mathrm{xx}}^{1.6}$.
With reducing the hole carrier density of EuCd2Sb2 films at 2.0 K,
$\rho_{\mathrm{yx,AHE}}$ increases at first, decreases, and then increases
again as shown in Fig. 3(e). As confirmed in Fig. 3(f), modulating carriers
with electrostatic gating also follows this trend.

Fig. 4:  (a) Carrier density dependence of
$\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ in EuCd2Sb2 films, compared to
bulks of EuCd2Sb2 Su et al. (2020) and EuCd2As2 Cao et al. (2021).
Electrostatic gating data obtained for low carrier samples ($p<1\times
10^{19}$ cm-3) are also plotted with curves, colored by the sign of
$V_{\mathrm{G}}$. (b) Magnified energy dispersion. (c)
$\sigma_{\mathrm{xy,AHE}}$ and density of states (DOS) calculated as a
function of energy. (d) $\sigma_{\mathrm{xy,AHE}}$ plotted for carrier density
converted from DOS.

In order to discuss the carrier density dependence of AHE in the dirty metal
regime Nagaosa et al. (2010); Onoda et al. (2006), we show the carrier density
dependence of $\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ obtained by
measuring EuCd2Sb2 films at 2.0 K, as shown in Fig. 4(a). Remarkably,
$\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ exhibits a large peak at about
$p=1\times 10^{19}$ cm-3 and the same trend is confirmed by continuously
modulating the carrier density with electrostatic gating. We also confirm that
a similar large peak is observed in the plots of $\sigma_{\mathrm{xy,AHE}}$
and $\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}$ (See Supplementary Materials [39]).
$\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ values previously reported for
bulk single crystals Su et al. (2020) also follow the same trend as revealed
by the set of film measurements. Therefore, the observed peak is ascribed to
successful adjustment of the Fermi level to $E_{\mathrm{W}}$.

In order to understand the enhancement of
$\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ more quantitatively, we perform
first-principles calculations of band structure and intrinsic
$\sigma_{\mathrm{xy,AHE}}$. As shown in Fig. 4(c), $\sigma_{\mathrm{xy,AHE}}$
exhibits a sharp peak at about $E-E_{\mathrm{F}}=-0.02$ eV, which corresponds
to the energy position of W2 and W3 in Fig. 4(b). The energy can be converted
to carrier density by integrating density of states from $E-E_{\mathrm{F}}=0$
eV. As confirmed in Fig. 4(d), $\sigma_{\mathrm{xy,AHE}}$ exhibits clear
carrier density dependence consistent with the experimental data,
demonstrating that the Fermi level of EuCd2Sb2 films passes through the energy
position of W2 and W3.

In summary, we have systematically investigated carrier density dependence of
the intrinsic anomalous Hall conductivity by adjusting the Fermi level in
simple magnetic Weyl semimetal EuCd2Sb2 films. The peak observed for
$\sigma_{\mathrm{xy,AHE}}/\sigma_{xx}^{1.6}$ has been also reproduced by
first-principles calculations of energy-dependent intrinsic anomalous Hall
conductivity. Our findings have provided transport evidence of intrinsic
anomalous Hall conductivity originating in WPs. The present work paves the way
for further exploring the potential of WPs-based exotic magnetotransport using
film techniques.

This work was supported by JST PRESTO Grant No. JPMJPR18L2 and JST CREST Grant
No. JPMJCR16F1, Japan and by Grant-in-Aid for Scientific Research (B) No.
JP21H01804 from MEXT, Japan, and by TEPCO Memorial Foundation, Japan. The data
that support the findings of this study are available from the corresponding
author upon reasonable request.

## References

  * Burkov et al. (2011) A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011).
  * Wang et al. (2017) S. Wang, B. C. Lin, A. Q. Wang, D. P. Yu, and Z. M. Liao, Adv. Phys.: X 2, 518 (2017).
  * Armitage et al. (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
  * Lv et al. (2021) B. Q. Lv, T. Qian, and H. Ding, Rev. Mod. Phys. 93, 025002 (2021).
  * Miyasato et al. (2007) T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y. Onose, N. Nagaosa, and Y. Tokura, Phys. Rev. Lett. 99, 086602 (2007).
  * Nagaosa et al. (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Xiao et al. (2010) D. Xiao, M. C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * Thouless et al. (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Burkov and Balents (2011) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).
  * Lu et al. (2015) H. Z. Lu, S. B. Zhang, and S. Q. Shen, Phys. Rev. B 92, 045203 (2015).
  * (11) J. Noky, J. Gooth, C. Felser, and Y. Sun, Phys. Rev. B 98, 241106 (R) (2018).
  * (12) L. Muechler , E. Liu, J. Gayles, Q. Xu, C. Felser, and Y. Sun, Phys. Rev. B 101, 115106 (2020).
  * Shen et al. (2020a) J. Shen, Q. Zeng, S. Zhang, H. Sun, Q. Yao, X. Xi, W. Wang, G. Wu, B. Shen, Q. Liu et al., Adv. Funct. Mater. 30, 2000830 (2020a).
  * Shen et al. (2020b) J. Shen, Q. Yao, Q. Zeng, H. Sun, X. Xi, G. Wu, W. Wang, B. Shen, Q. Liu, and E. Liu, Phys. Rev. Lett. 125, 086602 (2020b).
  * Zhou et al. (2020) H. Zhou, G. Chang, G. Wang, X. Gui, X. Xu, J. X. Yin, Z. Guguchia, S. S. Zhang, T. R. Chang, H. Lin, W. Xie, M. Z. Hasan, and S. Jia, Phys. Rev. B 101, 125121 (2020).
  * Thakur et al. (2020) G. S. Thakur, P. Vir, S. N. Guin, C. Shekhar, R. Weihrich, Y. Sun, N. Kumar, and C. Felser, Chem. Mater. 32, 1612 (2020).
  * Kuroda et al. (2017) K. Kuroda, T. Tomita, M. T. Suzuki, C. Bareille, A. A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S. Akebi et al., Nat. Mater. 16, 1090 (2017).
  * Khadka et al. (2020) D. Khadka, T. R. Thapaliya, S. H. Parra, X. Han, J. Wen, R. F. Need, P. Khanal, W. Wang, J. Zang, J. M. Kikkawa et al., Sci. Adv. 6, eabc1977 (2020).
  * Chen et al. (2021a) T. Chen, T. Tomita, S. Minami, M. Fu, T. Koretsune, M. Kitatani, I. Muhammad, D. Nishio-Hamane, R. Ishii, F. Ishii et al., Nat. Commun. 12, 572 (2021a).
  * Chen et al. (2021b) X. Chen, H. Xie, Q. Zhang, Z. Luo, L. Shen, and Y. Wu, arXiv:2107.00959 (2021b).
  * Kiyohara et al. (2016) N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl. 5, 064009 (2016).
  * Soh et al. (2019) J. R. Soh, F. de Juan, M. G. Vergniory, N. B. M. Schröter, M. C. Rahn, D. Y. Yan, J. Jiang, M. Bristow, P. Reiss, J. N. Blandy et al., Phys. Rev. B 100, 201102(R) (2019).
  * Su et al. (2020) H. Su, B. Gong, W. Shi, H. Yang, H. Wang, W. Xia, Z. Yu, P. J. Guo, J. Wang, L. Ding et al., APL Mater. 8, 011109 (2020).
  * Artmann et al. (1996) A. Artmann, A. Mewis, M. Roepke, and G. Michels, Z. Anorg. Allg. Chem. 622, 679 (1996).
  * Goryunov et al. (2012) Y. Goryunov, V. Fritsch, H. V. Löhneysen, and A. Nateprov, J. Phys. Conf. Ser. 391, 012015 (2012).
  * Soh et al. (2018) J. R. Soh, C. Donnerer, K. M. Hughes, E. Schierle, E. Weschke, D. Prabhakaran, and A. T. Boothroyd, Phys. Rev. B 98, 064419 (2018).
  * Nakazawa et al. (2019) Y. Nakazawa, M. Uchida, S. Nishihaya, S. Sato, A. Nakao, J. Matsuno, and M. Kawasaki, APL Mater. 7, 071109 (2019).
  * Ohno et al. (2021a) M. Ohno, M. Uchida, R. Kurihara, S. Minami, Y. Nakazawa, S. Sato, M. Kriener, M. Hirayama, A. Miyake, Y. Taguchi, R. Arita, M. Tokunaga, and M. Kawasaki, Phys. Rev. B 103, 165144 (2021a).
  * Ohno et al. (2021b) M. Ohno, M. Uchida, Y. Nakazawa, S. Sato, M. Kriener, A. Miyake, M. Tokunaga, Y. Taguchi, and M. Kawasaki, APL Mater. 9, 051107 (2021b).
  * Uchida et al. (2021) M. Uchida, S. Sato, H. Ishizuka, R. Kurihara, T. Nakajima, Y. Nakazawa, M. Ohno, M. Kriener, A. Miyake, K. Ohishi et al., Sci. Adv. 7, eabl5381 (2021).
  * Kresse and Furthmüller (1996a) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996a).
  * Kresse and Furthmüller (1996b) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996b).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Perdew et al. (1996) J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).
  * Dudarev and Botton (1998) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
  * Pizzi et al. (2020) G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune et al., J. Phys.: Condens. Matter 32, 165902 (2020).
  * Wu et al. (2018) Q. S. Wu, S. N. Zhang, H. F. Song, M. Troyer, and A. A. Soluyanov, Comput. Phys. Commun. 224, 405 (2018).
  * (39) See Supplementary Material [URL] for detailed characterization, which includes Refs. [6, 23, 44].
  * Cao et al. (2021) X. Cao, J.-X. Yu, P. Leng, C. Yi, Y. Yang, S. Liu, L. Kong, Z. Li, X. Dong, Y. Shi et al., arXiv:2103.09395 (2021).
  * Takahashi et al. (2018) K. S. Takahashi, H. Ishizuka, T. Murata, Q. Y. Wang, Y. Tokura, N. Nagaosa, and M. Kawasaki, Sci. Adv. 4, eaar7880 (2018).
  * (42) A. H. Mayo, H. Takahashi, M. S. Bahramy, A. Nomoto, H. Sakai, and S. Ishiwata, Phys. Rev. X 12, 011033 (2022).
  * Shen et al. (2021) J. Shen, J. Gao, C. Yi, Q. Zeng, S. Zhang, J. Yang, X. Zhang, B. Wang, J. Cong, Y. Shi et al., arXiv:2106.02904 (2021).
  * Onoda et al. (2006) S. Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. Lett. 97, 126602 (2006).
  * Sanjeewa et al. (2020) L. D. Sanjeewa, J. Xing, K. M. Taddei, D. Parker, R. Custelcean, C. dela Cruz, and A. S. Sefat, Phys. Rev. B 102, 104404 (2020).
  * (46) K. Momma and F. Izumi, J. Appl. Crystallogr., 44, 1272 (2011).


# CaCu3Ru4O12: a high Kondo-temperature transition metal oxide

D. Takegami  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Str. 40, 01187 Dresden, Germany  C. Y. Kuo  Max Planck Institute for Chemical
Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany  National
Synchrotron Radiation Research Center, 101 Hsin-Ann Road, 30076 Hsinchu,
Taiwan  K. Kasebayashi  Department of Physics and Electronics, Osaka
Prefecture University 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
J.-G. Kim  Department of Physics, POSTECH, Pohang 37673, Korea  C. F. Chang
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187
Dresden, Germany  C. E. Liu  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany  Department of
Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan  C. N. Wu
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187
Dresden, Germany  Department of Physics, National Tsing Hua University,
Hsinchu 30013, Taiwan  D. Kasinathan  S. G. Altendorf  K. Hoefer  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden,
Germany  F. Meneghin  A. Marino  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany  Y. F. Liao  K. D. Tsuei
C. T. Chen  National Synchrotron Radiation Research Center, 101 Hsin-Ann Road,
30076 Hsinchu, Taiwan  K.-T. Ko  Department of Physics, POSTECH, Pohang 37673,
Korea  A. Günther  Experimental Physics V, University of Augsburg, 86135
Augsburg, Germany  S. G. Ebbinghaus  Institute of Chemistry, Martin Luther
University Halle-Wittenberg, 06120 Halle, Germany  J. W. Seo  D. H. Lee  G.
Ryu  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40,
01187 Dresden, Germany  A. C. Komarek  Max Planck Institute for Chemical
Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany  S. Sugano  Y.
Shimakawa  Institute for Chemical Research, Kyoto University, Uji, Kyoto
611-0011, Japan  A. Tanaka  Department of Quantum Matter, ADSM, Hiroshima
University, Higashi-Hiroshima 739-8526, Japan  T. Mizokawa  Department of
Applied Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan  J. Kuneš
Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria  L. H. Tjeng
<EMAIL_ADDRESS>Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Str. 40, 01187 Dresden, Germany  A. Hariki
<EMAIL_ADDRESS>Department of Physics and Electronics, Osaka
Prefecture University 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria

###### Abstract

We present a comprehensive study of CaCu3Ru4O12 using bulk sensitive hard and
soft x-ray spectroscopy combined with local-density approximation (LDA) +
dynamical mean-field theory (DMFT) calculations. Correlation effects on both
the Cu and Ru ions can be observed. From the Cu $2p$ core level spectra we
deduce the presence of magnetic Cu2+ ions hybridized with a reservoir of
itinerant electrons. The strong photon energy dependence of the valence band
allows us to disentangle the Ru, Cu, and O contributions and thus to optimize
the DMFT calculations. The calculated spin and charge susceptibilities show
that the transition metal oxide CaCu3Ru4O12 must be classified as a Kondo
system and that the Kondo temperature is in the range of 500-1000 K.

Transition metal oxides show a wide variety of spectacular physical properties
such as superconductivity, metal-insulator and spin-state transitions,
unusually large magneto-resistance, orbital ordering phenomena, and
multiferroicity Cava (2000); Khomskii (2014); Keimer  _et al._ (2015).
Remarkably, heavy fermion or Kondo behavior is hardly encountered in oxides.
While quite common in the rare-earth and actinide intermetallics Stewart
(1984); Coleman (2006); Pfleiderer (2009); Wirth and Steglich (2016), one may
find perhaps only in the oxide LiV2O4 Kondo  _et al._ (1997); Urano  _et al._
(2000); Shimoyamada  _et al._ (2006) indications for heavy fermion physics.

The discovery of the transition metal oxide CaCu3Ru4O12 (CCRO) showing Kondo-
like properties therefore created quite an excitement Kobayashi  _et al._
(2004); Krimmel  _et al._ (2008, 2009); Kao  _et al._ (2017). The crystal
structure of this A-site ordered perovskite is shown in the inset of Fig. 1.
However, the Kondo interpretation has also met fierce reservations. It has
been argued that the specific heat coefficient $\gamma$ does not deviate much
from the band structure value, suggesting a minor role of the electronic
correlations Xiang  _et al._ (2007). Other interpretations of the mass
enhancement have been put forward Tanaka  _et al._ (2009a, b). Electron
spectroscopy studies have also not converged on the position or even the
presence of the putative Kondo peak Sudayama  _et al._ (2009); Hollmann  _et
al._ (2013); Liu  _et al._ (2020).

Figure 1: Magnetic susceptibility of CaCu3Ru4O12 reproduced from Refs.
Kobayashi  _et al._ (2004); Krimmel  _et al._ (2008); Kato  _et al._ (2009);
Cheng  _et al._ (2013) and of CaCu3Ti4O12 from Ref. Shimakawa (2008). The
black dashed line shows the Curie paramagnetic behavior for a $S=1/2$ Cu2+ ion
scaled by a factor of three. Inset: Crystal structure of CaCu3Ru4O12
visualized by VESTA Momma and Izumi (2011). The blue, red, gray, and indigo
blue spheres represent Cu, O, Ru, and Ca atoms, respectively.

Here we address the CCRO problem from a different perspective. Figure 1 shows
the magnetic susceptibility of CCRO together with that of CaCu3Ti4O12 (CCTO),
as reproduced from Refs. Kobayashi  _et al._ (2004); Krimmel  _et al._ (2008);
Kato  _et al._ (2009); Cheng  _et al._ (2013) and Ref. Shimakawa (2008),
respectively. One can observe that CCTO follows, far above its 25 K Néel
temperature, almost a text-book Curie-Weiss law that can be understood in
terms of paramagnetic $S=1/2$ Cu2+ ions. By contrast, one can also see that
CCRO shows a completely different behavior with a magnetic susceptibility that
is an order of magnitude smaller. There are also no indications at all for
magnetic order. If one believes that the system is non-magnetic and that Kondo
physics does not take place in CCRO, then the Cu ions in CCRO have to be
monovalent with the non-magnetic full-shell $3d^{10}$ configuration or
trivalent, which can be a nonmagnetic band insulator like NaCuO2.

We show here that the Cu ions are definitely divalent and thus have spin
degree of freedom. We have carried out a detailed photoelectron spectroscopy
study using a wide range of photon energies in order to establish the presence
of correlation effects on both the Cu and Ru ions as well as to disentangle
the Ru, Cu, and O contributions to the valence band. This allowed us to tune
the double counting corrections in the LDA+DMFT calculations accordingly,
making these calculations predictive for the low energy physics. We then were
able to determine how the Cu2+ magnetic moments can be screened. In
particular, we will show that, in going from high to low temperatures, this
screening takes place already at 500-1000 K, and that we thus must classify
CCRO as a Kondo system with a very high Kondo temperature.

Hard x-ray photoemission (HAXPES) measurements were carried out at the Max-
Planck-NSRRC HAXPES end-station Weinen  _et al._ (2015) at the Taiwan
undulator beamline BL12XU of SPring-8 in Japan. The photon energy was set to
$h\nu=6.5$ keV and the overall energy resolution was $\approx$ 270 meV as
determined from the Fermi cutoff of a gold reference sample. Soft x-ray
(resonant) photoelectron (PES) and absorption (XAS) spectroscopy experiments
were performed at the NSRRC-MPI TPS 45A Submicron Soft x-ray Spectroscopy
beamline at the Taiwan Photon Source (TPS) in Taiwan. The overall energy
resolution when using 1.2 keV, 931 eV, and 440 eV photons was $\approx$ 150
meV, 125 meV, and 60 meV, respectively. Photoemission measurements in the
vicinity of the Ru $4d$ Cooper minimum, i.e. at photon energies of $200$ eV,
$150$ eV, and $100$ eV, were performed at the PLS-II 4A1 micro-ARPES beamline
of the Pohang Light Source (PLS) in Korea. The overall energy resolution was
$\approx$ 55 meV. Polycrystalline samples of CCRO were synthesized by solid-
state reactions Ebbinghaus  _et al._ (2002). Clean sample surfaces were
obtained by cleaving sintered samples in situ in ultra-high vacuum preparation
chambers with pressures in the low $10^{-10}$ mbar range. The measurements at
SPring-8 and TPS were carried out at $80$ K, and the measurements at PLS at
$100$ K. We have used three different batches of CCRO samples for our
spectroscopic measurements, all giving the same results, providing confidence
in the reliability of the data, see Appendix A.

Our calculations employ the LDA+DMFT scheme Metzner and Vollhardt (1989);
Georges  _et al._ (1996); Kotliar  _et al._ (2006). We start with density
functional calculations for the experimental crystal structure Krimmel  _et
al._ (2008), see Fig. 1, using the Wien2k code Blaha  _et al._ and construct
the multi-band Hubbard model on the basis spanned by the Cu 3$d$, Ru 4$d$, and
O 2$p$ Wannier functions obtained with the wannier90 package Kuneš  _et al._
(2010); Mostofi  _et al._ (2014). The on-site Coulomb interactions on the Cu
and Ru sites are approximated with the density-density form with parameters
($U$, $J$)=(8.5 eV, 0.98 eV) for Cu 3$d$ electrons and (3.1 eV, 0.7 eV) for Ru
4$d$ electrons, which are typical values for Cu and Ru systems Petukhov  _et
al._ (2003); Hollmann  _et al._ (2013); Gorelov  _et al._ (2010); Pchelkina
_et al._ (2007). The continuous-time quantum Monte Carlo (CT-QMC) method with
the hybridization expansion Werner  _et al._ (2006); Boehnke  _et al._ (2011);
Hafermann  _et al._ (2012) was used to solve the auxiliary Anderson impurity
model (AIM). The double-counting corrections arising in LDA+X methods Karolak
_et al._ (2010); Kotliar  _et al._ (2006), which fix the charge transfer
energies on the Cu and Ru sites, were treated as adjustable parameters and
their values fixed by comparison to the present valence band and core-level
photoemission data as well as previous angle-resolved PES (ARPES) data. The
valence spectra were obtained by analytic continuation of self-energy using
the maximum entropy method Wang  _et al._ (2009); Jarrell and Gubernatis
(1996). The Cu 2$p$ and Ru $3d$ core-level XPS were calculated using the
method of Refs. Hariki  _et al._ , 2017a, 2020; Ghiasi  _et al._ , 2019.

Figure 2: Valence band resonant photoemission of CaCu3Ru4O12, with the
experimental spectra taken at the Cu $2p$ ($L_{3}$) resonance (h$\nu=931.2$
eV) and at 10 eV below the resonance (h$\nu=921.2$ eV). The inset displays the
experimental Cu-$L_{2,3}$ x-ray absorption spectrum.

We have carried out XAS and valence band resonant PES measurements in the
vicinity of the Cu $L_{2,3}$ edge, see Fig. 2. The peak positions and line
shape of the spectra are characteristic for divalent Cu Tjeng  _et al._ (1991,
1992); Hollmann  _et al._ (2013); Eskes  _et al._ (1990). We can exclude that
the Cu in CCRO is monovalent or trivalent since the spectral features of
Cu1+/Cu3+ oxides are positioned at quite different energies Tjeng  _et al._
(1992); Hollmann  _et al._ (2013); Chin  _et al._ (2021). We thus can conclude
that the Cu ions in CCRO possess a spin degree of freedom and that some form
of screening must take place as to make their magnetic susceptibility to
deviate dramatically from the Curie-Weiss law.

Figure 3: (a) Experimental Cu $2p$ core-level x-ray photoemission spectrum of
Li2CuO2 reproduced from Ref. Böske  _et al._ (1998). (b) Theoretical spectrum
from the CuO4 cluster model. (c) Experimental Cu $2p$ core-level HAXPES
spectrum of CaCu3Ru4O12. (d) Theoretical spectrum from the LDA+DMFT method.

Figure 3 shows the Cu $2p$ core level HAXPES spectrum of CCRO together with
that of Li2CuO2 as reproduced from Ref. Böske  _et al._ (1998). Here we took
Li2CuO2 as a reference system which contains CuO4 plaquettes that are weakly
coupled Böske  _et al._ (1998) and thus can serve for a comparison with CCRO
which also contains rather isolated CuO4 plaquettes, see Fig. 1. The spectra
share the gross features: the main peak (B) at around 932-933 eV binding
energy and the satellite (C) at 942 eV for the Cu $2p_{3/2}$ component.
However, the fine structure differs considerably: the main peak of CCRO
consists of two peaks (A and B) Tran  _et al._ (2006); Sudayama  _et al._
(2009) unlike the single peak (B) of Li2CuO2.

The Cu $2p$ core level spectrum of Li2CuO2 is typical for a Cu2+ oxide Ghijsen
_et al._ (1988). Thanks to the weak coupling between the CuO4 plaquettes, it
can be explained accurately with a full multiplet single CuO4-cluster
calculation Tanaka and Jo (1994); par , as shown in Fig. 3b. In contrast, the
two-peak structure (A and B) of the CCRO main peak cannot be captured by the
cluster model. This is indicative of a screening process Taguchi  _et al._
(2005); Panaccione  _et al._ (2008); Tran  _et al._ (2006); Sudayama  _et al._
(2009) which is present in CCRO but absent in Li2CuO2. Since the CuO4
plaquettes in CCRO are quite isolated from each other, a non-local screening
mechanism due to inter-Cu-cluster hopping is not expected to play an important
role. We rather relate the screening process to the metallic state of CCRO.
The LDA+DMFT calculations shown in Fig. 3d, reproduce the fine structure of
the main peak very well. We thus indeed can infer that CCRO contains
correlated magnetic Cu2+ ions, which experience screening by conduction
electrons. How strong or complete the screening is will be discussed below.

Figure 4: Valence band spectra of CaCu3Ru4O12: (a) experimental results
measured at different photon energies, (b) LDA+DMFT spectral intensities for
the Cu 3$d$, Ru 4$d$, and O 2$p$ states. The spectral broadening is taken into
account using a 200 meV Gaussian to simulate the experimental resolution. (c)
LDA+DMFT spectral intensities of the Cu $3d$ orbitals in CaCu3Ru4O12. The Cu
orbitals are defined in the local axis of the CuO4 plane, as shown in the
inset. Figure 5: (top panel) Close-up of the Fermi level region of the
CaCu3Ru4O12 spectrum together with the gold spectrum. (bottom panel) Division
of the CaCu3Ru4O12 spectra by the gold spectrum. Figure 6: (a) Temperature
dependence of the Cu $xy$ spectral intensities in the LDA+DMFT method. The
inset shows the imaginary part of the self-energy of the Cu $xy$ orbital. (b)
LDA+DMFT hybridization densities $-1/\pi\ {\rm Im}\Delta(\omega)$ of the Cu
$xy$ orbital. The inset shows the hybridisation densities in a wide energy
range for selected temperatures (300 K, 900 K, 2000 K).

Figure 4a shows the experimental valence band spectra of CCRO in a broad
energy range measured at various photon energies. Our motivation here is to
make use of the different photon-energy dependence of the photoionization
cross sections Trzhaskovskaya  _et al._ (2001, 2006); Yeh and Lindau (1985) to
distinguish the Ru $4d$, Cu $3d$, and O $2p$ contributions to the spectra. The
photon energy of 6.5 keV in HAXPES is much higher than the previously used
photon energies of 1486.6 eV and 920 eV Tran  _et al._ (2006); Sudayama  _et
al._ (2009); Hollmann  _et al._ (2013) and makes Ru $4d$ have the largest
cross section relative to Cu $3d$ and O $2p$. The low photon energies of 200,
150, and 100 eV are close to the Cooper minimum of the Ru $4d$ cross section
Yeh and Lindau (1985), so that with these photon energies the Ru $4d$ signal
gets maximally suppressed, enabling us to observe better the Cu $3d$
contribution. The ratio between the O $2p$ and Cu $3d$ cross sections also
becomes continuously larger with lowering the photon energy Yeh and Lindau
(1985). See also Appendix B displaying the photon energy dependence of the
cross sections in more detail.

In the set of valence band spectra, Fig. 4a, we can identify features labeled
A, B, C, D, and E. Features A and D are very strong at 6.5 keV and diminish
almost completely between 100-200 eV. This strongly suggests that A and D are
derived from the Ru $4d$ orbitals. Features B and E are visible throughout the
set, while the intensity of C is enhanced at 100 eV, the lowest photon energy
of the set. This observation indicates that the features B and E are related
to the Cu $3d$ while C is likely of O $2p$ origin. We point out that the
positions of these five features do not change with the photon energy and
coincide with the previously reported soft x-ray studies Tran  _et al._
(2006); Sudayama  _et al._ (2009); Hollmann  _et al._ (2013).

The LDA+DMFT results are presented in Fig. 4b. They corroborate the above
assignment. The calculated Ru $4d$ spectrum matches the experimental features
A and D, while the theoretical Cu $3d$ spectrum explains well the features B
and E, the former being the Cu $|d^{9}\underline{L}\rangle$ and the latter the
Cu $|d^{8}\rangle$ final state Tjeng  _et al._ (1991, 1992); Hollmann  _et
al._ (2013); Eskes  _et al._ (1990). The feature C is captured by the
theoretical O $2p$ spectra. The calculations reveal that the spectrum around
the Fermi level is dominated by hybridized Ru $4d$ and O $2p$ bands. In Fig.
4c, the Cu $3d$ spectrum is decomposed into $xy$, 3$z^{2}$-$r^{2}$,
$x^{2}$-$y^{2}$, $yz$, and $zx$ components. As shown in the inset, the Cu $3d$
$xy$ orbital points to the surrounding four oxygen sites. Therefore, the Cu
$3d$ $xy$ orbital hybridize with the Ru $4d$ orbitals (via the O $2p$
orbitals) most strongly among the five Cu $3d$ orbitals. The Cu $3d$ $xy$
spectral density near the Fermi level is quite low compared to the Ru $4d$ and
O $2p$ one, see Fig. 4b, and most of its weight is above the Fermi level.

In order to experimentally detect the Cu contribution around the Fermi level,
we focus on the larger peak at positive energies and the spectra taken at 150
eV which is almost the photon energy to minimize the Ru $4d$ signal. The top
panel of Fig. 5 displays a close-up of the spectra along with the
corresponding gold reference spectrum taken under the same conditions. In
order to look for the possible presence of states above the Fermi level, we
divide the CCRO spectrum by the corresponding gold spectrum. The results are
shown in the bottom panel of Fig. 5. We can identify clearly the presence of a
sharp peak at about 0.07-0.08 eV above the Fermi level, very consistent with
the results of the LDA+DMFT calculations.

Zooming to the vicinity of the Fermi level 111In order to reach lower
temperatures we have used a model without the empty Ru $4d$ $e_{g}$ states
here. The equivalence of Cu $3d$ $xy$ spectra in the models with and without
Ru $4d$ $e_{g}$ states is established in Appendix D., the calculated Cu $3d$
$xy$ spectrum in Fig. 6a reveals sharp temperature-dependent peaks. To put the
present results in the context of Anderson impurity and periodic Anderson
models we show the Cu $3d$ $xy$ hybridization densities in Fig. 6b. The global
view in the inset shows a strongly asymmetric situation with the Fermi energy
located in the tail of the hybridization density. The hybridization density
exhibits only a minor temperature dependence. A sharp hybridization peak below
the Fermi level, responsible for the peak around $-0.08$ eV in the Cu spectra
also observed with angle-resolved PES (ARPES) around the $H$ point in the
Brillouin zone Liu  _et al._ (2020), is an exception. This explains the
different temperature behavior of the peaks in Fig. 6a. Damping of the peaks
above the Fermi level with increasing temperature is not accompanied by
changes of the hybridization function and thus reflects the Anderson/Kondo
impurity physics controlled by the Kondo temperature. Damping of the $-0.08$
eV peak arises from temperature-induced changes of the hybridization function,
i.e., involves the feedback from the Cu ions (the temperature dependence of Ru
contribution is negligible in the studied range). It is therefore an
Anderson/Kondo lattice effect related to somewhat lower coherence temperature
Burdin  _et al._ (2000); Pruschke  _et al._ (2000). The inset of Fig. 6a shows
the inverse quasi-particle lifetime. The region of long lifetime (dip in the
inverse lifetime) between -$0.1$ eV and $+0.05$ eV around the Fermi level
found at lower temperatures, marks limits of the Fermi liquid theory. Between
600–700 K the scattering rate around Fermi level increases forming a peak in
$\operatorname{Im}\Sigma$. Eventually (1500–2000 K) the scattering rate grows
quasi-uniformly in entire low-energy regime (-0.5–0.5 eV) with increasing
temperature.

Figure 7: Local susceptibility $\chi_{\rm loc}(T)$ (blue) and the dynamical
spin susceptibility $\chi_{\rm loc}(\omega={+0})$ (red), calculated by the
LDA+DMFT method.

Next, we discuss the local spin susceptibility $\chi_{\rm loc}$ at the Cu
site. The top panel of Fig. 7 displays the $\chi_{\rm loc}$ obtained with
LDA+DMFT. It exhibits the Curie behavior at high temperatures, and turns into
a broad maximum at lower temperatures which is characteristic for Kondo
screening. The deviation from the Curie behavior starts around 1000 K,
suggesting a relatively high Kondo scale $T_{K}$. The calculated $\chi_{\rm
loc}$ reproduces quite well the experimental susceptibility Kobayashi  _et
al._ (2004) although an exact match can not be expected between the local and
the uniform spin susceptibility, due to the contribution of the itinerant Ru
4$d$ \- O $2p$ states. The dynamical local susceptibility $\chi_{\rm
loc}(\omega)$ is, nevertheless, directly related to the spin-lattice
relaxation rate $1/T_{1}$ measured in nuclear magnetic resonance (NMR)
experiments Kato  _et al._ (2009); Krimmel  _et al._ (2008)

$\frac{1}{T_{1}}\propto T\lim_{\omega\to+0}{\rm Im}\frac{\chi_{\rm
loc}(\omega)}{\omega}.$

The calculated temperature dependence of $1/T_{1}$ is shown in Fig. 7. It
shows an approximately linear increase at low-temperatures, which flattens
into a constant behavior at around 1000 K. It is well-known that the former is
characteristic of a Fermi liquid, while the latter of a fluctuating local
moment. For CCRO, the NMR experiment was performed up to 700 K so far, where
the absence of the constant behavior was brought up as an evidence against the
Kondo physics in this compound. Our result however suggests that CCRO
manifests its local moment signature in NMR above the reported temperature.
The corresponding spin–spin correlation functions $\chi_{\rm spin}(\tau)$, see
Appendix C, reflect the presence of an instantaneous Cu 3$d$ moment at all
temperatures, which rapidly disappears on a short-time scale at temperatures
below $T_{K}$. In contrast, the charge correlations are temperature
independent. Rapid charge fluctuations present at all temperatures rather
reflect the Cu-O bonding, not the hybridization with the states in the
vicinity of the Fermi level.

The presence of magnetic Cu ions immersed in an itinerant band leads indeed to
the emergence of Kondo physics, as demonstrated by our LDA+DMFT results. An
important aspect for the long standing discussions about CCRO is our finding
that the Kondo temperature is quite high, namely at around 700 K (between 500
and 1000 K). We would like to note that the parameters and double-counting
corrections in our LDA+DMFT calculations have been tuned as to reproduce the
available experimental (HAX)PES and the recent ARPES Liu  _et al._ (2020)
spectra, details can be found in Appendix D. In this respect it is worth
mentioning that we have also included on-site Coulomb interactions at the Ru
site. This was necessary to explain the presence of a satellite structure in
the Ru $3d$ core level PES spectrum, see Appendix D. It turned out that the
inclusion of $U_{\rm Ru}$ in the Ru $4d$ shell has also a substantial
influence on the low energy properties of CCRO. We have calculated the local
spin susceptibility $\chi_{\rm loc}$ at the Cu site with $U_{\text{Ru}}=3.1$
eV and $U_{\text{Ru}}=0.0$ eV, see Appendix D, indicating that correlations on
the Ru site influence the screening process on the Cu site. This can be traced
back to the influence of the Ru $U$ on the shape of the Ru $4d$ \- O $2p$
band. In Appendix D, we show the Cu $xy$ spectral intensities calculated with
$U_{\rm Ru}=0$ eV on the Ru 4$d$ shell, where the agreement to the available
PES data is surrendered.

The high Kondo temperature we have found from our LDA+DMFT calculations
implies that the contribution of the Kondo screening process to the low
temperature specific heat is modest, thus explaining why band structure
calculations can seemingly reproduce the experimentally observed $\gamma$
value of the specific heat rather well since the main contribution comes from
the uncorrelated Ru-O derived bands. The high Kondo temperature also implies
that one needs to go to very high temperatures to see the appearance of local
Cu moments, as demonstrated in Fig. 7. We can thus infer that upon going from
high to low temperatures the Kondo screening process is practically completed
already at 300 K, and that lowering the temperature further would not produce
significant changes in the electronic and magnetic properties. It is
understandable that there has been a controversy concerning Kondo physics in
CaCu3Ru4O12 since its signatures in low temperature measurements are weak.

Nevertheless, Kondo physics is present, as evidenced by the low-temperature
disappearance of the Cu2+ magnetic susceptibility for which we were able to
provide a quantitatively explanation using our LDA+DMFT calculations that
include fine tuning of the parameters from a detailed comparison to bulk
sensitive photoemission data. Our findings indicate that the material class
CaCu3M4O12 indeed provides a unique opportunity to explore Kondo phenomena in
transition metal compounds, where one may achieve lower Kondo temperatures by
suitably varying the M constituent.

The authors thank A. Sotnikov and J. Fernández Afonso, Mathias Winder for
fruitful discussions. A.H. and J.K. are supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 646807-EXMAG). A.H. was supported by JSPS
KAKENHI Grant Number 21K13884. The computations were performed at the Vienna
Scientific Cluster (VSC). The research in Dresden is supported by the Deutsche
Forschungsgemeinschaft (DFG) through Grant No. 320571839 and SFB 1143
(project-id 247310070). The experiments in Taiwan and Korea were facilitated
by the Max Planck-POSTECH-Hsinchu Center for Complex Phase Materials. S.S. and
Y.S. are supported by JSPS KAKENHI Grants-in-Aid for Scientific Research (No.
19H05823 and No. 20H00397) and by a JSPS Core-to-Core Program (A) Advanced
Research Networks.

##  Appendix A Sample consistency

In order to ensure that the experimental spectra and their features are
intrinsic to the material, we have utilized three different batches of samples
synthesized by three different groups: A.C. Komarek’s group from Max Planck
Institute for Chemical Physics of Solids, Dresden, A. Günther from the
University of Augsburg, and Y. Shimakawa’s group from the Kyoto University.
Figure 8 shows the comparison of the valence band PES and XAS spectra taken
from these three batches of samples.

Figure 8: Valence band PES (left) and Cu-$L_{2,3}$ XAS (right) of CaCu3Ru4O12
samples synthesized by the different groups used in this work.

The results match perfectly, and thus confirm that the data presented in this
paper are not sample specific or due to extrinsic contributions

##  Appendix B Photoionization cross sections

Figure 9: Photoionization cross section values of Cu $3d$, Ru $4d$, and O $2p$
interpolated from the data tabulated in Refs. Yeh and Lindau (1985);
Trzhaskovskaya  _et al._ (2001, 2006). The vertical lines indicate the photon
energies used for the photoemission data shown in Fig. 4.

For the valence band of CaCu3Ru4O12 the most relevant contributions are Cu
$3d$, Ru $4d$, and O $2p$. Figure 9 shows the photon energy dependence of the
photoionization cross sections as interpolated from the data tabulated in
Refs. Yeh and Lindau (1985); Trzhaskovskaya  _et al._ (2001, 2006). The
vertical lines highlight the photon energies used in this study. At high
photon energies (HAXPES, 6.5 keV), the Ru $4d$ provides a larger signal than
the Cu $3d$ and the O $2p$. Going to lower photon energies, the Cu $3d$
becomes gradually stronger, and with a crossover at $1.2$ keV, it becomes the
dominant contributor. At around $100$ eV, there is a minimum in the Ru $4d$
cross section, also known as the Cooper minimum, which provides the ideal
condition for studying the Cu $3d$ contributions. At this photon energy, the
Cu $3d$ signal is enhanced by a factor $\approx$ 70 with respect to the Ru
$4d$. For lower photon energies, the Ru $4d$ quickly gains strength and
becomes again the dominant contributor, making these low photon energies less
useful when searching for the Cu $3d$ signal close to the Fermi level. As for
the O $2p$ contribution, we observe that it is highly suppressed for high
energies, but becomes only competitive with the Cu $3d$ for energies below
$\approx$ 50 eV.

##  Appendix C LDA+DMFT computational method

Below we describe the LDA+DMFT scheme Metzner and Vollhardt (1989); Georges
_et al._ (1996); Kotliar  _et al._ (2006) employed to analyze the experimental
data. We start with density functional calculations for the experimental
crystal structure of CCRO Krimmel  _et al._ (2008) using the Wien2k code Blaha
_et al._ . Then we construct the multi-band Hubbard model on the basis spanned
by the Cu 3$d$, Ru 4$d$, and O 2$p$ Wannier functions from the LDA bands using
wien2wannier and wannier90 packages Kuneš  _et al._ (2010); Mostofi  _et al._
(2014). The multi-band Hubbard model is augmented with the local electron-
electron interaction within the Cu 3$d$ and Ru 4$d$ shells giving the
Hamiltonian,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{\textit{{k}}}\begin{pmatrix}\textit{{c}}^{{\dagger}}_{\textit{{k}}}&\textit{{r}}^{{\dagger}}_{\textit{{k}}}&\textit{{p}}^{{\dagger}}_{\textit{{k}}}\end{pmatrix}\begin{pmatrix}h^{cc}_{\textit{{k}}}-\mu_{\rm
cr}^{\rm Ru}&h^{dp}_{\textit{{k}}}&h^{cp}_{\textit{{k}}}\\\
h^{rc}_{\textit{{k}}}&h^{rr}_{\textit{{k}}}-\mu_{\rm dc}^{\rm
Cu}&h^{rp}_{\textit{{k}}}\\\
h^{pc}_{\textit{{k}}}&h^{pr}_{\textit{{k}}}&h^{pp}_{\textit{{k}}}\end{pmatrix}\begin{pmatrix}\textit{{c}}_{\textit{{k}}}\\\
\textit{{r}}_{\textit{{k}}}\\\ \textit{{p}}_{\textit{{k}}}\end{pmatrix}$
$\displaystyle+$ $\displaystyle\sum_{i}W^{dd}_{i,{\rm
Cu}}+\sum_{i}W^{dd}_{i,{\rm Ru}}.$

Here, $\textit{{c}}^{{\dagger}}_{\textit{{k}}}$ is an operator-valued vector
whose elements are Fourier transforms of $c_{\gamma i}$, that annihilate the
Cu 3$d$ electron in the orbital $\gamma$ in the $i$-th unit cell. Similarly
$\textit{{r}}^{{\dagger}}_{\textit{{k}}}$ and
$\textit{{p}}^{{\dagger}}_{\textit{{k}}}$ are those for Ru 4$d$ and O 2$p$
electrons. The on-site Coulomb interaction $W_{i,{\rm Cu}}^{dd}$ and
$W_{i,{\rm Ru}}^{dd}$ on Cu and Ru sites is approximated with the density-
density form with parameters ($U$, $J$)=(8.5 eV, 0.98 eV) for Cu 3$d$
electrons and (3.1 eV, 0.7 eV) for Ru 4$d$ electrons, which are typical values
for Cu and Ru systems Petukhov  _et al._ (2003); Hollmann  _et al._ (2013);
Gorelov  _et al._ (2010); Pchelkina  _et al._ (2007). The double-counting
terms $\mu_{\rm dc}^{\rm Ru}$, $\mu_{\rm dc}^{\rm Cu}$, which correct for the
$d$–$d$ interaction present in the LDA step Karolak  _et al._ (2010); Kotliar
_et al._ (2006), renormalize the $p$-$d$ splitting and thus the charge-
transfer energy. We have fixed the double-counting values to $\mu_{\rm
dc}^{\rm Cu}=70.1$ eV and $\mu_{\rm dc}^{\rm Ru}=16.2$ eV ($\mu_{\rm dc}^{\rm
Ru}=13.4$ eV for the model without Ru $e_{g}$ states) by comparison to the
photoemission spectroscopy data discussed in Appendix D. The CT-QMC method
with the hybridization expansion Werner  _et al._ (2006); Boehnke  _et al._
(2011); Hafermann  _et al._ (2012); Hariki  _et al._ (2015) was used to solve
the auxiliary Anderson impurity model (AIM) in the DMFT self-consistent
calculation. The valence spectra were obtained by analytic continuation of
self-energy $\Sigma(\varepsilon)$ using the maximum entropy method Wang  _et
al._ (2009); Jarrell and Gubernatis (1996). The hybridization function
$\Delta(\varepsilon)$ for the Cu 3$d$ orbital $\gamma$ ($|d_{\gamma}\rangle$)
is given by Georges  _et al._ (1996); Kotliar  _et al._ (2006)

$\displaystyle\Delta_{\gamma}(\varepsilon)=\langle d_{\gamma}|\varepsilon-
h_{0}-\Sigma(\varepsilon)-G^{-1}(\varepsilon)|d_{\gamma}\rangle,$

where $G(\varepsilon)$ and $h_{0}$ are the local Green’s function and the one-
body part of the on-site Hamiltonian at the Cu site, respectively.
$\Delta(\varepsilon)$ does not depend on the spin, and small off-diagonal
elements between different orbitals are neglected. The Cu 2$p$ core-level
spectrum was calculated using the method of Refs. Hariki  _et al._ , 2017a,
2020, where the AIM with the DMFT hybridization density is extended to include
the Cu 2$p$ core orbitals Hariki  _et al._ (2017a); Ghiasi  _et al._ (2019).
The configuration interaction scheme with 25 bath states representing the DMFT
hybridization density is employed to evaluate the 2$p$ spectra.

Figure 10: Correlation function $\chi(\tau)$ of the spin and charge (inset)
channel calculated by the LDA+DMFT method, where $\beta$=1/${k_{\rm B}T}$ is
the inverse temperature.

Figure 10 shows the local spin $\chi_{\rm spin}(\tau)$ and charge $\chi_{\rm
charge}(\tau)$ correlation functions at the Cu site. $\chi_{\rm spin}(\tau)$
is given as $\chi_{\rm
spin}(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$ Křápek  _et al._
(2012); Hariki  _et al._ (2017b) and $\chi_{\rm charge}(\tau)$ is given by
$\chi_{\rm charge}(\tau)=\langle\delta\hat{n}(\tau)\delta\hat{n}(0)\rangle$
Ylvisaker  _et al._ (2009), where
$\delta\hat{n}=\hat{n}-\langle\hat{n}\rangle$ with the Cu $d$-occupation
number operator $\hat{n}$. These quantities are calculated by the impurity
Anderson model with the DMFT hybridization using the CT-QMC method. The local
susceptibility $\chi_{\rm loc}(T)$ is obtained by integrating the $\chi_{\rm
spin}(\tau)$ at temperature $T$ with respect to the imaginary time $\tau$,

$\displaystyle\chi_{\rm loc}(T)$ $\displaystyle=\int_{0}^{1/T}d\tau\langle
m_{z}(\tau)m_{z}(0)\rangle.$

A rigid instantaneous Cu 3$d$ spin moment is present for all temperatures, as
imprinted in the temperature-independent value of $\chi_{\rm spin}(\tau=0)$,
see Fig. 10. The spin moment survives on a long-time scale at high
temperatures (see e.g. at 2000 K), giving the Curie behavior in $\chi_{\rm
loc}$ shown in the main text, while it disappears on short timescales at low
temperatures due to the Kondo screening by the Ru 4$d$ \- O $2p$ bands, giving
the Pauli-like behavior in $\chi_{\rm loc}$. In contrast to the temperature
dependence in $\chi_{\rm spin}(\tau/\beta)$, the local charge correlation
function $\chi_{\rm charge}$ is substantially suppressed at all temperatures,
see the inset of Fig. 10, indicating the frozen charge fluctuation of the Cu
3$d$ electrons in CCRO, as expected in the Kondo regime. Thus the calculated
spin and charge correlation functions support the Kondo behavior of Cu 3$d$
electrons in the studied material. We point out that the $\chi_{\rm
spin}(\tau=0)$ value is reduced to 0.7 from 1.0 which corresponds to the
$S=1/2$ Kondo limit. The reduction comes from the mixture of the
$|d^{10}\underline{L}\rangle$ configuration by the Cu-O hybridization, where
$|\underline{L}\rangle$ represents an O 2$p$ hole, but it does not show a
temperature dependence. This behavior characterizes the effect of charge-
transfer in the Cu-O subsystem. The energy scale of the Kondo physics
therefore builds on the ligand-metal hybridization in addition to the Coulomb
interaction $U$ at the Cu site.

##  Appendix D Parameters in the LDA+DMFT simulation

In the calculations we used two versions of the model described in Appendix C.
First model contains all Ru $4d$ states, while the second contains only the Ru
$t_{2g}$ states present at and below the Fermi level. The first model, which
we consider as the primary one, is needed to capture the full spectrum, in
particular the feature D of Fig. 4a. The second model is used for numerical
convenience as it allows us to perform LDA+DMFT calculations at lower
temperature (below 300 K). It is employed to compute the low-energy Cu
spectral intensities (Fig. 6) and response functions (Fig. 7) including
temperatures below 300 K. Its results are validated by comparison to the first
model at the higher temperatures.

The parameters $\mu_{\rm dc}^{\rm Cu}$ and $\mu_{\rm dc}^{\rm Ru}$ of our
theory (Appendix C) are fixed to reproduce (i) positions of the peaks A–E in
the valence valence photoemission spectra of Fig. 4, (ii) positions of the
low-energy peak at 0.07–0.08 eV above, Fig. 5, and around 0.08 eV below Liu
_et al._ (2020) the Fermi energy, (iii) the splitting of the Cu 2$p_{3/2}$ XPS
main line, see Fig. 3, and (iv) observation of a shoulder in Ru 3$d$ core-
level XPS spectrum, see Fig. 12.

Figure 11: (a) The Cu $xy$ spectral intensities in LDA+DMFT results for
different double counting $\mu^{\rm Cu}_{\rm dc}$ values. The positions of the
low-energy peak in the experimental PES data are indicated by vertical dashed
lines. The calculation is performed at 300 K. (b) Local susceptibility
$\chi_{\rm loc}(T)$, (c) dynamical spin susceptibility $\chi_{\rm
loc}(\omega={+0})$ at the Cu site. Here the model with all Ru 4d states is
employed. The dashed lines in panels (a) and (b) show the results obtained
with $U_{\rm Ru}=0.0$ eV. (d) Hybridization densities
$-1/\pi\textrm{Im}\Delta(\omega)$ of the Cu $xy$ orbital in LDA+DMFT with
$U_{\rm Ru}=3.1$ eV (solid) and with $U_{\rm Ru}=0.0$ eV (dashed). The inset
shows the hybridization densities near $E_{F}$. Figure 12: (a) Experimental Ru
$3d$ core-level photoemission spectrum of CaCu3Ru4O12 (black line) and
theoretical spectra from the LDA+DMFT calculation with $U_{\rm Ru}$ = 3.1 eV
(red line) and $U_{\rm Ru}$ = 0.0 eV (blue line). (b) (top) Local spin
susceptibility and (bottom) screened spin moment $M_{\rm
scr}(T)=\sqrt{T\chi_{\rm loc}(T)}$ Křápek  _et al._ (2012); Hariki  _et al._
(2017b) on the Ru site computed by the LDA+DMFT method. The horizontal dashed
line indicates the atomic value of the Ru4+ ion.

Before analysing $\mu_{\rm dc}^{\rm Cu}$ and $\mu_{\rm dc}^{\rm Ru}$ we
briefly discuss why we need to include the electron-electron interaction on
the Ru site Cao  _et al._ (2008); Kikugawa  _et al._ (2009); Liu  _et al._
(2018). To this end we have compared calculations with $U_{\rm Ru}=3.1$ eV
from previous DFT studies for Ru oxides (with the same formal Ru valence as in
CCRO) Gorelov  _et al._ (2010); Pchelkina  _et al._ (2007) and with $U_{\rm
Ru}=0.0$ eV. Fig. 12a shows that finite $U_{\rm Ru}$ is needed to yield the
shoulder/satellite feature (iv) in the Ru 3$d$ core-level XPS spectra.
However, finite $U_{\rm Ru}$ affects also the behavior of the Cu $xy$ state,
see dashed lines in Figs. 11ab, through dynamical renormalization (band
narrowing) of Ru bands, which modifies the environment (hybridization density)
of Cu, see Fig. 11d. With $U_{\rm Ru}=0.0$ eV we could not find $\mu_{\rm
dc}^{\rm Cu}$ to fulfill (i–iii). We would like to note that the Ru spin
response is far from the Curie form despite the presence of correlations in
the Ru $4d$ shell, see Fig. 12b. Correspondingly, also the screened
(effective) Ru moment is substantially smaller than the atomic value ($\sim
2\mu_{B}$) for a Ru4+ ion.

Figure 13: (a) Cu 2$p_{3/2}$ core-level photoemission spectra calculated by
the LDA+DMFT method with selected $\mu^{\rm Ru}_{\rm dc}$ values (with
$\mu^{\rm Cu}_{\rm dc}=70.1$ eV). (b) The spectra by LDA+DMFT with selected
$\mu^{\rm Cu}_{\rm dc}$ values (with $\mu^{\rm Ru}_{\rm dc}=16.2$ eV). The
experimental spectrum is shown for a comparison. Figure 14: LDA+DMFT results
for the models (left) with and (right) without the Ru 4d $e_{g}$ states at
selected temperatures. (top) Cu $xy$ spectral intensities, (middle) the
hybridization densities of the Cu $xy$ orbitals, and (bottom) local
susceptibility $\chi_{\rm loc}(T)$ and the dynamical spin susceptibility
$\chi_{\rm loc}(\omega={+0})$ at the Cu site. $\mu^{\rm Cu}_{\rm dc}=70.1$ eV
is employed in the two models, and $\mu_{\rm dc}^{\rm Ru}=16.2$ eV ($\mu_{\rm
dc}^{\rm Ru}=13.4$ eV) for the model with (without) Ru 4$d$ $e_{g}$ states.

The value of $\mu^{\rm Cu}_{\rm dc}$ affects the positions of the Cu $d^{8}$
satellite, feature $E$, and feature B in Fig.4 as well as the low-energy peaks
(ii). All of these are well reproduced by its chosen value of $\mu^{\rm
Cu}_{\rm dc}=70.1$ eV, see Fig. 11a and Fig. 4. The small screened spin
susceptibility due to the Kondo screening is found around the optimal
$\mu^{\rm Cu}_{\rm dc}$ value, see Figs. 11bc. The $\mu^{\rm Cu}_{\rm dc}$
also affects the Cu–O charge-transfer energy and thus the splitting between
the main line and charge-transfer satellite in Cu 2$p$ XPS, see Fig. 13b. On
the other hand, the behavior of Cu, including variation of $\mu_{\rm dc}^{\rm
Cu}$, has minor effect on its hybridization function, except for the peak just
below Fermi energy. As a result the splitting of Cu 2$p$ XPS main line (A,B)
is independent of $\mu_{\rm dc}^{\rm Cu}$, see Fig. 13b.

The same is not true for $\mu^{\rm Ru}_{\rm dc}$. Affecting primarily the Ru–O
charge-transfer energy, its variation modifies the Cu $3d$ $xy$ hybridization
function around Fermi level, which has a sizable effect on the splitting in
the Cu 2$p$ XPS main line (A,B), see Fig. 13a. We can thus use (iii) together
with the Ru features in the global spectrum (i) to establish $\mu^{\rm
Ru}_{\rm dc}=16.2$ eV.

Finally, we discuss the relationship of models with and without Ru $4d$
$e_{g}$ states. While $\mu^{\rm Cu}_{\rm dc}$ is the same for the two models,
$\mu^{\rm Ru}_{\rm dc}$ must be different since the models include interaction
between all Ru $4d$ orbitals in one case and between the $t_{2g}$ only in the
other one. The matching $\mu^{\rm Ru}_{\rm dc}$ for the model without Ru $4d$
$e_{g}$ states is 13.2 eV. Both models yield the very similar temperature
dependencies of the Cu spectral functions as well as hybridization densities
and local susceptibilities, as shown in Fig. 14.

## References

  * Cava (2000) R. J. Cava, J. Am. Ceram. Soc. 83, 5 (2000).
  * Khomskii (2014) D. Khomskii,  _Transition Metal Compounds_ (Cambridge University Press, 2014).
  * Keimer  _et al._ (2015) B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).
  * Stewart (1984) G. R. Stewart, Rev. Mod. Phys. 56, 755 (1984).
  * Coleman (2006) P. Coleman, arXiv:cond-mat/0612006 (2006).
  * Pfleiderer (2009) C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009).
  * Wirth and Steglich (2016) S. Wirth and F. Steglich, Nat. Rev. Mater. 1, 16051 (2016).
  * Kondo  _et al._ (1997) S. Kondo, D. C. Johnston, C. A. Swenson, F. Borsa, A. V. Mahajan, L. L. Miller, T. Gu, A. I. Goldman, M. B. Maple, D. A. Gajewski, E. J. Freeman, N. R. Dilley, R. P. Dickey, J. Merrin, K. Kojima, G. M. Luke, Y. J. Uemura, O. Chmaissem, and J. D. Jorgensen, Phys. Rev. Lett. 78, 3729 (1997).
  * Urano  _et al._ (2000) C. Urano, M. Nohara, S. Kondo, F. Sakai, H. Takagi, T. Shiraki, and T. Okubo, Phys. Rev. Lett. 85, 1052 (2000).
  * Shimoyamada  _et al._ (2006) A. Shimoyamada, S. Tsuda, K. Ishizaka, T. Kiss, T. Shimojima, T. Togashi, S. Watanabe, C. Q. Zhang, C. T. Chen, Y. Matsushita, H. Ueda, Y. Ueda, and S. Shin, Phys. Rev. Lett. 96, 026403 (2006).
  * Kobayashi  _et al._ (2004) W. Kobayashi, I. Terasaki, J. ichi Takeya, I. Tsukada, and Y. Ando, J. Phys. Soc. Jpn. 73, 2373 (2004).
  * Krimmel  _et al._ (2008) A. Krimmel, A. Günther, W. Kraetschmer, H. Dekinger, N. Büttgen, A. Loidl, S. G. Ebbinghaus, E.-W. Scheidt, and W. Scherer, Phys. Rev. B 78, 165126 (2008).
  * Krimmel  _et al._ (2009) A. Krimmel, A. Günther, W. Kraetschmer, H. Dekinger, N. Büttgen, V. Eyert, A. Loidl, D. V. Sheptyakov, E.-W. Scheidt, and W. Scherer, Phys. Rev. B 80, 121101 (2009).
  * Kao  _et al._ (2017) T.-H. Kao, H. Sakurai, S. Yu, H. Kato, N. Tsujii, and H.-D. Yang, Phys. Rev. B 96, 024402 (2017).
  * Xiang  _et al._ (2007) H. Xiang, X. Liu, E. Zhao, J. Meng, and Z. Wu, Phys. Rev. B 76, 155103 (2007).
  * Tanaka  _et al._ (2009a) S. Tanaka, N. Shimazui, H. Takatsu, S. Yonezawa, and Y. Maeno, J. Phys. Soc. Jpn. 78, 024706 (2009a).
  * Tanaka  _et al._ (2009b) S. Tanaka, H. Takatsu, S. Yonezawa, and Y. Maeno, Phys. Rev. B 80, 035113 (2009b).
  * Sudayama  _et al._ (2009) T. Sudayama, Y. Wakisaka, K. Takubo, T. Mizokawa, W. Kobayashi, I. Terasaki, S. Tanaka, Y. Maeno, M. Arita, H. Namatame, and M. Taniguchi, Phys. Rev. B 80, 075113 (2009).
  * Hollmann  _et al._ (2013) N. Hollmann, Z. Hu, A. Maignan, A. Günther, L.-Y. Jang, A. Tanaka, H.-J. Lin, C. T. Chen, P. Thalmeier, and L. H. Tjeng, Phys. Rev. B 87, 155122 (2013).
  * Liu  _et al._ (2020) H. Liu, Y. Cao, Y. Xu, D. J. Gawryluk, E. Pomjakushina, S.-Y. Gao, P. Dudin, M. Shi, L. Yan, Y.-f. Yang, and H. Ding, Phys. Rev. B 102, 035111 (2020).
  * Kato  _et al._ (2009) H. Kato, T. Tsuruta, M. Matsumura, T. Nishioka, H. Sakai, Y. Tokunaga, S. Kambe, and R. E. Walstedt, J. Phys. Soc. Jpn. 78, 054707 (2009).
  * Cheng  _et al._ (2013) J.-G. Cheng, J.-S. Zhou, Y.-F. Yang, H. D. Zhou, K. Matsubayashi, Y. Uwatoko, A. MacDonald, and J. B. Goodenough, Phys. Rev. Lett. 111, 176403 (2013).
  * Shimakawa (2008) Y. Shimakawa, Inorg. Chem. 47, 8562 (2008).
  * Momma and Izumi (2011) K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011).
  * Weinen  _et al._ (2015) J. Weinen, T. C. Koethe, C. F. Chang, S. Agrestini, D. Kasinathan, Y. Liao, H. Fujiwara, C. Schüßler-Langeheine, F. Strigari, T. Haupricht, G. Panaccione, F. Offi, G. Monaco, S. Huotari, K.-D. Tsuei, and L. H. Tjeng, J. Electron. Spectrosc. Relat. Phenom. 198, 6 (2015).
  * Ebbinghaus  _et al._ (2002) S. Ebbinghaus, A. Weidenkaff, and R. Cava, J. Solid State Chem. 167, 126 (2002).
  * Metzner and Vollhardt (1989) W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).
  * Georges  _et al._ (1996) A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * Kotliar  _et al._ (2006) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006).
  * (30) P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz,  _WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001), ISBN 3-9501031-1-2_.
  * Kuneš  _et al._ (2010) J. Kuneš, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held, Comput. Phys. Commun. 181, 1888 (2010).
  * Mostofi  _et al._ (2014) A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 185, 2309 (2014).
  * Petukhov  _et al._ (2003) A. G. Petukhov, I. I. Mazin, L. Chioncel, and A. I. Lichtenstein, Phys. Rev. B 67, 153106 (2003).
  * Gorelov  _et al._ (2010) E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. I. Lichtenstein, and E. Pavarini, Phys. Rev. Lett. 104, 226401 (2010).
  * Pchelkina  _et al._ (2007) Z. V. Pchelkina, I. A. Nekrasov, T. Pruschke, A. Sekiyama, S. Suga, V. I. Anisimov, and D. Vollhardt, Phys. Rev. B 75, 035122 (2007).
  * Werner  _et al._ (2006) P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97, 076405 (2006).
  * Boehnke  _et al._ (2011) L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann, and O. Parcollet, Phys. Rev. B 84, 075145 (2011).
  * Hafermann  _et al._ (2012) H. Hafermann, K. R. Patton, and P. Werner, Phys. Rev. B 85, 205106 (2012).
  * Karolak  _et al._ (2010) M. Karolak, G. Ulm, T. Wehling, V. Mazurenko, A. Poteryaev, and A. Lichtenstein, J. Electron. Spectrosc. Relat. Phenom. 181, 11 (2010), proceedings of International Workshop on Strong Correlations and Angle-Resolved Photoemission Spectroscopy 2009.
  * Wang  _et al._ (2009) X. Wang, E. Gull, L. de’ Medici, M. Capone, and A. J. Millis, Phys. Rev. B 80, 045101 (2009).
  * Jarrell and Gubernatis (1996) M. Jarrell and J. Gubernatis, Phys. Rep. 269, 133 (1996).
  * Hariki  _et al._ (2017a) A. Hariki, T. Uozumi, and J. Kuneš, Phys. Rev. B 96, 045111 (2017a).
  * Hariki  _et al._ (2020) A. Hariki, M. Winder, T. Uozumi, and J. Kuneš, Phys. Rev. B 101, 115130 (2020).
  * Ghiasi  _et al._ (2019) M. Ghiasi, A. Hariki, M. Winder, J. Kuneš, A. Regoutz, T.-L. Lee, Y. Hu, J.-P. Rueff, and F. M. F. de Groot, Phys. Rev. B 100, 075146 (2019).
  * Tjeng  _et al._ (1991) L. H. Tjeng, C. T. Chen, J. Ghijsen, P. Rudolf, and F. Sette, Phys. Rev. Lett. 67, 501 (1991).
  * Tjeng  _et al._ (1992) L. H. Tjeng, C. T. Chen, and S.-W. Cheong, Phys. Rev. B 45, 8205 (1992).
  * Eskes  _et al._ (1990) H. Eskes, L. H. Tjeng, and G. A. Sawatzky, Phys. Rev. B 41, 288 (1990).
  * Chin  _et al._ (2021) Y.-Y. Chin, Z. Hu, Y. Shimakawa, J. Yang, Y. Long, A. Tanaka, L. H. Tjeng, H.-J. Lin, and C.-T. Chen, Phys. Rev. B 103, 115149 (2021).
  * Böske  _et al._ (1998) T. Böske, K. Maiti, O. Knauff, K. Ruck, M. S. Golden, G. Krabbes, J. Fink, T. Osafune, N. Motoyama, H. Eisaki, and S. Uchida, Phys. Rev. B 57, 138 (1998).
  * Tran  _et al._ (2006) T. T. Tran, K. Takubo, T. Mizokawa, W. Kobayashi, and I. Terasaki, Phys. Rev. B 73, 193105 (2006).
  * Ghijsen  _et al._ (1988) J. Ghijsen, L. H. Tjeng, J. van Elp, H. Eskes, J. Westerink, G. A. Sawatzky, and M. T. Czyzyk, Phys. Rev. B 38, 11322 (1988).
  * Tanaka and Jo (1994) A. Tanaka and T. Jo, J. Phys. Soc. Jpn. 63, 2788 (1994).
  * (53) The parameters for the CuO4 calculations have been tuned based on the values taken from ref. Eskes  _et al._ (1990): $U_{dd}=7.8$, $U_{pd}=9.0$, $\Delta=4.0$, $pd\sigma=-1.44$, $pd\pi=0.625$, $T_{pp}=1.3$. Additionally, we have set the ionic crystal field paramaters to $10Dq=1.0$, $Ds=0.025$, $Dt=0.05$. All values are in eV.
  * Taguchi  _et al._ (2005) M. Taguchi, A. Chainani, K. Horiba, Y. Takata, M. Yabashi, K. Tamasaku, Y. Nishino, D. Miwa, T. Ishikawa, T. Takeuchi, K. Yamamoto, M. Matsunami, S. Shin, T. Yokoya, E. Ikenaga, K. Kobayashi, T. Mochiku, K. Hirata, J. Hori, K. Ishii, F. Nakamura, and T. Suzuki, Phys. Rev. Lett. 95, 177002 (2005).
  * Panaccione  _et al._ (2008) G. Panaccione, F. Offi, P. Torelli, G. Vanko, O. Tjernberg, P. Lacovig, A. Guarino, A. Fondacaro, A. Nigro, M. Sacchi, N. B. Brookes, and G. Monaco, Phys. Rev. B 77, 125133 (2008).
  * Trzhaskovskaya  _et al._ (2001) M. Trzhaskovskaya, V. Nefedov, and V. Yarzhemsky, At. Data Nucl. Data Tables 77, 97 (2001).
  * Trzhaskovskaya  _et al._ (2006) M. Trzhaskovskaya, V. Nikulin, V. Nefedov, and V. Yarzhemsky, At. Data Nucl. Data Tables 92, 245 (2006).
  * Yeh and Lindau (1985) J. J. Yeh and I. Lindau, Atomic Data and Nuclear Data Tables 32, 1 (1985).
  * Note (1) In order to reach lower temperatures we have used a model without the empty Ru $4d$ $e_{g}$ states here. The equivalence of Cu $3d$ $xy$ spectra in the models with and without Ru $4d$ $e_{g}$ states is established in Appendix D.
  * Burdin  _et al._ (2000) S. Burdin, A. Georges, and D. R. Grempel, Phys. Rev. Lett. 85, 1048 (2000).
  * Pruschke  _et al._ (2000) T. Pruschke, R. Bulla, and M. Jarrell, Phys. Rev. B 61, 12799 (2000).
  * Hariki  _et al._ (2015) A. Hariki, A. Yamanaka, and T. Uozumi, J. Phys. Soc. Jpn. 84, 073706 (2015).
  * Křápek  _et al._ (2012) V. Křápek, P. Novák, J. Kuneš, D. Novoselov, D. M. Korotin, and V. I. Anisimov, Phys. Rev. B 86, 195104 (2012).
  * Hariki  _et al._ (2017b) A. Hariki, A. Hausoel, G. Sangiovanni, and J. Kuneš, Phys. Rev. B 96, 155135 (2017b).
  * Ylvisaker  _et al._ (2009) E. R. Ylvisaker, J. Kuneš, A. K. McMahan, and W. E. Pickett, Phys. Rev. Lett. 102, 246401 (2009).
  * Cao  _et al._ (2008) G. Cao, O. Korneta, S. Chikara, L. DeLong, and P. Schlottmann, Solid State Communications 148, 305 (2008).
  * Kikugawa  _et al._ (2009) N. Kikugawa, L. Balicas, and A. Peter Mackenzie, Journal of the Physical Society of Japan 78, 014701 (2009).
  * Liu  _et al._ (2018) Y. Liu, H. P. Nair, J. P. Ruf, D. G. Schlom, and K. M. Shen, Phys. Rev. B 98, 041110 (2018).