††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Observation of spin-dependent dual ferromagnetism in perovskite ruthenates

Sungsoo Hahn  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Byungmin Sohn  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Minjae Kim<EMAIL_ADDRESS>Department of Physics, Pohang
University of Science and Technology (POSTECH), Pohang, Korea  Department of
Chemistry, Pohang University of Science and Technology (POSTECH), Pohang,
Korea  Jeong Rae Kim  Center for Correlated Electron Systems, Institute for
Basic Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Soonsang Huh  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Younsik Kim  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Wonshik Kyung  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Minsoo Kim  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Donghan Kim  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Youngdo Kim  Center for Correlated Electron Systems, Institute for Basic
Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  Tae Won Noh  Center for Correlated
Electron Systems, Institute for Basic Science, Seoul 08826, Korea  Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea  Ji
Hoon Shim  Department of Physics, Pohang University of Science and Technology
(POSTECH), Pohang, Korea  Department of Chemistry, Pohang University of
Science and Technology (POSTECH), Pohang, Korea  Changyoung Kim
<EMAIL_ADDRESS>Center for Correlated Electron Systems, Institute for
Basic Science, Seoul 08826, Korea  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea

(August 28, 2024)

###### Abstract

We performed in-situ angle-resolved photoemission spectroscopy (ARPES) and
spin-resolved ARPES (SARPES) experiments to investigate the relationship
between electronic band structures and ferromagnetism in SrRuO3 (SRO) thin
films. Our high quality ARPES and SARPES results show clear spin-lifted band
structures. The spin polarization is strongly dependent on momentum around the
Fermi level, whereas it becomes less dependent at high-binding energies. This
experimental observation matches our dynamical mean-field theory (DMFT)
results very well. As temperature increases from low to the Curie temperature,
spin-splitting gap decreases and band dispersions become incoherent. Based on
the ARPES study and theoretical calculation results, we found that SRO
possesses spin-dependent electron correlations in which majority and minority
spins are localized and itinerant, respectively. Our finding explains how
ferromagnetism and electronic structure are connected, which has been under
debate for decades in SRO.

The study on ferromagnetism has been at the core of solid state physics. For
example, the nature and origin of ferromagnetism in various materials have
been debated for decades. While some are explained within the picture of local
magnetism, others are believed to have an itinerant nature [1, 2, 3, 4]. A
fundamental difficulty in its interpretation is that magnetism requires both
local and itinerant characters; perfectly itinerant electrons may not have
ferromagnetism and local magnetism needs a certain degree of delocalization
for the exchange interaction [5, 6]. Electronic structure or, more
specifically, spin-polarized single-particle spectral functions can provide
vital information on the issue. For example, observation of dispersive spin-
splitting bands should be a strong evidence for itinerant magnetism while
momentum-independent broad spectral function suggests local magnetism [7, 8].
However, results of spin-polarized electronic structure studies are hard to be
found.

Figure 1: Electronic structures of SrRuO3 (SRO) thin films. (a) Schematic
Fermi surfaces of SRO thin films without folded bands. Under finite
magnetization (M), $\alpha$, $\beta$, and $\gamma$ bands split into majority-
(red) and minority-spin (blue) bands. (b) Constant energy maps at the Fermi
level ($E_{F}$) (left) and binding energy of $40$ meV (right) are measured by
angle-resolved photoemission spectroscopy (ARPES) at 10 K. Each map is
obtained by integrating over an energy window of $\pm$ $4$ meV around the
corresponding energy. Pseudocubic Brillouin zone is marked with white dashed
lines. Majority- and minority-spin $\alpha$ ($\alpha_{maj}$ and
$\alpha_{min}$), folded majority- and minority-spin $\alpha^{\prime}$
($\alpha^{\prime}_{maj}$ and $\alpha^{\prime}_{min}$), majority- and minority-
spin $\beta$ ($\beta_{maj}$ and $\beta_{min}$), and $\gamma$ bands are shown
with black dashed lines. (c-d) High-symmetry cuts along (c) $\Gamma$-${\it M}$
and (d) $\Gamma$-${\it X}$ lines. Lorentzian functions are used to fit the
peak positions as marked with black circles. Each cut is denoised with a deep
learning-based statistical method [9]. The insets show the cut direction in
the Brillouin zone.

In the study of ferromagnetism in terms of electronic structures, a perovskite
strontium ruthenate SrRuO3 (SRO) can be a good candidate material. It is a
proto-typical ferromagnet and is frequently used as ferromagnetic (FM)
electrodes [10, 11] for transition metal oxide electronic devices [12]. Its
ferromagnetism is believed to stem from itinerant electrons [13], which has
been utilized in the interpretation of the anomalous Hall effect [14, 15] and
optical conductivity [16]. In fact, the non-integer magnetic moment of the Ru
ion of 1.5 $\mu_{B}$ [17, 18], smaller than expected in the localized electron
picture, appears to secure the itinerant magnetism. On the other hand, the
large on-site Coulomb interaction of U $\sim$ $2$ eV [19, 20, 21] suggests
that there should be a significant portion of local magnetism as well in this
material. This issue can be investigated through electronic structure studies
as mentioned above.

Figure 2: Spin-resolved electronic structures of SRO thin films. SARPES
measurements were conducted at 20 K. (a) Spin-integrated and resolved constant
energy maps at $E_{F}$ and $E_{F}$ \- $450$ meV measured by ARPES and SARPES.
The pseudocubic Brillouin zone is indicated by gray dashed lines. Majority-
and minority-spin $\beta$ ($\beta_{maj}$ and $\beta_{min}$) bands are marked
with black dashed lines. Both spin polarizations are observed near $E_{F}$,
whereas only a single spin polarization is observed in the high binding energy
(HBE) region. (b) Momentum-dependent spin polarization along the high-symmetry
lines ($\Gamma$-${\it X}$ and $\Gamma$-${\it M}$ as marked with green dashed
lines in (a)). Red and black circles represent the spin polarizations at
$E_{F}$ and $E_{F}$ \- $450$ meV, respectively. Integration windows of $E_{F}$
\- $25$ meV $\pm$ 25 meV and $E_{F}$ \- $450$ meV$\pm$ 50 meV are used to
obtain spin polarizations near $E_{F}$ and in the HBE region, respectively.
The positions of $\alpha^{\prime}_{min}$ and $\beta_{min}$ bands are marked
with blue shaded lines. (c) Experimental and (d) theoretical results of spin-
resolved energy distribution curves at the $\Gamma$ point (left) and (-
$5\pi$/8, $5\pi$/8) ($\bf{A}$ and $\bf{B}$ points, respectively, in (a))
(right). Plotted in the bottom half of each panel is the spin polarization,
$I$ = $(I_{maj}-I_{min})/(I_{maj}+I_{min})$, where
$I_{maj}\leavevmode\nobreak\ (I_{min})$ is the intensity of spin-polarized
photoelectrons for the up- (down-) spin electrons obtained at each energy
step.

In principle, the issue may be addressed with spin-dependent electronic
structure information from spin- and angle-resolved photoemission spectroscopy
(SARPES) measurements [22]. Verification of such characteristics in a spin-
and momentum-resolved way would be an important step towards understanding the
ferromagnetism in SRO as well as other materials [23]. In this Letter, we
report the results of in-situ angle-resolved photoemission spectroscopy
(ARPES) and SARPES studies of epitaxial SRO thin films. 15 unit-cell (uc) SRO
thin films grown on (LaAlO3)0.3(Sr2TaAlO6)0.7 (LSAT) are found to have a
moderate coercive field strength [24], allowing us to do ${\it in}$-${\it
situ}$ magnetization with a permanent magnet. Our ARPES measurements show
clear spin-split band structures. Spin polarization of the spin-split bands
are obtained with SARPES and compared with the result of dynamical mean-field
theory (DMFT) calculation. We found that both localized and itinerant
electrons contribute to ferromagnetism, exhibiting spin-dependent electronic
couplings which lead to itinerant and localized electrons for minority and
majority spins, respectively.

Figure 3: Temperature-dependent spin-polarized band dispersions. (a) The
$\Gamma$-${\it X}$ cut (left) below 10 K and (right) above TC (160 K). Each
cut is denoised with a deep learning-based statistical method [9]. (b)
Calculated ARPES spectra of (left) ferromagnetic (FM) and (right) paramagnetic
(PM) states based on dynamical mean-field theory (DMFT) calculation along the
$\Gamma$-${\it X}$ line. (c) Momentum distribution curves (MDCs) at $E_{F}$
along the $\Gamma$-${\it M}$ line for various temperatures between 10 and 160
K. Black dashed lines are guides to the eye for the temperature-dependent
behavior of $\alpha^{\prime}$, $\beta$, and $\gamma^{\prime}$ bands. (d)
Temperature dependence of the full width at half maximum (FWHM) of
$\beta_{min}$ band peaks at $E_{F}$ \- $20$ meV. The values are obtained from
the Lorentzian fit of MDC peaks.

Details of experimental methods and DMFT+DFT calculation [25, 26] are
described in the Supplementary Materials (SM) [38]. Figure 1(a) schematically
shows the Fermi surface topology of SRO in which three $t_{2g}$ bands
($\alpha$, $\beta$, and $\gamma$) cross the Fermi Level ($E_{F}$) [27, 28].
The bands in the non-magnetic state (left) spin split into six bands as in the
right figure of Fig. 1(a). Although SRO is FM and the band splitting is
expected to exist, the spin-split bands have not yet been experimentally
observed in SRO [28]. With ${\it in}$-${\it situ}$ ARPES measurements on a
high-quality thin film, we carefully examined the fermiology and successfully
observed spin-split band structures, i.e. majority- and minority-spin bands.

Shown in Fig. 1(b) are constant energy maps of SRO thin films at $E_{F}$ (left
half) and at $E_{F}$ \- $40$ meV (right half). The bands are named in
accordance with those in previous reports [28, 27]. $\alpha^{\prime}$ (folded
$\alpha$) and $\beta$ bands split into majority- and minority-spin bands as
shown in Fig. 1(b). Hole-like majority- and minority-spin bands of
$\alpha^{\prime}$ ($\alpha^{\prime}_{maj}$ and $\alpha^{\prime}_{min}$,
respectively) are observed around the $\Gamma$ point while electron-like
majority- and minority-spin $\beta$ bands ($\beta_{maj}$ and $\beta_{min}$,
respectively) are located outside of $\alpha^{\prime}$ Fermi surfaces. Note
that oxygen octahedra in the real systems are distorted, which induces
$\sqrt{2}$$\times$$\sqrt{2}$ or 2$\times$2 enlarged unit cells [29, 30, 31].
As a result, folded bands are observed in ARPES data in addition to the bands
in Fig. 1(a). The folded bands are marked with a prime symbol in Fig. 1(b)
(See SM [38] for a schematic Fermi surface with folded bands).

Figures 1(c) and (d) show high-symmetry cuts along the $\Gamma$-$M$ and
$\Gamma$-$X$ directions as marked in the insets. Each cut clearly shows the
majority- and minority-spin $\alpha^{\prime}$ and $\beta$ bands. In order to
obtain the band dispersions, we fitted momentum distribution curves (MDCs)
with Lorentzian functions and determined the MDC peak positions as shown with
black circles in the figures. We speculate that $\alpha^{\prime}$ ($\beta$)
band is better observed in the $\Gamma$-$M$ ($\Gamma$-$X$) cut due to the
symmetry-related matrix element effects [32, 33].

Figure 4: Spin-dependent itinerant and localized ferromagnetism. (a)
Temperature-dependent magnetic moment (red solid line), energy splitting
between spin-polarized bands (green squares), and spin polarization near
$E_{F}$ \- $450$ meV (HBE) at the $\Gamma$ point (blue circles), respectively.
Energy splitting is obtained from the difference between
$\alpha^{\prime}_{maj}$ and $\alpha^{\prime}_{min}$ in Fig. 3(c). (b) A
schematic description for temperature-dependent spin-polarized bands. Black
dashed lines connect the band bottom of each band. A red band (blue band)
represents the majority- (minority-) spin band, and a purple band represents
indistinguishable spin bands. (c, d) A schematic description for movements of
electrons in real space in FM and PM phases. The majority (minority) spins are
localized (itinerant) in the FM phase, whereas spin-degenerate electrons are
localized.

To confirm the spin-polarized nature of the bands, we performed ${\it
in}$-${\it situ}$ SARPES on the SRO thin film. Figure 2(a) shows spin-
dependent constant energy maps at $E_{F}$ and $E_{F}$ \- $450$ meV with the
corresponding constant energy map measured with ARPES. At $E_{F}$, positive
spin polarization (red) is observed near the $\Gamma$ point, but it turns
negative (blue) away from the $\Gamma$ point. As Fig. 2(b) shows, we found
that the $\alpha^{\prime}_{min}$ and $\beta_{min}$ bands are negatively spin-
polarized. The spin polarization turns positive again when the momentum
increases beyond the two bands. Surprisingly, the momentum dependence of spin
polarization almost vanishes at high-binding energy (HBE) of $E_{F}$ \- $450$
meV; the spin polarization is entirely positive regardless of the electron
momentum.

The energy-dependent behavior of the spin polarization can be better seen in
spin-dependent energy distribution curves and spin polarization $I$ =
$(I_{maj}-I_{min})/(I_{maj}+I_{min})$ from $\Gamma$ (A) and (- $5\pi$/8,
$5\pi$/8) (B) plotted in Fig. 2(c). At B, a negative spin polarization is
observed near $E_{F}$ which is due to the contribution from
$\alpha^{\prime}_{min}$ and $\beta_{min}$ bands. On the other hand, the near
$E_{F}$ spin polarization is positive at A due to the contribution from the
$\alpha^{\prime}_{maj}$ band. In the HBE region, the spin polarization remains
positive regardless of the momentum which is amassed to form majority spins.
This behavior is consistent with what is observed in Fig. 2(a). The
experimental results are corroborated with DMFT calculation results in Fig.
2(d); the spin polarization changes its sign near $E_{F}$, whereas it is
constantly positive at HBEs. Based on the present and previous DMFT
calculation results [34], we assert that the momentum independent positive
spin polarization at HBEs stems from the localized character of the majority-
spin band. A localized electron band is expected to be broad and weakly
dispersive [35, 36]. Therefore, the majority-spin band is also expected to
provide momentum-independent spin polarization. The broader and incoherent
$\beta_{maj}$ band in Fig. 1(d) indeed indicates its localized character,
different from the $\beta_{min}$ band.

The spin-split bands are expected to have strong temperature-dependent
behaviors [37, 22], especially approaching the Curie temperature ($T_{C}$).
Figure 3(a) shows the high-symmetry cuts along the $\Gamma$-$X$ direction at
10 and 160 K, below and above $T_{C}$ (See SM [38] for the $\Gamma$-$M$ cut).
Both $\beta_{maj}$ and $\beta_{min}$ are observed at 10 K, whereas it is
difficult to distinguish the two bands at 160 K. Overall, the temperature-
dependent behavior is that the spin-split gap decreases, and bands become
broader as the temperature increases. Eventually, the spin-split bands become
indistinguishable near $T_{C}$. Figure 3(b) shows band dispersions of bulk SRO
along the $\Gamma$-$X$ high-symmetry cut from DMFT calculation. $\beta_{maj}$
and $\beta_{min}$ bands are seen in the FM state while the two bands are
degenerate in the paramagnetic (PM) state.

In order to investigate the temperature dependence more closely, we plot in
Fig. 3(c) $\Gamma$-$M$ MDCs at $E_{F}$ for various temperatures. The momentum
space distance between $\alpha^{\prime}_{maj}$ and $\alpha^{\prime}_{min}$
peaks becomes smaller as the temperature increases. Eventually, the two
$\alpha^{\prime}$ peaks merge each other and become indistinguishable near
$T_{C}$. Moreover, the peaks tend to be broader and incoherent with increasing
temperature, as shown in the temperature-dependent full width at half maximum
(FWHM) plot of $\beta_{min}$ in Fig. 3(d).

We may compare the temperature-dependent SAPRES results with the magnetization
of films. Plotted in Fig. 4(a) is the magnetic moment per Ru ion (from SQUID
measurements, solid red line), $\alpha^{\prime}$ band splitting (green
squares) and spin polarization at the $\Gamma$ point (blue circles) (See SM
[38] for details on the band splitting energy). The $T_{C}$ is 156 K, which is
obtained from the SQUID data. It is shown that the spin polarization follows
the magnetic moment curve, vanishing near the $T_{C}$. The $\alpha^{\prime}$
band splitting has a temperature dependence similar to that of the magnetic
moment in the low-temperature region, but cannot be precisely determined near
$T_{C}$ as the two bands cannot be discerned.

Cartoonistic summary of the observed temperature-dependent behavior of the
spin-resolved electronic structure is given in Fig. 4(b). Overall, the
majority-spin band is more incoherent than the minority-spin band. In the low-
temperature region, the majority- and minority-spin bands have a large spin-
split gap, $\Delta$. As the temperature increases, the bands become incoherent
and broader with decreasing $\Delta$. Above $T_{C}$, the two spin bands become
more incoherent and cannot be distinguished. At present, it is not clear if
the two bands immediately merge at $T_{C}$ or remain split above $T_{C}$ [16].
The temperature-dependent behavior in the real space is schematically shown
for the FM (below $T_{C}$) and PM (above $T_{C}$) phases.

An interesting aspect of the temperature dependence is that the majority-spin
band is more incoherent than the minority-spin band, supported both
experimentally and theoretically [34]. In general, bands become incoherent as
the electronic correlation gets stronger [35, 36]. With self-energy analysis
in our DMFT calculation (See SM [38] for details), we found that majority-spin
bands are more incoherent and correlated than minority-spin bands. Thus,
majority-spin electrons have shorter mean-free path, $\it{l}$($\omega$), in
comparison to minority-spin electrons, which follows
$l(\omega)=v^{\ast}(\omega)\times\tau(\omega)$, where $v^{\ast}(\omega)$ and
$\tau(\omega)$ = (-Z Im$\Sigma(\omega))^{-1}$ are renormalized quasiparticle
velocity and quasiparticle lifetime, respectively, as depicted in Fig. 4(c).
Meanwhile, in the PM state, the majority- and minority-spin bands merge, and
the band structure becomes very incoherent. Therefore, we expect that the
electron mean-free path in the PM state is sufficiently short as schematically
shown in Fig. 4(d). This view is also supported by DMFT calculation results
(See SM [38] for details).

In summary, we have investigated the connection between the ferromagnetism and
electronic structure in strontium ruthenate thin films. By combining ARPES and
SARPES studies, we report the clear band structure and its temperature- and
spin-dependent behaviors, which show a good agreement with DMFT calculation
results. In particular, we demonstrate that majority- and minority-spin
electrons are acting differently and induce the dual ferromagnetism having
(spin-dependent) locality and itinerancy at the same time. A decades-long
problem in the magnetism in SRO may be explained within our picture, which
sheds light on the connection between the electronic structure and magnetism.
Our work proposes a new way to explore the nature of ferromagnetism in
correlated materials.

We thank Y. Ishida for fruitful discussion. This work is supported by the
Institute for Basic science in Korea (Grant No. IBS-R009-D1, IBS-R009-G2). The
work at Postech was supported by NRF (Grant No. NRF-2021R1A2C2010972)

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