Room-temperature ferrimagnetism and polar phase in strained La_2CoRuO_6 films through 3d-4d cation engineering

Domain: biology
Published: 2026-03-13
Editorial outcome: major_revision

Abstract
Materials with room-temperature magnetic ordering and switchable polarization are essential for
spintronic devices. Although 3d transition metal oxides exhibit potential, their Curie temperature
(T ) remains unsatisfactory, and coexistence of magnetic and polar order has not been realized in
C
4d/5d oxides. Here, through epitaxial strain and 3d-4d cation ordering engineering, a ferrimagnetic
insulating state (T ~ 623 K) is achieved in La CoRuO films, coexisting with switchable short-
C 2 6
range polar nanodomains. Atomic-scale investigations and density functional theory calculations
reveal that compressive strain enhances lattice distortions. These distortions, combined with high-
spin state of Co2+ ions and ordered BN-site cations, significantly enhance Co-O-Ru
antiferromagnetic superexchange, inducing the ferrimagnetic insulating state. Concurrently, the
gradient BO octahedral rotations with inhomogeneous evolution trigger B-site ions’
6
displacements, driving the formation of polar nanodomains. Our work fills the experimental gap
in realizing magnetic and polar order coexistence in 4d/5d oxides and opens new avenues for
designing high-T multiferroics.

Introduction
Multiferroics, characterized by the coexistence of magnetic and electrically switchable polar
orders with cross-coupling capabilities, hold great promise for next-generation spintronic devices1-
3. Spin-orbit coupling (SOC) is certainly one of the key ‘glues’ that links magnetism and
polarization together4-7. For instance, it mediates the magnetoelectric coupling by translating
symmetry-breaking into charge dipoles or controlling magnetic textures under broken space
inversion symmetry6. Strong SOC is generally essential for enhancing the spin response to electric
stimuli in multiferroics8,9, thereby facilitating ultra-low power consumption, high speed, and high
density in spintronic devices10. However, the strength of SOC is proportional to the atomic number,
therefore it is naturally weak for 3d transition metals. Incorporating heavy elements, such as 4d or
5d transition metals, offers a promising route to enhance the effective SOC. Nevertheless,
experimental studies on the coexistence of magnetic and polar orders in transition metal oxides
containing these heavy elements have never been explored (Supplementary Table S1).

Double perovskite (DP) A BB'O oxides provide enhanced structural flexibility compared to
2 6
simple perovskites11-12. The presence of two distinct transition metal elements at the B/B' sites
offers additional degrees of freedom, creating an ideal platform for designing innovative
multiferroics and exploring the underlying multiferroic coupling behaviors13-15. Crucially, the spin
state associated with the B' site electronic configuration can be profoundly modified by SOC,
particularly when B' ions belong to the 4d or 5d series. Many members in this family exhibit high
magnetic ordering temperatures16-18. A notable example is Sr CrOsO , a ferrimagnetic (FiM)
2 6
insulator with ultra-high Curie temperature16 (T ~ 725 K). Achieving ferroelectricity in such high-
C
T DP materials would represent a significant technological breakthrough. Interestingly, recent
C
theoretical calculations19,20 predicted that the 3d-5d DP Zn FeOsO could be a room-temperature
2 6
(RT) multiferroic, exhibiting a large ferroelectric polarization and FiM ordering. In addition, this
compound is predicted to exhibit strong magnetoelectric coupling, primarily due to the enhanced
SOC effect of the 5d-Os element19. These findings highlight 3d-4d/5d DP oxides as a highly
promising class of materials for exploring new multiferroics.

In this regard, La CoRuO (LCRO) is particularly fascinating because its intrinsic high degree of
2 6
B-site cation ordering enables a pristine and well-organized Co-O-Ru interaction network, while
the 4d Ru ion simultaneously delivers strong SOC21,22. In the bulk form, LCRO crystallizes in a
monoclinic distorted perovskite structure (P2 /n; a = 5.572Å, b = 5.628 Å, c = 7.884 Å, and β =
1
89.99°), and exhibits an antiferromagnetic (AFM) insulating behavior21. It is well known that the
spin states of B/B'-site ions have crucial implications for DP materials' electrical and magnetic
ground states13,16,17,19,20. Cobalt oxides, in particular, have attracted significant attention because
the Co ion can accommodate various spin states and are highly responsive to external stimuli23-25.
It is worth noting that in LCRO, there exists a stable high-spin (HS) state of Co2+ (𝑡5 𝑒2, S = 3/2).
2𝑔 𝑔
Previous studies have demonstrated that HS Co2+ ions show a remarkable ability to regulate
magnetoelectric ground states26-28. For example, vacuum annealing of compressively strained
LaCoO epitaxial film can yield a LaCoO brownmillerite phase with zigzag-ordered oxygen
3 2.5
vacancies, exhibiting ferromagnetic insulating (FMI) behavior with T up to ~ 284 K due to the
C
exotic HS Co2+ state26. Additionally, the FMI state and anomalous Hall effect emerge in
La CoO epitaxial films supported by the HS Co²⁺ state27. Our previous work also demonstrated
2 4±x
that HS Co2+ ions resulting from oxygen vacancies play a significant role in stabilizing a robust
FMI state in tensile-strained LaCoO epitaxial films28.

Although the LCRO bulk has been extensively studied, its epitaxial film growth has not been
reported yet. Lattice distortions in strained films can strongly modify the orbital occupancy of
cations, the strength of superexchange coupling, and related interactions, often leading to
unexpected magnetoelectric properties28-31. Herein, we successfully realize the coexistence of an
RT FiM insulating state (T ~ 623 K) and switchable polar nanodomains in highly B-site cation
C
ordered LCRO epitaxial films. Combined scanning transmission electron microscopy (STEM)
investigations and density functional theory (DFT) calculations attribute this emergent coexistence
of magnetic and polar orders to the shrinkage of lattice volume and the gradient suppression of
oxygen octahedral rotations (OOR) with inhomogeneous evolution. Lattice volume reduction
enhances the magnetic interactions along the nearest-neighbor Co-O-Ru path while weakening the
next-nearest-neighbor coupling, leading to the emergence of the FiM insulating state.
Simultaneously, the origin of switchable short-range polar nanodomains is attributed to the
gradient rotations of BO octahedra, which trigger B-site cation displacement and promotes a
6
structural transition from the centrosymmetric non-polar P2 /n phase to the polar 4mm or P2
1 1
phase via the breaking of the sliding surface n. These structural and electronic features are further
corroborated by our first-principles calculations. This work provides the evidence of coexisting
RT magnetic and switchable polar orders in a 4d-oxide, which establishes a prerequisite for
achieving multiferroicity in 3d-4d/5d double perovskites, thereby providing a new platform for
discovering emerging multiferroic materials.

Results
Highly ordered B-site cations in strained LCRO epitaxial films
The DP LCRO films with thickness of approximate 30 nm grown on (111)-and (001)-oriented
SrTiO (STO) substrate were fabricated by pulsed laser deposition (PLD). Typical out-of-plane X-
3
ray diffraction (XRD) θ-2θ scans indicate the epitaxial growth and high phase purity of the films
(Fig. 1a and Supplementary Fig. 1a, b). The (111) reflection peaks were distinctly observed,
indicating the formation of the checkerboard type B-site ordered DP phase with Co and Ru ions
arranged alternately along the [111] direction. The rocking curve and oscillating peaks of X-ray
reflectivity (XRR) verify the high crystallinity and uniform chemical compositions of the films
(Supplementary Fig. 1c, d and Supplementary Fig. 2). Ex-situ atomic force microscopy (AFM)
images further confirm that all LCRO films have atomically smooth surface (Supplementary Fig.
1a, b). Additionally, to further examine the texture, the in-plane crystalline information of LCRO
films was analyzed by phi-scans of the (101) plane. The threefold symmetry for the (111)-oriented
film and fourfold symmetry for the (001)-oriented film reveal perfect in-plane orientation registry,
indicating a “cube-on-cube” growth mode and single-crystal properties (Supplementary Fig. 3a,
b). This observation was further verified by the high-angle annular dark-field (HAADF) STEM
images acquired along the [110] zone axis of the LCRO films (Fig. 1b and Supplementary Fig. 6a,
b). The HAADF-STEM images also show that the LCRO films exhibit high-quality epitaxy and
sharp interfaces with the substrates. Besides, to determine the lattice strain of the films, reciprocal
space mappings (RSMs) were measured and are displayed in Supplementary Fig. 4a, b. The RSMs
suggest that LCRO epitaxial films are coherently strained to the substrates with negligible strain
relaxation. Therefore, based on XRD θ-2θ and RSMs results, the structural parameters of the
LCRO films were determined, including lattice parameters (monoclinic symmetry) a, b, c,
mismatch strain (ε), and lattice volume (Supplementary Table S2). Compared to the LCRO bulk21,
the lattice volumes in these films are effectively reduced. To further confirm the strain exerted by
the substrate on the film, the geometric phase analysis31 (GPA) was performed based on the STEM
images, showing the strain distribution at the interface with nanometer-scale spatial resolution (Fig.
1b). The corresponding in-plane (IP, ε ) and out-of-plane (OP, ε ) strain distribution maps suggest
xx yy
that the IP compressive strain and OP tensile strain within the film are consistent. The mean square
root values of the strain are also consistent with those derived from RSMs.

The fascinating physical properties of DPE materials are closely related to the degree of B-site cation
ordering32-34. To investigate the degree of B-site cation ordering in the LCRO epitaxial films,
aberration-corrected HAADF-STEM28,31,33 imaging was employed. Figure 1c presents a projected
structure and HAADT-STEM image of LCRO along the [110] zone axis. In the projected structure,
Co and Ru ions alternate in columns along the [110] direction, and this pattern is repeated in every
other column along the [001] direction. The corresponding fast Fourier transform (FFT) pattern in
the inset reveals distinct satellite reflections (highlighted by the circles), which are indicative of
long-range B-site ordering due to the periodic alternation of Co and Ru at the octahedral sites.
Similar features were also observed in the (001)-oriented LCRO film, where the c-axis is oriented
perpendicular to the substrate surface (Supplementary Fig. 6). These results suggest that B-site
ordering is robust and independent of crystallographic orientation. To further validate the B-site
ordering, we conducted quantitative intensity analysis of the atomic columns in the HAADF-
STEM image. Each atomic column was fitted using a two-dimensional ellipse Gaussian function
via the CalAtom algorithm to accurately extract the column intensity. The resulting intensity map
is shown in Fig. 1d (Supplementary Fig. 5), corresponding to the region in Fig. 1c. It is evident
that A-site (La) columns exhibit uniform contrast, while the B-site columns display alternating
intensities in a checkerboard-like arrangement. Given that the HAADF-STEM is the Z (atomic
number)-contrast image, with atomic column intensity scaling approximately with Z1.3, the contrast
variation reflects the alternating presence of lighter Co (Z = 27) and heavier Ru (Z = 44) atoms at
the B-sites. Additionally, a line profile taken along the [110] direction at the marked position in
Fig. 1e demonstrates the periodic Co/Ru arrangement, further confirming a high degree of cation
ordering. Collectively, these results from atomic-resolution STEM imaging and intensity mapping
provide compelling evidence for long-range B-site ordering in both (111)- and (001)-oriented
LCRO epitaxial films.

Emergent high-temperature ferrimagnetic insulating state
The magnetic properties of LCRO epitaxial films were characterized by a combination of the
superconducting quantum interference device (SQUID) and the polarized neutron reflectometry
(PNR)35-37. Interestingly, we observed, within the temperature range of 5 ~ 400 K, the magnetic
field dependence of magnetization exhibits hysteretic and saturation behavior, indicating the
existence of robust magnetic ordering in the films (Fig. 2a, b and Supplementary Fig. 7a, b). At
300 K, both the (111)- and (001)-oriented films display a similar saturation magnetization (M )
of
s
approximately 0.16 𝜇 /f.u and a coercive field (H ) of about 200 Oe (Fig. 2c). Since the values of
𝐵 c
M and H are small as compared to typical ferromagnetic perovskite oxide thin films13,15,24,27,28
s c
and according to the previous report regarding the transformation of AFM to FiM in A CoRuO
2 6
bulk22, this magnetic ordering can be reasonably classified as a FiM state (FiM ordering can be
further corroborated later by DFT calculation). The M values extracted from magnetic hysteresis
s
(M-H) loops at different temperatures are summarized in Fig. 2d, showing a decrease with
increasing temperature. The temperature-dependent magnetization curves (M-T) indicate a FiM
transition temperature exceeding 400 K (Supplementary Fig. 8) with the high-temperature data
precisely determining the T of the (111)-oriented LCRO film to be approximately 623 K
C
(Supplementary Fig. 8b), which is well above room temperature. Additionally, an upturn in
magnetization is present at low temperatures. This feature could be attributed to ferromagnetic
clusters resulting from static anti-site disorder (Co-O-Co and Ru-O-Ru), consistent with
observations in related bulk A CoRuO double perovskites22.
2 6

We further conducted PNR measurements on the (111)-oriented LCRO film to quantitatively
determine the magnetization depth profile across the film thickness. Figure 3a displays the specular
neutron reflectivity for spin-up (R++) and spin-down (R--) polarized neutrons as a function of the
wave vector transfer q (=4πsin). The experimental setup and sample geometry are shown in
i/
the illustration35,36,37 ( is the incident angle, and  is the neutron wavelength). The spin
i
asymmetry (SA) [= (R++ - R--) / (R++ + R--)] and its corresponding fit were calculated from the PNR
data and the results from modeling, as shown in the upper right corner of Fig. 3a. A clearly weak
magnetization has been confirmed from the tiny differences between R++ and R−− and the
corresponding SA oscillation as a function of q36,37. The depth profiles of the nuclear and magnetic
scattering length densities (SLDs), which are related to the atomic density and magnetization in
the films, are plotted in Fig. 3b. Although the magnetization shows a slight difference at the
interface with a value of 0.14 (-0.07, +0.05) 𝜇 /f.u., it is essentially uniform across the film
𝐵
thickness with a value of 0.16 (-0.02, +0.02) 𝜇 /f.u. at RT, which is consistent with the result from
𝐵
the M-H loop. The slight weakening trend of interfacial magnetism is correlated with the B-O bond
length reduction due to shrinkage of unit cell volume indicated by enhanced nuclear SLD at the
interface (Fig. 3b and Supplementary Fig. 2c). This interfacial effect agrees well with a previous
literature38. Meanwhile, this further confirms that the FiM behavior stems from the LCRO epitaxial
films themselves. Through temperature-dependent resistivity (ρ) measurements, we also found that
LCRO films exhibit insulating behavior, with the resistivity monotonically decreasing upon
increasing temperature (Supplementary Fig. 9a). The plots of 𝑙𝑛ρ vs T-1/4 reveal that the charge
transport process can be described by VRH model in the entire temperature range (Supplementary
Fig. 9b).

According to the previous report22, we can infer that the emergence of FiM in the LCRO epitaxial
film is considered to be the result of the competition between the interactions along the exchange
paths of nearest-neighbor Co-O-Ru and next-nearest-neighbor Co-O-O-Co. Our subsequent DFT
calculations will provide further insight into the physical origin of this FiM state. It has been
reported that the magnetic ground state is strongly correlated with oxygen octahedral distortions
in the A CoRuO (A = La, Pr, Nd and Sm) and other rare earth-based DP materials22,34,39. A
2 6
transition from AFM to FiM has been observed in these DP bulk materials by introducing large
BO octahedral distortions, where lattice distortions were created by applying hydrostatic pressure
6
or A-site doping systematically22,39-42. Similarly, in the LCRO bulk, where octahedral distortions
are minimal, the next-nearest-neighbor interaction dominates, giving rise to strong AFM couplings
within the Co-sublattice and Ru-sublattice22,42. Whereas, in the LCRO epitaxial films, where the
strain effectively reduces the lattice volume (Supplementary Table S2), BO octahedral distortions
6
increase, and the Co-O-Ru AFM interaction becomes stronger with the simultaneous weakening
of next-nearest-neighbor coupling, and hence a FiM state emerges. This also matches well with
the case of Fe-Os and Co-Os based DP materials, where lattice distortion tunes the strength of the
magnetic interactions in different exchange pathways40,43.

Microstructure of LCRO epitaxial films revealed by STEM measurements
To probe the microstructural changes caused by strain, we employed integrated differential phase
contrast (iDPC) STEM imaging26, which enables the simultaneous visualization of both cation and
anion positions with atomic-level spatial precision. A representative cross-sectional iDPC-STEM
image across the LCRO/STO interface is shown in Fig. 3c, which reveals clear distinctions in
oxygen coordination environments between the substrate and film. In the STO substrate, oxygen
columns align in straight chains, consistent with the 180° Ti-O-Ti bond angle characteristic of its
Pm-3m symmetry28. In contrast, the LCRO epitaxial film exhibits an evident zigzag configuration
of Co-O-Ru chains accompanied by visibly distorted oxygen sublattices. This sharp contrast
signifies pronounced OOR within the LCRO film.

To quantify this OOR distortion, we fitted the atomic positions of A-site (La), B-site (Co/Ru), and
oxygen atoms in the iDPC-STEM image (Fig. 3d). Two key metrics were extracted: (1) the rotation
angle between the horizontal plane and the Co/RuO octahedral plane (α), and (2) the O-Co/Ru-O
6
bond angle (β). Atomically resolved maps of these parameters are shown in Fig. 3e and 3f,
respectively, with plane-averaged and layer-resolved profiles along the film growth direction
presented in Fig. 3g. It can be seen that the rotation angle α increases progressively with the
distance away from the interface, reaching values up to ~ 10°, but remains smaller than the bulk
LCRO value of 13.4°. Correspondingly, the O-Co/Ru-O bond angles β deviate significantly from
180°, confirming strong oxygen octahedral tilting beyond the interfacial region. In addition, abrupt
changes in α within the first few unit cells from the interface indicates a strong substrate-induced
clamping effect, arising from lattice and symmetry mismatch between the film and substrate
(Supplementary Fig. 10). This is similar to the phenomenon previously observed in La CoMnO
2 6
films grown on the (111)-oriented substrates33. With increasing distance away from the interface
increases, the rotation angle α gradually relaxes and becomes stable, reflecting the decay of the
clamping effect. In addition, we note that the oxygen octahedral rotations display a steep gradient
within the first 2-3 nm from the interface, followed by a more gradual gradient change farther from
the interface. According to previous reports, the increase of OOR promotes the strength of nearest-
neighbor Co-O-Ru interaction in A CoRuO DP materials22,39. In contrast, the STEM results above
2 6
suggest that the long-range suppression of OOR in the film is responsible for their small saturation
magnetic moment. The suppression of BO octahedral rotations at the interface also contributes to
6
the observed slight weakening trend of interfacial magnetization by reducing the strength of
nearest-neighbor Co-O-Ru interaction (Fig. 3b). Furthermore, we also noticed that the response of
the BO octahedral to strain exhibits inhomogeneous characteristics throughout the entire film
6
thickness (Fig. 3e-f). Overall, these findings indicate that epitaxial strain not only significantly
suppresses octahedral rotations but also induces the gradient, spatially non-uniform ORR
distribution.

Electronic structure investigation and DFT calculations of ferrimagnetic insulating state
In order to have a deeper understanding of the emergent FiM behavior, the electronic structure and
local chemical environment of the LCLRO thin films were further examined using X-ray absorption
spectroscopy (XAS) and scanning transmission electron microscopy-electron energy loss
spectroscopy (STEM-EELS). The O K-edge XAS spectrum (Fig. 4a) reveals excitation of O 1s
electron to O 2p states hybridized with adjacent unoccupied Co 3d states (Peaks A, B), Ru 4d states
(Peaks A, C), La 5d states (Peaks D), Co 4sp states (Peak E) and Ru 5sp states (Peak E) above the
Fermi energy (E)23,25,28,44. This reflects the complex hybridization environment and corroborates
f
the presence of mixed Co-O and Ru-O covalent bonding. Among them, the O K-edge XAS
spectrum of SrRuO epitaxial film was used to verify the overlap of peaks. Notably, in comparison
3
to previous reports23,44,45, peak B corresponds to Co2+. Meanwhile, the disappearance of the pre-
peak of the EELS spectrum on the O-K edge also supports this point24,26 (Supplementary Fig. 11b).
To further confirm the oxidation state and spin configuration of Co ions, Fig. 4b presents the Co
2p XAS spectra of the LCRO films, alongside reference spectra from standard Co samples with
known oxidation states28. We found that Co ions are divalent with the HS configuration (𝑡5 𝑒2, S
2𝑔 𝑔
= 3/2). At the microscale, spatially resolved Co L-edge EELS was analyzed in detail. By comparing
the L /L intensity ratio of Co and the L -L peak-to-peak energy separation, it was further
3 2 3 2
confirmed that the Co ions exhibit the characteristics of a divalent HS state28,46,47 (Fig. 4c and
Supplementary Fig. 11c). The Ru ions in the DP LCRO films are in the Ru4+ state (𝑡4 𝑒0),
2𝑔 𝑔
as
demonstrated by the Ru M-edge XAS spectrum (Supplementary Fig. 11a). Additionally, the La M-
edge EELS spectrum confirms the trivalent oxidation state of La ions, with no evidence of valence
fluctuation, supporting the overall charge neutrality of the LCRO system, in which the Co²⁺/Ru⁴⁺
configuration dominates (Supplementary Fig. 11d).

In-depth DFT+U calculations were further executed to elucidate the correlation between epitaxial
strain and ferrimagnetic insulating state. For bulk LCRO, the AFM ground state was obtained with
proper Uₑ (4 eV for Co’s 3d orbitals and 2 eV for Ru’s 4d orbitals), as shown in Supplemental
ff
Fig. 12. Each Co2+ ion possesses seven electrons in the 3d orbitals, forming a HS state with five
spin-up and two spin-down electrons in the 𝑡 orbitals (Fig. 5a). Each Ru4+ has four electrons in
2𝑔
the 4d orbitals, with three spin-down and one spin-up electron, all in the 𝑡 orbitals.
2𝑔
Consequently, Co2+ exhibits a large magnetic moment 2.66 𝜇 (close to its ideal limit 3 𝜇 ),
𝐵 𝐵
while Ru4+ has a small moment of 1.43 𝜇 (close to its ideal limit 2 𝜇 ). The DFT+U band gap
𝐵 𝐵
is 1.28 eV (Fig. 5d and Supplementary Fig. 12). In addition, this choice of U leads to lattice
eff
constants (a = 5.545 Å, b = 5.615 Å, c = 7.846 Å), which are closely match the experimental values
(a = 5.576 Å, b = 5.640 Å, c = 7.888 Å)21. To investigate the magnetic properties of strained LCRO
films, we systematically calculated the energies of various magnetic configurations across different
B-site cation ordering patterns under the experimental strain conditions (Fig. 5b and
Supplementary Fig. 13). Our calculations reveal that the G-type ordering pattern consistently
exhibits the lowest formation energy, confirming the checkerboard arrangement as the ground state
in agreement with experimental observations (Fig. 1c-e and Supplementary Fig. 6b-d). Moreover,
among all magnetic configurations examined, the G-type pattern with FiM ordering possesses the
lowest total energy. This FiM ground state remains stable not only under the specific experimental
strain corresponding to the in-plane lattice constraint (a = b = 5.52 Å in the P2
1
/n notation) but
also shows enhanced stability across a broad strain range (Fig. 5c). Furthermore, the calculated
structural parameters, including lattice volume shrinkage (from 245.18 Å³ to 241.73 Å³) and
reduced Co/Ru-O bond lengths, are in excellent agreement with experimental measurements
(Supplementary Table S2). The shrinkage of lattice volume under in-plane compressive strain
notably enhances the nearest-neighbor Co-O-Ru superexchange (J ) interaction, reinforcing its
1
antiferromagnetic character, while concurrently driving a sign reversal in the next-nearest-
neighbor coupling (J , J ) to favor ferromagnetic alignment22,34,39. This reconfiguration of
2 3
exchange couplings promotes parallel spin alignment within each of the Co2+ (S = 3/2) and Ru4+
(S = 1) sublattices, which remain antiferromagnetically locked via J . The resulting imbalance
1
between the Co2+ and Ru4+ sublattice moments yields the net ferrimagnetic moment of 1 𝜇 /f.u.
𝐵
observed in our strained films (Fig. 5a). This net moment 1 𝜇 /f.u. should be considered as the
𝐵
theoretical upper limit of ideally strained LCRO films, while the experimental value is lower due
to the microscale non-uniform strain or anti-site disorder between Co and Ru sites. Meanwhile,
the band gap is largely reduced to 0.55 eV (Fig. 5d), in qualitative agreement with the
semiconducting transport as observed in experiment (Supplementary Figure S9). These
conclusions are further confirmed by our additional benchmark calculations using the parameter-
free SCAN and r²SCAN meta-GGA functionals, which consistently reproduce the ferrimagnetic
ground state and insulating character under strain (Supplementary Fig.14 and Fig.15).

Investigation of an unexpected polar phase with nanodomains
The OOR may create polar structure due to strong lattice-charge coupling13,14,15. The macroscopic
polarization of the LCRO film was initially characterized by polarization-electric field (P-E) loop
measurements. A characteristic “leaf-shaped” P-E hysteresis observed at low temperature
(Supplementary Fig. 16b) could be consistent with a polar response. However, its non-ideal shape
likely results from polar nanodomain (or dipole glass state48) and significant persistent leakage, as
evidenced by the accompanying current-voltage characteristics (Supplementary Fig. 16a).
Therefore, this macroscopic measurement alone cannot serve as conclusive evidence for polar
switching. To overcome the limitations of macroscopic measurements and directly probe the polar
behavior at the nanoscale, we further performed detailed piezoresponse force microscopy (PFM)
investigations at room temperature14. We observed a clear butterfly loop and a corresponding phase
hysteresis loop (Supplementary Fig. 17), which confirms the switchable of the polarization.
Subsequently, our PFM domain switching experiments further successfully demonstrated
reversible and stable domain writing and erasing. Clear reversal of the PFM phase contrast upon
applying bias voltages of opposite polarities through the conductive tip confirms that the
spontaneous polarization can be reliably switched by an external electric field (Fig. 6a). To further
investigate the macroscopic polar order and crystal symmetry, we employed optical second-
harmonic generation (SHG) measurements36,49. As schematically illustrated in Fig. 6c, the
rotational anisotropy of the SHG response was characterized using a normal-incidence
configuration. Detailed SHG measurements were conducted across a temperature range of 300 to
500 K (Supplementary Fig. 18). The SHG polarimetry displayed a characteristic double-lobe
structure for measurements in both the horizontal (σ = 0°) and vertical (σ = 90°) polarization
directions, which are parallel and perpendicular to the [110] direction of the sample, respectively
(Fig. 6b and Supplementary Fig. 18). Furthermore, the intensity of SHG response exhibited
remarkable thermal stability, indicating that the proportion of polar domains does not decrease
with increasing temperature (Fig. 6c). Quantitative fitting of the SHG data further confirmed a
distinct domain configuration comprising mixed a₁/a₂ phases, oriented at approximately 45°
relative to the [110] zone axis (Supplementary note 1). The results of fitting indicate that the space
group structure of LCRO has changed from the non-polar centrosymmetric P2 /n in bulk form to
1
the polar 4mm or P2 phase in these epitaxial films through the breaking of the sliding surface n14.
1
The microscopic origin of polar nanodomains in LCRO films were further confirmed through
atomic-scale quantitative analysis using aberration-corrected STEM37. Figure 6d displays a
representative HAADF-STEM image taken from the interior of the LCRO film, where atomic
positions of La and Co/Ru are identified. To evaluate the local polarization, we quantified the
displacement of B-site cations (Co/Ru) relative to the centroid of their surrounding four nearest A-
site (La) cations (Fig. 6d and Supplementary Fig. 19). These displacements define the magnitude
and direction of the polarization vector at the unit-cell level. These calculated displacement vectors
are overlaid on the HAADF-STEM image, with arrow length and color corresponding to the
polarization and magnitude orientation, respectively. To correlated polarization with structural
distortions, we mapped the local c/a ratio, as shown in Fig. 6e, and statistically analyzed the
distribution of polarization angles in Fig. 6f. The polarization vector maps reveal a distribution of
nanoscale polar regions of varying sizes and orientations, consistent with nanoscale structural
variations, as evidenced in the c/a ratio mapping shown in Fig. 6e. Despite the apparent spatial
disorder, the polarization vectors within these polarized domains show a strong preferential
alignment along the [2-20] direction, as confirmed by the predominant angular distribution
between 0° and 15° in Fig. 6f. Within these nanoregions, distinct polar displacements of Co/Ru
atoms (δz ≈ 0.17 Å) are evident, as illustrated in Fig. 6g. Together, these results provide
𝐶𝑜/𝑅𝑢
direct atomic-scale evidence of local polar distortions and demonstrate the existence of nanoscale
polarization disorder within the LCRO films. The collective evidence from the atomically resolved
polar distortions and the macroscopic "leaf-shaped" P-E hysteresis loop consistently points to the
formation of a dipole glass or relaxor-like ferroelectric state in the LCRO films48, where nanoscale
polarization fragmentation suppresses long-range ferroelectric order while permitting local
switchability.

To further understand the origin of polar nanodomains, we performed the Berry phase calculations
to estimate the macroscopic polarization. However, for the ideally strained films, the net
polarization is zero. This is not surprise because its point space group of P2 /n is nonpolar. In other
1
words, a uniform strain can not lead to the polarity observed in experiment. In fact, accompanying
the microscale non-uniform strain, the gradient of OOR with inhomogeneous evolution can be
responsible for the polarity, which induces the displacements of Co/Ru ions with differing
magnitudes and directions (Fig. 6d). To confirm this assumption, we introduced the gradient of
OOR in our DFT structure, by slightly changing the rotation angles of half oxygen octahedra. Then
the output macroscopic polarization becomes nonzero. The Berry phase calculation of a model
structure with proper oxygen octahedral rotations gradient yields a locally polarization of
approximately (0.288, -0.257, 0.149) µC/cm ². The inhomogeneous distortive forces on each ions
are shown in Fig. 6h. This DFT model captures the essential polar mechanism whereby a spatially
varying pattern of oxygen octahedral rotations breaks inversion symmetry. The nanoscale domain
fragmentation observe experimentally arises from intrinsic structural complexities that extend
beyond the idealized model, including local strain variations and the distinct multi-mode
distortions of CoO and RuO octahedra (Fig. 6a-d). The system shows features analogous to a
6 6
hybrid improper ferroelectric, in which the polar state arises from structural distortions13-15.

In summary, by synergistically utilizing subtle strain engineering and B-site cation ordering, we
have achieved a room-temperature coexistence of FiM insulating state and polar nanodomains,
indicating ferroelectric ordering in DP LCRO epitaxial films. The B-site cation ordering in the
films is robust and independent of crystal orientation. Moreover, strain promotes significant lattice
distortions, including the shrinkage of lattice volume and long-range suppression of OOR.
Combining atomic-scale quantitative electron microscopy analysis with DFT calculations, we
unravel the origin of this FiM insulating behavior in which the cooperation of the shrinkage of
lattice volume and HS state of Co2+ contributes to the enhanced strength of the Co-O-Ru
superexchange interaction. The inconsistent response of BO₆ octahedra to strain leads to gradient
OOR with inhomogeneous evolution, driving the formation of switchable polar nanodomains with
a mixed a /a configuration. Insights from this study may point to effective strategies for designing
1 2
materials with coexisting magnetic and polar orders at high temperature. The material we
developed can serve as a starting point to explore a promising platform for multiferroics with
strong magnetoelectric coupling through further engineering.

Methods
Thin film preparation. High-quality La CoRuO (LCRO) thin films were epitaxially grown on
2 6
SrTiO (STO) and 0.5 wt% Nb-doped SrTiO (Nb-STO) substrates. Films were grown at a
3 3
temperature of 770 ℃ at an oxygen partial pressure of 150 mTorr, where the pulsed laser deposition
was carried out by a KrF excimer laser (248 nm) at a laser fluence of 1.47 J cm-2 and 2 Hz. After
growth, the films were cooled down to room temperature at a rate of 10 ℃/min under an oxygen
pressure of 100 Torr to avoid oxygen vacancies.

Structural characterizations. The crystal structures and surface morphologies of the films were
characterized by high-resolution X-ray diffraction (XRD) with Cu K radiation (λ = 1.5405 Å)
Sα
(Empyrean, PANalytical) and atomic force microscopy (AFM) (MFP-3D origin, Asylum Research
AFM), respectively.

Physical properties characterizations. Macroscopic magnetic measurements were performed
with an MPMS3 SQUID-VSM magnetometer (Quantum Design). All measurements were
performed by applying in-plane magnetic fields. The M-T curves were recorded during the sample
warm-up process after field-cooling at 1 kOe. Transport measurements were measured by a
Quantum Design Physical Property Measurement System (PPMS). Precision Premier II
Ferroelectric Tester (Radiant Technologies) was used to measure P-E hysteresis loops. Pt
electrodes were plated on the films for measurement. Piezoresponse force microscopy (PFM)
measurements were conducted by Asylum Research MFP-3D origin with an HVA 220 high-
voltage amplifier.

SHG measurements. The polarization-dependent SHG measurements were conducted by using a
custom laser-scanning microscope system43, with a Ti:sapphire mode-locking femtosecond laser
(MaiTai SP, Spectra-Physics) serving as the fundamental excitation source (1ω). The laser emitted
35 fs pulses at 800 nm center wavelength, with average power adjustable from 50 to 200 mW.
SHG signals were detected by a photomultiplier tube (PMT) equipped with an optical bandpass
filter to suppress residual 800 nm reflected light. For temperature-dependent SHG studies, an open
heating stage was employed, and as soon as the desired temperature was reached, the temperature
was kept for ~5 minutes before starting the measurement to ensure the accuracy and stability of
the sample temperature. During the entire measurement process, the focal spot position and sample
morphology were verified at each temperature points to ensure consistent incident power and
measurement location.

Scanning transmission electron microscopy. Scanning transmission electron microscopy
investigations, including HAADF-STEM imaging, iDPC-STEM imaging, and EELS
measurements, were performed using an aberration-corrected FEI Titan Cubed Themis G2
microscope operated at 300 kV. The instrument is equipped with an X-FEG electron source, a
monochromator, dual aberration correctors, a Gatan GIF Quantum ER energy filter with
DualEELS capability, and a Super-X EDXS detector system. For HAADF imaging, a collection
angle of 48-200 mrad was used. iDPC-STEM images were acquired using a four-quadrant
segmented DF4 detector under the same accelerating voltage. The EELS spectra were recorded
using DualEELS mode, which allows simultaneous acquisition of low-loss and core-loss spectra.
The zero-loss peak was used to calibrate the energy scale, and plural scattering effects were
corrected accordingly to ensure accurate interpretation of the electronic structure.

X-ray spectroscopic measurements. The X-ray absorption spectroscopy (XAS) measurements
are performed at the I06 beamline of the Diamond Light Source at room temperature in a total
electron yield mode. The spectra of Co L-edge, Ru M-edge and O K-edge are measured by
changing the incident angle of the linearly polarized x-ray beam.

Polarized Neutron Reflectivity Measurement. Polarized neutron reflectivity (PNR)
measurements on a La CoRuO epitaxial film were conducted on the Multipurpose Reflectometer
2 6
(MR) beamline at the Chinese Spallation Neutron Source (CSNS). The sample with a thickness of
about 30-nm La CoRuO thin film was grown on STO (111) substrate with a size of 10 mm by 5
2 6
mm by 0.5 mm. The room temperature neutron reflectivities were measured under an in-plane
magnetic field of 1 T as a function of the wave vector transfer along the film surface normal. R++
and R-- represent the reflectivities from the spin-up and spin-down polarized neutrons, respectively.
PNR data were fitted using GenX software.

Density functional theory calculations. The first-principles calculations are based on density
functional theory (DFT) within the generalized gradient approximation (GGA) utilizing the revised
Perdew-Becke-Erzenhof functional for solids (PBEsol) implemented in the Vienna ab initio
Simulation Package50-53 (VASP). The Co’s 3p63d74s2 and Ru’s 4p64d75s1 were treated as valence
electrons. To describe the correlated 3d electrons of Co and 4d electrons of Ru, the GGA+U
method was adopted54. The U value was tested from 3 to 5 eV for Co and from 2 to 4 eV for Ru
eff
in bulk LCRO, as summarized in Supplementary Fig. 14. Then, U = 4 eV for Co’s 3d orbitals
eff
and U = 2 eV for Ru’s 4d orbitals are adopted, respectively. To initialize the meta-GGA
eff
calculations, wavefunctions and charge densities from pre-converged GGA+U calculations were
employed as a starting point to mitigate convergence difficulties55-56. A cutoff energy of 500 eV is
used for the plane-wave bases, and a Γ-centered 5×5×1 (5×5×5 for LCRO bulk) k-point mesh for
the Brillouin zone integration is adopted for the strained LCRO thin films. A convergence threshold
of 10−6 eV was used for the electronic self-consistency loop. The lattice constants and internal
atomic coordinates were relaxed until the Hellman-Feynman force on each atom is less than 0.005
eV/Å. The ferroelectric polarization was calculated by using the Berry phase method57. And the
amplitudes of the octahedral rotation modes were constrained to specific values, and the positions
of all other ions (La, Co, Ru) were relaxed during these calculations.

Data availability. All the experimental/calculation data that support the findings of this study are
available from the corresponding authors upon reasonable request.

Code availability. All the codes used for this study are available from the corresponding authors
upon reasonable request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The study of Li et al promises room temperature multiferroicity in strained La2CoRuO6, opening routes to low power\nspintronic applications. The conclusions are drawn on the basis of advanced experimental techniques as well as density\nfunctional theory calculations. The paper is well written, even though the DFT appears rather disconnected from the\nexperimental results. It could be added here and there to support experimental results rather than just a final section at the\nend of the manuscript.\nAlthough the paper reads quite well and the presented results are sound, I feel a bit frustrated at the end of the manuscript. It\ndoes not deliver what is promised in the title: although the compound is certainly polar, the are no evidence of ferroelectricity.\nShowing polar displacements is not sufficient to claim it is ferroelectric (see Nature 533, 68 2016 for a similar study but with\npolar metals). In the present manuscript, there are no FE loops (polarization reversal), just hints of polar domains, and DFT\ncalculations do not provide any quantification of the spontaneous polarization – we only know that Berry phase calculations\nhave been performed! Furthermore, polar displacements are induced by the symmetry breaking induced by slightly different\noctahedra rotations amplitudes between consecutive planes along the growth direction. Nevertheless, as shown on figure\n3g, the variation of angles between consecutive planes does not seem to be very large, so I expect that the B cation off\ncentring to be small, and so might be the spontaneous polarization.\nUnless authors can provide significant improvement regarding the FE properties of their compound, I cannot recommend\npublication of the present manuscript.\nI append below other remarks and suggestions to be considered prior to a resubmission at Nature Communications or\nelsewhere:\n- I do not understand how the DFT calculations are performed with the varying octahedral rotation amplitudes presented in\nFig. 6.d. It seems that different rotations amplitudes are frozen, but are other atoms allowed to relax in position? If not, the\nonly polarization that can emerge is an electronic polarization. The method is not discussed in the manuscript even though\nthis is a key element of the paper.\n- Since the B cation displacements might be very small, I expect the polarization to be rather low. Could the authors use DFT\nand Born effective charges to estimate locally the polarization in their supercell?\n- Furthermore, the polarization being induced by different rotation amplitude between consecutive planes, I expect that the\nreversal of the polarization to go through the reversal of tilt patterns. Rotations are known to be associated with very large\nenergy gains, one to two orders of magnitude larger than polarization in oxides. The reversal of the polarization therefore\nappears to be unlikely. Could the authors check this possibility with their DFT by computing the energy barrier for instance?\n- DFT results are highly sensitive to the choice of exchange-correlation parameters. Nowadays, parameter free calculations\nwith meta-GGA are able to capture the correct trend in terms of physical properties of transition metal oxide compounds.\n(PRB 111, 144406 2025; PRB 102 145112 2020, PRB 100 035119 2019 for instance). Could the authors benchmark their\nresults with parameter-free calculations?\n- Does DFT predict a checkerboard pattern for the ground state of the compound?\n- The gap appears to be rather small (0.55 eV from DFT). Can the author rule out any leakage current preventing the\napplication of an E-field to reverse the polarization?", "reviewer_2": "In recent years, theoretical and experimental investigations have aimed to achieve a multiferroic state in oxide materials.\nHowever, the lack of materials with significant ferroelectric polarisation, magnetic moment and high critical temperatures\ncurrently prevents the practical application of this state. In their study, the authors investigate La₂CoRuO₆ thin films for this\npurpose. The double perovskite (DP) structure can accommodate two magnetic cations, which order in the structure due to\ntheir different ionic sizes. A combination of exchange interactions and epitaxial strain renders the system ferrimagnetic, while\nlocal fluctuations in the rotation angle of the oxygen octahedra give rise to an electric dipole and polar nano-domains. These\nclaims stem from the authors' analysis of a variety of experimental and theoretical data.\nCurrently, many of these claims are overstated.\nThe term 'multiferroism', used in the title, indicates the coexistence of two hysteresis properties; here, these should be\nmagnetism and ferroelectricity. Currently, there is no indication of a ferroelectric state.\nAb initio calculations performed by the authors indicate the absence of a polar state in coherently strained structures.\nThe experimental data shown in Figure 3g demonstrate a progressive increase in the oxygen octahedron rotation (OOR) angle,\nfrom 0° in the cubic STO substrate to the bulk value of LCRO. This behaviour is characteristic of an epitaxial thin film, in\nwhich the OORs are coupled across the film/substrate interface, and has been extensively discussed in the literature.\nThe authors achieve a polar state in their calculations by introducing this progressive increase in the OOR angle (Figure 6d).\nHowever, given the direction of the gradient in the OOR angle, the polarisation has a clearly defined crystallographic\norientation. In contrast, Figure 4c shows that the polar axis orientation in their films aligns with different crystallographic\ndirections and is therefore unrelated to the previously discussed mechanism. To explain this discrepancy, the authors\nintroduce 'a gradient suppression of oxygen octahedral rotation (OOR) with inhomogeneous evolution'. Since OORs are\ncoupled via the sharing of oxygen anions between adjacent octahedra, it is unclear how the system can accommodate such\nlocal distortions on a very short length scale in the absence of a driving mechanism.\nThe data shown in Figures 3e and 3f reveal deviations in angles measured over a column of many atoms. While these\nangles appear random in the [110] view, they should also be random in the perpendicular [-110] direction.\nIt should also be noted that if the polarization is coupled to local OOR fluctuations, it may be difficult to switch the\npolarization orientation.\nIn any case, these polar nano-domains do not constitute evidence of a ferroelectric state.\nRegarding ferrimagnetism, the temperature dependence of the magnetisation shown in Figures 2d and S8 does not\nresemble the characteristic dependence observed in other ferromagnets or ferrimagnets (see, for instance, Ref. 39). The\nauthors do not explain the origin of this difference.\nPresentation:\nThe text should be revised by a colleague who is proficient in English.\nFor instance:\n'a ferrimagnetic insulating state is achieved above 400 K': what is the physical state of the system below 400 K?\n'A gradient BO6 octahedral rotation with inhomogeneous evolution triggers': what does 'gradient' mean here? 'a' is singular,\n'rotations' is plural and 'triggers' is singular.\nWhat lattice symmetry is used to denote the directions of the LCRO films? Pseudocubic? This is not mentioned, which leads\nthe reader to make errors. Figure 1a: the (001) peak of LCRO cannot be close to the (001) peak of STO because their lattice\nparameters are very different (see Table S2). The notation also appears to be inconsistent: the 111 peak of a pseudocubic\nperovskite sits at ~40°. The RSMs in Figure S4 also suffer from this problem. Do the zone axes in the STEM images follow\nthe layer orientation shown in Figure S6e or are they in pseudocubic notation?\nFigure 1d: What do the numbers 580 and 1200 refer to in the legend of this plot?\nWhat unit do the authors use for magnetisation, and what does f.c. stand for?\nFigure S1a: the labels of the peaks are incorrect.\nIn Figure S3, the text in the figure is confusing. The top panel in (a) displays the (101) reflections of STO. Why is there a\nlabel for LCRO(111)? The sample is LCRO/(111)STO.\nAnalysis of the resistivity data: 'Linear fitting curves using the three-dimensional Mott-VRH model show good agreement with\nthe measured resistivity data': the linear fitting is done by plotting the log of the resistivity against the inverse of the\ntemperature. The actual phrasing is incorrect.", "reviewer_3": "The present MS reports an experimental and theoretical study of the onset of ferrimagnetism and local polar order in\nepitaxial thin films of the double perovskite La2CoRuO6 grown on [001]- and [111]-oriented Nb-doped SrTiO3 substrates.\nThe authors find that the in-plane compressive strain leads to enhanced lattice distortions that combine with the high spin\nstate of Co2+ and an ordered B-site cation arrangement that act to strengthen the Co-O-Ru antiferromagnetic exchange\ninteraction and to induce a ferrimagnetic state, while the presence of a gradient in the BO6 octahedral rotations is suggested\nto drive the onset of polar nanodomains. Overall, I find the experimental work to be well carried out, in particular, by\nproviding an extended characterization of the system with techniques ranging from x-ray diffraction and absorption\nspectroscopy, neutron reflectometry and TEM, supplemented by first principles calculations. As I see it, the key result is the\npresentation of a new approach to controlling the exchange coupling in the double perovskites by tuning the B-cation\ncomposition and atomic ordering. However, the main result claimed in the paper, the synthesis of room-temperature\nmultiferroicity in strained La2CoRuO6 films is not fully demonstrated, since the presence of long-range ferroelectric order,\nferroelectric switching and a coupling between magnetic and ferroelectric order parameters are not established.\nI have in addition a number of smaller points of criticism that the authors could consider:\n- The strain distribution maps in Fig. 1b show an abrupt change along the [111] direction and non-uniform variations along\nthe perpendicular direction; the text does not make clear what is the reference point used to plot the data, presumably the\nSrTiO3 lattice parameters, this needs to be mentioned explicitly. The authors should also indicate to which portion of the\nTEM image the strain field corresponds to. I find it interesting that the authors obtain a similar strain state for the La2CoRuO6\nfilms grown along the [001] and [111] directions (Table S2 and Fig. S4). From symmetry, I would expect different film\ndistortions, tetragonal for the (001) films and trigonal for the (111). How do the authors explain that the (111) films exhibit a\ntetragonal symmetry and how is this seen in the TEM images?\n- The values mentioned for the saturation magnetization are in fact the magnetization at the highest field applied. It is clear\nfrom the M-H curves that the magnetization is not saturated. Presumably, the diamagnetic contribution from the substrate\nwas removed and the question arises as to how the high field slopes shown in Fig.2 were determined (it seems surprisingly\nidentical for the various temperatures).\n- The PNR data suggest a reduced magnetization at the SrTiO3 interface, which would not be surprising (somehow,\ncorrelated with an increase in the nuclear density, which is more surprising). The small magnetic moment corresponds to a\nsmall magnetic density and to a very small difference between the spin up and down reflectivities and to a large scatter in\nthe data points at high momentum transfer. The question here is what the error bars associated with the magnetic moment\nvalues from PNR and how correlated the interfacial moment is with the nuclear density. I suspect that these parameters are\ndetermined mostly from the data at high momentum transfer and may have very larger error bars associated with them.\n- In page 12, the authors discuss the presence of ferrimagnetism as arising from an increase in the Ru-O-Co exchange\ninteractions, but it is not clear at all what is the link between the exchange coupling and the imbalance between the cation\nmagnetic moments.\n- Concerning the oxygen octahedral rotations, the authors speak of a gradient along the out of plane and in the plane\ndirections, based on the results presented in Fig.3e-g; in fact, the rotation and bond angle seem to vary in a random manner.\nIt is difficult to see that the average of the values shown in Fig.3e gives Fig.3g. Looking at the Fig.3g, I would be tempted to\nconclude that rotation relation occurs within 5 atomic planes. The variations at outer planes are probably within the scatter\nerror (which should be provided).\n- The authors use SHG to determine the presence of polar order in the system. In fact, the authors state that the SHG signal\ngives the IP spontaneous polarization. My understanding is that SHG cannot directly measure spontaneous polarization, it\nmay simply prove the presence of a break of inversion symmetry in the system. From the TEM data, the authors identify\nnanodomains where polar distortions are present. However, the criterion for ferroelectricity is more stringent, and requires\nnot only that a material is polar, but also that the polarization is switchable by an applied electric field.\n- I find the presentation of the DFT results not satisfactory. For example, the authors do not explain what is the difference\nbetween the AFM and FiM configurations in Fig.S14 or how the FiM state is implemented in the calculations. The\nintroduction of a gradient in the octahedral rotation seems to result in a polar state, however, this situation seems to be far\nfrom the experimental case, where the rotations appear random and where the size of the polar nanodomains do not quite\nmatch with the scale of the gradient suggested by Fig.3g (in fact, I would argue there is no clear gradient in the data shown\nhere).\nIn conclusion, I find that the present work is still in a preliminary phase as far as demonstrating multiferroic behavior in the\nsystem studied here."}, "author_response": "We appreciate the reviewers’ time and effort for reviewing our manuscript. The\nreviewers’ comments and suggestions were very constructive and helped us to improve\nour manuscript. The following is point-to-point response to their comments. We have\nproperly addressed all the questions from the reviewers and made all necessary changes\nin the revised manuscript. The corresponding changes have also been highlighted by\nyellow in the revised manuscript (main text and Supplementary Materials).\nReviewer #1\n1. The study of Li et al promises room temperature multiferroicity in strained\nLa CoRuO , opening routes to low power spintronic applications. The conclusions are\n2 6\ndrawn on the basis of advanced experimental techniques as well as density functional\ntheory calculations. The paper is well written, even though the DFT appears rather\ndisconnected from the experimental results. It could be added here and there to support\nexperimental results rather than just a final section at the end of the manuscript.\nResponse: We sincerely thank the reviewer for their thoughtful comments and\nconstructive suggestions, which have been invaluable in improving the quality of our\nmanuscript. We are particularly grateful for the positive assessment of our work as\n“well written” and the recognition of the soundness of our results. We fully agree that\na tighter integration of the DFT results with the experimental findings would strengthen\nthe manuscript. Following the reviewer’s suggestions, we have carefully refined the\nstructure of manuscript, as shown below.\nFig. R1 (update as Fig. 5) DFT calculation of ferrimagnetic insulating state in\n(111)-oriented LCRO epitaxial films (a) Schematic diagram of strain-induced\n1\nmagnetic transition and interactions. J is interaction between the nearest-neighbor Co-\n2 3\nRelative energies of different magnetic orders for three B-site cation ordering patterns,\nwith the FiM state of the G-type cation ordering pattern set as the energy reference. (c)\nRelative energy versus biaxial strain. The reference energy corresponds to the AFM\nstate. The horizontal axis denotes the lattice constant in the original P2 /n notation,\n1\nwhich is along the [110] direction of pseudocubic lattice. (d) Density of states of LCRO\nbulk and experimental strained film.\nFirstly, in the revised manuscript, we have systematically reorganized the theoretical\ncontent that was originally confined to the final Discussion section. Specifically, the\nDFT calculations on magnetism, together with the newly supplemented results on the\nDFT predicted checkerboard pattern for the ground state (Fig. R1b), have been\nconsolidated into Fig. R1 (now presented as Figure 5 in the revised manuscript). This\nreconstructed figure is now positioned immediately following the experimental results\non the ferrimagnetic insulating state. This rearrangement allows the theoretical insight\nto directly accompany and elucidate the experimental findings, providing a coherent\nand timely discussion on the emergence of the ferrimagnetic insulator in B-site highly\nordered La CoRuO films and its underlying mechanism.\n2 6\nSecondly, we have consolidated the original SHG and STEM data on polar\nnanodomains, the newly supplemented PFM domain switching results, and the Berry\nphase calculations into a new Fig. R2 (now presented as Figure 6 in the revised\nmanuscript), which now comprehensively presents the evidence and theoretical basis\nfor the polar phase.\nFig. R2 (update as Fig. 6) Observation of polar phase with nanodomains. (a) The\nschematic diagram of the PFM experimental measurements. The PFM amplitude and\nphase images (5×5 μm) of the (111)-oriented LCRO thin film indicate that the\npolarization is switchable. (b) Polar plots of the input polarization angle-dependent\npoints. (c) Measured SHG intensity as a function of temperature for different\npolarization states. The illustration is the schematic configuration of the optical SHG\nexperiment. Here, E , the electric field intensities at the fundamental frequency ω. The\nω\n1ω fundamental light and 2ω SHG signal are colored in red and purple, respectively.\n(d) HAADF STEM image of (111)-oriented LCRO film, overlaid with the polar\ndisplacement vector. Regions with the same displacement vectors are circled as\nnanodomains (bright yellow dashed lines). (e) Corresponding maps of c/a ratio. (f)\nHistogram of polarization orientations projected along the [2-20] direction. (g)\nMagnified image of selected areas from (d) and the corresponding structural model\nschematic. The displacement of Co/Ru cations are quantified by the centroid of their\nsurrounding four La cations (green circles) (h) The structural model of LCRO bulk and\n(111)-oriented film, under the dual effects of lattice strain and the suppression of\noxygen octahedral rotations, with the rotational gradient changes in the film inducing\nthe displacement of B-site (Co/Ru) ions (red arrow represents the direction and\nmagnitude of the displacement in each unite cell) and forming a polar phase.\nThe above structural adjustments ensure that the theoretical calculations are logically\nand visually integrated with the corresponding experimental evidences. This\nreorganization significantly improves the narrative flow and strengthens the connection\nbetween theory and experiment throughout the manuscript. We thank the reviewer\nagain for the valuable suggestions on improving the overall clarity and readability of\nour work.\n2. Although the paper reads quite well and the presented results are sound, I feel a bit\nfrustrated at the end of the manuscript. It does not deliver what is promised in the title:\nalthough the compound is certainly polar, the are no evidence of ferroelectricity.\nShowing polar displacements is not sufficient to claim it is ferroelectric (see Nature\n533, 68 2016 for a similar study but with polar metals). In the present manuscript, there\nare no FE loops (polarization reversal), just hints of polar domains, and DFT\ncalculations do not provide any quantification of the spontaneous polarization – we\nonly know that Berry phase calculations have been performed!\nResponse: We thank the reviewer for this critical comment and fully agree that\ndemonstrating switchable polarization is essential to claim ferroelectricity. The polar\nnature alone is indeed insufficient, as shown in the reference (Nature 533, 68 (2016))\nmentioned by the reviewer. Therefore, we have made our utmost efforts to strengthen\nthis core claim through a combination of additional experiments and theoretical analysis.\n(b) ferroelectric hysteresis loops of LCRO (purple) thin film grown on (111)-oriented\nSTO (blue) substrate measured at 77 K. The top electrode is platinum with a thickness\nof 20 nm.\nFirstly, we conducted direct polarization-electric field (P-E) measurements to seek\nmacroscopic evidence of ferroelectric switching. At room temperature, the narrow\nbandgap (~ 0.55 eV) of LCRO films leads to significant leakage currents. Although\ncooling to 77 K (Fig. R3a) did partially suppress the leakage current, a well-saturated\nsquare hysteresis loop can still not be obtained. Instead, we observed a\ncharacteristic “leaf-shaped” P-E loop (Fig. R3b). We attribute this behavior to the\nfollowing combined factors: (1) Persistent leakage current remains even at low\ntemperature (i.e., 77 K); (2) The polarization is strongly coupled to oxygen octahedral\nrotation (OOR) fluctuations, making its switching inherently difficult; (3) As revealed\nby the STEM results, polar domains are highly fragmented into nanoscale regions,\nforming a relaxor-ferroelectric-like state that suppresses coordinated long-range\nswitching.\nThe observed “leaf-shaped” hysteresis is consistent with a dipole glass state, a\nphenomenon reported in a recent study of SrTiO homoepitaxial films, where a similar\n3\ndipole glass state was induced by local symmetry manipulation (Phys. Rev. Lett. 131,\n246801 (2023)). While not constitutive proof on its own, such a characteristic loop\nshape has been reported in other emergent ferroelectric systems facing similar\nchallenges (Nat. Commun. 10, 1409 (2019); Nat. Commun. 16, 6417 (2025)). We\ntherefore present these data as part of a cumulative body of evidence, while\nacknowledging the limitations of the macroscopic measurement.\nSecondly, considering the challenges of macroscopic measurement, we turn to\npiezoelectric force microscopy (PFM) measurements, providing direct nanoscale\nevidence to further illustrate the switching of polarization. As shown in Fig. R4, our\nPFM domain switching experiments successfully demonstrate reversible and stable\ndomain writing and reading. The clear reversal of the PFM phase contrast upon\napplying bias pulses of opposite polarities is a definitive proof of ferroelectricity at the\n4\nmicro-scale. This is a widely accepted methodology when P-E loops are inconclusive,\nas seen in similar works like room-temperature ferroelectricity in Bi₂O₂Se thin\nfilm (Nano Letters, DOI: 10.1021/acs.nanolett.5c04402) and ferroelectricity in four-\nunit-cell freestanding BiFeO\n3\nfilm (Nature, 570, 87-90 (2019)).\nFig. R4 (added as Fig. 6a) (a) The schematic diagram of the PFM experimental\nmeasurements. The PFM amplitude and phase images (5×5 μm) of the (111)-oriented\nLCRO thin film indicate that the polarization is switchable.\nFinally, we employed the Berry phase method to analysis the ferroelectricity. The\npolarization intensity calculated by the Berry phase method is approximately (0.288, -\n0.257, 0.149) µC/cm2. Noting that this quantitative value depends on the preset\ndistortion amplitude. Even though, its order of magnitudes is consistent with\nexperimental observation. The ionic contribution, characterized by the point charge\nmodel, reaches (0.079, -0.487, 0.032) µC/cm2, which is in the same order of magnitude\nwith the total polarization\nIn conclusion, we once again sincerely thank the reviewer’s constructive comments,\nwhich guided us to improve our manuscript. Although macroscopic ferroelectric\nhysteresis is not ideal, we have provided comprehensive evidence, including PFM\ndomain switching, SHG, and STEM structural analyses, to strongly demonstrate the\ncoexistence of switchable polar domains and strong ferrimagnetism at room\ntemperature.\nTo more accurately present our findings and avoid overstatement, we have revised the\nmanuscript’s title to: “Room-temperature ferrimagnetism and polar phase in highly\nstrained La CoRuO epitaxial films through engineered 3d-4d cation ordering”.\n2 6\nCorresponding discussions on ferroelectricity and multiferroic have also been revised\nthroughout the text.\nadded Fig. R3 as Supplementary Fig. 16 and Fig. R4 as Fig. 6a, added and revised\nthe sentence of the main text as following:\nAdded sentence:\n(1). (See Page 20, Line 429-441) “The macroscopic polarization of LCRO film was\ncharacterized by polarization-electric field (P-E) loop measurements, where a\ncharacteristic “leaf-shaped” P-E hysteresis at low temperature suggests a dipole glass\nstate48 (Supplementary Fig.16b). This non-ideal loop shape primarily results from the\nleakage current that remains even at low temperature (Supplementary Fig.16a). To\novercome the limitations of macroscopic measurements and directly probe the polar\nbehavior at the nanoscale, we further performed detailed piezoresponse force\nmicroscopy (PFM) investigations at room temperature14. Our PFM domain switching\nexperiments successfully demonstrated reversible and stable domain writing and\nerasing. Clear reversal of the PFM phase contrast upon applying bias voltages of\nopposite polarities through the conductive tip confirms that the spontaneous\npolarization can be reliably switched by an external electric field (Fig. 6a).”\nAdded reference：[48] Li, T., Deng, S. H., Qi, H. et al. High-temperature ferroic glassy\nstates in SrTiO -based thin films. Phys. Rev. Lett. 131(24), 246801 (2023).\n3\n(2). (See Page 22, Line 478-482) “The collective evidence from the atomically resolved\npolar distortions and the macroscopic “leaf-shaped” P-E hysteresis loop consistently\npoints to the formation of a dipole glass or relaxor-like ferroelectric state in the LCRO\nfilms48, where nanoscale polarization fragmentation suppresses long-range ferroelectric\norder while permitting local switchability.”\n(3). (See Page 22, Line 492-494) “The Berry phase calculation of a model structure\nwith proper oxygen octahedral rotations gradient yields a locally polarization of\napproximately (0.288, -0.257, 0.149) µC/cm².”\n(4). (See Methods, Page 25, Line 561-565) Precision Premier II Ferroelectric Tester\n(Radiant Technologies) was used to measure P-E hysteresis loops. Pt electrodes were\nplated on the films for the measurements. Piezoresponse force microscopy (PFM)\nmeasurements were conducted by Asylum Research MFP-3D origin with an HVA 220\nhigh-voltage amplifier.\nRevised sentence:\n(1) Original title: “Room-temperature multiferroism in highly strained La CoRuO\n2 6\nepitaxial films through engineered 3d-4d cation ordering”.\nRevised title: “Room-temperature ferrimagnetism and polar phase in highly strained\n2 6\n(2) Original text: “Multiferroics with Curie temperatures (T ) exceeding room-\nC\ntemperature are ideal platforms for realizing the next-generation spintronic devices with\ncoupled degrees of freedom. Among various materials candidates, 3d transition metal\noxides (TMOs) exhibit great potential for multifunctional integration, however, most\nof their T are still unsatisfactory to date, and multiferroics incorporating 4d/5d\nC\nelements with strong spin-orbit coupling remain to be largely unexplored.”\nRevised text: (See Abstract, Page 2, Line 46-52) “Materials exhibiting room-\ntemperature magnetic ordering and switchable polarization are highly desirable for\ndeveloping next-generation spintronic devices with coupled degrees of freedom. While\n3d transition metal oxides (TMOs) exhibit great potential for multifunctional\nintegration, however, most of their T are still unsatisfactory to date, and the\nC\nexperimental realization of magnetic and polar order coexistence has remained\nunexplored in 4d/5d oxides, where strong spin-orbit coupling could enhance\nmagnetoelectric interactions.”\n(3) Original text: “Our work fills in the gap in experimental studies of 4d/5d TMOs-\nbased multiferroics and opens up new avenues for designing and exploring high-T\nC\nmultiferroics.”\nRevised text: (See Abstract, Page 2, Line 63-65) “Our work fills the experimental gap\nin realizing coexistence of magnetic order and switchable polarization in 4d/5d oxides\nand opens new avenues for designing high-T multiferroics.”\nC\n(4) Original text: “Multiferroics, which enables cross-coupling between magnetic,\nelectric and other orders, are promising for next-generation spintronic devices1-3.”\nRevised text: (See Introduction, Page 3, Line 69-71) “Multiferroics, characterized by\nthe coexistence of magnetic and electrically switchable polar orders with cross-\ncoupling capabilities, hold great promise for next-generation spintronic devices1-3.”\n(5) Original text: “Nevertheless, multiferroics containing these heavy elements have\nbeen rarely explored (Supplementary table S1).”\nRevised text: (See Introduction, Page 3, Line 80-82) “Nevertheless, experimental\nstudies on the coexistence of magnetic and polar orders in transition metal oxides\ncontaining these heavy elements have never been explored (Supplementary Table S1).”\n(6) Original text: “This work provides the first experimental demonstration of\nmultiferroicity in the 4d/5d oxides and offers a new strategy for exploring novel\nmultiferroics in 3d-4d/5d DP oxides.”\nevidence of coexisting room-temperature magnetic and switchable polar orders in a 4d-\noxide, which establishes a prerequisite for achieving multiferroicity in 3d-4d/5d double\nperovskites, thereby providing a new platform for discovering emerging multiferroic\nmaterials.”\n(7) Original text: “Insights from this study may point to effective strategies for\ndesigning high-Tc multiferroic materials. The material we developed can serve as an\nexcellent starting point to explore a promising platform for strong magnetoelectric\ncoupling through further engineering.”\nRevised text: (See Conclusion, Page 24, Line 532-536) “Insights from this study may\npoint to effective strategies for designing materials with coexisting magnetic and polar\norders at high temperature. The material we developed can serve as an excellent starting\npoint to explore a promising platform for multiferroics with strong magnetoelectric\ncoupling through further engineering.”\n3. Furthermore, polar displacements are induced by the symmetry breaking induced by\nslightly different octahedra rotations amplitudes between consecutive planes along the\ngrowth direction. Nevertheless, as shown on figure 3g, the variation of angles between\nconsecutive planes does not seem to be very large, so I expect that the B cation off\ncentering to be small, and so might be the spontaneous polarization.\nResponse: We thank the reviewer for this insightful comment. We fully agree with the\nreviewer’s intuition regarding the small magnitude of the polarization, which aligns\nconsistently with our experimental observations and theoretical understanding. As\nderived from the P-E measurements, the maximum polarization ( ) is indeed very\nmax\nsmall, approximately 0.6 µC/cm2 (Fig. R3b), rendering its macroscopic detection\n𝑃\nparticularly challenging.\nFurthermore, we compared the polarization values of some unconventional\nferroelectrics reported so far, such as o-RMnO films (P < 0.5 μC/cm2, < 50 K, Nat.\n3 r\nCommun. 11, 2207 (2020)), and multiferroics γ-BaFe O (P ~ 0.2 µC/cm2, close to\n2 4 r\nroom-temperature, Nat. Commun. 13, 7968 (2022)), as well as\nLa NiMnO /La CoMnO superlattice films (P ~ 0.13 µC/cm² at 300 K, Nat. Commun.\n2 6 2 6 r\n15, 5549 (2024)). Therefore, we hypothesize that the observed small polarization is\nattributable to this polar order originates from a hybrid improper mechanism. This\nmechanism is actively discussed in recent literature for similar systems, as reported in\nLa NiMnO /La CoMnO superlattice films (Nat. Commun. 15, 5549 (2024); Phys. Rev.\n2 6 2 6\nLett. 135, 046801 (2025); Small Methods 7, 2201246 (2023)). Under epitaxial strain,\nvarious oxygen octahedrons (CoO and RuO ) will develop different degrees of rotation\n6 6\nand Jahn-Teller distortion due to the different chemical properties, resulting in the\ndifferent inversion symmetry of B-site positions. As shown in our STEM results, the\ngradient rotation of the BO octahedron exhibits a non-uniform (inhomogeneous)\n6\n8\nevolution. Thus, in this case, the relatively complex lattice environment drives the polar\ntrilinear term. In fact, in perovskite superlattices, various experimental factors like\nlattice distortions, strain, and cation ordering can break inversion symmetry and induce\nsuch hybrid polarity.\n4. Unless authors can provide significant improvement regarding the FE properties of\ntheir compound, I cannot recommend publication of the present manuscript.\nResponse: We sincerely thank the reviewer for her/his constructive suggestions\nregarding the strengthening of ferroelectric evidence. As detailed in our response to\nComment 2, we have undertaken comprehensive efforts to address this point with\nperforming macroscopic P-E measurements, providing direct nanoscale evidence of\npolarization switching via PFM, enriching our theoretical analysis with additional DFT\ncalculations, and revising the manuscript title accordingly. We hope these\ncomprehensive revisions may address the reviewer’s concern.\nIn addition, we would like to emphasize the novelty of this work in the following\naspects, and further elaborate on its significance and impact on high-temperature\nferromagnetic insulators and the multiferroic field.\n(1) Realization of a high-temperature ferrimagnetic insulator via B-site cation\nordering and high-spin state of Co2+: Intrinsic ferromagnetic insulators (FMIs) are\nexceedingly rare in nature, as long-range electron hopping in ferromagnetic systems\ntypically favors metallic behavior. Most known FMIs, such as EuS (T = 16 K) and\nC\nCrGeTe (T = 66 K), exhibit Curie temperatures far below room temperature (see\nTable R1), severely limiting their practical applicability in spin-wave-based electronics.\nIn this context, our work achieves a remarkable advance: we have synthesized a highly\nB-site-ordered La CoRuO epitaxial film, in which Co2+ ions uniformly adopt a stable\n2 6\nhigh-spin state (a configuration proven to significantly enhance magnetic coupling and\nelevate ordering temperatures) (Nat. Commun. 12, 1853 (2021)；Nano Res.17, 7250-\n7255 (2024)；Nat. Commun. 14, 3638 (2023)). Through the synergistic combination of\ncationic ordering and strain engineering, we stabilized a robust ferrimagnetic insulator\nwith a Curie temperature exceeding 400 K, presenting a promising candidate material\nfor next-generation spin-electronic devices.\nTable R1 Survey of Curie temperature for ferromagnetic insulators.\nFMIs Tc Reference\nEuTiO 3.4 K Nat. Commun. 13, 2364. (2022)\nCrGeTe 66 K Phys. Rev. Lett. 127, 217203. (2021)\n3\nLaCoO 85 K Adv. Sci. 10, 2303630. (2023)\n0.1 0.9 3\nBiMnO 105 K Phys. Rev. B 72, 020406. 2005\nLa 0.4 Sr 0.6 Mn 0.6 Ir 0.4 O 3 152 K 15414 (2022)\nACS Appl. Mater. Interfaces 13, 8863-\nLa 0.9 Ba 0.1 MnO 3 185 K 8870 (2021)\nLa NiMnO 280 K Phys. Rev. B 77, 174423. (2008)\n2 6\nLaCoO 2.5 284 K Nat. Commun. 12, 1853. (2021)\nH-LaCoO 400 K Nano Lett. 25 ,10478-10486. (2025)\n2.5\nLa Sr MnO 560 K Phys. Rev. Lett. 109, 157207. (2012)\n0.67 0.33 3\nLaCoO -MgO 594 K Phys. Rev. Lett. 134, 016702. (2025)\n3\n[LaFeO /LaNiO ] 608 K Nat. Commun. 16, 3691. (2025)\n3 3 n\nLa CoO 820 K Nano Res.17, 7250-7255. (2024)\n2 4\nWe conducted a comprehensive survey of ferromagnetic insulators reported in previous\nstudies, which have been extensively investigated for their potential applications in\nspintronics, magnetic storage devices, and other advanced technologies. However, most\npreviously reported ferromagnetic insulating materials exhibit low Curie temperatures,\nsignificantly limiting their practical applications. Detailed statistical analysis of these\nmaterials is provided in Table R1.\n(2) First demonstration of coexisting ferrimagnetism and polar phase with\nswitched polarization in an oxide containing 4d metal ion: The coexistence of\nmagnetic and polar orders is inherently challenging due to the conflicting electronic\nstructure requirements for magnetism and ferroelectricity. Our work provides the first\nexperimental demonstration of room-temperature ferrimagnetism coupled with\nswitchable polar domains in a 4d-containing double perovskite. Given that 4d/5d\nelements possess strong spin-orbit coupling, a key factor in enhancing magnetoelectric\ncoupling, our discovery establishes a new material platform and creates the necessary\nprerequisites for exploring strong magnetoelectric effects and developing multiferroic\nfunctions in such promising spin-orbit coupled oxides.\nWe believe that these shared achievements, through sophisticated materials engineering\nto enable high-T ferrimagnetic insulating state with switchable polarization states,\nC\n10\nrepresent a significant step forward and will generate broad interest in the fields of\nmultiferroic, complex oxides, and spintronics.\nI append below other remarks and suggestions to be considered prior to a resubmission\nat Nature Communications or elsewhere:\n5. I do not understand how the DFT calculations are performed with the varying\noctahedral rotation amplitudes presented in Fig. 6.d. It seems that different rotations\namplitudes are frozen, but are other atoms allowed to relax in position? If not, the\nonly polarization that can emerge is an electronic polarization. The method is not\ndiscussed in the manuscript even though this is a key element of the paper.\nResponse: We thank the reviewer for this insightful comment regarding the\nmethodology of our DFT calculations. As presented in Fig. R2h (updated as Fig. 6h in\nthe revised manuscript), the amplitudes of the oxygen octahedral rotation modes were\nconstrained to specific values, and the positions of all other ions (La, Co, Ru) were\nrelaxed during these calculations. The polarization arises from the symmetry breaking\ninduced by the differing rotation amplitudes between adjacent layers. Therefore, both\nthe ionic displacements and the bias of electronic cloud will contribute to the net\npolarization.\nTo quantify this, we have now decomposed the total polarization from the Berry phase\ncalculation into ionic and electronic components. Our results reveal a total polarization\nof approximately (0.288, -0.257, 0.149) µC/cm2, with an ionic contribution of about\n(0.089, -0.445, 0.047) µC/cm2. We have now revised the manuscript to include a\ndetailed description of this approach in the DFT methods section.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded new sentence (see Methods, Page 27, Line 625-627) in the revised main text as\nfollowing:\n“And the amplitudes of the octahedral rotation modes were constrained to specific\nvalues, and the positions of all other ions (La, Co, Ru) were relaxed during these\ncalculations.”\n6. Since the B cation displacements might be very small, I expect the polarization to be\nrather low. Could the authors use DFT and Born effective charges to estimate locally\nthe polarization in their supercell?\nResponse: We agree with the reviewer that the polarization is likely to be small due to\nthe subtle symmetry breaking induced by the octahedral rotations. We have further\nperformed additional calculations to quantitatively estimate the local polarization.\nspecific amplitudes, while the positions of the A-site and B-site atoms were optimized\nthrough structural relaxation. And we employed the Berry phase method to analysis the\npolarity. The results consistently confirm that the polarization is indeed very small. The\nBerry phase calculation yields a polarization of approximately (0.288, -0.257, 0.149)\nµC/cm2, with an ionic contribution of about (0.089, -0.445, 0.047) µC/cm2. Therefore,\nboth the ionic displacements and the bias of electronic cloud will contribute to the net\npolarization.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded new sentence (see Page 22, Line 490-491) in the revised main text as following:\n“The Berry phase calculation of a model structure with proper oxygen octahedral\nrotational gradient yields a locally polarization of approximately (0.288, -0.257, 0.149)\nµC/cm².”\n7. Furthermore, the polarization being induced by different rotation amplitude between\nconsecutive planes, I expect that the reversal of the polarization to go through the\nreversal of tilt patterns. Rotations are known to be associated with very large energy\ngains, one to two orders of magnitude larger than polarization in oxides. The reversal\nof the polarization therefore appears to be unlikely. Could the authors check this\npossibility with their DFT by computing the energy barrier for instance?\nResponse: We thank the reviewer for this profound and insightful comment. We agree\nwith the reviewer that the octahedral rotations may associate with large energy gains.\nConsequently, a full reversal of the polarization involving a complete switching of the\ntilt pattern would indeed be expected to face a prohibitively high energy barrier.\nHowever, reliably simulating this specific energy barrier within our current DFT\nframework is not feasible. The polarization in our system originates from the symmetry\nbreaking caused by differing oxygen octahedral rotation amplitudes between adjacent\nlayers. This difference likely stems from either local strain inhomogeneity or the\ndissimilar response of various oxygen octahedra to strain, which motivates and couples\nmulti-mode distortions. In our DFT calculation, the ferroelectric state is constructed by\nmimicking the experimental gradient of oxygen octahedral rotation. This ferroelectric\nstate is higher in energy than the nonpolar state with uniform rotation. Thus, an energy\nwell will be obtained during the ferroelectric switching simulation, instead of an energy\nbarrier. Presently, no straightforward DFT approach can accurately capture this specific\nbarrier within a tractable supercell that fully accounts for the real material environment.\nNevertheless, our experimental PFM measurements demonstrate that polarization\nswitching does occur at the nanoscale under a sufficiently high local electric field (Fig.\nR4, added as Fig. 6a). This suggests that the switching mechanism in our system may\n12\nnot require a global reversal of the established tilt pattern, but could proceed through\nmore localized reorganization pathways or involve the motion of existing domain walls.\n8. DFT results are highly sensitive to the choice of exchange-correlation parameters.\nNowadays, parameter free calculations with meta-GGA are able to capture the correct\ntrend in terms of physical properties of transition metal oxide compounds.\n(PRB 111, 144406 2025; PRB 102 145112 2020, PRB 100 035119 2019 for instance). Could the\nauthors benchmark their results with parameter-free calculations?\nResponse: We thank the reviewer for this profound and insightful comment. We have\ncarefully studied the recommended literature and performed new benchmark\ncalculations using the SCAN and r²SCAN meta-GGA functionals to investigate the\nmagnetic ground state under strain and the electronic structure of both the bulk and\nstrained phases. Our new results are summarized as follows:\nElectronic Structure: As shown in Fig. R5, the SCAN functional predicts larger band\ngaps for both the bulk and strained phases compared to our original GGA+U results.\nIn contrast, the r²SCAN functional yields reduced band gaps, which align more closely\nwith our experimental observations.\nFig. R5 (added as Supplementary Fig. 15) A comparison of computed density of\nstates of LCRO bulk and experimental strained film in (a) SCAN and (b) r2SCAN\nfunctional.\nMagnetic Ground State: For the strained structure, both SCAN and r²SCAN\ncalculations confirm the original GGA+U finding: the ferrimagnetic (FiM) state\nremains the lowest-energy magnetic ground state.\nSCAN and (b) r2SCAN functional.\nThese comprehensive benchmark calculations confirm the robustness of our primary\nconclusion regarding the magnetic ground state. Furthermore, they demonstrate that the\nr2SCAN functional provides an improved description of the electronic structure, in\nbetter agreement with experiment. These new results have been incorporated into the\nrevised supplementals.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R5 as Supplementary Fig. 15, Fig. R6 as Supplementary Fig. 14, added\nnew sentence and references in the revised main text as following:\n(1). (See Page 19, Line 412-415) “These conclusions are further confirmed by our\nadditional benchmark calculations using the parameter-free SCAN and r²SCAN meta-\nGGA functionals, which consistently reproduce the ferrimagnetic ground state and\ninsulating character under strain (Supplementary Fig.14 and Fig.15).”\n(2). (See Methods, Page 27-28, Line 617-619) “To initialize the meta-GGA\ncalculations, wavefunctions and charge densities from pre-converged GGA+U\ncalculations were employed as a starting point to mitigate convergence difficulties55-56.”\nAdded new references:\n[55] Rezaei, N., Alaei, M., Oganov, A. R. Evaluating SCAN and r2SCAN meta-GGA\nfunctionals for predicting transition temperatures in antiferromagnetic materials. Phys.\nRev. B. 111(14), 144406 (2025).\n[56] Varignon, J., Bibes, M., Zunger, A. Mott gapping in 3d ABO perovskites without\n9. Does DFT predict a checkerboard pattern for the ground state of the compound?\ninvestigated different types of B-site cation ordering patterns of the experimental\nstrained film, specifically comparing the G-type pattern (i.e. the checkerboard pattern)\nwith A-type and C-type patterns (Fig. R7 added as Supplementary Fig. 13). Our DFT\ncalculations, summarized in Fig. R8 (added as Fig. 5b), conclusively show that the G-\ntype ordering pattern possesses the lowest energy across all considered magnetic\nconfigurations. This confirms that the G-type pattern is the most stable and\nenergetically favorable B-site order, representing the predicted ground state for the\ncompound. These new calculations and the accompanying figure have been added to\nthe revised supplementals.\nFig. R7 (added as Supplementary Fig. 13) The different magnetic orderings for the\nthree B-site cation ordering patterns under experimental strain. In our calculations,\nthese states were implemented by initializing collinear arrangements of the Co and Ru\nspins followed by structural relaxation, which yielded stable magnetic ground states.\nFig. R8 (added as Fig. 5b in the revised manuscript) Relative energies of different\nmagnetic orders for three B-site cation ordering patterns, with the FiM state of the G-\ntype cation ordering pattern set as the energy reference.\nadded Fig. R7 as Supplementary Fig. 13 and Fig. R8 as Fig. 5b, added new sentence\n(see Page 17-18, Line 385-393) in the revised main text as following:\n“To investigate the magnetic properties of strained LCRO films, we systematically\ncalculated the energies of various magnetic configurations across different B-site cation\nordering patterns under the experimental strain conditions (Fig. 5b and Supplementary\nFig. 13). Our calculations reveal that the G-type ordering pattern consistently exhibits\nthe lowest formation energy, confirming the checkerboard arrangement as the ground\nstate in agreement with experimental observations (Fig. 1c-e and Supplementary Fig.\n6b-d). Moreover, among all magnetic configurations examined, the G-type pattern with\nFiM ordering possesses the lowest total energy.”\n10. The gap appears to be rather small (0.55 eV from DFT). Can the author rule out\nany leakage current preventing the application of an E-field to reverse the polarization?\nResponse: We thank the reviewer for this excellent question. Our DFT-calculated\nbandgap of 0.55 eV indeed points to a narrow-gap semiconductor system. This\ncharacteristic does present challenges for applying high electric fields in macroscopic\nP-E loop measurements. In addition, our experimental transport data also supports the\nsmall bandgap. The film exhibits a room-temperature resistivity of 1-2 Ω·cm,\nconfirming its semiconducting nature but indicating limited insulation. This is further\ncorroborated by low-temperature (77 K) measurements, where a leakage current\ndensity of ~10-5 A/cm2 is observed at a modest field of 10 kV/cm (Fig. R3a, added as\nSupplementary Fig. 16a).\nThe high leakage levels at low temperatures and low fields strongly suggest that\nobtaining an ideal square P-E loop is practically impossible at higher application fields.\nThis is why we encountered the “leaf-shaped” P-E hysteresis loop mentioned in our\nmanuscript (Fig. R3b, added as Supplementary Fig. 16b).\nCrucially, this limitation of macroscopic measurements is why we relied on\npiezoresponse force microscopy (PFM) measurements to demonstrate polarization\nswitching. The PFM technique applies a highly localized electric field directly through\na sharp conductive tip, which minimizes the leakage path and allows for the poling and\nswitching of individual nanoscale domains, even in leaky matrices. Therefore, the clear\nand reversible domain switching observed in our PFM experiments provides direct\nevidence of field-induced polarization reversal (Fig. R4, added as Fig. 6a), which is\nnot obscured by the macroscopic leakage current that hinders the P-E measurements.\n1. In recent years, theoretical and experimental investigations have aimed to achieve a\nmultiferroic state in oxide materials. However, the lack of materials with significant\nferroelectric polarisation, magnetic moment and high critical temperatures currently\nprevents the practical application of this state. In their study, the authors investigate\nLa₂CoRuO₆ thin films for this purpose. The double perovskite (DP) structure can\naccommodate two magnetic cations, which order in the structure due to their different\nionic sizes. A combination of exchange interactions and epitaxial strain renders the\nsystem ferrimagnetic, while local fluctuations in the rotation angle of the oxygen\noctahedra give rise to an electric dipole and polar nano-domains. These claims stem\nfrom the authors' analysis of a variety of experimental and theoretical data.\nCurrently, many of these claims are overstated.\nThe term 'multiferroism', used in the title, indicates the coexistence of two hysteresis\nproperties; here, these should be magnetism and ferroelectricity. Currently, there is\nno indication of a ferroelectric state.\nResponse: We sincerely thank the reviewer’s insightful comments and for recognizing\nthe significance of our work within the broader context of multiferroic oxide research.\nWe fully agree with the opening statement that achieving practical multiferroic\nmaterials with significant polarization, magnetism, and high critical temperatures\nremains a formidable challenge, and we also appreciate the reviewer’s assessment of\nour experimental and theoretical data.\nThe reviewer correctly points out that our initial claims, particularly regarding\nferroelectricity, were overstated. We acknowledge that the data from SHG and STEM\nalone, while indicative of a non-centrosymmetric polar structure, do not constitute\ndefinitive proof of a switchable ferroelectric state.\nIn direct response to this central issue, we have undertaken substantial efforts to\nstrengthen the evidence for ferroelectricity. This includes attempting macroscopic\npolarization-electric field (P-E) loop measurements and, most critically, performing\ndetailed piezoresponse force microscopy (PFM) experiments to demonstrate the\nreversible switching of polar domains at the nanoscale.\nFirstly, we conducted direct polarization-electric field (P-E) measurements to seek\nmacroscopic evidence of ferroelectric switching. At room temperature, the narrow\nbandgap (~ 0.55 eV) of LCRO films leads to significant leakage currents. Although\ncooling to 77 K (Fig. R9a) did partially suppress the leakage current, a well-saturated\nsquare hysteresis loop can still not be obtained. Instead, we observed a\ncharacteristic “leaf-shaped” P-E loop (Fig. R9b). We attribute this behavior to the\n17\nfollowing combined factors: (1) Persistent leakage current remains even at low\ntemperature (i.e., 77 K); (2) The polarization is strongly coupled to oxygen octahedral\nrotation (OOR) fluctuations, making its switching inherently difficult; (3) As revealed\nby the STEM results, polar domains are highly fragmented into nanoscale regions,\nforming a relaxor-ferroelectric-like state that suppresses coordinated long-range\nswitching.\nFig. R9 (added as Supplementary Fig. 16) (a) In-plane Leakage current density and\n(b) ferroelectric hysteresis loops of LCRO (purple) thin film grown on (111)-oriented\nSTO (blue) substrate measured at 77 K. The top electrode is platinum with a thickness\nof 20 nm.\nThe observed “leaf-shaped” hysteresis is consistent with a dipole glass state, a\nphenomenon reported in a recent study of SrTiO homoepitaxial films, where a similar\n3\ndipole glass state was induced by local symmetry manipulation (Phys. Rev. Lett. 131,\n246801 (2023)). While not constitutive proof on its own, such a characteristic loop\nshape has been reported in other emergent ferroelectric systems facing similar\nchallenges (Nat. Commun. 10, 1409 (2019); Nat. Commun. 16, 6417 (2025)). We\ntherefore present these data as part of a cumulative body of evidence, while\nacknowledging the limitations of the macroscopic measurement.\nSecondly, considering the challenges of macroscopic measurement, we turn to\npiezoelectric force microscopy (PFM) measurements, providing direct nanoscale\nevidence to further illustrate the switching of polarization. As shown in Fig. R10, our\nPFM domain switching experiments successfully demonstrate reversible and stable\ndomain writing and reading. The clear reversal of the PFM phase contrast upon\napplying bias pulses of opposite polarities is a definitive proof of ferroelectricity at the\n18\nmicro-scale. This is a widely accepted methodology when P-E loops are inconclusive,\nas seen in similar works like room-temperature ferroelectricity in Bi₂O₂Se thin\nfilm (Nano Letters, DOI: 10.1021/acs.nanolett.5c04402) and ferroelectricity in four-\nunit-cell freestanding BiFeO\n3\nfilm (Nature, 570, 87-90 (2019)).\nFig. R10 (added as Fig. 6a) (a) The schematic diagram of the PFM experimental\nmeasurements. The PFM amplitude and phase images (5×5 μm) of the (111)-oriented\nLCRO thin film indicate that the polarization is switchable.\nFinally, we employed the Berry phase method to analysis the ferroelectricity. The\npolarization intensity calculated by the Berry phase method is approximately (0.288, -\n0.257, 0.149) µC/cm2. Noting that this quantitative value depends on the preset\ndistortion amplitude. Even though, its order of magnitudes is consistent with\nexperimental observation. The ionic contribution, characterized by the point charge\nmodel, reaches (0.079, -0.487, 0.032) µC/cm2, which is in the same order of magnitude\nwith the total polarization\nIn conclusion, we once again sincerely thank the reviewer’s constructive comments,\nwhich guided us to improve our manuscript. Although macroscopic ferroelectric\nhysteresis is not ideal, we have provided comprehensive evidence, including PFM\ndomain switching, SHG, and STEM structural analyses, to strongly demonstrate the\ncoexistence of switchable polar domains and strong ferrimagnetism at room\ntemperature.\nTo more accurately present our findings and avoid overstatement, we have revised the\nmanuscript’s title to: “Room-temperature ferrimagnetism and polar phase in highly\nstrained La CoRuO epitaxial films through engineered 3d-4d cation ordering”.\n2 6\nCorresponding discussions on ferroelectricity and multiferroic have also been revised\nthroughout the text.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R9 as Supplementary Fig. 16 and Fig. R10 as Fig. 6a, added and revised\nthe sentence of the main text as following:\nAdded sentence:\n19\n(1). (See Page 20, Line 429-441) “The macroscopic polarization of LCRO film was\ncharacterized by polarization-electric field (P-E) loop measurements, where a\ncharacteristic “leaf-shaped” P-E hysteresis at low temperature suggests a dipole glass\nstate48 (Supplementary Fig.16b). This non-ideal loop shape primarily results from the\nleakage current that remains even at low temperature (Supplementary Fig.16a). To\novercome the limitations of macroscopic measurements and directly probe the polar\nbehavior at the nanoscale, we further performed detailed piezoresponse force\nmicroscopy (PFM) investigations at room temperature14. Our PFM domain switching\nexperiments successfully demonstrated reversible and stable domain writing and\nerasing. Clear reversal of the PFM phase contrast upon applying bias voltages of\nopposite polarities through the conductive tip confirms that the spontaneous\npolarization can be reliably switched by an external electric field (Fig. 6a).”\nAdded reference：[48] Li, T., Deng, S. H., Qi, H. et al. High-temperature ferroic glassy\nstates in SrTiO -based thin films. Phys. Rev. Lett. 131(24), 246801 (2023).\n3\n(2). (See Page 22, Line 478-482) “The collective evidence from the atomically resolved\npolar distortions and the macroscopic “leaf-shaped” P-E hysteresis loop consistently\npoints to the formation of a dipole glass or relaxor-like ferroelectric state in the LCRO\nfilms48, where nanoscale polarization fragmentation suppresses long-range ferroelectric\norder while permitting local switchability.”\n(3). (See Page 22, Line 492-494) “The Berry phase calculation of a model structure\nwith proper oxygen octahedral rotations gradient yields a locally polarization of\napproximately (0.288, -0.257, 0.149) µC/cm².”\n(4). (See Methods, Page 25, Line 561-565) Precision Premier II Ferroelectric Tester\n(Radiant Technologies) was used to measure P-E hysteresis loops. Pt electrodes were\nplated on the films for the measurements. Piezoresponse force microscopy (PFM)\nmeasurements were conducted by Asylum Research MFP-3D origin with an HVA 220\nhigh-voltage amplifier.\nRevised sentence:\n(1) Original title: “Room-temperature multiferroism in highly strained La CoRuO\n2 6\nepitaxial films through engineered 3d-4d cation ordering”.\nRevised title: “Room-temperature ferrimagnetism and polar phase in highly strained\nLa CoRuO epitaxial films through engineered 3d-4d cation ordering”\n2 6\n(2) Original text: “Multiferroics with Curie temperatures (T ) exceeding room-\nC\ntemperature are ideal platforms for realizing the next-generation spintronic devices with\ncoupled degrees of freedom. Among various materials candidates, 3d transition metal\n20\noxides (TMOs) exhibit great potential for multifunctional integration, however, most\nof their T are still unsatisfactory to date, and multiferroics incorporating 4d/5d\nC\nelements with strong spin-orbit coupling remain to be largely unexplored.”\nRevised text: (See Abstract, Page 2, Line 46-52) “Materials exhibiting room-\ntemperature magnetic ordering and switchable polarization are highly desirable for\ndeveloping next-generation spintronic devices with coupled degrees of freedom. While\n3d transition metal oxides (TMOs) exhibit great potential for multifunctional\nintegration, however, most of their T are still unsatisfactory to date, and the\nC\nexperimental realization of magnetic and polar order coexistence has remained\nunexplored in 4d/5d oxides, where strong spin-orbit coupling could enhance\nmagnetoelectric interactions.”\n(3) Original text: “Our work fills in the gap in experimental studies of 4d/5d TMOs-\nbased multiferroics and opens up new avenues for designing and exploring high-T\nC\nmultiferroics.”\nRevised text: (See Abstract, Page 2, Line 63-65) “Our work fills the experimental gap\nin realizing coexistence of magnetic order and switchable polarization in 4d/5d oxides\nand opens new avenues for designing high-T multiferroics.”\nC\n(4) Original text: “Multiferroics, which enables cross-coupling between magnetic,\nelectric and other orders, are promising for next-generation spintronic devices1-3.”\nRevised text: (See Introduction, Page 3, Line 69-71) “Multiferroics, characterized by\nthe coexistence of magnetic and electrically switchable polar orders with cross-\ncoupling capabilities, hold great promise for next-generation spintronic devices1-3.”\n(5) Original text: “Nevertheless, multiferroics containing these heavy elements have\nbeen rarely explored (Supplementary table S1).”\nRevised text: (See Introduction, Page 3, Line 80-82) “Nevertheless, experimental\nstudies on the coexistence of magnetic and polar orders in transition metal oxides\ncontaining these heavy elements have never been explored (Supplementary Table S1).”\n(6) Original text: “This work provides the first experimental demonstration of\nmultiferroicity in the 4d/5d oxides and offers a new strategy for exploring novel\nmultiferroics in 3d-4d/5d DP oxides.”\nRevised text: (See Introduction, Page 6, Line 134-138) “This work provides the first\nevidence of coexisting room-temperature magnetic and switchable polar orders in a 4d-\noxide, which establishes a prerequisite for achieving multiferroicity in 3d-4d/5d double\nperovskites, thereby providing a new platform for discovering emerging multiferroic\nmaterials.”\n21\n(7) Original text: “Insights from this study may point to effective strategies for\ndesigning high-Tc multiferroic materials. The material we developed can serve as an\nexcellent starting point to explore a promising platform for strong magnetoelectric\ncoupling through further engineering.”\nRevised text: (See Conclusion, Page 24, Line 532-536) “Insights from this study may\npoint to effective strategies for designing materials with coexisting magnetic and polar\norders at high temperature. The material we developed can serve as an excellent starting\npoint to explore a promising platform for multiferroics with strong magnetoelectric\ncoupling through further engineering.”\n2. Ab initio calculations performed by the authors indicate the absence of a polar state\nin coherently strained structures. The experimental data shown in Figure 3g\ndemonstrate a progressive increase in the oxygen octahedron rotation (OOR) angle,\nfrom 0° in the cubic STO substrate to the bulk value of LCRO. This behaviour is\ncharacteristic of an epitaxial thin film, in which the OORs are coupled across the\nfilm/substrate interface, and has been extensively discussed in the literature.\nThe authors achieve a polar state in their calculations by introducing this progressive\nincrease in the OOR angle (Figure 6d). However, given the direction of the gradient in\nthe OOR angle, the polarisation has a clearly defined crystallographic orientation. In\ncontrast, Figure 4c shows that the polar axis orientation in their films aligns with\ndifferent crystallographic directions and is therefore unrelated to the previously\ndiscussed mechanism. To explain this discrepancy, the authors introduce “a gradient\nsuppression of oxygen octahedral rotation (OOR) with inhomogeneous evolution”.\nSince OORs are coupled via the sharing of oxygen anions between adjacent octahedra,\nit is unclear how the system can accommodate such local distortions on a very short\nlength scale in the absence of a driving mechanism.\nThe data shown in Figures 3e and 3f reveal deviations in angles measured over a\ncolumn of many atoms. While these angles appear random in the [110] view, they\nshould also be random in the perpendicular [-110] direction.\nResponse: We sincerely thank the reviewer for raising this crucial point regarding the\nrelationship between our DFT predictions and experimental observations. We fully\nagree with the reviewer that the apparent discrepancy between the uniformly oriented\npolarization predicted by our idealized DFT model (with a perfectly coherent OOR\ngradient) and the multi-directional polar axes observed experimentally requires a\nthorough and careful explanation.\nThe resolution to this apparent contradiction lies in recognizing the inherent structural\ncomplexity of real epitaxial films, which extends beyond the idealized model used in\nour DFT calculations. Our experimental characterization clearly demonstrates that the\n22\nfilm does not experience perfectly uniform strain, nor does it exhibit a perfectly smooth\nOOR gradient. Data from microstructural strain analysis (Fig. 1b) and atomic-scale\noctahedral angle mapping (Fig. 3c, e, f) reveal significant local strain variations and an\ninhomogeneous evolution of OOR, particularly between CoO and RuO octahedra due\n6 6\nto their distinct chemical characteristics.\nWe fully concur with the reviewer’s observation regarding the random angular\ndeviations shown in Fig. 3e and 3f. These local variations across different\ncrystallographic directions indeed serve as the “driving mechanism” that disrupts the\nideal long-range OOR coupling. The system naturally accommodates such short-\nlength-scale distortions through its local energy landscape, which is governed by\nvarying strain fields and distinct cation chemistry. Consequently, while the global OOR\ngradient establishes a background favoring polar order, the local structural\ninhomogeneities reshape its expression, causing the polarization to manifest as\nfragmented nano-domains with varying orientations rather than as a single uniform\npolarization vector.\nThis realistic scenario is analogous to the physics in [La NiMnO /La CoMnO ]\n2 6 2 6 n\nsuperlattices, where multi-mode epitaxial growth creates a complex strain landscape\n(Nat. Commun. 15, 5549 (2024); Phys. Rev. Lett. 135, 046801 (2025); Small Methods\n7, 2201246 (2023)). In such systems, the local strain is affected by the microstructures,\nand the response of various oxygen octahedrons to strain is different; thus, the multi-\nmode distortions of the oxygen octahedron are motivated and coupled with each other\nto produce polarization.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (See Page 23, Line 495-499) in the main text as following:\nOriginal text: “This DFT scenario supports the experimental observation of the polarity\n(Fig. 4).”\nRevised text: “This DFT scenario captures the essential polar mechanism, while the\nexperimentally observed nanoscale domain fragmentation results from intrinsic\nstructural complexities such as local strain variations and multi-mode oxygen\noctahedral distortions that extend beyond the idealized model (Fig. 6a-d).”\n3. It should also be noted that if the polarization is coupled to local OOR fluctuations,\nit may be difficult to switch the polarization orientation.\nResponse: We thank the reviewer for this insightful comment. We fully agree that\nswitching the polarization in a system strongly coupled to oxygen octahedral rotations\n(OOR) is energetically more challenging than in conventional ferroelectrics. This\ninherent difficulty also explains the characteristic “leaf-shaped” macroscopic P-E\nhysteresis loop and the relatively small polarization value (~ 0.6 µC/cm²) measured in\n23\nour films, which reflect the hindered long-range switching under an electric field (Fig.\nR9).\nGiven these challenges at the macroscopic level, we turned to nanoscale piezoresponse\nforce microscopy (PFM) measurements to probe local polarization reversibility. Our\nPFM experiments did observe reversible polarization switching (Fig. R10), but it\nrequired a relatively high tip bias (on the order of 10 V) to achieve. Furthermore, the\nphase hysteresis loops indicated a switching angle of approximately 150° rather than a\nperfect 180°. These observations are consistent with the behavior of a system where\npolarization is strongly coupled to OOR\nThus, while the switching barrier is non-negligible, our PFM results confirm that the\npolarization in LCRO films can be reversibly reoriented under a sufficiently high local\nelectric field, supporting the existence of a switchable polar phase.\n4. In any case, these polar nano-domains do not constitute evidence of a ferroelectric\nstate.\nResponse: We sincerely thank the reviewer for their constructive suggestions regarding\nthe strengthening of ferroelectric evidence. As detailed in our response to Comment 1,\nwe have undertaken comprehensive efforts to address this point, including performing\nmacroscopic P-E measurements, providing direct nanoscale evidence of polarization\nswitching via PFM, enriching our theoretical analysis with additional DFT calculations,\nand revising the manuscript title (“Room-temperature ferrimagnetism and polar phase\nin highly strained La CoRuO epitaxial films through engineered 3d-4d cation\n2 6\nordering”) to more accurately present our findings and avoid overstatement. We hope\nthese comprehensive revisions satisfactorily address the reviewer’s concern.\n5. Regarding ferrimagnetism, the temperature dependence of the magnetisation shown\nin Figures 2d and S8 does not resemble the characteristic dependence observed in other\nferromagnets or ferrimagnets (see, for instance, Ref. 39). The authors do not explain\nthe origin of this difference.\nResponse: We thank the reviewer for raising this point regarding the temperature\ndependence of magnetization. We fully agree that the M-T data of our LCRO films\nexhibits a different characteristic from typical ferromagnets or ferrimagnets with well-\ndefined Curie transitions below 400 K. This is because the ferrimagnetic-to-\nparamagnetic transition temperature of our films exceeds 400 K, which is beyond the\nupper measurement limit of our SQUID instrument. Consequently, we were unable to\ndirectly observe the complete transition within the measurable temperature range.\nHowever, we would like to note that the M-T behavior observed in our LCRO films\nfrom 2 K to 400 K is actually consistent with other perovskite oxide films reported to\n24\nhave ordering temperatures above 400 K, such as H-LaCoO and SmFeO (Nano\n2.5 3-δ\nLett. 25, 10478-10486 (2025); Sci. Adv. 8, eabm8550 (2022)).\nTo further clarify this issue, we conducted additional measurement on the M-T data of\nLCRO-111 oriented thin film over 400 K (Fig. R11b). However, the new M-T data of\nLCRO films were collected by piecing high-temperature (400 - 675 K) and low-\ntemperature (10 - 400 K) curves together. Due to limitations of our experimental\nequipment, we do not have a sample holder capable of meeting the requirements for\ntesting from 2 to 700 K on one SQUID instrument. Therefore, the different background\nsignals for the instruments and sample holders caused the observed anomaly at around\ndetermined to be approximately 623 K.\nFig. R11 (added as Supplementary Fig. 8b) (a) (a-b) Temperature-dependent\nmagnetization (M-T) curves were measured during LCRO epitaxial films warming after\na field cooling of 1 kOe along the in-plane direction. (b) The curves from 5 K to 675 K\nare composed of two segments: 5 K to 400 K and 400 K to 675 K. The inset is an\nenlarged M-T curve from 400 K to 675 K.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R11 as Supplementary Fig. 8 and revised the sentence (see Page 10, Line\n228-232) of the main text as following:\nOriginal text: “The temperature-dependent magnetization curves (M-T) indicate a FiM\ntransition temperature exceeding 400 K (Supplementary Fig. 8), which is well above\nroom temperature.”\nRevised text: “The temperature-dependent magnetization curves (M-T) indicate a FiM\ntransition temperature exceeding 400 K (Supplementary Fig. 8) with the high-\ntemperature data precisely determining the T of the (111)-oriented LCRO film to be\nC\napproximately 623 K (Supplementary Fig. 8b), which is well above room temperature.”\n25\n6. Presentation:\nThe text should be revised by a colleague who is proficient in English.\nFor instance:\n'a ferrimagnetic insulating state is achieved above 400 K': what is the physical state of\nthe system below 400 K?\n'A gradient BO6 octahedral rotation with inhomogeneous evolution triggers': what\ndoes 'gradient' mean here? 'a' is singular, 'rotations' is plural and 'triggers' is singular.\nResponse: We thank the reviewer’s comprehensive review and constructive comments.\nWe have thoroughly checked and made corrections to the language expression,\ngrammar, and spelling in the manuscript. In the manuscript, as shown in Figure 3g, the\n“gradient” refers to the gradual increase in the rotations of the BO octahedra as it\n6\nmoves away from the interface.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised the sentence (see Abstract, Page 2) of the main text as following:\nOriginal text:\n(1) “a ferrimagnetic insulating state above 400 K is achieved in La CoRuO double-\n2 6\nperovskite epitaxial films”\n(2) “A gradient BO6 octahedral rotation with inhomogeneous evolution triggers”\nRevised text:\n(1) “a ferrimagnetic insulating state with T above 400 K is achieved in La CoRuO\nC 2 6\ndouble-perovskite epitaxial films”\n(2) “the gradient BO octahedral rotations with inhomogeneous evolution trigger”\n6\n7. What lattice symmetry is used to denote the directions of the LCRO films?\nPseudocubic? This is not mentioned, which leads the reader to make errors. Figure 1a:\nthe (001) peak of LCRO cannot be close to the (001) peak of STO because their lattice\nparameters are very different (see Table S2). The notation also appears to be\ninconsistent: the 111 peak of a pseudocubic perovskite sits at ~40°. The RSMs in Figure\nS4 also suffer from this problem. Do the zone axes in the STEM images follow the layer\norientation shown in Figure S6e or are they in pseudocubic notation?\nResponse: We thank the reviewer for this critical question. The reviewer is correct that\npseudocubic notation is used to denote the directions in the LCRO films, and the zone\naxes in the STEM images are also given in pseudocubic notation.\nTo clarify this clearly, we have modified Supplementary Fig. 6e (Fig. R12 update as\nSupplementary Fig. 6e) to provide a schematic diagram of the LCRO thin film\nstructure. As shown in the updated figure, the tetragonal-like monoclinic crystal\nstructure (outlined in black) can be described as being composed of eight pseudocubic\n26\nperovskite subcells (in orange), following the relations: a = b = √2a , c = 2c , where\npc pc\npc represents pseudocubic lattice parameter.\nFig. R12 (update as Supplementary Fig. 6e) The tetragonal-like new-monoclinic\ncrystal structure (black lines) is formed by eight orange pseudocubic perovskite subcells\nand can be approximated with c = 2c , a = b = √2a as the monoclinic lattice\npc STO\nparameters.\nAccordingly, in Supplementary Table S2, according to Bragg’s law and XRD θ-2θ\nresults, the out-of-plane (OP) lattice parameter c (3.962 Å) and c (monoclinic lattice\npc\nparameters) can be approximated with c = 2c = 7.924Å. Similarly, RSM results\npc\ndemonstrate that LCRO films are fully strained to the STO substrate along in-plane (IP)\ndirection. So, the IP lattice parameter a = b = √2a (a a = 3.905 Å) = 5.52 Å. To\npc pc = STO\nfacilitate direct comparison with the bulk LCRO material (monoclinic P2 /n symmetry,\n1\na = 5.572Å, b = 5.628 Å, c = 7.884 Å), we intentionally converted the measured\npseudocubic parameters into a corresponding monoclinic-like setting using the standard\ntransformation for double perovskite structures: a = b = √2a , c = 2c\npc pc.\nTo avoid confusion, we have updated the XRD and RSM annotations following the\nconventions of double perovskite literature (Phys. Rev. B. 105, 024408 (2022); Nat.\nCommun. 15, 5549 (2024)) and added a clarifying note beneath Supplementary Table\nS2.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nmade comprehensive revisions as following:\n(1) We have updated all relevant figures (Fig. 1a, Supplementary Fig. 1 and\nSupplementary Fig. 4) to explicitly indicate pseudocubic notation with “pc” subscripts\nwhere appropriate.\n(2) The caption for Supplementary Table S2 has been modified to clarify that the\nmonoclinic lattice parameters were derived from pseudocubic values using the\nestablished transformation, with the conversion method explicitly stated.\nXRD θ-2θ and RSMs results, including lattice parameters a, b, c, mismatch strain (ε),\nand lattice volume.”\nRevised text: “We calculated the structural parameters of the LCRO films based on\nXRD θ-2θ and RSMs results, including lattice parameters (monoclinic lattice\nparameters) a, b, c, mismatch strain (ε), and lattice volume. Under pseudocubic\napproximation, double-perovskite La CoRuO lattice can be described as a = b = √2a\n2 6 pc\n(a a = 3.905 Å), c = 2c (pc: pseudo-cubic).”\npc = STO pc\n(3) We have updated Fig. R12 as Supplementary Fig. 6e, which now includes an\nenhanced schematic illustrating the relationship between the pseudocubic subcells and\nthe resulting monoclinic structure.\n8. Figure 1d: What do the numbers 580 and 1200 refer to in the legend of this plot?\nResponse: We thank the reviewer for pointing out this question. In Figure S5, each\natomic column was fitted using a two-dimensional ellipse Gaussian function via the\nCalAtom algorithm to accurately extract the column intensity. The resulting intensity\nmapping of La (blue stars), Co (green circles) and Ru (red circles) ions, corresponding\nto HAADF image in Fig. 1c. In Figure 1d, 580, 580, and 1200 represent the total number\nof Co, Ru, and La atoms marked correspondingly in Figure S5.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded the label “Number of Atoms” to the statistical chart in Figure 1d to help readers\nunderstand the meaning of the numbers 580 and 1200.\n9. What unit do the authors use for magnetisation, and what does f.c. stand for?\nResponse: We thank the reviewer for this critical comment. Sorry for mistakenly\nwriting f.u. as f.c., we have made all the modifications to the magnetization unit in the\nrevised manuscript.\nThe magnetization unit for the M-H and M-T curves should be /f.u. (f.u. = formula\nunit), where f.u. represents a complete La CoRuO unit cell (i.e., 2 La atoms, 1 Co atom,\n2 6\n𝜇𝐵\nexpressing the magnetization of double-perovskite thin films, as shown in reference\n𝜇𝐵\nPhys. Rev. Mater. 2, 084401 (2018), Appl. Phys. Lett. 90, 072903 (2007), Phys. Rev. B.\n91, 134107 (2015), Phys. Rev. Lett. 135, 046801 (2025) et al.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\ncorrected all the magnetization unit of “ /f.c.” to “ /f.u.” in the revised manuscript.\n𝜇𝐵 𝜇𝐵\n28\n10. Figure S1a: the labels of the peaks are incorrect.\nResponse: We thank the reviewer for this pointing out. We have revised the labels of\nthe peaks in Figure S1.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nupdated Fig. R13 as new Supplementary Fig. 1 in the revised manuscript as following:\nFig. R13 (update as Supplementary Fig. 1) (a-b) XRD θ-2θ scans of LCRO epitaxial\nfilms with different crystallographic orientations. The insets are AFM height images,\nroot-mean-square surface roughness is about 467.458 pm and 507.819 pm for (111) and\n(001) orientations, respectively. (c-d) The ω-scans of out-plane peak of (111)- and\n(001)-oriented LCRO epitaxial films, respectively.\n11. In Figure S3, the text in the figure is confusing. The top panel in (a) displays the\n(101) reflections of STO. Why is there a label for LCRO(111)? The sample is\nLCRO/(111)STO.\nResponse: We thank the reviewer for careful reading and bringing this to our attention.\nIn the previous manuscript, the labels of LCRO (111) and LCRO (001) indicate samples\ngrown on (111)- and (001)-oriented STO substrates, respectively. Following the\nreviewer’s suggestion, we removed the labels in Figure S3 and added the corresponding\nsample information in the figure captions to avoid confusion for reading.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nupdated Fig. R14 as new Supplementary Fig. 3 and revised the corresponding figure\ncaptions as following:\nsubstrates and (111)-oriented (a) and (001)-oriented (b) LCRO epitaxial films,\ndemonstrating a cube-on-cube epitaxial growth of LCRO with respect to the substrate.\n12. Analysis of the resistivity data: 'Linear fitting curves using the three-dimensional\nMott-VRH model show good agreement with the measured resistivity data': the linear\nfitting is done by plotting the log of the resistivity against the inverse of the temperature.\nThe actual phrasing is incorrect.\nResponse: We thank the reviewer for the careful reading and constructive comments.\nIn response to the Reviewer’s suggestion, we have revised the relevant phrasing.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised the sentence (See Page 12, Line 262-264) in the main text as following:\nOriginal text: “Linear fitting curves using the three-dimensional Mott-VRH\nmodel show good agreement with the measured resistivity data (Supplementary Fig. 9a,\nb).”\nRevised text: “The plots of ρ vs T-1/4 reveal that the charge transport process can be\ndescribed by VRH model in the entire temperature range (Supplementary Fig. 9b).”\n𝑙𝑛\n1. The present MS reports an experimental and theoretical study of the onset of\nferrimagnetism and local polar order in epitaxial thin films of the double perovskite\nLa2CoRuO6 grown on [001]- and [111]-oriented Nb-doped SrTiO3 substrates. The\nauthors find that the in-plane compressive strain leads to enhanced lattice distortions\nthat combine with the high spin state of Co2+ and an ordered B-site cation\narrangement that act to strengthen the Co-O-Ru antiferromagnetic exchange\ninteraction and to induce a ferrimagnetic state, while the presence of a gradient in the\nBO6 octahedral rotations is suggested to drive the onset of polar nanodomains. Overall,\nI find the experimental work to be well carried out, in particular, by providing an\nextended characterization of the system with techniques ranging from x-ray diffraction\nand absorption spectroscopy, neutron reflectometry and TEM, supplemented by first\nprinciples calculations. As I see it, the key result is the presentation of a new approach\nto controlling the exchange coupling in the double perovskites by tuning the B-cation\ncomposition and atomic ordering. However, the main result claimed in the paper, the\nsynthesis of room-temperature multiferroicity in strained La2CoRuO6 films is not fully\ndemonstrated, since the presence of long-range ferroelectric order, ferroelectric\nswitching and a coupling between magnetic and ferroelectric order parameters are not\nestablished.\nResponse: We sincerely thank the reviewer for her/his positive assessment of our work,\nnoting that “the experimental work is well carried out, in particular, by providing an\nextended characterization of the system…” and recognizing that “the key result is the\npresentation of a new approach to controlling the exchange coupling in the double\nperovskites by tuning the B-cation composition and atomic ordering.” We agree with\nthe reviewer that our initial claim of room-temperature multiferroicity was overstated,\nas we did not fully demonstrate long-range ferroelectric order or ferroelectric switching.\nIn direct response to this central issue, we have undertaken substantial efforts to\nstrengthen the evidence for ferroelectricity. This includes attempting macroscopic\npolarization-electric field (P-E) loop measurements and, most critically, performing\ndetailed piezoresponse force microscopy (PFM) experiments to demonstrate the\nreversible switching of polar domains at the nanoscale.\nFirstly, we conducted direct polarization-electric field (P-E) measurements to seek\nmacroscopic evidence of ferroelectric switching. At room temperature, the narrow\nbandgap (~ 0.55 eV) of LCRO films leads to significant leakage currents. Although\ncooling to 77 K (Fig. R15a) did partially suppress the leakage current, a well-saturated\nsquare hysteresis loop can still not be obtained. Instead, we observed a\ncharacteristic “leaf-shaped” P-E loop (Fig. R15b). We attribute this behavior to the\n31\nfollowing combined factors: (1) Persistent leakage current remains even at low\ntemperature (i.e., 77 K); (2) The polarization is strongly coupled to oxygen octahedral\nrotation (OOR) fluctuations, making its switching inherently difficult; (3) As revealed\nby the STEM results, polar domains are highly fragmented into nanoscale regions,\nforming a relaxor-ferroelectric-like state that suppresses coordinated long-range\nswitching.\nFig. R15 (added as Supplementary Fig. 16) (a) In-plane Leakage current density and\n(b) ferroelectric hysteresis loops of LCRO (purple) thin film grown on (111)-oriented\nSTO (blue) substrate measured at 77 K. The top electrode is platinum with a thickness\nof 20 nm.\nThe observed “leaf-shaped” hysteresis is consistent with a dipole glass state, a\nphenomenon reported in a recent study of SrTiO homoepitaxial films, where a similar\n3\ndipole glass state was induced by local symmetry manipulation (Phys. Rev. Lett. 131,\n246801 (2023)). While not constitutive proof on its own, such a characteristic loop\nshape has been reported in other emergent ferroelectric systems facing similar\nchallenges (Nat. Commun. 10, 1409 (2019); Nat. Commun. 16, 6417 (2025)). We\ntherefore present these data as part of a cumulative body of evidence, while\nacknowledging the limitations of the macroscopic measurement.\nSecondly, considering the challenges of macroscopic measurement, we turn to\npiezoelectric force microscopy (PFM) measurements, providing direct nanoscale\nevidence to further illustrate the switching of polarization. As shown in Fig. R16, our\nPFM domain switching experiments successfully demonstrate reversible and stable\ndomain writing and reading. The clear reversal of the PFM phase contrast upon\napplying bias pulses of opposite polarities is a definitive proof of ferroelectricity at the\n32\nmicro-scale. This is a widely accepted methodology when P-E loops are inconclusive,\nas seen in similar works like room-temperature ferroelectricity in Bi₂O₂Se thin\nfilm (Nano Letters, DOI: 10.1021/acs.nanolett.5c04402) and ferroelectricity in four-\nunit-cell freestanding BiFeO\n3\nfilm (Nature, 570, 87-90 (2019)).\nFig. R16 (added as Fig. 6a) (a) The schematic diagram of the PFM experimental\nmeasurements. The PFM amplitude and phase images (5×5 μm) of the (111)-oriented\nLCRO thin film indicate that the polarization is switchable.\nFinally, we employed the Berry phase method to analysis the ferroelectricity. The\npolarization intensity calculated by the Berry phase method is approximately (0.288, -\n0.257, 0.149) µC/cm2. Noting that this quantitative value depends on the preset\ndistortion amplitude. Even though, its order of magnitudes is consistent with\nexperimental observation. The ionic contribution, characterized by the point charge\nmodel, reaches (0.079, -0.487, 0.032) µC/cm2, which is in the same order of magnitude\nwith the total polarization\nIn conclusion, we once again sincerely thank the reviewer’s constructive comments,\nwhich guided us to improve our manuscript. Although macroscopic ferroelectric\nhysteresis is not ideal, we have provided comprehensive evidence, including PFM\ndomain switching, SHG, and STEM structural analyses, to strongly demonstrate the\ncoexistence of switchable polar domains and strong ferrimagnetism at room\ntemperature.\nTo more accurately present our findings and avoid overstatement, we have revised the\nmanuscript’s title to: “Room-temperature ferrimagnetism and polar phase in highly\nstrained La CoRuO epitaxial films through engineered 3d-4d cation ordering”.\n2 6\nCorresponding discussions on ferroelectricity and multiferroic have also been revised\nthroughout the text.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R15 as Supplementary Fig. 16 and Fig. R16 as Fig. 6a, added and revised\nthe sentence of the main text as following:\nAdded sentence:\n33\n(1). (See Page 20, Line 429-441) “The macroscopic polarization of LCRO film was\ncharacterized by polarization-electric field (P-E) loop measurements, where a\ncharacteristic “leaf-shaped” P-E hysteresis at low temperature suggests a dipole glass\nstate48 (Supplementary Fig.16b). This non-ideal loop shape primarily results from the\nleakage current that remains even at low temperature (Supplementary Fig.16a). To\novercome the limitations of macroscopic measurements and directly probe the polar\nbehavior at the nanoscale, we further performed detailed piezoresponse force\nmicroscopy (PFM) investigations at room temperature14. Our PFM domain switching\nexperiments successfully demonstrated reversible and stable domain writing and\nerasing. Clear reversal of the PFM phase contrast upon applying bias voltages of\nopposite polarities through the conductive tip confirms that the spontaneous\npolarization can be reliably switched by an external electric field (Fig. 6a).”\nAdded reference：[48] Li, T., Deng, S. H., Qi, H. et al. High-temperature ferroic glassy\nstates in SrTiO -based thin films. Phys. Rev. Lett. 131(24), 246801 (2023).\n3\n(2). (See Page 22, Line 478-482) “The collective evidence from the atomically resolved\npolar distortions and the macroscopic “leaf-shaped” P-E hysteresis loop consistently\npoints to the formation of a dipole glass or relaxor-like ferroelectric state in the LCRO\nfilms48, where nanoscale polarization fragmentation suppresses long-range ferroelectric\norder while permitting local switchability.”\n(3). (See Page 22, Line 492-494) “The Berry phase calculation of a model structure\nwith proper oxygen octahedral rotations gradient yields a locally polarization of\napproximately (0.288, -0.257, 0.149) µC/cm².”\n(4). (See Methods, Page 25, Line 561-565) Precision Premier II Ferroelectric Tester\n(Radiant Technologies) was used to measure P-E hysteresis loops. Pt electrodes were\nplated on the films for the measurements. Piezoresponse force microscopy (PFM)\nmeasurements were conducted by Asylum Research MFP-3D origin with an HVA 220\nhigh-voltage amplifier.\nRevised sentence:\n(1) Original title: “Room-temperature multiferroism in highly strained La CoRuO\n2 6\nepitaxial films through engineered 3d-4d cation ordering”.\nRevised title: “Room-temperature ferrimagnetism and polar phase in highly strained\nLa CoRuO epitaxial films through engineered 3d-4d cation ordering”\n2 6\n(2) Original text: “Multiferroics with Curie temperatures (T ) exceeding room-\nC\ntemperature are ideal platforms for realizing the next-generation spintronic devices with\ncoupled degrees of freedom. Among various materials candidates, 3d transition metal\n34\noxides (TMOs) exhibit great potential for multifunctional integration, however, most\nof their T are still unsatisfactory to date, and multiferroics incorporating 4d/5d\nC\nelements with strong spin-orbit coupling remain to be largely unexplored.”\nRevised text: (See Abstract, Page 2, Line 46-52) “Materials exhibiting room-\ntemperature magnetic ordering and switchable polarization are highly desirable for\ndeveloping next-generation spintronic devices with coupled degrees of freedom. While\n3d transition metal oxides (TMOs) exhibit great potential for multifunctional\nintegration, however, most of their T are still unsatisfactory to date, and the\nC\nexperimental realization of magnetic and polar order coexistence has remained\nunexplored in 4d/5d oxides, where strong spin-orbit coupling could enhance\nmagnetoelectric interactions.”\n(3) Original text: “Our work fills in the gap in experimental studies of 4d/5d TMOs-\nbased multiferroics and opens up new avenues for designing and exploring high-T\nC\nmultiferroics.”\nRevised text: (See Abstract, Page 2, Line 63-65) “Our work fills the experimental gap\nin realizing coexistence of magnetic order and switchable polarization in 4d/5d oxides\nand opens new avenues for designing high-T multiferroics.”\nC\n(4) Original text: “Multiferroics, which enables cross-coupling between magnetic,\nelectric and other orders, are promising for next-generation spintronic devices1-3.”\nRevised text: (See Introduction, Page 3, Line 69-71) “Multiferroics, characterized by\nthe coexistence of magnetic and electrically switchable polar orders with cross-\ncoupling capabilities, hold great promise for next-generation spintronic devices1-3.”\n(5) Original text: “Nevertheless, multiferroics containing these heavy elements have\nbeen rarely explored (Supplementary table S1).”\nRevised text: (See Introduction, Page 3, Line 80-82) “Nevertheless, experimental\nstudies on the coexistence of magnetic and polar orders in transition metal oxides\ncontaining these heavy elements have never been explored (Supplementary Table S1).”\n(6) Original text: “This work provides the first experimental demonstration of\nmultiferroicity in the 4d/5d oxides and offers a new strategy for exploring novel\nmultiferroics in 3d-4d/5d DP oxides.”\nRevised text: (See Introduction, Page 6, Line 134-138) “This work provides the first\nevidence of coexisting room-temperature magnetic and switchable polar orders in a 4d-\noxide, which establishes a prerequisite for achieving multiferroicity in 3d-4d/5d double\nperovskites, thereby providing a new platform for discovering emerging multiferroic\nmaterials.”\n35\n(7) Original text: “Insights from this study may point to effective strategies for\ndesigning high-Tc multiferroic materials. The material we developed can serve as an\nexcellent starting point to explore a promising platform for strong magnetoelectric\ncoupling through further engineering.”\nRevised text: (See Conclusion, Page 24, Line 532-536) “Insights from this study may\npoint to effective strategies for designing materials with coexisting magnetic and polar\norders at high temperature. The material we developed can serve as an excellent starting\npoint to explore a promising platform for multiferroics with strong magnetoelectric\ncoupling through further engineering.”\nI have in addition a number of smaller points of criticism that the authors could\nconsider:\n2. The strain distribution maps in Fig. 1b show an abrupt change along the [111]\ndirection and non-uniform variations along the perpendicular direction; the text\ndoes not make clear what is the reference point used to plot the data, presumably the\nSrTiO3 lattice parameters, this needs to be mentioned explicitly. The authors should\nalso indicate to which portion of the TEM image the strain field corresponds to. I find\nit interesting that the authors obtain a similar strain state for the La2CoRuO6 films\ngrown along the [001] and [111] directions (Table S2 and Fig. S4). From symmetry, I\nwould expect different film distortions, tetragonal for the (001) films and trigonal for\nthe (111). How do the authors explain that the (111) films exhibit a tetragonal symmetry\nand how is this seen in the TEM images?\nResponse: We thank the reviewer for these insightful observations and questions\nregarding our strain analysis and structural characterization. This has helped us to\nimprove the clarity and rigor of our discussion. Below, we provide a point-by-point\nresponse.\n(1) Reference for strain maps and analyzed region:\nThe reviewer is correct that the STO substrate lattice parameters were used as the\nreference for the strain analysis. Furthermore, the strain field map corresponds precisely\nto the entire HAADF-STEM image area in Fig. 1b. These clarifications have been\nincorporated into the revised caption for Fig. 1b.\n(2) Symmetry of (001)- and (111)-oriented films:\nWe fully agree with the reviewer that different symmetries are expected for the (001)-\nand (111)-oriented films. Specifically, a tetragonal-like distortion for the (001)\norientation and a trigonal distortion for the (111) orientation. Our experimental data\nfully confirms this distinction. The XRD phi-scan results (Supplementary Fig. 3a, b)\n36\nunambiguously show a fourfold symmetry for the (001)-oriented film and a threefold\nsymmetry for the (111)-oriented film, as previously stated.\nThe reference to a “similar strain state” in the context of Table S2 and Fig. S4 was\nintended to indicate that both films are fully and coherently strained to the STO\nsubstrate along in-plane direction. This was supported by reciprocal space mapping\n(RSM): (1) For the (111)-oriented film, the RSM around the STO (321) cubic reflection\n(Supplementary Fig. 4a) shows a coherently strained film peak; (2) For the (001)-\noriented film, the RSMs around the STO (103) family of reflections (Supplementary\nFig. 4b) confirm the same in-plane coherency.\nFrom these measurements, we determined that the in-plane pseudocubic lattice\nparameter (a ) for both films is 3.905 Å, equal to the STO substrate parameter (3.905\npc\nÅ). The different out-of-plane pseudocubic parameters (c ) were obtained from XRD\npc\nθ-2θ scans.\nWe sincerely apologize for the potential confusion caused by Table S2. To enable a\ndirect comparison of lattice distortion and unit cell volume with the bulk LCRO\nmaterial (which has a monoclinic P2₁/n structure), we deliberately converted the\nmeasured pseudocubic lattice parameters into a corresponding like-monoclinic setting\nusing the established transformation for the double perovskite structure: a = b = √2 a ,\npc\nc = 2c (Fig. R17 update as Supplementary Fig. 6e). The values in Table S2 are\npc\nthe results of this conversion and are presented for comparative purposes; they do not\nimply that the (111)-oriented film exhibits a tetragonal symmetry. A note has been\nadded to the table caption in the revised manuscript to explicitly state this. Besides, we\ndid not conclude a tetragonal symmetry for the (111) film from TEM. Its trigonal\nsymmetry is confirmed by XRD phi-scan (Supplementary Fig. 3a, b).\nFig. R17 (update as Supplementary Fig. 6e) The tetragonal-like new-monoclinic\ncrystal structure (black lines) is formed by eight orange pseudocubic perovskite subcells\nand can be approximated with c = 2c , a = b = √2a as the monoclinic lattice\npc STO\nparameters.\nadded Fig. R17 as Supplementary Fig. 6e, revised the captions of Fig.1b and\nSupplementary Table S2 as following:\nThe caption of Fig.1b:\nOriginal text: “HAADF STEM image near the interface of (111)-oriented LCRO film,\ncorresponding GPA analysis of and on the local atomic image, respectively.”\n𝜀𝑥𝑥 𝜀𝑦𝑦\nRevised text: “HAADF STEM image near the interface of (111)-oriented LCRO film\nand corresponding full-field GPA analysis of and , referenced to STO\nsubstrate.”\n𝜀𝑥𝑥 𝜀𝑦𝑦\nThe caption of Supplementary Table S2:\nOriginal text: “We calculated the structural parameters of the LCRO films based on\nXRD θ-2θ and RSMs results, including lattice parameters a, b, c, mismatch strain (ε),\nand lattice volume.”\nRevised text: “We calculated the structural parameters of the LCRO films based on\nXRD θ-2θ and RSMs results, including lattice parameters (monoclinic lattice\nparameters) a, b, c, mismatch strain (ε), and lattice volume. Under pseudocubic\napproximation, double-perovskite La CoRuO lattice can be described as a = b = √2a\n2 6 pc\n(a a = 3.905 Å), c = 2c (pc: pseudo-cubic).”\npc = STO pc\n3. The values mentioned for the saturation magnetization are in fact the magnetization\nat the highest field applied. It is clear from the M-H curves that the magnetization is\nnot saturated. Presumably, the diamagnetic contribution from the substrate was\nremoved and the question arises as to how the high field slopes shown in Fig.2 were\ndetermined (it seems surprisingly identical for the various temperatures).\nResponse: We thank the reviewer for pointing this out. In the previous manuscript, to\nshowcase a clear ferromagnetic hysteresis, the low-field M-H curves with a magnetic\nfield range of 0 - 0.6 T were enlarged from the M-H curves with a maximum testing\nfield up to 1 T, which were obtained by subtracting the diamagnetic signal of the SrTiO\n3\nsubstrate from the raw M-H curves. Therefore, the M-H curves presented in the previous\nmanuscript had a slight slope at high fields, which may be misleading to the reviewer.\nWe apologize for our previous imprecision. In the revised supplemental information,\nwe provided the M-H curve with the maximum magnetic field up to 1T. It can be seen\nthat the magnetization tends to be saturation when the magnetic field approaches 0.8 T.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nupdated Fig. R18 as Supplementary Fig. 7.\nepitaxial films with different crystallographic orientations measured by applying a\nmagnetic field along the in-plane direction at different temperatures.\n4. The PNR data suggest a reduced magnetization at the SrTiO3 interface, which would\nnot be surprising (somehow, correlated with an increase in the nuclear density, which\nis more surprising). The small magnetic moment corresponds to a small magnetic\ndensity and to a very small difference between the spin up and down reflectivities and\nto a large scatter in the data points at high momentum transfer. The question here is\nwhat the error bars associated with the magnetic moment values from PNR and how\ncorrelated the interfacial moment is with the nuclear density. I suspect that these\nparameters are determined mostly from the data at high momentum transfer and may\nhave very larger error bars associated with them.\nResponse: We thank the reviewer for the careful reviews. We address the two main\nquestions, as shown below:\n(1). The small magnetic moment corresponds to a small magnetic density and to a very\nsmall difference between the spin up and down reflectivities and to a large scatter in\nthe data points at high momentum transfer. The question here is what the error bars\nassociated with the magnetic moment values from PNR. I suspect that these parameters\nare determined mostly from the data at high momentum transfer and may have very\nlarger error bars associated with them.\nWe fully agree with the reviewer that the small magnetic moment leads to a weak spin\nasymmetry (SA) signal and that the data exhibits larger scatter at high momentum\ntransfer (e.g., q > 0.04 Å-1). This is a common challenge when quantifying small\nmagnetic moments with PNR. The magnetic moments of LCRO bulk and interface are\n0.16 (-0.02, +0.02) /f.u. and 0.14 (-0.07, +0.05) /f.u. derived from a χ²-based\nfitting algorithm by GenX package. As pointed out by the reviewer, the magnetic\n𝜇𝐵 𝜇𝐵\nmoment of interface layer indeed shows a relatively large error due to its thin layer\nthickness which mainly corresponds to the reflectivities at high momentum transfer.\nHowever, the LCRO film bulk shows much smaller error bars due to the fact that the\nSA shows a clearly oscillatory behavior for q < 0.04 Å-1, providing a strong constraint\nfor the fitting. We have added the error bars of the magnetic moments in the revised\nmanuscript accordingly.\n(2). How correlated the interfacial moment is with the nuclear density?\nThe interfacial moment could be correlated with the nuclear density via the B-O bond\nlength. As shown above, the interfacial magnetism (i.e., 0.14 (-0.07, +0.05) /f.u.)\nshows a slight weakening trend in comparison to the film bulk (i.e., 0.16 (-0.02, +0.02)\n𝜇𝐵\n/f.u.). Meanwhile, the interfacial nuclear SLD is found to be larger than that in the\nfilm bulk, which is mainly attributed to their different atom densities (the number of\n𝜇𝐵\nunit cells per unit volume). The interfacial atom density is found to be 0.00795 (-\n0.00011, +0.00011) unit cells/Å3, which is larger than that of the film bulk, i.e., 0.00775\n(-0.00006, +0.00005) unit cells/Å3. Thus, the interface layer shows a reduced unit cell\nvolume compared to the film bulk. As pointed out in a previously report for a similar\nLaCoO /STO system (Phys. Rev. Lett. 122, 187202 (2019)), the reduced the interfacial\n3\nunit cell volume could decrease the B-O bond length, thus increasing crystal field\nsplitting and favoring a low-spin state at the interface. In our case, the enhanced nuclear\natom density indicates shrinkage of unit cell volume at the interface due to lattice and\nsymmetry mismatch. This could reduce the B-O bond length inducing a low-spin state\nand further a slight weakening trend in the interfacial magnetism.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised the sentences in the main text as following:\nPage 11, Line 251-258:\nOriginal text: “Magnetization is essentially uniform across the film thickness, although\nit decreases slightly at the interface, with a value of 0.16 /f.u. at RT, which is\nconsistent with the result from the M-H loop.”\n𝜇𝐵\nRevised text: “Although the magnetization shows a slight difference at the interface\nwith a value of 0.14 (-0.07, +0.05) /f.u., it is essentially uniform across the film\nthickness with a value of 0.16 (-0.02, +0.02) /f.u. at RT, which is consistent with the\n𝜇𝐵\nresult from the M-H loop. The slight weakening trend of interfacial magnetism is\n𝜇𝐵\ncorrelated with the B-O bond length reduction due to shrinkage of unit cell volume\nindicated by enhanced nuclear SLD at the interface (Fig. 3b and Supplementary Fig.\n2c). This interfacial effect agrees well with a previous literature38.”\nmismatch to control magnetism in a ferroelastic heterostructure. Phys. Rev. Lett.\n122(18),187202 (2019).\nPage 14, Line 315-318:\nOriginal text: “The observed slight weakening of interfacial magnetism likely\noriginates from further suppression of BO6 octahedral rotations (from 0.16 /f.u. up\nto 0.14 /f.u., as observed in Fig.3b).”\n𝜇𝐵\n𝜇𝐵\nRevised text: “The suppression of BO octahedral rotations at the interface also\n6\ncontributes to the observed slight weakening trend of interfacial magnetization by\nreducing the strength of nearest-neighbor Co-O-Ru interaction (Fig. 3b).”\n5. In page 12, the authors discuss the presence of ferrimagnetism as arising from an\nincrease in the Ru-O-Co exchange interactions, but it is not clear at all what is the link\nbetween the exchange coupling and the imbalance between the cation magnetic\nmoments.\nResponse: We thank the reviewer for this insightful question, which allows us to clarify\nthe microscopic origin of the ferrimagnetic (FiM) state. The link lies in how epitaxial\nstrain selectively modifies the competing exchange pathways, thereby altering the spin\nconfiguration from the antiferromagnetic (AFM) ground state of the bulk.\nFig. R19 (Fig. 5a in revised manuscript) (a) Schematic diagram of strain-induced\nmagnetic transition and interactions. J is interaction between the nearest-neighbor Co-\n2 3\nAs shown in Fig. R19, in bulk LCRO, which possesses a B-site checkerboard order,\nthe magnetic order is AFM. In this structure, the nearest-neighbor (NN) interaction is\nexclusively the Co-O-Ru super-exchange (J₁), which is antiferromagnetic. This forces\nthe Co and Ru sublattices to align antiparallel. The next-nearest-neighbor\n41\ninteractions are the Co-O-O-Co (J₂) and Ru-O-O-Ru (J₃) super-exchange, which are\nalso antiferromagnetic in the bulk, leading to zero net magnetization (Fig. R19).\nHowever, our DFT calculations reveal that the in-plane compressive strain in the thin\nfilm dramatically changes this balance. The in-plane compressive strain in the thin film\nmodifies the crystal lattice, notably shortening the Co-O-Ru bond lengths. This change\nenhances the nearest-neighbor Co-O-Ru super-exchange (J₁), which robustly maintains\nthe antiparallel alignment between the Co and Ru sublattices. Concurrently, the strain-\ninduced distortion alters the crystal field and the geometry of the longer-range super-\nexchange pathways, specifically the Co-O-O-Co (J₂) and Ru-O-O-Ru (J₃) interactions.\nThese combined modifications in J₁, J₂, and J₃ under strain make the ferrimagnetic (FiM)\nconfiguration energetically favorable (Fig. R19).\nIn the stabilized FiM state, the Co2+ (S = 3/2) sublattice aligns ferromagnetically, and\nthe Ru4+ (S = 1) sublattice also aligns ferromagnetically. These two sublattices are then\nlocked antiparallel to each other by the strong antiferromagnetic J₁ interaction. The\ndifferent magnetic moments of Co2+ (3 ) and Ru4+ (2 ) then yield the net\nmagnetization (1 ) observed in our strained films. This net magnetization should be\n𝜇𝐵 𝜇𝐵\nconsidered as the theoretical upper limit of ideally strained LCRO films, while the\n𝜇𝐵\nexperimental value is lower due to the microscale non-uniform strain or anti-site\ndisorder between Co and Ru sites. We have revised the discussion in the manuscript to\npresent this mechanism more clearly and accurately.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded and revised the sentences in the main text as following:\nPage 12, Line 268-269:\nNew sentence: “Our subsequent DFT calculations will provide further insight into the\nphysical origin of this FiM state.”\nPage 18, Line 399-407:\nOriginal text: “The shrinkage of lattice volume gives rise to significantly enhanced Co-\nO-Ru superexchange interaction22,34,38, stabilizing the nearest-neighbor\nantiferromagnetic coupling (J ) while weakening the next-nearest-neighbor coupling\n1\n(J , J ), thus leading to a net FiM moment 1 /f.u. (Fig. 6a).”\n2 3\n𝜇𝐵\nRevised text: “The shrinkage of lattice volume under in-plane compressive strain\nnotably enhances the nearest-neighbor Co-O-Ru superexchange (J ) interaction,\n1\nreinforcing its antiferromagnetic character, while concurrently driving a sign reversal\nin the next-nearest-neighbor coupling (J , J ) to favor ferromagnetic alignment22,34,38.\n2 3\nThis reconfiguration of exchange couplings promotes parallel spin alignment within\neach of the Co2+ (S = 3/2) and Ru4+ (S = 1) sublattices, which remain\n42\nantiferromagnetically locked via J . The resulting imbalance between the Co2+ and Ru4+\n1\nsublattice moments yields the net ferrimagnetic moment of 1 /f.u. observed in our\nstrained films (Fig. 5a).”\n𝜇𝐵\n6. Concerning the oxygen octahedral rotations, the authors speak of a gradient along\nthe out of plane and in the plane directions, based on the results presented in Fig.3e-g;\nin fact, the rotation and bond angle seem to vary in a random manner. It is difficult to\nsee that the average of the values shown in Fig.3e gives Fig.3g. Looking at the Fig.3g,\nI would be tempted to conclude that rotation relation occurs within 5 atomic planes.\nThe variations at outer planes are probably within the scatter error (which should be\nprovided).\nResponse: We thank the reviewer for this insightful comment regarding the\nstatistical analysis of our octahedral rotation data. Following the reviewer’s suggestion,\nwe have re-examined our data, particularly Fig. 3g, and confirm that it is indeed\nderived from Fig. 3e. In Fig. 3e, the rotation angles in each horizontal plane alternate\nbetween positive and negative values from one unit cell to the next, which is consistent\nwith the LCRO structure shown in Fig. 1c. As the interface in the (111)-oriented epitaxial\nfilm is not perpendicular to the (001) plane (the horizontal plane in the image), this could\nlead to virtual randomness in the rotation angles. As the reviewer correctly points out,\nthe unit cells within the first 2-3 nm from the interface show a relatively less uniform\ndistribution of rotation angles, corresponding to the suppressed rotation near the\ninterface. This is consistent with our claim that the most significant change in the\naverage rotation angle occurs within this region, which corresponds to approximately\n5-8 atomic planes in the pseudocubic structure.\nAdditionally, to improve the accuracy of our data and in response to the reviewer’s\nsuggestion, we have added error bars (representing the standard deviation) to Fig. 3g\n(Fig. R20), which better illustrate the error range. It can be seen that beyond the\ninterface region, the statistical average of the rotation angles gradually increases,\neventually saturating at a value nearly double that of the interface region. While the\nvariations in these \"outer planes\" are smaller, they are statistically noticeable and lie\noutside the measurement error. To be more accurate, we have revised our expressions\nbased on the reviewer’s suggestions, now stating that the oxygen octahedral rotations\nexhibit a steep gradient near the interface, followed by a more gradual gradient change\nfurther from the interface.\nresolved analysis oxygen octahedral rotations angle along the film growth direction.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded error bars representing the standard deviation in the revised Fig. 3g and added\nadditional discussions (Page 14, Line 310-312) in the main text as following:\n“In addition, we note that the oxygen octahedral rotations display a steep gradient\nwithin the first 2-3 nm from the interface, followed by a more gradual gradient change\nfarther from the interface.”\n7. The authors use SHG to determine the presence of polar order in the system. In fact,\nthe authors state that the SHG signal gives the IP spontaneous polarization. My\nunderstanding is that SHG cannot directly measure spontaneous polarization, it may\nsimply prove the presence of a break of inversion symmetry in the system. From the\nTEM data, the authors identify nanodomains where polar distortions are present.\nHowever, the criterion for ferroelectricity is more stringent, and requires not only that\na material is polar, but also that the polarization is switchable by an applied electric\nfield.\nResponse: We thank the reviewer for this insightful comment. We fully agree with the\nreviewer that SHG signals confirm the breaking of inversion symmetry and that the\nobservation of static polar nanostructures via TEM, while indicative, does not fulfill the\nstringent criterion for ferroelectricity, which requires the demonstration of polarization\nswitching by an electric field.\nAs comprehensively detailed in our responses to the previous Comment 1 (Reviewer 3)\nand now incorporated into the revised manuscript, we have directly addressed this\ncrucial requirement. Our piezoresponse force microscopy (PFM) experiments provide\ndirect, nanoscale evidence of reversible polarization switching, thereby demonstrating\nthat the polar state is not merely static but is indeed electrically switchable (Fig. R16).\nrevised the statements (Page 20, Line 441-449) about SHG measurements in the main\ntext as following:\nOriginal text: “By measuring SHG intensities in both horizontal (σ = 0°) and vertical\n(σ = 90°) polarization directions (parallel and perpendicular to the [110] direction of the\nsample, respectively), the IP spontaneous polarization of the film are yielded.”\nRevised text: “To further investigate the macroscopic polar order and crystal symmetry,\nwe employed optical second-harmonic generation (SHG) measurements36,49. As\nschematically illustrated in Fig. 6c, the rotational anisotropy of the SHG response was\ncharacterized using a normal-incidence configuration. Detailed SHG measurements\nwere conducted across a temperature range of 300 to 500 K (Supplementary Fig. 17).\nThe SHG polarimetry displayed a characteristic double-lobe structure for\nmeasurements in both the horizontal (σ = 0°) and vertical (σ = 90°) polarization\ndirections, which are parallel and perpendicular to the [110] direction of the sample,\nrespectively (Fig. 6b and Supplementary Fig. 17).”\n8. I find the presentation of the DFT results not satisfactory. For example, the authors\ndo not explain what is the difference between the AFM and FiM configurations in\nFig.S14 or how the FiM state is implemented in the calculations. The introduction of a\ngradient in the octahedral rotation seems to result in a polar state, however, this\nsituation seems to be far from the experimental case, where the rotations appear\nrandom and where the size of the polar nanodomains do not quite match with the scale\nof the gradient suggested by Fig.3g (in fact, I would argue there is no clear gradient in\nthe data shown here).\nResponse: We thank the reviewer for this insightful comment. We address the two\nmain questions, as shown below:\n(1). For example, the authors do not explain what is the difference between the AFM\nand FiM configurations in Fig.S14 or how the FiM state is implemented in the\ncalculations\nWe thank the reviewer for raising these points. The AFM (antiferromagnetic) and FiM\n(ferrimagnetic) configurations differ in their net magnetic moments. The AFM\nconfiguration possesses a net magnetic moment of zero, whereas the FiM configuration\nexhibits a non-zero net moment. In our calculations, the FiM state was implemented by\ninitializing the magnetic moments of the two transition metal ions in opposite directions.\nStructural relaxation performed under this constraint consistently yielded a stable\nstructure with a net magnetic moment.\nS14) to detail the differences between the magnetic configurations and added a new\nSupplementary Fig. 13 (Fig. R21) summarizing the B-site ordering patterns and their\nassociated magnetic orderings, with a note in the caption on their computational\nimplementation under strain.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig R21 as Supplementary Fig. 13, revised caption (Supplementary Fig. 12,\nFig.S14 in origin manuscript) in the main text as following:\nCaption (Supplementary Fig. 12):\nOriginal text: “Supplementary Fig. 12 (a) U test of LCRO bulk. (b) The three\neff\nmagnetic configurations considered in the bulk LCRO.”\nRevised text: “Supplementary Fig. 12 (a) U test of LCRO bulk. (b) The three\neff\nmagnetic configurations considered in the bulk LCRO. The fundamental distinction\namong the three magnetic orderings lies in their net magnetic moments: zero for AFM,\nand distinct non-zero values for FiM and FM.”\nFig. R21 (added as Supplementary Fig. 13) The different magnetic orderings for the\nthree B-site cation ordering patterns under experimental strain. In our calculations,\nthese states were implemented by initializing collinear arrangements of the Co and Ru\nspins followed by structural relaxation, which yielded stable magnetic ground states.\n(2). The introduction of a gradient in the octahedral rotation seems to result in a polar\nstate, however, this situation seems to be far from the experimental case, where the\nrotations appear random and where the size of the polar nanodomains do not quite\nmatch with the scale of the gradient suggested by Fig.3g (in fact, I would argue there\nis no clear gradient in the data shown here).\nWe also thank the reviewer for the insightful comment regarding the octahedral rotation\ngradient and its connection to our experimental observations. We acknowledge the\ncomputational challenge in precisely modeling the full thickness of our film, which is\nseveral tens of nanometers. To make the problem tractable, our DFT calculation\nemployed a thinner model with an introduced OOR gradient. This approach is designed\nto capture the essential qualitative physics, demonstrating that such a model structure\nwith proper oxygen octahedral rotational gradient can break inversion symmetry and\ninduce a polarization, as confirmed by Berry phase calculations. It provides a plausible\nmechanistic origin for the polar behavior, even if the specific spatial arrangement in the\nreal film is more complex. To make this progression clearer, we have included an\nenlarged view of the relevant data (Fig. R22) in our response, which more explicitly\nshows the variation in OOR angles along the growth direction.\nFig. R22 Enlarged view of the octahedral rotation gradient in the computational model.\nRegarding the experimental data in Fig. 3g (Fig. R23g in response), we wish to clarify\nthe observed trend in octahedral rotation (OOR). Analysis near the interface (Fig.\nR23b-c) reveals strong suppression of OOR angles, with a maximum value of only ~\n5°. In regions farther from the interface (~ 30 unit cells, Fig. R23e-g), the angles\nincrease statistically to approximately 10° (a doubling in magnitude), yet still notably\nlower than the bulk value of ~ 13.4°. This progressive recovery of rotations over tens\nof unit cells demonstrates that the octahedral coupling at the interface exerts a long-\nrange effect, distinct from the previously reported short-range interfacial modulation (4\n~ 6 unit cells) (Nat. Commun. 5, 5710 (2014), Nat. Mater. 15, 432 (2016), Nat. Mater.\n15, 425 (2016)). Furthermore, the fact that the OOR angle remains suppressed even at\n30 uc strongly suggests that it may continue to increase toward the bulk value in thicker\n47\nfilms. These results indicate the presence of a gradient in octahedral rotation,\ncharacterized by a gradual increase with distance from the substrate.\nWe agree with the reviewer that the gradient is not perfectly smooth, which is a key\npoint for changing the magnetic and polar orders. Our experimental data reveals that\nthe system is not under perfectly uniform strain, nor does it possess a perfectly smooth\nOOR gradient. As seen in the microstructural strain analysis (Fig. 1b) and the precise\noctahedral angle mapping (Fig. R23b-g), the film exhibits significant local strain\nvariations and a highly inhomogeneous evolution of the OOR. This inhomogeneity\narises particularly from the different chemical responses of the CoO and RuO\n6 6\noctahedra to the local strain environment.\nFig. R23 (a) Atomically resolved iDPC image of (111)-oriented LCRO film near the\nLCRO/STO interface. (b-c) Distribution maps of calculated BO octahedral rotation\n6\nangle (b) and O-Co/Ru-O bond angle (c), inside the orange box in (a). (d) Enlargement\nof the yellow rectangle region in (a) with fitted position of oxygen ions (orange circles),\nB-site ions (blue circles) and A-site ions (red circles). (e-f) Distribution maps of\ncalculated BO octahedral rotation angle and O-Co/Ru-O bond angle, inside the yellow\n6\n48\nbox in (a). (g) Plane-averaged and layer-resolved analysis oxygen octahedral rotations\nangle along the film growth direction.\nIt is precisely this local structural disorder that explains the apparent discrepancy\nbetween the ideal DFT model and the experimental observation of multi-directional\npolar nano-domains. The global OOR gradient establishes a background polar\ninstability, but the local inhomogeneities fragment this instability, causing the\npolarization to manifest as small, randomly oriented nano-domains rather than a single\nuniform vector. This realistic scenario is analogous to the physics\nin [La NiMnO /La CoMnO ] superlattices, where multi-mode epitaxial growth\n2 6 2 6 n\ncreates a complex strain landscape (Nat. Commun. 15, 5549 (2024); Phys. Rev. Lett.\n135, 046801 (2025); Small Methods 7, 2201246 (2023)). In such systems, the local\nstrain is affected by the microstructures, and the response of various oxygen\noctahedrons to strain is different; thus, the multi-mode distortions of the oxygen\noctahedron are motivated and coupled with each other to produce polarization.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (See Page 23, Line 495-499) in the main text as following:\nPage 23, Line 495-499\nOriginal text: “This DFT scenario supports the experimental observation of the polarity\n(Fig. 4).”\nRevised text: “This DFT scenario captures the essential polar mechanism, while the\nexperimentally observed nanoscale domain fragmentation results from intrinsic\nstructural complexities such as local strain variations and multi-mode oxygen\noctahedral distortions that extend beyond the idealized model (Fig. 6a-d).”\n9. In conclusion, I find that the present work is still in a preliminary phase as far as\ndemonstrating multiferroic behavior in the system studied here.\nResponse: We sincerely thank the reviewer for this assessment. In response to the\nessential points raised during the review process, we have undertaken a comprehensive\nrevision of our manuscript. This includes providing direct evidence of polarization\nswitching through detailed PFM studies, substantially refining our structural and\nmagnetic analysis, and moderating our claims throughout the text, including a change\nof the title to more accurately present our findings.\nWe believe these revisions have significantly strengthened the manuscript. It presents,\nto the best of our knowledge, the first experimental demonstration of a room-\ntemperature ferrimagnetic insulator coexisting with a switchable polar phase in a 3d-4d\ncation-ordered double perovskite. The presence of the Ru-4d ion, with its significant\nspin-orbit coupling, makes this system a distinctive platform for exploring new\n49\nmechanisms of magnetoelectric coupling. We hope the reviewer finds our responses\nand the revised manuscript satisfactory and would be willing to consider it for\npublication in Nature Communications.\nIn addition, we would like to emphasize the novelty of this work in the following\naspects, and further elaborate on its significance and impact on high-temperature\nferromagnetic insulators and the multiferroic field.\n(1) Realization of a high-temperature ferrimagnetic insulator via B-site cation\nordering and high-spin state of Co2+: Intrinsic ferromagnetic insulators (FMIs) are\nexceedingly rare in nature, as long-range electron hopping in ferromagnetic systems\ntypically favors metallic behavior. Most known FMIs, such as EuS (T = 16 K) and\nC\nCrGeTe (T = 66 K), exhibit Curie temperatures far below room temperature (see\nTable R1), severely limiting their practical applicability in spin-wave-based electronics.\nIn this context, our work achieves a remarkable advance: we have synthesized a highly\nB-site-ordered La CoRuO epitaxial film, in which Co2+ ions uniformly adopt a stable\n2 6\nhigh-spin state (a configuration proven to significantly enhance magnetic coupling and\nelevate ordering temperatures) (Nat. Commun. 12, 1853 (2021)；Nano Res.17, 7250-\n7255 (2024)；Nat. Commun. 14, 3638 (2023)). Through the synergistic combination of\ncationic ordering and strain engineering, we stabilized a robust ferrimagnetic insulator\nwith a Curie temperature exceeding 400 K, presenting a promising candidate material\nfor next-generation spin-electronic devices.\nTable R1 Survey of Curie temperature for ferromagnetic insulators.\nFMIs Tc Reference\nEuTiO 3.4 K Nat. Commun. 13, 2364. (2022)\nCrGeTe 66 K Phys. Rev. Lett. 127, 217203. (2021)\nLaCoO 85 K Adv. Sci. 10, 2303630. (2023)\n0.1 0.9 3\nBiMnO 105 K Phys. Rev. B 72, 020406. 2005\nLa 0.4 Sr 0.6 Mn 0.6 Ir 0.4 O 3 152 K 15414 (2022)\nACS Appl. Mater. Interfaces 13, 8863-\nLa 0.9 Ba 0.1 MnO 3 185 K 8870 (2021)\nLa NiMnO 280 K Phys. Rev. B 77, 174423. (2008)\n2 6\nLaCoO 2.5 284 K Nat. Commun. 12, 1853. (2021)\n2.5\nLa Sr MnO 560 K Phys. Rev. Lett. 109, 157207. (2012)\n0.67 0.33 3\nLaCoO -MgO 594 K Phys. Rev. Lett. 134, 016702. (2025)\n3\n[LaFeO /LaNiO ] 608 K Nat. Commun. 16, 3691. (2025)\n3 3 n\nLa CoO 820 K Nano Res.17, 7250-7255. (2024)\n2 4\nWe conducted a comprehensive survey of ferromagnetic insulators reported in previous\nstudies, which have been extensively investigated for their potential applications in\nspintronics, magnetic storage devices, and other advanced technologies. However, most\npreviously reported ferromagnetic insulating materials exhibit low Curie temperatures,\nsignificantly limiting their practical applications. Detailed statistical analysis of these\nmaterials is provided in Table R1.\n(2) First demonstration of coexisting ferrimagnetism and polar phase with\nswitched polarization in an oxide containing 4d metal ion: The coexistence of\nmagnetic and polar orders is inherently challenging due to the conflicting electronic\nstructure requirements for magnetism and ferroelectricity. Our work provides the first\nexperimental demonstration of room-temperature ferrimagnetism coupled with\nswitchable polar domains in a 4d-containing double perovskite. Given that 4d/5d\nelements possess strong spin-orbit coupling, a key factor in enhancing magnetoelectric\ncoupling, our discovery establishes a new material platform and creates the necessary\nprerequisites for exploring strong magnetoelectric effects and developing multiferroic\nfunctions in such promising spin-orbit coupled oxides.\nWe believe that these shared achievements, through sophisticated materials engineering\nto enable high-T ferrimagnetic insulating state with switchable polarization states,\nC\nrepresent a significant step forward and will generate broad interest in the fields of\nmultiferroic, complex oxides, and spintronics.\nResponse to Reviewers’ Comments\nWe appreciate the reviewers’ time and effort in reviewing our resubmission. The\nreviewers’ valuable comments and suggestions were very professional and helped us to\nfurther improve our manuscript. In this second round, we carefully addressed the\nadditional comment provided by Reviewer #2 and made all necessary changes in the\nrevised manuscript. All the revisions in the revised manuscript are highlighted in yellow.\nReviewer #1 :\n1. The authors have satisfactorily addressed all the concerns raised in my original\nreport. The manuscript has greatly benefited from the additional experiments and\ncalculations, which demonstrate the existence of switchable polar domains at room\ntemperature, despite the remaining experimental challenges associated with potential\nleakage currents. This work represents a significant advance in the field of room-\ntemperature multiferroics with strong spin–orbit interactions. I also appreciate the\nmodification of the title, which now more accurately reflects the main findings of the\npaper. Overall, I find the manuscript to be solid and well supported, and fully deserving\nof publication in Nature Communications.\nResponse: We are deeply grateful to the Reviewer for the final recommendation for\npublication in Nature Communication. The reviewer’s expert scrutiny during the\nreview process has been invaluable in elevating the quality and impact of our\nmanuscript.\nReviewer #2 :\nRevised Manuscript Commentary\nAfter a thorough examination of the submitted manuscript, I have identified several\nsubstantive concerns that remain insufficiently addressed by the authors’ response.\n1. Lack of Convincing Experimental Evidence for Ferroelectricity\nThe original article fails to provide unequivocal experimental proof of a ferroelectric\nstate. In their rebuttal, the authors acknowledge the absence of clear ferroelectric\nhysteresis loops that would demonstrate a polar, switchable dipole within the films.\nInstead, they observe a large leakage current.\nA particularly misleading element is Figure R3, wherein the authors present a\n“polarization versus electric field” plot labeled as a ferroelectric hysteresis loop. The\nI–V trace reveals a complete lack of switching current, indicating that the displayed\n“polarization” curves are merely the integrated response of a resistive-capacitive (RC)\ncircuit rather than evidence of ferroelectric switching. Consequently, no ferroelectric\nsignature can be inferred from these data.\nregarding the macroscopic P-E measurement. In the revised manuscript, we have\nsignificantly moderated and clarified the presentation of this result. It is now described\nas a preliminary electrical characterization with clear limitations: the observed low-\ntemperature “leaf-shaped” P-E loop may imply a polar response, but its non-ideal shape\nlikely results from polar nanodomain (i.e., dipole glass state) and significant leakage\nbackground. Therefore, this data alone cannot serve as conclusive evidence for\nferroelectricity or polarization switching.\nThis adjustment allows us to shift the focus of our argument from reliance on a single\nmacroscopic electrical measurement to a more robust evidence framework based on\nmultiple complementary experimental techniques. As detailed in our response to\nComment 2, our initial PFM domain manipulation results have now been supplemented\nwith local polarization switching characterizations on a single point, which reveal clear\nphase hysteresis and butterfly-shaped amplitude loops. These PFM results collectively\npoint to local polarization switching. More importantly, this local polar behavior is\nsupported and corroborated by an atomic-scale STEM (revealing the non-\ncentrosymmetric structure and polar nanodomains) and macroscopic SHG (confirming\nsymmetry breaking).\nConsequently, the revised manuscript establishes a central claim, demonstrating the\ncoexistence of a room-temperature ferrimagnetic order with a polar phase in LCRO\nfilms. This is why we revised the title in the previous version of the manuscript from\n“Room-temperature multiferroism in highly strained La CoRuO epitaxial films\n2 6\nthrough engineered 3d-4d cation ordering” to “Room-temperature ferrimagnetism and\npolar phase in highly strained La CoRuO epitaxial films through engineered 3d-4d\n2 6\ncation ordering”. This conclusion is not based on the macroscopic P-E loop. It is\ngrounded instead in the convergent evidence from the combination of PFM, STEM,\nand SHG results. Thus, we believe this refined and more precise narrative, honed\nthrough the review process, effectively addresses the reviewer’s concerns regarding the\nstrength of the evidence. We sincerely thank the reviewer for this valuable feedback,\nwhich was crucial in helping us clarify the interpretation of this key data.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (Page 20-21, Line 432-439) in the revised main text as following:\nOriginal text: “The macroscopic polarization of LCRO film was characterized by\npolarization-electric field (P-E) loop measurements, where a characteristic “leaf-\nshaped” P-E hysteresis at low temperature suggests a dipole glass state48\n(Supplementary Fig. 16b). This non-ideal loop shape primarily results from significant\npersistent leakage that remained even at low temperature (Supplementary Fig. 16a).”\nRevised text: “The macroscopic polarization of the LCRO film was initially\ncharacterized by polarization-electric field (P-E) loop measurements. A characteristic\n2\n“leaf-shaped” P-E hysteresis observed at low temperature (Supplementary Fig. 16b)\ncould be consistent with a polar response. However, its non-ideal shape likely results\nfrom polar nanodomain (or dipole glass state48) and significant persistent leakage, as\nevidenced by the accompanying current-voltage characteristics (Supplementary Fig.\n16a). Therefore, this macroscopic measurement alone cannot serve as conclusive\nevidence for polar switching.”\n2. Interpretation of Piezoresponse Force Microscopy (PFM) Results\nFigure R4 displays PFM measurements that the authors claim substantiate\nferroelectric switching. However, the data are more consistent with surface charging\neffects than with genuine ferroelectric domains. Specifically:\nAmplitude Signal: The left-hand panel shows a pronounced dependence of the PFM\namplitude on the applied writing voltage. In a bona fide ferroelectric material, the\namplitude of the piezoresponse is largely independent of the writing voltage; only the\nphase component varies with the polarity of the written domain. The observed\namplitude contrast can be attributed to electrostatic charging rather than to a reversal\nof spontaneous polarization.\nThus, the PFM results do not constitute credible evidence of ferroelectric domain\nformation or reversible polarization.\nResponse: We fully agree with the reviewer’s point that PFM measurements can be\nsusceptible to artifacts from surface charge deposition, especially in materials with a\nvery low polarization magnitude, as estimated for our LCRO film. We acknowledge\nthis critical consideration in the interpretation of PFM data.\nIn our experiments, we followed standard procedures to mitigate surface charge effects.\nThe sample surface was carefully grounded during PFM measurements, as\nrecommended in the operational manual for the MFP-3D system (Asylum Research/\nOxford Instruments). Despite these precautions, we recognize that a complete exclusion\nof electrostatic contributions can be challenging for low-polarization materials.\nFig. R1 (added as Supplementary Fig. 17) (a) PFM amplitude-voltage butterfly loop\n(b) and phase-voltage hysteresis loop of (111)-oriented LCRO film.\npolarization switching characterization was further performed at single points on the\n(111)-oriented LCRO thin film. The measured results exhibit butterfly-like amplitude\nloop and clear phase hysteresis loop (Fig. R1 added as Supplementary Fig. S17),\nwhich are characteristic of a piezoelectric response and indicative of polarization\nswitching.\nFig. R2 (a-c) PFM measurements, including amplitude images, phase images, phase-\nvoltage hysteresis loops and amplitude-voltage butterfly loops in (a)\nLa NiMnO /La CoMnO double-perovskite superlattices, (b) 6 uc freestanding BiFeO\n2 6 2 6 3\nthin film and (c) 2.4 nm epitaxial Bi OSe thin films.\n2 2\nFurthermore, we note that similar PFM signatures have been commonly reported in\nprevious studies on materials with small polarization, such as ultrathin oxide films or\nsystems with hybrid improper polarization. In these works, the PFM amplitude images\nmay also show a degree of voltage dependence and are not always “perfect”.\nNevertheless, the collective evidence from phase-voltage hysteresis loops, amplitude-\nvoltage butterfly loops, and amplitude images and phase images, has been accepted as\nvalid confirmation of switchable polar domains. Pertinent examples include:\n4\n(1) A strain-driven oxygen octahedral distortion strategy for hybrid improper\nferroelectricity with remnant polarization of ~ 0.16 μC/cm2 in La NiMnO /La CoMnO\n2 6 2 6\ndouble-perovskite superlattices (Figure R2a, Nat. Commun., 15(1), 5549 (2024)).\n(2) Ferroelectricity in ultrathin freestanding BiFeO thin films (Figure R2b,\n(3) The discovery of room-temperature ferroelectricity in ultrathin Two-Dimensional\nBi O Se films grown by molecular beam epitaxy. (Figure R2c, Nano Letters, DOI:\n2 2\n10.1021/acs.nanolett.5c04402).\nTherefore, we present our PFM results as part of a convergent and mutually reinforcing\nbody of evidence. The existence of a room-temperature polar phase in LCRO films is\nconvergently supported by the combination of local PFM switching behavior, atomic-\nscale STEM imaging of polar domains, and macroscopic SHG symmetry analysis,\ndespite the inherent limitations of PFM measurements. As Reviewer #3 also noted: “the\nTEM and SHG results do suggest the presence of polar domains”.\nIn summary, while our PFM data may not be flawless in isolation, they provide\nsupportive evidence for local polarization switching. The revised manuscript focuses\non establishing the coexistence of room-temperature ferrimagnetism and a polar phase\nin LCRO films, as reflected in the title. The core claim of a room-temperature polar\nphase is firmly grounded in the definitive evidence from SHG and STEM, with PFM\noffering supplementary, consistent insights at the local scale.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R1 as Supplementary Fig. 17 and revised sentence (Page 21, Line 442-\n446) in the revised main text as following:\nOriginal text: “Our PFM domain switching experiments successfully demonstrated\nreversible and stable domain writing and erasing.”\nRevised text: “We observed a clear butterfly loop and a corresponding phase hysteresis\nloop (Supplementary Fig. 17), which confirms the switchable of the polarization.\nSubsequently, our PFM domain switching experiments further successfully\ndemonstrated reversible and stable domain writing and erasing.”\n3. Discrepancy Between Long-Range Order Predicted by DFT and Local Structural\nDisorder\nDensity-functional theory (DFT) calculations predict a ferroelectric ground state\ncontingent upon a coherent, long-range gradient in the rotation of oxygen octahedra.\nExperimentally, however, the films appear to possess substantial local disorder that\ndisrupts the requisite long-range ordering. While such local distortions may generate\nisolated dipoles, the random orientation and strong coupling of these dipoles to\nstructural irregularities render collective polarization reversal implausible. The\n5\nauthors’ assertion that their PFM measurements confirm ferroelectric switching of\nsuch domains corroborates the suspicious of the feasibility of dipole reversal.\nResponse: We sincerely thank the reviewer for this insightful comment regarding the\nrelationship between our DFT predictions and the experimental observations. We fully\nagree that a careful explanation of the apparent discrepancy between the idealized\nmodel and the realistic film structure is essential.\nWe first wish to clarify the purpose of our DFT calculation featuring an oxygen\noctahedral rotation (OOR) gradient. This model serves as a proof-of-principle, aiming\nto demonstrate the fundamental physical mechanism that a spatially varying pattern of\noctahedral rotations can collectively break inversion symmetry and induce polarity. It\nis not intended as a perfect, atom-by-atom representation of the experimental film,\nwhich naturally possesses greater complexity.\nThe experimental data reveals significant local strain variations and an inhomogeneous\nevolution of OOR (Fig. 1b, 3c, e, f). This spatial inhomogeneity arises largely from the\ndistinct chemical characteristics of the CoO and RuO octahedra, leading to different\n6 6\nlocal responses to the epitaxial strain. Critically, the DFT model’s core conclusion—\nthat non-uniform OOR can drive a polar instability—remains valid even in this more\ncomplex, locally varied environment. The collective effect of these inhomogeneous\nrotations still enables the formation of the polar nanodomains directly confirmed by the\nSTEM images. Therefore, the theory and experiment are consistent in identifying the\nsame driving mechanism, while the experiment reflects its manifestation in a realistic\nmaterial with intrinsic chemical disorder.\nRegarding the switchability of such polarization, we acknowledge its coupling to\nstructural distortions, making reversal distinct from conventional ferroelectrics.\nHowever, this does not preclude local reorientation under an electric field. Our PFM\nobservations of reversible switching are consistent with behavior reported in other\noxide systems where polarization is strongly coupled to octahedral rotations. For\ninstance, a strain-driven oxygen octahedral distortion strategy for hybrid improper\nferroelectricity in La NiMnO /La CoMnO superlattices was demonstrated alongside\n2 6 2 6\nclear PFM switching (Nat. Commun. 15, 5549 (2024)). More broadly, the geometric\nferroelectricity literature describes systems where polarization arises from geometric\nstructural constraints and is subsequently shown to be switchable (Nat. Mater. 14, 407-\n413 (2015); Adv. Funct. Mater. 30, 2003623 (2020)).\nThus, our PFM evidence for local polarization reversal is plausible within this\nestablished context. Ultimately, the existence of a switchable polar phase is supported\nby a convergent set of independent measurements: (1) STEM reveals the atomic-scale\nnon-centrosymmetric structure; (2) SHG confirms the macroscopic polar symmetry;\n(3) PFM indicates local reversibility. We have refined the discussion in the manuscript to\nmore clearly articulate this relationship between the theoretical mechanism, the\n6\nexperimental structural landscape, and the collective evidence for a polar state. We\nthank the reviewer for prompting this important clarification.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (Page 24, Line 502-509) in the revised main text as following:-\nOriginal text: “This DFT scenario captures the essential polar mechanism, while the\nexperimentally observed nanoscale domain fragmentation results from intrinsic\nstructural complexities such as local strain variations and multi-mode oxygen\noctahedral distortions that extend beyond the idealized model (Fig. 6a-d).”\nRevised text: “This DFT model captures the essential polar mechanism whereby a\nspatially varying pattern of oxygen octahedral rotations breaks inversion symmetry. The\nnanoscale domain fragmentation observed experimentally arises from intrinsic\nstructural complexities that extend beyond the idealized model, including local strain\nvariations and the distinct multi-mode distortions of CoO and RuO octahedra (Fig.\n6 6\n6a-d). The system shows features analogous to a hybrid improper ferroelectric, in which\nthe polar state arises from structural distortions13-15.”\n4. Unexplained Low-Temperature Magnetization Upturn\nThe magnetization versus temperature curve exhibits an upturn at low temperatures.\nThis feature is not addressed in the authors’ response, leaving its origin ambiguous.\nPotential contributions—such as impurity phases, magnetic clustering, or spin-glass\nbehavior—should be investigated and discussed.\nResponse: We thank the reviewer for highlighting this subtle feature in our\nmagnetization data. The low-temperature upturn observed in the M-T curve of our\nstrained La CoRuO film is indeed an important point for discussion. Upon careful\n2 6\ninvestigation of the broader A CoRuO (A = rare earth) double-perovskite family, we\n2 6\nfind that this feature is not unique to our film but appears to be a characteristic of the\nferrimagnetic (FiM) members within this series.\nSpecifically, bulk La CoRuO , which is antiferromagnetic (AFM), does not exhibit this\n2 6\nlow-temperature upturn. However, in compounds where the A-site is occupied by\nsmaller rare-earth ions (Pr3+, Nd3+, Sm3+) that stabilize a FiM ground state, their\nreported M-T data consistently show a similar magnetization increase at low\ntemperatures (J. Phys.:Condens. Matter, 31,385801(2019)). The origin of this feature\nin these bulk compounds is not yet fully understood, but it has been attributed by some\nauthors to the formation of ferromagnetic (FM) clusters arising from static anti-site\ndisorder (J. Appl. Phys., 116, 193901 (2014); J. Alloys Compd., 773, 770 (2019)).\nGiven that our highly strained La CoRuO film also exhibits a FiM state induced by\n2 6\nepitaxial strain and engineered cation ordering, it is plausible that a similar mechanism\nis at play. As noted in the studies on these bulk analogues, localized magnetic clusters\n7\narising from anti-site disorder can give rise to such a low-temperature magnetization\nupturn. The contribution of these clusters becomes measurable only at low temperatures\nwhen the primary FiM matrix approaches saturation.\nWe have added a necessary discussion regarding this low-temperature upturn in the\nrevised manuscript. However, this feature does not affect our central conclusion that\nthe film possesses a robust ferrimagnetic order with a Curie temperature well above\nroom temperature.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded new sentence (see Page 10, Line 232-235) in the revised main text as following:\nAdded sentence:\n“Additionally, an upturn in magnetization is present at low temperatures. This feature\ncould be attributed to ferromagnetic clusters resulting from static anti-site disorder (Co-\nO-Co and Ru-O-Ru), consistent with observations in related bulk A CoRuO double\n2 6\nperovskites22.”\n5. Conclusion\nGiven the unresolved issues outlined above—namely, the absence of definitive\nferroelectric hysteresis, the misinterpretation of PFM data, the inconsistency between\ntheoretical predictions and experimental structural disorder, and the unexplained\nmagnetic anomaly—I remain unconvinced that the manuscript meets the standards\nrequired for publication in a peer-reviewed scientific journal. Further experimental\nvalidation and comprehensive analysis are necessary before the work can be\nconsidered suitable for dissemination.\nResponse: We sincerely thank the reviewer for their thorough assessment, which has\nprompted a comprehensive revision of our work. We have carefully addressed the core\nconcerns regarding the ferroelectric evidence, PFM interpretation, theory-experiment\nconsistency, and low-temperature magnetic anomaly.\nIn response, we have conducted additional experiments, refined our analyses, and\nmoderated our claims throughout the manuscript. This includes a more precise\npresentation of the polar phase, supported by converging evidence from PFM,\nSTEM, and SHG. To accurately reflect this characterization, the previous revised\nmanuscript title has been revised to “Room-temperature ferrimagnetism and polar phase\nin highly strained La CoRuO epitaxial films through engineered 3d-4d cation ordering”.\n2 6\nWe believe the revised manuscript now clearly establishes, for the first time, the\ncoexistence of a room-temperature ferrimagnetic insulator with a polar phase in a 3d-\n4d cation-ordered double perovskite. We are grateful for the reviewer’s insightful\ncomments and hope the reviewer finds that our detailed responses and the revised\n8\nmanuscript, which now presents a more precise and well-supported narrative, meet the\nhigh standards for publication.\nIn addition, we would like to emphasize the novelty of this work in the following\naspects, and further elaborate on its significance and impact on high-temperature\nferromagnetic insulators and the multiferroic field.\n(1) High-temperature ferrimagnetic insulator realized through B-site ordering\nand Co2+ high-spin state: Realizing high-temperature magnetic insulators is a\nfundamental challenge, as ferromagnetic order typically promotes metallic conductivity\nthrough enhanced electron hopping. While a few intrinsic ferromagnetic insulators are\nknown, their magnetic ordering temperatures generally remain far below 300 K,\nlimiting their utility for practical room-temperature spintronic devices. (see Table R1).\nOur work demonstrates a robust, high-temperature ferrimagnetic insulator in epitaxial\nLa CoRuO films. The achievement is enabled by two key design principles: the\n2 6\nestablishment of a high degree of B-site cation ordering and the stabilization of Co2+ in\na high-spin state. By combining this cation ordering with epitaxial strain engineering,\nwe realized a ferrimagnetic ground state with a Curie temperature of 623 K. This\nmaterial system thus represents a significant advance in the design of candidate\nmaterials for room-temperature spintronics.\nTable R1 Survey of Curie temperature for ferromagnetic insulators.\nFMIs Tc Reference\nEuTiO 3.4 K Nat. Commun. 13, 2364. (2022)\nCrGeTe 66 K Phys. Rev. Lett. 127, 217203. (2021)\nLaCoO 85 K Adv. Sci. 10, 2303630. (2023)\n0.1 0.9 3\nBiMnO 105 K Phys. Rev. B 72, 020406. 2005\nLa 0.4 Sr 0.6 Mn 0.6 Ir 0.4 O 3 152 K 15414 (2022)\nACS Appl. Mater. Interfaces 13, 8863-\nLa 0.9 Ba 0.1 MnO 3 185 K 8870 (2021)\nLa NiMnO 280 K Phys. Rev. B 77, 174423. (2008)\n2 6\nLaCoO 2.5 284 K Nat. Commun. 12, 1853. (2021)\n2.5\nLa Sr MnO 560 K Phys. Rev. Lett. 109, 157207. (2012)\n0.67 0.33 3\n3\n[LaFeO /LaNiO ] 608 K Nat. Commun. 16, 3691. (2025)\n3 3 n\nLa CoO 820 K Nano Res.17, 7250-7255. (2024)\n2 4\nWe conducted a comprehensive survey of ferromagnetic insulators reported in previous\nstudies, which have been extensively investigated for their potential applications in\nspintronics, magnetic storage devices, and other advanced technologies. However, most\npreviously reported ferromagnetic insulating materials exhibit low Curie temperatures,\nsignificantly limiting their practical applications. Detailed statistical analysis of these\nmaterials is provided in Table R1.\n(2) First demonstration of coexisting room-temperature ferrimagnetism and polar\nphase in a 4d-oxide: The simultaneous stabilization of magnetic and polar orders is\nnotoriously difficult, as their underlying mechanisms often impose competing\nelectronic requirements. In this work, we experimentally demonstrate the coexistence\nof a room-temperature ferrimagnetic state with electrically switchable polar domains in\na double perovskite incorporating a 4d transition metal ion. Elements from the 4d series\nexhibit strong spin-orbit coupling, a property considered key for enabling substantial\nmagnetoelectric interactions. Therefore, this material not only establishes a new\nplatform for room-temperature multifunctionality but also creates the essential\npreconditions for exploring strong magnetoelectric coupling and engineering\nmultiferroic responses in this promising class of spin-orbit-coupled oxides.\nWe believe that these shared achievements, through sophisticated materials engineering\nto enable high-T ferrimagnetic insulating state with a polar phase, represent a\nC\nsignificant step forward and will generate broad interest in the fields of multiferroic and\nspintronics.\nReviewer #3:\n1. In this resubmission, the authors address in detail the various issues raised earlier\nby the reviewers. The most critical aspect, in my opinion, related to the demonstration\nof ferroelectric order in the system and the claim for multiferroicity. The authors have\naddressed these aspects, one on the hand by carrying out ferroelectric characterization\n(P-E switching and PFM), and on the other, by toning down the claims of the original\nsubmission. As the authors admit, the P-E loops are rather inconclusive, while the PFM\nmeasurements, while suggesting ferroelectric order, could also suffer from artifacts,\nsuch as charge deposition, in particular considering the very low estimated\nferroelectric polarization. However, the TEM and SHG results do suggest the presence\n10\nof polar domains, so the evidence points to the presence of ferroelectric order in the\nsystem. The revised version is much improved compared to the original, with a better\npresentation of the first principle results and of the STEM data. Overall, I find the\npresent work to be of high quality and original, highlighting how controlling the cation\ncontent and arrangement in the double perovskites can be explored to tune the\nelectronic and functional properties of this class of materials. Hence, I can recommend\npublication of this MS in Nature Communications.\nResponse: We sincerely appreciate the Reviewer’s final decision and thoughtful\nrecognition of our revisions. The reviewer’s constructive feedback throughout the\nreview process has been essential in strengthening the overall quality and presentation\nof our work.\nWe appreciate the reviewer’s time and effort in reviewing our re-submission. The\nreviewer’s valuable comments and suggestions were very professional and helped us to\nfurther improve our manuscript.\nReviewer #1 :\nThe authors have thoroughly and convincingly addressed the comments raised by\nreferees during the previous review round. In particular, they now demonstrate the\nexistence of polar domains using several complementary experimental techniques. In\naddition, they have appropriately moderated the tone of their claims, for instance by\nrevising the title of the manuscript.\nWhile certain measurements, when considered in isolation—such as PFM data or the\nabsence of a clear P-E hysteresis loop—might suggest that the system is not\nferroelectric, the manuscript clearly shows that a broad range of other experimental\napproaches provides strong and consistent evidence that the system is indeed polar.\nTaken together, these results support the presence of well-defined polar domains and\nmake the overall interpretation robust.\nIn this revised version, the authors further reinforce their conclusions by introducing\nseveral new arguments and by highlighting clear similarities with other polar and\nferroelectric systems based on double perovskites. A similarly careful and well-\nsupported analysis is presented to explain the small upturn observed in the\nmagnetization curve, which is now satisfactorily accounted for.\nIn my opinion, the authors have done an excellent job and have provided all reasonable\nevidence supporting the claims made in the manuscript. This work represents a\nsignificant advance in the field of oxide-based electronics, and I therefore consider the\nmanuscript to be fully suitable for publication in Nature Communications.\nResponse: We are deeply grateful to the Reviewer for the final recommendation for\npublication in Nature Communication. The reviewer’s expert scrutiny during the\nreview process has been invaluable in elevating the quality and impact of our\nmanuscript."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have satisfactorily addressed all the concerns raised in my original report. The manuscript has greatly benefited from the additional experiments and calculations, which demonstrate the existence of switchable polar domains at room temperature, despite the remaining experimental challenges associated with potential leakage currents. This work represents a significant advance in the field of room-temperature multiferroics with strong spin–orbit interactions. I also appreciate the modification of the title, which now more accurately reflects the main findings of the paper. Overall, I find the manuscript to be solid and well supported, and fully deserving of publication in Nature Communications.", "reviewer_2": "Revised Manuscript Commentary\nAfter a thorough examination of the submitted manuscript, I have identified several substantive concerns that remain insufficiently addressed by the authors’ response.\n1. Lack of Convincing Experimental Evidence for Ferroelectricity\nThe original article fails to provide unequivocal experimental proof of a ferroelectric state. In their rebuttal, the authors acknowledge the absence of clear ferroelectric hysteresis loops that would demonstrate a polar, switchable dipole within the films. Instead, they observe a large leakage current.\nA particularly misleading element is Figure R3, wherein the authors present a “polarization versus electric field” plot labeled as a ferroelectric hysteresis loop. The I–V trace reveals a complete lack of switching current, indicating that the displayed “polarization” curves are merely the integrated response of a resistive‑capacitive (RC) circuit rather than evidence of ferroelectric switching. Consequently, no ferroelectric signature can be inferred from these data.\n2. Interpretation of Piezoresponse Force Microscopy (PFM) Results\nFigure R4 displays PFM measurements that the authors claim substantiate ferroelectric switching. However, the data are more consistent with surface charging effects than with genuine ferroelectric domains. Specifically:\n• Amplitude Signal: The left‑hand panel shows a pronounced dependence of the PFM amplitude on the applied writing voltage. In a bona fide ferroelectric material, the amplitude of the piezoresponse is largely independent of the writing voltage; only the phase component varies with the polarity of the written domain. The observed amplitude contrast can be attributed to electrostatic charging rather than to a reversal of spontaneous polarization.\nThus, the PFM results do not constitute credible evidence of ferroelectric domain formation or reversible polarization.\n3. Discrepancy Between Long‑Range Order Predicted by DFT and Local Structural Disorder\nDensity-functional theory (DFT) calculations predict a ferroelectric ground state contingent upon a coherent, long‑range gradient in the rotation of oxygen octahedra. Experimentally, however, the films appear to possess substantial local disorder that disrupts the requisite long‑range ordering. While such local distortions may generate isolated dipoles, the random orientation and strong coupling of these dipoles to structural irregularities render collective polarization reversal implausible. The authors’ assertion that their PFM measurements confirm ferroelectric switching of such domains corroborates the suspicious of the feasibility of dipole reversal.\n4. Unexplained Low‑Temperature Magnetization Upturn\nThe magnetization versus temperature curve exhibits an upturn at low temperatures. This feature is not addressed in the authors’ response, leaving its origin ambiguous. Potential contributions—such as impurity phases, magnetic clustering, or spin‑glass behavior—should be investigated and discussed.\nConclusion\nGiven the unresolved issues outlined above—namely, the absence of definitive ferroelectric hysteresis, the misinterpretation of PFM data, the inconsistency between theoretical predictions and experimental structural disorder, and the unexplained magnetic anomaly—I remain unconvinced that the manuscript meets the standards required for publication in a peer‑reviewed scientific journal. Further experimental validation and comprehensive analysis are necessary before the work can be considered suitable for dissemination.", "reviewer_3": "In this resubmission, the authors address in detail the various issues raised earlier by the reviewers. The most critical aspect, in my opinion, related to the demonstration of ferroelectric order in the system and the claim for multiferroicity. The authors have addressed these aspects, one on the hand by carrying out ferroelectric characterization (P-E switching and PFM), and on the other, by toning down the claims of the original submission. As the authors admit, the P-E loops are rather inconclusive, while the PFM measurements, while suggesting ferroelectric order, could also suffer from artifacts, such as charge deposition, in particular considering the very low estimated ferroelectric polarization. However, the TEM and SHG results do suggest the presence of polar domains, so the evidence points to the presence of ferroelectric order in the system. The revised version is much improved compared to the original, with a better presentation of the first principle results and of the STEM data. Overall, I find the present work to be of high quality and original, highlighting how controlling the cation content and arrangement in the double perovskites can be explored to tune the electronic and functional properties of this class of materials. Hence, I can recommend publication of this MS in Nature Communications."}, "author_response": "Response to Reviewers’ Comments\nWe appreciate the reviewers’ time and effort in reviewing our resubmission. The\nreviewers’ valuable comments and suggestions were very professional and helped us to\nfurther improve our manuscript. In this second round, we carefully addressed the\nadditional comment provided by Reviewer #2 and made all necessary changes in the\nrevised manuscript. All the revisions in the revised manuscript are highlighted in yellow.\nReviewer #1 :\n1. The authors have satisfactorily addressed all the concerns raised in my original\nreport. The manuscript has greatly benefited from the additional experiments and\ncalculations, which demonstrate the existence of switchable polar domains at room\ntemperature, despite the remaining experimental challenges associated with potential\nleakage currents. This work represents a significant advance in the field of room-\ntemperature multiferroics with strong spin–orbit interactions. I also appreciate the\nmodification of the title, which now more accurately reflects the main findings of the\npaper. Overall, I find the manuscript to be solid and well supported, and fully deserving\nof publication in Nature Communications.\nResponse: We are deeply grateful to the Reviewer for the final recommendation for\npublication in Nature Communication. The reviewer’s expert scrutiny during the\nreview process has been invaluable in elevating the quality and impact of our\nmanuscript.\nReviewer #2 :\nRevised Manuscript Commentary\nAfter a thorough examination of the submitted manuscript, I have identified several\nsubstantive concerns that remain insufficiently addressed by the authors’ response.\n1. Lack of Convincing Experimental Evidence for Ferroelectricity\nThe original article fails to provide unequivocal experimental proof of a ferroelectric\nstate. In their rebuttal, the authors acknowledge the absence of clear ferroelectric\nhysteresis loops that would demonstrate a polar, switchable dipole within the films.\nInstead, they observe a large leakage current.\nA particularly misleading element is Figure R3, wherein the authors present a\n“polarization versus electric field” plot labeled as a ferroelectric hysteresis loop. The\nI–V trace reveals a complete lack of switching current, indicating that the displayed\n“polarization” curves are merely the integrated response of a resistive-capacitive (RC)\ncircuit rather than evidence of ferroelectric switching. Consequently, no ferroelectric\nsignature can be inferred from these data.\nregarding the macroscopic P-E measurement. In the revised manuscript, we have\nsignificantly moderated and clarified the presentation of this result. It is now described\nas a preliminary electrical characterization with clear limitations: the observed low-\ntemperature “leaf-shaped” P-E loop may imply a polar response, but its non-ideal shape\nlikely results from polar nanodomain (i.e., dipole glass state) and significant leakage\nbackground. Therefore, this data alone cannot serve as conclusive evidence for\nferroelectricity or polarization switching.\nThis adjustment allows us to shift the focus of our argument from reliance on a single\nmacroscopic electrical measurement to a more robust evidence framework based on\nmultiple complementary experimental techniques. As detailed in our response to\nComment 2, our initial PFM domain manipulation results have now been supplemented\nwith local polarization switching characterizations on a single point, which reveal clear\nphase hysteresis and butterfly-shaped amplitude loops. These PFM results collectively\npoint to local polarization switching. More importantly, this local polar behavior is\nsupported and corroborated by an atomic-scale STEM (revealing the non-\ncentrosymmetric structure and polar nanodomains) and macroscopic SHG (confirming\nsymmetry breaking).\nConsequently, the revised manuscript establishes a central claim, demonstrating the\ncoexistence of a room-temperature ferrimagnetic order with a polar phase in LCRO\nfilms. This is why we revised the title in the previous version of the manuscript from\n“Room-temperature multiferroism in highly strained La CoRuO epitaxial films\n2 6\nthrough engineered 3d-4d cation ordering” to “Room-temperature ferrimagnetism and\npolar phase in highly strained La CoRuO epitaxial films through engineered 3d-4d\n2 6\ncation ordering”. This conclusion is not based on the macroscopic P-E loop. It is\ngrounded instead in the convergent evidence from the combination of PFM, STEM,\nand SHG results. Thus, we believe this refined and more precise narrative, honed\nthrough the review process, effectively addresses the reviewer’s concerns regarding the\nstrength of the evidence. We sincerely thank the reviewer for this valuable feedback,\nwhich was crucial in helping us clarify the interpretation of this key data.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (Page 20-21, Line 432-439) in the revised main text as following:\nOriginal text: “The macroscopic polarization of LCRO film was characterized by\npolarization-electric field (P-E) loop measurements, where a characteristic “leaf-\nshaped” P-E hysteresis at low temperature suggests a dipole glass state48\n(Supplementary Fig. 16b). This non-ideal loop shape primarily results from significant\npersistent leakage that remained even at low temperature (Supplementary Fig. 16a).”\nRevised text: “The macroscopic polarization of the LCRO film was initially\ncharacterized by polarization-electric field (P-E) loop measurements. A characteristic\n2\n“leaf-shaped” P-E hysteresis observed at low temperature (Supplementary Fig. 16b)\ncould be consistent with a polar response. However, its non-ideal shape likely results\nfrom polar nanodomain (or dipole glass state48) and significant persistent leakage, as\nevidenced by the accompanying current-voltage characteristics (Supplementary Fig.\n16a). Therefore, this macroscopic measurement alone cannot serve as conclusive\nevidence for polar switching.”\n2. Interpretation of Piezoresponse Force Microscopy (PFM) Results\nFigure R4 displays PFM measurements that the authors claim substantiate\nferroelectric switching. However, the data are more consistent with surface charging\neffects than with genuine ferroelectric domains. Specifically:\nAmplitude Signal: The left-hand panel shows a pronounced dependence of the PFM\namplitude on the applied writing voltage. In a bona fide ferroelectric material, the\namplitude of the piezoresponse is largely independent of the writing voltage; only the\nphase component varies with the polarity of the written domain. The observed\namplitude contrast can be attributed to electrostatic charging rather than to a reversal\nof spontaneous polarization.\nThus, the PFM results do not constitute credible evidence of ferroelectric domain\nformation or reversible polarization.\nResponse: We fully agree with the reviewer’s point that PFM measurements can be\nsusceptible to artifacts from surface charge deposition, especially in materials with a\nvery low polarization magnitude, as estimated for our LCRO film. We acknowledge\nthis critical consideration in the interpretation of PFM data.\nIn our experiments, we followed standard procedures to mitigate surface charge effects.\nThe sample surface was carefully grounded during PFM measurements, as\nrecommended in the operational manual for the MFP-3D system (Asylum Research/\nOxford Instruments). Despite these precautions, we recognize that a complete exclusion\nof electrostatic contributions can be challenging for low-polarization materials.\nFig. R1 (added as Supplementary Fig. 17) (a) PFM amplitude-voltage butterfly loop\n(b) and phase-voltage hysteresis loop of (111)-oriented LCRO film.\npolarization switching characterization was further performed at single points on the\n(111)-oriented LCRO thin film. The measured results exhibit butterfly-like amplitude\nloop and clear phase hysteresis loop (Fig. R1 added as Supplementary Fig. S17),\nwhich are characteristic of a piezoelectric response and indicative of polarization\nswitching.\nFig. R2 (a-c) PFM measurements, including amplitude images, phase images, phase-\nvoltage hysteresis loops and amplitude-voltage butterfly loops in (a)\nLa NiMnO /La CoMnO double-perovskite superlattices, (b) 6 uc freestanding BiFeO\n2 6 2 6 3\nthin film and (c) 2.4 nm epitaxial Bi OSe thin films.\n2 2\nFurthermore, we note that similar PFM signatures have been commonly reported in\nprevious studies on materials with small polarization, such as ultrathin oxide films or\nsystems with hybrid improper polarization. In these works, the PFM amplitude images\nmay also show a degree of voltage dependence and are not always “perfect”.\nNevertheless, the collective evidence from phase-voltage hysteresis loops, amplitude-\nvoltage butterfly loops, and amplitude images and phase images, has been accepted as\nvalid confirmation of switchable polar domains. Pertinent examples include:\n4\n(1) A strain-driven oxygen octahedral distortion strategy for hybrid improper\nferroelectricity with remnant polarization of ~ 0.16 μC/cm2 in La NiMnO /La CoMnO\n2 6 2 6\ndouble-perovskite superlattices (Figure R2a, Nat. Commun., 15(1), 5549 (2024)).\n(2) Ferroelectricity in ultrathin freestanding BiFeO thin films (Figure R2b,\n(3) The discovery of room-temperature ferroelectricity in ultrathin Two-Dimensional\nBi O Se films grown by molecular beam epitaxy. (Figure R2c, Nano Letters, DOI:\n2 2\n10.1021/acs.nanolett.5c04402).\nTherefore, we present our PFM results as part of a convergent and mutually reinforcing\nbody of evidence. The existence of a room-temperature polar phase in LCRO films is\nconvergently supported by the combination of local PFM switching behavior, atomic-\nscale STEM imaging of polar domains, and macroscopic SHG symmetry analysis,\ndespite the inherent limitations of PFM measurements. As Reviewer #3 also noted: “the\nTEM and SHG results do suggest the presence of polar domains”.\nIn summary, while our PFM data may not be flawless in isolation, they provide\nsupportive evidence for local polarization switching. The revised manuscript focuses\non establishing the coexistence of room-temperature ferrimagnetism and a polar phase\nin LCRO films, as reflected in the title. The core claim of a room-temperature polar\nphase is firmly grounded in the definitive evidence from SHG and STEM, with PFM\noffering supplementary, consistent insights at the local scale.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded Fig. R1 as Supplementary Fig. 17 and revised sentence (Page 21, Line 442-\n446) in the revised main text as following:\nOriginal text: “Our PFM domain switching experiments successfully demonstrated\nreversible and stable domain writing and erasing.”\nRevised text: “We observed a clear butterfly loop and a corresponding phase hysteresis\nloop (Supplementary Fig. 17), which confirms the switchable of the polarization.\nSubsequently, our PFM domain switching experiments further successfully\ndemonstrated reversible and stable domain writing and erasing.”\n3. Discrepancy Between Long-Range Order Predicted by DFT and Local Structural\nDisorder\nDensity-functional theory (DFT) calculations predict a ferroelectric ground state\ncontingent upon a coherent, long‑range gradient in the rotation of oxygen octahedra.\nExperimentally, however, the films appear to possess substantial local disorder that\ndisrupts the requisite long‑range ordering. While such local distortions may generate\nisolated dipoles, the random orientation and strong coupling of these dipoles to\nstructural irregularities render collective polarization reversal implausible. The\n5\nauthors’ assertion that their PFM measurements confirm ferroelectric switching of\nsuch domains corroborates the suspicious of the feasibility of dipole reversal.\nResponse: We sincerely thank the reviewer for this insightful comment regarding the\nrelationship between our DFT predictions and the experimental observations. We fully\nagree that a careful explanation of the apparent discrepancy between the idealized\nmodel and the realistic film structure is essential.\nWe first wish to clarify the purpose of our DFT calculation featuring an oxygen\noctahedral rotation (OOR) gradient. This model serves as a proof-of-principle, aiming\nto demonstrate the fundamental physical mechanism that a spatially varying pattern of\noctahedral rotations can collectively break inversion symmetry and induce polarity. It\nis not intended as a perfect, atom-by-atom representation of the experimental film,\nwhich naturally possesses greater complexity.\nThe experimental data reveals significant local strain variations and an inhomogeneous\nevolution of OOR (Fig. 1b, 3c, e, f). This spatial inhomogeneity arises largely from the\ndistinct chemical characteristics of the CoO and RuO octahedra, leading to different\n6 6\nlocal responses to the epitaxial strain. Critically, the DFT model’s core conclusion—\nthat non-uniform OOR can drive a polar instability—remains valid even in this more\ncomplex, locally varied environment. The collective effect of these inhomogeneous\nrotations still enables the formation of the polar nanodomains directly confirmed by the\nSTEM images. Therefore, the theory and experiment are consistent in identifying the\nsame driving mechanism, while the experiment reflects its manifestation in a realistic\nmaterial with intrinsic chemical disorder.\nRegarding the switchability of such polarization, we acknowledge its coupling to\nstructural distortions, making reversal distinct from conventional ferroelectrics.\nHowever, this does not preclude local reorientation under an electric field. Our PFM\nobservations of reversible switching are consistent with behavior reported in other\noxide systems where polarization is strongly coupled to octahedral rotations. For\ninstance, a strain-driven oxygen octahedral distortion strategy for hybrid improper\nferroelectricity in La NiMnO /La CoMnO superlattices was demonstrated alongside\n2 6 2 6\nclear PFM switching (Nat. Commun. 15, 5549 (2024)). More broadly, the geometric\nferroelectricity literature describes systems where polarization arises from geometric\nstructural constraints and is subsequently shown to be switchable (Nat. Mater. 14, 407-\n413 (2015); Adv. Funct. Mater. 30, 2003623 (2020)).\nThus, our PFM evidence for local polarization reversal is plausible within this\nestablished context. Ultimately, the existence of a switchable polar phase is supported\nby a convergent set of independent measurements: (1) STEM reveals the atomic-scale\nnon-centrosymmetric structure; (2) SHG confirms the macroscopic polar symmetry;\n(3) PFM indicates local reversibility. We have refined the discussion in the manuscript to\nmore clearly articulate this relationship between the theoretical mechanism, the\n6\nexperimental structural landscape, and the collective evidence for a polar state. We\nthank the reviewer for prompting this important clarification.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nrevised sentence (Page 24, Line 502-509) in the revised main text as following:-\nOriginal text: “This DFT scenario captures the essential polar mechanism, while the\nexperimentally observed nanoscale domain fragmentation results from intrinsic\nstructural complexities such as local strain variations and multi-mode oxygen\noctahedral distortions that extend beyond the idealized model (Fig. 6a-d).”\nRevised text: “This DFT model captures the essential polar mechanism whereby a\nspatially varying pattern of oxygen octahedral rotations breaks inversion symmetry. The\nnanoscale domain fragmentation observed experimentally arises from intrinsic\nstructural complexities that extend beyond the idealized model, including local strain\nvariations and the distinct multi-mode distortions of CoO and RuO octahedra (Fig.\n6 6\n6a-d). The system shows features analogous to a hybrid improper ferroelectric, in which\nthe polar state arises from structural distortions13-15.”\n4. Unexplained Low‑Temperature Magnetization Upturn\nThe magnetization versus temperature curve exhibits an upturn at low temperatures.\nThis feature is not addressed in the authors’ response, leaving its origin ambiguous.\nPotential contributions—such as impurity phases, magnetic clustering, or spin‑glass\nbehavior—should be investigated and discussed.\nResponse: We thank the reviewer for highlighting this subtle feature in our\nmagnetization data. The low-temperature upturn observed in the M-T curve of our\nstrained La CoRuO film is indeed an important point for discussion. Upon careful\n2 6\ninvestigation of the broader A CoRuO (A = rare earth) double-perovskite family, we\n2 6\nfind that this feature is not unique to our film but appears to be a characteristic of the\nferrimagnetic (FiM) members within this series.\nSpecifically, bulk La CoRuO , which is antiferromagnetic (AFM), does not exhibit this\n2 6\nlow-temperature upturn. However, in compounds where the A-site is occupied by\nsmaller rare-earth ions (Pr3+, Nd3+, Sm3+) that stabilize a FiM ground state, their\nreported M-T data consistently show a similar magnetization increase at low\ntemperatures (J. Phys.:Condens. Matter, 31,385801(2019)). The origin of this feature\nin these bulk compounds is not yet fully understood, but it has been attributed by some\nauthors to the formation of ferromagnetic (FM) clusters arising from static anti-site\ndisorder (J. Appl. Phys., 116, 193901 (2014); J. Alloys Compd., 773, 770 (2019)).\nGiven that our highly strained La CoRuO film also exhibits a FiM state induced by\n2 6\nepitaxial strain and engineered cation ordering, it is plausible that a similar mechanism\nis at play. As noted in the studies on these bulk analogues, localized magnetic clusters\n7\narising from anti-site disorder can give rise to such a low-temperature magnetization\nupturn. The contribution of these clusters becomes measurable only at low temperatures\nwhen the primary FiM matrix approaches saturation.\nWe have added a necessary discussion regarding this low-temperature upturn in the\nrevised manuscript. However, this feature does not affect our central conclusion that\nthe film possesses a robust ferrimagnetic order with a Curie temperature well above\nroom temperature.\nAction taken: In light of the Reviewer’s feedback and the above discussion, we have\nadded new sentence (see Page 10, Line 232-235) in the revised main text as following:\nAdded sentence:\n“Additionally, an upturn in magnetization is present at low temperatures. This feature\ncould be attributed to ferromagnetic clusters resulting from static anti-site disorder (Co-\nO-Co and Ru-O-Ru), consistent with observations in related bulk A CoRuO double\n2 6\nperovskites22.”\n5. Conclusion\nGiven the unresolved issues outlined above—namely, the absence of definitive\nferroelectric hysteresis, the misinterpretation of PFM data, the inconsistency between\ntheoretical predictions and experimental structural disorder, and the unexplained\nmagnetic anomaly—I remain unconvinced that the manuscript meets the standards\nrequired for publication in a peer‑reviewed scientific journal. Further experimental\nvalidation and comprehensive analysis are necessary before the work can be\nconsidered suitable for dissemination.\nResponse: We sincerely thank the reviewer for their thorough assessment, which has\nprompted a comprehensive revision of our work. We have carefully addressed the core\nconcerns regarding the ferroelectric evidence, PFM interpretation, theory-experiment\nconsistency, and low-temperature magnetic anomaly.\nIn response, we have conducted additional experiments, refined our analyses, and\nmoderated our claims throughout the manuscript. This includes a more precise\npresentation of the polar phase, supported by converging evidence from PFM,\nSTEM, and SHG. To accurately reflect this characterization, the previous revised\nmanuscript title has been revised to “Room-temperature ferrimagnetism and polar phase\nin highly strained La CoRuO epitaxial films through engineered 3d-4d cation ordering”.\n2 6\nWe believe the revised manuscript now clearly establishes, for the first time, the\ncoexistence of a room-temperature ferrimagnetic insulator with a polar phase in a 3d-\n4d cation-ordered double perovskite. We are grateful for the reviewer’s insightful\ncomments and hope the reviewer finds that our detailed responses and the revised\n8\nmanuscript, which now presents a more precise and well-supported narrative, meet the\nhigh standards for publication.\nIn addition, we would like to emphasize the novelty of this work in the following\naspects, and further elaborate on its significance and impact on high-temperature\nferromagnetic insulators and the multiferroic field.\n(1) High-temperature ferrimagnetic insulator realized through B-site ordering\nand Co2+ high-spin state: Realizing high-temperature magnetic insulators is a\nfundamental challenge, as ferromagnetic order typically promotes metallic conductivity\nthrough enhanced electron hopping. While a few intrinsic ferromagnetic insulators are\nknown, their magnetic ordering temperatures generally remain far below 300 K,\nlimiting their utility for practical room-temperature spintronic devices. (see Table R1).\nOur work demonstrates a robust, high-temperature ferrimagnetic insulator in epitaxial\nLa CoRuO films. The achievement is enabled by two key design principles: the\n2 6\nestablishment of a high degree of B-site cation ordering and the stabilization of Co2+ in\na high-spin state. By combining this cation ordering with epitaxial strain engineering,\nwe realized a ferrimagnetic ground state with a Curie temperature of 623 K. This\nmaterial system thus represents a significant advance in the design of candidate\nmaterials for room-temperature spintronics.\nTable R1 Survey of Curie temperature for ferromagnetic insulators.\nFMIs Tc Reference\nEuTiO 3.4 K Nat. Commun. 13, 2364. (2022)\nCrGeTe 66 K Phys. Rev. Lett. 127, 217203. (2021)\nLaCoO 85 K Adv. Sci. 10, 2303630. (2023)\n0.1 0.9 3\nBiMnO 105 K Phys. Rev. B 72, 020406. 2005\nLa 0.4 Sr 0.6 Mn 0.6 Ir 0.4 O 3 152 K 15414 (2022)\nACS Appl. Mater. Interfaces 13, 8863-\nLa 0.9 Ba 0.1 MnO 3 185 K 8870 (2021)\nLa NiMnO 280 K Phys. Rev. B 77, 174423. (2008)\n2 6\nLaCoO 2.5 284 K Nat. Commun. 12, 1853. (2021)\n2.5\nLa Sr MnO 560 K Phys. Rev. Lett. 109, 157207. (2012)\n0.67 0.33 3\n3\n[LaFeO /LaNiO ] 608 K Nat. Commun. 16, 3691. (2025)\n3 3 n\nLa CoO 820 K Nano Res.17, 7250-7255. (2024)\n2 4\nWe conducted a comprehensive survey of ferromagnetic insulators reported in previous\nstudies, which have been extensively investigated for their potential applications in\nspintronics, magnetic storage devices, and other advanced technologies. However, most\npreviously reported ferromagnetic insulating materials exhibit low Curie temperatures,\nsignificantly limiting their practical applications. Detailed statistical analysis of these\nmaterials is provided in Table R1.\n(2) First demonstration of coexisting room-temperature ferrimagnetism and polar\nphase in a 4d-oxide: The simultaneous stabilization of magnetic and polar orders is\nnotoriously difficult, as their underlying mechanisms often impose competing\nelectronic requirements. In this work, we experimentally demonstrate the coexistence\nof a room-temperature ferrimagnetic state with electrically switchable polar domains in\na double perovskite incorporating a 4d transition metal ion. Elements from the 4d series\nexhibit strong spin-orbit coupling, a property considered key for enabling substantial\nmagnetoelectric interactions. Therefore, this material not only establishes a new\nplatform for room-temperature multifunctionality but also creates the essential\npreconditions for exploring strong magnetoelectric coupling and engineering\nmultiferroic responses in this promising class of spin-orbit-coupled oxides.\nWe believe that these shared achievements, through sophisticated materials engineering\nto enable high-T ferrimagnetic insulating state with a polar phase, represent a\nC\nsignificant step forward and will generate broad interest in the fields of multiferroic and\nspintronics.\nReviewer #3:\n1. In this resubmission, the authors address in detail the various issues raised earlier\nby the reviewers. The most critical aspect, in my opinion, related to the demonstration\nof ferroelectric order in the system and the claim for multiferroicity. The authors have\naddressed these aspects, one on the hand by carrying out ferroelectric characterization\n(P-E switching and PFM), and on the other, by toning down the claims of the original\nsubmission. As the authors admit, the P-E loops are rather inconclusive, while the PFM\nmeasurements, while suggesting ferroelectric order, could also suffer from artifacts,\nsuch as charge deposition, in particular considering the very low estimated\nferroelectric polarization. However, the TEM and SHG results do suggest the presence\n10\nof polar domains, so the evidence points to the presence of ferroelectric order in the\nsystem. The revised version is much improved compared to the original, with a better\npresentation of the first principle results and of the STEM data. Overall, I find the\npresent work to be of high quality and original, highlighting how controlling the cation\ncontent and arrangement in the double perovskites can be explored to tune the\nelectronic and functional properties of this class of materials. Hence, I can recommend\npublication of this MS in Nature Communications.\nResponse: We sincerely appreciate the Reviewer’s final decision and thoughtful\nrecognition of our revisions. The reviewer’s constructive feedback throughout the\nreview process has been essential in strengthening the overall quality and presentation\nof our work.\nWe appreciate the reviewer’s time and effort in reviewing our re-submission. The\nreviewer’s valuable comments and suggestions were very professional and helped us to\nfurther improve our manuscript.\nReviewer #1 :\nThe authors have thoroughly and convincingly addressed the comments raised by\nreferees during the previous review round. In particular, they now demonstrate the\nexistence of polar domains using several complementary experimental techniques. In\naddition, they have appropriately moderated the tone of their claims, for instance by\nrevising the title of the manuscript.\nWhile certain measurements, when considered in isolation—such as PFM data or the\nabsence of a clear P-E hysteresis loop—might suggest that the system is not\nferroelectric, the manuscript clearly shows that a broad range of other experimental\napproaches provides strong and consistent evidence that the system is indeed polar.\nTaken together, these results support the presence of well-defined polar domains and\nmake the overall interpretation robust.\nIn this revised version, the authors further reinforce their conclusions by introducing\nseveral new arguments and by highlighting clear similarities with other polar and\nferroelectric systems based on double perovskites. A similarly careful and well-\nsupported analysis is presented to explain the small upturn observed in the\nmagnetization curve, which is now satisfactorily accounted for.\nIn my opinion, the authors have done an excellent job and have provided all reasonable\nevidence supporting the claims made in the manuscript. This work represents a\nsignificant advance in the field of oxide-based electronics, and I therefore consider the\nmanuscript to be fully suitable for publication in Nature Communications.\nResponse: We are deeply grateful to the Reviewer for the final recommendation for\npublication in Nature Communication. The reviewer’s expert scrutiny during the\nreview process has been invaluable in elevating the quality and impact of our\nmanuscript."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have thoroughly and convincingly addressed the comments raised by referees during the previous review round. In particular, they now demonstrate the existence of polar domains using several complementary experimental techniques. In addition, they have appropriately moderated the tone of their claims, for instance by revising the title of the manuscript.\nWhile certain measurements, when considered in isolation—such as PFM data or the absence of a clear P-E hysteresis loop—might suggest that the system is not ferroelectric, the manuscript clearly shows that a broad range of other experimental approaches provides strong and consistent evidence that the system is indeed polar. Taken together, these results support the presence of well-defined polar domains and make the overall interpretation robust.\nIn this revised version, the authors further reinforce their conclusions by introducing several new arguments and by highlighting clear similarities with other polar and ferroelectric systems based on double perovskites. A similarly careful and well-supported analysis is presented to explain the small upturn observed in the magnetization curve, which is now satisfactorily accounted for.\nIn my opinion, the authors have done an excellent job and have provided all reasonable evidence supporting the claims made in the manuscript. This work represents a significant advance in the field of oxide-based electronics, and I therefore consider the manuscript to be fully suitable for publication in Nature Communications."}, "author_response": "Response to Reviewers’ Comments\nWe appreciate the reviewer’s time and effort in reviewing our re-submission. The\nreviewer’s valuable comments and suggestions were very professional and helped us to\nfurther improve our manuscript.\nReviewer #1 :\nThe authors have thoroughly and convincingly addressed the comments raised by\nreferees during the previous review round. In particular, they now demonstrate the\nexistence of polar domains using several complementary experimental techniques. In\naddition, they have appropriately moderated the tone of their claims, for instance by\nrevising the title of the manuscript.\nWhile certain measurements, when considered in isolation—such as PFM data or the\nabsence of a clear P-E hysteresis loop—might suggest that the system is not\nferroelectric, the manuscript clearly shows that a broad range of other experimental\napproaches provides strong and consistent evidence that the system is indeed polar.\nTaken together, these results support the presence of well-defined polar domains and\nmake the overall interpretation robust.\nIn this revised version, the authors further reinforce their conclusions by introducing\nseveral new arguments and by highlighting clear similarities with other polar and\nferroelectric systems based on double perovskites. A similarly careful and well-\nsupported analysis is presented to explain the small upturn observed in the\nmagnetization curve, which is now satisfactorily accounted for.\nIn my opinion, the authors have done an excellent job and have provided all reasonable\nevidence supporting the claims made in the manuscript. This work represents a\nsignificant advance in the field of oxide-based electronics, and I therefore consider the\nmanuscript to be fully suitable for publication in Nature Communications.\nResponse: We are deeply grateful to the Reviewer for the final recommendation for\npublication in Nature Communication. The reviewer’s expert scrutiny during the\nreview process has been invaluable in elevating the quality and impact of our\nmanuscript."}]

Decision letter