Sliding ferroelectric metal with ferrimagnetism

Domain: physics
Published: 2025-12-14
Editorial outcome: major_revision

Abstract
In this work, we develop a general strategy for constructing two-dimensional sliding ferroelectric ferrimagnetic metals from antiferromagnetic metallic bilayers, achieving triply-coupled switching in which the ferroelectric polarization, spin splitting, and net magnetic moment are simultaneously reversed via ferroelectric switching. As a prototype, we design a sliding bilayer ferroelectric metal with ferrimagnetic order based on Fe5GeTe2, a van der Waals ferromagnet near room temperature. The system undergoes a ferroelectric phase transition from a nonpolar antiferromagnetic phase to a ferroelectric ferrimagnetic phase driven by relative interlayer sliding. The breaking of in-plane mirror symmetry in the ferroelectric metallic states lifts the non-relativistic spin degeneracy present in the nonpolar phase, resulting in a sizable net magnetic moment. Moreover, the coexistence of metallic conductivity, ferroelectricity, and ferrimagnetism gives rise to pronounced sign-reversible transport responses near the Fermi level, all of which can be electrically modulated through ferroelectric switching.

Introduction
Ferroelectric (FE) materials have long been studied for their rich physics and potential applications1–8. Among them, two-dimensional (2D) FE metals have attracted considerable attention due to their switchable out-of-plane FE polarization combined with metallic conductivity9–13. This rare combination challenges the conventional understanding that ferroelectricity and metallicity are mutually exclusive, and opens up new opportunities for designing multifunctional devices in nanoelectronics and spintronics. However, most synthesized 2D materials intrinsically preserve inversion symmetry (P) and do not support the FE property. Fortunately, sliding ferroelectricity, proposed in 201714, provides a viable mechanism to induce ferroelectricity in otherwise nonpolar (NP) multilayer systems via interlayer sliding, offering a pathway to 2D FE metals15,16. To date, the intrinsically non-magnetic multilayer 1T’-WTe2 remains the only experimentally confirmed system that simultaneously exhibits sliding ferroelectricity and metallicity11, and integrating magnetism into such systems remains a major theoretical and experimental challenge.

Recent advances in the field of sliding ferroelectricity with collinear compensated magnetic order (including fully compensated ferrimagnets17–20 and altermagnets21,22) have still mainly focused on 2D insulating/semiconducting systems, which allow for the reversal of both FE polarization and spin splitting via FE switching. However, these systems usually maintain an open gap and zero net magnetization throughout the FE transition and lack metallic states near the Fermi level, limiting their potential for charge transport applications. In contrast, when collinear compensated magnetic order coexists with sliding ferroelectricity in metallic systems, the interplay among broken P (due to ferroelectricity) and time-reversal symmetry (T) (due to magnetism), and Fermi-level metallic states can give rise to a sizable net magnetic moment and induce pronounced anomalous transport phenomena, including anomalous Hall effect (AHE), anomalous Nernst effect (ANE), and magneto-optical effects (MOEs). We refer to such systems as 2D sliding FE ferrimagnetic (FiM) metals. In these systems, electric-field-driven interlayer sliding enables triply-coupled switching—simultaneous control of the FE polarization, net magnetic moment, and spin-polarized band structures—offering a promising route toward nonvolatile tuning of anomalous transport properties. 2D sliding FE FiM metals that combine sliding ferroelectricity, ferrimagnetism, and giant, electrically switchable anomalous transport—preferably at room temperature—would mark a major milestone in materials design. However, neither such a design strategy nor corresponding materials have been reported to date, making their realization an important open challenge.

In this paper, focusing on AFM bilayer metals, which possess nonrelativistic spin degeneracy under a [C2||Mz] symmetry operation, we propose a general strategy to construct the 2D sliding FE FiM metal, where the interplay of metallicity and lifted spin degeneracy leads to FiM order with a sizable net magnetic moment that can be efficiently modulated by FE switching. Taking bilayer Fe5GeTe2 as a model system23–25, we demonstrate that interlayer sliding induces a transition from a NP AFM state without net magnetization to a FE FiM phase with a significant net magnetization of up to 0.19 μB per unit cell, which exhibits a pronounced linear magnetoelectric coupling coefficient. Importantly, apart from out-of-plane FE polarization, the interlayer-sliding-driven FE switching can simultaneously switch the sign of the net magnetization and spin-splitting without altering the Néel vector. Such a triply-coupled switching can further enable reversible control of considerable anomalous transport properties by an electric field, opening pathways for electrically tunable spintronic devices.

Results
Triply-coupled switching in sliding FE FiM metals
Symmetry plays an essential role in understanding the properties of matter. The out-of-plane FE polarization in sliding FE FiM metals primarily arises from the potential difference between adjacent layers caused by interlayer sliding, which breaks the symmetry operation that connect the two opposite spins in the NP phase, such as [C2||Mz]. This broken symmetry enables the simultaneous control of the FE polarization, net magnetic moment, and spin-polarized band structures, leading to a triply-coupled switching behavior.

Here, we consider a FE transition from an NP A-type AFM (intralayer ferromagnetic (FM) and interlayer AFM) phase to a FE FiM phase in a bilayer system, as schematically illustrated in Fig. 1a. We begin by constructing a bilayer NP phase from a monolayer ferromagnet. In the NP AFM phase, the two spin sublattices located in the top and bottom layers are connected by a combined spin symmetry operation, [C2||O], where O denotes a real-space symmetry operation relating opposite spin sublattices in different layers, such as Mz. This operation guarantees both nonrelativistic spin degeneracy and a vanishing net magnetization. However, the combined operation [C2||O] only exists under specific stacking configurations. Interlayer sliding breaks this symmetry, leading to a transition into a FiM phase with spin-split band structures. Therefore, the key to identifying a suitable real material lies in the requirement that the stacking configuration, which is connected by the symmetry operation O relating the opposite spin sublattices of the two layers, must be breakable by interlayer sliding. Conversely, cases where O=P must be excluded, as it cannot be broken by interlayer sliding. These conditions provide the necessary screening criteria for the material search.

We then analyze the evolution of magnetization. In general, the net magnetization of a magnet is given by
M=μB ∫EF (D↑(E) - D↓(E)) dE, (1)
where μB is the Bohr magneton, EF is the Fermi level, and D↑(E) and D↓(E) denote the spin-resolved density of states (DOS) for spin-up and spin-down electrons, respectively. Depending on electronic band structures of the system, two distinct scenarios can arise upon the FE transition (see Fig. 1b):
(i) For insulators/semiconductors, assuming the band gap remains open across the FE transition, it follows that the integrated DOS (iDOS) below the Fermi level consistently satisfies iDOS↑(EF) = iDOS↓(EF). As such, even though the symmetry connecting the sublattices is broken during interlayer sliding, the integer filling condition ensures full magnetic compensation and a vanishing net magnetization, yielding a FE fully compensated ferrimagnetic (FE fc-FiM) insulating/semiconducting state26.
(ii) For metals, although the NP phase exhibits degenerate iDOS due to the preserved nonrelativistic spin degeneracy, the sliding-induced FE transition lifts this degeneracy. As a result, the unequal iDOS below the Fermi level (iDOS↑(EF) ≠ iDOS↓(EF)) lead to a sizable net magnetic moment, referred to as FE FiM metal.

Based on this understanding, we propose a general strategy for realizing 2D sliding FE FiM metals. Starting from a bilayer (or multilayer) AFM metal constructed from certain monolayer metallic magnetic materials, interlayer sliding ferroelectricity is introduced to break the symmetry connecting the two spin sublattices. This symmetry breaking induces spin splitting at the Fermi level, leading to unequal iDOS below the Fermi level (iDOS↑(EF) ≠ iDOS↓(EF)), and consequently a finite net magnetization, forming a 2D sliding FE FiM metal.

Remarkably, in such systems, it is possible to achieve an interesting triply-coupled switching, i.e., simultaneous switching of the net magnetic moment MN, the out-of-plane polarization Pz, and the spin splitting ΔES (denoted by S)—without reversing the Néel vector—purely through interlayer sliding. Considering an initial state ψik(MN, Pz, S), one can always find a final state ψfk(-MN, -Pz, -S) that is related to the initial state by a combined operation [C2||Mz], i.e.,
[C2||Mz]ψik(MN, Pz, S) = ψfk(-MN, -Pz, -S). (2)

Such a coupling between FE polarization and spin-polarized bands induced by ferrimagnetism, when further entangled with metallicity, can give rise to large anomalous transport properties that are controllable by an electric field.

Material realization with triply-coupled switching
To explore potential triply-coupled switching in real materials, we focus on a typically synthesized 2D metallic system, Fe5GeTe2, to construct a sliding FE structure. Fe5GeTe2 is a disordered vdW metal with intrinsic FM order near room temperature23–25. Starting from the FM monolayer with the so-called UUU configuration, where the Fe5 atom is positioned above a Ge atom (see Supplementary Fig. 1 and refs. 27–31), we considered eight distinct stackings: AA0, AB, AC, BA, AA0-1, AB-1, AC-1, and BA-1 (see Supplementary Fig. 2). As summarized in Supplementary Note 1, these eight stackings naturally fall into two sliding families, with in each of which the four stackings can be transformed by in-plane sliding. The first family consists of AA0, AB, AC, and BA, all of which favor A-type AFM ground state. In contrast, the second family includes AA0-1, AB-1, AC-1, and BA-1, where AA-1 remains A-type AFM ground state, while its sliding counterparts (AB-1, AC-1, and BA-1) adopt FM ground state. Therefore, we primarily focus on the AA, AB, AC, and BA stackings, which exhibit lower total energies and A-type AFM ground state, and are most relevant to the present study.

The structure of AC stacking belongs to the [C2||Mz] symmetry which enforces a NP phase. By sliding the top layer relative to the bottom one along the in-plane vectors t=(b-a)/6 and t=(a-b)/6 (see Fig. 2a), the Mz mirror symmetry is broken, resulting in two FE phases with degenerate energy but opposite Pz, corresponding to AB and BA stackings (No. 156, P3m1). The optimized lattice constants and atomic positions of AA, AB, BA, and AC stackings are listed in Supplementary Tables 1–4. The energy barrier of interlayer sliding between AB and BA stackings is estimated to be 15.4 meV per unit cell through the climbing image nudge-elastic-band (CL-NEB) method (see Fig. 2b). During the sliding process, although the relative atomic displacements within each layer are negligible6, the asymmetric atomic arrangement between the top and bottom layers induces an interlayer charge transfer, giving rise to a Pz14,15,32, which can be well defined by the classical formula in electrodynamics13
Pz = 1/S ∫ (ρions + ρvalence) d3r, (3)
The Pz of AB stacking is calculated to be 0.087 pC/m (see Fig. 2c). Considering the effective bilayer thickness (19.54 Å) of Fe5GeTe2, this corresponds to a converted polarization of 44.5 μC/m2, which is comparable to those of other reported two-dimensional ferroelectrics, such as VCl2 (23.6 μC/m2), VBr2 (22.3 μC/m2), and VI2 (12.2 μC/m2)33, VAl2S4 (0.08 pC/m)34, as well as 1T0-ReS2 (0.07 pC/m)35. Notably, ferroelectricity in 1T0-ReS2 has been experimentally confirmed using the piezoresponse force microscopy technique35, supporting the experimental feasibility of the predicted FE behavior in Fe5GeTe2. To further elucidate the charge transfer characteristics of the AB and BA stackings, Bader charges were calculated32, as shown in Supplementary Tables 5 and 6. In the AB stacking, the bottom and top layers exhibit Bader charges of +0.0051e and -0.0051e, respectively, signifying charge transfer from the top to the bottom layer. This confirms a negative charge center in the bottom layer and polarization along the +z direction. Conversely, the Bader charges of the bottom and top layers of BA stacking are -0.0051e and +0.0051e, respectively, giving rise to a FE polarization oriented along the -z direction.

We further confirm the magnetic ground states of both NP and FE phases by considering FM, AFM1 (A-type AFM—intralayer FM and interlayer AFM), and five other AFM orderings (AFM2-6, see Supplementary Fig. 4). Among them, the AFM1 is found to be energetically favored for both cases (see Supplementary Table 7). More importantly, while the NP phase exhibits zero net magnetization, the FE AB and BA stackings show a sizable net magnetic moment of up to 0.19 μB per unit cell with opposite signs, as shown in Fig. 2c. All atomic magnetic moments for both the NP and FE phases are listed in Supplementary Table 8. More details about the exchange parameters and Curie temperature of the FE FiM Fe5GeTe2 bilayer, in comparison with the FM Fe5GeTe2 monolayer, are provided in Supplementary Note 2.

To gain deeper insight into the origin of this sizable net magnetization, we then analyze the spin-resolved electronic structure and its evolution across the FE transition. As shown in Fig. 3b, the NP phase with the AC stacking exhibits nonrelativistic spin-degenerate band structures, with several bands crossing the Fermi level along the high-symmetry paths of the Brillouin zone (BZ). This spin degeneracy is protected by the [C2||Mz] symmetry that connects the two spin sublattices. Accordingly, the iDOS for spin-up and spin-down channels remains equal (iDOS↑(EF) = iDOS↓(EF)), resulting in a vanishing net magnetic moment as described by Eq. (1). In contrast, for the FE phase with AB stacking, the interlayer sliding breaks the [C2||Mz] symmetry. As a result, the two spin sublattices are no longer connected by any symmetry operation, leading to spin-splitting electronic bands throughout the BZ. Given the metallic nature of the system, this spin splitting induces a polarization in the spin-resolved iDOS below the Fermi level, with iDOS↑(EF) < iDOS↓(EF) (see Fig. 3a), which gives rise to a net magnetic moment of -0.19 μB per unit cell. As detailed in Supplementary Table 8, the spin-up Fe1–Fe5 atoms possess smaller local magnetic moments compared to the spin-down Fe6–Fe10 atoms, confirming the FiM character of the FE state.

For the other FE phase with BA stacking, it can be regarded as being derived from the AB stacking by applying a combined operation [C2||Mz]. According to Eq. (2), this FE switching operation simultaneously reverses both Pz and S without altering the Néel vector. The projected 2D spin-splitting maps (see Fig. 3d) clearly illustrate the opposite S of the AB and BA stackings, consistent with the reversal of the spin-resolved band structures induced by FE switching (Fig. 3c). As a consequence, the spin-resolved iDOS below the Fermi level becomes inverted, with iDOS↑(EF) > iDOS↓(EF) in contrast to the AB case. This reversal leads to an opposite net magnetic moment direction, while maintaining the same magnitude (see Fig. 2c).

Furthermore, we examine the effect of an external electric field on the magnetic behavior of the FE FiM phase. As depicted in Fig. 3e, the net magnetization exhibits an approximately linear response to a small external electric field. The corresponding magnetoelectric coupling coefficient can be evaluated using17,36
α = μ0 ΔM / E (4)
where μ0 denotes the vacuum permeability, ΔM is the change in magnetization, and E is the applied electric field. Based on this expression, we estimate the linear magnetoelectric coupling coefficient to be approximately 7.13 × 10⁻¹⁴ G·cm2/V for both AB- and BA-stacked FE phases. This value markedly exceeds those reported in conventional systems such as Fe thin films (2.9 × 10⁻¹⁴ G · cm2/V)36. Remarkably, it is comparable to that of the prototypical sliding FE fc-FiM bilayer VS2, which reaches a similar value (9.8 × 10⁻¹⁴ G · cm2/V) when the interlayer distance is artificially reduced by 0.6 Å17. These results underscore the strong magnetoelectric response inherent in this FE FiM metallic system, even without external structural tuning.

Moreover, the interlayer coupling also plays a significant role in modulating the net magnetization of the system. To quantify this effect, the interlayer distance was systematically varied by ±15% with respect to the equilibrium value (d0 = 3.09 Å) to simulate interlayer compression and expansion, i.e., d = d0(1±σ), where σ ∈ [-15%, +15%] (see Supplementary Fig. 6). Within this range, we first examine the magnetic ground state of the AB-stacked FE phase under different interlayer distances. As shown in Supplementary Fig. 7a, the energy difference between the FM and AFM1 configurations (ΔE = EFM - EAFM1, in meV per unit cell) remains positive, confirming that the AFM1 configuration is consistently the magnetic ground state across all interlayer distances. Furthermore, as illustrated in Fig. 3f, upon varying the interlayer distance, the two FE phases exhibit opposite magnetic moments with nearly identical magnitudes. In both phases, reducing the interlayer distance strengthens the interlayer coupling, which in turn enhances the net magnetic moment. This behavior demonstrates that the magnetoelectric coupling in Fe5GeTe2 bilayer can be effectively tuned by the interlayer interaction strength.

Notably, sliding ferroelectricity has been experimentally realized, first observed in few-layer WTe2 in 201811, where switchable interlayer polarization was confirmed by conductance hysteresis and piezoresponse measurements. The demonstrated electric-field-driven reversal of polarization establishes a clear precedent for realizing sliding ferroelectricity in van der Waals metals. Given the similarly layered structure of Fe5GeTe2, our proposed sliding FE FiM metal could be experimentally accessible with current techniques37–41, making its experimental validation highly probable in the near future.

Considerable sign-reversible anomalous transport properties
Owing to its metallic nature, the system may exhibit spontaneous transport responses, particularly the AHE and ANE, which are typically negligible or entirely absent in conventional AFM insulators within the bandgap. These anomalous transport responses are highly dependent on the Néel vector orientation or specific magnetic symmetries42–44. As summarized in Supplementary Note 3, both FE and NP phases exhibit in-plane magnetic anisotropy with the small magnetocrystalline anisotropy energy (<1 meV), which implies that the magnetization orientation is highly sensitive to experimental conditions and can be easily reoriented by an external magnetic field or through spin-orbit torque effects45,46. For example, a magnetization reorientation transition between in-plane and out-of-plane magnetization has been experimentally demonstrated in four-layer Fe3GeTe245, further confirming the ease of magnetic axis manipulation arising from the small magnetocrystalline anisotropy energy revealed in our calculations.

We conducted detailed magnetic symmetry analyses to determine symmetry-imposed anomalous transport responses in both the NP and FE phases, as summarized in Supplementary Tables 13 and 14 as well as Note 4. For the transient NP phase, anomalous transport responses may arise only when the Néel vector possesses an in-plane component, whereas they vanish for an out-of-plane orientation—a characteristic feature of magnets protected by the combined [C2||Mz] symmetry47 (see Supplementary Fig. 9, Table S13, and Note 4 for more details). For the ground-state FE phase, anomalous transport responses are forbidden when the Néel vector lies along the in-plane ⟨100⟩ or ⟨110⟩ crystallographic direction families, while they are allowed for all other orientations (see Supplementary Table 14). Specifically, when the Néel vector rotates within the x-y plane, anomalous transport responses periodically appear and disappear as the magnetic point group alternates between the symmetry-allowed (m0, 1) and symmetry-forbidden (m) configurations. A similar symmetry evolution occurs in the z-x plane, where the anomalous transport responses are depressed when the Néel vector is aligned with the ±x axis (⟨100⟩ direction), but are allowed for other directions, e.g., the out-of-plane direction associated with the magnetic point group 3m0. As representative examples, the main text focuses on the anomalous transport properties of the FE phase with the magnetization oriented out-of-plane. The cases with in-plane Néel-vector orientations are presented in Supplementary Figs. 10 and 11.

As shown in Fig. 4c–f, for the AB-stacked FE phase, the anomalous Hall conductivity (AHC) reaches approximately 0.49 e2/h at the Fermi level, approaching the half-quantized value. The anomalous Nernst conductivity (ANC) near the Fermi level attains a peak of 0.36 kB e/h (~0.62 AK⁻¹ m⁻¹) at 100 K and 0.34 kB e/h (~0.57 AK⁻¹ m⁻¹) at 300 K, exceeding that of noncollinear antiferromagnet Mn3Y (Y=Ge, Ga, Sn) (~0.3 AK⁻¹ m⁻¹)48. Moreover, the Kerr rotation angle reaches 0.14 deg, which is significantly larger than that of Mn3Y (~0.02 deg)49, while the Faraday rotation reaches 1.36 × 10⁵ deg·cm⁻¹, surpassing that of FM Y3Fe5O12 (~0.02 × 10⁵ deg·cm⁻¹)50,51. As summarized in Supplementary Note 3, the calculated anomalous transport properties are well converged. Furthermore, owing to the reversal of spin splitting S across the FE transition, the anomalous transport properties can be controlled by an external electric field (see Fig. 4a, b). Specifically, the AB- and BA-stacked FE phases are connected by TMz symmetry under which the anomalous transport properties are odd, leading to identical magnitudes but opposite signs in the AHC, ANC, as well as the Kerr and Faraday rotation angles (see Fig. 4c–f). This property allows for the nonvolatile control of these anomalous transport responses through electric field-driven FE switching, which is highly desirable for future spintronic applications. Finally, we emphasize that the anomalous transport properties are protected by symmetry. Variations in the interlayer distance only slightly affect their magnitudes but do not alter their qualitative existence, as detailed in Supplementary Note 5.

Methods
The first-principles calculations were performed using the Vienna ab initio simulation package (VASP)52,53 with generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) functional54. The projector-augmented wave (PAW) method55 was adopted and the cutoff energy was set as 500 eV. The first Brillouin zone (BZ) was sampled with a Monkhorst-Pack k-mesh of 9 × 9 × 1. As suggested by previous studies30,46,56,57, the GGA functional without applying a Hubbard U term provides a more reliable description of ground-state electronic and magnetic properties of the Fe5GeTe2 system. For the calculation of the out-of-plane FE polarization and magnetocrystalline anisotropy energy (MAE), a denser 15 × 15 × 1 Monkhorst-Pack k-mesh was employed. The energy and force convergence criteria were set as 10⁻⁶ eV and 0.01 eV/Å, respectively. A vacuum layer with a thickness of about 18 Å was taken to avoid artificial interactions between adjacent slabs and the interlayer van der Waals interaction was treated based on the DFT-D3 method of Grimme with zero-damping function58,59. The dipole correction was applied along the z direction to eliminate the spurious interaction between periodic images and the FE transition pathway was obtained by the climbing image nudged elastic band (CL-NEB) method60,61. To investigate the anomalous transport properties, the maximally localized Wannier functions (MLWFs) for the s-, p-, d-orbitals of Fe, Ge and Te atoms in Fe5GeTe2 were constructed by using the Wannier9062,63, the intrinsic AHC and ANC were calculated using the WannierTools package64.

Anomalous Hall conductivity and anomalous Nernst conductivity
The intrinsic AHC and the ANC were calculated on an ultra-dense k-mesh 251 × 251 × 1 using the Berry phase theory65,
σxy = -e2/h ∫ (d2k / (2π)2) Ωnxy(k) fnk, (5)
αAxy = -e / (T ∫ (d2k / (2π)2)) Ωnxy(k) × [(εnk - μ)fnk + kBT ln(1 + e-(εnk - μ)/kBT)] (6)
where n, k, T, μ, and kB are band index, crystal momentum, temperature, chemical potential, and Boltzmann constant, respectively. (x, y) denote the Cartesian coordinates. fnk = 1 / (exp((εnk - μ)/kBT) + 1) is the Fermi-Dirac distribution. Ωnxy(k) is the band-resolved Berry curvature, given by:
Ωnxy(k) = -2 Im ∑n0≠n (⟨ψnk | bvx | ψn0k ⟩ ⟨ψn0k | bvy | ψnk ⟩) / (ωnk - ωn0k)2 (7)
Here, bvx, bvy are the velocity operators, and ψnk (ℏωnk = εnk) is the eigenvector (eigenvalue) at band index n and momentum k.

Optical conductivity
The optical conductivity is calculated via the Kubo-Greenwood formula63,66:
σαβ = (ie2 / Ωc ℏ) ∑k ∑n,m (f(εmk) - f(εnk)) (⟨ψnk | bvx | ψmk ⟩ ⟨ψmk | bvy | ψnk ⟩) / ((εmk - εnk) (εmk - εnk + ℏω + iη)) (8)
where α, β denote Cartesian directions (x, y), Nk is the total number of k-points for sampling the BZ (here, an ultra-dense k-mesh 251 × 251 × 1 was used), Ωc is the unit cell volume, ℏω is the photon energy, and η is the broadening parameter.
For topologically trivial systems, the complex magneto-optical Kerr and Faraday angles can be described as67–70,
ϕK = θK + iεK = i ωd / c (σxy / σxx), (9)
ϕF = θF + iεF = ωd / 2c (n+ - n-), (10)
Here, θK,F and εK,F are the Kerr (Faraday) rotation angle and ellipticity. c, ω, and d are the speed of light in vacuum, the frequency of incident light, and the effective thickness of the thin film, respectively. σxx and σxy are the diagonal and off-diagonal components of the optical conductivity tensor for the magnetic thin film. σsxx = i(1-n2) ω / 4π represents the diagonal optical conductivity of a nonmagnetic substrate (commonly SiO2). The eigenvalues of the dielectric tensor of the magnetic film are given by n2± = 1 + 4πi(σxx ± iσxy) / ω.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Z. Guo, et al., propose a first-principles of a sliding Fe GeTe van der Waals bilayer. They demonstrate the emergence of\n5 2\nboth ferroelectricity and ferrimagnetism in such a metallic stacking and they show that this system displays a significant\nmagnetoelectric coupling. In addition, they calculated magnetotransport properties.\nCreating ferroelectrics from sliding 2D layers is an interesting topic but is not a novel idea (10.1038/s41467-022-35339-6,\n10.1021/acs.nanolett.3c04869, 10.1073/pnas.2115703118). Adapting this method to metallic and magnetic materials to form\nlow-dimensional magnetoelectrics is more original, with only few reported papers (10.1103/dmzg-ck2t). I would nonetheless\nrecommend the current paper for publication in a more focused journal (npj Comp. Mater. or Appl. Mater) rather than in Nat.\nCommun.\nIn addition, I would recommend the authors to develop or consider some of the following points:\n(1) I do not understand the chosen designation “tri-state” to describe the coupled functional properties of ferroelectric\npolarization, magnetization and spin polarization. In my opinion, the given system only displays a standard double state -\nP/+P, P=0 being not stable.\nThe calculated value of P seems rather small. Can the authors compare it to other 2D FE? I also recommend taking care of\nthe units (p.3, it is written 0.075 pC/m2 instead of pC/m). Can the authors provide more information about how the electric\npolarization sets up in their system? How would it be switched? Would it imply some in-plane atomic displacement?\n(2) More details about the set-up of the ferrimagnetism would also be appreciable. Combining information from Table SV\nwith Fig. S1 could certainly help the reader and some explanations about the charge disproportionation originating from the\nemergence of P would be interesting to know.\n(3) About the computational methodology: did the author carefully verify the k-mesh convergence of the spin-orbit\nproperties? Can the authors comment if the use of other (corrected) functionals or methods, such as the DFT+U, would\nqualitatively affect the results or not? Have the calculated magnetic orderings been already confirmed in the literature? Are\nother magnetic orderings (in-plane AFM) likely to happen, or not? Any idea how the Curie temperature (or at least the\nHeisenberg exchanges) are modified in the proposed system in comparison with FM Fe GeTe ?\n5 2\n(4) Can the authors add 1 or 2 sentences to discuss the experimental realization and manipulation of sliding ferroelectrics?", "reviewer_2": "The current manuscript presents a computational / theoretical study of a novel class of materials, sliding ferroelectric metals\nwith non-zero magnetic moment (due to ferrimagnetic alignment). As a prototypical system, the authors considered a bilayer\nof Fe5GeTe2.\nThe presented results are interesting, and the paper is well written. In addition, this work will likely motivate further\nexperimental research to confirm the presented results. I recommend the publication of this work, but I suggest that the\nauthors address the following issues.\n1) It is mentioned in the text that “the NP and FE phases exhibit the magnetic easy axis along the [100] and [001] direction,\nrespectively”. At the same time, the calculated energy difference between [100] and [001] directions is fairly small (-0.38 meV\n/ unit cell). Did the authors check if these results are not within the error bar of the calculations?\n2) This is a closely related question. Did the authors check convergence of their MAE calculations with respect to the k-\nmesh? The 9x9x1 mesh mentioned in supplementary materials may not be sufficient for good convergence of MAE.\n3) I recommend explicitly defining the “ME” (magnetoelectric) abbreviation in the following line: “The corresponding ME\ncoupling coefficient can be evaluated using”.\n4) Delete repeated word in the following line: “being derived from the AB stacking configuration configuration”.", "reviewer_3": "In this manuscript, the authors propose a mechanism based on symmetry analysis to design a ferroelectric metal with\nferrimagnetism, which can achieve tristate switching. In this process, the ferroelectric polarization, spin splitting, and net\nmagnetization are simultaneously reversed through ferroelectric switching. The authors further validate their proposed\nmechanism by designing a system using first-principles calculations. The manuscript has several unresolved issues that\nhave not been thoroughly investigated, and thus it cannot be accepted by Nature Communications.\n1. The interlayer charge transfer should be considered as ferroelectric state switching?\n2. Can the ferroelectric switching affect the intralayer ferromagnetic order? For instance, can the FM state in Fe5GeTe2\nchange to AFM?\n3. When the authors compare the total energy levels of different structures (Table IV in the Supplementary material), it can be\nobserved that the energy of the interlayer ferromagnetic (FM) state is higher than that of the interlayer ferromagnetic (AFM)\nstate when spin-orbit coupling (SOC) is not included in the calculations. Therefore, the ground state of the system should be\nferrimagnetic. However, when the SOC is included in the calculations, the energy of FM-001 is 0 eV, the energy of c-FiM-\n100 is 6.42 meV. This indicates that the ferromagnetic state (FM-001) has lower energy than the c-FiM (FiM-100). Otherwise,\nthe authors would need to compare the energy of FM-001 and c-FiM-100 in the manuscript. Only by comparing the energy\nlevels of these two states can the subsequent calculations be meaningful; otherwise, the first-principles calculations\nverification would lack validity.\n4. In Fig. 2, the magnetic configuration of the Fe5GeTe2 bilayer is down-up for the initial state, the author should compare\nthe energies of the down-up and up-down for the final state to support the neel vector do not change as the FE switching.\n5. The authors should consider other interlayer stacking configurations (AB’). Such as, Nano Lett. 2024, 24, 11, 3541–3547,\nNano Lett. 2023, 23, 13, 6226–6232, npj Comput Mater 2025, 11, 264.\n6. The magnetic exchange mechanism in the interlayer and intralayer should be comprehensively discussed.\n7. Are the magnetic configuration, AHE, and ANE affected by the interlayer distance."}, "author_response": "Response to Reviewer #1\nGeneral comment: “Z. Guo, et al., propose a first-principles of a sliding Fe GeTe van der Waals bilayer. They demonstrate the emergence of\n5 2\nboth ferroelectricity and ferrimagnetism in such a metallic stacking and they show that this system displays a\nsignificant magnetoelectric coupling. In addition, they calculated magnetotransport\nproperties.\nCreating ferroelectrics from sliding 2D layers is an interesting topic but is not a novel\nidea (10.1038/s41467-022-35339-6, 10.1021/acs.nanolett.3c04869,\n10.1073/pnas.2115703118). Adapting this method to metallic and magnetic materials\nto form low-dimensional magnetoelectrics is more original, with only few reported\npapers (10.1103/dmzg-ck2t). I would nonetheless recommend the current paper for\npublication in a more focused journal (npj Comp. Mater. or Appl. Mater) rather than in\nNat. Commun.”\nResponse: We sincerely thank the reviewer for their careful evaluation and insightful\ncomments. We respectfully clarify that the paper (DOI: 10.1103/dmzg-ck2t) suggested\nby the reviewer actually concerns a semiconducting/insulating sliding system, whereas\nour study focuses on a metallic sliding system. The underlying sliding and\nmagnetization mechanisms in these two cases are fundamentally distinct, as illustrated\nin Fig. 1 of our manuscript.\nSpecifically, in the semiconducting/insulating case (Case 1), an antiferromagnetic\n(AFM) semiconductor possessing the [𝒞 ∥ 𝒪] symmetry exhibits both spin\n2\ndegeneracy and a vanishing net magnetization in its pristine configuration. Upon\ninterlayer sliding, although this symmetry is broken, the integer filling and the presence\nof an energy gap still enforce a zero net magnetic moment, leading to a fully\ncompensated ferrimagnetic (fc-FiM) state.\nIn contrast, in the metallic case (Case 2), interlayer sliding breaks the [𝒞 ∥ 𝒪]\n2\nsymmetry and consequently lifts the spin degeneracy near the Fermi level. This\ndisruption leads to an imbalance in the spin-resolved density of states between the two\nspin channels, thereby generating a finite net magnetic moment, leading to a\nferrimagnetic (FiM) state. Such a mechanism is intrinsic to metallic systems and is\n1 / 52\nfundamentally distinct from that of semiconducting/insulating counterparts as reported\nin paper (DOI: 10.1103/dmzg-ck2t).\nFig. 1. (a) Schematic illustration of the sliding FE transition from a NP AFM phase to\na FE FiM phase. The red and blue arrows represent the atomic magnetic moments. The\ninterlayer sliding breaks the spin group symmetry [𝒞 ∥ 𝒪] and induces a net magnetic\n2\nmoment and out-of-plane polarization, leading to a FE FiM phase. (b) Panels (i) and (ii)\ncorrespond to the insulating/semiconducting and metallic states, respectively. The red\nand blue shaded areas represent the spin-resolved DOS for the spin-down and spin-up\nchannels, respectively. The net magnetization is given by the difference between\niDOS↑(E ) and iDOS↓(E ).\nF F\nBeyond the above-mentioned conceptual advance, we believe the manuscript meets the\nbroad interest and high impact expected for Nature Communications, as it introduces\nseveral additional innovations:\n2 / 52\nTriple-Coupled Switching – We demonstrate, for the first time, that\nferroelectric polarization, net magnetization, and spin splitting can be\nsimultaneously switched in a sliding FE FiM metal through FE switching—a\nphenomenon that does not exist in sliding ferroelectric\nsemiconductors/insulators.\nSign-Reversible Anomalous Transport in Sliding FE FiM Metals – Unlike\npreviously studied sliding FE semiconductors or insulators—where anomalous\ntransport is absent inside the band gap and requires external carrier doping or\nelectrostatic gating to activate—we demonstrate that the sliding FE FiM\nmetallic state in bilayer Fe GeTe inherently hosts pronounced anomalous\n5 2\ntransport responses at the Fermi level. Owing to the strong spin-orbit coupling\n(SOC) and intrinsic ferrimagnetism, the FE phase exhibits sizable anomalous\nHall, anomalous Nernst, and magneto-optical effects that are sign-reversible\nthrough FE switching. Importantly, Fe GeTe is one of the rare layered materials\n5 2\nthat combines metallicity, robust room-temperature magnetism, and strong SOC,\nmaking it a highly promising and experimentally accessible platform for\nrealizing sliding FE FiM metal with electrically controllable transport\nfunctionalities.\nFor these reasons, we respectfully maintain that our manuscript represents a conceptual\nbreakthrough with broad interdisciplinary implications, and we believe it meets the\noriginality and significance required for publication in Nature Communications.\nTechnical question 1: “I do not understand the chosen designation “tri-state” to\ndescribe the coupled functional properties of ferroelectric polarization, magnetization\nand spin polarization. In my opinion, the given system only displays a standard double\nstate -P/+P, P=0 being not stable.\nThe calculated value of P seems rather small. Can the authors compare it to other 2D\nFE? I also recommend taking care of the units (p.3, it is written 0.075 pC/m2 instead of\npC/m). Can the authors provide more information about how the electric polarization\nsets up in their system? How would it be switched? Would it imply some in-plane\natomic displacement?”\nResponse: We appreciate the reviewer’s insightful comment.\n3 / 52\n1. We agree that the term “tri-state” may cause confusion, as it could be interpreted as\nimplying three distinct polarization states (+P, 0, and –P). What we actually intended to\ndescribe is the simultaneous switching of the net magnetization (M ), the out-of-plane\nN\npolarization (P ), and the spin splitting (S) — all reversed without changing the Néel\nz\nvector, purely through interlayer sliding. To avoid ambiguity, we have replaced the\nexpression “tri-state switching” with “triple-coupled switching” throughout the revised\nmanuscript.\n2. We thank the reviewer for this valuable suggestion and sincerely apologize for the\ntypographical error in the manuscript. The reported value should be 0.075 pC/m instead\nof 0.075 pC/m2. This value is relatively small; therefore, to ensure accuracy, we\nrecalculated the polarization using a denser 15 × 15 × 1 k-point mesh, obtaining a\nconsistent result of 0.087 pC/m. Considering the effective bilayer thickness (19.54 Å)\nof Fe GeTe , this corresponds to a converted polarization of 44.5 μC/m², which is larger\n5 2\nthan those of other reported two-dimensional ferroelectrics, such as VCl (23.6 μC/m²),\n2 2\nwell as VAl S (0.08 pC/m) [Appl. Phys. Lett. 124, 212903 (2024)]. Moreover, it is\n2 4\ncomparable to the bilayer sliding ferroelectric 1T′-ReS (0.07 pC/m), whose\n2\nferroelectricity has been experimentally confirmed using the piezoresponse force\nmicroscopy technique [Phys. Rev. Lett. 128, 067601 (2022)].\n3. The origin of ferroelectric polarization of sliding ferroelectric bilayers primarily\narises from atomic rearrangement induced by interlayer sliding, while the relative\natomic displacements within each individual layer remain negligible [Adv. Mater. 36,\n2301472 (2024)]. The sliding between adjacent layers causes a redistribution of\nelectronic charge density, driven by the asymmetry in atomic arrangement between the\ntop and bottom layers. This results in localized electron accumulation and depletion,\nwhich in turn breaks spatial inversion symmetry, thereby generating sliding\nferroelectricity [Adv. Mater. 36, 2301472 (2024)].\nAs for the FE states of our system, the breaking of the 𝑀 symmetry renders the top\n𝑧\nand bottom layers nonequivalent, leading to an uncompensated interlayer charge\ntransfer that gives rise to an out-of-plane polarization P . To further elucidate the charge\nz\ntransfer characteristics of the AB and BA stackings, Bader charges were calculated, as\nshown in Tables S5 and S6. In the AB stacking, the bottom and top layers exhibit net\nBader charges of +0.0051 e and -0.0051 e, respectively, signifying charge transfer from\nthe top to the bottom layer. This confirms a negative charge center in the bottom layer\nand polarization along the +z direction. Conversely, the Bader charges of the bottom\nand top layers of BA stacking exhibit net Bader charges of -0.0051 e and +0.0051 e,\nrespectively, giving rise to a ferroelectric polarization oriented along the −z direction.\n4 / 52\nFollowing the suggestions, we have made the following revisions in the main text:\nIn the second paragraph of the section ''Material Realization with Triple-Coupled\nSwitching'':\n“…During the sliding process, although the relative atomic displacements within each\nlayer are negligible [10], the asymmetric atomic arrangement between the top and\nbottom layers induces an interlayer charge transfer, giving rise to a P [23, 24, 58],\nz\nwhich can be well defined by the classical formula in electrodynamics [22].…”\n“…The P of AB stacking is calculated to be 0.087 pC/m (see Fig. 2(c)). Considering\nz\nthe effective bilayer thickness (19.54 Å) of Fe GeTe , this corresponds to a converted\n5 2\npolarization of 44.5 µC/m2, which is comparable to those of other reported two-\ndimensional ferroelectrics, such as VCl (23.6 µC/m2), VBr (22.3 µC/m2), and VI\n2 2 2\n(12.2 µC/m2) [16], VAl S (0.08 pC/m) [59], as well as 1T′-ReS (0.07 pC/m) [60].\n2 4 2\nNotably, ferroelectricity in 1T′-ReS has been experimentally confirmed using the\n2\npiezoresponse force microscopy technique [60], supporting the experimental feasibility\nof the predicted FE behavior in Fe GeTe . To further elucidate the charge transfer\n5 2\ncharacteristics of the AB and BA stackings, Bader charges were calculated [58], as\nshown in Tables S5 and S6 of the SI. In the AB stacking, the bottom and top layers\nexhibit net Bader charges of +0.0051 e and -0.0051 e, respectively, signifying charge\ntransfer from the top to the bottom layer. This confirms a negative charge center in the\nbottom layer and polarization along the +z direction. Conversely, the Bader charges of\nthe bottom and top layers of BA stacking are -0.0051 e and +0.0051 e, respectively,\ngiving rise to a FE polarization oriented along the -z direction.…”\nTable S5. Bader charges (in units of e) for the AB-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.34819 -0.11227 0.11288 -0.19176 0.11138 0.03018 0.16538 0.23752\n5 / 52\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.35666 -0.11092 0.10209 -0.17193 0.101 0.03557 0.15393 0.24181\nTable S6. Bader charges (in units of e) for the BA-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.35545 -0.11125 0.10278 -0.1723 0.10159 0.03482 0.15426 0.24038\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.34937 -0.11261 0.13964 -0.21831 0.11105 0.03071 0.16563 0.23845\nTechnical question 2: “More details about the set-up of the ferrimagnetism would also\nbe appreciable. Combining information from Table SV with Fig. S1 could certainly help\nthe reader and some explanations about the charge disproportionation originating from\nthe emergence of P would be interesting to know.”\nResponse: We thank the reviewer for the valuable suggestion.\n1. For the set-up of the ferrimagnetism, please note that symmetry plays an essential\nrole in determining the properties of matter. In sliding FE ferrimagnetic metals, the out-\nof-plane FE polarization primarily originates from the potential difference between\nadjacent layers induced by interlayer sliding. This sliding breaks the symmetry\noperations that connect the two opposite spins in the NP phase, such as 𝑀 .\n𝑧\nOnce these symmetries are broken, the system allows simultaneous control over the FE polarization,\nthe net magnetic moment, and the spin-polarized band structures, giving rise to a triple-\ncoupled switching behavior.\nHere, we consider a FE transition from a NP A-type AFM phase—with intralayer FM\nand interlayer AFM order—to a FE FiM phase in a bilayer system, as schematically\nillustrated in Fig. 1(a). The NP AFM phase can be viewed as constructed from two FM\n6 / 52\nmonolayers coupled antiferromagnetically across the interlayer. In this configuration,\nthe two spin sublattices located in the top and bottom layers are connected by a\ncombined spin–space symmetry operation, denoted [𝒞 ∥ 𝒪] , where 𝒪 represents a\n2\nreal-space operation that relates opposite spin sublattices in different layers, such as 𝑀 .\n𝑧\nThis combined operation enforces spin degeneracy and ensures a vanishing net\nmagnetization.\nHowever, the combined operation [𝒞 ∥ 𝒪] only exists for specific stacking\n2\nconfigurations. Interlayer sliding destroys this symmetry, driving a transition to a FiM\nphase characterized by spin-split band structures and a finite net magnetization.\nTherefore, identifying suitable materials requires that the stacking configuration\nconnected by the real-space operation 𝒪 between opposite spin sublattices can be\nbroken by interlayer sliding. In contrast, the case where 𝒪 corresponds to inversion (P)\nshould be excluded, as it cannot be broken by sliding. These symmetry-based\nconditions establish the key screening criteria for discovering real materials that can\nexhibit sliding-induced ferrimagnetism.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the first and second paragraph of the section ''Triple-Coupled Switching in Sliding\nFE FiM Metals'':\n“…Symmetry plays an essential role in understanding the properties of matter. The out-\nof-plane FE polarization in sliding FE FiM metals primarily arises from the potential\ndifference between adjacent layers caused by interlayer sliding, which breaks the\nsymmetry operation that connects the two opposite spins in the NP phase, such as\n[𝒞 ∥ 𝑀 ] . This broken symmetry enables the simultaneous control of the FE\n2 𝑧\npolarization, net magnetic moment, and spin-polarized band structures, leading to a\ntriple-coupled switching behavior.\nHere, we consider a FE transition from a NP A-type AFM (intralayer ferromagnetic\n(FM) and interlayer AFM) phase to a FE FiM phase in a bilayer system, as\nschematically illustrated in Fig. 1(a). We begin by constructing a bilayer NP phase from\na monolayer ferromagnet. In the NP AFM phase, the two spin sublattices located in the\ntop and bottom layers are connected by a combined spin symmetry operation, [𝒞 ∥ 𝒪],\n2\nwhere 𝒪 denotes a real-space symmetry operation relating opposite spin sublattices in\n7 / 52\ndifferent layers, such as 𝑀 . This operation guarantees both nonrelativistic spin\n𝑧\ndegeneracy and a vanishing net magnetization. However, the combined operation\n[𝒞 ∥ 𝒪] only exists under specific stacking configurations. Interlayer sliding breaks\n2\nthis symmetry, leading to a transition into a FiM phase with spin split band structures.\nTherefore, the key to identifying a suitable real material lies in the requirement that the\nstacking configuration, which is connected by the symmetry operation 𝒪 relating the\nopposite spin sublattices of the two layers, must be breakable by interlayer sliding.\nConversely, cases where 𝒪 = 𝑃 must be excluded, as it cannot be broken by interlayer\nsliding. These conditions provide the necessary screening criteria for the material\nsearch.…”\n2. Following the reviewer’s comment, we have added an explanation of how\npolarization (P ) gives rise to charge disproportionation through interlayer sliding. To\nz\nclarify this point, we performed Bader charge analysis for both AB and BA stacking\nconfigurations (please see Tables S5 and S6). In the AB stacking, the bottom and top\nlayers carry net Bader charges of +0.0051 e and −0.0051 e, respectively, indicating\ncharge transfer from the top to the bottom layer. This establishes a negative charge\ncenter in the bottom layer and a polarization pointing along the +z direction. However,\nin the BA stacking configuration, the situation is reversed: the bottom and top layers\ncarry net Bader charges of −0.0051 e and +0.0051 e, respectively, resulting in a\npolarization oriented along the −z direction. This analysis indicates that both the\npolarization direction and the charge-transfer characteristics are reversed between the\nAB and BA stackings during interlayer sliding.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the second paragraph of the section ''Material Realization with Triple-Coupled\nSwitching'':\n“…To further elucidate the charge transfer characteristics of the AB and BA stackings,\nBader charges were calculated [58], as shown in Tables S5 and S6 of the SI. In the AB\nstacking, the bottom and top layers exhibit net Bader charges of +0.0051 e and -0.0051\ne, respectively, signifying charge transfer from the top to the bottom layer. This confirms\na negative charge center in the bottom layer and polarization along the +z direction.\nConversely, the Bader charges of the bottom and top layers of BA stacking are -0.0051\ne and +0.0051 e, respectively, giving rise to a FE polarization oriented along the -z\ndirection.…”\nTable S5. Bader charges (in units of e) for the AB-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.34819 -0.11227 0.11288 -0.19176 0.11138 0.03018 0.16538 0.23752\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.35666 -0.11092 0.10209 -0.17193 0.101 0.03557 0.15393 0.24181\nTable S6. Bader charges (in units of e) for the BA-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.35545 -0.11125 0.10278 -0.1723 0.10159 0.03482 0.15426 0.24038\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.34937 -0.11261 0.13964 -0.21831 0.11105 0.03071 0.16563 0.23845\nTechnical question 3: “About the computational methodology: did the author carefully\nverify the k-mesh convergence of the spin-orbit properties? Can the authors comment\nif the use of other (corrected) functionals or methods, such as the DFT+U, would\nqualitatively affect the results or not? Have the calculated magnetic orderings been\nalready confirmed in the literature? Are other magnetic orderings (in-plane AFM) likely\nto happen, or not? Any idea how the Curie temperature (or at least the Heisenberg\nexchanges) are modified in the proposed system in comparison with FM Fe GeTe ?”\n5 2\n9 / 52\nResponse: We thank the reviewer for these constructive questions regarding our\ncomputational methodology. Our responses are summarized as follows:\n1. For the issue of k-point convergence of the spin-orbit properties:\nWe have carefully verified the k-point convergence of the magnetocrystalline\nanisotropy energy (MAE) in both the FE and nonpolar (NP) phases. As shown in Fig.\nS8, the MAE results are well converged when the k-point mesh exceeds 15 × 15 × 1,\nwith the in-plane orientation identified as the magnetic easy plane, which remains\nisotropic within the plane. As also noted by Reviewer 2, the MAE is indeed quite small\n(~1 meV). To further ensure the robustness of our results for the FE phase, we\nadditionally tested eight alternative van der Waals (vdW) correction schemes,\nsummarized in Table S11. In all cases, the calculations consistently indicate that the in-\nplane orientation remains the magnetic easy plane.\nFurthermore, we carefully examined the anomalous transport properties of the FE\nphases using extremely dense k-point meshes. In the first-principles calculations\nperformed within VASP, a 15 × 15 × 1 Monkhorst–Pack mesh was employed. For the\nevaluation of anomalous transport coefficients, much denser k-point grids were used\nwithin Wannier90 and WannierTools, including 251 × 251 × 1, 501 × 501 × 1, and 1001\n× 1001 × 1 meshes. As shown in Figs. S12 and S13, the AHC, ANC, and MO Kerr and\nFaraday rotation angles calculated using 9 × 9 × 1 (DFT) and 251 × 251 × 1 (Wannier\ninterpolation) display identical overall trends and nearly the same peak magnitudes near\nthe Fermi level. These results confirm that the chosen k-point meshes are sufficiently\nconverged for accurately describing spin–orbit–coupled transport properties.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the first paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties'':\n“…As summarized in Section F of the SI, both FE and NP phases exhibit in-plane\nmagnetic anisotropy with small magnetic anisotropy energy (< 1 meV), which implies\nthat the magnetization orientation is highly susceptible to experimental conditions and\ncan be easily reoriented by an external magnetic field or through spin-orbit torque\neffects [69, 70]. For example, a magnetization reorientation transition between in-\nplane and out-of-plane magnetization has been experimentally demonstrated in four-\n10 / 52\nlayer Fe GeTe [69], further confirming the ease of magnetic axis manipulation arising\n3 2\nfrom the small magnetic anisotropy energy revealed in our calculations.…”\nAlso, we have added Section ''F. The k-Point Convergence of the Spin-Orbit\nProperties'' in the revised SI.\n“…We examined the k-point convergence of the MAE (∆E = E −E ) in both\nMAE [100] [001]\nFE and NP phases. As shown in Fig. S8, the MAE is well converged when the k-point\nmesh exceeds 15×15×1, with the in-plane orientation identified as the magnetic easy\nplane, which remains isotropic within the plane. The magnitude of the MAE is indeed\nquite small (approximately 1 meV). To further verify the accuracy of the calculated\nMAE for the FE phase, we additionally tested eight alternative van der Waals correction\nschemes, the results of which are summarized in Table S11. Across all tested cases, the\nin-plane orientation is consistently identified as the magnetic easy plane, with\ncomparable MAE values of less than 1 meV. Additionally, we carefully examined the\nanomalous transport properties of the FE phases using extremely dense k-point meshes.\nAs shown in Figs. S12 and S13, AHC, ANC, and MO Kerr and Faraday rotation angles\nexhibit consistent overall trends and nearly identical peak magnitudes near the Fermi\nlevel across increasingly dense k-meshes, confirming that the calculated spin-orbit-\ncoupled transport properties are well converged.…”\nFig. S8. MAE (ΔE = E – E ) as a function of k-mesh density (n × n × 1) for\nMAE [100] [001]\n11 / 52\nboth the FE (AB stacking) (a) and NP (AC stacking) (b) phases. The results demonstrate\nconvergence beyond a 15 × 15 × 1 mesh, with the in-plane orientation being\nenergetically favored as the magnetic easy plane, which remains isotropic within the\nplane.\nFIG. S12. Convergence of (a) AHC and (b) ANC for the AB stacking and BA stacking.\nPanels (i)–(iv) correspond to calculations with increasing k-mesh density: (i) 9 × 9 × 1\n(DFT), 251 × 251 × 1 (Wannier); (ii) 15 × 15 × 1 (DFT), 251 × 251 × 1 (Wannier); (iii)\n15 × 15 × 1 (DFT), 501 × 501 × 1 (Wannier); and (iv) 15 × 15 × 1 (DFT), 1001 × 1001\n× 1 (Wannier). The consistent peak positions and overall trends near the Fermi level\nconfirm the convergence of anomalous transport properties.\n12 / 52\nFIG. 13. Convergence of (c) MO Kerr rotation and (d) Faraday rotation angles for the\nAB stacking and BA stacking. Panels (i)–(iv) correspond to calculations with increasing\nk-mesh density: (i) 9×9×1 (DFT), 251×251×1 (Wannier); (ii) 15×15×1 (DFT), 251 ×\n251 × 1 (Wannier); (iii) 15 × 15 × 1 (DFT), 501 × 501 × 1 (Wannier); and (iv) 15 × 15\n× 1 (DFT), 1001 × 1001 × 1 (Wannier). The consistent peak positions and overall trends\nnear the Fermi level confirm the convergence of magneto-optical properties.\n13 / 52\nTable S11. MAE (ΔE = E – E , in meV) of the AB-stacked bilayer Fe GeTe\nMAE [100] [001] 5 2\ncalculated using different van der Waals (vdW) correction methods.\nMethod ΔE (meV)\nMAE\nrev-vdW-DF2 −0.69\nvdW-DF-cx −1.00\nvdW-DF3-opt2 −0.69\nPBE + rVV10L −0.57\nTkatchenko–Scheffler −0.51\nDFT-ulg −0.64\ndDsC dispersion correction −0.52\nDFT-D3 (Becke–Johnson damping) −0.45\n2. For the issue of the DFT+U:\nPrevious studies have shown that applying the DFT+U correction does not accurately\ndescribe the Fe GeTe system. As reported in Ref. [J. Phys. Chem. Lett. 13, 4877\n5 2\n(2022)], the DFT+U method leads to incorrect ground-state properties, such as lattice\nparameters and magnetic moments, whereas the GGA functional successfully\nreproduces the experimental results. Furthermore, as described in Ref. [Nano Lett. 21,\n5599 (2021); J. Phys. Chem. C 127, 17194 (2023)], the lattice parameters obtained from\nPBE functional deviate from experimental measurements by only −0.07 Å. Accordingly,\nthose works adopted the PBE functional throughout, as it provides a balanced and\nphysically reasonable description of both structural and magnetic properties.\nMoreover, Ghosh et al. systematically compared standard density functional theory\n(GGA), GGA + U, and GGA + dynamical mean field theory (DMFT) methods for\nFe GeTe [npj Comput. Mater. 9, 86 (2023)]. They demonstrated that the Fe GeTe\nn 2 5 2\nsystem exhibits delocalized Fe–3d states (bandwidth ≈ 8 eV, U ≈ 4 eV, i.e., U/W ≈\neff\n14 / 52\n0.5). Owing to these broad Fe-d bands, the electrons are highly delocalized, and\napplying a static Hubbard-U correction fails to reproduce the correct lattice parameters\nand magnetic moments, while the standard GGA already captures both structural and\nmagnetic behaviors reasonably well. Therefore, for the metallic and weakly correlated\n(U/W<<1) Fe GeTe system, the use of static Hubbard-U correction is not justified, and\n5 2\nthe GGA functional provides a reliable and balanced description of its ground-state\nelectronic and magnetic properties.\nIn our calculations, the spin moments on Fe5 and Fe10 are significantly suppressed, in\ngood agreement with previous results [Phys. Rev. B 104, 104427 (2021); Phys. Rev. B\n109, L220404 (2024)]. The lattice parameters and magnetic moments obtained using\nGGA are also consistent with earlier studies [Phys. Rev. B 104, 104427 (2021); Phys.\nRev. B 109, L220404 (2024)]. Therefore, the GGA functional was adopted for all\ncalculations in this work.\nFollowing the suggestions, we have revised the relevant part of the SI as follows:\nIn the section ''A. First-Principles Calculation'' of the SI:\n“…As suggested by previous studies [5–8], the GGA functional without applying a\nHubbard U term provides a more reliable description of ground-state electronic and\nmagnetic properties of the Fe GeTe system.…”\n5 2\n3. For the issue of magnetic orderings:\nWe systematically compared multiple possible spin configurations, including FM,\nAFM1 (intralayer FM and interlayer AFM), and five other AFM orderings (see Fig. S4).\nThe relative total energies of these configurations with respect to the AFM1 state are\nsummarized in Table S7. For both the FE and NP phases, the AFM1 configuration is\nidentified as the lowest-energy state, whereas all other AFM configurations (AFM2–\nAFM6) lie considerably higher in energy. These results clearly indicate that the ground\nstate of FE and NP phases is characterized by intralayer ferromagnetism and interlayer\nantiferromagnetism (AFM1).\nPlease note that although the magnetic ordering—namely, intralayer ferromagnetism\ncombined with interlayer antiferromagnetism—in the UUU-type bilayer Fe GeTe\n5 2\n15 / 52\ndiscussed in our work has not been experimentally reported, the same magnetic\nordering has been experimentally confirmed in UDU-type bilayer Fe GeTe systems\n5 2\n(Nano Lett. 21, 5599–5605 (2021)).\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the third paragraph of the section ''Material Realization with Triple-Coupled\nSwitching':\n“…We further confirm the magnetic ground states of both NP and FE phases by\nconsidering FM, AFM1 (A-type AFM—intralayer FM and interlayer AFM), and five\nother AFM orderings (AFM2-6, see Fig. S4 of the SI). Among them, the AFM1 is found\nto be energetically favored for both cases (see Table S7 of the SI).…”\nFig. S4. Magnetic configurations of bilayer Fe GeTe with (a) AB and (b) NP phases,\n5 2\nincluding FM, AFM1, and five other AFM orderings (AFM2–AFM6).\nTable S7. Relative energies of different magnetic configurations for the AB (FE) and\nAC (NP) phases. Energies are given in meV per unit cell with respect to the AFM1 state.\nThe magnetic moment directions illustrated in Fig. S4 correspond to the initial spin\n16 / 52\norientations adopted in the calculations.\nPhase FM AFM1 AFM2 AFM3 AFM4 AFM5 AFM6\nFE (AB) 1.68 0 384.38 0.0016 620.87 47 223.84\nNP (AC) 21.1 0 624.97 0.0004 46.85 46.82 625.46\n4. For the issue of the Curie temperature\nTo further compare the FE FiM bilayer system with the FM Fe GeTe monolayer and\n5 2\nto analyze the variation in their Curie temperatures, we introduce the following\nHeisenberg spin Hamiltonian for the Fe GeTe bilayer [Nat. Commun. 16, 6141 (2025)]:\n5 2\n𝐻 = 𝐻 top +𝐻bot +𝐻 , (1)\nintra intra inter\n𝐻 𝑡𝑜𝑝(𝑏𝑜𝑡) = −∑ 𝐽𝑖𝑛𝑡𝑟𝑎𝑆 ∙𝑆 , (2)\n𝑖𝑛𝑡𝑟𝑎 〈𝑖,𝑗〉∈𝑡𝑜𝑝(𝑏𝑜𝑡) 𝑖𝑗 𝑖 𝑗\n𝐻 = −∑ ∑ 𝐽𝑖𝑛𝑡𝑒𝑟𝑆 ∙𝑆 . (3)\n𝑖𝑛𝑡𝑒𝑟 𝑖∈𝑡𝑜𝑝 𝑗∈𝑏𝑜𝑡 𝑖𝑗 𝑖 𝑗\ntop(bot)\nHere, 𝐻 describes the intralayer magnetic coupling within the top (bottom)\nintra\nFe GeTe layer, while 𝐻 represents the interlayer magnetic coupling. 𝐽𝑖𝑛𝑡𝑟𝑎 and\n5 2 inter 𝑖𝑗\n𝐽𝑖𝑛𝑡𝑒𝑟 denote the intralayer and interlayer exchange parameters, 𝑆 (𝑆 ) is the spin vector\n𝑖𝑗 𝑖 𝑗\nat site 𝑖 (𝑗). Using the TB2J code [Comput. Phys. Commun. 264, 107938 (2021)], we\ncalculated the exchange parameters for both the Fe GeTe monolayer and AB-stacked\n5 2\nbilayer, considering magnetic atom pairs within 10 Å of each other in the unit cell.\nAs summarized in Tables S9 and S10, the dominant intralayer exchange parameters in the\nbottom layer arise mainly from Fe1-Fe2 (27.4662 meV), Fe1-Fe3 (15.9339 meV), and\nFe2-Fe4 (14.1976 meV), while those in the top layer primarily involve Fe6-Fe7\n(21.0638 meV), Fe6-Fe8 (17.0270 meV), and Fe7-Fe9 (14.2569 meV). These values\nare comparable to the major exchange parameters in the monolayer, i.e., Fe1-Fe2\n(22.6871 meV), Fe1-Fe3 (16.9821 meV), and Fe2-Fe4 (13.5174 meV), indicating that\nthe strong intralayer ferromagnetic (FM) character is well preserved in the bilayer\nsystem. In contrast, the interlayer exchange parameters 𝐽𝑖𝑛𝑡𝑒𝑟are rather small due to the\n𝑖𝑗\nvan der Waals nature of Fe GeTe bilayer and overall exhibit negative values, revealing\n5 2\nan AFM coupling between layers. However, such weak interlayer coupling is expected\nto have only a minor impact on the Curie temperature. To confirm this, we further\nestimated the Curie temperatures (T ) of both the monolayer and bilayer Fe GeTe using\nc 5 2\nMonte Carlo simulations performed within the VAMPIRE package [J. Phys.: Condens.\n17 / 52\nMatter 26, 103202 (2014)], based on the TB2J-derived exchange parameters. The\ncalculated T for the FM monolayer Fe GeTe is approximately 373 K, in excellent\nc 5 2\nagreement with previous reports [J. Phys. Chem. Lett. 13, 4877 (2022); Phys. Rev. B\n109, L220404 (2024)]. Owing to the weak interlayer exchange, the T of the AB-stacked\nc\nbilayer shows only a slight reduction, remaining nearly unchanged at about 369 K (see\nFig. S5).\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the third paragraph of the section ''Material Realization with Triple-Coupled\nSwitching':\n“…More details about the exchange parameters and Curie temperature of the FE FiM\nFe GeTe bilayer, in comparison with the FM Fe GeTe monolayer, are provided in\n5 2 5 2\nSection E of the SI.…”\nAlso, we have added Section ''E. Magnetic Exchange Parameters and Curie\nTemperature'' in the revised SI:\n“To elucidate the magnetic mechanism of the FE FiM bilayer system, we construct the\nHeisenberg spin Hamiltonian as follows [23]:\n…𝐻 = 𝐻 top +𝐻bot +𝐻 , (1)\nintra intra inter\n𝐻 𝑡𝑜𝑝(𝑏𝑜𝑡) = −∑ 𝐽𝑖𝑛𝑡𝑟𝑎𝑆 ∙𝑆 , (2)\n𝑖𝑛𝑡𝑟𝑎 〈𝑖,𝑗〉∈𝑡𝑜𝑝(𝑏𝑜𝑡) 𝑖𝑗 𝑖 𝑗\n𝐻 = −∑ ∑ 𝐽𝑖𝑛𝑡𝑒𝑟𝑆 ∙𝑆 . (3)\n𝑖𝑛𝑡𝑒𝑟 𝑖∈𝑡𝑜𝑝 𝑗∈𝑏𝑜𝑡 𝑖𝑗 𝑖 𝑗\ntop(bot)\nHere, 𝐻 describes the intralayer magnetic coupling within the top (bottom)\nintra\nFe GeTe layer, whereas 𝐻 represents the interlayer magnetic coupling. 𝐽𝑖𝑛𝑡𝑟𝑎 and\n5 2 inter 𝑖𝑗\n𝐽𝑖𝑛𝑡𝑒𝑟 are the intralayer and interlayer exchange parameters, and 𝑆 (𝑆 ) is the spin\n𝑖𝑗 𝑖 𝐽\nvector at site 𝑖(𝑗). Using the TB2J code [24], we calculated the exchange parameters\nfor both the Fe GeTe monolayer and AB-stacked bilayer, considering magnetic atom\n5 2\npairs within 10 Å of each other in the unit cell. As summarized in Tables S9 and S10,\nthe dominant intralayer exchange parameters in the bottom layer arise mainly from\n18 / 52\nFe1-Fe2 (27.4662 meV), Fe1-Fe3 (15.9339 meV), and Fe2-Fe4 (14.1976 meV), while\nthose in the top layer primarily involve Fe6-Fe7 (21.0638 meV), Fe6-Fe8 (17.0270\nmeV), and Fe7-Fe9 (14.2569 meV). These values are comparable to the major\nexchange parameters in the monolayer, i.e., Fe1-Fe2 (22.6871 meV), Fe1-Fe3 (16.9821\nmeV), and Fe2-Fe4 (13.5174 meV), indicating that the strong intralayer ferromagnetic\n(FM) character is well preserved in the bilayer system. In contrast, the interlayer\nexchange parameters 𝐽𝑖𝑛𝑡𝑒𝑟 are rather small due to the van der Waals nature of\n𝑖𝑗\nFe GeTe bilayer and overall exhibit negative values, revealing an antiferromagnetic\n5 2\n(AFM) coupling between layers. However, such weak interlayer coupling is expected to\nhave only a minor impact on the Curie temperature. To confirm this, we further\nestimated the Curie temperatures (T ) of both the monolayer and bilayer Fe GeTe\nc 5 2\nusing Monte Carlo simulations performed within the VAMPIRE package [25], based\non the TB2J-derived exchange parameters. The calculated T for the FM monolayer\nc\nFe GeTe is approximately 373 K, in excellent agreement with previous reports [6, 26].\n5 2\nOwing to the weak interlayer exchange, the T of the AB-stacked bilayer shows only a\nc\nslight reduction, remaining nearly unchanged at about 369 K (see Fig. S5).…”\nTable S9. Magnetic exchange parameters (in meV) of Fe GeTe monolayer, with\n5 2\natomic indices corresponding to those shown in Fig. S1.\nMonolayer\n19 / 52\nFe1-Fe2 Fe1-Fe3 Fe1-Fe4 Fe1-Fe5 Fe2-Fe3\n𝐽 22.6871 16.9821 0.6135 0.1793 4.6012\n𝑖𝑗\nFe2-Fe4 Fe2-Fe5 Fe3-Fe4 Fe3-Fe5 Fe4-Fe5\n𝐽 13.5174 0.6021 10.5736 0.2178 2.6612\n𝑖𝑗\nTable S10. Magnetic exchange parameters (in meV) of the AB stacked of Fe GeTe\n5 2\nbilayer, with atomic indices corresponding to those shown in Fig. S2.\nBottom layer\n20 / 52\nFe1-Fe2 Fe1-Fe3 Fe1-Fe4 Fe1-Fe5 Fe2-Fe3\n𝐽 27.4662 15.9339 0.3877 0.1651 4.1628\n𝑖𝑗\nFe2-Fe4 Fe2-Fe5 Fe3-Fe4 Fe3-Fe5 Fe4-Fe5\n𝐽 14.1976 0.1568 9.3803 0.1181 2.1576\n𝑖𝑗\nTop layer\nFe6-Fe7 Fe6-Fe8 Fe6-Fe9 Fe6-Fe10 Fe7-Fe8\n𝐽 21.0638 17.0270 0.5662 0.1663 3.6553\n𝑖𝑗\nFe7-Fe9 Fe7-Fe10 Fe8-Fe9 Fe8-Fe10 Fe9-Fe10\n𝐽 14.2569 0.9473 9.9398 0.2116 3.0788\n𝑖𝑗\nInterlayer coupling\nFe1-Fe10 Fe2-Fe7 Fe2-Fe9 Fe2-Fe10 Fe3-Fe9\n𝐽 -0.0409 -0.0235 -0.0206 0.0183 -0.0085\n𝑖𝑗\nFe3-Fe10 Fe4-Fe7 Fe4-Fe8 Fe4-Fe9 Fe4-Fe10\n𝐽 -0.0282 0.0243 0.0125 -0.0384 0.0185\n𝑖𝑗\nFe5-Fe6 Fe5-Fe7 Fe5-Fe8 Fe5-Fe9 Fe5-Fe10\n𝐽 0.0004 -0.0176 -0.0129 -0.0084 -0.0012\n𝑖𝑗\n21 / 52\nFig. S5. Temperature dependence of the specific heat 𝐶(𝑇) for (a) monolayer and (b)\nAB-stacked Fe GeTe bilayer. The sharp peaks at 𝑇 = 373 K and 𝑇 = 369 K mark\n5 2 𝑐 𝑐\nthe Curie temperatures of the monolayer and bilayer systems, respectively. The 10,000\nequilibration-time steps and 10,000 averaging steps were used for Monte Carlo method.\nQuestion 4: “Can the authors add 1 or 2 sentences to discuss the experimental\nrealization and manipulation of sliding ferroelectrics?”\nResponse: We appreciate the reviewer’s constructive suggestion. In fact, the\nexperimental realization of sliding ferroelectricity has already been demonstrated and\nis not unfamiliar. The earliest observation can be traced back to few-layer WTe in 2018\n2\n[Nature 560, 336–339 (2018)], where sliding ferroelectricity was detected through\nconductance hysteresis and piezoresponse measurements, directly confirming\nswitchable interlayer polarization. The controlled reversal of polarization under the\napplied field provides clear experimental evidence of the electric-field-driven\nmanipulation of sliding ferroelectricity [Sci. Adv. 5, eaax5080 (2019); Nat. Phys. 16,\n1028 (2020); Science 372, 1458 (2021); Proc. Natl. Acad. Sci. U.S.A. 118,\ne2115703118 (2021); Nat. Nanotechnol. 17, 390 (2022)].\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the last paragraph of the section ''Material Realization with Triple-Coupled\nSwitching':\n“…Notably, sliding ferroelectricity has been experimentally realized, first observed in\nfew-layer WTe in 2018 [20], where switchable interlayer polarization was confirmed\n2\n21 / 52\nby conductance hysteresis and piezoresponse measurements. The demonstrated\nelectric-field-driven reversal of polarization establishes a clear precedent for realizing\nsliding ferroelectricity in van der Waals metals. Given the similarly layered structure\nof Fe GeTe , our proposed sliding FE FiM metal is experimentally accessible with\n5 2\ncurrent techniques [62–66], making its experimental validation highly feasible in the\nnear future.…”\n22 / 52\n-------------------------------------------------------------------------------------------------------\nResponse to Reviewer #2\n-------------------------------------------------------------------------------------------------------\nGeneral comment: “The current manuscript presents a computational / theoretical\nstudy of a novel class of materials, sliding ferroelectric metals with non-zero magnetic\nmoment (due to ferrimagnetic alignment). As a prototypical system, the authors\nconsidered a bilayer of Fe5GeTe2.\nThe presented results are interesting, and the paper is well written. In addition, this work\nwill likely motivate further experimental research to confirm the presented results. I\nrecommend the publication of this work, but I suggest that the authors address the\nfollowing issues.”\nResponse: We thank the reviewer for summarizing our work. We are honored that our\nwork has aroused the interest of the reviewer and are grateful for the reviewers’ high\nopinion of our work. At the same time, we also appreciate the valuable questions raised\nby the reviewer regarding our manuscript. In the following, we will address the\nquestions one by one.\nTechnical question 1: “It is mentioned in the text that “the NP and FE phases exhibit\nthe magnetic easy axis along the [100] and [001] direction, respectively”. At the same\ntime, the calculated energy difference between [100] and [001] directions is fairly small\n(-0.38 meV / unit cell). Did the authors check if these results are not within the error\nbar of the calculations?”\nResponse: We sincerely thank the reviewer for this insightful comment, which greatly\nhelped us improve the quality and clarity of the manuscript. To clarify this point, we\ncarefully re-examined the MAE using a series of k-point meshes ranging from 5 × 5 ×\n1 to 25 × 25 × 1. As shown in Fig. S8, the MAE values become well converged beyond\na 15 × 15 × 1 grid. Both the FE and NP phases consistently exhibit a very small energy\ndifference (~1 meV per unit cell) between the [100] and [001] magnetization directions,\nindicating a weak in-plane magnetic anisotropy. We further tested eight different vdW\ncorrection schemes to verify the reliability of the calculated MAE for the FE phase. All\nof them yielded similarly small MAE values (less than 1 meV). As summarized in Table\n23 / 52\nS11, the small MAE implies that the magnetization orientation is highly susceptible to\nexperimental conditions and can be easily reoriented by an external magnetic field or\nthrough spin-orbit torque effects [Nano Lett. 21, 5599 (2021)]. To ensure completeness,\nwe further performed a magnetic point-group analysis of the FE phase for both in-plane\nand out-of-plane Néel-vector orientations to determine which anomalous transport\nresponses are symmetry-allowed or forbidden, as summarized in Table S13. As\nrepresentative examples, we explicitly analyzed the AHC, ANC, MO Kerr and Faraday\nrotation angles for the FE phase under the out-of-plane Néel-vector orientations ([001]\ndirection), as shown in Fig. 4 of the main text. The corresponding results for the in-\nplane Néel-vector orientations ([120] direction) are presented in Fig. S11 of the revised\nSI.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the first paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties'':\n“…As summarized in Section F of the SI, both FE and NP phases exhibit in-plane\nmagnetic anisotropy with the small magnetic anisotropy energy (< 1 meV), which\nimplies that the magnetization orientation is highly susceptible to experimental\nconditions and can be easily reoriented by an external magnetic field or through spin-\norbit torque effects [69, 70]. For example, a magnetization reorientation transition\nbetween in-plane and out-of-plane magnetization has been experimentally\ndemonstrated in four-layer Fe GeTe [69], further confirming the ease of magnetic axis\n3 2\nmanipulation arising from the small magnetic anisotropy energy revealed in our\ncalculations.…”\nIn the second paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties'':\n“…We conducted detailed magnetic symmetry analyses to determine symmetry-\nimposed anomalous transport responses in both the NP and FE phases, as summarized\nin Tables S12 and S13 as well as Section G of the SI. For the transient NP phase,\nanomalous transport responses may arise only when the Néel vector possesses an in-\n24 / 52\nplane component, whereas they vanish for an out-of-plane orientation—a\ncharacteristic feature of magnets protected by the combined [𝒞 ∥ 𝑀 ] symmetry [71]\n2 𝑧\n(see Fig. S9, Table S12, and Section G of the SI for more details). For the ground-state\nFE phase, anomalous transport responses are forbidden when the Néel vector lies along\nthe in-plane ⟨100⟩ or ⟨110⟩ crystallographic direction families, while they are allowed\nfor all other orientations (see Table S13 of the SI). Specifically, when the Néel vector\nrotates within the x-y plane, anomalous transport responses periodically appear and\ndisappear as the magnetic point group alternates between the symmetry-allowed (m′,\n1) and symmetry-forbidden (m) configurations. A similar symmetry evolution occurs in\nthe z-x plane, where the anomalous transport responses are depressed when the Néel\nvector is aligned with the ±x axis (⟨100⟩ direction), but are allowed for other directions,\ne.g., the out-of-plane direction associated with the magnetic point group 3m′. As\nrepresentative examples, the main text focuses on the anomalous transport properties\nof the FE phase with the magnetization oriented out-of-plane. The cases with in-plane\nNéel-vector orientations are presented in Figs. S10 and S11 of the SI.…”\nAlso, we have added Section ''F. The k-Point Convergence of the Spin-Orbit\nProperties'' in the revised SI.\n“…We examined the k-point convergence of the MAE (∆E = E −E ) in both\nMAE [100] [001]\nFE and NP phases. As shown in Fig. S8, the MAE is well converged when the k-point\nmesh exceeds 15×15×1, with the in-plane orientation identified as the magnetic easy\nplane, which remains isotropic within the plane. The magnitude of the MAE is indeed\nquite small (approximately 1 meV). To further verify the accuracy of the calculated\nMAE for the FE phase, we additionally tested eight alternative van der Waals correction\nschemes, the results of which are summarized in Table S11. Across all tested cases, the\nin-plane orientation is consistently identified as the magnetic easy plane, with\ncomparable MAE values of less than 1 meV.…”\n25 / 52\nFig. S8. MAE (ΔE = E – E ) as a function of k-mesh density (n × n × 1) for\nMAE [100] [001]\nboth the FE (AB stacking) (a) and NP (AC stacking) (b) phases. The results demonstrate\nconvergence beyond a 15 × 15 × 1 mesh, with the in-plane orientation being favored as\nthe magnetic easy plane, which remains isotropic within the plane.\nFig. S11. (a) AHC, (b) ANC, (c) MO Kerr rotation angle, and (d) Faraday rotation angle\nof the AB stacking for magnetization along the [120] direction.\n26 / 52\nTable S11. MAE (ΔE = E – E , in meV) of the AB-stacked bilayer Fe GeTe\nMAE [100] [001] 5 2\ncalculated using different van der Waals (vdW) correction methods.\nMethod ΔE (meV)\nMAE\nrev-vdW-DF2 −0.69\nvdW-DF-cx −1.00\nvdW-DF3-opt2 −0.69\nPBE + rVV10L −0.57\nTkatchenko–Scheffler −0.51\nDFT-ulg −0.64\ndDsC dispersion correction −0.52\nDFT-D3 (Becke–Johnson damping) −0.45\nTable S13. Magnetic point groups of the FE phase for different Néel vector orientations\nin the x–y (θ = π/2) and z–x (φ = 0) planes. Here, φ is the azimuthal angle from the x-\naxis and θ is the polar angle from the z-axis. “Y” and “N” indicate whether anomalous\ntransport is symmetry-allowed or forbidden, respectively.\nSystem 0 15 30 45 60 75 90 105 120 135 150 165 180\nm 1 m’ 1 m 1 m’ 1 m 1 m' 1 m\nx-y\nN Y Y Y N Y Y Y N Y Y Y N\nz-x 3m’ 1 1 1 1 1 m 1 1 1 1 1 3m’\nY Y Y Y Y Y N Y Y Y Y Y Y\nTechnical question 2: “This is a closely related question. Did the authors check\nconvergence of their MAE calculations with respect to the k-mesh? The 9x9x1 mesh\nmentioned in supplementary materials may not be sufficient for good convergence of\nMAE.?”\nResponse: We sincerely thank the reviewer for raising this important point. In response,\nwe have carefully re-examined the convergence of the MAE with respect to k-point\nsampling, performing calculations using meshes ranging from 5 × 5 × 1 to 25 × 25 × 1.\nAs shown in Fig. S8, the MAE becomes well converged beyond a 15 × 15 × 1 mesh.\nWe acknowledge that the previously used 9 × 9 × 1 grid was insufficient for\nquantitatively reliable MAE calculations, and we appreciate the reviewer’s constructive\nsuggestion, which has significantly improved the accuracy of our analysis. Following\nthe reviewer’s valuable suggestion, we have revised the manuscript to include the\nupdated converged MAE results, ensuring the accuracy of our conclusions.\nIt is worth noting that although the magnetic easy axis of the FE phase changes from\nout-of-plane to in-plane after recalculating the MAE, this refinement does not affect our\nconclusions, such as the emergence of sliding ferroelectric ferrimagnetism and the\ntriple-coupled (ferroelectric polarization, spin splitting, and net magnetization)\nswitching behavior through FE switching. For the transport properties, we realized that\nit would be insufficient to present the case only under the out-of-plane Néel vector\norientation, as done in our previous manuscript. To ensure completeness, we have\nadditionally examined the magnetic symmetry analyses of the FE phase for different\nNéel vector orientations, including both in-plane (i.e., x-y plane) and out-of-plane\nconfigurations (i.e., z-x plane). This symmetry analysis directly reveals whether\nanomalous transport responses are allowed or forbidden. The results show that the\nanomalous transport properties are symmetry-allowed for both the out-of-plane and\nmost in-plane directions, except for several special crystallographic directions (e.g.,\n⟨100⟩ and ⟨110⟩ families). As representative examples, the main text focuses on the\nanomalous transport properties of the FE phase with the magnetization oriented out-of-\nplane (Fig. 4). The cases with in-plane Néel-vector orientations are presented in Fig.\nS11 of the SI.\nImportantly, these different Néel-vector orientations are experimentally accessible.\nThe energy difference between the in-plane and out-of-plane magnetization orientations\nremains rather small (~1 meV per unit cell), suggesting that the magnetization\n28 / 52\norientation is highly sensitive to experimental conditions and can be readily reoriented\nby an external magnetic field or via spin–orbit torque effects. Notably, a magnetization\nreorientation transition between in-plane and out-of-plane magnetization has been\nexperimentally demonstrated in Fe GeTe [Nature 563, 94–99 (2018)], further\n3 2\nconfirming the ease of magnetic axis manipulation arising from the small MAE\nrevealed in our calculations. As reported in Ref. [Nano Lett. 21, 5599–5605 (2021)], in\nthe UDU-type Fe GeTe bilayer, the in-plane magnetization should be rotated to the\n5 2\nperpendicular direction gradually under an increasing perpendicular magnetic field.\nFollowing the reviewer’s suggestion above, the corresponding revisions have been\nmade in the same manner as for Technical question 1.\nTechnical question 3: “I recommend explicitly defining the “ME” (magnetoelectric)\nabbreviation in the following line: “The corresponding ME coupling coefficient can be\nevaluated using”.”\nResponse: We thank the reviewer for this helpful suggestion. We have revised the\nmanuscript to explicitly define the abbreviation “ME” as “magnetoelectric” when it\nfirst appears in the text. In addition, we have carefully checked all abbreviations\nthroughout the manuscript to ensure they are properly defined upon first use.\nTechnical question 4: “Delete repeated word in the following line: “being derived from\nthe AB stacking configuration configuration”.”\nResponse: We thank the reviewer for the valuable suggestion. We thank the reviewer\nfor pointing this out. We have corrected the typo by deleting the repeated word\n“configuration” in the revised manuscript, and we have also carefully checked the entire\nmanuscript to eliminate similar errors.\n29 / 52\n-------------------------------------------------------------------------------------------------------\nResponse to Reviewer #3\n-------------------------------------------------------------------------------------------------------\nGeneral Comment: “In this manuscript, the authors propose a mechanism based on\nsymmetry analysis to design a ferroelectric metal with ferrimagnetism, which can\nachieve tristate switching. In this process, the ferroelectric polarization, spin splitting,\nand net magnetization are simultaneously reversed through ferroelectric switching. The\nauthors further validate their proposed mechanism by designing a system using first-\nprinciples calculations. The manuscript has several unresolved issues that have not been\nthoroughly investigated, and thus it cannot be accepted by Nature Communications.”\nResponse: We thank the reviewer for their thoughtful summary and constructive\nassessment of our work. We have carefully considered each of the comments and\nrevised the manuscript accordingly, as detailed below.\nTechnical question 1: “The interlayer charge transfer should be considered as\nferroelectric state switching?”\nResponse: We thank the reviewer for the valuable suggestion. Generally, the origin of\nferroelectric polarization is the atomic rearrangement caused by sliding, while the\nrelative displacements within each layer remain negligible. The sliding between\nadjacent layers leads to a redistribution of electronic clouds. The driving force of the\ninterlayer charge transfer is the asymmetry in atomic distribution between the top and\nbottom layers, which induces localized accumulation/depletion of electrons. This\nasymmetric redistribution breaks spatial symmetry and thus gives rise to sliding\nferroelectricity.\nSince this mechanism was first proposed in 2017, interlayer charge-transfer–driven\nsliding ferroelectric polarization has been confirmed [ACS Nano 11, 6382 (2017)]. For\ninstance, vertical ferroelectric polarization was theoretically predicted in bilayer and\nmultilayer h-BN, AlN, ZnO, MoS , and GaSe, while in 2018 Fei et al. observed robust\n2\nout-of-plane polarization at room temperature in bilayer and trilayer 1T′-WTe through\n2\na carefully designed gating device [Nature 560, 336 (2018)].\nAs for the FE states of our system, the breaking of the 𝑀 symmetry renders the top\n𝑧\n30 / 52\nand bottom layers nonequivalent, leading to an uncompensated interlayer charge\ntransfer that gives rise to an out-of-plane polarization. To further elucidate the charge\ntransfer characteristics of the AB and BA stackings, Bader charges were calculated, as\nshown in Tables S5 and S6. In the AB stacking, the bottom and top layers exhibit net\nBader charges of +0.0051 e and -0.0051 e, respectively, signifying charge transfer from\nthe top to the bottom layer. This confirms a negative charge center in the bottom layer\nand polarization along the +z direction. Conversely, the Bader charges of the bottom\nand top layers of BA stacking are -0.0051 e and +0.0051 e, respectively, giving rise to\na ferroelectric polarization oriented along the −z direction.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the second paragraph of the section ''Material Realization with Triple-Coupled\nSwitching'':\n“…During the sliding process, although the relative atomic displacements within each\nlayer are negligible [10], the asymmetric atomic arrangement between the top and\nbottom layers induces an interlayer charge transfer, giving rise to a P [23, 24, 58],\nz\nwhich can be well defined by the classical formula in electrodynamics [22].…”\n“…To further elucidate the charge transfer characteristics of the AB and BA stackings,\nBader charges were calculated [58], as shown in Tables S5 and S6 of the SI. In the AB\nstacking, the bottom and top layers exhibit net Bader charges of +0.0051 e and -0.0051\ne, respectively, signifying charge transfer from the top to the bottom layer. This confirms\na negative charge center in the bottom layer and polarization along the +z direction.\nConversely, the Bader charges of the bottom and top layers of BA stacking are -0.0051\ne and +0.0051 e, respectively, giving rise to a FE polarization oriented along the -z\ndirection.…”\nTable S5. Bader charges (in units of e) for the AB-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\n31 / 52\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.34819 -0.11227 0.11288 -0.19176 0.11138 0.03018 0.16538 0.23752\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.35666 -0.11092 0.10209 -0.17193 0.101 0.03557 0.15393 0.24181\nTable S6. Bader charges (in units of e) for the BA-stacked Fe GeTe bilayer. Atoms\n5 2\nFe1–Fe5, Ge1, Te1 and Te2 belong to the bottom layer, while Fe6–Fe10, Ge2, Te3 and\nTe4 belong to the top layer.\nFe1 Fe2 Fe3 Fe4 Fe5 Ge1 Te1 Te2\n-0.35545 -0.11125 0.10278 -0.1723 0.10159 0.03482 0.15426 0.24038\nFe6 Fe7 Fe8 Fe9 Fe10 Ge2 Te3 Te4\n-0.34937 -0.11261 0.13964 -0.21831 0.11105 0.03071 0.16563 0.23845\nTechnical question 2: “Can the ferroelectric switching affect the intralayer\nferromagnetic order? For instance, can the FM state in Fe5GeTe2 change to AFM?”\nResponse: We thank the reviewer for the valuable suggestion. To address it, we\nsystematically compared multiple possible spin configurations, including FM,\nAFM1 (intralayer FM and interlayer AFM), and five other AFM orderings (see Fig. S4).\nThe relative total energies of these configurations with respect to the AFM1 state are\nsummarized in Table S7. For both the FE and NP phases, the AFM1 configuration is\nidentified as the lowest-energy state, whereas all other AFM configurations (AFM2–\nAFM6) lie considerably higher in energy. These results clearly indicate that the ground\nstate is characterized by intralayer ferromagnetism and interlayer antiferromagnetism\n(i.e., AFM1).\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the third paragraph of the section ''Material Realization with Triple-Coupled\n32 / 52\nSwitching':\n“…We further confirm the magnetic ground states of both NP and FE phases by\nconsidering FM, AFM1 (A-type AFM—intralayer FM and interlayer AFM), and five\nother AFM orderings (AFM2-6, see Fig. S4 in the SI). Among them, the AFM1 is found\nto be energetically favored for both cases (see Table S7 of the SI).…”\nFig. S4. Magnetic configurations of bilayer Fe GeTe with (a) AB and (b) NP stackings,\n5 2\nincluding FM, AFM1, and five other AFM orderings (AFM2-AFM6).\nTable S7. Relative energies of different magnetic configurations for the FE (AB) and\nNP (AC) phases. Energies are given in meV per unit cell with respect to the AFM1 state.\nThe magnetic moment directions illustrated in Fig. S4 correspond to the initial spin\norientations adopted in the calculations.\nPhase FM AFM1 AFM2 AFM3 AFM4 AFM5 AFM6\nFE (AB) 1.68 0 384.38 0.0016 620.87 47 223.84\nNP (AC) 21.1 0 624.97 0.0004 46.85 46.82 625.46\n33 / 52\nTechnical question 3: “When the authors compare the total energy levels of different\nstructures (Table IV in the Supplementary material), it can be observed that the energy\nof the interlayer ferromagnetic (FM) state is higher than that of the interlayer\nferromagnetic (AFM) state when spin-orbit coupling (SOC) is not included in the\ncalculations. Therefore, the ground state of the system should be ferrimagnetic.\nHowever, when the SOC is included in the calculations, the energy of FM-001 is 0 eV,\nthe energy of c-FiM-100 is 6.42 meV. This indicates that the ferromagnetic state (FM-\n001) has lower energy than the c-FiM (FiM-100). Otherwise, the authors would need\nto compare the energy of FM-001 and c-FiM-100 in the manuscript. Only by comparing\nthe energy levels of these two states can the subsequent calculations be meaningful;\notherwise, the first-principles calculations verification would lack validity.”\nResponse: We sincerely thank the reviewer for the careful reading and for pointing out\nthis issue. We believe that the confusion arises from the formatting of Table SIV in the\nSI of the previous version of the manuscript. To clarify, the energy values listed in the\n“MGS (meV/unit cell)” column represent the relative energies between different\nmagnetic configurations, while the “MAE (meV/unit cell)” column reports the\nmagnetocrystalline anisotropy energies corresponding exclusively to the AFM (for NP)\nand FiM (for FE) configurations. In other words, the MAE values do not correspond to\nthe FM states; rather, they describe the magnetic anisotropy of the respective ground-\nstate AFM or FiM phases.\nCombining the insightful suggestions from both Reviewer #1 and Reviewer #2, we have\ncarefully re-examined the MAE using increasingly dense k-point meshes ranging from\n5 × 5 × 1 to 25 × 25 × 1. As shown in Fig. S8, the MAE becomes well converged beyond\na 15 × 15 × 1 mesh. Both the FE and NP phases consistently exhibit a very small energy\ndifference (~1 meV per unit cell) between the [100] and [001] magnetization directions,\nindicating a weak in-plane magnetic anisotropy. We further tested eight different vdW\ncorrection schemes to verify the reliability of the calculated MAE for the FE phase. All\nof them yielded similarly small MAE values (less than 1 meV). As summarized in Table\nS11, the small MAE implies that the magnetization orientation is highly susceptible to\nexperimental conditions and can be easily reoriented by an external magnetic field or\nthrough spin-orbit torque effects [Nano Lett. 21, 5599 (2021)].\n34 / 52\nTo ensure completeness, we further performed a magnetic point-group analysis of the\nFE phase for both in-plane and out-of-plane Néel-vector orientations to determine\nwhich anomalous transport responses are symmetry-allowed or forbidden, as\nsummarized in Table S13. The results show that the anomalous transport properties are\nsymmetry-allowed for both the out-of-plane and most in-plane directions, except for\nseveral special crystallographic directions (e.g., ⟨100⟩ and ⟨110⟩ families). As\nrepresentative examples, the main text focuses on the anomalous transport properties\nof the FE phase with the magnetization oriented out-of-plane (Fig. 4). The cases with\nin-plane Néel-vector orientations are presented in Fig. S11 of the SI.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the first paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties'':\n“…As summarized in Section F of the SI, both FE and NP phases exhibit in-plane\nmagnetic anisotropy with the small magnetic anisotropy energy (< 1 meV), which\nimplies that the magnetization orientation is highly susceptible to experimental\nconditions and can be easily reoriented by an external magnetic field or through spin-\norbit torque effects [69, 70]. For example, a magnetization reorientation transition\nbetween in-plane and out-of-plane magnetization has been experimentally\ndemonstrated in four-layer Fe GeTe [69], further confirming the ease of magnetic axis\n3 2\nmanipulation arising from the small magnetic anisotropy energy revealed in our\ncalculations.…”\nIn the Second paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties'':\n“…We conducted detailed magnetic symmetry analyses to determine symmetry-\nimposed anomalous transport responses in both the NP and FE phases, as summarized\nin Tables S12 and S13 as well as Section G of the SI. For the transient NP phase,\nanomalous transport responses may arise only when the Néel vector possesses an in-\nplane component, whereas they vanish for an out of-plane orientation—a characteristic\n35 / 52\nfeature of magnets protected by the combined [𝒞 ∥ 𝑀 ] symmetry [71] (see Fig. S9,\n2 𝑧\nTable S12, and Section G of the SI for more details). For the ground-state FE phase,\nanomalous transport responses are forbidden when the Néel vector lies along the in-\nplane ⟨100⟩ or ⟨110⟩ crystallographic direction families, while they are allowed for all\nother orientations (see Table S13 of the SI). Specifically, when the Néel vector rotates\nwithin the x-y plane, anomalous transport responses periodically appear and disappear\nas the magnetic point group alternates between the symmetry-allowed (m′, 1) and\nsymmetry-forbidden (m) configurations. A similar symmetry evolution occurs in the z-\nx plane, where the anomalous transport responses are depressed when the Néel vector\nis aligned with the ±x axis (⟨100⟩ direction), but are allowed for other directions, e.g.,\nthe out-of-plane direction associated with the magnetic point group 3m′. As\nrepresentative examples, the main text focuses on the anomalous transport properties\nof the FE phase with the magnetization oriented out-of-plane. The cases with in-plane\nNéel-vector orientations are presented in Figs. S10 and S11 of the SI.…”\nAlso, we have added Section ''F. The k-Point Convergence of the Spin-Orbit\nProperties'' in the revised SI.\n“…We examined the k-point convergence of the MAE (∆E = E −E ) in both\nMAE [100] [001]\nFE and NP phases. As shown in Fig. S8, the MAE is well converged when the k-point\nmesh exceeds 15×15×1, with the in-plane orientation identified as the magnetic easy\nplane, which remains isotropic within the plane. The magnitude of the MAE is indeed\nquite small (approximately 1 meV). To further verify the accuracy of the calculated\nMAE for the FE phase, we additionally tested eight alternative van der Waals correction\nschemes, the results of which are summarized in Table S11. Across all tested cases, the\nin-plane orientation is consistently identified as the magnetic easy plane, with\ncomparable MAE values of less than 1 meV.…”\n36 / 52\nFig. S8. MAE (ΔE = E – E ) as a function of k-mesh density (n × n × 1) for\nMAE [100] [001]\nboth the FE (a) and NP (b) phases. The results demonstrate convergence beyond a 15 ×\n15 × 1 mesh, with the in-plane orientation being favored as the magnetic easy plane,\nwhich remains isotropic within the plane.\nFig. S11. (a) AHC, (b) ANC, (c) MO Kerr rotation angle, and (d) Faraday rotation angle\nof the AB stacking for magnetization along the [120] direction.\n37 / 52\nTable S11. MAE (ΔE = E – E , in meV) of the AB-stacked bilayer Fe GeTe\nMAE [100] [001] 5 2\ncalculated using different van der Waals (vdW) correction methods.\nMethod ΔE (meV)\nMAE\nrev-vdW-DF2 −0.69\nvdW-DF-cx −1.00\nvdW-DF3-opt2 −0.69\nPBE + rVV10L −0.57\nTkatchenko–Scheffler −0.51\nDFT-ulg −0.64\ndDsC dispersion correction −0.52\nDFT-D3 (Becke–Johnson damping) −0.45\nTable S13. Magnetic point groups of the FE phase for different Néel vector orientations\nin the x–y (θ = π/2) and z–x (φ = 0) planes. Here, φ is the azimuthal angle from the x-\naxis and θ is the polar angle from the z-axis. “Y” and “N” indicate whether anomalous\ntransport is symmetry-allowed or forbidden, respectively.\nSystem 0 15 30 45 60 75 90 105 120 135 150 165 180\nm 1 m' 1 m 1 m' 1 m 1 m' 1 m\nx-y\nN Y Y Y N Y Y Y N Y Y Y N\nz-x 3m' 1 1 1 1 1 m 1 1 1 1 1 3m'\nY Y Y Y Y Y N Y Y Y Y Y Y\nTechnical question 4: “In Fig. 2, the magnetic configuration of the Fe5GeTe2 bilayer\nis down-up for the initial state, the author should compare the energies of the down-up\nand up-down for the final state to support the neel vector do not change as the FE\nswitching.”\nResponse: We sincerely thank the reviewer for this valuable suggestion. In a fixed FE\nstate, the “down–up” and “up–down” magnetic configurations are related by time-\nreversal operation and therefore have degenerate total energies. Consequently, their\ntotal energies cannot be directly compared for the final state.\nNevertheless, rotating the Néel vector from 0° to 180° inevitably requires overcoming\nan energy barrier, which cannot be achieved without an external driving mechanism.\nThe reversal of ferroelectric polarization (P → −P) is realized through interlayer sliding;\nhowever, such sliding alone does not provide the necessary driving force to flip the\nintralayer Néel vector across the energy barrier.\nTechnical question 5: “The authors should consider other interlayer stacking\nconfigurations (AB’). Such as, Nano Lett. 2024, 24, 11, 3541–3547, Nano Lett. 2023,\n23, 13, 6226–6232, npj Comput Mater 2025, 11, 264.”\nResponse: We sincerely thank the reviewer for this valuable suggestion. We have\ncarefully studied the suggested references (Nano Lett. 2024, 24, 11, 3541–3547;\nNano Lett. 2023, 23, 13, 6226–6232; npj Comput. Mater. 2025, 11, 264), as well as other\nrelated works, and have appropriately cited these excellent papers in the revised\nmanuscript. In addition, following the reviewer’s advice, we have considered the\nadditional interlayer stackings. Beyond the stackings (AA′, AB, AC, and BA)\ndiscussed in the main text, we have also examined additional interlayer stackings (AA′-1,\nAB-1, AC-1, and BA-1), as illustrated in Fig. S2. Specifically, the AA′ stacked bilayer\nFe GeTe is constructed by flipping the top layer of a directly stacked bilayer, which\n5 2\nresults in two Fe atoms being aligned across the interlayer. The AB, AC, and BA\nstackings can then be obtained by sliding the top layer of AA′ by the translation vectors\n39 / 52\nt = (1/3a + 2/3b), t = 1/2(a + b), and t = (2/3a + 1/3b), respectively.\nFor the AA′-1 stacking, the top layer is similarly flipped, but this leads to two Ge atoms\nbeing aligned across the interlayer. The AB-1, AC-1, and BA-1 stackings are\ncorrespondingly generated from AA′-1 by applying the same translation vectors, t =\n(1/3a + 2/3b), t = 1/2(a + b), and t = (2/3a + 1/3b), respectively. We further investigated\nthe sliding energy barriers of both stacking families to determine their ground-state\nconfigurations. As shown in Figs. S3 (a) and (d), both AB/BA and AB-1/BA-1 stackings\nexhibit the lowest total energies, while the AC and AC-1 stackings are only slightly\nhigher than the AB and AB-1 stackings, but still much lower in energy than the AA′ and\nAA′-1 stackings. The transition from AB (AB-1) to BA (BA-1) through the AC (AC-1)\nphase requires overcoming an energy barrier of approximately 15.4 meV (18.9 meV).\nWe also evaluated the stacking energies of both AA′ and AA′-1 stackings across the\nentire two-dimensional lateral-shift space using a 6 × 6 grid. As shown in Figs. S3(b)\nand (e), the AB (AB-1) and BA (BA-1) stackings are confirmed to reside at the global\nenergy minima within the lateral-shift spaces. We further calculated the energy\ndifferences between the A-type AFM and FM states across the full two-dimensional\nlateral-shift space using a 6 × 6 grid (see Figs. S3 (c) and (f)). The results reveal that\nthe AA′, AB, AC, and BA stackings all favor A-type AFM ground state. In contrast, for\nthe AA′-1, AB-1, AC-1, and BA-1 stackings, the AA′-1 configuration exhibits an AFM\nground state, whereas its sliding counterparts (AB-1, AC-1, and BA-1) exhibit a\nferromagnetic ground state. Therefore, we mainly focus on the AA′, AB, AC, and BA\nstackings in the main text, as they exhibit A-type AFM ground state that is directly\nrelevant to the scope of this work.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the first paragraph of the section ''Material Realization with Triple-Coupled\nswitching'':\n“…Starting from the FM monolayer with the so-called UUU configuration, where the\nFe5 atom is positioned above a Ge atom (see Fig. S1 of the supporting information (SI)\nand Refs. [53 57]), we considered eight distinct stackings: AA′, AB, AC, BA, AA′-1, AB-\n1, AC-1, and BA-1 (see Fig. S2 of the SI). As summarized in Section D of the SI, these\neight stackings naturally fall into two sliding families, within each of which the four\nstackings can be transformed by in-plane sliding. The first family consists of AA′, AB,\n40 / 52\nAC, and BA, all of which favor A-type AFM ground state. In contrast, the second family\nincludes AA′-1, AB-1, AC-1, and BA-1, where AA′-1 remains A-type AFM ground state,\nwhile its sliding counterparts (AB-1, AC-1, and BA-1) adopt FM ground state.\nTherefore, we primarily focus on the AA′, AB, AC, and BA stackings, which exhibit\nlower total energies and A-type AFM ground state, and are most relevant to the present\nstudy.…”\nIn the second paragraph of the section ''Material Realization with Triple-Coupled\nswitching'':\n“…This bilayer structure of AC stacking belongs to space group Aem2 (No. 39), where\nthe presence of the [𝒞 ∥ 𝑀 ] symmetry enforces a NP phase. By sliding the top layer\n2 𝑧\nrelative to the bottom one along the in-plane vectors t =(b−a)/6 and t = (a−b)/6, the\n𝑀 mirror symmetry is broken, resulting in two FE phases with degenerate energy but\n𝑧\noppositely P , corresponding to AB and BA stackings (No. 156, P3m1).…”\nz\nAlso, we have added Section ''D. Stacking Configurations of Bilayer Fe GeTe '' in\n5 2\nthe revised SI.\n“…We considered eight distinct stackings, including AA′, AB, AC, BA, AA′-1, AB-1, AC-\n1, and BA-1 (see Fig. S2). Among them, the AA′ stacked bilayer Fe GeTe is constructed\n5 2\nby flipping the top layer of a directly stacked bilayer, resulting in Fe5 and Fe10 atoms\nbeing aligned across the interlayer. The AB, AC, and BA stackings can then be derived\nfrom AA′ by translating the top layer along the in-plane vectors t = (1/3a + 2/3b), t =\n1/2(a + b), and t =(2/3a+1/3b), respectively. For the AA′-1 stacking, the top layer is\nsimilarly flipped, but this leads to Ge1 and Ge2 atoms being aligned across the\ninterlayer. The corresponding AB-1, AC-1, and BA-1 stackings are then generated from\nAA′-1 by applying the same translation vectors, t = (1/3a + 2/3b), t = 1/2(a + b), and t\n= (2/3a + 1/3b), respectively. We further evaluated the sliding energy barriers of these\nstackings to identify their energetically favorable configurations. As shown in Figs.\n41 / 52\nS3(a) and (d), both AB/BA and AB-1/BA-1 stackings exhibit the lowest total energies,\nwhile the AC and AC-1 stackings are only slightly higher than the AB and AB-1\nstackings but remain much lower in energy than the AA′ and AA′-1 stackings.\nConsequently, the structural transition from the AB (AB-1) to BA (BA-1) stacking is\nmost favorably mediated by the AC (AC-1) intermediate stacking, encountering an\nenergy barrier of roughly 15.4 meV (18.9 meV). We also evaluated the stacking energies\nof both AA′ and AA′-1 stackings across the entire two-dimensional lateral-shift space\nusing a 6×6 grid. As shown in Figs. S3(b) and (e), the AB (AB-1) and BA (BA-1)\nstackings are confirmed to reside at the global energy minima within the lateral-shift\nspaces. By examining the total-energy differences between A-type AFM and FM states\nacross the entire two-dimensional lateral-shift space using a 6 ×6 grid (see Figs. S3(c)\nand (f)), we find that the AA′, AB, AC, and BA stackings favor an A-type AFM ground\nstate. In contrast, the AA′-1 stacking remains A-type AFM, whereas its sliding\ncounterparts (AB-1, AC-1, and BA-1) become ferromagnets.…”\n42 / 52\nFig. S2. Stacking structures of bilayer Fe GeTe , including the AA′, AB, AC, BA, AA′-\n5 2\n1, AB-1, AC-1, and BA-1 configurations.\n43 / 52\nFig. S3. Sliding energy barrier between the top and bottom layers of (a) AA′ stacked\nand (d) AA′-1 stacked Fe GeTe bilayer. Two-dimensional stacking-energy landscapes\n5 2\n(ΔE = E – E ) of the (b) AA′, AB, AC, and BA stackings and (e) AA′-1, AB-1, AC-\nstack AB\n1, and BA-1 stackings, obtained from a 6 × 6 grid of lateral shifts, showing global\nminima at the AB/BA and AB-1/BA-1 positions, respectively. Energy differences\nbetween FM and AFM states (ΔE = E – E ) across the full space of lateral shifts\nFM AFM\nfor (c) AA′, AB, AC, and BA stackings and (f) AA′-1, AB-1, AC-1, and BA-1 stackings.\nTechnical question 6: “The magnetic exchange mechanism in the interlayer and\nintralayer should be comprehensively discussed.”\nResponse: We thank the reviewer for the valuable suggestion.\nTo further explore the magnetic exchange mechanism of the FE FiM Fe GeTe bilayer,\n5 2\nwe introduce the following Heisenberg spin Hamiltonian for the Fe GeTe bilayer [Nat.\n5 2\nCommun. 16, 6141 (2025)]:\n𝐻 = 𝐻 top +𝐻bot +𝐻 , (1)\nintra intra inter\n𝐻 𝑡𝑜𝑝(𝑏𝑜𝑡) = −∑ 𝐽𝑖𝑛𝑡𝑟𝑎𝑆 ∙𝑆 , (2)\n𝑖𝑛𝑡𝑟𝑎 〈𝑖,𝑗〉∈𝑡𝑜𝑝(𝑏𝑜𝑡) 𝑖𝑗 𝑖 𝑗\n𝐻 = −∑ ∑ 𝐽𝑖𝑛𝑡𝑒𝑟𝑆 ∙𝑆 . (3)\n𝑖𝑛𝑡𝑒𝑟 𝑖∈𝑡𝑜𝑝 𝑗∈𝑏𝑜𝑡 𝑖𝑗 𝑖 𝑗\ntop(bot)\nHere, 𝐻 describes the intralayer magnetic coupling within the top (bottom)\nintra\nFe GeTe layer, while 𝐻 represents the interlayer magnetic coupling. 𝐽𝑖𝑛𝑡𝑟𝑎 and\n5 2 inter 𝑖𝑗\n𝐽𝑖𝑛𝑡𝑒𝑟 denote the intralayer and interlayer exchange parameters, 𝑆 (𝑆 ) is the spin\n𝑖𝑗 𝑖 𝐽\nvector at site 𝑖 (𝑗). Using the TB2J code [Comput. Phys. Commun. 264, 107938 (2021)],\nwe calculated the exchange parameters for both the Fe GeTe monolayer and AB-\n5 2\nstacked bilayer, considering magnetic atom pairs within 10 Å of each other in the unit\n44 / 52\ncell. As summarized in Tables S9 and S10, the dominant intralayer exchange parameters\nin the bottom layer arise mainly from Fe1-Fe2 (27.4662 meV), Fe1-Fe3 (15.9339 meV),\nand Fe2-Fe4 (14.1976 meV), while those in the top layer primarily involve Fe6-Fe7\n(21.0638 meV), Fe6-Fe8 (17.0270 meV), and Fe7-Fe9 (14.2569 meV). These values\nare comparable to the major exchange parameters in the monolayer, i.e., Fe1-Fe2\n(22.6871 meV), Fe1-Fe3 (16.9821 meV), and Fe2-Fe4 (13.5174 meV), indicating that\nthe strong intralayer ferromagnetic (FM) character is well preserved in the bilayer\nsystem. In contrast, the interlayer exchange parameters 𝐽𝑖𝑛𝑡𝑒𝑟are rather small due to the\n𝑖𝑗\nvan der Waals nature of Fe GeTe bilayer and overall exhibit negative values, revealing\n5 2\nan AFM coupling between layers. However, such weak interlayer coupling is expected\nto have only a minor impact on the Curie temperature. To confirm this, we further\nestimated the Curie temperatures (T ) of both the monolayer and bilayer Fe GeTe using\nc 5 2\nMonte Carlo simulations performed within the VAMPIRE package [J. Phys.: Condens.\nMatter 26, 103202 (2014)], based on the TB2J-derived exchange parameters. The\ncalculated T for the FM monolayer Fe GeTe is approximately 373 K, in excellent\nc 5 2\nagreement with previous reports [J. Phys. Chem. Lett. 13, 4877 (2022); Phys. Rev. B\n109, L220404 (2024)]. Owing to the weak interlayer exchange, the T of the AB-stacked\nc\nbilayer shows only a slight reduction, remaining nearly unchanged at about 369 K (see\nFig. S5).\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the third paragraph of the section ''Material Realization with Triple-Coupled\nSwitching':\n“…More details about the exchange parameters and Curie temperature of the FE FiM\nFe GeTe bilayer, in comparison with the FM Fe GeTe monolayer, are provided in\n5 2 5 2\nSection E of the SI.…”\nAlso, we have added Section ''E. Magnetic Exchange Parameters and Curie\nTemperature'' in the revised SI.\n45 / 52\n“To elucidate the magnetic mechanism of the FE FiM bilayer system, we construct the\nHeisenberg spin Hamiltonian as follows [23]:\n…𝐻 = 𝐻 top +𝐻bot +𝐻 , (1)\nintra intra inter\n𝐻 𝑡𝑜𝑝(𝑏𝑜𝑡) = −∑ 𝐽𝑖𝑛𝑡𝑟𝑎𝑆 ∙𝑆 , (2)\n𝑖𝑛𝑡𝑟𝑎 〈𝑖,𝑗〉∈𝑡𝑜𝑝(𝑏𝑜𝑡) 𝑖𝑗 𝑖 𝑗\n𝐻 = −∑ ∑ 𝐽𝑖𝑛𝑡𝑒𝑟𝑆 ∙𝑆 . (3)\n𝑖𝑛𝑡𝑒𝑟 𝑖∈𝑡𝑜𝑝 𝑗∈𝑏𝑜𝑡 𝑖𝑗 𝑖 𝑗\ntop(bot)\nHere, 𝐻 describes the intralayer magnetic coupling within the top (bottom)\nintra\nFe GeTe layer, whereas 𝐻 represents the interlayer magnetic coupling. 𝐽𝑖𝑛𝑡𝑟𝑎 and\n5 2 inter 𝑖𝑗\n𝐽𝑖𝑛𝑡𝑒𝑟 are the intralayer and interlayer exchange parameters, and 𝑆 (𝑆 ) is the spin\n𝑖𝑗 𝑖 𝐽\nvector at site 𝑖(𝑗). Using the TB2J code [24], we calculated the exchange parameters\nfor both the Fe GeTe monolayer and AB-stacked bilayer, considering magnetic atom\n5 2\npairs within 10 Å of each other in the unit cell. As summarized in Tables S9 and S10,\nthe dominant intralayer exchange parameters in the bottom layer arise mainly from\nFe1-Fe2 (27.4662 meV), Fe1-Fe3 (15.9339 meV), and Fe2-Fe4 (14.1976 meV), while\nthose in the top layer primarily involve Fe6-Fe7 (21.0638 meV), Fe6-Fe8 (17.0270\nmeV), and Fe7-Fe9 (14.2569 meV). These values are comparable to the major\nexchange parameters in the monolayer, i.e., Fe1-Fe2 (22.6871 meV), Fe1-Fe3 (16.9821\nmeV), and Fe2-Fe4 (13.5174 meV), indicating that the strong intralayer ferromagnetic\n(FM) character is well preserved in the bilayer system. In contrast, the interlayer\nexchange parameters 𝐽𝑖𝑛𝑡𝑒𝑟 are rather small due to the van der Waals nature of\n𝑖𝑗\nFe GeTe bilayer and overall exhibit negative values, revealing an antiferromagnetic\n5 2\n(AFM) coupling between layers. However, such weak interlayer coupling is expected to\nhave only a minor impact on the Curie temperature. To confirm this, we further\nestimated the Curie temperatures (T ) of both the monolayer and bilayer Fe GeTe\nc 5 2\nusing Monte Carlo simulations performed within the VAMPIRE package [25], based\non the TB2J-derived exchange parameters. The calculated T for the FM monolayer\nc\nFe GeTe is approximately 373 K, in excellent agreement with previous reports [6, 26].\n5 2\nOwing to the weak interlayer exchange, the T of the AB-stacked bilayer shows only a\nc\nslight reduction, remaining nearly unchanged at about 369 K (see Fig. S5).…”\nTable S9. Magnetic exchange parameters (in meV) of Fe GeTe monolayer, with\n5 2\natomic indices corresponding to those shown in Fig. S1.\nMonolayer\n46 / 52\nFe1-Fe2 Fe1-Fe3 Fe1-Fe4 Fe1-Fe5 Fe2-Fe3\n𝐽 22.6871 16.9821 0.6135 0.1793 4.6012\n𝑖𝑗\nFe2-Fe4 Fe2-Fe5 Fe3-Fe4 Fe3-Fe5 Fe4-Fe5\n𝐽 13.5174 0.6021 10.5736 0.2178 2.6612\n𝑖𝑗\nTable S10. Magnetic exchange parameters (in meV) of the AB stacked of Fe GeTe\n5 2\nbilayer, with atomic indices corresponding to those shown in Fig. S2.\nBottom layer\n47 / 52\nFe1-Fe2 Fe1-Fe3 Fe1-Fe4 Fe1-Fe5 Fe2-Fe3\n𝐽 27.4662 15.9339 0.3877 0.1651 4.1628\n𝑖𝑗\nFe2-Fe4 Fe2-Fe5 Fe3-Fe4 Fe3-Fe5 Fe4-Fe5\n𝐽 14.1976 0.1568 9.3803 0.1181 2.1576\n𝑖𝑗\nTop layer\nFe6-Fe7 Fe6-Fe8 Fe6-Fe9 Fe6-Fe10 Fe7-Fe8\n𝐽 21.0638 17.0270 0.5662 0.1663 3.6553\n𝑖𝑗\nFe7-Fe9 Fe7-Fe10 Fe8-Fe9 Fe8-Fe10 Fe9-Fe10\n𝐽 14.2569 0.9473 9.9398 0.2116 3.0788\n𝑖𝑗\nInterlayer coupling\nFe1-Fe10 Fe2-Fe7 Fe2-Fe9 Fe2-Fe10 Fe3-Fe9\n𝐽 -0.0409 -0.0235 -0.0206 0.0183 -0.0085\n𝑖𝑗\nFe3-Fe10 Fe4-Fe7 Fe4-Fe8 Fe4-Fe9 Fe4-Fe10\n𝐽 -0.0282 0.0243 0.0125 -0.0384 0.0185\n𝑖𝑗\nFe5-Fe6 Fe5-Fe7 Fe5-Fe8 Fe5-Fe9 Fe5-Fe10\n𝐽 0.0004 -0.0176 -0.0129 -0.0084 -0.0012\n𝑖𝑗\n48 / 52\nTechnical question 7: “Are the magnetic configuration, AHE, and ANE affected by the\ninterlayer distance.”\nResponse: We thank the reviewer for the valuable suggestion. Based on the comments,\nthe interlayer distance was decreased and increased by 15% with respect to the\nequilibrium value (𝑑 = 3.09 Å) to simulate interlayer compression and expansion, i.e.,\n0\n𝑑 = 𝑑 (1±𝜎%), where 𝜎 = [−15,+15].\nphase under different interlayer distances. As shown in Fig. S7(a), the energy difference\nbetween the FM and A-type AFM1 configurations (Δ𝐸 = 𝐸 −𝐸 ) remains\nFM AFM1\npositive, confirming that the AFM1 configuration is consistently the magnetic ground\nstate across all interlayer distances. Furthermore, we evaluated the MAE using a dense\n15×15×1 k-mesh, as presented in Fig. S7(b). The results indicate that there is weak\nin-plane magnetic anisotropy throughout the entire range of 𝜎, implying that variations\nin the interlayer distance do not qualitatively affect the magnetic configuration.\nFurthermore, variations of interlayer distance do not alter the symmetry or magnetic\npoint group of the system. Thus, the existence of anomalous transport responses under\ndifferent magnetization directions remains the same as in the equilibrium case (see\nTable S13). As examples, we calculated the AHC and ANC of the ferroelectric AB\nphase as functions of σ for magnetization aligned along the [001] and [120] directions.\nAs shown in Figs. S14-S16, both the AHC and ANC persist under all interlayer\ndistances due to the preserved symmetry, with only slight variations in their values, as\nexpected.\nFollowing the suggestions, we have made the following revisions to the main text:\nIn the seventh paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties':\n“…To quantify this effect, the interlayer distance was systematically varied by ±15%\nwith respect to the equilibrium value (d = 3.09 Å) to simulate interlayer compression\n0\nand expansion, i.e., d = d (1 ± σ), where σ ∈ [−15%,+15%] (see Fig. S6 of the SI).\nphase under different interlayer distances. As shown in Fig. S7(a) of the SI, the energy\ndifference between the FM and AFM1 configurations (∆E = EFM − EAFM1) remains\npositive, confirming that the AFM1 configuration is consistently the magnetic ground\nstate across all interlayer distances.…”\nIn the last paragraph of the section ''Considerable Sign-Reversible Anomalous\nTransport Properties':\n“…Finally, we emphasize that the anomalous transport properties are protected by\nsymmetry. Variations in the interlayer distance only slightly affect their magnitudes but\ndo not alter their qualitative existence, as detailed in Section H of the SI.…”\nAlso, we have added Section ''H. Anomalous transport properties under different\ninterlayer distances'' in the revised SI.\n“We evaluate the MAE under different interlayer distances, as presented in Fig. S7. The\nresults reveal a weak in-plane magnetic anisotropy throughout the entire range of\ninterlayer distance variation (σ), indicating that changes in σ do not qualitatively affect\nthe magnetic configuration. Moreover, variations in the interlayer distances do not alter\n49 / 52\nthe symmetry or magnetic point group of the system. Therefore, the presence or absence\nof anomalous transport responses under different magnetization directions remains the\nsame as in the equilibrium state. As examples, we calculate the AHC and ANC of the\nAB stacking as a function of σ for magnetization aligned along the [001] and [120]\ndirections. As shown in Figs. S14-S16, both the AHC and ANC persist across all\ninterlayer distances owing to the preserved symmetry, with only slight variations in\ntheir values, as expected.…”\nFig. S7. Calculated (a) energy differences (Δ𝐸 = 𝐸 −𝐸 ) and (b) MAE (ΔE\nFM AFM1 MAE\n= E – E ) as a function of the relative interlayer variation.\n[100] [001]\nFig. S14. AHC of the AB stacking under different interlayer distances for magnetization\n50 / 52\nalong the [001] and [120] directions.\nFig. S15. ANC of AB stacking under different interlayer distances for magnetization\nalong the [001] direction.\n51 / 52\nFig. S16. ANC of AB stacking under different interlayer distances for magnetization\nalong the [120] direction.\n52 / 52"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I reviewed the response provided by Zuo, et al. The authors correctly and convincingly answered all my questions.\nI would recommend some minor corrections on the text:\nMaterial Realization with Triple-Coupled Switching → replace “triple-coupled” by “triply-coupled”?\n“which implies that the magnetization orientation is highly susceptible to experimental conditions” → replace “susceptible”\nby “sensitive”?\nUnambiguously write everywhere when the energy is given in eV per unit cell (e.g. Fig. S7 or S8).\n“Given the similarly layered structure of Fe5GeTe2 , our proposed sliding FE FiM metal is experimentally accessible with\ncurrent techniques [62–66], making its experimental validation highly feasible in the near future.” → I would use a more\nconditional form: “is” replaced by “could be” or “should be”, “feasible” replaced by “probable”?", "reviewer_2": "In the revised version of the paper, authors properly addressed all the issues I raised in my original report. In particular, the\nMAE values are now recalculated and are well converged. Therefore, I recommend publication of this work in the current\nform.", "reviewer_3": "In this revised manuscript, the authors have provided a detailed elaboration and refinement of their research content.\nHowever, the authors have deliberately avoided addressing the questions raised by the reviewers. Specifically, they\nremoved TABLE IV from the appendix of the first draft, in which they indicated that the energy of FM-001 is 0, a value that\nrequires detailed explanation. Additionally, the energy of the FiM-100 state is 6.42 meV, but it is unclear whether this value\nis relative to FM-001 or FiM-001.Addtionally,the authors recalculated the magnetic anisotropy energy (MAE) and compared\nthe effects of various van der Waals (vdW) correction methods on the MAE (as shown in Table S11). However, they failed to\nprovide an explanation regarding the ferromagnetic (FM) ground state of the system after considering spin-orbit coupling\n(SOC). Instead, they deleted the original table. The authors must offer a reasonable explanation and directly address this\nissue.Specifically, the impact of SOC on the energy of the ferromagnetic state. Otherwise, the manuscript cannot be\naccepted by NC."}, "author_response": "Response to Reviewer #1\nGeneral comment: “I reviewed the response provided by Zuo, et al. The authors\ncorrectly and convincingly answered all my questions. I would recommend some minor\ncorrections on the text:”\nResponse: We are grateful for your thoughtful and detailed feedback. Your comments\nhave been invaluable in helping us improve the quality and clarity of our manuscript.\nTextual refinement 1: “Material Realization with Triple-Coupled Switching → replace\n“triple-coupled” by “triply-coupled”?”\nResponse: We thank the reviewer for the valuable suggestion. We have replaced\n“triple-coupled” with “triply-coupled” in the revised manuscript.\nTextual refinement 2: ““which implies that the magnetization orientation is highly\nsusceptible to experimental conditions” → replace “susceptible” by “sensitive”?”\nResponse: We thank the reviewer for the valuable suggestion. We have replaced\n“susceptible” with “sensitive” to improve clarity and accuracy of expression.\nTextual refinement 3: “Unambiguously write everywhere when the energy is given\nin eV per unit cell (e.g. Fig. S7 or S8).”\nResponse: We thank the reviewer for the valuable suggestion. We have carefully\nchecked the manuscript and supporting information and explicitly stated “energy in\nmeV per unit cell” wherever relevant (including Figures S7 and S8).\nTextual refinement 4: ““Given the similarly layered structure of Fe5GeTe2, our\n1 / 8\nproposed sliding FE FiM metal is experimentally accessible with current techniques\n[62–66], making its experimental validation highly feasible in the near future.” → I\nwould use a more conditional form: “is” replaced by “could be” or “should be”,\n“feasible” replaced by “probable”?”\nResponse: We sincerely thank the reviewer for this valuable suggestion. Following the\nreviewer’s advice, we have revised the sentence to adopt a more cautious and\nconditional tone. The revised version now reads as follows:\n“Given the similarly layered structure of Fe GeTe₂, our proposed sliding FE FiM metal\n5\ncould be experimentally accessible with current techniques [62–66], making its\nexperimental validation probable in the near future.”\n2 / 8\n-------------------------------------------------------------------------------------------------------\nResponse to Reviewer #2\n-------------------------------------------------------------------------------------------------------\nGeneral comment: “In the revised version of the paper, authors properly addressed all\nthe issues I raised in my original report. In particular, the MAE values are now\nrecalculated and are well converged. Therefore, I recommend publication of this work\nin the current form.”\nResponse: We sincerely thank the reviewer for the positive assessment and\nrecommendation for publication.\n3 / 8\n-------------------------------------------------------------------------------------------------------\nResponse to Reviewer #3\n-------------------------------------------------------------------------------------------------------\nGeneral comment: “In this revised manuscript, the authors have provided a detailed\nelaboration and refinement of their research content. However, the authors have\ndeliberately avoided addressing the questions raised by the reviewers. Specifically, they\nremoved TABLE IV from the appendix of the first draft, in which they indicated that\nthe energy of FM-001 is 0, a value that requires detailed explanation. Additionally, the\nenergy of the FiM-100 state is 6.42 meV, but it is unclear whether this value is relative\nto FM-001 or FiM-001.Addtionally,the authors recalculated the magnetic anisotropy\nenergy (MAE) and compared the effects of various van der Waals (vdW) correction\nmethods on the MAE (as shown in Table S11). However, they failed to provide an\nexplanation regarding the ferromagnetic (FM) ground state of the system after\nconsidering spin-orbit coupling (SOC). Instead, they deleted the original table. The\nauthors must offer a reasonable explanation and directly address this issue. Specifically,\nthe impact of SOC on the energy of the ferromagnetic state. Otherwise, the manuscript\ncannot be accepted by NC.”\nResponse: We sincerely thank the reviewer for acknowledging that the revised\nmanuscript provides a more detailed elaboration and refinement of our research content.\nWe truly appreciate the reviewer’s careful evaluation and constructive feedback.\n4 / 8\nFormer Table IV in the first draft. Magnetic ground states (MGS) and\nmagnetocrystalline anisotropy energies (MAE) of various stackings.\nMGS (meV/unit cell) MAE (meV/unit cell)\nSystem\nState Energy Direction Energy\nFM 9.89 [001] 0\nNP (AA’\nstacking)\nA-type AFM 0 [100] -0.39\nFM 2.42 [001] 0\nFE (AB and\nBA stacking)\nc-FiM 0 [100] 6.42\nUpdated Table IV. Magnetic ground states (MGS) and magnetocrystalline anisotropy\nenergies (MAE) of various stackings.\nMGS (meV/unit cell) MAE (meV/unit cell)\nSystem\nState Energy Direction Energy\nFM 9.89 AFM [001] 0\nNP (AA’\nstacking)\nA-type AFM 0 AFM [100] -0.39\nFM 2.42 FiM [001] 0\nFE (AB and\nBA stacking)\nc-FiM 0 FiM [100] 6.42\n5 / 8\nFirst, we would like to clarify that, as mentioned in our Round 2 response letter, the\nformer Table IV in the first draft may have caused some confusion due to its formatting.\nSpecifically, the energy values listed in the “MGS (meV/unit cell)” column represent\nthe relative energies between different magnetic configurations, whereas the “MAE\n(meV/unit cell)” column reports the magnetocrystalline anisotropy energies\ncorresponding exclusively to the AFM (for NP stacking) and FiM (for FE stacking)\nconfigurations. In other words, the MAE values do not correspond to the FM state;\nrather, they describe the magnetic anisotropy of the respective ground-state AFM or\nFiM state (please see the updated Table IV).\nSecond, we would like to emphasize that the removal of the former Table IV was not\nintended to avoid addressing the reviewer’s concerns. The former table was generated\nusing a 9×9×1 k-mesh. Following the valuable suggestions from Reviewer #1 and\nReviewer #2 to ensure numerical convergence, we recalculated the MAE with a denser\n15×15×1 k-mesh in the Round 2 manuscript. Therefore, the former Table IV was\nremoved solely because it was based on the 9×9×1 k-mesh, rather than being\ndeliberately omitted.\nThird, after carefully re-examining your previous review report, we realized that we\nhad inadvertently overlooked one of your valuable comments — “Otherwise, the\nauthors would need to compare the energy of FM-001 and c-FiM-100 in the manuscript.”\nWe sincerely apologize for this oversight. We fully agree with your point and greatly\nappreciate this important suggestion. Indeed, analyzing the impact of SOC on the\nenergies of the FM and FiM states is essential for a comprehensive understanding of\nthe system. Once again, we thank you for this insightful comment.\nTo directly address the reviewer’s concern regarding the influence of SOC on the\nenergies of the FM and FiM states of the FE phase, we recomputed the total energies of\nboth configurations — with magnetization aligned along the [100] and [001] directions\n— using a systematic series of k-meshes ranging from 9×9×1 to 25×25×1. The results\nare summarized in the Supplementary Table 12 of the revised Supplementary\nInformation. These calculations consistently demonstrate that the FM state is not the\n6 / 8\nground state under SOC across all tested k-point meshes.\nSupplementary Table 12. Total energies of FiM and FM configurations of the FE\nphase with magnetization along the [100] and [001] directions under SOC, calculated\nusing different k-point meshes. All energies are given in meV per unit cell, with the\nenergy of the FiM [001] state set as the reference. The lowest-energy configuration for\neach k-mesh is highlighted with an asterisk (*).\n9×9×1 10×10×1 12×12×1 15×15×1\nFiM [100] 6.42 -0.42* -3.85* -0.58*\nFiM [001] 0* 0 0 0\nFM [100] 2.35 3.15 -1.37 2.64\nFM [001] 2.4 3.51 0.2 3.15\n17×17×1 20×20×1 22×22×1 25×25×1\nFiM [100] -1.26* -1.08* -0.35* -1.02*\nFiM [001] 0 0 0 0\nFM [100] 1.69 2.02 1.81 1.91\nFM [001] 2.86 2.94 2.78 2.83\nFollowing the suggestion, we have made the following revisions to the supporting\ninformation:\nIn the section ''The k-Point Convergence of the Spin-Orbit Properties'' of the\nsupporting information:\n7 / 8\n“…We would like to point out that, to further examine the influence of spin-orbit\ncoupling (SOC) on the energies of the FM and FiM states of the FE phase, we have\ncalculated the total energies of both states with magnetization aligned along the [100]\nand [001] directions, using a systematic series of k-point meshes ranging from 9×9×1\nto 25×25×1. As summarized in Supplementary Table 12, the FiM state, rather than the\nFM state, is the ground state under SOC across all tested k-point meshes. …”\nMoreover, to clearly indicate that the MAE is evaluated for the FiM state, we have\nchanged the expression from MAE (ΔE = E - E ) to (ΔE = E - E\nMAE [100] [001] MAE FiM[100] FiM\n) in this version.\n[001]\n8 / 8"}, {"round": 3, "reviewer_comments": {"reviewer_3": "The manuscript has been revised accordingly, with the necessary computational details added and the tables clarified.\nTherefore, the manuscript can be accepted by Nature Communications."}, "author_response": "Response to Reviewer #3\nGeneral comment: “The manuscript has been revised accordingly, with the necessary\ncomputational details added and the tables clarified. Therefore, the manuscript can be\naccepted by Nature Communications.”\nResponse: We sincerely thank you for the positive evaluation and appreciative\ncomments. We are pleased that the revisions, including the added computational\ndetails and clarified tables, have satisfactorily addressed all concerns."}]

Decision letter