Voltage-controlled topological spin textures in the monolayer limit

Domain: chemistry
Published: 2026-02-19
Editorial outcome: major_revision

Abstract
The physics of phase transitions in low-dimensional systems has long been a subject of significant research interest. Long-range magnetic order in the strict two-dimensional limit, whose discovery circumvented the Mermin-Wagner theorem, has rapidly emerged as a research focus. However, the demonstration of a non-trivial topological spin textures in two-dimensional limit has remained elusive. Here, we demonstrate the out-of-plane electric field breaks inversion symmetry while simultaneously modulating the electronic band structure, enabling electrically tunable spin-orbit interaction for creation and manipulation of topological spin textures in monolayer CrI3. The realization of ideal two-dimensional topological spin textures may offer not only an experimental testbed for probing the Berezinskii–Kosterlitz–Thouless mechanism, but also potential insights into unresolved quantum phenomena including superconductivity and superfluidity. Moreover, voltage-controlled spin-orbit interaction offers a novel pathway to engineer two-dimensional spin textures with tailored symmetries and topologies, while opening avenues for skyrmion-based next-generation information technologies.

Introduction
The pursuit of low-dimensional (2D) magnetic materials and their unique topological spin textures dates back to the last century. Early theoretical studies recognized that while the Mermin-Wagner theorem prohibits long-range magnetic order in low-dimensional systems (d<3) with continuous symmetry1, this limitation could be overcome through substantial magnetic anisotropy. This is evidenced by recent realizations of Ising-like 2D magnets2,3, such as monolayer ferromagnetic CrI3 and Cr2Ge2Te6, which have since sparked extensive research interest in this field. The Berezinskii–Kosterlitz–Thouless (BKT) theory predicts the stabilization of topological spin textures in rigorously two-dimensional systems4,5. Through the breaking of inversion symmetry, the existence of skyrmions has been observed in van der Waals nanosheets via the topological magneto-optical effect6,7. However, conclusive experimental verification of topological spin textures in atomically thin monolayers remains a significant challenge. The topological spin textures in real-space and band structures of non-trivial topology in momentum space, have attracted enormous attention due to their elegant Berry curvature physics8–16. Topologically protected, particle-like spin textures—particularly skyrmions, which exhibit robust stability and controlled motion under electrical stimuli (e.g., applied currents)—represent promising candidates for information carriers in next-generation memory and logic devices17–20. Among the key challenges in developing future energy-efficient devices is achieving magnetic state switching via electric fields21,22. Despite early demonstrations of electric-field switching of skyrmions through STM tips23,24, reports on their translation into practical tunnel junction nanodevices remain scarce25.

Here, we demonstrate the electrical creation of the topological spin textures in monolayer CrI3, which we detect through the unconventional topological magnetic circular dichroism. The gate-controlled modulation of both Dzyaloshinskii-Moriya interaction (DMI) and magnetic anisotropy energy (MAE) drives a reversible phase transition from ferromagnetic order to stabilized topological spin states. Electrostatic gating enables continuous modulation of the MAE from perpendicular to in-plane configurations, with the sign reversal of MAE permitting voltage-controlled stabilization of topological spin textures across a well-defined biasing regime. Our results provide a platform for understanding topological magnetic quasiparticles and their interactions in two-dimensional systems. This may contribute to elucidating the mechanisms of the BKT transition and related fundamental quantum phenomena such as superfluidity and superconductivity26,27. Besides, the electric-field-controlled creation and annihilation of skyrmions break through the limitations of the energy-intensive current-driven paradigm in racetrack memory23–25, which may revolutionize next-generation memory devices and information technologies.

Results
The scalar spin chirality of topological magnetic structures has manifested itself as an emergent magnetic field (Be)14, which gives rise to a transversal electrical response (Fig. 1a) known as the conventional topological Hall effect (THE) at zero-frequency limits28,29. The emergent magnetic field arising from skyrmion potentially gives rise to the topological magneto-optical effect (TMOE)30,31, which can be considered as the optical-frequency counterpart of the conventional THE (Fig. 1b). Tunable optical-frequency permit the identification of topological magnetic structures in diverse material systems—including metals, semiconductors and insulators—thereby extending the utility of the topological magneto-optical effect beyond the metallic regimes accessible to conventional THE. To investigate magnetoelectric coupling in the ideal 2D limit, we engineered monolayer CrI3 devices supporting both perpendicular electric fields and electrostatic doping. Monolayer CrI3 was exfoliated mechanically from bulk crystals onto a hBN flake on 285-nm-thick SiO2/Si substrates in a nitrogen-filled glovebox (see “Methods”). A micrograph of the monolayer CrI3 and the device is shown in Supplementary Fig. 1a,b. Supplementary Fig. 1c illustrates the structure of a representative monolayer CrI3 device. This vertically stacked configuration consists of a monolayer of CrI3, hBN flakes and graphene contacts, in which the hBN flakes act as the dielectric layers for providing electrostatic doping and electric field. The dielectric layer of hBN has a thickness of 9.8 nm (Supplementary Fig. 2) and a dielectric constant of 2.9 (Supplementary Fig. 3; see “Methods”), consistent with the previous reports32,33. Reflective magnetic circular dichroism (RMCD) measurements were performed on a designated region of the device, using a laser with a wavelength of 633 nm and a fixed power of 3 µW (see “Methods”). Figure 1c shows a representative RMCD signal of the monolayer CrI3 device as a function of the out-of-plane magnetic field (µ0H) at zero gate voltage, with black arrows indicating the direction of the magnetic field sweep. The results of monolayer CrI3 are in agreement with the previous reports1. Namely, monolayer CrI3 exhibits ferromagnetism with a coercive field (μ0Hc) of approximately 0.04 T and a Curie temperature (Tc) close to 40 K (Supplementary Fig. 4). And the absence of a high-field step in the RMCD loop (Supplementary Fig. 5a) excludes interlayer antiferromagnetic coupling and validates the monolayer34. Furthermore, the polarization-resolved and temperature-dependent Raman measurements of spin-lattice coupling further confirm the monolayer feature35,36, in which the A1g phonons are only detected at 127.8 cm-1 in cross- (XY) and co-linear (XX) polarization configurations and significantly increased in XY configuration below 40 K. This temperature-dependent evolution of the Raman features and the RMCD signals provides explicit evidence that the CrI3 sample is a monolayer (Supplementary Note 1, Supplementary Figs. 4, 5).

Figure 1d shows the RMCD signal as a function of the out-of-plane magnetic field under an applied voltage of 8V, revealing a pair of anomalous RMCD peaks near the spin-switching transition field. The polar RMCD signal intensity (IRMCD) is approximately proportional to σxy(ω), as expressed by the following equation37–39

IRMCD = 2Z0 d σxy(ω) / (1 + Z0 d σxx(ω))2 (1)

Here, Z0 is the impedance of free space, ns denotes the refractive index of the substrate, and d represents the thickness of the monolayer CrI3, approximately 0.7 nm; σxx(ω) and σxy(ω) are the longitudinal and transverse components of the optical Hall conductivity, respectively.

Analogous to the analytical framework for the conventional topological Hall effect, σxy(ω) can be separated into two primary components: σxyA(ω) originating from the ferromagnetic spin texture, and σxyT(ω) from the topological spin textures31, which can be determined by evaluating the solid angle traced by the spin moment n as it winds within the xy-plane15, given by

σxyT(ω) / Be = 2 / n(3) ∂n/∂x × ∂n/∂y z^ (2)

For a skyrmion, Be is quantized: the integral over the 2D space is simply 2π times the topological charge (Q) of skyrmions, Q = 1/4π ∫ ∂n/∂x × ∂n/∂y dxdy. Therefore, spin textures with non-zero Q can induce differences in the reflectivity between the left- and right-handed circularly polarized lights, even in the absence of a net magnetic moment (Fig. 1a). We refer to this effect as reflective topological circular dichroism (RCDT), which can also lead to the rotation of the polarization direction of linearly polarized reflected light, known as the topological Kerr effect. The RMCD loops illustrated in Fig. 1e can be expressed as the sum of two components,

IRMCD = IRCD(M) + IRCD(T) (3)

Here, IRCD(M) is the intensity of conventional reflective circular dichroism (RCDM), which serves as a measure of the out-of-plane net magnetization of the ferromagnetic background (highlighted in purple, see Fig. 1c–f), while IRCD(T) is the intensity of RCDT and is correlated with the density of Q (highlighted in orange, Fig. 1d–f). The upward and downward RMCD peaks suggest the generation of the topologically nontrivial skyrmions with +Q and -Q, respectively15,16, in the vicinity of the spin-switching field40. Figure 1e illustrates the relationship between the magnetic field and the RMCD signal at an applied voltage of 12V. The RCDM signal is decreased, signifying that the out-of-plane magnetization is reduced, while the σxyT(ω) is proportional to the magnetic field induced by topological spin textures31, which can be determined by evaluating the solid angle traced by the spin moment n as it winds within the xy-plane15. Remarkably, under a voltage of 16V, the RCDM hysteresis completely vanishes, leaving only a significant RCDT, accompanied by a giant topological circular dichroism (Fig. 1f), akin to the giant topological Hall effect observed in skyrmion lattices15,16. This indicates that the system transitions to a small in-plane anisotropy, while the density of skyrmions significantly increases with the assistance of an out-of-plane magnetic field. Similar voltage-controlled RMCD phenomena were also observed in another monolayer CrI3 device (Supplementary Fig. 6).

Figure 2a presents the RMCD intensity as a function of gate voltage (Vg) and magnetic field (μ0H), sweeping from positive to negative values. In this magnetic phase diagram, white regions indicate near-zero RMCD intensity, corresponding to an in-plane ferromagnetic state. Darker red and blue areas represent two out-of-plane ferromagnetic states with strong RMCD signals of opposite signs. Light blue and red regions denote canted ferromagnetic states with opposite out-of-plane components. Notably, a distinct red stripe appears near the critical magnetic field for the spin-up to spin-down transition, identified as a topological magnetic phase with positive topological charge. In the non-topological phase region, increasing voltage drives a smooth transition from out-of-plane to in-plane ferromagnetism, suggesting a voltage-induced evolution of the perpendicular magnetic anisotropy toward in-plane anisotropy (highlighted by the blue arrows)42. The corresponding phase diagram, obtained by sweeping the magnetic field from negative to positive values, is shown in Supplementary Fig. 7a. Similar to Fig. 2a, a distinct blue stripe appears near the critical magnetic field for the spin-down to spin-up transition, representing a topological magnetic phase with negative topological charge. Figure 2b displays the intensities of the topological circular dichroism RCDT as a function of Vg and μ0H, revealing a critical voltage for the emergence of the topological magnetic orders near 5V. Additionally, the critical magnetic field distinguishing the ferromagnetic and topological magnetic phases varies with the applied voltage. RCDT is obtained by subtracting RCDM from the total RMCD, as described in Eq. (3) and Fig. 1e. Figure 2c shows the RCDT loops measured at several representative voltages, extracted from Fig. 2b. As the voltage increases, the RCDT peaks with positive and negative topological charges shift linearly toward higher magnetic fields, with slopes of 22 Oe V−1 and 13 Oe V−1, respectively (Fig. 2d). This linear shift demonstrates precise and continuous voltage-controlled tuning of the topological magnetic phases. This behavior arises from the reduction of the perpendicular anisotropy induced by higher voltages, thus necessitating a stronger out-of-plane magnetic field to stabilize the skyrmions. However, above 14V, the positions of the RCDT peaks abruptly shift toward lower magnetic fields. It suggests a transition from the perpendicular magnetic anisotropy to in-plane one, thereby causing a sudden shift in the magnetic field required to stabilize the topological magnetic structure. Figure 2e illustrates the electrically controlled RCDT under three selected magnetic fields with opposite directions, extracted from Fig. 2b. The RCDT signal remains near zero at voltages below approximately 5V, corresponding to the ferromagnetic ordering. As the voltage increases to a critical value around 5V, the RCDT signal rises sharply, indicating an electrical switch from trivial ferromagnetic to topological magnetic states. With voltage further increasing, the RCDT signal gradually rises and saturates, suggesting an increase in skyrmion density. Near 14V, the RCDT signal undergoes a sudden change, suggesting that a fundamental shift in the magnetic anisotropy may trigger the nucleation of the skyrmions with new characteristics, thereby causing a sharp variation in the RCDT signal. We refer to the skyrmions induced by an out-of-plane magnetic field under perpendicular anisotropy as Type-I skyrmions, and those formed under a small in-plane magnetic anisotropy induced by an out-of-plane magnetic field as Type-II skyrmions (Figs. 2a and 2e). This abrupt behavior at 14V is consistent with the voltage-induced shifts observed in RCDT peak positions (Fig. 2d) and intensities (Supplementary Fig. 7b). Furthermore, the increasing voltage results in a corresponding rise in RCDT, attributed to an increase in the density of type-II skyrmions. The voltage can continuously regulate the density and morphology of the topological magnetic quasiparticles, and one of the key reasons is that the voltage enables continuous adjustment of the magnetic anisotropy energy, even changing its sign42,43, leading to a gradual transition from perpendicular anisotropy to in-plane anisotropy (Supplementary Fig. 8c).

The stability of the skyrmions is governed by two critical parameters: the DMI and magnetic anisotropy, both originating from the fundamental mechanism of the spin–orbit coupling (SOC)43–45. Previous theoretical predictions suggested that skyrmions could be generated in a monolayer magnet through either out-of-plane electric fields or geometric defect-induced inversion symmetry breaking46,47. However, the exclusive consideration of DMI without incorporating magnetic anisotropy modulation may critically hinder skyrmion stabilization. This fundamental constraint most likely underlies the absence of experimentally observed skyrmions in the monolayer limit. As illustrated in Supplementary Fig. 9a, the application of an out-of-plane electric field in monolayer CrI3 breaks inversion symmetry, leading to the emergence of a significant DMI48. The magnetic anisotropy energy (MAE) in CrI3 monolayers is intimately associated with the spin and orbital structures near the Fermi level. First-principles calculations have demonstrated that the MAE is primarily governed by the indirect SOC, with the direct SOC contribution being negligible49. By the perturbation theory, the MAE can be succinctly expressed as50

MAE = ξ2 Σμ,o,σ,σ0 |<o,σ|LIz|μ,σ0>|2 - Σμ,o,σ,σ0 |<o,σ|LIx|μ,σ0>|2 (4)

Here, u and o denote the unoccupied and occupied states, respectively, Eu/o,σ represent the band energy of these states, and the spin indices σ/σ′ range over ±1, corresponding to the two orthogonal spin states at the k-point. Previous theoretical investigations have established that |<o,σ|LIz|μ,σ0>|2 - |<o,σ|LIx|μ,σ0>|2 is negative51. In addition to the contribution from the orbital angular momentum, the relative spin polarization (σ, σ′) of the states near the Fermi level must also be considered. When CrI3 is slightly n-doped, the two lowest conduction bands act as the unoccupied and occupied states, with their parallel spin alignment (σσ′=1, right panel in Supplementary Fig. 9b) resulting in a negative MAE, which favors an in-plane magnetic anisotropy. In stark contrast, under slight p-doping, the dominant contributions originate from the two highest valence bands with oppositely aligned spins (σσ′=−1, left panel in Supplementary Fig. 9b), resulting in a positive MAE that favors perpendicular magnetic anisotropy. In our experimental observations, the application of a positive voltage results in electron doping and further induces a transition from perpendicular magnetic anisotropy to in-plane anisotropy in monolayer CrI3, while a negative voltage (hole doping) strengthens the perpendicular out-of-plane magnetization (Supplementary Fig. 8). Theoretically, MAE(K) must satisfy K ≤ π2⋅D2/16⋅J for enabling the formation of a skyrmion lattice43–45, implying that the skyrmions are stabilized only within a regime of weak anisotropy. Here, D denotes the Dzyaloshinskii–Moriya interaction energy, while J represents the exchange interaction energy. Supplementary Fig. 9c illustrates the relationship between the factor η=π2⋅D2/16⋅J, MAE(K), and gate voltage, where the D term is induced by the electric field and the K term arises from electrostatic doping42,48. Using a simple parallel-plate capacitor model52,53, the corresponding voltage values can be inferred from the electric field strength and doping concentration within the CrI3 monolayer (see “Methods”). This approach enables a comparative analysis of the voltage ranges for skyrmion stabilization between theoretical predictions and experimental observations. The theoretically predicted voltage thresholds for type-I (0 < K < η) and type-II (K < 0) skyrmion formation closely match the experimental voltages for both skyrmion types (highlighted in light yellow and purple areas).

The evolution of the magnetic structures in monolayer CrI3 under out-of-plane magnetic fields at different voltages was simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation (see “Methods”). The parameters for atomic-scale spin dynamics simulations were selected based on the criteria for the skyrmion formation as illustrated in Supplementary Fig. 9c. Figure 3a presents the atomic-scale spin dynamics simulation results at a voltage of 16V (in-plane anisotropy, K<0), depicting the evolution of the magnetic domains as the out-of-plane magnetic field B/J is swept from +0.214 to -0.214. As the magnetic field decreases from positive saturation to zero, the system evolves from an up-magnetized single-domain state to stripe domains. With increasing negative field, these domains fragment into skyrmions with positive topological charge (Q>0), which subsequently annihilate at higher negative fields, establishing a down-magnetized state. The reverse sweep (-0.214 → +0.214, Fig. 3b) shows analogous but inverted behavior, with stripe domains forming negative-charge skyrmions (Q<0) that disappear upon reaching positive saturation. As shown in Fig. 3c, the Mz versus magnetic field curve shows no hysteresis, a characteristic resulting from the in-plane anisotropy (K<0) induced by electron doping42. Figure 3d shows the topological charge density versus magnetic field. During positive-to-negative magnetic field sweeps, a positive topological charge peak emerges at the negative critical field. Conversely, negative-to-positive sweeps generate a negative peak at. This antisymmetric behavior correlates well with the field-dependent RCDT shown in Fig. 1f. Analysis of the magnetic structure in Fig. 3d inset reveals a Néel-type skyrmion configuration at fields near the topological charge peak. Figure 3e schematically demonstrates the variation of the sum of Mz and the topological charge with the out-of-plane magnetic field, which aligns with the experimentally observed RMCD loop featuring anomalous peaks and no residual signal of RCDM (Fig. 1f). This confirms the core conclusion: the total RMCD signal intensity IRMCD can be expressed as Eq. (3) contributing to the net out-of-plane magnetization Mz, and the density of the topological charge Q. Supplementary Fig. 10a,b illustrate the evolution of the magnetic structure under an out-of-plane magnetic field at a voltage of 8V (perpendicular magnetic anisotropy, K>0). The evolution of the magnetic structure resembles that observed at 16V. However, due to the perpendicular anisotropy, the Mz hysteresis loop shows a distinct opening (Supplementary Fig. 10c), indicating finite remanent magnetization. This observation is consistent with the residual RMCD signal detected at approximately 8V (Fig. 1d). In contrast to the in-plane anisotropy at 16V, the perpendicular anisotropy at 8V yields lower skyrmion density, indicated by a weaker topological charge peak (Fig. 3d, Supplementary Fig. 10d), consistent with the voltage-dependent RCDT peak evolution (Fig. 2c, Supplementary Fig. 7b).

To further explore the properties of the voltage-induced two-dimensional topological magnetic phase, Fig. 4a displays the RMCD intensities as a function of the magnetic field at various temperatures under a 16V voltage. As the temperature increases, the topological magnetic phase undergoes a transition to an in-plane ferromagnetic state at approximately 24 K, followed by a transition from the in-plane ferromagnetism to a paramagnetic state around 27 K. The Curie temperature (Tc) of monolayer CrI3 under a 16V voltage is lower than that of intrinsic monolayer CrI3 (Supplementary Fig. 4) because the electron doping reduces the MAE (Fig. 3c). On the contrary, applying a voltage of -14V slightly increases the Tc of CrI3 (Supplementary Fig. 11) due to the enhancement of MAE through the hole doping42. This behavior is attributed to the fact that the Tc of monolayer CrI3 is primarily governed by magnetic anisotropy induced by the SOC effects51. Figure 4b,c show the phase diagrams of topological magnetic orders in monolayer CrI3 as functions of temperature and magnetic field. The results indicate that the magnetic field range for stabilizing the magnetic skyrmions decreases with increasing temperature, consistent with atomic-scale spin dynamics simulations (Supplementary Fig. 12, Supplementary Notes 2). Additionally, the magnetic fields corresponding to the maximum density of the topological quasiparticles—the positions of the RMCD peaks with positive topological charge (upward peaks) and negative topological charge (downward peaks)—decrease linearly with temperature, exhibiting slopes of 11.5 Oe K−1 and 10.4 Oe K−1, respectively (Fig. 4d). Furthermore, the half-peak width of the RMCD peaks, which characterizes the robustness of topological quasiparticles against magnetic fields, also gradually diminishes as the temperature rises. This behavior closely resembles the field-temperature phase diagrams of the skyrmion lattices in bulk Gd2PdSi3 and GdRu2Ge215,16. Figure 4e illustrates the critical temperature behavior of the topological phase transition by comparing the temperature dependence of RMCD under different magnetic fields (horizontal cuts of the phase diagrams in Fig. 4b,c). The critical temperature shifts from approximately 15 K at ±0.04 T to 24 K at ±0.02 T. This indicates that as the out-of-plane magnetic field increases, the critical temperature decreases. The evolution of the topological magnetic phase induced by a 14V voltage, as a function of temperature and magnetic field, is similar to that observed at 16V (Supplementary Figs. 13, 14). However, compared to the 16V case, a slight perpendicular magnetic anisotropy is persisted. Therefore, in addition to the significant RCDT peaks, an open RCDM loop was also observed (detailed discussion is provided in Supplementary Notes 3).

In summary, we demonstrated a voltage-control of the topologically protected spin textures in monolayer CrI3. By utilizing the voltage-induced synergistic effects between DMI and MAE, we achieved the generation and continuous manipulation of the skyrmion quasiparticles at the two-dimensional limit, providing a powerful tool for controlling the topological magnetic states at a single atomic layer limit. The results highlight the potential of the voltage-driven modulation of the spin-orbit interactions in creating a diverse range of two-dimensional magnetic textures with distinct topological properties. The precise electric-field control in engineering topological magnetic quasiparticles opens up new possibilities for ultra-high-density, ultra-low-power, and high-speed spintronic applications, while also broadening the scope of topological magneto-optical effects, thereby advancing the development of cutting-edge magneto-optical information storage and processing technologies.

Methods
Sample fabrication
Monolayer CrI3 was mechanically exfoliated from bulk crystals using PDMS films in an inert glovebox. The bulk CrI3 crystals were synthesized via chemical vapor transport from elemental precursors with a Cr molar ratio of 1:3. The exfoliated flakes of hBN, CrI3 and graphene were sequentially transferred onto pre-patterned Au electrodes on SiO2/Si substrates to form heterostructures. These samples were then in-situ loaded into an optical cryostat within the glovebox for magneto-optical-electric joint measurements. Throughout the fabrication and measurement processes, the CrI3 samples were kept isolated from the atmosphere.

Magneto-optical measurement
Polar RMCD and Raman measurements were conducted using a magneto-optical-electric joint-measurement scanning imaging system (MOEJSI)54. This system is based on a Witec Alpha 300R Plus low-wavenumber confocal Raman microscope, integrated with a closed-cycle superconducting magnet (7 T) and a cryogen-free optical cryostat (10 K), which features a specially designed sample mount and electronic transport measurement setup.

Raman signals were recorded using the Witec Alpha 300R Plus, coupled with a closed-cycle He optical cryostat and a superconducting magnet. A 633 nm laser was directed into the microscope via a photonic crystal fiber and directed into the spectrometer with an 1800 g mm−1 grating. Polarization-resolved Raman spectra were obtained by rotating the analyzer placed before the fiber. The excitation laser at 633 nm (1.96 eV) was set to an intensity of approximately 0.3 mW, with a typical integration time of 120 s.

For polar RMCD measurements, a 633 nm laser (~3 µW) modulated by a photoelastic modulator (PEM, 50 kHz) was reflected by a non-polarizing beamsplitter (R/T=30/70) and focused onto the sample using a 50× objective (NA=0.55, Zeiss). The reflected beam, collected by the same objective, passed through the beamsplitter and was detected by a photomultiplier tube (PMT), coupled to a lock-in amplifier, the Witec scanning imaging system, a superconducting magnet, and a voltage source meter.

Estimation of the electric field strength and doping concentration
We evaluate the doping level and electric field strength of monolayer CrI3 in our devices under electrostatic gating using a parallel-plate capacitor model. A simple heterostructure has been employed in this study: graphene/hBN/CrI3/graphene (Fig. 1c). The doping density of CrI3 (nCr) and the electric field (ECr) of CrI3 are obtained under the applied voltage (V) between the two graphene electrodes.

ECr = V / (dCr + εCr/εBN * dBN) (5)

nCr = εBN * V / (4πk * (dCr * εBN + dBN * εCr)) (6)

Here, dCr and dBN represent the thicknesses of CrI3 and hBN, respectively, εCr and εBN are the dielectric constants of CrI3 and hBN, and k is the electrostatic constant. The dielectric constant of hBN is measured to be 2.9 using a ferroelectric analyzer54, while the dielectric constant of CrI3 is 7.55. The thickness of hBN is determined to be 9.8 nm using optical contrast methods, while the thickness of a monolayer of CrI3 is 0.7 nm. By calculating Eqs. (4, 5), along with the theoretical predictions of the electric field–DMI function48 and the doping concentration–magnetic anisotropy function42, we are able to compare the experimentally observed voltage window for the emergence of topological magnetic order with the theoretically predicted critical voltage for its onset (Supplementary Fig. 9c).

Optical contrast measurement
White light is normally incident on the hBN/SiO2/Si structure and focused into a small spot using a 50×, NA=0.45 microscope objective. The reflected light is then collected, passes through a beamsplitter, and is directed onto the entrance slit of a spectrometer. The spectrally resolved reflection signal is captured by a CCD at the spectrometer’s exit aperture. In our experiments, we measure the optical contrast of the three-layer structure, which is defined as:

C = (RSiO2 - RSiO2+hBN) / RSiO2 (7)

Where RSiO2 represents the reflection coefficient at normal incidence of the SiO2/Si substrate, which is covered with hBN. The normalized change in reflectivity of the hBN layer relative to the underlying substrate56.

Atomic-scale spin dynamics simulations
The numerical simulation were performed by solving the atomic-scale LLG equation (1+α2) dm/dt = -γ mi × Hi - γα mi × (mi × Hi) with periodic boundary conditions, where γ = 1.76 × 1011 T−1 s−1 is the gyromagnetic ratio, α is the Gilbert damping constant (set to 1.0 to significantly reduce the evolution time) and Hi = -1/μs * δH/δmi is the effective field with the magnetic moment μs = 2.95 μB, respectively. The Hamiltonian of CrI3 monolayer reads H = -Σ<i,j> (J mi · mj + D · mi × mj) - K Σi (mi · z)2 - Σi mi · B, where S=3/2 is the magnitude of spin, the J is the Heisenberg exchange parameter, D is the Dzyaloshinskii-Moriya interaction (DMI) vector with both the in-plane component (Dxy) and out-plane component (Dz) to stabilize the skyrmion, K is the anisotropy parameter and B is the external magnetic field, respectively. Noteworthily, the in-plane component of the DMI vector is perpendicular to the line between the two Cr atoms. In the case of perpendicular magnetic anisotropy (8V), J is set to -1.962 meV for ferromagnetic coupling, Dxy = -0.189J, Dz = -0.119J and K = 0.019J. As for the case of in-plane anisotropy (16V), the J is set to -1.992 meV, Dxy = -0.210J, Dz = -0.132J and K = -0.013J. We derived these parameters from previous studies42,48, and, based on the skyrmion formation conditions in Supplementary Fig. 9c, selected 8V and 16V for atomic-scale spin dynamics simulations. Moreover, the thermal fluctuation is introduced with a stochastic field HT = η(step) * sqrt(2kBTα / (γμs Δt)), where η(step) is a random vector from a standard normal distribution, kB is the Boltzmann constant, T=2 K is the temperature, and Δt = 10 fs is the time interval. The system is allowed to reach equilibrium, with 10^5 steps of relaxation at each magnetic field point.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reports the emergence of skyrmions when electrostatically dope a monolayer CrI3. The main experimental\nsignatures are the topological magneto-optical signals in RMCD, with supporting analysis through theoretical simulations.\nThe results and claims to electrically manipulate the magnetic anisotropy and skyrmion states will attract a lot of interest in\nthe field of spintronics. My main concern is that such topological RMCD signatures are missing in an earlier well-established\nwork (https://www.nature.com/articles/s41565-018-0135-x) that also studies the monolayer CrI3 within similar doping levels.\nI also have a number of technical questions that the authors will need to first address.\n1. As noted above, the earlier Nature Nano paper reports the doping dependence of RMCD in a monolayer of CrI3. How do\nthe authors explain or understand this discrepancy? Is the reported topological RMCD signal reproducible in multiple\nmonolayer devices?\n2. The manuscript didn't show the estimated doping level using experimental data, even though it's mentioned that such\nvalues were extracted and compared to theory. The doping level needs to be explicitly stated along with the gate-dependent\nresults.\n3. Relevant to 2, the corresponding Fermi level shift should also be estimated using the doping level to compare with earlier\ntheory work like ref 40.\n4. Even if the authors claim that the magnetic anisotropy is completely in-plane with 16V backgate, the magnetization under\nan out-of-plane magnetic field should still undergo a polarizing transition to a finite RMCD signal, which is not the case in\nFig. 1f. The lack of RMCD signal at higher fields is not consistent with the authors' current explanation, and it will need\nfurther discussion or measurements (e.g., under in-plane field).\n5. The authors assign the spin textures to be type-I and type-II skyrmions. Their specific distinction is not clear to me, and the\nassignment doesn't seem very justified. Are they assigned due to the kinks in Fig. 2d and 2e at 14V? There is insufficient\ndiscussion on the physical significance or mechanism for the results in 2d&e.\n6. Further data interpretation of spin textures based on their temperature dependences can be done to deepen the physical\nunderstanding.", "reviewer_2": "The current manuscript presents a study on the RMCD measurements on a gated monolayer CrI3 device. The authors\nobserved that as the voltage turns on, the saturated RMCD signal in the hysteresis loop decreases, and a peak emerges\nright at the switching of magnetization, resembling the topological Hall effect at the DC limit. The authors interpret this as\nevidence of the existence of the skyrmion, and the role of gating is to tune magnetic anisotropy and DMI continuously with\nelectric field. The authors also did theoretical analysis to support this interpretation.\nThe paper is well written. Nevertheless, I have several questions before I can accept the topological Hall effect (THE)\ninterpretation.\n• The standard method to extract THE in a bulk sample is to compare the magnetization with anomalous Hall effect, then\nextract the difference between the two as the THE. In work, there is no direct measurement of magnetization.\n• The direct measurement of magnetization of a 2D magnet is very difficult, but not impossible (for example: Nat Commun 13,\n5369 (2022).). I am not suggesting the authors have to perform other kinds of measurements to measure magnetization, but\nthey should try very hard to rule out the other potential origins for this RMCD signal, before interpreting it as THE. For\nexample, could the domain structure causes the peak? Did the authors do spatial imaging?\n• The other thing that is a bit strange is that if the reduced saturated magnetization is due to the canting from out of plane to\nin-plane, the spin should cant back to c-axis when the field increases. I don’t see that in the data.\n• Finally, is it possible that the reduced RMCD signal is not due to the magnetization, rather because of electric field\nmodulated the band structure, so that the RMCD spectrum changes?", "reviewer_3": "In this manuscript, the authors demonstrated a voltage-control of magneto-optical effect observed in monolayer CrI3. By\napplying a voltage on the sample, a THE like RMCD loop has been observed, and the authors attributed the peak and valley\non hysteresis loop as the feature of skyrmions. The results are interesting and provide a way to manipulate the magnetism of\nCrI3. Before the manuscript be recommended for publication in Nature Communications, the following items should be\naddressed.\n1. For the device used in the manuscript, a hBN with 9.8 nm has been used as a dielectric layer, when a 5V voltage is\napplied on the sample, there may exist a tunneling current. How to exclude the effect of electric current?\n2. By applying different voltage, the authors state that there are two different kinds of skyrmions, what is the difference of\nthese two types of skyrmions?\n3. The authors demonstrate the evolution of magnetic structures, is it impossible to observe the magnetic structures\n(domains and skyrmions) directly?\n4. The authors attribute the peaks on the RMCD loops as the generation of skyrmions, how the features of the ellimilate\nprocesses?\n5. For 2D materials, topological effect have also been observed in CrVI6 (Nature Physics, https://doi.org/10.1038/s41567-\n024-02465-5) and Fe3GeTe2 (ACS Nano, https://doi.org/10.1021/acsnano.3c11480), the authors are encouraged to include\na brief review of topological magneto-optical effects of 2D materials part."}, "author_response": "Response to Reviewers' Comments\nReviewer #1 (Remarks to the Author):\nThis manuscript reports the emergence of skyrmions when electrostatically dope a\nmonolayer CrI . The main experimental signatures are the topological magneto-optical\n3\nsignals in RMCD, with supporting analysis through theoretical simulations. The results\nand claims to electrically manipulate the magnetic anisotropy and skyrmion states will\nattract a lot of interest in the field of spintronics. My main concern is that such\ntopological RMCD signatures are missing in an earlier well-established work\n(https://www.nature.com/articles/s41565-018-0135-x) that also studies the monolayer\nCrI within similar doping levels. I also have a number of technical questions that the\n3\nauthors will need to first address.\nReply: We appreciate the reviewer's time in assessing our work and their\nacknowledgment of its potential to attract broad interest within the spintronics\ncommunity. In response, we have elaborated on the uniqueness of this manuscript by\nciting the works recommended by the reviewers. Furthermore, we have fabricated a\nnew monolayer CrI device as advised, which confirmed the reproducibility of the\n3\nexperimental results, and conducted additional theoretical calculations to address the\nprimary concerns raised. These insightful suggestions have significantly enhanced the\nquality of the manuscript.\n1) As noted above, the earlier Nature Nano paper reports the doping dependence of\nRMCD in a monolayer of CrI . How do the authors explain or understand this\n3\ndiscrepancy? Is the reported topological RMCD signal reproducible in multiple\nmonolayer devices?\nReply: We thank the reviewer for their comment regarding why this work has been able\nto uniquely observe the topological RMCD signal. It is noteworthy that the devices used\nin the previous study published in Nature Nanotechnology employed a symmetric dual-\ngate architecture. Moreover, that investigation deliberately excluded electric field\neffects, focusing solely on doping effects in isolation. In contrast, the present study\nutilizes an asymmetric single-gate device design and establishes that the emergence of\nthe topological RMCD signal necessitates both doping-mediated modulation of\nmagnetic anisotropy and electric field-induced breaking of inversion symmetry.\nFurthermore, the asymmetric device configuration inherently breaks inversion\nsymmetry, which may promote the establishment of the Dzyaloshinskii–Moriya\ninteraction (DMI). Taken together, these factors suggest that the absence of DMI—a\nfundamental prerequisite for the stabilization of topological spin textures—likely\nconstitutes the principal reason why topological RMCD signals remained undetected in\nearlier works. More importantly, we fabricated a new device and also observed the\ntopological RMCD signal under an appropriate voltage (Fig. R1). The experimental\ndata for Device 2 has been added to the supplementary materials (Supplementary Fig.\n6), with corresponding descriptions highlighted in red in the main text.\nFig. R1. a, Optical microscope image of CrI nanosheets exfoliated from bulk CrI and\n3 3\nadhered to the PDMS surface, and the region enclosed by the red dashed line represents\nthe atomic-thick CrI area. b, False-color optical micrograph of the device 2, with the\n3\njunction region represented by the area enclosed by a white dashed line. c-g, RMCD\nversus magnetic field at different gate voltages at 10 K. The black arrows indicate the\ndirection of the field sweep.\n2) The manuscript didn't show the estimated doping level using experimental data, even\nthough it's mentioned that such values were extracted and compared to theory. The\ndoping level needs to be explicitly stated along with the gate-dependent results.\nReply: We thank the reviewer for their valuable suggestions. We have now indicated\nthe doping density (top axes) in Fig. R2c and have replaced the corresponding\nSupplementary Fig. 9c in the supplementary materials.\nFig. R2. a, A schematic representation of the electric-field-induced breaking of spatial\ninversion symmetry in monolayer CrI , leading to the emergence of the DMI. b, A\n3\nschematic illustration of the band-edge electronic structure in p-doped or n-doped CrI ,\n3\nwhere the direction of orbital angular momentum is indicated by color coding, and spin-\nup and spin-down states are denoted by arrows. c, The voltage (doping density)\ndependence of the factor η, the magnetic anisotropy energy (K), and the RCD intensity\nT\n(at -0.04 T) is investigated. D is induced by the electric field, while K is caused by\ndoping\nelectrostatic doping. Both contributions are derived from prior theoretical frameworks\nand quantified using interpolation methods. Notably, K shows negligible voltage\nE\ndependence, indicating that the electric field has minimal effect on magnetic anisotropy.\n3) Relevant to 2, the corresponding Fermi level shift should also be estimated using the\ndoping level to compare with earlier theory work like ref 40.\nReply: We sincerely thank the reviewer for their helpful suggestions. According to ref\n40, we estimated a Fermi level shift of approximately 0.33 eV at 16 V.\n4) Even if the authors claim that the magnetic anisotropy is completely in-plane with\nundergo a polarizing transition to a finite RMCD signal, which is not the case in Fig.\n1f. The lack of RMCD signal at higher fields is not consistent with the authors' current\nexplanation, and it will need further discussion or measurements (e.g., under in-plane\nfield).\nReply: We sincerely thank the reviewer for this highly professional comment, which\nprompted us to re-examine our data with greater care. To address the reviewer's\nobservation, Fig. R3 presents an enlarged section of Fig. 1f. As the magnetic field\nincreases, the RMCD signal exhibits a nearly linear rise with a very small slope,\napproaching a positive finite value. This trend is consistent with a weak out-of-plane\nmagnetic susceptibility resulting from in-plane anisotropy. It is this weak out-of-plane\nmagnetization that likely led to the impression that the RMCD signal approaches zero\nunder stronger out-of-plane magnetic fields in Fig. 1f. Our RMCD results (Fig. 1c-f)\nare consistent with the previously reported voltage-induced weak anomalous Hall\nsignal observed even under high magnetic fields (the anomalous Hall curve at 2.8 V in\nFig. R4), a phenomenon attributed to the gate-voltage-driven transition of magnetic\nanisotropy from perpendicular to in-plane (Nat Electron 6, 28–36, 2023)).\nFig. R3. A magnified view of Fig. 1f. The gray line represents the experimental data,\nwhile the solid red line corresponds to its linear fit.\n[FIGURE REDACTED]\nFig. R4. Hysteresis loops of R at 2 K with varying V (Nat Electron 6, 28–36, 2023).\nAHE G\n5) The authors assign the spin textures to be type-I and type-II skyrmions. Their specific\ndistinction is not clear to me, and the assignment doesn't seem very justified. Are they\nassigned due to the kinks in Fig. 2d and 2e at 14 V? There is insufficient discussion on\nthe physical significance or mechanism for the results in 2d&e.\nReply: Indeed, the kinks observed at 14 V in Fig. 2d and 2e can also be used to\ndistinguish between Type-I and Type-II skyrmions, as the voltage around 14 V marks\nthe critical transition point between perpendicular magnetic anisotropy and in-plane\nmagnetic anisotropy (Fig. R2c). To prevent readers from assuming that Type-I and\nType-II skyrmions differ in topological nature, we have explicitly provided the\nfollowing definition in the article: \"Type-I skyrmions are topological spin structures\nstabilized by an out-of-plane magnetic field in systems with perpendicular magnetic\nanisotropy, whereas Type-II skyrmions are formed under in-plane magnetic anisotropy\nwith the assistance of an out-of-plane field.\" This definition has also been included in\nthe caption of Fig. 2. The main purpose of introducing this definition is to facilitate\nunderstanding of how the evolution of magnetic anisotropy with voltage affects the\nformation of skyrmions under an out-of-plane magnetic field throughout the process.\nFor example, the kinks in the topological peak magnetic field position shown in Fig. 2d\nis caused by the transition in magnetic anisotropy direction. This is because skyrmion\nformation primarily results from the competition among magnetic anisotropy, the\nDzyaloshinskii–Moriya interaction (DMI), and the out-of-plane magnetic field. A\nqualitative change in magnetic anisotropy leads to an abrupt shift in the required out-\nof-plane magnetic field strength for skyrmion stabilization, accompanied by a sudden\nchange in skyrmion density (Fig. 2e). The physical mechanisms underlying Fig. 2d and\n2e have been addressed in the section detailing skyrmion formation mechanisms (Fig.\nR2). Moreover, theoretical simulations of the field-assisted skyrmion formation process\nunder both anisotropy conditions confirm consistency with the experimental\nobservations (Fig. 3 and Supplementary Fig. 10).\n6) Further data interpretation of spin textures based on their temperature dependences\ncan be done to deepen the physical understanding.\nReply: We thank the reviewer for their valuable comments, which are instrumental in\nimproving this manuscript. Both temperature and out-of-plane magnetic field act as\neffective fields and play a crucial role in the formation and annihilation of topological\nspin textures. Fig. R5a shows the evolution of magnetic domains under a temperature\nof T/J = 0.0044 as the out-of-plane magnetic field is swept from B/J = -0.214 to B/J =\n0.214. The results indicate that skyrmions form at an out-of-plane magnetic field of B/J\n= 0.086 and undergo complete annihilation at B/J = 0.197. When the temperature is\nincreased to T/J = 0.0879, skyrmions also form at B/J = 0.086, but become fully\nannihilated at a lower magnetic field of B/J = 0.171 (Fig. R5b). The methodology\nemployed in the atomic-scale spin simulations remains consistent with that used in the\ntheoretical simulations at 16 V (Fig. 3 and methods), with the exception that\ntemperature is treated as a variable. Fig. R5c and 5d present the phase diagrams of the\ntopological charge density in monolayer CrI as functions of temperature and magnetic\n3\nfield. The results indicate that the magnetic field required for the initial formation of\nskyrmions remains unchanged with increasing temperature (highlighted by the black\ndashed line), whereas the magnetic field at which skyrmions undergo complete\nannihilation decreases as the temperature rises (highlighted by the green dashed line).\nThis behavior is consistent with the variation of the topological RMCD signal with\nmagnetic field and temperature observed experimentally (Fig. 4b and 4c). This\nconsistency provides further support for the interpretation that the topological RMCD\npeaks originate from skyrmions. New theoretical simulation data along with the\nrelevant text highlighted in red have been included in the Supplementary Note 2 and\nSupplementary Fig 12.\nFig. R5. a, A series of representative magnetic domain configurations at T/J = 0.0044\nduring a magnetic field (B/J) sweep from negative to positive maximum. A 100×100\nsupercell was used for the magnetic simulation of the two-dimensional spin lattice, with\ncolors mapping the out-of-plane magnetic moment components (M ). b, A series of\nz\nrepresentative magnetic domain configurations at T/J = 0.0879 during a magnetic field\n(B/J) sweep from negative to positive maximum. c, The topological charge (per 10,000\nspin sites) as a function of temperature and magnetic field (sweeping from negative to\npositive). d, The time-reversal process corresponding to (c) (sweeping from positive to\nnegative). The black arrows indicate the direction of the magnetic field sweep.\nReviewer #2 (Remarks to the Author):\nThe current manuscript presents a study on the RMCD measurements on a gated\nmonolayer CrI device. The authors observed that as the voltage turns on, the saturated\nswitching of magnetization, resembling the topological Hall effect at the DC limit. The\nauthors interpret this as evidence of the existence of the skyrmion, and the role of gating\nis to tune magnetic anisotropy and DMI continuously with electric field. The authors\nalso did theoretical analysis to support this interpretation.\nThe paper is well written. Nevertheless, I have several questions before I can accept the\ntopological Hall effect (THE) interpretation.\nReply: We thank the reviewer for taking the time to assess our work and for recognizing\nits value. The reviewer’s remarks and comments were very helpful in clarifying certain\ncrucial aspects of our analysis, particularly regarding the topological Hall effect (THE)\ninterpretation. The revised manuscript clarifies these points and includes new analysis.\nWe hope the revisions and the response below fully address the reviewer’s points.\n1)The standard method to extract THE in a bulk sample is to compare the magnetization\nwith anomalous Hall effect, then extract the difference between the two as the THE. In\nwork, there is no direct measurement of magnetization.\nReply: We thank the reviewer for raising an important point regarding the interpretation\nof the magnetization measurement. Actually, conventional RMCD hysteresis loop\nmeasurements without topological peaks have been widely recognized as an effective\nmethod for magnetization characterization in the field of two-dimensional magnetic\nmaterials (purple shaded area in Fig. R1), similar to how the anomalous Hall effect is\nemployed for magnetization measurement. Therefore, extracting the topological\nRMCD signal by subtracting the conventional RMCD background is analogous to\nisolating the topological Hall effect in bulk materials by removing the anomalous Hall\ncomponent (Fig. R1). Moreover, this perspective is further supported by our theoretical\nsimulations (Fig. 3). Furthermore, as you noted below, directly measuring the magnetic\nmoment in two-dimensional magnets remains a significant challenge at liquid He\ntemperature — particularly for those with atomic-scale thickness. Even when such\nmeasurements are achieved, it is difficult to correlate them quantitatively with the\nRMCD signal, thereby providing little substantive support for the extraction of the\ntopological RMCD component.\nFig. R1. RMCD versus magnetic field at different gate voltages at 10 K. The black\narrow indicate the direction of the field sweep, while the purple arrows with spheres\nschematically represent the magnetic anisotropy. The purple shaded area corresponds\nto the conventional RMCD signal, which is proportional to the out-of-plane\nmagnetization, while the orange peak represents the topological RMCD contribution.\n2)The direct measurement of magnetization of a 2D magnet is very difficult, but not\nimpossible (for example: Nat Commun 13, 5369 (2022).). I am not suggesting the\nauthors have to perform other kinds of measurements to measure magnetization, but\nthey should try very hard to rule out the other potential origins for this RMCD signal,\nbefore interpreting it as THE. For example, could the domain structure causes the peak?\nDid the authors do spatial imaging?\nReply: We thank the reviewer for their insightful comments on the correlation between\ntopological RMCD signals and skyrmions. As you rightly noted, direct measurement of\nmagnetic moments or magnetization in two-dimensional magnets remains highly\nchallenging. While defect-mediated spin states allow detection in magnetic\nnanosheets—which may exhibit quasi-bulk behavior—direct probing in atomically thin\nlayers is considerably more difficult.\nReal-space observation of skyrmion remains exceptionally difficult for several\nreasons. First, the magnetic signal from a monolayer is extremely weak and requires\nultra-high-sensitivity detectors. Second, our calculations indicate a skyrmion diameter\nbelow 10 nm, posing severe spatial resolution challenges. Although scanning NV\nmagnetometry offers high sensitivity, achieving ∼10 nm resolution remains challenging;\nthe current best spatial resolution for imaging magnetic domains in 2D materials is\naround 50 nm (Science 374, 1140-1144, 2021). Third, such experiments require low-\ntemperature conditions and in situ electric and magnetic field control. Additionally, CrI\n3\nis extremely air-sensitive and bare CrI flakes drastically degrade within few seconds\n3\nin ambient air, which exponentially increases the difficulty of the experiments.\nA comprehensive experimental platform capable of real-space imaging of skyrmions\nin monolayer CrI would need to simultaneously meet all these demanding criteria—a\n3\ngoal that entails methodological advances well beyond the scope of this study.\nTo further verify that the topological RMCD peaks indeed originate from skyrmions,\nwe conducted additional theoretical simulations on the evolution of magnetic structures\nunder various temperatures and magnetic fields at 16 V. Fig. R2a shows the evolution\nof magnetic domains under a temperature of T/J = 0.0044 as the out-of-plane magnetic\nfield is swept from B/J = -0.214 to B/J = 0.214. The results indicate that skyrmions form\nat an out-of-plane magnetic field of B/J = 0.086 and undergo complete annihilation at\nB/J = 0.197. When the temperature is increased to T/J = 0.0879, skyrmions also form\nat B/J = 0.086, but become fully annihilated at a lower magnetic field of B/J = 0.171\n(Fig. R2b). The methodology employed in the atomic-scale spin simulations remains\nconsistent with that used in the theoretical simulations at 16 V (Fig. 3 and methods),\nwith the exception that temperature is treated as a variable. Fig. R2c and 2d present the\nphase diagrams of the topological charge density in monolayer CrI as functions of\n3\ntemperature and magnetic field. The results indicate that the magnetic field required for\nthe initial formation of skyrmions remains unchanged with increasing temperature\n(highlighted by the black dashed line), whereas the magnetic field at which skyrmions\nundergo complete annihilation decreases as the temperature rises (highlighted by the\ngreen dashed line). This behavior is consistent with the variation of the topological\nRMCD signal with magnetic field and temperature observed experimentally (Fig. 4b\nand 4c). The dependence of the topological RMCD signal on temperature and magnetic\nfield closely mirrors the evolution of the topological charge density under identical\nconditions. This consistency provides further support for the interpretation that the\ntopological RMCD peaks originate from skyrmions. New theoretical simulation data\nalong with the relevant text highlighted in red have been included in the Supplementary\nNote 2 and Supplementary Fig 12.\nFig. R2. a, A series of representative magnetic domain configurations at T/J = 0.0044\nduring a magnetic field (B/J) sweep from negative to positive maximum. A 100×100\nsupercell was used for the magnetic simulation of the two-dimensional spin lattice, with\ncolors mapping the out-of-plane magnetic moment components (M ). b, A series of\nz\nrepresentative magnetic domain configurations at T/J = 0.0879 during a magnetic field\n(B/J) sweep from negative to positive maximum. c, The topological charge (per 10,000\nspin sites) as a function of temperature and magnetic field (sweeping from negative to\npositive). d, The time-reversal process corresponding to (c) (sweeping from positive to\nnegative). The black arrows indicate the direction of the magnetic field sweep.\nIt is worth emphasizing that in bulk magnets, skyrmions have been shown to\ngenerate topological magneto-optical Kerr peaks (Fig. R3b) analogous to topological\nRMCD signals (Fig. R3d and Fig. 4b), as both the magneto-optical Kerr effect and\nRMCD share the same physical origin—the transverse optical Hall conductivity (Phys.\nRev. B 75, 214416, 2007; Phys. Rev. Lett. 108, 087403, 2012; Phys. Rev. B 67, 235203,\n2003). Moreover, consistency has been demonstrated between the topological Hall\neffect in the DC limit and the topological magneto-optical peak in the same bulk magnet\nsystem (Fig. R3b and c).\nFig. R3. a, The AlB -type crystal structure of Gd PdSi (left panel) and the schematic\n2 2 3\nillustration of the triangular skyrmion lattice (SkL) (right panel). b, Magnetic phase\ndiagram with magnetic field (H) parallel to the c axis and a contour map of topological\nKerr rotation angle θ at 0.3 eV. IC-1 and IC-2 represent incommensurate spin-state\nK\nphases, and PM represents the paramagnetic phase. The open circles represent phase\nboundaries determined by magnetization measurements. c, Magnetic-field dependence\nof the d.c. Hall conductivity σ (left axis) and magnetization M (right axis) for H || c at\nxy\nThe red shaded area denotes the SkL phase (Nat. Commun. 14, 5416, 2023). d, The\ntopological RMCD intensity as a function of temperature and magnetic field (sweeping\nfrom negative to positive).\n3)The other thing that is a bit strange is that if the reduced saturated magnetization is\ndue to the canting from out of plane to in-plane, the spin should cant back to c-axis\nwhen the field increases. I don’t see that in the data.\nReply: We sincerely thank the reviewer for this highly professional comment, which\nprompted us to re-examine our data with greater care. To address the reviewer's\nobservation, Fig. R4 presents an enlarged section of Fig. 1f. As the magnetic field\nincreases, the RMCD signal exhibits a nearly linear rise with a very small slope,\napproaching a positive finite value. This trend is consistent with a weak out-of-plane\nmagnetic susceptibility resulting from in-plane anisotropy. It is this weak out-of-plane\nmagnetization that likely led to the impression that the RMCD signal approaches zero\nunder stronger out-of-plane magnetic fields in Fig. 1f. Our RMCD results (Fig. 1c-f)\nare consistent with the previously reported voltage-induced weak anomalous Hall\nsignal observed even under high magnetic fields (the anomalous Hall curve at 2.8 V in\nFig. R5), a phenomenon attributed to the gate-voltage-driven transition of magnetic\nanisotropy from perpendicular to in-plane (Nat Electron 6, 28–36, 2023).\nFig. R4. A magnified view of Fig. 1f. The gray line represents the experimental data,\nwhile the solid red line corresponds to its linear fit.\n[FIGURE REDACTED]\nFig. R5. Hysteresis loops of R and magnetoresistance at 2 K with varying V (Nat\nAHE G\nElectron 6, 28–36, 2023).\n4) Finally, is it possible that the reduced RMCD signal is not due to the magnetization,\nrather because of electric field modulated the band structure, so that the RMCD\nspectrum changes?\nReply: We gratefully acknowledge the reviewer for their insightful comments. We have\nclarified that the electronic band structure, magnetization, and RMCD signal are\nfundamentally interrelated. Specifically, electric-field-induced doping modifies the\nband structure, which in turn alters the magnetization and correspondingly affects the\nRMCD signal.\nThe RMCD signal arises from the differential absorption of right- and left-circularly\npolarized (RCP and LCP) light within the material. RCP light carries an angular\nmomentum of +1, while LCP light carries -1. Fig. R6 presents the electronic band\nstructures of monolayer CrI under both undoped and electron-doped conditions (Nano\nright.\nIn undoped (intrinsic) CrI , which exhibits perpendicular magnetic anisotropy (out-\n3\nof-plane magnetization), RCP excitation induces electronic transitions that obey the\noptical selection rule requiring angular momentum conservation. Only the transition\nfrom J = –3/2 to J = –1/2 (green arrow in Fig. R6a) is allowed; all others are forbidden.\nIn contrast, LCP excitation would require a decrease in angular momentum by 1, which\nviolates conservation, resulting in negligible absorption. This strong contrast in\nabsorption produces a large RMCD signal.\nUnder n-doping, the two lowest conduction bands—serving as the relevant\nunoccupied and occupied states—exhibit parallel spin alignment (σσ′ = 1, Fig. R6b).\nThis alignment yields a negative magnetic anisotropy energy (MAE), stabilizing an in-\nplane magnetic anisotropy (in-plane magnetization). For a detailed explanation of how\nn-doping drives this band structure change and consequent magnetic reorientation, see\nNote R1. Concurrently, n-doping shifts the Fermi level near the conduction band\nminimum (Fig. R6b). The black fill indicates that the lower conduction band states\nbecome occupied. Under RCP excitation, Pauli blocking strongly suppresses the J = –\n3/2 → –1/2 transition compared to the undoped case, markedly reducing RCP\nabsorption. Thus, the n-doping–induced Fermi level shift not only switches the\nmagnetic anisotropy (magnetization) from out-of-plane to in-plane but also strongly\nsuppresses the RMCD signal. This confirms that the RMCD signal is intrinsically\nlinked to both the magnetization and the underlying electronic structure.\nNote R1: The magnetic anisotropy energy (MAE) in CrI monolayers is intimately\n3\nassociated with the spin and orbital structures near the Fermi level. Under electron\ndoping, the shift of the Fermi level toward the conduction band edge alters the magnetic\nanisotropy from the perpendicular type (perpendicular magnetization)—characteristic\nof the undoped case—to an in-plane anisotropy (in-plane magnetization). First-\nprinciples calculations have demonstrated that the MAE is primarily governed by the\nindirect SOC, with the direct SOC contribution being negligible (2D Mater. 4, 035002,\n2017). By the perturbation theory (Phys. Rev. B. 47, 14932, 1993), the MAE can be\nsuccinctly expressed as\nM A E 2\n, , ,\n, L Iz ,\nE\n,\n,\n,\nL Ix ,\n2\n   \n \n  \n= \n\n\n −\n−\n\n\nHere, u and o denote the unoccupied and occupied states, respectively, E represents\nu/o,σ\nthe band energy of these states, and the spin indices σ/σ′ range over ±1, corresponding\nto the two orthogonal spin states at the k-point. Previous theoretical investigations have\nestablished that |⟨o,σ|LI|u,σ'⟩| 2 -|⟨o,σ|LI|u,σ'⟩| 2 is negative (Phys. Rev. B. 101, 134418,\nz x\n2020). In addition to the contribution from the orbital angular momentum, the relative\nspin polarization (σ, σ′) of the states near the Fermi level must also be considered. When\nCrI is n-doped, the two lowest conduction bands act as the unoccupied and occupied\n3\nstates, with their parallel spin alignment (σσ′=1, Fig. R6b) resulting in a negative MAE,\nwhich favors an in-plane magnetic anisotropy (in-plane magnetization).\nFig. R6. a, b, A schematic illustration of the band-edge electronic structure in undoped\nor n-doped CrI , where the direction of orbital angular momentum is indicated by color\n3\ncoding, and spin-up and spin-down states are denoted by black arrows. The green arrow\nindicates the allowed electronic transition pathway under right-circularly polarized\n(RCP) light excitation in accordance with the selection rules, the purple arrow denotes\nthe magnetic anisotropy or magnetization direction, and E represents the Fermi energy\nF\nlevel.\nReviewer #3 (Remarks to the Author):\nIn this manuscript, the authors demonstrated a voltage-control of magneto-optical\neffect observed in monolayer CrI . By applying a voltage on the sample, a THE like\nhysteresis loop as the feature of skyrmions. The results are interesting and provide a\nway to manipulate the magnetism of CrI . Before the manuscript be recommended for\n3\npublication in Nature Communications, the following items should be addressed.\nReply: We are greatly grateful to the reviewer for their interest in and recognition of\nthe significance of our manuscript. We also appreciate the time and effort they have\ndedicated to evaluating our work. In response to their comments, we have performed\nadditional experiments and theoretical simulations to further address the main concern\nraised. These insightful suggestions have significantly enhanced the quality of the\nmanuscript.\n1) For the device used in the manuscript, a hBN with 9.8 nm has been used as a\ndielectric layer, when a 5 V voltage is applied on the sample, there may exist a tunneling\ncurrent. How to exclude the effect of electric current?\nReply: We thank the reviewer for their helpful comments. To characterize the tunneling\ncurrent, the time-dependent variation of the current in the device was measured under\na 8 V bias (Fig. R1). The tunneling current was found to be on the order of 10⁻⁸ mA,\nwhich is approximately eight orders of magnitude lower than the current levels\npreviously shown to manipulate magnetic order (ACS nano 18, 20055-20064, 2024).\nThus, we conclude that any influence of the current on the magnetic order in our device\ncan be ruled out.\nFig. R1. Leakage-time characteristics were measured at 10 K for the Gr/monolayer\nCrI /hBN/Gr devices.\n3\n2) By applying different voltage, the authors state that there are two different kinds of\nskyrmions, what is the difference of these two types of skyrmions?\nReply: We appreciate the reviewers' constructive feedback. The kinks observed at 14\nV in Fig. 2d and 2e can be used to distinguish between Type-I and Type-II skyrmions,\nas the voltage around 14 V marks the critical transition point between perpendicular\nmagnetic anisotropy and in-plane magnetic anisotropy (Fig. R2c). To prevent readers\nfrom assuming that Type-I and Type-II skyrmions differ in topological nature, we have\nexplicitly provided the following definition in the article: \"Type-I skyrmions are\ntopological spin structures stabilized by an out-of-plane magnetic field in systems with\nperpendicular magnetic anisotropy, whereas Type-II skyrmions are formed under in-\nplane magnetic anisotropy with the assistance of an out-of-plane field.\" This definition\nhas also been included in the caption of Fig. 2. The main purpose of introducing this\ndefinition is to facilitate understanding of how the evolution of magnetic anisotropy\nwith voltage affects the formation of skyrmions under an out-of-plane magnetic field\nthroughout the process. For example, the kinks in the topological peak magnetic field\nposition shown in Fig. 2d is caused by the transition in magnetic anisotropy direction.\nThis is because skyrmion formation primarily results from the competition among\nmagnetic anisotropy, the Dzyaloshinskii–Moriya interaction (DMI), and the out-of-\nplane magnetic field. A qualitative change in magnetic anisotropy leads to an abrupt\nshift in the required out-of-plane magnetic field strength for skyrmion stabilization (Fig.\n2d), accompanied by a sudden change in skyrmion density (Fig. 2e).\nFig. R2. a, A schematic representation of the electric-field-induced breaking of spatial\ninversion symmetry in monolayer CrI , leading to the emergence of the DMI. b, A\n3\nschematic illustration of the band-edge electronic structure in p-doped or n-doped CrI ,\n3\nwhere the direction of orbital angular momentum is indicated by color coding, and spin-\nup and spin-down states are denoted by arrows. c, The voltage (doping density)\ndependence of the factor η, the magnetic anisotropy energy (K), and the RCD intensity\nT\n(at -0.04 T) is investigated. D is induced by the electric field, while K is caused by\ndoping\nelectrostatic doping. Both contributions are derived from prior theoretical frameworks\nand quantified using interpolation methods. Notably, K shows negligible voltage\nE\ndependence, indicating that the electric field has minimal effect on magnetic anisotropy.\n3) The authors demonstrate the evolution of magnetic structures, is it impossible to\nobserve the magnetic structures (domains and skyrmions) directly?\nReply: We thank the reviewer for their helpful suggestions. Real-space observation of\ndomain and skyrmion evolution remains extremely challenging with current\nexperimental technologies, and the air-sensitivity of CrI flakes exponentially increases\n3\nthe experimental difficulties, as noted by Reviewer #2. Firstly, the magnetic signal from\na monolayer is exceedingly weak, necessitating detectors with ultra-high sensitivity.\nSecondly, our theoretical calculations suggest that the skyrmion diameter is below 10\nnm, rendering spatial resolution at this scale particularly difficult to achieve. Although\nscanning NV magnetometry measurements offer high sensitivity, achieving a spatial\nresolution of 10 nm remains extremely challenging. To the best of our knowledge, the\nhighest spatial resolution currently achieved for observing magnetic domains in two-\ndimensional magnets is only about 50 nm (Science 374,1140-1144, 2021). Thirdly, such\nmeasurements require a low-temperature environment coupled with the capability to\napply in situ electric and magnetic fields. An experimental system capable of real-space\nimaging of skyrmion formation in monolayer CrI must simultaneously satisfy all these\n3\ndemanding criteria—a task that will likely require the development of novel\nmethodologies and extends far beyond the scope of the present study.\n4) The authors attribute the peaks on the RMCD loops as the generation of skyrmions,\nhow the features of the eliminate processes?\nReply: We thank the reviewer for their comments on the corresponding features of the\nskyrmion generation and annihilation processes reflected in the RMCD hysteresis loops.\nWe have revealed that the emergence and disappearance of topological peaks\ncorrespond to the generation and annihilation of skyrmions through theoretical\nsimulations of the out-of-plane magnetic field evolution of magnetic domains in\nmonolayer CrI under 8 V and 16 V. This observation is consistent with previously\n3\nreported findings in non-centrosymmetric van der Waals bulk materials, where the\nemergence and disappearance of magneto-optical Kerr peaks were also associated with\nthe generation and annihilation of skyrmions (Nat. Phys. 20, 1145–1151, 2024).\nIt is worth emphasizing that in bulk magnets, skyrmions have been shown to\ngenerate topological magneto-optical Kerr peaks (Fig. R3b) analogous to topological\nRMCD signals (Fig. R3d and Fig. 4b), as both the magneto-optical Kerr effect and\nRMCD share the same physical origin—the transverse optical Hall conductivity (Phys.\nRev. B 75, 214416, 2007; Phys. Rev. Lett. 108, 087403, 2012; Phys. Rev. B 67, 235203,\n2003). Moreover, consistency has been demonstrated between the topological Hall\neffect in the DC limit and the topological magneto-optical peak in the same bulk magnet\nsystem (Fig. R3b and 3c).\nFig. R3. a, The AlB -type crystal structure of Gd PdSi (left panel) and the schematic\n2 2 3\nillustration of the triangular skyrmion lattice (SkL) (right panel). b, Magnetic phase\ndiagram with magnetic field (H) parallel to the c axis and a contour map of topological\nKerr rotation angle θ at 0.3 eV. IC-1 and IC-2 represent incommensurate spin-state\nK\nphases, and PM represents the paramagnetic phase. The open circles represent phase\nboundaries determined by magnetization measurements. c, Magnetic-field dependence\nof the d.c. Hall conductivity σ (left axis) and magnetization M (right axis) for H || c at\nxy\nThe red shaded area denotes the SkL phase (Nat. Commun. 14, 5416, 2023). d, The\ntopological RMCD intensity as a function of temperature and magnetic field (sweeping\nfrom negative to positive).\nTo further verify that the topological RMCD peaks indeed originate from\nskyrmions, we conducted additional theoretical simulations on the evolution of\nmagnetic structures under various temperatures and magnetic fields at 16 V. Fig. R4a\nshows the evolution of magnetic domains under a temperature of T/J = 0.0044 as the\nout-of-plane magnetic field is swept from B/J = -0.214 to B/J = 0.214. The results\nindicate that skyrmions form at an out-of-plane magnetic field of B/J = 0.086 and\nundergo complete annihilation at B/J = 0.197. When the temperature is increased to T/J\n= 0.0879, skyrmions also form at B/J = 0.086, but become fully annihilated at a lower\nmagnetic field of B/J = 0.171 (Fig. R4b). The methodology employed in the atomic-\nscale spin simulations remains consistent with that used in the theoretical simulations\nat 16 V (Fig. 3 and methods), with the exception that temperature is treated as a variable.\nFig. R4c and R4d present the phase diagrams of the topological charge density in\nmonolayer CrI as functions of temperature and magnetic field. The results indicate that\n3\nthe magnetic field required for the initial formation of skyrmions remains largely\nunchanged with increasing temperature (highlighted by the black dashed line), whereas\nthe magnetic field at which skyrmions undergo complete annihilation decreases as the\ntemperature rises (highlighted by the green dashed line). This behavior is consistent\nwith the variation of the topological RMCD signal with magnetic field and temperature\nobserved experimentally (Fig. 4b and 4c). The dependence of the topological RMCD\nsignal on temperature and magnetic field closely mirrors the evolution of the\ntopological charge density under identical conditions. This consistency provides further\nsupport for the interpretation that the topological RMCD peaks originate from\nskyrmions. New theoretical simulation data along with the relevant text highlighted in\nred have been included in the Supplementary Note 2 and Supplementary Fig 12.\nFig. R4. a, A series of representative magnetic domain configurations at T/J = 0.0044\nduring a magnetic field (B/J) sweep from negative to positive maximum. A 100×100\nsupercell was used for the magnetic simulation of the two-dimensional spin lattice, with\ncolors mapping the out-of-plane magnetic moment components (M ). b, A series of\nz\nrepresentative magnetic domain configurations at T/J = 0.0879 during a magnetic field\n(B/J) sweep from negative to positive maximum. c, The topological charge (per 10,000\nspin sites) as a function of temperature and magnetic field (sweeping from negative to\npositive). d, The time-reversal process corresponding to (c) (sweeping from positive to\nnegative). The black arrows indicate the direction of the magnetic field sweep.\n5) For 2D materials, topological effect have also been observed in CrVI (Nature\n3 2\nNano, https://doi.org/10.1021/acsnano.3c11480), the authors are encouraged to\ninclude a brief review of topological magneto-optical effects of 2D materials part.\nReply: We sincerely thank the reviewer for their helpful suggestions. Indeed,\ntopological magneto-optical effects have been previously observed in bulk van der\nWaals materials; however, voltage-controlled manipulation of skyrmions in the\nmonolayer limit remains challenging and has not yet been reported. We have added\nrelevant content to the introduction section, which is located at line nine."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors made efforts to address my questions. However, they failed to address my main concerns in Question 1, and my key technical Question 4(that underlies the credibility of their data interpretation). Also, their estimation of the doping level in response to Question 2 also seems questionable. Based on the lack of reliable responses, I cannot recommend its publication.\nIn the response of Question 1, the authors argued that the data in the mentioned Nature Nano paper does not break inversion symmetry (only doping, no electric field) and thereofore differ from their data. However, the corresponding E-field measurements were presented in a Nature Materials paper (https://www.nature.com/articles/s41563-018-0040-6) for a monolayer, which still lacks the \"peaks\" discussed in this manuscript, and therefore, there is still a discrepancy.\nIn Question 4, I don't see any saturation of mangetization, in the figure that the authors indicate there should be. I don't think the authors provide a proper and direct response to this question, and my original concerns remain\nIn question 2, the authors didn't provide the device parameters (dielectric thickness, etc.) and just directly provide the estimate of up to 3*10^12/cm^2 with electrostatic doping. I don't think such high doping level is realistic because the dielectric breakdown limit will be first reached before such high doping.It indicates the authors lack proper estimation of their device structure that undermines the credibility.", "reviewer_2": "The authors have addressed most of my concern. I still disagree with the statement that RMCD can be used as a representation of magnetization. I understand that it has been commonly used in the community. However, in the context of this work, the main result is that the RMCD is not proportional to magnetization (due to topological Hall effect.) Furthermore, as the author discussed, the magnitude of RMCD depends not only on the magnetization, but also on the electronic structures. Therefore, I suggest the authors refrain from making such a statement, as it is logically inconsistent with the observation of topological Hall effect. The comparison with theory modelling is reasonable and sufficient.", "reviewer_3": "The authors have addressed all my concers, I recommend the maunscript to be published as it is."}, "author_response": "RESPONSE TO REVIEWERS' COMMENTS\nReviewer #1 (Remarks to the Author):\nThe authors made efforts to address my questions. However, they failed to address my\nmain concerns in Question 1, and my key technical Question 4(that underlies the\ncredibility of their data interpretation). Also, their estimation of the doping level in\nresponse to Question 2 also seems questionable. Based on the lack of reliable responses,\nI cannot recommend its publication.\nIn the response of Question 1, the authors argued that the data in the mentioned Nature\nNano paper does not break inversion symmetry (only doping, no electric field) and\nthereofore differ from their data. However, the corresponding E-field measurements\nwere presented in a Nature Materials paper (https://www.nature.com/articles/s41563-\n018-0040-6) for a monolayer, which still lacks the \"peaks\" discussed in this manuscript,\nand therefore, there is still a discrepancy.\nReply: We thank the reviewer for their time and comments. We respectfully disagree\nwith the overall assessment. We are concerned that there may be a misunderstanding of\nboth the relevant literature and our work, as the critique seems to overlook the\nfundamental conclusions of the cited studies. We would be grateful if the reviewer could\nreconsider our points in light of a careful re-reading of the references.\nThe Nature Nanotechnology paper used dual-gated devices to isolate the effect of\ndoping by canceling out the electric field. This work intentionally focused solely on\ndoping effects while deliberately avoiding electric field effects (Fig. R1, highlighted\nby blue underlines). Due to the absence of an electric field to break inversion symmetry,\nthe Dzyaloshinskii-Moriya interaction (DMI) is absent. Therefore, no skyrmion-related\nRMCD peak was observed.\nOppositely, the Nature Materials paper, also using dual-gated devices, specially\nexcluded doping to isolate the effect of the electric field. By doing so, the\nstudy focused solely on electric field effects while avoiding doping effects (Fig. R2,\nhighlighted by blue underlines). Due to the absence of doping to modulate magnetic\nanisotropy, no skyrmion-related RMCD peak is observed either, even though the\nelectric field breaks inversion symmetry and generates DMI. As a supplementary note,\nit should be clarified that the Nature Materials paper on electric-field control\nreferenced by the reviewer actually investigated bilayer CrI , not the monolayer.\nfundamentally different because of the interlayer antiferromagnetic coupling in the\nbilayer. The distinct magnetic behaviors and switching mechanisms between\nbilayer and monolayer systems render such a comparison scientifically invalid. To\nprevent any potential misunderstanding, we kindly request that the reviewer\ncarefully re-examine the references they have cited.\n[FIGURE REDACTED]\nFig. R1. A partial screenshot of the Nature Nanotechnology article supplied by the\nreviewer (Jiang, S. et al. Nature Nanotech 13, 549–553, 2018).\n[FIGURE REDACTED]\nFig. R2. A partial screenshot of the Nature Material article supplied by the reviewer\n(Jiang, S. et al. Nature Mater 17, 406–410, 2018).\nIt must be emphasized that skyrmions can only form when the magnetic anisotropy\nenergy (MAE, K) is smaller than the DMI-dominated parameter η. As shown the phase\npicture in Fig. R3, it must be in the area of K < η highlighted by the red square.\ndoping\nTherefore, the realization of skyrmions requires doping to modulate the MAE, while\nsimultaneously applying an electric field to break inversion symmetry and induce the\nDMI. Only when the specific conditions of MAE and DMI—K <η—are met can\ndoping\nskyrmions be stabilized. For the skyrmion-stabilization mechanism arising from the\nsynergistic effect of electric field and doping, please refer to the relevant passages in\nthe manuscript, which are marked in red.\nOur work focuses not on electric field or doping effects alone, but on the regime\nwhere both coexist and act synergistically. This combined approach is essential for\nstabilizing skyrmions in our system.\nTherefore, the absence of skyrmion-related features in the cited literature does\nnot represent a contradiction, but rather confirms that neither electric field nor\ndoping effects alone can easily generate skyrmions. Indeed, those studies were not\ndesigned to achieve the synergistic regime we have explored.\nFig. R3. Mechanism for generating magnetic skyrmions in monolayer CrI by\n3\ncombined electric-field-induced inversion symmetry breaking and doping-controlled\nmagnetic anisotropy (Please see the Supplementary Fig. 9).\nIn question 2, the authors didn't provide the device parameters (dielectric thickness,\netc.) and just directly provide the estimate of up to 3*10^12/cm^2 with electrostatic\ndoping. I don't think such high doping level is realistic because the dielectric breakdown\nlimit will be first reached before such high doping.It indicates the authors lack proper\nestimation of their device structure that undermines the credibility.\nReply: The reviewer questions the rationality of our estimated doping concentration\n(~3×1012 cm-2), suggesting it is unrealistic due to dielectric breakdown limitations. We\nbelieve this concern to lack factual basis and likely constitutes subjective speculation.\nOur estimated maximum doping concentration in Fig. R3 (Supplementary Fig.\n9c) is consistent with the experimental data reported in the Nature\nNanotechnology reference cited by the reviewer (Fig. R4), where the use of hBN as\nthe dielectric layer achieved a similar doping level of ~2.8×1013 cm-2 in monolayer CrI .\nreviewer. Moreover, we disagree with the comment regarding the absence of our device\nparameters. In fact, the methodology section of our original manuscript explicitly states\nthe specific device parameters used to estimate the doping concentration, with the\nrelevant paragraph highlighted in red. We respectfully ask the reviewer to read carefully\nagain.\n[FIGURE REDACTED]\nFig. R4. The magnetic characteristics of monolayer CrI as a function of gate voltage\n3\n(bottom axis) and induced doping density (top axis) (Fig, 2c, Jiang, S. et al. Controlling\nmagnetism in 2D CrI by electrostatic doping. Nature Nanotech 13, 549–553, 2018).\nindicate there should be. I don't think the authors provide a proper and direct response\nto this question, and my original concerns remain.\nReply: In systems with in-plane magnetic anisotropy, achieving out-of-plane\nmagnetization and saturation generally requires very strong magnetic fields, as\nmagnetizing along the hard axis is inherently challenging. As shown in Fig. R5, aside\nfrom the topological magneto-optical effect (the RMCD peak) originating from the\nskyrmion, the remainder of the RMCD signal—highlighted by the green shading and\nresulting from the out-of-plane magnetization of the magnetic background—gradually\ndecreases with increasing voltage. This reveals that the magnetic anisotropy of CrI\n3\nshifts from out-of-plane to in-plane, which makes out-of-plane magnetization\nincreasingly difficult. The voltage-induced reorientation of CrI 's magnetic anisotropy\n3\nto the in-plane direction makes out-of-plane magnetization difficult (as indicated by the\ngreen-highlighted loop at 18 V). Therefore, the observed lack of saturation is not only\na normal phenomenon but also strong evidence that the magnetic anisotropy has been\nswitched from out-of-plane to in-plane.\nFig. R5. Voltage-controlled RMCD loop in other monolayer CrI\n3\ndevice (device 2). The\nhysteresis loop highlighted in green reveals that the magnetic anisotropy of CrI is\n3\ncontinuously tuned by voltage, gradually shifting from out-of-plane to in-plane.\nTo corroborate this point, we have added references from Nature Materials and\nNature Electronics which show that under high fields, when voltage induces an out-of-\nplane to in-plane magnetic anisotropy transition, the out-of-plane magnetization\nremains difficult to saturate, with the signal intensity staying far below the 0 V baseline\n(Figs. R6, R7). These results are consistent with our reports that voltage induces an out-\nof-plane to in-plane magnetic anisotropy transition in monolayer CrI (green-\n3\nhighlighted loop in Fig. R5) and provide direct support for our interpretation. A\ncomprehensive analysis of our experimental results, together with prior findings, leads\nus to conclude that the unsaturated phenomenon mentioned by the reviewers is not an\nincomprehensible or physically contradictory effect, but rather a normal manifestation\nof voltage-controlled magnetic anisotropy.\n[FIGURE REDACTED]\nFig. R6. The out-of-plane hysteresis loops of the polar MOKE at varying gate voltages\ndemonstrate that the voltage induces a transition of the magnetic anisotropy from out-\nof-plane to in-plane (Tan, A.J. et al. Magneto-ionic control of magnetism using a solid-state\nproton pump. Nat. Mater. 18, 35–41 2019).\n[FIGURE REDACTED]\nFig. R7. Continuous manipulation of magnetic anisotropy in a van der Waals\nferromagnet via electrical gating. R hysteresis loops at 2 K with varying\nAHE\nV reveal that the applied voltage drives a transition from out-of-plane to in-plane\nG\nmagnetic anisotropy (Tang, M. et al. Continuous manipulation of magnetic anisotropy in a van\nder Waals ferromagnet via electrical gating. Nat. Electron. 6, 28–36, 2023).\nReviewer #2 (Remarks to the Author):\nThe authors have addressed most of my concern. I still disagree with the statement that\nRMCD can be used as a representation of magnetization. I understand that it has been\ncommonly used in the community. However, in the context of this work, the main result\nis that the RMCD is not proportional to magnetization (due to topological Hall effect.)\nFurthermore, as the author discussed, the magnitude of RMCD depends not only on the\nmagnetization, but also on the electronic structures. Therefore, I suggest the authors\nrefrain from making such a statement, as it is logically inconsistent with the observation\nof topological Hall effect. The comparison with theory modelling is reasonable and\nsufficient.\nReply: We sincerely appreciate the valuable comments. From your comments, we feel\nthat the reviewer is rigorous in academic research. We fully acknowledge that\ndescribing RMCD as strictly proportional to magnetization is not entirely accurate.\nAlthough RMCD primarily originates from magnetization in most collinear magnets, it\ncan also generate RMCD signals in systems with topological magnetic structures or\ncertain special magnetic configurations even in the absence of net magnetization.\nAccordingly, we have revised the relevant statements in the updated manuscript. To\nprovide necessary clarification, in the manuscript we have decomposed the total RMCD\nin monolayer CrI into two components: RCD and RCD , as defined in Equations (1)-\n(3) and the corresponding text. RCD originates from the magnetization of the\nM\nferromagnetic background, while RCD arises from the emergent magnetic field\nT\ngenerated by the topological magnetic structure. Therefore, we understand that\ndescribing the total RMCD as proportional to magnetization is imprecise, particularly\nin systems possessing topological magnetic structures. Once again, we appreciate your\ninsightful comments.\nReviewer #3 (Remarks to the Author):\nThe authors have addressed all my concers, I recommend the maunscript to be\npublished as it is.\nReply: We sincerely appreciate your support in the publication of our manuscript in\nNature Communications."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors this time are able to address my concerns and clarify that they will need a combined doping and E-field dependence to stabilize the skyrmion states, which was not clear to me at least in their earlier draft and reply. Just for the author's information, the mentioned E-field dependence of bilayer CrI paper also contains E-field 3 dependence of a monolayer device in their supplementary information, instead of only focusing on the bilayer scenario.\nAs for my second question, I made a typo by typing 3*10^13 as 3*10^12. However, the authors indeed now give the specific device geometry information in their updated manuscript. While the acheived doping is large, it's still within the reported maximum dielectric breakdown voltage of hBN. I therefore have no issue with their reported numbers.\nI can now recommend its publication."}, "author_response": "Reply: We sincerely appreciate your support in the publication of our manuscript in\nNature Communications."}]

Decision letter