Introduction

In non-centrosymmetric magnetic materials with broken global inversion symmetry, the interplay among Heisenberg exchange interaction, symmetry-enabled Dzyaloshinskii-Moriya interaction (DMI), and the uniaxial magnetic anisotropy can stabilize chiral spin textures such as skyrmions1. These topological magnetic structures hold great potential for achieving high-density storage and low-power computing2. Skyrmions can provide an effective gauge field that exerts a Lorentz-like force on moving electrons, giving rise to a transverse Hall current- a phenomenon known as the topological Hall effect (THE)3,4. This effect is a key signature for identifying and investigating the topological magnetic textures.

On the other hand, in centrosymmetric magnetic materials preserving global inversion symmetry, skyrmions could also emerge through mechanisms such as magnetic frustration5,6 the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction7, etc., hence giving rise to the THE. This phenomenon has been observed across various materials, including ferromagnetic semiconductors8,9, oxides10,11,12,13,14,15, transition metals16,17,18,19,20, magnetic topological insulators12,21,22, magnetic heterostructure23,24,25, etc. Additionally, the global inversion symmetry in such centrosymmetric systems does not preclude local inversion symmetry breaking that allows for alternating DMIs on the atomic scale, hence the formation of skyrmions and the ensuing THE26,27. Despite these theoretical predictions, direct experimental observation of such intrinsic electronic responses in centrosymmetric systems remains elusive.

This article reports a giant THE in a centrosymmetric Fe3GeTe2 (FGT) twisted lattice respecting a global inversion symmetry, observed almost without an external magnetic field. As a representative member of the developing van der Waals (vdW) magnets28,29,30,31,32,33,34,35, FGT36,37,38,39 is a ferromagnetic metal with magnetic iron atoms arranged on a triangular lattice, preserving the spatial inversion symmetry. Twisting FGT to form an FGT/FGT homostructure creates a twisted lattice exhibiting alternating AA and AB stacking regions. Our calculation by using the Levy and Fert’s model shows that twisting FGT can induce very strong intralayer DMI. This twisted FGT system introduces an excellent platform for studying emergent magnetic orderings driven by the competition among the exchange interaction, DMI, RKKY interaction, Coulomb interaction, and others40,41.

Moreover, the itinerant electrons in FGT allow for exploring the topological nature of the emergent non-coplanar magnetic ordering through transport measurement. Unlike magnetic insulators such as CrI342,43,44,45 and CrSBr46,47, however, the strong metallic bonds in FGT pose a significant challenge for fabricating twisted lattices using conventional tear-and-stack techniques. To overcome this difficulty, we developed a novel PCL-hBN tear-and-stack method48,49,50, leveraging the extreme stickiness of PCL to pick up the specific region of target materials like FGT while avoiding adhesion to hBN. This technique successfully fabricated a twisted FGT system with a twist angle from 0° to 5° (see Methods).

When the twist angle falls within the range of 0.45° to 0.75°, we observe a giant THE that is entirely absent at other angles, signaling the emergence of skyrmions. This effect is more pronounced in thinner FGT twisted systems, highlighting the strong interfacial influence at play. Furthermore, our micromagnetic simulations confirm that THE arises from intralayer DMI induced by twisting, which stabilizes the FGT twisted system into a skyrmion lattice within the specific twist angle range of 0.45° to 0.75°.

Our simulations further indicate that different magnetic ground states, including ferromagnetic metal and magnetic stripe phases, can be stabilized depending on the twist angle and the interlayer thickness. The twist-angle-dependent THE demonstrated in this work provides a compelling example of how topological spin texture can be created and manipulated using vdW ferromagnets. This discovery holds significant potential for various spintronics applications, such as employing a twisted FGT system as a magnetic quantum simulator or in high-density data storage.

Results

Twisted device fabrications and emergent topological Hall effect

In light of the strong adhesion between PCL and FGT and the weaker interaction between hBN and FGT, we developed a combined PCL-hBN method to fabricate twisted FGT devices through an optimized tear-and-stack process, as illustrated in Fig. 1a-d. The process begins with preparing the PCL/hBN stamp (Fig. 1a), which picked up the h-BN nanoflake from a SiO2/Si substrate.

Fig. 1: Twisted FGT device fabrication and emergent topological Hall effect.
Fig. 1: Twisted FGT device fabrication and emergent topological Hall effect.The alternative text for this image may have been generated using AI.
Full size image

Illustration of twisted FGT device fabrication. There are four major steps: Pick up h-BN nanoflake by PCL method (a), PCL/hBN covers half of FGT nanoflake and lift to tear the FGT nanoflake (b), Rotate to fine control the twist angle and drop down to stack the FGT/FGT homostructure (c), Hall-bar geometry electrodes are patterned onto the twisted region (d). e Schematic of a twisted FGT homostructure. f Typical optical image of a real twisted FGT device. The white scale bar represents 5 µm. The Hall electrodes are patterned onto the twisted overlapping region. g Transverse Hall resistance Rxy as a function of an out-of-plane magnetic field H. It shows the typical ferromagnetic hysteresis loops at low temperatures. h Rxy versus magnetic field H for a twisted FGT device at a twist angle of 0.6°. At low temperatures, two asymmetric Rxy hump unexpectedly appear at both positive and negative fields in addition to the anomalous Hall loop, manifesting an emergent topological Hall effect by twisting.

Next, the PCL/hBN stamp is aligned with a selected FGT nanoflake on a separate SiO2/Si substrate, ensuring that the hBN covers only half of the FGT nanoflake. Gradually increasing the temperature of the substrate to 62 °C allows the PCL polymer fringe to cross the entire FGT nanoflake fully. Then, the PCL/hBN stamp is lifted after the temperature has decreased to 30 °C to tear the FGT nanoflake. After this process, half of the FGT nanoflake covered by hBN remains on the SiO2 substrate due to the weak adhesion of FGT to hBN. In contrast, the uncovered half FGT adheres to the PCL stamp due to the strong adhesive force of PCL (Fig. 1b). This results in the half FGT nanoflake (top FGT layer) being attached to the PCL stamp adjacent to the hBN. In contrast, the last half of the FGT nanoflake (bottom FGT layer) remains on the original SiO2/Si substrate, completing the tear process.

The next step involves finely rotating the substrate to introduce a twist angle between the top FGT layer on the stamp and the bottom FGT layer on the SiO2/Si substrate. Finally, the top FGT layer is carefully aligned on the bottom FGT layer and is gradually attached to the bottom FGT layer to complete the stacking process. Then, the temperature is increased above the melting point of PCL to detach the top FGT layer from the PCL stamp. After successful detachment, the substrate with FGT homostructure is dipped into tetrahydrofuran for 10 minutes to remove the PCL residual. As a result, the top FGT layer is well stacked onto the bottom FGT layer (Fig. 1c). This innovative technique overcomes the limitations of the conventional PC tear-and-stack method for rigid vdW metals, offering a significant advancement for the 2D vdW materials community.

Finally, Hall electrodes are patterned onto the overlapping region to facilitate the investigation of the transverse Hall response in the twisted region (Fig. 1d), forming the central focus of this study. Figure 1e, 1f present the schematic of the twisted FGT homostructure and the typical optical image of fabricated devices, respectively. The white scale bar represents approximately 5 μm.

In general, the transverse Hall resistance \({R}_{{xy}}\) consists of three components \({R}_{{xy}}={r}_{0}B+{r}_{s}M+{R}_{T}\). The first term represents the normal Hall effect arising from the Lorentz force induced by the magnetic field. The second term corresponds to the anomalous Hall effect (AHE), originating from spin-orbit coupling, where \({r}_{0}\) and \({r}_{s}\) are their respective coefficients. The third term \({R}_{T}\) is the THE, which results from the Berry phase accumulation in real space and is a hallmark of topological magnetic textures. When itinerant electrons traverse a topologically nontrivial spin texture, their spins undergo adiabatic evolution, acquiring an additional Berry phase due to the local spin configuration51,52. This Berry phase acts as an effective magnetic flux, giving rise to an anomalous velocity component transverse to the direction of electron motion. Consequently, the scalar spin chirality—defined as \({\chi }_{{ijk}}={\vec{S}}_{i}\cdot ({\vec{S}}_{j}\times {\vec{S}}_{k})\)—produces a fictitious magnetic field that deflects the itinerant electrons, resulting in the observable topological Hall response. Since the normal Hall effect is linearly proportional to the applied magnetic field and the AHE can be determined from the magnetization, \({R}_{T}\) can, therefore, be extracted from the experimental data.

FGT is a hard ferromagnet of perpendicular magnetic anisotropy (See Supplementary Note 436,37,39). As shown in Fig. 1g, a single pristine FGT nanoflake exhibits only a typical AHE at low temperatures, characterized by a ferromagnetic hysteresis loop in the magnetic field-dependent transverse resistance (Rxy-H curve)53,54. In contrast, a twisted FGT device with a twist angle of 0.6° (Fig. 1h) strikingly displays two asymmetric humps in addition to the typical anomalous Hall loop. These Rxy humps are phenomenologically identical to the THE observed in the topological spin texture of a single magnetic system. This emergent THE is exclusive to metallic magnets, as it relies on electronic conduction, which is detectable through transport measurements. In the following sections, we systematically investigate this emergent quantum effect experimentally and elucidate its underlying mechanism induced by twisting.

Twist angle dependence of the emergent giant topological Hall effect

Next, we investigate the dependence of the exotic THE on the twist angle. Figure 2 presents the normalized Rxy (scaled by its maximum) as a function of the magnetic field (H) for various twisted FGT devices with different twist angles. All samples were fabricated under identical conditions, with the twist angle being the only controlled variable among the devices. As shown in Fig. 2a, the sample with a zero-twist angle exhibits the typical anomalous Hall loop at 20 K, consistent with the behavior of a pristine FGT nanoflake. Similarly, samples with twist angles of 0.15° and 0.3° display no topological Hall signals. Remarkably, however, a distinct topological Hall hump emerges in devices with twist angles of 0.45°, 0.6°, and 0.75°. As the twist angle increases beyond this range, the topological Hall hump disappears, and the pristine anomalous Hall loop reappears. Figure 2b demonstrates the same behavior even more prominently at a temperature of 60 K, confirming that the observed THE is indeed induced by twisting.

Fig. 2: Twist angle dependence of the new topological Hall effect.
Fig. 2: Twist angle dependence of the new topological Hall effect.The alternative text for this image may have been generated using AI.
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a Normalized Rxy as a function of magnetic field H under 20 K for different twisted FGT samples with individual twist angles. Topological Hall humps exist for twist angles between 0.45° and 0.75° in addition to the anomalous Hall loops. The other twisted FGT samples only have the typical anomalous Hall loops as that of single pristine FGT nanoflakes. b Similar Rxy versus H curves at different twist angles, but under the temperature of 60 K. The twist angle dependence of the new topological Hall effect exhibits universality at all low temperatures listed above. c The color map of the THE Peak ratio [defined as (topological Hall hump) / (topological Hall hump + anomalous Hall magnitude) × 100%] in the temperature-twist angle phase space. The inset indicates that only twist angles of 0.45°, 0.6°, and 0.75° can produce the topological Hall effect among the whole twist angle range from 0° to 5°, large enough compared to the previous twisted case for vdW magnetic insulators42,43,44. The color map shows the magic angles regime between 0.45° and 0.75°, where the topological Hall effect emerges. Meanwhile, these results also demonstrate reliable reproducibility for different temperatures and individual twisted devices. Note that the THE ratio can reach about 50 %, showing a colossal topological Hall effect comparable to its intrinsic gigantic AHE governed by large Berry curvatures of band topology.

Before going further, we would like to discuss several important physical quantities for the THE in our twisted FGT system. Firstly, the supercell lattice \({L}_{m}\left(\theta \right)\) is related to the lattice constant a and a twist angle \(\theta\). It can be expressed in the following formula: \({L}_{m}\left(\theta \right)=\frac{a}{2\sin \left(\theta /2\right)}\). For θ=0.45°, 0.6°, and 0.75°, this yields a supercell lattice of approximately 50.8, 38.1, and 30.5 nm, respectively. Secondly, the topological Hall resistivity \({\rho }_{{xy}}^{{THE}}\) can be related to the carrier density and the effective filed generated by the skyrmion lattice. Within this framework, \({\rho }_{{xy}}^{{THE}}\) can be written as \({\rho }_{{xy}}^{{THE}}={R}_{0}P{\varPhi }_{0}{N}_{{sk}}\), where \({R}_{0}\) is the ordinary Hall coefficient, \(P\) the spin polarization, \({\varPhi }_{0}\) the magnetic flux quantum, and \({N}_{{sk}}\) the skyrmion density24,55. Although this relation is oversimplified for thin-film magnetic layers with interfacial DMI, it remains useful for order-of-magnitude estimates24,56. In our twisted FGT with \(\theta\) = 0.45°, the magnitude of \({\rho }_{{xy}}^{{THE}}\) at 60 K is 0.71 μΩ∙cm, and the corresponding \({R}_{0}\) is estimated to be 1.51 × 10−9 m3/C. Using the reported spin polarization of FGT, \(P\) is 0.53 at 60 K57. Then skyrmion density \({N}_{{sk}}\) is expected to be 2.15 × 1015 m−2. For a triangular skyrmion lattice, this density is related to the skyrmion lattice constant \(D\) via \(\frac{1}{{N}_{{sk}}}=\frac{\sqrt{3}}{2}{D}^{2}\), from which we estimate \(D\) to be about 23 nm. These characteristics emerging on top of the ground-state magnetic configuration suggest a specific microscopic mechanism behind THE, which will be discussed in our theoretical analysis and calculation section afterwards.

To provide a concise representation, we define THE peak ratio as (topological Hall hump)/(topological Hall hump + anomalous Hall magnitude) × 100 %. This ratio is plotted as a color map in Fig. 2c, showing its dependence on temperature and twist angle. The THE peak ratio reaches values as high as 50 %, indicating a giant THE, even comparable in magnitude to the colossal AHE37,38,58 previously reported for FGT. The latter arises from the large Berry curvatures associated with its topological nodal-line band structures38,39,59. As highlighted in the inset, we extended our examination of the twist angle up to 5°, comparable to the twist angle range adopted in the previous study of vdW magnetic insulators42,43,44.

Notably, the twist-angle-dependent THE persists across a wide temperature range from 20 to 70 K (Fig.2c). These results not only confirm the universal dependence of the emergent THE on the twist-angle dependence but also demonstrate its reproducibility across different temperatures and individual devices at small twist angles.

Thickness dependence of topological Hall effect

The observed dependence on the twist angle suggests that this extraordinary THE is closely tied to the interfacial effect created by twisting. This inspires us to examine the thickness dependence of such behavior to investigate further the interfacial nature of the underlying interactions driving this phenomenon. We successfully achieved twisted FGT devices as thin as ~6 nm using our current fabrication method. Figure 3d illustrates the normalized Rxy (scaled by its maximum) as a function of an out-of-plane magnetic field for several twisted FGT devices with the same twist angle of 0.75° but varying thicknesses, all measured at 50 K. The data clearly show a pronounced THE in the ~6 nm-thick device, consistent with the results in Fig. 2. However, as the thickness increases, the topological Hall hump becomes weaker in the ~10 nm-thick device before vanishing entirely in a ~ 20 nm-thick sample.

Fig. 3: Thickness dependence of the twisting-induced topological Hall effect.
Fig. 3: Thickness dependence of the twisting-induced topological Hall effect.The alternative text for this image may have been generated using AI.
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d Normalized Rxy versus magnetic field H under 50 K for 0.75° twisted FGT devices of different thicknesses. The topological Hall loop is significant for a ~ 6 nm-thick twisted device, suppressed for a ~ 10 nm-thick twisted device and vanishing for a ~ 20 nm-thick twisted device. a–c, e, f, Similar results as in (d), but under the temperature of 20, 30, 40, 60, and 70 K. Considering that the out-of-plane magnetic interaction expands to ~5 nm in FGT, this suppression upon increasing thickness underlines the interfacial effect at the twisting interface, for accounting for the artificially created topological Hall effect by twisting.

This behavior aligns with the intuitive expectation that the interfacial twist effect dominates (diminishes) in thinner (thicker) homostructures and is substantially suppressed in thick samples. The thickness dependence observed in Fig. 3d supports this hypothesis, with the disappearance of THE in the thickest 20 nm device, where the Hall response reverts to the typical AHE seen in pristine single FGT nanoflakes. Figure 3a–c, 3e, f display the same thickness-dependent trend across different temperatures.

Considering that the out-of-plane magnetic interaction in FGT expands to ~5 nm36, this suppression of THE with increasing thickness highlights the critical role of the interfacial effect at the twisting interface. These observations shed significant light on the physical mechanism underlying this intriguing twist-induced THE, which we will further explore in the next section.

Theoretical analysis

To theoretically explain the emergent THE in the twisted FGT lattice, we analyze its underlying physical mechanism from an energy perspective. Since the THE is observed over a continuous twist angle range rather than at the discretized “magic angles” (Fig.2c), electron-electron interactions that dominate when the kinetic energy quenches at “magic angles”, can be excluded as the primary mechanism for the skyrmion formation. Moreover, the absence of THE in pristine FGT as shown in Fig. 1g rules out the possibility of the superimposition of two ferromagnet with different coercive field. As detailed in the Supplementary Information, this twisted FGT lattice allows both intralayer and interlayer DMIs, which can be obtained by an analytical derivation based on the Lévy–Fert model within the perturbation theory framework (Supplementary Note 1). Our calculation shows that the intralayer DMI is about four orders of magnitude larger than the interlayer one. Meanwhile, the magnitude of the intralayer DMI is almost a constant independent of position inside the twisted FGT. Figure 4a shows the magnitude of the interlayer DMI as a function of twisting angle \(\theta\) for three different thicknesses. It turns out that this intralayer DMI is significantly enhanced by twisting. Mathematically, this intralayer DMI can be expressed as \({\vec{D}}_{{ij}}\cdot {\vec{S}}_{i}\times {\vec{S}}_{j}\) where \({\vec{D}}_{{ij}}={D}_{0}(\theta ){\left(-1\right)}^{l}\hat{z}\times {\hat{e}}_{{ij}}\) with \(l=1\) or \(2\) referring to the top or bottom layers, \(\hat{z}\) being the unit vector of the plane normal, and \({\hat{e}}_{{ij}}\) the lattice vector connecting site \(i\) and \(j\).

Fig. 4: Micromagnetic simulation for FGT twisted system.
Fig. 4: Micromagnetic simulation for FGT twisted system.The alternative text for this image may have been generated using AI.
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a Magnitude of the intralayer DMI \(D\) as a function of twisting angle \(\theta\) for three different thicknesses \(6\) nm, 10 nm, and 20 nm. The inset illustrates the direction of the intralayer DMI on top (blue) and bottom (red) layer. b Phase diagram represented by the skyrmion density inside the entire simulated region versus twisting angle \(\theta\) and thickness. Here, SKX, FM, and stripe refer to the skyrmion lattice, ferromagnetic state, and stripe state, respectively. c Spin configurations of the bottom layer represented by the z-component of the magnetization (\({m}_{z}\)) at each site for representative twisting angle \(\theta\) (labeled on the top) and thickness (labeled on the right side).

In the presence of the DMI, the twisted FGT lattice can be modeled using the continuum free energy (Supplementary Note 2):

$$H=\int d{{{\boldsymbol{r}}}}^{2}\left\{{\sum }_{l}[J{a}_{0}^{2}{(\nabla {\vec{{{\boldsymbol{m}}}}}_{l})}^{2}-K{({m}_{l}^{z})}^{2}]+{J}_{t}{\vec{{{\boldsymbol{m}}}}}_{1}\cdot {\vec{{{\boldsymbol{m}}}}}_{2}\right.\\ \left. \,+{D}_{0}\left(\theta \right){a}_{0}{\sum }_{l}{(-1)}^{l}[\hat{\hat{{{\boldsymbol{y}}}}}\cdot ({\vec{{{\boldsymbol{m}}}}}_{l}\times {\partial }_{x}{\vec{{{\boldsymbol{m}}}}}_{l})-\hat{\hat{{{\boldsymbol{x}}}}}\cdot ({\vec{{{\boldsymbol{m}}}}}_{l}\times {\partial }_{y}{\vec{{{\boldsymbol{m}}}}}_{l})]\right\}$$
(1)

where \(J{a}_{0}^{2}\) is the exchange stiffness with \({a}_{0}\) the lattice constant, \(K\) is the uniaxial anisotropy, \({J}_{t}\) is the interlayer exchange interaction, and \({D}_{0}\left(\theta \right)\) is the intralayer DMI constant obtained using the Lévy and Fert’s model (Supplementary Note 1), which flips signs between layers and depends on the twisting angle \(\theta\). Given the ferromagnetic nature of FGT, we resort to micromagnetic simulations to determine the ground-state magnetic configuration. Specifically, we discretize the energy in Eq. (1) on a triangular superlattice and numerically minimize the total energy at finite temperatures (see Methods). Figure 4b presents the phase diagram of the twisted FGT lattice representing by the skyrmion numbers versus the twist angle \(\theta\) and interlayer thickness \(d\), with the spin configurations at some representative \(\theta\) and \(d\) plotted in Fig. 4c. Generally, we identify three different phases on the \(\theta -d\) plane, e.g., the magnetic stripe phase at small twisting angles, the ferromagnetic (FM) phase at large twisting angles, and the skyrmion lattice (SKX) phase interpolating the two. Meanwhile, we also find the magnetic stripe can coexist with the skyrmion lattice at some region.

When \(d=6{{\rm{nm}}}\), the system is a skyrmion lattice at \(0.45\le \theta \le 0.75\). In principle, the topological Hall resistance is proportional to the spatial topological charge of the magnetic configuration: \({\rho }_{{xy}}^{T}={PQ}/{en}\), where \(P\) is the electron polarization, \(Q\) is the topological charge of the magnetic ground state, and \(e\) and \(n\) are the electron charge and density, respectively9,16. Moreover, the THE peaks are typically proportional to the skyrmion density inside the skyrmion lattice. This phase diagram aligns quantitatively with the color map of the THE peak ratio shown in Fig. 2c. Although the spins in opposite layers rotate oppositely due to the reversed DMI, the topological charges in different layers are identical27. Moreover, the high density of the skyrmion lattice provides evidence for the giant THE peaks in the experimental data. Our simulations thus confirm that the giant THE can be ascribed to the emergence of skyrmion lattice at specific twisting angles.

Notably, the DMI plays a crucial role in facilitating the formation of skyrmions and spin stripes at small twist angles. Since the DMI depends heavily on the interlayer coupling induced by twisting, increasing the thickness of the twisted FGT system overall reduces the DMI as shown in Fig. 4a, thereby suppressing THE. Consequently, the THE only appears at a small twisting angle region indicated by the SKX in the phase diagram (Fig. 4b). Furthermore, competitions among the exchange interactions, uniaxial anisotropy, and DMI underpin the formation of skyrmions. Thermal agitations at high temperatures effectively reduce the uniaxial anisotropy, potentially enhancing the THE and broadening the range for the skyrmion formation, as observed in Fig. 2c.

Discussion

In our study, the emergence of THE in twisted FGT and micromagnetic simulation provide insights into how local inversion-symmetry breaking modifies the magnetic ground state through twisting. On this basis, we clarify several important issues on the emergent giant THE in our twisted FGT devices. First, the temperature dependence of the THE in twisted FGT suggests that the formation of topological spin textures is governed by a competition of magnetic anisotropy, exchange interactions, and the DMI (See Supplementary Note 5). Second, we estimated the skyrmion size from the magnitude of THE in the twisted system24,55,56, which is consistent with the theoretical calculations (See Supplementary Note 6). Moreover, we find that the electronic mean free path in FGT is much smaller than the skyrmion size, which further supports our interpretation that the THE originates from topological spin textures (See Supplementary Note 7). Furthermore, we considered the cases in our experiments, where the “two-channel AHE”60,61,62 is difficult to be straightforwardly applicable in our twisted FGT devices (See Supplementary Note 8).

In conclusion, we have observed a twist-angle-dependent THE in an FGT twisted lattice that preserves a general global inversion symmetry. The local inversion symmetry breaking induces an intralayer DMI, driving the transition from a ferromagnetic ground state to a skyrmion state and eventually to a spin stripe state as the twist angle decreases. The THE emerges as the system enters the skyrmion state, a phenomenon further corroborated by the micromagnetic simulations. Our findings demonstrated the potential to create and manipulate skyrmions and other spin states by engineering twisted magnetic systems. This study presents the first report of a twisted metallic vdW magnet enabled by our advanced fabrication technique and the first observation of an emergent THE at small twist angles. These results surpass previous studies on rare cases of twisted vdW magnetic insulators, highlighting the promise of twisted vdW magnetic metals for novel, current-driven spintronics applications.

Methods

FGT single-crystal growth

High-quality FGT single crystals were grown using the chemical vapour transport (CVT) method, following our previous work63. High-purity chemical elements were purchased from Thermo Fisher (Fe 99.9%, Ge 99.999%, and Te 99.999%). These elements were placed into a quartz tube with low water and oxygen concentrations (below 1 ppm); all processes were carried out inside a glove box. Iodine was used as the transport agent for the CVT method. After all the chemical powders were placed into the quartz tube, the tube was evacuated and sealed. The sealed tube was then placed in the middle of a two-zone furnace. The temperatures of the cold and hot zones of the two-zone furnace were raised to 700 and 750 °C, respectively. After two weeks, high-quality FGT single crystals were synthesized.

Twisted FGT homostructure fabrication

The detailed descriptions of the fabricating method have been presented in the main text and Fig. 1. Here, we would like to elaborate on the development history of the combined PCL-hBN technique for ease of understanding of the experimental technical part. The best method to fabricate a twisted homostructure is the so-called tear-and-stack technique. Unlike CrI3, FGT is difficult to be exfoliated and picked up by a traditional PC transfer polymer, not to mention tearing it apart and stacking it with fine control. To address these problems, we develop a strongly adhesive dry transfer technique based on the newly invented PCL polymer48, which can pick up many vdW magnets effectively with nearly 100% probability under a very low operation temperature of around 60 °C.

Adopting this PCL technique to FGT led to our first attempt to access twisted FGT homostructure in 202049, where a plateau-like magnetoresistance was observed in this twisted FGT homostructure, even with no spacer in between. In this paper, PCL polymer gradually attaches to an FGT nanoflake, with the polymer fringe stopping in the middle of the nanoflake. Then, it picks up half of the nanoflake and drops it down on the remaining half after rotation. Although doable, there is still a low possibility of success, and it is difficult to control the whole process finely regarding the stopping fringe in the middle of FGT, preventing a further efficient and systematic investigation. More seriously, this rough tear-and-stack method can only be applied to a large-area exfoliated nanoflake, which generally hosts a large thickness, e.g., 40 nm. This, thus, would smear out the intriguing quantum effects of a twisting interface effect, if any.

The PCL method is further developed into a PCL and hBN combined technique, enabling the reliable fabrication of different twisted homostructures of diverse materials with fine control50. Such a new twisted fabrication method realizes the tear-and-stack process by taking advantage of the uniqueness of PCL, which is extremely sticky and can pick up the target material like FGT, for example. At the same time, it is more challenging for hBN: we used commercial hBN for this work. Most importantly, one can readily achieve fine angle control with a resolution of 0.015° and twisted fabrication on a small nanoflake of several micrometres wide and several nanometers thin. These long-lasting technique developments48,49,50 eventually allow us to confidently investigate the emergent quantum phenomena in twisted metallic FGT homostructures for the first time. This new universal twisting fabrication method developed for diverse vdW magnets is extremely suitable for making twisted metallic vdW magnets such as twisted FGT homostructures.

Transport measurement

The substrate with twisted FGT is coated with poly (methyl methacrylate) (PMMA) through a spin coating process. A pattern of electrodes is drawn on the PMMA using electron beam lithography. For the electrodes, an electron beam evaporator is used to deposit titanium and gold, with thicknesses of 10 and 85 nm, respectively. Magnetoresistance and Hall resistance of twisted FGT are measured using the standard lock-in technique with SR830 (Stanford Research System). The device is placed inside a cryostat to maintain cryogenic temperatures. The magnetic field is applied perpendicular to the surface of the twisted FGT sample.

Magnetic simulations

To perform the micromagnetic simulations, we discretize the free energy in Eq. 1 on a triangular lattice with the lattice constant \({A}_{0}=2{{\rm{nm}}}\), and the system size is taken as \({L}_{x}=200{A}_{0}\), \({L}_{y}=100\sqrt{3}{A}_{0}\). The intralayer DMI at different twisting angles is calculated by using the Lévy and Fert’s model as detailed in the Supplementary Note 1. The other parameters are taken as \(J=2.0\times {10}^{-21}\) Joule, \({J}_{t}=3.2\times {10}^{-23}\) Joule, \(K=1\times {10}^{-24}\) Joule, which are fitted from experimental data.

Subsequently, the magnetic ground state can be calculated by numerically minimizing the total energy, which can be implemented by different methods, such as the steepest descent. In this work, the magnetic ground state at different twist angles is obtained using VAMPIRE under the field-cool program (\(B=0.1\)Tesla) and Monte Carlo integrator. During the simulation, the cooling time is set to 100 picoseconds with a time-step of 1 femtosecond, and the temperature is fixed at 20 K.