Introduction

The two-dimensional (2D) material family has drawn significant attention due to its high specific surface area and distinctive electronic performance1. The extraordinarily low bending stiffness of nature permits itself to be artificially engineered, for example, to be folded, bent, and rolled2. Meanwhile, the interlayer van der Waals force provides opportunities to achieve superlubrication between layers forming incommensurate contact, through interlayer twist in homojunctions and stacking of heterojunctions3. Interestingly, by superimposing one crystalline material with another relative lattice constant or spatial displacement, which may be affected by the interlayer relative position as well as the substrate misalignment, periodic moiré superlattice structures with a period much longer than the intrinsic lattice constants will emerge4. Such moiré periodic potential can influence the electronic band structure and the charge carrier mobility of 2D-layered materials5,6. These create opportunities for successfully and effectively tuning their electronic performance through mechanical deformations7.

2D moiré superlattice attracts significant interests due to the emerging and intriguing properties originated from the moiré patterns, such as low energy Van Hove singularities8,9, superconductivity10, the fractal quantum Hall effect11, topological valley transport12, and so on13,14—which are not exhibited in the individual layers themselves—were then studied extensively. Notably, atomic reconstruction with spontaneous electric polarization, including sliding ferroelectricity15, and moiré ferroelectricity16, was first theoretically proposed in 2017, and proved in bilayer hexagonal boron nitrogen (h-BN)17,18,19. By direct growth20, tear-rotate-stack method21, or mechanical force manipulation22, a 2D moiré superlattice with ferroelectricity can be obtained. The occurrence of 2D moiré ferroelectricity not only makes information memory with much higher density and lower energy consumption, which breaks the limitation of the size effect23, possible, but also allows various properties that are hard to exist simultaneously in one system to couple with each other24. However, such stacking configurations generally occur in multilayer systems with two or more elements and similar unit cell sizes, for instance, inequivalent stacking occurs via interlayer sliding and marginally twisted angles (\({{{\rm{\theta }}}}\, < \,2^\circ\))25. Although inversion symmetry breaking could be triggered in across-layer (n > 3) mono-element systems, e.g., graphene26,27,28, that substantial experimental evidence remains largely desired.

The nature of moiré ferroelectricity, including the structure of the moiré supercell, electrostatic potential, etc., is usually visualized using several high-resolution microscopic techniques, such as transmission electron microscopy (TEM)25, scanning tunneling microscopy (STM)29, piezoelectric force microscopy (PFM)4, electrostatic force microscopy (EFM)18, kelvin probe force microscopy (KPFM)6 and conductive atomic force microscopy (C-AFM)30, etc.31,32. They can be manipulated locally by the application of an external electric field31,33,34, force35, temperature36, etc. However, in contrast to the ideal equilateral triangle moiré (regular moiré) patterns induced by heterostructures with different twist angles between neighboring layers, irregular moiré supercells including distorted triangles and strips are more common in practical measurements because contamination or additional strain fields are generally unavoidable during fabrications or characterizations. Due to the strain fields in moiré ferroelectric materials, various types of domain layouts can be manipulated more easily37,38, allowing polarization exist in the form of different shapes, sizes, and densities. This makes information storage with various configurations, for example, strips and triangles, possible.

In this study, we propose a mechanical-force-induced polarization switching pathway in twisted 3R-h-BN (t-BN) which has been proved as topological ferroelectrics in our previous work39. Theoretical simulations are conducted to elucidate the atomic mechanism underlying the shift of moiré patterns under uniaxial strain. Then the dynamics of the moiré patterns under the external strain field are demonstrated in experiments by piezoresponse force microscopy (PFM) via a nanoprobe. In contrast to ideal regular moiré patterns, irregular polarization moirés functioned by external strain field get less limitation of topological protect or domain pinning when the shear direction is not aligned with the strain direction, indicative of the chances in precisely manipulating interfacial polarization, which is critical for understanding the polarization switching in interfacial ferroelectrics.

Results

We conducted classical force field molecular dynamics (MD) simulations to reveal the mechanisms governing moiré pattern evolution during interlayer sliding under strain. A detailed methodology can be found in the “Experimental methods” section. The twisted t-BN bilayer at a twist angle of 1° was built with periodic boundary conditions (PBC). Although the twist angle used in this study is larger than the extremely small angles (e.g., 0.01°) observed in experiments, it is sufficient to reveal the underlying physical mechanisms of strain dependence. The chosen angle allows for capturing key features such as atomic reconstruction and provides insight into the behavior of the system while maintaining computational feasibility. The atomic model is found in Fig. S1A in Supplementary Information. Uniaxial strain was applied to both layers by stretching the in-plane periodic boundaries. Notably, the complex strain field and corresponding moiré potential map observed in experiments40 are beyond the present simulation studies. The atomic structures under uniaxial strain are depicted in Fig. 1A. Blue and red triangles represent adjacent AB/BA ferroelectric domains, colored according to the local polarization registry index (LPRI) previously proposed41. The two domain walls (labeled DW1 and DW2 in Fig. 1A) behave differently according to the strain (see Fig. 1B). This is further illustrated by the profiles of the domain walls (see Fig. S2 in Supplementary Information). It is important to note that the width and height of the domain walls in Fig. 1A are defined based on their geometric characteristics. For instance, under strain along the AC direction, the reduction in the height of DW1 occurs because the stretching smoothens the atomic buckling at the domain walls. Conversely, due to the positive Poisson’s ratio of the structure, stretching one side of the cell leads to compression on the other side, resulting in the thickening of DW2. The changes in the dimensions of the simulation cell are provided in Fig. S1B. The atomic structures of these two domain types are presented in Fig. S3, further illustrating the strain-induced differences between DW1 and DW2. The areas of the nodes also increase with strain, indicating a breakdown of C3 symmetry, which could facilitate sliding in a specific direction.

Fig. 1: Moiré dynamics of t-BN under uniaxial strain via MD simulations.
figure 1

A Moiré patterns of the t-BN bilayer with a twist angle of \({{{\rm{\theta }}}}=1^\circ\) under strain along the armchair (AC) and zigzag (ZZ) directions. The AC and ZZ directions are defined from the triangular moiré supercell, as indicated by the arrows. The polarization map is shown using the Local Polarization Registry Index (LPRI), and the distribution of interlayer distance (\(h\)) is illustrated. Model sizes are provided in Fig. S1. B The domain wall width as a function of strain. Domain walls DW1 and DW2 are labeled in (A). C Static friction stress as a function of strain for shear along the AC and ZZ directions. D Evolution of the moiré pattern during the shear process. From I to V, the top BN layer shifts by one periodicity: bAC = a = 2.5 Å for shear along AC, and bZZ = 3/2 × a/√3 = 2.2 Å for shear along ZZ, where \({{{\rm{a}}}}\) is the lattice constant of h-BN. The five steps correspond to the typical forces observed in the force traces (see Fig. S4 in Supplementary Information).

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The pinning of the strained moiré patterns is investigated through quasi-static simulations of the shear stress between two BN layers. The force traces are provided in Fig. S4 in the Supplementary Information, where the static friction stress is determined from the first peaks. A decrease in the pinning force due to strain is observed when the shear direction is perpendicular to the uniaxial strain direction (see Fig. 1C). Pinning becomes more pronounced when the shear and strain directions align. When shear is applied along the strain direction, atomic movements are restricted to the shear direction. Conversely, when shear is applied perpendicular to the strain direction, the interlayer coupling is weaker, allowing atoms to traverse lower energy paths. This behavior is further supported by the evolution of the registry in the structure during interlayer sliding. We analyzed the global registry index (GRI) during the sliding process, as shown in Fig. S5. The GRI ranges from 0 to 1, corresponding to optimal AB/BA stacking and the least favorable AA stacking, respectively. This method of analyzing the registry has been widely used to demonstrate potential corrugation and structural superlubricity during interlayer sliding in layered materials42,43,44. The figure reveals a trend similar to that of the force traces in Fig. S4, showing that potential corrugation decreases with increasing strain when the sliding direction is misaligned with the strain direction.

The shift of the moiré pattern during sliding is illustrated in Fig. 1D. Shear stress along the AC and ZZ can produce different moiré patterns. From I to V, moiré edges undergo distortion and reformation periodically, with the moiré pattern shifting by one superlattice constant (i.e., the lattice constant of a moiré). Shear under strain induces more significant deformations of moiré edges compared to the un-strained bilayers. Moiré patterns differ between shear processes under different strains. This is because the shift of the moiré pattern is perpendicular to the shear force for marginally twisted van der Waals (vdW) bilayers45. The shifts of the moiré pattern for the four systems shown in Fig. 1D are available in Supplementary Movies 1–4. The dynamics of the moiré pattern during interlayer sliding is further characterized by the local registry index (LRI) (see Fig. S6). By combining the LPRI patterns, the evolution of domain walls can be clearly visualized and demonstrated.

Then, we conducted experiments in t-BN using PFM. The schematic of t-BN shown in Fig. 2A is fabricated by tear-and-rotate technique with the aid of a curved PDMS. The curvature is fabricated to be ~113 m−1, applying a continuous strain clamping field. The detailed process is given in Figs. S7 and S8, in Supplementary Information. Although some tiny particles were left on the sample after the transfer process (Fig. S7D), it exhibits a relatively flat and clean surface (Fig. 2B), which largely eliminates topographic crosstalk that could interfere with the analysis, allowing for the clear observation of moiré patterns within the samples. To investigate the stacking order of the moiré patterns, electrostatic force microscopy (EFM) is employed to study how the surface potential varies with the supercell size. A representative EFM image, shown in Fig. 2C, demonstrates a gradual deformation in the supercell size, which is attributed to strain-induced variations in the twist angles between the layers. The color contrast in the EFM image is indicative of the surface potential differences. Figure 2D displays an enlarged view of the region obtained from Fig. 2C by zooming in on the appropriate area. The line profile extracted along the diagonal direction labeled by a dash is presented in Fig. 2E, providing a quantitative representation of these potential variations. The PFM amplitude image corresponding to the EFM image is given in Fig. 2F. The supercell sizes are measured to be around 500 nm, indicating a twist angle of ~0.01°. These measurements demonstrate the moiré ferroelectricity19, where adjacent moiré regions exhibit opposite out-of-plane polarization. They are identified as AB and BA domains labeled in Fig. 2D–F according to the sinusoidal function6. The new periodicity of the moiré patterns shown in the top right is disrupted by the increase of thickness, and those presented in the left are attributed to the function of wrinkle labeled by a white dash in Fig. 2B. The triangular domain networks also provide supportive evidence for our model shown in Fig. 1A, where the twist angle is slightly larger than that in the experiments. The non-equilateral triangles form due to the strain field, with two typical types modeled by applying normal strain to both layers, as illustrated in Fig. 1A.

Fig. 2: Structure and moiré potential of t-BN.
figure 2

A Schematic of a t-BN sample produced through a tear-and-stack method with the aid of a curved PDMS. B Topography of the t-BN surface. C EFM phase image of the moiré pattern extracted from the black square in (B). D EFM image of the enlarged region zoomed from the region 4 in (C). E Line profile of the EFM image labeled by the black dashed line in (D). F PFM image corresponding to (D).

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The dynamic characteristics of the moiré is then observed by employing a PFM tip to exert mechanical force along a consistent direction (sliding-disturb measurement). The measurement was performed by repeatedly scanning the target region under an AC voltage of 0.8 V with approximately 30 nN (~2.4 MPa) compressive force. Similar measurements were taken in a sample without curved PDMS, revealing the key role of the curved PDMS, and largely excluding the effect of the applied AC voltage (Fig. S9, Supplementary Information). This lateral movement induces a shear force that influences the relative positioning of the upper BN layer (BN2) in relation to the bottom BN layer (BN1). A series of PFM amplitude images selected from different regions in Fig. 2C (R1–R3) capture the evolution of moiré patterns after several lateral sliding operations by nanoprobe. To discern the supercell topologies within the moiré pattern, one can count the number of nodes present along the boundary surrounding a particular supercell. Some representative nodes are indicated by dashed triangles and circles, with the direction of force and edges marked by blue and black arrows, respectively.

Throughout the sliding-disturb experiments, dynamic PFM responses of t-BN are observed. Three types of irregular moiré patterns are highlighted in Fig. 3. For type I, the moiré pattern consists of an approximate hexagon formed by six triangles, highlighted by gray masks in Fig. 3A. This pattern remains largely unchanged throughout the sliding-disturb measurements, exhibiting only slight shape changes. The mechanical force induced by the PFM tip cannot overcome the pinning force of the periodic moiré pattern. This is because the interlayer friction in periodic moirés under minimal rotation is close to that of the commensurate stacking mode45. Friction force decreases when heterostrain is introduced to the system, as indicated by the displacement of the light blue triangle moiré. The length of the edges increases with the applied force (process I–V), suggesting interlayer sliding or a reduction of the rotation angle46. The shift of the moiré edge, denoted by a black arrow, is approximately perpendicular to the direction of the applied force.

Fig. 3: Moiré patterns’ variation under the sliding-disturb measurements.
figure 3

A Evolution of type I moiré patterns. When the disturbing mechanical force is applied, those approximately equilateral triangular moirés remain largely unchanged (light gray masks) accompanied with slight migration perpendicular to the strain direction. B Dynamics of type II moiré domains. The distorted and stretched triangular moirés are partially pinned by the grains or particles in the sample. They shift and merge vertically to the direction of strain. C Kinetics of type III moirés. The irregular long strips are formed under extreme strain and shift forward and backward, accompanied with annihilation and reconstruction under the function of continuous mechanical force.

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Type II is a group of distorted and stretched triangular moirés, which are locally pinned by particles (Fig. 3B)46. The number of moirés around a node remains unchanged, and the elongated side length of triangular moirés shifts with the node, marked by the yellow arrow. The motion of the node makes the moiré domains gradually eliminate into smaller and smaller triangles (green triangles), causing adjacent nodes to merge into one finally. The distinguish merging phenomenon is created by the pinning centers, tiny particles, after carefully checking the corresponding topography (Fig. S10, Supplementary Information). The change of topography before and after sliding is given in Fig. S11 in Supplementary Information. Specifically, we can observe that node #1 (black dashed circle in process I–IV in Fig. 3B) is pinned by particle 1 (Fig. S10A, C) on the sample. Coupled with moiré domains, node #1 moves, gets close to the adjacent node #2, and finally merges into one node #3 (process V in Fig. 3B), forming a larger hexagon and a new local pinning center.

Type III is those inhomogeneous long parallel strips or one-dimensional domain walls, as shown in Fig. 3C38. Those unique moirés form due to large strain at the expense of collapsing triangular moirés. Clear shifts along the direction of the long sides of these strips are observed, and the distance between the adjacent parallel lines also experiences dynamic evolutions. The labeled blue dashes are the boundary position of the previous process, for example, blue dashes in III is the boundary position in II labeled by black dashes, and the yellow and black arrows highlight the shifts and deformations. It is not difficult to find that the number of boundaries fluctuates during the sliding-disturb measurements, which might be due to the competition between lattice mismatch and commensurate condition.

From the above discussion, we can find that the curved PDMS offers a global effect (type III) while the particles introduced during transfer process give a relatively local influence (type II). When heterostrain is brought into the moiré system by external mechanical force, type II moirés can shift, distort, and merge along with the particle pinning center, and the type III moirés move, dissociate, and reconstruct accompanied by the fluctuating number of boundaries. Remarkably, if the nodes are pushed back in the opposite direction, they revert to their original positions, indicating the system’s reversible behavior (Fig. S12 in Supplementary Information). This demonstrates that controlled force application can engineer the t-BN structure to induce specific transitional states. Similar measurements are repeated as well, with a bubble on the surface as a remark, which is given in Fig. S13 in Supplementary Information.

We provide direct evidence on the polarization switching of t-BN under strain from both theoretical modeling and experimental measurements, which is schematically illustrated in Fig. 4 as well. The experimental results confirm the theoretical predictions in two key aspects when the moiré patterns are relatively regular (corresponding to types I and II). First, the pinning of an elongated moiré pattern can be easier than that of an unstrained periodic moiré pattern when the direction of the shear force and strain is not aligned. Second, the observed moving direction of the moiré edge matches our expectations, demonstrating that it is not aligned with the shear force. It should be noted that the strain field induced deformation of supercells is more complex than the simple normal strain studied in the modeling, leading to the formation of one-dimensional domain walls (corresponding to type III). Additionally, the shift of the domain walls under strain becomes more uncontrollable in the presence of multilayers, edge contacts, contaminants, bubbles, defects, etc.25,37,40.

Fig. 4: Schematic for moiré patterns’ variation under shear and force modulation.
figure 4

Moiré patterns can be elongated along the stretch direction, while the polarization move vertical to the disturbing force direction.

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In summary, we find an effective pathway to controllably manipulate the t-BN moiré patterns at the nanoscale. By taking sliding-disturb measurements, three types of moiré patterns/edges with similar evolution dynamics are observed by PFM: (I) almost unchanged patterns due to large pinning force, (II) merged nodes and eliminated moirés, (III) collapsing moirés and formed parallel edges. Theoretical calculations of the moiré pattern evolution under strain confirm the experimental results regarding the reduction of the pinning force and the shift directions of the moiré patterns, shed light on the regulation of polarization switching through interlayer sliding. By introducing an external strain field, moiré polarization could be switched by applying a disturbed mechanical force, which offers a promising platform for modulating the moiré ferroelectricity of sophisticated twisted van der Waals structures.

Methods

Sample preparation

The curved Polydimethylsiloxane (PDMS) is fabricated onto a glass with specified dimensions. A Poly (bisphenol A carbonate) (PC) film is then affixed to the curved PDMS, creating a glass/PDMS/PC structure. Subsequently, a 100 nm thick evaporated Au thin film is transferred onto the substrate at 90 °C to enhance the adhesion of the PC film, resulting in the glass/PDMS/PC/Au substrate. Graphite with more than a hundred nanometers is transferred onto the substrate as the bottom electrode, followed by the sequential placement of hexagonal boron nitride (h-BN) flakes. These h-BN flakes are exfoliated on SiO2 (300 nm)/Si substrates and cut using a sharp nanoscale tip with a high scanning frequency (~50 Hz). The target region measures approximately 100 µm ∗ 50 µm, with a thickness of ~2 nm verified by atomic force microscopy (AFM). The substrate with the bottom electrode is used to pick up the two parts of h-BN in parallel using the tear-and-rotate technique. It’s important to note that tiny rotation angles are typically unavoidable during the transfer process, even with efforts to minimize them to nearly 0°. Dry transfer techniques are employed throughout the fabrication process. Slow transfer rates and in-situ heating (90 °C for 15 min) help mitigate the formation of bubbles and wrinkles. Although some contaminants remain on the sample, they serve as pinning centers during sliding-disturb measurements, adding intrigue to the dynamic evolution observed.

SPM measurements

SPM measurements are carried out on commercial SPM system Cypher S and Park AFM NX10 containing the functions of AFM, PFM, EFM, SKPM, etc. The Pt coated conductive probes (NSC18, MicroMasch) are utilized during the entire measurement. The electromechanical responses in PFM were detected by driving the tip at a vertical resonance frequency of ~340 kHz and an AC voltage of 0.8 V. The surface potential was detected ~20 nm far away from the sample surface by conductive probes with +3 V positive charges.

Theoretical calculations

Molecular dynamics (MD) simulations of the t-BN bilayer at a twist angle of 1° were performed using the LAMMPS package with lateral PBC47. In the vertical direction, a vacuum size of 50 nm was applied to prevent spurious interactions between adjacent bilayer images. The intralayer and interlayer interactions were described using the Tersoff48 intralayer potential and the registry-dependent interlayer potential (ILP), respectively49,50. The structures were relaxed under uniaxial strain by stretching one boundary while allowing the other boundary to compress freely. To simulate the h-BN substrate, the lower layer was coupled by a spring in the normal direction with a spring constant of \({{{\rm{k}}}}=2.656{{{\rm{N}}}}/{{{\rm{m}}}}\). The structure was minimized in the following steps: Initially, the geometry of the atoms was optimized with a fixed supercell size using the Fire algorithm51, with a force tolerance of 10−5 eV/Å. Subsequently, the supercell dimensions were optimized using the conjugate gradient (CG) algorithm52 with a force tolerance of 10−3 eV/Å. This two-step energy minimization procedure was repeated five times to ensure well-converged results. Finally, the atomic coordinates were further relaxed using the Fire algorithm with a force tolerance of 10−5 eV/Å. This methodology aligns with the approach used in a recent study19. The optimized box sizes after minimization are shown in Fig. S2. Quasi-static interlayer sliding simulations were conducted to study the static friction stress between the t-BN bilayer. The atoms in the upper layer were coupled to a fictitious stage with a spring constant of \({{{{\rm{k}}}}}_{||}=100\,{{{\rm{N}}}}/{{{\rm{m}}}}\,\) along the sliding direction to mimic the bending of the PFM tip. The in-plane movement of the lower layer was restricted to facilitate interlayer sliding. The stage was incrementally moved in steps of 0.012 Å. The simulation model is illustrated in Fig. S6. At each step, optimization cycles similar to the relaxation process described above were performed. The shear force was recorded based on the spring expansion. The force traces are shown in Fig. S3.