Moiré-induced modulation of out-of-plane elasticity in twisted hBN

Abstract

Moiré superlattices in twisted van der Waals materials provide a versatile platform where interlayer interactions generate periodic variations of atomic registry and local physical properties. Here, we reveal that these moiré patterns also modulate the effective out-of-plane elastic response of twisted hexagonal boron nitride (t-hBN). Using contact-mode atomic force microscopy and force-spectroscopy nanomechanics, we show that the effective out-of-plane Young’s modulus exhibits a reproducible modulation by up to 10% across a single moiré unit cell, with softer AA and saddle-point regions and stiffer AB/BA domains. The resulting contrast produces a measurable topographic modulation even under ambient conditions, enabling the moiré superlattice to be visualized as a manifestation of its elastic landscape. These findings demonstrate that the effective out-of-plane elastic response is modulated by the moiré superlattice, extending the understanding of twisted two-dimensional materials beyond their electronic and optical properties and opening routes toward mechanically tunable moiré systems.

Twisted hBN moiré patterns are shown to modulate the effective out-of-plane elastic response, enabling AFM visualization through a stacking-dependent elastic landscape.

Introduction

In twisted layered materials (t-LMs), moiré superlattices (MSs) can modulate most of the sample physical properties, determining unconventional behaviors such as correlated insulator¹ and superconductive² states in graphene (Gr), ferroelectricity in hexagonal boron nitride (hBN)^3–5, and transition metal dichalcogenides (TMDs)^6,7, and orbital magnetism³ in t-Gr aligned with hBN. Such unique behaviors stem from the lattice interactions at the interface between the twisted layers. Specifically, at marginal twist angles (θ_TW < 1°), the MS interfacial atomic arrangement is shaped by a 3d lattice reconstruction^4–7 that gives rise to different stacking domains which in the case of t-hBN with parallel alignment of the untwisted layers, are called AA, saddle points (SP) and AB/BA, depending on the relative position of boron and nitrogen atoms at the interface^8,9. Their physical properties have been widely investigated by a multitude of scanning probe microscopy (SPM) techniques. Among them, electrostatic force microscopy^10,11, kelvin probe force microscopy^4,10, tapping mode phase imaging^12,13, and piezoresponse force microscopy^9,14 have shown how t-hBN parallel MSs are characterized by an array of staggered triangular AB/BA polarization domains, whose typical in-plane dimension can vary from the μm down to the nm scale, depending on θ_TW⁹. Interestingly, recent simulations on t-hBN^4,15 and on t-TMDs¹⁶ predict how MSs can also modulate the out-of-plane interfacial interlayer distance. According to these simulations, AA and SP domains show an interlayer distance a few tens of picometers (pm) larger than for AB/BA regions. In principle, these vertical displacements could be used to visualize a MS by simply looking at the topography of a t-LM via atomic force microscopy (AFM), a technique with the full capability of measuring such out-of-plane variations in the pm-scale^17–19. However, no AFM topographical visualizations of MSs have been shown, so far, to the best of our knowledge. In fact, several contributions explicitly reported no detection of MS details in the AFM topography of different t-LMs^10,20–22.

In this work, t-hBN MSs are detected by reversible topographical contrast using contact-mode AFM. Through a combination of force-controlled imaging and nanomechanical spectroscopy, we demonstrate that this contrast arises from stacking-dependent elasticity. Our analysis reveals that domains with AA or saddle-point registry are softer than AB/BA regions by up to ~10%, leading to localized deformation under AFM tip load.

These findings show that the effective out-of-plane elastic response can be quantitatively mapped and is modulated across moiré stacking domains, opening new avenues for studying and manipulating stacking configurations through mechanical means^23–25. Importantly, this mechanical readout of moiré structures requires no electronic measurement, low-temperature environment, or applied bias—making it directly compatible with ambient AFM techniques.

Results and discussion

AFM contact-mode moiré topographical contrast

We consider an 8 nm top hBN flake on a 4 nm bottom hBN crystal, supported by a Si substrate covered by 90 nm SiO₂ (see “Methods” and Supplementary Information (SI), Section 1 for more details). The two flakes are aligned at a nominal twist angle of θ_TW ~ 0° (more details on the sample fabrication in the “Methods” section). When two hBN flakes are stacked together and marginally twisted, the misalignment of the interfacial atoms results in a periodic array of stacking domains, i.e., a MS. Two general alignments are possible (depending on the interfacial alignment of top and bottom layers), i.e., parallel and antiparallel^5,9. In the parallel case, the MS typically appears as in Fig. 1a, with AA, SP, and triangular AB/BA stacking regions: their unique atomic registry is reported in Fig. 1b (SP can be considered transitory regions between AA and AB/BA domains)^5,9,10. In the antiparallel case, instead, the MS is typically hexagonal (Fig. 1e) and dominated by AA’, AB’, and BA’ regions whose atomic arrangement is shown in Fig. 1f^5,9,10.

Fig. 1. Contact-mode atomic force microscopy (AFM) topography of parallel and antiparallel moiré superlattice (MS) in a twisted-hexagonal boron nitride (t-hBN) sample.

a Sketch of a parallel alignment in a t-hBN moiré pattern emerging from an inter-layer marginal rotation of θ_TW. The four typical domains after lattice reconstruction are highlighted above the twisted rigid layers (green and yellow lattices): AA, SP, AB, and BA. b Interfacial atomic registry for the AA, AB, and BA stacking regions. c Contact-mode topography of a t-hBN parallel MS characterized by a super-hexagonal structure (white dashed line). AFM cantilever: Arrow UHF AuD, parameters: spring constant k ≈ 30 N·m⁻¹, set point ≈ 30 nm. d Height profile of the black line shown in (c). The corresponding position of each stacking domain is labelled. e Sketch of an antiparallel alignment in a t-hBN moiré pattern. The three typical domains are reported: AA’, AB’, and BA’. f Interfacial atomic registry for the AA’, AB’, and BA’ stacking regions. g Contact-mode topography of a t-hBN antiparallel MS. AFM cantilever: PPP-NCH AuD, parameters: k ≈ 35 N·m⁻¹, set point ≈ 10 nm. h Height profile of the black line shown in (g). The corresponding position of each stacking domain is illustrated. Source data are provided as a Source Data file.

In Fig. 1c, we report a contact-mode AFM image obtained on our t-hBN sample on a parallel MS area. The pattern perfectly resembles the expected features from Fig. 1a, i.e., a super-hexagonal structure (see white dashed line) with six internal triangular regions. The typical dimension (moiré wavelength) of these domains is 120 nm (Fig. 1d). Hence, we estimate a twist angle of about 0.1°⁹. Similarly, in Fig. 1g, we extend the same contact-mode experimental observation to an antiparallel MS region. Hexagonal shapes are visible, matching the expected geometry (Fig. 1e).

Despite the qualitative agreement between the expected and measured shapes for a parallel and antiparallel t-hBN MS, recent simulations performed on a t-hBN bilayer^4,15 predict the opposite topographical evolution for both scenarios. In the parallel case, AA regions are expected to have the highest topography (i.e., larger interlayer separation), AB/BA domains by the lowest, and SPs by an intermediate behaviour. Similarly, in the antiparallel case, AA’ regions are predicted to show the lowest interlayer distance, the opposite of our measurements (see Fig. 1g, h).

These two puzzling observations may be explained by considering that in contact-mode the tip is physically touching the surface of the sample. Likely, the measured inverted contrast in topography of Fig. 1d, h could be due to a stacking-dependent local deformability, not considered in simulations (carried out in absence of external forces). If local deformation is considered, this would be maximum for AA domains and minimum for AB/BA regions, with SP areas showing an intermediate trend.

Moiré stacking-dependent deformation

To rationalize the contribution of different deformability of stacking domains, we explore what happens when we change the applied force in contact-mode. If the contrast emerges from a stacking-dependent deformation, imaging at “low” forces should allow to recover the pristine/undeformed topography predicted by simulations^4,15. To test this, we image in attractive contact-mode (F = −60 nN, see SI, Section 2) a parallel and antiparallel MS area, to minimize the possible sample deformation. As shown in Fig. 2a, g, operating in the attractive forces range results in the theoretically predicted contrast¹⁵, with AA (AB’) domains higher than AB/BA (AA’) regions in the case of a parallel (antiparallel) MS. Despite this qualitative agreement, we generally expect a lower moiré contrast of the top hBN flake with respect to the actual topographical modulation happening at the twisted interface.

Fig. 2. Evolution of the topographical contrast with an increasing applied atomic force microscopy (AFM) tip force.

a–f Contact-mode images of the same twisted-hexagonal boron nitride (t-hBN) parallel moiré superlattice (MS) area probed at different tip forces. The applied forces are reported in the top-left corner of each panel. In (c), we show a representative super-hexagonal structure for the MS. g–l Contact-mode images of the same t-hBN antiparallel MS area probed at the same forces used in (a–f). m–p Cross-sections for the line profiles reported in (a–d). q Relative stacking area evolution with respect to the applied force for AA, SP, and AB/BA parallel domain. Two regimes of evolution are highlighted in yellow and blue colour. AFM cantilever for this figure (a–l) images: Arrow UHF AuD, parameters: k = 30 N·m⁻¹. Source data are provided as a Source Data file.

Instead, when we move to standard repulsive contact-mode (Fig. 2), with forces ranging from about 400 to 2000 nN (F = k·d, where d is the cantilever deflection), the topographical contrast is inverted. This is characterized, in the case of parallel MS, by low AA stacking domains (compare with Fig. 1c, d), and in the case of antiparallel MS, by high AA’ regions (compare with Fig. 1g, h). In Fig. 2m–p, the height varies along the lines shown in Fig. 2a–d, respectively. For comparative analysis, the AB/BA average heights are set to zero. Notably, while the topography contrast obtained in the attractive force range is characterized by a prominent AA region (Fig. 2m), about 15 pm above the AB/BA level (as predicted by simulations), when we apply an increasing repulsive force, the AA domain starts to be gradually compressed below the AB/BA level (Fig. 2n–p). The compression is evolving with the applied force, moving from about −10 pm in Fig. 2n (F = 420 nN), to about −30 pm in Fig. 2p (F = 1050 nN). We consider this experimental observation the fingerprint of a stacking-dependent deformability of the MS, which can only stem from a local stacking-dependent out-of-plane elasticity.

Before quantifying the local Young’s modulus, it is also worth analysing the in-plane structure of the parallel MSs presented in Fig. 2a–f. Remarkably, a clear geometrical evolution is visible, which we also prove to be reversible upon the force applied by the AFM tip (SI, Section 3). To quantify such behaviour, we calculate the relative stacking area of each parallel domain with respect to the applied force (top-left corner of each panel of Fig. 2a–f). Specifically, we first measure the area of the super-hexagon (see Fig. 1c and white dashed line in Fig. 2c) representing the MS. Then, we measure the normalized area of each stacking regions, i.e., AA, SP, and AB/BA, relative to the super-hexagon coverage. The results are reported in Fig. 2q and are addressing a non-trivial evolution characterized by two distinct regimes. A first “dynamic” regime (yellow area in Fig. 2q), where the relative stacking areas are changing, and higher applied forces increase the relevance of AA and SP regions with respect to AB/BA domains, followed by a second “static” regime (blue area in Fig. 2q), where the evolution saturates. Interestingly, the general trend reported in Fig. 2q strongly resembles the recently reported evolution of the same relative stacking areas with respect to the twist angle⁹.

To fully understand the elastic response of the moiré superlattice under stress, simulations would be essential to clarify the associated energy landscape. As a possible interpretation of these experimental results, we can conjecture the following two options: (i) the areal coverage of each stacking domain is generally determined by energy minimization. Likely, under the applied tip force, an additional mechanical energy should be considered in the energy balance, redefining the minimum. (ii) atomic reconstruction is typically considered only at the interface between top and bottom flake. However, as reported in a recent contribution²⁶, atomic reconstruction has an intrinsic 3D nature evolving along the whole top flake thickness. Accordingly, in Fig. 2a–l, we could be probing this out-of-plane geometrical evolution of the moiré superlattice due to different applied forces, i.e., indentation depths.

Moiré stacking-dependent effective out-of-plane Young’s modulus

As definitive evidence of the stacking-dependent elastic response in parallel MSs of t-hBN, we directly measure, via AFM force-distance curves, the out-of-plane Young’s modulus across all distinct stacking configurations. We select AFM force-spectroscopy (FS-AFM) technique²⁷ because of its simplicity and wide applicability on every AFM microscope. Nonetheless, the same result should be possible via bimodal AFM^28–32 or other SPM techniques for nanomechanical characterizations^33–35. FS-AFM is based on the measurement of the force applied by the tip on the sample as a function of their relative distance (force-distance curves) from which, according to specific contact mechanics models, it is possible to retrieve the out-of-plane local elasticity of the sample²⁷. In SI, Section 4, we present the main contact mechanics model used in FS-AFM and in this contribution, i.e., the Hertz model^36,37. Importantly, this model relies on a direct relationship between the tip radius R and the Young’s modulus of the sample, from which once R is known, the elasticity can be obtained, and vice versa.

We start our nanomechanical FS-AFM experiment by a proper choice of the t-hBN sample region. Visualizing the MS already in topography (Figs. 1c, g and 2a–l) allows us to easily locate the MS on the t-hBN surface with no need for other more complex SPM techniques. Figure 3a shows the contact-mode topography of the selected t-hBN area. There are two patterns along the two terraces. On the top part of the topography map (top terrace), triangles are clearly visible, which define the AB/BA domains of a parallel stacking alignment (see also Figs. 1c and 2a–f).On the bottom part of the image (bottom terrace), hexagons are present, typical of an antiparallel alignment^5,9.

Fig. 3. Moiré modulation of effective Young’s modulus in a twisted-hexagonal boron nitride (t-hBN) parallel moiré superlattice (MS).

a Contact-mode image showing two terraces separated by a trilayer sub-step (masked to avoid scale saturation to the colour palette). The bottom terrace is characterized by an antiparallel alignment (hexagonal shapes). The top terrace by a parallel alignment (triangular shapes). A circle with a radius of about 80 nm is shown around an AA domain. This corresponds to the experimental distance between force-distance curves achieved on different parallel stacking regions (SI, Section 8). Atomic force microscopy (AFM) cantilever: Arrow UHF AuD, parameters: k = 30 N·m⁻¹, deflection set point = 7 nm, R = 31 nm. b Young’s modulus histograms for AA, SP, AB, and BA domains measured by force-distance curves located at the positions highlighted by coloured stars in this figure (a). For each stacking domain, the average Young’s modulus is reported at the center of each fitted Gaussian. The reported Young’s modulus values correspond to the average obtained from approximately N ≈ 280 force–indentation curves per stacking domain; the associated uncertainty represents the standard error of the mean (σ/√N), while the width of each distribution reflects the intrinsic variability of the measurements. Among all the force-distance curves achieved in each region, we pinpoint only those characterized by a very good Hertz-fit (i.e., selecting a chi-square χ² threshold <2). These uncertainties do not include systematic uncertainty in the absolute modulus scale. The data distribution is plotted with greater detail in Fig. S6. Source data are provided as a Source Data file.

The choice of this t-hBN area is convenient since it provides in the same scanned area both interfacial alignments (parallel and antiparallel). As already reported in literature, such parallel-to-antiparallel transition typically happens when the top crystal of hBN lies on an underneath flake presenting a monolayer-step^9,10. This is true also for a trilayer-step as in our case (SI, Section 5).

Notably, the central area of the antiparallel hexagons corresponds to AA’ domains^5,10. This is the natural stacking for bulk hBN^4,10, therefore justifying the assumption of the same vertical elasticity, i.e., 34 GPa³⁸. Hence, we first perform force-distance curves on this AA’ stacking to determine the tip radius R from the known elasticity value, via the Hertz model (SI, Section 4). Then with the known value of R (31 nm), we repeat the FS-AFM measurements on the parallel stacking domains (AA, SP, AB, and BA) with unknown Young’s modulus. In Fig. 3a, we mark the central position of all these stacking regions (AA’, AA, SP, AB, and BA) by coloured stars. Figure 3b shows the statistical analysis emerging from the FS-AFM experiment. For each domain, we provide the corresponding histogram of the measured Young’s modulus. Despite the similarity and partial overlap between the four histograms, the mean values show a systematic evolution across the stacking domains. Specifically, AA regions (blue star) are the softest, with an average elasticity of (34.0 ± 0.3) GPa. Then we find the SP regions (green star), with a mean elasticity of (35.9 ± 0.4) GPa. Finally, AB and BA domains (brown and violet stars) show basically the same Young’s modulus of about 38 GPa (with a tiny difference possibly related to opposite polarization effects³⁹). We note that the absolute Young’s modulus values extracted from Hertzian analysis are subject to systematic uncertainties arising from tip radius calibration, the choice of reference modulus, and the assumptions of the contact mechanics model. These effects can introduce deviations on the order of tens of percent in the absolute values, as also illustrated by independent calibration measurements (SI, Section 8). However, since all measurements are performed using the same cantilever and analysis protocol, these systematic contributions are expected to act primarily as a common scaling factor and therefore are not expected to alter the relative comparison between different stacking domains. Accordingly, the key result of this work is the reproducible modulation of the effective out-of-plane elastic response across the moiré superlattice. We further note that the reported uncertainties correspond to the statistical error on the mean (σ/√N), which reflects the precision of repeated measurements. This quantity is significantly smaller than the width of the measured distributions (σ), which instead captures the intrinsic variability of the force–indentation data and is not reflected in the reported error bars. Accordingly, the results are interpreted in terms of relative trends across stacking domains rather than precise absolute values. We further support this conclusion in SI, Section 6, where we report a one-way ANOVA followed by Tukey HSD post-hoc tests on the effective Young’s modulus distributions of the four stacking domains. In SI, Section 7, we report the average force-indentation curves for each stacking region, from which we can retrieve the same quantitative conclusion. In SI, Section 8, we confirm the precision of the AFM x-y piezo in performing hundreds of force-distance curves on different domains that are typically less than 80 nm apart (see Fig. 3a).

We stress that the vertical elasticity modulation along the parallel MS is a consequence of the atomic relaxation, which is changing the local atomic registry for each stacking domain⁵. Starting from a natural AA’ stacking, characterized by a Young’s modulus of 34 GPa, we therefore observe that, relative to AA′, the effective modulus increases by up to about 10% in the AB/BA regions and decreases by about 5% in the AA domains. These results are in line with a recent study focusing on a twisted graphene sample³³, although some dependence on twist angle and material may be expected. By contrast, we experimentally find that the top-layer thickness does not play a significant role in the out-of-plane elastic modulation: repeating the same AFM nanomechanical analysis on an independent twisted hBN sample with a thinner top layer (≈5 nm) yields an essentially identical stacking-dependent Young’s modulus modulation (SI, Section 9). This reproducibility supports the robustness of the relative stacking-dependent modulation despite systematic uncertainties affecting the absolute modulus scale.

Finally, we do not expect any influence on the reported elasticity quantification (Fig. 3b) from the possibility of a sp2-to-sp3 hBN rehybridization³⁸. This effect, indeed, strongly depends on the hBN total thickness and is showing a maximum impact for few layers of hBN, only. This is a completely different regime with respect to our actual sample thickness, roughly 220 nm (SI, Section 1).

Having shown that the effective out-of-plane elasticity is not constant across the stacking regions, we can rationalize the results shown in Figs. 1 and 2 as a direct consequence of the different local elastic response of the t-hBN stacking domains to the applied force in contact-mode. As anticipated, AA and SP domains are slightly softer than AB/BA regions. Therefore, they deform more upon an applied force. This tip force-induced deformation effectively provides an inverted topographical contrast in the t-hBN MS with respect to the numerical predictions^4,16.

AFM thermal noise screening of moiré topographical contrast

Finally, we return to Fig. 1c, g, from which this investigation started. Several recent contributions explicitly reported no experimental detection of MS details in the SPM topography of different t-LMs^10,20–22. To investigate this issue further, we image the same t-hBN antiparallel MS region with four different cantilevers, each characterized by a spring constant varying from about 0.5 to 20 N·m⁻¹ (Fig. 4). During this contact-mode imaging, the setpoint was fixed to keep the applied tip force constant (F = 100 nN). In Fig. 4a, we illustrate the topography map for the softest cantilever (0.56 N·m⁻¹). The image appears flat, with no moiré details captured by the AFM probe. Then we start increasing the spring constant by changing the cantilever. In Fig. 4b, the image is obtained with a k = 3.6 N·m⁻¹ cantilever. Faint moiré topographical features start to appear. Figure 4c, d confirm the same trend: higher spring constants (k = 7.6 and k = 20 N·m⁻¹, respectively) provide a progressively increasing moiré topographical contrast.

Fig. 4. Cantilever spring constant-dependent imaging of a twisted-hexagonal boron nitride (t-hBN) anti-parallel moiré pattern.

a–d Contact-mode images of a t-hBN antiparallel moiré superlattice (MS) obtained with cantilevers of k = 0.56, 3.9, 7.6, and 20 N ∙ m⁻¹, respectively. Below each topography image the corresponding root mean square (RMS) is reported. Atomic force microscopy (AFM) cantilevers: Scanasyst-air-hr (a), PPP-FM AuD (b), PPP-NCSTAuD (c), Arrow-UHF AuD (d). In (a), we have used a different z-scale bar in order not to saturate the image. Source data are provided as a Source Data file.

This set of contact-mode measurements demonstrates that stiff cantilevers enhance the topography contrast of a MS. The reason for such a behavior lies in the spring constant-dependent thermal noise⁴⁰ (TN) of the cantilever. TN is ubiquitous in AFM images, since it emerges from the unavoidable interaction between the cantilever and the thermal bath of molecules (ambient) in which the probe is immersed⁴⁰.

Remarkably, TN plays an important role in the AFM topographical imaging only if its order of magnitude matches the extent of the height variations measured. In contact-mode, TN can be calculated through the equipartition theorem applied to a cantilever probe that is supported by a hard surface^17,40:

$$TN\equiv \sqrt{<\mathrm{\Delta }{z}_{th}^2>}=\sqrt{\frac{k_{b}T}{3}}\cdot k^{-0.5}$$ 1

with $<\mathrm{\Delta }{z}_{th}^2>;$, $k_b$, and $T$ equal to the average of the squared thermal fluctuations in the cantilever deflection, the Boltzmann constant and the temperature, respectively. Below each panel of Fig. 4, we report the corresponding experimental topographical root-mean-square (RMS) (the RMS of the height profile). Clearly, increasing the cantilever stiffness decreases the topographical RMS. In SI, Section 10, we plot this experimental RMS data as a function of k. A power law fitting, i.e., $y=a\cdot x^b$, yields an exponent b = −0.45 ± 0.09. The agreement with the theoretical exponent of b = −0.5 (see Eq. (1)) is remarkable. In SI, Section 10, we also show how this agreement (related to the b-exponent) is independent from the actual method used to flatten the images of Fig. 4.

From this analysis, we conclude TN is the dominant source of noise in the contact-mode images shown in Fig. 4.

In this work, we revealed that the effective out-of-plane Young’s modulus of t-hBN is modulated by the moiré superlattice, providing a quantitative nanomechanical view of stacking-dependent elasticity in layered materials. Leveraging this elastic modulation, we successfully observed (parallel and antiparallel) MSs directly in the topography channel of contact-mode AFM images in air-ambient and at room temperature, going beyond already moiré topographical imaging in ultra-high vacuum at low temperatures by scanning tunnelling microscopy^41,42. Our AFM results identify TN as the main reason why such topographical information has not previously been reported in the literature to date. Contact-mode AFM does not require any sample perturbation (e.g., applied voltages) to visualize the MS. Additionally, samples’ elasticities significantly influence other SPM images, such as those obtained by piezoresponse force microscopy⁴³. A careful quantification of their magnitude on moiré patterns could facilitate the acquisition of cross-talk-free images of t-LMs, enhancing the reliability of nanoscale measurements.

Finally, the discovery that a local force can trigger a change of stacking, effectively modulating the MS geometry for both parallel and antiparallel alignments, is extremely intriguing for realizing tunable devices based on t-LMs. This becomes even more compelling when considering that such modulation can be reversible, as demonstrated in our experiment where the t-hBN MS was locally reconfigured through the elastic response to nanoscale pressure exerted by an AFM tip. This finding not only enables the exploration of the elastic-electrostatic energy balance of moiré patterns³⁹, but also provides a tool for real-time modification of t-LMs functional properties, such as ferroelectricity in the case of t-hBN^4,10,14,44. Furthermore, we foresee that such a framework could be extended to other t-LMs, potentially enabling optical property modulation and expanding the range of applications of t-LMs.

Methods

Sample fabrication

Twisted hBN samples are fabricated by micromechanical exfoliation of bulk hBN crystals grown by the atmospheric pressure high temperature method, as previously described^45,46. Briefly, the process starts by mixing high purity elemental boron with nickel and chromium with mass ratios of 4:48:48, respectively. The mixture is then heated at 200 °C/h under nitrogen and hydrogen at partial pressures of 787 torr and 60 torr, respectively, to 1550 °C, to produce a homogeneous molten solution. After 24 h, the solution is slowly cooled at 1 °C/h, to 1500 °C, then at 50 °C/h to 1350 °C, then 100 °C/h to room temperature. The hBN solubility is decreased as the temperature is reduced, causing crystals to precipitate on the surface of the metal. The hBN flakes were exfoliated from the metal with thermal release tape. The free-standing hBN crystalline flakes are typically greater than 20 μm thick.

The exfoliated flakes are deposited onto Si substrates with a 90 nm SiO₂ layer using micromechanical cleavage (MC). To control the interlayer twist angle, we identify either large flakes (> 50 μm) selectively torn during transfer⁴⁷ or adjacent hBN flakes originating from the same parent crystal during MC, based on the orientation of their faceted edges as observed under optical microscopy.

Twisted hBN heterostructures with controlled θ_TW, are assembled using polycarbonate (PC) films supported on polydimethylsiloxane (PDMS) stamps⁴⁷. The PC/PDMS stamp is brought into contact with the hBN-covered substrate at 110 °C using a micromanipulator, ensuring partial contact with a target flake or a pair of neighboring flakes. Upon retraction, the contacted portion is lifted onto the PC film. The twist angle θ_TW (accurate to ±0.01°, as limited by the resolution and wobble of the rotation stage) is then applied by rotating the sample stage. The flake on PC is subsequently aligned to the second flake and brought into contact at 110 °C and then subsequently dropped down at 180 °C. Residual PC is removed by sequential immersion in chloroform for 30 min followed by annealing at 450 °C on a hot plate.

Atomic force microscopy

AFM measurements were performed at room temperature (~ 23 °C, RH ~ 40%), in air-ambient, using a Multimode 8 (Bruker) microscope. The microscope is equipped with an active vibration isolation system (TS-150). We used different cantilevers for AFM imaging/spectroscopy: Scanasyst-air-hr (Bruker), PPP-FMAuD (NanoAndMore), PPP-NCSTAuD (NanoAndMore), PPP-NCHAuD (NanoAndMore), Arrow-UHFAuD (NanoAndMore). Both tip and sample were electrically grounded during the AFM measurements, ruling out possible electrical contributions to the imaging. To avoid damaging the tip, the cantilevers sensitivity calibration procedure was performed at the end of the experiments. The deflection sensitivity was obtained by performing 10 force-distance curves on mica (without changing the laser spot position onto the cantilever) and calculating the average inverse slope in the contact region. The spring constants were calculated via the TN method. Despite the air-ambient AFM imaging/spectroscopy, any contribution coming from capillary tip-sample interactions was considered negligible due to applied forces typically in the range of hundreds of nN, much larger than standard capillary forces of at least one order of magnitude less^17,48.

All the AFM image processing was performed by Gwyddion⁴⁹. For the calculation of the RMS in Fig. 4, all the four images were flattened in the same way: plane subtraction, align rows and 2nd order polynomial correction.

Author contributions

A.A. initiated and supervised the project. S.C. performed the experimental work, developed the model and analysed the data. L.N. prepared the samples. S.C., L.N., G.V., and A.A. discussed the data. T.P. and J.H.E. were responsible for the hexagonal boron nitride crystal growth. All authors contributed to the final manuscript.

Peer review

Peer review information

Nature Communications thanks Ruben Millan-Solsona and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work has been financially supported by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme “METAmorphoses”, grant agreement no. 817794 (awarded to A.A.). T.P. and J.H.E. acknowledge support for hBN crystal growth from the U.S. National Science Foundation, award number 2413808.

Data availability

The Source Data files can be accessed at https://doi.org/10.5281/zenodo.21626883.

Competing interests

The authors declare no competing interests

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-76466-8.