## Introduction

Periodic modulation of interlayer stacking in two-dimensional van der Waals (vdWs)-based systems provide a unique control of their physical properties1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18 not available in commonplace epitaxial heterostructures. Prominent examples include twisted bilayer graphene (TBG), graphene on lattice-mismatched hBN substrate1,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18, and hetero-bilayers of transition metal dichalcogenides9,19. In particular, the moiré superlattice in TBG at the “magic” twist angle θ ≈ 1° gives rise to strong electron correlations and superconductivity2,20. On the other hand, a network of solitons forming in TBG at smaller θ2,11,13,14,21,22, radically alters its electronic13,21,22,23,24 and plasmonic14,23,25 properties. Here we report on nano-imaging of soliton superlattices in hexagonal boron nitride (hBN). The solitons appear to originate at a misfit atomic plane located ~15 nm beneath the surface of hBN crystals. Utilizing scattering-type scanning near-field optical microscope (s-SNOM), we uncovered that networks of these buried soliton superlattices are registered in infrared (IR) spectral features associated with dipole-active phonons of hBN. We modeled the near-field IR contrast of the solitons in terms of local hardening and broadening of the phonon modes, which we related to the distribution of the elastic strain in the system.

## Results

### Topography of hBN domain patterns

We begin with the topographic images of the hBN domain patterns (Fig. 1a–e) obtained with the atomic force microscope (AFM). The patterns extend over macroscopic areas (>104 μm2) and have periods Λ1, Λ2 ~ 300–800 nm. The domains vary from nearly perfectly hexagonal (Fig. 1b) to highly distorted, diamond-like motifs (Fig. 1e). The domain boundaries are demarcated by the dips in the topography, which are about w ~ 90 nm wide (Fig. S2 of Supplementary Information). Macroscopic domain patterns are not impacted by defects on the sample surface such as wrinkles26: the white lines of elevated topography in Fig. 1a–d. Similarly, the domain boundaries do not disturb the 14-nm-periodic moiré pattern formed by monolayer graphene deposited on top of hBN (Fig. 1e, inset). Both observations suggest that the long-range superlattices form at some depth d below the surface of the hBN crystal. Based on a theoretical model for the strain distribution around a soliton, we estimate d ~ w/7 ~ 15 nm (Supplementary Information Section 2).

### Origin of hBN soliton superlattices

For an in-depth look at the solitons, we focus on two representative cases (Fig. 1b, e). The locations of the domain-wall junctions in these images, connected by straight lines, are plotted in Fig. 2c–h. These latter plots reveal somewhat deformed lattices whose primitive periods Λ1, Λ2 (Fig. 1c) vary in space. We can relate such variations to the interlayer strain and rotation in the system, or more precisely, to the averaged (coarse-grained) strain tensor $$\bar u_{\alpha \beta }$$ and twist angle $$\bar \theta$$. Figure 2 illustrates the calculated color maps of $$\bar u_{\alpha \beta }({\mathbf{r}})$$ and $$\bar \theta \left( {\mathbf{r}} \right)$$ superimposed on the soliton meshes. Here α, β {x, y} and r = (x, y) is the in-plane position. Note that the angles at all the soliton junctions are close to 120°, which is analogous to the Plateau law of foam films27 (see Supplementary Information Section 3 for details). For domains shaped as unilateral hexagons, we find an isotropic and predominantly tensile average strain $$\bar u_{xx},\bar u_{yy} \sim 0.05\%$$ with the average twist angle $$\bar \theta \sim 10^{ - 4}\,{\mathrm{rad}} \approx 0.01^\circ$$. For diamond-like domains, the strain is anisotropic, almost uniaxial. A couple of remarks on these results are in order. First, the coarse-grained strain $$\bar u_{\alpha \beta }$$ should not be confused with the local strain uαβ, which is to be discussed below. Second, the calculation of $$\bar u_{\alpha \beta }$$ and $$\bar \theta$$ requires as an input the Burgers vector bj of the solitons. These vectors connect the nearest identical atoms (e.g., borons) in the same atomic plane. They have magnitude b = 0.25 nm each and angular directions separated by 120°. However, since our AFM does not resolve the hBN crystal orientation, we do not know these directions. To generate Fig. 2 we chose one of these directions along the x-axis of the plot (Fig. 2a, b). If the Burgers vectors are rotated through some common angle, qualitatively similar $$\bar u_{\alpha \beta }({\mathbf{r}})$$ and $$\bar \theta \left( {\mathbf{r}} \right)$$ maps are obtained.

### Nano-IR imaging of hBN superlattices

We proceed to the survey of the results obtained by the IR nano-imaging. We have carried out these scanning nano-spectroscopy experiments in the frequency region of hBN phonon polaritons28,29,30,31,32. In our s-SNOM apparatus the metalized AFM tip was illuminated by IR light thus generating a strong enhancement of the electric field underneath the tip (Fig. 3a)4,33. Such an antenna-based nano-IR setup solves the problem of the photon-polariton momentum mismatch33,34,35 and enables local spectroscopy of polariton modes in hBN with ~25 nm spatial resolution (see the “Methods” section). Employing tunable quantum cascade lasers, we acquired nano-IR images at more than 50 different discrete frequencies. We found that the domain patterns revealed by the AFM topography are also prominent in the nano-IR images (Fig. 3b–d). We obtained nano-IR data with both monochromatic tunable quantum cascade lasers (Fig. 3b–d, f) and also broadband difference frequency generation sources (Fig. 3e) using Fourier transform spectroscopy. The strongest contrast is observed at frequency ω = 1368 cm−1 (Fig. 3d), which corresponds to the phonon polariton resonance of hBN (Fig. 3e). The contrast systematically weakens as ω is shifted away from ω = 1368 cm−1. In all the images the back-scattering amplitude s(ω, r) is enhanced at the centers of the domains (s = sd(ω)) and reduced at the solitons (s = ssol(ω)). We therefore conclude that the lattice dynamics is modified near the solitons where the strain is concentrated. The frequency dependence of the ratio sd(ω)/ssol(ω) highlights subtle yet systematic variations of this modified lattice response. This ratio exhibits an asymmetric maximum, with a broader low-frequency side (Fig. 3f).

### Elucidating nano-IR response

Quantitative modeling of the nano-IR response is a challenging task that goes beyond the scope of the present work. To analyze the sd(ω)/ssol(ω) spectra, we restricted ourselves to the simplified approach, similar those in recent nano-IR study of wrinkled and strained hBN crystals26 and an earlier work on nano-indented SiC36. We assumed that the hBN crystal can be described by a permittivity tensor εα(ω, r) varying only as a function of in-plane coordinates. Here α = (||) is the in-plane (out-of-the-plane) component. We further assumed that the near-field amplitude s(ω, r) depends only on the local value of εα(ω, r). We adopted the standard Lorentzian model for the hBN permittivity,

$$\frac{{\varepsilon _\alpha \left( \omega \right)}}{{\varepsilon _{\infty ,\alpha }}} = 1 + \frac{{\omega _{{\mathrm{LO}},\alpha }^2 - \omega _{{\mathrm{TO}},\alpha }^2}}{{\omega _{{\mathrm{TO}},\alpha }^2 - \omega ^2 - i\omega {\mathrm{\Gamma }}_\alpha }},$$
(1)

where ωTO,α (ωLO,α) is the transverse (longitudinal) optical phonon frequency and Γα is the broadening35, which we treated as adjustable parameters. We numerically simulated the near-field scattering amplitude s(ω) modeling the tip as an elongated conducting spheroid and taking into account the presence of the quartz substrate underneath hBN10. Through these simulations we found that the nano-IR contrast can be attributed to the broadening of the phonon resonance, from Γ = 6.5 cm−1 at the domain centers to 7.35 cm−1 at the walls. The dashed green trace in Fig. 3f illustrates the effect of this extra broadening alone. In addition, a minute blue shift of ωTO, from 1365.5 to 1365.8 cm−1 helps to better account for the spectral form of sd(ω)/ssol(ω), as shown by the red line in Fig. 3f (for more details of these fits, see Supplementary Information Section 4).

Our analysis of the nano-IR line-form in Fig. 3 is phenomenological. The microscopic approach can be developed by relating the phonon frequency shift ΔωTO, to the local strain. Notably, the frequency shift caused by the strain is not a single number. As shown by recent first-principles calculations26, a uniaxial strain splits the degenerate TO mode of hBN into two separate modes of orthogonal in-plane polarization. Moreover, from symmetry consideration we can predict that for an arbitrary strain, the fractional frequency shifts of these two modes should be:

$$\frac{{{\mathrm{\Delta }}\omega _{{\mathrm{TO}}, \bot }}}{{\omega _{{\mathrm{TO}}, \bot }}} = - \frac{{A + B}}{2}\left( {u_{xx} + u_{yy}} \right) \pm \sqrt {\left[ {\frac{{A - B}}{2}\left( {u_{xx} - u_{yy}} \right)} \right]^2 + \left( {Cu_{xy}} \right)^2}$$
(2)

where A, B, C ~ 1are constants (Supplementary Information Section 5). Effectively, the strain turns hBN from the uniaxial hyperbolic material into a bi-axial one, analogous to molybdenum trioxide37,38. Because of different polarization, the two modes have different coupling to the in-plane field, which complicates the modeling. Even more arduous task is to take into account the realistic three-dimensional strain distribution around the solitons. The dependence of the strain on the in-plane coordinates is relatively weak because the characteristic width w ~ 90 nm of the strained region near the surface is large (Figs. 1 and S2). Nevertheless, the dependence of the strain on the depth z is strong because of the anisotropic character of the strain (Fig. 3a and Supplementary Information Section 2). Leaving a quantitative study for future work, we limit ourselves to the following estimate. Qualitatively, the nano-IR amplitude measured above the soliton can be viewed as the sum of signals from all the underlying hBN layers, each with a shifted (and split) ωTO,. Assuming the characteristic strain variation of δu ~ b/w ~ 0.3%, the nano-IR line-shape is expected to acquire an additional inhomogeneous broadening ΔΓ ~ ωTO,δu ~ 4cm−1. This estimate is of the same order of magnitude as the fit parameter quoted above.

## Discussion

The confinement of both the topographic and nano-IR contrast to narrow regions along the domain walls is consistent with the notion that the individual layers in hBN are not rigid but instead behave as deformable atomic membranes prone to incommensurate–commensurate transitions. The key signature of these transitions is the formation of structural solitons where mechanical strain is accumulated11,12,13,14,15. Previously, the transmission electron microscopy and second harmonic generation experiments have identified isolated solitons39 in few-layer-thin hBN specimens. Our work demonstrates that the solitons can form in the interior of a bulk vdW crystal, not just at the surface or in few-layer systems, and that they can organize themselves in large regular superlattices. We have also demonstrated that nano-IR imaging can be applied to map the local strain field in a polar crystal26,36.

The observed residual strain concentrated in the solitons, could originate from multiple sources, one of which is the difference in the thermal expansion coefficients of hBN and graphene deposited on top of it (−8 × 10−6/K for graphene40 and −3 × 10−6/K for hBN41). A rough estimate of the lattice mismatch that could develop during the cooling of the sample is [(−8)–(−3)] × 10−6/K×(1150–20) K = 0.6%, which is close to the critical mismatch for the commensurate–incommensurate transition in an hBN bilayer42.

Finally, we mention several predictions testable by atomic-resolution or stacking-sensitive probes. Unlike TBG, where the lowest-energy stacking is AB stacking and the domains are triangular1,6,14,18, the hexagonal domains in hBN are of AA′ type (Figs. 2a and 3e), the lowest-energy stacking in hBN43. For the domain-wall junctions in hBN, we expect two distinct stacking types, AB1′ and AB2′, that alternate in space (Fig. 2a). Future studies may also seek to observe changes in the electronic structure in hBN moiré superlattices44. More broadly, it will be fruitful to explore in operando tuning of the twist angle45 and attendant lattice, electronic2,46, plasmonic15 and possibly excitonic8,9 responses of various vdW materials.

## Methods

### Sample synthesis

The hBN flakes were prepared on quartz substrates using mechanical exfoliation. hBN/Quartz samples were then transferred into chemical vapor deposition chamber and annealed at 1150 °C at low pressure with continuous argon flow of 50 standard cubic centimeters per minute (s.c.c.m.) for 30 min. After that, CH4:H2 at 5:5 s.c.c.m. were injected into the chamber for 300 min at pressures below 12 mbar—a process used for graphene growth39. Finally, samples are cooled to room temperature in argon flow.

### Nano-IR imaging

The IR nano-imaging experiments were performed using s-SNOM (neaspec.com) equipped with continuous wave mid-IR quantum cascade lasers (daylightsolutions.com). The s-SNOM is based on AFM with curvature radius ~ 25 nm operating in the tapping mode with a tapping frequency around 270 kHz. A pseudo-heterodyne interferometric detection module was implemented to extract both the scattering amplitude s and the phase of the near-field signal. In the current work, we discuss the amplitude of the signal. In order to subtract the background signal, we demodulated the near-field signal at the third harmonics of the tapping frequency. All the IR nano-imaging experiments were performed in ambient conditions. We used quantum cascade lasers with tunable frequency and a broad-band difference frequency generation laser systems.