# Moiré pattern multiplicity driven by electronic effects in two-dimensional CrCl3/Au heterostructures

Eugenio Gambari, Hugo Le Du, Mathieu Lizée, Arindam Mukherjee, Laurent Limot, Fabrice Scheurer, Marie d'Angelo, François Debontridder, Tristan Cren, Marie Hervé

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Eugenio Gambari, Hugo Le Du, Mathieu Lizée, Arindam Mukherjee, Laurent Limot, et al. Moiré pattern multiplicity driven by electronic effects in two-dimensional CrCl3/Au heterostructures. 2025. [hal-05415372v1](https://hal.science/hal-05415372v1)

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Preprint submitted on 14 Dec 2025 (v1), last revised 21 Dec 2025 (v2)

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Moiré pattern multiplicity driven by electronic
effects in two-dimensional CrCl<sub>3</sub>/Au
heterostructures.

Eugenio Gambari,† Hugo Le Du,† Mathieu Lizée,‡ Arindam Mukherjee,† Laurent

Limot, ¶ Fabrice Scheurer, ¶ Marie D'angelo,† François Debontridder,† Tristan

Cren,† and Marie Hervé*,†

†Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, 4 place
Jussieu, Paris, 75005, France

‡Fritz Haber Institute of the Max Planck Society, Berlin, Germany

¶ Université de Strasbourg, CNRS, Institut de Physique et Chime des Matériaux de Strasbourg, UMR7504, Strasbourg, F-67000, France

E-mail: marie.herve@sorbonne-universite.fr

Abstract

Moiré patterns are a central motif in van der Waals heterostructures arising from
the superposition of two-dimensional (2D) incommensurate lattices. These patterns
reveal a wealth of correlated effects, influencing electronic, magnetic, and structural
phenomena. While diffraction techniques typically resolve multiple moiré wave-vectors
corresponding to the incommensurate nature of the underlying lattices, Scanning Tun-
neling Microscopy (STM) often reveals only a dominant superperiod. In this work,
we address this apparent discrepancy through an STM study of a twisted monolayer of
CrCl₃ on Au(111). We observe the coexistence of several moiré patterns at a fixed twist
angle, whose relative intensity depends on the tunneling bias. Fourier analysis of STM
data uncovers hidden higher-order moiré components not visible in STM topographic
images, while spectroscopy maps reveal that the spectral weight of each pattern varies
with electron energy. Our results establish that STM selectively probes on the same
area distinct moiré modulations depending on electronic confinement, providing a uni-
fied framework that reconciles real space and reciprocal space observations of complex
moiré superstructures.

Introduction

Moiré patterns in two-dimensional materials are ubiquitous. They are found in van der Waals heterostructures such as Gr/Ir, Gr/Pt, ¹⁻⁵ MoS₂/Au⁶,⁷ when the lattice parameters of the two materials are incommensurate, or in twisted bilayers of van der Waals materials such as graphene, CrI₃, or WSe₂. ⁵,⁸⁻¹⁶ For specific orientations, moiré patterns can lead to the emergence of flat bands ¹⁷,¹⁸ giving rise to many exciting physical phenomena. Recently, superconductivity has been discovered in materials that are not intrinsically superconducting, such as twisted bilayer graphene or WSe₂. ¹⁰,¹⁴ Wigner crystals have been observed in moiré lattices of WSe₂/WS₂. ¹⁶ Non-collinear magnetic orders have been reported in materials in which they are usually collinear, such as in twisted bilayers of CrI₃. ¹¹⁻¹³

Moiré patterns are a direct manifestation of the quasiperiodic structure induced by the
superposition of two incommensurate periodic lattices. In reciprocal space this leads to an
infinite number of satellite peaks surrounding the Bragg peaks, as observed by diffraction
techniques. 2,7,19 However, other experimental probes accessing to the real space order, such
as scanning probe microscopy techniques, tend to reveal a single dominant moiré pattern
that is commonly interpreted as a commensurate superperiod. 20–25

The use of a periodic superlattice description in *ab-initio* calculations is also quite com-
mon. For example this is used for the prediction of a periodic non-collinear magnetic order
in twisted bilayers of CrI<sub>3</sub>.<sup>26</sup> It sometimes happens that the same system studied by diffrac-
tion and STM leads to contradictory observations, as in the case of MoS₂/Au.⁷ The two approaches, in real and reciprocal space, seem to differ conceptually. In this paper, we link them through a real space study exhibiting the coexistence of several moiré patterns in a same location. In particular, our analysis reveals that STM is selectively sensitive to different coexisting moiré patterns, depending on the energy of the probed electrons.

When examining a moiré pattern arising from the superposition of two 2-dimensional periodic lattices, as observed in van der Waals heterostructures, the mathematical representation of the moiré pattern in real space can become quite complex, especially when higher-order moiré patterns are involved.¹⁻³,⁵,⁸,¹⁹,²⁷ In contrast, the description in reciprocal space is quite straightforward,¹,⁸,²⁸ a moiré pattern is generated when two periodic functions with spatial frequencies $Q_1$ and $Q_2$ are multiplied together. This yields a beating frequency $Q_{\text{moiré}} = Q_1 - Q_2$, corresponding to the moiré wave vector. For two 2-dimensional atomic lattices, the general reciprocal lattice vectors of lattice 1 and 2, $Q_1^{i,j}$ and $Q_2^{k,l}$ can be expressed as linear combinations of the primitive reciprocal lattice vectors $Q_1^{1,0}$, $Q_1^{0,1}$, $Q_2^{1,0}$ and $Q_2^{0,1}$, as:

$$Q_1^{i,j} = iQ_1^{1,0} + jQ_1^{0,1} \qquad Q_2^{k,l} = kQ_2^{1,0} + lQ_2^{0,1} \qquad (1)$$

This leads to an infinite number of possible Fourier components of the moiré patterns, where the wave vectors can be expressed as:

$$Q_{\text{moiré}}^{i,j,k,l} = Q_1^{i,j} - Q_2^{k,l} \qquad (2)$$

Electron diffraction experiments often reveal multiple moiré patterns in the Fourier space for a given specific orientation between the atomic lattices²,⁷,¹⁹ whereas STM experiments often identify a single dominant moiré pattern.²,⁷ A possible reason for that could be that electron diffraction integrate over large surface areas while STM reveal a local order. Pham et al.²² showed that the moiré contrast observed in STM topographs depends on the tunneling
bias, suggesting that the modulation is electronic rather than structural. However, STM topography alone are not straightforward to interpret in real space since several distinct moiré modulations may coexist at the same energy and within the same real-space region, with often one dominant component that hides less intense modulations. We will show that an efficient way to reveal the coexistence of multiple moiré patterns consist in combining energy-resolved spectroscopy with Fourier-space analysis. Here, we study a CrCl₃ monolayer on Au(111) and show that the moiré landscape is far richer than what topography alone reveals. Fourier analysis of STM data uncovers hidden higher-order moiré components not visible in STM topographic images, while spectroscopy maps reveal that the spectral weight of each pattern varies with electron energy. Our results establish that STM selectively probes on a same area distinct moiré modulations depending on electronic confinement. These findings provide a unified framework for complex moiré superstructures, and, more generally, imply that the emergent electronic, magnetic, and correlated properties of moiré materials cannot be fully understood within a single-period description.

## Results and discussion

CrCl₃ monolayer islands were grown on a clean Au(111) substrate by molecular beam epitaxy (MBE).⁸ Figure 1a displays a large-scale STM topography of the surface. The bare atomic terraces of the Au(111) substrate are partially covered by CrCl₃ monolayer islands, presenting a width larger than 100 nm. These islands can grow with various crystalline orientations with respect to the Au(111) surface, leading to the formation of different moiré patterns (see Supporting Information S1). We can sort the island orientations into two main sets, labeled as α and β. The dense crystallographic directions of the CrCl₃<sup>α</sup> islands (Figure 1a) are aligned with those of the underlying Au(111) lattice (<11̄0><sub>Au</sub>), while the CrCl₃<sup>β</sup> islands (Figure 1b) are twisted by about 30°. In the CrCl₃<sup>α</sup> islands, a dominant moiré pattern with a periodicity of 6.2 nm is observed (Figure 1a). By contrast, a much shorter periodicity of approximately
1.6 nm is observed in $CrCl_3^β$ islands (Figure 1b). In a prior study, we conducted a detailed analysis of the moiré pattern and the associated topological defects found in the $CrCl_3^α$ islands.⁸ In the present work, we focus on the $CrCl_3^β$ islands, showing that the observed superstructure stems from the superposition of multiple moiré patterns. Figure 2 shows a detailed analysis of the moiré patterns in $CrCl_3^β$ islands. The crystallographic orientation of these domains varies slightly from island to island, between about 30° and 35° relative to the <110>$_{Au}$ direction. Two limiting cases are shown in Figures 2a and 2b: one island showing a 30° twist and the other a 35° one. For these configurations, as well as for intermediate angles, the observed superstructures arise from the superposition of multiple moiré patterns, as revealed by reciprocal space analysis. Figures 2c and 2d are obtained by performing the 2D Fourier Transform (FT) of the STM topographies in Figures 2a and 2b, respectively.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** STM topography of $CrCl_3$ islands deposited on Au(111). (a) Large scale STM topography displaying the two types of $CrCl_3$ islands: $CrCl_3^α$ and $CrCl_3^β$. Their atomic lattices are twisted by 30° with respect to each other and their surfaces show two distinct moiré patterns. (b) STM topography of a $CrCl_3^β$ island, the moiré pattern has a periodicity of 1.6 nm. Tunneling parameters: (a) V = 1.2 V, I = 100 pA; (b) V = 1.4 V, I = 100 pA.</figcaption></figure>

These FT images reveal numerous spots. Figures 2e and 2f present the same Fourier-transform images as in Figures 2c and 2d, with Bragg spots and moiré satellites highlighted. Red circles indicate the main Bragg spots corresponding to the dense atomic directions of $CrCl_3$, while black circles denote higher-order Bragg spots rotated by 30° relative to the dense
atomic axes. The remaining spots correspond to satellites from multiple moiré patterns.
The superposition of the CrCl₃ and Au(111) hexagonal lattices generates a hexagonal moiré
superlattice in real space, which manifests in reciprocal space as six satellite spots forming
a hexagon around each CrCl₃ Bragg peak. In Figure 2e, where the CrCl₃ lattice is twisted
by 30° relative to <110>ₐu, three moiré patterns (highlighted by green, red, and black
hexagons) are visible both at the Brillouin zone center and around the Bragg peaks. For a
domain twisted by 35° (Figure 2f), four distinct moiré patterns appear, marked by orange,
pink, green, and red hexagons. Each vertex of the moiré hexagons in the FT images originates
from the conjunction of one CrCl₃ and one Au(111) Bragg spot, denoted (i,j)CrCl₃ and
(k,l)ₐu. To identify which Bragg spots generate a given moiré component, we superimposed
the reciprocal lattices of CrCl₃ (violet) and Au(111) (blue), as shown in Figures 2g and 2h. In
Figure 2g, the CrCl₃ lattice is twisted by 30° relative to <110>ₐu, and by 35° in Figure 2h.
This angular relationship is illustrated by the brown dashed arrows indicating the Q<sub>Au</sub><sup>1,0</sup> and
Q<sub>CrCl₃</sub><sup>1,0</sup> directions. For clarity, only one quadrant of reciprocal space is shown; the complete
construction is provided in the Supporting Information S2. By measuring the size and
orientation of a given Q<sub>moiré</sub> in Figures 2e and 2f, we can determine which pair of Bragg
vectors, from Au and CrCl₃, is involved. As highlighted by green hexagons in Figures 2e
and 2f, the dominant moiré pattern arises from the same combination of Au(111) and CrCl₃
lattice vectors, although its size and orientation differ between the two domains. In the 30°
twisted domain (Figure 2e), the green highlighted moiré spots are aligned with the higher
order CrCl₃ Bragg spots (marked by black circles). The associated Q<sub>moiré</sub> is shown as a
green arrow in Figure 2g and is oriented at 30° from Q<sub>CrCl₃</sub><sup>1,0</sup>. In the 35° domain (Figure 2f),
the same moiré spots line up with the first order Bragg spots marked by the red circles, and
the corresponding Q<sub>moiré</sub> (the green arrow in Figure 2h) lies parallel to Q<sub>CrCl₃</sub><sup>1,0</sup>. We found
that the dominant moiré present in these twisted domains correspond to the conjunction of
the (2,1)CrCl₃ and (1,1)ₐu spots. The graphical constructions in Figures 2e and 2f illustrate
that a 5° twist of the CrCl₃ lattice results in an approximately 30° rotation of the two moiré
patterns. By analyzing in this manner all moiré patterns observed in the 30° and 35° twisted
domains, we identified three and four pairs of Bragg spots, respectively, as indicated by the
colored arrows in Figures 2g and 2h. The corresponding Bragg spots are listed in Table 1.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Analysis of multiple moiré patterns observed in CrCl₃β islands. (a),(b) STM topographies of β islands with atomic lattices rotated by 30° and 35° relative to the <110>Au direction. Tunneling parameters: V = 1.5 V, I = 100 pA. (c),(d) Corresponding Fourier transform images. (e),(f) Same FT images as in (c),(d), with highlighted Bragg spots: the CrCl₃ spots are marked by red and black circles, while the moiré spots are located at the vertices of colored hexagons. (g),(h) Graphical constructions of the CrCl₃ (violet) and Au (blue) reciprocal lattices for twist angles of 30° and 35°. Colored arrows indicate the Bragg spot pairs responsible for the distinct moiré patterns.</figcaption></figure>
<table><caption>**Table 1.** Summary of CrCl₃ and Au Bragg spot pairs generating different moiré components in 30° and 35° twisted domains.</caption><thead><tr><th>Twist angle</th><th>Green<br>(2,1)<sub>CrCl<sub>3</sub></sub>/(1,1)<sub>Au</sub></th><th>Black<br>(2,1)<sub>CrCl<sub>3</sub></sub>/(1,1)<sub>Au</sub></th><th>Red<br>(4,0)<sub>CrCl<sub>3</sub></sub>/(2,1)<sub>Au</sub></th><th>Orange<br>(5,2)<sub>CrCl<sub>3</sub></sub>/(2,2)<sub>Au</sub></th><th>Pink<br>(6,1)<sub>CrCl<sub>3</sub></sub>/(3,2)<sub>Au</sub></th></tr></thead><tbody><tr><td>30°</td><td>Detected</td><td>Detected</td><td>Detected</td><td>Not detected</td><td>Not detected</td></tr><tr><td>35°</td><td>Detected</td><td>Not detected</td><td>Detected</td><td>Detected</td><td>Detected</td></tr></tbody></table>

Slight variations in the orientation of CrCl₃ lead to the appearance or disappearance of specific moiré patterns. For example, the moiré marked in black in Figure 2e, resulting from the conjunction of (4,2)CrCl₃ and (2,2)Au, disappears when the CrCl₃ lattice is twisted from 30° to 35°. In the 35° domain (Figure 2f), two additional moiré patterns emerge (in orange and pink) and originate from the pairs (5,2)CrCl₃/(2,2)Au and (6,1)CrCl₃/(3,2)Au respectively. For both orientations of CrCl₃ presented in Figure 2, the various overlapping moiré patterns have similar periodicities, making them challenging to distinguish in real space images. The topographic contrast in Figures 2a and 2b is dominated by a moiré stemming from the (2,1)CrCl₃ and (1,1)Au conjunction (the green arrows in Figures 2g and 2h). For these two twist angles, disentangling the contributions of overlapping moiré patterns requires reciprocal space analysis, as shown in Figures 2e and 2f.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** (a) STM topography of a CrCl₃β island twisted of approximately 32° relative to the <110>$_{Au}$ direction, showing a weak, long-wavelength moiré modulation (~20 nm) superimposed on the dominant ~1.6 nm moiré pattern. Tunneling parameters: V = 1.2 V, I = 100 pA. (b) Graphical construction of CrCl₃ and Au reciprocal lattices. The Bragg spot pair generating the longer period moiré is circled in pink.</figcaption></figure>
At intermediate twist angles, such as 32°, the Bragg spots associated with the pink moiré, (6,1)CrCl₃ and (3,2)Au, approach one another in reciprocal space (see Figure 3b). As a consequence, the corresponding moiré wave vector becomes so small that its spots are buried within the central region of the FT, where background contributions from other electronic structure effects dominate. In real space, this moiré manifests as a long-wavelength modulation, clearly distinguishable from the shorter-period moirés, despite its comparatively weak spectral weight. Figure 3a shows an STM topography of a region where the CrCl₃ lattice is twisted by approximately 32°. Alongside the main moiré modulation, with a real-space period of about 1.6 nm, a fainter superstructure with a larger period of approximately 20 nm is also observed.

We now show that the observed moiré patterns originate from electronic effects rather than from any modulation of the out-of-plane atomic positions of CrCl₃. STM topographs mix contributions from electronic states over a broad energy window and may therefore superimpose several moiré modulations of electronic origin. To overcome this limitation, we use tunneling spectroscopy to isolate the local density of states at a well-defined energy, enabling each moiré component to be individually identified in Fourier space. To this end, we performed scanning tunneling spectroscopy over a 120 nm × 120 nm region (STM topography in Figure 4a), where the CrCl₃ lattice is twisted by 31° relative to the <110>$_{Au}$ direction. Differential conductance (dI/dV) maps were acquired across an energy range from -0.1 eV to 1.8 eV. The FT of the dI/dV map at 0.33 eV (Figure 4c) reveals three sets of moiré spots, highlighted in green, red, and pink in Figure 4e. These results demonstrate that the moiré patterns in the CrCl₃<sup>β</sup> islands manifest not only in topographic images but also in the spectroscopic dI/dV signal.

To examine how the spectral weight of each moiré component evolves with energy, we quantified the FT intensity at the corresponding moiré spot positions for each bias voltage. The resulting energy dependence is shown in Figure 4d. The spectral weights of the three moiré patterns vary strongly with energy. At 0.33 eV, the energy at which the FT in Figure 4e
was obtained, the pink moiré, originating from the conjunction of $(6,1)CrCl_3$ and $(3,2)Au$, dominates, despite being absent in the topographic channel. Each moiré pattern exhibits distinct resonances at specific energies, indicating a purely electronic origin. A similar conclusion was previously reached for $CrCl_3^\alpha$ islands$^8$ where a single moiré was reported. In contrast, for the twisted $CrCl_3^\beta$ islands, multiple moiré components coexist, each displaying a distinct energy dependence, further reinforcing our interpretation of their purely electronic origin.

As a working hypothesis, we propose that when the wave vectors of the $CrCl_3$ electronic Bloch states match the moiré periodicity, a nesting-like effect may occur, leading to a spatial modulation of the electronic wave function. Density Functional Theory (DFT) calculations could provide further insight into this intriguing phenomenon. However, performing DFT calculations for this system is beyond the reach of the present study, primarily because the multiple coexisting moiré periods require an extremely large real-space supercell to be treated explicitly.
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** (a) STM topography of 31° twisted CrCl₃<sup>β</sup> domain. Tunneling parameters: V = 1.8 V, I = 200 pA. (b) Reciprocal lattice construction of CrCl₃ and Au. (c), (e) FT of dI/dV map at 0.33 eV, having distinct moiré components (green, red, pink). (d) Energy dependence of FT intensities extracted from the dI/dV maps for different moiré components.</figcaption></figure>

To summarize, we investigated monolayer CrCl₃ islands deposited on Au(111). The CrCl₃
lattice is rotated by approximately 30° relative to the close-packed directions of Au(111).
The incommensurability between CrCl₃ and Au(111) lattices gives rise to a quasiperiodic
pattern resulting from the superposition of several moiré orders. We show that multiple
moirés can be observed directly for some orientations, and even when one appears domi-
nantly in topography, additional moiré orders emerge in reciprocal space. Importantly, the
spectral weight of these distinct moiré components varies considerably with electron energy,
demonstrating that the observed quasiperiodic pattern arises from an electronic, rather than
structural, modulation.
When simulating electronic or magnetic properties of moiré systems, it is quite generic to assume a single moiré periodicity in the form of a supercell. For instance, this results in the prediction of a non-collinear magnetic order with a well-defined period²⁶ that is driven by the considered moiré, considering additional moiré would certainly result in a more complex spin texture. Our results suggest that, in practice, multiple magnetic periods are likely to coexist. This would give rise to a quasiperiodic spin texture that could be probed by spin-resolved STM for instance. Similarly, in systems where mechanisms such as chiral topological superconductivity are associated with a specific moiré period,²⁹ it becomes natural to ask how the electronic properties evolve when additional moiré periodicities are considered. The coexistence of multiple moiré patterns may therefore lead to a richer and more complex phenomenology than previously anticipated.

## Experimental

All experiments were performed under ultra high vacuum (UHV) conditions. The Au(111) single crystal was cleaned by repeated cycles of argon-ion sputtering and annealing at 450 °C. Anhydrous $CrCl_3$ powder was carefully degassed prior to deposition and evaporated from a Knudsen cell onto the clean Au(111) surface at room temperature. To obtain flat $CrCl_3$ islands, the sample was subsequently annealed at 150 °C for a few minutes.

STM topography measurements were carried out at 4.2 K in UHV using electrochemically etched tungsten tips in both a home-built low temperature STM and a commercial Omicron LT-STM. Scanning tunneling spectroscopy (STS) measurements were performed in situ using the home-built setup. The tunneling setpoints were typically 200 pA and 1.8 V.

The spectroscopic grid (Figure 4) was recorded over an area of 120 nm × 120 nm, with the STM tip positioned every 234 pm to acquire an I(V) spectrum between -0.1 V and 1.8 V. Differential conductance curves (dI/dV) were obtained by numerical differentiation of the raw I(V) data. Individual dI/dV maps were extracted at each bias voltage and Fourier
transformed. The spectral weight of each moiré spot was then quantified as a function of
the bias voltage.

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Acknowledgement

The authors thanks Pascal David for technical support. This work was supported by the French Agence Nationale de la Recherche through the contract ANR GINET2-0 (ANR-20-CE42-0011) and the ANR MASCOTE (ANR-24-CE30-1342). E.G. acknowledged the GDR-NS-CPU for funding it research stay in IPCMS.

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