[1]Marie Hervé

[1]Sorbonne Universite, CNRS, Institut des Nanosciences de Paris, UMR7588, 4
place Jussieu, Paris, 75005, France 2]Université de Liège,
Nanomat/Q-mat/CESAM, Sart Tilman, City, B-4000, Belgium 3]Université de Liège,
TOM/Q-mat/CESAM, Sart Tilman, City, B-4000, Belgium 4]Fonds de la Recherche
Scientifique (FRS-FNRS), Bruxelles,Belgium 5]Université de Strasbourg, CNRS,
Institut de Phsyique et Chime des Matériaux de Strasbourg, UMR7504,
Strasbourg, F-67000, France

# Higher order topological defects in a moiré lattice

Eugenio Gambari  Sebastian Meyer  Sacha Guesne  Pascal David  François
Debontridder  Laurent Limot  Fabrice Scheurer  Christophe Brun  Bertrand Dupé
Tristan Cren<EMAIL_ADDRESS>*  [  [  [  [

###### Abstract

Topological defects are ubiquitous, they manifest in a wide variety of systems
such as liquid crystals, magnets or superconductors. The recent quest for non-
abelian anyons in condensed matter physics stimulates the interest for
topological defects since they can be hosted in vortices in quantum magnets or
topological superconductors. In addition to these vortex defects, in this
study we propose to investigate edge dislocations in 2D magnets as new
building blocks for topological physics since they can be described as
vortices in the structural phase field. Here we demonstrate the existence of
higher order topological dislocations within the higher order moiré pattern of
the van der Waals 2D magnet CrCl3 deposited on Au(111). Surprizingly, these
higher order dislocations arise from ordinary simple edge dislocations in the
atomic lattice of CrCl3. We provide a theoretical framework explaining the
higher order dislocations as vortex with a winding Chern number of 2. We
expect that these original defects could stabilize some anyons either in a 2D
quantum magnet or within a 2D superconductor coupled to it.

###### keywords:

Moiré pattern, Topological defects, Van der Waals materials, Scanning
Tunneling microscopy, Scanning Tuneling Spectroscopy, Density functional
theory, Transition metal surfaces

##  1 Introduction

Van der Waals materials are emerging as extremely versatile building blocks
for many fields of research, both at the fundamental and applied level [1, 2].
They offer tremendous possibilities for fine-tuning spintronics [3, 4],
superconducting [5, 6, 7], nanoelectronics [1, 2], optical devices [8, 9]. The
toolbox of these heterostructures is continuously growing with for instance
the recent discovery of ferromagnetic order in the family of chromium
trihalides down to the monolayer limit, $\textrm{CrX}_{\textrm{3}}$ (X = I,
Br, Cl) [10, 11, 12, 13, 14]. Due to the absence of magnetic anisotropy in the
plane when the $\textrm{CrCl}_{\textrm{3}}$ monolayer is decoupled from its
substrate it behaves as a 2D XY quantum magnet [14] that can host non-abelian
anyons in the center of magnetic vortex cores [15]. Their integration in van
der Waals heterostructure is expected to lead to a wealth of new exotic
effects. Stacking chromium trihalides with transition metal dichalcogenides
that exhibits a strong Ising spin-orbit coupling appears as a versatile and
promising route towards the development of hybrid magnetic/superconducting
systems [5, 6]. For instance, topological superconductivity has recently been
shown to appear in $\textrm{CrBr}_{\textrm{3}}$/$\textrm{NbSe}_{\textrm{2}}$
heterostructures [5, 6]. The proposed mechanisms leading to a topological
order in these hybrid structures was intimately related to the presence of a
moiré pattern [6]. Several studies have related the observation of a moiré
structure to the mismatch between the $\textrm{CrX}_{\textrm{3}}$ monolayers
and their substrate [16, 14, 6, 17]. The orientational degree of freedom may
add a supplementary possibility to tune the moiré, as observed in twisted
$\textrm{CrX}_{\textrm{3}}$ bilayers [18, 19, 20]. The physics of moiré is
quite complex, a same structure might exhibit several moiré patterns of
different orders, as observed in Gr/Ir or Gr/Pt systems [21, 22, 23, 24, 25,
26, 27]. Higher order structures appear when, in a particular direction,
$n_{1}$ lattice periods of layer 1 almost match with $n_{2}$ periods of layer
2. Correspondingly, in the reciprocal space, a higher order moiré appears when
a small enough wave vector
$\textbf{q}_{moire}=m_{1}\textbf{q}_{1}-m_{2}\textbf{q}_{2}$ can be define
with $\textbf{q}_{i}$ being the Bragg vectors of the two lattices and
$|\textbf{q}_{moire}|\ll\textrm{Min}(|\textbf{q}_{1}|,|\textbf{q}_{2}|)$.
Ordinary moiré correspond to the case where $m_{1}=m_{2}=1$. Interestingly,
the $\textrm{CrX}_{\textrm{3}}$ compounds have a unit cell almost twice the
one of dense metals, that would naturally lead to a second order moiré where
$m_{1}=1$ and $m_{2}=2$.

In the $\textrm{CrX}_{\textrm{3}}$ family, till now, the magnetisation was
shown to be essentially collinear [28, 29, 10, 11, 14]. However, in the
$\textrm{CrI}_{\textrm{3}}$ bilayers, some hint of non-collinear magnetism was
found. It has been related to the moiré pattern responsible of a non-
negligible spatial modulation of the magnetic exchange interaction[18, 20, 30,
31, 32]. This confers an additional interest for the
$\textrm{CrX}_{\textrm{3}}$ family. Indeed, a non-collinear magnetic order is
predicted to be a major ingredient for inducing a topological order in
superconducting/magnetic heterostructures [33, 34, 35, 36, 37]. Topological
superconductors may host Majorana bound states localized on topological
defects such as superconducting vortex cores or spin-orbit vortices [38], but
other kind of defects such as dislocations could also be of interest since
edge dislocations are topological defects analogous to vortices [39, 40].
Interestingly, a moiré pattern in van der Waals materials seems to be a good
candidate for exploring such physics since in graphene/Ir(111), graphene
bilayers and h-BN/Ru [41, 42, 43, 44], simple edge dislocations in the atomic
lattice lead to dislocations in the moiré pattern.

Here we report that $\textrm{CrCl}_{\textrm{3}}$ deposited on a Au(111)
substrate produces large two-dimensional islands that exhibit moiré pattern.
In this particular case where the $\textrm{CrCl}_{\textrm{3}}$ lattice period
is very close to twice the one of Au(111), the moiré pattern is of order 2. We
show that the presence of an edge in the $\textrm{CrCl}_{\textrm{3}}$ lattice
leads to a corresponding second order edge dislocation in the moiré. We
demonstrate that this dislocation is a topological defect of Chern winding
number of 2. These observations are explained within a unified model. We show
that a higher order moiré pattern is a way to access to topological defects of
higher Chern winding number. This unified model can be used as a predictive
tool to build Van der Waals heterostructures in order to engineer topological
defects of the desired Chern winding numbers.

##  2 Results

###  2.1 Superstructures in $\textrm{CrCl}_{\textrm{3}}$ monolayer on Au(111)

Crystalline chromium trichloride ($\textrm{CrCl}_{\textrm{3}}$) thin films
were grown using molecular beam epitaxy (MBE) on a clean Au(111) substrate.
Figure 1.a depicts a large scale STM topography of 0.6 monolayers (ML) of
$\textrm{CrCl}_{\textrm{3}}$ measured at 4 K. The dashed red line highlights
the presence of 1 ML thick islands of $\textrm{CrCl}_{\textrm{3}}$. Two
distinct types of islands denoted as $\textrm{CrCl}_{\textrm{3}}^{\alpha}$ and
$\textrm{CrCl}_{\textrm{3}}^{\beta}$ were observed. Notably, these islands
display two different preferred in-plane crystalline orientations rotated by
30° with respect to each other (see Supplementary note and figure S1.1 and
S1.2) .

Throughout the manuscript, we will exclusively focus on discussing the
$\textrm{CrCl}_{\textrm{3}}^{\alpha}$ islands. In these islands, the dense
atomic directions of $\textrm{CrCl}_{\textrm{3}}$ aligns along the dense
atomic directions of the Au substrate, the
$\textrm{$<$1$\bar{1}$0$>$}_{\textrm{Au}}$ direction. In these islands, our
STM study shows that $\textrm{CrCl}_{\textrm{3}}$ is restoring the lattice
parameter of a free standing $\textrm{CrCl}_{\textrm{3}}$ monolayer (see
Supplementary note S1.1 and Supplementary figure S1.1 and S1.2 ). Remarkably,
upon cooling down the sample to 4K, a superstructure emerges within these
$\textrm{CrCl}_{\textrm{3}}^{\alpha}$ islands, which is not visible at room
temperature (see Supplementary figure S2). Figure 1.b shows an STM image
featuring two islands labelled A and B, which form a twin boundary (see
Supplementary figure S4). Both islands exhibit an hexagonal superstructure
with a periodicity of 6.2 nm. Their orientation differ by an angle of 13° (see
green and dark dashed lines). Interestingly, the rotation of the
superstructure does not seem, at first sight, to be correlated with a rotation
of the atomic lattice. Figure 1.c presents the Fourier filtering of the STM
topography of figure 1.b revealing the atomic structure of
$\textrm{CrCl}_{\textrm{3}}$. Fourier filtering is obtained by selecting the
Bragg spots marked in red and green in the fast Fourier transform (FFT) of
Figure 1.c. The white dashed line serves as a visual guide, highlighting the
dense atomic direction of $\textrm{CrCl}_{\textrm{3}}$ in both islands. At
first sight, no misalignment of the crystal axis is detectable between these
two islands. Additionally, within individual islands, the superstructure can
exhibit a distortion, while the atomic lattice seems to remain unaffected.
This is shown in Figure 1.d and 1.e. Figure 1.d displays an STM topography
acquired inside an island. The pink dashed line is a guide for the eye showing
that the superstructure is not straight but slightly undulate. Figure 1.e
shows the corresponding Fourier filtered image, revealing the atomic lattice
of $\textrm{CrCl}_{\textrm{3}}$. The white dashed line that follows the dense
atomic direction seems to be insensitive to the change of orientation in the
superstructure. Nevertheless, we conducted modelization of a moiré pattern in
this system (see Supplementary note and figure S5) and show that this
superstructure is indeed a moiré effect which shows an unusual high
sensitivity to subtle variations in the atomic lattice orientation. The change
of orientation of the moiré pattern shown in Figure 1.b and d is explained by
a deviation angle in the atomic lattice of less than 1°. This system is rather
rich with plenty of different moiré patterns induced by very little deviations
of the $\textrm{CrCl}_{\textrm{3}}$ lattice with respect to Au(111). It could
be a good plateform to explore the influence of a moiré on the magnetic
texture.

Figure 1: Superstructure in $\textrm{CrCl}_{\textrm{3}}$/Au(111). (a) Large
scale STM topography of 0.6 monolayer of $\textrm{CrCl}_{\textrm{3}}$
deposited on a Au(111) substrate. (b) Zoom in the yellow frame of (a) showing
2 structural domains where the superstructure orientation differs by an angle
of 13°. The green and dark dashed lines are guides for the eyes to follow the
orientation of the superstructure in the two domains, the red dashed line
shows the separation between the two islands. (c) Fourier filtered image
revealing the atomic lattice of $\textrm{CrCl}_{\textrm{3}}$ in the two
structural domains. The white dashed line is a guide for the eye to follow the
orientation of the atomic lattice. (d) STM topography acquired in the pink
frame of (a) showing that the superstructure is undulating inside a
$\textrm{CrCl}_{\textrm{3}}$ island, while no rotation of the atomic lattice
can be detected in the Fourier filtered image (e). The pink and white dashed
lines are eye-guides to follow the orientation of the superstructure and the
atomic lattice inside the island. For all the STM images, the tunneling
parameters are: Bias voltage, $\textrm{U}_{\textrm{0}}=2$ V; Tunneling
current, $\textrm{I}_{\textrm{t}}=200$ pA

###  2.2 Role of Au(111) on the electronic structure

From our measurements, we deduce that CrCl3 restores its lattice constant as
in monolayer without substrate (see Supplementary note S1.1 and Supplementary
figure S1.2), which is why we need to clarify the role of Au(111) on the
electronic structure of the system. Scanning tunneling microscopy imaging
reflects the local density of states and does not reflect directly the
structure of the surfaces. It contains also some information on the local
electronic structure that can eventually be modulated along the moiré pattern.
We have performed tunneling spectroscopy experiments in order to elucidate the
origin of the observed corrugation in the topographic measurements. Figure 2.a
shows scanning tunneling spectroscopy data for different positions above the
moiré pattern of CrCl3/Au(111) (from red, center, to blue, edge of the
pattern). Regardless of the position, the measured conductance shows the same
features, that is, no intensity below 0.2 eV; then there is a small increase
at 0.2 eV followed by a plateau and then a strong increase after 0.3 eV,
eventually followed by a plateau above 1.2 eV. The behavior of the conductance
above 1.2 eV is strongly dependent on the tip position over the moiré pattern.
Figure 2.c to f display 4 conductance maps revealing the spatial variation of
the local density of states over the moiré. Figure 2.c, recorded at 0.2 eV,
shows a pattern with a lower density of states in the center of the protrusion
found in the topography sector (Figure 2.a). At 0.9 eV, the contrast is
reversed, the highest density of state being in center of the protrusions
(Figure 2.d). At $1.3$ eV the contrast cancels (Figure 2.e), it can be seen in
the tunneling spectra of Figure 2.a that it corresponds to a crossing point
where all spectra cross. Above $1.3$ eV a second contrast inversion is
observed with a lower density of states in center over the protrusions (Figure
2.f). These measurements show that the moiré pattern produces quite clear
spectroscopic features and thus we expect a strong effect due to the
structural modulation of the moiré on the electronic and magnetic properties
of the CrCl3 layer. This is corroborated by the fact that the moiré pattern is
not showing up in the topographic images at room temperature. We have
performed a temperature ramp from 4 K to 130 K and we observed a gradual
decrease of the moiré corrugation as the temperature is increased. We
attribute this to the fact that the moiré seen in the topography is mainly of
spectroscopic origin, i.e. there is no real corrugation (see Supplementary
note and figure S3). The fact that the moiré reflects mainly a modulation of
the local density of states suggests the importance to investigate the
electronic band structure by density functional theory.

We therefore calculate the vacuum density of states (VacDOS) following the
Tersoff & Hamann model [45, 46, 47] to compare and interpret the spectroscopy
data. In Figure 2.b, we see the results for an unsupported CrCl3 monolayer
(UML) as the green shaded area (for further information, see the Methods
section and Supplementary note and figure S6). Without the Au(111) substrate,
we would expect peaks to occur at the Fermi energy and more significantly
below the Fermi energy, where the experimental data do not show any states. A
gap appears in between $0.2$ eV and $1.4$ eV, which is absent in the
measurement of Fig. 2 (a). When adding Au underneath of CrCl3, the VacDOS
changes drastically, shifting the band gap to the Fermi level and below (black
curve). Note that due to the computational effort within the DFT calculation,
we adapt a perfect registry between CrCl3 and Au(111). Nevertheless, the
calculated VacDOS (Figure 2.b - dark curve) exhibits qualitatively the same
trends as in the measurement (cf. panel (a) and (b)): the band gap and the
shoulder/peak structures at 0.2 eV and 1.7 eV. We attribute the differences
between the calculations and the measurements to the differing structures
within the calculations. From this comparison, we deduce that Au has an
influence on the electronic structure of CrCl3 and that this system is not
solely driven by Van-der Waals forces. This is in accordance with reference
[48], where the authors compared the binding energies and concluded that for
CrCl3/Au(111) these energies lie at the threshold of VdW and ionic bonds.

Figure 2: Role of Au(111) on the electronic structure. Comparison between
experimentally measured spectroscopy data and calculated vacuum density of
states of CrCl3/Au(111). (a) Scanning tunneling spectroscopy measurements on
different positions of the moiré pattern as indicated in the inset. (c), (d),
(e) and (f) dI/dV mapping of the moiré pattern taken at 4 different energies.
(setpoint: $\textrm{U}_{\textrm{0}}=2$ V, $\textrm{I}_{\textrm{t}}=400$ pA,
these data were recorded using a lock in, a modulation of 20mV rms was added
to the bias voltage, the full spectroscopic map was recorded from 2 V to -1 V)
((b) Density functional theory calculations of the vacuum density of states 3
Å above the surface of a CrCl3 unsupported monolayer (UML, green area) in
comparison with CrCl3/Au(111) (black line). Note that in the calculations, it
is assumed that CrCl3 is in perfect registry with Au(111), that is the moiré
is not taken into account.

###  2.3 Higher order edge dislocations of the moiré pattern

Very peculiar defects have been observed in this moiré pattern. They consist
in higher order edge dislocations. One of those particular defect is shown in
Figure 3.a. On the left side of the dislocation, as indicated by the green
arrows, three rows of the moiré pattern merge into a single row on the right
side, resulting in the disappearance of a pair of rows. Notably, we have
observed dislocations with similar characteristics in multiple locations of
the sample. For each dislocation, a pair of rows vanishes, which represents an
inherent characteristic of this particular moiré pattern.

To further characterize the dislocation, we analyze its Burgers vector (Figure
3.b) and coordination (Figure 3.c). The Burgers vector of the dislocation is
found to be twice the lattice vector of the moiré pattern:
$\boldsymbol{b}_{moir\acute{e}}=-2\boldsymbol{a}_{moir\acute{e}}^{1}$. The
coordination of the lattice cells surrounding the dislocation is determined to
be 5-5-8, i.e. two cells are five fold coordinated while one is height fold,
this is in contrast with ordinary 5-7 dislocations. To gain insight into the
atomic structure surrounding the dislocation, we provide a magnified view
around the dislocation in the STM topography (Figure 3.d). By Fourier
filtering the Bragg spots, the atomic lattice of $\textrm{CrCl}_{\textrm{3}}$
surrounding the dislocation becomes visible. Remarkably, at the exact position
of the dislocation in the moiré pattern, we also observe an edge dislocation
in the atomic lattice of $\textrm{CrCl}_{\textrm{3}}$. Specifically, one row
of atoms on the left side of the dislocation transforms into two rows on the
right side, as indicated by the green arrows. Prior studies in the literature
has suggested that a dislocation or defect within a moiré pattern faithfully
replicates the corresponding dislocation or defect at the atomic level [42,
43, 44, 49]. However, in this case, the dislocation at the atomic level is
characterized by a Burgers vector
$\boldsymbol{b}_{{CrCl}_{3}}=\boldsymbol{a}_{{CrCl}_{3}}^{1}$ (as shown by the
red arrow in Figure 3.d) and a 5-7 defect. This is in contrast, with the
dislocation in the moiré pattern where the Burgers vector is opposite to the
one in the atomic lattice and 2 times larger than the moiré lattice vector.

To understand the particularity of this dislocation in the moiré pattern, we
modelize our system by introducing a 5-7 defect in the atomic lattice of
$\textrm{CrCl}_{\textrm{3}}$ and convolute it with a defect-free atomic
lattice of Au(111). Figure 3.e illustrates the result of this simulation,
which successfully reproduces the experimental observations of a dislocation
with a Burgers vector
$\boldsymbol{b}_{moir\acute{e}}=-2\boldsymbol{a}_{moir\acute{e}}^{1}$.

Figure 3: Higher order edge dislocations. (a) STM topograhy showing a double
dislocation in the moiré pattern. The dashed green lines and arrows are guide
for the eyes to follow the dislocation. ($\textrm{U}_{\textrm{0}}=2$ V,
$\textrm{I}_{\textrm{t}}=200$ pA) (b) Drawing of the Burgers circuit around
the dislocation, and its coordination (c). (d) Zoom around the dislocation in
the moiré pattern. The STM topography was Fourier filtered to reveal the
atomic lattice. It shows a simple dislocation in the atomic lattice of
$\textrm{CrCl}_{\textrm{3}}$ at the exact position of the dislocation in the
moiré pattern. (e) Numerical simulation of the dislocation in the moiré
pattern.

###  2.4 Analytical description

To understand the physical origin of the observed moiré and its peculiar
dislocations, we conducted an analytical description of the moiré pattern. A
conventional approach to describe a moiré pattern involves employing a
continuous model [21]. The moiré may be described as the product between two
lattice functions, denoted as $\textrm{f}_{\textrm{1}}(x,y)$ and
$\textrm{f}_{\textrm{2}}(x,y)$ which are the superposition of plane waves on
top of a constant background with an associated spatial frequency $Q_{1}$ and
$Q_{2}$ respectively. To illustrate this with the simplest case, we consider
the scenario of waves in each lattice propagating along the x-direction. The
product between two lattice functions $f_{1}(x,y)\times
f_{2}(x,y)=\left(1+a\cos(Q_{1}x))\times(1+a\cos(Q_{2}x)\right)$ can be
rewritten as:

$f_{1}\times
f_{2}=1+\frac{a^{2}}{2}\left(\cos[(Q_{1}+Q_{2})x]+\cos[(Q_{1}-Q_{2})x]\right)+2a\cos[\frac{(Q_{1}+Q_{2})}{2}x]\cos[\frac{(Q_{1}-Q_{2})}{2}x]$
(1)

We obtain the addition of 3 terms: a constant background, and 2 periodic
function. In the following, to describe the moiré we will consider the first
periodic function. It is a sum of two cosines with one slowly varying term
$\cos[(Q_{1}-Q_{2})x]$ that stems for the moiré term. The resulting spatial
beating frequency, $Q_{1}-Q_{2}$ corresponds to the spatial frequency of the
moiré, $Q_{moire}$. For the case of a moiré formed between two hexagonal
lattices as for $\textrm{CrCl}_{\textrm{3}}$/Au(111), each lattice needs to be
treated as the superposition of two waves propagating in directions at 120°
one from the other. This renders the mathematical description in the real
space heavy. A simpler way to describe a moiré is to treat the problem in the
reciprocal space where the moiré wave vector is expressed as the difference
between the wave vectors of each atomic lattice. We employed the intelligible
model described by Zeller et al. and Le Ster et al. [21, 50]. Within this
framework, not only the 1st order, but also the higher order Fourier component
of the moiré need to be considered.

The general reciprocal vectors for both the Au and
$\textrm{CrCl}_{\textrm{3}}$ atomic lattices, $\boldsymbol{Q}_{Au}^{i,j}$ and
$\boldsymbol{Q}_{{CrCl}_{3}}^{k,l}$ are expressed as linear combinations of
the primitive reciprocal lattice vectors, namely $\boldsymbol{Q}_{Au}^{1,0}$,
$\boldsymbol{Q}_{Au}^{0,1}$, $\boldsymbol{Q}_{{CrCl}_{3}}^{1,0}$,
$\boldsymbol{Q}_{{CrCl}_{3}}^{0,1}$:

$\displaystyle\boldsymbol{Q}_{Au}^{i,j}$
$\displaystyle=i\boldsymbol{Q}_{Au}^{1,0}+j\boldsymbol{Q}_{Au}^{0,1}$ (2)
$\displaystyle\boldsymbol{Q}_{{CrCl}_{3}}^{k,l}$
$\displaystyle=k\boldsymbol{Q}_{{CrCl}_{3}}^{1,0}+l\boldsymbol{Q}_{{CrCl}_{3}}^{0,1}.$
(3)

Figure 4: Reciprocal lattice of $\textrm{CrCl}_{\textrm{3}}$/Au(111). The blue
dots corresponds to the 1st (a) and 2nd (b) order moiré spots.

Here, $|\boldsymbol{Q}_{Au}^{1,0}|$ $=|\boldsymbol{Q}_{Au}^{0,1}|=2\pi/a_{Au}$
and
$|\boldsymbol{Q}_{{CrCl}_{3}}^{1,0}|=|\boldsymbol{Q}_{{CrCl}_{3}}^{0,1}|=2\pi/a_{{CrCl}_{3}}$,
where ${a_{Au}}=0.288$ nm and $a_{{CrCl}_{3}}=0.604$ nm corresponds to the
lattice parameters of Au and $\textrm{CrCl}_{\textrm{3}}$, respectively. The
possible Fourier components of the moiré pattern’s reciprocal vectors are
expressed as the difference between the corresponding reciprocal vectors of Au
and $\textrm{CrCl}_{\textrm{3}}$, leading to:

$\boldsymbol{Q}_{moir\acute{e}}^{i,j,k,l}=\boldsymbol{Q}_{Au}^{i,j}-\boldsymbol{Q}_{{CrCl}_{3}}^{k,l}.$
(4)

When the dense atomic directions of $\textrm{CrCl}_{\textrm{3}}$ align with
the $\textrm{$<$1$\bar{1}$0$>$}_{\textrm{Au}}$ directions,
$\boldsymbol{Q}_{Au}^{1,0}$ is colinear with
$\boldsymbol{Q}_{{CrCl}_{3}}^{1,0}$, as depicted in Figure 4.a. Let us
illustrate why one should here consider higher order moiré by first showing
that a usual first order moiré is doesn’t correspond to our observation. The
first order moiré wave vector formed between the (1,0) Bragg spots of Au and
$\textrm{CrCl}_{\textrm{3}}$ is expressed as:

$\boldsymbol{Q}_{moir\acute{e}}^{1,0,1,0}=\boldsymbol{Q}_{Au}^{1,0}-\boldsymbol{Q}_{{CrCl}_{3}}^{1,0}$
(5)

This results in a moiré period in real space:

$L_{moir\acute{e}}^{1,0,1,0}=2\pi/|\boldsymbol{Q}_{moir\acute{e}^{1,0,1,0}}|=0.55~{}\textrm{nm}.$
(6)

This period is notably smaller than 0.604 nm, the lattice parameter of
$\textrm{CrCl}_{\textrm{3}}$. As a result, a moiré pattern between the (1,0)
Bragg spots of Au and $\textrm{CrCl}_{\textrm{3}}$ is meaningless. This
emanates from the fact that $a_{{CrCl}_{3}}$ exceeds the lattice parameter of
Au by more than a factor two. Noteworthy, $\textrm{CrCl}_{\textrm{3}}$/Au(111)
is the paragon case where higher order moiré is popping up. Extending the
reciprocal lattice of $\textrm{CrCl}_{\textrm{3}}$ to the second order (as
depicted in Figure 4.b) reveals that the shortest moiré wave vector is formed
between the (1,0) Bragg spot of Au and the (2,0) Bragg spot of
$\textrm{CrCl}_{\textrm{3}}$, expressed as:

$\boldsymbol{Q}_{moir\acute{e}}^{1,0,2,0}=\boldsymbol{Q}_{Au}^{1,0}-\boldsymbol{Q}_{{CrCl}_{3}}^{2,0}.$
(7)

In the real space, this yields a moiré pattern period:

$L_{moir\acute{e}}^{1,0,2,0}=2\pi/|\boldsymbol{Q}_{moir\acute{e}^{1,0,2,0}}|=6.2~{}\text{nm},$
(8)

which corresponds exactly to the period we measure in our experiments.

Figure 5: Analytical description of the dislocation in the moiré pattern. (a)
First order periodic lattice simulated by a simple plane wave in which a
dislocation has been introduced in the form of phase difference which
correspond to a Berry phase: $\Delta\varphi(x,y)$. (b) First order moiré
pattern when a simple dislocation exists in one of the two periodic lattices:
a simple dislocation appears in the moiré. The dislocation in the moiré
pattern exhibits the same Berry phase as in the primary lattice. (c) Second
order primary lattice: the Berry phase of the dislocation is doubled
($2\Delta\varphi(x,y)$) with respect to the first order shown in (a). (d) The
resulting second order moiré pattern shows an additional double row with a
Berry phase ($2\Delta\varphi(x,y)$) .

We will now explain the emergence of the peculiar dislocations (Figure 3). As
a dislocation is a punctual defect that cannot be modeled easily in reciprocal
space, the problem needs to be partially treated in real space. In order to
avoid a heavy mathematical description and maintain conceptual clarity, we
will consider only a dislocation in a simple planar wave that can be modeled
by the expression $\cos[(Q_{1}x+\Delta\varphi(x,y)]$, with
$\Delta\varphi(x,y)$ being a Berry phase associated to the defect. If one
considers a dislocation located at the origin, $\Delta\varphi(x,y)$ can just
be taken as the angle of the position vector $\boldsymbol{r}=(x,y)$ to the
abscissa, i.e. $\Delta\varphi(x,y)=\arg(x+iy)$. The resulting dislocation
pattern is shown in Figure 5.a.

Consider now two lattices forming a first order moiré, with a dislocation in
one of the lattices. It naturally results in a dislocation in the moiré
pattern, as shown in Figure 5.b. This can easily be explained using eq.1. The
moiré term is $\cos[((Q_{1}-Q_{2})x+\Delta\varphi(x,y)]$ which also exhibits a
Berry phase that naturally gives a dislocation in the moiré pattern. The
question that immediately arises is what about our higher order moiré? The
second order moiré is a beating between $\cos(2Q_{1}x)$ and $\cos(Q_{2}x)$,
however, in the presence of a dislocation the phase $Q_{1}x$ has to be
replaced by $(Q_{1}x+\Delta\varphi(x,y))$. This results in a beating between
$\cos[2(Q_{1}x+\Delta\varphi(x,y))]$ and $\cos(Q_{2}x)$ that leads to a moiré
term $\cos[(2Q_{1}-Q_{2})x+2\Delta\varphi(x,y)]$. As we can see, the resulting
Berry phase is now $2\Delta\varphi(x,y)$, the double of the Berry phase in the
primary defect. Figure 5.c,d shows the resulting dislocation patterns in the
second order lattice and its moiré pattern.

In Figure 5 one can see that the additional rows in the moiré pattern and the
primary lattice are on the same side (bottom part of the images). This is at
odds with what we observe in our experiments as shown in Figure 3.a,d. There,
the additional rows in the atomic and moiré lattices are located at the
opposite sides of the dislocation. This peculiarity is explained in Figure 6.
Let us first assume that the additional row in the primary lattice is located
at the bottom of the image as shown in Figure 5.a,c. To first order, it
corresponds to a term $\cos(Q_{1}x+\Delta\varphi(x,y))$. Then, the second
order moiré pattern is given by the expression
$\cos((2Q_{1}-Q_{2})x+2\Delta\varphi(x,y))$. In Figure 6 we show that
depending on the sign of $(2Q_{1}-Q_{2})$, the dislocation in the moiré
pattern can appear either on the same side of the dislocation in the primary
lattice ($2Q_{1}-Q_{2}>0$, Figure 6.a), either in the opposite side
($2Q_{1}-Q_{2}<0$, Figure 6.b). To demonstrate this, let us consider the phase
term $(2Q_{1}-Q_{2})x+2\Delta\varphi(x,y)$ over a circuit surrounding the
dislocation (see Figure 6.a,b). In the case where $2Q_{1}>Q_{2}$ (Figure 6.a),
along the blue arc, from A to B, the optical path difference $\Delta\theta$ is
the sum of two positive contributions $\Delta\theta_{1}=(2Q_{1}-Q_{2})\Delta
x+2\pi$, while for the green arc, from B to A, the optical path difference is
the sum of a negative term and a positive one
$\Delta\theta_{2}=-(2Q_{1}-Q_{2})\Delta x+2\pi$. Hence,
$|\Delta\theta_{1}|-|\Delta\theta_{2}|=4\pi$, which means that the blue arc
encompass two additional fringes as compared to the green arc. Now, in the
case where $2Q_{1}<Q_{2}$ (Figure 6.b), the situation is just reversed. On the
blue circuit the two phase shifts are opposite while on the green arc they
add, this lead to the appearance of two additional rows in the top part of
Figure 6.b, while the in the primary lattice the dislocation is still located
at the bottom (Figure 5). The situation depicted in Figure 5.b is in perfect
agreement with what we observe in Figure 5.a, d and is consistent with the
fact that $2Q_{\textrm{CrCl3}}<Q_{\textrm{Au}}$. The double dislocation in
CrCl3/Au(111) is carrying a Berry phase of $-4\pi$, which correspond to a
Chern winding number of 2, which is equivalent to a double vortex.

Figure 6: Dislocation’s Berry phase. Dislocation in the 2nd order moiré when
$2Q_{1}>Q_{2}$ (a) and $2Q_{1}<Q_{2}$ (b)

##  3 Conclusion & Outlook

We have observed a higher order moiré pattern in a monolayer of CrCl3 on a
Au(111) surface that is driven by a large lattice mismatch: two periods of
Au(111) matching almost one period of CrCl3. The moiré is revealed in STM
topography images but it originates mainly from an electronic structure
modulation due to a strong coupling to the substrate which is not expected in
such a van der Waals material. This interaction with the substrate was
ascertained by DFT calculations that reveal a substantial shift of the
chemical potential in CrCl3 as compared to a free standing monolayer. The in
situ grown samples exhibit single edge dislocations in the CrCl3 monolayer
that manifest as a double dislocations in the moiré pattern. This doubling
effect is a direct consequence of the second order moiré. The dislocations in
the moiré pattern are structural topological defects which can be described,
in the continuous limit, as double vortices carrying Berry phases of $4\pi$.
Our model can be used with many different systems and may serve to identify
interesting ones that might host topological defects with a well controlled
Chern number.

Due to the substantial coupling to a substrate with a strong Rashba spin-orbit
coupling, this system could be an interesting platform for the stabilization
of exotic non-collinear magnetic textures induced by the periodic potential of
the moiré. The dislocations/vortices observed in the moiré pattern could be
potentially interesting to pin topological spin textures that may host non
conventional excitations such as non abelian anyons. It has been recently
proposed in CrBr3/NbSe2, that the moiré was responsible of the appearance of
topological superconductivity in NbSe2 [6]. We may therefore expect that the
topological defects in the moiré pattern of CrBr3 could stabilize Majorana
bound states.

##  4 Method

###  4.1 Experimental Section

The experiments were performed under ultra high vacuum. The Au(111) single
crystal was cleaned by cycles of argon-ion sputtering and annealing to 450°C.
Starting from an anhydrous powder CrCl3 was carefully degazed before being
evaporated using a Knudsen cell on the clean Au(111) substrate at room
temperature. To obtain flat CrCl3 islands, the sample was post annealed to
about 150°C for few minutes. STM experiments were realized in a home built low
temperature STM and on a LT omicron commercial STM. Experiments were performed
at 4K.

###  4.2 Density functional theory calculations

The vacuum LDOS calculations in Figure 2.b have been done using the FLAPW
method implemented in the code FLEUR [51]. Shown are the two VacDOS
calculations of a unsupported CrCl3 monolayer (UML) and the CrCl3 monolayer on
top of a Au(111) surface. The presented VacDOS is calculated approximately
$3\,\text{\AA}$ above the surface. Since the unit cell of the moiré pattern
with its substrate is too large for being feasible in first principles
calculations, the atomic positions of both Cr and Cl atoms are placed in a
perfect hexagonal structure where, a lattice constant in between Au bulk and
CrCl3 has been applied (tests have shown negligible dependence of the VacDOS
vs. small variations of the lattice parameter, distances and atomic positions
of the van der Waals material). The lattice parameters and distances used are
presented in Tab. 1

Table 1: Lattice parameters in Å used for vacuum LDOS in Figure 2.b. $a_{\textrm{in-plane}}$ denotes the in-plane lattice constant of the hexagonal lattice, $\Delta d_{\textrm{Cl-Cr}}$ is the vertical distance between the Cl layers and the Cr layer, $\Delta d_{\textrm{CrCl\textsubscript{3}-Au(111)}}$ denotes the vertical distance between CrCl3 and the Au(111) surface. System |  $a_{\textrm{in-plane}}$ (Å) |  $\Delta d_{\textrm{Cl-Cr}}$ (Å) |  $\Delta d_{\textrm{CrCl\textsubscript{3}-Au(111)}}$ (Å)
---|---|---|---
CrCl3 UML | 5.909 | 1.289 | $-$
CrCl3/Au(111) | 5.909 | 1.336 | 4.233

Supplementary information

The online version contains supplementary material available at

Acknowledgements We Acknowledged Johann Coraux for his thoughtful reading of
the paper.

## Declarations

  * •

Funding: This work was supported by the French Agence Nationale de la
Recherche through the contract ANR GINET2-0 (ANR-20-CE42-0011). Computing time
was provided by the Consortium d’Équipements de Calcul Intensif (FRS-FNRS
Belgium GA 2.5020.11) and the LUMI CECI/Belgium for awarding this project
access to the LUMI supercomputer, owned by the EuroHPC Joint Undertaking,
hosted by CSC (Finland) and the LUMI consortium through LUMI CECI/Belgium,
ULiege-NANOMAT-SKYRM-1. Sebastian Meyer is a Postdoctoral Researcher [CR] of
the Fonds de la Recherche Scientifique – FNRS. Bertrand Dupé is a Research
Associate [CQ] of the Fonds de la Recherche Scientifique – FNRS.

  * •

Conflict of interest/Competing interests: Not applicable

  * •

Ethics approval and consent to participate: Not applicable

  * •

Consent for publication: Not applicable

  * •

Data availability: Not applicable

  * •

Materials availability: Not applicable

  * •

Code availability: Not applicable

  * •

Author contribution: M.H. and T.C proposed and conceived the research, E.G.,
S.G., M.H., T.C, L.L., C.B. and F.S. performed the experiments with the
technical assistance of P.D. and F.D., E.G., S.G. and M.H analyzed the data,
E.G., T.C and M.H. performed the numérical simulation and derived the
analytical model. S.M. and B.D. performed the DFT calculations. M.H., T.C and
S.M. wrote the original manuscript. All the authors discussed the results.

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