††thanks: These authors contributed equally††thanks: These authors contributed
equally

# One-Dimensional Moiré Physics and Chemistry in Heterostrained Bilayer
Graphene

Gabriel R. Schleder<EMAIL_ADDRESS>John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138,
United States  Brazilian Nanotechnology National Laboratory (LNNano), CNPEM,
13083-970 Campinas, São Paulo, Brazil  Michele Pizzochero
<EMAIL_ADDRESS>John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, United States  Efthimios
Kaxiras<EMAIL_ADDRESS>John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138, United States
Department of Physics, Harvard University, Cambridge, MA 02138, United States

###### Abstract

Twisted bilayer graphene (tBLG) has emerged as a promising platform to explore
exotic electronic phases. However, the formation of moiré patterns in tBLG has
thus far been confined to the introduction of twist angles between the layers.
Here, we propose heterostrained bilayer graphene (hBLG), as an alternative
avenue to access twist-angle-free moiré physics via lattice mismatch. Using
atomistic and first-principles calculations, we demonstrate that uniaxial
heterostrain can promote isolated flat electronic bands around the Fermi
level. Furthermore, the heterostrain-induced out-of-plane lattice relaxation
may lead to a spatially modulated reactivity of the surface layer, paving the
way for the moiré-driven chemistry and magnetism. We anticipate that our
findings can be readily generalized to other layered materials.

Figure 1: Different types of moiré patterns. a) Examples of mechanisms that
lead to moiré patterns in one and two dimensions. b) The supercell matching
approach used in this work to model heterostrained bilayers. The local
stackings are highlighted in the insets, periodically changing from AB to
displaced AA-like regions, separated by domain walls or saddle-point-like (SP-
like) regions.

## Introduction.

The stacking of a pair of graphene layers with a relative twist angle between
them introduces an in-plane moiré pattern and a consequent modulation of the
lattice potential. Depending on the magnitude of the twist angle, the
resulting twisted bilayer graphene (tBLG) exhibits a wide range of intriguing
electronic properties, such as superconductivity, correlated insulating and
topological semimetallic phases [1, 2, 3]. These observations have sparked
considerable interest in the investigation of the physical and chemical
properties of moiré materials, an emerging field dubbed “twistronics” [4, 5,
6, 7, 8]. In addition to twisting, there are several means to create moiré
patterns in 2D materials, such as shear and strain. Furthermore, it is
possible to achieve one-dimensional moiré patterns between 2D materials by
enforcing a periodic modulation selectively in one direction, as illustrated
in Fig. 1a. Different approaches to achieve these moiré patterns have been
proposed, including vertical and lateral heterostructures of lattice-mismatch
heterostrained materials [9, 10, 11, 12], nanotubes [13, 14], substrate-driven
moiré patterns, or chemically functionalized surfaces [15, 16, 17]. However,
earlier theoretical [18, 19, 20, 21] and experimental [22, 23, 24, 25, 26]
works restrict their focus mainly on the twist angle as a knob to tune moiré
materials, with the individual effect and properties of heterostrain currently
lacking a detailed understanding.

In this Letter, we expand the class of moiré materials beyond twisted systems
by introducing heterostrained bilayer graphene (hBLG), where uniaxial strain
is exerted only in one of the two layers. This heterostrain creates a 1D moiré
pattern in the absence of a twist angle. Using first-principles and atomistic
simulations, we demonstrate that hBLG can feature isolated electronic bands
with vanishing dispersion near the Fermi level, similarly to twisted bilayer
graphene near the magic angle. Additionally, we show that the out-of-plane
lattice relaxation induced by heterostrain results in a moiré chemical
reactivity of the surface layer. Our findings establish heterostrained bilayer
graphene as a platform to access and engineer moiré physics in 2D materials
without relying on twist angles.

## Structural relaxation in hBLG.

In Fig. 1b, we illustrate the supercell-matching approach that we adopt to
generate heterostrained models. Although we focus on graphene, this approach
can be used for any periodic crystal. We start with an unstrained rectangular
cell of graphene and repeat it $N$ times in the direction along which strain
will be introduced. For the strained layer, we repeat it $N\pm 1$ times, such
that this layer is strained to match the lattice constant of the unstrained
one. The resulting model will thus have a strain of $\varepsilon=\pm 1/N$. The
strain can be exerted either in the zigzag or armchair direction. We optimize
the atomic structures by means of a registry-dependent atomistic force-field
potential, as discussed in more detail in Supplementary Note 1. From the
optimized atomic structure, we determine the maximum interlayer distance and
external curvature of the unstrained layer. These results are summarized in
Fig. 2, and figures of structures before and after relaxation are presented in
the Supporting Note 4.

The structural relaxation is the result of a competition between the
minimization of the local elastic strain and the expansion of the regions
exhibiting the lowest-energy interlayer stacking configuration (AB- or Bernal-
stacking) and concurrent contraction of regions exhibiting the higher-energy
interlayer stacking configuration (AA-stacking). For tensile heterostrain, we
observe that the atomic structure retains a planar configuration and features
approximately the same interlayer distance as unstrained bilayer graphene. For
compressive heterostrain, the atomic structure acquires a pronounced curvature
that increases with the magnitude of the heterostrain exerted. For low values
of compressive heterostrain, the atomic structure relaxes by homogeneously
increasing the curvature, such that the local strain is minimized and
therefore their periodic structure has a sinusoidal shape. Above a critical
strain value, the homogeneous curvature cannot minimize local forces,
resulting in a rippled lattice [27, 28, 29], where the strain spatially
localizes inducing a local increase of the interlayer distance, as displayed
in Fig. 2a. This regime starts at values of heterostrain of approximately -1%
in the zigzag direction, and at a larger value in the armchair direction,
proportional to the ratio of armchair/zigzag lattice vectors length. For both
directions, the interlayer distances increase continuously by over 200%, and
the curvature increases from zero (planar) to about 0.17 Å-1 (equivalent to a
radius of 6 Å). Interestingly, the unstrained layer accommodates part of the
strain, and the transition from the homogenous sinusoidal curvature and the
rippled regimes are seen in Fig. 2c,d, where the curvature increases with
increasing compressive strain.

Figure 2: Structural properties of hBLG. a) Side-view of heterostrained
bilayer graphene for a representative heterostrain value of $\epsilon=$ -5 %;
also shown are the lattice sites of maximum curvature and interlayer distance.
b) Maximum interlayer distance $\Delta z$ and c) layer curvature as a function
of the heterostrain applied. Positive and negative values correspond to
tensile and compressive strains, respectively. Structural properties of the
unstrained bottom layer: d) the $z$-axis difference, corresponding to the
layer height, and the e) layer curvature. Figure 3: Electronic structure of
hBLG. a) Pristine bilayer graphene rectangular cell and Brillouin zone. b)
Moiré Brillouin zone obtained from the two individual layers. c) Summary of
electronic properties of the relaxed systems: valence–conduction band gap
$\Delta_{vc}$, gap to the band below or above it $\Delta_{v-1}$ or
$\Delta_{c+1}$, and the width of the valence and conduction moiré bands
$w_{v}$ and $w_{c}$. d) Band structures of hBLGs including relaxation effects.
The results for rigid hBLGs, without relaxation effects, are presented in the
Supporting Information.

## Moiré flat bands in hBLG.

Motivated by the effects of strain engineering in graphene [30, 31, 32], we
next investigate the electronic structure of the different hBLG systems. As a
reference, in Fig. 3a we show the rectangular simulation cell and Brillouin
zone of pristine bilayer graphene. In the rectangular cell, the high-symmetry
points where the Dirac cones occur, are folded into a single point between the
$\Gamma-X$ path [33, 34]. Therefore, to study the low-energy physics of these
systems we focus on the effects around this point by means of density
functional theory (DFT) calculations, as detailed in Supporting Note 2. For
heterostrain in the x-direction (zigzag), the moiré Brillouin zone is
displayed in Fig. 3b, interpreted as formed by the vectors connecting the high
symmetry points of each layer. In Supporting Note 2, we discuss heterostrain
applied in the y-direction (armchair), which leads to metallic band
structures. Fig. 3c summarizes the main electronic properties for relaxed
systems extracted from the band structures: the valence–conduction band gap
$\Delta_{vc}$, gap to the band below or above it $\Delta_{v-1}$ or
$\Delta_{c+1}$, and the width of the valence and conduction moiré bands
$w_{v}$ and $w_{c}$. The results for the rigid systems are presented in the
Supporting Information.

In the middle panels of Fig. 3d we show the band structures of unstrained AA
and AB bilayer graphene in the rectangular supercell ($N=20$) to illustrate
band folding effects (the unfolded bands are shown in Supporting Note 2). In
line with previous works [33], AA-stacked bilayer graphene possesses a pairs
of Dirac cones crossing the Fermi energy, while AB-stacked bilayer graphene
possesses two sets of parabolic bands, one of which touches at the Fermi
level. Farther from the Fermi level, additional crossings are observed due to
the folding of these bands.

In Fig. 3d, we show the band structures of the heterostrained systems
including relaxation effects, i.e., the strain is non-uniformly distributed
along the strained direction minimizing the elastic energy of the system. The
band structures for the rigid systems are presented in the Supporting
Information. The magnitude of the strain ranges from $\varepsilon=-5\%$
(compressive) to $\varepsilon=+5\%$ (tensile). It has been shown that below a
critical strain of 1%, the heterostrained systems distributes the strain to
the second layer in order to maintain a uniform AB-stacking configuration
[35], while above this critical value the systems have the non-uniform
stacking profile seen here.

For both compressive and tensile strain, two sets of isolated valence and
conduction bands form, having both a gap between them ($\Delta_{vc}$) and a
gap to the band below or above it ($\Delta_{v-1}$ or $\Delta_{c+1}$). The
strain-induced gaps arise from the potential difference between the layers,
resulting in a pseudoscalar effective potential as a transverse electric field
[36]. The width of these moiré bands ($w_{v}$ and $w_{c}$) decreases with the
magnitude of the strain, yet features are sensitive to the magnitude and
direction of the strain exerted. For compressive heterostrain, smaller strains
lead to flatter bands, and simultaneously $\Delta_{vc}$ increases while
$\Delta_{v-1}$ and $\Delta_{c+1}$ decreases, interpreted as valence and
conduction bands moving away from each other. Hence, intermediate values of
strain lead to isolated bands. For tensile heterostrain, which retains the
planarity of the lattice, larger strains lead to bands similar to AB graphene,
while weaker strains also lead to flatter bands, but now simultaneously
$\Delta_{vc}$ decreases while $\Delta_{v-1}$ and $\Delta_{c+1}$ increases,
interpreted as valence and conduction bands moving closer to each other.
Therefore, smaller strains present flatter isolated bands.

In summary, the results for both rigid and relaxed systems shown in Fig. 3c
suggests that both low-tensile and intermediate-compressive heterostrains are
promising routes for experimentally achieving flat bands. This is of
particular relevance not only to display spin-charge separation or Peierls
distortions, but also to enable strongly correlated physics in one dimension,
hosting phases such as Luttinger liquids, band insulators, bond ordered waves,
and Mott insulators [37].

## Moiré chemistry in hBLG.

Figure 4: Chemical modification of hBLG. a) (Top) Side view illustration of
the atomic structure for the hydrogen atom chemisorption on hBLG; the inset
shows a perpendicular view with a chemisorbed H atom. (Bottom) Binding energy
of an H atom along the moiré pattern of hBLG compared to monolayer graphene
(horizontal dashed line). The horizontal axis matches the atomic structure on
top. b) Side and top views of the spin density upon chemisorption of an H atom
at the most stable site of hBLG.

The heterostrain-induced lattice relaxation in hBLG has important consequences
on its surface chemistry. We investigate the reactivity of hBLG using the
first-principles methodology outlined in Supporting Note 3. We consider the
adsorption of a hydrogen atom (see Fig. 4a), the simplest yet the most
promising ad-species for tailoring the properties of graphene and related
nanostructures [38]. We assess the stability of such adatom at various sites
of the surface graphene layer along the one-dimensional moiré pattern. To that
end, we define the binding energy as, $E\textsubscript{b}$, as

$E\textsubscript{b}=(E\textsubscript{Gr}+E\textsubscript{H})-E\textsubscript{H+Gr},$
(1)

where $E\textsubscript{Gr}$ and $E\textsubscript{H+Gr}$ are the total energies
of the pristine and hydrogenated nanoribbon, respectively, and
$E\textsubscript{H}$ is the energy of an isolated hydrogen atom. According to
this expression, positive values of $E\textsubscript{b}$ indicate exothermic
processes.

In Fig. 4a, we give the evolution of $E\textsubscript{b}$ along the moiré
pattern of hBLG for the representative model shown in Fig. 2a, and compare it
with the corresponding value for free-standing graphene. The binding energy in
hBLG strongly depends on the adsorption site. Its maximum value
($E\textsubscript{b}=1.41$ eV) occurs at the lattice site of hBLG that
exhibits the most pronounced curvature, leading to an enhancement of the
stability of the hydrogen adatom by over 75% compared to free-standing
graphene. This is because the local surface curvature induced by the
heterostrain weakens the $\pi$-bonds and promotes a facile addition of the
hydrogen atom and accompanying $sp^{2}$ $\rightarrow$ $sp^{3}$ orbital
rehybridization of the functionalized carbon atom [39]. Analogous effects were
previously observed for the case of graphene supported on various substrates,
such as silicon carbide [40] or iridium [41]. Hence, we suggest that the
exposure of hBLG to atomic hydrogen may lead to selective hydrogenation of the
convex regions, possibly realizing the long-sought patterned chemical
functionalization of graphene that has been studied in the context of, e.g.,
band-gap engineering [41, 42].

The addition of the hydrogen adatom introduces a magnetic moment of 1
$\mu\textsubscript{B}$ in otherwise non-magnetic hBLG. Such magnetic moment
originates from the removal of a $p_{z}$ orbital from the $\pi$-electron cloud
as a consequence of the formation of the covalent C-H bond. This is in line
with Lieb’s theorem for the repulsive Hubbard model at half-filling, according
to which a lattice imbalance translates to a spin imbalance [43]. In Fig. 4b,
we show the spin density (i.e., the difference in charge density between spin-
majority and spin-minority channels) upon chemisorption of the hydrogen atom
at the stable site. Analogously to free-standing graphene, the spin density
features a $\sqrt{3}\times\sqrt{3}$ $R$ $30$ localization pattern, as it
mainly resides at the majority sublattice and decays away from the adatom [44,
45, 46]. We thus anticipate that patterned hydrogenation of hBLG may lead to a
periodic array of magnetic moments, as illustrated in Fig. 4b, that can be
regarded as a spin-half chain hosted in a moiré system [47].

## Summary and Conclusions.

We have proposed heterostrain bilayer graphene (hBLG) as an interesting
platform to achieve twist angle-free moiré patterns through the application of
uniaxial lattice deformations. This could be achieved experimentally by
different means, such as using substrates, nanoindentation, mechanical
deformations, pressure, piezoelectricity, or having suspended layers [31]. We
have shown that fully relaxed hBLG exhibits isolated bands around the Fermi
level which become dispersionless at values of strain of approximately 1%.
Similar ideas could be exploited for other types of systems, such as
metamaterials, photonic crystals [48], and artificial lattices. The
application of heterostrain leads to a substantial out-of-plane lattice
relaxation, which can be exploited to accomplish patterned chemical
functionalization [49] of the surface layers through, e.g., hydrogenation,
with implications for $\pi$-electron magnetism. To conclude, we anticipate
that the approach devised in this work can be qualitatively extended to other
van der Waals bilayers, thus opening new avenues for the exploration of moiré
physics and chemistry.

## Acknowledgments.

This research used resources of the National Energy Research Scientific
Computing Center (NERSC), operated under Contract No. DE-AC02-05CH11231 using
NERSC award BES-ERCAP0020773. G.R.S. acknowledges funding from the Army
Research Office under Cooperative Agreement Number W911NF-21-2-0147. M.P. is
financially supported by the Swiss National Science Foundation (SNSF) through
the Early Postdoc.Mobility program (Grant No. P2ELP2-191706). E.K.
acknowledges funding from the STC Center for Integrated Quantum Materials, NSF
Grant No. DMR-1231319; NSF DMREF Award No. 1922172; and the Army Research
Office under Cooperative Agreement Number W911NF-21-2-0147.

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