# Designing Moiré Patterns by Strain

Federico Escudero<EMAIL_ADDRESS>Departamento de Física,
Universidad Nacional del Sur, Av. Alem 1253, B8000, Bahía Blanca, Argentina
Instituto de Física del Sur, Conicet, Av. Alem 1253, B8000, Bahía Blanca,
Argentina  IMDEA Nanoscience, Faraday 9, 28049 Madrid, Spain  Andreas Sinner
Institute of Physics, University of Opole, 45-052 Opole, Poland  Zhen Zhan
Pierre A. Pantaleón<EMAIL_ADDRESS>IMDEA Nanoscience, Faraday 9, 28049
Madrid, Spain  Francisco Guinea  IMDEA Nanoscience, Faraday 9, 28049 Madrid,
Spain  Donostia International Physics Center, Paseo Manuel de Lardizábal 4,
20018 San Sebastián, Spain  Ikerbasque, Basque Foundation for Science, 48009
Bilbao, Spain

###### Abstract

Experiments conducted on two-dimensional twisted materials have revealed a
plethora of moiré patterns with different forms and shapes. The formation of
these patterns is usually attributed to the presence of small strains in the
samples, which typically arise during their fabrication. In this work we find
that the superlattice structure of such systems actually depends crucially on
the interplay between twist and strain. For systems composed of honeycomb
lattices, we show that this can lead to the formation of practically any moiré
geometry, even if each lattice is only slightly distorted. As a result, we
show that under strain the moiré Brillouin zone is not a stretched irregular
hexagon, but rather a primitive cell that changes according to the geometry of
the strained moiré vectors. We identify the conditions for the formation of
hexagonal moiré patterns arising solely due to shear or biaxial strain, thus
opening the possibility of engineering moiré patterns solely by strain.
Moreover, we study the electronic properties in such moiré patterns and find
that the strain tends to suppress the formation of the flat moiré bands, even
in the strain-induced hexagonal patterns analogous to those obtained by the
twist only.

##  I Introduction

The recent discovery of correlated electronic states and superconductivity in
twisted bilayer graphene (TBG)Andrei  _et al._ (2021); Cao  _et al._ (2018a,
b) has sparked a great interest in twisted moiré systems. Theoretical works in
TBG Lopes dos Santos  _et al._ (2007); Shallcross  _et al._ (2008, 2010);
Bistritzer and MacDonald (2011); de Laissardière  _et al._ (2010); Mele
(2010); Lopes dos Santos  _et al._ (2012), and transition metal dicalcogenides
(TMDs) Xian  _et al._ (2021); Angeli and MacDonald (2021); Naik and Jain
(2018); Wu  _et al._ (2018); Tang  _et al._ (2020); Regan  _et al._ (2020);
Wang  _et al._ (2020); Ni  _et al._ (2019); Xian  _et al._ (2019); Kögl  _et
al._ (2023), have demonstrated that the moiré patterns in these systems can
give rise to narrow bands that are largely responsible for the correlated
effects Cao  _et al._ (2018a, b); Yankowitz  _et al._ (2019); Lu  _et al._
(2019); Polshyn  _et al._ (2019); Sharpe  _et al._ (2019); Serlin  _et al._
(2020); Chen  _et al._ (2020); Saito  _et al._ (2020); Zondiner  _et al._
(2020); Wong  _et al._ (2020); Stepanov  _et al._ (2020); Xu  _et al._ (2020);
Choi  _et al._ (2021); Rozen  _et al._ (2021); Cao  _et al._ (2021); Stepanov
_et al._ (2021); Oh  _et al._ (2021); Xie  _et al._ (2021); Berdyugin  _et
al._ (2022); Turkel  _et al._ (2022); Huang  _et al._ (2022). The form of
these moiré patterns, however, can be highly sensitive to the presence of
strain in the system Kazmierczak  _et al._ (2021). This can have significant
effects on the electronic properties, e.g., by preventing the bands from
becoming flat around the magic angle, or by splitting the van Hove
singularities Huder  _et al._ (2018); Bi  _et al._ (2019); Mesple  _et al._
(2021); Wang  _et al._ (2023). Although the strain in superlattices
configurations typically arises randomly during the fabrication of the samples
Hsieh  _et al._ (2023); Cazeaux  _et al._ (2023), recent experimental advances
have opened the possibility of inducing and controlling, in a precise way,
different types of strain fields Peña  _et al._ (2023). This provides a
promising platform for designing moiré patterns, and tune the electronic
properties, through the interplay between twist and strain Kapfer  _et al._
(2023a).

In superlattice configurations, the effect of the strain is usually magnified
in the resulting moiré pattern Cosma  _et al._ (2014). Local variations of
strain in the samples can, indeed, lead to large changes in the moiré pattern
Engelke  _et al._ (2023). This is consistent with several recent experimental
studies where the creation of different types of moiré lattice defects have
been reported. Examples include domain walls between different stacking
domains in TBG Alden  _et al._ (2013), hexagonal boron nitride Woods  _et al._
(2021), or TMDs Shabani  _et al._ (2021). On the other hand, the effect of
strains in monolayer graphene and other non-twisted bidimensional materials
has been extensively studied Pozrikidis (2008); Vozmediano  _et al._ (2010a);
Cosma  _et al._ (2014); Amorim  _et al._ (2016); Naumis  _et al._ (2017), and
important insights on the role of strains in twisted bilayer graphene have
been described in Huder  _et al._ (2018); Qiao  _et al._ (2018); Bi  _et al._
(2019); Engelke  _et al._ (2023); Cazeaux  _et al._ (2023); Ahmed  _et al._
(2023). Interestingly, highly anisotropic moiré patterns in strained twisted
bilayer graphene have been reported in many experiments Alden  _et al._
(2013); Qiao  _et al._ (2018); Woods  _et al._ (2021); Mendoza  _et al._
(2021); Shabani  _et al._ (2021); Kazmierczak  _et al._ (2021); de Jong  _et
al._ (2022, 2023); Kapfer  _et al._ (2023b); Tilak  _et al._ (2022a). In
addition to anisotropies, almost every experiment in multilayer graphene Alden
_et al._ (2013); Jiang  _et al._ (2017); Mendoza  _et al._ (2021); Craig  _et
al._ (2023); Engelke  _et al._ (2023); Cazeaux  _et al._ (2023); Hesp  _et
al._ (2023) and TMDs Bai  _et al._ (2020); Shabani  _et al._ (2021); Tilak
_et al._ (2022a), have shown the existence of moiré patterns with different
geometries. In particular, recent experiments have shown the existence of
unconventional rectangular moire patterns in TMDs Tilak  _et al._ (2022a) and
multilayer graphene Hesp  _et al._ (2023).

Inspired by these findings, in this work we study how the interplay between
twist and strain can modify the geometrical properties of the moiré patterns.
We find that by selectively applying strain to the system one can change the
moiré patterns to practically any geometry, even at very small strain
magnitudes that only slightly distort each lattice. Exploiting a unique
transformation that determines the relative angle and length between the moiré
vectors, we develop a general theoretical scheme which allows one to describe
any strained moiré geometry. We discuss different experimentally relevant
types of strain, such as uniaxial heterostrain, shear strain and biaxial
strain. We obtain and discuss the formation of special moiré geometries, such
as the square moire patterns. We also show that hexagonal moiré patterns,
analogous to those obtained with only a twist, can be formed solely by the
application of shear or biaxial strain, thus opening the possibility of
engineering moiré patterns only by strain. Finally, we observe that the
typical irregular hexagonal cell, commonly used to describe strained honeycomb
lattices, is no longer the moiré Brillouin zone (mBZ) of the strained
superlattice. Instead, we identify a family of mBZ, with distinct geometries,
that reflect the symmetries of the superlattice.

Our geometrical analysis of strained moiré patterns overlaps with those
recently presented in Ref. Kögl  _et al._ (2023), where various types of
strain have also been examined. However, despite the similarities, our
theoretical scheme is build upon finding a unique transformation that directly
determines the geometrical properties of the moiré vectors. This allows us to
analytically study, in greater detail, what combinations of twist and strain
result in any particular moiré pattern, thus providing a firm platform from
which one can actually design moiré patterns. In addition, we develop a
comprehensive account of the strain effects in both real and reciprocal space,
and in particular discuss how these can strongly reshape the moiré BZ, which
has not been addressed before in the literature. We thus believe that our work
complements previous theoretical studies by providing a detailed account of
how the moiré patterns can be actually designed by strain.

Furthermore, our geometrical analysis is complemented by the studies of the
electronic properties. We find that the modification of the moiré patterns by
the strain plays a crucial role in the formation of flat bands around a magic
angle. We attribute this to an interplay between the shift of the Dirac points
in each deformed lattice, due to geometric and energetic effects, and the
moiré potential that couples them. The strain influences both by breaking
almost all the symmetries in the system, effectively preventing the lowest
moiré bands to flatten across the BZ. We find that this occurs even in the
hexagonal moiré patterns that arise due to strain only, and that on the moiré
scale look practically identical to those obtained with the twist only. In
these cases the strain reorganizes the charge density in the system, and leads
to the splitting and appearance of multiple high-order van Hove singularities.

The rest of the paper is organized as follows: In Section II we discuss
geometrical properties of strained moiré patterns. We describe in details how
different patterns can be achieved through the interplay between twist and
different types of strain. We also obtain how the first moiré Brillouin zone
changes under strain. In Section III we discuss the electronic properties of
the strained moiré patterns, using an extension of the continuum model in the
presence of strain. We calculate the band structure, the density of states,
and the charge density profile under different types of strain, and compare
them to the case of TBG without strain. Finally, our conclusions follow in
Section IV.

##  II Geometrical properties of strained moiré patterns

###  II.1 General considerations

We choose the lattice vectors of a honeycomb lattice as
$\mathbf{a}_{1}=a\left(1,0\right)$ and
$\mathbf{a}_{2}=a\left(1/2,\sqrt{3}/2\right)$, where $a$ is the lattice
constant ($a\simeq 2.46\>\textrm{Å}$ in graphene). In a honeycomb twisted
bilayer configuration, the usual rotation by $\pm\theta/2$, and a further
application of strain, yields
$\tilde{\mathbf{a}}_{i,\pm}=\left(\mathbf{1}+\mathcal{E}_{\pm}\right)\mathrm{R}\left(\pm\theta/2\right)\mathbf{a}_{i}$,
where $\mathrm{R}\left(\theta\right)$ is the rotation matrix and
$\mathcal{E}_{\pm}$ is the strain tensor. At small deformations (that is, to
leading order in $\mathcal{E}_{\pm}$), the reciprocal vectors can be obtained
as
$\tilde{\mathbf{b}}_{i,\pm}\simeq\left(\mathbf{1}-\mathcal{E}_{\pm}\right)\mathrm{R}\left(\pm\theta/2\right)\mathbf{b}_{i}$.
In what follows, we only restrict our discussion to small twist angles and the
practical case in which the forces act oppositely in each layer,
$\mathcal{E}_{+}=-\mathcal{E}_{-}=\mathcal{E}/2$. Then, for a general strain
tensor of the form
$\mathcal{E}=\sum_{ij}\epsilon_{ij}\left(\mathbf{e}_{i}\otimes\mathbf{e}_{j}\right)$
(where $i,j=x,y$), the moiré lattice vectors can be obtained as
$\mathbf{g}_{i}=\tilde{\mathbf{b}}_{-,i}-\tilde{\mathbf{b}}_{+,i}$, which
implies $\mathbf{g}_{i}=\mathbf{T}\mathbf{b}_{i}$, where

$\displaystyle\mathbf{T}$
$\displaystyle=\left(\mathbf{1}+\mathcal{E}/2\right)\mathrm{R}\left(-\theta/2\right)-\left(\mathbf{1}-\mathcal{E}/2\right)\mathrm{R}\left(\theta/2\right).$
(1)

We are interested in how the combination of rotation and strain changes the
geometry of the moiré patterns. The angle $\beta$ between the moiré vectors
can be determined from the symmetric transformation
$\mathbf{F}=\mathbf{T}^{\mathrm{T}}\mathbf{T}$ acting on the reciprocal
vectors $\mathbf{b}_{i}$,

$\cos\beta=\frac{\mathbf{F}\mathbf{b}_{1}\cdot\mathbf{b}_{2}}{\sqrt{\left(\mathbf{F}\mathbf{b}_{1}\cdot\mathbf{b}_{1}\right)\left(\mathbf{F}\mathbf{b}_{2}\cdot\mathbf{b}_{2}\right)}}.$
(2)

We can separate $\mathbf{F}=\mathbf{F}_{0}+\mathbf{F}_{\epsilon}$, where
$\mathbf{F}_{0}$ is the contribution due to pure rotations, and
$\mathbf{F}_{\epsilon}$ is the contribution due to the combination of rotation
and strain:

$\displaystyle\mathbf{F}_{0}$
$\displaystyle=4\sin^{2}\left(\theta/2\right)\mathbf{1},$ (3)
$\displaystyle\mathbf{F}_{\epsilon}$
$\displaystyle=\sin\theta\left(\begin{array}[]{cc}-2\epsilon_{xy}&\epsilon_{xx}-\epsilon_{yy}\\\
\epsilon_{xx}-\epsilon_{yy}&2\epsilon_{xy}\end{array}\right)+\cos^{2}\left(\theta/2\right)\mathcal{E}^{2}.$
(6)

Since $\mathbf{F}_{0}$ is a spherical tensor, a transformation by
$\mathbf{F}_{0}$ alone does not change $\beta$. This is, of course, the
situation without strain, where the honeycomb layers are only rotated and the
moiré vectors have always the same angle $\beta=2\pi/3$. However, under strain
the vectors are also transformed by the non-spherical tensor
$\mathbf{F}_{\epsilon}$, which changes the angle of $\mathbf{b}_{i}$ and hence
modifies the geometrical properties of the moiré pattern. Note that the second
term in Eq. (6) describes the possibility of obtaining moiré patterns without
rotations, i.e., purely by strain San-Jose  _et al._ (2012); Georgoulea  _et
al._ (2022).

Equations (1) to (6) constitute the central results of the geometrical part of
our study. They possess the versatility to describe a wide range of moiré
structures, relying solely on the transformation matrix $\mathbf{F}$. This
matrix can be constructed using an arbitrary strain tensor, rotation matrix,
and even lattice geometries with appropriately chosen lattice vectors. These
equations provide a concise and straightforward representation, that can also
be employed to reproduce the results presented in Ref. Kögl  _et al._ (2023).

One crucial aspect of the modification of moiré patterns under strain is that
it requires significantly smaller strain magnitudes compared to the strain
needed to modify a monolayer honeycomb lattice. This can be observed by
examining the strain required to change the angle between the corresponding
lattice vectors. Consider, for instance, the case of uniaxial heterostrain
along the $\phi=0$ direction, i.e., $\epsilon_{xx}=\epsilon$,
$\epsilon_{yy}=-\nu\epsilon$ and $\epsilon_{xy}=0$, where $\nu$ is the Poisson
ratio. As in Eq. (2), we can obtain the angle $\alpha$ between the strained
reciprocal vectors $\tilde{\mathbf{b}}_{\pm}$ through the symmetrical
transformation $\mathbf{T}_{\pm}^{\mathrm{T}}\mathbf{T}_{\pm}$, where
$\mathbf{T}_{\pm}=\left(\mathbf{1}\mp\mathcal{E}/2\right)\mathrm{R}\left(\pm\theta/2\right)$.
Then, at low twist angle and to leading order in $\epsilon$ we get

$\displaystyle\cos\alpha$
$\displaystyle\simeq-\frac{1}{2}\mp\frac{3}{16}\epsilon\left(\nu+1\right),$
(7) $\displaystyle\cos\beta$
$\displaystyle\simeq-\frac{1}{2}+\frac{3}{16}\epsilon\left(\nu+1\right)\frac{2\sqrt{3}}{\theta}.$
(8)

Thus at small values of $\theta$ one needs much smaller strain magnitudes to
modify $\beta$ than to modify $\alpha$. In fact, for experimentally relevant
values $\epsilon\lesssim 10\%$, and sufficiently low twist angles Bai  _et
al._ (2020); Kazmierczak  _et al._ (2021); Peña  _et al._ (2023), one can, in
principle, vary the angle $\beta$ to any value between $0$ and $\pi$. In
comparison, for the same strain range, the actual angle between the lattice
vectors in the monolayer varies only by just a few degrees Pereira  _et al._
(2009) (see also Naumis  _et al._ (2017) and references therein). This means
that, under the right strain parameters, the moiré patterns can be practically
changed to any desired geometry, even if the underlying honeycomb lattices are
only slightly distorted Hesp  _et al._ (2023); Kögl  _et al._ (2023). Such
behavior is possible because the moiré pattern arises from the twist angle or
the lattice mismatch between the two monolayers, and any small distortion is
enhanced by the moiré Cosma  _et al._ (2014).

Figure 1: Construction of the moiré vectors for the case of uniaxial
heterostrain with $\theta=5^{\circ}$, $\epsilon=5\%$ and $\phi=60^{\circ}$.
The strained reciprocal lattice vectors $\tilde{\mathbf{b}}_{1,\pm}$ and
$\tilde{\mathbf{b}}_{2,\pm}$ in top and bottom layer are shown in red and
blue, from which the moiré vectors
$\mathbf{g}_{i}=\tilde{\mathbf{b}}_{i,-}-\tilde{\mathbf{b}}_{i,+}$ are
obtained. The superlattice spanned by these vectors is shown on the right. In
this case the vector $\mathbf{g}_{2}$ is not the shortest one that can be
taken, since it can be translated by $\mathbf{g}_{1}$ to obtain a shorter
moiré vector $\mathbf{g}^{\prime}_{2}=\mathbf{g}_{2}+\mathbf{g}_{1}$. The
superlattice vector $\mathbf{g}^{\prime}_{2}$ arises from constructing the
second moiré vector by taking the difference between the reciprocal vectors
$\left(\tilde{\mathbf{b}}_{1,\pm}+\tilde{\mathbf{b}}_{2,\pm}\right)$. Figure
2: Moiré patterns generated by uniaxial heterostrain in a twisted honeycomb
lattice with $\theta=2{{}^{\circ}}$, for the case of equal length moiré
vectors. Panel (a): Strain parameters vary from left to right as: $\epsilon=0$
(no strain); $\epsilon\simeq 1.64\%,\phi\simeq-9.40{{}^{\circ}}$;
$\epsilon\simeq 4.44\%,\phi\simeq-0.90{{}^{\circ}}$; and
$\epsilon\simeq-2.28\%,\phi\simeq-22.7{{}^{\circ}}$. The corresponding angles
between the primitive moiré vectors, shown in black, are
$\beta_{R}=60{{}^{\circ}},90{{}^{\circ}},140{{}^{\circ}},30{{}^{\circ}}$. The
Wigner–Seitz cells of the moiré superlattices are shown in white. The insets
underneath each panel (in blue) show the strain magnitude in a scale of $5\%$,
and the strain angle relative to the non-rotated lattice orientation. Panel
(b): The evolution of the moiré pattern within the Wigner–Seitz cell for
angles (from left to right)
$\beta_{R}=40{{}^{\circ}},60{{}^{\circ}},80{{}^{\circ}},90{{}^{\circ}},100{{}^{\circ}},120{{}^{\circ}},140{{}^{\circ}}$.
Bars underneath (thick line) indicate the magnitude of the strain $\epsilon$
in a scale of $5\%$ (thin line). As $\epsilon$ increases, the stretch of the
AA stacking within the primitive cell increases. As a result, the shape of two
moiré patterns with the same periodicity is not the same if they correspond to
different strain magnitudes.

It is important to note that, under strain, the moiré vectors obtained by
using a unique construction (e.g., by the difference between
$\tilde{\mathbf{b}}_{-,i}$ and $\tilde{\mathbf{b}}_{+,i}$) may not be the
smallest ones (the so called primitives). Consequently, for arbitrary strain
parameters the moiré vectors may not reflect the symmetries of the
corresponding moiré geometry. For example, a square moiré pattern can result
from equal length moiré vectors with an angle $\beta=\pi/2$ between them, but
also from moiré vectors with an angle $\beta=\pi/4$ and relative length
$\left|\mathbf{g}_{1}\right|/\left|\ \mathbf{g}_{2}\right|=\sqrt{2}$. In
general, for arbitrary strain parameters, the primitive moiré vectors are
obtained by appropriately changing the set of reciprocal lattice vectors from
which they are constructed (e.g., by taking the difference between the
deformed vectors $\mathbf{b}_{1}$ and $\mathbf{b}_{1}+\mathbf{b}_{2}$, rather
than $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$; see Fig. 1). These different
constructions of $\mathbf{g}_{i}$ eventually reflect the underlying symmetries
of the honeycomb lattices.

Furthermore, although the form of the moiré patterns between nonstrained
honeycomb lattices are uniquely determined by the periodicity of the
superlattice, the same is not true under strain. Two set of symmetric moiré
vectors, which technically describe the same superlattice, may actually
correspond to different forms of the moiré pattern. The reason is that the
_stretch_ of the AA stacking, which is periodically repeated by the moiré
vectors, increases under strain Kapfer  _et al._ (2023a); Mendoza  _et al._
(2021); Hesp  _et al._ (2023). The effect of the strain on the moiré patterns
thus acts not only on the modification of the moiré periodicity, but also on
how the form of the stacking shape is repeated. This behavior is, in a way,
similar to the usual description of crystal structures through a basis within
a primitive cell and a set of Bravais vectors that repeat such a basis.

Given the reciprocal vectors $\mathbf{g}_{i}$, the primitive moiré lattice
vectors $\mathbf{g}_{i}^{R}$ are most easily obtained by the relation
$\mathbf{g}_{i}\cdot\mathbf{g}_{j}^{R}=2\pi\delta_{ij}$, which implies
$\mathbf{g}_{i}^{R}=\mathbf{T}^{-\mathrm{T}}\mathbf{a}_{i}$, where
$\mathbf{T}$ is given by Eq. (1). Thus, the geometrical properties of the
primitive moiré vectors are determined by the inverse transformation
$\mathbf{F}^{-\mathrm{T}}=\mathbf{F}^{-1}$. In particular, the angle between
the primitive vectors is $\beta_{R}=\pi-\beta$, where $\beta$ is the angle in
reciprocal space given by Eq. (2).

In what follows we discuss in details the geometrical properties of the moiré
patterns under three important kinds of strain: the uniaxial heterostrain, the
biaxial strain, and the shear strain. It is worth mentioning that our
formalism, applied here to moiré heterostructures which arise from honeycomb
lattices, can be directly extended to other geometries by appropriately
modifying the lattice vectors and the strain tensor Liu  _et al._ (2018).

###  II.2 Uniaxial heterostrain

The uniaxial heterostrain refers to a type of strain that is applied along a
unique axis, and acts oppositely in each honeycomb lattice. From the
experimental point of view it is widely regarded as the most relevant kind of
strain in TBG. It was first introduced both theoretically and experimentally
in Ref. Huder  _et al._ (2018), and then further investigated in Refs. Qiao
_et al._ (2018); Bi  _et al._ (2019); Mesple  _et al._ (2021); Gao  _et al._
(2021); Kazmierczak  _et al._ (2021). The developed approach can,
nevertheless, be directly extended to other types of strain, as discussed in
the next sections.

The strain tensor of uniaxial heterostrain with magnitude $\epsilon$, along an
angle $\phi$ relative to the $x$ axis, reads

$\displaystyle\mathcal{E}$
$\displaystyle=\mathrm{R}^{\mathrm{T}}\left(-\phi\right)\left(\begin{array}[]{cc}\epsilon&0\\\
0&-\nu\epsilon\end{array}\right)\mathrm{R}\left(-\phi\right)$ (11)
$\displaystyle=\epsilon\left[\begin{array}[]{cc}\cos^{2}\phi-\nu\sin^{2}\phi&\left(1+\nu\right)\sin\phi\cos\phi\\\
\left(1+\nu\right)\sin\phi\cos\phi&\sin^{2}\phi-\nu\cos^{2}\phi\end{array}\right].$
(14)

The transformation matrix $\mathbf{F}$ then becomes

$\displaystyle\mathbf{F}$
$\displaystyle=4\sin^{2}\left(\frac{\theta}{2}\right)\mathbf{1}+\epsilon\left(1+\nu\right)\sin\left(\theta\right)\mathrm{R}\left(2\phi\right)\sigma_{x}$
$\displaystyle+\cos^{2}\left(\frac{\theta}{2}\right)\frac{\epsilon^{2}}{2}\left[\left(1+\nu^{2}\right)\mathbf{1}+\left(1-\nu^{2}\right)\mathrm{R}\left(2\phi\right)\sigma_{z}\right],$
(15)

where $\sigma_{i}$ are the Pauli matrices. From here one can readily see that
the solutions of Eq. (2) for the strain magnitude always scale with the twist
angle as $\sim\tan\left(\theta/2\right)$. Indeed, by writing
$\epsilon=\epsilon^{\prime}\tan\left(\theta/2\right)$ it follows that
$\mathbf{F}\propto\sin^{2}\left(\theta/2\right)$ and, consequently, that the
angle equation for $\beta$ as function of $\epsilon^{\prime}$ is independent
of the twist angle. Thus, for any $\beta$ and $\phi$, the solutions of Eq. (2)
for the strain magnitude have the form
$\epsilon\propto\tan\left(\theta/2\right)$. What this general result reflects
is that the lower the twist angle, the weaker strain is needed to modify the
geometry of the moiré superlattices.

####  II.2.1 Equal length moiré vectors

In the following, to simplify our analysis, we focus on the moiré patterns
formed by equal-length moiré vectors, i.e., on the structures with
$\left|\mathbf{g}_{1}\right|=\left|\mathbf{g}_{2}\right|$. This choice allows
for analytical solutions, which can be used to analyze the geometrical
effects. The consideration of moiré vectors with different lengths is a
straightforward extension of our analysis, as described in a following
section. From Eq. (15), the equal-length moiré vector condition for non-zero
strain is given by

$\epsilon_{\mathrm{eq}}=\frac{4}{1-\nu}\cot\left(\frac{\pi}{3}-2\phi\right)\tan\left(\frac{\theta}{2}\right).$
(16)

Since $\epsilon_{\mathrm{eq}}\propto\tan\left(\theta/2\right)$ and thus
$\mathbf{F}\propto\sin^{2}\left(\theta/2\right)$, Eq. (2) for
$\epsilon=\epsilon_{\mathrm{eq}}$ does not depend on $\theta$. This is a
rather remarkable result: it means that the strain direction $\phi$ needed to
obtain the equal length moiré vectors, with an angle $\beta$ between them, is
independent of the twist angle. The twist angle only modifies the needed
strain magnitude, and the resulting (equal) length of the moiré vectors,
$\left|\mathbf{g}_{i}\right|^{2}=\mathbf{F}\mathbf{b}_{i}\cdot\mathbf{b}_{i}\propto\sin^{2}\left(\theta/2\right)$
(as in the unstrained case). Note that Eq. (16) is not invariant under the
transformation $\phi\rightarrow\phi+\pi/3$ because we are not taking here
other solutions into account, that are obtained by appropriately changing the
construction of the primitive moiré vectors (see Appendix A).

As detailed in Appendix B, by solving Eq. (2) for $\phi$ one can get the
required strain parameters, in order to obtain the equal length moiré vectors
with the angle $\beta$ between them. At low twist angles, the corresponding
strain magnitudes are relatively small and well within the experimental range
Peña  _et al._ (2023). Some moiré patterns that can be formed under uniaxial
heterostrain are shown in Fig. 2. In general, the moiré patterns are quite
sensitive to the values of the strain parameters, in the sense that small
changes in $\epsilon$ and $\phi$ can, in contrast, result in significant
changes in the geometry of the moiré vectors Alden  _et al._ (2013); Engelke
_et al._ (2023). Thus, the precise control over the magnitude and direction of
the applied uniaxial heterostrain is crucial for designing moiré patterns
through the strain manipulation. It is worth noting that this control has
already been achieved experimentally. Reference Peña  _et al._ (2023)
describes a methodology for process-induced strain engineering, where the
strain magnitude and direction in TBG can be controlled.

Fig. 2 also shows that the orientation of the Wigner–Seitz cell, and of the
stretched AA stacking within it, change depending on the strain magnitude.
This is because the strain modifies not only the angle between the moiré
vectors, but also their orientation with respect to the non-strain case. For
instance, in the strained case with $\beta_{R}=120^{\circ}$, the hexagonal
primitive cell is rotated with respect to the same cell in the non-strain
case. In general, the stretch of the AA stacking occurs along the direction of
the moiré vector $\mathbf{g}_{1}^{R}\pm\mathbf{g}_{2}^{R}$, where $+$ ($-$)
when $\beta_{R}<90^{\circ}$ ($\geq 90^{\circ}$). Such direction always
coincides with one corner of the Wigner–Seitz cell. The angle $\phi_{s}$ of
the stretching can thus be estimated as

$\cos\phi_{s}=\frac{\mathbf{g}_{1}^{R}\pm\mathbf{g}_{2}^{R}}{\left|\mathbf{g}_{1}^{R}\pm\mathbf{g}_{2}^{R}\right|}\cdot\mathbf{e}_{x}.$
(17)

Note that $\phi_{s}$ generally differs from the strain angle $\phi$, i.e., the
observed stretch of the AA stacking does not reflect the direction along which
the uniaxial heterostrain is applied. It only reflects the magnitude of the
applied strain. Since when $\epsilon\propto\tan\left(\theta/2\right)$ one has
$\mathbf{T}\propto\sin\left(\theta/2\right)$ [cf. Eqs. (14) and (1)], it
follows that for any strain direction $\phi$ that yields an angle $\beta_{R}$
between the moiré vectors, the corresponding stretch angle $\phi_{s}$ is
independent of the twist angle $\theta$. The above analysis may allow one to
estimate the strain properties of twisted bilayer honeycomb samples by
analysing only the shape of the AA regions.

####  II.2.2 Special moiré patterns

Figure 3: Moiré patterns under uniaxial heterostrain with unequal length
superlattice vectors, for twist angle $\theta=2{{}^{\circ}}$. The strain
parameters are (a) $\epsilon=1\%,\phi=0{{}^{\circ}}$ and (b)
$\epsilon=3.5\%,\phi=20{{}^{\circ}}$. The corresponding relative length and
angle between the moiré vectors are (a)
$\left|\mathbf{g}_{1}^{R}\right|/\left|\mathbf{g}_{2}^{R}\right|\simeq
1.11,\beta_{R}\simeq 72.5^{\circ}$ and (b)
$\left|\mathbf{g}_{1}^{R}\right|/\left|\mathbf{g}_{2}^{R}\right|\simeq
2.57,\beta_{R}\simeq 97.8^{\circ}$. The respective strain magnitude $\epsilon$
(in blue) and the strain angle direction $\phi$ relative to the non-rotated
lattice orientation is shown in a scale of $4\%$ underneath each panel.

Some special moiré patterns that may be accomplished deserve further
discussion. One case is the square moiré pattern shown in Fig. 2a). Squared-
like moiré patterns have already been experimentally observed Jiang  _et al._
(2017); Tilak  _et al._ (2022b); Yu  _et al._ (2022), and theoretically
predicted Kögl  _et al._ (2023). While their shape has been attributed to
highly distorted moiré patterns, our model indicates that this geometry can be
alternatively obtained by the right combination of twist angle and strain.
Another interesting case occurs when $\beta_{R}=120^{\circ}$, where one can
have the same hexagonal moiré periodicity as in the non-strain (where
$\beta_{R}=60^{\circ}$), albeit with a stretched AA stacking within the
primitive cell (see Fig. 2b).

A particularly relevant case is the critical limit in which the moiré vectors
become collinear. This can lead to quasi-unidimensional channels that have
been predicted Cosma  _et al._ (2014); Sinner  _et al._ (2023) and observed in
several experiments Alden  _et al._ (2013); Woods  _et al._ (2021); Mendoza
_et al._ (2021); Shabani  _et al._ (2021); Kazmierczak  _et al._ (2021); de
Jong  _et al._ (2022, 2023); Kapfer  _et al._ (2023b); Tilak  _et al._
(2022a); Hsieh  _et al._ (2023); Craig  _et al._ (2023); Peña  _et al._
(2023); Engelke  _et al._ (2023); Bai  _et al._ (2020); Liu  _et al._ (2018).
Plugging $\beta=\\{0,\pi\\}$ into Eq. (49) yields a critical strain parameter
Sinner  _et al._ (2023)

$\epsilon_{c}=\pm\frac{2}{\sqrt{\nu}}\tan\left(\frac{\theta}{2}\right).$ (18)

This expression for $\epsilon_{\mathrm{c}}$ is actually quite general, i.e. it
always leads to the collinear moiré vectors, regardless of the strain angle
$\phi$ Sinner  _et al._ (2023); Kögl  _et al._ (2023). Technically, this is
because at this critical strain the determinant of the matrix $\mathbf{F}$
vanishes, which means that it becomes non-invertible and the moiré vectors are
no longer linearly independent.

####  II.2.3 Arbitrary strain parameters

The general situation of arbitrary strain parameters (within the limit of
small deformations) is, in many ways, qualitatively very similar to the
special case of equal length moiré vectors. By fixing, for example, the strain
angle to $\phi=0$, one can still obtain many different geometries in which the
angle between the moiré vectors can be tuned solely by changing the strain
magnitude. Examples of such moiré patterns are shown in Fig. 3. There one sees
that, although the length of one moiré vector may be more than double the
length of the other one, the moiré patterns follows a similar behavior to the
simpler ones analyzed in Fig. 2. Thus our discussion in the previous section
is readily generalized to arbitrary strain parameters. In particular, the
symmetry of the moiré superlattice, and the magnitude of the strain, are
always reflected in the shape of the Wigner–Seitz cell, and the stretch of the
AA stacking within it. Furthermore, the direction of the AA stretching also
follows, in general, the direction of the moiré vector
$\mathbf{g}_{1}^{R}\pm\mathbf{g}_{2}^{R}$ [cf. Eq. (17)].

In the case of pure uniaxial heterostrain, without a twist, the transformation
given by Eq. (15) reduces to

$\mathbf{F}=\frac{\epsilon^{2}}{2}\left[\left(1+\nu^{2}\right)\mathbf{1}+\left(1-\nu^{2}\right)\mathrm{R}\left(2\phi\right)\sigma_{z}\right].$
(19)

Since the second term is not a spherical tensor, the resulting moiré pattern
is not hexagonal.

###  II.3 Shear strain

Figure 4: Hexagonal moiré patterns generated only by: (a) twist angle
$\theta=5^{\circ}$; (b) shear strain with magnitude
$\epsilon_{s}=2\sin\left(\theta/2\right)\simeq 8.7\%$; (c) biaxial strain with
magnitude $\epsilon_{b}=2\sin\left(\theta/2\right)\simeq 8.7\%$. From Eqs.
(28) and (32), all cases have the same moiré periodicity. The figures in the
bottom row visualize the enlarged Wigner–Seitz cells. Here, the vicinity of
the AA, AB and BA stacking positions looks different for each case. This
difference, however, becomes smaller (and practically unnoticeable at the
moiré scale) as the twist and strain decrease. Panel (d) shows schematically
the corresponding deformations in the bottom (left) and top (right) lattices
due to (from top to bottom panels) rotation, shear strain, and biaxial strain.
The effects are exaggerated for better visualization.

In a honeycomb lattice, shear strain occurs when forces act parallel to its
surface but in opposite directions. This leads to a distortion of the lattice.
In simpler terms, shear strain in a honeycomb lattice is like the “sliding”
forces that deform the lattice without altering its overall volume, cf. Fig.
4d). This kind of strain has been studied in graphene and transition metal
dicalcohenides Hovden  _et al._ (2014); Liu  _et al._ (2012); Lee  _et al._
(2017); Wang  _et al._ (2017).

The strain tensor due to shear forces applied perpendicularly to an angle
direction $\varphi$ is given by

$\displaystyle\mathcal{E}$
$\displaystyle=\mathrm{R}^{\mathrm{T}}\left(-\varphi\right)\left(\begin{array}[]{cc}0&\epsilon_{xy}\\\
\epsilon_{yx}&0\end{array}\right)\mathrm{R}\left(-\varphi\right)$ (22)
$\displaystyle=\epsilon_{s}\left(\begin{array}[]{cc}-\sin 2\varphi&\cos
2\varphi\\\ \cos 2\varphi&\sin 2\varphi\end{array}\right),$ (25)

where $\epsilon_{xy}=\epsilon_{yx}=\epsilon_{s}$ is the shear strain
magnitude. For a twisted bilayer lattice, this leads to the transformation

$\displaystyle\mathbf{F}$
$\displaystyle=\left[4\sin^{2}\left(\frac{\theta}{2}\right)+\epsilon_{s}^{2}\cos^{2}\left(\frac{\theta}{2}\right)\right]\mathbf{1}$
$\displaystyle\qquad-2\sin\left(\theta\right)\epsilon_{s}\mathrm{R}\left(2\varphi\right)\sigma_{z}.$
(26)

The second term implies that the combined effect of twist and shear strain can
change the geometry of the moiré patterns, similar to the effect of uniaxial
heterostrain. The main difference lies in how the distortion of each honeycomb
lattice gives rise to a particular moiré geometry. Thus, although the moiré
patterns for different strain types may appear similar, their electronic
properties can be substantially different (cf. Fig. 8). In particular, shear
forces can also lead to a critical case in which the moiré vectors become
collinear. This occurs when the determinant of the transformation given by Eq.
(26) vanishes, which implies

$\epsilon_{s,c}=\pm 2\tan\left(\frac{\theta}{2}\right),$ (27)

independently of the shear angle $\varphi$. This critical shear strain is
$\sim\sqrt{\nu}\simeq 0.4$ times smaller than the one required for the case of
uniaxial heterostrain.

An interesting situation occurs in the case of pure shear forces without a
twist angle, where Eq. (26) reduces to
$\mathbf{F}=\epsilon_{s}^{2}\mathbf{1}$. This transformation acts as in the
twisted non-strain case, where
$\mathbf{F}=4\sin^{2}\left(\theta/2\right)\mathbf{1}$, with the resulting
moiré pattern being always hexagonal. This means that one can form moiré
patterns without any twist between the layers, just by applying opposite shear
forces in each lattice, thus opening the possibility of engineering
superlattice heterostructures purely by strain (cf. Fig 4b). The shear angle
$\varphi$ only changes the orientation of the moiré pattern. Interestingly,
the moiré superlattice with pure shear strain can have the same periodicity as
that of TBG with twist angle $\theta$ if the strain magnitude satisfy

$\epsilon_{s}=2\sin\left(\frac{\theta}{2}\right).$ (28)

For example, a strain magnitude $\epsilon_{s}\sim 1.8\%$ yields a moiré
periodicity $L\sim 13.4$ nm, corresponding to an equivalent twist angle
$\theta\sim 1.05^{\circ}$.

###  II.4 Biaxial strain

In the case of biaxial strain the forces are equally applied along the $x$ and
$y$ directions, and in opposite directions in each layer. The corresponding
strain tensor reads $\mathcal{E}=\epsilon_{b}\mathbf{1}$, thus yielding the
transformation matrix

$\displaystyle\mathbf{F}$
$\displaystyle=\left[4\sin^{2}\left(\frac{\theta}{2}\right)+\epsilon_{b}^{2}\cos^{2}\left(\frac{\theta}{2}\right)\right]\mathbf{1}.$
(29)

Since $\mathbf{F}$ is always a spherical tensor, a biaxial strain cannot
change the moiré geometry: any combination of strain and twist always results
in a hexagonal moiré pattern. This is, of course, expected because the biaxial
strain does not distort the hexagonal lattices, it only changes the size of
the primitive cell. The effect of twist and strain, in this case, is to only
modify the orientation and length of the superlattice vectors.

The change of orientation can be measured in relation to the direction of the
moiré vectors in the case of no strain, where, according to our reference
convention, the second moiré vector in reciprocal space is always along the
$x$ axis,
$\mathbf{g}_{2}=8\pi\sin\left(\theta/2\right)/\sqrt{3}a\mathbf{e}_{x}$ [cf.
Eq. (1)]. In the case of biaxial strain, this moiré vector becomes
$\mathbf{g}_{2}=\left(8\pi/\sqrt{3}a\right)\sin\left(\theta/2\right)\left[\mathbf{e}_{x}+\epsilon\cot\left(\theta/2\right)\mathbf{e}_{y}/2\right]$,
so its angle $\alpha_{\epsilon}$ with respect to the $x$ axis reads

$\cos\alpha_{\epsilon}=\frac{1}{\sqrt{1+\frac{\epsilon^{2}}{4}\cot^{2}\left(\theta/2\right)}}.$
(30)

By comparing Eqs. (3) and (29) one can obtain the combinations of strain
magnitude $\epsilon_{b}$ and twist angle $\theta_{\epsilon}$ that gives the
same moiré periodicity as with only a twist angle $\theta$,

$\sin^{2}\left(\theta_{\epsilon}/2\right)=\frac{\sin^{2}\left(\theta/2\right)-\epsilon_{b}^{2}/4}{1-\epsilon_{b}^{2}/4}.$
(31)

This condition does not, however, guarantee that both moiré patterns would be
align, since their orientation may differ due to the strain effect. This can
be important when one seeks an alignment between two (or more) moiré patterns
arising from a combination of rotation and lattice mismatch.

A relevant example occurs in heterostructures of TBG/hBN in which hBN acts as
a substrate of TBG Yankowitz  _et al._ (2012); Xue  _et al._ (2011); Decker
_et al._ (2011). In this case, the lattice mismatch between graphene
($a_{g}=2.46\;\textrm{Å}$) and hBN ($a_{h}=2.50\;\textrm{Å}$) can be accounted
as a biaxial strain with magnitude $\epsilon_{b}\sim 1-a_{T}/a_{B}=0.016$. If
the twist angle in TBG is $\theta_{T}$, and the twist angle between hBN and
the graphene layer directly on top is $\theta_{b}$, a _moiré_ _alignment_
implies that both moiré patterns have the same orientation and periodicity.
Since in TBG the layers are only rotated, the orientation condition is
obtained from Eq. (30) by setting $\cos\alpha_{\epsilon}=\pm 1/2$, which gives
$\theta_{b}\simeq\epsilon_{b}/\sqrt{3}\sim 0.53^{\circ}$. Then the equal
periodicity condition, Eq. (31), implies that the twist angle in TBG should be
$\theta_{T}\simeq\sqrt{\theta_{b}^{2}+\epsilon_{b}^{2}}\sim 1.06^{\circ}$, in
agreement with previous calculations Cea  _et al._ (2020); Long  _et al._
(2022, 2023). We emphasize that this is the _only_ twist angle in TBG for
which one can have a perfect moiré pattern alignment (or a single moiré) with
a hBN substrate. As this is only a geometrical condition, it is quite
remarkable that it occurs practically at the magic angle where the bands in
TBG tend to become flat.

In the particular case of pure biaxial strain, with no twist, Eq. (29) reduces
to $\mathbf{F}=\epsilon_{b}^{2}\mathbf{1}$. In fact, according to Eq. (1) one
simply has $\mathbf{g}_{i}=\epsilon_{b}\mathbf{b}_{i}$, i.e., the moiré
vectors are just the reciprocal vectors scaled by the biaxial strain
magnitude. Thus, in contrast to the cases of only a twist or shear strain, the
moiré BZ for only biaxial strain has the same orientation as the BZ of the
honeycomb lattices (see Fig. 7). Similarly to the case of pure shear strain,
the resulting moiré pattern has the same hexagonal periodicity as with only a
twist angle $\theta$ when

$\epsilon_{b}=2\sin\left(\frac{\theta}{2}\right).$ (32)

However, the moiré orientation with only biaxial strain is rotated
$90^{\circ}$ with respect to the case of only a twist angle, see Eq. (30).

A comparison between hexagonal moiré patterns formed by only a twist, and only
shear or biaxial strain, can be seen in Fig. 4. Although in all situations the
moiré patterns look practically the same at the moiré scale, the local
distortions of each honeycomb lattice can be significantly different. Note
that only in the cases of a pure twist or a pure biaxial strain, the moiré
patterns have $\mathcal{C}_{3}$ rotational symmetry.

###  II.5 Deformation of the Brillouin zone

In the reciprocal space, the most symmetrical primitive cell is given by the
first Brillouin zone, which is constructed by considering the set of points
that can be reached from the origin without crossing a Bragg plane (lines in
the 2D case). The moiré patterns discussed in the previous sections imply that
such cell would drastically change its shape under the application of strain.
Consider, for example, the hexagonal BZ of a honeycomb lattice. In terms of
the reciprocal vectors, it can be obtained by the union of the points

$\displaystyle\mathbf{q}_{1}$
$\displaystyle=-\frac{2\mathbf{g}_{1}+\mathbf{g}_{2}}{3},$
$\displaystyle\mathbf{q}_{2}$ $\displaystyle=\mathbf{q}_{1}+\mathbf{g}_{1},$
(33) $\displaystyle\mathbf{q}_{3}$
$\displaystyle=\mathbf{q}_{1}+\mathbf{g}_{1}+\mathbf{g}_{2},$

and their negatives. This construction holds in general for the moiré pattern
of a twisted bilayer superlattice without strain, since then the two lattices
are only relatively rotated. However, when the lattices are deformed, the
construction through the six vectors $\mathbf{q}_{i}$ yields a deformed
hexagon which is no longer the first BZ. The same holds for the moiré
superlattice.

Although the construction through Eq. (33) still gives a unit cell in
reciprocal space, such cell does not reflect the symmetries of the strained
moiré patterns. Specifically, we refer to the symmetries relating the AA and
AB stacking positions of the moiré patterns, as seen in Fig. 2. The correct
construction of the moiré BZ (mBZ) under strain requires a generalization of
Eq. (33) for the case in which the lattice vectors can have any angle and
length. Following our previous discussion, we will focus on the situations in
which the lattice vectors have equal length. In that case, the points that
determine the mBZ are given by

$\displaystyle\mathbf{Q}_{1}$
$\displaystyle=-\frac{\left(1+2\chi\right)\mathbf{g}_{1}-\lambda\mathbf{g}_{2}}{2\left(1+\chi\right)},$
$\displaystyle\mathbf{Q}_{2}$ $\displaystyle=\mathbf{Q}_{1}+\mathbf{g}_{1},$
(34) $\displaystyle\mathbf{Q}_{3}$
$\displaystyle=\mathbf{Q}_{1}+\mathbf{g}_{1}-\lambda\mathbf{g}_{2},$

where
$\chi=\left|\mathbf{g}_{1}\cdot\mathbf{g}_{2}\right|/\left|\mathbf{g}_{1}\cdot\mathbf{g}_{1}\right|$
and
$\lambda=\mathrm{sign}\left(\mathbf{g}_{1}\cdot\mathbf{g}_{2}\right)+\delta_{0,\mathbf{g}_{1}\cdot\mathbf{g}_{2}}$
(see the appendix C for details). It is easy to see that the points in Eq.
(33) reduce to those given in the above equation only for a hexagonal lattice
with $\beta=2\pi/3$. Note that for $\beta=\pi/2$ the six points are reduced to
four because $\mathbf{Q}_{1}=-\mathbf{Q}_{3}$, thus resulting in a square mBZ.
In Fig. 5 we show the evolution of the mBZ with the applied strain. A
comparison is made with the deformed hexagon calculated with the points
$\mathbf{q}_{i}$ given by Eq. (33).

Figure 5: Evolution of the Brillouin zone for different angles $\beta$ between
equal length superlattice vectors (shown in black). Each respective mBZ
constructed with vectors from Eqs. (34) is shown in blue, while the deformed
hexagons constructed using vectors from Eqs. (33) is shown in red. Both
constructions coincide only in the non-strain limit where
$\beta=120{{}^{\circ}}$. With strain, the deformed hexagons do no longer
capture the full symmetry of the moiré patterns. In particular, only the mBZ
is symmetric around $\beta=90{{}^{\circ}}$, since it corresponds to the same
moiré pattern rotated by $180{{}^{\circ}}$.

The mBZ, and its counterpart in real space (the Wigner–Seitz cell shown in
Fig. 2), provides a direct visualization of the geometrical properties of the
moiré patterns under strain. This becomes clear by analyzing the shapes of the
mBZ in Fig. 5, which follow a distinct pattern depending on the angle $\beta$.
In contrast, the deformed hexagon cell only reflects the magnitude of the
strain in the system (i.e., the larger the strain, the longer the deformed
hexagon gets), similarly as how the AA stacking stretches in real space (see
Fig. 2b). This behavior has been used to characterize the moiré patterns under
strain, e.g., by reshaping the deformed hexagons to a regular form Bi  _et
al._ (2019). We believe, however, that the alternative way of looking at the
moiré patterns, by considering the mBZ or the Wigner–Seitz cell in real space,
gives a clearer representation of the strained superlattice geometry. As noted
in Sec. II.1, the underlying distortion of the honeycomb lattices, and thus of
the magnitude of the strain, is reflected in the stretch of the AA stacking
within the primitive cell. Furthermore, the reshaping of the mBZ may
complement the approach in Ref. Wang  _et al._ (2023), where strain-induced
open Fermi surfaces in a distorted honeycomb cell were proposed to explain the
unusual magnetotransport experiments in Ref. Finney  _et al._ (2022). However,
the impact of mBZ reshaping due to strain on magnetotransport experiments
remains an open question.

##  III Electronic properties of strained moiré lattices

Figure 6: Evolution of the band structure of heteroaxially strained TBG with a
twist angle $\theta=1.05^{\circ}$, for different angles $\beta$ between equal
length moiré vectors. The strain parameters are: a) $\epsilon=0$ (no strain),
b) $\epsilon\simeq 0.4\%$, $\phi=-12.0^{\circ}$ and c) $\epsilon\simeq 0.8\%$,
$\phi=-9.40^{\circ}$. The angles between the reciprocal moiré vectors are
$\beta=120^{\circ},105^{\circ}\text{ and }90^{\circ}$, respectively.
Underneath each 3D plot we show the corresponding mBZ, and to their side the
corresponding density plot of the lower (bottom panel) and upper (top panel)
middle band. The respective mBZ constructed with vectors defined in Eq. (34)
are shown in white.

###  III.1 Effective continuum models

While the shape and form of strained moiré patterns only reflect the
geometrical differences between deformed lattices, the electronic properties
reflect other important consequences of the strain, such as the shift of the
Dirac points, the influence of the moiré potential that couples them, and the
splitting of the van Hove singularities, among others Mesple  _et al._ (2021);
Huder  _et al._ (2018). For sufficiently low twist angles, these properties
can be captured by a direct extension of the continuum model Lopes dos Santos
_et al._ (2007); Bistritzer and MacDonald (2011); Lopes dos Santos  _et al._
(2012) in the presence of strain Mesple  _et al._ (2021); Huder  _et al._
(2018); Bi  _et al._ (2019).

First, we note that under strain the mBZ and the position of the Dirac points
in each lattice change. As a result, the latter in general do not coincide
with the high symmetry points at borders of the mBZ. At small deformations,
the new positions of the Dirac points in the $\zeta$ valley of the $\ell=\pm$
layer are given by

$\mathbf{D}_{\ell,\zeta}\simeq\left(\mathbf{1}-\ell\mathcal{E}/2\right)\mathrm{R}\left[\ell\theta/2\right]\mathbf{K}_{\zeta}-\ell\zeta\mathbf{A},$
(35)

where $\mathbf{K}_{\zeta}=-\zeta\left(2\mathbf{b}_{1}+\mathbf{b}_{2}\right)/3$
is the Dirac point in the undeformed honeycomb lattice, and

$\mathbf{A}=\frac{\sqrt{3}}{2a}\beta\left(\epsilon_{xx}-\epsilon_{yy},-2\epsilon_{xy}\right)$
(36)

is the strain-induced gauge potential ($\beta_{G}\simeq 3.14$ is the Grüneisen
parameter) Vozmediano  _et al._ (2010b). The two terms in Eq. (35) represent
the combined effect of the strain on the position of the Dirac points: the
first term gives the shift due to the geometrical distortion of the lattice,
while the second term gives the shift due to the change in the hopping
energies. In addition, strains can also lead to scalar (or deformation)
potentials Mañes (2007); Suzuura and Ando (2002); Vozmediano  _et al._ (2010a)

$V=g\left(\epsilon_{xx}+\epsilon_{yy}\right).$ (37)

We use $g=4$ eV for monolayer graphene Choi  _et al._ (2010). The
aforementioned potential is incorporated into the diagonal elements of the
Dirac Hamiltonian, resulting in a vertical energy displacement of the Dirac
cones within each monolayer. This phenomenon resembles the responses observed
under the influence of a perpendicular electric field Moon  _et al._ (2014).

In a TBG configuration, at low twist angles and strain magnitudes the low
energy physics is dominated by states near the shifted Dirac points
$\mathbf{D}_{\ell,\zeta}$. The continuum model Hamiltonian under strain, for
the $\zeta$ valley, then reads

$H_{\zeta}=\left[\begin{array}[]{cc}h_{-,\zeta}\left(\mathbf{q}\right)-V\mathbf{1}&U^{\dagger}\left(\mathbf{r}\right)\\\
U\left(\mathbf{r}\right)&h_{+,\zeta}\left(\mathbf{q}\right)+V\mathbf{1}\end{array}\right].$
(38)

Here $h_{\ell,\zeta}\left(\mathbf{q}\right)$ is the Dirac Hamiltonian in the
$\ell$ layer,

$h_{\ell,\zeta}\left(\mathbf{q}\right)=-\hbar
v_{F}\bm{\sigma}_{\zeta}\cdot\mathrm{R}^{\mathrm{T}}\left(\ell\theta/2\right)\left(1+\ell\mathcal{E}/2\right)\left(\mathbf{q}-\mathbf{D}_{\ell,\zeta}\right),$
(39)

where $v_{F}$ is the Fermi velocity and
$\bm{\sigma}_{\zeta}=\left(\zeta\sigma_{x},\sigma_{y}\right)$. The coupling
between the layers is given by the matrix $U\left(\mathbf{r}\right)$ in the
non-diagonal terms of $H_{\zeta}$; its leading order Fourier expansion is

$U\left(\mathbf{r}\right)=U_{0}+U_{1}e^{i\zeta\mathbf{g}_{1}\cdot\mathbf{r}}+U_{1}^{\mathrm{T}}e^{i\zeta\left(\mathbf{g}_{1}+\mathbf{g}_{2}\right)\cdot\mathbf{r}},$
(40)

where $\mathbf{g}_{i}$ are the strained moiré vectors and

$U_{0}=\left(\begin{array}[]{cc}u_{0}&u_{1}\\\
u_{1}&u_{0}\end{array}\right),\quad
U_{1}=\left(\begin{array}[]{cc}u_{0}&u_{1}\omega^{-\xi\zeta}\\\
u_{1}\omega^{\xi\zeta}&u_{0}\end{array}\right),$ (41)

where $\omega=e^{i2\pi/3}$, and $u_{1}$, $u_{0}$ are the $AB$ and $AA$ hopping
energies, respectively. For the numerical calculations, we use
$u_{1}=u_{2}=90$ meV and $\hbar v_{F}/a=2.135$ eV. In the matrix $U_{1}$ we
have introduced a factor $\xi=\pm 1$ that accounts for the phase of the three
leading order momentum transfers between the shifted Dirac points in each
layer. This phase is contingent upon the specific type of strain. In
particular, $\xi=1$ in Fig. 7a) and c) for TBG and pure biaxial strain,
respectively, and $\xi=-1$ in Fig. 7b) for pure shear strain.

Figure 7: Reciprocal space representation of superlattice structures with
hexagonal unit cells. Large hexagons represent the BZ of each monolayer (in
red and blue). The moiré BZ are represented by the small black hexagons. The
figure shows the structures for: a) pure twist angle, b) pure shear strain and
c) pure biaxial strain. The corresponding mBZ and the hopping processes
between the Dirac cones of each graphene monolayer are displayed at the
bottom. Arrows indicate the direction of the momentum transfers between Dirac
points.

###  III.2 Electronic structure: twist and strain

We first consider the case of TBG with uniaxial heterostrain Huder  _et al._
(2018); Bi  _et al._ (2019). The numerical results for the band structure are
shown in Fig. 6. As can be seen, even for relative low strain magnitudes the
band structure can greatly differ from the one in the non-strain case. The
discussion of several aspects is in order. First, we note that under strain
the positions of the shifted Dirac points define a periodicity which does not
coincide anymore with the corners of the mBZ. Indeed, according to Eq. (35),
the difference $\Delta\mathbf{D}=\mathbf{D}_{-}-\mathbf{D}_{+}$ between two
shifted Dirac points corresponding to, e.g., the non-deformed position
$\mathbf{K}=-\left(2\mathbf{b}_{1}+\mathbf{b}_{2}\right)/3$, is given by

$\Delta\mathbf{D}=-\frac{2\mathbf{g}_{1}+\mathbf{g}_{2}}{3}+2\mathbf{A},$ (42)

where $\mathbf{g}_{i}=\mathbf{T}\mathbf{b}_{i}$ are the strained moiré
vectors. Clearly, the vector $\Delta\mathbf{D}$ only coincides with the corner
$\mathbf{Q}_{1}$ of the mBZ [cf. Eq. (34)] when the angle between
$\mathbf{g}_{1}$ and $\mathbf{g}_{2}$ is $120^{\circ}$ and $\mathbf{A}=0$,
i.e., the non-strain case. Note that even in the case of pure shear strain
with no twist, where the moiré geometry is the same as in the only-twist case,
the vector $\Delta\mathbf{D}$ would still be shifted from the hexagonal mBZ
due to the non-zero gauge field
$\mathbf{A}_{\mathrm{shear}}\propto\left(0,-2\epsilon_{xy}\right)$. This is
expected because the honeycomb lattices are distorted due to the strain, and
therefore the hopping energies are no longer the same as in the only twist
case. It should be also noted that any relation between the Dirac points and
the borders of the moiré BZ is further blurred at low twist angles, where the
Dirac points are strongly coupled by the moiré potential.

Besides the actual shift in momentum due to strain induced gauge and
deformation fields, there is also an additional energy shift of the Dirac
points, which gets larger as the strain increases. As a result, the lowest
bands around the magic angle still have two distinct Dirac points in the
presence of strain. A close inspection reveals that such suppression of the
flat bands occurs even when the gauge and deformation fields are not taken
into account (cf. Appendix D), thus hinting that it is mainly due to how the
strain influences the coupling of the Dirac points by the moiré potential.
Although a concrete explanation of this behavior is still lacking, it may hint
that the origin of flat bands in TBG is intrinsically related to the
symmetries of the system, particularly those relating the moiré potential
$U\left(\mathbf{r}\right)$ (which always has a hexagonal symmetry), and the
three momentum transfers $\mathbf{q}_{i}$ (whose hexagonal symmetry is in
general broken by the strain). Note that, although the strain breaks
$\mathcal{C}_{3z}$, $\mathcal{C}_{2x}$ and $\mathcal{C}_{2y}$ rotational
symmetries, the inversion symmetry $\mathcal{C}_{2z}$ remains intact, as seen
in Fig. 6.

###  III.3 Electronic structure: Pure strain

Figure 8: Band structures of hexagonal moiré patterns generated by: (a) only
twist angle $\theta=1.05^{\circ}$; (b) only shear strain with the magnitude
$\epsilon_{s}=2\sin\left(\theta/2\right)\simeq 1.83\%$; (c) only biaxial
strain with the magnitude $\epsilon_{b}=2\sin\left(\theta/2\right)\simeq
1.83\%$. The momentum path in each case is shown in Fig. 7. From Eqs. (28) and
(32), all case have the same moiré periodicity $L\simeq 13.4$ nm. Panels e)-g)
display the corresponding total charge density profile of the lower middle
band. 3D plots in panels d) and h) show the bands for the moiré structures
realized by the shear and biaxial strain, respectively.

Next we examine the scenario of hexagonal moiré structures emerging solely
from strain (cf. Fig. 4). These cases are interesting because, when compared
to the situation of hexagonal patterns arising from only a twist, they reflect
the direct effect of strain in the electronic properties. In particular, by
using the relations (28) and (32), we are able to compare the electronic
structures of cases that share the same moiré periodicity. In Fig. 8 we
present the results for the band structure, density of states and charge
density. For comparison, we also include the results for TBG without strain.

Remarkably, although all cases shown have the same hexagonal moiré
periodicity, their electronic properties differ substantially. The strain thus
plays a decisive role in how the Dirac points in each lattice couple through
the moiré potential. This can be attributed to the actual distortion of each
lattice under strain, which, as seen in Fig. 4, results in different behaviors
around AA or AB stacking positions, even if at the moiré scale they all look
the same. Within the continuum model, these differences are mainly reflected
in how are the three leading hopping processes between the Dirac points in
each lattice, cf. Fig. 7. In particular, we only observe flat bands, and a
corresponding peak in the density of states, in unstrained TBG. With strain,
these flat bands disappear, and a splitting and emergence of multiple high-
order van Hove singularities takes place. We also observe that in TBG the
difference between the charge density at the center and at the edges of the
mBZ is more significant than in the two cases involving only strain. This
contrast implies potential variations in the electrostatic interactions within
purely strained systems when compared to those observed in TBG Guinea and
Walet (2018); Cea  _et al._ (2022).

##  IV Conclusions

We have presented a general theoretical scheme that describes the strain
effects in twisted two-dimensional materials. We have shown that the interplay
between twist and strain can lead to the formation of practically any moiré
geometry. The strain plays a central role in this by distorting the lattices
and thus modifying the resulting relative length and angle between the moiré
vectors. Due to the magnifying effect of the moiré pattern formations, this
effect becomes significant even at very small strain magnitudes, where each
layer’s lattice is barely deformed. Thus the plethora of moiré patterns
observed in experiments can be directly attributed to the presence of small
strain in the samples. Our considerations, however, go far beyond the mere
diagnosis of such intrinsic effects and offer a platform to actually design
moiré patterns by strain. Indeed, we have described in details the necessary
conditions to form any desired moiré geometry, simply by selectively changing
the twist and strain parameters. In particular, we have specified the
conditions to form special moiré geometries, such as square moiré patterns, or
hexagonal moiré patterns induced solely by strain. Furthermore, we have
identified that the modifications of the moiré geometry due to the strain lead
to significant deformations of the moiré Brillouin zone (mBZ). In contrast to
previous studies we have found that, when subject to strain, the mBZ is not a
deformed stretched hexagon, but rather a primitive cell that reflects new
symmetries of the strained moiré vectors. This might have important
implications, in particular with respect to identifying the high symmetry
points in band structures. We have rounded up our studies by analyzing the
electronic properties of the above strained moiré pattern. We have found that
the strain almost invariably suppresses the formation of moiré flat bands,
even in those hexagonal patterns formed only by strain. It also tends to split
and induce higher order van Hove singularities, as well as to modify the
charge density profile.

###### Acknowledgements.

We thank Vincent Renard, Niels R. Walet, Adrian Ceferino and Gerardo Naumis
for discussions. IMDEA Nanociencia acknowledges support from the “Severo
Ochoa” Programme for Centres of Excellence in R&D (Grant No. SEV-2016-0686).
F.E. acknowledges support from a research fellowship of CONICET, and PIP-
CONICET 2021-2023 Grant No. 11220200100941CO. Z.Z. acknowledges support
funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 101034431 and
from the “Severo Ochoa” Programme for Centres of Excellence in R&D
(CEX2020-001039-S / AEI / 10.13039/501100011033). P.A.P and F.G. acknowledge
funding from the European Commission, within the Graphene Flagship, Core 3,
grant number 881603 and from grants NMAT2D (Comunidad de Madrid, Spain),
SprQuMat and (MAD2D-CM)-MRR MATERIALES AVANZADOS-IMDEA-NC.

##  Appendix A Hexagonal symmetry in strained moiré patterns

As discussed in Sec. II.1, and shown in an example in Fig. 1, under strain the
construction of the moiré vectors using the usual recipe
$\mathbf{g}_{i}=\tilde{\mathbf{b}}_{-,i}-\tilde{\mathbf{b}}_{+,i}$ does not
always yield the smallest (primitive) vectors of the superlattice. A
comprehensive analysis of all moiré geometries that can be formed under strain
should account for which construction of the moiré vectors give the smallest
one. The analysis can be simplified by using symmetry arguments.

First we note that the smallest primitive moiré vectors are, in general, given
by one of these vectors: $\mathbf{g}_{1}$, $\mathbf{g}_{2}$, or
$\left(\mathbf{g}_{1}+\mathbf{g}_{2}\right)$, where
$\mathbf{g}_{i}=\mathbf{T}\mathbf{b}_{i}$ [cf. Eq. (1)]. Choosing the first
two gives, of course, the usual set
$\left\\{\mathbf{g}_{1},\mathbf{g}_{2}\right\\}$, from which one can carry out
the analysis of the strained moiré geometries as done in the manuscript. But
there are also two other possible sets:
$\left\\{\mathbf{g}_{1},\mathbf{g}_{1}+\mathbf{g}_{2}\right\\}$ and
$\left\\{\mathbf{g}_{2},\mathbf{g}_{1}+\mathbf{g}_{2}\right\\}$ (see Fig. 1).
One of these two sets can be primitive when a translation of one moiré vector
$\mathbf{g}_{i}$ by the other one $\mathbf{g}_{j}$ results in a smaller vector
(i.e., when
$\left|\mathbf{g}_{1}+\mathbf{g}_{2}\right|<\left|\mathbf{g}_{i}\right|$). The
appropriate construction of the primitive moiré vectors by one of these three
sets preserves the symmetries of the system (in the present case, the
hexagonal symmetry of the underlying honeycomb lattices).

Figure 9: Symmetries of the square moiré pattern solutions for
$\theta=2^{\circ}$, as given by Eq. (16). The depicted cases imply
$\epsilon\simeq 0.94\tan\left(\theta/2\right)$, with $\phi\simeq-9.4^{\circ}$
(reciprocal moiré vectors $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$), and
$\phi^{\prime}=\phi+60^{\circ}$ (reciprocal vectors $\mathbf{g}^{\prime}_{1}$
and $\mathbf{g}^{\prime}_{2}$). First case leads straightforwardly to the
perpendicular moiré vectors (cf. Fig. 2a). In the latter case though, the
vectors $\mathbf{g}^{\prime}_{1}$ and $\mathbf{g}^{\prime}_{2}$ are not
primitive, and the angle between them is not $90^{\circ}$. The resulting
superlattice is still a square though, as seen by the translation of
$\mathbf{g}^{\prime}_{2}$ by $\mathbf{g}^{\prime}_{1}$ that yields the vector
$\left(\mathbf{g}^{\prime}_{2}+\mathbf{g}^{\prime}_{1}\right)$ perpendicular
to $\mathbf{g}^{\prime}_{1}$. Therefore, the symmetric case with
$\phi^{\prime}=\phi+60^{\circ}$ gives, as expected, a square moiré pattern
too, except that the respective primitive moiré vectors are to be constructed
from the difference between the strained reciprocal vectors $\mathbf{b}_{1}$
and $\mathbf{b}_{1}+\mathbf{b}_{2}$ in each lattice.

To see this, consider the equal length moiré vectors condition for the case of
uniaxial heterostrain. For the usual set
$\left\\{\mathbf{g}_{1},\mathbf{g}_{2}\right\\}$, such condition implies the
strain magnitude given by Eq. (16). For the other two sets
$\left\\{\mathbf{g}_{1},\mathbf{g}_{1}+\mathbf{g}_{2}\right\\}$ and
$\left\\{\mathbf{g}_{2},\mathbf{g}_{1}+\mathbf{g}_{2}\right\\}$, the equal
length condition can be stated as

$\displaystyle\mathbf{F}\left[\mathbf{b}_{1}-\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)\right]\cdot\left[\mathbf{b}_{1}+\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)\right]$
$\displaystyle=0,$ (43)
$\displaystyle\mathbf{F}\left[\mathbf{b}_{2}-\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)\right]\cdot\left[\mathbf{b}_{2}+\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)\right]$
$\displaystyle=0,$ (44)

where we have used that $\mathbf{F}=\mathbf{T}^{\mathrm{T}}\mathbf{T}$ is a
symmetric transformation. Solving for the strain magnitude gives

$\displaystyle\left|\mathbf{g}_{1}\right|=\left|\mathbf{g}_{1}+\mathbf{g}_{2}\right|$
$\displaystyle\rightarrow\epsilon_{\mathrm{eq,2}}=\frac{4}{\nu-1}\cot\left(2\phi\right)\tan\left(\theta/2\right),$
(45)
$\displaystyle\left|\mathbf{g}_{2}\right|=\left|\mathbf{g}_{1}+\mathbf{g}_{2}\right|$
$\displaystyle\rightarrow\epsilon_{\mathrm{eq,3}}=\frac{4}{\nu-1}\cot\left(\frac{\pi}{3}+2\phi\right)\tan\left(\theta/2\right).$
(46)

Comparing with Eq. (16) we see that

$\displaystyle\epsilon_{\mathrm{eq}}\left(\phi+\pi/3\right)$
$\displaystyle=\epsilon_{\mathrm{eq,3}}\left(\phi\right),$ (47)
$\displaystyle\epsilon_{\mathrm{eq}}\left(\phi-\pi/3\right)$
$\displaystyle=\epsilon_{\mathrm{eq,2}}\left(\phi\right),$ (48)

thus restoring the hexagonal symmetry. Geometrically, the obtained result
means that for given parameters
$\left(\theta,\epsilon_{\mathrm{eq}},\phi\right)$, one of the equal length
moiré vector is no longer primitive after the transformation
$\phi\rightarrow\phi\pm\pi/3$. Rather, the new primitive vector is found by
appropriately changing the moiré vector construction (e.g., by using other set
than the usual one $\left\\{\mathbf{g}_{1},\mathbf{g}_{2}\right\\}$). We
emphasize that this is only a change in the superlattice description, due to
how the moiré vectors are constructed. The observed moiré geometry, arising
from the superposition of two strained honeycomb lattices with primitive
vectors
$\tilde{\mathbf{a}}_{i,\pm}=\left(\mathbf{1}+\mathcal{E}_{\pm}\right)\mathrm{R}\left(\pm\theta/2\right)\mathbf{a}_{i}$,
always reflects the honeycomb symmetries of the underlying lattices, such that
any translation $\phi\rightarrow\phi\pm\pi/3$ leads to the same moiré pattern
(up to an overall rotation of the system). For this reason it is more
convenient to study, as done in the manuscript, the strained moiré patterns by
using only the set of vectors
$\left\\{\mathbf{g}_{1},\mathbf{g}_{2}\right\\}$, and generalizing the
obtained results by taking into account the missing solutions corresponding to
translations $\phi\rightarrow\phi+\pi/3$. These latter solutions would then
correspond to the ones obtained by considering the other sets of possible
primitive moiré vectors.

##  Appendix B Analytical solutions for equal length moiré vectors

In the case of uniaxial heterostrain (Sec. II.2), by solving the angle
equation (2) for $\phi$ one can get the needed strain parameters to obtain
equal length moiré vectors with an angle $\beta$ between them. Taking into
account the symmetrical solutions, we find

$\displaystyle\epsilon_{s,r}$
$\displaystyle=\frac{4s}{1-\nu}\frac{f_{r}}{\sqrt{1-f_{r}^{2}}}\tan\left(\theta/2\right),$
(49) $\displaystyle\phi_{s,r}$ $\displaystyle=-\frac{s}{2}\arccos
f_{r}+\frac{\pi}{3}\left(n+\frac{1}{2}\right),$ (50)

where

$\displaystyle f_{r}\left(\nu,\cos\beta\right)$
$\displaystyle=\left(\frac{1-\nu}{1+\nu}\right)\frac{2+\cos\beta+r\sqrt{3}\left|\sin\beta\right|}{1+2\cos\beta}.$
(51)

Here $s,r=\pm 1$, and $n$ is an integer. The solutions are given in terms of
four roots, which correspond to four equivalent strain directions that yield
the same angle $\beta$. For both $r=\pm 1$ one has two strain angles $\phi$
which are related by $\phi_{-,r}+\phi_{+,r}=\pi/3+n\pi$. Consequently there is
always two strain angles, $\phi_{+}$ and $\phi_{-}=\pi/3-\phi_{+}$, with
corresponding strain magnitudes $\pm\epsilon_{r}$, which give the same moiré
pattern. Each angle $\phi_{\pm}$ is, in turn, symmetrical under the exchange
$\phi_{\pm}\rightarrow\phi_{\pm}+\pi/3$, due to the honeycomb symmetry of the
lattice. The $r=1$ roots correspond to the moiré patterns formed through the
lateral contraction of the honeycomb lattices, as measured by the Possion’s
ratio, and thus correspond to larger strain magnitudes. While the $r=-1$ roots
are solutions for any angle $\beta$, the roots $r=1$ are only solutions for
certain $\beta$. The corresponding equal length of the moiré vectors reads

$\displaystyle\frac{\left|\mathbf{g}_{i}\right|^{2}}{\left|\mathbf{b}_{i}\right|^{2}}$
$\displaystyle=\frac{\left(1+\nu\right)^{2}f_{r}^{2}-\left(1-\nu^{2}\right)f_{r}+\left(1-\nu\right)^{2}}{\left(1-f_{r}^{2}\right)\left(1-\nu\right)^{2}}4\sin^{2}\left(\frac{\theta}{2}\right).$
(52)

It is important to note that the strain angle $\phi$ is measured with respect
to the orientation of the (non-deformed) honeycomb lattice. Upon rotation of
both hexagonal monolayers by $\pm\theta/2$, the actual strain direction
relative to each lattice is $\pm\theta/2+\phi$. Although the axis from which
$\phi$ is measured depends on the chosen frame of reference (i.e., the lattice
vectors $\mathbf{a}_{i}$), the actual direction of the strain, in relation to
the orientation of the honeycomb primitive cell (hexagon), is always fixed.

##  Appendix C Construction of the moiré Brillouin zone

Figure 10: Construction of the mBZ for equal length lattice vectors
$\mathbf{g}_{1}$ and $\mathbf{g}_{2}$, with angles between them
$\beta=70^{\circ}$ (left) and $\beta=110^{\circ}$ (right). The Bragg lines are
shown in light gray, whose interceptions determine the mBZ (shown in black).
If $\beta<90^{\circ}$, the interceptions are between the Bragg lines
associated with the vectors $\mathbf{g}_{1}$, $\mathbf{g}_{2}$,
$\mathbf{g}_{1}-\mathbf{g}_{2}$ (and their negatives), whereas if
$\beta>90^{\circ}$ the interceptions are between the Bragg lines of
$\mathbf{g}_{1}$, $\mathbf{g}_{2}$, $\mathbf{g}_{1}+\mathbf{g}_{2}$. Note
that, up to a rotation, both cases have the same mBZ, since they represent the
same lattice. The transition at which
$\left|\mathbf{g}_{1}-\mathbf{g}_{2}\right|$ becomes larger (or smaller) than
$\left|\mathbf{g}_{1}+\mathbf{g}_{2}\right|$ occurs at the critical square
case $\beta=90^{\circ}$, where
$\left|\mathbf{g}_{1}+\mathbf{g}_{2}\right|=\left|\mathbf{g}_{1}-\mathbf{g}_{2}\right|$
and the six points of the mBZ are reduced to four.

Consider two equal length vectors $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$ with
angle $\beta$ between them. We set, without loss of generality, the vector
$\mathbf{g}_{1}$ on the $x$ axis,

$\displaystyle\mathbf{g}_{1}$ $\displaystyle=g\left(1,0\right),$ (53)
$\displaystyle\mathbf{g}_{2}$
$\displaystyle=g\left(\cos\beta,\sin\beta\right).$ (54)

For any reciprocal vector
$\mathbf{R}\left(m_{1},m_{2}\right)=m_{1}\mathbf{g}_{1}+m_{2}\mathbf{g}_{2}$,
the corresponding Bragg line, which we shall denote as
$l\left(m_{1},m_{2}\right)$, crosses $\mathbf{R}$ perpendicularly at
$\mathbf{R}/2$. Since the mBZ has a mirror symmetry at $\beta=\pi/2$ by a
reflection at the $x$ axis, it is sufficient to consider $\beta<\pi/2$. In
that case the six intersections are between the set of two Bragg lines

$\displaystyle l\left(1,0\right);\;$ $\displaystyle l\left(0,1\right),$ (55)
$\displaystyle l\left(0,1\right);\;$ $\displaystyle l\left(-1,1\right),$ (56)
$\displaystyle l\left(-1,0\right);\;$ $\displaystyle l\left(-1,1\right),$ (57)

and their negatives (see Fig. 10). Now, for an arbitrary vector
$\mathbf{f}=\left(f_{x},f_{y}\right)$ in the $xy$ plane, a perpendicular
vector is
$\mathbf{n}=\mathbf{e}_{z}\times\mathbf{f}=\left(-f_{y},f_{x}\right)$, whose
angle with the $x$ axis is $\alpha=\arctan\left(n_{y}/n_{x}\right)$. A
perpendicular line to $\mathbf{f}$ that crosses $\mathbf{f}/2$ then reads
$y=\left(f_{x}/f_{y}\right)\left(f_{x}-x\right)+f_{y}$. Therefore, since
$\mathbf{g}_{1}$ is all in $x$, the three lines that we need for the mBZ
construction are

$\displaystyle l\left(1,0\right)$ $\displaystyle:x_{1}=\frac{g}{2},$ (58)
$\displaystyle l\left(0,1\right)$
$\displaystyle:y_{2}=\frac{1}{\tan\beta}\left(\frac{g}{2}\cos\beta-x\right)+\frac{g}{2}\sin\beta,$
(59) $\displaystyle l\left(-1,1\right)$
$\displaystyle:y_{3}=\frac{\cos\beta-1}{\sin\beta}\left(g\frac{\cos\beta-1}{2}-x\right)+\frac{g}{2}\sin\beta.$
(60)

This leads to the three intersections points

$\displaystyle\mathbf{I}_{1}$
$\displaystyle=\frac{g}{2}\left[1,\frac{1}{\tan\beta}\left(\cos\beta-1\right)+\sin\beta\right],$
(61) $\displaystyle\mathbf{I}_{2}$
$\displaystyle=\frac{g}{2}\left[2\cos\beta-1,-\frac{1}{\tan\beta}\left(\cos\beta-1\right)+\sin\beta\right],$
(62) $\displaystyle\mathbf{I}_{3}$
$\displaystyle=\frac{g}{2}\left[-1,\frac{1}{\tan\beta}\left(\cos\beta-1\right)+\sin\beta\right].$
(63)

We can write these points in terms of the vectors $\mathbf{g}_{i}$ as

$\displaystyle\mathbf{I}_{1}$
$\displaystyle=\frac{\mathbf{g}_{1}+\mathbf{g}_{2}}{2}\left(1+\frac{\mathbf{g}_{1}\cdot\mathbf{g}_{2}}{\mathbf{g}_{1}\cdot\mathbf{g}_{1}}\right)^{-1},$
(64) $\displaystyle\mathbf{I}_{2}$
$\displaystyle=-\mathbf{I}_{1}+\mathbf{g}_{2},$ (65)
$\displaystyle\mathbf{I}_{3}$ $\displaystyle=\mathbf{I}_{1}-\mathbf{g}_{1}.$
(66)

The case $\beta>\pi/2$ is obtained by a mirror reflection of $\mathbf{g}_{2}$
around $\mathbf{g}_{1}$, thus leading to Eq. (34) after changing the notation
of the interception points.

##  Appendix D Electronic properties without strain fields

Figure 11 shows the electronic structure of TBG under uniaxial heterostrain,
but with zero gauge and scalar strain fields. Parameters are the same as in
Fig. 6. Even in absence of gauge fields there is a distortion of the energy
bands. As the strain increases, the mBZ is distorted, the Dirac cones are
shifted and the remote bands are pushed to a region close to the narrow bands.

Figure 11: Electronic structure of TBG with $\mathbf{A}=0$ and $V=0$. Other
parameters are the same as in Fig. 6c).

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