# Dislocations in twistronic heterostructures

V. V. Enaldiev  Abrikosov Center for Theoretical physics, MIPT, Dolgoprudnyi,
Moscow Region 141701, Russia  Kotelnikov Institute of Radio-engineering and
Electronics of the RAS, Mokhovaya 11-7, Moscow 125009, Russia

###### Abstract

Long-period moiré superlattices at the twisted interface of van der Waals
heterostructures relax into preferential-stacking domains separated by
dislocation networks. Here, we develop a mesoscale theory for dislocations in
the networks formed in twistronic bilayers with parallel (P) and antiparallel
(AP) alignment of unit cells across the twisted interface. For P bilayers we
find an exact analytical displacement field across partial dislocations and
determine analytic dependences of energy per unit length and width on
orientation and microscopic model parameters. For AP bilayers we formulate a
semi-analytical approximation for displacement fields across perfect
dislocations, establishing parametric dependences for their widths and
energies per unit length. In addition, we find regions in parametric space of
crystal thicknesses and moiré periods for strong and weak relaxation of moiré
pattern in multilayered twistronic heterostructures.

Study of van der Waals heterostructures, featuring moiré superlattice at the
twisted interface between constituent layers, became a rapidly growing
research area boosted by discovery of novel quantum phenomena [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Moiré superlattices in the twistronic
structures are due to periodic spatial variation of local stacking
arrangement, with a period determined by misorientation angle and/or lattice
mismatch between layers at the interface. Long period moiré superlattices,
specific for small-angle interfacial twist, minimize energy of lattice
transforming moiré pattern into arrays of preferential stacking domains
separated by network of domain walls [16, 5, 17, 18, 19, 20, 21, 22, 23, 24].

Vertical assembly of twistronic heterostructures, composed of non-inversion
symmetric blocks, such as transition metal dichalcogenide (TMD) or hexagonal
boron nitride (hBN) monolayers, results in different domain structures for
parallel (P) and antiparallel (AP) alignment of unit cells across the twisted
interface. For P alignment, relaxed moiré pattern consists of triangular
domains characterized by the lowest energy AB (BA) stacking [17, 18, 20, 25],
that corresponds to vertical alignment of A (B) and B (A) sublattices in two
layers at the interface (see Fig. 1). We accept stacking nomenclature used for
graphene-based heterostructures, where AB and BA stackings correspond to
alignment of Bernal graphene bilayers [16, 26, 5, 27], whereas for TMD
structures the stackings possess atomic structure of rhombohedral polytype
[28, 29, 30, 31].

For twistronic heterostructures with AP alignment, relaxed moiré pattern
consists of hexagonal domains with energetically favourable ABBA-stacking [18,
20], characterized by simultaneous vertical overlap of A and B sublattices in
one layer with B and A sublattices of neighboring layer across the interface.
Such ABBA stacking, shown on the inset in Fig. 1, is identical to alignment of
adjacent layers in bulk hexagonal TMD and BN crystals [30, 32, 33].

Figure 1:  (Left) Map of adhesion energy densities (2) for twistronic bilayers
with P alignment. Non-equivalent minima of the adhesion energy occur at
$\bm{u}=(0,a/\sqrt{3})$ (AB stacking) and $\bm{u}=(0,2a/\sqrt{3})$ (BA
stacking). Purple line indicates path across a partial dislocation with
Burgers vector $\bm{b}_{\rm P}=(0,a/\sqrt{3})$. (Right) same for AP twistronic
bilayers. Energetically favorable ABBA stacking corresponds to
$\bm{u}=(0,-a/\sqrt{3})$. Red/blue line highlights path across a screw/edge
perfect dislocation with $\bm{b}_{\rm AP}=(a,0)$. Middle inset sketches
reciprocal vectors used in Eq. (2).

Domain walls separating preferential stacking domains in P (AP) twistronic
heterostructures can be identified with extended crystalline defects known as
partial (perfect) dislocations [30, 27, 34], characterized by Burgers vector
$|\bm{b}_{\rm P}|=a/\sqrt{3}$ ($|\bm{b}_{\rm AP}|=a$). In case of equilateral
domains (triangular for P and hexagonal for AP) forming at the interface with
undistorted moiré pattern, the dislocations have screw type, with the Burgers
vectors aligned with dislocation lines, whereas, in general case, the
partial/perfect dislocations possess both screw and edge components of
displacement field [26, 18, 34].

Here, we develop a theory for partial/perfect dislocations forming in small-
angle P/AP twistronic bilayers of graphene, hBN and MX2 TMD (M=Mo, W; X=S,
Se), combining elasticity theory and microscopic models for adhesion energy
between constituent layers. For P bilayers we find an exact analytic
distribution for displacement field across partial dislocations determining
explicit dependences of their widths and energies per unit length on
orientation and microscopic model parameters. For AP bilayers we formulate a
semi-analytical approximation for displacement field across perfect
dislocations providing parametric and orientation dependences of widths and
energy per unit length, which are compared with the exact ones. Using an
energetical criterium we find out dependence of minimal moiré superlattice
period versus number of layers for transformation of moiré pattern into domain
structures in multilayered twistronic heterostructures.

Mesoscale model for dislocations. Using smoothness of the dislocations on
atomic scale [18, 35] we formulate a continuous mesoscale theory based on the
energy functional,

$\mathcal{E}=\int_{-\infty}^{\infty}\left[W_{\rm ad}(\bm{u})+W_{\rm
el}(u_{ij})\right]d\xi,$ (1)

accounting for sum of elastic and adhesion energy densities across dislocation
axis ($\xi=0$) set by an angle $\alpha$ ($-\pi/2\leq\alpha\leq\pi/2$) with
respect to zigzag crystallographic direction (see Fig. 1). The adhesion energy
density [33, 28, 30],

$\displaystyle W_{\rm ad}(\bm{u})=\sum_{\begin{subarray}{c}n=1,2,3\\\
l=1,2,3\end{subarray}}\left[w^{(s)}_{n}\cos\left(\bm{G}^{(n)}_{l}\bm{u}\right)+w^{(a)}_{n}\sin\left(\bm{G}^{(n)}_{l}\bm{u}\right)\right],$
(2)

is determined by an interlayer lateral offset, $\bm{u}=(u_{x},u_{y})$, counted
from AA stacking for P and AP bilayers, and three smallest triads of
reciprocal lattice vectors, $|\bm{G}^{(1)}_{1,2,3}|=4\pi/a\sqrt{3}\equiv G$,
$|\bm{G}^{(2)}_{1,2,3}|=G\sqrt{3}$, $|\bm{G}^{(3)}_{1,2,3}|=2G$, with
orientation shown on middle inset in Fig. 1. For P bilayers this choice of
origin for the interlayer offsets leads to $w^{(a)}_{1,2,3}=0$ [28, 30] and,
consequently, $W_{\rm ad}(\bm{u})=W_{\rm ad}(-\bm{u})$, with two inequivalent
minima – AB and BA. The elastic energy density,

$W_{\rm el}=\frac{\lambda}{4}u_{ii}^{2}+\frac{\mu}{2}u_{ij}u_{ji},$ (3)

is determined by strain tensor $u_{ij}=\tfrac{1}{2}\left(\tfrac{\partial
u_{j}}{\partial x_{i}}+\tfrac{\partial u_{i}}{\partial x_{j}}\right)$ and Lamé
parameters, $\lambda$ and $\mu$, of a monolayer. Note that $u_{ij}$
characterizes relative strain between the two layers which is incorporated in
$W_{\rm el}$ (3) by halving elastic energy of a single layer. Material-
dependent energy parameters in Eq. (1) are gathered in Table 1.

Table 1: Elastic and adhesion parameters (in eV/nm2) for studied materials. Last column shows values of fitting $\eta$ in Eq. (8).  |  | $\lambda$ | $\mu$ | $w_{1}^{(s)}$ | $w_{2}^{(s)}$ | $w_{3}^{(s)}$ |  $w_{1}^{(a)}$, |  $w_{3}^{(a)}$, | $\eta$
---|---|---|---|---|---|---|---|---|---
|  |  |  | $\times 10^{3}$ | $\times 10^{3}$ | $\times 10^{3}$ | $\times 10^{3}$ | $\times 10^{3}$ |
MoS2 | P | 520 | 443 | 151.9 | -3.71 | -1.85 |  |  | 1.9
AP | 138.9 | -3.19 | -1.57 | 23.8 | -0.26 |
WS2 | P | 328 | 453 | 166.4 | -4.8 | -2.4 |  |  | 2.05
AP | 151.7 | -4.09 | -2.01 | 29.55 | -0.36 |
MoSe2 | P | 260 | 306 | 155.3 | -4.25 | -2.12 |  |  | 2.017
AP | 138.6 | -3.5 | -1.71 | 32.51 | -0.37 |
WSe2 | P | 185 | 303 | 132.6 | -2.75 | -1.38 |  |  | 1.76
AP | 112.6 | -2.12 | -1.01 | 33.63 | -0.31 |
BN | P | 627 | 736 | 89.5 | -0.31 | 2.11 |  |  | 1.51
AP | 62.1 | -0.57 | -0.2 | 49.05 | -0.42 |
Gr |  | 617 | 1007 | 77.5 | -0.71 | -0.18 |  |  | 1.455
Figure 2:  (a) Exact (5) (solid), approximated (6) (dotted) and fitted (black
dashed) displacement fields across screw and edge partial dislocations in
bilayer graphene. Fits are made with Eq. (6) and rescaling $w^{(s)}_{1}\to\eta
w^{(s)}_{1}$. (b) Orientation dependences of partial dislocation energies per
unit length. Dashed lines show fits of the exact values with $\mathcal{E}_{\rm
P}(\alpha)$ (8). (c) Orientation dependence of partial dislocation widths,
$\mathcal{W}_{\rm P}(\alpha)/\sqrt{\eta}$.

First, we consider a partial dislocation, characterized by a Burgers vector
$\bm{b}_{P}=(0,a/\sqrt{3})$, in P bilayers (see inset in Fig. 1). Since
crossover between AB and BA domains occurs along a straight line [36, 37, 30],
depicted by purple in Fig. 1, displacement field, $\bm{u}=(0,u_{y})$, across
the partial dislocation is determined by the only component parallel to
$\bm{b}_{\rm P}$. To determine $u_{y}(\xi)$ we minimize functional (1), which
leads to the following equation,

$\frac{\left[\mu+\left(\lambda+\mu\right)\sin^{2}\alpha\right]}{-4G}\frac{\partial^{2}u_{y}}{\partial\xi^{2}}=w_{1}^{(s)}\sin\left(\tfrac{3Gu_{y}}{4}\right)\cos\left(\tfrac{Gu_{y}}{4}\right)\quad\\\
+\sin\left(\tfrac{3Gu_{y}}{2}\right)\left[2w_{3}^{(s)}\cos\left(\tfrac{Gu_{y}}{2}\right)+\tfrac{3}{2}w_{2}^{(s)}\right],$
(4)

with boundary conditions, $u_{y}(-\infty)=a/\sqrt{3}$ and
$u_{y}(+\infty)=2a/\sqrt{3}$, accounting for crossover from AB (at
$\xi=-\infty$) to BA (at $\xi=+\infty$) stacking domains (see Fig. (1)).
Integrating Eq. (4) with the help of a table integral [38], one finds an
analytic solution:

$\frac{\pi\xi}{a}\left(r_{-}+1\right)\sqrt{\frac{-8w_{3}^{(s)}\left(1+\frac{2}{3}r_{+}\right)\left(1-2r_{-}\right)}{\mu+\left(\lambda+\mu\right)\sin^{2}\alpha}}=\\\
{\rm sign}(2u_{y}-a\sqrt{3})\left[(3+2r_{-})\Pi(\varphi,\nu,k)-{\rm
F}(\varphi,k)\right].$ (5)

Here, ${\rm F}(\varphi,k)$ and $\Pi(\varphi,\nu,k)$ are the incomplete
elliptic integrals of the first and third kinds, with the following arguments:

$\displaystyle\varphi=$
$\displaystyle\arcsin\sqrt{\frac{\left(1-2r_{-}\right)\left[\cos\left(\tfrac{Gu_{y}}{2}\right)+1\right]}{4\cos\left(\tfrac{Gu_{y}}{2}\right)-2-4r_{-}}},$
$\displaystyle\nu=$ $\displaystyle\frac{8(r_{-}+1)}{2r_{-}-1},$ $\displaystyle
k=$
$\displaystyle\sqrt{\frac{8\left(r_{+}-r_{-}\right)}{(3+2r_{+})(1-2r_{-})}},$

and

$r_{\pm}=-\tfrac{w^{(s)}_{2}}{2w^{(s)}_{3}}\pm\tfrac{1}{2}\sqrt{\tfrac{2w^{(s)}_{2}}{w^{(s)}_{3}}-\tfrac{w^{(s)}_{1}}{w^{(s)}_{3}}+\left(\tfrac{w^{(s)}_{2}}{w^{(s)}_{3}}\right)^{2}}.$

Displacement field (5) across a partial dislocation, exemplified in Fig. 2(a)
for bilayer graphene, demonstrates growth of the dislocation width from screw
($\alpha^{\circ}=0^{\circ}$) to edge ($\alpha^{\circ}=90^{\circ}$)
orientations. Similar width dependences exhibit partial dislocations in the
other materials, indicating increase of lattice rigidity from shear to
hydrostatic stresses, specific for screw and edge dislocations, respectively.
Moreover, such a behaviour agrees with orientation dependences of energy per
unit length of partial dislocations, shown in Fig. 2(b), where screw possess
the minimal energy compared to the most energetically expensive edge ones.
These energy dependences are obtained by numerically computing integral (1)
with the exact displacement field (5). Among studied materials, partial
dislocations in P WS2 have the highest energies per unit length, whereas for P
hBN bilayers the dislocation formation costs the least energies.

To determine analytic dependences of the dislocation energies on microscopic
parameters and orientation, we obtain an approximated solution of Eq. (4) in
the limit $w^{(s)}_{2,3}\to 0$ [37]:

$u_{y}(\xi)=a\sqrt{3}\left\\{\tfrac{1}{2}+\tfrac{1}{\pi}\arctan\left[\tfrac{\tanh\left(\frac{2\xi}{\mathcal{W}_{\rm
P}(\alpha)}\right)}{\sqrt{3}}\right]\right\\},$ (6)

where,

$\mathcal{W}_{\rm
P}(\alpha)=\frac{2a}{\pi}\sqrt{\frac{\mu+(\lambda+\mu)\sin^{2}\alpha}{2w_{1}^{(s)}}},$
(7)

is the partial dislocation width. In Fig. 2(a) we show the approximated
displacement field (6) across screw (red dotted) and edge (blue dotted)
partial dislocations in bilayer graphene which demonstrate a reasonable
agreement with the exact distributions (5) in virtue of
$w^{(s)}_{1}\gg|w_{2,3}^{(s)}|$ (see Table 1). Moreover, we find that
rescaling the adhesion parameter, $w^{(s)}_{1}\to\eta w^{(s)}_{1}$, in Eqs.
(6), (7) with a single orientation-independent parameter, $\eta$, one attains
a perfect overlap with the exact displacement field (5) for any orientation,
see black dashed lines in Fig. 2(a). The values of $\eta$ for each bilayer are
obtained from fit of the exact energy dependences (Fig. 2(b)) with the help of

$\mathcal{E}_{\rm
P}(\alpha)=\tfrac{3a}{\sqrt{2}\pi}\left(1-\tfrac{\pi}{3\sqrt{3}}\right)\sqrt{\eta
w_{1}^{(s)}\left[\mu+(\lambda+\mu)\sin^{2}\alpha\right]},$ (8)

which is obtained from functional (1) for displacement field (6) with
$w^{(s)}_{1}\to\eta w^{(s)}_{1}$. Results of the fitting are gathered in Table
1 and shown by black dashed in Fig. 2(b), where one finds a perfect matching
between $\mathcal{E}_{\rm P}(\alpha)$ (8) and exact numerical data for every
studied material. Therefore, using Eq. (6) we can identify $\mathcal{W}_{\rm
P}(\alpha)/\sqrt{\eta}$ with the widths of exact partial dislocations (4),
plotted in Fig. 2(c). Confirming the above-mentioned orientation dependences,
the displayed width behaviours demonstrate that partial dislocations are the
thickest for bilayer graphene, as the latter possess the highest lattice
rigidity and the lowest adhesion (see Table 1). We also note that the obtained
$\mathcal{W}_{\rm P}(\alpha)/\sqrt{\eta}$-dependence for bilayer graphene is
in a good quantitative agreement with that of extracted experimentally in Ref.
[16].

Next, we consider perfect dislocations, separating identical ABBA stacking
domains in AP-bilayers of TMD and hBN. We specify Burgers vector for a single
perfect dislocation, $\bm{b}_{AP}=(a,0)$, as shown on the inset in Fig. 1.
Minimizing functional (1) we obtain the following system:

$\begin{pmatrix}\frac{\mu}{4}+\frac{\lambda+\mu}{4}\cos^{2}\alpha&\frac{\lambda+\mu}{8}\sin(2\alpha)\\\
\frac{\lambda+\mu}{8}\sin(2\alpha)&\frac{\mu}{4}+\frac{\lambda+\mu}{4}\sin^{2}\alpha\end{pmatrix}\begin{pmatrix}\tfrac{\partial^{2}u_{x}}{\partial\xi^{2}}\\\
\tfrac{\partial^{2}u_{y}}{\partial\xi^{2}}\end{pmatrix}=\begin{pmatrix}G_{x}\sin\left(G_{x}u_{x}\right)\left[w_{1}\cos\left(G_{y}\widetilde{u}_{y}-\varphi_{1}\right)+w_{2}^{(s)}\cos\left(3G_{y}\widetilde{u}_{y}\right)\right]\\\
G_{y}\cos\left(G_{x}u_{x}\right)\left[w_{1}\sin\left(G_{y}\widetilde{u}_{y}-\varphi_{1}\right)+3w_{2}^{(s)}\cos\left(3G_{y}\widetilde{u}_{y}\right)\right]\end{pmatrix}\\\
-\begin{pmatrix}G_{x}\sin\left(2G_{x}u_{x}\right)\left[w_{2}^{(s)}+2w_{3}\cos\left(2G_{y}\widetilde{u}_{y}-\varphi_{3}\right)\right]\\\
2G_{y}\cos\left(2G_{x}u_{x}\right)\left[w_{3}\sin\left(2G_{y}\widetilde{u}_{y}-\varphi_{3}\right)-w_{3}^{(a)}\cos\left(2G_{y}\widetilde{u}_{y}\right)\right]+G_{y}w_{1}^{(s)}\sin\left(2G_{y}\widetilde{u}_{y}\right)+G_{y}w_{3}\sin\left(4G_{y}\widetilde{u}_{y}+\varphi_{3}\right)\end{pmatrix},$
(9)

where we introduced $(G_{x},G_{y})=\tfrac{2\pi}{a}(1,\tfrac{1}{\sqrt{3}})$,
$\widetilde{u}_{y}=u_{y}+\tfrac{a}{2\sqrt{3}}$,
$w_{1,3}=\sqrt{w_{1,3}^{(s)2}+w_{1,3}^{(a)2}}$, and
$\varphi_{1,3}=\arctan\left(\tfrac{w_{1,3}^{(a)}}{w_{1,3}^{(s)}}\right)$. The
system (9) is supplemented with boundary conditions,
$\bm{u}(-\infty)=(0,-a/\sqrt{3})$ and
$\bm{u}(+\infty)-\bm{u}(-\infty)=\bm{b}_{AP}$, describing crossover between
neighboring ABBA domains. Numerical solution of system (9), obtained in Ref.
[30] and shown by solid lines in Figs. 1 and 3(a), involves variation of both
displacement components across screw and edge perfect dislocations. Our aim is
to find approximated distributions for the displacement fields, extracting
explicit analytical parametric dependences for widths and energies per unit
length of perfect dislocations in studied AP bilayers. To this end we notice
that the variation range of displacement component orthogonal to $\bm{b}_{AP}$
is confined within interval $a/2\sqrt{3}$, as
$\max(u_{y}-u_{y}(\pm\infty))=u_{y}(0)+a/\sqrt{3}<a/2\sqrt{3}$, see Fig. 3(a).
Therefore, considering $(u_{y}/a)+1/\sqrt{3}$ as a small parameter, we look
for an approximated solution setting $u_{y}=-a/\sqrt{3}$ and neglecting
$\partial^{2}u_{y}/\partial\xi^{2}$ in the first line of the system (9). This
leads to an equation for $u_{x}$ only, with the following solution:

$\displaystyle
u_{x}(\xi)=a-\frac{a}{\pi}\operatorname{arccot}\left(\frac{\sinh\left(\frac{2\xi}{\mathcal{W}_{\rm
AP}(\alpha)}\right)}{\sqrt{1-4\frac{w_{2}^{(s)}-w_{3}^{(s)}+\sqrt{3}w_{3}^{(a)}}{w_{1}^{(s)}-2w_{2}^{(s)}+\sqrt{3}w_{1}^{(a)}}}}\right),$
(10)

where

$\mathcal{W}_{\rm AP}(\alpha)=\\\
\frac{a}{\pi\sqrt{2}}\sqrt{\frac{\mu+(\lambda+\mu)\cos^{2}\alpha}{w_{1}^{(s)}-6w_{2}^{(s)}+4w_{3}^{(s)}+\sqrt{3}\left(w_{1}^{(a)}-4w_{3}^{(a)}\right)}}$
(11)

is approximated width of the perfect dislocation. To find $u_{y}(\xi)$ we
substitute $u_{x}(\xi)$ (10) into the second line of system (9), and
numerically determine $u_{y}$ from the resulting equation. The approximated
and exact displacement fields across perfect screw and edge dislocations,
shown in Fig. 3(a) for AP MoS2 bilayers, demonstrate good agreement between
each other. Note that the approximation is more accurate for screw-type
dislocations as in this case the used condition $|(u_{y}/a)+1/\sqrt{3}|\ll 1$
is better satisfied.

In Fig. 3(b) we show that energies per unit length of the exact and
approximated displacement fields across perfect dislocations differ by less
than 5% in whole interval of interest $|\alpha|\leq 90^{\circ}$ for each AP
bilayer. In particular, for screw-type dislocations, where the exact and
approximated energies almost coincide, the parametric dependence reads

$\mathcal{E}_{\rm
AP}=\tfrac{2a}{\pi}\sqrt{2\mu\left(w_{1}^{(s)}-2w_{2}^{(s)}+\sqrt{3}w_{1}^{(a)}\right)},$
(12)

resulted from Eq. (1) with (10) and $u_{y}=0$.

Similar to the partials, widths of perfect dislocations increase from screw to
edge orientation, as shown in Fig. 3(c), where we plotted the approximated
dependences (11). Although the latter underestimate the widths of exact
displacement fields, they can be used as lower boundary values.

Figure 3: (a) Exact (solid) and approximated (dashed) displacement fields
across screw and edge perfect dislocations in AP MoS2 bilayers. (b)
Orientation dependences of perfect dislocation energies per unit length for
the exact and approximated displacement fields. (c) Orientation dependences of
perfect dislocation widths (11) for the approximated displacement field.

Finally, we estimate minimal moiré superlattice period, necessary to form
dislocation networks at the twisted interface of $N+M$-layered
heterostructures sketched on inset in Fig. 4. Moiré superlattices take the
form of domain arrays when energy gain from expansion of the preferential
stacking areas in moiré supercells negotiates energy costs on creation of
dislocation networks. The energy costs per supercell, comprising three partial
screw dislocations of $\ell$-length (for P) and three perfect screw
dislocations of $\ell/\sqrt{3}$-length (for AP) [30], can be evaluated as
$3\,\mathcal{E}_{\rm P}\ell\sqrt{(N+M)/2}$ and $\sqrt{3}\,\mathcal{E}_{\rm
AP}\ell\sqrt{(N+M)/2}$, respectively, where $\mathcal{E}_{\rm P/AP}$ is energy
per unit length of partial/perfect screw dislocation (8), (12) and the factor
$\sqrt{(N+M)/2}$ accounts for an increase of the shear module
$\mu\to(N+M)\mu/2$. The energy gain, $W_{\rm
ad}(\bm{u}_{0})\sqrt{3}\ell^{2}/2$, takes into account formation of
preferential stacking domains ($\bm{u}_{0}=(0,-a/\sqrt{3})$) with a supercell
area. Balance between the costs and gain is attained at the minimal moiré
period

$\ell_{*}^{\rm P/AP}=\frac{\mathcal{E}_{\rm P/AP}\gamma_{\rm P/AP}}{W_{\rm
ad}(\bm{u}_{0})}\sqrt{N+M},\gamma_{\rm P}=\sqrt{6},\gamma_{\rm AP}=\sqrt{2}.$
(13)

In parameter space $(N,M,\ell)$, characterising twistronic heterostructures,
Eq. (13) defines a surface, displayed in Fig. 4 for several heterostructures,
which distinguishes a strong relaxation regime of moiré superlattice
$\ell>\ell_{*}(N,M)$, associated with formation of domain structures, from a
weak relaxation regime, $\ell<\ell_{*}(N,M)$, where relaxation of moiré
pattern is negligible. For heterostructures, consisting on $N$-layer stack
marginally twisted with respect to bulk crystal, the two regimes are divided
by the line $\ell=\ell_{*}(2N,0)$ (Fig. 4), where exchange $N\to 2N$ assumes
no relaxation in the bulk crystal.

Figure 4:  Surface $\ell=\ell_{*}(N,M)$ (13) divides parameter space
$(N,M,\ell)$ characterising each twistronic heterostructures on regions
$\ell>\ell_{*}(N,M)$, featuring relaxation of moiré superlattice into domain
structure, and $\ell<\ell_{*}(N,M)$ with almost non-relaxed moiré pattern. For
twistronic heterostructures with twisted interface between $N$-layered and
bulk crystals parametric space, $(N,\ell)$, is divided along line
$\ell=\ell_{*}(2N,0)$.

Acknowledgements. The author acknowledges fruitful discussions with Roman
Gorbachev and Sarah Haigh.

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