# Twistronics of Janus transition metal dichalcogenide bilayers

Mattia Angeli  John A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA  Gabriel R. Schleder
John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, Massachusetts 02138, USA  Efthimios Kaxiras  John A.
Paulson School of Engineering and Applied Sciences, Harvard University,
Cambridge, Massachusetts 02138, USA  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA

###### Abstract

Twisted multilayers of two-dimensional (2D) materials are an increasingly
important platform for investigating quantum phases of matter, and in
particular, strongly correlated electrons. The moiré pattern introduced by the
relative twist between layers creates effective potentials of long-wavelength,
leading to electron localization. However, in contrast to the abundance of 2D
materials, few twisted heterostructures have been studied until now. Here we
develop a first-principle continuum theory to study the electronic bands
introduced by moire patterns of twisted Janus transition metal dichalcogenides
(TMD) homo- and hetero-bilayers. The model includes lattice relaxation,
stacking-dependent effective mass, and Rashba spin-orbit coupling. We then
perform a high-throughput generation and characterization of DFT-extracted
continuum models for more than a hundred possible combinations of materials
and stackings. Our model predicts that the moiré physics and emergent
symmetries depend on chemical composition, vertical layer orientation, and
twist angle, so that the minibands wavefunctions can form triangular,
honeycomb, and Kagome networks. Rashba spin-orbit effects, peculiar of these
systems, can dominate the moiré bandwidth at small angles. Our work enables
the detailed investigation of Janus twisted heterostructures, allowing the
discovery and control of novel electronic phenomena.

##  I Introduction

Strong interactions between particles are a central pillar of quantum physical
phenomena, leading to many different phases of matter, such as superconductors
and fractional quantum Hall states [1, 2]. These phenomena are investigated in
condensed matter systems in the so-called quantum materials [3, 4, 5], where
quantum effects dominate their properties. These properties can in turn be
tuned through materials-engineering "knobs" that pertain to each type of
material, such as chemical composition, structural changes, and external
perturbations.

Recently, two-dimensional few-layer systems, such as graphene and transition
metal dichalcogenides (TMDs), have proven to be one of the leading platforms
to study many of these effects, owing to their low dimensionality. The
possibility to manipulate interlayer interactions of van der Waals (vdW)
materials by changing the twist angle between two layers has been triggered by
the breakthroughs in magic-angle graphene multilayers [6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16]. In these systems, the relative twist angle results in a moiré
superlattice, which acts as a long-wavelength modulating potential changing
the electronic, vibrational, and structural properties [17, 18, 19]. Flat
bands, non-trivial topology, emergent symmetries, enhanced correlations, and
strong electron-phonon coupling can arise as a result of the charge
localization induced by the moiré superlattice [20, 21, 22, 23, 24, 25, 26,
27, 28].
Indeed, recent experiments have moved beyond graphene. As an example, work on
twisted TMDs quickly uncovered a plethora of novel phenomena such as
correlated insulating states, quantum anomalous Hall states, superconductivity
and moiré excitons [29, 30, 31, 32, 33, 34]. Given the huge number of possible
combinations to create these moiré materials [35, 36] and the experimental
challenges in achieving them, only a small subset of systems have been studied
to date. Nonetheless, by creating moiré structures with different properties
in their individual constituents, new physical properties are possible in the
interacting system.

Transition-metal dichalcogenide layers with different chalcogen atoms on
either side of the metal plane [37], referred to as "Janus" layers [38], have
attracted intense interest due to the special properties introduced by the
broken mirror symmetry of the corresponding pure layers [37]. Compared to
$\text{MX}_{2}$ monolayers with $\text{D}_{3v}$ symmetry such as
$\text{MoS}_{2}$, Janus group-VI chalcogenides MXY
($\text{X},\text{Y}=\text{S, Se, Te};\;\;\text{X}\neq\text{Y}$) are non-
centrosymmetric compounds with $\text{C}_{3v}$ symmetry [13]. The difference
in electronegativity between the two chalcogen layers leads to a built-in
electric dipole oriented in the out-of-plane direction, generating strong
Rashba spin–orbit couplings (SOC). Furthermore, the intrinsic electric field
provides an additional degree of freedom with which to tune the van der Waals
interaction between adjacent layers. As such, these systems present new
ingredients and properties to the already known tuning knobs available for
probing moiré-driven phenomena.

In this paper, we present results from an $ab\;initio$ continuum model
designed for twisted Janus TMDs. The model extends the one-band continuum
Hamiltonian of Refs. [39, 40] to better capture the rich physics of polar
TMDs. A set of inexpensive bilayer calculations is used to compute the
electronic and structural potentials which are then used to include lattice
relaxation and compute the bandstructure. The effective mass approximation of
the band edges is relaxed by the inclusion of a moiré-dependent term. The flat
bands of these systems are thoroughly analyzed and characterized in terms of
the symmetries and centers of the corresponding Wannier orbitals. Finally, we
create a database of relevant physical parameters that can be used as a
reference, to reproduce the continuum model bands of more than a hundred
twisted bilayers, and also to guide experimental and theoretical efforts
toward those systems which exhibit the most interesting physics.

The work is organized as follows: in Sec. II we start by briefly introducing
the geometrical features of twisted bilayers. In Sec. II.1 we focus on the
continuum model Hamiltonian and the methodology employed to obtain its
potentials from $ab\;initio$ calculations. In Sec. II.2 we discuss a set of
twisted bilayers in which Rashba SOC is negligible and the moiré bands develop
emergent symmetries. In Sec. II.3 we focus on those bilayers in which Rashba
effects are important. The last section (Sec. III) is devoted to discussing
some of the exotic physics that could be observed in these systems. The
parameters that characterize the continuum models of more than a hundred of
twisted bilayers are listed in Tables LABEL:tab:homo and LABEL:tab:hetero.

##  II Results

Figure 1: Moiré superlattices created by twisting TMDs bilayers. Two different
configurations are possible, 3R or 2H. From an aligned reference, these two
configurations differ by a 180° rotation of one layer. The high-symmetry
stacking regions of each superlattice are highlighted with dashed circles and
triangles, together with the corresponding top and side views.
In 3R twisted bilayers, perfect layer alignment is found in the AA regions.
MX/XM are Bernal-stacked areas in which a metal atom is on top/below a pair of
chalcogens. These regions form a hexagonal pattern. At the interface between
Bernal domains lie the domain wall (DW) regions. Three of these areas are
highlighted in dotted green. As pointed out by black lines, the DWs form a
Kagome network. In 2H twisted bilayers, the horizontal displacement of two
layers overlaps in the AB regions where metal and chalcogens lie on top of
each other. In the XX/MM Bernal domains, only the chalcogen/metal layers
align.

Twisted heterostructures between TMD monolayers, occur in two distinct
configurations [41], known as 2H and 3R. The two configurations differ by a
$180^{\circ}$ rotation of the top layer around a vertical axis passing through
the position of the transition metal atom, and are not related by any shift of
one layer with respect to the other. In Fig. 1 we show 3R and 2H twisted
bilayers. In the 3R configuration, the perfectly aligned areas called AA are
surrounded by six Bernal-stacked regions (MX and XM) that form a honeycomb
network. In the MX/XM areas, a metal atom (M) on one layer is directly on
top/below a chalcogen atom (X) on the other layer. The two regions are related
by a reflection that exchanges the two layers. In 2H bilayers, metal/chalcogen
atoms are on top of each other in the MM/XX Bernal stacked regions. Vertical
alignment between the layers is obtained in the AB regions, where different
atoms are on top of each other. In both configurations, the interface regions
between Bernal domains are the so-called domain walls (DWs) which form a
Kagome network.
In general, the geometry of the moiré pattern is characterized by the presence
of high-symmetry regions orderly distributed to form hexagonal Bravais
lattices [42]. The AA/AB regions form a triangular network while two adjacent
MX/XM or XX/MM regions are the basis of a honeycomb lattice. Finally, the DWs
centers correspond to the three-site basis of a Kagome lattice.

Figure 2: Homobilayers band structures of WSSe at different stackings,
projected on each monolayer (blue to orange). The energy is shifted to the
valence band maximum of each configuration. Inset 1 highlights the K-valley of
an asymmetrically ordered bilayer (Se-S-Se-S), presenting both maximal layer
polarization and strong spin-orbit spin-splitting. Inset 2 highlights the
$\Gamma$-valley of a symmetrically ordered bilayer (Se-S-S-Se), where strong
interlayer interaction especially between $p_{z}$ and $d_{z^{2}}$ orbitals
leads to a large band splitting, isolating the valence band maximum
(antibonding state) from other states.

Unlike in non-polar TMDs, the electronic properties of Janus bilayers depend
on how the chalcogen layers are vertically ordered [43]. This ordering will
determine the interface and interlayer interactions between the layers,
dictating where the valence band maximum and conduction band minimum (VBM/CBM)
are located in momentum space.
As an example, in Fig. 2 the bandstructure of AA and MX stacked WSSe is shown
for the three possible orderings. In WSSe bilayers with the Se-S-Se-S
interface configuration, named only by the chalcogen planes, there is an
obvious layer polarization caused by the intrinsic dipole moment of the layers
and the asymmetry of the order. At the interface, the potential energy on the
side of the S atomic plane is larger, so the energy of the top layer is larger
than in the other layer. Since the layers are weakly coupled, this results in
the bottom (top) layer contributing to the CBM (VBM).
On the contrary, in bilayers with symmetric orders (S-Se-Se-S and Se-S-S-Se),
the effective dipole in each layer points in opposite directions and the
VBM/CBM combine states from both layers. Therefore, the electronic structure
of twisted Janus TMDs bilayers and their consequent physical properties will
sensitively depend not only on the chemical compositions, but also on the
interface orderings.

###  II.1 Twisted bilayers moiré Hamiltonian

We derive the moiré Hamiltonian from first principles following the approach
outlined in Ref. [44]. The procedure requires a relatively inexpensive set of
bilayer untwisted calculations. These calculations are intended to extract how
the Bloch states near a band edge are affected by the misalignment between the
layers. This information is then used to derive a continuum model Hamiltonian
which targets the moiré bands that form in twisted bilayers. This model is
further augmented to include lattice relaxation [45], an effective mass
correction, and Rashba spin-orbit coupling terms.

For twisted semiconductors, the moiré bands closest to the band gap can be
derived from the untwisted parabolic band extrema (or valleys) in the original
Brillouin zone. In particular, in Janus bilayers, we can identify two distinct
situations.
In the first one, when the two layers are dissimilar or weakly coupled, the
moiré physics derives mostly from one layer only. This condition is not
relevant only for heterobilayers, but also for asymmetrically ordered
homobilayers (X-Y-X-Y). In this case, the bilayer’s effective dipole shifts in
energy each layer’s band edge, so that the VBM/CBM are mostly layer polarized
states.
A different second situation, is that of symmetrically ordered homobilayers
(X-Y-Y-X or Y-X-X-Y) where the system recovers the mirror symmetry that
relates the chalcogen planes, so that the layer’s effective dipoles cancel
out. This further implies they are $\text{D}_{3}$ centrosymmetric structures,
in which Rashba spin splittings are forbidden by symmetry. Therefore, the main
differences between asymmetrically ordered homobilayers or heterobilayers and
symmetrycally ordered homobilayers are the presence or not of Rashba effects
and the degree of layer polarization.

In momentum space, the valence band maximum is located either at the Brillouin
zone corners (K/K’) or at $\Gamma$. In the former case with the VBM at K/K’,
the strong Ising spin-orbit splitting isolates in energy one state per valley,
realizing a situation similar to that of non-polar TMD heterobilayers [39].
In the latter case with the VBM at $\Gamma$, the interlayer interaction is
particularly important as out of plane $d_{z^{2}}/p_{z}$ orbitals are involved
at the band edge. This leads to a large band splitting where the VBM is the
layer antibonding state that, depending on the degree of layer polarization,
is separated by its bonding counterpart by several tens or hundreds of meV.
When Rashba SOC vanishes due to symmetry, the physics is then similar to that
of non-polar TMD homobilayers [40] where the $\Gamma$ band edge is spin-
degenerate.
The conduction band minimum is, either at K/K’ or the Q points [46]. The
physics of the moiré bands forming around the Q points will be discussed in
future work.

Figure 3: Continuum model moiré and effective mass potentials, bandstructure,
and charge density distribution are shown for three representative examples,
each displaying the different possible physics of twisted Janus TMDs. The
examples are (a) 3R MoSSe, (b) 3R WSTe, and (c) 3R WSSe twisted homobilayers
with twist angles $4.0^{\circ}$, $3.0^{\circ}$ and $1.5^{\circ}$,
respectively. The moiré $\Delta(\textbf{r})$ and effective mass potential
$\varepsilon(\textbf{r})$ follow the same real space distribution (encoded in
colors). The charge distribution of the topmost Bloch state (denoted by a
star) is shown. The charge is localized around the highly-symmetric moiré
potential maxima that form, in each example, triangular, honeycomb, and Kagome
networks.

####  II.1.1 Continuum model Hamiltonian

The low energy physics of twisted hetero/homobilayers can be modeled by means
of a low energy continuum model Hamiltonian. In both symmetrically and
asymmetrically ordered twisted bilayers the conduction and valence band edges
are isolated in energy by at least 50-100 meV.
The corresponding moiré physics is well described by the one-band continuum
model Hamiltonian [39, 40]:

$\mathcal{H}=-\frac{\hbar^{2}\textbf{k}^{2}}{2m^{*}_{0}}+\Delta(\textbf{r})+\varepsilon_{m^{*}}(\textbf{r})-\alpha_{R}(\boldsymbol{\sigma}\crossproduct\textbf{k})\cdot\hat{z},$
(1)

where $m^{*}_{0}$ and $\alpha_{R}$ are the average (over all the stacking
configurations) effective mass and Rashba parameter, $\Delta(\textbf{r})$ is
the moiré potential, $\varepsilon_{m^{*}}(\textbf{r})$ is the effective mass
correction, r tracks the local stacking configuration, and
$\boldsymbol{\sigma}$ are Pauli matrices.
$\Delta(\textbf{r})$ and $\varepsilon_{m^{*}}(\textbf{r})$ are scalar
functions that capture the energy and effective mass of holes/electrons in the
VBM/CBM as a function of the relative in-plane displacement r between the two
aligned layers. To capture the stacking-dependent electronic details necessary
for the determination of these potentials, we perform density functional
theory (DFT) calculations on non-twisted bilayers. Accordingly, we refer to
this model as a first principle effective Hamiltonian. The DFT calculations
are performed with the aim of sampling how the physics locally changes with
stacking: the top layer is shifted relative to the bottom layer over a
9$\crossproduct$9 grid in the unit cell.
These calculations are performed with Quantum Espresso [47, 48] using the
generalized gradient approximation with the weak vdW forces acting between the
layers taken into account by the nonlocal vdW functional vdW-DF2-C09 [49, 50].
All calculations include spin-orbit coupling. For each displacement, the in-
plane position of the atoms is fixed while the interlayer distance is relaxed.
The total energy, band edge energy, and effective mass are extracted for each
bilayer stacking, and then used in the continuum model. To find the effective
mass and Rashba coupling, we fit the values obtained from the aligned bilayers
band edges to the $\varepsilon\textbf{k}^{2}+\alpha|\textbf{k}|$ function. The
quadratic term tracks the effective mass $\varepsilon$ while the linear term
takes into account the Rashba spin splitting $\alpha$. In general, we found
that, unlike for the effective mass, $\alpha$ is only weakly dependent on
stacking. We therefore consider in the continuum model only the average (over
stacking configurations) Rashba parameter $\alpha_{R}$.

####  II.1.2 Lattice relaxation

To evaluate in-plane relaxation effects, we first compute the monolayers’ bulk
and shear moduli $\mathcal{G}$ and $\mathcal{K}$ (see Table 1). For the
interlayer energy, we extract from the bilayers’ DFT calculations their total
energies, which are then combined in the generalized stacking fault energy
functional (GSFE) $\Omega(\textbf{r})$ [45, 51, 52]. The GSFE tracks how the
interlayer energy depends on the relative stacking between the layers.
Then, we minimize the total mechanical energy of the moiré pattern obtaining
the top/bottom layer relaxation displacements $\textbf{u}_{t/b}$. In general,
the relaxation distorts the layers to maximize the area of the lowest-energy
stacking between layers. This is achieved via small in-plane deformations
characterized by strain fields that locally change the layer alignment [23,
45]. Out-of-plane relaxation is implicitly included at the level of the DFT
calculations, with each stacking allowed to move to its ideal interlayer
separation.
Finally, we incorporate the effect of atomic relaxations on the electronic
Hamiltonian of Eq. (1) through a linear mapping that updates the unrelaxed
configuration $\textbf{r}\to\textbf{r}+\textbf{u}_{t}+\textbf{u}_{b}$ [53].

Table 1: Bulk ($\mathcal{K}$) and shear moduli ($\mathcal{G}$), in meV per unit cell, of the six Janus TMDs monolayers considered. | MoSSe | WSSe | MoSTe | WSTe | MoSeTe | WSeTe
---|---|---|---|---|---|---
$\mathcal{K}$ | 49.7 | 45.8 | 39.7 | 42.7 | 41.7 | 39.1
$\mathcal{G}$ | 34.2 | 30.7 | 25.4 | 29.6 | 27.7 | 27.7

####  II.1.3 Fourier expansion

The two-dimensional periodicity of the aligned bilayers implies that the
potentials $\Delta(\textbf{r})$, $\varepsilon_{m^{*}}(\textbf{r})$, and
$\Omega(\textbf{r})$ are periodic functions that can be Fourier transformed in
momentum space. All the information relative to the moiré pattern, from its
structural relaxation to its effect on the bilayer effective masses, is
therefore conveniently stored in few Fourier coefficients.
The number of these coefficients can be reduced by taking into account
symmetry. Three-fold rotations with respect to the out-of-plane z-axis
($\text{C}_{3z}$) require that the potentials and their Fourier transforms are
invariant under $120^{\circ}$ rotations. This, and the fact that the
potentials are real functions, reduces the number of independent coefficients
to two: an amplitude and phase $\phi$ [39, 40].
Therefore, the potentials can be expressed as follows:

$\displaystyle\Omega(\textbf{r})=\sum_{l=1}^{3}\sum_{j=1}^{6}W_{l}\exp(i\textbf{g}_{j}^{l}\cdot\textbf{r}+\phi_{W}^{l}),$
(2a)
$\displaystyle\Delta(\textbf{r})=\sum_{l=1}^{3}\sum_{j=1}^{6}V_{l}\exp(i\textbf{g}_{j}^{l}\cdot\textbf{r}+\phi_{V}^{l}),$
(2b)
$\displaystyle\varepsilon(\textbf{r})=\sum_{j=1}^{6}U_{1}\exp(i\textbf{g}_{j}^{1}\cdot\textbf{r}+\phi_{U}^{1}),$
(2c)

where $\textbf{g}_{j+1}^{l}=\text{C}_{6z}\textbf{g}_{j}^{l}$ is the $l$-th
shell of six moiré g vectors ordered with increasing $|\textbf{g}|$. Unlike
for the effective mass correction, the moiré potential and GSFE have to be
expanded up to the third momentum shell to accurately capture the moiré
physics.

###  II.2 Moiré bands without Rashba effects: symmetric homobilayers and
emergent symmetries

We start by focusing on symmetrically ordered homobilayers (X-Y-Y-X or Y-X-
X-Y). In these cases, the system recovers the mirror symmetry that relates the
chalcogen planes. Being centrosymmetric structures, Rashba spin splittings are
forbidden by symmetry.
We consider a simplified version of the Hamiltonian in Eq. 1 and solve for the
Bloch states by expanding in plane waves:

$\bra{\textbf{k}+\textbf{g}^{\prime}}\mathcal{H}\ket{\textbf{k}+\textbf{g}}=-\delta_{\textbf{g},\textbf{g}^{\prime}}\frac{\hbar^{2}|\textbf{k+g}|^{2}}{2m_{0}^{*}}+\Delta(\textbf{g}-\textbf{g}^{\prime})+\varepsilon(\textbf{g}-\textbf{g}^{\prime}),$
(3)

where k is a wavevector in the moiré Brillouin-zone. We then use the continuum
model to calculate the bandstructure of twisted Janus TMDs at angles where
full microscopic calculations are prohibitive.

####  II.2.1 Emergent symmetries

An interesting feature of moiré materials is that the potentials in Eq. (2)
can develop symmetries additional to those of the underlying twisted lattice,
with profound consequences on the electronic and elastic properties of the
system [54, 23, 40, 25, 55]. In symmetrically ordered 3R bilayers with
$\text{D}_{3}$ symmetry, two bilayers stacked by r and
$\text{C}_{2z}\textbf{r}=-\textbf{r}$ are mapped into each other by a
$z\leftrightarrow-z$ mirror operation and hence have the same band and total
energies. This property, which is peculiar to 3R homobilayers, further implies
that $\Delta(\textbf{r})=\Delta(\text{C}_{2z}\textbf{r})$,
$\varepsilon_{m^{*}}(\textbf{r})=\varepsilon_{m^{*}}(\text{C}_{2z}\textbf{r})$,
and $\Omega(\textbf{r})=\Omega(\text{C}_{2z}(\textbf{r}))$, i.e., that the
potentials are inversion-symmetric functions. In terms of their expansion in
momentum space, this constraints the phases $\phi$ in Eq.(2) to be either $0$
or $\pi$ [40].
The moiré Hamiltonian in Eq. (1) and the GSFE inherit this additional
symmetry. This, combined with the three-fold rotations, promotes them to be
six-fold ($\text{D}_{6}$) symmetric functions. This allows the moiré
bandstructure to have distinct features, such as Dirac nodes, which would
otherwise be forbidden by symmetry.
It is important to stress that the moiré Hamiltonian in Eq. (1) and also the
GSFE develop the emergent $\text{C}_{2z}$ symmetry of the group $\text{D}_{6}$
despite the lattice having only three-fold $\text{D}_{3}$ symmetry.

####  II.2.2 Bandstructure symmetry analysis

We employ the Topological Quantum Chemistry theory [56] to identify the
symmetries and centers of the Wannier orbitals underlying the moiré bands.
This procedure involves computing the symmetry of the Bloch states,
classifying them in terms of the irreducible representations (irreps) of the
little groups at the corresponding high symmetry points, and then comparing
the list of irreps with the Elementary Band Representations (EBR) listed on
the Bilbao Crystallographic server [57]. The centers of the Wannier orbitals
for all the twisted bilayers’ considered are listed in Tables LABEL:tab:homo
and LABEL:tab:hetero.

In Fig. 3, the valence band structure of three different twisted bilayers is
shown along with their moiré/effective mass potentials and charge
distribution. They are three representative examples of the different physics
that can occur in twisted Janus TMDs. As can be seen, the maxima of the
potentials are located around high symmetry regions of the moiré pattern. As a
consequence, the charge tends to localize around these maxima, as shown by the
charge distribution of the topmost moiré bands.
Thus, the moiré pattern acts as a long-wavelength confining potential,
localizing holes/electrons around triangular, honeycomb, and Kagome lattice
sites. This behavior can be understood [39, 40, 45] by making a harmonic
approximation to the moiré potential near its minima/maxima, thus identifying
the harmonic oscillators wavefunctions as Wannier functions. The Bloch states
symmetry analysis agrees with this picture. The Wannier orbitals generating
the topmost set of moiré bands transform as a fully symmetric $s$ state while
the set of bands below is spanned by $p_{x}\pm ip_{y}$.
Thus, Fig. 3 summarizes the richness of the physics found in TMDs Janus
homobilayers, and in Table LABEL:tab:homo we present the characterization of
each system in terms of which moiré superlattice it forms, along with the
corresponding continuum model parameters. Depending on valley, chalcogen
ordering, 3R/2H twist configuration, and chemical composition, the system
realizes a platform to simulate triangular, honeycomb, and Kagome flat bands,
respectively shown in Fig. 3(a-c). The first case is usual, and commonly known
for twisted hexagonal systems including TMDs. The last two cases are a
peculiarity of 3R symmetrically ordered twisted bilayers and the emergent six-
fold symmetry of the continuum model Hamiltonian. In particular, in Fig. 3(b),
the charge localizes around the honeycomb network formed by the MX/XM regions.
The topmost set of bands are non-degenerate at $\Gamma$, form a Dirac node at
K, and are topologically equivalent to the $\pi$ bands of graphene. The bands
in Fig. 3(c) are even more intriguing as they exhibit an extremely narrow band
degenerate at $\Gamma$ and Dirac nodes. These bands are those of the
geometrically frustrated Kagome lattice, an exotic situation that can host
interesting correlated phenomena such as magnetic frustration and spin liquid
physics [58, 59]. The charge distribution, which is localized around the
Kagome network formed by the domain walls centers, further confirms this
picture.

###  II.3 Moiré bands with Rashba effects: asymmetric homobilayers and
heterobilayers

Figure 4: $\Gamma$-valley moiré valence bands of 3R MoSSe/WSSe with Se-S-S-Se
ordering, at $\theta=3^{\circ}$. The $\text{S}_{x}$ spin component is encoded
in colors. The in-plane spin texture of the Bloch states around $\Gamma$ in
the topmost moiré band is shown in the inset. The spin distribution in
momentum space follows a spiral pattern winding around $\Gamma$.

We now discuss the physics of heterobilayers and asymmetrically ordered
homobilayers. In these cases, due to the dissimilarity between the layers or
chalcogen ordering, the system is non-centrosymmetric. This further implies
that Rashba SOC is not prevented to arise around the valence band edge at
$\Gamma$. By expanding in plane waves the full Hamiltonian in Eq. 1, we
obtain:

$\displaystyle\bra{\textbf{k}+\textbf{g},s}\mathcal{H}\ket{\textbf{k}+\textbf{g}^{\prime},s^{\prime}}=$
$\displaystyle-\delta_{\textbf{g},\textbf{g}^{\prime}}\delta_{\textbf{s},\textbf{s}^{\prime}}\frac{\hbar^{2}|\textbf{k+g}|^{2}}{2m_{0}^{*}}$
$\displaystyle+\delta_{\textbf{s},\textbf{s}^{\prime}}\Delta(\textbf{g}-\textbf{g}^{\prime})$
$\displaystyle+\delta_{\textbf{s},\textbf{s}^{\prime}}\varepsilon(\textbf{g}-\textbf{g}^{\prime})$
$\displaystyle+\delta_{\textbf{g},\textbf{g}^{\prime}}\alpha_{R}[{\scriptstyle(k_{y}+g_{y})\sigma_{x}-(k_{x}+g_{x})\sigma_{y}}],$

where $s$ is the spin index, $\sigma_{x,y,z}$ are Pauli matrices, and
otherwise the same notation as before.

In Fig. 4 we show the spin-resolved bandstructure of a MoSSe/WSSe
heterobilayer twisted at 3 degrees. Given the 3R configuration and the
symmetry of chalcogen layers’ ordering, the bandstructure resembles that of
the honeycomb network moiré materials (Fig. 3b), with the first set of bands
similar to that of graphene. The second set of 4 bands resembles that of
$p_{x}\pm ip_{y}$ orbitals on a honeycomb lattice. However, in heterobilayers,
the fourfold Dirac degeneracy at the K points is lifted. First of all, the
moiré potential is not six-fold symmetric because the MX regions, with a
molybdenum atom on top of a pair of chalcogens, has a slightly different
energy than the XM regions, where the chalcogens are on top of a tungsten
atom. The imbalance between the MX/XM honeycomb lattice sites corresponds to a
mass term in the Dirac Hamiltonian [60]. On top of this, the Rashba spin-orbit
coupling further spoils the Dirac nodes by lifting the spin degeneracy. The
spin texture distribution in momentum space follows a spiral pattern winding
around $\Gamma$.
The continuum model parameters for the Janus TMDs heterobilayers, all having
triangular moiré superlattices, are listed in Table LABEL:tab:hetero, while
the parameters for the asymmetric homobilayers are part of Table
LABEL:tab:homo. In particular, the Rashba parameters are of the order of tens
or a few hundred meV/Å, greatly exceeding the moiré bandwidth at small angles.
Therefore, in the small angle regime, Rashba effects are likely to affect, if
not dominate, the correlated physics of these systems.

##  III Discussion

In conclusion, we have presented an extended continuum model able to
accurately calculate the electronic structure of twisted Janus TMDs. The model
features terms that properly take into account lattice relaxation, the
stacking dependence of the effective mass, and Rashba spin-orbit coupling. The
application of this methodology to more than a hundred twisted bilayers
reveals the richness of their physics, and that it can be carefully tuned by
choosing atomic composition, stacking structure, interface ordering, and twist
angle.
In symmetrically ordered homobilayers, the layers’ inversion symmetry prevents
Rashba spin splittings and allows for the emergence of symmetries and Dirac
nodes in the moiré bands. Thus, they are able to realize not only common
triangular moiré superlattices, but also honeycomb and Kagome networks,
potentially displaying rich correlated-electronic phenomena and enabling their
use as a platform of quantum simulators [35]. The honeycomb bands that form in
most 3R twisted homobilayers provide a convenient realization of artificial
graphene, but with a fine-structure constant that can be tuned at will simply
by varying twist angle [61]. The Kagome lattice models of 3R WSSe and MoSSe
are expected to host different phases, including spin-liquids [58, 59].
Furthermore, by manipulating spin-orbit coupling with gate electric fields or
by proximity effects, these bands can exhibit topologically non-trivial states
[62, 63, 64, 65]. Since all systems have bandwidths that can be adjusted
simply by varying twist angles, they provide an enticing toolbox to study the
exotic properties of strongly correlated physics in hexagonal Bravais lattices
[66, 67, 68, 69, 70].
In heterobilayers and asymmetrically ordered homobilayers, Rashba spin-orbit
coupling constants can greatly exceed the systems’ bandwidth, therefore being
expected to have an important impact on the correlated states. Indeed, this
situation could provide opportunities for observing non-trivial phenomena such
as the intrinsic spin Hall effect [71], non-centrosymmetric superconductivity
[72], and Majorana fermions [73]. Furthermore, the presence of a magnetic
substrate can strongly impact the non-coplanar magnetic texture of the moiré
unit cell [74], providing an additional degree of freedom to tune the system’s
properties.
Finally, with the data employed in the continuum model calculations and shown
in Tables LABEL:tab:homo and LABEL:tab:hetero, we present a step toward the
creation of a twistronics database, containing data from different systems and
able to calculate properties for any, including arbitrarily small, twist
angles. This information can serve as a basis for guiding both theoretical and
experimental studies, relevant to different research strategies, such as
direct and inverse design, and artificial intelligence investigations [36,
75]. We expect our results to have implications for the theoretical and
experimental study of correlated physics in moiré materials, which could pave
the way for the use of these systems for magnetism, spintronics, and other
nanoscale devices.

Table 2:  Homobilayer continuum model parameters. The first line for each material corresponds to the valence band edge (VBM), which occurs at $\Gamma$, and the second line corresponds to the conduction band edge (CBM), which occurs at $K$. The exceptions to this rule are marked by an asterisk, in which case both the VBM and CBM occur at $K$. The GSFE ($\omega$) and moiré potential ($\Delta$) fourier coefficients (modulus and phase $\phi$) are listed up to the third moiré G-vectors shell. The effective mass ($m_{0}^{*}$) and potential ($\varepsilon$) fourier coeffient are in bare electron mass units. The Rashba coefficient ($\alpha_{R}$) is in meV/Å. The moiré superlattice ($MSL$) formed by the topmost/lowest valence/conduction band’s localized charge distribution is listed in the last column, where T stands for triangular, H for honeycomb, and K for Kagome. Material |  Stack ordering |  | $W_{1,2,3}$ | $\phi_{W}^{1,2,3}$ | $V_{1,2,3}$ | $\phi_{V}^{1,2,3}$ | $m_{0}^{*}$ | $U_{1}$ | $\phi_{U}^{1}$ | $\alpha_{R}$ | MSL
---|---|---|---|---|---|---|---|---|---|---|---
3R MoSSe |  Se-S-Se-S $\bigg{|}$ |  | (8.7,0.2,0.1) | (1.0,0.2, 2.6) | (3.6,0.1,0.1) | (74.6,62.5,10.6) | -0.1 | 0.01 | 163.5 | 66.3 | T @ MX
| (8.7,0.2,0.1) | (1.0,0.2, 2.6) | (3.7,0.14,0.16) | (92.3,81.0,87.3) | 0.92 | 0.01 | 2.6 | 0.0 | T @ MX
S-Se-Se-S $\bigg{|}$ |  | (7.4,1.0,0.1) | (0.0,180.0,180.0) | (12.0,0.1,1.9) | (180.0,-180.0,0.0) | -0.41 | 0.02 | 0.0 | 0.0 | K @ DW
| (7.4,1.0,0.1) | (0.0,180.0,180.0) | (5.9,0.6,0.2) | (0.0,180.0,0.0) | 0.37 | 0.08 | 0.0 | 0.0 | H @ MX/XM
Se-S-S-Se $\bigg{|}$ |  | (6.1,0.2,0.2) | (0.0,0.0, 0.0) | (7.8,0.2,0.3) | (0.0,0.0,0.0) | -0.4 | 0.01 | 0.0 | 0.0 | T @ AA
| (6.1,0.2,0.2) | (0.0,0.0, 0.0) | (0.5,1.1,0.51) | (0.0,180.0,0.0) | 0.65 | 0.01 | 180.0 | 0.0 | H @ MX/XM
2H MoSSe |  Se-S-Se-S $\bigg{|}$ |  | (7.9,0.1,0.1) | (131.9, 1.2, 85.4) | (4.9,0.1,0.1) | (125.9,156.9,140.5) | -0.36 | 0.0 | 0.0 | 16.6 | T @ MM
| (7.9,0.1,0.1) | (131.9, 1.2, 85.4) | (0.7,0.3,0.1) | (122.2,175.5,73.0) | 0.83 | 0.01 | 144.0 | 0.0 | T @ XX
S-Se-Se-S $\bigg{|}$ |  | (7.5,1.3,0.4) | (143.7, 180.0, 87.5) | (11.9,0.96,1.3) | (58.5,175.5,116.0) | -0.42 | 0.02 | 180.0 | 0.0 | T @ AB
| (7.5,1.3,0.4) | (143.7, 180.0, 87.5) | (5.9,0.6,0.2) | (115.0,8.6,38.8) | 0.29 | 0.06 | 150.0 | 0.0 | T @ XX
Se-S-S-Se $\bigg{|}$ |  | (8.9,1.4,0.2) | (129.2, 180.0, 46.7) | (35.0,14.5,7.7) | (60.5,0.8,128.0) | -0.62 | 0.08 | 127.6 | 0.0 | T @ XX
| (8.9,1.4,0.2) | (129.2, 180.0, 46.7) | (0.7,0.0,0.2) | (24.0,7.3,64.8) | 0.5 | 0.03 | 122.9 | 0.0 | T @ MM
3R WSSe |  Se-S-Se-S*$\bigg{|}$ |  | (8.3,0.2,0.1) | (0.0,0.0, -2.4) | (4.9,0.6,0.9) | (92.9,-162.7,-56.7) | -2.64 | 0.0 | 0.0 | 0.0 | T @ XM
| (8.3,0.2,0.1) | (0.0,0.0, -2.4) | (3.8,0.2,0.2) | (-79.1,39.7,93.9) | 2.07 | 0.01 | -2.5 | 0.0 | T @ XM
S-Se-Se-S $\bigg{|}$ |  | (7.6,1.3,0.6) | (0.0,-180.0, -180.0) | (8.9,2.8,0.5) | (180.0,180.0,0.0) | -0.7 | 0.02 | 0.0 | 0.0 | K @ DW
| (7.6,1.3,0.6) | (0.0,-180.0, -180.0) | (2.6,0.8,1.5) | (0.0,0.0,0.0) | 1.19 | 0.02 | 0.0 | 0.0 | H @ XM/XM
Se-S-S-Se $\bigg{|}$ |  | (6.0,0.1,0.1) | (0.0,0.0,0.0) | (7.9,0.2,0.06) | (0.0,0.0,-120.0) | -0.82 | 0.01 | 0.0 | 0.0 | T @ AA
| (6.0,0.1,0.1) | (0.0,0.0,0.0) | (2.4,2.6,1.3) | (0.0,180.0,0.0) | 1.9 | 0.01 | 180.0 | 0.0 | H @ XM/XM
2H WSSe |  S-Se-S-Se*$\bigg{|}$ |  | (8.3,0.2,0.1) | (0.0, 0.0, -2.0) | (0.7,0.4,0.3) | (-168.0,-71.0,142.5) | -2.63 | 0.0 | 0.0 | 0.0 | T @ XX
| (8.3,0.2,0.1) | (0.0, 0.0, -2.0) | (1.2,0.1,0.2) | (-160.5,-66.1,-134.9) | 2.1 | 0.01 | 115.0 | 0.0 | T @ AB
S-Se-Se-S $\bigg{|}$ |  | (7.2,0.1,0.7) | (148.1, -109.9, 60.5) | (3.6,1.2,1.5) | (93.4,178.9,29.7) | -0.72 | 0.03 | 125.0 | 0.0 | T @ M
| (7.2,0.1,0.7) | (148.1, -109.9, 60.5) | (2.8,0.8,0.2) | (113.4,5.0,-45.1) | 1.3 | 0.05 | 132.3 | 0.0 | T @ MM
Se-S-S-Se $\bigg{|}$ |  | (7.8,1.9,0.5) | (127.7, 180.0, 75.5) | (20.2,20.6,9.5) | (-66.6,0.0,-125.7) | -0.99 | 0.07 | 125.0 | 0.0 | T @ XX
| (7.8,1.9,0.5) | (127.7, 180.0, 75.5) | (0.8,0.1,0.1) | (-23.7,164.1,60.0) | 2.0 | 0.06 | 55.0 | 0.0 | T @ MM
3R MoSTe |  Te-S-Te-S $\bigg{|}$ |  | (10.9,1.8,0.5) | (-7.4,180.0, 177.4) | (4.9,0.7,0.5) | (-29.0,26.4,-171.9) | -0.2 | 0.0 | 0.0 | 0.0 | T @ AA
| (10.9,1.8,0.5) | (-7.4,180.0, 177.4) | (3.7,0.1,0.1) | (92.1,-86.9,81.9) | 0.96 | 0.01 | -2.5 | 0.0 | T @ MX
Te-S-S-Te $\bigg{|}$ |  | (6.4,0.2,0.2) | (0.0,0.0, 0.0) | (12.6,0.1,0.4) | (0.0,-67.1,2.2) | -1.0 | 0.02 | 0.0 | 0.0 | T @ AA
| (6.4,0.2,0.2) | (0.0,0.0, 0.0) | (0.5,1.1,0.5) | (0.0,180.0,0.0) | 0.68 | 0.01 | 180.0 | 0.0 | H @ MX/XM
S-Te-Te-S $\bigg{|}$ |  | (9.0,1.6,0.3) | (0.0,180.0, 180.0) | (1.5,0.4,0.2) | (180.0,0.0,-165.2) | -0.19 | 0.01 | 180.0 | 0.0 | H @ MX/XM
| (9.0,1.6,0.3) | (0.0,180.0, 180.0) | (4.3,1.6,0.2) | (0.0,180.0,0.0) | 0.1 | 0.08 | 0.0 | 0.0 | H @ MX/XM
2H MoSTe |  S-Te-S-Te*$\bigg{|}$ |  | (9.9,2.4,1.4) | (136.2, 180.0, 53.6) | (5.2,0.4,0.2) | (-128.5,153.4,-137.9) | -2.3 | 0.0 | 0.0 | 0.0 | T @ MM
| (9.9,2.4,1.4) | (136.2, 180.0, 53.6) | (4.8,4.3,2.0) | (114.5,0.0,-128.4) | 1.2 | 0.1 | -107.7 | 0.0 | T @ AA
Te-S-S-Te $\bigg{|}$ |  | (10.9,2.0,0.7) | (112.2, 180.0, 20.8) | (43.5,13.5,12.4) | (90.5,0.0,-138.9) | -1.49 | 0.17 | -104.4.0 | 0.0 | T @ AB
| (10.9,2.0,0.7) | (112.2, 180.0, 20.8) | (1.36,0.98,0.68) | (49.8,180.0,27.5) | 1.07 | 0.01 | 40.7 | 0.0 | T @ XX
S-Te-Te-S $\bigg{|}$ |  | (9.4,1.4,0.56) | (155.3, 180.0, 129.0) | (1.21,0.22,0.28) | (7.6,3.0,14.6) | -0.3 | 0.01 | 155.0 | 0.0 | T @ MM
| (9.4,1.4,0.56) | (155.3, 180.0, 129.0) | (3.64,0.33,0.36) | (122.8,172.0,55.5) | 0.13 | 0.09 | 178.9 | 0.0 | T @ MM
3R WSTe |  Te-S-Te-S*$\bigg{|}$ |  | (9.5,1.86,0.8) | (142.4,180.0, 114.7) | (1.1,0.65,1.1) | (152.4,-21.3,-64.8) | -2.02 | 0.01 | 40.0 | 0.0 | T @ XM
| (9.5,1.86,0.8) | (142.4,180.0, 114.7) | (1.64,0.53,0.75) | (-143.1,9.0,75.5) | 0.96 | 0.01 | -2.5 | 0.0 | T @ AA
Te-S-S-Te $\bigg{|}$ |  | (9.6,2.8,2.0) | (0.0,180.0, 0.0) | (14.7,33.6,13.4) | (180.0,0.0,180.0) | -1.3 | 0.09 | 0.0 | 0.0 | H @ MX/XM
| (9.6,2.8,2.0) | (0.0,180.0, 0.0) | (1.1,2.2,1.4) | (0.0,180.0,0.0) | 1.52 | 0.0 | 0.0 | 0.0 | H @ MX/XM
S-Te-Te-S $\bigg{|}$ |  | (8.6,1.1,0.16) | (0.0,180.0, 0.0) | (2.5,0.5,0.3) | (130.0,0.0,-135.2) | -0.3 | 0.01 | 180.0 | 0.0 | T @ MX
| (8.6,1.1,0.16) | (0.0,180.0, 0.0) | (6.7,2.6,0.8) | (0.0,180.0,180.0) | 0.42 | 0.07 | 0.0 | 0.0 | H @ MX/XM
2H WSTe |  S-Te-S-Te*$\bigg{|}$ |  | (9.5,1.9,0.8) | (142.5, 180.0, 114.7) | (1.1,0.6,1.1) | (152.5,-21.0,-64.4) | -2.0 | 0.01 | 40.2 | 0.0 | T @ MM
| (9.5,1.9,0.8) | (142.5, 180.0, 114.7) | (1.6,0.5,0.75) | (-141.1,9.0,75.4) | 1.6 | 0.02 | 112.7 | 0.0 | T @ AA
Te-S-S-Te $\bigg{|}$ |  | (9.3,1.3,0.2) | (116.2, 180.0, 0.0) | (37.7,8.9,2.9) | (-82.5,0.0,-171.1) | -1.52 | 0.13 | 110.4 | 0.0 | T @ XX
| (9.3,1.3,0.2) | (116.2, 180.0, 0.0) | (2.85, 0.4,0.2) | (-62.8,0.0,-107.5) | 1.54 | 0.02 | 45.7 | 0.0 | T @ MM
S-Te-Te-S $\bigg{|}$ |  | (9.4,1.4,0.6) | (155.3, 180.0, 129.0) | (0.2,0.4,0.2) | (140.6,24.0,14.6) | -0.4 | 0.02 | 70.0 | 0.0 | T @ XX
| (9.4,1.4,0.6) | (155.3, 180.0, 129.0) | (6.7,0.3,0.3) | (148.8,-169.0,20.5) | 0.36 | 0.12 | 176.9 | 0.0 | T @ AB
3R WSeTe |  Te-Se-Te-Se $\bigg{|}$ |  | (11.8,1.7,1.25) | (2.4,180.0, 8.6) | (30.7,1.9,5.5) | (-177.0,-168.4,35.7) | -0.4 | 0.0 | 0.0 | 0.0 | T @ MX
| (11.8,1.7,1.25) | (2.4,180.0, 8.6) | (1.6,0.5,0.8) | (-143.1,9.0,75.5) | 0.96 | 0.01 | -2.5 | 0.0 | T @ MX
Te-Se-Se-Te $\bigg{|}$ |  | (9.6,2.8,2.0) | (0.0,180.0, 0.0) | (14.7,33.6,13.4) | (180.0,0.0,180.0) | -1.3 | 0.09 | 0.0 | 0.0 | H @ MX/XM
| (9.6,2.8,2.0) | (0.0,180.0, 0.0) | (1.1,2.2,1.36) | (0.0,180.0,0.0) | 1.52 | 0.0 | 0.0 | 0.0 | H @ MX/XM
Se-Te-Te-Se $\bigg{|}$ |  | (11.2,0.3,0.4) | (0.0,180.0, 0.0) | (3.9,5.2,5.7) | (180.0,0.0,0.0) | -0.9 | 0.07 | 0.0 | 0.0 | T @ AA
| (11.2,0.3,0.4) | (0.0,180.0, 0.0) | (1.4,1.8,1.6) | (0.0,180.0,0.0) | 1.6 | 0.01 | 180.0 | 0.0 | H @ MX/XM
2H WSeTe |  Se-Te-Se-Te $\bigg{|}$ |  | (10.3,0.8,1.0) | (151.2, -172.0, -103.7) | (24.5,4.7,6.0) | (-55.5,-6.0,-155.4) | -0.3 | 0.13 | 123.4 | 107.4 | T @ XX
| (10.3,0.8,1.0) | (151.2, -172.0, -103.7) | (3.6,0.5,0.6) | (-78.9,-165.9,20.4) | 1.7 | 0.03 | 94.7 | 0.0 | T @ XX
Te-Se-Se-Te $\bigg{|}$ |  | (12.0,0.7,0.5) | (160.0, -167.0, -48.0) | (9.9,2.7,1.6) | (-9.0,34.5,90.0) | -0.3 | 0.03 | 70.0 | 0.0 | T @ AB
| (12.0,0.7,0.5) | (160.0, -167.0, -48.0) | (5.2, 0.4,0.3) | (-153.9,174.0,-58.5) | 0.9 | 0.12 | 166.7 | 0.0 | T @ AB
Se-Te-Te-Se $\bigg{|}$ |  | (9.1,0.7,0.2) | (128.3, 180.0, -43.0) | (21.1,3.4,1.4) | (-65.3,-177.0,-142.1) | -0.9 | 0.07 | 125.0 | 0.0 | T @ XX
| (9.1,0.7,0.2) | (128.3, 180.0, -43.0) | (22.5,23.3,11.0) | (-62.5,78.0,-107.5) | 1.7 | 0.01 | 76.9 | 0.0 | T @ MM
3R MoSeTe |  Te-Se-Te-Se $\bigg{|}$ |  | (11.4,2.1,1.3) | (-3.8,180.0, 9.6) | (55.7,6.1,5.4) | (177.7,-173.9,36.9) | -0.17 | 0.15 | 0.0 | 70.3 | T @ MX
| (11.4,2.1,1.3) | (-3.8,180.0, 9.6) | (22.4,8.5,3.8) | (19.1,2.0,1.1) | 0.4 | 0.07 | 3.5 | 0.0 | T @ MX
Te-Se-Se-Te $\bigg{|}$ |  | (11.8,0.1,1.0) | (0.0,0.0, 0.0) | (6.3,9.3,14.4) | (180.0,180.0,180.0) | -0.8 | 0.06 | 0.0 | 0.0 | T@MX
| (11.8,0.1,1.0) | (0.0,0.0, 0.0) | (0.8,0.6,0.7) | (0.0,180.0,0.0) | 3.9 | 0.0 | 0.0 | 0.0 | H @ MX/XM
Se-Te-Te-Se $\bigg{|}$ |  | (8.9,1.1,0.2) | (0.0,180.0, 0.0) | (5.6,1.5,0.5) | (180.0,0.0,180.0) | -0.1 | 0.03 | 0.0 | 0.0 | H @ MX/XM
| (8.9,1.1,0.2) | (0.0,180.0, 0.0) | (1.75,0.6,0.8) | (0.0,0.0,0.0) | 4.6 | 0.02 | 180.0 | 0.0 | T @AA
2H MoSeTe |  Se-Te-Se-Te $\bigg{|}$ |  | (11.5,0.7,0.6) | (146.2, -167.0, -61.9) | (32.5,4.0,3.9) | (-45.5,-178.0,105.4) | -0.02 | 0.09 | 143.4.0 | 104.2 | T @ AB
| (11.5,0.7,0.6) | (146.2, -167.0, -61.9) | (18.1,9.2,4.4) | (130.1,1.7,-101.4) | 4.5 | 0.03 | 145.7 | 0.0 | T @ XX
Te-Se-Se-Te $\bigg{|}$ |  | (11.9,0.0,0.9) | (0.0, 0.0, 0.0) | (6.5,9.5,14.5) | (180.0,180.0,180.0) | -0.8 | 0.06 | 0.0 | 0.0 | T @ MM
| (11.9,0.0,0.9) | (0.0, 0.0, 0.0) | (0.6, 0.4,0.1) | (-161.9,2.0,-138.5) | 4.9 | 0.01 | -107.7 | 0.0 | T @ MM
Se-Te-Te-Se $\bigg{|}$ |  | (10.6,0.9,0.2) | (157.7, 180.0, -53.0) | (7.9,1.6,0.5) | (13.0,2.0,-11.0) | -0.02 | 0.0 | 0.0 | 0.0 | T @ AB
| (10.6,0.9,0.2) | (157.7, 180.0, -53.0) | (6.6,0.5,0.8) | (75.5,51.9,-41.5) | 4.7 | 0.07 | 20.9 | 0.0 | T @ AB

Table 3:  Heterobilayer continuum model parameters. All symbols have the same meaning as in Table LABEL:tab:homo. For each material there are now three lines, corresponding to the values for the respective band edges: $\Gamma$ at the VBM, $K$ at the VBM, and $K$ at the CBM.  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---
Material |  Stack ordering |  | $W_{1,2,3}$ | $\phi_{W}^{1,2,3}$ | $V_{1,2,3}$ | $\phi_{V}^{1,2,3}$ | $m_{0}^{*}$ | $U_{1}$ | $\phi_{U}^{1}$ | $\alpha_{R}$ | MSL
3R MoSSe/WSSe |  S-Se-S-Se $\bigg{|}$ |  | (7.4,1.5,0.6) | (-3.7,177.9, 54.6) | (5.3,6.6,2.8) | (121.3,3.8,-125.6) | -0.6 | 0.02 | -20.5 | 25.4 | T @ XM
| (7.4,1.5,0.6) | (-3.7,177.9, 54.6) | (3.7,0.6,0.4) | (104.6,166.0,-99.5) | -2.3 | 0.01 | -2.3 | 0.0 | T @ XM
| (7.4,1.5,0.6) | (-3.7,177.9, 54.6) | (3.6,0.3,0.8) | (-106.2, -157.9, 61.5) | 2.1 | 0.02 | -173.0 | 0.0 | T @ XM
S-Se-Se-S $\bigg{|}$ |  | (11.5,0.7,0.6) | (146.2, -167.0, -61.9) | ( 9.3 , 5.2 , 1.7 ) | ( -165.8 , 1.8 , -2.4 ) | -0.75 | 0.0 | 0.0 | 60.8 | T @ MX
| (11.5,0.7,0.6) | (146.2, -167.0, -61.9) | (4.0,0.5,1.0) | (-61.7,13.9,24.2) | -2.7 | 0.0 | 0.0 | 0.0 | T @ MX
| (11.5,0.7,0.6) | (146.2, -167.0, -61.9) | (2.5,0.3,0.8) | (108.1,0.3,-23.0) | 1.6 | 0.03 | -12.9 | 0.0 | T @ MX
Se-S-S-Se $\bigg{|}$ |  | (10.0,2.1,0.9) | (-0.7,180.0,179.2) | (38.3,14.8,11.1) | (-178.9,0.7,0.8) | -1.0 | 0.1 | 0.0 | 44.5 | T @ MX
| (10.0,2.1,0.9) | (-0.7,180.0,179.2) | (8.1,0.8,1.6) | (-59.9,180.0,140.4) | -2.68 | 0.0 | 0.0 | 0.0 | T @ MX
|  | (10.0,2.1,0.9) | (-0.7,180.0,179.2) | (6.2,0.0,1.1) | (81.8,-164.0,-129.4) | 1.8 | 0.01 | 0.0 | 0.0 | T @ MX
|  Se-S-Se-S $\bigg{|}$ |  | ( 9.7 , 2.1 , 0.7 ) | ( -5.3 , -179.9 , 162.1 ) | ( 9.3 , 5.2 , 1.7 ) | ( -165.8 , 1.8 , -2.4 ) | -0.75 | 0.0 | 0.0 | 60.8 | T @ MX
|  | ( 9.7 , 2.1 , 0.7 ) | ( -5.3 , -179.9 , 162.1 ) | (8.0,2.4,1.9) | (-54.5,180.0,141.4) | -2.7 | 0.0 | 0.0 | 0.0 | T @ MX
|  | ( 9.7 , 2.1 , 0.7 ) | ( -5.3 , -179.9 , 162.1 ) | (6.0,1.1,1.0) | (113.4,-3.9,-90.4) | 1.8 | 0.01 | -4.4 | 0.0 | T @ MX
2H MoSSe/WSSe |  S-Se-S-Se $\bigg{|}$ |  | ( 4.8 , 0.1 , 0.1 ) | ( 132.1 , 0.5 , -78.1 ) | ( 2.8 , 0.1 , 0.2 ) | ( 116.9 , 22.2 , -102.9 ) | -0.55 | 0.0 | 0.0 | 29.8 | T @ XX
| ( 4.8 , 0.1 , 0.1 ) | ( 132.1 , 0.5 , -78.1 ) | ( 0.8 , 0.1 , 0.2 ) | ( -26.4 , 97.8 , -70.6 ) | -2.28 | 0.0 | 0.0 | 0.0 | T @ AB
| ( 4.8 , 0.1 , 0.1 ) | ( 132.1 , 0.5 , -78.1 ) | ( 1.2 , 0.0 , 0.1 ) | ( -70.9 , -28.5 , 109.7 ) | -2.06 | 0.0 | 0.0 | 0.0 | T @ MM
S-Se-Se-S $\bigg{|}$ |  | ( 7.9 , 1.3 , 0.6 ) | ( 143.7 , 179.7 , 123.6 ) | ( 8.9 , 1.8 , 2.9 ) | ( -26.5 , -0.6 , -3.9 ) | -1.04 | 0.0 | 0.0 | 64.5 | T @ AB
| ( 7.9 , 1.3 , 0.6 ) | ( 143.7 , 179.7 , 123.6 ) | ( 0.8 , 0.1 , 0.2 ) | ( -26.4 , 97.8 , -70.6 ) | -2.28 | 0.0 | 0.0 | 0.0 | T @ AB
| ( 7.9 , 1.3 , 0.6 ) | ( 143.7 , 179.7 , 123.6 ) | ( 1.2 , 0.0 , 0.1 ) | ( -70.9 , -28.5 , 109.7 ) | -2.06 | 0.0 | 0.0 | 0.0 | T @ MM
Se-S-S-Se $\bigg{|}$ |  | ( 3.3 , 0.1 , 0.3 ) | ( 129.1 , 27.8 , -10.7 ) | ( 4.9 , 0.3 , 0.2 ) | ( 123.2 , -76.4 , -122.0 ) | -0.73 | 0.0 | 0.0 | 67.1 | T @ MM
| ( 3.3 , 0.1 , 0.3 ) | ( 129.1 , 27.8 , -10.7 ) | ( 0.8 , 0.1 , 0.2 ) | ( -26.4 , 97.8 , -70.6 ) | -2.28 | 0.0 | 0.0 | 0.0 | T @ AB
| ( 3.3 , 0.1 , 0.3 ) | ( 129.1 , 27.8 , -10.7 ) | ( 1.2 , 0.0 , 0.1 ) | ( -70.9 , -28.5 , 109.7 ) | -2.06 | 0.0 | 0.0 | 0.0 | T @ MM
|  Se-S-Se-S $\bigg{|}$ |  | ( 8.3 , 1.8 , 0.7 ) | ( 137.1 , 180.0 , 34.5 ) | ( 6.8 , 5.0 , 2.4 ) | ( -32.8 , -0.9 , -145.8 ) | -0.69 | 0.0 | 0.0 | 70.1 | T @ XX
|  | ( 8.3 , 1.8 , 0.7 ) | ( 137.1 , 180.0 , 34.5 ) | ( 0.8 , 0.1 , 0.2 ) | ( -26.4 , 97.8 , -70.6 ) | -2.28 | 0.0 | 0.0 | 0.0 | T @ AB
|  | ( 8.3 , 1.8 , 0.7 ) | ( 137.1 , 180.0 , 34.5 ) | ( 1.2 , 0.0 , 0.1 ) | ( -70.9 , -28.5 , 109.7 ) | -2.06 | 0.0 | 0.0 | 0.0 | T @ XX
3R MoSTe/WSTe |  S-Te-S-Te $\bigg{|}$ |  | ( 10.6 , 2.0 , 0.7 ) | ( 8.3 , 180.0 , -150.6 ) | ( 5.1, 0.9 , 2.0 ) | ( -164.6 , 172.5 , 175.9 ) | -0.65 | 0.02 | -4.5 | 68.1 | T @ MX
| ( 10.6 , 2.0 , 0.7 ) | ( 8.3 , 180.0 , -150.6 ) | ( 4.9 , 0.3 , 1.0 ) | ( 42.4 , -24.0 , -52.8 ) | -1.83 | 0.01 | -70.0 | 0.0 | T @ AA
| ( 10.6 , 2.0 , 0.7 ) | ( 8.3 , 180.0 , -150.6 ) | ( 3.5 , 0.4 , 0.8 ) | ( -6.6 , 27.5 , -0.5 ) | 1.64 | 0.02 | 25.4 | 0.0 | T @ XM
S-Te-Te-S $\bigg{|}$ |  | ( 9.6 , 1.8 , 0.5 ) | ( 0.0 , 180.0 , 176.6 ) | ( 2.4 , 0.5 , 0.1 ) | ( -12.7 , 1.9 , 20.9 ) | -0.55 | 0.01 | 180.0 | 186.4 | T @ MX
| ( 9.6 , 1.8 , 0.5 ) | ( 0.0 , 180.0 , 176.6 ) | ( 7.2 , 0.5 , 0.7 ) | ( -37.4 , 161.2 , 66.6 ) | -2.1 | 0.01 | 166.0 | 0.0 | T @ MX
| ( 9.6 , 1.8 , 0.5 ) | ( 0.0 , 180.0 , 176.6 ) | ( 2.3 , 0.3 , 0.15 ) | ( 11.9 , -180.0 , -12.6 ) | 0.6 | 0.07 | -3.0 | 0.0 | T @ MX
Te-S-S-Te $\bigg{|}$ |  | ( 12.5 , 2.5 , 1.0 ) | ( -0.3 , 180.0 , 180.0 ) | ( 29.0 , 4.3 , 2.6 ) | ( -177.4 , 2.1 , -3.5 ) | -1.6 | 0.15 | 0.0 | 32.6 | T @ MX
| ( 12.5 , 2.5 , 1.0 ) | ( 0.4 , 180.0 , 180.0 ) | ( 13.6 , 0.5 , 1.0 ) | ( -31.4 , -166.3 , 123.6 ) | -2.13 | 0.01 | 130.0 | 0.0 | T @ MX
| ( 12.5 , 2.5 , 1.0 ) | ( 0.4 , 180.0 , 180.0 ) | ( 10.4, 1.8 , 1.1 ) | ( 44.7 , 2.5 , -74.7 ) | 1.15 | 0.01 | 28.5 | 0.0 | T @ MX
|  Te-S-Te-S $\bigg{|}$ |  | ( 10.6 , 2.0 , 0.7 ) | ( 8.1 , 180.0 , -151.7 ) | ( 5.1 , 0.9 , 2.0 ) | ( -166.8 , 175.0 , 175.8 ) | -0.65 | 0.02 | -4.9 | 69.1 | T @ MX
|  | ( 10.6 , 2.0 , 0.7 ) | ( 8.1 , 180.0 , -151.7 ) | ( 3.2 , 0.9 , 1.3 ) | ( -66.4 , 0.0 , 31.33 ) | -2.1 | 0.01 | 55.2 | 0.0 | T @ MX
|  | ( 10.6 , 2.0 , 0.7 ) | ( 8.1 , 180.0 , -151.7 ) | ( 3.9 , 0.3 , 0.95 ) | ( 2.7 , -22.5 , -13.7 ) | 1.1 | 0.02 | 55.3 | 0.0 | T @ MX
2H MoSTe/WSTe |  S-Te-S-Te $\bigg{|}$ |  | ( 10.1 , 1.1 , 0.9 ) | ( 132.3 , 180.0 , 61.6 ) | ( 1.8 , 0.9 , 0.9 ) | ( 55.0 , -4.1 , -102.1 ) | -0.4 | 0.0 | 0.0 | 98.9 | T @ MM
| ( 10.1 , 1.1 , 0.9 ) | ( 132.3 , 180.0 , 61.6 ) | ( 0.5 ,1.5 , 1.0 ) | ( -73.3 , 1.4 , -103.6 ) | -1.83 | 0.01 | 43.3 | 0.0 | T @ XX
| ( 10.1 , 1.1 , 0.9 ) | ( 132.3 , 180.0 , 61.6 ) | ( 1.1 , 0.7 , 0.5 ) | ( -118.6 , 31.5 , -142.9 ) | 1.66 | 0.03 | 24.0 | 0.0 | T @ MM
S-Te-Te-S $\bigg{|}$ |  | ( 9.7 , 1.6 , 0.8 ) | ( 153.3 , 180.0 , 130.6 ) | ( 3.0 , 0.5 , 0.2 ) | ( 134.6 , 1.2 , -19.5 ) | -0.55 | 0.01 | 160.0 | 227.4 | T @ XX
| ( 9.7 , 1.6 , 0.8 ) | ( 153.3 , 180.0 , 130.6 ) | ( 7.1 , 0.7 , 0.9 ) | ( -36.4 , 171.2 , 81.6 ) | -2.1 | 0.01 | 166.0 | 0.0 | T @ MM
| ( 9.7 , 1.6 , 0.8 ) | ( 153.3 , 180.0 , 130.6 ) | ( 2.3 , 0.3 , 0.15 ) | ( 11.9 , -180.0 , -12.6 ) | 0.6 | 0.07 | -3.0 | 0.0 | T @ MM
Te-S-S-Te $\bigg{|}$ |  | ( 10.9 , 1.7 , 0.9 ) | ( 117.3 , 178.0 , 46.6 ) | ( 29.0 , 5.7 , 3.6 ) | ( -177.4 , 1.0 , -2.3 ) | -1.6 | 0.15 | 0.0 | 32.6 | T @ MM
| ( 10.9 , 1.7 , 0.9 ) | ( 117.3 , 178.0 , 46.6 ) | ( 9.6, 0.3 , 1.5 ) | ( 98.0 , -113.5 , -47.1 ) | -2.14 | 0.0 | 0.0 | 0.0 | T @ XX
| ( 10.9 , 1.7 , 0.9 ) | ( 117.3 , 178.0 , 46.6 ) | ( 4.1, 1.4 , 0.2 ) | ( 115.0 , -9.4 , 60.2 ) | 1.14 | 0.01 | 71.6 | 0.0 | T @ MM
|  Te-S-Te-S $\bigg{|}$ |  | ( 9.9 , 1.8 , 0.7 ) | ( 137.3 , 180.0 , 100.6 ) | ( 27.8 , 20.5 , 13.7 ) | ( -1.8 , -33.0 , 8.7 ) | -0.65 | 0.02 | -6.1 | 69.8 | T @ AB
|  | ( 9.9 , 1.8 , 0.7 ) | ( 137.3 , 180.0 , 100.6 ) | (5.0 , 0.2 , 1.3 ) | ( 167.0 , -171.0 , -157.8 ) | -2.1 | 0.01 | 56.2 | 0.0 | T @ XX
|  | ( 9.9 , 1.8 , 0.7 ) | ( 137.3 , 180.0 , 100.6 ) | ( 2.5 , 0.4 , 1.1 ) | ( 140.9 , 1.5 , -143.7 ) | 1.1 | 0.02 | -33.8 | 0.0 | T @ AB
3R MoSeTe/WSeTe |  Se-Te-Se-Te $\bigg{|}$ |  | ( 13.3 , 2.4 , 0.9 ) | ( 2.3 , 180.0 , -170.0 ) | ( 16.9 , 2.2 , 1.6 ) | ( 3.4 , 142.5 , 176.7 ) | -0.7 | 0.04 | 11.1 | 109.0 | T @ XM
| ( 13.3 , 2.4 , 0.9 ) | ( 2.3 , 180.0 , -170.0 ) | ( 12.6 , 2.2 , 0.7 ) | ( 82.3 , -5.6 , -39.6 ) | -2.0 | 0.0 | 0.0 | 0.0 | T @ XM
| ( 13.3 , 2.4 , 0.9 ) | ( 2.3 , 180.0 , -170.0 ) | ( 2.6 , 1.8 , 0.6 ) | ( -17.8 , 2.8 , 5.0 ) | 1.9 | 0.03 | 4.9 | 0.0 | T @ XM
Se-Te-Te-Se $\bigg{|}$ |  | ( 12.0 , 2.2 , 0.7 ) | ( -0.5 , 180.0 , -178.6 ) | ( 22.2 , 16.2 , 9.9 ) | ( -3.7 , -26.9 , -4.6 ) | -0.7 | 0.01 | -7.4 | 70.9 | T @ AA
| ( 12.0 , 2.2 , 0.7 ) | ( -0.5 , 180.0 , -178.6 ) | ( 10.8 , 0.6 , 0.9 ) | ( -34.4 , 164.2 , 99.6 ) | -2.35 | 0.01 | 151.0 | 0.0 | T @ AA
| ( 12.0 , 2.2 , 0.7 ) | ( -0.5 , 180.0 , -178.6 ) | ( 6.9 , 0.6 , 0.4 ) | ( 21.1 , -177.0 , -162.6 ) | 1.0 | 0.06 | -11.0 | 0.0 | T @ MX
Te-Se-Se-Te $\bigg{|}$ |  | ( 15.3 , 2.4 , 1.0 ) | ( 0.0 , 180.0 , 180.0 ) | ( 14.9 , 5.2 , 5.6 ) | ( 170.0 , 142.5 , 176.7 ) | -0.7 | 0.03 | 3.1 | 73.0 | T @ AA
| ( 15.3 , 2.4 , 1.0 ) | ( 0.0 , 180.0 , 180.0 ) | ( 10.4 , 0.9 , 0.2 ) | ( -53.3 , 148.6 , 16.6 ) | -2.4 | 0.01 | 141.7 | 0.0 | T @ AA
| ( 15.3 , 2.4 , 1.0 ) | ( 0.0 , 180.0 , 180.0 ) | ( 9.9 , 1.5 , 1.1 ) | ( 60.6 , 180.0 , -70.0 ) | 1.57 | 0.01 | 20.0 | 0.0 | T @ XM
|  Te-Se-Te-Se $\bigg{|}$ |  | ( 15.3 , 3.5 , 0.9 ) | ( -2.1 , 180.0 , 168.7 ) | ( 17.1 , 7.9 , 2.0 ) | ( -3.0 , 5.0 , 5.8 ) | -0.65 | 0.01 | -8.9 | 72.1 | T @AA
|  | ( 15.3 , 3.5 , 0.9 ) | ( -2.1 , 180.0 , 168.7 ) | ( 7.5 , 1.1, 0.3 ) | ( 135.4 , 0.5 , 20.33 ) | -2.4 | 0.0 | 0.0 | 0.0 | T @ XM
|  | ( 15.3 , 3.5 , 0.9 ) | ( -2.1 , 180.0 , 168.7 ) | ( 8.0 , 1.3 , 0.3 ) | (134.7 , 0.5 , -4.7 ) | 1.5 | 0.02 | -32.3 | 0.0 | T @ AA
2H MoSeTe/WSeTe |  Se-Te-Se-Te $\bigg{|}$ |  | ( 13.4 , 2.3 , 0.9 ) | ( 2.3 , 180.0 , -169.0 ) | ( 16.9 , 2.2 , 1.6 ) | ( 3.4 , 142.5 , 176.7 ) | -0.7 | 0.04 | 11.1 | 109.0 | T @ XX
| ( 13.4 , 2.3 , 0.9 ) | ( 2.3 , 180.0 , -169.0 ) | ( 12.6 , 2.2 , 0.7 ) | ( 82.3 , -5.6 , -39.6 ) | -2.0 | 0.0 | 0.0 | 0.0 | T @ XX
| ( 13.4 , 2.3 , 0.9 ) | ( 2.3 , 180.0 , -169.0 ) | ( 2.6 , 1.8 , 0.6 ) | ( -17.8 , 2.8 , 5.0 ) | 1.9 | 0.03 | 4.9 | 0.0 | T @ XX
Se-Te-Te-Se $\bigg{|}$ |  | ( 12.0 , 2.0 , 0.9 ) | ( 148.6 , 180.0 , 122.5 ) | ( 3.3 , 2.4 , 1.4 ) | ( 25.6 , 2.0 , -40.6 ) | -1.24 | 0.04 | 155.4 | 29.0 | T @ AB
| ( 12.0 , 2.0 , 0.9 ) | ( 148.6 , 180.0 , 122.5 ) | ( 8.0 , 0.1 , 0.3 ) | ( 148.7 , 164.9 , -55.6 ) | -2.36 | 0.01 | 23.0 | 0.0 | T @ XX
| ( 12.0 , 2.0 , 0.9 ) | ( 148.6 , 180.0 , 122.5 ) | ( 5.3 , 0.1 , 0.4 ) | ( 132.9 , 30.9 , 4.2 ) | 1.0 | 0.07 | 164.0 | 0.0 | T @ AB
Te-Se-Se-Te $\bigg{|}$ |  | ( 9.9 , 2.0 , 0.8 ) | ( 126.0 , 180.0 , 51.2 ) | ( 25.7 , 6.6 , 5.1 ) | ( -62.8 , -4.8 , 164.7 ) | -0.9 | 0.1 | 127.1 | 65.5 | T @ MM
| ( 9.9 , 2.0 , 0.8 ) | ( 126.0 , 180.0 , 51.2 ) | ( 4.5 , 0.4 , 0.4 ) | ( 130.3 , 145.6 , -16.2 ) | -2.4 | 0.01 | 107.7 | 0.0 | T @ XX
| ( 9.9 , 2.0 , 0.8 ) | ( 126.0 , 180.0 , 51.2 ) | ( 2.6 , 0.2 , 0.4 ) | ( 127.6 , -2.7 , -58.0 ) | 1.44 | 0.01 | 67.0 | 0.0 | T @ MM
|  Te-Se-Te-Se $\bigg{|}$ |  | ( 12.0 , 3.0 , 1.8 ) | ( 135.8 , 180.0 , 59.8 ) | ( 2.1 , 1.3 , 0.5 ) | ( 21.7 , 172.0 , 55.8 ) | -0.61 | 0.03 | 120.3 | 153.0 | T @ XX
|  | ( 12.0 , 3.0 , 1.8 ) | ( 135.8 , 180.0 , 59.8 ) | ( 7.4 , 4.2, 2.3 ) | ( 112.4 , 3.2 , -166.3 ) | -2.4 | 0.01 | 46.9 | 0.0 | T @ XX
|  | ( 12.0 , 3.0 , 1.8 ) | ( 135.8 , 180.0 , 59.8 ) | ( 4.5 , 1.4 , 1.2 ) | (160.7 , -3.3 , -82.7 ) | 1.45 | 0.02 | 94.0 | 0.0 | T @ AB

##  IV Acknowledgements

We are grateful to Daniel T. Larson, Stephen Carr, Hao Tang and Ziyan Zhu for
helpful discussions and comments.
This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract No. DE-AC02-05CH11231 using NERSC award BES-ERCAP0020773. This work
used the Extreme Science and Engineering Discovery Environment (XSEDE)
Stampede2 at the Texas Advanced Computing Center through allocation TG-
DMR120073, which is supported by National Science Foundation grant number
ACI-1548562. MA and GRS acknowledge funding from the National Science
Foundation under Award No. DMR-1922172 and the Army Research Office under
Cooperative Agreement Number W911NF-21-2-0147. EK acknowledges funding from
the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319;
NSF DMREF Award No. 1922172; and the Army Research Office under Cooperative
Agreement Number W911NF-21-2-0147.

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