# Proximity spin-orbit and exchange coupling in ABA and ABC
trilayer graphene van der Waals heterostructures

Klaus Zollner<EMAIL_ADDRESS>Institute for
Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany
Martin Gmitra  Institute of Physics, Pavol Jozef Šafárik University in Košice,
04001 Košice, Slovakia  Jaroslav Fabian  Institute for Theoretical Physics,
University of Regensburg, 93053 Regensburg, Germany

(August 28, 2024)

###### Abstract

We investigate the proximity spin-orbit and exchange couplings in ABA and ABC
trilayer graphene encapsulated within monolayers of semiconducting transition-
metal dichalcogenides and the ferromagnetic semiconductor Cr2Ge2Te6. Employing
first-principles calculations we obtain the electronic structures of the
multilayer stacks and extract the relevant proximity-induced orbital and spin
interaction parameters by fitting the low-energy bands to model Hamiltonians.
We also demonstrate the tunability of the proximity effects by a transverse
electric field. Using the model Hamiltonians we also study mixed spin-
orbit/exchange coupling encapsulation, which allows to tailor the spin
interactions very efficiently by the applied field. We also summarize the
spin-orbit physics of bare ABA, ABC, and ABB trilayers, and provide, along
with the first-principles results of the electronic band structures, density
of states, spin splittings, and electric-field tunabilities of the bands,
qualitative understanding of the observed behavior and realistic model
parameters as a resource for model simulations of transport and correlation
physics in trilayer graphene.

spintronics, trilayer graphene, heterostructures, proximity spin-orbit
coupling, proximity exchange

##  I Introduction

Two-dimensional (2D) van der Waals (vdW) materials are vital building blocks
in the design of ultracompact electronic and spintronic devices Han  _et al._
(2014); Fabian  _et al._ (2007); Gong and Zhang (2019); Li  _et al._ (2019);
Cortie  _et al._ (2019). In this context, proximity-induced phenomena Zutic
_et al._ (2019); Sierra  _et al._ (2021) were found to be of great importance,
since 2D materials influence each others electronic and spin properties in vdW
heterostructures. Recently, it has been demonstrated that superconductivity Li
_et al._ (2020); Moriya  _et al._ (2020); Trainer  _et al._ (2020), magnetism
Zollner  _et al._ (2016, 2018, 2019a, 2020a); Zhang  _et al._ (2015, 2018);
Vila  _et al._ (2021); Wang  _et al._ (2015); Xu  _et al._ (2018); Yang  _et
al._ (2013); Zhong  _et al._ (2020); Averyanov  _et al._ (2018); Ciorciaro
_et al._ (2020), and spin-orbit coupling (SOC) Gmitra and Fabian (2015);
Gmitra  _et al._ (2016); Zollner and Fabian (2019, 2021a); Song  _et al._
(2018); Zhang  _et al._ (2014); Avsar  _et al._ (2017, 2014); Alsharari  _et
al._ (2018); Frank  _et al._ (2016), can be induced on demand in such weakly
glued vdW multilayers. In addition, gating, twisting, stacking, and straining
are efficient tunability knobs to tailor these spin interactions Zollner  _et
al._ (2019a); Song  _et al._ (2018); David  _et al._ (2019); Avsar  _et al._
(2017); Ghiasi  _et al._ (2019); Benitez  _et al._ (2020); Luo  _et al._
(2017); Safeer  _et al._ (2019); Herling  _et al._ (2020); Zollner and Fabian
(2021b); Naimer  _et al._ (2021), while the individual materials also preserve
a great degree of autonomy.

In twisted vdW heterostructures Carr  _et al._ (2017); Hennighausen and Kar
(2021); Ribeiro-Palau  _et al._ (2018); Carr  _et al._ (2020) the interlayer
interaction can be controlled by the twist angle. In this context, bilayer
graphene (BLG) was the first model playground for gate- and twist-tunable
correlated physics Cao  _et al._ (2018a, b); Arora  _et al._ (2020); Stepanov
_et al._ (2020); Lu  _et al._ (2019); Sharpe  _et al._ (2019); Ribeiro-Palau
_et al._ (2018), as well as for layer-dependent proximity-induced spin
interactions Gmitra and Fabian (2017); Amann  _et al._ (2021); Zollner  _et
al._ (2020b); Lin  _et al._ (2022); Wang  _et al._ (2019); Zollner  _et al._
(2018); Island  _et al._ (2019); Cardoso  _et al._ (2018); Tiwari  _et al._
(2021); Zollner and Fabian (2021c). At very small twist angles ($\approx
1^{\circ}$), a sharp peak arises in the density of states (DOS) of twisted
BLG, which is associated with flat bands Bistritzer and MacDonald (2011) in
the dispersion. By doping the twisted BLG, it can become insulating,
ferromagnetic Sharpe  _et al._ (2019), or superconducting Cao  _et al._
(2018a, b); Arora  _et al._ (2020); Stepanov  _et al._ (2020); Lu  _et al._
(2019); Nimbalkar and Kim (2020); Saito  _et al._ (2020). As a logical next
step, also twisted BLG-BLG structures were considered Shen  _et al._ (2020);
Liu  _et al._ (2020); Cao  _et al._ (2020); Burg  _et al._ (2019), where the
flat bands are additionally tunable by a gate field.

In addition, recent experiments Lin  _et al._ (2022) could demonstrate that
the proximity-induced Rashba and valley Zeeman SOC in twisted-BLG/WSe2
heterostructures, induces orbital magnetism without the need for a rotational
alignment to a hexagonal boron-nitride substrate Sharpe  _et al._ (2019);
Serlin  _et al._ (2020). Moreover, the transition-metal dichalcogenide (TMDC)
WSe2 can help to stabilize superconductivity in twisted BLG Arora  _et al._
(2020), emphasizing the role of the dielectric environment. For the
interpretation of such experimental results, it is also important to have
qualitative and quantitative knowledge about the proximity effects in TMDC/BLG
heterostructures Zollner and Fabian (2021c).

Unlike mono- and bilayer graphene, trilayer graphene (TLG) has not yet been
systematically investigated for spin proximity effects. The spin-orbit physics
of pristine TLG in ABA and ABC stackings was investigated by ab-initio
calculations Konschuh (2011) and tight-binding modeling Kormányos and Burkard
(2013), revealing many subtleties of the low-energy bands stemming from the
presence of $d$-orbitals which give rise to spin-orbit splittings on the order
of 10 $\mu$eV. Depending on the stacking order of the three layers, either ABA
or ABC, very different electronic structures can be realized with distinct
features in electronic and spin transport Jhang  _et al._ (2011); Ghosh and
Misra (2012) and gate tunable SOC Chen  _et al._ (2012). TLG is also important
from the topological perspective, since the ABC structure potentially hosts
quantum spin Hall and quantum valley Hall states Rehman and Abid (2017), while
there is evidence for a giant topological magnetic moment in ABA TLG Ge  _et
al._ (2021). Also interesting is the energetics of the different stackings,
see, for example, Ref. Guerrero-Avilés  _et al._ (2021)

More recently, TLG has emerged as a novel platform for correlated electrons
Zhu  _et al._ (2020); Park  _et al._ (2021); Chen  _et al._ (2019a, 2020, b);
Zhou  _et al._ (2021a); Polshyn  _et al._ (2020), as there is one more
independent layer, leading to a wider range of magic angles. As a consequence,
the tunability of the electronic and superconducting properties are superior
to the ones in BLG, as recently demonstrated Park  _et al._ (2021); Phong  _et
al._ (2021); Zhou  _et al._ (2021b); Qin and MacDonald (2021); Cao  _et al._
(2021) and theoretically explained Chou  _et al._ (2021). Remarkably, a zero-
field superconducting diode, signalling an interplay of spin-orbit physics and
time-reversal symmetry broken phase, has been demonstrated in twisted-TLG/WSe2
heterostructures Siriviboon  _et al._ (2021); Lin  _et al._ (2021); Scammell
_et al._ (2021). For the interpretation and quantitative theoretical
understanding of such experiments it is important to have a deeper microscopic
knowledge of the spin proximity effects in TLG. Providing such a resource is
the goal of our manuscript.

In particular, we investigate, by performing first-principles calculations,
the electronic structures of ABA and ABC TLG encapsulated within strong SOC
semiconductors MoSe2 and WSe2 and ferromagnetic monolayers Cr2Ge2Te6 (CGT). We
provide the essential band structure information, low-energy band dispersions,
spin splittings of the low-energy bands, spin textures, and the behavior of
the Dirac bands in the presence of a transverse electric field. To provide
reference for the encapsulated systems, we summarize the essential low energy
physics of bare ABA and ABC TLG, also including ABB TLG for completeness.
While pristine graphene exhibits intrinsic SOC of 10–20 $\mu$eV, encapsulated
TLG displays spin splittings of some meV, strongly tunable by the electric
field.

In terms of modeling, we focus on the proximity-induced spin-orbit and
exchange couplings, and demonstrate that in order to reproduce the density
functional theory (DFT) data, it is sufficient to modify only the spin
interactions of the outer layers of encapsulated TLG. The effective low-energy
Hamiltonians, which are fitted to the TLG dispersions to obtain reasonable
parameter sets, nicely reproduce the first-principles data. Moreover, we show
that the model reproduces the data also in the presence of a transverse
electric field, whereby modifying only a few effective couplings is sufficient
for a quantitative comparison.

Employing the model, with fitted proximity spin-orbit and exchange couplings
separately, we then provide predictions for ex-so-tic heterostructures Zollner
_et al._ (2020b); exs  which comprise both a strong SOC monolayer and a
ferromagnetic monolayer, encapsulating ABA and ABC TLG. The interplay between
spin-orbit and exchange coupling is imprinted onto the corresponding low-
energy band structures and most markedly pronounced by the different band
dispersions at $K$ and $K^{\prime}$. We also show that the band dispersions
respond sensitively to the applied electric field, which can serve as a knob
to change the spin polarization of the low-energy states in ABC TLG.

In all the studied cases we provide both qualitative understanding, based on
the atomic arrangement and hybridization across the layers, and quantitative
modeling with realistic parameters fitted to the DFT results, which should be
useful for model simulations of TLG.

The paper is organized as follows: In section II, we first discuss the
electronic properties of bare TLG with ABA, ABC, and ABB stackings, under the
influence of SOC and an external electric field, to have reference results for
the encapsulated structures. In section III, we present the encapsulated TLG
geometries, address the structural setup, and summarize the calculation
details for obtaining the electronic structures. In section IV, we introduce
the model Hamiltonians that capture the low energy physics of encapsulated TLG
(including orbital, SOC, and exchange terms), which are used to fit the DFT-
calculated dispersions. In section V, we then show and discuss the DFT-
calculated electronic structures, along with the model Hamiltonian fits, of
TMDC and CGT encapsulated TLG. In section VI, we turn to the electric field
tunability of the relevant low energy bands of TMDC encapsulated TLG. In
section VII, we discuss the interplay of spin interactions in CGT/TLG/TMDC
heterostructures, on the model level, based on our results for TMDC and CGT
encapsulated TLG. Finally, in section VIII we conclude the manuscript.

##  II Bare trilayer graphene

Recent investigations on twisted-graphene/BLG heterostructures, that contain
all three high-symmetry TLG stacking types (ABA, ABC, and ABB), reveal highly
interesting topological properties Polshyn  _et al._ (2020); Rademaker  _et
al._ (2020); Park  _et al._ (2020); Ma  _et al._ (2021). For example, in Ref.
Polshyn  _et al._ (2020) the electrical control of magnetism, which arises due
to strong correlations by twisting graphene on top of BLG, has been
demonstrated.

To have reference results for evaluating the proximity effects in encapsulated
graphene trilayers, we briefly review the essential electronic properties of
bare ABA, ABC, and ABB stackings, focusing on the energy dispersions of the
lowest energy bands and the spin-orbit splittings. The electronic band
structures of the bare trilayers were obtained by performing first-principles
calculations employing a full potential linearized augmented plane wave
(FLAPW) code based on density functional theory (DFT), as implemented in
WIEN2k Blaha  _et al._ (2001). This code was already used to calculate SOC in
monolayer and bilayer graphene, Gmitra  _et al._ (2009); Konschuh  _et al._
(2010, 2012), yielding results consistent with recent experimental findings
Sichau  _et al._ (2019); Banszerus  _et al._ (2020).

In our implementation, exchange-correlation effects are treated with the
generalized-gradient approximation (GGA) Perdew  _et al._ (1996), including
the DFT-D3 vdW dispersion correction Grimme  _et al._ (2010). We used a very
dense $k$-point grid of $120\times 120\times 1$ to accurately determine the
Fermi level. The Muffin-tin radius of carbon atoms is $R_{\textrm{C}}=1.34$
and the plane wave cutoff parameter $RK_{\textrm{MAX}}=9.5$. For all TLG
stackings, we use the lattice parameter of bare graphene $a=2.46$ Å, with
fixed interlayer distances of $d=3.3$ Å. In order to avoid interactions
between periodic images of our slab geometries, we add a vacuum of about $20$
Å in the $z$ direction. An earlier account of WIEN2k TLG calculations can be
found in Ref. Konschuh (2011), while multi-orbital tight-binding modeling of
ABC TLG in the presence of SOC was performed in Ref. Kormányos and Burkard
(2013).

###  II.1 ABA

Figure 1: (a) Lattice structure of the ABA TLG. Due to $z$-mirror symmetry,
the sublattice atoms with the same color (red, blue, green, and grey) belong
together. Grey rhombuses define the unit cells of the layers. (b) The DFT-
calculated band structure in the vicinity of the $K$ point. The bands are
color-coded by their atomic projections as defined in (a). (c) The energy
level diagram of states at the $K$ point without SOC, with SOC, and with SOC
in the presence of a perpendicular electric field of 0.2 V/nm.

In Fig. 1 we show the lattice structure, the calculated energy dispersion, and
a schematic energy level diagram for ABA TLG. Due to $z$-mirror symmetry, four
nonequivalent atoms are present in the geometry, see Fig. 1(a). The low-energy
band structure, see Fig. 1(b), shows graphene-like Dirac states, as well as
BLG-like parabolic bands, consistent with literature Partoens and Peeters
(2006); Latil and Henrard (2006); Koshino and McCann (2009a); Aoki and
Amawashi (2007). The Dirac bands are exclusively formed by orbitals of the
atoms in the outermost graphene layers. In contrast, the parabolic bands are
formed by atoms B1, A2, and B3, which form non-dimer interlayer pairs.

Without SOC, the bands remain spin-degenerate, see the scheme in Fig. 1(c).
Including SOC, the degeneracy is lifted due to the absence of space-inversion
symmetry and the low-energy bands are split by about 20 $\mu$eV (corresponding
to the intrinsic SOC of pristine graphene monolayers Gmitra  _et al._ (2009)).
The high-energy states — not shown in Fig. 1(b) but included in the energy
level diagram in Fig. 1(c) — which are formed by atoms A1, B2, and A3, are
about 500 meV away from the Dirac point at the Fermi level. The reason is that
these atoms are coupled by direct interlayer hopping, pushing the
corresponding bands away from the Fermi level, similar to bare BLG Konschuh
_et al._ (2012). The spin-orbit splittings of the high-energy bands are just a
few $\mu$eV. The diagram in Fig. 1(c) also indicates the effect of an applied
transverse electric field which introduces a potential difference between the
outermost layers. The field essentially reshuffles the energy levels, but
mainly the band offsets of the Dirac and parabolic low-energy bands change,
while the spin-orbit splittings remain nearly unchanged for the electric field
as large as 0.2 V/nm. In appendix A, we show and discuss the evolution of the
ABA TLG dispersion and DOS for electric fields up to 1 V/nm. We also provide
model Hamiltonian fit results there, that nicely reproduce the dispersion.

###  II.2 ABC

Figure 2: (a) Lattice structure of the ABC TLG. Due to space inversion
symmetry, the sublattice atoms of the same color (red, blue, and green)
contribute equally to the energy bands. Grey rhombuses define the unit cells
of the layers. (b) The DFT-calculated band structure in the vicinity of the
$K$ point. The bands are color-coded by their atomic projections as defined in
(a). The inset in (b) shows a zoom to the low-energy bands near the Fermi
level, indicating the cone-like touching point, where a gap of about 25
$\mu$eV arises due to SOC. (c) The energy level diagram of the low and high-
energy states at the $K$ point without SOC, with SOC, and with SOC in the
presence of a perpendicular electric field of 0.2 V/nm.

In Fig. 2 we show the lattice structure, the calculated energy dispersion, and
an energy-level diagram for ABC TLG. This structure has spatial inversion
symmetry and only three nonequivalent atoms are present in the geometry, see
Fig. 2(a). From the band structure at the $K$ point, see Fig. 2(b), we find
two flat low-energy bands which are formed by the outer-layer atoms B1 and A3,
although A2 and B2 become prominent at larger momenta away from $K$. The four
high-energy bands, which are split off from the Fermi level by more than 0.35
eV, are built by sublattices A1, A2, B2, and B3. The presented dispersion is
consistent with earlier reports Latil and Henrard (2006); Koshino and McCann
(2009b); Aoki and Amawashi (2007); Zhang  _et al._ (2010); Lu  _et al._
(2006).

At the $K$ point, the low-energy bands are twofold spin degenerate, while the
high-energy bands are even fourfold degenerate, in the absence of SOC, see
Fig. 2(c). Due to spatial inversion symmetry, all bands remain spin
degenerate, even when SOC is turned on. However, a tiny (about 0.7 $\mu$eV)
spin-orbit gap opens at the high-energy levels and removes the fourfold
degeneracy. Zooming into the Fermi level, we find a cone-like touching feature
of the low-energy bands along the $K\rightarrow M$ high-symmetry line, see
Fig. 2(b). When SOC is included, the Dirac cones are split by 25 $\mu$eV,
which corresponds to the spin-orbit gap, caused almost solely by $d$ orbitals
of a pristine graphene monolayer Gmitra  _et al._ (2009).

When a perpendicular electric field is applied across the ABC TLG, the
inversion symmetry, along with the spin degeneracy of the bands, gets lifted,
see Fig. 2(c). The field introduces a potential difference between the
outermost layers, leading to further gap openings in the spectrum. By
increasing the field amplitude, the low-energy spin splittings at the $K$
point first increase, then saturate at about $25\leavevmode\nobreak\ \mu$eV,
and finally decrease again Konschuh (2011). For a field of 0.2 V/nm, the
splittings of the low-energy bands at the $K$ point are about
$15\leavevmode\nobreak\ \mu$eV, see Fig. 2(c). Near the Dirac cone vertex, the
band splittings are $25\leavevmode\nobreak\ \mu$eV, again corresponding to the
intrinsic SOC of monolayer graphene Gmitra  _et al._ (2009). Such intrinsic
splittings, that get exposed by an applied electric field and that do not
depend on the field above some crossover value, are also present in BLG
Konschuh  _et al._ (2012) and lead to a marked spin relaxation anisotropy as
already experimentally detected Leutenantsmeyer  _et al._ (2018).

###  II.3 ABB

Figure 3: (a) Lattice structure of the ABB TLG. Due to the absence of spatial
inversion and $z$-mirror symmetry, all sublattice atoms are different (red,
green, blue, purple, black, orange). Grey rhombuses define the unit cells of
the layers. (b) The energy level diagram of states at the $K$ point without
SOC, with SOC, and with SOC in the presence of a transverse electric field of
0.2 V/nm. (c) The DFT-calculated band structure in the vicinity of the $K$
point without SOC. From left to right, we project onto the different basis
atoms from the three graphene layers, with the color code as defined in (a).

In Fig. 3 we show the lattice structure, the calculated dispersion, and an
energy level diagram for ABB TLG. Due to the absence of spatial inversion and
$z$-mirror symmetry, all sublattice atoms are different, see Fig. 3(a). The
band structure, see Fig. 3(c), features parabolic high-energy bands at around
600 meV at the $K$ point, predominantly formed by B2 atoms. The reason is that
B2 is interlayer-coupled to both surrounding layers, splitting these states
off to high energies. The low-energy bands near the $K$ point are almost
exclusively formed by B1, with a small contribution from A1 and B3. The reason
is that B1 does not couple directly to the other layers, so the corresponding
band remains close to the Dirac point. The intermediate bands are formed by A2
and A3 near the $K$ point, because they are only connected by one interlayer
coupling. The energy level diagram, see Fig. 3(b), shows that the bands remain
spin-degenerate without SOC. Including SOC lifts the degeneracy, because
spatial inversion symmetry is absent. The low-energy valence (conduction) band
is split by about 25 $\mu$eV (8 $\mu$eV). A transverse electric field further
opens the gap at the $K$ point, while swapping the spin-orbit splittings—and
the orbital composition of the bands themselves—of the low-energy valence and
conduction bands (see also Fig. 18 in appendix B). The high-energy bands
remain nearly the same, except that band offsets are tunable by the field. In
Appendix B we show the evolution of the ABB TLG dispersion and DOS for
electric fields up to 1 V/nm. We also provide model Hamiltonian fit results
there, that nicely reproduce the dispersion.

For the presented TLG dispersions, we have fixed all the interlayer distances
to $d=3.3$ Å. In Appendix C we compare the dispersions and DOS for ABA, ABC,
and ABB TLG with relaxed interlayer distances. Especially the ABB dispersion
and DOS are markedly modified, since the interlayer distances become highly
asymmetric.

###  II.4 Electric Field Effects

Figure 4: Top: Zooms to the ABC TLG bands in the vicinity of the $K$ point. We
compare DFT data (symbols) with model Hamiltonian fits (solid lines) employing
parameters from Table 3 in appendix A. In the dispersion, we specify the band
gap at the Fermi level. Bottom: The corresponding calculated density of states
(DOS). From left to right, we increase the transverse electric field from 0 to
1 V/nm.

An important factor for correlation physics in TLG is the flatness of the
dispersion, associated with van Hove singularities (VHS) in the DOS. Recently,
superconductivity in ABC TLG was demonstrated Zhou  _et al._ (2021b), which is
certainly strongly related with the electric field tunability of these VHS.
Below we describe the electric tunability of ABC TLG, while in appendices A
and B we show and discuss the electric field behavior of ABA and ABB TLG
dispersions and their corresponding DOS.

In Fig. 4 we show the evolution of the dispersion and DOS for ABC TLG for
electric fields up to 1 V/nm. The field opens a band gap at the Fermi level,
since the low-energy bands are formed by atoms B1 and A3 (see Fig. 2) of the
outermost graphene layers, now located in different potentials. Increasing the
field up to 1 V/nm opens a sizable gap of about 90 meV in the spectrum. Also
in the high-energy bands, gaps are introduced by the field. However, because
they are split-off in energy from the Fermi level, they are not so important
for transport experiments. Viewing the corresponding DOS, we find VHS
associated with the low-energy bands. As the electric field separates these
bands in energy, they further flatten and the VHS become strongly pronounced.

Given the similarity of the electric field behavior of the low energy bands of
BLG Konschuh  _et al._ (2012) and ABC TLG, one can expect, for example,
similar tunability of the valley g-factor as demonstrated for BLG Moulsdale
_et al._ (2020); Lee  _et al._ (2020). Moreover, one can also expect a spin-
orbit valve operation in ABC TLG, as experimentally and theoretically
demonstrated in BLG/TMDC heterostructures Gmitra and Fabian (2017); Island
_et al._ (2019).

In Fig. 4 we also compare DFT data and model Hamiltonian fits. Employing the
parameters from Table 3 in appendix A, we can perfectly fit the dispersions
with applied electric field. The model Hamiltonian will be introduced in the
following section, when discussing encapsulated TLG heterostructures.

##  III Encapsulated Trilayer Graphene

###  III.1 Geometry Setup

Figure 5: Geometries of the encapsulated TLG heterostructures. Left: TMDC
(MoSe2 and WSe2) encapsulated ABA and ABC TLG. Right: CGT encapsulated ABA and
ABC TLG. The relaxed interlayer distances are indicated. We also specify the
direction of a positive electric field.

In the following we focus on TLG in ABA and ABC stacking, which are of most
current interest. For both stacking types, we consider $4\times 4$ supercells
of TLG to be encapsulated by $3\times 3$ monolayer supercells of bottom MoSe2
and top WSe2. In the case of TMDC encapsulation, we use the lattice constant
of pristine TLG ($2.46\leavevmode\nobreak\ \textrm{\AA}$) and the MoSe2 and
WSe2 lattice constants are barely strained (below 0.3% Schutte  _et al._
(1987); James and Lavik (1963)) to $3.28\leavevmode\nobreak\ \textrm{\AA}$.
Moreover, in the case of TMDC encapsulation, the twist angles between all
layers is $0^{\circ}$. Here, we do not investigate different twist angles.
Note that the electronic structures of TMDCs are very sensitive to strain
Zollner  _et al._ (2019b), especially in terms of band gap. In contrast, the
Dirac states of graphene are quite robust against biaxial strain Si  _et al._
(2016); Choi  _et al._ (2010). Moreover, upon twisting, proximity SOC in
graphene/TMDC heterostructures can be strongly modified, which has been
investigated in Refs. David  _et al._ (2019); Li and Koshino (2019); Naimer
_et al._ (2021). Especially in Ref. Naimer  _et al._ (2021), DFT calculations
have shown that straining graphene, while leaving the TMDC unstrained,
determines the position of the Dirac states within the TMDC band gap. This
band offset can be tuned by gating, thereby influencing proximity SOC. We
believe that these findings would also apply for our heterostructures of TMDC
encapsulated TLG.

In addition, we consider $5\times 5$ supercells of TLG to be encapsulated
within $\sqrt{3}\times\sqrt{3}$ supercells of ferromagnetic CGT. In the case
of CGT encapsulation, we also keep the lattice constant of pristine TLG but
stretch the CGT lattice constant by roughly 4% from
$6.8275\leavevmode\nobreak\ \textrm{\AA}$ Carteaux  _et al._ (1995) to
$7.1014\leavevmode\nobreak\ \textrm{\AA}$, for better comparability of the
encapsulation cases. Note that, similar to the TMDCs, CGT also shows a strain-
tunable band gap Li and Yang (2014); Chen  _et al._ (2015); Liu  _et al._
(2021). Again, the twist angle is crucial for proximity exchange in
graphene/CGT heterostructures Zollner and Fabian (2021b). In our geometries,
the relative twist angle between top (bottom) CGT and the top (bottom)
graphene layer is $30^{\circ}$. In Ref. Zollner and Fabian (2021b), it has
been explicitly shown by DFT calculations, that mainly twisting influences
proximity exchange, while straining determines the position of the graphene
Dirac states within the CGT band gap. This band offset is again tunable by
gating, thereby influencing proximity exchange. Also these findings should be
applicable for our TLG structures.

Initial atomic structures are set up with the atomic simulation environment
(ASE) Bahn and Jacobsen (2002) and visualized with VESTA software Momma and
Izumi (2011), see Fig. 5. Our choice of the encapsulating monolayers is based
on the fact that TMDCs and CGT are semiconductors, providing strong spin-orbit
and exchange couplings to graphene, correspondingly Gmitra  _et al._ (2016);
Gmitra and Fabian (2015); Zollner  _et al._ (2018, 2020b); Zollner and Fabian
(2021c). Another important factor for our choice is the lattice matching, such
that the different materials can be combined in commensurate supercells for
periodic DFT calculations, without straining them beyond reasonable limits.

###  III.2 Computational Details

The electronic structure calculations and structural relaxations of the TLG-
based vdW heterostructures are performed by DFT Hohenberg and Kohn (1964) with
Quantum ESPRESSO Giannozzi  _et al._ (2009). Self-consistent calculations are
performed with the $k$-point sampling of $18\times 18\times 1$ to get
converged results for the proximity-induced exchange and SOC. We use an energy
cutoff for the charge density of $520$ Ry, and the kinetic energy cutoff for
wavefunctions is $65$ Ry for the scalar relativistic pseudopotentials with the
projector augmented wave method Kresse and Joubert (1999) with the Perdew-
Burke-Ernzerhof exchange correlation functional Perdew  _et al._ (1996). In
the case of CGT encapsulation, we perform open shell calculations that provide
the spin polarized ground state and proximity exchange coupling. In addition,
a Hubbard parameter of $U=1$ eV is used for Cr $d$-orbitals, similar to recent
calculations Gong  _et al._ (2017); Zollner  _et al._ (2018); Zollner and
Fabian (2021c). In the case of TMDC encapsulation, we use the relativistic
versions of the pseudopotentials, to capture (proximity) SOC effects.

For the relaxation of the heterostructures, we add DFT-D2 vdW corrections
Grimme (2006); Barone  _et al._ (2009) and use quasi-newton algorithm based on
trust radius procedure. Dipole corrections Bengtsson (1999) are also included
to get correct band offsets and internal electric fields. In order to simulate
quasi-2D systems, we add a vacuum of $20$ Å, to avoid interactions between
periodic images in our slab geometries. To determine the interlayer distances,
the atoms of TLG and the TMDCs are allowed to relax only in their $z$
positions (vertical to the layers), and the atoms of CGT are allowed to move
in all directions, until all components of all forces are reduced below
$10^{-3}$ [Ry/$a_{0}$], where $a_{0}$ is the Bohr radius. The obtained
interlayer distances are summarized in Fig. 5 and are similar to previous
reports Gmitra and Fabian (2017); Zollner  _et al._ (2018, 2020b). Since we
have assumed perfectly aligned individual layers, the full heterostructures
still have $C_{3}$ symmetry after relaxation.

##  IV Model Hamiltonians

Here we present the Hamiltonians that we employ to model the low-energy bands
of the (encapsulated) ABA and ABC TLG. The basis states are
$|\textrm{C}_{\textrm{A1}},s\rangle$, $|\textrm{C}_{\textrm{B1}},s\rangle$,
$|\textrm{C}_{\textrm{A2}},s\rangle$, $|\textrm{C}_{\textrm{B2}},s\rangle$,
$|\textrm{C}_{\textrm{A3}},s\rangle$, $|\textrm{C}_{\textrm{B3}},s\rangle$,
providing 12 eigenvalues $\varepsilon_{1-12}$, for both spin species
$s=\\{\uparrow,\downarrow\\}$ of each C atom. In this basis the Hamiltonian
comprises several terms Konschuh (2011); Zhang  _et al._ (2010); Koshino and
McCann (2009b); Kormányos and Burkard (2013); Koshino (2010)

$\mathcal{H}=\mathcal{H}_{\textrm{orb}}+\mathcal{H}_{\textrm{soc}}+\mathcal{H}_{\textrm{ex}}+E_{D}.$
(1)

The orbital terms for ABA and ABC TLG consist of $p_{z}$ intra- and interlayer
hoppings

$\displaystyle\mathcal{H}_{\textrm{orb}}^{\textrm{ABA}}=$
$\displaystyle\begin{pmatrix}\Delta+V_{1}&\gamma_{0}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{1}&\gamma_{5}&0\\\
\gamma_{0}f^{*}(\bm{k})&\eta+V_{1}&\gamma_{3}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&0&\gamma_{2}\\\
\gamma_{4}f(\bm{k})&\gamma_{3}f^{*}(\bm{k})&V_{2}&\gamma_{0}f(\bm{k})&\gamma_{4}f(\bm{k})&\gamma_{3}f^{*}(\bm{k})\\\
\gamma_{1}&\gamma_{4}f(\bm{k})&\gamma_{0}f^{*}(\bm{k})&\Delta+V_{2}&\gamma_{1}&\gamma_{4}f(\bm{k})\\\
\gamma_{5}&0&\gamma_{4}f^{*}(\bm{k})&\gamma_{1}&\Delta-
V_{1}&\gamma_{0}f(\bm{k})\\\
0&\gamma_{2}&\gamma_{3}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{0}f^{*}(\bm{k})&\eta-
V_{1}\end{pmatrix}\otimes s_{0},$ (2)
$\displaystyle\mathcal{H}_{\textrm{orb}}^{\textrm{ABC}}=$
$\displaystyle\begin{pmatrix}\Delta+V_{1}&\gamma_{0}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{1}&0&0\\\
\gamma_{0}f^{*}(\bm{k})&\eta+2V_{1}&\gamma_{3}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{6}&0\\\
\gamma_{4}f(\bm{k})&\gamma_{3}f^{*}(\bm{k})&\Delta+V_{2}&\gamma_{0}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{1}\\\
\gamma_{1}&\gamma_{4}f(\bm{k})&\gamma_{0}f^{*}(\bm{k})&\Delta+2V_{2}&\gamma_{3}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})\\\
0&\gamma_{6}&\gamma_{4}f(\bm{k})&\gamma_{3}f^{*}(\bm{k})&\eta-
V_{1}&\gamma_{0}f(\bm{k})\\\
0&0&\gamma_{1}&\gamma_{4}f(\bm{k})&\gamma_{0}f^{*}(\bm{k})&\Delta-
V_{1}\end{pmatrix}\otimes s_{0}.$ (3)

The spin-orbit coupling and exchange terms are the same for both stackings in
the above basis:

$\displaystyle\mathcal{H}_{\textrm{soc}}+\mathcal{H}_{\textrm{ex}}=$
$\displaystyle\begin{pmatrix}(\tau\lambda_{\textrm{I}}^{\textrm{A1}}-\lambda_{\textrm{ex}}^{\textrm{A1}})s_{z}&2\textrm{i}\lambda_{\textrm{R1}}s_{-}^{\tau}&0&0&0&0\\\
-2\textrm{i}\lambda_{\textrm{R1}}s_{+}^{\tau}&(-\tau\lambda_{\textrm{I}}^{\textrm{B1}}-\lambda_{\textrm{ex}}^{\textrm{B1}})s_{z}&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&(\tau\lambda_{\textrm{I}}^{\textrm{A3}}-\lambda_{\textrm{ex}}^{\textrm{A3}})s_{z}&2\textrm{i}\lambda_{\textrm{R3}}s_{-}^{\tau}\\\
0&0&0&0&-2\textrm{i}\lambda_{\textrm{R3}}s_{+}^{\tau}&(-\tau\lambda_{\textrm{I}}^{\textrm{B3}}-\lambda_{\textrm{ex}}^{\textrm{B3}})s_{z}\\\
\end{pmatrix}.$ (4)

We use the linearized version of the nearest-neighbor structural function
$f(\bm{k})=-(\sqrt{3}a/2)(\tau k_{x}-\textrm{i}k_{y})$, with the graphene
lattice constant $a$ and the Cartesian wave vector components $k_{x}$ and
$k_{y}$ are measured from $\pm K$ for the valley indices $\tau=\pm 1$. The
Pauli spin matrices are $s_{i}$, with $i=\\{0,x,y,z\\}$, and
$s_{\pm}^{\tau}=(s_{x}\pm\textrm{i}\tau s_{y})/2$. Here, $\gamma_{j}$,
$j=\\{0,1,2,3,4,5,6\\}$, describe intra- and interlayer hoppings in TLG, see
Fig. 6 for an overview. The coupling $\gamma_{0}$ is the nearest neighbor
intralayer hopping amplitude between A and B sublattice within each graphene
layer. The parameter $\gamma_{1}$ describes the direct interlayer hopping
between A and B sublattice of adjacent graphene layers. Similarly, parameters
$\gamma_{2}$, $\gamma_{5}$, and $\gamma_{6}$ are also vertical interlayer
hoppings, but between the outermost layers in TLG. In addition, the hoppings
$\gamma_{3}$ and $\gamma_{4}$ are interlayer couplings, but not vertical. They
connect next nearest neighbor atoms of adjacent graphene layers. Vertical
hoppings in TLG couple only two atoms, hence they appear without structural
function in the Hamiltonian, while the other hoppings couple an atom to three
corresponding partner atoms, hence they appear linear in momentum.

Figure 6: Schematic illustration of the ABA (left) and ABC (right) TLG
lattices, showing the relevant intra- and interlayer hoppings $\gamma_{j}$,
$j=\\{0,1,2,3,4,5,6\\}$ (dashed lines). In addition, the bottom (top) graphene
layer is placed in the potential $V_{1}$ ($-V_{1}$), while the middle layer is
placed in potential $V_{2}$. The asymmetries $\Delta$ and $\eta$ arise due to
vertical hoppings.  Figure 7: (a) DFT-calculated band structure of the
MoSe2/ABA-TLG/WSe2 heterostructure along the $M-K-\Gamma$ path. The color of
the bands corresponds to the $s_{z}$ spin expectation value. We specify the 12
relevant energy bands, $\varepsilon_{1-12}$, corresponding to TLG, which seem
to be pairwise spin-degenerate. (b) Zoom to the calculated low-energy bands
(symbols) around the $K$ point, corresponding to the band structure in (a),
with a fit to the model Hamiltonian (solid lines). The bands are spin-split
due to proximity-induced SOC. (c) The spin expectation values of the 8 low-
energy bands as labeled in (b). The $s_{x}$ and $s_{y}$ values are multiplied
by a factor of 5 for better visualization.

In general, due to the proximity effect the three layers experience different
energy potentials, which we describe by assigning $V_{2}$ to the middle layer
and $\pm V_{1}$ to the outer layers. In addition, asymmetries $\Delta$ and
$\eta$ arise due to the vertical hoppings ($\gamma_{1}$ and $\gamma_{6}$ in
ABC TLG; $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{5}$ in ABA TLG). The
combination of parameters $V_{1}$, $V_{2}$, $\Delta$, and $\eta$ can then
describe the on-site energies of the individual sublattices of the different
layers within TLG. The parameters $\lambda_{\textrm{I}}$
($\lambda_{\textrm{ex}}$) describe the intrinsic or proximity-induced spin-
orbit or exchange couplings of the corresponding layer and sublattice atom
($\textrm{C}_{\textrm{A1}},\textrm{C}_{\textrm{B1}},\textrm{C}_{\textrm{A3}},\textrm{C}_{\textrm{B3}}$).
The parameters $\lambda_{\textrm{R}l}$ represent the Rashba terms of the
bottom and top layers $l=\\{1,3\\}$. The middle layer, formed by atoms
$\textrm{C}_{\textrm{A2}}$ and $\textrm{C}_{\textrm{B2}}$, is far away from
the proximitizing TMDC and CGT encapsulation layers. Therefore, we can neglect
the corresponding proximity SOC and exchange parameters, as we will see from
the band structure fits. Actually, in the ABC TLG orbital Hamiltonian, the
additional factors of 2 in front of $V_{1}$ and $V_{2}$ in the diagonal
entries arise due to SOC Konschuh (2011) and are relevant for the spin-orbit
gaps in the high-energy bands, see the energy level diagram in Fig 2. To
capture doping effects from the calculations, we introduce another parameter
$E_{D}$, which leads to an energy shift of the model band structure.

Finally, to extract the fit parameters from the DFT, we employ a least-squares
routine, taking into account band energies, splittings, and spin expectation
values. In each case, we first fix the orbital parameters from fitting the
dispersion and then fit band splittings and spin expectation values to find
the spin-orbit and exchange parameters.

##  V Band Structure and Fit Results

###  V.1 TLG encapsulated within WSe2 and MoSe2

Monolayer TMDCs are direct band gap semiconductors with strong SOC. Due to
proximity of graphene to a TMDC, a significant amount of SOC is introduced in
graphene, leading to a splitting of the Dirac states on the order of 1 meV. In
the case of TMDC-encapsulated TLG, only the outer graphene layers experience
proximity coupling, due to the short rangeness of the proximity effect in vdW
heterostructures. This will lead to spin splittings in the TLG band structure,
associated with bands originating from the top and bottom layer sublattice
atoms.

####  V.1.1 MoSe2/ABA-TLG/WSe2 stacks

Figure 8: (a) DFT-calculated band structure of the MoSe2/ABC-TLG/WSe2
heterostructure along the $M-K-\Gamma$ path. The color of the bands
corresponds to the $s_{z}$ spin expectation value. We specify the 12 relevant
energy bands, $\varepsilon_{1-12}$, corresponding to TLG, which seem to be
pairwise spin-degenerate. (b) Zoom to the calculated low-energy bands
(symbols) around the $K$ point, corresponding to the band structure in (a),
with a fit to the model Hamiltonian (solid lines). The bands are spin-split
due to proximity-induced SOC. (c) The spin expectation values of the 4 low-
energy bands as labeled in (b). The $s_{x}$ and $s_{y}$ values are multiplied
by a factor of 5 for better visualization.  Figure 9: First-principles
calculated spin-orbit fields around the $K$ point of the MoSe2/ABC-TLG/WSe2
heterostructure, corresponding to the four low-energy bands $\varepsilon_{5}$
\- $\varepsilon_{8}$ in Fig. 12 at zero electric field. The dashed white lines
represent the edges of the hexagonal Brillouin zone with the $K$ point at the
center ($k_{x}=k_{y}=0$). The $s_{z}$ spin expectation value determines the
color, while in-plane spins are depicted as arrows. Here, we enhanced in-plane
spin expectation values by a factor of 10 for better visualization.

We start our discussion by considering the dispersion of the MoSe2/ABA-
TLG/WSe2 heterostructure, see Fig. 7(a). We find that the ABA TLG band
structure, featuring the aforementioned 12 energy bands near the $K$ point, is
nicely preserved within this heterostructure. Here, we are mainly interested
in the 8 low-energy bands, see Fig. 7(b), which resemble a combination of
single- and bilayer-graphene low-energy spectra. Similar to BLG, the parabolic
high-energy bands located at around $\pm 0.5$ eV are mainly formed by states
from dimer atoms A1, B2, and A3 (see Fig. 21 in appendix D) and are split-off
from the Fermi level due to the interlayer hopping $\gamma_{1}$. The low-
energy bands are mainly formed by the non-dimer atoms B1, A2, and B3, but one
of them also contains contributions from A1 and A3 atoms.

The band structure and spin expectation values, see Fig. 7(b,c), are nicely
fitted by our model Hamiltonian employing the parameters in Table 1. The
proximity-induced SOC parameters are also as expected, giving valley-Zeeman
and opposite Rashba couplings for top and bottom graphene layers. Because the
bottom graphene layer couples to MoSe2, the values of the corresponding SOC
parameters ($\lambda_{\textrm{I}}$ and $\lambda_{\textrm{R}}$) are smaller
compared to the values for the top graphene layer, which couples to WSe2. This
result is in agreement with findings from single-layer-graphene/TMDC
heterostructures Gmitra  _et al._ (2016). In addition, the middle layer
experiences no proximity SOC, in agreement to BLG/TMDC results Zollner  _et
al._ (2020b), where only adjacent layers are proximitized. The two Rashba SOCs
are opposite, because the bottom (top) graphene effectively feels the presence
of a strong spin-orbit substrate (capping) layer, leading to opposite
distortion of the corresponding $p_{z}$ orbitals.

####  V.1.2 MoSe2/ABC-TLG/WSe2 stacks

Next, we analyze the dispersion of the MoSe2/ABC-TLG/WSe2 heterostructure, see
Fig. 8(a). Again, the bands near the $K$ point closely resemble the ones from
bare ABC-stacked TLG, but now with proximity SOC due to the surrounding TMDC
layers. In contrast to ABA TLG, here we have 8 high-energy bands at around
$\pm 0.4$ eV, which are again formed by the dimer atoms in the structure, i.
e., by the atoms A1, A2, B2, and B3 (see Fig. 22 in Appendix D). The remaining
4 low-energy bands are formed by non-dimer atoms B1 and A3, which are coupled
by the hopping $\gamma_{6}$. In addition, they feature a cone-like band
touching along the $K\rightarrow M$ direction. Again, the bands and spin
expectation values can be nicely reproduced by our model, see Fig. 8(b,c),
employing the fit parameters from Table 1. Similar to BLG, the low-energy
bands of ABC TLG can be strongly tuned by an electric field (in terms of band
gap, see Fig. 4) Konschuh (2011), since only two sublattice atoms from
different layers contribute. The electric field tunability of the
heterostructure dispersion will be investigated later. Similar to the ABA
heterostructure, proximity-induced SOCs are of valley-Zeeman type and the
outermost layers experience opposite Rashba couplings. The spin-orbit fields
(see Fig. 9), corresponding to the low-energy bands $\varepsilon_{5-8}$, are
mainly $s_{z}$ polarized, but show a Rashba texture especially around the
cone-like touching points along $K\rightarrow M$ direction. The band touching
should actually determine the Fermi level. However in Fig. 8, we have a small
amount of doping ($E_{D}\neq 0$), due to the finite $k$-point sampling of the
Brillouin zone in the DFT calculation. We therefore fix $E_{D}$ from the DFT,
before we apply our fitting procedure.

###  V.2 TLG encapsulated within magnetic Cr2Ge2Te6

Figure 10: (a) Band structure of the CGT/ABA-TLG/CGT heterostructure along the
$M-K-\Gamma$ path. Bands in solid red (dashed blue) correspond to spin up
(spin down). The two CGT layers have parallel magnetizations along $z$
direction. (b) Zoom to the calculated low-energy bands (symbols) around the
$K$ point, corresponding to the band structure in (a), with a fit to the model
Hamiltonian (solid lines). (c,d) Same as (a,b) but for antiparallel
magnetization of the two CGT layers (bottom layer along $z$, top layer along
$-z$).  Figure 11: (a) Band structure of the CGT/ABC-TLG/CGT heterostructure
along the $M-K-\Gamma$ path. Bands in solid red (dashed blue) correspond to
spin up (spin down). The two CGT layers have parallel magnetizations along $z$
direction. (b) Zoom to the calculated low-energy bands (symbols) around the
$K$ point, corresponding to the band structure in (a), with a fit to the model
Hamiltonian (solid lines). (c,d) Same as (a,b) but for antiparallel
magnetization of the two CGT layers (bottom layer along $z$, top layer along
$-z$).

We now turn to proximity exchange, and consider TLG in ABA and ABC stacking
encapsulated by the ferromagnetic semiconductor CGT. Again, the outermost
graphene layers will experience proximity couplings, now stemming from the
exchange interaction. In general, top and bottom graphene layers can feel
different exchange fields. We investigate both parallel and antiparallel
magnetizations of the encapsulating CGT layers and analyze the consequences
for the TLG low-energy bands.

####  V.2.1 CGT/ABA/CGT stacks

We first address the ABA TLG, sandwiched between two CGT layers. In Fig. 10(a)
we show the calculated global band structure of the CGT/ABA-TLG/CGT
heterostructure for parallel CGT magnetizations. Especially the ABA TLG low-
energy bands can be nicely recognized near the Fermi level. The high-energy
conduction bands of TLG are located within the CGT conduction bands. In
addition, we find spin-polarized bands close to the Fermi level near the $M$
point, originating from the CGT layers. Zooming in on the Fermi level near the
$K$ point, we again recognize the 8 low-energy bands originating from ABA TLG,
see Fig. 10(b). Similar to the TMDC encapsulation, the bands are also
significantly split, but now due to the exchange coupling originating from the
ferromagnetic CGT layers. The bands can be nicely reproduced by our model
Hamiltonian employing the fit parameters listed in Table 1. In general,
orbital parameters are barely affected by the surrounding materials
proximitizing TLG, as can be seen by comparing with the fit results from the
TMDC encapsultion. However, proximity-induced exchange couplings,
$\lambda_{\textrm{ex}}$, are now necessary to capture the essential band
structure features. From the fit, we find uniform exchange parameters of about
$-3.4$ meV for all C sublattice atoms of the two outer graphene layers. This
is in agreement with previous considerations for encapsulated BLG Zollner and
Fabian (2021c); Zollner  _et al._ (2020b).

What happens if we switch the magnetization direction of one CGT layer? In
Fig. 10(c), we show the calculated global band structure of the CGT/ABA-
TLG/CGT heterostructure for antiparallel CGT magnetizations. In general, the
band structure remains the same, but bands originating from the topmost CGT
layer have switched their spin-polarization. Most important are the
consequences for the TLG low-energy bands, see Fig. 10(d). For parallel
magnetizations, the bands were spin split due to uniform proximity-induced
exchange couplings. By switching the magnetization direction of the top CGT
layer, the proximity exchange of the topmost graphene layer also adapts and
switches sign, as reflected in the fitted parameters, see Table 1. Since the
low-energy bands of TLG are equally formed by carbon atoms from top and bottom
graphene layers, now with opposite proximity exchange couplings which
effectively cancel each other, the bands remain spin degenerate. Again, this
is similar as in CGT encapsulated BLG Zollner and Fabian (2021c).

####  V.2.2 CGT/ABC/CGT stacks

Now, we turn to the CGT encapsulated ABC heterostructure. The band structure
and fit results are summarized in Fig. 11, again considering parallel or
antiparallel magnetizations of the CGT layers. The low-energy bands of ABC TLG
are located near the Fermi level, similar to the TMDC encapsulation. The
touching point of the bands is now along the $K\rightarrow\Gamma$ direction,
see Fig. 11(b). In contrast, for the TMDC-encapsulated case, the touching was
along the $K\rightarrow M$ direction, see Fig. 8(b). This is related to the
different supercell sizes of ABC TLG in the different encapsulation scenarios,
and the backfolding of the graphene $k$ points into the heterostructure
Brillouin zone Zhou  _et al._ (2013). A similar observation can be made
considering the ABA TLG band structures for the different encapsulations.
Actually, the touching point should determine the Fermi level, but due to the
finite $k$-point sampling in our DFT calculation a small doping of about 5 meV
appears.

Most important is the low-energy band structure, featuring 4 Dirac-like bands,
see Fig. 11(b). In the parallel magnetization case, the bands are split due to
uniform proximity-induced exchange coupling in the outermost graphene layers
of ABC TLG. The model parameters from Table 1 perfectly reproduce the
dispersion in the vicinity of the $K$ point, featuring exchange parameters of
about $-3.2$ meV. Looking at the fitted orbital parameters, we find that
$\gamma_{3}$ and $\gamma_{6}$ have the opposite sign, but are nearly equal in
value, compared to the TMDC-encapsulated ABC structure. This is again related
to the different supercells and the backfolding, as already explained.

What happens when we switch the magnetization direction of one CGT layer? In
the antiparallel case, see Fig. 11(c,d), the overall band structure is similar
to the parallel magnetization. Again, bands originating from the topmost CGT
layer have switched their spin-polarization. Most important are the
consequences for the ABC TLG low-energy bands. They remain nearly spin
degenerate, since the proximity exchange couplings from top and bottom
graphene layer are again opposite in sign but nearly equal in value. In
addition, a sizable gap opens at the former touching point, which originates
from the layered antiferromagnetic proximity exchange in the outermost
graphene layers of ABC TLG. This is similar to the observations we made in CGT
encapsulated BLG Zollner and Fabian (2021c), where cancellation of proximity
exchange or proximity SOC leads to a gap opening without an external electric
field. Therefore, also in the TMDC encapsulated scenario, we could open the
gap if proximity SOCs from top and bottom graphene layers are opposite. This
happens, for example, when employing the same TMDC layers for the
encapsulation of ABC TLG, but with a relative twist angle of $60^{\circ}$
between them.

Given the similarity of the low energy physics of BLG and ABC TLG, the
recently proposed spin valve device concepts Zhai  _et al._ (2021); Wu  _et
al._ (2021); Cardoso  _et al._ (2018), which employ proximitized BLG, should
be also applicable to ABC TLG. Most important, since the low energy band
splittings depend on the magnetic configuration of the outer magnetic layers,
as well as on the applied transverse electric field, one can efficiently
control the in-plane conductance in doubly proximitized ABC-TLG.

Table 1:  The fit parameters of the model Hamiltonian $\mathcal{H}$ for the TMDC and the CGT encapsulated TLG structures. The arrows indicate the magnetization direction of CGT.  system | WSe2 | WSe2 | CGT $\uparrow$ | CGT $\downarrow$ | CGT $\uparrow$ | CGT $\downarrow$
---|---|---|---|---|---|---
ABC | ABA | ABC | ABC | ABA | ABA
MoSe2 | MoSe2 | CGT $\uparrow$ | CGT $\uparrow$ | CGT $\uparrow$ | CGT $\uparrow$
$\gamma_{0}$ [eV] | 2.5307 | 2.5482 | 2.5048 | 2.5200 | 2.5412 | 2.5375
$\gamma_{1}$ [eV] | 0.4014 | 0.4028 | 0.3899 | 0.3898 | 0.3612 | 0.3585
$\gamma_{2}$ [eV] | 0 | -0.0119 | 0 | 0 | -0.0099 | -0.0101
$\gamma_{3}$ [eV] | 0.3292 | 0.3102 | -0.3287 | -0.3341 | -0.2926 | -0.2926
$\gamma_{4}$ [eV] | -0.1838 | -0.1765 | -0.1755 | -0.1778 | -0.1723 | -0.1727
$\gamma_{5}$ [eV] | 0 | 0.0181 | 0 | 0 | 0.0076 | 0.0049
$\gamma_{6}$ [eV] | 0.0125 | 0 | -0.0113 | -0.0116 | 0 | 0
$V_{1}$ [meV] | 0.027 | 0.077 | 0.116 | -0.170 | -0.530 | -0.076
$V_{2}$ [meV] | -11.022 | -6.278 | -12.045 | -7.469 | -3.264 | -2.952
$\Delta$ [meV] | 7.961 | 22.084 | 11.095 | 6.937 | 10.162 | 7.583
$\eta$ [meV] | -3.121 | 6.572 | -2.831 | -3.076 | 4.494 | 4.484
$\lambda_{\textrm{R1}}$ [meV] | 0.233 | 0.215 | 0 | 0 | 0 | 0
$\lambda_{\textrm{R3}}$ [meV] | -0.475 | -0.459 | 0 | 0 | 0 | 0
$\lambda_{\textrm{I}}^{\textrm{A1}}$ [meV] | 0.209 | 0.204 | 0 | 0 | 0 | 0
$\lambda_{\textrm{I}}^{\textrm{B1}}$ [meV] | -0.204 | -0.206 | 0 | 0 | 0 | 0
$\lambda_{\textrm{I}}^{\textrm{A3}}$ [meV] | 1.125 | 0.995 | 0 | 0 | 0 | 0
$\lambda_{\textrm{I}}^{\textrm{B3}}$ [meV] | -0.983 | -1.011 | 0 | 0 | 0 | 0
$\lambda_{\textrm{ex}}^{\textrm{A1}}$ [meV] | 0 | 0 | -3.393 | -3.393 | -3.210 | -3.210
$\lambda_{\textrm{ex}}^{\textrm{B1}}$ [meV] | 0 | 0 | -3.393 | -3.393 | -3.210 | -3.210
$\lambda_{\textrm{ex}}^{\textrm{A3}}$ [meV] | 0 | 0 | -3.349 | 3.349 | -3.210 | 3.210
$\lambda_{\textrm{ex}}^{\textrm{B3}}$ [meV] | 0 | 0 | -3.349 | 3.349 | -3.210 | 3.210
$E_{D}$ [meV] | -4.171 | 0 | 5.540 | 10.605 | 0 | 0
dipole [debye] | -0.0152 | -0.0109 | -0.0110 | 0.0010 | 0.0087 | 0.0101

##  VI Electric Field Tunability

Finally, we consider the electric field tunability of the low-energy bands of
encapsulated TLG. The main effect of the field is that we place the outermost
graphene layers in different potentials, thereby opening band gaps in the
spectrum; see for example Fig. 4 for bare ABC TLG. In the following, we
restrict ourselves to the case of TMDC encapsulation, since for the CGT case
one can expect similar outcomes. More precisely, as we have seen above, the
most striking difference between the CGT and TMDC encapsulation cases is that
proximity exchange coupling, rather than proximity SOC, is responsible for the
band splittings. From the model Hamiltonian perspective, see Table 1, orbital
parameters are barely different for the two encapsulation scenarios, while
proximity exchange couplings, $\lambda_{\textrm{ex}}$, are about 3-times
larger in magnitude compared to proximity SOCs, $\lambda_{\textrm{I}}$.
Therefore, one can expect a similar electric field behavior of the TLG low
energy bands in both encapsulations. In addition, below we restrict ourselves
to consider small field values of $\pm$0.25 V/nm, to capture the main effects
of the field on the dispersion. In Fig. 5, we specify the direction of a
positive electric field.

Figure 12: Top: Electric field evolution of the low-energy bands of the
MoSe2/ABC-TLG/WSe2 heterostructure near the $K$ point. The color of the bands
corresponds to the $s_{z}$ spin expectation value. From left to right, we tune
the electric field from $-0.25$ V/nm to 0.25 V/nm. The calculated low-energy
bands (symbols) match well with the model Hamiltonian fits (solid lines).
Bands are plotted with respect to the Dirac point energy $E_{D}$. For the $\pm
0.25$ V/nm cases, we fixed the parameters from Table 1, and refitted the
potential $V_{1}$ and asymmetry $\eta$, since mainly these parameters are
affected by a small electric field (see Table 3). We obtained
$V_{1}=6.259\leavevmode\nobreak\ (-3.839)$ meV and
$\eta=-6.782\leavevmode\nobreak\ (-0.771)$ meV for the negative (positive)
field. Bottom: The corresponding electric field evolution of the DOS, as
calculated from the model Hamiltonian.

###  VI.1 MoSe2/ABC-TLG/WSe2 stacks

In Fig. 12 we show the electric field evolution of the MoSe2/ABC-TLG/WSe2 low-
energy bands, including model fits. Since our model Hamiltonian very
accurately reproduces the low-energy dispersions, we also show the
corresponding DOS, calculated from the model dos . When an electric field is
applied across ABC TLG, a band gap opens at the cone-like band touching along
the $K\rightarrow M$ line. The field separates the outer graphene layers in
energy by the potential $V_{1}$. In order to fit the low-energy bands, when an
electric field is applied, we keep the parameters from Table 1 fixed, but
refit the parameters $\eta$ and $V_{1}$. The reason is that the applied field
is small ($\pm 0.25$ V/nm) and especially these two orbital parameters are
relevant for the atoms B1 and A3, forming the low-energy bands. With this
approach, the band structures with electric field are almost perfectly
reproduced by the model.

In addition, we note that for negative field, the conduction band is split
much less than the valence band and vice versa for the positive field. This
can be also seen by looking at the calculated DOS. Such a switching of band
splittings reminds us of the BLG physics Gmitra and Fabian (2017); Zollner
_et al._ (2020b); Island  _et al._ (2019); Zollner and Fabian (2021c). As
theoretically proposed Gmitra and Fabian (2017) and experimentally
demonstrated Island  _et al._ (2019), a transverse electric field, applied
across a BLG/TMDC heterostructure, can efficiently tune SOC of conduction
electrons, making it a potential platform for a spin-transistor. For ABC TLG,
the low-energy bands are formed by atoms B1 and A3, which experience different
magnitudes of proximity SOC, as the corresponding spin-orbit parameters in
Table 1 show. For positive electric field, the bottom graphene layer, which
couples to MoSe2, sits in a lower potential than the top graphene layer, which
couples to WSe2. Therefore, the valence band is mainly formed by atoms B1,
experiencing proximity SOC of about 0.2 meV, while the conduction band is
formed by atoms A3, having proximity SOC of about 1 meV. For negative electric
field, the situation is reversed allowing to fully electrically swap between
the two magnitudes of SOC in the low-energy bands of ABC TLG. Such a swapping
between spin-orbit splittings allows to electrically control spin relaxation
in proximitized ABC TLG.

###  VI.2 MoSe2/ABA-TLG/WSe2 stacks

Figure 13: Top: Electric field evolution of the low-energy bands of the
MoSe2/ABA-TLG/WSe2 heterostructure near the $K$ point. The color of the bands
corresponds to the $s_{z}$ spin expectation value. From left to right, we tune
the electric field from $-0.25$ V/nm to 0.25 V/nm. The calculated low-energy
bands (symbols) match well with the model Hamiltonian fits (solid lines). For
the $\pm 0.25$ V/nm cases, we fixed the parameters from Table 1, and refitted
the potentials $V_{1}$ and $V_{2}$, since mainly these parameters are affected
by a small electric field (see Table 2). We obtained
$V_{1}=13.748\leavevmode\nobreak\ (-12.021)$ meV and
$V_{2}=-4.984\leavevmode\nobreak\ (-6.195)$ meV for the negative (positive)
field. Bottom: The corresponding electric field evolution of the DOS, as
calculated from the model Hamiltonian.

The impact of a transverse electric field on the ABA bands and DOS is more
complicated than for the ABC case, see Fig. 13. Here, we have 8 low-energy
bands that are present near the Fermi level, and which are formed by almost
all carbon atoms from the three layers. Nevertheless, similar to ABC TLG, the
electric field introduces a potential difference between the outermost
graphene layers. In the model Hamiltonian, we capture this with parameter
$V_{1}$. In addition, the middle layer can in general also sit in some nonzero
potential, that we capture by $V_{2}$. Therefore, in order to fit the bands
when an electric field is applied, we fix the parameters from Table 1, but
refit the parameters $V_{1}$ and $V_{2}$. Again, the band structures with
electric field are also nicely reproduced by the model Hamiltonian, allowing
us to calculate the DOS dos .

With the small field amplitudes that we employ, the low-energy dispersion
barely changes. Nevertheless, the Dirac-like bands are further separated in
energy and now strongly anticross with the BLG-like bands. The parabolic bands
still touch along the $K\rightarrow\Gamma$ direction and the spectrum is not
yet gapped for fields of $\pm 0.25$ V/nm, as can be seen from the DOS.

##  VII Interplay of spin interactions

One important conclusion of the previous discussion is that one can very well
reproduce DFT data with our model Hamiltonians. The inclusion of an electric
field, transverse to the monolayers, is straightforward, as it can be
parametrized by essentially two orbital parameters. Considering also that the
proximity effects are short ranged, we can realistically study more complex
structures, such as ex-so-tic CGT/TLG/TMDC heterostructures exs .

To be specific, we investigate TLG encapsulated by strong spin-orbit
semiconductor WSe2 on the top, and ferromagnetic semiconductor CGT on the
bottom. To calculate the dispersion from the model Hamiltonian, we use the
orbital parameters from Table 1 of TMDC-encapsulated ABA and ABC TLG. In
addition, we employ valley Zeeman and Rashba SOC for the top graphene layer
($\lambda_{\textrm{I}}^{\textrm{A3}}=-\lambda_{\textrm{I}}^{\textrm{B3}}=1$
meV, $\lambda_{\textrm{R3}}=-0.4$ meV), as well as exchange coupling for the
bottom graphene layer
($\lambda_{\textrm{ex}}^{\textrm{A1}}=\lambda_{\textrm{ex}}^{\textrm{B1}}=-3.5$
meV). This is a reasonable choice, according to our findings. To study the
electric field behavior, also on the model level, we additionally employ the
fitted parameters for the different electric fields ($V_{1}$ and $V_{2}$ for
ABA; $V_{1}$ and $\eta$ for ABC), see Fig. 12 and Fig. 13. The combination of
SOC and proximity exchange breaks time-reversal symmetry. Therefore, we
calculated the dispersion near both the $K$ and $K^{\prime}$ valleys.

###  VII.1 ABC

The model results for ABC TLG are summarized in Fig. 14. For zero field, both
sublattice atoms B1 and A3 are equally contributing to all four low-energy
bands. The band splittings result now from the interplay of proximity spin-
orbit and exchange couplings, originating from the individual layers.
Consequently, at $K$ ($K^{\prime}$), the two spin interactions are additive
(subtractive) and band splittings are about 4.5 meV (2.5 meV). Moreover, we
find that the ABC low-energy bands exhibit no gap near $K$ valley, while there
is a gap near $K^{\prime}$ valley, see Fig. 14, potentially important for the
realization of a valley-polarized quantum anomalous Hall effect Vila  _et al._
(2021).

When an electric field is introduced, the bands are separated in energy, since
the outermost layers are now in different potentials. In addition, the bands
are not anymore equally formed by the two sublattice atoms. Still, the
different spin interactions are at interplay, either in an additive or
subtractive way. Consequently, for positive field both bands are spin split at
$K$, while at $K^{\prime}$ the valence band is strongly split and the
conduction band stays nearly degenerate. Reversing the direction of the
electric field, the bands are flipped with respect to the Fermi level, since
also the potential difference is now opposite compared to the positive field.
The fact that different magnitudes of splittings arise in different bands and
valleys is important for the gate control of spin-relaxation Gmitra and Fabian
(2017), potentially resulting in valley- and spin-polarized currents Vila  _et
al._ (2021).

Figure 14: Electric field evolution of the low-energy bands of a CGT/ABC-
TLG/WSe2 heterostructure near the $K$ (top row) and $K^{\prime}$ (bottom row)
points. The color of the bands corresponds to the $s_{z}$ spin expectation
value. From left to right, we tune the electric field from $-0.25$ V/nm to
0.25 V/nm. Next to the bands, we list the spin splitting at the
$K$/$K^{\prime}$ point. The dispersions are calculated from the model,
assuming reasonable parameters as explained in the text.

###  VII.2 ABA

In Fig. 15, we show the calculated model low-energy dispersion of the ABA TLG,
when the two spin interactions are at interplay. For zero field, we observe a
similar behavior as for the ABC TLG, where near $K$ the band splittings are
larger than near $K^{\prime}$, due to additive/subtractive effects of the spin
interactions. In addition, looking at the field evolution ($-0.25\rightarrow
0.25$ V/nm) of the Dirac-like bands, we observe a swapping of the nearly spin-
degenerate bands from valence into conduction side at the $K^{\prime}$ valley,
and vice versa for the strongly split bands. Similar observations can be made
at the $K$ valley. The BLG-like parabolic bands remain more or less the same
with the applied field.

Figure 15: Electric field evolution of the low-energy bands of a CGT/ABA-
TLG/WSe2 heterostructure near the $K$ (top row) and $K^{\prime}$ (bottom row)
points. The color of the bands corresponds to the $s_{z}$ spin expectation
value. From left to right, we tune the electric field from $-0.25$ V/nm to
0.25 V/nm. The bands are calculated from the model, assuming reasonable
parameters as explained in the text.

##  VIII Summary

We have investigated systematically the electronic band structure of TLG in
different stacking and encapsulation configurations. Starting with bare ABA,
ABC, and ABB TLG, we have provided the essential ingredients of the orbital
and spin-orbital effects, and both quantitative and qualitative understanding
of the SOC at the sub-meV level. Important for ongoing investigations, we also
report on the electric-field tunability of the electronic DOS and VHS.

The bulk of our investigation is devoted to encapsulated TLG. We deal
separately with proximity SOC and proximity exchange. The proximity SOC is
studied by encapsulating ABA and ABC TLG within WSe2 and MoSe2. We show how
the two semiconducting monolayers affect the low-energy bands of TLG, inducing
relatively strong (meV) spin splittings that are tunable by electric field. We
also show that the effective model Hamiltonian, which considers sizeable SOC
on the outer layers, performs great in explaining the DFT results.

Proximity exchange is analyzed for the stacks of TLG encapsulated within CGT
monolayers. The proximity exchange, also on meV scale, modifies the low-energy
bands of TLG, differently for parallel and antiparallel CGT magnetizations.
The magnetic tunability should be a useful way to control correlated phases,
especially in ABC TLG which exhibits a large DOS at low energies.

The nice comparison between the DFT results and the model makes us confident
in applying the model itself in more complex situations, such as studying ex-
so-tic CGT/TLG/TMDC heterostructures exs . The interplay of spin-orbit and
exchange coupling leads to a different band structure at $K$ and $K^{\prime}$.
In addition, the layer polarization then allows for a strong tunability of the
bands with a transverse electric field, which is able to close or open spin
splittings, or even switch the spin polarization, for a particular band in
encapsulated ABC TLG.

In all our investigated DFT cases we provide fitting parameters to the
effective model as a useful resource for subsequent investigations of
transport and correlated physics based on realistic model simulations.

###### Acknowledgements.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) SFB 1277 (Project No. 314695032), SPP 2244 (Project No.
443416183), and the European Union Horizon 2020 Research and Innovation
Program under contract number 881603 (Graphene Flagship). M. G. acknowledges
financial support provided by Slovak Research and Development Agency provided
under Contract No. APVV-20-0150 and by the Ministry of Education, Science,
Research and Sport of the Slovak Republic provided under Grant No. VEGA
1/0105/20.

##  Appendix A Electric field effects of bare ABA and ABC trilayer graphene

Table 2:  The fit parameters of the orbital model Hamiltonian $\mathcal{H}_{\textrm{orb}}^{\textrm{ABA}}$ for ABA TLG with electric field. The fitted band structure results are summarized in Fig. 16.  el. field [V/nm] | 0 | 0.25 | 0.5 | 0.75 | 1.0
---|---|---|---|---|---
$\gamma_{0}$ [eV] | 2.5692 | 2.5693 | 2.5695 | 2.5693 | 2.5688
$\gamma_{1}$ [eV] | 0.3699 | 0.3700 | 0.3700 | 0.3700 | 0.3701
$\gamma_{2}$ [eV] | -0.0101 | -0.0101 | -0.0100 | -0.0099 | -0.0098
$\gamma_{3}$ [eV] | 0.2829 | 0.2821 | 0.2798 | 0.2772 | 0.2742
$\gamma_{4}$ [eV] | -0.1643 | -0.1640 | -0.1638 | -0.1642 | -0.1635
$\gamma_{5}$ [eV] | 0.0150 | 0.0130 | 0.0129 | 0.0130 | 0.0132
$V_{1}$ [meV] | 0 | -19.146 | -38.608 | -57.927 | -78.221
$V_{2}$ [meV] | -19.991 | -14.814 | -13.980 | -13.846 | -14.715
$\Delta$ [meV] | 20.566 | 24.193 | 24.452 | 24.158 | 23.983
$\eta$ [meV] | 6.965 | 12.842 | 13.189 | 12.757 | 12.506
dipole [debye] | 0 | 0.0250 | 0.0499 | 0.0749 | 0.0996
Table 3:  The fit parameters of the orbital model Hamiltonian $\mathcal{H}_{\textrm{orb}}^{\textrm{ABC}}$ for ABC TLG with electric field. The fitted band structure results are summarized in Fig. 4.  el. field [V/nm] | 0 | 0.25 | 0.5 | 0.75 | 1.0
---|---|---|---|---|---
$\gamma_{0}$ [eV] | 2.5539 | 2.5560 | 2.5656 | 2.5884 | 2.6198
$\gamma_{1}$ [eV] | 0.3690 | 0.3689 | 0.3689 | 0.3687 | 0.3685
$\gamma_{3}$ [eV] | 0.2917 | 0.2948 | 0.2960 | 0.2905 | 0.2816
$\gamma_{4}$ [eV] | -0.1646 | -0.1642 | -0.1618 | -0.1571 | -0.1516
$\gamma_{6}$ [eV] | 0.0108 | 0.0113 | 0.0117 | 0.0115 | 0.0111
$V_{1}$ [meV] | 0 | -3.050 | -10.124 | -18.172 | -25.409
$V_{2}$ [meV] | -37.243 | -37.184 | -36.066 | -34.365 | -32.529
$\Delta$ [meV] | 10.288 | 9.746 | 9.001 | 9.816 | 9.094
$\eta$ [meV] | -1.121 | -1.551 | -1.664 | 0.172 | 0.467
dipole [debye] | 0 | 0.0286 | 0.0552 | 0.0812 | 0.1077
Figure 16: Top: Zooms to the ABA TLG bands in the vicinity of the $K$ point.
We compare DFT data (symbols) with the model fits (solid lines) employing
parameters from Table 2. In the dispersion, we list the band gap at the Fermi
level. Bottom: The corresponding calculated density of states (DOS). From left
to right, we increase the transverse electric field from 0 to 1 V/nm.

The following electric field results for bare ABA and ABC TLG have been
obtained with Quantum ESPRESSO Giannozzi  _et al._ (2009). We used the same
structural input as for WIEN2k Blaha  _et al._ (2001), as described in the
main text. Self-consistent calculations are performed with the $k$-point
sampling of $240\times 240\times 1$ to accurately determine the Fermi level
and the DOS. We used an energy cutoff for the charge density of $450$ Ry, and
the kinetic energy cutoff for wavefunctions is $55$ Ry for the scalar
relativistic pseudopotentials with the projector augmented wave method Kresse
and Joubert (1999) with the Perdew-Burke-Ernzerhof exchange correlation
functional Perdew  _et al._ (1996). Moreover, we added vdW Grimme (2006);
Barone  _et al._ (2009) and dipole corrections Bengtsson (1999).

Since we are mainly interested in the electric field behavior of the
dispersion and the DOS, we neglect SOC, since the induced splittings are on
the $\mu$eV level (see main text). In any case, the plane wave and
pseudopotential method, implemented in Quantum ESPRESSO Giannozzi  _et al._
(2009), cannot give correct spin-orbit splittings in graphene, since the
relevant $d$-orbitals are missing Gmitra  _et al._ (2009); Konschuh  _et al._
(2010). Nevertheless, on an orbital level, the dispersions are the same as
calculated with WIEN2k Blaha  _et al._ (2001), and we can safely study the
electric field tunability.

The fit parameters from Table 2 and Table 3, nicely reproduce the DFT-
calculated band structures in Fig. 4 for ABC TLG and Fig. 16 for ABA TLG,
employing the orbital model Hamiltonians,
$\mathcal{H}_{\textrm{orb}}^{\textrm{ABA}}$ and
$\mathcal{H}_{\textrm{orb}}^{\textrm{ABC}}$, from the main text. The electric
field behaviour of the ABC TLG dispersion and DOS was already discussed in the
main text. For the ABA TLG, the electric field also introduces a potential
difference between the outermost layers, separating the monolayer graphene
bands in energy. The BLG bands stay more or less the same with the applied
field, see Fig. 16. In addition, a small band gap is introduced at the Fermi
level. However, the gap first increases and then decreases again, when tuning
the field from 0 to 1 V/nm. From the DOS, we find VHS associated with the
parabolic BLG bands. With increasing field, we find that these bands even
further flatten near the Fermi level, which comes along with increasing VHS in
the DOS.

The fitted electric field results, summarized in Table 2 and Table 3 are
valuable for further theoretical considerations, e. g., gate-tunable transport
simulations of bare ABA and ABC TLG. If one wants to add SOC, this can be
easily done on a model level, employing the intrinsic SOC of bare graphene,
$\lambda_{\textrm{I}}=12\leavevmode\nobreak\ \mu$eV, for all sublattice atoms
Gmitra  _et al._ (2009); Konschuh  _et al._ (2010). The Rashba SOC would be
even smaller (at maximum 5 $\mu$eV) for our investigated field values Konschuh
(2011); Konschuh  _et al._ (2012). Certainly, one can also add proximity spin-
orbit or exchange couplings, as we describe it for our encapsulated
structures.

##  Appendix B Bare ABB trilayer graphene

In addition to ABA and ABC TLG, there is also another stacking sequence, the
ABB one. All those stackings are present in recent investigations on twisted-
graphene/BLG heterostructures Polshyn  _et al._ (2020); Rademaker  _et al._
(2020); Park  _et al._ (2020); Ma  _et al._ (2021). In the same manner as
above and in the main text, we calculated the dispersion, DOS, and analyze the
electric field behavior of ABB TLG. A reasonable orbital Hamiltonian for ABB
TLG can be constructed from combining the knowledge about AA and AB stacked
BLG, together with the ABC and ABA TLG Konschuh  _et al._ (2012); Konschuh
(2011); Rakhmanov  _et al._ (2012); Rozhkov  _et al._ (2016). We propose the
following ABB TLG orbital Hamiltonian

$\displaystyle\mathcal{H}_{\textrm{orb}}^{\textrm{ABB}}=$
$\displaystyle\begin{pmatrix}\Delta+V_{1}&\gamma_{0}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&\gamma_{1}&0&\gamma_{5}\\\
\gamma_{0}f^{*}(\bm{k})&V_{1}&\gamma_{3}f(\bm{k})&\gamma_{4}f^{*}(\bm{k})&0&0\\\
\gamma_{4}f(\bm{k})&\gamma_{3}f^{*}(\bm{k})&\Delta+V_{2}&\gamma_{0}f(\bm{k})&\gamma_{1}^{\prime}&\gamma_{7}f(\bm{k})\\\
\gamma_{1}&\gamma_{4}f(\bm{k})&\gamma_{0}f^{*}(\bm{k})&\Delta+V_{2}&\gamma_{7}f(\bm{k})&\gamma_{1}^{\prime}\\\
0&0&\gamma_{1}^{\prime}&\gamma_{7}f^{*}(\bm{k})&\Delta-
V_{1}&\gamma_{0}f(\bm{k})\\\
\gamma_{5}&0&\gamma_{7}f^{*}(\bm{k})&\gamma_{1}^{\prime}&\gamma_{0}f^{*}(\bm{k})&\Delta-
V_{1}\end{pmatrix}\otimes s_{0}.$ (5)

Figure 17: Schematic illustration of the ABB TLG lattice, showing the relevant intra- and interlayer hoppings $\gamma_{j}$, $j=\\{0,1,3,4,5,7\\}$ (dashed lines). In addition, the bottom (top) graphene layer is placed in the potential $V_{1}$ ($-V_{1}$), while the middle layer is placed in potential $V_{2}$. The asymmetry $\Delta$ arise due to vertical hoppings. In general, the direct interlayer couplings $\gamma_{1}$ and $\gamma_{1}^{\prime}$ can be different. However, when both interlayer distances are the same (in our case fixed to $d=3.3$ Å), $\gamma_{1}=\gamma_{1}^{\prime}$ holds.  Figure 18: (a) DFT-calculated low-energy bands of ABB TLG with SOC (black solid line) and additionally with a transverse electric field of 0.2 V/nm (red dashed line). (b) The spin splitting of valence (red) and conduction (blue) bands. Solid lines are with SOC only and dahed lines are with SOC in the presence of the electric field.  Table 4:  The fit parameters of the model Hamiltonian $\mathcal{H}_{\textrm{orb}}^{\textrm{ABB}}$ for ABB TLG with electric field, assuming $\gamma_{1}=\gamma_{1}^{\prime}$. The fitted band structure results are summarized in Fig. 19.  el. field [V/nm] | 0 | 0.25 | 0.5 | 0.75 | 1.0
---|---|---|---|---|---
$\gamma_{0}$ [eV] | 2.5662 | 2.5682 | 2.5713 | 2.5737 | 2.5754
$\gamma_{1}$ [eV] | 0.3698 | 0.3697 | 0.3693 | 0.3687 | 0.3683
$\gamma_{3}$ [eV] | 0.2726 | 0.2723 | 0.2716 | 0.2706 | 0.2694
$\gamma_{4}$ [eV] | -0.1716 | -0.1699 | -0.1722 | -0.1760 | -0.1793
$\gamma_{5}$ [eV] | 0.0229 | 0.0228 | 0.0246 | 0.0264 | 0.0275
$\gamma_{7}$ [eV] | -0.1087 | -0.1101 | -0.1053 | -0.0966 | -0.0874
$V_{1}$ [meV] | 0 | -12.982 | -31.824 | -48.449 | -64.292
$V_{2}$ [meV] | -51.533 | -53.755 | -60.330 | -66.823 | -71.386
$\Delta$ [meV] | 21.367 | 21.737 | 20.917 | 19.660 | 18.542
dipole [debye] | 0 | 0.0245 | 0.0500 | 0.0764 | 0.1029
Figure 19: Top: Zooms to the ABB TLG bands in the vicinity of the $K$ point.
We compare DFT data (symbols) with the model fits (solid lines) employing
parameters from Table 4. Bottom: The corresponding calculated density of
states (DOS). From left to right, we increase the transverse electric field
from 0 to 1 V/nm.

A schematic illustration of the couplings in ABB TLG is shown in Fig. 17. The
hoppings between the bottom and middle graphene layer are the same as for the
other TLG structures. However, the hoppings between the middle and the top
graphene layer are adapted from the AA BLG structure Rozhkov  _et al._ (2016).
Since the ABB TLG has no $z$-mirror or inversion symmetry, connecting the top
and bottom layer, the direct interlayer hoppings $\gamma_{1}$ and
$\gamma_{1}^{\prime}$ can be in principle different. Only when the interlayer
distances are the same, also the hopping amplitudes are the same. The coupling
$\gamma_{7}$, connecting opposite sublattices from different layers, is
similar to the coupling $\gamma_{4}$, connecting same sublattices from
different layers. Nevertheless, they can be very different in amplitude.

The pseudospin character and the spin-orbit splittings in the bare ABB TLG
dispersion have been already discussed in the main text. In Fig. 18, we show a
zoom to the ABB TLG low-energy bands, further emphasizing the effect of a
transverse electric field on the dispersion and the spin splittings. Applying
the field, apparently flips the low-energy band splittings at the $K$ point
with respect to the Fermi level. This is not surprising, considering the
projected band structure in Fig. 3(c). Without the field, the low-energy
conduction (valence) band has a pronounced contribution from B1 (B3) atoms
near the anticrossing points away from $K$. With the electric field, applied
along positive $z$ direction, a potential difference between the outermost
layers is introduced, such that the bottom layer is now in the lowest
potential. Therefore, the band characters of valence and conduction bands
flip, along with the spin splittings near the $K$ point.

In Fig. 19, we summarize the electric field behavior of the band structure and
DOS for the ABB TLG. The dispersion can be nicely reproduced by the orbital
model Hamiltonian, employing the fit results from Table 4.

##  Appendix C Effects of relaxation on bare TLG dispersions

In the main text, when discussing bare TLG, we have fixed the interlayer
distances between the graphene layers to $d=3.3$ Å, which is an approximation.
We now allow the TLG structures to minimize their energy by relaxing the C
atom positions. For the relaxation we employ DFT-D2 vdW corrections Grimme
(2006); Barone  _et al._ (2009) and use quasi-newton algorithm based on trust
radius procedure. To determine the interlayer distances, the carbon atoms are
allowed to relax only in their $z$ positions (vertical to the layers), until
all components of all forces are reduced below $10^{-4}$ [Ry/$a_{0}$], where
$a_{0}$ is the Bohr radius.

After relaxation, interlayer distances are slightly reduced to about $d=3.24$
Å. The only exception is the interlayer distance between BB-layers in ABB TLG,
which is relaxed to about $d=3.50$ Å. With the relaxed interlayer distances it
is now also reasonable to compare the total energies of all the stackings. We
find that the ABA TLG is energetically most favorable. However, the ABC one is
only about $60\leavevmode\nobreak\ \mu$eV higher in energy. The ABB TLG is 25
meV higher in energy than the ABA one, making this stacking less favorable to
be observed naturally. In Fig. 20, we compare the band structure and DOS
results for the TLG structures with and without the relaxation. The most
drastic changes can be seen in the ABB TLG dispersion, since interlayer
distances are now highly asymmetric. For the ABC and ABA TLG structures, the
dispersion is barely different, except for the high-energy bands.

Figure 20: Top: Zooms to the TLG dispersions in the vicinity of the $K$ point. We compare unrelaxed (black) and fully relaxed (red) structures employing DFT-D2 vdW corrections. Bottom: The corresponding calculated density of states (DOS). The red lines can be reproduced (not shown) by model Hamiltonian parameters summarized in Table 5.  Table 5:  The fit parameters of the model Hamiltonians for the relaxed TLG structures. The dispersions are shown in Fig. 20. TLG | ABC | ABA | ABB
---|---|---|---
$\gamma_{0}$ [eV] | 2.5470 | 2.5687 | 2.5845
$\gamma_{1}$ [eV] | 0.4103 | 0.4128 | 0.4133
$\gamma_{1}^{\prime}$ [eV] | - | - | 0.2593
$\gamma_{2}$ [eV] | - | -0.0067 | -
$\gamma_{3}$ [eV] | 0.3243 | 0.3128 | 0.2755
$\gamma_{4}$ [eV] | -0.1794 | -0.1954 | -0.2018
$\gamma_{5}$ [eV] | - | 0.0089 | 0.0167
$\gamma_{6}$ [eV] | 0.0133 | - | -
$\gamma_{7}$ [eV] | - | - | -0.0077
$V_{1}$ [meV] | 0 | 0 | 0
$V_{2}$ [meV] | -45.161 | -20.831 | -27.767
$\Delta$ [meV] | 11.105 | 31.501 | 9.040
$\eta$ [meV] | -2.838 | 5.069 | -

In Table 5, we summarize the fit results, when the structures are relaxed.
Especially the direct interlayer couplings $\gamma_{1}$ and
$\gamma_{1}^{\prime}$ are strongly renormalized, because of the different
interlayer distances. The high-energy bands are split off exactly due to these
couplings, explaining the differences in the dispersions with and without
relaxation.

##  Appendix D Pseudospin character of TMDC encapsulated TLG dispersions

In Fig. 21, we show the pseudospin character of the relevant ABA TLG bands of
the TMDC encapsulated heterostructure. Similar as in Fig. 1, the parabolic
BLG-like bands are formed by atoms B1, A2, and B3, which form the non-dimer
interlayer pairs. The Dirac bands are formed by orbitals from the outermost
graphene layers.

In Fig. 22, we show the pseudospin character of the ABC TLG bands of the TMDC
encapsulated heterostructure. Consistent with Fig. 2, the flat low-energy
bands are formed by the outer-layer atoms B1 and A3, while high-energy bands
are formed by all other atoms.

Figure 21: Zooms to the TLG bands in the vicinity of the $K$ point of the
MoSe2/ABA-TLG/WSe2 heterostructure. From left to right, we project onto the
sublattice atoms (A = red-filled circles, B = blue-open circles) of the
bottom, middle, and top graphene layers. The sublattice atoms are indicated in
the sketch of the ABA TLG geometry in the lower panels.  Figure 22: Zooms to
the TLG bands in the vicinity of the $K$ point of the MoSe2/ABC-TLG/WSe2
heterostructure. From left to right, we project onto the sublattice atoms (A =
red-filled circles, B = blue-open circles) of the bottom, middle, and top
graphene layers. The sublattice atoms are indicated in the sketch of the ABC
TLG geometry in the lower panels.

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