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

# Highly Tunable Ground and Excited State Excitonic Dipoles in Multilayer
2H-MoSe2

Shun Feng  Aidan Campbell  Institute of Photonics and Quantum Sciences, SUPA,
Heriot-Watt University, Edinburgh EH14 4AS, UK  Mauro Brotons-Gisbert
<EMAIL_ADDRESS>Institute of Photonics and Quantum Sciences,
SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK  Daniel Andres-Penares
Institute of Photonics and Quantum Sciences, SUPA, Heriot-Watt University,
Edinburgh EH14 4AS, UK  Hyeonjun Baek  Institute of Photonics and Quantum
Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK  Takashi
Taniguchi  International Center for Materials Nanoarchitectonics, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Kenji
Watanabe  Research Center for Functional Materials, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Bernhard Urbaszek
Institute of Condensed Matter Physics, Technische Universität Darmstadt, 64289
Darmstadt, Germany  Iann C. Gerber  Université de Toulouse, INSA-CNRS-UPS,
LPCNO, 135 Avenue de Rangueil, 31077 Toulouse, France  Brian D. Gerardot
<EMAIL_ADDRESS>Institute of Photonics and Quantum Sciences, SUPA,
Heriot-Watt University, Edinburgh EH14 4AS, UK

###### Abstract

The fundamental properties of an exciton are determined by the spin, valley,
energy, and spatial wavefunctions of the Coulomb bound electron and hole. In
van der Waals materials, these attributes can be widely engineered through
layer stacking configuration to create highly tunable interlayer excitons with
static out-of-plane electric dipoles, at the expense of the strength of the
oscillating in-plane dipole responsible for light-matter coupling. Here we
show that interlayer excitons in bi- and tri-layer 2H-MoSe2 crystals exhibit
electric-field-driven coupling with the ground ($1s$) and excited states
($2s$) of the intralayer A excitons. We demonstrate that the hybrid states of
these distinct exciton species provide strong oscillator strength, large
permanent dipoles (up to $0.73\pm 0.01$ enm), high energy tunability (up to
$\sim$ 200 meV), and full control of the spin and valley characteristics such
that the exciton g-factor can be manipulated over a large range (from -4 to
+14). Further, we observe the bi- and tri-layer excited state ($2s$)
interlayer excitons and their coupling with the intralayer excitons states
($1s$ and $2s$). Our results, in good agreement with a coupled oscillator
model with spin (layer)-selectivity and beyond standard density functional
theory calculations, promote multilayer 2H-MoSe2 as a highly tunable platform
to explore exciton-exciton interactions with strong light-matter interactions.

##  I Introduction

A range of exotic collective effects are predicted to arise from dipolar
interactions Butov (2004); Lahaye  _et al._ (2009), which have a quadratic
dependence on the magnitude of the static electric dipoles (_p_). For example,
strong dipolar interactions may result in exciton crystals, which exhibit
ordering due to a balance between exciton kinetic energy and many-body Coulomb
interactions Astrakharchik  _et al._ (2007); Capogrosso-Sansone  _et al._
(2007); Slobodkin  _et al._ (2020); Astrakharchik  _et al._ (2021); Bondarev
_et al._ (2021), or lead to nonlinear exciton switches which can reach the
quantum limit when the strength of the interaction is larger than the
exciton’s radiative linewidth Chang  _et al._ (2014); BUT (2017); Delteil  _et
al._ (2019). Hence, in the solid-state, much emphasis has been placed on
engineering interlayer excitons with large _p_ in pioneering III-V
heterostructures High  _et al._ (2012) and more recently in transition metal
dichalcogenide (TMD) heterostructures Fogler  _et al._ (2014); Erkensten  _et
al._ (2021); Lagoin and Dubin (2021) which host excitons with huge binding
energies and thus small Bohr radii Chernikov  _et al._ (2014); He  _et al._
(2014) that enable high exciton densities Wang  _et al._ (2019); Kulig  _et
al._ (2018). In TMD heterostructure devices, tunable interlayer excitons with
large $p$ have been realized in homobilayers Arora  _et al._ (2017, 2018a);
Peimyoo  _et al._ (2021); Lorchat  _et al._ (2021); Leisgang  _et al._ (2020);
Horng  _et al._ (2018); Wang  _et al._ (2018); Kipczak  _et al._ (2022);
Paradisanos  _et al._ (2020); Sung  _et al._ (2020); Shimazaki  _et al._
(2020) and heterobilayers Rivera  _et al._ (2015); Ciarrocchi  _et al._
(2019); Jauregui  _et al._ (2019); Tang  _et al._ (2021), even at the single
exciton level Kremser  _et al._ (2020); Baek  _et al._ (2020); Li  _et al._
(2020). However, many of the exotic collective effects underpinned by strong
dipolar interactions remain to be observed, motivating further exploration of
interlayer excitons and ways to manipulate their spin and optical properties.
For example, it is desirable to increase the electron-hole spatial separation
beyond the interlayer distance, but not at the cost of vanishing oscillator
strength. This goal is intrinsically difficult for bare interlayer exciton
states (e.g. in TMD heterobilayers with Type II interfaces). Recent
experimental and theoretical efforts Leisgang  _et al._ (2020); Peimyoo  _et
al._ (2021); Gerber  _et al._ (2019) suggest that each ‘bare exciton’ is
actually composed of a mixture of other exciton wave functions, and their
mixture can be tuned by detuning energy between transitions. Here we use the
term bare exciton to denote the majority of each transition for simplicity.
While previous studies focused on hybrid interlayer excitons formed in
adjacent ‘natural’ homobilayers Peimyoo  _et al._ (2021); Lorchat  _et al._
(2021); Leisgang  _et al._ (2020), an open question remains if larger hybrid
interlayer dipoles, with the electron and hole highly confined in separate
layers, can be generally obtained in multilayer TMD platforms. Further, due to
the decreasing oscillator strengths of excited Rydberg-like states Chernikov
_et al._ (2014); He  _et al._ (2014); Stier  _et al._ (2018), the observation
of excited states of interlayer excitons has proven elusive to date.
Characterisation of the excited state spectrum of interlayer excitons provides
additional information about their basic properties and a potential means to
further engineer dipolar interactions by taking advantage of their larger Bohr
radii, which has been crucial to realize optical nonlinearity Walther  _et
al._ (2018) and excitonic analogues of spatially ordered structures in
ultracold atomic gases Schauß  _et al._ (2012). To address these issues, we
take advantage of hybridised excitons in bilayer (2L) and trilayer (3L)
2H-MoSe2 composed of the bare exciton states of the interlayer excitons (both
$1s$ and $2s$) and the ground ($1s$) and excited ($2s$) states of the
intralayer A-excitons. We simultaneously demonstrate strong oscillator
strength and wide tunability of the fundamental properties (spin, valley,
energy, and spatial wavefunction) of the hybrid exciton species in multilayer
MoSe2.

Figure 1: Intralayer and interlayer excitons in a terraced 2L/3L 2H-MoSe2
sample. a, Sketch of the sample and device used in this work: a 2H-MoSe2
crystal with 2L- and 3L-thick terraces in a dual-gated device configuration.
Electrons and holes (red and blue circles, respectively) can be localised
either in the bottom (L1), middle (L2), or top (L3) MoSe2 layers, giving rise
to different species of strongly bound intralayer and interlayer excitons. The
vertical white arrows indicate the direction of the permanent electric dipoles
for the different interlayer excitons. The interlayer stackings are
highlighted as $R^{h}_{h}$ between L1 and L3 and $H^{h}_{h}$ between L1 (L2)
and L2 (L3), where $h$ denotes the hexagon center in the crystal lattice of
each layer. b, Spin–valley configurations of the intralayer and interlayer
excitons depicted in panel a: intralayer A and B excitons (purple shaded
ovals), and bilayer and trilayer interlayer excitons (red and grey shaded
ovals, respectively). The layer labels (L1-L3) on each panel indicate the
layer origin of the corresponding electronic states in the conduction (CB) and
valence (VB) bands. The purple (green) bands correspond to spin up (down).

##  II Results and discussion

###  II.1 Device structure and introduction to excitons in multilayer 2H-MoSe2

Fig. 1a shows a sketch of our device, which consists of a terraced 2H-MoSe2
flake with bilayer (2L) and trilayer (3L) regions encapsulated by hexagonal
boron nitride (h-BN) layers with nearly identical thicknesses ($\sim$18 nm).
Graphene layers act as electrical contacts for the MoSe2 crystal and the top
and bottom gates (see Suppl. Note S1 for further details of the device
fabrication). In our experiments, the MoSe2 contact is grounded while we apply
voltages to the top and bottom gates, labeled as VT and VB, respectively. This
configuration allows us to apply a vertical electric field with a magnitude VE
(where VE = VB = -VT) while keeping the carrier concentration in the MoSe2
sample constant at charge neutrality. Similar to other TMDs, each layer in our
terraced 2H-MoSe2 flake hosts tightly bound intralayer excitons with both
ground ($1s$) and excited ($2s$, $3s$, etc) states Chernikov  _et al._ (2014).
Ground and excited intralayer exciton states present the same exact spin-
valley configuration. In the case of the lowest energy excitons in MoSe2 (the
so called A excitons, XA), both the ground and excited exciton states
originate from Coulomb-bound electron-hole pairs in the lower- (higher-) lying
conduction (valence) band at $\pm$K (see Fig. 1b), respectively, which endows
them with the same optical selection rules: excitons at $\pm$K couple to
$\sigma^{\pm}$-polarised light Xiao  _et al._ (2012). In addition to XA, there
are intralayer B excitons (XB) composed of the electron (hole) at the top
(bottom) of the conduction (valence) band at $\pm$K respectively, with a
considerable energy difference and opposite spin index compared to XA.

Beyond intralayer exciton states, multilayer TMDs also host interlayer
excitons, in which the layer-localised electron wave functions can bind to
holes with wave functions confined predominantly within adjacent layers or
spread across several layers, giving rise to excitons with spatially extended
wave functions Arora  _et al._ (2017); Wang  _et al._ (2018); Horng  _et al._
(2018); Leisgang  _et al._ (2020); Lorchat  _et al._ (2021); Peimyoo  _et al._
(2021). Interlayer and intralayer excitons have been shown to coexist in
multilayer Arora  _et al._ (2018a) and bilayer 2H-MoSe2 Horng  _et al._
(2018). Figure 1a shows a sketch of the possible intra- and interlayer exciton
spatial configurations in our terraced 2L/3L 2H-MoSe2, where we have assumed
that, despite the possible spatial spread of the carrier wave functions, the
carriers are predominantly confined to a single layer. This assumption leads
to three different bare exciton species: i) intralayer excitons within each
individual layer (i.e., ground and excited states of the A and B exciton
series in MoSe2 Arora  _et al._ (2015)); ii) interlayer excitons with the
carriers residing in adjacent layers (IX2L), in which the electron and hole
occupy the upper spin-orbit-split conduction band and the topmost valence band
of each layer, respectively (see Fig. 1b); and iii) interlayer excitons in
which the carriers reside in the outermost layers of the 3L MoSe2 region
(IX3L), where the electron (hole) occupies the lowermost (topmost) conduction
(valence) band (see Fig. 1b). In the 3L MoSe2 region, IX2L can be formed with
carriers from the middle MoSe2 layer (L2) and carriers from either the bottom
(L1) or top (L3) MoSe2 layers. Note that, despite the spatially indirect
character of IX2L and IX3L, these interlayer exciton species exhibit momentum-
direct (intravalley) optical transitions within $\pm$K. Moreover, the vertical
displacement of the electron and hole wave functions of the bare interlayer
excitons results in an out-of-plane static electric dipole (negligible for
intralayer excitons), with a dipole polarity and magnitude that depend on the
positions and the spatial separation of the electron and hole in the
multilayer, respectively (see Fig. 1a). Such out-of-plane permanent electric
dipoles of interlayer excitons in other TMD multilayers, homostructures, and
heterostructures have been shown to lead to large shifts of the exciton
transition energies via the the quantum confined Stark effect Rivera  _et al._
(2016); Jauregui  _et al._ (2019); Leisgang  _et al._ (2020); Baek  _et al._
(2020); Lorchat  _et al._ (2021); Peimyoo  _et al._ (2021). Finally, the
natural 2H stacking of our terraced 2L/3L MoSe2 flake results in a different
relative stacking configuration between the different layers. While IX2L
involves MoSe2 layers with a relative 2H stacking, IX3L originates from MoSe2
layers with a relative R${}^{h}_{h}$-type stacking, where _h_ denotes the
hexagon centre of the crystal lattice in each layer (see Fig. 1a). Hence, each
IX species is endowed with distinct spin–layer–valley configurations (see Fig.
1b) that can be optically probed. Finally, we note that the relative
R${}^{h}_{h}$-stacking between L1 and L3 is distinct from 3R-stacked
homobilayers Paradisanos  _et al._ (2020); Shi  _et al._ (2017). For 3R-type
bilayers the two layers are laterally shifted so that the hexagon centres in
each layer are not vertically aligned, rendering a different symmetry and
resulting in a forbidden hole tunneling Gong  _et al._ (2013).

Figure 2: Layer hybridised excitons in 2L 2H-MoSe2. a, VE dependence of the
first derivative of the reflectance contrast spectra with respect to photon
energy (d($\Delta$R/R0)/dE) in our 2L 2H-MoSe2 in the spectral range 1.68 -
1.78 eV. b, Calculated energies of the different hybrid IX2L-X${}^{A}_{2s}$
exciton states as a function of VE, which we label as hX1, hX2, hX3, and hX4
from low to high energy at $V_{E}>0$ V, respectively. The colour of the solid
lines denotes the contribution of the different bare exciton states to each
hybrid exciton. c, Schematics of the spin, valley, and layer configuration of
the exciton states responsible for the exciton hybridisation shown in panel a
for negative and positive applied VE (left and right panels, respectively).
The exciton hybridisation is attributed to a second-order effective coupling
between IX2L and the intralayer A exciton facilitated via the A and B exciton
admixture (depicted by the glowing double arrows). d, The energy position of
IX2L and X${}^{A}_{2s}$ as a function of applied electric field, as obtained
from $GW$+BSE calculations. The labels identify the simplified bare exciton
states. e Normalized theoretical oscillator strengths (red vertical line) and
absorption spectra (black line) at 0 eV/Å and 0.03 eV/Å focusing on the energy
range of IX2L and X${}^{A}_{2s}$. The numerical precision of our calculations
is estimated to be of the order of $\pm$5 meV, see computational details in
Supplementary Note S4.

###  II.2 Electric field dependent excitonic transitions in 2L 2H-MoSe2

To investigate the layer-dependent and intra-/interlayer nature of the
different exciton species in the terraced 2H-MoSe2 flake, we perform
differential reflection contrast ($\Delta$R/R0) spectroscopy at cryogenic
temperature (4 K) as a function of VE at charge neutrality, where $\Delta$R =
Rs \- R0, and Rs (R0) is the intensity of the light reflected by the flake
(substrate). We focus first on the 2L region of our MoSe2 sample.
Supplementary Fig. S1 shows a representative reflection contrast spectrum of
our 2L 2H-MoSe2 at $V_{E}=$ 0 V, where several excitonic resonances with
different linewidths and absorption strengths are observed (see Suppl. Fig.
S2. for the extracted exciton linewidths). The strongest exciton resonances at
low ($\sim$1.63 eV) and high energy ($\sim$1.86 eV) correspond to the ground
states of intralayer A (X${}^{A}_{1s}$) and B (X${}^{B}_{1s}$) neutral
excitons in 2L MoSe2, respectively. Two additional weaker exciton resonances,
separated only by $\sim$25 meV, are also observed in the energy range between
the A and B intralayer 1$s$ exciton states, which can be attributed to IX2L
($\sim$1.715 eV) and the first excited exciton state X${}^{A}_{2s}$ of the
neutral A exciton ($\sim$1.74 eV), in agreement with previous experimental and
theoretical results on bulk Arora  _et al._ (2018a) and 2L 2H-MoSe2 Horng  _et
al._ (2018). More precisely, we perform $GW$+BSE calculations (see Suppl. Note
S4 for computational details) and obtain IX2L and X${}^{A}_{2s}$ peaks 0.11
and 0.17 eV above the X${}^{A}_{1s}$ one, respectively (see Suppl. Fig. S3),
which qualitatively agrees with our experiment in which IX2L and
X${}^{A}_{2s}$ are 0.07 and 0.11 eV above X${}^{A}_{1s}$. To corroborate the
intra-/interlayer character of the IX2L and X${}^{A}_{2s}$ states and unravel
their layer-dependent properties, we show in Fig. 2a a density plot of the VE-
dependence of the first derivative of the reflectance contrast spectra with
respect to photon energy (d($\Delta$R/R0)/dE) in the spectral range 1.68 -
1.78 eV, which helps to visualise these exciton transitions (see Suppl. Fig.
S4 for comparison with bare $\Delta$R/R0 spectra). At VE = 0 V, we observe the
two excitonic resonances attributed to IX2L and X${}^{A}_{2s}$. The
application of a VE leads to a stark contrast in the behaviour of the two
resonance peaks. For small positive applied VE, the resonance energy of the
high energy peak remains almost constant, while the low energy peak splits
into two exciton branches which shift symmetrically towards lower and higher
energies with a linear dependence with VE. This phenomenon can be interpreted
in terms of the DC Stark energy ($\Delta U$) tuning experienced by interlayer
excitons under applied out-of-plane electric fields: $\Delta U=-pE$, with $E$
the strength of the vertical electric field and $p=ed$ the out-of-plane
electric dipole moment (where $e$ represents the electron charge and $d$ the
electron-hole distance). The absolute values of the energy shifts of the two
IX2L branches allow us to estimate an average electron-hole spatial separation
of $d=0.34\pm 0.01$ nm for IX2L, which is comparable to the reported values of
0.39-0.47 nm for IXs in other TMD homostructures such as 2L MoS2 Leisgang  _et
al._ (2020); Lorchat  _et al._ (2021), and 0.63 nm (0.26 nm) for momentum
direct (indirect) IXs in twisted bilayer MoSe2 Sung  _et al._ (2020). This
result unambiguously demonstrates the presence of a sizeable static electric
dipole in the out-of-plane direction and corroborates the interlayer exciton
nature of IX2L Arora  _et al._ (2018a); Horng  _et al._ (2018). Moreover, the
symmetrical but opposite energy shifts of the two IX2L branches reveal the
presence of IX2L excitons with different polarities (i.e., with static
electric dipoles aligned parallel and anti-parallel to the applied electric
field), as sketched in Fig. 1a. For positive VE, IX2L excitons shifting to
lower (higher) energies originate from Coulomb-bound electron-hole pairs in
which the hole is spatially located in the top (bottom) layer of the 2L MoSe2,
i.e., with static electric dipoles pointing up (IX${}_{2L}^{\uparrow}$) and
down (IX${}_{2L}^{\downarrow}$), respectively. Note that for negative VE, the
behaviour of IX${}_{2L}^{\uparrow(\downarrow)}$ is reversed. Therefore, the
Stark shifts of IX${}_{2L}^{\uparrow(\downarrow)}$ allow us to unravel their
layer configuration.

Further, the large DC Stark tuning of IX2L allows us to explore the possible
hybridisation between IX2L and the energetically close X${}^{A}_{2s}$ by
reducing their energy detuning via the applied VE. For $|V_{E}|\approx$ 2 V,
IX2L and X${}^{A}_{2s}$ show an energy anti-crossing characteristic of coupled
systems, suggesting the hybridisation of the two exciton species. We note that
this observation is reproduced across the 2L 2H-MoSe2 sample. Supplementary
Fig. S5 shows the results for a different spatial location in the 2L 2H-MoSe2.
In order to estimate the magnitude of the coupling between IX2L and
X${}^{A}_{2s}$, we employ a phenomenological model in which the hybridisation
between the exciton states is treated as a coupling between oscillators with
resonance energies corresponding to the bare exciton states (see Suppl.
Section S3 for details). In our model, we take into account the spin, valley,
and layer degrees of freedom of each exciton species, which leads to eight
exciton resonances with different spin, valley, and layer properties: two IX2L
with opposite polarities (IX${}_{2L}^{\uparrow(\downarrow)}$) and momentum-
direct transitions at $\pm$K each, and X${}^{A}_{2s}$ localised in the top
(X${}_{2s}^{A,L_{2}}$) and bottom (X${}_{2s}^{A,L_{1}}$) MoSe2 layers with
momentum-direct optical transitions at $\pm$K each. Regarding the
IX2L-X${}^{A}_{2s}$ hybridisation, we include it in our model as a
phenomenological interlayer coupling of holes at $\pm$K, with a magnitude
which we assume to be independent of VE. In our calculations, the energies of
the exciton states at $V_{E}=$ 0 V and the slope of the DC Stark shift for
IX${}_{2L}^{\uparrow(\downarrow)}$ are set to match the corresponding
experimental values, while the value of the coupling strength between the
exciton states is left as a free parameter that can be tuned to fit our
experimental data. Figure 2b shows the calculated energies of the resulting
hybrid exciton states as a function of VE, which we label as hX1, hX2, hX3,
and hX4 from low to high energy at V${}_{E}>0$ V, respectively. The colour of
the solid lines in Fig. 2b denotes the contribution of the different bare
exciton states to each hybrid exciton. The phenomenological model captures
well the hybridisation-induced renormalisation of the exciton resonance
energies with increasing electric field, allowing us to estimate a
IX2L-X${}^{A}_{2s}$ coupling strength $\kappa_{2L-2s}\approx$ 5.2 meV, which
is slightly smaller but of the same order of magnitude as the linewidths of
the exciton states at $V_{E}=0$ V (see Suppl. Fig. S2).

The physical origin of the observed hybridisation between an interlayer
exciton and the first excited state of an intralayer exciton is intriguing,
and to the best of our knowledge has not been been reported in any other
homobilayer TMD system. We discard the possibility of spin-conserving electron
hopping between the electron states involved in IX2L and X${}^{A}_{2s}$, since
such interlayer electron hopping is forbidden in 2L 2H-MoSe2 due to the
$C_{3}$ symmetry of the $d_{z^{2}}$ orbitals of the conduction band states at
$\pm$K Gong  _et al._ (2013); Hagel  _et al._ (2021). However, a recent
theoretical work has shown that the application of a vertical electric field
can lead to the hybridisation of IX2L and X${}_{1s}^{B}$ in 2L 2H-MoSe2 via
spin-conserving interlayer hole tunneling Hagel  _et al._ (2022), in agreement
with previous experimental and theoretical results for 2L 2H-MoS2 Leisgang
_et al._ (2020); Sponfeldner  _et al._ (2022). Moreover, similar to 2L
2H-MoS2, the theoretical results in Ref. Hagel  _et al._ (2022) also suggest a
weak admixture between the A and B excitons in 2L 2H-MoSe2, which leads to a
non-vanishing second-order effective hybridisation of IX2L and the intralayer
X${}^{A}_{1s}$ exciton. Since X${}^{A}_{1s}$ and X${}^{A}_{2s}$ have the same
nature (i.e., same exact spin and valley configuration), we tentatively
attribute the observed IX2L-X${}^{A}_{2s}$ hybridisation to a second-order
effective coupling between IX2L and the intralayer A exciton facilitated via
the A and B exciton admixture (see Fig. 2c) Sponfeldner  _et al._ (2022);
Hagel  _et al._ (2022). This result agrees well conceptually with the
theoretical modeling in Ref. Hagel  _et al._ (2022), and leads to a layer
selective coupling between IX2L and the intralayer A exciton, as also captured
by our phenomenological model (see Fig. 2b). Our estimated value of the
coupling strength $\kappa_{2L-2s}\approx$ 5.2 meV is very similar to the
coupling strength reported for IX2L and X${}^{A}_{1s}$ excitons in 2H-MoS2,
and significantly smaller than the direct coupling between the IX2L and
X${}_{1s}^{B}$ excitons in the same 2H-MoS2 sample Leisgang  _et al._ (2020);
Sponfeldner  _et al._ (2022), which supports our hypothesis.

To further corroborate our hypothesis, we calculate $GW$+BSE-based estimates
of the excitonic transition energies and their corresponding oscillator
strengths at various external electric field values (see Figs. 2d to 2e). For
vertical applied positive electric fields, the IX2L excitons experience a
significant Stark shift, which supports their interlayer nature. The
X${}^{A,L{{}_{1}}}_{2s}$ remains relatively unchanged, while the
X${}^{A,L_{2}}_{2s}$ exhibits an energy shift with the applied electric field,
indicative of the layer-selective mixing with the IX2L state. In fact, our
calculations reveal a sizeable mixing between IX2L and X${}^{A,L_{2}}_{2s}$
even at zero applied electric field. Table 1 in Suppl. Section S4 summarises
the oscillator strengths of the different bare optical transitions
contributing to each excitonic resonance in Figs. 2d and 2e, corresponding to
0 and 0.03 V/Å applied electric fields, respectively.

Figure 3: Layer hybridised excitons in 3L MoSe2. a, VE dependence of
d($\Delta$R/R0)/dE in the 3L MoSe2 region of our sample. b, Energies of the
different hybrid IX3L-X${}^{A}_{1s}$ exciton states as a function of VE, where
the colour of the solid lines denotes the contribution of the different bare
exciton states to each hybrid exciton. c, Schematics of the spin, valley, and
layer configuration of the exciton states responsible for the exciton
hybridisation shown in panel a for negative and positive applied VE (left and
right panels, respectively). The exciton hybridisation is attributed to direct
spin-conserving interlayer hole tunnelling between L1 and L3.

###  II.3 Electric field dependent excitonic transitions in 3L 2H-MoSe2

To explore the potential for multilayer TMDs to host IX with dipole moments
even larger than IX2L and with greater tunability, we optically probe the 3L
region of our terraced 2H-MoSe2 sample at charge neutrality as a function of
VE. Figure 3a shows the VE-dependence of d($\Delta$R/R0)/dE in the spectral
range 1.58 - 1.78 eV (with the values in the energy range 1.66 - 1.78 eV
multiplied by a factor 4 for visualisation purposes). The reflectance spectrum
at VE = 0 V is markedly different to the one observed in the 2L region (see
Fig. 2a). We observe two exciton transitions in the energy range corresponding
to X${}^{A}_{1s}$ with an energy splitting of $\sim$11 meV, and two exciton
resonances in the energy range of IX2L with an energy splitting of $\sim$ 14.5
meV. The resonances at low energy can be attributed to X${}^{A}_{1s}$ excitons
localised in the different layers of our sample, in which the lower average
permittivity environment of L1 and L3 (h-BN/L1,(3)/MoSe2) compared to L2
(MoSe2/L2/MoSe2) results in a dielectric-induced energy blue shift for
X${}^{A}_{1s}$ excitons in L1 and L3 (X${}_{1s}^{A,L_{1}(L_{3})}$) compared to
X${}^{A}_{1s}$ excitons in L2 (X${}_{1s}^{A,L_{2}}$) Raja  _et al._ (2017),
similar to what has been observed for 3L MoS2 Leisgang  _et al._ (2020).

To unravel the nature of the excitons in the energy range corresponding to
IX2L, we focus on their behaviour as a function of VE. As shown in Fig. 3a,
the application of a vertical electric field gives rise to a Stark-effect-
induced splitting of each resonance into two distinct exciton branches which
shift symmetrically towards lower and higher energies with a linear dependence
with VE (see comparison between $\Delta$R/R0 and d$(\Delta$R/R0)/d$E$ in
Suppl. Fig. S6). For the lower exciton resonance, we estimate an average
electron-hole spatial separation of $d=0.36\pm 0.01$ nm, which agrees very
well with the value found for IX in the 2L region of our sample, confirming
its IX2L nature. The energy splitting of the two branches belonging to the
higher energy resonance yields an average electron-hole separation of
$d=0.73\pm 0.01$ nm, which exceeds the interlayer distance of TMD homobilayers
Liu  _et al._ (2014) and is approximately twice the IX2L dipole size. We
therefore identify this resonance as IX3L. Furthermore, the larger Stark shift
of IX3L allows us to tune their energy in resonance to X${}^{A}_{1s}$ (see
Fig. 3a). For $|V_{E}|\sim$ 4 V we observe a clear avoided crossing between
IX3L and X${}_{1s}^{A,L_{1}/L_{3}}$, indicative of the hybridisation between
the exciton species with carriers hosted in the outermost layers of the 3L
MoSe2 sample. Supplementary Fig. S7 shows the results for a different spatial
location in the 3L 2H-MoSe2. We note that, contrary to the results for the 2L
region, we observe negligible oscillator strength of X${}^{A}_{2s}$ in the 3L
MoSe2 region, which prevents clear observation of coupling between this
exciton state and the two IX species. Therefore, we focus on the clear
IX3L-X${}^{A}_{1s}$ hybridisation.

Similar to the 2L case, we simulate the VE-dependent energy dispersion of the
hybridised excitons in the 3L MoSe2 system using a phenomenological model of
coupled oscillators, in which the exciton coupling is both spin- and layer-
selective (see Fig. 3b). In this case, we include three different
X${}^{A}_{1s}$ excitons (i.e., one in each layer) and two IX3L with opposite
polarities (IX${}_{3L}^{\uparrow(\downarrow)}$) and momentum-direct
transitions at $\pm$K each. Spin conserving hole tunneling results in a layer-
selective coupling between IX3L and X${}^{A}_{1s}$; the polarity of IX3L is
locked to the layer degree of freedom of X${}_{1s}^{A}$ (e.g.
IX${}_{3L}^{\uparrow(\downarrow)}$ only couples to
X${}_{1s}^{A,L_{1}(L_{3})}$). However, the energy degeneracy of
X${}_{1s}^{A,L_{1}}$ and X${}_{1s}^{A,L_{3}}$ prevents clear observation of
such layer-selective coupling experimentally. Figure 3b shows the results of
the best fit of the model to our experimental data. The colour of the solid
lines denotes the contribution of the different bare exciton states to each
hybrid exciton. Overall, the phenomenological model captures well the
hybridisation-induced renormalisation of the exciton resonance energies with
increasing electric field, allowing us to estimate a IX3L-X${}^{A}_{1s}$
coupling strength $\kappa_{3L-1s}\approx$ 7.5 meV, which is slightly larger
but of the same order of magnitude as $\kappa_{2L-2s}$ in the 2L region.
Furthermore, it is worth noting that the R-type relative stacking of the
layers $L_{1}$ and $L_{3}$ leads to some differences between the
IX3L-X${}^{A}_{1s}$ and IX2L-X${}^{A}_{2s}$ couplings. The $R$-type stacking
between $L_{1}$ and $L_{3}$ results in IX3L with electron-hole pairs with the
same spin-valley configurations as XA (see Fig. 1a), which allows the
hybridisation of the two exciton species via direct spin-conserving interlayer
hole tunnelling in the presence of an additional layer between $L_{1}$ and
$L_{3}$ Shimazaki  _et al._ (2020); Zheng  _et al._ (2022). For the specific
stacking registry of $R^{h}_{h}$ between $L_{1}$ and $L_{3}$, a recent work
calculated an 11 meV tunnel splitting of the valence band (with monolayer BN
as $L_{2}$) Shimazaki  _et al._ (2020), supporting our interpretation. On the
contrary, due to the 2$H$ relative stacking between adjacent layers, the
direct hole tunneling between the valence band states in IX2L and XA has to
compete with a sizable detuning equal to the spin splitting at the valence
band edges (150 meV in MoSe2 Zheng  _et al._ (2022)), and is thus only
facilitated by the admixture of XA and XB. Finally, we note that the energy
detuning between IX2L and the energy degenerate X${}_{1s}^{A,L_{1}}$ and
X${}_{1s}^{A,L_{3}}$ exciton resonances is slightly smaller in the 3L region
as compared to the 2L one. As a consequence, we are able to observe
experimental signatures of the coupling between these two exciton species at
$V_{E}\sim 6$ V (see Fig. 3a and Suppl. Section S6), which corroborate our
results for the 2L 2H-MoSe2.

Figure 4: Magneto-optical properties of layer-hybridised excitons in 2L MoSe2
and 3L MoSe2. a, Zeeman splitting of hX4 at three different applied VE. The
blue dots represent the experimental values, while the blue solid lines show
linear fits of the experimental data, from which we are able to estimate the
effective $g$-factor of this hybrid exciton at each applied VE. b, VE-driven
evolution of the $g$-factor of the hybrid excitons hX3 (-5 to 0 V, red shaded
area) and hX4 (0 to 5 V, blue shaded area) in bilayer MoSe2 (bottom panel).
The top panel shows the VE-dependent contributions of each bare exciton state
$|C_{3,4}^{IX(X)}|^{2}$ to the corresponding hybrid excitons. c, Zeeman
splitting of IX3L at measured at $V_{E}=0$ V.

###  II.4 Electric-field-dependent magneto-optical properties of hybrid
excitons in MoSe2

Beyond tuning of the exciton resonance energies, oscillator strengths, and
effective permanent electric dipoles, here we explore if the electric-field-
dependent control of the exciton nature enables precision tuning of the
effective Landé $g$-factors. The application of a vertical magnetic field $B$
results in the Zeeman splitting of the optical transitions of each exciton at
$\pm$K, with an energy splitting $\Delta E(B)=g\mu_{0}B$, with $\mu_{0}$ the
Bohr magneton. Optical transitions at $\pm$K can be independently probed by
$\sigma^{\pm}$-polarised light, respectively, which enables us to perform
circularly polarised reflectance contrast measurements to estimate the
experimental Zeeman splitting $\Delta E=E^{\sigma^{+}}-E^{\sigma^{-}}$, with
$E^{\sigma^{\pm}}$ the energy of the $\sigma^{\pm}$-polarised transition. We
focus first on the hybrid IX2L-X${}^{A}_{2s}$ exciton states in our 2L region.
Figure 4a shows the measured Zeeman splittings (blue dots) for hX4 at three
different applied VE values. Supplementary Figure S8 shows the linecuts for
the $\sigma^{\pm}$-resolved d($\Delta$R/R0)/d$E$ at $V_{E}=0$ V for 5 T. The
blue solid lines represent linear fits of the experimental data, from which we
estimate the effective $g$-factor at each applied VE. We observe the effective
$g$-factor of hX4 is tuned from a negative value ($-1.9\pm 0.5$) to a
relatively large positive value ($11.8\pm 0.4$). To explore this effect in
more detail, we employ the same experimental procedure and extract the
effective $g$-factors of hX3 and hX4 in the range of applied VE in which the
oscillator strength of each transition and their energy detuning with respect
to other transitions enable a reliable estimate of $E^{\sigma^{\pm}}$, i.e.,
$V_{E}<0$ for hX3 and $V_{E}>0$ for hX4 (as indicated by the red and blue
shaded areas in Fig. 4b). The results highlight a continuous and smooth
transition of the $g$-factor of the two hybrid exciton states from $-1.9\pm
0.5$ to $11.8\pm 0.4$ by changing VE from 0 V to $\pm 4.5$ V. Such evolution
of the effective $g$-factor arises from the VE-dependent hybridisation between
IX2L and X${}^{A}_{2s}$, and can be quantitatively explained by our
phenomenological model of coupled oscillators. In this model, the wave
function of each hybrid exciton is expressed as a superposition of the bare
intralayer and interlayer exciton wave functions (see Suppl. Section S3). In
the case of hX3 and hX4, these exciton wave functions are expressed as
follows:

$\displaystyle\ket{hX_{3}(V_{E})}=C_{3}^{IX}(V_{E})\ket{IX_{2L}^{\uparrow}}+C_{3}^{X}(V_{E})\ket{X_{2s}^{A,L_{1}}}$
(1)

and

$\displaystyle\ket{hX_{4}(V_{E})}=C_{4}^{IX}(V_{E})\ket{IX_{2L}^{\downarrow}}+C_{4}^{X}(V_{E})\ket{X_{2s}^{A,L_{2}}},$
(2)

where $C_{i}^{IX(X)}(V_{E})$ represents the VE-dependent amplitude of the bare
interlayer (intralayer) exciton state in hybrid exciton state $\ket{hX_{i}}$,
with $|C_{i}^{IX}(V_{E})|^{2}+|C_{i}^{X}(V_{E})|^{2}=1$. Therefore, the
effective $g$-factor of each hybrid exciton state ($g_{hX_{i}}$) can be
expressed as

$\displaystyle
g_{hX_{i}}(V_{E})=|C_{i}^{IX}(V_{E})|^{2}g_{IX_{2L}}+|C_{i}^{X}(V_{E})|^{2}g_{X_{2s}},$
(3)

where $g_{IX_{2L}(X_{2s})}$ represents the $g$-factor of the bare IX2L (X2s)
state. The solid line in Fig. 4b represents a fit of the experimental data to
Eq. (3), in which we have used the $C_{i}^{IX(X)}(V_{E})$ values shown in the
top panel of Fig. 4b (obtained from the fits in Fig. 2b), and $g_{IX_{2L}}$
and $g_{X_{2s}}$ have been left as free fitting parameters. Our model captures
well the hybridisation-induced evolution of the $g$-factors and allows us to
estimate of the effective $g$-factors of the bare states:
$g_{X_{2s}}\approx-4$ and $g_{IX_{2L}}\approx 12.5$. We note that, although to
the best of our knowledge $g_{X_{2s}}$ has not been previously reported for 2L
MoSe2, our estimated value of $g_{X_{2s}}$ is in very good agreement with
reported experimental ($-3.6\pm 0.1$ Arora  _et al._ (2018b)) and theoretical
(-3.7 Deilmann  _et al._ (2020)) values for this excited exciton state in
monolayer MoSe2 and other TMD systems such as bulk WSe2 ($-3.3\pm 0.6$ Arora
_et al._ (2018a)). Regarding $g_{IX_{2L}}$, the estimated sign and value are
also in good agreement with the value predicted by a simplistic “atomic
picture”, in which the $g$-factor of the bands hosting the electron-hole pairs
are assumed to be equal to the sum of their spin, orbital, and valley magnetic
moments Aivazian  _et al._ (2015). Within this model, the spin-valley
configuration of IX2L results in a
$g_{IX_{2L}}=2(\frac{m_{0}}{m^{*}_{c}}+\frac{m_{0}}{m^{*}_{v}})+4\approx 9.7$,
with m0 the free electron mass, and m${}^{*}_{c}$ and m${}^{*}_{v}$ the
experimentally reported electron and hole effective masses for the bottom
conduction band and top valence band in monolayer MoSe2, respectively Goryca
_et al._ (2019). The relatively large and positive value of the $g$-factor
estimated with this atomic picture model provides an additional confirmation
of the spin-valley configuration and interlayer nature of IX2L. Also, we note
that the small discrepancy between the experimental and the calculated value
for $g_{IX_{2L}}$ might arise from a combination of several factors, including
the limitation of this simple model to estimate accurately the $g$-factor of
the relevant bands Arora (2021) and the absence of experimental values for the
electron effective mass of the top conduction band in 2L MoSe2, which
theoretical calculations predict to be slightly smaller than for the bottom
conduction band in Mo-based TMDs such as 2H-MoSe2 Kormányos  _et al._ (2015)
and 2H-MoS2 Brotons-Gisbert  _et al._ (2018). Nevertheless, we note that the
extracted $g$-factors of the bare IX2L and X2s states allow us to include the
effects of the Zeeman splitting in the calculated VE-dependent evolution of
the hybrid IX2L-X2s states, for which we find a very good agreement with the
experimental results (see Suppl. Note S5 and Suppl. Fig. S12).

Finally, we investigate the magneto-optical properties of IX3L in the 3L
region of the sample. Figure 4c shows the experimental Zeeman splitting for
this exciton species at $V_{E}=0$ V, from which we estimate a $g$-factor
$g_{IX_{3L}}=-4.9\pm 0.1$. The extracted value for $g_{IX_{3L}}$ is very
similar to the experimentally reported $g$-factor of X${}^{A}_{1s}$ in 3L
MoSe2 Arora (2021), which confirms the identical spin-valley configuration of
these two exciton species and corroborates that the giant dipole IX3L indeed
originates from electron and hole in L1 and L3 (or vice versa), respectively.
The identical spin-valley configuration of X${}^{A}_{1s}$ and IX3L has an
important consequence on the magneto-optical properties of the hybrid
X${}^{A}_{1s}$-IX3L states: contrary to the hybrid X${}^{A}_{2s}$-IX2L states
in 2L MoSe2, hybrid X${}^{A}_{1s}$-IX3L do not feature a VE-tunable
$g$-factor, since the two bare exciton states already exhibit a similar
$g$-factor.

###  II.5 Excited state interlayer excitons in multilayer 2H-MoSe2

Although excited (Rydberg) exciton states have weaker oscillator strength than
their corresponding ground states Chernikov  _et al._ (2014), in this section
we show that the hybridisation of interlayer excitons with intralayer
transitions leads to clear spectroscopic signatures of excited Rydberg states
for both IX2L and IX3L. Figures 5a and 5b show d2($\Delta$R/R0)/d$E^{2}$
spectra as a function of VE for 2L and 3L 2H-MoSe2 regions, respectively,
corresponding to spatial locations where the interlayer excited exciton states
present appreciable oscillator strengths. The data shown in the the spectral
ranges 1.797 - 1.830 eV and 1.665 - 1.800 eV in Figs. 5a and 5b have been
multiplied by a factor 2 for visualisation purposes. In the 2L region (Fig.
5a), for $4\lesssim V_{E}\lesssim 6$ V we resolve an additional exciton
transition in the energy range $\sim$1.7 eV with a linear Stark shift parallel
to the IX${}_{2L}^{\uparrow}$. Extrapolation of the linear energy shift to
$V_{E}=0$ V gives an estimated energy of $\sim$ 1.77 eV for this excitonic
transition with clear interlayer nature. We note that the relative energy
position of this excitonic peak with respect to both IX2L and X${}^{A}_{2s}$
agrees well with the first excited Rydberg state of IX2L predicted by our
$GW-$BSE results (see Figs. 2e and Suppl. Fig. S3). This, together with the
dipole moment of 0.30 $\pm$ 0.01 enm estimated from the linear Stark shift
(almost identical to the one for IX2L), allows us to attribute the observed
resonance to the first excited Rydberg state of IX${}_{2L}^{\uparrow}$ (i.e.,
IX${}_{2s,2L}^{\uparrow}$). Notably, for $|V_{E}|\sim 3$ V and energies $\sim$
1.82 eV, we observe two additional resonances with opposite Stark shift slopes
that also extrapolate to an energy of $\sim$1.77 eV at $V_{E}=0$ V, which we
attribute to IX${}^{\downarrow}_{2s,2L}$ ($V_{E}>0$ V) and
IX${}^{\uparrow}_{2s,2L}$ ($V_{E}<0$ V), corroborating our peak assignment. We
note that the IX2s,2L state is a general feature rather than a location
dependent feature of our sample, as we also observe IX2s,2L in the main
locations of 2L and 3L MoSe2 shown in Figs. 2-4, as depicted in Suppl. Fig.
S10. Further, we observe a very weak transition near 1.72 eV for $4\lesssim
V_{E}\lesssim 6$ V with a Stark shift parallel to IX2s,2L (see Suppl. Fig.
S9), which we tentatively ascribe as the $3s$ Rydberg state of IX2L.
Extrapolation to VE = 0 V gives an estimated energy for the $3s$ state of
$\sim$20 meV higher than the $2s$ at zero applied electric field.

Similar to Fig. 5a, the results in Fig. 5b also display clear excited Rydberg
states in our 3L 2H-MoSe2 sample. In addition to the coexisting interlayer
exciton species IX2L and IX3L already shown in Fig. 3, we observe two
additional transitions with Stark-induced linear energy shifts parallel to the
IX3L, which we attribute to the two opposite polarities of the $2s$ excited
Rydberg state of IX3L (i.e., IX${}^{\uparrow(\downarrow)}_{2s,3L}$).
Extrapolation of the measured linear energy shifts of
IX${}^{\uparrow(\downarrow)}_{2s,3L}$ to V${}_{E}=0$ V gives an estimated
energy $\sim$ 40 meV above the IX3L ground state (near the X${}^{A}_{2s}$
transition).

Figure 5: Observation of excited states of IX2L and IX3L. a, $V_{E}$
dependence of d2($\Delta$R/R0)/dE2 in a second location of the 2L 2H-MoSe2
sample in the spectral range 1.67 - 1.83 eV. b, d2($\Delta$R/R0)/dE2 in
another location of the 3L 2H-MoSe2 sample in the spectral range 1.58 - 1.8
eV. c Calculated energies of the different exciton states of 2L 2H-MoSe2
including IX2s,2L with same spin and layer configuration as IX2L. d Calculated
energies of the different exciton states in 3L 2H-MoSe2 including IX2s,3L with
the same spin and layer configuration as IX3L.

Finally, we simulate the VE-dependent energy dispersions of the hybridised
ground and excited exciton Rydberg states using the phenomenological coupled
oscillator model previously described. We build on the models used for Figs.
2b and 3b and add the the observed excited Rydberg states, which we assume to
have identical Stark shifts and the same spin- and layer-selective couplings
than their corresponding ground states. Figures 5c and 5d show the simulated
energy dispersions corresponding to the results in Figs. 5a and 5b,
respectively. As can be observed in these figures, the simulated energy
dispersions capture well the $V_{E}$-induced hybridisation and energy
dispersion of the different excitonic transitions (see Suppl. Fig. S15 for the
the simulated absorption spectrum as a function of VE corresponding to Figs.
5b using this model and suppl. section 6 for simulation details), supporting
our interpretation of their different intra-/interlayer origin, ground/excited
state character, and spin and valley configurations.

##  III Summary and Outlook

Our work reports the observation of giant Stark splitting of the interlayer
excitons in 2L and 3L 2H-MoSe2. First, we observe hybridisation between IX2L
and X${}^{A}_{2s}$. In addition to their spectral evolution, a hybridisation
driven g-factor evolution of the coupled excitons is resolved and understood
via a unified coupled oscillator model. The ability to drive the exciton
Zeeman splitting from negative to positive through zero has potential in
electrically tunable valleytronics and spin-dependent exciton-exciton
interactions. Next, we report the giant excitonic trilayer dipole IX3L, which
exhibits distinct spin-layer selection rules for hybridisation with
X${}^{A}_{1s}$. A salient feature of the large IX3L dipole is that it can be
Stark tuned to become the ground state (e.g. lower energy than
X${}^{A}_{1s}$), promising for applications in exciton transport. Finally, by
harnessing the exciton hybridisation affects, we successfully probe the
excited state IX for the first time in a TMD system. Future theory and
experimental effort is encouraged to better understand the excited IX states,
including comparisons with the hydrogen model and the magnetic field and
doping dependence to reveal the g-factor, effective mass Goryca  _et al._
(2019) and Roton-like Liu  _et al._ (2021) properties of the interlayer
Rydberg excitons. Altogether, the results promote a new TMD homostructure
candidate for applications with enhanced exciton-exciton interactions with
strong light-matter coupling. Beyond 3L 2H-MoSe2, a strategy of further
engineering IX dipoles by tuning the layer number, including thicker
multilayer (> 3L) 2H-TMDs or heterostrutures with multilayer TMD components
and hBN spacers, is encouraged.

##  IV Acknowledgements

S.F. and A.C. contributed equally to this work. This work was supported by the
EPSRC (grant nos. EP/P029892/1 and EP/L015110/1), the ERC (grant no. 725920).
S.F. is supported by a Marie Skłodowska-Curie Individual Fellowship
H2020-MSCA-IF-2020 SingExTr (No. 101031596). M.B.-G. is supported by a Royal
Society University Research Fellowship. B.D.G. is supported by a Wolfson Merit
Award from the Royal Society and a Chair in Emerging Technology from the Royal
Academy of Engineering. K.W. and T.T. acknowledge support from the Elemental
Strategy Initiative conducted by the MEXT, Japan (Grant Number
JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and
21H05233). I.C.G acknowledges the CALMIP initiative for the generous
allocation of computational time, through Project No. p0812, as well as GENCI-
CINES, GENCI-IDRIS, GENCI-CCRT for Grant No. A012096649.

*

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