# Review: Spatial inhomogeneities, moiré potential and moiré excitons.

Alberto de la Torre  Quantum Materials and Sensing Institute, Northeastern
University, Burlington, Massachusetts, 01803, USA  Department of Physics,
Northeastern University, Boston, Massachusetts, 02115, USA  Dante M. Kennes
Institute for Theory of Statistical Physics, RWTH Aachen University,and JARA
Fundamentals of Future Information Technology, 52062 Aachen, Germany  Max
Planck Institute for the Structure and Dynamics of Matter, Center for Free
Electron Laser Science, 22761 Hamburg, Germany  Ermin Malic  Fachbereich
Physik, Philipps-Universität Marburg, 35032 Marburg, Germany  Department of
Physics, Chalmers University of Technology, Gothenburg, Sweden  Swastik Kar
Quantum Materials and Sensing Institute, Northeastern University, Burlington,
Massachusetts, 01803, USA  Department of Physics, Northeastern University,
Boston, Massachusetts, 02115, USA  Department of Chemical Engineering,
Northeastern University, Boston, Massachusetts, 02115, USA

###### Abstract

In this short review, we provide an overview of recent progress in deploying
advanced characterization techniques to understand the effects of local
inhomogeneities in moiré heterostructures over multiple length scales.
Particular emphasis is placed on correlating the impact of twist angle
misalignment, nano-scale disorder, and atomic relaxation on the moiré
potential and its collective excitations, particularly moiré excitons.
Finally, we discuss future technological applications leveraging based on
moiré excitons.

###### Contents

  1. I Introduction
  2. II Effect of spatial inhomogeneities on moiré heterostructures
    1. II.1 Determining the twist angle of moiré heterobilayers via non-linear optical techniques
    2. II.2 Nano-imaging of moiré exciton using photoemission microscopes
    3. II.3 Atomic relaxation and its influence on moiré excitons
  3. III Outlook
  4. IV Acknowledgements

##  I Introduction

Figure 1: A subset of advanced characterization techniques deployed to study
the spatial variation of moiré heterostructures. From left to right, in order
of increasing resolution, Second Harmonic Generation (SHG) ($\sim 1\mu$m),
photoelectron momentum microscopy ($\sim 100$ nm), scanning near-infrared
optical microscopy (SNOM, $\sim 20$ nm), and laser scanning tunneling
microscopy and spectroscopy (laser-STM/S, $\sim 1$ pm). Adapted with
permission from Psilodimitrakopoulos  _et al._ (2021); Zhang  _et al._ (2022);
Mogi  _et al._ (2019); Li  _et al._ (2024)

In condensed matter, moiré patterns emerge when two or more atomically thin
layers are artificially stacked with a slight angular misalignment or, in the
case of heterolayers, are stacked featuring a small lattice constant mismatch
Du  _et al._ (2023b). Remarkably, the moiré interference patterns create a
superstructure, the moiré lattice, with a new periodicity that is usually much
larger than the underlying atomic lattices. This superlattice can dramatically
alter the electronic band structure of the material, leading to the formation
of mini-band structures and flat bands that directly affect the transport
behaviour. Some key discoveries in this field of moiréengineering include the
formation of flat bands Bistritzer and MacDonald (2011), promoting the
relative relevance of correlations, but also the engineering of topological
properties Choi  _et al._ (2021), controlling polaritonsZhang  _et al._ (2021)
or networks of solitonsFu  _et al._ (2020) and excitons, the key quasi-
particle we will focus upon in our review. We refer the reader to recent
reviews of moiré physics for an in-depth discussion of moiré physics:Behura
_et al._ (2021); Alexeev  _et al._ (2019); Mueller and Malic (2018); Huang
_et al._ (2022); Ciarrocchi  _et al._ (2022); Zhang  _et al._ (2023a)

Moiré excitons refer to photo-excited electron-hole bound states that form
within (intra-) or between (inter-) layers with the potential defined by the
moiré super-pattern Tran  _et al._ (2019). Moiré excitons exhibit distinct
energy levels and behaviors compared to their semiconductor-monolayer or
three-dimensional counterparts. For a review of exciton physics in transition
metal dichalcogenides and their applications, we refer the reader to Ref.
Mueller and Malic (2018). Moiré excitons can host a rich tapestry of phenomena
with valuable insights into novel quantum phenomena and many-body effects in
low-dimensional systems, including exciton condensation, superfluidity, and
topologically protected statesShimazaki  _et al._ (2020); Choi  _et al._
(2021). Moiré excitons also offer opportunities for tuning the electronic band
structure of materials, enabling tailored properties for semiconductors,
insulators, and topological materials, paving the way for technological
innovationCiarrocchi  _et al._ (2022). Additionally, they contribute a new
dimension to valleytronics, allowing for control over valley polarization and
dynamics in van der Waals heterostructuresUnuchek  _et al._ (2019); Jin  _et
al._ (2019). Furthermore, moiré excitons play a pivotal role in realizing
topological phases and the quantum anomalous Hall effect, leading to the
development of innovative electronic and spintronic devicesLi  _et al._
(2021b). Interactions between moiré excitons and antiferromagnetic ordering
enable the manipulation of spin-dependent properties and spin information in
optoelectronic devices Ramos  _et al._ (2022). Moreover, moiré excitons
present opportunities for controlling carrier dynamics and charge transport
properties in heterostructures, resulting in high-performance optoelectronic
devices with improved efficiency and reduced energy consumption. Excellent,
complementary reviews of moiré excitons are Huang  _et al._ (2022) focusing on
early optical experiments and selection rules and Ciarrocchi  _et al._ (2022)
which highlights device architectures and engineering approaches to tailor
excitons toward future applications.

Controlling and understanding moiré superlattices, arising from the
periodicity of moiré patterns, is fundamental to engineer moiré excitons and
their dynamics phenomena Brem  _et al._ (2020); Merkl  _et al._ (2020);
Meneghini  _et al._ (2024). However, the excitonic properties of moiré
heterostructures are extremely sensitive to local deviations from the ideal
Lego-like description. In this short overview, we embark on a journey through
multiple length scales and how new theoretical descriptions and advanced
characterization techniques (Figure 1) have revealed the effects of twist
angle misalignment, local inhomogeneities, and atomic relaxation in the
physics of moiré excitons.

##  II Effect of spatial inhomogeneities on moiré heterostructures

Excitonic states in moiré van der Waals superlattices offer exciting prospects
for novel optoelectronics and quantum technologiesCiarrocchi  _et al._ (2022).
The strong exciton physics in TMD-based materials has been extensively
investigated using different optical spectroscopy techniques, such as
absorption spectroscopy, photoluminescence and photocurrent measurements, and
optical pump-probe measurements Huang  _et al._ (2022). While these techniques
provide access to spectral and temporal degrees of freedom, they lack
information about the real space properties of excitonic TMDs. However, local
spatial modulations of the moiré superstructure due to strain fields, atomic
defects Guo  _et al._ (2021), and twist angle change extremely influence their
optoelectronic properties. Correlating the averaged optical and transport
properties with these local variations is needed to unambiguously resolve the
role of the moiré energy and length scale. In the following, we review recent
applications of advanced characterization techniques to disentangling emergent
excitonic phenomena from moiré potential disorder in order of spatial
resolution.

Figure 2:  a) Normalized reflections spectroscopy of WS2/MoSe2 bilayers as a
function of twist angle. (b) Schematic of the Brillouin zones for each of the
two ML (red, blue) and the moiré heterostructures (black) for a small twist
angle ($6^{\circ}$) and a large twist angle ($21.8^{\circ}$). (c) Schematic of
the alignment of the main crystallographic directions function of twist
angle,$\theta$, keeping the armchair direction of the bottom ML fixed and
rotating the armchair direction of the top ML. The simulated interference SHG
pattern reflects the twist angle by the rotation of the lobes and the
modulation of the overall intensity. d) Micrograph of MoSe2 (LM) and Graphene
(LG) for different stacking configurations and the associated second-harmonic
generation (SHG) spatial imaging map in the highlighted dashed red box. e)
Normalized phase spectra at different positions in the sample indicating the
excitonic responses for the WSe2, MoSe2 monolayers, and the MoSe2/WSe2
heterobilayer. f) Near field image away from resonance and phase resolve map
as a function of energy revealing the spatial dependence of the excitonic
energies. Panels adapted with permission from Psilodimitrakopoulos  _et al._
(2021); Zhang  _et al._ (2020a, 2023b, 2022)

###  II.1 Determining the twist angle of moiré heterobilayers via non-linear
optical techniques

In the fabrication of heterobilayer devices, the twist angle, the effective
misalignment between the main crystallographic directions of the two stack
monolayers has emerged as a novel degree of freedom that controls the moiré
potential and enables tunability of the optical emission and absorption
spectrum of TMD heterobilayers Huang  _et al._ (2022) (Fig. 2-(a)). For
example, in closely aligned WS2/MoSe2 heterobilayers, corresponding to
$\theta=0^{\circ}$ (AA stacking) and $\theta=60^{\circ}$ (AB stacking), the
almost perfect overlap between the two Brillouin zones leads to direct band
gaps between inter- and intra-layer transitions. However, even small
deviations from the AA or AB stacking lead to large changes in the band
overlap in momentum space. Figure 2-(b) shows the evolution of the lower
($\approx 1.6$ eV) and upper hybrid excitons ($\approx 1.7$ eV) with twist
angle as measured from reflection spectroscopyZhang  _et al._ (2020a)
revealing changes to the excitonic spectrum of more than $100$ meV as a
function of the twist angle. Thus, it is fundamental to precisely determine
the stacking angle between the heterobilayers for future devices based on
moiré excitons.

In this context, the observation of strong non-linear responses dependent on
the twist angle has emerged as a unique and versatile tool to characterize the
formation of the moiré excitons Jin  _et al._ (2019); Tran  _et al._ (2019).
Similarly, changes of symmetry and nonlinear responses associated by the
interlayer coupling and charge transfer associated with the moiré potential
have also been detected via non-linear responses Shree  _et al._ (2021b); Deng
_et al._ (2022); Yang  _et al._ (2020). For strong electric fields, the
interaction of light can be modeled by a resulting polarization given by:

$\begin{split}P_{i}(t)=\varepsilon_{0}(\chi^{(1)}_{i}E_{i}(t)+\chi^{(2)}_{ij}E_{i}(t)E_{j}(t)+\\\
+\chi^{(3)}_{ijk}E_{i}(t)E_{j}(t)E_{k}(t)+...)\end{split}$ (1)

with $\chi^{(}i)$ as the linear and non-linear optical susceptibility tensors
and $\varepsilon_{0}$ as the permittivity of vacuum. These high-order
processes are governed by the crystallographic symmetry and the electronic
band structure, carrying fundamental information about the wavefunctions
involved in the two-photon absorption processMa  _et al._ (2021). Terms with
an even number of powers on the electric field $E$ are only allowed in
inversion symmetry-breaking compounds. The second-order term carries special
importance as it is responsible of second-harmonic generation (SHG), in which
the constructive interference of the two oscillating electric fields with
frequency $\omega$ can give rise to a polarization density at twice the
frequency $P(2\omega)$Zhao  _et al._ (2018). In a normal-incidence SHG
experiment, a single wavelength or broadband light pulse with controlled
polarization is focused onto the sample by a microscopic objective. Emission
from the sample is first collimated by the objective and a short-pass filtered
to eliminate contributions from the fundamental before being focused onto, a
generally fixed, photodiode or CCD camera. The angular dependence of the
intensity of the $2\omega$ response with respect to the main crystallographic
symmetries and the polarization of the incoming and outgoing beam enables the
resolution of the underlying structural point group.

For 2D transition metal dichalcogenide (TMD) monolayers, a strong SHG response
with maxima along the armchair or zig-zag direction can be measured due to
their inversion symmetry-breaking nature (point group $D_{3h}$). When a
heterobilayer is formed, the SHG signal from each layers coherently interferes
resulting in a pattern that is extremely sensitive to small twist angle
misalignment from perfect AA and AB stacking, as shown in Fig.2 Hsu  _et al._
(2014); Psilodimitrakopoulos  _et al._ (2021). Note that a twist angle of
$\theta=60^{\circ}$ results in a completely destructive interference, fully
suppressing the SHG response. Moreover, strong electric dipole SHG from
centrosymmetric bilayer MoS2 with $2-H$ stacking has been observed Shree  _et
al._ (2021a); Zhang  _et al._ (2023b) due to a resonance enhancement
associated with inter- and intra-layer excitons, a phenomenon also expected to
occur in heterobilayers hosting moiré excitons. Indeed, strong twist angle
enhancement has been observed in MoS2/WSe2 devices at the intralayer
hybridized exciton energies Paradisanos  _et al._ (2022). Comprehensive
reviews of the enhancement of non-linear processes in twisted and
heterostructures of TDMs can be found in Ref. Wen  _et al._ (2019); Zhou  _et
al._ (2022).

Optical contrast of the moiré heterobilayers beyond subwavelength resolution
($\sim$ 20 nm) can be achieved by scanning near-field optical microscopy
(SNOM) Vincent (2021). In a scattering SNOM (s-SNOM), the metallic AFM tip
locally confines the electric light field. Thus, the excitonic spectrum of TMD
heterostructures could be resolved by measuring the scattered light amplitude
and phase as a function of position and energy McLeod  _et al._ (2014). s-SNOM
has been recently deployed to monolayers of MoSe2 and WSe2 and MoSe2/WSe2
heterobilayers to extract the excitonic resonance energy and interlayer
hybridization with nanometer resolution Zhang  _et al._ (2022). These early
measurements have open the way for local measurements and control of moiré
excitons. For example, the AFM tip from a SNOM has been used to introduce a
vertical deformation in heterostructures of WSe2/WS2 at cryogenic temperatures
and concomitantly measure a shift on the moiré intralayer exciton as a
function of uniaxial pressure Zhao  _et al._ (2021), as shown in Fig. 2.

Optical characterization techniques with real space imaging capabilities are
thus fundamental to studying the properties of moiré heterostructure and
excitons. Moreover, their large versatility enables a multi-messenger approach
in which the combination of different wavelengths and scanning techniques
characterize the excitation spectrum over a broad range of energies Sternbach
_et al._ (2023).

###  II.2 Nano-imaging of moiré exciton using photoemission microscopes

Figure 3: (a) Schematic on the ARPES process in the exemplary WSe2-MoS2
heterostructure after optical excitation of excitons. A second extreme
ultraviolet laser pulse breaks the exciton, and the ejected electrons are
detected while the holes remain in the material. (b)-(c) Electronic band
alignment in the heterostructure with the formation of interlayer excitons via
electron or hole transfer, depending on the excitation condition.
Energy–momentum cuts of the ARPES signal along the K–$\Gamma$ direction in the
Brillouin zone (d) before and (e) after optical excitation of excitons in the
WSe2-MoS2 heterostructure. The insets show the zoomed-in momentum-space images
around the valence band minimum (f,g) and around the interlayer exciton energy
(h). Note that the interlayer exciton signal has been enhanced by a factor of
50. (i) The ARPES signal from the ejected electrons exhibits the negative
dispersion of the remaining holes. Figures adapted with permission from Ref.
Bange  _et al._ (2024); Karni  _et al._ (2022).

Angle-resolved photoemission spectroscopy (ARPES) is a technique that
explicitly accesses the material-specific electronic band structure Sobota
_et al._ (2021). Its extension to the time domain (tr-ARPES) is sensitive to
the transient dynamics of carriers with momentum resolution. In a tr-ARPES
measurement, a pump pulse, typically in the visible or near-infrared range,
excites electrons at a specific energy in the conduction band. Then, a second
extreme ultraviolet probe pulse is applied to eject electrons from the
material (Fig. 3 (a)) – according to Einstein’s photoelectric effect.
Measuring the kinetic energy and the emission angle of the ejected electrons
allows us to extract the energy and momentum of the initially excited
electrons in the material Karni  _et al._ (2023). Moreover, the successful
development of high-repetition-rate extreme ultraviolet light sources pawed
the way for tr-ARPES measurements on TMD materials. This powerful technique
has been demonstrated to visualize momentum-dark excitons in TMD monolayers
directly Madéo  _et al._ (2020) and the momentum-resolved charge transfer
dynamics in twisted TMD heterostructures Schmitt  _et al._ (2022); Bange  _et
al._ (2024). While the first ARPES studies were performed on exciton dynamics
in bulk TMD materials Bertoni  _et al._ (2016); Wallauer  _et al._ (2020);
Dong  _et al._ (2021); Grubišić Čabo  _et al._ (2015), the more recent
results, enabled by the development of time-of-flight momentum microscopes,
have focused on the exciton physics of TMD monolayers Madéo  _et al._ (2020);
Wallauer  _et al._ (2021); Kunin  _et al._ (2023). Typically, tr-ARPES
experiments do not spatially resolve the carrier dynamics, but they average
over large sample areas of typically 10$\mu$m or more. Thus, they cannot
access spatial inhomogeneities, dielectric disorder, strain gradients, or
local lattice reconstructions. Recently, ultrafast dark field momentum
microscopy has been introduced as a new technique for simultaneous nano-
imaging and nano-spectroscopy of materials Schmitt  _et al._ (2023). The
technique has a temporal resolution of 55 fs and a spatial resolution of
$\approx 100$ nm and has been applied to study the spatial heterogeneity in a
WSe2-MoS2 heterostructure as well as its impact on the energy and the
formation time of interlayer excitons. For a more in-depth discussion of the
techniques’ details we refer the reader to other recent reviews Karni  _et
al._ (2023); Sobota  _et al._ (2021).

For excitonic materials, such as TMDs, tr-ARPES measurements provide access to
exciton binding energies and their momentum in the reciprocal space. Here, the
excitation pulse first creates excitons in the material. In contrast, the
energy of the second pulse is used to break up the Coulomb-bound electron-hole
pairs, photo-ejecting the electrons while leaving a hole density behind (Fig.
3 (a)). The ARPES signal of the photoemitted electrons reflects the excitonic
band structure since the Coulomb-bound electrons and holes present a strongly
correlated system Meneghini  _et al._ (2023). The excitonic ARPES signal
appears at the momentum corresponding to the ejected electron’s reciprocal-
space valley. It is spectrally located one exciton binding energy below the
conduction band. The momentum-resolved shape of the signal is characterized by
the negative curvature of the valence band of the remaining hole. This
negative dispersion from the photoemitted electron is the hallmark of the
excitonic origin of the ARPES signal Christiansen  _et al._ (2019); Man  _et
al._ (2021); Meneghini  _et al._ (2023).

Figure 4:  (a) Theoretical prediction for the hybrid exciton dynamics in the
MoS2 homobilayer showing the excitation of the nearly purely intralayer KK
excitons (0 fs), the formation of strongly hybridized $\Gamma_{\text{hyb}}$K
excitons (200 fs), and the thermalized hybrid exciton distribution (600 fs),
respectively. The dashed lines denote the shifted split valence bands of the
hybrid hole at the K and the $\Gamma$ point. (b) Momentum-integrated energy-
distribution for the heterostructure and in comparison also for the WSe2
monolayer, respectively. (c) Charge-transfer dynamics from optical excitation
of excitons in the WSe2 monolayer (K${}_{\text{W}}$K${}_{\text{W}}$, orange
line) via phonon-driven scattering into the hybrid dark excitons
(K${}_{\text{W}}$$\Lambda_{\text{hyb}}$, grey line) to the energetically
lowest interlayer exciton states (K${}_{\text{W}}$K${}_{\text{Mo}}$, red
line). The points represent experimental data and solid lines the theoretical
prediction. (d) Direct comparison of the charge transfer dynamics when the
heterostructure is excited resonant to the excitons either in the WSe2 layer
(green circles, 1.7 eV) or in the MoS2 layer (black circles, 1.9 eV). While
the first excitation scenario leads to an electron transfer only, the latter
gives rise to a combined electron- and hole-transfer. (e) Time evolution of
the energy distribution filtered at the momentum region of the K valley in
MoS2 layer for the excitation at 1.9 eV. As the interlayer exciton is formed,
the photoelectron energy shows an unintuitive blue-shift. (f) Spatial heatmaps
illustrating the variation of the interlayer exciton (ILX) energy and the
energy difference $\Delta E_{\text{exp}}$ to the optically excited intralyer
exciton, respectively. (g) The interlayer exciton formation time plotted as a
function of the energy difference $\Delta E_{\text{exp}}$. The inset shows a
theoretical prediction confirming qualitatively the measured correlation
between the formation time and the energy difference. Figures adapted with
permission from Ref. Schmitt  _et al._ (2022, 2023); Meneghini  _et al._
(2023); Bange  _et al._ (2024)

In the following, we first review the exciton dynamics in TMD homo- and
heterobilayers, which have attracted huge interest in current research. The
latter exhibits a type-II band alignment favoring the tunneling of an electron
or hole into the opposite layer and resulting in the formation of spatially
separated interlayer excitons Yong  _et al._ (2019), where the Coulomb-bound
electrons and holes are located in different layers (Fig. 3 (b)-(c)). Finally,
we discuss how the deployment of ultrafast dark field momentum microscopy has
revealed local inhomogeneities of the dark excitons.

ARPES signatures of the electron and hole bound in an interlayer exciton in a
twisted WSe2-MoS2 heterostructure have been observed [Fig. 3(d)-(i)] thanks to
the high signal-to-noise ratio and high-quality samples with narrow initial
valence band line widths Karni  _et al._ (2022). The unexcited band structure
of the heterostructure is characterized by two spin-split valence band maxima
of the WSe2 layer at the K valley and a single broadband at lower energies
corresponding to the valence band of the MoS2 layer (Fig. 3 (d)). After
optically exciting the material resonant at the intralayer exciton in WSe2,
clear signatures of interlayer excitons about 1 eV above the valence band
maximum are found (Fig. 4 (e)). Interestingly, the signal around the WSe2
valence band maximum becomes slightly depleted at the K and K’ points (Fig. 4
(f)). This was assigned to the holes created in WSe2 during the
photoexcitation. Finally, the interlayer exciton signature at 1 eV exhibits a
negative dispersion resembling the negative curvature of the WSe2 valence band
(Fig. 3(i)). This anomalous dispersion has been theoretically predicted
Christiansen  _et al._ (2019) and is a hallmark of the excitonic origin of the
observed ARPES signal.

Recently, it was shown that tr-ARPES is an ideal technique to visualize hybrid
exciton states directly Meneghini  _et al._ (2023). Tunneling between two TMD
layers leads to a mixture of intra- and interlayer exciton states, resulting
in hybrid excitons that inherit properties of both exciton species. Here,
electrons and/or holes are in a superposition of both layers. The formation of
such hybrid states is facilitated in naturally stacked TMD homobilayers,
whereas the dominating excitons in TMD heterostructures are mostly of
interlayer character. Thus, the performed study focuses on MoS2 homobilayers,
where hybridized momentum-dark $\Gamma_{\text{hyb}}$K excitons are the
energetically lowest states. Here, the hole at the $\Gamma$ valley is strongly
delocalized between the layers. During the ARPES process, the hybrid exciton
breaks up into an ejected free electron from the K valley and a superposition
of hybrid holes. This gives rise to a double-peak signal reflecting the
superposition of the remaining holes between the two hybrid valence bands at
the $\Gamma$ point.

Figure 4 (a) illustrate how the momentum- and energy-resolved ARPES signal
evolves in time. During the optical excitation, the spectrum is characterized
by a pronounced single peak stemming from the optically excited intralayer
exciton (Fig. 4 (a)). On a sub-100 fs timescale, these excitons scatter via
emission of phonons to the energetically lower hybridized
$\Gamma_{\text{hyb}}$K excitons and the ARPES signal shows signatures of both
KK intralayer and hybridized $\Gamma_{\text{hyb}}$K excitons. After approx.
600 fs, an equilibrium distribution is reached with $\Gamma_{\text{hyb}}$K
excitons carrying nearly the entire population. Interestingly, their ARPES
signature is characterized by two peaks, one slightly above the KK exciton and
one red-shifted by more than 600 meV.

Figure 4-(b) shows the momentum-integrated energy distribution of the tr-ARPES
signal in WSe2-MoS2 heterostructures Bange  _et al._ (2024). The highest
contribution at early delay times occurs at around 1.7 eV (orange dashed line)
corresponding to the energy of the optically excited
K${}_{\text{W}}$K${}_{\text{W}}$ intralayer excitons in the WSe2 layer.
Interestingly, one observes a signal transition to a second long-lived peak at
the lower energy of about 1.1 eV (red dashed line) occurring on a femtosecond
timescale. This peak has been identified as the photoemitted electronic
contribution of the K${}_{\text{W}}$K${}_{\text{Mo}}$ interlayer exciton.
Here, the first/second capital letter denotes the position of the
hole/electron in the reciprocal space (K, $\Lambda$ valley), while the
subindex stands for the layer (W or Mo). To support this assignment, the same
measurement was performed on a WSe2 monolayer. No spectral weight is observed
in the momentum and energy space region associated with the interlayer
excitons.

The observed charge transfer dynamics, Fig. 4(c), involves a phonon-driven
two-step scattering process Meneghini  _et al._ (2022). The optically excited
intralayer K${}_{\text{W}}$K${}_{\text{W}}$ excitons in the WSe2 layer cannot
efficiently scatter directly to the energetically lowest
K${}_{\text{W}}$K${}_{\text{Mo}}$ interlayer exciton states (hole at the K
valley of the WSe2 layer and electron in the K valley of the MoSe2) [Fig.
3-(b)]. The K${}_{\text{W}}$K${}_{\text{W}}$ excitons rather scatter first
with phonons to the momentum-dark K$\Lambda$ excitons, where the electrons at
the $\Lambda$ valley are delocalized over both layers. In a second step, these
hybridized excitons scatter into the K${}_{\text{W}}$K${}_{\text{Mo}}$
interlayer excitons, where the electron has been transferred to the MoSe2
layer. The direct scattering process is prevented by the weak wave function
overlaps at the K valley. In contrast, the electronic wave function at the
$\Lambda$ valley has large contributions at the outer selenium atoms,
resulting in much more efficient hybridization of K$\Lambda$ states Merkl  _et
al._ (2020).

Tr-ARPES has also revealed the hole transfer process mechanism across the
WSe2-MoS2 interface Meneghini  _et al._ (2023). The
K${}_{\text{Mo}}$-K${}_{\text{Mo}}$ intralayer excitons in the MoS2 layer can
be resonantly excited at 1.9 eV (Fig. 4-(d)). However, this still leads to a
partial electron transfer due to the off-resonant excitation of the
energetically lower K${}_{\text{W}}$-K${}_{\text{W}}$ excitons (1.7 eV) in the
WSe2 layer. To distinguish the impact of the electron and hole transfer, the
temporally resolved formation of K${}_{\text{W}}$-K${}_{\text{Mo}}$ interlayer
excitons have been studied for both scenarios, i.e. excitation at 1.7 eV (only
electron transfer expected) and excitation at 1.9 eV (electron and hole
transfer expected). For the electron-only transfer process, the interlayer
excitons are formed quickly ($\sim$ 35 fs), and the ARPES signal saturates on
the sub-200 fs timescale (Fig. 4-(d)). In contrast, the joint electron- and
hole-transfer is found to be much slower, with a formation time of
approximately 3 ps. This can be traced back to the different relaxation
pathways of optically excited K${}_{\text{Mo}}$-K${}_{\text{Mo}}$ and
K${}_{\text{Mo}}$-K${}_{\text{Mo}}$ excitons. The degree of the hybridization
of the intermediate hybrid states and the energetic offset to the initial and
final states determine the efficiency of the charge transfer process. While
the first scatter via hybridized $\Gamma_{\text{hyb}}$-K${}_{\text{Mo}}$
excitons, the latter scatter via hybridized
K${}_{\text{W}}$-$\Lambda_{\text{hyb}}$ states towards to energetically lowest
K${}_{\text{W}}$-K${}_{\text{Mo}}$ interlayer excitons.

Assuming that ARPES only provides information on the ejected electrons, one
would not expect any difference whether the measured photoelectrons stem from
the break-up of the intralayer K${}_{\text{Mo}}$-K${}_{\text{Mo}}$ or
interlayer K${}_{\text{W}}$-K${}_{\text{Mo}}$ excitons as the electron remains
in the MoSe2 layer, while the hole transfers across the interface (Fig.
4-(d)). Interestingly, one finds a non-intuitive blue shift of about 170 meV
of the ARPES signal for increasing delay times (Fig. 3-(e)). During this time,
the optically excited intralayer excitons scatter into the energetically lower
interlayer states, so one would rather naively expect a red-shift in the ARPES
signal. Auger processes and energy renormalization could be excluded as
possible explanations for the unexpected blue-shift. The unintuitive shift
towards higher energies is a hallmark of the correlated nature of the
electron-hole pairs.

Figure 5:  (a) Large STM topography in a $1.7^{\circ}$ WSe2/MoSe2
heterostructure showing a spatially dependent moiré periodicity. (b)
Calculation of the stacking energies for the relaxed atomic energy reflecting
the relation between the moiré pattern and strain. (c) Schematic of the laser-
STM to measrue a near-58∘ twisted WS2 bilayer sample. A bias between the STM
tip and the sample in the presence of a 520 nm continuous wave laser enables
measurements of the photocurrent with near-atomic resolution. (d) Illustration
of the possible local stacking structures in twisted WS2. (e) Calculated
topographic variation twisted WS2 bilayers at $B^{W/W}$ and AB stacking sites.
(f) Comparison of the experimentally observed spatial dependence of the charge
transfer excitonic photocurrent sign and (g) GW-Bethe–Salpeter equation
calculations. (h)-(j) Intralayer exciton lifetime dependence at the top and
bottom of a strain induced ripple in WS2. Adapted with permission from Shabani
_et al._ (2021); Li  _et al._ (2024); Mogi  _et al._ (2022)

Typically, tr-ARPES experiments do not spatially resolve the carrier dynamics,
but they rather average over large sample areas of typically 10$\mu$m or more.
Thus, they cannot access spatial inhomogeneities, dielectric disorder, strain
gradients, or local lattice reconstructions. Recently, ultrafast dark field
momentum microscopy has been introduced as a new technique for simultaneous
nano-imaging and nano-spectroscopy of materialsSchmitt  _et al._ (2023). The
technique has a temporal resolution of 55 fs and a spatial resolution of 500
nm and has been applied to study the spatial heterogeneity in a WSe2-MoS2
heterostructure as well as its impact on the energy and the formation time of
interlayer excitons. Interestingly, distinct spatial heterogeneity is found
even in seemingly flat areas of the sample exhibiting high-quality
photoemission spectra. Positioning the dark-field aperture on the momenta of
the interlayer excitons provides access to the spatially and spectrally
resolved evolution of the interlayer exciton energy. The nanoscale variation
of the latter as well as its difference $\Delta E_{\text{exp}}$ to the energy
of the optically excited intralayer exciton is shown in spatial heat maps in
Fig. 4-(f). Interestingly, the experiment reveals a correlation between the
interlayer exciton formation time $\tau$ and the energy difference $\Delta
E_{\text{exp}}$, i.e. $\tau$ gets larger with the increasing $\Delta
E_{\text{exp}}$ (Fig.4-(g)). This observation is qualitatively confirmed by
the many-particle theory (inset), which could trace this correlation back to
nanoscale variations of the strength of interlayer hybridization in the
WSe2-MoS2 bilayers. This results in a changed efficiency for the phonon-driven
charge transfer to the intermediate hybridized states and thus has a direct
impact on the formation time of interlayer excitons.

###  II.3 Atomic relaxation and its influence on moiré excitons

The emergence of two-dimensional van der Waals materials has raised the
intriguing idea of mixing and matching layers of different materials to
realize novel properties in a given stack very early on. With the rush to
build new Lego-like heterostructures in which to combine the vastly different
electronic and magnetic ground states Geim and Grigorieva (2013), two
fundamental real space properties of van der Waals materials have been
predominantly ignored: (i) their lattice constants are extremely dependent on
local changes of composition and (ii) the atomistic positions in van der Waals
materials are not rigid, but rather follow a membrane-like behavior. The
second point becomes most prominent in large moiré unit cells formed by small
twist angles between adjacent layers or by small lattice constant mismatches
in heterostructures. In this scenario, the relaxation of atomistic positions
in and out of the plane can minimize the energy of the system, and the
reconstructed atomistic registry might be very different from that expected
for unrelaxed, rigid heterobilayers with severe consequences for electronic,
excitonic and mechanical properties Carr  _et al._ (2018a); Halbertal  _et
al._ (2022); Susarla  _et al._ (2022). Indeed, today, it is known that atomic
corrugation plays a crucial role in the celebrated flat bands and, in
particular, the energy gaps of twisted bilayer graphene at the magic angle
Uchida  _et al._ (2014); Lucignano  _et al._ (2019). Recently, even the role
of moiré phonons, as vibrational stacking re-arrangements, have been
highlighted as an unconventional way to boost exciton diffusion Rossi  _et
al._ (2023). Thus, it is fundamental to characterize experimentally the local
stacking arrangements. Piezoresponse force microscopy (PPM) is an extension of
atomic force microscopy that has emerged as a universal and easily implemented
technique to map the real space variation of the moiré potential McGilly  _et
al._ (2020). PPM measures the amplitude and phase of the electromechanical
response of a material to local deformations induced by an AC bias between its
surface and the AFM tip. PPM maps the moiré superlattice by recording the
changes in the flexoelectric response of heterobilayers associated with the
inherent strain gradients of moiré heterostructures. This manifestation of the
moiré potential as polar domain walls in a non-polar background is
irrespective of the inversion symmetry of the constituent mono-layers McGilly
_et al._ (2020).

Figure 6:  (a) Schematic comparison of the lego-like and continuous layer-
dependent interpretation of 2D heterostructures. (b)-(c) STM topography of 3
and 8 ML, respectively, of PdTe2 on Bi2Se3 heterostructures. (d)-(e)
Corresponding calculated moiré pattern after. (f) Calculated moiré potential
wavelength dependence on the number of PdTe2. Adapted with permission from
Halbertal  _et al._ (2023)

While scanning tunneling microscopy and spectroscopy experiments (STM/S) lack
the momentum resolution to resolve the reciprocal space information of the
electronic band structure of moiré heterobilayers, STM/S has emerged as the
only techniques with the required atomic and energy resolution to observe
changes to the excitonic landscape due to atomic relaxation. For the STM/STS
technique and its application to quantum materials, we refer the reader to
other previous reviews Bian  _et al._ (2021); Yin  _et al._ (2021). Early STM
results on moiré heterostructures including Ref. Zhang  _et al._ (2017,
2020b); Li  _et al._ (2021a) were reviewed in Ref. Huang  _et al._ (2022). We
now discuss examples of the effects of local strain in the electronic
structure of moiré heterobilayers. In small-angle heterostructures of
WSe2/MoSe2 STM experiments revealed spatially varied moiré period up ranging
from $5$ to $~{}17nm$ wavelengths over a $500\times 500$ nm2 Shabani  _et al._
(2021). The resulting strain associated with the moiréstructure induces
systematic large variations on the moiré potential depths and local changes to
the near $E_{F}$ states, see Fig. 5. In $3^{\deg}$ twisted bilayers of WS2 STS
revealed the existence of flat bands associated with localized states that
cannot be accounted for with first-principle calculations, but that can be
recovered when the atomic displacements due to strain are includedMolino  _et
al._ (2023). STM has also been deployed to observe changes of more than $10\%$
in the moiré period of heterostructures of MBE grown multilayer PdTe2 on
Bi2Se3 (Figure 6) Halbertal  _et al._ (2023).

In studying intra- and interlayer moiré excitons, it is also vital to
understand and characterize the precise atomistic registry as it will modify
the specific interactions and relaxation pathways. Laser-STM experiments in
which STM/STS measurements are performed while a continuous-wave laser shines
on the sample, enabling resolving the electronic structure of moiré excitons
with sub-nm resolution. Laser-STM experiments in twisted bilayer WS2 revealed
nanoscale modulation in the moiré excitons Li  _et al._ (2024) (Figure 5). The
STM topography maps in these heterostructures are characterized by a moiré
pattern with a 9 nm period. STS measurements show clear differences in the
electronic band structure at BW/W and AB stacking sites reflecting the deep
moire potential in twisted WS2. Upon turning the laser on, the emerging
excitonic photocurrent displays different behavior for both sites. When the
bias voltage between tip and sample is set in the region $-2V<V_{bias}<1V$,
the photocurrent is concomitantly positive in the BW/W sites and a negative in
the AB stacking sites (Figure 5), indicating a dominant electron and hole
contribution respectively. This phenomenology was explained by GW-
Bethe–Salpeter equation calculations, including the moiré potential and
electron-hole Coulomb interactions, which indicate the formation of in-plane
charge transfer (ICT) moiré excitons. Thus,the STM tip-position photocurrent
can be explained as a result of the lateral separation of electrons and holes
in an ICT excitons Li  _et al._ (2024).

Optical pump-probe STM setups with picosecond resolution could enable studying
the atomic dependence of exciton dynamics Liang  _et al._ (2023). The
challenge in tr-STM measurements is to remove thermal fluctuations associated
with the optical pump-induced expansion of the atomic tip and sample, which
can result in large artifacts in the tunneling currents. The development of
shaken-pulse-pair-excited STM, in which the pump beam is electronically
modulated by an external trigger acting as the reference signal for lock-in
amplifier detectionMogi  _et al._ (2019); Takeuchi  _et al._ (2002), has
enabled the required high-signal-to-noise ratios and time resolution to study
non-equilibrium dynamics with atomic resolution Iwaya  _et al._ (2023).
Recently, tr-STM has been deployed to study the spatial dependence exciton
dynamics in $7.6^{\circ}$ heterobilayers of WS2/WSe2 Mogi  _et al._ (2022)
near grain boundaries and strain-induced ripples (see Figure 5). The authors
revealed a $\times 10$ larger exciton lifetime between the top and bottom of
the ripple, suggesting that interaction with the substrate phonon modes can
lead to shorter excitonic lifetimes. Similarly, substrate-induced strain near
grain boundaries also induced faster excitonic relaxation. The effect
diminishes rapidly in space, with measurements $~{}8$ nm away from the grain
boundary, where long exciton lifetimes are recovered.

Theoretical modeling of the atomic relaxation requires taking the multi-
layered nature of vdW heterostructures into account properly, as it has been
shown that the strain fields emerging from moiré defects can penetrate deep
into a vdW material stack Cook  _et al._ (2022); Halbertal  _et al._ (2023).
However, a theoretical description of vdW heterostructures with large moiré
periods and/or many layers on the atomic scale is computationally out of
reach, and as a consequence, sophisticated coarse-grained models were
developed that address atomic relaxation at large length scales. If the moiré
lattice spans many unit cells of a single layer, the local atomic stacking
registry and in- and out-of-plane strain fields, can be described by
generalized forces acting on a stack of membranes (i.e., one models each
single layer as one of these membranes) Carr  _et al._ (2020, 2018b);
Halbertal  _et al._ (2023); Zhu  _et al._ (2020); Cazeaux  _et al._ (2020);
Halbertal  _et al._ (2021). In such membrane models, each layer of the vdW
stack is allowed to relax individually. The total energy functional is then a
sum of the generalized stacking fault energy, which accounts for lattice
mismatch at any of the stack’s interfaces, and the elastic energy, which
accounts for the in- and out-of-plane strain of each of the membranes. The
parameters for both the elastic and the stacking fault energy are readily
obtained from _ab-initio_ simulations of the individual constituent layers and
interfaces Halbertal  _et al._ (2023). The multi-layered relaxation process in
vdW heterostructures can heavily affect electronic and, thus, excitonic
properties. Therefore, the theoretical modeling of relaxation, e.g., by the
aforementioned membrane models, is an essential ingredient also in describing
these properties in moiré structures even far away from the interface
Halbertal  _et al._ (2023); Mullan  _et al._ (2023).

These experimental and theoretical studies exemplify that it is essential to
understand the relaxation mechanisms of moiré heterostructures toward fully
understanding their emergent excitonic properties. In a second step one might
consider using long-ranged relaxation even for engineering-by-stacking
purposes.

##  III Outlook

New applications are emerging as our understanding of moiré physics
progresses. In the following, we highlight a few recent applications:

Optics, photonics, and optoelectronic devices: moiré excitons have been shown
to play a pivotal role in developing tunable optoelectronic devices across
various heterostructures and materials Behura  _et al._ (2021); Guo  _et al._
(2021); Brem  _et al._ (2020); Liu  _et al._ (2022). Moiré excitons serve as a
platform for optoelectronic modulation, offering efficient control over light
absorption, emission, and transmission essential for devices like optical
switches, photodetectors, and modulators, ensuring high-speed operation and
low energy consumption Ramos  _et al._ (2022); Yu  _et al._ (2017). For
example, moiré excitons enhance device efficiency in WS2/graphene
heterostructures by promoting efficient carrier extraction and utilization
with high sensitivity and signal-to-noise ratios Trovatello  _et al._ (2022).
Integration of moiré excitons into integrated photonic circuits enables the
development of compact, scalable systems for telecommunications, sensing, and
quantum information processing. Specifically, MoS2/FePS3 heterostructures have
potential for new integrated photonicsDuan  _et al._ (2022), offering
multifunctional devices with advanced performance and functionality Onga  _et
al._ (2020). Novel excitonic devices exploiting strong light-matter
interactions and long-range effects will result in novel excitonic
transistors, photodetectors, LEDs, and solar cells with enhanced efficiency
and tunability.

Quantum Information Processing: The long coherence times and strong
interactions of moiré excitons position moiré devices as promising candidates
for quantum information processingHuang  _et al._ (2022). Moreover, moiré
excitons serve as valuable sources of quantum light, capable of generating
non-classical light states like single photons and entangled photon pairs Baek
_et al._ (2020); Yu  _et al._ (2017). Researchers aim to leverage these
characteristics for quantum computing, communication, and cryptography,
harnessing coherence and entanglement properties to encode and process quantum
information effectively. Their utilization in semiconductor moiré
superlattices, particularly in Kekulé/moiré superlattices, offers new
prospects for practical quantum devicesYe  _et al._ (2023). Additionally,
moiré excitons show potential in magnetic field sensing applications,
especially in antiferromagnet-semiconductor heterostructuresOnga  _et al._
(2020). Their sensitivity to external magnetic fields enables high-resolution
and sensitive detection, making them suitable for various magnetic imaging,
navigation, and magnetometry applications.

Other devices and applications: moiré excitons offer significant potential for
enhancing the efficiency and functionality of various energy and detection
devices Ciarrocchi  _et al._ (2022); Du  _et al._ (2023a). Their strong light-
matter interaction in energy harvesting and photovoltaics allows for efficient
photon absorption, leading to higher photoconversion efficiencies Duan  _et
al._ (2022). Additionally, their tunability enables optimization across
different wavelengths, making them suitable for diverse photonic applications.
In broadband photodetection, MoSe2/FePS3 heterostructures demonstrate a broad
spectral response, enhancing versatility for applications like imaging and
spectroscopy Onga  _et al._ (2020). These photodetectors operate self-driven
due to efficient carrier generation and transport, resulting in low power
consumption and enhanced portability. Moreover,moiré exciton-based
photodetectors exhibit fast response times and low noise performance, crucial
for high-speed data acquisition and sensitive detection of weak optical
signals in noisy environments. Enhanced light-matter interaction in moiré
excitons, particularly in Kekulé/moiré superlattices and 2D TMD materials,
improves optical absorption and emission properties, benefiting applications
like optical sensing and imaging Ye  _et al._ (2023). Furthermore, moiré
excitons facilitate efficient hot carrier transfer in WS2/graphene
heterostructures, enabling rapid carrier dynamics and high-speed device
operation essential for photodetectors and communication systems.

These applications and recent understanding of the spatial variation of moiré
require a new paradigm for understanding the effects of spatial
inhomogeneities and atomic relaxation on the electronic properties of moiré
devices. A tandem effort of new advanced characterization techniques and new
machinery to calculate the structural variation of the moiré potential and
collective excitations is needed towards new excitonic devices. For an
extensive review of the applications of moiré optoelectronics we refer the
reader to Ref. Du  _et al._ (2023b) and Ref. Ciarrocchi  _et al._ (2022).

##  IV Acknowledgements

We thank Lennart Klebl for his valuable input. D.M.K gratefully acknowledges
support from the DFG through SPP 2244 (Project No. 443274199) and by the Max
Planck-New York City Center for Nonequilibrium Quantum Phenomena. E.M. has
received funding from the Deutsche Forschungsgemeinschaft (DFG) via CRC 1083.
SK acknowledges support provided by the National Science Foundation through
the ExpandQISE award No. 2329087 and the Massachusetts Technology
Collaborative through award number 22032.

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