# Collective charge excitations between moiré-minibands in twisted WSe2
bilayers from resonant inelastic light scattering

Nihit Saigal  Institute of Physics, University of Münster, Wilhelm-Klemm-Str.
10, 48149 Münster, Germany  Lennart Klebl  Institute of Theoretical Physics,
University of Hamburg, Notkestrasse 9, 22607 Hamburg, Germany  Hendrik Lambers
Institute of Physics, University of Münster, Wilhelm-Klemm-Str. 10, 48149
Münster, Germany  Sina Bahmanyar  Institute of Physics, University of Münster,
Wilhelm-Klemm-Str. 10, 48149 Münster, Germany  Veljko Antić  Institute of
Physics, University of Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany
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  Tim O.
Wehling  Institute of Theoretical Physics, University of Hamburg, Notkestrasse
9, 22607 Hamburg, Germany  The Hamburg Centre for Ultrafast Imaging, 22761
Hamburg, Germany  Ursula Wurstbauer  Institute of Physics, University of
Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany  Center for Soft
Nanoscience (SoN), Busso-Peus-Str. 10, 48149 Münster, Germany  wurstbauer@uni-
muenster.de

###### Abstract

We establish low-temperature resonant inelastic light scattering (RILS)
spectroscopy as a tool to probe the formation of a series of moiré-bands in
twisted WSe2 bilayers by accessing collective inter-moiré-band excitations
(IMBE). We observe resonances in such RILS spectra at energies in agreement
with inter-moiré band (IMB) transitions obtained from an _ab-initio_ based
continuum model. Transitions between the first and second IMB for a twist
angle of about $8^{\circ}$ are reported and between first and second, third
and higher bands for a twist of about $3^{\circ}$. The signatures from IMBE
for the latter highlight a strong departure from parabolic bands with flat
minibands exhibiting very high density of states in accord with theory. These
observations allow to quantify the transition energies at the K-point where
the states relevant for correlation physics are hosted.

Twisted van der Waals (vdW) bilayers present an unique condensed matter
platform to realize and control electronic correlation effects [1, 2, 1]. The
large scale superlattice created by the superposition of the two layers at a
slight rotational mismatch defines a reciprocal mini-Brillouin zone with
nearly dispersionless (flat) bands as long as the layers hybridize
sufficiently. The drastically reduced kinetic energy, potentially together
with van Hove physics, results in a very high density of states (DOS) in those
flat bands favorable for driving electrons into the correlated regime. In
several graphene-based systems correlated and ordered electronic phases are
experimentally well-established [6, 7, 8, 9, 10]. In transition metal
dichalcogenide (TMDC) based vdW stacks the absence of complications like
topological obstructions have facilitated high-level microscopic many-body
studies from early on [11] and explained the emergence of ordered, insulating
and also different flavors of correlated metallic states of matter [12, 13].
Experimentally, the aforementioned correlation effects have been realized [14,
15], while superconductivity has remained elusive with currently one report of
an (unclear and controversial) zero resistance state [14] even though there
are multiple theoretical studies [16, 17, 18]. These emergent phases in
semiconductor-based moiré bilayers are attributed to strongly interacting
electronic K/K′-states.

Figure 1:  (a) Scheme of the hBN encapsulated tWSe2 bilayer. (b) Stick-ball
model of a moiré- superlattice from a tWSe2 bilayer with the moiré unit cell
indicated. (c) RILS spectra on a 3o tWSe2 bilayer. Spectra are offset for
clarity. The A1g, E${}^{1}_{2g}$ phonons are indicated by a black arrow. The
broad RILS signal are collective electronic inter-moiré band excitations
(IMBE). The vertical red solid lines mark resonance energies from
theoretically predicted transition energies. The tilted cyan line indicate
emission signatures. [T = 4K, PLaser = 1 mW]. (d) Calculated energy dispersion
of the 7 highest moiré mini bands around the K, K’ states for a 3o tWSe2
bilayer. The minibands are labelled m1, m2, …m7. Exemplary inter-miniband
transitions are sketched.

In this letter, we experimentally access moiré minibands at the valence band
maximum (VBM) around the K/K′ valley by studying their collective electronic
inter moiré band excitations (IMBE) by resonant inelastic light scattering
(RILS) experiment as summarized in Fig. 1. Accessing the moiré bands at the
K-points is challenging for the combined reason of twist angle variations and
reconstruction in realistic devices together with the VBM at the $\Gamma$ and
K-points being close in energy [19]. The morphology of twisted bilayers,
particularly at small twist angles, is such that variations in twist angle,
but also reconstruction (in plane as well as corrugation) plays an important
role. This leads to periodically patterned areas different to those expected
from a rigid moiré lattice-picture [20, 21, 22, 23]. While these patterns
result in rich optical interband spectra [23] and even host coherent many-body
states of excitons [24], it makes the interpretation of spectroscopic
signatures and the direct spectroscopy of moiré bands e.g. by angle-resolved
photoemission spectroscopy (ARPES) difficult with only a few reports on
selected materials combinations [25, 26]. Recent $\mu-$ARPES and STM studies
demonstrate the formation of moiré-bands at the valence band maximum (VBM) at
the $\Gamma-$point of twisted WSe2 bilayers [27, 22, 28]. Accessing moiré
bands around the K,K’ states and their properties is still lacking.

We remedy this standing problem by reporting on collective electronic
excitations between moiré bands formed in twisted and hBN encapsulated WSe2
bilayers (Fig. 1(a)) at the VBM around the K/K’ states by resonant excitation
at the K/K′-point. In this way IMBEs at the K/K′ VBM can be probed by RILS
spectroscopy at low temperatures providing experimental access to the moiré
minibands caused by the geometrical superlattice in twisted bilayers as
sketched in Fig. 1(b). Typical RILS spectra taken on a $\Theta\thickapprox
3^{\circ}$ twisted WSe2 bilayer at $T=4$K are shown in a water fall
representation in Fig. 1(c). A rich spectrum of rather wide and dispersive
RILS modes in addition to the sharp nearly degenerate optical active A1g,
E${}^{1}_{2g}$ phonon modes appears at an energy range between $\thickapprox$
20mev and 120meV. Those excitation bands are in very good quantitative
agreement with suitable IMB transitions extracted from the calculated
electronic bands in the mini-BZ (mBZ) in the vicinity of the K/K′ points. We
obtain the band structures of tWSe2 from a continuum model [1, 2, 3, 4] with
parameters adjusted to _ab-initio_ simulations (for details see SI [32]). Fig.
1(d) summarizes the first seven moiré minibands with examples of vertical IMB
transitions starting from the Fermi surface in the highest moiré band m1.

We find that at the highest VBM minibands are rather flat with low energy
separation of a few tens of meV for a twist angle around $3^{\circ}$, while
the bands become nearly parabolic with a separation of several tens of meV for
a twist angle of about $8^{\circ}$ as will be discussed in more details below.
Combined theoretical and experimental efforts, probing the moiré-bands via
collective single-particle-like IMBE by RILS spectroscopy analogue to
intersubband excitations in 2D charge carrier systems hosted e.g. in GaAs
quantum wells [33, 34, 35] provids a promising approach to study the band
structure of twisted transition metal dichalcogenides at twist angles where
correlation physics play an important role [1, 12, 13, 16, 17, 18].

The hBN encapsulated tWSe2 bilayers in this study have been prepared by
micromechanical cleavage using adhesive tape and viscoelastic dry transfer on
top of Si/SiO2 substrates using micro-manipulators with an estimated twist
uncertainty of about $\pm 0.5^{o}$. To check for sufficient interlayer
coupling and twist angle we employ non-resonant Raman spectroscopy (see SI
[32]). An optical micrograph of a 3o tWSe2 bilayer is displayed in inset of
Fig. 2. Three different types of WSe2 samples have been prepared with a twist
of 3o, 8o and a natural homobilayer and characterized by low-temperature non-
resonant Raman and photoluminescence (PL) measurements. For all measurements,
the sample are mounted on the cold-finger of a closed-cycle refrigerator at a
temperature of $T=4$K. Position control is provided by x-y-z piezo actuators.
The light from either a green solid state laser (2.33 eV) or a continuously
tunable Ti:sapphire laser (linewidth of about 50 kHz) is focused with a
cryogenic large-NA (NA = 0.82) objective lens to spot size of less than
2$\mu$m. The emitted/scattered light is guided to the entrance slit of a
triple grating spectrometer.

Figure 2:  (a) PL spectrum shows weak emission from direct K/K′ excitons (XKK)
and a intense broad emission band from dark, charged and defect bound
excitons. The spectral range around (XKK) utilized for RILS spectroscopy is
grey shaded and the region where modes in RILS are expected red shaded [T =
4K, PLaser = 1mW, ELaser = 2.33eV]. Inset: Optical micrograph of a hBN
encapsulated 3o tWSe2 sample with active region marked marked by black line.
[scale bar 20$\mu$m]

A typical PL spectrum for 3o tWSe2 is displayed in Fig. 2. The emission is
dominated by a broad asymmetric emission band peaking at around 1.65eV
assigned to momentum and/or spin forbidden transitions from dark excitons that
can only couple to light by interaction with suitable phonons [36]. In
addition, charged excitons (trions), defect bound as well as charge-transfer
excitons are supposed to contribute on the low-energy side of the emission
band [36, 37]. A less intense emission line occurs at higher-energy
$\sim$1.74eV (grey-shaded spectral range). It originates from radiative
recombination of Coulomb-coupled electron hole pairs residing at the same K or
K′ valley and hence includes a real VB state at the K/K′-points. This is
important for resonant inelastic light scattering on collective charge
excitations of a 2D hole system residing at the K/K′ points of the moiré VBM
[33, 35]. In RILS experiments the incident photon energies ELaser is closely
tuned to the energy of the XKK-exciton observed in PL. The sample is excited
by linearly polarized light in back-scattering geometry. Due to the large NA
of the objective lens a bundle of in-plane momenta $q_{\parallel}<2\omega_{\rm
L}/c\sin\theta_{\rm max}$ with $\theta_{\rm max}\thickapprox 55^{o}$ is
transferred to the hole system. Following the band-structure calculations
shown in Fig. 1(d), the IMBE are expected in an energy range between
$\sim$20meV and $\sim$120meV. The resonance condition is given by the energy
of the XKK emission line. Following the consideration above, the energy range
where we expect the IMBE in RILS is superimposed by the dark emission band
marked by a red box in Fig. 2.

Figure 3:  RILS spectra on (a) 3o tWSe2, (b) natural bilayer (NBL) and (c)
WSe2 8o tWSe2. The straight white arrows indicate phonon modes and the white
curly braces in (a) and (c) IMBE with moiré band assignment according to
caclulations. In the natural bilayer spectra (b) in addition to the vertical
phonon line only shifting emission signals marked with tilted cyan lines
emerge. [T = 4K, Plaser = 500$\mu$W]. (d) Calculated electronic band structure
for (8${}^{o}\pm 0.5)^{o}$ considering realistic twist variations. (e)
Calculated electronic band structure for (3${}^{o}\pm 0.3)^{o}$. Twist
variations can result in broadened mini-bands causing significantly broadened
IMBE in RILS experiments. Breakdown of wave- conservation together with finite
transferred in-plane momentum $q_{\parallel}$ allows also for non-vertical
transitions as indicated by the arrows. (f) Calculated joint density of states
jDOS for IMB transitions in dependence of the Fermi-level EF for 3o tWSe2 for
vertical transitions. EF is given by the color code.

In Fig. 3, we contrast RILS spectra in a false color representation taken on
3o tWSe2 (a), natural bilayer (NBL) WSe2 (b) and 8o tWSe2 (c). RILS
intensities are plotted vs the relative energy (laser energy is at zero) with
the intensity encoded in the color scheme. The well known Raman active phonon
modes occur at the same energies as for the non-resonant measurements, however
the intensities follow the expected resonance enhancement. In the RILS spectra
of the twisted bilayers 3o tWSe2 and 8o tWSe2 displayed in Figs. 3(a) and (c),
highly resonance, intense and rather broad RILS modes interpreted as as IMBE
dominate the RILS spectra. In addition to the phonon and IMBE resonances, a
few features occur with energies depending on the incoming laser energy (tiled
cyan lines) that are interpreted as defect or 0D-moiré potential localized
emission signatures.

For the NBL sample despite the sharp phonon resonance, no clear RILS modes
occur. The spectra are dominated by typical emission signatures. These lines
are likely due to emission from interlayer excitons as expected for NBL layer
[19]. The absence of RILS modes for NBL is expected due to the absence of
moiré lattice and minibands. The energy separation of the SOI-split valence
band is in the order of a few hundreds of meV [38] and hence far beyond the
investigated energy range. For the 8o tWSe2, instead, one weaker and one
dominant broad RILS resonance appear at an energy range of about 60meV to
120meV . For a quantitative comparison with theory, the related band structure
for the moiré mBZ around the VBM at the K-point is displayed in Fig. 3 (d) in
the experimentally relevant energy range (for extended range see SI LABEL:SI).
For such a large twist, the bands are nearly parabolic with larger energy
separation. In comparison with theory, the weaker RILS mode at $\thickapprox$
60meV could be interpreted as a collective intraband transition within the
m1-band from the Fermi-edge to the local VBM. The more intense, broader RILS
mode at $\thickapprox 75-115$ meV is interpreted as an IMBE
m${}_{1}\rightarrow$m2 transition with the broadening due to non-vertical
transitions provided by the finite transferred in-plane momentum
$q_{\parallel}\leq q_{\rm max}$. The good quantitative agreement justifies our
interpretation of the modes as being collective intra- and interband
excitations allowing to study the band-structure of the mBZ at the K-points.
To consider realistic twist-angle variations to be in the order of $\pm
0.3^{o}$. While the impact of twist angle variations is shown from calculated
bands to be negligible small for (8$\pm 0.3)^{o}$ tWSe2, the impact is
significant in the mini-band energy and dispersion for (3$\pm 0.3)^{o}$ tWSe2
as plotted in Fig. 3(e). Also here non-vertical transitions are expected due
to finite momentum transfer $q_{\parallel}$ in addition to defect induced
breakdown of momentum conservation [39]. Twist angle variation together with
non-vertical transitions are supposed to broaden IMBE signatures in RILS. The
RILS modes are a convolution of jDOS and the momentum distribution. and a
combined effect is most likely the reason that we do not observe well
separated IMBE for the 3o tWSe2. The extracted joint density of states (jDOS)
in dependence of the position of EF for a twist angle of 3o for the individual
IMB transitions is plotted in Fig. 3(f). The jDOS demonstrate EF-dependent IMB
transitions m${}_{1}\rightarrow$m2, m${}_{1}\rightarrow$m3 and so on.
Comparison between RILS spectra and the jDOS for 3$\pm 0.3^{o}$ tWSe2 (Fig.
3(a,f)) provides a very good quantitative agreement with only the
m${}_{1}\rightarrow$m2 transition ($\thickapprox 20-25$meV) being clearly
separated. Also the m${}_{1}\rightarrow$m3 transition ($\thickapprox
50-67$meV) can be identified and the m${}_{1}\rightarrow$ m${}_{4},5,6,7$
transitions ($\thickapprox 70-110$ meV) are overlapping due to the discussed
reasons of twist angle variations and finite momentum transfer
$q_{\parallel}$. The experimental mode energies are in very good quantitative
agreement with the transition from the jDOS. While the m${}_{1}\rightarrow$m2
and m${}_{1}\rightarrow$m4 transitions are nearly independent of $E_{F}$,
position and particularly the intensity of the m${}_{1}\rightarrow$m2 as well
as m${}_{1}\rightarrow$m5,6,7 are strongly dependent on EF.

For this reason, excitation-power dependent RILS measurements for the 3o tWSe2
sample are done in order to vary EF by the photo-induced holes from $\Delta$p
$\thickapprox 10^{9}$cm-2 to $\Delta$p $\thickapprox 10^{10}$cm-2 and
$\Delta$p $\thickapprox 10^{12}$cm-2. Due to the strong band non-parabolicity
and the unknown intrinsic doping it was not possible to estimate the change in
EF from $\delta$p. In agreement with expectation from jDOS, the RILS intensity
of mode assigned to IMBE m${}_{1}\rightarrow$m2 is significantly reduced for
the low density measurements (data shown in SI [32]). This comparison strongly
supports our interpretation that the observed broad RILS signatures are indeed
collective IMBE between moiré bands at the VBM.

To conclude, we establish that low-temperature RILS experiments on collective
”single particle like” [33, 35, 40] IMBE on twisted WSe2 bilayers is a
powerful method to experimentally study the moiré band formation selectively
at the VBM around the K/K′ states, their energetic separation and twist angle
dependence. We find several IMBE for a 3o tWSe2 in an energy range between
20meV and 110 meV, while for a 8o tWSe2 only one clear IMBE and presumably one
collective intraband excitation emerge in a similar energy range. RILS modes
are absent for natural bilayers. These observation are in very good
quantitative agreement with calculated band structures. In comparison with
theory, we identify that twist angle variations significantly alters the
energy dispersion for smaller twist angle. The observation of collective IMBE
by RILS in a semiconducting vdW heterostructure demonstrates the potential to
access the collective low-lying excitation spectra as unique fingerprints of
individual quantum phases by RILS similar to correlated phases e.g. in the
fractional quantum Hall effect regime [41, 42]. We would like to emphasize
that collective excitation probe the weakly disordered parts of the sample,
while disorder often results in the most dominant signatures in emission
experiments and disturb transport investigations.

###### Acknowledgements.

The authors gratefully acknowledge the German Science Foundation (DFG) for
financial support via Grants WU 637/7-1, WE 5342/5-1 and the Priority Program
SPP 2244 “2DMP” - 443273985, 443274199 as well as the computing time granted
through JARA on the supercomputer JURECA [43] at Forschungszentrum Jülich. LK
acknowledges support from the DFG through FOR 5249 (QUAST, Project No.
449872909). TW acknowledges support by the Cluster of Excellence ‘CUI:
Advanced Imaging of Matter’ of the DFG (EXC 2056, Project ID 390715994). DMK
acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - Cluster of
Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 -
390534769. We acknowledge support by the Max Planck-New York City Center for
Nonequilibrium Quantum Phenomena.

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Supplemental Materials:
Collective charge excitations between moiré-minibands in twisted WSe2 bilayers
from resonant intelatic light scattering

##  I Methods and experimental procedure

### Sample preparation

The tWSe2 samples were prepared by first mechanically exfoliating the
monolayer WSe2 and multilayer hBN crystals onto viscoelastic PDMS stamps
followed by their deterministic dry transfer (DDT) using a home-built setup
with x-y-z and rotational micro-manipulators. To fix the stacking (twist)
angle between the WSe2 monolayers, their crystalline edges were determined
from optical images on the PDMS stamps. Following this, the top WSe2 monolayer
was stacked on top of the bottom WSe2 monolayer at the desired twist angle by
using a rotation stage attached to the sample stage in the DDT setup. After
preparation of the full heterostructure including the hBN encapsulation
layers, the samples were annealed under a home-built vacuum annealing chamber
at 100 - 120 oC for 5 - 6 hours.

### Non-resonant photoluminescence (PL) and Raman spectroscopy

After preparation of the samples, they were characterized by low temperature
non-resonant PL and Raman spectroscopies. The samples were mounted on the cold
finger equipped with x-y-z piezo stages of a dilution refrigerator. The
samples were cooled down to 4K using pulse tube refrigeration operating with
helium gas. For non-resonant characterization measurements, a 532 nm (2.33 eV)
solid state laser, cleaned up using a grating based monochromator with a 1nm
bandpass, was used for excitation of the samples. The laser was focused on the
samples with a low temperature compatible objective lens with an NA of 0.82
mounted on a homebuilt stage attached to the 4K stage of the refrigerator. The
focused spot on the sample had a diameter of $\sim$ 2$\mu$m. The emission and
scattering signals from the sample were collected by the same objective lens
in a back scattering geometry and dispersed using a single spectrometer with a
600 l/mm grating for PL and an 1800 l/mm grating for Raman spectroscopy. The
focal length of the spectrometer was 750 mm and the spectral resolution (532
nm, 1800 l/mm grating) was $\sim$ 0.4 cm-1. The Rayleigh scattered light was
filtered out by using a 532 nm long pass filter in front of the spectrometer.
The dispersed light was collected and measured using a liquid nitrogen cooled
CCD camera. Position and laser power dependent PL and position dependent Raman
measurements were used to select the region of the sample for subsequent
resonant inelastic light scattering measurements.

### Resonant Inelastic Light Scattering spectroscopy

Following the characterization by low temperature PL and Raman spectroscopy,
inelastic light scattering spectra under resonant excitation conditions were
measured. The samples were mounted inside a dilution refrigerator as described
above. The samples were excited at energies between 1.698 eV (730 nm) and
1.771 eV (700 nm) using a continuously frequency tunable Ti:sapphire laser
stabilized by a ring cavity with a linewidth of about 50kHz. The luminescence
background from the Ti:sapphire crystal was removed by passing the excitation
laser beam through a monochromator (1nm bandpass). The light emitted or
scattered from the sample surface were again collected in back scattering
geometry and sent to either a single spectrometer with a 600 l/mm grating or
to a triple spectrometer with a grating combination of 300 l/mm, 300 l/mm and
600 l/mm. The triple spectrometer was used in a subtractive mode where the
second stage of the spectrometer is used to suppress the dispersion generated
by the first stage by appropriate alignment of the gratings. This meant that
at the exit slit of the second stage, light could be filtered out based on the
angle of entrance in the spectrometer. Thus, the first two stages served as a
bandpass filter and filtered out any stray light not following the path of
light scattered from the sample, as well as the Rayleigh scattered background.

##  II Theotetical caclulations

### Continuum Hamiltonian

In the small-angle approximation $\Theta\sim 0^{\circ}$, the dispersion of
tWSe2 can be obtained by constructing a continuum model (following Refs. [1,
2, 3, 4]). As the electrons in a WSe2 monolayer are spin-valley locked, only
one valley per layer contributes to each spin sector of the moiré Hamiltonian

$H_{\uparrow}(\boldsymbol{k})=\begin{pmatrix}-\frac{\hbar^{2}(\boldsymbol{k}-K_{1})^{2}}{2m^{\star}}+\Delta_{1}(\boldsymbol{r})&\Delta_{T}(\boldsymbol{r})\\\
\Delta_{T}^{\dagger}(\boldsymbol{r})&-\frac{\hbar^{2}(\boldsymbol{k}-K_{2})^{2}}{2m^{\star}}+\Delta_{2}(\boldsymbol{r})\end{pmatrix}\,$
(S1)

where $m^{\star}=0.43\,m_{e}$ is the effective mass, $K_{l}$ the original
$K$-point of layer $l=1,2$, $\Delta_{T}(\boldsymbol{r})$ the interlayer
potential, and $\Delta_{l}(\boldsymbol{r})$ the intralayer potential. The
other spin sector $H_{\downarrow}(\boldsymbol{k})$ is obtained from time
reversal symmetry. The inter- and intralayer potentials are parameterized by
the following simple formulæ:

$\Delta_{l}(\boldsymbol{r})=2V\,\sum_{\boldsymbol{g}\in\triangle}\cos\big{(}\boldsymbol{g}\cdot\boldsymbol{r}+\sigma_{z}^{ll}\psi\big{)}\,,\qquad\Delta_{T}(\boldsymbol{r})=w\,\sum_{\boldsymbol{g}\in\triangle}e^{-i\boldsymbol{g}\cdot\boldsymbol{r}}\,.$
(S2)

Here, $\sigma_{z}$ is the Pauli $z$ matrix and $\boldsymbol{g}$ are three
reciprocal moiré lattice vectors that are connected by $C_{3}$ rotations.
Hence we index the summation over those three reciprocal moiré lattice vectors
as $\boldsymbol{g}\in\triangle$. The $K$\- and $K^{\prime}$-points in the
moiré Brillouin zone (BZ) coincide with $K_{1}$ and $K_{2}$, respectively.

The intralayer potential strength $V=9\,\mathrm{meV}$, the interlayer hopping
$w=18\,\mathrm{meV}$, and the intralayer potential parameter
$\psi=128^{\circ}$ are taken to match density functional theory results [3].
Note that the twist angle $\Theta$ enters Eq. (S1) through geometry in the
momentum argument $\boldsymbol{k}$ as well as the length of the inverse moiré
lattice vectors $\boldsymbol{g}$ in Eq. (S2).

For numerical diagonalization of the Hamiltonian Eq. (S1), the expansion in
inverse moiré lattice vectors must be truncated. We take into account the
following inverse moiré lattice vectors $\boldsymbol{G}$

$\boldsymbol{G}\in\\{\boldsymbol{H}~{}|~{}\|\boldsymbol{H}\|\leq
3\|\boldsymbol{g}\|~{}\forall~{}\boldsymbol{g}\in\triangle\\}\,,$ (S3)

i.e., a distance shell cutoff with length $3$, which is sufficient for
convergence of the resulting band structures.

### Joint Density of States

The joint density of states (jDOS) data in Fig. 3 (f) of the main text is
obtained numerically from the model discussed above. In essence, we implement
the following formula:

$\mathrm{jDOS}(\omega)=\mathrm{Im}\sum_{\boldsymbol{k},b_{1},b_{2}}\frac{f\big{(}\epsilon_{b_{1}}(\boldsymbol{k})\big{)}-f\big{(}\epsilon_{b_{2}}(\boldsymbol{k})\big{)}}{\omega-i\eta+\epsilon_{b_{1}}(\boldsymbol{k})-\epsilon_{b_{2}}(\boldsymbol{k})}\,,$
(S4)

where $\epsilon_{b}(\boldsymbol{k})$ is the dispersion of band $b$ with
$\boldsymbol{k}$ in the mini BZ,
$f(x)=\big{(}1+\exp[\beta(x-\mu)]\big{)}^{-1}$ the Fermi function, and
$\eta=5\,\mathrm{meV}$ a Lorentzian broadening. The inverse temperature is set
to $\beta=10^{2}\,\mathrm{meV^{-1}}$. As we perform massive scans in frequency
$\omega$ and chemical potential $\mu$ (Fermi energy) for significant
$\boldsymbol{k}$-meshes ($96\times 96$ regularly spaced points in the
primitive zone), we make use of the parallel nature of Eq. (S4) using a custom
GPU implementation.

##  III Additional Figures

### Non-resonant Raman measurements

Figure S1:  (c) Non-resonant Raman spectra on hBN encapsulated WSe2 monolayer
(a), hBN encapsulated natural bilayer (NBL) WSe2 (b), hBN encapsulated 3o
tWSe2 (c) and hBN encapsulated 8o tWSe2. All spectra show the optical phonon
modes A1g, E${}^{1}_{2g}$ and in addition the NBL and twisted bilayer show the
B${}^{1}_{2g}$ phonon mode. The features marked by an asterisk (*) are due to
multiphonon scattering processes [T = 4K, Plaser = 100$\mu$W, Elaser =
2.33eV].

### Excitation Power dependent RILS measurements

Figure S2:  (a) False color plots of RILS spectra for 3o tWSe2 at three
different excitation powers of (a) 1$\mu$W, (b) 50$\mu$W and (c) 1mW,
respectively. The corresponding photo-excited hole-densities are
$\sim{10^{9}}$ cm-2, $\sim{10^{10}}$ cm-2 and $\sim{10^{12}}$ cm-2,
respectively.The phonon-lines are marked by straight arrows and emission
features by tilted cyan solid lines. White braces indicate the energy
positions of IMBE. Particularly the m${}_{1}\rightarrow$ m2 transition is
enhanced with increasing doping and hence position of EF in agreement with
jDOS(EF) [T = 4K].

Excitation-power dependent RILS measurements are performed for the 3o tWSe2
sample in order to vary EF by the photo-induced holes. RILS spectra from a
different set of experiments are compared in Fig. S2 for (a) 1$\mu$W
corresponding to a change in the hole density by $\delta$p $\thickapprox
10^{9}$cm-2, (b) 50$\mu$W ($\delta$p $\thickapprox 10^{1}0$cm-2) and (c) 1mW
($\delta$p $\thickapprox 10^{1}2$cm-2), respectively. Due to the strong band
non-parabolicity and the unknown intrinsic doping it is not possible to
estimate the change in EF from $\delta$p. While it is not possible within the
experimental certainty to clearly identify a shift in the energy, the
intensity of the RILS mode assigned to IMBE m${}_{1}\rightarrow$m2 is
significantly reduced for the low density measurements as expected from the
jDOS. Similarly, for the high energy transition that seems to be most
pronounced for the higher power measurements indicated by an even broader
overlapping emission band towards 100meV and beyond. This is most pronounced
for 50$\mu$W. For the 1mW spectra some spectral weight shifts back to the
intermediate transitions regions m${}_{1}\rightarrow$m${}_{4},5$. This also
agrees with the jDOS from the band structure calculation and strongly supports
our interpretation that the observed broad RILS signatures are indeed
collective IMBE between moiré bands at the VBM.

### Calculated mini-band structure

Figure S3:  Calculated energy dispersion of the highest moiré mini bands down
to -300meV around the K, K’ states for a 3o tWSe2 bilayer (a) and a 8o tWSe2
bilayer (b).

## References

  * Wu  _et al._ [2018] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard Model Physics in Transition Metal Dichalcogenide Moir\’e Bands, Physical Review Letters 121, 026402 (2018).
  * Pan  _et al._ [2020] H. Pan, F. Wu, and S. Das Sarma, Band topology, Hubbard model, Heisenberg model, and Dzyaloshinskii-Moriya interaction in twisted bilayer ${\mathrm{}}WSe{}{}_{2}$, Physical Review Research 2, 033087 (2020).
  * Devakul  _et al._ [2021] T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature Communications 12, 6730 (2021).
  * Ryee and Wehling [2023] S. Ryee and T. O. Wehling, Switching between Mott-Hubbard and Hund Physics in Moiré Quantum Simulators, Nano Letters 23, 573 (2023).


# Interface superconductivity in FeSe on SrTiO3 thin films grown by the PLD
technique

T Kobayashi, H Ogawa, F Nabeshima, and A Maeda  Dept. of Basic Science,
University of Tokyo, Meguro, 153-8902, Tokyo, Japan  kobayashi-
<EMAIL_ADDRESS>
###### Abstract

In this study, we fabricate 7–25-nm-thick films of FeSe/STO using pulsed laser
deposition (PLD). The grown films exhibit superconductivity with an onset
$T_{\mathrm{c}}$ that is much higher than that of bulk FeSe under ambient
pressure. The observed $T_{\mathrm{c}}$ values are exceptionally high in terms
of the strain vs. $T_{\mathrm{c}}$ relationship of the same material
established so far. Furthermore, $T_{\mathrm{c}}$ increases as the film
thickness decreases. This is in good agreement with the results reported for
films grown by molecular beam epitaxy (MBE) that exhibit interface
superconductivity. These results indicate the realization of interface
superconductivity in PLD-grown FeSe/STO. Furthermore, our PLD technique
requires no post-annealing to realize interface superconductivity, which is
different from MBE techniques. Because the PLD technique has the advantage
that various interfaces can be fabricated easily by simply altering the target
materials, our results open novel routes to study interface superconductivity
toward higher $T_{\mathrm{c}}$ by systematic control of the interface.

##  1 Introduction

The iron chalcogenide superconductor FeSe has garnered significant attention
for its high tunability of the superconducting transition temperature
($T_{\mathrm{c}}$). Although the onset $T_{\mathrm{c}}$
($T_{\mathrm{c}}^{\mathrm{onset}}$) of bulk FeSe is 9 K at ambient pressure
[1], it is highly enhanced by various methods. For instance,
$T_{\mathrm{c}}^{\mathrm{onset}}$ increases to 38 K by applying hydrostatic
pressure [2, 3]. The doping of electrons by intercalation [4, 5] or the
electric field effect [6, 7, 8, 9, 10, 11, 12] increases
$T_{\mathrm{c}}^{\mathrm{onset}}$ to 50 K. In particular, zero resistivity
below 46 K is realized in [11, 12]. Furthermore, angle-resolved photoemission
spectroscopy (ARPES) measurements reported that a monolayer FeSe film on
$\mathrm{SrTiO_{3}}$ (STO) [13, 14] exhibited gap opening at the Fermi level
below 65 K, which has been interpreted as a manifestation of
superconductivity. The $T_{\mathrm{c}}$ enhancement in monolayer FeSe on STO
(FeSe/STO) is attributed to interfacial effects, such as electron doping from
the STO substrate[15] and electron-phonon interaction with phonons of STO
[16]. Thus, the superconductivity in FeSe/STO is expected to be the first
interface-enhanced superconductivity, where the existence of the interface
plays a crucial role [17, 18].

In contrast to the ARPES results, resistivity measurements reported a
$T_{\mathrm{c}}^{\mathrm{onset}}$ of 40–45 K [13, 19, 20] and a much lower
zero-resistance temperature ($T_{\mathrm{c}}^{\mathrm{zero}}$), and these are
lower than those of electron-doped FeSe systems [11, 12]. This discrepancy in
the reported values of $T_{\mathrm{c}}$ observed by different measurement
techniques suggests that the exact feature of interface superconductivity in
FeSe/STO is still far from being fully understood. Therefore, a more detailed
investigation of the interfacial effects is essential. In this context, the
artificial control of the interface is a promising approach. For feasible
control of the interface, pulsed laser deposition (PLD) has advantages in
fabricating various heterostructures at low cost because different materials
can be deposited by simply switching the target materials. However, all
studies on the interface superconductivity of FeSe published so far were
performed using the molecular beam epitaxy (MBE) technique. This is because it
is difficult to grow high-quality thin films of compounds that contain highly
volatile Se using other growth techniques. Indeed, none of the previous
studies by PLD succeeded in fabricating FeSe films that exhibited the
interface superconductivity.

In this paper, we report the successful growth of FeSe/STO which exhibits
interface superconductivity, using the PLD technique for the first time. We
fabricate 7–25-nm-thick FeSe films on atomically flat STO substrates. The
grown films exhibit a $T_{\mathrm{c}}^{\mathrm{onset}}$ higher than 20 K,
which is much higher than that of bulk FeSe [1]. The enhanced $T_{\mathrm{c}}$
cannot be explained in terms of the in-plane strain effect on the bulk
superconductivity of FeSe [21, 22, 23, 24]. $T_{\mathrm{c}}$ increases with
decreasing film thickness, which is in good agreement with those reported for
MBE-grown FeSe/STO [25, 26]. These results suggest that we realized interface
superconductivity in FeSe/STO, grown using the PLD technique for the first
time.

##  2 Experimental methods

All the FeSe thin films in this study were grown on insulating STO (001)
substrates using PLD with a KrF laser. Prior to deposition, the STO substrate
was annealed at 1000∘C in air and rinsed with water to obtain a
$\mathrm{TiO_{2}}$-terminated surface with a step-terrace structure [27]
(figure 1(a)). Reflection high-energy electron diffraction (RHEED) images of
the substrates exhibited the ($2\times 1$) reconstructed patterns [13] under
vaccum at a growth temperature of 500${}^{\circ}\mathrm{C}$ (figure 1(b)).
Films were deposited by ablating the $\mathrm{Fe_{1.1}Se}$ target [21] at a
growth rate of 0.2–0.5 nm/min. The RHEED patterns of the grown films displayed
streaks, indicating a flat surface of the films (figure 1(c)). Because RHEED
oscillation in FeSe/STO was sometimes not observed in this study, the RHEED
intensity of the films on LaAlO3, which were grown simultaneously as the films
on STO, was observed during the deposition to control the film thickness.
Below, we present a series of four samples with different thicknesses of 7,
10, 16, and 25 nm, as typical representatives.

The orientation and crystal structure of the grown films were characterized
using X-ray diffraction (XRD) measurements with Cu K$\alpha$ radiation at room
temperature. The thicknesses of the films were estimated from the Laue fringe
around the (001) reflection peak or using a Dektak 6M stylus profiler.
Resistivity was measured using a physical property measurement system (PPMS),
from 2 to 200 K.

##  3 Results and discussion

Figure 1: (a) Atomic force microscope topographic image of an STO substrate
after the surface treatment. Clear step terrace structures are observed. (b)
RHEED image of an STO substrate just before the deposition. (c) RHEED image of
a grown FeSe film. Figure 2: (a) XRD patterns of the grown films. Asterisk
represents reflections from Ag paste. (b) Enlarged plot around the (001)
reflections. (c) Rocking curves of the (001) peaks of the grown films. Figure
3: (a), (b) Temperature dependence of the resistivity of the grown films. (c)
$T_{\mathrm{c}}^{\mathrm{onset}}$ as a function of strain of the films
together with the data in [22]. (d) Thickness dependence of
$T_{\mathrm{c}}^{\mathrm{onset}}$ of the films. Data for the films grown by
MBE[25] are also plotted (black triangles).

Figure 2(a) illustrates the XRD patterns of four films with different
thicknesses. All the films indicated only the (00l) reflection peaks of
tetragonal FeSe, except for the peaks derived from the substrates and Ag paste
attached to the films for the resistivity measurement. These results indicate
the single-phase nature of FeSe and c-axis orientation of the films. Figure
2(b) illustrates an enlarged plot of the (001) reflection peaks. All the films
showed clear Laue fringes, indicating a flat surface. Figure 2(c) illustrates
the rocking curves of the (001) reflection peaks of the films. All films
showed a full width at half maximum (FWHM) smaller than 0.16∘, indicating high
crystalline quality. All the films were under tensile strain with their a-axis
constant larger than 3.81 Å (cf. bulk FeSe has $a$ = 3.77 Å). The structural
properties of the films are summarized in table 1.

Table 1:  Characteristics of the grown films.

  * Sample | Thicknsess (nm) |  $a$ (Å) |  $c$ (Å) |  $T_{\mathrm{c}}^{\mathrm{onset}}$ (K) |  $T_{\mathrm{c}}^{\mathrm{zero}}$(K)
---|---|---|---|---|---
#1 | 7 | 3.81 | 5.48 | 28 | 17
#2 | 10 | 3.82 | 5.47 | 30 | 19
#3 | 16 | 3.83 | 5.47 | 28 | 17
#4 | 25 | 3.82 | 5.48 | 23 | 13

Figures 3(a) and (b) illustrate the temperature dependence of resistivity
$\rho(T)$ of the films. All films exhibited a metallic behavior. The 7-nm-
thick film exhibited a higher resistivity than the other films possibly
because of the deterioration caused by air exposure. Remarkably, all samples
exhibited a superconducting transition with a
$T_{\mathrm{c}}^{\mathrm{onset}}$ higher than 20 K. These
$T_{\mathrm{c}}^{\mathrm{onset}}$ values were much higher than those of the
bulk at ambient pressure. Furthermore, $T_{\mathrm{c}}$ increased as the
thickness of the films decreased, except for the 7-nm-thick film, as will be
discussed later.

It is well known that in-plane strain strongly affects the superconducting
properties of FeSe [21, 22, 24, 23]. FeSe thin films under compressive strain
exhibit enhanced $T_{\mathrm{c}}^{\mathrm{onset}}$ up to 12 K when
$\epsilon\equiv(a_{\mathrm{film}}-a_{\mathrm{bulk}})/a_{\mathrm{bulk}}<-1.0\%$
[22], where $a_{\mathrm{film}}$ and $a_{\mathrm{bulk}}$ are the a-axis
constants of the films and bulk [28], respectively. However, tensile strain
decreases $T_{\mathrm{c}}$ and completely suppresses superconductivity when
$\epsilon$ is greater than +0.6% [22]. Similar results were obtained in
subsequent studies on bulk crystals [24, 23]. We plot
$T_{\mathrm{c}}^{\mathrm{onset}}$ of the films grown in this study in the
strain-vs-$T_{\mathrm{c}}^{\mathrm{onset}}$ plot [22] in figure 3(c). The
present data largely deviate from the well-established strain-$T_{\mathrm{c}}$
relation. The $\epsilon$ values in the grown films, which are higher than
$0.9\%$, should lead to non-superconductivity, according to the
strain-$T_{\mathrm{c}}$ relation. Thus, these high $T_{\mathrm{c}}$ values
cannot be explained by the established strain effect. This result suggests
that interface effects are responsible for the high $T_{\mathrm{c}}$ of the
films.

The interface superconductivity in the grown films is also suggested by the
thickness dependence of $T_{\mathrm{c}}^{\mathrm{onset}}$ (figure 3(d)).
Except for the 7-nm-thick film, $T_{\mathrm{c}}^{\mathrm{onset}}$ increases as
the thickness of the film decreases, as aforementioned. This negative
correlation between the film thickness and $T_{\mathrm{c}}$ is in good
agreement with the results obtained for MBE-grown FeSe/STO, which exhibited
interface superconductivity [25, 26]. The negative correlation between film
thickness and $T_{\mathrm{c}}$ was interpreted as follows [25]: Because FeSe
layers apart from the interface are considered not to be subject to
interfacial effects [13], it is likely that the very thin layer in the
vicinity of the interface becomes superconducting. When the thickness of the
film is increased, the number of doped electrons at the superconductive
interface of FeSe/STO is reduced, which results in a reduction of
$T_{\mathrm{c}}$. Although the validity of this interpretation is still open,
the results of the thickness dependence, together with the extraordinarily
high $T_{\mathrm{c}}$ in terms of the strain-$T_{\mathrm{c}}$ relation
indicate the interface superconductivity in the films grown on STO. Note that
the thickness dependence of $T_{\mathrm{c}}$ is in sharp contrast to that of
FeSe thin flakes [29, 30, 31] and FeSe on bilayer graphens [32] , where
$T_{\mathrm{c}}$ decreases as the thickness decreases. This difference is
consistent with the fact that the interface between FeSe and STO plays a
crucial role in realizing enhanced superconductivity in the grown films.

Compared with MBE-grown films [25, 26], all films other than the 7-nm-thick
sample exhibited approximately the same value of
$T_{\mathrm{c}}^{\mathrm{onset}}$ (figure 3(d)). Remarkably, our films
exhibited a high $T_{\mathrm{c}}$ without post-annealing. MBE growth requires
a post-annealing process to remove excess Se [33] which suppresses the
superconductivity. The excess Se is due to the Se-rich growth conditions of
the MBE techniques. However, approximately the same composition is transferred
from the Fe1.1Se target to a film during PLD growth, which results in much
less excess Se in the grown film. This feature of the PLD technique enables us
to obtain a superconducting FeSe/STO without post-annealing, which is
different from MBE.

The 7-nm-thick film exhibited a lower $T_{\mathrm{c}}$ than the 10-nm-thick
film, possibly owing to degradation by air exposure. Thus, deposition with a
protection layer is necessary for films thinner than 10 nm to obtain a higher
$T_{\mathrm{c}}$ , which is now under way.

##  4 Conclusion

In conclusion, we fabricated FeSe films on STO with a thickness of 7–25 nm
which exhibited superconductivity with $T_{\mathrm{c}}^{\mathrm{onset}}$ much
higher than that of bulk FeSe under ambient pressure. The observed
$T_{\mathrm{c}}$ in all the films was exceptionally high in terms of the
strain-$T_{\mathrm{c}}$ relation of the same material established so far.
$T_{\mathrm{c}}$ increased with decreasing film thickness, which is in good
agreement with that of MBE-grown films that exhibited interface
superconductivity [25, 26]. These results indicate that we successfully
realized interface superconductivity in PLD-grown FeSe/STO. Furthermore, the
PLD growth method has the advantage of realizing interface superconductivity
without post-annealing, which is different from MBE techniques. PLD has the
advantage of fabricating various heterostructures because different materials
can be deposited by simply changing the target materials. Thus, our results
accelerate a study of interface superconductivity toward higher
$T_{\mathrm{c}}$ by systematically controlling the interface.

We would like to acknowledge K. Ueno and H. Okuma at the University of Tokyo
for their technical support of the XRD and AFM measurements.

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