# Polarons shape the interlayer exciton emission of MoSe2/WSe2 heterobilayers

P. Soubelet<EMAIL_ADDRESS>Walter Schottky Institut and TUM School
of Natural Sciences, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany.  A. Delhomme  Walter Schottky Institut and TUM School of
Natural Sciences, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany.  A. V. Stier  Walter Schottky Institut and TUM School of
Natural Sciences, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany.  J. J. Finley  Walter Schottky Institut and TUM School of
Natural Sciences, Technische Universität München, Am Coulombwall 4, 85748
Garching, Germany.

(28th August 2024)

###### Abstract

We present time-resolved and CW optical spectroscopy studies of interlayer
excitons (IXs) in 2$H-$ and 3$R-$stacked MoSe2/WSe2 heterobilayers and obtain
evidence for the strong participation of hot phonons in the underlying photo-
physics. Photoluminescence excitation spectroscopy reveals that excess energy
associated with optical excitation of intra-layer excitons and relaxation to
IXs affects the overall IX-PL lineshape, while the spectrally narrow emission
lines conventionally associated with moiré IXs are unaffected. A striking
uniform line-spacing of the sharp emission lines is observed together with
temperature and excitation level dependent spectra suggesting an entirely new
picture that photo-generated phonons lead to phonon-replicas shaping the IX-
emission. Excitation power and time resolved data indicate that these features
are polaronic in nature. Our experimental findings modify our current
understanding of the photophysics of IXs beyond current interpretations based
on moiré-trapped IXs.

Electron-phonon (e-ph) interactions in bulk-semiconductors play a major role
in charge and exciton dynamics where they dominate carrier transport and
optoelectronic properties [1, 2, 3]. In particular, optical phonons most
strongly couple to the lattice polarization, allowing the formation of
polarons, quasiparticles mediated by the e-ph interaction, and transforming
the spectral form of the optical susceptibility. [1, 2, 3]. For example, e-ph
interactions modulate the line shape of interband emission spectra via the
emergence of phonon sidebands [4, 5, 6, 7, 8]. These effects are expected to
be particularly strong in two-dimensional (2D) semiconductors such as
monolayer (ML) semiconducting transition metal dichalcogenides (TMDs). These
materials feature a direct bandgap in the visible range and efficient light-
matter interaction leading to strongly bound excitons [9, 10, 11, 12, 13, 14,
15, 16, 17]. Moreover, the strength of e-ph interactions are enhanced due to
the reduced dimensionality and weaker dielectric screening [1]. Experimental
and theoretical investigations of the impact of polarons in 2D systems are
surprisingly scarce [1, 18, 19, 20], although the fingerprints of polarons
have been reported in Raman [21, 22] and ARPES [23] experiments.

When two TMD monolayers are vertically stacked to form van der Waals
heterobilayers (HBs) their excitonic photophysics becomes even richer.
Interlayer hybridization establishes an in-plane periodic moiré potential with
a periodicity only determined by the twist angle ($\theta$) and the respective
lattice parameters [24, 25, 26]. This new potential folds acoustic phonon
branches leading to moiré phonons [27, 28, 29, 30] and tailors collective
excitations and the optical response of the system by forming topological
exciton bands [31, 32]. Moreover, for HBs having type-II band alignment, such
as tungsten diselenide (WSe2) and molybdenum diselenide (MoSe2), the lowest
energy excitons form via charge transfer between the layers. They are,
therefore, spatially indirect interlayer excitons (IXs), with the electrons
located at the bottom of the MoSe2 conduction band and the holes at the top of
the WSe2 valence band [33]. In this case, the periodic moiré potential
confines the IXs, establishing TMD HBs as a promising platform for the study
of strongly correlated phenomena and quantum optoelectronics applications [34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 37, 45, 46].

Since IX typically lie several hundred millielectronvolts lower in energy than
intra-layer counterparts, their formation involves the absorption and emission
of phonons to dissipate the excess photon energy. [47, 48, 49, 50, 51, 52, 53,
54]. Hereby, a large out-of-thermal-equilibrium phonon population is produced
by near resonant excitation of intra-layer excitons [47, 48, 49, 50, 52, 55,
53, 56]. Moreover, when an IX is formed, it strains the lattice, changing the
equilibrium separation between the constituent TMD layers. Electron-phonon
interactions then couple the relative motion of the layers, described through
acoustic phonon modes, with the IXs, leading to the formation of IX-polarons
[57, 58]. In this letter, we provide compelling experimental proof that the
narrow IX emission lines in MoSe2/WSe2 HBs, conventionally ascribed to the
recombination of individual moiré-trapped IXs, arise from IXs dressed with an
integer number ($N$) of non-thermal acoustic phonons.

##  I results and discussion

Figure 1: a Schematic of the investigated hBN-encapsulated WSe2/MoSe2 HBs.
Sketch of the IX formation in a) real space and b) momentum space. An
electron-hole pair is excited via the absorption of a photon with energy
$E_{ex}$ (red arrow) within the light cone (red dotted lines). During the
charge transfer and energy relaxation process, the electron-hole pair strongly
interacts with the lattice by absorbing and emitting phonons
($\hbar\omega_{phn}$). c) and d) show the normalized IX emission intensity for
the 3$R-$ and 2$H-$ sample, respectively. The general shape of each spectrum
can be approximated by two Gaussians (orange and violet shaded regions).
Insets in c and d: Schematics of the spin-valley configuration for the 3$R-$
and 2$H-$ sample. The corresponding excitons are indicated.

To investigate the physics governing the IX formation and emission, we
fabricated HBs via the well-known tear-and-stack method [59] to assemble
twisted samples with $\theta$ approximately 4.5∘ and 57∘. These twist angles
were chosen to allow investigations of near 3$H-$ and 2$R-$stacked optical
selection rules, maintaining significant oscillator strength while avoiding
lattice reconstruction effects typically arising at small $\theta$s [60, 61,
62, 63, 64]. A schematic of the hexagonal boron nitride (hBN) encapsulated HBs
used in this work is shown in Figure 1a. Details regarding the fabrication,
characterization and optical selection rules of the samples are provided in
the Supplemental Material (SM) III.1. Although local variations across the
samples are present, all key results pertaining to IX-polaron physics were
found to be reproducible and independent of the specific location on the
samples and twist angle.

The IX formation in real and momentum space is depicted in Figures 1a and b,
respectively. It involves the absorption of a photon of energy $E_{ex}$ within
the light cone (red dashed line in Fig.1b) and forms an intralayer electron-
hole pair in one of the component monolayers. The charge transfer between MLs
occurs over fs-timescales through the $\Sigma-$($\Gamma-$) hybridized states
of the WSe2(MoSe2) Brillouin zones [48, 51, 54]. This process generates
phonons ($\hbar\omega_{phn}$), leading to the dissipation of hundreds of
millielectronvolts from the $intra$-layer exciton to the IX state [47, 48, 49,
50, 52, 55, 53]. In particular, the charge transfer process triggers the
emission of the interlayer breathing mode [56, 65, 66]. Note that the IX is a
spatially and momentum-indirect state, since it is situated at the edge of the
mini-Brillouin zone $\sim\pi/d_{moir\acute{e}}$, where $d_{moir\acute{e}}$ is
the moiré periodicity (see Fig.1b) [67, 48]. Therefore, the interaction with
the lattice is intrinsic to the radiative recombination of IXs, as
investigated in detail below.

Typical photoluminescence (PL) spectra recorded from the HBs are shown in
Figure 1c and d, consistent with reports in the literature. PL spectroscopy
was performed in a He exchange gas cryostat ($T=4.2$ K) and the optical CW
power ($P_{ex}$) from a wavelength-tunable Ti:Sa laser focused to a
diffraction-limited spot ($NA=0.82$) was $100\,$nW. The PL spectra for the
3$R$ and 2$H$ samples in figures 1c and d were obtained by resonant optical
pumping of the MoSe2 intralayer exciton at $E_{ex}=1.602\,eV$ and $1.616\,eV$,
respectively. The emission is from the singlet(triplet) IX configuration for
the 3$R$(2$H$) sample [68, 69]. For details regarding the exciton complexes
and selection rules see SM III.1.2. Typically, IX emission lineshapes are
composed of multiple peaks, the details of which depend significantly on the
experimental conditions. However, usually, $\sim 10\,$meV broad lobes are
attributed to quantized states within the moiré potential (as indicated by the
orange and violet shades in Fig.1c and d) [70, 71, 67, 72, 73]. On top of
these broad emission features and specifically at low to medium excitation
powers, a number of narrow emission lines (sub meV linewidth) are
superimposed. These lines are frequently ascribed to the emission of
individual or few IXs confined at local moiré sites that, due to local sample
inhomogeneities, lead to emission at varying energy [69, 74, 47]. We note that
at very low excitation power, even narrower emission lines ($\sim 100$ $\mu$eV
of linewidth) have been observed exhibiting single-photon emission
characteristics [75]. Furthermore, these sharp emission lines have recently
been interpreted as interlayer donor-acceptor emission centers. [76]. In the
following, we show that the MoSe2/WSe2 IX PL, as those presented in Fig.1c and
d, is shaped by phonon sidebands (orange and violet lobes in Fig.1c and d))
and the central dip in the overall spectrum between the broad lobes
corresponds to the zero phonon line (ZPL). Moreover, these phonons are
coherently generated by the IX during the interlayer charge transfer and
energy relaxation process. Therefore, the low(high) energy lobe is formed by
processes in which the IX energy is reduced(increased) by the
emission(absorption) of acoustic phonons forming exciton-polarons [58, 57].
The sharp PL lines are phonon replicas from IX-polarons dressed with an
integer number of $N-$phonons. While high energy sidebands are frequently
suppressed in semiconductor emission [4, 8], HBs are a particular case in
which the phonons involved in the polaron formation are of low energy, of the
order of the thermal energy and, consequently, stimulated processes must be
considered.

Figure 2: a) False color plot of the IX PL spectra for a series of excitation
energies $E_{ex}$. b) Selected IX PL spectra of the 3$R-$ (top) and 2$H-$
sample (bottom), plotted on an energy scale relative to the ZPL (peak labeled
$N=0$). Spectra in green correspond to the IX emission obtained by exciting
the sample just below the intralayer exciton of WSe2 at $E_{ex}=1.653$
eV(1.687 eV) for the 3$R-$(2$H-$) sample and the blue spectra are excited just
above the resonance, at $E_{ex}=1.685$ eV(1.720 eV) for the 3$R-$(2$H-$)
sample. Vertical dotted lines mark the position of the sharp PL features.
Black arrows mark the position of peaks that were not discernible. c) Peak
energy as function of peak number for the 3$R-$ (top) and the 2$H-$ (bottom)
sample. The peaks are uniformly spaced with an energy interval of $\sim 0.8$
meV.

To show that phonons are integral to the IX-PL, we perform PL excitation (PLE)
experiments, varying $E_{ex}$ across the MoSe2 and the WSe2 intralayer
exciton. As such, we vary the excess energy that needs to be dissipated in the
IX formation process. Figure 2a shows false colour PLE maps for each twisted
HB sample. Both samples display an enhancement of the IX emission intensity
when the intralayer excitons are pumped resonantly (horizontal dotted lines).
The general lineshape of the IX emission depends on $E_{ex}$, highlighted by
the selected spectra presented in Fig. 2b that shows the IX-PL for $E_{ex}$
just above (blue) and below (green) the intralayer exciton resonances, plotted
on a relative energy axis with respect to the ZPL. We observe that the
intensities of the low and high energy sidebands are affected by $E_{ex}$;
pumping above the intralayer exciton resonance enhances the high energy
sideband as compared to the low energy sideband. Furthermore, inspecting the
PLE map in Fig. 2a, the high energy sideband blueshifts with increasing
$E_{ex}$ as compared to the low energy sideband. We qualitatively explain this
observation considering that the phonon population that gives rise to the
phonon sidebands depends on the excess energy between $E_{ex}$ and the IX.
Strikingly, the narrow IX emission features are completely independent of
$E_{ex}$ and, therefore, only depend on the IX recombination process and not
on the IX formation process (electron-hole excitation, charge transfer and
relaxation processes). Note that the PLE experiments were repeated at higher
$P_{ex}$ across multiple spots on each sample, and the observed behavior
remained consistent (see SM III.2 for details).

Remarkably, a careful inspection of the sharp polaron peaks across all our
experiments (see SM III.3.1) shows that the overwhelming majority of
observable peaks are uniformly spaced in energy. This observation cannot be
explained with local moiré potential variations[69], or filling of a single
moiré trap with dipolar excitons [77], but rather points towards a specific
phonon mode modulating the spectral properties of the IX emission. The uniform
energy spacing is plotted versus an integer number $N$ in Figure 2c, relative
to the ZPL, depicted as peak number $0$. In semiconductors, the vibrational
modes that lead to sideband emission are usually lattice modes [4, 8]. The
observed spacing $\sim 0.8\,$meV is an energy that is smaller than the
interlayer breathing mode energy of MoSe2/WSe2 HBs [56, 29] suggesting that
the phonon is located at the $ZA$ acoustic branch [30, 28]. In general,
phonons involved in the IX emission are located at the edge of the mini-
Brillouin zone to absorb the momentum mismatch of the IX relative to the light
cone (see Fig.1b). Additionally, as the folded phonon dispersion flattens
towards the edge of the mini-Brillouin zone, it displays a maximum in the
phonon density of states, enabling efficient e-ph coupling. However, the size
of the mini-Brillouin zone depends on $\theta$ and, therefore, our samples
stacked at 4.5∘ and 57∘ should display different energy spacing, which we do
not observe. A second option for the involved vibrational modes are local and
quasi-local modes [78, 79]. These modes are associated with defects and might
imply that the presence of the moiré localized IX strongly distorts the
lattice and decouples the involved phonons from the general lattice
dispersion. In such a case, the polarization of the lattice and therefore the
phonon energy may weakly depend on $\theta$, but should have a dependence on
the IX density, i.e. on $P_{ex}$. As we show below, such a power dependence
was not observed. The precise identification of the phonon mode is out of the
scope of this paper in which we limit the discussion to the polaronic nature
of the IX emission.

Electron-phonon coupling strength is usually parameterized via the Huang-Rhys
factor[80, 81, 82]. In SM III.3.2 we calculated the e-ph coupling obtaining an
estimated value of $3.1\pm 0.4$ and $1.5\pm 0.2$ for the samples stacked in
the $3R$ and $2H$ configurations, respectively.

Figure 3: a) Temperature dependent PL spectra for the 2$H-$sample. On top, the
region of $\pm 12\,$meV from the ZPL is marked with orange(violet) to define
the low(high) energy sideband. b) PL intensity integrated on the spectral
regions defined in a. In orange(violet) is the low(high) energy sideband
intensity. Upper panel correspond to the 3$R-$sample and the bottom panel to
the 2$H-$sample.

To characterize the phonon influence on the IX emission, we performed
temperature ($T$) dependent PL spectroscopy. In all experiments, the CW laser
power was set to 300 nW and $E_{exc}=1.96$ eV. The spectra recorded for the
2$H-$sample are plotted in Figure 3a (the 3$R-$sample is shown in SM III.4);
all spectra are normalised to the intensity of the $T=7.2\,$K spectrum. The
vertical grey dotted line in the figure marks the position of the ZPL. With
increasing $T$, the high energy sideband rapidly decreases in intensity, while
the low energy sideband initially maintains its intensity and then decreases
only above a threshold of $\sim 20\,$K. This data is presented in Figure 3b,
where we plot the integrated intensity of each lobe as a function of $T$ for
both samples. The narrow emission lines progressively fade out, and cannot be
distinguished anymore from the broad background above $30\,K$. Importantly, we
note that they do not shift in energy throughout this $T-$range. Further
increasing $T$ above $35\,K$, the emission redshifts and successively
decreases in intensity to $\sim 5\%$ of the magnitude at $7\,$K.

The fade-out of the phonon replicas with increasing $T$ is a distinct
fingerprint of polaronic states [4, 22]. As the phonon population dressing the
IX emission is generated through charge transfer and energy relaxation
processes, IX and phonons are both out of thermal equilibrium and spatially
well-overlapped, specifically enhancing the phonon-IX interactions. The
observed $T-$dependence of the high-energy sideband is consistent with
stimulated processes. The $N^{th}$ polaron feature on the high-energy sidelobe
is an IX dressed with a positive number $N>0$ of phonons. By increasing $T$,
the phonon lifetime is reduced monotonically [83, 84] and the IX emission
intensity of the high energy sideband reduces accordingly. The low energy lobe
is composed of additional spontaneous emission of phonons, a process
independent of $T$. Consequently, the low-energy sideband maintains its
intensity until a threshold temperature, above which the thermal energy of the
crystal affects the radiative lifetime of the IX.

Figure 4: a) Power dependent PL spectra for the 2$H-$sample. The region of
$\pm 12\,$meV around the ZPL is marked with orange(violet) to define the
low(high) energy sideband. b) Integrated intensity of the high energy sideband
divided by the integrated intensity of the low energy sideband for the 3$R-$
(top) and the 2$H-$ (bottom) sample as a function of excitation power. The
transition from a low to a high power regime can be modeled with a rate
equation model (solid grey).

To support the scenario of non-thermal phonons generated during the IX
formation process and the relationship between spontaneous and stimulated
emission described above, we perform power-dependent and time-resolved
spectroscopy. In Figure 4a, we present the CW power-dependent PL for the
2$H-$sample, varying $P_{ex}$ from $12\,$nW to $9000\,$nW, at 7 K with the CW
laser at $E_{ex}=1.96$ eV. The data corresponding to the 3$R-$sample is shown
in SM III.5. In both cases, the sharp polaron peaks coalesce into a broad
emission line with increasing $P_{ex}$. Within the discernible power range,
the position of the narrow emission lines does not blueshift, similar to the
absence of any $T-$dependent shift observed above. The power dependent PL is
accompanied by an increasing dominance of the high energy sidelobe as compared
to the low energy sidelobe. This apparent blueshift of the overall IX emission
has been interpreted as dipolar IX-IX interactions in the literature [69, 40,
47, 85]. We show in the SM (SM III.5) that the sidelobes barely shift in
energy, consistent with recent literature taking into account a more
comprehensive view on IX interactions beyond simple dipolar repulsive
interactions [86].

We explain the observed $P_{ex}$ dependence using a phenomenological model,
making the following assumptions: i) The phonons contributing to the sidebands
are generated during the charge transfer and energy relaxation processes.
Consequently, the phonon occupation number is $n=n_{T}+n_{P}$, where $n_{T}$
is the thermal occupation and $n_{P}=C\cdot P_{ex}$ are the optically
generated phonons, proportional to $P_{ex}$ via a proportionality factor $C$.
ii) The $N^{th}$ emission line corresponds to $N-$ phonon absorption (for
$N>0$) and $N-$ phonon emission (for $N<0$). Note that, strictly speaking, the
$N^{th}$ emission line fulfills the following condition: $N=\alpha-\beta$,
where $\alpha$($\beta$) corresponds to phonon absorption(emission) processes
[4] and, therefore, it is not possible to fully isolate absorption and
emission processes. However, the approximation allows us to determine the
relative intensity between the low and high-energy sideband. For instance, the
PL intensity of the $N=1$ peak is then composed by the emission from all IXs
dressed with one phonon, whose abundance is proportional to $n$. Therefore,
its PL intensity is $I_{N=1}\propto n/\tau_{N=1}$, where $\tau_{N=1}$ is the
radiative lifetime of this polaron. Analogously, the PL intensity of the
$N=-1$ peak is, $I_{N=-1}\propto(n+1)/\tau_{N=-1}$, where $\tau_{N=-1}$ is its
radiative lifetime and the term $+1$ is the phonon spontaneous emission
probability. The ratio between these quantities becomes

$\frac{I_{N=1}}{I_{N=-1}}=\frac{\tau_{N=-1}}{\tau_{N=1}}\frac{n_{T}+n_{P}}{n_{T}+n_{P}+1}.$
(1)

The expression 1 leads to two interesting limits that describe the ratio
$I_{N=1}/I_{N=-1}$ for low and high $P_{ex}$:

$\lim_{n\to
0}\frac{I_{N=1}}{I_{N=-1}}=\frac{\tau_{N=-1}}{\tau_{N=1}}\frac{n_{T}}{n_{T}+1}$
(2)

and

$\lim_{n\to\infty}\frac{I_{N=1}}{I_{N=-1}}=\frac{\tau_{N=-1}}{\tau_{N=1}}.$
(3)

At low excitation power, the intensity between low and high energy sideband is
proportional to the radiative recombination rates and the thermal occupation.
In contrast, at high power, only the radiative recombination rates determine
the relative intensities. Further details of the model and, particularly, the
extension to $|N|>1$ are discussed in the Supplemental Material III.5.2.
Without loss of generality and considering the difficulty to isolate
individual polaron peaks in the spectra, we analyze the total integrated
intensity of the high and low energy sideband $I_{h}/I_{l}$ as a function of
power. Figure 4b shows this analysis for the 3$R-$(2$H-$) sample in the
top(bottom) panel. In both cases, the experimental data grows from a constant
value, in agreement with expression 2. By increasing $P_{ex}$, the 2$H-$sample
shows a monotonic increase and saturates to a value we identify from equation
3. Fitting equation 1 to the data of the the 2$H$ sample yields
$\tau_{l}/\tau_{h}=4.2\pm 0.1$ and $n_{T}=0.27$. This thermal population
number is consistent with a lattice temperature of $6\,$K, very close to the
experimental conditions. The deduced proportionality constant $C$ is $C=(9\pm
1)\times 10^{-4}$ nW-1 and, consequently, $P_{ex}=10\,\mu$W of optical power
produces an occupation number of $n_{P}\sim 10$, much higher than the thermal
occupation number and consistent with an effective $T$ of $\sim 100\,$K for
this particular mode (see SM III.5.2 for details). In the case of the
3$R-$sample, our data does not reach the limit $n\sim n+1$ and, therefore, the
fit to the data assumes $\tau_{l}/\tau_{h}=4.2$ as in the 2$H-$sample. In this
case, we obtain $n_{T}=0.23$, a thermal population corresponding to $5.5\,$K
and again in good agreement with experimental conditions.

Our phenomenological model predicts that the high-energy sideband might
display, on average, a faster radiative decay time than the low-energy
sideband. To test this prediction, we performed time and spectrally resolved
PL spectroscopy on the 3$R-$sample. The measurements were performed at
$T=4\,K$ with a diode laser at $635\,nm$, chopped with a TTL signal to obtain
excitation pulses of $1\,ns$ pulse length with a $0.5\,ns$ rise(fall) time. To
avoid the re-excitation of the sample before it reaches the equilibrium, the
repetition rate was set to $1/200\,$ns. Figure 5a shows the PL spectra
measured with the pulsed laser at an average power of $P_{ex}=20\,$nW and
$P_{ex}=20\,\mu$W. As before (see 4a), there is a clear blueshift of the
overall emission, but the ZPL and sharp lines observed in both spectra are
independent of excitation power. To time and spectrally resolve the IX
emission, we used a monochromator fiber-coupled to a single photon detector
resulting in a spectral resolution of $\sim 0.75$ meV. The time-resolved PL
shown in the false colour map 5b was obtained with an average
$P_{ex}=20\,\mu$W.

Figure 5: a) IX emission of the 2$H-$sample sample using a pulsed laser
excitation (pulse duration $1\,ns$, $E_{ex}=632$ nm) for $P_{ex}=20\,nW$ and
$20\,\mu W$. The ZPL is marked by a grey dotted line. b) False colour map of a
time resolved IX emission spectrum for $P_{ex}=20\,\mu W$. c) PL time traces
for $1.385\,eV$ (orange), $1.388\,eV$ (black, ZPL) and $1.394\,eV$ (violet). A
biexponential decay is observed. In grey, the instrument response function
(IR). d) Fast (black) and slow (red) decay time obtained from a bi-exponential
fit to the data in b).

Clearly, the majority of the IX-emission decays in the first few $ns$ after
the excitation, while a slower decay extends out into the 10s of $ns$.
Strikingly, the ZPL is visible at $\sim 1.388$ meV, particularly in the long
decay trail with higher intensity. We investigate time-dependent PL traces at
constant energy above and below the ZPL in Figure 5c. A clear bi-exponential
time dependence is observed, with a fast decay that depends on the PL energy.
The time constants for a bi-exponential fit to all-time traces are summarized
in Figure 5d, clearly showing an abrupt change of the fast decay across the
ZPL, varying from $\sim 1.6\,$ns below ZPL to $\sim 0.6\,$ns above ZPL.
Although the latter approaches our instrument response time,
$\tau_{l}/\tau_{h}\sim 2.6$ is in very good agreement with the value obtained
from the power-dependent PL fits of our rate equation model discussed above.
This internal consistency suggests that the fast decay time of the IX emission
is dominated by polaronic decay, where the non-thermal generation of phonons
are due to the IX-formation process. Meanwhile, the slower time constant
gradually changes across the spectral range with a peak at $1.394\,$eV, 6 meV
above the ZPL, supporting the fact that IX-emission depends on phonons and
revealing a longer intrinsic IX lifetime of $\sim 40\,$ns, consistent with
literature [87, 88, 89, 90, 91, 92, 93].

In summary, we have studied the photo-emission of two MoSe2/WSe2 HBs, with
twist angles of $\theta=4.5^{\circ}$ and $\theta=57^{\circ}$. Our results
strongly suggest that the narrow lines observed in the luminescence of these
HBs are attributable to polaronic sidebands. To validate these assumptions, we
performed temperature-dependent PL, excitation power-dependent PL, and
spectrally and temporally resolved PL experiments. In all cases, we found
compelling evidence supporting the validity of the polaronic picture we
proposed. The IX formation involves a complex interplay between the optically
excited electron-hole pair and the lattice, resulting in a multiple phonon
emission and absorption to relax the excess energy from $E_{ex}$ to the IX
energy [47, 48, 49, 50, 52, 55, 53]. As result of these processes, IX
polarises the HB lattice and open a path to interact with phonon acoustic
modes [57, 58]. Considering the low energy spacing between peaks, we propose
that the involved phonon belongs to the ZA phonon branch [30, 28]. Moreover,
as IXs are momentum indirect with a wavevector mismatch of
$\pi/d_{moir\acute{e}}$, we assess that the phonon is located at the edge of
the mini-Brillouin zone. Our power dependent PL suggests that by increasing
the optical excitation power, the generated phonon distribution is
significantly higher than the thermal population. This phonon population
dominates the IX dynamics and emission at high $P_{ex}$ and allows the
observation of the phonon absorption sideband in the IX emission. Our results
point to an enhanced interpretation of IX emission, emphasising the need for a
deeper understanding of the moiré potentials influence on the coupled IX and
lattice dynamics.

##  II Acknowledgements

We gratefully acknowledge the German Science Foundation (DFG) for financial
support via the SPP-2244 (DI 2013/5-1, FI 947/7-2, FI 947/7-1 and FA 971/8-1)
the clusters of excellence MCQST (EXS-2111) and e-conversion (EXS-2089). We
additionally acknowledge M. M. Glazov for the fruitful discussions on this
project. P.S. acknowledge the financial support of the DFG through the Walter
Benjamin program.

The Authors declare no Competing Financial or Non-Financial Interests.

##  III Supplemental material

###  III.1 Samples fabrication and characterization

####  III.1.1 Samples Fabrication

Monolayer (ML) MoSe2 and WSe2 were obtained from commercial bulk crystals via
mechanical exfoliation. The hexagonal crystal structure of a TMD lacks of
inversion symmetry and then the crystallographic directions are defined unless
a $60^{\circ}$ rotation. Thereby, Both HBs were produced from the same
monolayer MoSe2 and WSe2 and each monolayer was divided in two through the
tear-and-stack method [59], as sketched in figure 6a. During the fabrication
process one of the resulting MLs was rotated by 180∘ as depicted by arrows in
figure 6a and the micrographs in figure 6b and c. During the stacking of each
sample, the MLs were aligned to assemble two samples with a known twist angle
$\theta$ of approximately 4.5∘ and 57∘. In MoSe2/WSe2 HBs stacked near
0∘(60∘), atoms within each layer try to adjust the stacking landscape forming
reconstructed commensurate regions [60, 61, 62, 63, 64]. Therefore, we used
twist angles above 2.5∘ and below 59∘ for 2$H-$ and 3$R-$ stacking,
respectively, to avoid reconstruction [61, 62, 63]. The samples were
encapsulated between thin hexagonal boron nitride (hBN) flakes using dry
transfer techniques based on polycarbonate films, similar to Ref. 94.

Figure 6: a) Sketch of the WSe2 (red) and MoSe2 (green) monolayer and the site
in which they were cut during the tear-and-stack process. a) and b) Optical
micrographs of the HBs with near 3$R-$ and 2$H-$ stacking, respectively.
Green(red) lines indicate the MoSe2(WSe2) monolayer and the scale bar
correspond to 8 $\mu$m. Arrows in a, b and c depict the orientation of the
flakes. d) and e) Polar plot of the polarization-resolved SHG intensity in the
monolayer region measured for each sample. The relative angle between lobes in
d suggest a stacking angle of 4.5∘ for the sample in b and the plot in e a
stacking angle of 57∘ for the sample in c. f) and g) Integrated PL map in the
spectral range of the interlayer exciton emission. for the samples presented
in b and c, respectively.

The substrates used in this work correspond to piezoelectric strain actuators
as those used in Refs.95, 96, 97, 98, 99 and, therefore, have evaporated gold
on the surface. However, it is important to note that we do not present in
this work any effect related with strain as well as we do not expect any
effect of the gold substrate on the TMDs emission, in agreement with Ref.100.

The stacking angle was confirmed by performing polarized resolved Second
Harmonic Generation (SHG) experiments [101]. Figure 6d and e, present the
polar plot of the polarized resolved SHG from which the stacking angle was
determined.

Figure 6f and g present the interlayer exciton integrated emission for the
3$R-$ and the 2$H-$sample, respectively. While the shape of the emission
slightly depends on the position in the sample, all measurements presented in
this work correspond to regions in which the emission was spectrally sharp and
homogeneous. We observed that, in all such regions, the tendencies and
features observed and presented are consistent.

####  III.1.2 Optical selection rules.

Figure 7: a) and b) Schematics of the spin-valley configuration for samples
with stacking angle $\theta\rightarrow 0^{\circ}$ and $\theta\rightarrow
60^{\circ}$, respectively, the corresponding exciton complexes are plot over
the electronic dispersion of each sample. c) Polarized resolved interlayer
exciton PL (top) and circular polarization degree (bottom) for the samples
stacked at 4.5∘ (left) and 57∘ (right), respectively. The spectra are
normalized to the maximum intensity of the co-polarized PL in each case.

In MoSe2/WSe2 HBs, the 3$R-$stacking positions the WSe2
$K$($K^{\prime}$)-valley at the same point in the k-space as the MoSe2
$K$($K^{\prime}$)-valley and the 2$H-$stacking aligns the WSe2
$K$($K^{\prime}$)-valley with the MoSe2 $K^{\prime}$($K$)-valley. However, the
optical selection rules are governed not only by the valley and spin degrees
of freedom but also by the Bloch phase factor within the three-fold rotational
symmetry $\hat{C}_{3}$ [102, 68, 103]. The band alignment of the 3$R$ and 2$H$
HBs are illustrated in the Figure 7a and b, respectively. Additionally, the
sketches includes the exciton complexes observed in this work that were also
previously reported in the literature, they are a singlet interlayer exciton
(IXs) in both kind of stacking and the triplet interlayer exciton (IXt) in the
2$H$ case [68, 69].

Figure 7c shows in the upper panels the circularly co- and cross-polarized
photoluminescence (PL) for the 3$R-$ and the 2$H-$sample at left and right
panels, respectively. These measurements were performed with a CW 700 nm laser
and an excitation power of 40$\,\mu$W. Under the PL spectra, the lower panels
of Figure 7c show the circular polarization degree, defined as
$\eta=(I^{+}-I^{-})/(I^{+}+I^{-})$, were $I^{+}$($I^{-}$) is the PL intensity
in the co-(cross-)polarized experiments. Although we are not using the
conventional 3$R-$ and 2$H-$stacking, the sample stacked at 4.5∘(57∘) exhibits
circularly cross-polarized(co-polarized) photoluminescence (PL), consistent
with an interlayer emission from moiré sites in samples with 3$R-$(2$H-$)type
structures [102, 104, 69, 68, 105, 40].

###  III.2 PLE experiments at different excitation power

Figure 8 displays the PLE experiments for the 3$R-$ and 2$H-$sample,
respectively. Each figure compare the experiments performed at
$P_{ex}=1\,\mu$W and $P_{ex}=10\,\mu$W. In this range of energies, while the
sharp peaks are clearly distinguished, the observations are independent of
$P_{ex}$ and are the same as those discussed in the main text.

Figure 8: a) and b) PLE experiments at different excitation power for the
sample stacked in 3$R$ and 2$H$ configuration, respectively. On left and right
panels are the experiments performed at $P_{ex}=0.1\,\mu$W and
$P_{ex}=1\,\mu$W, respectively. Top panels: False color map of the IX emission
as function of $E_{ex}$. The horizontal dotted line marks the selected spectra
displayed in the bottom panels. Vertical dotted lines mark the ZPL.

###  III.3 Characterization of the phonon replicas

####  III.3.1 Additional information regarding the normal distribution of
peaks in the PL

Figure 9: a) and b) present the interlayer exciton emission in three different
spots on each sample, 3$R-$ and 2$H-$sample, respectively. On each panel, the
emmission peaks are marked with dotted vertical lines and the zero order with
a red line. c) and d) Peak energy as function of peak order for the spectra in
a) and b), respectively.

Figure 9a and b present, for the 3$R$ and the 2$H$ HBs, respectively, the PL
spectra taken at different positions of each sample. While the overall
emission lineshape varies along the sample, the narrow lines keep the spacing
of $\sim$0.8 meV. This shows that the phonon mode involved in the emission
presents a defined energy that rise from the moiré lattice, averaging many
sites. The presence of higher disorder in the lattice is then expressed by the
broadening of those lines and not by the peaks separation. For instance, the
third site of the 3$R$ sample displays a PL in which the peaks are
comparatively much broader and are only distinguishable as shoulders in the
overall emission.

It is important to note that all experiments presented in the main text were
performed on the first and second spot on each sample and, for this reason,
the ZPL was observed in PLE, temperature dependent PL or power dependent PL.
For this reason, the ZPL of the third spot on each sample is not properly
defined. In those cases the peak marked as ZPL is merely an estimate.

####  III.3.2 Additional information regarding electron-phonon coupling factor

The phonon-assisted PL equations can be solved analytically to calculate the
steady-state intensity of phonon replicas of arbitrary order [4, 8]. However,
these theoretical descriptions are based on some assumptions that do not apply
to IXs in HBs. First of all the calculations are based on the fact that,
generally, the thermal energy is much lower than that of the phonon modes
involved in the electron-phonon coupling. Consequently, only spontaneous
emissions are considered in the process, resulting in red-shifted phonon
sidebands. In our case, the electron hole couples produce a cascade process,
emitting phonons to relax the excess energy from the $intra$-layer exciton
energy to the IX state. Therefore, not only the thermal energy is higher than
the phonons implied in the e-p coupling, but also the stimulated processes
cannot be neglected.

As we lack of theoretical background including the stimulated processes in the
polaron description, we calculate the electron phonon coupling as usual, using
only the low energy phonon sideband. Polaron spectra are frequently described
through a summation of Lorentzian profiles of the form

$\sum_{N}=\frac{A_{N}\left(\frac{\Gamma_{N}}{2}\right)^{2}}{(\omega-\omega_{N})^{2}+\left(\frac{\Gamma_{N}}{2}\right)^{2}},$
(4)

where $\omega_{N}$ is the central frequency of the $N$th peak and $\Gamma_{N}$
its linewidth. In the strong coupling regime [80, 81, 82], the individual peak
intensity $A_{N}$ is well described by a Poisson distribution function

$A_{N}=A_{0}\frac{e^{-\alpha}\alpha^{N}}{N!}$ (5)

where $\alpha$ is the Huang-Rhys factor that characterize the strength of the
e-ph coupling and $A_{0}$ is the peak intensity of the ZPL. By fitting
expression 4 to the spectra from Fig.2b it is possible to obtain $A_{N}$ and
calculate $\alpha$ through expression 5.

Figure 9a and b presents, for the 3$R-$ and the 2$H-$sample, respectively, the
fitting of the spectra with multiple Lorentzian curves. Note that in our case,
it presents some difficulties because the distance between phonon replicas is
similar to their linewidth. The peaks used for the calculation are marked with
a grey rectangular shape. Figure 9c and d display the extracted peak
intensities that, although noisy, can be fitted with a Poisson distribution
curve resulting in a Huang-Rhys factor of $3.1\pm 0.4$ and $1.5\pm 0.2$ for
the 3$R-$ and the 2$H-$sample, respectively. Note that in expression 5
$\alpha$ is the e-ph coupling in three dimensions, we have scaled it by a
factor $3\pi/4$ for 2D systems [106, 21].

Figure 10: a) and b) PL spectra for the 3$R$ and 2$H$ samples, respectively.
On each spectra, the fitting of Lorentzian curves is presented in a colour
scheme varying from yellow to violet. The experiments are in black and the
fitting is superimposed in red. The grey curve in the 2$H$ sample correspond
to the background. c) and d) Extracted intensities of the peaks marked with a
rectangle in figure a and b. In grey, is the Poisson distribution fitting that
results in a Huang-Rhys factor of $3.1\pm 0.4$ and $1.5\pm 0.2$ for the 3$R-$
and the 2$H-$sample, respectively. Note that in the 3$R-$sample, we did not
consider the intensity of the peaks -3 and -6 because they presented an
unexpected intensity. far above the rest of the distribution.

###  III.4 Additional information regarding the temperature series

Figure 11: Temperature dependent PL spectra for the 3$R-$sample (left) and the
2$H-$sample (right).

In this section we complement the information regarding the temperature
dependent PL presented in the main text by adding the data corresponding to
the 3$R-$sample. The experiments were performed with a CW laser, $P_{ex}=300$
nW and $E_{exc}=1.96$ eV. The spectra, normalized to their maximum intensity
at 7 K, are presented in Figure 11, the left(right) panel present the data for
the 3$R-$(2$H-$)sample. The grey dotted lines mark the position of the ZPL.
Both samples display the same behaviour; at low temperature high and low
energy sidebands show similar intensity but, by increasing the temperature,
the high energy sideband is strongly suppressed. The narrow emission lines
progressively fade out by increasing the temperature and, above $20$ K for the
high energy sideband and $30$ K for the lower one, they are almost
indistinguishable.

###  III.5 Additional information regarding the power series

####  III.5.1 Experiments

In this section we complement the information regarding the power dependence
experiments presented in the main text by adding the data corresponding to the
sample stacked at 4.5∘. In both samples, the experiments were performed at 7
K, with a CW laser and $E_{exc}=1.96$ eV. Figure 12a present the power
dependent PL, where left(right) panel correspond to the 3$R-$(2$H-$)sample. In
both cases, the distinct emission peaks coalesce into a broad emission line by
increasing $P_{ex}$. Additionally, there is a clear blueshift of the overall
emission by increasing $P_{ex}$, in agreement with previous reports.

Figure 12: a) Power dependent PL spectra for the 3$R-$sample (left) and the
2$H-$sample (right). On top of each panel, the region of $\pm 12\,$meV from
the ZPL is marked with orange(violet) to define the low(high) energy sideband.
b) Centroid energy of the total spectra (black) and the low and high energy
sideband in orange and violet, respectively.

A careful observation of Figure 12a reveals that the observed blueshift is due
to a variation in the relative intensities between the low and high energy
sidebands. Figure 12b present the centroid energy of the spectra in Fig. 12a
the upper(lower) panel correspond to the 3$R-$(2$H-$)sample. Black square dots
show, in both cases, the blueshift of the total IX emission centroid. This
observation is completely different when we consider the centroid of each
sideband, presented in orange(violet) dots for the lower(higher) energy
sideband. In this case, it is clear that each sideband is spectrally fixed,
and the $P_{ex}$ blueshift is currently an intensity increase of the upper
energy sideband respect to the lower energy one.

####  III.5.2 Phenomenological description

We described the power dependence PL through a phenomenological model that
accounts phonon replicas of arbitrary order under the following assumptions:
i) The phonons that contribute to the sidebands are generated during the
charge transfer and energy relaxation processes. Consequently, the phonon
occupation number is $n=n_{T}+n_{P}$, were $n_{T}$ is the thermal occupation
and $n_{P}$ are the optically generated phonons, ii) The $N^{th}$ emission
line corresponds to $N-$ phonon absorption (for $N>0$) and $N-$ phonon
emission (for $N<0$). Note that, strictly speaking, the $N^{th}$ emission line
fulfills the following condition: $N=\alpha-\beta$, where $\alpha$($\beta$)
corresponds to phonon absorption(emission) processes [4] and, therefore, it is
not possible to fully isolate absorption and emission processes. However, this
approximation allows us to determine the relative intensity between the low
and high energy sideband. It is well established in rate equation models that
the number of radiative decays from a particular excited level and then its
emission intensity in the spectra are proportional to $1/\tau$, where $\tau$
is the radiative lifetime of the transition. On the other hand, as the peaks
we describes are phonon replicas of the ZPL, their abundance is a function of
the phonon occupation factor. Following the peak labeling of the main texts,
$N>0$ are IX emission in which the IX is dressed with $N$ phonons and its
intensity in PL experiments is $n^{N}/\tau_{N}$. In the case $N<0$, however,
there is always the possibility of an spontaneous emission and, therefore, the
intensity of a phonon emission replica is $(n+1)^{-N}/\tau_{N}$. The ratio
between positive and negative phonon replicas of the same order is then

$\frac{I_{N>0}}{I_{N<0}}=\frac{\tau_{N<0}}{\tau_{N>0}}\left(\frac{n}{n+1}\right)^{N}=\frac{\tau_{N<0}}{\tau_{N>0}}\left(\frac{n_{T}+n_{P}}{n_{T}+n_{P}+1}\right)^{N}.$
(6)

Consequently, the limit $n_{P}\rightarrow 0$ becomes

$\lim_{n_{P}\to
0}\frac{I_{N>0}}{I_{N<0}}=\frac{\tau_{N<0}}{\tau_{N>0}}\left(\frac{n_{T}}{n_{T}+1}\right)^{N}$
(7)

and the limit $n_{P}\rightarrow\infty$

$\lim_{n_{P}\to\infty}\frac{I_{N>0}}{I_{N<0}}=\frac{\tau_{N<0}}{\tau_{N>0}}.$
(8)

Figure 13a shows for some $N$s the function
$\frac{\tau_{N<0}}{\tau_{N>0}}\left(\frac{n_{T}+n_{P}}{n_{T}+n_{P}+1}\right)^{N}$
considering $n_{T}=0.25$ and $n_{P}$ varying from 0 to 100.

Figure 13: a)
$\frac{\tau_{N<0}}{\tau_{N>0}}\left(\frac{n_{T}+n_{P}}{n_{T}+n_{P}+1}\right)^{N}$
considering $n_{T}=0.25$ and $n_{P}$ varying from 0 to 100 for some $|N|$
varying from 1 to 6. b) Thermal and power dependent occupation factors and
mode temperature derived from the $n_{P}$ and proportionality constant derived
in the main text.

In the main text, we did not make any distinction between modes of different
order and only considered the difference between high and low energy sideband.
As result, we obtained an average behaviour of the different phonon replicas
that, as expressed by eq. 8, allowed us to estimate the ratio between the low
and high energy sideband lifetime. On the other hand, the low excitation power
limit depends on $N$ (see eq. 7) and, therefore, the thermal population
derived from it is underestimated.

Figure 13b presents, considering the parameters obtained in the main text,
i.e. the average behaviour of eq. 6, the comparison between the thermal and
power dependent occupation factor. The right axes displays the mode
temperature, showing that at a relative low $P_{exc}$ the mode temperature is
much higher than that of the lattice ($\sim 7$ K).

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