# Supplementary Information for:
Observation of ultrafast interfacial Meitner-Auger energy transfer in a van
der Waals heterostructure

Shuo Dong1∗ Samuel Beaulieu1,2 Malte Selig3 Philipp Rosenzweig4 Dominik
Christiansen3 Tommaso Pincelli1 Maciej Dendzik1,5 Jonas D. Ziegler6 Julian
Maklar1 R. Patrick Xian1,7 Alexander Neef1 Avaise Mohammed4 Armin Schulz4 Mona
Stadler8 Michael Jetter8 Peter Michler8 Takashi Taniguchi9 Kenji Watanabe10
Hidenori Takagi4,11,12 Ulrich Starke4 Alexey Chernikov6,13 Martin Wolf1 Hiro
Nakamura4,14 Andreas Knorr3 Laurenz Rettig1∗ & Ralph Ernstorfer1,15∗

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195
Berlin, Germany

Université de Bordeaux - CNRS - CEA, CELIA, UMR5107, F33405, Talence, France

Nichtlineare Optik und Quantenelektronik, Institut für Theoretische Physik,
Technische Universität Berlin, 10623 Berlin, Germany

Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany

Department of Applied Physics, KTH Royal Institute of Technology, Hannes
Alfvéns väg 12, 114 19 Stockholm, Sweden

Department of Physics, University of Regensburg, Regensburg D-93053, Germany

Department of Engineering, University of Cambridge, Trumpington Street,
Cambridge CB2 1PZ, United Kingdom

Institute of Semiconductor Optics and Functional Interfaces, Research Center
SCoPE and IQST, University of Stuttgart, 70569 Stuttgart, Germany

International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

Research Center for Functional Materials, National Institute for Materials
Science, 1-1 Namiki, Tsukuba 305-0044, Japan

Department of Physics, University of Tokyo, 113-0033 Tokyo, Japan

Institute for Functional Matter and Quantum Technologies, University of
Stuttgart, 70569 Stuttgart, Germany

Institute for Applied Physics, Dresden University of Technology, Dresden,
01187, Germany

Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701,
USA

Institut für Optik und Atomare Physik, Technische Universität Berlin, 10623
Berlin, Germany

This file includes:

Supplementary Notes

Supplementary Methods

Supplementary Equation (1) to (88)

Supplementary Fig.1 to Supplementary Fig.14

Supplementary Table 1

References

Supplementary Notes

Characterization of the ML-WSe2/graphene heterostructure

Raman measurements were carried out at room temperature using a 532 nm laser
with a power of 1 mW and a spot size of 5 to 10 $\mathrm{\mu m}$. As shown in
Supplementary Fig. 1a, ML-WSe2 is confirmed by an intense peak at 250 cm-1
which comes from essentially degenerate A1g and E2g lattice vibration modes[1,
2]. Photoluminescence (PL) measurements are performed using another system
with a 532 nm excitation laser, power of 1 mW and spot size of 1 $\mu$m at
room temperature. In Supplementary Fig. 1b, ML-WSe2 on graphene presents two
weak PL peaks (778 nm and 914 nm) only slightly above the background. The peak
at 778 nm is close in energy to the A-exciton transition energy[3, 4]. The
origin of the peak at higher wavelength is unknown, and may come from the
existence of in-gap defect states. The weakness of the PL signals is
consistent with the quenching of PL known to occur for ML-TMDCs adjacent to
graphene[5].

The sample is protected by the Se capping layer before sending to our lab.
After introducing the sample into our ultrahigh vacuum (UHV) photoemission
end-station, we have annealed the sample for 15 minutes at 400∘C through
direct current heating to remove the Se capping. After annealing, we recorded
a low energy electron diffraction (LEED) pattern with the incident beam energy
of 95 eV, to verify the surface cleanliness and ordering (Supplementary Fig.
1c). The six outer sharp LEED spots come from the bottom ML graphene layer
(yellow box) and the inner six arc-shaped diffraction spots originate from the
top ML-WSe2 layer (red box). The occurrence of a well-oriented hexagonal
pattern of WSe2 spots aligned to the graphene pattern attests the epitaxial
nature of our heterostructure and single-domain lattice orientation. A certain
level of strain-induced misalignment between nanoislands of WSe2 with respect
to the graphene layer is evident from the azimuthal widths of the diffraction
spots.

Our sample exhibits areas without WSe2, since the top WSe2 layer consists of
spatially uniform small islands with some distance between the islands.
However, the relevant processes discussed in our manuscript, _i.e._ , both
interlayer hot electron injection with 1.2 eV pump and Meitner-Auger type
interlayer energy transfer with 1.55 eV pump, are based on the photoemission
signals from the heterostructure, but will not occur at an isolated graphene
or WSe2 layer. For example, the observation of the excited state population at
WSe2 with below-bandgap excitation is not possible for an isolated monolayer
WSe2. With near-resonant pump, the deep-lying holes would not be excited in a
pure graphene sample. Therefore, the signal discussed in our work is based on
the heterostructure area.

Additionally, the doping level of epitaxial graphene is different from the
pristine graphene without WSe2. We discuss possible reasons for this
observation: static charge (electron) transfer from graphene to WSe2, or the
decrease of electrons at the graphene/SiC interface with W or Se intercalation
as the referee suggested. In the latter case that a substantial part of the Se
or W elements is intercalated below the graphene layer, the intensity of the
diffraction pattern of SiC, _i.e._ , 6$\times$6 spots around the graphene (10)
and diamond-shape distributed spots around $\sqrt{3}\times\sqrt{3}$ SiC, would
be dramatically reduced. As we observed a clear SiC pattern in our LEED image
(Supplementary Fig.1c), this scenario seems unlikely. Second, we would expect
graphene bilayer growth with Se or W intercalation, which is also not
observed. Finally, there is no core level energy shift of SiC peaks after the
growth of WSe2[7], which demonstrates the band alignment at the graphene/SiC
interface (band bending of p-n junction) is not influenced by the top layer.
These observations rule out a significant contribution of W or Se
intercalation to the graphene’s doping level. Therefore, we conclude on a
static charge transfer from graphene to WSe2 is the major contribution to the
modification of the doping of graphene.

Figure 1: Optical characteristics and surface analysis of the heterostructure
sample. a, Raman measurement of ML-WSe2/graphene (Gr) at room temperature. An
intense peak at 250 cm-1 belongs to ML-WSe2, whereas other peaks belong to the
Gr/SiC substrate. b, Photoluminescence measurement of the ML-WSe2/Gr
heterostructure. The peak originating from the A-exciton of ML-WSe2 is marked
with the dashed red line. c, LEED pattern of the ML-WSe2/Gr heterostructure at
95 eV after annealing.

Delayed population rises at $\mathrm{K_{WSe_{2}}}$ and $\mathrm{Q_{WSe_{2}}}$
with the below-bandgap excitation

Upon 1.2 eV excitation, the time trace at the $\mathrm{K_{Gr}}$ points shown
in Fig.2f is by fitting with a single exponential decay function convolved
with the instrument response function (IRF), $I(t)=H(t-t_{0})\times(A\cdot
exp(-(t-t_{0})/\tau)+C)\otimes\mathrm{IRF}$. Here, $H(t)$ is the Heaviside
step function, $\mathrm{IRF}$ is Gaussian envelope function, $A$ is the
amplitude and $C$ is the offset. In contrast, as the $\mathrm{K_{WSe_{2}}}$
and $\mathrm{Q_{WSe_{2}}}$ valleys are populated by interlayer charge transfer
(ICT) processes, the corresponding time traces are fitted with an exponential
growth function to describe the ICT process adding with a single exponential
decay for relaxation. The delayed population rise between graphene and WSe2,
$\Delta t=51\pm 9\leavevmode\nobreak\ \mathrm{fs}$, is obtained by taking the
time difference between the peak at $\mathrm{K_{Gr}}$ and the time delay when
the population at the $\mathrm{K_{WSe_{2}}}$ valley reaches its maximum.

Exclusion of two/multiple-photon absorption in WSe2

Upon 1.2 eV excitation, we could rule out the two/multiple-photon absorption
of WSe2 based on three experimental observations. First, we find no excited
state population of WSe2 in the energy range higher than the conduction band
minimum from 1.0 to 2.5 eV as shown in Supplementary Fig.2a. The valence band
maximum of WSe2 is identified at $E-E_{F}=-1.1$ eV by the EDC analysis in
Fig.3d (main text). In the scenario of two-photon absorption, the photo-
induced carriers would be directly populated at $E-E_{F}=1.3$ eV. This direct
optical excitation is not observed in our measurement. The three-photon
absorption, corresponding to the excitation at $E-E_{F}=2.5$ eV, is more
difficult to happen and is neither observed. Second, the delayed population
rise in $\mathrm{K_{WSe_{2}}}$ and $\mathrm{Q_{WSe_{2}}}$ valleys compared to
the rise of hot carriers in graphene, as shown in Fig.2f (main text), is
strong evidence of the absence of the direct optical excitation in WSe2. The
hot carrier rise in graphene demonstrates the arrival of the pump pulse.
However, the excited-state population of the WSe2 layer appears $\sim$50 fs
after the pump excitation. Finally, by integrating the photoemission
intensity, we present the population dynamics of electron (ROI1 in
Supplementary Fig.2a) and hole components (ROI2 in Supplementary Fig.2a) in
WSe2 upon 1.2 eV excitation. Different from the hot electron dynamics (blue
curve in Supplementary Fig.2b), the intensity of holes (red curve in
Supplementary Fig.2b) stays constant without any signature of optical
excitation. The absence of hole dynamics could also be explained by an
ultrafast interlayer hole transfer on a time scale much shorter than our
temporal resolution. However, combined with the first two observations, we
exclude the occurrence of two/multiple photon absorption. With near-resonant
excitation, the two-photon absorption is also excluded since no excited-state
population is found at $E-E_{F}=2.0$ eV, which corresponds to the energy of
excited-state electrons by absorbing two pump photons.

Figure 2: Carrier dynamics upon the below-bandgap excitation a, Two
dimensional energy-momentum cut with 1.2 eV pump. b, By integrating the
photoemission intensity in ROI1 and ROI2 in a, the population dynamics of
electrons (blue curve) and holes (red curve) are extracted, respectively. The
conduction band of WSe2 is filled via interlayer hot electron injection. The
population of holes stay constant without the signature of optical excitation.
The time traces are normalized to total photoemission intensity at negative
time.

Identification of the Fermi level and distribution of excited-state carriers
at $\mathrm{K_{WSe_{2}}}$

In both experiments, the energy axis calibration has been performed using the
position of the Fermi level of graphene, which is obtained from the energy
distribution curve (EDC) at the Dirac point of graphene (Supplementary Fig. 3a
and d). Before optical excitation, the EDCs at the $\mathrm{K_{Gr}}$ point
(Supplementary Fig. 3b,e) are fitted with a Fermi-Dirac distribution function
at 300 K convolved with the IRF ($\sim$150 meV FWHM) determined by the energy
resolution of the spectrometer and the bandwidth of the probe pulses[6]. The
chemical potentials are set to be zero for both experimental conditions to
remove the XUV-probe-induced space charging effect in each measurement. The
EDCs in Supplementary Fig. 3b,e are integrated over a momentum window, $\Delta
k=0.1$ $\mathrm{\SIUnitSymbolAngstrom}^{-1}$ and selected at negative time
delay. Based on the energy reference obtained from the Fermi level fits, the
energy positions of the conduction band minima at the $\mathrm{K_{WSe_{2}}}$
point can be obtained from EDCs showing the excited state carrier
distributions of WSe2 upon resonant and off-resonant excitation as displayed
in Supplementary Fig. 3c and f, respectively. The photoemission intensity has
been normalized by the total electron count of the spectrum. The energy
difference of the carrier distribution, more specifically, the smaller kinetic
energy of excited carriers with 1.55 eV pump, arises from the exciton
formation upon the resonant excitation. By fitting with a single Gaussian
lineshape on top of a empirical second-order polynomial background, the
energies of excited-state carriers are extracted as $0.63\leavevmode\nobreak\
\textrm{eV}$ upon the resonant A-exciton excitation (Supplementary Fig. 3c)
and $0.73\leavevmode\nobreak\ \textrm{eV}$ upon the below-bandgap excitation
(Supplementary Fig. 3f). The background has been removed in the main text
(side figures in Fig.3a-b).

Figure 3: Fermi level calibration. a,d, The 2D photoemission intensity spectra
as a function of energy and momentum at time zero with 1.55 eV and 1.2 eV
pump, respectively. b,e, EDCs of $\mathrm{K_{Gr}}$ at negative time delay
fitted with a Fermi-Dirac distribution convolved with the energy IRF for 1.55
eV and 1.2 eV pump, respectively. The chemical potentials are aligned to zero
by rigidly shifting the energy axis in both cases. c, f EDCs at
$\mathrm{K_{WSe_{2}}}$ integrated within the first 100 $\mathrm{fs}$ obtained
with 1.55 eV and 1.2 eV pump, respectively. The dashed lines represent the
center of the excited-state carrier distributions extracted by a fitting
procedure (see text).

Identification of the Dirac point energy

To identify the energy position of the Dirac point, we selected an
energy-$k_{y}$ cut ($\Delta t<0\leavevmode\nobreak\ \mathrm{fs}$) at Dirac
point and along the green dashed line in Supplementary Fig. 4a. The small
titling angle allows us to see both valence bands clearly in Supplementary
Fig. 4b. We track the graphene valence band dispersion by fitting the momentum
distribution curves of occupied bands with two Voigt lineshape functions.
Then, each graphene valence band is fitted to a linear dispersion and the
Dirac point is estimated at the intersection of two lines (red and black),
$E-E_{F}=-0.10\pm 0.05\leavevmode\nobreak\ \mathrm{eV}$, in a reasonable
agreement with the previous characterization of similar heterostructures[7].
This energy/momentum cut is different than the one presented in the main text
(Fig.3) along the $\mathrm{\Gamma}$-K direction, which is featured by the
suppression of one side of the cone due to photoemission matrix element
effects (sublattice interference)[8]. In the main text, we choose this cut
direction because it allows us to clearly resolve the excited state dynamics
from both layers.

Figure 4: Experimental determination of the Dirac energy. a, 2D momentum
distribution map $I(k_{x},k_{y})$ at $E-E_{F}=-0.2$ eV and $E-E_{F}=-1.3$ eV.
At the boundary of the Brillouin zone, it shows the quasi-triangular-shaped
$\pi$ band of graphene. b, Energy/momentum cut $I(E,k)$ along the green dash
line in a showing the conical band dispersion of graphene. Red and black
markers indicate the band positions extracted from momentum distribution curve
fits. Lines are linear fits of the band positions, yielding the energy
position of the Dirac point of $E_{D}=-0.1\pm 0.05$ eV.

Identification of the Fermi velocity

The Fermi velocity along $\Gamma$-K direction is extracted as $v_{F}=(1.8\pm
0.1)\cdot 10^{6}\leavevmode\nobreak\ m/s$ as shown in Supplementary Fig. 5
using the same band dispersion tracking method in the above paragraph. The
energy/momentum spectrum is selected at negative time delay ($\Delta
t<0\leavevmode\nobreak\ \mathrm{fs}$). The Fermi velocities in the direction
perpendicular to $\Gamma$-K with a slightly tilting angle (Supplementary Fig.
4b) are $v_{F}=(1.1\pm 0.1)\cdot 10^{6}\leavevmode\nobreak\ m/s$ (red) and
$v_{F}=(1.5\pm 0.1)\cdot 10^{6}\leavevmode\nobreak\ m/s$ (black),
respectively. The Fermi velocity of graphene has been found to to be in the
range of $1\cdot 10^{6}$ to $3\cdot 10^{6}$ $\mathrm{m/s}$, depending on the
dielectric constant of the environment[9]. For epitaxially grown
heterostructures, the dielectric constant of the embedding graphene layer
between the bottom substrate and top TMDC layer could be modified by the
coverage sizes of TMDC layer and substrate material, as the dielectric
constant is determined by
$\epsilon=(\epsilon_{top}$+$\epsilon_{substrate})/2$. At the same time, the
Fermi velocity is also sensitive to the graphene’s doping level[10]. We would
like to note that because of the steep band dispersion of the graphene band,
the momentum of transiently excited intraband electron-hole pairs is small.
Therefore, it requires a relative small excitonic COM momentum $\mathbf{Q}$ to
fulfill energy and momentum conservation, which favors the IET process.

Figure 5: Experimental determination of the Fermi velocity. The
energy/momentum cut $I(E,k)$ along $\Gamma$-K direction with band positions
(red markers) and the linear fit of the band dispersion (red line).

The near-unity efficiency of IET

To estimate the efficiency of energy transfer, we performed the same
measurement (1.55 eV excitation) on bare ML-WSe2, which is prepared by scotch-
tape exfoliation and transferred on top of thin hexagonal boron nitrid (hBN)
with conductive TiO2 substrate (Supplementary Fig. 6a). The efficiency of the
energy transfer process is commonly defined by the lifetime of the ’donor’
material (here WSe2) with and without the ’acceptor’ material (here graphene)
as: $\eta_{ET}=(\tau_{ML}-\tau_{hetero})/\tau_{ML}$, where $\tau_{ML}$
represents the exciton lifetime of the bare ML-WSe2 and $\tau_{hetero}$ is the
exciton lifetime in the WSe2/graphene heterostructure. The excited-state
population dynamics at the $\mathrm{K_{WSe_{2}}}$ valley within each system
are presented in Supplementary Fig. 6b. The lifetimes of the ML sample
$\tau_{ML}=1616\pm 345$ fs is extracted by fitting with an exponential decay
function convolved with the IRF. The exciton lifetime of the heterostructure
$\tau_{hetero}=67\pm 7$ fs is obtained by solving the system of rate equations
as described in the main text. Thus, we obtain for the interlayer energy
transfer efficiency, $\eta_{ET}=96\pm 1\%$. This near-unity transfer
efficiency is supported by the underlying conservation of energy and momentum.
We note that the different sample fabrication methods of the bare ML and
heterostructure may have influence on the exciton lifetime. However,
picosecond to sub-nanosecond exciton lifetimes in ML samples, consistent with
our observations, have been reported for samples fabricated with various
methods[11, 12]. Therefore, we believe that the comparison with the exciton
lifetimes in the bare ML WSe2 sample provides a reasonable estimate of the
transfer efficiency.

Figure 6: Estimation of the IET efficiency. a, Schematic of the bare ML-WSe2
sample (blue slab) with the bottom hBN layer (yellow) mounted on a Nb:TiO2
substrate (grey). b, Time traces of the excited-state carriers at the
$\mathrm{K_{WSe_{2}}}$ valleys of the bare ML sample (black) and
heterostructure (red) sample, respectively.

Valence bands shifting and broadening effects

After photoexcitation, we observe shifting and broadening effects of the WSe2
valence band as shown in the 2D difference spectrum (Fig. 3c) and EDCs at
$\mathrm{K_{WSe_{2}}}$ (Fig. 3d). To extract the transient lineshape, we fit
the EDC of the top two VBs (VB1 and VB2) with two Gaussian functions on top of
an empirical second-order polynomial background (BG), $I(E)=A_{1}\cdot
exp(-\frac{(E-E_{1})^{2}}{2\omega_{1}^{2}})+A_{2}\cdot
exp(-\frac{(E-E_{2})^{2}}{2\omega_{2}^{2}})+\mathrm{BG}$, where $E_{1}$,
$E_{2}$ are peak positions and $\omega_{1}$, $\omega_{2}$ are the peak width.
Supplementary Fig. 7a-d present representative fitting results at four time
delays, $\Delta t<0$ fs, $\Delta t=0$ fs, $\Delta t=200$ fs and $\Delta
t=1000$ fs. Because of the large spectral overlap between VB1 and VB2, the
fitting is performed with the same shifting, $\Delta E_{1}(t)=\Delta
E_{2}(t)$, and broadening parameter ,
$\Delta\omega_{1}(t)=\Delta\omega_{2}(t)$, for the two peak functions,
assuming the VBs respond to the interfacial coupling and the excitation-
induced modification in the same way. The extracted time dependent peak shift
and linewidth parameter are shown in Supplementary Fig. 7e and f,
respectively. The band shifting reflects the electronic band gap
renormalization due to the ICT-induced hot electrons[13, 14]. Thus, a relative
time delay can be observed compared with the excited-state population dynamics
in the conduction bands. In contrast, the linewidth is a measure of the
photohole self-energy, which depends on the many-body interactions with
photoexcited carriers and phonons[15]. It follows more closely the transient
of overall excited carriers in the system (grey dashed curve in Supplementary
Fig. 7f).

Figure 7:  Transient lineshape of the valence bands of WSe2. a-d, The EDCs at
the $\mathrm{K_{WSe_{2}}}$ valley present the spectral features of the first
two valence bands at selected time delays, $\Delta t<0$ fs, $\Delta t=0$ fs,
$\Delta t=200$ fs and $\Delta t=1000$ fs. The EDCs are fitted with two
Gaussian functions describing VB1 (green dashed curve) VB2 (blue dashed
curve). The second-order polynomial background is shown as yellow dashed
lines. e, Transient peak position of VB1 (green). f Peak linewidth both of VBs
as function of time (yellow). The population dynamic of excited-states in the
WSe2 conduction band (grey) is shown in e-f as a reference.

Meitner-Auger type IET-induced hot electrons near the Fermi level

The Meitner-Auger-like interlayer energy transfer involves an intraband
excitation of the deep-lying valence electrons to the photo-generated hot
holes below the Fermi level. We observe the deep-lying hot holes, as shown in
Fig.3e(main text). However, to separate the contribution of Meitner-Auger IET
induced electrons from the photo-generated hot holes of graphene near the
Fermi level requires an accurate description of the intrinsic energy-momentum
dynamics in the graphene layer. The carrier dynamics in graphene after
photoexcitation include multiple contributions: intraband carrier-carrier
scattering, interband Auger heating, phonon-mediated cooling, and Auger
recombination. These components contribute to the energy-dependent dynamics
along the graphene bands[16, 17]. This is particularly important for states
close to the Fermi level, which are subject to hot carrier accumulation from
the highly excited states, carrier redistribution due to the thermalization,
and carrier annihilation via Auger recombination. Together with the
complication of interlayer charge and energy transfer, it is very challenging
to quantitatively disentangle these dynamics with our current energy and time
resolution.

Although a quantitative separation of the IET-induced electrons from the
photo-generated hot holes remains difficult, some information can be gained
from the energy distribution of hot carriers below and above the Fermi level
in Fig.3e. In particular, comparing the near-resonant and below-bandgap
excitation conditions proves insightful. The ratio of the relative intensity
increase above the Fermi level ($E-E_{F}=0-1$ eV, red area) to the signal
depletion below EF ($E-E_{F}=-1-0$ eV, blue area), $A_{el}/A_{h}$ can be
regarded as a measure of the electron-hole imbalance in the system, and we
find a significantly stronger imbalance $A_{el}/A_{h}=1.6$ for near-resonate
excitation, compared to $A_{el}/A_{h}=1.1$ for below-bandgap excitation. This
supports the occurrence of MA-type IET process. Compared with the almost
identical spectral weights of electrons and holes using below-bandgap
excitation, the unequal spectral areas with the 1.55 eV pump demonstrate the
carrier redistribution due to the strong interlayer interactions. While we
would like to note that the spectral intensities are subject to photoemission
matrix elements, therefore cannot be directly interpreted as electron and hole
occupations. These probe-related effects can be reasonably assumed to be
identical for the two excitation conditions, thus not affecting our
conclusions.

Pump fluence dependent Meitner-Auger interlayer energy transfer

The pump fluence plays an important role on the MA-type IET process, because
of the deep-lying valence electrons are excited to the photo-generated hot
holes. The high pump fluence generates large density of hot holes, _i.e._ ,
more vacancies for IET-induced intraband transition, which benefits the MA-
type interlayer coupling. On the other hand, the weak pump fluence would
suppress the MA-type IET process with the reduced photo-generated hole
density. To check this effect, we perform the same measurement with the near-
resonant excitation (1.55 eV) but weak pump fluence,
$F=0.9\leavevmode\nobreak\ \mathrm{mJ/cm^{2}}$, around half of the pump
fluence applied in the manuscript, $F=1.7\leavevmode\nobreak\
\mathrm{mJ/cm^{2}}$. We find the decreased intensity of the deep-lying hole
population, as shown in the Supplementary Fig.8. From the energy distribution
curve (EDC) of graphene band with the weak pump fluence (blue curve), a
depletion signal around $E-E_{F}=-1.8$ eV could be observed but weaker than
that with the high pump fluence.

Figure 8: Pump fluence dependent Meitner-Auger interlayer energy transfer. The
momentum-integrated spectrum of the graphene band when pumping with a weak
pump fluence (blue). The EDC of graphene with the high pump fluence (black;
applied in the main text) is also shown as a comparison.

Doping level dependence of interlayer energy transfer process

As we discussed in the main text, the optical excitation prepares the hot
holes near the Fermi level, which enables the intraband transition under the
mechanism of Meitner-Auger-type IET. The doping level of graphene plays an
essential role in the energy position of the photo-generated hot holes,
therefore, influencing the MA-type IET. To clarify the influence of the doping
level, we calculate the Meitner-Auger type transfer rate by increasing the
Fermi energy to $E_{F}=0.2\,\mathrm{eV}$. As shown in Supplementary Fig.9a, a
dramatically suppression of the Meitner-Auger IET is observed with the
transfer rate on the order of $0.06\cdot 10^{-3}\,\mathrm{meV}$, much smaller
than that of $3\,\mathrm{meV}$ based on our real sample system. Compared with
the Förster-type coupling (Supplementary Fig.9c), the maximum of the transfer
rate is even larger than that of the MA-type coupling.

However, we would like to note that MA-type IET could also happen in the
conduction bands with the n-doped graphene composition. The natural doping in
the graphene layer provides electrons in the conduction band, below the Fermi
level and above the Dirac point. After compensating with the photo-generated
hot holes near the Fermi level, the net electron population could be excited
to highly excited states via the MA-type IET process. By adding the equation
of motion for the conduction band electron occupation, we find a rate of the
MA transfer in the conduction band as follows.

$\displaystyle\Gamma_{\mathbf{Q}}=|W_{\mathbf{Q}}|^{2}\frac{4}{\hbar
v_{F}}\left(Q-\frac{E_{\mathbf{Q}}}{\hbar
v_{F}}\right)\left(f^{c}_{Q-\frac{E_{\mathbf{Q}}}{\hbar
v_{F}}}-f^{c}_{\mathbf{Q}}\right)\theta\left(Q-\frac{E_{\mathbf{Q}}}{\hbar
v_{F}}\right)$ (1)

with $f^{c}_{\mathbf{k}}=(\exp(\hbar v_{F}|\mathbf{k}|-\mu)/k_{B}T)+1)^{-1}$
and chemical potential $\mu$. It is similar to the transfer rate in the
valence band (details see section ’Meitner-Auger-like energy transfer’), but
with the electron occupation $f^{c}_{\mathbf{k}}$ in the conduction band.

As shown in Supplementary Fig.9b. the maximum transfer rate reaches
$1.6\,\mathrm{meV}$. Therefore, although the Förster-type IET is faster than
the intraband transition in the valence band, the MA-type IET in the
conduction band is the dominant process with highly n-doped graphene. All the
calculations are performed at room temperature.

Figure 9: Energy transfer rate of a heretostructure with the highly n-doped
graphene. a-b, The calculated Meitner-Auger transfer rate with
$E_{F}=0.2\,\mathrm{eV}$ in the valence band and conduction band,
respectively. c, The calculated Förster-type transfer rate in the same
condition.

Supplementary Methods

Characteristics of pump beams

In this work, we use two different pump beams which wavelengths are centred at
800 nm and 1030 nm, respectively (Supplementary Fig. 10). The pulse duration
of 1030nm pump line is $\sim$ 200 fs FWHM, while the transform-limited pulse
duration of 800 nm pump is $\sim$ 35 fs FWHM. In the measurement, the pump
fluence of 800 nm is $F_{800}=1.7\leavevmode\nobreak\ \mathrm{mJ/cm^{2}}$ and
that of 1030 nm is $F_{1030}=5.3\leavevmode\nobreak\ \mathrm{mJ/cm^{2}}$, with
the consideration of effective pump-probe overlap profile based on the
formula, $a=\frac{1}{\pi(\omega_{pump}^{2}+\omega_{probe}^{2})}$, in the work
of Harb _et al_[18]. The beam size of the pump is $\omega_{pump}=248\pm 20$
$\mathrm{\mu m}$ and that of probe pulse is $\omega_{probe}=80\pm 5$
$\mathrm{\mu m}$. Here, $\omega_{pump}$ and $\omega_{probe}$ are the
respective beam widths of the pump and probe beams.

Figure 10: Pump spectra. Excitation spectra of the two light sources used for
pumping as a function of wavelength.

Separating the interlayer charge and energy transfer by a rate equation model

As discussed in the main text, we observed the photoemission signatures of
both ICT and IET upon resonant A-exciton excitation. To extract the
corresponding transfer rates, $\Gamma_{ICT}$ and $\Gamma_{IET}$, we develop a
multi-state coupled rate equation model describing the interlayer charge and
energy flow, as well as the hot carrier relaxations. In Fig.4a, the time trace
of hot carriers in the CBM of WSe2 (black curve) includes the dynamics of
photo-generated excitons $N_{T}^{ex}$ and ICT-induced quasi-free electrons
$N_{T}^{el}$. The VB1 shifting (green curve in Fig.4a) mainly reflects the
dynamic of $N_{T}^{el}$. Therefore, it provides the possibility to disentangle
the dynamics of these two kinds of quasiparticles. Here, subscript _T_
represents TMDC. Simultaneously, the deep valence band holes in graphene
$N_{Gr}^{h}$ are populated by the IET process and recombine with the rate
$\Gamma_{h}$, as shown in Fig.4b. Finally, the dynamics of hot electrons in
graphene $N_{Gr}^{el}$ contains the ICT-induced charge flow (input and output
towards WSe2) and a decay process with the rate of $\Gamma_{el}$. With these
considerations, the complete dynamics across the interface can be described
with the following set of coupled rate equations:

$\displaystyle\dot{N}_{T}^{ex}=-\Gamma_{IET}N_{T}^{ex}+S(t)$ (2)
$\displaystyle\dot{N}_{T}^{el}=-\Gamma_{ICT}N_{T}^{el}+\Gamma_{ICT}N_{Gr}^{el}$
(3)
$\displaystyle\dot{N}_{Gr}^{el}=+\Gamma_{ICT}N_{T}^{el}-\Gamma_{ICT}N_{Gr}^{el}-\Gamma_{el}N_{Gr}^{el}+S(t)$
(4)
$\displaystyle\dot{N}_{Gr}^{h}=-\Gamma_{h}N_{Gr}^{h}+\Gamma_{IET}N_{T}^{ex}$
(5)

Here, $S(t)$ represents the optical excitation as a Gaussian-shaped pump
envelope function. By numerically solving the system of coupled differential
equations, and a global fit of the solution to the data, we obtain the IET
transfer time $\tau_{IET}=67\pm 7$ fs, and the ICT transfer time
$\tau_{ICT}=118\pm 18$ fs ($\tau=\hbar/\Gamma$). At the same time, we get the
relaxation times of electrons in graphene, $\tau_{Gr}^{el}=84\pm 7$ fs, and
that of the deep valence holes, $\tau_{Gr}^{h}=7\pm 4$ fs. The fitting results
are shown in Fig.4a and b.

Microscopic calculation of IET mechanisms

We perform microscopic calculations of the IET process mediated by the
Meitner-Auger, Förster and Dexter type mechanisms.

###  0.1 Meitner-Auger-like energy transfer

A schematic illustration of the Meitner-Auger type (MA) interlayer transfer is
depicted in the main text in Fig.4f. Here, an exciton in the TMDC recombines
non-radiatively; its energy excites an electron deep in the valence band of
graphene to states close to the Dirac point but in the valence band.

Starting point for the calculation of the MA-type interlayer coupling is the
Hamiltonian

$H_{F}=\sum_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}},\lambda,\lambda^{\prime},\nu,\nu^{\prime}}V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q^{\prime}},\mathbf{k^{\prime}}}\lambda^{\dagger}_{\mathbf{k}}\nu^{\dagger}_{\mathbf{q}}\nu^{\prime}_{\mathbf{q^{\prime}}}\lambda^{\prime}_{\mathbf{k^{\prime}}}.$
(6)

As a convention, we use $\lambda^{(^{\prime})}$ as band indices and
$\mathbf{k}^{(^{\prime})}$ as momenta in WSe2 layer and $\nu^{(^{\prime})}$ as
band indices and $\mathbf{q}^{(^{\prime})}$ as momenta in graphene. The
appearing matrix element is formally given as

$V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q^{\prime}},\mathbf{k^{\prime}}}=\int_{\mathbb{R}^{3}}d^{3}r\int_{\mathbb{R}^{3}}d^{3}r^{\prime}\Psi^{\lambda*}_{\mathbf{k}}(\mathbf{r})\Psi^{\nu*}_{\mathbf{q}}(\mathbf{r^{\prime}})V(\mathbf{r},\mathbf{r^{\prime}})\Psi^{\nu^{\prime}}_{\mathbf{q^{\prime}}}(\mathbf{r^{\prime}})\Psi^{\lambda^{\prime}}_{\mathbf{k^{\prime}}}(\mathbf{r}).$
(7)

The band indices in the TMD are restricted to interband transitions
$\lambda\neq\lambda^{\prime}$ but the band indices in graphene are taken as
the valence band $\nu=\nu^{\prime}=v$. The remaining integrals can be
evaluated within a $\mathbf{k}\cdot\mathbf{p}$ expansion. Last we introduce
exciton operators in WSe2
$P_{\mathbf{Q}}^{\mu}=\sum_{\mathbf{q}}\varphi_{\mathbf{q}}^{\mu}c^{\dagger}_{\mathbf{q}+\frac{m_{e}}{m_{h}+m_{e}}\mathbf{Q}}v_{\mathbf{q}-\frac{m_{h}}{m_{h}+m_{e}}\mathbf{Q}}$
with quantum state $\mu$ and COM momentum $\mathbf{Q}$. The final Hamiltonian
reads

$H=\sum_{\mathbf{k},\mathbf{Q},\mu}W^{\mu}_{\mathbf{Q}}P_{\mathbf{Q}}^{\dagger\mu}v_{\mathbf{k-Q}}^{\dagger}v_{\mathbf{k}}+h.c.\,,$
(8)

with the coupling element

$W_{\mathbf{Q}}=\frac{1}{e}V_{\mathbf{Q}}\mathbf{d}^{cv}\cdot\mathbf{Q}\varphi^{*\mu}(\mathbf{r}=0).$
(9)

In the following we restrict ourselves to the lowest bound excitons $\mu=1s$.
From this Hamiltonian we calculate the equation of motion for the exciton
occupation in the TMD $N_{\mathbf{Q}}=\langle
P^{\dagger}_{\mathbf{Q}}P_{\mathbf{Q}}\rangle$ and the electron occupation in
the valence band of graphene $f_{\mathbf{k}}=\langle
v^{\dagger}_{\mathbf{k}}v_{\mathbf{k}}\rangle$ by exploiting Heisenberg
equation of motion.

The resulting equations of motion read

$\displaystyle\partial_{t}N_{\mathbf{Q}}=\frac{2\pi}{\hbar}\sum_{\mathbf{k}}|W_{\mathbf{Q}}|^{2}$
$\displaystyle\left(f_{\mathbf{k}}(1-f_{\mathbf{k-Q}})-N_{\mathbf{Q}}(f_{\mathbf{k-Q}}-f_{\mathbf{k}})\right)\delta(\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{k-Q}}-E_{\mathbf{Q}})$
(10)
$\displaystyle\partial_{t}f_{\mathbf{k}}=\frac{2\pi}{\hbar}\sum_{\mathbf{Q}}|W_{\mathbf{Q}}|^{2}$
$\displaystyle\left(f_{\mathbf{k-Q}}(1-f_{\mathbf{k}})-N_{\mathbf{Q}}(f_{\mathbf{k}}-f_{\mathbf{k-Q}})\right)\delta(\epsilon_{\mathbf{k-Q}}-\epsilon_{\mathbf{k}}-E_{\mathbf{Q}})$
(11) $\displaystyle+\frac{2\pi}{\hbar}\sum_{\mathbf{Q}}|W_{\mathbf{Q}}|^{2}$
$\displaystyle\left(N_{\mathbf{Q}}(f_{\mathbf{k-Q}}-f_{\mathbf{k}})-f_{\mathbf{k}}(1-f_{\mathbf{k-Q}})\right)\delta(\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{k-Q}}-E_{\mathbf{Q}})$
(12)

###  0.2 Estimation of the decay rate of WSe2 excitons

From the Boltzmann equation we can identify the decay rate of WSe2 excitons as

$\Gamma_{\mathbf{Q}}=8\pi\sum_{\mathbf{k}}(f_{\mathbf{k-Q}}-f_{\mathbf{k}})\delta(\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{k-Q}}-E_{\mathbf{Q}}),$
(13)

where we have added a factor of 4 to account for the valley and spin degree of
freedom in graphene. Analyzing the Dirac distribution, we find that
$\mathbf{k}$ accounts for electrons close the Dirac point, and
$\mathbf{k}-\mathbf{Q}$ for electrons deep in the valence band. In order to
get a simple expression for the decay rate, we assume that the electrons close
to the Dirac point have much smaller momenta than the electrons deep in the
valence band, i .e. $\mathbf{k}\ll\mathbf{Q}$ and
$\mathbf{k+Q}\approx\mathbf{Q}$.

This way, the Dirac distribution and the $\mathbf{k}$ can be evaluated
analytically yielding

$\Gamma_{\mathbf{Q}}=|W_{\mathbf{Q}}|^{2}\frac{4}{\hbar
v_{F}}\left(Q-\frac{E_{\mathbf{Q}}}{\hbar
v_{F}}\right)\left(f_{\mathbf{Q}}-f_{Q-\frac{E_{\mathbf{Q}}}{\hbar
v_{F}}}\right)\theta\left(Q-\frac{E_{\mathbf{Q}}}{\hbar v_{F}}\right)$ (14)

The rate depends on the matrix element of the MA transfer, the density of
states in graphene and on the occupation difference of the involved states in
graphene which accounts for the Pauli blocking. The heavyside function
$\theta\left(Q-\frac{E_{\mathbf{Q}}}{\hbar v_{F}}\right)$ accounts for the
fact, that a minimal momentum is required to fulfil the energy and momentum
conservation during the intervalley transfer.

Fig. 4c in the main text illustrates the MA rate of WSe2 excitons to graphene
for the photo-induced hole vacancies at different energy of $\mu_{Gr}^{h*}$.
We adjusted the graphene dispersion to the results from the ARPES measurement.
From $\tau=\hbar/\Gamma$ we find scattering times of 270 fs
($\mu_{Gr}^{h*}=-0.3\,\mathrm{eV}$), 210 fs
($\mu_{Gr}^{h*}=-0.4\,\mathrm{eV}$), and 175 fs
($\mu_{Gr}^{h*}=-0.5\,\mathrm{eV}$).

###  0.3 Origin of the finite center-of-mass (COM) momentum

The MA mediated IET requires nonzero COM momentum of the exciton. For example,
the required COM momentum is $\sim$ 1.3 nm-1 as shown in Fig.4c (main text)
with $\mu_{Gr}^{*}=-0.3\leavevmode\nobreak\ \mathrm{eV}$, which corresponds to
a kinetic energy of approximately 100 meV based on the effective mass of the
exciton $m_{ex}=0.65m_{e}$.[19] Then, where does the energy (momentum) come
from? In the following, we discuss the possible origins of COM momentum of the
exciton which is quantified by the kinetic energy assuming the parabolic
excitonic band dispersion.

At room temperature, the mean kinetic energy of the excitons is 25.6 meV which
is not enough to explain the required energy. Therefore, we calculate the
temporal evolution of the exciton energy-momentum occupation during the
optical pump for detuned excitation[20]. The equation of motion for the
excitonic coherence in the rotating frame reads

$\dot{P}_{\mathbf{0}}(t)=\frac{1}{i\hbar}\left(E_{\mathbf{0}}-\hbar\omega_{L}-i\gamma\right)P_{\mathbf{0}}(t)+\mathbf{d}\cdot\mathbf{E}(t),$
(15)

where the first term accounts for the detuning of the excitonic transition
energy $E_{\mathbf{0}}$ from the light pulse energy $\hbar\omega_{L}$.
$\gamma$ accounts for the dephasing of the excitonic coherence with
contributions from radiative and exciton phonon coupling[21]. The last term
accounts for the optical excitation with the dipole element $\mathbf{d}$ and
the exciting electric field $\mathbf{E}(t)$. The equation of motion for the
incoherent exciton occupation reads

$\displaystyle\dot{N}_{\mathbf{Q}}=\Gamma^{Form}_{\mathbf{Q}}|P_{\mathbf{0}}|^{2}+\sum_{\mathbf{K}}\Gamma^{in}_{\mathbf{Q,K}}N_{\mathbf{K}}-\sum_{\mathbf{K}}\Gamma_{\mathbf{Q,K}}^{out}N_{\mathbf{Q}}.$
(16)

The first term accounts for the formation of incoherent exciton occupation
from phonon induced dephasing from the excitonic coherence. The last two terms
account for the thermalization of incoherent excitons[22]. The coupling
element of the exciton formation reads

$\Gamma_{\mathbf{Q}}^{Form}=\frac{2}{\hbar}\sum_{\pm,\alpha}|g_{\mathbf{Q}}|^{2}\left(\frac{1}{2}\pm\frac{1}{2}+n_{\mathbf{Q}}^{\alpha}\right)\frac{\gamma}{(E_{\mathbf{Q}}-\hbar\omega_{L}\mp\hbar\Omega^{\alpha})^{2}+\gamma^{2}}.$
(17)

with the energy $\hbar\Omega_{\mathbf{Q}}^{\alpha}$ and the occupation
$n_{\mathbf{Q}}^{\alpha}$ of phonons in the branch $\alpha$ with momentum
$\mathbf{Q}$. The $\pm$ summation accounts for phonon emission/absorption
processes.

Supplementary Fig. 11a illustrates the snapshots of the exciton occupation
directly at the maximum of the pump pulse as a function of kinetic energy with
selected detuning pump photon energy above the excitonic transition energy.
The temperatures are set as room temperature for all the calculation. With
increasing pump photon energy, the amount of injected excitons decreases due
to the non-resonant excitation, eq. 15. However, at larger detunings, excitons
occupy larger energy states due to the excess energy of the pump pulse, which
is provided by acoustic and optical phonon transitions. The Supplementary Fig.
11b illustrates the exciton occupations but normalized to the maximum. Here it
is even more obvious that the excitons obtain higher energies as the detuning
increases. Interesting, for larger detunings two maxima can be observed, where
the higher one originates from the formation of excitons via acoustic phonon
scattering. The lower peak originates from the formation of excitons via
optical phonon emission but also from relaxation of excitons from the higher
peak via optical phonon emission.

Supplementary Fig. 11c illustrates snapshots of the exciton occupation
directly at the maximum of the pump pulse as a function of kinetic energy for
selected detunings below the excitonic transition energy. Similar to the
previous scenario, the density of injected excitons decreases with increasing
detuning due to the non-resonant excitation, eq. 15. In Supplementary Fig.
11d, the energy-dependent exciton occupations are normalized to their maximum.
Interestingly, for pumping with larger negative detunings, the exciton
distribution broadens such that the relative exciton occupation at large
kinetic energies increases. The reason is, that for larger detunings the
Lorentzian in equation 17 flattens which results in higher occupation of hot
exciton at large energy range. As a consequence, for the near-resonant
excitation below the excitonic transition, a substantial amount of excitons is
formed at energies above 100 meV which contribute to the Meitner-Auger
scattering. To conclude, non-resonant excitation of the exciton, both above
and below the resonance, introduce hot excitons with high kinetic energy,
which are subjected to the Meitner-Auger IET.

In our experiment, the pump photon energy is
$\hbar\nu_{pump}=1.55\leavevmode\nobreak\ \mathrm{eV}$ and the A-exciton
transition energy is $E_{ex}=1.63\leavevmode\nobreak\ \mathrm{eV}$ determined
by the energy difference of excited-state particles at CBM and VBM. After the
photoexcition which prepares the coherent excitons with zero COM momentum
(Supplementary Fig. 11e), the phonon-assisted dephasing process transfers the
coherent excitons to incoherent exciton population which gain the finite COM
momenta (Supplementary Fig. 11f). This dephasing process has been observed by
our previous study[23]. The subsequent thermalization of excitons at the
excitonic states (Supplementary Fig. 11g) also contribute to the nonzero COM
momenta which is already included in our calculation.

Figure 11:  Finite COM momentum of excitons at the $\mathrm{K_{WSe_{2}}}$
valley. a, The calculated energy-dependent hot exciton occupation with detuned
pump photon energy above the resonant excitonic transition energy. b,
Normalized hot exciton distributions in a. c The hot exciton occupation with
detuned pump photon energy below the resonant excitonic transition energy and
d is the corresponding normalized hot exciton distributions. Schematic
illustrations of involved ultrafast dynamics: e, photoexcitation creates the
coherent exciton. The dash curve represents the coherent excitonic state. f,
The phonon-assisted dephasing process transfers coherent excitons to
incoherent excitons, at the same time, increases the COM momentum of excitons.
g, The following thermalization also contributes to the finite COM momentum.

###  0.4 Förster coupling

To calculate the Förster rate from WSe2 to graphene, we start with the
Hamiltonian

$H_{F}=\sum_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}},\lambda,\lambda^{\prime},\nu,\nu^{\prime}}V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q^{\prime}},\mathbf{k^{\prime}}}\lambda^{\dagger}_{\mathbf{k}}\nu^{\dagger}_{\mathbf{q}}\nu^{\prime}_{\mathbf{q^{\prime}}}\lambda^{\prime}_{\mathbf{k^{\prime}}}.$
(18)

As a convention, we use $\lambda^{(^{\prime})}$ as band indices and
$\mathbf{k}^{(^{\prime})}$ as momenta in WSe2 layer and $\nu^{(^{\prime})}$ as
band indices and $\mathbf{q}^{(^{\prime})}$ as momenta in graphene. The
appearing matrix element reads

$V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q^{\prime}},\mathbf{k^{\prime}}}=\int_{\mathbb{R}^{3}}d^{3}r\int_{\mathbb{R}^{3}}d^{3}r^{\prime}\Psi^{\lambda*}_{\mathbf{k}}(\mathbf{r})\Psi^{\nu*}_{\mathbf{q}}(\mathbf{r^{\prime}})V(\mathbf{r},\mathbf{r^{\prime}})\Psi^{\nu^{\prime}}_{\mathbf{q^{\prime}}}(\mathbf{r^{\prime}})\Psi^{\lambda^{\prime}}_{\mathbf{k^{\prime}}}(\mathbf{r}).$
(19)

Here, $\Psi^{\lambda/\nu}_{\mathbf{k}/\mathbf{q}}$ account for the electronic
Bloch waves in WSe2 and graphene. The appearing Coulomb potential shall take
into account the dielectric environment of the heterostructure, including the
WSe2 and graphene layer which are distanced by a gap with dielectric constant
$\epsilon_{R}$ and width $z$ (closely stacked structures have $z=0$)[24].
Additionally we take substrates below and above the structure into account.

We can evaluate the matrix element by Fourier transforming the Coulomb
potential and calculating the real space integrals within a $\mathbf{k\cdot
p}$ expansion[25]. We introduce exciton operators in WSe2
$P_{\mathbf{Q}}^{\mu}=\sum_{\mathbf{q}}\varphi_{\mathbf{q}}^{\mu}c^{\dagger}_{\mathbf{q}+\frac{m_{e}}{m_{h}+m_{e}}\mathbf{Q}}v_{\mathbf{q}-\frac{m_{h}}{m_{h}+m_{e}}\mathbf{Q}}$
with quantum state $\mu$ and COM momentum $\mathbf{Q}$ as well as pair
operators in graphene
$R_{\mathbf{Q}}^{\mathbf{q}}=c^{\dagger}_{\mathbf{q}+\frac{1}{2}\mathbf{Q}}v_{\mathbf{q}-\frac{1}{2}\mathbf{Q}}$.
The Hamiltonian then reads

$H_{F}=\sum_{\mathbf{Q},\mathbf{q},\mu}F_{\mathbf{Q}}^{\mu}(z)P^{\dagger\mu}_{\mathbf{Q}}R_{\mathbf{Q}}^{\mathbf{q}}+h.c.$
(20)

The appearing coupling element reads

$F_{\mathbf{Q}}^{\mu}(z)=\frac{1}{e^{2}\sqrt{A}}V_{\mathbf{Q}}(z)\varphi^{\mu}(\mathbf{r}=0)\mathbf{Q}\cdot\mathbf{d_{T}}^{cv}\mathbf{Q}\cdot\mathbf{d}_{G}^{vc}$
(21)

with $\mathbf{d}_{T}^{vc}$ the dipole element in WSe2, $\mathbf{d}_{G}^{vc}$
the dipole element in graphene, $\varphi^{\mu}(\mathbf{r})$ the excitonic wave
function in real space with quantum number $\mu$ in WSe2. We restrict our
analysis to the lowest lying excitons $\mu=1s$.

The Förster induced transition rate is given as [26]

$\Gamma_{\mathbf{Q}}(z)=4\pi\sum_{\mathbf{q}}|F_{\mathbf{Q}}(z)|^{2}\delta\left(E^{\mathbf{q}}_{\mathbf{Q}}-E_{\mathbf{Q}}^{1s}\right),$
(22)

where we have already added a factor of 2 to account for the electron spin in
graphene. We analytically treat the summation over the delta function, where
the area which appears in equation (21) cancels. We arrive at

$\Gamma_{\mathbf{Q}}(z)=\frac{|F_{\mathbf{Q}}(z)|^{2}AE_{\mathbf{Q}}^{1s}}{2\hbar^{2}v_{F}^{2}}.$
(23)

$v_{F}$ is the Fermi velocity in graphene. The area $A$ cancels with the area
in $|F_{\mathbf{Q}}(z)|$. In a last step we average over the angle dependence
of $|F_{\mathbf{Q}}(z)|^{2}$, und sum the result over the $K$ and $K^{\prime}$
point in graphene, which is already included in the $\mathbf{q}$ summation in
eq. 22. This way we arive at the final expression

$\Gamma_{\mathbf{Q}}(z)=\frac{|V_{\mathbf{\mathbf{Q}}}(z)|^{2}|\varphi^{1s}(\mathbf{r}=0)|d_{T}^{2}d_{G}^{2}E_{\mathbf{Q}}^{1s}Q^{4}}{8\hbar^{2}v_{F}^{2}e^{2}}.$
(24)

The Coulomb potential $V_{\mathbf{\mathbf{Q}}}(z)$ is given as

$V_{\mathbf{Q}}(z)=\frac{e^{2}}{\epsilon_{0}|\mathbf{Q}|\epsilon_{\mathbf{Q}}(z)},$
(25)

where the momentum dependent dielectric function $\epsilon_{\mathbf{Q}}(z)$
accounts for the dielectric screening from the surrounding[24]. As input
parameters, we require the thickness of graphene and WSe2 layers and their
respective dielectric constants. Note, that in the limit of infinitely thin
films and a uniform background, our results coincides with our previous
one[22]. The required parameters are listed in table 1.

Fig. 4d in the main text illustrates the Förster transfer rate as a function
of COM momentum and for different WSe2 \- graphene distances. For
$\mathbf{Q}=0$ we find a vanishing Förster rate followed by a monotonous
increase. The large $\mathbf{Q}$ behavior is dictated by the interplay of the
momentum dependence of the Coulomb potential and the factor $Q^{4}$. For the
closest stacking, i.e. 0.0 nm we find a peak transition rate of about 0.08
meV.

Now, we would like to emphasize the difference between MA and Förster transfer
theoretically. The MA and Förster transfer are related and can be derived from
the same Hamiltonian since the physical origin of both mechanisms is the
Coulomb interaction between the involved two-dimensional materials. However,
the MA-type IET can be described as monopole-dipole interaction which is often
neglected in the Coulomb potential and different from the dipole-dipole
coupling term corresponding to the Förster-type IET. To illustrate this, we
present the detailed derivations of Dexter, Förster, and MA transfer using a
multipole expansion of the Coulomb interaction. We start from the Coulomb
Hamiltonian involving the typical Coulomb matrix element. Here, we use a
simple Coulomb potential
$V(\mathbf{r-r^{\prime}})=\frac{e_{0}^{2}}{4\pi\epsilon_{0}\epsilon}\frac{1}{|\mathbf{r-r^{\prime}}|}$
with a constant uniform background. However, in the manuscript, we use a more
complex nonlinear dielectric function valid for the TMDC-graphene interface
and treat it in the momentum space. The Hamiltonian involving all transfer
processes (Förster, Dexter, MA) reads

$\displaystyle H$
$\displaystyle=\sum_{\begin{subarray}{c}\lambda,\lambda^{\prime},\nu,\nu^{\prime}\\\
\mathbf{k,k^{\prime},q,q^{\prime}}\end{subarray}}V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}\lambda_{\mathbf{k}}^{\dagger}\nu^{\dagger}_{\mathbf{q}}\nu^{\prime}_{\mathbf{q}^{\prime}}\lambda^{\prime}_{\mathbf{k}^{\prime}}$
(26) $\displaystyle
V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}$
$\displaystyle=\int d^{3}r\int d^{3}r^{\prime}\leavevmode\nobreak\
\Psi^{*}_{\lambda,\mathbf{k}}(\mathbf{r})\Psi^{*}_{\nu,\mathbf{q}}(\mathbf{r^{\prime}})V(\mathbf{r-r^{\prime}})\Psi_{\nu^{\prime},\mathbf{q}^{\prime}}(\mathbf{r}^{\prime})\Psi_{\lambda^{\prime},\mathbf{k}^{\prime}}(\mathbf{r})\;,$
(27)

where $(\lambda^{(^{\prime})},\mathbf{k}^{(^{\prime})})$ stand for graphene
quantum numbers and $(\nu^{(^{\prime})},\mathbf{q}^{(^{\prime})})$ for WSe2
quantum numbers. We provide now a detailed derivation of all these processes:

The spatial vector $\mathbf{r}$ can be decomposed into a lattice vector
$\mathbf{R}_{n}$ pointing to the $n$th unit cell and a vector $\mathbf{r}_{n}$
defined locally in the $n$th unit cell:
$\mathbf{r}\rightarrow\mathbf{r}_{n}+\mathbf{R}_{n}$. Then the spatial
integral is changed into a sum over all unit cells and an integration over one
unit cell. For electronic Bloch functions of the form
$\Psi_{\lambda,\mathbf{k}}=\xi_{\mathbf{k}}(\mathbf{r})u_{\lambda,\mathbf{k}}(\mathbf{r})$,
with lattice periodic function $u_{\lambda,\mathbf{k}}(\mathbf{r})$ and
envelope $\xi_{\mathbf{k}}(\mathbf{r})$, we obtain

$\displaystyle
V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}$
$\displaystyle=\sum_{\mathbf{R}_{n},\mathbf{R}_{n}^{\prime}}\xi^{*}_{\mathbf{k}}(\mathbf{R}_{n})\xi^{*}_{\mathbf{q}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{q}^{\prime}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{k}^{\prime}}(\mathbf{R}_{n})$
$\displaystyle\times\int_{UC}d^{3}r_{n}\int_{UC}d^{3}r_{n}^{\prime}\leavevmode\nobreak\
u^{*}_{\lambda,\mathbf{k}}(\mathbf{r})u^{*}_{\nu,\mathbf{q}}(\mathbf{r^{\prime}})V(\mathbf{r}_{n}-\mathbf{r}^{\prime}_{n}+\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n})u_{\nu^{\prime},\mathbf{q}^{\prime}}(\mathbf{r}^{\prime})u_{\lambda^{\prime},\mathbf{k}^{\prime}}(\mathbf{r})\;.$
(28)

We assumed that the envelope is spatially constant over one unit cell and
exploited the periodicity of the functions
$u_{\lambda,\mathbf{k}}(\mathbf{r}).$ Next, we Taylor expand the Coulomb
potential at two points $(\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime})$:

$\displaystyle\frac{1}{|\mathbf{r}_{n}-\mathbf{r}_{n}^{\prime}+\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}|}$
$\displaystyle=\frac{1}{|\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}|}$
$\displaystyle+\frac{(\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime})}{|\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}|^{3}}\cdot(\mathbf{r}_{n}^{\prime}-\mathbf{r}_{n})$
$\displaystyle+\frac{\mathbf{r}_{n}\cdot\mathbf{r}_{n}^{\prime}}{|\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}|^{3}}-3\frac{(\mathbf{r}_{n}^{\prime}\cdot(\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}))((\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime})\cdot\mathbf{r}_{n})}{|\mathbf{R}_{n}-\mathbf{R}_{n}^{\prime}|^{5}}\;.$
(29)

The first line is the zeroth-order and corresponds to monopole-monopole
interaction between the two parts of the heterostructure. The second line is
the monopole-dipole interaction which corresponds to an intraband-interband
coupling. This term is often neglected in a rotating wave approximation when
applied to gaped structures, such as a semiconductor heterostructure. However,
in the case of graphene, this term can not be ignored since the vanishing
bandgap allows energetically favourite intraband transitions (if not Pauli-
blocked). Here, a coupling of intraband excitation in graphene and excitons in
TMDCs can occur. A typical effect is carrier multiplication[27, 28]. In our
study, we include this term which gives rise to the MA coupling, as a monopole
(graphene)-dipole (TMDC) interaction. The last line, appearing in the first
order of both arguments $\mathbf{r}_{n}$, $\mathbf{r}_{n}^{\prime}$,
constitutes the typical dipole-dipole interaction coupling (Förster-type
transfer).

Coming back to the Hamiltonian and inserting the last line of Taylor expansion
(Eq. (29)) into Eq. (28), we obtain the Förster-type coupling,

$\displaystyle
V^{\lambda\nu\bar{\nu}\bar{\lambda}}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}$
$\displaystyle=\frac{1}{4\pi\epsilon_{0}\epsilon}\sum_{\mathbf{R}_{n},\mathbf{R}^{\prime}_{n}}\xi^{*}_{\mathbf{k}}(\mathbf{R}_{n})\xi^{*}_{\mathbf{q}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{q}^{\prime}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{k}^{\prime}}(\mathbf{R}_{n})$
$\displaystyle\times\left(\frac{\mathbf{d}^{\lambda\bar{\lambda}}_{\mathbf{k,k^{\prime}}}\cdot\mathbf{d}^{\nu\bar{\nu}}_{\mathbf{q,q^{\prime}}}}{|\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n}|^{3}}-3\frac{\mathbf{d}^{\lambda\bar{\lambda}}_{\mathbf{k,k^{\prime}}}\cdot(\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n})\mathbf{d}^{\nu\bar{\nu}}_{\mathbf{q,q^{\prime}}}(\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n})}{|\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n}|^{5}}\right)$
(30)

with the dipole matrix elements of both materials defined as
$\mathbf{d}^{\lambda\bar{\lambda}}_{\mathbf{k,k^{\prime}}}=e_{0}\int_{UC}d^{3}r\leavevmode\nobreak\
u^{*}_{\lambda,\mathbf{k}}(\mathbf{r})\mathbf{r}u_{\bar{\lambda},\mathbf{k}^{\prime}}(\mathbf{r})$
where $\bar{\lambda}\neq\lambda$. As explicitly expressed, the Förster-type
transfer is referred to as a dipole-dipole coupling.

Next, we insert the second line of Eq. (29) into Eq. (28) to investigate the
MA-type coupling:

$\displaystyle
V^{\lambda\nu\nu^{\prime}\lambda^{\prime}}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}$
$\displaystyle=\frac{e_{0}^{2}}{4\pi\epsilon_{0}\epsilon}\sum_{\mathbf{R}_{n},\mathbf{R}_{n}^{\prime}}\xi^{*}_{\mathbf{k}}(\mathbf{R}_{n})\xi^{*}_{\mathbf{q}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{q}^{\prime}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{k}^{\prime}}(\mathbf{R}_{n})\frac{(\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n})}{|\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n}|^{3}}$
$\displaystyle\times\left(\int_{UC}d^{3}r_{n}\leavevmode\nobreak\
u^{*}_{\lambda,\mathbf{k}}(\mathbf{r}_{n})u_{\lambda^{\prime},\mathbf{k}^{\prime}}(\mathbf{r}_{n})\int_{UC}d^{3}r^{\prime}_{n}\leavevmode\nobreak\
u^{*}_{\nu,\mathbf{q}}(\mathbf{r}_{n}^{\prime})\mathbf{r}_{n}^{\prime}u_{\nu^{\prime},\mathbf{q}}(\mathbf{r}_{n}^{\prime})\right.$
$\displaystyle\left.-\int_{UC}d^{3}r_{n}\leavevmode\nobreak\
u^{*}_{\lambda,\mathbf{k}}(\mathbf{r}_{n})\mathbf{r}_{n}u_{\lambda^{\prime},\mathbf{k}^{\prime}}(\mathbf{r}_{n})\int_{UC}d^{3}r^{\prime}_{n}\leavevmode\nobreak\
u^{*}_{\nu,\mathbf{q}}(\mathbf{r}_{n}^{\prime})u_{\nu^{\prime},\mathbf{q}}(\mathbf{r}_{n}^{\prime})\right)\;.$
(31)

Here, the second line of Eq. (31) describes the interaction of a TMDC
interband transition ($r_{n}^{\prime}$-integral, $(\nu,\mathbf{q})$ WSe2
quantum numbers) and an intraband graphene transition ($r_{n}$-integral,
$(\lambda,\mathbf{k})$ graphene quantum numbers). Note, the third line of Eq.
(31) describes an interband transition in the graphene layer and an intraband
transition in the WSe2 layer. It is not feasible in our experiment and also
unlikely from theoretical considerations. Thus, we neglect this term. Finally,
we obtain

$\displaystyle
V^{\lambda\nu\bar{\nu}\lambda}_{\mathbf{k},\mathbf{q},\mathbf{q}^{\prime},\mathbf{k}^{\prime}}$
$\displaystyle=\frac{e_{0}}{4\pi\epsilon_{0}\epsilon}\sum_{\mathbf{R}_{n},\mathbf{R}_{n}^{\prime}}\xi^{*}_{\mathbf{k}}(\mathbf{R}_{n})\xi^{*}_{\mathbf{q}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{q}^{\prime}}(\mathbf{R}_{n}^{\prime})\xi_{\mathbf{k}^{\prime}}(\mathbf{R}_{n})\frac{(\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n})}{|\mathbf{R}_{n}-\mathbf{R}^{\prime}_{n}|^{3}}\mathbf{d}^{\nu\bar{\nu}}_{\mathbf{q,q^{\prime}}}\delta_{\mathbf{k,k^{\prime}}}\;,$
(32)

where we used the orthogonality of the lattice periodic functions to solve the
$r_{n}$-integral. As explicitly shown in Eq. (32), only one dipole element is
involved in MA-type transfer, distinguishable from the Förster-type dipole-
dipole coupling.

###  0.5 Distance dependence of Meitner-Auger-like and Förster-type energy
transfer

To study the distance dependence of the Meitner-Auger transfer and compare it
to the known dependence of the Förster process, we calculate MA- and Förster-
type transfer rates as a function of interlayer distance $z$. In Supplementary
Fig. 12 we show the $z$-dependence of the Meitner-Auger transfer and also the
Förster result, from which we observe a very different behaviour at long
distances. For the Förster transfer, we obtain an exponential decay
($\exp(-z)$) in the near-field and the well-known $z^{-4}$ dependence in the
far-field. In contrast to the Förster transfer, the Meitner-Auger transfer
decays solely exponentially. In principle, these two processes could be
distinguished by the different behaviours at long distances. However, both
processes decay exponentially at short distances, where they can not be
distinguished. To formally study the $z$-dependence of both mechanisms, we
assume a thermal distribution of the initial states. The corresponding
$Q$-dependent rate is then averaged over this thermal distribution (cp. for
instance, the approach of reference 14 in the main text). To study the
behaviour analytically, we assume a uniform dielectric environment.

The detailed computations are shown in the following: The Meitner-Auger rate
reads $\Gamma_{\mathbf{Q}}=|W_{\mathbf{Q}}|^{2}\frac{4}{\hbar
v_{F}}(Q-E_{\mathbf{Q}}/\hbar v_{F})(f_{\mathbf{Q}}-f_{E_{\mathbf{Q}}-Q/\hbar
v_{F}})\theta(Q-E_{\mathbf{Q}}/\hbar v_{F})$, where $W_{\mathbf{Q}}$ carries a
general Coulomb potential depending on the dielectric environment. In the
manuscript we consider a complex non-linear dielectric function to accurately
describe the two-dimensional materials on the substrate and evaluate the
matrix element in momentum space. However, to have analytical insights, we
assume now a Coulomb potential of the form
$V(\mathbf{r-r^{\prime}})=\frac{1}{4\pi\epsilon\epsilon_{0}}\frac{1}{|\mathbf{r-r^{\prime}}|}$
with uniform background. We can then start from the Coulomb matrix element Eq.
(32) forming $W_{\mathbf{Q}}$ in the Meitner-Auger rate. We perform a
coordinate transformation with the new coordinates
$\mathbf{s}=(\mathbf{R}_{n}+\mathbf{R}_{n}^{\prime}+\mathbf{z})/2$ and
$\mathbf{S}=\mathbf{R}_{n}-\mathbf{R_{n}^{\prime}-\mathbf{z}}$. Equation (32)
becomes

$\displaystyle
V^{\lambda\nu\bar{\nu}\lambda}_{\mathbf{k,q,q^{\prime},k^{\prime}}}=\frac{1}{4\pi\epsilon_{0}\epsilon}\sum_{\mathbf{s,Q}}e^{-i\mathbf{Q}\cdot(\mathbf{s+z})}\frac{\mathbf{s+z}}{|\mathbf{s+z}|^{3}}\delta_{\mathbf{Q,k-k^{\prime}}}\delta_{\mathbf{Q,q-q^{\prime}}}\mathbf{d}^{\nu\bar{\nu}}_{\mathbf{q,q-Q}}\;.$
(33)

The dipole element can be written as
$\mathbf{d}^{\nu\bar{\nu}}_{\mathbf{q,q-Q}}=d^{\nu\bar{\nu}}_{\mathbf{q,q-Q}}\mathbf{e}$
with polarization vector $\mathbf{e}$. The Hamiltonian Eq. (26) reads

$\displaystyle H$ $\displaystyle=\sum_{\begin{subarray}{c}\lambda,\nu\\\
\mathbf{q,Q,k}\end{subarray}}\frac{1}{4\pi\epsilon_{0}\epsilon}d^{\nu\bar{\nu}}_{\mathbf{q-Q}/2,\mathbf{q+Q}/2,}a(\mathbf{Q,z})\leavevmode\nobreak\
\lambda^{\dagger}_{\mathbf{k+Q}/2}\nu^{\dagger}_{\mathbf{q-Q}/2}\bar{\nu}^{\dagger}_{\mathbf{q+Q}/2}\lambda^{\dagger}_{\mathbf{k-Q}/2}$
(34) $\displaystyle\text{with}\quad a(\mathbf{Q,z})$ $\displaystyle=\int
d^{2}s\leavevmode\nobreak\
\frac{e^{-i\mathbf{Q}\cdot(\mathbf{s+z})}}{|\mathbf{s+z}|^{3}}(\mathbf{s+z})\cdot\mathbf{e}\;.$
(35)

The integral can be solved and reads

$\displaystyle a(\mathbf{Q},z)=-2\pi
i\frac{e^{-Qz}}{Q}\mathbf{Q}\cdot\mathbf{e}\;.$ (36)

To evaluate the $z$-dependence of the Meitner-Auger rate, we perform a thermal
average according to our previous work[26]:

$\displaystyle\frac{1}{\tau_{T}}=\langle\frac{1}{\tau}\rangle_{T}=\frac{1}{\Omega}\int
d^{2}Q\leavevmode\nobreak\ e^{-\beta
E_{\mathbf{Q}}}\frac{1}{\tau_{\mathbf{Q}}}=\frac{2}{\hbar\Omega}\int
d^{2}Q\leavevmode\nobreak\ e^{-\beta E_{\mathbf{Q}}}\Gamma_{\mathbf{Q}}$ (37)

with $\Omega=\int d^{2}Q\leavevmode\nobreak\ \exp(-\beta E_{\mathbf{Q}})$ and
$\beta=1/k_{B}T$. To calculate the integral, we set the out-scattering and in-
scattering occupation $f_{\mathbf{Q}}$ and $f_{E_{\mathbf{Q}}-Q/\hbar v_{F}}$
to one and zero, which correspond to a fairly rough approximation. The matrix
element $|W_{\mathbf{Q}}|^{2}$ is determined by the function
$|a(\mathbf{Q},z)|^{2}=2\pi^{2}\exp(-2Qz)$ after an angle average. The
solution of the integral reads

$\displaystyle\frac{1}{\tau_{T}}$
$\displaystyle=\frac{\pi^{2}}{2\hbar\lambda(T)}e^{-\frac{E_{1s}(E_{1s}\lambda^{2}(T)+2\hbar
v_{F}z)}{\hbar^{2}v_{F}^{2}}}\left[-2\hbar\lambda(T)v_{F}z\right.$
$\displaystyle\left.+e^{\frac{(E_{1s}\lambda^{2}(T)+\hbar
v_{F}z)^{2}}{\hbar^{2}v_{F}^{2}\lambda^{2}(T)}}\sqrt{\pi}\left(2E_{1s}\lambda^{2}(T)z+\hbar
v_{F}(\lambda^{2}(T)+2z^{2})\right)\text{erfc}\left(\frac{E_{1s}\lambda(T)}{\hbar
v_{F}}+\frac{z}{\lambda(T)}\right)\right]\;,$ (38)

where we introduced the thermal wavelength $\lambda(T)=\hbar/\sqrt{2Mk_{B}T}$.
In Supplementary Fig. 12 we show the $z$-dependence of the Meitner-Auger
transfer and also the Förster result. In contrast to the Förster tansfer, the
Meitner-Auger transfer decays exponentially. From the function
$a(Q,z)\propto\exp(-Qz)$ we see that the main contribution stems from momenta
$Q$ at $Q\approx 0$. But to account for energy and momentum conservation, the
Meitner-Auger transfer requires large $Q$, ensured by the Heaviside function
in $\Gamma_{\mathbf{Q}}$. We see from the exponential function in front of the
parenthesis, that the strength of the exponential decay is determined by the
slope of the linear graphene band structure. For the Förster transfer the
function $a(Q,z)$ is proportional to $Q\exp(-Qz)$ resulting in the different
far-field behavior. Interestingly, the different far-field behavior can be
traced back to the linear band structure of graphene, reflected by the
Heaviside function in $\Gamma_{\mathbf{Q}}$, and not to the difference in the
interaction. When we artificially set the TMDC band gap to zero, the Meitner-
Auger transfer shows the same $z$-dependence as the Förster transfer in the
near- and far-field. At last, we want to stress, that to calculate Eq. (38) we
assumed thermalized electron occupations in graphene, which is obviously not
the case for a transient experiment.

Compared with a heterostructure made by the exfoliation and stacking
technique, the epitaxially grown heterostructure provides a shorter interlayer
distance, which benefits the interfacial energy transfer. We have identified
the closely stacked WSe2 and graphene layer without any significant spatial
gap with atomic force microscopy in a similar heterostructure[7]. The
$z$-dependence of the energy transfer rates is shown in Supplementary Fig.12.
It demonstrates a larger transfer energy rate at a shorter z-distance for both
MA and Förster-type IET processes. A different IET mechanism could be dominant
with increased interlayer distance. However, not only the interlayer distance
$z$ determines the interlayer coupling, but also the rotational orientation
between two layers. The band structure alignment of an epitaxially grown
heterostructure is favoured in a few discrete orientations, for example,
twisting angle of $0\deg$ and $60\deg$ between the WSe2 and graphene layer.
The heterostructures prepared by the exfoliation method provide the
flexibility for engineering the band structure alignment by turning the
twisting angel. Therefore, the heterostructure samples prepared by these
methods provide their specific way for us to understand the interlayer
interaction.

Figure 12: Meitner-Auger and Förster transfer rate as function of layer
distance. The Förster transfer rate decays exponentially at short distance and
goes over into a $z^{-4}$ dependence (blue). The Meitner-Auger transfer rate
continues its exponential decay (red). The strength of the exponential decay
of Meitner-Auger rate depends on the slope of the linear graphene band
structure

###  0.6 Dexter Coupling

The IET process could also mediated by Dexter-type two-particle exchange,
whose transfer rate is determined by the wave function overlap[29]. Starting
point for the calculation is the Hamiltonian

$H_{D}=\sum_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}c^{\dagger}_{\mathbf{k}}v^{\dagger}_{\mathbf{q}}v_{\mathbf{k^{\prime}}}c_{\mathbf{q^{\prime}}}+h.c.,$
(39)

with the same conventions for the notation as for the calculation of the
Förster transfer. The appearing coupling element is defined as

$V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}=\int_{\mathbb{R}^{3}}d^{3}r\int_{\mathbb{R}^{3}}d^{3}r^{\prime}\Psi^{c*}_{\mathbf{k}}(\mathbf{r})\Psi^{v*}_{\mathbf{q}}(\mathbf{r^{\prime}})V(\mathbf{r},\mathbf{r^{\prime}})\Psi^{v}_{\mathbf{k^{\prime}}}(\mathbf{r^{\prime}})\Psi^{c}_{\mathbf{q^{\prime}}}(\mathbf{r}).$
(40)

The Coulomb potential is translational invariant in the in-plane direction,
i.e.
$V(\mathbf{r},\mathbf{r^{\prime}})=V(\mathbf{r}_{\parallel}-\mathbf{r^{\prime}}_{\parallel},z,z^{\prime})$.
Fourier transforming the Coulomb potential w.r.t. the in-plane components,
writing the electronic wave functions as Bloch waves and decomposing the
spatial coordinates into one component inside the unit cell and one which
addresses the unit cells $\mathbf{r}\rightarrow\mathbf{r}+\mathbf{R}$, yields
for the coupling element

$V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}=\frac{1}{A}\sum_{\mathbf{K}}\int_{uc}dz\int_{uc}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}},$
(41)

with

$\chi^{\lambda}(z)=\frac{1}{V_{uc}}\int_{uc}d^{2}r_{\parallel}u^{*\lambda}_{TMD}(\mathbf{r}_{\parallel},z)u^{\lambda}_{Graphene}(\mathbf{r}_{\parallel},z)$
(42)

To evaluate the coupling element further, we restrict ourselves to the case
with vanishing interlayer spacing. We decompose the $z$ and $z^{\prime}$
integration into two integrals over WSe2 and graphene

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}=\frac{1}{A}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}\times$
$\displaystyle\left(\int_{WSe_{2}}dz\int_{WSe_{2}}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\right.$
$\displaystyle+\int_{WSe_{2}}dz\int_{Graphene}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})$
$\displaystyle+\int_{Graphene}dz\int_{WSe_{2}}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})$
$\displaystyle\left.+\int_{Graphene}dz\int_{Graphene}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\right)$
(43)

Given that the Coulomb potential varies only weakly with each layer, we can
replace the $z/z^{\prime}$ dependence by the position of the layer
$z=z_{WSe_{2}},z_{Graphene}$ in the Coulomb potential. This way we arrive at

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}=\frac{1}{A}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}\times$
$\displaystyle\left(\chi^{c}_{WSe_{2}}\chi^{v}_{WSe_{2}}V_{\mathbf{K}}(z=z_{WSe_{2}},z^{\prime}=z_{WSe_{2}})\right.$
$\displaystyle+\chi^{c}_{WSe_{2}}\chi^{v}_{Graphene}V_{\mathbf{K}}(z=z_{WSe_{2}},z^{\prime}=z_{Graphene})$
$\displaystyle+\chi^{c}_{Graphene}\chi^{v}_{WSe_{2}}V_{\mathbf{K}}(z=z_{Graphene},z^{\prime}=z_{WSe_{2}})$
$\displaystyle\left.+\chi^{c}_{Graphene}\chi^{v}_{Graphene}V_{\mathbf{K}}(z=z_{Graphene},z^{\prime}=z_{Graphene})\right),$
(44)

with

$\chi^{\lambda}_{WSe_{2}/Graphene}=\int_{WSe_{2}/Graphene}dz\chi^{\lambda}(z),$
(45)

i.e. the contribution of the wave function overlap of the band $\lambda$ in
the individual layers. Assuming, that the integration in both layers
contributes equally to the wave function overlap between WSe2 and graphene in
conduction and valence band, i.e.
$\chi^{\lambda}_{WSe_{2}}=\chi^{\lambda}_{Graphene}=\frac{1}{2}\chi^{\lambda}$,
we obtain the final expression for the matrix element

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}=\frac{1}{4A}\chi^{c}\chi^{v}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}V^{Dex}_{\mathbf{K}}$
(46)

with

$\displaystyle
V^{Dex}_{\mathbf{K}}=\left(V_{\mathbf{K}}(z=z_{WSe_{2}},z^{\prime}=z_{WSe_{2}})\right.+V_{\mathbf{K}}(z=z_{WSe_{2}},z^{\prime}=z_{Graphene})$
$\displaystyle+V_{\mathbf{K}}(z=z_{Graphene},z^{\prime}=z_{WSe_{2}})\left.+V_{\mathbf{K}}(z=z_{Graphene},z^{\prime}=z_{Graphene})\right).$
(47)

Reinserting the result back into the Hamiltonian (and indexing the operators
according to the layer, since due to the momentum conservation our convention
breaks down) yields

$H_{D}=\sum_{\mathbf{K},\mathbf{k},\mathbf{q}}\frac{1}{4A}\chi^{c}\chi^{v}V^{Dex}_{\mathbf{K+k-q}}c^{\dagger
WSe_{2}}_{\mathbf{k+K}}v^{\dagger
Gr}_{\mathbf{q-K}}v^{WSe_{2}}_{\mathbf{k}}c^{Gr}_{\mathbf{q}}+h.c..$ (48)

So far, the momenta are defined w.r.t. the $\Gamma$ point in graphene and
WSe2. Redefining the coordinates $\mathbf{k}\rightarrow K^{W}+\mathbf{k}$ and
$\mathbf{q}\rightarrow K^{G}+\mathbf{q}$ expresses them w.r.t. the $K$ point
in WSe2/graphene. A projection on excitonic wave functions in WSe2 yields

$H_{D}=-\sum_{\mathbf{K},\mathbf{q},\nu}\left(\frac{1}{4\sqrt{A}}\chi^{c}\chi^{v}\sum_{\mathbf{k}}\varphi^{*\nu}_{K^{W}+\mathbf{k}}V^{Dex}_{K^{W}-K^{G}+\mathbf{K+k-q}}\right)P^{\dagger\lambda}_{\mathbf{K}}R^{K^{G}+\mathbf{q}}_{\mathbf{K}}+h.c..$
(49)

In the Dexter coupling element, the momentum distance between the $K$ points
in graphene and the TMD directly enters. We have $|K^{G}-K^{W}|\approx$ 3.6
nm-1. As a first approximation we can ignore the COM and relative momenta
inside the Coulomb potential $V^{Dex}_{K^{W}-K^{G}+\mathbf{K+k-q}}\approx
V^{Dex}_{K^{W}-K^{G}}$ and get a first estimate for the Dexter coupling

$H_{D}=-\sum_{\mathbf{K},\mathbf{q},\nu}D_{\mathbf{K}}P^{\dagger\nu}_{\mathbf{K}}R^{K^{G}+\mathbf{q}}_{\mathbf{K}}+h.c.,$
(50)

with

$D_{\mathbf{K}}=\frac{1}{4\sqrt{A}}\chi^{c}\chi^{v}\varphi^{*\nu}(\mathbf{r=0})V^{Dex}_{K^{W}-K^{G}}.$
(51)

Similar to the Förster transfer, we can evaluate the Dexter induced scattering
rate for WSe2 excitons to graphene

$\Gamma_{\mathbf{Q}}=2\pi\sum_{\mathbf{q}}|D_{\mathbf{Q}}|^{2}\delta(E_{\mathbf{Q}}^{\mathbf{q}}-E^{1s}_{\mathbf{Q}})$
(52)

with yields

$\Gamma_{\mathbf{Q}}=\frac{|D_{\mathbf{Q}}|^{2}AE_{\mathbf{Q}}}{4\hbar^{2}v_{F}^{2}}.$
(53)

Supplementary Fig. 13a illustrates the Dexter transfer rate from WSe2 to
graphene as a function of the wave function overlap for the same structure as
considered for the Förster transfer. Assuming an overlap between the TMD and
graphene wave functions of $\chi=$0.039 (see estimation below, Supplementary
Fig. 14a), we arrive at a Dexter rate of 1.0 $\cdot$10-6meV. This number is
small due to the mismatch of the $K$ points of WSe2 and graphene and due to
the small overlap of the wave functions which enters with the fourth power. As
long as the COM momentum $\mathbf{Q}$ is much smaller compared to the distance
between the $K$ points, the Dexter rate is independent of the COM momentum.

Figure 13: Estimation for the Dexter transfer rate. a, Dexter transfer rate
from WSe2 to graphene as a function of the overlap of the electronic wave
functions of graphene and WSe2. b, Dexter transfer rate from WSe2 to graphene
as a function of the excitonic COM momentum $\mathbf{Q}$, with the
$\mathrm{K_{WSe_{2}}}$ valley and graphene being shifted on top of each
others.

###  0.7 Maximum estimation for the Dexter process

The reason for the very weak Dexter process is the large momentum mismatch
between the $K$ points in WSe2 and graphene. This generates a momentum-
bottleneck. To get an estimation of the maximally possible Dexter rate (and
Dexter-like processes, such as Dexter-two-phonon processes), we remove this
bottleneck artificially and move the $K$ points of both layers on top of each
other. The Hamiltonian reads

$H_{D}=-\sum_{\mathbf{K},\mathbf{q},\nu}D_{\mathbf{q,K}}P^{\dagger\lambda}_{\mathbf{K}}R^{\mathbf{q}}_{\mathbf{K}}+h.c..$
(54)

with the coupling element

$D_{\mathbf{q,K}}=\frac{1}{4\sqrt{A}}\chi^{c}\chi^{v}\sum_{\mathbf{k}}\varphi^{*\nu}_{\mathbf{k}}V^{Dex}_{\mathbf{K+k-q}}.$
(55)

The relaxation rate of excitons to graphene is then given by

$\Gamma_{\mathbf{K}}=\frac{A}{2\pi\hbar}\int_{0}^{2\pi}d\phi\frac{E_{\mathbf{K}}}{\hbar^{2}v_{F}^{2}}|D_{\frac{E_{\mathbf{K}}}{\hbar
v_{F}}\left(cos\phi,sin\phi\right)^{T},\mathbf{K}}|^{2}.$ (56)

Supplementary Fig. 13b illustrates the maximum estimation of the Dexter rate
as the function of the COM momentum. We find a relatively weak rate in the
order of $10^{-3}\leavevmode\nobreak\ \mathrm{meV}$ due to the poor wave
function overlap. If we would set $\chi=1$, we would obtain 70 meV at the
maximum. Last, we calculate the thermal average of the Dexter rate, i.e.
integrate the momentum dependent Dexter rate together with a normalized
Boltzmann distribution at 300 K. We obtain 1.5 $\cdot
10^{-4}\leavevmode\nobreak\ \mathrm{meV}$.

We would like to note the computation of orbital overlaps is very difficult.
Practically, we do not calculate the overlap explicitly but compute the Dexter
rate as function of wavefunction overlap. Note, the overlap is not a "totally"
unknown parameter in our sample, since we can use the trARPES result as an
input. With the below-bandgap excitation (outside the range of the MA
transition and therefore constituting as an independent result), we observe an
interlayer electron scattering from graphene to WSe2. By comparing with our
calculation of the phonon-assisted charge tunneling rate as function of
wavefunction overlap, we could extract an overlap value of around 4%. It can
be used to calculate the Dexter-type energy transfer rate in a reasonable
range. This wavefunction overlap value is comparable to the ab-initio
calculated value of the neighboured atoms overlap in graphene, 7%[30], and the
interlayer overlap in a MoSe2/WSe2 heterostructure, 1%[24].

Besides the wavefunction overlap, we present the details of the Dexter
transfer calculation and the applied approximations in the following. We start
from the general Coulomb Hamiltonian involving all transfer processes
(Förster, Dexter, Meiner-Auger)

$\displaystyle H$
$\displaystyle=\sum_{\begin{subarray}{c}\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\\
\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}\\\
l_{1},l_{2},l_{3},l_{4}\end{subarray}}V^{\lambda_{1}\nu_{2}\lambda_{3}\lambda_{4}l_{1}l_{2}l_{3}l_{4}}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}\lambda_{1,\mathbf{k}_{1}}^{\dagger
l_{1}}\lambda^{\dagger
l_{2}}_{2,\mathbf{k}_{2}}\lambda^{l_{3}}_{3,\mathbf{k}_{3}}\lambda^{l_{4}}_{4,\mathbf{k}_{4}}$
(57) $\displaystyle
V^{\lambda_{1}\nu_{2}\lambda_{3}\lambda_{4}l_{1}l_{2}l_{3}l_{4}}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}$
$\displaystyle=\int d^{3}r\int d^{3}r^{\prime}\leavevmode\nobreak\
\Psi^{*l_{1}}_{\lambda_{1},\mathbf{k}_{1}}(\mathbf{r})\Psi^{*l_{2}}_{\lambda_{2},\mathbf{k}_{2}}(\mathbf{r^{\prime}})V(\mathbf{r-r^{\prime}})\Psi^{l_{3}}_{\lambda_{3},\mathbf{k}_{3}}(\mathbf{r}^{\prime})\Psi^{l_{4}}_{\lambda_{4},\mathbf{k}_{4}}(\mathbf{r})$
(58)

with band $\lambda_{i}$, wave vector $\mathbf{k}_{i}$ and layer $l_{i}$. In
Eq. (57) we can perform the sum over the band and layer indices. The
combination $l_{1}=l_{3}$ and $l_{2}=l_{4}$ and $\lambda_{1}=\lambda_{4}=c$
and $\lambda_{2}=\lambda_{3}=v$ corresponds to the Dexter Hamiltonian:

$\displaystyle
H=\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}V^{cvvcGWGW}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}c^{\dagger
G}_{\mathbf{k}_{1}}v^{\dagger
W}_{\mathbf{k}_{2}}v^{G}_{\mathbf{k}_{3}}c^{W}_{\mathbf{k}_{4}}+\text{H.c.}=\sum_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}c^{\dagger}_{\mathbf{k}}v^{\dagger}_{\mathbf{q}}v_{\mathbf{k^{\prime}}}c_{\mathbf{q^{\prime}}}+\text{H.c.}$
(59)

where in the second step, we define $\mathbf{k}^{(^{\prime})}$ as wave vectors
in graphene and $\mathbf{q}^{(^{\prime})}$ as wave vectors in WSe2, such that
we can drop the layer index for graphene ($G$) and WSe2 ($W$). The Hamiltonian
Eq. (59) describes the electron transfer from the conduction band of WSe2 to
graphene, together with the valence electron transfer from graphene to WSe2.
The matrix element reads

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}$
$\displaystyle=\frac{1}{A}\sum_{\mathbf{K}}\int d^{3}r\int
d^{3}r^{\prime}\leavevmode\nobreak\
\Psi^{*}_{c,\mathbf{k}}(\mathbf{r})\Psi^{*}_{v,\mathbf{q}}(\mathbf{r^{\prime}})V_{\mathbf{K}}(z,z^{\prime})\Psi_{v,\mathbf{q}^{\prime}}(\mathbf{r}^{\prime})\Psi_{c,\mathbf{k}^{\prime}}(\mathbf{r})\;,$
(60)

where we denote the in-plane Fourier transformed Coulomb potential by
$V_{\mathbf{K}}(z,z^{\prime})$.

Now, we calculate the Dexter transfer rate. First, we set the interlayer
distance to zero. This yields an upper limit of the Dexter rate in the
heterostructure using the proper Coulomb potential. From Eq. (60) we perform
the standard steps of shifting the integrals into the first unit cell
$\mathbf{r}_{\parallel}^{(^{\prime})}\rightarrow\mathbf{r}_{\parallel,n}^{(^{\prime})}+\mathbf{R}_{n}^{(^{\prime})}$
and summing over all unit cells $\sum_{n,n^{\prime}}$. After inserting Bloch
functions we obtain

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}$
$\displaystyle=\frac{1}{A}\sum_{\mathbf{K}}\int_{UC}dz\int_{UC}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}\;,$
(61)

where the Kronecker delta is obtained from the Bloch wave envelopes. In Eq.
(61) we define the orbital overlap
$\chi^{\lambda}(z)=\frac{1}{V_{uc}}\int_{UC}d^{2}r_{\parallel}u^{*\lambda}_{W}(\mathbf{r}_{\parallel},z)u^{\lambda}_{G}(\mathbf{r}_{\parallel},z)$
of WSe2 ($W$) and graphene ($G$) wavefunction. The
$z^{(^{\prime})}$-integration runs over the two materials, such that Eq. (61)
can also be written as

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}$
$\displaystyle=\frac{1}{A}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}\left(\int_{W}dz\int_{W}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})+\int_{W}dz\int_{G}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\right.$
$\displaystyle\left.+\int_{G}dz\int_{W}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})+\int_{G}dz\int_{G}dz^{\prime}\chi^{c}(z)\chi^{v}(z^{\prime})V_{\mathbf{K}}(z,z^{\prime})\right)\;.$
(62)

To treat the $z$-integrations in Eq. (62), we assume that the Coulomb
potential varies only weakly in the out-of-plane direction (due to the atomic
thickness of the two materials). The approach presented here is also performed
for the Rytova-Keldysh potential in 2D materials[31, 32, 33] and serves as a
standard assumption[34, 35, 36]. Then, we can use the Coulomb potential
directly at the material position and take it out of the integral. We obtain
Eq. (62)

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}$
$\displaystyle=\frac{1}{A}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}\left(\chi^{c}_{W}\chi^{v}_{W}V_{\mathbf{K}}(z=z_{W},z^{\prime}=z_{W})\right.$
$\displaystyle\left.+\chi^{c}_{W}\chi^{v}_{G}V_{\mathbf{K}}(z=z_{W},z^{\prime}=z_{G})+\chi^{c}_{G}\chi^{v}_{W}V_{\mathbf{K}}(z=z_{G},z^{\prime}=z_{W})+\chi^{c}_{G}\chi^{v}_{G}V_{\mathbf{K}}(z=z_{G},z^{\prime}=z_{G})\right)$
(63)

with $\chi^{\lambda}_{W/G}=\int_{W/G}dz\chi^{\lambda}(z)$ describing the
wavefunction overlap of the bands $\lambda$ of the individual layers (since
$\chi^{\lambda}=\int_{UC}d^{2}r_{\parallel}u_{W}^{*}(\mathbf{r}_{\parallel},z)u_{G}(\mathbf{r}_{\parallel},z)$).
Next, we assume that both layers contribute equally to the wavefunction
overlap. This has the consequence that the interlayer and intralayer Coulomb
potential contribute equally to the Dexter rate. In an extreme scenario, we
calculate the Dexter-type transfer rate with the wavefunction overlap
occurring only in one material. It means two times of the intralayer potential
will appear instead of the interlayer potential. The Dexter-type transfer rate
only increases by a factor of $\sim$1.5, indicating that the transfer process
depends mainly on the wavefunction overlap value (which enters to the power of
4 in the Dexter rate) but not where the overlap occurs. By using
$\chi^{\lambda}_{W}=\chi^{\lambda}_{G}=\frac{1}{2}\chi^{\lambda}$ we obtain
the final expression

$\displaystyle
V^{cvvc}_{\mathbf{k},\mathbf{q},\mathbf{k^{\prime}},\mathbf{q^{\prime}}}$
$\displaystyle=\frac{1}{4A}\chi^{c}\chi^{v}\sum_{\mathbf{K}}\delta_{\mathbf{k,q^{\prime}+K}}\delta_{\mathbf{q,k^{\prime}-K}}V^{Dex}_{\mathbf{K}}$
(64) $\displaystyle V^{Dex}_{\mathbf{K}}$
$\displaystyle=V_{\mathbf{K}}(z=z_{W},z^{\prime}=z_{W})+V_{\mathbf{K}}(z=z_{W},z^{\prime}=z_{G})+V_{\mathbf{K}}(z=z_{G},z^{\prime}=z_{W})+V_{\mathbf{K}}(z=z_{G},z^{\prime}=z_{G})\;.$
(65)

When we reinsert the matrix element into the Hamiltonian Eq. (59) and shift
the origin of the wave vector on the K points of the respective materials,
i.e. $\mathbf{k}\rightarrow\mathbf{K}^{W}+\mathbf{k}$ and
$\mathbf{q}\rightarrow\mathbf{K}^{G}+\mathbf{q}$ we obtain

$\displaystyle
H_{D}=\sum_{\mathbf{K},\mathbf{k},\mathbf{q}}\frac{1}{4A}\chi^{c}\chi^{v}V^{Dex}_{\mathbf{K}^{W}-\mathbf{K}^{G}+\mathbf{K+k-q}}c^{\dagger
W}_{\mathbf{K}^{W}+\mathbf{k+K}}v^{\dagger
G}_{\mathbf{K}^{G}+\mathbf{q-K}}v^{W}_{\mathbf{K}^{W}+\mathbf{k}}c^{G}_{\mathbf{K}^{G}+\mathbf{q}}+\text{H.c.}$
(66)

Finally, we find that the Fourier transformed Coulomb potential
$V^{Dex}_{\mathbf{K}^{W}-\mathbf{K}^{G}+\mathbf{K+k-q}}$ depends on the
momentum difference
$\mathbf{K}_{W}-\mathbf{K}_{G}+\mathbf{K}+\mathbf{k}-\mathbf{q}$, where
$\mathbf{K}_{W}$ and $\mathbf{K}_{G}$ are the K points of WSe2 and graphene
respectively, $\mathbf{k}$ and $\mathbf{q}$ are the electronic momenta in WSe2
and graphene, respectively and $\mathbf{K}$ is the Coulomb induced momentum
transfer. From $V^{Dex}_{\mathbf{K}^{W}-\mathbf{K}^{G}+\mathbf{K+k-q}}$ we see
that if a large momentum mismatch between the valleys occurs, the Dexter
contribution is reduced. To obtain an upper limit estimate, we approximate
$\mathbf{K}_{W}-\mathbf{K}_{G}+\mathbf{K}+\mathbf{k}-\mathbf{q}\approx\mathbf{K}_{W}-\mathbf{K}_{G}$,
to be the dominant distance.
The computation of the Dexter rate is based on the above approximations and
the trARPES result of ICT.

Microscopic calculation of interlayer phonon-assisted tunneling process

In this section we will derive an expression for the phonon-assisted tunneling
of carriers between the layers. The Hamiltonian of phonon scattering and
tunneling can be generally written as

$\displaystyle H=H_{0}+H_{1},$ (67)

with $H_{0}$ accounting for the dispersion of electrons and phonons.

$\displaystyle
H_{0}=\sum_{a}\epsilon^{a}a^{\dagger}_{a}a_{a}+\sum_{b}\hbar\omega^{b}b_{b}^{\dagger}b_{b}.$
(68)

The first term accounts for the dispersion of carriers with operators
$a^{(\dagger)}_{a}$ and the second term accounts for the dispersion of phonons
with operators $b^{(\dagger)}_{b}$. The quantum numbers $a,b$ account for
layer and momentum of the carriers. The interaction Hamiltonian $H_{1}$ reads

$H_{1}=\sum_{ab}t^{ab}a^{\dagger}_{a}a_{b}+\sum_{abc}g^{abc}a^{\dagger}_{a}a_{b}(b_{c}+b^{\dagger}_{-c}),$
(69)

where the first term represents the tunneling and the second term the
scattering of carriers with phonons. Here the notation $-c$ implies, that the
momentum has to be inverted, but all other quantum numbers stay the same.

While we are interested in the second order processes of phonon-assisted
tunneling, we apply a canocical transformation to the Hamiltonian

$H^{\prime}=e^{-S}He^{S}=H_{0}+\underbrace{\left(H_{1}+[H_{0},S]\right)}_{\text{first
order}}+\underbrace{\frac{1}{2}[H_{1},S]}_{\text{second order}},$ (70)

and claim that the first order in the interaction vanishes. This holds true
for the choice

$S=\sum_{ab}\alpha_{ab}t^{ab}a^{\dagger}a_{b}+\sum_{abc}g^{abc}a^{\dagger}_{a}a_{b}(\beta_{abc}b_{c}+\gamma_{abc}b^{\dagger}_{-c}),$
(71)

with coefficients

$\displaystyle\alpha_{ab}$
$\displaystyle=\frac{1}{\epsilon^{b}-\epsilon^{a}},$ (72)
$\displaystyle\beta_{abc}$
$\displaystyle=\frac{1}{\epsilon^{b}-\epsilon^{a}+\hbar\omega^{c}},$ (73)
$\displaystyle\gamma_{abc}$
$\displaystyle=\frac{1}{\epsilon^{b}-\epsilon^{a}-\hbar\omega^{c}}.$ (74)

The second order Hamiltonian is now given as

$H_{2}=\frac{1}{2}[H_{1},S].$ (75)

Restricting ourselves only to the tunneling-phonon contribution (besides this,
also higher order tunneling terms, two-phonon processes as well as attractive
electron-electron interaction through phonon interaction are contained in this
Hamiltonian) we obtain

$\displaystyle H$
$\displaystyle=\frac{1}{2}\sum_{abcd}t^{db}g^{adc}a^{\dagger}_{a}a_{b}((\frac{1}{\epsilon^{b}-\epsilon^{d}}-\frac{1}{\epsilon^{d}-\epsilon^{a}+\hbar\omega^{c}})b_{c}+(\frac{1}{\epsilon^{b}-\epsilon^{d}}-\frac{1}{\epsilon^{d}-\epsilon^{a}-\hbar\omega^{-c}})b_{-c}^{\dagger})$
$\displaystyle-\frac{1}{2}\sum_{abcd}t^{ad}g^{dbc}a^{\dagger}_{a}a_{b}((\frac{1}{\epsilon^{d}-\epsilon^{a}}-\frac{1}{\epsilon^{b}-\epsilon^{d}+\hbar\omega^{c}})b_{c}+(\frac{1}{\epsilon^{d}-\epsilon^{a}}-\frac{1}{\epsilon^{b}-\epsilon^{d}-\hbar\omega^{-c}})b_{-c}^{\dagger})$
(76)

Now we insert the compounds: $a=(\mathbf{k}_{a},\lambda_{a},l_{a})$ for
electrons, where $\mathbf{k}_{a}$ accounts for the momentum, $\lambda_{a}$
accounts for the band and $l_{a}$ accounts for the layer quantum number. For
phonons we insert the compounds $c=(\mathbf{k}_{c},l_{c},\xi_{c})$, with
momentum $\mathbf{k}_{c}$, layer $l_{c}$ and branch $\xi_{c}$ and apply the
selection rules from the matrix elements:

$\displaystyle
t_{\mathbf{k}_{b}\mathbf{k}_{d}}^{\lambda_{b}\lambda_{d}l_{b}l_{d}}=t^{\lambda_{b}\lambda_{d}l_{b}l_{d}}\delta_{\mathbf{k}_{b}\mathbf{k}_{d}}^{l_{b}\bar{l_{d}}}\delta^{\lambda_{b}\lambda_{d}},$
(77)

i.e. the tunneling conserves momentum, and band but changes the layer index,
and

$\displaystyle
g_{\mathbf{k}_{a}\mathbf{k}_{d}\mathbf{k}_{c}}^{\lambda_{a}\lambda_{d}l_{a}l_{d}l_{c}}=g_{\mathbf{k}_{c}}^{\lambda_{a}\lambda_{d}l_{a}l_{d}l_{c}}\delta^{l_{a}l_{d}}\delta_{\mathbf{k}_{c},\mathbf{k}_{a}-\mathbf{k}_{d}}^{l_{d}l_{c}}\delta^{\lambda_{a}\lambda_{d}}.$
(78)

i.e. phonon scattering conserves the layer index, but changes the momentum of
the carriers.

As a result we obtain

$\displaystyle H$ $\displaystyle=\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda
l\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger\bar{l}}\lambda_{\mathbf{k}}^{l}\left(\underbrace{t^{\lambda
l\bar{l}}g_{\mathbf{K}}^{\lambda\bar{l}\xi}(\alpha^{\lambda
l\bar{l}}_{\mathbf{k}}+\gamma^{\lambda\bar{l}\xi}_{\mathbf{k},\mathbf{K}})}_{s_{\mathbf{k,K}}^{\lambda
l\bar{l}\xi}}b_{\mathbf{K}}^{\bar{l}\xi}+\underbrace{t^{\lambda
l\bar{l}}g_{\mathbf{K}}^{\lambda\bar{l}\xi}(\alpha^{\lambda
l\bar{l}}_{\mathbf{k}}+\beta^{\lambda\bar{l}\xi}_{\mathbf{k},\mathbf{K}})}_{\tilde{s}_{\mathbf{k,K}}^{\lambda
l\bar{l}\xi}}b_{-\mathbf{K}}^{\dagger\bar{l}\xi}\right)$
$\displaystyle-\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda
l\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger\bar{l}}\lambda_{\mathbf{k}}^{l}\left(\underbrace{t^{\lambda\bar{l}l}g_{\mathbf{K}}^{\lambda
l\xi}(\alpha^{\lambda l\bar{l}}_{\mathbf{k+K}}+\gamma^{\lambda
l\xi}_{\mathbf{k},\mathbf{K}})}_{u_{\mathbf{k,K}}^{\lambda
l\bar{l}\xi}}b_{\mathbf{K}}^{l\xi}+\underbrace{t^{\lambda\bar{l}l}g_{\mathbf{K}}^{\lambda
l\xi}(\alpha^{\lambda
l\bar{l}}_{\mathbf{k+K}}+\beta_{\mathbf{k},\mathbf{K}}^{\lambda
l\xi})}_{\tilde{u}_{\mathbf{k,K}}^{\lambda
l\bar{l}\xi}}b_{-\mathbf{K}}^{\dagger l\xi}\right),$ (79)

with

$\displaystyle\alpha^{\lambda ij}_{\mathbf{k}}$
$\displaystyle=\frac{1}{\epsilon_{\mathbf{k}}^{\lambda
i}-\epsilon_{\mathbf{k}}^{\lambda j}}$ (80) $\displaystyle\beta^{\lambda
i\xi}_{\mathbf{k},\mathbf{K}}$
$\displaystyle=\frac{1}{\epsilon_{\mathbf{k}+\mathbf{K}}^{\lambda
i}-\epsilon_{\mathbf{k}}^{\lambda i}+\hbar\omega_{-\mathbf{K}}^{i\xi}}$ (81)
$\displaystyle\gamma^{\lambda i\xi}_{\mathbf{k},\mathbf{K}}$
$\displaystyle=\frac{1}{\epsilon_{\mathbf{k}+\mathbf{K}}^{\lambda
i}-\epsilon_{\mathbf{k}}^{\lambda i}-\hbar\omega_{\mathbf{K}}^{i\xi}}$ (82)

Both lines describe phonon assisted tunneling from $\mathbf{k},b$ to
$\mathbf{k}+\mathbf{K},\bar{b}$. However, in the first line first the
tunneling and next the phonon scattering takes place, whereas in the second
line, first the phonon scattering and next the tunneling takes place.
Considering the intermediate states$\mathbf{k},\bar{b}$ and
$\mathbf{k}+\mathbf{K},b$ being much larger in energy, in both lines, all
contributions have the same sign and add up. The relative sign between both
lines ($-$) is compensated by the opposite signs of the appearing $u$ and $s$
functions. To further evaluate the phonon assisted tunnel Hamiltonian, we
carry out the summation over the layer

$\displaystyle H$ $\displaystyle=\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda
G\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
W}\lambda_{\mathbf{k}}^{G}\left(\underbrace{t^{\lambda
GW}g_{\mathbf{K}}^{\lambda W\xi}(\alpha^{\lambda
GW}_{\mathbf{k}}+\gamma^{\lambda
W\xi}_{\mathbf{k},\mathbf{K}})}_{s_{\mathbf{k,K}}^{\lambda
GW\xi}}b_{\mathbf{K}}^{W\xi}+\underbrace{t^{\lambda GW}g_{\mathbf{K}}^{\lambda
W\xi}(\alpha^{\lambda GW}_{\mathbf{k}}+\beta^{\lambda
W\xi}_{\mathbf{k},\mathbf{K}})}_{\tilde{s}_{\mathbf{k,K}}^{\lambda
GW\xi}}b_{-\mathbf{K}}^{\dagger W\xi}\right)$
$\displaystyle-\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda
W\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
G}\lambda_{\mathbf{k}}^{W}\left(\underbrace{t^{\lambda
GW}g_{\mathbf{K}}^{\lambda W\xi}(\alpha^{\lambda
WG}_{\mathbf{k+K}}+\gamma^{\lambda
W\xi}_{\mathbf{k},\mathbf{K}})}_{u_{\mathbf{k,K}}^{\lambda
WG\xi}}b_{\mathbf{K}}^{W\xi}+\underbrace{t^{\lambda GW}g_{\mathbf{K}}^{\lambda
W\xi}(\alpha^{\lambda
WG}_{\mathbf{k+K}}+\beta_{\mathbf{k},\mathbf{K}\xi}^{\lambda
W})}_{\tilde{u}_{\mathbf{k,K}}^{\lambda WG\xi}}b_{-\mathbf{K}}^{\dagger
W\xi}\right)$
$\displaystyle+\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
G}\lambda_{\mathbf{k}}^{W}\left(\underbrace{t^{\lambda
WG}g_{\mathbf{K}}^{\lambda G\xi}(\alpha^{\lambda
WG}_{\mathbf{k}}+\gamma^{\lambda
G\xi}_{\mathbf{k},\mathbf{K}})}_{s_{\mathbf{k,K}}^{\lambda
WG\xi}}b_{\mathbf{K}}^{G\xi}+\underbrace{t^{\lambda WG}g_{\mathbf{K}}^{\lambda
G\xi}(\alpha^{\lambda WG}_{\mathbf{k}}+\beta^{\lambda
G\xi}_{\mathbf{k},\mathbf{K}})}_{\tilde{s}_{\mathbf{k,K}}^{\lambda
WG\xi}}b_{-\mathbf{K}}^{\dagger G\xi}\right)$
$\displaystyle-\frac{1}{2}\sum_{\mathbf{k}\mathbf{K}\lambda\xi}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
W}\lambda_{\mathbf{k}}^{G}\left(\underbrace{t^{\lambda
WG}g_{\mathbf{K}}^{\lambda G\xi}(\alpha^{\lambda
GW}_{\mathbf{k+K}}+\gamma^{\lambda
G\xi}_{\mathbf{k},\mathbf{K}})}_{u_{\mathbf{k,K}}^{\lambda
GW\xi}}b_{\mathbf{K}}^{G\xi}+\underbrace{t^{\lambda WG}g_{\mathbf{K}}^{\lambda
G\xi}(\alpha^{\lambda
GW}_{\mathbf{k+K}}+\beta_{\mathbf{k},\mathbf{K}}^{\lambda
G\xi})}_{\tilde{u}_{\mathbf{k,K}}^{\lambda GW\xi}}b_{-\mathbf{K}}^{\dagger
G\xi}\right)$ (83)

Here, the first two lines involve phonons in WSe2 whereas the last two lines
involve phonons in graphene. The $-$ signs in front of line two and line four
are compensated by the opposite signs of the $s$ and $u$ functions. To
investigate the different contributions to the tunnel-phonon coupling in more
detail, we evaluate the functions $s$ and $u$ in the different terms: (a) we
fix the momenta in WSe2 to the vicinity of the $K$ point. (b) While all
scattering processes conserve the energy, the approximation (a) settles the
energy and momentum range of involved carriers. (c) In the next step, we
approximate the prefactors $\alpha^{\lambda ij}_{\mathbf{k}}$ by their values
in the region of interest, where we find $\alpha^{cGW}_{\mathbf{k}\approx
K_{W}}=\frac{1}{1\text{eV}}$ and $\alpha^{cWG}_{\mathbf{k}\approx
K_{G}}=\frac{1}{250\text{meV}}$, which are read out from the DFT calculation
in the main manuscript. To evaluate the prefactors $\beta^{\lambda
i\xi}_{\mathbf{k},\mathbf{K}}$ and $\gamma^{\lambda
i\xi}_{\mathbf{k},\mathbf{K}}$ we first realize, that that $\Delta E_{K_{G}}$
appears whenever WSe2 phonons are involved, and $\Delta E_{K_{W}}$ appears
whenever graphene phonons are involved. $\Delta E_{K_{G}}\approx$250 meV is
large in comparison to typical phonon energies of 30 meV in WSe2 and $\Delta
E_{K_{W}}\approx$1 eV is large in comparison to typical phonon energies of 200
meV in graphene. Consequently we ignore the appearing phonon energies in
$\beta^{\lambda i\xi}_{\mathbf{k},\mathbf{K}}$ and $\gamma^{\lambda
i\xi}_{\mathbf{k},\mathbf{K}}$. As a result, the Hamiltonian simplifies to

$\displaystyle H$
$\displaystyle=\sum_{\mathbf{k}\mathbf{K}\lambda,\xi}\underbrace{\frac{t^{\lambda
GW}g_{\mathbf{K}}^{\lambda
W\xi}}{\epsilon^{W}_{K_{gr}}-\epsilon^{G}_{K_{gr}}}}_{h^{W\xi}_{\mathbf{K}}\approx
const.}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
W}\lambda_{\mathbf{k}}^{G}\left(b_{\mathbf{K}}^{W\xi}+b_{-\mathbf{K}}^{\dagger
W\xi}\right)+h.c.\,$
$\displaystyle+\sum_{\mathbf{k}\mathbf{K}\lambda\xi}\underbrace{\frac{t^{\lambda
WG}g_{\mathbf{K}}^{\lambda
G\xi}}{\epsilon^{G}_{K_{W}}-\epsilon^{W}_{K_{W}}}}_{h^{G}_{\mathbf{K}}\approx
const.}\lambda_{\mathbf{k}+\mathbf{K}}^{\dagger
W}\lambda_{\mathbf{k}}^{G}\left(b_{\mathbf{K}}^{G\xi}+b_{-\mathbf{K}}^{\dagger
G\xi}\right)+h.c.\,.$ (84)

For a carrier in graphene, the relaxation rate to WSe2 via phonon-assisted
tunneling is given as

$\displaystyle\Gamma^{G}_{\mathbf{k}}$
$\displaystyle=2\pi\sum_{\pm,\mathbf{K},\xi,i\in\\{W,G\\}}|h^{i\xi}_{\mathbf{K}}|^{2}\left(\frac{1}{2}\pm\frac{1}{2}+n^{i\xi}_{\mathbf{K}}\right)\delta(\epsilon_{\mathbf{k}}^{G}-\epsilon_{\mathbf{k+K}}^{W}\mp\hbar\omega^{i\xi}_{\mathbf{K}})$

Assuming $h^{i\xi}_{\mathbf{K}}\approx h^{i\xi}$ and
$\hbar\omega^{i\xi}_{\mathbf{K}}\approx\hbar\omega^{i\xi}$, we obtain

$\Gamma^{G}_{\mathbf{k}}=A\sum_{i\in\\{W,G\\},\xi,\pm}\frac{m^{W}}{\hbar^{2}}|h^{i}|^{2}\left(\frac{1}{2}\pm\frac{1}{2}+n^{i\xi}\right)1_{\epsilon^{G}_{\mathbf{k}}\mp\hbar\omega^{i\xi}-\epsilon^{W}_{\mathbf{0}}>0},$
(86)

which is constant for graphene electrons which have at least the energy of the
conduction band plus the phonon energy in graphene. This reflects the constant
density of states in WSe2. The area $A$ cancels with the area in the phonon
coupling element which is contained in $h^{i}$.

In contrast for carriers initially located in WSe2, we get

$\displaystyle\Gamma^{W}_{\mathbf{k}}$
$\displaystyle=2\pi\sum_{\pm,\mathbf{K},\xi,i\in\\{W,G\\}}|h^{i\xi}_{\mathbf{K}}|^{2}\left(\frac{1}{2}\pm\frac{1}{2}+n^{i\xi}_{\mathbf{K}}\right)\delta(\epsilon_{\mathbf{k}}^{W}-\epsilon_{\mathbf{k+K}}^{G}\mp\hbar\omega^{i\xi}_{\mathbf{K}}).$
(87)

With similar approximation as above, we end up at

$\Gamma^{W}_{\mathbf{k}}=A\sum_{i\in\\{W,G\\},\xi,\pm}\frac{\epsilon_{\mathbf{k}}\mp\hbar\omega^{i\xi}}{\hbar^{2}v^{2}_{F}}|h^{i}|^{2}\left(\frac{1}{2}\pm\frac{1}{2}+n^{i\xi}\right),$
(88)

To calculate the relaxation rates, we assume a potential barrier of $E_{B}=$5
eV corresponding to the energy of the WSe2 conduction band w.r.t. to the
vacuum level. Then we approximate the tunneling element as $t=\chi E_{B}$,
with $\chi$ being the wavefunction overlap[24]. For graphene, we include two
optical phonon branches with energies of $\hbar\omega=$200 meV and the
coupling strength of $g=$200 meV[37]. For WSe2, we include two optical phonon
branches with energies of $\hbar\omega=$30 meV and the coupling strength of
$g=$10 meV[38]. Supplementary Fig. 14a illustrates the approximate relaxation
rates of electrons from graphene to WSe2 ($\Gamma^{G}$) and WSe2 to graphene
($\Gamma^{W}$) as a function of the overlap of the wavefunctions between WSe2
and graphene. We find for both a quadratic increase as a function of the
overlap, since the latter enters quadratic in both relaxation rates. The
difference between both relaxation rates arise from different final densities
of states of the carrier relaxation. In the experiment, a delayed rise of the
WSe2 signal w.r.t. to the graphene signal of about 50 fs (13 meV) was found,
which indicates an overlap of 4.0%. The order of magnitude of this value
appears reasonable, since the overlap between two neighboring graphene atoms
is 7% as an example[30].

Figure 14: Tunnel transfer for electrons and holes. a, The ICT-induced
relaxation rate of electrons in graphene towards WSe2 (blue) as a function of
the electronic wavefunction overlap between the involved conduction bands. The
relaxation rate of electrons in WSe2 towards graphene (pink). The dashed lines
indicate the overlapping values which can be expected from the experiment
results. b, The relaxation rate of holes in graphene, which have larger
energies compared to the valence band maximum in WSe2 (blue) and the
relaxation rate of holes in WSe2 (pink).

To evaluate the phonon-assisted tunneling rates for holes, we assume similar
phonon coupling elements in conduction and valence band in WSe2[38] and
graphene but account for the different dispersion of the valence band[43].
Supplementary Fig. 14b illustrates the tunneling rates of holes. We find
qualitatively similar tunneling rates as for electrons. However, the tunneling
of holes from WSe2 to graphene is stronger compared to the electrons. This
arises from the larger density of states of graphene for the involved final
states. The reason for this, is that the Fermi energy of the system is closer
to the conduction band minimum compared to the valence band maximum, cp. Fig.
2a in the manuscript.

Table 1: Parameters used in the computation. ∗ exemplary value at $z=0$ determined numerically by evaluating the Wannier equation for WSe2 on a SiC substrate. [40, 21]. ∗∗ taken as double distance between the chalcogen atoms. The Fermi velocity taken the experimental result. Param. |  | Param. |  | Ref.
---|---|---|---|---
$\hbar$ | 0.658 eV fs | $d_{G}$ | 0.25 e nm | [41]
$e$ | 1 e | $v_{F}$ | 1.8 nm fs-1 | exp
$\epsilon_{0}$ | 5.5$\cdot$10-2 e2eV-1nm-1 | $d_{WSe_{2}}$ | 0.32 e nm | [21]
$k_{B}$ | 8.6$\cdot$10-5 eV K-1 | $|\varphi_{WSe_{2}}(\mathbf{r}=0)|$ | 0.36 nm-1 | ∗
$\epsilon_{SiC}$ | 9.6 | $E^{1s}_{WSe_{2}}$ | 1.7 eV | [42]
|  | $M_{WSe_{2}}$ | 3.7 eVfs2nm-2 | [kormanyos2015k]
$\epsilon_{G}$ | 6[44] | $\epsilon_{WSe_{2}}$ | 13.36 |
$a_{G}$ | 0.33 nm [45] | $a_{WSe_{2}}$ | 0.67 nm |  ∗∗ [46]

## References

## References

  * [1] Zhao, W. _et al._ Lattice dynamics in mono-and few-layer sheets of WS2 and WSe2.  _Nanoscale_ 5, 9677–9683 (2013).
  * [2] Terrones, H. _et al._ New first order Raman-active modes in few layered transition metal dichalcogenides.  _Sci. Rep._ 4, 4215 (2014).
  * [3] Tonndorf, P. _et al._ Photoluminescence emission and Raman response of monolayer MoS2, MoSe2, and WSe2.  _Opt. Express_ 21, 4908–4916 (2013).
  * [4] He, K. _et al._ Tightly bound excitons in monolayer WSe2.  _Phys. Rev. Lett._ 113, 026803 (2014).
  * [5] Froehlicher, G., Lorchat, E. & Berciaud, S.  Charge versus energy transfer in atomically thin graphene-transition metal dichalcogenide van der Waals heterostructures.  _Phys. Rev. X_ 8, 011007 (2018).
  * [6] Maklar, J. _et al._ A quantitative comparison of time-of-flight momentum microscopes and hemispherical analyzers for time-resolved ARPES experiments  _Rev. Sci. Instrum._ 91, 123112 (2020).
  * [7] Nakamura, H. _et al._ Spin splitting and strain in epitaxial monolayer WSe2 on graphene.  _Phys. Rev. B_ 101, 165103 (2020).
  * [8] Gierz, I., Henk, J., Höchst, H., Ast, C. R. & Kern, K.  Illuminating the dark corridor in graphene: Polarization dependence of angle-resolved photoemission spectroscopy on graphene.  _Phys. Rev. B_ 83, 121408 (2011).
  * [9] Hwang, C. _et al._ Fermi velocity engineering in graphene by substrate modification.  _Sci. Rep._ 2, 590 (2012).
  * [10] Siegel, D. A. _et al._ Many-body interactions in quasi-freestanding graphene.  _PNAS_ 108, 11365–11369 (2011).
  * [11] Yuan, L., Wang, T., Zhu, T., Zhou, M. & Huang, L.  Exciton dynamics, transport, and annihilation in atomically thin two-dimensional semiconductors.  _J. Phys. Chem. Lett._ 8, 3371–3379 (2017).
  * [12] Wang, H., Zhang, C. & Rana, F.  Ultrafast dynamics of defect-assisted electron–hole recombination in monolayer MoS2.  _Nano Lett._ 15, 339–345 (2015).
  * [13] Dendzik, M. _et al._ Observation of an excitonic Mott transition through ultrafast core-_cum_ -conduction photoemission spectroscopy.  _Phys. Rev. Lett._ 125, 096401 (2020).
  * [14] Liu, F., Ziffer, M. E., Hansen, K. R., Wang, J. & Zhu, X.  Direct determination of band-gap renormalization in the photoexcited monolayer MoS2.  _Phys. Rev. Lett._ 122, 246803 (2019).
  * [15] Ruppert, C., Chernikov, A., Hill, H. M., Rigosi, A. F. & Heinz, T. F.  The role of electronic and phononic excitation in the optical response of monolayer WS2 after ultrafast excitation.  _Nano Lett._ 17, 644–651 (2017).
  * [16] Tomadin, A., Brida, D., Cerullo, G., Ferrari, A. C. & Polini, M.  Nonequilibrium dynamics of photoexcited electrons in graphene: Collinear scattering, auger processes, and the impact of screening.  _Phys. Rev. B_ 88, 035430 (2013).
  * [17] Liang, Y. & Yang, L.  Carrier plasmon induced nonlinear band gap renormalization in two-dimensional semiconductors.  _Phys. Rev. Lett._ 114, 063001 (2015).
  * [18] Harb, M. _et al._ Carrier relaxation and lattice heating dynamics in silicon revealed by femtosecond electron diffraction.  _J. Phys. Chem. B_ 110, 25308–25313 (2006).
  * [19] Hong, J., Senga, R., Pichler, T. & Suenaga, K.  Probing exciton dispersions of freestanding monolayer WSe2 by momentum-resolved electron energy-loss spectroscopy.  _Phys. Rev. Lett._ 124, 087401 (2020).
  * [20] Selig, M. _et al._ Ultrafast dynamics in monolayer transition metal dichalcogenides: interplay of dark excitons, phonons, and intervalley exchange.  _Phys. Rev. Res._ 1, 022007 (2019).
  * [21] Selig, M. _et al._ Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides.  _Nat. Commun._ 7, 13279 (2016).
  * [22] Selig, M. _et al._ Dark and bright exciton formation, thermalization, and photoluminescence in monolayer transition metal dichalcogenides.  _2D Mater._ 5, 035017 (2018).
  * [23] Dong, S. _et al._ Measurement of an excitonic wave function.  _arXiv:2012.15328_ (2020).
  * [24] Ovesen, S. _et al._ Interlayer exciton dynamics in van der Waals heterostructures.  _Commun. Phys._ 2, 1–8 (2019).
  * [25] Katsch, F., Selig, M. & Knorr, A.  Theory of coherent pump–probe spectroscopy in monolayer transition metal dichalcogenides.  _2D Mater._ 7, 015021 (2019).
  * [26] Selig, M., Malic, E., Ahn, K. J., Koch, N. & Knorr, A.  Theory of optically induced Förster coupling in van der Waals coupled heterostructures.  _Phys. Rev. B_ 99, 035420 (2019).
  * [27] Rana, F.  Electron-hole generation and recombination rates for coulomb scattering in graphene.  _Phys. Rev. B_ 76, 155431 (2007).
  * [28] Wendler, F., Knorr, A. & Malic, E.  Carrier multiplication in graphene under landau quantization.  _Nat. Commun._ 5, 3703 (2014).
  * [29] Dexter, D. L.  A theory of sensitized luminescence in solids.  _J. Chem. Phys._ 21, 836–850 (1953).
  * [30] Reich, S., Maultzsch, J., Thomsen, C. & Ordejon, P.  Tight-binding description of graphene.  _Phys. Rev. B_ 66, 035412 (2002).
  * [31] Chernikov, A. _et al._ Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2.  _Phys. Rev. Lett._ 113, 076802 (2014).
  * [32] Wu, F., Qu, F. & Macdonald, A. H.  Exciton band structure of monolayer MoS2.  _Phys. Rev. B_ 91, 075310 (2015).
  * [33] Robert, C. _et al._ Optical spectroscopy of excited exciton states in MoS2 monolayers in van der Waals heterostructures.  _Phys. Rev. Mater._ 2, 011001 (2018).
  * [34] Rytova, N. S.  Screened potential of a point charge in a thin film.  _arXiv:1806.00976_ (2018).
  * [35] Keldysh, L.  Coulomb interaction in thin semiconductor and semimetal films.  _J. Exp. Theor. Phys._ 29, 658 (1979).
  * [36] Cudazzo, P., Tokatly, I. V. & Rubio, A.  Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphene.  _Phys. Rev. B_ 84, 085406 (2011).
  * [37] Piscanec, S., Lazzeri, M., Mauri, F., Ferrari, A. & Robertson, J.  Kohn anomalies and electron-phonon interactions in graphite.  _Phys. Rev. Lett._ 93, 185503 (2004).
  * [38] Jin, Z., Li, X., Mullen, J. T. & Kim, K. W.  Intrinsic transport properties of electrons and holes in monolayer transition-metal dichalcogenides.  _Phys. Rev. B_ 90, 045422 (2014).
  * [39] Kormányos, A., Zólyomi, V., Drummond, N. D. & Burkard, G.  Spin-orbit coupling, quantum dots, and qubits in monolayer transition metal dichalcogenides.  _Phys. Rev. X_ 4, 011034 (2014).
  * [40] Berghäuser, G. & Malic, E.  Analytical approach to excitonic properties of MoS2.  _Phys. Rev. B_ 89, 125309 (2014).
  * [41] Malic, E., Winzer, T., Bobkin, E. & Knorr, A.  Microscopic theory of absorption and ultrafast many-particle kinetics in graphene.  _Phys. Rev. B_ 84, 205406 (2011).
  * [42] Christiansen, D. _et al._ Phonon sidebands in monolayer transition metal dichalcogenides.  _Phys. Rev. Lett._ 119, 187402 (2017).
  * [43] Kormányos, A., Zólyomi, V., Drummond, N. D. & Burkard, G.  Spin-orbit coupling, quantum dots, and qubits in monolayer transition metal dichalcogenides.  _Phys. Rev. X_ 4, 011034 (2014).
  * [44] Bessler, R., Duerig, U. & Koren, E.  The dielectric constant of a bilayer graphene interface.  _Nanoscale Adv._ 1, 1702–1706 (2019).
  * [45] Ye, S. _et al._ Thickness-dependent strain effect on the deformation of the graphene-encapsulated Au nanoparticles.  _J. Nanomater._ 2014 (2014).
  * [46] Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R.  Theory of neutral and charged excitons in monolayer transition metal dichalcogenides.  _Phys. Rev. B_ 88, 045318 (2013).