# Symmetry-dependent dielectric screening of optical phonons in monolayer
graphene

Loïc Moczko  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Sven
Reichardt  Department of Physics and Materials Science, University of
Luxembourg, 162a avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg
Aditya Singh  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Department
of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016,
India  Xin Zhang  Present address: State Key Laboratory of Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing 100083, China  Université de Strasbourg, CNRS, Institut de Physique et
Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Luis
E. Parra López  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Joanna L.
P. Wolff  Université de Strasbourg, CNRS, Institut de Physique et Chimie des
Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Aditi Raman
Moghe  Université de Strasbourg, CNRS, Institut de Physique et Chimie des
Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Etienne Lorchat
Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux
de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Rajendra Singh
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New
Delhi 110016, India  Kenji Watanabe  Research Center for Functional Materials,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
Takashi Taniguchi  International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
Hicham Majjad  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France
Michelangelo Romeo  Université de Strasbourg, CNRS, Institut de Physique et
Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France
Arnaud Gloppe  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France  Ludger
Wirtz  Department of Physics and Materials Science, University of Luxembourg,
162a avenue de la Faïencerie, L-1511 Luxembourg, Luxembourg  Stéphane Berciaud
<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de
Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg,
France

###### Abstract

Quantised lattice vibrations (i.e., phonons) in solids are robust and
unambiguous fingerprints of crystal structures and of their symmetry
properties. In metals and semimetals, strong electron-phonon coupling may lead
to so-called Kohn anomalies in the phonon dispersion, providing an image of
the Fermi surface in a non-electronic observable. Kohn anomalies become
prominent in low-dimensional systems, in particular in graphene, where they
appear as sharp kinks in the in-plane optical phonon branches. However, in
spite of intense research efforts on electron-phonon coupling in graphene and
related van der Waals heterostructures, little is known regarding the links
between the symmetry properties of optical phonons at and near Kohn anomalies
and their sensitivity towards the local environment. Here, using inelastic
light scattering (Raman) spectroscopy, we investigate a set of custom-designed
graphene-based van der Waals heterostructures, wherein dielectric screening is
finely controlled at the atomic layer level. We demonstrate experimentally and
explain theoretically that, depending exclusively on their symmetry
properties, the two main Raman modes of graphene react differently to the
surrounding environment. While the Raman-active near-zone-edge optical phonons
in graphene undergo changes in their frequencies due to the neighboring
dielectric environment, the in-plane, zone-centre optical phonons are
symmetry-protected from the influence of the latter. These results shed new
light on the unique electron-phonon coupling properties in graphene and
related systems and provide invaluable guidelines to characterise dielectric
screening in van der Waals heterostructures and moiré superlattices.

##  I Introduction

Phonons in insulators and semiconductors, such as diamond and silicon,
respectively [1, 2], are routinely exploited for calibration and sensing
purposes and generally considered as weakly sensitive to the local
environment. In the particular case of metals, strong coupling between
electrons and optical phonons with specific nesting wavevectors can give rise
to Kohn anomalies in the phonon dispersion [3]. Kohn anomalies may drive phase
transitions such as charge density waves and also provide an image of the
Fermi surface [4]. In addition, Kohn anomalies may be very sensitive to
screening effects, through the screened electron-phonon coupling. However, the
fundamental links between symmetries and Kohn anomalies deserve further
attention. These links become prominent and experimentally accessible in
reduced dimensions, for instance in low-dimensional carbon-based materials,
e.g. carbon nanotubes [5] and graphene [6, 7]. Interestingly, the rise of two-
dimensional (2D) materials beyond graphene (such as transition metal
dichalcogenides (TMD) [8] and hexagonal boron nitride (hBN) [9]) and of the
concept of van der Waals engineering [10, 11] make it possible to devise
quantum materials and devices, wherein dielectric screening can be finely
controlled at the atomic monolayer level [12, 13], to uncover and exploit the
subtle interplay between electronic and vibrational degrees of freedom.

Here, taking monolayer graphene as a model system, we show that the two Kohn
anomalies that occur at the centre ($\mathbf{\Gamma}$) and edges ($\mathbf{K}$
and $\mathbf{K}^{\prime}$ points) of its Brillouin zone, respectively (Fig.
1a), behave differently when the local dielectric environment is changed, just
because the associated phonon modes behave differently under symmetry
operations. For this purpose, we investigate, using micro-Raman scattering
spectroscopy [14], the G-, 2D- and 2D′-mode features in monolayer graphene
embedded in a carefully chosen variety of environments, evolving from the
unscreened, freely suspended case to a van der Waals heterostructure, where
graphene is tightly coupled to a TMD monolayer and fully encapsulated in
between thin films of hBN. We unravel a progressive upshift of the 2D-mode
feature of up to 26 cm-1 that we attribute to the screening of the Kohn
anomaly that affects non-degenerate zone-edge in-plane transverse optical
phonons (TO branch at $\mathbf{K}$ and $\mathbf{K}^{\prime}$, with
$A^{\prime}_{1}$ symmetry). Importantly, we find that screening arises for the
most part from the first adjacent layer and that a TMD monolayer screens more
efficiently than bulk hBN. Measurements of the dispersion of the 2D-mode
frequency reveal that the effect of screening is more pronounced as the
wavevector of the 2D-mode phonon approaches $\mathbf{KK}^{\prime}$.
Remarkably, we demonstrate that the zone-centre in-plane optical phonons
(degenerate LO/TO branch at $\mathbf{\Gamma}$, with $E_{\mathrm{2g}}$
symmetry) giving rise to the Raman G-mode feature are, to experimental
accuracy, insensitive to dielectric screening. These contrasting behaviours
are explained theoretically by considering the symmetry properties of low-
energy electronic and electron-phonon Hamiltonians in graphene and showing
that the zone-centre phonon frequency is protected from changes in the
screening by a Ward identity, which leads to a cancellation of many-body
renormalization effects in the interatomic force constants. Our results shed
light on electron-phonon physics and demonstrates how symmetry is crucial to
understand the protection of certain observables, including the frequency of
optical phonons. From a practical standpoint, the protection of the G-mode
frequency combined with the high sensitivity of the 2D-mode feature with
respect to dielectric screening provide an invaluable missing element for
advanced characterization of graphene and more broadly of near-field coupled
van der Waals materials.

##  II Raman spectroscopy of graphene in various dielectric environments

Raman scattering spectroscopy is known as a unique probe of the physical
properties of graphene and van der Waals materials [14, 15], able to yield
decisive insights into their nanoscale environment [16], symmetry properties
[17, 18] and sensitivity towards external perturbations [19]. Monolayer
graphene stands out as the first [20, 21] and most studied 2D material,
largely because of the strong coupling between its electronic states and its
optical phonon modes. More precisely, the Fermi surface of neutral graphene
(i.e., the $\mathbf{K},\>\mathbf{K^{\prime}}$ points) enables Kohn anomalies
for phonon wavevectors $q=0$ and $\mathbf{q}=\mathbf{KK}^{\prime}$ , which
appear as sharp kinks in the phonon dispersion [6, 7] (Fig. 1a,b).

Figure 1: Symmetry-dependent screening-induced changes in the Raman spectrum
of graphene. (a) Dispersion of the higher energy phonon branches in graphene
along the high-symmetry lines of its first Brillouin zone (see inset), adapted
from [22]. The Kohn anomalies at $\mathbf{K}$ and $\mathbf{\Gamma}$ are
highlighted with red dashed lines. (b) In-plane atomic displacements
associated with the $E_{\mathrm{2g}}$ and $A^{\prime}_{1}$ phonons. (c)
Sketches of the Raman G-, 2D-, and 2D′\- modes processes in the momentum-
energy space. (d) Typical Raman spectra of hBN-capped graphene (black dots)
and hBN-capped WSe2/graphene (orange dots). Data are shown together with their
fit (solid lines). The layered structures are sketched with the encapsulating
hBN films in light yellow. (e) Optical image of the corresponding sample
(Sample 1). The contours of the relevant regions of the sample are indicated.
FLG denotes a few-layer graphene flake. (f,g) Hyperspectral Raman maps of the
2D-mode and G-mode frequencies in the sample shown in (e), respectively.

The Raman spectrum of graphene (Fig. 1c,d) is dominated by two main features.
First, the one-phonon, non-resonant G-mode feature (frequency
$\omega_{\mathrm{G}}$ near 1580 cm-1) arising from degenerate LO/TO
$E_{\mathrm{2g}}$ phonons at $\mathbf{\Gamma}$ [23]. Second, strong electron-
phonon coupling favors resonant multiphonon processes [24, 25, 26], the most
famous of which being the 2D mode, with a pair of near zone-edge
($\mathbf{K}$, $\mathbf{K^{\prime}}$) TO phonons with opposite momenta that
warrants conservation of energy and momentum (Fig. 1c). The 2D-mode feature is
widely known as a hallmark of monolayer graphene [20, 21]; it is also
dispersive (frequency $\omega_{\mathrm{2D}}$ between 2500 and 2700 cm-1 for
near-IR to near UV incoming photons) and thereby enables probing both the
electron and phonon dispersions in various environments [27, 28]. A
conceptually similar mode involving a pair of LO phonons near
$\mathbf{\Gamma}$ gives rise to the fainter 2D′ mode feature near 3240 cm-1
(Ref. 25, 26, Fig. 1c and Appendix B).

The sensitivity of the G- and 2D-mode features upon strain [29], doping [30,
31, 32, 33, 34] and defects [35] has been extensively documented, whereas
studies of dielectric screening effects are scarce [36, 28, 37]. Here,
crucially, in order to isolate the role of dielectric screening, we have
focused on samples that have been thoroughly characterised by means of
hyperspectral Raman mapping and that display negligible unintentional doping,
built-in strain and defect density (Supplemental Material, Sections S1 and S2
[38]).

Figure 1e shows an optical image of a graphene monolayer, partly covered by a
tungsten diselenide (WSe2) monolayer and fully encapsulated in hBN (Sample 1,
see Appendix A for details on the experimental methods and the Supplemental
Material [38]). Typical Raman spectra associated with both regions of the
sample (Fig. 1d) evidence a clear shift of $\omega_{\mathrm{2D}}$, while at
the same time, the G-mode and the 2D′-mode features remain unaffected. An
hyperspectral map of $\omega_{\mathrm{2D}}$ is presented in Fig. 1f and
readily confirms that hBN-capped graphene/WSe2 exhibits higher values of
$\omega_{\mathrm{2D}}$ than hBN-capped graphene. In order to quantitatively
characterize the role of the environment, we spatially average
$\omega_{\mathrm{2D}}$ and $\omega_{\mathrm{G}}$ over an area of a few tens of
µm2 in regions (i) and (ii) in Fig. 1e. We find a spatially averaged upshift
of $\omega_{\mathrm{2D}}$ of $7.3\pm 3~{}\mathrm{cm}^{-1}$, when comparing
hBN-capped WSe2/graphene to hBN-capped graphene. Noteworthy, the spatial
averages of ($\omega_{\mathrm{G}}=1583.7\pm 0.8~{}\mathrm{cm}^{-1}$ and
$\omega_{\mathrm{G}}=1583.5\pm 0.5~{}\mathrm{cm}^{-1}$ in hBN-capped graphene
and graphene/WSe2, respectively) are nearly identical. These contrasting
behaviours directly exclude a simple interpretation in terms of distinct
levels of strain and/or doping in both regions, as one would expect a G-mode
upshift of $\approx 3.3~{}\mathrm{cm}^{-1}$ in the case of biaxial compressive
strain [39, 40] and a much larger upshift shift in excess of
$10~{}\mathrm{cm}^{-1}$ in the case of electron or hole doping [41, 34]. A
more plausible explanation is that the presence of one single layer of TMD
screens the Kohn anomaly at $\mathbf{K}$ and $\mathbf{K^{\prime}}$ [42],
leading to a sizeable upshift of $\omega_{\mathrm{2D}}$ [36, 37], as
schematised in Fig. 2.

##  III Impact of dielectric screening near the Kohn anomaly at K

To get deeper insights into the influence of the dielectric environment, we
now consider Sample 2, made of a monolayer graphene deposited onto a SiO2
substrate patterned with microscopic circular pits and partly covered by a
WSe2 monolayer. Fig. 2a shows an optical image of this sample, where one can
identify four regions of interest, namely, freely suspended graphene (i),
graphene on SiO2 (ii), suspended graphene/WSe2 (iii), and SiO2-supported
graphene/WSe2 (iv). The interest of such a sample is threefold. First, it
makes it possible to study suspended graphene as a pristine, minimally
screened sample [43, 44]. Second, the screening induced by one single TMD
monolayer can be easily identified in suspended WSe2/graphene. Third, by using
SiO2 as a bulk substrate with similar dielectric constant as bulk hBN [45,
46], we will be able to determine to which extent surface roughness affects
the observed 2D-mode upshift introduced above.

Figure 2: Probing dielectric screening of the Kohn anomaly at the K and K′
points. (a) Optical image of a graphene/WSe2 heterostructure deposited on an
Si/SiO2 substrate with pre-patterned holes (Sample 2). The white dashed
contour indicates the WSe2 monolayer. The scalebar is $6~{}\mu\mathrm{m}$.
Four regions of interest corresponding to freely suspended graphene (i),
graphene on SiO2 (ii), suspended graphene/WSe2 (iii), and SiO2-supported
graphene/WSe2 (iv) are identified. (b) Hyperspectral Raman map of the graphene
2D-mode frequency $\omega_{\mathrm{2D}}$. The mapped area corresponds to the
dark yellow rectangle in (a). (c) Raman 2D-mode spectra of the sample shown in
(a) for different laser energies: 1.58 eV (black), 1.96 eV (red) and 2.33 eV
(green). Data are shown together with their fit. (d) 2D-mode frequency
$\omega_{\mathrm{2D}}$ in the different regions of the sample as a function of
the (symbol-coded) laser photon energy. Data are shown relative to
$\omega_{\mathrm{2D}}^{0}$, the value recorded on suspended graphene. (e)
Sketch of the TO phonon dispersion in the vicinity of the $\mathbf{K}$ point
for unscreened (dashed black line) and screened graphene (solid orange line).
The vertical arrows indicate the expected screening-induced phonon upshift for
phonon wavevectors corresponding to increasing laser photon energy. Dielectric
screening flattens the Kohn anomaly (kink at $\mathbf{K}$).

Figure 2b shows an hyperspectral map of $\omega_{\mathrm{2D}}$ (under laser
excitation at 2.33 eV), that, again, evidences a clear progressive upshift
when moving from suspended graphene to SiO2-supported graphene/WSe2, i.e.,
from region (i) to (iv) (Fig. 2b,c). In stark contrast, the spatially averaged
$\omega_{\mathrm{G}}$ are nearly identical on all four regions, spread only
within a 2 cm-1 window (see Supplemental Material, Section S2 [38]), in
keeping with the trend observed in Fig. 1.

Remarkably, by investigating the dispersion of $\omega_{\mathrm{2D}}$ using
laser excitation at 2.33 eV, 1.96 eV and 1.58 eV in the four regions
identified above (Fig. 2c), we observe that the screening-induced 2D-mode
upshift augments as the laser photon energy decreases. In Fig. 2d, we plot, in
each region, the laser photon energy dependent upshift of
$\omega_{\mathrm{2D}}$, with color-coded symbols. Under laser excitation at
2.33 eV (resp. 1.58 eV), $\omega_{\mathrm{2D}}$ upshifts by up to
$20~{}\mathrm{cm}^{-1}$ (resp. $27~{}\mathrm{cm}^{-1}$) when moving from
region (i) to (iv). These observations are consistent with the screening of
the Kohn anomaly illustrated in Fig. 2e. Indeed, as the laser photon energy
increases, the momentum of the phonons involved in the 2D-mode process (taken
relative to $\mathbf{K}$) also increases and the screening-induced upshift of
phonon frequency fades away.

Thus far, the 2D-mode upshift has been interpreted as being the result of a
change of the phonon dispersion when the screening is varied [42, 36]. In
general though, the 2D mode process strongly relies on the resonant interplay
between the electronic and phonon dispersions (in a simple one-dimensional
description with linear bands and no trigonal warping effects, one gets
$\omega_{\mathrm{2D}}=2\omega_{\mathbf{K}}+2\frac{v_{\mathrm{TO}}}{v_{\mathrm{F}}}\omega_{\mathrm{L}}$,
with $\omega_{\mathbf{K}}$, $v_{\mathrm{TO}}$, $v_{\mathrm{F}}$ and
$\omega_{\mathrm{L}}$, the TO phonon frequency at $\mathbf{K}$, the phonon and
electron band velocities, and the laser photon frequency, respectively). As a
result, the observed 2D-mode upshift may also be affected by the partial
compensation between the screening-induced reduction of the phonon and
electron band velocities [27, 28, 24]. Nevertheless, the results in Fig. 2d,
which reveal the dispersive character of the relative dielectric shift, also
indicate that in the energy range probed in our study, this dispersive effect
appears only as a correction to main effect that is a rigid upshift of
$\omega_{\mathbf{K}}$ when the dielectric screening increases. Further
measurements at lower $\omega_{\mathrm{L}}$ would allow probing 2D-mode
phonons closer to the $\mathbf{K}$ point [47] and thereby get more insights
into the upshift of $\omega_{\mathrm{2D}}$.

##  IV Evidence for symmetry-dependent dielectric screening of optical phonons

Figure 3: Upshift of the 2D-mode frequency and invariance of the G-mode
frequency in various dielectric environments.  Spatially averaged 2D-mode (a)
and G-mode (b) frequencies measured on a variety of graphene-based van der
Waals heterostructures (see sketches below the graphs; the bare graphene
monolayer and graphene/WSe2 heterostructure -first and fourth samples-
correspond to the freely suspended systems discussed in Fig. 2). Samples are
ordered by increasing expected dielectric screening experienced by graphene.
(c) Correlation between the spatially averaged 2D- and G-mode frequencies
shown in (a) and (b). For reference, the short dashed, dot-dashed and dotted
lines correspond to the expected correlations in the presence of biaxial
strain, hole and electron doping, respectively. The reference point for
undoped and unstrained graphene is taken for the suspended graphene monolayer.
All measurements were performed under laser excitation at 2.33 eV.

The spatially averaged values of $\omega_{\mathrm{2D}}$ and
$\omega_{\mathrm{G}}$ obtained from Sample 1 (Fig. 1) and Sample 2 (Fig. 2) as
well as on another sample (BN/MoSe2/graphene, Sample 3 introduced in the
Supplemental Material, Section S2 [38], see also Ref. 48) are gathered in Fig.
3a and b, respectively. The areas of interest are ordered by increasing
expected dielectric screening. Errorbars correspond to the standard deviation
of the spatially-averaged values of $\omega_{\mathrm{2D}}$ and
$\omega_{\mathrm{G}}$.

While the upshift of $\omega_{\mathrm{2D}}$ due to the bulk SiO2 substrate
itself is only of $5.9\pm 2.3~{}\mathrm{cm}^{-1}$, we find much larger
upshifts of $14.8\pm 2.8~{}\mathrm{cm}^{-1}$ and $16.0\pm
1.2~{}\mathrm{cm}^{-1}$ when SiO2 is replaced by van der Waals materials,
namely bulk hBN and monolayer WSe2, respectively. In spite of their similar
dielectric constants [45, 46], bulk hBN leads to a much larger shift of
$\omega_{\mathrm{2D}}$ that bulk SiO2. As SiO2 is rougher than a layered
material such as hBN, this observation suggests that graphene remains partly
decoupled from SiO2, whereas hBN, as an atomically flat substrate, warrants
optimal coupling and drastically reduces surface roughness and disorder in
graphene [49]. This hypothesis is corroborated by the particularly narrow
2D-mode full width at half maximum of only 16 cm-1 observed in hBN-capped
graphene and hBN-capped graphene/WSe2 (Appendix C and Ref. 50, 51) and by a
comparative study of SiO2/graphene/TMD versus SiO2/TMD/graphene
heterostructures (Appendix D.2). Noteworthy, the screening-induced upshift of
$\omega_{\mathrm{2D}}$ is slightly larger in suspended graphene/monolayer WSe2
than in graphene deposited on bulk hBN (Fig. 3a and Supplemental Material,
Section S2 [38]), illustrating the remarkably strong interaction between a TMD
monolayer and graphene, as suggested otherwise by recent electron transport
measurements [52]. Altogether, this set of results demonstrates that the
screening of the Kohn anomaly at $\mathbf{K}$ is intimately linked to the
interfacial coupling between graphene and the nearest atomic layer. This
conclusion is bolstered by our demonstration that a TMD bilayer does only lead
to a marginal increase of $\omega_{\mathrm{2D}}$ compared to a TMD monolayer
(Appendix D.1).

Let us stress that the screening of optical phonons also stands out as an
extremely short-range effect (typically, the sub-nanometre van der Waals gap
between coupled 2D layers), whereas it was shown that excitons could undergo
the effects of dielectric screening at larger distances of several nanometres
(compare Fig. 1c and 1d Ref. 37 and see also Ref. 53). We attribute this
difference to the fact that for the excitons, the excited electron interacts
both with the electrostatic potential of the hole as well as with the
potential of the screening cloud originating from all over space, and in
general from several neighboring layers (Appendix D.3 for details).

We now analyze the contrasting behaviour introduced in Fig. 1 between Raman
modes involving optical phonons at or near $\mathbf{\Gamma}$ (G- and
2D′-modes, respectively) and near $\mathbf{K},~{}\mathbf{K}^{\prime}$
(2D-mode). As demonstrated in Fig. 3a,b, even in very different dielectric
environments, $\omega_{\mathrm{G}}$ remains constant at $1582\pm
2~{}\mathrm{cm}^{-1}$ (fluctuations that can be ascribed to built-in strain
and unintentional doping, see Supplemental Material, Section S2 [38]), whereas
$\omega_{\mathrm{2D}}$ upshifts by more than $20~{}\mathrm{cm}^{-1}$. The
invariance of $\omega_{\mathrm{G}}$ is further evidenced on the
$\omega_{\mathrm{2D}}$ versus $\omega_{\mathrm{G}}$ correlation plot displayed
in Fig. 3c. A linear fit to the plotted data yields a slope
$\partial\omega_{\mathrm{2D}}/\partial\omega_{\mathrm{G}}$ above 10, that
largely exceeds the expected slopes in the presence of strain [39, 40] or
doping [41, 34]. This behaviour is a priori counter intuitive as a Kohn
anomaly is also present at $\mathbf{\Gamma}$, suggesting that a sizeable
upshift of $\omega_{\mathrm{G}}$ (and $\omega_{\mathrm{2D^{\prime}}}$) may be
in order when graphene is screened by a partner material.

##  V Theoretical argument

The insensitivity of the $\mathbf{\Gamma}$-point phonon frequency to changes
in the dielectric environment can be understood from a symmetry point of view.
For this, we focus on the contribution of the low-energy $\pi$-band
excitations to the part of the phonon frequency that is sensitive to screening
[22, 54]:

$\omega^{2}_{\lambda}|_{\pi\text{-bands}}=\frac{2}{M_{\mathrm{C}}}\int_{\mathrm{BZ}}\frac{\mathrm{d}^{2}k}{A_{\mathrm{BZ}}}\sum_{\mathrm{spin}}\frac{\left(g_{\mathrm{bare}}^{\lambda}\right)^{*}_{\mathbf{k},\pi^{*},\pi}\left(g_{\mathrm{screened}}^{\lambda}\right)_{\mathbf{k},\pi^{*},\pi}}{\varepsilon_{\mathbf{k},\pi^{*}}-\varepsilon_{\mathbf{k},\pi}}.$
(1)

Here, $\lambda=x,y$ is a label for the phonon polarization,
$g^{\lambda}_{\mathrm{bare}}$ and $g^{\lambda}_{\mathrm{screened}}$ are the
unscreened and screened electron-phonon coupling matrix elements,
respectively, $\varepsilon_{\mathbf{k},\pi^{(*)}}$ denote the band energies,
$M_{\mathrm{C}}$ is the mass of a carbon atom, and the integral runs over the
first Brillouin zone, whose area is denoted as $A_{\mathrm{BZ}}$. While the
exact definitions of the electron-phonon matrix elements are given in Appendix
E, we note that the screened electron-phonon coupling and the $\pi$-band
transition energies are not independent of one another. The lattice symmetry
of the graphene lattice leads to the effective $\pi$-band Hamiltonian being of
the Dirac-Weyl type, while the $E_{2\mathrm{g}}$ symmetry of the phonon
implies that it transforms like a vector under rotations, in contrast to the
non-degenerate phonon at $K$, which transforms like a pseudo-scalar under
rotations. This ”vector-like” character of the phonon mode in combination with
the Dirac-Weyl nature of the lead to the effective electron-phonon interaction
Hamiltonian being of the same form as the electron-photon Hamiltonian in
Quantum Electrodynamics (QED), in which the vector-like phonon is coupled to
an electronic current (Appendix E). Crucially, for Dirac electrons, this
current is conserved, which on the quantum mechanical level implies that any
correlation functions involving this current obey the Schwinger-Dyson
equations (see e.g., Ref. [55]). Noting that for the $\pi$-band electrons, the
screened electron-phonon coupling can be expressed in terms of a correlation
function involving the conserved current, an application of the Schwinger-
Dyson equation leads to a Ward identity [55, 56] for the screened electron-
phonon coupling. For electrons near the $K$ valley, it reads:

$\left(g_{\mathrm{screened}}^{\lambda}\right)_{\mathbf{k},\pi^{*},\pi}\propto
v_{\mathrm{F}}\,\left[\chi^{\dagger}_{\mathbf{k}\pi^{*}}\cdot\bm{\sigma}\cdot\chi_{\mathbf{k},\pi}\right]\wedge\mathbf{e}_{\lambda},$
(2)

where $\mathbf{e}_{\lambda=x,y}$ is the effective polarization vector of the
phonon, $\bm{\sigma}$ is the vector of Pauli matrices, and $\chi$ denotes a
two-component Weyl spinor in sublattice space, and
$\mathbf{a}\wedge\mathbf{b}=a_{x}b_{y}-a_{y}b_{x}$ is the two-dimensional
anti-symmetric product. Most importantly, the only screening-dependent, i.e.,
substrate-sensitive, quantity here is the effective Fermi velocity
$v_{\mathrm{F}}$. Notably, the bare electron-phonon coupling is naturally
independent of screening and $v_{\mathrm{F}}$, while the electronic
transitions energies in the denominator of Eq. (1), are proportional to the
very same $v_{\mathrm{F}}$:
$\varepsilon_{\mathbf{k},\pi^{*}}-\varepsilon_{\mathbf{k},\pi}\propto
v_{\mathrm{F}}|\mathbf{k}|$. As such, the screened electron-phonon coupling
for the $\mathbf{\Gamma}$-point optical phonon as well as the electron
$\pi$-band slope and thus all screening effects from the dielectric
environment in Eq. (1) compensate each other. This exact cancellation for the
low-energy $\pi$-band electrons is a result purely of the lattice symmetry,
leading to the Dirac-like effective electronic and electron-phonon
Hamiltonians, and of current conservation, which is tied to a global $U(1)$
gauge symmetry. It is in excellent agreement with our experimental findings,
which show no noticeable change of $\omega_{G}$ when the dielectric
environment is changed. Note that for acoustic phonons, a stronger version of
the gauge symmetry exists, which slightly simplifies the discussion in that
case [57, 58] (Appendix E).

Consistently, the 2D′ mode, involving a pair of LO phonons near $\bm{\Gamma}$,
albeit being of similar nature as the 2D mode [25, 26], is, to experimental
accuracy, unaffected by changes in the dielectric environment, due to the
protection of the LO phonon frequency at $\bm{\Gamma}$ (Appendix B). Since the
phonons involved in the 2D′ mode are not strictly at $\bm{\Gamma}$, this
result illustrates the robustness of the symmetry argument introduced above.

We want to stress once more that, while a more complicated and generalized
form of the Ward identity can always be derived for the screened electron-
phonon vertex in any system, its usefulness in the case of graphene is due to
the symmetry of the graphene lattice, which ensures that the electron-phonon
interaction for the zero-momentum optical in-plane optical phonons takes on
the specific form of an electronic current coupled to a phonon field that
transforms like a vector under rotations. This fact makes the Ward identity
useful to derive insights into the influence of the dielectric environment on
the G-mode phonon. Conversely, for non-degenerate phonons, which necessarily
do not transform like a vector under rotations and hence cannot couple to the
electronic current, the more general form of the Ward identity, cannot be used
to derive any straight insight. In particular, we thus expect that the Raman
G-mode frequency will no longer be protected if it becomes non-degenerate, as
is, for instance, the case in carbon nanotubes and in uniaxially strained
graphene. The latter point also specifically applies to the frequency of the
$A^{\prime}_{1}$ phonon at $\mathbf{K}$ in graphene, which, while also
influenced by a Kohn anomaly, is not protected from changes in the screening
environment. As a result, there is no symmetry-based reasoning that protects
the 2D-mode frequency.

The symmetry-based arguments discussed above could in principle also affect
the Raman scattering intensities. Experimentally, we observe that the
integrated intensity ratio between the 2D- and G-mode features
($I_{\rm{2D}}/I_{\rm{G}}$) is indeed reduced when comparing freestanding
graphene with a TMD/graphene heterostructure (see Supplemental Material,
Section S3 for a detailed analysis [38]). Although computing the Raman
scattering intensities of the G- and 2D-modes as a function of screening
remains a challenge, the aforementioned observation is in qualitative
agreement with the stronger sensitivity of the $A^{\prime}_{1}$ phonons at
$\mathbf{K}$ (and of its coupling to electrons) to dielectric screening.

##  VI Conclusion and outlook

We have demonstrated that the Raman-active optical phonons in graphene are an
invaluable gauge of dielectric screening and, importantly of the microscopic
details of interlayer coupling within van der Waals heterostructures. The
presence of a surrounding medium screens the Kohn anomaly near the
$\mathbf{K}$ point, yielding a sizeable upshift of the Raman 2D-mode feature
by up to $1\ \%$ when comparing a suspended graphene monolayer with a
monolayer graphene/monolayer TMD heterostructure encapsulated in hexagonal
boron nitride. Conversely, $E_{\mathrm{2g}}$ phonons, involved in the Raman
G-mode and 2D′-mode process are symmetry-protected from screening effects such
that their frequency is invariant in all the experimental configurations
examined here.

As an outlook, we believe that the impact of our work can be threefold.

First, the theoretical aspects of our work sheds new light on the deep
connections between crystal symmetries, the resulting electronic and
vibrational properties, and quantum mechanical conservation laws, which are
relevant well-beyond graphene science. In particular, our work provides a
general definition of the exact electron-phonon coupling in terms of
electronic correlation functions. This is a novel viewpoint on electron-phonon
physics, beyond simple models or sophisticated first principles simulations,
that will significantly advance ongoing debates on the exact description of
the electron-phonon interaction in quantum materials.

Second, in recent years the correlation between the frequencies of the 2D- and
G-mode features has enabled an ever more advanced characterization of
graphene-based samples. The comprehensive study undertaken here demonstrates
that the observed variability in the 2D-mode frequency can also stem for a
large part from changes in the dielectric environment. Thus, using the well-
documented sensitivity of the G- and 2D-modes features to strain and doping
[59, 39, 34, 60, 50], one may straightforwardly isolate the effect of the
dielectric environment, which manifests itself as a clear rigid shift of the
2D-mode frequency only. An illustration of such a multiparameter analysis is
presented in the Supplemental Material, Section S2.1 [38]. Going further, our
results provide invaluable guidelines to decipher hyperspectral Raman maps
[37], to better understand proximity effects at van der Waals heterointerfaces
[13] and more broadly to develop advanced Raman scattering-based methods in
moiré superlattices [16].

Finally, our work reveals that the impact of dielectric screening on optical
phonons in graphene is chiefly determined by the microscopic details of the
interfacial coupling and is extremely short-range, illustrating once again the
unique physical properties of van der Waals materials. The extreme sensitivity
of the Raman features of graphene towards dielectric screening could be used
to develop two-dimensional sensors, able to quantitatively probe interfacial
coupling in van der Waals materials with sub-nanometre out-of-plane accuracy,
as well as built-in high-pressure gauges [61] as an alternative to ruby [62].

###### Acknowledgements.

We thank T. Sohier for fruitful discussions. We are grateful to the StNano
clean room staff for technical support. We acknowledge financial support from
the Agence Nationale de la Recherche (under grants ATOEMS ANR-20-CE24-0010 and
VANDAMME ANR-21-CE09-0022). This work of the Interdisciplinary Thematic
Institute QMat, as part of the ITI 2021 2028 program of the University of
Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR 10 IDEX 0002),
and by SFRI STRAT’US project (ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024
under the framework of the French Investments for the Future Program. S.B.,
A.R.M., and A.S. acknowledge support from the Indo-French Centre for the
Promotion of Advanced Research (CEFIPRA). S.B. acknowledges support from the
Institut Universitaire de France (IUF). K. W. and T. T. acknowledge support
from the Elemental Strategy Initiative conducted by the MEXT, Japan, and the
CREST (JPMJCR15F3), JST. S.R. acknowledges Fonds National de la Recherche
(FNR) Luxembourg project “RESRAMAN” (Grant No. C20/MS/14802965) and L.W.
acknowledges support through the FNR project INTER/19/ANR/13376969/ACCEPT.

Figure 4: Invariance of the 2D′-mode frequency. (a) Sketch of the Raman
2D′-mode process. (b) Spectra of the 2D′-mode feature in the different
configurations considered. The sample and its areas of interest are introduced
in Fig. 2 of main text. (c) Averaged values of 2D′-, 2D- and G-mode
frequencies in each dielectric environment, respectively.

##  Appendix A Experimental methods

All samples were made from mechanically exfoliated bulk crystals of graphite,
TMDs (here WSe2, MoSe2 and MoS2) and hBN. The van der Waals heterostructures
were deposited onto Si/SiO2 substrates using dry transfer methods, either
using directly a PDMS poly(dimethylsiloxane) stamp [63] or through the ”pick-
up and lift” method, using an additional PC (polycarbonate) layer that is
later dissolved in chloroform [64, 65]. Suspended structures were prepared by
depositing graphene and WSe2 layers onto Si/SiO2 substrates patterned by
$\approx 5~{}\mu\rm m$ in diameter pits fabricated using photolithography
followed by reactive ion etching [43]. Raman and photoluminescence (PL)
measurements were performed in ambient conditions using a home-built scanning
confocal setup as described elsewhere [39, 37], using single mode, linearly
polarised lasers emitting at 2.33 eV, 1.96 eV and 1.58 eV. The laser intensity
was maintained low enough (i.e., well below $1~{}\rm{mW}/\mu\rm{m}^{2}$ for
Raman measurements, with even lower intensities for PL measurements) to
minimize heating and, importantly, photoinduced doping effects [37, 48]. No
polarization analysis was performed on the Raman scattered and PL photons. All
samples discussed here have been characterised by means of hyperspectral
mapping (Supplemental Material, Sections S1 and S2 [38]), allowing unambiguous
identification of graphene [20, 21] and TMD monolayers [66], respectively. PL
maps allow to identify coupled and decoupled TMD/graphene regions based on the
strong PL quenching due to non-radiative transfer of TMD excitons to graphene
[37]. From the spatially-resolved Raman measurements, we can map the built-in
strain and non-intentional doping (Supplemental Material, Section S2 [38] and
Ref. 41, 67). All samples discussed in this manuscript display small built-in
strain of at most a few $10^{-2}\,\%$, while unintentional doping remains
below $10^{-12}~{}\rm{cm}^{-2}$. In these conditions the shifts of
$\omega_{\mathrm{G}}$ and $\omega_{\mathrm{2D}}$ due to built-in strain and
unintentional doping are negligible compared to the dominant effect of
dielectric screening (Fig. 3a) and are responsible for the small scatter in
the spatially averaged $\omega_{\mathrm{G}}$ (Fig. 3b).

##  Appendix B Impact of dielectric screening on the Raman 2D’ mode

The 2D′ mode is a resonant process that involves two LO phonons with opposite
momenta near the $\mathbf{\Gamma}$ point and appears near 3240 $\rm{cm^{-1}}$.
It is the intra-valley analogue of the 2D mode [25, 26]. The fully resonant
part of the Raman 2D′ mode process is sketched in Fig. 4a. Since the electron-
phonon coupling at $\mathbf{\Gamma}$ is smaller than at $\mathbf{K}$, the
2D′-mode feature is less intense than the 2D-mode feature [6, 57].

In this Appendix, we investigate the evolution of the 2D′ mode frequency
$\omega_{\mathrm{2D^{\prime}}}$ in various dielectric environments to further
confirm the behaviour evidenced in Fig. 1d of the main text (Sample 1). For
this purpose, we consider the same sample and areas of interest as in Fig. 2
of the main text (see also Fig. S2d-f [38]), namely suspended graphene (i),
SiO2-supported graphene (ii), suspended graphene/WSe2 (iii) and SiO2-supported
graphene/WSe2 (iv).

In the most screened case (i.e., in this sample, SiO2-supported
graphene/WSe2), we measure an almost negligible spatially averaged blueshift
$\Delta\omega_{\mathrm{2D^{\prime}}}=2.6\pm 1.4$ cm-1 with respect to the
unscreened case of suspended graphene, while, within experimental accuracy, we
do not resolve any sizeable shift when comparing the other remaining cases to
suspended graphene. We can thus consider that
$\Delta\omega_{\mathrm{2D^{\prime}}}$ is negligibly small and in any event
more than one order of magnitude smaller than the 2D-mode upshift
$\Delta\omega_{\mathrm{2D}}$ presented in Fig. 2. These results are in
agreement with the theoretical discussion on the symmetry protection of the
optical phonon frequency at $\Gamma$ presented in the main text and in Section
E below and further demonstrate that the screening of the Kohn anomaly that
occurs for zone-edge TO phonons at $\mathbf{K}$ can be used as a sensitive
probe of dielectric screening.

##  Appendix C 2D-mode linewidth

The FWHM of the 2D-mode feature of graphene $\Gamma_{\mathrm{2D}}$ can provide
additional information about the dielectric environment of the graphene
monolayer. Fig. 5a shows $\langle\Gamma_{\mathrm{2D}}\rangle$ the spatially
averaged FWHM of the 2D-mode of graphene depending of its environment, ordered
by increasing expected dielectric screening. A sharpening of the 2D mode can
be observed as screening increases. In particular, hBN-capping gives greatly
reduced linewidths of the 2D mode down to $16\pm 0.5$ $\mathrm{cm}^{-1}$ for
the sharpest spectra (see Fig. 5d). This sharpening of the 2D-mode feature
points towards an increased homogeneity of the dielectric environment of
graphene, when it is tightly coupled to a layered material [50, 51].
Interestingly, this reduction of $\Gamma_{\mathrm{2D}}$ is correlated to the
blueshift of $\omega_{\mathrm{2D}}$, as visible in Fig. 5c, highlighting the
impact of the interface quality on the overall dielectric environment of
graphene. These observations are consistent with our discussion in the main
text on the impact of surface roughness on the screening-induced blueshift of
the 2D-mode feature, as further discussed in the next section.

Figure 5: Analysis of the 2D-mode linewidth. (a) Spatially averaged FWHM of
the Raman 2D mode of graphene depending of its dielectric environment, ordered
by increasing expected dielectric screening. (b) Spatially averaged FWHM of
the Raman G mode of graphene depending of its dielectric environment, ordered
by increasing expected dielectric screening. (c) Observed correlation between
$\langle\Gamma_{\mathrm{2D}}\rangle$ and $\langle\omega_{\mathrm{2D}}\rangle$.
(d) Observation of a narrow 2D-mode feature in hBN-capped graphene/WSe2. The
solid line is a Lorentzian fit.

##  Appendix D Fine tuning of the dielectric environment

###  D.1 Monolayer versus bilayer TMD

To further explore the impact of interfacial coupling on the Raman 2D-mode
feature, we compare different areas of another graphene/WSe2 sample (Sample 4,
not discussed in Fig. 3) with various degrees of coupling between graphene
WSe2.

Figure 6: Supplementary data on a spatially inhomogneous sample. (a) Optical
image of another graphene/WSe2 sample on Si/SiO2 with different regions of
interest identified (Sample 4). (b, c) Photoluminescence intensity and 2D-mode
frequency maps of the sample shown in (a, respectively. (D) Typical PL spectra
for each region of interest. (E) Typical Raman 2D-mode spectra for each region
of interest. Note that although the substrate is patterned with holes and
venting channels, the sample is not partly suspended. Figure 7: Spatially
averaged parameters for Sample 4. (a) Spatially averaged 2D-mode frequency
$\langle\omega_{\mathrm{2D}}\rangle$ and integrated PL intensity $\langle
I_{\mathrm{PL}}\rangle$ and (b) spatially averaged 2D-mode FWHM
$\langle\Gamma_{\mathrm{2D}}\rangle$ and integrated PL intensity $\langle
I_{\mathrm{PL}}\rangle$ for the 5 different regions of interest in Sample 4,
introduced in Fig. 6.

Fig. 6a shows an optical image of the considered sample, where we highlight
five regions of interest. These regions were identified by considering the
WSe2 PL intensity map presented in Fig. 6b. Regions 1, 2 and 3 correspond to
graphene/WSe2 heterostructures with different interlayer coupling, as
illustrated by their respective PL emission spectra shown Fig. 6d (see also
Fig. 7b). The energy shift visible between region 1 and regions 2 and 3 can
also be seen as an effect of dielectric screening due to a smaller interlayer
distance between the two layers in regions 2 and 3 with respect to region 1.
Fig. 6e and Fig. 7a prove the strong sensitivity of $\omega_{\mathrm{2D}}$ to
the coupling between the TMD layer and graphene, as $\omega_{\mathrm{2D}}$
increases when the PL intensity diminishes. Note that in this specific sample,
the graphene reference gives $\langle\omega_{\mathrm{2D}}\rangle\approx
2670.5\pm 1$ $\mathrm{cm}^{-1}$. The sharpening of the 2D mode discussed in
Appendix C is also observed here depending on the interlayer coupling (see
Fig. 7b). Regions 4 and 5 of the same sample are formed by bilayer WSe2
(2L-WSe2) interfaced with graphene and present two different levels of
coupling (see Fig. 6). Note that the PL intensity of bilayer WSe2 is lower
than in the monolayer limit because of the indirect to direct bandgap
transition occurring in the monolayer limit [68]. These two areas also show a
coupling-dependent blueshift of $\omega_{\mathrm{2D}}$ and reduction of
$\Gamma_{\mathrm{2D}}$ (see Fig. 7c,e). More interestingly, the obtained
values of $\omega_{\mathrm{2D}}$ and $\Gamma_{\mathrm{2D}}$ for
graphene/2L-WSe2 are similar to the ones observed for graphene/1L-WSe2, which
suggests that dielectric screening essentially stems from the first layer in
contact with graphene, highlighting the short-range character of the
phenomenon.

###  D.2 Impact of surface roughness

Figure 8: Impact of surface roughness. Spatially averaged screening-induced
upshift of the 2D-mode feature in a set of graphene/TMD heterostructures
deposited onto SiO2, with the graphene layer either sandwiched between SiO2
and the TMD or deposited on top of the SiO2/TMD structure.

To investigate on the effect of surface roughness of the used substrate, we
compare the observed upshift of the 2D mode frequency in an additional set of
graphene/TMD heterostructures deposited onto SiO2 substrates. Fig. 8 presents
$\Delta\omega_{\mathrm{2D}}$, the difference between the 2D-mode frequency in
a graphene/TMD heterostructure and in the graphene reference of the same
sample. We observe that $\Delta\omega_{\mathrm{2D}}$ is appreciably higher
when the graphene layer is not directly in contact with SiO2. We understand
this as originating from the thicker nature of TMDs, which may conform less
than graphene to the microscopic irregularities of the SiO2 surface. Thus,
having the TMD layer in contact with SiO2 gives a flatter surface from the
view point of graphene, and balancing the dielectric environment towards a
stronger contribution from the TMD than in the opposite situation.

###  D.3 Screening of phonons versus screening of excitons

As discussed above and in the main manuscript, the screening of optical
phonons stands out as an extremely short-range effect (typically, the sub-
nanometre van der Waals gap between coupled 2D layers) in contrast to the case
of excitons, where longer range effects have been observed [37, 53]. We
attribute this difference to the fact that for the excitons, the excited
electron interacts both with the electrostatic potential of the hole as well
as with the potential of the screening cloud originating from all over space,
and in general from several neighboring layers. Mathematically, the effective
potential seen by the electron is given in terms of the dielectric response
function $\varepsilon^{-1}(\mathbf{r},\mathbf{r}^{\prime})$ convoluted with
the electrostatic potential of the hole, i.e., the dielectric response
function is sampled over a large volume. By contrast, for the phonon, the
interatomic force constants that define the dynamical matrix can be written as
the projection of $\varepsilon^{-1}(\mathbf{r},\mathbf{r}^{\prime})$ at both
points $\mathbf{r}$ and $\mathbf{r}^{\prime}$ onto the change of the lattice
potential with respect to the ionic displacements (cf. chap. 4 of Ref. [22]).
As the change of the lattice potential with respect to the atomic
displacements falls off much faster in space (as $1/r^{2}$) than the
electrostatic potential of the hole ($1/r$), it is much more localised and
hence the dielectric response function is only sampled mostly over the layer
in which the phonon resides in and its immediate neighboring layer.

##  Appendix E Symmetry-protected zone-centre optical phonons

In this Appendix, we provide all the necessary details and definitions to
understand Eqs. (1) and (2) from the main text. Firstly, Eq. (1) holds for the
following definition of the bare and screened electron-phonon coupling matrix
elements. The definition of the bare electron-phonon coupling matrix elements
is

$g^{\mathbf{q},\lambda;\mathrm{(bare)}}_{\mathbf{k},\pi^{*},\pi}\equiv\int\mathrm{d}^{3}r\,\psi^{*}_{\mathbf{k}+\mathbf{q},\pi^{*}}(\mathbf{r})\psi_{\mathbf{k},\pi}(\mathbf{r})\frac{\partial
V_{\mathrm{lat}}}{\partial R_{\mathbf{q},\lambda}}(\mathbf{r}),$ (3)

where $\partial/\partial
R_{\mathbf{q},\lambda}\equiv\sum_{n,\kappa,i}u^{\mathbf{q},\lambda}_{n,\kappa,i}\partial/\partial
R_{n,\kappa,i}$ is the directional derivative with respect to ionic
displacements according to a phonon displacement pattern
$u^{\mathbf{q},\lambda}_{n,\kappa,i}$, where $\mathbf{q}$ is the phonon wave
vector, $\lambda$ the phonon branch, $n$ denotes the $n$th unit cell, $\kappa$
runs over the atoms within one unit cell, and $i$ runs over the Cartesian
directions. $\psi_{\mathbf{k},\pi^{(*)}}(\mathbf{r})$ denotes the wave
function of an electron with wave vector $\mathbf{k}$ in the $\pi^{(*)}$-band
of graphene and $V_{\mathrm{lat}}(\mathbf{r})$ denotes the lattice potential.

The exact screened electron-phonon coupling matrix elements is defined as (see
chap. 4 of [22] for a derivation and motivation):

$g^{\mathbf{q},\lambda}_{\mathbf{k},\pi^{*},\pi}\equiv\int\mathrm{d}^{3}r\int\mathrm{d}^{3}r^{\prime}\,\psi^{*}_{\mathbf{k}+\mathbf{q},\pi^{*}}(\mathbf{r})\psi_{\mathbf{k},\pi}(\mathbf{r}^{\prime})\partial\mathcal{V}^{\mathbf{q},\lambda}(\mathbf{r},\mathbf{r}^{\prime}),$
(4)

where
$\partial\mathcal{V}^{\mathbf{q},\lambda}(\mathbf{r},\mathbf{r}^{\prime})$
denotes the irreducible vertex function. In density functional (perturbation)
theory, it is just the local change of the self-consistent potential. In
general, however, it can be defined as

$\begin{split}\partial\mathcal{V}^{\mathbf{q},\lambda}(\mathbf{r},\mathbf{r}^{\prime})\equiv&-\int\mathrm{d}t\,\frac{\partial}{\partial
R_{\mathbf{q},\lambda}}G^{-1}(\mathbf{r},t;\mathbf{r}^{\prime},0)\\\
=&+\int\mathrm{d}t\int\mathrm{d}^{3}r_{1}\int\mathrm{d}t_{1}\int\mathrm{d}^{3}r_{2}\int\mathrm{d}t_{2}\int\mathrm{d}^{3}r_{3}\int\mathrm{d}t_{3}\,G^{-1}(\mathbf{r},t;\mathbf{r}_{1},t_{1})\\\
&\quad\times\frac{\partial V_{\mathrm{lat}}}{\partial
R_{\mathbf{q},\lambda}}(\mathbf{r}_{3})\Lambda(\mathbf{r}_{3},t_{3};\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})G^{-1}(\mathbf{r}_{2},t_{2};\mathbf{r}^{\prime},0),\end{split}$
(5)

where $G(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})\equiv(-i)\langle
0|\mathcal{T}\\{\hat{\psi}(\mathbf{r}_{1},t_{1})\hat{\psi}^{\dagger}(\mathbf{r}_{2},t_{2})\\}|0\rangle$
is the exact one-electron Green’s function (we set $\hbar\equiv 1$), and
$\Lambda(\mathbf{r}_{3},t_{3};\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})$ is
the reducible vertex function defined as

$\Lambda(\mathbf{r}_{3},t_{3};\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})\equiv\langle
0|\mathcal{T}\left\\{\hat{\rho}(\mathbf{r}_{3},t_{3})\hat{\psi}(\mathbf{r}_{1},t_{1})\hat{\psi}^{\dagger}(\mathbf{r}_{2},t_{2})\right\\}|0\rangle\Big{|}_{\mathrm{connected}}.$
(6)

Here, the subscript “connected” refers to the fully connected part of the
correlation function, $\mathcal{T}$ denotes the time-ordering symbol,
$\hat{\psi}(\mathbf{r},t)$ the electron field operator in the Heisenberg
picture, and $\hat{\rho}(\mathbf{r},t)$ the operator of the electronic charge
density in the Heisenberg picture. As alluded to earlier, we note that our
definition of the screened electron-phonon coupling reduces, for instance, to
the one used in density functional perturbation theory, when the corresponding
one-electron Green’s function is used (see chap. 4 of [22] for a proof).

For the special case of graphene and for the $\pi$ bands in the vicinity of
the $K,K^{\prime}$-points in the first Brillouin zone, we can work with the
effective low-energy expansion of the electronic Hamiltonian and the electron-
phonon interaction Hamiltonian, both of which are determined by the lattice
symmetry [69]. In the following, we will ignore the spin degree of freedom and
focus on electrons near the $K$-point. Analogous results hold for electrons
near the $K^{\prime}$-point. The low-energy effective Hamiltonian for the
$\pi$-band electrons near the $K$-point reads

$\hat{H}_{\mathrm{el,eff}}=\sum_{\alpha,\beta\in\\{A,B\\}}\int\mathrm{d}^{3}r\,\hat{\psi}^{\dagger}_{\alpha}(\mathbf{r})\left(-iv_{\mathrm{F}}\bm{\nabla}\cdot\bm{\sigma}_{\alpha,\beta}\right)\hat{\psi}_{\beta}(\mathbf{r}),$
(7)

where $v_{\mathrm{F}}$ is the exact band slope, including all screening
effects from the substrate, $\bm{\sigma}$ is the vector of Pauli matrices in
sublattice space, and the pseudo-spinor field operator is defined as

$\hat{\psi}_{\alpha}(\mathbf{r})\equiv\sum_{n}\frac{1}{\sqrt{N}}\psi_{p_{z}}(\mathbf{r}-\bm{\tau}_{\alpha}-\mathbf{R}_{n})\hat{\psi}(\mathbf{r})$
(8)

in terms of the wave functions $\psi_{p_{z}}(\mathbf{r})$ of the $p_{z}$
orbitals centered at the position $\tau_{\alpha}$ of the $n$th of the $N$ unit
cells with origin at $\mathbf{R}_{n}$.

The effective bare electron-phonon interaction Hamiltonian for the
$\mathbf{q}\to\mathbf{0}$ optical phonons ($\lambda=$LO,TO) is likewise
determined by symmetry [70, 69] and can be written as

$\hat{H}^{\mathbf{q},\lambda}_{\mathrm{el-
ph,eff}}=F_{0}\int\mathrm{d}^{3}r\,\mathrm{e}^{i\mathbf{q}\cdot\mathbf{r}}\left(\hat{\mathbf{u}}^{\mathbf{q},\lambda}_{A}-\hat{\mathbf{u}}^{\mathbf{q},\lambda}_{B}\right)\wedge\hat{\mathbf{J}}(\mathbf{r}),$
(9)

where
$\hat{\mathbf{J}}(\mathbf{r})=\sum_{\alpha,\beta\in\\{A,B\\}}\hat{\psi}^{\dagger}_{\alpha}(\mathbf{r})\bm{\sigma}_{\alpha,\beta}\hat{\psi}(\mathbf{r})$
is the electronic current operator for a Weyl fermion,
$\mathbf{a}\wedge\mathbf{b}\equiv a_{x}b_{y}-a_{y}b_{x}$ the two-dimensional
wedge product, and $\hat{\mathbf{u}}^{\mathbf{q},\lambda}_{\alpha}$ is the
vectorial lattice displacement operator for the atoms on sublattice
$\alpha=A,B$ projected on phonon mode $(\mathbf{q},\lambda)$. Since this is
the bare electron-phonon interaction Hamiltonian, the constant $F_{0}$ is by
definition independent of the substrate screening.

It follows that the non-trivial part of the bare electron-phonon coupling
matrix element can be written as

$g^{\mathbf{q},\lambda;\mathrm{(bare)}}_{\mathbf{k},\pi^{*},\pi}\propto\sum_{\alpha,\beta\in\\{A,B\\}}\left(\chi^{\alpha}_{\mathbf{k}+\mathbf{q},\pi^{*}}\right)^{*}\chi^{\beta}_{\mathbf{k},\pi}\left(\mathbf{u}^{\mathbf{q},\lambda}_{A}-\mathbf{u}^{\mathbf{q},\lambda}_{B}\right)\wedge\bm{\sigma}_{\alpha,\beta},$
(10)

where the two-dimensional pseudo-spinors $\chi_{\mathbf{k},\pi^{(*)}}$ obey
the (screening-independent) Weyl equation

$\mathbf{k}\cdot\sum_{\beta}\bm{\sigma}_{\alpha,\beta}\chi^{\beta}_{\mathbf{k},\pi^{(*)}}=\begin{subarray}{c}(+)\\\
-\end{subarray}|\mathbf{k}|\chi^{\alpha}_{\mathbf{k},\pi^{(*)}}.$ (11)

The screened electron-phonon matrix elements for the $\pi$-bands can be
written in similar form:

$g^{\mathbf{q},\lambda}_{\mathbf{k},\pi^{*},\pi}\propto\sum_{\alpha,\beta\in\\{A,B\\}}\left(\chi^{\alpha}_{\mathbf{k}+\mathbf{q},\pi^{*}}\right)^{*}\chi^{\beta}_{\mathbf{k},\pi}\left(\mathbf{u}^{\mathbf{q},\lambda}_{A}-\mathbf{u}^{\mathbf{q},\lambda}_{B}\right)\wedge\bm{\partial\mathcal{V}}^{\mathbf{q}}_{\alpha,\beta}(\mathbf{k}),$
(12)

with the vector- and matrix-valued irreducible vertex function

$\bm{\partial\mathcal{V}}^{\mathbf{q}}_{\alpha,\beta}(\mathbf{k})\equiv\int\mathrm{d}t\int\mathrm{d}t_{1}\int\mathrm{d}t_{2}\sum_{\alpha^{\prime},\beta^{\prime}\in\\{A,B\\}}G^{-1}_{\alpha,\alpha^{\prime}}(\mathbf{k}+\mathbf{q};t,t_{1})\bm{\Lambda}^{\mathbf{q}}_{\alpha^{\prime},\beta^{\prime}}(\mathbf{k};t_{1},t_{2})G^{-1}_{\beta^{\prime},\beta}(\mathbf{k};t_{2},0),$
(13)

which is in turn given in terms of the reducible vertex function

$\begin{split}\bm{\Lambda}^{\mathbf{q}}_{\alpha^{\prime},\beta^{\prime}}(\mathbf{k};t_{1},t_{2})\equiv&\int\mathrm{d}^{3}r_{1}\int\mathrm{d}^{3}r_{2}\int\mathrm{d}t_{3}\,\mathrm{e}^{-i(\mathbf{k}+\mathbf{q})\cdot\mathbf{r}_{1}}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}_{2}}\\\
&\quad\times\langle
0|\mathcal{T}\left\\{\hat{\mathbf{J}}(\bm{0},t_{3})\hat{\psi}_{\alpha}(\mathbf{r}_{1},t_{1})\hat{\psi}^{\dagger}_{\beta}(\mathbf{r}_{2},t_{2})\right\\}|0\rangle\Big{|}_{\mathrm{connected}}.\end{split}$
(14)

and the matrix-valued electron Green’s function

$G^{-1}_{\alpha,\beta}(\mathbf{k};t_{1},t_{2})\equiv\int\frac{\mathrm{d}\omega}{2\pi}\mathrm{e}^{-i\omega(t_{1}-t_{2})}\left[\omega\delta_{\alpha,\beta}-v_{\mathrm{F}}\mathbf{k}\cdot\bm{\sigma}_{\alpha,\beta}+i\,\mathrm{sgn}(\omega)0^{+}\right],$
(15)

wherein $\mathrm{sgn}$ denotes the sign function and $0^{+}$ a positive
infinitesimal.

For the reducible vertex function, a Ward identity exists, which relates the
reducible vertex function to the one-particle Green’s function. Note that this
Ward identity is a result of global $U(1)$ gauge invariance, or,
equilvalently, the result of electron number conservation. In the limit
$\mathbf{q}\to\mathbf{0}$, the mentioned Ward identity reads [55, 56]

$\bm{\Lambda}^{\mathbf{q}\to\mathbf{0}}_{\alpha,\beta}(\mathbf{k};t_{1},t_{2})\propto\frac{\partial}{\partial\mathbf{k}}G_{\alpha,\beta}(\mathbf{k};t_{1},t_{2}).$
(16)

It follows immediately, that the irreducible vertex function is proportional
to the momentum derivative of the inverse Green’s function:

$\bm{\partial\mathcal{V}}^{\mathbf{q}\to\mathbf{0}}_{\alpha,\beta}(\mathbf{k})\propto\frac{\partial}{\partial\mathbf{k}}G^{-1}_{\alpha,\beta}(\mathbf{k};t_{1},t_{2})\propto
v_{\mathrm{F}},$ (17)

which directly implies Eq. (2) of the main text. As mentioned in the main
text, this proportionality means that all effects of screening, encoded in the
exact value of the Fermi velocity, cancel out in squared phonon frequency,
leading to the robustness of the G mode frequency with respect to changes in
the screening environment.

As a side point, we want to emphasize that the scaling of the screened
electron-phonon coupling with the exact Fermi velocity if not obvious for
optical phonons. For acoustic phonons, it is often argued [57, 58] that a
local $U(1)$ gauge symmetry exists, i.e., the action is invariant under a
position-dependent change of phase of the electron field and the adding of the
gradient of a position-dependent function to the phonon displacement field
(akin to the standard local U(1) gauge transformation in quantum
electrodynamics). While this local $U(1)$ symmetry does indeed exist for
acoustic phonons, it is not required by any more fundamental physics principle
or necessary to derive the scaling of the screened electron-phonon phonon
coupling. Moreover, for optical phonons, their finite frequency breaks the
local $U(1)$ gauge symmetry, just as a finite photon mass would break the
local gauge symmetry in electrodynamics. As such, the arguments used for the
scaling of the electron-phonon coupling of acoustic phonons cannot be used in
our case, requiring our more detailed theoretical reasoning.

## References

  * Solin and Ramdas [1970] S. A. Solin and A. K. Ramdas, Raman spectrum of diamond, Phys. Rev. B 1, 1687 (1970).
  * Temple and Hathaway [1973] P. A. Temple and C. E. Hathaway, Multiphonon Raman spectrum of silicon, Phys. Rev. B 7, 3685 (1973).
  * Kohn [1959] W. Kohn, Image of the Fermi surface in the vibration spectrum of a metal, Phys. Rev. Lett. 2, 393 (1959).
  * Baroni  _et al._ [2001] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001).
  * Piscanec  _et al._ [2007] S. Piscanec, M. Lazzeri, J. Robertson, A. C. Ferrari, and F. Mauri, Optical phonons in carbon nanotubes: Kohn anomalies, Peierls distortions, and dynamic effects, Phys. Rev. B 75, 035427 (2007).
  * Piscanec  _et al._ [2004] S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, Kohn anomalies and electron-phonon interactions in graphite, Phys. Rev. Lett. 93, 185503 (2004).
  * Lazzeri  _et al._ [2008] M. Lazzeri, C. Attaccalite, L. Wirtz, and F. Mauri, Impact of the electron-electron correlation on phonon dispersion: Failure of LDA and GGA DFT functionals in graphene and graphite, Phys. Rev. B 78, 081406 (2008).
  * Wang  _et al._ [2018] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons in atomically thin transition metal dichalcogenides, Rev. Mod. Phys. 90, 021001 (2018).
  * Roy  _et al._ [2021] S. Roy, X. Zhang, A. B. Puthirath, A. Meiyazhagan, S. Bhattacharyya, M. M. Rahman, G. Babu, S. Susarla, S. K. Saju, M. K. Tran, L. M. Sassi, M. A. S. R. Saadi, J. Lai, O. Sahin, S. M. Sajadi, B. Dharmarajan, D. Salpekar, N. Chakingal, A. Baburaj, X. Shuai, A. Adumbumkulath, K. A. Miller, J. M. Gayle, A. Ajnsztajn, T. Prasankumar, V. V. J. Harikrishnan, V. Ojha, H. Kannan, A. Z. Khater, Z. Zhu, S. A. Iyengar, P. A. d. S. Autreto, E. F. Oliveira, G. Gao, A. G. Birdwell, M. R. Neupane, T. G. Ivanov, J. Taha-Tijerina, R. M. Yadav, S. Arepalli, R. Vajtai, and P. M. Ajayan, Structure, properties and applications of two-dimensional hexagonal boron nitride, Advanced Materials 33, 2101589 (2021).
  * Geim and Grigorieva [2013] A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures, Nature 499, 419 (2013).
  * Lau  _et al._ [2022] C. N. Lau, M. W. Bockrath, K. F. Mak, and F. Zhang, Reproducibility in the fabrication and physics of moiré materials, Nature 602, 41 (2022).
  * Raja  _et al._ [2017] A. Raja, A. Chaves, J. Yu, G. Arefe, H. M. Hill, A. F. Rigosi, T. C. Berkelbach, P. Nagler, C. Schüller, T. Korn, C. Nuckolls, J. Hone, L. E. Brus, T. F. Heinz, D. R. Reichman, and A. Chernikov, Coulomb engineering of the bandgap and excitons in two-dimensional materials, Nature Communications 8, 15251 (2017).
  * Lorchat  _et al._ [2020] E. Lorchat, L. E. P. López, C. Robert, D. Lagarde, G. Froehlicher, T. Taniguchi, K. Watanabe, X. Marie, and S. Berciaud, Filtering the photoluminescence spectra of atomically thin semiconductors with graphene, Nat. Nanotechnol. 15, 283 (2020).
  * Ferrari and Basko [2013] A. C. Ferrari and D. M. Basko, Raman spectroscopy as a versatile tool for studying the properties of graphene, Nat. Nanotechnol. 8, 235 (2013).
  * Zhang  _et al._ [2015] X. Zhang, X.-F. Qiao, W. Shi, J.-B. Wu, D.-S. Jiang, and P.-H. Tan, Phonon and Raman scattering of two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material, Chem. Soc. Rev. 44, 2757 (2015).
  * Gadelha  _et al._ [2021] A. C. Gadelha, D. A. A. Ohlberg, C. Rabelo, E. G. S. Neto, T. L. Vasconcelos, J. L. Campos, J. S. Lemos, V. Ornelas, D. Miranda, R. Nadas, F. C. Santana, K. Watanabe, T. Taniguchi, B. van Troeye, M. Lamparski, V. Meunier, V.-H. Nguyen, D. Paszko, J.-C. Charlier, L. C. Campos, L. G. Cançado, G. Medeiros-Ribeiro, and A. Jorio, Localization of lattice dynamics in low-angle twisted bilayer graphene, Nature 590, 405 (2021).
  * Chakraborty  _et al._ [2012] B. Chakraborty, A. Bera, D. V. S. Muthu, S. Bhowmick, U. V. Waghmare, and A. K. Sood, Symmetry-dependent phonon renormalization in monolayer MoS2 transistor, Phys. Rev. B 85, 161403 (2012).
  * Froehlicher  _et al._ [2015] G. Froehlicher, E. Lorchat, F. Fernique, C. Joshi, A. Molina-Sánchez, L. Wirtz, and S. Berciaud, Unified description of the optical phonon modes in N-layer MoTe2, Nano Lett. 15, 6481 (2015).
  * Ryu  _et al._ [2010] S. Ryu, L. Liu, S. Berciaud, Y.-J. Yu, H. Liu, P. Kim, G. W. Flynn, and L. E. Brus, Atmospheric oxygen binding and hole doping in deformed graphene on a SiO2 substrate, Nano Lett. 10, 4944 (2010).
  * Ferrari  _et al._ [2006] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, Raman spectrum of graphene and graphene layers, Phys. Rev. Lett. 97, 187401 (2006).
  * Graf  _et al._ [2007] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, and L. Wirtz, Spatially resolved Raman spectroscopy of single- and few-layer graphene, Nano Letters 7, 238 (2007).
  * Reichardt [2018] S. Reichardt, Many-body perturbation theory approach to raman spectroscopy and its application to 2D materials, PhD. thesis, RWTH Aachen University & University of Luxembourg (2018).
  * Basko [2009] D. M. Basko, Calculation of the Raman G peak intensity in monolayer graphene: role of Ward identities, New Journal of Physics 11, 095011 (2009).
  * Maultzsch  _et al._ [2004] J. Maultzsch, S. Reich, and C. Thomsen, Double-resonant Raman scattering in graphite: Interference effects, selection rules, and phonon dispersion, Phys. Rev. B 70, 155403 (2004).
  * Basko [2008] D. M. Basko, Theory of resonant multiphonon Raman scattering in graphene, Phys. Rev. B 78, 125418 (2008).
  * Venezuela  _et al._ [2011] P. Venezuela, M. Lazzeri, and F. Mauri, Theory of double-resonant Raman spectra in graphene: Intensity and line shape of defect-induced and two-phonon bands, Phys. Rev. B 84, 035433 (2011).
  * Mafra  _et al._ [2007] D. L. Mafra, G. Samsonidze, L. M. Malard, D. C. Elias, J. C. Brant, F. Plentz, E. S. Alves, and M. A. Pimenta, Determination of LA and TO phonon dispersion relations of graphene near the Dirac point by double resonance Raman scattering, Phys. Rev. B 76, 233407 (2007).
  * Berciaud  _et al._ [2013] S. Berciaud, X. Li, H. Htoon, L. E. Brus, S. K. Doorn, and T. F. Heinz, Intrinsic line shape of the Raman 2D-mode in freestanding graphene monolayers, Nano Lett. 13, 3517 (2013).
  * Mohiuddin  _et al._ [2009] T. M. G. Mohiuddin, A. Lombardo, R. R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, N. Marzari, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Grüneisen parameters, and sample orientation, Phys. Rev. B 79, 205433 (2009).
  * Pisana  _et al._ [2007] S. Pisana, M. Lazzeri, C. Casiraghi, K. S. Novoselov, A. K. Geim, A. C. Ferrari, and F. Mauri, Breakdown of the adiabatic Born-Oppenheimer approximation in graphene, Nat. Mater. 6, 198 (2007).
  * Yan  _et al._ [2007] J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Electric field effect tuning of electron-phonon coupling in graphene, Phys. Rev. Lett. 98, 166802 (2007).
  * Das  _et al._ [2008] A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C. Ferrari, and A. K. Sood, Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor, Nat. Nanotechnol. 3, 210 (2008).
  * Chen  _et al._ [2011] C.-F. Chen, C.-H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, Controlling inelastic light scattering quantum pathways in graphene, Nature 471, 617 (2011).
  * Froehlicher and Berciaud [2015] G. Froehlicher and S. Berciaud, Raman spectroscopy of electrochemically gated graphene transistors: Geometrical capacitance, electron-phonon, electron-electron, and electron-defect scattering, Phys. Rev. B 91, 205413 (2015).
  * Cançado  _et al._ [2011] L. G. Cançado, A. Jorio, E. H. M. Ferreira, F. Stavale, C. A. Achete, R. B. Capaz, M. V. O. Moutinho, A. Lombardo, T. S. Kulmala, and A. C. Ferrari, Quantifying defects in graphene via Raman spectroscopy at different excitation energies, Nano Lett. 11, 3190 (2011).
  * Forster  _et al._ [2013] F. Forster, A. Molina-Sanchez, S. Engels, A. Epping, K. Watanabe, T. Taniguchi, L. Wirtz, and C. Stampfer, Dielectric screening of the Kohn anomaly of graphene on hexagonal boron nitride, Phys. Rev. B 88, 085419 (2013).
  * Froehlicher  _et al._ [2018] G. Froehlicher, E. Lorchat, and S. Berciaud, Charge versus energy transfer in atomically thin graphene-transition metal dichalcogenide van der Waals heterostructures, Phys. Rev. X 8, 011007 (2018).
  * [38] See Supplemental Material for details on data analysis, advanced optical characterization of built-in strain and unintentional doping, discucussion on the impact of dielectric screening on the Raman scattering intensities.
  * Metten  _et al._ [2014] D. Metten, F. Federspiel, M. Romeo, and S. Berciaud, All-optical blister test of suspended graphene using micro-Raman spectroscopy, Phys. Rev. Applied 2, 054008 (2014).
  * Androulidakis  _et al._ [2015] C. Androulidakis, E. N. Koukaras, J. Parthenios, G. Kalosakas, K. Papagelis, and C. Galiotis, Graphene flakes under controlled biaxial deformation, Scientific Reports 5, 18219 (2015).
  * Lee  _et al._ [2012a] J. E. Lee, G. Ahn, J. Shim, Y. S. Lee, and S. Ryu, Optical separation of mechanical strain from charge doping in graphene, Nat. Commun. 3, 1024 (2012a).
  * Endlich  _et al._ [2013] M. Endlich, A. Molina-Sánchez, L. Wirtz, and J. Kröger, Screening of electron-phonon coupling in graphene on Ir(111), Phys. Rev. B 88, 205403 (2013).
  * Berciaud  _et al._ [2009] S. Berciaud, S. Ryu, L. E. Brus, and T. F. Heinz, Probing the intrinsic properties of exfoliated graphene: Raman spectroscopy of free-standing monolayers, Nano Lett. 9, 346 (2009).
  * Faugeras  _et al._ [2015] C. Faugeras, S. Berciaud, P. Leszczynski, Y. Henni, K. Nogajewski, M. Orlita, T. Taniguchi, K. Watanabe, C. Forsythe, P. Kim, R. Jalil, A. K. Geim, D. M. Basko, and M. Potemski, Landau level spectroscopy of electron-electron interactions in graphene, Phys. Rev. Lett. 114, 126804 (2015).
  * Geick  _et al._ [1966] R. Geick, C. H. Perry, and G. Rupprecht, Normal modes in hexagonal boron nitride, Phys. Rev. 146, 543 (1966).
  * Volksen  _et al._ [2010] W. Volksen, R. D. Miller, and G. Dubois, Low dielectric constant materials, Chemical Reviews 110, 56 (2010).
  * Venanzi  _et al._ [2023] T. Venanzi, L. Graziotto, F. Macheda, S. Sotgiu, T. Ouaj, E. Stellino, C. Fasolato, P. Postorino, V. Mišeikis, M. Metzelaars, P. Kögerler, B. Beschoten, C. Coletti, S. Roddaro, M. Calandra, M. Ortolani, C. Stampfer, F. Mauri, and L. Baldassarre, Probing enhanced electron-phonon coupling in graphene by infrared resonance raman spectroscopy, Phys. Rev. Lett. 130, 256901 (2023).
  * López  _et al._ [2021] L. E. P. López, L. Moczko, J. Wolff, A. Singh, E. Lorchat, M. Romeo, T. Taniguchi, K. Watanabe, and S. Berciaud, Single- and narrow-line photoluminescence in a boron nitride-supported MoSe2/graphene heterostructure, Comptes Rendus. Physique 22, 77 (2021).
  * Xue  _et al._ [2011] J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride, Nat. Mater. 10, 282 (2011).
  * Neumann  _et al._ [2015] C. Neumann, S. Reichardt, P. Venezuela, M. Drögeler, L. Banszerus, M. Schmitz, K. Watanabe, T. Taniguchi, F. Mauri, B. Beschoten, S. V. Rotkin, and C. Stampfer, Raman spectroscopy as probe of nanometre-scale strain variations in graphene, Nat. Commun. 6, 8429 (2015).
  * Neumann  _et al._ [2016] C. Neumann, L. Banszerus, M. Schmitz, S. Reichardt, J. Sonntag, T. Taniguchi, K. Watanabe, B. Beschoten, and C. Stampfer, Line shape of the Raman 2D peak of graphene in van der Waals heterostructures, Phys. Status Solidi B 253, 2326 (2016).
  * Banszerus  _et al._ [2019] L. Banszerus, T. Sohier, A. Epping, F. Winkler, F. Libisch, F. Haupt, K. Watanabe, T. Taniguchi, K. Müller-Caspary, N. Marzari, F. Mauri, B. Beschoten, and C. Stampfer, Extraordinary high room-temperature carrier mobility in graphene-WSe2 heterostructures, arXiv preprint arXiv:1909.09523 (2019).
  * Tebbe  _et al._ [2023] D. Tebbe, M. Schütte, K. Watanabe, T. Taniguchi, C. Stampfer, B. Beschoten, and L. Waldecker, Tailoring the dielectric screening in WS2–graphene heterostructures, npj 2D Materials and Applications 7, 29 (2023).
  * Giustino [2017] F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. 89, 015003 (2017).
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An Introduction to Quantum Field Theory_ (Westview Press, Boulder, 1995).
  * Srednicki [2007] M. Srednicki,  _Quantum field theory_ (Cambridge University Press, Cambridge, 2007).
  * Basko and Aleiner [2008] D. M. Basko and I. L. Aleiner, Interplay of Coulomb and electron-phonon interactions in graphene, Phys. Rev. B 77, 041409 (2008).
  * Sohier  _et al._ [2014] T. Sohier, M. Calandra, C.-H. Park, N. Bonini, N. Marzari, and F. Mauri, Phonon-limited resistivity of graphene by first-principles calculations: Electron-phonon interactions, strain-induced gauge field, and Boltzmann equation, Phys. Rev. B 90, 125414 (2014).
  * Lee  _et al._ [2012b] J. E. Lee, G. Ahn, J. Shim, Y. S. Lee, and S. Ryu, Optical separation of mechanical strain from charge doping in graphene, Nat. Commun. 3, 1024 (2012b).
  * Metten  _et al._ [2016] D. Metten, G. Froehlicher, and S. Berciaud, Monitoring electrostatically-induced deflection, strain and doping in suspended graphene using raman spectroscopy, 2D Materials 4, 014004 (2016).
  * Pimenta Martins  _et al._ [2023] L. G. Pimenta Martins, R. Comin, M. J. S. Matos, M. S. C. Mazzoni, B. R. A. Neves, and M. Yankowitz, High-pressure studies of atomically thin van der Waals materials, Applied Physics Reviews 10, 011313 (2023).
  * Chijioke  _et al._ [2005] A. D. Chijioke, W. J. Nellis, A. Soldatov, and I. F. Silvera, The ruby pressure standard to 150 GPa, Journal of Applied Physics 98, 114905 (2005).
  * Castellanos-Gomez  _et al._ [2014] A. Castellanos-Gomez, M. Buscema, R. Molenaar, V. Singh, L. Janssen, H. S. J. van der Zant, and G. A. Steele, Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping, 2D Mater. 1, 011002 (2014).
  * Wang  _et al._ [2013] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, One-dimensional electrical contact to a two-dimensional material, Science 342, 614 (2013).
  * Zomer  _et al._ [2014] P. J. Zomer, M. H. D. Guimarães, J. C. Brant, N. Tombros, and B. J. van Wees, Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride, Applied Physics Letters 105, 013101 (2014).
  * Mak  _et al._ [2010] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin MoS2: a new direct-gap semiconductor, Phys. Rev. Lett. 105, 136805 (2010).
  * Metten  _et al._ [2013] D. Metten, F. Federspiel, M. Romeo, and S. Berciaud, Probing built-in strain in freestanding graphene monolayers by Raman spectroscopy, Phys. Status Solidi B 250, 2681 (2013).
  * Zhao  _et al._ [2013] W. Zhao, Z. Ghorannevis, L. Chu, M. Toh, C. Kloc, P.-H. Tan, and G. Eda, Evolution of electronic structure in atomically thin sheets of WS2 and WSe2, ACS Nano 7, 791 (2013).
  * Mañes [2007] J. L. Mañes, Symmetry-based approach to electron-phonon interactions in graphene, Phys. Rev. B 76, 045430 (2007).
  * Ando [2006] T. Ando, Anomaly of optical phonon in monolayer graphene, Journal of the Physical Society of Japan 75, 124701 (2006).
  * Basko  _et al._ [2009] D. M. Basko, S. Piscanec, and A. C. Ferrari, Electron-electron interactions and doping dependence of the two-phonon raman intensity in graphene, Phys. Rev. B 80, 165413 (2009).
  * Yoon  _et al._ [2009] D. Yoon, H. Moon, Y.-W. Son, J. S. Choi, B. H. Park, Y. H. Cha, Y. D. Kim, and H. Cheong, Interference effect on Raman spectrum of graphene on SiO2/Si, Phys. Rev. B 80, 125422 (2009).
  * Metten  _et al._ [2015] D. Metten, G. Froehlicher, and S. Berciaud, Doping- and interference-free measurement of I2D/IG in suspended monolayer graphene blisters, Phys. Status Solidi B 252, 2390 (2015).

Supplemental Material for: Symmetry-dependent dielectric screening of optical
phonons in monolayer graphene

##  Appendix S1 Data analysis

###  S1a Fitting procedure

Figure S1: Examples of fits of 2D-mode spectra. Top: fit of the 2D-mode
spectrum of suspended graphene with a double modified lorentzian profile. The
two features are identified in dashed grey lines together with their
corresponding central frequencies $\omega_{\mathrm{2D}_{\pm}}$. Bottom: fit of
the 2D-mode spectrum of SiO2-supported graphene with a Voigt profile.

To extract the relevant spectral parameters of each considered
heterostructure, we followed a specific fitting procedure. For all samples,
the Raman G-mode feature of graphene was fit using a Lorentzian profile to
directly extract the peak position $\omega_{\mathrm{G}}$ and full width at
half maximum $\Gamma_{\mathrm{G}}$. The 2D′-mode feature was also fit to a
Lorentzian profile. Except for the case of suspended graphene (see below) and
unless otherwise specified, the 2D-mode feature was fit to a Voigt profile.
$\Gamma_{\mathrm{2D}}$ is then taken to be equal to the FWHM of the Voigt
profile, $\omega_{\mathrm{2D}}$ is taken as its central position.

To determine $\omega_{\mathrm{2D}}$ in suspended graphene, we rely on the
phenomenological discussion proposed in Ref. 28 and fit the 2D-mode spectra
with a double modified Lorentzian profile [25]
$f(\omega)=f_{+}(\omega)+f_{-}(\omega)$ with
$f_{\pm}(\omega)\propto\left[(\omega-\omega_{\mathrm{2D}_{\pm}})^{2}+\frac{\gamma}{4(2^{2/3}-1)}\right]^{-3/2}$,
$\gamma$ being a common parameter for the two sub-features. Since the
$\mathrm{2D}_{-}$ feature has much larger spectral weight than the
$\mathrm{2D}_{+}$ feature, we define the 2D-mode frequency as
$\omega_{\mathrm{2D}}=\omega_{\mathrm{2D}_{-}}$. Fig. S1 presents examples of
the two types of fits considered.

###  S1b Spatial averaging

Spatial averages of the fitting parameters presented in Fig. 3 of the main
text are obtained by considering a given region of interest (see Fig. S2 for a
depiction of all the considered areas). The regions of interest are first
identified using the optical images of our samples. Graphene/TMD areas are
further identified by examining the PL intensity of the TMD layer, which is
strongly quenched due to near-field transfer of TMD excitons to graphene. To
ensure that only well-coupled graphene/TMD heterostructures are considered,
only regions displaying a PL quenching factor above 10 at room temperature are
considered.

The error bars associated with the spatially averaged values are given by the
standard deviation of each parameter in a specific area.

Figure S2: Identification of the areas of interest. Top row: Optical image, PL
intensity map and $\omega_{\mathrm{2D}}$ map of Sample 1 with hBN-capped
graphene and graphene/WSe2 areas. Middle row: Optical image, PL intensity map
and $\omega_{\mathrm{2D}}$ map of Sample 2 with SiO2-supported graphene and
graphene/WSe2 areas together with their suspended counterparts. Bottom row:
Optical image, PL intensity map and $\omega_{\mathrm{2D}}$ map of Sample 3
with hBN/graphene and hBN/MoSe2/graphene areas.

##  Appendix S2 Determining built-in strain and residual doping

###  S2a Built-in strain and unintentional doping

Figure S3: Correlation plots. (a, b, c) $\omega_{\mathrm{2D}}$ vs
$\omega_{\mathrm{G}}$ correlations for the three samples shown in Fig. 3 of
the main text (Samples 2, 3 and 1 in Fig. S2, respectively). The plotted lines
give the expected correlation slopes for electron doping
($\partial\omega_{\mathrm{2D}}/\partial\omega_{\mathrm{G}}\approx 0.2$), hole
doping ($\partial\omega_{\mathrm{2D}}/\partial\omega_{\mathrm{G}}\approx
0.55$) and strain
($\partial\omega_{\mathrm{2D}}/\partial\omega_{\mathrm{G}}\approx 2.2$). (d,
e, f) $\Gamma_{\mathrm{G}}$ versus $\omega_{\mathrm{G}}$ correlations for the
same samples.

For each type of heterostructure, we examine the $\omega_{\mathrm{2D}}$ versus
$\omega_{\mathrm{G}}$ and $\Gamma_{\mathrm{G}}$ versus $\omega_{\mathrm{G}}$
correlations, as shown in Fig. S3. As extensively documented elsewhere [41,
67, 39, 50, 34], such plots provide quantitative information on the residual
charge carrier density (or non-intentional doping) and built-in strain field
of the considered graphene area. First, the
$\Gamma_{\mathrm{G}}$-$\omega_{\mathrm{G}}$ correlation allows to estimate the
doping level based on robust models for doping induced zone-centre optical
phonon renormalisation [70, 31, 30]. For all cases, we find spatially averaged
values $\langle\Gamma_{\mathrm{G}}\rangle>10$ $\mathrm{cm}^{-1}$ and no clear
$\omega_{\mathrm{G}}$-$\Gamma_{\mathrm{G}}$ correlation (see Fig. S3D-F and
Fig. 5 in the main text). These observations indicate that unintentional
doping leads to Fermi level shifts typically less than 100 meV away from the
Dirac point [34].

The $\omega_{\mathrm{2D}}$-$\omega_{\mathrm{G}}$ correlation allows us to
separate the effect of doping and strain and hence resolve small variations of
built-in strain in the studied hBN-based heterostructures (see Fig. S3b,c and
main text for details). These variations lead to a scatter in the measured
values of $\omega_{\mathrm{2D}}$ and hence to slightly increased errorbars in
the precise determination of the blueshift of $\omega_{\mathrm{2D}}$ due to
dielectric screening. However, despite these small variations, the
$\omega_{\mathrm{2D}}$-$\omega_{\mathrm{G}}$ correlations differs only by a
rigid (screening-induced) shift of $\omega_{\mathrm{2D}}$ when comparing bare
graphene to graphene/TMD heterostructures, consolidating the hypothesis of low
built-in strain (typically $\sim 10^{-2}\,\%$) in all the samples investigated
here.

Let us finally stress that in the presence of an inhomogeneous dielectric
environment, our demonstration that the correlation between the 2D- and G-mode
frequencies is essentially a vertical line (see Fig. 3 in the main text)
enables a straightforward identification of domains associated with distinct
dielectric environments within a given sample. As illustrated in Fig. S3a-c,
these domains appear as vertically offset clouds of points in the
$(\omega_{\mathrm{G}},\omega_{\mathrm{2D}})$ plane. Note that such a vertical
offset in the $(\omega_{\mathrm{G}},\omega_{\mathrm{2D}})$ plane would only be
only achievable by a combination of unrealistically large strain and doping
levels and thus is a unique signature of a change of dielectric environment.
At the same time, the magnitude of these vertical shifts is virtually
independent of the level of strain and doping. In each of these domains, some
scattering may be seen due to nanoscale strain and doping gradients that
follow the well-known correlations in the
$(\omega_{\mathrm{G}},\omega_{\mathrm{2D}})$ plane [59, 34, 39, 50], allowing
a quantitative determination of the residual charge carrier density and strain
level within an identified dielectric environment. Such a multiparameter
analysis can be further consolidated using the analysis of
$\Gamma_{\mathrm{G}}$ (Fig. S3d-f) and of the integrated intensity ratio
between the 2D-mode and G-mode intensities (Sec. S3 and Fig. S6).

###  S2b Photoinduced doping

Figure S4: Assessing photoinduced doping. Evolution of $\omega_{\mathrm{G}}$
with increasing laser intensity (a, b, c) and $\Gamma_{\mathrm{G}}$ versus
$\omega_{\mathrm{G}}$ correlation under increasing laser intensity (D, E, F)
for Sample 2, 3 and 1, respectively (see Fig. S2). The laser photon energy is
2.33 eV.

Interaction with polar substrates and molecular adsorbates [19] as well as
interlayer charge transfer between TMD and graphene can lead to photoinduced
doping of the graphene layer [37] and experimentally translate into a sizeable
upshift and narrowing of the G-mode feature, which may affect our results.
Because of this, we performed laser intensity-dependent measurements on our
samples to determine the threshold of such unwanted photoinduced doping. As
shown in Fig. S4, photodoping is observable in SiO2-supported samples and is
negligible in suspended graphene and in hBN-supported or hBN-capped samples
[48]. All the data presented in the main text were acquired with laser
intensities low enough to avoid any spurious photoinduced doping of the
graphene layer (typically $\sim 100~{}\mu\rm W/\mu\rm m^{2}$).

##  Appendix S3 Analysis of the Integrated Raman intensities

In this section, we experimentally address the impact of dielectric screening
on the integrated intensities of the Raman G- and 2D-modes, denoted $I_{\rm
G}$ and $I_{\rm{2D}}$, respectively. As discussed in the main manuscript and
in Appendix E, the electron-phonon coupling strength (and hence the Raman
intensities) are directly affected by screening. As a logical consequence, the
strong effect of dielectric screening on the TO phonon dispersion at and near
$\mathbf{K}$ could, at least on the qualitative level, result in a reduction
of the 2D-mode intensity. In order to test this scenario experimentally, we
have carefully analysed $I_{\rm G}$ and $I_{\rm{2D}}$ in the sample, discussed
in Fig. 2 of the main manuscript (graphene with and without a WSe2 capping
layer, with suspended and supported areas), where the effects of screening can
be seen most prominently. Let us recall that in this sample, as in all the
samples investigated here, the impact of unintentional doping and built-in
strain on the Raman features is negligible (Sec. S2) and, in particular, a
reduction of the $I_{\rm{2D}}/I_{\rm{G}}$ ratio in WSe2/graphene due to doping
[71, 34] can safely be ruled out.

Additionally, optical interference effects due to multiple reflections in our
layered structures (Fabry-Perot effects) are well-known to affect the
intensity of the Raman scattering features and must be taken into account
properly [72, 39, 60]. Figure S5 shows the computed Raman intensity
enhancement factors solely due to optical interference for TMD/graphene versus
bare graphene, within the geometry indicated in the inset of Fig. S5a. The
main effects due to optical interference are the following:

  * •

The Raman intensities of the G- and 2D-modes are expected to be significantly
larger, by more than one order of magnitude, in the supported regions (marked
with $\\#$ in Fig. S5), compared to the suspended regions (marked with $\ast$
in Fig. S5).

  * •

The presence of a WSe2 monolayer should lead to a slight increase of the Raman
intensities by a bit less than a factor of 2 (Fig. S5a,b).

  * •

$I_{\rm{2D}}/I_{\rm{G}}$ is expected to be larger in the suspended regions
than in the supported regions (Fig. S5c). This is a purely optical effect due
to the fact that the enhancement factor of $I_{\rm G}$ reaches lower values
than that of $I_{\rm{2D}}$ in the suspended areas, because the wavelength of
the outgoing photons for the G-mode is shorter than those of the 2D-mode
photons [73].

  * •

Only a marginal increase of $I_{\rm{2D}}/I_{\rm{G}}$ by a few percent is
expected in TMD/graphene compared to bare graphene (see (Fig. S5c). This is
below the standard deviation in the spatially averaged values measured over a
given region of interest (Fig. S6).

Figure S5: Optical interference effects. Computed interference enhancement
factors for the G-mode intensity $I_{\rm{G}}$ (a), the 2D-mode intensity
$I_{\rm{2D}}$ (b) and the $I_{\rm{2D}}/I_{\rm{G}}$ ratio. The geometry
considered is indicated in the inset of panel (a). Specifically, the SiO2
thickness of our sample is $h=285~{}\rm{nm}$ and the supported and suspended
geometries are highlighted with $\\#$ and $\ast$ symbols, respectively. The
suspended geometry in our experiments corresponds to a hole depth of
$d=250~{}\rm{nm}$. The cases of bare graphene (blue lines) and WSe2/graphene
(orange lines) are considered.

The spatially averaged values $\langle I_{\rm G}\rangle$, $\langle
I_{\rm{2D}}\rangle$ and $\langle I_{\rm{2D}}/I_{\rm{G}}\rangle$ for the map
shown in Fig. 2a of the main text (also shown in Fig. S2f) are shown in Figure
S6 together with the standard deviations as errorbars. For convenience, we
have normalised the Raman intensities to the value of $\langle I_{\rm
G}\rangle$ measured on bare suspended graphene. First and foremost, our
experimental observations are in very good qualitative agreement with the
trends predicted by our interference calculations. Second, we note a
remarkable reduction of $\langle I_{\rm{2D}}/I_{\rm{G}}\rangle$ by a factor of
approximately 1.5 when moving from suspended graphene to suspended
WSe2/graphene. A qualitatively similar, yet smaller reduction of 10$\%$ is
also observed when comparing WSe2/graphene on SiO2 and graphene on SiO2. We
also note that these changes arise from a decrease in the 2D-mode intensity,
whereas the G-mode intensity remains essentially constant when moving from
graphene to WSe2/graphene. These changes can tentatively by attributed to
dielectric screening.

Figure S6: Spatially averaged Raman intensities. Spatially averaged values of
the integrated Raman intensities $\langle I_{\rm G}\rangle$ (a), $\langle
I_{\rm{2D}}\rangle$ (b) and integrated intensity ratio $\langle
I_{\rm{2D}}/I_{\rm{G}}\rangle$ (c) for the Raman map shown in Fig. 2a of the
main manuscript. The data in (a) and (b) are normalised with respect to the
value of $\langle I_{\rm G}\rangle$ in bare suspended graphene. The semi-
transparent vertical bars correspond to the computed values based solely on
optical interference effects (see Fig. S5). The height of the bars accounts
for the experimental uncertainty in the hole depth $d$ and for a possible
slack of the suspended graphene and WSe2/graphene membranes (see inset in Fig.
S5a).

To close this section, we wish to stress that from a theoretical standpoint,
it is by no means trivial to compute the changes in the absolute Raman
scattering intensity of the G- and 2D-modes as a function of screening.
Indeed, the Raman processes involved in these two modes are very different and
may be affected by other effects, notably resonance effects and quantum
interference effects [25, 23, 26, 33], that are not directly related to
dielectric screening and may overshadow the sole contribution of the
dielectric screening. A comprehensive theoretical analysis of the Raman
susceptibility of graphene in various dielectric environments goes well-beyond
the scope of the present study.