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

Loïc Moczko, Sven Reichardt, Aditya Singh, Xin Zhang, Elise Jouaiti, Luis Enrique Parra Lopez, Joanna L. P. Wolff, Aditi Raman Moghe, Étienne Lorchat, Rajendra Singh, et al.

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To cite this version:

Loïc Moczko, Sven Reichardt, Aditya Singh, Xin Zhang, Elise Jouaiti, et al. "Symmetry-dependent dielectric screening of optical phonons in monolayer graphene." Physical Review X, 2025, 15 (2), pp. 021043. 10.1103/PhysRevX.15.021043. hal-04982208

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# Symmetry-Dependent Dielectric Screening of Optical Phonons in Monolayer Graphene

Loïc Moczko<sup>1</sup>, Sven Reichardt<sup>1,2</sup>, Aditya Singh<sup>1,3</sup>, Xin Zhang<sup>1,*</sup>, Elise Jouaiti<sup>1</sup>, Luis E. Parra López<sup>1</sup>

Joanna L. P. Wolff<sup>1</sup>, Aditi Raman Moghe<sup>1</sup>, Etienne Lorchat<sup>1</sup>, Rajendra Singh<sup>3</sup>, Kenji Watanabe<sup>4</sup>, Takashi Taniguchi<sup>5</sup>

Hicham Majjad<sup>1</sup>, Michelangelo Romeo<sup>1</sup>, Arnaud Gloppe<sup>1</sup>, Ludger Wirtz<sup>1,2</sup> and Stéphane Berciaud<sup>1,†</sup>

<sup>1</sup>Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France

<sup>2</sup>Department of Physics and Materials Science, University of Luxembourg, 162a avenue de la Faîencerie, L-1511 Luxembourg, Luxembourg

<sup>3</sup>Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

<sup>4</sup>Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

<sup>5</sup>International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

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(Received 20 October 2023; revised 19 March 2025; accepted 28 March 2025; published 7 May 2025)

Quantized 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 nonelectronic 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 hetero-structures, 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 lines of graphene react differently to the surrounding environment. While the 2D line, which is due to near-zone-edge optical phonons, undergoes changes due to the neighboring dielectric environment, the in-plane, zone-center 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 characterize dielectric screening in van der Waals heterostructures and moiré superlattices.

DOI: 10.1103/PhysRevX.15.021043

Subject Areas: Condensed Matter Physics, Materials Science, Nanophysics

## 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 wave vectors 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

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*Present address: State Key Laboratory of Semiconductor Physics and Chip Technologies, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China.</footnote>

<footnote>†Contact author: stephane.berciaud@ipcms.unistra.fr</footnote>
(hBN) [9] and of the concept of van der Waals engineering [10,11] makes 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 Raman lines connected to the two Kohn anomalies occurring at the center ($\Gamma$) and edges ($K$ and $K'$ points) of its Brillouin zone, respectively [Fig. 1(a)], behave differently when the local dielectric environment is changed because the associated phonon modes behave differently under symmetry operations. For this purpose, using micro-Raman scattering spectroscopy [14], we

investigate 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, which we attribute to the screening of the Kohn anomaly that affects nondegenerate zone-edge in-plane transverse optical phonons (TO branch at $K$ and $K'$, with $A'_1$ 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

<figure><img src="image_4.png" /><figcaption>FIG. 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 Ref. [15]. The Kohn anomalies at $K$ and $\Gamma$ are highlighted with red dashed lines. (b) In-plane atomic displacements associated with the $E_{2g}$ and $A'_1$ phonons. (c) Sketches of the Raman G-, 2D-, and 2D'-mode processes in the momentum-energy space. (d) Typical Raman spectra of hBN-capped graphene (black dots) and hBN-capped $WSe_2$/graphene (orange dots). Data are shown together with their fits (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 panel (e), respectively.</figcaption></figure>
effect of screening is more pronounced as the wave vector of the 2D-mode phonon approaches $\mathbf{K} - \mathbf{K}'$. Remarkably, we demonstrate that the zone-center in-plane optical phonons (degenerate LO/TO branch at $\mathbf{\Gamma}$, with $E_{2g}$ symmetry) giving rise to the Raman G-mode feature are, to experimental accuracy, insensitive to dielectric screening. These contrast- ing behaviors are explained theoretically by considering the symmetry properties of low-energy electronic and electron- phonon Hamiltonians in graphene and showing that the zone-center phonon frequency is protected from changes in the screening by a Ward identity, which leads to a cancella- tion of many-body renormalization effects in the interatomic force constants.

Our results shed light on electron-phonon physics and demonstrate 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 provides an invaluable missing element for the advanced characterization of graphene and, more broadly, of near-field coupled van der Waals materials. Beyond graphene, the theoretical framework we introduce is general enough to be useful for other cases with anomalous phonons or electron-phonon couplings.

## 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 [14,16], able to yield decisive insights into their nanoscale environment [17], symmetry properties [18,19], and sensitivity towards external perturbations [20]. Monolayer graphene stands out as the first [21,22] 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 $\mathbf{K}, \mathbf{K}'$ points) enables Kohn anomalies for phonon wave vectors $q = 0$ and $\mathbf{q} = \mathbf{K} - \mathbf{K}'$, which appear as sharp kinks in the phonon dispersion [6,7] [Figs. 1(a) and 1(b)].

The Raman spectrum of graphene [Figs. 1(c) and 1(d)] is dominated by two main features: first, the one-phonon, nonresonant G-mode feature (frequency $\omega_G$ near $1580 \text{ cm}^{-1}$) arising from degenerate LO/TO $E_{2g}$ phonons at $\mathbf{\Gamma}$ [23]; second, strong electron-phonon coupling favor- ing resonant multiphonon processes [24–26], the most famous of which being the 2D mode, with a pair of near zone-edge ($\mathbf{K}, \mathbf{K}'$) TO phonons with opposite momenta that warrants conservation of energy and momentum [Fig. 1(c)]. The 2D-mode feature is widely known as a hallmark of monolayer graphene [21,22]; it is also dis- persive (frequency $\omega_{2D}$ between 2500 and 2700 $\text{cm}^{-1}$ 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 involv- ing a pair of LO phonons near $\mathbf{\Gamma}$ gives rise to the fainter 2D'- mode feature near $3240 \text{ cm}^{-1}$ [see Refs. [25,26], Fig. 1(c), and Appendix B].

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

Figure 1(e) shows an optical image of a graphene monolayer, partly covered by a tungsten diselenide ($WSe_2$) monolayer and fully encapsulated in hBN (Sample 1; see Appendix A for details on the experimental methods, as well as the Supplemental Material [38]). Typical Raman spectra associated with both regions of the sample [Fig. 1(d)] show a clear shift of $\omega_{2D}$; at the same time, the G-mode and the 2D'-mode features remain unaffected. A hyperspectral map of $\omega_{2D}$ is presented in Fig. 1(f), and it readily confirms that hBN-capped graphene/WSe$_2$ exhibits higher values of $\omega_{2D}$ than hBN- capped graphene. In order to quantitatively characterize the role of the environment, we spatially average $\omega_{2D}$ and $\omega_G$ over an area of a few tens of $\mu\text{m}^2$ in regions (i) and (ii) in Fig. 1(e). We find a spatially averaged upshift of $\omega_{2D}$ of $7.3 \pm 3 \text{ cm}^{-1}$ when comparing hBN-capped WSe$_2$/gra- phene to hBN-capped graphene. It is noteworthy that the spatial averages ($\omega_G = 1583.7 \pm 0.8 \text{ cm}^{-1}$ and $\omega_G = 1583.5 \pm 0.5 \text{ cm}^{-1}$ in hBN-capped graphene and graphene/WSe$_2$, respectively) are nearly identical. These contrasting behaviors directly exclude a simple interpreta- tion in terms of distinct levels of strain and/or doping in both regions, as one would expect a G-mode upshift of approximately $3.3 \text{ cm}^{-1}$ in the case of biaxial compressive strain [39,40] and a much larger upshift in excess of $10 \text{ cm}^{-1}$ in the case of electron or hole doping [34,41]. A more plausible explanation is that the presence of one single layer of TMD screens the Kohn anomaly at $\mathbf{K}$ and $\mathbf{K}'$, leading to a sizable upshift of $\omega_{2D}$ accompanied by a change in the slope of the phonon dispersion near $\mathbf{K}$, as schematized in Fig. 2(f) [36], and/or that a screening-induced change in the Fermi velocity (slope of the linear electronic bands) alters the resonance condition [25,26,28,37].

## III. IMPACT OF DIELECTRIC SCREENING NEAR THE KOHN ANOMALY AT K

To obtain deeper insights into the influence of the dielectric environment, we now consider Sample 2, which is made of a monolayer graphene deposited onto a $\text{SiO}_2$ substrate patterned with microscopic circular pits and partly
<figure><img src="image_5.png" /><figcaption>FIG. 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 prepatterned holes (Sample 2). The white dashed contour indicates the WSe₂ monolayer. The scale bar is 6 µm. 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 ω₂D. The mapped area corresponds to the dark yellow rectangle in panel (a). (c) Raman 2D-mode spectra of the sample shown in panel (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 ω₂D on one selected spot in each of the four different regions of the sample as a function of the laser photon energy. (e) Same data as in panel (d), with color-coded laser photon energies, shown relative to ω₀²D, the value recorded on suspended graphene. The error bars in panels (d) and (e) correspond to the uncertainty in the fitted values of ω₂D. (f) Sketch of the TO phonon dispersion in the vicinity of the **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 wave vectors corresponding to increasing laser photon energy. Dielectric screening flattens the Kohn anomaly (kink at **K**).</figcaption></figure>

covered by a WSe₂ monolayer. Figure 2(a) 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 is possible to study suspended graphene as a pristine, minimally screened sample [42,43]. 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 a similar dielectric constant as bulk hBN [44,45], we will be able to determine to what extent surface roughness affects the observed 2D-mode upshift introduced above.

Figure 2(b) shows a hyperspectral map of ω₂D (under laser excitation at 2.33 eV), which, again, gives evidence of

a clear progressive upshift when moving from suspended graphene to SiO₂-supported graphene/WSe₂, i.e., from region (i) to (iv) [Figs. 2(b) and 2(c)]. In stark contrast, the spatially averaged ω_G is nearly identical on all four regions, spread only within a 2-cm⁻¹ window (see Supplemental Material, Sec. S2 [38]), in keeping with the trend observed in Fig. 1.

Remarkably, by investigating the dispersion of ω₂D using laser excitation at 2.33 eV, 1.96 eV, and 1.58 eV on single spots in each of the four regions identified above [Fig. 2(c)], we observe that the screening-induced 2D-mode upshift augments as the laser photon energy decreases. In each region, we plot the 2D-mode dispersion [Fig. 2(d)] and the laser photon energy-dependent upshift of ω₂D, with color-coded symbols [Fig. 2(e)]. Under laser excitation at
2.33 eV (respectively, 1.58 eV), $\omega_{2D}$ upshifts by up to 20 cm$^{-1}$ (respectively, 27 cm$^{-1}$) when moving from region (i) to region (iv). These observations are consistent with the screening of the Kohn anomaly illustrated in Fig. 2(f). Indeed, as the laser photon energy increases, the momentum of the phonons involved in the 2D-mode process (taken relative to **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 [36,46]. 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 finds $\omega_{2D} = 2\omega_K + 2(v_{TO}/v_F)\omega_L$, with $\omega_K$, $v_{TO}$, $v_F$, and $\omega_L$, the TO phonon frequency at **K**, the phonon and electron band velocities, and the laser photon frequency, respectively [24,27,28]. As a result, the observed 2D-mode upshift may also be affected by the partial compensation between the screening-induced reduction of the phonon [36] and electron [43,47] band velocities. Nevertheless, the results in Figs. 2(d) and 2(e), as well as the analysis developed in the Supplemental Material (Sec. S3 [38]), demonstrate that the large screening-induced 2D-mode upshift revealed in Fig. 2 cannot be understood without considering a significant upshift of the phonon frequency at **K** and an associated reduction of $v_{TO}$ due to the smearing of the Kohn anomaly at **K**. The dispersive character of the relative dielectric shift

evidenced in Fig. 2(e) appears only as a correction to the main effect that likely arises from changes in the ratio $v_{TO}/v_F$ as the laser photon energy varies. These findings shed new light on the physical origin of the screening-induced upshift of the Raman 2D mode and will stimulate further theoretical works to model the latter at the *ab initio* level (see also Appendix F). On the experimental side, measurements at lower $\omega_L$ would allow for probing of 2D-mode phonons closer to the **K** point [48], thereby obtaining additional insights into the dispersive effects unveiled in Fig. 2(e).

## IV. EVIDENCE FOR SYMMETRY-DEPENDENT DIELECTRIC SCREENING OF OPTICAL PHONONS

The spatially averaged values of $\omega_{2D}$ and $\omega_G$ obtained from Sample 1 (Fig. 1) and Sample 2 (Fig. 2), as well as on another sample (BN/MoSe$_2$/graphene, Sample 3 introduced in the Supplemental Material, Secs. S2 and S4 [38]; see also Ref. [49]), are gathered in Figs. 3(a) and 3(b), respectively. The areas of interest are ordered by increasing expected dielectric screening. Error bars correspond to the standard deviation of the spatially averaged values of $\omega_{2D}$ and $\omega_G$.

While the upshift of $\omega_{2D}$ due to the bulk $SiO_2$ substrate itself is only of $5.9 \pm 2.3 \text{ cm}^{-1}$, we find much larger upshifts of $14.8 \pm 2.8 \text{ cm}^{-1}$ and $16.0 \pm 1.2 \text{ cm}^{-1}$ when $SiO_2$ is replaced by van der Waals materials, namely, bulk

<figure><img src="image_6.png" /></figure>

FIG. 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/WSe$_2$ 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 panels (a) and (b). For reference, the short-dashed, dot-dashed, and dotted lines correspond to the expected correlations in the presence of biaxial strain, and 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. Each spatial average is performed over 15 to 30 spectra per region of interest. The error bars correspond to the standard deviation associated with each spatially averaged value.
hBN and monolayer WSe$_2$, respectively. In spite of their similar dielectric constants [44,45], bulk hBN leads to a much larger shift of $\omega_{2D}$ than bulk SiO$_2$. As SiO$_2$ is rougher than a layered material such as hBN, this observation suggests that graphene remains partly decoupled from SiO$_2$, whereas hBN, as an atomically flat substrate, warrants optimal coupling and drastically reduces surface roughness and disorder in graphene [50]. 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/WSe$_2$ (Appendix C and Refs. [51,52]). The impact of the SiO$_2$ roughness is further investigated through a comparative study of SiO$_2$/graphene/TMD versus SiO$_2$/TMD/graphene heterostructures (Appendix D 2). It is noteworthy that the screening-induced upshift of $\omega_{2D}$ is slightly larger in suspended graphene/monolayer WSe$_2$ than in graphene deposited on bulk hBN [Fig. 3(a) and Supplemental Material, Secs. S2 and S4 [38]], illustrating the remarkably strong interaction between a TMD monolayer and graphene, as suggested otherwise by recent electron transport measurements [53]. Altogether, this set of results demonstrates that the screening of the Kohn anomaly at 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 only leads to a marginal increase of $\omega_{2D}$ 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 subnanometer 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 nanometers [compare Figs. 1(c) and 1(d) of Ref. [37]]; see also Ref. [54]). We attribute this difference to the fact that, for the excitons, the excited electron interacts with the electrostatic potential of the hole as well as with the potential of the screening cloud originating from the whole space and, in general, from several neighboring layers (see Appendix D 3 for details).

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

(and $\omega_{2D}$) may be in order when graphene is screened by a partner material.

## V. THEORETICAL ARGUMENT

The insensitivity of the $\Gamma$-point phonon frequency to changes in the dielectric environment can be understood from a symmetry point of view. For this purpose, we focus on the contribution of the low-energy $\pi$-band excitations to the part of the phonon frequency that is sensitive to screening [15,55]:

$$ \omega_{\lambda|\pi\text{-bands}}^2 = \frac{2}{M_C} \int_{\text{BZ}} \frac{d^2k}{A_{\text{BZ}}} \sum_{\text{spin}} \frac{(g_{\text{bare}}^{\lambda})_{\mathbf{k},\pi^{*},\pi} (g_{\text{screened}}^{\lambda})_{\mathbf{k},\pi^{*},\pi}}{\varepsilon_{\mathbf{k},\pi^{*}} - \varepsilon_{\mathbf{k},\pi}} \quad (1) $$

Here, $\lambda = x, y$ is a label for the phonon polarization; $g_{\text{bare}}^{\lambda}$ and $g_{\text{screened}}^{\lambda}$ are the unscreened and screened electron-phonon coupling matrix elements, respectively; $\varepsilon_{\mathbf{k},\pi^{*}}$ denote the band energies; $M_C$ is the mass of a carbon atom; and the integral runs over the first Brillouin zone, whose area is denoted as $A_{\text{BZ}}$. 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 $E_{2g}$ symmetry of the phonon implies that it transforms like a vector under rotations. These transformation properties can either be inferred from the Mullikan notation label or by noting that a 60° rotation is represented in $E_{2g}$ by the standard $2 \times 2$ rotation matrix used to rotate a two-dimensional vector while being represented in the $A'_1(=A_{1g})$ representation of the TO phonon at K by the number 1. Visually, the symmetry behavior is most apparent when the phonon displacement patterns are represented by arrows (see, e.g., Fig. 4.3 in Ref. [15]); then, it becomes clear that the rotated arrows in the $E_{2g}$ case are rotated as a vector and are pointing in a rotated direction, while the transformed arrows in the $A'_1$ case remain, in effect, unaffected and only pick up a minus sign from the $\exp(i\mathbf{q} \cdot \mathbf{R})$ factor. This “vectorlike” character of the phonon mode in combination with the Dirac-Weyl nature of the effective $\pi$-band Hamiltonian leads to the effective electron-phonon interaction Hamiltonian being of the same form as the electron-photon Hamiltonian in quantum electrodynamics (QED), in which the vectorlike 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. [56]). 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 [56,57] for the screened electron-phonon coupling. For electrons near the K valley, it reads

$$ (g_{\text{screened}}^2)_{\mathbf{k},\pi^*,\pi} \propto v_F [\chi_{\mathbf{k}\pi^*}^\dagger \cdot \boldsymbol{\sigma} \cdot \chi_{\mathbf{k},\pi}] \wedge \mathbf{e}_\lambda, \quad (2) $$

where $\mathbf{e}_{\lambda=x,y}$ is the effective polarization vector of the phonon, $\boldsymbol{\sigma}$ is the vector of Pauli matrices, $\chi$ 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 antisymmetric product. Most importantly, the only screening-dependent, i.e., substrate-sensitive, quantity here is the effective Fermi velocity $v_F$. Notably, the bare electron-phonon coupling is naturally independent of screening and $v_F$, while the electronic transition energies in the denominator of Eq. (1) are proportional to the very same $v_F$: $\epsilon_{\mathbf{k},\pi^*} - \epsilon_{\mathbf{k},\pi} \propto v_F |\mathbf{k}|$. As such, the screened electron-phonon coupling for the $\Gamma$-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 purely a result 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 [58] (Appendix E).

Consistently, the 2D' mode, involving a pair of LO phonons near $\Gamma$ (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 $\Gamma$ (Appendix B and Supplemental Material, Sec. S4.2 [38]). Since the phonons involved in the 2D' mode are not strictly at $\Gamma$, 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 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 nondegenerate phonons, which do not necessarily 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 nondegenerate, 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 $A'_1$ 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. In the absence of any protection, however, a phonon branch featuring a Kohn anomaly is very sensitive to the immediate dielectric environment. This case can be intuitively understood from the perspective of the interatomic force constants. A phonon branch undergoing a Kohn anomaly features a kink in its dispersion. Such a nonanalytic feature, however, can only be produced by a Fourier series with infinitely many terms, i.e., slowly decaying interatomic force constants. These long-range force constants make the phonon branch of the Kohn anomaly susceptible to dielectric screening by the environment. We illustrate this case with the example of the linear carbon chain, which we discuss in Appendix F.

The screening of the long-range force constants is consistent with the previous observation of an upshift of the 2D line on a dielectric substrate [36] and with the observation of a considerably higher-lying Kohn anomaly at $\mathbf{K}$ for graphene on a metallic substrate [46]. In both cases, it was argued that a screening-induced reduction of the electron-phonon coupling comes with a higher phonon frequency of the highest optical branch at $\mathbf{K}$ [see schematic drawing in Fig. 2(f)]. The argument is based on the strong “interplay of the Coulomb interaction and electron-phonon interaction in graphene” [59] and the numerical observation that electron-electron interaction (which is susceptible to screening effects) plays an important role in the exact calculation of the electron-phonon coupling in graphene [7]. The finding is also consistent with a previous report on the (more pronounced) reduction of the electron-phonon coupling in graphene upon doping [60].

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 ($I_{2D}/I_G$) is indeed reduced when comparing freestanding graphene with a TMD/graphene heterostructure (see Supplemental Material, Sec. S5 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 sensitivity of the $A'_1$ 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 **K** point, yielding a sizable upshift of the Raman 2D line by up to 1% when comparing a suspended graphene monolayer with a monolayer graphene/monolayer TMD heterostructure encapsulated in hexagonal boron nitride. Conversely, $E_{2g}$ phonons, involved in the Raman G-mode and 2D'-mode processes 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, 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 subnanometer out-of-plane accuracy, as well as built-in high-pressure gauges [61] as an alternative to ruby [62].

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-mode features to strain and doping [34,39,51,63,64], 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, Sec. S2.1 [38]. Furthermore, 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 [17].

Finally, the theoretical aspects of our work shed 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 novel viewpoint on electron-phonon physics is 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. For instance, the formulation of the electron-phonon coupling in terms of electronic correlation

functions opens up other possible ways to tackle its quantitative description, e.g., with Green's function-based methods from the field of strongly correlated electrons. In addition, our formulation makes it possible to identify exact constraints on the electron-phonon coupling that any model or first-principles description needs to obey.

## ACKNOWLEDGMENTS

We thank T. Sohier for fruitful discussions. We are grateful to the STnano cleanroom staff for technical support. We acknowledge financial support from the Agence Nationale de la Recherche (under the following grants: ATOEMS ANR-20-CE24-0010, VANDAMME ANR-21-CE09-0022, TEXTURES ANR-22-CE09-0008, INFERNO ANR-22-CE42-0015, and EXODUS ANR-23-QUAC-0004). The 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 the 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). 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.

## APPENDIX A: EXPERIMENTAL METHODS

All samples were made from mechanically exfoliated bulk crystals of graphite, TMDs (here, $WSe_2$, $MoSe_2$, and $MoS_2$), and HBN. The van der Waals heterostructures were deposited onto $Si/SiO_2$ substrates using dry transfer methods—either directly using a PDMS poly (dimethylsiloxane) stamp [65] or through the “pick-up and lift” method—with an additional polycarbonate (PC) layer that is later dissolved in chloroform [66,67]. Suspended structures were prepared by depositing graphene and $WSe_2$ layers onto $Si/SiO_2$ substrates patterned by pits of approximately 5 µm in diameter fabricated using photolithography followed by reactive ion etching [42]. Raman and photoluminescence (PL) measurements were performed in ambient conditions using a homebuilt scanning confocal setup as described elsewhere [37,39], employing single-mode, linearly polarized lasers emitting at 2.33 eV, 1.96 eV, and 1.58 eV. The laser intensity was maintained low enough (i.e., well below $1 \text{ mW/µm}^2$ for Raman measurements, with even lower intensities for PL measurements) to minimize heating and, importantly,
<figure><img src="image_7.png" /><figcaption>FIG. 4. Invariance of the 2D'-mode frequency. (a) Sketch of the Raman 2D'-mode process. (b) Spectra of the 2D'-mode feature recorded at a laser photon energy of 2.33 eV in the different configurations indicated. The sample and its areas of interest are introduced in Fig. 2 of the main text. (c) Spatially averaged values of the 2D'-, 2D-, and G-mode frequencies in each dielectric environment, respectively. The averages were typically performed over five selected points in each region of interest, and the error bars correspond to the standard deviation associated with each spatially averaged value.</figcaption></figure>

photoinduced doping effects [37,49]. No polarization analysis was performed on the Raman scattered and PL photons. All samples discussed here have been characterized by means of hyperspectral mapping (Supplemental Material, Secs. S1 and S2 [38]), allowing unambiguous identification of graphene [21,22] and TMD monolayers [68], respectively. PL maps allowed us to identify coupled and decoupled TMD/graphene regions based on the strong PL quenching due to nonradiative 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, Sec. S2 [38] and Refs. [41,69]). All samples discussed in this work display small built-in strain of at most a few $10^{-2}\%$, while unintentional doping remains below $10^{-12} \text{ cm}^{-2}$. Under these conditions, the shifts of $\omega_G$ and $\omega_{2D}$ due to built-in strain and unintentional doping are negligible compared to the dominant effect of dielectric screening [Fig. 3(a)] and are responsible for the small scatter in the spatially averaged $\omega_G$ [Fig. 3(b)]. Note that three consistent data sets are shown for Sample 2 in Fig. 2(b) (with spatial averages in Fig. 3), Figs. 2(c)–2(e), and 4, respectively. More details on these data sets, as well as additional data on another sample, are available in the Supplemental Material, Sec. S4 [38].

## 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 $\Gamma$ point and appears near $3240 \text{ cm}^{-1}$. It is the intravalley analog of the 2D mode [25,26]. The fully resonant part of the Raman 2D' mode process is sketched in Fig. 4(a). Since the electron-phonon coupling at $\Gamma$ is smaller than at $K$, the 2D'-mode feature is less intense than the 2D-mode feature [6,59].

In this appendix, we investigate the evolution of the 2D'-mode frequency $\omega_{2D'}$ in various dielectric environments to further confirm the behavior evidenced in Fig. 1(d) of the main text (Sample 1). For this purpose, we consider a second set of data recorded in the same selected areas of Sample 2, already discussed in Fig. 2 of the main text [see also Fig. S2(d)–S2(f) and Fig. S5 [38]], namely, suspended graphene (i), $SiO_2$-supported graphene (ii), suspended graphene/WSe$_2$ (iii), and $SiO_2$-supported graphene/WSe$_2$ (iv).

In the most screened case (i.e., in this sample, $SiO_2$-supported graphene/WSe$_2$), we measure an almost negligible, spatially averaged blueshift $\Delta\omega_{2D'} = 2.6 \pm 1.4 \text{ cm}^{-1}$ with respect to the unscreened case of suspended graphene, while, within experimental accuracy, we do not resolve any sizable shift when comparing the other remaining cases to suspended graphene. We can thus consider that $\Delta\omega_{2D'}$ is negligibly small and, in any event, more than 1 order of magnitude smaller than the 2D-mode upshift $\Delta\omega_{2D}$ presented in Figs. 2 and 3 (see also Supplemental Material, Sec. S4.2 [38]). 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 Appendix E below, and they further demonstrate that the screening of the Kohn anomaly that occurs for zone-edge TO phonons at $K$ can be used as a sensitive probe of dielectric screening.

## APPENDIX C: THE 2D-MODE LINEWIDTH

The FWHM of the 2D-mode feature of graphene $\Gamma_{2D}$ can provide additional information about the dielectric environment of the graphene monolayer. Figure 5(a) shows $\langle\Gamma_{2D}\rangle$, 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
<figure><img src="image_8.png" /><figcaption>FIG. 5. Analysis of the 2D-mode linewidth. (a) Spatially averaged FWHM of the Raman 2D mode of graphene depending on its dielectric environment, ordered by increasing expected dielectric screening. (b) Spatially averaged FWHM of the Raman G mode of graphene depending on its dielectric environment, ordered by increasing expected dielectric screening. (c) Observed correlation between $\langle\Gamma_{2D}\rangle$ and $\langle\omega_{2D}\rangle$. (d) Observation of a narrow 2D-mode feature in hBN-capped graphene/WSe$_2$. The solid line is a Lorentzian fit.</figcaption></figure>

mode can be observed as screening increases. In particular, hBN capping gives greatly reduced linewidths of the 2D mode down to $16 \pm 0.5 \text{ cm}^{-1}$ for the sharpest spectra [see Fig. 5(d)]. 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 [51,52]. Interestingly, this reduction of $\Gamma_{2D}$ is correlated with the blueshift of $\omega_{2D}$, as visible in Fig. 5(c), 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 appendix.

# APPENDIX D: FINE-TUNING OF THE DIELECTRIC ENVIRONMENT

## 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/WSe$_2$ sample (Sample 4, not discussed in Fig. 3) with various degrees of coupling between graphene and WSe$_2$.

Figure 6(a) shows an optical image of the considered sample, where we highlight five regions of interest. These regions were identified by considering the WSe$_2$ PL intensity map presented in Fig. 6(b). Regions 1–3 correspond to graphene/WSe$_2$ heterostructures with different interlayer coupling, as illustrated by their respective PL emission spectra shown Fig. 6(d) [see also Fig. 7(b)]. 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. Figures 6(e)

<figure><img src="image_9.png" /><figcaption>FIG. 6. Supplementary data on a spatially inhomogeneous sample. (a) Optical image of another graphene/WSe$_2$ sample on Si/SiO$_2$ with different regions of interest identified (Sample 4). (b),(c) Photoluminescence intensity and 2D-mode frequency maps, respectively, of the sample shown in panel (a). (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, there is no suspended area in this sample.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>FIG. 7. Spatially averaged parameters for Sample 4. (a) Spatially averaged 2D-mode frequency $\langle\omega_{2D}\rangle$ and integrated PL intensity $\langle I_{PL}\rangle$ and (b) spatially averaged 2D-mode FWHM $\langle\Gamma_{2D}\rangle$ and integrated PL intensity $\langle I_{PL}\rangle$ for the five different regions of interest in Sample 4, introduced in Fig. 6.</figcaption></figure>

and 7(a) prove the strong sensitivity of $\omega_{2D}$ to the coupling between the TMD layer and graphene, as $\omega_{2D}$ increases when the PL intensity diminishes. Note that, in this specific sample, the graphene reference gives $\langle\omega_{2D}\rangle \approx 2670.5 \pm 1 \text{ cm}^{-1}$. The sharpening of the 2D mode discussed in Appendix C is also observed here, depending on the interlayer coupling [see Fig. 7(b)]. Regions 4 and 5 of the same sample are formed by bilayer WSe$_2$ (2L-WSe$_2$) interfaced with graphene, and they present two different levels of coupling (see Fig. 6). Note that the PL intensity of bilayer WSe$_2$ is lower than in the monolayer limit because of the indirect-to-direct band-gap transition occurring in the monolayer limit [70]. These two areas also show a coupling-dependent blueshift of $\omega_{2D}$ and a reduction of $\Gamma_{2D}$ [see Figs. 7(c) and 7(e)]. More interestingly, the obtained values of $\omega_{2D}$ and $\Gamma_{2D}$ for graphene/2L-WSe$_2$ are similar to the ones observed for graphene/1L-WSe$_2$, which suggests that dielectric screening essentially stems from the first layer in contact with graphene, highlighting the short-range character of the phenomenon.

## 2. Impact of surface roughness

An atomically thin material such as graphene deposited onto a bulk three-dimensional substrate (such as SiO$_2$) may naturally exhibit a surface roughness that is slightly smaller than that of the underlying substrate [71] while still remaining far from the atomically smooth limit. Sample processing may, however, have adverse consequences in terms of sample roughness, unintentional doping, and sensitivity to the environment, as demonstrated in Refs. [20, 72], where graphene was thermally annealed and/or exposed to various atmospheric environments. In our study, no special treatment has been performed to optimize the interfacial coupling between graphene and SiO$_2$, and in the most general case, it is a well-established fact that graphene and related systems conform much better to partner van der Waals materials within van der Waals heterostructures, leading to minimal surface roughness [50] and atomically smooth heterointerfaces [10].

To investigate the effect of surface roughness of the substrate used, we compare the observed upshift of the 2D-mode frequency in an additional set of graphene/TMD heterostructures deposited onto an SiO$_2$ substrate. Such geometries are particularly relevant for scalable devices made from atomically thin heterobilayers, which often require a bulk dielectric substrate that is not a van der Waals material. Figure 8 presents $\Delta\omega_{2D}$, the difference between the 2D-mode frequency in a graphene/TMD heterostructure and in the graphene reference of the same sample. We observe that $\Delta\omega_{2D}$ is appreciably higher when the graphene layer is *not* directly in contact with SiO$_2$. We understand this finding as originating from the thicker nature of TMDs, which may conform less than graphene to the microscopic irregularities of the SiO$_2$ surface. Thus, having the TMD layer in contact with SiO$_2$ gives a flatter surface from the viewpoint of graphene, which yields a better interfacial coupling and stronger dielectronic screening from the TMD than in the opposite situation.

<figure><img src="image_11.png" /><figcaption>FIG. 8. Impact of surface roughness. Spatially averaged screening-induced upshift of the 2D-mode feature in a set of graphene/TMD heterostructures deposited onto SiO$_2$, with the graphene layer either sandwiched between SiO$_2$ and the TMD or deposited on top of the SiO$_2$/TMD structure.</figcaption></figure>
### 3. Screening of phonons versus screening of excitons

As discussed above and in the main text, the screening of optical phonons stands out as an extremely short-range effect (typically, the subnanometer van der Waals gap between coupled 2D layers) in contrast with the case of excitons, where longer-range effects have been observed [37,54]. We attribute this difference to the fact that, for the excitons, the excited electron interacts both with the electrostatic potential of the hole and with the potential of the screening cloud originating from the whole 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 $\epsilon^{-1}(\mathbf{r}, \mathbf{r}')$ 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 $\epsilon^{-1}(\mathbf{r}, \mathbf{r}')$ at both points $\mathbf{r}$ and $\mathbf{r}'$ onto the change of the lattice potential with respect to the ionic displacements (cf. Chap. 4 of Ref. [15]). 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 localized, and hence the dielectric response function is sampled mostly over the layer in which the phonon resides and its immediate neighboring layer.

## APPENDIX E: SYMMETRY-PROTECTED ZONE-CENTER OPTICAL PHONONS

In this appendix, we provide all the necessary details and definitions to understand Eqs. (1) and (2) from the main

text. First, 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

$$g_{\mathbf{k},\pi^*,\pi}^{\mathbf{q},\lambda;(\text{bare})} \equiv \int d^3\mathbf{r} \psi_{\mathbf{k}+\mathbf{q},\pi^*}(\mathbf{r}) \psi_{\mathbf{k},\pi}(\mathbf{r}) \frac{\partial V_{\text{lat}}}{\partial R_{\mathbf{q},\lambda}}(\mathbf{r}), \quad (\text{E1})$$

where $\partial/\partial R_{\mathbf{q},\lambda} \equiv \sum_{n,k,i} u_{n,k,i}^{\mathbf{q},\lambda} \partial/\partial R_{n,k,i}$ is the directional derivative with respect to ionic displacements according to a phonon displacement pattern $u_{n,k,i}^{\mathbf{q},\lambda}$, where $\mathbf{q}$ is the phonon wave vector, $\lambda$ is 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. Here, $\psi_{\mathbf{k},\pi^*}(\mathbf{r})$ denotes the wave function of an electron with wave vector $\mathbf{k}$ in the $\pi^*$ band of graphene, and $V_{\text{lat}}(\mathbf{r})$ denotes the lattice potential.

The exact screened electron-phonon coupling matrix element is defined as (see Chap. 4 of Ref. [15] for a derivation and motivation)

$$g_{\mathbf{k},\pi^*,\pi}^{\mathbf{q},\lambda} \equiv \int d^3\mathbf{r} \int d^3\mathbf{r}' \psi_{\mathbf{k}+\mathbf{q},\pi^*}^*(\mathbf{r}) \psi_{\mathbf{k},\pi}(\mathbf{r}') \partial V^{\mathbf{q},\lambda}(\mathbf{r},\mathbf{r}'), \quad (\text{E2})$$

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

$$\begin{aligned} \partial V^{\mathbf{q},\lambda}(\mathbf{r}, \mathbf{r}') &\equiv -\int dt \frac{\partial}{\partial R_{\mathbf{q},\lambda}} G^{-1}(\mathbf{r}, t; \mathbf{r}', 0) \\ &= +\int dt \int d^3\mathbf{r}_1 \int dt_1 \int d^3\mathbf{r}_2 \int dt_2 \int d^3\mathbf{r}_3 \int dt_3 G^{-1}(\mathbf{r}, t; \mathbf{r}_1, t_1) \frac{\partial V_{\text{lat}}}{\partial R_{\mathbf{q},\lambda}}(\mathbf{r}_3) \Lambda(\mathbf{r}_3, t_3; \mathbf{r}_1, t_1; \mathbf{r}_2, t_2) G^{-1}(\mathbf{r}_2, t_2; \mathbf{r}', 0), \end{aligned} \quad (\text{E3})$$

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

$$\Lambda(\mathbf{r}_3, t_3; \mathbf{r}_1, t_1; \mathbf{r}_2, t_2) \equiv \langle 0 | \mathcal{T} \{ \hat{\rho}(\mathbf{r}_3, t_3) \hat{\psi}(\mathbf{r}_1, t_1) \hat{\psi}^{\dagger}(\mathbf{r}_2, t_2) \} | 0 \rangle |_{\text{connected}}. \quad (\text{E4})$$

Here, the subscript “connected” refers to the fully connected part of the correlation function, $\mathcal{T}$ denotes the

time-ordering symbol, $\hat{\psi}(\mathbf{r}, t)$ is the electron field operator in the Heisenberg picture, and $\hat{\rho}(\mathbf{r}, t)$ is 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 Ref. [15] for a proof).

For the special case of graphene and for the $\pi$ bands in the vicinity of the $K, K'$ 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 [73]. 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' point. The low-energy effective Hamiltonian for the π-band electrons near the K point reads

$$ \hat{H}_{\text{el,eff}} = \sum_{\alpha,\beta \in \{A,B\}} \int d^3 r \hat{\psi}_{\alpha}^{\dagger}(\mathbf{r}) (-i v_F \nabla \cdot \boldsymbol{\sigma}_{\alpha,\beta}) \hat{\psi}_{\beta}(\mathbf{r}), \quad (\text{E5}) $$

where $v_F$ is the exact band slope, including all screening effects from the substrate; $\boldsymbol{\sigma}$ is the vector of Pauli matrices in sublattice space; and the pseudospinor field operator is defined as

$$ \hat{\psi}_{\alpha}(\mathbf{r}) = \sum_{n} \frac{1}{\sqrt{N}} \psi_{p_z}(\mathbf{r} - \boldsymbol{\tau}_{\alpha} - \mathbf{R}_n) \hat{\psi}(\mathbf{r}) \quad (\text{E6}) $$

in terms of the wave functions $\psi_{p_z}(\mathbf{r})$ of the $p_z$ orbitals centered at the position $\boldsymbol{\tau}_{\alpha}$ of the nth of the N unit cells with an origin at $\mathbf{R}_n$.

The effective bare electron-phonon interaction Hamiltonian for the $\mathbf{q} \to \mathbf{0}$ optical phonons ($\lambda = \text{LO, TO}$) is likewise determined by symmetry [73,74] and can be written as

$$ \hat{H}_{\text{el-ph,eff}}^{\mathbf{q},\lambda} = F_0 \int d^3 r e^{i\mathbf{q} \cdot \mathbf{r}} (\hat{\mathbf{u}}_A^{\mathbf{q},\lambda} - \hat{\mathbf{u}}_B^{\mathbf{q},\lambda}) \wedge \hat{\mathbf{J}}(\mathbf{r}), \quad (\text{E7}) $$

where $\hat{\mathbf{J}}(\mathbf{r}) = \sum_{\alpha,\beta \in \{A,B\}} \hat{\psi}_{\alpha}^{\dagger}(\mathbf{r}) \boldsymbol{\sigma}_{\alpha,\beta} \hat{\psi}(\mathbf{r})$ is the electronic current operator for a Weyl fermion, $\mathbf{a} \wedge \mathbf{b} = a_x b_y - a_y b_x$ is the two-dimensional wedge product, and $\hat{\mathbf{u}}_\alpha^{\mathbf{q},\lambda}$ is the vectorial lattice displacement operator for the atoms on sublattice $\alpha = A, B$ projected on phonon mode $(\mathbf{q}, \lambda)$. Since this case is the bare electron-phonon interaction Hamiltonian, the constant $F_0$ is, by definition, independent of the substrate screening.

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

$$ g_{\mathbf{k},\pi^+,\pi}^{\mathbf{q},\lambda;(\text{bare})} \propto \sum_{\alpha,\beta \in \{A,B\}} (\chi_{\mathbf{k}+\mathbf{q},\pi}^{\alpha})^* \chi_{\mathbf{k},\pi}^{\beta} (\mathbf{u}_A^{\mathbf{q},\lambda} - \mathbf{u}_B^{\mathbf{q},\lambda}) \wedge \boldsymbol{\sigma}_{\alpha,\beta}, \quad (\text{E8}) $$

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

$$ \mathbf{k} \cdot \sum_{\beta} \boldsymbol{\sigma}_{\alpha,\beta} \chi_{\mathbf{k},\pi^{(*)}}^{\beta} = \begin{pmatrix} + \\ - \end{pmatrix} |\mathbf{k}| \chi_{\mathbf{k},\pi^{(*)}}^{\alpha}. \quad (\text{E9}) $$

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

$$ g_{\mathbf{k},\pi^+,\pi}^{\mathbf{q},\lambda} \propto \sum_{\alpha,\beta \in \{A,B\}} (\chi_{\mathbf{k}+\mathbf{q},\pi^+}^{\alpha})^* \chi_{\mathbf{k},\pi}^{\beta} (\mathbf{u}_A^{\mathbf{q},\lambda} - \mathbf{u}_B^{\mathbf{q},\lambda}) \wedge \partial \mathcal{V}_{\alpha,\beta}^{\mathbf{q}}(\mathbf{k}), \quad (\text{E10}) $$

with the vector- and matrix-valued irreducible vertex function

$$ \partial \mathcal{V}_{\alpha,\beta}^{\mathbf{q}}(\mathbf{k}) = \int dt \int dt_1 \int dt_2 \sum_{\alpha',\beta' \in \{A,B\}} G_{\alpha,\alpha'}^{-1}(\mathbf{k} + \mathbf{q}; t, t_1) \Lambda_{\alpha',\beta'}^{\mathbf{q}}(\mathbf{k}; t_1, t_2) G_{\beta',\beta}^{-1}(\mathbf{k}; t_2, 0), \quad (\text{E11}) $$

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

$$ \Lambda_{\alpha',\beta'}^{\mathbf{q}}(\mathbf{k}; t_1, t_2) = \int d^3 r_1 \int d^3 r_2 \int dt_3 e^{-i(\mathbf{k}+\mathbf{q})\cdot\mathbf{r}_1} e^{i\mathbf{k}\cdot\mathbf{r}_2} \langle 0 | \mathcal{T} \{ \hat{\mathbf{J}}(0, t_3) \hat{\psi}_{\alpha}(\mathbf{r}_1, t_1) \hat{\psi}_{\beta}^{\dagger}(\mathbf{r}_2, t_2) \} | 0 \rangle |_{\text{connected}} \quad (\text{E12}) $$

and the matrix-valued electron Green's function

$$ G_{\alpha,\beta}^{-1}(\mathbf{k}; t_1, t_2) = \int \frac{d\omega}{2\pi} e^{-i\omega(t_1-t_2)} [\omega\delta_{\alpha,\beta} - v_F\mathbf{k} \cdot \boldsymbol{\sigma}_{\alpha,\beta} + i\operatorname{sgn}(\omega)0^+], \quad (\text{E13}) $$

wherein 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, equivalently, the

result of electron number conservation. In the limit $\mathbf{q} \to \mathbf{0}$, the mentioned Ward identity reads [56,57]

$$ \Lambda_{\alpha,\beta}^{\mathbf{q} \to 0}(\mathbf{k}; t_1, t_2) \propto \frac{\partial}{\partial \mathbf{k}} G_{\alpha,\beta}(\mathbf{k}; t_1, t_2). \quad (\text{E14}) $$
It follows immediately that the irreducible vertex function is proportional to the momentum derivative of the inverse Green's function:

$$ \partial \mathcal{V}_{\alpha, \beta}^{q \to 0}(\mathbf{k}) \propto \frac{\partial}{\partial \mathbf{k}} G_{\alpha, \beta}^{-1}(\mathbf{k}; t_1, t_2) \propto v_F, \quad (E15) $$

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 the squared phonon frequency, leading to the robustness of the G-mode frequency with respect to changes in the screening environment [75].

As a side point, we want to emphasize that the scaling of the screened electron-phonon coupling with the exact Fermi velocity is not obvious for optical phonons. For acoustic phonons, it is often argued [58,59] 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 addition 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 indeed exists for acoustic phonons, it is not required by any more fundamental physics principle nor is it necessary to derive the scaling of the screened electron-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 more detailed theoretical reasoning.

## APPENDIX F: COMPARISON WITH THE KOHN ANOMALY IN THE MONATOMIC CARBON CHAIN

In order to qualitatively understand the potential sensitivity of the Kohn anomaly at the high-symmetry point **K**

to the dielectric environment, we briefly discuss a prototypical 1D system with a similar feature in its optical phonon dispersion: a chain of carbon atoms in the cumulene structure (where all bonds have equal length). Figure 9(a) shows the *ab initio* dispersion [76] of the longitudinal phonon branch (calculated with density-functional theory in the local-density approximation, using the quantum-espresso code [78]). Similarly to graphene, the dispersion of cumulene displays a kink at the Brillouin zone edge, which can be interpreted as a Kohn anomaly. (Actually, the phonon at the Brillouin zone edge becomes soft, owing to the Peierls instability of the cumulene chain versus dimerization of the C-C $\pi$ bonds [79]. Parenthetically, we remark that a similar soft-mode behavior can be observed for the Kohn anomaly at **K** in graphene under tension [80]. Conversely, if one “compresses” the linear carbon chain along its axis, the “soft” phonon at the Brillouin zone boundary becomes “hard,” but the strong kink in the phonon dispersion, corresponding to a 1D Kohn anomaly, remains.)

The kink in the phonon dispersion of the highest optical branch at $\Gamma$ can only be caused by interatomic force constants that are finite over a large distance in the direction along the chain since otherwise their Fourier transform would be analytic and the phonon dispersion kinkless. Indeed, a Fourier transform of the phonon dispersion yields slowly decaying long-range force constants of alternating signs in the direction along the chain, as shown in Fig. 9(b) (see also Problem 4 of Chap. 4 in Ref. [81]). The alternating sign in force constants is caused by long-range charge fluctuations in the $\pi$-electron density upon displacing one atom from its equilibrium position. The Kohn anomaly is clearly associated with these long-range forces within the carbon chain and similarly in graphene at the $\Gamma$ and **K** points. Note that the long-range character of the force constants in graphene had already been pointed out in the first *ab initio* calculations of phonons of graphene [82].

We can understand the sensitivity of the in-chain or in-plane long-range force constants to the presence of the

<figure><img src="image_12.png" /><figcaption>FIG. 9. Kohn anomaly in the one-dimensional carbon chain and associated forces and electric fields. (a) Calculated phonon dispersion of a one-dimensional chain of equidistant carbon atoms. (b) Corresponding interatomic force constants in the direction along the chain. Inset: zoom of the force constants from the fifth nearest neighbor onward.</figcaption></figure>
substrate by considering the involved electric fields. At short distances (first, second, and third nearest neighbors), the interatomic force constants are dominated by the part of the covalent bond that stems from the sp² hybridized electrons (σ bonds), where dielectric screening by the environment is not expected to play a major role. For two atoms further apart, however, the slowly decaying interatomic forces are caused by the induced charge oscillations in the π electron density that lead to electrostatic interaction between the induced dipole at the site of the displaced atom and the induced dipole at the nth nearest neighbor. This electrostatic interaction will be partially screened by the dielectric environment provided by the substrate. Since the phonon mode at a Kohn anomaly is associated with long-range force constants, the Kohn anomaly itself and its corresponding phonon branch are sensitive to screening effects from the environment as long as they are not protected by symmetry, as, for instance, in the case of the Γ LO/TO phonons in graphene.

We want to emphasize that this is a very qualitative argumentation. The effect of dielectric screening on the coupling between the π electrons and the phonon branch of the Kohn anomaly at K had been shown in Ref. [36] through a GW calculation: The atomic displacement along the phonon eigenvector leads to a band-gap opening at the high-symmetry point K. This gap opening is reduced by dielectric screening. Since the electron-phonon coupling determines the slope of the highest optical branch around K [83], the dielectric screening influences this slope. A quantitative proof of the effect of dielectric screening on the phonon dispersion around the Kohn anomaly would require a calculation on the level of the total-energy random-phase approximation [84]. Unfortunately, because of the extremely dense k-point mesh required for a semi-metallic system, this calculation is currently not feasible for graphene.

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