**Theoretical assignment and calculation of phonon modes.** Bulk GeS is a layered material crystallizing in a distorted orthorhombic structure (space group D2h16) with eight atoms per unit cell; see Fig. 1(a)-(c). The lattice constants are experimentally determined to \(a\) = 4.30, \(b\) = 3.64, and \(c\) = 10.47 Å [27], in agreement with ab-initio calculations of fully relaxed lattice constants [28]. Each Ge atom is bonded to three S atoms, and each atomic layer stack along the \(c\) axis as well as the unit cell contains two adjacent double layers. The puckered lattice of layered GeS possesses an anisotropic crystal structure with two distinct orthogonal directions: An armchair atomic chain prolongs along the \(a\) axis (Fig. 1(b)) and a zigzag-type connection is formed along the \(b\) axis (Fig. 1(c)).

The unit cell with its eight atoms results in 24 branches of the vibrational spectrum. According to the group theory analysis it has 24 irreducible zone-center phonon modes denoted by Γ = 4Ag + 2Au + 2B1g + 4B1u + 4B2g + 2B2u + 2B3g + 4B3u. They consist of 21 optical modes, two of them are inactive, 12 are Raman active and seven are infrared (IR) active. The Raman active modes are 4Ag, 2B1g, 4B2g, and 2B3g, whereas the IR active modes are 3B1u, B2u, and 3B3u. The three acoustic modes are described by the irreproducible representations B1u, B2u, and B3u. In the backscattering Raman geometry, the six modes 4Ag and 2B1g (4B2g and 2B3g) are detected when the laser light propagates along (perpendicular to) the \(c\) axis of the GeS crystal [11, 26, 29]. The frequencies \({\omega }_{\text{c}\text{a}\text{l}\text{c}}\) and energies \(\hslash {\omega }_{\text{c}\text{a}\text{l}\text{c}}\) of the phonon modes at the high symmetry points of the Brillouin zone are presented in Table 1, whereby \(\hslash\) is the reduced Planck constant. Moreover, in Fig. 1(d) the numerically calculated phonon dispersion of bulk GeS is presented. For that purpose, the ab-initio plane-wave density functional theory implemented in the QUANTUM ESPRESSO code [30] was used with a nonlocal van der Waals density functional (vdw-DF3-opt1) [31]. The wave function (kinetic energy) and density cut-offs were set to ∼1020 eV (75 Ry) and ∼8200 eV (600 Ry), respectively. The Monkhorst-Pack scheme of 4x11x10, for the k-sampling grid, was chosen. Self-consistent calculations were performed with an energy convergence criterion of ∼\(1.36\times {10}^{-7}\) eV (\(1\times {10}^{-8}\) Ry) and, for the relaxation of atomic positions to their equilibrium, with a force convergence criterion of ∼\(1.3\times {10}^{-3}\) eV/Å (\(5\times {10}^{-5}\) Ry/bohr). Using density functional perturbation theory the dynamical matrices were established on a 3x5x5 regular mesh q-grid. Based on these matrices, interatomic force constants (IFC) in real space were calculated. The phonon dispersion shown in Fig. 1(d) finally followed from the IFC and the Phonopy code [32].

Table 1

Symmetry and intensity assignments of the phonon modes observed in the experiment, with their frequencies derived from the experiment, calculation, and literature. The energetical distances of the resonance profile maxima from the exciton resonance are listed in the fourth column, values in brackets, for the Raman forbidden modes p3, p4, p5, and p6.

Notation | Intensity | Symmetry | \({\omega }_{\text{e}\text{x}\text{p}}\) [cm− 1] | \({\omega }_{\text{c}\text{a}\text{l}\text{c}}\) [cm− 1] | \(\hslash {\omega }_{\text{c}\text{a}\text{l}\text{c}}\) [meV] | \({\omega }_{\text{l}\text{i}\text{t}}\) [cm− 1] |

p1 | weak | 2LA(X) | 105 | 105 | 13.0 | |

| strong | Ag2(Γ) | 116 | 111.8 | 13.9 | 11629, 11611 |

p2 | weak | 2LA(Z) | 122 | 125 | 15.5 | |

g1 | strong | 2LA(Y) | 167 | 2x83 = 166 | 20.6 | |

g2 | medium | 2B2g1(Z) | 175 | 2x88 = 176 | 21.8 | |

| medium | B1g2(Γ) | 218 | 215 | 26.7 | 21929, 21811 |

p3 | strong | B1u2(Γ), IR | 231 (27 meV) | 232 | 28.8 | |

p4 | weak | B2u2(Γ), IR | 237 (31 meV) | 239 | 29.6 | |

| weak | Ag3(Γ) | 243 | 246 | 30.5 | 24429, 24411 |

| very weak | Ag4(Γ) | 276 | 276 | 34.2 | 27629, 27711 |

p5 | strong | B3u2(Γ), IR | 282 (33 meV) | 285 | 35.3 | 27526 |

p6 | strong | B1u3(Γ), IR | 327 (41 meV) | 329 | 40.8 | 32526 |

d1 | strong | B1u3(Γ) + Ag1(Γ) | 378 327 + 52 = 379 | | | |

d2 | strong | Ag2(Γ) + B1u3(Γ) | 440 116 + 327 = 443 | | | |

d3 | weak | B1g2(Γ) + B3u2(Γ) | 500 218 + 282 = 500 | | | |

d4 | medium | B1u3(Γ) + B1u2(Γ) | 558 327 + 231 = 558 | | | |

d5 | medium | B3u2(Γ) + B1u3(Γ) | 615 282 + 327 = 609 | | | |

d6 | medium | 2xB1u3(Γ) | 654 2x327 = 654 | | | |

We attribute the phonon modes to two groups: Phonon modes at non-\({\Gamma }\)-symmetry points as well as Raman active and Raman forbidden phonon modes at the \({\Gamma }\)-point with frequencies below 350 cm− 1 are assigned to the group one. The second-order scattering processes at the \({\Gamma }\)-point including Raman and IR active phonons with frequencies above 350 cm− 1 constitute the second group. Let us start to describe the phonon modes of the first group.

The phonon modes p1 and p2 with \({\omega }_{\text{c}\text{a}\text{l}\text{c}}=\) 105 and 125 cm−1 are attributed to second-order scattering of longitudinal acoustic phonons 2LA(X) and 2LA(Z), respectively. Further second-order processes labelled by g1 and g2 have the frequencies 166 and 176 cm− 1, respectively. They are positioned in the phonon frequency gap, compare Fig. 1(d), and are assigned to 2LA(Y) and 2B2g1(Z), respectively. The symmetry assignments for these second-order modes are consistent with the crystal momentum conservation principle. At higher frequencies the infrared active modes B1u2(Γ) at 232 cm− 1 and B2u2(Γ) at 239 cm− 1 are labelled by p3 and p4, respectively. In infrared reflectivity experiments performed at room temperature, the latter mode has been observed in bulk GeS [26]. The modes p5 at 285 cm− 1 and p6 at 329 cm− 1 are infrared active and have the symmetries B3u2(Γ) and B1u3(Γ), respectively. Similar infrared active modes were identified in BP which is also characterized by a puckered crystal structure [23].

The Raman lines in the second group stem from second-order scattering processes which are realized by superpositions of intensive first-order phonon modes from the Γ-point reaching frequencies above 350 cm− 1. We derive \({\omega }_{\text{c}\text{a}\text{l}\text{c}}=\) 378 cm−1 (d1) for the combined mode B1u3(Γ) + Ag1(Γ), and 440 cm−1 (d2) for Ag2(Γ) + B1u3(Γ). In both cases, Raman and IR active phonon modes determine the scattering process. Furthermore, we find at 500 cm−1 the d3 mode with the symmetry assignment B1g2(Γ) + B3u2(Γ), at 558 cm− 1 d4 with B1u3(Γ) + B1u2(Γ), and at 615 cm− 1 d5 with B3u2(Γ) + B1u3(Γ). The superposition of two phonons, each with symmetry B1u3(Γ), has a frequency of 654 cm− 1 (d6).

**Observation of Raman forbidden phonon modes at resonant exciton excitation and responses on external stimuli.** In the following we focus on the PL and RC as well as Raman scattering for non- and quasi-resonant excitation of the bright exciton in a layered GeS flake. The PL and RC spectra measured at 7 K are demonstrated in Fig. 2(a). The PL spectrum was excited non-resonantly at 2.330 eV. The laser light propagated along the \(c\) axis of the GeS flake (laser light wave vector \({{k}}_{\text{e}\text{x}\text{c}}\) is parallel to \({c}\)), while its linear polarization (electric field vector \({{ϵ}}_{\text{e}\text{x}\text{c}}\)) was parallel to the \(a\) axis. The PL exhibits five significant lines originating from the exciton (X) and most probably localized (impurity or defect) states [11] denoted by L1, L2, L3, and L4 in the low-energy part of the PL spectrum. By comparison, in the RC spectrum only a single resonance is observed, whose energy coincides with the maximum of the X PL line positioned at about \({E}_{\text{X}}=1.776\) eV. The full width at half maximum (FWHM) of the X emission is about 15 meV. Recent works have confirmed that the X feature in the PL and RC spectra has the same origin related to a direct transition at the \({\Gamma }\)-point of the Brillouin zone [10, 11, 14]. For GeS flakes having a thickness of several tens of nm the band gap switches from indirect (characteristic for bulk and monolayer GeS) to the direct type [10]. The resonance in the RC spectrum and the nonzero PL efficiency underline the presence of bright \({\Gamma }\)-excitons in our GeS flakes.

Characteristic polarization properties of the X PL as well as of Raman active phonon modes are depicted in Fig. 2(b), for quasi-resonant exciton excitation at 1.867 eV. As clearly seen, the X emission is polarized along the armchair direction of the GeS crystal (\({ϵ}\left|\right|{a}\)), while it is suppressed along the zigzag direction (\({ϵ}\left|\right|{b}\)). The angular dependence of the polarized exciton PL intensity \({I}_{\text{X}}^{\text{P}\text{L}}\left(\varphi \right)\) is presented in Fig. 2(c), where the rotation angle \(\varphi =0^\circ\) (\(\varphi =90^\circ\)) corresponds to \({ϵ}\left|\right|{a}\) (\({ϵ}\left|\right|{b}\)). This angular behavior may be described by Malus law \({I}_{\text{X}}^{\text{P}\text{L}}\left(\varphi \right)\propto {\text{cos}}^{2}\varphi\) [10]. This spatial anisotropy of the exciton emission implies the presence of transition dipole moments aligned only along the armchair direction. Indeed, along the \(a\) axis the Ge and S atoms are closely spaced giving rise to a polar bonding and – within this electric field – to an alignment of the excitonic carriers so that their dipole moment is parallel to \({a}\). Thus, the measurement of the X emission polarization allows for determining the armchair and zigzag crystallographic directions.

In Fig. 2(b), the Raman active phonon modes A2g, B21g and A3g are observed in the Stokes range from 1.853 to 1.830 eV. Considering the spectral positions \({E}_{\text{p}\text{h}}\) of the Raman lines with respect to the laser excitation energy \({E}_{\text{e}\text{x}\text{c}}\), the phonon energies \(\hslash {\omega }_{\text{e}\text{x}\text{p}}={E}_{\text{e}\text{x}\text{c}}-{E}_{\text{p}\text{h}}\) correspond to our theoretical calculation and to previous reports [11, 29]: 116 cm−1 \(\widehat{=}\) 14.4 meV (A2g), 218 cm− 1 \(\widehat{=}\) 27.0 meV (B21g), and 243 cm− 1 \(\widehat{=}\) 30.1 meV (A3g). They also exhibit distinct polarization properties. The A2g and A3g modes are detected in the \({ϵ}\left|\right|{a}\) polarization configuration, while the B21g peak is only allowed for \({ϵ}\left|\right|{b}\). The respective angular dependences are shown in Fig. 2(d). They indicate that the polarization axes of the A2g and A3g phonon modes are oriented along the armchair direction like the polarized X emission. In contrast to this, the polarization axis of the B21g mode is tilted by 90° so that it is parallel to the zigzag crystallographic direction; its intensity is proportional to \({\text{sin}}^{2}\varphi\).

In what follows, we study the detection of Raman forbidden (dark) phonon modes for resonantly exciting the \({\Gamma }\)-exciton. Figure 3(a) shows the Raman spectra as function of the excitation energy. The Stokes scattering spectra were measured at 7 K in the range from \({\Delta }E={E}_{\text{e}\text{x}\text{c}}-E=\) 11 to 89 meV and the incident laser light (propagating along the \(c\) axis) was polarized along the armchair direction of the GeS flake (\({{ϵ}}_{\text{e}\text{x}\text{c}}\left|\right|{a}\)). The excitation energy was tuned – on the one hand – from 1.874 to 1.795 eV corresponding to energies from 98 to 19 meV above the neutral exciton X, as indicated in Fig. 3. Due to the limited emission range of the laser source (DCM-based dye laser), the exciton resonance at 1.776 eV could not be addressed directly. Nevertheless, we excited the GeS flake – on the other hand – at 1.736 and 1.748 eV, *i.e.*, 40 and 28 meV below the exciton resonance, respectively. At these quasi-resonant excitation conditions, we are able to identify in the Raman scattering spectra 18 lines, among which 14 lines have not been observed hitherto. They are labelled by p1 to p6, g1, g2, and d1 to d6. Their experimentally evaluated frequencies \({\omega }_{\text{e}\text{x}\text{p}}\) are given in Table 1. In comparison with the theoretically assigned phonon modes, the observed lines can be attributed to acoustic phonon modes at non-\({\Gamma }\) symmetry points (p1, p2, g1), IR active (Raman forbidden) phonon modes (p3, p4, p5, and p6), and second-order scattering processes including Raman and IR active phonons (g2, d1 – d6).

Interestingly, the intensities of – in particular – the Raman forbidden lines are significantly enhanced. This resonant behavior is also outlined in Fig. 3(b) which contains the resonance profiles of the different Raman lines, namely the intensities of the Raman lines as function of the excitation energy. The Raman forbidden lines become strongly intensified at about \({E}_{\text{e}\text{x}\text{c}}=\) 1.803 eV (p3), 1.807 eV (p4), 1.809 eV (p5), and 1.817 eV (p6). The absolute error in determining the maxima amounts to \(\pm 3\) meV. Comparing these values with the \({\Gamma }\)-exciton energy \({E}_{\text{X}}\) and the phonon energies listed in Table 1, it becomes clear that the intensities of the Raman forbidden lines are intensified when the excitation energy \({E}_{\text{e}\text{x}\text{c}}\) is equal to \({E}_{\text{X}}+\hslash {\omega }_{\text{e}\text{x}\text{p}}\). Additionally, energetically scanning the Raman lines through the exciton resonance leads to a drastic increase in the exciton emission, as shown in Fig. 3(c). Accordingly, both the incident as well as scattered photons are in resonance with states in which the exciton is involved (double resonance). This is also the case for the second-order phonon modes d1 to d6. The peaks g1 (2nd order acoustic phonons at Y-point) and g2 (2nd order optical phonons at Z-point) also seem to be enhanced in their intensities; it is not a definite observation due to the limited number of excitation energies. In contrast to that, the Raman active phonon modes at the \({\Gamma }\)-point, *e.g.*, B1g2 and Ag3, do not significantly increase in intensity. The Raman scattering lines all have in common that their spectral positions \({E}_{\text{p}\text{h}}\) and linewidths remain constant.

A further common feature of the phonon modes p1 to p6 as well as g1 and g2 is their optical anisotropy. As depicted in Fig. 4, their integrated Raman intensities are maximum when the electric field vector of the scattered light is oriented along the armchair crystallographic direction (\({ϵ}\left|\right|{a}\)), while the intensities become negligibly small for \({ϵ}\left|\right|{b}\). Hereby, the polarization of the incident light was polarized along the \(a\) axis. This angular dependence agrees with that of the X emission.

We moreover study the temperature dependence of particularly the Raman lines with frequencies below 350 cm− 1. Changing the temperature from 7 to 100 K yields the Raman spectra depicted in Fig. 5(a). The intensities of both the Raman active as well as Raman forbidden lines decrease with increasing temperature. For obtaining details about their thermal behavior, their integral intensities \({I}_{\text{p}\text{h}}\) are shown as a function of the inverse temperature 1/*T* in Fig. 5(b). These Arrhenius plots allow for determining the deactivation energies of the scattering processes. Accordingly, each intensity dependence is fitted by \({I}_{\text{p}\text{h}}\propto \text{e}\text{x}\text{p}(-{E}_{\text{d}}/{k}_{\text{B}}T)\) with the thermal deactivation energy \({E}_{\text{d}}\) and the Boltzmann constant \({k}_{\text{B}}\). The Raman forbidden lines p5 and p6 possess the highest deactivation energies of about (\(5.5\pm 1.0\)) meV, while the phonon lines g1 and g2 (at Y- and Z-symmetry points) and Ag2 (at \({\Gamma }\)-point, but Raman active) are quenched at lower thermal energies. For instance, the g1 mode has the smallest thermal deactivation energy of about 1.2 meV. The Arrhenius plots are reproducible for different GeS flakes; the values of \({E}_{\text{d}}\) vary slightly (\(\pm 0.5\) meV) which may be related to the efficiency and spectral dispersion of the X emission. Furthermore, the exciton energy is red-shifted when the temperature is increased; see reflectivity spectra in Fig. 5(c). In average, the X energy is thermally decreased by 1.8 meV per 10 K, for temperatures varying from 10 to 120 K. Thus, the difference between \({E}_{\text{X}}\) and \({E}_{\text{e}\text{x}\text{c}}\) is not constant, but it becomes enhanced. Consequently, \({E}_{\text{e}\text{x}\text{c}}\) set at 1.795 eV meets different excitation conditions at low *T* (quasi-resonance) and high *T* (weak quasi-resonance). This thermally induced shift out of the resonance leads to an additional shrinkage of the Raman line intensities and in turn to smaller values of \({E}_{\text{d}}\).

**Mechanism of the resonantly enhanced phonon scattering.** Striking features of the IR active \({\Gamma }\)-point phonon modes observed in the Raman spectra are that (i) their intensities are enhanced for (quasi)-resonantly exciting the \({\Gamma }\)-exciton, in particular their resonance profiles are peaked at \({E}_{\text{X}}+\hslash {\omega }_{\text{e}\text{x}\text{p}}\) (incoming resonance), (ii) they result in an enhanced X PL indicating also an outgoing resonance, (iii) the Raman forbidden phonon modes are only detected for copolarized incident and scattered photons (\({{ϵ}}_{\text{e}\text{x}\text{c}}\left|\right|{ϵ}\left|\right|{a}\)), (iv) the linewidths do not show any dispersive behavior with changing excitation energy, (v) the temperature dependences yield deactivation energies of about 5.5 meV for the IR active phonon modes, while the Raman active and non-\({\Gamma }\)-point phonons are drastically quenched by increasing temperature, and (vi) the intensities of the phonon lines increase practically linearly with increasing laser power, see Fig. 5(d). These properties of the Raman forbidden phonon modes observed at practically resonantly addressing the exciton in GeS flakes will allow us to evaluate the scattering mechanism.

In non-resonant Raman scattering, only Raman active phonons are detected, while IR active phonon modes remain optically non-accessible (dark). It is based on the parity selection rule. Activating the IR active phonon modes for Raman scattering requires a non-zero transition dipole momentum. This criterion related to a breakdown of the parity selection rule and, in turn, the optical observation of the inelastic scattering by IR active phonons are realized when the incident photon energy is close to an electronic transition and the incident (\({{ϵ}}_{\text{e}\text{x}\text{c}}\)) and scattered (\({ϵ}\)) photon polarizations are parallel to each other [33]. This scattering does not follow the selection rules imposed on the Raman tensor by the symmetry of the \({\Gamma }\)-point phonons. Different microscopic mechanisms for such a selection rule relaxation have been proposed and will be discussed in the following considering our results for the GeS flakes.

When the scattering volume is in proximity to the sample surface, electric fields due to band bending may enable forbidden optical phonon scattering [34]. In our experiments, the intensities of all phonon lines depend practically linearly on the laser power whose increase mainly corresponds to an enhancement in the number of photogenerated carriers. Their presence would alter surficial electric fields – if they were present – or they would even screen them partially so that a strongly non-linear power dependence would be expected [35, 36]. Alternatively, considering an electron-phonon interaction based on the Fröhlich mechanism, which is inversely proportional to the dielectric constant, an increase in the carrier concentration would suppress the Fröhlich interaction and, in turn, the phonon line intensities with increasing laser power. This is not the case in our experiments.

Another possible mechanism (for non-zero matrix elements of the Fröhlich interaction) is contributed by extrinsic scattering of an exciton bound to an impurity (being the intermediate scattering state) which does not impose restrictions to the scattering wave vector \({q}\) so that it is independent of the scattering geometry. In this case, the maxima of the resonance profiles would be shifted to energies lying below the resonance energy of the \({\Gamma }\)-exciton. This energy difference would correspond to the binding energy of the impurity-bound exciton. Moreover, since the phonon momentum would be not fixed, the optical phonon lines should be dispersively broadened as a function of the excitation energy [37]. However, the widths of the phonon lines (Raman forbidden and allowed) are practically invariant, for applying different excitation energies close to the \({\Gamma }\)-exciton resonance.

The forbidden optical phonon scattering may arise from intraband matrix elements of the Fröhlich electron-phonon interaction in the frame of (a) a third-order process including a LO-phonon-induced intraband scattering process or (b) a fourth-order process including a scattering process of the electron with an optical and acoustic phonon, both for the \({\Gamma }\)-exciton. In general, the longitudinal optical (LO) phonon scattering is contributed by short-range deformation potential and the long-range Fröhlich interaction [38]. The Fröhlich interaction is induced by the electric field created by longitudinal phonons in polar materials. As the electronegativity difference between the constituents Ge and S amounts to 0.6 eV, GeS has a quite strong polarity (polar covalent bonding) [10]. Moreover, the scattering lines are copolarized; thus, we anticipate that the Fröhlich interaction dominates against the deformation potential and that the resonantly activated modes p3 to p6 are LO phonons with energies \(\hslash {{\omega }}_{\text{L}\text{O}}\) (= \(\hslash {{\omega }}_{\text{e}\text{x}\text{p}}\)).

The intrinsic intraband Fröhlich interaction (a) yields a resonance profile with a maximum at \({E}_{\text{X}}+\hslash {{\omega }}_{\text{L}\text{O}}/2\) [39]. However, the resonance profiles of the forbidden Raman scattering lines are most intensive at about \({E}_{\text{X}}+\hslash {{\omega }}_{\text{L}\text{O}}\). This is actual a clear indicator that mechanism (b) plays the dominant role in our experiments on the GeS flakes [39, 40]. In BP the LO resonant Raman scattering is proposed to be mediated by an intraband Fröhlich interaction including the dark and bright excitons whose states are energetically different due to strong spin-orbit splitting [23]. In GeS both excitons at the \({\Gamma }\)-point are practically degenerate [41], so that an intraband transition – in particular with energies of a few tens of meV – between the bright and dark exciton is improbable. Thus, we conclude that the activation of the Raman forbidden phonon modes is mediated by (quasi)-resonantly exciting the \({\Gamma }\)-exciton scattered twice by the electron-LO phonon and electron-acoustic phonon interaction. This mechanism is presented by the Feynman diagram in Fig. 6(a). Instead of an acoustic phonon, an impurity could be involved. Due to the slight variation in the resonance profile maxima, we propose the involvement of an acoustic phonon which shifts the profiles slightly by \(\hslash s\left|{{q}}_{\text{a}\text{c}}\right|\), where \(s\) is the sound velocity of the acoustic (ac) phonon. Since GeS exhibits giant piezoelectricity due to its characteristic puckered symmetry [17], we propose the involvement of a piezoelectric acoustic phonon in the scattering process. Taking into account a shift of \(\le 1\) meV and a scattering vector of \({q}_{\text{a}\text{c}}\approx 1/2a\), the sound velocity of the piezoelectric acoustic phonon in the armchair direction is approximately \(1.3\times {10}^{3}\) m/s. The large momentum transfer which occurs in the scattering event enhances the scattering cross section, despite the high order (4th ) of perturbation theory involved. The large wavevector \({{q}}_{\text{a}\text{c}}\) of the acoustic phonon significantly raises the intraband Fröhlich contribution. Moreover, owing to the relaxation of the phonon wave vector it is likely that a double resonance appears, which also leads to a high Raman scattering efficiency. In our case, the incident photon resonantly excites the \(|\text{X}+\text{L}\text{O}>\) state (incoming resonance), the electron of the exciton is twice scattered so that the bright exciton | \(\text{X}\) > is the intermediate state whose recombination yields the scattered photon (outgoing resonance) and the system goes back into its initial vacuum state | 0 >. The respective scheme is depicted in Fig. 6(b).

To estimate the strength of this Fröhlich interaction we follow the approach described in Ref. 42. We consider the presence of Wannier excitons which is confirmed by Pastorino et al. [43] showing that the wave function of the first bright exciton in GeS is highly delocalized and spreads over several atomic layers. The large spatial extension of the exciton is a further central property of forbidden LO scattering based on Fröhlich interaction and (quasi-)resonant exciton excitation. The scattering intensity is proportional to \({\left({q}_{\text{L}\text{O}}{ a}_{\text{B}}\right)}^{2}\) with the exciton Bohr radius \({a}_{\text{B}}\) [33]. Thus, forbidden phonons are observed only if the exciton Bohr radius is much larger than the lattice constant, which is the case in GeS. Accordingly, the Fröhlich coupling constant, for the electron-optical phonon interaction part, is given by:

$${\alpha }_{\text{e}-\text{L}\text{O}}=\frac{{e}^{2}}{\hslash }\frac{1}{4\pi {\epsilon }_{0}}\left(\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_{\text{s}\text{t}}}\right)\sqrt{\frac{{m}^{*}}{2\hslash {{\omega }}_{\text{L}\text{O}}}}$$

Here, \(e\) is the electron (e) charge, \({\epsilon }_{0}\) the vacuum dielectric constant, \({\epsilon }_{\infty }\) (\({\epsilon }_{\text{s}\text{t}}\)) the optical (static) dielectric constant, and \({m}^{*}\) the effective electron mass. For the bulk values (considering \({ϵ}\left|\right|{a}\)) \({\epsilon }_{\infty }=14.8\), \({\epsilon }_{\text{s}\text{t}}=25.1\) [44], \({m}^{*}=0.22 {m}_{0}\) [45], and the phonon modes whose energies range from 20 to 40 meV, the coupling constant takes the values 0.34 to 0.24, respectively. These values lie within the relatively weak regime characteristic for the formation of large exciton-polarons [46]. The presence of spatially extended exciton-polarons is illustrated by the comparably small deactivation energies evaluated from the temperature dependences. It is also worthwhile to mention that the neutral exciton binding energy in GeS [41] is significantly larger than the thermal deactivation energies. Accordingly, for quasi-resonant excitation of a large, but thermally robust exciton in GeS flakes a fourth-order scattering process is responsible for observing the actually Raman forbidden LO phonon modes p3, p4, p5 and p6 with the symmetries B1u2, B2u2, B3u2 and B1u3.