Plasmonic hot electron induced layer dependent anomalous Fröhlich interaction in InSe

Despite the great promise of InSe for electronic and optoelectronic applications, Fröhlich interaction plays an important role in electrical transport due to the polar nature of it, which can become more significant in reduced dimensionality. Here, we report on how the dimensionality influences the strength and nature of the Fröhlich polaronic effect in InSe with the aid of plasmonic hot electrons injection. Polar optical phonons couple to hot electrons via the Fröhlich interaction in InSe and enable us to monitor them in conventional Raman measurements. We observed that the intensity of these phonon modes initially increases gradually with decreasing layer number and then drops drastically from 7 L to 6 L (transition from quasi-direct to indirect bandgap at room temperature). Additionally, a gradual decrease of intensity of the polar modes with further decreasing layer number is observed due to the increasing indirect bandgap nature of InSe suggesting reduced Fröhlich coupling below this thickness. Electron-phonon coupling plays a central role in a variety of low dimensional structures. The authors unveiled the subtle interaction between optical phonons and charge carriers by applying Raman spectroscopy to plasmonic nanostructures fabricated on atomically thin InSe. These findings provide insights to develop future high-performance nanoelectronic devices.

A mong all 2D semiconductors, InSe (a member of metal monochalcogenides) has emerged as an outstanding candidate for post-silicon electronic devices. It possesses a high electron mobility (~10 3 cm 2 V −1 s −1 ) due to its small effective mass (~0.14 m e ) at room temperature (RT) 1 . Even though 2D black phosphorus has a comparable mobility 2 , it is extremely unstable in environmental conditions 3 , which limits the possibility of practical applications. The large surface-to-volume ratio and the atomic thickness of 2D materials make them ideal for electrostatic control and significant device downscaling for highdensity integration 4 . Moreover, the ultraclean, dangling bond free smooth surfaces of these materials are less sensitive to carrier scattering. They can thus outperform existing silicon devices in the regime of scaling limitation. Therefore, InSe provides a promising playground for studying low dimensional phenomena and for developing high mobility nano-electronics 4,5 .
One of the limiting factors of intrinsic mobility are the phonons, quanta of lattice vibration perturbing the electron potential in a lattice. In polar materials, optical phonons couple strongly with elementary charges due to long-range Coulomb interactions induced by the macroscopic polarization field resulting from the atomic displacement, known as Fröhlich interaction 6 . Therefore, scattering with optical phonons limits the mobility in 2D semiconductors at RT significantly 7,8 . Intriguingly, 2D Fröhlich interaction is markedly different from its bulk analog as reported in a recent theoretical work 9 . The Fröhlich interaction in 2D materials diverges from the 3D one at zero phonon-momentum limit creating van Hove singularities, where it transforms into a very large finite value compared to the standard electron-phonon coupling (EPC). As a consequence, due to the reduced dielectric screening, the enhanced macroscopic field created by the phonon-assisted polarization density may lead to a stronger EPC in ultrathin materials resulting in a layer-dependent Fröhlich potential and hence device performance. Therefore, the question of how the dimensionality influences the Fröhlich interaction in InSe has great significance and importance for both fundamental understanding and device application. Very recently, Ma et al. 7 and Li et al. 10 laid the theoretical ground for understanding the 2D model of Fröhlich potential and layer-dependent Fröhlich interaction in InSe by means of intrinsic carrier mobility. To the best of our knowledge, no experimental investigation of layerdependent Fröhlich interaction in 2D semiconductors, in particular for InSe, has been reported until now.
To accomplish this goal, a systematic investigation of layerdependent Fröhlich interaction in InSe via the injection of plasmonic hot electrons was performed. Plasmonic nanostructures have outstanding light-trapping and electromagnetic fieldconfining properties and can generate hot electrons with energies up to 4 eV when excited at localized surface plasmon resonance (LSPR) 11 . These highly energetic electrons can escape from the plasmonic structures and be collected by the conduction band (CB) of adjacent semiconductors depending on the energy barrier at the interface. The small Schottky barrier (0.1-0.9 eV) of InSe with noble metals is, therefore, an ideal system for the collection of hot electrons from the plasmonic structures 12,13 . In polar semiconductors, the electric field carried by the free carrier plasmons (here the hot electrons collected by the CB of InSe) interact with the long-range macroscopic field of the longitudinal optical phonons (LO) breaking the Raman selection rules 14 . Therefore, Raman spectroscopy can directly measure coupled modes 15 enabling the investigation of the polar LO phonons via plasmonic hot electrons doping as a function of InSe layer number.
InSe studied in this work belongs to the commercially available rhombohedral γ-type crystal structure with an ABC stacking sequence and no centrosymmetry 16 . The unit cell consists of three tetra-atomic layers, in which each tetra-atomic layer (Se-In-In-Se) creates the monolayer of InSe 17 . The thickness of each monolayer is 0.8 nm. Bulk InSe is a direct bandgap (E g ) semiconductor 18 . However, when thinned down to below 20 layer (20 L) the valence band maxima forms a 'Mexican hat' like structure centered at the Γ point with the energy difference between the two maxima and the center minimum smaller than the thermal activation energy at RT 19 . Therefore, it remains a quasi-direct bandgap semiconductor down to a certain thickness at RT. Like other layered semiconductors, its bandgap can be tuned upward by as large as~1.0 eV from 1.2 eV in bulk to 2.1 eV in monolayer 18 . One notable characteristic of InSe is that it has large optical anisotropy near E g : the electronic transitions associated with polarization of incident light parallel to the c axis (E || c) are two order of magnitude larger than that perpendicular to c axis (E⊥c) 18,20 . In contrary the electronic transition at around 2.42 eV (E 1 electronic transition) is optically allowed in the normal incident geometry (E⊥c). The E 1 gap also exhibits a layer dependence varying from 2.42 to 3.04 eV from bulk to monolayer, respectively 18 .
Being a member of the R3m space group with four atoms per monolayer, there are twelve normal modes of vibrations in γ-InSe. Group theory predicts 4A 1 and 4E modes at the Γ point for γ-InSe and apart from the acoustic A 1 and E ones all other modes are Raman and infrared active, but with quite different intensities 21 . Among them, two modes with A 1 and E symmetry are strongly polar and therefore lead to strong infrared absorption. However, at resonant Raman conditions, when the excitation energy matches the E 1 electronic transition, the q-dependent A 1 (LO) mode around 200 cm −1 is enhanced dramatically 22 . When performing Raman measurements far below the E 1 transition in a plasmonic environment, in which the excitation energy approaches the LSPR (surface enhanced Raman scattering (SERS) conditions), we were able to observe three phenomena. These are: (i) appearance of two polar modes A 1 (LO) and E (LO) around 200 and 210 cm −1 for InSe, (ii) the intensity of both polar modes are layer dependent with the E (LO) mode having a stronger sensitivity to layer thickness, and (iii) a drastic intensity reduction of the polar modes for the InSe layer number decreasing from 7 to 6 L, which correlates directly to the transition point from quasi-direct to indirect bandgap at RT 19,23 . Using the combination of different experiments on a large number of samples (>20), we confirm that the appearance of both polar LO modes is due to the Fröhlich coupling with the injected plasmonic hot electrons. The experimental evidence was further confirmed by finite element method (FEM) simulations of the plasmonic coupling between metal nanostructures and InSe using COMSOL.
Our work provides a clear picture of layer-dependent Fröhlich interaction induced EPC in InSe and offers great prospect in the design and optimization of next generation high mobility nanoelectronics/optoelectronics. It also opens up the possibility of applications in photovoltaics 24 and photocatalysis 25 .

Results and discussion
Resonant Raman induced Fröhlich interaction in InSe. Figure 1a presents a sample sketch with metal nanotriangles (NTs) on top of InSe flakes. Here, we used Ag, Au, and Al for the coupling between InSe and the plasmon of the metal NTs. Hexagonal arrays of the metal NTs have LSPR in the visible range, which can be tuned via changing the material, size, shape, or dielectric environment 26,27 .
A typical scanning electron microscope (SEM) image of the sample after nanosphere lithography (NSL) using Ag is shown in Fig. 1b. As can be seen, Ag formed NTs homogeneously covering InSe without damaging the flake. Figure 1c-f are the optical images and atomic force microscope (AFM) topographies of one of the InSe samples with different layer thicknesses on a 300 nm SiO 2 /Si substrate. Like other layered semiconductors, InSe has a well-known layer-dependent optical bandgap 18,28 . We thus determined the InSe layer number using the photoluminescence (PL) peak position. The details of thickness determination via PL peak position are presented in the Supplementary note 1. The PL spectrum of 7 L InSe is shown in the inset of Fig. 1e. Using Voigt fitting the PL peak is determined to be centered at (1.35 ± 0.01) eV corresponding to 7 L InSe in good agreement with the literature 28 . Before turning our attention to the plasmonic hot electron induced Fröhlich interaction, we investigated nonresonant and near-resonant Raman scattering spectra of InSe. This helps in the evaluation of polar modes in InSe since, at resonance excitation, Raman scattering is dominated by the exciton-LO phonon Fröhlich interaction 22 . Figure 1g displays both nonresonant and resonant Raman spectra of 7 L InSe measured at 1.94 and 2.41 eV excitation, respectively. In the nonresonant Raman spectrum, we observed three prominent first order Raman modes around 114, 178, and 225 cm −1 assigned to A 1 1g , E 2g , and A 2 1g modes, respectively, (see Fig. 1h) according to the irreducible representation of the symmetry group C 5 3v (γ-InSe) 29 . There is another in-plane E 2g mode around 40 cm −1 for InSe, which is inaccessible in our experiment. Additionally, we can also observe a very weak contribution of the polar A 1 (LO) around 200 cm −1 , while the increasing background around 300 cm −1 is the contribution from Si 2nd order phonon scattering. Interestingly, under nearresonant Raman condition when excited by the 2.41 eV laser, the energy of which is close to the E 1 gap (around 2.42 eV) of InSe, together with the nonpolar Raman modes, we observed a dramatic enhancement of the polar modes (asymmetric feature around 200 cm −1 ). The peak centered at 200 cm −1 can be assigned to A 1 (LO) and the shoulder on the higher frequency side around 210 cm −1 to E (LO), both of them are more active in the infrared absorption than in the ordinary Raman spectra 21,30,31 . However, under near-resonant Raman excitation q-dependent scattering dominates and dictates the Raman symmetry selection rules 32 . Therefore, infrared active polar phonons are enhanced in the Raman spectra of InSe with strong A 1 (LO) and weaker E (LO) contributions in backscattering geometry.
Plasmonic hot electron induced Fröhlich interaction in InSe. Figure 1i shows the Raman spectra of 7 L InSe (E g = 1.35 eV) without and with Ag NTs. While the Raman spectra of InSe without Ag NTs show first order Raman modes at 2.33 and 1.94 eV excitations, no Raman modes are observed under 1.58 eV excitation. This could be due to the excitation energy, which is far away from the E 1 and the Raman scattering is less efficient near weakly allowed E g . In contrast, the Raman spectra of InSe/Ag NTs system exhibit significant modifications. First, we observed the strongest SERS enhancement at 1.58 eV excitation. This is unexpected taking our recent work into account since we showed that hexagonally packed Ag NTs of the same geometry on a SiO 2 / Si substrate have a LSPR around 1.9 eV and are thus resonant with 1.94 eV excitation 26 . However, the present investigation clearly shows that the 1.58 eV excitation is closer to resonance if not at resonance with LSPR of Ag NTs suggesting a shift of the LSPR to lower energies due to the change of dielectric environment in the presence of few layer InSe at the bottom 27 . Thanks to the resonant SERS effect, together with first order Raman modes we can now observe the polar modes in InSe at 1.58 eV excitation. The broad feature between 150 and 190 cm −1 is the combination Polystyrene spheres of 400 nm size were used for nanosphere lithography (NSL). Ag, Au, and Al were used for nanotriangles (NTs) formation. b A typical scanning electron microscope (SEM) image of the InSe sample after Ag NT deposition. Silicon with native oxide was used as substrate for SEM to avoid charging effects. c, d Optical images and e, f atomic force microscope (AFM) topographies of one typical InSe sample with different thicknesses before and after NSL with Ag, respectively. Inset of (e) displays the photoluminescence spectra of 7 layer (7 L) InSe with a Voigt fitting. The red dashed lines in (f) marks the area of the thin flake as shown in the optical images. g Resonant and nonresonant Raman spectra acquired with 2.41 and 1.94 eV, respectively, of 7 L InSe. Acquisition time for the Raman measurements were 10 s × 2 and 30 s × 10, respectively, and the laser power in both measurements were~100 μW. h Elongation patterns of the phonon vibrations visible in the Raman spectra. i Raman spectra of 7 L InSe before and after NSL with Ag acquired with 1.58, 1.94, and 2.33 eV excitations. The broad spectral feature between 150 and 190 cm −1 is due to the presence of E 2g , and contribution of In 2 O 3 (green dashed line and asterisk). The gray asterisk indicates a feature assigned to amorphous Se. Acquisition time for the measurements were 5 s × 2 and the laser power was 10 μW. of in-plane E 2g mode of InSe and the contribution from In 2 O 3 (a byproduct of oxidation in ultrathin InSe) 33,34 .
To understand the role of InSe on the LSPR of metal NTs, we performed FEM simulations using COMSOL multiphysics 5.6 in 3D platform. Figure 2a presents the model used to simulate the experimental geometry studied in this work. Details of the 3D model can be found in the methods section and Supplementary note 2. The transmittance spectra of the three metal NTs (Au, Ag, Al) on 7 L of InSe computed in the spectral range from 360 nm (3.44 eV) to 1200 nm (1.03 eV) are presented in Supplementary note 2 (Fig. S2a). The distribution of the electric field intensity enhancement M around the Ag and Al NTs calculated at 1.58 and 1.94 eV excitations, respectively, are shown in Fig. 2b. The M distribution for Au NTs at 1.58 eV excitation can be found in Supplementary note 2 (Fig. S2b). As expected, the corners of the NTs generate the strongest electric field enhancement in agreement to previous works 26,35 . The absolute values of the calculated maximum M for Ag, Au, and Al NTs at three different wavelengths are displayed in Fig. 2c. From Fig. 2c and Supplementary note 2 (Fig. S2a), we can see that Au and Ag NTs show strongest enhancement under 1.58 eV excitation compared to the other two energies. Therefore, excitation with 1.58 eV shall produce a stronger plasmonic field and generate highly energetic hot electrons in both metal NTs. This result thus confirms that the strongest enhancement in the Raman spectra of InSe with Ag or Au NTs should appear at 1.58 eV excitation (Raman spectrum of InSe/Au NTs at 1.58 eV excitation is presented in Supplementary note 2 (Fig. S2c)). Under this excitation condition the two polar modes (A 1 (LO) and E (LO)) are observed in the Raman spectra.
From the simulation results shown in Fig. 2c and Supplementary note 2 it is clear that the LSPR of Al NTs shifts towards 638 nm (1.94 eV) and one should expect the strongest plasmonic enhancement effect at 1.94 eV excitation. Figure 2d presents Raman spectra of InSe with Al NTs acquired at all three excitation wavelengths. Indeed we observe the strongest plasmonic enhancement at 1.94 eV excitation in terms of the appearance of the two polar modes. No Raman features are visible at 1.58 eV since this excitation energy is nonresonant for both the InSe and the LSPR of Al NTs. Please note that the Raman spectra of InSe at 2.33 eV excitation is the combination of plasmonic enhancement and near-resonant Raman effect as discussed above.
One notable difference between the near-resonant Raman spectra in Fig. 1g and the SERS spectra at 1.58 eV excitation in Fig. 1i is the relative intensity ratio between A 1 (LO) and E (LO). In the case of near-resonant Raman excitation, the A 1 (LO) mode appears much stronger compared to the E (LO) one in agreement with literature 21,36,37 . On the contrary, the E (LO) mode is more enhanced than the A 1 (LO) mode in resonant SERS spectra, in which the probability of generated hot electron in NTs collected by InSe is the maximum, consistently in the present study. This is likely due to hot carrier induced EPC of the E (LO) polar mode being much stronger in InSe as suggested in a recent DFT investigation 38 . Additionally, our simulation shows that the stronger in-plane plasmonic field component at 1.58 eV excitation (see Supplementary note 2 (Fig. S2d)) couples to the in-plane E (LO) mode better than A 1 (LO) mode resulting in enhanced relative intensity.
The Raman spectra presented in Figs. 1i and 2d clearly indicate that the plasmonic coupling between InSe and metal NTs is the sole contributor to the observation of the polar modes in the Raman spectrum excited by 1.58 and 1.94 eV, which would be nonresonant without NTs. There are two possible plasmonic phenomena, which can contribute to the observation of the polar modes: (i) the local field strongly confined at the corners of NTs as shown in Fig. 2b and Supplementary note 2 (Fig. S2d) and (ii) injection of the plasmonic hot electrons. A similar behavior was also observed for the case of a monolayer MoS 2 /Au nanodisc system in our previous work 39 . The strongly confined plasmonic field in a monolayer MoS 2 sandwiched by a nanodisc and a sharp metallic tip generated hot electrons, which excite an infrared active A 2u mode under 1.58 eV excitation energy, far below the bandgap of the monolayer MoS 2 .
To test our hypothesis, we partially encapsulated InSe by a few layers of hBN (~10 L) and deposited Ag NTs using the same procedure. The plasmonic field decays exponentially through the hBN capping layer with a decay constant (see Supplementary note 3) and hBN blocks InSe from being doped by plasmonic hot electrons. Therefore, we are able to determine the influence of the two effects mentioned above on hBN capped InSe directly.   Fig. 1i. However, the intensities of the nonpolar first order Raman modes display weaker enhancement for hBN encapsulated InSe with Ag NTs than without Ag NTs, since we did not observe any Raman features in the absence of Ag NTs at 1.58 eV excitation as demonstrated in Fig. 1i. The weak enhancement of the nonpolar Raman modes can be due to the screening of the plasmonic field by the hBN interlayer. The effect of this weak plasmonic coupling can also be observed through the enhanced PL (by a factor of 5.4) from hBN encapsulated InSe after Ag NTs deposition (see Supplementary note 3), which also leads to an increased background in the red spectrum in Fig. 3a.
For bare InSe with Ag NTs, on the other hand, we did not observe any PL. This is likely due to the efficient exciton dissociation known to occur at the semiconductor/metal interface 37 . However, the hBN capping blocks the exciton dissociation from InSe to the Ag NTs. Therefore, we can observe the PL enhancement via plasmonic coupling. The Raman spectra taken from the hBN encapsulated InSe with Ag NTs show negligible contribution from the polar modes, especially the E (LO) mode. This is a stark contrast to the nonpolar Raman modes in hBN capped InSe. The appearance of a much weaker A 1 (LO) mode in hBN capped InSe indicates that the contribution of local plasmonic field gradient induced selection rules (as discussed above) may aid to the weak appearance of A 1 (LO) mode. On the other hand, the vanishing E (LO) mode suggests that the plasmonic hot electron doping plays the critical role in activating the polar modes in nonresonant Raman spectra for InSe. Figure 3b, c depict approximated band alignment diagrams for the heterostructures shown in Fig. 3a indicating the injection of the plasmonic hot electrons mechanism in InSe. The values needed to construct the band alignment were taken from the literature 13,40,41 . After excitation at LSPR, the plasmon can decay radiatively or non-radiatively by transferring the energy to the conduction electrons in the metal nanostructures. This process creates highly energetic hot electrons, which can escape from the metal and be collected by a semiconductor in contact with the plasmonic structures. The energy transferred to the hot electrons from the noble metal nanostructures at LSPR excitation is higher than 1 eV and can reach up to 4 eV 11 . Therefore, hot electrons generated in Ag NTs can easily overcome the barrier height φ B (~0.4 eV) and dope InSe. When encapsulated by hBN, this wide bandgap insulator creates a blocking layer for the hot electrons to be collected by InSe (Fig. 3c). As a result, polar modes in InSe are weak in the red spectrum of Fig. 3a.
One of the critical sources of the Fröhlich interaction in polar semiconductors is the intraband EPC stemming from the scattering of charges within the same valence or conduction band (CB) by LO phonons. Injection of plasmonic hot electrons in InSe meets such criteria since electrons escaping the metal nanostructures are collected by the CB of InSe. These in turn interact with the long-range macroscopic polarization density associated with either the creation or annihilation of LO phonons in InSe 7 . The charge transport driven by an electric field through InSe can also be influenced by such a macroscopic field, which consequently reduces the mobility.
Even though the electronic transition at the bandgap in InSe is weak at normal incidence, it can still induce resonant Raman scattering in a strong plasmonic field. Hence, the strong intensity of the polar phonons observed in 7 L InSe may also be activated via this resonance phenomenon. To examine this scenario, we prepared a 1 L InSe with Ag NTs sample. The PL spectra of the flake confirming the layer thickness is presented in Supplementary note 4. The bandgap of 1 L InSe at RT is 2.1 eV, i.e. 1.58 eV excitation is well below the electronic transition. Therefore, if the appearance of the polar phonons would only stem from the coupling between these phonons and the weakly allowed E g transition in a strong plasmonic field, we should not observe them for 1 L InSe. Instead, the polar phonons in 1 L InSe when covered with Ag NTs are clearly observed (Fig. 3d).
Furthermore, by conducting a temperature-dependent Raman investigation on two selected thicknesses (4 L: E g ≥ 1.58 eV and 7 L: E g < 1.58 eV at RT) of InSe with Ag NTs at 1.58 eV excitation, it becomes clear that with decreasing temperature the bandgap of 4 L (7 L) InSe moves away (towards) from the excitation energy of 1.58 eV, i.e. the resonance condition is tuned correspondingly. The result for 4 L InSe is presented in Fig. 3e and for 7 L InSe in Supplementary note 5. As depicted in Fig. 3e, the Raman signatures of polar E (LO) and nonpolar A 2 1g phonons in 4 L InSe are visible in the spectra and their intensities remain constant throughout the temperature range (233-313) K. Due to strong increasing Rayleigh background the A 1 1g mode around 114 cm −1 could not be observed under ×50, 0.45 NA objective used for the temperature-dependent study. The constant intensity of the E (LO) mode over the whole temperature range indicates that the activation of polar phonons in InSe at 1.58 eV excitation does not depend on the E g transition.
On the other hand, the temperature-dependent Raman spectra of 7 L InSe/Ag NTs at 1.58 eV excitation presented in Supplementary note 5 reveal that the intensity of polar phonons is enhanced when the temperature-dependent bandgap approaches the excitation energy at lower temperature. This is because with decreasing temperature the bandgap of 7 L InSe approaches the excitation energy of 1.58 eV meaning resonance with E g is established. This phenomenon suggests that the fundamental bandgap E g plays a role in enhancing the polar LO modes. However, as we showed in this work hot electron injection is the prerequisite for observing the polar LO modes, thus Fröhlich interaction in InSe takes place.
Layer dependent evolution of Fröhlich interaction in InSe. Figure 4 depicts the layer dependent Raman spectra of InSe with Ag NTs with corresponding PL spectra acquired prior to NTs deposition. As can be seen, the intensity of the polar modes (especially the E (LO) mode) gradually increases from bulk to 7 L of InSe, where it reaches a maximum. This is likely due to the resonant condition since, with decreasing layer number the bandgap of InSe approaches the excitation energy of 1.58 eV. After that, there is a drastic drop of intensity of the E (LO) mode at 6 L InSe, which then continues to decrease gradually with further decreasing layer number. The corresponding PL spectra in Fig. 4b clearly reveal that the transition from direct to indirect bandgap at RT also occurs from 7 to 6 L InSe. Details of the layer dependent bandgap evolution in InSe are discussed in Supplementary note 1.
In a polar semiconductor, free carrier plasmons couple to LO phonons via the interaction between the macroscopic electric field associated with LO phonons and the electric field induced by the free carriers. The strength of this coupling together with several other factors depends on the nature of the bandgap since conduction-electron plasmons couple strongly to the LO phonon in a direct bandgap semiconductor due to the formation of an electron gas with small effective mass and large dielectric constant 15 . According to recent DFT investigations of layer dependent optoelectronic properties of InSe, the electron effective mass in the CB minima also increases with decreasing layer number due to strong interlayer coupling associated with the P z hybridized orbitals of both In and Se atoms 10,42 . Therefore, one can expect a reduced electron-LO phonon coupling in thinner flakes of InSe. Even though this hypothesis could explain the gradual decrease of the Fröhlich interaction below 6 L InSe, the sudden drop of intensity of the polar modes from 7 to 6 L (the transition point of direct to indirect bandgap) suggests that the nature of the bandgap plays a crucial role as well. To understand the role of electron effective mass, change of dielectric environment, and nature of bandgap on layer-dependent Fröhlich coupling in InSe, more detailed theoretical studies are needed, which are beyond the scope of this work.
Note that the bandgap is increasing with decreasing layer number in InSe, which also increases the barrier height at the InSe/Ag interface. This may also influence the hot electron injection probability. However, considering the fact that the hot electrons can gain large energy in Ag nanostructures 11 , a barrier height of <1 eV to InSe can easily be overcome. Additionally, the LSPR of Ag NTs shall move to higher energy with decreasing thickness of InSe, especially for very thin flakes, resulting in different coupling between InSe and Ag NTs.
To understand how the LSPR compares at different thicknesses we calculated the local electric field enhancement due to Ag NTs for 2 nm InSe (representing 2 L InSe) at 1.58 eV excitation energy. The simulation results are presented in Supplementary note 6. The local electric field enhancement of Ag NTs on 2 L InSe and 7 L InSe are comparable, i.e. the sharp intensity decrease of polar phonons at ultrathin flakes cannot be explained by a change of the LSPR of Ag NTs. Therefore, the decreasing intensity of the polar phonons below 6 L InSe is due to the reduced Fröhlich coupling as is clearly indicated in Fig. 4a. This is a very important result and critical since the Fröhlich interaction strongly suppresses the mobility of the charge carriers thus reducing any device performance. The reported sequence of procedure, experimental results, simulations and cross referenced interpretations provide a clear picture of the layer-dependent Fröhlich interaction mechanism occurring in InSe, which is relevant in future device applications of this promising 2D material. We also performed laser power dependent Raman measurement on bare InSe on SiO 2 substrate under 1.58 eV excitation to investigate if it is possible to observe the polar modes with increasing laser power. The Raman results are presented in Supplementary note 7. As can be seen with increasing laser power we are able to monitor the first order Raman modes in InSe. However, no polar modes were visible. This could be due to the strong optical anisotropy at the band edge of E g at normal incidence in InSe resulting in less efficient electron pumping to the CB.

Conclusion
We studied the Fröhlich interaction in InSe by monitoring the strength of polar LO modes induced by injected plasmonic hot electrons. We deposited three metal (Al, Ag, and Au) nanostructures on InSe and excited the heterostructure system near LSPR. The laser lines (1.58 and 1.94 eV) used in this work are far away from the electronic transition energy at E 1 , i.e. far away from the resonant-Raman condition for InSe. We observed the strong appearance of polar LO modes of InSe in resonant SERS condition, which are otherwise very weak in ordinary Raman conditions. The combination of the strong local field at the corners of the NTs and the hot electrons generated due to near LSPR excitation activates and enhances these polar phonon modes enabling their observation in the Raman measurements. We performed FEM simulations to calculate the plasmonic coupling between InSe and metal NTs. We observed an excellent agreement between the simulation and experiments showing that the polar LO modes are only observed when the excitation energy approaches the LSPR (the condition for the largest concentration of hot electrons excited in the metal nanostructures). To elucidate our findings further, we conducted a series of experiments including a repetition of the SERS experiments on hBN encapsulated InSe, which confirmed that the plasmonic hot electron doping is the origin of the polar modes appearing in the spectra. Finally, we investigated the layer-dependent Fröhlich interaction in InSe. We found that intensity of polar modes initially increases with decreasing layer number indicating increasing Fröhlich coupling. The sudden decrease of intensity of the polar modes from 7 to 6 L coincides with the transition from direct to indirect bandgap at RT. The polar mode intensities gradually decrease with further decreasing layer number. The reduction of the intensity of the polar modes suggests that the Fröhlich coupling is smaller for thinner flakes below 7 L due to the indirect bandgap nature. Our results provide an experimental picture of layerdependent Fröhlich coupling in InSe, which is vital for any potential device application of this promising material.

Methods
Sample preparation. The InSe material used in this work is undoped and was purchased from 2D Semiconductors, USA. Thickness dependent InSe samples were prepared using a PDMS assisted deterministic dry transfer method onto SiO 2 (300 nm)/silicon substrates 43 . After transferring the flakes, self-assembled monolayers of polystyrene spheres of 400 nm size were used for NSL creating a hexagonally packed metal (Ag, Au, Al) arrays of NTs with~120 nm side length and~50 nm height on the sample surface. Details of the deposition technique can be found elsewhere 26 . Prior to the dry transfer, PL spectra were acquired for each flake for thickness determination.
PL and Raman measurements. A Horiba Xplora plus system equipped with an electron multiplying charge couple detector (EMCCD) was used for both PL and Raman measurements at normal incidence. PL measurements were carried out using a 532 nm (2.33 eV) excitation. Raman spectra of InSe were acquired in backscattering geometry using 532 nm (2.33 eV), 638 nm (1.94 eV), and 785 nm (1.58 eV) excitations and a 1200 l/mm grating was used to disperse the Raman signals onto the EMCCD. A ×100, 0.9 NA objective was used to excite and collect the signals for both PL and Raman measurements. The laser powers of 100 and 10 µW were employed at the sample surface for PL and Raman measurements, respectively. The spectral acquisition times were different for different measurement sets and stated in the respective result sections.
Resonant Raman spectra of multilayer InSe at E 1 transition (2.42 eV) were acquired under 514.7 nm (2.41 eV) excitation using a Horiba LabRam HR800, a symphony CCD detector, and 600 l/mm grating. A ×100, 0.9 NA objective was used to excite and collect the signal and the laser power was 100 µW at the sample surface.
Temperature-dependent Raman measurement were carried out using a liquid nitrogen cooled Linkam stage and 1.58 eV laser excitation in the Xplora plus system with the aid of a ×50, 0.45 NA objective.
Finite element method simulation. FEM simulations of InSe/metal NTs were performed using COMSOL multiphysics 5.6 in 3D platform in the wavelength domain. The dimension of the NTs were set according to the experimental conditions (side length: 120 nm, height: 50 nm). In order to avoid a singularity problem and to have an optimal approach to real conditions, edges and corner of the NTs were rounded with a radius of curvature of 5 nm. The 7 L InSe on 300 nm SiO 2 is modeled as a 7 nm thick InSe layer. The optical constants of InSe 44 , Ag 45 , Au 45 , and Al 46 were taken from the literature.
SEM and AFM measurements. SEM images were taken using an SEM (FEI Nova NanoSEM 200) with an accelerating voltage of 10 kV, and 5 mm working distance in immersion mode (high-resolution mode). AFM topography images were acquired using a AIST-NT scanning probe microscope in intermittent contact mode. Commercially available Si cantilevers were used for the AFM measurements.

Data availability
All data generated and analyzed during this investigation are available from the corresponding author upon reasonable request.