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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.


Introduction
Among all 2D semiconductors, InSe (a member of metal monochalcogenides, MMC) 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 me) at room temperature (RT) 1 .
Even though 2D black phosphorus (BP) 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 high-density 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 nanoelectronics 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 analogue 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 ultra-thin 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 layer-dependent Fröhlich interaction in 2D semiconductors, in particular for InSe, has been reported until now.
To accomplish this goal, a systematic investigation of layer-dependent Fröhlich interaction in InSe via plasmonic hot electron doping was performed. Plasmonic nanostructures have outstanding light-trapping and electromagnetic field-confining 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  InSe studied in this work belongs to the commercially available rhombohedral γ-type crystal structure with a 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 (Eg) semiconductor 18 . However, when thinned down to below 20 L the valence band (VB) maxima forms a 'Mexican hat' like structure centered at the point with the energy difference between the two maxima and the center minimum smaller than the thermal activation energy at room temperature (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 Eg : the electronic transitions associated with polarization of incident light parallel to the c axis ( ∥ ) are two order of magnitude larger than that perpendicular to c axis ( ⊥ ) 18 . In contrary the electronic transition at around 2.42 eV (E1 electronic transition) is optically allowed in the normal incident geometry ( ⊥ ). The E1 gap also exhibits a layer dependence varying from 2.42 eV to 3.04 eV from bulk to monolayer, respectively 18 .
Being a member of the 3 space group with four atoms per monolayer, there are twelve normal modes of vibrations in γ-InSe. Group theory predicts 4A1 and 4E modes at the point for γ-InSe and apart from the acoustic A1 and E ones all other modes are Raman and infrared active, but with quite very different intensities 20 . Among them, two modes with A1 and E symmetry are strongly polar and therefore lead to strong infrared absorption. However, at resonant Raman conditions, when the excitation energy matches the E1 electronic transition, the q dependent A1 (LO) mode around 200 cm -1 is enhanced dramatically 21 . When performing Raman measurements far below the E1 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 A1 (LO) and E (LO) around 200 cm -1 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 7L to 6L, which correlates directly to the transition point from quasi-direct to indirect bandgap at RT 19,22 . 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 interaction via plasmonic hot electron doping. The experimental evidence was further confirmed by finite element method simulations of the plasmonic coupling between metal nanostructures and InSe using COMSOL.
Our work provides the first 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 nano-electronics/optoelectronics. It also opens a new prospective for very interesting applications like photovoltaics 23 and photocatalysis 24 .

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 SiO2 (300 nm)/silicon substrates 25 . After transferring the flakes, selfassembled monolayers of polystyrene spheres (PS) of 400 nm size were used for nanosphere lithography (NSL) creating a hexagonally packed metal (Ag, Au, Al) arrays of nanotriangles (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, photoluminescence (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 x, 0.9 NA objective was used to excite and collect the signals for both PL and Raman measurements. The laser powers of 100 µW 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 below.
Resonant Raman spectra of multilayer InSe at E1 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 x, 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 x, 0.45 NA objective.

Finite Element Method Simulation
Finite element method (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 7L InSe on 300 nm SiO2 is modelled as a 7 nm thick InSe layer. The optical constant of InSe 27 , Ag 28 , Au 28 , and Al 29 were taken from the literature.

SEM and AFM measurements
SEM images were taken using a scanning electron microscope (FEI Nova NanoSEM 200) with an accelerating voltage of 10 kV, and 5 mm working distance in immersion mode (highresolution 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. which can be tuned via changing the material, size, shape, or dielectric environment 26,30 .

Resonant Raman induced Fröhlich interaction in InSe
A typical SEM image of the sample after NSL using Ag is shown in Figure 1b. As can be seen, Ag formed nanotriangles homogeneously covering InSe without damaging the flake.
x y z dominates and dictates the Raman symmetry selection rules 35 . Therefore, infrared active polar phonons are enhanced in the Raman spectra of InSe with strong A1 (LO) and weaker E (LO) contributions in backscattering geometry. 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.

Plasmonic hot electron induced Fröhlich interaction in InSe
One notable difference between the near-resonant Raman spectra in Figure 1g and the SERS spectra at 1.58 eV excitation in Figure 1i is the relative intensity ratio between A1 (LO) and E (LO).
In the case of near-resonant Raman excitation, the A1 (LO) mode appears much stronger

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 41 . Additionally, our simulation shows that the stronger in-plane plasmonic field component at 1.58 eV excitation (see Figure SI2d) couples to the in-plane E (LO) mode better than A1 (LO) mode resulting in enhanced relative intensity.
The Raman spectra presented in Figure 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 eV and 1.94 eV, which would be non-resonant 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 Figure 2b and 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 Figure SI3) 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.  Figure SI3), which also leads to an increased background in the red spectrum in Figure 3a.   Even though the electronic transition at the bandgap in InSe is weak at the normal incidence, it can still induce resonant Raman scattering in a strong plasmonic field. Hence, the strong intensity of the polar phonons observed in 7L InSe may also be activated via this resonance phenomenon. To examine this scenario, we prepared a 1L InSe with Ag NTs sample. The PL spectra of the flake confirming the layer thickness is presented in Figure SI4.  Figure 3d and for 7L InSe in Figure SI5.
As depicted in Figure 3d  On the other hand, the temperature dependent Raman spectra of 7L InSe/Ag NTs at 1.58 eV excitation presented in Figure SI5 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 7L InSe approaches the excitation energy of 1.58 eV meaning resonance with Eg is established. This phenomenon suggests that the fundamental bandgap Eg plays a role in enhancing the polar LO modes. However, as we showed in this work hot electron doping is the prerequisite for observing the polar LO modes, thus Fröhlich interaction in InSe takes place. 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 2L InSe) at 1.58 eV excitation energy. The simulation results are presented in Figure SI6.     Layer dependent (a) Raman excited at 1.58eV and (b) corresponding PL spectra of InSe. There is a drastic drop of intensity of the polar LO modes from 7L to 6L corresponding to the thickness where the direct to indirect bandgap transition of InSe occurs.