Theoretical Study of Electron-Surface Optical Phonon Interaction in Monolayer Transition Metal Dichalcogenides Deposited on SiC and Hexagonal BN Dielectric Substrates in the Vicinity of the Points K+ K- of the Brillouin Zone

1. Introduction

In the past few years, there has been a growing interest in Transition Metal Dichalcogenides (TMDCs), specifically in the form of monolayer (ML) van der Waals materials. This has sparked significant research interest for various applications in electronics and optoelectronics [1,2,3].

In comparison to conventional semiconductors, such as GaAs, the Coulomb interaction between conduction electrons and valence holes, as well as the oscillator strengths of excitons in MLs TMDCs, are significantly higher due to the two-dimensional confinement of charge carriers, heavy effective masses, and weak screening in 2D systems [4,5,6,7,8,9]. That is why scientists studying the physics of semiconductor nanosystems have been very interested in 2D materials during the past few years, such as graphene, hexagonal boron nitride MLs, TMDC MLs, and the heterostructures they generate [10,11]. The ML TDMCs that have been investigated the most are MoS₂, MoSe₂, WS₂, and WSe₂ [12,13].

The transport of carriers in ML 2D materials under low fields is influenced by multiple scattering mechanisms, including interactions with acoustic and optical phonons. Additionally, scattering can occur due to polar coupling with the substrate beneath or with dielectrics, introducing another factor involving remote optical phonons. Thus, Polar optical phonons, situated at the interface, play a significant role in scattering carriers in TMDCs through the Fröhlich coupling. [14,15,16,17]. Therefore, it is crucial to comprehend these scattering events by examining the coupling between surface optical phonons (SOP) and electrons in TMDCs. Developing models that can elucidate experimental results becomes essential. This coupling is typically characterized by interactions between electronic excitations and phonons, giving rise to various intriguing effects in a crystal, including the formation of polarons [14,15,16,17,18,19,20].

Several studies in the literature proved the importance of the role played by the SOP coupling in the optical proprieties in ML TDMCs deposited on polar substrates. As an example, we referenced Suvodeep Paul et al. [21], who showed that the creation of a ${\displaystyle \frac{W{S}_2}{hBN}}$ heterostructure leads to the coupling between electrons in $W{S}_2$ and polar phonons in $hBN$. This coupling governs the enhancement of defect-bound excitons and biexcitons. Additionally, they have performed an extensive resonant Raman analysis, varying both polarization and magnetic field, which provided further confirmation of the electron-phonon coupling in the ${\displaystyle \frac{W{S}_2}{hBN}}$ heterostructure.

Likewise, Colin M. Chow et al. [22], observed a resonant Raman scattering phenomenon through cross-material exciton-phonon coupling at van der Waals interfaces. They noted that the sensitivity of excitons in monolayer materials to their phononic environments, such as those provided by $Si{O}_2$, $hBN$, and sapphire, can be exploited to deepen our understanding of atomically thin devices. Else, Bastian Miller et al. [23], investigated exciton-phonon coupling in charge-tunable single-layer MoS₂ devices using polarization-resolved Raman spectroscopy. They found a strong defect-mediated coupling between the long-range oscillating electric field of the longitudinal optical phonon in the dipolar medium and the exciton.

Sanjay Gopalan et al. [24] also explored the impact of the dielectric environment on electronic transport in monolayers TMDCs. By employing ab initio methods, they calculated the low-field carrier mobility in free-standing layers, considering the effects of dielectric screening on electron-phonon interactions induced by the bottom oxide and gate insulator, as well as scattering from hybrid interface optical-phonon/plasmon excitations. Their findings revealed that using insulators with a high dielectric constant can greatly improve carrier mobility.

This paper is structured as follows: first, we theoretically investigate the interaction between electrons and surface optical phonons in ML TMDCs on polar substrates, such as silicon carbide ($SiC$) and hexagonal boron nitride ($hBN$). Furthermore, we present a theoretical examination of the polaronic oscillator strength in ML TMDCs on polar substrates. Finally, we investigate the temperature dependence of the polaronic scattering rate in ML TDMCs on $SiC$ and h$BN$ polar substrates.

2. Electron-Surface Optical Phonon Interaction in ML TMDCs on $\mathit{S}\mathit{i}\mathit{C}$ and h$\mathit{B}\mathit{N}$ Dielectric Substrates

A monolayer transition metal dichalcogenide (TMDC) consists of a central layer of metal M atoms arranged in a triangular lattice, flanked by two layers of chalcogen X atoms positioned on the same triangular lattice. The triangular Bravais lattice is defined by the basis vectors:

${\overrightarrow{a}}_1=a_0\left(1,\mathrm{0,0}\right)$ and ${\overrightarrow{a}}_2={\displaystyle \frac{a_0}{2}}\left(1,\sqrt{3},0\right)$ (see Figure 1a).

Figure 1b illustrates the reciprocal lattice, defined in relation to the triangular Bravais lattice and characterized by the vectors:

${\overrightarrow{b}}_1={\displaystyle \frac{4\pi }{\sqrt{3}{a}_0}}\left({\displaystyle \frac{\sqrt{3}}{2}},-{\displaystyle \frac{1}{2}},0\right)$ and ${\overrightarrow{b}}_2={\displaystyle \frac{4\pi }{\sqrt{3}{a}_0}}\left(0,1,0\right)$ where $a_0$ is the lattice constant.

The two-dimensional Brillouin zone of the TMDCs exhibits a hexagonal shape, featuring high-symmetry points denoted as $\Gamma ,K,andM,$ each defined as follows:

$\mathsf{\Gamma }=\left(0,0\right)$ , $K={\displaystyle \frac{4\pi }{3a_0}}\left(1,0\right)$, $M={\displaystyle \frac{4\pi }{3a_0}}\left(0,{\displaystyle \frac{\sqrt{3}}{2}}\right)$

The effective 2×2 Hamiltonian characterizing the states of the conduction and valence bands with the parallel spins $s=+{\displaystyle \frac{1}{2}}$ in the vicinity of the point $K_{+}$ is represented by the following expression [25,26,27]:

${\mathcal{H}}_{+}=\left(\begin{array}{cc}{\displaystyle \frac{E_g}{2}}& \gamma \left(k_x-i{k}_y\right)\\ \gamma \left(k_x+i{k}_y\right)& -{\displaystyle \frac{E_g}{2}}\end{array}\right)$ (1)

Where,$k=\left(k_x,k_y\right)$ denotes the two-dimensional wave vector of the electron measured from the point $K_{+}$, the parameter $\gamma$ is directly proportional to the interband matrix element of the momentum operator, $\gamma ={\displaystyle \frac{P^2}{m^{*}}}$ where $m^{*}$ is the electron effective mass and $E_g$ represents the width of the band gap.

The Hamiltonian describing a pair of spin sublevels with $s=-{\displaystyle \frac{1}{2}}$ in the same valley has the form of Equation (1) through the substitution $E_g\to E_g+∆$ , where $∆$ represents the sum of the spin–orbit splittings of the conduction and valence bands. The effective Hamiltonian in the $K_{-}$, valley is derived from Equation (1) through the substitution $\left(k_x\pm i{k}_y\right)\to \left(k_x\mp i{k}_y\right)$.

The energy spectrum of the electron derived from the Hamiltonian in Equation (1) has the Dirac form:

${\epsilon }_{\lambda ,k}=\lambda {\epsilon }_k=\sqrt{{\left({\displaystyle \frac{E_g}{2}}\right)}^2+{\gamma }^2{k}^2}$ (2)

Here, $\lambda =+$ and $\lambda =-$ correspond to the conduction and valence bands, respectively.

In our study, we use the assumption of homogeneous and defect-free interfaces between transition metal dichalcogenide monolayers and dielectric substrates as a simplification often used in theoretical models and simulations to make the problem tractable.

In this work we have investigated the electron surface optical phonon (SOP) interaction in ML TDMCs on $SiC$ and $hexagonalBN$ dielectric substrates across the long-range Fröhlich coupling. Indeed, the long-range Fröhlich coupling model provides a robust framework for understanding electron-SOP interactions in TMDCs on polar substrates. However, it relies on several approximations; like for example Born-Oppenheimer Approximation [28]. Short-range interactions [29] (e.g., electron-phonon interactions in non-polar materials) are not considered. It assumes a constant effective mass for the electron, non-linear interactions and multi-phonon processes are typically neglected [30]. Impurities, defects, and other forms of disorder that can affect the electron-phonon interaction in real materials are usually not included in the idealized Fröhlich model [28,31]. The phonon dispersion is typically assumed to be linear, which is an approximation that might not hold for all phonon modes or substrates [32]. Often, a single dominant phonon mode is considered, neglecting the possible contribution of multiple phonon modes [29].

To simplify, our analysis, we consider the phonon spectrum as isotropic, implying that phonons exhibit either longitudinal or transverse polarization. The Fröhlich Hamiltonian introduces an interaction term wherein an electron scatters from $\overrightarrow{k}$ to ${\overrightarrow{k}}^{\text{'}}=\overrightarrow{k+q}$ involving the emission or absorption of a phonon. In both cases the total momentum is conserved and is expressed as follows:

${\mathcal{H}=H}_{ph}+H_{e-ph}$ (3)

The term $H_{ph}$ denotes the phonon energies, incorporating both the Longitudinal Optical (LO) and Surface Optical (SO) modes, and can be expressed as:

$H_{ph}=\sum _{q,\nu }\mathrm{\hslash }{\omega }_{\nu }{a}_q^{+}{a}_q$ (4)

In this context, $a_q^{+}$, $a_q$ represent the creation and annihilation operators, respectively, for the phonon characterized by the wave vector $q$ , while ${\omega }_{\nu }$ refers to the frequency of the phonon.

The second term $H_{e-ph}$ is the Hamiltonian of interaction between electron and phonon [33]:

$H_{e-ph}=\sum _{q,\nu }{M}_{q,\nu }\left(a_{-q}^{+}{+a}_q\right)e^{-iqr}$ (5)

The Fröhlich Hamiltonian is given as follow:

$\mathcal{H}=\sum _{q,\nu }\mathrm{\hslash }{\omega }_{\nu }{a}_q^{+}{a}_q+\sum _{q,\nu }{M}_{q,\nu }\left(a_{-q}^{+}{+a}_q\right)e^{-iqr}$(6)

The interaction between carriers in monolayer Transition Metal Dichalcogenides (TMDCs) and surface optical phonons is described by the second term in Equation (6).

The coupling element in the Fröhlich Hamiltonian $M_{q,\nu }$ represents the interaction between the electron in TMDCs and surface optical phonon of the polar substrates. This matrix element is expressed as [34,35,36]:

$V_{SOP}{=M}_{q,SO}=\left|\overrightarrow{k}-\overrightarrow{k+q}\right|\sqrt{{\displaystyle \frac{e^2{F}_{\nu }^2}{2NAq}}}{e}^{-q{z}_0}$ (7)

In the given context, $F_{\nu }^2$ represents the magnitude of the polarization field, which is determined by the Fröhlich coupling [37]:

$F_{\nu }^2={\displaystyle \frac{\mathrm{ħ}{\omega }_{SO,\nu }}{2\pi }}\left({\displaystyle \frac{1}{{\epsilon }_{\mathrm{\infty }}+{\epsilon }_{env}}}-{\displaystyle \frac{1}{{\epsilon }_0+{\epsilon }_{env}}}\right)$ (8)

While ${\epsilon }_0$and ${\epsilon }_{\infty }$are the low- and high-frequency dielectric constants of the polar substrate, (refer to Table 1), $z_0$ represents the internal distance between the TMDCs and polar substrate (refer to Table 2). The term $ħ{\omega }_{SO,\nu }$ denotes the energy of SO phonon of the polar substrates with two branches $\nu =1,2.$

The SOP energies are extracted from the bulk longitudinal optical (LO) phonons as follows [38]:

$\mathrm{ħ}{\omega }_{SO}=\mathrm{ħ}{\omega }_{LO}{\left({\displaystyle \frac{1+\frac{1}{{\epsilon }_0}}{1+\frac{1}{{\epsilon }_{\mathrm{\infty }}}}}\right)}^{{\displaystyle \frac{1}{2}}}$ (9

The screening of the Coulomb interaction by the polar dielectric environment is considered through${\epsilon }_{env}$. Given the weak screening of the electric field perpendicular to the plane of the ML TMDCs, ${\epsilon }_{env}$is set to1 [44].

On polar substrates, surface optical phonons (SOP) induce an electric field that interacts with the electrons in the neighboring ML TMDCs. Using Equations (7) and (8), the SOP coupling is expressed as:

${W=\sum _{\overrightarrow{q}}\left|⟨{\psi }_k|V_{SOP}|{\psi }_{k+q}⟩\right|}^2={\displaystyle \frac{NA}{{\left(2\pi \right)}^2}}\iint {\displaystyle \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}}{\displaystyle \frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}}{e}^{-2q{z}_0}qdqd{\theta }_q$ (10)

The summation is performed over one spin and one valley, where $A={\displaystyle \frac{\sqrt{3}}{2}}{a}^2$ is the area of the two-atom unit cell.

In our analysis in the present case, we have followed the same theoretical method presented in our previous calculations [14,15,16]. So, to study the interactions between electron and surface optical phonons in the ML TMDCs, we have specifically considered the electronic states $\left|{\psi }_k〉\right.$ and $\left|{\psi }_{k+q}〉\right.$ , with electron energies $E_k={\epsilon }_k$ and $E_{k+q}={\epsilon }_{k+q}$ , respectively. We have also considered the effective 2×2 Hamiltonian characterizing the states of the conduction and valence bands with the parallel spins $s=\pm {\displaystyle \frac{1}{2}}$ in the vicinity of the point ${K_{+}(K}_{-})$of the hexagonal Brillouin zone.

The space of polaronic states results from a tensor product between the two subspaces of electronic and phononic states. Thus, we consider new states called polaronic states given by:

$\left\{\left|{\psi }_{k+q},0q〉\right.,\left|{\psi }_k,1q〉\right.\right\}$ (11)

The Polaron electron energies $E_{\pm }^e$ for the states $\left|{\psi }_{\pm }〉\right.$ in ML TMDCs on polar substrates are given bellow [14,15,16]:

$E_{\pm }^e={\displaystyle \frac{1}{2}}\left(E_{k+q}+E_k+\mathrm{\hslash }{\omega }_{LO}\right)\pm \sqrt{{\left[{\displaystyle \frac{1}{2}}\left(E_{k+q}-E_k+\mathrm{\hslash }{\omega }_{LO}\right)\right]}^2+{\displaystyle \frac{NA}{{\left(2\pi \right)}^2}}\iint {\displaystyle \frac{1-\cos \left({\theta }_k-{\theta }_q\right)}{2}}{\displaystyle \frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}}{e}^{-2q{z}_0}qdqd{\theta }_q}$ (12)

Figure 2 depicts the SO coupling strength between the electronic states $\left|{\psi }_k〉\right.$ and $\left|{\psi }_{k+q}〉\right.$ versus the wave vector $k$in ML WS₂ on $SiC$ and h$BN$ polar substrate. As shown in Figure 2, it is evident that the coupling with SOP is significantly influenced by the type of polar substrate.

We show in Figure 3, the Polaron electron energies versus the wave vector $k$in ML $W{S}_2,WS{e}_2,Mo{S}_{2}andMoS{e}_2$ on $SiC$ and h$BN$ polar substrate. For comparison purposes, we have included in the same figures the energies of the non-interacting states $\left|{\psi }_{k+q},0q〉\right.$and $\left|{\psi }_k,1q〉\right.$. For example, in the case of WS₂ these noninteracting levels cross near $k~2.2{nm}^{-1}$ in the case of $SiC$ and near $k~2.85{nm}^{-1}$ in the case of $hBN$ indicating resonant coupling (see Table 3). These crossings indicate that the separation between electronic levels is equal to $\hslash {\omega }_{LO}$ for both $SiC$ and $hBN$ cases, where $\hslash {\omega }_{LO}=123.2meVand\hslash {\omega }_{LO}=103.7mev$ respectively. In fact, the electronic level crossings are clearly replaced by significant anticrossings with energy levels approximately at ~94 meV and ~70 meV for $SiC$ and $hBN$polar substrates, respectively. It can be also observed in Figure 3, the enhancement of the Rabi splitting of the electron levels when shifting from $hBN$ to $SiC$ (refer Table 4).

In these anticrossings, the wave functions of the levels become mixed, allowing for multiple transitions such as $E_k\to E_{\pm }^e$, $E_k\to E_k+\hslash {\omega }_{LO}$ and $E_k\to E_{k+q}$. This demonstrates that the interaction between electrons and surface polar phonons cannot be considered a weak coupling. The coupling between electrons and SOP leads to the Rabi splitting of the electron levels. Hence, the calculations indicate the possibility of energetically resonant coupling between the electronic sub-levels and surface vibration modes in ML TMDCs on the studied polar substrates. Furthermore, the two resulting polaron states can be expressed as:

$\left|{\psi }_{\pm }〉\right.={\alpha }_{\pm }\left|{\psi }_{k+q},0q〉\right.+{\beta }_{\pm }\left|{\psi }_k,1q〉\right.$(13)

The weight of electronic component${\alpha }_{\pm }$ and the weight of the one-phonon component ${\beta }_{\pm }$ of the polaron states ± vary with the polaron energies $E_{\pm }^e$ . The expressions detailing these dependencies are as follows [14,15,16]:

${\left|{\alpha }_{\pm }\right|}^2={\displaystyle \frac{{\left(E_{\pm }^e-\mathrm{\hslash }{\omega }_{LO}\right)}^2}{{\left(E_{\pm }^e-\mathrm{\hslash }{\omega }_{LO}\right)}^2+\frac{NA}{{\left(2\pi \right)}^2}\iint \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}\frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}{e}^{-2q{z}_0}qdqd{\theta }_q}}$ (14)

${\left|{\beta }_{\pm }\right|}^2={\displaystyle \frac{\frac{NA}{{\left(2\pi \right)}^2}\iint \frac{1-\cos \left({\theta }_q\right)}{2}\frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}{e}^{-2q{z}_0}qdqd{\theta }_q}{{\left(E_{\pm }^e-\mathrm{\hslash }{\omega }_{LO}\right)}^2+\frac{NA}{{\left(2\pi \right)}^2}\iint \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}\frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}{e}^{-2q{z}_0}qdqd{\theta }_q}}$ (15)

where $\sqrt{{\displaystyle \frac{NA}{{\left(2\pi \right)}^2}}\iint {\displaystyle \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}}{\displaystyle \frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}}{e}^{-2q{z}_0}qdqd{\theta }_q}$ is the SO coupling strength between the electronic states $\left|{\psi }_k,1q〉\right.and\left|{\psi }_{k+q},0q〉\right.$[14,15,16,17].

Figure 4, depicts the weight of the electronic components and the one-phonon components of the lower polaron state $\left|{\psi }_{-}〉\right.$ in ML TMDCs for $SiC$ and h$BN$ polar substrates as a function of the wave vector $k$in ML TMDCs. It can be seen for example in the case of $W{S}_2$ on $SiC$ polar substrate (see Table 3, Figure 4) that; when the wave vector $k$ approaches $k~2.2{nm}^{-1},$ the value of the weight of the one-phonon component for the lower polaron state $\left|{\psi }_{-}〉\right.$ is much larger compared to that of the electronic component $\left({\beta }_{\pm }\gg {\alpha }_{\pm }\right)$. This result demonstrates that the SOP situated at the interface of the ML $W{S}_2$ on $SiC$ polar substrate plays a crucial role in the resonant coupling between the noninteracting states $\left|{\psi }_k,1q〉\right.$ and $\left|{\psi }_{k+q},0q〉\right.$ allowing the formation of the polaron states. It can be noted that the same result has been proved for the other cases of ML $\mathrm{T}\mathrm{M}\mathrm{D}\mathrm{C}\mathrm{s}$ on $SiC$ and h$BN$ polar substrates (see Table 3 and Figure 4).

3. Polaronic Oscillator Strength of ML TMDCs on SiC and hBN Polar Substrates

In the following section, we theoretically investigate the polaronic oscillator strength (OS) which is another crucial quantity. Drawing an analogy to the oscillator strengths of interband transitions in quantum dots, we have computed the OS for interband transitions in ML TMDCs on polar substrates. In the strong confinement limit, the OS is linked to the overlap integral of the polaronic states, ${\left|\left({\psi }_{-}\right)\right|}^2$ by [45,46]:

$f_{Osc}={\displaystyle \frac{{\left|\left({\psi }_{-}\right)\right|}^2{E}_p}{2E_{PL}}}$ (16)

where $E_P$ is the Kane energy and $E_{PL}$ is the emission energy for one-phonon of the ML TMDCs on polar substrates, which is given by:

$E_{PL}=\left(\mathrm{\hslash }{\omega }_{LO}=E_g+E_{-}^e\right)$(17)

where$E_{-}$ is the lower polaron energy of the exciton, $\hslash {\omega }_{LO}$ is the emitted photon energy and $E_g$ is the energy gaps of the ML TMDCs.

We have calculated the OS for the lower polaron state $\left|{\psi }_{-}〉\right.$ which is a linear combination of the two states $\left|{\psi }_{k+q},0q〉\right.$and $\left|{\psi }_k,1q〉\right.$:

$\left|{\psi }_{-}〉\right.={\alpha }_{-}\left|{\psi }_{k+q},0q〉\right.+{\beta }_{-}\left|{\psi }_k,1q〉\right.$(18)

Figure 5 shows the polaronic OS of $W{S}_2$ on $SiC$ and h$BN$ polar substrates versus the wave vector $k$in ML $W{S}_2$.

As a result, we proved theoretically; that the polaronic OS is especially sensitive to the phonon mode of the surrounding dielectrics. In fact this result is due to the emission energy for one-phonon of the ML TMDCs for $SiC$ and h$BN$ polar substrates, which is given by: $E_{PL}=\left(\hslash {\omega }_{LO}=E_g+E_{-}\right)$. Hence, the highest polaronic oscillator strength corresponds to the highest optical phonon energy of the polar substrates $E_{PL}=\left(\hslash {\omega }_{LO}\right)$ . This is by analogy to polaronic oscillator strength in ML TMDCs which is also much higher as compared with conventional semiconductors [6,7,8,9].

Using the same method, we can easily prove that for the other ML TMDCs such as: $\mathrm{W}{\mathrm{S}\mathrm{e}}_2$, $\mathrm{M}\mathrm{o}{\mathrm{S}}_2$, and $\mathrm{M}\mathrm{o}{\mathrm{S}\mathrm{e}}_2$:

$f_{Osc}\left[\mathrm{M}\mathrm{L}\mathrm{T}\mathrm{D}\mathrm{M}\mathrm{C}\mathrm{s}/\mathrm{S}\mathrm{i}\mathrm{C}\right]>f_{Osc}\left[\mathrm{M}\mathrm{L}\mathrm{T}\mathrm{D}\mathrm{M}\mathrm{C}\mathrm{s}/\mathrm{h}\mathrm{B}\mathrm{N}\right]$. (19)

This result is due to the $SiC$ dielectric constant as well as the $SiC$ phonon energy compared to that of $hBN$ $\left(ħ{\omega }_{LO}\left(SiC\right)=123.2meV>ħ{\omega }_{LO}\left(hBN\right)=103.7meV\right)$, (see Table 1), so the polar field created near the interface should be the highest in the case of $SiC$ substrate, compared to $hBN$ substrate. This result leads to the highest polaronic OS in ML TDMCs on $SiC$ polar substrate. Similarly, it can be concluded from the Figure 6 that:

$f_{Osc}\left(\mathrm{W}{\mathrm{S}}_2\right)>f_{Osc}\left(\mathrm{W}{\mathrm{S}\mathrm{e}}_2\right)>f_{Osc}\left(\mathrm{M}\mathrm{o}{\mathrm{S}}_2\right)>f_{Osc}\left(\mathrm{M}\mathrm{o}{\mathrm{S}\mathrm{e}}_2\right)$. (20)

Hence, this result can be explained by the difference in electron effective masses in ML TDMCs. This result actually confirms the decrement in the polaronic OS for heavy electron in the nearby ML TDMCs / dielectric substrate interface. Otherwise, in the case of light electron, the polaronic OS increases considerably at the strong confinement regime.

4. Polaronic Scattering Rate of ML TMDCs on SiC and hBN Polar Substrates

Now we consider the temperature dependence of the polaronic scattering rate due to the SO phonon. The polaronic scattering rate (SO phonon scattering rate) is given as follow: [17]

${\displaystyle \frac{1}{{\tau }_{Polaron}}}={\displaystyle \frac{2\pi }{\mathrm{ħ}}}\sum _q{\left|M_{k,k+q}\right|}^2\left[1-\cos \left({\theta }_k-{\theta }_{k+q}\right)\right]\times$$\times \left\{N_q\delta \left(E_k-E_{k+q}+\mathrm{ħ}{\omega }_q\right)+\left(N_q+1\right)\delta \left(E_k-E_{k+q}-\mathrm{ħ}{\omega }_q\right)\right\}$ (21)

Here $N_q$ is the Bose-Einstein phonon occupation number, ${\theta }_k$is a directional angle of wave vector $\overrightarrow{k}$ , ${\left|M_{k,k+q}\right|}^2$ is given by:

${\left|M_{k,k+q}\right|}^2={\left|⟨{\psi }_k|V_{SOP}|{\psi }_{k+q}⟩\right|}^2=\left[{\displaystyle \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}}{\displaystyle \frac{4{\pi }^2{e}^2{F}^2}{N\frac{\sqrt{3}}{2}{a}^{2}q}}{e}^{-2q{z}_0}\right]$ (22)

The summation $\sum _q$ is replaced by the integral ${\displaystyle \frac{NA}{4{\pi }^2}}\iint \mathrm{q}\mathrm{d}\mathrm{q}\mathrm{d}\mathsf{\theta }$ (sum over a spin and a valley), where$A={\displaystyle \frac{\sqrt{3}}{2}}{a}^2$ is the area of the elementary cell that contains two atoms.

Thus,

${\displaystyle \frac{1}{{\tau }_{Polaron}}}={\displaystyle \frac{2\pi }{\mathrm{ħ}}}{\displaystyle \frac{NA}{{\left(2\pi \right)}^2}}\iint {\displaystyle \frac{1-\cos \left({\theta }_k-{\theta }_{k+q}\right)}{2}}{\displaystyle \frac{4{\pi }^2{e}^2{F}_{\nu }^2}{NAq}}{e}^{-2q{z}_0}\left[1-\cos \left({\theta }_k-{\theta }_{k+q}\right)\right]\times$$\times \left\{N_q\delta \left(E_k-E_{k+q}+\mathrm{ħ}{\omega }_q\right)+\left(N_q+1\right)\delta \left(E_k-E_{k+q}-\mathrm{ħ}{\omega }_q\right)\right\}qdqd{\theta }_q$ (23)

Figure 7 shows the temperature dependence of the SO phonon scattering rate in ML TDMCs on $SiC$ and h$BN$ polar substrates. We clearly see that at temperatures higher than the room temperature, the SO phonon scattering rate increases with temperature, whereas at low temperatures, SO phonon scattering rate is not significant. Similarly, it can be concluded from the Figure 7 that:

$1/{\tau }_{Polaron}\left(\mathrm{W}{\mathrm{S}}_2\right)>1/{\tau }_{Polaron}\left(\mathrm{W}{\mathrm{S}\mathrm{e}}_2\right)>1/{\tau }_{Polaron}\left(\mathrm{M}\mathrm{o}{\mathrm{S}}_2\right)>1/{\tau }_{Polaron}\left(\mathrm{M}\mathrm{o}{\mathrm{S}\mathrm{e}}_2\right)$. (24)

From Figure 7, we can also observe that the SO phonon scattering rate in ML TDMCs depends strongly on the dielectric constant of the polar substrate, thus:

$1/{\tau }_{Polaron}\left(\mathrm{M}\mathrm{L}\mathrm{T}\mathrm{D}\mathrm{M}\mathrm{C}\mathrm{s}/\mathrm{S}\mathrm{i}\mathrm{C}\right)>1/{\tau }_{Polaron}\left(\mathrm{M}\mathrm{L}\mathrm{T}\mathrm{D}\mathrm{M}\mathrm{C}\mathrm{s}/\mathrm{h}\mathrm{B}\mathrm{N}\right)$ (25)

Finally, choosing a suitable dielectric as a substrate one can achieve the highest SO phonon scattering rate in ML TDMCs.

To test experimentally electron-surface optical phonon (SOP) interaction in transition metal dichalcogenides (TMDs) deposited on silicon carbide or hexagonal boron nitride dielectric substrates we propose several experiments such as Photoluminescence (PL) Spectroscopy, Raman Spectroscopy, Time-Resolved Photoluminescence (TRPL), Electroluminescence (EL) Spectroscopy and Angle-Resolved Photoemission Spectroscopy (ARPES) [22,47,48,49,50].

5. Conclusions

In conclusion, our results indicate that the SOP coupling in ML TDMCs depends strongly upon the polar substrate. Moreover, the resonant coupling between electronic sub-levels and surface polar vibration modes, leads to the Rabi splitting of electron levels in single-layer TMDCs. Hence, the polarization field induces a robust, resonant mixing between electronic states and surface vibration modes when their energies become comparable. This resonant coupling has the potential to diminish the probability of nonradiative recombination processes and enhance the efficiency of photoluminescence (PL). Finally, we demonstrate that, the highest polar field created near the interface leads to the highest polaronic OS and the enhancement of SO phonon scattering rates in ML TDMCs. Thus, polaronic OS and polaronic scattering rates depend strongly upon the choice of the polar substrate. The presence of polaronic states may yield significant implications for energy relaxation in ML TMDCs. Moreover, the interaction between electrons and surface optical phonons is a crucial factor influencing the physical characteristics of 2D semiconductors, particularly influencing processes like phonon-assisted hot carrier relaxation.