Electron-phonon dynamics in 2D carbon based-hybrids XC (X = Si, Ge, Sn)

TL;DR

This study investigates electron-phonon interactions in 2D hybrid materials SiC, GeC, and SnC, revealing mode-specific scattering behaviors and ultrafast hot carrier relaxation times relevant for optical applications.

Contribution

It provides ab initio insights into electron-phonon coupling effects and relaxation dynamics in 2D SiC, GeC, and SnC hybrids, highlighting their potential for optical technologies.

Findings

Electron-phonon scattering is dominated by specific phonon modes in each material.

Hot carrier thermalization occurs within 90-120 fs after illumination.

The materials show temperature-dependent electronic band gap behavior.

Abstract

The effect of the presence of electron-phonon (e-ph) coupling in the SiC, GeC and SnC hybrids is studied in the framework of the ab initio perturbation theory. The electronic bang gap thermal dependence reveals a normal monotonic decrease in the SiC and GeC semiconductors, whereas SnC exhibits an anomalous behavior. The electron line widths were evaluated and the contributions of acoustic and optical phonon modes to the imaginary part of the self-energy were determined. It has been found that the e-ph scattering rates are globally controlled by the out-of-plane acoustic transverse mode ZA in SiC while both ZA and ZO are overriding in GeC. In SnC, the out-of-plane transverse optical mode ZO is the most dominant. The relaxation lifetime of the photo-excited electrons shows that the thermalization of the hot carrier occurs at 90 fs, 100 fs and 120 fs in SiC, GeC and SnC, respectively. The present study properly describes the subpicosecond time scale after sunlight illumination using an approach that requires no empirical data. The results make the investigated structures suitable for providing low cost and high-performance optical communication and monitoring applications using 2D materials.

Figures

Equations

$$G_{n\mathbf{k}}(\omega ,T)={\left[\omega -{ϵ}_{n\mathbf{k}}-{\Sigma }_{n\mathbf{k}}^{Fan}(\omega ,T)-{\Sigma }_{n\mathbf{k}}^{DW}(T)\right]}^{-1}$$

$${\Sigma }_{n\mathbf{k}}^{Fan}=\sum _{n^{\prime },\mathbf{q}\lambda }\frac{|g_{n{n}^{\prime }\mathbf{k}}^{\mathbf{q}\lambda }{|}^2}{N}\left[\frac{N_{\mathbf{q}\lambda }(T)+1-f_{n^{\prime }\mathbf{k}-\mathbf{q}}}{i\omega -{ϵ}_{n^{\prime }\mathbf{k}-\mathbf{q}}-{\omega }_{\mathbf{q}\lambda }}+\frac{N_{\mathbf{q}\lambda }(T)+f_{n^{\prime }\mathbf{k}-\mathbf{q}}}{i\omega -{ϵ}_{n^{\prime }\mathbf{k}-\mathbf{q}}+{\omega }_{\mathbf{q}\lambda }}\right],$$

$${\Sigma }_{n\mathbf{k}}^{DW}=-\frac{1}{2}\sum _{n^{\prime }\mathbf{q}\lambda }\frac{{\Lambda }_{n{n}^{\prime }\mathbf{k}}^{\mathbf{q}\lambda }}{N}\left[\frac{2N_{\mathbf{q}\lambda }(T)+1}{{ϵ}_{n\mathbf{k}}-{ϵ}_{n^{\prime }\mathbf{k}}}\right],$$

$$g_{n{n}^{\prime }\mathbf{k}}^{\mathbf{q}\lambda }=⟨{\psi }_{n^{\prime },\mathbf{k}-\mathbf{q}}|{\partial }_{\mathbf{q}\lambda }V|{\psi }_{n,\mathbf{k}}⟩,$$

$$E_{n\mathbf{k}}(T)\approx {ϵ}_{n\mathbf{k}}+Z_{n\mathbf{k}}(T)\left[{\Sigma }_{n\mathbf{k}}^{Fan}({ϵ}_{n\mathbf{k}},T)+{\Sigma }_{n\mathbf{k}}^{DW}(T)\right]$$

$$Z_{n\mathbf{k}}(T)={\left[-{\frac{\partial R{\sum }_{n\mathbf{k}}^{Fan}(\omega )}{\partial \omega }|}_{\omega ={ϵ}_{n\mathbf{k}}}\right]}^{-1}.$$

$${\tau }_{n\mathbf{k}}^{el-ph}=\frac{2{\Gamma }_{n\mathbf{k}}}{\hslash },$$

$$E_{n\mathbf{k}}(T)={ϵ}_{n\mathbf{k}}+\Delta E_{n\mathbf{k}}(T)$$

$$\Delta E_{n\mathbf{k}}(T)\approx {\Delta E_{n\mathbf{k}}(T)|}_{har}+{\Delta E_{n\mathbf{k}}(T)|}_{anhar}.$$

$$g^2{F}_{n\mathbf{k}}(\omega )=\frac{1}{N}\sum _{n^{\prime }\mathbf{q}\lambda }\left[\frac{|g_{n{n}^{\prime }\mathbf{k}}^{\mathbf{q}\lambda }{|}^2}{{ϵ}_{n\mathbf{k}}-{ϵ}_{n^{\prime }\mathbf{k}-\mathbf{q}}}-\frac{1}{2}\frac{{\Lambda }_{n{n}^{\prime }\mathbf{k}}^{\mathbf{q}\lambda }}{{ϵ}_{n\mathbf{k}}-{ϵ}_{n^{\prime }\mathbf{k}-\mathbf{q}}}\right]\delta (\omega -{\omega }_{\mathbf{q}\lambda })$$

$${\Delta E_n(T)|}_{har}=\int d\omega g^2{F}_{n\mathbf{k}}(\omega )\left[N_{\mathbf{q}\lambda }(T)+\frac{1}{2}\right].$$

$$F_g=F_{CBM}-F_{VBM},$$

Full text

2D binary compounds, ab-initio calculations; electronic properties; electron lifetime; phonon modes; scattering rate.

I Introduction

The graphene synthesis revealed exceptional properties for this material such as, high ambient temperature mobility, ambipolar effect, Klein tunneling, anomalous quantum Hall effect, etc 24 . This Breakthrough suggested many opportunities for the creation of other two dimensional (2D) materials. Among them, stanene, germanene and silicene are of particular interest because of their compatibility with current silicon technology sii . Similar to graphene, the electrons in these 2D materials behave as massless Dirac fermions and exhibit a linear dispersion-band near the Fermi level due to their $sp^{2}$ electronic configuration 75 . The successful growth of Si, Sn and Ge- 2D monolayers has led to extensive investigation into the search for new hexagonal materials, such as phosphorene, borophene, aluminene and ZnO, with great potential for applications.

The materials consisting of column IV-elements of the periodic table are gapless in the absence of spin orbit coupling. This fact makes them less interesting for optoelectronic applications such as photovoltaic cells and light emitting diodes, which require a material with a nonzero band gap 18 . To overcome this deficiency, columns IV-IV hybrid materials have been introduced topsakal ; grr . The combination of silicon and carbon atoms creates silicene/graphene monolayer (SiC). This hybrid has a 2D planar honeycomb structure rather than a buckled one due to the strong $\pi$-bond through its perpendicular pz orbitals** 19 **-111 . On the other hand, germanene/graphene hybrid (GeC) films are another column IV binary compound which can be prepared using deposition techniques such as laser ablation 211 and radio frequency reactive sputtering in Ar/CH4 212 . This material possesses excellent electro-catalytic properties and is promising for fuel cell and lithium–oxygen battery applications GeCaa . Both GeC and SiC sheets are direct band gap semiconductors and show strong excitonic effects with high binding energy 1 ; 2 . Meanwhile, the SnC monolayer, resulting from the combination of graphene and stanene, exhibits an indirect band gap. All these 2D graphene-based materials are good candidates for optoelectronic applications and are very promising for nanotechnology applications requiring catalytic performance for reduction and oxidation.

In a large group of semiconductors, a monotonic decrease of the energy gap has been observed as the temperature increases Eliashyambo10 ; Eliashyambo11 . This well understood phenomenon does not occur in some exceptional materials where band gaps exhibit two kinds of anomalous temperature dependence. In the non-monotonic anomalous case, the gap first increases at low temperatures and then decreases at high temperature Eliashyambo12 . In the monotonic anomalous case, the gap increases continuously with temperature. This behavior is observed for some perovskites Eliashyambo14 , copper halides Eliashyambo15 and lead chalcogenides Eliashyambo16 ; Eliashyambo17 . Despite the abundance of experimental evidence, the understanding of anomalous gap dependence is explained by different, and sometimes even contradictory arguments, such as** the role played by harmonic and anharmonic contributions Eliashyambo17 ; abc . **

The temperature dependence of the electronic energies originates partially from the volume expansion bardeen and results mainly from the electron-phonon (e-ph) interactions at constant volume ttte .** In the latter case, the major contribution of the effect of e-ph interaction proves to be the most difficult to compute from first principles due to the huge number of k-points in the Brillouin zone required for convergence reee . **This can be avoided by using schemes based on Wannier functions. The spatial localization of the electronic and lattice Wannier functions allows to compute only a limited set of electronic and vibrational states as well as the corresponding e-ph interaction matrix elements from first principles. This requires low costs of a standard phonon dispersion calculation for a large number of $k-$points in the Brillouin zone barr .

Attention to the study of properties related to electron-phonon couplings (EPC)** using ab-initio computation has become more and more common for 2D materials. The obtained results subsequently explain many phenomena observed in honeycomb materials. **The presence of Kohn anomalies in graphene and in the low buckled silicene is associated with their highest-branched optical modes grapp ; eph1 . However, a negligible EPC is observed in germanene which shows very small square of the EPC matrix-element eph1 . In stanene, the electron-phonon process is mainly dominated by the out-of-plane optical mode ($ZO$) leading to a hot carrier thermalization occurring at 250 fs nouss . In silicene, the low energy of out-of-plane acoustic phonon mode $(ZA)$, derived from the weak sp2 bond between its Si atoms, contributes to the high scattering rate and significant degradation in electron transport Mos2 . With access to the first-principles EPC, the effect of the scattering rate on the performance of MoS2 and WSe2 FET for a channel length of 10 nm shows a ballistic of $83\%$ and $75\%$ for WSe2 and MoS2 devices respectively MWs .

Therefore, understanding the EPC from imaginary self energy, electron life time and carrier mobility in 2D hybrids is highly desirable and essential in order to facilitate their further applications. Considering the interest for graphene-based hybrids: GeC, SiC and SnC, this work studies the electron-phonon coupling involved in the temperature effect on the gap behavior. Using an ab initio approach based on the density functional theory (DFT), we investigate the behavior of the electronic bang gap thermal dependence in the three carbon based hybrids in the presence of electron-phonon interaction at constant volume. With the exception of SnC, which shows a non-monotonic anomaly, both GeC and SiC exhibit no anomalous behavior in their gap variation. To calculate the electron lifetime in these 2D materials, the imaginary part of electron’s self energy around the Fermi level and the corresponding scattering rate are determined in detail. The phonons contribution to the linewidth for an electron initially at the conduction band minimum (CBM) and the valence band maximum (VBM), as well as the variation of their corresponding lifetime with respect to the electron energy are discussed.** **,

The present paper is organized as follows. In section II, we present the computational method required for the calculation of the reported data. In section III, we start by a comparative description of the temperature effect on the gap value. Then, we report a complete description of the behavior of SiC, GeC and SnC electron linewidth with respect to the electron energy and the temperature. From the plot of the scattering rate, we compute the electron lifetime and elucidate the role of phonon modes. Finally, in section IV, we end with a conclusion.

II Computational method

In this work, three planar carbon-based hybrids are studied. The unit cell of each crystal contains 2 atoms: C and X (X=Si, Ge and Sn). As displayed in Fig.1 , each carbon atom is bonded to three X-atoms forming hexagonal structures.

Employing norm-conserving pseudo-potentials, the generalized gradient approximation of Perdew-Zunger exchange-correlation functional is used considering a plane-wave basis set with a kinetic-energy cutoff of 60 Ry gga . The electronic states are computed using a k-sampling grid of $32\times 32\times 1$ in the Brillouin zone based on the Monkhorst-Pack scheme. The subsequent computational recipe is based on the scheme proposed in ttte .

The vibrational frequencies, $\omega_{q\lambda}$, and the derivatives of the self-consistent Kohn-Sham potential with respect to the atomic displacements are calculated on $16\times 16\times 1$ q-point grids using density functional perturbation theory (DFPT) barr as implemented in the Quantum-Espresso code manu30 . These parameters are needed to evaluate the e-ph coupling matrix elements.

The many body perturbation theory $(MBPT)$ is used to describe the behavior of electronic states as a function of temperature mbpt . The electron-phonon coupling is perturbatively treated yambo47 . We consider the Fan and Debye-Waller self-energy terms corresponding respectively to the first and second order of the Taylor expansion in the displacement of the nuclei. The Green function is written as ttte :

[TABLE]

where $\epsilon_{n}$ is the Kohn-Sham ground-state eigenenergies of the frozen atoms, $\sum^{Fan}$ is the Fan contribution to the self energy given by:

[TABLE]

and $\Sigma^{DW}$ is the Debye-Waller term expressed as follows loiui :

[TABLE]

with $N_{\mathbf{q}\lambda}$ and $f_{n^{\prime}}$ represent the distribution functions of Bose-Einstein and Fermi-Dirac, while $\omega_{\lambda}$ and $N$ are the phonon frequencies and the number of $\mathbf{q}-$points taken randomly to better map out the phonon transferred momentum yambo47 . $\Lambda_{nn^{\prime}\mathbf{k}}^{\mathbf{q}\lambda}$ is the second order electron-phonon matrix elements.

The first order electron-phonon matrix elements $g_{nn^{\prime}\mathbf{k}}^{\mathbf{q}\lambda}$, which represent the probability amplitude for an electron to be scattered with emission or absorption of phonons are given by:

[TABLE]

where $n^{\prime}$ and $n$ are initial and final electron band indices with wavevectors $\mathbf{k}$ and $\mathbf{k-q}$, respectively, and $\partial_{\mathbf{q}\lambda}$ is the derivative of the self-consistent potential associated with a phonon of the wavevector $\mathbf{q}$ in the branch $\lambda$. To calculate the imaginary part of the electron’s self-energy using the Electron-Phonon coupling using Wannier functions code (EPW) epw , the matrix elements were evaluated over a grid of $360\times\ 360\times\ 1$ $\mathbf{q}$ points.

Assuming the quasiparticle approximation $(QPA),$ one can expand in first-order the self-energy frequency dependence around the bare energies. It follows that the temperature dependent quasi-particle energy is given by ttte :

[TABLE]

where the renormalization factor is expressed as follows:

[TABLE]

We note that the imaginary part of the Fan’s self-energy ($\sum_{n\mathbf{k}}^{Fan}$) provides information on the rate of e-ph processes responsible for hot carrier thermalization, which is related to the intrinsic quasiparticule lifetime. The real part, however, informs us about the energy renormalization shift. Thus, in order to obtain the associated e-ph scattering rate, one should first compute the linewidth ($\Gamma_{n\mathbf{k}})$ epw :

[TABLE]

which just means that the quasiparticle lifetime corresponds to the inverse of the scattering rate $\tau^{-1}$.

Eq.4 states that the single particle energy $E_{n\mathbf{k}}$ depends on temperature as follows ttte :

[TABLE]

The energy $\Delta E_{n\mathbf{k}}(T)$ involves two contributions, namely:

[TABLE]

The first term results from pure electron-phonon interactions and represents the harmonic contribution at constant volume. The second term corresponds to the effects flowing from the lattice expansion (variable volume). According to equation 8, it is obvious that the harmonic and anharmonic terms have some contribution in the energy shifting. At constant volume, the effect of the electron-phonon interaction is usually the major contribution, so by setting $\left.\Delta E_{n\mathbf{k}}(T)\right|_{anhar}<<\left.\Delta E_{n\mathbf{k}}(T)\right|_{har},$ only the harmonic effect will be considered.

In order to understand the gap behavior as function of temperature, one should first compute the generalized Eliashberg spectral function. The spectral function $g^{2}F_{n\mathbf{k}}(\omega)$ is expressed as follows ttte :

[TABLE]

$\Lambda_{nn^{\prime}\mathbf{k}}^{\mathbf{q}\lambda}$ gives information about the phonon mode contributions to the electron-phonon harmonic term $\left.\Delta E_{n}(T)\right|_{har}$ given by:

[TABLE]

Now that the correction to the single particle state $E_{n\mathbf{k}}$ is related to the Eliashberg function, one can estimate the correction to the band gap, $E_{g}(T)$, by defining band-edge Eliashberg function, $F_{g}$, which is simply defined ttte :

[TABLE]

where the sub-indices $CBM$ and $VBM$ refer to a given conduction and valence state. The equation 10 reveals that a positive (negative) region of $g^{2}F$ would then lead to an increasing (decreasing) gap as function of temperature in contrast to a negative region stands for a decreasing $E_{g}(T)$.

III Results and discussion

III.1 Temperature effect on the band gap

This part investigates the electron-phonon interaction in order to calculate and describe the temperature dependence of the electronic gap $(E_{g})$ for SiC, GeC and SnC hybrids. Fig.(2) displays the variation of the band gap with respect to the temperature ($T$) for the three carbon-based hybrids.

The value of the bandgap of both SiC and GeC show a monotonic decrease as a function of temperature. It follows that these two materials are semiconductors with normal behavior. Meanwhile, SnC shows a different behavior. Indeed, by varying the temperature from $0K$ to $100K$ , the band gap of the SnC configuration also increases as plotted in Fig.(2). Thus, SnC has an anomalous trend in this temperature range. However, for a temperature above $100K$, the gap of SnC starts to decrease. This gap behaviour makes SnC hybrid a semi-conductor of non-monotonic anomaly kind.

The projected phonon density of state in Fig.(3) reveals that the heavier is the atom the smaller is his contribution to the optical modes. In the three hybrids, all optical modes are mainly dominated by carbon atoms, while the acoustical modes are mostly dominated by the heaviest atoms, namely Si, Ge and Sn in SiC, GeC and SnC respectively.

To shed more light on the gap behavior for the three binary compounds, Fig.(4) plots the generalized Eliashberg spectral function** **for the CBM and VBM states to visualize the e-ph coupling strength for a given state $|n\mathbf{k}\rangle$ in SiC, GeC and SnC hybrids. Both SiC and GeC show a conventional semiconductor behavior for all phonon frequencies in agreement with the normal plot of their $E_{g}(T)$ previously highlighted in Fig.(2). When increasing the temperature, the band gap of SiC and GeC exhibits the same behavior like the bulk Si and Ge cardona . However, the Eliashberg function describing the SnC hybrid has two relevant regions. More precisely, the function behaves as in conventional semiconductor for all acoustic modes and higher frequency optical modes, while it shows an anomalous trend for lower frequency optical modes. The pace of the Eliashberg function explains the anomalous non-monotonic behavior of the SnC band gap as a function of temperature.

Generally, the Eliashberg function, defined in eq(9) is positive for the valence band maximum and negative for conduction band minimum. This opposite trend leads to the reduction of the gap. Regarding equation 10, a positive region of $g^{2}F$ corresponds to a temperature increasing of the gap energy** **while its negative region stands for a decreasing of $E_{g}(T)$. For the carbon based hybrids, one deduces that in SiC, the VBM and CBM Eliashberg functions are all normal for all phonon modes leading to a gap decreasing. For GeC, the CBM contributions are negative for all phonons while for the VBM, the contribution for the Eliashberg function are positive except for phonon with frequency around $250cm^{-1}$. Therefore, the resulting Eliashberg function arising from VBM and CBM contribution show a gap decreasing. Finally, the curve associated to the SnC sheet shows a strong anomalous behavior for the VBM due to the contribution of the high optical phonon modes. This anomalous behavior leads to the non monotonic variation of the gap when increasing the temperature.

III.2 Electron linewidth vs temperature

The following part investigates the electron linewidth arising from the electron-phonon interaction and determines the electron lifetime for all structures. Fig.(5) displays the pooling of the electronic band structure of stanene and the imaginary part of the self energy $\Gamma_{n\mathbf{k}}$ on a fine electron wavevector k-grid for two different temperatures namely, $0K$ and $300K$. The temperature $300K$ is chosen rather than other temperatures in order to investigate hot carrier behaviour at room temperature.

At $0K$, the variation of electron linewidth as a function of the electron energy is plotted in Fig.(5) along k-points on the high symmetry line for SiC, GeC and SnC respectively. The band structures of SiC, GeC and SnC show that the states in SnC are more closer from each other compare to SiC and GeC explaining the smaller linewidth for SnC with respect to SiC and GeC, which is more observable in the unoccupied states.

Recall that the imaginary part $\Gamma_{n\mathbf{k}}$ of the Fan’s self-energy provides information on the rate of e-ph processes responsible for hot carrier thermalization gius . The following focus on the carriers thermalization and their lifetime around the Fermi level for the three monolayers.

At $300K$, Figs.(5-d,-e,-f) show that the imaginary part of the self energy is more sensitive** **to the temperature in the conduction bands compared to valence bands. This is due to the fact that, the carrier in the conduction band are free to move which makes them very susceptible to any external perturbation such as temperature. This sensitivity also depends on the gap value as it gets increased when the gap value is decreasing. Indeed, when increasing the temperature, the electron linewidth in the SiC structure changes less significantly with respect to its two counterparts GeC and SnC having smaller gaps. Moreover, compared to GeC, the linewidth in SnC is more sensitive to temperature at the vicinity of the $M$-point. However, far of this symmetric point and around the Fermi level, the thermal sensitivity is quite similar in both GeC and SnC. The very small density of state makes the imaginary self energy very small and unchangeable in term of temperature dependence compared to other energy levels. The obtained results are in good accordance with the high thermal sensitivity observed for bulk silicon and bulk germanium due to their small gap values of $1.17eV$ and $0.74eV$ respectively sige . Finally, the used method gives accurate variation of the linewidth with respect to the electron energy for the three carbon based hybrids as this method allows to extract information about the linewidth for each energy level. This is promising for determining the exact relaxation time and engineering hot carriers cells hot .

III.3 Scattering rate and lifetime

Based on eq(6), the electron-phonon scattering rate is plotted with respect to the electron energy in Fig.(6) for the three hybrids SiC, GeC and SnC. Only the electron-phonon contribution is considered as it is the most dominant near the Fermi level compared to the electron-electron interaction. Indeed, as reported in eph111 ; eph2 , hot electrons essentially interact with phonon because of the small density of state and the low energy range around the Fermi level.

Fig.(6) shows a very small change of the scattering rate in term of temperature for the SiC hybrid. The shifting is more observable in GeC compared to SiC, and the displacement of the scattering rate is much more visible for SnC. All these observations are in good agreement with the results obtained for bulk germanium, bulk silicon sige and diamond diam .

Hot carrier relaxation time, that is inversely proportional to the imaginary part of the self energy, shows in Fig.(7) a high value near the band edges. Moreover, electronic energy regions with many bands results in a faster electron-phonon relaxation time of hot electrons by phonon emission. This behaviour is due to the large number of possible transitions. The hot electron lifetime plot corresponding to the SiC structure indicates that hot carrier relaxation near the band edge is characterized by relaxation time around 90 fs while faster relaxation time of 10-20 fs are found at energy more than 75 meV away from the band edge. For GeC, the analysis of Fig.(6-d) reveals hot electron lifetime around 100 fs near the Fermi level at room temperature. For an energy ranging from 200 meV up to the Fermi level, the relaxation time is much shorter at less than 50 fs. Finally around the edge of the band of SnC, the gap is characterized by a relaxation time limiting the thermalization of the rate of hot electrons in the interval ranging from 100 fs to 120 fs. At 100 meV away, the hot electron relaxation time ranges in the interval [10,90] fs . In the three carbon-based hybrids, the relaxation time observed near the band edges are slower than the 400 fs found for graphene grappp .

III.4 Linewidth and scattering rate phonon dependency

In general, the electron scattering rate arises from the contribution of acoustic and optical phonons together. Fig.(8) provides a quantitative analysis of the contributions from the individual phonon modes. In SiC hybrid, the scattering rate is mainly dominated by the out of plane acoustic mode $(ZA)$ over all energy levels. More precisely, $ZA$ mode is followed by the transversal acoustic mode ($TA)$ mode at VBM, while $ZA$ is followed by the longitudinal acoustic mode ($LA)$ mode at CBM. For higher energy levels, $ZA$ is followed by the optical transversal mode $(TO)$. When the Si atom is replaced by the much heavier Ge atom, the optical modes contribution is getting much important. As shown in Figs.(8-c,-d), the electron scattering is mostly due to the presence of $TO$ followed by** $ZA$ **modes for higher electron energy ranges in GeC. At the band edges of GeC, the CBM arise essentially from both $ZA$ and $ZO$ mode while the VBM is dominated by $ZA$ and $TA$. Unlike SiC, the major contribution arises from the optical modes in SnC. Indeed, both the VBM and the CBM are  mainly dominated by $ZO$ mode followed by $TA$ mode. For higher energy in the conduction band, $ZO$ is followed by $ZA$.

To shed more light on the $k$-dependence of the imaginary part of the self energy in the three binary compounds, Figs.(9) and (10) display the linewidth variation over the high symmetry line of BZ and electron state being at the $CBM$ and $VBM$. In these figures, the linewidth is projected over acoustic and optical phonon modes to (i) show their different contribution** depending on the high symmetry directions related to real space and to **(ii) deduce information regarding the linewidth and which phonon plays crucial role in this direction.

III.4.1 VBM

For electron state at the $VBM$, the examination of acoustic modes contributions plotted in Fig.(9) shows that the $LA$ mode is lower compared to $TA$ and $ZA$ in all configurations. Over the $\Gamma-K$ and $M-\Gamma$ directions, the highest contribution in the linewidth is attributed to the $ZA$ mode for SiC and also for GeC but with smaller ampleness. This result matches very well with their counterparts silicene and germanene ficheti . In SnC, Fig.(9-e) reveals that the linewidth arise mainly from both $ZA$ and $TA$ modes with a $ZA$ magnitude very small compared to GeC and SiC. At $K$-point, all acoustic modes have no contribution in the electron linewidth due to the weak interaction occurring because of the lack of states at the band edges as shown in Fig.(9).

For optical phonon, the curves plotted in Figs.(9-b,-d and -f) show that in SiC, the linewidth is dominated by $ZO$ mode all over the high symmetry line, followed by $LO$ and $TO$. In both GeC and SnC, the main contribution arises also from the out of plane mode $ZO$.

III.4.2 CBM

For electron state at the CBM, Fig.(10) plots the electron linewidth projected over phonon modes. In SiC, the largest contribution comes from $ZA$ mode over the high symmetry line, however the linewidth is shared into three parts over the high symmetry line as displayed for GeC. Indeed, the linewidth is dominated by the $TO$ and $ZA$** **in the $\Gamma-K$ region and only by $ZA$ in the interval $M-\Gamma$. While, the major contribution is attributed to the $LO$ mode along $K-M$. In SnC, the electron linewidth is dominated by $ZO$ mode for all the BZ high symmetry line. It is worth noting that in SiC and GeC, the linewidth is zero for all phonon modes at $K$-point. However, for both $TO$ and $ZO$ mode, the linewidth is not zero in SnC. This difference is due to the indirect gap of SnC as the band edge is located at $K$-point for VBM and at $\Gamma$-point for CBM.

Finally, by evaluating the linewidth over the typical edge of the Brillouin zone in SiC, GeC and SnC, one can deduce that there exists a large variation in the magnitude of the electron-phonon coupling along the different crystallographic directions.

III.5 Conclusion

Using ab-initio calculations, we reported a theoretical description of the electron-phonon coupling effects on the gap, linewidth and scattering rate in two-dimensional SiC, GeC and SnC materials. In addition to the study of the thermal sensitivity of the linewidth, we investigated its k-resolved behavior. This type of study is important for transport phenomena where phonons play a crucial role and therefore has an impact on the engineering concepts of devices based on nanostructures. While the band gap in SiC and GeC showed a normal monotonic dependence to temperature, its behavior was rather anomalous in the SnC semiconductor. It has been found that the out-of-plane acoustic transverse mode $ZA$ efficiently couples to the electronic states at the SiC band edge, while the linewidth mainly arises from the $ZA$ and $ZO$ modes in GeC. In SnC, the optical mode, namely $ZO$, was predominant. The hot carrier thermalization losses in SiC, GeC and SnC under solar lighting were also calculated. It was found that, under sunlight illumination, the hot carriers thermalize within 90 fs, 100 fs and 120 fs for SiC, GeC and SnC respectively. This work highlights the microscopic origin of the thermalization of hot electrons, which is difficult to deduce from experiments and opens the way toward the study of materials for renewable energies by means of ab intio calculations of hot electrons.

Acknowledgment 1

L. B. Drissi and N. B-J. Kanga would like to acknowledge ”Académie Hassan II des Sciences et Techniques-Morocco” for financial support. L. B. Drissi, F. Djeffal and S. Haddad acknowledge the Extended African Network for Advanced 2D Materials funded by ICTP, Trieste, Italy for networking. L. B. Drissi, S. Lounis and F. Djeffal thank also the Arab German Young Academy of Sciences and Humanities (AGYA) sponsored by the Federal Ministry of Education and Research (BMBF) for the membership scheme. S. L. acknowledges support from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (ERC-consolidator Grant No. 681405 DYNASORE).