The chemical functionalization of the planar GeC sheet by fluorine disrupts its native π-bonding network, inducing a re-hybridization of the carbon and germanium atoms to an sp³ state. Consequently, the planar structure of GeC transforms into a puckered conformation. The calculations in this work focus on this fully fluorinated system, specifically in its stable chair conformer. As depicted in Figure 1, this structure features fluorine atoms bonded to carbon on one side of the sheet and to germanium on the opposite side, creating an alternating “1-up/1-down” arrangement.

Figure 1. Fully fluorinated GeC atomic structure: (a) Top view and (b) slide view. The lattice parameter a = 3.33 Å, the bond length Ge-C d = 2.02 Å, the buckling Δ = 0.60 Å.

In the following, we apply this formalism to investigate the effect of full fluorination on the e-ph interaction in the GeC sheet. We conduct a detailed analysis of how different phonon modes contribute to the electron linewidth, tracking these contributions as a function of temperature and electronic energy along the high-symmetry lines of the Brillouin zone. The results of this investigation are presented in Figure 2, which displays the electronic band structure overlaid with

Figure 2. Imaginary part of the electron self-energy Γ n , k due to electron-phonon coupling on top of the band structure of the fully fluorinated GeC at (a) 300 K and (b) 0 K.

the imaginary part of the self-energy, calculated on a fine k-grid for both 0 K and 300 K. It is worth noting that understanding the thermalization of hot carriers depends on analyzing the rate of e-ph processes. This rate is quantitatively described by the imaginary part of the electron self-energy Γ n , k expressed in Equation (2). The evaluation of this electron linewidth, as detailed in [41], requires a simultaneous consideration of the e-ph matrix elements, energy conservation, and phase space occupation.

As shown in Figure 2(b), a non-negligible electron linewidth emerges for states located away from the band gap, specifically impacting electrons in the higher conduction bands and holes in the lower valence bands. Consistent with the 0K condition, where thermal excitations are frozen and phonon absorption is suppressed, this observed e-ph interaction originates purely from spontaneous phonon emission. This entire analysis, which links specific scattering rates to carrier energy levels, is enabled by plotting the linewidth as a function of energy along the high-symmetry path.

As expected, the linewidth is negligible at the Γ-point for both the VBM and the CBM. This minimal scattering is a direct result of the limited phase space available for transitions, specifically the very low density of electronic states at the band edges. A small δ-function width of 0.005 eV was employed in the calculations to accurately resolve these small linewidth values.

At 300 K, Figure 2(a), the linewidth becomes significantly enhanced, particularly near the Γ-K path, reflecting the thermal activation of phonon absorption channels. Regions with relatively flat dispersion, such as the deep valence bands below −4 eV, exhibit higher linewidths due to the larger density of electronic states and reduced carrier velocity. In contrast, states around the band gap show negligible linewidth, consistent with the absence of available electronic states for scattering. These observations confirm the strong temperature dependence of the electron-phonon coupling in fully fluorinated GeC and highlight the dominant role of thermally activated phonon modes in carrier relaxation at room temperature.

The choice to investigate hot carrier behavior in fully fluorinated GeC at 300 K, rather than at another temperature, is motivated by our aim to study this behavior under room-temperature conditions.

Plotting the electron linewidth over the band structure provides a direct visualization of how electron-phonon interactions vary across different electronic states. It highlights regions where electrons experience stronger scattering and shorter lifetimes. This approach gives valuable insight into the strength and distribution of electron-phonon coupling within the material.

In this present work, the electron-electron interaction is neglected, as the e-ph contribution is the dominant scattering mechanism near the Fermi level. This assumption is justified for fully fluorinated GeC, where the very low electron density and the small energy range around the band edge ensure that electrons interact primarily with phonons. This approach is well-supported by previous studies [42][43], including key work on hot carriers in silicon.

The total e-ph scattering rate, which arises from the combined contributions of both acoustic and optical phonons, is therefore the focus of our analysis. A critical factor influencing this rate is temperature. An increase in temperature enhances lattice vibrations, which in turn expands the available phase space for scattering events.

Consequently, this process amplifies the scattering rates in both the valence and conduction bands. This trend is clearly quantified in Figure 3(a), which plots the scattering rate versus electron energy. A direct comparison of the 0 K and 300 K data Figure 3(a) vividly illustrates this temperature-induced increase, providing a precise quantification of the electron linewidth’s dependence on temperature. This temperature-driven increase in scattering is also directly reflected in the linewidths presented in Figure 2, as the linewidth itself is a quantitative measure of the total scattering rate.

The hot carrier relaxation time τ n , k e − p h , which is inversely proportional to the imaginary part of the self-energy, is fundamentally dictated by the available phase space for scattering. Consequently, electronic energy windows populated with a higher density of bands facilitate a larger number of possible transitions, resulting in a faster (shorter) e-ph relaxation time. This general behavior is consistent with previous findings for pristine and semi-fluorinated GeC [14][44].

This principle is clearly illustrated by our results, as depicted in Figure 3(b). Near the band edges, where the density of states is limited, τ n , k e − p h exhibits a correspondingly high value, indicating slower relaxation. Our analysis quantifies the hot electron lifetime at 50 - 100 fs near the Fermi level at room temperature. Conversely, for hot carriers with excess energy 200 meV or more from the Fermi level, the relaxation time shortens dramatically to less than 20 fs as more transition pathways become accessible.

Interestingly, this 50 - 100 fs lifetime is nearly identical to the ≈100 fs relaxation time recently reported for unfluorinated GeC [14]. This suggests that full fluorination does not significantly alter the hot carrier dynamics, an observation we

Figure 3. Electron scattering rate at 0 K and 300 K and corresponding lifetime.

attribute to the conservation of the material’s direct band gap nature. This contrasts sharply with semi-fluorinated GeC with F bonded to C, which was found to have much slower relaxation [44]. That system’s indirect band gap inherently restricts the efficient e-ph scattering pathways that are available in its direct-gap for pristine and fully-fluorinated counterparts.

This investigation into carrier relaxation is therefore critical for assessing the material’s practical applications. Since the calculated linewidth is inversely related to the relaxation time, it provides a direct probe of the hot-carrier dynamics. Such information is fundamentally essential for the engineering of hot-carrier devices, where the primary objective is to extract carriers before they lose their excess energy through thermalization.

To identify the specific e-ph scattering pathways responsible for this rapid relaxation, we performed a quantitative analysis of the contributions from individual phonon modes to the scattering rates, as shown in Figure 4 for 0 K and 300 K. The analysis reveals a clear distinction between the contributions of acoustic and optical modes.

For the acoustic modes, the out-of-plane acoustic (ZA) modes provide a substantial contribution in the conduction band at both temperatures 0 K and 300 K, dominating the other acoustic modes. In contrast, the situation in the valence band is temperature-dependent: the longitudinal acoustic LA mode contribution is dominant at 0 K, whereas the ZA mode becomes dominant at room temperature 300 K. This strong contribution from the temperature-sensitive ZA mode is consistent with observations made for other 2D materials like silicene and germanene [45].

Regarding the optical modes, their contributions appear less sensitive to temperature variations. In the conduction band, the LO2, LO3, and TO2 modes are the main contributors at both temperatures. Similarly, in the valence band near the band edge, the longitudinal optical LO2 mode provides the dominant contribution and is also not significantly affected by temperature.

Based on these results, we can deduce that optical modes have a higher overall contribution to the scattering rates compared to acoustic modes. The ability to compute these detailed scattering rates in the absence of experimental data highlights the benefit of the present approach for predicting material properties.

Figure 4. Temperature-dependent contribution of individual phonon modes (acoustic and optical) to the e-ph scattering rate at 0 K and 300 K.

Beyond the total scattering rates at the band extrema, a more detailed understanding requires examining the momentum k-dependence of these interactions. To this end, Figure 5 and Figure 6 display the e-ph coupling linewidth projected onto longitudinal and transversal phonon modes along the high symmetry lines of the Brillouin Zone at 0 K and 300 K.

At the VBM, as shown in Figure 5, the contributions from acoustic and optical modes are distinct. For acoustic phonons, the linewidth at 0 K is mainly due to the LA mode across the entire BZ (Figure 5(a)). At 300 K, however, the configuration changes significantly, and the major contribution shifts to the ZA mode (Figure 5(c)), confirming its high sensitivity to temperature. In contrast, the optical phonons (Figure 5(b), Figure 5(d)) exhibit much larger linewidths, dominating the acoustic contributions over the whole BZ except at Γ point. At 0 K, the coupling is dominated by the LO2 mode, which represents the primary phonon emission process responsible for thermalization process. At 300 K, the contribution from the TO1 mode increases substantially due to the larger phonon population, becoming the dominant coupling partner. Thus, the overall temperature sensitivity at the VBM arises from both the TO1 optical mode and the ZA acoustic mode.

At the CBM, detailed in Figure 6, a different set of mechanisms is dominant. Among the acoustic modes, the ZA mode provides the strongest coupling and is, again, highly sensitive to temperature. However, the primary interaction is with the optical modes. At 0 K, the phonon emission process is clearly dominated by

Figure 5. Momentum-resolved electron-phonon coupling linewidth at the VBM along the high symmetry lines of the Brillouin Zone. Panels (a) and (b) show the contributions from acoustic and optical phonon modes at 0 K, respectively, while panels (c) and (d) present the corresponding contributions at 300 K.

Figure 6. Momentum-resolved electron-phonon coupling linewidth at the CBM along the high symmetry lines of the Brillouin Zone. Panels (a) and (b) show the contributions from acoustic and optical phonon modes at 0 K, respectively, while panels (c) and (d) present the corresponding contributions at 300 K.

the longitudinal optical LO3 mode, particularly around the K point. This LO3 mode remains the most dominant contributor at 300K and, notably, is not very sensitive to temperature. Based on these findings, we conclude that the main thermalization process at the CBM is governed by the LO3 optical mode, while the overall temperature sensitivity of the coupling is driven by the ZA acoustic mode.

At the VBM, as can be seen in Figure 7(a), the acoustic mode contributions are characterized by the out-of-plane acoustic ZA mode. The linewidth of the ZA mode exhibits a near-linear dependence on temperature, establishing it as the dominant acoustic contributor at elevated temperatures. In contrast, the contributions from the transverse TA and longitudinal LA acoustic modes are significantly smaller and display only a weaker temperature dependence. For the optical branches at the VBM, Figure 7(b) shows that, their overall linewidths are substantially larger than their acoustic counterparts. At temperatures below 150 K, the linewidth is primarily governed by the LO2 mode, followed by LO3. As temperature increases from 150 K to 400 K, the TO1 and ZO2 modes demonstrate greater temperature sensitivity, and the coupling hierarchy shifts. In this higher temperature regime, the dominant contributions come from LO2, followed by TO1 and ZO2. Nevertheless, the LO2 mode remains the principal optical contributor across the entire 0 - 400 K range, which is consistent with the findings in Figure 5.

Turning to the CBM, as depicted in Figure 7(c) shows that the ZA mode is again the most dominant acoustic contribution, with a linewidth that grows sharply with temperature, while the TA and LA modes remain subdominant. The optical mode behavior at the CBM (Figure 7(d)) is distinct. Here, the longitudinal LO3 mode provides the highest contribution to the linewidth by a significant

Figure 7. Mode-resolved e-ph coupling linewidth as a function of temperature at the electronic band edge. (a), (b) Acoustic and Optical mode contributions at the VBM, respectively. (c), (d) Acoustic and Optical mode contributions at the CBM, respectively.

margin. However, this dominant LO3 mode, along with the LO2 mode, displays the lowest sensitivity to temperature variations compared to the other optical branches.

A direct comparison reveals notable differences between the band edges. The absolute acoustic linewidth at the CBM is appreciably larger than at the VBM. Moreover, the ZA mode at the CBM is more temperature-sensitive than its VBM counterpart. The primary source of optical coupling also differs significantly. Indeed, while the LO2 mode governs the VBM, the LO3 mode overwhelmingly dominates at the CBM.