Raman Linewidth Contributions from Four-Phonon
and Electron-Phonon Interactions in Graphene

Graphene has been studied Novoselov et al. (2004); Geim and Novoselov (2007); Akinwande et al. (2019) as an emerging atomically thin electronic and optoelectronic material and for thermal management Bonaccorso et al. (2010); Ghosh et al. (2008). Weak interactions between some acoustic phonon polarizations especially the flexural modes with the electronic and optical phonon excitations can give rise to hot electrons and overpopulated optical phonons Freitag et al. (2009); Chae et al. (2009) and limit the heat spreading contribution from low frequency phonons in graphene electronic and optoelectronic devices Sullivan et al. (2017). Meanwhile, the increased population of hot charge carriers can enhance the responsivity of graphene-based photodetectors Freitag et al. (2013). A detailed understanding of electron-phonon and phonon-phonon interaction is essential to understanding the transport properties and device performance of graphene and other 2D systems.

Raman spectroscopy provides an useful probe of the electron-phonon and phonon-phonon interactions in solid-state materials such as graphene Menéndez and Cardona (1984); Beechem et al. (2007); Ferrari and Basko (2013), and it has important implications for phonon anharmonicity. The Raman peak shift and linewidth depend on the coupling of the Raman-active optical phonon mode with electrons and other phonon polarizations. Prior first-principles studies Bonini et al. (2007) have investigated the contributions from electron-phonon and phonon-phonon scatterings to the linewidths of $G$ peak caused by Raman scattering of a zone-center optical phonon in graphene. The intrinsic phonon linewidth $\gamma^{\rm in}$ is expressed as $\gamma^{\rm in}=\gamma^{\rm e-ph}+\gamma^{\rm ph-ph}$ with $\gamma^{\rm e-ph}$ and $\gamma^{\rm ph-ph}$ representing contributions from the electron-phonon (e-ph) and phonon-phonon interactions (ph-ph), respectively. It was predicted that $\gamma^{\rm e-ph}$ decreases while $\gamma^{\rm ph-ph}$ increases with temperature ($T$), and the descending trend of $\gamma^{\rm e-ph}$ would dominate up to 800 K Bonini et al. (2007). Opposite to this theoretical prediction, prior experiments show a monotonically increased linewidths with $T$ in graphite, few-layer graphene, and supported monolayer graphene Kang et al. (2010); Berciaud et al. (2010); Lin et al. (2011); Montagnac et al. (2013). This contradiction between theory and experiment underscores the need for an in-depth examination of the relative strength of intrinsic electron-phonon and phonon-phonon interactions in graphene.

In this study, we employ first-principles methods that explicitly consider higher-order phonon anharmonicity based on recent advances Feng and Ruan (2016); Feng et al. (2017). Specifically, we account for both three-phonon scattering contribution ($\gamma^{\rm 3ph}$) and four-phonon scattering contribution ($\gamma^{\rm 4ph}$) in the calculation of $\gamma^{\rm ph-ph}=\gamma^{\rm 3ph}+\gamma^{\rm 4ph}$ without involving fitting parameters that were used in several prior studies Lin et al. (2011); Apostolov et al. (2012); Cuscó et al. (2016). We further utilize a recently developed temperature dependent effective potential (TDEP) formalism Hellman et al. (2013), which can be combined with four-phonon scattering for a unified treatment Ravichandran and Broido (2018); Xia (2018), to capture the phonon renormalization effect in graphene. While a prior work Yang et al. (2020) suggested that the four-phonon scattering channel is generally important and even dominant in the zone-center optical phonon linewidth in 3D dielectric crystals, our results show that considering the effect of temperature is necessary for accurately predicting $\gamma^{\rm 4ph}$ in pristine graphene and can allow accurate prediction of the Raman peak shift at finite temperatures. In particular, $\gamma^{\rm 4ph}$ in graphene would be greatly overestimated if the effect of temperature on the phonon self-energy is neglected. The calculated linewidth and peak shift agree well with previous experiments of supported graphene and our own measurements of clean suspended monolayer graphene. By considering not only EPI as in previous work Hasdeo et al. (2016) but also the $T$ dependence of EPI in addition to full anharmonicity, our calculations predict that $\gamma^{\rm e-ph}$ changes nonmonotonically with increasing doping level and temperature.

All first-principles calculations are implemented in VASP package Kresse and Hafner (1993) and QUANTUM-ESPRESSO package Giannozzi et al. (2009). The ShengBTE code incorporating four-phonon scattering Li et al. (2014); Han et al. (2021) is then used to calculate ph-ph scattering rates. The EPW package Poncé et al. (2016) is used to calculate e-ph scattering rates. To consider the phonon renormalization effect, we use TDEP to compute $T$-dependent effective interatomic force constants (IFCs). Further computational details are presented in the Supplemental Material sup .

We first present our results of phonon anharmonicity in pristine graphene. Figure 1 presents the first principles predicted Raman $G$ peak frequency, which is equivalent to the frequency of the zone-center $E_{2g}$ mode. The results show good agreement with the available experiments Calizo et al. (2007); Lin et al. (2011) and our own measurement of monolayer graphene sample grown by chemical vapor deposition and suspended over a circular hole Sullivan et al. (2017), which validates our choice of using the TDEP method to capture the temperature effects in graphene. This $T$ dependence of the Raman $G$ peak is a signature of anharmonicity that comes from both phonon-phonon interactions and lattice thermal expansion. The TDEP method intrinsically includes the impact of higher-order phonon-phonon interactions on the phonon frequency Kim et al. (2018). For the thermal expansion contribution, we directly use the first-principles-predicted lattice constants from Ref. Mounet and Marzari (2005) in our calculations. Consistent with Ref. Bonini et al. (2007), and despite the negative thermal expansion of graphene, the overall frequency shift still decreases with increasing $T$.

We next shift our focus to the calculation of phonon linewidth, which is related to the phonon-phonon scattering rate $\tau^{-1}$ as $\gamma^{\rm ph-ph}=\frac{\tau^{-1}}{2\pi}$. Figure 2(a) presents our calculated $\gamma^{\rm 3ph}$ and $\gamma^{\rm 4ph}$, which are expressed in the full width at half maximum (FWHM). The results convey two important insights. First, $\gamma^{\rm 4ph}$ is dominant over $\gamma^{\rm 3ph}$, even at room temperature. With rising $T$, $\gamma^{\rm 3ph}$ only slightly increases while $\gamma^{\rm 4ph}$ grows dramatically. Based on this finding, neglection of four-phonon scattering is the main cause of the opposite $T$ dependence calculated in Ref. Bonini et al. (2007) compared to experiments, as given in Fig. 2(b) (dash dot black curve). Second, we note that the modification of the phonon self-energy with $T$ is necessary for accurate calculation of $\gamma^{\rm 4ph}$, which exhibits a different $T$ dependence compared to $\gamma^{\rm 3ph}$. Without considering the phonon renormalization effect, the calculation would overestimate the four-phonon scattering rates especially at higher temperatures, as shown by the comparison between the red and orange lines in Fig. 2(a). In comparison, $\gamma^{\rm 3ph}$ is relatively insensitive to temperatures (see comparison between the dashed and solid blue lines in Fig. 2(a)). Similarly, a recent work on graphene Gu et al. (2019) based on an optimized Tersoff potential suggests that fourth-order IFCs show much stronger dependence on $T$ than third-order IFCs.

For pristine graphene, the EPI contribution to FWHM is given by Fermi’s golden rule Bonini et al. (2007):

where $\omega_{\mathbf{q}}$ is the phonon frequency, the sum is on $N_{\mathbf{k}}$ electron vectors $\mathbf{k}$. $g_{(\mathbf{k}+\mathbf{q})j,\mathbf{k}i}$ is the e-ph coupling matrix element for a phonon with wave vector $\mathbf{q}$ exciting an electronic state $|\mathbf{k}i\rangle$ with wavevector $\mathbf{k}$ into the state $|(\mathbf{k}+\mathbf{q})j\rangle$. $f_{\mathbf{k}i}(T)$ is the Fermi-Dirac occupation for an electron with energy $\epsilon_{\mathbf{k}i}$, $\delta$ is the Dirac delta used to describe the energy selection rule.

With the above results, Figure 2(b) shows our calculated intrinsic phonon linewidth $\gamma^{\rm in}$ of pristine graphene. The red solid line in Fig. 2(b) shows that $\gamma^{\rm in}$ calculated without temperature corrections would be well above the experimental data. With four-phonon scattering and phonon renormalization accounted for, the obtained renormalized linewidth agrees reasonably well with prior experiments and our own measurements of suspended monolayer graphene. While the absolute FWHM values of our calculation results fall slightly below the experimental data for pristine graphene, this discrepancy could be explained by the finite instrument resolution of the spectrometers used in the experiments, which is on the order of a few cm^-1 in our measurements and would broaden the measured linewidth. In contrast to the dash dot black line calculated in previous studies Bonini et al. (2007), the phonon linewidth for the $E_{2g}$ mode is not completely dominated by $\gamma^{\rm e-ph}$. Rather, its descending trend would be compensated and then outweighed by the growing $\gamma^{\rm ph-ph}$, which is mainly due to the increasing four-phonon scattering rates at higher temperatures. Thus, at all temperatures $\gamma^{\rm in}$ exhibits an increasing trend. Our calculations successfully explain this $T$ dependence observed in experiments.

Our calculations above also clearly show the contribution of the EPI to the $G$ band linewidth of pristine graphene, and especially it dominates the $\gamma^{\rm in}$ below 500 K. Prior calculations have demonstrated that the EPI in graphene can be tuned over a wide range by changing the carrier density through the application of a gate voltage Yan et al. (2007); Park et al. (2007); Si et al. (2013); Park et al. (2014); Kim et al. (2016); Yang et al. (2021). While it is not possible to apply a gate voltage to our suspended monolayer graphene sample, we investigate this effect by calculating the $G$ band linewidth arising from the e-ph scattering for graphene with doping of either electrons or holes.

Figure 3(a, b) shows the calculated $T$-dependent $\gamma^{\rm e-ph}$ of the $G$ band at different carrier densities $n$ for graphene. The contribution from the intrinsic ph-ph scattering ($\gamma^{\rm 3ph}$+$\gamma^{\rm 4ph}$), which is independent of carrier density, is also provided for comparison. It can be seen that $\gamma^{\rm e-ph}$ is significantly decreased with increasing $n$. This indicates that as the carrier density increases, the contribution of e-ph scattering to the linewidth decreases as compared to the contribution from ph-ph scattering. It is also seen in Fig. 3(a) that the $T$ dependence of the $\gamma^{\rm e-ph}$ is strongly dependent on the carrier concentration. For low electron densities, e.g., $n=\rm 2.1\times 10^{11}~cm^{-2}$, the calculated $\gamma^{\rm e-ph}$ decreases with increasing $T$, whereas it increases with $T$ for carrier densities above $\sim\rm 1.1\times 10^{12}\ cm^{-2}$. From Eq. 1, the temperature dependence of $\gamma^{\rm e-ph}$ is governed by $f_{\mathbf{k}i}(T)-f_{\mathbf{k}j}(T)$, which are closely associated with the sharpness of the Fermi function and the position of the Fermi energy relative to the threshold of the onset energy for vertical transitions of an electron from a $\pi$ valence band to a $\pi^{*}$ conduction band state Yan et al. (2007); Park et al. (2007). This energy corresponds to $\hbar\omega_{G}\approx 0.2$ eV. For low carrier density regimes ($E_{F}<\hbar\omega_{G}/2$), as the Fermi function is smeared out with increasing $T$, the number of empty electron states available for transition by absorbing the $G$ phonon is reduced, thus causing the $\gamma^{\rm e-ph}$ to decrease with increasing $T$. As the carrier density increases to reach $E_{F}>\hbar\omega_{G}/2$, the smoothing of the Fermi function with increasing $T$ makes part of the occupied electronic states available, and consequently $\gamma^{\rm e-ph}$ increases with $T$. These analyses are also applicable to the case of holes.

To examine the carrier dependence closely, Fig. 3(c) displays $\gamma^{\rm e-ph}$ of the $G$ band varying with $E_{F}$. The carrier density with respect to the Fermi energy is also included in Fig. 3(c) (blue curve). It is remarkable in Fig. 3(c) that the variation of $\gamma^{\rm e-ph}$ with doping level varies with $T$. Note that our calculation at 10 K is in reasonable agreement with recent experimental measurements Yan et al. (2007), and that at 300 K coincides with the previous theoretical prediction Lazzeri and Mauri (2006). The difference between experiment Yan et al. (2007) and our calculation at 10 K is mainly due to the local density variations in graphene samples. Following the Pauli exclusion principle, near the ground state ($T\rightarrow 0$ K) the vertical transitions can be allowed only when $E_{F}<0.1$ eV (corresponding to $\sim\rm 1.1\times 10^{12}\ cm^{-2}$), as illustrated in the inset of Fig. 3(d) on the left, where the energy selection rules are easily satisfied. When $E_{F}>0.1$ eV, the energy selection rules fully prohibit the $\pi\rightarrow\pi^{*}$ transitions, as illustrated in the inset of Fig. 3(d) on the right. Hence, as seen in Fig. 3(c), at $T=10$ K the $\gamma^{\rm e-ph}$ suddenly drops at $E_{F}=0.1$ eV. As $T$ increases, however, the smoothing of the Fermi function makes part of the occupied electronic states available, thus the $\gamma^{\rm e-ph}$ is smeared out also. As a consequence, the threshold of the Fermi energy, above which $\gamma^{\rm e-ph}$ vanishes, increases with $T$. It can be seen in Fig. 3(c) that at $T=300$ K the threshold of the Fermi energy increases to $\sim 0.2$ eV from $0.1$ eV at $T=10$ K, and it reaches $\sim 0.3$ eV at $T=600$ K. Note also that as $E_{F}$ increases, $\gamma^{\rm e-ph}$ exhibits a nearly symmetric reduction relative to the position of the charge-neutral Dirac point, which is closely related to the symmetry of the electron density of states (DOS) near the Dirac point, as shown in Fig. 3(d).

Figure 4 shows the calculated total linewidth of the $G$ mode at different carrier concentrations in comparison to available experimental data Lin et al. (2011); Berciaud et al. (2010). Our calculations show that the FWHM of the $G$ mode is extremely sensitive to the carrier density and can vary over a wide range as the carrier density changes. It is clear that after considering a residual charge density of $\rm 2.16\times 10^{12}~\mathrm{cm^{-2}}$ our calculation can well explain another Raman measurement Berciaud et al. (2010). This finding indicates that the variations of the $G$ band linewidth among reported experimental data can be attributed to the e-ph scattering contribution, which strongly depend on the carrier density. On the other hand, the $T$ dependence of the FWHM exhibits a strong doping dependence, which is strongly connected to the interplay between the ph-ph scattering and e-ph scattering. At low carrier densities, e.g., $n=\rm 2.1\times 10^{11}~\mathrm{cm^{-2}}$, the calculated linewith increases slowly with $T$, since e-ph processes make a dominant contribution to $\gamma$ and it partially compensates the increase of the linewidth due to ph-ph processes. At higher carrier densities, e.g., $n=\rm 2.16\times 10^{12}~\mathrm{cm^{-2}}$, the FWHM is completely dominated by the ph-ph processes, and it thus increase rapidly as $T$ increases. These results reveal the significance of the e-ph scattering in determining the $G$ band linewith and its temperature dependence in graphene.

In conclusion, we have investigated the $G$ band frequency shift and linewith of graphene by including phonon renormalization and the anharmonic three-phonon, four-phonon, and electron-phonon scattering contributions, using first principles. We reveal that four-phonon scattering, which was neglected in the past, plays an indispensable role in the $G$ band linewith of pristine graphene, although it is greatly weakened by phonon renormalization. When combining both ph-ph and e-ph scattering, our prediction successfully explains previous measurements and our own measurements. Our calculation also shows that the e-ph coupling contribution and its temperature dependence significantly varies with doping levels. By calculating the $G$ band linewidth at different carrier densities, we suggest that the variations among previous experiments results can be attributed to the e-ph coupling contribution. Our work provides important insights into the understanding phonon-phonon interaction and electron-phonon interaction of graphene and other 2D materials.