4. Results and Discussion
4.6 Temperature Dependence of Energies and Lifetimes of Zone-Center Phonons
The final stage of the analysis of our k-VACS results will now focus on the temperature dependence of the phonon energies and lifetimes of the ZO and E2g at the BZ center Γ-point. As we showed in section 4.2, the system size greatly affects the PSD spectrum quality, mainly through the shape details of the peaks, but has little effect on the position of its center. Therefore, for this last part of this work, in order to have the maximum accuracy possible in the determination of the peak widths, we have used the MD data for the larger 40x40 simulation box. Henceforth, the results presented will refer to this larger system. The k-VACS spectra at 𝑘 = (0,0,0) were collected for every temperature step and manually fitted with Lorentzian functions for the two peaks present at each spectrum. The temperature dependence of the peak positions and FWHM will be the focus of this last section of the present work.
4.6.2 Phonon Energies
The change in zone-center phonon frequencies for hBN are presented in Figure 29. On the left we see the change in the ZO frequency with temperature with respect to its value at T=300 K. It is readily visible that phonon energies feature a clear linear trend, so the curve slope was calculated from linear fitting of the data and was estimated at 𝜆𝛧𝛰,𝑡ℎ𝑒𝑜𝑟𝑦= −0.0254 𝑐𝑚−1𝐾−1. On the right we see the respective change in frequency with increasing temperature for the Raman-active E2g mode of hBN,
Figure 29: (Left) Change with temperature of the ZO phonon energy of 1-L hBN, with respect to the energy at T=300 K, linear fitting, and curve slopes. (Right) Respective change of the E2g phonon energy, as compared with Raman data by Seremetis et al. [70]. The curve slopes were also calculated from linear fittings.
along with experimental data from Raman scattering measurements [70]. The slope of the k-VACS simulations data was determined at 𝜆𝐸2𝑔,𝑡ℎ𝑒𝑜𝑟𝑦 = −0.0516 𝑐𝑚−1𝐾−1, while the experimental slope was calculated at 𝜆𝐸2𝑔,𝑒𝑥𝑝 = −0.0193 𝑐𝑚−1𝐾−1, almost half as large in magnitude as our calculations. This is an indication that the anharmonicity modelled by the Tersoff-2010 potential is too large; the phonon modes soften too fast with increasing temperature, which indicates that the strength of the interatomic force constants weakens significantly with increasing temperature. This claim will be further supported by the temperature dependence of the lifetimes, which will be presented shortly after.
Similarly with hBN, the phonon energies in graphene decrease linearly with temperature. The temperature dependence of the ZO and E2g modes for graphene is also presented in Figure 30. On the left, the change of the ZO mode frequency with temperature is presented, along with a linear fitting, from which we obtain the slope at 𝜆𝛧𝛰,𝑡ℎ𝑒𝑜𝑟𝑦= −0.0357 𝑐𝑚−1𝐾−1, which has a value very close to the slope we obtained for hBN above. On the right we see the change of the E2g mode frequency with temperature, also compared with available Raman scattering measurements [71], and linear fittings with their respective slopes. Again, in this case both the theoretical and experimental slopes in graphene are very close to the respective slopes for hBN, and the k-VACS slope obtained at 𝜆𝐸2𝑔,𝑡ℎ𝑒𝑜𝑟𝑦= −0.0504 𝑐𝑚−1𝐾−1 is about
Figure 30: (Left) Change with temperature of the ZO phonon energy of graphene, with respect to the energy at T=300 K, linear fitting, and curve slopes. (Right) Respective change of the E2g phonon energy, as compared with Raman data by Yang et al. [71]. The curve slopes were also calculated from linear fittings.
two times as large as the corresponding slope at 𝜆𝐸2𝑔,𝑒𝑥𝑝 = −0.02083 𝑐𝑚−1𝐾−1 for the experimental curve. Again, the overestimation of the theoretical mode softening for graphene as well, is strongly suggesting that the Tersoff-2010 potential and the parameters used have induced large anharmonicities to both systems.
4.6.2 Phonon Decay Rates and Lifetimes
Particle lifetimes is an intrinsically quantum-mechanical concept; in general, they arise from interactions with other particles, in the case that these interactions are sufficiently weak as not to induce significant changes in the energy spectral structure of the corresponding non-interacting system. It has been shown [62, 72] that, by employing a Green’s Functions’ and Feynman’s perturbative diagrammatic expansions formalism, the particles of the interacting system can then be re-interpreted as non-interacting Quasi-Particles (QP) “dressed” in a virtual cloud of other interacting particles and having finite lifetimes. The quasi-particles then have a re-normalized complex energy given by
𝜔(𝒌, 𝑗) → {𝜔(𝒌, 𝑗) + 𝛥(𝒌, 𝑗)} + 𝑖𝛤(𝒌, 𝑗) (122) Here 𝛥(𝒌, 𝑗) and 𝛤(𝒌, 𝑗) represent the real and complex part of the interacting system’s self-energy;
𝛥(𝒌, 𝑗) induces a frequency shift to the non-interacting particle’s energy, while 𝛤(𝒌, 𝑗) is linearly proportional to its decay rate. More specifically in the case of phonons, their main interactions are with other phonons due to the anharmonicity of the interatomic potentials (3rd and higher-order terms in its Taylor expansion). This is especially true for semiconducting and insulating materials like hBN, although for metallic and semi-metallic systems like graphene, interactions with the electrons close to the Fermi surface also contribute to their lifetimes. In any case, here we consider the decay rates and lifetimes arising only to the anharmonic phonon-phonon interactions (Γph-ph and τph-ph), as the electronic degrees of freedom are not explicitly considered in MD calculations.
Exact formulae for the phonon self-energies have been deduced by Maradudin [73] almost 50 years ago, but their calculation is tedious and still only treats interactions up to a finite order of anharmonicity. A significant simplification of the formulae of Maradudin is obtained by the use of the Klemens model [74], which treats zone center optical phonons (𝒌 = 0). The model considers only 3- phonon interactions where the anharmonic phonon decays into two daughter phonons of equal frequency 𝜔1 = 𝜔2 = 𝜔0/2 and opposite momenta, 𝒌1 = −𝒌2, by conservation of energy and momentum, called symmetric decay. A generalization of Klemens’ simple 3-phonon model is when the daughter phonons do not have the same frequency (asymmetric decay). The Feynman diagram of the 3-phonon asymmetric decay is shown in Figure 31. Balkanski et al. [75] have shown that in the case of Silicon a further generalization of
Figure 31: Feynman diagram of the 3-phonon asymmetric decay of an anharmonic phonon (generalized Klemens model).
Klemens’ model that includes 4-phonon interactions is needed to properly describe the temperature dependence of the shifts and widths of Raman peaks, Chatzakis et al. showed that in the case of graphite (and by extension in graphene), just the asymmetric 3-phonon decay is enough to describe the temperature dependence of the E2g phonon decay rate.
According to the asymmetric 3-phonon decay process of the generalized Klemens model, the decay rate Γph-ph is proportional to the term (1 + 𝑛1+ 𝑛2), where n1 and n2 are the Bose-Einstein phonon occupation numbers of the daughter phonons, namely
𝛤𝑝ℎ−𝑝ℎ(𝑇) = 𝛤0(1 + 1 𝑒−
ħ𝜔1 𝑘𝐵𝑇− 1
+ 1
𝑒−
ħ𝜔2 𝑘𝐵𝑇− 1
) (123)
where conservation of energy implies 𝜔1+ 𝜔2 = 𝜔(𝒌, 𝑗). Note that at large temperatures, where 𝑇 ≫ħ𝜔
𝑘𝐵 (classical limit), 𝛤𝑝ℎ(𝑐𝑙𝑎𝑠𝑠𝑖𝑐𝑎𝑙)(𝑇) ∝ 𝑎𝑇, whence the familiar phonon lifetime law
𝜏𝑝ℎ(𝑐𝑙𝑎𝑠𝑠𝑖𝑐𝑎𝑙)∝ 𝑇−1 (124)
The temperature dependence of the decay rate (or more specifically the FWHM of the PSD peaks, which is equal to half the decay rate of eq. (123)) of the zone center phonons in graphene are
presented in Figure 32. On the left we present the decay rate of the ZO phonon, while on the right we can see the decay rate of the E2g phonon, along with Time-Resolved Raman
Figure 32: (Left) Temperature dependence of the decay rate of the ZO phonon for graphene.
(Right) Temperature dependence of the decay rate of the E2g phonon for graphene, compared with Time-Resolved Raman experimental data on HOPG by Chatzakis et al. [76].
experimental data on Highly Oriented Pyrolytic Graphite (HOPG) by Chatzakis et al. [76], along with a theoretical fitting of the experimental data by eq. (123). The parameters used in that work were 𝛤0= 2.5 𝑐𝑚−1, ħ𝜔1 = 535 𝑐𝑚−1 and ħ𝜔2 = 1050 𝑐𝑚−1. In a previous work [77], Density Functional Perturbation Theory (DFPT) calculations by Bonini et al. [77] revealed that besides significant 4- phonon, electron-phonon and lattice expansion contributions to the phonon decay rate of graphite, 44% of the 3-phonon decay process was due to the symmetric decay channel and 55% was due to the asymmetric channel. Nevertheless, the experimental work of Chatzakis et al. presented above showed that the asymmetric channel alone is more adequate to describe experimental data.
The phonon lifetimes of the zone center modes in graphene across the temperature range are presented in Figure 33. On the left we see the results for the ZO mode and on the right the results for the E2g mode, along with the lifetimes corresponding to the experimental data of Chatzakis et al. [76].
The equation that converts the FWHM decay rates of phonons with their lifetimes is 𝐹𝑊𝐻𝑀(𝑐𝑚−1) = 1
2𝜋𝑐(𝑐𝑚/𝑠)∙ 1
𝜏(𝑠) =5.306
𝜏(𝑝𝑠) (125)
From the comparison of our results with the experimental data we first note again that the anharmonicity of the system is strongly overestimated by the interatomic potential, as the slope of the k-VACS curves are quite larger than both the experimental data and of the theoretical fit. The k- VACS curves of the decay rates also exhibit a linear trend, in accordance with the classical limit case of eq. (124), which is also extrapolated to the low-temperature, non-classical limit. The decay rates close to room temperature and the lifetimes in general, however, are remarkably close to experiment.
Besides, from the flat part of the theoretical fit of eq. (123) in Figure 32, we deduce that below around 100 K, quantum effects become dominant and our classical description becomes less and less reliable.
Figure 33: (Left) Temperature dependence of the ZO phonon lifetime of graphene. (Right) Temperature dependence of the E2g phonon lifetime of graphene, compared with the experimental Time-Resolved Raman data of Chatzakis et al. [76].
In Figure 34 we present the temperature dependence of the zone center phonon decay rates in hBN. On the left, the dependence of the ZO phonon is presented, while on the right we see the dependence of the Raman-active E2g phonon, along with Time-Resolved Raman experimental data at T=300 K by Katsiaounis et al. [78]. To our knowledge, no other data on the dependence of the E2g
phonon decay rate is to be found in the literature as of yet. Also, the corresponding lifetimes are
Figure 34: (Left) Temperature dependence of the ZO phonon decay rate of hBN. (Right) Temperature dependence of the E2g phonon decay rate of hBN, compared with the experimental Time-Resolved Raman data at T=300 K of Katsiaounis et al. [78].
presented in Figure 35 for the ZO (left) and for the E2g (right) phonons, along with the experimentally determined lifetime of the E2g phonons at room temperature corresponding to the Time-Resolved Raman data by Katsiaounis et al. [78]. The agreement of our results with experiment at room temperature is again remarkable. However, we expect large deviations at lower and at higher temperatures, in accordance with the previous analysis for graphene.
Figure 35: (Left) Temperature dependence of the ZO phonon lifetime of hBN. (Right) Temperature dependence of the E2g phonon lifetime of hBN, compared with the experimental Time-Resolved Raman data at T=300 K of Katsiaounis et al. [78].
We see that in both systems, the decay rates are overestimated and does not capture the low temperature flattening of the experimental curves, which is purely due to quantum effects, while the softening of the zone center modes with temperature is also overestimated. The above point towards an overestimation of the anharmonicity and a lack of proper description of the interatomic bonding by the potential used. However, the agreement with experiment at room temperature of the decay rates and the general trend of the lifetimes is most satisfactory, given the fact that we have calculated quantities that are inherently quantum mechanical, with purely classical methods, and with a very simple and relatively fast potential. We thus expect that with potentials specifically designed to capture the anharmonicity of these systems, future work will show a great agreement with experiments.
Conclusions
In this work, we have employed the k-space Autocorrelation Sequence (k-VACS) method, in order to study the temperature dependence of the phonon properties of monolayer graphene and hBN from atomic trajectory and velocity data obtained from Molecular Dynamics simulations, using the Tersoff-2010 potential. The dependence of the quality of the kVACS spectra on the parameters of the MD simulations was studied and it was found that larger simulation cells produce spectra with higher quality Lorentzian curves, due to the reduction of the effects of the Periodic Boundary Conditions with increasing system size. The phonon energies and lifetimes were extracted from the positions and the widths of the kVACS peaks by fitting with Lorentzian functions and the phonon dispersion curves for both materials were calculated, while the phonon Densities of State were directly calculated from the Fourier transforms of the velocities in the time domain. From the comparison of the calculated dispersion curves with experimental data one can get an estimate for the accuracy of bond-stretching and bending parameters of the interatomic potential used for the pristine materials, while for distorted systems, one can study the effect of the generalized distortion parameter (crystallite size, dopant concentration, strain, etc.) on the phonon energies and lifetimes. The temperature dependency of energies and lifetimes was also studied and comparison with available experimental data revealed the accuracy of the anharmonicity of the interatomic potential. Although the mode softening rate, the rate of change of phonon decay rates with temperature, as well as the energies of the LO+TO branches in hBN and the energies of the ZO branch in graphene, as produced with the Tersoff-2010 potential, were all largely overestimated, the overall performance of the potential was more than satisfactory. All this despite its lack of complexity, in comparison with other, more sophisticated but computationally much slower potentials routinely used with such systems. The importance of the calculation of phonon lifetimes as crucial parameters in applications of electrical and thermal transport phenomena should be also stressed; these parameters, which are inherently quantum mechanical in nature, are nevertheless hereby calculated by a method that is purely based in classical physics. Finally, as a closing remark one should note that the greatest advantage of this method is that it can accurately describe any system that can be modelled by MD, periodic or not, as long as it is in thermodynamic equilibrium. It can thus be used as a powerful tool to study systems of theoretically arbitrary complexity, as are the ones that pose the greatest scientific and technological interest.
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