Time dynamics of photoexcited electron-hole pairs is important for a number of technologies, in particular solar cells. We combined ultrafast pump-probe Raman scattering and photoemission to directly follow electron-hole excitations as well as the G-phonon in graphite after an excitation by an intense laser pulse. This phonon is known to couple relatively strongly to electrons. Cross-correlating effective electronic and phonon temperatures places new constraints on model-based fits. The accepted two-temperature model predicts that G-phonon population should start to increase as soon as excited electron-hole pairs are created and that the rate of increase should not depend strongly on the pump fluence. Instead we found that the increase of the G-phonon population occurs with a delay of ~65 fs. This time-delay is also evidenced by the absence of the so-called self-pumping for G phonons. It decreases with increased pump fluence. We show that these observations imply a new relaxation pathway: Instead of hot carriers transferring energy to G-phonons directly, the energy is first transferred to optical phonons near the zone boundary K-points, which then decay into G-phonons via phonon-phonon scattering. Our work demonstrates that phonon-phonon interactions must be included in any calculations of hot carrier relaxation in optical absorbers even when only short timescales are considered.
The ultrafast dynamics and relaxation of hot carriers and phonons in a variety of materials investigated by pump-probe techniques has attracted a lot of attention in the last decade1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. Rapid development of pulsed lasers combined with increased utilization of optical absorbers16,17,18 stimulated spectroscopic investigations of materials on picosecond and sub-picosecond timescales under optically driven non-equilibrium conditions. Time-resolved experiments use intense laser pulses to create non-equilibrium states in materials and track their relaxation in real time. They provide information about unoccupied electronic states, electron-electron (e-e) interactions1,4,6, electron-phonon (e-ph) coupling9,14,19, phonon-phonon (ph-ph) coupling20,21, etc. Here we show that combining time-resolved photoemission with time-resolved Raman (TRR)22 scattering provides insights that are much more difficult to obtain otherwise. Angle-resolved photoemission spectroscopy (ARPES) views the process from electron perspective tracking electron-hole (e–h) excited state occupation as a function of time. Raman scattering used as a probe reveals the phonon perspective providing vibration frequency, lifetime, and occupation number of Raman-active phonons. The fundamental physical law of detailed balance allows extracting the phonon occupation number n from each Raman-active phonon directly from the ratio of the energy loss (Stokes) and energy gain (anti-Stokes) one-phonon Raman scattering intensities23. The anti-Stokes side, where the phonon intensities are proportional to n, is especially sensitive.
Up to now the data have been described in terms of the two-temperature (2T) model where hot electrons thermalize with a few “hot” phonons relatively quickly (few hundred fs) via strong e-ph coupling2,13,24. In this description, hot electrons and hot phonons are characterized by their temperatures. Other e-ph and ph-ph interactions are assumed to be much weaker resulting in the whole system reaching the final thermal equilibrium at much longer picosecond timescales.
Our investigation focused on graphite as a model electron-phonon system. TRR as well as trARPES has been previously applied to graphene, graphite, and carbon nanotubes1,2,3,20,25,21,26,27,28, but they have never been applied to one compound simultaneously as we have done in this paper. G-phonons in graphite and related materials are believed to play the role of “hot” phonons in the 2T description due to strong coupling to electrons. We observed G-phonon population (data points in Fig. 1(a)) as a function of time in the sub-picosecond region by Raman scattering and electron-hole population (black points in Fig. 1(b)) by ARPES with nearly the same time-resolution and pump fluence. The unexpected result is that G-phonon generation is delayed by about 65 fs from the prediction of the 2T model (green dashed line in Fig. 1(a)). Furthermore, the ARPES measurements show that the e-h pairs decay faster than expected from direct coupling to the G-phonon, and the G-phonon generation time constant is reduced with a higher pump power in contradiction with the 2T model. This suggests that hot carriers couple to G-phonons not directly but via an intermediate excitation that is not Raman active (red lines in Fig. 1(a,b)). We argue that K-phonons play this role (Fig. 1(c)). Our results agree with LDA calculation that predicts that e-h pairs predominantly decay into K-phonons29.
The Raman spectrum of graphite measured with TRR setup shown in Fig. 2(a) has two main features: the G peak at ~1580 cm−1 and the 2D peak at ~2700 cm−1 (Fig. 2(b)). The G peak is the zone center E2g phonon. The 2D peak is due to the two-phonon scattering at K points called double resonance30,31,32. These peaks measured by the 395 nm ultrafast probe pulse sit on a luminescence background and have large linewidths due to the finite bandwidth of the pulse. The absence of the defect-induced D peak (~1350 cm−1) indicates good quality of our sample. Figure 2(c) displays the anti-Stokes spectra at various delay times. We fit the phonon spectrum at each time delay with a Gaussian (Fig. 2(d)) and then extracted the integrated intensity as a function of time (Fig. 2(e) blue dots and inset).
The time evolution of the G-phonon can be fit with two exponential functions −exp[−(t − t0)/τgrowth] + exp[−(t − t0)/τdecay] convolved with the instrument response. The time constant τgrowth is 220 ± 20 fs for the quick buildup of G-phonons and τdecay is 2400 ± 83 fs for the slow decay (Fig. 2(e) inset). The fitting parameter t0 = 2 ± 15 fs. The G-phonon temperature is determined from the G peak intensity ratio on the Stokes and anti-Stokes sides IS/IAS = exp(ħΩ/kBTG), where ħΩ is the G-phonon energy, kB is the Boltzmann’s constant, and TG is the phonon temperature (Fig. 1(a)). TG is over 3000 K (over 1650 K for pump fluence 0.15 mJ/cm2 as shown in Fig. 1(a)) around t = 400 fs. It decreases to room temperature with a time constant of 2.4 ps, which agrees with previous results20,21. The slow decay has been attributed to the energy transfer from G-phonons to other phonon modes via anharmonic coupling25,33. The apparent oscillation around t = 600 fs in Fig. 3d might be due to imperfect laser power stability. We did not observe this oscillation in other datasets.
Figure 3 shows electronic dynamics probed by trARPES. Electronic temperature Tel was determined by fitting the slope of the high energy tail in the momentum-integrated ARPES spectra (Fig. 3 inset). Tel increases rapidly after an optical excitation and reaches 3400 K at t = 100 fs. Tel fits well to a bi-exponential decay with decay times of τfast = 0.35 ps and τslow = 3.5 ps. The decay of Tel has a similar timescale to the one observed by previous photoemission experiments1,2,5. We evaluated the ratio of the electronic heat capacity to G-phonon heat capacity (see supplementary) from the electronic temperature and G-phonon temperature. It is one of the parameters in our theoretical model as will be discussed later. The 2D Raman peak also reflects electronic temperature, though it is less precise than trARPES results due to self-pumping and greater statistical error (see Supplementary).
Figure 4 shows G-phonon dynamics with pump fluence 2.1 mJ/cm2 and 0.15 mJ/cm2 respectively. The phonon dynamics are similar for these two pump powers. Maximum phonon temperature decreases with lower pump fluence as expected. The generation of G phonons is slower with low pump fluence (τgrowth = 210 fs) than for the high pump fluence (τgrowth = 100 fs).
We now briefly discuss the electronic background in the Raman spectra, which has been assigned to hot carrier luminescence34. Figure 2(d) inset shows that the background starts to increase at t = −200 fs and reaches its maximum at t ~ 0 fs, when the G-phonons begin to appear. The background then decreases to the initial level at t = 300 fs, while the phonon intensity keeps increasing from t = 0 fs to t = 300 fs. This behavior reflects changes in electronic temperature and is consistent with previous reports34,35,36,37. However, it is not possible to read out instantaneous electronic temperatures from the observed intensities.
Our ultrafast pump pulse generates hot electrons and holes, which quickly thermalize via e-e interaction on femtosecond timescales1,2. These e-h pairs decay into phonons with strong e-ph coupling on longer timescales11,13,14,15; The hot phonons in turn slowly decay into other phonons, which carry energy out of the system20. Here we focused on graphite where there are no experimental complications due to the substrate and sample size issues. Our results also apply to graphene since its high energy electronic structure is similar. The states with energy less than the hot-phonon frequency are less relevant for the phonon dynamics but are relevant for other physics such as high electric and thermal conductivity resulting from massless Dirac fermions.
Previous experiments on graphite and graphene reported that hot carriers transfer most of their energy to hot optical phonons within 500 fs7,13,14,38,39. They observed a quick buildup of G-phonon occupation through e-ph coupling3,20, but they were not able to clearly resolve the short-time delay region and identify which transition is dominant. Due to finite time resolution, the G-phonons generated directly from e-h recombination are supposed to appear at negative times. Given our 90 fs time resolution, the increase of phonon population would be detected at ~−35 fs (see the green dashed line in Fig. 1). The abrupt appearance of G-phonons near t = 0 therefore suggests that G-phonon emission occurs with a time delay. The presence of a time delay is also clear when the probe pulse is used without the pump. In this case photons in the early-time part of a probe pulse pump the system, and the photons coming later but still belonging to the same pulse serve as an excitation for Raman scattering (This process is called self-pumping20). Due to the short time duration of the probe pulse (~50 fs), self-pumping integrates the excitations within the first 35 fs, which suggests that G-phonons are emitted more than 35 fs after a pump pulse, not about 10 fs as expected from direct decay of e-h pairs into this phonon.
Another experimental result contradicting the scenario of direct e-h recombination is the slower G-phonon buildup with lower pump fluence. The opposite is expected, because e-ph coupling was previously reported to increase with decreasing electronic temperature33,40.
Slow thermalization of carriers cannot explain these results. If hot carriers couple to G-phonons directly, G-phonons can be emitted regardless of whether carriers are thermalized or not as long as hot carriers occupy the states near EF. The inverse Auger process that dominates e-e scattering leads to a rapid accumulation of hot carriers near EF within 30 fs4.
The presence of the time delay and the unusual fluence dependence lead us to propose an alternative relaxation pathway where e-h pairs decay primarily into K-point optical phonons. These in turn decay into the G-phonon via strong ph-ph coupling (Fig. 1(c)). Consistent with this scenario, lifetimes calculated from e-ph coupling for K-phonons are 176 fs, 3 times faster than G-phonons41,42. Furthermore, strong 4-phonon scattering at high temperature has been reported43,44 with the inferred lifetimes for the decay of K-phonons into G-phonons of about 132 fs (latter based on the G-phonon linewidth of over 40 cm−1 at T > 2500 K44. The inverse process where K phonons decay to G phonons has the same time constant.). Thus the overall timescale for this indirect process, 300 fs, is consistent with the observed phonon buildup (~220 fs). The fact that 2D peak is more intense than 2D′ peak (double resonance for Γ phonons) also suggests a stronger e-ph coupling for K-phonons31,29,45.
In order to understand how K-phonons may be responsible for the late appearance of G-phonons, we simulate the optical phonon temperature based on the above scenario where anharmonic hot phonons are involved. Our model assumes that carriers thermalize rapidly into a Fermi-Dirac distribution with a well-defined temperature and that G phonons and K phonons thermalize among themselves. The electronic system, G-phonons, and K-phonons are characterized by their temperatures Tel, TG, and TK and are linked by e-ph coupling, Γe−ph(Tel, TG/K), and ph-ph coupling between G and K-phonons, Γph−ph(TG, TK). This leads to three coupled differential equations:
where I(t) is the laser irradiance, Cel and CG/K are the heat capacity of electrons and G/K-phonons respectively, T0 is room temperature, and τ = 2.4 ps is the measured decay rate of optical phonons into low-energy phonons through anharmonicity. I(t), Γe−ph(Tel, TG/K), and Γph−ph(TG, TK) contain adjustable parameters to fit experimental results (see supplementary for details). Combining ARPES and Raman results constrains the fits so that the same adjustable parameters have to describe the ARPES and the Raman data. Specifically, having to obtain the ARPES simulation close to the experiment constrains Cel.
The simulation without coupling between G and K phonons (Γph−ph = 0) is equivalent to the 2T model. It always gives a curve that is shifted from the Raman data to earlier times (Fig. 1(a)) because the 2T model predicts that the G-phonons are generated starting at t ~ −35 fs due to the finite width of the pulse. Note that the coupling constant λG in Γe−ph only determines the time constant of the G-phonon generation, but it has no influence on when TG starts to rise. As a result, the phonon dynamics in the early stage of relaxation is clearly not consistent with the 2T model.
We call the simulation with nonzero Γph−ph the anharmonic hot phonon model (AHP model), because it includes anharmonic couling between the G and K phonons. Due to energy and momentum conservation, the two K-phonons can decay into one G-phonon and one Γ phonon in the branch of lower energy, or decay into one G-phonon and one K-phonon in the branch of lower energy (Fig. 5). It agrees much better (Fig. 1(a,b)) with the data. Within this picture, pump of lower power (0.15 mJ/cm2) gives a slower G-phonon buildup (see Fig. 4) consistent with decreased anharmonicity at lower phonon temperature. So the inclusion of ph-ph scattering is essential for explaining the observed time delay.
Some previous experiments and calculations in the literature agree with out result. They are not consistent with the dominance of e-h channel in the G-phonon population buildup. For example, phonons are still emitted efficiently in the early stage even when doping puts EF well below or above half of G-phonon energy28. As the interband transition that contributes to G-phonon emission is suppressed with doping, the G-phonons would not be generated as fast as in the undoped samples by direct e-h recombinations, which suggests that G-phonons are emitted by other processes. However, all these results are consistent with our scenario.
To conclude, we applied TRR and trARPES to investigate e-ph relaxation in graphite within a few hundred femtoseconds. Our simulations combined with experimental results show that instead of electrons coupling to G-phonons directly, K-phonons are emitted via strong e-ph scattering and then transfer energy to G-phonons via ph-ph scattering. This relaxation pathway is also consistent with previously published experiments and calculations28,29,44,41, though the mechanism was not explicitly discussed.
Our findings have a broader implication for understanding of e–h relaxation processes in other materials such as light absorbers where maximizing photoexcited carrier lifetime is an important engineering challenge. For example in graphite and graphene, carrier recombination could be slowed down significantly by reducing the coupling between the K-point phonons and G-phonons. Thus it is essential to understand and control ph-ph interactions in addition to e-ph ones in order to design better optical absorbers and optoelectronic devices.
Our TRR setup (Fig. 2(a)) uses a 20 kHz 790 nm (1.57 eV) laser pulse from an amplified mode-locked Ti:sapphire laser as the pump pulse. The pulse width is about 50 fs. Second harmonic generation at 395 nm (3.14 eV) was used as the probe light source for Raman scattering. The cross-correlation measurement of the pump and probe pulses gave the time resolution of 90 fs. The scattered light was collected by a pair of parabolic mirrors and then analyzed on a McPherson custom triple spectrometer equipped with a water-cooled CCD detector. The spectral resolution is limited by the time-energy Heisenberg uncertainty principle for the probe pulse. The experiment was performed in air at room temperature. The fluence of the pump was from 0.15 mJ/cm2 to 5 mJ/cm2, and the probe fluence was 500 μJ/cm2. TrARPES used the same laser system with 1.57 eV for the pump with a fluence of 130 μJ/cm2 and 6.3 eV for the probe (Fig. 2(a)). The time resolution for trARPES based on a cross-correlation measurement was 110 fs.
Time zero in TRR was determined by measuring the sum frequency generation (SFG) intensity of pump and probe pulses on graphite. SFG that originates from the weak nonlinear dielectric polarization of graphite is greatly enhanced when pump and probe pulses arrive at graphite at the same time, which allows us to determine time zero accurately. The SFG intensity at various delay time fits well to a Gaussian function that gives an accuracy of ±4 fs in the determination of time zero. The 0.1 μm translation resolution of the delay stage only adds 0.67 fs to the time uncertainty. Therefore the total uncertainty of time zero is less than 5 fs.
How to cite this article: Yang, J.-A. et al. Novel Electron-Phonon Relaxation Pathway in Graphite Revealed by Time-Resolved Raman Scattering and Angle-Resolved Photoemission Spectroscopy. Sci. Rep. 7, 40876; doi: 10.1038/srep40876 (2017).
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The time-resolved Raman work at the University of Colorado was supported by the NSF under Grant No. DMR-1410111. The time-resolved ARPES work was supported by NSF grant DMR-1508785, with the equipment purchased in part through NSF MRI-1546961. We thank Justin Griffiths, Justin Waugh, Tom Nummy, Haoxiang Li, and Kyle Gordon for help with the experiments.
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