Abstract
Graphene is emerging as a viable alternative to conventional optoelectronic, plasmonic and nanophotonic materials. The interaction of light with charge carriers creates an outofequilibrium distribution, which relaxes on an ultrafast timescale to a hot FermiDirac distribution, that subsequently cools emitting phonons. Although the slower relaxation mechanisms have been extensively investigated, the initial stages still pose a challenge. Experimentally, they defy the resolution of most pumpprobe setups, due to the extremely fast sub100 fs carrier dynamics. Theoretically, massless Dirac fermions represent a novel manybody problem, fundamentally different from Schrödinger fermions. Here we combine pumpprobe spectroscopy with a microscopic theory to investigate electron–electron interactions during the early stages of relaxation. We identify the mechanisms controlling the ultrafast dynamics, in particular the role of collinear scattering. This gives rise to Auger processes, including charge multiplication, which is key in photovoltage generation and photodetectors.
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Introduction
Photonics encompasses the generation, manipulation, transmission, detection and conversion of photons. Applications of photonics are nowadays ubiquitous, affecting all areas of everyday life. Photonic devices, enabled by a continuous stream of novel materials and technologies, have evolved with a steady increase in functionalities and reduction of device dimensions and fabrication costs. Graphene has decisive advantages^{1}, such as wavelengthindependent absorption, tunability via electrostatic doping, large chargecarrier concentrations, low dissipation rates, high mobility and the ability to confine electromagnetic energy to unprecedented small volumes^{1,2}. These unique optoelectronic properties make it an ideal platform for a variety of photonic applications^{1}, including fast photodetectors^{3,4}, transparent electrodes in displays and photovoltaic modules^{1}, optical modulators^{5}, plasmonic devices^{2,6}, microcavities^{7}, ultrafast lasers^{8}, to cite a few.
Understanding the interaction of light with graphene is pivotal to all these applications. In the first instance, absorbed photons create optically excited (‘hot’) carriers. Their nonequilibrium dynamics can be very effectively studied by ultrafast pumpprobe spectroscopy. In this technique, an ultrashort laser pulse creates a strongly outofequilibrium (nonthermal) distribution of electrons in conduction band and holes in valence band. The optically excited carriers then relax, eventually reaching thermal equilibrium with the lattice. The relaxation dynamics, due to various processes, including electron–electron (e–e) and electron–phonon (e–ph) scattering, as well as radiative electron–hole (e–h) recombination, is then accessed by a timedelayed probe pulse (see Fig. 1). The timeevolving hotelectron distribution inhibits, due to Pauli blocking, the absorption of the probe pulse at lower photon energies with respect to the pump, thus yielding an increase in transmission (‘photobleaching’ (PB)). Thus, monitoring this transient absorption enables the direct measurement of the distribution function in real time. Here we are interested in the early stages of the dynamics, during which two main processes occur: (i) the initial peak produced by the pump laser broadens, due to e–e collisions, converging towards a hot FermiDirac (FD) shape in an ultrashort timescale^{9,10,11}. (ii) Then, optical phonon emission^{12} becomes predominant and drives a cooling in which the FD distribution shifts towards the Dirac point.
In order to experimentally access the very first stages of relaxation and to disentangle the role of e–e scattering from other mechanisms, it is necessary to probe at an energy lower than that of the pump and, at the same time, achieve the highest possible temporal resolution. Pumpprobe spectroscopy has been extensively employed to investigate relaxation processes in carbonbased materials: a variety of different samples have been studied, including thin graphite^{13}, few^{14,15,16} and multilayer^{17} graphene, but very few did experiments on singlelayer graphene (SLG)^{18,19,20,21}. Furthermore, the temporal resolution reported in earlier literature, either in degenerate (i.e., pump and probe with same photon energy) or twocolour pumpprobe, was in most cases ≥100 fs^{14,15,16,17,18,21}. This prevented the direct observation of the intrinsically fast e–e scattering processes. The electron distribution in graphene was also mapped by timeresolved photoemission spectroscopy^{22}; however, its evolution on the critical sub100fs timescale was not resolved. Thus, earlier studies mostly targeted the phononmediated cooling of a thermalized (but still hot) electron distribution, established within the pump pulse duration. To date, only Breusing et al.^{20} probed SLG with a resolution~10 fs. However, this was a degenerate experiment, probing the relaxation dynamics on a limited spectral window.
The thrust of this work is to investigate the role of e–e interactions on the initial stages of the nonequilibrium dynamics. Even at equilibrium, e–e interactions are responsible for a wealth of exotic phenomena in graphene^{23}. They reshape the Dirac bands^{24,25} and substantially enhance the quasiparticle velocity^{25}. Angleresolved photoemission spectroscopy showed electron–plasmon interactions in doped samples^{24}, and a marginal Fermiliquid behaviour in undoped ones^{26}. Manybody effects were also revealed in optical spectra both in the infrared (IR)^{27} and in the ultraviolet ^{28}, where strong excitonic effects were measured^{28}. In the nonequilibrium regime, an extremely fast e–e relaxation on a timescale of tens fs was theoretically suggested in refs 29, 30, 31, 32. However, these pioneering approaches relied solely on numerical methods and, as discussed below, did not take full advantage of the symmetries of the scattering problem. This resulted in an uncontrolled treatment of crucially important ‘collinear’ scattering events (i.e., with the scattering particles’ incoming and outgoing momenta on the same line), thus characterized by a high degree of symmetry. A deeper theoretical understanding of collinear scattering events and, most importantly, their phase space, requires more work.
Here we combine extreme temporal resolution broadband pumpprobe spectroscopy with a microscopic semianalytical theory based on the semiclassical Boltzmann equation (SBE) to investigate e–e collisions in graphene during the very early stages of relaxation. We identify the fundamental processes controlling the ultrafast dynamics, in particular the significant role of Auger processes, including charge multiplication.
Results
Samples and pumpprobe experiments
SLG is grown by chemical vapour deposition (CVD) and transferred onto 100 μm quartz substrates (see Methods). These are selected because they induce negligible artifacts in the pumpprobe experiments, as further verified here by measuring the uncoated substrates (see Methods). We perform twocolour pumpprobe spectroscopy using fewopticalcycle pulses (see Methods). We impulsively excite interband transitions with a 7 fs visible pulse at 2.25 eV (2–2.5 eV bandwidth) and probe with a 13 fs IR pulse (1.2–1.45 eV bandwidth), as well as a redshifted 9 fs IR pulse (0.7–1.2 eV bandwidth). The density of photoexcited electrons is~10^{13} cm^{−2} (see Methods), while the equilibrium carrier density before photoexcitation corresponds to ~200 meV hole doping (see Methods). The availability of such short IR probe pulses allows us to follow the electron population as it evolves towards a FD distribution. Our instrumental response function (full width at half maximum of the pumpprobe crosscorrelation) is <15 fs^{33}, with a crucially important order of magnitude improvement compared with previous twocolour studies^{16,17}. This allows us to directly probe the e–e dynamics.
Figure 2a plots the two dimensional (2d) map of the differential transmission (ΔT/T) as a function of pumpprobe delay in the 1.2–1.45 eV range. Even with our time resolution, Fig. 2b shows an almost pulsewidthlimited rise of the PB signal in the nearIR. This immediately points to an ultrafast e–e relaxation, taking place over a timescale comparable to our instrumental response function. The PB signature is nearly featureless as a function of probe wavelength, as expected in SLG, given the linear energymomentum dispersion of massless Dirac fermions (MDFs). The selected time traces at different probe energies undergo a biexponential decay, with a first time constant 150–170 fs, and a second longer one τ_{2}>1 ps. As in previous studies, we assign the first decay to the cooling of the hotelectron distribution via interaction with optical phonons^{16,17}, and the longer one to relaxation of the thermalized electron and phonon distributions by anharmonic decay of hot phonons^{34}. By varying the excitation intensity we observe a linear dependence of the PB peak on pump fluence, while its dynamics is nearly fluenceindependent.
A deeper insight into the e–e thermalization process can be obtained from the inset of Fig. 2b, showing a delay in the onset of the PB maximum at longer probe wavelengths. In addition, by comparing ΔT/T at selected probe delays (Fig. 2c) we observe that, at early times (~10 fs), the spectrum has a positive slope, peaking at high photon energy. Starting from ~20 fs, it progressively flattens and changes to a negative slope, which persists and increases at longer delays. To understand the data, we recall that ΔT/T is proportional to the transient electron distribution^{17,20} at time t. The sub10 fs 2.25 eV pump pulse creates an electron distribution peaking at~1.12 eV above the Fermi level (red line in Fig. 5e), while the probepulse samples the 0.6–0.72 eV interval. At early times, we, therefore, observe the tail of this distribution, with a positive slope. On the other hand, a thermal FD distribution, even with a typical doping usually present in asprepared SLG (~100–200 meV)^{35}, peaks at low photon energies, yielding a differential transmission with a negative slope. The transition from the nonthermal to the thermal regime, completed within~50 fs, is responsible for the change of slope of the ΔT/T spectrum. Figure 2d plots the ΔT/T dynamics measured with the second redshifted IR probe pulse. The high temporal resolution combined with the low photon energy allows us to observe an even clearer delay in the PB peak formation. In particular, the maximum ΔT/T is reached after 40, 60 and 80 fs for probe wavelengths of 1,050 nm (1.2 eV), 1,350 nm (0.92 eV) and 1,550 nm (0.8 eV), respectively. To the best of our knowledge, this is the first experiment setting a timescale for the outofequilibrium carrier thermalization in SLG with a direct measurement. Tuning the probe to longer wavelengths also allows us to follow the subsequent e–ph cooling of the carrier distribution. In fact, τ_{1} becomes longer (~400 fs at 1,550 nm) when probing at smaller photon energies, consistent with a distribution moving towards the Dirac point, before dissipating the excess energy into the phonon bath (Fig. 5e).
Semiclassical Boltzmann analysis
The ultrashort time needed for a hotelectron distribution to thermalize is a consequence of e–e collisions^{9,10,11,29,30,31,32}. We now theoretically investigate the possible Coulombmediated twobody collisions in SLG (Fig. 3). These include intra and interband scattering^{10,11}, ‘impact ionization’ or, equivalently, ‘carrier multiplication’ (CM, i.e., increase in the number of carriers in conduction band from valencebandassisted Coulomb scattering), and Auger recombination (i.e., decrease in the number of carriers in conduction band from valencebandassisted Coulomb scattering). In a CM process, e.g., electrons in valence band are ‘ejected’ from the Fermi sea and promoted to unoccupied states in conduction band. CM in graphene could thus have a pivotal role in the realization of very efficient photovoltaic devices and photodetectors with ultrahigh sensitivity. The findings of González et al.^{9} imply that, due to severe kinematic constraints for 2d MDFs, illustrated in Fig. 4a, these processes can only take place in a collinear scattering configuration. However, it was not noticed that CM and Auger recombination processes should be nearly irrelevant in graphene as they occur in a 1d manifold (as incoming and outgoing momenta of the scattering particles lie on the same line) embedded in 2d space. Indeed, Fig. 4c demonstrates that the phase space for CM and Auger recombination in 2d MDF bands vanishes. We will return to this issue later, in connection with the calculation of the e–e contribution to the collision integral in the SBE^{36}.
We use this knowledge of Coulombmediated collisions in the framework of the SBE, the standard tool to investigate electron dynamics in metals and semiconductors^{36,37}. We consider the equations of motion for the electron [f_{s,μ}(k)] and phonon [] distributions, including transverse and longitudinal optical phonons at the Γ and K points of the Brillouin zone^{38,39,40}. Here s=+1 (s=−1) labels the conduction (valence) band while the valley degreeoffreedom, μ=±1, indicates whether the electron wavevector k is measured from K or K'. The electron distribution is independent of the spin degreeoffreedom. The SBE describes (i) Coulomb scattering between electrons and (ii) phononinduced electronic transitions, where the energy of an electron decreases (increases) by the emission (absorption) of a phonon. In particular, the phonons at Γ are responsible for intravalley transitions, while intervalley transitions involve K phonons. Emission of optical phonons is crucial in the cooling stage of the dynamics and is possible because the photoexcited electrons energy (~1 eV) is much higher than the typical phonon one (~150–200 meV) (ref. 38). Finally, ph–ph interactions, arising from the anharmonicity of the lattice, are taken into account phenomenologically^{30,32}, by employing a linear relaxation term, parametrized by a decay rate γ_{ph}/.
We neglect acoustic phonons, as they are expected to modify the electron dynamics on a timescale^{34,41}>1 ps. Even when scattering between electrons and acoustic phonons is assisted by disorder, the socalled ‘supercollision’ process^{42}, relaxation times become of the order of~1–10 ps (ref. 42). These are still too long compared with those considered here. We note that radiative recombination was observed either in oxygen treated graphene under continuous excitation^{43}, not relevant for our case, or following ultrafast excitation in pristine graphene^{44}, but with quantum efficiency ~10^{−9} (ref. 44). We thus neglect radiative recombination.
To simulate the experiments, we solve the SBE with an initial condition given by the superposition of a FD distribution in equilibrium with the lattice at T=300 K and a Gaussian peak (dip) in conduction (valence) band, centred at =±1.125 eV, with a width of 0.09 eV. The width of the pump pulse determines the width of the Gaussian peak (dip) because dephasing effects during the pump pulse, as those reported for example, in Leitenstorfer et al.^{45}, contribute negligibly to the subsequent dynamics. Malić et al.^{30} argued that the anisotropy introduced in the electron distribution by the pump pulse disappears in a few fs due to e–e scattering. We thus enforce circular symmetry in our SBE to deal with timedependent distribution functions, f_{μ}(ε_{k,s}), not dependent on the polar angle θ_{k} of k, but on k and s=±1 only, through the MDF band energy ε_{k,s}=svk, where 10^{6} m s^{−1} is the Fermi velocity. Thus, the e–e contribution to the SBE for the electron distribution becomes:
where C_{μ}(ε_{1},ε_{2},ε_{3}) is the Coulomb kernel (see Methods) describing exchange of (momentum and) energy from ε_{1} and ε_{2} (incoming states) to ε_{3} and ε_{4}=ε_{1}+ε_{2}−ε_{3} (outgoing states) during a twobody (intravalley) collision. Energy and momentum conservation are automatically enforced in equation (1).
Circular symmetry allows us to treat the angular integrations in the Coulomb kernel analytically, taking particular care of the contributions arising from the subtle collinear scattering processes described above (see Methods). The contribution of intra and interband processes can then be cast into an integration over the allowed total momentum Q, see Fig. 4c.
CM and Auger recombination require additional care. As described in Fig. 4c, their phase space in the case of 2d MDFs vanishes if momentum and energy are conserved. This statement holds true for infinitely sharp bare bands with strictly linear dispersions. Nonlinear corrections to the MDF Hamiltonian in powers of momentum (measured from the Dirac point), however, appear due to lattice effects (for example, trigonal warping)^{23}. Their impact at the energy set by our pump laser is discussed in ref. 48. Nonlinearities appear also due to the inclusion of e–e interactions^{23}. These give rise to a selfenergy correction to the bare MDF bands, whose real part is responsible for the Fermi velocity enhancement^{25}. This correction becomes significantly large at low doping (≲10^{10} cm^{−2}) (refs 23, 25), while here we are interested in the nonequilibrium dynamics of a substantial population of photoexcited electrons (~10^{13} cm^{−2}). Most importantly, any effect giving a finite width to the quasiparticle spectral function, such as e–e interactions^{24}, opens up a phase space for Auger processes. To calculate the Auger contribution to C_{μ}(ε_{1},ε_{2},ε_{3}) we take into account these electronlifetime effects by a suitable limiting procedure (see Methods). We stress that the final result is independent of the precise mechanism limiting the electron lifetime.
Crucially, we go beyond the Fermi golden rule by including screening in the matrix element of the Coulomb interaction at the random phase approximation (RPA) level^{47}. To this end, we introduce the screened potential^{47} W(q,ω; t)=v_{q}/ε(q,ω; t), where q and ω are the momentum and energy transfer in a scattering event, respectively, v_{q}=2πe^{2}/(q) is the 2d Fourier transform of the Coulomb potential, is an average dielectric screening, depending on the media around the sample^{23}. The RPA dynamical dielectric function is ε(q,ω; t)=1−v_{q}χ^{(0)}(q,ω; t), where the noninteracting timedependent polarization function χ^{(0)}(q,ω; t) depends on the distribution function f_{μ} at time t:
Where the prefactor 2 accounts for spin degeneracy, while the chirality factor , which depends on the polar angle θ_{k} of k, is defined in Methods.
Collinear scattering has also a key role in the theory of screening of 2d MDFs. It takes place on the ‘light cone’ ω=vq when k+q is either parallel or antiparallel to k in equation (2). This implies a strong peak in the imaginary part of χ^{(0)}(q,ω;t) (that physically represents the spectral density of e–h pairs) close to the light cone^{23}, where diverges like ω^{2}−v^{2}q^{2}^{−1/2}. Dynamical screening at the RPA level suppresses Auger scattering. A different approximation, in which the impact of Auger processes is maximal, is the ‘static’ screening model, which consists in evaluating χ^{(0)}(q,ω;t) at ω=0. In this case, the collinear contribution to the screened potential vanishes. Note that χ^{(0)}(q,0; t) still depends on time through f_{μ}. RPA is a very good starting point to deal with screening in metals and semiconductors^{47}, but is certainly not exact. We thus modify the RPA dynamical screening function to interpolate the strength of Auger processes between its maximum (static screening) and minimum (dynamical screening). We thus introduce a third approximate screening model by cuttingoff the singularity of χ^{(0)}(q,ω; t) in the region (ω−vq)≤Λ of width 2Λ near the light cone. This regularized polarization function, , leads to a regularized screened potential . The smearing of the singularity of the polarization function deriving from the use of the cutoff parameter Λ is a result of nonlinear corrections to the graphene band structure^{46,48}.
We stress that static and dynamical screening models are free of fitting parameters and are applicable from the IR to the optical domain. However, the theory is expected to work better in the IR limit, as it is based on the lowenergy MDF Hamiltonian and thus neglects bandstructure effects, which become nonnegligible at high energy. We also note that Λ is a free parameter, not fixed by any fundamental constraint such as a sumrule. In the calculations based on the regularized screening model we set Λ=20 meV without fitting the numerical results to yield the best agreement with experiments. More details on our theoretical approach and other screening models are presented elsewhere^{46}. In particular, ref. 46 demonstrates that the results presented here are robust against wide changes in the parameter space.
Discussion
Figure 5 shows that the theory with dynamical screening compares (dotted curves in Fig. 5) poorly with experiments, in predicting both the prompt PB onset and its subsequent decay. While with static (and regularized dynamical) screening the calculated ΔT/T time traces are in good agreement with experiments, Fig. 5a–c, the dynamics is much slower in the presence of dynamical screening. This is best seen at low probe energies, Fig. 5b. The dependence of the ΔT/T maximum on probe wavelength, Fig. 5c, further highlights the large discrepancy between experiments and theory with dynamical screening. We trace back this discrepancy to the fact that, as stated above, dynamical screening completely suppresses Auger processes. We thus conclude that these processes are a crucially important relaxation channel for the nonequilibrium electron dynamics in graphene. Figure 5c shows that it would have not been possible to draw this conclusion without comparing theory with experiments in the lowenergy regime, i.e, for energies ε<0.6 eV. Note that, even though the numerical results in Fig. 5 refer to ~200 meV pdoping, they are robust with respect to the sign of the chemical potential. This is expected, as the density of photoexcited carriers created by the pump pulse is much larger than the equilibrium carrier density.
A closer inspection reveals that the thermalization of the initial hotelectron distribution, Fig. 5e, is accompanied by a very fast relaxation of the chemical potentials of conduction and valence bands, over few tens of fs. Auger processes are the only channel coupling the two bands, see Fig. 3c, and are thus responsible for this ultrafast relaxation, as e–ph scattering becomes dominant on much longer time scales of hundreds fs^{14,15}. Thus, sufficiently hot carriers provide enough energy for the promotion of electrons from valence to conduction band, resulting in CM, as shown in Fig. 5f. CM is crucial for graphene's application in photovoltaics and photodetectors, as it can maximize the number of carriers created for a given excitation/absorption process^{49}. Even though some evidence of CM was reported by Tani et al.^{50}, here we experimentally access the time window and the spectral coverage where direct spectroscopic signatures of CM can be obtained. In Fig. 5f we also illustrate the ratio between the energy stored in the electronic subsystem at time t and that at time t=0 (Methods), which can be understood as the ‘efficiency’ with which the energy transferred by photons is maintained by electronic degreesoffreedom (before being fully transferred to the lattice). As we can see from this plot, this is larger than ~50% for t<100 fs.
We emphasize that dynamical screening, when cured by cuttingoff the singularity of the polarization function along the light cone with the parameter Λ, improves the agreement with experiments when compared with static screening only. Indeed, the static theory underestimates screening as it misses collinear contributions to the polarization function. This explains why ΔT/T calculated with static screening: (i) increases too fast in the early stages, the electron dynamics being initially boosted by the poorly screened Coulomb repulsion, and (ii) slows down too much in the subsequent stages, when poorly screened carriers begin to accumulate close to the Dirac point.
The SBE neglects intra and interband quantum coherences and phase memory (nonMarkovian) effects. The effect of quantum coherences can be taken into account by employing the semiconductor Bloch equations^{29} or, to a greater degree of accuracy, quantum kinetic equations^{37}. Including quantum coherences is expected to significantly affect the results only on the very short timescale below 10 fs (see, in particular, Fig. 5b). Indeed, the processes that occur on the sub10 fs timescale are not significantly affecting the overall dynamics. The striking 20 to 50 fs risetime in the PB signal at lower photon energies and the subsequent decays are sufficient to discriminate between different solutions of the SBE including or not Auger processes (see, in particular, Fig. 5c for ε<0.5 eV). These spectroscopic signatures cannot be affected by coherences. For this reason, we argue that our assessment of the importance of Auger scattering in graphene does not depend on neglecting quantum coherences in the model. A similar argument can be made about phase memory, which is also expected to persist for ultrashort time scales, comparable with the experimental resolution^{51}.
In conclusion, we performed timeresolved spectroscopy on SLG with an unprecedented combination of temporal resolution and spectral tunability, allowing us to track the early stages of electron thermalization. A microscopic theory based on the SBE, including collinear scattering and screening, reproduces the experiments. The ultrafast relaxation dynamics can only be explained by considering CM and Auger recombination as fundamental mechanisms driven by electron–electron interactions.
We note that collinear Coulomb collisions in the intraband scattering channel yield logarithmically enhanced quasiparticle decay rates and transport coefficients (such as viscosities and conductivities)^{52}. Angleresolved ultrafast measurements of the hotelectron distribution may shed light on these important processes.
Ultrashort light pulses could be used to create a superhot plasma of ultrarelativistic fermions (MDFs) and bosons (e.g., phonons) in graphene or in other Dirac materials, thereby creating conditions analogue to those in early universe cosmology, but within a smallscale, tabletop experiment. Understanding the impact of collinear scattering on the ultrafast thermalization of MDFs can thus be of pivotal importance to achieve a deeper understanding of hot nuclear matter^{53}.
Methods
Graphene growth and transfer
SLG is first grown on copper foils by CVD^{54}. A ~25 μm thick Cu foil is loaded in a 4inch quartz tube and heated to 1,000 °C with an H_{2} gas flow of 20 cubic centimeters per minute (sccm) at 200 mTorr. The Cu foils are annealed at 1,000 °C for 30 min. The annealing process not only reduces the oxidized foil surface, but also extends the graphene grain size. The precursor gas, a mixture of H_{2} and CH_{4} with flow rates of 20 and 40 sccm, is injected into the CVD chamber while maintaining the reactor pressure at 600 mTorr for 30 min. The carbon atoms are then adsorbed onto the Cu surface, and nucleate SLG via grain propagation. Finally, the sample is cooled rapidly to room temperature under a hydrogen atmosphere at a pressure of 200 mTorr. The quality and number of layers of the grown samples are investigated by Raman spectroscopy^{55,56}. The Raman spectrum of graphene grown on Cu does not show any D peak, indicating the absence of structural defects. The 2D peak is a single sharp Lorentzian, which is the signature of SLG^{55}. We then transfer a 10 × 10 mm^{2} region of SLG onto quartz substrates (100 μm thick) as follows^{54}. Poly(methyl methacrylate) (PMMA) is spincoated on the one side of graphene samples. The graphene film formed on the other side of Cu foil, where PMMA is not coated, is removed by using oxygen plasma at a pressure of 20 mTorr and a power of 10 W for 30 s. Cu is then dissolved in a 0.2 M aqueous solution of ammonium persulphate. The PMMA/graphene/Cu foil is then left floating until all Cu is dissolved. The remaining PMMA/graphene film is cleaned by deionized water to remove residual salt. Finally, the floating PMMA/graphene layer is picked up using the target quartz substrate and left to dry under ambient conditions. After drying, the sample is heated to 180 °C for 20 min to flatten out any wrinkles^{57}. The PMMA is then dissolved in acetone, leaving the graphene adhered to the quartz substrate. A portion of the substrate is not covered with graphene, thus allowing the measurement of the nonlinear response of the substrate by a simple transverse translation of the sample. This contribution is measured to be negligible.
The transferred graphene is then inspected by optical microscopy, Raman spectroscopy and absorption microscopy. After transfer, the 2D peak is still a single sharp Lorentzian, indicating that SLG has indeed been transferred. The absence of D peak proves that no structural defects are induced during the transfer process. Raman measurements over a large number of points indicate a ~200 meV pdoping^{35,58,59}.
Pumpprobe spectroscopy
The transient absorption spectroscopy setup is driven by a regeneratively amplified modelocked Ti:Sapphire laser (Clark Instrumentation) that delivers 150 fs pulses at 780 nm with 500 μJ energy at 1 kHz repetition rate. The laser drives three optical parametric amplifiers (OPAs), from which the visible pump pulses and the two nearIR probe pulses are generated. These are then compressed to the transformlimited duration by means of custom made chirped mirrors (visible OPA), a fused silica prism pair (IR OPA 1) and an adaptive pulse shaper based on a deformable mirror (IR OPA 2) (ref. 33). The pump and probe pulses are synchronized by a motorized translation stage and spatially overlapped on the sample. After the sample, the IR OPA 1 probe pulse is focused onto the entrance slit of a spectrometer equipped with a 1024 pixel linear Si photodiode array (Entwicklungsbuero Stresing). The IR OPA 2 probe pulse is instead detected by an InGaAs CCD (chargecoupled device) spectrometer (Bayspec Super Gamut). Both these devices allow a full 1 kHz readout of the spectra. By recording pumpon and pumpoff probe spectra, we extract ΔT/T as a function of pumpprobe delay (t) as ΔT/T(λ,t)=[T_{on}(λ,t)−T_{off}(λ,t)]/T_{off}(λ,t). The system has a sensitivity better than ΔT/T=10^{−4}. The pump intensity is reduced to avoid any sample saturation or highorder nonlinear effects (I_{pump}<1 mJ cm^{−2}). By moving from multichannel to singlewavelength detection, we are able to reduce the fluence by a factor 20, and observe a substantially unchanged dynamics. ΔT/T is lower than 0.007 at the PB maximum, and the signal from the substrate is found to be negligible. The density of photoexcited electrons is calculated to be 10^{−13} cm^{−2} given the ~10 nJ pump pulse energy on a spot size of radius 80 μm, corresponding to a fluence of 50 μJ cm^{−2}.
Semiclassical Boltzmann equation
The SBE for the electron distribution,
includes collision integrals due to e–e and e–ph scattering. The magnitude of the e–ph couplings is discussed in Piscanec et al.^{38}, Lazzeri et al.^{39}, Basko et al.^{40} Here we use the values for the phonons at Γ and , for the longitudinal and transverse phonons at K, respectively.
The equation for the phonon distribution,
includes the collision term due to e–ph scattering and the linear relaxation with ps^{−1} (ref. 32). The equilibrium phonon distribution function consists of the BoseEinstein thermal factor evaluated at the νth phonon branch , assumed dispersionless in the present treatment. This approximation is well justified as changes slowly with respect to the electron dispersion.
The Coulomb kernel in equation (1) reads:
where E ≡ ε_{1}+ε_{2}, k_{2} ≡ Q−k_{1}, k_{4} ≡ Q−k_{3}, and S is the sample area. The polar angle of k_{1} does not matter, while the modulus of k_{1} is equal to ε_{1}/(v). The Dirac delta distributions follow from the conservation of total energy E and momentum Q. In equation (5) we introduce the infinitesimal η in the argument of the third delta to relax energy conservation, which is restored by taking the limit η → 0. As shown in Fig. 4c, when η=0 the summand in equation (5) vanishes for Auger processes. In this case, it is important to first perform the summations over Q and k_{3}, and then take the limit η →0.
The squared matrix element (where the integers 1…4 indicate the dependence on s_{i} and k_{i} for i=1…4) includes a summation over spin degreesoffreedom and direct and exchange^{60} contributions to e–e scattering. It reads , where
is the matrix element of the Coulomb interaction in the eigenstate representation of the MDF Hamiltonian^{23}, with the socalled ‘chirality factor’^{23} and ω=(ε_{1}−ε_{3})/. Note that Coulomb scattering occurs only within a valley μ.
The contribution due to Auger processes to the Coulomb kernel can be calculated analytically:
The strength of e–e interactions is parametrized^{23} by the dimensionless finestructure constant α_{ee} ≡ e^{2}/(v). We use α_{ee}=0.9, as appropriate for graphene with one side exposed to air and the other to SiO_{2} (ref. 23).
ΔT/T is calculated as a function of wavelength λ=2πc/ω from the electron distribution via the following relation^{20}:
where is the finestructure constant and n_{F}(E) is the FD distribution. Here μ=±1 is not summed over and can be chosen at will, as the electron distribution is identical for the two valleys.
The CM is calculated as:
where is the electron population per unit cell in conduction band at time t, ν(ε)=2εA_{0}/[2π(v)^{2}] being the MDF densityofstates, and nm^{2} the area of the elementary cell.
The efficiency ε(t) at time t is calculated as ε(t) ≡ E_{el}(t)/E_{el}(0), where
is the energy stored in the electronic subsystem at time t.
The numerical solution of the SBE is performed with a fourthorder RungeKutta method. The electron energies are discretized on a mesh with a 25meV step. The screening function and the Coulomb kernel are updated in time at multiples of the integration step, depending on the speed of the relaxation, with more frequent updates (for example, every 2 fs at the beginning of the timeevolution). The numerical results are stable with respect to changes in these choices.
Additional information
How to cite this article: Brida, D. et al. Ultrafast collinear scattering and carrier multiplication in graphene. Nat. Commun. 4:1987 doi: 10.1038/ncomms2987 (2013).
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Acknowledgements
We thank Frank Koppens, Alfred Leitenstorfer, Leonid Levitov, Allan MacDonald, Justin Song and Giovanni Vignale for useful discussions. We acknowledge funding from MIUR ‘FIRB  Futuro in Ricerca 2010’  Project PLASMOGRAPH (Grant No. RBFR10M5BT), ERC grants NANOPOT and STRATUS (ERC2011AdG No. 291198), EU grants RODIN and GENIUS, a Royal Society Wolfson Research Merit Award, EPSRC grants EP/K017144/1, EP/K01711X/1, EP/GO30480/1 and EP/G042357/1, and the Cambridge Nokia Research Centre. Free software (www.gnu.org, www.python.org) was used.
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A.C.F., G.C. and M.P. conceived and coordinated the study. D.B., C.M., A.L. and G.C. carried out the pumpprobe experiments. D.B., C.M. and G.C. designed and built the experimental apparatus. Y.J.K., A.L., S.M., R.R.N., K.S.N. and A.C.F. prepared the samples. D.B. Analysed the experimental data. A.T. and M.P. developed the modelling and carried out the numerical simulations. D.B., A.T., G.C., M.P. and A.C.F. interpreted the data and the simulations and wrote the manuscript.
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Brida, D., Tomadin, A., Manzoni, C. et al. Ultrafast collinear scattering and carrier multiplication in graphene. Nat Commun 4, 1987 (2013). https://doi.org/10.1038/ncomms2987
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DOI: https://doi.org/10.1038/ncomms2987
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