Direct Access to Auger recombination in Graphene
Marius Keunecke, David Schmitt, Marcel Reutzel, Marius Weber, Christina Möller, G. S. Matthijs Jansen, Tridev A. Mishra, Alexander Osterkorn, Wiebke Bennecke, Klaus Pierz, Hans Werner Schumacher, Davood Momeni Pakdehi, Daniel Steil, Salvatore R. Manmana, Sabine Steil, Stefan Kehrein, Hans Christian Schneider, Stefan Mathias
DDirect Access to Auger Recombination in Graphene
Marius Keunecke, ∗ David Schmitt, Marcel Reutzel,
1, †
Marius Weber, Christina Möller, G. S. Matthijs Jansen, Tridev A. Mishra, Alexander Osterkorn, Wiebke Bennecke, Klaus Pierz, Hans Werner Schumacher, Davood Momeni Pakdehi, Daniel Steil, Salvatore R. Manmana, Sabine Steil, Stefan Kehrein, Hans Christian Schneider, and Stefan Mathias
1, ‡ I. Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany TU Kaiserslautern, Erwin-Schrödinger-Str. 46, 67663 Kaiserslautern, Germany Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
Auger scattering channels are of fundamental importance to describe and understand the non-equilibriumcharge carrier dynamics in graphene. While impact excitation increases the number of carriers in the conductionband and has been observed experimentally, direct access to its inverse process, Auger recombination, has so farbeen elusive. Here, we tackle this problem by applying our novel setup for ultrafast time-resolved photoelectronmomentum microscopy. Our approach gives simultaneous access to charge carrier dynamics at all energiesand in-plane momenta within the linearly dispersive Dirac cones. We thus provide direct evidence for Augerrecombination on a sub-10 fs timescale by identifying transient energy- and momentum-dependent populationsfar above the excitation energy. We compare our results with model calculations of scattering processes in theDirac cone to support our experimental findings.
Non-equilibrium light-matter interaction processes havebeen studied in graphene as a prototype system for funda-mental energy dissipation channels of non-thermal and hotcharge carriers in two-dimensional systems . In order toaccess these out-of-equilibrium properties on the femtosec-ond timescale, the optically-excited non-thermal charge car-rier distributions are commonly probed using ultrafast op-tical and photoemission spectroscopy.Due to the linear band dispersions with a vanishing densityof states at the Dirac point combined with the weak screen-ing of the Coulomb interaction in the two-dimensional ma-terial, many-particle electron-electron (e-e) interactions areparticularly strong. As a result, the excited charge car-riers thermalize to a hot Fermi-Dirac distribution on the ≈
50 fs timescale . Concurrently, scattering with op-tical phonons leads to an azimuthal thermalization and fur-ther cooling of the Fermi-Dirac distribution on a timescale of200 femtoseconds to a few picoseconds . Of particu-lar interest in the carrier thermalization and cooling dynamicsin graphene are Auger processes, where one carrier changesfrom the valence to the conduction band and vice versa, whilethe other involved carrier remains in the same band. Here,one distinguishes impact excitation (IE) and its inverse pro-cess Auger recombination (AR), which increase and decreasethe number of carriers in the conduction band, respectively.For example, Auger processes can lead to a population in-version in the conduction and the valence band, highlightingits potential as a gain medium in optoelectronic devices .However, while IE has been clearly identified experimentallyin graphene and other materials , to the best of our knowl-edge, a direct experimental verification and analysis of ARprocesses have remained elusive.In this manuscript, we provide direct experimental evidencefor AR on timescales as short as 10 fs. We show that ARinduces an energy- and momentum-dependent population inthe conduction band at energies higher than that reached bythe optical excitation itself. Moreover, depending on the ef- FIG. 1. Possible two-body e-e scattering events of non-thermalcharge carriers in n-doped graphene that are able to create popula-tion at energies higher than reached by the excitation itself. Excita-tion with 1.2 eV pump photons (black arrows) resonantly populatesstates around E − E F = . ficiency of AR vs. other scattering processes, we find dis-tinct temporal shifts of transient energy- and momentum-dependent carrier populations in the conduction band. Ourfindings are supported by model calculations of two-bodyCoulomb scattering processes in graphene.We specifically selected an n-doped epitaxial graphenesample for our study, because it is known that in thiscase strong optical excitation of charge carriers just above a r X i v : . [ c ond - m a t . o t h e r] N ov the Fermi-level is most advantageous for dominant Auger re-combination processes . Thus, the Dirac cone is located0.4 eV below the Fermi-level and excitation with 1.2 eV pho-tons results in a non-equilibrium electron distribution withits peak located at 0.2 eV due to resonant optical excitation( ≈ ± , we expect such high-energy electrons to be adetectable signature of AR in our time-resolved momentummicroscopy experiment. Moreover, the temporal evolution ofsuch a high-energy electron population will give us quantita-tive information on the Auger recombination scattering rateitself.We stress that the identification of electrons at energieshigher than those reached by the direct optical excitationmight not be sufficient proof for the presence of AR. Strongoptical excitation, as we carry out in our experiment, apartfrom creating a pronounced non-equilibrium situation alsoconsiderably increases the kinetic energy of the carriers.Coulomb scattering processes will therefore instantly con-tribute towards establishing a quasi-equilibrium at a higher“temperature.” This involves cascaded intra- and interbandscattering processes that also generate population at higherenergies in the conduction band [green and blue arrows inFig. 1 (b)]. The most important aspect for our experimentalstudy is now to realize that the build-up of high-energy elec-tron populations by multiple cascaded scattering processes ex-hibits distinct temporal structures that distinguishes it fromdirect band-crossing Auger recombination processes. In theformer case, build-up of population at higher energies requiresmore and more individual intra- and interband scattering pro-cesses: the higher the energy, the longer it takes to create pop-ulation. In contrast, for AR, energies up to about 1.2 eV can bereached within a single Auger scattering event, and potentiallyat much earlier times than expected when cascaded multipleintra- and interband scattering events are necessary to reach these energies. FIG. 2. Time-of-flight momentum microscopy data obtained on theDirac cone of n-doped graphene when excited with p -polarized IRpump pulses and probed with EUV light ( ∆ t =
10 fs). (a) 3D illus-tration of the collected dataset, showing the linear-dispersive valence( π ) and conduction ( π ∗ ) bands (marked by dashed lines), the darkcorridor (DC), the anisotropic optical excitation (black dashed ar-rows), and the photon-dressed sideband (SB). The shape of the Diraccone is indicated by a dotted line. (b) Selected ( k x , k y )-momentummaps at E − E F = − . . . E − E F = . In order to study the temporal structure of charge car-rier populations at energies higher than those reached bythe direct optical excitation, we now turn to our experimen-tal data collection and analysis. In our time-resolved pho-toemission experiment, we use a multidimensional data col-lection scheme that consists of a time-of-flight momentummicroscope in combination with a 1 MHz table-top ex-treme ultraviolet (EUV) high-harmonic generation beamline(p-polarized, h ν = 26.5 eV, pulse length ≈
20 fs, angle of in-cidence = 68°) . An exemplary time-resolved measurementis shown in Fig. 2 (a), which illustrates an ( E , k x , k y )-resolveddata set that is collected at ∆ t = +
10 fs after the maximum in-tensity of the pump pulse envelope (p-polarized, h ν = 1.2 eV,incident fluence: 6.5 mJ/cm , pulse length: 37 ± . Clearly visible is the ring-like struc-ture of the Dirac cone (black dashed lines as a guide to the eye)below the Fermi-level, and also (at lower intensity) above theFermi-level due to prior excitation with the 1.2 eV pump pulse[compare also with the ( E , k y )-map in Fig. 3 (a)]. Note thatin photoemission some parts of the Dirac cone are not visi-ble due to the so-called "dark corridor of graphene" (markedwith "DC") , and that the Dirac cone shows a deviationfrom the perfect circular shape in the momentum maps, called"trigonal warping" . In addition, it is known and also seenin our experiment that excitation with 1.2 eV laser pulses isanisotropic (indicated with black dashed arrows for better vis-ibility), which can be explained by a polarization-dependentmatrix element for the optical excitation . Hence, in ourcase, the excitation is primarily in ± k y -direction. All replicafeatures marked with "SB" (for sideband) are induced by thepump pulse, as we discuss in Refs. 32 and 37. However, thesefeatures (i.e., all features not marked with black dashed linesin Fig. 2) do not influence or contribute significantly to theobserved electron dynamics (see SI).Turning to the middle panel of Fig. 2 (b) we find thatthe highest density of charge carriers in the conduction bandabove E F is observed at E − E F = . E − E F ≈ . ∆ t = +
10 fs in Fig. 2), we can exclude significantcontributions of electron-phonon scattering.Figure 3 presents a detailed analysis of the sub-50-fs tempo-ral structure of these high-energy populations, which shouldbe distinct for AR vs. inter- and intraband scattering pro-cesses. In our analysis, we focus on a momentum slice forwhich k x =
0, where the conduction band is directly popu-lated in ± k y -direction by the optical excitation [compare blackdashed arrows in Fig. 2]. In Fig. 3 (a), we show the respective( E , k y )-cut and indicate the regions-of-interest (ROI) with col-ored boxes that are further analyzed in Fig. 3 (b). At this point,we make use of p -polarized pump light that generates photon-dressed sidebands (SB) and thus provides a direct calibrationof the time axis : An analysis of the SB intensity yieldsa direct cross-correlation of the pump and probe laser pulses,implicating that maximum SB intensity is reached when thepump and probe pulses are in temporal overlap [i.e., ∆ t = FIG. 3. Population dynamics in the conduction band of graphene. (a)( E , k y )-cut through the multidimensional data for k x =
0. The regionsof interest evaluated in (b) and (c) are indicated by colored boxes.The dotted line at E − E F = . ∆ t max ), obtained by Gaussian fitting ofthe time traces in the respective ROI between -20 fs and +30 fs, isindicated by colored arrows. The grey dashed line corresponds to across-correlation of pump an probe pulses, extracted from the time-dependent sideband yield. (c) Energy-dependence of ∆ t max ; errorbars correspond to the 1 σ standard deviation of the fit. For increasing E − E F , ∆ t max first increases, saturates, and finally decreases againfor energies where AR is expected to become the dominant scatteringevent. The brown dashed line is a guide to the eye through the data. region where the resonant optical excitation occurs [black boxin Fig. 3 (a)]. However, as we detail in the SI, we can excludea significant contribution of this sideband yield to the mea-sured dynamics in our further analysis.In Fig. 3 we evaluate the occupation of the conduction bandas a function of ∆ t separately for each ROI indicated in the( E , k y )-resolved data set. Interestingly, already for E − E F = . ∆ t = ∆ t max ≈ ± due to e-e scattering processes redistributing the charge carrierdensity already during the pump pulse, which is in agreementwith earlier reports . When evaluating these time traces forincreasing energies above the resonant excitation peak, we ob-serve that the maximum population is reached at even larger ∆ t max (blueish traces). At first glance, this is counter-intuitivein terms of excited state lifetimes when considering the re-laxation of charge carriers towards the Fermi level, see, e.g.Refs. 40 and 41. However, this behavior should indeed be ex-pected for the generation of this population via cascaded intra-and interband scattering processes: because of energy conser-vation and the phase space that is available for inter- and in-traband scattering events, more and more scattering events arenecessary to reach higher and higher energies. Consequently, ∆ t max must increase towards higher energies. However, atthe highest energies where we still detect electron population, ∆ t max decreases again (green and red trace). The temporal be-havior of the charge carriers at these high energies impliesthat these carriers have not been excited via multiple scat-tering events. Instead, they must have been excited to theseenergies in a single scattering event, which, because of en-ergy conservation, can only be AR. The time to reach maxi-mum population for the highest measured energy can be usedfor a quantitative estimation of the average Auger recombi-nation time τ AR , and at E − E F = . ∆ t max =10 ± τ AR < 10 fs. In Ref. 17,it was found that the maximum carrier multiplication due toIE was reached within 26 fs of the optical excitation. Thisrequires the average scattering time for IE to be well belowthis value, placing it also on the 10-fs timescale. Although adirect comparison of our work with Ref. 17 is hindered by themany parameters which influence Auger scattering times (e.g.phase space, Pauli blocking and screening), we conclude thata sub-10-fs AR time is reasonable.In order to obtain theoretical support of the experimentalfingerprints of AR processes, we carried out simulations oftwo-body Coulomb scattering processes in graphene at thelevel of Boltzmann scattering integrals (see SI for details).Exemplary results of these simulations are shown in Fig. 4.To keep the simulation and comparison simple, we initiatedthe dynamics in the simulation ( ∆ t = E − E F = . − . E − E F > . E − E F > . y FIG. 4. Computed time-dependent carrier densities integrated overthe energy range E − E F > . density in both cases. We do not observe a subsequent de-crease of the density in this energy range because we do notinclude electron-phonon interactions that would lead to sucha “cooling” behavior. However, the direct comparison clearlyshows that AR processes play a large enough role at high en-ergies such that its signatures can be picked up by the exper-iment. While our timescales are slightly longer than those inthe experiment due to our choice of the screening parameter,we conclude from the calculations that it is the large influ-ence of AR processes on the high-energy dynamics that makesthem detectable, even though the total carrier density at high-energies is extremely small.In conclusion, we have reported on the first direct ex-perimental observation of Auger recombination in graphene.We show that strong optical excitation of charge carriers inn-doped graphene will lead to significant AR, resulting in amacroscopic number of highly excited charge carriers at ener-gies higher than reached by the optical excitation itself. De-pending on the dominant scattering processes, a distinct tem-poral structure on a sub-50-fs timescale has been identifiedand can be used in the future for further studies on primarythermalization events in graphene. For example, it will behighly interesting to systematically vary the doping level inorder to control the relative contribution of AR, IE or othere-e scattering processes, and thereby tune the response to theoptical excitation. I. ACKNOWLEDGEMENTS
This work was funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) -217133147/SFB 1073, projects B07 and B03. G.S.M.J. andM.R. acknowledge funding by the Alexander von HumboldtFoundation. S.S. acknowledges the Dorothea Schlözer Post-doctoral Program for Women. D.M.P acknowledges supportfrom the Joint Research Project "GIQS" (18SIB07). Thisproject has received funding from the EMPIR programmeco-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation program.
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Shen, Physical Review Letters ,117403 (2012). upplemental Information:Direct Access to Auger Scattering in Graphene
Marius Keunecke, ∗ David Schmitt, Marcel Reutzel,
1, †
Marius Weber, Christina Möller, G. S. Matthijs Jansen, Tridev A. Mishra, Alexander Osterkorn, Wiebke Bennecke, Klaus Pierz, Hans Werner Schumacher, Davood Momeni Pakdehi, Daniel Steil, Salvatore R. Manmana, Sabine Steil, Stefan Kehrein, Hans Christian Schneider, and Stefan Mathias
1, ‡1
I. Physikalisches Institut, Georg-August-Universität Göttingen,Friedrich-Hund-Platz 1, 37077 Göttingen, Germany TU Kaiserslautern, Erwin-Schrödinger-Str. 46, 67663 Kaiserslautern, Germany Institut für Theoretische Physik, Georg-August-Universität Göttingen,Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Physikalisch-Technische Bundesanstalt,Bundesallee 100, 38116 Braunschweig, Germany a r X i v : . [ c ond - m a t . o t h e r] N ov he supplemental material contains additional TR-MM data excluding contributions of two-photon transitions, and sideband yield to the identification of AR. In addition, we provide detailson the simulations. I. EXCLUDING CONTRIBUTIONS OF TWO-PHOTON EXCITATION PROCESSES
In order to exclude the possible contribution of two-photon excitation processes to the occu-pation of high-energy regions of the conduction band [ E − E F ≥ . . ± . E − E F = . . ± .
2. We can therefore conclude that two-photon absorption does not contribute to the signalat 0.8 eV, and that the majority of the detected charge carriers must have been excited through e-escattering processes. Furthermore, when evaluating energies above 0.8 eV up to 1.4 eV, i.e. theenergy window that provides direct evidence for AR, we also extract a slope of 1 . ± . ∗ [email protected] † [email protected] ‡ [email protected] IG. 1. Fluence dependence of the charge carrier occupation in the conduction band obtained with p -polarized pump light. (a) Excitation diagram for possible two-photon processes (crossed arrows) that wouldterminate at E − E F = . E − E F = . ∆ t = K -point. The colored boxes indicate the regions of interest evaluated in (b). (c) Fluence dependence ofthe photoemission yield in the sideband (black), at E − E F = . E − E F ≥ . II. EXCLUDING THE INFLUENCE OF SIDEBAND INTENSITIES ON CARRIER SCATTER-ING ANALYSIS ( p -POLARIZED PUMP LIGHT) From Fig. 3 (a) of the main text, one can see that the sideband trace crosses the main Dirac conewhere optical excitation proceeds resonantly (black ROI). We calibrate ∆ t = ∆ t max . Such a scenariois not be expected if significant contributions of the black ROI would stem from sideband yield;instead, the dynamics measured in the black ROI are dominantly determined by the charge carrierdynamics in the conduction band. If still considering minor contributions of the sideband to theblack ROI, ∆ t max = ± s -polarized pump light.3 IG. 2. Identification of AR in the ultrafast, energy- and momentum-resolved, redistribution of non-thermalcharge carriers in the conduction band of graphene. The same data processing as in Fig. 3 of the main text iscarried out, but for s -polarized pump light. (a) ( E , k y )-cut at k x =
0. The regions of interest evaluated in (b)and (c) are indicated by colored boxes. The dotted line at E − E F = . ∆ t max ) inthe respective ROI is indicated with a colored arrow. Note that the SB labelled time trace is obtained froma weak sideband signal. (c) Energy-dependence of ∆ t max obtained by Gaussian fitting of the time tracesbetween -20 fs and +30 fs; error bars correspond to the 1 σ standard deviation of the fit. The sideband trace(grey) is used for the calibration of the time-axis (note that its plotting on the E − E F -axis is not physicaland therefore marked with a ∗ ). For increasing E − E F , ∆ t max first increases, saturates, and finally decreasesagain for energies where AR is expected to become the dominant scattering event. The brown dashed lineis a guide to the eye through the data. II. ADDITIONAL DATA FOR S-POLARIZED PUMP PULSES
Figure 2 shows TR-MM data obtained on n-doped graphene for s -polarized pump light (1.2 eV,17.3 mJ/cm ). The data is evaluated as done in the main text for the p -polarized case in Fig. 3.Similar as discussed in the main text, we observe a distinct shift of ∆ t max = 9 ± s -polarized pump-ing, no contributions from the laser-assisted photoelectric effect [S1] can be expected, as detailedin Ref. [S2]. The fact that the s -polarized data and analysis in Fig. 2 shows the same energy-dependent delay further supports that the intensity of the sideband in the p-polarized case does notcontribute significantly to our analysis procedure.5 V. SIMULATION OF SCATTERING DYNAMICSIV.1. Boltzmann Scattering
We outline the general treatment of electron-electron scattering for a solid in the single particlepicture. We calculate the time evolution of the electronic distribution functions f ν k in state | ν , k i ,where ν is the band index, by using the Boltzmann scattering integral [S3] ddt f k ( t ) = L n ¯ h ( π ) n Z d n q Z d n l (cid:2) V ν , µ µ , µ ( q ) (cid:3) ϒ ( ∆ E , t , Γ )[ − f ν k ][ − f µ l + q ] f µ l f µ k + q − f ν k f µ l + q [ − f µ l ][ − f µ k + q ] (1)which describes the two-fermion scattering transition l → l + q and k + q → k mediated by thescreened Coulomb interaction. The energy difference ∆ E between initial and final states entershere via ϒ ( ∆ E , t , Γ ) = ¯ h Γ + ∆ E (cid:16)h ∆ E sin (cid:16) ∆ E ¯ h ( t − t ) (cid:17) − Γ cos (cid:16) ∆ E ¯ h ( t − t ) (cid:17)i e − Γ ¯ h ( t − t ) + Γ (cid:17) . (2)where Γ is the imaginary part of the self-energy. This expression approaches ϒ → ¯ h δ ( ∆ E ) atlonger times and for Γ →
0, which conserves kinetic energy, but at short times it leads to anincrease of the kinetic energy of our system during the building up of correlations. [S4] In orderto have a computationally feasible model, the self energy will be assumed as constant Γ . A finiteconstant Γ also leads to an increase in energy and we choose Γ = . IV.2. Graphene
For the numerical solution of the dynamical equation for the distribution we employ some sim-plifications, which are described below. The bandstructure is linearized around the Γ point[S6]with a slope of 650 meV/nm, and the matrix elements are calculated using the linearized disper-sions. The Coulomb matrix elements can then be written as V ν , µ µ , µ = V Dq g ν , µ µ , µ (3)and we use a statically screened Coulomb potential V Dq . We have assumed that an additionalmomentum dependent factor in Eq. (3) can be dropped, [S6] and we use a background dielectric6onstant of 4 and a constant screening parameter κ = − . The coefficient g ν , µ µ , µ is determinedby g ν , µ µ , µ = (cid:18) + c ν , µ e ( k ) ∗ e ( k ) | e ( k ) e ( k ) | (cid:19) (cid:18) + c µ , µ e ( k ) ∗ e ( k ) | e ( k ) e ( k ) | (cid:19) . (4)The prefactor c a , b equals 1 for a = b (intraband processes) and − a = b (interband processes),further e ( k ) = − a √ ( ik x + k y ) . These matrix elements describe all scattering channels near theDirac cone. To quantify the impact of Auger processes we are interested in index combinations µ = µ = µ = ν and all their permutations. Within these eight possible combinations, Augerrecombination and impact excitation are included, both types of Auger scattering having basicallythe same strength.We have made two further important simplifications: In order to focus on electron electronscattering, we did not include electron-phonon scattering processes, which would result in a de-crease of the kinetic energy. The excitation is included in a purely phenomenological way byinstantaneously creating the excited carrier density. IV.3. Numerical Details
Finally we provide a short review of the used numerical scheme and the boundary conditions.We implemented Eq. (1) for a Cartesian k grid, which has the advantage that it does not introducenumerical errors in the carrier density conservation. Therefore we are able to calculate the scat-tering properties of the complete 2D k-space with an arbitrary excitation. Due to the numericalcostly right-hand side of Eq.(1) we used parallelization techniques and a Dormand-Prince dif-ferential equation solver with adaptive step size. Finally the k-space was sampled with 51 gridpoints in each direction and with a maximal momentum of k max = . − . This approach usinga Cartesian k grid conserves carrier density by construction, which is important as we are lookingrelatively small effects at high energies (> 800 meV). On the other hand, it is numerically quitecostly. The initial distribution is chosen as a Fermi-Dirac distribution f eq ± = / ( + e [( E ± − µ ) / ( k B T )] ) with a chemical potential of 400 meV to model the relaxed electrons introduced by doping.We assume an instantaneous excitation of the form δ f = Ae − b ( k − k e ) | sin ( k x ) | (5)with the amplitude A , the width of the excitation b and the center of the excitation k e , whichfollows the spacial shape of the measured excitation and the calculations from Ref. [S7]. In or-der to distinguish between scattering effects and building up of correlations, we let the system7 a) (b) FIG. 3. Distribution functions after 300 fs, for the full calculation (a), and the calculation without Augerprocesses. The red/blue part corresponds to the lower/upper band. The full calculation reaches a quasi-equilibrium with a chemical potential of about 300 meV. Neglecting Auger processes in the calculationprevents the system from reaching a quasi equilibrium. with the doped carrier density relax and then initialize the calculation by changing the electronicdistributions by ± δ f in the upper/lower band with positive/negative energy dispersion.To eliminate Auger type processes, we exclude the Coulomb matrix elements relevant for thesetransitions by setting g µ , νµ , µ =
0. We can thus compare the scattering dynamics with and withoutAuger processes. In order to replicate the photoemission experiments to some extent, we integratethe time-dependent distributions over the ( , k y ) axis for the energy interval 0 . . S1] G. Saathoff, L. Miaja-Avila, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, Laser-assistedphotoemission from surfaces, Physical Review A , 022903 (2008).[S2] M. Keunecke, M. Reutzel, D. Schmitt, A. Osterkorn, T. A. Mishra, C. Möller, W. Bennecke, G. S. M.Jansen, D. Steil, S. R. Manmana, S. Steil, S. Kehrein, and S. Mathias, Electromagnetic dressing of theelectron energy spectrum of Au(111) at high momenta, Phys. Rev. B , 161403 (2020).[S3] M. Bonitz, D. Kremp, D. C. Scott, R. Binder, W. D. Kraeft, and H. S. Köhler, Numerical analysisof non-Markovian effects in charge-carrier scattering: one-time versus two-time kinetic equations, J.Phys.: Cond. Mat. , 6057 (1996).[S4] K. Morawetz and H. S. Köhler, Formation of correlations and energy-conservation at short times, Eur.Phys. J. A , 291 (1999).[S5] H. Haug and C. Ell, Coulomb quantum kinetics in a dense electron gas, Phys. Rev. B , 2126 (1992).[S6] E. Malic and A. Knorr, Graphene and carbon nanotubes: ultrafast optics and relaxation dynamics(John Wiley & Sons, 2013).[S7] T. Winzer and E. Malic, The impact of pump fluence on carrier relaxation dynamics in opticallyexcited graphene, J. Phys.: Cond. Mat. , 054201 (2013)., 054201 (2013).