Impact of doping on the carrier dynamics in graphene

We present a microscopic study on the impact of doping on the carrier dynamics in graphene, in particular focusing on its influence on the technologically relevant carrier multiplication in realistic, doped graphene samples. Treating the time- and momentum-resolved carrier-light, carrier-carrier, and carrier-phonon interactions on the same microscopic footing, the appearance of Auger-induced carrier multiplication up to a Fermi level of 300 meV is revealed. Furthermore, we show that doping favors the so-called hot carrier multiplication occurring within one band. Our results are directly compared to recent time-resolved ARPES measurements and exhibit an excellent agreement on the temporal evolution of the hot carrier multiplication for n- and p-doped graphene. The gained insights shed light on the ultrafast carrier dynamics in realistic, doped graphene samples.

by graphene Bloch equations 1 corresponding to a coupled set of differential equations for the occupation probability ρ ( ) λ t k in the state k and the band λ = (v, c), the microscopic polarization p k (t) that is a measure for the optical transition probability between both bands, and the phonon occupation ( ) n t j q with the momentum q for different optical and acoustic phonon modes j 29 : is the Rabi frequency and Ω ( ) λλ t k accounts for intraband transitions 12 . The many-particle interactions are treated within the second-order Born-Markov approximation 1,28,30,31 , which yields a Boltzmann-like scattering equation for the carrier occupation with the time-and momentum-dependent scattering rates Γ ( ) λ , , t k in out accounting for Coulomb-and phonon-induced processes. At the same time, the microscopic polarization is damped by the many-particle-induced diagonal dephasing γ k (t) and is driven by the off-diagonal dephasing term U k (t). In analogy, the equation of motion for the phonon occupation ( ) n t j q is obtained and contains phonon emission and absorption rates Γ ( ) , , t j q em abs . The finite phonon lifetime 32 γ ph is considered by a coupling to a phonon bath n B j at room temperature. The explicit form of the time-and momentum-dependent scattering rates is discussed in the supplementary material. More details on the diagonal and off-diagonal dephasing terms can be found in Malic et al. 29 .
A finite Fermi level E F breaks the symmetry between the valence and conduction band around the Dirac point, cp. Fig. 1 where + stands for the hole and − for the electron occupation at the temperature T. Another important aspect of doping is the increased screening of the Coulomb interaction. The bare Coulomb potential V q appearing in the Coulomb-induced scattering rates Γ ( ) λ , , t k in out is screened via the dynamic dielectric function ε(q, ω) that is defined by the Lindhard equation 28,33 , cf. the supplementary material for a more detailed discussion. Since this many-particle-induced screening is directly influenced by carrier occupations in the conduction and valence bands, doping plays a crucial role and has a significant influence on the ultrafast carrier dynamics in graphene.
With the presented microscopic approach, we can track the relaxation dynamics of non-equilibrium charge carriers in time, and energy including the temporal evolution of the carrier density after the In the presence of a finite Fermi level, carrier multiplication (CM) and hot carrier multiplication (hCM) need to be distinguished. While the first is induced by Auger processes bridging the valence and the conduction band (red arrows) and increasing the number of charge carriers in the conduction band, the second corresponds to Coulomb-induced intraband scattering (orange arrows) and increases the number of carriers above the Fermi level.
Scientific RepoRts | 5:16841 | DOI: 10.1038/srep16841 optical excitation. First, we focus on the electron and hole dynamics in highly doped graphene and then we discuss the impact of doping on the appearance of carrier multiplication.

Results and Discussion
Electron and hole dynamics. Here, we discuss how the doping-induced symmetry breaking between electrons in the conduction band and holes in the valence band influences the dynamics of optically excited charge carriers in realistic doped graphene samples. Figure 2 illustrates the angle-averaged occupation probability ρ λ k for (a) electrons and (b) holes for an initial Fermi level of 300 meV as a function of the carrier energy for different times after the optical excitation. Note that for symmetry reasons, the physical picture remains the same in p-doped graphene samples but electrons and holes switch their roles, respectively. The system is excited by a 10 fs pulse with a photon energy of 1.5eV and a pump fluence of 0.3 μJcm −2 . The characteristics of the excitation pulse correspond to typical values that can be realized by standard pulsed lasers 34 . The pulse is centered at 0fs and gives rise to a well pronounced non-equilibrium distribution for electrons and holes around the carrier energy of 0.75eV, cf. Fig. 2. For both electrons and holes, the efficient carrier-carrier and carrier-phonon scattering leads to an ultrafast thermalization of the system towards a hot Fermi distribution already after some tens of femtoseconds. Then, a slower phonon-induced carrier cooling occurs that drives the electron and hole occupations towards their initial thermal Fermi distributions. Due to the increased number of available scattering partners in the conduction band of n-doped graphene, the Coulomb-driven carrier thermalization occurs faster for electrons. Here, a hot thermalized Fermi distribution is already reached after 30 fs, while at the same time the holes exhibit still a non-equilibrium distribution, cf. the purple lines in Fig. 2.

Carrier multiplication (CM). Now, we study the impact of an initial Fermi level E F on the
Coulomb-induced multiple carrier generation, which is generally defined as the ratio between the number of overall generated electron-hole pairs and the optically excited charge carriers where n is the total carrier density, n T the initial thermal carrier background, and n opt the optically excited carrier density.  where L 2 is the graphene area and σ s (σ v ) denotes the spin (valley) degeneracy. On the other side, electric transport phenomena are also of great interest, where hot electrons around the Fermi level are relevant giving rise to a hot carrier multiplication, cf. Fig. 1. This situation will be discussed in the next section.
Treating the full set of graphene Bloch equations, we have microscopic access to the temporal evolution of the carrier density including the contributions of carrier-light, carrier-carrier, and carrier-phonon interactions. Figure 3(a) illustrates the temporal evolution of CM as a function of the initial Fermi level E F at a fixed absorbed pump fluence of 0.3 μJcm −2 . The surface plot reveals that doping clearly reduces the CM efficiency: The lower E F , the higher is the CM factor reaching values of up to approximately 1.7 for undoped graphene at the considered pump fluence, cf. Fig. 3(b). CM can be observed for E F of up to 300 meV. It occurs on a timescale of up to 150 fs for undoped graphene and becomes significantly shorter for increasing doping, as illustrated in Fig. 3(c).
The observed CM in the low-doping case can be explained by the strongly efficient impact excitation (IE) prevailing over the inverse process of Auger recombination (AR), which is a result of the large gradient in carrier occupation around the Dirac point, cf. Fig. 2(a). For undoped graphene, the probability for IE can be written as ρ v (1 − ρ c ) ≈ 1, whereas the probability for AR is given by ρ c (1 − ρ v ) ≈ 0 (since ρ v ≈ 1 and ρ c ≈ 0). In this case, IE is significantly favored by Pauli blocking during the initial dynamics. This is reflected by the corresponding rates γ IE and γ AR that are shown in Fig. 4(a). We observe that the IE rate is clearly higher for a time range of approximately 100 fs determining the strength of the appearing CM. During the carrier relaxation both rates converge to the same value and end up in an equilibrium, where no more carriers are generated. The timescale of the CM is determined by the duration of the imbalance between IE and AR rates in combination with the interplay with competing channels of carrier-phonon scattering, which transfer energy from the electronic system to the lattice. With an increasing doping, Auger scattering becomes more and more Pauli blocked resulting in overall lower rates, cf. Fig. 4(b). For Fermi levels higher than 300 meV, AR becomes the dominant relaxation channels and CM does not appear anymore.

Hot carrier multiplication (hCM). Now
and the associated hot carrier multiplication is highly relevant for transport phenomena 16,21 , cf. Fig. 1. Note that we obtain symmetric results for n-and p-doped graphene, since the contribution of both electrons and holes to the carrier density is considered. For undoped graphene, both definitions of carrier density [Eqs. (5) and (6)] and carrier multiplication are equivalent for symmetry reasons. The surface plot in Fig. 5(a) illustrates hCM as a function of the Fermi level E F and time at a fixed absorbed pump fluence of ε abs = 0.3 μJcm −2 . In contrast to the behavior of CM, we observe a clear increase of hCM with doping. We reach hCM factors of up to approximately 2 (at the considered pump fluence) on a timescale of about 200 fs for highly doped graphene with E F = 300 meV, cf. Fig. 5(b,c). There is nearly a linear dependence between hCM and doping: The smaller E F , the less pronounced is hCM, and the faster it decays. The probability for intraband IE processes is given by ρ h,c (1 − ρ e,c ), which is initially large compared to the probability for intraband AR processes ρ e,c (1 − ρ h,c ), cf. Fig. 1. With the increasing Fermi level, the intraband Auger processes are shifted into a region of higher density of states making them more efficient, as reflected by the much higher scattering rates displayed in Fig. 4(b). The initial strong imbalance between γ  AR and γ  IE gives rise to a pronounced hCM. Besides the discussed doping dependence, CM or hCM are strongly sensitive to the excitation regime. A detailed discussion is provided in the supplementary material, where a semi-analytical approach is presented focusing on the purely Coulomb-induced CM and hCM.
Direct comparison to experimental data. After having presented the theoretical results on the doping dependence of the carrier multiplication, we perform a direct comparison with recently performed time-resolved ARPES measurements on n-and p-doped graphene samples 16 . We explicitly take into account the experimental conditions, such as the Fermi level and the excitation strength. Considering that our microscopic theory does not contain any fitting parameters, we obtain an excellent agreement between theory and experiment, cf. the inset in Fig. 5(c). There is a clearly higher hCM for n-doped graphene reaching values of up to 2.2 in the theory and more than 3 in the experiment. In contrast, for p-doped graphene only a small hCM of 1.4 or 1.2 is obtained in theory and experiment, respectively. This pronounced difference is not due to the type of doping (n, p), as one might assume 16 . It can be clearly explained by the differences in the applied fluence ε abs and the actual Fermi level E F . Note that at the exactly same conditions with respect to E F and ε abs , we obtain the same hot carrier multiplication for both n-and p-doped samples. However, the experiment has been performed for: (i) n-doped graphene with the Fermi level E F = 380 meV and an absorbed pump fluence Temporal evolution of Auger rates including impact excitation (IE) and the inverse process of Auger recombination (AR) for (a) undoped and (b) highly n-doped graphene (ε = 300 meV). In the latter case, also the corresponding intraband Auger rates (γ  IE , γ  AR ) describing Coulomb-induced scattering bridging the states below and above the Fermi level are illustrated (dashed lines), cf. Fig. 1. In undoped graphene, there is a clear asymmetry between both Auger processes in favor of IE in the first 100 fs. In the doped case, γ AR is significantly larger than γ IE , however, for intraband Auger rates, the situation is opposite and γ  IE clearly prevails over γ  AR resulting in a pronounced hot carrier multiplication.
Scientific RepoRts | 5:16841 | DOI: 10.1038/srep16841 of ε abs = 0.5 μJcm −2 and (ii) p-doped graphene with E F = 240 meV and ε abs = 1.5 μJcm −2 . As shown in Fig. 5(b), hot carrier multiplication increases almost linearly with the Fermi level. Furthermore, it is strongly suppressed in the strong excitation regime, i.e. the larger the pump fluence, the less efficient is the hCM, as illustrated in Fig. S3 in the supplementary material. As a result, the n-doped graphene sample shows a much more pronounced hCM, since its Fermi level E F is significantly higher and since the experiment has been performed at a clearly smaller pump fluence compared to the p-doped graphene sample.
While the theory qualitatively reproduces well the experimental data, quantitatively, it underestimates the CM value in the first 50 fs as well as its temporal decay in the n-doped case. This might be due to the applied semi-Markovian description of the CM processes 1 . A full non-Markovian approach would allow for a dynamical build-up of the screening of the Coulomb interaction and would lead to more allowed scattering channels on short time scales. Another possible reason for the deviation in the first 50 fs is that the experimental estimate of the carrier density is based on the introduction of a carrier temperature. However, the latter can only be well defined once a thermalized carrier distribution is reached, which occurs during that timescale. The faster temporal decay of the CM in the experiment could also be explained by the occurrence of additional scattering channels in a non-Markovian description or might suggest an additional impurity-induced scattering channel that accelerates the CM decrease.
In conclusion, we have presented a microscopic study of the carrier dynamics in doped graphene samples, in particular focusing on the impact of a finite Fermi level on the (hot) carrier multiplication. We reveal the appearance of Auger-induced carrier multiplication up to Fermi levels of 300 meV. In the case of the hot carrier multiplication occurring within one band doping is even advantageous, since it increases the phase space by providing a large number of available scattering partners. Finally, we have directly compared our results to recent time-resolved ARPES measurements finding an excellent agreement and providing a microscopic explanation for the observed different behavior in n-and p-doped graphene samples. Our results contribute to a better understanding of the ultrafast carrier dynamics in realistic graphene samples and give valuable insights into the technologically relevant carrier multiplication in graphene.  Fig. 4 illustrating now the temporal evolution of the doping-dependent hCM for a fixed absorbed pump fluence of ε abs = 0.3 μJcm −2 . The inset in (c) shows a direct comparison between theoretically predicted (solid lines) and experimentally measured time-resolved ARPES data (dashed lines) for: (i) n-doped graphene (E F = 380 meV, ε abs = 0.5 μJcm −2 ) and (ii) p-doped graphene (E F = 240 meV, ε abs = 1.5 μJcm −2 ). The experimental data is taken from Johannsen et al. 16 .