Tracking photocarrier-enhanced electron-phonon coupling in nonequilibrium

Light-modulated electron-phonon coupling (EPC) is significant in many intriguing phenomena including light-enhanced superconductivity, polaron formation, and hidden charge orders, which provides a powerful strategy to engineer materials’ functionalities on demand. Here we explore EPC in photoexcited graphene during the ultrafast photocarrier dynamics with a phonon bath. Via analysing energy transport between electrons and phonons, light-induced EPC enhancement by more than one order of magnitude is demonstrated, which originates from the dynamic distribution of photoexcited carriers out of equilibrium. Excellent agreements between theory and experiment have been achieved, justifying the validity of the present approach for extracting excited-state dynamic properties. Our result unravels a crucial impact of photoexcitation on EPC by modulating the density and distribution of photocarriers, and provides a useful strategy for tracking ultrafast EPC in real time.


INTRODUCTION
Electron-phonon coupling (EPC) is one of the fundamental topics in condensed matter physics and is highly relevant to many intriguing phenomena 1 such as the conventional superconductivity 2 , phonon-assisted optical absorption 3,4 , and formation of charge density waves 5,6 . The EPC properties of materials play an important role in both fundamental physics and realistic applications such as power transmission 7-10 , carrier transport 11,12 and solar cells [13][14][15][16] . With the rapid development on femtosecond laser technologies, light-modulated EPC attracts extensive attentions recently. Pomarico et al. 17 and Sentef et al. 18 suggested that the EPC in graphene can be strengthened by light due to the presence of distorted lattice and phonon nonlinearity. The possibility of light-induced superconductivity in cuprates [19][20][21] and molecular solids 22,23 was investigated via coherent optical control of lattice vibrations 17,21,24 . Moreover, laser-excited surface plasmon is demonstrated as a rational way to enhance the EPC in metallic crystals 25 . These studies reveal advantages of photoexcitation to control quantum properties in nonequilibrium phases and might inspire the ways to engineer materials' functionalities on demand.
In general, light modulation of EPC requires effective lightmatter and electron-phonon (e-ph) interactions. Due to the complex interactions and excitation dynamics beyond the ground state, the characterization and mechanism of the light-modulated EPC are not fully understood. The impact of photoexcitation on EPC and its time evolutions are still elusive, since in literature the EPC was often treated in a static picture even when ultrafast photoexcitation is present 26,27 . Thanks to its band structure, graphene represents a superior candidate for the study of lightmodulated EPC dynamics 28 . The gapless Dirac cone not only gives rise to the broadband and ultrafast optical responses [29][30][31] , but also provides resonant EPC channels between Dirac fermions and phonons. The linear dispersion of Dirac cone guarantees an ineffective electrostatic screening [32][33][34][35][36] , which benefits the EPC enhancement 37,38 . Another advantage of graphene is that abundant carrier dynamic processes such as ultrafast carrier cooling [39][40][41][42] and carrier multiplication 43-47 offer a good platform for studying nonequilibrium EPC by e-ph scatterings [48][49][50][51][52][53] , where the state of carriers is expected to have an impact on EPC properties 26,54,55 .
Here we study the nonequilibrium EPC in photoexcited graphene with the help of newly developed real-time timedependent density-functional theory (rt-TDDFT) schemes [56][57][58][59] . We find a strong light-induced EPC enhancement by monitoring the EPC strength λ and matrix g 2 Γ F during ultrafast photocarrier relaxation. The good agreements between the EPC strength extracted here and the data fitted to experimental Raman spectroscopy proves the validity of the present dynamic approach for extracting excited-state properties. Different from existing models which rely on coherent lattice vibrations and distortions 17,18 , light-enhanced EPC identified here demonstrates that the photoexcited carriers, whose nonequilibrium distribution offers additional channels for e-ph scatterings, are crucial for modulating EPC on demand. Our results suggest a way towards dynamic optical control of EPC and related quantum phenomena, such as superconductivity [19][20][21][22][23][24] , polaron formation 60,61 , charge density waves 5,6 and quantum Hall effects 50,[62][63][64] .

RESULTS AND DISCUSSION Carrier and phonon dynamics
After laser illumination, the photocarriers are produced in the Dirac bands by generation of photoexcited electron-hole (e-h) pairs (Fig. 1b). Within hundreds of femtoseconds, the photocarriers cool down through the energy and momentum redistribution, where the e-ph interaction plays an important role 39,42,45,48,49,52 . Because of the e-h symmetry, the relaxation process is illustrated only by the population of electronic photocarriers n(t) in Fig. 1c, n(t) = ∫ E in π* bands n e (E − E f , t) dE, where n e (E − E f , t) is the energy distribution of excited electron carriers (EECs) on π* bansd and E f is the Fermi level. Since the n here is proportional to the electron temperature T e (Supplementary Note 8 and Supplementary Fig. 3), the evolution of n(t) is equivalent to the change of T e in describing the carrier relaxation.
In the absence of phonons (i.e., clamped ions), the carrier population after photoexcitation (t > 30 fs) remains almost unchanged over time due to the lack of decay channels. In contrast, in the presence of phonons, n(t) shows a biexponential decay, which includes two distinct time constants τ 1~7 fs and τ 2~6 50 fs. The τ 1 corresponds to the full width at half maximum (FWHM) of laser pulse and implies a rapid recombination of e-h pairs after photoexcitation. While τ 2 represents the relaxation time of residual hot carriers interacting with optical phonons. The τ 2 obtained from our simulations agrees well with that extracted from experimental measurements (100 fs-1 ps) 39,42,48,65 . In the presence of phonons, n(t) presents a phonon-resonant oscillation along with its decaying behaviour. As shown in the inset of Fig. 1c, the Fourier analysis of population n(t) contains the E 2g phonon@Γ frequency (0.2 eV) and its doubling (0.4 eV). The comparison of photocarrier evolution with and without phonons confirms the EPC effect, which provides significant cooling channels for photocarrier relaxation.
During the photocarrier relaxation, the EPC leads to asymmetric phonon peaks in the simulated Raman spectrum (Fig. 1d) through Fano resonance 53,66,67 . The cross term of the e-ph scattering amplitudes gives the Fano interference and affects the spectrum configuration, which is also observed in the Raman spectroscopy of Weyl semimetals 68,69 . The asymmetry is proportional to the EPC strength and quantified by the value of Fano factor |q|, obtained by fitting the spectrum with the lineshape A(E)~(q + ε) 2 /[γ(q 2 + 1) (1 + ε 2 )]. Here ε = 2(E − ħω ph )/γ, ħω ph and γ are the phonon energy and peak width, respectively. In Fig. 1d, both the harmonic (0.2 eV) and anharmonic (0.4 eV) peaks of E 2g phonons@Γ [70][71][72] show the obvious asymmetry with the Fano factor |q harmonic |~0.04 and |q anharmonic |~1.3, respectively, implying a strong EPC effect in photoexcited graphene. More phonon dynamic information can be found in Supplementary Note 10.

Light-enhanced EPC
The strong EPC effects introduce a measurable energy transport between the photocarriers and phonons 7,9 , characterized by the energy transport rate P(t) = Σ k,i ∂E k,i (t)/∂t. Here k and i are the k point and band index, respectively; E k,i (t) = n k,i (t)ε k,i (t) is the carrier energy on the ith band at wavevector k in the Brillouin zone, where ε k,i (t) is the Kohn-Sham energy level and n k,i (t) is the population of carriers. With a~300 K lattice temperature, the time average of the rate amplitude P ¼ PðtÞ h i t at t~50 fs is estimated with and without the laser illumination. The results are plotted as a function of doping concentration in Fig. 2.
The energy transport at the ground state is also investigated for comparison, in the same way, but without laser illumination. The value of ground state P in undoped case is consistent with the experimental measurement by Freitag et al. 10,50 . The doping concentration is tuned by adding electrons/holes and described by the Fermi level shift ±(E f − E D ) ( Fig. 2a-e). Here E f is the Fermi level and E D is the energy level of Dirac point. For undoped case, E f = E D . The + (−) sign labels the increasing (decreasing) of electron concentration. Similar to the anomalous phonon softening phenomenon 36,73-82 , P first increases and then decreases with the increase of doping concentration. It exhibites a logarithmic singularity when the Fermi level shift reaches the half of phonon frequency ħω ph , i.e., |E f − E D | ≈ 0.1 eV. Figure 2a-e illustrates the electronic origin of this anomaly. The doping concentration affects the energy transport process and renormalizes the phonon energy by adjusting the e-h excitation as well as their couplings with phonons. Due to the zero band gap of Dirac cone, e-h carriers can be vertically excited by optical phonons, if their excitation energy E e-h is close to the phonon energy ħω ph and higher than the twice of the Fermi level shift (E e-h ≈ ħω ph > 2|E f − E D |). This phonon-induced excitation enables the electrons to respond immediately to the change of phonons, which provides screening and enhances the EPC resonance. The EPC resonance contributes to phonon softening and benefits the energy transport between carriers and phonons. However, when the increase of doping concentration makes the Fermi level shift greater than the half of phonon energy (2|E f − E D | > ħω ph ), the phonon-induced excitation is switched off due to restrictions of Pauli blocking. The e-h carriers can hardly participate in the EPCinduced energy transport, and the states below the Fermi level can no longer contribute to the screening. For this reason, the doping concentration dependence of P, as well as phonon softening, shows the opposite trendency before and after 2|E f − ED| = ħω ph .
In contrast, under a~4 μJ • cm −2 laser illumination, the excited carrier has an effective electron temperature~2500 K. The logarithmic singularity of P is completely washed out, because  of the Fermi level broadening introduced by photoexcitation 74 . More importantly, the magnitude of P in the excited state is highly enhanced, implying EPC effect is significantly promoted by photoexcitation.
The different energy transport rate P indicates different EPC strength with and without photoexcitation. Tse et al. derived a generic expression for the EPC-mediated energy transport in graphene using Green function theory 9,83,84 , which implies that one can extract the microscopic EPC strength λ directly from the macroscopic energy transport rate P (Supplementary Note 2): where α = 12ħ|E f − E D |E f 2 /(ħω ph ) 5 , the Fermi level E f = sgn(n)ħv f (π|n|) 1/2 . The v f and n are the Fermi velocity and carrier density, respectively. The N B = Mu 2 ω ph 2 /2ħω ph is the number of phonons, M and u are the mass and displacement of the carbon atom.
] is the Fermi-Dirac (F-D) distribution at the energy |E f − E D | and temperature T. The same Fermi level E f is used here to extract the EPC strength λ in the ground and excited state. The ground state λ in the absence of laser illumination is calculated for benchmarking the validity of this approach. Similar to the effective mass picture in band theory, the effects of photoexcitation are naturally included in the difference of effective λ between the ground state and excited states 81 .
In Fig. 2g, the EPC strength λ is obtained from the time average of energy transport rate P by Eq. (1) and plotted as a function of doping concentration represented by E f − E D . Owing to the increasing screening, λ is proportional to the doping concentration. Without laser excitation, the system is near the ground state, thus the doping concentration dependence of λ is consistent with that calculated by the density-functional perturbation theory (DFPT) (See details in Supplementary Note 1).  10 . The blue symbols and lines show anomalous phonon softening measured by Raman spectroscopy in ref. 36 , ref. 74 and ref. 81 . g The EPC strength λ as a function of E f −E D with and without laser illumination. The dashed blue and green lines represent experimental results from ref. 75 and ref. 82 .
The deviation at high |E f − E D | may be due to the lack of nonadiabatic effects in DFPT, which plays a key role in EPC at high doping levels. Indeed, our results including the non-adiabatic effects show a better agreement with the experiments (Supplementary Note 3) 75,82 .
By fitting the relationship between λ and the Fermi level shift |E f − E D |, the effective EPC matrix of E 2g phonon@Γ in graphene g 2 Γ F can be obtained according to the Eliashberg theory 83,84 (Supplementary Note 1): where g 2 Γ F = Σ π* i,j |g ik,jk | 2 /4 is the square sum of EPC matrix elements over the two degenerate π* bands and g ik,jk is the electron-phonon matrix element 80,85 2  .04 36,80,85 and the experimental values g 2 Γ F = 0.043 eV 2 in ref. 75 and 0.047 eV 2 in ref. 80

(Supplementary Note 3).
Under laser illumination, the excited-state EPC is dynamic. The effective λ averaged at t~50 fs after photoexcitation is shown in Fig. 2g. The λ shows a similar doping dependence with that in the ground state, while its magnitude is significantly enhanced by~3 times. The fitted g 2 Γ F in excited state is~0.15 eV 2 . The threefold increase of g 2 Γ F agrees well with the experimental measurement by time-and angleresolved photoemission spectroscopy (tr-ARPES) 17 , demonstrating that the EPC strength is significantly enhanced by laser excitation, which is different from the continuous wave case (Supplementary Note 7).
In order to gain insights into the mechanism of light-induced EPC enhancement, the population distribution of excited carriers and their contribution to λ are investigated at different doping concentrations (represented by E f − E D ). Note that with charge doping, the total carrier distribution on π* band n t (E − E f ) includes two parts, n t (E − E f ) = n d (E − E f ) + n e (E − E f ). The n d (E − E f ) represents the static charge doping, which is proportional to the Fermi level shift |E f -E D |, and independent of photoexcitation. While n e (E − E f ) is contributed by the EECs. The distribution of EECs averaged at t~50 fs after photoexcitation is shown in Fig. 3, the EECs now are in the nonequilibrium state. The results are similar on π band with hole doping, because of the e-h symmetry.
For both cases, the population of excited carriers decreases with the doping level due to Pauli blocking.
Without laser illumination, n e (E − E f ) is contributed by the excitation of optical phonons thanks to the gapless Dirac bands, as discussed in Fig. 2a. It is located in the low energy region (E − E f < 0.6 eV) and forms a thermal F-D distribution. In contrast, under laser illumination, the n e (E − E f ) averaged at t~50 fs is mainly contributed by the photoexcitation and has the nonequilibrium distribution. The nonthermal n e (E − E f ) mainly occupies high energy levels around E − E f = 0.8 eV, which corresponds to the half of the photon energy ħω ≈ 1.55 eV due to the e-h symmetry of Dirac cone. The different energy distribution enables us to divide the excited carriers into low and high energy regions according to their excitation source, the low (high) energy carriers are excited by phonons (photons), which are labelled by black (red) arrows in Fig. 3a, b. The contribution of EECs to λ at different energy ranges is shown in Fig. 3c. It is clear that the carriers excited by photons (high energy range) and phonons (low energy range) have the dominant contributions to λ in cases with and without laser illumination, respectively. Similar results can also be seen from the photocarrier distribution averaged at t~50 fs and the effective EPC matrix elements g 2 Γ F as a function of pump fluence J L , FWHM and wavelength λ L in Fig. 4. Under different laser conditions, the EPC strength is proportional to the distribution of photocarriers (Supplementary Note 9). The energy-resolved λ and distribution dependence of g 2 Γ F confirm that the light-induced EPC enhancement is dominated by the photocarriers, whose nonthermal distribution plays a significant role.
The mechanism for the light-induced EPC enhancement can be understood as schematically shown in Fig. 3d. Upon photoexcitation, nonequilibrium photocarriers generated at the high energy levels have wider distribution due to the large density of state (DOS). The broadening distribution enhances the phonon-assisted intraband transitions (PAITs, as shown in the red rectangle in Fig. 3d), which provide the additional e-ph interaction channels and improve the EPC. The PAITs are induced by the electronic broadening effects 86 (process I) and scattering between different k points (process II). As shown in Supplementary Fig. 2, the PAITs (blue and black lines in Supplementary  Fig. 2) contribute~80% of the EPC channels, demonstrating a dominant role of PAITs of nonequilibrium photocarriers on EPC enhancement. showing the EPC channels in photoexcitation state. Regions with light blue shades are filled with electrons. The deep blue arrow shows the phonon-assisted interband transitions. Without illumination, electrons have the interband transitions by interacting with the phonons, which provide the conventional EPC channels. The green arrow shows the photoexcitation. The red rectangle and red arrows label the additional EPC channels originate from the phonon-assisted intraband transitions (PAITs) of carriers out of equilibrium, which include the electronic broadening effects (process I) and scattering between different k points (process II).

Real-time tracking on the light-enhanced EPC dynamics
To further illustrate the impact of photocarrier distribution on EPC dynamics, we choose to study the undoped graphene under a laser fluence of~44 μJ • cm −2 and analyse the ultrafast EPC during photocarrier relaxation. Figure 5a shows the distribution of EECs at different time n e (E − E f , t). For comparison, the F-D distribution with an electron temperature T e~5 000 K is marked by grey dash line. As time goes on, the peak of excited-carrier distribution moves from the high energy region to the low energy region and finally approaches the F-D distribution, beaconing the cooling down process of hot carriers.
The corresponding time evolution of effective EPC matrix element g 2 Γ F is shown in Fig. 5b (Also in Supplementary Fig. 1 for doping case). With photoexcitation (t ≤ 30 fs), the carriers are  excited to high energy levels E − E f~1 .0 eV. The wider distribution of the nonthermal photocarriers in energy and momentum space provides additional channels for e-ph interactions which enable stronger EPC. The g 2 Γ F is significantly enhanced by 2~3 orders of magnitude ( g 2 Γ F > 1 eV 2 ). After the pulse duration (t > 30 fs), the photocarriers relax towards equilibrium, the energy and momentum range of carrier distribution is suppressed and thus the EPC channels become limited. The light-enhanced g 2 Γ F decreases again over time. Comparing to experiment, we successfully rebuild the time evolution of EPC strength using Eqs. (1) and (2), the only required input is the data of carrier states (i.e., population and energy distribution of carriers) from tr-ARPES measurements 42 .05 eV 2 ) within a picosecond timescale as the hot carriers cool down to equilibrium.
The dynamic g 2 Γ F implies that ultrafast EPC in the excited state is dependent on the photocarrier distribution, which enables not only the real time measurement, but also precise manipulation of EPC by optical technologies. Direct evaluation of EPC strength has been recently achieved in ultrafast spectroscopy experiments 26,27 . The optical modulation can be achieved by coherent control of photocarrier dynamics. For example, the EPC enhancement is expected to be more significant by suppressing other carrier relaxation channels such as electron-electron scattering. The similar analysis may also be used to observe and manipulate the microscopic motions of Cooper pair generation 87,88 , superconducting gap fluctuation 89,90 and quantum phase transitions 91 .
Our study offers a dynamic analysis on evaluating and tracking nonequilibrium EPC in excited states. The nonequilibrium distribution of photocarriers has a vital impact on the light-enhanced EPC, which provides channels for e-ph scatterings. We believe that such effects are general and measurable in other quantum materials including 2D/3D Dirac or Weyl semimetals, transition metal dichalcogenides 68,86,92-94, and even the conventional metals 95 , as long as the wide distribution and the phonon-assisted transitions are available for nonequilibrium photocarriers. More interestingly, since the EPC is connected to multifarious photocarrier dynamics, the topological property of photocarriers such as chiral selection and valley polarization 96-98 may bring some physics and technology applications of light-modulated EPC.

METHODS TDDFT calculations
The calculations are performed using the time-dependent ab initio package (TDAP) as implemented in SIESTA 99 , which has been developed lately to successfully describe photoexcitation dynamics in solids 5,100,101 . In our simulations, numerical atomic orbitals with double zeta polarization are used as the basis set. The k sampling in the Brillouin zone is Gammacentred 144 × 144 × 1 mesh. The electron-nuclear interactions are described by norm-conserving pseudopotentials with an energy cutoff of 200 Ry in the local-density approximation. For the laser parameters, a Gaussian laser pulse F = F 0 cos(ωt)exp[−(t − t 0 ) 2 /2σ 2 ] with a wavelength of 800 nm (ħω ≈ 1.55 eV) is adopted. The polarization is along the zigzag direction (defined as x direction in Fig. 1a). The pulse peak is located at t 0 = 20 fs, and the full width at half maximum (FWHM) σ is 7 fs. For phonon dynamics, the coherent E 2g phonon@Γ (ħω ph ≈ 0.2 eV), as the most dominant optical phonon in graphene 70,102 , is considered via introducing an initial weak stretch of the carbon-carbon bond 55 (by 1~2% of the equilibrium bond length). More details about the algorithm and parameters can be found in ref. [103][104][105] .

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.