Self-energy dynamics and the mode-specific phonon threshold effect in Kekulé-ordered graphene

Abstract Electron-phonon interaction and related self-energy are fundamental to both the equilibrium properties and non-equilibrium relaxation dynamics of solids. Although electron-phonon interaction has been suggested by various time-resolved measurements to be important for the relaxation dynamics of graphene, the lack of energy- and momentum-resolved self-energy dynamics prohibits direct identification of the role of specific phonon modes in the relaxation dynamics. Here, by performing time- and angle-resolved photoemission spectroscopy measurements on Kekulé-ordered graphene with folded Dirac cones at the Γ point, we have succeeded in resolving the self-energy effect induced by the coupling of electrons to two phonons at Ω1 = 177 meV and Ω2 = 54 meV, and revealing its dynamical change in the time domain. Moreover, these strongly coupled phonons define energy thresholds, which separate the hierarchical relaxation dynamics from ultrafast, fast to slow, thereby providing direct experimental evidence for the dominant role of mode-specific phonons in the relaxation dynamics.

: Schematic illustration of Li intercalation of bilayer graphene. Schematic illustration of bilayer graphene on a buffer layer before Li intercalation (left), and the Li-intercalated trilayer graphene with Kekulé order (right). Energy and time resolution. The Fermi edge measured from the graphene at 80 K shows an energy width of 33 ± 1 meV by fitting with the Fermi-Dirac distribution function (Fig. S3a). Therefore, the overall instrumental energy resolution is extracted to be √ 33 2 − 29 2 ≈ 16 meV after removing the thermal broadening of 29 meV at 80 K. The time resolution is determined by the cross correlation between pump and probe laser pulses, which is reflected from the rising edge of the measured TrARPES trace (shown by red symbols in Fig. S3b). The TrARPES data is fitting by a Gaussian function convolved with the product of the step function and a single-exponential function, where ∆t is the width of the rising edge and τ is the relaxation time. Therefore, the extracted time resolution is determined to be 480 ± 70 fs. Determination of coupled phonon energies by extraction of Eliashberg function. By fitting ReΣ with the Eliashberg function 1 , we can determine the coupled phonon energies from the experimental results. Broken curves in Fig. S4a are components of the extracted Eliashberg function, which show two obvious peaks at energy of -54 and -177 meV (black broken curves) and a broad background (gray). These two phonons also correspond to those observed in ImΣ (Fig. S4c), with an additional jump at -82 meV (gray arrow), indicating possible coupling to a third phonon. Fitting ReΣ along the Γ-M* (Γ-K) direction ( Fig. S4b) also gives similar results, with peak energies at -56 and -173 meV and corresponding feature in ImΣ as shown in Fig. S4d. Figure S4e,f shows the calculated ReΣ and ImΣ taking into account the coupling of electrons to the in-plane and out-of-plane phonon modes (indicated by red and black arrows).
Assuming intraband scattering, the retarded electron-phonon self-energy within the non-selfconsistent Migdal approximation is computed 2 according to Here, η is an infinitesimally small positive number (we fix η = 10 −4 a.u.), f kα denotes occupation of the Bloch state |ψ kα , while n qν is the occupation of corresponding phonon mode (given by the Bose distribution). We evaluated eq. (1) for T = 80 K in accordance with the experiment. Note that the electron-phonon matrix elements g ν αα ( k, q) are transformed to the Bloch basis. We used a N k = 400 × 400 sampling of the Brillouin zone, ensuring convergence.
The calculation results agree very well with the experimental data assuming a zero value of ReΣ at the Fermi energy. Test calculations excluding out-of-plane coupling reveal that the feature denoted by the black arrow disappears, thus confirming that Ω 2 is an out-of-plane mode. Phonon threshold effect in the momentum space. The phonon threshold effect is also observed in other regions of the momentum space. Figure S5b shows the differential image along Γ-M* direction at delay time of 700 fs. Phonon-window effect at -177 meV (red arrow in Fig. S5b) is also clearly observed. Indeed, such phonon threshold effect is also observed in the entire momentum space measured (see intensity maps in Fig. S5c-n), with strong differential signal within -177 meV, and negligible signal beyond the phononwindow. Dynamic evolution of the kink at -Ω 1 upon pump excitation. Figure S6 shows more analysis to reveal the dynamic evolution of the kink. The differential image in Fig. S6c suggests that there is a shift of dispersion near the kink energy (pointed by red arrow). This is further confirmed by the extracted dispersion ( Fig. S6d) and in particular the zoomed-in dispersion (Fig. S6e). Such change in the dispersion near the kink energy is confirmed by the MDCs shown in Fig. S6f, and results in a reduction of the peak amplitude in ReΣ (Fig. S6h) and broadening of FWHM (Fig. S6g). We note that similar change in the self-energy has been reported in BSCCO 3, 4 as ultrafast quenching of electron-phonon interaction in the superconducting state upon laser excitation 3 . Our Li-intercalated graphene is unlikely superconducting at the measurement temperature of 80 K. Therefore, the observed apparent shift of the kink is interpreted as phonon threshold induced change in the self-energy due to dynamical spectral weight transfer. Difference curves of ReΣ and |ImΣ| at different delay times. To better resolve the dynamical change of the self-energy with changing of delay time, we show in Fig. S7 the difference curves of ReΣ and |ImΣ| at different delay times, which are obtained by subtracting the self-energy extracted at -2.1 ps from that at different delay times. The differential ReΣ clearly shows that the real part of the self-energy decreases at 0.7 ps and then gradually recover with increasing of delay time, while the |ImΣ| shows a broadening behavior at 0.7 ps and gradually recover. Hierarchical relaxation and more data on the relaxation time. Figure S8a is a plot of the evolution of momentum-integrated differential intensity with energy and delay time, which only shows differential signal within energy of Ω 1 , indicating the energy threshold between "ultrafast" and "fast" regime. Figure S8b shows the relaxation curves at different energy regimes. The relaxation curve integrated over 177-300 meV in the "ultrafast" regime is shown by blue dotted curve, from which electrons relax within the time resolution and leads to a Gaussian-shaped peak. The relaxation in the "fast" regime is shown by yellow dotted curves, and the relaxation time of energy range between 78 and 177 meV is 337 fs, which suggests that the relaxation time in the "ultrafast" regime should be even faster than 337 fs. For electrons in the "slow" regime, the relaxation curves (red dotted curve) show an additional component (red shading area) with relaxation time larger than 7 ps, which clearly sets the energy threshold between the "fast" and "slow" regime. The extracted relaxation time for τ 1 and τ 2 are plotted as black and red symbols in Fig. S8c. This determines the phonon HamiltonianĤ ph = qν ω qνb † qνb qν (b qν is the phonon annihilation operator with respect to momentum q and mode ν).
The electron-phonon coupling Hamiltonian has the generic form whereĉ † km (ĉ km ) is the electron creation (annihilation) operator with respect to the sublattice site m, while The electron-phonon coupling matrix element is defined as where ∆Ĥ qν denotes the change of the electronic Hamiltonian upon atom displacements associated to mode ( qν). For the electron-phonon Hamiltonian ∆Ĥ qν we consider two mechanisms: (i) modulation of hopping (deformation potential), (ii) dipole coupling to an internal electric field. Mechanism (i) gives rise to coupling of the in-plane phonon modes to the electrons, while (ii) affects out-of-plane modes. Direct (linear) coupling to out-of-plane modes is excluded for free-standing graphene due to mirror symmetry; in a heterostructure, however, the coupling becomes finite.
The deformation potential contribution can be extracted from the tight-binding model 8 . The only parameter is the relative variation of the hopping upon atom movement: q 0 = (1/J)∂J/∂b (b is the distance between two carbon atoms). A straightforward derivation yields Here, v mm ( L) denotes the vector connecting atom m in the unit cell and atom m in the same ( L = 0) or neighboring ( L = 0) unit cell. The sum over L includes all lattice vectors such that | v mm ( L)| = a CC (a CC is the distance between two neighboring carbon atoms). For graphene J L = J, while J L = J 1 or J L = J 2 for Kekulé-ordered graphene.
For modeling the interaction of electrons with the out-of-plane modes, we consider each Li atom in the heterostructure as a point charge giving rise to an electric field. The effective field in the z direction E 0 is identical for each carbon atom, and is treated as a parameter. The coupling matrix element becomes To determine the contribution of the specific phonon modes in the electron-phonon coupling of electrons at the Fermi level (E F ), we computed the averaged coupling Here, |ψ kα stand for the Bloch states with energy ε α ( k); ρ(ε) denotes the density of states.
We fixed q 0 = 0.1 a.u. and E 0 = 0.03 a.u. by comparing to resulting photoemission spectra to the experiment. For graphene, this gives rise to predominant in-plane coupling of the LO and TO mode near Γ and the degenerate LA and LO mode at K (Fig. S9a), while the out-of-plane mode ZA mode near K couples strongest (Fig. S9c). For Kekulé-ordered graphene (Fig. S9b,d), the TO mode folded from K (extended Brillouin zone) onto Γ dominates the in-plane electron-phonon coupling (labelled as Ω 1 ). The calculated electron-phonon coupling strength is roughly twice as big as the strongest coupling in regular graphene.
Similarly, we find that the out-of-plane phonon mode originating from the ZA mode folded to Γ (labelled as