High-resolution angle-resolved photoemission studies of quasiparticle dynamics in graphite

We obtained the spectral function of the graphite $H$ point using high-resolution angle-resolved photoelectron spectroscopy (ARPES). The extracted width of the spectral function (inverse of the photohole lifetime) near the $H$ point is approximately proportional to the energy as expected from the linearly increasing density of states (DOS) near the Fermi energy. This is well accounted for by our electron-phonon coupling theory considering the peculiar electronic DOS near the Fermi level. We also investigated the temperature dependence of the peak widths both experimentally and theoretically. The upper bound for the electron-phonon coupling parameter is 0.23, nearly the same value as previously reported at the $K$ point. Our analysis of temperature-dependent ARPES data at $K$ shows that the energy of a phonon mode of graphite has a much higher energy scale than 125 K, which is dominant in electron-phonon coupling.

Figure 1

(Color online) (a) The structure of graphite. Spheres are carbon atoms. Graphite shows a layered structure, which has a stacking order of $ABAB\cdots$. (b) First Brillouin zone (BZ) of graphite. The symbols represent high-symmetry points. (c) Calculated electronic band structure along $A\text{−}L\text{−}H\text{−}A$. (d) Approximated band structure. Each corner of hexagon is $H$ point and $z$ direction is energy.

Figure 2

(Color online) (a) The lowest-order Feynman diagram for EPC. $g$ is electron-phonon coupling constant. $k$ and $k^{\prime }$ are crystal momenta of holes. $q$ is momentum of phonon. (b) Schematic diagram of the EPC process as shown in panel (a). Photohole $k$ makes a transition to $k^{\prime }$ emitting phonon of $q$. $\hslash {\omega }_0$ is the emitted phonon energy. (c) Schematic diagram for scattering in $k_z$ direction. (d) The imaginary part of the self-energy vs binding energy predicted by our qualitative theory (see the text).

Figure 3

(Color online) (a) The Feynman diagram for the lowest order for electron-hole pair creation process. Photohole $k$ makes a transition to a hole $k^{\prime }$, and an electron $k_e$ and a hole $k_h$ are created, conserving momenta and energy. (b) Schematic diagram for electron-hole pair creation in phase space. (c) Schematic diagram for electron-hole pair creation considering the scattering along $k_z$ direction. (d) Hatched area represents the possible $\Delta k$ and $\Delta \omega$ for electron-hole pair creation. Dashed line represents the possible $\Delta k$ and $\Delta \omega$ for photohole transition. These two areas slightly touch each other at the line; therefore there is no available phase space.

Figure 4

(Color online) (a) ARPES data taken at $H$ point of graphite along $L\text{−}H\text{−}A$ direction. (b) The EDC at the $k$ point marked the arrow in panel (a). Circles are the experimental data and thick line is the fit for which finite escape depth effect in photoemission process has been considered (see the text). (c) Contributions from different $k_z$ points due to the finite escape depth effect. (d) The model fitting function with finite escape effect considered but without lifetime effect. (e) ARPES data at $K$ from epitaxially grown graphene on $6H\text{-SiC}$. (f) The EDC [at the $k$ point marked by the arrow in panel (e)] shows symmetric line shape unlike that from graphite. The EDCs can be fitted with a single Lorentzian and constant background (thick line).

Figure 5

(Color online) (a) Extracted half-width at half maximum (HWHM) as a function of the binding energy for NB (empty triangles) and BB (filled triangles). (b) Calculated partial electronic densities of states (pDOSs) for NB and BB. (c) Constructed ${\Sigma }^{″}$ from the data in panel (a) for the low-energy region and pDOS in panel (b). pDOS is scaled so that it matches the experimental HWHM at 0.9 eV. (d) ${\Sigma }^{\prime }$ using Hilbert transform of ${\Sigma }^{″}$. The EPC parameter is $\approx 0.23$.

Figure 6

(Color online) ARPES data taken along $M\text{−}K\text{−}\Gamma$ at different temperatures. (a), (b), (c), (d), and (e) were taken at 225, 175, 125, 75, and 25 K, respectively.

Figure 7

(Color online) HWHM vs binding energy at different temperatures: 25, 75, 125, 175, and 225 K. HWHM is extracted from NB band along $M\text{−}K$ near Fermi energy.

Figure 8

(Color online) HWHMs extracted from ARPES data at different temperatures are fitted by calculated imaginary part of self-energy. Panels (a), (b), (c), and (d) are for temperatures of 25, 75, 125, and 175 K, respectively. The solid line in each panel is the best fit to the experimental data by calculated imaginary part of self-energy. The calculated imaginary part of self-energy includes electron-phonon and electron-electron interaction terms. The electron-phonon coupling constant and electron-electron interaction prefactor were used as fitting parameters (see the text).