Electron-phonon coupling and electron self-energy in electron-doped graphene: Calculation of angular-resolved photoemission spectra

We obtain analytical expressions for the electron self-energy and the electron-phonon coupling in electron-doped graphene using electron-phonon matrix elements extracted from density functional theory simulations. From the electron self-energies we calculate angle-resolved photoemission spectra (ARPES). We demonstrate that the measured kink at $\approx -0.2\mathrm{eV}$ from the Fermi level is actually composed of two features, one at $\approx -0.195\mathrm{eV}$ due to the twofold-degenerate $E_{2g}$ mode, and a second one at $\approx -0.16\mathrm{eV}$ due to the $A_1^{\prime }$ mode. The electron-phonon coupling extracted from the kink observed in ARPES experiments is roughly a factor of 5.5 larger than the calculated one. This disagreement can be only partially reconciled by the inclusion of resolution effects. Indeed, we show that a finite resolution increases the apparent electron-phonon coupling by underestimating the renormalization of the electron velocity at energies larger than the kink positions. The discrepancy between theory and experiments is thus reduced to a factor of $\approx 2.5$. From the linewidth of the calculated ARPES we obtain the electron relaxation time. A comparison with available experimental data in graphene shows that the electron relaxation time detected in ARPES is almost two orders of magnitudes smaller than that measured by other experimental techniques.

Figure 1

(Color online) Lowest-order contribution to the electron self-energy due to the electron-phonon interaction. The dotted (continuous) line represents the phonon (electron) self-energy.

Figure 2

(Color online) Real $\left({\Sigma }^{\prime }\right)$ and imaginary $\left({\Sigma }^{″}\right)$ parts of the electron self-energy in graphene (continuous line). Dashed lines refer to self-energy parts obtained using a constant density of states. The Fermi level is ${ϵ}_f=0.4\mathrm{eV}$.

Figure 3

MDC scans for different energies (top) and ARPES maximum position as a function of energy and momentum (bottom). The Fermi level is ${ϵ}_f=0.4\mathrm{eV}$; the experimental resolution is not included in the calculation. A $1\mathrm{meV}$ broadening is used for illustration purposes.

Figure 4

(Color online) MDC linewidth (HWHM) of the two models’ self-energies. The curve in the upper panel is obtained using a finite resolution, while that in the lower one is without resolution effects. In the absence of resolution effects, the linewidth is equal to $-\beta {\Sigma }^{″}\left(ϵ\right)$ (see Fig. 1). The Fermi level is ${ϵ}_f=0.4\mathrm{eV}$.

Figure 5

Position of the maximum in MDC curves using a finite resolution. The Fermi level is ${ϵ}_f=0.4\mathrm{eV}$.

Figure 6

(Color online) Shift of the maximum in MDC curves due to the electron-phonon coupling, ${\Sigma }^{\prime }(\mathbf{k},{ϵ}_{\mathbf{k}}^{\max })$, with (b) and without (a) the experimental resolution. The derivative of the curve at $k_f$ in the absence of a finite resolution is shown as a dotted black line in (a) and (b). The dashed red line is a linear fit to ${\Sigma }^{\prime }(\mathbf{k},{ϵ}_{\mathbf{k}}^{\max })$ in the presence of a finite resolution in the $k$ range $-0.195<k-k_f<-0.04\mathrm{eV}∕\beta$.

Figure 7

Electron-phonon coupling in electron-doped graphene. The continuous line labeled DFT refers to Eq. (15), the empty squares are experimental data from Fig. 5 in Ref. 9, while the empty circles represents the apparent electron-phonon coupling extracted from fits to the calculated ARPES spectra using Eq. (30) and including resolution effects. The curves $\mathrm{DFT}\times 5.5$ and $\mathrm{DFT}\times 2.2$ indicate Eq. (30) multiplied by the prefactors 5.5 and 2.2, respectively.