Decay and frequency shift of both intervalley and intravalley phonons in graphene: Dirac-cone migration

By considering analytical expressions for the self-energies of intervalley and intravalley phonons in graphene, we describe the behavior of $D$, $2D$, and $D^{\prime }$ Raman bands with changes in doping ($\mu$) and light-excitation energy ($E_L$). Comparing the self-energy with the observed $\mu$ dependence of the $2D$ bandwidth, we estimate the wave vector $q$ of the constituent intervalley phonon at $\hslash vq\simeq E_L/1.6$ ($v$ is the electron's Fermi velocity) and conclude that the self-energy makes a major contribution (60$\%$) to the dispersive behavior of the $D$ and $2D$ bands. The estimate of $q$ is based on a concept of shifted Dirac cones in which the resonance decay of a phonon satisfying $q>\omega /v$ ($\omega$ is the phonon frequency) into an electron-hole pair is suppressed when $\mu <(\hslash vq-\hslash \omega )/2$. We highlight the fact that the decay of an intervalley (and intravalley longitudinal optical) phonon with $q=\omega /v$ is strongly suppressed by electron-phonon coupling at an arbitrary $\mu$. This feature is in contrast with the divergent behavior of an intravalley transverse optical phonon, which bears a close similarity to the polarization function relevant to plasmons.

Figure 1

(a) The $\mu$ dependence of $G$ band broadening (frequency ${\omega }_G$). (b) An electron-hole pair between two valleys. The Dirac cone at the $K$ ($K^{\prime }$) point is indicated in black (red). With the migration of the Dirac cone, the electron-hole pair creation process is viewed as a direct transition. (c) The $\mu$ dependence of the broadening of an intervalley phonon is different from that of the $G$ band. Note that spectral broadening of an intravalley phonon can be discussed with the replacement $K^{\prime }\to K$ (or $K\to K^{\prime }$).

Figure 2

(a) 3D plot of $-\mathrm{Im}{\Pi }_{\mu }(q,{\omega }_{\mathrm{D}})$. The variables $vq$ and $\mu$ are given in units of eV. The cross section of (a) for different $vq$ values (b), and for different $\mu$ values (c). In (c), $vq$ is proportional to the light-excitation energy $E_L$.

Figure 3

(a) A 3D plot of $\mathrm{Re}{\Pi }_{\mu }(q,{\omega }_{\mathrm{D}})$. The variables $vq$ and $\mu$ are given in eV. (b) The cross section of (a) for different $vq$ values and (c) for different $\mu$ values. Note that $\mathrm{Re}{\Pi }_{\mu }(q,\omega )$ does not include the $q$ dependence of the bare frequency.

Figure 4

Projection of Dirac cones. (a) Slight doping (red) causes a softening, while (b) heavy doping (blue) causes hardening as well as softening.

Figure 5

3D plot of ${\Pi }_{\mu }(q,{\omega }_{\mathrm{D}})$ for an intravalley LO phonon [(a) and (b)] and TO phonon [(c) and (d)]. Panels (a) and (c) show the imaginary part, and panels (b) and (d) show the real part. The variables $vq$ and $\mu$ are given in eV.