Phonon switching and combined Fano-Rice effect in optical spectra of bilayer graphene

Recent infrared measurements of phonon peaks in gated bilayer graphene reveal two striking signatures of electron-phonon interaction: an asymmetric Fano lineshape and a giant variation in the peak intensity as a function of the applied gate voltage. In this Rapid Communication we provide a unified theoretical framework which accounts for both these effects and unveils the occurrence of a switching mechanism between symmetric $\left(E_g\right)$ and antisymmetric $\left(E_u\right)$ phonon modes as dominant channel in the optical response. A complete phase diagram of the optical phonon response is also presented, as a function of both the charge density and the band gap.

Figure 1

(a) Scheme of the band structure. (b)–(d) Relevant diagrams entering the optical conductivity for $\Delta =0$: dashed, solid, and wavy lines represent the photon, the electron, and the phonon Green’s functions, respectively. Squares and circles are the current and the electron-phonon-scattering matrices $\widehat{j}$, ${\widehat{V}}_{\text{A}}$, ${\widehat{V}}_{\text{A}}^{†}$, respectively.

Figure 2

(Color online) $W_{\text{A}}$, $W_{\text{A}}^{\prime }$ intensities of the $E_u$ mode as a function of the charge concentration $n$ for $\Delta =0$ and $\eta =0$ and 30 meV. Also shown are experimental data from Ref. 15 for the bare intensity $W_{\text{exp}}$ and the $\omega$-integrated one $W_{\text{exp}}^{\prime }$.

Figure 3

(Color online) (a)–(c) Strength $p_{\nu }$ of the different electron-phonon contributions to the optical conductivity. The above color scale is in units of ${10}^3{\Omega }^{-1}{\text{cm}}^{-2}$. (d) Relative phonon weight $R_{\text{AS}}$. (e)–(f) Fano-asymmetry factors $q_{\text{A}}$ and $q_{\text{S}}$. Also shown in these plots is the $\Delta \text{−}\mu$ location of experimental data: (Ref. 16) (●) and (Ref. 15) (◼). The corresponding optical spectra are shown in panels (g) ($\mu =0$ and increasing $\Delta$ from A to E) and (h) (increasing doping from 1 to 21), respectively. Curves are displaced vertically for clarity. (i) Different electron-phonon contributions $W_{\nu }^{\prime }$ (open symbols) to total $\omega$-integrated spectral weight (solid line) calculated for $\Delta \text{−}\mu$ of Ref. 15. (j) corresponding Fano factors $q_{\nu }$ from Eq. (3). The size of the open symbols in panels (i) and (j) is proportional to $p_{\nu }$. Crosses are experimental data from Ref. 15.