Kohn anomalies and nonadiabaticity in doped carbon nanotubes

The high-frequency Raman-active phonon modes of metallic single-walled carbon nanotubes (SWCNT) are thought to be characterized by Kohn anomalies (KAs) resulting from the combination of SWCNT intrinsic one-dimensional nature and a significant electron-phonon coupling (EPC). KAs are expected to be modified by the doping-induced tuning of the Fermi energy level ${ϵ}_F$, obtained through the intercalation of SWCNTs with alkali atoms or by the application of a gate potential. We present a density-functional theory (DFT) study of the phonon properties of a (9,9) metallic SWCNT as a function of electronic doping. For such study, we use, as in standard DFT calculations of vibrational properties, the Born-Oppenheimer (BO) approximation. We also develop an analytical model capable of reproducing and interpreting our DFT results. Both DFT calculations and this model predict, for increasing doping levels, a series of EPC-induced KAs in the vibrational mode parallel to the tube axis at the $\mathbf{\Gamma }$ point of the Brillouin zone, usually indicated in Raman spectroscopy as the $G^{-}$ peak. Such KAs would arise each time a new conduction band is populated. However, we show that they are an artifact of the BO approximation. The inclusion of nonadiabatic effects dramatically affects the results, predicting KAs at $\mathbf{\Gamma }$ only when ${ϵ}_F$ is close to a band crossing $E_X$. For each band crossing, a double KA occurs for ${ϵ}_F=E_X\pm \hslash \omega ∕2$, where $\hslash \omega$ is the phonon energy. In particular, for a $1.2\mathrm{nm}$ metallic nanotube, we predict a KA to occur in the so-called $G^{-}$ peak at a doping level of about $N_{\mathrm{el}}∕C=\pm 0.0015$ atom $({ϵ}_F\approx \pm 0.1\mathrm{eV})$ and, possibly, close to the saturation doping level $(N_{\mathrm{el}}∕C\sim 0.125)$, where an interlayer band crosses the ${\pi }^{*}$ nanotube bands. Furthermore, we predict that the Raman linewidth of the $G^{-}$ peak significantly decreases for $\mid {ϵ}_F\mid ⩾\hslash \omega ∕2$. Thus, our results provide a tool to determine experimentally the doping level from the value of the KA-induced frequency shift and from the linewidth of the $G^{-}$ peak. Finally, we predict KAs to occur in phonons with finite momentum $\mathbf{q}$ not only in proximity of a band crossing but also each time a new band is populated. Such KAs should be observable in the double-resonant Raman peaks, such as the defect-activated $D$ peak, and the second-order peaks $2D$ and $2G$.