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 epsilon(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 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 Gamma only when epsilon(F) is close to a band crossing E(X). For each band crossing, a double KA occurs for epsilon(F)=E(X)+/- h omega/2, where h omega is the phonon energy. In particular, for a 1.2 nm metallic nanotube, we predict a KA to occur in the so-called G(-) peak at a doping level of about N(el)/C=+/- 0.0015 atom (epsilon(F)approximate to +/- 0.1 eV) and, possibly, close to the saturation doping level (N(el)/C similar to 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 parallel to epsilon(F)parallel to >= h 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 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.