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 $2G$.

Figure 1

Variation of the LO axial frequencies with electron doping at $315\mathrm{K}$, calculated from phonon-EZF (full diamonds) or through full-nanotube frozen-phonon (open circles) case. The solid line is a guide to the eyes. The metal and/or carbon composition corresponding to the doping level is indicated on the top of the figure. Inset: variation with electronic doping of the phonon-EZF LO axial frequency (full diamonds) compared to a SWCNT full DFPT dynamical matrix calculation (open squares) at $1578\mathrm{K}$.

Figure 2

Variation of the electronic doping per C atom as a function of the Fermi energy ${ϵ}_F$. Solid line, analytical expression [Eq. (2)] with $E_{11}^M∕2=0.83\mathrm{eV}$; diamonds, DFT calculations, within phonon EZF, of the Fermi energy in a (9,9) SWCNT at $315\mathrm{K}$. The $E_{11}^M∕2$ and $E_{22}^M∕2$ levels are indicated by vertical dotted lines. Inset: Calculated electronic bands of the (9,9) SWCNT around the $\mathbf{K}$ point. ${ϵ}_F^0$ is taken as the zero energy; $E_{11}^M∕2$ and $E_{22}^M∕2$ are the minima of the second and third (hyperbolic) conduction bands. The valence bands (below ${ϵ}_F^0$) and the conduction bands (above ${ϵ}_F^0$) are usually referred to as the $\pi$ and ${\pi }^{*}$ bands, respectively. The bands crossing at ${ϵ}_F^0$ have the index $\nu =0$, while the valence and conduction bands immediately below and above have indices $\nu =\pm 1$ [see Eq. (1)].

Figure 3

(Color online) Variation, within the adiabatic approximation, of the LO axial (black) and TO tangential (red and/or dark gray) $\Gamma$ frequencies with the Fermi level at $T=315\mathrm{K}$, calculated from DFT phonon EZF (full diamonds), from the integral model [solid line, Eq. (11)], and from the analytical model at $T=0\mathrm{K}$ [dashed line, Eqs. (17, 22, 23)]. The metal and/or carbon composition corresponding to the Fermi level is indicated on the top of the figure.

Figure 4

Schematic representation of electronic transitions allowed by conservation of energy and momentum in electron-phonon scattering processes involving a phonon with momentum $\mathbf{q}$ and energy $\hslash {\omega }_{\mathbf{q}}$. An abrupt change in the phonon dispersion, i.e., a Kohn anomaly, occurs when, by change of the Fermi energy, one of this transitions becomes allowed or forbidden by the Pauli exclusion principle. In practice, this occurs when the Fermi level crosses a tip of one of the arrows corresponding to allowed transitions. Top panel: transitions at $\mathbf{q}=0$ $\left(\mathbf{\Gamma }\right)$. Bottom panel: transitions at a finite $\mathbf{q}$ point.

Figure 5

(Color online) Lower panel: variation of the LO axial ($G^{-}$ peak in Raman experiments, black) and TO tangential ($G^{+}$ peak, red and/or dark gray) $\Gamma$ frequencies with the Fermi level at $T=315\mathrm{K}$ (solid line) and at $77\mathrm{K}$ (LO axial, dashed line), calculated from the nonadiabatic integral model [Eq. (25)]. Top panel: variation, at the same temperatures, of the EPC contribution to the FWHM of the LO axial mode with the doping level. The metal and/or carbon composition corresponding to the Fermi level is indicated on the top of the figure.

Figure 6

(Color online) Lower panel: variation, with respect to zero doping, of the LO axial frequencies with the Fermi level at $T=315\mathrm{K}$ (black) and $T=77\mathrm{K}$ (red and/or dark gray), calculated from the nonadiabatic integral model [Eq. (25)]. The phonon momentum is equal to $\mathbf{q}=\left(0.1\right)\left(\frac{2\pi }{a_0}\right)$. Top panel: same, at $\mathbf{q}=\left(0.05\right)\left(\frac{2\pi }{a_0}\right)$. The dotted vertical line indicates the minimum of the second conduction band. The energy values ${\mu }_{1-}$, ${\mu }_{1+}$, ${\mu }_{2-}$, and ${\mu }_{2+}$ correspond to the levels schematically shown in Fig. 4. A similar behavior is expected for the highest optical phonon near the $\mathbf{K}$ point, which is responsible for the Raman $D$ band. The metal and/or carbon composition corresponding to the Fermi level is indicated on the top of the figure.