Diameter dependence of TO phonon frequencies and the Kohn anomaly in armchair single-wall carbon nanotubes

We present resonant Raman scattering experiments on nanotube samples enriched in metallic armchair single-wall carbon nanotubes (SWCNTs). We establish the transverse optical $\left({\mathcal{A}}_{\mathrm{TO}}\right)$ phonon frequency for the (5,5) through (10,10) armchair species, ranging in diameter from 0.68 to 1.36 nm. The frequencies show a strong diameter dependence similar to that previously observed in semiconducting nanotubes. We show that the ${\mathcal{A}}_{\mathrm{TO}}$ frequencies in armchair SWCNTs are dramatically upshifted from those of semiconducting SWCNTs. Furthermore, using electrochemical doping, we demonstrated that the ${\mathcal{A}}_{\mathrm{TO}}$ frequencies in armchair SWCNTs are independent of the position of the Fermi level. These results suggest that the upshift is a result of a Kohn anomaly involving a forward-scattering mechanism of electrons close to the Fermi level. This is in contrast to the well-known Kohn anomaly that dominates the downshift of the ${\mathcal{A}}_{\mathrm{LO}}$ and ${\mathcal{E}}_{2g}$ phonons in nonarmchair metallic SWCNTs and graphene, respectively.

Figure 1

Optical absorption spectra of aqueous 1% (wt/vol) DOC solutions of armchair-enriched samples, prepared via DGU [(a) orange trace], DNA-IEX [(b) black and (c) blue traces], and ATPE [(d) magenta and (e) red traces] methods. All spectra are offset and scaled for clarity.

Figure 2

(a) RBM and (b) G-mode spectra of the various samples in aqueous 1% (wt/vol) DOC solution excited in resonance with the particular armchair SWCNT given in the legend. Excitation energies for both RBM and G-mode spectra for a given chirality are as indicated in the figure. For clarity, all spectra are normalized to the maximum peak intensity.

Figure 3

${\mathcal{A}}_{\mathrm{TO}}^M$ frequency (blue squares) and its fit to Eq. (1) (solid line) as a function of diameter. Below $\approx 1$ nm ${\mathcal{A}}_{\mathrm{TO}}^M$ can clearly be distinguished from the LO (${\mathcal{A}}_{\mathrm{LO}}^S$, triangles) and TO (${\mathcal{A}}_{\mathrm{TO}}^S$, circles) phonons in semiconducting SWCNT (taken from Refs. [19] and [38]). Fits to Eq. (1) for ${\mathcal{A}}_{\mathrm{LO}}^S$ and ${\mathcal{A}}_{\mathrm{TO}}^S$ are shown as dashed and dashed-dotted lines, respectively, using the parameters given in Table 2.

Figure 4

Illustration of how variations in the Fermi energy, $E_F$, affect a Kohn anomaly involving a backscattering of electrons (green vertical arrow), where $\partial E\left(k\right)/\partial k$ changes sign, and a forward-scattering process (orange diagonal arrow), where no sign change in $\partial E\left(k\right)/\partial k$ occurs. (a) and (b) show the electronic band structure of a semimetal with linear bands equivalent to that of metallic SWCNTs. In (a), the undoped case, both processes are possible. In (b), as $E_F$ is changed by more than half the phonon energy, the backscattering process is blocked while the forward-scattering remains possible.

Figure 5

(a) RBM spectra with excitation at 465.8 nm (top) and 501.7 nm (bottom) of the (6,6)-enriched thin film used for electrochemical doping. (b) $G$-band spectra of a (6,6)-enriched thin film as a function of electrochemical potential taken at 465.8 nm laser excitation. Potential is varied from $-1$ V (bottom) to +1 V (top), in 55 mV steps. Spectra were decomposed into five Lorentzian components labeled blue (1498 ${\mathrm{cm}}^{-1}$), cyan (1532 ${\mathrm{cm}}^{-1}$), orange (1567 ${\mathrm{cm}}^{-1}$), red (1582 ${\mathrm{cm}}^{-1}$), and green (1592 ${\mathrm{cm}}^{-1}$), where the noted peak positions are taken from the spectrum measured at $V=-1$ V. (c) Peak positions of the five Lorentzian components fitted to the spectra in (b) as a function of electrochemical potential. Dashed line indicates the 1540 ${\mathrm{cm}}^{-1}$ expected ${\mathcal{A}}_{\mathrm{LO}}^S$ frequency for semiconducting species of the same diameter as the (6,6). (d) A schematic diagram of the electrochemical cell used to measure spectra in (b).