Photoreflectance of $0.4\text{‐}\mathrm{nm}$ single-walled carbon nanotubes

The photoreflectance spectra of the $0.4\text{‐}\mathrm{nm}$ diameter single-walled carbon nanotubes (SWNTs) have been measured. The sharp features obtained from the experimental spectra have been identified to be the van Hove singularities associated with tubes of different chiralities, and fitted by the first-order derivatives of both real and imaginary parts of the dielectric functions, such that the band structures of the SWNTs are further clarified.

Figure 1

The photoreflectance spectrum (solid line) from a bundle of the $0.4\text{‐}\mathrm{nm}$ SWNTs. The absorption spectrum (dotted line) is drawn at the bottom for comparison. The upper inset is the schematic diagram of the PR measurement setup. BS is short for beam splitter. The illustrated sample is a photo taken from the side view of a hexagonal zeolite crystal filled with SWNTs.

Figure 2

The (a) real part ${\epsilon }_1$ and (b) imaginary part ${\epsilon }_2$ of the dielectric functions of three chiralities of SWNTs from the Liu and Chan calculation (Ref. 3). The Seraphin coefficients of the (4, 2) and (5, 0) tubes are plotted in the (c) and (d) panels, respectively.

Figure 3

The first derivative fitting for the PR (a) peak $A$, (b) peak $B$, and (c) peak $A^{\prime }$. The dots are the experimental data and the lines are the fitted curves.

Figure 4

The band structure for the (5, 0) tube calculated by Liu and Chan (Ref. 3). The optical transitions corresponding to $A$ and $A^{\prime }$ are indicated by the arrows. The Fermi level $E_F$ is at 0.