Direct observation of cross-polarized excitons in aligned single-chirality single-wall carbon nanotubes

Optical properties of single-wall carbon nanotubes (SWCNTs) for light polarized parallel to the nanotube axis have been studied extensively, whereas their response to light polarized perpendicular to the nanotube axis has not been well explored. Here, by using a macroscopic film of highly aligned single-chirality (6,5) SWCNTs, we performed a systematic polarization-dependent optical absorption spectroscopy study. In addition to the commonly observed angular-momentum-conserving interband absorption of parallel-polarized light, which generates $E_{11}$ and $E_{22}$ excitons, we observed a small but unambiguous absorption peak whose intensity is maximum for perpendicular-polarized light. We attribute this feature to the lowest-energy cross-polarized interband absorption processes that change the angular momentum along the nanotube axis by $\pm 1$, generating $E_{12}$ and $E_{21}$ excitons. Unlike previous observations of cross-polarized excitons in polarization-dependent photoluminescence and circular dichroism spectroscopy experiments, our direct observation using absorption spectroscopy allowed us to quantitatively analyze this resonance. Specifically, we determined the energy and oscillator strength of this resonance to be 1.54 and 0.05, respectively, compared with the values for the $E_{11}$ exciton peak. These values, in combination with a comparison with theoretical calculations, in turn led to an assessment of the environmental effect on the strength of Coulomb interactions in this aligned single-chirality SWCNT film.

Figure 1

Illustration of the lowest-energy allowed optical interband transitions in a semiconducting SWCNT. The numbers shown for the four subbands, two in the conduction band and two in the valence band, are their subband indices. $E_{ij}$ ($i=j$) denotes an allowed optical transition for parallel ($\parallel$) polarization, whereas $E_{ij}$ ($i\ne j$) indicates an allowed optical transition for perpendicular ($\perp$) polarization.

Figure 2

Absorbance spectrum in the near-infrared and visible range for the (6,5)-purified aqueous suspension of SWCNTs with an estimated chirality purity of 99.3%. See Appendix pp1 for more details on chirality purity determination.

Figure 3

Illustration of the geometry of the polarization-dependent transmission experiments performed on an aligned SWCNT film. The incident beam is linearly polarized along the horizontal axis, and the nanotube alignment direction is rotated from the horizontal axis by angle $\beta$.

Figure 4

(a) Polarization-dependent attenuation spectra for the aligned (6,5) SWCNT film for polarization angles ($\beta$) of $0^{\circ }, {30}^{\circ }, {45}^{\circ }, {60}^{\circ }$, and ${90}^{\circ }$ with respect to the nanotube alignment direction. (b) Comparison of attenuation spectra for $0^{\circ }$ ($A_{\parallel }$, black line) and ${90}^{\circ }$ ($A_{\perp }$, red line). $A_{\perp }$ is multiplied by 3.2. They match except in the spectral region of $E_{12}/E_{21}$. (c) Comparison of the $0^{\circ }$ ($A_{\parallel }$) and ${90}^{\circ }$ ($A_{\perp }$) spectra. The blue line indicates $3.2A_{\perp }-A_{\parallel }$. (d) A normalized spectral difference $(3.2A_{\perp }-A_{\parallel })/A_{\parallel }$, which shows a prominent peak due to the $E_{12}/E_{21}$ exciton.

Figure 5

Spectral analysis for the polarization-dependent extinction spectra for the aligned (6,5) SWCNT film using Eq. (1) as the fit function. The experimental spectra (black), overall fit (red dashed line), and individual components (blue lines) are shown for polarization angles of (a) $0^{\circ }$, (b) ${45}^{\circ }$, (c) ${60}^{\circ }$, and (d) ${90}^{\circ }$.

Figure 6

Detailed polarization dependence of the individual spectral components deduced from the fits: (a) $E_{11}$ and $E_{22}$, (b) $E_{11}$ phonon sideband, (c) $E_{12}/E_{21}$, and (d) polynomial baseline for polarization angles of $0^{\circ }, {30}^{\circ }, {45}^{\circ }, {60}^{\circ }$, and ${90}^{\circ }$.

Figure 7

Integrated peak intensity as a function of polarization angle $\beta$ extracted for (a) $E_{11}$, (b) $E_{22}$, (c) $E_{11}$ phonon sideband, and (d) $E_{12}/E_{21}$.

Figure 8

Simulated nanotubes' angular distribution $f\left(\theta \right)$, based on Eq. (2). The three traces correspond to $\sigma ={25}^{\circ }, \sigma ={32}^{\circ }$, and $\sigma \to \infty$, respectively.

Figure 9

(a) Nematic order parameter $S$ as a function of standard deviation angle $\sigma$ based on Eq. (3). (b) Nematic order parameter $S$ as a function of absorption ratio between parallel and perpendicular polarization based on Eq. (9).

Figure 10

Polarization angle dependence of the integrated intensity of the $E_{11}$ peak. Black dashed curve: theoretical calculation assuming $S=0.52$. Red open circles: experimental data. The experimental observation is well reproduced by the theoretical curve. Blue dash-dotted curve: theoretical calculation assuming perfect alignment, i.e., $S=1$.

Figure 11

Fit peak comparison of $E_{11}$ and $E_{22}$ for $0^{\circ }$ and $E_{12}/E_{21}$ for ${90}^{\circ }$. The $E_{12}/E_{21}$ peak is multiplied by 10.

Figure 12

Absorbance spectrum for the SWCNT suspension used for making the film studied in this study. Two spectral regions of interest—(i) and (ii)—are expanded in the bottom two panels.

Figure 13

Spectral fitting analysis performed to determine the relative peak intensities of the $E_{11}$ peaks of (6,5), (9,1), and metallic SWCNTs in the sample.

Figure 14

(a) An ensemble of spheroidal molecules in 3D space. (b) Detailed illustration of a molecule in a 3D global coordinate system. The alignment direction is along the $z$ axis. (c) Schematic of a 2D ensemble of carbon nanotubes. The alignment direction is along the $y$ axis.