Comprehensive model of the optical spectra of carbon nanotubes on a substrate by polarized microscopy

Polarized optical microscopy and spectroscopy are progressively becoming key methods for the high-throughput characterization of individual carbon nanotubes (CNTs) and other one-dimensional nanostructures, on substrate and in devices. The optical response of CNTs on substrate in cross polarization experiments is usually limited by the polarization conservation of the optical elements in the experimental setup. We developed a theoretical model taking into account the depolarization by the setup and the optical response of the substrate. We show that proper modelization of the experimental data requires to take into account both noncoherent and coherent light depolarization by the optical elements. We also show how the nanotube signal can be decoupled from the complex reflection factor of the antireflection substrate, which is commonly used to enhance the optical contrast. Finally, we describe an experimental protocol to extract the depolarization parameters and the complex nanotube susceptibility, and how it can improve the chirality assignment of individual carbon nanotubes in complex cases.

Figure 1

Polarization-based optical spectroscopy of carbon nanotubes (CNT). (a) Simplified schema of the experimental setup. (b) Cross polarizers configuration. Analyzer $P_2$ is set at an angle $\pi /2-\delta$ with respect to the polarizer $P_1$ and the CNT (green line) is set at around $\pi /4$ with respect to the polarizers. (c) Typical experimental spectrum of a CNT on substrate measured with $\delta =0.02$ rad.

Figure 2

Experimental spectra of carbon nanotubes for three different angles $\delta$: ${10}^{-2}$ (black), $-5\times {10}^{-3}$ (blue), and $-5\times {10}^{-2}$ (red) rad. (a) Case A. (b) Case B. In both cases, the substrate is 100 nm ${\mathrm{SiO}}_2$/Si.

Figure 3

Evolution of the real (black line) and imaginary (blue line) parts of (a) the field reflection coefficient $r_E$ and (b) amplification factor $r_A$. Green and purple dashed lines represent the coefficients of Si and ${\mathrm{SiO}}_2$ substrates as references. (c) Reflectance $|r_E{|}^2$ and (b) amplification factor $|r_A{|}^2$ for three ${\mathrm{SiO}}_2$ thicknesses: 100 (black), 300 (blue), and 500 (red) nm. Green and purple dashed lines represent the coefficients of Si and ${\mathrm{SiO}}_2$ substrates as references. [(e) and (f)] Contrast obtained on different substrates (bulk Si, bulk ${\mathrm{SiO}}_2$, and ${\mathrm{SiO}}_2$/Si with different ${\mathrm{SiO}}_2$ thicknesses from 80 to 500 nm) considering four excitonic transitions [Eq. (19)].

Figure 4

(a) Raw spectrum on 100 nm ${\mathrm{SiO}}_2$/Si substrate for different angles $\delta$: 0.05 (black), 0.02 (blue), 0.005 (green), $-0.02$ (orange), and $-0.05$ (red) rad. (Inset) Intensity as a function of the angle for 3 wavelengths (450, 580, and 740 nm), fitted by Eq. (20). [(b) and (c)] Parameters extracted from the fitted curves: (b) extinction ratio $|f_t{|}^2$ and (c) tilt angle ${\delta }_0$. Vertical dotted lines show the energies used in inset.

Figure 5

Contrast evolution of the tubes in (a) case A and (b) case B for one energy corresponding to an absorption peak was selected. (Black line) Fit from the ideal optics model. (Orange line) Fit from the noncoherent depolarization model. (Red line) Fit from the complete model. (Green line) Scattering term extracted from the complete model. [(c) and (d)] Depolarization parameter $f_t$ extracted from the fits [(orange) noncoherent and (red) complete models] compared to the data extracted beside the tube (blue). [(e) and (f)] ${\delta }_0$ from the complete model (red) compared to the values measured beside the tube (blue).

Figure 6

[a and b] Susceptibility extracted for (a) case A and (b) case B: $\text{Re}\left(A\right)$ (blue line) and $\text{Im}\left(A\right)$ (red line) extracted from the complete model. The black line represent $\text{Re}\left(A\right)$ obtained from Kramers-Kronig transformation. [c and d] ${\left|A\right|}^2$ extracted from the complete model for (c) case A and (d) case B. The indexation of the resonances is described in the text.