Direct Observation of Dark Excitons in Individual Carbon Nanotubes: Inhomogeneity in the Exchange Splitting

We report the direct observation of spin-singlet dark excitons in individual single-walled carbon nanotubes through low-temperature micro-magneto-photoluminescence spectroscopy. A magnetic field ($B$) applied along the tube axis brightened the dark state, leading to the emergence of a new emission peak. The peak rapidly grew in intensity with increasing $B$ at the expense of the originally dominated bright exciton peak and became dominant at $B>3\mathrm{T}$. This behavior, universally observed for more than 50 tubes of different chiralities, can be quantitatively modeled by incorporating the Aharonov-Bohm effect and intervalley Coulomb mixing. The directly measured dark-bright splitting values were 1–4 meV for tube diameters 1.0–1.3 nm. Scatter in the splitting value emphasizes the role of the local environment surrounding a nanotube in determining its excitonic fine structure.

Figure 1

(a) Schematic energy diagram for spin-singlet excitons in SWNTs. Left: four possible electron-hole configurations—direct ($KK\&K^{\prime }{K}^{\prime }$, solid) and indirect ($K{K}^{\prime }\&K^{\prime }K$, dashed). Right: two lowest singlet exciton states—symmetric ($+$) and antisymmetric ($-$) combinations of the $KK\&K^{\prime }{K}^{\prime }$ direct excitons, “bright” and “dark”, respectively, with splitting ${\Delta }_x$. Indirect excitons, not shown, lie at higher energies and are degenerate and optically inactive. (b) Schematic of the experimental setup used. C: cryostat, M: magnet, L: laser, Obj: objective lens, S: spectrometer, FPA: focal plane array, QWP: quarter-wave plate, HWP: half-wave plate, LPF: long pass filter, and WP: Wollaston prism. (c) Typical spectra simultaneously showing the $s$ (top) and $p$ (bottom) components of the PL. The $y$ axis represents the vertical dimension of the 2D array detector.

Figure 2

(a) Typical magnetic-field-dependent PL spectra for a single tube, showing the appearance of a “dark exciton” peak at a lower energy with respect to the main bright emission peak when a magnetic field is applied parallel to the tube axis. The peak grows with the field at the expense of the bright emission peak and becomes dominant at fields $>3\mathrm{T}$. (b) Typical magnetic field dependence for the case of a perpendicular magnetic field. Clearly, no such magnetic brightening of the dark exciton peak is seen. (c) Linear fit of the data in (a) using Eq. (1), to obtain ${\Delta }_x$ from the offset and $\mu$ from the slope (see text for details).

Figure 3

(a) Measured zero-field dark-bright splitting, ${\Delta }_x$, as a function of tube diameter for 45 tubes. The data show scatter even for tubes with the same chirality. Inset shows the same data plotted against emission wavelength before the assignment of respective chiralities. Structural assignment was done based on room temperature photoluminescence excitation spectroscopy data taking inhomogeneity of emission wavelength into account, as shown representatively for (9,7) species. (b) The data in (a) are averaged over 5 diameter ranges and plotted against average diameter to highlight the trend of decreasing ${\Delta }_x$ with increasing diameter.

Figure 4

$\mathrm{\text{Dark}}(\delta )/\mathrm{\text{bright}}(\beta )$ oscillator-strength ratio for 9 different nanotubes versus zero-field dark-bright splitting (${\Delta }_x$) normalized to the finite-field splitting ($\Delta$). They all fall onto a single curve given by Eq. (3). The temperature of all nanotubes was taken to be 10 K, which was obtained by fitting the intensity ratio against $B_{\parallel }$ to Eqs. (2, 3), as shown in the inset. This value is very close to the measured temperature of 5–7 K.