Scaling of Excitons in Carbon Nanotubes

Light emission from carbon nanotubes is expected to be dominated by excitonic recombination. Here we calculate the properties of excitons in nanotubes embedded in a dielectric, for a wide range of tube radii and dielectric environments. We find that simple scaling relationships give a good description of the binding energy, exciton size, and oscillator strength.

Figure 1

(a) Binding energy of first optically active exciton, versus $\epsilon$, in four semiconducting zigzag tubes: $(13,0)$, $(19,0)$, $(25,0)$, and $(31,0)$. (b) Scaling of binding energy of first and second exciton (red dots) in semiconducting tubes with all possible chirality (156 tubes with $d=1.0–2.5\mathrm{nm}$, $\epsilon =2–15$). Here $R$ and $m$ are in a.u. The black solid line is the best fit to Eq. (7) for $\epsilon =4–15$ (3% rms error over this range), corresponding to $\alpha =1.40$ and $A_b=24.1\mathrm{eV}$.

Figure 2

(a) First exciton rms $e\mathrm{\text{−}}h$ separation $L_1$ in four zigzag tubes $(13,0)$, $(19,0)$, $(25,0)$, and $(31,0)$. The solid lines are the best linear fits. The slope $k$ scales approximately as $m^{-1}$, the product $k{m}_1$ (in a.u.) equals 0.167, 0.160, 0.157, and 0.155 for tubes with diameter 1.0, 1.5, 2.0, and 2.5 nm, respectively. (b) The linear scaling of $L_1/R$ with $\epsilon /mR$ in semiconducting tubes of all possible chirality with $d=1.0–2.5\mathrm{nm}$, and $\epsilon =2–15$. The solid line is the best fit to Eq. (9) to the results in the range $\epsilon \ge 4$, giving $A_L=2.13$ and $B_L=0.174$ (for $m$ and $R$ in a.u.). The rms discrepancy between the fit and the full calculations is 2% for the subset of data having $\epsilon \ge 4$ or $\epsilon /m{R}^2\ge 9.5$.

Figure 3

Absorption spectra ${\epsilon }_2$ of $(19,0)$ tube in dielectric environment (a) $\epsilon =\infty$ (equivalent to no $e\mathrm{\text{−}}h$ interaction), (b) $\epsilon =10$, (c) $\epsilon =4$. ($E_s$ is the unknown self-energy shift.) Note expanded scales for dotted lines in continuum region in (b) and (c). The fractional spectral weight transfer to the first exciton is $I_1/I_0=0$, 0.71, and 0.95, respectively. Spectra are broadened with a Gaussian width of 0.0125 eV. (d) The scaling of spectral weight transfer to the first exciton, $I_1$, in all semiconducting tubes with $d=1.0–2.5\mathrm{nm}$ and $\epsilon =2–15$ and all possible chiralities. The best fit to Eq. (10) (rms difference 3.5%) is obtained with $A_I=1.22{\mathrm{eV}}^2{\mathrm{nm}}^2$ and $B_I=1.61$.