$E_{33}$ and $E_{44}$ optical transitions in semiconducting single-walled carbon nanotubes: Electron diffraction and Raman experiments

By combining, on the same freestanding single-walled carbon nanotubes, electron diffraction and Raman experiments, we were able to obtain the resonance energy of unambiguously $(n,m)$-identified single-walled carbon nanotubes. We focus on the analysis of the first optical transition of metallic tubes $\left(E_{11}^M\right)$ and the third and fourth transitions of semiconducting tubes ($E_{33}^S$ and $E_{44}^S$, respectively) in comparison with calculated values using a nonorthogonal tight-binding approach. For semiconducting tubes, we find that the calculated energies $E_{33}^S$ and $E_{44}^S$ have to be corrected by non-diameter-dependent (rigid) shifts of about $0.43\mathrm{eV}$ and $0.44\mathrm{eV}$, respectively, for tubes in the $1.4–2.4\text{−}\mathrm{nm}$-diameter range. For metallic tubes in the $1.2–1.7\text{−}\mathrm{nm}$-diameter range, we show that a rigid shift $\left(0.32\mathrm{eV}\right)$ of the calculated transition energy also leads to a good estimation of $E_{11}^M$. The rather large and non-diameter-dependent shifts for the third and fourth transitions in semiconducting tubes question a recent theoretical study, which relates the shifts to electron-electron correlation and exciton binding energy and suggest that the exciton binding is very small or missing for the higher transitions $E_{33}^S$ and $E_{44}^S$, contrary to the lower transitions $E_{11}^S$ and $E_{22}^S$.

Figure 1

(Color online) From top to bottom: radial breathing modes of the (15,6) $(E_{\mathit{\text{laser}}}=1.70\mathrm{eV})$, (16,7) $(E_{\mathit{\text{laser}}}=1.57\mathrm{eV})$, and (12,12) $(E_{\mathit{\text{laser}}}=1.64\mathrm{eV})$. Solid red line is a fit by a Lorentzian.

Figure 2

(Color online) $E_{11}^M$, $E_{33}^S$, and $E_{44}^S$ optical transitions vs diameter. The experimental transition energies are obtained from Raman (red circles from this work and Ref. 15) and Rayleigh measurements (red diamonds from Ref. 14). Black solid squares, black open squares, and black stars stand for normalized calculated $E_{ii}^N$ transitions for semiconducting type I $[(2n+m)\mathrm{mod}3=1]$, type II $[(2n+m)\mathrm{mod}3=2]$, and metallic SWNTs $[(2n+m)\mathrm{mod}3=0]$. The large blue symbols are the normalized calculated $E_{ii}^N$ transition energies for the $(n,m)$ SWNTs investigated by Raman spectroscopy and Rayleigh scattering.

Figure 3

(Color online) The predicted KM correction (see text) is displayed in the diameter range $1.2–2.4\mathrm{nm}$. Black dashed, blue dotted, and red dot-dashed lines stand for KM corrections for the first metallic, third semiconducting, and fourth semiconducting transitions, respectively. The horizontal red solid line (black solid line) at $\Delta E=0.44\mathrm{eV}$ $\left(0.32\mathrm{eV}\right)$ stands for the value of the average rigid shift applied to the $E_{33}^S$ and $E_{44}^S$ $\left(E_{11}^M\right)$ calculated transitions. The blue squares and red triangles (black diamonds) are the differences $\Delta E_{33}^S=E_{33}^{S\left(\mathit{expt}\right)}-E_{33}^{S\left(\mathit{NTB}\right)}$ and $\Delta E_{44}^S=E_{44}^{S\left(\mathit{expt}\right)}-E_{44}^{S\left(\mathit{NTB}\right)}$ $(\Delta E_{11}^M=E_{11}^{M\left(\mathit{expt}\right)}-E_{11}^{M\left(\mathit{NTB}\right)})$, respectively. Here “exp” stands for Raman data (this work and Ref. 15) or Rayleigh data (Ref. 14).

Figure 4

(Color online) Deviation of $\Delta E$ from the KM correction vs $1∕d$ for the third semiconducting (red dots), fourth semiconducting (black squares), and first metallic transitions (green diamonds). The blue solid line (black dotted line) is the linear fit of the average deviation vs $1∕d$ for the third and fourth semiconducting transitions (first metallic transition).