Temperature Dependence of the Band Gap of Semiconducting Carbon Nanotubes

The temperature dependence of the band gap of semiconducting single-wall carbon nanotubes (SWNTs) is calculated by direct evaluation of electron-phonon couplings within a “frozen-phonon” scheme. An interesting diameter and chirality dependence of $E_g(T)$ is obtained, including nonmonotonic behavior for certain tubes and distinct “family” behavior. These results are traced to a strong and complex coupling between band-edge states and the lowest-energy optical phonon modes in SWNTs. The $E_g(T)$ curves are modeled by an analytic function with diameter- and chirality-dependent parameters; these provide a valuable guide for systematic estimates of $E_g(T)$ for any given SWNT. The magnitudes of the temperature shifts at 300 K are smaller than 12 meV and should not affect $(n,m)$ assignments based on optical measurements.

Figure 1

Calculated and fitted $E_g(T)$. Solid dots and lines correspond to $\nu =2$ SWNTs, whereas open dots and dashed lines correspond to $\nu =1$ SWNTs.

Figure 2

Dimensionless $g^{2}F$ (lower panels) and phonon spectra (upper panels) for (a) $(10,0)$ and (b) $(11,0)$ SWNTs. Lower panels: Solid black lines are the total $g^{2}F$ and red dashed lines are the contributions from the shape-deformation modes (SDMs) near $\Gamma$. The $g^{2}F$ is obtained from Gaussian broadening ($\sigma =23\mathrm{K}$) of individual phonon contributions. Shapes for each SDM are also shown in the figure. Upper panels: Phonon frequencies ($\Omega$) measured in both ${\mathrm{c}\mathrm{m}}^{-1}$ and K. SDM branches are highlighted by red dashed lines.