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.