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 non-monotonic 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. Magnitudes of the temperature shifts at 300 K are smaller than 12 meV and should not affect $(n,m)$ assignments based on optical measurements. RBC acknowledges financial support from the John Simon Guggenheim Memorial Foundation and Brazilian funding agencies CNPq, FAPERJ, Instituto de Nanoci{\^e}ncias, FUJB-UFRJ and PRONEX-MCT. Work partially supported by NSF Grant No. DMR00-87088 and DOE Contract No. DE-AC03-76SF00098. Computer resources were provided by NERSC and NPACI.