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Atomistic quantum simulation is performed to study the band gap variation of single walled (SW) and double walled (DW) carbon nanotubes (CNTs) with uniaxial and torsional strain. For DW CNT the band gap variation with uniaxial or torsional strain depends on the band gap variation of each CNT from which DW CNT is constructed. The band gap of DW CNT is the minimum of band gaps of two CNTs it constituted with for every value of strain. The bonding energy between inner and outer tubes atoms has insignificant effect on band gap.