Band gaps of primary metallic carbon nanotubes

Primary metallic, or small gap semiconducting nanotubes, are tubes with band gaps that arise solely from breaking the bond symmetry due to the curvature. We derive an analytic expression for these gaps by considering how a general symmetry breaking opens a gap in nanotubes with a well defined chiral wrapping vector. This approach provides a straightforward way to include all types of symmetry breaking effects, resulting in a simple unified gap equation as a function of chirality and deformations.