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 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.