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Among the quasi one-dimensional transition metal tetrachalcogenides(MSe$_4$)$_n$I (M=Nb,Ta), the $n=3$ compounds are the only ones not displayingcharge density waves. Instead, they show structural transitions with puzzlingtransport behavior. They are semiconductors at the lowest temperatures, buttheir transport gaps are significantly smaller than those inferred from ARPESand optical conductivity. Recently, a metallic polytype of (TaSe$_4$)$_3$I hasbeen found with ferromagnetism and superconductivity coexisting at lowtemperature, in contrast to previous reports. In this work we present detailedab-initio and tight binding band structure calculations for the different(MSe$_4$)$_n$I reported structures. We obtain good agreement with the observedtransport gaps, and explain how ARPES and optics experiments effectively probea gap between different bands due to an approximate translation symmetry,solving the controversy. Finally, we show how small extrinsic hole doping cantune the Fermi level through a Van Hove singularity in (TaSe$_4$)$_3$I anddiscuss the implications for magnetism and superconductivity.

Band structures of (NbSe$_4$)$_3$I and (TaSe$_4$)$_3$I: Reconciling transport, optics and ARPES