  Crystallographic shear phases of niobium oxide form an interesting family of
compounds that have received attention both for their unusual electronic and
magnetic properties, as well as their performance as intercalation electrode
materials for lithium-ion batteries. Here, we present a first-principles
density-functional theory study of the electronic structure and magnetism of
H-Nb$_2$O$_5$, Nb$_{25}$O$_{62}$, Nb$_{47}$O$_{116}$, Nb$_{22}$O$_{54}$, and
Nb$_{12}$O$_{29}$. These compounds feature blocks of niobium-oxygen octahedra
as structural units, and we show that this block structure leads to a
coexistence of flat and dispersive energy bands, corresponding to localised and
delocalised electronic states. Electrons localise in orbitals spanning multiple
niobium sites in the plane of the blocks. Localised and delocalised electronic
states are both effectively one-dimensional and are partitioned between
different types of niobium sites. Flat bands associated with localised
electrons are present even at the GGA level, but a correct description of the
localisation requires the use of GGA+U or hybrid functionals. We discuss the
experimentally observed electrical and magnetic properties of niobium suboxides
in light of our results, and argue that their behaviour is similar to that of
$n$-doped semiconductors, but with a limited capacity for localised electrons.
When a threshold of one electron per block is exceeded, metallic electrons are
added to existing localised electrons. We propose that this behaviour of shear
phases is general for any type of $n$-doping, and should transfer to doping by
alkali metal (lithium) ions during operation of niobium oxide-based battery
electrodes. Future directions for theory and experiment on mixed-metal shear
phases are suggested.
