  Vanadium dioxide (VO$_2$) undergoes a metal-insulator transition (MIT) at 340
K with the structural change between tetragonal and monoclinic crystals as the
temperature is lowered. The conductivity $\sigma$ drops at MIT by four orders
of magnitude. The low-temperature monoclinic phase is known to have a lower
ground-state energy. The existence of a $k$-vector ${\boldsymbol k}$ is
prerequisite for the conduction since the ${\boldsymbol k}$ appears in the
semiclassical equation of motion for the conduction electron (wave packet).
Each wave packet is, by assumption, composed of the plane waves proceeding in
the ${\boldsymbol k}$ direction perpendicular to the plane. The tetragonal
(VO$_2$)$_3$ unit cells are periodic along the crystal's $x$-, $y$-, and
z-axes, and hence there are three-dimensional $k$-vectors. The periodicity
using the non-orthogonal bases does not legitimize the electron dynamics in
solids. There are one-dimensional ${\boldsymbol k}$ along the c-axis for a
monoclinic crystal. We believe this decrease in the dimensionality of the
$k$-vectors is the cause of the conductivity drop. Triclinic and trigonal
(rhombohedral) crystals have no $k$-vectors, and hence they must be insulators.
The majority carriers in graphite are "electrons", which is shown by using an
orthogonal unit cell for the hexagonal lattice.
