Origin of the metal-insulator transition in TaS2A signature feature of correlated electron materials is the occurrence of metallic and insulating phases, and of transitions between them. These metal-insulator transitions (MIT), can be caused by temperature, pressure, doping, or by other external influences. Two possible mechanisms for such an MIT are a) the Peierls instability and b) the Mott-Hubbard transition.
The Peierls instability
The Peierls instability  causes a metal to become an insulator by the action of a lattice distor tion which doubles the crystal unit cell (see Fig. V.10). The energy cost incurred by the elastic distortion is more than compensated by a lowering of the electronic energy, due to the opening of a Peierls gap at the electronic Fermi level. Note that the situation is similar to that for the spin Peierls effect, discussed in Chapter I, where it is the lowered magnetic energy for spin dimers which drives the lattice distortion. Both the charge Peierls instability and the spin Peierls effect are intimately connected to the motion of lattice atoms, hence the relevant time scale will be that of lattice vibrations (i.e., 10–100 fs).
Fig. V.10. The Peierls instability . Under special circumstances,
it may become energetically favorable for a metallic crystal
to undergo a spontaneous lattice distortion, forming atomic
dimers, and doubling the crystal unit cell. In reciprocal space, the
Brillouin zone is halved, and a gap opens at the Fermi level, causing
a lowering of the electronic energy, which more than compensates
for the cost in elastic energy. The presence of the gap causes
the material to become insulating.
The Mott-Hubbard transition
The Hubbard Model
The Hubbard Model  is the simplest model of interacting particles in a lattice and the star ting point of many descriptions of correlated electron systems. It is based on the “Hubbard Hamiltonian”:
Fig. V.i6. Average site occupancy vs. the chemical potential
μ, for the Hubbard model without electron hopping (W = 0). The
jump in μ by the value U at half-filling implies the existence of an
energy gap, and hence insulating behavior.
Where the operators c†jσ and ciσ are electron creation and annihilation operators, n = c†c is the number operator, and the sums run over the spin directions σ = ↑ and ↓ and the N lattice sites of the model.
where β = 1/kBT, and is the chemical potential. Plotting
b) Non-interacting electrons (U = 0) In this case, it is convenient to use the reciprocal-space representation of the electron operators:
Fig. V.i7. Energy eigenvalues for the Hubbard Hamiltonian for
non-interacting electrons (U = 0). The points are for a model with
N = 8 lattice sites. Half-filling of such a continuous band implies
where k takes the values kn = 2pn/N assuming periodic boundary conditions in one dimension. The Hubbard Hamiltonian now has the form:
where the last expression follows from per forming the lattice sums. The energy levels of this Hamiltonian show a “band” behavior (see Fig. V.i7). As N goes to infinity, we obtain a (gapless) continuum of states, with bandwidth 4W, which, at half-filling, implies metallic behavior. We thus find that the Hubbard Hamiltonian describes an insulator, in the case W = 0, and a metal, for U = 0. Between these two limits, i.e., for intermediate U/W, there must occur a metal-insulator transition: the Mott transition. %ENDCONTAINER%