Figure 95: Small crystals as a model to describe a Mott insulator.RTcritical ResistanceinsulatormetalFigure 96: T vs. R;
starting with resistance above the critical resistance - insulator
starting with resistance below the critical resistance - metal.
IfNis the number of atoms per unit cell there are 2 Nstates in each band:
In a one dimensional conductor it is possible to label the electronic states withspinandk-vector.
Everykstate corresponds to a certain translational symmetry. So if there areNatoms in a row with
periodic boundary conditions, there areNtranslational symmetries.
On the one hand everykvector corresponds to a symmetry, on the other hand there are 2 values of
spin (up and down) for every state, which gives all in all 2 Nstates for everyband. If the crystal is
getting bigger the number of states in each band does not change because there is an increase of the
symmetry and also an increase of thekstates so again: 2 Nstates.
Last but not least it is very important how many atoms are in the unit cell. If there is an even number
of atoms, the material will be an insulator or a semiconductor (because there are just filled and empty
Figure 97: One dimensional lattice of atoms with lattice constanta