In 1987, J. LABBE and J. BOK1 were first to propose that a logarithmic singularity in the density of states, could be a mechanism for high critical temperature superconductivity. This model explained: high Tc, the variation of the critical temperature with the density of holes in the CuO planes with a maximum when the Fermi level is exactly at the singularity, an anomalous isotope effect which is observed experimentally. We have calculated the coherence length xi in the framework of this model and we are able to explain the observed low value in the a-b plane. For a 2D model we find that xi = aD/DELTA where a is the lattice constant. DELTA the gap in the a-b plane and D the width of the singularity. We also show that including a repulsive interaction between second nearest neighbors, the Van Hove singularity is no longer at half-filling of the band but for 0.85 electron per Cu atom or 0.15 hole. This explains the separation between half-filling (magnetic state) and the concentration which puts the Fermi level at the singularity (superconductivity with maximum critical temperature). This also explains that the singularity always happens on the hole side of the half-filling.
