Van Hove singularity and isotope effect in high-Tccopper oxides

Recently it was proposed that a very small oxygen-isotope effect in the high-${\mathit{T}}_{\mathit{c}}$ Cu oxides can be understood in terms of a logarithmic singularity in the density of states, N(E), near the Fermi energy, along with a conventional BCS phonon-mediated pairing. In this paper we show that a relatively realistic N(E), derived from a tight-binding model on two-dimensional lattices, contains a logarithmic term plus a constant term. The calculated result indicates that the additional constant term in N(E) makes the minimum for the isotope-mass exponent α well exceed the corresponding experimental value (α≃0–0.2). The interplanar hopping is shown to smear out the sharp peak of N(E) so that α shifts further towards the standard BCS value of 1/2. So it appears to us that an explanation of the isotope effect as a function of doping in terms of a Van Hove singularity is not tenable.