Scaling of transition temerature and CuO_2 plane buckling in the cuprate   superconductors
  Recently it is seen\cite{Nature},that both $\rm{T}_c$ and the buckling of the $CuO_2$ planes goes through a maximum at the same doping level. We show that only for optimal doping concentration the Fermi surface touches the M($0,\pi$) point in the BZ, where the matrix element for interlayer pair tunneling amplitude is largest, so that the gain in delocalization energy by tunneling(in pairs) along the $c$ axis is largest. Buckling of the planes on the other hand modulates the separation between the planes and thereby modulates the interlayer pair tunneling amplitude. That is why both $\rm{T}_c$ and buckling angle(Oxygen atom displacement out of the plane) scales the same way with doping concentration. We have calculated $\rm{T}_c$ and buckling angle for various doping concentration. The agreement with experiment is remarkably good. We also point out the possible reason for large(about 1 percent) change of the buckling mode phonon frequency, accross the transition temperature. scattering 
Manas Sardar and Satadeep Bhattacharjee