Interaction of Zhang-Rice singlets with the buckling phonons in Cuprates

Starting from a fully interacting three-band model we derive an effective single-band Hamiltonian which describes the interaction of Zhang-Rice (ZR) singlets[1] with the buckling phonons. Our approach is based on exact diagonalization of small clusters which contain both Cu and O atoms[2]. We study the parametric dependence of the electron-phonon coupling. The coupling to the phonon mode with $B_{1g}$ symmetry is one order of magnitude stronger than the coupling to the $A_{1g}$ phonon mode and there is an anisotropy in the interaction. For the $B_{1g}$ phonon, the antinodes contribute to the interaction more than the nodes, in agreement with weak coupling approaches[3]. By increasing the O-O hopping, the coupling amplitude becomes smaller in antinodal points. We also find that a simpler model, which considers the modulation of the ZR hopping by the corresponding bond phonons, captures the symmetry of the electron-phonon interaction. Reference: [1] F. C. Zhang and T. M. Rice, PRB {\bf 37}, 3759 (1988) [2] H. Eskes {\em et.al.}, Physica C {\bf 160}, 424 (1989) [3] T. P. Devereaux {\em et.al.}, PRL {\bf 93}, 117004 (2004)