Bond Stretching Phonon Softening of Underdoped Copper-Oxide Superconductors

The anomalous properties of the bond stretching (BS) phonon mode are analyzed in the underdoped region of cuprates by studying the t-t?-J-Holstein model and the Hubbard-Holstein model with both numerical and analytical approaches. It is shown that a BS phonon softening occurs at an intermediate wavevector along the horizontal direction of the Brillouin zone (BZ) in agreement with recent INXS experiments as the result of the interaction among strongly correlated holes and antiferromagnetic fluctuations. An explanation of the single vs. double mode scenario is also discussed.