Isotope effect on the superconducting critical temperature of cuprates in the presence of charge order

TL;DR

This study models the isotope effect on the superconducting transition temperature in cuprates, revealing its dependence on doping levels and charge order, and aligns well with experimental data.

Contribution

It introduces a theoretical framework combining the $t$-$J$ model and electron-phonon interactions to explain isotope effects in cuprates, emphasizing the role of charge order.

Findings

Isotope effect $eta$ has a minimum at optimal doping.

$eta$ increases in underdoped and overdoped regions.

Good agreement with experimental data in YBCO.

Abstract

Using the large-$N$ limit of the $t$-$J$ model and allowing also for phonons and the electron-phonon interaction we study the isotope effect $\alpha$ for coupling constants appropriate for YBCO. We find that $\alpha$ has a minimum at optimal doping and increases strongly (slightly) towards the underdoped (overdoped) region. Using values for the electron phonon interaction from the local density approximation we get good agreement for $\alpha$ as a function of $T_c$ and doping $\delta$ with recent experimental data in YBCO. Our results strongly suggest that the large increase of $\alpha$ in the underdoped region is (a) caused by the shift of electronic spectral density from low to high energies associated with a competing phase (in our case a charge density wave) and the formation of a gap, and (b) compatible with the small electron phonon coupling constants obtained from the local density approximation. We propose a similar explanation for the anomalous behavior of $\alpha$ in Sr doped La$_2$CuO$_4$ near the doping 1/8.