On the theory of supermodulation of the superconducting order parameter created by structural supermodulation of apex distance in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$

TL;DR

This paper presents a theoretical study supporting the idea that super-exchange interactions, specifically involving Cu 4s and 3d orbitals, explain the supermodulation of the superconducting order parameter observed in optimally doped Bi2212, aligning with recent SJTM experiments.

Contribution

The study introduces a simplified five-orbital model to explain supermodulation effects and supports the super-exchange pairing mechanism in cuprates, aligning theory with recent experimental observations.

Findings

Calculated electron-pair density dependence on apical oxygen displacement matches experiments

Supports super-exchange as the pairing mechanism in cuprates

Confirms hybridization of Cu 4s with conduction band increases T_c

Abstract

Recently using Scanning Josephson Tunneling Microscopy (SJTM) in the group of S\'eamus Davis a super-modulation of the superconducting order parameter induced by super-modulation of the distance $\delta$ between planar Cu and apical O was observed in [O'Mahony et. al., PNAS Vol. 119(37), e2207449119 (2022)]. The authors conclude that: "concurrence of prediction from strong correlation theory ... with these observations indicates that ... super-exchange is the electron pairing mechanism of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$". In our current theoretical study we use the LCAO approximation, Hilbert space spanned on 5 atomic orbitals: Cu$4s$, Cu$3d_{x^2-y^2}$, O$2p_x$, O$2p_y$, O$2p_z$. For the only super-exchange amplitude $J_{sd}$ we use the Kondo double electron exchange between Cu$4s$ and Cu$3d_{x^2-y^2}$ orbitals and its anti-ferromagnetic sign is determined by adjacent to the Cu ion in-plane O orbitals. Within this approximations we have calculated: "Measured dependence of \dots electron-pair density $n_p$ on the displacement $\delta$ of the apical O atoms from the planar Cu atoms" depicted in Fig. 5(C) of O'Mahony \textit{et. al.} and obtained an acceptable accuracy. As a whole, the analyzed SJTM experiment is one of the best confirmations of J. R\"{o}hler [J. R\"{o}hler, Physica B: Cond. Matter Vol. 284-288, 104 (2000)] idea that hybridization of Cu4$s$ with conduction band leads to increasing of $T_c$. The lack of an alternative explanations for SJTM data $n_p$ versus $\delta$ and shape-$T_{c,max}$ correlations for the description of the critical temperature of optimally doped cuprates for several decades on the background of a simple view gives a hint that the long sought pairing mechanism has possibly been found and the Kondo exchange interaction as a property of strongly correlated quantum matter deserves further attention in the physics of layered cuprates.