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$_{8+x}$

Abstract

Recently using Scanning Josephson Tunneling Microscopy (SJTM) in the group of Séamus Davis a super-modulation of the superconducting order parameter induced by super-modulation of the distance $δ$ 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 $δ$ 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. Röhler [J. Röhler, 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 $δ$ 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.

Figures (1)

• Figure 1: On the upper left inset the isotope effect for mercury Reynolds:50 (1950) is reproduced. The approximately 10% accuracy of the logarithmic derivative $Q_{T,M}\equiv \mathrm{d} \ln T_c/\mathrm{d} \ln M\approx\frac{1}{2}$ triggered the development of phonon mediated pairing theory. On the central large figure: the experimentally determined electron pair density $\overline{n}_p$ versus the apex distance $\delta$ according to Ref. Omahony:22 entitled On the electron pairing mechanism of cooper-oxide high temperature superconductivity. The dashed line is the linear regression of the experimental data with slope approximately defined by \ref{['Q_experimental']}. The continuous line is the theoretical approximation within the s-d-LCAO theory according to the derivative of the theoretically calculated order parameter according to \ref{['Q_theoretical']}. The s-d-LCAO pairing theory gives correct order and evaluation of the slope of the linear regression. On the right lower inset the correlation $T_c$ versus $\delta$ for different compounds after Pavarini:01 is given. The correlation coefficient is much better in $\mathcal{F}$-$\delta$ plot Pavarini:01 which gives a hint that dependence is via the Cu4s ion $T_c(\mathcal{F}(\epsilon_s(t_{as}(\delta))))$ as demonstrated by the s-d-LCAO pairing theory.