Interlayer couplings in cuprates: structural origins, analytical forms, and structural estimators

TL;DR

The paper addresses how interlayer couplings in cuprates arise from multiple microscopic hopping pathways and how these couplings depend on structural distortions. By combining DFT with maximally localized Wannier functions, the authors identify three dominant mechanisms—interlayer $p_{\sigma}$-$p_{\sigma}$, $p_z$-$p_z$, and $d_{z^2}$-$p_{\sigma}$ hoppings—and derive analytic, structure-based estimators for the effective interlayer coupling that depend only on crystal structure. They validate these estimators against DFT for several cuprates, notably YBCO7, and show good agreement across the Brillouin zone, including correct nodal-antinodal trends. This work provides a transferable framework to predict EICs from structural data, improving the interpretability and transferability of tight-binding models and enabling material design toward higher Tc in layered oxides.

Abstract

We quantitatively identify the multiple distinct microscopic mechanisms contributing to effective interlayer couplings (EICs) by performing first-principle calculations for two prototype superconducting cuprate families, pristine and doped Bi$_2$Sr$_2$CaCuO$_2$O$_{8+x}$ and Pr$_{x}$Y$_{1-x}$Ba$_2$Cu$_3$O$_7$. The major mechanisms are mediated by interlayer oxygen $p_σ$-$p_σ$ and $p_z$-$p_z$ hoppings as well as interlayer copper $d_{z^2}$-oxygen $p_σ$ hoppings. Furthermore, we show how EICs are closely related to structural distortions such as layer bucklings and bond length changes. This allows us to provide analytical formulae that permit direct estimation of the key interatomic hoppings and the EICs based only on the crystal structure. Finally, we benchmark our method on YBa$_2$Cu$_3$O$_7$ to estimate the strength and anisotropy of the EIC.

Figures (12)

• Figure 1: Experimental and theoretical Fermi surfaces. (a) ARPES-measured Fermi surface of untwinned YBa$_2$Cu$_3$O$_{6.85}$ and the associated tight-binding fit adapted from existing experiments zabolotnyy2007momentum. Note that the origin $\Gamma (0,0)$ is not at the center but at the top right. (b) DFT Fermi surface of YBa$_2$Cu$_3$O$_{7}$ at $k_z=0$, horizontal and vertical axes are $k_x$ and $k_y$ ranging from $-\pi/a$ to $\pi/a$. (c) ARPES-measured Fermi surface of $x=26$% hole-doped BSCCO measured at $T=104$K, adapted from experiments he2021superconducting. (d) DFT Fermi surface of $x=25$% hole-doped BSCCO.
• Figure 2: (a) Illustration of the direct interlayer hopping $t_{d_{x^2}d_{x^2}}^\perp$ between two $d_{x^2-y^2}$ orbitals from different layers. (b) The unfolded Wannier band structure of YBCO7 where small hoppings weaker than 0.05 eV are truncated. The opacity of bands represents the projection weight to the planar $d_{x^2-y^2}$ orbitals. The blue dashed circle on the left highlights the band splitting around the antinodal region originating from EIC, while the black dashed circle on the right highlights the nodal region. (c) Illustration of an effective interlayer coupling mechanism through the interlayer $p_{\sigma}$-$p_{\sigma}$ hopping. (d) Band structure modified from (b) by setting $t_{p_{\sigma}p_{\sigma}}^\perp=0$. (e) Illustration of an EIC mechanism through the interlayer $p_z$-$p_z$ hopping. (f) Band structure modified from (d) by additionally setting $t_{p_zp_z}^\perp=0$.
• Figure 3: (a) Illustration of an EIC mechanism through the interlayer $d_{z^2}$-$p_{\sigma}$ hopping. (b) The unfolded Wannier band structure modified from Fig. \ref{['fig:dppd']}(f) by additionally setting $t_{d_{z^2} p_{\sigma}}^\perp=0$. The opacity of bands represents the projection weight to the planar $d_{x^2-y^2}$ orbitals. The blue dashed circle highlights how the band splitting changes around the antinodal region compared to Fig. \ref{['fig:dppd']}(f).
• Figure 4: Relations between strongly-varying hoppings and structural properties. Different data points are from symmetry-independent atoms in undoped BSCCO, 25% hole-doped BSCCO, YBCO7, Y-site Pr-doped YBCO7, and Ba-site Pr-doped YBCO7. (a) In-plane hopping between $d_{x^2}$ and $p_z$ orbitals $t_{d_{x^2} p_z}^\parallel$ as illustrated in Fig. \ref{['fig:dppd']}(e) versus the corresponding in-plane oxygen buckling strength. The dashed line represents a linear relation fit. (b) Inter-layer hopping between $p_z$ orbitals from different layers $t_{p_z p_z}^\perp$ as illustrated in Fig. \ref{['fig:dppd']}(e) versus the interlayer oxygen-oxygen bond length. (c) Inter-layer hopping between $p_{\sigma}$ orbitals from different layers $t_{p_{\sigma} p_{\sigma}}^\perp$ as illustrated in Fig. \ref{['fig:dppd']}(c) versus the interlayer oxygen-oxygen bond length. (d) Inter-layer hopping $t_{d_{z^2} p_{\sigma}}^\perp$ as illustrated in Fig. \ref{['fig:dpdpd']}(a) versus the corresponding bond length. The dashed lines in (b-d) are given by exponential fits of the data points. Detailed analytical forms of all analytical fits are tabulated in Appendix \ref{['app:formula']}.
• Figure 5: Multiple hopping pathways for the mechanism mediated by interlayer $d_{z^2}$-$p_{\sigma}$ hopping as illustrated in Fig. \ref{['fig:dpdpd']}(a). These four hopping pathways are classified by the directions of the upper-layer $d_{x^2}$-$p_{\sigma}$-$d_{z^2}$ hopping and the lower-layer $p_{\sigma}$-$d_{x^2}$ hopping, namely (a) $\pm x$ and $\pm x$; (b) $\pm y$ and $\pm y$; (c) $\pm x$ and $\pm y$; (d) $\pm y$ and $\pm x$ directions. The background light blue and red atoms and bonds show the bilayer structure of BSCCO. The black circles and diamonds represent a path involving Cu and O atoms.