Diagonal superexchange in a simple square CuO$_2$ lattice

TL;DR

This paper analyzes diagonal superexchange in a CuO$_2$ lattice using a multiorbital $pd$ framework to resolve conflicting signs reported for next-nearest-neighbor exchange. By exact diagonalization of a CuO$_6$ cluster and symmetry-resolved accounting of orbital channels, it separates $J_{tot}$ into $J_{AFM}$ and $J_{FM}$ and shows that, in a perfect square lattice, the diagonal exchange is purely AFM with a small magnitude $J_{tot}(\vec{R}_{11}) \approx 0.002$ eV, because the $B_{1g}$ FM channel is symmetry-forbidden. The nearest-neighbor exchange remains larger, $J_{tot}(\vec{R}_{01}) \approx 0.099$ eV, dominated by the $A_{1g}$ channel. The work clarifies how lattice symmetry suppresses FM diagonal contributions and argues that real materials with broken square symmetry could exhibit nonzero FM diagonal exchange, affecting interpretations of spin excitations and magnetic inhomogeneity in cuprates.

Abstract

Many microscopic models with the interaction between the next-nearest neighbours as a key parameter for cuprate physics have inspired us to study the diagonal superexchange interaction in a CuO$_2$ layer. Our investigation shows that models with extended hopping provide a correct representation of magnetic interactions only in a hypothetical square CuO$_2$ layer, where the diagonal superexchange interaction with the next-nearest neighbors always has the AFM nature. The conclusions are based on the symmetry prohibition on FM contribution to the diagonal superexchange between the next-nearest neighbors for a simple square CuO$_2$ layer rather than for a real CuO$_2$ layer, where diagonal AFM superexchange may be overestimated. We also discuss the reasons for magnetic frustration effects and high sensitivity of spin nanoinhomogeneity to square symmetry breaking.

Figures (4)

• Figure 1: Paths P$_0$, P$_1$ and P$_2$ of the superexchange interactions $J_{tot} \left( {\vec{R}_{01} } \right)$ and $J_{tot} \left( {\vec{R}_{11} } \right)$ for the first and the second neighbor ions with the participation of $2p$ oxygen orbitals forming $\sigma$ overlapping with $3d$ copper ions, and $90^\circ$(P$_1$) or small $\pi$(P$_2$) - overlapping between themselves. Here, the interactions $J_{tot} \left( {\vec{R}_{01} } \right)$ are of AFM nature, but the magnitude of the $J_{tot} \left( {\vec{R}_{11} } \right)$ interaction with the second neighbors is still unknown.
• Figure 2: (a): A configuration space of the unit cell of CuO$_2$ layer. The cross denotes the occupied hole eigenstates $\left| {b^s_{1g} } \right\rangle$ in the $N_0 \left( {d^9 } \right)$ sector. Ellipses correspond to the virtual $e_g$ electron-hole pairs with the $J_{FM}^{} \left( {\vec{R}_{ij} } \right)$ and $J_{AFM} \left( {\vec{R}_{ij} } \right)$ contributions to the total exchange interaction $\hat{H}_S$. (b): A lattice diagram of the direct and diagonal superexchange interactions in the cell representation Eq.(\ref{['eq:4']}) for the square CuO$_2$ layer.
• Figure 3: Diagram of the CuO$_2$ layer in the symmetry cell representation of the $a_{1g}$ and $b_{1g}$ oxygen orbitals (see Eq.\ref{['eq:4']}) for diagonal: (a) AFM and (b) FM superexchange interactions. The color indicates the orbitals involved into the virtual electron-hole pairs at the next-neighboring copper ions. On the right side (b) it is clearly seen that due to the zero diagonal $pp$ and $pd$ overlapping, the FM contribution from a virtual pair with $B_{1g}$ symmetry is impossible.
• Figure 4: Paths $P_0$, $P_1$ and $P_2$ of the superexchange interactions $J_{tot} \left( {\vec{R}_{01} } \right)$ and $J_{tot} \left( {\vec{R}_{11} } \right)$ for the nearest and next-nearest neighbors. Here, the oxygen $2p$ orbitals $\pi$-overlap with the $t_{2g}$ magnetic ions, and form $90^\circ \left( {P_1 } \right)$ and $\sigma \left( {P_2 } \right)$ - overlapping between themselves. The interactions $J_{tot}^{\left( {P_1 } \right)} \left( {\vec{R}_{11} } \right)$ and $J_{tot}^{\left( {P_2 } \right)} \left( {\vec{R}_{11} } \right)$ are comparable in magnitude for magnetic materials with the partially occupied $t_{2g}$ shell.