5d-mediated indirect exchange and effective spin Hamiltonians in Ce triangular-lattice delafossites

TL;DR

This work tackles the challenge of quantifying anisotropic intersite exchange in Ce-based triangular-lattice delafossites by extending the ab initio force-theorem in Hubbard-I (FT-HI) to include 5d-mediated indirect exchange via a static mean-field treatment of 4f–5d Coulomb interactions. The generalized FT-HI framework, implemented in MagInt/Wien2k, yields a full set of NN and NNN exchange parameters that decompose into 4f superexchange and 5d-mediated contributions, revealing a material-dependent competition among exchange channels across CsCeSe_2, KCeS_2, and RbCeO_2. For CsCeSe_2 and KCeS_2, the dfIE channel dominates or strongly influences the NN anisotropy, aligning the predicted spin Hamiltonians with observed yz-stripe order and providing good INS agreement when included. In contrast, RbCeO_2 is governed mainly by SE, with dfIE remaining non-negligible, illustrating how chemical substitution tunes exchange mechanisms. Overall, the method delivers quantitative insight into exchange anisotropy and magnetic excitations, and can be extended to other rare-earth insulators and intermetallics.

Abstract

Anisotropic intersite exchange interactions in frustrated rare-earth magnets are difficult to assess both theoretically and experimentally. Here, we propose an ab initio force-theorem framework combining the quasi-atomic Hubbard-I approach to 4f correlations with a static mean-field treatment of the on-site intershell Coulomb interaction between rare-earth 4f and 5d states to simultaneously capture both 4f superexchange and 5d-mediated indirect exchange. Applying it to the triangular lattice Ce delafossites CsCeSe$_2$, KCeS$_2$, and RbCeO$_2$, we find that the indirect exchange dominates in the selenide, the superexchange in the oxide, while both mechanisms contribute almost equally in the sulfide. The magnetic exciation spectra of CsCeSe$_2$ and KCeS$_2$ evaluated from the calculated spin Hamiltonains are in good qualitative and quantitative agreement with experimental data.

Figures (3)

• Figure 1: The calculated nearest-neighbor (NN) intersite exchange interaction $J$ (in meV, left $y$-axis) together with the NN exchange anisotropy terms $\Delta$, $J_{\pm\pm}$, $J_{z\pm}$, and next-nearest-neighbor $J'$ (in units of $J$, right $y$-axis) calculated by FT-HI method. The values obtained using the generalized FT-HI approach with the 5$d$-mediated indirect exchange included are shown as solid bars, whereas the hatched bars represent the superexchange-only values.
• Figure 2: (a). Top view of the $yz$-stripe order (left) and the fully-polarized state (right). (b) Top view of the reciprocal triangular lattice. The first Brillouin zone is shaded, the $K=\Gamma-M-Y$ path along which the INS in applied field is calculated is indicated by dashed red line, the thinner lines show the two equivalent $\Gamma-M'-Y'$ paths. (c) The calculated INS intensity along the path $[0,K,3.5]$ in applied field of 6 Tesla using full IEI with the dfIE included. The experimental dispersion extracted from the experimental INS spectrum Xie2024prl at the same field is shown by the black dashed line. (d) The same data as in (c) calcualted using only SE IEI. (e) The zero-field INS calculated along $[0,K,3]$ using full IEI. (f). The experimental INS data at zero field Xie2024prl. The black lines were calculated with their experimental IEI fit. The figure is reproduced from Ref. Xie2024prl under CC-BY 4.0 license.
• Figure 3: (a). Spherically-averaged INS intensity of KCeS$_2$ calculated using the full IEI Hamiltonian with the dfIE included (a) and using the superexchange IEI only (b).