From localized 4f electrons to anisotropic exchange interactions in ferromagnetic CeRh6Ge4

Abstract

CeRh6Ge4 is a cerium-based ferromagnetic material exhibiting a quantum critical behavior under pressure. We derive effective exchange interactions, using the framework of density functional theory combined with dynamical mean-field theory. Our results reveal that the nearest-neighbor ferromagnetic interaction along the c axis is isotropic in spin space, leading to a formation of spin chains. On the other hand, the inter-chain coupling is highly anisotropic: The in-plane moment weakly interacts ferromagnetically in the a--b plane to stabilize the ferromagnetic state, whereas the z-component couples antiferromagnetically, contributing to its destabilization. The magnetic anisotropy of the interchain interactions as well as of the local 4f wavefunctions characterizes the magnetic properties underlying the ferromagnetic transition and the quantum critical behavior in CeRh6Ge4.

Figures (6)

• Figure 1: The crystal structure of CeRh6Ge4.a Side view showing the coordination of Ce and Rh. b View along $c$ showing the symmetry of the Ce site. c Spin configuration proposed in neutron scattering and $\mu$SR experiments Shu2021.
• Figure 2: Electronic structure of CeRh6Ge4.a Fully relativistic DFT bandstructure (blue) with a 108 band tight binding fit (black). b Corresponding density of states of CeRh6Ge4. The maximum of the very sharp Ce $4f$ density of states of 76.2 states/eV/f.u. is not shown. c Brillouin zone of CeRh6Ge4 with the high symmetry paths shown in a. d Single-particle excitation spectrum $A(\bm{k},\omega)$ in DFT+DMFT calculated at $T=0.01$ eV. e Corresponding $\bm{k}$-summed spectrum $A(\omega)$.
• Figure 3: CEF level schemes of $4f$ electrons in CeRh6Ge4.a Scheme derived from experiment Shu2021. b CEF level scheme obtained by our DFT calculations.
• Figure 4: The susceptibilities in momentum space and their temperature dependence. a. The eigenvalues of the susceptibility matrix, $\chi_{\lambda}(\bm{q})$ computed at $T=0.01$ eV. b The magnetic susceptibility $\chi_{\xi}(\bm{q})$ within $\ket{\pm1/2}$ states. c The temperature dependence of the inverse of the ferromagnetic susceptibility, $1/\chi_{\xi}(\bm{0})$ with $\xi=x, y$. The vertical dashed lines indicate the energy of the CEF level splitting $\Delta_1$ and $\Delta_2$ given in Fig. \ref{['fig:CEF']}a. The solid lines show the fitting by the high-$T$ and low-$T$ expressions in Eqs. \ref{['eq:chi_high']} and \ref{['eq:chi_low']}, respectively. The inset shows a zoom-up of the low-temperature region. d The temperature dependence of the inverse of the susceptibility $\chi_J$ that corresponds to experiments.
• Figure 5: Effective interactions between the magnetic moments in the $\ket{\pm1/2}$ states.a The momentum-dependent effective interaction $I_{\xi}(\bm{q})$. b Linear-scale plot of the effective inter-site interaction $I_{\xi}(i, j)$ as a function of the distance $|\bm{r}_{ij}| = |\bm{R}_i - \bm{R}_j|$. c A log-log plot with the line showing $I_{\xi}(i, j) \propto |\bm{r}_{ij}|^{-3}$. The vertical dashed lines show the Ce-Ce distances with the labels $n_1 n_2 n_3$ indicating $\bm{r}=n_1 \bm{a} + n_2 \bm{b} + n_3 \bm{c}$.