Exotic 4f Correlated Electronic States of Ferromagnetic Kondo Lattice Compounds ReRh$_6$Ge$_4$ (Re=Ce, Ho, Er, Tm)

TL;DR

This study uses density functional theory with spin-orbit coupling and Hubbard U to investigate the ferromagnetic ReRh6Ge4 (Re = Ce, Ho, Er, Tm) family. It predicts the magnetic easy axes (Ce: ab plane; Ho/Er: ac plane; Tm: along c) and reveals distinct topological features along the Gamma–A direction: triply degenerate nodal points for Tm due to preserved C3v symmetry, and Weyl points for Ce, Ho, and Er where C3v is broken by the magnetic axis. The 4f electrons evolve from localized in Ce/Ho/Er to itinerant in Tm, driving a growth of the Fermi surface and changing band contributions near EF, with Rh 4d and Ge 4p states dominating near EF. These results connect magnetic order with electronic structure and topology, offering concrete predictions for future experimental verification and for understanding pressure-driven quantum critical behavior in CeRh6Ge4.

Abstract

CeRh$_6$Ge$_4$ stands out as the first stoichiometric metallic compound with a ferromagnetic quantum critical point, thereby garnering significant attention. Ferromagnetic Kondo lattice compounds ReRh$_6$Ge$_4$ (Re=Ce, Ho, Er, Tm) have been systematically investigated with density functional theory incorporating Coulomb interaction U and spin-orbital coupling. We determined the magnetic easy axis of CeRh$_6$Ge$_4$ is within the ab plane, which is in agreement with previous magnetization measurements conducted under external magnetic field and muSR experiments. We also predicted the magnetic easy axes for the other three compounds. For TmRh$_6$Ge$_4$, the magnetic easy axis aligns along the c axis, thus preserving the $C_3$ rotational symmetry of the c axis. Especially, there are triply degenerate nodal points along the $Γ-A$ direction in the band structure including spin-orbital coupling. A possible localized to itinerant crossover is revealed as $4f$ electrons increase from CeRh$_6$Ge$_4$ to TmRh$_6$Ge$_4$. Specifically, the $4f$ electrons of TmRh$_6$Ge$_4$ contribute to the formation of a large Fermi surface, indicating their participation in the conduction process. Conversely, the $4f$ electrons in HoRh$_6$Ge$_4$, ErRh$_6$Ge$_4$ and CeRh$_6$Ge$_4$ remain localized, which result in smaller Fermi surfaces for these compounds. These theoretical investigations on electronic structure and magnetic properties shed deep insight into the unique nature of $4f$ electrons, providing critical predictions for subsequent experimental studies.

Figures (5)

• Figure 1: (a) Top view and (b) perspective view along the c-axis of the crystal structure of $ReRh_6Ge_4$ ($Re=Ce,Ho,Er,Tm$). The red sites are rare-earth ions, and they form triangular lattice in the $ab$ plane. The blue sites are $Rh$ ions, and the yellow sites are $Ge$ ions. (c) The first Brillioun zone of $ReRh_6Ge_4$, and the red points are the high symmetry points.
• Figure 2: The total ground state energy along different magnetic axis. (a) The ground state energy of CeRh$_6$Ge$_4$ when the magnetic axis is within the $ab$ plane. $\varphi$ is the angle with the $a$ axis, so the red line and green line represent the $a$ axis and the $b$ axis respectively. (b)-(d) The ground state energy of HoRh$_6$Ge$_4$, ErRh$_6$Ge$_4$ and TmRh$_6$Ge$_4$ when the magnetic axis is within the $ac$ plane. $\theta$ is the angle with the $c$ axis, so the blue line and the red line correspond to the $c$ axis and the $a$ axis, respectively.
• Figure 3: Band structure of (a) CeRh$_6$Ge$_4$, (b) HoRh$_6$Ge$_4$, (c) ErRh$_6$Ge$_4$ and (d) TmRh$_6$Ge$_4$ along high symmetry paths in the BZ. The calculations are carried in the ferroamgnetic phase with the SOC, and by fixed $U=6$$eV$. These compounds are all metallic, with bands crossing the Fermi surface (red dashed lines). The blue lines are the $Re$-$4f$ orbital projected band structure.
• Figure 4: Density of states of (a) CeRh$_6$Ge$_4$, (b) HoRh$_6$Ge$_4$, (c) ErRh$_6$Ge$_4$ and (d) TmRh$_6$Ge$_4$. The calculations are carried in the ferromagnetic phase with the SOC, and by fixing $U=6$$eV$. The total DOS is displayed with black line, and the partial DOS of $Ce$-$4f$ orbitals, $Rh$-$4d$ orbitals and $Ge$-$4p$ orbitals are displayed with blue lines, shaped purple lines and orange lines.
• Figure 5: The cross session of the $xz$ plane and the Fermi surface of (a) CeRh$_6$Ge$_4$, (b) HoRh$_6$Ge$_4$, (c) ErRh$_6$Ge$_4$ and (d) TmRh$_6$Ge$_4$ calculated with the SOC and $U=6$$eV$. The orange line ($\alpha$), yellow line ($\alpha'$), dark blue line ($\beta$) and blue line ($\beta'$) represent the hole type bands, while the brown line ($\gamma$), red line ($\gamma'$), dark green line ($\delta$) and green line ($\delta'$) represent the hole type bands.