Magnetic structure and critical behavior of GdRhIn5 : Resonant x-ray diffraction and renormalization group analysis

The magnetic structure and fluctuations of tetragonal $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_{5}$ were studied by resonant x-ray diffraction at the Gd ${L}_{\mathit{II}}$ and ${L}_{\mathit{III}}$ edges, followed by a renormalization group analysis for this and other related Gd-based compounds, namely ${\mathrm{Gd}}_{2}\mathrm{Ir}{\mathrm{In}}_{8}$ and $\mathrm{Gd}{\mathrm{In}}_{3}$. These compounds are spin-only analogs of the isostructural Ce-based heavy-fermion superconductors. The ground state of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_{5}$ shows a commensurate antiferromagnetic spin structure with propagation vector $\stackrel{P\vec}{\ensuremath{\tau}}=(0,\frac{1}{2},\frac{1}{2})$, corresponding to a parallel spin propagation along the $\stackrel{P\vec}{a}$ direction and antiparallel propagation along $\stackrel{P\vec}{b}$ and $\stackrel{P\vec}{c}$. The spin direction lies along $\stackrel{P\vec}{a}$. A comparison between this magnetic structure and those of other members of the ${R}_{m}{(\mathrm{Co},\mathrm{Rh},\mathrm{Ir})}_{n}{\mathrm{In}}_{3m+2n}$ family ($R=\text{rare}$ earth, $n=0,1$; $m=1,2$) indicates that, in general, $\stackrel{P\vec}{\ensuremath{\tau}}$ is determined by a competition between first- $({J}_{1})$ and second-neighbor $({J}_{2})$ antiferromagnetic (AFM) interactions. While a large ${J}_{1}∕{J}_{2}$ ratio favors an antiparallel alignment along the three directions (the $G$-AFM structure), a smaller ratio favors the magnetic structure of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_{5}$ ($C$-AFM). In particular, it is inferred that the heavy-fermion superconductor $\mathrm{Ce}\mathrm{Rh}{\mathrm{In}}_{5}$ is in the frontier between these two ground states, which may explain its noncollinear spiral magnetic structure. The critical behavior of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_{5}$ close to the paramagnetic transition at ${T}_{N}=39\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ was also studied in detail. A typical second-order transition with the ordered magnetization critical parameter $\ensuremath{\beta}=0.35$ was experimentally found, and theoretically investigated by means of a renormalization group analysis. Although the Gd $4{f}^{7}$ electrons define a half-filled, spherically symmetrical shell, leading to a nearly isotropic spin system, it is argued that a significant spin anisotropy must be claimed to understand the second order of the paramagnetic transition of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_{5}$ and the related compound ${\mathrm{Gd}}_{2}\mathrm{Ir}{\mathrm{In}}_{8}$.