Ferromagnetic Quantum Critical Point in Heavy-Fermion Iron Oxypnictide $\mathrm{Ce}({\mathrm{Ru}}_{1-x}{\mathrm{Fe}}_x)\mathrm{PO}$

We have performed ${}^{31}\mathrm{P}$-NMR measurements on $\mathrm{Ce}({\mathrm{Ru}}_{1-x}{\mathrm{Fe}}_x)\mathrm{PO}$ in order to investigate ferromagnetic (FM) quantum criticality, since a heavy-fermion (HF) ferromagnet CeRuPO with a two-dimensional structure turns into a HF paramagnet by an isovalent Fe substitution for Ru. We found that $\mathrm{Ce}({\mathrm{Ru}}_{0.15}{\mathrm{Fe}}_{0.85})\mathrm{PO}$ shows critical fluctuations down to $\sim 0.3\mathrm{K}$, as well as the continuous suppression of Curie temperature and the ordered moments by the Fe substitution. These experimental results suggest the presence of a FM quantum critical point (QCP) at $x\sim 0.86$, which is a rare example among itinerant ferromagnets. In addition, we point out that the critical behaviors in $\mathrm{Ce}({\mathrm{Ru}}_{0.15}{\mathrm{Fe}}_{0.85})\mathrm{PO}$ share a similarity with those in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, where the local criticality of $f$ electrons has been discussed. We reveal that $\mathrm{Ce}({\mathrm{Ru}}_{1-x}{\mathrm{Fe}}_x)\mathrm{PO}$ is a new system to study FM quantum criticality in HF compounds.

Figure 1

(a) $T$ dependence of $M_i/H$ evaluated from the Knight shift determined at the peaks of $H\perp c$ and $H\parallel c$ spectra obtained at $\simeq 0.5\mathrm{T}$ for $\mathrm{Ce}({\mathrm{Ru}}_{1-x}{\mathrm{Fe}}_x)\mathrm{PO}$ ($x=0.25$, 0.5, 0.75, 0.85, 0.87, 0.9, 0.95, and 1). The Knight shift $K_{\perp }$ and $K_c$ are also represented in the right axis. $M_i/H$ is estimated from the equation $M_i/H=(K_i-K_{\mathrm{orb}})/A_{\mathrm{hf},\mathrm{iso}}$ where the isotropic hyperfine coupling constant ${}^{31}A_{\mathrm{hf},\mathrm{iso}}=0.2\mathrm{T}/{\mu }_B$ and $T$-independent orbital part of the Knight shift $K_{\mathrm{orb}}=0.09\%$ derived at $x=1$ were used. Below $x=0.85$, $K_i$ increases below $T_{\mathrm{C}}$ due to the appearance of the internal field at the P site. The arrows indicate $T_{\mathrm{C}}$. (Inset) $T$ dependence of $M_i/H$ above $x=0.87$. $M_{\perp }/H$ shows a peak at around $T_{\max }\sim 5\mathrm{K}$ (indicated by the arrows) instead of the FM order, suggestive of the PM HF ground state at low fields. (b) $T$ dependence of $M_i/H$ (upper panel) and in-plane dynamical susceptibility $S_{\perp }$ probed by $1/T_1$ (lower panel) at $x=0.25$. $T_{\mathrm{C}}$ (dashed line) is unambiguously determined by the peak of $S_{\perp }$.

Figure 2

$T$ dependence of low-energy in-plane spin fluctuations $S_{\perp }$ at $\simeq 0.5\mathrm{T}$. (See in the text.) Below $x=0.85$, $T$ dependence of $S_{\perp }$ shows a peak at $T_{\mathrm{C}}$ (indicated by the arrows). Above $x=0.9$, $S_{\perp }$, as well as $K_{\perp }$, along both directions becomes almost a constant at low temperatures, indicative of the formation of a HF ground state, although $S_{\perp }$ continue to increase down to 0.08 K at $x=0.87$. (Inset) $x$ dependence of $S_{\perp }$ at 0.2 K. $S_{\perp }$ diverges toward $x\sim 0.86$. A dashed curve is a guide to the eyes.

Figure 3

(a) $H$ dependence of magnetization $M_{\perp }(H)$ in $H\perp c$ at 0.1 K above $x=0.87$ and at 0.5 K for $x=0.85$ just above $T_{\mathrm{C}}$ using the relation of $M_{\perp }(H)=K_{\perp }(H)H/A_{\mathrm{hf},\mathrm{iso}}$. Dashed curves are guides to the eyes. Above $x=0.87$, the $M_{\perp }(H)$ suddenly increases with increasing $H$ and deviates from linear relation in the field range of 1–5 T, which is a definition of a metamagnetic behavior, although $M_{\perp }(H)$ increases linearly at $x=0.85$. (b) $H$ dependence of $(1/T_{1}T{)}_{H\perp c}$ at 0.1 K above $x=0.87$ and at 0.5 K for $x=0.85$. At $x=1$, the low-field PM state with HF character is separated from the high-field polarized PM state by a broad peak of $(1/T_{1}T{)}_{H\perp c}$ at $H_{\mathrm{M}}\simeq 4.3$ T. A similar broad peak $(1/T_{1}T{)}_{H\perp c}$ was observed at $x=0.95$, but is changed to a shoulder structure at $x=0.9$. On the other hand, $(1/T_{1}T{)}_{H\perp c}$ is monotonically suppressed by a magnetic field and no anomaly is observed at $x=0.85$ and 0.87. We define $H_{\mathrm{M}}$ from the $H$ dependence of $M_{\perp }$ and $(1/T_{1}T{)}_{H\perp c}$ as shown by the dashed lines. $H_{\mathrm{M}}$ continuously decreases with decreasing $x$. Dashed curves are guides to the eyes.

Figure 4

$x\mathrm{\text{−}}T\mathrm{\text{−}}H$ phase diagram in $\mathrm{Ce}({\mathrm{Ru}}_{1-x}{\mathrm{Fe}}_x)\mathrm{PO}$ determined by NMR results. $T_{\mathrm{C}}$ in CeRuPO is continuously suppressed by Fe substitution, and approaches zero at $x\sim 0.86$, suggestive of the presence of a FM QCP at $x\sim 0.86$. Furthermore, $H_{\mathrm{M}}$ observed in CeFePO decreases with decreasing $x$ and seems to merge with the FM QCP at $x\sim 0.86$. (Inset) $x$ dependence of ordered magnetic moment $⟨{\mu }_{\mathrm{ord}}⟩$ estimated from $K_c$. $⟨{\mu }_{\mathrm{ord}}⟩$ continuously approaches zero toward $x\sim 0.86$, which is consistent with $x$ dependence of $T_{\mathrm{C}}$. Dashed curve is a guide to the eyes. The second-order FM transition is suggested from the development of the critical fluctuations down to low temperatures as well as the continuous suppression of $T_{\mathrm{C}}$ and the ordered moments $⟨{\mu }_{\mathrm{ord}}⟩$.

Figure 5

$T$ dependence of $M_{\perp }/H$ ($K_{\perp }$) (a) and $S_{\perp }$ (b) measured at the smallest field we could measure (${\mu }_{0}H\simeq 0.07\mathrm{T}$) at $x=0.85$. Static susceptibility related with $q=0$ becomes saturated below 3 K, although $S_{\perp }$ continues to develop toward $T_{\mathrm{C}}$, suggesting that spin fluctuations would possess AFM ($q\ne 0$) components as well as the FM ($q=0$) components near the FM QCP. These behaviors were in sharp contrast with those observed in $x<0.75$ as shown in Fig. 1b, but were actually observed in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ near an $H$-induced AFM QCP, in which the possibility of the local criticality has been discussed.