Pressure-induced unconventional superconductivity in the heavy-fermion antiferromagnet ${\mathrm{CeIn}}_3$: An ${}^{115}\mathrm{In}$-NQR study under pressure

We report on pressure-induced unconventional superconductivity (SC) in the heavy-fermion (HF) antiferromagnet ${\mathrm{CeIn}}_3$ by means of nuclear-quadrupole-resonance (NQR) studies conducted under a high pressure. The temperature $\left(T\right)$ and pressure $\left(P\right)$ dependences of the In-NQR spectra have revealed a first-order quantum-phase transition (QPT) from antiferromagnetism (AFM) to paramagnetism (PM) at a critical pressure $P_{\mathrm{c}}=2.46\mathrm{GPa}$ at which AFM disappears with a minimum value of $T_{\mathrm{N}}\left(P_{\mathrm{c}}\right)=1.2\mathrm{K}$. High-energy x-ray scattering measurements under $P$ show a progressive decrease in the lattice density without any change in the crystal structure, whereas an increase in the NQR frequency $\left({\nu }_{\mathrm{Q}}\right)$ indicates an increase in the hybridization between $4f$ electrons and conduction electrons, which stabilizes the HF-PM state. This competition between the AFM phase where $T_{\mathrm{N}}$ is reduced and the formation of the HF-PM phase triggers the first-order QPT at $P_{\mathrm{c}}=2.46\mathrm{GPa}$. Despite the lack of an AFM quantum critical point in the $P\text{−}T$ phase diagram, we highlight the fact that unconventional SC occurs in both phases of AFM and PM. The measurements of the nuclear spin-lattice relaxation rate $1∕T_1$ in the AFM phase have provided evidence for the uniformly coexisting $\mathrm{AFM}+\mathrm{SC}$ phase. Remarkably, the significant increase in $1∕T_1$ upon cooling in the AFM phase has revealed the development of low-lying magnetic excitations down to $T_{\mathrm{c}}$ in the AFM phase; it is indeed relevant to the onset of the uniformly coexisting AFM+SC phase. In the HF-PM phase where AFM fluctuations are not developed, $1∕T_1$ decreases without the coherence peak just below $T_{\mathrm{c}}$, followed by a power-law-like $T$ dependence that indicates an unconventional SC with a line-node gap. Remarkably, $T_{\mathrm{c}}$ has a peak around $P_{\mathrm{c}}$ in the HF-PM phase as well as in the AFM phase. In other words, an SC dome exists with a maximum value of $T_{\mathrm{c}}=230\mathrm{mK}$ around $P_{\mathrm{c}}$, indicating that the origin of the pressure-induced HF SC in ${\mathrm{CeIn}}_3$ is not relevant to AFM spin fluctuations but to the emergence of the first-order QPT in ${\mathrm{CeIn}}_3$. These phenomena observed in ${\mathrm{CeIn}}_3$ should be understood in terms of the first-order QPT because these new phases of matter are induced by applying $P$. When the AFM critical temperature is suppressed at the termination point of the first-order QPT, $P_{\mathrm{c}}=2.46\mathrm{GPa}$, the diverging AFM spin-density fluctuations emerge at the critical point from AFM to PM. The results with ${\mathrm{CeIn}}_3$ leading to a new type of quantum criticality deserve further theoretical investigations.

Figure 1

Schematic phase diagrams of Ce-based heavy-fermion compounds: (a) for ${\mathrm{CePd}}_2{\mathrm{Si}}_2$ (Refs. 5, 6, 7), ${\mathrm{CeIn}}_3$ (Refs. 5, 7, 8, 9, 10), and ${\mathrm{CeRh}}_2{\mathrm{Si}}_2$ (Refs. 12, 13); the dashed line indicates the crossover; (b) for ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ (Refs. 2, 14, 15, 16, 17) and ${\mathrm{CeRhIn}}_5$ (Refs. 18, 19, 20).

Figure 2

Crystal and spin structures of ${\mathrm{CeIn}}_3$ below $T_{\mathrm{N}}$.

Figure 3

(Color online) (a) NQR spectra for ${\mathrm{CeIn}}_3$ above and below $T_{\mathrm{N}}=10.2\mathrm{K}$ at $P=0$. The solid arrow indicates the position of the $3{\nu }_{\mathrm{Q}}$ transition. (b) The $T$ dependence of the peak of the $3{\nu }_{\mathrm{Q}}$ spectrum in a range of $P=0–1.90\mathrm{GPa}$. The solid arrows point to $T_{\mathrm{N}}$.

Figure 4

(Color online) (a) Plots of $H_{\mathrm{int}}\left(t\right)∕H_{\mathrm{int}}\left(0\right)$ vs $T∕T_{\mathrm{N}}\left(P\right)$ for the AFM phase. Here, $H_{\mathrm{int}}\left(0\right)$ is a saturated value extrapolated to $T=0$. The solid curve indicates the molecular-field theory (MFT) with $S=1∕2$. (b) $H_{\mathrm{int}}\left(0\right)$ vs $T_{\mathrm{N}}$ plot in a $P$ range of $P=0–1.90\mathrm{GPa}$. The dotted line is a guide to the eye.

Figure 5

Temperature dependence of the $1{\nu }_{\mathrm{Q}}$-NQR spectrum at (a) $P=2.28$ and (b) $P=2.43\mathrm{GPa}$ just below $P_c=2.46\mathrm{GPa}$. Open and solid circles indicate the respective spectra above and below $T_{\mathrm{N}}$. The dotted line and solid arrows point to the respective frequencies where the peak in NQR spectrum is observed for the PM and AFM phases.

Figure 6

(Color online) $1{\nu }_{\mathrm{Q}}$-NQR spectra for $P=2.37$ and $2.43\mathrm{GPa}$ just below $P_{\mathrm{c}}=2.46\mathrm{GPa}$ and for $P=2.50\mathrm{GPa}$ just above $P_{\mathrm{c}}$. Solid curves are the simulations assuming an inevitable $P$ distribution inside the sample in the pressure cell (see text). In the vicinity of $P_{\mathrm{c}}$, NQR spectra result from the AFM and PM phases, which are separated by the first-order quantum phase transition. Here, the NQR spectral intensity is normalized by a peak intensity of the NQR spectrum of the PM phase.

Figure 7

(Color online) (a) Temperature dependence of the NQR intensity $\times$ temperature for the $1{\nu }_{\mathrm{Q}}$ transition around $P_{\mathrm{c}}$. Arrows indicate $T_{\mathrm{N}}$ at $P=2.28$, 2.32, and $2.43\mathrm{GPa}$. The solid curve is a simulation assuming that $T_{\mathrm{N}}$ is distributed due to the inevitable $P$ distribution in the pressure cell (see text). (b) Temperature dependence of NQR $\mathrm{intensity}\times \mathrm{temperature}$ at $P=2.43$ and $2.50\mathrm{GPa}$. Solid and dotted curves are simulations assuming first-order $\left[T_{\mathrm{N}}\right(P_{\mathrm{c}})=1.2\mathrm{K}]$ and second-order $\left[T_{\mathrm{N}}\right(P_{\mathrm{c}})=0\mathrm{K}]$ phase transitions, respectively.

Figure 8

(Color online) Pressure $\left(P\right)$ dependence of the volume fraction of the AFM phase at $T=0.1\mathrm{K}$ in the vicinity of $P_{\mathrm{c}}$. The solid curve is a simulation assuming a Gaussian-type $P$ distribution inside the sample in the $P$ cell (see text). The vertical dotted line indicates the first-order quantum phase transition for $\sigma =0$ provided that a $P$ distribution is absent. Arrow points to $P_{\mathrm{c}}=2.46\mathrm{GPa}$.

Figure 9

Pressure dependence of $T_{\mathrm{N}}$ in ${\mathrm{CeIn}}_3$. $T_{\mathrm{N}}$ above $P=2.28\mathrm{GPa}$ are determined from the analysis of the $T$ and $P$ dependences of the NQR spectrum. The solid curve is a guide to the eye. Dotted line indicates $P_{\mathrm{c}}=2.46\mathrm{GPa}$ where the first-order quantum phase transition separates AFM and PM.

Figure 10

(Color online) (a) Temperature dependence of $1∕T_1$ at $P=0$, 1.79, 2.17, 2.43, and $2.65\mathrm{GPa}$. The dotted line indicates a relation of $1∕T_1=\mathrm{const}$. Dotted and solid arrows point to $T_{\mathrm{N}}$ and $T^{*}$, respectively. (b) Temperature dependence of $1∕T_{1}T$ at $P=0$, 1.79, 2.17, 2.43, and $2.65\mathrm{GPa}$. The dotted line indicates the relation of $1∕T_{1}T=\mathrm{const}$. Dotted and dashed arrows indicate $T_{\mathrm{N}}$ and $T_{\mathrm{FL}}$, respectively.

Figure 11

(Color online) The $P\text{−}T$ phase diagram of the magnetic properties for ${\mathrm{CeIn}}_3$ obtained from the present NQR experiments. Solid circles indicate $T_{\mathrm{N}}$, while open triangles denote a crossover temperature $T^{*}$ from a localized to an itinerant regime of $4f$ electrons; open squares indicate a Fermi temperature $T_{\mathrm{FL}}$ below which a HF state is established. The inset shows the detailed phase diagram in the vicinity of $P_{\mathrm{c}}$ in a semilogarithmic scale. The vertical dotted line indicates the first-order phase boundary with a minimum value of $T_{\mathrm{N}}=1.2\mathrm{K}$ between the AFM and PM.

Figure 12

(Color online) (a) Pressure dependence of the lattice volume for ${\mathrm{CeIn}}_3$. (b) $P$ dependence of ${\nu }_{\mathrm{Q}}$ obtained from the local-density-approximation band calculation for ${\mathrm{CeIn}}_3$ and ${\mathrm{LaIn}}_3$. ${\mathrm{LaIn}}_3$ corresponds to the $4f$-localized model of ${\mathrm{CeIn}}_3$. Both are calculated with the same lattice constant.

Figure 13

(Color online) Pressure dependence of ${\nu }_{\mathrm{Q}}$ at $T=10\mathrm{K}$. The dotted line is a guide to the eye. The inset shows $1{\nu }_{\mathrm{Q}}$-NQR spectra at $P=0$, 1.79, and $2.59\mathrm{GPa}$.

Figure 14

Temperature dependence of ${\chi }_{\mathrm{ac}}$ in a $P$ range of $2.17–2.65\mathrm{GPa}$. Arrows indicate an onset temperature $T_{\mathrm{c}}^{\mathrm{onset}}$ below which SC diamagnetism starts to appear.

Figure 15

(Color online) Detailed $P\text{−}T$ phase diagram for PM, AFM, and SC for ${\mathrm{CeIn}}_3$ in the vicinity of the first-order QPT at $P_{\mathrm{c}}=2.46\mathrm{GPa}$, which is denoted by a vertical dashed line. Solid circles and triangles indicate $T_{\mathrm{N}}$ and $T_{\mathrm{c}}$, respectively. The open square is a crossover temperature towards the HF state. $T_{\mathrm{c}}^{\mathrm{onset}}$ is determined by ${\chi }_{\mathrm{ac}}$ in a $P$ range of $2.17–2.65\mathrm{GPa}$. Arrows point to $P=2.17$, 2.28, 2.43, and $2.50\mathrm{GPa}$ where the $1∕T_1$ shown in Fig. 16 was measured (see text).

Figure 16

(Color online) Temperature dependence of ${}^{115}(1∕T_{1}T)$ for ${\mathrm{CeIn}}_3$ at (a) $P=2.17$ and $2.28\mathrm{GPa}$, (b) $2.43\mathrm{GPa}$, and (c) $2.50\mathrm{GPa}$. Open and solid symbols indicate the respective data for PM and AFM (see the text). The solid arrows indicate the respective SC transition temperature $T_{\mathrm{c}}^{\mathrm{PM}}$ and $T_{\mathrm{c}}^{\mathrm{AFM}}$ for PM and AFM. The dotted and dashed arrows indicate $T_{\mathrm{N}}$ and $T_{\mathrm{FL}}$ below which the HF state becomes valid, characterized by the $T_{1}T=\mathrm{const}$ law (dotted line).

Figure 17

Temperature dependence of ${}^{115}(1∕T_{1}T)$ (open circles) and ${\chi }_{\mathrm{ac}}$ (solid circles) for ${\mathrm{CeIn}}_3$ at $P=2.28\mathrm{GPa}$ where the SC emerges. Arrows point to $T_{\mathrm{c}}^{\mathrm{MF}}$ below which ${}^{115}(1∕T_{1}T)$ decreases and $T_{\mathrm{c}}^{\mathrm{onset}}$ below which the SC diamagnetism appears.

Figure 18

(Color online) Temperature dependence of $1∕T_1$ at $P=2.50\mathrm{GPa}$ just above $P_{\mathrm{c}}=2.46\mathrm{GPa}$. Solid, dashed, and dotted arrows indicate $T_{\mathrm{c}}$, $T_{\mathrm{FL}}$, and $T^{*}$, respectively. Solid, dashed, and dotted lines indicate the respective relations of $1∕T_1\propto T^3$, $1∕T_1\propto T$, and $1∕T_1=\mathrm{const}$.

Figure 19

(Color online) (a) Temperature dependences of $1∕T_{1}T$ in a range $P=2.17–2.65\mathrm{GPa}$. The $1∕T_{1}T$ data are offset for clarity. The solid and open symbols indicate the respective data of $1∕T_{1}T$ measured below and above $P_{\mathrm{c}}$. Solid, dotted, and dashed arrows indicate $T_{\mathrm{N}}$, $T_{\mathrm{c}}$, and $T_{\mathrm{FL}}$, respectively. Dotted lines indicate a relation of $1∕T_{1}T=\mathrm{const}$. (b) The detailed $P\text{−}T$ phase diagram of ${\mathrm{CeIn}}_3$ in the vicinity of $P_{\mathrm{c}}$. Shaded region indicates the unconventional magnetic state where the low-lying spin-density fluctuations develop down to $T_{\mathrm{c}}$ upon cooling. All these phases of matter are determined by the present NQR measurements under $P$ (see text). The arrow points to a value of $P$ where $1∕T_{1}T$ in (a) was measured. (c) Pressure dependence of $(1∕T_{1}T{)}^{0.5}$ for $P_{\mathrm{c}}<P$ which is in proportion to the effective density of states at the Fermi level of HF band. Solid line is a guide to the eye.