Localized-to-itinerant transition preceding antiferromagnetic quantum critical point and gapless superconductivity in CeRh_0.5Ir_0.5In₅

The hyperfine coupling between nuclear and electronic spins relates the measured $1/T_{1}$ to the underlying dynamical spin susceptibility as discussed in Methods. Figure 1 shows the pressure-temperature phase diagram of the CeRh_1-xIr_xIn₅ system. If substituting Ir for Rh acts as a positive chemical pressure, we would expect the hyperfine coupling constant [${}^{115}A(1)$] at the In(1) site (Fig. 2a) of CeRh_0.5Ir_0.5In₅ to be smaller than that of the host material CeRhIn₅ because ${}^{115}A(1)$ = 25 kOe $\mu_{\rm B}^{-1}$ in CeRhIn₅ decreases with increasing pressure but becomes constant at ${}^{115}A(1)$ $\sim$ 7 kOe $\mu_{\rm B}^{-1}$ above $P$ = 1 GPa. Such a pressure dependent ${}^{115}A(1)$ will affect the information inferred from 1/$T_{1}$ Curro and, therefore, it is important to determine ${}^{115}A(1)$ for CeRh_0.5Ir_0.5In₅ before proceeding to details of its $T_{1}$ results under pressure. Figure 2b shows the frequency-swept ¹¹⁵In-nuclear magnetic resonance (NMR) spectra at a constant field. The spectrum is consistent with previously reported spectra of CeRhIn₅ Curro . From these data, we establish the temperature dependence of the ¹¹⁵In(1)-NMR center line plotted in Fig. 2c. With these results, we calculate (Methods) the temperature dependence of the Knight shift ${}^{115}K(1)_{\rm c}$ (%) and compare it to dc susceptibility $\chi_{\rm c}$ (emu mol^-1) in Fig. 2d. As clearly shown in the figure, the relation $K(T)$ $\propto$ ${}^{115}A(1)\chi(T)$ holds above $T$ = 30 K. A breakdown of this linear relationship at low temperature is common in heavy electron systems CurroYoung , but a previous In-NMR study suggested that ${}^{115}A(1)$ is temperature-independent in CeIrIn₅ in such temperature regime Kambe1 . In the inset of Fig. 2d we plot ${}^{115}K(1)_{\rm c}$ versus $\chi_{\rm c}$ and obtain the hyperfine coupling constant as ${}^{115}A(1)$ = 7.64 kOe $\mu_{\rm B}^{-1}$. This closely corresponds to the value of ${}^{115}A(1)$ in CeRhIn₅ under a pressure of 1 GPa (Supplementary Note 1 and Supplementary Figure 1), and thus, it substantiates the suggestion that Ir substitution with $x$ = 0.5 (chemical pressure) is equivalent to the application of a physical pressure greater than $P$ = 1 GPa to CeRhIn₅. Therefore, we reasonably can ignore any significant pressure dependence of ${}^{115}A(1)$ in inferring the pressure evolution of physical properties from 1/$T_{1}$ in CeRh_0.5Ir_0.5In₅.

We consider the origin for the $\nu_{\rm Q}$ jump. Figure 5a shows results of X-ray diffraction measurements that give the pressure dependence of $a$- and $c$-axis lengths; the lattice parameters decrease smoothly with increasing pressure to $P$ = 4 GPa, without any signature of a structural transition. From these data, we calculate the EFG using the first-principles Hiroshima Linear-Augmented-Plane-Wave (HiLAPW) codes Oguchi . As expected, the calculated $\nu_{\rm Q}^{\rm HiLAPW}$ increases monotonically with applied pressure (Fig. 5b). This and the lack of any anomaly in 1/$T_{1}$ at ($P^{\rm*},T^{\rm*}$) rule out a change in lattice contribution to $\nu_{\rm Q}$ as the origin of the $\nu_{\rm Q}$ jump. Furthermore, Fig. 5b compares the pressure dependence of $\nu_{\rm Q}^{\rm HiLAPW}$ for CeRh(Ir)In₅ and LaRh(Ir)In₅. $\nu_{\rm Q}^{\rm HiLAPW}$ for CeRh(Ir)In₅ is uniformly greater than that for LaRh(Ir)In₅, because CeRh(Ir)In₅ has the additional EFG from hybridized $f$ electrons, unlike non-magnetic LaRh(Ir)In₅. This result is consistent with previous band calculations Harima ; CurroBandCalc .

We conclude from these results that the jump in $\nu_{\rm Q}(P)$ at ($P^{\rm*}$, $T^{\rm*}$) is due to a pronounced increase in Kondo hybridization at ($P^{\rm*}$, $T^{\rm*}$) and that the larger $d\nu_{\rm Q}/dP$ above ($P^{\rm*}$, $T^{\rm*}$) reflects that increased hybridization. Because increased Kondo hybridization transfers $f$ spectral weight from localized to itinerant degrees of freedom, and hence an increase in Fermi surface volume, the pronounced jump in $\nu_{\rm Q}$ signals the experimental observation of a small (localized) to large (itinerant) Fermi surface. As the principal axis of the EFG at the In(2) site is perpendicular to that of the In(1) site, the $\nu_{\rm Q}$ change at both sites suggests that the entire Fermi-surface volume changes at ($P^{*}$, $T^{*}$).

Figures 6a and 6b are plots of $1/T_{1}T$ vs. $T$ for the antiferromagnetic phase below $P$ = 1.12 GPa and for the paramagnetic phase above $P$ = 1.24 GPa, respectively. The solid curves in Fig. 6b are least-squares fits to 1/$T_{1}T$ = $a/(T+\theta)^{0.5}$ + 1.44 just above $T_{\rm c}(P)$, with $a$ and $\theta$ as parameters. Approaching the antiferromagnetic QCP from $P$ = 2.53 GPa, $\theta$ decreases with decreasing pressure, as can be seen in Fig. 6b. The pressure dependences of $\theta$, $T_{\rm N}$, and $T_{\rm c}$ are plotted in Fig. 6c. From a linear fit of $\theta(P)$, the antiferromagnetic QCP ($\theta$ = 0 K) for CeRh_0.5Ir_0.5In₅ is obtained at $P_{\rm c}^{\rm AF}$ = 1.2 GPa, where the highest $T_{\rm c}$ is realized. The present results clearly indicate that spin fluctuations play a significant role for superconductivity in CeRh_0.5Ir_0.5In₅.

To obtain the relative DOS, for simplicity we assume that $T_{1}$ below $T_{\rm c}$ is predominantly determined by itinerant quasi particles. Previously, an analysis within the context of two-fluid phenomenological theory has deduced that the 4$f$ local moments also contribute to relaxation PinesPNAS ; Curro2fluid . Nonetheless, as shown in Supplementary Note 3 and Supplementary Figures 3 and 4, such a model reproduces essentially the same result as we obtained here.

As shown in Fig. 8b, in CeRh_0.5Ir_0.5In₅, remarkably, the fraction of ungapped excitations strongly depends on pressure, reaching a maximum at the antiferromagnetic QCP. In the coexistent state at $P$ = 0 Yamaguchi , this fraction is 50 %, but increases to 96% at $P_{\rm c}^{\rm AF}$ and then decreases to 55% with the increasing pressure at $P$ = 2.53 GPa. The highest $T_{\rm c}$ around the antiferromagnetic QCP of CeRh_0.5Ir_0.5In₅ is realized unexpectedly with the largest fraction of gapless excitations. The present observation is completely different from that for CeRhIn₅ under pressure; in CeRhIn₅, the relative fraction of gapless excitations (88%) in the coexistent state is rapidly suppressed to almost zero as it approaches the QCP Yashima2007 .

From the phase diagram shown in Fig. 8, it is clear that the localized to itinerant transition ($T^{\rm*}(P)$) does not occur exactly at the antiferromagnetic QCP; in the limit of zero temperature, $P_{\rm c}^{\rm*}$ precedes $P_{\rm c}^{\rm AF}$. One possibility would be that the Fermi-surface change across the $T^{\rm*}(P)$ boundary marks a line of abrupt changes of the Ce valence that terminates near $T$ = 0 in a critical end point. A model that considers this possibility, however, appears to exclude the $T^{\rm*}(P)$ boundary from extending into the antiferromagnetic state MiyakeWatanabe ; MiyakeWatanabe2014 , contrary to our results. In contrast, a breakdown of the Kondo effect gives rise to a small to large Fermi surface change across $T^{\rm*}(P)$ QSiT1 . This idea leads to a $T$ = 0 phase diagram QSiPhysB ; ColemanJLTP similar to the results of Fig. 8. Associated with the Kondo breakdown and development of a large Fermi surface, soft charge fluctuations can emerge without a change in formal valence of Ce ions Komijani . Within experimental uncertainty of $\pm$1.5 %, there is no detectable difference between CeRhIn₅ and CeIrIn₅ at 10 K in their spectroscopically determined Ce valence Sundermann , even though their Fermi volumes differ - a result that, together with the phase diagram of Fig. 8, supports a Kondo breakdown interpretation as do thermopower measurements on CeRh_0.58Ir_0.42In₅ Luo . Notably, the $T^{\rm*}$ boundary has no notable effect on the evolution of $T_{\rm c}(P)$. In passing, we mention a possibility of a more general case, a Lifshitz transition. Watanabe and Ogata Ogata , and Kuramoto $et$ $al$ Kuramoto pointed out that, even though the Kondo screening remains, a competition between the Kondo effect and the RKKY interaction can lead to a topological Fermi surface transition (Lifshitz transition) below the antiferromagnetic QCP Ogata ; Kuramoto , which is also consistent with our observation.

The large $1/T_{1}T$ below $T_{\rm c}$ at pressures near the antiferromagnetic QCP (Figs. 6 and 7) is quite remarkable, and its temperature independence implies a large DOS that remains ungapped in the superconducting state even though this is the pressure range where $T_{\rm c}$ is a maximum. Such an anomalously high value of ungapped DOS in the superconducting state has never been found in other QCP materials such as high $T_{\rm c}$ cuprates Asayama and the iron-pnictide superconductors Zhou ; Oka , even though magnetic fluctuations are also strong around their QCP. For the Ce115 family, no significant gapless excitations have been observed so far around a QCP Kohori .

In general, gapless excitations are expected from impurity scattering in $d$-wave superconductors MiyakeVarma ; Hirschfeld ; Maebashi ; Haas . Though there are no intentionally added impurities in our crystal, the random replacement of 50 % Rh by Ir results in a broadening of the In(1) NQR line by a factor of $\sim$ 5 compared to CeIrIn₅ Yamaguchi ; ZhengIr . Such randomness increases the resistivity at 4 K (just above $T_{\rm N}$) from $\sim$ 4 $\mu\Omega$cm in CeRhIn₅ to over 20 $\mu\Omega$cm in CeRh_0.5Ir_0.5In₅ Nicklas . Quantum critical fluctuations can further enhance that scattering Maebashi to make part of a multi-sheeted Fermi surface gapless Barzykin . Such scattering concomitantly leads to pair breaking, resulting in a large reduction of $T_{\rm c}$ Abrikosov ; MiyakeVarma ; Hirschfeld ; Maebashi ; Haas ; Barzykin , which is inconsistent with our observations. The relative DOS at the pressure-induced QCP is almost zero in CeRhIn₅ Yashima2007 but is enhanced to 96% in CeRh_0.5Ir_0.5In₅ at $P_{\rm c}^{\rm AF}$. If we assume that the symmetry of CeRh_0.5Ir_0.5In₅ is also $d$-wave, $T_{\rm c}$ should be reduced to zero with such a significant residual DOS at $E_{\rm F}$ Maki . In contrast to this expectation, the maximum $T_{\rm c}$ = 1.4 K for CeRh_0.5Ir_0.5In₅ remains at 61% of $T_{\rm c}$ = 2.3 K for CeRhIn₅ at their respective QCPs. Hence, the present results suggest that superconductivity near $P_{\rm c}^{\rm AF}$ in CeRh_0.5Ir_0.5In₅ is more exotic than $d$-wave.

Model calculations of superconductivity in a two-dimensional Kondo lattice show that near an antiferromagnetic QCP $d$- and $p$-wave spin-singlet superconducting states are nearly degenerate, with an odd frequency $p$-wave spin-singlet state being favored when entering the large Fermi surfaces region to take advantage of the nesting condition with the vector ${\bf Q}$ = ($\pi$,$\pi$) Otsuki . A $p$-orbital wave function with spin-singlet pairing symmetry satisfies Fermi statistics in the odd-frequency channel Berezinskii ; Coleman ; Balatsky ; Fuseya , and this odd-frequency pairing is more robust against non-magnetic impurity scattering than even-frequency pairing Yoshioka . Indeed, for a scattering strength that kills $d$-wave superconductivity completely, spin-singlet odd-frequency pairing will survive with a $T_{\rm c}$ that is approximately 60 % of that in the absence of scattering Yoshioka . Motivated by these theoretical results, we suggest that odd-frequency spin-singlet pairing is realized in CeRh_0.5Ir_0.5In₅ in the vicinity of its critical pressures. Its robust $T_{\rm c}$ in the presence of substantial disorder scattering that gives rise to a large residual density of states at $P_{\rm c}^{\rm AF}$ where quantum critical fluctuations are strongest and the presence of a nearby change from small to large Fermi surface at $P_{\rm c}^{*}$ are fully consistent with our proposal. We stress that the unique aspect of both strong fluctuations and large Fermi surface is not shared by the end members, CeIrIn₅ or CeRhIn₅. Knight shift and experiments that directly probe the gap symmetry will be useful to test this possibility.

In summary, we reported systematic ¹¹⁵In-NQR measurements on the heavy fermion antiferromagnetic superconductor CeRh_0.5Ir_0.5In₅ under pressure and find that an antiferromagnetic QCP is located at $P_{\rm c}^{\rm AF}$ = 1.2 GPa, at which $T_{\rm c}(P)$ reaches its maximum. The pressure and temperature dependence of $\nu_{\rm Q}$ reveal a pronounced increase in hybridization that signals a change from small to large Fermi surface in the limit of zero temperature at $P_{\rm c}^{\rm*}$ = 0.8 GPa which is notably lower than $P_{\rm c}^{\rm AF}$. Thus, our work sheds new light on the quantum phase transition in $f$-electron systems. There is a strong enhancement of the quasiparticle DOS in the superconducting state around $P_{\rm c}^{\rm AF}$ where the Fermi surface is large. The robustness of $T_{\rm c}$ under these conditions can be understood if the superconductivity is odd-frequency $p$-wave spin singlet. Traditionally, Hall coefficient and quantum oscillation experiments have been used to probe the Fermi surface change. Our work demonstrates that the NQR frequency can be used as a powerful tool to examine the change in Fermi surface volume for heavy electron systems. In particular, the NQR technique does not require single crystals and is not limited by sample quality or pressure, and thus will open a new venue to understand strongly correlated electron superconductivity.

Single crystals of CeRh_0.5Ir_0.5In₅ were grown from an In flux as reported in a previous study Pagliuso . All experiments were performed with the same batch of crystals used in the previous NQR study Yamaguchi . As documented in detail in Ref. Yamaguchi , there is no phase separation into Rh-rich and Ir-rich parts even in the coarsely crushed powder. In fact, no excess peaks in the NMR/NQR spectrum are found and the spectrum can be reproduced by a Gaussian function. Moreover, $T_{1}$ is of single component, which also indicates that Ir is randomly distributed. For NMR Knight shift measurements, two single crystals, sized 2 mm - 4 mm - 0.5 mm and 2 mm - 3 mm - 0.4 mm, were used. For NQR measurements, the crystals were moderately crushed into grains to allow RF pulses to penetrate easily into the samples. Small and thin single-crystal platelets cleaved from an as-grown ingot were used for X ray and dc-susceptibility measurements.

For NQR, the nuclear spin Hamiltonian can be expressed as, $\mathcal{H}_{\rm Q}$ = $(h\nu_{\rm Q}/6)[3{I_{z}}^{2}-I(I+1)+\eta({I_{x}}^{2}-{I_{y}}^{2})]$, where $h$ is Planck’s constant and $I$ = 9/2 for the In nucleus is the nuclear spin; $\nu_{\rm Q}$ and the asymmetry parameter $\eta$ are defined as $\nu_{\rm Q}$ = $\frac{3eQV_{zz}}{2I(2I-1)h}$ and, $\eta$ $=$ $\frac{V_{xx}-V_{yy}}{V_{zz}}$, respectively, and $Q$ and $V_{\alpha\beta}$ are the nuclear quadrupole moment and EFG tensor, respectively. In CeRh_0.5Ir_0.5In₅, there are two inequivalent In sites, one in the CeIn [In(1)] layer and another in the (Rh,Ir)In₄ [In(2)] layer (see Fig. 2a). The principle axis of the EFG at the In(1) [In(2)] site is parallel [perpendicular] to the $c$ axis. The ¹¹⁵In-NQR spectra for In(1) $\pm$1/2 $\leftrightarrow$ $\pm$3/2 transition line (Supplementary Note 2 and Supplementary Figure 2), $\pm$7/2 $\leftrightarrow$ $\pm$9/2 transition line (Supplementary Figure 5), the In(2) $\pm$3/2 $\leftrightarrow$ $\pm$5/2 transition line and $T_{1}$ for the In(1) site ($\eta=0$) (Supplementary Note 4 and Supplementary Figures 6 and 7) were obtained as reported in an earlier study Yamaguchi ; recovery . Here, $T_{1}$ probes the dynamic spin susceptibility through the hyperfine coupling constant $A_{\bf q}$ as $1/T_{1}\propto T\sum_{\bf q}\left|A_{\bf q}\right|^{2}\chi^{\prime\prime}({\bf q},\omega_{0})/\omega_{0}$, where $\omega_{0}$ is the NQR frequency MoriyaJPSJ and ${\bf q}$ is a wave vector for antiferromagnetic order and/or quantum critical fluctuation in CeRh_0.5Ir_0.5In₅. Meanwhile, $1/T_{1}$ $\propto$ $N(E_{\rm F})^{2}$$k_{\rm B}T$ holds in a Pauli paramagnetic metal, i.e., in a heavy Fermi liquid state (Korringa law). Here, $N(E_{\rm F})$ is the density of states at $E_{\rm F}$.

A NiCrAl/BeCu piston-cylinder-type pressure-cell filled with Daphne (7474) oil was used. The $T_{\rm c}$ of Sn was measured to determine the pressure. Measurements below 1 K were performed using a ³He refrigerator.

A diamond anvil cell filled with a CeRh_0.5Ir_0.5In₅ single crystal and Daphne oil were used for room temperature X-ray measurements under pressure; the pressure was determined by measuring the fluorescence of ruby. All measurements were made at zero magnetic field.

The EFG is calculated using the all electron full-potential linear augmented plane wave method implemented in the Hiroshima Linear-Augmented-Plane-Wave (HiLAPW) code with generalized gradient approximation including spin-orbit coupling Oguchi .

To estimate the hyperfine coupling constant for CeRh_0.5Ir_0.5In₅, ${}^{115}A(1)$, we measure the ¹¹⁵In(1)-NMR spectrum and dc susceptibility. For NMR, the nuclear spin Hamiltonian is expressed as $\mathcal{H}$ = $-^{115}$$\gamma\hbar\bf{I}\cdot\bf{H}$$(1+K)+\mathcal{H}_{\rm Q}$, where the gyromagnetic ratio ${}^{115}\gamma$ = 9.3295 MHz T^-1, $\bf{H}$ is the external magnetic field, and $K$ is the Knight shift. ¹¹⁵In-NMR spectra have nine transitions from $I_{\rm z}$ = (2$m$$+$1)/2 to (2$m$$-$1)/2 where $m$ = $-$4, $-$3, $-$2, $-$1, 0, 1, 2, 3, 4 for In(1) and In(2) sites, respectively, with $K$, $\nu_{\rm Q}$, and $\eta$ as parameters. At ambient pressure, $\nu_{\rm Q}$ and $\eta$ at the In(1) [In(2)] site are 6.35 MHz (17.37 MHz) and 0 (0.473), respectively Yamaguchi . The Knight shift for In(1), ${}^{115}K(1)_{\rm c}(T)$, was calculated from the peak in the ¹¹⁵In(1)-NMR center line ($m$ = 0) taken by sweeping the RF frequency at a fixed field $H$ = 12.950 T and $\chi_{\rm c}$ (emu mol^-1) is obtained by dc susceptibility measurements at $H$ = 3 T using a superconducting quantum interference device (SQUID) with the vibrating sample magnetometer (VSM). The magnetic field $H$ is fixed perpendicular to the CeIn layer ($c$ axis).

We would like to thank T. Kambe for help with the susceptibility measurement, Y. Fuseya, Y. Kuramoto, H. Kusunose, J. Otsuki, K. Miyake, S. Watanabe, T. Oguchi and Q. Si for discussion. This work was supported in part by research grants from MEXT (No. JP19K03747, JP23102717, and JP25400374), NSFC grant No. 11634015 and MOST Grant (No. 2016YFA0300300, No. 2017YFA0302904, and No. 2017YFA0303103) and, at Los Alamos, was performed under the auspices of the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering.

G.-q.Z planned the project. J.L.S and J.D.T synthesized the single crystal. S.K, T.O, A.S, Y.K, and K.U performed NQR measurements. S.K and Y.K performed NMR and susceptibility measurements. J.G, S.C, and L.L.S performed X-ray measurement. K.M performed band structure calculation. G.-q.Z, S.K and J.D.T wrote the manuscript. All authors discussed the results and interpretation.