Magnetic field driven complex phase diagram of antiferromagnetic heavy-fermion superconductor Ce3PtIn11

We present the results of our comprehensive investigation on the antiferromagnetic heavy-fermion superconductor Ce3PtIn11 carried out by means of electrical transport, heat capacity and ac magnetic susceptibility measurements, performed on single-crystalline specimens down to 50 mK in external magnetic fields up to 9 T. Our experimental results elucidate a complex magnetic field – temperature phase diagram which contains both first- and second-order field-induced magnetic transitions and highlights the emergence of field stabilized phases. Remarkably, a prominent metamagnetic transition was found to occur at low temperatures and strong magnetic fields. In turn, the results obtained in the superconducting phase of Ce3PtIn11 corroborate an unconventional nature of Cooper pairs formed by heavy quasiparticles. The compound is an almost unique example of a heavy fermion system in which superconductivity may coexist microscopically with magnetically ordered state.

Investigation of anomalous physical properties at quantum critical point (QCP) in heavy fermion (HF) systems has been at the forefront of contemporary condensed matter research as it may enlighten the fascinating physics involved with these unusual observations at this precarious point of instability [1][2][3][4] . QCP is associated with continuous quantum phase transition (QPT) which, unlike classical phase transition, is driven by tuning non-thermal parameters 5,6 . In this framework, Ce based HF systems are of particular interest as some of them exhibit superconductivity along with enhanced quasiparticle effective mass, known as heavy fermion superconductors (HFSC) [7][8][9][10][11][12][13] , where the pairing mechanism is believed to be mediated by magnetic fluctuations rather than phonons as in case of conventional superconductors 14 . In addition, due to unstable nature of 4f orbitals, magnetic ordering in these systems can be tuned by controlling parameters of non-thermal origin, like pressure, chemical doping or magnetic field, leading the system to a QCP where various unusual features such as non-Fermi liquid (NFL) behavior, unconventional superconductivity, etc., are witnessed [15][16][17][18] . The majority of the vast number of Ce based HF systems reported in the literature, posses just one position for Ce ions in their crystallographic unit cell. The ground states in these compounds basically result from the interplay between two competitive interactions, namely Ruderman-Kittel-Kasuya-Yosida (RKKY) and Kondo interactions, which can qualitatively be understood in terms of the Doniach phase diagram 19 . However, the scenario does not appear trivial when there are more than one Ce sites in the crystal structure. The theoretical study by Benlagra et al. on the interplay between two different Kondo effects originating from two inequivalent Kondo sublattices dictates that in different ranges of the conduction-band filling these interactions can be competitive or cooperative, which leads to a fairly complex phase diagram 20 .
Investigation of novel Ce based compounds hosting multiple inequivalent Ce sites has received considerable attention from the scientific research community in recent times, as they showcase diverse unusual ground state properties at low temperatures. Different local environment of Ce ions leads to different hybridization strengths, which spark the possibility of having distinctly different ground states for each individual inequivalent Ce ion. For instance, separate AFM orderings of two Ce sublattices has been reported in Ce 5 Ni 6 In 11 21 , while Ce 9 Ru 4 Ga 5 contains three independent Ce sites where two of them order antiferromagnetically at T N = 3.7 K and a third one exhibits valence fluctuations 22 . Most remarkably, in the compounds Ce 3 TIn 11 (T = Pt and Pd), bearing two inequivalent Ce sites, a coexistence of AFM order and HFSC has recently been discovered [23][24][25] , which set a new playground for comprehensive studies on mutual relationship between magnetism and superconductivity in Ce-based HF systems. The compound Ce 3 PtIn 11 belongs to the homologous series of phases Ce n T m In 3n + 2m (T stands for d-electron transition metal), which encompasses a large variety of intriguing materials including CeCoIn 5 11 , CeRhIn 5 26 , Ce 2 PdIn 8 13,27,28 , Ce 2 CoIn 8 29 . It crystallizes in a tetragonal structure with space group P4/mmm, which harbors two inequivalent Ce sites 30 . Interestingly, Ce 3 PtIn 11 exhibits two successive AFM phase transitions at T N1 = 2.2 K and T N2 = 2 K, followed by a superconducting transition at T c = 0.32 K 23,24 . It was arguably speculated that the magnetic ordering is associated with one of the Ce sublattices, while the other one is responsible for HF behavior and superconductivity 24 . Remarkably, under hydrostatic pressure the compound exhibits a quantum critical behavior with QCP located at a critical pressure p c = 1.3 GPa, where T N → 0 and T c becomes maximum, with normal state electrical resistivity showing NFL behavior 24 .
The spectacular low-temperature properties of Ce 3 PtIn 11 motivated us to study the effect of applied magnetic field on the complex AFM ordering and the superconducting state. In particular, our research was aimed to explore field effect on the two AFM transitions and construct a relevant magnetic phase diagram. Electrical transport and thermodynamic measurements were performed on single crystalline specimens at temperatures down to 50 mK. Here, we report the results of our investigations on Ce 3 PtIn 11 , which conjointly indicate a complex H − T phase diagram and unconventional SC state coexisting with the AFM ordering.

Results and Discussions
Crystal structure. We carried our a comprehensive investigation to determine the precise crystal structure of Ce 3 PtIn 11 using an anisotropic model. Even though the crystal structure of this compound was reported previously 31 , here we present the results of our X-ray diffraction measurements performed on a tiny crystal which offers very little absorption and high transmission factors of the X-ray beam. This allowed us to reach a high refinement quality of the observed pattern. The crystallographic data are presented in Table 1. The lattice parameters are in good agreement with the previous study 31 . The anisotropic displacement parameters (see Table 2) indicated that the structure is well defined. The atomic positions and interatomic distances are presented in Tables 2  and 3, respectively. AFM ordering. In order to have an in-depth understanding of the complex AFM ordering in Ce 3 PtIn 11 , we performed heat capacity measurements, C(T), in different magnetic fields applied along the crystallographic c direction in the tetragonal unit cell. Figure 1a displays the temperature evolution of C/T taken in zero field, which manifests two successive AFM transitions at T N1 = 2.08 K and T N2 = 1.93 K. It is worth noting that both critical temperatures are slightly lower than those reported by Prokleška et al. 23 . We tentatively anticipate that this discrepancy could be associated with very tiny structural features, like atomic disorder, vacancies, local distortion, which may affect the electronic state of the Ce ion (labeled in ref. 24 as Ce2) responsible for the AFM ordering in Ce 3 PtIn 11 . Another important remark that should be highlighted is the absence of any anomaly in C/T(T) at 10 K (see Fig. 1a), which proves that the sample investigated was free from CeIn 3 impurity phase.
The heat capacity of antiferromagnetic Kondo lattices in the ordered state is often modeled by the formula 32,33 where the first term signifies the contribution from the heavy quasi-particles and the second term represents AFM magnons contribution with Δ SW being the gap in the spin wave spectrum. The approach assumes a spin wave dispersion relation of the form ω = Δ + Dk SW 2 2 , where D is the spin wave stiffness. The coefficient c in Eq. 1 is related to D as − c D 3 32,33 . The least-squares fitting of the above formula to the experimental data of Ce 3 PtIn 11 measured below T N2 is shown in Fig. 1a by the red solid line. The so-obtained parameters are γ AFM = 1.33 (1) J/mol K 2 , c = 76(2) mJ/mol K 4 and Δ SW = 5.9(2) K. The value of Δ SW is in fairly good agreement with that of estimated by Custers et al. (Δ SW = 7.35 K 24 ). Furthermore, this gap value is comparable with the energy scale of the critical field (4.8 T) of the metamagnetic transition at 0.5 K (see below), which is coming out to be 3.6 K.
As can be inferred from Fig. 1b, with increasing magnetic field the two transitions first shift towards lower temperatures, elucidating AFM nature of the ordering. Remarkably, in a field of about 4 T, these two lambda-type features merge into a single sharp peak of a first-order character, that appears at a temperature which is decreasing with ramping field up to 5 T. In stronger fields, the latter singularity again splits into two anomalies at temperatures which continue to decrease with increasing field. In addition to the sharp anomalies seen, we observed another very broad anomaly near 1.4 K in the specific heat data measured in an applied field of 5.5 T. This feature can be associated with the metamagnetic transition (see below).
The temperature dependence of the electrical resistivity, ρ(T), of Ce 3 PtIn 11 , measured with the current flowing within the basal ab plane of the tetragonal unit cell of the compound, is presented in Fig. 2a. In the interval 30-100 K, the resistivity decreases with increasing temperature in a logarithmic manner signifying the dominance of Kondo type spin-flip scattering processes. At higher temperatures, ρ(T) forms a broad minimum, and susequently an increase of the resistivity with rising T is seen. This feature is most likely a direct consequence of phonon contribution competing with the Kondo interactions 34,35 . Near 15 K, ρ(T) forms a broad maximum that can be attributed to a crossover from incoherent to coherent Kondo regime that is a generic feature of Ce-based Kondo lattices, including the phases Ce n T m In 3n + 2m . For instance, similar feature has been seen in CeCoIn 5 (n = 1 and m = 1) 11 , Ce 2 PdIn 8 (n = 2 and m = 1) 13 and Ce 3 PdIn 11 (n = 3 and m = 1) 31 . Successively, a distinct drop in ρ(T) is seen at 2.1 K caused by the reduction in the spin-disorder scattering due to the AFM ordering below T N1 . Another sharp resistivity drop down to zero occurs at T c = 0.23 K, and signals the onset of the superconducting state. As displayed in Fig. 2a, at temperatures T c < T < T N1 , the ρ(T) data can be modeled by the formula 36 where the second term accounts for Fermi liquid contribution, and the third term represents scattering of conduction electrons on AFM spin-wave excitations with an energy gap Δ SW in the magnon spectrum (here, phonon contribution was neglected because of very low temperature range considered). The coefficient b in this expression is related to the spin-wave stiffness D as . Least-squares fitting of Eq. 2 to the experimental data (see the figure) yielded:  (1) 13 (1) In (2) 10 (1) 10 (1) 20 (1) 0 0 0 0.5 0.5 0.2778 (1) 13 (1) In (3) 10 (1) 18 (1) 12 (1) 0 0 0 0.5 0 0.1379 (1) 13 (1) In (4) 11 (1) 11 (1) 18 (1)   so-obtained value of Δ SW is somewhat larger than that derived from the heat capacity data (see above). However, considering difference in methods/experiments involved, these values are fairly comparable. In turn, the quite large value of A manifests the significance of electron-electron scattering in this compound, as generally expected for HF systems.
In accordance with the heat capacity results, the AFM transition seen in ρ(T) gradually shifts towards lower temperatures with increasing strength of transverse (applied along the crystallographic c axis) magnetic field (see Fig. 2b). Notably, in fields 0 T ≤ μ 0 H ≤ 2 T and 7 T ≤ μ 0 H ≤ 9 T, two distinct anomalies in dρ/dT(T) are observed (note the arrows in the inset to Fig. 2b), highlighting two separate phase transitions. It is worth pointing out that the field variations of the critical temperatures coincide very well with those derived from the heat capacity data. In addition, another observation which demands special note is the sudden drop in ρ(T) at the ordering temperature observed in applied fields of 7 T and 9 T that gives rise to sharpening of the peak in dρ/dT(T). This finding is in perfect concert with the features seen in C(T) (compare Fig. 1b) suggesting a first order transition.
In order to gain further insight on the field-dependent critical behavior observed in the specific heat data, the transverse magnetoresistance (MR =  Fig. 3a. Far below T N1 , MR is positive and increases with ramping field in a quadratic manner (cf. Figure 3b) up to a critical field μ 0 H c , at which a pronounced peak is observed. The positive value of MR is consistent with the AFM ordering in the system, and the MR ∝ (μ 0 H c ) 2 dependence can be attributed to the influence of magnetic field on the energy dispersion of the AFM spin waves, as predicted by Yamada and Takada within the random phase approximation 37 . In turn, the MR singularities at μ 0 H c manifest metamagnetic-like phase transitions. Remarkably, the value of μ 0 H c shows a non-monotonic temperature dependence: on rising T up Ce (1)-In (1) 3.2623 (12) Ce (1)-In (2) 3.32177 (14) Ce (1)-In (3) 3.3287 (12) Ce (2)-In (3) 3.3065 (9) Ce (2)-In (4) 3.32177 (14) Pt (1)-In (1) 2.7821 (7) In (1)-In (1) b 2.982 (2) In (1)-In (2) 3.2591 (15) In (1)-In (1) a 3.32177 (15) In (2)-In (1) a 3.2590 (15) In (2)-In (3) 3.3320 (15) In (3)-In (4) 3.3065 (9) In (3)-In (3) a 3.32177 (15)  to 1.5 K, it increases but with further increasing temperature, it slightly decreases. Close to T N2 , the feature in MR is quite broadened and then disappears. Figure 3c shows the high-field MR data measured at temperatures 0.5 K (one can expect that in these conditions the scattering of conduction electrons on spin fluctuations is strongly damped). As can be inferred from the figure, above the metamagnetic transition, MR has a linear dependency with field. This feature accounts for some unusual kind of cyclotron motion of the charged particles. It is worth recalling that linear MR may arise in systems with small concentrations of charge carriers having small effective masses, in regime of the electrical transport involving only the lowest Landau level 38 . Clearly, this mechanism should be ruled out for Ce 3 PtIn 11 which is a HF compound. Another possibility for linear MR arises for gapless materials with linear energy spectrum 39,40 . In the case of the compound studied, the spin wave gap Δ SW = 10.1 K was found (from electrical resistivity), and hence also the latter scenario cannot be justified. Actually, the physical origin of the linear contribution to  the magnetoresistance of Ce 3 PtIn 11 , being dominant at T = 0.5 K (see Fig. 2c) remains unclear. It is worthwhile mentioning that similar behavior of MR was found before for antiferromagnetic Ce 2 PdGa 12 41 but also for this compound no explanation of this unusual feature was given. Above 1.5 K, the MR isotherms change their overall shapes in strong fields. Figure 4a displays the transverse magnetoresistance of single-crystalline Ce 3 PtIn 11 measured as described above at few temperatures in the paramagnetic state. At each temperature, MR is negative and its absolute value decreases with increasing temperature. Such a behavior of MR is expected for a Kondo compound due to freezing-out of the spin-flip scattering by external magnetic field. Remarkably, as shown in Fig. 4b, all the MR isotherms taken at T ≥ 8 K can be projected onto a single curve by plotting the MR data as a function of μ , where the parameter T * is the characteristic temperature, usually considered as an approximate measure of the Kondo temperature 42 . This so-called Schlottmann-type scaling was applied to Ce 3 PtIn 11 yielding T * = 12 K. Figure 5 presents the magnetic field -temperature phase diagram of Ce 3 PtIn 11 , constructed based on the results of thermodynamic and electrical transport measurements. Interestingly, the phase boundary constructed from the MR data, i.e., describing metamagnetic transition (MMT), evidences a first order like phase transition which is quite remarkable. Now, in order to properly understand the field evolution of the AFM ordering temperature and emergence of new field-stabilized magnetic phases, the phase diagram can be divided into three regions as pointed out in the inset of Fig. 5. Initially, two AFM transitions occurring at T N1 and T N2 shift to lower temperatures with increasing magnetic field strength. Then, in the field range 4 T ≤ μ 0 H ≤ 5 T these two transitions merge into a single feature at T M further decreasing with ramping field. This finding is quite consistent with the expectation that with increasing field, Zeeman energy is increased and when it exceeds the energy of the intersite-coupling strength, the long-range ordering is turned into a field-induced ferromagnetic state. However, with magnetic field above 5 T, one observes another peculiarity, namely the latter transition again splits into two well separated anomalies seen at T M1 and T M2 . The positions of these singularities systematically decrease with raising field, at least up to 9 T. Notably, the height of the peak at T M1 systematically decreases with increasing field, while the peak at T M2 rapidly sharpens on going from μ 0 H = 5.5 T to μ 0 H ≥ 6 T. Furthermore, at high fields, the shape of the latter anomaly, observed both in the specific heat (see Fig. 1b) and the electrical resistivity (see Fig. 2b), resembles to a first order like transition. This may hint towards possible rearrangement of the Fermi surface sparking the possibility of a field induced Lifshitz transition in this system. Further detailed investigations probing the Fermi surface geometry such as quantum oscillations or angle-resolved photoemmision spectroscopy are needed to verify that conjecture.

Magnetic phase diagram.
Remarkably, the H − T phase diagram constructed for Ce 3 PtIn 11 bears striking similarities with the magnetic phase diagrams of intensively studied HF antiferromagnets CeRhIn 5 and Ce 2 RhIn 8 43 . In particular, a common feature is the presence of both first-and second-order field-induced magnetic transitions. Therefore, the phase diagram of Ce 3 PtIn 11 turns out to be quite remarkable as it may suggest the existence of competing order parameters in this material. In order to explore this peerless feature, further investigation involving neutron diffraction and muon spin relaxation/reorientation will be essential. Another direction for future studies would be investigating the thermodynamic and transport properties of Ce 3 PtIn 11 in other magnetic field orientations, in order to shed some light on the expected anisotropy of the field stabilized magnetic phases emerging in this material. Superconducting phase. Figure 6 depicts the real and imaginary components of the dynamic magnetic susceptibility of Ce 3 PtIn 11 measured in zero steady magnetic field with an ac field of 10 μT oscillating with frequency of 113 Hz. Both a clear diamagnetic signal in χ′(T) and a pronounced upturn in χ″(T) below T c = 0.23(1) K manifest the onset of bulk superconductivity in the specimen measured. Remarkably, the so-derived value of T c is in perfect agreement with the electrical resistivity data described below. However, it is distinctly smaller than T c = 0.32 K reported in the literature 23,24 . Possible source of this discrepancy may be related to some tiny structural features, as suggested in the previous section in the context of the magnetic critical temperatures. Clarification of this intriguing issue requires comprehensive crystallographic investigations of single crystals of Ce 3 PtIn 11 grown in different batches. Figure 7a shows the ultra-low temperature dependence of the electrical resistivity of Ce 3 PtIn 11 measured with electric current flowing in the tetragonal ab plane and external magnetic field applied along the crystallographic c axis. Clearly, with increasing field strength the superconducting transition gradually broadens and shifts to lower temperatures. The critical temperature, defined at the midpoint of the drop in ρ(T), is equal to T c = 0.23(2) K, which is in good agreement with the ac magnetic susceptibility data. Plotting the change of T c in the magnetic field one can derive the temperature variation of the upper critical field in the specimen studied. From the results presented in Fig. 7b, the initial slope of the μ 0 H c2 (T) dependence near μ 0 H = 0 is found out to be where φ 0 = h/2e is the flux quantum, one can estimate Ginzburg-Landau (GL) coherence length in Ce 3 PtIn 11 to be ξ GL = 88 Å.
The key characteristics of the superconducting state in Ce 3 PtIn 11 are gathered in Table 4, where they are compared with those reported for Ce 2 PdIn 8 13 and CePt 3 Si 44 . A close resemblance of these various superconducting parameters with those of the well established HF superconductors hints towards a possible unconventional origin of the superconductivity that emerges in Ce 3 PtIn 11 even within the AFM ordered state.

Conclusions
In summary, the results of our detailed investigation of AFM ordering and superconducting phase in single-crystalline Ce 3 PtIn 11 elucidate a likely coexistence of AFM and superconductivity in this compound. The specific heat and electrical transport data collected in various applied magnetic fields conjointly establish a complex magnetic phase diagram with several distinct field stabilized magnetic phases. It sparks the possibility of finding competing order parameters near the critical field in this compound. This observation is quite remarkable considering the uniqueness of such phase diagram in the existing literature. Furthermore, the magnetotransport data collected at low temperatures revealed a metamagnetic transition followed by a linear field dependence. This feature implies an unusual kind of cyclotron motion.  The electrical resistivity and ac magnetic susceptibility data obtained in the superconducting state in Ce 3 PtIn 11 indicated unconventional superconductivity with the key parameters similar to those reported for well-established heavy-fermion superconductors. Thus, Ce 3 PtIn 11 turns out to be one of the rare examples of HF systems where superconductivity coexists with bulk magnetic order. Further detailed investigations involving microscopic techniques such as muon spin rotation, neutron diffraction and photoemission spectroscopy are called for in order to address the unusual and unique features witnessed in Ce 3 PtIn 11 .

Methods
Single crystals of Ce 3 PtIn 11 were grown using In flux, as outlined by Kratochvílová et al. 31 . The crystals selected for physical properties measurements were examined by x-ray diffraction (XRD) employing an Oxford Diffraction four-circle single crystal diffractometer equipped with a CCD detector and using graphite-monochromatized Mo-Kα radiation. The raw data were treated with the CrysAlis Data Reduction Program (version 1.171.38.34a). The intensities of the reflection were corrected for Lorentz and polarization effects. The crystal structures were solved by direct methods and refined by full-matrix least-squares method using SHELXL-2014 program 45 . The atoms were refined using anisotropic displacement parameters. Their chemical composition was checked by energy-dispersive X-ray (EDX) analysis using a FEI scanning electron microscope equipped with an EDAX PV9800 microanalyzer. The XRD and EDX results confirmed the expected stoichiometry and the crystal structure of the compound, in line with the literature data 23,24 .
The electrical resistivity was measured over the temperature interval 0.4 to 300 K and in magnetic fields up to 9 T using a standard ac four-probe technique implemented in a Quantum Design PPMS platform. In order to probe the superconducting state, a Cryogenic Ltd. 3 He-4 He dilution refrigerator was employed to carry out electrical resistivity measurements down to 50 mK in applied fields up to 1.2 T. Furthermore, the ac magnetic susceptibility was measured in the same dilution fridge. Heat capacity measurements were performed in the temperature range 0.35-20 K in fields up to 9 T using relaxation method and the PPMS equipment.