Coexistence of Antiferromagnetism and Superconductivity in Heavy Fermion Cerium Compound Ce₃PdIn₁₁

Abstract

Many current research efforts in strongly correlated systems focus on the interplay between magnetism and superconductivity. Here we report on coexistence of both cooperative ordered states in recently discovered stoichiometric and fully inversion symmetric heavy fermion compound Ce₃PdIn₁₁ at ambient pressure. Thermodynamic and transport measurements reveal two successive magnetic transitions at T₁ = 1.67 K and T_N = 1.53 K into antiferromagnetic type of ordered states. Below T_c = 0.42 K the compound enters a superconducting state. The large initial slope of dB_c2/dT ≈ – 8.6 T/K indicates that heavy quasiparticles form the Cooper pairs. The origin of the two magnetic transitions and the coexistence of magnetism and superconductivity is briefly discussed in the context of the coexistence of the two inequivalent Ce-sublattices in the unit cell of Ce₃PdIn₁₁ with different Kondo couplings to the conduction electrons.

Introduction

The vast majority of cerium intermetallic compounds investigated to date have been compounds which have only one crystallographic site for the Ce-ion. As such, the physical properties of these dense Kondo lattices are determined by the competition between the magnetic inter-site interaction, i.e., the indirect RKKY exchange interaction and the demagnetizing on-site Kondo interaction. This competition is well described by the Doniach diagram¹; the ground state depends on the relative value of the respective RKKY and Kondo energies, and , with N_F being the conduction band density of states at the Fermi level and J_cf the coupling constant (hybridization) between 4f (Ce) and conduction electrons. For small values of the compound’s ground state is magnetic while, for high values of , the Kondo effect dominates and the ground state is nonmagnetic. The value of J_cf itself depends critically on the local environment of the 4f electron. This aspect plays a crucial role in compounds exhibiting inequivalent crystallographic Ce-sites per unit cell constituting different Ce–sublattices. The different local symmetry of each inequivalent Ce-ion results in a different hybridization strength and hence each corresponding Ce–sublattice is characterized by its own Kondo temperature. Therefore, it gives the possibility of having different ground states for each sublattice. Examples are for instance Ce₇Ni₃, Ce₅Ni₆In₁₁ and Ce₃Pd₂₀Si₆. The first-mentioned compound crystallizes in the hexagonal Th₇Fe₃ structure in which the Ce-ions reside in three inequivalent sites: (i) one atom in sublattice Ce1 (Wyckoff site 2b: 3 Ni and 12 Ce neighbors); (ii) three in sublattice Ce2 (6c: 4 Ni, 11 Ce) and (iii) three in sublattice Ce3 (6c: 4 Ni, 11 Ce but different distances than for Ce2). From thermodynamic and magnetic studies it was suggested that at low-T the Ce1 ions are responsible for antiferromagnetic (AFM) order at T_N = 1.8 K, the Ce2 ions are associated with heavy fermion (HF) behavior (T_K ≈ 4 K) and the Ce3 ones give rise to intermediate valence contributions². The second compound, Ce₅Ni₆In₁₁, has a base centered orthorhombic unit cell. There are ten Ce-ions per unit cell. They occupy two different Ce-sites. Eight Ce-ions sit at Wyckoff position 8p (Ce1) while the remaining two occupy the 2c site (Ce2). The compound has been classified as an intermediate HF system with γ ≈ 145 mJ/(mol_Ce · K²) and exhibits two successive AFM magnetic phase transitions at T_N2 = 1.10 K and T_N1 = 0.63 K which have been attributed to the Ce1 and Ce2 sublattices, respectively³. The analysis of the entropy indicated a weak interaction between the two magnetic sublattices below T_N2. The latter compound, Ce₃Pd₂₀Si₆, has recently attracted much interest because of its unusual quantum critical behavior⁴. The compound crystallizes in the cubic structure of space group . It harbors two different crystallographic Ce-sites, both with cubic symmetry but one with coordination 16 (tetrahedral 8c: all Pd) and one with coordination 18 (octahedral 4a: 12 Pd, 6 Si). At T = 0.5 K, Ce₃Pd₂₀Si₆ undergoes a quadrupolar phase transition and at T_N = 0.31 K a transition in an AFM ordered state⁵. Each ordered state has been associated with a particular sublattice, meaning that the quadrupolar state and the dipolar coexist at low temperatures.

Yet, the field of Kondo lattice compounds with multiple distinct local moments (4f (Ce)) per unit cell is still uncharted terrain. This despite the fact that interplay between different ground states, which in these compounds might come from interaction between sublattices with different electronic or magnetic order, often give rise to new interesting phenomena. Indeed recent theoretical work of Benlarga et al. examines the question of a Kondo lattice with two distinct Kondo ions⁶. Their model describes a unit cell with two inequivalent Ce-sites. Each experiences different hybridization which reflects in different (sublattice) Kondo temperatures T_K1≠T_K2, respectively. For simplicity of the discussion it was assumed T_K1>T_K2 . Long-range magnetic ordering was induced by allowing for Ce1-Ce2 interaction and a schematic phase diagram has been constructed. Particular interesting becomes the situation T_K1>T>T_K2 where one enters a region in the phase diagram of so-called partial Kondo screening. Partial refers to the fact that only 4f (Ce1) moments are effectively screened by Kondo effect forming heavy particle states. Hence, as has been pointed out by the authors, any magnetic phase in this regime will manifest characteristics of a weakly polarized HF phase coexisting with properties of typical local-moment magnetism originating from the strongly polarized and little screened 4f (Ce2)-moments. The compound Ce₇Ni₃ mentioned earlier shows exactly such behavior. Although not explicitly discussed, in a broader sense one can speculate that under certain conditions the HF sublattice becomes superconducting while the second sublattice remains magnetically ordered.

In this work we present first low-temperature magnetization, specific heat and resistivity measurements on the recently^7,8 discovered HF system Ce₃PdIn₁₁. The compound belongs to the Ce_nT_mIn_3n+2m class of layered materials which comprises a numerous amount of compounds including CeCoIn₅, CeRhIn₅ and Ce₂RhIn₈ ^9,10,11. Ce₃PdIn₁₁ can be regarded as “intermediate step” between the cubic CeIn₃ which orders antiferromagnetically at T_N = 10.1 K¹² and tetragonal Ce₂PdIn₈ being a HF superconductor with T_c between 0.45 and 0.7 K depending on the crystal quality^13,14,15,16. It crystallizes in the typical tetragonal structure (space group P4/mmm) based on the AuPt₃-type (CeIn₃-block) and PtHg₂-type (Tin₂) units alternating along the [001]-direction as depicted in the Fig. 1. The unit cell possesses two inequivalent crystallographic Ce-sites denoted by Ce1 and Ce2. What distinguishes Ce₃PdIn₁₁ from previously investigated multi-site compounds is that the Ce1–Ce1 nearest-neighbor distance is exactly equal to the Ce2–Ce2 nearest-neighbor distance. That means, the distance as a parameter to explain the various behaviors of the sublattices like in other compounds plays no role¹⁷. Instead, the ground state properties of the sublattices are solely determined by the respective hybridization of the Ce-ions in our compound. A second point which outrivals Ce₃PdIn₁₁ from previous multi-site Ce-compounds is that the local environment of both Ce-ions are not uniquely related to Ce₃PdIn₁₁ but already exists in closely related compounds. As displayed in Fig. 1, Ce2 resides in the 1a Wyckoff site (C_4v symmetry) featuring CeIn₃ environment, whereas the Ce1-site occupies the 2g position (D_4h symmetry). Its surrounding atoms are identical with those of the Ce-ions in Ce₂PdIn₈. Preliminary results on Ce₃PdIn₁₁ revealed two successive magnetic transitions at T₁ ≈ 1.7 K and T_N ≈ 1.5 K and a pronounced enhancement of the Sommerfeld coefficient⁸. Our extended low-T experiments presented here show that below T_c = 0.42 K the compound enters a HF superconducting state, suggesting coexistence of long-range magnetic order and HF superconductivity in Ce₃PdIn₁₁ at ambient pressure.

Figure 1

The crystal structure of Ce₃PdIn₁₁ (middle) pointing out the two inequivalent Cerium sites Ce1 and Ce2.

The red shaded part emphasizes the CeIn₃ building blocks. The lattice parameters yield a = 4.6896(11) Å and c = 16.891(3) Å. To show the local environment of the Ce2-ion more clearly, the unit cell has been extended along the [001]-direction (semi-transparent part). The unit cells of CeIn₃ (left) and Ce₂PdIn₈ (right) are displayed for comparison.

Results

The electrical resistivity ρ(T) of Ce₃PdIn₁₁ with current applied along different crystallographic axes is shown in Fig. 2. At high temperatures above 80 K the resistivity ρ(T) is weakly temperature dependent with dρ/dT>0, passes through a shallow maximum at T_max ≈ 60 K (j ‖ [100]) with lowering temperature and then rapidly decreases signaling the formation of coherent Bloch states at low temperatures. At T = 2 K the resistivity reaches ρ(2 K) ~ 16 μΩ cm, 11 μΩ cm and 10 μΩ cm for j ‖ [100], ‖[001] and ‖[011], respectively. The overall behavior is similar to that reported from a previous study performed on polycrystalline material⁷. The room temperature resistivity value corresponds well to ours, i.e., ρ = 55 μΩ cm has been reported compared to 52 μΩ cm measured for j ‖ [100] measured in our study. The maximum structure at T_max is at 30 K and more pronounced in the polycrystal. The low temperature resistivity value, however, differs significantly being worse for the polycrystal (ρ(2 K) ≈ 34 μΩ cm) which is an indication that our single crystal is of higher quality. From our single crystal data we observe that the resistivity is fairly isotropic despite the complex tetragonal structure.

Figure 2

The electrical resistivity ρ(T) of Ce₃PdIn₁₁ with current j || [100] (triangles), ||[110] (open squares) and ||[001] (circles) from room temperature down to 2 K.

The residual resistivity ratio RRR = ρ_300K/ρ_2K equals 5.

The magnetic susceptibility χ(T) displays similar isotropic behavior. At T = 4 K the ratio between parallel and perpendicular [001]-axis yields χ_[001]/χ_[100] = 1.2. This suggests a strong influence of the cubic CeIn₃ structure unit on the magnetic correlations in this compound. In Fig. 3 we plotted the inverse susceptibility 1/χ = B/M as function of temperature along both crystallographic directions in applied field of B = 1 T. For temperatures T >85 K and B ‖ [100] and for T >100 K and B ‖ [001] the susceptibility follows Curie-Weiss law. As inferred from Fig. 3 the data 1/χ(T) for both directions are parallel resulting in an equal effective moment of μ_eff = 2.43 μ_B per Ce-ion. The value is close to the Hund’s rule value of 2.54 μ_B for a free Ce³⁺ ion. In addition, because of the nearly magnetocrystalline isotropy of Ce₃PdIn₁₁ the extrapolated Weiss temperatures are close, yielding and . The upper inset of Fig. 3 displays the low-temperature susceptibility for B ‖ [001] in greater detail. Below about 6 K χ(T) gradually deviates from the high-temperature behavior and exhibits a broad hump with a maximum at T = 2.4 K. In fact, Ce-based compounds (J = 5/2) commonly exhibit a low-T maximum in χ(T). This is expected from the theory of orbitally degenerate Kondo impurities¹⁸. However, we have to account for the crystal electric field (CEF) level scheme. In tetragonal crystal symmetry the J = 5/2 line splits into three doublets, , and Γ₆. Assuming that the Ce1-ion in Ce₃PdIn₁₁ has similar CEF scheme as Ce in Ce₂PdIn₈, it follows that the first excited level is located at about Δ_CEF ≈ 60 K¹³. The situation for Ce2-ion is more unclear. It exhibits tetragonal symmetry contrary to cubic symmetry of the Ce-ion in CeIn₃. The first excited CEF level is at Δ_CEF ≈ 140 K¹⁹. In a rough approximation we may assume a similar level splitting of the J = 5/2 line for Ce2. Because the Kondo temperature in Ce₂PdIn₈ and CeIn₃ is much smaller than the CEF splitting being of the order of T_K ≈ 10 K^13,19 only the ground state doublet is relevant and the effective degeneracy of the moments becomes J = 1/2. In that case the Kondo effect produces a maximum in susceptibility at T = 0. We therefore attribute the maximum structure to the presence of short-range AFM correlations and determine the transition into long-range AFM ordered state in accordance with work of Fisher²⁰: for a mean-field transition, the dc susceptibility should show a cusp at T_N, whereas for classical critical behavior the derivative d(χT)/dT should diverge at T_N with a specific exponent (i.e., ) so that T_N occurs at an inflection point below a rounded maximum. From specific heat measurements two inflection points in χ at T₁ = 1.67 K and T_N = 1.53 K can be expected⁸. We observe only one at approximately 1.45 ± 0.1 K as displayed in the lower inset of Fig. 3. The value coincides well with the lower specific heat jump at T_N. The upper T₁-anomaly seems not to be reflect in χ(T) (B ‖ [001]) but it is not excluded that it might be visible for perpendicular direction. Alternatively, the T₁–signature is too small and beyond resolution of our current measurement setup.

Figure 3

Temperature dependence of the inverse of the susceptibilities 1/χ(T) of Ce₃PdIn₁₁ for the [100] (triangles) and [001] (circles) directions in an applied field of B = 1 T.

The solid (red) lines are Curie-Weiss fits to the data. The upper left inset shows the low temperature susceptibility for B ‖ [001] in more detail. The lower right inset displays the corresponding derivative d(χT)/dT.

We continued our investigation down to lower temperatures by means of specific heat (C/T) and resistivity experiments. Measurements were performed on the same single crystal to avoid sample dependent effects. In order to trace the associated anomalies and to complement the methods among each other the results of C(T)/T and ρ(T) in zero field are displayed one below the other in Fig. 4. The synthesis of the respective non-magnetic La and Y–compounds was unsuccessful so far. In order to retrieve the electronic part of the specific heat, C_el/T, we approximated the specific heat data in the interval 10<T<25 K by C(T)/T = C_phon/T + γ, where the Debye expression, , accounts for the phononic and the Sommerfeld term (γ) for the the electronic contribution. The fit yields a characteristic temperature of Θ_D ~ 182 K and γ = 0.890 J/(mol · K²). C_phon/T was subtracted from the total specific heat data and we obtained C_el/T presented in Fig. 4a. In our treatment we have disregarded the contribution arising from CEF. A more accurate analysis including this term is not possible because of the limited fitting interval and because of the fact that the exact CEF level scheme for Ce₃PdIn₁₁ is unknown. The CEF contribution, C_CEF/T, is somehow captured by C_phon/T and γ. To give an estimate about C_CEF/T we might take the CEF scheme of Ce₂PdIn₈ for comparison. The first excited level (Δ_CEF ≈ 60 K) gives rise to a Schottky-type of anomaly with a maximum at around 25 K (C_CEF/T ≈ 0.17 J/(mol_Ce · K²))¹⁴. Below it falls off exponentially and at 10 K reaches a value of C_CEF/T ≈ 0.05 J/(mol_Ce · K²). We can assume that similar values are reached for Ce₃PdIn₁₁. This means, that our simplified fitting of the heat capacity data leads to an overestimation of the phononic and electronic contribution due to the inclusion of the CEF contribution. Towards lower temperatures, because C_CEF/T and C_phon/T become small, the error in C_el/T displayed in Fig. 4 becomes negligible. Figure 4a shows three distinct features. Upon cooling down a first anomaly appears at T₁ = 1.67 K which is immediately followed up by a second one at 1.53 K, being marked as T_N and visible by a λ-like structure in the C_el/T data. Like in susceptibility, the higher transition T₁ is not observed in ρ(T) within the accuracy of the measurement. T_N is noticeable (Fig. 4b). It is manifested by a subtle change of the slope as shown more clearly by the temperature derivative d(ρ/ρ_2K)/dT (inset of Fig. 4b). More intriguing is the third feature appearing in C_el/T at approximately 0.4 K. The jump marks the entrance into a superconducting phase, as it is evident from the corresponding resistivity data. The transition is fairly broaden which makes a proper analysis impossible. It resembles the first results on UPt₃, where after improving sample quality it turned out that the broad specific heat jump actually consists of two very close superconducting jumps²¹. A more likely explanation is that the broadening results from an artifact caused by a strong increase of C_el/T of a normal state background. Consequently the expected decrease becomes compensated. Such an effect can be observed for instance in CeIr(In_1−xCd_x)₅ ²². To extract some numbers for comparison, we define the SC transition temperature being at half of the jump height, T_c = 0.42 K, which corresponds roughly with the onset of a second step in the resistivity data. The measured specific heat discontinuity, ΔC divided by the linear term in the specific heat at the transition (γT_c) equals 0.73 (γ = 0.55 ± 0.03 J/(mol_Ce · K²)). This value is well below that expected from BCS theory (ΔC/γT_c = 1.43). However, such downsized number is not unusual when the transition is particularly broad, see for instance ref. 23,24.

Figure 4

Zero field low-temperature results on Ce₃PdIn₁₁.

(a) the specific heat C_el/T vs. T (open symbols) in units of J/(mol_Ce · K²). The transitions are clearly visible at T₁ = 1.67 K, T_N = 1.53 K and T_c = 0.42 K. The solid line represents the function with values for A and Δ given in the text. The right ordinate displays the presumable magnetic entropy S_Ce per mol_Ce in units of Rln2. The inset compares S_Ce with the spin-1/2 Kondo model of Desgranges and Schotte with T_K = 12 K (red curve)³⁰. (b) The temperature dependence of the electrical resistivity normalized to 2 K. The current was applied j ⊥ [001]. Inset: temperature derivative of ρ/ρ_2K in the temperature region around the magnetic transitions. The error bar indicates scattering of the data points.

Finally, we estimated the magnetic entropy by cutting off the superconducting transition and extrapolating the low-temperature tail of the T_N–transition towards zero. For this we employed a fit expression accounting for the cooperative effect of spin-wave excitations²⁵, with A = 0.0109 J/(mol · K²) per Ce and Δ = 2.74 K (solid line in Fig. 4a). It is fair to state that a second-order mean-field BCS-type of fit²⁶ gives similar result due to the limited fitting range of 0.5–1.2 K. S_Ce(T) is plotted as closed symbols in Fig. 4a. The entropy deliberated at T₁ is only 0.2Rln2 per mol_Ce. This is well below the value (Rln2) associated with the lifting of the degeneracy of the ground state doublet suggesting magnetic ordering with substantially reduced magnetic moments.

B – T Magnetic Phase Diagram

The influence of an applied magnetic field on the magnetic transitions was studied by means of specific heat experiments. The magnetic transitions are most pronounced in this measurement. The subtraction of the phonon and CEF terms has been omitted because we are only interested in the change of the transition temperatures. Experiments were conducted on a sample from a different batch than the one shown in Fig. 4. Figure 5 cumulates the results. We first focus on the behavior for B ‖ [001]. From Fig. 5a it can be seen that in small fields B < 2.5 T the transition T₁ shifts to lower temperatures while T_N remains almost unaffected. At B ≈ 3 T both transitions seem to merge. In higher fields B >4 T we observe again two transitions. Whether or not these two transitions indeed correspond to the low-field transitions as suggested by the figure is subject of further investigation. Our attention was devoted to the measurements in B = 5 T and above. In this field region the transition at T_N shows a sharp λ-like shape, typical for a first-order type transition. Such increase in sharpening and peak size has also been found for the field-induced transition in CeRhIn₅ ¹¹.

Figure 5

Specific heat as function of temperature for various values of applied magnetic fields (a) B || [001] and (b) B ⊥ [001]. The temperature – magnetic field phase diagram (c) for B ‖ [001] mapped out by the thermal response technique (see text). The green circles and black diamonds show T₁ and T_N, respectively, as obtained from the data present in (a). Inset depicts the B − T phase diagram for B ⊥ [001] axis.

Figure 5b shows the specific heat for applied field perpendicular to [001]-axis. The T₁-transition and the successive ordering at T_N are clearly observed in low fields. T₁ first shows a tiny increase, likely caused by a realignment of the moments, before gradually decreasing with increasing field. The lower transition temperature remains almost unchanged for B <5 T and moves monotonically to lower temperatures in larger fields. The constructed B − T phase diagrams are presented Fig. 5c. The phase diagram for B ‖ [001] has been measured in more detail by the method of analyzing the response of the compound to thermal heat pulses. By this method a heat pulse resulting in a ΔT = 1 K temperature change of the sample (mounted quasi-adiabatically) to the bath temperature T_bath was applied. The subsequent thermalization of the sample temperature to a temperature T was monitored over time t. The color plot shows the derivative d(ΔT)/dt as a function of the bath temperature and field (yellow: small change in dT/dt; blue/violet: large change in dT/dt). The thermal relaxation result is complemented by data points of T₁ and T_N obtained from specific heat experiments (T₁: circles, T_N: diamonds). The inset displays the respective phase diagram for B ⊥ [001].

From Fig. 5c it becomes more evident that the two transitions, T₁ and T_N, merge at ≈3 T. Moreover, in fields >4 T the T_N anomaly sharpens as inferred by the rapid change in color code. Such a behavior indicative of a pronounced first-order type of transition and hence corroborates our observation already made in C/T. An intriguing aspect is that the sensitivity of the magnetic transitions with respect to the applied field is weakest for B ⊥ [001] and strongest for B ‖ [001]. For the two other magnetically ordered compounds of the Ce_nT_mIn_3n+2m class of materials, CeRhIn₅ and Ce₂RhIn₈, it is vice versa¹¹.

The Superconducting State

The observation of ambient pressure bulk superconductivity below T_N gives Ce₃PdIn₁₁ a unique place within the Ce_nT_mIn_3n+2m family. So far pressure or doping was necessary to induce SC in a magnetically ordered compound. Corresponding examples are CeRhIn₅, Ce₂RhIn₈, Ce(Rh_1−xIr_x)In₅ or CeCoIn₅ doped with Sn and Cd^22,27,28. Figure 6 shows the superconducting phase diagram of Ce₃PdIn₁₁ for B ‖ [001] and j ‖ [100]. The jump in resistivity in various external fields is plotted in the inset of Fig. 6. The data are normalized to the value at 1.5 K (ρ(T)/ρ(1.5 K)). We defined the second step in the resistivity drop as T_c as discussed earlier. Within the accuracy of the measurement, the value of T_c fairly well corresponds with that obtained from the transition temperature in the specific heat. As expected, the application of magnetic fields reduces T_c, giving rise to a rather large change of the initial slope of the upper critical field dB_c2/dT ≈ −8.6 T/K. Extrapolation of the field dependent transition temperatures to T → 0 using a simple mean-field type expression B_c2(T) = B_c2(0)[1 − (T/T_c)²] yields B_c2(0) ≈ 2.3 T.

Figure 6

Temperature dependence of the upper critical field of B_c2 as derived from resistivity experiments on Ce₃PdIn₁₁.

Typical curves are shown in the inset. Current was j ⊥ [001] and the field was applied parallel to the [001]-axis. The red solid line yields –dB_c2/dT = 8.6 T/K.

Discussion

Our experimental results show that Ce₃PdIn₁₁ undergoes two successive magnetic transitions into antiferromagnetic-type ordered states at T₁ = 1.67 K and T_N = 1.53 K. At lower temperatures the compound becomes superconducting (T_c = 0.42 K). Beyond that the data reveal that the magnetic ground state exhibits a reduced magnetic moment and suggest a heavy fermion superconducting state. Hence, Ce₃PdIn₁₁ takes a unique place being the first known full inversion symmetrical stoichiometric Ce-based compound which shows coexistence of magnetic order and superconductivity at ambient pressure. As we mentioned in the introduction the existence of two inequivalent Ce-sites in Kondo lattice systems can lead to the observation of unusual ground states and complex phase diagrams (e. g., ref. 2, 3, 4). Such an expectation found some general justification in the framework of the Kondo lattice model describing the consequences of the coexistence of two inequivalent local moment Ce sublattices on the ground state properties, when linked to different Kondo couplings to the conduction electrons⁶. Since Ce₃PdIn₁₁ belongs to this class of Kondo systems, we will try in the following to provide an explanation of the novel phenomena in the context of two Ce sublattices.

The tetragonal structure of Ce₃PdIn₁₁ comprises two crystallographically inequivalent Ce-sublattices. The Ce-ions on the Ce1-sublattice (2 ions per f.u.) exhibit a Ce₂PdIn₈ environment and the Ce-ions on the Ce2-sublattice (1 ion per f.u.) possess a CeIn₃ surrounding. Due to different local environments, each sublattice will have its individual CEF scheme and Kondo scale (T_K1 and T_K2). As a result complex interacting mechanisms are expected in Ce₃PdIn₁₁. In a first approach we can gain some insight by a quantitative analysis of C(T)/T and S(T). As shown in Fig. 4 the electronic specific heat coefficient γ of Ce₃PdIn₁₁ is found to be 0.55 ± 0.03 J/(mol_Ce · K²) when extrapolated to zero temperature. This large γ places Ce₃PdIn₁₁ into the heavy fermion category. However, since there are two kinds of Ce-ions (Ce1 and Ce2) with different site symmetries and neighboring ions it is quite likely that the two do not contribute equally to the density of states at the Fermi level, i.e., γ = 0.55 J/(mol_Ce · K²) is an average value. The same argument accounts for the magnetic entropy S_Ce which is depicted in the same figure. The average S_Ce per mol of cerium deliberated at T₁ equals 0.2Rln2, that is one-fifth of the value expected for a ground state doublet. Thus there is a considerable reduction of the magnetic entropy. We relate the deficiency of S_Ce (T₁) to the presence of the Kondo effect which partially lifts the twofold degeneracy above T₁. In such a case it can be shown that S_Ce(T₁) = S_K(T₁/T_K), where S_K(T₁/T_K) is the Kondo entropy at T₁ ²⁹. The relation holds if the first exited level and . From earlier discussion of the temperature dependence of the susceptibility we concluded that Δ_CEF ≈ 60 K. Thus the former condition is satisfied in Ce₃PdIn₁₁. Typical Kondo scales in the Ce_nT_mIn_3n+2m class of materials are in the range between 2 and 10 K suggesting that also the second condition is fulfilled¹⁰. Further, using the Bethe ansatz for spin-1/2 Kondo model, Desgranges and Schotte calculated the specific heat and the corresponding magnetic entropy³⁰. By inserting the experimentally found S_Ce into the above given relation we obtain the ratio T₁/T_K. The estimated Kondo temperature for Ce₃PdIn₁₁ hence yields 12 K. This Kondo temperature is an average of T_K1 and T_K2. The value agrees well with the estimated Kondo temperature from the relation ³¹. The inset in Fig. 4a compares S_Ce with the calculated values of the spin-1/2 Kondo model³⁰. We observe that below T₁ the experimental data fall under the theoretical curve as expected (entropy is absorbed in the transition). For the S_Ce values exceed the calculated ones which is an indication for the presence of additional contributions to the specific heat, most likely originating from excited CEF levels which have not been subtracted from the C/T data.

The small entropy release of ~0.2Rln2 at T₁ implies a strong screening of the 4f-moment which gives rise to a small ordered moment in the magnetic state. This is consistent with the observation of a small signal in χ(T) at the magnetic transition temperatures. The nature of these two phase transitions at T₁ and T_N needs to be studied further in particular with regard to the two Ce-sites. We notice that with an increasing field parallel to [001] both T₁ and T_N decrease which is consistent with the fact that both phase transitions are of antiferromagnetic type. It is thus expected that the magnetic moments of both Ce–sublattices are mutually interacting (RKKY-type interaction). The long-range magnetic interaction on the other hand competes with the Kondo interaction. In Ce₃PdIn₁₁ T_K will be in turn determined by the effective strength and type of interplay of the two sublattice Kondo temperatures T_K1 and T_K2 ⁶. The complexity of the interactions may account for the intriguing magnetic phase diagram of Ce₃PdIn₁₁ being distinct different from the other two magnetically ordered compounds of the Ce_nT_mIn_3n+2m family (CeRhIn₅ and Ce₂RhIn₈) which have only one Ce-site per f.u.¹¹.

Next we briefly discuss the superconducting state. A remarkable close analogy is found between the SC properties of Ce₃PdIn₁₁ and Ce₂PdIn₈. The SC transition temperature of our compound is T_c = 0.42 K while for Ce₂PdIn₈ values between 0.45 and 0.7 K have been reported^{13,14,15,16,32}. In addition, the upper critical field yields B_c2 = 2.3 T for Ce₃PdIn₁₁ and 2.5 T for Ce₂PdIn₈ which is very similar³². Difference is solely observed in the initial slope of B_c2 being dB_c2/dT ≈ −8.6 T/K and dB_c2/dT ≈ −13.5 T/K, respectively.

Before we discuss such a similarity to Ce₂PdIn₈ and to avoid confusion we want to exclude the possibility of having Ce₂PdIn₈ impurity phase in our samples. As we recently reported, our single crystal Ce₃PdIn₁₁ samples have been carefully analyzed by X-ray diffraction and Scanning Electron Microscopy with an Energy Dispersive X-ray detector (see Methods section) and clearly excludes the presence of any Ce₂PdIn₈ inclusions.

The large observed SC discontinuity of ΔC/T_c ≈ 0.89 J/(mol · K²) in Ce₃PdIn₁₁ corresponds to a SC jump ΔC/(γT_c) = 0.73 in Ce₃PdIn₁₁ and amounts to 51% of the BCS-value. Such a high ratio suggests that a large portion of the sample is in the superconducting state. One possibility to understand the observed similarities of the superconducting properties could be the fact that Ce₃PdIn₁₁ has a layer structure and the Ce1-site in Ce₃PdIn₁₁ exhibits a Ce₂PdIn₈ environment (see above). However this needs further microscopic proof.

In summary, at ambient pressure Ce₃PdIn₁₁ undergoes magnetic transitions into an AFM state at T₁ = 1.67 K and an order-to-order transition into a second AFM state at T_N = 1.53 K. The entropy analysis shows an magnetic entropy release at T₁ of only 0.2Rln2 indicating that the ordered moment is small. The strong reduction of the magnetic moment is due to Kondo effect. Below T_c = 0.42 K the compound becomes superconducting. The size of the jump in specific heat and the large value of initial slope dB_c2/dT ≈ −8.6 T/K indicate that the Cooper pairs are formed of heavy quasiparticles. In this sense Ce₃PdIn₁₁ is unique being the first known full inversion symmetrical stoichiometric heavy fermion material based on Ce exhibiting coexistence of both attributes, magnetism and superconductivity at ambient pressure. In fact, the crystal structure of Ce₃PdIn₁₁ possesses two inequivalent Ce-sites. The Ce1-site exhibits Ce₂PdIn₈ surrounding and the Ce2-site has CeIn₃-like environment. Each Ce-sublattice is expected to have a different Kondo temperature and different RKKY interaction within the sublattice. The interplay between these two sublattices in Ce₃PdIn₁₁ is likely strong and gives rise the complex ground state. To unravel and to gain deeper insight in the particular mechanisms at work, microscopic methods are indispensable. For instance ¹⁰⁵Pd and ¹¹⁵In NQR and NMR can provide information about the magnetic moment on the Ce-ion but also about the superconducting gap structure which is an important quantity to reveal the mechanism behind Cooper-pairing. On the other hand neutron experiments can resolve the magnetic structure. Finally we feel that our experimental results would stimulate further experimental and theoretical efforts to explore the nature of electronic correlation and their theoretical understanding in this class of compounds.

Methods

Single crystal growth

Plate-like single crystals with mass up to ≈1 mg were grown by indium self-flux method⁸. High-purity starting materials (Ce 99.9%, Pd 99.995% and In 99.999%) in the ratio 3:25:50 were placed in high purity alumina crucibles and sealed under vacuum in quartz glass tubes together with another crucible filled with quartz wool plug for later flux removal. The crucibles were heated up to a temperature of 750 °C for several hours (8–10 h) and slowly (3 °C/h) cooled down to 550 °C. At this temperature, the material was decanted to separate the single crystals from the remaining In-flux.

The quality of the single crystals was checked by X-ray Laue diffraction in reflection geometry. The chemical composition was verified employing a Tescan Mira I LMH scanning electron microscope (SEM). The instrument is equipped with a Bruker AXS energy dispersive X-ray detector (EDX). Elemental mapping has confirmed composition homogeneity⁸. Several pieces were pulverized and examined at room temperature by means of powder X-ray diffraction (Bruker D8 Advance diffractometer with Cu-Kα radiation). The obtained diffraction patterns were refined by Rietveld analysis using FullProf Suite. In addition, in order to refine the structural parameters more precisely and to confirm phase homogeneity, several samples were investigated by single crystal X-ray diffraction using a X-ray diffractometer Gemini with Mo-K radiation. The crystal structure was resolved by charge flipping program Superflip and adjacent refinement was done by full-matrix least-squares based on F² using Jana2006. Experiments were performed on samples from different batches.

Experimental Details

The physical properties of the Ce₃PdIn₁₁ samples were attained employing standard equipment. The magnetic susceptibility was determined using a MPMS 7 T SQUID magnetometer (Quantum Design) with an applied field of 1 T. Additionally, in the temperature range between 0.5 and 4 K magnetization measurements were performed using a Hall-probe which is sensitive to the dipole field created by the sample magnetization. In order to obtain absolute values the Hall probe data were scaled in the overlapping region (2–10 K) with the data measured by the MPMS. The method is still in testing stage. Resistivity was measured utilizing standard four point a. c. technique. Low-temperature measurements up to T ≈ 2 K were performed in a Leiden Cryogenics MCK72 dilution refrigerator. The setup is equipped with a 9 T magnet. Measurements above 0.4 K up to room temperature were done in a Physical Property Measurement System (PPMS) from Quantum Design. The low-T range of the PPMS guarantees sufficient overlap of low- and high-temperatures data sets. In similar manner specific heat experiments were conducted in PPMS down to 0.45 K and were continued to lower temperatures using a Quantum Design dilution refrigerator heat capacity puck mounted to the MCK72 dilution refrigerator.