Unusual d-electron heavy-fermion behaviour in f-electron ferromagnetic antiperovskite Gd3SnC

Inverted structures of common crystal lattices, referred to as antistructures, are rare in nature due to their thermodynamic constraints imposed by the switched cation and anion positions in reference to the original structure. However, a stable antistructure formed with mixed bonding characters of constituent elements in unusual valence states can provide unexpected material properties. Here, we report a heavy-fermion behaviour of ferromagnetic gadolinium lattice in Gd 3 SnC antiperovskite, contradicting the common belief that ferromagnetic gadolinium cannot be a heavy-fermion system. The specic heat shows an unusually large Sommerfeld coecient of ~ 1114 mJ ⋅ mol − 1 ⋅ K − 2 with a logarithmic behaviour of non-Fermi-liquid state. We demonstrate that the heavy-fermion behaviour in the non-Fermi-liquid state appears to arise from the hybridized electronic states of gadolinium 5d-electrons participating in metallic Gd–Gd and covalent Gd–C bonds. These results accentuate unusual chemical bonds in CGd 6 octahedra with the dual characters of gadolinium 5d-electrons for the emergence of heavy-fermions.


Introduction
The crystal structure constructed with chemical bonds of constituent elements determines the electronic structure of a material and its properties. In a xed symmetry, electronically inverted structures with switched crystallographic positions of cations and anions from the original structure often have a reversed compositional ratio, which necessarily forms a different coordinated substructure with additional bonding characters and can lead to an exotic electronic structure. However, in general, such antistructures composed of the same constituent elements in the reversed compositional ratio are thermodynamically unstable. Indeed, due to the thermodynamic constraints, antistructures found in several classes of materials are constructed by subsidiary chemical bonds of different constituent elements from original structures. This appears, for example, in the Cd-de cient Cd 3 As 2 anti uorite 1 , which has a strong covalent bonding nature in a reversed compositional ratio from the ionic CaF 2 uorite.
It is noteworthy that the Cd 3 As 2 anti uorite exhibits exotic properties: a three-dimensional topological Dirac band 1-3 along with negative magnetoresistance 4 , anomalous Hall effect 5 , and pressure-induced superconductivity 6 . ABX 3 perovskites, as the largest family of crystalline materials, also have their counterpart antistructures, A 3 BX antiperovskites 7 . In contrast to the perovskite compounds constructed by the ionic bonds of constituent elements in stable valence states, the antiperovskites have mixed bonding characters with ionic, covalent, and metallic bonds in unusual valence states 8 . However, oxide antiperovskites (A 3 BO), antistructures of common oxide perovskites (ABO 3 ), are hardly stabilized in ambient conditions as oxygen tends to form ionic bonds with counter-cationic A and B elements due to its large ionicity 9 . This constraint may be lifted for s-block and transition metal-based antiperovskites, in which additional covalent and metallic bonds come into play for their crystallographic stability 7,10 . The boundary for such stable antiperovskite compounds can be extended to f-block lanthanides, which form a crystal with various elemental groups of d-block transition metals and p-block post-transition metals 7 . Those compounds may exhibit diverse electronic properties because of the multiple bonding characters with enhanced covalency and/or metallicity. Furthermore, considering the fact that f-electrons often contribute to strongly correlated electronic properties, a variety of unconventional quantum phenomena may be anticipated in f-block antiperovskites, which can exhibit exotic electronic band structures arising from the mixed chemical bonds. However, thermodynamically stable f-block element-based antiperovskites and their novel properties have been hardly demonstrated up to now 7,11 . Herein, we demonstrate the unprecedented heavy-fermion behaviour of the ferromagnetic (FM) gadolinium lattice in the Gd 3 SnC antiperovskite structure and verify the crucial role of the metallic Gd-Gd and covalent Gd-C bonds of CGd 6 octahedral skeleton on an unusually hybridized Gd 5d-electrons responsible for the emergent heavyfermions.

Results
Synthesis and structural analysis. Figure 1a shows that the Gd 3 SnC crystallizes in the antiperovskite structure (space group: Pm m) with face-centred Gd and body-centred C atoms in a Sn cubic lattice, as con rmed by Rietveld analysis of powder X-ray diffraction (XRD) pattern (Supplementary Table 1). Singlephase bulk Gd 3 SnC was synthesized via the melt-solidi cation process at one atmosphere. This indicates the thermodynamic stability of Gd 3 SnC antiperovskite, which gets additional support from the calculated phonon dispersions having no imaginary frequency ( Supplementary Fig. 1). In contrast to the ABX 3 perovskite, which has cation-centred BX 6 octahedra with ionic bonds, Gd 3 SnC has anion-centred CGd 6 octahedra with strong covalent Gd-C and metallic Gd-Gd bonds. This mixed bonding character of Gd bonds in CGd 6 octahedra results in different Gd orbital levels from those of the Gd element and its ionic compounds. As shown in Fig. 1b, while Gd metal and Gd 2 O 3 have ordinary Gd orbital levels of Gd-Gd metallic and Gd-O ionic bonds, respectively, Gd 3 SnC has unusual energy levels for Gd orbitals due to the coexistence of metallic and covalent bonds. Considering the shorter Gd-Gd bond length (3.48 Å) in CGd 6 octahedra compared to that of Gd metal (~ 3.6 Å), one expects an enhanced metallicity and thus additional Fermi level (E F ) crossings for Gd bands in Gd 3 SnC. On the other hand, the electronegativity difference between Gd and C is much smaller than that between Gd and O, resulting in the formation of covalent bonds between Gd and C. The energy difference between bonding and antibonding states of the covalent bonds is expected to be signi cantly smaller than that of ionic bonds between Gd and O. These mixed bonding characters of Gd valence orbitals in CGd 6 octahedra can lead to a peculiar molecular orbital diagram with both metallic and covalent bonds as shown in Fig. 1b. Such observation indicates that Gd 3 SnC antiperovskite may exhibit unusual physical properties that are hardly found in elemental Gd and conventional Gd-related compounds.
Tricritical ferromagnetic behaviour. Figure  Gd metal with a T C of 297 K, different aspects of the FM order are observed 12 . A sharp increase in the magnetization (M) near the T C indicates a rst-order transition, which is further evidenced by the hysteresis in the temperature(T) dependence of electrical resistivity (ρ) (Fig. 2e) and the negative slope in the M 2 dependence of H/M ( Supplementary Fig. 2e). In addition, no structural phase transition is observed around the T C , indicating that the FM ordering occurs while preserving the antiperovskite structure ( Supplementary Fig. 3). We con rm the saturation magnetic moment of ∼6.8 µ B per Gd obtained from the M-H curve (Fig. 2b), indicating this FM state comes mostly from the seven Gd 4felectrons. A notable aspect of M (T) is a slight decrease in the ZFC curve below ~ 20 K (inset of Fig. 2a), which suggests a possible formation of singlets such as Kondo singlet 13,14 , as will be discussed in the following sections. This unusual behaviour in M indicates that the near E F states of Gd, which do not have contributions from fully localized f-orbitals, may be different from those of the familiar FM Gd metal and Gd-based compounds 12,15 .
The most peculiar aspect of the FM state in Gd 3 SnC is the tricritical behaviour that appears near the boundary between the rst-and the second-order phase transitions 16 . Figure 2c shows a modi ed Arrott plot 17 of the tricritical mean-eld model for the FM order with β Arrott = 0.25 and γ Arrott = 1 in (H/M) 1/γArrott and M 1/βArrott , respectively. Compared to other magnetic ordering models, it is apparent from the relative slopes (dashed red lines in Fig. 2c and Supplementary Fig. 2a-d) that the FM transition belongs to the universality class of the tricritical mean-eld model (Fig. 2d). A clear crossover from a rst-order to second-order transition is seen in the T dependent ρ and M curves under different H as shown in Fig. 2e,f, respectively. Three representative features in the data verify the crossover between FM orders: (i) disappearance of the peak in ρ at T C , (ii) suppression of the hysteresis in the ρ-T curve during a thermal cycle (insets of Fig. 2e), and (iii) smooth increase of M upon cooling (Fig. 2f). Such tricritical behaviour often emerges in heavy-fermion systems such as UGe 2 18 and YbRh 2 Si 2 19 due to the competition between Kondo coupling (T K ) and Ruderman-Kittel-Kasuya-Yosida interaction 20 .
Heavy-fermions with non-Fermi liquid behaviour. More unusual properties of Gd 3 SnC may be found in the heat capacity (C) measurement results (Fig. 3). As a way to determine the unexpected behaviour, we compare the C of Gd 3 SnC to those of FM Gd metal and non-magnetic La 3 SnC antiperovskite, as shown in Fig. 3a. In contrast to the small C values of Gd metal and La 3 SnC antiperovskite in the low-T region, an extremely large C value with a Sommerfeld coe cient (γ) of ~ 1114 mJ mol − 1 K − 2 at 90 mK is observed for the Gd 3 SnC. The dramatic low-T rise of C is reminiscent of a nuclear Schottky anomaly, most likely from the Sn nucleus (the natural abundance for non-zero nuclear spin 13 C is below 1%). However, our modelling, which considers 16 % natural abundance of 117 Sn and 119 Sn, rules out the Schottky anomaly by showing that the required internal eld is about − 1287 kOe, a value much larger than any reported values [21][22][23] (Supplementary Fig. 4d). In addition, the Gd nuclear spin cannot be the culprit as we do not see a similar rise in C for Gd metal. Therefore, we attribute the logarithmic rise of C for Gd 3 SnC below ~ 0.13 K (Supplementary Fig. 4a) to a non-Fermi-liquid (NFL) behaviour of itinerant conduction electrons 24- Fig. 3a). A T-linear dependence of ρ in the low-T region also supports the NFL behaviour of itinerant conduction electrons in Gd 3 SnC (left panel in Fig. 3b). The transition from T-linear to T 2 behaviour of ρ upon applying 140 kOe (right panel in Fig. 3b and Supplementary Fig. 5) is suggestive of a magnetic origin of the NFL behaviour 27,28 . Thus, we assume that an exotic quantum phenomenon coupled with a correlation between itinerant and localized electrons in the Gd 3 SnC antiperovskite appears as heavy fermions.
As Gd f-electron levels locate deep in the binding energy, the heavy-fermion with the NFL behaviour should originate from the electrons of other orbitals, which is revealed from the analysis of the C of FM Gd 3 SnC. The magnetic contribution (C mag ) to the C of FM Gd 3 SnC is obtained by subtracting the C of non-magnetic La 3 SnC without 4f-electrons (La metal with the con guration of [Xe]4f 0 5d 1 6s 2 ) from those of Gd 3 SnC with seven 4f-electrons (Gd metal with the con guration of [Xe]4f 7 5d 1 6s 2 ). We nd two distinct features in C mag : a large C mag in the high-T region (C mag,high , Fig. 3c) and a small C mag in the low-T region (C mag, low , inset of Fig. 3c). While the C mag,high is attributed to the FM order of 4f 7 -electrons, which agrees well with the prediction of the renormalization-group approach 29 in the scheme of tricritical ferromagnetism ( Supplementary Fig. 2f), the origin of C mag,low is ambiguous. From the T-dependent magnetic entropy (ΔS) obtained by integrating C mag (T) (Supplementary Fig. 4b), we can assign which electrons are responsible for C mag,low . Figure 3d shows a large difference in ΔS between the high-and low-T regions: 0.5Rln8 and 0.05Rln2, respectively. Since 0.5Rln8 re ects the contribution from seven 4felectrons (4f 7 ) involved in FM ordering, one may notice that the much smaller value of 0.05Rln2 has no correlation with 4f-electrons but is ascribed to 5d-electrons, which are responsible for the heavy-fermion with the NFL behaviour.
Dual character of Gd 5d electrons. This unconventional and anomalous d-electron heavy-fermion state in f-block Gd-based Gd 3 SnC is, if true, the rst case of FM-assisted heavy-fermion behaviour in 5d-electron systems. Density functional theory (DFT) calculations along with angle-resolved photoemission spectroscopy (ARPES) measurements provide a clear picture for the formation of the heavy-fermion state, which is induced not by 4f-electrons but by 5d-electrons of Gd element in the context of mixed bonding characters of the antiperovskite structure. Figure 4a shows ARPES data along the Γ-X-Γ direction taken above (125 K) and below (90 and 13 K) the T C . The 125 K data shows a fast-dispersing band centred at the X point. As the T is lowered to 90 K, the band shifts down due to the exchange splitting of both 4fand 5d-electrons ( Supplementary Fig. 6d). In addition to the fast-dispersing band (green curve), there is a weakly dispersing band near E F (red curve in Fig. 4a, Supplementary Fig. 6a-c). Upon cooling to 13 K, the fast-dispersing band shifts further down, and the weakly dispersing band near E F disappears. These changes in the electronic structure might be related to the hybridization of fast and weakly dispersing bands below 20 K. The T-dependent electronic structure change is also seen in the k x -k y plane of Fermi surface maps in Fig. 4b; the Fermi surface pocket becomes larger as the T decreases. This enlarged Fermi surface is a typical phenomenon as found in various heavy-fermion systems 27,30−32 . The calculated partial density of states (DOS) clearly demonstrates that the Gd 5d-orbitals are almost solely responsible for the states near E F , while Sn 5p and C 2p derived states are located away from E F (Fig. 4c).
Meanwhile, we con rm that the contribution of Gd 4f-orbitals into the band structure near E F is negligible but responsible for the FM ordering ( Supplementary Fig. 7).
We note that the calculated band structure shows an anomalous at band feature near E F , which is also a signature of heavy-fermion behaviour (Fig. 4d,e). The experimentally observed atness of the fastdispersing band at the X point (13 K in Fig. 4a) is indeed well-reproduced by band calculations (highlighted as yellow colour in Fig. 4e). The fat band analysis reveals that the band hybridization triggers the appearance of a at band feature near E F . Importantly, the two hybridized bands originate from the only Gd 5d-orbitals: t 2g bands from the Gd-Gd bonds (red curves) and e g bands from the mostly Gd-C bonds (green curves). Thus, we believe that the at t 2g band observed at 90 K (Fig. 4a) is lifted above E F after the band hybridization process (Supplementary Fig. 6e). Finally, we emphasize the importance of antiperovskite structure with mixed bonding nature, which is the key to imparting the unusual heavy-fermion behaviour to Gd 3 SnC. The crystal orbital Hamiltonian population (COHP) analysis

Summary
In summary, we discovered the d-electron heavy-fermion state in the f-electron FM Gd 3 SnC antiperovskite.
The mixed bonding nature of the antiperovskite structure, composed of metallic Gd-Gd bonds and covalent Gd-C bonds, is found to be the key ingredient for realizing the heavy-fermions of Gd 5delectrons. The present ndings are against the knowledge that the FM lanthanide elements are not suitable for heavy-fermion states, implying that the properties of the elements in the common crystal structure can be completely changed by adopting the antistructured lattice framework. This work will trigger further exploratory studies on the antistructures to challenge the common knowledge in the elemental properties of materials.

Methods
Crystal growth and structural analysis. Stoichiometric Polycrystalline Gd 3 SnC ingot rods were used for the single-crystal growth by the oating zone (FZ) melting method. We mixed Gd, Sn metals, and graphite chips in a 3:1:1 molar ration to fabricate a polycrystal ingot rod. The mixed sample was melted using the arc melting furnace in a high purity argon atmosphere (Ar > 99.9999%). The single crystal was grown by four-mirror FZ melting method with the rod-shaped ingots under high purity argon (Ar > 99.9999%) with pressurized conditions in 0.3 MPa. The feed and the seed rod were rotated at 50 rpm, and the growing speed was 2 mm per hour. The crystal structure was determined using the high-resolution X-ray diffractometer and Rietveld analysis using the GSAS-II program.
Physical property measurements. ρ, M, and C measurements were performed using a physical property measurement system (PPMS). The ρ and C at ambient pressure were measured down to 0.06 K using a diluted helium-3 refrigerator. To measure the electrical properties of samples, the electrical contacts in the four-probe con guration were adopted that made by Ag epoxy onto a sample. To prevent movement of samples by H (up to 140 kOe) in the chamber, samples were xed using torr seal epoxy onto the PPMS puck. To measure the M of the sample, a vibrating sample magnetometer (VSM) was used. For C measurement, the standard relaxation method was adopted. Every process was performed in an Ar-lled glove box, and N-grease covered samples for ρ and C measurements to prevent degradation of DFT Calculations. First-principles DFT calculations were performed using the generalized gradient approximation with the Perder-Burke-Ernzerhof functional with spin-orbit coupling (SOC) and the projector augmented plane-wave method implemented in the Vienna ab initio simulation program code [33][34][35] . The 4f-, 5s-, 5p-, 5d-and 6s-electrons of Gd, the 5s-and 5p-electrons of Sn, and the 2s-and 2p-electrons of C were used as valence electrons. The plane-wave-basis cut-off energy was set to 600 eV. On-site Coulomb interaction values of U = 3 eV were used for the Gd 5d-electrons ( Supplementary Fig. 7). Self-consistency was carried out using an a × b × c unit cell containing 5 atoms, and a 12 × 12 × 12 k-point mesh was used. Structural relaxation was performed until the Hellmann-Feynman forces were less than 1 × 10 − 5 eV Å −1 , respectively. The crystal structures and charge density distribution were visualized with the Visualization for Electronic and Structural Analysis code 36 .

Declarations Data Availability
The data that support the plots in this paper and other ndings of this study are available from the corresponding author upon reasonable request.   Red curves are single power-law ts (ρ = ρ0 + ATn) of the data. c, C versus T for Gd3SnC and La3SnC.
The inset is an enlarged view of the low-T region. Cmag for Gd3SnC is estimated by subtracting the C of the non-magnetic La3SnC (Supplementary Fig. 4b). d, T-dependent ΔS − ΔS0 for Gd3SnC per mole of Gd.
of Gd 5d-orbitals. Band thickness represents the contribution of each orbital. e, Near Fermi energy band structure around the X point, showing a band hybridization between fast-dispersing and weakly dispersing bands. f, Partial COHP (pCOHP) for Gd-Gd, Gd-C, and Gd-Sn bonds. pCOHP for other bonds can be found in Supplementary Fig. 8. g-i, Schematic illustration of the mixed bonding state for Gd 5dorbitals.

Supplementary Files
This is a list of supplementary les associated with this preprint. Click to download. 20210317Gd3SnCSupplementary.docx