Superconductivity in chromium nitrides Pr3Cr10-xN11 with strong electron correlations

Abstract Exploration of superconductivity in Cr-based compounds has attracted considerable interest because only a few Cr-based superconductors (CrAs, A2Cr3As3 and ACr3As3 (A = K, Rb, Cs, Na)) have been discovered so far and they show an unconventional pairing mechanism. We report the discovery of bulk superconductivity at 5.25 K in chromium nitride in Pr3Cr10-xN11 with a cubic lattice structure. A relatively large upper critical field of Hc2(0) ∼ 12.6 T is determined, which is larger than the estimated Pauli-paramagnetic pair-breaking magnetic field. The material has a large electronic specific-heat coefficient of 170 mJ K−2 mol−1—about 10 times larger than that estimated by the electronic structure calculation, which suggests that correlations between 3d electrons are very strong in Pr3Cr10-xN11, and thus quantum fluctuations might be involved. Electronic structure calculations show that the density of states at the Fermi energy are contributed predominantly by Cr 3d electrons, implying that the superconductivity results mainly from the condensation of Cr 3d electrons. Pr3Cr10-xN11 represents a rare example of possible unconventional superconductivity emerging in a 3D system with strong electron correlations. Nevertheless, clarification of the specific pairing symmetry needs more investigation.


INTRODUCTION
The 3d transition-metal oxides or pnictides exhibit rich quantum phases with novel quantum states, such as long-range magnetic orders, charge or spin density waves, metal-insulator transitions, high-T c superconductivity and colossal magnetoresistance. In particular, unconventional high-T c superconductivity has been discovered in cuprates as well as in iron-based superconductors [1,2]. Many compounds with 3d transition-metal elements can become superconducting at low temperatures. However, it is relatively difficult to find a superconducting material in chromium-based compounds because most Cr-based compounds have strong magnetism, which generally is not in favor of superconductivity. In fact, CrAs, A 2 Cr 3 As 3 and ACr 3 As 3 (A = Na, K, Rb, Cs) are the only Cr-based superconductors so far [3][4][5][6].
Superconductivity in CrAs was discovered in 2014 [3,4]. CrAs undergoes a first-order antifer-romagnetic transition with a double helical spin structure at T N ≈ 265 K [3,7,8]. A bulk superconductivity with T c ≈ 2 K emerges above a pressure P c ≈ 8 kbar, where the antiferromagnetic order is completely suppressed. Both the NMR and neutron-scattering measurements for CrAs under high pressure revealed that there are strong magnetic fluctuations and line nodes may exist in the superconducting gap function [9][10][11][12][13], as in Sr 2 RuO 4 or some heavy fermion superconductors [14,15]. After the discovery of superconductivity in CrAs, superconductivity has also been found in the quasi-1D compounds A 2 Cr 3 As 3 (A = K, Rb, Cs) [3,4]. The upper critical field H c2 of K 2 Cr 3 As 3 is about three times larger than the Pauli-paramagnetic limit.
In this paper, we report a novel Cr-based nitride superconductor with a cubic structure, Pr 3 Cr 10-x N 11 , which was first synthesized and characterized by Broil et al. in 1995 [16]. The compound crystallizes in space group Fm-3m (No. 225) 22 Natl Sci Rev, 2020, Vol. 7, No. 1 RESEARCH ARTICLE Figure 1. (a) The face-centered cubic cell of Pr 3 Cr 10-x N 11 with three kinds of building blocks: Pr 6 N 9 , Cr 8 N 13 and Cr 6 , respectively. The blocks are shown in a shrunken form. There are two Cr positions: Cr1 and Cr2. (b) Typical Rietveld refinement of Pr 3 Cr 10-x N 11 under ambient conditions. The vertical bars represent the calculated Bragg reflection positions of the diffraction peaks for Pr 3 Cr 10-x N 11 . The difference between the observed (scatters) and the fitted patterns (line) is shown at the bottom of the diffraction peaks.
with lattice constant a = 12.891Å. It contains 192 atoms in a face-centered cubic (FCC) cell with three kinds of building blocks, as illustrated in Fig. 1a. The building blocks are Pr 6 N 9 , Cr 8 N 13 and Cr 6 . Previous studies showed that there are Cr vacancies in the lattice structure. However, no low-temperature physical properties were reported [16,17]. Our key finding of this study is the observation of superconductivity in Pr 3 Cr 10-x N 11 with T c ∼ 5.25 K. The high-quality samples have a shielding fraction of 85% at 2 K from zero-field-cooled (ZFC) magnetic susceptibility and a prominent superconducting peak in the specific-heat measurement. A relatively large upper critical field is found at the zero-temperature limit, H c2 (0) ∼ 12.6 T, which is larger than the Pauli-paramagnetic pair-breaking field. From electronic structure calculations, we find that the density of states (DOS) at the Fermi energy are predominately contributed by Cr 3d electrons. The present results demonstrate that Pr 3 Cr 10-x N 11 is the first Cr-based superconductor discovered in chromium nitrides and it represents a rare example that possibly unconventional superconductivity emerges in a 3D system with strong electron correlations.

Sample and characterization
Polycrystalline samples of Pr 3 Cr 10-x N 11 were prepared by direct reactions of the corresponding binary nitrides with solid reaction. Synthesized powder shows a dark-brown color and is air-sensitive, as it is easily oxidized to Pr 2 O 3 within a few hours. Figure 1b shows the General Structure Analysis System (GSAS) refinement of Pr 3 Cr 10-x N 11 under ambient conditions, which indicates that Pr 3 Cr 10-x N 11 crystallized in an FCC structure with space group Fm-3m. All the Pr 3 Cr 10-x N 11 reflections can be well indexed based on a cubic cell with lattice parameter a = 12.8521Å, which is consistent with those reported in the literature (a = 12.891Å) [16], indicating their ideal composition. A small amount of raw reactant Cr 2 N was marked with a red vertical bar. Figure 2a shows the temperature-dependent resistivity for Pr 3 Cr 10-x N 11 from 1.8 to 300 K at zero field. The normal-state resistivity is metallic, with no phase transition observed. At low temperatures, a sharp superconducting transition is observed with onset T c of about 5.25 K, as shown in the inset of Fig. 2a. The T c of Pr 3 Cr 10-x N 11 is higher than that of CrAs with T max ∼ 2 K under pressure and is close to K 2 Cr 3 As 3 with T c ∼ 6.1 K [12].

Temperature-dependent resistivity in Pr 3 Cr 10-x N 11
The new superconductor Pr 3 Cr 10-x N 11 shows a relatively large upper critical field. Figure 2b shows resistivity data in magnetic fields up to 9 T. As the field increases, the transition temperature T c shifts to a lower temperature and the transition width is gradually broadened, similar to the iron-based superconductors [18,19]. The upper critical field H c2 (T) curve obtained from the field-dependent transition temperatures shows a remarkably high critical field of 12.6 T using the formula H c2 (T) = H c2 (0)(1 − t 2 ), where t is the reduced temperature t = T/T c and of 22 T using the Werthamer−Helfand−Hohenberg (WHH) theory [20]. On the other hand, the Pauliparamagnetic limit for the upper critical field is H P = 1.84T c ≈ 9.6 T in the case of an isotropic full  superconducting gap without considering spinorbit coupling [21,22]. The H c2 (0) in Pr 3 Cr 10-x N 11 is 130% as large as H P . Usually, the high superconducting upper critical field can be originated from multi-band effects, the strong-coupling effect, the spin-triplet pairing and the strong spin-orbit coupling effect in a low-dimensional system [23][24][25]. The origin of large H c2 (0) in Pr 3 Cr 10-x N 11 needs to be further studied. The obtained μ 0 H c 2 (0) allows us to estimate the Ginzburg-Landau coherence length ξ = 51Å according to the relationship: Wb is the magnetic flux quantum.

Magnetic-susceptibility measurements in Pr 3 Cr 10-x N 11
The bulk superconductivity in Pr 3 Cr 10-x N 11 was confirmed by magnetic-susceptibility measurements. Figure 3a shows susceptibility χ at low temperatures with ZFC and field cool (FC) under a magnetic field of 30 Oe. χ starts to drop below T c and the diamagnetic signal tends to saturate at low temperatures. The shielding fraction estimated from the ZFC magnetic susceptibility at 2 K is 85%, confirming bulk superconductivity in the sample. The normal-state susceptibility χ increases with decreasing temperature in Pr 3 Cr 10-x N 11 , showing a Curie-Weiss behavior as shown in the inset of Fig. 3a. Such behavior is different from that of isostructural material La 3 Cr 10-x N 11 , which shows a Pauli paramagnetism with a nearly temperatureindependent susceptibility [16]. Since La 3+ ion has no occupied 4f electrons while each Pr 3+ ion has two occupied 4f electrons, it is natural to attribute the Curie-Weiss behavior of χ (T) in Pr 3 Cr 10-x N 11 to the magnetic moments of Pr 3+ 4f electrons. Using a Curie-Weiss fit with formula χ (T) = χ 0 + C/(T-θ ), we obtained the effective moment of each Pr ion at about 3.6 μ B , which is very close to the calculated moment of 3.5 μ B for Pr 3+ by Hund's rule. The negative value of θ indicates the correlations between Pr ions are antiferromagnetic.
Further confirmation of superconductivity is shown by the magnetic hysteresis of the sample measured at 2 K in Fig. 3b, which displays the typical magnetic hysteresis curve for a type-II superconductor. From the inset of Fig. 3b, the lower critical magnetic fields H c1 of 170 Oe can be obtained. Figure 4a shows the specific-heat coefficient C/T as a function of T 2 from 2 to 10 K at the zero field. The bulk nature of superconductivity is confirmed by a pronounced anomaly around T c = 5.25 K, consistently with the resistivity and susceptibility measurements. Extrapolating C/T to zero temperature gives a residual value of γ 0 = 0.061 mJ g −1 K −2 . As indicated previously, there is raw reactant phase Cr 2 N in the sample. We measured the specific heat of Cr 2 N and found that it can be well fitted by C = γ T + βT 3 below 10 K with γ = 22 mJ mol −1 K 2 and β = 0.0373 mJ mol −1 K 4 . Given the residual specific-heat γ 0 origin from the Cr 2 N phase, we can then obtain the specific heat of pure Pr 3 Cr 10-x N 11 by subtracting that of Cr 2 N from the total specific heat, as shown in the inset of Fig. 4a.

Specific-heat measurements
Above T c , the good linear T 2 dependence of C/T indicates that the normal-state specific heat consists of two parts of contributions: the electronic part, which is proportional to T, and the phonon part, which is proportional to T 3 at low temperatures. By fitting the normal-state specific heat C with the formula C = γ n T + βT 3 , we find that γ n = 173 mJ K −2 mol −1 and β = 1.186 mJ K −4 mol −1 . The normal-state electronic coefficient γ n is proportional to the DOS at the Fermi level. Assuming that most of the DOS comes from the Cr 3d electrons, the value of γ n per mole Cr is γ n =  17.3 mJ K −2 mol Cr −1 for Pr 3 Cr 10-x N 11 , which is much larger than the corresponding value for CrAs (∼7 mJ K −2 mol −1 ) and MnP (∼8.3 mJ K −2 mol −1 ) at ambient pressure [11,26] and it is slightly less than that for K 2 Cr 3 As 3 (∼23.3 mJ K −2 mol Cr −1 ) and KCr 3 As 3 (∼27.1 mJ K −2 mol Cr −1 ) [14,27,28]. The relatively large γ n for Pr 3 Cr 10-x N 11 indicates strong correlations of Cr 3d electrons. The Debye temperature θ D obtained from β is 339 K. In Fig. 4b, the normalized specific-heat jump at T c is found to be C/γ n T c = 1.51. This value is much smaller than those of K 2 Cr 3 As 3 (∼2.5) and LaNiAsO (∼1.9), which are regarded as strongcoupling superconductivity [14,27,29]. The normalized C/γ n T c reflects the coupling strength between the conducting electrons and the pairing glue. We can then estimate the electron-phonon coupling constant λ = 0.6 from the modified McMillian formula [25,30,31]:

RESEARCH ARTICLE
where μ * is a Coulomb pseudopotential and ω is a logarithmic averaged phonon frequency. ω can be determined from the specific-heat jump at T c using the formula: C/γ n T c = 1.43[1 + 53(T c /ω)2ln(ω/3T c )]. Taking μ * = 0.10 and T c = 5.25 K, we obtained ω = 320 K and λ = 0.6. For such a small electron-phonon coupling constant, the large H c2 (0) value is not likely due to the strongcoupling effect.

Theoretical calculations
In order to examine which atomic species contribute most around the Fermi level (E F ), we have plotted the local density of states (LDOS) for Pr 3 Cr 9.5 N 11 as shown in Fig. 5. Electronic structure calculations show that the DOS at the Fermi energy are contributed predominantly by Cr 3d electrons, implying that the superconductivity results mainly from the condensation of Cr 3d electrons similar to that in CrAs [32]. The primitive cell of Pr 3 Cr 10-x N 11 is shown in the inset of Fig. 5, in which there are two types of nonequivalent Cr atoms, labeled as Cr1 and Cr2, respectively. Both the Cr1 and Cr2 atoms have large contributions around E F , mainly originating from the 3d orbitals of Cr atoms. In contrast, the vast majority of states for Pr and N atoms are far from the Fermi level. A higher density of Cr2 vacancies, such as in Pr 3 Cr 9 N 11 , does not change the results very much. According to our calculations, the total DOS of Pr 3 Cr 9.5 N 11 at the Fermi level N(E F ) is 7.37 states/(eV * f.u.). As a result, the corresponding calculated electronic specific-heat coefficient γ e is 17.4 mJ K −2 mol −1 . Experimentally, the measured γ e for Pr 3 Cr 10-x N 11 is about 173 mJ K −2 mol −1about 10 times more than the band calculations, which indicates the strong mass-enhancement effect.

CONCLUSION
Superconductivity in Pr 3 Cr 10-x N 11 shows several novel characters. First, electronic structure calculations show that most of DOS at Fermi energy is contributed by Cr 3d electrons, suggesting that the superconductivity is originated from the condensation of Cr 3d electrons. So Pr 3 Cr 10-x N 11 is the first Cr-based superconductor discovered in nitrides. A few other known Cr-based superconductors (CrAs, A 2 Cr 3 As 3 and ACr 3 As 3 (A = K, Rb, Cs, Na)) are arsenide. Superconductivity in CrAs emerges in the vicinity of a quantum critical point and antiferromagnetic spin fluctuations associated with the quantum criticality could act as an important glue medium for Cooper pairing. Superconductivity in the quasi-1D compounds A 2 Cr 3 As 3 (A = K, Rb, Cs) shows non-s-wave pairing. Both of them show an unconventional pairing mechanism. So the Cr d-electrons play an important role in electron correlations and possibly unconventional superconductivity in Pr 3 Cr 10-x N 11 with Cr d-electrons.
Second, Pr 3 Cr 10-x N 11 has a relatively large upper critical field H c2 (0) ∼ 12.6 T, exceeding the Pauli limit of the paramagnetic pair-breaking field; this is rare in 3D-structure superconductors. The upper critical field provides very important information on the superconducting pairing. This behavior resembles that unconventional superconducting in K 2 Cr 3 As 3 in which H c2 is three times the Pauliparamagnetic limit that is regarded as evidence of spin-triplet superconductivity.
Last but not least, the measured γ e for Pr 3 Cr 10-x N 11 is about 173 mJ mol −1 K −2 , equivalent to 17.3 mJ K −2 mol Cr −1 . This γ value is close to that of K 2 Cr 3 As 3 (mJ K −2 mol Cr −1 ), indicating enhanced electron correlations in Pr 3 Cr 10-x N 11. The experimental value of γ e is about 10 times higher than the electronic calculations. This large renormalization factor cannot be explained by electron-phonon interactions and thus quantum fluctuations might be involved.
In conclusion, we report the experimental result for a novel Cr-based superconductor nitrides Pr 3 Cr 10-x N 11 with a cubic lattice structure. Bulk superconductivity with T c ∼ 5.25 K is observed from the resistivity, susceptibility and specific-heat measurements. Further theoretical and experimental studies are needed to determine the pairing symmetry and the corresponding mechanism, especially the role of Cr 3d electrons, for the observed superconductivity in Pr 3 Cr 10-x N 11 .

Sample preparation
Polycrystalline samples of Pr 3 Cr 10-x N 11 were prepared by direct reactions of the corresponding binary nitrides, staring with PrN (99%) and a mixture of the chromium nitrides (CrN and Cr 2 N) in a mass ratio of 4:6. The operations were all performed in an Ar-filled glovebox. Cold-pressed pellets of the mixtures were sealed in an evacuated quartz tube (<10 −4 Pa). The pellets were gradually heated in 1 day to 1000 • C, held at that temperature for 50 hours, then cooled to the room temperature in the furnace. The products were again ground in a glovebox, pressed into pellets, wrapped in Ta foil and heated in an evacuated quartz tube at 1165 • C for 120 hours. It is worth noting that, after this treatment, the samples were usually still contaminated by the binary nitrides. The treatment of these samples with diluted hydrochloric acid only dissolved the rare-earth nitrides. In between these treatments, the pellets were ground to a fine powder, the decomposed rare-earth nitride was dissolved in hydrochloric acid and fresh rare -earth nitride was added again to the mixture to maintain the proper composition. The reaction temperature was carefully selected to avoid decomposition of the products, meanwhile obtaining a good crystallization. In our study, we found that Pr 3 Cr 10-x N 11 partially decomposed above 1200 • C. Synthesized powder shows a dark-brown color and is air-sensitive, as it is easily oxidized to Pr 2 O 3 within a few hours.

Measurements
The electrical transport measurement was carried out on a physical-property measurement system (PPMS-9, Quantum Design). The resistivity was measured by a standard four-probe method, employing silver-paste contacts cured at room temperature, used for resistivity measurements, with the electric current applied in an arbitrary direction. The magnetic susceptibility was measured in a Quantum Design SQUID VSM. The specific-heat measurements were performed up to 9 T in a PPMS.

Theoretical modeling
We carried out first-principles electronic structure calculations on Pr 3 Cr 10-x N 11 . The first-principles calculations were performed using the projector augmented-wave method [33], as implemented in the VASP package [34]. The generalized gradient approximation of the Perdew-Burke-Ernzerhof type [35] was adopted for the exchange-correlation function.