Superconductivity in non-centrosymmetric ThCoC2

Superconductivity in compounds whose crystal structure lacks inversion symmetry are known to display intriguing properties that deviate from conventional BCS superconducting behavior. Here we report magnetization, resistivity, and heat capacity measurements on polycrystalline samples of ThCoC2, which has been reported to crystallize in the non-centrosymmetric CeNiC2 prototype structure, and show clear evidence of bulk superconductivity in ThCoC2 with a critical temperature of Tc = 2.65 K. From the specific heat data we find a Sommerfeld coefficient of γ = 8.38 mJ mol−1 K−2 and a Debye temperature of ΘD = 449 K. Interestingly the Hc2 superconducting phase diagram displays positive curvature, and the specific heat at low temperature deviates from conventional exponential temperature dependence, which is suggestive of possible unconventional superconducting behavior in ThCoC2, similar to that seen in the isostructural and isoelectronic non-centrosymmetric superconductor LaNiC2.


Introduction
One of the highlights of Solid State Chemistry in recent decades has been the discovery of many types of ternary transition metal carbides that crystallize in different novel prototype structures and exhibit interesting magnetic and electrical properties. These compounds by definition are composed of carbon, a transition metal, and a highly electropositive multivalent metal such as a lanthanide (Ln), Sc, Y, or Th. The complete ionization of the electropositive metal to stable ions Ln 3+ , Y 3+ , or Th 4+ leads to a negatively charged transition metal-carbon sub-network, which can be regarded as an organometallic net [1][2][3]. An important characteristic of these carbide compounds is the presence of single and/or multiple metal-carbon and carbon-carbon bonds. Compounds of particular interest in this paper are those that crystallize in the CeNiC 2 prototype structure, which is base centered orthorhombic space group Amm2 (No. 38) [4]. The structure is formed from the stacking of alternating two-dimensional (2D) NiC 2 and distorted hexagonal rare-earth-metal sheets, with the structure of ThCoC 2 , shown in figure 1. This kind of prototype structure is selective in electron count with the transition metals limited to group VIII Fe, Co, and Ni atoms [5]. In these compounds the stoichiometric formula is given by LnTC 2 , YTC 2 , or ThCoC 2 (T = Fe, Ni, or Co) and interestingly have short bond lengths between adjacent carbon atoms in the range of 1.32-1.47 Å, which is suggestive of carbon-carbon double bonds in the organometallic net, as shown figure 1.
More than 30 compounds have been found to crystallize in the same CeNiC 2 prototype structure, particularly from a series of rare-earth (R) carbides RNiC 2 and RCoC 2 [6][7][8][9][10][11][12][13]. Neutron diffraction studies on these series (R = Pr, Nd, Tb, Dy, Ho, Er, and Tm) [8][9][10][11][12][13] found different magnetic order between the series, with RCoC 2 compounds generally becoming ferromagnets and RNiC 2 compounds ordering antiferromagnetically in different spin configurations [13]. The studies found that the magnetic order arises from the 4f electrons confined on the rare earth sites, whereas the Co and Ni sites remain non-magnetic, and is also evident by a very weak temperature-dependent magnetic susceptibility in YCoC 2 [8], YNiC 2 [7], and LaNiC 2 [14]. Interestingly LaNiC 2 becomes superconducting below 2.7 K [14] and is therefore classified as a non-centrosymmetric (NCS) superconductor since the CeNiC 2 prototype structure lacks an inversion center. This structural characteristic brings important implications for the physics in a superconducting state. Most superconducting materials possess an inversion center in their structures and have the Cooper pairs classified as either an even-parity spin-singlet, or an odd-parity spintriplet state in accordance to the Pauli exclusion principle and parity conservation. However, the classification of the Cooper pairs into even or odd-parity states relies on the presence of an inversion center in the crystal structure. The absence of inversion symmetry introduces an antisymmetric spin-orbit coupling (ASOC) that lifts the spin degeneracy with spin-split Fermi surfaces, and importantly allows a mixing of spin-singlet and spin-triplet pairing states into a so called mixed-parity pairing state. Exotic properties can therefore emerge in parity violated superconductors such as time reversal symmetry (TRS) breaking, nodal gap structure, and other unconventional behavior attributed to the lack of inversion symmetry [15][16][17][18][19][20]. Indeed, it was recently demonstrated that LaNiC 2 shows TRS breaking in muon spin relaxation (µSR) measurements [21], and evidence for nodal gap structure in magnetic penetration depth measurements [22]. Therefore, in this work we examine the compound ThCoC 2 that is reported to be iso-structural [23] and iso-electronic to LaNiC 2 , and show that ThCoC 2 is likely a new non-centrosymmetric superconductor with a critical temperature of T c = 2.65 K. The results extracted from specific heat, magnetization, and resistivity measurements suggest that the superconducting state is similar to that seen in LaNiC 2 and possibly unconventional in origin.

Experiment
The polycrystalline ThCoC 2 sample was synthesized by arc melting together the stoichiometric amounts of the elements Th (3N), Co (4N), and graphite C (5N) on a water cooled Cu hearth in an arc-furnace under UHP (6N) Argon atmosphere and using a Zr getter. The samples were flipped over and re-melted 5 times to ensure good homogeneity, with minimal resultant weight loss (<0.7%) from arc melting. The sample was subsequently wrapped in tantalum foil and sealed in a quartz tube under vacuum for annealing treatment at 1100 • C for 14 days to improve sample quality and remove any unreacted magnetic impurities. Powder x-ray diffraction (XRD) patterns were obtained using a Rigaku MultiFlex diffractometer equipped with a monochromator providing Cu Kα (λ = 1.540 56 Å) radiation. The diffraction pattern analysis and phase identification were done using MDI Jade software. Magnetization measurements were performed down to T = 1.8 K using a commercial VSM-SQUID magnetometer by Quantum Design. Electrical resistivity and heat capacity measurements were obtained using a physical property measurement system (PPMS) also by Quantum Design. The specific heat of a sample piece polished flat (∼17 mg) was measured in the temperature range of 0.45-20 K with a He 3 calorimeter using the relaxation method. The resistivity sample was polished into a flat block (2.6 mm × 0.8 mm × 0.25 mm), and fine platinum wires were spot-welded to the sample as voltage and current leads for the standard four-probe method. The AC resistivity was measured down to 0.6 K with an applied current of 10 mA using a Linear Research Inc. model LR700 AC resistance-bridge and controlled by Lab View data acquisition software. Figure 2 shows the XRD pattern and corresponding Miller indices for the annealed polycrystalline ThCoC 2 sample. The peaks are indexed to the orthorhombic CeNiC 2 prototype structure with space group Amm2 (38), and the refinement of the lattice parameters yields a = 3.8053 Å, b = 4.5321 Å, and c = 6.1424 Å, in good agreement with results reported in literature [23]. These results suggest that the ThCoC 2 sample is phase pure at least within the standard detection limit for powder XRD method (∼5%), and is consistent with the CeNiC 2 prototype structure. To our knowledge the investigation of ThCoC 2 compound has been rather limited [23], and therefore it would be highly desirable to have additional in-depth structure determination by single crystal x-ray or neutron diffraction experiments to definitively demonstrate ThCoC 2 crystallizes in the non-centrosymmetric CeNiC 2 prototype structure, as has been shown for many of the other tertiary rare-earth carbides RTC 2 by neutron diffraction studies [8][9][10][11][12][13].

Results and discussion
The temperature dependent magnetization using zerofield cooled (ZFC) and field cooled (FC) method using an applied 50 Oe magnetic field shows a clear diamagnetic transition onset around 2.55 K in figure 3. It should be noted that the as-cast samples slowly decomposed in air and had a small residual ferromagnetic signal, likely from small inclusions of unreacted Cobalt or a Cobalt binary, which makes the annealing treatment critical to stabilize the sample and eliminate any magnetic impurities. The hysteresis between ZFC and FC regimes indicate ThCoC 2 is a type II superconductor. Without correcting for demagnetization or sample size effects, we estimate the superconducting volume fraction (ZFC) to be 110% of perfect diamagnetism, which suggests possible bulk superconductivity. The Meissner flux expulsion (FC) is about 5% of the diamagnetic flux expulsion, a characteristic of strong flux pinning. The inset in figure 3 displays 4-quadrant M versus H data at T = 1.8 K and shows weak type II superconducting behavior.
Sample resistivity from 0.6-300 K is presented in figure 4, with the figure inset showing the superconducting transition suppressed with applied magnetic field. The polycrystalline ThCoC 2 sample is a very good metal with resistivity increasing linearly above T > 100 K at a rate of 0.33 µ cm K −1 , and an extremely low residual resistivity of 0.37 µ cm at T = 4 K. The resultant residual resistivity ratio (RRR = ρ 300 /ρ 4 ) is thus a very large RRR = 218, indicating excellent alloy quality of the sample. The zero magnetic field data shows a sharp superconducting transition centered at T c = 2.65 K and a zero-resistance temperature of T c = 2.55 K. The sharp superconducting transition ( T C ∼ 0.2 K) is consistent with the magnetization measurements and is also indicative of good sample quality.
While the magnetization and resistivity data suggest bulk superconductivity in ThCoC 2 , an anomaly in the specific heat measurement is necessary for confirmation. Figure 5 displays the specific heat divided by temperature (C/T ) versus T 2  in various applied magnetic fields. A jump in the specific heat is clearly observed with the midpoint of the transition at T c = 2.53 K and a small transition interval T C ∼ 0.3 K . The consistency between the magnetization, resistivity, and heat capacity transitions is clear evidence of bulk superconductivity in ThCoC 2 . The expected suppression of the superconducting transition with applied magnetic field is also seen in figure 5, along with a broadening of the transition that is not unusual for polycrystalline samples.
Analysis of the superconducting state first requires a comparison to the normal state specific heat C n . The normal state specific heat is comprised of the electronic contribution C e and the lattice contribution C lattice , such that C n = C e + C lattice . The electronic contribution is the linear Fermi liquid term C e = γ T , with γ being the Sommerfeld coefficient. The lattice contribution to the heat capacity at low temperature is usually described just by the cubic Debye model term βT 3 ; however an increasing deviation above T c requires the addition of the next higher order phonon dispersion term, such that C lattice = βT 3 + δT 5 . Fitting the normal state specific heat data to the expression C n = γ T + βT 3 + δT 5 results in the fit parameters γ = 8.38 mJ mol −1 K −2 , β = 0.0856 mJ mol −1 K −4 , and δ = 5.59 × 10 −4 mJ mol −1 K −6 . The β value corresponds to a Debye temperature of D ∼ 449 K. The Sommerfeld coefficient γ suggests a low density of states at the Fermi level typical of other transition metal superconductors, with corresponding LaNiC 2 compound having γ = 7.83 mJ mol −1 K −2 and D ∼ 496 K for comparison [14].
Subtraction of the phonon contribution C lattice from the total specific heat allows analysis of just the electronic contribution C e , displayed here as C e /T versus T in figure 6. The analysis of the specific heat anomaly shows the magnitude of the jump at T c to be C e /γ n T c ∼ 0.86, which is significantly smaller than the weak-coupling BCS limit of 1.43. Since the superconducting transition is a second order phase transition, thermodynamics requires the entropies of the normal and superconducting states to be equal at the transition temperature T c . This equality has the convenient geometric representation in figure 6 that the two enclosed areas above and below the γ = 8.38 mJ mol −1 K −2 baseline must be equal. The entropy balance is demonstrated in figure 6 with good agreement between the upper area of 6.07 mJ mol −1 K −1 , and lower area of 6.09 mJ mol −1 K −1 after the low temperature data is extrapolated to the origin. This thermodynamic consistency shows the lack of a residual heat capacity from impurity phases and demonstrates the excellent sample quality with full superconducting volume.
The microscopic theory of superconductivity predicts the electronic specific heat of conventional fully gapped s-wave superconductors to display a low-temperature exponential behavior of the form exp[− (0)/k B T ]. The electronic specific heat for ThCoC 2 does not appear to follow this exponential temperature dependence that is expected for conventional BCS superconductors. This deviation from exponential behavior is visualized in figure 7, which displays the logarithmic graph of the electronic specific heat versus the inverse reduced temperature (T c /T ). Clearly the data shows a marked deviation at low temperature from a linear plot in figure 7 that would be expected for conventional superconductivity. Indeed it was shown that the specific heat of the analog LaNiC 2 compound also deviates from conventional exponential behavior with a low-temperature T 3 dependence [14] that is consistent with nodes in the energy gap, in addition to other evidence of non-conventional behavior of time reversal symmetry breaking Figure 6. Temperature dependence of the electronic specific heat contribution divided by temperature (C e /T ). The calculated upper (6.07 mJ mol −1 K −1 ) and lower (6.09 mJ mol −1 K −1 ) area between the data and the value demonstrates entropy balance. in µSR measurements [21], and evidence for nodal gap structure with T 2 dependence in magnetic penetration depth measurements [22]. It is worth noting that earlier conflicting reports on LaNiC 2 conclude conventional BCS behavior in heat capacity [24] and Nuclear quadrupole resonance (NQR) measurements [25]; however the recent report by Bonalde et al [22] shows that is likely a result of the measurements sensitivity to magnetic Fe impurities introduced by lower quality Ni (3N) and the insufficient low temperature evaluation of the superconducting state in those studies. They correctly point out that temperatures below 0.3T c are needed to properly determine the structure of the energy gap, in which they measured magnetic penetration depth down to ∼0.017T c in LaNiC 2 . The ThCoC 2 samples examined here are comparative to the higher quality LaNiC 2 samples, with the characteristic measurements indicating excellent ThCoC 2 sample quality without the signature of impurities being readily apparent. It is therefore not unreasonable to expect possible unconventional behavior for ThCoC 2 in future lower temperature examinations of the superconducting state.
The upper critical field H C2 as a function of temperature from the resistivity and specific heat data is plotted in figure 8, showing good agreement between the measurements. Unexpectedly the superconducting phase diagram has positive curvature instead of the conventional dome like plot separating the superconducting and normal state domains. Both the midpoint of the resistivity transition and the zero resistance values are plotted to emphasize this unusual behavior, showing positive curvature close to T c then transitioning into an apparent linear behavior below T < 1.4 K. An accurate estimate of the upper critical field H C2 (0) at absolute zero is difficult without conventional modeling, but it is evident that superconductivity in ThCoC 2 is quickly suppressed in magnetic field with H C2 only 3000 Oe at T = 0.6 K. Interestingly, examples of positive curvature in the upper critical field diagram include the borocarbides LnNi 2 B 2 C [26,27], magnesium diboride MgB 2 [28], the non-centrosymmetric Li 2 (Pd, Pt) 3 B [29], and the non-centrosymmetric heavyfermion superconductor CeRhSi 3 [30]. This kind of behavior can be attributed to interband coupling that can occur within a  multiband model [27,31], with the small gap contributing at low temperature. In fact, a recent study on the superconducting state of LaNiC 2 also displays positive curvature near T c and an enhancement of H c2 (0) [31]. Chen et al systematically measured the low temperature London penetration depth, heat capacity, and resistivity, and argue the superconducting state in LaNiC 2 is best described by a two-gap BCS model, which is the likely result of a moderate value of the ASOC that splits the spin degenerate bands. Within this scenario the analog non-centrosymmetric superconductor ThCoC 2 could be a new example of a multiband compound, making additional low temperature characterizations of the superconducting state necessary to determine the exact nature of the energy gap.

Conclusion
In summary, polycrystalline samples of the reported noncentrosymmetric ThCoC 2 compound were characterized by magnetization, resistivity, and heat capacity measurements that unambiguously show bulk superconductivity with a critical temperature of T c = 2.65 K. The superconducting state shows interesting behavior with a deviation from conventional BCS exponential temperature dependence in the specific heat, and positive curvature in the superconducting phase diagram. These results, in conjunction with those of analog noncentrosymmetric superconductor LaNiC 2 , suggest possible unconventional superconducting behavior and warrant future low temperature investigations of the superconducting state in ThCoC 2 . The limited reports on ThCoC 2 also make it a good candidate for precise structure analysis by such methods as single crystal x-ray or neutron diffraction experiments.