Nodal Superconducting Gap in FeS
Abstract
Low temperature specific heat has been measured in superconductor FeS with T = 4.55 K. It is found that the low temperature electronic specific heat C/T can be fitted to a linear relation in the low temperature region, but fails to be described by an exponential relation as expected by an swave gap. We try fittings to the data with different gap structures and find that a model with one or two nodal gaps can fit the data. Under a magnetic field, the field induced specific heat coefficient =[C(H)C(0)]/T shows the Volovik relation , suggesting the presence of nodal gap(s) in this material.
 Subject Areas

Condensed Matter Physics, Superconductivity
I Introduction
The discovery of superconductivity in iron pnictides has opened a new era for the study on unconventional superconductorsKamihara2008 (). Until now, superconductivity has been found in several relatives of the FeAsbased superconductors, including the FeSebased families and the iron phosphidesstewart (); RGreene (); review (). In either FeAsbased or FeSebased compounds, the superconducting transition temperature can be raised to much higher than 40 KREN (); llsun (), the socalled McMillan limitmcmillan (). This allows us to categorize the iron based superconductors as an unconventional superconductor with the pairing mechanism beyond the electronphonon coupling. In the research on the iron based superconductors, one of the core issues is about the superconducting pairing mechanism. Early on, theoretical pictures concerning the s pairing gap was proposedMazinPRL (); KurokiPRL (), this model has got many experimental supportsHanaguriQPI (); Neutron (); CuImpurity () and is specially satisfied by the systems with the presence of both electron and hole pockets. Recently, superconductivity has been reportedLinJH () in FeS which has an isostructure of the FeSe superconductorWuMK (). Earlier band structure calculations have revealed that the electronic structures of the FeSe and FeS systems are quite similar to each otherSingh (). Usually the orbital of the element sulfur has a narrower bandwidth comparing with that of selenium. Therefore, it is interesting to know the superconducting gap structure and pairing mechanism in this newly discovered superconductor.
Superconductivity is induced by the condensation of Cooper pairs formed by electronboson interactions (or strong local magnetic superexchange), such as electronphonon coupling in the BardeenCopperSchrieffer (BCS) theory. The superconducting gap that protects the superconducting condensate from exciting the quasiparticles is momentum dependent and is intimately related to the underlying pairing interactions. A wave gap has been observed in cuprates in revealing the unconventional mechanismTsuei (). While accidental nodes are sometimes observed in iron pnictide superconductors reflecting a basic feature of the extended swaveMazinPRL (); KurokiPRL (); hirschfeldReview (). At the gap nodes, the magnitude of a superconducting gap is zero and the signs are opposite at two sides of the nodal point. The nodal gaps, if enforced by symmetry, would suggest a pairing mechanism with repulsive interactions, such as that in the cuprates. Therefore to determine the gap structures and know whether the gap is nodal or nodeless is an effective method to check or even prove the mechanism of superconductivity. Theoretically it is understood that the lowenergy quasiparticle (QP) excitations for nodal gap superconductors is much stronger than that of the convention superconductors with swave gaps and the magnetic field induced generation of QP density of states (DOS) has very different field dependence. Since the discovery of ironbased superconductors, nodal gap(s) were suggested or indicated in several superconducting systems, such as LaFePOLaFePO (), BaFeAsPBaFeAsP (), and KFeAsKFeAs1 (); KFeAs2 (); KFeAs3 (). Although early works about the specific heat and thermal conductivity in FeSe suggested multiple nodeless gapsFeSeSH (); FeSeHT (), the recent STM works on FeSe single crystal and FeSe film supported that there are line nodes in those samplesQKXue (); Hanaguri ().
In this paper, we report the investigations on low temperature specific heat in FeS crystals. A linear relationship between C/T and T is observed in the low temperature limit, this is deviating from the exponential relation expected for swave gaps and indicates nodal superconducting gap(s). It is further found that the fitting to the electronic specific heat by a model with two nodal gaps is better than that by adopting either a single s or dwave gaps or a mixture of swave and dwave gaps. Furthermore, magnetic field induced electronic specific heat shows consistency with the Volovik relation approximatelyvolovik (), which was predicted specially for wave superconductors with line nodes. The analysis of low temperature specific heat reveals the presence of nodal superconducting gap(s) in this system, which is in accordance with the recent heat transport experimentFeSHT ().
Ii Experimental Method
The FeS crystals were fabricated by a hydrothermal method which has been presented beforeHLin (). The resistivity and low temperature specific heat were measured with a Quantum Design instrument PPMS16T, and the magnetization was detected by a Quantum Design instrument SQUIDVSM. The specific heat measurements were done with the thermal relaxation method by He3 accessory of PPMS16T with an advanced single crystal quatz platform which has a quite weak magnetic field effect. All the data in this paper were measured on samples from the same batch FeS crystals.
Iii Results and Discussion
Fig. 1 shows the temperature dependence of resistivity at zero field. As can be seen in the lower right inset, a sharp resistive transition occurs with the onset transition temperature of 4.55 K, and the zero resistivity appears at about 4.15 K. The normal state resistivity shows a monotonous decreasing with temperature from 300K to 4.55K, which indicates metallic conductivity. The residual resistivity ratio (RRR) defined as is about 60, indicating a good quality of the sample. This result is similar to the former reported valueHLin (). The upper left inset shows the temperature dependence of magnetic susceptibility measured in the zerofieldcooled (ZFC) and the fieldcooled (FC) mode. A rough estimate on the magnetic signal yields a full superconducting screening volume. At an external magnetic field of 5Oe the diamagnetic transition starts near 4.5K, which is accordance with the resistivity measurement very well.
The raw data of specific heat at 0T and 0.75T are shown in Fig. 2. An obvious superconducting anomaly appears at about 4.5K, which is also consistent with the resistive and magnetization data. The residual specific heat coefficient at 0K and zero external magnetic field is about . Although this value is rather low, but comparing with the normal state Sommerfeld constant , it is quite sizable. As the magnetic field is increased to 0.75T, the superconducting anomaly is already suppressed to below 0.4K and the extracted is about 3.80.2 . We try to fit the normal state specific heat data with the Debye model in low temperature region, however, the results hardly match the data well if using only the term . We thus use the following model that contains three terms to fit the data from 0.4K to 15K. The fitting formula reads
(1) 
The fitting curve as shown by the solid line in Fig.2 is better than the simple model and yields parameters , , and . Using the relation between Debye temperature and the fitting parameter , , where = 6.02 mol is the Avogadro constant, Z = 2 is the number of atoms in one unit cell, we can calculate the Debye temperature of FeS, which is about 215K. The Debye temperature derived here for FeS is close to = 210K obtained previously for FeSe single crystalsFeSeSH (). The seems to be smaller than that of other ironbased superconductors, for example, in FeSe single crystals = 5.730.19 FeSeSH (). According to the density functional calculationsSingh (), it is found that the DOS in FeSe and FeS are quite close to each other. The smaller finds here may suggest weaker electronboson coupling or correlation effect in FeS. We present the specific heat under different magnetic fields in Fig. 3(a). By applying several magnetic fields up to 0.75T, the superconducting anomaly shifts to lower temperatures gradually and is hardly observed above 0.4K when the magnetic field is over 0.75T.
The superconducting electronic contribution can be obtained by deducting the phonon contribution below . From Fig. 3 it is clear that the normal state specific heat does not depend on the magnetic field. Since a weak magnetic field 0.75T is enough to suppress the superconductivity in FeS, we can obtain the electronic specific heat through formula C=C(0T)C(0.75T). The electronic specific coefficient C derived in this way is shown in Fig. 3(b). It is found that the superconducting anomaly is not high, although the transition is rather sharp. Considering the entropy conservation near the superconducting transition, we determined the superconducting transition temperature T = 4.3K from the the specific heat, and the jump is . This leads to the ratio 0.65. The value is much lower than the predicted one by the BCS theory in the weak coupling regime which gives =1.43. This is also rather lower than 1.65 obtained for FeSe single crystalsFeSeSH (). The specific heat anomaly in FeS is much lower than that in KFeAs although the transition temperature are close to each otherHardy (). This value of reveals that FeS system may be a superconductor with rather weak electronboson coupling and correlation effect. In KFeAs the Hund’s rule coupling becomes stronger, this is assumed to be the origin of the correlation effectHardy (). One can see that a tail appears in the specific heat data above T, which may be caused by the superconducting fluctuation. The inset of Fig. 3(b) shows the enlarge view of the low temperature specific heat data by fitting to a straight line. Interestingly, the electronic specific heat coefficient varies linearly with temperature in the low temperature regime. This relation is deviated from an exponential dependence as expected for an swave gap and can be explained by models containing nodal gaps. To investigate the details concerning superconducting gap(s), we use the BCS formula to fit our electronic specific heat data in the superconducting state
(2) 
where . We try to use four different gap structures to fit the electronic specific heat data: (1) single swave gap ; (2) single anisotropic gap with dwave feature ; (3) mixture of two components with one swave and one dwave gap; (4) mixture of two dwave gaps. If more than one gap is concerned in the calculation, a linear combination of the two components is used to get the electronic specific heat, namely with the fraction contributed by the component of . The fitting parameters are listed in table 1. In Fig. 3(b) we can see that a simple swave or dwave can not fit the data, although the fitting with a single dwave gap is better. A model with the mixture of two dwave gaps, as shown by the black solid line, gives the best fit, albeit a slight deviation in the low temperature region. The model with a mixture of swave and dwave, but with large fraction of dwave component could also describe the data roughly. The detailed fitting reveals that there should be anisotropic gaps with nodes in the FeS system. From previous studies in FeSeFeSeSH (), it was shown that the electronic specific heat could be described by a model with one anisotropic swave component, but with no sign of nodes in FeSe. However, recent scanning tunneling spectroscopy (STS) measurements on FeSe crystalsHanaguri () or thick filmQKXue () show that line nodes exist in this system. These different gap structures of FeSe need more time to be resolved.
To make the conclusion more convincing, we also derived the magnetic field dependence of . The is obtained by extending the low temperature specific heat to 0K at different magnetic fields. Fig. 4 shows the relation between and magnetic field at T = 0K. It is clear that is close to the Volovik relation, namely , which indicates again the presence of gap nodes. Actually the observations of and present selfconsistent evidence for the existence of line nodes, which gives a further enforcement on the gap structure comparing with the earlier thermal transport measurementFeSHT (). We should emphasize, however, that we could not distinguish whether the gap nodes are induced by the sign change of the gap, like in a dwave case, or it is induced by the accidental nodes as theoretically predicted for the FeAs and FeSe based superconductorshirschfeldReview (). This needs to be resolved by more investigations.
model  (meV)  p  (meV)  1p 

0.5  100%      
0.6  100%      
0.6(s)  23%  0.5(d)  77%  
0.6  91%  1.0  9% 
In ironbased superconductors a power law correlation , named as the BNC relation, was discovered for many iron based superconductorsBNC (). In addition, another relation between condensation energy and superconducting transition temperature was also reportedcondensation () with , which was explained by the quantum critical phenomenonZaanen (). So this inspires us to investigate whether the new superconductor FeS is also satisfied with these relations. Interestingly we find that the power law relation is roughly obeyed if we put the data of FeS on the general plot. From the electronic specific heat data of FeS, we calculate the condensation energy through integrating the entropy of superconducting state and normal state by the formula below
(3) 
The calculated condensation energy of FeS is about 7.6mJ/mol. In Fig. 5 we present the condensation energy of FeS and other ironbased superconductors measured previously in our groupcondensation () together with that of element La. Intriguingly, the data of FeS falls into the power law relation perfectly. This may hint on that the FeS superconductor has similar mechanism as other FeAs or FeSebased superconductors. However, both the specific heat anomaly and in FeS are much lower than that in FeSe crystalsFeSeSH (). We must point out that either the BNC scalingBNC () or the relation between the condensation energy and condensation () do not apply to the KFeAs since it has a very large specific heat anomaly, which may suggest a strong renormalization of electron mass in that systemHardy (). In order to have comparison with a conventional superconductor, we also calculate the condensation energy of element La with T = 5K which is similar to that of FeSLacondensationenergy (). Although the T of La is slightly higher than FeS, the condensation energy of La is about 54mJ/mol, which is almost 710% of FeS. Therefore the data of La element deviates clearly from the power law that is satisfied by many iron based superconductors. This may suggest the different mechanism of the iron based superconductors and the conventional metal superconductors.
Iv Summary
In summary, we have successfully grown the crystals of the newly discovered superconductor FeS and measured the low temperature specific heat. The electronic specific heat was obtained by removing the phonon contribution. A linear relation between the electronic specific heat and temperature was found in the low temperature region. By fitting to the theoretical models with different gap structures, we find that the model with nodal gap(s) fits the data best. The magnetic field induced specific heat shows the consistency with the Volovik relation . All these suggest the existence of nodes on the gap structure. Our results will stimulate the investigations on the superconductor FeS and helps to resolve the intimate superconductivity mechanism in all iron based superconductors.
Acknowledgements.
We appreciate the discussions with S. Y. Li. This work is supported by the NSF of China, the Ministry of Science and Technology of China (973 projects: 2011CBA00102, 2012CB821403, 2010CB923002), and PAPD.hhwen@nju.edu.cn
References
 (1) Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, IronBased Layered Superconductor La(OF)FeAs (x=0.050.12)with T=26 K, J. Am. Chem. Soc. 130, 3296 (2008).
 (2) G. R. Stewart, Superconductivity in iron compounds, Rev. Mod. Phys. 83, 1589(2011)
 (3) J. Paglione and R. Greene, Hightemperature superconductivity in ironbased materials, Nat. Phys, 6, 645(2010).
 (4) H. H. Wen and S. Li, Materials and novel superconductivity in iron pnictide superconductors, Annu. Rev. Condens. Matter Phys. 2, 121 (2011).
 (5) Z. Ren, W. Lu, J. Yang, W. Yi, X. Shen, Z. Li, G. Che, X. Dong, L. Sun, F. Zhou, Z. X. Zhao, Superconductivity at 55 K in ironbased Fdoped layered quaternary compound Sm(OF)FeAs, Chin. Phys. Lett. 25, 2215 (2008)
 (6) L. Sun, X. Chen, J. Guo, P. Gao, Q. Huang, H. Wang, M. Fang, X. Chen, G. Chen, Q. Wu, C. Zhang, D. Gu, X. Dong, L. Wang, K. Yang, A. Li, X. Dai, Hokwang Ma and Z. Zhao, Reemerging superconductivity at 48 kelvin in iron chalcogenides, Nature 483, 6 (2012).
 (7) W. L. McMillan, Transition Temperature of StrongCoupled Superconductors, Phys. Rev. 167, 331 (1968).
 (8) I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Unconventional Superconductivity with a Sign Reversal in the Order Parameter of LaFeAsOF, Phys. Rev. Lett. 101, 057003 (2008).
 (9) K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki, Unconventional Pairing Originating from the Disconnected Fermi Surfaces of Superconducting LaFeAsOF, Phys. Rev. Lett. 101, 087004 (2008).
 (10) T. Hanaguri, S. Niitaka, K. Kuroki and H. Takagi, Unconventional swave superconductivity in Fe(Se,Te), Science 328, 474(2010).
 (11) A. Christianson, E. Goremychkin, R. Osborn, S. Rosenkranz, M. Lumsden, C. Malliakas, I. Todorov, H. Claus, D. Chung, M. Kanatzidis, R. Bewley and T. Guidi, Unconventional superconductivity in BaKFeAs from inelastic neutron scattering, Nature 456, 930 (2008).
 (12) H. Yang, Z. Wang, D. Fang, Q. Deng, QiangHua Wang, Y. Xiang, Y. Yang and HaiHu Wen, Ingap quasiparticle excitations induced by nonmagnetic Cu impurities in Na(FeCoCu)As revealed by scanning tunnelling spectroscopy, Nature Communication 4, 2749 (2013).
 (13) X. Lai, H. Zhang, Y. Wang, X. Wang, X. Zhang, J. Lin,and F. Huang, Observation of Superconductivity in Tetragonal FeS, J. Am. Chem. Soc. 137, 10148 (2015).
 (14) F. Hsu, J. Luo, K. Yeh, T. Chen, T. Huang, P. Wu, Y. Lee, Y. Huang, Y. Chu, D. Yan and MawKuen Wu, Superconductivity in the PbOtype structure ¦ÁFeSe, PNAS 105, 14262(2008).
 (15) L. Zhang, D. J. Singh, and M. H. Du, Density functional study of FeS, FeSe, and FeTe: Electronic structure, magnetism, phonons, and superconductivity, Phys. Rev. B. 78, 134514 (2015)
 (16) C. Tsuei and J. Kirtley, Pairing symmetry in cuprate superconductors, Rev. Mod. Phys. 72, 969 (2000).
 (17) P. Hirschfeld, M. Korshunov and I. Mazin, Gap symmetry and structure of Febased superconductors, Reports on Progress in Physics, 74, 124508 (2011).
 (18) J. D. Fletcher, A. Serafin, L. Malone, J. G. Analytis, J.H. Chu, A. S. Erickson, I. R. Fisher, and A. Carrington, Evidence for a NodalLine Superconducting State in LaFePO, Phys. Rev. Lett. 102, 147001(2009).
 (19) Y. Zhang, Z. R. Ye, Q. Q. Ge, J. Chen, F. Jiang, M. Xu, B. P. Xie, and D. L. Feng, Nodal superconductinggap structure in ferropnictide superconductor BaFe(AsP), Nat. Phys 8, 371 (2012).
 (20) K. Okazaki, Y. Ota, Y. Kotani, W. Malaeb, Y. Ishida, T. Shimojima, T. Kiss, S. Watanabe, C.T. Chen, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa, Y. Kohori, K. Hashimoto, T. Shibauchi, Y. Matsuda, H. Ikeda, H. Miyahara, R. Arita, A. Chainani, S. Shin, Octetline node structure of superconducting order parameter in KFeAs, Science, 337, 13141317 (2012).
 (21) H. Fukazawa, Y. Yamada, K. Kondo, T. Saito, Y. Kohori, K. Kuga, Y. Matsumoto, S. Nakatsuji, H. Kito, P. M. Shirage, K. Kihou, N. Takeshita, C.H. Lee, A. Iyo, and H. Eisaki, Possible Multiple Gap Superconductivity with Line Nodes in Heavily HoleDoped Superconductor KFe2As2 Studied by 75As Nuclear Quadrupole Resonance and Specific Heat, J. Phys. Soc. Jpn. 78, 083712 (2009).
 (22) K. Hashimoto, A. Serafin, S. Tonegawa, R. Katsumata, R. Okazaki, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, H. Ikeda, Y. Matsuda, A. Carrington, and T. Shibauchi, Evidence for superconducting gap nodes in the zonecentered hole bands of KFe2As2 from magnetic penetrationdepth measurements, Phys. Rev. B 82, 014526 (2010).
 (23) J.Y. Lin, Y. S. Hsieh, D. A. Chareev, A. N. Vasiliev, Y. Parsons, and H. D. Yang, Coexistence of isotropic and extended swave order parameters in FeSe as revealed by lowtemperature specific heat, Phys. Rev. B 84, 220507(R) (2011).
 (24) J. K. Dong, T. Y. Guan, S. Y. Zhou, X. Qiu, L. Ding, C. Zhang, U. Patel, Z. L. Xiao, and S. Y. Li, Multigap nodeless superconductivity in FeSe: Evidence from quasiparticle heat transport, Phys. Rev. B 80, 024518 (2009).
 (25) F. Hardy, A. Böhmer, D. Aoki, P. Burger, T. Wolf, P. Schweiss, R. Heid, P. Adelmann, Y. Yao, G. Kotliar, J. Schmalian, and C. Meingast, Evidence of Strong Correlations and CoherenceIncoherence Crossover in the Iron Pnictide Superconductor KFeAs, Phys. Rev. Lett. 111, 027002 (2013).
 (26) C.L. Song, Y.L. Wang, P. Cheng, Y.P. Jiang, W. Li, T. Zhang, Z. Li, K. He, L. Wang, J.F. Jia, H.H. Hung, C. Wu, X. Ma, X. Chen, Q.K. Xue, Direct Observation of Nodes and Twofold Symmetry in FeSe Superconductor, Science, 332, 1410 (2011).
 (27) T. Watashige, Y. Tsutsumi, T. Hanaguri, Y. Kohsaka, S. Kasahara, A. Furusaki, M. Sigrist, C. Meingast, T. Wolf, H.v.Lihneysen, T. Shibauchi, and Y. Matsuda, Evidence for TimeReversal Symmetry Breaking of the Superconducting State near TwinBoundary Interfaces in FeSe Revealed by Scanning Tunneling Spectroscopy, Phys. Rev. X 5, 031022 (2015).
 (28) G. E. Volovik, Superconductivity with lines of gap nodes: density of states in the vortex, JETP LETTERS 58, 469 (1993).
 (29) T. P. Ying, X. F. Lai, X. C. Hong, Y. Xu, L. P. He, J. Zhang, M. X. Wang, Y. J. Yu, F. Q. Huang, and S. Y. Li, Nodal superconductivity in FeS: Evidence from quasiparticle heat transport, arXiv:1511.07717 (2015).
 (30) H. Lin, Y. Li, J. Xing, J. Liu, X. Zhu, H. Yang, HaiHu Wen, Multiband superconductivity and large anisotropy in FeS crystals, arXiv:1511.08716 (2015).
 (31) S. L. Bud’ko, N. Ni, P. C. Canfield, Jump in specific heat at the superconducting transition temperature in Ba (FeCo)As and Ba(FeNi)As single crystals, Phys. Rev. B 79, 220516(2009).
 (32) J. Xing, S. Li, B. Zeng, G. Mu, B. Shen, J. Schneeloch, R. D. Zhong, T. S. Liu, G. D. Gu, and HaiHu Wen, Powerlawlike correlation between condensation energy and superconducting transition temperatures in iron pnictide/chalcogenide superconductors: Beyond the BCS understanding, Phys. Rev. B, 89, 140503(R) (2014).
 (33) J. Zaanen, Specificheat jump at the superconducting transition and the quantum critical nature of the normal state of pnictide superconductors, Phys. Rev. B 80, 212502(2009).
 (34) P. Pan, D. Finnemore, A. Bevolo, H. Shanks, B. Beaudry, F. Schmidt and G. Danielson, Heat capacity of highpurity lanthanum, Phys. Rev. B 21, 2809(1980).