Coexistence of isotropic and extended s-wave order parameters in FeSe as revealed
by low-temperature specific heat
J.-Y. Lin,1Y. S. Hsieh,1D. A. Chareev,2A. N. Vasiliev,3Y. Parsons,4and H. D. Yang5
1Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan 2Institute of Experimental Mineralogy, Chernogolovka, Moscow Region 142432, Russia 3Department of Low Temperature Physics, Moscow State University, Moscow 119991, Russia
4Department of Physics, University of California, Santa Barbara, California 93106, USA 5Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
(Received 8 September 2011; revised manuscript received 1 December 2011; published 27 December 2011) The comprehensive low-temperature specific-heat C(T ) data identify both an isotropic s-wave and extended s-wave order parameters coexisting in a superconducting single-crystal FeSe with Tc= 8.11 K. The isotropic gap
0= 1.33 meV on the hole Fermi sheets and the extended s-wave gap = e(1+αcos2φ) with e= 1.13 meV and α = 0.78 on the electron Fermi sheets. The extended s wave is rather anisotropic, but the low-energy quasiparticle excitations demonstrate no sign of the accidental nodes. The coefficient γ (H ) manifesting the quasiparticle contribution to C is a nonlinear function of the applied magnetic field H in the mixed state in accord with the anisotropic multiorder parameters.
DOI:10.1103/PhysRevB.84.220507 PACS number(s): 74.25.Bt, 74.25.Op, 74.70.Xa
The discovery of Fe-based high-temperature superconduc-tivity in 2008 has been the most significant event in the field since cuprate superconductors and MgB2.1Alongside the multiband electronic structure similar to that of MgB2 (and in contrast to the one-band physics in cuprates),2,3 there is keen competition between superconductivity and magnetism in Fe-based superconductors.4 These new superconductors
are likely owing to unique superconducting mechanisms, and invite rethinking on other superconductors, including cuprates. As always, the superconducting order parameter is one of the core elements toward full understanding of the superconducting mechanism. The initial proposal was that the Cooper pairs are glued by the spin excitations generated from Fermi-surface nesting.5 This pairing mechanism would lead
to nearly isotropic order parameters with opposite signs on hole and electron pockets.5 Early on, the angular resolved photoemission spectroscopy reported nearly isotropic order parameters on both hole and electron Fermi sheets, consistent with the above scenario.6 However, later nuclear magnetic
resonance and thermal conductivity experiments reported existence of nodal order parameters, at least in certain 122 Fe-based superconductors.7–9Moreover, a recent angular
resolved specific-heat study of FeSe0.45Te0.55suggests a very anisotropic order; whether it is with nodes or not remains unclear.10Indeed, very recent model calculations demonstrate,
within reasonable physical parameter range, either d- or s-wave order parameter could be stabilized.11–13For the latter case, the hole bands could host an isotropic order parameter opposite to an anisotropic one on the electron bands. The critical temperature Tc of FeSe scales with the density of
states (DOS) at the Fermi energy N (EF) compared to those
of other Fe-based superconductors.14 Although with lower Tc, FeSe has the simplest structure, and this very simplicity
could provide the most appropriate venue of understanding the superconducting mechanism of Fe-based superconductors.15
In this Rapid Communication, we resolve the currently debated issue of the order parameter in FeSe by the com-prehensive low-temperature specific-heat data of high-quality
single-crystalline FeSe. These data unambiguously prove the coexistence of an isotropic s wave and a very anisotropic extended s-wave in an Fe-based superconductor. Furthermore, this anisotropic order parameter has no accidental nodes. This work also better elucidates the anisotropy of both γ(H ) manifesting the quasiparticle contribution to C and the upper critical field Hc2 of FeSe, both of which were rarely appropriately explored in the literature.
FeSe single crystals were grown in evacuated quartz ampoules using a KCl/AlCl3 flux. The samples prepared by the KCl/AlCl3 flux method were found with no impurities. The structure of tetragonal P 4/nmm was demonstrated at room temperature by x-ray diffraction. The homogeneity range of tetragonal FeSe1−xis from FeSe0.96to FeSe0.965.16The present sample has the mass of 2.27 mg and its composition is FeSe0.963 as proved by an average of the measurements of the energy-dispersive x-ray spectroscopy performed on a CAMECA SX100 (15 keV) analytical scanning electron microscope. The low-temperature specific heat C(T ,H ) was measured with a3He heat-pulsed thermal relaxation calorimeter in the temperature range from 0.5 to 15 K by applying magnetic fields with H⊥c and H//c up to H = 9 T, respectively.
Figure 1 shows both the zero field and the mixed state specific heat C(T ,H ) of FeSe plotted as C/T versus T2with the magnetic field H varying from 0 to 9 T. The original C versus T data at H = 0 are shown in the inset of Fig.1. The specific heat jump associated with the superconducting tran-sition is obvious. The superconducting trantran-sition temperature Tcis 8.11 K as determined by the local entropy balance. C/T
approaches zero at T= 0, as will be seen more clearly in Fig.3. The coefficient γ at the linear-T term of C(T ) should be absent in the superconducting state. This absence of the so-called “residual γ ” indicates high quality of the present sample with a complete superconducting volume, and help to avoid many complications encountered in the previous specific-heat studies on Fe-based superconductors.17 The data in H allow
the normal state Cn(T ) to be determined more reliably. Cn(T ) below 10 K can be well described by the simple expression
FIG. 1. (Color online) C/T vs T2with H from 0 to 9 T. The solid line represents the normal state Cn(T )= γnT+βT3determined from the normal state data of all fields. The inset presents the original C vs T data taken at H= 0.
Cn(T ) = γnT+βT3 where γnT is the normal electronic contribution and βT3represents the phonon contribution. The
resultant γn = 5.73 ± 0.19 mJ/mol K2 and β = 0.421 ±
0.002 mJ/mol K4 (the corresponding Debye temperature is 210 K). Both are consistent with previous results from the polycrystalline samples.18The superconducting electronic
contribution Ces(T ) can be obtained by Ces(T )= C(T )-βT3. Ces(T )/T versus T /Tcfor H = 0 is shown in Fig.2. The
en-tropy conservation required for a second-order phase transition is fulfilled as shown in the inset of Fig.3. This check warrants the thermodynamic consistency for both the measured data and the determination of Cn(T ). By the balance of entropy around the transition, the dimensionless specific-heat jump δC/γnTc= 1.65 at Tc is determined compared with the BCS
value of 1.43. Without any model fitting, this value has already implied a moderate or stronger coupling in FeSe.
The data are obviously not reconcilable with the conven-tional s-wave order parameter due to the significant quasiparti-cle contribution at low T . [For a typical example of Ces(T )/T of a conventional superconductor, see Ref.19.] Actually, the quasiparticle contribution to Ces(T ) at low T is even larger than that of the well-known two-gap superconductor MgB2.20,21To elucidate the order parameter of FeSe, data in Fig. 2 were fit into various models. The cases of d wave, extended s wave, two-gap (S±), and s+ (extended s wave) (s + ES) are shown in Figs. 2(a), 2(b), 2(c), and 2(d), respectively. The order parameters used to fit the data are = 0cos2φ for
d wave and = e(1+αcos2φ) (where α denotes the gap
anisotropy) for the extended s wave. In the two-gap model, two (assumed) isotropic order parameters L and S are introduced as in the previous works.22–24 In the s+ ES case,
= 0 for an isotropic s wave is assumed. These four
cases are all allowed from the symmetry aspect and model calculations.11–13 Considering the whole temperature range,
FIG. 2. (Color online) Superconducting electronic Ces/Tfit by (a) d wave; (b) extended s wave; (c) two gap (S±); (d) s+ extended s-wave models. For the formula of the order parameters in each case, see text. The insets denote the deviations of the fits from the experimental data.
FIG. 3. (Color online) The comparison of various models focused on the very low-temperature regime. The inset shows the entropy conservation required for a second-order phase transition and justifies the determination of Cn(T ) in Fig.1. The error bars of data near 0.5 and 2 K were depicted. The absolute error at low T is about 6% as denoted by the error bar. The relative error can be seen from the slight scattering of the data. (We performed two measurements at each temperature.) The uncertainty from the subtracted phonon term is negligible since the phonon contribution itself is small at∼0.5 K. On the other hand, the absolute error at T ∼ 2 K is about 2%, and the uncertainty from the subtracted phonon term contributes about one third of the error bar.
the s+ES and the two-gap models lead to more satisfactory fits of the data than either the d wave or extended s wave does. In order to distinguish between the s+ ES and two-gap models, further insight into the quasiparticle excitations below 0.25 T /Tc is provided by Fig. 3. Thanks to the precision
of the measurements, the fitting is surprisingly selective. Conclusively, the s+ Es model leads to the most preferable fit among alternatives. Since α = 0.78, the results illustrate a very anisotropic order parameter existing in FeSe. This conclusion is consistent with the very recent angular resolved specific-heat measurements on FeSe0.45Te0.55.10Furthermore, this work excludes the scenarios of the d-wave line nodes as shown in Fig.3, an issue which could not be concluded in the previous literature.10From the theoretical side, the robustness
TABLE I. The parameters derived from the fits in Figs.2 and3. For the s-wave+ e(coskx+ cosky) model, kr denotes
the radius of the electron pockets in the unit of 1/a where a is the lattice constant.
Order parameter Energy gap (meV) Weight (%) α
d-wave 0= 1.93 Extended s-wave e= 1.25 0.64 Two-gap L= 1.55 71 S= 0.45 29 s-wave+ 0= 1.33 33 Extended s-wave e= 1.13 67 0.78 s-wave+ 0= 1.62 63 kr= 1.12 e(coskx+cosky) e= 1.49 37
of the s+ ES model has been confirmed very recently.25 An
αapproaching 1 (or >1) would generate too much low-energy excitations to reconcile with the present experimental data (fitting not shown).
Intriguingly, there have been two scanning tunneling microscopy (STM) studies of Fe(Se,Te) and FeSe, respectively (see Refs. 26 and 27). These two separate works reported nodes in FeSe and no node in Fe(Se,Te), respectively. Since the minimum of the present anisotropic gap is∼0.2 meV, the observation of no accidental node might not contradict the ex-perimental data of STM in Fe(Se,Te). The gap maxima e(1+ α)= 2.01 meV is also close to the maxima gap ∼2.2 meV in FeSe by STM. To have a quantitative comparison to the nodal scenario in Ref.27, the zero field Ces(T ) was fit to an isotropic gap 0(representing the order parameter on the hole pockets)
combined with an extended s-wave order parameter =
e(coskx + cosky), which results in nodes on the electron
pockets. The fitting curve was denoted as the green line in Fig.3, which is apparently inferior to that of s+ES, albeit
slightly better than that of the nodal d wave. Consideration of = 1coskxcosky + 2(coskx + cosky) for the whole
FIG. 4. (Color online) (a) The mixed state quasiparticle contri-bution γ (H ) for H //c and H⊥c data. The solid lines denote the phenomenological linear fits above H = 1 T. The dashed line on the top denotes the value of γn. (b) Hc2(T ) with H //c and H⊥c. The inset shows data in the whole range.
unfolded Brillouin zone can not lead to any satisfactory fit. Therefore, the scenario of accidental nodes is as unlikely in FeSe. However, if the nodes cross the portion of the Fermi surface with the low DOS, the quasiparticle contribution to Ces(T ) will be small and the nodal scenario could reconcile with the low-temperature data in Fig.3. The resultant fitting parameters for various models are summarized in Table I. With a simplified two-band model in Fig.2(d), rather than the more realistic four-band model, how much the ratio of γs/γES = 33%:67% reflects the exact physical parameters remains to be seen. However, this apparently imperfect match might undermine the effect of Fermi-surface nesting on the magnetic ordering and help pairing states win over the spin density wave in FeSe.
It is noted that the recent angular resolved thermal con-ductivity study proposes a nodal structure in the 122 system, together with other work as mentioned above. Evidence of nodes in these pnictides might not contradict the present results since the model calculations have demonstrated that the gap symmetry is very sensitive to the model parameters and may vary among the pnictide family.11–13 As for the previously
focused two-gap model in Fig. 2(c), the values of 2L/kTc
and 2s/kTcare different from those of FeTe0.55Se0.45,28 but are nearly identical to those of FeSe probed by Andreev spectroscopy at 4 K.29 The qualitative results of multigap
nodeless superconductivity in FeSe were reported earlier,30
although the extended s-wave feature and the magnitude of the order parameters were elusive then.
Figure4(a)shows the mixed state linear coefficient γ (H ) as obtained from the empirical fit of the in-field data between 0.5 and 2 K by C(T ,H )/T = γ (H ) + aT + bT2where γ (H ), a, and b are fitting parameters. To illustrate the anisotropy
in FeSe, γ (H ) with H //c and H⊥c [C(T ,H⊥c)/T data
not shown] was depicted in Fig.4(a). Previously, γ (H ) was thought to be linear with respect to H for the isotropic s-wave pairing and proportional to H1/2for line nodes (see discussions in Refs.17and18). In general, γ (H ) is very similar to that of BaFe2(As0.7P0.3)2for H up to 35 T.31In low fields, γ (H ) shows a pronounced curvature. In high fields, γ (H ) is quasi-linear. The dashed lines are the linear fits for H //c and H⊥c data above H = 1 T. Knowing that γn = 5.73/mJ/mol K2,
the upper critical field can be estimated as Hc2,H //c≈ 13.1 T and Hc2,H⊥c ≈ 27.9 T. The anisotropy in Hc2 is about 2.1. Although this ratio is moderate, FeSe is already one of the most anisotropic Fe-based superconductors.32 For the whole field range, γ (H ) with either H //c or H⊥c is qualitatively in accord with S± or s+ ES.33,34 It is likely that γ (H ) in
Fig.1(d)of Ref.33would resemble the present results even more if the s+ ES was used in the calculations. Figure4(b)
shows Hc2(T ) determined by the local entropy balance with
TABLE II. Some fundamental properties of FeSe. γn: normal state electronic coefficient; D: Debye temperature; λ: electron-boson coupling constant; Hc2,H //c: upper critical field for H //c; Hc2,H⊥c: upper critical field for H⊥c.
γn(mJ/mol K2) D(K) δC/γnTc λ Hc2,H //c(T) Hc2,⊥(T)
5.73 210 1.65 1.55 13.1 27.9
H //c and H⊥c. For both sets of Hc2(T ), in contrast with
the linear T dependence near Tc for conventional
supercon-ductors, the results show a pronounced positive curvature with respect to T . This feature generally manifests multigap order parameters.22(Other interpretations can be seen in Ref.35.) It
is noted that the anisotropic Hc2(T ) was rarely explored in the literature. The present Hc2(T ) in Fig.4(b)shows a much more pronounced negative curvature with respect to T in high fields as compared with that in Ref.36.
Finally, γn of this work is compared with the bare γo = 2.24 mJ/mol K2from the band-structure calculations.14Since
γn= (1 + λ)γo, where λ is the total coupling strength between quasiparticle and bosons, λ= 1.55 is estimated. This coupling strength is slightly stronger than the moderate coupling. The electron-phonon coupling strength λep= 0.17 was calculated in Ref.14. Therefore, the quasiparticles in FeSe mainly couple with other gluons rather than phonons such as those from spin fluctuations. The value of λ= 1.55 is in semiquantitative agreement with both the magnitude of δC/γnTcand the energy
gap from fitting in TableI. The fundamental properties of FeSe are summarized in TableII.
To conclude, this work selects nodeless s+ ES from several alternative theoretical models. Furthermore, other precious normal state and mixed state physical properties are revealed. The low-temperature specific heat on high-quality single crystals, if well executed, is indeed a powerful tool for probing the order parameter in Fe-based superconductors.
This work was supported by National Science Council of Taiwan, ROC, under Grants No. NSC98-2112-009-005-MY3 and No. NSC100-2112-M110-004-MY3; by grant of the President of the Russian Federation No. MK-1557.2011.5, by the State Contract No. 11.519.11.6012 of Russian Ministry of Science and Education, and by the Russian Founda-tion for Basic Research under Grants No. 11-519-11-6012, No.10-02-90409, and No. 10-02-00021. Discussions with A. B. Vorontsov, W. Ku, and Y. K. Bang are appreciated. We would like to thank B. C. Lai, T. N. Dokina, A. A. Virus, K. V. Van, and A. N. Nekrasov for technical support.
1Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.
Chem. Soc. 130, 3296 (2008).
2I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du,Phys. Rev.
Lett. 101, 057003 (2008).
3D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003
(2008).
4C. de la Cruz, Q. Huang, J. W. Lynn, Jiying Li, W. Ratcliff II, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, and P. Dai,Nature (London) 453, 899 (2008).
5I. I. Mazin,Nature (London) 464, 183 (2010).
6T. Kondo, A. F. Santander-Syro, O. Copie, Chang Liu, M. E. Tillman, E. D. Mun, J. Schmalian, S. L. Bud’ko, M. A. Tanatar,
P. C. Canfield, and A. Kaminski, Phys. Rev. Lett. 101, 147003 (2008).
7J.-Ph. Reid, M. A. Tanatar, X. G. Luo, H. Shakeripour, N. Doiron-Leyraud, N. Ni, S. L. Bud’ko, P. C. Canfield, R. Prozorov, and L. Taillefer,Phys. Rev. B 82, 064501 (2010).
8Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, S. Kasahara, T. Shibauchi, Y. Matsuda, and T. Terashima,Phys. Rev. B 81, 020503(R) (2010).
9M. Yamashita, Y. Senshu, T. Shibauchi, S. Kasahara, K. Hashimoto, D. Watanabe, H. Ikeda, T. Terashima, I. Vekhter, A. B. Vorontsov, and Y. Matsuda,Phys. Rev. B 84, 060507(R) (2011).
10B. Zeng, G. Mu, H. Q. Luo, T. Xiang, I. I. Mazin, H. Yang, L. Shan, C. Ren, P. C. Dai, and H.-H. Wen,Nat. Commun. 1, 112 (2010).
11T. A. Maier, S. Graser, D. J. Scalapino, and P. J. Hirschfeld,Phys.
Rev. B 79, 224510 (2009).
12A. V. Chubukov, M. G. Vavilov, and A. B. Vorontsov,Phys. Rev. B
80, 140515(R) (2009).
13A. Nicholson, W. Ge, X. Zhang, J. Riera, M. Daghofer, A. M. Ole´s, G. B. Martins, A. Moreo, and E. Dagotto,Phys. Rev. Lett. 106, 217002 (2011).
14A. Subedi, L. Zhang, D. J. Singh, and M. H. Du,Phys. Rev. B 78,
134514 (2008).
15F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang, P. M. Wu, Y.-C. Lee, Y.-L. Huang, Y.-Y. Chu, D.-C. Yan, and M.-K. Wu, Proc. Natl. Acad. Sci. USA 105, 14262 (2008).
16H. Okamoto,J. Phase Equilib. 12, 383 (1991).
17F. Hardy, P. Burger, T. Wolf, R. A. Fisher, P. Schweiss, P. Adelmann, R. Heid, R. Fromknecht, R. Eder, D. Ernst, H. v. L¨ohneysen, and C. Meingast,Euro. Phys. Lett. 91, 47008 (2010).
18T. M. McQueen, Q. Huang, V. Ksenofontov, C. Felser, Q. Xu, H. Zandbergen, Y. S. Hor, J. Allred, A. J. Williams, D. Qu, J. Checkelsky, N. P. Ong, and R. J. Cava,Phys. Rev. B 79, 014522 (2009).
19C. P. Sun, J.-Y. Lin, S. Mollah, P. L. Ho, H. D. Yang, F. C. Hsu, Y. C. Liao, and M. K. Wu,Phys. Rev. B 70, 054519 (2004).
20H. D. Yang, J.-Y. Lin, H. H. Li, F. H. Hsu, C. J. Liu, S.-C. Li, R.-C. Yu, and C.-Q. Jin, Phys. Rev. Lett. 87, 167003 (2001).
21F. Bouquet, R. A. Fisher, N. E. Phillips, D. G. Hinks, and J. D. Jorgensen,Phys. Rev. Lett. 87, 047001 (2001).
22C. L. Huang, J.-Y. Lin, C. P. Sun, T. K. Lee, J. D. Kim, E. M. Choi, S. I. Lee, and H. D. Yang,Phys. Rev. B 73, 012502 (2006).
23C. L. Huang, J.-Y. Lin, Y. T. Chang, C. P. Sun, H. Y. Shen, C. C. Chou, H. Berger, T. K. Lee, and H. D. Yang,Phys. Rev. B 76, 212504 (2007).
24F. Bouquet, Y. Wang, R. A. Fisher, D. G. Hinks, J. D. Jorgensen, A. Junod, and N. E. Phillips,Europhys. Lett. 56, 856 (2001).
25V. Stanev, B. S. Alexandrov, P. Nikoli´c, and Z. Teˇsanovi´c,Phys.
Rev. B 84, 014505 (2011).
26T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi,Science 328,
474 (2010).
27C. L. Song, Y.-L. Wang, P. Cheng, Y.-P. Jiang, W. L., T. Zhang, Z. Li, K. He, Lili Wang, J.-F. Jia, H.-H Hung, Congjun Wu, Xucun Ma, X. Chen, and Q.-K. Xue,Science 332, 1410 (2011).
28C. C. Homes, A. Akrap, J. S. Wen, Z. J. Xu, Z. W. Lin, Q. Li, and G. D. Gu,Phys. Rev. B 81, 180508(R) (2010).
29Ya. G. Ponomarev, S. A. Kuzmichev, M. G. Mikheev, M. V. Sudakova, S. N. Tchnesnokov, T. E. Shanygina, O. S. Volkova, A. N. Vasiliev, and Th. Wolf,JETP 113, 459 (2011).
30J. K. Dong, T. Y. Guan, S. Y. Zhou, X. Qiu, L. Ding, C. Zhang, U. Patel, Z. L. Xiao, and S. Y. Li,Phys. Rev. B 80, 024518 (2009).
31J. S. Kim, P. J. Hirschfeld, G. R. Stewart, S. Kasahara, T. Shibauchi, T. Terashima, and Y. Matsuda,Phys. Rev. B 81, 214507 (2010); (unpublished).
32D. C. Johnston,Adv. Phys. 59, 803 (2010). 33Y. K. Bang,Phys. Rev. Lett. 104, 217001 (2010). 34Y. Wang and P. J. Hirschfeld (unpublished).
35J. T. Kao, J.-Y. Lin, and C. Y. Mou, Phys. Rev. B 75, 012503
(2007).
