Superfluid response in heavy fermion superconductors

Abstract

Motivated by a recent London penetration depth measurement [H. Kim, et al., Phys. Rev. Lett. 114, 027003 (2015)] and novel composite pairing scenario [O. Erten, R. Flint, and P. Coleman, Phys. Rev. Lett. 114, 027002 (2015)] of the Yb-doped heavy fermion superconductor CeCoIn₅, we revisit the issue of superfluid response in the microscopic heavy fermion lattice model. However, from the literature, an explicit expression for the superfluid response function in heavy fermion superconductors is rare. In this paper, we investigate the superfluid density response function in the celebrated Kondo–Heisenberg model. To be specific, we derive the corresponding formalism from an effective fermionic large-N mean-field pairing Hamiltonian whose pairing interaction is assumed to originate from the effective local antiferromagnetic exchange interaction. Interestingly, we find that the physically correct, temperature-dependent superfluid density formula can only be obtained if the external electromagnetic field is directly coupled to the heavy fermion quasi-particle rather than the bare conduction electron or local moment. Such a unique feature emphasizes the key role of the Kondo-screening-renormalized heavy quasi-particle for low-temperature/energy thermodynamics and transport behaviors. As an important application, the theoretical result is compared to an experimental measurement in heavy fermion superconductors CeCoIn5 and Yb-doped Ce_1−xYb_xCoIn₅ with fairly good agreement and the transition of the pairing symmetry in the latter material is explained as a simple doping effect. In addition, the requisite formalism for the commonly encountered nonmagnetic impurity and non-local electrodynamic effect are developed. Inspired by the success in explaining classic 115-series heavy fermion superconductors, we expect the present theory will be applied to understand other heavy fermion superconductors such as CeCu₂Si₂ and more generic multi-band superconductors.

Keywords

heavy fermion superconductor / Kondo lattice system / superfluid density

References

[1] M. Tinkham, Introduction to Superconductivity, New York: McGraw-Hill, 1996

[2] T. Xiang, d-wave superconductor, Beijing: Science Publisher, 2007 (in Chinese)

[3] C. P. Poole, R. Prozorov, H. A. Farach, and R. J. Creswick, Superconductivity, 3rd Ed., Amsterdam: Elsevier, 2014

[4] W. N. Hardy, D. A. Bonn, D. C. Morgan, R. Liang, and K. Zhang, Precision measurements of the temperature dependence of lin YBa₂Cu₃O_6.95: Strong evidence for nodes in the gap function, Phys. Rev. Lett. 70(25), 3999 (1993)

[5] M. S. Kim, J. A. Skinta, T. R. Lemberger, A. Tsukada, and M. Naito, Magnetic penetration depth measurements of Pr_2−xCe_xCuO_4−d films on Buffered substrates: Evidence for a nodeless gap, Phys. Rev. Lett. 91(8), 087001 (2003)

[6] R. Prozorov and V. G. Kogan, London penetration depth in iron-based superconductors, Rep. Prog. Phys. 74(12), 124505 (2011)

[7] R. J. Ormeno, A. Sibley, C. E. Gough, S. Sebastian, and I. R. Fisher, Microwave conductivity and penetration depth in the heavy fermion superconductor CeCoIn₅, Phys. Rev. Lett. 88(4), 047005 (2002)

[8] S. Özcan, D. M. Broun, B. Morgan, R. K. W. Haselwimmer, J. L. Sarrao, S. Kamal, C. P. Bidinosti, P. J. Turner, M. Raudsepp, and J. R. Waldram, London penetration depth measurements of the heavy-fermion superconductor CeCoIn₅ near a magnetic quantum critical point, Europhys. Lett. 62(3), 412 (2003)

[9] E. E. M. Chia, D. J. Van Harlingen, M. B. Salamon, B. D. Yanoff, I. Bonalde, and J. L. Sarrao, Nonlocality and strong coupling in the heavy fermion superconductor CeCoIn₅: A penetration depth study, Phys. Rev. B 67(1), 014527 (2003)

[10] K. Hashimoto, Y. Mizukami, R. Katsumata, H. Shishido, M. Yamashita, H. Ikeda, Y. Matsuda, J. A. Schlueter, J. D. Fletcher, A. Carrington, D. Gnida, D. Kaczorowski, and T. Shibauchi, Anomalous superfluid density in quantum critical superconductors, Proc. Natl. Acad. Sci. USA 110(9), 3293 (2013)

[11] C. J. S. Truncik, W. A. Huttema, P. J. Turner, S. Özcan, N. C. Murphy, P. R. Carrière, E. Thewalt, K. J. Morse, A. J. Koenig, J. L. Sarrao, and D. M. Broun, Nodal quasiparticle dynamics in the heavy fermion superconductor CeCoIn5 revealed by precision microwave spectroscopy, Nat. Commun. 4, 2477 (2013)

[12] L. Shu, D. E. MacLaughlin, C. M. Varma, O. O. Bernal, P. C. Ho, R. H. Fukuda, X. P. Shen, and M. B. Maple, Landau renormalizations of superfluid density in the heavy-fermion superconductor CeCoIn₅, Phys. Rev. Lett. 113(16), 166401 (2014)

[13] H. Kim, M. A. Tanatar, R. Flint, C. Petrovic, R. Hu, B. D. White, I. K. Lum, M. B. Maple, and R. Prozorov, Nodal to nodeless superconducting energy-gap structure change concomitant with Fermi-surface reconstruction in the heavy-fermion compound CeCoIn₅, Phys. Rev. Lett. 114(2), 027003 (2015)

[14] C. Petrovic, P. G. Pagliuso, M. F. Hundley, R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux, Heavy-fermion superconductivity in Ce- CoIn₅ at 2.3 K, J. Phys.: Condens. Matter 13(17), 337 (2001)

[15] R. Movshovich, M. Jaime, J. D. Thompson, C. Petrovic, Z. Fisk, P. G. Pagliuso, and J. L. Sarrao, Unconventional superconductivity in CeIrIn₅ and CeCoIn₅: Specific heat and thermal conductivity studies, Phys. Rev. Lett. 86(22), 5152 (2001)

[16] K. An, T. Sakakibara, R. Settai, Y. Onuki, M. Hiragi, M. Ichioka, and K. Machida, Sign reversal of field-angle resolved heat capacity oscillations in a heavy fermion superconductor CeCoIn₅ and d_x2−y2 pairing symmetry, Phys. Rev. Lett. 104(3), 037002 (2010)

[17] K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Angular position of nodes in the superconducting gap of quasi-2D heavy-fermion superconductor CeCoIn₅, Phys. Rev. Lett. 87(5), 057002 (2001)

[18] T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai and Y. Onuki, Unconventional heavy-fermion superconductor CeCoIn₅: dc magnetization study at temperatures down to 50 mK, Phys. Rev. B 65, 180504(R) (2002)

[19] Y. Kohori, Y. Yamato, Y. Iwamoto, T. Kohara, E. D. Bauer, M. B. Maple, and J. L. Sarrao, NMR and NQR studies of the heavy fermion superconductors CeTIn₅ (T=Co and Ir), Phys. Rev. B 64(13), 134526 (2001)

[20] S. Ernst, S. Wirth, F. Steglich, Z. Fisk, J. L. Sarrao, and J. D. Thompson, Scanning tunneling microscopy studies on CeCoIn₅ and CeIrIn₅, Phys. Status Solidi B 247(3), 624 (2010)

[21] C. Stock, C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic, Spin resonance in the d-wave superconductor CeCoIn₅, Phys. Rev. Lett. 100(8), 087001 (2008)

[22] M. P. Allan, F. Massee, D. K. Morr, J. Van Dyke, A. W. Rost, A. P. Mackenzie, C. Petrovic, and J. C. Davis, Imaging Cooper pairing of heavy fermions in CeCoIn₅, Nat. Phys. 9(8), 468 (2013)

[23] B. B. Zhou, S. Misra, E. H. da Silva Neto, P. Aynajian, R. E. Baumbach, J. D. Thompson, E. D. Bauer, and A. Yazdani, Visualizing nodal heavy fermion superconductivity in CeCoIn₅, Nat. Phys. 9(8), 474 (2013)

[24] J. Van Dyke, F. Massee, M. P. Allan, J. C. Davis, C. Petrovic, and D. K. Morr, Direct evidence for a magnetic f-electron mediated pairing mechanism of heavyfermion superconductivity in CeCoIn₅, Proc. Natl. Acad. Sci. USA 111(32), 11663 (2014)

[25] Y. Xu, J. K. Dong, L. I. Lum, J. Zhang, X. C. Hong, L. P. He, K. F. Wang, Y. C. Ma, C. Petrovic, M. B. Maple, L. Shu, and S. Y. Li, Universal heat conduction in Ce_1−xYb_xCoIn₅: Evidence for robust nodal d-wave superconducting gap, Phys. Rev. B 93(6), 064502 (2016)

[26] O. Erten, R. Flint, and P. Coleman, Molecular pairing and fully gapped superconductivity in Yb-doped Ce- CoIn₅, Phys. Rev. Lett. 114(2), 027002 (2015)

[27] C. M. Varma, K. Miyake, and S. Schmitt-Rink, London penetration depth of heavy-fermion superconductors, Phys. Rev. Lett. 57(5), 626 (1986)

[28] P. Coleman, A. M. Tsvelik, N. Andrei, and H. Y. Kee, Co-operative Kondo effect in the two-channel Kondo lattice, Phys. Rev. B 60(5), 3608 (1999)

[29] P. Coleman and N. Andrei, Kondo-stabilised spin liquids and heavy fermion superconductivity, J. Phys.: Condens. Matter 1(26), 4057 (1989)

[30] Y. Liu, H. Li, G. M. Zhang, and L. Yu, d-wave superconductivity induced by short-range antiferromagnetic correlations in the two-dimensional Kondo lattice model, Phys. Rev. B 86(2), 024526 (2012)

[31] Y. Liu, G. M. Zhang, and L. Yu, Pairing symmetry of heavy fermion superconductivity in the two-dimensional Kondo–Heisenberg lattice model, Chin. Phys. Lett. 31(8), 087102 (2014)

[32] J. P. Hu and H. Ding, Local antiferromagnetic exchange and collaborative Fermi surface as key ingredients of high temperature superconductors, Sci. Rep. 2, 381 (2012)

[33] D. J. Scalapino, A common thread: The pairing interaction for unconventional superconductors, Rev. Mod. Phys. 84(4), 1383 (2012)

[34] P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi, and F. C. Zhang, The physics behind high temperature superconducting cuprates: The plain vanilla version of RVB, J. Phys.: Condens. Matter 16(24), R755 (2004)

[35] P. A. Lee, N. Nagaosa, and X. G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78(1), 17 (2006)

[36] Y. Zhong, L. Zhang, H. T. Lu, and H. G. Luo, Fermionology in the Kondo–Heisenberg model: the case of CeCoIn₅, Eur. Phys. J. B 88(9), 238 (2015)

[37] P. Coleman, Introduction to Many Body Physics, Chapters 15 to 18, Cambridge: Cambridge University Press, 2015

[38] C. Pfleiderer, Superconducting phases of f-electron compounds, Rev. Mod. Phys. 81(4), 1551 (2009)

[39] L. Shu, D. E. MacLaughlin, W. P. Beyermann, R. H. Heffner, G. D. Morris, O. O. Bernal, F. D. Callaghan, J. E. Sonier, W. M. Yuhasz, N. A. Frederick, and M. B. Maple, Penetration depth, multiband superconductivity, and absence of muon-induced perturbation in superconducting PrOs₄Sb₁₂, Phys. Rev. B 79(17), 174511 (2009)

[40] X. Y. Tee, H. G. Luo, T. Xiang, D. Vandervelde, M. B. Salamon, H. Sugawara, H. Sato, C. Panagopoulos, and E. E. M. Chia, Penetration depth study of LaOs₄Sb₁₂: Multiband s-wave superconductivity, Phys. Rev. B 86(6), 064518 (2012)

[41] T. Senthil, M. Vojta, and S. Sachdev, Weak magnetism and non-Fermi liquids near heavy-fermion critical points, Phys. Rev. B 69(3), 035111 (2004)

[42] Y. Zhong, K. Liu, Y. F. Wang, Y. Q. Wang, and H. G. Luo, Half-filled Kondo lattice on the honeycomb lattice, Eur. Phys. J. B 86(5), 195 (2013)

[43] L. Zhang, Y. F. Wang, Y. Zhong, and H. G. Luo, Extended s-wave pairing symmetry on the triangular lattice heavy fermion system, Eur. Phys. J. B 88(10), 267 (2015)

[44] A. Ramires and P. Coleman, Supersymmetric approach to heavy fermion systems, Phys. Rev. B 93(3), 035120 (2016)

[45] N. Read and D. Newns, On the solution of the Coqblin- Schrieffer Hamiltonian by the large-N expansion technique, J. Phys. C 16, 3273 (1983)

[46] P. Coleman, Mixed valence as an almost broken symmetry, Phys. Rev. B 35(10), 5072 (1987)

[47] M. Z. Asadzadeh, M. Fabrizio, and F. Becca, Superconductivity from spoiling magnetism in the Kondo lattice model, Phys. Rev. B 90(20), 205113 (2014)

[48] P. Coleman and A. H. Nevidomskyy, Frustration and the Kondo effect in heavy fermion materials, J. Low Temp. Phys. 161(1–2), 182 (2010)

[49] G. Kotliar and J. Liu, Superexchange mechanism and dwave superconductivity, Phys. Rev. B 38(7), 5142 (1988)

[50] A. Koitzsch, I. Opahle, S. Elgazzar, S. V. Borisenko, J. Geck, V. B. Zabolotnyy, D. Inosov, H. Shiozawa, M. Richter, M. Knupfer, J. Fink, B. Büchner, E. D. Bauer, J. L. Sarrao, and R. Follath, Electronic structure of Ce- CoIn5 from angle-resolved photoemission spectroscopy, Phys. Rev. B 79(7), 075104 (2009)

[51] X. W. Jia, Y. Liu, L. Yu, J. F. He, L. Zhao, W. T. Zhang, H. Y. Liu, G. D. Liu, S. L. He, J. Zhang, W. Lu, Y. Wu, X. L. Dong, L. L. Sun, G. L. Wang, Y. Zhu, X. Y. Wang, Q. J. Peng, Z. M. Wang, S. J. Zhang, F. Yang, Z. Y. Xu, C. T. Chen, and X. J. Zhou, Growth, characterization and fermi surface of heavy fermion Ce- CoIn₅ superconductor, Chin. Phys. Lett. 28(5), 057401 (2011)

[52] L. Dudy, J. D. Denlinger, L. Shu, M. Janoschek, J. W. Allen, and M. B. Maple, Yb valence change in Ce_1−xYb_xCoIn₅ from spectroscopy and bulk properties, Phys. Rev. B 88(16), 165118 (2013)

[53] A. Polyakov, O. Ignatchik, B. Bergk, K. Götze, A. D. Bianchi, S. Blackburn, B. Prévost, G. Seyfarth, M. Côté, D. Hurt, C. Capan, Z. Fisk, R. G. Goodrich, I. Sheikin, M. Richter, and J. Wosnitza, Fermi-surface evolution in Yb-substituted CeCoIn₅, Phys. Rev. B 85(24), 245119 (2012)

[54] H. Hegger, C. Petrovic, E. G. Moshopoulou, M. F. Hundley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Pressure-induced superconductivity in Quasi-2D CeRhIn₅, Phys. Rev. Lett. 84(21), 4986 (2000)

[55] T. Park, F. Ronning, H.-Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Hidden magnetism and quantum criticality in the heavy fermion superconductor CeRhIn₅, Nature (London) 440, 65 (2006)

[56] P. J. Hirschfeld and N. Goldenfeld, Effect of strong scattering on the low-temperature penetration depth of a dwave superconductor, Phys. Rev. B 48(6), 4219 (1993)

[57] I. Kosztin and A. J. Leggett, Nonlocal effects on the magnetic penetration depth in d-wave superconductors, Phys. Rev. Lett. 79(1), 135 (1997)

[58] E. Abrahams, J. Schmalian, and P. Wölfle, Strongcoupling theory of heavy-fermion criticality, Phys. Rev. B 90(4), 045105 (2014)

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg