Analytical assessment of some characteristic ratios for s-wave superconductors

Abstract

We evaluate some thermodynamic quantities and characteristic ratios that describe low- and hightemperature s-wave superconducting systems. Based on a set of fundamental equations derived within the conformal transformation method, a simple model is proposed and studied analytically. After including a one-parameter class of fluctuations in the density of states, the mathematical structure of the s-wave superconducting gap, the free energy difference, and the specific heat difference is found and discussed in an analytic manner. Both the zero-temperature limit T = 0 and the subcritical temperature range T≤T_c are discussed using the method of successive approximations. The equation for the ratio R₁, relating the zero-temperature energy gap and the critical temperature, is formulated and solved numerically for various values of the model parameter. Other thermodynamic quantities are analyzed, including a characteristic ratio R₂, quantifying the dynamics of the specific heat jump at the critical temperature. It is shown that the obtained model results coincide with experimental data for low-T_c superconductors. The prospect of application of the presented model in studies of high-T_c superconductors and other superconducting systems of the new generation is also discussed.

Keywords

superconductivity / characteristic ratios / fluctuation of the DoS

References

[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Iron-based layered superconductor La[O_1−xF_x]FeAs (x= 0.05–0.12) with T_c= 26 K, J. Am. Chem. Soc. 130(11), 3296 (2008)

[2] H. Hosono and K. Kuroki, Iron-based superconductors: Current status of materials and pairing mechanism, Physica C 514, 399 (2015)

[3] M. Krzyzosiak, R. Gonczarek, A. Gonczarek, and L. Jacak, Conformal Transformation Method in Studies of High-T_c Superconductors – Beyond the Van Hove Scenario in: Superconductivity and Superconducting Wires, Eds. D. Matteri and L. Futino, Nova Science Publishers, Hauppage, New York, 2010, Ch. 5

[4] R. Gonczarek and M. Krzyzosiak, Model of Superconductivity in the Singular Fermi Liquid in: Progress in Superconductivity Research, Ed. O. A. Chang, Nova Science Publishers, Hauppage, New York, 2008, Ch. 6

[5] R. Gonczarek and M. Krzyzosiak, Conformal transformation method and symmetry aspects of the group C_4v in a model of high-T_c superconductors with anisotropic gap, Physica C 426(431), 278 (2005)

[6] R. Gonczarek, L. Jacak, M. Krzyzosiak, and A. Gonczarek, Competition mechanism between singlet and triplet superconductivity in the tight-binding model with anisotropic attractive potential, Eur. Phys. J. B 49(2), 171 (2006)

[7] R. Gonczarek, M. Krzyzosiak, L. Jacak, and A. Gonczarek, Coexistence of spin-singlet s- and d-wave and spin-triplet p-wave order parameters in anisotropic superconductors, phys. stat. sol. (b) 244, 3559 (2007)

[8] R. Gonczarek, M. Krzyzosiak, and A. Gonczarek, Islands of stability of the d-wave order parameter in s-wave anisotropic superconductors, Eur. Phys. J. B 61(3), 299 (2008)

[9] D. Kasinathan, K. W. Lee, and W. E. Pickett, On heavy carbon doping of MgB₂, Physica C 424(3–4), 116 (2005)

[10] J. Kortus, O. V. Dolgov, R. K. Kremer, and A. A. Golubov, Band filling and interband scattering effects in MgB₂: Carbon versus aluminum doping, Phys. Rev. Lett. 94(2), 027002 (2005)

[11] W. S. Agrestini, C. Metallo, M. Filippi, L. Simonelli, G. Campi, C. Sanipoli, E. Liarokapis, S. De Negri, M. Giovannini, A. Saccone, A. Latini, and A. Bianconi, Substitution of Sc for Mg in MgB₂: Effects on transition temperature and Kohn anomaly, Phys. Rev. B 70(13), 134514 (2004)

[12] H. Mori, T. Okano, M. Kamiya, M. Haemori, H. Suzuki, S. Tanaka, Y. Nishio, K. Kajita, and H. Moriyama, Bandwidth and band filling control in organic conductors, Physica C357–360, 103 (2001)

[13] M. Mulak and R. Gonczarek, Structures of thermodynamic functions for S-paired fermi systems in parametric equations approach, Acta Phys. Pol. A 89(5–6), 689 (1996)

[14] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill Book Company, 1971, §51

[15] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976, Ch. 34 Superconductivity

[16] C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, Inc. NY, 1966, Ch. 11

[17] J. Spałek, Introduction to Condensed Matter Physics, Wydawnictwo Naukowe PWN SA, Warszawa, 2015, Ch. 17

[18] H. Ibach and H. Lüth, Solid State Physics. An Introduction to Principles of Material Science, Berlin Heidelberg: Springer-Verlag, 1995, Ch. 10.5

[19] M. Cyrot and D. Pavuna, Introduction to Superconductivity and High-T_c Materials, World Scientific Publ. Co. (London, New Jersey, Singapore, Hong Kong, Bangalore, Beijing, 1992), Ch. 7.1

[20] A. P. Durajski, R. Szcze¸śniak, and Y. Li, Non-BCS thermodynamic properties of H₂S superconductor, Physica C 515, 1 (2015)

[21] A. P. Durajski and R. Szcze¸śniak, Estimation of the superconducting parameters for silane at high pressure, Mod. Phys. Lett. B 28(07), 1450052 (2014)

[22] R. Szcze¸śniak, A. P. Durajski, M. W. Jarosik, Specific heat and thermodynamic critical field for calcium under the pressure at 120 GPa, Mod. Phys. Lett. B 26(08), 1250050 (2012)

[23] B. Lorenz, J. Cmaidalka, R. L. Meng, and C. W. Chu, Thermodynamic properties and pressure effect on the superconductivity in CaAlSi and SrAlSi, Phys. Rev. B 68(1), 014512 (2003)

[24] A. P. Durajski, Quantitative analysis of nonadiabatic effects in dense H₃S and PH₃ superconductors, Sci. Rep. 6(1), 38570 (2016)

[25] A. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, and S. I. Shylin, Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system, Nature 525(7567), 73 (2015)

[26] M. Einaga, M. Sakata, T. Ishikawa, K. Shimizu, M. I. Eremets, A. P. Drozdov, I. A. Troyan, N. Hirao, and Y. Ohishi, Crystal structure of the superconducting phase of sulfur hydride, Nat. Phys. 12(9), 835 (2016)

[27] Y. Li, J. Hao, H. Liu, Y. Li, and Y. Ma, The metallization and superconductivity of dense hydrogen sulfide, J. Chem. Phys. 140(17), 174712 (2014)

[28] R. Gonczarek, M. Gładysiewicz, and M. Mulak, On possible formalism of anisotropic fermi liquid and BCS– type superconductivity, Int. J. Mod. Phys. B 15(05), 491 (2001)

[29] F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B 37(7), 3759 (1988)

[30] R. Szcze¸śniak and A. P. Durajski, The thermodynamic properties of the high-pressure superconducting state in the hydrogen-rich compounds, Solid State Sci. 25, 45 (2013)

[31] R. Szcze¸śniak and A. P. Durajski, Superconducting state above the boiling point of liquid nitrogen in the GaH₃ compound, Supercond. Sci. Technol. 27(1), 015003 (2013)

[32] D. Y. Xing, M. Liu, Y. G. Wang, and J. Dong, Analytic approach to the antiferromagnetic van Hove singularity model for high-T_c superconductors, Phys. Rev. B 60(13), 9775 (1999)

[33] E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Band-structure trend in holedoped cuprates and correlation with T_c max, Phys. Rev. Lett. 87(4), 047003 (2001)

[34] O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen, LDA energy bands, low-energy hamiltonians, t′ t′′ t^(k), and J^, J. Phys. Chem. Solids 56(12), 1573 (1995) O. K. Andersen, S. Y. Savrasov, O. Jepsen, and A. I. Liechtenstein, Out-of-plane instability and electronphonon contribution tos- and d-wave pairing in hightemperature superconductors; LDA linear-response calculation for doped CaCuO₂ and a generic tight-binding model, J. Low Temp. Phys. 105(3–4), 285 (1996)

[35] R. Gonczarek and M. Krzyzosiak, Some universal relations between the gap and thermodynamic functions plausible for various models of superconductors, phys. stat. sol. (b) 238, 29 (2003)

[36] R. Szczee¸śniak, A. P. Durajski, and L. Herok, Theoretical description of the SrPt₃P superconductor in the strongcoupling limit, Phys. Scr. 89(12), 125701 (2014)

[37] R. Szczee¸śniak and A. P. Durajski, The Energy Gap in the (Hg_1−xSnx)Ba₂Ca₂Cu₃O_8+y Superconductor, Journal of Superconductivity and Novel Magnetism 27(6), 1363 (2014)

[38] R. Szczee¸śniak and A. P. Durajski, Thermodynamics of the superconducting state in calcium at 200 GPa, Journal of Superconductivity and Novel Magnetism 25(2), 399 (2012)

[39] M. Krzyzosiak, R. Gonczarek, A. Gonczarek, and L. Jacak, Applications of the conformal transformation method in studies of composed superconducting systems, Front. Phys. 11(6), 117407 (2016)

[40] R. Baquero, D. Quesada, and C. Trallero-Giner, BCSuniversal ratios within the Van Hove scenario, Physica C 271(1–2), 122 (1996)

[41] M. Mulak and R. Gonczarek, Discontinuous phase transitions in S-paired Fermi systems, Acta Physica Polonica A 92, 1177 (1997)

[42] R. Gonczarek, M. Krzyzosiak, and M. Mulak, Valuation of characteristic ratios for high-T_c superconductors with anisotropic gap in the conformal transformation method, J. Phys. A 37(18), 4899 (2004)

[43] R. Gonczarek, M. Gładysiewicz, and M. Mulak, Equilibrium states and thermodynamical properties of d-wave paired HTSC in the tight-binding model, phys. stat. sol. (b) 233, 351 (2002)

[44] R. Gonczarek and M. Mulak, Enhancement of critical temperature of superconductors implied by the local fluctuation of EDOS, Phys. Lett. A 251(4), 262 (1999)

[45] J. Bouvier and J. Bok, The gap symmetry and fluctuations in high T_c superconductors, Eds. J. Bok, G. Deutscher, D. Pavuna, and S. Wolf, Plenum Press, New York, 1998, p. 37

[46] R. Baquero, D. Quesada, and C. Trallero-Giner, BCSuniversal ratios within the Van Hove scenario, Physica C 271(1–2), 122 (1996)

[47] R. Gonczarek, M. Krzyzosiak, A. Gonczarek, and L. Jacak, New classes of integrals inherent in the mathematical structure of extended equations describing superconducting systems, Int. J. Mod. Phys. B 29(17), 1550117 (2015)

[48] R. Gonczarek, M. Krzyzosiak, A. Gonczarek, and L. Jacak, On new families of integrals in analytical studies of superconductors within the conformal transformation method, Adv. Condens. Matter Phys. 2015, 1 (2015)

[49] A. Bianconi and M. Filippi, Feshbach Shape Resonances in Multiband high T_c Superconductors in: Symmetry and Heterogeneity in High Temperature Superconductors, Ed. A. Bianconi, NATO Science Series (II): Mathematics, Physics and Chemistry- Vol. 2014, Springer 2006, Ch. 1.2

[50] J. K. Ren, X. B. Zhu, H. F. Yu, Y. Tian, H. F. Yang, C. Z. Gu, N. L. Wang, Y. F. Ren, and S. P. Zhao, Energy gaps in Bi₂Sr₂CaCu₂O_8+d cuprate superconductors, Sci. Rep. 2(1), 248 (2012)

[51] T. J. Reber, S. Parham, N. C. Plumb, Y. Cao, H. Li, Z. Sun, Q. Wang, H. Iwasawa, M. Arita, J. S. Wen, Z. J. Xu, G. D. Gu, Y. Yoshida, H. Eisaki, G. B. Arnold, and D. S. Dessau, Pairing, pair-breaking, and their roles in setting the T_c of cuprate high temperature superconductors, arXiv: 1508.06252v1 (2015)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature