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Self-Consistent Study of the Superconducting Gap in the Strontium-doped Lanthanum Cuprate

This work is aimed at numerically investigating the behavior of the Fermi energy in Strontium-doped Lanthanum Cuprate, using a numerical zero temperature elastic scattering cross-section procedure in the unitary collision regime. The main task is to vary the zero temperature superconducting energy gap from its zero value in the normal state, to the highest value of 60 meV. We find that there are two different reduced phase space regimes for the first harmonic line node's order parameter. The first one, represented by a Fermi energy with a value of - 0.4 meV, where the rest of the material parameters, and the degrees of freedom of the normal to the superconducting phase transition are not sensitive to the self-consistent variation of the zero temperature superconducting energy gap. A different case is found when in the self-consistent numerical procedure, the Fermi energy takes a value of - 0.04 meV, indicating that the fermion-dressed quasiparticles have material parameters strongly sensitive to the numerical changes in the zero temperature gap, resulting in a reduced phase space, where the input and output zero superconducting energy values, and the degrees of freedom are separated by the self-consistent numerical analysis. The first scenario considers that when the Fermi energy and the nearest hopping terms have the same order of magnitude, the physics can be described by a picture given by nonequilibrium statistical mechanics. A second scenario indicates, that when the Fermi energy parameter and the hopping term have different order of magnitude; the physical picture tends to be related to the nonrelativistic quantum mechanical degrees of freedom coming from quasi-stationary quantum energy levels, with a damping term seen in the probability density distribution function, that is described in the configuration space. Henceforth, it is concluded that the use of the zero temperature elastic scattering cross-section links the phase and configuration spaces through the inverse scattering lifetime, and helps to clarify the role of the degrees of freedom in Strontium-doped Lanthanum Cuprate. Finally, we think that the self-consistent numerical procedure with the reduced phase space, induces nonlocality in the inverse scattering lifetime.

Strontium-Doped Lanthanum Cuprate, Reduce Phase Space, Configuration Space, Phase Space, Zero Temperature Elastic Scattering Cross-Section, Lifetime, Mean Free Path, Numerical Modelling

APA Style

Pedro Contreras, Dianela Osorio, Anjna Devi. (2023). Self-Consistent Study of the Superconducting Gap in the Strontium-doped Lanthanum Cuprate. International Journal of Applied Mathematics and Theoretical Physics, 9(1), 1-13.

ACS Style

Pedro Contreras; Dianela Osorio; Anjna Devi. Self-Consistent Study of the Superconducting Gap in the Strontium-doped Lanthanum Cuprate. Int. J. Appl. Math. Theor. Phys. 2023, 9(1), 1-13. doi: 10.11648/j.ijamtp.20230901.11

AMA Style

Pedro Contreras, Dianela Osorio, Anjna Devi. Self-Consistent Study of the Superconducting Gap in the Strontium-doped Lanthanum Cuprate. Int J Appl Math Theor Phys. 2023;9(1):1-13. doi: 10.11648/j.ijamtp.20230901.11

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Bednorz, J. & Müller. K. (1986). Possible high Tc superconductivity in the BaLaCuO system. Z. Physik B - Condensed Matter 64, 189–193. doi: 10.1007/BF01303701.
2. M. Kastner, R. Birgeneau, G. Shirane, and Y. Endoh. 1998. Magnetic, transport, and optical properties of mono layer copper oxides Rev. Mod. Phys. 70, 897. doi: 10.1103/RevModPhys.70.89.
3. Wu, M., Ashburn, J., Torng, C., Hor, P., Meng R., Gao, L. Huang, Z., Wang, Y. & Chu, C. (1987). Superconductivity at 93 K in a new mixed-phase Y-Ba-Cu-O compound system at ambient pressure. Phys. Rev. Lett. 58 (9), 908 doi: 10.1103/PhysRevLett.58.908.
4. Bardeen, J., Cooper, L. & Schrieffer, J. (1957). Microscopic Theory of Superconductivity, Phys. Rev. 106 (1), 162. doi: 10.1103/PhysRev.106.162.
5. Sheadem. T. (1994). Introduction to High Tc Superconductivity. Plenum Press.
6. Waldran, I. (1996). Structure of Cuprate Superconductors. Wiley.
7. Cava. I. (2000). Oxide Superconductors. J. Am. Ceram. Soc., 83:5, doi: 10.1111/j.1151-2916.2000.tb01142.x.
8. Xiao, G. Streitz, F., Gavrin, Du, A. & Chien C. (1987). Effect of transition-metal elements on the superconductivity of Y-Ba-Cu-O, Phys. Rev. B.85 (16), 8782. doi: 10.1103/PhysRevB.35.8782.
9. Ambegaokar, V. & Griffin, A. (1965) Theory of the Thermal Conductivity of Superconducting Alloys with Paramagnetic Impurities, Phys. Rev. 137 (4A), A1151. doi: 10.1103/PhysRev.137.A1151.
10. Momono, M. & Ido, M. (1996) Evidence for nodes in the superconducting gap of La2−xSrxCuO4. T2 dependence of electronic specific heat and impurity effects, Physica C 264, 311, doi: 10.1016/0921-4534(96)00290-0.
11. Sun, Y. & Maki, K. (1995). Transport Properties of D-Wave Superconductors with Impurities, EPL 32, 355.
12. Larkin, A. (1965). Vector pairing in superconductors of small dimensions. JETP Letters. Vol. 2 (5), 105. ISSN: 0370-274X.
13. Pethick, C. & Pines, D. (1986). Transport processes in heavy-fermion superconductors. Phys. Rev. Lett. 57 (1), 118, doi: 10.1103/PhysRevLett.57.118.
14. Takeya, J., Ando, Y, Komiya, S., & Sun, XF. (2002). Low-temperature electronic heat transport in La2−xSrxCuO4. Single crystals: unusual low-energy physics in the normal and superconducting states. Phys. Rev. Lett. 88 (7): 077001. doi: 10.1103/phys.rev.lett.88.077001.
15. Scalapino, D. (1995) The case for d pairing in the cuprate superconductors, Physics Reports. 250 (6), 329 doi: 10.1016/0370-1573(94)00086-I.
16. Hussey, N. (2002). Low-energy quasiparticles in High-Tc cuprates, Adv. in Phys, 51:8, 1685. doi: 10.1080/00018730210164638.
17. Yamase, H. Sakurai, Y. Fujita, M. et al. (2021) Fermi surface in La-based cuprate superconductors from Compton scattering imaging. Nat Commun 12, 2223. doi: 10.1038/s41467-021-22229-6.
18. Photopoulos, R. and Frésard, R. (2019), Cuprate Superconductors: A 3D Tight-Binding Model for La-Based Cuprate Superconductors Ann. Phys. 531, 1970044. doi: 10.1002/andp.201970044.
19. Walker. M. B. (2001). Fermi-liquid theory for anisotropic superconductors. Phys. Rev. B. 64 (13) 134515, doi: 10.1103/PhysRevB.64.134515.
20. Pitaevskii, L: (2008). Superfluid Fermi liquid in a unitary regime, Phys. Usp. 51, 603 doi: 10.1070/PU2008v051n06ABEH006548.
21. Contreras, P., Osorio, D. & Devi, A. (2022). The effect of nonmagnetic disorder in the superconducting energy gap of strontium ruthenate, Physica B: Condensed Matter. Vol. 646, 414330. doi: 10.1016/j.physb.2022.414330.
22. Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill.
23. Pitaevskii, L., Lifshitz, EM & Sykes, J. (1981). Physical Kinetics, Vol. 10, Pergamon Press.
24. Dorfman, J. Van Beijeren, H. & Kirkpatrick, T. (2021). Contemporary Kinetic Theory of Matter. Cambridge University Press. doi: 10.1017/9781139025942.
25. Kaganov, M. & Lifshitz, I. (1989). Quasiparticles: Ideas and Principles of Quantum Solid State Physics. 2nd edition. Moscow “Nauka”.
26. Mineev, V. & Samokhin, K. (1999). Introduction to Unconventional Superconductivity. Gordon and Breach Science Publishers.
27. Schachinger, E. & Carbotte, J. (2003). Residual absorption at zero temperature in d-wave superconductors. Phys. Rev. B 67, 134509. doi: 10.1103/PhysRevB.67.134509.
28. Lifshitz, I., Gredeskul S. & and Pastur, L. (1988) Introduction to the theory of disordered systems. John Wiley and Sons.
29. Edwards, S. (1958). A new method for the evaluation of electric conductivity in metals, Philosophical Magazine, 3 (33) 1020. doi: 10.1080/14786435808243244.
30. Ziman, J. (1979). Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems, Cambridge University Press.
31. Contreras, P., Walker, M. B. & Samokhin, K. (2004). Determining the superconducting gap structure in Sr2RuO4 from sound attenuation studies below Tc. Phys. Rev. B, 70: 184528. doi: 10.1103/PhysRevB.70.184528.
32. Contreras, P. (2011). Electronic heat transport for a multiband superconducting gap in Sr2RuO4. Rev. Mex. Fis. 57 (5) 395.
33. Contreras, P. et al., (2014). A numerical calculation of the electronic specific heat for the compound Sr2RuO4 below its superconducting transition temperature. Rev. Mex. Fis. 60 (3) 184.
34. Contreras, P. & Osorio, D. (2021) Scattering Due to Non-magnetic Disorder in 2D Anisotropic d-wave High Tc Superconductors. Engineering Physics 5, 1, doi: 10.11648/j.ep.20210501.11.
35. Contreras, P. & Moreno, J. (2019). A non-linear minimization calculation of the renormalized frequency in dirty d-wave superconductors. Canadian Journal of Pure and Applied Sciences. Vol. 13 (2), 4765 ISSN: 1920-3853.
36. Tsuei, C. & Kirtley, J. (2000). Pairing symmetry in cuprate superconductors. Rev. Mod. Phys., 72: 969. doi: 10.1103/RevModPhys.72.969.
37. Landau, L. & Lifshitz, E. (1980). Statistical Physics. Pergamon Press.
38. Contreras, P. & Osorio, D. (2023). A Tale of the Scattering Lifetime and the Mean Free Path. arXiv: 2301.05322 [cond- mat.supr-con] doi: 10.485550/arXiv.230105322.
39. Blatt, F. (1957). Theory of mobility of electrons in solids, Academic Press.
40. Schrieffer, J. (1970). What is a quasiparticle? Journal of Research of the National Bureau of Standards, Vol. 74A (4), 537.
41. Davydov, A. (1965). Quantum Mechanics. Pergamon Press.
42. Kvashnikov, I. (2003). The theory of systems out of equilibrium, 3rd Vol. Moscow State University Press.
43. Miyake, K. & Narikiyo, O. (1999). Model for Unconventional Superconductivity of Sr2RuO4. Effect of Impurity Scattering on Time-Reversal Breaking Triplet Pairing with a Tiny Gap. Phys. Rev. Lett. 83, 1423. doi: 10.1103/PhysRevLett.83.1423.
44. Contreras, P., Osorio, D. & Ramazanov, S. (2022). Nonmagnetic tight- binding effects on the γ-sheet of Sr2RuO2. Rev. Mex. Fis 68 (2) 1, doi: 10.31349/RevMexFis.68.020502.
45. Carruthers, P. & Zachariasen, F. (1983). Quantum collision theory with phase-space distributions. oxides Rev. Mod. Phys. 55 (1), 245 doi: 10.1103/RevModPhys.55.245.
46. Brandt, N. & Chudinov, S. (1975). Electronic structure of metals, Mir Publishers.
47. Contreras, P. Osorio, D. & Beliayev, E. (2022) Dressed behavior of the quasiparticles lifetime in the unitary limit of two unconventional superconductors. Low Temp. Phys. 48, 187, doi: 10.1063/10.0009535.
48. Contreras, P. Osorio, D. & Beliayev, E. (2022) Tight-Binding Superconducting Phases in the Unconventional Compounds Strontium-Substituted Lanthanum Cuprate and Strontium Ruthenate. American Journal of Modern Physics. Vol. 11 (2) 32, doi: 10.11648/j.ajmp.20221102.13.
49. Yoshida, T. et al. (2012). Pseudogap, Superconducting Gap, and Fermi Arc in High-Tc Cuprates Revealed by Angle-Resolved Photoemission Spectroscopy. Journal of the Physical Society of Japan, 81: 011006, doi: 10.1143/JPSJ.81.011006.
50. Kaganov, M. & Contreras, P. (1994) Theory of the anomalous skin effect in metals with complicated Fermi surfaces. Journal of Experimental and Theoretical Physics, 79: 985, 1994. ISSN: 0080-4630.
51. Kaganov, M., Lyubarskiy, G. & Mitina, A. (1997) The theory and history of the anomalous skin effect in normal metals, Physics Reports, Vol. 288 (1–6), 291, doi: 10.1016/S0370-1573(97)00029-X.
52. Baker, G. (2022) Non-local electrical conductivity in PdCoO2 (Ph.D. Thesis). University of British Columbia. doi: 10.14288/1.0421263.
53. G. Baker, et al., (2022) Non-local microwave electrodynamics in ultra-pure PdCoO2 arXiv preprint arXiv: 2204.14239 doi: 10.48550/arXiv.2204.14239.
54. Torkhov, N. et al., (2022) Conversion of the anomalous skin effect to the normal one in thin-film metallic microwave systems. Phys. Scr. 97 095809 doi: 10.1088/1402-4896/ac837d.
55. Contreras, P., Osorio, D. & Tsuchiya, S. (2022) Quasi-point versus point nodes in Sr2RuO2, the case of a flat tight binding  sheet. Rev. Mex. Fis 68 (6), 060501 1–8. doi: 10.31349/RevMexFis.68.060501.