Condensation Energy in a Superconductor for All Temperatures
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Condensation Energy in a Superconductor for All Temperatures J. Ortega1 · F. Zúñiga1,2 · M. de Llano1 Received: 28 April 2020 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract An exact analytical expression for the condensation energy Econd (T) of a phonondriven superconductor for all absolute temperatures T and for any coupling strength is introduced so as to calculate the Helmholtz free energy difference Fs (T) − Fn (T) between superconducting (s) and normal (n) states. This is achieved via a boson– fermion ternary gas theory—called the generalized Bose–Einstein condensation (GBEC) theory—which includes two-hole Cooper pairs, two-electron ones as well as single, free/unbound electrons. The GBEC formalism turns out to be quite useful in dealing with nonzero T values of Econd (T) and reproduces several well-known experimental results. An expression for the condensation energy per atom is also calculated and applied to aluminum and niobium, and both results are compared with experimental data. Keywords Condensation energy · BCS theory · Generalized Bose–Einstein condensation (GBEC) theory · Two-electron and two-hole Cooper pairs
1 Introduction Conventional superconductors (SCs) are examples of low-energy quantum phenomena exhibiting macroscopic electrodynamic and thermodynamic phenomena. Here, we study one such macroscopic effect, namely the condensation energy in a phonon-driven SC for all temperatures T . The condensation energy of a SC is equivalent to the Helmholtz free energy difference between normal and superconducting phases and can thus give an explicit expression for the energy difference at the phase transition. This is done by calculating the condensation energy by applying the generalized Bose–Einstein * F. Zúñiga fzf1@st‑andrews.ac.uk 1
Posgrado en Ciencia e Ingeniería de Materiales, Universidad Nacional Autónoma de México, 04510 Mexico City, Mexico
2
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK
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Vol.:(0123456789)
Journal of Low Temperature Physics
condensation (GBEC) [1–5] theory which is a quantum statistical theory containing the Bardeen–Cooper–Schrieffer (BCS) as a special case in that it reproduces both its characteristic electronic gapped spectrum for all temperatures T and all interelectronic couplings, including the more common condensation energy at T = 0 . The GBEC theory includes two-hole Cooper pairs (CPs) besides the more common two-electron CPs plus free/unbound electrons [6]. Thus, without abandoning electron–phonon dynamics this viewpoint seems to be an attractive alternative to study the condensation energy for all T values. It is worth mentioning that the GBEC theory yields substantially higher transition critical temperatures Tc compared with BCS theory. Surprisingly, the electron–phonon dynamics was also seen to drive the high Tc values reported for the H3 S sulfide in 2015 with Tc = 203 K [7] and the recently discovered lanthanum hydride (LaH)10
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