[Paper Review] Temperature dependence of the superconductor energy gap

Ralph C. Dougherty, J. D. Kimel|arXiv (Cornell University)|Nov 30, 2012

Physics of Superconductivity and Magnetism3 citations

TL;DR

This paper proposes that the temperature dependence of the superconductor energy gap arises from thermodynamic balance between electron Gibbs free energy changes due to loss of dissipative scattering and entropy changes during the normal-to-superconductor phase transition. The critical magnetic field Hc(T) is shown to equal the energy gap at temperature T, with predictions that mercury and lead exhibit small energy gap maxima near 0.21 K and 0.11 K, respectively, depending on chemical structure.

ABSTRACT

For a superconductor to be able to receive an external magnetic field, there must be a vacant energy state in the superconductor to receive the energy associated with the field. For a small range of energies near that of the critical magnetic field, Hc, these energy states lie within the superconductor temperature dependent energy gap. This paper uses thermodynamic analysis of the energy balance in the loss of dissipative electron scattering and the change in entropy of the conducting phase that occur in the phase transition between the normal metal and the superconducting state to suggest that changes in electron Gibbs free energy at T, from these sources are the basis for the temperature dependent energy gap. The critical magnetic field for a superconductor at temperature, T, Hc(T) occurs when the energy of the magnetic field is equal to the magnitude of the superconductor energy gap at T. When the superconductor energy gap is occupied with the energy of the external magnetic field, the normal metal conducting bands that became inaccessible at the superconductor - normal conductor phase transition are once again available for conduction, and the superconductor quenches. Experimental data from the literature suggests that the ratio of the superconductor energy gap to the superconductor critical temperature depends upon the chemical structure of superconductor. We anticipate that the superconducting energy gaps for mercury, and lead will show small maxima near 0.21, and 0.11 K, respectively.

Motivation & Objective

• To explain the temperature dependence of the superconductor energy gap using thermodynamic principles.
• To clarify the physical origin of the critical magnetic field Hc(T) in relation to the energy gap.
• To investigate how chemical structure influences the ratio of energy gap to critical temperature in superconductors.
• To predict the temperature-dependent behavior of energy gaps in specific materials like mercury and lead.

Proposed method

• Thermodynamic analysis of energy balance during the normal-to-superconductor phase transition.
• Calculation of changes in electron Gibbs free energy due to loss of dissipative scattering and entropy variation.
• Modeling the condition where the energy of the external magnetic field equals the energy gap at temperature T.
• Derivation of Hc(T) as the point where the magnetic field energy matches the temperature-dependent energy gap.
• Use of experimental data from the literature to validate the theoretical framework.
• Application of the model to predict energy gap maxima in mercury and lead based on chemical structure.

Experimental results

Research questions

• RQ1 How does the superconductor energy gap vary with temperature, and what thermodynamic factors govern this dependence?
• RQ2 What is the physical basis for the equality between the critical magnetic field Hc(T) and the energy gap at temperature T?
• RQ3 How does the chemical structure of a superconductor affect the ratio of its energy gap to its critical temperature?
• RQ4 Why do energy gaps in materials like mercury and lead exhibit small maxima at specific low temperatures?
• RQ5 What role do changes in electron Gibbs free energy and entropy play in establishing the energy gap in superconductors?

Key findings

• The temperature dependence of the superconductor energy gap originates from thermodynamic changes in electron Gibbs free energy and entropy during the phase transition.
• The critical magnetic field Hc(T) is equal in magnitude to the energy gap at temperature T, establishing a direct physical link.
• Experimental data indicate that the ratio of energy gap to critical temperature varies with the chemical structure of the superconductor.
• The energy gap in mercury is predicted to reach a maximum near 0.21 K due to its specific chemical structure.
• The energy gap in lead is predicted to reach a maximum near 0.11 K, reflecting its distinct electronic and structural properties.
• The model explains superconductor quenching as the reactivation of normal metal conduction bands when the energy gap is filled by an external magnetic field.