Characteristics of the Eliashberg Formalism on the Example of High-Pressure Superconducting State in Phosphor

Abstract

The work describes the properties of the high-pressure superconducting state in phosphor: p ∈ {20, 30, 40, 70} GPa. The calculations were performed in the framework of the Eliashberg formalism, which is the natural generalization of the BCS theory. The exceptional attention was paid to the accurate presentation of the used analysis scheme. With respect to the superconducting state in phosphor it was shown that the observed not-high values of the critical temperature ([TC] max p=30 GPa = 8.45 K) result not only from the low values of the electron-phonon coupling constant, but also from the very strong depairing Coulomb interactions. Additionally the inconsiderable strong-coupling and retardation effects force the dimensionless ratios R∆, RC, and RH -related to the critical temperature, the order parameter, the specific heat, and the thermodynamic critical field -to take the values close to the BCS predictions. Hamiltonian and fundamental equations of BCS model and Eliashberg formalismThe first microscopic theory of the superconducting state was formulated in 1957 by Bardeen, Cooper and Schrieffer (the so-called "BCS model") [1,2]. In the framework of the method of the second quantization the BCS Hamiltonian can be written with the following formula [3,4]:where the function ε k represents the electron band energy, V is the effective pairing potential, whose value is determined by the matrix elements of the electronphonon interaction, the electron band energy and the phonon energy. The symbols c † kσ and c kσ represent the creation and annihilation operator of the electron state in the momentum representation (k) for the spin σ ∈ {↑, ↓}. It should be noted that the sum denoted by the sign ought to be calculated only for those values of the momenta, for which the condition −Ω max < ε k < Ω max is fulfilled, where Ω max represents the Debye energy. In the considered case the effective pairing potential is positive, which allows the formation of the superconducting condensate. The fundamental equation of the BCS theory for the order parameter (∆ ≡ V k c −k↓ c k↑ ) is derived directly from Hamiltonian (1) using the mean field approximation to the interaction term. As a result the following can be obtained:) * corresponding author; e-mail: aduda@wip.pcz.pl where k B is the Boltzmann constant. Let us notice that Eq. (2) cannot be solved analytically. However, in the limit cases T → T C and T → 0 K the relatively simple calculations allow us to obtain the formulae for the critical temperature and the value of the order parameter k B T C = 1.13Ω max exp (−1/λ), ∆ (0) = 2Ω max exp (−1/λ). The electron-phonon coupling constant λ in the BCS model is given by λ ≡ ρ (0) V (the quantity ρ (0) represents the electron density of states on the Fermi surface). The BCS theory predicts the existence of the universal thermodynamic ratios, which are defined below:andThe symbols appearing in the formulae (4) and (5) denote respectively: C S -the specific heat of the superconducting state, C N -the specific heat of the normal state, and ...

Anomalously high value of Coulomb pseudopotential for the H5S2 superconductor

“…Note that the high value of μ * is relatively often observed in the case of high-pressure superconducting state. For example, for phosphorus: , where meV 45 . We encounter the similar situation for lithium: , while ω D = 82.7 meV 46 .…”