The superconducting state in vanadium characterizes with the critical temperature (T c ) equal to 5.3 K. The Coulomb pseudopotential, calculated with the help of the Eliashberg equations, possesses anomalously high value μ (3 max ) = 0.259 or μ (10 max ) = 0.368 ( max denotes the maximum phonon frequency). Despite the relatively large electron-phonon coupling constant (λ = 0.91), the quantities such as the order parameter ( ), the specific heat (C), and the thermodynamic critical field (H c ) determine the values of the dimensionless ratios not deviating much from the predictions of the BCS theory: R = Keywords Electron-phonon superconductivity in vanadium · Eliashberg theory · Semi-analytical approach · Coulomb pseudopotential · Thermodynamic propertiesThe classical Eliashberg theory is used to quantitative description of the superconducting state, which is mediated by the electron-phonon interaction [1]. The strong-coupling and the retardation effects omitted in BCS model [2,3] are included in the Eliashberg theory by assumption that the electron band energy is directly renormalized by the frequency-dependent order parameter ( n = (iω n )), the wave function renormalization factor (Z n = Z (iω n )), and the energy shift function (χ n = χ (iω n )). The symbol ω n = (π/β) (2n − 1) denotes the Matsubara frequency, β = 1/ k B T , where k B is the Boltzmann constant. In most cases, the materials, in which we observe the phonon induced superconducting state, characterize with the very wide electron band (W max , where W denotes the half-width of the band, and max is the Debye frequency). For this reason, the analysis of the superconducting state can be conducted with the help of the Eliashberg equations for the half-filled electron band [4,5]:whereas the order parameter is given by the formula n = φ n /Z n . The electron-phonon pairing kernel has the form of λ (iω n − iω m ) = 2 +∞ 0 dω α 2 F (ω)ω (ω n −ω m ) 2 +ω 2 . The spectral function α 2 F (ω) is usually calculated numerically by using the ab initio approach, where the electron band energy, the

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When considering a Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron–phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio λωD/εF ≈ 0.46, where λ is the electron–phonon coupling constant, ωD is the Debye frequency, and εF represents the Fermi energy. Due to nonadiabatic effects, the phonon–induced superconducting state in Li-hBN is characterized by much lower values of the critical temperature (T LOVC C ∈ {19.1, 15.5, 11.8} K, for μ* ∈ {0.1, 0.14, 0.2}, respectively) than would result from calculations not taking this effect into account (T ME C∈ {31.9, 26.9, 21} K). From the technological point of view, the low value of T C limits the possible applications of Li-hBN. The calculations were carried out under the classic Migdal–Eliashberg formalism (ME) and the Eliashberg theory with lowest-order vertex corrections (LOVC). We show that the vertex corrections of higher order (λ3) lower the value of T LOVC C by a few percent.

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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 ...