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Register a new userTemperature of superconducting transition for very strong coupling in antiadiabatic limit of Eliashberg equations
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M.V. Sadovskii
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It is shown that the famous Allen -- Dynes asymtotic limit for superconducting transition temperature in very strong coupling region $T_{c}>frac{1}{2pi}sqrt{lambda}Omega_0$ (where $lambdagg 1$ - is Eliashberg - McMillan electron - phonon coupling constant and $Omega_0$ - the characteristic frequency of phonons) in antiadiabatic limit of Eliashberg equations $Omega_0/Dgg 1$ ($Dsim E_F$ is conduction band half-width and $E_F$ is Fermi energy) is replaced by $T_c>(2pi^4)^{-1/3}(lambda DOmega_0^2)^{1/3}$, with the upper limit for $T_c$ given by $T_c<frac{2}{pi^2}lambda D$.
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The discovery of record - breaking values of superconducting transition temperature $T_c$ in quite a number of hydrides under high pressure was an impressive demonstration of capabilities of electron - phonon mechanism of Cooper pairing. This lead to an increased interest to foundations and limitations of Eliashberg - McMillan theory as the main theory describing superconductivity in a system of electrons and phonons. Below we shall consider both elementary basics of this theory and a number of new results derived only recently. We shall discuss limitations on the value of the coupling constant related to lattice instability and a phase transition to another phase (CDW, bipolarons). Within the stable metallic phase the effective pairing constant may acquire arbitrary values. We consider extensions beyond the traditional adiabatic approximation. It is shown that Eliasberg - McMillan theory is also applicable in the strong antiadiabatic limit. The limit of very strong coupling, being most relevant for the physics of hydrides, is analyzed in details. We also discuss the bounds for $T_c$ appearing in this limit.
The standard Eliashberg - McMillan theory of superconductivity is essentially based on the adiabatic approximation. Here we present some simple estimates of electron - phonon interaction within Eliashberg - McMillan approach in non - adiabatic and even antiadiabatic situation, when characteristic phonon frequency $Omega_0$ becomes large enough, i.e. comparable or exceeding the Fermi energy $E_F$. We discuss the general definition of Eliashberg - McMillan (pairing) electron - phonon coupling constant $lambda$, taking into account the finite value of phonon frequencies. We show that the mass renormalization of electrons is in general determined by different coupling constant $tildelambda$, which takes into account the finite width of conduction band, and describes the smooth transition from the adiabatic regime to the region of strong nonadiabaticity. In antiadiabatic limit, when $Omega_0gg E_F$, the new small parameter of perturbation theory is $lambdafrac{E_F}{Omega_0}simlambdafrac{D}{Omega_0}ll 1$ ($D$ is conduction band half -- width), and corrections to electronic spectrum (mass renormalization) become irrelevant. However, the temperature of superconducting transition $T_c$ in antiadiabatic limit is still determined by Eliashberg - McMillan coupling constant $lambda$. We consider in detail the model with discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature $T_c$ is derived, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We also analyze the contribution of such phonons into the Coulomb pseudopotential $mu^{star}$ and show, that antiadiabatic phonons do not contribute to Tolmachevs logarithm and its value is determined by partial contributions from adiabatic phonons only.
The weak-coupling limits of the gap and critical temperature computed within Eliashberg theory surprisingly deviate from the BCS theory predictions by a factor of $1/sqrt{e}$. Interestingly, however, the ratio of these two quantities agrees for both theories. Motivated by this result, here we investigate the weak-coupling thermodynamics of Eliashberg theory, with a central focus on the free energy, specific heat, and the critical magnetic field. In particular, we numerically calculate the difference between the superconducting and normal-state specific heats, and we find that this quantity differs from its BCS counterpart by a factor of $1/sqrt{e}$, for all temperatures below $T_{c}$. We find that the dimensionless ratio of the specific-heat discontinuity to the normal-state specific heat reduces to the BCS prediction given by $Delta C_{V}(T_{c})/C_{V,n}(T_c)approx1.43$. This gives further evidence to the expectation that all dimensionless ratios tend to their universal values in the weak-coupling limit.
The influence of antiadiabatic phonons on the temperature of superconducting transition is considered within Eliashberg - McMillan approach in the model of discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature $T_c$ is proposed, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We study the contribution of such phonons into the Coulomb pseudopotential $mu^{star}$. It is shown, that antiadiabatic phonons do not contribute to Tolmachevs logarithm and its value is determined by partial contributions from adiabatic phonons only. The results obtained are discussed in the context of the problem of unusually high superconducting transition temperature of FeSe monolayer on STO.
The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional electron-phonon superconductors, including the retardation of the interaction and the effect of the Coulomb pseudopotential, to predict the critical temperature $T_c$ and other properties. McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature predicted by this theory, which depends essentially on the electron-phonon spectral function $alpha^2F(omega)$, using $alpha^2F$ for low-$T_c$ superconductors. Here we show that modern machine learning techniques can substantially improve these formulae, accounting for more general shapes of the $alpha^2F$ function. Using symbolic regression and the sure independence screening and sparsifying operator (SISSO) framework, together with a database of artificially generated $alpha^2F$ functions, ranging from multimodal Einstein-like models to calculated spectra of polyhydrides, as well as numerical solutions of the Eliashberg equations, we derive a formula for $T_c$ that performs as well as Allen-Dynes for low-$T_c$ superconductors, and substantially better for higher-$T_c$ ones. The expression identified through our data-driven approach corrects the systematic underestimation of $T_c$ while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen-Dynes formula for the prediction of higher-temperature superconductors and for the estimation of $lambda$ from experimental data.
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