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Register a new userLimits of Eliashberg Theory and Bounds for Superconducting Transition Temperature
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M.V. Sadovskii
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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.
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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.
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 $T_c<frac{2}{pi^2}lambda D$.
We show that vertex corrections to Migdals theorem in general induce an odd-frequency spin-triplet superconducting order parameter, which coexists with its more commonly known even-frequency spin-singlet counterpart. Fully self-consistent vertex-corrected Eliashberg theory calculations for a two dimensional cuprate model, isotropically coupled to an Einstein phonon, confirm that both superconducting gaps are finite over a wide range of temperatures. The subordinate $d$-wave odd-frequency superconducting gap is found to be one order of magnitude smaller than the primary even-frequency $d$-wave gap. Our study provides a direct proof of concept for a previously unknown generation mechanism of odd-frequency superconductivity as well as for the generic coexistence of both superconducting states in bulk materials.
Specific heat and resistivity measurements were performed on polycrystalline samples of the solid-solution Y_xLu_(1-x)Ni2B2C in order to determine thermodynamic properties such as the specific-heat difference, the thermodynamic critical field H_c(T), as well as the upper critical field H_{c2}(T). These properties were analyzed within the Eliashberg theory including anisotropy effects, yielding electron-phonon coupling anisotropy parameters <a_k^2> ranging beween 0.02 and 0.03 for the whole series, and Fermi velocity anisotropy parameters of <b_k^2> = 0.245-0.3. Excellent agreement between theory and experiment was achieved for these parameters, the Sommerfeld constant and model phonon spectra determined from specific heat measurements. An analysis of the previously investigated boronitride La3Ni2B2N3 for comparison revealed the electron-phonon anisotropy to be of great significance in describing its thermodynamic properties and the calculations yielded <a_k^2> ~ 0.08 and <b_k^2> ~ 0.245. The T_c-behavior within the series Y_xLu_(1-x)Ni2B2C is discussed in terms of coupling and impurity effects, and the density of states at the Fermi level N(0).
The newly discovered iron pnictide superconductors apparently present an unusual case of interband-channel pairing superconductivity. Here we show that, in the limit where the pairing occurs within the interband channel, several surprising effects occur quite naturally and generally: different density-of-states on the two bands lead to several unusual properties, including a gap ratio which behaves inversely to the ratio of density-of-states; the weak-coupling limit of the Eliashberg and the BCS theory, commonly taken as equivalent, in fact predict qualitatively different dependence of the $Delta_{1}/Delta_{2}$ and $Delta/T_{c}$ ratios on coupling constants. We show analytically that these effects follow directly from the interband character of superconductivity. Our results show that in the interband-only pairing model the maximal gap ratio is $sqrt{N_{2}/N_{1}}$ as strong-coupling effects act only to reduce this ratio. This suggests that if the large experimentally reported gap ratios (up to a factor 2) are correct, the pairing mechanism must include more intraband interaction than is usually assumed.
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