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Register a new userAntiadiabatic phonons, Coulomb pseudopotential and superconductivity in Eliashberg - McMillan theory
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M.V. Sadovskii
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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.
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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.
We examine the Coulomb pseudopotential $mu^*$ in the McMillan equation applying to the superconductivity of heavily doped semiconductors. Systematic calculation using the first-principles calculation suggests that $mu^*$ should be considered as a variable quantity depending on carrier density $n$ in semiconductors, although it is usually considered as a constant about 0.1. To clarify $n-$dependence of $mu^*$, we solve the McMillan equation inversely for $mu^*$ by combining the result of the first-principles calculation and that of experiments. It indicates that $mu^*$ decreases with $n$ and becomes negative under $n sim 5 times 10^{-21}[{rm cm^{-3}}]$. This reduction is explained by the effect of plasmon which may play an important role in the superconductivity of low carrier systems such as heavily doped semiconductors.
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.
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 extend the two leading methods for the emph{ab initio} computational descrip tion of phonon-mediated superconductors, namely Eliashberg theory and density fu nctional theory for superconductors (SCDFT), to include plasmonic effects. Furth ermore, we introduce a hybrid formalism in which the Eliashberg approximation fo r the electron-phonon coupling is combined with the SCDFT treatment of the dynam ically screened Coulomb interaction. The methods have been tested on a set of we ll-known conventional superconductors by studying how the plasmon contribution a ffects the phononic mechanism in determining the critical temperature (tc). Our simulations show that plasmonic SCDFT leads to a good agreement between predict ed and measured tcs, whereas Eliashberg theory considerably overestimates the plasmon-mediated pairing and, therefore, tc. The hybrid approach, on the other hand, gives results close to SCDFT and overall in excellent agreement with exper iments.
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