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Register a new userThe functional-integral approach to Gaussian fluctuations in Eliashberg theory
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The Eliashberg theory of superconductivity is based on a dynamical electron-phonon interaction as opposed to a static interaction present in BCS theory. The standard derivation of Eliashberg theory is based on an equation of motion approach, which incorporates certain approximations such as Migdals approximation for the pairing vertex. In this paper we provide a functional-integral-based derivation of Eliashberg theory and we also consider its Gaussian-fluctuation extension. The functional approach enables a self-consistent method of computing the mean-field equations, which arise as saddle-point conditions, and here we observe that the conventional Eliashberg self energy and pairing function both appear as Hubbard-Stratonovich transformations. An important consequence of this fact is that it provides a systematic derivation of the Cooper and density-channel interactions in the Gaussian fluctuation response. We also investigate the strong-coupling fluctuation diamagnetic susceptibility near the critical temperature.
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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.
We present a first-principles approach to describe magnetic and superconducting systems and the phenomena of competition between these electronic effects. We develop a density functional theory: SpinSCDFT, by extending the Hohenberg-Kohn theorem and constructing the non-interacting Kohn- Sham system. An exchange-correlation functional for SpinSCDFT is derived from the Sham Schluter connection between the SpinSCDFT Kohn-Sham and a self-energy in Eliashberg approximation. The reference Eliashberg equations for superconductors in the presence of magnetism are also derived and discussed.
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 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 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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