Eliashberg Equations Research Articles

Perturbation Theory on the Superconductivity of Heavy Fermion Superconductor UPd2Al3

We investigate the superconducting mechanism and the transition temperature of heav y fermion superconductor UPd_2Al_3 on the basis of a single band two-dimensional Hubbard model on triangular lattice, whi ch represents the most heavy band of UPd_2Al_3. Both normal and anomalous self-energies are calculated up to third order with respect to the Coulomb repulsion U between itinerant electrons. The superconducting transition temperature is obtained by solving the Eliashberg's equation. Reasonable transition temperatures are obtained for moderately large U. It is fou nd that the momentum and frequency dependence of spin fluctuations given by RPA-like terms gives rise to the d-wave pairing state, while the vertex correction terms are important for obtaining reasonable transition tempe ratures. These results seem to show that the superconductivity in UPd_2Al_3 can be explained by the perturbation theory with respect to U.

Superconductivity near itinerant ferromagnetic quantum criticality.

Superconductivity mediated by spin fluctuations in weak and nearly ferromagnetic metals is studied close to the zero-temperature magnetic transition. We solve analytically the Eliashberg equations for p-wave pairing and obtain the quasiparticle self-energy and the superconducting transition temperature T(c) as a function of the distance to the quantum critical point (QCP). We show that the reduction of quasiparticle coherence and lifetime due to scattering by quasistatic spin fluctuations is the dominant pair-breaking process, which leads to a rapid suppression of T(c) to a nonzero value near the QCP. We point out the differences and similarities of the problem to that of paramagnetic impurities in superconductors.

Scaling of Coulomb pseudo-potential in narrow-band superconductors

The standard Eliashberg theory is extended to incorporate the narrow-band effects, that is, the vertex corrections and frequency dependence of the effective interaction between electrons. From the extended Eliashberg equation, the Coulomb pseudo-potential μ * is extracted. It is shown that even for narrow-band superconductors, where the Fermi energy ϵ F is comparable with the phonon frequency ω ph, the Coulomb pseudo-potential is a pertinent parameter, and is still given by μ *= μ/[1+ μln( ϵ F/ ω ph)], provided ω ph is appropriately scaled.

Constraints fromTcand the isotope effect inMgB2

With the constraint that ${T}_{c}=39\mathrm{K},$ as observed for ${\mathrm{MgB}}_{2},$ we use the Eliashberg equations to compute possible allowed values of the isotope coefficient \ensuremath{\beta}. We find that while the observed value ${\ensuremath{\beta}}_{\mathrm{obs}}=0.32\ifmmode\pm\else\textpm\fi{}0.01$ can be obtained in principle, it is difficult to reconcile a recently calculated spectral function with such a low observed value.

Infrared absorption and reflection spectra of high-density electron–hole systems

We analyze infrared absorption and reflection spectra of high-density electron–hole (e–h) systems. The BCS-like energy gap (BEG) is found in the calculated spectra. We incorporate the large quantum fluctuation with the quasi-static Eliashberg equation for e–h systems which allows us to calculate the renormalized band energy, the BEG, and the Coulomb renormalization factor. We find that the collective fluctuation significantly modifies the spectra, and that the strong visible-light excitation distinctly stabilizes the e–h BCS state.

Study of Superconductivity in Borocarbides on the Basis of Perturbation Theory

The borocarbide YNi 2 B 2 C with quasi-2-dimensional structure encounters a superconducting transition at T c = 15.4 K, the pairing symmetry of which is not clarified; most of experiments suggest an extended s -wave state, while some claim a d -wave. In this paper, we investigate a possibility of a d -wave state, solving the Eliashberg equation within the third order perturbation theory for a repulsive Hubbard model with the band structure reproducing the main Fermi surface. It is shown that if the electron correlation gives the dominant contribution to the superconducting pairing interaction, the d x y -symmetry is the most favorable state corresponding to the (π, 0) peak structure of the spin susceptibility.

Perturbation Analysis of Pairing Symmetry and Transition Temperature in Quasi-One-Dimensional Organic Superconductors

We investigate pairing symmetry in quasi-one-dimensional organic superconductors, solving Eliashberg's equation within third order perturbation theory for a repulsive Hubbard model. According to the calculation results, spin-singlet state is expected to give higher transition temperature than spin-triplet one and the vertex corrections reduce the transition temperature for the spin-singlet case. It is shown that the contributions of antiferromagnetic spin fluctuations dominate strongly the momentum dependence of the effective pairing interaction for both of the spin-triplet and the spin-singlet pairing cases.

Superconductivity of Rb 3 C 60 : breakdown of the Migdal-Eliashberg theory

In this paper, through an exhaustive analysis within the Migdal-Eliashberg theory, we show the incompatibility of experimental data of Rb$_3$C$_{60}$ with the basic assumptions of the standard theory of superconductivity. For different models of the electron-phonon spectral function $\alpha^2F(\Omega)$ we solve numerically the Eliashberg equations to find which values of the electron-phonon coupling $\lambda$, of the logarithmic phonon frequency $\Omega_{ln}$ and of the Coulomb pseudopotential $\mu^*$ reproduce the experimental data of Rb$_3$C$_{60}$. We find that the solutions are essentially independent of the particular shape of $\alpha^2F(\Omega)$ and that, to explain the experimental data of Rb$_3$C$_{60}$, one has to resort to extremely large couplings: $\lambda=3.0\pm 0.8$. This results differs from the usual partial analyses reported up to now and we claim that this value exceeds the maximum allowed $\lambda$ compatible with the crystal lattice stability. Moreover, we show quantitatively that the obtained values of $\lambda$ and $\Omega_{ln}$ strongly violate Migdal's theorem and consequently are incompatible with the Migdal-Eliashberg theory. One has therefore to consider the generalization of the theory of superconductivity in the nonadiabatic regime to account for the experimental properties of fullerides.

Superconductivity in the presence of pseudogap: strong-coupling approach

We discuss the structure of the pseudogap, which originates from the charge-density wave (CDW). For imperfect nesting the density of states at Fermi level remains finite despite the opening of the CDW gap. Its depth depends on the concentration of holes and decreases with doping. This property can significantly affect the competition between CDW and superconductivity. We apply the Eliashberg equations and obtain analytical results for the superconducting transition temperature in the presence of the pseudogap of arbitrary magnitude and depth. We demonstrate that the isotope effects for the superconducting transition temperature ( α) and the pseudogap ( α ∗ ) are inversely correlated with each other. In particular, doping dependence of the superconducting isotope effect observed in underdoped high-temperature superconductors, can originate from opening of the normal-state pseudogap, provided that α ∗<1/2 .

Crib-shaped triplet-pairing gap function for an orthogonal pair of quasi-one-dimensional Fermi surfaces inSr2RuO4

The competition between spin-triplet and singlet pairings is studied theoretically for the tight-binding $\ensuremath{\alpha}--\ensuremath{\beta}$ bands in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4},$ which arise from two sets of quasi-one-dimensional Fermi surfaces. Using the multiband fluctuation exchange approximation, where we incorporate an anisotropy in the spin fluctuations as suggested from experiments, we show that (i) the triplet can dominate over the singlet (which turns out to be extended s), and (ii) the triplet gap function optimized in the Eliashberg equation has an unusual, very nonsinusoidal form, whose time-reversal-broken combination exhibits a crib-shaped amplitude with dips.

Charge-Density-Wave Pseudogap and Two-Dimensional Superconductivity Within Strong-Coupling Description

We have considered the role of charge-density-pseudogap for phonon-mediated super-conductivity on two-dimensional lattice. The propagators that enter generalized Eliashberg equations have been renormalized to account for quasi-particle energies related to the formation of the pseudogap, which has been assumed to be of d-wave symmetry. We have evaluated the superconducting transition temperature Tc as a function of doping. It occurs that Tc for d-wave symmetry well reflects experimental behavior. Our results for the isotope shift exponent α show that at low doping, the presence of the pseudogap may contribute to α > 1/2 as well as to α > 1/2 values.

S+d pairing in orthorhombic phase of copper-oxides

A microscopical theory of electronic spectrum and superconductivity is formulated within the two-dimensional anisotropic t- J model with t x ≠ t y and J x ≠ J y . Renormalization of electronic spectrum and superconductivity mediated by spin-fluctuations are investigated within the Eliashberg equation in the weak coupling approximation. The gap function has d + s symmetry with the extended s-wave component being proportional to the asymmetry t y − t x . Some experimental consequences of the obtained results are discussed.

Solution of real-axis Eliashberg equations with different pari symmetries and tunneling density of states

The real-axis direct solution of the Eliashberg equations for the retarded electron-boson interaction in the half-filling case and in the presence of impurities is obtained for six different symmetries of the order parameter: s, s + i d, s + d, d, anisotropic- s and extended- s. The spectral function is assumed to contain an isotropic part α is 2F(Ω) and an anisotropic one α an 2F(Ω) such that α is 2F(Ω) = g·α an 2F(Ω) , where g is a constant, and the Coulomb pseudopotential μ ∗ is set to zero for simplicity. The density of states is calculated for each symmetry at T = 2, 4, 40 and 80 K. The resulting curves are compared to those obtained by analytical continuation of the imaginary-axis solution of the Eliashberg equations and to the experimental tunneling curves of optimally-doped Bi 2212 crystals.

Resistivity and electron-phonon coupling in YNi 2B 2C single crystals

In this work, we present the results of precise measurements of the resistivity of YNi 2B 2C single crystals with T c = 15.5K as a function of the temperature, and analyze the experimental data in the framework of the Bloch-Grüneisen theory and electron-phonon coupling. The transport electron-phonon spectral function that best fits the resistivity data is then inserted in the real-axis Eliashberg equations, which are directly solved to determine the normalized tunneling conductance both in s- and d-wave symmetry.

S- and d-wave solution of Eliashberg equations with finite bandwidth

In this work, we discuss the results of the direct solution of the Eliashberg equations with finite bandwidth, in the cases of s- and d-wave symmetry for the pair wave function and in the presence of scattering from impurities. We show that the reduction of the critical temperature T c due to the finite bandwidth depends on the value of the bandwidth itself, but is almost independent of the symmetry of the order parameter. The same happens for the shape of Z( ω) and Δ(ω). Moreover, we discuss the effect of the finite bandwidth on the shape of the quasiparticle density of states. The results clearly indicate that the infinite bandwidth approximation leads to an underestimation of the electron-boson coupling constant.

REAL-AXIS SOLUTION OF ELIASHBERG EQUATIONS IN VARIOUS ORDER-PARAMETER SYMMETRIES AND TUNNELING CONDUCTANCE OF OPTIMALLY-DOPED HTSC

In the present work we calculate the theoretical tunneling conductance curves of SIN junctions involving high-T c superconductors, for different possible symmetries of the order parameter (s, d, s + i d, s + d, anisotropics and extendeds). To do so, we solve the real-axis Eliashberg equations in the case of an half-filled infinite band. We show that some of the peculiar characteristics of HTSC tunneling curves (dip and hump at eV &gt; Δ, broadening of the gap peak, zero bias and so on) can be explained in the framework of the Migdal-Eliashberg theory. The theoretical d I/ d V curves calculated for the different symmetries at T=4 K are then compared to various experimental tunneling data obtained in optimally-doped BSCCO, TBCO, HBCO, LSCO and YBCO single crystals. To best fit the experimental data, the scattering by non-magnetic impurities has to be taken into account, thus limiting the sensitivity of this procedure in determining the exact gap symmetry of these materials. Finally, the effect of the temperature on the theoretical tunneling conductance is also discussed and the curves obtained at T = 2 K are compared to those given by the analytical continuation of the imaginary-axis solution.

ELECTRON-PHONON COUPLING ORIGIN OF THE RESISTIVITY IN YNi2B2C SINGLE CRYSTALS

Resistivity measurements from 4.2 K up to 300 K were made on YNi 2 B 2 C single crystals with T c = 15.5 K . The resulting ρ(T) curve shows a perfect Bloch-Grüneisen (BG) behavior, with a very small residual resistivity which indicates the low impurity content and the high cristallographic quality of the samples. The value λ tr = 0.53 for the transport electron-phonon coupling constant was obtained by using the high-temperature constant value of d ρ/ d T and the plasma frequency reported in literature. The BG expression for the phononic part of the resistivity ρ ph (T) was then used to fit the data in the whole temperature range, by approximating [Formula: see text] with the experimental phonon spectral density G(Ω) multiplied by a two-step weighting function to be determined by the fit. The resulting fitting curve perfectly agrees with the experimental points. We also solved the real-axis Eliashberg equations in both s- and d-wave symmetries under the approximation [Formula: see text]. We found that the value of λ tr here determined in single-band approximation is quite compatible with T c and the gap Δ experimentally observed. Finally, we calculated the normalized tunneling conductance, whose comparison with break-junction tunnel data gives indication of the possible s-wave symmetry for the order parameter in YNi 2 B 2 C .

Calculated Nb superconducting transition temperature under hydrostatic pressure

A full-potential linear muffin-tin orbital method (FP-LMTO) based linear-response approach is used to calculate the electron-phonon coupling in Nb under hydrostatic pressure. The superconducting transition temperature Tc is calculated using the Eliashberg equation. The calculated Tc agrees nicely with the experimental result at ambient pressure, but the agreement is only fair at high pressures. The Tc measured anomaly at 60–70GPa is understood in terms of the 2.5 order Lifshitz transition and its origin is traced back to the qualitative changes in the Fermi surface topology.

Calculated Nb superconducting transition temperature under hydrostatic pressure

A full-potential linear muffin-tin orbital method (FP-LMTO) based linear-response approach is used to calculate the electron–phonon coupling in Nb under hydrostatic pressure. The superconducting transition temperature Tc is calculated using the Eliashberg equation. The calculated Tc agrees nicely with the experimental result at ambient pressure, but the agreement is only fair at high pressures. The Tc measured anomaly at 60–70 GPa is understood in terms of the 2.5-order Lifshitz transition and its origin is traced back to the qualitative changes in the Fermi surface topology.

-wave nonadiabatic superconductivity

The inclusion of nonadiabatic corrections to the electron-phonon interaction leads to a strong momentum dependence in the generalized Eliashberg equations beyond Migdal's limit. For a s-wave symmetry of the order parameter, this induced momentum dependence leads to an enhancement of when small momentum transfer is dominant. Here we study how the d-wave symmetry affects the above behavior. We find that the nonadiabatic corrections depend only weakly on the symmetry of the order parameter provided that only small momentum scatterings are allowed for the electron-phonon interaction. In this situation, We show that also for a d-wave symmetry of the order parameter, the nonadiabatic corrections enhance . We also discuss the possible interplay and crossover between s- and d-wave depending on the material's parameters.