Kondo impurities in strong coupling superconductors

Generalized Eliashberg equations including the effects of Kondo impurities in strong coupling superconductors are given. The Kondo effect is treated using the Matsuura, Ichinose, Nagaoka (MIN) theory. The transition temperatureT _c is calculated from the general equations by using the square well model for the electron-phonon interaction. Deviations of the present results from those of MIN are significant. In general, the depression ofT _c by the impurities is softened.On leave of absence from Department of Physics, Srinakharinwirot University, Prasarnmitr, Bangkok, Thailand.     

Continuous change from weak to strong coupling behavior in quasi-one-dimensional superconductors

The current-induced breakdown of superconductivity in whiskers from the In-Pb system has been investigated. For indium-rich alloys the characteristics show the well-known voltage step structure due to the appearance of phase-slip centers. From the linear portion following the first voltage step in the voltage-current characteristics at fixed temperatures, the quasiparticle diffusion length and the time-averaged supercurrent above the critical current have been calculated. In the intermetallic phase the interpretation becomes more involved. For pure lead there is no separation of the linear portions by voltage steps. A comparison with theoretical work is given.     

Thermodynamics of strong coupling superconductors including the effect of anisotropy

The thermodynamics of several elemental superconductors is computed from isotropic Eliashberg theory formulated on the imaginary frequency axis. A symmary of the available experimental literature is presented and a comparison with theory is given. The small disagreements that are found are all in the direction expected from anisotropy effects. We calculate the effect of a small amount of model anisotropy on the critical temperature, critical field, and high-temperature specific heat from an exact solution of the anisotropic Eliashberg equations. These are the first such results below the critical temperature; unlike previous analytical work, we include retardation, anisotropy in the mass enhancement, and the effect of the Coulomb repulsion in enhancing anisotropy, all of which are significant. We derive a new formula independent of any model anisotropy for the rate of decrease with impurity lifetime of the critical temperature. Finally we demonstrate how the commonly used formulas of Markowitz and Kadanoff and of Clem may give entirely misleading estimates of the gap anisotropy when used to interpret certain experiments.Research supported by the Natural Sciences and Engineering Research Council of Canada.     

On the critical temperature of strong coupling superconductors with paramagnetic impurities

We calculate the effects of paramagnetic impurities on the critical temperature of several intermediate and strong coupling superconductors by numerically solving the Eliashberg equations, and compare the results with the results of Abrikosov-Gorkov theory. For all the superconductors considered, the values of critical impurity concentration obtained from this exact numerical calculation are found to be considerably larger than the values given by Abrikosov-Gorkov theory. By generalizing a previous work of Leavens and Carbotte, we derive simple analytical expressions for the critical temperature and the critical concentration that give results much closer to the exact values than the corresponding BCS values.     

Crossover between weak and strong coupling in 2D superconductors

We study the crossover between weak coupling and strong coupling in two-dimensional (2D) superconductors atT⊋0 in the weak coupling regime, described by the BCS state, and we calculate the order parameter and the chemical potential atT=0 andT⊋0 for the model of a 2D superconductor. In the strong coupling regime the energy excitation has the scale of the pairs generated by the attractive interaction. The order parameter and the chemical potential atT⊋0 are calculated, and we show that there is a continuous transition between the two regimes.     

Temperature dependence ofH c2 andк 1 in strong coupling superconductors

According to the law of corresponding states, the properly scaled critical fields of a BCS superconductor are universal functions of the reduced temperatureT/T _c . In realistic superconductors strong-coupling effects can cause deviations from this universal behavior. We present numerical results for the strong-coupling anomalies in the temperature dependence of the upper critical fieldH _c2 and the Ginzburg-Landau parameter κ₁. The calculations are based on the Eliashberg theory for strong-coupling superconductors and are performed for several realistic superconductors as well as for a series of model superconductors with increasing coupling strength. It is demonstrated that the magnitude of the deviations can be estimated from the magnitude of the ratioT _c /〈ω〉, where〈ω〉 is an average phonon frequency. This rule does not apply to amorphous superconductors. The reason for this is shown to be due to the large weight ofα ² F(ω) at low frequencies which has been observed in these materials.     

Specific heat difference functional derivative within strong coupling theory

A formal expression is derived for the functional derivative of the specific heat difference with respect to the electron-phonon spectral density valid for strong coupling superconductors and for all temperatures. A numerical solution of the Eliashberg equations is required to obtain results for the functional derivative. Moreover, a previous weak coupling approximation has been extended into the strong coupling regime and analytic expressions for the functional derivative atT _c have been derived. Finally, the way in which this work encompasses the previous weak coupling results is discussed.     

On the influence of Fermi surface anisotropy onH c2 of weak and strong coupling superconductors

The influence of Fermi surface anisotropy on the upper critical fieldH _c2(T) of bulk type-II strong and weak coupling superconductors is investigated. It is found that anisotropic superconducting parameters can cause an enhancement ofH _c2(0) of more than 30% and can thus explain the 30% difference between measured values ofH _c2(0) and the prediction of weak coupling theories which recently has been found in the system Nb _75±x Pt _25x .Research supported by Sonderforschungsbereich 125.     

Spectral shape dependence in the very strong coupling regime of Eliashberg theory

The effect of ² F() shape dependence on several physical properties of superconductors is studied at various values of the strong coupling indexT _c/ _ln . Our results indicate that the degree of shape dependence of each property is sensitive to the value ofT _c/ _ln . Generally, for the region we examine, 0.25T _c/ _ln 1.3, the dependence on shape is found to be higher than in the conventional strong coupling regimeT _c/ _ln 0.2. However, with the exception of the mass enhancement parameter , the amount of shape dependence does not increase steadily withT _c/ _ln and there appears to be regions of maximum shape sensitivity.     

Approximating the electron-phonon coupling constant in superconductors from thermodynamic data

A method is discussed for estimating the electron-phonon coupling constant _ep from the jump in heat capacity at the superconducting transition which can be enhanced above the BCS value by strong coupling effects. Although based on a simple model of the superconducting state, reasonable agreement is found with published experimental data for a variety of crystalline superconductors, none of which is strongly anisotropic.     

Spin-fluctuation theory of strong coupling corrections to the free energy in superfluid 3He

Previous spin-fluctuation theories yielding the strong coupling corrections \- _i to the five coefficients of the fourth-order invariants in the free energy functional are extended. First, the superfluid part of the susceptibility is calculated up to order ⁴ for all momenta and frequencies and the contribution arising from p-wave fluctuations of the order parameter is included. Then the frequency sums yielding the \- _i are calculated by taking into account the full momentum and frequency dependence of the superfluid susceptibility and the spin fluctuation propagator. The results for the \- _i are plotted vs. a cutoff q_c on the momentum integration for spin-fluctuation parameters N(0)I = 0.75 and = 0.95. The cutoff takes into account in a rough way the effect of additional terms in the free energy functional which were neglected in previous theories. These additional terms are due to the implicit dependence of the superfluid susceptibility on the spin-fluctuation parameter I via the gap parameter . The gap equation providing the relation between and I is derived in the weak coupling approximation. The cutoffs obtained by fitting the experimental values of the three combinations of \- _i (arising from the measured specific heat discontinuities on the melting curve) are comparable to the cutoff obtained from the spin-fluctuation contribution to the weak coupling freeenergy [q _c = 0.3(2k _F ) for = 0.75]. The corrections due to the momentum and frequency dependence of the superfluid susceptibility and the spinfluctuation propagator are large and point in the direction of better agreement with experiment: The ratio % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiOuamaaBa% aaleaacaaIXaaabeaakiabg2da9iabfs5aejqbek7aIzaaraWaaSba% aSqaaiaaiwdaaeqaaOGaai4laiaacIcacqqHuoarcuaHYoGygaqeam% aaBaaaleaacaaIYaaabeaakiabgUcaRiabfs5aejqbek7aIzaaraWa% aSbaaSqaaiaaisdaaeqaaOGaaiykaaaa!47E0!\[R_1 = \Delta \bar \beta _5 /(\Delta \bar \beta _2 + \Delta \bar \beta _4 )\]decreases from 2 to about 1.2. However, in view of the large discrepancy with experiment (experimentally, R ₁= 0.14), we conclude that spin-fluctuation theory in its present form (without taking into account renormalization effects) cannot account quantitatively for the measured specific heat discontinuities on the melting curve.     

Temperature of a decoherent oscillator with strong coupling

The temperature of an oscillator coupled to the vacuum state of a heat bath via Ohmic coupling is non-zero, as measured by the reduced density matrix of the oscillator. This study shows that the actual temperature, as measured by a thermometer, is still zero (or, in the thermal state of the bath, the temperature of the bath). The decoherence temperature is due to 'false-decoherence', with a correlation between the oscillator and the heat bath causing the decoherence, but the heat baths state dragged along with the state of the oscillator.     

Collective modes and a theory of response ofd-wave superconductors

We present a theory of the response ofd-wave superconductors to weak applied fields, by taking account of the Coulomb interaction and all the collective degrees of freedom as well as crystal symmetry. We choose two representative phases: the d phase, which has point nodes in the energy gap, and theY _2–1 phase, which has line as well as point nodes. The former is a self-consistent solution for cubic as well as spherical symmetries and the latter is one for spherical, cubic, and hexagonal symmetries. We obtain obviously gauge-invariant expressions for the order-parameter fluctuations and the currents, having forms common not only to thed-wave states, but also to thep-wave states studied earlier. We also investigate the collective excitations; in the long-wavelength limit for spherically symmetric systems, there are, on the frequency-temperature plane, seven branches for eachd-wave phase considered, in addition to the common plasma mode and orbital Goldstone modes resulting from the spontaneous breakdown of the rotational invariance. In theY _2–1-phase two eigenmodes are found to become gapless at a finite temperature, below which they are purely imaginary. This implies instability of the phase. The effect of crystal anisotropy on the collective spectra is also studied.A preliminary report on the present work was published inJpn. J. Appl. Phys. Suppl. 26-3, 167 (1987).     

On the theory of resistive states in superconductors

The contribution R _s of a superconductor S to the resistance of a superconductor-normal metal (S-N) system is calculated. The case when the characteristics of metals S and N are different (different mean free path, density of states, etc.) is considered. It is shown that the value and temperature dependence of R _s differ from those obtained earlier. The resistance of the S-N-S system and of a system consisting of alternating S-N layers is calculated. The excess current I _exc in contacts S₁-c-S₂ (S_1,2 are different superconductors, c denotes a narrow constriction) is determined. The influence of magnetic field and of paramagnetic impurities on the value of the I _exc is found.     

Relevance of electron-phonon coupling in high-T c superconductors

The experimental observation of phonon renormalization in high-T_c superconductors gives clear evidence for the action of electron-phonon coupling. But in spite of the estimated coupling strength 1–3, these effects do not allow for a unique identification of the coupling mechanism. This situation is contrasted with that for the low-T_c superconductors where tunneling spectroscopy provides a hard proof for electron pairing via phonons.     

Studies of the strong coupling and weak coupling methods in FSI analysis

The weak coupling methods in fluid–structure interaction analysis are newly classified into three types; the weak coupling method for solving structures with interfaces, the weak coupling method for solving fluids with interfaces, and the weak coupling method for solving both fluids and structures with interfaces. The consistent added matrices of these weak coupling methods are derived from the condensation of the strong coupling formulation. Some approximations for the consistent added matrices, which can avoid the matrix coupling, are proposed. The reasons for convergence difficulty in the weak coupling methods are clarified. A number of numerical results are presented to investigate the convergence properties and computational efficiency of these methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Enhanced electron-phonon coupling and phonon-plasmon mixing in the high-temperature superconductors

It is shown, using La₂CuO₄ as an example, that the already large (nonlocal) electron-phonon coupling as calculated in the adiabatic approximation is further strongly enhanced for certain phonons in the nonadiabatic regime, where dynamical screening comes into play. At the same time, a low-lying plasmon mode of the electronic system appears for wave vectors in the (0,0,1) direction which mixes strongly with the phonons of appropriate symmetry and leads in addition to the phonons to very large changes of the crystal potential. Such a coexistence of phononlike and plasmonlike modes in the same frequency range is argued to be a specific feature of the high-temperature superconductors(HTSC's) related to their layered structure and weak screening of the strong ionic forces in c direction. It leads to a significant increase of the retarded attractive interaction.