Temperature dependence of the energy gap in Bi2223 metal oxide superconductor

Spectroscopic studies of the silver-optimum-doped Bi2223 contacts show that the temperature dependence of the parameter Δ follows the BCS curve. However, the tunnel measurements performed for the same series of specimens did not reveal any temperature dependence of the energy gap Δ. The feature observed in the tunnel density of states was retained at temperatures T>T _c , manifesting the presence of the temperature-independent pseudogap E _p . The difference between the data obtained with tunnel spectroscopy and Andreev reflection spectroscopy is explained by the fact that the latter measures the true superconducting energy gap Δ_s(T), whereas the peaks of the tunneling conductivity are related to the total energy gap Δ of cuprates, which includes both the parameter Δ_s and the pseudogap $E_p :\Delta \approx \sqrt {\Delta _s^2 + E_p^2 } $ .     

A precursor superconductivity approach to magnetic field effects in the pseudogap phase

We demonstrate that the observed dependences of T_c and T^∗ on small magnetic fields can be readily understood in a precursor superconductivity approach to the pseudogap phase. In this approach, the presence of a pseudogap at T_c (but not at T^∗) and the associated suppression of the density of states lead to very different sensitivities to pair-breaking perturbations for the two temperatures. Our semi-quantitative results address the puzzling experimental observation that the coherence length ξ is weakly dependent on hole concentration x throughout most of the phase diagram. We present our results in a form which can be compared with the recent experiments of Shibauchi et al. and argue that orbital effects contribute in an important way to the H dependence of T^∗.     

Theory of high-T c superconductivity based on the fermion-condensation quantum phase transition

A theory of high-temperature superconductivity based on the combination of the fermion-condensation quantum phase transition and the conventional theory of superconductivity is presented. This theory describes maximum values of the superconducting gap, which can be as big as Δ₁～0.1ε _F , with ε _F being the Fermi level. We show that the critical temperature 2T _c ?Δ₁. If the pseudogap exists above T _c , then 2T*?Δ₁ and T* is the temperature at which the pseudogap vanishes. A discontinuity in the specific heat at T _c is calculated. The transition from conventional superconductors to high-T _c ones as a function of the doping level is investigated. The single-particle excitations and their lineshape are also considered     

Point-contact study of the heavy-fermion superconductor CeCoIn5

We have investigated the differential conductance spectra of the point contacts between the heavy-fermion superconductor CeCoIn₅ and Pt. Many of them show a double-maximum structure that indicates the superconducting energy gap Δ. The Δ values derived using Blonder-Tinkham-Klapwijk model, however, varies from 0.47 to 0.77 meV, and yet they are within the scatter of the reported values. The evolution of Δ below T_c is slow as compared with that of BCS gap probably reflecting the unconventional superconductivity in CeCoIn₅.     

Electron tunneling study on superconductivity of (Li0.65Na0.35)0.9Mo6O17

Quasiparticle tunneling measurement has been carried out for a single crystal of (Li_0.65Na_0.35)_0.9Mo₆O₁₇ through the point contact with Al electrode. The observed energy gap is consistent with the BCS value and the shape of the tunneling density-of-states confirms the intrinsic origin of the occurence of the superconductivity in the present compound. The result supports the idea that the same electron system exhibits two conflicting phenomena, the superconductivity and the sharp resistivity upturn above T_c with decreasing temperature.     

Quasiparticles in a strongly correlated liquid with the fermion condensate: applications to high-temperature superconductors

A model of a strongly correlated electron liquid based on fermion condensation (FC) is extended to high-temperature superconductors. Within our model, the appearance of FC presents a boundary separating the region of a strongly interacting electron liquid from the region of a strongly correlated electron liquid. We study the superconductivity of a strongly correlated liquid and show that, under certain conditions, the superconductivity vanishes at temperatures T > T _c ≈ T _node, with the superconducting gap being smoothly transformed into a pseudogap. As a result, the pseudogap occupies only a part of the Fermi surface. The gapped area shrinks with increasing the temperature and vanishes at T = _T*. The single-particle excitation width is also studied. The quasiparticle dispersion in systems with FC can be represented by two straight lines, characterized by the effective masses and, intersecting near the binding energy that is on the order of the superconducting gap. It is argued that this strong change of the quasiparticle dispersion upon binding can be enhanced in underdoped samples because of strengthening the FC influence. The FC phase transition in the presence of the superconductivity is examined, and it is shown that this phase transition can be considered as driven by the kinetic energy.     

Doping dependence of the superconducting gap by Andreev reflection in Au/La2−xSrxCuO4 point-contact junctions

We studied the doping dependence of the superconducting gap in La_2−xSr_xCuO₄ (LSCO) by means of Andreev reflection measurements in Au/LSCO point-contact junctions. The Andreev reflection features were found to disappear at T_c^A close to the bulk T_c. The fit of the conductance curves with the BTK-Tanaka-Kashiwaya model gives good results if a (s+d)-wave gap symmetry is used. The low-temperature dominant isotropic gap component Δ_s follows very well the T_c vs. x curve, while the gap-like features observed by angle-resolved photoemission spectroscopy and tunneling scale with T^∗. This confirms the different origin of these two energy scales at T<T_c.     

Superconducting energy gap in Bi2Sr2Ca2Cu3O10+δ (Bi2223) single crystals

Current-voltage characteristics of S-I-S tunnel break junctions fabricated from pure undoped Bi2223 single crystals (T _c =110 K) were measured. High quality of the crystals enabled production of good tunnel junctions with a low or almost zero leakage current and well developed gap structure in the tunneling spectra. The peak-to-peak energy gap values 2Δ_p-p in different crystals and the tunnel junctions ranged from 80 to 105 meV. The tunneling conductance in the superconducting state was normalized to that in the normal state and compared to a smeared BCS density of states. A simple fit of the data gave the average value of Δ=38.5 meV and reduced gap 2Δ/kT _c ?8, consistent with a very strong coupling mechanism.     

Tunneling in single-layer Bi2Sr2CuO6+δ single crystals in high magnetic field

In tunneling experiments with high-quality single crystals of a single-layer cuprate superconductor Bi₂Sr₂CuO_6+δ using the break junction and point-contact techniques at T<T _c, the coexistence of the superconducting-state gap and the normal-state gap was observed. The values of the superconducting energy gap 2Δ_p?p are in the range from 13.4 to 15 meV (Δ_p?p=6.7–7.5 meV). The values of 2Δ_p?p are similar for two samples with T _c=4 K and for two samples with T _c=9–10 K and are independent of the carrier concentration. The normal-state gap, with the magnitude approximately equal to 50 meV, persists at T<T _c and in the magnetic field H?H _c2 up to 28 T. After the transition of the sample to the normal state, the intensity of the tunneling conductance rapidly decreases with increasing magnetic field strength and temperature. The observed large broadening of the tunneling spectra and large zero-bias conductances can be caused by a strong angular dependence of the superconducting gap. The tunneling results are in full agreement with the data of the angle-resolved photoemission spectroscopy measurements.     

Strong coupling superconductivity and prominent superconducting fluctuations in the new superconductor Bi4O4S3

Electric transport and scanning tunneling spectrum(STS)have been investigated on polycrystalline samples of the new superconductor Bi4O4S3.A weak insulating behavior in the resistive curve has been induced in the normal state when the superconductivity is suppressed by applying a magnetic field.Interestingly,a kink appears on the temperature dependence of resistivity near 4 K at all high magnetic fields above 1 T when the bulk superconductivity is completely suppressed.This kink associated with the upper critical field as well as the wide range of excess conductance at low fields and high temperatures is explained as the possible evidence of strong superconducting fluctuation.From the tunneling spectra,a superconducting gap of about 3 meV is frequently observed yielding a ratio of 2Δ/kB TC~16.6.This value is much larger than the one predicted by the BCS theory in the weak coupling regime(2Δ/kB TC~3.53),which suggests the strong coupling superconductivity in the present system.Furthermore,the gapped feature persists on the spectra until 14 K in the STS measurement,which suggests a prominent fluctuation region of superconductivity.Such a superconducting fluctuation can survive at very high magnetic fields,which are far beyond the critical fields for bulk superconductivity as inferred both from electric transport and tunneling measurements.     

Unusual behavior of nuclear relaxation in CeCu2Si2 “possible evidence for triplet superconductivity”

Nuclear relaxation of ⁶³Cu in the superconducting state of the Kondo-lattice system CeCu₂Si₂ has been studied with the use of the ⁶³Cu nuclear quadrupole resonance technique under zero field and down to 65mK. The nuclear spin-lattice relaxation rate (1/T₁) decreases drastically just below T_c=0.67 K down to 0.5T_c without the apparent enchanced behavior and then is found to be almost temperature independent below 0.3T_c. These results suggest that the superconductivity in CeCu₂Si₂ is not in the usual BCS regime. The analysis based upon the existing triplet pairing model with an anisotropic energy gap describes well the behavior from T_c down to 0.5T_c, while the temperature independence below 0.3T_c remains unexplained.     

Superconducting behavior of amorphous Zr70Cu30

Measurement of the Meissner penetration depth, λ(T) were made in amorphous Zr₇₀Cu₃₀ samples. The results indicate that this amorphous alloy behaves as a BCS superconductor with 2Δ(0)?kT_c = 3.8, where Δ(0) is the superconducting energy gap at T=0 and T_c the critical temperature. It is also concluded that the low energy excitation, TLS, characteristics of amorphous material does not contribute to T_c.     

Tunneling spectroscopy of layered superconductors: intercalated Li0.48(C4H8O)xHfNCl and De-intercalated HfNCl0.7

Tunneling measurements have been carried out on layered superconductors of the β(SmSI)-type – Li_0.48(THF)_xHfNCl (THF?=?C₄H₈O) and HfNCl_0.7 – by means of break-junction and scanning tunneling spectroscopy. Break-junction technique reveals Bardeen-Cooper-Schrieffer (BCS) – like gap structures with typical gap values of 2Δ (4.2 K) = 11–12 meV for Li_0.48(THF)_xHfNCl with the highest T_c = 25.5 K. Some of our measurements revealed multiple gaps and dip-hump structures, the largest gap 2Δ (4.2 K) ≈ 17–20 meV closing at T_c. This was shown both by break-junction and scanning-tunneling spectroscopy. From these experiments it stems that the highest obtained gap ratio 2Δ/k_BT_c ~ 8 substantially exceeds the BCS weak-coupling limiting values: ≈3.5 and ≈4.3 for s-wave and d-wave order parameter symmetry, respectively. Such large 2Δ/k_BT_c ratios are rather unusual for conventional superconductors but quite common to high-T_c cuprates, as well as to organic superconductors. Our studies allowed to collect much more evidence concerning the huge pairing energy in those materials and to investigate in detail the complexity of their superconducting gap spectra. An origin of the observed phenomena still remains to be clarified.     

d-Wave superconductivity via buckling-like phonon mode

We follow the classic strong-coupling theory of superconductivity through electron-phonon interaction with a buckling-like phonon mode. We find a nonzero d-wave order parameter in the sense of the Eliashberg theory. We derive a zero temperature gap Δ(0,π) at the gap edge versus the electron-phonon coupling strength g² relation. We find that large enough value for Δ(0,π) as compared to those of high-T_c superconductors cannot be realized in the electron-buckling-like-phonon coupling on the CuO₂ planes.     

On the limits of consistency of Eliashberg theory and the density of states of high-T c superconductors

We make a detailed study of the Eliashberg theory in the coupling region where some fundamental qualitative deviations from the conventional BCS-like behavior begin to appear. These deviations are identified as the onset of a cross-over from BCS superconductivity to Bose condensation. We point out that the beginning of this cross-over occurs when the gap δ _g becomes comparable to the boson energies Ω_ph. This condition traduces the physical constraint that the distance the paired electron covers during the absorption of the virtual boson, cannot be larger than the coherence length. The frontier region of couplings is of the order of λ ≈ 3, and high-T _c, materials are concerned. A clear qualitative indication of the occurrence of a crossover regime should be a dip structure above the gap in the density of states of excitations, and this is one of the most robust characteristics of the high-T _c, superconducting state. Comparing our results with tunneling and photoemission experiments we conclude that high-T _c materials (cuprates and fullerides) are indeed at the beginning of a cross-over from BCS superconductivity to Bose condensation, even though the fermionic nature still prevails. If the Uemura plot is relevant, then the dip should also be present in the other materials that are close to the cross-over regime like heavy Fermion and organic superconductors. In all these materials Ginzburg Landau equations are irrelevant.     

Broad-band infrared ellipsometry measurements of the c-axis response of underdoped YBa2Cu3O7−δ: Spectroscopic distinction between the normal-state pseudogap and the superconducting gap

We present broad-band infrared ellipsometry measurements of the c-axis dielectric response of underdoped YBa₂Cu₃O_7−d single crystals. Our data provide a clear spectroscopic distinction between the normal-state pseudogap and the superconducting gap. In particular, they establish that different energy scales are underlying the respective redistributions of spectral weight. Furthermore, our data are suggestive of a mutual competition between the two gaps and thus of an extrinsic nature of the pseudogap with respect to superconductivity.     

Pseudogap and its temperature dependence in YBCO from the data of resistance measurements

The temperature dependence of the excess conductivity Δσ for Δσ = A(1 ? T/T^*)exp(Δ^*/T) (YBCO) epitaxial films is analyzed. The excess conductivity is determined from the difference between the normal resistance extrapolated to the low-temperature range and the measured resistance. It is demonstrated that the temperature dependence of the excess conductivity is adequately described by the relationship Δσ = A(1 ? T/T^*)exp(Δ^*/T). The pseudogap width and its temperature dependence are calculated under the assumption that the temperature behavior of the excess conductivity is associated with the formation of the pseudogap at temperatures well above the critical temperature T _c of superconductivity. The results obtained are compared with the available experimental and theoretical data. The crossover to fluctuation conductivity near the critical temperature T _c is discussed.     

Influence of the charge concentration on the density of states in a two-dimensional superconducting system

The gap and the density of states of high-T_c superconductors have been a subject of paramount interest. In order to explain the observed experimental behavior several pairing mechanisms in high-temperature superconductivity have been considered, by theoretical calculations. In this work, within the BCS scheme, a two-band model with energy band overlapping is introduced. The gap parameter and the density of states in a two-dimensional superconducting system are studied as functions of the charge concentration. This model is applied to Bi2212 in order to obtain numerical results.     

Fractal superconductivity near localization threshold

We develop a semi-quantitative theory of electron pairing and resulting superconductivity in bulk “poor conductors” in which Fermi energy E_F is located in the region of localized states not so far from the Anderson mobility edge E_c. We assume attractive interaction between electrons near the Fermi surface. We review the existing theories and experimental data and argue that a large class of disordered films is described by this model.Our theoretical analysis is based on analytical treatment of pairing correlations, described in the basis of the exact single-particle eigenstates of the 3D Anderson model, which we combine with numerical data on eigenfunction correlations. Fractal nature of critical wavefunction's correlations is shown to be crucial for the physics of these systems.We identify three distinct phases: ‘critical’ superconductive state formed at E_F = E_c, superconducting state with a strong pseudo-gap, realized due to pairing of weakly localized electrons and insulating state realized at E_F still deeper inside a localized band. The ‘critical’ superconducting phase is characterized by the enhancement of the transition temperature with respect to BCS result, by the inhomogeneous spatial distribution of superconductive order parameter and local density of states. The major new feature of the pseudo-gapped state is the presence of two independent energy scales: superconducting gap Δ, that is due to many-body correlations and a new “pseudo-gap” energy scale Δ_P which characterizes typical binding energy of localized electron pairs and leads to the insulating behavior of the resistivity as a function of temperature above superconductive T_c. Two gap nature of the pseudo-gapped superconductor is shown to lead to specific features seen in scanning tunneling spectroscopy and point-contact Andreev spectroscopy. We predict that pseudo-gapped superconducting state demonstrates anomalous behavior of the optical spectral weight. The insulating state is realized due to the presence of local pairing gap but without superconducting correlations; it is characterized by a hard insulating gap in the density of single electrons and by purely activated low-temperature resistivity ln R(T) ∼ 1/T.Based on these results we propose a new “pseudo-spin” scenario of superconductor-insulator transition and argue that it is realized in a particular class of disordered superconducting films. We conclude by the discussion of the experimental predictions of the theory and the theoretical issues that remain unsolved.     

Phase fluctuations and pseudogap phenomena

This article reviews the current status of precursor superconducting phase fluctuations as a possible mechanism for pseudogap formation in high-temperature superconductors. In particular we compare this approach which relies on the two-dimensional nature of the superconductivity to the often used T-matrix approach. Starting from simple pairing Hamiltonians we present a broad pedagogical introduction to the BCS–Bose crossover problem. The finite temperature extension of these models naturally leads to a discussion of the Berezinskii–Kosterlitz–Thouless superconducting transition and the related phase diagram including the effects of quantum phase fluctuations and impurities. We stress the differences between simple Bose–BCS crossover theories and the current approach where one can have a large pseudogap region even at high carrier density where the Fermi surface is well-defined. Green's function and its associated spectral function, which explicitly show non-Fermi liquid behavior, is constructed in the presence of vortices. Finally different mechanisms including quasi-particle–vortex and vortex–vortex interactions for the filling of the gap above T_c are considered.