Theory of High Temperature Superconductivity: A Conventional Approach

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This muddy situation possibly originated from the indirect nature of the experimental evidence, as well as experimental issues such as sample quality, impurity scattering, twinning, etc. This summary makes an implicit assumption : superconductive properties can be treated by mean field theory. It also fails to mention that in addition to the superconductive gap, there is a second gap, the pseudogap. The cuprate layers are insulating, and the superconductors are doped with interlayer impurities to make them metallic. The superconductive transition temperature can be maximized by varying the dopant concentration.

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The Sr impurities also act as electronic bridges, enabling interlayer coupling. Proceeding from this picture, some theories argue that the basic pairing interaction is still interaction with phonons , as in the conventional superconductors with Cooper pairs. While the undoped materials are antiferromagnetic, even a few percent of impurity dopants introduce a smaller pseudogap in the CuO 2 planes which is also caused by phonons. The gap decreases with increasing charge carriers, and as it nears the superconductive gap, the latter reaches its maximum.

The reason for the high transition temperature is then argued to be due to the percollating behaviour of the carriers - the carriers follow zig-zag percolative paths, largely in metallic domains in the CuO 2 planes, until blocked by charge density wave domain walls , where they use dopant bridges to cross over to a metallic domain of an adjacent CuO 2 plane. The transition temperature maxima are reached when the host lattice has weak bond-bending forces, which produce strong electron-phonon interactions at the interlayer dopants.

The symmetry of the order parameter could best be probed at the junction interface as the Cooper pairs tunnel across a Josephson junction or weak link. But, even if the junction experiment is the strongest method to determine the symmetry of the HTS order parameter, the results have been ambiguous. Kirtley and C. Tsuei thought that the ambiguous results came from the defects inside the HTS, so that they designed an experiment where both clean limit no defects and dirty limit maximal defects were considered simultaneously.

But, since YBCO is orthorhombic, it might inherently have an admixture of s symmetry. Despite all these years, the mechanism of high- T c superconductivity is still highly controversial, mostly due to the lack of exact theoretical computations on such strongly interacting electron systems. However, most rigorous theoretical calculations, including phenomenological and diagrammatic approaches, converge on magnetic fluctuations as the pairing mechanism for these systems. The qualitative explanation is as follows:. In a superconductor, the flow of electrons cannot be resolved into individual electrons, but instead consists of many pairs of bound electrons, called Cooper pairs.

In conventional superconductors, these pairs are formed when an electron moving through the material distorts the surrounding crystal lattice, which in turn attracts another electron and forms a bound pair. This is sometimes called the "water bed" effect. Each Cooper pair requires a certain minimum energy to be displaced, and if the thermal fluctuations in the crystal lattice are smaller than this energy the pair can flow without dissipating energy. This ability of the electrons to flow without resistance leads to superconductivity. In a high- T c superconductor, the mechanism is extremely similar to a conventional superconductor, except, in this case, phonons virtually play no role and their role is replaced by spin-density waves.

Just as all known conventional superconductors are strong phonon systems, all known high- T c superconductors are strong spin-density wave systems, within close vicinity of a magnetic transition to, for example, an antiferromagnet. When an electron moves in a high- T c superconductor, its spin creates a spin-density wave around it. This spin-density wave in turn causes a nearby electron to fall into the spin depression created by the first electron water-bed effect again. Hence, again, a Cooper pair is formed. When the system temperature is lowered, more spin density waves and Cooper pairs are created, eventually leading to superconductivity.

Note that in high- T c systems, as these systems are magnetic systems due to the Coulomb interaction, there is a strong Coulomb repulsion between electrons.

Theory of High Temperature Superconductivity a Conventional Approach

This Coulomb repulsion prevents pairing of the Cooper pairs on the same lattice site. The pairing of the electrons occur at near-neighbor lattice sites as a result. This is the so-called d -wave pairing, where the pairing state has a node zero at the origin. Examples of high- T c cuprate superconductors include La 1. From Wikipedia, the free encyclopedia. Superconductive behavior at temperatures much higher than absolute zero. Main article: Iron-based superconductor. Main article: Resonating valence bond theory.

Main article: List of superconductors. Ars Technica. Archived from the original on 4 March Retrieved 2 March Ford; G. The rise of the superconductors. Boca Raton, Fla. Bibcode : ZPhyB.. Georg Bednorz, K. Retrieved Leggett Nature Physics. Bibcode : NatPh April 23, Bibcode : EL Bibcode : Natur. Nature News. Archived from the original on 18 August Retrieved 18 August Phys Rev B. Bibcode : PhRvB.. Superconductor: The race for the prize Television Episode. Room-Temperature Superconductivity. Cambridge International Science Publishing, Cambridge.

Bibcode : cond. JSAP International. Archived PDF from the original on 16 August Bibcode : Sci Archived from the original on 3 July Maurice; Zhang, F. Physica C. Bibcode : PhyC.. Physical Review B. Bibcode : NatPh.. Archived from the original on 23 December Retrieved 23 December Handbook of High-Temperature Superconductor Electronics.

CRC Press. Physical Review Letters. Bibcode : PhRvL.. The data from Fig. There is a strong reduction or gapping of the near E F scattering rate as temperature is lowered into the paired state, along with a shift of this scattering to high energies beyond the dashed line.

High-temperature superconductivity - Wikipedia

Dramatic growth of self-energies towards the antinodal regime. In e , the states near E F are almost dispersionless, i. All data were taken with 24 eV photons, and the sample was the same as used for Fig. We use the conventional Nambu—Gorkov formalism for superconductivity 5 , 6 for theoretically describing these spectra:. That both the superconducting state and normal state spectra can be so well described within this semi-conventional approach is in itself a surprising finding: the generally broad structures and large backgrounds observed in ARPES spectra of cuprates have often been described as being so far outside the realm of conventional physics that such semi-conventional quasiparticle-like approaches were considered untenable 7.

The data also show that the overall value of the scattering rate at all energies in this high-temperature regime increases smoothly with temperature. Both of these are the expected behavior of the strange metal 21 or Marginal Fermi Liquid state 22 , or more precisely the Power Law Liquid state These large scattering rates are the reason for the overall broad spectra shown in Fig.

These large normal state scattering rates are a critical aspect of the strongly correlated state of the cuprates, which we will come back to later. This effect is absent at the highest temperature and strongest at the lowest temperatures. In addition to the superconducting gap in the spectral function growing as we move to the antinode we see that the quasiparticle dispersion becomes flatter and flatter from node to antinode, with a correspondingly stronger dispersion kink or renormalization effect. In Fig. More details of the effects of the self-energies.

Upon cooling from high temperature, the main evolution of the parameters begins at T pair and not at T C. The superconducting gap is 40 meV. This effective k range is significantly larger for the renormalized band blue than for the bare band red. The renormalization at the antinode is so huge that the quasiparticle dispersion at the gap edge is almost completely flat.

Such a huge renormalization factor and mass enhancement is very unconventional, partly because obtaining such a strong coupling via an Eliashberg electron—phonon interaction would typically cause a different instability such as a charge density wave that will compete with the superconductivity. Our data and discussion below argue that in this case the superconductivity is intimately related to and self-consistently drives the huge coupling, so this is very different from the conventional theory. This effect is illustrated in Fig. As for the entire Brillouin zone Fig.

Within this picture the huge 6. This large mass enhancement is beyond the simple concepts of undressing of the normal state correlations as discussed in the previous works 7 , 15 , as in those ideas the normal state correlations were removed, not converted. The conversion effect described here also has similarities to the Kondo effect observed in heavy Fermion materials, in which high-temperature incoherent correlations give rise to a highly massive coherent state at low temperature 2.

Conversion of electronic correlations. Hence, the large diffusive scattering above T C is converted to a strong kink effect and mass renormalization—an effect that is much larger for the antinode than the node because the normal state diffusive scattering is much larger for the antinode than the for the node. Our observations point toward the possibility for a positive feedback on the pairing process that can markedly enhance and stabilize superconductive pairing.

The idea is illustrated in Fig. This feedback effect should favor an anisotropic e. Over the past years, ARPES has made great progress at revealing certain aspects of the electronic interactions characterized by the self-energies of the cuprates. In particular, ARPES has discovered and provided key details of the unusual incoherent scattering or peak broadening 23 , 25 , 29 that is consistent with the strange-metal transport 21 or Marginal Fermi Liquid state 22 , or more precisely the Power Law Liquid state In contrast, using the standard language of peak tracking of quasiparticles, ARPES has described dispersion kinks or mass enhancements in the superconducting state in the nodal 24 , 30 — 32 and antinodal regime 25 — 28 , which have been generally interpreted as indicating rather conventional electron-—boson coupling, of the type for example that may act to pair electrons in a conventional non strongly-coupled electronic material.

However, prior to the present work, ARPES analysis only focus on one-dimensional energy cuts EDCs or one-dimensional momentum cuts MDCs 17 , 23 , 33 , but not both together—an issue that is not important when the peaks are sharp and the scattering rates are low like in a Fermi Liquid. In strongly correlated superconductors like cuprates, the complications of the peak broadening, the renormalization effect, and the spectral gap make a quantitative extraction of the band dispersions and self-energies from ARPES possible only in certain special cases.

In particular, along the nodal direction where there is a linearly dispersing band with relatively high velocity, the self-energies can be extracted through MDC analysis 23 , 30 — 32 , but this method fails for mid-zone or antinodal cuts, in which more complicated structures come in with the spectral gap and a strong kink feature below T pair 25 — Whereas near the antinode, the sharp quasiparticle peak below T C allows the relatively flat band dispersion to be extracted by EDC analysis, the EDC method fails when the gap starts to disappear near or above T C 26 — Thus, the MDC and EDC methods can only provide qualitative result in certain cases, and in the general case they may give very different results from each other Therefore, previous studies have only been able to focus on certain parts of the Brillouin zone or certain temperature ranges below or above T C and these studies only extracted qualitative feature from part of the self-energy either the real or imaginary part within a certain energy range 24 — 28 , 30 — The self-energies extracted from spectra covering a wide temperature range and the full Brillouin zone provide a more comprehensive picture of the electronic interactions Figs.

The extracted self-energies shown in Figs. Moreover, we present the evolution of the electron correlation effects from different parts of the Brillouin zone, with these strong renormalization effects providing a strong enhancement to the pairing state as shown in Fig. The strength of our observed coupling is unprecedented in superconductors. The typical theory for dealing with strong coupling is Eliashberg theory.

The standard Eliashberg equations have been derived for conventional superconductors where the Cooper pairing is mediated by an electron—boson interaction. Thus, the standard Eliashberg theory is adiabatic and semi-perturbative, i. On the other hand, the maximum electron—phonon interaction strength for driving superconductivity is limited owing to the stability of the lattice. Therefore, a renormalization factor Z up to 6. Our data also question the concept of adiabiticity, i. Regardless of the specific mechanism for the pairing, the strength of the positive feedback effect depends upon the strength of the incoherent normal state scattering, and presumably also the details of this incoherent scattering.

Of course, understanding the details of this diffusive normal state self-energy at high temperatures also has remained elusive, capturing the attention of physicists from a great range of disciplines 9 , 11 , Even more, seeing how this strong diffusive scattering can be converted to strong coherent effects that can enhance and stabilize superconductivity opens a potential new path for engineering such effects into other materials, possibly with higher transition temperatures. We used 9 eV photons for the temperature-dependent set of data in Fig. These two photon energies are selected to minimize the matrix element of the bonding band and enhance the antibonding band.

The energy resolution was 5. The count rate nonlinearity of the electron detector is corrected A weak energy-dependent background, presumably from elastically scattered electrons, is subtracted from each spectrum before fitting. This background is determined by measuring the counts at the edge of the spectrum as shown in Supplementary Fig.

The data that support the plots within this paper and other finding of this study are available from the corresponding author on reasonable request. We thank D. We call this a Luttinger Liquid. After this experimental clarification the theoretical comprehension will hardly keep us waiting long. This experiment has also solved the old problem of the nature of charge carriers created by doping of a single Mott-Hubbard band, cf. Dogma II. Now we know that charge carriers of the normal state are standard Landau quasi-particles [, ] for which we have conventional Cooper pairing in the superconducting phase.

January 13, World Scientific Book - 9in x 6in The pairing mechanism of overdoped cuprates superconductivity 43 In short, in our opinion the experimental validation of the WiedemannFranz law in overdoped cuprates [] is a triumph of the Landau [,] and Migdal concept of Fermi quasi-particles and Landau spirit of trivialism in general and provides a refutation of the spin-charge separation in cuprates [].

Hence, the problem of deriving the Wiedemann-Franz law for strongly correlated electrons in the CuO2 plane has just been set in the agenda.

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According to the Fermi liquid theory [] interactions between the particles create an effective self-consistent Hamiltonian. As Kadanoff [] has pointed out, this idea was much developed by Landau [] and Anderson []. Unfortunately, for high-Tc cuprates a link is still missing between the Landau quasiparticle concept and the one due to Slater that even scattering matrix elements can be calculated from first principles. The impact of Dogma V, then, is that the two-dimensional state has separation of charge and spin into excitations which are meaningful only within their two-dimensional substrate; to hop coherently as an electron to another plane is not possible, since the electron is a composite object, not an elementary excitation.

In other words, the two constant energy curves due to the bilayer splitting are described by the same equation Eq. This experiment, crucial for Dogma V, cf. It is only one of the details when one concentrates on the material-specific effects in high-Tc superconductors. The Heitler—London approach is well-known in quantum chemistry [—], and has been successfully used for a long time in the physics of magnetism []. We hope that realistic first-principles calculations aiming at the exchange integrals J of the CuO2 plane can be easily carried out.

Should they validate the January 13, World Scientific Book - 9in x 6in The pairing mechanism of overdoped cuprates superconductivity 45 correct antiferromagnetic sign and the correct order of magnitude of Jsd , we can consider the theory of high-Tc superconductivity established.

We stress that the two-electron exchange, analyzed here, is completely different from the double exchange considered in reference [, ]. The fit to the extended Van Hove singularity as observed, e. According to this possible interpretation, the broadly discussed maximum of the absorption in the MIR range is due to 3d-4s interband transition: one electron in the conduction band is excited by the light to the empty Cu4s band.

It seems that, up to now, there is no natural explanation of this MIR optical adsorbtion for a review see [81]. The derived gap anisotropy function 2. These inter- and intra-atomic processes occur on energy scales unusually large for solid state physics. However, the subsequent treatment of the lattice Hamiltonian can be performed completely within the framework of the traditional BCS theory.

Taking into account the typical ARPES-derived bandwidths, which are much bigger than Tc we come to the conclusion that the BCS trial wave function [] is applicable for the description of superconductivity in the layered cuprates with an acceptable accuracy if Tc does not significantly exceed room temperature. It is worth adding also a few remarks on the normal properties of the layered cuprates. Among all debated issues in the complex physics of the cuprates, the most important one is perhaps that of the normal-phase kinetics.

The long-standing problem is whether the paring interaction dominates or totally determines the mechanism of Ohmic resistance in the normal phase, as is the case for conventional superconductors. Within the present theory this question can be formulated as follows: does the s-d exchange interaction dominate the scattering of the normal-state charge carriers above Tc?

This is a solvable kinetic problem whose rigorous treatment will be given elsewhere. Here we shall restrain ourselves in providing only a qualitative discussion. For backscattering i. In this sense, cuprates repeat the qualitative feature of the conventional superconductors, with a maximal gap corresponding to maximal scattering on the Fermi surface.

All layered cuprates are strongly anisotropic and two-dimensional models give a reasonable starting point to analyze the related electronic processes. The thermodynamic fluctuations of this electric field and related fluctuations of the electric potential and charge density constitute an intensive scattering mechanism analogous to the blue-sky mechanism of light scattering by density fluctuations.

It has recently been demonstrated [] that the experimentally observed linear resistance can be rationalized in terms of the plane capacitor scenario; density fluctuations in the layered conductors are more important than the nature of the interaction. The resistance of the normal phase may not be directly related to the pairing mechanism and these problems can be solved separately. Nevertheless it will be interesting to check whether the anisotropic scattering in cuprates [—] can be explained within the framework of the s-d pairing Hamiltonian.

The present theory can also predict a significant isotope effect in the cuprates. Even though the Jsd pairing amplitude does not depend on the atomic mass, the charge carriers reside the ionic CuO2 2D lattice, thereby rendering polaron effects, as in any ionic crystal, possible. For the lighter oxygen isotope the lattice polarization is more pronounced, leading to enhanced effective mass and density of states, and reducing the transfer integrals. Overall, the isotope effect in the CuO2 plane is due to the isotope effect of the density of states.

This rationale can be quantitatively substantiated. At that the superfluid density remains unchanged. The calculation of the isotope effect on Tc requires an evaluation of the polaron effects on the conduction band. Although this is a feasible problem, it is beyond the scope of the present work. It would be worthwhile attempting to apply the approach, used in this chapter, for modeling triplet and heavy-fermion superconductivity as well. It is tempting to speculate about the relevance of the s-d exchange to January 13, 48 World Scientific Book - 9in x 6in superconductivity Theory of High Temperature Superconductivity — A Conventional Approach the pairing mechanism even of the iron-based superconductors [].

The wide s-band resulting from s-p hybridization is completely empty, which is somewhat unusual for compounds containing transition ions. It is omnipresent in the physics of the transition ions but in order for it to become the pairing mechanism in perovskites it is necessary that the s- and d-levels be close.

In other words, a virtual population of the s-level is at least needed in order to make the Jsd amplitude operative. Indeed, the conduction d-band is, actually, a result of the s-p-d hybridization in the two-dimensional CuO2 plane. With the above remarks, one can speculate that among the perovskites the layered ones are more favorable for achieving higher Tc see discussion in the next subsection.

However, the energy difference between these two Cu shell configurations is very small. Thus, post factum the success of Cu and O looks quite deterministic: the CuO2 plane is a tool to realize a narrow d-band with a strong s-p-d hybridization. It was mentioned earlier that Jsd is one of the largest exchange amplitudes, but the 4s and 3d orbitals are orthogonal and necessarily require an intermediary whose role is played by the O2p orbital.

The Jsd amplitude is omnipresent for all transition ion compounds, the hybridization of 3d, 4s and 2p is however specific only for the CuO2 plane. How this qualitative picture can be employed to predict new superconducting compounds is difficult to assess immediately. We believe, however, that this picture, working well for the overdoped regime, is robust enough against the inclusion of all the accessories inherent to the physics of optimally doped and underdoped cuprates: cohabitation of superconductivity and magnetism [, ], stripes [], pseudo-gap [, ], interplay of magnetism and superconductivity at individual impurity atoms [], apex oxygen, CuO2 plane dimpling, doping in chains [], the 41 meV resonance [], etc.

Perhaps some of these ingredients can be used in the analysis of triplet superconductivity in the copper-free layered perovskite Sr2 RuO4 []. It is also likely that the superconductivity of the RuO2 plane is a manifestation of a ferromagnetic exchange integral J. The two-electron exchange mediates superconductivity and magnetism in heavy Fermion compounds [—] as well.

We suppose that lattice models similar to the approach here will be of use in revealing the electronic processes in these interesting materials. Two-electron exchange may even contribute to the 30 K Tc of the cubic perovskite Ba0. The significance of this correlation to the physics of HTS was, however, emphasized in a handbook on high-Tc superconductivity edited by Shrieffer and Brooks [2] see Fig. Half-filled square — nonbonding subband. The explanation of this correlation is the crucial test for the theory of HTS.

Here we shall emphasize that the missing link, in fact, has already been found [], and the work by Pavarini et al. Perhaps the simplest possible interpretation, though one could search for alternatives, is given within the framework of the present theory. In order for the Schubin—Zener—Kondo exchange amplitude Jsd to operate as a pairing interaction of the charge carriers, the Cu4s orbital needs to be significantly hybridized with the conduction band. Cu3d and Cu4s are orthogonal orbitals and their hybridization is indirect.

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  • Suppose that those January 13, World Scientific Book - 9in x 6in The pairing mechanism of overdoped cuprates superconductivity 51 levels are not so close to each other. Hence, we conclude that the correlations reported in Ref. With first-principles electronic structure calculations available for many cuprates, it is worthwhile performing a LCAO fit to them [27] and using experimental values of Tc to extract the pairing amplitude Jsd for all those compounds. The ab initio calculation of the Kondo scattering amplitude parameterized by Jsd is an important problem which has to be set in the agenda of computational solid state physics.

    We expect that it will be a weakly material dependent parameter of the order of the s-d exchange amplitude in Kondo alloys, but perhaps slightly bigger as for the Cu ion the 3d and 4s levels are closer compared to many other ions. The final qualitative conclusion that can be extracted from the correlations reported by Pavarini et al. The natural explanation is: because its s-parameter is not small enough below its critical value.

    Thereby, the correlation reported by Pavarini et al. January 13, 52 2. The technological success in preparing the second generation of high-Tc superconducting cables by depositing thin-layer superconducting ceramics on a flexible low-cost metallic substrate is crucial for the future energy applications. On the other hand atomic-layer engineering of superconducting oxides will trigger progress in materials science and electronics.

    One can envision multi-functional all-oxide electronics, e. In this chapter we presented a traditional theory for superconductivity in overdoped, and possibly also optimally doped cuprates. Nonetheless, let us use the example of QED to illustrate the essence of our contribution. QED appeared as a synthesis between perturbation theory and relativity. Both components had been known well before the QED conception. Similarly, both the BCS theory and the exchange interaction have been known for ages, so the point in the agenda was how to conceive out of them the theory of high-Tc cuprates.

    The first step will definitely be 3 The gap-anisotropy fit in Fig. This set of parameters corresponds to band calculations but gives a factor 2—3 wider conduction band. A realistic fit is deemed to be a subject of a collaboration with experimentalists. Having a big variety of calculated variables the parameters of the theory can be reliably fitted. Another research direction is the first-principles calculation of the transfer amplitudes and two-electron exchange integrals. The level of agreement with the fitted values will be indicative for the completeness of our understanding. In addressing more realistic problems, the properties of a single space-homogeneous CuO2 plane will be a reasonable starting point.

    Concluding, we believe that there is a true perspective for the theoretical physics of cuprate superconductors to become an important ingredient of their materials science. Magnetism and superconductivity are among the most important collective phenomena in condensed matter physics. And, remarkably, magnetism of transition metals and high-Tc superconductivity of cuprates seem to be two faces of the same ubiquitous two-electron exchange amplitude.

    January 13, World Scientific Book - 9in x 6in This page intentionally left blank superconductivity January 13, World Scientific Book - 9in x 6in superconductivity Chapter 3 Specific heat and penetration depth 3. Despite the strong coupling effects and influence of disorder, which are all essential as a rule, for a qualitative analysis it is particularly useful to start with the weak-coupling BCS approximation for clean superconductors. In this case, very often model factorizable pairing potentials give an acceptable accuracy for the preliminary analysis of the experimental data.

    The aim of the present chapter is twofold. Firstly, we shall derive an explicit interpolation formula for the temperature dependence of the specific heat C T. The formula is formally exact for factorizable pairing kernels which are consequence of the approximative separation in superconducting order parameter derived in BCS weak-coupling approximation by Pokrovskii [,]. That is why we believe that the suggested formula can be useful for the analysis of experimental data when only gap anisotropy and band structure are known.

    The nontrivial results [] is that this separation of the variables is asymptotically correct in the BCS weak-coupling limit for an arbitrary kernel which is generally non factorizable. In fact, a factorizable kernel is a fairly unnatural property which, however, can occur if the pairing interaction is local, intra-atomic and located in a single atom in the unit cell. This is the special case of the s-d interaction at the copper site s in the CuO2 plane [60], considered in the previous chapter; The separability ansatz, though, shall be employed here to obtain a general interpolation formula formally exact for factorizable kernels.

    We apply the ansatz 3. Details on the derivation of the trial function approximation Eq. Differentiating Eq. This BCS formula 3. The Gorter-Casimir two fluid model has very simple physical grounds. According to the general idea by Landau [], the order parameter is an adequate notion for description of second order phase transitions, regardless of the concrete particle dynamics. Again, at Tc the general formulas Eq. Let us evaluate the upper limit which can give a VHS. Another simulation of strong coupling effects can be demonstrated by simple model density of states, corresponding to the case of layered cuprates 1.

    The model is applicable with a remarkable accuracy [] to MgB2 —a material which is in the limelight in the physics of high-Tc superconductivity over the past years. For the normal specific heat we have 2 3. For MgB2 determination of the two gaps has been carried out by directional point-contact spectroscopy [] in single crystals. One can see that for model evaluations the temperature dependence of the gap ratio could be neglected.

    For application of the two-band model to the specific heat of MgB2 the reader is referred to Ref. The analysis of the specific heat for MgB2 gives perhaps the best corroboration of the BCS results due to Pokrovskii [] and Moskalenko []. Solving the Eliashberg equation and performing first-principle calculations for the specific heat of MgB2 Golubov et al. On the other hand, Eqs. Unfortunately, the groups solving the Eliashberg equation have not compared their results to the classical results of the BCS theory for anisotropic superconductors [] in order to analyze several percent strong-coupling corrections to the specific heat jump for MgB2.

    For anisotropic superconductors, functions of the gap have to be averaged independently on the Fermi surface; this is the interpretation of the general formulas Eq. For illustration, we now apply this general formula to three typical cases and the results are shown in Fig.

    The latter two models are often applied to analyze the behavior of CuO2 or MgB2 superconductors. The theoretical curve is convoluted with a Gaussian kernel Eq. The experimental data [] are digitized from Ref. Consider now the low temperature behavior of the specific heat per unit area for a 2D d-wave superconductors. Here we have taken into account January 13, World Scientific Book - 9in x 6in Specific heat and penetration depth 4 nodal points.

    In such a way Eq. This result together with Eq.

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    The penetration depth has a similar linear low temperature behavior for d-wave superconductors. Very often fluctuations of stoichiometry and crystal defects make the theory of homogeneous crystal inapplicable close to the critical region. Let Tc r be a weakly fluctuating Gaussian field of the space vector r. Hence, the simplest possible empirical model is to apply a Gaussian kernel to the theoretically calculated curve. The result is depicted at Fig. In order to reach the analogous quality of the fit of C T for cuprates we have to take into account simultaneously the gap anisotropy and the VHS in the general expressions Eq.

    An analogous to Eq. Such a precise investigation of fluctuations in the magnetization of Nb and Sn in the past led to the discovery of twinning plane superconductivity. For analytical GL results for twinning plane superconductivity see Ref. Here we wish to emphasize that a large body of experimental data for Bc2 T are strongly influenced by the disorder.

    Various spurious curvatures of Bc2 T have been reported merely as a result of disorder of the crystals. In such a way the thermodynamic behavior is in agreement with the spectroscopic data. This is a good hint in favor of the Landau—Bogoliubov quasiparticle picture January 13, World Scientific Book - 9in x 6in superconductivity 69 Specific heat and penetration depth applied to high-Tc cuprates.

    A plot of the function F x is shown in Fig. For fast calculations one has to take only several terms of the expansions Eq. The BCS order parameter equation Eq. Recently, Abrikisov [] has derived the same equation for the temperature dependence of the amplitude of spindensity waves in cuprates. January 13, World Scientific Book - 9in x 6in superconductivity 71 Specific heat and penetration depth 1 2 0. For 2D d-wave superconductors the Pokrovskii equation 3. As a last problem, let us derive the factorizable kernel 3. The comparison of Eq. The experimental points for MgB2 circles are digitized from Ref.

    Generally speaking, the separability ansatz is a low-Tc approximation; Tc should be much smaller than all other energy parameters: energy cutoff, January 13, 74 World Scientific Book - 9in x 6in superconductivity Theory of High Temperature Superconductivity — A Conventional Approach Debye frequency for phonon superconductors, exchange integrals for exchange mediated superconductivity, the Fermi energy and the bandwidths. Room temperature superconductivity is not yet discovered, but the good message is that we have still a simple approximation acceptably working for all superconductors. For theoretical models the accuracy of the separable approximation can be easily probed when investigating the angle between the order parameter at different temperatures, e.

    The performed analysis shows that the separation of the variables Eq. The factorizable kernel gives a simple solution to the gap equation, the nontrivial detail being that this separability can be derived by the BCS gap equation. The factorizable kernel has also been discussed by Markowitz and Kadanoff [] and employed, e. Factorizable kernels are now used in many works on exotic superconductors. However, in none of them is mentioned that the separability of the superconducting order parameter is an immanent property of the BCS theory [, ].

    The accuracy of the separable approximation is higher if the other eigenvalues of the pairing kernel are much smaller than the maximal one. This is likely to be the situation for the s-d model for layered cuprates [60], where the s-d pairing amplitude Jsd is much bigger than the phonon attraction and the other interatomic exchange integrals. In order for us to clarify this important approach to the theory of superconductivity, we have given here a rather methodical derivation of the Pokrovsky theory.

    January 13, World Scientific Book - 9in x 6in superconductivity 75 Specific heat and penetration depth 3. This shift of all conduction electrons explains why for the penetration depth the influence of VHS is less essential than the influence on the heat capacity. If the magnetic field B is parallel to the surface of a bulk superconductor this formula gives B2. The constancy of the electrochemical potential in the superconductor gives the change of the electric potential, i. For the temperature dependence of the electrochemical potential of the normal phase we have [Ref.

    For discussions of possible experimental setups see Ref. Some experimental points for MgB2 circles are digitized from Ref. In such a way the electrodynamic behavior of a superconductor can be expressed in terms of the functions, defined for description of its thermodynamic behavior. Using Eqs. For a two-band superconductor, Eqs. The graphs of ri y and the corresponding gi z functions are given in Figs. For a 2D d-wave superconductor the general formula Eq.

    Finally, we have a good working BCS-like formula. Our calculations are depicted in Fig. In this model calculation we have taken into account only one band responsible for superconductivity. The experimental points circles from Ref. Note that the model with vertical line nodes predicts spontaneous breaking of the symmetry of the penetration depth in the ab-plane. According to the conclusions by Zhitomirsky and Rice [] their model with horizontal line nodes see also Ref.

    For illustration, in Fig. From aesthetic point of view our preferences are for the recent model for the gap anisotropy by Deguchi et al. The theoretical prediction corresponding to Eq. There is one detail that is worth focusing on: for the s-d model for high-Tc superconductivity [60] the kernel is indeed separable because the contact interaction is localized in a single atom in the lattice unit cell. The nature of superconductivity for those superconductors is completely different: high-Tc and low-Tc, phononand exchange-mediated, singlet and triplet Cooper pairs.

    In all those cases the derived formulas work with an acceptable accuracy; in some cases we have even quantitative agreement and for high-Tc cuprates we have shown what the BCS analysis can give. Often after the synthesis of a new superconductor single crystals are not available and only the data for heat capacity C T can help the theory to distinguish between different models for the gap anisotropy even before detailed spectroscopic investigation is performed.

    January 13, World Scientific Book - 9in x 6in This page intentionally left blank superconductivity January 13, World Scientific Book - 9in x 6in superconductivity Chapter 4 Plasmons and the Cooper pair mass 4. But, for the layered and extremely anisotropic Bi-based with weak coupling between the conducting CuO2 planes, the attenuation of the plasmons will be negligible.

    The predicted slow decay may encourage experimentalists to look for plasmons in their samples. Due to the strong anisotropy of highTc superconductors, the bulk plasmons are known as Josephson plasma resonances. One may ask then what is the doping dependence [] of this Hall anomaly and how the vortexlattice melting [] affects the Hall behavior? Alas, due to the complexity of the vortex matter many related problems are still not answered satisfactorily if at all. It is quite possible that the sign reversal of the temperature dependence of the Hall effect could be closely related to charging of the vortices [—].

    There is no doubt [] that the experimental solution of this enigma would provide the key towards understanding the various electromagnetic phenomena. On the other hand, the currently existing theoretical models often lead to conflicting results, thus making it difficult to discriminate between all those competing explanations.

    In such a situation we feel it is appealing to accelerate the selection by looking for simplicity in experiments with artificial structures where many of the complications typical for the real systems are avoided. The aim of the present chapter is to propose an experiment for determination of the vortex charge employing transport measurement in a layered metal-insulator-superconductor MIS system. We shall require that the quality of the insulator-superconductor interface be extremely high and the insulator layer be very thin.

    Such a layered MIS structure incorporating a high-Tc film can be manufactured by the contemporary technology of atomic-level engineering of superconducting oxide multilayers and superlattices [, —]. In fact, structures of the kind are now being in use for purposes of the fundamental research [—] in the physics of high-Tc superconductors, therefore the vortex charge problem can find its solution thanks to the technological progress. The chapter is organized as follow: in Sec. In Sec. An overview is made in Sec. It is finally concluded in Sec.

    These important parameters enter the theories of a number of phenomena related to electrodynamics of superconductors and can be simultaneously determined by standard electronic measurements. We notice that q 2D r has the same sign as the charge of the Cooper pairs in the superconductor. On the other hand, the Bernoulli potential keeps the Cooper pairs on a circular orbit inside the vortex. In order to derive the total charge related to the vortex we have to integrate the charge density up to some maximum radius, r!

    According to our model, the charge related to vortices is localized not in the vortex core but in the adjacent conducting layers: superconducting CuO2 planes in a real high-Tc crystal or the normal layer in the model MIS system. With this we close the electrostatic consideration of the vortex charge, but rhe reader is reffered to a number of ingenious experiments related to electrostatics of vortices which are suggested in Ref. We believe, however, that January 13, World Scientific Book - 9in x 6in Plasmons and the Cooper pair mass superconductivity 91 the standard transport measurement have some advantage even if they are related to observations of pA-range and below.

    The next important step is to address the vortex flow regime of the superconducting film when a strong enough dc current density jy is applied through the superconducting film. This condition will create small dissipation and give rise to an electric field Ey parallel to the current density. In a coordinate system moving with the vortex drift velocity vv the electric field is zero. Along this line let us recall the fact that airplanes fly thanks to the Bernoulli theorem that holds true for a unviscous dissipationless fluid, but the significant part of the ticket price covers the dissipated energy.

    In the present model we used the hydrodynamic approach applicable for extreme type-II superconductors and January 13, 92 World Scientific Book - 9in x 6in superconductivity Theory of High Temperature Superconductivity — A Conventional Approach completely neglected the influence of the geometrically small vortex core. However the states in vortex core can have some influence in the total charge of vortex core [, ]. This problem, certainly, is only of an academic interest and is irrelevant for the oxide superconductors.

    The contact potential difference between the normal and the superconducting phase see Eq.

    High Temperature Superconducting Materials

    Again, in the mobile coordinate system the domain structure is static and the mean electric field is zero. For comparison with Eq. Having derived the formulae, Eq. The contemporary technology of layer-by-layer growth of oxide superconductors opens the possibility for realization of such a layered structure—a superconducting film protected by an insulating plate.

    Moreover, we consider that a MIS plane capacitor is one of the simplest possible systems employed in the fundamental research towards further technical applications. The voltage Vy applied through the Ag electrodes in circuit 1 creates a drift of the vortices with mean velocity vv.

    Due to the Bernoulli effect the superfluid currents around every vortex create a change in the electric potential on the superconducting surface. The Bernoulli potential of the vortex leads to an electric polarization on the normal Au surface. The charge qv , related to the vortex, has the same drift velocity vv.

    The corresponding current Ix in circuit 2 can be read by a sensitive ammeter. The quality of the SrTiO3 plate should be high enough so as to allow detection of the interface Hall current without being significant perturbed by the leakage currents between circuits 1 and 2. January 13, 94 World Scientific Book - 9in x 6in superconductivity Theory of High Temperature Superconductivity — A Conventional Approach could become a standard tool in studying the quality of the insulatorsuperconductor interface. It is now straightforward to work out the vortex charge at liquid-helium temperature, i.

    In this case the substitution of the above mentioned set of parameters in Eq. Further, Eq. For conventional superconductors similar evaluations show that effect is less but still observable. One can consider, for example, a thin Nb metal film grown by molecular beam epitaxy, and an Al layer after oxidation in natural condition could give a good insulator layer. All technologies for planar Josephson junctions provide as a rule metal-insulator interface of sufficient quality. Only the insulator layer should be thick enough to prevent leakage tunneling.

    The example analyzed above shows that the proposed experiment is in principle possible to be carried out but we find it difficult to anticipate all problems that could arise in the course of it. For instance, due to a good capacitance cross-talk the noise created by the vortex motion in the superconducting layer will be transmitted to the normal layer thus disturbing the detection of the small Hall current. We believe, however, that similar problems could be surmounted, given the challenge of the novel physics underlying the vortex charge.

    Furthermore, it is quite possible that the January 13, World Scientific Book - 9in x 6in Plasmons and the Cooper pair mass superconductivity 95 charge, concentrated in the vortex core, is comparable to the charge outside, so only a detailed analysis within the microscopic theory can shine a light on the latter point. In order to verify whether a hydrodynamic approach based upon the Bernoulli effect suffices to quantitatively describe the predicted vortex-charge interface current one needs independent methods to determine the effective mass of the Cooper pairs.

    In the next section we will analyze similar experiments employing artificial MIS structures. The uncertainty, however, immediately disappears during the first earthquake when a dynamical problem should be solved. They are contained in the experimental parameters, such as the penetration depth Eq. In order to determine the effective mass one has to investigate some dynamic phenomenon, which is time-dependent. The subtle point is that the latter are already dynamic effects even if the electric fields are static. The set-up proposed to determine the vortex charge, Fig.

    Probably the most simple method to accomplish the task would be to use the same MIS structure without making any contacts on the superconducting layer and to investigate the surface Hall current [] as described in the next subsection. This is an electrostatic effect and the superconducting film is in vortex-free state. The dissipation is zero and the superconductor is in thermodynamic equilibrium. A symmetric layered structure is grown by capping of the superconducting film with an insulator layer. Two normal metal layers are evaporated on the protecting insulator layer and on the back side of the substrate thus achieving a plane capacitor configuration.

    The normal-metal electrodes are circles with radius R and a cartoon of the experimental set-up in Corbino geometry is shown in Fig. Exploiting the axial symmetry of 2D the geometry Eq. The ac magnetic moment can be detected by the electromotive voltage dM t , 4. The core ingredient is a layered MIS structure see text in the field of a plane capacitor Corbino geometry; schematically, not to be scaled. The ac voltage generator creates current I through the plane capacitor, and the dc current source generates opposite oriented magnetic poles in the drive coils and a radial magnetic field Br in the plane of the superconducting film.

    The experimental difficulties might be related with the careful compensation of the mutual inductance M12 between the solenoid and the ac generator charging the MIS plane capacitor. The rigorous analysis of the experiment requires the knowledge of the break-through voltages of the MIS structure and the noise induced in the detecting coil, but in any case this auxiliary experiment would be easier to perform than the detection of vortex charge currents.

    In the following we will also provide an elementary derivation of the formula for the surface Hall current Eq. Let us trace the trajectory of a London superconducting electron i. In this case in the initial Eq. One plate of the capacitor is the bulk high-Tc crystal and the other one is the Au layer. In order to avoid frozen January 13, World Scientific Book - 9in x 6in Plasmons and the Cooper pair mass superconductivity 99 vortices the constant magnetic field of the dc drive coil must be applied after cooling down to low temperatures. An ac voltage should be applied to the plane capacitor, a lock-in ammeter will measure the polarization current, and the induced due to the effect ac magnetic moment can be detected by a lock-in voltmeter connected to the detector coil.

    For derivation of the above formula Eq.