**1. Introduction**

26

[58] J. M. Harris, Y. F. Yan, O. K. C. Tsui, Y. Matsuda, and N. P. Ong. Hall angle evidence for the superclean regime in 60 k YBa2Cu3O6+*y*. *Phys. Rev. Lett.*, 73:1711–1714, Sep 1994. [59] A. Leo, G. Grimaldi, R. Citro, A. Nigro, S. Pace, and R. P. Huebener. Quasiparticle scattering time in niobium superconducting films. *Phys. Rev. B*, 84:014536, Jul 2011. [60] Manlai Liang and Milind N. Kunchur. Vortex instability in molybdenum-germanium

superconducting films. *Phys. Rev. B*, 82:144517, Oct 2010.

288 Superconductors – Materials, Properties and Applications

The discovery and further development of superconductivity is extremely interesting because of its pragmatic (practical) and purely academic reasons. At the same time, the superconductivity science is very remarkable as an important object for the study in the framework of the history and methodology of science, since all the details are well documented and well-known to the community because of numerous interviews by participants including main heroes of the research and the fierce race for higher critical temperatures of the superconducting transition, *Tc*. Moreover, the whole science has well-documented dates, starting from the epoch-making discovery of the superconducting transition by Heike Kamerlingh-Onnes in 1911 [1–7], although minor details of this and, unfortunately, certain subsequent discoveries in the field were obscured [8–11]. As an illustrative example of a senseless dispute on the priority, one can mention the controversy between the recognition of Bardeen-Cooper-Schrieffer (BCS) [12] and Bogoliubov [13] theories.

If one looks beyond superconductivity, it is easy to find quite a number of controversies in different fields of science [14, 15]. Recent attempts [16–18] to contest and discredit the Nobel Committee decision on the discovery of graphene by Andre Geim and Kostya Novoselov [19, 20] are very typical. The reasons of a widespread disagreement concerning various scientific discoveries consist in a continuity of scientific research process and a tense competition between different groups, as happened at liquefying helium and other cryogenic gases [9, 21–24] and was reproduced in the course of studying graphite films [25, 26]. At the same time, the authors and the dates of major discoveries and predictions in the science of superconductivity are indisputable, fortunately to historians and teachers.

Macroscopic manifestations of the superconducting state and diverse properties of the plethora of superconductors are consequences of main fundamental features: (i) zero

©2012 Gabovich et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Gabovich et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

resistivity found already by Kamerlingh-Onnes (sometimes the existence of persistent currents discovered by him in 1914 is considered more prominent and mysterious [27]), (ii) expulsion of a weak magnetic field (the Meissner effect [28]), and (iii) the Josephson effects [29–37], i.e. the possibility of dc or ac super-currents in circuits, containing thin insulating or normal-metal interlayers between macroscopic superconducting segments. Of course, the indicated properties are interrelated. For instance, a macroscopic superconducting loop with three Josephson junctions can exhibit a superposition of two states with persistent currents of equal magnitudes and opposite polarity [38].

first, liquid-nitrogen temperatures were achieved and, second, the predominant *dx*<sup>2</sup>−*y*<sup>2</sup> - order parameter symmetry (at least in hole-doped oxides) made possible applications in electronics

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

291

While studying high-*Tc* cuprates, superconductivity was shown to compete with charge density waves (CDWs), so that the observed properties in the superconducting state must be modified by CDWs [123–128]. It should concern Josephson currents phenomenon too

Of course, other superconducting materials found after the discovery of high-*Tc* oxide materials are also very remarkable, because of their non-trivial electron spectra, so that Josephson currents through junctions involving those materials should possess interesting features. We mean, in particular, MgB2 with *Tc* ≤ 40 K [135] and a multiple energy-gap structure [136, 137], as well as Fe-based pnictides and chalcogenides with *Tc* ≤ 56 K and concomitant spin density waves (SDWs) suspected to have deep relations with

In this paper, we present our theoretical studies of dc Josephson currents between conventional superconductors and partially CDW-gapped materials with an emphasis on cuprates, although the gross features of the model can be applied to other CDW superconductors as well. The next Section 2 contains the justification of the approach and the formulation of the problem, whereas numerical results of calculations, as well as the detailed discussion, are presented in Section 3. Section 4 contains some general conclusions concerning dc Josephson currents across junctions involving partially gapped CDW superconductors.

A more involved case of Josephson junctions between two CDW superconductors with

Coherent properties of Fermi liquids in the paired state are revealed by measurements of dc or ac Josephson tunnel currents between two electrodes possessing such properties. The currents depend on the phase difference between superconducting order parameters of the electrodes involved [30, 31, 119]. Manifestations of the coherent pair tunneling are more complex for superconductors with anisotropic order parameters than for those with an isotropic energy gap. In particular, it is true for *d*-wave superconductors, where the order parameter changes its sign on the Fermi surface (FS) [119, 138–143]. As was indicated above, high-*Tc* oxides are usually considered as such materials, where the *dx*<sup>2</sup>−*y*<sup>2</sup> pairing is usually assumed at least as a dominating one [117, 144–152]. However, conventional *s*-wave contributions were also detected in electron tunneling experiments [153–160] and, probably, in nuclear magnetic resonance (NMR) and nuclear quadrupole resonance measurements [161]. Therefore, only a minority of researchers prefer to accept the isotropic *s*-wave (or extended *s*-wave) nature of superconductivity in cuprates [162–175]. Notwithstanding the existing fundamental controversies, the *d*-wave specificity of high-*Tc* oxide superconductivity has

various symmetries of superconducting pairing will be treated elsewhere.

**2.1.** *d***-wave versus** *s***-wave order parameter symmetry**

already been used in technical devices [95, 116, 118–120, 122].

and quantum computation more diverse [37, 113–122].

superconductivity in those materials [78].

**2. Theoretical approach**

[129–134], although this topic has not been properly developed so far.

We note that those findings, reflecting a cooperative behavior of conducting electrons (later interpreted in terms of a quantum-mechanical wave function [12, 39–43]), had to be augmented by the observed isotope dependence of *Tc* [44, 45] in order that the first successful semi-microscopic (it is so, because the declared electron-phonon interaction was, in essence, reduced to the phenomenological four-fermion contact one) BCS theory of superconductivity [12] would come into being. Sometimes various ingenious versions of the BCS theory, explicitly taking into account the momentum and energy dependences of interaction matrix elements, as well as the renormalization of relevant normal-state properties by the superconducting reconstruction of the electron spectrum [46–50], are called "the BCS theory". Nevertheless, such extensions of the initial concept, explicitly related to Ref. [12] and results obtained therein, are inappropriate. This circumstance testifies that one should be extremely accurate with scientific terms, since otherwise it may lead to reprehensible misunderstandings [51].

Whatever be a theory referred to as "the BCS one" or as "the theory of superconductivity" [52], we still lack a true consistent microscopic picture scenario (scenarios?) of superconducting pairing in different various classes of superconductors. As a consequence, all existing superconducting criteria [53–72] are empirical rather than microscopic, although based on various relatively well-developed theoretical considerations. Hence, materials scientists must rely on their intuition to find new promising superconductors [73–78], although bearing also in mind a deep qualitative theoretical reasoning [43, 79–83].

It is no wonder that unusual transport properties of superconductors together with their magnetic-field sensibility led to a number of practically important applications. Namely, features (i) and (ii) indicated above made it possible to manufacture large-scale power cables, fly-wheel energy storage devices, bearings, high field magnets, fault current limiters, superconductor-based transformers, levitated trains, motors and power generators [84–93]. At the same time, the Josephson (weak-coupling) feature (iii) became the basis of small-scale superconducting electronics [88, 94–98], which also uses the emergence of half-integer magnetic flux quantization in circuits with superconducting currents [99, 100]. Smartly designed SQUID devices with several Josephson junctions and a quantized flux serve as sensible detectors of magnetic field and electromagnetic waves, which, in their turn, are utilized in industry, research, and medicine [95–98, 101]. Recently oscillatory effects inherent to superfluid 3He [102–104] and 4He [103–105], which are similar to the Josephson one, were used to construct superfluid helium quantum interference devices (SHeQUIDs) [106].

High-*Tc* oxide superconductors found in 1986 [107] and including large families of materials with *Tc* ≤ 138 K [108–112] extended the application domain of superconductivity, because, first, liquid-nitrogen temperatures were achieved and, second, the predominant *dx*<sup>2</sup>−*y*<sup>2</sup> - order parameter symmetry (at least in hole-doped oxides) made possible applications in electronics and quantum computation more diverse [37, 113–122].

While studying high-*Tc* cuprates, superconductivity was shown to compete with charge density waves (CDWs), so that the observed properties in the superconducting state must be modified by CDWs [123–128]. It should concern Josephson currents phenomenon too [129–134], although this topic has not been properly developed so far.

Of course, other superconducting materials found after the discovery of high-*Tc* oxide materials are also very remarkable, because of their non-trivial electron spectra, so that Josephson currents through junctions involving those materials should possess interesting features. We mean, in particular, MgB2 with *Tc* ≤ 40 K [135] and a multiple energy-gap structure [136, 137], as well as Fe-based pnictides and chalcogenides with *Tc* ≤ 56 K and concomitant spin density waves (SDWs) suspected to have deep relations with superconductivity in those materials [78].

In this paper, we present our theoretical studies of dc Josephson currents between conventional superconductors and partially CDW-gapped materials with an emphasis on cuprates, although the gross features of the model can be applied to other CDW superconductors as well. The next Section 2 contains the justification of the approach and the formulation of the problem, whereas numerical results of calculations, as well as the detailed discussion, are presented in Section 3. Section 4 contains some general conclusions concerning dc Josephson currents across junctions involving partially gapped CDW superconductors.

A more involved case of Josephson junctions between two CDW superconductors with various symmetries of superconducting pairing will be treated elsewhere.
