Abstract:
We study the k-dependence of the gap function of a bilayer superconductor, using standard mean-field techniques applied to a 2D extended Hubbard model, in the presence of coherent interlayer pair-tunneling and quenched coherent single-particle tunneling. The intralayer pairing potential thus defined is expandable in a finite number of basis functions for the irreducible representations of the point-group of the perfectly square lattice, C_{4v}. This gives rise to a competition between s- and d-wave symmetry, as the chemical potential is increased from the bottom to the top of a realistic band for most cuprates. It allows for mixed-symmetry paired state at temperatures below T_c, but never at T_c on a square lattice. Inclusion of the interlayer pair-tunneling into the effective pairing potential leads to highly non-trivial k-space structures, such as pronounced maxima along the Fermi line not seen in the absence of interlayer pair-tunneling. We show how such a gap structure evolves with temperature and with band filling, and how it affects various observables. In particular, a nonuniversal value of the normalized jump in the specific heat at T_c will be evidenced, at variance with the conventional universal BCS result.

Abstract:
We have found the mechanism of the electron Cooper pair formation via the electron interaction by means of the spin-electron acoustic waves. This mechanism takes place in metals with rather high spin polarization, like ferromagnetic, ferrimagnetic and antiferromagnetic materials. The spin-electron acoustic wave mechanism leads to transition temperatures 100 times higher than the transition temperature allowed by the electron-phonon interaction. Therefore, spin-electron acoustic waves give the explanation for the high-temperature superconductivity. We find that the transition temperature has strong dependence on the electron concentration and the spin polarization of the electrons.

Abstract:
According to the theory of relativity, the relativistic Coulomb's force between an electron pair is composed of two parts, the main part is repulsive, while the rest part can be attractive in certain situations. Thus the relativistic attraction of an electron pair provides an insight into the mechanism of superconductivity. In superconductor, there are, probably at least, two kinds of collective motions which can eliminate the repulsion between two electrons and let the attraction being dominant, the first is the combination of lattice and electron gas, accounting for traditional superconductivity; the second is the electron gas themselves, accounting for high $T_c$ superconductivity. In usual materials, there is a good balance between the repulsion and attraction of an electron pair, the electrons are regarded as free electrons so that Fermi gas theory plays very well. But in some materials, when the repulsion dominates electron pairs, the electron gas will has a behavior opposite to superconductivity. In the present paper the superconducting states are discussed in terms of relativistic quantum theory in details, some significant results are obtained including quantized magnetic flux, London equation, Meissner effect and Josephson effect.

Abstract:
The magnon exchange mechanism of ferromagnetic superconductivity (FM-superconductivity) was developed to explain in a natural way the fact that the superconductivity in $UGe_2$, $ZrZn_2$ and $URhGe$ is confined to the ferromagnetic phase.The order parameter is a spin anti-parallel component of a spin-1 triplet with zero spin projection. The transverse spin fluctuations are pair forming and the longitudinal ones are pair breaking. In the present paper, a superconducting solution, based on the magnon exchange mechanism, is obtained which closely matches the experiments with $ZrZn_2$ and $URhGe$. The onset of superconductivity leads to the appearance of complicated Fermi surfaces in the spin up and spin down momentum distribution functions. Each of them consist of two pieces, but they are simple-connected and can be made very small by varying the microscopic parameters. As a result, it is obtained that the specific heat depends on the temperature linearly, at low temperature, and the coefficient $\gamma=\frac {C}{T}$ is smaller in the superconducting phase than in the ferromagnetic one. The absence of a quantum transition from ferromagnetism to ferromagnetic superconductivity in a weak ferromagnets $ZrZn_2$ and $URhGe$ is explained accounting for the contribution of magnon self-interaction to the spin fluctuations' parameters. It is shown that in the presence of an external magnetic field the system undergoes a first order quantum phase transition.

Abstract:
As an extension of our previous work on the holon pairing instability in the t-J Hamiltonian [Phys. Rev. B {\bf 66}, 054427 (2002)], we examine the orbital symmetries of holon pairing order parameters in high $T_c$ superconductivity by examining the energy poles of t-matrix. We find that both $s$- and d-wave symmetries in holon pair order parameter occur at low lying energy states corresponding to the higher energy poles of t-matrix while only the s-wave symmetry appears at the lowest energy pole and that this results in the d-wave symmetry in the Cooper pair order which is a composite of the holon pair of s-wave symmetry and the spinon pair of d-wave symmetry below $T_c$. Finally we demonstrate that there exists no time-reversal symmetry breaking in association with the Cooper pair order parameter.

Abstract:
We study the depairing effect due to Coulomb interactions in Na$_x$CoO$_2$. We consider the electron-phonon coupling and the Coulomb interactions, and determine $T_c$ for s-wave superconductivity by solving the linearized Eliashberg equation. When we consider shear phonons as well as breathing phonons, $T_c$ is enhanced by Suhl-Kondo (SK) mechanism. Since SK mechanism is strong against Coulomb interactions, $T_c$ remains finite even if strong Coulomb interactions cancel out the attractive force due to breathing phonons. The orbital degree of freedom is important to understand the mechanism of superconductivity in Na$_x$CoO$_2$.

Abstract:
In their article, Zhang et al. [Phys. Rev. B 86, 024516 (2012)] present a remarkable result for A$_x$(S)$_y$TiNCl compounds ($\alpha$-phase TiNCl partially intercalated with alkali A and optionally co-intercalated molecular species S), finding the superconducting transition temperature T$_C$ scales with $d$$^{-1}$, where the spacing $d$ between TiNCl layered structures depends on intercalant thickness. Recognizing that this behavior indicates interlayer coupling, Zhang et al. cite, among other papers, the interlayer Coulombic pairing mechanism picture [Harshman et al., J. Phys.: Condens. Matter 23, 295701 (2011)]. This Comment shows that superconductivity occurs by interactions between the chlorine layers of the TiNCl structure and the layers containing A$_x$, wherein the transverse A$_x$-Cl separation distance {\zeta} is smaller than $d$. In the absence of pair-breaking interactions, the optimal transition temperature is modeled by T$_{C0}$ $\propto$ ({\sigma}/$A$)$^{1/2}$$\zeta$$^{-1}$, where {\sigma}/$A$ is the fractional charge per area per formula unit. Particularly noteworthy are the rather marginally-metallic trends in resistivities of A$_x$(S)$_y$TiNCl, indicating high scattering rates, which are expected to partially originate from remote Coulomb scattering (RCS) from the A$_x$ ions. By modeling a small fraction of the RCS as inducing pair-breaking, taken to cut off exponentially with {\zeta}, observations of T$_C$ < T$_{C0}$ are quantitatively described for compounds with {\zeta} < 4 {\AA}, and T$_C$ $\approx$ T$_{C0}$ for Na$_{0.16}$(S)$_y$TiNCl with propylene carbonate and butylene carbonate co-intercalants for which {\zeta} > 7 {\AA}. Since a spatially separated alkali-ion layer is not formed in Li$_{0.13}$TiNCl, the observed T$_C$ of 5.9 K is attributed to an intergrowth phase related to TiN (T$_C$ = 5.6 K).

Abstract:
It is assumed that in some sense the High-T$_c$ superconductivity is similar to the quantum chromodynamics (QCD). This means that the phonons in High-T$_c$ superconductor have the strong interaction between themselves like to gluons in the QCD. At the experimental level this means that in High-T$_c$ superconductor exists the nonlinear sound waves. It is possible that the existence of the strong phonon-phonon interaction leads to the confinement of phonons into a phonon tube (PT) stretched between two Cooper electrons like a hypothesized flux tube between quark and antiquark in the QCD. The flux tube in the QCD brings to a very strong interaction between quark-antiquark, the similar situation can be in the High-T$_c$ superconductor: the presence of the PT can essentially increase the binding energy for the Cooper pair. In the first rough approximation the PT can be approximated as a nonrelativistic string with Cooper electrons at the ends. The BCS theory with such potential term is considered. It is shown that Green's function method in the superconductivity theory is a realization of discussed Heisenberg idea proposed by him for the quantization of nonlinear spinor field. A possible experimental testing for the string approximation of the Cooper pair is offered.

Abstract:
We analyze the effect of the non-vanishing range of electron-electron repulsion on the mechanism of unconventional superconductivity. We present asymptotically exact weak-coupling results for dilute electrons in the continuum and for the 2D extended Hubbard model, as well as density-matrix renormalization group results for the two-leg extended Hubbard model at intermediate couplings, and approximate results for the case of realistically screened Coulomb interactions. We show that $T_c$ is generally suppressed in some pairing channels as longer range interactions increase in strength, but superconductivity is not destroyed. Our results confirm that electron-electron interaction can lead to unconventional superconductivity under physically realistic circumstances.

Abstract:
We use the Hubbard type model to describe the coexistence between superconductivity (SC) and ferromagnetism (F). Our Hamiltonian contains single-site and two-site interactions. All inter-site interactions will have included the inter-site kinetic correlation: $$, within the Hartree-Fock approximation. To obtain the SC transition temperature $T_{SC}$ and Curie temperature $T_{C}$ we use the Green's functions method. The numerical results show that the singlet SC is eliminated by F, but the triplet SC is either enhanced or depleted by F, depending on the carrier concentration and direction of a superconducting spin pair with respect to magnetization. The kinetic correlation is capable of creating superconductivity. We find that the ferromagnetism created by change of the bandwidth can coexist with singlet superconductivity. In the case of triplet superconductivity the ferromagnetism creates different critical SC temperatures for the $A_{1}$ and $A_{2}$ phase (the pair's spin parallel and antiparallel to magnetization, respectively).