Abstract A brief outline of the generalized BCS pairing theory is presented. The pairing of carriers can be due to the exchange of lattice-phonons or due to the exchange of electronic charge-density and spin-density excitations. It is argued that anisotropic physical properties in the normal as well as superconducting states in the new high-Tc materials are crucual in the development of any realistic theoretical approach, and in comparison of experimental results with correct BCS predictions involving superconductivity in layered crystals. The possibility of the break-down of the mean-field approximation is also discussed. As of now, the generalized BCS pairing approach is the only realistic microscopic theory available which may be applied to high-Tc superconductors.
The bilinear current density induced in a superconducting metal by a laser beam with frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ has been calculated. The calculation is done within the framework of the BCS theory of superconductivity at temperature $T=0$ \ifmmode^\circ\else\textdegree\fi{}K. It is shown that in the superconducting state of the metal the component of the induced current density, varying with the difference frequency $\ensuremath{\Omega}={\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ and the wave vector Q=${\mathrm{q}}_{1}$-${\mathrm{q}}_{2}$, where ${\mathrm{q}}_{1}$ and ${\mathrm{q}}_{2}$ are the wave vectors of the fundamental fields in the metal, differs considerably from the corresponding component in the normal state of the metal when $\ensuremath{\hbar}\ensuremath{\Omega}$ is of the order of the energy gap $2\ensuremath{\Delta}$. In this paper only that special case is considered where the wave vector Q is such that $\ensuremath{\hbar}Q{v}_{f}\ensuremath{\ll}2\ensuremath{\Delta}$, ${v}_{f}$ being the Fermi velocity of the electrons. If the collision frequency $\ensuremath{\Omega}_{c}^{}{}_{}{}^{S}$ of the electrons in the superconducting state is small compared to $\ensuremath{\Omega}$, there is a sharp peak at $\ensuremath{\hbar}\ensuremath{\Omega}=2\ensuremath{\Delta}$ in the energy flux of the light wave of frequency $\ensuremath{\Omega}$ reflected from the surface of the superconductor. For $\ensuremath{\hbar}\ensuremath{\Omega}\ensuremath{\gg}2\ensuremath{\Delta}$, the reflectivities are the same for both the normal and the superconducting states of the metal.
A model for the lowest-order optical nonlinearity in InSb is presented. A method for ensuring the cancellation of sizeable contributions throughout the Brillouin zone is incorporated in the model. A calculation, based on the band structure of InSb at $k=0$, successfully accounts for the dispersion of the optical nonlinearity near the band gap, measured by a second-harmonic-generation (SHG) experiment in InSb. General results are also presented on the interpretation of SHG measurements in absorbing media, with special attention devoted to interference experiments. From a comparison of the experimental dispersion with the theory, values are obtained for momentum matrix elements near $k=0$.
Anisotropy and the wave-vector dependence of the energy gap function determine many important properties of a superconductor. Starting from first principles, we present here a complete analysis of possible symmetries of the superconducting gap function ${\mathrm{E}}_{\mathrm{g}}$(k) at the Fermi surface in high-${\mathrm{T}}_{\mathrm{c}}$ layered superconductors with either a simple orthorhombic or a tetragonal unit cell. This is done within the framework of Gorkov's mean-field theory of superconductivity in the so-called 'layer representation' introduced by us earlier. For N conducting cuprate layers, J=1,2,\dots{},N, in each unit cell, the spin-singlet order parameters ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{JJ}\ensuremath{'}}$(k) can be expanded in terms of possible basis functions of all the irreducible representations relevant to layered crystals, which are obtained here. In layered materials, the symmetry is restricted to the translational lattice periodicity in the direction perpendicular to the layers and the residual point group and translational symmetries for the two-dimensional unit cell in each layer of the three-dimensional unit cell. We derive an exact general relation to determine different branches of the energy gap function ${\mathrm{E}}_{\mathrm{g}}$(k) at the Fermi surface in terms of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{JJ}\ensuremath{'}}$(k), which include both intralayer and interlayer order parameters. For N=2, we also obtain an exact expression for quasiparticle energies ${\mathrm{E}}_{\mathrm{p}}$(k), p=1,2, in the superconducting state in the presence of intralayer and complex interlayer order parameters as well as complex tunneling matrix elements between the two layers in the unit cell, which need not be equivalent. The form of the possible basis functions are also listed in terms of cylindrical coordinates ${\mathrm{k}}_{\mathrm{t}}$,\ensuremath{\varphi},${\mathrm{k}}_{\mathrm{z}}$ to take advantage of the orthogonality of functions with respect to \ensuremath{\varphi} integrations. In layered materials, with open Fermi surfaces in the ${\mathrm{k}}_{\mathrm{z}}$ direction, there is orthogonality of basis functions with respect to ${\mathrm{k}}_{\mathrm{z}}$ also (-\ensuremath{\pi}\ensuremath{\leqslant}${\mathrm{k}}_{\mathrm{z}}$d\ensuremath{\leqslant}\ensuremath{\pi}).mOur results show that in orthorhombic systems, planar ${\mathrm{d}}_{{\mathrm{k}}_{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{k}}_{\mathrm{y}}^{2}}$-like (${\mathrm{B}}_{1\mathrm{g}}$) and ${\mathrm{d}}_{\mathrm{kx}}$${\mathrm{k}}_{\mathrm{y}}$-like (${\mathrm{B}}_{2\mathrm{g}}$) symmetries are always mixed, respectively, with the planar s-wave-like (${\mathrm{A}}_{1\mathrm{g}}$) and ${\mathrm{A}}_{2\mathrm{g}}$-like symmetries of the corresponding tetragonal system. There is also the possibility of a weak modulation of ${\mathrm{E}}_{\mathrm{g}}$(k) as a function of ${\mathrm{k}}_{\mathrm{z}}$(\ensuremath{\sim}cos ${\mathrm{k}}_{\mathrm{z}}$d). In addition, in the presence of interlayer pairings which may or may not have the same symmetry as the intralayer order parameters, even in tetragonal systems the nodes of the ${\mathrm{d}}_{{\mathrm{k}}_{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{k}}_{\mathrm{y}}^{2}}$-like intralayer gap function will be shifted. In view of this, some suggestions for analyzing experimental data are also presented.
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