Dipolon theory of energy gap parameters in high-temperature superconductors at zero temperature

Many-body field-theoretic techniques involving dipolon propagator and electron Green's function have been used to derive the general expressions at zero temperature for the renormalized energy gap parameter $\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ the gap renormalization parameter $Z(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ and energy band renormalization parameter $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ for momentum $\stackrel{\ensuremath{\rightarrow}}{k}$ and frequency $\ensuremath{\omega}.$ The present theory takes into account explicitly the dressed dipolons as mediators of superconductivity, the screened Coulomb repulsion, and nonrigid electron energy bands considering retardation and damping effects and electron-hole asymmetry. For superconducting cuprates it has been shown by symmetry considerations that, in the lowest order approximations, there exists two energy gap parameters, one being antisymmetric (as) with respect to the exchange of the ${k}_{x}$ and ${k}_{y}$ components of vector $\stackrel{\ensuremath{\rightarrow}}{k}$ and the other being symmetric (s) with respect to the exchange of ${k}_{x}$ and ${k}_{y}.$ The antisymmetric solution is a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ wave $($i.e., $\ensuremath{\propto}$ $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})])$ which changes sign with respect to the exchange of ${k}_{x}$ and ${k}_{y},$ and the symmetric solution which is highly asymmetric s wave or equivalently, a combination of a symmetric s wave and absolute value of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ wave $($i.e. $\ensuremath{\propto}$ $|$ $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})]$ $|$ $)$ which does not change sign with respect to the exchange of ${k}_{x}$ and ${k}_{y}.$ Our self-consistent calculations of the real and imaginary parts of $\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ $Z(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega}),$ and $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{k},\ensuremath{\omega})$ verify the existence of these two (different) solutions and lead to the conclusion that the antisymmetric solution of the gap parameter corresponds to the observed regular (reg) superconducting energy gap whereas the symmetric solution corresponds to the observed pseudo (pse) energy gap. Calculations have been made for ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ as well as ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}.$ Explicitly, for ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ superconductor our calculated values of the antisymmetric and symmetric energy gap parameters are ${\ensuremath{\Delta}}_{0}^{\mathrm{as}}=22\ifmmode\pm\else\textpm\fi{}15 \mathrm{meV}$ and ${\ensuremath{\Delta}}_{1}^{s}=30\ifmmode\pm\else\textpm\fi{}10 \mathrm{meV}$ with ${\ensuremath{\Delta}}_{0}^{s}=1.5\ifmmode\pm\else\textpm\fi{}1.5 \mathrm{meV},$ where ${\ensuremath{\Delta}}^{\mathrm{as}}(\stackrel{\ensuremath{\rightarrow}}{k})={\ensuremath{\Delta}}_{0}^{\mathrm{as}}$ $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})]$ and ${\ensuremath{\Delta}}^{s}(\stackrel{\ensuremath{\rightarrow}}{k})={\ensuremath{\Delta}}_{0}^{s}+{\ensuremath{\Delta}}_{1}^{s}|$ $[\mathrm{cos}{(k}_{x})\ensuremath{-}\mathrm{cos}{(k}_{y})]$ $|,$ in agreement with the corresponding experimental results ${\ensuremath{\Delta}}_{0}^{\mathrm{reg}}=16.5\ifmmode\pm\else\textpm\fi{}1.5 \mathrm{meV},$ ${\ensuremath{\Delta}}_{1}^{\mathrm{pse}}\ensuremath{\approx}24 \mathrm{meV},$ and ${\ensuremath{\Delta}}_{0}^{\mathrm{pse}}=\ensuremath{-}0.5\ifmmode\pm\else\textpm\fi{}2.5 \mathrm{meV}.$ For ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ superconductor our calculated values of the antisymmetric and the symmetric energy gap parameters are ${\ensuremath{\Delta}}_{0}^{\mathrm{as}}=24\ifmmode\pm\else\textpm\fi{}13 \mathrm{meV}$ and ${\ensuremath{\Delta}}_{1}^{s}=29\ifmmode\pm\else\textpm\fi{}15 \mathrm{meV}$ with ${\ensuremath{\Delta}}_{0}^{s}\ensuremath{\approx}0$ which also agree with the corresponding experimental results ${\ensuremath{\Delta}}_{0}^{\mathrm{reg}}=16.5\ifmmode\pm\else\textpm\fi{}1.5 \mathrm{meV}$ and ${\ensuremath{\Delta}}_{1}^{\mathrm{pse}}=30\ensuremath{-}40 \mathrm{meV}$ with ${\ensuremath{\Delta}}_{0}^{\mathrm{pse}}=\ensuremath{-}0.5\ifmmode\pm\else\textpm\fi{}2.5 \mathrm{meV}.$ Our calculations also indicate that the bosonic energy relevant to superconductivity in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ superconductor is about 60 meV arising from ${O}_{1,2}$ dipolon excitations and that the broad bands observed in optical experiments are due to ${O}_{1,1},$ ${O}_{1,2},$ and ${O}_{3}$ dipolon excitations. Relative contributions from the various dipolon modes have been analyzed. One finds that the symmetric longitudinal modes of oxygen dipolons contribute dominantly in energy gap parameters. The uncertainties in the calculated values of the various parameters are due to uncertainties in the values of the polarizability particularly of oxygen ions, the shielding parameter, the repulsive Coulomb energy, and due to the calculational errors. The origin of the experimentally deduced ${T}^{*}$ values has been discussed in terms of the present theory which reveals that ${T}^{*}$ is greater than ${T}_{c}$ and that they have the same physical origin.