Ginzburg-Landau-Gorkov theory for high-temperature superconductors

We use Gorkov’s theory to calculate the coefficients of the Ginzburg-Landau theory from a narrow two-dimensional tight-binding band. We consider s-wave, d-wave, and coupled s-d-wave pairing states of square symmetry. The resulting energy gap Δ, the thermodynamic critical field $H_c$, the coherence length ξ, and the magnetic penetration depth ${\lambda }_L$ depend on band width, band filling, and pairing state. The simple s-wave state yields the highest Δ and the highest $H_c$ and thus appears to be the most likely candidate. Scattering of carriers by phonons and impurities leads to multiplication of ${\xi }^2$ and ${\lambda }_L^{\mathrm{-}2}$ by χ(x), where χ is the Gorkov function and x=λ+(Γ/π$k_B$ $T_c$) (here λ is the electron-phonon coupling constant and Γ the scattering rate due to impurities). For small band widths and low carrier densities, we obtain qualitative agreement with the measured values of the coherence length and penetration depth in the ab plane for ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$. .AE