Gapless High-Current Superconductivity

The idea of gaplessness, already familiar in several areas of superconductivity, is considered in connection with high-current superconductivity. A calculation is made of the critical temperature $T_c$ at which a normal metal with zero current becomes thermodynamically unstable against decay into a high-current super-conducting phase. It is pointed out that at $T_c$ all the various suggested forms of high-current electron-electron interaction, both retarded and nonretarded, are equivalent, so that a calculation of $T_c$ does not suffer from ambiguity of choice of interaction. It is found that a transition temperature exists when the Bardeen, Cooper, and Schrieffer (BCS) parameter $N\left(0\right)\mathrm{}{V}_0>~0.43$ [assuming $N\left(0\right)\mathrm{}{V}_{c0\mathrm{}}=0.1\mathrm{}$]. In fact, for $0.43<N\left(0\right)\mathrm{}{V}_0<0.63\mathrm{}$, two values of $T_c$ exist, since at sufficiently low temperatures, the normal phase is thermodynamically stable against high-current superconductivity. When $N\left(0\right)\mathrm{}{V}_0=0.43\mathrm{}$, $T_c=\frac{0.259\hslash \omega }{k_B}$. At $T=0\mathrm{}$, a separate calculation is made of the $N\left(0\right)\mathrm{}{V}_0$ required to obtain high-current superconductivity with a finite thermal energy gap $2({\epsilon }_0-p_F{v}_0)$. This calculation, using a nonretarded interaction, includes the correction of an algebraic error in the 1959 paper by Parmenter. The result, $N\left(0\right)\mathrm{}{V}_0>~0.70$, is in close agreement with the recent work of Hone, who made use of a retarded interaction in calculating that one must have $N\left(0\right)\mathrm{}{V}_0>~0.67$ in order to get finite-thermal-gap high-current superconductivity. This suggests that retardation effects are not crucial in determining the occurrence or non-occurrence of high-current superconductivity. It is pointed out that experimental measurements of Morin and Maita have suggested that some transition-metal superconductors have large enough values of $N\left(0\right)\mathrm{}{V}_0$ for both the gapless form and the finite-thermal-gap form of high-current superconductivity to occur. Either form, if it exists, will exhibit the same form of electrical instability that is known to occur under exceptional conditions in conventional low-current superconductivity.