Constraints on the temperature dependence of the magnetic penetration depth of superconducting ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_{8+\mathit{x}}$ from measurements of the low-temperature specific heat

We have measured the low-temperature (2–10 K) specific heat of polycrystalline ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_{8+\mathit{x}}$ in zero applied field and in an applied magnetic field of 1 T. We find no field-induced enhancement of the specific heat, to within the 1% resolution of our calorimeter, in a field of 1 T up to 10 K, in contrast to the ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$ system. The results suggest that the enhancements which have been observed in the specific heat result from a temperature dependence of the pinning of vortices in the mixed state. In analyzing our data, we use a model for the enhancement of the specific heat, and find quantitative agreement. Using this model, we place constraints on the allowed temperature dependence of the ab-plane magnetic penetration depth.