Low-Frequency ac Measurements of the Entropy Flux Associated with the Moving Vortex Lines in a Low-κType-II Superconductor

We present here very detailed ac measurements of the convection heat flux ${\overrightarrow{\mathrm{J}}}_2^h$ and the induction electric field $\overrightarrow{E}$ associated with the moving vortex lines in a low-$\kappa$ type-II superconductor (Pb-4.56-at.% In). The measurements were made using an original low-frequency current-modulation method based on a second-sound technique. ${\overrightarrow{\mathrm{J}}}_2^h$ and $\overrightarrow{E}$ have been measured as functions of the applied magnetic field $H_e$ and the applied electric current density and frequency $\nu$ ($300<\mathrm{}\nu <1200\mathrm{}$Hz) in the temperature range between 1.3 and 2.1 °K. The great accuracy of this ac method allows us to observe some interesting details in the ${\overrightarrow{\mathrm{J}}}_2^h\left(H_e\right)$ and $\overrightarrow{E}\left(H_e\right)$ curves. In particular, ${\overrightarrow{\mathrm{J}}}_2^h\left(H_e\right)$ reverses its direction at low fields ($H_e\gtrsim H_{c1}$), while $\overrightarrow{E}\left(H_e\right)$ remains in the same direction in the whole of the mixed state. This reversal effect suggests that, due to the presence of nonlocal effects in our low-$\kappa$ alloy ($\kappa \approx 1.25$), there are vortex lines of opposite sign in the superconductor. Furthermore, we have observed that ${\overrightarrow{\mathrm{J}}}_2^h\left(H_e\right)$ vanishes only for fields larger than the upper critical field $H_{c2}\mathrm{}\left(T\right)\mathrm{}$. The transported entropy $S_D$ per unit lenght of flux line per flux quantum computed from these measurements was found to be independent of the applied current density and frequency. In the high-field region ($H_e\lesssim H_{c2}$) experimental values of $S_D$ are in good agreement with the theoretical predictions of Caroli and Maki in the dirty limit. The calorimetric entropy per vortex line was obtained theoretically from the free-energy forms in the mixed state and compared with $S_D$.