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How do the specific heat and thermal conductivity of a superconductor change near its transition temperature?

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Figure 1: Specific heat in the normal (Cen) and superconducting (Ces) states of a classic superconductor as a function of absolute temperature. The two functions are identical at the transition temperature (Tc) and above Tc. — superconductivity Figure 1: Specific heat in the normal (C_en) and superconducting (C_es) states of a classic superconductor as a function of absolute temperature. The two functions are identical at the transition temperature (T_c) and above T_c.

Near the transition temperature ( $T_{c}$ ), the thermal properties of a superconductor undergo distinct changes as the material shifts between states:

Specific Heat: The electronic specific heat in the superconducting state ( $C_{e s}$ ) is generally smaller than in the normal state ( $C_{e n}$ ) at low temperatures. However, as $T_{c}$ is approached, $C_{e s}$ becomes larger than $C_{e n}$ . For classic superconductors, it drops abruptly to $C_{e n}$ at $T_{c}$ , while high- $T_{c}$ superconductors exhibit a cusp-like shape near the transition. This increase in $C_{e s}$ occurs because the energy gap decreases as the temperature rises, leading to an increase in the number of thermally excited electrons, which requires the absorption of heat.1
Thermal Conductivity: The thermal conductivity in the normal state ( $K_{n}$ ) approaches the thermal conductivity in the superconducting state ( $K_{s}$ ) as the temperature nears $T_{c}$ . This convergence occurs for both pure and impure materials and suggests that the energy gap for each electron approaches zero as the temperature reaches the transition point.1