Specific heat and low-lying excitations in the mixed state for a type-II superconductor

Low temperature behavior of the electronic specific heat $C\left(T\right)$ in the mixed state is by the self-consistent calculation of the Eilenberger theory. In addition to the $\gamma T$-term ($\gamma$ is a Sommerfeld coefficient), $C\left(T\right)$ has a significant contribution of $T^2$-term intrinsic in the vortex state. We identify the origin of the $T^2$ term as (i) V-shape density of states in the vortex state and (ii) the Kramer-Pesch effect of vortex-core shrinking upon lowering $T$. These results for both full-gap and line-node cases reveal that the vortex core is a richer electronic structure beyond the normal core picture.

Figure 1

(Color online) (a) Free energy difference $F_{\mathrm{s}}-F_{\mathrm{n}}$ and (b) specific heat $C\left(T\right)∕T$ as a function of $T∕T_{\mathrm{c}}$ for various fields $B∕B_0=0,0.3,0.4,0.5,0.6$ from bottom to top. $B_0={\varphi }_0∕{\xi }^2$.

Figure 2

(Color online) Energy dependence of the total density of states $N\left(E\right)$ is calculated at $T∕T_{\mathrm{c}}=0.01$. Plotted data are for $B∕B_0=0,0.3,0.4,0.5,0.6$ from bottom to top at $E=0$, showing characteristic V-shape cusp structures near the Fermi level $E=0$.

Figure 3

(Color online) Spectral evolution $N(E,{\mathbf{r}}_{\mathrm{NN}})$, where ${\mathbf{r}}_{\mathrm{NN}}$ is along the nearest neighbor vortex direction of the hexagonal vortex lattice. The vortex center is situated at $\mid {\mathbf{r}}_{\mathrm{NN}}\mid =0$. $N(0,0)$ is truncated over 20.

Figure 4

(Color online) (a) Temperature dependence of the vortex-core radius $\rho$ and (b) temperature dependence of the zero-energy density of states $N(E=0,T)$ are plotted for $B∕B_0=0,0.3,0.4,0.5,0.6$ from bottom to top. In (a) data for $B=0$ and normal state are not shown. Lines are guides for the eye.

Figure 5

(Color online) Specific heat $C\left(T\right)∕T$ for $B∕B_0=0.3$ (solid circles) with the estimated contributions $1.9{\alpha }_{E}T∕T_{\mathrm{c}}$, ${\alpha }_{\mathrm{K}}T∕T_{\mathrm{c}}$, and $(1.9{\alpha }_E+{\alpha }_{\mathrm{K}})T∕T_{\mathrm{c}}$.