Interface energy of two-band superconductors

Using the Ginzburg-Landau theory for two-band superconductors, we determine the surface energy ${\sigma }_s$ between coexisting normal and superconducting states at the thermodynamic critical magnetic field. Close to the transition temperature, where the Ginzburg-Landau theory is applicable, we demonstrate that the two-band problem maps onto an effective single band problem. While the order parameters of the two bands may have different amplitudes in the homogeneous bulk, near $T_c$ the Josephson-type coupling between the bands leads to the same spatial dependence of both order parameters near the interface. This finding puts into question the possibility of intermediate, so-called type-1.5 superconductivity, in the regime where the Ginzburg-Landau theory applies.

Figure 1

(Color online) The function, ${\rm Y}\left(\kappa \right)$, of the one-band problem calculated numerically (full points) compared to theoretical limit of Eq. (2) (dashed lines). The inset shows an enlargement of the large $\kappa$ domain.

Figure 2

(Color online) Schematic representation of the interface between the normal and the superconducting states.

Figure 3

(Color online) (a) The reduced field and order parameters obtained numerically by minimizing the interface energy functional, $\Sigma$, for ${\kappa }_1=0.45$, ${\kappa }_2=5$, $u=0.6$, and $t=0.07$. The reduced order parameter ${\psi }_1$ is shown by a solid line and the dashed line is ${\psi }_2$. (b) The closeup of the order parameters for $6.5<z/\lambda <7.5$.

Figure 4

(Color online) (a) The reduced field and order parameters obtained numerically by minimizing the interface energy functional, $\Sigma$, for ${\kappa }_1=0.45$, ${\kappa }_2=5$, $u=0.6$, and $t=0.01$. The reduced order parameter ${\psi }_1$ is shown by a solid line and the dashed line is ${\psi }_2$. (b) The closeup of the order parameters for $6.5<z/\lambda <7.5$.

Figure 5

(Color online) (a) The reduced field and order parameters obtained numerically by minimizing the interface energy functional, $\Sigma$, for ${\kappa }_1=3$, ${\kappa }_2=4$, $u=0.6$, and $t=0.2$. The reduced order parameter ${\psi }_1$ is shown by a solid line and the dashed line is ${\psi }_2$. (b) The closeup of the order parameters for $6.5<z/\lambda <7.5$.

Figure 6

(Color online) (a) The reduced field and order parameters obtained numerically by minimizing the interface energy functional, $\Sigma$, for ${\kappa }_1=3$, ${\kappa }_2=4$, $u=0.6$, and $t=0.01$. The reduced order parameter ${\psi }_1$ is shown by a solid line and the dashed line is ${\psi }_2$. (b) The closeup of the order parameters for $6.5<z/\lambda <7.5$.