Stable fractional flux vortices and unconventional magnetic state in two-component superconductors

In the framework of London theory we study the unconventional magnetic state in two-component superconductors with a finite density of fractional flux vortices stabilized near the surface. We show that the process of vortex entry into the two-component superconductor consists of several steps, while the external magnetic field increases from zero. At the first stage only vortices in one of the order parameter components penetrate and sit at the equilibrium position near the surface. When the magnetic field is increased further, vortices in the second-order parameter component eventually enter the superconductor. Such a complex partial vortex penetration leads to the modification of a Bean-Livingston barrier and a magnetization curve as compared to conventional single-component superconductors. We discuss the possibility of experimental identification of protonic superconductivity in the projected superconducting state of liquid metallic hydrogen and hydrogen-rich alloys with the help of the partial vortex penetration effect.

Figure 1

The sketch of the system under consideration. It consists of a two-component superconductor occupying the half space $x>0$ bounded by the $\mathit{yz}$ plane. The fractional vortices are situated at distances $x_A$ and $x_B$ from the boundary. The boundary conditions are taken into account by placing image antivortices at the proper points behind the boundary plane.

Figure 2

Left-hand panel: Gibbs energy of the molecule consisting of two fractional vortices near the surface of the superconductor. The parameters are ${\xi }_A={\xi }_B$ and ${\varphi }_A={\varphi }_B$. The magnetic field is taken to be $H_0=5H_{\lambda }$. In this plot the local energy minima are clearly seen at $x_B=0$, $x_A=x_A^{*}$ and $x_A=0$, $x_B=x_B^{*}$. Right-hand panel: The dependence of the fractional vortex position $x_A^{*}$ on the magnetic field $H_0$ for different values of ${\varphi }_A/{\varphi }_B=1,2,3,4$ (from top to bottom curve). All lengths are normalized to the scale $\lambda$.

Figure 3

Gibbs energy of the molecule consisting of two fractional vortices near the surface of the superconductor. (a) Equal coherence lengths of two condensates ${\xi }_A/{\xi }_B=1$. In this case the Bean-Livingston barrier is suppressed simultaneously for fractional vortices of both types. (b) Different coherence lengths of two condensates ${\xi }_A/{\xi }_B=5$. In this case the barrier is suppressed for the fractional vortex in the $A$ condensate while for the $B$ condensate it still exists.

Figure 4

Qualitative behavior of magnetization curve of a superconductor modified due to the surface barrier. Left-hand panel: Single-gap superconductor. Right-hand panel: Two-gap superconductor.