Vortex matter in mesoscopic two-gap superconductor square

In mesoscopic superconductors with low intrinsic pinning, the boundary plays the most important role in the stabilization of the vortex patterns. Especially in the case of symmetric sample shape, very distinct vortex locations inside the sample are defined, for the different vorticity states. The study of two-component superconductors with the Ginzburg-Landau equation implies the introduction of a coupling term between the two condensates, changing the linear part of the potential. This article presents the analysis of the impact of the competition between the coupling term and the geometry of confinement on the vortex patterns of thin mesoscopic square samples made from a two-gap superconductor. For a simple case presented here, it was found that the appearance of the noncomposite vortices is accompanied by an unusual shape of the component of the order parameter associated with the passive band as a function of the temperature. This shape is a distinct fingerprint for materials with noncomposite vortices.

Figure 1

Diagrams of the vortex phase boundaries for one-component (a) and two-component (b) superconductors. The one-component case is parameter free. For the two-component diagram, the values of the parameters used were $\frac{m_1}{m_2}=1$, $\frac{{\beta }_2}{{\beta }_1}=1$, $\frac{S}{{\xi }_{\gamma }^2}=0.05$, and $\frac{S}{{\xi }_2^2\left(T_c\right)}=20$. On the diagrams, the almost vertical lines are the boundaries for the vorticity phases, and the almost horizontal lines, inside the vorticity areas, are boundaries for the broken-symmetry phases. The dashed red line is the nucleation phase boundary for the second component with $\gamma =0$. The small square diagrams over the regions represent the respective vortex patterns of that region. In these square diagrams, the noncomposite vortices in the active band (${\Psi }_1$) and passive band (${\Psi }_2$) are represented by green and red circles, respectively, and the composite vortices are represented in black as in the case of only one band on the diagram (a). The filled small circles, empty small circles, and the big filled circles correspond to vorticity 1, $-$1, and 2, respectively.

Figure 2

3D view of the condensate in the common logarithm scale (log${}_{10}$). (a), (b), and (c) correspond to phases (h), (i), and (g) of the diagram 1b and to the parameters ($\Phi /{\Phi }_0$,$S/{\xi }_1^2$) = (4.5, 60), (6, 50), and (6.6, 52), respectively. For each vertical set of plots, the top plot corresponds to $|{\Psi }_1{|}^2$, the second to $|{\Psi }_2{|}^2$, and the third to the sum. The remaining parameters are the same as in Fig. 1b.

Figure 3

A zoom, in the center of the sample, of the passive band-density distribution for the vortex phase area (i) of Fig. 1b.

Figure 4

(Color online) Plots of the root mean square of the two components of the order parameter, $|{\Psi }_1{|}_{\mathrm{rms}}=\sqrt{(\int |{\Psi }_1{|}^{2}dS)}/S$ and $|{\Psi }_2{|}_{\mathrm{rms}}=\sqrt{(\int |{\Psi }_2{|}^{2}dS)}/S$, and the distance(in units of the sample size $a$) between vortices in the different bands, as defined in the text, along the lines in the $T$-$H$ phase space(that pass through regions of vorticity two, these lines are shown in the small plots inside the plots) for different parameters of the system. Vertical dashed lines show the transition point between symmetric and broken vortex patterns. We note that there are plots with one and plots with two vertical dashed line. In the case of one vertical line this represents the symmetry breaking of both of the condensates. In the case of two vertical lines the left and the right lines are the points were the ${\Psi }_1$ and ${\Psi }_2$ condensates break symmetry, respectively.

Figure 5

Density plots of the $\ln ({\Psi }_1/{\Psi }_2)$ for the MgB${}_2$ parameters expressed in the text. The regions with the lighter green(lighter color) and darker blue(darker color for black and white picture) colors identify were ${\Phi }_1$ and ${\Phi }_2$ are more close to zero, respectively. (a), (b) and (c) correspond to the $\Phi /{\Phi }_0=$ 4.5, 6 and 7, respectively, with $\frac{S}{{\xi }_1^2\left(T\right)}=100$