Phase diagram of vortex matter of type-II superconductors

We propose a model to construct the phase diagram for type-II superconductors through molecular-dynamics simulation. In this model, the Abrikosov vortices interact with long-range repulsion and short-range attraction. We are able to obtain the phase diagrams consisting of the vortex lattice, the intermediate-mixed phase (IMP), and the disordered vortex phase in the $B$-$T$, $B$-$\kappa$, and $B$-$q$ planes, where $B$ is induction, $\kappa$ is the Ginzburg-Landau parameter, and $q$ is the relative strength of the attraction to repulsion interactions between vortices. The IMP is in the form of a superlattice of either vortex bubbles or parallel vortex stripes, agreeing excellently with experimental results that have not yet been explained by existing models.

Figure 1

Phase diagram of vortex matter as a function of field and temperature. The insets show the structures of the vortex phases in regimes I–IV.

Figure 2

Vortex phases for different magnetic fields at fixed temperature $T=0$ and $N_{\mathrm{v}}=400$. Panel (a) shows the vortex lattice for $B=0.8B_{c20}$. Panels (b) and (c) show the superlattice of vortex stripes for $B=0.59B_{c20}$ and the superlattice of vortex bubbles for $B=0.43B_{c20}$, respectively. Panel (d) shows the disordered phase for $B=0.04B_{c20}$.

Figure 3

Structures of the IMP as a function of temperature at $B=0.43B_{c20}$: (a) superlattice of vortex bubbles at $T=0.5T_c$; (b) vortex stripe phase at $T=0.6T_c$; (c) liquid phase at $T=0.75T_c$.

Figure 4

(a) Phase diagram of vortex matter as function of field $B$ and $\kappa$, (b) phase diagram of vortex matter as function of $B$ and $q$ for $T=0$, $N_{\mathrm{p}}=2N_{\mathrm{v}}$, $f_{\mathrm{pv}}=0.15f_0$, and $r_{\mathrm{p}}=0.2\lambda$. The phases in both (a) and (b) are the disordered vortex phase (I), the hexagonally ordered bubble phase (II), the ordered stripe phase (III), and the vortex lattice (IV). The inset in (a) shows the intervortex interaction force $F^{\mathrm{vv}}$ (r) as a function of the distance $r$ between vortices for different $\kappa$ (see text for a more detailed discussion).

Figure 5

Vortex phases for different $q$ for $B=0.65B_{c20}$, $T=0$, $N_{\mathrm{p}}=2N_{\mathrm{v}}$, $f_{\mathrm{pv}}=0.15f_0$, and $r_{\mathrm{p}}=0.2\lambda$: (a) the vortex lattice for $q=0$; (b) the vortex stripe phase for $q=2.3$; (c) the vortex bubble phase for $q=3$.

Figure 6

Time evolution of the formation of the ordered bubble phase, starting from a random disordered state, as shown in (a), and the corresponding structure factors with $B=0.43B_{c20}$ and $T=0$. The vortex configurations are (b) $t=2$, (c) $t=10$, (d) $t=20$, (e) $t=30$, and (f) $t=50$. The corresponding structure factor $S(k)$ are (g) $t=0$, (h) $t=2$, (i) $t=10$, (j) $t=20$, (k) $t=30$, and (l) $t=50$.