Vortex-chain phases in layered superconductors

Layered superconductors in tilted magnetic field have a very rich spectrum of vortex lattice configurations. In the presence of in-plane magnetic field, a small $c$-axis field penetrates in the form of isolated vortex chains. The structure of a single chain is mainly determined by the ratio of the London $\left(\lambda \right)$ and Josephson $\left({\lambda }_J\right)$ lengths, $\alpha =\lambda ∕{\lambda }_J$. At large $\alpha$ the chain is composed of tilted vortices (tilted chains) and at small $\alpha$ it consists of a crossing array of Josephson vortices and pancake stacks (crossing chains). We studied the chain structures at intermediate $\alpha$’s and found two types of behavior. (i) In the range $0.4\lesssim \alpha \lesssim 0.5$ a $c$-axis field first penetrates in the form of pancake-stack chains located on Josephson vortices. Due to attractive coupling between deformed stacks, their density jumps from zero to a finite value. With further increase of the $c$-axis field the chain structure smoothly evolves into modulated tilted vortices and then transforms via a second-order phase transition into the tilted straight vortices. (ii) In the range $0.5\lesssim \alpha \lesssim 0.65$ a $c$-axis field first penetrates in the form of kinks creating kinked tilted vortices. With increasing the $c$-axis field this structure is replaced via a first-order phase transition by the strongly deformed crossing chain. This transition is accompanied by a large jump of pancake density. Further evolution of the chain structure is similar to the higher anisotropy scenario: it smoothly transforms back into the tilted straight vortices.

Figure 1

(Color online) Crossing (left) and tilted (right) vortex chains. Upper pictures show three-dimensional views and lower pictures show the structures of isolated chains.

Figure 2

(Color online) (a) The dependence of the maximum pancake displacement in the crossing (defined in the inset) on the ratio $\alpha =\lambda ∕{\lambda }_J$ for different periods $a$ in $x$ direction. The plot shows that crossing becomes unstable near $\alpha =0.69$. The inset shows crossing configuration near instability point. (b) The dependence of crossing energy on the ratio $\alpha$. For comparison, small-$\alpha$ calculation is shown in both plots.

Figure 3

(Color online) Main plot shows the dependence of the maximum displacement(defined in the inset) divided by pancake separation on the parameters $\alpha$ for $N=14$ and $a=1.52{\lambda }_J$. The chain structures are illustrated at marked points. In the configuration pictures circles show positions of the pancake vortices and horizontal lines mark locations of the Josephson vortices. One can see that the system evolves from weakly deformed chain $(\alpha =0.25)$ via strongly deformed chain $(\alpha =0.4)$ to modulated tilted chain $(\alpha =0.46)$. The last structure transforms via a second-order phase transition at $\alpha =0.48$ into tilted straight vortices.

Figure 4

(Color online) (a) The dependence of the local part of energy of the chain configuration on the ratio $\alpha$ for $N=10$ and $a=3.4{\lambda }_J$. For comparison, we also show analytical estimates for the crossing and tilted chain configuration. (b) Phase diagram in the plane (pancake separation $a$)-(ratio $\alpha$) for the fixed separation between JVs, $N=10$. The solid line shows computed transition into the tilted chain state. The dashed line shows location of the transitional region where the energy of the crossing configuration is equal to the energy of the tilted configuration.

Figure 5

(Color online) The dependence of the excess pancake energy per stack in units of ${\epsilon }_0$, $U$, on the separation between pancakes $a$ for $\alpha =0.5$ and different $N$. The position of minimum of this energy corresponds to the maximum equilibrium separation $a_m$ (marked for $N=8$ plot).

Figure 6

(Color online) Chain phase diagrams in the plane $a∕{\lambda }_J-N$ for two values of the ratio $\alpha$, 0.4 and 0.5. The solid line indicate phase transition into the tilted-chain phase. The dotted line shows the maximum equilibrium separation $a_m$. The dashed line shows crossover “reconnection” line at which the maximum displacement exceeds $a∕4$ and weakly deformed crossing configuration crosses over into the strongly corrugated configuration. For $\alpha =0.5$ we also show $a_m$ for the tilted-chain phase analytically calculated from Eq. (39).

Figure 7

(Color online) The dependence of the excess pancake energy per stack on pancake separation $a$ for $N=16$ and $\alpha =0.6$. The two branches correspond to the two different starting states at large $a$, crossing chain and kinked lines. One can see that the kinked lines have lower energy at very large $a$, $a>14.81$. Crossing chain smoothly transforms back into tilted chain with decrease of $a$. The transformation is completed at a second-order phase transition point at $a_t=2.5{\lambda }_J$. Chain configurations at four points marked by arrows are shown below. The evolution of chain configuration along this energy curve is also illustrated by an animation (See Ref. 26).

Figure 8

(Color online) The density dependence of the pancake part of the thermodynamic potential per unit area, $\delta E-\mu n$ (in units of ${ϵ}_0∕{\lambda }_J$) at different chemical potentials $\mu$ corresponding to different values of the magnetic field strength for $N=14$ and $\alpha =0.6$. Kinks in the curves separate regions of tilted/kinked lines (low $n$) and crossing chains (high $n$). The equilibrium density in units of $1∕{\lambda }_J$ is given by the global minimum of this energy. One can see that at $\mu \approx -0.152$ the system experiences a first-order phase transition with very large density jump.

Figure 9

(Color online) The upper plot shows the phase diagram in the $N\text{−}a$ plane for $\alpha =0.6$. The left plot in the lower plat illustrates meanings of the phase lines using the plot of the thermodynamic potential $\delta E-\mu n$ vs $a∕{\lambda }_J$ at the transition point (main plot) and the maximum displacement $u$ divided by lattice spacing $a$ vs $a∕{\lambda }_J$ (inset). The lines 1 and $1^{\prime }$ correspond to the two limiting pancake separations at the transition point between which the jump occurs. The line 2 indicates crossing of the energy curves for the kinked and crossing chain. The line $2^{\prime }$ shows the position of a continuous transition into the tilted chain (illustrated by the inset in the lower panel). Dotted line slightly above $1^{\prime }$ line shows the position of the maximum equilibrium separation $a_m$. We also show the crossover line $a∕{\lambda }_J=N$ above which well-defined kinks appear. The right plot in the lower part shows blowup of the phase diagram above the $2^{\prime }$ line.

Figure 10

(Color online) Phase diagram in $a\text{−}\alpha$ plane for $N=14$. Solid nose-shaped line shows the phase transition into the tilted-chain state. Dashed line shows location of the maximum equilibrium separation, which terminates at some point. We also show by dotted line location of the stability boundaries obtained from inflection points of the dependencies $U\left(a\right)$ [see criterion (38)].

Figure 11

(Color online) Left panel: The dependencies of the excess pancake energy per stack $U$ on separation $a$ for $N=14$ and different values of $\alpha$. With increasing $\alpha$ the first-order transition from tilted to crossing chain vanishes and the crossing chain do not realize at all. Also the dependence $U\left(a\right)$ becomes monotonic which corresponds to termination of $a_m\left(\alpha \right)$ line in Fig. 10. Right panel: The dependence of the relative maximum displacement $u$ on separation $a$ for $N=14$ and $\alpha =0.63$. The maximum value of $u$, corresponding to tilted chains, is given by $(1-1∕14)∕2\approx 0.464$. One can see that there is a range of $a$ where the chains become only slightly modulated.

Figure 12

Left panel shows plot of the pancake part of the chain energy per unit area in units ${\epsilon }_0∕{\lambda }_J$, $\delta E$, vs pancake density $n$ at very small $n$. Slope of this energy at $n\to 0$ determines the energy of an isolated kink and the numerical constant $C_{kv}$. Calculations were made for two sets of parameters, $(N=10,\alpha =0.52)$ and $(N=14,\alpha =0.6)$, and for two system sizes for each set. Right picture illustrates the numerically calculated kink structure. Arrows show in-plane currents and gray level plot codes distribution of the cosine of phase difference between the neighboring layers (dark area below corresponds to the Josephson vortex.)