Density functional theory for the freezing transition of the vortex-line liquid with periodic layer pinning

Possible phases and phase transitions of interlayer Josephson vortex lines are explored by the density functional theory. With the aid of liquid correlation functions, the present theory exposes explicitly the interference between the tendency of crystallization driven by the intervortex repulsions and the underlying layer structure. In strong magnetic fields $B⩾\sqrt{3}{\varphi }_0∕2\gamma s^2$, the freezing transition can be continuous, provided the layer pinning is beyond a threshold value. For magnetic fields around $B=\sqrt{3}{\varphi }_0∕8\gamma s^2$, a vortex smectic with more vortices in alternate layers (i.e., period $m=2$) is stabilized by the layer pinning beyond another, larger threshold value; the smectic freezes into lattice via a first-order transition and transforms into liquid via a second-order transition upon temperature sweeping. No thermodynamically stable smectic with period $m⩾3$ can be found even for extremely large layer pinning. In a wide regime of magnetic field, although the layer pinning enhances the lattice order to higher temperatures at commensurate magnetic fields and thus produces a meandering phase boundary in the $H\text{−}T$ phase diagram, the freezing transition of vortex lattice is one step and first order. In even lower magnetic fields, the effect of layer pinning is small and the phase boundary becomes monotonic, where the anisotropy scaling of the melting line is recovered. Taking material parameters into account, it is found that the $m=2$ smectic is realized in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ around $H\simeq 40T$, but not in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$. Continuous melting transitions can be observed in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$ in magnetic field of several teslas where the Josephson vortex lattice is in the dense limit.

Figure 1

Typical liquid DPCF $C(k,0)$ for null layer pinning as a function of rescaled wave number $k$. Inset: temperature dependence of DPCF at three wave numbers $k_a,k_b$, and $k_c$ corresponding to the first, second, and third shells of RLVs of the equilateral triangular vortex-line lattice. $a_0=\sqrt{2∕\left(\sqrt{3}{\rho }_0\right)}$.

Figure 2

Josephson vortex lattices commensurate with layer modulation. The $x$ axis is rescaled with the anisotropy parameter $\gamma$. The Josephson vortex lattices are labeled in sequence as $A1,B1,A2,A3$, and $B2$, …, where $A$ and $B$ are for the type of unit cell, and the number denotes the $c$-axis period of vortices in lattice phase in units of layer separation.

Figure 3

Two-step freezing of liquid, a second-order transition into an intermediate smectic, and a first-order one into the lattice structure $A2$, upon temperature sweeping. The magnetic field and layer pinning are fixed at $B=\sqrt{3}{\varphi }_0∕\left[2\gamma {\left(ms\right)}^2\right]$ with $m=2$ and $\beta V_{\mathrm{p}}=1$.

Figure 4

${\rho }_{0}C(k_a,0)-\beta V_{\mathrm{p}}$ phase diagram for $B=\sqrt{3}{\varphi }_0∕\left[2\gamma {\left(ms\right)}^2\right]$ with $m=2$.

Figure 5

Possible $B\text{−}T$ phase diagrams for materials of (a) weak, (b) moderate, and (c) strong layer pinning. L, S, and KT abbreviate liquid, smectic, and Kosterlitz-Thouless phases, respectively, and $A1,B1$ and $A2$ for lattices with specified structures. The thick (thin) curves denote first-order (continuous) phase boundaries. The intermediate phase at the highest magnetic fields refers to the KT phase found in computer simulations.[12] The enhancement of the melting temperature at commensurate magnetic fields is only schematic and not to scale.