Observation of Well-Ordered Metastable Vortex Lattice Phases in Superconducting ${\mathrm{MgB}}_2$ Using Small-Angle Neutron Scattering

The vortex lattice (VL) symmetry and orientation in clean type-II superconductors depends sensitively on the host material anisotropy, vortex density and temperature, frequently leading to rich phase diagrams. Typically, a well-ordered VL is taken to imply a ground-state configuration for the vortex-vortex interaction. Using neutron scattering we studied the VL in ${\mathrm{MgB}}_2$ for a number of field-temperature histories, discovering an unprecedented degree of metastability in connection with a known, second-order rotation transition. This allows, for the first time, structural studies of a well-ordered, nonequilibrium VL. While the mechanism responsible for the longevity of the metastable states is not resolved, we speculate it is due to a jamming of VL domains, preventing a rotation to the ground-state orientation.

Figure 1

SANS diffraction patterns showing the different triangular VL phases in ${\mathrm{MgB}}_2$ with $\mathbf{H}\parallel \mathbf{c}$. The orientation of the crystalline $a$ axis is shown in panel (a) and the $a^{*}$ axis in panel (e). The three columns show the VL configurations at 18 K and 0.5 T [(a) and (d)], 2 K and 0.5 T [(b) and (c)], and 2 K and 1.1 T [(e) and (f)]. In cases where not all VL reflections were rocked through the Bragg condition, symmetry equivalent positions, obtained by mirroring through and/or rotating around $q=0$, are indicated by open circles. In the high symmetry $F$ and $I$ phases a single-domain VL is observed aligned with Bragg reflections along respectively $\mathbf{a}$ and ${\mathbf{a}}^{*}$. In the intermediate $L$ phase two VL domains orientations, indicated by $2\times 6=12$ peaks, are observed rotated away from $\mathbf{a}$ by an angle $\phi$ in opposite directions as shown in panel (f). For each field and temperature the ground state VL is obtained following a damped field oscillation with an initial amplitude of 25 mT.

Figure 2

VL orientation angle as a function of applied field. Panel (a) shows the results of measurements at 2 K following a FC procedure and a zero field cooling followed by a FR. The arrow indicates the FL transition for the ground-state VL. Panel (b) shows the FR VL orientation at temperatures ranging from 2 to 20 K. All lines are guides to the eye.

Figure 3

Ground-state VL phase diagram for ${\mathrm{MgB}}_2$ with $\mathbf{H}\parallel \mathbf{c}$. The FL transition is indicated by squares and the LI transition by circles. The lines are guides to the eye with dashes indicating extrapolations to higher temperatures and fields. Diamonds indicate measured values of $H_{c2}$.

Figure 4

Free energy curves for the VL phases in ${\mathrm{MgB}}_2$. Panel (a) shows the progression of free energy curves calculated using Eqn. (1) with $K_{12}=1$ and increasing $K_6$. The corresponding ground-state VL phase is indicated by each curve. The FL and LI transitions occur at $K_6=\pm 4$. Panel (b) shows free energy curves corresponding to the sequence of diffractions patterns at 0.5 T shown in Fig. 1. Solid circles represent ground-state VL configurations while open circles represent metastable phases.