Metastability and hysteretic vortex pinning near the order-disorder transition in ${\mathrm{NbSe}}_2$: Interplay between plastic and elastic energy barriers

We studied thermal and dynamic history effects in the vortex lattice (VL) near the order-disorder transition in clean ${\mathrm{NbSe}}_2$ single crystals. Comparing the evolution of the effective vortex pinning and the bulk VL structure, we observed metastable superheated and supercooled VL configurations that coexist with a hysteretic effective pinning response due to thermal cycling of the system. A novel scenario, governed by the interplay between (lower) elastic and (higher) plastic energy barriers, is proposed as an explanation for our observations: Plastic barriers, which prevent the annihilation or creation of topological defects, require dynamic assistance to be overcome and to achieve a stable VL at each temperature. Conversely, thermal hysteresis in the pining response is ascribed to low energy barriers, which inhibit rearrangement within a single VL correlation volume and are easily overcome as the relative strength of competing interactions changes with temperature.

Figure 1

In-phase linear AC susceptibility ${\chi }^{\prime }\left(T\right)$ recorded during a FCC process (black curve), after shaking the system at $T_{\mathrm{sh}}<T_{\mathrm{po}}$ (red curve), and during various FCW from different minimum temperatures $T_{min}$ (green and blue curves). Arrows indicate the direction of temperature variation; vertical dotted lines indicate relevant temperatures (see text). FCW processes display lower effective pinning than FCC without any dynamic assistance. Inset: Normalized height of the PE [Eq. (4), see text] measured in FCW processes as a function of the lowest temperature reached, $T_{min}$. The dotted line is a guide to the eye.

Figure 2

Rocking curves from SANS measurements after different thermal and dynamic histories recorded at the base temperature $3\mathrm{K}$ (a) and at $4.65\mathrm{K}$ (b): FCC (black open squares), VL shaken at $5.6\mathrm{K}$ (red open circles), and FCW from $3\mathrm{K}$ (blue open triangles). Continuous lines are Lorentzian fits with full width at half maximum $\mathrm{\Delta }\theta$.

Figure 3

Maximum BP intensity, $I_{max}\left(T\right)$, recorded during cooling/warming processes after different dynamic histories: VL shaken in the BG phase (red dots), FCC (black squares), and FCW (blue triangles). Continuous lines show the expected $F^2\left(T\right)$-like dependence from Eq. (3), using an effective vortex core size (see text). Arrows indicate the direction of temperature evolution in the three processes (labelled FCC, FCW, and BG) and the intensities corresponding to RCs shown in Fig. 2 [labelled (a) and (b)]. A vertical dashed line marks $T_{\mathrm{po}}$.

Figure 4

AC response measured during cooling (open symbols) and immediately warming (full symbols) VL configurations with intermediate disorder. Each of these configurations (dark yellow, orange, and green symbols) was prepared by shaking the VL at a different temperature in the transitional region, near $T_{\mathrm{po}}$ (represented by arrows in dotted lines). Arrows in continuous lines indicate the direction of temperature evolution and numbers in parenthesis, temporal order. Continuous lines reproduce FCC (black), FCW (blue), and BG (red, shaken well below $T_{\mathrm{po}}$) curves for reference. Each warming response is still intermediate between FCW and BG responses.

Figure 5

AC response measured while thermally cycling an initially ordered VL. Straight arrows indicate the direction of temperature evolution and red curved arrows mark turning points. In each sequence, (1) a fresh BG was warmed to $T_{max}$ (red symbols), (2) cooled to $4.5\mathrm{K}$ (open symbols), and (3) warmed again (full symbols). There is a temperature $T^{*}$ (vertical dashed line) such that, if $T_{max}<T^{*}$, the BG response is recovered [panel (a)], but memory is lost when $T_{max}>T^{*}$ [panel (b)].