Dynamic ordering of driven vortex matter in the peak effect regime of amorphous MoGe films and $2\mathrm{H}\text{−}\mathrm{Nb}{\mathrm{Se}}_2$ crystals

Dynamic ordering of driven vortex matter has been investigated in the peak effect regime of both amorphous MoGe films and $2\mathrm{H}\text{−}\mathrm{Nb}{\mathrm{Se}}_2$ crystals by mode locking (ML) and dc transport measurements. ML features allow us to trace how the shear rigidity of driven vortices evolves with the average velocity. Determining the onset of ML resonance in different magnetic fields and/or temperatures, we find that the dynamic ordering frequency (velocity) exhibits a striking divergence in the higher part of the peak effect regime. Interestingly, this phenomenon is accompanied by a pronounced peak of dynamic critical current. Mapping out field-temperature phase diagrams, we find that divergent points follow well the thermodynamic melting curve of the ideal vortex lattice over wide field and/or temperature ranges. These findings provide a link between the dynamic and static melting phenomena which can be distinguished from the disorder induced peak effect.

Figure 1

Magnetic-field dependence at $1.96\mathrm{K}$ of critical current $I_c$ (solid circles) and dynamic critical current $I_{c,\mathit{dyn}}$ (open circles) for an $\alpha \text{−}\mathrm{Mo}\mathrm{Ge}$ film (MG 1). In the inset, an $I\text{−}V$ curve measured at $7.2\mathrm{T}$ is given. The solid line represents the linear extrapolation of the flux-flow behavior to the zero voltage. As indicated by an arrow, the intersectional point defines $I_{c,\mathit{dyn}}$.

Figure 2

A set of dc $I\text{−}V$ curves for an $\alpha \text{−}\mathrm{Mo}\mathrm{Ge}$ film (MG 2) measured at $2.0\mathrm{T}$ in $4.2\mathrm{K}$ with superimposed ac currents (amplitude 3.3, 2.6, 2.1, 1.6, 1.3, 1.0, 0.82, 0.65, 0.52, 0.41, 0.33, and $0.26\mathrm{mA}$ from left to right). A curve denoted as DC indicates the pure dc $I\text{−}V$ curve. Application of ac induces clear mode-locking current steps at equidistant voltages denoted by $p∕q=1∕1$ and $1∕2$.

Figure 3

Frequency dependence of maximum current width $\Delta I_{\mathit{max}}$ of the fundamental ML step for an $\alpha \text{−}\mathrm{Mo}\mathrm{Ge}$ film (MG 2) measured (a) at $2.0\mathrm{T}$, far below the peak effect regime, and (b) at fields of 3.25, 3.40, and $3.45\mathrm{T}$ in the peak effect regime in $4.2\mathrm{K}$. A solid curve represents an empirical function for the ML width given in text (Ref. 26). Dotted lines represent the linear low-frequency behavior of the ML width. Arrows mark the onset frequency $f_c$ for ML resonance.

Figure 4

Magnetic-field dependence of (a) critical current $I_c$ and dynamic critical current $I_{c,\mathit{dyn}}$ and (b) the onset frequency $f_c$ for an $\alpha \text{−}\mathrm{Mo}\mathrm{Ge}$ film (MG 2) at $4.2\mathrm{K}$. Peak points of the critical current and the dynamic critical current are indicated by arrows in (a). The inset of (b) displays the plot of $f_c$ on a linear frequency scale.

Figure 5

(Color online) Magnetic field-temperature phase diagrams for (a) $\alpha \text{−}\mathrm{Mo}\mathrm{Ge}$ films and (b) $\mathrm{Nb}{\mathrm{Se}}_2$ crystals. The magnetic field and temperature are normalized by superconducting transition temperature $T_c$ and the Ginzburg-Landau second critical field $H_{c2}^{GL}\left(0\right)(\equiv S{T}_c)$. Plotted are reduced second critical fields $h_{c2}$ (small solid circles), divergence fields $h_M$ of the ordering frequency (solid squares), and peak fields $h_p$ of the critical current (open circles). In (a), black and red colored symbols correspond to results for MG 1 and MG 2, respectively. In (b), black (red) colored symbols are for NS 1 (NS 2). In (a), a solid black curve represents the mean-field curve for the dirty limit in WHH theory (Ref. 45). For comparison, the linear dependence of $h_{c2}$ on $1-t$ is displayed by a broken line in (a) [also in (b)]. A dotted curve in (a) is a guide for the eyes. Blue curves in (a) and (b) represent thermodynamic melting curves obtained from a quantitative GL model (Ref. 28) for MG 1 and NS 1 crystals, respectively.