Critical behavior of nonequilibrium depinning transitions for vortices driven by current and vortex density

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In Fig. 1a,c, we show the current-voltage (I–V) characteristics at 4.1 K in 1.27 and 3.8 T, respectively, on a linear-linear scale. In the insets, the I–V data in the main panels, including additional data, are plotted on a log-log scale. We define \(I_{d}\) as a threshold current at which the vortices start to move, using a \(10^{-8}\) V criterion⁹. The location of \(I_{d}\) = 0.375 mA for each field is marked with a vertical arrow and dashed line in the main panel and inset, respectively. Upward curvature of the I–V characteristics just above \(I_{d}\) in the main panels indicates that the vortex flow immediately after the depinning is plastic flow, as described later.

Figure 1

The critical divergence of the relaxation time \(\tau (I)\) at the current-driven depinning transition (\(I_{c}\)). (a, c) I–V characteristics at 4.1 K in (a) 1.27 T and (c) 3.8 T on a linear-linear scale. Inset: I–V data plotted on a log–log scale. A vertical arrow and dashed line in each main panel and inset, respectively, indicate the depinning current \(I_{d}\) = 0.375 mA defined using a \(10^{-8}\) V criterion. Other lines are guide to the eye. (b, d) The critical divergence of \(\tau (I)\) at the I-driven depinning transition in (b) 1.27 T and (d) 3.8 T. The open and solid red circles represent \(\tau\) for \(I\le\) 0.37 mA and \(I\ge\) 0.39 mA, respectively, showing a power-law divergence at 0.375 ± 0.005 mA(\(\equiv I_{c}\)) from both sides, as marked with a vertical dashed line. Inset: \(\log {\tau }\) versus \(\log {|I-I_{c}|}\) in (b) 1.27 T and (d) 3.8 T, where symbols are the same as in the main panel. Both the dotted and full red lines indicate the power-law fits by \(\tau \propto |I-I_{c}|^{-\nu }\) with (b) \(\nu\) = 1.34 ± 0.13 and (d) \(\nu\) = 1.02 ± 0.15. The error bars in \(\tau\) mainly correspond to the fitting errors resulting from the uncertainty in determining a.

Shown in Fig. 2a is the B dependence of \(I_{d}\), which is interpreted as a phase diagram of the pinned and moving vortex phases: A full line \(I_{d}(B)\) corresponds to the transition line between the pinned phase (\() and the moving phase (\(>I_{d}(B)\)). The peak effect is clearly observed in the field range \(B=\) 2–5.6 T\((=B_{m})\), where the peak field \(B_{p} =\)4.6 T marks the structural transition from the Bragg glass to the vortex glass at equilibrium. Because of the non-monotonic dependence of \(I_{d}\) on B, three fields, such as \(B =\)1.27 T(\(\equiv B_{cL}\)), 3.8 T(\(\equiv B_{cH}\)), and 5.24 T, give the same \(I_{d} =\)0.375 mA.

Figure 3a,b, respectively, show the time-dependent voltage, V(t) and \(V
(1)

Here \(V^{0}\) and \(V^{\infty }\) are the initial and steady-state voltages, respectively, and \(\tau\) is the characteristic time at which the relaxation crosses over from a power-law decay with an exponent a to an exponential decay, as shown in Fig. 3c. Hence, a is relevant very close to the transition where \(\tau \rightarrow \infty\)^5,13,17. In Fig. 3c, we replot all the data shown in Fig. 3a,b as \(\log {V}\)–\(\log {t}\) plots. We find that as I approaches closer to 0.37-0.42 mA, the replotted data fall on nearly a straight line with a slope of \(-a=-0.55\), as indicated with a dashed line. The clear and systematic discrepancy between the fitting lines and the experimental curves at \(t<\) 1 ms is inevitable from the functional form of Eq. (1). To extract the values of \(\tau\), the fitting in the larger t regime where the relaxation crosses over from a power-law decay to an exponential decay is important and the deviation from the fitting at smaller \(t(<\) 1 ms) does not affect the extracted values of \(\tau\). The obtained value of \(a=0.55\pm 0.1\) is almost consistent with the DP theory^36,37, which predicts that the fraction of active (moving) particles, in the case of the depinning transition, obeys the power-law time dependence at the critical point with an exponent \(a\approx 0.45\) for the DP class or \(a\approx 0.5\) for the conserved DP class.

The full lines in Fig. 3a,b,c display the results of the fits to Eq. (1) using \(a=0.55\). In Fig. 1b, the values of \(\tau\) thus obtained are plotted as a function of I with open and solid red circles for \(I\le\)0.37 mA and \(I\ge\)0.39 mA, respectively. They show a power-law divergence at 0.375 ± 0.005 mA(\(\equiv I_{c}\)) from both sides, as marked with a vertical dashed line. As far as we know, this is the first experimental observation of the critical divergence of \(\tau\) in the pinned phase (\(I) of the depinning transition. The inset of Fig. 1b displays the plots of all the values of \(\tau\) against \(|I-I_{c}|\) on a double logarithmic scale, i.e., \(\log {\tau }\) versus \(\log {|I-I_{c}|}\), where symbols are the same as in the main panel. Both the dotted and full red lines in the main panel and inset indicate the power-law fits by \(\tau \propto |I-I_{c}|^{-\nu }\) with \(\nu\)=1.34 ± 0.13, which is, within error bars, in agreement with the theoretical value of \(\nu =1.295\pm 0.006\) expected for the absorbing phase transition in the DP universality class in 2D³⁶ and \(\nu =1.225\pm 0.029\) for the conserved 2D DP class.