Collective Dynamics and Strong Pinning near the Onset of Charge Order in ${\mathrm{La}}_{1.48}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$

The dynamics of charge-ordered states is one of the key issues in underdoped cuprate high-temperature superconductors, but static short-range charge-order (CO) domains have been detected in almost all cuprates. We probe the dynamics across the CO (and structural) transition in ${\mathrm{La}}_{1.48}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$ by measuring nonequilibrium charge transport, or resistance $R$ as the system responds to a change in temperature and to an applied magnetic field. We find evidence for metastable states, collective behavior, and criticality. The collective dynamics in the critical regime indicates strong pinning by disorder. Surprisingly, nonequilibrium effects, such as avalanches in $R$, are revealed only when the critical region is approached from the charge-ordered phase. Our results on ${\mathrm{La}}_{1.48}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$ provide the long-sought evidence for the fluctuating order across the CO transition, and also set important constraints on theories of dynamic stripes.

Figure 1

(a) $R_c$ vs $T$. $T_c$ and $T_{\mathrm{LTT}}$ were determined as shown; superconducting fluctuations vanish at $T\sim 30\mathrm{K}$ [38, 39, 40]. $T_{\mathrm{CO}}$ and $T_{\mathrm{SO}}$ are from Refs. [24, 25, 26]. Inset: Transition region. (b) Relaxation and avalanches in $R_c$ after warming from $\sim 40$ to $T=70.59\mathrm{K}$. $T$ becomes stable at $t=t^{\prime }$, but $R_c$ relaxes for hours after $t^{\prime }$. Inset: The relaxation, with avalanches removed, fitted to a stretched exponential function (dashed line), with $\beta =0.2$ and ${\tau }_{\text{relax}}=32\mathrm{s}$. (c) The cumulative distribution $N_c$ of avalanche sizes $\mathrm{\Delta }R$ for the data in (b) ($T=70.59$) and at $T=70.7\mathrm{K}$, obtained from ten separate measurements [34]. Dashed lines are fits $N_c\sim (\mathrm{\Delta }R{)}^{-({\tau }_T-1)}$ with ${\tau }_T\approx 1.8$, as shown. Inset: The occurrence of avalanches in (b) after $t^{\prime }$; the dashed line is a logarithmic fit.

Figure 2

(a) $R_c$ vs $H$ after warming to $T=71.10\mathrm{K}$. The arrows and numbers show the direction and the order of field sweeps. The MR exhibits return-point memory; e.g., $R_c$ at 5 T has the same value after the first subloop (2–3) has closed as in the initial sweep. Inset: Subloops shifted vertically for comparison. (b) $N_c(\mathrm{\Delta }R)$ at $T=70.7\mathrm{K}$ [34]. The solid line is a fit $N_c\sim (\mathrm{\Delta }R{)}^{-({\tau }^{\prime }-1)}$ with ${\tau }^{\prime }=1.666\pm 0.002$. Inset: The number of avalanches ($N$) vs $H$ (centers of 0.25 T-wide bins). The solid line is a Gaussian fit, centered at $H_c=(9.5\pm 0.1)\mathrm{T}$ with a 2.2 T half-width. (c) $N_c(\mathrm{\Delta }R)$ for $T=70.7\mathrm{K}$ and fixed $H$ (1 T bins), as shown. Traces are offset vertically for clarity. Solid lines are power-law fits; dashed lines are extrapolations of those fits. At $H=H_c$, $N_c\sim (\mathrm{\Delta }R{)}^{-(\tau -1)}$ with $\tau =1.54\pm 0.04$.

Figure 3

The average number of avalanches per MR measurement (red diamonds), after warming, vs $T$. To increase the statistics, i.e., the number of data sets used, only avalanches for $H\le 9\mathrm{T}$ were included. $d\ln R_c/dT$ (blue line) has a minimum at the suppression of avalanche occurrence. The features of $d\ln R_c/dT$ marked by the vertical dashed lines correspond to $T_{d3}$ and $T_{d2}$, the temperatures of the LTT-LTLO and LTLO-LTO transitions, respectively [32].