Charge-order dynamics in underdoped La 1.6− x Nd 0.4 Sr x CuO 4 revealed by electric pulses

Charge density modulations or charge orders (COs) are observed in all families of hole-doped cuprate high-temperature superconductors, 1 but their relevance for the unconventional properties of the normal state and superconductivity is still an open question. [2][3][4][5][6] According to one broadly-considered scenario, fluctuations of the incipient CO could be favorable or even contribute to the pairing mechanism. 7,8 Therefore, the existence of CO fluctuations and the nature of their dynamics are some of the key issues in the physics of cuprates. Although detecting CO fluctuations has been a challenge because of the remarkable stability of the CO and its short-range nature, both believed to be due to the pinning by disorder, they have been reported recently in several cuprates over a wide range of doping. [9][10][11][12][13] However, relatively little is known about their dynamics.

We report a technique to study the cuprate CO dynamics, in which we apply electrical pulses to drive the CO system out of equilibrium and then study its response using charge transport measurements. Similar studies have been used previously to probe the dynamics of conventional charge-densitywave (CDW) systems, such as 1T-TaS 2 [14][15][16] and some organic conductors. 17,18 More generally, electrical control and switching of resistive states by electric pulsing in strongly correlated materials is of great interest for the development of next generation of solid-state devices. 19 However, one of the main challenges has been to distinguish between the effects of Joule heating and nonthermal effects of the electric field. 20,21 We study La 1.48 Nd 0.4 Sr 0.12 CuO 4 , in which CO is in the form of stripes, 22 and demonstrate that, for small enough perturbation, pulsed current injection allows access to nonthermallyinduced resistive metastable states. The results are consistent with strong pinning of the fluctuating CO by disorder. Our findings pave the way for similar studies in various stripeordered materials, such as other cuprates and nickelates.

La 2-x-y Sr x (Nd,Eu) y CuO 4 and La 2-x Ba x CuO 4 compounds are cuprates that exhibit strongest CO correlations. The striped CO is stabilized by the anisotropy within the CuO 2 a) Electronic mail: dragana@magnet.fsu.edu planes that is present only in the low-temperature tetragonal (LTT) crystallographic phase. Stripes are rotated by 90 • from one CuO 2 layer to next 22 and, just like in other cuprates, this CO is most pronounced for hole doping x ≈ 1/8, corresponding to a minimum in bulk superconducting transition temperature T c . In La 1.48 Nd 0.4 Sr 0.12 CuO 4 , the onset of the apparent static CO occurs at T = T CO T LTT 71.65 K, where T LTT is the transition temperature from the low-temperature orthorhombic (LTO) to LTT phase, with the transition consisting of a 45 • rotation of the tilting axis of the oxygen octahedra surrounding the Cu atoms. 23 The LTO-LTT transition region is also characterized by the presence of an intermediate, lowtemperature less-orthorhombic (LTLO) phase, in which the rotation of the octahedral tilt axis is not complete. The structural transition region is manifested as a jump in the c-axis resistance R c (T ), accompanied by a thermal hysteresis (Fig. 1), which is attributed to the first-order nature of the structural FIG. 1. R c vs T across the structural and CO transition regions of La 1.48 Nd 0.4 Sr 0.12 CuO 4 with a hysteresis loop (independent of sweep rates for 0.01-1 K/min); T CO T LTT 71.65 K. The features of dR c /dT of the warming branch marked by the vertical dotted lines correspond to T d3 = T LTT and T d2 , the temperatures of the LTT-LTLO and LTLO-LTO transitions, respectively. 24 Inset: R c vs T . R c → 0 at T c 3.5 K; T SO 50 K is the onset for spin stripe order. [25][26][27] transition. 28 La 1.48 Nd 0.4 Sr 0.12 CuO 4 is an ideal candidate for electrical pulse studies because evidence for metastable states, collective behavior, and criticality, signatures of fluctuating CO, were found 10 in R c (T ) in the regime across the CO (and structural) transition following the application of a magnetic field (H) or a large change in T as an external perturbation. Surprisingly, those effects were revealed only when the transition region was approached from the CO phase. The measurements were performed using a small, constant electric field ∼ 0.06 V/cm. Here, in contrast, we apply current pulses of different amplitude I p and duration τ at a constant T , and measure the initial and final resistances with a low current I = 10 µA before and after each pulse.

The single crystal of La 1.48 Nd 0.4 Sr 0.12 CuO 4 was grown using the traveling-solvent floating-zone technique. We measure R c on a bar-shaped sample with dimensions 0.24 mm×0.41 mm×1.46 mm (a × b × c), between the voltage contacts placed at a distance ≈ 0.25 mm. The contacts are made by attaching gold leads (≈ 25 µm thick) using the DuPont 6838 Ag-paste, followed by a heat treatment at 450 • C in the flow of oxygen for 30 min. The resulting contact resistances are less than 0.5 Ω at both room temperature and ∼ 70 K. The sample, one Cernox thermometer (CX-1070-BG-HT, serial X92666), and two surface-mount metal-film resistors (as heaters) are placed on the same sapphire platform on top of the 16-pin DIP plug made of G-10 [Fig. 2(a)]. The precise T control at the sample is achieved by a Lake Shore 336 temperature controller using the heaters and the Cernox thermometer; temperature reading from the Cernox is used as the nominal sample T . A Si diode is fixed beneath the 16-pin DIP plug as a secondary thermometer to monitor T stability. For better T control, a probe thermometer and a vaporizer temperature are also monitored during the measurement. The probe thermometer, also a Cernox, is quite far from the sample and is controlled by a probe heater. The probe thermometer is used to sweep or maintain temperature coarsely. The vaporizer temperature is monitored to keep the flow of liquid helium constant during the measurement; this is obtained by finetuning the needle-valve opening of the variable-temperature insert and pumping the sample space using a roughing pump. R c is measured using either a Keithley 6221 current source and 2182A nanovoltmeter in delta mode or SR 7265 lock-in amplifiers using a standard four-probe ac method (∼ 157 Hz). Relatively longer pulses (τ ≥ 1 ms) are generated using the Keithley instruments, controlled with a home-made Lab-VIEW program, or using the LDP-3811 precision current source. For shorter pulses (τ ≥ 2 µs), LDP-3811 is used together with the lock-ins [Fig. 2(b)]. The results did not depend on the choice of instrumentation. The output of the LDP-3811 actually consists of two pulses separated by 100 ns; hereafter, we refer to this sequence as a "pulse" (e.g., a 20-µs, 20-mA "pulse" consists of two 10-µs, 20-mA pulses, the second one starting 100 ns after the first pulse ends). Current pulses are applied after a measurement T is reached by following either the "warm-up" or the "cool-down" protocols.

In the warm-up protocol, the sample is first cycled across the hysteresis by warming up to 90 K and cooling back down to 40 K, followed by warming to a temperature slightly lower than the intended temperature using the probe heater at a rate of 1 K/min. Then, using the heaters near the sample, the measurement T is reached at a slower rate, typically 0.1 K/min, to avoid overshooting of T . In the cool-down protocol, the measurement T is approached from above: first, the sample is cycled across the hysteresis by cooling down to 40 K and warming up to 90 K using probe heater at a rate of 1 K/min, then the probe heater is used at a rate of 1 K/min to reach a temperature slightly higher than the intended temperature and, finally, metal-film resistors are used to reach the measurement T at a slower rate, typically 0.1 K/min, without an overshoot. Hereafter, unless stated otherwise, the pulses are applied after the warm-up protocol.

Figure 2(c) shows a representative effect of a single current pulse on R c . The pulse induces switching to a stable, lower resistance state, with ∆R c defined as the drop in R c after the pulse. Similar measurements are performed with different I p and τ at various T , with each measurement carried out after either a warm-up or a cool-down protocol. ∆R c has a maximum at T ≈ T CO T LTT [Fig. 2(d)], where the difference δ R c between the warming and cooling branches of the main hysteresis loop in Fig. 1 is also maximum. Notably, the resistance drops are observed only after the warm-up protocol, i.e. when the measurement T is approached from a CO phase, consistent with the asymmetry observed in the prior study 10 of CO dynamics in La 1.48 Nd 0.4 Sr 0.12 CuO 4 , and suggesting that current pulses induce switching in a CO system into different metastable states. The question is whether such pulse-induced metastable states are a) caused by nonthermal effects of the current, or b) they result from the Joule heating of the CO system during the pulse and its subsequent cooling to the bath T . In the latter scenario, ∆R c would be observed simply because the system follows the hysteretic R c (T ) behavior.

To explore the possibility of heating, we apply a "heat pulse"', i.e. we increase and then decrease T by a fixed ∆T (Fig. 3(a), lower-right inset; also Supplementary Fig. 2). We find that ∆R c depends only on ∆T , and it does not depend on the number of subsequent heat pulses with the same ∆T . ∆R c increases with ∆T and saturates for ∆T 1 K [Fig. 3(a)]. Importantly, ∆R c ∝ ∆T at low ∆T (Fig. 3(a), upper-left inset), indicating that ∆R c vanishes as ∆T → 0. These results are indeed consistent with the presence of a thermal hysteresis in R c (T ). For example, if a heat pulse is applied at T = 71.65 K after a warm-up protocol (Fig. 3(b), top), R c follows the main warming branch of the hysteresis (black trace, arrow marked 1), followed by cooling along a subloop marked by (blue) arrow 2, resulting in a lower R c once back at the initial T = 71.65 K. Any subsequent heat pulse with the same ∆T will keep R c on the same subloop (blue-red, arrows 2 and 3), as the system exhibits return-point memory. The return-point memory was found also in the magnetoresistance hysteresis in the same material. 10 Figure 3(b), bottom shows the subloop (black-blue, arrows 1 and 2) obtained when the heat pulse is applied at T = 71.65 K after a cool-down protocol. In that case, ∆R c = 0 is expected after a heat pulse, as observed.

The effects of electric pulses are different from those of heat pulses. First, we examine the dependence of ∆R c on the power applied to the sample during a single pulse, P ≈ I p 2 R c , and on Black curve: δ R c , the difference between the warming and cooling branches of the hysteresis loop (Fig. 1). For all pulses, the maximum ∆R c is observed at T ≈ T CO T LTT after the warm-up protocol.

the energy injected into the system, E ≈ Pτ, where I I p and R c is the resistance state before applying the pulse. (P and E are thus calculated for the fraction of the sample volume where R c is measured. The dependence of ∆R c on I p is shown in Supplementary Fig. 3.) It is obvious that, for each τ, there is a threshold power below which no resistance drop is observed, followed by an increase in ∆R c , and then a tendency towards saturation at the highest P [Fig. 4(a)]. Similar behavior is observed as a function of injected energy [Fig. 4(b)], with an important difference that the data for all different τ and I p scale with E and exhibit the same threshold energy ∼ (4 -10) × 10 -7 J. The scaling of ∆R c with E indicates poor thermal coupling of the electronic system to the environment during τ, such that the system cannot reach thermal equilibrium with the bath during the application of a pulse. Although some heating might be expected, especially for high values of E, the existence of a threshold, absent in the case of heat pulses (Fig. 3(a) inset), suggests that nonthermal processes dominate at low E.

Next, we apply electric pulses multiple times. Figure 5(a) shows the data obtained with 2.5-ms, 4-mA pulses applied four times following the initial warm-up protocol (i.e. the warm-up protocol was performed only before the first pulse). The first three pulses cause observable drops in R c , and further pulse application does not result in any change of R c . This behavior is significantly different from the effect of heat pulses. For example, the "first drop" in R c , produced by the first pulse, is about 0.034 Ω; if the entire effect of the pulse was Joule heating, this would correspond to an increase in temperature by ∆T ∼ 0.04 K [Fig. 3(a)] and there should be no change in R c after additional pulses are applied, in contrast to our findings. This provides additional evidence that an electric pulse in this case causes predominantly nonthermal effects.

To explore the conditions necessary to overcome the thermal regime, we determine both the first drops and the "residual drops", i.e. ∆R c produced by all subsequent pulses, as a function of I p for a fixed τ [Fig. 5(b)] and as a function of τ for a fixed I p [Fig. 5(c)]. The "total drop" is defined as the sum of the first and residual drops. In both cases, we see similar behavior: the first and total drops increase rapidly with I p (and τ), followed by a much weaker dependence or 5(c)], before vanishing at higher values of I p and τ. This indicates that, although Joule heating might dominate at large I p and long τ, for small perturbations the situation is different: here pulsed current injection allows access to nonthermallyinduced resistive states. The similarities in the dependence of various drops ∆R c on I p and on τ signify that it is again the energy injected into the system that plays a major role, i.e. that determines the size of the resistance drops. Indeed, it is only for longer pulses (τ 1 ms) that ∆R c , for a fixed E, starts to depend also on τ (Supplementary Fig. 4), indicating that the system is no longer thermally isolated from the bath.

Finally, by using multiple-pulse current injection in the regime where nonthermal effects dominate, we probe the current-induced resistive metastable states as a function of T . We find that the first, residual, and total drops all have a sharp peak at T ≈ T CO T LTT = 71.65 K, and that the drops are observed only after a warm-up protocol 5(d)]. The asymmetry the observed nonequilibrium states is analogous to that found 10 by studying sharp resistance drops or avalanches resulting from a change of the applied H. In addition, the avalanches were observed only above a threshold H ∼ 2 T, which was thus identified as the minimum depinning field for the CO domains. (The stripe correlation length in La 1.48 Nd 0.4 Sr 0.12 CuO 4 is ∼ 11 nm. 29 ) Our study of the pulsed current injection has revealed that there is indeed a threshold energy that needs to be injected into the system to induce switching into another resistive metastable state. In contrast to the magnetoresistance study that showed 10 two peaks in the avalanche occurrence, a stronger one in the LTLO phase and a weaker one in the LTT phase, tentatively attributed to the onset of precursor nematic order and CO, respectively, we find only one peak, sharp and somewhat asymmetric, such that ∆R c is more pronounced on the LTT side of the transition. In addition, there is no evidence of metastable states in the LTO phase. All the results are consistent with pinning of the fluctuating CO, with fluctuations becoming weaker away from the transition, in agreement with general expectations.

We have established that pulsed current injection is a viable and effective method for probing the CO domain dynamics in cuprates. Previous attempts to detect collective stripe motion in cuprates 30 and nickelates 31,32 using high electric fields, i.e. by measuring current-voltage characteristics, found only nonlinear transport effects that could be attributed to Joule heating. The effects of current pulses were either estimated 30 or explored 31 for very long pulse duration (τ ≈ 200 ms). However, heating effects are generally not easy to estimate because they depend on a variety of factors in a given experimental set-up, including the sample substrate and the cooling power of the cryostat. In contrast, we have demonstrated a systematic way to investigate the effects of pulsed current injection and experimentally identify the regime in which nonthermal effects dominate. This has allowed us to detect signatures of the fluctuating CO in La 1.48 Nd 0.4 Sr 0.12 CuO 4 , thus paving the way for similar studies in other materials.