Measurement of spin dynamics in a layered nickelate using x-ray photon correlation spectroscopy: Evidence for intrinsic destabilization of incommensurate stripes at low temperatures

We study the temporal stability of stripe-type spin order in a layered nickelate with X-ray photon correlation spectroscopy and observe fluctuations on time scales of tens of minutes over a wide temperature range. These fluctuations show an anomalous temperature dependence: they slow down at intermediate temperatures and speed up both upon heating and cooling. This behavior appears to be directly connected with spatial correlations: stripes fluctuate slowly when stripe correlation lengths are large and become faster when spatial correlations decrease. A low-temperature decay of nickelate stripe correlations, reminiscent of what occurs in cuprates due to a competition between stripes and superconductivity, hence occurs via loss of both spatial and temporal correlations.

cannot play a role; the mechanism for low-temperature decay of nickelate stripes has to be a different one. On the other hand, the mechanism at work in nickelates may also play a role in cases where superconductivity is present.
In order to learn how stripe order decays upon cooling when superconductivity as competing phase is absent, we study for a nickelate system not only spatial but also temporal correlations of spin stripes. We find slow fluctuations of the stripe order pattern over a wide temperature range. Like spatial stripe correlations, the fluctuations show an unusual For our study, we used a single crystal of La 2−x Sr x NiO 4 (LSNO) with x = 0.28 [26]. We studied stripe order by resonant soft x-ray diffraction with the photon energy tuned to the Ni 2p → 3d (L 3 ) resonance energy, which makes the experiment directly sensitive to spatial modulation of electronic degrees of freedom [11]. In layered nickelates the stripe-related superstructure peaks with the lowest momentum transfer occur at wave vectors (2 , 0, 1) for charge order and (1 − , 0, 0) for spin order (SO) where is the temperature dependent incommensurability value [13,27]. The notation refers to the commonly used F 4/mmm unit cell with a = b = 5.43Å and c = 12.68Å. We found to vary in the range between 0.292 and 0.298. Dynamics is investigated using X-ray photon correlation spectroscopy (XPCS) [28][29][30][31] with coherent X-rays of the same photon energy. Static experiments were carried out at the P04 beam line of PETRA III at DESY, the XPCS study was carried out at beamline 12.0.2 at the Advanced Light Source (ALS) in Berkeley. In both experiments an in-vacuum CCD was used to detect the scattered intensity [26].
The stripe periodicity described by the inverse of changes with temperature [7,10]. For the doping levels of the sample studied here, the increase of in particular upon heating eventually shifts the charge order peaks out of the Ewald sphere of our experiment [26], while the spin order peak remains reachable at all temperatures. We therefore focus on the dynamics of the spin component of stripe order.
We start by discussing the static properties. Line cuts through the spin order peak along different directions in momentum space are presented in Fig. 1(a-c), the intensity integrated along different directions, I H,K,L as symbols in Fig. 1(d) and the correlation lengths extracted from the peak widths, ξ H,K,L , in Fig. 1(e) [26]. Spin order sets in around 120 K with a superstructure peak, which is broad in all directions. The peak growths and sharpens up to about 60 K and then starts to decay again upon further cooling. The intensity integrated along one of the reciprocal space directions at 20 K is less than a third of its 60-K value.
The decrease of I is accompanied by a peak broadening, i.e., a loss of spatial coherence. The broadening occurs to a similar extent in all reciprocal space directions with the correlation length along the L-direction being about 10 times shorter than along the other two directions, reflecting the weak correlations between the different nickel oxide planes.
A decreasing correlation length can be either due to the formation of dislocations in the order pattern or due to a shrinking of the ordered regions into smaller and smaller patches.
An indication what kind of mechanism is at work here can be obtained from the total ordered volume reflected in the integrated intensity of the scattering peak. We estimated this quantity by taking the area of the line cut along H multiplied by the peak widths along K and L. The result is presented as grey line in Fig. 1(d). While the integrated intensity varies less than the intensity of the line cuts, also the integrated quantity decays towards low temperatures. In our experiment, we are sensitive to only one of the two possible stripe orientations, namely that one where the stripe propagation vector lies in the scattering plane. A change of integrated intensity could in principle be achieved by a redistribution of the ordered volume between the two stripe directions, but since there is no obvious symmetry breaking mechanism that would favor one direction over the other, such a scenario appears unlikely. Possible scenarios that could explain the observed behavior of a loss of integrated intensity and correlation length at low temperatures are (a) the total stripe ordering is actually shrinking towards low temperatures via the formation of regions that are disordered or have a completely different kind of order or (b) defects form within the stripe-ordered patches that either reduce the ordered volume or imply a spatial phase shift, which, by destructive interference, reduces the diffracted intensity. Finally, (c), fluctuations of the stripe order may reduce the detected intensity.
In order to address this last point, we studied the dynamics of stripe order. When illuminated with X-rays with a longitudinal and lateral coherence length that matches the illuminated sample volume, the SO peak breaks up into a myriad of speckles. The speckle pattern reflects disorder in the sample. It is caused by interference between stripe order in different sample regions. Depending on their spatial arrangement they contribute to the diffraction signal with a different phase factor, which leads to constructive and destructive interference resulting in a characteristic speckle pattern on the detector. Any change of the spatial arrangement of these regions or their internal stripe order causes a change of the speckle pattern; its temporal evolution hence directly reveals the dynamics within the stripe order on the time scale of the measurement. An XPCS experiment uses this effect by recording a series of speckle patterns and analyzing the changes between them. The experimental scheme is depicted in Fig. 2(a); a representative series of speckle patterns obtained at 69 K is shown in Fig. 2(b): The later speckle patterns were recorded, the more they differ from those taken at the beginning of that series, thus indicating a temporal change of the stripe order. To quantify this evolution, we determined the intensity autocorrelation function g 2 which leads to the intermediate scattering function or autocorrelation function . . . τ denotes the integration over the whole set of frames recorded for one temperature.
We restrict our analysis to the central part of the SO peak with highest intensity. There we found no indications for different temporal behavior in different regions, which is why we integrate the autocorrelation function over different values of Q near the peak center [26]. the results for τ are summarized in Fig. 2(e). τ shows a pronounced non-monotonous temperature dependence and changes quite strongly. As compared to its value at 100 K, the fluctuation time grows by more than a factor of four when cooling to 72 K, decreases upon further cooling and appears to approach a constant level below 30 K.
The observation that fluctuations are slowest in a temperature range where the corre-   [30] and, generally, one can expect fluctuation times to increase when the correlation volume of the fluctuations gets larger [33].
In the present case, however, the fluctuations are not determined by the correlation length alone. In Fig. 3(a) we plot the fluctuation time, τ , vs. the correlation length ξ H (interpolated from the results in Fig. 1(e)). The data show no clear trend; in particular fast fluctuations (small τ ) at low temperatures occur in the presence of shorter correlation lengths than the similarly fast fluctuations at high temperatures. This observation suggests that both temperature and correlation length define the fluctuation times.
A model that quantitatively relates fluctuation times to both temperature and correlation length is the so-called activated dynamical scaling (ADS) [34]. This model has been developed to describe systems where two states of almost degenerate energy form, separated by a distribution of energy barrier heights determined by randomness [35]. The ADS model predicts the fluctuation time to follow τ ∝ exp(Cξ z /T ) where C is a constant, ξ the correlation length, and z the so-called dynamical critical exponent, which is typically around 2. In this model k B Cξ z acts as an effective energy barrier height for thermally activated fluctuations (k B is the Boltzmann constant).
An ADS-like model indeed reproduces the peak in τ (ξ, T ) around 72 K. The orange line in Fig. 3(b) shows this for ξ H ; we obtain similar curves for ξ K and ξ L . Since the bare ADS model leads to zero fluctuation times (infinitely fast dynamics) for very high and very low temperatures, we include an additive offset, τ 1 , to account for the finite fluctuation time that we observe for all temperatures. We further modified the ADS model by assuming an additional contribution to the correlation length, ξ 0 H that has no influence on dynamics (possibly related to structural defects in the sample). The curve in Fig. 3(b) shows the experimental data with a modified ADS curve, τ = τ 0 exp(C(ξ H − ξ 0 H ) z /T ) + τ 1 with ξ 0 H = 140Å and τ 1 = 420 s; z is 2.2. Already with such simple assumptions we can model our experimental data fairly well [36]. shorter length scales would appear in the outer wings of the diffraction peak profile, where in our experiment the intensity was too low to determine temporal correlations. In fact, microdiffraction experiments from stripe order found a distribution of stripe domains over a wide range of length scales [38]. It is hence to be expected that smaller domains and disorder on shorter length scales exist. With the observed relation between correlation length and fluctuation time, short-length correlations could be connected with much faster fluctuations [39].
We studied here only fluctuations of spin stripes. Since both spin and charge order show a similar low-temperature decay [26], one may assume that our results reflect a property of the stripe order as a whole. We note, however, that in a similar experiment for La 1.875 Ba 0.125 CuO 4 no charge stripe fluctuations could be observed [40].
As the low-temperature decay of stripe order and the concomitant speeding up of fluctuations in this nickelate cannot be caused by a competing superconducting phase, there must be another mechanism at work here. One possibility discussed in the literature is a competition between incommensurate stripe order and the periodic potential of the underlying lattice [41][42][43]. While 'ideal' stripe order favors a stripe periodicity given by the doping level, the lattice potential (and possibly the disorder potential of the randomly distributed dopant ions) may lead to a locking in of stripes to commensurate positions and a broadening of the diffraction peak [44]. A low-temperature change of towards commensurate 1/3 as found here [26] seems to accompany the low-temperature decay in most observations [9,10,12,13,25], suggesting that both effects are related.
One may wonder to what extent the effect observed here might matter for the interplay between superconductivity and stripes or density wave order in cuprates. In a delicate energy balance between superconductivity and stripe/density wave order even weak additional effects might play a role. When incommensurate static stripe order already intrinsically decays into fluctuating disorder at low temperatures, this decay may help to tip the balance towards the formation of superconductivity as a competing phase.
In summary, we observed fluctuations in the incommensurate spin-stripe pattern of