Single laser pulse driven thermal limit of the quasi-two dimensional magnetic ordering in Sr$_2$IrO$_4$

Upon femtosecond-laser stimulation, generally materials are expected to recover back to their thermal-equilibrium conditions, with only a few exceptions reported. Here we demonstrate that deviation from the thermal-equilibrium pathway can be induced in canonical 3D antiferromagnetically (AFM) ordered Sr$_2$IrO$_4$ by a single 100-fs-laser pulse, appearing as losing long-range magnetic correlation along one direction into a glassy condition. We further discover a `critical-threshold ordering' behavior for fluence above approximately 12 mJ/cm$^2$ which we show corresponds to the smallest thermodynamically stable $c$-axis correlation length needed to maintain long-range quasi-two-dimensional AFM order. We suggest that this behavior arises from the crystalline anisotropy of the magnetic-exchange parameters in Sr$_2$IrO$_4$, whose strengths are associated with distinctly different timescales. As a result, they play out very differently in the ultrafast recovery processes, compared with the thermal equilibrium evolution. Thus, our observations are expected to be relevant to a wide range of problems in the nonequilibrium behavior of low-dimensional magnets and other related ordering phenomena.


I. INTRODUCTION
Understanding the mechanisms behind laser manipulation of magnetism in materials is indispensable to many problems in both fundamental and applied science [1][2][3][4][5]. Phenomenologically, the classic three-temperature model [5] works quite well in explaining the experimental observations on the evolution of the spins upon external femtosecond-laser stimuli in many systems [1,[5][6][7][8]. In this model, the spin, electron and lattice baths are coupled through a set of mutual interaction constants which govern the energy-flow rates when the system is stimulated into nonthermal equilibrium conditions. Eventually, the spin sector thermalizes back to equilibrium condition after the externally deposited energy equilibrates among the charge, spin and lattice reservoirs [5]. Here we report an exceptional observation of single-laser-pulseinduced anisotropic partial recovery of the magnetic ordering in Sr 2 IrO 4 , leading to permanent reduction of the interplane magnetic correlation. We propose that this unusual recovery behavior could be explained by the noncooperation of different terms of the exchange interactions dictating the spin ordering. When these exchanges differ significantly in strength, they are unfolded along the time axis [9,10] into different time scales in the ultrafast recovery process. Because of this 'time-window mis- * liuxr@shanghaitech.edu.cn match', the weaker exchange terms could be quenched, leading to a deviation from the quasiequilibrium relaxation pathway. Sr 2 IrO 4 is a layered Mott insulator with each Ir site hosting a 1/2 pseudospin [11].
The highly anisotropic crystalline structure leads to strong magnetic anisotropy( Fig.1(a)). The system starts to develop three-dimensional anteferromagnetic(AFM) ordering below T N ∼ 240 K [12], from the cooperation of a strong intraplane exchange J of approximately 60 meV and a weak inter-plane exchange J c of approximately 16 µeV [13,14]. The magnetic ordering in Sr 2 IrO 4 has been well observed with x-ray resonant magnetic scattering (XRMS) measurements, appearing as Bragg-peaks in the magnetic scattering channel when the incident x-ray energy is tuned to ∼11.216 keV at the Ir-L 3 resonance edge [15]. This high x-ray resonant energy allows the access to a large reciprocal space, thus the flexibility in selecting low x-ray incident angle scattering geometry to best meet the x-ray and laser penetration match requirement [16,17]. Our experiment was set up such that our ∼100 nm epitaxial thin film sample was stimulated with 1 eV laser (∼100 fs) pulse by pulse, and the AFM ordering peak was continuously monitored with XRMS at an x-ray incident angle of less than 5 • to track the recovery of the magnetic ordering at a readout frequency of 1 Hz (see Ref. [18]). Response of the magnetic Bragg-peak height to a single laser pulse (arrives at t=0) as a function of laser fluence. The black dots are the average of the last 100 s of each curve. (e) Multiple-shot evolution of the magnetic-peak height with moderate laser fluence. Multiple shots arrive in the shaded region and each creates a small intensity drop until the decrease is fully compensated by an initial recovery. Figure 1(b) shows the (1 0 16) 3D AFM ordering peak in Sr 2 IrO 4 observed on the pixelated area detector at 80 K[19], and Fig.1(c) depicts the detector surface projection in the reciprocal space. In order to catch the true first laser-shot response from pristine thermalequilibrium condition, each dataset is collected after the sample goes through a fresh thermal cycle by warming above T N and then cooling back again to 80 K with the laser turned off. Then x-ray is turned on to continuously monitor the magnetic Bragg-peak intensity. In between, laser stimuli are controlled to come in pulse by pulse. The arrival time of the first laser pulse is defined as time zero. The evolution of the experimentally observed magneticpeak height as a function of the laser fluence is shown in Fig.1(d).

II. EXPERIMENTAL OBSERVATIONS
After a single-laser-pulse stimulus to the magnetic ordering prepared from thermal-equilibrium evolution, the system does not recover to the initial state, evidenced by the permanent suppression of the magnetic Bragg-peak height as shown in Fig.1(d). The degree of the suppression depends on the laser fluence, i.e., the degree of the damage to the initial ordered spin network. More details of the response to the pulsed laser stimulation are shown in Fig.1(e) with a laser fluence of 8.4 mJ/cm 2 . At this fluence, the first laser pulse leads to an approximately 33% drop in the magnetic Bragg-peak height, and the system settles at this condition without further evolution within an hour of continuous x-ray measurement. A second pulse arriving after one hour causes further suppression to the measured peak height, but with a much smaller drop. After 5∼6 pulses, the system recovery becomes repeatable and more stimulation does not cause further permanent peak-height reduction.
These experimental observations clearly demonstrate that the recovery of the magnetic ordering in Sr 2 IrO 4 depends on the history of laser stimulation, and the response can be classified into two distinct stages. In the first stage of the initial few pulses, the degree of magnetic ordering keeps to be permanently suppressed and the system does not recover to the condition before pulse arrival. In the second stage, the system does recover but only to a state prepared by multiple initial laser pulses, which, of course is different from the thermal-equilibrium evolution. The laser-pulse-induced suppression of the magnetic ordering can be erased with a full thermal cycle and our measurements are fully reproducible (see Ref. indicating the observed response is not due to sample damage but intrinsic to the spin system in Sr 2 IrO 4 . Phenomenologically, the observed shot-by-shot dependence is similar to the photoinduced metastable insulator-tometal phase transition in La 2/3 Ca 1/3 MnO 3 [20]. The laser stimulation drives the system into a nonthermalequilibrium condition, and the deviation depends on the history of laser stimulation, a non-Markovian-type behavior. Such deviation is obviously beyond the classic three-temperature model [5][6][7][8]. To explore the microscopic origin of such evolution, detailed reciprocal space scans along the in-plane (H scans) and interplane (L scans) directions are performed across the magnetic peak before and after the first laser-pulse arrival under various laser fluences, and the results are shown in Fig.2. Note that the peaks in the H scan, a measure of the in-plane spin ordering, are very sharp and identical before and after the laser pulses for all laser fluences applied. From the width of the peaks in H scans, the in-plane spin-ordering correlation length is estimated to be of a macroscopic scale about 0.9 µm. Thus, within a layer, the magnetic ordering is always fully restored to the thermal-equilibrium condition. The observed permanent suppression in the first stage is due to the incomplete recovery of the spin correlation along the interplane direction, which is evidenced by the broadening of the peaks in L scans. With the spin-correlation length reduced, the scattering intensity is transferred from the center to the tails, leading to a permanent suppression of the peak height as observed. These results suggest that the laser pulse leads to highly anisotropic response between the in-plane and interplane AFM ordering correlations in the recovery process.
To quantify such a response, the L scans of the mag-netic Bragg peak measured at thermal-equilibrium condition and after the very first laser pulse stimulus with varying fluence were analyzed to extract the evolution of the interlayer spin-ordering texture. It turns out that, for all experimental conditions, the L scans can be well described by a single Lorentzian function as Fig.3(a,b), and Ref.
[18]), with d to be the interlayer distance and q z the interlayer direction relative-momentum transfer. The evolution of the extracted correlation length ξ z is shown in Fig.3(c). As a function of the laser fluence, ξ z , as well as the peak height ( Fig.3(d)), generally follows an exponential decay. On the other hand, the total area under the magnetic peak is conserved ( Fig.3(e)). The significant data fluctuation mostly comes from slight misalignment in the H direction where the peak width corresponds to 0.01 • in instrument rotational angle. Such conservation indicates that the size and direction of the local ordered magnetic moments are the same as those of the pristine condition [21,22]. Thus the incomplete recovery is solely due to the reduction in the interplane correlation. It is interesting to notice that the exponential decay is offset from zero. Both the correlation length and the peak height show a tendency to saturation at higher laser fluence, and the interplane spin correlation cannot be completely destroyed.

III. DISCUSSION
Our observations reveal a few interesting aspects of the recovery of the magnetic-ordering texture in this highly magnetically anisotropic layered Sr 2 IrO 4 . First, starting from the thermal-equilibrium condition, each of the first few femtosecond-laser pulses induces permanent suppression to the magnetic ordering, and the degree of the suppression depends on the states prepared by the preceding pulses. Second, the system does eventually stabilize into conditions that the recovery of the magnetism becomes repeatable upon consecutive laser stimulation. This justifies the validity of experiments of stroboscopic mode in probing the spin dynamics [1,5,17]. Third, although the degree of the partial recovery keeps degrading with increased laser fluence, it saturates to a robust nonzero value, indicating a minimum interplane partial recovery is intrinsically protected. The observed history and laser fluence-dependence hint at the nature of the magnetic recovery in Sr 2 IrO 4 upon ultrafast laser stimulation.
The history dependence evidences that the demagnetization has certain local character. In the ultrafast process, the demagnetization does not completely wipe out all traces of magnetic order, and in the recovery the spins do not have long enough time to reach a global bath temperature. As a result, there are unperturbed (or less-perturbed) local spin clusters which preserve the memory of the prior state. Thus the prevailing global energy-dissipation picture in the classic threetemperature model [5][6][7][8] is oversimplified.
Notably, in both the initial and the second stages, the in-plane spin correlation always recovers to the thermalequilibrium evolution condition of micron size. This differentiates our observation from the conventional fast quenching of a high-temperature state where the correla-tions along all directions are expected to change [23][24][25]. The full in-plane recovery and the permanent loss of the interplane spin correlation are intimately related to the individual terms in the exchange interaction governing the magnetic dynamics along different directions. Although the 3D magnetic ordering in Sr 2 IrO 4 is jointly determined by both the intraplane and interplane magnetic couplings [14,26,27], in the ultrafast recovery process they act differently. The strong in-plane exchange of tens of meV drives a quick re-establishment of the inplane correlation within a few picoseconds [17]. We suggest that in a such short duration, the additional energy introduced by laser pumping into the spin system cannot be efficiently absorbed by the lattice reservoir [28,29]. Instead, the spin sector is still highly excited in picosecond time scale. Associated with the weak interlayer exchange J c of approximately 16 µeV, a time window much longer than picoseconds is needed to allow them to fully dissipate. Such a process is cut off when the macroscopic intralayer spin correlation is established, since there is an enormous energy barrier to flip a whole layer.
Following the above arguments, we can understand the saturation plateau at high laser fluence as shown in Fig.3(c,d). In our earlier report [17], we have demonstrated that the laser pulse completely destroys the magnetic order in the initial hundreds of femtoseconds, when the laser fluence is stronger than approximately 12 mJ/cm 2 . Thus, beyond such laser fluence, the system completely loses its memory of the history and ll ,ij S li · S l j at T = 80 K with exchange terms J = 60 meV , Jc = 16.4 µeV . Different curves correspond to different anisotropic exchange ∆ strengths, and the consequent gap sizes are used as an index.
the recovery follows the same path, regardless of further laser-fluence increasing. This is also consistent with the observed multishot evolution where, at high fluence, the laser pulses after the very first shot drive marginal further suppression to the magnetic Bragg-peak height (see Fig. S9 in Supplemental Material [18]). Without the assistance from the remnant order, reestablishment of the global interplane correlation completely lags behind the intraplane recovery. Thus the system recovery enters a quasi-two-dimensional regime. For such condition, Mermin and Wagner [30] prove that spontaneous two-dimensional ferromagnetic or antiferromagnetic long-range order at finite temperature is highly susceptible to spin thermal fluctuations, and the nonzero third-dimension correlation is critical to suppress the thermal fluctuation to realize long-range two-dimensional ordering.
We confirm such an explanation for our observed plateau by calculating the magnetic correlation function in a few layers of square lattice from an effective anisotropic Heisenberg spin-1/2 model, whose dynamics is solved with the equation-of-motion technique and mean-field approximation [31](see Ref. [18]). With the realistic parameters from experiment and published literature [13,14], the self-consistent ordered magnetic moment S z and the magnetic correlation function S − S + are obtained as functions of the model slab thickness. As the reported exchange anisotropy for Sr 2 IrO 4 is quite controversial [14,[32][33][34], the calculated S z middle , the ordered magnetic moment at the middle of the slab, is shown in Fig.4 as a function of the calculated magnon gap size. As expected, both the anisotropic exchange ∆ strengths and the layer thickness are critical for true long-range magnetic ordering. When the gap size approaches approximately 1 meV, the magnetic-order parameter S z middle stabilizes around 4-5 layers, which agrees with the interplane correlation length of the saturation plateau we observe in experiment (Fig.3(c,d)). Thus, we realize a spin thermal fluctuation limit in real material with laser-pulse stimulation. Furthermore, this result indicates that indeed the interplane recovery-time window is set by the in-plane spin correlation. The longer interplane correlation established at lower fluence is assisted by memory of the spin network, which is only partially destroyed below the high-fluence threshold.

IV. CONCLUSION
In conclusion, a history and laser-fluence dependence of the partial-recovery process of the 3D AFM ordering in Sr 2 IrO 4 was observed, which is related to the distinctly different timescales for the interplane and intraplane recoveries in the nonthermal-equilibrium ultrafast process. The noncooperation of the different exchange interactions in the ultrafast process drives the system to deviate from the quasiequilibrium relaxation pathway. Light-induced deviation from thermal-equilibrium evolution has been rarely observed, and previous reports are associated with the lattice [35] or charge [20,[36][37][38] degrees of freedom. Our results extend the direct observation of deviation from recovery to thermal-equilibrium condition into the spin sector. Furthermore, we suggest that such time-window mismatch could generally happen in complex systems during ultrafast nonthermal evolution, and our observations could be relevant to a wide range of problems in the nonequilibrium-behavior of lowdimensional magnets and related ordering phenomena. For example, the laser-induced 'hidden quantum state' in layered 1T-TaS 2 could be one special case [38,39].

V. ACKNOWLEDGMENTS
We thank Yi Zhu for the assistance of experimental setup. The experimental work by X. L. and R. W

I. SAMPLE SYNTHESIS AND CHARACTERIZATION
Sr 2 IrO 4 thin film samples with thickness of ∼100 nm grown with PLD method were used [1]. Sr 2 IrO 4 is crystallized in I4 1 /acd structure with single IrO 2 layers separated by SrO layers [2]. Each structural unit cell contains two Ir in one layer and four IrO 2 layers along c direction. The antiferromagnetic(AFM) ordering sets in at T N ≈ 240K for bulk crystal. In our thin film samples, the magnetic susceptibility measurement gave slightly lower T N as shown in Fig. S1. The AFM ordering shares the same unit cell at that of the structure [3]. In the tetragonal Sr 2 IrO 4 , two twined magnetic domains are expected to produce two sets of magnetic reflection peaks at (1 0 4n), (0 1 4n+2) and (1 0 4n+2), (0 1 4n) respectively, where lattice reflections are forbidden. The magnetic ordering peaks from both domains were observed in the long range L-scan shown in

II. CHARACTERIZATION OF LASER SPOT AND ESTIMATION OF THE EFFECTIVE AVERAGE FLUENCE
Discrete laser pulses with a duration of ∼ 100 fs was used to pump the sample. The pump laser photon energy was selected to be 1 eV derived from a Ti: Sapphire laser system with an optical parametric amplifier, corresponding to the resonant excitation from J ef f = 3 2 to unoccupied J ef f = 1 2 states [4]. Thus the pumping largely creates double occupancy of the J ef f = 1 2 states and leaves a hole in the J ef f = 3 2 manifold. We have shown that 1eV pumping can efficiently break the long range AFM ordering [5]. The laser system runs at 1 KHz and can be controlled to deliver a single-shot laser pulse on demand.
To properly characterize the laser fluence, the laser power density profile was measured, as shown in Fig.  S3. The profile can be be modeled as an isotropic Gaussian pulse with the fitted σ to be 350µm. In the experiment, the laser incident angle was 47 • . Thus the laser on-sample footprint was elongated along one direction with σ = σ/sin(47 • ) = 479µm. The effective laser fluence under the X-ray spot can be calculated as, where A is the overlapped area of the X-ray at the laser spot center on the sample surface, f is the running frequency as 1 kHz, and P is the laser power measured during the experiments. At P = 1mW with A = 2.41 × 10 −4 cm 2 , the average fluence within the overlapped region of the single laser shot and X-ray beam spot on the sample is 0.113 mJ/cm 2 . As the X-ray to the laser spot center overlap was done by referring to a video camera monitor, we expect certain miss-alignment. In Fig. 3 in the main text, the error bar given for fluence is 10% by assuming possible ±60µm miss-alignment.
At ∼ 1eV, the penetration depth of the pumping laser for Sr 2 IrO 4 is estimated [5] to be ∼100 nm.

III. SCHEMATIC OF EXPERIMENT
The X-ray resonant magnetic scattering(XRMS) measurements were conducted at the Advanced Photon Sources(APS) using beamline 7-ID-C. The data were collected at Ir L 3 absorption edge of 11.216 KeV. A horizontal scattering geometry was used.(see Fig. S4) The laser pulse came in at a large angle of 47 • relative to the sample surface to allow a more homogeneous excitation along the sample depth direction.
To amplify the magnetic scattering signal [6], the scattering experiment was performed in the a-c plane with the incident X-ray came in at a shallow angle of 4.87 • relative to the sample surface. Its polarization was almost parallel to the sample surface c-direction. A Pilatus CCD with pixel size of 172 µm 2 was used in the experiment to monitor the scattered X-ray signal. It was placed ∼ 1m away from the sample, which gives an angular resolution of 0.01 • per pixel. The sample was cooled down to 80 K with cryostat, well below the Néel ordering temperature. During the experiment, a full thermal cycle was done by warming the sample up to 280 K and then slowly cooling down to 80 K with a cooling rate of 0.05 K/s. The laser induced suppression of the magnetic peak height was fully recovered after a thermal cycle, as shown in the cross-sample scan in Fig. S5. The entire process is repeatable, ruling out the irreversible sample damage issue.

IV. LASER EFFECT ON STRUCTURAL PEAK
To check the laser shot effect on the crystal structure, (0 0 16) structure peak height was monitored with single laser pulse stimulation. As shown in Fig. S6, the fluctuation in the structural peak height, mainly due to X-ray beam instability, is uncorrelated with the laser stimulation. Thus laser induces minimum effect to the lattice at 1Hz frequency of which our data was taken, and the suppression of the magnetic peak height is intrinsic to the spin sector.

V. X-RAY EFFECT ON MAGNETIC PEAK
We checked the X-ray effect on magnetic peak by monitoring the (1 0 16) magnetic Bragg peak height after X-ray was initially turned on after a full thermal cycle, without any optical pumping on sample. A gradual reduction of the peak height about 7% was noticed after the X-ray exposure of the sample, as shown in Fig. S7. Then the peak height stabilized after a few minutes.  Fig. S4) across the sample while monitoring the (1 0 16) magnetic peak height. Three scans were taken for: pristine thermal equilibrium condition(blue curve), after single laser pulse excitation(red curve), and after a full thermal cycle process(black curve). The laser pumping leads to a drastic suppression of the scattering intensity, which is fully recovered after a thermal cycle.

VI. FORMULA FOR THE DIFFRACTION PROFILE OF THE L-SCAN OF MAGNETIC BRAGG PEAK
Since the in-plane AFM ordering correlation length is fully restored, we focus on the the inter-plane correlation. the AFM ordered iso-spins are still pointing to the crystal a-direction as in the thermal equilibrium state while their inter-plane correlation is described by an exponential decay as e − |zm−zn | ξ with z m(n) to be the c-direction coordinate. Accordingly, the magnetic reflection intensity can be written as, where F is the magnetic scattering factor for Ir sites, and j and k represents the j-th and k-th plane along cdirection. N 3 is the total plane number along c-direction. ξ z is the c-direction magnetic correlation length. The inplane structure factors are simplified to δ-functions due to the fact that the in-plane correlation lengths are orders of magnitude larger than the c-direction correlation length(see main text). The summation can be analytically carried out as: where d is the inter-layer distance.
The relative momentum transfer can be defined as q z = Q z − G with G indexing the Bragg points. When Thus, with inter-layer ordering correlation defined as e − |zm −zn| ξz , the X-ray scattering profile is of a Lorentzian shape. The peak height at q z = 0 should be proportional to the correlation length ξ z , while the whole integrated intensity is constant. All these predictions agree well with our observations, suggesting a quite homogeneous statistical distribution of the c-direction spin ordering.

VII. FITTING PROCEDURE OF THE MAGNETIC BRAGG PEAK
All the magnetic peaks were fitted based on the Eqn. 6 plus a linear background intensity, as shown in Eqn. 7. For a set of magnetic Bragg peaks studied with the same fluence of laser pulse, firstly we fit the magnetic Bragg peaks of pristine thermal limit(before laser excitation), and extract a background intensity; Then we fit the magnetic Bragg peaks after the single shot excitation with the same background intensity.(see Fig. S8b) The fitting was done by least-squares fitting. And here the reduced Chi-square χ 2 ν is defined as: where N is the number of data points, N varys is the number of variables in the fit, y exp i is the measured data, y model i (v) is the model calculation and v is the set of variables in the model to be optimized in the fit, and i is the estimated uncertainty in the data. Representative fitting goodness is listed in Table. SI. And we plot three of the typical fitting results in Fig.  S8a.