Abstract
The coupling of ordered electronic phases with lattice, spin, and orbital degrees of freedom are of central interest in strongly correlated systems. Their interplay has been intensively studied from femtosecond to picosecond time scales, while their dynamics beyond nanoseconds are usually assumed to follow lattice cooling. Here, we report an unusual slowing down of the recovery of an electronic phase across a firstorder phase transition. Following optical excitation, the recovery time of both transient optical reflectivity and Xray diffraction intensity from the chargeordered superstructure in a La_{1/3}Sr_{2/3}FeO_{3} thin film increases by orders of magnitude as the sample temperature approaches the phase transition temperature. In this regime, the recovery time becomes much longer than the lattice cooling time. The combined experimental and theoretical investigation shows that the slowing down of electronic recovery corresponds to the pseudocritical dynamics that originates from magnetic interactions close to a weakly firstorder phase transition.
Introduction
The interactions between electronic, spin, and structural degrees of freedom in correlated materials are the basis of emergent phenomena, including hightemperature superconductivity, metaltoinsulator phase transitions, and colossal magnetoresistance^{1,2,3}. Strong correlations amongst these degrees of freedom hold promise for engineering material properties using targeted optical excitation to create hidden phases that do not exist in thermal equilibrium. These hidden phases can live as short as a few picoseconds, such as transient superconductivity in cuprates^{4,5}, or can be longlived metastable states, such as those in photoexcited manganites^{6} and dichalcogenides^{7}. Understanding and possibly controlling how driven quantum systems achieve equilibrium is of central interest in nonequilibrium physics^{8} and for elucidating the lifetime of emergent photoinduced phenomena at hierarchical time scales.
The chargeordered state, where patterns of charge density spontaneously emerge below a critical temperature T_{c}, is one of the most interesting collective electronic phases and plays a critical role in determining the properties of many correlated materials^{9}. Thanks to the distinct dynamics associated with different interaction mechanisms, complex interactions amongst multiple degrees of freedom, such as charge ordering (CO), lattice distortion, and spin ordering, can be effectively disentangled on ultrafast time scales, motivating studies of femtosecondtopicosecond (fsps) responses^{10,11,12,13,14}. Far below T_{c}, CO can be quenched by ultrafast optical excitation and typically recovers on ps time scales, during which its relation with nonequilibrium structural distortions has been experimentally studied^{15,16,17,18}. When the system temperature is close to T_{ c }, the initiation and recovery of photoinduced changes are drastically different from the case far below T_{c}^{10}. For example, a critical slowing down of charge density waves on ps timescales was observed in cuprates^{19}, molybdenum oxides^{20,21}, and chalcogenides^{11,22}. The recovery of charge ordering in layered organic salts can approach nanosecond (ns) timescales due to electronic instabilities^{23}. On longer time scales, after the charge, spin, and lattice degrees of freedom have had sufficient time to exchange energy to reach the same temperature, the evolution of electronic properties usually follows the cooling of the system through thermal exchange with the environment. Thermal recovery hereafter refers to the film lattice cooling due to the heat transport to the substrate. Although the glasslike recovery of antiferromagnetic spin order can be extremely slow, electronic recovery does not necessarily follow the recovery of spin order^{24}. On long timescales beyond those typically associated with thermal recovery, electronic slowing down has not been studied, and its microscopic mechanisms and the relation to other degrees of freedom are not clear. Understanding the role of correlations on these mesoscopic timescales is essential to extending the lifetime of exotic electronic phases beyond thermal cooling of the system.
Here, we report on an unusually slow recovery of a collective electronic phase that becomes significantly longer than the lattice cooling time near T_{c} in photoexcited La_{1/3}Sr_{2/3}FeO_{3} (LSFO) thin films. Using timeresolved optical spectroscopy and Xray diffraction, we directly track three quantities in the time domain: the optical reflectivity, the superlattice Xray diffraction peaks related to the order parameter of CO, and the lattice constant of LSFO (Fig. 1a). When the sample temperature increases towards T_{c}, we observe concurrent slowing down of the recovery of both transient optical reflectivity and CO superlattice diffraction peak intensity, well beyond the fewns thermal recovery of the independently characterized outofplane lattice constant . Thus, the time scale for lattice cooling does not determine the time scale of electronic recovery. Moreover, the Xray diffraction measurements reveal important structural information: no significant change of the CO domain size is observed, which suggests a mesoscopic nucleation and growth scenario is not the dominant mechanism for the slowing down. By density functional theory plus U (DFT + U) calculations, we find that the magneticexchangedriven phase transition is weakly first order, in which phase separation of two competing CO configurations occurs close to T_{c}. The recovery of CO following pathways along the temperaturedependent potential energy surfaces to the ground state qualitatively explains the observed slowdown. The scaling of the time constants as a function of temperature is ascribed to pseudocritical dynamics in a weakly firstorder transition^{25}, driven by magnetic exchange interactions.
Results
Static sample characterization
The perovskite oxide La_{1/3}Sr_{2/3}FeO_{3} is a prototypical material in which the oxygenmediated superexchange interaction between Fe sites drives a firstorder metaltoinsulator (MIT) phase transition^{26,27,28}. The MIT, paramagnetic to antiferromagnetic (AFM) transition, and CO transition occur concurrently at a transition temperature of around 200 K. Seventy nanometer LSFO thin film samples were grown by ozoneassisted molecular beam epitaxy on (111) SrTiO_{3} (STO) substrates. The T_{c} of the LSFO films was experimentally identified as the temperature at which charge ordering emerges (Fig. 1b). Upon cooling below T_{c}, charge disproportionation between the Fe sites leads to ordered planes of Fe^{5+}–Fe^{3+}–Fe^{3+} stacked along the [111] direction in the pseudocubic representation^{26}, accompanied by a sharp hysteretic rise in resistivity (Fig. 1b). While the reported valence states of Fe ions vary from 3+ to 5+ ^{27,28}, an accompanying periodic structural distortion gives rise to CO superlattice (n ± 1/3, n ± 1/3, n ± 1/3; n is a positive integer) peaks in Xray diffraction measurements (Supplementary Fig. 2)^{28,29}. Unlike organic Mott insulators^{23}, the formation of CO in LSFO is a firstorder phase transition driven by magnetic interactions^{26,27,28}.
Transient electronic and structural dynamics
The transient electronic and structural dynamics were measured by timeresolved optical reflectivity and Xray diffraction at the Center of Nanoscale Materials and the Advanced Photon Source at Argonne National Laboratory (see Methods). Calibrated sample temperatures are used throughout this paper (see Supplementary Note 3). In both optical and Xray measurements, the photon energy of the pump laser is above the optical absorption edge of the LSFO thin film (2.2 eV)^{30}. The penetration depth of the pump laser is comparable to the optical probing depth of 30 nm^{30}, but less than the thickness of the film (70 nm). The mismatch between pump and probe penetration depth in the Xray measurements is not a concern on the time scale of interest here (ns to μs), which is well beyond the time scale of the lattice thermalization within the film^{31}.
The changes of optical reflectivity measured at 1.1 μm following optical excitation with an absorbed pump fluence of 2.9 mJ cm^{−2} are shown at various temperatures in Fig. 2a. The recovery process can be fit by an exponential decay function \({\mathrm{\Delta }}R\left( t \right)\sim A_1\exp \left( {  {\textstyle{t \over {t_{{\mathrm{fast}}}}}}} \right) + A_2\exp \left( {  {\textstyle{t \over {t_{{\mathrm{slow}}}}}}} \right)\) with time constants t_{fast} and t_{slow}. Far below T_{c}, e.g., at T = 176 K, the fitting yields t_{fast} = t_{slow} ~2.3 ns, thus, a single exponential decay is sufficient to describe the recovery dynamics. However, Fig. 2b shows that, as the sample temperature approaches T_{c}, t_{slow} increases by orders of magnitude, while t_{fast} does not change significantly. This observation indicates that t_{slow} is related to a nontrivial recovery mechanism, while t_{fast} is consistent with thermal recovery as discussed later. Measurements performed above T_{c} show a similar, severalns recovery to that observed far below T_{c}, suggesting the recovery of optical reflectivity above T_{c} is mainly driven by thermal recovery. The decrease in reflectivity at 1.1 μm upon optical excitation and the slowing down of the recovery are universal across the probed optical spectrum from 0.9 to 1.3 μm (Supplementary Note 1). The change of optical reflectivity is consistent with the spectral weight transfer of the optical conductivity from around 1 eV to low energy (<0.5 eV) as the film temperature increases^{32}. This spectral range is ascribed to the transition from O 2p to Fe^{3+}/Fe^{5+} e_{g↑} states and is closely related to the macroscopic Drude response across the metaltoinsulator phase transition^{32}.
In order to understand the slowing down of the recovery of the electronic degree of freedom, we studied the recovery of COinduced structural distortion, as well as that of the lattice constant, by monitoring the \(\frac{4}{3}\frac{4}{3}\frac{4}{3}\) and 222 Xray diffraction peaks, respectively. Because the lattice remains rhombohedral (R3c space group) across T_{c}, the timedependent lattice constant provides an independent measurement of the film temperature as the lattice returns to thermal equilibrium. At T = 121 K, the CO recovery follows the lattice dynamics. Radial scans of both \(\frac{4}{3}\frac{4}{3}\frac{4}{3}\) and 222 peaks were measured as a function of delay between the pump laser and probe Xray pulses. The intensity of the superstructure peak decreases as CO melts, while the peak center shifts to lower HKL, indicating a superlattice expansion (Fig. 3a). The expansion strain of 0.04% measured by 222 peak at 100 ps corresponds to the film temperature increase of 24 K, calculated using the thermal expansion coefficient of 1.55 × 10^{−5} K^{−1} ^{33} and Poisson’s ratio of 0.32^{34} (See Supplementary Note 2). Figure 3b shows that the photoinduced strain measured by the shifts of the \(\frac{4}{3}\frac{4}{3}\frac{4}{3}\) and 222 peaks relax at the same rate. The intensity of the superstructure peak (not shown) and transient expansion of the lattice can both be fit by a single exponential function with recovery time constants, τ_{CO} = 3.1 ns and τ_{lattice} = 2.8 ns, in good agreement with the recovery time constant of optical reflectivity at a temperature far from the CO phase transition. These observations show that, at temperatures far below T_{c}, the recovery of the electronic degree of freedom is determined by the cooling rate of the thin film. Figure 3c shows the radial scans of the CO peak before and after laser excitation for a temperature closer to T_{c} (T = 196 K). The CO peak intensity is significantly suppressed upon laser excitation. Due to the mismatch between optical excitation depth and film thickness, the CO only partially melts and the measured diffraction peak is a sum of the residual unmelted CO and the recovering CO states. Nevertheless, no discernible change in the CO peak width was observed along the outofplane direction (Fig. 3c) or along the inplane direction (Supplementary Fig. 8), indicating a nearly constant CO domain size during the recovery of CO state upon photoexcitation. Compared with the CO intensity recovery measured at 121 K in Fig. 3d, the recovery time of the CO peaks increases two orders of magnitude from 3.5 ns at 121 K to 263 ns at 196 K. Meanwhile, the recovery time of 222 peak shift only increases by a factor of 1.5. The relaxation of the film lattice temperature is welldescribed by onedimensional thermal transport to the substrate^{31,35,36}. At tens of ns, the relaxation can be modeled by an exponential decay plus an offset. A careful evaluation of the timedependent film temperature shows that the lattice cooling at longer time scales cannot explain the observed slowing down of CO recovery (Supplementary Note 2).
To quantify the dynamical scaling of slowing down, we plot the recovery time constants of the CO and lattice diffraction peaks as a function of the film temperatures in Fig. 3e. Approaching T_{c}, τ_{CO} increases by two orders of magnitude, while τ_{lattice} remains below 7 ns across the CO phase transition, which clearly indicates the recovery of the CO phase does not follow the recovery of the lattice. Under similar pump laser fluence, the concurrent slowing down of the recovery of both the optical reflectivity and the CO superlattice peak around 190 K suggests that the dielectric constant at 1.1 μm is related to the charge ordering, although the probe photon energy of 1.13 eV is much higher than the COinduced energy gap of 0.13 eV^{32}. This observation indicates the macroscopic optical properties across a wide infrared spectrum range in LSFO are highly correlated with the formation of charge ordering on longerthanthermal recovery time scales. Measurements at other pump fluences are consistent with this observation (Supplementary Note 3). The time constant of the slowing down as a function of temperature is fit to a power law τ = τ_{0} (1−T/T_{c})^{−Δ} where T_{c} = 200 K, Δ is the scaling exponent, and τ_{0} is a constant^{25}. The best fit yields Δ = 1.25 ± 0.10, shown by the magenta curve in Fig. 3e. The time constant of optical reflectivity recovery is fit by the same function and shown as the blue curve in Fig. 2b, with Δ = 1.06 ± 0.16. In conventional secondorder critical phenomena, the measured scaling exponent Δ is equivalent to vz (v and z are two critical indices^{37}) and approximately agrees with 1.3 and 1.37 for threedimensional (3D) Ising^{38} and Heisenberg^{39} models, respectively, and differs from 2.16 for a twodimensional model^{40,41}. However, we point out that the phase transition in LSFO is firstorderlike, because a latent heat is present at the transition in bulk samples^{42}. In addition, our electrical transport measurements (Fig. 1b) and the antiferromagnetic order parameter reported previously^{29} are weakly hysteretic as a function of temperature. Thus, our measurements show a pseudocritical phenomenon near a weakly firstorder phase transition^{25}, rather than conventional secondorder critical phenomena^{37}. We also note that the attempted fit for the nucleation and growth model does not agree with the scaling of the time constant as a function of temperatures (Supplementary Note 4). Therefore, the observed scaling cannot be explained by the reduction of nucleation rate of CO domains as system temperature approaches T_{c}, consistent with no discernible changes of coherence length of CO domains upon optical excitation.
Firstprinciples calculations
To further understand the origin of the slowing down, we calculated the total energy of LSFO using the first principles DFT + U method, which illustrates the recovery pathways of coupled degrees of freedom during a firstorder phase transition. First, we performed the structural relaxation using U = 5 eV and J = 1 eV. We found that the ground state of LSFO is charge ordered with an associated structural distortion as a result of antiferromagnetic ordering, consistent with the previous DFT + U calculations^{43}. The resulting structural distortion is characterized by an average Fe–O bondlength difference δ_{a} = 0.06 Å (inset, Fig. 1b) between different Fe sites, with an average AFM moment of 3.7 μ_{B}. Since DFT + U is a zerotemperature theory, we simulate the effect of temperature by varying the interaction parameters (U and J) to tune the resulting magnetic moment, which is controlled by the sample temperature experimentally. Without any magnetic interactions, i.e., U = J = 0, corresponding to a sample temperature is above T_{c}, the energy surface shows no CO or accompanying structural distortion. The calculation thus agrees with a magnetizationdriven CO in LSFO^{27}. The structural pathway as a function of order parameter δ_{a} is determined by interpolating the CO structure relaxed using U = 5 eV and J = 1 eV and the nonCO structure relaxed using U = J = 0 eV. We then vary U values from U = 5 eV for the lowtemperature CO state to U = 0 eV for the high temperature state without CO, while fixing the U / J ratio to 5. This is used to study the qualitative features of the energy landscapes between two local minima by simulating an increase of sample temperature. The structural pathway is fixed, while we explore the energy landscapes qualitatively due to the change in temperature by varying U. While the resulting magnetic moments and energy landscapes can be changed by tuning U, the calculated moments for reasonable U (3−5 eV) are on the order of 3 μ_{B}, comparable with the experimental values^{44,45}. At T « T_{c}, our calculations show that the large AFM spin exchange energy between Fe^{3+} ions dominates the energetics and gives rise to one minimum energy state at δ_{a} = 0.06 Å, accounting for Fe^{5+}–Fe^{3+}–Fe^{3+} order or smalllargelarge oxygen octahedra, as illustrated by diamonds in Fig. 4a. As the AFM moment was reduced, simulating an increase in sample temperature, we discovered that a metastable state starts to emerge at δ_{a} = −0.06 Å, with Fe^{4+}–Fe^{4+}–Fe^{3+} ordering or smallsmalllarge oxygen octahedra. By reducing the value of U to 3.7 eV, which corresponds to a magnetic moment of 3.3 μ_{B} per Fe, the energy of this metastable state at δ_{a} = −0.06 Å becomes degenerate with the state at δ_{a} = 0.06 Å, giving rise to the coexistence of two competing CO states with nonzero energy barrier. While the magnetic moments and relative energies of two CO states may sensitively depend on the value of U, the position of the two CO states, i.e., the values of δ_{a} at two local energy minima, are not sensitive to U.
Discussion
The DFT + U calculation shows the energetics of the system is driven by magnetic exchange. Comparing with a typical firstorder (Fig. 4b) or secondorder phase transition, the energy surfaces exhibit unique characteristics that govern the electronic dynamics. First, the energy barrier E_{b} is small, but persists with nonzero AFM moments, which is consistent with the signature of a weakly firstorder instead of a typical secondorder phase transition. Second, the existence of two nearly degenerate energy states and the rapid quench following the heat pulse provides the opportunity for the system to enter the regime of spinodal decomposition^{46}. In this scenario, following the initial laser excitation, the system is ‘quenched’ with δ_{a} = 0 (disordered state) from temperatures above T_{c} due to relatively fast cooling of the film lattice temperature on the ns timescale. At this point, the system is in the unstable ‘spinodal’ regime where the second derivative of the free energy is less than zero. There is no energy barrier to the formation of the CO states, and the kinetics is limited only by diffusion^{46}. Furthermore, this process slows down in the vicinity of spinodal points where the second derivative of the free energy is zero (Fig. 4c) and the diffusion constant becomes vanishingly small^{25}. During this process, the domain size does not change, consistent with our observation. At later stages, beyond our measurement time window, processes similar to Ostwald ripening can occur that will increase the coherence length of CO domains^{46}. A direct observation of the associated kinetics of spinodal decomposition needs timeresolved resonant Xray diffraction microscopy with sufficient spatiotemporal resolution which is beyond the scope of this work. At low temperatures (T « T_{c}), as schematically shown in (Fig. 4c), the deep potential well is associated with a strong restoring force that gives rise to a fast recovery from the excited state following a welldefined recovery pathway.
In summary, we observed that the recovery of an electronic phase slows down, becoming longer than the thermally driven processes in a photoexcited LSFO thin film. The multimodal probes via transient optical reflectivity and CO superlattice diffraction allow direct correlation between optical properties and the longrange electronic ordering in the time domain. The electronic recovery is significantly different from the relaxation of the average lattice parameter, providing decisive evidence of the recovery of the electronic phase that is not determined by lattice cooling. First principles DFT + U calculations elucidate a microscopic picture of magneticinteraction driven slowing down and suggest a pseudocritical phenomenon close to a weakly firstorder phase transition. Our combined experimental and theoretical investigation provides experimental verification and mechanistic insight on an unconventional critical behavior and the interplay of multiple degrees of freedom on unusually long time scales at an electronic phase transition.
Methods
Sample preparation and characterization
Epitaxial La_{1/3}Sr_{2/3}FeO_{3} (LSFO) thin films were grown using ozoneassisted molecular beam epitaxy (MBE) on (111)oriented SrTiO_{3} (STO) substrates. Prior to growth, trichloroethylene was used to remove organic contaminants from the substrate surface. The codeposited elemental materials La, Sr, and Fe were evaporated from effusion cells under an ozone environment with partial pressure of 3 × 10^{−6} mbar, with the substrate temperature maintained at 680 °C. The evaporation rates were determined for each material from a quartz crystal thickness monitor that was calibrated to within 2% from Rutherford backscattering measurements. The film thickness and surface symmetry were monitored in real time from reflection highenergy electron diffraction (RHEED) intensity oscillations. Brief anneal periods of ~30 s followed the completion of each unit cell layer. After the LSFO deposition, the samples were cooled down to room temperature in an environment of 3 × 10^{−6} mbar of O_{3}. The 70nm thick LSFO sample was characterized by static Xray diffraction and the results are shown in Supplementary Fig. 2. The integrated intensity of the CO peak was measured as a function of the sample temperature, showing the CO emerges at 200 K, and fourpoint probe measurement of the sheet resistance shows a hysteresis loop at the transition temperature, shown in Fig. 1b.
Experimental setup
The ultrafast optical pumpprobe experiment is shown schematically in Supplementary Fig. 1a. An optical parametric amplifier (OPA) was pumped by a femtosecond Ti:Al_{2}O_{3} laser at 1 kHz repetition rate. The output wavelength of the OPA was doubled to λ = 420 nm to excite the LSFO sample, with a pulse duration of 100 fs. A Nd:YAG laser was electronically synchronized with the femtosecond laser to perform asynchronous optical sampling measurements. The output of the YAG laser with a pulse duration of ~100 fs at the wavelength of 1064 nm was focused into a sapphire plate to generate white light with wavelength from 900 nm to 1300 nm. The probing white light was refocused to be smaller than and spatially overlapped with the pump laser beam on the LSFO sample surface. The reflected white light was analyzed by a spectrometer. The sample temperature was controlled from 78 K to room temperature. In the timeresolved hard Xray diffraction experiment shown in Supplementary Fig. 1b, the pump laser pulse was derived from the third harmonic generation (THG) of a high repetition rate (54 kHz) Nd:YAG laser, with 355 nm central wavelength and ~10 ps pulse duration. Use of a highrepetitionrate laser is essential to achieve high signaltonoise ratio for probing timeresolved CO Xray diffraction. Xray pulses at 12 keV photon energy and ~100 ps pulse duration were focused by KirkpatrickBaez mirrors to a beam size of ~50 µm, smaller than the focused pump laser beam size of ~220 µm. An area detector (Pilatus100K, DECTRIS Ltd.) gated at 54 kHz was used to detect the diffraction intensity. The LSFO sample was mounted on a sixcircle diffractometer (Huber GmbH.) in a cryostat with temperature control from 30 K to 300 K.
DFT calculations
The DFT + U calculations were performed using the Vienna abinitio simulation package (VASP). We adopted the generalized gradient approximation (GGA) exchangecorrelation functional for simulations using an energy cutoff of 600 eV and a kpoint mesh of 8×8×2 aligning the zaxis along the [111] direction. We use the supercell of LSFO to accommodate both the antiferromagnetic spin configuration and the octahedral expansion or collapse. The supercell elongated along the [111] direction contains 2 La atoms, 4 Sr atoms, 6 Fe atoms, and 18 O atoms allowing the expansion or the collapse of 6 FeO_{6} octahedral volumes. The convergence of the total energy calculation was reached if the consecutive energy difference was within 10^{−4} eV and the atomic forces of all ions were required to be smaller than 0.01 eV/Å for ionic relaxations. The “+U” interaction Hamiltonian was adopted using the rotationally invariant form and the DFT Hamiltonian part did not include the spinexchange splitting (i.e., the DFT energy part was a function of spin unpolarized charge density, and the spin exchange interaction is entirely treated within the correlated Fe d orbitals) aligning with the spirit of DFT + dynamical mean field theory, as implemented in VASP using the option of “LDAUTYPE = 4”. The widely used spinDFT + U implementation, i.e., the spin polarization accounts for both the charge density and the correlated orbitals, often overestimates the spinexchange interaction in several other systems^{47,48,49}. Especially for this LSFO material, the resulting magnetic moments and energy curves computed using spinDFT + U were insensitive to the change of U and J parameters since the DFT part alone already accounts for the large portion of the spinexchange interaction.
Code availability
The codes for numerical calculations are available from the corresponding author upon reasonable request.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
References
 1.
Dagotto, E. Complexity in strongly correlated electronic systems. Science 309, 257–262 (2005).
 2.
Basov, D., Averitt, R., van der Marel, D., Dressel, M. & Haule, K. Electrodynamics of correlated electron materials. Rev. Mod. Phys. 83, 471–542 (2011).
 3.
Millis, A. J. Lattice effects in magnetoresistive manganese perovskites. Nature 392, 147–150 (1998).
 4.
Fausti, D. et al. Lightinduced superconductivity in a stripeordered cuprate. Science 331, 189–191 (2011).
 5.
Mitrano, M. et al. Possible lightinduced superconductivity in K3C60 at high temperature. Nature 530, 461–464 (2016).
 6.
Zhang, J. et al. Cooperative photoinduced metastable phase control in strained manganite films. Nat. Mater. 15, 956–960 (2016).
 7.
Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electronic crystal. Science 344, 177–180 (2014).
 8.
Eisert, J., Friesdorf, M. & Gogolin, C. Quantum manybody systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).
 9.
Rao, C. N. R. Charge, spin, and orbital ordering in the perovskite manganates, Ln1_{xA}x_{M}nO3 (Ln=Rare Earth, A=Ca or Sr). J. Phys. Chem. B 104, 5877–5889 (2000).
 10.
Averitt, R. D. & Taylor, A. J. Ultrafast optical and farinfrared quasiparticle dynamics in correlated electron materials. J. Phys. Condens. Matter 14, R1357–R1389 (2002).
 11.
Yusupov, R. et al. Coherent dynamics of macroscopic electronic order through a symmetry breaking transition. Nat. Phys. 6, 681–684 (2010).
 12.
Giannetti, C. et al. Ultrafast optical spectroscopy of strongly correlated materials and hightemperature superconductors: a nonequilibrium approach. Adv. Phys. 65, 58–238 (2016).
 13.
Lee, W. S. et al. Phase fluctuations and the absence of topological defects in a photoexcited chargeordered nickelate. Nat. Commun. 3, 838 (2012).
 14.
Beaud, P. et al. A timedependent order parameter for ultrafast photoinduced phase transitions. Nat. Mater. 13, 923–927 (2014).
 15.
Huber, T. et al. Coherent structural dynamics of a prototypical chargedensitywavetometal transition. Phys. Rev. Lett. 113, 026401 (2014).
 16.
Eichberger, M. et al. Snapshots of cooperative atomic motions in the optical suppression of charge density waves. Nature 468, 799–802 (2010).
 17.
Schmitt, F. et al. Transient electronic structure and melting of a charge density wave in TbTe_{3}. Science 321, 1649–1652 (2008).
 18.
Porer, M. et al. Nonthermal separation of electronic and structural orders in a persisting charge density wave. Nat. Mater. 13, 857–861 (2014).
 19.
Eesley, G. L., Heremans, J., Meyer, M. S., Doll, G. L. & Liou, S. Relaxation time of the order parameter in a hightemperature superconductor. Phys. Rev. Lett. 65, 3445–3448 (1990).
 20.
Demsar, J., Biljaković, K. & Mihailovic, D. Single particle and collective excitations in the onedimensional charge density wave solid K_{0.3}MoO_{3} probed in real time by femtosecond spectroscopy. Phys. Rev. Lett. 83, 800–803 (1999).
 21.
Kise, T. et al. Ultrafast spin dynamics and critical behavior in halfmetallic ferromagnet: Sr_{2}FeMoO_{6}. Phys. Rev. Lett. 85, 1986–1989 (2000).
 22.
Demsar, J., Forró, L., Berger, H. & Mihailovic, D. Femtosecond snapshots of gapforming chargedensitywave correlations in quasitwodimensional dichalcogenides 1 T−TaS_{2} and 2 H−TaSe_{2}. Phys. Rev. B 66, 041101 (2002).
 23.
Iwai, S. et al. Photoinduced melting of a stripetype chargeorder and metallic domain formation in a layered BEDTTTFbased organic salt. Phys. Rev. Lett. 98, 097402 (2007).
 24.
Zhou, S. Y. et al. Glasslike recovery of antiferromagnetic spin ordering in a photoexcited manganite Pr0.7Ca0.3MnO3. Sci. Rep. 4, 4050 (2014).
 25.
Ikeda, H. Pseudocritical dynamics in firstorder transitions. Prog. Theor. Phys. 61, 1023–1033 (1979).
 26.
Li, J. Q., Matsui, Y., Park, S. K. & Tokura, Y. Charge ordered states in La_{1x}Sr_{x}FeO_{3}. Phys. Rev. Lett. 79, 297–300 (1997).
 27.
McQueeney, R. J. et al. Stabilization of charge ordering in La_{1/3} Sr_{2/3} FeO_{3 − δ} by magnetic exchange. Phys. Rev. Lett. 98, 126402 (2007).
 28.
HerreroMartin, J., Subias, G., Garcia, J. & Blasco, J. & Concepción Sánchez, M. Evidence for chargedensitywave nature of the chargeordered phase in La_{1/3}Sr_{2/3}FeO_{3}. Phys. Rev. B 79, 045121 (2009).
 29.
Okamoto, J. et al. Quasitwodimensional dspin and phole ordering in the threedimensional perovskite La_{1/3}Sr_{2/3}FeO_{3}. Phys. Rev. B 82, 132402 (2010).
 30.
Scafetta, M. D., Xie, Y. J., Torres, M., Spanier, J. E. & May, S. J. Optical absorption in epitaxial La_{1−x}Sr_{x}FeO_{3} thin films. Appl. Phys. Lett. 102, 081904 (2013).
 31.
Shayduk, R. et al. Nanoscale heat transport studied by highresolution timeresolved Xray diffraction. New J. Phys. 13, 093032 (2011).
 32.
Ishikawa, T., Park, S. K., Katsufuji, T., Arima, T. & Tokura, Y. Optical spectroscopy of chargeordering transition in La_{1/3}Sr_{2/3}FeO_{3}. Phys. Rev. B 58, R13326–R13329 (1998).
 33.
Fossdal, A. et al. Crystal structure and thermal expansion of La_{1 x}Sr_{x}FeO_{3δ} materials. J. Am. Ceram. Soc. 87, 1952–1958 (2004).
 34.
Romero, M., Escamilla, R., Marquina, V. & Gómez, R. Structural and mechanic properties of RFeO_{3} with R=Y, Eu and La perovskites: a firstprinciples calculation. Eur. Phys. J. D. 69, 177 (2015).
 35.
Wen, H. et al. Structural and electronic recovery pathways of a photoexcited ultrathin VO_{2} film. Phys. Rev. B 88, 165424 (2013).
 36.
Wen, H. et al. Electronic origin of ultrafast photoinduced strain in BiFeO_{3}. Phys. Rev. Lett. 110, 037601 (2013).
 37.
Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977).
 38.
Ito, N. Nonequilibrium critical relaxation of the threedimensional Ising model. Phys. Stat. Mech. Its Appl. 192, 604–616 (1993).
 39.
Peczak, P. & Landau, D. P. Monte Carlo study of critical relaxation in the 3D Heisenberg model. J. Appl. Phys. 67, 5427–5429 (1990).
 40.
Nightingale, M. P. & Blöte, H. W. J. Dynamic exponent of the twodimensional ising model and monte carlo computation of the subdominant eigenvalue of the stochastic matrix. Phys. Rev. Lett. 76, 4548–4551 (1996).
 41.
Dunlavy, M. J. & Venus, D. Critical slowing down in the twodimensional Ising model measured using ferromagnetic ultrathin films. Phys. Rev. B 71, 144406 (2005).
 42.
Gao, F. et al. Charge order suppression and weak ferromagnetism in La_{1∕3}Sr_{2∕3}FeO_{3} nanoparticles. Appl. Phys. Lett. 91, 072504 (2007).
 43.
SahaDasgupta, T., Popović, Z. S. & Satpathy, S. Density functional study of the insulating ground states in CaFeO3 and La_{1∕3}Sr_{2∕3}FeO_{3} compounds. Phys. Rev. B 72, 045143 (2005).
 44.
Battle, P. D., Gibb, T. C. & Lightfoot, P. The crystal and magnetic structures of Sr_{2}LaFe_{3}O_{8}. J. Solid State Chem. 84, 237–244 (1990).
 45.
Yang, J. B. et al. Charge disproportionation and ordering in La_{1/3}Sr_{2/3}FeO_{3δ}. J. Phys. Condens. Matter 15, 5093–5102 (2003).
 46.
Binder, K. Theory of firstorder phase transitions. Rep. Prog. Phys. 50, 783–859 (1987).
 47.
Hausoel, A. et al. Local magnetic moments in iron and nickel at ambient and Earth’s core conditions. Nat. Commun. 8, 16062 (2017).
 48.
Park, H., Millis, A. J. & Marianetti, C. A. Density functional versus spindensity functional and the choice of correlated subspace in multivariable effective action theories of electronic structure. Phys. Rev. B 92, 035146 (2015).
 49.
Chen, J., Millis, A. J. & Marianetti, C. A. Density functional plus dynamical meanfield theory of the spincrossover molecule Fe(phen)_{2} (NCS)_{2}. Phys. Rev. B 91, 241111 (2015).
Acknowledgements
We acknowledge G. Doumy and A. M. March for the assistance of using the highrepetition rate laser at APS beamline 7IDC.Y.Z., J.H., A.B., and H.W. acknowledge support of U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. H.P. acknowledges support of the startup funding from the University of Illinois at Chicago and Argonne National Laboratory (by the U.S. Department of Energy, Office of Science program) and the computing resources provided on Blues, a highperformance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. H.W. and J.F. acknowledge the support of data analysis by DOEBES grant No. DESC0012375. The use of the Advanced Photon Source and Center for Nanoscale Materials is supported by DOEBES, under Contract No. DEAC0206CH11357.
Author information
Affiliations
Contributions
H.W. and A.B. conceived the project. Y.Z., D.W., J.F., P.R., H.W. did the Xray measurements. C.R. and R.S. did the optical measurements. J.H. and A.B. made the samples. H.P. performed firstprinciples calculations. Y.Z. and H.W. wrote the paper with contributions from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhu, Y., Hoffman, J., Rowland, C.E. et al. Unconventional slowing down of electronic recovery in photoexcited chargeordered La_{1/3}Sr_{2/3}FeO_{3}. Nat Commun 9, 1799 (2018). https://doi.org/10.1038/s41467018041994
Received:
Accepted:
Published:
Further reading

Critical Slowing Down at the Abrupt Mott Transition: When the FirstOrder Phase Transition Becomes Zeroth Order and Looks Like Second Order
Physical Review Letters (2020)

Quasiparticle relaxation dynamics in URu2−xFexSi2 single crystals
Physical Review B (2019)

Transient quantum isolation and critical behavior in the magnetization dynamics of halfmetallic manganites
Physical Review B (2019)

Superlatticeinduced ferroelectricity in chargeordered La1/3Sr2/3FeO3
Proceedings of the National Academy of Sciences (2019)

Subthreshold firing in Mott nanodevices
Nature (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.