Dynamical limits by downsizing for the photo-induced switching in a molecular material revealed by time-resolved X-ray diffraction

Cooperative molecular switching at the solid state is exemplified by spin crossover 2 phenomenon in crystals of transition metal complexes. Time-resolved studies with temporal 3 resolutions that separate molecular level dynamics from macroscopic changes, afford clear 4 distinction between the time scales of the different degrees of freedom involved. In this work 5 we use 100 ps X-ray diffraction to follow simultaneously the molecular spin state and the 6 structure of the lattice during the photoinduced low spin to high spin transition in microcrystals 7 of [Fe III (3-MeO-SalEen) 2 ]PF 6 . We show the existence of a delay between the crystalline 8 volume increase driven by the propagation of collective volumic strain waves, and the 9 cooperative macroscopic switching of molecular state. Such behaviour is different from the 10 expectation that phase transformation only requires atomic displacements in the unit cell, that 11 can occur simultaneously with propagation of a volumic strain. Model simulations and 12 discussions of the physical picture explain the phenomenon with thermally activated kinetics 13 governed by local energy barriers separating the molecular states.


Introduction
Dynamical processes induced by a laser pulse in materials are intrinsically multi-scale in time 2 and space. The difference of time scale is significant between the electronic processes, typically 3 occurring within femtoseconds, the coherent atomic displacements ranging from few 10's of 4 femtoseconds to picoseconds reflecting the period of optical phonons, the volume expansion, 5 and finally the slower kinetics dictated by activation energies. It is noteworthy that the by laser-induced internal stress, generated by ultrafast lattice heating and/or instantaneous 10 electronic change [Thomsen1986, Wright1994]. Recovery of the mechanical equilibrium with 11 the sample environment occurs through wave propagation over the relevant length of the 12 system (crystal size, light penetration depth, …). The associated acoustic time scale falls in the 13 picosecond range for nanometers, and in the nanosecond range for micrometers in many 14 materials of interest. Moreover, there may exist activation barriers at the local scale, which can 15 further slow down the atomic rearrangements. As opposed to conventional time-averaging 16 experiments, ultrafast time-resolved experiments are typically used to delineate the dynamics 17 of the different degrees of freedom, such as change of electronic distribution, atomic 18 reorganizations, as well as cell deformations [Lorenc2009]. They are key to understanding the 19 non-equilibrium dynamics on material scale, and to harnessing the mechanisms that govern the 20 properties of materials [Bargheer2004, Baum2007, Braun2007, Morrison2014]. 21 The opportunity to photo-induce a phase transition with a laser pulse [Nasu1997, Nasu2004] 22 has opened up a vast field of research ranging from the melting of charge, spin and structural 23 orders in electronically correlated materials [Zhang2014, Basov2017] to molecular switching 24 in the solid state [Lorenc2009, Cailleau2010, VanderVeen2013a, VanderVeen2013b]. 1 Recently, some attention has been turned to phase transitions that explore the possibility of 2 combining the ultra-fast with the ultra-small [Sagar2016, Park2017, Ridier2019]. We 3 previously reported the size effect on an elastically-driven cooperative dynamical response in 4 a switchable molecular crystal, a spin crossover (SCO) iron complex [Bertoni2016a]. Owing 5 to a positive elastic feedback from the expanding lattice on the volume-changing bistable 6 molecules, the number of switched molecules is significantly enhanced and the lifetime of their 7 photo-induced state is prolonged. The time scale of such dynamics showed to be scaling with 8 the size, namely becoming shorter in micro-and nano-crystals than in bulk single-crystals (> 9 100 μm). This suggested a central role of propagating volume expansion for the cooperative 10 transformation at the scale of a crystallite [Bertoni2016b]. 11 At thermal equilibrium, volume change and spin state switching occur together. In contrast, the 12 out-of-equilibrium dynamics triggered by a laser pulse implies a sequence of processes. To 13 establish the respective evolution of the elastic deformation of crystalline volume and the 14 switching of molecular spin state, a simultaneous measurement of the two purportedly coupled 15 parameters is required. To address this challenge, we used time-resolved X-ray diffraction (tr- 16 XRD) on small crystallites. Our findings were corroborated with extended Monte Carlo 17 simulations.

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Phase transition and volume change in bulk single crystal 20 In this study, we focus on an SCO compound [Fe III (3-MeO-SalEen)2]PF6. The SCO materials 21 serve as prototypical examples of cooperative switching between two molecular electronic 22 states, Low Spin (LS) and High Spin (HS). The sample was previously identified as suitable 1 candidate for the photo-induced out-of-equilibrium studies in solid state from the angle of 2 elastic properties that lead to strong cooperativity [Bertoni2016a]. The crystalline structure of 3 this material was characterized in previous studies (S1, [Tissot2011, Tissot2012]). At the 4 molecular level, the switching of electronic state of Fe III system, from LS (S = 1/2) to HS (S = 5 5/2) causes an increase of molecular volume, due to elongation of the Fe-Ligand bonds by 6 around 0,15 Å between LS and HS states. The molecules in this compound are arranged in a 7 closed packed network [Tissot2011] (figure 1a). It is known that the relative stability between 8 different macroscopic phases in such SCO crystals is ensured by the elastic intermolecular 9 interactions of various strengths, resulting in more or less cooperative transformations 10 [Rat2017, Buron2012]. In the present case, a strongly first order phase transition is observed 11 around 162 K, with a thermal hysteresis of 3 K, between the low temperature LS phase and the 12 high temperature HS phase [Tissot2011]. This phase transition is isostructural, as it does not 13 imply any change of symmetry (same space group P-1 and Z=2 for each phase), in a way similar 14 to the gas-liquid transition [Chernyshov2004]. The volume is a totally symmetric parameter 15 and plays central role for an isostructural phase transition. At thermal equilibrium, both the unit 16 cell volume and the concentration of HS molecules show correlated jumps at the phase 17 transition. This is usually observed for two totally symmetric degrees of freedom involved in 18 changes associated with a phase transition without symmetry change [Landau1980]. 19 We performed XRD measurements on a single crystal (see Methods and S2) between 100 K 20 and 250 K, in order to quantify accurately the volume jump (volume discontinuity at the first 21 order phase transition). The measured temperature evolution of the unit cell volume is 22 displayed in figure 1b. It shows a significant volume jump of 1.6% (22 Å 3 per unit cell) at 23 transition temperature T↑ = 166 K (both consistent with the values reported in the previous 24 studies [Tissot2011]). Below and above this discontinuity, the thermal expansion is also 25 significant in both LS and HS phases. The same measurement allowed for accurate 1 determination of the thermal dependence of all six unit cell parameters for each triclinic phase. 2 Their extrapolated evolution can be described with a linear function of temperature T for both 3 phases (S2). The volume expansion coefficients for HS and LS phases were found to be 4 respectively 0.31 Å 3 /K and 0.16 Å 3 /K (figure 1b). It is a considerable thermal expansion, 5 leading to volume increase of 33 Å 3 between 100 and 250 K, in comparison with 22 Å 3 jump 6 originating from the transition. 7 Powder XRD study at thermal equilibrium of micro-crystals 8 In the following, we will discuss measurements performed on micro-crystals of [Fe III (3-MeO-9 SalEen)2]PF6 embedded in a polymer thin film (see Methods). The small crystallites are plate- 10 shaped, with average dimensions 3.5 µm x 0.35 µm x 0.13 µm [Tissot2012]. The size is very 11 much dependent on the synthesis conditions. Smaller crystallites are possible to obtain, but 12 these were chosen to ensure diffraction patterns of sufficient quality for a quantitative analysis. 13 The crystallites were dispersed in polyvinylpyrrolidone (PVP) polymer matrix and the 14 composite films were spin-coated on a glass substrate, as described in [Bertoni2012]. 15 The powder XRD measurements were performed at ESRF, ID09 beamline. XRD images were 16 recorded on a 2D detector in quasi-grazing reflection geometry at 0.2° incidence angle (see 17 Methods). This experimental geometry was used to reduce the diffuse background due to 18 scattering from the glass substrate and thereby enhance the diffraction from the thin film 19 sample. A typical diffraction image is shown in figure 1c. The diffraction rings from the 20 polycrystalline sample are clearly visible, with a patterning due to preferred orientation of the 21 micro-crystals. Indeed, since the micro-crystals are plate-shaped, they tend to align with the 22 shortest dimension (crystallographic c axis) perpendicular to the film surface. Several micro- 23 crystals can also stack in-depth. 1c) was removed, and the important peak broadening was ascribed mainly to a large X-ray 4 footprint at the small angle incidence. The latter made the analysis challenging, yet as detailed 5 below, the full-pattern refinement was possible. This allowed for retrieval of key parameters, 6 namely the phase fraction and the volume change very accurately. 7 To this end, we applied the Pawley approach [Pawley1981] using the Topas software 8 [Coelho2018], and a similar method to the one recently applied for powder diffraction studies 9 of photo-induced structural changes [Azzolina2019, Mariette2021]. The refinements details 10 are provided in Methods. In short, this method constrains the Bragg peak positions according 11 to the refined unit cell parameters, but allows the Bragg peak intensities to vary freely. At room 12 temperature, the sample is fully in the HS phase. At 100 K, it was not possible to fit the pattern 13 satisfactorily considering LS phase only, therefore a biphasic state had to be considered, in 14 contrast with the bulk single crystal. We ensured the stability of the refinement further by 15 parametrization of the unit cell and peak intensities, as detailed hereafter. Firstly, to describe 16 the volume evolution of both phases as a function of temperature T, the respective six unit cell 17 parameters were forced to follow the thermal expansion determined with the single crystal 18 study and shown in the figure S2. Consequently, the temperature T was the only parameter 19 required to describe the evolution of all unit cell parameters. Secondly, the relative Bragg peak 20 intensities Ihkl were fixed for both phases: Ihkl,HS and Ihkl,LS were estimated from the refinements 21 of the patterns at room temperature (RT = 293 K) and 100 K. Thirdly, the complete powder 22 patterns of a biphasic state (LS, HS) were fitted with XHS as the only refined parameter. In such 23 a case, XHS accounts both for intensity changes and peak shifts, since the thermal expansion is 24 phase-dependent.

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The refined XHS is plotted in figure 1f as a function of temperature. A clear change of slope is 1 observed around the transition temperature in the bulk single crystal. Above this temperature, 2 the sample is almost fully in the HS phase. The observed XHS evolution correlates very well 3 with the M*T product (M, magnetisation and T, temperature) measured by SQUID 4 (Superconducting QUantum Interference Device, see Methods) probing the evolution of the 5 fraction of molecules in the HS state. The spin state conversion in these micro-crystals shows 6 two peculiar features: a gradual conversion (compared to abrupt in bulk), and incomplete 7 conversion at low temperature. The residual XHS at T = 100 K is equal to (34 +/-3) % is 8 consistent with the previous reports [Bertoni2012]. These two features are explained in terms 9 of size/strain effects and non-homogeneous stress due to surface interaction with polymers, 10 resulting in a broad regime of phase coexistence [Tissot2012, Laisney2020]. The refined XHS 11 also allows to calculate the average volume: 13 Concurrence of the gradual spin-state conversion and thermal expansion leads to a very smooth 14 evolution of the average volume. Yet, the average volume Vaverage changes slope around 160 K 15 (figure 1e). 16 Thus parametrized powder pattern refinement allows extracting accurate values for both 17 XHS(T) and Vaverage(T) simultaneously, and the ensuing discussion is hinged upon it. We should 18 nonetheless mention some underlying assumptions. First hypothesis is that the LS and HS 19 phases can be treated as separate diffracting domains. Explicitly, XHS quantifies the fraction of 20 ordered HS domains, rather than counting the HS molecules. The excellent correlation between 21 XHS and the fraction obtained with SQUID (counting individual molecules) supports our 22 assumption. The second hypothesis is that other structural distortions on temperature within a 23 given phase have negligible contribution to the changes of Ihkl. This is substantiated with 1 measurements on a single crystal (figure S2) and therefore seems reasonable too. 2 Time-resolved XRD study of photoinduced dynamics 3 The tr-XRD images were recorded at 100 K with the setup described above. The crystallites 4 were excited with 1 ps pump laser at 800 nm, as in previous optical studies [Bertoni2016a]. 5 The experimental time resolution was limited by the X-ray pulse duration to 100 ps. The details 6 about the setup and data reduction are given in Methods. induced changes can be seen on these patterns for all positive delays, and they are emphasized 9 in the difference patterns (figure 2b). The comparison with steady state diffraction patterns 10 measured at low and high temperature allows for a qualitative description. The shape of the 11 difference patterns can be explained by a weight transfer of the diffracted intensity from LS-12 to HS-Bragg peaks. However, the peak shift to smaller q due to volume expansion would 13 produce a similar difference pattern. Therefore, separating these two possible contributions 14 requires more quantitative analysis as described below. A closer inspection of the difference 15 patterns also gives some insight into the dynamics. First changes occur within the 100 ps time 16 resolution. Thereafter, the difference patterns change shape, suggesting a sequence of processes 17 with structurally distinct signatures. 18 In order to analyze the underlying structural dynamics, the same method of full-pattern 19 refinement as in the temperature study was applied to the tr-XRD patterns. The refinement 20 results are shown in figure 2c. The parametrized model was similar to that used in the 21 temperature study. However, in addition to XHS, the lattice temperature becomes an adjustable 22 parameter Tlattice, to account for the heating and non-equilibrium lattice expansion (see 1 Methods). 2 The time evolution of the structural parameters obtained for the highest excitation density (380 3 µJ/mm²) reveals multistep dynamics. A small increase of ΔXHS estimated at 7%, accompanied 4 by a small volume increase, is observed at the early step (i.e., within 100 ps time resolution). 5 Even if the initial volume rise cannot be accurately determined due to the 100 ps time 6 resolution, the maximum of volume expansion is sufficiently pronounced and well resolved at 7 around 300 ps. At this delay, for high excitation density, the refinement yields a value of ΔTlattice the nonlinear increase of ΔXHS sets in with a threshold [Bertoni2016a]. However, no significant 22 increase of average volume is observed when ΔXHS reaches its maximum. 23 The validity of such analysis of non-equilibrium dynamics whereby Ihkl and the lattice 1 expansion are a-priori assumed, can be questioned. Both affect mainly the refined XHS. In  Monte-Carlo simulations of mechanoelastic model 10 In order to rationalize the experimental results, we applied the mechanoelastic model, also 11 referred to as ball-and-spring model, that had been successfully used in previous studies to 12 simulate the out-of-equilibrium self-amplification [Enachescu2017, Bertoni2019]. In this two-13 dimensional model, molecules are mimicked with spheres (balls) whose radii reflect LS or HS 14 state (10% increase in radius from LS to HS molecules, as observed on the Fe-Ligand bond 15 length). The balls are connected by springs, which replicate elastic interactions occurring in the 16 solid state through the lattice (figure 3a). When molecules (balls) switch, they induce a local 17 deformation of neighboring springs through a change of radii, that will propagate to the 18 adjacent molecules (balls). Hence, the elastic force network accounts for both short-and long- 19 range interactions in the model. The probabilities for switching a given molecule from LS to 20 HS state, and the reverse, depend on the temperature and the local pressure through Arrhenius-21 like activation. 22 The elastic stresses originate not only from the molecular swelling upon LS to HS switch, but 1 also from the lattice heating. The latter leads to the so-called thermo-elastic stresses that trigger 2 thermal expansion [Thomsen1986, Matsuda2015, Ruello2015]. It originates from the energy 3 transfer between the photoexcited molecules and the thermal bath of lattice phonons 4 [Lorenc2012]. Because the simulations rely on a harmonic potential, such phenomenon cannot 5 be reproduced by a temperature increase alone, which only modifies the probability of 6 switching. For this work, the Monte-Carlo (MC) model was extended to account for the thermal 7 expansion by allowing distances between molecules to increase without changing the spin state. 8 To factor this aspect of non-equilibrium thermodynamics in MC simulations, the springs were 9 allowed to stretch beyond the HS and LS equilibrium distances at a given temperature ( figure   10 3b). This mimics thermal lattice expansion. In practice, we started with a simulation assuming 11 a homogeneous increase of temperature, with 15% increase for HS and LS spring lengths. 12 Following this very first MC step, the spring lengths recover equilibrium positions  Discussion on the physical picture of photo-induced multistep dynamics 17 We now consider the implications of the observed multistep structural dynamics and propose 18 a comprehensive scheme for the macroscopic photo-switching. In particular, we discuss the 19 origin of a significant delay between the volume expansion process and the transformation 20 from LS to HS phase.

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The initial step of photo-excitation by a 100 fs laser pulse was described previously 22 [Camarata2014]. Direct photo-excitation from LS to HS molecular state is forbidden, so the 23 800 nm light is used to photo-excite a LS molecule to a short-lived intermediate singlet state, 1 followed by an ultrafast intersystem crossing towards the HS state [Hauser1986]. The latter is 2 structurally relaxed on the ps time scale. The switched molecules are uniformly distributed in 3 each microcrystal, since the thickness of microcrystals is much smaller than the penetration 4 depth of laser light (approximately 5 μm at 800 nm [Bertoni2016a]). Initial ΔXHS is small and 5 it scales linearly with the excitation density [Bertoni2016a], here estimated at 7% for the high 6 excitation density (figure 2c). Importantly, the absorbed photon energy (1.55 eV) is much 7 higher than the energy difference between HS and LS ground states (tens of meV). In  By contrast, a linear elastic response for the propagating volume is observed. The delay by 23 almost two temporal decades between the two processes is relatively long. It is fair to assume 24 that achieving a phase transition at the macroscopic scale requires more time than does the 25 switching of independent molecules [Boukheddaden2000]. Also, it is noteworthy that no 1 additional volume increase is observed when ΔXHS reaches maximum. It is possible that part 2 of the thermal energy stored in the lattice is spent on phase transformation. Consequently, the 3 lattice temperature and volume would decrease and counterweight the increase of volume 4 originating from phase transformation.   16 In conjunction with the earlier optical studies that revealed size dependent switching time 17 [Bertoni2016a], and a priori expectation that in the ever so smaller nanoparticles the 18 macroscopic switching of the molecular state can be indefinitely fast, the new results point to 19 a bottleneck in the switching dynamics. Hence, our findings define the ultimate time scale for 20 macroscopic transformation of molecular state in nanoscale objects for this class of solids.

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They do not obey the rule "the smaller the faster" [Ridier2019]. This finding could motivate 22 an optimised material design, scalable with size dependent dynamics and intrinsic energetics. 23 A new perspective for the design of functional spin-crossover nano systems could reside in 24 chemical leverage of the energy barriers by tuning the ligand field, or other methods.

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Single crystal X-ray diffraction 2 Temperature-dependent X-ray diffraction (XRD) study was performed on [Fe III (3-MeO-  The Fe III compound was prepared in the form of a powder of pure microcrystals as previously 12 reported [Tissot2012]. The microcrystals (typical dimensions of 3.5 µm, 0.35 µm, 0.13 µm 13 from TEM) were processed in polyvinylpyrrolidone (PVP, MM = 45000 gmol -1 ) thin films 14 formed by spin-coating on glass substrate [Bertoni2012]. 15 X-ray diffraction on polycrystalline sample 16 Experimental setup 17 X-ray diffraction experiment was performed at ESRF, ID09 beamline during hybrid mode (⅞ th 18 continuous filling + 1 isolated electron bunch). The isolated electron bunch was used in the 19 experiment. The ID09 setup was discussed in detail previously [Cammarata2009]. Briefly, fast 20 rotating choppers were used to isolate X-ray single pulses (each 100 ps long) at 40 Hz repetition 21 rate. The X-rays energy was centered at 18 keV (λ = 0.6888 Å) with 1.5% bandwidth. The 1 beam size at sample position was 0.1 (horizontal) x 0.02 (vertical). Diffracted X-rays were 2 integrated on a Rayonix MX170-HS CCD detector. Each image was recorded with 1000 shots 3 exposure. 4 For the thermal study at equilibrium, sample was cooled down using a nitrogen cryostream 5 700Plus from Oxford Cryosystems and diffraction images were measured with 1 K steps from 6 293 K to 100 K. 7 For the time-resolved study, a synchronized laser (800 nm) at 40 Hz was used to excite the  14 [-50 ps, 0 ps, 50 ps, 100 ps, 200 ps, 300 ps, 700 ps, 1 ns, 1.5 ns, 3 ns, 5 ns, 7 ns, 10 ns, 30 ns, 15 100 ns, 300 ns, 1 µs, 3 µs, 10 µs, 30 µs, 100 µs, 300 µs, 1 ms] 16 Measurements were performed with shuffled delays, to detect any drift effects during the 17 experimental time. 18 Data Reduction 19 The diffraction images were azimuthally integrated using pyFAI [Ashiotis2015]. Background 20 was subtracted using scikit-ued python library [Rene2017]. Negative reference patterns (-5 ns) 21 were used to calculate differential patterns and averaging with trx python library 22 [Cammarata2017]. 23 Powder patterns refinement 1 Full powder pattern refinements were performed after the data reduction step described above. 2 Peak profiles and sample displacement were described from a fundamental parameter approach 3 using expressions derived in [Rowles2017] with the TOPAS software [Coelho2018]. The   lines are the result of powder pattern refinements (see Methods) and grey lines correspond to 7 the residual (difference between experimental and refined patterns). Bottom: Time-resolved 8 differential pattern measured at 99 K from -50 ps to 1 ms after photo-excitation under high 9 fluence 380 μJ/mm 2 (multiplied by 4 for clarity), showing the characteristic shape for volume 1 expansion (peak shifts towards lower q values). Red and blue vertical lines indicate the position 2 of major HS and LS Bragg peaks, respectively. c) Time evolution of the volume (top) and 3 relative HS fraction (middle) as extracted from powder pattern refinement. Bottom: time 4 evolution of the relative photo-induced HS fraction extracted from optical spectroscopy 5 (adapted from [Bertoni2016a]). In orange: high excitation density (380 µJ/mm² and 410 6 µJ/mm² for XRD and OD measurements, respectively), in dark green: low excitation density 7 (60 µJ/mm² and 100 µJ/mm² for XRD and OD measurements, respectively). of the surface (equivalent to volume in 2D box, top) following high (20%, orange) and low 9 (5%, dark green) fractions of switched sites at MC's step zero. Dotted grey lines correspond to 10 previous model, without initial spring length increase. In that case, the increase of HS fraction 11 and the volume were not separated in time. between expansion (300 ps, or less for nano-crystallites) and the switching (ns time-scale), due 12 to incompressible Arrhenius-type kinetics sketched in the insert.