Unprecedented Bistability in Spin-Crossover Solids Based on the

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Unprecedented Bistability in Spin-Crossover Solids Based on the Retroaction of the High Spin Low-Spin Interface with the Crystal Bending Miguel Paez-Espejo, Mouhamadou Sy, and Kamel Boukheddaden J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b04802 • Publication Date (Web): 30 Aug 2018 Downloaded from http://pubs.acs.org on August 31, 2018

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Journal of the American Chemical Society

Unprecedented Bistability in Spin-Crossover Solids Based on the Retroaction of the High Spin Low-Spin Interface with the Crystal Bending Miguel Paez-Espejo, Mouhamadou Sy, and Kamel Boukheddaden∗ Groupe d’Etudes de la Matière Condensée, UMR 8635, CNRS-Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue des Etats Unis, 78035 Versailles, France E-mail: [email protected]

Abstract There has been in recent years a continuous increase in spatiotemporal investigations of the dynamics of the first-order transitions in spin-crossover (SCO) solids. In single crystals, this phenomenon proceeds via a single domain nucleation and propagation, characterized in some systems with the presence of two equivalent and symmetric interface orientations, between the high-spin (HS) and low-spin (LS) phases, due to the anisotropic structural change of the unit cell at the transition. The present investigations bring an experimental evidence of the reversible driving of the translational and rotational degrees of freedom of the HS-LS interface. In addition to its rectilinear displacement, the interface rotates between two stable angles, 60o and 120o . It is demonstrated that while the translation motion is accompanied with a crystal’s length change, the interface rotation is controlled by the crystal’s bending. These results are well explained in the frame of an elastic theoretical description in which the effect of the crystal bending, on the stability of the interface’s orientation, is simulated by applying

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a moment of forces on the crystal. It is found that the interface orientation becomes unstable beyond a threshold load value, announcing the emergence of a bistability in SCO solids, taking place at constant HS fraction and volume. This work underlines the sensitive character of the interface orientation to any macroscopic crystal bending, an idea that can be used to develop a new generation of robust stress sensors, working at constant volume, thus avoiding deterioration problems due to crystal fatigue.

Introduction Today, material science is an interdisciplinary area at the interface between physics, chemistry and biology focusing on new materials with novel and/or multi-functionalities that are promising for future development of the high-tech applications. Based on the Moore law, predicting that the capacity of the electronic devices doubles every two years in average, it is not difficult to conceive that the usual miniaturization techniques of the devices will reach their limitation in few decades. That’s why the molecular electronic emerges as a powerful research area to propose alternative solutions well adapted to the constant need for highly efficient devices at very small scales. In this context, the spin-crossover (SCO) materials (whose fundamentals aspects are under clarification) present serious potentialities in the development of new generation of electronic devices, 1–7 but also as very accurate sensors, displays as well as optical memories. These materials combine thermo-, piezo, magneto- and photo-switching features, leading to original physical properties. In octahedral symmetry, transition metal complexes with 3d4 -3d7 , as well as 3d8 when molecular symmetry is lower than Oh , may present electronic configurations adopting two electronic ground states according to the splitting of the d orbitals in the eg and t2g subsets. When the energy gap between them, called the 00 ligand field00 , ∆, is greater than the electronic repulsion energy, Π, the electrons tend to occupy the lower energy orbitals, t2g and the metal complex adopts the low-spin (LS) state. In contrast, when ∆ Π, the d electrons follow the Hund’s rule and the metal complex adopts the high-spin (HS) state. When ∆ and Π have similar values, and 2


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if in addition the energy difference between the HS and the LS states, ∆E, becomes of the same order of magnitude of the thermal energy, kB T , the application of an external stimulus (temperature, light, pressure) may induce a change in the spin state. However, the complete character of the transition between the LS and the HS states, even in weak-cooperative systems, cannot be understood without invoking the crucial role of the entropy difference between the LS and HS states. This entropy difference, originating from the large change in electronic and vibrational properties of the two states, constitutes the driving force of this phenomenon. Nevertheless, this view is limited, since it only explains the intra-molecular aspects of the spin-crossover phenomenon, which also appears in the solid state as a firstorder phase transition. Now, it is well admitted that the thermally induced spin crossover (SCO) transition between the low spin (LS) and the high spin (HS) states of Fe(II) complexes with suitable ligands is a typical example of switchable molecular solids (SMs). Such materials have been studied 8–10 for many years due to their promising applications as materials for information storage. The bistability of SMSs originates from an intra-molecular vibronic coupling 11,12 between the electronic structure and the vibrational one. The latter can be enhanced in the solid state by inter-molecular interactions. Indeed, elastic interactions 13–15 are recognized as one of the basic ingredients of the SCO transition, and lead to various behaviors, going from gradual, simple Boltzmann distribution between two states (in non cooperative systems), to rather abrupt hysteretic thermal spin transitions, characteristic of a first-order phase transition, 16–18 in the case of strongly cooperative interactions. The occurrence of two-step transitions has been assigned in the past to the coexistence of interactions with opposite signs, 18,19 but was clarified recently as the result of the existence of elastic frustration in the lattice. 20–22 Recent developments of discrete atomistic models based on deformable lattices 23–35 , so as to mimic the spatio-temporal features of the nucleation and growth process of spin phases, which were revealed by optical microscopy investigations 36–47 at the thermal transition of SCO single crystals, have boosted the research on the dynamics of the first-order transition

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in SCO complexes. Imaging spin-transition in single crystals was initiated by Jeftic and Hauser in the 90s, 36 who showed optical microscope images of [Fe(ptz)6 ](BF4)2 (Fe-ptz) large single crystals at low temperature during low-temperature photo-excitation and further images were reported by Ogawa et al. 37 during the photo-excitation of [Fe(2-pic)3 ]Cl2 .EtOH by a femtosecond laser pulse, showing inhomogeneous features occurring transiently during the phototransformation or subsequent relaxation of the metastable HS state. The real start of optical microscope investigations occurred later, when the quantitative character of the method was recognized in terms of colorimetric analysis. 48 Since the first studies of optical microscopy in cooperative spin crossover single crystals undergoing first order phase transitions, as a result of collective volume change between the LS and the HS states, some compounds have shown an orientation of their HS/LS interface during these transitions. 41,49 In the SCO single crystal, [{Fe(NCSe)(py)2 }2 (m-bpypz)], subject of this review, the interface orientation was assigned to the existence of an anisotropic deformation of the SCO unit cell during the transformation between the LS and the HS states. 45,47 Optical microscopy investigations on this sample, showed the existence of two possible symmetric and stable orientations of the HS/LS interface, that appear spontaneously in the beginning of the transformation process inside the thermal hysteresis region. However, once one front orientation takes place in the beginning of the process, it remains stable and invariant during the front dynamics, except in some rare cases when the interface meets some defect along the front propagation direction. In a previous work, 47 we reported the reversible control of the dynamics of the HS/LS interface with light (causing the local photo-heating of the crystal) inside the bistable area of the thermal hysteresis. There, we showed that weak modulated light intensity allows the control of the HS fraction by enslaving the interface position along its propagation direction without any effect on its orientation angle. Thus, the light controlled only the translational motion of the HS/LS interface. During this dynamical process, the crystal

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length changed according to that of the HS fraction, which then led to conclude that the crystal elongation/contraction controls the translation motion of the HS/LS interface and vice-versa. In the present work, we bring the first example of reversible control of a new degree of freedom of the interface motion, that is the interface orientation. Contrary to the interface translation which is induced/accompanied by the change of the crystal length (and so the volume), the re-orientation of the interface requires the crystal bending, as we prove it theoretically in the present manuscript, and keeps the crystal volume almost unchanged. However, the experimental realization of this effect, consisting to apply a moment of forces on a single crystal of 100 µm size under a microscope, faces up to serious technical challenges. So the idea was to take advantage of the existence of two interacting interface fronts in the same single crystal to produce the crystal bending and to observe the interface re-orientation. We could realize this requirement in which the interplay between the photocontrol dynamics of two fronts led to a spontaneous and reversible re-orientation of one of the HS/LS interfaces during their relative motion, without the presence of any observable defect. This dynamical switching is unique. To this experimental fact, we provide the explanation of the physical mechanisms at the origin of the interface tilting on the one hand, and that of its switching on the other. The understanding of the interaction between the macroscopic deformation of the crystal and the HS/LS front orientation is at the heart of this problem, which allowed us to propose an original method to control this new type of bistability, related to the switching of the orientation of the HS/LS domain wall between two stable states. The manuscript is organized as follows: Sec. II summarizes the experimental data of optical microscopy showing the translational and orientation motion of the domain wall under illumination and Sec. III introduces the anisotropic elastic model adapted for the description of the unit cell deformation of the current system. Sec. IV reports the results of Monte Carlo simulations of the anisotropic electro-elastic model and in Sec. V, we conclude and underline the possible extensions of this work.

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Experimental evidence of the bistability of the domain wall rotation The thermal spin-transition We selected a single crystal (length × width × thickness ' 89.5 × 12.2 × 10 µm3 ) of the SCO compound [{Fe(NCSe)(py)2 }2 (m-bpypz)], 50 (abbreviated as Fe(NCSe)) where py = pyridine and bpypz = 3,5-bis(2-pyridyl)pyrazolate, undergoing a thermal first-order transition with hysteresis between 108.1 K and 114.4 K (see Fig. 3). This crystal is very robust and presents an exceptional resilience upon repeated switching. 41,45,47 The spatiotemporal character of the HS-LS domain wall propagation, is recorded by optical microscopy setup. All images were obtained using a microscope Nikon Eclipse LV100 (objective ×50, numerical aperture NA= 0.45) connected to a digital camera Coreview Dalsa Falcon 1.4 M100 HG color.

The real-time dynamics of the spin transition of the present single crystal is provided in movies S1 and S2 of the SI for heating and cooling processes. A selected set of snapshots, displayed in Fig. 1, shows that on cooling, the LS phase appeared inside the crystal and then propagates towards the crystal borders with two HS/LS interfaces oriented at ∼ 60o and 120o with respect to the crystal’s length direction. In contrast, on heating, the HS phase appeared from the left tip of the crystal with an interface oriented at 60o throughout the propagation process. Quantitative data are derived from OM measurements by tracking, on each pixel, the temporal evolution of the transmitted light intensity of the RGB (Red, Blue, Green) levels (1 pixel × 1 pixel ∼ 0.142 × 0.142 µm2 ). During the transformation stages of the crystal, we automatically recorded the PC screen by using the CoreView software, and simultaneously recorded snapshots every ∼ 0.1 seconds in average. To minimize the signal/noise ratio, we

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Figure 1: Selected OM snapshots of the crystal [(F e(N CSe)(py)2 )2 (m − bpypz)] during the spin transition between the HS (light area) and LS (dark phase) along the cooling and heating branches of the thermal hysteresis. On cooling, the LS nucleates inside the crystal and then propagates towards the crystal borders with two HS/LS interfaces oriented at ∼ 60o and 120o . On heating, the HS phase emerges from the left part of the crystal with one unique domain wall oriented at ∼ 60o with respect to the crystal length. Temperature scan rate is r = 0.5 K.min−1 and crystal’s size is (length × width × thickness = 89.5 µm × 12.2 µm × 10 µm.)

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Figure 2: Temporal evolution of the HS/LS interface position, showing a quasi-linear behavior, along the cooling (blue circles and black pentagons) and heating (magenta triangles) branches, depicted in the snapshots of Fig. 1. The average interface velocities on cooling are 8.6 µm.s−1 (blue circle) and 7.8 µm.s−1 (black pentagon) for the left (2) and right (1) fronts. On heating, the front propagation exhibits two successive regimes with velocities 3.6 µm.s−1 and 5.9 µm.s−1 .

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followed the optical density (OD = log10

I0 , where I0 = bright field intensity and It = It

transmitted intensity) which merely scales with the HS state fraction. The analysis was made for the three types of pixels (Red, Green, Blue), and all three revealed to be more or less sensitive to the spin transition, but we only report here the data of the green pixels, which ensured the best ratio signal/noise.

Using a specific Matlab program, designed for

the treatment and image processing of OM data, 42 we could monitor the interface position as a function of time, as illustrated in Fig. 2. It is interesting to note the slight sigmoidal character of the curves of the interface position with time for this cooling process, for which the average velocity of the left (resp. right) interface is 8.6 µm.s−1 (resp. 7.8 µm.s−1 ). In contrast, on heating, the only interface propagating along the crystal showed two successive linear regimes with estimated velocities 3.6 µm.s−1 and 5.9 µm.s−1 . The occurrence of the second regime is attributed here to the existence of a spontaneous weak crystal bending accompanying the spin transition. The key important feature of these transmission images is the straight and narrow character of the borderlines which implies that the interface is planar and oriented perpendicular to the large faces of the crystal, as sketched in the schematic view of Fig. S1, in which the crystallographic axes of Fe(NCSe)with respect to the crystal habits are given. 30,51 On the other hand, the structural data of Fe(NCSe), listed in Table S1, in the LS and HS states, show an opposite variation of the length (b parameter) and the width (a0 parameter) along the spin transition. This strong structural anisotropy at the transition is identified as one of the most important ingredients in the non-conventional mechanical resilience of this system. This property seems to be general since, another example of robust SCO single crystal has been brought forth to light in Ref. 52

Light-control of the HS-LS interface orientation The experimental procedure was as follows: first, the crystal was prepared in the LS state, just below the thermal hysteresis loop (see Fig. 3a), at 108 K under a weak intensity of the microscope I ' 2.47 mW with the laser switched off. In the second step, we modulated 9



a

b

c

d

Figure 3: a) Thermal dependence of the HS fraction, nHS , of Fe(NCSe), derived from optical density analysis of the OM data of videos S1 and S2 of the SI. b) Snapshots of the HS-LS interface dynamics under modulated light excitation (wavelength λ = 690 nm), revealing the translation and rotation (from 60o to 121o ) of the central HS-LS interface. From top to bottom, the light intensity is maximum in the first image, then decreases in the second and third images, and then increases again in the last image. The intensity of the laser spot (diameter = 4 µm) is modulated at frequency, ν = 0.3 Hz. Notice the appearance of a second interface on the right tip causing the slight bending of the crystal. c) Time dependence of the light intensity I(t) (red line) and the interface position (blue curve). Remark the dephasing between the two signals. d) Time dependence of the interface HS/LS tilt angle, θ(t) (blue line) and the laser intensity I(t) (red curve).

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(at controlled frequency, ν = 0.3 Hz) the intensity (varying in the interval 52µm ≤ I ≤ 3.6 mW) of the laser spot of diameter Φ ' 4 µm and wavelength λ = 690nm, which was placed at the left tip of the crystal, in the LS phase (see Fig. 3b), and we followed the interface response by mean of optical microscopy. After 2.7 seconds of light excitation, a HS phase emerged as a result of photo-heating, which brings the temperature of the area impacted by the laser, beyond the upper transition temperature, ∼ 114.4 K reported in the thermal hysteresis of Fig. 3a. Then, a significant HS phase (light area in the images of Fig. 3b) appears from the left part of the crystal and propagates rapidly to the right with a well oriented domain wall. Concomitantly, a small HS domain also appears in the opposite tip only in the ascendant phase of the light intensity excitation. Hence, the total surface of the HS domain is slaved to the intensity modulation by increasing in the ascending branch of the light modulation due to already explained photo-heating effects. In the descending branch of the light modulation, the HS fraction decreases due to the efficient crystal cooling by the exchange gas (heat bath) which plays the role of a restoring force for the crystal temperature. We selected in Fig. 3b, a few snapshots of the crystal during continuous light illumination with a modulated amplitude at frequency 0.3 Hz with a maximum intensity of 3.6 mW. The complete spatiotemporal behavior is shown in the SI (movie S3). The pictures taken at times, 12.7 s, 13 s, 13.8 s, 15.7 s from top to bottom, reveal a clear rotation of the HS/LS domain wall from 120o to 60o during the translational motion. The reversible and reproducible character of the dynamics of the front re-orientation has been checked several times and can be visualized in movie S3. Figs. 3c and 3d display the time evolution of excitation I(t) (red line) with the response of the interface, in terms of position, X(t) and orientation (blue lines), respectively, obtained after image processing of movie S3, the details of which are given in Figs. S1 and S2 of the SM. Two points can be noticed. First, the HS-LS domain wall position, X(t), that linearly connects to the HS fraction (nHS ∝ X(t)) performs periodic oscillations at the same frequency as the light excitation, with a delay ∆t ' 1.3 s as we have already reported in a previous report

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a

b

c

d

Figure 4: (a) Time dependence of the position (in blue) of the second front which appeared (in Fig. 3 ) at the right side of the single crystal Fe(NCSe). The red curve is the excitation signal. (b) Temporal evolution of the crystal bending during the interface propagation. The maximum reached deflection is ∼ 1.5µm ± 0.4µm. In both previous figures, the red curve is the intensity of light excitation signal. c) Time dependence of the macroscopic crystal bending (in blue) and the reorientation of fronts showing two signals in opposite phases. d) Flow diagram θ vs Xbending showing the reorientation of interface 1 driven by the crystal bending. Here, the time is a hidden variable. A clear ”limit cycle” is obtained revealing the existence of a bistability. The blue arrows indicate the starting point. on the reversible photo-control of the interface position in Ref. 53 Second and original result, the dynamics of the front tilting, depicted in Fig. 3d, follows a periodic square-shaped signal with rapid switching (in less than 0.4 s) or pulses at the extreme positions of the front excursion. An interesting observation which can be drawn from the visualization of the movie S3 lies with the direct connection between the front re-orientation and the interplay 12


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between the dynamics of left and right front interfaces, which will be denoted hereafter front 1 and 2, respectively. Interestingly, this interplay is also at the origin of the macroscopic crystal bending, that we evaluate using a home-made image processing software. So that, we first determine the time dependence of front 2 position’s and crystal bending, shown in Fig. S5a and S5b of SM and Fig. 4a, where we estimate the maximum crystal’s deflection along the vertical direction at ∆L ∼ ± 1.5 µm, corresponding to relative bending rate of ∆L/L ∼ 1.6%. To correlate the crystal bending with the interplay between the interfaces 1 and 2, we plotted on the same panel (Fig. 4b and S5c of the SM), the time dependence of the distance, d12 , between the two fronts and the crystal deflection, Xbending . We see that the bending starts to appear when the two fronts are at their minimum distance from each other; a configuration at which the LS phase, caught between the two fronts, experiences a large strain. Then, due to their mutual interaction, the two fronts move away from each other, releasing the elastic energy stored in the LS domain, thus causing an enhancement of the deflection rate, which reaches its maximum value at the remarkable distance, d12 ' 326µm, for which front 1 changes its orientation angle from 120o to 60o , as depicted by Fig. 4c. To well characterize the dependence of the interface reorientation with the crystal bending, we plotted in Fig. 4d; the flow diagram of the system in the phase space (θ, Xbending ) where the time becomes a hidden variable. Remarkably, a limit cycle is revealed during the spatiotemporal process of the interface propagation. Although presenting fluctuations due to those of the intensity of the pixels combined with the vibrations of the OM images, the obtained curves clearly allow to identify a hysteresis loop, characteristic of the bistability of the system in this space of variables. In particular, at the two switching deflection values, Xbending ' 1.4 and 0.6µm, the angle of the front 1 changes abruptly between the values 60o and ∼ 120o . On the basis of these results, one can reasonably conclude that the orientation of the HS-LS interface of a spin-crossover single crystal might be tuned using an appropriate micro-mechanical control of the crystal deflection. To our best knowledge, this is the first evidence of an effect of the macroscopic deformation of a spin-crossover crystal on the stability

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of the orientation of its HS/LS interface, which then confirms the long-range nature of the interactions between SCO units. On the other hand, it is worth mentioning that in this experiment, performed at 108 K, the usual LIESST (Light-Induced Excited Spin State Trapping) 9 effect, which operates at very low-temperature, below ∼ 60 K, is completely negligible. 47

Anisotropic Electroelastic Model To provide a realistic picture of the studied phenomenon, we consider the extension to the anisotropic version of our previous isotropic electro-elastic model, 54,55 accounting for the volume change between the spin transition units, which are modeled as a set of two-states fictitious spins, so as to mimic the LS and HS states of the SCO molecules, which are linked by springs, as it is represented in the zoom of Fig 5. In this anisotropic 2D model, the unit cell lattice constants are denoted by a and b along (1, 0) and (0, 1) directions and their expansion/contraction during the spin transition will be anisotropic, which constitutes one of the most important ingredients entering in the model. The total Hamiltonian, which includes electronic and elastic contributions, is given by

H=

X i

1 ∆ − kB T ln g σi + Helas , 2

(1)

where σ is the fictitious spin whose eigenvalues −1 and +1 are respectively associated to the LS and HS states of the molecule, ∆ is the ligand field gap, g is the usual degeneracy ratio between the HS and LS states, originating from the difference of vibrational entropies of the HS and LS states, resulting in a temperature-dependent field, 56 T is the temperature and kB is the Boltzmann constant.

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Figure 5: Example of a spatial electronic configuration of the SCO lattice for an orientation angle θ of the interface between the HS (red) ans LS (blue) phases in the (a, b) plane. The zoomed area is the local view of the elastic lattice. The indexes, i, j and k run over the central, nearest-neighbor and next-nearest-neighbor atoms, respectively The second term, Helas , in (1) is the elastic contribution of the lattice, which writes,

Helas = Vnn (|~r|) + Vnnn (|~r|) XXA (rij − Rl (σi , σj ))2 = 2 l=x,y i,j XB + (rik − Rd (σi , σk ))2 . 2 i,k

(2)

Here, A and B are the elastic constants connecting nearest- (nn) and next-nearest neighbors (nnn) atoms, respectively, while rij and rik are the respective nn and nnn instantaneous distances between sites. The indexes i and j (resp. k) run over nn (resp. nnn) sites. For the sake of simplicity, the elastic constants are considered as independent on the direction. To take into account for the molecular volume change in the Hamiltonian, the equilibrium bond lengths between two nn nodes i and j, Rl (σi , σj ), are set depending on their spin states as follows; Rl (+1, +1) = RlHH , Rl (−1, −1) = RlLL and Rl (+1, −1) = Rl (−1, +1) = RlHL = (RlLL + RlHH )/2, where RlHH , RlHL and RlLL are the respective equilibrium distances between HS-HS (HH), HS-LS (HL) and LS-LS (LL) sites. The index, l, denotes here the spatial 15



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direction x or y, where the equilibrium distances are considered as different. Next, Rx and Ry will be denoted a and b, respectively, which leads to the following equilibrium distances aHH , aHL , aLL (resp. bHH , bHL , bLL ) along x (resp. y) direction. The anisotropy of the unit cell change upon spin transition is considered by adopting different equilibrium distances in directions x and y. We denote by ∆a = aHH − aLL and ∆b = bHH − bLL , the lattice constant misfit between the HS and the LS lattices along a and b directions, respectively. We assume that the nnn equilibrium distance, along the diagonals, depends only on the spin states of the linked sites i and k. The equilibrium nn and nnn distances are given as function of the connected spin states, by the following general formula

∆a (σi + σj ) , 4 ∆b b(σi , σj ) = bHL + (σi + σj ) , 4 p a2 (σi , σk ) + b2 (σi , σk ). d(σi , σk ) = a(σi , σj ) = aHL +

(3)

Where, d is the equilibrium distance along the diagonals. These expressions clearly show the intricate nature of the coupling between the electronic and the elastic structures of Hamiltonian (2), whose analytical resolution is out of reach.

Simulation set up and results Although thermal properties of Hamiltonian (1) deserve to be studied, here we focus our investigations on the stability of the HS/LS interface that systematically appears during the first-order spin transitions. In usual isotropic elastic models studied for SCO materials, the unit cell change at the transition is considered as isotropic (i.e. a = b and ∆a = ∆b). In the case of rectangular-shaped crystal, this leads to a HS/LS interface, perpendicular to the length of the crystal. 54,55,57 However, this does not meet the experimental results of optical

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microscopy shown in Fig. 3, which exhibit tilted HS/LS interfaces, with two symmetric angles, oriented at 60o and 120o with respect to the crystal length. So, this section is devoted to the study of the energetic stability of the interface orientation in order to highlight the interrelationships between the preferential orientation, θmin , of the HS/LS interface and the ∆b , which also corresponds to the ratio between the HS/LS anisotropy ratio parameter, ∆a lattice misfit along the x and y directions. The scheme of Fig. 6 illustrates the possible situations of anisotropic unit cell’s change, obtained by only varying the lattice parameter bHH , which means changing ∆b. Hence, three situations are reached, corresponding to: (i) a contraction along the y direction and an ∆b expansion along the x direction for < 0, and anisotropic expansions along both directions ∆a ∆b ∆b (i.e. ∆b > 0 and ∆a > 0) with,(ii) 0 < < 1 or (iii) > 1. ∆a ∆a

Figure 6: Schematic representation of the anisotropic unit cell change between the LS (blue ∆b dots) and HS (red dots) states for the different regions as function of the ratio . The ∆a ∆b particular value, ∆a = 1 corresponds to the isotropic case. It is worthy to notice that the possible contraction of the unit cell along one (here bdirection, ∆b < 0) and its expansion along another direction during the LS to HS transition has been observed experimentally in several systems. 47,58–60 Since, we are working in 2D, the ∆V ∆a ∆b ∆b b relative volume change = + is negative when < − (= −1). However, if one V a b ∆a a ∆V ∆a ∆b ∆c considers the third direction, the total volume change ( = + + ) accompanying V a b c the LS to HS transformation is usually positive, which then guaranties that the HS state will have a bigger volume than that of the LS state. In the current study, the simulations are done at very low-temperature and they share the same spin state which we did not allow to change, thus focusing this study exclusively on 17



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the mechanical behavior of the system, and their effect on the electronic organization of the system. For that, we used the following parameter values for the elastic constants, A = 2 ×105 K/nm2 , B = 0.7×A; the nn equilibrium distances are taken as aLL = bLL = 1 nm in the LS state, and the nn equilibrium distance in the HS state along the x-direction was fixed to aHH = 1.01nm. Within these values, the bulk modulus in the LS state ( =

A ) aLL

is estimated to ∼ 15 GPa,

which is in excellent agreement with available experimental results. 30 The nn equilibrium distance along the y-direction, bHH is utilized here as a parameter allowing to tune the ratio ∆b . ∆a Initially, we set the lattice of 192 × 49 sites (i.e., SCO molecules) with all nn distances equal to those of the HS state (i.e. aHH and bHH ) at equilibrium and the fictitious spin state for the molecules are those of the LS state (i.e. σ = −1), for the first half and HS (i.e. σ = +1) for the second half of the lattice. Between the two spin states phases (red and blue in Fig. 5), we impose a narrow spin interface of one molecule without noise, oriented towards a fixed angle, θ, between the interface and the x-direction (or the bottom border) as we show in the schematic view of Fig. 5. Thus, initially all elastic energy excess is stored in the LS phase whose mechanical relaxation will drive also that of the HS phase. The lattice positions are then relaxed at fixed electronic configuration using an over-damped molecular dynamics procedure so as to reach the minimum of total elastic energy. Fig. 7 summarizes the time-dependence of the total elastic energy during the relaxation process, showing stretched exponential profiles, for which ∼ 20000 time steps are necessary to reach the mechanical equilibrium. On the other hand the relaxed elastic energy (corresponding to the stable mechanical state) does not vanish (Emin ' 100 K) due to the existence of a lattice misfit at the interface between the HS and LS phases. It is legitimate to question the origin of the stretched exponential profile of the total elastic energy of Fig. 7. The existence of stretched relaxation usually indicates dynamic heterogeneity, a concept used in the study of glassy systems

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61–63

, related non-exponential

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relaxations are universal and can be understood as a continuous sum of exponential decays with a distribution of relaxation times or local energy barrier heights. In the present case, the heterogeneity of the energy barriers, first originates from the heterogeneity of the spatial distribution of initial spin states, compare Fig. 5. When performing molecular dynamics simulations, the stress relaxation will depend on space. Indeed, the time-evolution of the sites positions depend on their location in the lattice. For example, sites located the LS phase, in the HS phase, around the HS-LS interface, or at the surface will behave differently due to their different environments. This causes an inhomogeneous stress in the lattice, which ”renormalizes” the local energy barriers, leading to a distribution of the local relaxation times, which in addition interfere due to the elastic interactions. This results in the stretched exponential behavior of the relaxation process.

Figure 7: Temporal evolution of the total elastic energy of the lattice during its mechanical relaxation. Initially all nn distances between molecules are set equal to aHH and bHH along the x and y directions, respectively. The mechanic equilibrium is reached after ∼ 20000 time steps. Notice the nonzero final elastic energy, denoting that a strain does remain in the crystal, due to the presence of the HS/LS interface. In practice, we performed extensive deterministic molecular dynamics calculations using a code written in CUDA, 64 which allows to exploit the performance of Nvidia cores. The calculation of the lattice positions is made by classical mechanics 20,27,65 by solving the Newton ~ elas , with a time step, dt2 = 0.002 a.u. with a strong numerical viscosity equations, F~ = −∇H to avoid oscillations. This was implemented simply by imposing |~vi | = 0 for the speed of all sites after each step of the simulation.

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The mechanical relaxation procedure is repeated for all orientation values of the HS/LS interface, θ ∈ (20o : 160o ) with a step of 2o . To find out the preferred interface orientation, denoted θmin , we plot the relaxed elastic energy as function of θ, leading to the results of Fig. 8. This plot is representative of the angular dependence of the completely relaxed energy, for the frustration ratio

∆b ∆a

= −2.0. The energy landscape shows a double well

structure with stable positions obtained for θ ∼ 50o and ∼ 130o , whereas the right interface (perpendicular to the crystal length) for which θ ∼ 90o corresponds to an unstable state. It is worth mentioning that the symmetric character of this macroscopic elastic energy profile does not mean any symmetry between the HS and LS atomic states, which are on the contrary asymmetric through the ligand field contribution and the anisotropic transformation of the unit cell. On the other hand, the existence of two equi-energetic orientations of the interface, leads to the emergence of a new bistability, related to the switching of the interface orientation between the two values of θmin (50o and 130o ). This is particularly interesting because the energy barrier between the two minimums is only 10 K (within the used lattice size), which explains the origin of the well defined and stable interface orientations observed experimentally during the coexistence, of HS and LS phases, regime.

The previous parameters, θmin , and the energy barrier, ∆E (see Fig. 8), which separates the two minimums, strongly depend on the ratio the case of a positive anisotropic ratio,

∆b ∆a

∆b , ∆a

as shown in Fig. 9, where are reported

= 1.4 and the special isotropic case,

∆b ∆a

= 0. We

see that the anisotropic deformation of the unit cell between the LS and HS states, generates a tilted macroscopic HS/LS interface. Compared with the case of negative anisotropy given in Fig. 8, the energy barrier for

∆b ∆a

∆b ∆a

= −2,

= 1.4 is only 1 K, while the value of the total

elastic energy at the minimums is ∼ 60 K larger. In contrast, for ∆b/∆a = 0, the elastic energy is flat in the interval 60o < θ < 120o with a weak elastic energy, because the latter depends on (∆b)2 and (∆a)2 . We performed intensive calculations of configuration diagrams hEi as function of θ for

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Figure 8: Interface orientation-dependence of the relaxed elastic energy for the negative anisotropic ∆b/∆a = −2.0, showing a symmetric double well structure. The minimums of elastic energy are obtained for the angle values, θ ' 50o and 130o and the maximum for θ ' 90o . Inset: electronic configurations of the lattice showing the interface orientations, corresponding to the three extrema of the elastic energy.

Figure 9: Interface orientation dependence of the relaxed elastic energy for ∆b/∆a = 1.4 (top) and ∆b/∆a = 0.0 (bottom) showing the vanishing of the energy barrier in the isotropic case. various anisotropy ratio

∆b ∆a

values, from which we derived the results of Fig. 10, which

present the behavior of θmin and ∆E as function of

∆b . ∆a

The anisotropy ratio

of θmin shows the existence of three regions. In the region of −4

Unprecedented Bistability in Spin-Crossover Solids Based on the

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