SPATIO-TEMPORAL ASPECTS OF THE DOMAIN PROPAGATION IN A SPIN-CROSSOVER LATTICE
WITH DEFECT
Rachid Traiche1,2, Hassane Oubouchou1,3, Mourad Zergoug4, and Kamel Boukheddaden1*.
1GEMAC, Université de Versailles Saint-Quentin-CNRS, Université Paris Saclay, 45 avenue des Etats-Unis, 78035 Versailles, France;
2Université Hassiba BenBouali-Chlef,, Hay Essalem Route National N 19, Chlef, 02000 Algerie,
3Laboratoire de physique des Matériaux, Faculté de Physique, Université des Sciences et de la Technologie Houari Boumedienne 16111, Bab ezzouar Alger ,Algeria.
4Centre Scientifique et Technique en Soudage et Controle, Division des Procédés Electrique et Magnétiques,16002 Route de Dely Ibrahim BP 64.
* Correspondence:kamel.boukheddaden@yahoo.fr, Tel.: +331-39-25-46-64
Abstract:We study the spatiotemporal formation and spreading of the high-spin state (HS) during the cooperative relaxation of the photo-induced metastable high spin (HS) state at low temperature of an elastic lattice, in the presence of a defect injected in the center of the lattice. For that, we designed a 2D rectangular-shaped lattice with discrete spins coupled by springs. The distances between the sites are spin-dependent which prevents any analytical resolution of the present problem. The elastic coupling between the spin-crossover (SCO) sites results in a long-range effective interactions between the spin states from which originates the complexity and the richness of this problem. The numerical resolution of the problem is performed using Monte Carlo simulations on the spin states and the atomic positions. The simulations are restricted to a lattice with a hole (simulating the defect) with a fixed size. The presence of the defect shows the dynamics of the spin-crossover transformations starts from one corner of the rectangular lattice and propagates along the width (shortest distance to the surface).Then a second regime of longitudinal propagation takes place, whose velocity slows down significantly in the vicinity of the defect. Then the interface leaves the defect, where it was pinned at low velocity and accelerates when it approaches the border of the latticeWe have also investigated the spatial dependence of the displacement field, from which we derived the spatial distribution of the divergence, which directly connects to the distribution of the internal strain and elastic energy.
Valuable information is derived from the time-dependence of the total elastic energy in particular around the defect.
Keywords: Spin Crossover; phase transition; interface propagation; defects; elasticity; Monte Carlo Simulations
1. Introduction
Spin-crossover solids (SCO) [1–3], which combines electronic, magnetic, mechanical and optical changes at the transition, are among the most studied switchable molecular solids from the theoretical point of view. The first models which have been proposed to give a simple view of the SCO phenomenon were based on two-level approaches, like regular solutions [4] or Ising-like [5,6]
approaches. Although they reproduced the main features of the SCO phenomenon, among which like the existence of the first-order transition and its dependence with the ligand field strength and
interaction parameters, they remained too qualitative, particularly on the true physical origin of physical mechanism of the SCO phenomenon. In these models, the first-order thermal transition is obtained through a phenomenological interaction parameter, electronic in nature. In addition, one of the most serious drawbacks lies with the independence of the transition temperature on the interaction parameter, even though the latter was a phenomenological one. In addition, these simple models discarded the volume change accompanying the spin transition, and so the effect of elastic stresses on the transition mechanism. At variance from these electronic two-state models, the continuous medium model developed by Spiering [7] showed that an elastic interaction could as well give rise to the observed first-order transitions. More recently, atomistic models based on deformable lattices [8–16] were introduced so as to mimic the spatiotemporal features of the SCO transition, revealed by optical microscopy investigations [17–19] on the study of the nucleation and growth mechanisms of SCO single crystals. These relevant models allowed to reproduce the main features of the spatio-temporal bahaviors od SCO solids, which can be summarized as follows: the transition from the high-spin to low-spin usually starts from a corner of the crystal; t and spreads over the whole crystal with a well-defined HS/LS interface whose shape depends on that of the crystal. In most of the cases, the interface propagates at a very slow velocity (≈2–10 μm/s), almost six-orders of magnitude smaller than that of sound. This slow velocity of transformation is sue to the anelastic character of the spin-transition which accompanied with a volume change at the macroscopic scale. Our previous optical microscopy investigations demonstrated the presence of stresses which strongly affect the transition temperature and may lead in some cases to irreversible damage of the crystals after several thermal cycling [19]. The experimental studies clearly underlined that the nucleation and growth process in a spin-crossover single crystal at the thermal transition is a multi-scale process driven by the propagation of mechanical stresses ahead of the transformation front-line. This specific character confirms the crucial role of the coupling between the spin state change and the local deformation (volume change) of the lattice. The latter delocalizes over long distances, leading to long-range interactions between the spin-states through lattice deformations. The electro-elastic model [20], that we solved for several types of crystal shapes [21–23] has proved its consistency in describing the interplay between the spin state propagation and the macroscopic lattice deformation.
The optical microscopy observations of the spin transition in single crystals allowed to suspect the presence of mechanical constraints that controls the spread of the HS / LS interface. For this purpose, the present work will be devoted to complete the study that we recently published in Ref. [28] on the spatiotemporal bahaviors of the HS and LS domains during relaxation of the photoinduced metastable HS state obtained after photoexcitation through LIESST (Light-Induced Excited Spin State Trapping) effect [24] at very low temperature. The study is restricted to a 2D rectangular lattice in the presence of a defect, which is in practice a hole, injected in center of the lattice. Here, we are mainly interested in (i) the effect of the hole on the HS/LS interface velocity during the relaxation processes ,(ii) the energetic aspects of this transformation at the presence of the defect and (iii) on the mechanical properties of this transformation through the analysis the divergence and the rotational of the displacement field which should clarify the role of the elastic strain in the pinning process of the HS/LS interface around the defect.
The paper is organized as follows: in Section 2, the model and simulation method are described;
section 3 is devoted to the presentation of the results of the simulations; ; in Section 4 we summarize the main conclusions and outline some possible developments of the present work.
2. The Model
To describe theoretically the SCO solids, we consider a model which couples electronic and elastic properties of a SCO lattice for which we have already discussed some of its thermodynamic properties in recent works [19, 21]. This model is based on the description of the HS and LS states of the SCO molecule by a two-states fictitious spins, with respective values = +1and = −1. The molecules (here, the sites) interact via elastic springs, as depicted in Figure 1.
Figure 1.Adapted from Ref. [15]. The electro-elastic configuration of a SCO molecule (red ball in the center) elastically-coupled to surrounding SCO molecules by four nearest-neighbors and four next-nearest- neighbors.
The elastic constant and the equilibrium distances between the sites, depend on the connected spin states, so that the total Hamiltonian of the system writes as:
= 1
2 [∆ − ln ( )] + − ( , ) +
( , )
[ − ( , )] (1)
( , )
The first term (1) is the energy gap separating the LS and HS states of an isolated molecule. It contains an energy contribution,, arising from the difference of ligand fields in the HS and LS states, and the entropic contribution,− ln resulting from the electro-vibrational degeneracy ratio,g, between the HS and LS states. The second and third contributions account for elastic interactions, between nearest- neihghbors (nn) and next-nearest-neighbors (nnn) of spin-crossover units, respectively.
The quantity
R
0( S
i, S
j)
(resp.R
0'( S
i, S
k)
stands for the equilibrium distance between the nn sitesi andj(respectively,iandk), connected by a spring, whose elastic constant isA(resp.B), while (resp.) is the instantaneous distance between the nn (resp. nnn) nodes and ( and ). Thus, the elastic energy contribution, ∑ , − , of Equation (1) describes the energy cost of the “bond length”,
r
ijr
ir
j
, when its value is different from its ideal equilibrium valueR
0( S
i, S
j)
.Let us denote by, , (= ) and , the respective nn distances between two HS-HS, HS-LS, and LS-LS neighboring SCO molecules. In addition, let us consider for simplicity that, = ( + )/2. The general expression of the equilibrium distance, , ,can be straightforwardly expressed as:
, = + + , (1a)
where:
= ( − )and = (2b)
Here, denotes for the lattice parameter misfit between the LS and HS lattices. According to these expressions, we then have: = (+1, +1), = (−1, −1), and = (+1, −1).
The main thermodynamic properties of this generalized Ising-like model accounting for elastic interactions between sites can be summarized as follows:
(i) the transition temperature, , is found as a temperature for which the average value of the effective field(∆ − ln )is equal to zero (< > = 0and< > = ) which gives:
= ∆
; (3)
(ii) the system undergoes a first-order transition from some critical value of the elastic energy . For a deeper analysis of the phase diagram of the present model, the reader can refer to [17].
Buoyed by these first results, we attack now the solution of Equation (1) using Monte Carlo (MC) simulations. All simulations presented in this work are obtained from the study of the general Equation (1). We have used the following realistic parameter values in our simulations:∆ = 450K for the ligand field energy, = 150, for the degeneracy ratio, = 1.2nm, = 1nm, = 1.1nm for the equilibrium nn distances (lattice parameters). The equilibrium distances of nnn distances have been taken equal to those of the nn multiplied by √2, owing to the adopted 2D square lattice symmetry. The elastic constant values are chosen equal to = = 23000K∙nm−2.
Within these parameter values, we have estimated the bulk modulus value, as ~6GPa, and the equilibrium temperature, ≈ 90K.
The present work is devoted to the study of nucleation and growth accompanying the spin transition processes during the relaxation of the photo-induced HS metastable state . These phenomena will be investigated in a 2D rectangular lattice in which we inject a structural defect (a hole). We aim, here, to study the effect of the presence of such a defect on the electro-elastic properties of the spin transition, for both equilibrium and non-equilibrium aspects.
3. Results and Discussion
We investigate the relaxation of the photo-induced HS metastable state at low temperatures of a planar rectangular lattice with = × = 60 × 20particles with a defect (a hole) in the center. We consider free boundary conditions in the simulations, and the calculations are performed atT= 10 K.
For so doing, we prepare all sites in the HS state, by initializing all spin values at = +1and all nn distances = = 1.2nm. We then perform usual MC simulations on the spin states and also on lattice positions to relax the total elastic energy. After each MC step, we measure the average value of the HS fraction and that of the average nn distances.
The MC procedure is performed on spins variables and atomic positions as follows: we chose randomly an atom and flip its spin state using the usual MC Metropolis algorithm. In any case (spin flip accepted or rejected), we visit randomly and sequentially the whole lattice sites and attempt to move each site by = 0.015nm in randomly chosen direction. If the new position is permitted by the usual Monte Carlo Metropolis algorithm, then we update the position of the atom; if it is forbidden, the atom is left in its original position. Then, we chose randomly a new site for which we perform the same procedure. The algorithm of relaxation of all lattice positions is repeated 10 times, for each spin flip, to reach the stable mechanical state. Next, we update each of the × spins, using the same procedure. We define the single MC step (MCS) as the time for which we have visited all the spins one time. So, in 1 MCS, each atomic positions has been updated 12,000 times. To reach the thermal equilibrium, the ”spin-lattice MCS”—was repeated 2.0 × 103times at each temperature. The first 103 times were used for reaching the equilibrium of the system, and the second 103MCS are used for the statistical analysis of the physical quantities of interest.
Figure 7, in which we presents some selected snapshots of the electro-elastic configuration of the lattice, summarizes the spatiotemporal features of the spin and structural transformations during the thermal relaxation of the lattice with defect published in [29]. One can easily see that the HS (red) to LS (blue) transformation starts from a corner of the lattice (snapshot B) with a curved interface. In the course of the process, the transformation spreads out over the whole lattice (snapshots C-F), with an arrow-shaped HS/LS interface, due to the presence of the defect. Indeed, oldest simulations with a perfect lattice showed regular and straight interfaces in this region. Finally a second new nucleation site appears in the top right corner of the lattice (snapshot B) which complicates the final steps of the relaxation.
Figure 1 After [29].Spatiotemporal configurations of the spin and lattice transformations during the HS to LS relaxation. Red points denote HS sites and blue dots correspond to LS sites. Remark the particular shape of the HS/LS interface, affected by the presence of the defect. See text for more explanations about the mode of propagation and the role of the defect on it. From A to H, Monte Carlo times were:
0,1944,2986,6709,11529,12949,13580,16451, MCS
To estimate the interface velocity, we determined elastic profiles of the interface along a reference line in the direction of propagation. The elastic, interface profile is obtained by plotting the spatial dependence, along the interface propagation direction (here, axis), of the distance,〈 〉, e between two successive sites, located along the reference line = /2. We illustrate in FIG. 2 some spatial profiles corresponding to the interface propagation of Figure 1 at several selected times. Far from the hole, the interface is sharp and its velocity (see FIG 3) is constant. When approaching the defect region, which can be considered here as an additional surface at which the system can relax the elastic energy excess, we do not see any significant effect on the curves of
< > ( )profiles, which globally keep the same shape, except at = 3241MCS, where the HS / LS interface crosses the hole’s region.
On the other hand, one can notice in FIG. 2 the presence on the left (resp. right) side of the elastic interface, of a region where < >< (resp. < >> ) which clearly indicate the specific nature of the interface which stores the stress due to the misfit of lattice parameter between the LS and the HS states. Far from the interface, this misfit becomes zero and the system recovers the equilibrium distances.
diffe The elastic profiles are also used to determine the instantaneous positions of the interface on the lattice, by following the coordinates of their centers, during the propagation process, which allows a reliable estimate of the interface velocity
.
The positions of the interface of FIG. 1 as function of the Monte Carlo time are reported in FIG 3.
We note the presence of five regimes of propagation. At the beginning, the growth starts from one corner of the rectangle along the width; this regime, which is quite short and was not depicted in FIG 1, is the regime of the interface creation. So, we denote as the ) first regime (1), the flow regime of longitudinal propagation at almost constant velocity = 0.01nm/MCS. Then a second regime (2) where the propagation significantly slows down, takes place when the HS/LS interface’s gets the hole. There, the velocity is nonzero but very small = 0.00025nm/MCS, and one can say that the interface can be considered as stapled around the hole. This phenomenon takes place at ~3000MCS and ends at = 7000MCS. The third regime (3) corresponds to the time region where the interface comes off from the hole, at relatively low velocity = 0.001,
nm/MCS. Finally, the last regime (4) corresponds to rapid motion of the interface, = 0.018 nm/MCS, when the elastic interface “feels” the surface border.
Figure 2: Elastic interface profiles along the propagation direction ( axis) for the atomic line = /2, plotted for different MC times during the relaxation process of FIG. 1.See text for more explanations..
Figure 3: Time dependence of the HS:LS interface position along the propagation direction ( axis) for the reference atomic line = /2showing four er regions of propagation.. The first
regime (blue line) corresponds to the interface propagation far from the defect. Around the defect, the interface is fixed to the hole, and its velocity is extremely small. It comes off in the
third regime. The last regime is driven by the presence of the lattice border.
Now, we discus the energetic aspects of this transformation. in figure 4 we show the Map of local elastic energy at several positions along the relaxation of figure1 , This energy is given by :
= ∑( , ) − ( , ) + ∑( , ) [ − ( , )] (4)
This Map explains well the two peaks obtained in the diagram of the total elastic energy stiring in the lattice of the figure 8 publiched in [29].this is due to the intersection of two interfaces, this interaction of two interfaces creates an energy excess in the lattice, which is observed in snapshots C and G.
FIG. 4: Maps of the local elastic energy at several positions along the relaxation of Fig.1. The elastic energy is mostly stored in the interface region.
Figure 8.[28]Time dependence of the total elastic energy stored in the lattice during the HS to LS relaxation of the HS fractions of Figure 6. The blue curve stands for the perfect lattice and red curve for that of the lattice with a defect. In the middle region, 4000–10,000 MCS, corresponding to the longitudinal propagation of the interface, the presence of the hole decreases the total strain in the lattice. Parameter values are the same as in Figure 6. See text for more explanations[28].
We also have studied the distribution of the local pressure, calculated according to equation (5), during the relaxation and the results are shown in Figure 1. This is written as:
= − ∑ − , − ∑ [ − ( , )]
(5)
The Map of local pressure. we can observe the location of regions under positive stress Indeed, when the system enters the HS to LS state, we observe a positive pressure distribution , in HS/LS interface and around the hole when the interface crossed it . Certainly due to local contraction.
Figure4 : FIG. 4: Maps of the local pressure at several positions along the relaxation of Fig.1.
Study of the displacement field
To reveal the lattice strain in the relaxation process, we focus on the study of the displacement field of lattice sites AH the configurations of Figure 1. We denote by ( , )the displacement field associated with the site coordinates (i, j) and defined by:
( , ) = ( , ) − ( , ) (6)
were
( , )and
( , )are the initial and final positions of the site vectors (i, j) in the lattice, in thisanalysis we take the positions of the HS state equilibrium as reference positions.
( , ) = ( , )
(7)
To reveal the lattice strain, we calculate the derivatives of the various components of the displacement field. For example, the derivative of the component along x is written as follows in the approximation of continuous medium:
=
( , ) ( , )(8)
, , Are calculated in the same way. These terms provide access to the strain tensor defined by:
= + (9)
Or α and β correspond to x and y respectively. The composition of the various elements of the strain tensor elucidates the mechanical effects induced by the growth areas. For example, the components provide information on the amount of relative change while components and are related to shear stresses. In
particular, the divergence and rotational displacement field are interesting to study quantities and expressions are given by:
[ ( )] = ( ) + ( )and [ ( )] = ( ) − ( )
(10)
The divergence of the displacement field is the trace of the strain tensor describes the relative volume expansion (here Surface because of the two-dimensional nature of the lattice). The rotational displacement field describes meanwhile, stresses shear which cause the distortion of the lattice during the transition.
Figures 6 and 7, shows the divergence and the rotational of the displacement field during the relaxation of the curve of figure 1, the first figure, the expansion is zero at the HS state (because it is the reference state) and becomes negative when the BS domains appear more precisely around the hole (snapshot C and D) which corresponds to the contraction of lattice around the hole when the interface HS / S crossed it. the second figure, can locate the most fragile regions of the lattice, mainly at the intersection between the HS / BS interface and the edges of the lattice, and around the hole (snapshot C and D) which explains that this region may be the potential center to initiate dislocations and / or cracks during the transition as observed by optical microscopy.
FIG. 16 (Color online) Maps of the divergence of the displacement at deferent positions along the relaxationof Fig. 1. The reference state is the HS phase, thus leading to a negative value of the
divergence, which reaches its maximum value in the saturated LS state.
FIG. 7: Maps of the rotational of the displacement field at several positions along the relaxation of Fig.1. The maximum values are observed at the edges of the system and/or at the interfaces between the LS and the HS domains, as depicted in the images A,B,C,D,E,FG,H. They outline the
presence of shear strains.
4. Conclusions
The effect of the presence of a defect in the center of the SCO lattice has been studied on an elastic spin-distortion model using the Monte Carlo method. The model accounts for the spin state and volume changes accompanying the spin transition. On the other hand, the study of the relaxation properties of the lattice with defect demonstrated that the defect plays the role of the pinning site for the front interface. Indeed, the results showed that the interface propagation is significantly altered around the defect, which then can be used as a way to stabilize or to control [26,27] the dynamic HS/LS elastic interface. And the region around the defect may be the potential center to initiate dislocations and / or cracks during the transition as observed by optical microscopy.
Further aspects related to the defect problem in SCO solids remain to be explored: the role of the size of the defect and its position in the lattice are important issues, which should be explored in the future.
Abbreviations
The following abbreviations are used in this manuscript:
SCO: Spin crossover MCS: Monte Carlo Steps MC: Monte Carlo HS: High-Spin LS: Low-Spin References
1. König,E.; Kanellakopulos, B. Mössbauer effect and magnetism down to 1.2 K in the triplet ground state of an iron(II)—Phenanthroline complex.Chem. Phys. Lett.1972,12, 485–488.
2. Gütlich, P. Spin Crossover in iron(II)-complexes.Struct. Bond.1981,44, 83–195.
3. Kahn, O; Launay, J.P. Molecular bistability: An overview.Chemtronics, 1988; Volume 3, p. 140.
4. Slichter, C.P.; Drickamer, H.G. Pressure-induced electronic changes in compounds of iron. J. Chem.
Phys.1972,56, 2142–2160.
5. Wajnflasz, J.; Pick, R. Transitions «Low spin»-«High spin» dans les complexes de Fe2+.J. Phys.1971,32, 91–92.
6. Doniach, S. Thermodynamic fluctuations in phospholipid bilayers. J. Chem. Phys.1978, 68, doi:10.1063/1.435647.
7. Spiering, H.; Willenbacher, N. The elastic interaction of high-spin and low-spin complex molecules in spin-crossover compounds.J. Phys. Condens. Matter1989,1, doi:10.1088/0953-8984/1/50/011.
8. Miyashita, S.; Konishi, Y.; Nishino, M.; Tokoro, H.; Rikvold, P.A. Realization of the mean-field universality class in spin-crossover materials.Phys. Rev. B2008,77, doi:10.1103/PhysRevB.77.014105.
9. Nishino, M.; Boukheddaden, K.; Konishi, Y.; Miyashita, S. Simple two-dimensional model for the elastic origin of cooperativity among spin states of spin-crossover complexes. Phys. Rev. Lett.2007, 98, doi:10.1103/PhysRevLett.98.247203.
10. Nishino, M.; Enachescu, C.; Miyashita, S.; Boukheddaden, K.; Varret, F. Intrinsic effects of the boundary condition on switching processes in effective long-range interactions originating from local structural change.Phys. Rev. BRapid Comm.2010,82, doi:10.1103/PhysRevB.82.020409.
11. Stoleriu, L.; Chakraborty, P.; Hauser, A.; Stancu, A.; Enachescu, C. Thermal hysteresis in spin-crossover compounds studied within the mechanoelastic model and its potential application to nanoparticles.
Phys. Rev. B2011,84, doi:10.1103/PhysRevB.84.134102.
12. Enachescu, C.; Stoleriu, L.; Stancu, A.; Hauser, A.Model for Elastic Relaxation Phenomena in Finite 2D Hexagonal Molecular Lattices.Phys. Rev. Lett.2009,102, doi:10.1103/PhysRevLett.102.257204.
13. Nicolazzi, W.; Pillet, S.; Lecomte, C.Two-variable anharmonic model for spin-crossover solids: A like- spin domains interpretation.Phys. Rev. B2008,78, doi:10.1103/PhysRevB.78.174401.
14. Nishino, M.; Enachescu, C.; Miyashita, S.; Rikvold, P.A.; Boukheddaden, K.; Varret, F. Macroscopic nucleation phenomena in continuum media with long-range interactions, Sci. Rep. Nat. PG2011, 1, doi:10.1038/srep00162.
15. Klinduhov, N.; Sy, M.; Boukheddaden, K. Vibronic theory of ultrafast intersystem crossing dynamics in a single spin-crossover molecule at finite temperature beyond the born-oppenheimer approximation.J.
Phys. Chem. Lett.2016,7, 722–727.
16. Espejo-Parez, M.; Sy, M.; Boukheddaden, K. Elastic frustration causing two-step and multi-step transitions in spin-crossover solids: Emergence of complex antiferroelastic structures. J. Am. Chem.
Soc.2016, doi:10.1021/jacs.6b00049.
17. Varret, F.; Slimani, A.; Boukheddaden, K.; Chong, C.; Mishra, H.; Collet, E; Haasnoot, J.; Pillet, S. The propagation of the thermal spin transition of [Fe(btr)2(NCS)2]·H2O single crystals, observed by optical microscopy.New J. Chem.2011,35, 2333–2340.
18. Chong, C.;Mishra, H.; Boukheddaden, K.; Denise, S; Bouchez, G.; Collet, E.; Ameline, J.C.; Naik, A.D.;Garcia, Y.; Varret, F.Electronic and structural aspects of spin transitions observed by optical microscopy. the case of [Fe(ptz)6](BF4)2.J. Phys. Chem. B2010,114, 1975–1984.
19. Slimani, A.; Varret, F.; Boukheddaden, K.; Chong, C.; Mishra, H.; Haasnoot, J.; Pillet, S. Visualization and quantitative analysis of spatiotemporal behavior in a first-order thermal spin transition: A stress- driven multiscale process.Phys. Rev. B2011,84, doi:10.1103/PhysRevB.84.094442.
20. Slimani, A.; Boukheddaden, K.; Varret, F.; Oubouchou, H.; Nishino, M.; Miyashita, S. Velocity of the high-spin low-spin interface inside the thermal hysteresis loop of a spin-crossover crystal, via photothermal control of the interface motion. Phys. Rev. Lett.2013,110, doi:10.1103/PhysRevLett.110.087208.
21. Slimani, A.; Boukheddaden, K.; Yamashita, K. Thermal spin transition of circularly shaped nanoparticles in a core-shell structure investigated with an electroelastic model. Phys. Rev. B2014, 89, doi:10.1103/PhysRevB.89.214109.
22. Slimani, A.; Boukheddaden, K.; Yamashita, K. Effect of intermolecular interactions on the nucleation, growth, and propagation of like-spin domains in spin-crossover materials. Phys. Rev. B2015, 92, doi:10.1103/PhysRevB.92.014111.
23. Boukhedadden, K.; Slimani, A.; Sy, M.; Varret, F.; Oubouchou, H.; Traiche,R. Physical properties of 2D spin-crossover solids from an electroelastic description: Effect of shape, size, and spin-distortion interactions. InMagnetic Structures of 2D and 3D Nanoparticles; Properties and Applications; Serge Levy, J.-C., Ed.; Pan Stanford Publishing: Berlin, Germany, 2016.
24. Decurtins, S.; Gutlich, P.; Kohler, C.P.; Spiering, H.; Hauser, A. Light-induced excited spin state trapping in a transition-metal complex—The hexa-1-propyltetrazole-iron(II) tetrafluoroborate spin- crossover system.Chem. Phys. Lett.1984,105, 1–4.
25. Bousseksou, A.; Nasser, J.; Linares, J.; Boukheddaden, K.; Varret, F. Ising-like model for the 2-step spin- crossover.J. Phys. I1992,2, 1381–1403.
26. Bedoui, S.; Lopes, M.; Nicolazzi, W.; Bonnet, S.; Zheng, S.; Molnar, G.; Bousseksou, A. Triggering a phase transition by a spatially localized laser pulse: Role of the strain. Phys. Rev. Lett.2012, 109, doi:10.1103/PhysRevLett.109.135702.
27. Sy, M.; Garrot, D.; Slimani, A.; Paez-Espejo, M.A.; Varret, F.; Boukheddaden, K.Angew. Chem. Int. Ed.
Enql.2016,128, 1787–1791.
28. Boukheddaden, K; Traiche.R; Oubouchou,H; Linares, J; Magnetochemistry 2016, 2(1), 17
