Role of surface vibrational properties on cooperative phenomena in spin-crossover nanomaterials

Mirko Mikolasek, Gautier Félix, Gábor Molnár, Férial Terki, William Nicolazzi, and Azzedine Bousseksou

Phys. Rev. B 90, 075402 – Published 4 August 2014

Abstract

The influence of surface/interface on the lattice dynamics of spin crossover nanoparticles has been investigated by a spring-ball model solved by Monte Carlo methods. The bond cohesion energy of the model has been extracted from Mössbauer spectroscopy measurements performed on the model compound ${Ni}_{3} [Fe {(CN)}_{6}]$ . We show that the coupling between bulk and surface vibrational properties, which drastically affects the mechanical properties of the whole particle below a characteristic size, has a major impact on the phase stability of the particles. In the case of free surfaces, the Debye temperature decreases with the size and the first-order nature of the spin transition disappears. On the other hand, a hardening of the surface bonds leads to increasing particle stiffness with the size reduction. In this case, a persistence of the hysteretic behavior in the spin transition curve is also predicted in good agreement with previous theoretical and experimental results.

Received 6 June 2014
Revised 17 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.075402

Authors & Affiliations

Mirko Mikolasek^1,2, Gautier Félix¹, Gábor Molnár¹, Férial Terki², William Nicolazzi^1,*, and Azzedine Bousseksou^1,†

¹Laboratoire de Chimie de Coordination, CNRS UPR-8241 and Université de Toulouse, UPS, INP, Toulouse France
²Institut Charles Gerhardt, Université Montpellier 2, place Eugène Bataillon, Montpellier France

^*willliam.nicolazzi@lcc-toulouse.fr
^†azzedine.bousseksou@lcc-toulouse.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 90, Iss. 7 — 15 August 2014

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Experimental data (blue crosses) and numerical simulation (purple circles) of the thermal evolution of the Lamb-Mössbauer factor $f$ in a logarithmic representation. The straight green line corresponds to a linear fit of experimental data obtained on the compound ${Ni}_{3} [Fe {(CN)}_{6}]$ using the Debye model in the high-temperature limit (from Ref. [31]). MC simulation has been realized on a system with $N = 30^{3}$ sites and the temperature sweep has been performed with a heating rate of $5 \times 10^{- 4} K {MCS}^{- 1}$ (Monte Carlo step) in the warming and cooling modes, the initial system configuration being the final configuration of the previous temperature. The sizes of circles are directly related to the MC numerical errors.
Reuse & Permissions
Figure 2
(a)Spatial distribution of the local mean-square displacements $〈 u_{i}^{2} 〉$ for a section of the cubic system of size $N = 40^{3}$ with free surfaces. (b) Corresponding averaged profile of $〈 u_{i}^{2} 〉$ for different system sizes $(N = 5^{3},$ $10^{3},$ $20^{3},$ $30^{3}$ , and $40^{3})$ . The center of the cube is taken as the reference to identify the layers between two opposite surfaces. For (a) and (b), MC simulations have been performed at $T = 200$ K with a cohesion energy $ε = 6000$ K. (c) Schematic representation of the surface effects on the lattice dynamics at the nanometric scale where $λ$ is the characteristic surface thickness. When $L \approx L_{c}$ , the vibrational properties of the system are completely different from the corresponding bulk material $(L ≫ L_{c})$ . (Insert) Schematic representation of the path along which the profile of $〈 u_{i}^{2} 〉$ has been determined.
Reuse & Permissions
Figure 3
(a) Size dependence of the Debye temperature $θ_{D}$ for a cubic lattice (blue crosses) with $ε = 6000$ $K$ . For each size, $θ_{D}$ has been determined from the MC estimation of $〈 u^{2} 〉$ at a fixed temperature $T = 200$ $K$ . These data have been fitted by equation (10) linking $θ_{D}$ and the melting temperature $T_{m}$ (green straight line). The dotted red line corresponds to the Debye temperature in the thermodynamic limit $(L ≫ L_{c})$ . (b) Application to the spin transition: Monte Carlo simulation of the thermal evolution of the high spin fraction for a $N = 4^{3}$ cubic system with free surfaces (blue straight line). The SCO properties of the corresponding bulk material (green dotted line) is also shown for comparison. The cohesion energies $A (σ_{i}, σ_{j})$ (lattice parameters $r_{〈 i, j 〉}^{0}$ ) are $A (1, 1) = 6100$ $K$ $(r_{HS} = 5.15$ $Å)$ , $A (- 1, - 1) = 6200$ $K$ $(r_{LS} = 5.1$ $Å)$ , and $A (- 1, 1) = A (1, - 1) = 6000$ $K$ $(r_{HL} = r_{LH} = 5.125$ $Å)$ for the HS-HS, LS-LS, and HS-LS/LS-HS bonds, respectively. The coding heating rate is $4 \times 10^{- 4}$ $K {MCS}^{- 1}$ .
Reuse & Permissions
Figure 4
(a) Simulation of the size evolution of the Debye temperature $θ_{D}$ for a core-shell nanoparticle where surface bond energies $ε_{surf} = 18 000$ $K$ have been increased in comparison with the cohesion energy of the core, which is the same as in Fig. 3. (Insert) Schematic representation of the simulated core-shell system. (b) Application to the spin transition: Monte Carlo simulation of the thermal evolution of the high spin fraction for a core-shell cubic system of size $N = 4^{3}$ . The reopening of the hysteresis loop induced by the modification of surfaces is compared with the SC0 properties of the corresponding bulk material (green dotted line). Model and simulation parameters are the same as in Fig. 3 except the cohesion energies of HS-HS, LS-LS, and HS-LS (LS-HS) bonds at the surface, which have been increased [ $A (1, 1) = 18 100$ $K, A (- 1, - 1) = 18 200$ $K$ , and $A (- 1, 1) = A (1, - 1) = 18 000$ $K$ ], keeping the difference in stiffness between the HS and LS phases constant.
Reuse & Permissions
Figure 5
Monte Carlo simulation of the thermal evolution of the high spin fraction for a core-shell cubic system of size $N = 4^{3}$ (blue straight line). The large reopening of the hysteresis loop is induced by an increase of the surface cohesion energy by making the assumption that the cohesion energy changes in a different way for the HS and LS states. The hysteresis loop of the nanoparticle is compared with the hysteresis loop of the corresponding bulk material (green dotted line). Model and simulation parameters are the same as in Fig. 4 except the cohesion energies of HS-HS, LS-LS, and HS-LS (LS-HS) bonds at the surface [ $A (1, 1) = 18 150$ $K, A (- 1, - 1) = 18 300$ $K$ , and $A (- 1, 1) = A (1, - 1) = 18 000$ $K$ ].
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics