On the Competition Between Relaxation and Photoexcitations in Spin Crossover Solids under Continuous Irradiation

Abstract

We report recent work on the competition between opposite kinetic processes: (i) photo-excitation (LIESST) and relaxation processes, and (ii) direct and reverse LIESST. The light-induced instability eventually occurring at the light-induced equilibrium temperature is attributed to a cooperative origin. The subsequent light-induced thermal hysteresis (LITH) and intensity threshold effect (LIOH) are adequately described through a mean-field macroscopic master equation assuming a linear photo-excitation term and a self-accelerated term for cooperative relaxation. Further analysis provides evidence for non-linear character of the photo-excitation terms, capable of inducing bistability in the light-driven quasi-static regime when the direct and reverse LIESST regimes compete. The behaviour of correlations under permanent photo-irradiation is also considered, both experimentally and theoretically, through a kinetic Ising-model including a photo-excitation term and accounting for both long- and short-range interactions. Paradoxical effects of light are reported, with various experimental features and theoretical model providing a qualitative explanation.

A novel instability effect is introduced, due to the non-linear competition between direct and reverse LIESST, for which the expected hysteresis with respect to wavelength is denoted Light Induced Spectral Hysteresis (LISH).

Appendix. The dynamic Ising model in the presence of photo-excitation

1.1 Introduction

The aim of this section is to present the Ising-like model which is very often used to describe the static and dynamic properties of the spin-crossover (SC) systems. We are interested here in these properties under photo-excitation. In particular we will study the photo-excitation effect on the relaxation curve of the HS fraction in order to analyze their effect on the developments of the correlations during the relaxation.

1.2 Ising-Like Model

The microscopic models developed for cooperative spin-crossover solids are based on the Ising-like hamiltonian, following the pioneering works of Wajnsflasz and Pick and Bousseksou et al. [41a, 41b]. Such a two-state model can be viewed as a simple Ising model under a temperature-dependant ‘‘effective’’ field which accounts for the different degeneracies of the levels [44].

In the Ising like model, the two states associated with the eigenstates of the fictitious spin ±1, have different degeneracies, denoted respectively, g₊ and g_- . In the spin crossover systems, the eigenvalues +1 and −1 of the fictitious spin correspond to the high-spin (HS) and low-spin (LS) molecular states respectively.

The Ising-like hamiltonian including long- and short-range interactions [59] writes:

$$ H = - J{\sum\limits_{{\left\{ {i,j} \right\}}} {s_{i} s_{j} - \Delta _{{eff}} {\sum\limits_i {s_{i} } }} } $$

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where: Δ _eff =(1/2) k _B T ln(g ₊ /g ₋ ) −Δ+G <s> is the effective field, <s>= m is the net fictitious magnetization, J and G are the short- and long-range interactions associated with short- and long-range elastic effects [60] respectively. 2Δ is the energy difference E(HS)-E(LS) for isolated molecules. g ₊ /g ₋ is the degeneracy ratio between the HS and LS states and T the temperature. The ratio g ₊ /g ₋ may be quite large (up to a few thousands) because it involves both the spin degeneracies and the density of vibrational levels [61] in the two spin states.

The static properties of this model in the case J>0 and J<0 have been studied analytically in [59]: thermal hysteresis loops with simple and double transitions can occur, due to the competing effect of short-range ‘‘ferro-’’ or ‘‘anti-ferromagnetic’’ interaction and long-range ‘‘ferromagnetic’’ interactions. The phase diagram of this model has been obtained in [59, 41a], where the conditions of the occurrence of the first order transitions of the Ising-like model have been analyzed.

1.3 Master Equation

Now, we are interested in the dynamical properties of such cooperative SC systems. So as to do it, we use the well known general stochastic formalism developed by Glauber [62]. In this stochastic approach, the spin-flips s _i →−s _i are induced by the thermal bath, with transition rates W(s _i ).

Following Glauber we consider P({s};t) the probability of observing the system in the configuration (s ₁ , ..., s _N )={s} at time t. The time evolution P({s};t) is given by the master equation:

$$ \frac{{\partial P{\left( {{\left\{ s \right\}};t} \right)}}} {{\partial t}} = - {\sum\limits_{j = 1}^N {W_{j} {\left( {s_{j} } \right)}P{\left( {{\left\{ s \right\}}_{j} ,s_{j} ;t} \right)}} } + {\sum\limits_{j = 1}^N {W_{j} {\left( { - s_{j} } \right)}P{\left( {{\left\{ s \right\}}_{j} , - s_{j} ;t} \right)}} } $$

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In the last formula, {s}_j denotes the configuration of all spins excepted spin s _j and the expectation value of the j-th spin is defined as: ${\left\langle {s_{j} } \right\rangle } = {\mathop \Sigma \limits_{{\left\{ s \right\}}} }s_{j} P{\left( {{\left\{ s \right\}};t} \right)},$, where the sum is taken over all spin configurations.

The detailed balance condition at equilibrium writes as:

$$ \frac{{W_{i} (s_{i} )}} {{W_{i} ( - s_{i} )}} = \frac{{P_{e} {\left( {{\left\{ s \right\}}_{i} , - s_{i} } \right)}}} {{P_{e} {\left( {{\left\{ s \right\}}_{i} ,s_{i} } \right)}}} = \frac{{e^{{ - gs_{i} {\sum\limits_\alpha {s_{{i + \alpha }} - bs_{i} } }}} }} {{e^{{gs_{i} {\sum\limits_\alpha {s_{{i + \alpha }} + bsi} }}} }} $$

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where P _e (s ₁ ,s ₂ , ...s _N ) ~ exp[-β E(s ₁ ,...,s _i , ...,s _N ) ] is the equilibrium probability of finding the system with the energy E(s ₁ ,...,s _i , ...,s _N ), i.e. in the spin configuration (s ₁ ,s ₂ , ...s _N ).

1.4 The dynamic choice

Several dynamic choices leading to the same equilibrium states are possible according to Eq. 13 which only provides the ratio of the probabilities of opposite transition rates. We have established recently [4] that the choice suited to the spin-crossover systems (above the tunneling regime) was of the Arrhenius-type [63], because the dynamical process in these systems is thermally activated over an intra- and/or inter-molecular energy barrier (see Fig. 18), and this fact is strongly correlated to the sigmoidal character of the relaxation curves in these systems.

Fig 18

Schematic representation of the potential wells of the (HS) and (LS) states of a spin crossover system. E _H and E _L denote their respective energies, and E _a ⁰ is the intramolecular energy barrier due to the vibronic interaction

Let’s assume that E _a ⁰ is the energy barrier corresponding to the saddle point energy of the double well configurational energy diagram of Fig. 18 when the HS and the LS fractions are equal, i.e. at equilibrium temperature. Therefore, we can re-write the Eq. 13 under the following form, which obeys the detailed balance equation:

$$ \frac{{W^{{{\text{Therm}}}}_{i} (s_{i} )}} {{W^{{{\text{Therm}}}}_{i} ( - s_{i} )}} = \frac{{e^{{ - \beta {\left( {E^{0}_{a} - E{\left( {s_{i} } \right)}} \right)}}} }} {{e^{{ - \beta {\left( {E^{0}_{a} - E{\left( { - s_{i} } \right)}} \right)}}} }} $$

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where βE(s _i ) = - g s _i Σ _α=1,q s _i+α - b s _i , with g=βJ, b=βΔ _eff , and α runs over the neighbours. We choose for the transition rate the general form W ^Therm (s _i ) ~ exp[-β (E _a ⁰ – E(s _i )], which can be re-written as:

$$ W^{{{\text{Therm}}}}_{i} {\left( {s_{i} } \right)} = \frac{1} {{2\tau }}{\left( {x - x's_{i} } \right)}{\prod\limits_{\alpha = 1}^q {{\left( {y - y's_{{i + \alpha }} } \right)}} } $$

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with the following notations: x= coshb, y = coshg, x’ = sinhb, y’ = sinhg.

Under photo-excitation, which is assumed to induce only the LS→HS transitions, we must introduce an additional optical transition rate W ^Opt given by:

$$ W^{{{\text{Opt}}}}_{i} {\left( {s_{i} } \right)} = I_{0} \sigma {\left( {1 - s_{i} } \right)} $$

(16)

where I ₀ is the intensity of the incident radiation and σ is the absorption cross-section, related to the quantum photo-process. σ is considered as independent of the lattice configuration {s}. Photo-excitation is considered here as a non-cooperative process, since it is written as a single site term. The exact formulation of the dynamic equations leads to the following evolution equations for the fictitious magnetization and the equal-time correlation:

$$ \frac{{{\text{d}}{\left\langle {s_{i} } \right\rangle }}} {{{\text{d}}t}} = \frac{{{\text{d}}m_{i} }} {{{\text{d}}t}} = - 2{\left\langle {s_{i} {\left[ {W^{{{\text{Therm}}}}_{i} {\left( {s_{i} } \right)} + W^{{{\text{Opt}}}}_{i} {\left( {s_{i} } \right)}} \right]}} \right\rangle }_{t} $$

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and

$$ \frac{{{\text{d}}{\left\langle {s_{i} s{}_{j}} \right\rangle }}} {{{\text{d}}t}} = \frac{{{\text{d}}r_{{ij}} }} {{{\text{d}}t}} = - 4{\left\langle {s_{i} s_{j} {\left[ {W^{{{\text{Therm}}}}_{i} {\left( {s_{i} ,s_{j} } \right)} + W^{{{\text{Opt}}}}_{i} {\left( {s_{i} ,s_{j} } \right)}} \right]}} \right\rangle }_{t} $$

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The right-hand sides of Eqs. 17, 18 involve the average of clusters of spins. In the present work, we are interested by the correlations effect during the transition, therefore we perform all calculations in the pair approximation. Indeed, in that case our approach represents the dynamical extension of the well known Bethe-Peierls approach of the equilibrium statistical mechanics. We now have to choose an explicit form for the probabilities P({s},t) as a function of the order parameters of the model and the spin variables (s ₁ , s ₂ ,...,s _N ) ={s}.

1.5 Pair Approximation

In the pair approximation, the probabilities (or the density matrix operators) P ₁ (s _j ; t) for the single spin s _j , and the pair probability P ₂ (s _i ; s _j ; t) associated with a pair of neighbouring spins s _i ; s _j ; are given by:

$$ P_{1} {\left( {s_{i} ;t} \right)} = \frac{1} {2}{\left( {1 + m_{i} {\left( t \right)}s_{i} } \right)} $$

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and

$$ P_{2} {\left( {s_{i} ,s_{j} ;t} \right)} = \frac{1} {4}{\left( {1 + m_{i} {\left( t \right)}s_{i} + m_{j} {\left( t \right)}s_{j} + r_{{ij}} {\left( t \right)}s_{i} s_{j} } \right)} $$

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with m _i = <s _i > and r _i,j = <s _i s _j >.

We now have to examine the probabilities of occupation of the configuration of clusters of q+1 spins, which are formed by the central spin surrounded by its q neighbours. The probability P _q+1 (s _i , {s _i+α }; t), of such a cluster is approximated by using an elegant formulation due to Mamada and Takano [64], in which the authors considered P _q+1 (s _i , {s _i+α }; t) ≈ P ₁ (s _i ,; t) ×...× P (s _i+α ,∣s _j ; t) ×...× P (s _i+qα ,∣s _j ; t), where P (s _i ∣s _j ; t) is the conditional probability at time t of s _j at fixed value of spin s _i . Using the identity P ₂ (s _i , s _j ; t) =P ₁ (s _j ; t) × P(s _j ∣s _i ; t), we obtain:

$$ P_{{q + 1}} {\left( {s_{1} ,{\left\{ {s_{{i + \alpha }} } \right\}};t} \right)} = P_{1} {\left( {s_{i} ;t} \right)}{\mathop \Pi \limits_{\alpha = 1}^q }\frac{{P_{2} {\left( {s_{i} ,s_{{i + \alpha }} ;t} \right)}}} {{P_{1} {\left( {s_{i} ;t} \right)}}} $$

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where the subscript α runs over the neighbouring spins of s _i . Inserting Eq. 21 in the right-hand side of Eqs. 17, 18, we obtain the closed set of motion equations of the system, presented in the next section.

1.6 Relaxation under photo-excitation

For this first attempt, the lattice is assumed to be spatially invariant; then we put m _i = m and r _ij = r _{| i-j|} = r. Substituting now the probability of flipping site i, W _i ^Therm (s _i ), W _i ^Opt (s _i ) and P _q+1 (s _i , {s _i+α }; t) for their corresponding expressions (15–21), we obtain, after some calculations, the following evolution equations for the long- and short-range order parameters, respectively:

$$ \begin{aligned} 2\tau \frac{{{\text{d}}m}} {{{\text{d}}t}} = {\left( {x + x'} \right)}{\left( {1 - m} \right)}{\left( {y + y'\frac{{m - r}} {{1 - m}}} \right)}^{q} - {\left( {x - x'} \right)}{\left( {1 + m} \right)}{\left( {y - y'\frac{{m + r}} {{1 + m}}} \right)}^{q} & \\ {\text{ }} + {\text{2I}}_{{\text{o}}} \omega {\left( {1 - m} \right)} & \\ \end{aligned} $$

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and

$$ \begin{aligned} \tau \frac{{{\text{d}}r}} {{{\text{d}}t}} &= {\left( {x + x'} \right)}{\left( {1 - m} \right)}{\left( {y\frac{{m - r}} {{1 - m}} + y'} \right)}^{{}} {\left( {y + y'\frac{{m - r}} {{1 - m}}} \right)}^{{q - 1}} \\ & - {\left( {x - x'} \right)}{\left( {1 + m} \right)}{\left( {y\frac{{m + r}} {{1 + m}} - y'} \right)}{\left( {y - y'\frac{{m + r}} {{1 + m}}} \right)}^{{q - 1}} + {\text{4}}I_{{\text{o}}} \omega {\left( {m - r} \right)} \\ \end{aligned} $$

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with (x+x’) = exp(b) and (x-x’) = exp(-b), where b, y and y’ are given by: b = β [Gm+(1/2)kT ln (g+/g-) - Δ], y = coshβJ, and y’ = sinhβJ.

For convenience, it is useful to re-express the latter equations in terms of the fractions of the high spin molecules (n _H) and the pairs HS-LS molecules (n _HL), respectively n _H (t) = (1+m(t))/2 and n _HL (t)=(1-r(t))/4 [59]. They give, in the low-temperature region in which we are looking for the relaxation of the HS fraction, the following non-linear equations:

$$ \begin{aligned} \frac{{{\text{d}}n_{{{\text{HS}}}} }} {{{\text{d}}t}} = - \frac{{n_{{{\text{HS}}}} }} {{2\tau _{o} }}e^{{ - {\left( {\beta E^{o}_{a} + b} \right)}}} {\text{ }}{\left( {e^{{ - \beta J}} + 2\sinh \beta J\frac{{n_{{{\text{HL}}}} }} {{n_{{{\text{HS}}}} }}} \right)}^{q} + {\text{2I}}_{{\text{o}}} \omega {\left( {1 - n_{{{\text{HS}}}} } \right)} & \\ {\text{ }} & \\ \end{aligned} $$

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and

$$ \begin{aligned} & \frac{{{\text{d}}n_{{{\text{HL}}}} }} {{{\text{d}}t}} = - \frac{{n_{{{\text{HS}}}} }} {{2\tau _{o} }}e^{{ - {\left( {\beta E^{o}_{a} + b} \right)}}} {\left( {e^{{ - \beta J}} + 2\cosh \beta J\frac{{n_{{{\text{HL}}}} }} {{n_{{{\text{HS}}}} }}} \right)}{\text{ }}{\left( {e^{{ - \beta J}} + 2\sinh \beta J\frac{{n_{{{\text{HL}}}} }} {{n_{{{\text{HS}}}} }}} \right)}^{{q - 1}} \\ & {\text{ }} + {\text{8I}}_{{\text{o}}} \omega {\left( {n_{{{\text{HS}}}} + 2n_{{{\text{HL}}}} - 1} \right)} \\ \end{aligned} $$

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The photo-excitation effect depends on the time t _w at which the light is switched on. This time has to be compared with the characteristic time t _c at which the correlation onsets in sizeable amount. t _c should be considered as an incubation time, defined as the time at which the transition rate exhibits a sizeable departure from the linear mean-field behaviour. The t _c value is of course determined numerically. We obtain the following results:

• 1. t _w <t _c : the lifetime of the metastable HS state is increased; however, the relaxation tail is reduced. The light drives the system to the mean-field behavior. We stress on the paradoxical effect that finally light speeds up relaxation in the late stage of the relaxation.

• 2. t _w >t _c : in a first stage, light slows down relaxation; even, above a threshold value of intensity, an increase of the HS population may be observed. In a further stage, i.e. after some incubation process, the light starts destroying the already established correlations and the final effect is to speed up relaxation.

  (Figure 10 curves (b), (c)). The paradoxical effect is remarkably rapid and efficient.

Systematic computations have shown that the above effects are exclusively associated with the short-range interaction. An extensive investigation including short- and long-range interactions is under progress.