10644
J. Phys. Chem. C 2007, 111, 10644-10651
Effects of Intra- and Intercenter Interactions in Spin Crossover: Application of the Density Matrix Method to the Nonequilibrium Low-SpinTHigh-Spin Transitions Induced by Light S. Klokishner*,† and J. Linares‡ Institute of Applied Physics of the Academy of Sciences of MoldoVa, Academy Strasse 5, MD-2028 KishineV, MoldoVa, and Groupe de la Matie´ re Condense´ e (UMR-8635), CNRS-UniVersite de Versailles, 45 aVenue des Etats-unis, 78035 Versailles Cedex, France ReceiVed: January 7, 2007; In Final Form: April 24, 2007
A microscopic model of nonequilibrium light-induced phase transitions in spin crossover systems based on the solution of the equations for the density operator is developed. The conditions of light-induced and thermally induced bistability are revealed. The results are compared with those obtained in the framework of the Glauber’s approach. The qualitative explanation of experimental data on a series of spin crossover systems is given.
Introduction Spin crossover compounds were a subject of many experimental and theoretical studies.1-4 For years, the research interest was focused on spin transitions driven by temperature (thermally induced spin crossover). The low-spin-high-spin (ls-hs) transition can be a gradual function of temperature, but in some cases, it is a very steep function so that magnetization, magnetic susceptibility, volume, and so forth change nearly discontinuously. A lot of effort has been put into the search for compounds which demonstrate bistability at high temperatures because, namely, these compounds are of primary importance for practical applications. A new stage in the area of spin crossover was opened by the discovery of the LIESST (light-induced excited-spin-state trapping) effect when the metastable hs state was populated by light.5 This finding gave the possibility of investigation of the hs-ls relaxation in spin crossover systems at low temperatures.2,6 Further developments in the field of spin crossover were connected with the studies of spin transitions under the action of light. First, a thermal hysteresis loop has been discovered by Kahn et al.7 and termed light-induced thermal hysteresis (LITH). Independently, thermal and optical hysteresis loops were observed by Varret et al.,8 and their cooperative origin was demonstrated experimentally. The optical hysteresis loop, recorded as a function of the beam intensity at a constant temperature, was titled as light-induced optical hysteresis (LIOH). Initially, the experimental data on systems with cooperative spin crossover effect were interpreted by the aid of a phenomenological macroscopic master equation,8,9 which takes into account both a linear term on the intensity of pumping and the nonlinear cooperative relaxation term introduced in refs 2, 5, and 6 for the explanation of the sigmoidal shape of the relaxation curves. Then, the dynamics of spin crossover was examined in ref 10 in the framework of the Glauber approach.11 In order to obtain a self-accelerated, pressure-dependent relaxation rate, evidenced by the experimental data12 in ref 10 as compared with ref 11, the expression for the rate of the transition from spin configuration {s} to {s′}, which is accompanied by a flip-flop of a single spin, was * To whom correspondence should be addressed. E-mail: klokishner@ yahoo.com. † Institute of Applied Physics of the Academy of Sciences of Moldova. ‡ CNRS-Universite de Versailles.
rewritten in a slightly different form. In ref 13, the LITH and LIOH effects were reproduced by introducing an additional phenomenological term describing external pumping in the Glauber-type equation. The approach developed in refs 11 and 13 is based on an Ising-like model. In the framework of this approach, the degeneracy of the hs level is usually considered as a fitting parameter, and it is taken within the limits of 100200 in order to reproduce the experimental dependencies for reasonable values of the energy gap between the hs and ls states. Meanwhile, the spin-orbital interaction acting within the excited 5T multiplet of the Fe(II) ion even being slightly reduced 2 promotes the hs-ls transition, and therefore, the effects of this interaction are in need of examination. Along with this, a microscopic model of the interaction between spin crossover molecules is required in order to provide a quantum mechanical estimation of the parameter characterizing the cooperative characteristics of the system. In spin crossover compounds, the phase transitions induced by light are nonequilibrium, and it is reasonable to describe these transitions by the density operator method, which is genuine for such effects and gives the possibility of their consideration at the microscopic level. The aim of the present paper is the elaboration of a new microscopic approach to the description of thermodynamic characteristics of a system of interacting spin crossover molecules under external stimuli and the elucidation of conditions for the observation of pronounced thermally and light-induced bistability in these systems. Hamiltonian of the System A crystal is considered to be built of N cells, each cell containing an Fe(II) molecule. The conventional TanabeSugano diagram14 shows that the Fe(II) ion with d6 configuration changes its ground state 5T2 S 1A1(for simplicity, we omit the subscript g in all of the irreducible representations of the Oh group in this paper) for a specific value of the ratio between the crystal field parameter Dq and the Racah parameter B. The mechanism for the low-spin-high-spin transition is related to the change of electronic configuration t62 f t42e2. The model includes the following interactions.
10.1021/jp070126j CCC: $37.00 © 2007 American Chemical Society Published on Web 06/20/2007
Intra- and Intercenter Interactions in Spin Crossover
J. Phys. Chem. C, Vol. 111, No. 28, 2007 10645
(i) The Spin-Orbit (SO) Coupling Operating within the Excited Cubic 5T2(t42e2) Term of Each Fe(II) Ion. The orbital triplet T2 can be regarded as the state with the fictitious orbital angular momentum l ) 1 so that the action of SO within this term can be described by the following single-ion Hamiltonian
HnSO
) -λb s nBl n
1 0 0 τn ) 0 .. 0
(1)
where n is the number of Fe(II) ions in the crystal and λ is the many-electron SO coupling parameter for the 5T2(t42e2) multiplet. The sign “minus” in the term - λs bnBln appears since the matrices of the operator of the orbital angular momentum defined in T1 and P bases differ in sign.1 The eigenvalues of the Hamiltonian (1) are expressed as follows
E(j) ) -λ[j(j + 1) - s(s + 1) - l(l + 1)]/2
(2)
( )
the following form
E(2) ) λ
E(1) ) 3λ
(3)
corresponding to j ) 3(A1,T2,T1), j ) 2(E,T1), j ) 1(T2). For a free Fe(II) ion, the many-electron parameter is λ ) -103 cm-1,15 and the typical values of the orbital reduction factor κ are 0.8-0.9;15 therefore, in a crystal, the energies E(j) satisfy the inequality
E(3) > E(2) > E(1)
(4)
and do not differ significantly from those for a free ion (κ ) 1). (ii) The Interaction of the Fe(II) Ions with the Spontaneous All-Round Full Symmetric Lattice Strain. As in most cases, the spin crossover transformation is not accompanied by the change in crystal symmetry, this strain is supposed to arise from the spin interconversion, and it is assumed to be a consequence of the occupation of the antibonding e orbitals in the Fe(II) ion under the spin transformation. In spin crossover compounds, the medium between the molecules is more easily effected by deformation than that inside of the molecule. In other words, it is “softer”. Therefore, in the subsequent examination, a difference is made between the molecular and intermolecular space. We introduce the internal molecular 1 ) (1xx + 1yy + 1zz)/x3 and external (intermolecular volume) 2 ) (2xx + 2yy + 2zz)/x3 strains and, corresponding to these full symmetric strains, the bulk moduli c1 and c2. In each molecule, we shall consider the interaction of the Fe(II) ion with the arising 1 strain. Within the framework of Kanamori’s approach,16,17 the contribution of uniform strains 1 and 2 in the crystal Hamiltonian can be set by the operator
Hst )
Nc1Ω021 2
+
Nc2(Ω - Ω0)22 2
+ V 1 1
∑n τn + V21N
(5)
where Ω0 and Ω are the molecular and unit cell volumes, respectively, N is the number of unit cells in the crystal, V1 ) (Vhs - Vls)/2, V2 ) (Vhs + Vls)/2, and Vhs and Vls are the constants of interaction of the Fe(II) ion with the full symmetric strain 1 in the hs and ls states, respectively. In eq 5, the first two terms describe the elastic energy of the deformed crystal, and the third and the fourth terms correspond to the coupling of the d electrons with the uniform deformation 1. In the basis of the hs and ls states, the matrix τn has the dimension 16 × 16 and
0 0 1 0 .. 0
0 0 0 1 .. 0
... ... ... ... ... ...
0 0 0 0 .. -1
(6)
It should be mentioned that the matrix τn has the same form in the basis of the single Fe(II) ion states obtained after the diagonalization of the SO interaction. In this case, the diagonal matrix elements of τn equal to 1 and -1 correspond to the states with the energies (3) arising from the hs multiplet and ls singlet, respectively. For a uniform crystal compression (or extension), the relation of 2 to 1 is roughly supposed to be
Here, j is the quantum number of the fictitious total angular momentum of the Fe(II) ion in the excited state. The SO coupling splits the 5T2 term into three levels with energies
E(3) ) -2λ
0 1 0 0 .. 0
c1 2 ≈ 1 c2
(7)
If we replace the molecular and intermolecular volumes by coupled parallel springs, the same relation will hold. With account of eq 7 after minimizing eq 5 over the strain 1, we find
A 1 ) - V1 N
∑n τn - AV2
(8)
where
A)
c2 c1[c2Ω0 + c1(Ω - Ω0)]
(9)
Then, if the value 1 is substituted back into eq 5, we obtain
Hst ) -B
J
τnτm ∑n τn - 2N ∑ n,m
(10)
where B ) AV1V2 and J ) AV21. Thus, the coupling to the strain gives rise to an infinite-range interaction between all molecules of the crystal. It should be mentioned that the model of the elastic continuum, introduced above, satisfactorily describes only the long-wave acoustic vibrations of the lattice. Therefore, the obtained intermolecular interaction, in fact, corresponds to the exchange via the field of long-wave acoustic phonons. If c1 ) c2, the Hamiltonian of intermolecular interaction acquires the form generally accepted in the theory of the cooperative Jahn-Teller effect.16,17 Provided that c1 . c2 and V1 ∼ c1, the interaction term in eq 10 corresponds to that obtained in the model of elastic interactions.2 Another term in eq 10 also arises from the interaction with the deformation and acts as an additional field applied to each spin crossover molecule. (iii) The Interaction of the Fe(II) Ions with Light has the form
V)
∑n Vn ) -∑n Bd nBE(t)
(11)
Here, B dn is the operator of the effective dipole moment of the Fe(II) ion, and B E(t) is the effective field of the light wave. This interaction induces the ls 1A1 f hs 5T2 transitions under the action of light. Actually, the direct transition 1A1 T 5T2 is spinforbidden, and the population of the state 5T2 by light involves the irradiation of the system with green light (λ ∼ 550 nm for
10646 J. Phys. Chem. C, Vol. 111, No. 28, 2007
Klokishner and Linares
Fe(II) complexes) into the 1A1 f 1T1 absorption band, then the nonradiative intersystem crossing from the excited 1T1 state to an intermediate low-lying 3T1 state, and from there to the 5T2 state.18 Thus, the operator of the effective dipole moment can be obtained by admixing the state 3T1 to both the states 5T2 and 1A by spin-orbital interaction (see, for instance, ref 19). In 1 the basis of states 1A1and 5T2, the 16 × 16 matrix of the effective dipole moment looks as follows
(
0 0 B d) 0 .. B d0
0 0 0 .. ...
0 0 ... .. B d0
0 ... 0 .. B d0
... 0 0 .. B d0
B d0 B d0 B d0 .. 0
)
∆ 2
∑n τn
(13)
∑n HSO n
and ∆ is the energy gap between the minima of the adiabatic potentials corresponding to hs and ls states of the Fe(II) ion in the cubic crystal field. Mean Field Approximation: Density Operator Equations The distinctive feature of spin transitions in the presence of an external light wave is their nonequilibrium character. In the case under examination, the external variable parameter is the intensity of pumping applied to the system. As it will be shown below, at a critical value of this intensity, the population of the hs states becomes appreciable for low temperatures under which the thermally induced spin crossover has not taken place yet. The light irradiation facilitates significant changes in the populations of the system levels when the frequency of light is close to that of the transition in a single ion. For a system of interacting spin crossover molecules in the field of the light wave, the solution of the problem can be obtained in the framework of the mean field approximation using the formalism of the density operator.20 Within the framework of this approximation, we reduce the problem of interacting spin crossover molecules to the one-center problem, substituting τnτm in the interaction Hamiltonian (10) with20
τnτm ) 〈τ〉τm + 〈τ〉τn - 〈τ〉2
(14)
where 〈τ〉 ) Tr(Fτn) and F is the density operator for the spin crossover system interacting with light. The Hamiltonian Hst decomposes into the sum of single-ion Hamiltonians
H ˜ st )
∑n Hstn
Hstn ) -(B + J τj)
∑n τn
(15)
Here, τj ) 〈τ〉. Then, taking into account eq 15, we obtain the total Hamiltonian for a single Fe(II) ion
Hn ) H0n + Vn
n H0n ) HSO n + Hst +
∂t
∆ τ 2 n
)
1
15
1
∑ (Vls,iFi,ls - Fls,iVi,ls) + T ip i)1
(Fels,ls - Fls,ls)
1
∂Fi,i 1 1 e - Fi,i) ) (Vi,lsFls,i - Fi,lsVls,i) + (Fi,i ∂t ip T1
(17)
∂Fi,ls Vi,ls 1 1 ) (F - Fi,i) - Fi,ls + ∆iFi,ls ∂t ip ls,ls T2 ip
(18)
Here, i numbers the 15 states arising from the splitting of the 2 multiplet by spin-orbital interaction (eq 3), Fls,i, Fi,i, and Fls,ls are the matrix elements of the density operator in the case when the external pumping is applied to the system, Vi,ls are the matrix elements of the interaction Vn between the spin crossover molecule and external perturbation, and T1 and T2 are the longitudinal and transversal life times, respectively. The diagonal matrix elements of the density operator give the relative populations of states, and thus, T1 is the time of relaxation due to spontaneous and nonradiative transitions. The time T2 characterizes the relaxation of the nondiagonal matrix elements and can be expressed through the halfwidths of levels participating in the light-induced transition. Finally, the energy gaps ∆i are defined as follows (see eqs 3-15) 5T
where
HSO )
∂Fls,ls
(12)
Here, B d0 ) 〈1A1|d B|5T2〉. The full Hamiltonian of the system is of the form
H ) HSO + Hst + V +
Regarding the operator Vn as a perturbation and using the eigenstates of the Hamiltonian H0n as a basis set for calculation of the matrix elements of the density operator, we obtain the following equations
(16)
∆1 ) ∆ - 2λ - 2B - 2J τj0 ≡ ∆i (i ) 1-7, gi ) 7) ∆2 ) ∆ + λ - 2B - 2J τj0 ≡ ∆i (i ) 8-12, gi ) 5) ∆3 ) ∆ + 3λ - 2B - 2J τj0 ≡ ∆i (i ) 13-15, gi ) 3) (19) Here, τj0 is the equilibrium or quasi-equilibrium value of the order parameter in the absence or presence of the external light, respectively, the values gi determine the degeneracies of the Fe(II) ion levels in the molecular field,
1 Fels,ls ) , Z
( )
∆i exp kT e ) Fi,i Z
(20)
are the equilibrium or quasi-equilibrium values of the diagonal matrix elements of the density operator, and 15
Z)
( ) ∆i
gi exp ∑ kT i)1
+1
(21)
is the partition function. Assuming that the time dependence of the electromagnetic pumping wave with the frequency ω is described as
B E(t) ) B E(0)Exp(-iωt) + c.c. the solution for the nondiagonal elements Fls,i of the density operator is presented in the form
Fi,ls(t) ) Fi,lsexp(-iωt)
(22)
Intra- and Intercenter Interactions in Spin Crossover
J. Phys. Chem. C, Vol. 111, No. 28, 2007 10647 temperature. The equation for τj0 ) 2nehs - 1 can be easily obtained from eqs 24-27 if we put γiE ) 0, and it has the following form
Substituting (22) into (18), one obtains
(
Vi,ls (Fi,i - Fls,ls) δi Fi,ls ) δ2i
)
ip T2
δi ) ∆i - pω (23)
,
2
p + 2 T2
Then, in the case of the stationary regime20 when the time of the external irradiation impulse exceeds the characteristic relaxation times T1 and T2, we obtain, with the aid of eqs 17 and 23, the following expressions for the diagonal matrix elements of the density operator
[
Fls,ls ) Fels,ls +
p2
γiEFeii
15
∑
T22 i)1
δ2i
+
p2 T22
(1 +
[
15
1+
[( )
Fii ) Feii δ2i +
γiE)
∑ i)1
]
×
p2 T22
]
-1
γiE (1 + γiE)
,
]
-1 p2 p2 2 i 2 i -2 + p γ T F × δ + (1 + γ ) E 2 ls,ls i E T22 T22 (24)
Here, as usual, i numbers the 15 states which originate from the hs 5T2 level; for the dimensionless parameter γiE, the following expression holds
γiE )
2T1T2 p2
(d Bi,lsB E )2
7exp(-∆1/kT) + 5exp(-∆2/kT) + 3exp(-∆3/kT) - 1 7exp(-∆1/kT) + 5exp(-∆2/kT) + 3exp(-∆3/kT) + 1 (28)
Here, the gaps ∆i are determined by eq 19. The same equation can be obtained by minimizing the free energy F(T) per one crossover molecule over τj0
F(T) ) -kTln[(7exp(-∆1/kT) + 5exp(-∆2/kT) + 3exp(-∆3/kT)) + 1] - J τj0 + J τj20/2 (29)
δ2i +
][
τj0 )
(25)
Here, B di,ls is the matrix element of the dipole moment. The mean field parameter τj is connected by a simple relation with the diagonal matrix elements of the density operator
The temperature dependence of the high-spin fraction is governed by the parameters ∆, J, B, and λ. In order to elucidate the conditions that favor or suppress the spin crossover transition, we start with the discussion of the values of the above-mentioned parameters. The values of the parameters J and B can be estimated microscopically. The matrix elements Vhs and Vls that enter in V1 and V2 can be expressed as the mean values of the derivatives of the crystal field energies in the ls and hs states, respectively
〈 |(
Vls ) ls
) | 〉x
∂W(r,R) ∂R
R)Rls
ls
Rls
,
3
〈 |(
Vhs ) hs
) | 〉x
∂W(r,R) ∂R
R)Rhs
hs
Rhs
(30)
3
Here, Rhs and Rls are the metal-ligand distances in these states, and W(r,R) is the potential energy of the interaction of the electrons of the iron ion with the atoms of the surrounding. In the point charge crystal field model, W(r,R) is the crystal field acting on the electronic shell of the iron ion. For an octahedral complex FeX6
V1 ) 10(Dqhs - 6Dqls)/x3 V2 ) 10(Dqhs + 6Dqls)/x3 (31)
15
τj )
Fii - Fls ∑ i)1
(26)
On the other hand, the hs fraction or the total population of the hs state 15
nhs )
Fii ∑ i)1
can be expressed as
2nhs - 1 ) τj
(27)
The obtained self-consistent equation for the hs fraction has a set of nontrivial solutions determined by different relations between the characteristic internal parameters of the system and the intensity of external pumping. Temperature Dependence of the High-Spin Fraction in the Case of the Absence of External Pumping In this section, we examine the case of thermally induced spin crossover when the external electromagnetic field is not applied and the system is in the equilibrium state. Thus, the changes in the population of the hs state are only governed by
While estimating V1 and V2 for the crystal field parameters, we take the typical values listed in ref 5, Dqhs ) 1176 cm-1 and Dqls ) 2055 cm-1, and obtain V1 ) -6.4 × 104 cm-1 and V2 ) 7.8 × 104 cm-1. Then, we suggest that the main change in the volume falls at the intermolecular space. This allows us to assume that c1 . c2. For c2 ≈ 0.1c1, V1 ) -6.4 × 104 cm-1, and, typical for spin crossover compounds, values Ω - Ω0 ∼ 103A3 and c2 ) 1011 dyne/cm2, the parameters J and B take on the values of 83 and -100 cm-1, respectively. The estimated value of the parameter J is close to that obtained in ref 9 from fitting the experimental data for the LITH loop in Fe0.85Co0.25(btr)2(NCS)2 × H2O. Then, we estimate the change of the unit cell volume under the ls f hs transition
Ωhl ) -
2x3 V ) 44A3 c1 1
This value also falls within the experimental limits.5 In such a way, the model gives reasonable values for the microparameter J and the change of the unit cell volume. Further, the value J ) 75 cm-1 is retained during the most part of the calculations. The term -2B (eq 19), which depends on the same parameters V1 and V2, is considered as redetermining the energy gap ∆ (see eq 19). The gap ∆ is varied in the range of 100 < ∆ < 800
10648 J. Phys. Chem. C, Vol. 111, No. 28, 2007
Figure 1. Temperature dependence of the hs fraction in the absence of external pumping with λ ) 0 and J ) 75 cm-1; (1) ∆ - 2B ) 100 cm-1, (2) ∆ - 2B ) 130 cm-1, (3) ∆ - 2B ) 150 cm-1, (4) ∆ - 2B ) 170 cm-1, (5) ∆ - 2B ) 200 cm-1.
Klokishner and Linares
Figure 3. Thermal variation of the hs fraction as a function of the gap ∆ - 2B and the parameter of cooperative interaction J in the absence of external pumping with λ ) -103 cm-1; (1) ∆ - 2B ) 410 cm-1 and J ) 90 cm-1, (2) ∆ - 2B ) 450 cm-1 and J ) 120 cm-1, (3) ∆ - 2B ) 500 cm-1 and J ) 150 cm-1.
Figure 2. Thermal variation of the hs fraction as a function of the gap ∆ - 2B in the absence of external pumping with λ ) -103 cm-1 and J ) 75 cm-1; (1) ∆ - 2B ) 380 cm-1, (2) ∆ - 2B ) 390 cm-1, (3) ∆ - 2B ) 450 cm-1, (4) ∆ - 2B ) 600 cm-1, (5) ∆ - 2B ) 800 cm-1.
Figure 4. Variation of the free energy as a function of the mean field parameter τj0 and temperature with λ ) -103 cm-1, J ) 75 cm-1, and ∆ - 2B ) 380 cm-1.
cm-1. For the SO coupling parameter λ, the experimental value -103 cm-1 for the free Fe(II) ion, which does not noticeably differ from that in a crystal, is used. In the calculations, the total degeneracy of the hs state 5T2 is taken as equal to 15, which corresponds to the real physical situation. Figure 1 illustrates the temperature dependence of the hs fraction in neglect of the SO interaction. The increase of the gap ∆ - 2B leads to the narrowing of the hysteresis loop, the transition temperature rises, and for a certain value of this gap, the hysteresis disappears completely. It is seen that already for ∆ - 2B ) 200 cm-1, the population of the hs state at high temperatures differs from 1, and the spin transition becomes incomplete. From Figure 1, it is also seen that for ∆ - 2B ) 100 and 130 cm-1, a metastable hs branch exists at all temperatures. The role of the SO interaction as a factor facilitating spin crossover is shown in Figure 2. Actually, this interaction splits the 5T2 term and leads to the decrease of the energy gap (∆ ˜ ) ∆ - 3|λ| - 2B in comparison with ∆ - 2B), which the system first overcomes when passing from the ls state to the hs one. As a result, the transition temperature diminishes, and for gap values inherent to spin crossover compounds, we obtain nhs values close to the experimental ones.2 From Figure 2, it is seen that starting from a gap value close to the observed ones, the spin transition changes its type; instead of a transition demonstrating a hysteresis loop, a gradual transition takes place. It should be also mentioned that for ∆ - 2B ) 390 cm-1 (Figure 2, curve 2), a metastable hs branch appears at low temperatures. Further, it will be shown that the free energy as a function of τj0 has a local minimum at τj0 ) 1 and low temperatures. In this
case, the metastability of the hs state is intrinsic, and if the initial state is set to be near the metastable point, the population nhs is expected to move first to the metastable value and then to relax to the low-temperature state. A similar metastable hs state at low temperatures that exists separately from the high-temperature hs state first was found for Co-Fe Prussian blue analogues in ref 21. In ref 21, a prediction was made that this metastable at a low-temperature hs state should play an important role in the study of photoirradiated processes in systems undergoing spin transitions. An insignificant increase of the cooperative interaction parameter J leads to the increase of the effective gap values right up to which hysteresis can be observed (Figure 3). The common feature of Figures 1 and 2 is the appearance of two branches in the temperature dependence of the hs fraction. In such a situation, the behavior of the system can be described as follows. Starting from any of the extreme temperatures, the system will continuously follow the upper line and will not visit the states which belong to the lower line. Thus, the behavior of the system depends on the way of changing the external variable parameter, which manages the system properties. In the case under consideration, temperature plays the role of such a parameter. Finally, we examine the dependence of the free energy (eq 29) on τj0 for different temperatures and ∆ - 2B ) 380 and 390 cm-1 (Figures 4 and 5), taking into account spin-orbital interaction. For ∆ - 2B ) 380 cm-1 in the temperature range of 20-75 K, the free energy clearly demonstrates two inequivalent minima and a barrier between these minima (Figure 4). Up to 50 K, the minima correspond to τj0 ) (1 and indicate both the coexistence of the ls and hs
Intra- and Intercenter Interactions in Spin Crossover
Figure 5. Variation of the free energy as a function of the mean field parameter τj0 and temperature with λ ) -103 cm-1, J ) 75 cm-1, and ∆ - 2B ) 390 cm-1.
phases in the system and the metastability of the hs state. With increasing temperature, the height of the barrier between these phases decreases, and the minima flatten and correspond to values of |τj0| < 1. The coexistence of two phases still takes place. Beginning from T ) 90 K, the free energy possesses one minimum, and all of the molecules of the system are in the hs state (Figure 2, curve 1). However, the limit nhs ) 1 is not achieved (Figures 2 and 4), and this is the result of a common action of a comparatively weak intermolecular interaction and SO coupling. Even a small increase in the value of ∆ - 2B leads to noticeable changes both in the behavior of the free energy and the hs fraction (Figure 5 and curve 2 of Figure 2). For ∆ - 2B ) 390 cm-1, the height of the barrier between the minima of the free energy significantly decreases at about 50 K; as a result, the hs metastable state is not observed for T > 50 K (Figure 2). At the same time, in the temperature range of 90 < T < 100 K, the minima corresponding to the ls and hs fraction become equivalent, which manifests in a narrow-width hysteresis (Figure 2). Effects of External Pumping: Light-Induced Thermal and Optical Hysteresis Now we analyze the effects caused by external light irradiation in a system of interacting spin crossover molecules. For calculation of the hs fraction, we employ eqs 24-27 and note e e that external pumping also affects the populations Fi,i and Fls,ls , which are contained in these equations. These populations can be considered as quasi-equilibrium ones under the assumption that the time of pumping exceeds the characteristic intrinsic times of relaxation. Supposing that this condition is fulfilled, e e and Fls,ls in a self-consistent way with due we determine Fi,i account of the interaction with the external field. It means that e e in eqs 19-21, τj0 will be replaced by τj, and the values Fi,i , Fls,ls , and ∆i become dependent on the intensity of pumping through the parameter τj, satisfying eq 26. At this point, we follow the method developed in ref 20 for consideration of the ferromagnetic phase transition induced by light in a crystal containing two level paramagnetic atoms. An approximation of the same type was applied in ref 10 where, in the framework of the Glauber approach, the expression for the probability Wj(sj) of the transition from spin configuration {s} to {s′} accompanied by one spin flip was written on the basis of the relation Wj(sj)/ Wj(-sj) obtained in the mean field approximation under the condition of thermodynamic equilibrium. Then, in Wj(sj)
nehs )
∑i Feii
J. Phys. Chem. C, Vol. 111, No. 28, 2007 10649
Figure 6. Thermal variation of the hs fraction as a function of the intensity I of external pumping with λ ) -103 cm-1, J ) 75 cm-1, ∆ - 2B ) 450 cm-1, and ω ) 450 cm-1; (1) I ) 0 mW/cm2, (2) I ) 20 mW/cm2, (3) I ) 200 mW/cm2.
was replaced by nhs, and this expression for Wj(sj) was employed in the examination of the cooperative dynamics of a spin crossover system in the presence of light. At this stage, in order to calculate the temperature and field dependence of the hs fraction, we discuss the values of the parameters T1, T2, and γiE. For T1, we accept the thermal activation law T1 ) T10exp(EA/kT) with the parameters T10 ) 10-5 s and EA ) 900 K, characteristic for FexM1-x(btr)2(NCS)2•H2O with M ) Co, Ni, and Zn.13 The transversal lifetime T2 was taken equal to 2 × 10-15 s.20 For the transition ls f hs, an averaged value of the matrix element B di,ls ) B d of the dipole moment is accepted, and the case of linear polarized light is examined so E)2 is substituted by (dE)2. Insofar as the pumping of that (d Bi,lsB the 5T2 level occurs through the transition 1A1 f 1T1, we assume that the dipole moment of this transition is approximately equal to that of the spin forbidden transition 5T2 f 1A1, overestimating, in such a way, the value of d. The extinction coefficient of the 1A1 f 1T1 transition (550 nm, 18200 cm-1), characteristic of spin crossover compounds, is on the order of ) 27.5 mole-1 cm-1; the full energy width of the half-maximum is ∆ω ) 2780 cm-1;22 therefore, as the oscillator strength of the transition is f ) 4.32 × 10 -9max, ∆ω ) 3.64 × 10 -4 and, consequently, d ≈ 10-19 esucm. When the intensity of external pumping I falls into the range of values 20-100 mW/cm2, which are close to the experimental ones,8,9 the energy dE takes on the values 4.6 × 10-6-1 × 10-5 cm-1. With the aid of the parameters defined above, the hs fraction was calculated as a function of temperature for different intensities of external pumping. Figure 6 illustrates the case of an effective gap ∆ - 2B ) 450 cm-1; when in the absence of external stimuli, the system undergoes a gradual thermally induced spin transition (curve 1). The variation of nhs in the presence of irradiation is determined by the internal parameters of the molecule in the mean field and the intensity of pumping I. It is seen that for I ) 20 mW/cm2, nonvanishing values of nhs appear in the range of temperatures T < 75 K (Figure 6, curve 2). A considerable growth of the intensity of pumping (Figure 6, curve 3) leads to the widening of the range of low temperatures at which the hs state is populated. However, this widening is on the order of 25 K, and it is smaller than expected because of the significant growth of the probability of depopulation of the hs state with the increase of temperature from 75 to 100 K. Actually, at 100 K, the longitudinal relaxation time T1 that characterizes the decay of the hs state becomes 20 times shorter than that at 75 K (T1(75 K) ) 1.6 s, T1(100 K) ) 0.08 s). It is seen that the low-temperature spin transition induced by light irradiation is abrupt, and this is the result of the steep decrease of the relaxation time T1. Figure 6 also shows that at
10650 J. Phys. Chem. C, Vol. 111, No. 28, 2007
Figure 7. Thermal variation of the hs fraction as a function of the intensity I of external pumping with λ ) -103 cm-1, J ) 75 cm-1, ∆ - 2B ) 390 cm-1, and ω ) 390 cm-1; (1) I ) 0 mW/cm2, (2) I ) 20 mW/cm2, (3) I ) 40 mW/cm2, (4) I ) 60 mW/cm2, (5) I ) 180 mW/ cm2.
Figure 8. Dependence of the hs fraction on the intensity of external irradiation for different temperatures with λ ) -103 cm-1, J ) 75 cm-1, ∆ - 2B ) 390 cm-1, and ω ) 390 cm-1; (1) T ) 60 K, (2) T ) 70 K, (3) T ) 80 K, (4) T ) 100 K.
low (T < 100 K) and higher temperatures (100 < T < 250 K), the transitions are light and thermally driven, respectively, and no hysteresis loop is observed. Qualitatively, another picture arises in the situation when the hs fraction as a function of temperature demonstrates a hysteresis loop in the absence of external irradiation (Figure 2, curve 2). In this case, starting from some critical value of the intensity I of light (Figure 7, curves 2 and 3), the irradiation of the system leads to a significant population of the hs state and to LITH in the range of low temperatures. It is worth noting that, at these temperatures in the absence of pumping, the system is in the ls state. As the intensity of pumping increases, the temperature of the lightinduced spin transition changes, and the hysteresis loop stimulated by light shifts toward higher temperatures (Figure 7). The higher the intensity of pumping, the higher the temperature, starting from which the population of the hs state approaches its upper bound (Figure 7), attainable in the absence of pumping. For I ) 20 and 40 mW/cm2, the trends of the nhs curves (Figure 7, curves 2 and 3) at high temperatures resemble those corresponding to thermally induced hysteresis (I ) 0, Figure 7, curve 1). With the growth of the intensity of pumping, both parts of the nhs curve approach each other, and at a critical value of the intensity of pumping, they merge in such a way that the parts with lower values of nhs form a closed curve with a droplet-like shape. Meanwhile, the parts of these curves with nhs g 0.8 flow together, giving a branch of nhs values slightly dependent on temperature. Further increase in the intensity of pumping leads to the situation when the system remains in the hs state at all temperatures. The described patterns of the hs
Klokishner and Linares
Figure 9. Variation of the hs fraction as a function of the intensity of external irradiation for different temperatures with λ ) -103 cm-1, J ) 120 cm-1, ∆ - 2B ) 450 cm-1, and ω ) 450 cm-1; (1) T ) 60 K, (2) T ) 65 K, (3) T ) 70 K.
fraction behavior are expected to be reproduced experimentally, by manipulating the temperature of the system preparation in the dark and switching the light on and off at different stages of the thermally induced transition from the ls to the hs state.13 Figures 8 and 9 demonstrate the variation of the hs fraction as a function of the intensity of external light for constant temperatures and are in accordance with the experimental finding that the interaction of a spin crossover system with light can result in LIOH.8 The intensity of light is taken within the experimentally accessible limits. A temperature increase results in a noticeable widening of the hysteresis loop, while the maximum value of the hs fraction diminishes with the increase of temperature, in correspondence with the fast decrease of the hs state lifetime. In such a way, the interplay between the intensity of pumping, nonradiative relaxation of the excited hs state, and intramolecular spin-orbital and intermolecular electron deformation interactions facilitates hysteresis loops in the thermal and light intensity dependence of the hs state population. It is this effect that was discovered in FexM1-x(btr)2(NCS)2•H2O (M ) Co, Ni, and Zn) crystals.8 Concluding Remarks In this article, the effects of external irradiation on a system containing interacting spin crossover molecules are discussed. The Hamiltonian of the system includes the spin-orbital interaction for the orbitally degenerate 5T2 state of the hs Fe(II) ion, long-range cooperative interaction, as well as the interaction of the system with light irradiation. The examination has been performed in the framework of the density operator method. The equations for the matrix elements of the density operator have been derived and expressed in terms of the averaged high-spin fraction and the intensity of external pumping as well as of the parameters characterizing intra- and intermolecular interactions. In the case of stationary pumping, a self-consistent equation for the population of the hs state as a function of temperature and intensity of pumping has been derived. For certain realistic values of the parameters of intraand intercluster interactions, the model developed clearly reproduces the LITH and LIOH observed in FexM1-x(btr)2(NCS)2•H2O (M ) Co, Ni, and Zn) compounds. Acknowledgment. The support of the ECONET program (Dossier N 102117UL) is highly appreciated. S.I.K also thanks the Supreme Council for Science and Technological Development of Moldova for financial support.
Intra- and Intercenter Interactions in Spin Crossover References and Notes (1) Gu¨tlich, P. P.; Jung, J. NuoVo Cimento Soc. Ital. Fis., D 1996, 18, 107. (2) Gu¨tlich, P.; Hauser, A; Spiering, H. Angew. Chem., Int. Ed. Engl. 1994, 33, 2024. (3) Zarembowitch, J. New J. Chem. 1992, 16, 255. (4) Gaspar, A. B.; Ksenofontov, V.; Serdyuk, M.; Gu¨tlich, P. Coord. Chem. ReV. 2005, 249, 2661. (5) (a) Decurtins, S.; Gu¨tlich, P.; Ko¨hler, C. P.; Spiering, H.; Hauser, A. Chem. Phys. Lett. 1984, 1, 139. (b) Decurtins, S.; Gu¨tlich, P.; Hasselbach, K. M.; Hauser, A.; Spiering, H. Inorg. Chem. 1985, 24, 2174. (6) (a) Hauser, A. Coord. Chem. ReV. 1991, 11, 275. (b) Hauser, A. Comments Inorg. Chem. 1995, 17, 17. (c) Hinek, R.; Spiering, H.; Gu¨tlich, P.; Hauser, A. Chem.sEur. J. 1996, 2, 1435. (7) (a) Letard, J. F.; Ginneau, P.; Rabardel, L.; Howard, J. A. K.; Goeta, A. E.; Chasseau, D.; Kahn, O. Inorg. Chem. 1998, 37, 4432. (b) Letard, J. F.; Ginneau, P.; Rabardel, L.; Howard, J. A. K.; Goeta, A. E.; Chasseau, D.; Kahn, O. Chem.sEur. J. 1996, 2, 1435. (8) Desaix, A.; Roubeau, O.; Jeftic, J.; Haasnoot, J. G.; Boukheddaden, K.; Codjovi, E.; Linares, J.; Nogues, M.; Varret, F. Eur. Phys. J. B 1998, 6, 183. (9) Varret, F.; Boukheddaden, K.; Jeftic, J.; Roubeau, O. Mol. Cryst. Liq. Cryst. 1999, 335, 561.
J. Phys. Chem. C, Vol. 111, No. 28, 2007 10651 (10) Boukheddaden, K.; Shteto, I.; Hoˆo, B.; Varret, F. Phys. ReV. 2000, 62, 14796. (11) Glauber, R. J. J. Math. Phys. 1963, 4, 294. (12) Schenker, S.; Hauser, A.; Wang, W.; Chan, I. Y. J. Chem. Phys. 1998, 109, 9870. (13) Boukheddaden, K.; Shteto, I.; Hoˆo, B.; Varret, F. Phys. ReV. 2000, 62, 14806. (14) Sugano, S.; Tanabe, Y.; Kamimura, H. Multiplets of Transition Metal Ions in Crystals; Academic Press: London, 1970. (15) Abragam, A.; Bleaney, B. Electron Paramagnetic Resonance of Transition Ions; Clarendon: Oxford, England, 1970. (16) Kanamori, J. J. Appl. Phys. 1960, 31, 145. (17) Gehring, G. A.; Gehring, K. A. Rep. Prog. Phys. 1975, 38, 1. (18) Poganiuch, P.; Decurtins, S.; Gu¨tlich, P. J. Am. Chem. Soc. 1990, 112, 3270. (19) Klokishner, S.; Linares, J.; Varret, F. J. Phys.: Condens. Matter 2001, 13, 595. (20) Andreev, A. V.; Emel’yanov, V. I.; Ll’ynskii, Yu. A. CooperatiVe Phenomena in Optics: Superiradiation, Bistabilty, Phase Transitions; Nauka: Moscow, 1988; p 288. (21) Miyashita, S.; Konishi, Y.; Toroko, H.; Nishino, M.; Boukheddaden, K.; Varret, F. Prog. Theor. Phys. 2005, 114, 719. (22) Hauser, A. J. Chem. Phys. 991, 94, 2741.
