ORIGIN OF THE SINGLE MOLECULE MAGNET BEHAVIOUR OF THE

[CuIILTbIII(hfac)2]2 CLUSTER

S.Klokishner, O. Reu, S. Ostrovsky, A. Palii

Institute of Applied Physics, Academy of Sciences of Moldova, Academy str.5, Kishinev, MD-2028, Moldova,

E-mail address:

P.Tregenna-Piggott

ETHZ and Paul Scherrer Institut, CH-5232 Villigen, Switzerland,

E-mail address:

A.Fishman

Institute of Metallurgy, Ural Branch of the Russian Academy of Sciences, 620016 Ekaterinburg, Russia

E-mail address:

1.INTRODUCTION

One of the most fascinating developments of the last decade in the field of molecule-based magnetism involves the discovery and characterization of single molecule magnets (SMMs) [1-18]. Magnetic bistability and slow magnetic relaxation at low temperatures are the distinctive features of these systems that may lead to future trends for wide applications in quantum and molecular electronics. The majority of known SMMs contain transition metal ions with orbitally non-degenerate ground states (spin clusters). The Mn12 cluster derivatives [Mn12O12(O2CR)16(H2O)x]n- (n=0,1,2; x=3,4] [1-9] , distorted cubane complexes with [MIVMIII4O3X]cores[10-12], tetranuclear vanadium complexes [V4O2(O2CR)7(L)2]n[13], and iron complexes[14] formulated as [Fe8O2(OH)12(L)6]8+ exemplify this type of SMMs. For these SMMs the energy barrier for magnetization reversal appears as a result of the combination of a large ground state spin of the cluster and a significant negative zero-field splitting. However, the leading anisotropy term, represents a second order correction with respect to the spin -orbit coupling and, hence,DS is usually small. The latter essentially constrains the possibility to increase the barrier exhibited by such molecules. Along with this the relaxation of magnetization in existing transition metal clusters with SMM properties is still very fast. For instance, for the Mn12Ac[1,2] cluster having a barrier of about 61 K the relaxation time at T=2K is 3.7·106 s. Meanwhile, the relaxation time acceptable for applications should be at least 4.7·108 s =15 years at room temperature.

In an attempt to increase the energy barrier for magnetization reversal researchers have turned to SMMs[15-18] that contain transition metal ions with unquenched orbital angular momenta in the ground state. We demonstrated [19-24] that for the SMMs of this kind, the first–order single ion anisotropy and the anisotropy of exchange interaction are responsible for the formation of the barrier for reversal of magnetization.

Considerable effort has been focused on the design of SMMs functioning at higher temperatures than those containing transition metal ions. Recently a new class of SMMs based on mixed 3d-4f complexes has been reported [25-27]. It was demonstrated that the cyclic 3d-4f tetranuclear compound [CuIILLnIII(hfac)2]2 (Ln=Tb)[25]exhibits a SMM behavior. The out-of-phase component of the alternating current (ac) susceptibility () measured for Cu2Tb2 complex shows frequency-dependent peaks, which are characteristic of SMMs. The reference complex Ni2Tb2 with the diamagnetic NiIIion in place of the CuIIion showed no signal in the same temperature range, thus demonstrating that the present SMM behavior is not intrinsic to the TbIII centers but arises from the 3d-4f ferromagnetic interaction in the tetranuclear complex [CuIILTbIII(hfac)2]2.

All cited articles dealing with the 3d-4f SMMs only contain the synthesis and experimental data, but no theoretical models able to explain the observed SMM behavior of these systems have been suggested. The aim of this work is to reveal the mechanisms underlying the single molecule magnet (SMM) behavior of the [CuIILTbIII(hfac)2]2 cluster containing 3d and 4f ions.

2. MODEL CALCULATIONS OF THE CRYSTAL-FIELD PARAMETERS FOR THE Tb3+ ION

The nearest surrounding of the Tb3+ ion in [CuIILLnIII(hfac)2]2 contains eight oxygen ions. The numerical values of the distances between the Tb3+ ion and the nearest –neighbor oxygen ions are given in Table 1. The bond lengthsare different. As a consequence the oxygen ligands are nonequivalent and one can expect a stronger interaction of the electrons of the Tb3+ ion with the ligands .Taking into account the real structure of the complex formed by theTb3+ ion and eight nearest-neighbor oxygen ions in [CuIILLnIII(hfac)2]the effective parametric crystal field Hamiltonian acting within the space of the orbitals of the rare-earth ion can be written in the following form

(1)

whereand are the crystal field parameters, are the tensor spherical operators. The values of the crystal field parameters can be estimated in the framework of the exchange-charge model [28-30]that takes into account two contributions to the energy of the valent electrons in the crystal field and, namely,the contribution arising from the interaction of the electrons with the point charges of the surrounding ions and the contribution coming from the overlap of the wave-functions with the ligand functions. The latter is referred to as the

Table 1Bond lengths (nm) between the Tb3+ ion and the nearest -neighbor oxygen ions [27]

0.2275 / 0.2467 / 0.2351 / 0.2541 / 0.2405 / 0.2376 / 0.2337 / 0.2408

contributionof exchange chargesIn the exchange-charge model of the crystal field the parameters and are represented as

, .(2)

The components and are determined as follows:

,

(3)

where is the effective charge of the th ligand with the spherical coordinates (the origin of the co-ordinates coincides with the position of the Tb3+ ion), > is the radial integral for the Tb3+ ion,in the subsequent calculations the values 0.755 a.u., 1.485 a.u., 5.691 a.u. [31] have been used . Finally,are the shielding factors [32] whose values [28] have been taken for calculation of the crystal field parameters and .

The parameters and are given by the following relations

, (4)

whereare dimensionless phenomenological parameters of the model which will be obtained below from a comparison of the calculated and experimental values of the magnetic susceptibility for the reference complex Ni2Tb2 ,

, (5)

, are the overlap integrals of the wave functions of the Tb3+ ion and 2s, 2p wave functions of the oxygen ion. Numerical values of the overlap integrals used in this work have been computed with the aid of the radial 4f wave functions of Tb3+ and functions of O2- given in ref. [31] , [33].

3. MAGNETIC SUSCEPTIBILITY OF THE [NiIILTbIII(hfac)2]2 CLUSTER

In order to inspect the role of the TbIII ion in the expression of the SMM behavior of the[CuIILTbIII(hfac)2]2 cluster first we examine the magnetic properties of the reference cluster[NiIILTbIII(hfac)2]2[27]. The latter system contains diamagnetic NiII ions, and thus itsmagnetic properties are solely determined by two

non-interacting TbIII –ions. The Hamiltonian of the TbIII ion in the external magnetic field looks as follows:

(6)

whereand are the orbital angular momentum and the spin of the TbIII ion, respectively, is the Bohr magneton. Insofar as the energy gaps between the terms of the free TbIII ion exceed significantly the splitting of the ground 7F6-multiplet in the crystal fieldthe following relation is appliedin calculation of the magnetic characteristics of theTbIII ion

, (7)

and the operator of the Zeeman interaction is rewritten in the form

(8)

where is the total angular of the TbIII ion , is the quantum number of the total angular momentum projection, =3/2 is the Lande factor for the ground 7F6 –multiplet of the TbIII ion.

At the first stage with the aid of formulae (3-5) we perform the calculation of the parameters of the crystal field acting on the TbIII ion and obtain the expressions for these parameters as functions of 2 exchange-charge model parameters and corresponding totwo groups of oxygen ligands picked out according to the distances between the and oxygen ions. The advantage of this approach based on the exchange charge model is the significant reduction of the number of the parameters of the model from 27 parameters and of the crystal field (1) for the Tb3+ion in [NiIITbIIILnIII(hfac)2]2 to only 2 parameters and that characterize the exchange charge contribution . Then, the best fit procedure isapplied, and the parameters and are considered as the fittingones.First, the energies and wavefunctions of the Stark levels,which arise from the splitting of the ground 7F6 –multiplet of the TbIII ion, arecomputed. Next, using the obtained wave functions and energy levels the components of the magnetic susceptibility and the susceptibility of the powder samplearecalculated with the aid of formulae

(9)

(10)

where

(11)

isthe partition function and are the energies of theion in the crystal and magnetic fields, the factor 2 in Eqs (9), (10) accounts for two Tb ions in the cluster. The procedure is repeated until for certain values of and the optimal coincidence between the calculated and experimental curves is achieved.

Fig. 1 displays the temperature dependence of measured over the temperature range 1.8-300 K, and the theoretical curve calculated for the set of the best fit parameters. For [NiIITbIIILnIII(hfac)2]2 the obtained value is almost constant over the wholetemperature range except for a decrease at temperatures lower than 50 K.

The best fit is achieved for, with the agreement criterion being equal to0.009 (=33 is the number of the experimental points). One can see that the theoretical curve calculated with this set of parameters is in a satisfactory agreement with the experimental data.

Fig.1. as a function of temperature for the [NiIITbIIILnIII(hfac)2]2 cluster
calculated with the set of the best fit parameters: :
experiment- circles, theory-solidline

The temperature dependences of , and calculated with the set of the best fit parameters are presented in Fig.2. It is seen that at low temperatures up to 5K all components abruptlyincrease. Then, the growth of and becomes slower, while the component passes through a maximum at and then starts decreasing. Fig.2 clearly demonstrates that in the low temperature range the values of, and differ significantly thus evidencing a strong rhombic component of the crystal field that prevents from SMM behavior. Another confirmation of the absence of the SMM behavior of the reference Ni2Tb2 complex comes from the fact that for all Stark levels originating from the ground 7F6 –multiplet the calculated mean value of the projection of the total angular momentum of the Tb-ion is vanishing. Thus, the levels of the Tb-ion in the crystal field do not form a barrier for magnetization reversal. The results obtained are usedfurther to examine the SMM properties of the cluster[CuIILTbIII(hfac)2]2.

Fig.2. Temperature dependences of, and for the

[NiIITbIIILnIII(hfac)2]2 cluster calculated with the set of

the best fit parameters: .

4. BARRIER FOR MAGNETIZATION REVERSAL IN THE CuII-TbIIIPAIR

While examining the magnetic behavior of the [CuIILTbIII(hfac)2]2complex the CuII –ion is considered as a spin ½. Due to the lack of magnetic mediators between the two Cu and two Tb ions in the titled complex the Cu-Cu and Tb-Tb magnetic interactions are negligibly small , and the magnetic characteristics of the [CuIILTbIII(hfac)2]2 can be calculated as a sum of contributions arising from the TbIII-CuII pairs.The Hamiltonian of the TbIII-CuII pairis written as

wherethe second term describes the exchange interaction between the Tb and Cuions, in derivation of this term for the Tb-ion the expression

(13)

valid within the ground 7F6 multiplet of this ion is employed,=3/2, . In (12) the components of the g-tensor, the components of the external magnetic field as well as the components of the copper ion spin are determined in the frame of reference related to the Tb-ion. The analytical relations for the components, and expressed through those determined in the co-ordinates related to the Cu-ion are too cumbersome to be given here. We only note that the Eyler angles describing the rotation from the Cu local frame to that related to the Tb ion are determined using the crystallographic data given in [34] and take on the values :. Theseangles and the g-tensor values for a Cu –ion in a square-planar surrounding of oxygen ligands [35] are taken for calculation of the g- tensor components in (12)and the following valuesare obtained.Then,weconstruct and diagonalizethe matrix of the Hamiltonian (12)in the basis of the wave functions that represent the direct product of spin-1/2 functions of the CuII ion and the wave functions of all Stark levels of the TbIII ion obtained from the previous calculations. The obtained eigenfunctions and eigenvalues are used forcalculation of the susceptibility of the[CuIILTbIII(hfac)2]2cluster as a function of temperature.A satisfactory agreement between the calculated and experimental values (Fig.3)is achievedfor the best fitparameters,.The least squares parameter is equal to 0.034.The exchange interaction betweenTbIII and CuII ions turns outa ferromagnetic one, and the reason for thisis the orthogonality of the 4f orbitals of the TbIII ion and 3d orbitals of the CuII ion. The value seems to be reasonable since it falls into the region of typical values of exchange interaction parameters in oxo-bridged clusters containing lanthanide and transition metal ions.

Fig.3.for the [CuIITbIIILnIII(hfac)2]2 cluster as a function of temperaturecalculated with the set of the best fit parameters: ,:

experiment- circles, theory-solidline

The calculated curve reproduces the main qualitative features of the experimental one: the value increases with temperature, reaches a maximum and then decreases.Fig.4 demonstrates the temperature dependence of the tensorcomponents.It can be seen that, at low temperatures,the system exhibits strong magnetic anisotropy. The difference between the and componentsbecomessmaller as compared with that for the Ni2Tb2 complex,whilethe temperature behavior of essentially differs: the value at is slightly lower than the maximum value attained at.

Finally,we examine the low-lying part of the energy pattern of the TbIII-CuII pair calculated in the absence of the external magnetic field with the best-fit parameters,

Fig.4.Temperature dependences of, and for the

[CuIITbIIILnIII(hfac)2]2 cluster calculated with the set of the best fit

parameters: , .

. We characterize each -th state of this pair by the expectation value of the total angular momentum projection. The value is calculated as a diagonal matrix element of the total angular momentum projection operator, in which the single-ion operators are defined in the frame of reference related to the Tb-ion. Figure 5 shows the low-lying energy levels of theTbIII-CuII pair as functions of. It is clearly seen that the energies tend to decrease with enhance of thus indicating the existence of the barrier for magnetization reversal.The height of the barrier is about 100cm-1. Therefore, we arrive at the conclusion that the proposed model not only provides a satisfactory description of the dcsusceptibility but it is also compatible with the observed SMM behavior of the [CuIITbIIILnIII(hfac)2]2 cluster.Thus, the ferromagnetic exchange interaction between Cu2+ and Tb3+ ionsthat couples their moments plays a crucial role in the formation of the SMM properties of the [CuIILTbIII(hfac)2]2cluster.

6. Concluding remarks

The model developed in this study represents a first attempt to reveal the underlying mechanism responsible for the SMM behavior of mixed 3d-4f clusters containing lanthanide ions with unquenched orbitalangular momenta and spins.The model includes the crystal field of real symmetry acting on the 4f-ions and the isotropic exchange interaction between transition metal and lanthanide ions.

The model was shown to account for the observed dc magnetic susceptibility of the [NiIITbIIILnIII(hfac)2]2 and[CuIITbIIILnIII(hfac)2]2 clusters. The interplay between the crystal field acting on the Tb3+ ion and theferromagnetic Heisenberg-type exchange between TbIII and CuII ionswas shown to produce an appreciable barrier for the reversal of magnetization. The proposed model provides a satisfactory agreement between the observed and calculated dc magnetic susceptibilities for the the [NiIITbIIILnIII(hfac)2]2 and[CuIITbIIILnIII(hfac)2]2 clusters and also confirms the ac susceptibility evidence for the SMM behavior of the [CuIITbIIILnIII(hfac)2]2 cluster.

Fig.5. Low-lying energy levels of the Tb-Cu pair as functions of calculated

with, .

At the same time it should be mentioned that the examined type of clusters needs a deeper magnetic characterization in order to extract the key magnetic parameters that govern the properties of this SMM. Single crystal studies, specific heat capacity data measurements and inelastic neutron scattering studies are under way.

ACKNOWLEDGEMENT

Financial support of the joint Russian-Moldovan grant program (grants No.06-03-90893:Moldova 06.09CRF) and Swiss National Science Foundation (grant IB7320-111004) is highly appreciated.

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