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ELECTRIC FIELD CONTROL OF MOLECULAR MAGNETIC SWITCHING

Andrew Palii,1,2 Sergey Aldoshin,1 Boris Tsukerblat,3 Juan Modesto Clemente-Juan,4

Alejandro Gaita-Ariño,4 Eugenio Coronado4

1 Chemistry Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel; e-mail: tsuker@bgu.ac.il 2 Institute of Applied Physics, Academy of Sciences of Moldova, Academy str. 5, 2028 Kishinev, Moldova 3 Instituto de Ciencia Molecular, Universidad de Valencia, Polígono de la Coma, s/n 46980 Paterna, Spain

ABSTRACT

We propose a toy model to describe the exchange coupling between the localized spins mediated by the itinerant electron in complex mixed-valence (MV) polyoxometalates hosting the metal ions. The model takes into account the key interactions that are common for all of such kind systems, namely, electron transfer in the MV moiety and magnetic exchange between the localized spins and the delocalized electrons. In the case of relatively strong exchange coupling the combined action of these two interactions is shown to give rise to a specific kind of double exchange coupling termed here “external core” double exchange. In the opposite case of relatively strong electron transfer the general Hamiltonian is shown to reduce to the effective Hamiltonian of the indirect exchange. A possibility to control the coupling between the localized spins by the external electric field as well as limitations and perspectives of the present model for molecular spintronics and spin qubits are discussed.

1. INTRODUCTION

Polyoxometalates (POMs) are molecular species of anionic metal oxides which possess outstanding properties and functionalities [1-5]. Along with the traditional areas of applications such as medicine [6], magnetism [7], water splitting [8], catalysis [9], during the last decade these systems also attract attention as the objects of the emerging field of molecular spintronics. A rich variety of sizes and shapes of POMs as well as their ability to accommodate electrons delocalized over metal sites predetermine functionality of these systems in the spintronics of the future. As molecular spintronics is dealing with the spin of the electrons (along with the charges used in conventional electronics) the MV POMs are especially promising when along with the itinerant electrons they host also localized spins. Such kind of reduced POMs accommodating metal ions with unfilled shells are advantageous owing to the possibility to accept various number of electrons which are the delocalized and able to mediate spin coupling between the localized spins. Since the itinerant electrons are quite sensitive to the external stimuli due to their mobility, one can hope to efficiently control the spin states of these system. Thus, due to a large dipole moment of MV systems in instantly localized states the external electric field is expected to be especially interesting as a factor controlling spin states in such kind of systems through the modulation of the magnetic exchange mediated by the itinerant electrons.

In this communication we propose a simplified descriptive model for a special class of POMs which combine in their structure spin-localized and spin-delocalized moieties and, therefore, can be referred to as partially delocalized MV systems. Two

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representative examples of bicapped MV POMs [10b] are shown in Figure 1. In the Keggin anion [PMo1MoVI

11MoV2O40(VO)2] the two (VO)2+ (S=1/2) are located on

opposite tetragonal sites (Fig.1a), while the extra electron

(a)

(b)

Figure 1. Structures of two bicapped MV POMs: Keggin anion [PMoVI

11MoV O40(VO)2] containing two (VO)2+ groups (a); [PMoVI11

MoVO40 {Ni(phen)2(H2O)}2] linked to two Ni2+ in opposite MoO6 sites (b).

is delocalized over the Mo network. A similar physical situation occurs in reduced POM [PMoVI

11MoVO40{Ni(phen)2(H2O)}2] in which two Ni2+ ions (S=1) occupy two opposite MoO6 sites (Figure 1b). In both cases the direct coupling between considerable remote localized spins is negligible, so that the effective magnetic interaction between them is only transmitted via the delocalized electron.

2. THE MODEL

The proposed toy model includes the main interactions that are common for the partially delocalized systems but intentionally ignores the specific features that would restrict the applicability of this minimal model. Thus, a complex multiroute spin-delocalized subsystem (peculiar for most POMs) will be mimicked by a MV dimer in which the extra electron is delocalized over two spinless cores B and C. The localized spins SA=SD=S0 (for example, metal sites with unfilled shells) are assumed to occupy the two terminal positions A and D which are linked with the two sites of the MV dimer (B and C) forming thus a linear tetramer as shown in Figure 2a. To quantify consideration of this model one can introduce the following Hamiltonian:

( ) CDCBABBCCB nJnJcccctH sSsS ˆ2ˆ2ˆ21

−−+= ∑±=

++

sssss . (1)

The first term in Eq. (1) represents the one-electron transfer over two orbitals located

on the sites B and C of the MV dimer B-C, +sic and sic are the creation and

annihilation operators, ˆBn and ˆCn are the operators of the site populations, 1ˆˆ =+ CB nn . The extra electron localized on the sites B ( 1 2Bs = ) or C ( 21=Cs ) is coupled to the terminal spins SA or SD through the Heisenberg-Dirac-Van-Vleck exchange interaction (last terms in Eq. (1) ) giving rise to the two spin states of the A-B pair

0* 1 2ABS S S≡ = ± , or 210* ±== SSSCD in the C-D pair.

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(a) (b) Figure 2. The model spin sites in partially delocalized system and EC

double exchange (a); scheme of spin sites and ferromagnetic coupling in the conventional model of the double exchange (b).

The Hamiltonian, Eq. (1) describes the spin polarization mechanism within which the spin in the MV unit keeps its orientation while travelling and, therefore, polarizes localized terminal spins which prove to be effectively coupled. In its physical sense the polarization mechanism under consideration is similar to the double exchange proposed to explain ferromagnetic ordering in MV perovskite [11, 12]. The double exchange mechanism assumes that the spin of the extra electron is coupled through a strong intra-center (“internal”) ferromagnetic (Hund type) exchange interaction with the spins of the ions (so-called “spin cores”), thus aligning them parallel to the spin of the moving electron which keeps its orientation while travelling. As distinguished from this conventional picture, the mechanism we propose here involves inter-center exchange coupling of the extra electron with the alien (“external”) spin cores which are the terminal centers A and D. We will refer this specific polarization mechanism to as external core (EC) double exchange. It is remarkable that as distinguished from the intracenter (Hund) exchange that is always ferromagnetic, the inter-center exchange between the extra electron and the alien spin core can be either ferro- or antiferromagnetic depending on the character of the bridging ligands, participating orbitals and topology of the system. Note also that the intracenter exchange in the conventional double exchange normally strongly exceeds the transfer integral, so the non-Hund configurations can be neglected. On the contrary, in the EC double exchange problem the transfer integral can be comparable or even larger than the intercenter exchange integral leading thus to a strong mixing of “Hund” and “non-Hund” configurations.

Mathematical similarity of the two mechanisms of spin polarization suggests to use here the angular momentum techniques adopted in the consideration of the double exchange for polynuclear MV clusters [12-14]. A convenient basis set in the case under consideration is given by the spin-coupled representation which can be designated as

( ) ( ) SABSDABeA MSSSMSSSsS ,,,, 21

0 ±=≡

and ( ) ( ) SCDSACDeD MSSSMSSSsS ,,,, 2

10 ±=≡ ,

where the two sets with the definite SAB and SCD and full spin S (S=1/2, 3/2,…2S0+1) belong to the two possible localizations of the extra electron (on the sites B and C respectively). Using the spin recoupling equations for a three-spin system (see, for example, [12-14]) one can obtain the following expressions for the matrix elements of the transfer Hamiltonian (first term in Eq. (1) ):

tMSSSSHMSSSS SmaxCDtrSmaxAB ==+==+= ,,21ˆ,,21 00 ,

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( ) ( )1221,,21ˆ,,21 000 ++=+=+= SStMSSSHMSSS SCDtrSAB , (2)

( ) ( )202

00 12211,,21ˆ,,21 ++−=−=+= SSttMSSSHMSSS SCDtrSAB .

The matrix of the full Hamiltonian is blocked accordingly to the values S of the allowed full spins of the system. Omitting the details of calculation we give here the final expressions for the energies as functions of the key parameters:

( ) ,212,21 000* tSJSSSSSE max ±−=+==+=±

( ) ( ) ( )

++−+±=<−

±22

02 421412

21 tStJSJJSSE max ,

( ) ( ) ( )

++++±=<+

±22

02 421412

21 tStJSJJSSE max . (3)

Since the energy levels are characterized by the total spin of the system, an obvious conclusion that one can draw from Eq. (3) is that the travelling electron couples the terminal localized spins in compliance with the qualitative consideration. A measure of this coupling as well as its magnetic manifestations can be determined by analyzing the energy pattern of the system. We will first consider the limiting cases in order to reveal the main peculiarities of this coupling and then pass to a more general case.

3. Strong exchange limit The indicative is the case of a strong exchange limit when t /|J |<< 1. For the sake of definiteness in subsequent calculations we assume that t > 0 and J < 0 (antiferromagnetic exchange). Expanding the energies in a series over the ratio t/|J| we find the following expressions (with accuracy up to linear terms):

( ) ( ) ( )1221,21 000 ++±=+=∗± SStSJSSSE (4)

with 1/ 2, ... 1, ,max maxS S S= − and

( ) ( ) ( ) ( )0 0 01 2, 1 1 2 2 1E S S S J S t S S∗± = − = − + ± + + (5)

with 1,...21 −= maxSS . Figure 3a shows the energy pattern in strong exchange limit for a particular case of S0=1. This pattern consists of the two exchange multiplets separated by the gap 3|J|, each split by the electron transfer in the pairs of levels with different total spin values S separated by the energy gaps

( ) ( ) ( )02 1 2 2 1 2 1 2 3S t S S t S∆ = + + = + . Such spin-dependence of the splitting caused by the electron delocalization is indicative for the double exchange interaction [12] with the only difference that the exchange interaction with the external spin cores can be antiferromagnetic (e. g., in the case under consideration). As a result in the most typical for POMs case of antiferromagnetic intercenter exchange the EC double exchange stabilizes the level with the total spin value Sgr=Smax ̶ 1=3/2 for which the spins of the two cores are antiparallel to the spin of the extra electron. The first excited state corresponds to the situation when the spins of the spin-cores (regarded as classical vectors) are tilted with respect to each other giving rise to the spin 1/2. The gap t/3 can be considered as a measure of the coupling between the localized spins in the limit of strong exchange, so that in this case the coupling is proportional to the transfer parameter.

283

(a)

(b)

Figure 3. Energy pattern in the cases of strong limit of exchange (a) and strong electron transfer (b) in a particular case of S0=1. The exchange interaction is assumed to be antiferromagnetic (arrows). The full gaps of the split multiplets are intentionally shown out of the scale appropriate for the perturbation theory (see text).

4. STRONG TRANSFER LIMIT

In the opposite case of strong transfer limit, |J|/t << 1, one can find approximate expressions for the energy levels by expansion of the exact expression, Eq. (3), over the ratio |J|/t. Alternatively, the first order expressions can be obtained by projecting the full Hamiltonian,

Eq. (1), onto the restricted Hilbert spaces corresponding to the zeroth order energies +t and −t (bonding and antibonding orbitals for the B-C dimer). Actually, this means that the perturbative Hamiltonian can be obtained from the exchange part of the initial Hamiltonian by substituting the occupation number operators by their expectation values 21ˆˆ == CB nn . The projected Hamiltonian represents the exchange Hamiltonian

( )eDeAex JH sSsS +−= (6) for the linear trimer with localized spins in which se is the spin operator of the extra electron connected to the terminal spins through the effective superexchange with

2effJ J= . Therefore, in the strong transfer limit the extra electron can be thought as a spin located at the inversion center (L) of the MV dimer and coupled to the terminal spins through the effective exchange interaction (Figure 4a). It is remarkable that the last proves to be twice smaller compared with the initial exchange due to the fact that actually the spin density in the MV unit is shared between the two centers. The eigenvalues of the Hamiltonian, Eq.(6), are characterized by the intermediate spin SAD and the total spin S of the system:

( ) ( ) ( ) ( )[ ]431121 −+−+−= ADADAD SSSSJS,SE . (7)

Finally, the energy levels in strong transfer limit can be found as:

( ) ,25,2 JtE −±= ( ) ( ) ,2323,2,223,1 JtEJtE +±=−±=

284

( ) ( )0,1 2 , 0,1 2 .E t E t J= ± = ± + (8) The results for a particular case of S0=1 are illustrated in Figure 3b. The energy pattern, Eq. (8), consists of the two levels with the energies +t and −t of the MV dimer, each split by the exchange

(a) (b)

Figure 4. Illustration for the origin of the energy pattern in the case of strong electron transfer limit (a) and influence of the external electric field (b).

interaction into five sublevels with the definite total spins. One can also conclude that in the limit of strong transfer that can be regarded as the case of indirect exchange the measure of the coupling between the localized spins mediated by the travelling electron is the effective exchange parameter.

5. THE GENERAL CASE The general case of arbitrary relationship between the t and J parameters can

be illustrated by the plot of energy levels E/t vs |J|/t in the case of S0=1 (Figure 5a) correlated with the temperature dependences of the magnetic susceptibility χT vs T (Figure 5b) at fixed J (J = −500 cm-1). Providing t = 0 the extra electron is coupled to one of the two spin-cores while the two spins S*=S0−1/2=1/2 (J < 0) and se=1/2 are not correlated. As a result the system up to room temperatures can be regarded as consisting of two independent spins and consequently, the ( ) ( )[ ++== ∗∗ 182 SSgTχ

( )]=+100 SS K/molemu375.1 ⋅ (g=2) is temperature independent. The electron transfer switch on the coupling between centers so that

(a)

(b)

Figure 5. Illustration to the general interrelation between J and tin A-B-C-D – molecule with S0=1: Correlation diagram for the energy levels (the levels with S=3/2, 1/2 and 5/2 are shown in blue, green and red colors, respectively) (a) and χT vs T dependences calculated with 1500 cmJ −= − and different values of t (b).

the low-temperature limit Tχ becomes K/molemu875.1 ⋅ in accordance with the above finding that the EC double exchange with J < 0 stabilizes the ground state with Sgr=Smax−1=3/2. For relatively weak transfer ( 150 cmt −= in Figure 5b) we are

285

dealing with the limit of strong J and consequently the gap between the ground state and the first excited level with S = 1/2 is approximately t/3. In this case under increase of the temperature the excited level becomes populated giving rise to the pronounced decrease of χT. Providing strong electron transfer (t=6000 cm-1 in Figure 5b) the energy gap between the ground and excited states is mainly determined by the parameter J rather than by t value (case of strong t limit), and so the further increase of t in this case produces only small changes of χT. Finally for the values of |J| which significantly exceed kBT (not shown in Figure 5b) the magnetic susceptibility calculated at t>>|J| obeys the Curie low with 1.875 emu K/molTχ = ⋅ . To conclude the discussion, it is to be noted that we have not considered the vibronic coupling which is known as a factor of localization reducing the transfer integral. That is why it can be said that it is parametrically included in the model as far as t is varied.

6. EFFECT OF THE ELECTRIC FIELD Since the external electric field is able to induce a large dipole moment in MV systems due to the presence of the tunnel levels of opposite parity, the redistribution of the electronic density induced by the field is expected to affect also the effective coupling between the localized spins in the system under consideration. In this respect it is especially interesting that one can achieve an efficient electric field control of the magnetic coupling between localized spins. To describe this concept one should add to the initial Hamiltonian, Eq. (1), the interaction of MV moiety with the external electric field ε. The Stark interaction is diagonal in the adopted spin-coupled representation

( ) ( ) ( ) ( ) WReSMSHSMSSMSHSMS SCDStarkSCDSABStarkSAB ≡−=−= 2ˆˆ e , (9)

where ABRR ≡ , e is the electronic charge, and W is the Stark energy in the field directed along the axis of the molecule. One can find the analytical expressions for the Stark levels which are numerated by the total spin:

( )

( ) ( ) ( )

( ) ( ) ( )

* 2 20 0 0

2 22 2 2 2 20

2 22 2 2 2 20

1 2, 2 1 2 ,

1 2 1 4 4 4 9 1 2 ,2

1 2 1 4 4 4 9 1 2 .2

max

max

max

E S S S S S JS W t

E S S J J S W t J W t S

E S S J J S W t J W t S

±

−±

+±

= + = = + = − ± +

< = ± + + + − + +

< = ± + + + + + +

(10)

The field dependence of the energy levels is illustrated in Figure 6a. The main effect of the field can be understood by analyzing the transformation of the levels at W=0 (described above)to the high field limit. One can see that the ground state possesses S=3/2 at any field but the energy gap δ separating this level from the first excited level with S=1/2decreases with the increase of the field (Figure 6b). The gap vanishes in the strong field limit which means that the ground levels represent a paramagnetic mixture of the spin states with S=1/2 and 3/2. The energies of the ground and excited levels comprising S=1/2 and 3/2 in this limit are defined by the approximate expressions 2 | |J W− which correspond to the situations when the extra electron if fully delocalized on the energetically

286

(a)

(b)

Figure 6. Electric field effects: (a) energy levels of the tetramer calculated with t/|J|=1 as functions of W/|J| (coloring as in Figure 5a); (b) field dependence of the gap δ for different values of t/|J|.

favorable site (ground state) or on the energetically unfavorable site (first excited state). Similarly, the two states comprising S=1/2, 3/2 and S=5/2 possess the energies| |J W . Therefore, the electric field suppresses the EC double exchange so that the two spins (S*= S0−1/2 and S0) become uncorrelated. The energy gap δ which determines the strength of the coupling between the localized spins, is a decreasing function of the field as shown in Figure 6b. Again, in the limit of strong transfer or/and weak field (t>>|J|, W) the system can be described by the three-spin indirect exchange Hamiltonian

eDeAex JJH sSsS ′′−′−=′ (11) that includes two different exchange parameters

( )( ) ( )( )2 1 , 2 1J J W t J J W t′ ′′≈ + ≈ − evaluated up to linear terms with respect to small ratio W/t. The extra electron in this limit can be viewed as localized spin which is shifted by the field from the inversion center (L). A pictorial representation of this situation is given in Figure 4b in which one can visualize the dipole moment induced by the field.

7. PERSPECTIVES OF THE MODEL FOR MOLECULAR SPINTRONICS AND SPIN QUBITS

Having solved analytically a complex system involving localized and delocalized electrons, with their spin and charge degrees of freedom, opens interesting perspectives for molecular spintronics and molecular spin qubit, as we will discuss below. Furthermore, the analytical prediction of how the application of electric and/or magnetic fields continuously will modify the Hamiltonian is potentially useful for employing these electrons in applications where a fine control of the evolution of the ground state is desired, such as adiabatic quantum computing [15].

Of course, before being applicable for such ambitious goals, our current approach would need to overcome important limitations. Let us start by discussing those. Firstly, from the point of view of MV systems in general: the effect of electric fields on electric charges is very intense compared with the effect of magnetic fields on magnetic moments. This can facilitate both external manipulation and inter-qubit communication but, since charge noise (and not only spin noise) will affect these spin qubits, they will also experience more decoherence. Secondly, from the point of view of the present theory: in the present model the molecule is isolated from its environment, which means it's not directly applicable to dynamical or open systems as

287

the ones we will discuss in this section. In particular, the possibility of coupling to electrodes is not considered, and neither are the coupling to phonons or to nuclear spins, or to a resonant cavity. This is why we do not aspire to model dynamical processes in this work. Nevertheless, since the present model can serve to understand the nature of the phenomena and interactions in the static regime, it can serve as foundation for the study of dynamical processes in the future.

The most immediate perspective would be using these systems for simple quantum algorithms such as quantum error correction. Since three S=1/2 spins (two localized spins and one delocalized spin) have an 8-dimensional Hilbert space, equivalent to three qubits, it is tempting to draw similarities to proposals for quantum gates or even quantum error correction [16]. Indeed, minimal quantum error correction can be achieved with simple 3-qubit codes, e. g. either bit-flip or phase-flip errors. For this one would need 7 distinguishable, coherent, addressable transitions that connect the 8 spin states of the Hilbert space [17]. A set of addressable transitions would be those not only allowed by EPR selection rules but also at such frequencies and fields that pulses can be effected in quick succession with realistic technology. The present model allows establishing the ratios and ranges of parameters that allow the transitions to be distinguishable and addressable by available EPR technologies. Further theoretical methods are currently being developed to study and optimize the coherence of these transitions versus both the spin bath [18] and the oscillator bath [19].

We need to note that the utility of this model extends much beyond molecules which in crystalline state present two localized spins coupled to a delocalized spin, which are relatively few. Indeed, in the currently rising field of single-molecule spintronics, molecules can be experimentally contacted in either a two- or a three-electrode setup. This means that it is possible apply not only a bias voltage to a given molecule, but it is often also possible to apply a gate voltage which allows the control the Fermi level, or, seen from the point of view of the molecule, the charge state and therefore the SOMO. Therefore, many molecules with two magnetic metal sites and an extended conducting HOMO or LUMO can be studied by the model presented here, since the extended orbital can function as a SOMO at the appropriate gate voltage. This is the reason why the analytical solution presented herein has interesting consequences in the context of molecular nano-spintronics. Indeed, both the experimental and theoretical states of the art are nearing this situation where two localized electron spins of a molecule are exchange-coupled to a third delocalized electron. Consider for example the recent theoretical study where precisely a system of this type was studied [20]. The situation with two localized electrons communicated by a delocalized electron is also the natural extension of the current experimental state of the art, where electronic spin transitions in a single molecule are detected by a change in a spin filtering effect, using two distinct parts of the same molecule, and this is being used to read-out and address a double nuclear spin qubit [21]. In that particular case, the progression from one single molecule magnet (SMM) to two transfer-coupled SMMs is an obvious way of advancing toward a higher number of qubits and more complex quantum operations, and the theoretical framework needs to be set up in advance. A further landmark experiment achieved supramolecular spin valves which relied on a weak exchange coupling between multiple localized spins of SMMs and delocalized electrons on a single-walled carbon nanotube [22]. The analytic model solved here allows to understand more in depth these challenging spintronics problems as a generalization of the comparatively

288

simple exchange transfer problem. This allows the analysis of the key interactions and a better understanding of the nature of these processes.

Beyond proper (chemical) molecules, the equations derived here are similarly useful to “physical molecules”, for example semiconductor multiple quantum dot [23], or intermediate systems such as carbon nanotube multiple quantum dots [24]. Being tunable, a physical quadruple quantum dot might actually be the most adequate to experimentally explore the validity of our model via a continuous variation between different parameter ranges. Indeed, coherent spin exchange via a quantum mediator has recently been achieved in this context using a triple quantum dot [25].

8. CONCLUDING REMARKS

To summarize, the following main results are to be mentioned: 1) we have proposed a toy models for the polarization mechanism describing the magnetic coupling between the arbitrary localized spins via delocalized electrons in complex MV POMs hosting metal ions. The proposed descriptive model leads to the exactly solvable problem that allowed us to qualitatively and quantitatively discuss the main features of whole class of the systems; 2) in the case of relatively strong exchange coupling the considered model predicts the existence of a specific kind of double exchange with participation of the external spin cores which seems to be peculiar for the partially delocalized MV systems; 3) in the opposite case of strong electron delocalization the general solution is reduced to the case of indirect exchange; 4) the model allows to quantify in general terms the effect of coupling. In particular, in the limit of strong exchange between localized and itinerant electrons (double exchange regime) the mediated coupling is determined by the transfer parameter, while in the limit of strong transfer (indirect exchange regime) the measure of the coupling is the exchange integral; 5) the consideration of the Stark effect illustrates a possibility to tune the coupling between localized spins by the electric field acting on the MV subsystem.

ACKNOWLEDGMENTS

A.P., S.A., B.T. and J.M.C. acknowledge support from the Ministery of Education and Science of Russian Federation (Agreement No. 14.W03.31.0001). B.T., J.M.C. and E.C. are grateful to COST Action CA15128: Molecular Spintronics (MOLSPIN). A.G.-A. acknowledges the European Union (ERC-CoG DECRESIM 647301) and the Spanish MINECO for a Ramón y Cajal Fellowship.

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