Mixed-Valence Triferrocenium Complex with Electric Field

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Mixed Valence Triferrocenium Complex with Electric Field Controllable Superexchange as a Molecular Implementation of Triple Quantum Dot Andrew V. Palii, Sergey M. Aldoshin, and Boris Tsukerblat J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09950 • Publication Date (Web): 08 Nov 2017 Downloaded from http://pubs.acs.org on November 11, 2017

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The Journal of Physical Chemistry

Mixed Valence Triferrocenium Complex with Electric Field Controllable Superexchange as a Molecular Implementation of Triple Quantum Dot Andrew Palii,1,2* Sergey Aldoshin,2 Boris Tsukerblat 3* 1

Institute of Problems of Chemical Physics, Chernogolovka, Moscow Region, Russia 2

3

Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau, Moldova

Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva, Israel E-mail: [email protected] (AP) ; [email protected] (BT)

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Abstract In this article we propose molecular implementation of the quantum logic gate originally realized by the linear triple-quantum dot array accommodating two electrons. To reach this goal we

propose to

employ the mixed-valence

triferrocenium complex exhibiting three III

isomeric forms with different oxidation degrees Fe -FeII-FeIII , FeIII-FeIII-FeII , FeII-FeIII-FeIII which correspond to three instant localizations of the two holes over three iron ions. The longrange interaction between the terminal metal sites is considered in the framework of the Hubbard-like Hamiltonian which accounts for the electron transfer,

inter- and intra-site

Coulomb repulsion and takes into account differences in the orbital energies of FeIII and FeII ions. The interaction of the electrons with the applied electric field is also included in the Hamiltonian. It is shown that due to long-range superexchange between the two electronic spins the ground state of trierrocenium complex is always a spin-singlet and the first excited level is a spin-triplet. The electric field is shown to increase the antiferromagnetic exchange coupling. The efficiency of the electric field control is especially pronounced due to the field induced transformation of the system from the isomeric form FeIII-FeII-FeIII to the form FeIII-FeIII-FeII. Estimations of the efficiency of the electric field control of the exchange coupling and entanglement show that the triferrocenium complex is emerging as potential candidate to act as a gate in quantum computing.


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1. Introduction To use the coupled semiconductor quantum dots (QDs) for quantum computation one needs to have a well-defined qubits (e.g., spins of the electrons confined to QDs), as well as a controllable source of entanglement, i.e., a mechanism through which the two qubits can be entangled to be able to produce the controlled NOT gate operation (the basic operation in quantum gate which inverts the binary signal). This goal was shown to be achievable through the time-dependent coupling of two spins by means of the exchange interaction in the double QD system1 with the exchange parameter being a function of the external electric and/or magnetic field. Later on it was suggested to use more distant (long-range) interaction between the spinqubits by creating the linear two-electron triple-QD arrays in which the long-range superexchange interaction between the two electrons occupying the outer dots occurs through the empty middle dot, which acts as a quantum mediator.2-6 Such long-range exchange interaction plays an important role in quantum computing making it possible to manipulate a distant quantum gate or qubit in one step, which is of higher operating efficiency and fault-tolerant capability than nearest-neighbor control 1 in exchange-based quantum gates.

Fig. 1. Schematic structure of mixed-valence triferrocenium complex as chemical analog of triple two-electron QD system. The adopted numbering of iron sites and molecular coordinate frame: site 2 is chosen as the origin of the frame and a is the distance between the nearest neighbor iron ions. The next step towards further miniaturization of the quantum logical gates is to use proper molecules instead of the so-called “physical molecules” composed of QDs (sometimes they are called “QD molecules” ). In search of true chemical molecules exhibiting the same 3 ACS Paragon Plus Environment


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mode of functionality as triple linear two-electron arrays of QDs we have paid attention on the linear mixed valence (MV) trimers comprising two extra electrons or holes delocalized over three or four spinless cores. This wide class of MV clusters can be exemplified by MV triferrocenium complex

7, 8

whose structure is shown in Fig. 1 (see review article

references therein). The three iron ions 2FeIII and FeII

9

and

arranged in a linear topology with the

distances a between neighboring ions. The low-spin FeIII and FeII ions possess crystal field electronic configurations in which this complex can be regarded as comprising two extra holes per three spin-less cores (d5 and d6). According to Ref. 1 a SWAP gate (a gate which realizes the interchanges of two qudits which is one of the basic operation) can be realized through the antiferromagnetic Heisenberg type exchange coupling between the two electrons in QDs systems. Conventionally, this coupling is described by the Hamiltonian Hˆ ex = −2 J S1 S2 , where S1 and S2 are the spin-1/2 operators (and thus the full spin of the system can be S= 0 and 1). The exchange parameter and consequently the singlet-triplet splitting 2 | J | is time-dependent and it is determined by the time dependence of the external stimuli (e.g. the external electric field) used to control the exchange coupling. Then the “state swap time” is defined as τ ∼ 1/ ( 2 | J |) . In the triple two-electron QD array the long-range superexchange Hˆ ex = −2 J S1 S3 between the electrons of the outer dots 1 and 3 is typically tuned through modulating the energy of the single occupied level of the central dot 2 by applying a gate voltage. 5, 6 Provided that the energy gap δ between the orbitals of the central and the outer QDs strongly exceeds the transfer integral t between the nearest neighboring dots, the effective transfer parameter t13 between the two terminal dots 1 and 3 is given by the second order expression t13 = − t12t23 δ = − t 2 δ and hence the parameter J of the kinetic

superexchange

is

given

by

the

approximate

fourth-order

expression5,6

J = −2 t132 U = − 2t 4 (U δ 2 ) , where U is approximately the on-dot Coulomb energy.

It is to be noted that being efficient in QD system, this way of control of superexchange coupling (which is local by it's nature), is difficult to implement in molecular system, in particular in triferrocenium complex. Indeed, due to much smaller size of the molecule as compared with that of triple arrays of QDs it is difficult to selectively address an external stimulus to a certain constituent atom, that is to change the energy of the central iron site without affecting the neighboring outer sites. For this reason here we will focus on the alternative way of tuning the exchange interaction in the triferrocenium complex, by considering the effect of the external homogeneous electric field applied along the axis passing through the three iron ions. Using the extended 4 ACS Paragon Plus Environment

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Hubbard-type Hamiltonian with the three parameters (transfer integral t, on-site Coulomb energy U and the energy difference δ between the isomers FeIII-FeII-FeIII and FeIII-FeIII-FeII) previously reported for this and similar systems we will demonstrate that the electric field of an attainable strength can be used for the efficient control of the exchange coupling, magnetic properties and quantum entanglement between electronic spins in triferrocenium complex. 2. The model Note that the results are independent of the sign of the charge, so the discussion will be done in terms of the two electrons. The model assumes that the two electrons are shared among three one-orbital sites, and consequently the consideration is based on the following Hubbardtype Hamiltonian:

(

Hˆ = ∑ (Ei + e Ri F ) ni ,σ + U ∑ ni ,↓ ni ,↑ + ∑ ∑ U i , j ni ,σ ni ,σ ′ + t ∑∑ ci+,σ ci +1,σ + ci++1,σ ci ,σ i ,σ

i

i < j σ ,σ ′

)

.

(1)

i =1, 3 σ

The first term in Eq. (1) represents the one-electron one-site part of the Hamiltonian, where Ei + e Ri F , is the orbital energy of the i-th iron site in the external homogeneous electric field

F, e is the charge of the electron. We assume that the on-site energies of the ground states Ei of the outer sites in zero electric field are equal and different from the energy of the central site, i.e. E1 = E3 ≡ E , E2 ≡ E ′ . The second term in Eq. (1) describes the on-site Coulomb repulsion, with the parameter U that is assumed to be the same for all three sites. The third term describes the intersite Coulomb repulsion which is characterized by the two different Coulomb parameters U12 = U 23 ≡ V , U13 ≡ V ′ . Finally, the last term in Eq. (1) describes the intersite electron hopping, where t = t12 = t23 is the parameter of the electron transfer between the nearest-neighboring sites, + while the direct transfer between the next neighbors is neglected, t13 = 0 . Operator ciσ ( ciσ )

creates (annihilates) electron with the spin projection σ on the ground orbital located on the site i, ni ,σ are the occupation numbers. The potential exchange terms are not included in the Hamiltonian, Eq. (1), while the kinetic exchange (which is usually more important) will be shown to appear due to the electron (hole) hopping processes. Now we will pass to the matrix representation of the Hamiltonian, Eq. (1). The basis we use includes the three spin-triplet states ψ i j ( S = 1) ( i, j = 1, 2,3, i ≠ j ) and six spin-singlet states

ψ i j ( S = 0 ) (i and j can be both equal and different from each other), where i and j indicate the iron. Note that the number of states with S = 1 is half of that with S = 0 because the spin-triplet states with double occupancy are forbidden due to the Pauli principle. Let us consider first the diagonal matrix elements of the Hamltonian, Eq. (1). We will consider the energy 5 ACS Paragon Plus Environment


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ψ 13 ( S ) Hˆ ψ 13 ( S ) = 2 E + V ′ of the isomer FeIII-FeII-FeIII as the reference energy. The energy difference δ between the isomers FeIII-FeII-FeIII and FeIII-FeIII-FeII) can be written as follows:

ψ 12 ( S ) Hˆ ψ 12 ( S ) − ψ 13 ( S ) Hˆ ψ 13 ( S ) = ψ 23 ( S ) Hˆ ψ 23 ( S ) − ψ 13 ( S ) Hˆ ψ 13 ( S ) = E′ − E + V − V ′ ≡ δ .

(2)

Finally, the energy gaps between the highest

excited

spin singlets

levels with single site occupancies and the

with double site occupancy are given

by the following

expressions:

ψ 11 ( S = 0 ) Hˆ ψ 11 ( S = 0 ) − ψ 13 ( S ) Hˆ ψ 13 ( S ) = ψ 33 ( S = 0 ) Hˆ ψ 33 ( S = 0 ) − ψ 13 ( S ) Hˆ ψ 13 ( S ) = U − V ′ ≈ U ,

(3)

ψ 22 ( S = 0 ) Hˆ ψ 22 ( S = 0 ) − ψ 13 ( S ) Hˆ ψ 13 ( S ) = 2 E ′ + U − 2 E − V ′ ≈ U .

(4)

For the sake of simplicity here we have neglected the small energies V ′ and 2 ( E ′ − E ) − V ′ as compared with the large on-site Coulomb repulsion energy U. The electron transfer operator (last term in Eq. (1)) mixes the states ψ i j ( S ) and ψ i k ( S ) with i ≠ j, k and k-j=1 (nearest neighbors) and also the spin singlets ψ j j ( S = 0 ) and

ψ jk (S = 0 )

with

k-j=1.

The

corresponding matrix elements can be calculated using the expressions of the two-electron states in terms of Slater determinants. Taking into account all above observations we arrive at the following energy matrices for spin triplets and spin singlets:

  Н ( S = 1) =   

ψ 12 (1)

ψ 13 (1)

ψ 23 (1)

δ −W

t

0

t

0

t

0

t

δ +W

  ,  

(5)

ψ 12 ( 0 ) ψ 13 ( 0 ) ψ 23 ( 0 ) ψ 11 ( 0 ) ψ 22 ( 0 ) ψ 33 ( 0 ) δ −W   t  0 Н ( S = 0) =   t 2   t 2   0 

t 0

0 t

t 2 0

t

δ +W

0

0

0

U − 2W

0

t 2

0

0

t 2

0

    t 2 t 2 . 0 0   U 0   0 U + 2W  t 2 0

0 0

(6)

In Eqs. (5) and (6) the parameter W = −e Fa having the sense of the Stark energy is introduced as the measure of the interaction with the electric field. The matrix of the Stark interaction 6 ACS Paragon Plus Environment

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e F ∑ Ri ni ,σ is defined with respect to the adopted molecular coordinate frame (Fig. 1). It i ,σ

should be also noticed that the field independent parts of the matrices Hˆ ( S = 1) and Hˆ ( S = 0 ) in Eqs. (5) and (6) can be regarded as a particular case (linear topology) of a more general energy matrices given in Ref.

2

which are related to the arbitrary topology of the triple lateral two-

electron QD system. The energy pattern of the triferrocenium complex is determined by the three parameters

δ, t and U. The energy difference δ between the isomers FeIII-FeII-FeIII and FeIII-FeIII-FeII was estimated to be 0.12 eV ≈ 968 cm−1.

7

As to the electron transfer parameter it is reasonable to

assume that for triferrocenium complex one can use the same value t = 50 meV (≈ 400 cm−1) as that reported for diferrocenylacetylene molecule.10 Finally, for the on-site Coulomb integral U we will use the value U = 4 eV (≈ 32000 cm−1).11

3. Electric field control of superexchange coupling The analysis of the dependences of the low-lying energy levels of the linear triferrocenium complex on the applied electric field is based on the diagonalization of the energy matrices in Eqs. (5), (6). The calculations show that as in the linear triple two-electron QD system the ground state of trierrocenium complex is always a spin-singlet and the first excited level is a spin-triplet. We will mainly focus on the elucidation of the field dependence of the singlet-triplet gap ∆ S −T = 2 | J | arising from the long-range superexchange between the two electronic spins. For the adopted set of the values of the parameters δ, t and U the singlet-triplet gap is found to be ∆ S −T ≈ 5 cm −1 at zero electric field. In the absence of the field the electrons are predominantly localized at the outer sites in order to minimize the Coulomb repulsion. The electric field polarizes the system resulting in a partial population of neighboring sites and consequently acts to overcome the Coulomb repulsion between the two electrons enhancing thus the antiferromagnetic exchange coupling. These qualitative consideration is confirmed by the results of calculations which are illustrated by Fig. 2a showing that the increase of the Stark energy W from 0 to 3000 cm−1 increases four times the singlet-triplet gap ∆ S −T . The energy pattern reveals several ranges of the field corresponding to different kinds of the field dependence of the levels (Figs. 2a, b). Thus, the increase of W from 0 to ≈500 cm−1 leads to a slow increase of the gap ∆ S −T from ≈ 5 to ≈7 cm−1 and further increase of W results in a more pronounced increase of this gap to the value of around 17 cm−1 at W = 1500 cm−1 (Fig. 2a). Note that a more significant increase of the singlet-triplet gap occurs in the vicinity of the



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critical value W = δ = 968 cm −1 at which the electric field is able to cause the transformation of the system from the isomeric form FeIII-FeII-FeIII to the isomeric form FeIII-FeIII-FeII which requires to overcome the Coulomb energy gap δ . This corresponds to the transition from the case of exchange between the distant spins (FeIII-FeII-FeIII) to the situation when the exchange coupling occurs between the nearest neighboring spins (FeIII-FeIII-FeII and FeII-FeIII-FeIII). At W >1500 cm−1 the gap ∆ S −T starts to increase more slowly again. Finally, the second pronounced increase of the gap ∆ S −T occurs in the vicinity of the critical value W = U − δ = 31000 cm −1 (Fig. 2b) when the electric field is strong enough to overcome the on-site Coulomb interaction and to transform the system from the isomeric form FeIII-FeIII-FeII to the fully polarized form for which one of the iron sites (terminal site) is doubly occupied while the remaining two sites are empty. For W >> 31000 cm −1 the dependence ∆ S −T (W ) becomes linear.

(b)

(a)

Fig. 2. Energy levels of the linear triferrocenium MV complex calculated as functions of the Stark energy W ranging from 0 to 3000 cm−1: (a) field dependences of the two low-lying levels with S=0 and 1 demonstrating fast increase of the singlet-triplet gap in the vicinity of W = δ = 968 cm −1 (vertical section); (b)field dependences of the two low-lying spin singlets and one spin triplet demonstrating fast increase of the singlet-triplet gap in vicinity of the value W = U − δ ≈ 31000 cm −1 (vertical section). To clarify the physical sense of the two mentioned critical values of W, we have plotted in Fig. 3 the two low-lying levels with S = 0 as functions of W for two ranges of W. It is seen that the spin singlets undergo two avoided crossings. First occurs at W = 968 cm −1 which corresponds to the value of δ (Fig. 3a), while the second one takes place at ≈ 31000 cm −1 which corresponds to the energy U − δ . The adiabatic increase of the electric field changes the distributions of the two electrons over the three sites from the distant distribution 1,3 (phase of 8 ACS Paragon Plus Environment

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long range superexchange, ground spin singlet) to the nearest neighbors distribution 1, 2 (phase of nearest neighbors exchange ground spin singlet) in course of the passage through the first avoided crossing point (Fig. 3a). Further increase of the field leads to the change from the nearest neighbors distribution to the on-site one (distribution 1, 1 corresponding to the on-site ground spin singlet phase) while passing through the second avoided crossing point (Fig. 3b). In the non-adiabatic regime one can expect a possibility to generate the states representing the mixtures of different electronic distributions through the Landau-Zener transitions, which are dependent on the sweep rates of the electric field. A remarkable feature of the energy pattern shown in Fig. 2a is that the change of the gap

∆ S −T is quite gradual even in the vicinity of the point W = δ . This is because the electron delocalization in the considered triferrocenium complex is rather strong precluding thus from the

(a)

(b)

Fig. 3. Field dependences of the low-lying spin-singlets of the linear triferrocenium MV complex showing avoided crossing points at (a) W = δ = 968 cm −1 (vertical section), and (b) W = U − δ ≈ 31000 cm −1 (vertical section). The pairs of numbers 1, 2 etc. show the

dominant electronic distributions. effective polarization of the electronic density by the electric field. In this context it is worth to mention that the degree of localization of the two electrons on the terminal sited is determined by the ratio t/δ. Thus in the limit of a weak transfer the electrons are predominantly localized on the terminal sites while providing relatively strong transfer the electronic density is homogenuously distributed over three sites. In the present case of triferrocenium complex we dealing with the moderate localization which accounts for gradual field dependence of singlet gap (Fig. 2). For more localized systems the increase of the singlet-triplet gap with the increase of W is expected to be more abrupt as evidences from Fig. 4 in which the hypothetical case of smaller value of t is presented.



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Let us estimate the values of the external electric field corresponding to the Stark energies |W| falling within the interval 0 ÷ 3000 cm−1 as shown in Figs. 2a and 3a. For the o

shortest Fe…Fe distance a = 6.8 A (typical Fe-Fe distance in such kind of complexes) this gives o

the electric field ranging from 0 to F = 5.5 ×10 6 V cm = 0.055 V A .

This estimation shows

that the maximal field is rather strong but attainable and does not exceed the dielectric strength of the crystal, while the value W = δ = 968 cm −1 corresponds to the field of 1.8 ×106 V cm . As to the value W=U− δ ≈ 31000 cm−1 at which the second pronounced increase of the singlet-triplet gap takes place, it corresponds to a very strong electric field of around 5.7 × 107 V cm that seems to be at the borderline of practical attainability.

Fig. 4. Field dependence of the two low-lying energy levels of hypothetic linear mixed valence complex with t = 50 cm −1 and the same values of δ and U parameters as those for triferrocenium complex for W ranging from 0 to 3000 cm−1 (vertical section for W = δ = 968 cm −1 is shown as well). The above described consideration along with the estimation of the actual parameters conceptually confirms a possibility of the electric field control of the exchange coupling in the system under consideration and, in particular, to achieve time-dependent exchange coupling induced by a time-varying electric field.

3. Electric field control of magnetic susceptibility and entanglement In context of the electric field control of superexchange between two electrons playing role of qubits, it is worthwhile to consider the quantum entanglement that is one of the main phenomena employed for realization of quantum logic operations. The quantum mechanical concept of entanglement means that the wave function of the system consisting of two parts cannot be presented as a product of the states of the constituent components. Therefore, in a 10 ACS Paragon Plus Environment

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complex system it is impossible to describe one part without knowledge of the state of the second part

(see detailed descriptions of the conceptual issues and related problems of

molecular magnetism in refs weak interaction.

12-22

). Entanglement in quantum systems is correlated by some

In order to achieve the implementation of some algorithms, one should

generate the entanglement of the states of two connected qubits. As a physical measure of the interaction between qubits the gap between the low lying levels of the system (which disappears in the absence of interaction) can be considered. As far as we are dealing with the two s=1/2 particles, the gap is actually the singlet-triplet separation. It is seen from Fig. 2 that for all W values not exceeding the value of ≈ 30000 cm −1 the two lowest energy levels with S = 0 and 1 are well separated from the rest of the levels. Providing further increase of the electric field, however, the second excited spin singlet approaches the first excited spin triplet as shown in Fig. 2b. Still, in a wide range of the values of the Stark energy W including most attainable values ranging from 0 to 3000 cm−1 (Fig. 2a) the low-lying part of the energy pattern comprises two isolated levels (S=0 and S=1), while the rest of the spin-levels are significantly higher in energy. Therefore, while analyzing the magnetic properties of the triferrocenium complex at all reasonable temperatures (up to room temperatures) we can restrict ourselves to considering only these two lowest levels which can be described by the Heisenberg exchange Hamiltonian with the parameter defined as J = − ∆ S −T 2 , were the singlet-triplet gap ∆ S −T (and consequently the parameter J ) is a function of the Stark energy W. We thus arrive at the conclusion that the mixed valence triferrocenium complex subjected to the action of not too strong electric field (W

Mixed-Valence Triferrocenium Complex with Electric Field

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