Spin electric effects in molecular antiferromagnets
Molecular nanomagnets show clear signatures of coherent behavior and have a wide variety of effective low-energy spin Hamiltonians suitable for encoding qubits and implementing spin-based quantum information processing. At the nanoscale, the preferred mechanism for control of quantum systems is through application of electric fields, which are strong, can be locally applied, and rapidly switched. In this work, we provide the theoretical tools for the search for single molecule magnets suitable for electric control. By group-theoretical symmetry analysis we find that the spin-electric coupling in triangular molecules is governed by the modification of the exchange interaction, and is possible even in the absence of spin-orbit coupling. In pentagonal molecules the spin-electric coupling can exist only in the presence of spin-orbit interaction. This kind of coupling is allowed for both and spins at the magnetic centers. Within the Hubbard model, we find a relation between the spin-electric coupling and the properties of the chemical bonds in a molecule, suggesting that the best candidates for strong spin-electric coupling are molecules with nearly degenerate bond orbitals. We also investigate the possible experimental signatures of spin-electric coupling in nuclear magnetic resonance and electron spin resonance spectroscopy, as well as in the thermodynamic measurements of magnetization, electric polarization, and specific heat of the molecules.
The control of coherent quantum dynamics is a necessary prerequisite for quantum information processing. This kind of control is achieved through coupling of the internal quantum degrees of freedom of a suitable micro- or mesoscopic system to an external classical or quantum field that can readily be manipulated on the characteristic spatial and temporal scales of the quantum system.
The molecular nanomagnets (MNs) Gatteschi et al. (2006); Gatteschi and Sessoli (2003) represent a class of systems that show rich quantum behavior. At low energies, the MNs behave as a large spin or a system of only few interacting spins. The behavior of this spin system can be designed to some degree by altering the chemical structure of the molecules, and ranges from a single large spin with high anisotropy barrier, to small collections of ferro- or antiferromagnetically coupled spins with various geometries and magnetic anisotropies. This versatility of available effective spin systems makes the MNs promising carriers of quantum information Leuenberger and Loss (2001a). While the interaction with magnetic fields provides a straightforward access to the spins in an MN, it is preferable to use electric fields for the quantum control of spins, since the electric fields are easier to control on the required short spatial and temporal scales. In this work, we explore the mechanisms of spin-electric coupling and study the ways in which an MN with strong spin-electric coupling can be identified.
Quantum behavior of MNs is clearly manifested in the quantum tunneling of magnetization Chudnovsky and Gunther (1988); Awschalom et al. (1992); Sessoli et al. (1993); Thomas et al. (1996); Friedman et al. (1996); Wernsdorfer et al. (1997); Tejada et al. (1997); del Barco et al. (2003). A prototypical example of quantum tunneling of magnetization is the hysteresis loop of an MN with a large spin and high anisotropy barrier. The height of the barrier separating the degenerate states of different magnetization leads to long-lived spin configurations with nonzero magnetic moment in the absence of external fields. The transitions between magnetization states in the MN driven through a hysteresis loop occur in tunneling events that involve coherent change of a many-spin state. These transitions have been observed as step-wise changes in magnetization in single-molecule ferromagnets Gatteschi et al. (1994); Thomas et al. (1996); Friedman et al. (1996); Sangregorio et al. (1997); Gatteschi et al. (2000). Similar tunneling between spin configurations are predicted in antiferromagnetic molecules Chiolero and Loss (1998); Meier and Loss (2001), and the observed hysteresis was explained in terms of the photon bottleneck and Landau-Zener transitionsLeuenberger and Loss (2000a); Chiorescu et al. (2000a, b); Waldmann et al. (2002). The transitions between spin states are coherent processes and show the signatures of interference between transition paths Loss et al. (1992); Leuenberger and Loss (2000b); Leuenberger et al. (2003), as well as the effects of Berry phase in tunneling Loss et al. (1992); Wernsdorfer and Sessoli (1999); Leuenberger and Loss (2001b); Leuenberger et al. (2003); González and Leuenberger (2007); González et al. (2008).
Spin systems within molecular nanomagnets offer a number of attractive features for studying the quantum coherence and for the applications in quantum information processing Leuenberger and Loss (2001a). A wide variety of spin states and couplings between them allows for encoding qubits. Chemical manipulation offers a way to modify the structure of low-energy spin states Troiani et al. (2005a). Coherence times of up to Ardavan et al. (2007) which can persist up to relatively high temperatures of the order of few Kelvin are sensitive to the isotopic composition of the molecule. A universal set of quantum gates can be applied in a system of coupled antiferromagnetic ring molecules, without the need for local manipulation Carretta et al. (2007). The presence of many magnetic centers with the coupled spins allows for the construction of spin cluster qubits that can be manipulated by relatively simple means Meier et al. (2003). In polyoxometalates, the spin structure of the molecule is sensitive to the addition of charge, and controlled delivery and removal of charges via an STM tip can produce useful quantum gates Lehmann et al. (2007). Chemical bonds between the molecules can be engineered to produce the permanent coupling between the molecular spins and allow for interaction between the qubits Timco et al. (2008).
Sensitivity of molecular state to the addition of charge was demonstrated in the tunneling through single molecules Heersche et al. (2006), and used to control the spin state of a MN Osorio et al. (2010). Transport studies of the MNs can provide a sensitive probe of their spin structure Romeike et al. (2006a, b); Leuenberger and Mucciolo (2006); González et al. (2008); Michalak et al. (2010).
The most straightforward and traditional way of controlling magnetic molecules is by applying an external magnetic field. With carefully crafted ESR pulses, it is possible to perform the Grover algorithm, or use the low-energy sector of the molecular nanomagnet as a dense classical memory Leuenberger and Loss (2001a). Unfortunately the approaches based on magnetic fields face a significant drawback in the large-scale quantum control application. Typically, the quantum manipulation has to be performed on the very short spatial and temporal scales, while the local application of rapidly varying magnetic field presents a challenging experimental problem. For that reason, the schemes for quantum computing tend to rely on modifying the spin dynamics that is caused by intramolecular interaction, rather than on the direct manipulation of spins Troiani et al. (2005b).
For the applications that require quantum control, the electric fields offer an attractive alternative for spin manipulation in the molecular nanomagnets Trif et al. (2008). One major advantage is that they can be applied to a very small volume via an STM tip Hirjibehedin et al. (2006); Bleszynski-Jayich et al. (2008), and rapidly turned on and off by applying voltage pulses to the electrodes placed close to the molecules that are being manipulated. Switchable coupling between different nanomagnets is essential for qubit implementation. At present, this can be implemented only locally, and the interaction is practically untunable. The use of microwave cavities can offer a solution to this problem. By placing the nanomagnets inside a microwave cavity, one can obtain a fully controllable, long-range interaction between themTrif et al. (2008). This coupling relies on the presence of a quantum electric field inside such a cavity, which mediates the interaction between distant nanomagnets. The interaction can be tuned by tuning each molecule in- or out-of-resonance with the cavity field using local electric or magnetic fieldsTrif et al. (2008). The spins, however, do not couple directly to the electric fields, classical or quantum, and therefore any electric spin manipulation is indirect, and involves the modification of molecular orbitals or the spin-orbit interaction.
The description of the molecular nanomagnets in terms of spins is an effective low-energy theory that does not carry information about the orbital states. However, it is still possible to predict the form of spin-electric coupling from symmetry considerations and single out the molecules in which such a coupling is possible. In particular, the molecules with the triangular arrangement of antiferromagnetically coupled spin- magnetic centers interact with external electric field through chirality of their spin structure Trif et al. (2008); Popov et al. (2009). The same coupling of chirality to the external electric field was derived for the triangular Mott insulators Bulaevskii et al. (2008).
While the symmetry of a molecule sets the form of spin-electric coupling, no symmetry analysis can predict the size of the corresponding coupling constant. The coupling strength will depend on the underlying mechanism that correlates the spin and orbital states, and on the detailed structure of low-energy molecular orbitals. To identify molecules that can be efficiently manipulated by electric fields, it is necessary to perform an extensive search among the molecules with the right symmetries and look for the ones that also have a large coupling constant. Unfortunately, this search has to proceed by ab-initio calculations of the coupling constants for a class of molecules of a given symmetry, or by an indiscriminate experimental scanning of all of the available molecules.
In this paper, we contribute to the search for molecules that exhibit strong spin-electric coupling. Based on the symmetry analysis, we identify the parameters of the spin Hamiltonian that can change in the magnetic field, and cause spin-electric coupling. We study the mechanisms that lead to this coupling and describe the experiments that can detect it.
We will consider the spin electric coupling in the language of effective model, namely either the spin Hamiltonian, or the Hubbard model. In reality the mechanism behind the spin-electric coupling involves either the modification of the electronic orbitals in an external field and the Coulomb repulsion of electrons, or the much weaker direct spin-orbit coupling to the external fields. A derivation of spin-electric coupling from this realistic picture would require the knowledge of electronic orbitals from an ab-initio calculation, and the distribution of electric field within the molecule. Both of these problems require substantial computational power, and can not be performed routinely. Since the electric field acts primarily on the orbital degrees of freedom, and the spin Hamiltonian carries no information about the orbital states, we provide a description in terms of a Hubbard model that still contains some information about the orbital states. We can then described the properties of the molecule that allow for strong spin-electric coupling in the language of orbitals that offers some intuitive understanding of the underlying mechanisms of interaction.
We identify the response of an MN with spin-electric coupling in the standard measurements of ESR, nuclear magnetic resonance (NMR), magnetization, polarization, linear magnetoelectric effect, and specific heat measurements.
In Sec. II we present a symmetry analysis of the spin-electric coupling in the ring-shaped molecules with antiferromagnetic coupling of spins. In Sec. IV, we describe the MNs using the Hubbard model, and relate the symmetry-based conclusions to the structure of molecular orbitals. In Sec V, we analyze the experimental signatures of spin-electric coupling, and present our conclusions in Sec. VI.
Ii Symmetry analysis of antiferromagnetic spin rings
Spin chains whose ground state multiplet consists of two quasi-degenerate S = 1/2 doublets represent suit- able candidates for the manipulation of the spin state by pulsed electric fields. Such a ground-state multiplet characterizes a number of frustrated spin rings, consisting of an odd number of half-integer spins. In the following we consider prototypical examples of such systems.
ii.1 Triangle of spins
The low-energy properties of most molecular nanomagnets (MNs) are well described in terms of spin degrees of freedom alone. Within the spin-Hamiltonian approach, the coupling of external electric fields to the molecule can be accounted by suitably renormalizing the physical parameters. In the following, we use the symmetry of the molecules to calculate the changes of spin-Hamiltonian parameters, to identify the system’s eigenstates, and to deduce the allowed transitions. Quantitative estimates of the parameters entering the spin Hamiltonian require the use of ab-initio calculations Bellini et al. (2006), or the comparison with experiments. The simplest example of a spin system which may couple to an external electric field in a non-trivial way is a triangle of s = 1/2 spins, like, for example, the Cu MNChoi et al. (2006). The schematics of such a spin system in the presence of an electric field is showed in Fig. 1. Its spin Hamiltonian, for the moment in the absence of any external fields (magnetic or electric), reads:
with and in the summation over . The first term in Eq. (1) represents the isotropic Heisenberg exchange Hamiltonian with the exchange couplings between the spins and , and the second term represents the Dzyalozhinsky-Moriya (DM) interaction due to the presence of spin-orbit interaction (SOI) in the molecule, with the DM vectors . The states of the spin triangle can be found by forming the direct product of the representations of three spins : , meaning there are eight states in total. The point group symmetry of the molecule is DChoi et al. (2006), i.e. the triangle is assumed to be equilateral. The D symmetry imposes the following restrictions on the spin Hamiltonian parameters: and , and . However, if lower symmetry is considered these restrictions will be relaxed. The spin states in a form adapted to the rotational symmetry C of the system are
where and . The states with opposite spin projection , i.e. with all spins flipped can be written in an identical way (not shown). These states are already the symmetry adapted basis functions of the point group D. Moreover, these are eigenstates of the chirality operator
with , and . The above states in Eq. (3) carry different total spin. There are two spin states, corresponding to , and a spin state corresponding to . Obviously, the states have .
In an even-spin system, double valued point groups, instead of single valued groups, are usually used in order to describe the states, the splittings and the allowed transitions (magnetic or electric)Tsukerblat (2006). In the presence of of spin-orbit interaction the splittings can be accounted for either by single group analysis (perturbatively), or by double group analysis (exact). In the following, we analyze the spectrum and the allowed transitions by both single valued point group analysis and double valued point group analysis.
Single valued group analysis of the spin triangle
In the single valued point group D, the states with form the basis of the two dimensional irreducible representation , while the states , and the transform as . The allowed electric transitions in the system are determined by the transformation properties of the basis states.
The simplest and possibly the dominant dependence of the spin Hamiltonian on the applied electric field comes via the modification of the exchange interactions, like depicted in Fig. 1. This gives rise to the following term in the spin Hamiltonian
where , with being vectors that describe the electric-dipole coupling of the bond to the electric field in leading order. There are three such vector parameters and thus nine scalar parameters in total. However, symmetry will allow to drastically reduce the number of free parameters by providing relations between them. The states of the unperturbed spin Hamiltonian form the multiplet , while the states form two multiplets . The electric dipole Hamiltonian is , with standing for the electron charge, being the coordinates of the -th electron and . The non-zero electric dipole matrix elements of in the D symmetric molecule are
proportional to the effective electric dipole parameter . The value of is not determined by symmetry, and has to be found by some other means (ab-initio, Hubbard modeling, experiments, etc). We mention that all the other matrix elements are zero, e.g. , etc. We see that the electric field acts only in the low-energy sector, which allows us to write the effective spin-electric coupling Hamiltonian acting in the lowest quadruplet as
where , with describing the rotation with an angle about the axis, and is the angle between in-plane component of the electric field and the bond . For we have
The low-energy spectrum in the presence of electric field and the related states can be expressed in terms of the spin Hamiltonian Eq. (5), so that we find anisotropic variations of the exchange coupling constants:
which depend on the angle and the projection of the electric field on the plane of the triangle. In the triangle the -operators can be written as
with ( are the Levi-Civita symbols)Trif et al. (2008); Bulaevskii et al. (2008). From the above relations we can conclude that (i) only the electric field component perpendicular to the bond and lying in the plane of the molecule gives rise to spin-electric coupling and (ii) there is only one free parameter describing the coupling of the spin system to electric fields and , where labels the triangle sites and .
The SOI in a D symmetric MN is constrained by the transformation properties of the localized orbitals. It reads
with being tensor operators transforming according to the irreducible representation Tsukerblat (2006). The non-zero matrix elements of this SOI Hamiltonian in the low-energy quadruplet read so that the SOI takes the following effective form
with and . An effective SOI Hamiltonian is obtained also from the DM SOI Hamiltonian in Eq. (1). The constraints and on the DM vectors due to D symmetry of the molecule, give rise to the same effective SOI in Eq. (14), with . Thus, as expected, the molecular SOI and the DM SOI give rise to the same effective SOI Hamiltonian acting in the low energy quadruplet. Like in the case of the electric dipole parameter , finding requires more than symmetry, like ab-initio methods or experiments. The transverse SOI, with interaction strength does not act within the low-energy space, and its effect will appear only in higher orders of perturbation theory in .
An external magnetic field couples to the spin via the Zeeman term , with being the -factor tensor in D. The full effective Hamiltonian describing the low-energy quadruplet in the presence of SOI, electric field and magnetic field read
Note that , and chirality and spin act as independent spin degrees of freedom. Furthermore, in the absence of SOI the chirality and the spin evolve independently. However, the SOI couples the two and provides with means for electric control of both spin and chirality. Vice-versa, magnetic fields can also couple to chirality due to SOI. Also, while magnetic fields (time-dependent) cause transitions between states of opposite spin projection but with the same chirality , the electric field does the opposite: it causes transitions between states of opposite chirality , but carrying the same . Full control of the lowest quadruplet is thus realized in the presence of both electric and magnetic fields, as can be seen in Fig. 2.
Double valued group states of the spin triangle
The double group representations allow to non-perturbatively describe the magnetic and electric transitions in the presence of spin-orbit interaction. The lowest quadruplet consists of two Kramers doublets, one of them transforming like , and the other one according to . Here represent pairs of eigenstates of a given angular momentum , with spin projection . For example, if , then . The higher energy states instead ( states), transform now not as , but as () and as (). Thus, the states mix with the states, but only the ones transforming according to the same representations, i.e. there is no mixing between and due to spin-orbit interaction. The magnetic dipole transitions take place between and , and within and , respectively, while electric dipole transitions take place only between and . The selection rules for the electric transitions are , while for the magnetic transitions these are . We see that within the double group analysis, i.e. in the presence of SOI, there are allowed electric dipole transitions also within the subspace.
Using both the single group and double group analysis we can pinpoint to the transitions that arise in the absence or only in the presence of SOI. Therefore, the electric dipole transitions present in the single-group are a consequence of the modified exchange interaction, and can arise even in the absence of SOI, while the ones that show up only in the double group analysis are a consequence of the SOI (or modification of SOI in electric field).
We now can establish several selection rules for the SOI, electric field and magnetic field induced transitions. Note that the above analysis was exact in SOI. However, it instructive to treat electric field, magnetic fields and SOI on the same footing. First, we find that the electric dipole transitions fulfill the selection rules and , meaning that electric field only couples states within the lowest quadruplet. The SOI transitions show a richer structure. We can separate the SOI interaction in two parts: the perpendicular SOI, quantified by in the DM interaction Hamiltonian, and the in-plane SOI, quantified by in the DM interaction Hamiltonian, respectively. By doing so, we find that the SOI terms obey the selections rules and , while for the terms we get the selection rules and . We see in-plane SOI ( terms) do not cause any splitting in the ground state and can lead to observable effects only in second order in perturbation theory in . Also, note that if symmetry is present, and thus there are no in-plane SOI effects at all. Modification of these terms due to an in-plane external electric field , however, lead to different selection rules: changes of terms lead to and , while modification of lead to and . The magnetic field transitions obey the selection rules and . Thus, we can make clear distinction between pure electric field transitions, SOI-mediated electric transitions and magnetic transitions. This distinction between the electric and magnetic field induced transitions could be used to extract the spin-electric coupling strength parameter from spectroscopic measurements.
ii.2 Spin triangle
The spin triangle has a more complex level structure than the triangle due to its higher spin. The spin Hamiltonian, however, is similar to the one in Eq. (1) for , and the reduction of the representation of three spins is , a total of spin states. The total number of irreducible representations is the same as in the case, and we need only to identify these basis states in terms of the spin states. The triangle states can be defined according to their transformation properties under three-fold rotations C in D and are of the following form
where , are the -fold rotation of order , and . The states represent all possible states ( states in total) with a given spin projection that cannot be transformed into each other by application of the rotation operator . These states are showed in Table 1.
The corresponding states with all spins flipped, namely with , can be written in a similar form (not shown). Having identified the symmetric states in terms of the spin states, we proceed to analyze the allowed transitions induced in the spin systems by magnetic and electric field, both within the single valued group and double valued group representations.
Single valued group states of the triangle
The above states are basis of the point group D, but not eigenstates of the total spin operator , i.e. they do not have definite total spin. However, linear combinations of states of a given total spin projection and a given ’chiral’ numbers become eigenstates of . The total spin eigenstates can be written as , where is the number of different states with a given . The coefficients are to be identified so that these states satisfy , with . The states with are all transforming according to the representation, while the states with are organized in doublets, being the bases of the two dimensional representation . However, as mentioned above, different combinations of symmetry adapted states carry different total spin . The magnetic and electric transitions are similar to the ones in the triangle, in the absence of SOI. The electric field causes transitions only between states with the same and , but opposite chirality (this is different from the triangle with spins in each of the vertices). As for the spin triangle, there are electric dipole transitions within the spin system even in the absence of SOI. The ground states is four-fold degenerate consisting of two eigenstates
We see that, as opposed to the triangle, the lowest states are given by linear combinations of the several symmetry adapted states (the states are obtained by flipping the spins in the states in Eqs. (18), (19). This, however, does not modify the conclusions regarding the electric and magnetic transitions in the absence of SOI, these being given by the same rules as in the triangle: electric-field induced transitions between the states of opposite chirality and the same spin projection . The lowest states are still organized as spin and chirality eigenstates that are split in the presence of SOI as in the previous case.
In the original spin Hamiltonian in Eq. (1) the electric field causes modification of the spin Hamiltonian parameters. As for the spin triangle, the strongest effect comes from modification of the isotropic exchange interaction, so that
with , where is the angle between the projection of the external electric field to the molecule’s plane and the bond, and . The effect of the electric field on the lowest quadruplet is found to be similar to the spin case. While the SOI splits the two chiral states without mixing them (at least in lowest order), the electric field, on the other hand, mixes the chiral states. The effective Hamiltonian acting in the lowest quadruplet reads
Above, , , with and , and stands for the SO splitting. However, in this situation the in-plane chirality operators cannot be written in a simple form as a function of the individual spin operators, as opposed to the triangle.
Double valued group states of the triangle
The double group representation allows to identify the couplings between different spin states induced by the SOI and to identify the allowed magnetic dipole transitions. Due to SOI, the electric field induced spin transitions will take place also outside the spin quadruplet. In the absence of extra degeneracies (induced, for example, by external magnetic fields), however, these transitions are strongly reduced due the gap of the order . We can then focus, as for the triangle, only on the lowest quadruplet. These states are organized in two Kramer doublets of the form , one transforming as and the other one as . Here again, represent angular momentum eigenstates with spin projection .
As in the case of the triangle, the electric field induced transitions take place between and , with the selection rules . Magnetic transitions instead take place both within and between and , satisfying the selection rules .
If we now treat the SOI, electric field and magnetic fields on the same footing, we arrive at the same selection rules as for the triangle, namely and for electric transitions, and for SOI transitions, and and for magnetic transitions, respectively.
ii.3 Spin pentagon
We now analyze the spin-electric coupling in a pentagonal molecule with a spin in each of the vertices, like depicted schematically in Fig. 3. As in the case of the spin triangle, an external electric field gives rise to modification of exchange interaction in Eq. (1). However, the net spin-electric coupling in the lowest spin sector can only be mediated by SOI. i.e. via the DM interaction (which can be also modified in the presence of the -field).
To make the analysis simpler, we assume in the following that the pentagonal spin molecule possesses a point group symmetry, thus no horizontal reflection plane . However, no generality is lost, since lower symmetry implies more allowed transitions in the spin system. If, for example, in the lower symmetric situation some transitions are forbidden, these transitions will be forbidden in the higher symmetry case. The Hamiltonian is given in Eq. (1) with . The states of the pentagon are found from the product of the individual spin representations , meaning there are spin states in total. As before, these states can be organized in a symmetry adapted basis in the following way
where with , are the -fold rotations of order . The states represent all possible states ( states in total) with a given spin projection that cannot be transformed into each other by application of the rotation operator . These states are showed in in Table 2
and the corresponding states with all spins flipped, i.e. states (not shown). In the absence of SOI there is no mixing of different states, i.e. the chirality is a good quantum number. In this case the chirality is quantified by the operator (the prefactor is chosen for convenience; see below). As in the spin triangles, the above states are not yet the eigenstates of the Hamiltonian and we have to solve the equation , with . The ground state is spanned, again, by four states, two Kramers doublets with spin . In the following we inspect the level structure of these four states in terms of the above symmetry adapted states.
Single valued group pentagon
We focus here only on the four lowest energy states, which are two pairs of states. The first (second) pair is given by linear combination of states with chirality () and spin projection . We obtain
so that . These states (for a given projection) form the basis of the two dimensional irreducible representation . We are now in positions to investigate the allowed electric dipole transitions within this lowest subspace. The in-plane electric dipole forms a basis of the irreducible representation in D. By calculating the product we see that the totally symmetric representation of D is absent. Therefore, there are no electric dipole transitions within the four dimensional subspace in the absence of SOI.
As in the previous two cases, the coupling of the spin Hamiltonian to electric field comes via modification of the spin Hamiltonian parameters. If only the modification of the isotropic exchange Hamiltonian is taken into account, the spin-electric Hamiltonian takes the same form as in Eq. (7), with , . The parameter quantifies the electric dipole coupling of each of the bonds and is the angle between the electric field and the bond . Note that is in principle non zero in D point group symmetry. However, the matrix elements of the spin-electric Hamiltonian within the lowest quadruplet are all zero, i.e. . This means that electric field has no effect on the lowest quadruplet, as found out also by purely symmetry arguments. Therefore, we may expect that the spin-electric coupling in pentagonal spin molecule is caused by SO effects.
Double valued group pentagon
Double valued group analysis allows identifying of the level structure and the allowed transitions in the presence of SOI and magnetic fields. The lowest four states in the double group are described by the two dimensional irreducible representations and , respectively. Since both the magnetic and electric dipoles transform as in , both electric and electric transitions will take place between the same pair of states. The products of the irreducible representations that labels the states in the low-energy quadruplet read: , and . These equalities imply the same selection rules in the lowest subspace as for the spin triangle case: () for electric dipole transitions, and ( and ), for the magnetic ones.
The main feature of pentagonal spin ring is the absence of electric dipole transitions in the lowest quadruplet in the absence of SOI. This is to be contrasted to the spin triangle case, where spin-electric coupling exists in the ground state even in the absence of SOI. This feature finds its explanation from the interplay between the selection rules for electric field transitions and the ones for the SOI. In fact, these selection rules are by no means different from the triangular spin rings. Since the ground state is spanned by four states with chirality and spin , we see that the condition for the electric field transitions implies no electric field coupling within the ground state! In the presence of SOI though, spin electric coupling is still possible, but it will be times smaller than in triangles. Spin-electric coupling can arise also via modification of the DM vectors in electric field. However, the selection rules for this transitions are, like for the triangle, and . This means direct splitting in the ground state, and thus we expect that for pentagon spin ring the electric dipole response will be much weaker.
Iii Hubbard model of a molecular nanomagnet
Spin-Hamiltonian models of molecular nanomagnets are based on the assumption that the spins on magnetic centers are the only relevant degrees of freedom. This assumption of fully quenched and localized orbitals allows for the relatively simple predictions of spin structure in the low-energy states of the molecule. However, since the orbital dynamics plays a crucial role in spin-electric coupling, spin-Hamiltonian models are unable to predict the corresponding coupling constants. In this Section, we relax the assumption of quenched and localized orbitals and treat the orbital degrees of freedom of electrons on magnetic ions within a Hubbard model. This provides an intuitive picture of spin-electric coupling in terms of the deformation of the molecular orbitals induced by the external field. Besides, in the limit of strong quenching of the orbitals, the Hubbard model reproduces a spin Hamiltonian, similar to the results found in the studies of cuprates Moriya (1960a); Shekhtman et al. (1992); Yildirim et al. (1994) and multiferroics Sergienko and Dagotto (2006); Dong et al. (2009). In particular, we find the relation between modifications of the electronic hopping matrix elements induced by the field and that of the spin-electric coupling in the spin Hamiltonian, thus providing a guide for the estimate of the size of spin-electric coupling in a molecule.
The outline of the present Section is the following. In Subsection III.1, we introduce the Hubbard model of a spin chain with the shape of regular -tangon, and derive the resulting symmetry constraints for the hopping parameters. In Subsection III.2 we assume a direct electron hopping between magnetic sites, and derive the spin Hamiltonian of a spin triangle from the Hubbard model, in the limit of large on-site repulsions; we thus express the coupling to electric fields in terms of the Hubbard-model parameters. In Subsection III.3, we introduce a Hubbard model of a magnetic coupling in the case where this is mediated by a non-magnetic bridge between the magnetic centers; also in this case, we find a connection between the modification of the bridge and spin-electric coupling.
iii.1 Parameters of the Hubbard model of molecular nanomagnets
Magnetic properties of molecular nanomagnets are governed by the spin state of few electrons in the highest partially occupied atomic orbitals, split by the molecular field. The spin density is localized on the magnetic centers Postnikov et al. (2006), and thus the low-energy magnetic properties are correctly described by quantum models of interacting localized spins Belinsky (2009a, b).
The response of molecular nanomagnets to electric fields, as a matter of principle, does not have to be governed by the electrons occupying the same orbitals that determine the molecule’s spin. However, the quantum control of single molecule magnets by electric fields depends on the electrons that both react to electric fields and produce the magnetic response. Therefore, the models of molecular nanomagnets that consider only few orbitals can provide useful information about the electric control of spins.
Hubbard model provides a simplified description of orbital degrees of freedom by including only one or few localized orbitals on each magnetic center. Furthermore, the interaction between electrons is accounted for only by introducing the energies of the atomic configurations with different occupation numbers. The Hubbard model of the MN is given by:
where () creates (annihilates) an electron with spin on the orbital localized on th atom, and is the corresponding number operator. Model parameters , describe the energy of spin up(down) electrons electrons on the site . Hopping parameters , describe the spin-independent and spin-dependent hopping between sites and .
We assume that the largest energy scale is the splitting between the energy of the highest occupied atomic orbital and lowest unoccupied one, induced by the molecular crystal field: this justifies the inclusion of one orbital only for each magnetic center. The on-site repulsion energy is the next largest energy scale in the problem, being larger than the hopping coefficients. Amongst these, processes involving states of different spin, mediated by spin-orbit interaction, are described by the and components of . The parameters , instead, describe the difference of the hopping matrix elements between spin-up and spin-down electrons. In the following, we shall consider both the case where electron hopping takes place directly between neighboring magnetic ions and that where the magnetic interaction is mediated by bridges of non-magnetic atoms. The Hubbard Hamiltonian can be approximated by a spin Hamiltonian model in the limit . The symmetry constraints on the spin Hamiltonian parameters can be deduced from those on the Hubbard model parameters Moriya (1960a). If the spin-independent hopping dominates (), the resulting spin Hamiltonian will contain the Heisenberg exchange terms and a small additional spin-anisotropic interaction. If , the size of spin-dependent interactions in the spin Hamiltonian will be comparable to the Heisenberg terms. Both these cases appear in the molecule nanomagnets Choi et al. (2006); Chiorescu et al. (2000a); Clemente-Juan et al. (2005); Luzon et al. (2008).
Symmetry of the molecule imposes constraints to the Hubbard model, thus reducing the number of free parameters. The on-site repulsion parameters are equal for all equivalent magnetic ions. In the molecules of the form of regular -tagon, all of the spin-independent hopping parameters are equal, due to the symmetry. The spin-dependent hopping elements are related by both the full symmetry of the molecule and the local symmetry of localized orbitals. For example, in the case of localized orbitals in a regular polygon that are invariant under the local symmetry group of the magnetic center,
with the convention that site coincides with site . In this case, there is only one free parameter that determines all of the matrix elements. Therefore, the regular tagon molecule in the absence of external electric and magnetic fields can be described by a Hubbard model, with five independent parameters: , , . In addition, the symmetry, if present will impose , thus reducing the number of free parameters to three.
iii.2 Hubbard model of the spin triangle: direct exchange
In this Subsection we give a brief description of the Hubbard model for a triangular molecule with symmetry. In this model we assume only direct coupling between the magnetic centers, thus no bridge in-between. Even so, this simplified model catches the main features of the effective spin Hamiltonian and gives the microscopic mechanisms for the spin-electric coupling. The Hamiltonian describing the electrons in the triangular molecule reads
where is the spin-orbit parameter (only one), is the on-site orbital energy, and is the on-site Coulomb repulsion energy. As stated before, typically , which allows for a perturbative treatment of the hopping and spin-orbit Hamiltonians. These assumptions agree well with the numerical calculations performed in Postnikov et al. (2006).
The perturbation theory program involves the unperturbed states of the system. The first set of unperturbed states are the one-electron states
while the three-electron states split in two categories: (i) the site singly occupied states
with for and , for , and (ii) the double-occupied sites
with and .
The states in Eqs. (28), (35) and (30) are degenerate with energies , and , respectively. Note that these state are eigenstates of the Hamiltonian in Eq. (27) only in the absence of tunneling and SOI.
The above defined states are not yet adapted to the symmetry of the system, i.e. they are not basis states of the corresponding irreducible representations of D point group. Finding these states is required by the fact that the symmetry of the molecule is made visible through the hopping and SOI terms in the Hubbard Hamiltonian. This is accomplished by using projector operatorsTsukerblat (2006). We obtain for the one-electron symmetry adapted states.
where and are one-dimensional and two-dimensional irreducible representations in D, respectively. Similarly, the symmetry adapted states with the singly-occupied magnetic centers read:
while the symmetry adapted states of the doubly-occupied magnetic centers read:
The tunneling and SOI mixes the singly-occupied and doubly-occupied states. Since both the tunneling and SOI terms in the Hubbard Hamiltonian transform as the totally symmetric irreducible representation in D, only states transforming according to the same irreducible representations mix. We obtain the perturbed in first order in and :