Exciton-induced electric dipole moment in organic ferromagnets

Based on an Anderson-like model including electron–lattice interaction and electron–electron (e–e) interactions, charge and spin properties of excitons in quasi-one-dimensional organic ferromagnets with spin radicals are investigated. The results demonstrate the appearance of an unusually large electric dipole moment in the magnetic molecule upon the formation of the exciton. The sign of the dipole moment depends on the spin of the excited electron relative to the magnetization of the spin radicals. The underlying mechanism is analysed based on the different charge distribution and lattice distortion in the two excitation modes with opposite spin. The origin is attributed to the preferred occupation on different domain walls of the exciton distortion for different spin-resolved excitonic levels. The experimental realization of the large dipole moment is discussed. Although the realization of a large dipole moment is impeded by the superposition state formed due to the degeneracy of two excitation modes, we propose an achievable route to break the symmetry and create controllable electric polarization by optical illumination. The dipole moment is robust even if the long-range e–e interaction is included. The effects of the system parameters, including the electron hopping between the main chain and the radicals and the e–e interaction, on the magnitude of the dipole moment are also discussed. This work indicates a novel way to realize organic multiferroic materials with controllable polarization, which can be induced by photon excitation.


Introduction
Organic spintronics is an attractive field which explores the role of electronic spin in functional devices with the utilization of various organic materials [1].Among these, organic ferromagnets (OFs) are promising candidates since they combine both the attractive properties of organics and ferromagnets, which makes the synthesis of flexible and low-cost ferromagnets with chemical processing possible.The synthesis of OFs has evoked much interest for several decades.It usually follows one of two ways, i.e. doping transition-metal ions into organics [2][3][4] or using spin radicals [5][6][7].The latter method has the advantage to produce pure OFs without heavy-metal atoms.For example, in 1987, Ovchinnikov and Spector [8] successfully synthesized poly-(1,4-bis(2,2,6,6-tetramethyl-4-piperidyl-1-oxyl)-butadiin) (poly-BIPO) by partially substituting hydrogen atoms in polyacetylene with spin radicals containing unpaired electrons.The radical spins develop ferromagnetic order in the ground state due to their interaction with the conjugated π orbitals of the main chain [9,10].A variety of OFs with different spin radicals has been fabricated up to now, such as m-polydiphenylcarbene (m-PDPC) [11] and nitrophenyl nitronylnitroxide (p-NPNN) [12].The stability of spin radicals was also examined and confirmed [13], which demonstrates a promising prospect for further applications although challenges in understanding the underlying physics still remain.
A main characteristic of organic polymers is that the excitations are composite states of electrons (holes) and lattice distortions [14,15], such as solitons, polarons, bipolarons, and polaronic excitons, due to the strong electron-lattice (e-l) interaction in organics.The charge and spin characteristics of these excitations are important for the mobility, spin transport, and luminescence properties of organics, and have been well studied in the past decades [15].However, in OFs with spin radicals, the presence of these radicals and hence the spin interactions introduce novel features.For example, previous studies have shown the disparity of charge and spin for polarons in OFs [16,17], which strongly affects the polaron dynamics and triggers many interesting phenomena, such as the intermittent rebound of polarons [18] and asymmetric polaron velocity with the reversal of the electric field or of the polaron spins [19].Nanoscale devices based on OFs have also been designed.By considering the spin coupling with radicals, spin filtering, and spin rectification were reported for these devices [20][21][22].
Excitons are crucial excitations in optoelectronic devices.They are typically described as Wannier-Mott excitons [23,24] in inorganic semiconductors and as Frenkel excitons [25] in organic semiconductors.The photoinduced Frenkel excitons in organics are characterized by zero charge and spin densities (singlet states), quasi-localized energy levels within the Peierls band gap, and a concomitant lattice distortion.Excitons in organics have been widely studied in the past decades, including their formation [25][26][27][28], dissociation [29][30][31], and reexcitation [32], as well as their fission, fusion, and diffusion [33][34][35][36][37][38].Benefitting from the development of femtosecond technology, it has become feasible to experimentally investigate transient excitonic processes [39,40], where novel properties which do not exist in the ground state can be revealed.For example, the celebrated photoinduced polarization inversion has been observed, where the electric dipole of a molecule was reversed when excitons are further excited to biexcitons after adsorbing one photon [41,42].On the other hand, excitons in OFs are rarely explored.In 2018, Zhang and Xie [43] investigated the photoinduced spin-lattice dynamics in OFs with radicals, where the dynamical processes of excitons formation were explored and pure spin transfer between the main chain and radicals was mentioned.In spite of this, the picture of excitons in OFs is still inadequate, especially in comparison to nonmagnetic polymers.In this paper, the charge and spin properties of excitons in OFs are studied theoretically within an Anderson-like model including the e-l interaction and a Hubbard electron-electron (e-e) interaction.By considering two kinds of excitation modes, characterized by exciting a spin-up or a spin-down electron, a new phenomenon of an exciton-induced electric dipole moment in OFs is revealed, which is much larger in magnitude than for other reported organic ferroelectrics.Thus, a strong organic multiferroic effect is achieved.This paper is organized as follows.In section 2, the model Hamiltonian and calculation method are introduced.In section 3, the numerical results are presented and analysed.A brief summary is given in section 4.

Model and method
The typical model structure of OFs with spin radicals is shown in the inset of figure 1.It consists of a π-conjugated carbon chain and spin radicals coupled to the odd sites of the main chain.Each radical has an unpaired electron and thus a net spin.Such a model has been widely used to study the electronic structure and spin transport of OFs with spin radicals [9,10,21,22].The Hamiltonian of the whole system is written as where H 0 represents the Hamiltonian of the main chain in the form of an extended Su-Schrieffer-Heeger model [44], The first term is the π-electron hopping along the main chain.t 0 and α are the hopping integral of the electrons along a uniform chain and the e-l coupling constant, respectively.y i is the lattice distortion with y i = u i +1 − u i , where u i is the displacement of the carbon atom in the main chain at site i. c † i,σ (c i,σ ) is the creation (annihilation) operator, indicating that an electron is created (annihilated) at site i with spin σ.The spin quantum number σ can assume the values ↑≡ 1 and ↓≡ −1.The second term is the elastic energy of the lattice with elastic coefficient K.The last term is the on-site e-e interaction between π electrons described by a Hubbard model with interaction strength U.
H 1 represents the Hamiltonian of the radicals and the electron hopping between the radicals and the main chain, where ε 1 is the on-site energy of electrons at the radicals relative to the carbon atoms in the main chain.iR refers to the radical site connecting to site i of the main chain.δ i,o signifies that the radicals only connect to the odd sites of the main chain with δ i,o = 1(δ i,o = 0) for odd (even) atoms.t 1 is the hopping integral of electrons between the main chain and the radicals.The last term is the Hubbard e-e interactions on the radical sites, where the same interaction strength U is assumed for simplicity.Within the Hartree-Fock approximation, the Hamiltonian of the e-e interaction can be rewritten as The stable state of the OF can be obtained by solving the electronic Schrödinger equation and lattice equilibrium equation self-consistently.The electronic Schrödinger equation reads where H e is the electronic Hamiltonian involved in H.In Wannier space, the µth electronic state with spin σ can be expanded as |ψ µ,σ ⟩ = i(iR) Z µ,i(iR),σ |i(iR), σ⟩, where Z µ,i(iR),σ are the eigenfunctions.Thus, the electronic eigenequations can be obtained in the following form Here, ε µ,σ is the µth eigenenergy with spin σ. ni(iR),σ is the average electron number on the site i (iR) with spin σ.It is given by where 'occ' .Refers to the occupied states.The occupation of each state is determined according to the excitation mode, as explained below.
The eigenequations are solved with a given initial lattice configuration y i and electron number ni(iR),σ .After that, a new electron number ni(iR),σ is obtained from equation (8) with the new eigenfunctions Z µ,i,σ .The lattice configuration can be calculated by minimizing the molecular total energy as a function of the atomic displacement, E {u i } = E e {u i } + E L {u i }.The former can be calculated from the average value of the electronic energy at ground state E e {u i } = ⟨H e ⟩, while the latter is the lattice elastic energy shown in equation (2).The lattice equilibrium condition requires ∂E{u i } ∂u i = 0, which leads to Then the lattice configuration y i is obtained as where N 1 is number of total sites in the main chain.Here, fixed-end boundary conditions are used for the main chain, e.g.u 1 = u N1 = 0. Note that equations ( 8) and (10) do not contain terms that are off-diagonal in µ since Z µ,i,σ are eigenfunctions.Equations ( 6)- (10) need to be solved self-consistently, until the convergence criterion between two steps, |u ′ i − u i | < 10 −6 Å, is met.After the iteration, the charge density ρ c,i(iR) and spin density ρ s,i(iR) are obtained, which are defined as ρ c,i(iR) = 1 − (n i(iR),↑ + ni(iR),↓ ) in units of the elementary charge e > 0 and ρ s,i(iR) = h 2 (n i(iR),↑ − ni(iR),↓ ).
For the numerical calculations, the parameters are adopted as those for poly-BIPO [9,10,21,45]: t 0 = 2.5 eV, α = 4.5 eV Å −1 , K = 21.0 eV Å −2 , ε 1 = 0, u = U/t 0 = 1.0, and t R = t 1 /t 0 = 0.3.In spite of this, the explored physics applies to all OFs with similar structure.The total number of sites of the OF is 120, which consists of 80 sites in the main chain and 40 radical sites.We have checked that the chain length used here is sufficiently long compared to the characteristic length of the excitons to eliminate finite-size effects.Note that the x axis only shows the site number of the main chain, where the radicals are connected to the odd-numbered sites.The same notation is used in the following figures.The inset shows the schematic structure of the OF.

Dipole moment of excitons in OFs
We first consider the ground state of the OF.The band structure, spin density, and charge density are shown in figure 1.The spectrum in figure 1(a) exhibits six subbands.The two high-lying (spin-resolved π * ) and the two low-lying (spin-resolved π) subbands are dominated by the π electrons in the main chain, which are separated by a large Peierls gap and weakly spin split near the gap.Two fully spin-polarized and nearly flat bands exist in the Peierls gap, which stem from the radicals, separated by a Hubbard gap of 1.25 eV.The three low-lying subbands belong to bonding states while the three higher-energy subbands are antibonding states.Assuming one electron per site, a half-filled situation holds, where the π bands and the lower spin-up flat band are fully occupied, while the upper spin-down flat band and the π * bands are empty.Hence, the majority electrons of the OF have spin up.As shown in figure 1(b), the spin-up (majority) electrons of the OF mainly distribute on the radical sites and exhibit ferromagnetic order.A ferrimagnetic spin density wave is observed on the main chain, where spins between neighbouring sites are opposite in sign and unequal in magnitude.This is different from the model of localized spin for radicals, where an antiferromagnetic spin density wave exists [46].This is because in our Anderson-like model, spin transfer from the radicals to the main chain occurs.The magnitude of net spins at each radical is about 0.48 in units of h.The whole OF remains neutral without any nonzero charge density, as shown in figure 1(c).This picture of the ground state is qualitatively consistent with previous studies about OFs with spin radicals [46,47].Now we consider the excitonic state of the OF.This state is obtained by lifting an electron of given spin from the highest occupied level to the lowest unoccupied level, which can be achieved in experiments by photon excitation, and letting the lattice relax.According to the spectrum shown in figure 1(a), there exists two different possible excitation processes, namely the excitation of a spin-up electron (ESU) from the highest occupied level (µ = 80) of the flat band to the lowest unoccupied level (µ = 81) of the π * band, and the excitation of a spin-down electron (ESD) from the highest occupied level (µ = 40) of the π band to the lowest unoccupied level (µ = 41) of the flat band, see figures 2(a) and (b).The two processes are degenerate in the excitation energy with energy of 1.67 eV due to the symmetry of the spectrum.However, the resulting occupations of the electron-hole (e-h) pair are different.For the ESU, the excited electron resides in the antibonding π * band and the hole in the flat band.On the other hand, for the ESD, the excited electron resides in the flat band and the hole in the bonding π band.Hence, the property of the exciton in the two cases needs to be investigated separately.Note that here only the lowest-energy spin-conserved excitation is considered.The spin-flip excitation between the two flat bands and the high-energy excitation between the two π bands are beyond the focus of this paper.
Figure 2 shows the electron occupation, lattice configuration, and single-particle energy spectrum of the excitons for the two excitation modes.Here, the smoothed lattice configuration is defined as ỹi = 1 4 (−1) i (2y i − y i +1 − y i −1 ).The characteristics of exciton-induced lattice distortion and shallow levels in the energy gap are clearly observed.They are identical for the ESU and ESD excitation processes.Note that due to the spin splitting of the energy bands, four spin-resolved shallow levels appear in the gap.The lattice distortion consists of two soliton-like domain walls located at sites 20-40 and at sites 40-60, where the phase of the dimerization changes by π.This picture is similar to that for nonmagnetic polymers in nature [23,25].By checking the data for the lattice configuration, we find that ỹi is not strictly asymmetric about the centre of the main chain, i.e. the centre of the 40th bond.The difference between the two mirrored bonds is on the order of 10 −3 Å.This is because the radicals are only connected to the odd sites and are thus not symmetric about the chain centre.
Let us consider the charge and spin density of the exciton.Here, the spin density is calculated by subtracting the spin density of the ground state from the one of the excitonic state.As shown in figure 3, localized nonzero charge density and spin density appear both on the main chain and on the radicals, whereas they are usually zero in nonmagnetic polymers without external electric field.The spin density remains degenerate for the ESU and ESD modes, it contains a spin-up peak at odd sites in the left moiety and a spin-down one at even sites in the right moiety of the main chain.A small negative peak appears for the radicals, which demonstrates an exciton-induced spin transfer from the radicals to the main chain.Interestingly, we notice that the charge density of the exciton for the two excitation modes takes on completely different spatial distributions, as shown in figure 3(a).For the ESU mode, the charge density consists of a negative peak at odd sites in the left moiety and a positive one at even sites in the right moiety.A nonzero small negative peak is also observed at the radical sites in the left moiety, which indicates that a charge transfer from radicals to the main chain occurs.Conversely, for the ESD mode, the positions of the positive and negative charge density peaks in the main chain are exchanged.The charge density at the radical sites is changed to positive values, which means that the charge transfer is reversed.
The above results show that the emergence of the exciton in the OF will trigger the appearance of charge and spin density waves as well as charge and spin transfer.Interestingly, the asymmetric charge density wave (CDW) causes an intrinsic dipole moment for the molecule.This picture can be investigated qualitatively by calculating the total charge and spin in the left and right moieties of the OF, including both the main chain Table 1.Total charges and spins in the left and right moieties and dipole moments induced by the exciton for the two excitation modes.The charges and spins on the main chain and the radical sites are also given.and the radical sites.Moreover, we define the dipole moment of the OF as P = e i(iR) ρ c,i(iR) x i(iR) , where x i(iR) is the atomic coordinate with x i(iR) = [i(iR) − n c ]a + u i by taking the chain centre n c = 40.5 as the origin and a is the lattice constant of the uniform main chain with a value of 1.22 Å.The results are shown in table 1.It is found that in the ESU mode, the total charge in the left and right moieties are −0.477 and 0.477, respectively, and the dipole moment is 9.391 e • Å.The total charge on the main chain (radicals) is 0.09 (−0.09), which signifies a slight charge transfer from the radicals to the main chain.For the ESD mode, an opposite charge in each moiety is obtained, in agreement with an opposite dipole moment of −9.391 e • Å as well as an opposite charge transfer.For the two modes, the spin distribution is identical, with up spins of 0.095 in the left moiety and down spins of −0.095 in the right moiety.We emphasize that the magnitude of this exciton-induced dipole moment is much larger than the one reported for other organic ferroelectrics, such as 0.44 e • Å for poly(vinylidene fluoride) (PVDF) molecules [48,49].The spontaneous exciton-induced dipole moment has not been reported for normal nonmagnetic polymers.

Main chain
The unusual charge and spin properties of the exciton in the OF are attributed to the wave function of the electronic states as well as their occupation.In normal polymers, the frontier states occupied by the e-h pair of the exciton are conjugated π and π * orbitals, where the dipole moment is absent due to the spatial symmetry of the π orbitals.However, in the OF with radicals, the occupied states are the π (π * ) and the flat-band states, and the spatial symmetry is not conserved.The reason is that the radicals are only connected to the odd sites, which, together with the Peierls dimerization, leads to a strong breaking of the mirror symmetry already in the ground state and even for infinitely long chains.In figure 4, we plot the probability distribution of the occupied excitonic levels.For the ESU mode, the two occupied frontier states, |ψ 81,↑ ⟩ and |ψ 40,↓ ⟩, cf figure 2(a), show a larger weight in the left moiety of the main chain and the radicals, specifically, at the odd sites.This is because the spin-up radicals hybridize with the odd sites of the main chain.The antiferromagnetic coupling in the main chain then favours spin-down electrons at odd sites and spin-up electron at even sites.Note that |ψ 81,↑ ⟩ is an antibonding state, which also tends to be localized at odd sites where a bonding electron does not want to be.Conversely, for the ESD mode, the occupied frontier states are the bonding state |ψ 80,↑ ⟩ and the antibonding state |ψ 41,↓ ⟩.The two states contribute larger proportion in the right moiety, specifically, at even sites.
In the following, we address the contribution from the excitonic levels to the charge and spin densities of the OF.The sum of the probability distributions contributed by the occupied excitonic levels from figure 4 for the two excitation modes is shown in figures 5(a) and (c).Obviously, they contribute a similar weight distribution as the charge density curve for each mode, where the region with larger weight corresponds to the centre of the negative charge peak in figure 3(a).The spin density contributed by the frontier states is the difference of the probability for different spins, as shown in figures 5(c) and (d).It demonstrates that both for the ESU and the ESD mode, surplus up spins emerge in the left moiety and down spins in the right moiety, which is consistent with the spin density of the OF, see figure 2(b).Note that the probability distributions in figure 5 are not the same as the charge and spin densities shown in figure 3, which means that other electronic states also contribute.Spin-charge disparity in OFs has been demonstrated in previous studies for the case of polarons [16,17,47], where the modification of the spin density contributed by the Fermi sea has been reported.In the following, we focus on the details of the dipole moment.
In principle, the dipole moment consists of a contribution from the lattice ions with positive charges and a contribution from the electrons with negative charges.The former is nonzero both in the ground state and in the excitonic state since the radical ions are only connected to the odd sites of the main chain.The value of the dipole moment from the lattice ions can be calculated from P l = e i(iR) x i(iR) .The calculated value is −22.37 e • Å in the ground state, which in fact only comes from the radical sites because the sites of the main chain are symmetric about the centre in dimerization.When the exciton is formed, the asymmetric lattice configuration of the main chain slightly modifies the lattice dipole moment to P l = −22.93 e • Å regardless of the ESU or ESD excitation.Thus, the distinct change of the molecular dipole moment mainly comes from the electronic part.For the electronic part, in the ground state, all electrons will contribute an opposite dipole moment to compensate that the one from the lattice ions because of the absence of a net dipole moment for the whole molecule.With the formation of the exciton, the electron dipole moment is modified dramatically by two factors, i.e. the change of occupation numbers and the lattice distortion.The latter will in turn affect all electronic states.To investigate the change of the dipole moment of each electronic state, we define the dipole operator of an electron with spin σ as [50] pe,σ = −e i(iR) Then the dipole moment of each state is calculated as The sum of P µ,σ over all occupied states is the total electronic dipole moment.We have found that the total electron dipole moment is 22.37 e • Å in the ground state and 32.32To get a clear picture of the exciton-induced change of electronic states, we plot the probability distribution of the four levels shown in figures 2(a) and (b) before the formation of the exciton.In figures 6(a) and (b), the two levels |ψ 81,↑ ⟩ and |ψ 40,↓ ⟩, which are occupied for the ESU mode, are delocalized throughout the molecule.They also have larger weight at odd sites but only a slight asymmetry about the chain centre is induced with a small dipole moment of 0.83 e • Å.The case of |ψ 80,↑ ⟩ and |ψ 41,↓ ⟩, which are occupied for the ESD mode, is shown in figures 6(c) and (d).The situation is similar, where the larger weight at even sites of the main chain only leads to a small negative dipole moment of −0.217 e • Å.Once the exciton is formed, the four levels turn into localized states bounded to the exciton distortion.The larger weight at odd (even) sites causes the distribution of the levels to be concentrated in the left (right) domain wall of the distortion.As a result, the asymmetry as well as the dipole moment is enhanced.The changes of the dipole moments of |ψ 81,↑ ⟩ and |ψ 40,↓ ⟩ are 4.82 and 3.23 e • Å, respectively.The sum is 8.05 e • Å and thus contributes the vast majority of the total electron dipole moment for the ESU mode.In contrast, the changes for |ψ 80,↑ ⟩ and |ψ 41,↓ ⟩ for ESD mode are −3.78 and −5.33 e • Å, respectively, where the sum is −9.11 e • Å, which exceeds the total electronic dipole moment.This means that low-energy states also contribute.
To investigate the contributions of different occupied states to the change of the total electronic dipole moment, we first calculate the change of the dipole moment of each occupied state from the ground state to

Effect of superposition state
The above results demonstrate that the formation of an exciton for either the ESU or the ESD mode induces a dipole moment of the molecule.However, due to the energy degeneracy between the ESU and ESD excitations, the optically excited state may be a superposition of the two modes, the stable state of the superposition may be different from the exciton state of the pure ESU or ESD mode, which can be calculated self-consistently through the equations ( 6)- (10) with the given electron occupation.
In figure 8, the lattice configuration of the superposition state for different weights of the ESU state is shown.It is found that when |c u | 2 decreases from 1.0, the excitonic lattice distortion deviates from the one of the pure ESU mode, where the distortion becomes deeper and broader.From values of 0.7-0.3, a soliton-antisoliton pair is obtained, where the distortion in the centre goes back to the dimerization value.With a further decrease of |c u | 2 , the distortion turns back to the excitonic one of the pure ESD mode.
In figure 9, we further compare the charge density, dipole moment, and total energy of each superposition state.The charge density and dipole moment show that a positive or negative dipole moment remains for the superposition state when the weight of the ESU mode is greater or less than 0.5, where the maximum value appears at 0.7 or 0.3.However, at the weight of 0.5, the total molecular dipole moment vanishes.The total molecular energy shown in figure 9(d) indicates that the superposition state with the weight of 0.5 is the lowest-energy state after structure relaxation, while the pure ESU and ESD are higher-energy states.Thus, in the presence of photon excitation, the system has the largest probability to relax into the superposition state without dipole moment.

Nondegenerate dipole moment with ε 1 ̸ = 0
To achieve a nonzero net dipole moment, the breaking of the degeneracy between the two modes is necessary.In real OFs, the unpaired electron usually comes from the heteroatoms, such as oxygen or nitrogen, where the effective on-site energy of the radical sites is typically different from the carbon atoms in the main chain.The on-site energy of the radicals relative to the main chain can be further adjusted by a gate voltage perpendicular to the chain.In the following, we take ε 1 = −0.05t0 and calculate the band structure in the ground state.As shown in figure 10(a), the flat bands are shifted downwards.This leads to a smaller energy gap for the spin-down levels as well as a lower excitation energy for the ESD mode with ∆E = 0.055 eV than for the ESU mode.The different gaps make the excitation of pure ESU or ESD mode possible.The charge density in the excitonic state for the two modes is shown in figure 10(b).The dipole moment for the ESD mode is −7.88 e • Å, whereas the value is 11.22 e • Å for the ESU mode.This means that one can realize positive or negative dipole moments in the OF by selecting specific wave lengths for the photon excitation, in this case, 733 nm for the ESU mode and 757 nm for the ESD mode.In the combination with the intrinsic magnetism of the OF, it is interesting to find that an organic multiferroic material is achieved by photon excitation [51].We now discuss the possibility of superposition states in this case.In experiment, there are two possible origins of a superposition state.Firstly, if the light source has a finite spectral width spanning the energy gap between the ESU and ESD modes, the formation of the superposition state can be triggered.The stable state of the superposition state in such nondegenerate case has been checked numerically (results not shown).The one with equal weight of the ESU and ESD modes is also the lowest-energy state but a quite small nonzero dipole moment remains due to the symmetry breaking.On the other hand, the excitation of a pure ESD or ESU mode can be expected since a laser source with sufficiently small spectral width is achievable in experiment.For example, the all-solid-state red laser is able to provide a wavelength of 758 ± 2 nm, which meets the requirement for the pure ESD excitation shown in figure 10(a).The available wavelength generated by such laser equipment varies from 213 to 4800 nm [52], which is sufficient for different excitation energy in OFs.Secondly, electronic transitions during the lattice relaxation after the excitation of pure ESU or ESD mode is another possible reason since the lattice relaxation is typically slow compared to electronic transitions.The probability of electron transitions actually relies on the time-dependent energy barrier during the lattice relaxation, which is complicated and beyond the calculations of this model.Considering the energy difference of about 0.46 eV between the spin-up and spin-down lowest unoccupied molecular orbitals (LUMOs), a dipole moment for a ESU or ESD excitation can be expected in the nondegenerate case.

Effects of t R and u
In OFs with radicals, the electron hopping t R between the main chain and radicals and the e-e u interaction discriminate between different molecules.In figure 11, we investigate the effects of the values of t R and u on the dipole moment of the excitonic state, for the case of ε 1 = 0.As shown in figures 11(a) and (b), the magnitude of the dipole moment increases with t R but decreases with u.The reason lies in the change of the lattice distortion with the two parameters.In figures 11(c) and (d), a weaker and broader lattice distortion of the exciton is observed for the case of larger t R and smaller u.This is because the lattice distortion originates from the Peierls dimerization of the main chain.A larger t R will enhance the hybridization between the main chain and the radicals.This will weaken the quasi-one-dimensional characteristics of the OF, which is essential for the Peierls dimerization, and so reduce the dimerization.The dependence of the lattice dimerization on u in polymers is non-monotonic and parameter dependent, as has been well studied in [53].Therein a stronger dimerization with increasing u has been found in the range of u < 2 with a moderate e-l coupling, similar to the present work.The stronger dimerization at a larger u can be observed in figure 11(d).A stronger dimerization, like the effect of a stronger e-l interaction, will lead to a narrower excitonic lattice distortion.This will weaken the separation of the two domain walls, as well as the separation of the positive and negative charge centres.Consequently, the dipole moment will be decreased.

Effect of long-range electronic interaction
The above results indicate a spatial separation of the positive and negative charge centres in the OF caused by the exciton.Thus, it is necessary to discuss the effect of long-rang e-e interaction, which has not been included in the initial Hamiltonian.In the following, an additional term of long-range e-e interaction is added to the Hamiltonian [53,54], The three terms represent the e-e interactions between two neighbour sites i and i + 1 of the main chain (V m ), between the radical site iR and the connected site i (V mR1 ), and between the radical site iR and sites i ± 1 (V mR2 ).Here, only the nearest-neighbour and the next-nearest-neighbour interactions are considered because the e-e interaction in polymers decreases exponentially with distance due to the screening by the ions and other electrons [53,54].The long-rang e-e interaction is also treated within the mean-field approximation.The e-e interaction strength between two sites i and j is estimated as where ε and β are the dielectric constant and the screening factor, respectively.The parameter values are taken from those of polyacetylene as ε = 5.7 and β = 1.7 [53,55].For simplicity, the distance between the radical site iR and the connected site i is set to a., The distance between iR and i ± 1 is then √ 2a.These choices lead to the values V m = V mR1 = 2V mR2 = 0.38 eV.
With the above parameters, the charge density in the ground state and the excitonic states is examined in figure 12.It is found that in the presence of long-range e-e interaction, a CDW arises in the main chain in the ground state, with positive charges at even sites.Negative charges appear on the radicals, which implies electron transfer from the main chain to the radicals.This is understandable since in the mean-field approximation, the long-range e-e interaction raises the on-site energy of the main chain relative to the radicals and causes an electron transfer from the odd sites to the connected radicals.Due to the electron hopping along the main chain, the positive charges at odd sites will be partially compensated by electron transfer from their neighbouring even sites.This also leads to the positive charges at even sites.The CDW is not mirror symmetric due to the asymmetric structure from the radicals connected only to odd sites, which leads to a small dipole moment of 0.8 e • Å.
Figures 12(b) and (c) show the charge density of the exciton for the ESU and ESD modes.We notice that the exciton-induced separation of charge centres still exists.However, the peaks of the charge density are modified for the different modes.For the ESU mode, the amplitude of the positive excitonic charge peak on the right is enhanced by the background CDW because of the same preferred occupation of positive charges at the even sites.Conversely, for the ESD mode, the negative charge peak on the right is weakened by the CDW due to the opposite preferred occupations at the even sites.As a result, the dipole moment for the ESU mode is increased to 13.68 e • Å, whereas it is decreased to −4.64 e • Å for the ESD mode.
In figures 12(d) and (e), we subtract the background CDW from the excitonic charge density.It is clearly seen that the local charge density for the ESU mode is larger than for the ESD mode.This means that besides the extended CDW, the long-range e-e interaction also causes an interesting local effect by enhancing the charge separation for the ESU mode.The reason can be understood as follows.The CDW is caused by the intersite repulsion of electrons, which causes a redistribution of electrons.The electron redistribution is also affected by the hopping integral.In the region of the exciton, the dimerization is reduced and the average hopping rate is thus enhanced-recall that the hopping rate is proportional to the hopping amplitude squared.Hence, the electron redistribution becomes easier in this region, which favours the formation of a CDW and enhances its amplitude within the excitonic region.For the ESU mode, this local effect is consistent with the excitonic effects on the charge distribution so that the dipole moment is enhanced.Conversely, for the ESD mode, the two effects are opposite and the dipole moment is reduced.Therefore, the long-range e-e interaction does not eliminate the exciton-induced dipole moment but breaks the degeneracy of the dipole moment for the two modes.

Conclusions
We have investigated the static charge and spin properties of photoinduced excitons in OFs with spin radicals.Unlike for nonmagnetic polymers, an unexpected very large electric dipole moment of the molecule is revealed upon the formation of the exciton.The sign of the dipole moment may be positive or negative, depending on the spin of the excited electron.We have explored the mechanism for the dipole moment based on the electronic occupations, the wave function of the electronic states, and the lattice distortion, for two different excitation modes (ESU and ESD) distinguished by the spin of the excited electron.The different occupied electronic levels for the ESU and ESD excitation modes lead to opposite shifts of the positive and negative charge centres and hence result in opposite dipole moments.This effect can be attributed to the different occupation probabilities between odd and even sites for a bonding or antibonding state with specific spin, where it is crucial that the spin radicals are only connected to the odd sites of the main chain.The low-energy states are also perturbed, and we have investigated the contribution of each state.
Due to the degeneracy of excitation energy for the ESU and ESD modes, superposition states with various weights of the two modes have also been analysed.We have found that the superposition state with equal weights for the ESU and ESD modes will eliminate the dipole moment.Nevertheless, we have proposed a promising experimental route to create controllable electric polarization by optical illumination, that is, organic multiferroicity.By changing the relative on-site energy of the radicals, we found that the degeneracy of excitation energies between the ESU and ESD modes is generically broken for real OFs and can be tuned by a transverse electric field, which makes it feasible to generate a net nonzero dipole moment in the OF with controllable sign.The effects of the electron hopping t R between the main chain and the radicals and the e-e interaction u have also been studied, where the magnitude of the dipole moment increases with t R and decreases with u.In addition, we have examined the effect of a long-range e-e interaction.The long-range e-e interaction does not eliminate the dipole moment but generates a background CDW in the main chain and a local effect on the excitonic charge density by enhancing or reducing the dipole moment.Our work proposes a new route to realize a strong multiferroic effect in organics.This work also reveals the intriguing properties of excitons in OFs with spin radicals, and thus broadens the field of exciton physics.
Since the lifetime of the exciton is hundreds of picoseconds [40], which is three orders of magnitude longer than the formation time, the exciton-induced dipole moment can be measured by using a pump-probe method.By shining a laser pulse on the OF with frequency matching the excitation condition as described in section 3.2, a controllable dipole moment is generated.As a result, the scattering of the probe light will be changed by the large polarization of the molecule.Another way to reveal the dipole moment is to measure the change of the conductance of the molecular solid at different excitation modes.It is expected that the large dipole moment may modify the resistance of the OF depending on the alignment with the external field, and thus a ferroelectric diode is possible as reported in other ferroelectrics [56,57].The change of resistance will also induce the phenomenon of multi-state magnetoresistance if a spin valve structure is constructed by using ferromagnetic electrodes, as demonstrated in many organic ferroelectric spin valves [58,59].The results proposed here will enrich and broaden the interesting field of organic exciton multiferroics [60], which is believed to open the way for potential applications in wireless nanodevices and sensors controlled by light.

Figure 1 .
Figure 1.(a) Single-particle energy spectrum, (b) spin density, and (c) charge density of the OF with radicals in the ground state.Note that the x axis only shows the site number of the main chain, where the radicals are connected to the odd-numbered sites.The same notation is used in the following figures.The inset shows the schematic structure of the OF.

Figure 2 .
Figure 2. Sketch of the electron occupation of the four frontier states with different spins for (a) the ESU and (b) the ESD excitation mode.(c) Lattice configuration and (d) energy spectrum of the exciton for the two modes.

Figure 3 .
Figure 3. (a) Charge density and (b) difference of spin densities induced by the exciton for the two excitation modes.Circles (squares) indicate odd (even) numbered sites.The same symbols in the following figures have the same meaning.
(13.54)  e • Å for the ESU (ESD) mode.The changes induced by the exciton are thus 9.95 e • Å for ESU and −8.83 e • Å for ESD.

Figure 7 .
Figure 7. Contribution to the change of the total electronic dipole moment from each (a) spin-up and (b) spin-down frontier state for the ESU mode.(c) and (d) in the same for the ESD mode.

Figure 8 .
Figure 8. Lattice configuration of the superposition state for various weights of the ESU mode.

Figure 9 .
Figure 9. (a) and (b) Charge density of the superposition state for various weights of the ESU mode.(c) Corresponding dipole moment of the superposition state for each weight.(d) Its energy difference relative to the pure ESU (ESD) state.

Figure 10 .
Figure 10.(a) Band structure in the ground state and (b) net charge density in the exciton state for an on-site energy of the radicals of ε1 = −0.05t0.The arrows indicate the specific wavelength required by the ESU and ESD excitation modes.

Figure 11 .
Figure 11.Dependence of the dipole moment for the two excitation modes of the value of (a) the hopping term tR and (b) the e-e interaction u.Here u is fixed to u = 1.0 in (a) and tR is fixed to tR = 0.3 in (b).(c), (d) Smoothed lattice distortions of the exciton for different values of tR and u, respectively.

Figure 12 .
Figure 12.Charge density in the presence of long-range e-e interaction for (a) the ground state, (b) the excitonic state for the ESU mode, and (c) the excitonic state for the ESU mode.(d), (e) Net charge density for the ESU and ESD modes, respectively, obtained by subtracting the ground-state density.Here, the on-site energy of the radicals is set to ε1 = 0.