Fused Azulenes: Possible Organic Multiferroics
FFused Azulenes: Possible Organic Multiferroics
Simil Thomas and S. Ramasesha ∗ Solid State and Structural Chemistry Unit,Indian Institute of Science, Bangalore 560 012, India
Karen Hallberg and Daniel Garcia
Centro At ´ o mico Bariloche and Instituto Balseiro,Comisi ´ o n Nacional de Energ ´ i a At ´ o mica and CONICET, 8400 Bariloche, Argentina Abstract
We present compelling theoretical results showing that fused azulene molecules are strong can-didates for exhibiting room temperature multiferroic behavior, i.e., having both, ferroelectric andferromagnetic properties. If this is experimentally proved, these systems will be the first organicmultiferroic materials with important potential applications.
PACS numbers: 71.10.Fd, 75.50.Xx, 77.55.Nv, 77.84.Jd a r X i v : . [ c ond - m a t . s t r- e l ] N ov . INTRODUCTION Organic electronic materials have come to the fore since the provocative suggestion ofLittle , that a high temperature excitonic superconductor can be realized in conjugatedmolecular systems. This has led to the discovery of organic systems which exhibit a widerange of properties ranging from superconductivity to ferromagnetism. In recent years, in thefield of materials, multiferroics have become increasingly important because of the possibilityof a wide range of applications. Most common multiferroics exhibit both, ferroelectric andferromagnetic (FM) properties. A few inorganic materials are known to be multiferroics andthe most widely known are ferrites and manganites with perovskite structures . A few purelyorganic ferroelectrics or exciton multiferroics and ferromagnets are known. However, tothe best of our knowledge there are no organic multiferroic materials. In the quest fororganic systems that are potential multiferroic materials, we have discovered theoreticallythat fused azulene, a π -conjugated system, can exhibit multiferroic behavior.A simple frustrated conjugated molecule that has been well studied is azulene (C H ),a molecule with fused five and seven-member rings. It also has a dipole moment ∼ . n + 2” rule . Azulene oligomers (n-azulene) can be made up inmultiple geometries, particularly in fused (n-C H ) azulene form (see Fig. 1(a)). Becauseof the 5 and 7 member rings of the base monomer all these oligomers are expected to havelong-range frustration and show a non trivial magnetic state with possible low energy spinexcitations. Besides, depending on the oligomer, the dipole moment of the azulene monomercan also align resulting in a molecular ferroelectric phase.To explore these molecules, we study the low-lying charge and spin energy gaps of azuleneoligomers using the long-range interacting Pariser-Parr-Pople (PPP) model and the densitymatrix renormalization group method (DMRG) technique . We show that the existence ofa magnetic state in this case does not require the presence of flat energy bands, since italso arises within a spin-1/2 antiferromagnetic Heisenberg model (AHM), thus, suggestingmagnetic frustration as the origin.We find that the fused azulene ground state is a singlet for oligomers with up to 5 azuleneunits, while with more azulene units the ground state spin increases. Our studies show thatwith more than 5 units and up to 11 azulene units the oligomers have a triplet ground state.From these results we predict that the ground state spin of the fused azulene increases with2he number of azulene units. In the thermodynamic or polymer limit (n → ∞ ) we expectfused azulene to be a (non-saturated) ferromagnet. We suggest that the ferromagnetism inthe model comes from the magnetically frustrated geometry of the chain.In addition, our charge-density calculations show that the ground state of the system hasferroelectric alignment of the electric dipoles of the monomeric units and that the charge gapis finite but small in the polymer limit of fused azulene. Fused azulenes are large but finitemolecules and do not have center of inversion This leads to the appearance of an electricdipole moment in one direction (sometimes referred to as a pyroelectric state).These results show that fused azulenes are excellent candidates to exhibit multiferroicbehavior. In the next section, we briefly discuss the model Hamiltonian used in our study.In the last section we present our results and discussions. II. MODEL HAMILTONIAN FOR FUSED AZULENES
The π -conjugated systems in fused azulene (Fig. 1(a)), azulene polymer (Fig. 1(c)) andoligoacene (Fig. 1(d)) are modeled by considering one p z orbital on each carbon atom. ThePPP model Hamiltonian is given by, H P P P = (cid:88) t (ˆ a † i,σ ˆ a j,σ + H.c. ) + U (cid:88) i ˆ n i (ˆ n i − (cid:88) i>j V ij (ˆ n i − n j −
1) (1)where ˆ a † i,σ (ˆ a i,σ ) creates (destroys) an electron on site i with spin σ and ˆ n i is the numberoperator and < i, j > implies summation over nearest neighbors.For the fused azulene case, the pentagons and the heptagons are treated as regular poly-gons of side 1.397 ˚A. The transfer integral between bonded sites is taken to be -2.4 eV.The Hubbard parameter U for Carbon is fixed at 11.26 eV. The inter-site interactions areparameterized using the Ohno formula. The PPP model with these standard parametershas been extensively used for successfully modeling the excitations in a host of conjugatedmolecules. This model is used throughout the paper unless otherwise stated.The relatively large value of U/t allows us to make the approximation of the active phasespace by a single electron per site. This allows checking the magnetic excitations by using3 spin-1/2 AHM. H H = J (cid:88) (cid:126)S i · (cid:126)S j ( J >
0) (2)We study this model to compare magnetic excitations in oligoacenes and fused azulenes.The DMRG method has proved to be ideally suited to study conjugated polymericsystems, within model Hamiltonian approaches. It has been shown that the DMRG methodis accurate even in the presence of long-range interactions if these interactions are diagonalin real space, as with the PPP model . Many of the interesting conjugated polymers consistof ring systems, such as phenyl, pyridine, thiophene, furan or pyrrole rings. Building thedesired oligomers of such ring systems by adding two sites at each infinite DMRG step is non-trivial and the general method was first illustrated for the case of poly-para-phenylenes .A similar approach has been used for building the molecular systems in this study. We FIG. 1. Structure of (a) fused azulene (b) DMRG ladder-like scheme for building fused azulene(c) Polyazulene and (d)oligoacene. have checked our finite DMRG calculations against the non-interacting model results for alloligomers and for different cut-offs in the number of dominant density matrix eigenvectors, m . Based on these studies, we find m = 500 is optimal.4 II. RESULTS AND DISCUSSION
We have carried out the DMRG calculations on the structures shown in Fig. 1(a)(fused azulene), (using a ladder-like building block (Fig. 1(b)), Fig. 1(c) (polyazulene) andFig. 1(d) (oligoacene). For fused azulene the energy per azulene unit linearly extrapolatesto − .
117 eV in the thermodynamic limit.The oligomers we consider have an even number of sites and a half-filled band, implyingstates with an integer total spin S . These systems have SU(2) symmetry so we take advan-tage of the degeneracy of the different spin projections M S for a total spin S to determine thetotal spin of the system in its ground state. Using the DMRG method it is straightforwardto calculate energy of states with fixed M S . The ground state spin of the system is S if thelowest energy states in the M S =0,1,..S subspaces are degenerate and a gap exists from theseto the M S =S+1 state.Within the PPP model we computed the lowest energy E(M S ) for spin projections M S =0 , , M S = E(M S ) − E(0). Fused azulene oligomers with less than 5 monomers(n ≤
5) have total spin S = 0. For larger oligomers, i.e., 5 < n ≤
11, the lowest M S = 1 statebecomes degenerate (within the accuracy of the calculation) with the lowest M S = 0 stateindicating a total spin S = 1 for the ground state. For even larger oligomers, we have strongindications that lowest energy states in higher M S sectors will become degenerate showingtransitions to larger spin values for the ground state. For example, from the behavior of thegap between the lowest M S = 2 and M S = 0 level, we see that this gap will also vanish whenthe system size increases to about 10 or 11 azulene units leading to spin S = 2 ground state.The gap ∆ between the lowest M S = 3 and M S = 0 levels also appears to vanish for evenlarger systems, making S = 3 the total spin of the ground state. Therefore, it is likely thatin the polymer limit, a (non-saturated) ferromagnetic ground state will result.Polymers with a triplet ground state are, to the best of our knowledge, not known.Monkman and coworkers predicted that a polymer will have a triplet ground state if thelowest singlet exciton energy is below 1 . eV . This conclusion was arrived at since tripletexciton in their systems were ∼ IG. 2. Spin gaps (in eV) vs n defined as ∆ M S = E ( M S ) − E (0), M S = 1 , is always positive, implying a singlet ground state. that the (HOMO-k) and (LUMO+k) levels come within an energy gap < ≤ n <
10, k=1 for 10 ≤ n <
15 and k=2 for 15 ≤ n <
20. However, the degeneracy is notsufficiently close to expect exchange correlation to yield a high spin ground state.An alternative mechanism for the existence of ferromagnetism for larger systems is mag-netic frustration stemming from the lattice geometry . To illustrate this we considered asimple spin-1/2 AHM for fused azulenes with a spin-1/2 entity at each site. As seen inFig. 2(b) the existence of high spin ground states are further confirmed with larger chainsusing this simplified model. While the number of monomers needed to increase the ground6tate spin is different compared to the PPP model, both models share the same type oftransition. The fact that the purely AHM reproduces these results is a clear indication thatthis FM state does not have its origin in Fermi level degeneracies.To further confirm the geometric nature of this FM state we have computed the spin gap∆ using the spin-1/2 AHM on the oligoacene lattice(Fig. 1(d)). This differs from fusedazulene in the fact that all the fused rings are hexagons. The geometry allows for a Ising-like non-frustrated antiferromagnetic ground state (which is impossible in systems with oddmembered rings). In this case, we found that the singlet was always the ground state, witha finite gap to the lowest triplet state which remained finite in the thermodynamic limit.Similar calculations were earlier performed on oligoacenes using the PPP model . It wasfound that the spin gap in the PPP model remained finite in the polymer limit. To exploreif extended frustration is essential for a ferromagnetic phase, we carried out calculationson polyazulene (Fig. 1(c)). In this case we found a singlet ground state for the largestsystem with eight azulene units in our study and a finite spin gap for the thermodynamicsystem. These results show the complex nature of the origin of the high spin states in thesegeometrically frustrated systems. Further studies are necessary to elucidate the nature ofthe ferromagnetism in these cases.It is well known that low-dimensional systems can undergo a distortion of the Peierls’type which could lead to stabilization of the singlet state relative to the triplet state . Sofor a real molecule it is important to see if distortions could destroy the magnetic state.Bond-order of a bond (i,j) in the ground state is defined as
15 12 < (cid:80) σ { a † i,σ a j,σ + a † j,σ a i,σ } > .Bonds with large (small) bond orders will have a tendency to become shorter (longer) inthe equilibrium geometry. We note that in both the singlet and the triplet states, thebond orders for the peripheral bonds are nearly uniform. This shows that the system isunlikely to undergo a structural distortion. We also note that the bond orders for the bondsthat connect the upper and lower polyene like chains are the smallest, implying that anydistortion of the system will only uniformly increase the distance between the upper andlower chains, which is unlikely to alter energy level ordering of the low-lying states.To determine the polarized nature of the ground state of fused azulene, we investigatethe charge distribution and charge gap in these oligomers within the PPP model. We definethe charge of a ring as (p-c) (p = number of sites in the ring; c = total charge in thering, (cid:80) i (cid:15) ring < n i > ) for oligomers with up to 10 azulene units. We show the net charge7 IG. 3. (a) Net charge (p-c) in each ring of fused azulenes with n=10 , in the seven- and five-member rings in the oligomers in Fig. 3(a). In all these oligomers,seven member rings have positive charge while five member rings have negative charge.The amplitude of the charge density wave in the system remains approximately the same,independent of the oligomer size, hence we expect a ferroelectric state in the polymer limit.In Fig. 3(b) we give the dipole moment ( (cid:126)µ = (cid:80) i ( < n i > − (cid:126)r i ) in the ground state. We notethat when the spin of the ground state changes, there is a drop in the dipole moment. Thisis due to increase in covalency of the ground state as the spin of the state increases. Ourresults show that the fused azulene system in the polymer limit also exhibits a spontaneouspolarization. Notice that fused azulene does not have a center of inversion, a necessarycondition for the emergence of electric dipolar moment. It consists of a sequence of azulenemonomers containing seven and five-member rings in a definite order. Thus the polymernecessarily has a 7-member ring on one end and a 5-member one on the other.The charge gap measures the energy required to create an independent electron-holepair in a system. The charge gap for a polymeric system can be obtained by extrapolat-8 IG. 4. Charge gap (in eV) for fused azulenes vs n . ing ∆ c ( n ) = E + ( n ) + E − ( n ) − E gs ( n ) (where n is the number of monomers andsuperscripts ‘+’ and ‘-’ refer to cation and anion while ‘0’ refers to the neutral species) tothe thermodynamic limit. The quantity ∆ c ( n ) gives the energy required to create a pairof free moving electron and hole in the ground state . The plot of the charge gap vs n isshown in Fig. 4. Extrapolating the charge gap to thermodynamic limit, we obtain the value∆ C ( ∞ ) = 0 .
403 eV. This is smaller than typical charge gaps found for conjugated polymersby almost an order of magnitude . The existence of a charge gap reinforces the conclusionobtained from the ring charge disproportionation on the existence of a ferroelectric state inthe thermodynamic limit.To conclude, the results presented in this paper for the ground state of fused azulenesusing the PPP and Heisenberg models, namely the increasing ground state magnetic mo-ment with system size, the presence of a finite but small charge gap for all lengths anda dipolar moment arising from positively-charged seven-membered and negatively chargedfive-membered rings, show that this system is both ferromagnetic and ferroelectric. Due tothe fact that these molecules can be large but finite, we expect that real polymers will behavemore like single molecular magnets and electrets. These are compelling results indicatingthat the fused azulene system could have multiferroic behavior. This would not only signalthese materials as the first organic multiferroics, but would be also of great importance for9uture organic device applications. ∗ [email protected] W. A. Little, Phys. Rev. , A1416 (1964)Z. Hiroi and M. Ogata, in
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