~ Solid State Communications, Vol. 78, No. 6, pp. 477-480, 1991. Printed in Great Britain.

0038-1098/9153.00+.00 Pergamon Press plc

SSH-HAMILTONIAN DESCRIPTION OF THE ELECTRONIC STRUCTURE AND VIBRATIONAL PROPERTIES OF POLYPARAPHENYLENE VINYLENE

Z. Shuai, D. Beljonne and J.L. Br6das

Service de Chimie des Mat6riaux Nouveaux et D~partement des Mat6riaux et Proc6d6s Universit6 de Mons, Place du Parc 21, B-7000 Mons (Belgium)

(Received by D. Van Dyck - December 7, 1990)

We present the parameterization of a Su-Schrieffer-Heeger (i.e., Hiickel plus sigma compressibility) Hamihonian to describe the molecular geometry and the electronic structure of undoped and doped polyparaphenylene vinylene, one of the currently most studied conjugated polymers. We then apply a perturbation technique to obtain a qualitative insight into the vibrational structure of the electron-lattice coupled system. Analysis is focused on the localized modes appearing around the bipolaron defects.

Polyparaphenylene vinylene [-C6H4-CH=CH-], PPV, has recently attracted a major interest in the field of conjugated polymers (1-5). Due to the availability of a soluble precursor route, thin films can be synthesized which present good environmental stability, mechanical properties, and optical quality. Upon doping, conductivities as large as 5x103 S/cm can be achieved (6). In the pristine state, the films exhibit interesting nonlinear optical properties, e.g. Z (3) values larger than lff 11 esu, have been measured through third-harmonic generation experiments at 1.06 lain (7)

The evolution of the electronic properties of conjugated polymers upon doping has been often described in the framework of one-electron Hamiltonians emphasizing the electron-lattice coupling (8-13). It is indeed well established that in conjugated molecules, the electronic structure of the n system and the geometric structure are strongly interconnected. As a result, Su-Schrieffer-Heeger Hamiltonians (8) which correspond to Hilckel Hamiltonians including sigma compressibility and bond-length dependent transfer (resonance) integrals, have been successfully applied to polyacetylene (8,9), polyparaphenylene (to), polypyrrole GO), polythiophene (it,121, or polyaniline (13). One of the practical interests of such Hamiltonians is to allow one to deal with chains containing a large number of backbone atoms (> I00) and a varying number of (doping- induced or photogenerated) defects.

The limitations of such Hamiltonians an: obvious and come from the explicit neglect of electron-electron interactions (14). Such interactions can however be partly introduced in an implicit way through the parameterization scheme. For instance, it has been shown that, in order to reproduce the experimental degree of bond-length alternation in polyacetylene, the transfer integrals for the single and double carbon-carbon bonds, have to assume values which differ much more from one another within an SSH approach than within a Pariser-Parr-Pople al313roach

where electron-electron interactions are included (14,15). It is also one of the goals of this work to assess the validity of a simple SSH method when applied to polyparaphenylene vinylene.

As in previous instances (10,12,13), the parameterization of the SSH Hamiltonian for PPV is considered in such a way as to reproduce: (i) the geometric structure of the chain in the ground state and (ii) the n-band electronic structure obtained from Valence Effective Hamihonian (VEIl) calculations (16). The ground state geometric structure, as derived from semiempirical MNDO (Modified Neglect of Diatomic Overlap) (17) and AMI (Austin Model I) (16) geometry optimizations on PPV oligomers and X-ray diffraction data on trans-stilbene (18), presents the following characteristics: the bond lengths within the benzene rings are all on the order of 1.40 A; within the vinylene linkages, single bonds and double bonds are about 1.45 ,~ and 1.35 A long, respectively (16-18). In agreement with the optical absorption 4at_a 03, the VEH bandgap for PPV is calculated to be 2.5 eV (16)

According to bond order-bond length relationships of Coulson-type (19), the transfer integral between adjacent sites is written as:

13(r) = - A . exp(-r/B) and the o compressibifity of the lattice takes the form:

fir) = C . ~ r ) . (r-ro+B) where r denotes the bond length and A, B, C, and r o are the parameters to be optimized, in the framework of the SSH- type Hamiltonian:

H=I 1 --~],,.°., (c:c,.÷h. c. ~ +-~],,.,.~ ecx,,x,.1

477

where (n,n') denotes the site summations over nearest neighbors (the factor 1/2 compensating for double counting). We have found that in order to be able to

478 PROPERTIES OF POLYPARAPHENYLENE VINYLENE Vol. 78, No. 6

reproduce the ground state geometric and electronic structure, it is essential to optimize two separate sets of parameters, one applying to the rings, the other to the vinylene linkages. The optimized parameters are:

ring: A=36.0 eV; B=0.6 ,~; C=7.1 ~-l; r0~l.56 A; vinylene: A=34.0 eV; Be0.6 ~; C=5.2 ~" ; r0=1.48 ~.

The ring parameters resemble those optimised in the case of polyparaphenylene while the vinylene p.~ameters are similar to those obtained for polyacetylene ~lu)

With these parameters, we reproduce very well the ground state geometry described above as well as the n electronic structure, which is illustrated in Figure 1. The band gap is calculated to be 2.5 eV which matches the optical absorption onset (5]; the width of the highest occupied band is obtained to be 2.3 eV, in excellent agreement with the VEH results (20). The total energy difference between the aromatic ground state geometric structure and the fully quinoid geometric structure is found to be 5 kcal/mol per unit cell, which is slightly lower than the 7 kcal/mol difference obtained earlier in the case of polyparaphenylene (10)

We now investigate the formation of bipolarons upon doping of PPV. Where the doubly charged defects appear on the chains, we allow for a geometry relaxation in the following way (8,10):

-I -

~-4

w

-5

-6

-7

0 0 5 - 1 . 0

k (pi la)

Figure 1. SSH ~ band structure of polyparaphenylene vinylene.

r1=1.4~-a Ith~ th 6N-n 1

r2=l • 4~+a I th-~ th 6N-n 1

r I

r3=l. 45~-~%~ th I th 6N-nl

Figure 2. Sketch of the locations of bonds r 1 to r 4 discussed in the text,

t /-/ 6N-n r4=l. 3 5A+aa th-= ~ th 1

The location of bond length r t to r 4 are shown in Figure 2. N indicates the number of unit cells over which the defect extends; n is the site location relative to one end of the defect, (6N-n) then being the separation from the other end. The value of 1 modulates the amplitude of the bond-length relaxation; (z I (=0.07 ,~) and t~ (=0.1 ,~) correspond to the maximum deformations that can be achieved, their values being chosen m be consistent with the relaxations calculated at the AM1 level (tt).The calculations have been carried out for phenyl-capped PPV chains containing 21 rings and 20 vinylenic linkages, i.e., 166 carbons. The results are provided in Table L

The relaxation energy obtained through bipolaron formation with respect to two vertical ionisation processes (21) is of the order of 0.42 eV. This value is almost identical to that found in polyparaphenylene (0.45 eV) (10) The defect appears to extend over about 6 units, which is in agreement with the AMI results O6). Furthermore, the bond lengths appearing in the center of the bipolaron compare very favorably with those optimized at AM1 level

Table 1. Characteristics of bipolaron formation on a PPV oligomer chain containing 166 carbons as a function of bipolaron width, N: relaxation energy E tel (in eV) relative to two vertical ionizations; optimized defect amplitude, 1; location o f HOMO level (in eV) relative to the Fermi energy; and optimized bond lengths, r t to r 4 (in A), in the center of the bipolaron.

N=5 N=6 N=7

E re~ 0.421 0.425 0.422 1 13 17 22 HOMO -0.841 -0.842 -0.865 rl 1.353 1.358 1.361 r2 1.446 1.442 1.438 r3 1.384 1.389 1.396 r4 1.414 1.412 1.404

on a five-ring oligomer (differences at most of the order of 0.01 ~). The bipolarons induce the appearance of two localized electronic states within the gap, located at about 0.4 eV from the band edges. They lead to novel subgap optical absorptions centered around 0.4 and 2.1 eV.

Next, we investigate the nature of the infrared active

Vol. 78, No. 6

vibrational (IRAV) modes which are induced by bipolaron formation. In order to do so, we add a dynamical term in the Hamiltonian:

PROPERTIES OF POLYPARAPHENYLENE VINYLENE 479

H--H,+Ho+iM%-'2 ~.o (~)2 (a)

where M denotes the mass of a (CH) unit. By expanding x n around the equilibrium position:

xa:xn°+SXn(t) we obtain the variation of 13 and f (up to 2nd order):

(x,,x,.) :p (x°,,x",.) ,p c~ (xo,xo.) +[jc2~ (x°,x~)

f(x., x..) :f(x.°, x°,) .f~1> (x.,x..) +ft2~ (x.,x)

The first order perturbation is equivalent to geometry optimization. The second order perturbation leads to the variation of the electronic levels following:

<1~ ~ HI 1) I v > 12

(b)

(c)

where 11/} and Iv) are LCAO molecular orbitals.

The variation in total energy can be written as:

AE=2E~(occ.) 8 (2)e~+Ho TM + -~M~.I (8~n) 2

As we are interested only in trends, we consider simply one-dimensional displacements of the atoms along the chain axis. The calculations have been carried out as before on a phenyl-capped chain containing 21 unit-ceUs, both for the ground state and for the doubly chargext (bipolaron) case. By comparing the symmetry of the modes, we find that the bipolaron induces five IRAV modes. These modes have their largest amplitudes in the center of the defect and decay very rapidly outside of the bipolaron.

The fast of these modes is the Goldstone mode, which reflects the translational invariance (~1=0). The other four modes are illustrated in Figure 3 and correspond to frequencies: ~,z=816 cm'l; ~3=1075 cm'l; ~4=1240 cm'l; and ~5=1292 cm "1. These frequencies arc consistently 150- 300 cm "l too low relative to the experimental values recently reported by Bradley and Friend (~2=1100 cm "1, ~3=1274 cm "1, ~4=1398 cm d and ~5=1470 cm "l) (22). These differences arc understandable given the simplicity of our Hamiltonian; we arc confident, however, that the characteristics of the modes are reliably reproduced.

The second IRAV mode corresponds to collective vibrations of the rings in opposition m collective vibrations of the vinylene linkages. The next three modes arc opposite

• (d)

Figure 3. Illustration of the IRAV doping-induced localized modes appearing around a bipolaron in polyparaphenylene vinylene~ Modes 2 trough 5 (see text) arc given from top to bottom, (a) through (d).

in character: (i) the third mode counteracts the appearance of the quinoid-type geometry within the defect, both within the rings and within the vinylcnc linkages; (ii) in the fourth mode, the qninoid-type resonance form is slightly cnhancezi within the vinylenc linkages and is alternatively reinforced and weakened within the rings; (iii) in the fifth mode, the quinoid-type geometry is strongly favored within the vinylene units but is weakened within all the rings.

Using a simple SSH-type approach to describe polyparaphenylenc vinylcne, we thus obtain a very satisfactory description of the geometric and electronic structure and of the vibrational properties in the ground state and upon doping (i.e., in the ionized state). In such a Hiickel framework, our msuhs are identical whether we refer to the formation of either a positive bipolaron upon p- type doping, or a negative bipolaron upon n-type doping, or

480

what is termed a polaron-exciton upon photogeneration of an electron-hole pair (in this case, one electron remains on the highest occupied molecular orbital, HOMO, and one electron is promoted to the lowest unoccupied molecular orbital, LUMO). Our results would thus predict that upon phoexcitation a neutral polaron-exciton is formed with a width of about 6 rings and a relaxation energy of about 0.42 eV. This is in total disagreement with the photoexcitation and luminescence measurements recently carried out by Bradley and co-workers (22). Indeed, the luminescence data tend to indicate that the Stokes shift is very small (22). Furthermore, by varying the polarisation of the probe beam in the photoinduced absorption experiments,

PROPERTIES OF POLYPARAPHENYLENE VINYLENE Vol. 78, No. 6

the photogenerated defects appear to have a much narrower spatial extent than the doping-induced bipolarons (22)

The SSH description of the excited states of polyparaphenylene vinylene appears thus to be inadequate and electron correlation effects do obviously play an important role. Work is in progress in our laboratory to provide a better description of the excited states of PPV by means of calculations explicitly including electron-electron interactions and allowing for configuration mixing.

Acknowledgements. This work has been partly supported by EEC-BRITE/EURAM contract 0148 (NAPOLEO).

.R. eferences.

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(13) M.C. dos Santos and J.L. Brtdas, Phys. Rev. Lett. 62, 2499 (1989). (14) D. Baeriswyl, D.K. Campbell, and S. Mazumdar, in Conducting Polymers, ed. by H. Kiess (Springer, Berlin, 1990). (15) J.L. Br~las and A.J. Heeger, Phys. Rev. Lett. 63, 2534 (1989). (16) J.L. Br~tas, D. Beljonne, Z. Shuai, and J.M. Toussaint, Synth. Met., in press. (17) H. Eckhardt, K.Y. Jen, L,W. Shacklette, and S. Lefrant, in Conjugated Polymeric Materials: Opportunities in Electronics, Optoelectronics, and Molecular Electronics, ed. by J.L. Br~ias and R.R. Chance (Kluwer, Dordrecht, 1990), p. 305. (18) C.J. Finder, M.G. Newton, and N.L. Allinger, Acta Crystallogr. B 30, 411 (1974). (19) L. Salem, The Molecular Orbital Theory of Conjugated Systems (Benjamin, New York, 1966). (20) J.L. Br~:las, R.R. Chance, R.H. Baughman, and R. Silbey, J. Chem. Phys. 76, 3673 (1982). (21) J.L. Brtdas and G.B. Street, Acc. Chem. Res. 18, 309 (1985). (22) D.D.C. Bradley and R.H. Friend, J. Molec. Electronics 5, 19 (1989).