J. Phys. Chem. 1992,96,2827-2830
2827
Transverse Polaron Bandwidth In trans-Polyacetylene E.M. ConweU,**t** H.-Y. Choi:q* and S. Jeyadev* Center for Photoinduced Charge Transfer, University of Rochester, Hutchison Hall, Rochester, New York 14627, and Xerox Webster Research Center, Webster. New York 14580 (Received: September 18, 1991; In Final Form: November 25, 1991) Measurements over the temperature range 150 < T 5 300 K have shown that polarons (radical anions or cations) in trans-polyacetylene move perpendicular to the chains by hopping. This indicates that the transverse polaron bandwidth W is less than kBTin this temperature range, in contradiction to a previous calculation of Wfor T = 0. We have recalculated the bandwidth, extending the calculation to temperatures above 0 K and taking into account the change in overlap of the valence electron wavefunctions when the polaron hops. Because this overlap is a function of chain length, the resulting bandwidth and hopping rate will be sample dependent. Using the available information on conjugation lengths in polyacetylene, we find that the bandwidth is consistent with hopping dominated transport in the range where it is observed.
I. Introduction Addition of an electron (hole) to a polyacetylene chain, according to the Su-Schrieffer-Heeger (SSH) Hamiltonian and its continuum version, creates a stable polaron or radical anion (cation) with chain relaxation extending over more than 20 sites.'-3 The polaron has two levels in the energy gap, occupied by three electrons in the case of a negatively charged polaron and one in the case of a positively charged polaron. In both cases the charges are screened by opposite charges in the valence band, leaving a net charge of &e, the electronic charge, on the polaron. The extension of the polaron over many sites on the chain and the fact that it can move freely along the chains are in sharp contrast to the properties of the much studied small polaron, found typically in ionic materials such as KCI, for e ~ a m p l e . The ~ small polaron is characterized by a wave function that extends no more than an interatomic separation in all directions; this is true only for the conjugated polymer polaron in the direction perpendicular to the chains. There is strong evidence from photoinduced absorption experiments on trans-polyacetylene that a sizable fraction of the excitations created by photons with energy greater than the bandgap are p o l a r ~ n s . These ~ polarons persist for nanoseconds.5 Photoconductivity measurements also provide evidence for the existence of polarons on a nanosecond time scale.6 Polaron transport may be by band motion, in which the chain deformation or phonon cloud essentially moves with the polaron, or by hopping. Polaron transport along the chains, which is band motion, has been studied by Jeyadev and Conwell.' They calculated the polaron mobility due to phonon scattering in transpolyacetylene. Polaron motion perpendicular to the chains must also be important for photoconductivity, because of the relatively short chain and conjugation lengths in polyacetylene, and has been observed in oriented samples.* It is expected, as is generally true for the small polaron: that in the low-temperature limit, motion perpendicular to the chains is band transport, while at higher temperatures phonon-assisted hopping dominates. The transverse photoconductivity was found by Walser et ale8to increase with increasing temperature over the measured range 160-300 K. The increase was very slow at the lower temperatures in the range and then more rapid. This behavior indicates that the transport is hopping over the measured range. The critical factor determining whether polaron motion is bandlike or hopping is the bandwidth. For temperatures where the bandwidth W >> kBT, band motion dominates. When kBT > W, the band loses its meaning and hopping d ~ m i n a t e s .The ~ bandwidth of polarons transverse to the chains in polyacetylene was first considered by Jeyadev and Schrieffer? to be denoted JS. They used the SSH Hamiltonianio to calculate the 0 K Franck-Condon factor for overlap of the nuclear (C-H) wave
-
University of Rochester. 'Xerox Webster Research Center.
functions of a polyacetylene chain with and without a polaron. It is the quantum fluctuations of the order parameter, the zeropoint oscillations in this case, that make the overlap finite and thus the transition possible. The JS result was a 0 K bandwidth of 0.03 eV.9 This suggests that band transport would persist to temperatures greater than 300 K, in contradiction to the experimental data. The JS calculation of the bandwidth did not allow for the fact that the wave functions of all the valence band electrons are different when there is a polaron on the chain from those of the equilibrium chaii without a polaron, Le., the dimerized chain. This decreases the overlap of the valence band wave functions and must be taken into account in calculating W. As will be seen,this factor substantially decreases the bandwidth, removing the contradiction between theory and experiment. The JS method of calculating W does not allow extension to temperatures above 0 K. To study transport at f ~ t temperatures, e we have converted the SSH Hamiltonian to a normal mode representation." This will be described briefly in the next section. Calculation of the Franck-Condon factor and of the valence electron overlap will be taken up in the following two sections. Finally, the bandwidth and its temperature variation will be evaluated and the implications discussed. 11. Formal Expression for Bandwidth To calculate the bandwidth, we need the transfer or hopping integral between the state with the polaron on chain I, which could be called the initial state, and the state with the polaron on a neighboring chain, 1 f 1, which could be called the final state. Since all other chains may be neglected, the wave functions of the initial and final states, \ki and qf,respectively, may be taken as I q i ) = l*/(~))l*/*~(d)) (11.1) (11.2)
where d stands for dimerized, the description of the chain which does not have the polaron. Within the Born-Oppenheimer ap(1) Campbell, D. K.; Bishop, A . R. Phys. Reu. E 1981, 24,4859. (2) Chance, R.; Boudreaux, D. S.;Brcdas, J.-L.; Silbey, R. In Handbook of Conducting Polymers; Skotheim, T. A., Ed.; Dekker: New York, 1986; Vol. 2, p 805. (3) Su,W.-P. In Handbook of Conducting Polymers; Skotheim, T. A., Ed.; Dekker: New York, 1986; Vol. 2, p 757. (4) For a review see: Appel, J. In Solid State Physics; Seitz, F., Turnbull, D., Ehrenreich, H., Eds.; Academic Press: New York, 1968; Vol. 21, p 193. (5) Rothberg, L. J.; Jedju, T. M.; Townsend, P. D.; Etemad, S.;Baker, G. L.Phys. Reu. Lett. 1990, 65, 100. (6) Bleier, H.; Donovan, K.; Friend, R. H.; Roth, S.;Rothberg, L.; Tubino, R.; Vardeny, Z.; Wilson, G. Synfh. Met. 1989, 28, D189. (7) Jeyadev, S.; Conwell, E. M. Phys. Reu. E 1987, 35, 6253. (8) Walser, A.; Seas, A.; Dorsinville, R.; Alfano, R. R.; Tubino, R. Solid Sfafe Commun. 1988, 67, 33. (9) Jeyadev, S.;Schrieffer, J. R. Phys. Reo. E 1984, 30, 3620. (10) Su, W.-P.; Schrieffer, J . R.; Heeger, A . J. Phys. Rev. E 1980, 22, 2209; 1983, 28, 1138(E). (11) Choi, H.-Y.; Conwell, E. M. Mol. Crysf.Liq. Cryst. 1991, 194, 23.
0022-365419212096-2827%03.00/0 0 1992 American Chemical Society
2828 The Journal of Physical Chemistry, Vol. 96,No. 7, 1992
proximation the wave functions may be written as the product of an electronic part $ and a nuclear part X,i.e.
Id(*) ) M ( m ) ) = I$/(P) )I$!(*) ) W(m) )
I*/(d)) = I*/(P) )
I
(11.4)
(11.5)
where /3 = l / k B T and H, is the operator for the interchain electron hop. Since the wave function $Ap) transverse to the chain is the same as that of any a electron, we take 4(*/*1(P)IH,l*i(P)) = -4t, (11.6) where -4t, is the transverse bandwidth of the ?r band for the particular hopping direction. That bandwidth will be taken from band structure calculations, as was done by JS. With eq 11.6 and the other equations of this section, eq 11.5 may be rewritten:
where
4/ = (dP(*)I$P(*)) and m' represents the number of phonons in the various oscillators for the state with total energy E,. The factor involving the X s is squared because chains I and l&l give identical contributions. The quantity in angular brackets is the Franck-Condon factor for the transition. We will calculate this factor first. 111. Franck-Condon Factor To convert the nuclear coordinates to a set of normal coordinates qk we must first obtain the dynamical matrix U for the system. Because of the electron-phonon coupling, this requires considering the electronic motion as well as the nuclear motion. The electronic part of the SSH Hamiltonian for motion along the chain is given bylo He = -E[to + (-1)".(J.n+1
+ $n)(CZ+lCn + HC)I
n
= c(p;/2M) n
+ f / zn,mE x n q t $ m
= $n -
+:
n
+ $n)'
(111.5)
+ I/2C(xn - G n ) G m ( x m - 6,)
(111.6)
n,m
where 6, is the deviation in the order parameter from the perfectly dimerized state due to the polaron. The dynamical matrix for this case may be written qm
- d2vp(i$iI)l W n W m
(111.7)
luil=ppipI
where V, is given by (111.5) with Ed((+,,))replaced by EP(($,,]), the electronic energy obtained from (111.1) when one electron is added to the half-filled band, giving rise to a polaron. To convert the coordinates xn to a set of normal-mode coordinates qk, we use the unitary transformation B that diagonalizes the dynamical matrix U . Thus, with BUB-l a diagonal matrix, the desired transformations are xn
= C(flk)-'qk
(111.8)
k
for the dimerized chain and xn
- 6 n = C(B!k)-'(qk - b k ) k
(111.9)
for the chain with the polaron. The matrices Bd and BP may be taken as having rows made up of the normalized eigenvectors of Udand UP, respectively. Note that, according to (111.8) and (111.9) the coordinates qk are not the same for the dimerized chain and the chain with the polaron. Because, as will be discussed, all but a few of the normal modes are, to all intents and purposes, the same for the chain with and without the polaron, we have made the approximation that the two sets of qk are the same. The introduction of normal modes allows the nuclear wave function to be written as a product of harmonic oscillator wave functions with frequency wk and occupation number mk. Thus we take VCP(m) ) = ?lx'(mA)
(III.10)
XP(mk)= fimt[(MUk/h)l/'(qk- 8 k ) l
(111.11)
where
fim(z) = (2mm!ai/2)-i~2e-z'/2Hm(z) (111.12)
H,(z) being the Hermite polynomial of order m. The index I has been dropped because the wave function (111.10) is the same whatever chain the polaron is on. For the chain without the polaron we may write, similarly
CXd(m?) = vIXd(m'k))
(111.2)
(111.1 3 )
where
where xn
Hp = C ( P ? / % M )
(111.1)
where +n = ( - 1 ) " ~is~ the staggered order parameter, un being the actual displacement of the nth C-H group, to the transfer or resonance integral, and CY the electron-phonon coupling constant. In the harmonic approximation the effective Hamiltonian for the nuclear coordinates on a perfectly dimerized chain may be writteng Hd
= Ed({$"))+ (K/2)E($n+1
where K is the spring constant, arising from the u electrons. &(($,j) in this case is the *-electron contribution obtained from (111.1) for the case that the *-band is exactly half-filled. In the state with one polaron an effective harmonic Hamiltonian may be written similarly as
(11.7)
I
I
vd
n
I
WT= -4t, $dl+l [ Ee-bErI (x;'(m')W(ml)) 12/ Ce-bE1]
taken as the ground-state energy &(($,I)of the *-electron system plus the harmonic lattice interaction,
(11.3)
Here $?(a) and I$?(*)represent the wave functions of the valence band electrons in the absence and presence, respectively, of the polaron, $,(p) the wave function of the electron bound on the polaron, and tn the number of phonons in the various normal modes. Because of the dependence of the nuclear wave functions on m, the system may exist in different energy states at a given temperature T. To obtain the bandwidth W, it is therefore necessary to average over all possible initial states. (No average or sum over final states is necessary; for the calculation of the bandwidth, only final states with energy equal to that of the initial state contribute.) The thermally averaged transverse bandwidth WTis given by WT = 4Ce-BE1(\kflH,(\k,) /Ce-bEI
Conwell et al.
(111.3)
the displacement of the nth C-H from its position, represented by the superscript d, in the perfectly dimerized chain. Ud is the dynamical matrix, whose elements are
IXd(m'k)) = fim,k[(MWk/h)1/2qkl
(111.14)
After integration over the normal coordinates, using (111.10)(111.14) and mk = mi as required for calculation of the bandwidth, we obtainI2
(Xd(m)lXP(m)) = ne-AX2/4Lm,(&2/2) (111.15) (111.4) Vd being the adiabatic potential in which a C-H moves. Vd is
k
(12) Gradshteyn, I. S.; Ryzhik, I. M. Tables of Integrals, Series and Products; Academic Press: Orlando, FL, 1980.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2829
Transverse Polaron Bandwidth in trans-Polyacetylene
2.00
where Ak
(111.16)
(MOk/h)lJ26k
1.60
and Lm,is the Laguerre polynomial of the mkth degree. To complete the calculation of the Franck-Condon factor, we must carry out the summations over i indicated in eq 11.7. The energy Ei of the chain vibrations in the ith state of the I chain is given by
E , ( / ) = Ehwk(mi k
+ 72)
(111.17)
where mk may take on any integer value from 0 to -. With (111.17) and (111.15) the term in angular brackets in (11.7) becomes a product over k for each chain. Using the notation z = e-uhuk, x = Ak2/2
m
zmkLmk(x)= exp[xr/(z - l)] (111.19) mt = 0
the right-hand equality coming from Gradshteyn and Ryzhik.Iz This leads finally to the Franck-Condon factor” FC
= [13e-uEil(xj‘(mi)lxf’(mi))12/Ce-uEl] i
i
FC = eXp[-CAk2(mk k
0.80
0.40
(111.18)
we find each term in the product over k to be of the form (1 - z)
1.20
+ y2)]
(111.20)
where Ak is defined in (111.16) and
mk =: (e8hwk-
(111.21)
The form (111.20) for FC was also obtained by Holstein in his classical calculations for the small p01aron.I~ In that case Ak2 is replaced by a factor involving the polaron binding energy and (1 - cos &a),a being the distance along the chain between hopping sites.
IV. Overlap of Valence Band Wave Functions There are two reasons for the valence electron wave functions to change in the presence of a polaron: the Coulomb repulsion (attraction) due to the negative (positive) charge on the polaron, and the change in the C-H positions, or chain relaxation, due to the polaron. Analytic expressions for the valence electron wave functions in the presence of a polaron have been obtained with the continuum model.I4 These wave functions take into account only the second of the two effects listed above because the SSH and continuum models do not explicitly include Coulomb interactions. Using these wave functions and the valence electron wave functions for the dimerized chain obtained from the continuum model, Su and YuI5 have calculated the overlap. They took the valence band wave functions in the presence and absence of the polaron as simple Hartree products of the respective one-electron wave functions. The result isI5
(IV.1) where f = 2toa/A
a being the distance along the chain between C-H’s, A the half-gap, and L the chain length or, more accurately, conjugation length. (IV.l) is clearly incorrect in the limit L 0 but, of course, is not applicable in that limit because the chain could not support a polaron. As L increases, 0 decreases rapidly. This is not unreasonable because as L increases there are more valence band wave functions to be affected by a polaron hop. A consequence
-
(13) Holstein, T. Ann. Phys. ( N Y ) 1959, 8, 343. (14) Campbell, D.; Bishop, A. Nucl. Phys. E 1982, 200, 297. (15) Su,2.-B.; Yu,L. Phys. Rev. E 1983, 27, 5199.
0.00 0
10
20
30
40
50
60
MODE NUMBER
Figure 1. Calculated values of 1AkI2vs k for a 60-site (CH), chain. The modes are plotted in order of ascending frequency. Thus acoustic modes are almost all below number 30, while almost all optical modes are above 30. The inset shows the dispersion for a chain with a polaron. uQ2= 4 K / M = 2.5 X IOI4/s and the 0 indicate localized modes.
of the rapid decrease of Q with L is that the bandwidth is much greater for hops between short chains, or more accurately short conjugation lengths. Investigations of resonant Raman scattering have characterized polyacetylene as having a bimodal distribution of conjugation lengths.I6 For a recent group of samples with varying molecular weight, fits to the Raman data gave the most probable length of the long segments as 200a, of the short segments 30a.I’ For the Gaussian distribution assumed, the standard deviation for the long segments was 100a, for the short segments lOa.” With the usual parameter values-to = 2.5 eV, a = 1.22 A and A = 0.7 eV-for L = lOOa Q = 0.074, while for L = 300 9 = 2.47. In calculating the bandwidth, it is necessary to allow for the chains 1 and If1 having different conjugation lengths. From the fit to the Raman spectra it was concluded that the relative proportions of long and short conjugation lengths depends on sample treatment. At a minimum, however, the fraction of long conjugation lengths was going up to 4/5 in a “very high quality sample”.17 We conclude that, at least for the samples of the Raman studies, most of the hopping that contributes significantly to transport would be between one short and one long chain. With the Q values cited above, a representative value of might be 0.074 X 2.47 = 0.18.
V. Determination of the Bandwidth To evaluate FC, we used the method of Terai and OnoI8 to calculate the dynamical matrics Udand UP. The calculations were done for a chain of 60 sites, thus 60 X 60 dynamical matrices. The q ’ s that resulted are shown in the inset of Figure 1. They are in excellent agreement with those obtained by Terai and Ono and others. With bk calculated from UP using (111.9) and the definition of Akr (111.16), we obtained the 1Ak12values shown in Figure 1. It is seen that the coupling is strongest to higher frequency acoustic and optical modes. Summation over k of Ak2 gave 10, leading to a Franck-Condon factor at T = 0 of 5.6 X This result is considerably smaller than that of JS, 0.09, which was obtained by a different method. We checked our result, however, by using the analytic expression obtained for FC at T = 0 by JS, involving the set of 6,‘s and the dynamical matrices. The temperature dependence of the bandwidth, which comes from that of FC, given in (111.20), is plotted in Figure 2. The small variation at low temperatures is due to the fact, noted earlier, that is sizable only for fairly high-energy modes, which are (16) Brivio, G. P.; Mulazzi, E. Phys. Reu. E 1984, 30, 876. (17) Shen, M. A.; Chien, J. C. W.; Perrin, E.; Lefrants, S.;Mulazzi. E. J . Chem. Phys. 1988,89, 7615. (18) Terai, A.; Ono, Y . J . Phys. SOC.Jpn. 1986, 55, 213.
J . Phys. Chem. 1992, 96, 2830-2836
2830
o.20 0.00
t
1
~
0
100
300
200
m
500
T(k)
Figure 2. Temperature variation of the transverse polaron bandwidth. not excited at low temperatures. Hopping may occur between equivalent chains or inequivalent chains. Although in one direction equivalent chains are spaced only 4 A apart, about the same as the distance between inequivalent chains, t , for the inequivalent chains is considerably larger.19 In further considerations we will deal only with in(19) See, for example: Mizes, H. A.; Conwell, E. M. Phys. Reu. B 1991, 43, 9053.
equivalent chains. For 4lt,l we take the maximum transverse bandwidth calculated for trans-polyacetylene by Vogl and Campbell,200.5 eV. Combining this factor with the other factors in (11.7), and assuming conjugation lengths of 30a and 100a, we find, for T = 0, WT = 5 X lo4 eV, corresponding to 6 K. For a more favorable, but less likely, hop, between two lengths of 30a, WT = 0.017 eV, corresponding to 200 K. We conclude that, when a suitable average over all pairs of conjugation lengths is taken, WT should certainly be less than 150 K, the lowest temperature at which measurements of transverse photoconductivity were taken.8 It is evident that the average hopping rate, and therefore conductivity or photoconductivity, transverse to the chains will be sample dependent. Samples with shorter average conjugation lengths should have higher transverse conductivity. Although we do not have the information on conjugation lengths required to obtain an accurate bandwidth or hopping rate for any given sample, we are able to determine the temperature dependence of the hopping rate, which does not depend on the 0,'s. Using the expression for the hopping rate obtained earlier," we have been able to calculate the temperature dependence of the hopping rate in the range -250 IT I 300 K.*' We find that it agrees well with the temperature dependence of the transverse photoconductivity measured in this range, where it is increasing rapidly. Registry No. trans-Poly(acetylene),25768-71-2. (20) Vogl, P.; Campbell, D. K. Phys. Rev. B 1990, 41, 12797. (21) Conwell, E. M.; Choi, H. Y.; Jeyadev, S., to be published.
Nonlinear Optical Properties of Polyanilines and Derivatives John A. Osaheni, Samson A. Jenekhe,* Department of Chemical Engineering and Center for Photoinduced Charge Transfer, University of Rochester, Rochester, New York 14627-0166
Herman Vanherzeele, Jeffrey S. Meth, Du Pont Central Research and Development, Wilmington, Delaware 19880-0356
Yan Sun, and Alan G. MacDiarmid Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 191 04-6323 (Received: September 23, 1991; In Final Form: November 27, 1991)
The third-order optical susceptibility x ( ~ ) ( - ~ w ; w , w , wof) polyanilines and derivatives has been systematically investigated by picosecond third harmonic generation spectroscopy on spin-coated thin films of the polymers in the wavelength range 0.9-2.4 pm (1.4-0.5 eV). It is shown that the magnitude of x ( ~ ) ( - ~ w ; w , w , wof) this class of polymers is as large as that of other conjugated polymers and that the optical nonlinearity depends on the oxidation level and the derivatization of the pphenylene rings. The dispersion of the optical nonlinearity is dominated by the three-photon resonance to the dipole allowed transition occuring at 1.8 eV, so that the excitonic transition is the major contributor to the optical nonlinearity of polyanilines. Polyemeraldine base, with an oxidation level of about 50%, has a larger optical nonlinearity than the fully oxidized form pemigraniline or poly(phenylani1ine) which we use as a model compound for the fully reduced form. The effects of derivatization are more complex. Methoxy substitutionof the phenyl ring increases the transition moment, which would increase the microscopic nonlinearity. However, these substituents also reduce the number density of the polymer repeat units, which in turn reduces the macroscopic nonlinearity.
-
Introduction Interest in conjugated polymers with large third-order nonlinear optical (NLO) properties for potential applications in photonic devices has grown t r e m e n d o ~ s l ybecause ~~ the third-order NLO effects in these materials are very rapid (subpicosecond). The ease of fabrication and molecular engineering of the materials is an additional advantage of polymers for future nonlinear optical *To whom correspondence should be addressed.
TABLE I: Intrinsic Viscosity of Polyanilines and Derivatives in MSA at 40 O C polymer 7,dL/g polymer 7 , dL/g PEMB 1.63 PDMAB 0.38 POTB 0.72 P4PAB 0.13 PMAB 0.47
devices. In order to synthesize new materials with larger thirdorder susceptibility or optimize existing ones, it is essential to
0022-365419212096-2830$03.00/0 0 1992 American Chemical Society
