- Email: [email protected]
Excitons and interband excitations in polyphenylenes
syR'il'll frl'lI[ I II|TaLS ELSEVIER
Synthetic Metals 73 (1995) 183-190
Excitons and interband excitations in polyphenylenes M.J. Rice 3, Yu.N. Gartstein a,b,l a Xerox Corporation, Wilson Center for Research and Technolgy, Webster, NY 14580, USA b Center for Photoinduced Charge Transfer, Chemistry Department, University of Rochester, Rochester, NY 14627, USA Received 26 January 1995; accepted 7 February 1995
Abstract
A microscopic model which elucidates the essential physics of the photoexcitations of the polyphenylenes is introduced. Recognizing that these excitations are derived from the local excitations of the phenylene monomer, it allows for the effects of the 'correlation-energy gaps' Un between charged and charge-neutral excitations, and the dipole-dipole interaction V~d between neighboring monomers. It leads to the generic appearance of three principal, dispersing, absorption bands whose relative spacings in energy are largely set by U,, and the electronic bandwidth W, thereby establishing direct experimental access to these microscopic parameters. There is a fourth, dipole-forbidden, singlet excitation band which can be rendered allowed by distortion of local D6h symmetry. A family of four distinct, tightly bound, triplet state excitons is also found. Keywords: Polyphenylenes; Excitons; Interband excitations; Model
1. Introduction
The undoped forms of the conducting polymers based on phenylene form a wide and interesting class of organic semiconductors [1]. They include poly(para-phenylene) (PPP), poly(phenylene vinylene) (PPV), and the atomically bridged polyphenylenes such as polyaniline (PAN) and poly(phenylene oxide) (PPO). They have relevant 7r-electron bandwidths W ranging from W< A to W<< A, respectively, where A is the mean spacing between the molecular orbital (MO) energy levels of the isolated phenylene monomer [ 2,3 ]. In this paper we introduce a simplified but analytically explicit microscopic model which we believe elucidates the essential physics of the excitations of these polymers, not only as a function of the frequency w but also of the wavevector Q of the exciting electric field. The model recognizes that the primary excitations are derived from the local (elg--* ezu) excitations of the phenylene monomer and, correspondingly, includes two effects which although are implicit in quantum chemical calculations are not ordinarily considered in the usual band models of the polymers [4]. These are the 'correlation-energy' gaps U, between charged and charge-neutral excitations [5], and the dipole--dipole interaction Vud between local excitations on neighboring Also at the Department of Thermal Physics, Uzbekistan Academy of Sciences, Kata~al 28, Tashkent 700135, Uzbekistan. 0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D 1 0 3 7 9 - 6 7 7 9 ( 9 5 ) 0 3 3 1 3-9
monomers [ 6]. U, is the energy required, in the limit W--* 0, to dissociate a local singlet state monomer excitation of symmetry species n into a well-separated electron and hole. This effect, which behaves as an effective intramonomer electronhole attraction, is crucial as it leads, for W ~ 0, to the transfer of oscillator strength from interband transitions to exciton excitation even before the long-range electron-hole Coulomb attraction Veh is allowed for. In fact, the symmetry of the local phenylene excitations is shown to lead to three dispersing excitons whose relative spacings in energy are largely set by U, and W, and corrected in interesting ways by Vdd and Veh, thereby establishing experimental access to the values of these parameters. Interestingly, there is also a fourth, dipoleforbidden, singlet exciton that can be rendered allowed by distortion of the local m o n o m e r D6/, symmetry. The significance of the inter-relation of these four excitons has not been recognized in the current debate [7 ] on the origin of the first absorption band in PPV. As in a molecular crystal, the effect of Vdj is to contribute to the stabilization and dispersion of the excitons and is clearly an important consideration in the limit of small W. While the complex conductivity o'(Q, ~o) that we calculate is in good qualitative agreement with that derived from the existing electron energy loss spectra (EELS) of the polymers [2,8], we call for new EELS studies on films of improved crystallinity in order to investigate the anisotropy of the conductivity and the exciton dispersion due to Vdd that we predict in the UV. Finally, we briefly consider
M.J. Rice, Yu.N.Gartstein/ SyntheticMetals 73 (1995) 183-190
184
X d = -- ~_,{tbb(b~+l,obm + h.c.) +taa(aJ+l.,fllj,,+h.c.)
j,r "...-
+
+ t~b( (b]+ ,.,~- b)_ ipl,,~)aj,- ~ + h.c.) }
describes the delocalization of the b and a orbitals due to the intermolecular hopping integrals t, while
-
(a)
(b) =
Fig. 1. (a) The orbital patterns of the e~gand e2, molecularorbitals and (b) the four componente~g--+e2~transitions of the phenylene monomer. the triplet state excitations of the model and find a family of four distinct triplet excitons. They are tightly bound on account of the largeness of their respective correlation-energy gaps U,, in relation to W. Relaxation of the lattice about such excitations is an important consideration. Our model, based on PPP as the prototypical polymer, consists of a chain of N(N--+ o0) interacting phenylene monomers. Each monomer is assumed to have two degenerate (elg) hole states, labeled a and c, with excitation energy Eh, and two degenerate (e2,) electron states, labeled b and d, with'excitation' energy E~. The orbital patterns of these states under an assumed D 6 h symmetry of the monomer are shown in Fig. 1 (a) [9]. If we introduce the fermion operators a]~., b]~, c}~ and ~ , which create, respectively, an electron or hole with spin polarization o- in the latter orbitals of the monomer j ( j = 1 ..... N), the four possible dipole-allowed elg-+e2u excitations of the monomer are described by the operators: B]~,=btoa]_~., D]~=~oZ:~_~, A]~.=~oa]~ and CJ~ = b]~.c~_,,. These unsymmetrized component excitations are illustrated in Fig. 1 (b). They may be combined to form the four (n = 1-4) following singlet excitations of the phenylene monomer [%10]: t PJay'(B]~,-D}o.)/2;
E,.(x)
(1)
*-~_,(A]o~+C}~,)/2; E..(y)
(2)
o"
P;,2-
o.
P]a=~,(B},~+D],~)/2;
Blu
o-
P tJ , 4D - Y ' ( A ~ - C J ~ ) / 2 ;
B2.
The first and second E~u excitations have transition dipole moments of magnitude P-o ( ~ aoe, where ao denotes the carbon-carbon bond length and e the electronic charge) in the monomer x and y directions, respectively, while the Blu and B2~ excitations are dipole-forbidden under D6h symmetry [ 9]. In terms of these operators the Hamiltonian ,g( defining the model is
,.~=
,~,{ Eh(a]oaj,~ + c~oc),.) + Ee(b~o-bi,~ + ~odj,D } jo"[- ~ d
where
"[- ,~'fee
(4)
(3)
-
E u.t'J.Pj° + vdo + Veh j,,
(5)
specifies the leading contributions arising from Coulomb interactions. The first term in Eq, (5) specifies the energy gain, Un, when an electron and a hole in a singlet state are simultaneously present on the same monomer in the symmetry combination n. Un actually corresponds to the energy required, in the limit t ~ 0, to dissociate the local excitations Pin into wellseparated electrons and holes [ 5 ]. Since in the hypothetical dissociation process the total number of intramonomer interelectron distances increases precisely by 1, Un can be qualitatively understood as the repulsive Coulomb interaction associated with this extra 'bond' 1. Since P]1 and P]2 are symmetry doublets, UI = U2. If we denote Ee + Eh by 2ao, it follows from the first terms of Eqs. (3) and (5) that the local monomer excitation energies are En = 2 a o - U,. We expect these to depend mildly on the electronic polarizability of the polymeric medium, while we expect both ~ and U~ to depend much more significantly on this quantity. For the benzene molecule, vapor phase molecular spectroscopy gives [ 12] E~ = E 2 --- 6.8 eV, E 3 = 6 eV and E 4 = 4.7 eV. For the polymer we may crudely estimate the U, from Un = 2c~o - E n with 2 a o ~ ( I - A - 2 E p ) , where L A and E, are the ionization potential, electron affinity and singlet excitation energies, respectively, of benzene and Ep the electronic polarization energy [ 11 ] of the singly charged monomer in the dielectric medium of the polymer. If, for PPP, we take Ep = 1.5 eV, we obtain U~ = Uz ~ 1 eV, [/3 ~ 1.8 eV and U4 ~ 3 eV. We note that for U1 v~/-/3 the first term of Eq. (5) leads to an intramonomer coupling of the B t and D t excitations, while for U2 ¢ U4 it leads to a coupling of the A* and C t excitations. We can expect that such couplings play a role in determining the form of the absorption spectrum of a given polyphenylene. The second term of Eq. (5)
Vdd= - EVom(eJ+,. +
l,o)(eL +P,°,)
jnm
describes the delocalization of the charge-neutral intramonomer excitations via the dipole-dipole interaction: If NOis the total number of electrons on the charge-neutral monomer there are No(No-1 )/2 interelectron distances on the monomer and this number remains the same if the monomeris excited. For the well-separated negatively and positively charged monomers the number is changed to N+ = (No- 1) (No - 2)/2 and N_ = (No+ 1)No/2, respectively. Thus, the net change in monomer interelectron distances following the hypothetical dissociation process is N÷ +N_-2No = 1. This energy cost is already included in E~ and Eh, but must be subtracted when an electron and hole reside on the same monomer.
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190 Vnm = R - 3 (/.tn "/X,, l - - 3 R - 2 ( / . t n - R ) ( / x m - R ) )
where R = Rj + ~- Rj is the relative coordinate of the jth and (j + 1)th site, and/x~ is the transition dipole moment of the local excitation n. For the actual case of PPP, the phenylene x-axis coincides with the polymer axis, so that V,m is diagonal with V~l = V > 0 for the E~u(x) excitation and V== - V / 2 for the E ~ ( y ) excitation (provided a planar configuration of the PPP chain is assumed). Following the discussion of Salem [13] we estimate V~0.25 eV if the known experimental value of the transition dipole moment of benzene is employed. As will be seen later, a nonvanishing V,,, will lead to dispersion of the absorption bands in the ultraviolet (UV) which could be measured via EELS experiments on highquality crystalline films. With only on-site and dipole-dipole interaction left in Eq. (5), our model resembles a simpler model introduced by Egri [ 14]. The third term in eq. (5) is the long-range Coulomb attraction Veh which is introduced as
185
In the present discussion of our model, the effects of electron-phonon interaction, lattice relaxation and disorder are not included at the microscopic level. As we shall see, the effect of disorder will be treated by the introduction of phenomenological lifetimes for the single electron and single hole states. We note that if H d = 0, ,U describes two dispersing Frenkei excitons, one derived mainly from the E~u(x ) monomer excitation and the other from the Elu(Y) excitation. We now illustrate the generic way in which this picture is changed for ~'~d 4=0 by calculating the complex conductivity o-~,(Q,~o) ( a = x or y) for the case of PPP. The case for PPV will be similar. Denoting by Q an arbitrary wavevector along the xdirection, the paramagnetic conductivity is given by the Kubo formula [ 16]: zo
~r,~,~(Q,w) = (iwANb) - ~ / d t ,/ 0
× exp(iwt) ([J~(t), J~_Q(O) ] )
Veh = V c ~ & p J 2 IJ - 11
(6)
j~l
where pj='Z~(a]oaj,~+c],~cj~-b]~bi,~-~oztj~ ) is the net charge density on the monomerj. In Eq. (6) no allowance is made for the different symmetries of the local electron and hole orbitals. The constant Vc is the interaction energy of charges on neighboring monomers and ECT = 2 a o - Vc may be considered to be the 'charge-transfer' exciton energy in the limit t--+ 0. An estimate of Vc is provided by Vc -- e2/eb, b being the monomer lattice spacing and • the dielectric constant. Reasonable estimates put Vc ~ 0.1 to 0.3 eV for PPP and PPV. For the atomically bridged polyphenylenes the hopping integrals in Z/d represent virtual hopping over the bridging atom. A more general model Hamiltonian for these polyphenylenes, which do not possess charge conjugation symmetry, would contain different site energies for the a, b, c and d excitations. For PPV, both the electron hopping in "g"d and the excitation hopping in ~ represent virtual hopping processes over the vinylene unit. Note, however, that in this case the mixing between benzene and vinylene states is rather strong. The calculation in Ref. [ 15] showed that the band structures of the relevant bands in planar PPP and PPV are very similar and the interband splitting in PPV is much smaller than the bandwidths. In the context of electron delocalization, therefore, the PPP-type band structure is expected to be a good approximation for the effective model of PPV. Of course, in a more detailed description the model can be extended to treat the vinylene states in PPV explicitly. We note that in Eq. (4) there is no intermonomer hopping between the c and d orbitals (cf. Fig. 1(a)) so that these hole and electron orbitals remain localized. For the explicit calculations in the present paper we will assume charge conjugation symmetry and take taa = tbb = tab = t, a constant, and Ee = E~ = ao.
(7)
where J~ = ~/exp(iQRj)Ji'~is the current fluctuation operator for the a-direction and A the transverse area corresponding to one chain. The interband current densities ~" are
~=ie{vo(P),l-Pj, I) +(Vl/2)~7~((b]+l,o-+b]_,,o-)af_o--h.c.)
}
(8)
(r
~' = ievo( P],2 - Pj,2)
(9)
where vo=tzOEl/eh arises from the transition dipole moments of the E~u excitations and vt arises from the 'oblique' interband hopping terms in ~?'d. For PPP taa = / b b = l a b ~ t, and for the planar polymer MO theory gives Vl = Vo/3 2. More generally, the ratio Vl/Vo may be regarded as a parameter of the model. For nonplanar PPP, the y-polarized absorption must be understood as the absorption for the polarizations perpendicular to the chain direction. We note from Eqs. (8) and (9) that intermonomer hopping breaks the local Etu symmetry of the displacement current, leading to anisotropic absorption. The algebraic evaluation of Eq. (7) is greatly simplified if two effects in ~ are ignored. The first effect concerns the terms in ,,g?'eedescribing the creation and destruction of pairs of excitations. The second effect concerns the oblique hopping terms in Z o after they have been retained in ~. The neglect of these two terms introduces errors of order (V/2ao) and (t/ao) 2, respectively. Since t < 1 eV, 2 a o ~ 7 eV and V< 0.25 eV typically, these corrections are small. Also, since the neglect of the pair-excitation terms corresponds to the neglect of local-field effects, it follows that the final state problem in the evaluation of Eq. (4) reduces to the problem of a single electron interacting with a single hole. The Kubo 2 In the MO theory/.to = eao, E~ = 2to, where to is the interatomic hopping integral between neighboring C(2pz ) orbitals and ao the C - C distance. For the planar PPP, t = to~3 and vl = 2aot=/3o/3.
186
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
formulae are then readily evaluated in terms of standard ladder diagrams only [16]. These are shown for o'x~(Q,~o) in Fig. A1 in Appendix A. The results can be expressed in a closed analytical form if the long-range Vch is set to be zero. (We will discuss the effect of finite V~h later in the paper.) For reasons of clarity we first present the results for v~ = 0. The latter does not affect the form of the excitation spectrum or the explicit form of rryy(Q,w). We obtain
~r~,~,(Q,w) = ( eZv~/2Abiw) [ F~(Q,w + iF) +F.(-Q,-
(w+iF))]
where F,, = (X~ + X~'- 2( U,,.Q -Vct, Q) X~lX.~)/D,
(10)
D~, = ( 1 - U~,QXl ) ( 1 - U,~,QX2) ,~ ~ - ( G , o ) 2x~x2
(11)
in which X~'= X'J(Q, w) are the propagators for the four component elg ~ e2~ monomer transitions and U~,Q and V,,,Q are matrix elements that we shall define shortly. The q u a n t i t y / ' = h ( 1 / % + 1 / %), where % and % are phenomenologically introduced momentum-independent lifetimes of the individual electron and hole states./'is supposed to simulate not only lifetime effects but also disorder-induced broadening and other microscopic processes, such as phonon excitation, that merely redistribute the oscillator strength of the absorption bands. Its importance is that it merges the exciton and interband absorption bands that we are about to find into a single dispersing band peaked at the exciton energy. This effect is deserving of further investigation as it has a bearing on the important question of the photogeneration of charge at the frequency of the exciton. According to Eqs. (1) and (8), o'=(Q,o) describes absorption and polarization due to the excitations derived from the B* and D t transitions (Fig. 1 ( b ) ) . The corresponding propagators are, respectively: X~I= Xs = N -
IE(2ot
0-
(12)
and X~=XD = (2c~o - W) -1. Here W o = W cos(Qb/2) and the bandwidth W = 4r Furthermore: U~,Q= ( U1 + U3)/2 + V cos(Qb) V~,Q= (U1 - U3)/2 + V c o s ( Q b ) Now, we note that if Vx.Q were zero, the B* and D t derived excitations would be decoupled. It follows from Eqs. ( 10)(12) that the B* derived excitations would then consist of a continuum of interband transitions with energies 2c¢o - WQ < o~~<2oro + WQ, with an excitonic bound state ( 'B t exciton' ) below this continuum with energy:
hwQA = 2c~o -- ( U~,Q+ W~) t/z
hOgQ,2 = 2C~o- Ux,Q
(14)
and one-half of the oscillator strength of the original El,(x) excitation. In general, however, the two types of excitations are coupled to each other by Vx,Q4=0 and lead generically to a lower energy and higher energy absorption band in Re cr~(Q,~o). This coupling is reflected in the function D~, defined in Eq. (11), the zeros of which as a function of ~o determine the energies htoQ and lifetimes of the B t and D t excitons. When v, v~0, the total interband current contributed by the excitation ~* _ t~* ~* ~'k,Q,~r--~'k+Q/2,oxt-k+Q/2, ,r is changed from Vo/ 2 to Vk,Q/2 = Vo/2 + vl cos(kb) cos(Qb/2), so that oscillator strength is added to these excitations for kb < ~r/2 and subtracted for kb > 7r/2. The net effect is to increase the oscillator strength of the lower energy band relative to the higher energy one (see Fig. 3, where Vl =0.25Vo). It is straightforward to obtain the formula replacing Eq. (10) for this case. Since it is algebraically cumbersome it is presented in Appendix A. We will use it to obtain the calculated plots of cr~(Q,w) to be shown in Figs. 3 and 4. Cryy(Q,w) arises from the excitations derived from the monomer A* and C* elg--->e2u transitions (Eq. (2), Fig. 1 (b)). The formulae ( t 0 ) and (11) in which Uy,Q = ( U 2 -q- U 4 ) / 2 - V cos ( Q b ) / 2
V~,Q= ( U 2 - U 4 ) / 2 - V cos(Qb ) /2 are simplified because both propagators are equal: X[ = XA = X~ =Xc = N - 1 ~ ( 2 C ¢ o - 2t cos (kb) - oJ) -1 k = ( (20¢° - w)2 _ (W/2) 2) - 1/2
4t cos(kb) cos(Qb/2) - to) - 1
k
= ( ( 2 a o - w)2_ W~) -1/2
2). As Q increases, therefore, increasingly more oscillator strength would be transferred to the exciton. On the other hand, the D t derived excitations would consist of a single dispersing Frenkel-like exciton ( ' D t exciton') with energy:
(13)
The fraction of the total oscillator strength of the B* excitations assumed by this exciton would be fQ=Ux,Q/ (U2Q+ W~)1/2 and is close to unity for U~.Q> W cos(Qb/
so that F.~.=2XA/( 1 -- FJy,QXA), where Oy,Q= U 2 - V cos(Qb), and the resulting absorption is not split into higher and lower energy bands. This is a consequence of the assumption of charge conjugation symmetry. In these excitations either the electron is in a band state while the hole is in a localized monomer state, or vice versa. In the limit F--* 0, we see that the original E~u(y) excitation has spread into a continuum of interband excitations with energies 2c¢o - W~ 2-%<~o,%<2C~o+W/2, together with an exciton below this continuum with energy: A(..OQ,3= 2C¢o_ ( ~.~2,Q_4_( W / 2 ) z ) 1 / 2
(15)
This 'E~u(y) exciton' corresponds to a bound state between an extended electron and a localized hole or vice versa. Its energy and dispersion are quite different from that of the B* exciton occurring in O-x~(Q,w). The fraction of the total oscillator strength taken by this exciton is fQ=(Jy.Q/ ((/~,Q + (W/2)2)1/2. Its energy lies in between those of the two Bt and D* excitons and we refer to it as the 'intermediate
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183--190
energy' exciton. Also, its energy is increased by V while the energy of the B t exciton is decreased by V. We note that the threshold of the E~u(y)-derived interband excitations is 2 a o - (W/2), while it is 2 a o - W for the B* interband excitations. Thus, there is a bandgap anisotropy of precisely (W/ 2) on going from the x- to the y-direction. Interestingly, we also note that it follows from the function Dy(Q,w) defined in Eq. (11) (or, alternatively, from the retarded propagator iO(t) ([ (e],4(t) --Pj,4(t) ), ( P ] 4 ( O ) Pj,4(O))]}) that there is a fourth, dipole-forbidden, singlet state 'exciton' with energy:
ho~4= 2ao - ( U24+ ( W / 2 ) 2) ~/2
Coupled Elu (x) and Blu r
1
20(O . . . . . . . . .
~ -
0)2
,'x\\\\\\
D+ .......
3 We note that for the atomically bridged polyphenylenes the 'fourth' singlet exciton is rendered allowed as a natural consequence of the breaking of charge conjugation symmetry, i.e. XA =/=XC-
0)3 .....
b
(16)
This resides below a continuum of dipole- forbidden interband excitations lying in the same range of energies as the allowed E~,(y)-derived interband excitations. These excitations, of course, are simply those derived from the B2, monomer excitation. Note the absence of dispersion in Eq. (16) for hw4, which reflects the absence of a transition dipole moment for this exciton under local D6h symmetry. Distortion of the local D6h symmetry, either by the zero-point motion of phonons of appropriate symmetry, or by static perturbation of the phenylene electron states, can render this exciton allowed, however. The former occurs in benzene, while the latter has been observed by the Fink group to occur in both PPS and PPO 3 [2]. Generally, it follows from Eq. (10) for Fy(Q,w) that the fourth exciton is rendered allowed whenever charge conjugation symmetry is broken, i.e. when XA q=Xc. The four singlet state excitation bands just described are schematically illustrated in Fig. 2 for the ideal c a s e / ' = 0. The typical ordering of the four exciton levels is oJQ,~< ~o4 < ~oQ.3 < oJQ.2. Note that the symmetries shown in this figure indicate the symmetries of the benzene excitations from which these polymer excitations are derived. The polymer excitations themselves in nonplanar PPP belong to the D2 point group and have B 3 and B2 symmetry, respectively, for polarizations along and perpendicular to the polymer axis. Our microscopic model thus leads to the generic appearance of three strong, dispersing, absorption bands in the spectra of phenylene-based polymers. The corresponding excitonic states, in order of increasing energy, are formed from two extended states, one localized state and one extended state, and two localized states, respectively. The polarization of the intermediate energy exciton is different from that of the other two. When the polymer axis makes a finite angle 0 with the phenylene x-axis, as in PPV, all three bands will appear in o-(Q,w) for a given polarization of the electric field vector, with weightings, calculable from ,,~, depending on the angle O. In the inset of Fig. 3, an experimental absorption spectrum [ 17] of PPV is shown, the threepeak structure of which we interpret as being due to these absorption bands. The weak higher-energy shoulder of the
187
0)4
031
Fig. 2. The four singlet excitation energy bands of the model phenylene polymer. The first two bands are derived from coupled E~.(x) and B~, excitations of benzene while the third and fourth bands are derived from E1,(y) and B2. excitations, respectively, wt, ~ , w3 and o)4 denote the four exciton levels, while 2ao is the ionization energy minus the electron affinity for the polymer in the limit of zero bandwidth, W= 0.
1.5
o
=E 1.0
2 4 6 Photon Erler~y (eV)
o
rr
0.5
0.0 2
4 Photon Energy (eV)
6
Fig. 3. Re o'~(0,~o) (full curve) and Re O'ry(0,o~) (broken curve) vs. photon energy hoJ calculated for the polyphenylene model employing the parameters given in the text. The inset shows the absorbance spectrum of 'standard' PPV (Ref. [21]).
lowest energy absorption band in the region 3.7 eV that is observed in the data of Halliday et al. [ 17] and which in dimethyloxy PPV becomes a separate but weak absorption band [ 17], we interpret as the B2,-derived exciton hoJ4. In the present discussion, however, we wish to concentrate on the three strongly allowed absorption bands. The relative spacings in energy of these three bands are largely set by Ut, U3 and W, but are corrected by Vdo (and also by Vch when Veh=~0). It is important to notice that in the absence of any Coulomb interaction ( U = V= 0), the intermediate energy peak (actually, a band absorption edge in this case) would fall precisely in the middle of the lower and higher energy peaks (see Eqs. ( 13)-(15) ). The shift of the intermediate peak from such a position indicates the significance of the Coulomb interaction and, in particular, of the U.. In Fig. 3 we have calculated o-~(O, oJ) and Cryy(O,w)
188
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190 2.0
1.5 .5 1.0 Ce b
o.___~3 05
0.0 0
2
4
6
8
10
Energy (eV) Fig. 4. Re tr~(Q,to) vs. energy hto for given values of the wavevector Q (in units of ~r/b) calculated for the polyphenylene model employing the parameters given in the text. For each Q the curves are displaced vertically by an arbitrary amount.
versus hto for the PPP-type model, employing the parameter values 2 % = 7 eV, U , = U = 0 . 8 eV (all n) W=4.2 eV, V=0.1 eV, v~=0.25Uo and / ' = 0 . 3 eV. The assumption U, = U (all n) is introduced here for simplicity. These trial values of the parameters yield absorption peaks at the three exciton energies of about 2.7, 4.8 and 6.1 eV, close to the three peaks of the experimental data in the inset of Fig. 3; other available data [2, 17] can be fitted with reasonable modification of the parameters. A finite value of Vaj is found to influence efficiently the relative position of the intermediate peak since it 'moves' this peak in the direction opposite to that of the lower and higher energy bands. Note that Vchv~0 would tend to compensate for such an effect for it does not affect the higher energy peak (Frenkel-like exciton), but moves the two other peaks downwards in energy. In independent calculations [ 18], performed on finite chains, we have included the long-range Coulomb interaction V~hexactly and found that the corresponding two exciton energies can be lowered by an amount of about 0.1-0.2 eV. In the same calculations we have also found that the different values of El, play an important role in determining the relative intensities of the absorption bands. Of course, it is impossible to extract unambiguously all the values of the microscopic parameters from only three experimental peak positions. More could be done by analysis of EELS data. Fig. 4 presents calculations of trxx(Q,to) versus h~o for increasing values of Qb/zr. For clarity the dispersing 'E~u(y) band' (having a 'negative' dispersion) is not shown. The calculated spectra are in good qualitative agreement with the EELS data of the Fink group [2,8] for PPP and PPV. The lack of good crystallinity of the materials studied in these experiments precludes a detailed fit of the present theory. EELS measurements on films of improved crystallinity would be able to test the detailed validity of our microscopic model. We note that in the current debate [ 7 ] on the nature of the first absorption band of PPV, the significance of this polymer's two other strong absorption bands in the UV has not
been recognized. From the spacing of all three bands [ 17] we are able to assess the relative importance of excitonic and interband excitation in the lowest energy band: the B* exciton assumes ~ ( U + V ) / W ~ 20% of the total oscillator strength of the delocalized electron-hole pair excitations and has a binding energy of about ( U + V)2/2W~ 0.1 eV. This places this exciton in the weak-coupling regime (W>> U). Finally, we note that it is straightforward to extend our model to calculate the energies of the triplet state excitons. To Eq. (5) for Z~e we add the following term describing the energy gain U,t when an electron and a hole, in a triplet state, are simultaneously present on the same monomer in the symmetry combination n:
-EU.,~.,.T~.m
(17)
jnm
The operators ~,m are the triplet state counterparts of the four singlet excitation operators P], defined in equations following and including Eq. ( 1 ). m = 0, + 1 denote the magnetic quantum numbers. The local monomer triplet state excitation energies are then E , t = 2 % - U , t . For benzene, Salem [9] tabulates E1,=E2t=4.5 eV (3E]u), E3,=3.6 e V (3Blu) and E4, = 4.7 eV (3B2u). If, once again, we assume the polarization energy Ep= 1.5 eV, we obtain U1, =U2,=3.8 eV, U3, = 4.7 eV and U4, = 3.6 eV. These values are significantly larger than the corresponding singlet state values El, since more work has to be performed to separate a local triplet exciton into a widely spaced electron and hole on account of the exchange interaction. If the long-range Coulomb interaction Vch is neglected, the triplet state T matrix may be calculated analytically from the summation of ladder diagrams. In analogy to the singlet case we find a Bt-like triplet exciton and a D*-like triplet exciton whose energies htolt(Q ) and hto3,(Q) are given by the solutions of ( 1 - U~XB) ( 1 - UtxXD ) -- V2~XBXo = 0
( 18 )
and an E~,(y)- and B2u-derived triplet exciton whose energies htO2t and htO4t are given by the solutions of ( 1 - U2tXA ) ( 1 -- U4,Xc) = 0
(19)
In Eqs. (18) and (19) XB, Xo, XA= Xc are the propagators already introduced for the singlet excitations, while
U,x= (u,,+ u3,)/2 V,x=(U~,-u3,)/2 It follows immediately from Eq. (19) that htozt = 2 % - ( U2,+ ( W / 2 ) 2 ) ~/z hto4,= 2 % - (U],+ ( W / 2 ) 2 ) l/z Both of these energies are independent of wavevector Q. Eq. (18) shows that the actual B* and D* triplet excitations are coupled via the matrix element V,~. In the absence of this coupling ( Ult = U3t) we have
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
189
O-x~(Q,w) = ( e=/2Abiw) [ F ( Q , w + iF)
hwtt(Q) = 2ao - ( U 2, + W~)'/2
+ F(-Q,-(~o+iF)
hw3, = 2 % - U1, Since, based on the benzene estimates, the U., are comparable to the electronic bandwidth W all four of the triplet excitons are strongly bound. Clearly, relaxation of the lattice [ 19] about such tightly bound excitons, which concentrate loss of bond order over only a very few monomers [ 20], is an important consideration for the triplet excitons. This consideration is also relevant to the tightly bound singlet excitons hw2 and h~o4. A detailed application of the model presented in this paper to discuss the absorption spectra of oligomers of the wide bandwidth polyphenylenes will be presented in a forthcoming paper [ 18]. The main purpose of the present paper has been to present the basic model and its salient results.
Acknowledgements
M.J.R. is indebted to G.A. Sawatsky for a revealing discussion. The authors are grateful to E.M. Conwell, P. Gomes da Costa, M. Stolka, H. Antoniadis, C.B. Duke and D.D.C. Bradley for stimulating discussions and to the Xerox Corporation for its support of this work. The courtesy of J. Fink, R.H. Friend and B.R. Hsieh in providing relevant experimental data is highly appreciated. Yu.N.G. was partially supported by a CAST grant of the National Research Council.
)]
The function F(Q,oJ) is
F :At--[- Dx-1 (f< U)(ab>2 + g: + h )
(AI)
where we have introduced the Q- and w-dependent quantities:
= N - l ~ V~.Q/( 2ao -- 2E~.Q -- ~o) k
= V~I (2aO -- OJ) f = Ux,Q( I - U=,Q) + V~Z,Q
(A2)
g = U.,Q( 1 - Ux,Q)
(A3)
Jr V2,Q<)(ab>
h = 2Vx.e Here we have defined Ek.e=2tcos(kb)cos(Qb/2) and vk.Q= Vo+2V~ cos(kb) cos(Qb/2). The functions Ux,Q and Vx,Qare defined in the text following Eq. (12). We also note that in Eqs. (A2) and (A3) the functions (Xab>=XB, = Xo are just the propagators for the B* and D* excitations introduced in the text. Finally, the function Dx(Q,oo) in Eq. (A1) is defined in Eq. (11) of the text. For Vl=0, the function F(Q,w) simplifies and leads to the result given for ~r=(Q,~o) in the text. Note from Eq. (A1) that when v~ S0, the zeros of Dx(Q,w) still give the exciton energies ho~Q.1 and hOJQ.2, as stated in the text.
References Appendix A
We present here the explicit formula for o'~(Q,oJ) when vl v~0. It is straightforward, but algebraically tedious, to derive this expression from Eq. (7) by summation of standard ladder diagrams [ 16]. These are illustrated in Fig. AI. We obtain
a a (~b (ii)
c c a c c a a~ a(~ab i a(~~d =
a
a
c
a
(iii)
=
+
ba
+
+
(iv) • = Ux,Q • = Vx,Q Fig. AI. (i) Feynmandiagramsfor thecalculationof ~r~(Q,oj). (ii)Ladder summation for the first term in (i). (iii) Ladder summation for the fourth term in (i). The second and third terms of (i) are given by ladder summations similar to (ii) and (iii). The vertices U:,,Qand Vx,~are defined in the text.
[ 1] For a review, see, e.g., R.L. Elsenbaumer and L.W. Shacklette, in T.A. Skotheim (ed.), Handbook of Conducting Polymers, Vol. 1, Marcel Dekker, New York, 1986, pp. 213-263. [2] G. Crecelius et al., Phys. Rev. B, 28 (1983) 1802. [3] J.L. Br6das et al., J. Chem. Phys., 76 (1982) 3673; J.L. Br6das et al., J. Chem. Phys., 77(1982) 371. [4] See, e.g., A.J. Heeger et al., Rev. Mod. Phys., 60 (1988) 781. [5] See, e.g., W.A. Harrison, Phys. Rev. B, 31 (1985) 2121, and the lucid discussion of R.W. Lof et al., Phys. Rev. Lett., 68 (1992) 3924 for crystalline C6o. [6] W.T. Simpson, J. Am. Chem. Soc., 73 (1951) 5363. [7] See, e.g., U. Rauscher et al., Phys. Rev. B, 42 (1990) 9830; M. Gailberger and H. B~sler, Phys. Rev. B, 44 (1991) 8643; K. Pichler et al., J. Phys.: Condens. Matter, 5 (1993) 715 5 ; C.H. Lee et al.,Phys. Rev. B, 49 (1994) 2396; J.M. Leng et al., Phys. Rev. Lett., 72 (1994) 156, and theoretical discussions of excitons in conducting polymers in S. Abe, J. Yu and W.P. Su, Phys. Rev. B, 45 (1992) 8264; H.X. Wang and S. Mukamel, J. Chem. Phys., 97 (1992) 8019; P. Gomes da Costa and E.M. Conwell, Phys. Rev. B, 48 (1993) 1993. [8] J. Fink, Advances in Electronics and Electron Physics, Vol. 75, Academic Press, New York, 1989, pp. 121-231; E. Pellegrin et al., Synth. Met., 41-43 (1991) 1353. [9] L. Salem, The Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading, MA, 1966, p. 392. [ 10] M. Goeppert-Mayer and A.L. Sklar, J. Chem. Phys., 6 (1938) 645. [11] See, e.g., C.B. Duke, in J.S. Miller (ed.), Extended Linear Chain Compounds, Vol. 2, Plenum, New York, 1982, pp. 59-125; C.B. Duke, Mol. Cryst. Liq. Cryst., 50 (1979) 63.
190
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
[ 12] R.W. Kiser, Introduction to Mass Spectroscopy and its Applications, Prentice-Hall, Englewood Cliffs, N J, 1965, p. 308; G. Briegleb, Angew. Chem. Int. Ed. EngL, 3 (1964) 617. [13] L. Salem, The Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading, MA, 1966, pp. 443-452. [ 14 ] I. Egri, J. Phys. C: Solid State Phys., 12 (197) 1843; see also T. Hibma et al., Chem. Phys. Lett., 23 (1973) 21. [ 15] C.B. Duke and W.K. Ford, Int. J. Quantum Chem., Symp., 17 (1983) 597.
[16] See, e.g., G.D. Mahan, Many Particle Physics, Plenum, New York, 1981, pp. 730-738. [17] N.F. Colaneri et al., Phys. Rev. B, 42 (1990) 11 670; R.H. Friend, personal communication; D. Halliday et al., Synth. Met., 55-57 (1993) 954; B.R. Hsieh, Polym. Bull., 25 ( 1991 ) 177. [ 18] Yu.N. Gartstein, M.J. Rice and E.M. Conwell, Phys. Rev. B, in press. [19] S.A. Brazovskii and N.N. Kirova, Pis'ma Zh. Eksp. Teor. Fiz., 33 (1981) 6. [20] M.J. Rice and S.R. Phillpot, Phys. Rev. Lett., 58 (1987) 937.
Synthetic Metals 73 (1995) 183-190
Excitons and interband excitations in polyphenylenes M.J. Rice 3, Yu.N. Gartstein a,b,l a Xerox Corporation, Wilson Center for Research and Technolgy, Webster, NY 14580, USA b Center for Photoinduced Charge Transfer, Chemistry Department, University of Rochester, Rochester, NY 14627, USA Received 26 January 1995; accepted 7 February 1995
Abstract
A microscopic model which elucidates the essential physics of the photoexcitations of the polyphenylenes is introduced. Recognizing that these excitations are derived from the local excitations of the phenylene monomer, it allows for the effects of the 'correlation-energy gaps' Un between charged and charge-neutral excitations, and the dipole-dipole interaction V~d between neighboring monomers. It leads to the generic appearance of three principal, dispersing, absorption bands whose relative spacings in energy are largely set by U,, and the electronic bandwidth W, thereby establishing direct experimental access to these microscopic parameters. There is a fourth, dipole-forbidden, singlet excitation band which can be rendered allowed by distortion of local D6h symmetry. A family of four distinct, tightly bound, triplet state excitons is also found. Keywords: Polyphenylenes; Excitons; Interband excitations; Model
1. Introduction
The undoped forms of the conducting polymers based on phenylene form a wide and interesting class of organic semiconductors [1]. They include poly(para-phenylene) (PPP), poly(phenylene vinylene) (PPV), and the atomically bridged polyphenylenes such as polyaniline (PAN) and poly(phenylene oxide) (PPO). They have relevant 7r-electron bandwidths W ranging from W< A to W<< A, respectively, where A is the mean spacing between the molecular orbital (MO) energy levels of the isolated phenylene monomer [ 2,3 ]. In this paper we introduce a simplified but analytically explicit microscopic model which we believe elucidates the essential physics of the excitations of these polymers, not only as a function of the frequency w but also of the wavevector Q of the exciting electric field. The model recognizes that the primary excitations are derived from the local (elg--* ezu) excitations of the phenylene monomer and, correspondingly, includes two effects which although are implicit in quantum chemical calculations are not ordinarily considered in the usual band models of the polymers [4]. These are the 'correlation-energy' gaps U, between charged and charge-neutral excitations [5], and the dipole--dipole interaction Vud between local excitations on neighboring Also at the Department of Thermal Physics, Uzbekistan Academy of Sciences, Kata~al 28, Tashkent 700135, Uzbekistan. 0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D 1 0 3 7 9 - 6 7 7 9 ( 9 5 ) 0 3 3 1 3-9
monomers [ 6]. U, is the energy required, in the limit W--* 0, to dissociate a local singlet state monomer excitation of symmetry species n into a well-separated electron and hole. This effect, which behaves as an effective intramonomer electronhole attraction, is crucial as it leads, for W ~ 0, to the transfer of oscillator strength from interband transitions to exciton excitation even before the long-range electron-hole Coulomb attraction Veh is allowed for. In fact, the symmetry of the local phenylene excitations is shown to lead to three dispersing excitons whose relative spacings in energy are largely set by U, and W, and corrected in interesting ways by Vdd and Veh, thereby establishing experimental access to the values of these parameters. Interestingly, there is also a fourth, dipoleforbidden, singlet exciton that can be rendered allowed by distortion of the local m o n o m e r D6/, symmetry. The significance of the inter-relation of these four excitons has not been recognized in the current debate [7 ] on the origin of the first absorption band in PPV. As in a molecular crystal, the effect of Vdj is to contribute to the stabilization and dispersion of the excitons and is clearly an important consideration in the limit of small W. While the complex conductivity o'(Q, ~o) that we calculate is in good qualitative agreement with that derived from the existing electron energy loss spectra (EELS) of the polymers [2,8], we call for new EELS studies on films of improved crystallinity in order to investigate the anisotropy of the conductivity and the exciton dispersion due to Vdd that we predict in the UV. Finally, we briefly consider
M.J. Rice, Yu.N.Gartstein/ SyntheticMetals 73 (1995) 183-190
184
X d = -- ~_,{tbb(b~+l,obm + h.c.) +taa(aJ+l.,fllj,,+h.c.)
j,r "...-
+
+ t~b( (b]+ ,.,~- b)_ ipl,,~)aj,- ~ + h.c.) }
describes the delocalization of the b and a orbitals due to the intermolecular hopping integrals t, while
-
(a)
(b) =
Fig. 1. (a) The orbital patterns of the e~gand e2, molecularorbitals and (b) the four componente~g--+e2~transitions of the phenylene monomer. the triplet state excitations of the model and find a family of four distinct triplet excitons. They are tightly bound on account of the largeness of their respective correlation-energy gaps U,, in relation to W. Relaxation of the lattice about such excitations is an important consideration. Our model, based on PPP as the prototypical polymer, consists of a chain of N(N--+ o0) interacting phenylene monomers. Each monomer is assumed to have two degenerate (elg) hole states, labeled a and c, with excitation energy Eh, and two degenerate (e2,) electron states, labeled b and d, with'excitation' energy E~. The orbital patterns of these states under an assumed D 6 h symmetry of the monomer are shown in Fig. 1 (a) [9]. If we introduce the fermion operators a]~., b]~, c}~ and ~ , which create, respectively, an electron or hole with spin polarization o- in the latter orbitals of the monomer j ( j = 1 ..... N), the four possible dipole-allowed elg-+e2u excitations of the monomer are described by the operators: B]~,=btoa]_~., D]~=~oZ:~_~, A]~.=~oa]~ and CJ~ = b]~.c~_,,. These unsymmetrized component excitations are illustrated in Fig. 1 (b). They may be combined to form the four (n = 1-4) following singlet excitations of the phenylene monomer [%10]: t PJay'(B]~,-D}o.)/2;
E,.(x)
(1)
*-~_,(A]o~+C}~,)/2; E..(y)
(2)
o"
P;,2-
o.
P]a=~,(B},~+D],~)/2;
Blu
o-
P tJ , 4D - Y ' ( A ~ - C J ~ ) / 2 ;
B2.
The first and second E~u excitations have transition dipole moments of magnitude P-o ( ~ aoe, where ao denotes the carbon-carbon bond length and e the electronic charge) in the monomer x and y directions, respectively, while the Blu and B2~ excitations are dipole-forbidden under D6h symmetry [ 9]. In terms of these operators the Hamiltonian ,g( defining the model is
,.~=
,~,{ Eh(a]oaj,~ + c~oc),.) + Ee(b~o-bi,~ + ~odj,D } jo"[- ~ d
where
"[- ,~'fee
(4)
(3)
-
E u.t'J.Pj° + vdo + Veh j,,
(5)
specifies the leading contributions arising from Coulomb interactions. The first term in Eq, (5) specifies the energy gain, Un, when an electron and a hole in a singlet state are simultaneously present on the same monomer in the symmetry combination n. Un actually corresponds to the energy required, in the limit t ~ 0, to dissociate the local excitations Pin into wellseparated electrons and holes [ 5 ]. Since in the hypothetical dissociation process the total number of intramonomer interelectron distances increases precisely by 1, Un can be qualitatively understood as the repulsive Coulomb interaction associated with this extra 'bond' 1. Since P]1 and P]2 are symmetry doublets, UI = U2. If we denote Ee + Eh by 2ao, it follows from the first terms of Eqs. (3) and (5) that the local monomer excitation energies are En = 2 a o - U,. We expect these to depend mildly on the electronic polarizability of the polymeric medium, while we expect both ~ and U~ to depend much more significantly on this quantity. For the benzene molecule, vapor phase molecular spectroscopy gives [ 12] E~ = E 2 --- 6.8 eV, E 3 = 6 eV and E 4 = 4.7 eV. For the polymer we may crudely estimate the U, from Un = 2c~o - E n with 2 a o ~ ( I - A - 2 E p ) , where L A and E, are the ionization potential, electron affinity and singlet excitation energies, respectively, of benzene and Ep the electronic polarization energy [ 11 ] of the singly charged monomer in the dielectric medium of the polymer. If, for PPP, we take Ep = 1.5 eV, we obtain U~ = Uz ~ 1 eV, [/3 ~ 1.8 eV and U4 ~ 3 eV. We note that for U1 v~/-/3 the first term of Eq. (5) leads to an intramonomer coupling of the B t and D t excitations, while for U2 ¢ U4 it leads to a coupling of the A* and C t excitations. We can expect that such couplings play a role in determining the form of the absorption spectrum of a given polyphenylene. The second term of Eq. (5)
Vdd= - EVom(eJ+,. +
l,o)(eL +P,°,)
jnm
describes the delocalization of the charge-neutral intramonomer excitations via the dipole-dipole interaction: If NOis the total number of electrons on the charge-neutral monomer there are No(No-1 )/2 interelectron distances on the monomer and this number remains the same if the monomeris excited. For the well-separated negatively and positively charged monomers the number is changed to N+ = (No- 1) (No - 2)/2 and N_ = (No+ 1)No/2, respectively. Thus, the net change in monomer interelectron distances following the hypothetical dissociation process is N÷ +N_-2No = 1. This energy cost is already included in E~ and Eh, but must be subtracted when an electron and hole reside on the same monomer.
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190 Vnm = R - 3 (/.tn "/X,, l - - 3 R - 2 ( / . t n - R ) ( / x m - R ) )
where R = Rj + ~- Rj is the relative coordinate of the jth and (j + 1)th site, and/x~ is the transition dipole moment of the local excitation n. For the actual case of PPP, the phenylene x-axis coincides with the polymer axis, so that V,m is diagonal with V~l = V > 0 for the E~u(x) excitation and V== - V / 2 for the E ~ ( y ) excitation (provided a planar configuration of the PPP chain is assumed). Following the discussion of Salem [13] we estimate V~0.25 eV if the known experimental value of the transition dipole moment of benzene is employed. As will be seen later, a nonvanishing V,,, will lead to dispersion of the absorption bands in the ultraviolet (UV) which could be measured via EELS experiments on highquality crystalline films. With only on-site and dipole-dipole interaction left in Eq. (5), our model resembles a simpler model introduced by Egri [ 14]. The third term in eq. (5) is the long-range Coulomb attraction Veh which is introduced as
185
In the present discussion of our model, the effects of electron-phonon interaction, lattice relaxation and disorder are not included at the microscopic level. As we shall see, the effect of disorder will be treated by the introduction of phenomenological lifetimes for the single electron and single hole states. We note that if H d = 0, ,U describes two dispersing Frenkei excitons, one derived mainly from the E~u(x ) monomer excitation and the other from the Elu(Y) excitation. We now illustrate the generic way in which this picture is changed for ~'~d 4=0 by calculating the complex conductivity o-~,(Q,~o) ( a = x or y) for the case of PPP. The case for PPV will be similar. Denoting by Q an arbitrary wavevector along the xdirection, the paramagnetic conductivity is given by the Kubo formula [ 16]: zo
~r,~,~(Q,w) = (iwANb) - ~ / d t ,/ 0
× exp(iwt) ([J~(t), J~_Q(O) ] )
Veh = V c ~ & p J 2 IJ - 11
(6)
j~l
where pj='Z~(a]oaj,~+c],~cj~-b]~bi,~-~oztj~ ) is the net charge density on the monomerj. In Eq. (6) no allowance is made for the different symmetries of the local electron and hole orbitals. The constant Vc is the interaction energy of charges on neighboring monomers and ECT = 2 a o - Vc may be considered to be the 'charge-transfer' exciton energy in the limit t--+ 0. An estimate of Vc is provided by Vc -- e2/eb, b being the monomer lattice spacing and • the dielectric constant. Reasonable estimates put Vc ~ 0.1 to 0.3 eV for PPP and PPV. For the atomically bridged polyphenylenes the hopping integrals in Z/d represent virtual hopping over the bridging atom. A more general model Hamiltonian for these polyphenylenes, which do not possess charge conjugation symmetry, would contain different site energies for the a, b, c and d excitations. For PPV, both the electron hopping in "g"d and the excitation hopping in ~ represent virtual hopping processes over the vinylene unit. Note, however, that in this case the mixing between benzene and vinylene states is rather strong. The calculation in Ref. [ 15] showed that the band structures of the relevant bands in planar PPP and PPV are very similar and the interband splitting in PPV is much smaller than the bandwidths. In the context of electron delocalization, therefore, the PPP-type band structure is expected to be a good approximation for the effective model of PPV. Of course, in a more detailed description the model can be extended to treat the vinylene states in PPV explicitly. We note that in Eq. (4) there is no intermonomer hopping between the c and d orbitals (cf. Fig. 1(a)) so that these hole and electron orbitals remain localized. For the explicit calculations in the present paper we will assume charge conjugation symmetry and take taa = tbb = tab = t, a constant, and Ee = E~ = ao.
(7)
where J~ = ~/exp(iQRj)Ji'~is the current fluctuation operator for the a-direction and A the transverse area corresponding to one chain. The interband current densities ~" are
~=ie{vo(P),l-Pj, I) +(Vl/2)~7~((b]+l,o-+b]_,,o-)af_o--h.c.)
}
(8)
(r
~' = ievo( P],2 - Pj,2)
(9)
where vo=tzOEl/eh arises from the transition dipole moments of the E~u excitations and vt arises from the 'oblique' interband hopping terms in ~?'d. For PPP taa = / b b = l a b ~ t, and for the planar polymer MO theory gives Vl = Vo/3 2. More generally, the ratio Vl/Vo may be regarded as a parameter of the model. For nonplanar PPP, the y-polarized absorption must be understood as the absorption for the polarizations perpendicular to the chain direction. We note from Eqs. (8) and (9) that intermonomer hopping breaks the local Etu symmetry of the displacement current, leading to anisotropic absorption. The algebraic evaluation of Eq. (7) is greatly simplified if two effects in ~ are ignored. The first effect concerns the terms in ,,g?'eedescribing the creation and destruction of pairs of excitations. The second effect concerns the oblique hopping terms in Z o after they have been retained in ~. The neglect of these two terms introduces errors of order (V/2ao) and (t/ao) 2, respectively. Since t < 1 eV, 2 a o ~ 7 eV and V< 0.25 eV typically, these corrections are small. Also, since the neglect of the pair-excitation terms corresponds to the neglect of local-field effects, it follows that the final state problem in the evaluation of Eq. (4) reduces to the problem of a single electron interacting with a single hole. The Kubo 2 In the MO theory/.to = eao, E~ = 2to, where to is the interatomic hopping integral between neighboring C(2pz ) orbitals and ao the C - C distance. For the planar PPP, t = to~3 and vl = 2aot=/3o/3.
186
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
formulae are then readily evaluated in terms of standard ladder diagrams only [16]. These are shown for o'x~(Q,~o) in Fig. A1 in Appendix A. The results can be expressed in a closed analytical form if the long-range Vch is set to be zero. (We will discuss the effect of finite V~h later in the paper.) For reasons of clarity we first present the results for v~ = 0. The latter does not affect the form of the excitation spectrum or the explicit form of rryy(Q,w). We obtain
~r~,~,(Q,w) = ( eZv~/2Abiw) [ F~(Q,w + iF) +F.(-Q,-
(w+iF))]
where F,, = (X~ + X~'- 2( U,,.Q -Vct, Q) X~lX.~)/D,
(10)
D~, = ( 1 - U~,QXl ) ( 1 - U,~,QX2) ,~ ~ - ( G , o ) 2x~x2
(11)
in which X~'= X'J(Q, w) are the propagators for the four component elg ~ e2~ monomer transitions and U~,Q and V,,,Q are matrix elements that we shall define shortly. The q u a n t i t y / ' = h ( 1 / % + 1 / %), where % and % are phenomenologically introduced momentum-independent lifetimes of the individual electron and hole states./'is supposed to simulate not only lifetime effects but also disorder-induced broadening and other microscopic processes, such as phonon excitation, that merely redistribute the oscillator strength of the absorption bands. Its importance is that it merges the exciton and interband absorption bands that we are about to find into a single dispersing band peaked at the exciton energy. This effect is deserving of further investigation as it has a bearing on the important question of the photogeneration of charge at the frequency of the exciton. According to Eqs. (1) and (8), o'=(Q,o) describes absorption and polarization due to the excitations derived from the B* and D t transitions (Fig. 1 ( b ) ) . The corresponding propagators are, respectively: X~I= Xs = N -
IE(2ot
0-
(12)
and X~=XD = (2c~o - W) -1. Here W o = W cos(Qb/2) and the bandwidth W = 4r Furthermore: U~,Q= ( U1 + U3)/2 + V cos(Qb) V~,Q= (U1 - U3)/2 + V c o s ( Q b ) Now, we note that if Vx.Q were zero, the B* and D t derived excitations would be decoupled. It follows from Eqs. ( 10)(12) that the B* derived excitations would then consist of a continuum of interband transitions with energies 2c¢o - WQ < o~~<2oro + WQ, with an excitonic bound state ( 'B t exciton' ) below this continuum with energy:
hwQA = 2c~o -- ( U~,Q+ W~) t/z
hOgQ,2 = 2C~o- Ux,Q
(14)
and one-half of the oscillator strength of the original El,(x) excitation. In general, however, the two types of excitations are coupled to each other by Vx,Q4=0 and lead generically to a lower energy and higher energy absorption band in Re cr~(Q,~o). This coupling is reflected in the function D~, defined in Eq. (11), the zeros of which as a function of ~o determine the energies htoQ and lifetimes of the B t and D t excitons. When v, v~0, the total interband current contributed by the excitation ~* _ t~* ~* ~'k,Q,~r--~'k+Q/2,oxt-k+Q/2, ,r is changed from Vo/ 2 to Vk,Q/2 = Vo/2 + vl cos(kb) cos(Qb/2), so that oscillator strength is added to these excitations for kb < ~r/2 and subtracted for kb > 7r/2. The net effect is to increase the oscillator strength of the lower energy band relative to the higher energy one (see Fig. 3, where Vl =0.25Vo). It is straightforward to obtain the formula replacing Eq. (10) for this case. Since it is algebraically cumbersome it is presented in Appendix A. We will use it to obtain the calculated plots of cr~(Q,w) to be shown in Figs. 3 and 4. Cryy(Q,w) arises from the excitations derived from the monomer A* and C* elg--->e2u transitions (Eq. (2), Fig. 1 (b)). The formulae ( t 0 ) and (11) in which Uy,Q = ( U 2 -q- U 4 ) / 2 - V cos ( Q b ) / 2
V~,Q= ( U 2 - U 4 ) / 2 - V cos(Qb ) /2 are simplified because both propagators are equal: X[ = XA = X~ =Xc = N - 1 ~ ( 2 C ¢ o - 2t cos (kb) - oJ) -1 k = ( (20¢° - w)2 _ (W/2) 2) - 1/2
4t cos(kb) cos(Qb/2) - to) - 1
k
= ( ( 2 a o - w)2_ W~) -1/2
2). As Q increases, therefore, increasingly more oscillator strength would be transferred to the exciton. On the other hand, the D t derived excitations would consist of a single dispersing Frenkel-like exciton ( ' D t exciton') with energy:
(13)
The fraction of the total oscillator strength of the B* excitations assumed by this exciton would be fQ=Ux,Q/ (U2Q+ W~)1/2 and is close to unity for U~.Q> W cos(Qb/
so that F.~.=2XA/( 1 -- FJy,QXA), where Oy,Q= U 2 - V cos(Qb), and the resulting absorption is not split into higher and lower energy bands. This is a consequence of the assumption of charge conjugation symmetry. In these excitations either the electron is in a band state while the hole is in a localized monomer state, or vice versa. In the limit F--* 0, we see that the original E~u(y) excitation has spread into a continuum of interband excitations with energies 2c¢o - W~ 2-%<~o,%<2C~o+W/2, together with an exciton below this continuum with energy: A(..OQ,3= 2C¢o_ ( ~.~2,Q_4_( W / 2 ) z ) 1 / 2
(15)
This 'E~u(y) exciton' corresponds to a bound state between an extended electron and a localized hole or vice versa. Its energy and dispersion are quite different from that of the B* exciton occurring in O-x~(Q,w). The fraction of the total oscillator strength taken by this exciton is fQ=(Jy.Q/ ((/~,Q + (W/2)2)1/2. Its energy lies in between those of the two Bt and D* excitons and we refer to it as the 'intermediate
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183--190
energy' exciton. Also, its energy is increased by V while the energy of the B t exciton is decreased by V. We note that the threshold of the E~u(y)-derived interband excitations is 2 a o - (W/2), while it is 2 a o - W for the B* interband excitations. Thus, there is a bandgap anisotropy of precisely (W/ 2) on going from the x- to the y-direction. Interestingly, we also note that it follows from the function Dy(Q,w) defined in Eq. (11) (or, alternatively, from the retarded propagator iO(t) ([ (e],4(t) --Pj,4(t) ), ( P ] 4 ( O ) Pj,4(O))]}) that there is a fourth, dipole-forbidden, singlet state 'exciton' with energy:
ho~4= 2ao - ( U24+ ( W / 2 ) 2) ~/2
Coupled Elu (x) and Blu r
1
20(O . . . . . . . . .
~ -
0)2
,'x\\\\\\
D+ .......
3 We note that for the atomically bridged polyphenylenes the 'fourth' singlet exciton is rendered allowed as a natural consequence of the breaking of charge conjugation symmetry, i.e. XA =/=XC-
0)3 .....
b
(16)
This resides below a continuum of dipole- forbidden interband excitations lying in the same range of energies as the allowed E~,(y)-derived interband excitations. These excitations, of course, are simply those derived from the B2, monomer excitation. Note the absence of dispersion in Eq. (16) for hw4, which reflects the absence of a transition dipole moment for this exciton under local D6h symmetry. Distortion of the local D6h symmetry, either by the zero-point motion of phonons of appropriate symmetry, or by static perturbation of the phenylene electron states, can render this exciton allowed, however. The former occurs in benzene, while the latter has been observed by the Fink group to occur in both PPS and PPO 3 [2]. Generally, it follows from Eq. (10) for Fy(Q,w) that the fourth exciton is rendered allowed whenever charge conjugation symmetry is broken, i.e. when XA q=Xc. The four singlet state excitation bands just described are schematically illustrated in Fig. 2 for the ideal c a s e / ' = 0. The typical ordering of the four exciton levels is oJQ,~< ~o4 < ~oQ.3 < oJQ.2. Note that the symmetries shown in this figure indicate the symmetries of the benzene excitations from which these polymer excitations are derived. The polymer excitations themselves in nonplanar PPP belong to the D2 point group and have B 3 and B2 symmetry, respectively, for polarizations along and perpendicular to the polymer axis. Our microscopic model thus leads to the generic appearance of three strong, dispersing, absorption bands in the spectra of phenylene-based polymers. The corresponding excitonic states, in order of increasing energy, are formed from two extended states, one localized state and one extended state, and two localized states, respectively. The polarization of the intermediate energy exciton is different from that of the other two. When the polymer axis makes a finite angle 0 with the phenylene x-axis, as in PPV, all three bands will appear in o-(Q,w) for a given polarization of the electric field vector, with weightings, calculable from ,,~, depending on the angle O. In the inset of Fig. 3, an experimental absorption spectrum [ 17] of PPV is shown, the threepeak structure of which we interpret as being due to these absorption bands. The weak higher-energy shoulder of the
187
0)4
031
Fig. 2. The four singlet excitation energy bands of the model phenylene polymer. The first two bands are derived from coupled E~.(x) and B~, excitations of benzene while the third and fourth bands are derived from E1,(y) and B2. excitations, respectively, wt, ~ , w3 and o)4 denote the four exciton levels, while 2ao is the ionization energy minus the electron affinity for the polymer in the limit of zero bandwidth, W= 0.
1.5
o
=E 1.0
2 4 6 Photon Erler~y (eV)
o
rr
0.5
0.0 2
4 Photon Energy (eV)
6
Fig. 3. Re o'~(0,~o) (full curve) and Re O'ry(0,o~) (broken curve) vs. photon energy hoJ calculated for the polyphenylene model employing the parameters given in the text. The inset shows the absorbance spectrum of 'standard' PPV (Ref. [21]).
lowest energy absorption band in the region 3.7 eV that is observed in the data of Halliday et al. [ 17] and which in dimethyloxy PPV becomes a separate but weak absorption band [ 17], we interpret as the B2,-derived exciton hoJ4. In the present discussion, however, we wish to concentrate on the three strongly allowed absorption bands. The relative spacings in energy of these three bands are largely set by Ut, U3 and W, but are corrected by Vdo (and also by Vch when Veh=~0). It is important to notice that in the absence of any Coulomb interaction ( U = V= 0), the intermediate energy peak (actually, a band absorption edge in this case) would fall precisely in the middle of the lower and higher energy peaks (see Eqs. ( 13)-(15) ). The shift of the intermediate peak from such a position indicates the significance of the Coulomb interaction and, in particular, of the U.. In Fig. 3 we have calculated o-~(O, oJ) and Cryy(O,w)
188
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190 2.0
1.5 .5 1.0 Ce b
o.___~3 05
0.0 0
2
4
6
8
10
Energy (eV) Fig. 4. Re tr~(Q,to) vs. energy hto for given values of the wavevector Q (in units of ~r/b) calculated for the polyphenylene model employing the parameters given in the text. For each Q the curves are displaced vertically by an arbitrary amount.
versus hto for the PPP-type model, employing the parameter values 2 % = 7 eV, U , = U = 0 . 8 eV (all n) W=4.2 eV, V=0.1 eV, v~=0.25Uo and / ' = 0 . 3 eV. The assumption U, = U (all n) is introduced here for simplicity. These trial values of the parameters yield absorption peaks at the three exciton energies of about 2.7, 4.8 and 6.1 eV, close to the three peaks of the experimental data in the inset of Fig. 3; other available data [2, 17] can be fitted with reasonable modification of the parameters. A finite value of Vaj is found to influence efficiently the relative position of the intermediate peak since it 'moves' this peak in the direction opposite to that of the lower and higher energy bands. Note that Vchv~0 would tend to compensate for such an effect for it does not affect the higher energy peak (Frenkel-like exciton), but moves the two other peaks downwards in energy. In independent calculations [ 18], performed on finite chains, we have included the long-range Coulomb interaction V~hexactly and found that the corresponding two exciton energies can be lowered by an amount of about 0.1-0.2 eV. In the same calculations we have also found that the different values of El, play an important role in determining the relative intensities of the absorption bands. Of course, it is impossible to extract unambiguously all the values of the microscopic parameters from only three experimental peak positions. More could be done by analysis of EELS data. Fig. 4 presents calculations of trxx(Q,to) versus h~o for increasing values of Qb/zr. For clarity the dispersing 'E~u(y) band' (having a 'negative' dispersion) is not shown. The calculated spectra are in good qualitative agreement with the EELS data of the Fink group [2,8] for PPP and PPV. The lack of good crystallinity of the materials studied in these experiments precludes a detailed fit of the present theory. EELS measurements on films of improved crystallinity would be able to test the detailed validity of our microscopic model. We note that in the current debate [ 7 ] on the nature of the first absorption band of PPV, the significance of this polymer's two other strong absorption bands in the UV has not
been recognized. From the spacing of all three bands [ 17] we are able to assess the relative importance of excitonic and interband excitation in the lowest energy band: the B* exciton assumes ~ ( U + V ) / W ~ 20% of the total oscillator strength of the delocalized electron-hole pair excitations and has a binding energy of about ( U + V)2/2W~ 0.1 eV. This places this exciton in the weak-coupling regime (W>> U). Finally, we note that it is straightforward to extend our model to calculate the energies of the triplet state excitons. To Eq. (5) for Z~e we add the following term describing the energy gain U,t when an electron and a hole, in a triplet state, are simultaneously present on the same monomer in the symmetry combination n:
-EU.,~.,.T~.m
(17)
jnm
The operators ~,m are the triplet state counterparts of the four singlet excitation operators P], defined in equations following and including Eq. ( 1 ). m = 0, + 1 denote the magnetic quantum numbers. The local monomer triplet state excitation energies are then E , t = 2 % - U , t . For benzene, Salem [9] tabulates E1,=E2t=4.5 eV (3E]u), E3,=3.6 e V (3Blu) and E4, = 4.7 eV (3B2u). If, once again, we assume the polarization energy Ep= 1.5 eV, we obtain U1, =U2,=3.8 eV, U3, = 4.7 eV and U4, = 3.6 eV. These values are significantly larger than the corresponding singlet state values El, since more work has to be performed to separate a local triplet exciton into a widely spaced electron and hole on account of the exchange interaction. If the long-range Coulomb interaction Vch is neglected, the triplet state T matrix may be calculated analytically from the summation of ladder diagrams. In analogy to the singlet case we find a Bt-like triplet exciton and a D*-like triplet exciton whose energies htolt(Q ) and hto3,(Q) are given by the solutions of ( 1 - U~XB) ( 1 - UtxXD ) -- V2~XBXo = 0
( 18 )
and an E~,(y)- and B2u-derived triplet exciton whose energies htO2t and htO4t are given by the solutions of ( 1 - U2tXA ) ( 1 -- U4,Xc) = 0
(19)
In Eqs. (18) and (19) XB, Xo, XA= Xc are the propagators already introduced for the singlet excitations, while
U,x= (u,,+ u3,)/2 V,x=(U~,-u3,)/2 It follows immediately from Eq. (19) that htozt = 2 % - ( U2,+ ( W / 2 ) 2 ) ~/z hto4,= 2 % - (U],+ ( W / 2 ) 2 ) l/z Both of these energies are independent of wavevector Q. Eq. (18) shows that the actual B* and D* triplet excitations are coupled via the matrix element V,~. In the absence of this coupling ( Ult = U3t) we have
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
189
O-x~(Q,w) = ( e=/2Abiw) [ F ( Q , w + iF)
hwtt(Q) = 2ao - ( U 2, + W~)'/2
+ F(-Q,-(~o+iF)
hw3, = 2 % - U1, Since, based on the benzene estimates, the U., are comparable to the electronic bandwidth W all four of the triplet excitons are strongly bound. Clearly, relaxation of the lattice [ 19] about such tightly bound excitons, which concentrate loss of bond order over only a very few monomers [ 20], is an important consideration for the triplet excitons. This consideration is also relevant to the tightly bound singlet excitons hw2 and h~o4. A detailed application of the model presented in this paper to discuss the absorption spectra of oligomers of the wide bandwidth polyphenylenes will be presented in a forthcoming paper [ 18]. The main purpose of the present paper has been to present the basic model and its salient results.
Acknowledgements
M.J.R. is indebted to G.A. Sawatsky for a revealing discussion. The authors are grateful to E.M. Conwell, P. Gomes da Costa, M. Stolka, H. Antoniadis, C.B. Duke and D.D.C. Bradley for stimulating discussions and to the Xerox Corporation for its support of this work. The courtesy of J. Fink, R.H. Friend and B.R. Hsieh in providing relevant experimental data is highly appreciated. Yu.N.G. was partially supported by a CAST grant of the National Research Council.
)]
The function F(Q,oJ) is
F :
(AI)
where we have introduced the Q- and w-dependent quantities:
(A2)
g = U.,Q( 1 - Ux,Q
(A3)
Jr V2,Q<)(ab>
h = 2Vx.e Here we have defined Ek.e=2tcos(kb)cos(Qb/2) and vk.Q= Vo+2V~ cos(kb) cos(Qb/2). The functions Ux,Q and Vx,Qare defined in the text following Eq. (12). We also note that in Eqs. (A2) and (A3) the functions (Xab>=XB,
References Appendix A
We present here the explicit formula for o'~(Q,oJ) when vl v~0. It is straightforward, but algebraically tedious, to derive this expression from Eq. (7) by summation of standard ladder diagrams [ 16]. These are illustrated in Fig. AI. We obtain
a a (~b (ii)
c c a c c a a~ a(~ab i a(~~d =
a
a
c
a
(iii)
=
+
ba
+
+
(iv) • = Ux,Q • = Vx,Q Fig. AI. (i) Feynmandiagramsfor thecalculationof ~r~(Q,oj). (ii)Ladder summation for the first term in (i). (iii) Ladder summation for the fourth term in (i). The second and third terms of (i) are given by ladder summations similar to (ii) and (iii). The vertices U:,,Qand Vx,~are defined in the text.
[ 1] For a review, see, e.g., R.L. Elsenbaumer and L.W. Shacklette, in T.A. Skotheim (ed.), Handbook of Conducting Polymers, Vol. 1, Marcel Dekker, New York, 1986, pp. 213-263. [2] G. Crecelius et al., Phys. Rev. B, 28 (1983) 1802. [3] J.L. Br6das et al., J. Chem. Phys., 76 (1982) 3673; J.L. Br6das et al., J. Chem. Phys., 77(1982) 371. [4] See, e.g., A.J. Heeger et al., Rev. Mod. Phys., 60 (1988) 781. [5] See, e.g., W.A. Harrison, Phys. Rev. B, 31 (1985) 2121, and the lucid discussion of R.W. Lof et al., Phys. Rev. Lett., 68 (1992) 3924 for crystalline C6o. [6] W.T. Simpson, J. Am. Chem. Soc., 73 (1951) 5363. [7] See, e.g., U. Rauscher et al., Phys. Rev. B, 42 (1990) 9830; M. Gailberger and H. B~sler, Phys. Rev. B, 44 (1991) 8643; K. Pichler et al., J. Phys.: Condens. Matter, 5 (1993) 715 5 ; C.H. Lee et al.,Phys. Rev. B, 49 (1994) 2396; J.M. Leng et al., Phys. Rev. Lett., 72 (1994) 156, and theoretical discussions of excitons in conducting polymers in S. Abe, J. Yu and W.P. Su, Phys. Rev. B, 45 (1992) 8264; H.X. Wang and S. Mukamel, J. Chem. Phys., 97 (1992) 8019; P. Gomes da Costa and E.M. Conwell, Phys. Rev. B, 48 (1993) 1993. [8] J. Fink, Advances in Electronics and Electron Physics, Vol. 75, Academic Press, New York, 1989, pp. 121-231; E. Pellegrin et al., Synth. Met., 41-43 (1991) 1353. [9] L. Salem, The Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading, MA, 1966, p. 392. [ 10] M. Goeppert-Mayer and A.L. Sklar, J. Chem. Phys., 6 (1938) 645. [11] See, e.g., C.B. Duke, in J.S. Miller (ed.), Extended Linear Chain Compounds, Vol. 2, Plenum, New York, 1982, pp. 59-125; C.B. Duke, Mol. Cryst. Liq. Cryst., 50 (1979) 63.
190
M.J. Rice, Yu.N. Gartstein / Synthetic Metals 73 (1995) 183-190
[ 12] R.W. Kiser, Introduction to Mass Spectroscopy and its Applications, Prentice-Hall, Englewood Cliffs, N J, 1965, p. 308; G. Briegleb, Angew. Chem. Int. Ed. EngL, 3 (1964) 617. [13] L. Salem, The Molecular Orbital Theory of Conjugated Systems, Benjamin, Reading, MA, 1966, pp. 443-452. [ 14 ] I. Egri, J. Phys. C: Solid State Phys., 12 (197) 1843; see also T. Hibma et al., Chem. Phys. Lett., 23 (1973) 21. [ 15] C.B. Duke and W.K. Ford, Int. J. Quantum Chem., Symp., 17 (1983) 597.
[16] See, e.g., G.D. Mahan, Many Particle Physics, Plenum, New York, 1981, pp. 730-738. [17] N.F. Colaneri et al., Phys. Rev. B, 42 (1990) 11 670; R.H. Friend, personal communication; D. Halliday et al., Synth. Met., 55-57 (1993) 954; B.R. Hsieh, Polym. Bull., 25 ( 1991 ) 177. [ 18] Yu.N. Gartstein, M.J. Rice and E.M. Conwell, Phys. Rev. B, in press. [19] S.A. Brazovskii and N.N. Kirova, Pis'ma Zh. Eksp. Teor. Fiz., 33 (1981) 6. [20] M.J. Rice and S.R. Phillpot, Phys. Rev. Lett., 58 (1987) 937.