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Stark profiles of singlet excitons in conjugated polymers
Chemical Physics ELSEVIER
Chemical Physics 210 (1996) 249-257
Stark profiles of singlet excitons in conjugated polymers Z,G. Soos *, D. Mukhopadhyay, M.H. Hennessy Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
Received 18 December 1995
Abstract
Electroabsorption (EA) spectra associated with overlapping vibronics of even- and odd-parity excitons are related in the Condon approximation to the linear absorption of discrete localized states. The EA intensity and profile depend strongly on the linewidth, F, with F -2 scaling of features due to Stark shifts and F-1 scaling of induced absorption and bleaching. Static disorder in polymer films broadens the linear absorption and reduces EA intensity compared to polydiacetylene (PDA) crystals, whose resolved spectra indicate overlapping vibrational sidebands. Similar excitations and transition moments in films conspicuously alter EA profiles and allow estimates of symmetry-breaking perturbations. EA spectra of PDA crystals and films are consistent with Pariser-Parr-Pople theory of linear and nonlinear optical spectra. The high sensitivity of EA also provides direct tests of excited-state displacements at induced transitions of even-parity excitons.
I. Introduction
Electroabsorption (EA) or Stark spectroscopy measures changes in the linear absorption, /(to, F ) I(to), in a static electric field F. Its high sensitivity has been recognized and exploited in atoms, molecules, molecular solids, inorganic complexes, semiconductors, and conjugated polymers. Since F mixes even- and odd-parity states in centrosymmetric systems, EA spectra probe the even-parity states responsible for large nonlinear optical (NLO) responses. Quantitative analysis has been difficult in condensed phases with congested excitations. Assignments of excited states are further hindered in conjugated polymers by sample morphology: with the notable exception of polydiacetylene (PDA) sin-
* Corresponding author.
gle crystals, conjugated polymers are only available as films. The linear absorption l(to) of PDA crystals is dominated by a singlet exciton, 1B, some 0.5 eV below the band edge for photoconduction. Weiser [1] showed that EA spectra of 1B excitons at low temperature closely follow the derivative, l'(to). The l'(to) profile is particularly convincing in view of resolved sidebands for single, double, and triple bond vibrations o f the c e n t r o s y m m e t r i c ( C R = C R C ~ - C ) , backbone of PDA crystals with different nonconjugated substituents R ( = PTS, DCHD, and PFBS). Several groups [2-5] have noted that EA spectra of films resemble the second derivative, l"(to). Except for a low-energy feature, polyacetylene (PA) films have an 1"(o)) profile [3]. PDA films with R = 4BCMU have both l'(to) and l"(to) contributions [5], but the fit is less compelling at lower resolution when only vestiges of vibronic structure remain. Such profiles are expected for extended states
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above the band gap of semiconductors [6], as well as in charge-transfer molecular crystals [7]. Thus EA profiles are consistent with different physical pictures, as summarized in Refs. [3,1]. Stark shifts leading to i F ( t o ) spectra in the 1B region indicate mixing with an even-parity state above or below the singlet exciton, respectively. Since shifts depend on both the location and transition dipole of the A state, one or the other must be known independently. The location of A states is decisively given by two-photon spectra, as recently shown by Lawrence et al. [8] for PDA-PTS single crystals. We assigned [9] the features below 1B to a 2A exciton and its vibrational sidebands. Overlapping linear and two-photon features at 2.00 eV imply strong mixing by the field that cannot be treated perturbatively. In this paper, we analyze Stark spectra of overlapping 1B and 2A vibronics and explain why EA intensities of films are ~ 100 times smaller than crystals at similar F. Stark profiles are not restricted to l'(to) or l"(to), but may resemble either in special cases. There is much information in EA(to) lineshapes, especially when combined with other experimental or theoretical knowledge about -rr-w * excitations. We follow the joint analysis [10] of NLO spectra of PA, PDA, and other conjugated polymers in terms of exact Pariser-Parr-Pople excitations [11] for oligomers with up to 14 w-electrons; excitations and transition moments are related to 1B of finite polyenes, where 2A is indeed below lB. The vibronic analysis in Section 2 is general for discrete states, however, and extends previous mixing [12,13,1] of 1B with a hypothetical A state at the band edge~ Such generalization becomes mandatory now that the two-photon spectra are available. We emphasize the sensitivity of EA profiles to the vibronic structure of overlapping excitons, as shown directly by experimental frequencies, where changes of ~ 0.02 eV can be significant. The resolution of current theories is at best ~ 0.1 eV, as discussed [14] recently in a joint analysis of NLO and EA spectra of PDA crystals and films. In that work, hereafter I, we related w-electron theory to PDA spectra and identified the displacements and transition moments of 1B and 2A. The analysis of Stark profiles is general for discrete states of centrosymmetric systems. I(to) is
polarized along the PDA backbone in crystals and this approximation is used throughout. Vibronic structure of 1B and 2A excitons is treated in the Condon approximation for displaced harmonic oscillators with ground-state vibrational frequencies. We consider an isotropic distribution of backbones in films, retain transition dipoles and vibronic structure for excitons with shifted energies, and introduce a distribution g(o~) to model static disorder. We find that broader l(to) reduce the EA intensity and alter the profile without invoking symmetry-breaking perturbations such as internal electric fields or site energies.
2. Electroabsorption of discrete states in centrosymmetric systems
2.1. General expression and linewidth dependence We consider centrosymmetric systems with an even ground state ]G> and discrete even- and oddparity excitations Is> and Ir>. The linear absorption
/(to) is z(.,) = E r
rr ~--- Er 77" (Ojr-- 09)2-[- C 2'
(1)
where Ixr is the transition moment and I ( o ~ r - - to,Fr) is a normalized line centered at o~r. The particular choice in (1) is a Lorentzian with FWHM of 2F~, but any profile can be used. The perturbation - ix. F in a static electric field mixes even- and odd-parity states. The linear absorption becomes
l(to,F) = E i . t r ( F ) 2 1 ( % ( F ) -- w,Fr) r
" ~ E txs(F ) 2 l(tos(F) - to,I'~ ). x
(2)
We have induced absorption at even-parity states and Stark shifts of all transitions.
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
The formal expression (2) holds for arbitrary fields. In practice, - I ~ " F is small and strong mixing is restricted to nearby Ir),ls) pairs. Taylor expansion to order F 2 gives the EA spectrum I(to,F) -
251
pair, neglect mixing with the ground state, and introduce w, = w , + 2A, V = ( r l / x l s ) " F.
(5)
The new energies and mixing angle 2 + are
l(to),
w,( F),w,( F)
air(to) EA(w) = ~
/,/,2
ato
+[/.tr(F) 2-
at.Or
(3)
S
with lj(to) = l(toj - to,Fj). Stark shifts in centrosymmetric systems are quadratic in F,
ator=O~r-tor(F)
= F. (a,-o%)'F/2
+ OJs) ++6,
e = f ~ - 7 + V 2 ' tan24, =
]d~2]lr(¢0))
+ ~,(F)21,(to),
= ½(o~r
(4)
and depend on the polarizability tensors of Ir ) and IG). Shifts of ~ 10 p.eV are inferred [1] for PDA crystals at F = 20 k V / c m . The sharpest lines are I 0 - 2 eV in crystals and ~ 0 . 1 eV in films. Similar expressions hold for 8to~, but are not needed to order F z because iXs(F) is proportional to F. Induced absorptions in (3) thus appear at the twophoton lines tos, while bleaching occurs at to,, and all coefficients go a s F 2 in lowest order. Stark shifts scale as l~(to) and dominate when linewidths are small compared to I r ) , l s ) splittings. The general expression does not reduce to 1'(o~), however, unless all shifts are identical. The nature of the discrete localized states Ir ) and Is) is left open in (3). The strong EA(to) dependence on F is consequently a general feature. Induced absorptions or bleachings in (3) decrease as F - l , while the maxima of l'(to) decrease as F -z. A 10-fold increase of F reduces l'(to) features 100fold. This explains the decreased amplitude and different Stark profiles in crystals and films. The argument also holds for asymmetric exciton profiles l(to) in many films.
V/A.
(6)
The transition intensity is partitioned as cosZqb for I r ( F ) ) and sin2+ for I s ( F ) ) . We introduce x = to (% + tos)/2 to measure frequency from the mean, assume a common F, express sinZqb in terms of V and A, and retain terms of order V 2 to obtain a special case of (3), EA(to;A,F)
M(A+x,F) 2A ax V2 +--~-£[I(A--x,F) - I ( A + x , F ) ] . V 2
This exact result holds for any splitting, including degenerate excited states. The Stark shift is - V2/2 A for 21AI >> ~ F ; it is positive for A < 0 (when tos < t%), negative for A > 0. The other terms are bleaching at x = - A and induced absorption at x = A. The V 2 factor containing the F dependence is common to all three terms in (7). Reduced variables
A =
5r/2
."
'A -~0.05F /'X,
,/~ ~
~-2
'/~
"g/A=r
°
2.2. Exact results for two-state mixing We will use the general expression (3) for simulations of Stark spectra in Section 3. To study EA(to) profiles analytically, we consider a single I r ) , l s )
(7)
"-4
-'2
0 -
u 2
t
l
4
(% + ~,.)/z
Fig. 1. Electroabsorption profiles, Eq. (7), for mixing an even and an odd state with splitting 2A and width 2F. The inset shows the individual components of Eq. (7) for A = F: a Stark-shift derivative and bleaching at - A, induced absorption at A.
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Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
readily show EA(to;A,F)FE/V 2 to be a function of A / F . The physical situation of fixed splitting and uariable linewidth is more conveniently shown with fixed F and variable A, thus removing the F dependence. The representative EA(to;A/F) curves in Fig. 1 have the expected I'(o~) profile for A = 5 F / 2 , with well-separated states and weak mixing. The opposite limit of A < F gives entirely different profiles that go exactly as I"(o) at A = 0. The inset shows the individual contributions to (7) for A = F, with moderate mixing. For 8-functions and A = 0 , t%.s(F) reduce to lines at t% _+ IxF that appear as peaks in the EA spectrum on either side of a strong bleaching. The profile depends on I~F/F, with F >> IxF in these polymers. When the splitting is small compared to the linewidth, we Taylor expand EA(to;A,F). The second term in (7) has only odd derivatives of l ( x , F ) and the l ' ( x , F ) contribution exactly cancels the first term of the Stark shift. The IAI < F limit of (7) is consequently EA( o.);]A[ < F ) / V 2
= ½I"(x,F) + O [ a l ' ( x , r ) ] .
(8)
As shown in Fig. 1, the profile is a second derivative about the mean excitation energy for nearly degenerate states. The argument holds for any absorption profile, even for pairwise mixing of vibronics Ir) and Is) in electronic states with identical geometries.
2.3. Static disorder in films The general expression (3) for an isolated system is appropriate for crystals with both E and F along the backbone. The simplest approximation for films is an isotropic distribution of backbones. This gives a reduction of (cos40) = 1 / 5 when all transition moments are parallel to the backbone, at angle 0 with EIIF. The reduction is (cos20 sinZ0cos2~b)= 1/15 for E _t_F. The predicted 3 : 1 scaling of the EA(to) for parallel and orthogonal E, F has been observed in PDA-4BCMU films [5]. Such scaling implies transition dipoles along the chain, as also found in two-photon polysilane spectra [15]. The smaller PA ratio of Ref. [3] 2.3 : 1 or Ref. [5] 2.6 : 1 indicates deviations [16] of the transition moments from the backbone. The transition moment [31~GIB of trans-
polyenes is predicted [17] and found [18] to be 15° towards the double bonds from the backbone. We model inhomogeneous broadening with normalized Gaussians at to r,
g(o,r= (~)-lexp
-- ( ogr -- Nr)2/2o-2,
(9)
and assume for simplicity the same width 2or > 2F for all lines. The film average for the linear spectrum (1) and EA spectrum (3) are term by term convolutions with (9). For ~r >> F, we simplify by setting F = 0, so that the Lorentzians become 8 functions and (1) reduces to Gaussians about the mean mr. For the EA spectrum, we integrate the I'(co) term by parts to obtain an integral over g'(co). The F = 0 limit then regains (3) for Gaussian profiles. The absorption peaks to [r), l(t%,F r) in crystals and l ( ~ r , O ' ) in films, are related by
I( ~, ,or ) / I ( w r,/'r) = 3 ~ 7r~-/2 "
(10)
The reduction (cos20) = 1 / 3 is the assumed isotropic distribution of backbones. The ('rr/2) I/2 factor is a correction due to different line shapes. The ~ 20% blue shift of 1B 0 - 0 in PDA films rules out exact cancellation of transition moments. We are assuming equal integrated 1B intensity in crystals and films. The absorption coefficients a of PDA crystals are an order of magnitude larger than in films [5], while the 300 K lines are 2 - 3 times narrower [9]. This agrees with (10) within the accuracy of estimating the large a in thin samples. Theoretical support comes from calculated [14] 1B oscillator strengths per site in oligomers, which converge to the polymer value [1]. The same widths necessarily apply to EA spectra. For El[F, induced absorption and bleaching follow (10) with ( c o s 4 0 ) = 1 / 5 replacing the 1 / 3 factor. The maxima of l~(co) are at _+F/vr3 for Lorentzians, while those of g'(to) are at __+or. The E[JF ratio is
I'( "~r H- O",o- ) / I ' ( oar +__Fr/3'/2,C ) C2 - 5o.2 ~/32~/27e.
(11)
The square root is due to the assumed profiles in crystals and films, and is again a small correction.
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
To first approximation, random orientations determine the relative EA intensities of crystals and films. In addition to similar Ixr for equal linear absorption in crystals and films, the ratio (11) assumes equal transition dipoles of 1B with excited A states in crystals and films. The NLO fits in I in fact required 10-30% variations of excited-state moments.
1.0
-
-
T
253 "1
1~
I - - ~ - - 7 " - -
0.5
.<
0.0
-0.5
I X3
3. Simulated electroabsorption spectra Linear and two-photon spectra of PDA-PTS crystals at 300 K fix the location of the 1B vibronics Ir> and 2A vibronics Ss), respectively, as summarized in Table 1 of I. The singlet exciton 1B 0 - 0 is 2,00 eV. Its sidebands at 0.20 and 0.26 eV are C = C and C-=C stretches, respectively. These backbone vibrations also appear in resonance Raman spectra, whose fits gives the displacements b relative to the ground state [19]. The even-parity states PDA-PTS crystals [8] at 1.80 and 2.03 eV are taken as [9] 2A 0 - 0 and its C = C and C=-C vibronics, with displacement a. The ~ 0.05 eV widths are comparable to the linear spectrum. The strong 2.7 eV feature is assigned to nA 0-0. The scaled NLO spectra of I also require transition moments IXlBEA and I'hB,A in units of IXlBG"
Overlapping vibronics at 2.0 eV imply degenerate Ir > and Is> and preclude a perturbation expansion in V = - I x " F. Given the states and transition dipoles, however, /(to,F) is readily found in the Condon approximation. The mixing of 2A and 1B vibrational levels goes as (2A,sl/zllB,r> = tZ2glBFsr( b - a ) ,
(12)
with Franck-Condon overlaps Fsr. These factors are known analytically for displaced harmonic oscillators with the same frequency. Two displaced modes yield similar matrix elements between liB,r) and IG,0> or InA, s'>, with s ' = 0,1 . . . . for C = C and C=-C. To describe r = 0, 1, 2, and 3 levels in 1B, we need 2A vibronics up to s = 7 and nA vibronics to s' = 3 for the displacements in I. Higher levels have negligible Fsr, but can easily be included. We use 10 vibrational levels for 1B and nA, 36 for 2A. The 57 × 57 Hamiltonian with off-diagonal elements I~" F given by (12) is diagonalized. Once F 2 dependencies have been checked, we obtain the EA spectrum (3).
-1.0
÷
_ _ _ _ _ . . a ~ ~ _ _ 1.6 1.8 2.0 2.2 2.4
(eV)
2.6
2.8
Fig. 2. Solid line: scales electroabsorption spectrum, Eq. (3), of PDA crystals based on inputs from linear, Raman, and NLO spectra, as described in the text, and Lorentzian profiles with F = 0.01 eV. Dashed line: scaled derivative of the linear absorption, Eq. (1), for the same parameters.
The solid and dashed lines in Fig. 2 are EA(to) and /'(to), respectively, based on Table 1 of I. To compare with 10 K spectra, we took a smaller F = 0.01 eV and 2A 0 - 0 at 1.81 eV. In PDA-PTS, PDA-DCHD, and PDA-PFBS crystals below 10 K, 1B 0 - 0 shifts to lower energy (1.935, 1.856, and 1.965 eV, respectively), the lines sharpen to 2 F 0.03 eV, and resolved sidebands are listed in Table 1 of Ref. [1]; their lower intensity implies slightly smaller b and a. The (dashed) l'(to) curve in Fig. 2 is based on (1) with the 300 K displacements. Two 0-1 and three 0 - 2 sidebands are resolved and easily seen on increasing the scale above 2.1 eV. The d e 2 / d E curves in Figs. 9-11 of Ref. [1] show similar resolution for l'(to) derived from reflectance data on crystals. The (solid) EA spectrum for F = 0.01 eV coincides well with l'(to), just as the related Ae 2 curves in Figs. 9-11 of Ref. [1] did. We emphasize that the calculated EA(to) and l'(to) curves in Fig. 2 are related in the same way as the experimental curves for PDA crystals in Ref. [1]. Only such comparisons are pertinent in view of experimental inputs about modes, displacements, and frequencies. These inputs from independent spectra also account for EA. The unsymmetric EA profiles in Fig. 2 result from approximate cancellation of large induced absorption and bleaching at nearly degenerate Ir>,ls> pairs, as expected from (7) and Fig. 1. The 2.00 eV feature, for example, depends
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Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
sensitively on the position of 2A 0-0: the l"(to) maxima are doubled and halved, respectively, on shifting from 1.81 to 1.80 and 1.79 eV. The resemblance between calculated EA(to) and l'(to) in Fig. 2 is retained in each case, although the explicit decomposition in (3) does n o t indicate a dominant Stark shift; the calculated profiles have major contributions from bleaching and induced absorption. The location of the 2A 0 - 0 and nA 0 - 0 are indicated in Fig. 2. The induced absorption at 1.81 eV is 0.9 )< 10 - 3 , about 0.1% of the 1B 0 - 0 amplitude. The small relative intensity of 2A 0 - 0 is independent of F, since EA scales as F 2. Detection of induced absorption at 2A 0 - 0 is a matter of resolution, with current spectra probably sensitive to signals over 5% of the 1B 0 - 0 amplitude. The NLO fits in I or in 13-carotene [20] suggested opposite displacements b and a in 1B and 2A, leading to small F00(Ibl + lal) in (12). The induced absorption at 2A 0 - 0 is 30 times smaller for Ibl + lal than for r b l - lat for these displacements. Thus EA spectra at 2A 0 - 0 provide sensitive tests for these excited-state parameters. We did not consider vibronic contributions to EA spectra in I, but emphasized the collective effects of all w-'n" * excitations. The stronger mixing of degenerate states in Fig. 2 increases the 1B Stark shift by an order of magnitude. Direct inclusion of F through (12) is presently restricted to vibronics of nearby electronic states chosen from the full ,rr--rr * spectrum of oligomers. The strong two-photon absorption at 2.7 eV gives a small EA response in Fig. 2 because, as discussed in I, the induced absorption also depends on excitation energies. The Stark shift of nA rationalizes the l'(to) profile at 1B 0 - 0 in Fig. 2 for degenerate vibronics. The two-level model predicts an l"(to) profile for A = 0, but neglects all other states. Coupling to a high-energy state approximately restores a first derivative when both tot(F) and tos(F) in (6) shift below tot. Linear absorption indicates 2F = 0.03 eV below [1] 10 K and =0.05 eV at 300 K i n PDAcrystals [9], while film widths have [5,9] 2or = 0.2 eV. An order of magnitude decrease of c~ is consistent with (10). A ratio cr/F = 5 reduces the EA intensity (11) by ~ 1/125 in films, consistent with AoL changes [5] in PDA-4BCMU crystals and films at the same F. These estimates set the scale for reduced EA
2
,
L
t
0
........
m 0 -1
A
--2
t
2.0
I
2.4
B i
(eV)
i
2.8
I
3.2
Fig. 3. Top: same as Fig. 2, with broader F = 0.1 eV for films leading to reduced amplitude according to Eq. (11). Bottom: electroabsorption spectrum, Eq. (3), and scaled 1'(~o), Eq. (i), for inputs based on linear and NLO spectra of PDA films, as described in the text. The observed bleaching at A in Ref. [5] is roughly twice the bleaching at B, and the low-energy features are offset.
intensity and, as anticipated from the resolved vibronics in Fig. 2, for altered profiles. There are two other changes: 1B 0 - 0 of PDA-4BCMU at 300 K shifts [21,22] from 1.99 eV in crystals to 2.35 eV in films and, although barely resolved, is slightly less intense than the 0-1 feature at 2.55 eV; ~ 20% larger b were needed in I for films. NLO spectra fix the location [9] of nA at 3.22 eV, again ~ 35% above 1B, but allow some flexibility for 2A 0-0, I~m2A, and a. These parameters are fixed by twophoton spectra in PDA-PTS crystals. To highlight the linewidth dependence, the upper curves in Fig. 3 are the crystal spectra (Fig. 2) with larger F = 0.1 eV and a blue shift of 0.35 eV; they represent PTS rather than 4BCMU inputs. The scale decreases by 500, as expected from (11), and the solid line depicting EA(to) now follows the dashed line, /'(co), only around 1B 0-0. The bleaching features labeled A and B in Fig. 3 are split by the vibrational spacing ( ~ 0.2 eV). There is partial agreement with the experimental PDA-4BCMU curve in Fig. 1 of Ref. [5]; however, the relative magnitudes of A and B disagree with experiment. The lower EA(to) and /'(to) curves in Fig. 3 are based on Lorentzians with F = 0.01 eV and transition moments, displacements and excitations for PDA-4BCMU films taken from I, except that 2A 0 - 0 is 1.86 rather than 1.90 eV and a is 30%
Z.G. Soos et a l . / Chemical Physics 210 (1996) 249-257
smaller. Each curve is then convolved with a Gaussian whose a = 0.10 eV is fixed by l(to). The EA features are properly located, the scale is again 500fold smaller, and the relative bleaching at A and B is closer to experiment. The maximum of EA(to) no longer occurs at / ' ( t o ) = 0, as observed [5] and expected for l"(to). The region above 2.8 eV disagrees with experiment. Since the location of the band edge is not known in films, deviations at high energy from I'(to) or l"(to) profiles have not been modeled. This underscores both the limited nature of EA fits in films, where the relevant electronic states are not known independently, and the sensitivity of Stark profiles to overlapping vibronics.
4. Discussion
The strong linewidth dependence of the EA spectrum (3) is a general feature of discrete localized states of centrosymmetric systems. The resulting EA profiles depend sensitively on the inputs, linewidths, and static disorder. EA intensities provide simpler tests. The absorption coefficient changes [1] by 400600 cm -1 in PDA crystals at F = 2 0 k V / c m . Broader lines in films indicate a 100-fold smaller EA signal, consistent with (11) for static disorder. Larger F is typically used in films, but reported Act values can readily be compared since EA scales as F 2. Published values give Act = 2 - 5 cm -1 at F = 20 k V / c m for PA [3], a polysilane [4], and PDA4BCMU. The agreement is encouraging for a joint w-electron analysis of crystals and films. Partial ordering in stretched [3] or oriented [13] films is more difficult to assess; linewidths are hardly reduced while there is considerable orientation. The location of singlet excitons, the magnitude of Stark shifts (4), and transition dipoles are problems in electronic structure. The vibronic structure of G, 1B, and 2A depend on their potential surfaces and, as discussed in I, require all-electron models. The 1B exciton and its sidebands appear in EA spectra of crystals, as shown in Fig. 2, even for overlapping 2A vibronics: only nearby states mix strongly for small IxF. EA profiles rather than positions reflect strong mixing, as illustrated for two states in Fig. 1. Although l'(to) is favored for small F, the result is not general for overlapping states in (3). Stark profiles at
255
allowed and induced transitions of excitons contain detailed information about transition dipoles and Franck-Condon overlaps. Static disorder includes variations of chain lengths, transfer integrals t~, or backbone conformation. Except for chain length, such perturbations break inversion symmetry. In "rr-electron models with arbitrary spin-independent interactions, however, electronhole symmetry [11] precludes mixing of the ionic 1B state with either covalent G or 2A states as long as t n are restricted to neighbors and all site energies en (Hiickel ct) are the same. Finite chains, disordered t~, and arbitrary backbone conformations still yield quadratic Stark shifts. We have an inhomogeneously broadened 1B at F = 0 and rigorously vanishing absorption at 2A and other two-photon-allowed singlets. Ionic A states and covalent B states may now appear in linear and two-photon spectra, respectively. Since ionic A states are above the band gap, they are not important in the exciton region. Covalent B states fall below the allowed 1B 0 - 0 in polyenes [17], but it will be some time before such corrections to two-photon spectra can be identified. There are many other l(to) changes, however, including different excitation energies, transition dipoles, and vibronic structure. Symmetry-breaking perturbations Vd are conveniently given by site energies, Vd = E ~ p ( 1 - no),
(13)
P
where n e is the w-electron number operator and the distribution G(e) is symmetric about ( e ) = 0 . Choices leading to internal fields F 0 or dipoles are special cases. Since even-odd selection rules no longer hold, the F = 0 spectrum l(to,Vd) already has features at both tor(Vd) and to,(Va). Once electronhole symmetry is lifted, transitions to covalent B states and ionic A states become allowed. Modest Vd is indicated for conjugated polymers by similar film and crystal spectra. The detection of site disorder depends on quantitative analysis of l(to) and EA(to), whose high sensitivity is then advantageous. The general EA spectrum (3) now has linear as well as quadratic Stark shifts and linear as well as quadratic contributions to induced absorption and bleaching. The new contributions go as F 2 Vaz in lowest order. While the EA intensity rules out large Va,
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Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
symmetry-breaking contributions somewhat larger than Ix F may eventually be extracted from film spectra. Linear Stark shifts generate l~'(to) terms that scale as F2Vd2. The physical picture is that F > 0 now gives both red and blue shifts of 1B, thus broadening the film's absorption, while quadratic Stark shifts (4) are unidirectional. The general case has many additional terms in F 2 Vd2, including bleaching terms in l~(to) and induced absorptions l~(to). Even if Vd is approximated by an internal field F 0, we have an additional parameter for EA fits in films. The general appearance of I"(to) profiles suggests linear Stark shifts, but the sensitivity of EA spectra in Fig. 3 to the location of 1B and 2A vibronics complicates the identification of Vd contributions. Accurate l(to) or EA(to) fits in films are hampered by the limited range of the exciton region and unknown band edges. The intense two-photon absorption at 2.7 eV in PDA-PTS crystals [8] is above the 2.5 eV band edge, where no signal is resolved. The nA state used to model NLO spectra was recognized [10] to be above the PDA band edge. By contrast EA contributions from band-edge states are seen in PDA crystals, especially by electroreflectance [1], as well as in oriented PDA films [13] and polysilane films [4,23]. The Franz-Keldysh effect in PDA crystals [24] at low temperature clearly implies a continuum above a well-defined band edge. These states may become localized in films, but are dense rather than discrete and consequently require more than a single [12,13] A state at the band edge. Accurate treatment of dense states is far more difficult than discrete states in the exciton region. Except for Va contributions, however, the same theoretical model should account for EA signatures of excitons in crystals and films. We have deliberately retained excitation energies, transition moments, and vibrational inputs from the linear, Raman, and NLO fits discussed in I. Similar linewidth and vibronic effects follow for other parameters. The high sensitivity of EA may precisely determine some values, such as the Franck-Condon factor for induced absorption at 2A 0 - 0 in PDA crystals. We have shown vibronic sidebands play a major role in EA spectra and have accounted for the lower intensity of films using exciton states derived from NLO spectra. EA spectra in the singlet exciton
region are another illustration that conjugated polymers can be modeled by molecular Pariser-Parr-Pople theory.
Acknowledgements We gratefully acknowledge support of this work by the National Science Foundation through DMR9300163.
References [1] G. Weiser, Phys. Rev. B 45 (1992) 14076. [2] J. Orenstein, G.L. Baker and Z. Vardeny, J. Phys. (Paris) Colloq. 3 (1983) C3-407. [3] S.D. Phillips, R. Worland, G. Yu, T.W. Hagler, R. Freedman, Y. Can, V. Yoon, J. Chiang, W.C. Walker and A.J. Heeger, Phys. Rev. B 40 (1989) 9751. [4] S. Jeglinski and Z.V. Vardeny, Synth. Met. 49/50 (1992) 509. [5] A. Horvath, G. Weiser, G.L. Baker and S. Etemad, Phys. Rev. B 51 (1995) 2751. [6] D.E. Aspnes and J.E. Rowe, Phys, Rev. B 5 (1972) 4022; D. Aspnes, in: Handbook on semiconductors, Vol. i, ed. T.S. Moss (North-Holland, Amsterdam, 1980). [7] L. Sebastian, G. Weiser and H. B~sler, Chem. Phys. 61 (1981) 125. [8] B. Lawrence, W.E. Torruellas, M. Cha, M.L. Sundheimer, G.I. Stegeman, J. Meth, S. Etemad and G.L. Baker, Phys. Rev. Letters 73 (1994) 597. [9] D. Mukhopadhyay and Z.G. Soos, in: Optical and photonic applications of electroactive and conducting polymers, eds. S.C. Yang and P. Chandrasekhar, SPIE Proc. 2528 (1995) 116. [10] S. Etemad and Z.G. Soos, in: Spectroscopy of advanced materials, eds. R.J.H. Clark and R.E. Hester (Wiley, New York, 1991) p. 87; Z.G. Soos, D.S. Galv,~o and S. Etemad, Advan. Mater. 6 (1994) 280. [11] Z.G. Soos and G.W. Hayden, in: Electroresponsive molecular and polymeric systems, ed. T.A. Skotheim (Marcel Dekker, New York, 1988) p. 197. [12] Y. Tokura, Y. Oowaki, T. Koda and R. Baughman, Chem. Phys. 88 (1984) 437; Y. Tokura, T. Koda, A. ltsubo, M. Miyabashi, K. Okuhara and M. Ueda, J. Chem. Phys. 85 (1986) 99. [13] Y. Kawabe, F. Jarka, N. Peyghambarian, D. Guo, S. Mazumdar, S.N. Dixit and F. Kajzar, Synth. Met. 49/50 (1992) 517. [14] D. Mukhopadhyay and Z.G. Soos, J. Chem. Phys. 104 (1996) 1600. [15] R.G. Kepler and Z.G. Soos, in: Relaxation in polymers, ed. T. Kobayashi (World Scientific, Singapore, 1994) p. 100. [16] T.W. Hagler, Chem. Phys. Letters 218 (1994) 195. [17] Q.Y. Shang, X. Dou and B. Hudson, Nature 352 (1991) 703.
Z.G. Soos et a l . / Chemical Physics 210 (1996) 249-257
[18] S. Ramasesha and Z.G. Soos, J. Chem. Phys. 80 (1984) 3278. [19] D.N. Batchelder and D. Bloor, J. Phys. C 15 (1982) 3005. [20] Z.G. Soos and D. Mukhopadhyay, J. Chem. Phys. 101 (1994) 5515. [21] M.J. Nowak, G.J. Blanchard, G.L. Baker, S. Etemad and Z.G. Soos, in: Conjugated polymeric materials: opportunities in electronics, optoelectronics and molecular electronics, eds.
257
J.L. Br6das and R.R. Chance, NATO ASI Series E, Vol. 182 (Kluwer, Dordrecht, 1989 p. 421. [22] M. Cha, W.E. Torruellas, G.I. Stegeman, H.X. Wang, A. Takahashi and S. Mukamel, Chem. Phys. Letters 228 (1994) 73. [23] R.G. Kepler and Z.G. Soos, Phys. Rev. B 43 (1991) 12530. [24] L. Sebastian and G. Weiser, Phys. Rev. Letters 46 (1981) 1156.
Chemical Physics 210 (1996) 249-257
Stark profiles of singlet excitons in conjugated polymers Z,G. Soos *, D. Mukhopadhyay, M.H. Hennessy Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
Received 18 December 1995
Abstract
Electroabsorption (EA) spectra associated with overlapping vibronics of even- and odd-parity excitons are related in the Condon approximation to the linear absorption of discrete localized states. The EA intensity and profile depend strongly on the linewidth, F, with F -2 scaling of features due to Stark shifts and F-1 scaling of induced absorption and bleaching. Static disorder in polymer films broadens the linear absorption and reduces EA intensity compared to polydiacetylene (PDA) crystals, whose resolved spectra indicate overlapping vibrational sidebands. Similar excitations and transition moments in films conspicuously alter EA profiles and allow estimates of symmetry-breaking perturbations. EA spectra of PDA crystals and films are consistent with Pariser-Parr-Pople theory of linear and nonlinear optical spectra. The high sensitivity of EA also provides direct tests of excited-state displacements at induced transitions of even-parity excitons.
I. Introduction
Electroabsorption (EA) or Stark spectroscopy measures changes in the linear absorption, /(to, F ) I(to), in a static electric field F. Its high sensitivity has been recognized and exploited in atoms, molecules, molecular solids, inorganic complexes, semiconductors, and conjugated polymers. Since F mixes even- and odd-parity states in centrosymmetric systems, EA spectra probe the even-parity states responsible for large nonlinear optical (NLO) responses. Quantitative analysis has been difficult in condensed phases with congested excitations. Assignments of excited states are further hindered in conjugated polymers by sample morphology: with the notable exception of polydiacetylene (PDA) sin-
* Corresponding author.
gle crystals, conjugated polymers are only available as films. The linear absorption l(to) of PDA crystals is dominated by a singlet exciton, 1B, some 0.5 eV below the band edge for photoconduction. Weiser [1] showed that EA spectra of 1B excitons at low temperature closely follow the derivative, l'(to). The l'(to) profile is particularly convincing in view of resolved sidebands for single, double, and triple bond vibrations o f the c e n t r o s y m m e t r i c ( C R = C R C ~ - C ) , backbone of PDA crystals with different nonconjugated substituents R ( = PTS, DCHD, and PFBS). Several groups [2-5] have noted that EA spectra of films resemble the second derivative, l"(to). Except for a low-energy feature, polyacetylene (PA) films have an 1"(o)) profile [3]. PDA films with R = 4BCMU have both l'(to) and l"(to) contributions [5], but the fit is less compelling at lower resolution when only vestiges of vibronic structure remain. Such profiles are expected for extended states
0301-0104/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0301-0104(96)00117-6
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Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
above the band gap of semiconductors [6], as well as in charge-transfer molecular crystals [7]. Thus EA profiles are consistent with different physical pictures, as summarized in Refs. [3,1]. Stark shifts leading to i F ( t o ) spectra in the 1B region indicate mixing with an even-parity state above or below the singlet exciton, respectively. Since shifts depend on both the location and transition dipole of the A state, one or the other must be known independently. The location of A states is decisively given by two-photon spectra, as recently shown by Lawrence et al. [8] for PDA-PTS single crystals. We assigned [9] the features below 1B to a 2A exciton and its vibrational sidebands. Overlapping linear and two-photon features at 2.00 eV imply strong mixing by the field that cannot be treated perturbatively. In this paper, we analyze Stark spectra of overlapping 1B and 2A vibronics and explain why EA intensities of films are ~ 100 times smaller than crystals at similar F. Stark profiles are not restricted to l'(to) or l"(to), but may resemble either in special cases. There is much information in EA(to) lineshapes, especially when combined with other experimental or theoretical knowledge about -rr-w * excitations. We follow the joint analysis [10] of NLO spectra of PA, PDA, and other conjugated polymers in terms of exact Pariser-Parr-Pople excitations [11] for oligomers with up to 14 w-electrons; excitations and transition moments are related to 1B of finite polyenes, where 2A is indeed below lB. The vibronic analysis in Section 2 is general for discrete states, however, and extends previous mixing [12,13,1] of 1B with a hypothetical A state at the band edge~ Such generalization becomes mandatory now that the two-photon spectra are available. We emphasize the sensitivity of EA profiles to the vibronic structure of overlapping excitons, as shown directly by experimental frequencies, where changes of ~ 0.02 eV can be significant. The resolution of current theories is at best ~ 0.1 eV, as discussed [14] recently in a joint analysis of NLO and EA spectra of PDA crystals and films. In that work, hereafter I, we related w-electron theory to PDA spectra and identified the displacements and transition moments of 1B and 2A. The analysis of Stark profiles is general for discrete states of centrosymmetric systems. I(to) is
polarized along the PDA backbone in crystals and this approximation is used throughout. Vibronic structure of 1B and 2A excitons is treated in the Condon approximation for displaced harmonic oscillators with ground-state vibrational frequencies. We consider an isotropic distribution of backbones in films, retain transition dipoles and vibronic structure for excitons with shifted energies, and introduce a distribution g(o~) to model static disorder. We find that broader l(to) reduce the EA intensity and alter the profile without invoking symmetry-breaking perturbations such as internal electric fields or site energies.
2. Electroabsorption of discrete states in centrosymmetric systems
2.1. General expression and linewidth dependence We consider centrosymmetric systems with an even ground state ]G> and discrete even- and oddparity excitations Is> and Ir>. The linear absorption
/(to) is z(.,) = E r
rr ~--- Er 77" (Ojr-- 09)2-[- C 2'
(1)
where Ixr is the transition moment and I ( o ~ r - - to,Fr) is a normalized line centered at o~r. The particular choice in (1) is a Lorentzian with FWHM of 2F~, but any profile can be used. The perturbation - ix. F in a static electric field mixes even- and odd-parity states. The linear absorption becomes
l(to,F) = E i . t r ( F ) 2 1 ( % ( F ) -- w,Fr) r
" ~ E txs(F ) 2 l(tos(F) - to,I'~ ). x
(2)
We have induced absorption at even-parity states and Stark shifts of all transitions.
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
The formal expression (2) holds for arbitrary fields. In practice, - I ~ " F is small and strong mixing is restricted to nearby Ir),ls) pairs. Taylor expansion to order F 2 gives the EA spectrum I(to,F) -
251
pair, neglect mixing with the ground state, and introduce w, = w , + 2A, V = ( r l / x l s ) " F.
(5)
The new energies and mixing angle 2 + are
l(to),
w,( F),w,( F)
air(to) EA(w) = ~
/,/,2
ato
+[/.tr(F) 2-
at.Or
(3)
S
with lj(to) = l(toj - to,Fj). Stark shifts in centrosymmetric systems are quadratic in F,
ator=O~r-tor(F)
= F. (a,-o%)'F/2
+ OJs) ++6,
e = f ~ - 7 + V 2 ' tan24, =
]d~2]lr(¢0))
+ ~,(F)21,(to),
= ½(o~r
(4)
and depend on the polarizability tensors of Ir ) and IG). Shifts of ~ 10 p.eV are inferred [1] for PDA crystals at F = 20 k V / c m . The sharpest lines are I 0 - 2 eV in crystals and ~ 0 . 1 eV in films. Similar expressions hold for 8to~, but are not needed to order F z because iXs(F) is proportional to F. Induced absorptions in (3) thus appear at the twophoton lines tos, while bleaching occurs at to,, and all coefficients go a s F 2 in lowest order. Stark shifts scale as l~(to) and dominate when linewidths are small compared to I r ) , l s ) splittings. The general expression does not reduce to 1'(o~), however, unless all shifts are identical. The nature of the discrete localized states Ir ) and Is) is left open in (3). The strong EA(to) dependence on F is consequently a general feature. Induced absorptions or bleachings in (3) decrease as F - l , while the maxima of l'(to) decrease as F -z. A 10-fold increase of F reduces l'(to) features 100fold. This explains the decreased amplitude and different Stark profiles in crystals and films. The argument also holds for asymmetric exciton profiles l(to) in many films.
V/A.
(6)
The transition intensity is partitioned as cosZqb for I r ( F ) ) and sin2+ for I s ( F ) ) . We introduce x = to (% + tos)/2 to measure frequency from the mean, assume a common F, express sinZqb in terms of V and A, and retain terms of order V 2 to obtain a special case of (3), EA(to;A,F)
M(A+x,F) 2A ax V2 +--~-£[I(A--x,F) - I ( A + x , F ) ] . V 2
This exact result holds for any splitting, including degenerate excited states. The Stark shift is - V2/2 A for 21AI >> ~ F ; it is positive for A < 0 (when tos < t%), negative for A > 0. The other terms are bleaching at x = - A and induced absorption at x = A. The V 2 factor containing the F dependence is common to all three terms in (7). Reduced variables
A =
5r/2
."
'A -~0.05F /'X,
,/~ ~
~-2
'/~
"g/A=r
°
2.2. Exact results for two-state mixing We will use the general expression (3) for simulations of Stark spectra in Section 3. To study EA(to) profiles analytically, we consider a single I r ) , l s )
(7)
"-4
-'2
0 -
u 2
t
l
4
(% + ~,.)/z
Fig. 1. Electroabsorption profiles, Eq. (7), for mixing an even and an odd state with splitting 2A and width 2F. The inset shows the individual components of Eq. (7) for A = F: a Stark-shift derivative and bleaching at - A, induced absorption at A.
252
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
readily show EA(to;A,F)FE/V 2 to be a function of A / F . The physical situation of fixed splitting and uariable linewidth is more conveniently shown with fixed F and variable A, thus removing the F dependence. The representative EA(to;A/F) curves in Fig. 1 have the expected I'(o~) profile for A = 5 F / 2 , with well-separated states and weak mixing. The opposite limit of A < F gives entirely different profiles that go exactly as I"(o) at A = 0. The inset shows the individual contributions to (7) for A = F, with moderate mixing. For 8-functions and A = 0 , t%.s(F) reduce to lines at t% _+ IxF that appear as peaks in the EA spectrum on either side of a strong bleaching. The profile depends on I~F/F, with F >> IxF in these polymers. When the splitting is small compared to the linewidth, we Taylor expand EA(to;A,F). The second term in (7) has only odd derivatives of l ( x , F ) and the l ' ( x , F ) contribution exactly cancels the first term of the Stark shift. The IAI < F limit of (7) is consequently EA( o.);]A[ < F ) / V 2
= ½I"(x,F) + O [ a l ' ( x , r ) ] .
(8)
As shown in Fig. 1, the profile is a second derivative about the mean excitation energy for nearly degenerate states. The argument holds for any absorption profile, even for pairwise mixing of vibronics Ir) and Is) in electronic states with identical geometries.
2.3. Static disorder in films The general expression (3) for an isolated system is appropriate for crystals with both E and F along the backbone. The simplest approximation for films is an isotropic distribution of backbones. This gives a reduction of (cos40) = 1 / 5 when all transition moments are parallel to the backbone, at angle 0 with EIIF. The reduction is (cos20 sinZ0cos2~b)= 1/15 for E _t_F. The predicted 3 : 1 scaling of the EA(to) for parallel and orthogonal E, F has been observed in PDA-4BCMU films [5]. Such scaling implies transition dipoles along the chain, as also found in two-photon polysilane spectra [15]. The smaller PA ratio of Ref. [3] 2.3 : 1 or Ref. [5] 2.6 : 1 indicates deviations [16] of the transition moments from the backbone. The transition moment [31~GIB of trans-
polyenes is predicted [17] and found [18] to be 15° towards the double bonds from the backbone. We model inhomogeneous broadening with normalized Gaussians at to r,
g(o,r= (~)-lexp
-- ( ogr -- Nr)2/2o-2,
(9)
and assume for simplicity the same width 2or > 2F for all lines. The film average for the linear spectrum (1) and EA spectrum (3) are term by term convolutions with (9). For ~r >> F, we simplify by setting F = 0, so that the Lorentzians become 8 functions and (1) reduces to Gaussians about the mean mr. For the EA spectrum, we integrate the I'(co) term by parts to obtain an integral over g'(co). The F = 0 limit then regains (3) for Gaussian profiles. The absorption peaks to [r), l(t%,F r) in crystals and l ( ~ r , O ' ) in films, are related by
I( ~, ,or ) / I ( w r,/'r) = 3 ~ 7r~-/2 "
(10)
The reduction (cos20) = 1 / 3 is the assumed isotropic distribution of backbones. The ('rr/2) I/2 factor is a correction due to different line shapes. The ~ 20% blue shift of 1B 0 - 0 in PDA films rules out exact cancellation of transition moments. We are assuming equal integrated 1B intensity in crystals and films. The absorption coefficients a of PDA crystals are an order of magnitude larger than in films [5], while the 300 K lines are 2 - 3 times narrower [9]. This agrees with (10) within the accuracy of estimating the large a in thin samples. Theoretical support comes from calculated [14] 1B oscillator strengths per site in oligomers, which converge to the polymer value [1]. The same widths necessarily apply to EA spectra. For El[F, induced absorption and bleaching follow (10) with ( c o s 4 0 ) = 1 / 5 replacing the 1 / 3 factor. The maxima of l~(co) are at _+F/vr3 for Lorentzians, while those of g'(to) are at __+or. The E[JF ratio is
I'( "~r H- O",o- ) / I ' ( oar +__Fr/3'/2,C ) C2 - 5o.2 ~/32~/27e.
(11)
The square root is due to the assumed profiles in crystals and films, and is again a small correction.
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
To first approximation, random orientations determine the relative EA intensities of crystals and films. In addition to similar Ixr for equal linear absorption in crystals and films, the ratio (11) assumes equal transition dipoles of 1B with excited A states in crystals and films. The NLO fits in I in fact required 10-30% variations of excited-state moments.
1.0
-
-
T
253 "1
1~
I - - ~ - - 7 " - -
0.5
.<
0.0
-0.5
I X3
3. Simulated electroabsorption spectra Linear and two-photon spectra of PDA-PTS crystals at 300 K fix the location of the 1B vibronics Ir> and 2A vibronics Ss), respectively, as summarized in Table 1 of I. The singlet exciton 1B 0 - 0 is 2,00 eV. Its sidebands at 0.20 and 0.26 eV are C = C and C-=C stretches, respectively. These backbone vibrations also appear in resonance Raman spectra, whose fits gives the displacements b relative to the ground state [19]. The even-parity states PDA-PTS crystals [8] at 1.80 and 2.03 eV are taken as [9] 2A 0 - 0 and its C = C and C=-C vibronics, with displacement a. The ~ 0.05 eV widths are comparable to the linear spectrum. The strong 2.7 eV feature is assigned to nA 0-0. The scaled NLO spectra of I also require transition moments IXlBEA and I'hB,A in units of IXlBG"
Overlapping vibronics at 2.0 eV imply degenerate Ir > and Is> and preclude a perturbation expansion in V = - I x " F. Given the states and transition dipoles, however, /(to,F) is readily found in the Condon approximation. The mixing of 2A and 1B vibrational levels goes as (2A,sl/zllB,r> = tZ2glBFsr( b - a ) ,
(12)
with Franck-Condon overlaps Fsr. These factors are known analytically for displaced harmonic oscillators with the same frequency. Two displaced modes yield similar matrix elements between liB,r) and IG,0> or InA, s'>, with s ' = 0,1 . . . . for C = C and C=-C. To describe r = 0, 1, 2, and 3 levels in 1B, we need 2A vibronics up to s = 7 and nA vibronics to s' = 3 for the displacements in I. Higher levels have negligible Fsr, but can easily be included. We use 10 vibrational levels for 1B and nA, 36 for 2A. The 57 × 57 Hamiltonian with off-diagonal elements I~" F given by (12) is diagonalized. Once F 2 dependencies have been checked, we obtain the EA spectrum (3).
-1.0
÷
_ _ _ _ _ . . a ~ ~ _ _ 1.6 1.8 2.0 2.2 2.4
(eV)
2.6
2.8
Fig. 2. Solid line: scales electroabsorption spectrum, Eq. (3), of PDA crystals based on inputs from linear, Raman, and NLO spectra, as described in the text, and Lorentzian profiles with F = 0.01 eV. Dashed line: scaled derivative of the linear absorption, Eq. (1), for the same parameters.
The solid and dashed lines in Fig. 2 are EA(to) and /'(to), respectively, based on Table 1 of I. To compare with 10 K spectra, we took a smaller F = 0.01 eV and 2A 0 - 0 at 1.81 eV. In PDA-PTS, PDA-DCHD, and PDA-PFBS crystals below 10 K, 1B 0 - 0 shifts to lower energy (1.935, 1.856, and 1.965 eV, respectively), the lines sharpen to 2 F 0.03 eV, and resolved sidebands are listed in Table 1 of Ref. [1]; their lower intensity implies slightly smaller b and a. The (dashed) l'(to) curve in Fig. 2 is based on (1) with the 300 K displacements. Two 0-1 and three 0 - 2 sidebands are resolved and easily seen on increasing the scale above 2.1 eV. The d e 2 / d E curves in Figs. 9-11 of Ref. [1] show similar resolution for l'(to) derived from reflectance data on crystals. The (solid) EA spectrum for F = 0.01 eV coincides well with l'(to), just as the related Ae 2 curves in Figs. 9-11 of Ref. [1] did. We emphasize that the calculated EA(to) and l'(to) curves in Fig. 2 are related in the same way as the experimental curves for PDA crystals in Ref. [1]. Only such comparisons are pertinent in view of experimental inputs about modes, displacements, and frequencies. These inputs from independent spectra also account for EA. The unsymmetric EA profiles in Fig. 2 result from approximate cancellation of large induced absorption and bleaching at nearly degenerate Ir>,ls> pairs, as expected from (7) and Fig. 1. The 2.00 eV feature, for example, depends
254
Z.G. Soos et al. / Chemical Physics 210 (1996) 249-257
sensitively on the position of 2A 0-0: the l"(to) maxima are doubled and halved, respectively, on shifting from 1.81 to 1.80 and 1.79 eV. The resemblance between calculated EA(to) and l'(to) in Fig. 2 is retained in each case, although the explicit decomposition in (3) does n o t indicate a dominant Stark shift; the calculated profiles have major contributions from bleaching and induced absorption. The location of the 2A 0 - 0 and nA 0 - 0 are indicated in Fig. 2. The induced absorption at 1.81 eV is 0.9 )< 10 - 3 , about 0.1% of the 1B 0 - 0 amplitude. The small relative intensity of 2A 0 - 0 is independent of F, since EA scales as F 2. Detection of induced absorption at 2A 0 - 0 is a matter of resolution, with current spectra probably sensitive to signals over 5% of the 1B 0 - 0 amplitude. The NLO fits in I or in 13-carotene [20] suggested opposite displacements b and a in 1B and 2A, leading to small F00(Ibl + lal) in (12). The induced absorption at 2A 0 - 0 is 30 times smaller for Ibl + lal than for r b l - lat for these displacements. Thus EA spectra at 2A 0 - 0 provide sensitive tests for these excited-state parameters. We did not consider vibronic contributions to EA spectra in I, but emphasized the collective effects of all w-'n" * excitations. The stronger mixing of degenerate states in Fig. 2 increases the 1B Stark shift by an order of magnitude. Direct inclusion of F through (12) is presently restricted to vibronics of nearby electronic states chosen from the full ,rr--rr * spectrum of oligomers. The strong two-photon absorption at 2.7 eV gives a small EA response in Fig. 2 because, as discussed in I, the induced absorption also depends on excitation energies. The Stark shift of nA rationalizes the l'(to) profile at 1B 0 - 0 in Fig. 2 for degenerate vibronics. The two-level model predicts an l"(to) profile for A = 0, but neglects all other states. Coupling to a high-energy state approximately restores a first derivative when both tot(F) and tos(F) in (6) shift below tot. Linear absorption indicates 2F = 0.03 eV below [1] 10 K and =0.05 eV at 300 K i n PDAcrystals [9], while film widths have [5,9] 2or = 0.2 eV. An order of magnitude decrease of c~ is consistent with (10). A ratio cr/F = 5 reduces the EA intensity (11) by ~ 1/125 in films, consistent with AoL changes [5] in PDA-4BCMU crystals and films at the same F. These estimates set the scale for reduced EA
2
,
L
t
0
........
m 0 -1
A
--2
t
2.0
I
2.4
B i
(eV)
i
2.8
I
3.2
Fig. 3. Top: same as Fig. 2, with broader F = 0.1 eV for films leading to reduced amplitude according to Eq. (11). Bottom: electroabsorption spectrum, Eq. (3), and scaled 1'(~o), Eq. (i), for inputs based on linear and NLO spectra of PDA films, as described in the text. The observed bleaching at A in Ref. [5] is roughly twice the bleaching at B, and the low-energy features are offset.
intensity and, as anticipated from the resolved vibronics in Fig. 2, for altered profiles. There are two other changes: 1B 0 - 0 of PDA-4BCMU at 300 K shifts [21,22] from 1.99 eV in crystals to 2.35 eV in films and, although barely resolved, is slightly less intense than the 0-1 feature at 2.55 eV; ~ 20% larger b were needed in I for films. NLO spectra fix the location [9] of nA at 3.22 eV, again ~ 35% above 1B, but allow some flexibility for 2A 0-0, I~m2A, and a. These parameters are fixed by twophoton spectra in PDA-PTS crystals. To highlight the linewidth dependence, the upper curves in Fig. 3 are the crystal spectra (Fig. 2) with larger F = 0.1 eV and a blue shift of 0.35 eV; they represent PTS rather than 4BCMU inputs. The scale decreases by 500, as expected from (11), and the solid line depicting EA(to) now follows the dashed line, /'(co), only around 1B 0-0. The bleaching features labeled A and B in Fig. 3 are split by the vibrational spacing ( ~ 0.2 eV). There is partial agreement with the experimental PDA-4BCMU curve in Fig. 1 of Ref. [5]; however, the relative magnitudes of A and B disagree with experiment. The lower EA(to) and /'(to) curves in Fig. 3 are based on Lorentzians with F = 0.01 eV and transition moments, displacements and excitations for PDA-4BCMU films taken from I, except that 2A 0 - 0 is 1.86 rather than 1.90 eV and a is 30%
Z.G. Soos et a l . / Chemical Physics 210 (1996) 249-257
smaller. Each curve is then convolved with a Gaussian whose a = 0.10 eV is fixed by l(to). The EA features are properly located, the scale is again 500fold smaller, and the relative bleaching at A and B is closer to experiment. The maximum of EA(to) no longer occurs at / ' ( t o ) = 0, as observed [5] and expected for l"(to). The region above 2.8 eV disagrees with experiment. Since the location of the band edge is not known in films, deviations at high energy from I'(to) or l"(to) profiles have not been modeled. This underscores both the limited nature of EA fits in films, where the relevant electronic states are not known independently, and the sensitivity of Stark profiles to overlapping vibronics.
4. Discussion
The strong linewidth dependence of the EA spectrum (3) is a general feature of discrete localized states of centrosymmetric systems. The resulting EA profiles depend sensitively on the inputs, linewidths, and static disorder. EA intensities provide simpler tests. The absorption coefficient changes [1] by 400600 cm -1 in PDA crystals at F = 2 0 k V / c m . Broader lines in films indicate a 100-fold smaller EA signal, consistent with (11) for static disorder. Larger F is typically used in films, but reported Act values can readily be compared since EA scales as F 2. Published values give Act = 2 - 5 cm -1 at F = 20 k V / c m for PA [3], a polysilane [4], and PDA4BCMU. The agreement is encouraging for a joint w-electron analysis of crystals and films. Partial ordering in stretched [3] or oriented [13] films is more difficult to assess; linewidths are hardly reduced while there is considerable orientation. The location of singlet excitons, the magnitude of Stark shifts (4), and transition dipoles are problems in electronic structure. The vibronic structure of G, 1B, and 2A depend on their potential surfaces and, as discussed in I, require all-electron models. The 1B exciton and its sidebands appear in EA spectra of crystals, as shown in Fig. 2, even for overlapping 2A vibronics: only nearby states mix strongly for small IxF. EA profiles rather than positions reflect strong mixing, as illustrated for two states in Fig. 1. Although l'(to) is favored for small F, the result is not general for overlapping states in (3). Stark profiles at
255
allowed and induced transitions of excitons contain detailed information about transition dipoles and Franck-Condon overlaps. Static disorder includes variations of chain lengths, transfer integrals t~, or backbone conformation. Except for chain length, such perturbations break inversion symmetry. In "rr-electron models with arbitrary spin-independent interactions, however, electronhole symmetry [11] precludes mixing of the ionic 1B state with either covalent G or 2A states as long as t n are restricted to neighbors and all site energies en (Hiickel ct) are the same. Finite chains, disordered t~, and arbitrary backbone conformations still yield quadratic Stark shifts. We have an inhomogeneously broadened 1B at F = 0 and rigorously vanishing absorption at 2A and other two-photon-allowed singlets. Ionic A states and covalent B states may now appear in linear and two-photon spectra, respectively. Since ionic A states are above the band gap, they are not important in the exciton region. Covalent B states fall below the allowed 1B 0 - 0 in polyenes [17], but it will be some time before such corrections to two-photon spectra can be identified. There are many other l(to) changes, however, including different excitation energies, transition dipoles, and vibronic structure. Symmetry-breaking perturbations Vd are conveniently given by site energies, Vd = E ~ p ( 1 - no),
(13)
P
where n e is the w-electron number operator and the distribution G(e) is symmetric about ( e ) = 0 . Choices leading to internal fields F 0 or dipoles are special cases. Since even-odd selection rules no longer hold, the F = 0 spectrum l(to,Vd) already has features at both tor(Vd) and to,(Va). Once electronhole symmetry is lifted, transitions to covalent B states and ionic A states become allowed. Modest Vd is indicated for conjugated polymers by similar film and crystal spectra. The detection of site disorder depends on quantitative analysis of l(to) and EA(to), whose high sensitivity is then advantageous. The general EA spectrum (3) now has linear as well as quadratic Stark shifts and linear as well as quadratic contributions to induced absorption and bleaching. The new contributions go as F 2 Vaz in lowest order. While the EA intensity rules out large Va,
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symmetry-breaking contributions somewhat larger than Ix F may eventually be extracted from film spectra. Linear Stark shifts generate l~'(to) terms that scale as F2Vd2. The physical picture is that F > 0 now gives both red and blue shifts of 1B, thus broadening the film's absorption, while quadratic Stark shifts (4) are unidirectional. The general case has many additional terms in F 2 Vd2, including bleaching terms in l~(to) and induced absorptions l~(to). Even if Vd is approximated by an internal field F 0, we have an additional parameter for EA fits in films. The general appearance of I"(to) profiles suggests linear Stark shifts, but the sensitivity of EA spectra in Fig. 3 to the location of 1B and 2A vibronics complicates the identification of Vd contributions. Accurate l(to) or EA(to) fits in films are hampered by the limited range of the exciton region and unknown band edges. The intense two-photon absorption at 2.7 eV in PDA-PTS crystals [8] is above the 2.5 eV band edge, where no signal is resolved. The nA state used to model NLO spectra was recognized [10] to be above the PDA band edge. By contrast EA contributions from band-edge states are seen in PDA crystals, especially by electroreflectance [1], as well as in oriented PDA films [13] and polysilane films [4,23]. The Franz-Keldysh effect in PDA crystals [24] at low temperature clearly implies a continuum above a well-defined band edge. These states may become localized in films, but are dense rather than discrete and consequently require more than a single [12,13] A state at the band edge. Accurate treatment of dense states is far more difficult than discrete states in the exciton region. Except for Va contributions, however, the same theoretical model should account for EA signatures of excitons in crystals and films. We have deliberately retained excitation energies, transition moments, and vibrational inputs from the linear, Raman, and NLO fits discussed in I. Similar linewidth and vibronic effects follow for other parameters. The high sensitivity of EA may precisely determine some values, such as the Franck-Condon factor for induced absorption at 2A 0 - 0 in PDA crystals. We have shown vibronic sidebands play a major role in EA spectra and have accounted for the lower intensity of films using exciton states derived from NLO spectra. EA spectra in the singlet exciton
region are another illustration that conjugated polymers can be modeled by molecular Pariser-Parr-Pople theory.
Acknowledgements We gratefully acknowledge support of this work by the National Science Foundation through DMR9300163.
References [1] G. Weiser, Phys. Rev. B 45 (1992) 14076. [2] J. Orenstein, G.L. Baker and Z. Vardeny, J. Phys. (Paris) Colloq. 3 (1983) C3-407. [3] S.D. Phillips, R. Worland, G. Yu, T.W. Hagler, R. Freedman, Y. Can, V. Yoon, J. Chiang, W.C. Walker and A.J. Heeger, Phys. Rev. B 40 (1989) 9751. [4] S. Jeglinski and Z.V. Vardeny, Synth. Met. 49/50 (1992) 509. [5] A. Horvath, G. Weiser, G.L. Baker and S. Etemad, Phys. Rev. B 51 (1995) 2751. [6] D.E. Aspnes and J.E. Rowe, Phys, Rev. B 5 (1972) 4022; D. Aspnes, in: Handbook on semiconductors, Vol. i, ed. T.S. Moss (North-Holland, Amsterdam, 1980). [7] L. Sebastian, G. Weiser and H. B~sler, Chem. Phys. 61 (1981) 125. [8] B. Lawrence, W.E. Torruellas, M. Cha, M.L. Sundheimer, G.I. Stegeman, J. Meth, S. Etemad and G.L. Baker, Phys. Rev. Letters 73 (1994) 597. [9] D. Mukhopadhyay and Z.G. Soos, in: Optical and photonic applications of electroactive and conducting polymers, eds. S.C. Yang and P. Chandrasekhar, SPIE Proc. 2528 (1995) 116. [10] S. Etemad and Z.G. Soos, in: Spectroscopy of advanced materials, eds. R.J.H. Clark and R.E. Hester (Wiley, New York, 1991) p. 87; Z.G. Soos, D.S. Galv,~o and S. Etemad, Advan. Mater. 6 (1994) 280. [11] Z.G. Soos and G.W. Hayden, in: Electroresponsive molecular and polymeric systems, ed. T.A. Skotheim (Marcel Dekker, New York, 1988) p. 197. [12] Y. Tokura, Y. Oowaki, T. Koda and R. Baughman, Chem. Phys. 88 (1984) 437; Y. Tokura, T. Koda, A. ltsubo, M. Miyabashi, K. Okuhara and M. Ueda, J. Chem. Phys. 85 (1986) 99. [13] Y. Kawabe, F. Jarka, N. Peyghambarian, D. Guo, S. Mazumdar, S.N. Dixit and F. Kajzar, Synth. Met. 49/50 (1992) 517. [14] D. Mukhopadhyay and Z.G. Soos, J. Chem. Phys. 104 (1996) 1600. [15] R.G. Kepler and Z.G. Soos, in: Relaxation in polymers, ed. T. Kobayashi (World Scientific, Singapore, 1994) p. 100. [16] T.W. Hagler, Chem. Phys. Letters 218 (1994) 195. [17] Q.Y. Shang, X. Dou and B. Hudson, Nature 352 (1991) 703.
Z.G. Soos et a l . / Chemical Physics 210 (1996) 249-257
[18] S. Ramasesha and Z.G. Soos, J. Chem. Phys. 80 (1984) 3278. [19] D.N. Batchelder and D. Bloor, J. Phys. C 15 (1982) 3005. [20] Z.G. Soos and D. Mukhopadhyay, J. Chem. Phys. 101 (1994) 5515. [21] M.J. Nowak, G.J. Blanchard, G.L. Baker, S. Etemad and Z.G. Soos, in: Conjugated polymeric materials: opportunities in electronics, optoelectronics and molecular electronics, eds.
257
J.L. Br6das and R.R. Chance, NATO ASI Series E, Vol. 182 (Kluwer, Dordrecht, 1989 p. 421. [22] M. Cha, W.E. Torruellas, G.I. Stegeman, H.X. Wang, A. Takahashi and S. Mukamel, Chem. Phys. Letters 228 (1994) 73. [23] R.G. Kepler and Z.G. Soos, Phys. Rev. B 43 (1991) 12530. [24] L. Sebastian and G. Weiser, Phys. Rev. Letters 46 (1981) 1156.