The calculated and experimental results for the radiative lifetime τrad of the S1 exciton and
τrad 13αxx,S1
1 3α
loss
xx,S1 ∆αS1 rrms
Polymer Calc. Exp. Calc. Exp. ˚A
PPVa 1.96 1.3±0.2 1.24 0.24 1.00 1.6±0.3 11.1
PPVb 1.34 1.3±0.1 1.24 0.20 1.04 1.4±0.3 11.1
PTc 2.2 2.5±0.3 0.70 0.12 0.58 0.6±0.2 8.6
PPPd 1.05 1.2±0.1 0.53 0.05 0.48 0.3±0.1 7.6
a experimental values averaged for 3 different dialkoxy-substituted PPVs
b experimental values obtained by extrapolation to 1/n = 0 of results for a series of OPVn’s with n=2-5
c experimental values averaged for 3 different alkyl-substituted PTs d experimental values for a dialkoxy-substituted PPP
TABLE 7.1: Exciton radiative lifetimes τrad (in ns), polarizabilities 13αxx,S1, po-
larizability corrections 13αlossxx,S
1, excess polarizability ∆αS1 (all in 10
3 ˚A3), and root-
mean-square electron-hole distances rrms(in ˚A) [from Equation (4.32)] for PPV, PT,
and PPP, in a dilute benzene solution.
we find very good agreement between the measured lifetimes and the ones calculated with Equation (7.7). For PPV the interpretation is less straightforward; as mentioned before, alkoxy-substitution substantially alters the fluorescence spectrum of PPVs. For the PPVs in the experiment we have the zero-phonon fluorescence maximum at 2.2 eV, while the extrapolation of the maxima for the OPVn’s is 2.5 eV. As the inverse of the radiative lifetime depends on the frequency of the emitted light to the third power [see Equation (7.7)], this shift can substantially alter the calculated value of the radiative lifetime. As we replace the alkoxy side-groups by hydrogen atoms in the calculation, a comparison with the extrapolation of the unsubstituted OPVn data is more relevant. Taking this into account, the agreement for PPV is also considered to be good.
The calculated excess polarizability for PT is in good agreement with experiment; that of PPV is slightly, but hardly significantly, smaller than the experimental value. For PPP there appears to be an overestimation of the excess polarizability. In all three systems, some rings are, by thermal effects, rotated over more than the equilibrium dihedral angle, leading to a reduction of the conjugation length and hence to more confined excitons with lower polarizabilities. Since the overlap of wave functions between adjacent rings (with a dihedral angle θ) scales as cos θ, the thermal fluctuations have a much larger effect in PPP, which has a non-zero equilibrium angle, than in PPV or PT.
It turns out that 99% or more of the first term in Equation (4.36), i.e., the coupling between the S1 exciton and the other excitons Si, in the calculation of αxx,S1 results from
the coupling to the S2 state. The second term, i.e., the reduction of the polarizability due
Eb (eV) fij S1 S2 f01 f12 PPVa 0.87 0.29 0.90 7.6 PPVb 0.87 0.29 0.80 7.6 PTc 1.00 0.33 0.72 5.5 PPPd 1.06 0.30 0.75 5.3
TABLE 7.2: Exciton binding energies Ei
b and oscillator strengths fij for PPV, PT,
and PPP, in a dilute benzene solution. See Table 7.1 for the explanation of the footnotes.
only. The correction αloss
xx,S1 of Equation (7.8) is 10-15% of αxx,S1, as can be seen in Table
7.1. The fact that the polarizability results almost exclusively from coupling of the S1 to
the S2 exciton justifies the use of three-level models (the ground state S0 and the exciton
states S1 and S2) for the explanation of excitonic properties [144]. Here, it enables us
to understand the difference in polarizability between PPV and PT/PPP, which results partly from the difference in energy between the S1 and S2 excitons, but mostly from the
larger oscillator strength fij = 2|µij,||kx|
2(Ei b− E
j
b)/3 in PPV. This oscillator strength as
well as that between the ground state S0 and the S1 exciton is given in Table 7.2.
To get an impression of the shape of the exciton wave functions, we have plotted in Fig- ure 7.4 the probabilities P (xe − xh) to find the electron and hole of the S1 exciton at a
separation xe− xh along the chain as defined in Equation (4.31). It is observed that the
root-mean-square electron-hole distance √< (xe− xh)2 >, listed in Table 7.1, increases
with increasing polarizability, as expected.
As we showed in the preceding Chapter, the exciton binding energy is rather sensitive to the choice of the dielectric constant ε. Therefore, the fact that the binding energies in a benzene solution (ε = 2.28) are larger than those for a bulk polymer (ε = 3.0) is not surprising. This means that the exciton binding energy of 0.87 eV given here for PPV in solution is not in contradiction with recently determined experimental values of 0.35± 0.15 eV for bulk alkoxy-substituted PPV [8] and 0.48± 0.14 eV for bulk unsubstituted PPV [9].
7.5
Conclusions
In conclusion, we have demonstrated that two important properties of excitons on conju- gated polymer chains, their polarizability and lifetime, can reliably be obtained from the solution of the many-body Bethe-Salpeter equation. This also strengthens the reliability of this approach for the prediction of other properties which are experimentally less acces- sible, like the exciton binding energy. More generally, this approach can open the way to a deeper understanding of the optical behaviour of conjugated polymers in terms of their
-40 -30 -20 -10 0 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.005 0.004 0.003 0.002 0.001 0.000
20 x
PPV
PT
PPP
Probability (Å
-1)
x
e-x
h(Å)
FIGURE 7.1: The probability P (xe− xh) as defined in Equation (4.31) to find the
electron and hole at a distance xe− xh along the polymer chain (offsets applied).
Especially in the tails (no offsets), which dominate the polarizability, the difference between the three polymers is clearly seen.
molecular structure and will be of help in the development of electro-optical devices based on these materials.
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Summary
The discovery, in the middle of the 1970s, of the fact that conjugated polymers (poly- mers with alternating single and double carbon-carbon bonds in their backbones) can be made electrically conductive by doping initiated a large number of studies, both experi- mental and theoretical, into the electronic and optical properties of these materials. The subsequent discovery of electroluminescence of a polyphenylenevinylene derivative has en- abled the fabrication of polymer light-emitting diodes (pLEDs); in the meantime the first commercially viable matrix displays of conjugated polymers have been demonstrated. Ap- plication of conjugated polymers in logic gates and integrated circuits (’chips’), mostly made of polythiophene derivatives, has also been achieved. While in these applications the conjugated polymers merely replace more traditional semiconductors, they have the advantage that they are much easier to process, offer a flexible choice of materials, and, once mass-produced, are much cheaper.
In spite of the rapid development of applications, much of the fundamental physics of conjugated polymers and the devices based upon them is still ill-understood. To a large extent, this lack of knowledge can be attributed to the fact that conjugated polymer systems are ill-defined. Chemical defects are present, large sidegroups or substituents, added to make the polymer soluble and processable, cause the chain to be buckled rather than planar, and the ordering of chains relative to each other is often poor. Each of these effects varies from sample to sample, depending on the precise parameters of the sample preparation.
A property that is heavily disputed in conjugated polymers is the exciton binding energy, the binding energy of an electron-hole pair, for which values ranging from 0.1 to 1.0 eV have been coined. In the low-binding-energy case, a solid-state band picture suffices to describe the electronic and optical excitations in conjugated polymers; for larger binding energies, a molecular exciton picture is more appropriate.
Up to now the modeling of the electronic and optical properties of conjugated polymers has mostly been performed with (often semi-emperical) quantum-chemistry methods, which can not treat extended systems, or with solid-state-based density-functional methods, which do not give reliable one-particle excitation energies and do not include excitonic effects. Only recently first-principles calculations predicting the electronic and optical
excitations of extended systems have become possible, obtaining good agreement with ex- periments for a wide range of properties and materials. The calculational method is in fact a combination of three methods: density-functional theory to predict the ground-state geometric structure of a material, the so-called GW method to calculate the one-particle spectrum (the energies required to add or remove electrons from the system), and the Bethe-Salpeter equation for the two-particle spectrum (which also includes the interaction between electrons and holes). Combined, these methods, applied to any system, represent the state-of-the-art of first-principles calculations. This is especially true for conjugated polymers as the systems are very large and the calculations computationally very expen- sive. At the start of this work, not a single calculation with these combined methods had yet been done for conjugated polymers; it was our aim to endeavor this, and to investigate to what extent qualitatively correct predictions could be made. If successful, this might open the way to, e.g., theoretical band gap engineering.
The system that is studied most extensively in this thesis is polythiophene; it was chosen as a starting point because of larger symmetry (and hence smaller computational work load) compared to, e.g., polyphenylenevinylene. We study polythiophene in three geometries: an isolated chain in vacuum, an isolated chain in a bulk polythiophene environment (where the environment is modeled by an ab-initio direction- and frequency-dependent dielectric tensor), and a polythiophene crystal. It turns out that the electronic and optical properties depend critically on the surroundings of the chain. The exciton binding energy in the three cases mentioned above is 1.86, 0.76, and 0.15 eV (with an accuracy of about 0.1 eV), respectively; the one-particle band gap decreases concomitantly. Surprisingly, the optical gap is more or less unaffected.
In the case of the isolated chain in vacuum the Coulomb interaction (calculated in the GW formalism) between two charges is not screened at long distances, leading to large binding energies and band gaps. By including this long-range screening through embedding of the chain in a dielectric medium, both the exciton binding energy and the band gap are decreased, and we obtain good agreement with experimentally determined exciton splittings and with the absorption gap. In the crystalline situation, we also include wave- function overlap between neighboring chains, which lowers the band gap by the splitting