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2009, 113, 5899–5901 Published on Web 03/25/2009
Transport in Charged Defect-Rich π-Conjugated Polymers Brian A. Gregg National Renewable Energy Laboratory, 1617 Cole BouleVard, Golden, Colorado 80401 ReceiVed: January 20, 2009; ReVised Manuscript ReceiVed: February 25, 2009
Some models of charge transport in π-conjugated polymers treat these materials as if they were electrical insulators. Although this may be appropriate for a few materials, many polymers are effectively doped p-type by a high density of charged defects. Herein, limits are estimated for the charged defect density above which the resulting electrostatic fluctuations may govern transport and for the corresponding free hole density above which space-charge-limited currents should not occur. These limits are lower than the experimentally observed values in many π-conjugated polymers, suggesting that these materials are more accurately described by models of doped semiconductors. This analysis also provides an explanation for two otherwise puzzling experimental observations, the low-field Poole-Frenkel mobility and the correlated energetic disorder. Theories of charge transport in excitonic semiconductors, XSCs, such as π-conjugated polymers, are a topic of active interest for both fundamental studies1-3 and for practical applications in electronics, displays, and photovoltaics.4,5 Excitonic semiconductors are those in which the electrostatic binding energy between electrons and holes is greater than the thermal energy, kT, resulting in the formation of excitons upon illumination rather than free electron hole pairs.6 Other properties of XSCs, such as their doping efficiencies and transport behavior, are also dominated by electrostatic forces.7 For example, the free charge carrier density, nf or pf, is a temperature- and field-dependent fraction of the total charge density, Ncd.7,8 The localized carrier wave functions and low dielectric constants of organic semiconductors render them excitonic at room temperature, whereas most inorganic semiconductors become excitonic only when cooled.8 Theoretical models of transport in thin films of π-conjugated polymers, π-CPs, are often derived from Ba¨ssler’s Gaussian Disorder Model, GDM.1-3,5,9,10 The GDM was originally developed to describe charge transport in insulating polymers containing electroactive small molecules, such as the hole transport layers, HTL, of electrophotography. HTLs must be highly insulating in order to function; they exhibit negligible dark currents at typical operating fields of ∼5 × 105 V/cm.11 In contrast, π-CPs are semiconducting with zero-field conductivities commonly in the range of 10-8-10-5 S/cm.12-14 Thus, it is not obvious a priori that the GDM can accurately describe π-CPs. The GDM postulates that each electroactive site has an energy that is uncorrelated to its neighbors due to its discrete local solvation, orientation, and dipolar effects (Figure 1A). Furthermore, it predicts Poole-Frenkel, PF, behavior of the carrier mobility (µ proportional to exp(F1/2), where F is the applied field) only when F > 3 × 105 V/cm. Experimentally however, the energy levels of π-CPs appear to be correlated, and PF mobilities are often observed at arbitrarily low fields.2,3,9 Several versions of correlated disorder models based on the GDM have thus been proposed along with various explanations for the 10.1021/jp900616g CCC: $40.75
Figure 1. (A) Schematic of an XSC conduction band showing uncorrelated Gaussian disorder. (B) Same with an ionized n-type dopant and the fluctuations caused by bound charges, Ncd. Activation energies for carrier generation and for mobility are shown.
origin of the correlations and the PF mobilities.3,9,15 Here, we propose an alternative explanation for both phenomena; they may result from the electrostatic fluctuations caused by charged defects. Heretofor, the presence and influence of charged defects in these materials have been mostly overlooked. The long-range energetic fluctuations caused by charged defects will perturb the short-range energetic disorder postulated by the GDM and impose correlations between the energy levels of neighboring electronic states near the charges (Figure 1B). Thus, correlated disorder is expected whenever Ncd is above a certain limit. Moreover, in the presence of such electrostatic fluctuations, the carrier mobility should exhibit approximately PF behavior even at the lowest field (see below) as F decreases the electrostatic barriers, inhibiting the flow of current. These conclusions are consistent with a recently published model of the effect of doping on the GDM.10 This paper did not suggest, as we do here, that many existing π-CPs are already so strongly doped by their defects that a model of doped semiconductors may be necessary to describe them correctly. An older treatment of transport in the presence of charged defects, the Poole-Frenkel model,16 is also relevant. It does 2009 American Chemical Society
5900 J. Phys. Chem. C, Vol. 113, No. 15, 2009
Letters
not consider the site to site energetic disorder but treats only the electrostatic influences on free carrier generation in a lowdielectric medium. The Coulomb attraction between incipient charge carriers and their counterions ensures that most carriers remain bound and only a thermally activated few (modeled by Boltzmann statistics) are free. These conditions naturally pertain to XSCs. The PF equation (written for mobile electrons and a single donor level) is
J ) qµnFnf ) qµnFnd exp((-Ea0 + (q3 /πε0ε)1/2F1/2)/kT) (1) where q is the electronic charge, µn is the electron mobility, nd is the effective n-type doping density (equivalent to Ncd for otherwise pristine materials), Ea0 is the binding (activation) energy of the charge, ε0 is the permittivity of free space, and ε is the dielectric constant. The PF model predicts that the current will increase as exp(F1/2) because of the increase in free carrier density as the applied field decreases the Coulomb well binding the incipient carrier to its counterion. Curiously, the PF model is now often assumed to apply only to the field dependence of the mobility (which was specifically excluded from the original model), while the change in nf and/or pf with field is ignored. The original PF model is unrealistic, however, because it treats only the change in the number of carriers emitted from their Coulomb wells7,15 but not the electrostatic fluctuations caused by the great majority of charges which remain bound. These fluctuations will decrease the carrier mobility by attracting mobile carriers into potential wells and, if the charge density and field are high enough, forcing them to surmount potential barriers (at low concentration and field, the carriers can go around barriers in three-dimensional systems). Assuming Coulomb potentials, this will result in a field dependence of the mobility of approximately exp(F1/2). Thus, both the number of free carriers and the carrier mobility should increase approximately with exp(F1/2).7,13 However, the mathematics is complex due to the coupling between the carrier density, the mobility, and the field, and this problem has not yet been solved for three-dimensional systems.15 Nevertheless, good agreement between experiment and theory is often achieved in both organic and inorganic semiconductors if the term multiplying F1/2 in eq 1 is divided by approximately 2.7 It seems clear that both energetic correlations and PF mobilities at low field can result from the presence of charged defects. The question then becomes whether Ncd is high enough to cause this behavior in some, or most, of the commonly studied π-CPs that are not purposely doped. To address this question, we estimate the value of Ncd above which the resulting electrostatic fluctuations begin to overlap, thus leaving no electrostatically unperturbed transport pathways. The distance between uniformly distributed charges is set equal to the distance at which the Coulomb energy of an isolated charge decays to kT
Ncd-1/3 )
q2 4πεε0kT
(2)
For Ncd to have a negligible effect on transport, its value must be much less than this limit
Ncd ,
(
4πεε0kT q2
)
3
) 5.9 × 1015ε3 cm-3
(3)
where the constants are evaluated at room temperature. The right-hand side equals 1.6 and 3.8 × 1017 cm-3 when ε ) 3 and 4, respectively. Unfortunately, there are few estimates of Ncd in π-CPs,13,14 although studies of controlled doping in crystalline small-molecule XSCs provide some relevant comparisons.7 Reported measurements of the equilibrium free hole density, pf, from which Ncd can be estimated, are also surprisingly rare considering its fundamental importance to understanding the semiconductor properties. Yet, pf has been estimated by several methods, for example, from the conductivity near zero field when the carrier mobility is known. This is best achieved with widely spaced electrodes to ensure that only bulk properties are measured. Reported values of pf for thin films of poly(thiophenes) and poly(phenylene vinylenes) are in the range of 1014-1017 cm-3.12-14 More studies are needed to confirm these values. Because XSCs are dominated by electrostatic forces, pf is only a small fraction of Ncd.7,8,10,13 This is equally true for organic semiconductors at room temperature and for inorganic semiconductors at lower temperature.8 The activation energy for free carrier production (Ean in Figure 1B) can be estimated by measuring the activation energy of the dark current, EaJ, and subtracting the activation energy of the mobility, Eaµ.7 If a typical value is assumed, Ean ≈ 120-180 meV, then only ∼1-0.1% of the carriers will be free at room temperature. Thus, Ncd ≈ 100-1000pf, or given the published estimates of pf, Ncd ≈ 1016-1019 cm-3. Comparing these values to eq 3 suggests that some of the cleanest π-CPs may have sufficiently low values of Ncd that the fluctuations caused by charged defects will influence, but not dominate, transport. Yet, in many existing materials, it seems likely that the insulator models may be invalid because they neglect an important factor controlling transport. Even when electrostatic fluctuations do not greatly perturb the mobility, pf should still increase with field and temperature. This is an unavoidable consequence of the attraction between a carrier and its countercharge being greater than kT. It is treated explicitly only in the PF model, though it seems also implicit in the doped GDM.10 On a related topic, excitons are thought to be strongly quenched by bulk charges. The electric field around an ion may overcome the exciton binding energy, resulting in dissociation followed by nonradiative recombination. In this case, Ncd-1/3 (10 nm for Ncd ) 1018 cm-3) might set an effective upper limit to the exciton diffusion length, Lex. This seems consistent with existing measurements of Lex in conducting polymers. Recently, chemical treatments designed to decrease Ncd were shown to substantially increase Lex (as well as the carrier mobility).14 We turn now to the subject of carrier injection. If pf were very low, as in the case of insulators, the measured dark currents in π-CPs would derive almost entirely from the carrier density injected by the electrodes, pinj. Space-charge-limited, SCL, currents would then result. Numerous studies have employed the SCL model to describe transport in π-CPs.5 The conditions under which SCL currents can occur are easily calculated. The maximum charge density that can be injected from a perfect contact is limited by the capacitance of the device, and this injected density must completely overwhelm the carrier density already present in the bulk to observe SCL currents
pinj ) Fεε0 /qd . pf
(4)
where d is the device thickness. If ε ) 4 and d ) 100 nm, pinj (cm-3) ) 2.2 × 1011F (V/cm). Thus, at moderate fields, SCL currents would require values of pf lower than any yet observed
Letters in π-CPs. Moreover, as discussed above, pf should increase approximately with exp(F1/2), which is substantially faster than the linear increase of pinj with F, making SCL currents also unlikely at higher fields. Experimentally, it is difficult to distinguish SCL currents (with a PF mobility) from true PF currents by the current-voltage behavior alone;13 eq 4 provides the clearest distinction. Nevertheless, for materials with very low values of pf, SCL currents can be observed and provide an accurate measure of the carrier mobility. As pf increases, the SCL model becomes first an approximation and then an increasingly poor approximation. A serious conceptual flaw remains, however; use of the SCL model when eq 4 is not satisfied ignores the effects of Ncd and pf, thus mistaking changes in these variables for changes in the mobility. For example, a beneficial decrease in Ncd and pf would be misinterpreted as a detrimental decrease in mobility. As with other insulator models, the SCL model may accurately describe only a small subset of existing π-CPs. Finally, we discuss some of the sources of charged defects in π-CPs. Chemical impurities will always be present, some of which may be charged. They may be classified into two types, those that are chemically bound to the polymer chain and thus probably cannot be removed and those that are not bound and can potentially be removed by precipitation, extraction, and so forth. More fundamental are the morphological defects. These may be classified into noncovalent defects, such as stacking faults, in which only low-energy bonds are perturbed, and covalent defects, in which high-energy covalent bonds are distorted.7 Because of the higher energies involved, covalent defects are more likely to be charged. The π-CPs have long (∼20-200 nm) backbones consisting of sp2-hybridized carbons. Significant mechanical stress may be applied to these long “wires” even by thermal fluctuations in solution, and far greater stresses occur during film deposition. In any morphology short of a single crystal, a significant number of these chains may contain covalent defects that distort an sp2 carbon from its equilibrium planar, trigonal geometry and thus introduce electronic states in the band gap.14 Some fraction of these morphological defects can be eliminated by careful deposition techniques and postdeposition anealling procedures. In summary, this paper describes the powerful influence of the charged defect density, Ncd, and the resulting free carrier density, pf, on the transport properties of π-conjugated polymers. Existing theoretical models based on insulators do not consider Ncd and pf and thus can accurately describe only those few π-CPs with very low conductivity. Some π-CPs exhibit an intermediate range of Ncd and pf values where their properties may be influenced by the charged defects but not wholly controlled by them. Finally, the apparently largest fraction of currently available π-CPs are those that, although not purposely doped,
J. Phys. Chem. C, Vol. 113, No. 15, 2009 5901 are in fact strongly doped by their defects. For these, models of doped semiconductors such as the Poole-Frenkel model or the doped Gaussian Disorder Model may be most appropriate. Any given material may exist in more than one group and thus require a different theoretical description, depending on its purity, deposition conditions, anealling processes, exposure to oxygen or water, and so forth. Routine measurements of Ncd and pf would help impose some order on the diversity of results obtained by different groups and promote a unified understanding of these versatile materials. Acknowledgment. This work was funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences under Contract No. DE-AC36-08GO28308 to NREL. References and Notes (1) Ba¨ssler, H. Phys. Status Solidi 1993, 175, 15. (2) Coropceanu, V.; Cornil, J.; da Silvo Filho, D. A.; Olivier, Y.; Silbey, R.; Bre´das, J.-L. Chem. ReV. 2007, 107, 926. (3) Kreouzis, T.; Poplavskyy, D.; Tuladhar, S. M.; Campoy-Quiles, M.; Nelson, J.; Campbell, A. J.; Bradley, D. D. C. Phys. ReV. B 2006, 73, 235201. (4) (a) Jenekhe, S., Chem. Mater. 2004, 16, (23). (b) Gu¨nes, S.; Neugebauer, H.; Sariciftci, N. S. Chem. ReV. 2007, 107, 1324. (5) Blom, P. W. M.; Mihailetchi, V. D.; Koster, L. J. A.; Markov, D. E. AdV. Mater. 2007, 19, 1551. (6) (a) Gregg, B. A. J. Phys. Chem. B 2003, 107, 4688. (b) Gregg, B. A.; Hanna, M. C. J. Appl. Phys. 2003, 93 (6), 3605. (7) (a) Gregg, B. A. J. Phys. Chem. B 2004, 108, 17285. (b) Gregg, B. A.; Chen, S.-G.; Cormier, R. A. Chem. Mater. 2004, 16 (23), 4586. (8) (a) Debye, P. P.; Conwell, E. M. Phys. ReV. 1954, 93 (4), 693. (b) Pearson, G. L.; Bardeen, J. Phys. ReV. 1949, 75 (5), 865. (9) (a) Novikov, S. V.; Dunlap, D. H.; Kenkre, V. M.; Parris, P. E.; Vannikov, A. V. Phys. ReV. Lett. 1998, 81 (20), 4472. (b) Rakhmanova, S. V.; Conwell, E. M Appl. Phys. Lett. 2000, 76, 3822. (c) Arkhipov, V. I.; Heremans, P.; Emilianova, E. V.; Adriaenssens, G. J.; Ba¨ssler, H. Appl. Phys. Lett. 2003, 82, 3245. (10) Arkhipov, V. I.; Heremans, P.; Emilianova, E. V.; Ba¨ssler, H. Phys. ReV. B 2005, 71, 045214. (11) Loutfy, R. O.; Hor, A. M.; Kazmaier, P.; Tam, M. J. Imaging Sci. 1989, 33 (5), 151. (12) (a) Dicker, G.; de Haas, M. P.; Warman, J. M.; de Leeuw, D. M.; Siebbeles, L. D. A. J. Phys. Chem. B 2004, 108, 17818. (b) Jain, S. C.; Geens, W.; Mehra, A.; Kumar, V.; Aernouts, T.; Poortmans, J.; Mertens, R.; Willander, M. J. Appl. Phys. 2001, 89, 3804. (c) Mozer, A. J.; Saraciftci, ¨ sterbacka, R.; Westerling, M.; Juska, N. S.; Lutsen, L.; Vanderzande, D.; O G. Appl. Phys. Lett. 2005, 86, 112104. (d) Mozer, A. J.; Sariciftci, N. S.; ¨ sterbacka, R.; Juska, G.; Brassat, L.; Ba¨ssler, H. Phys. ReV. Pivrikas, A.; O B. 2005, 71, 035214. (13) Gregg, B. A.; Gledhill, S. E.; Scott, B. J. Appl. Phys. 2006, 99, 116104. (14) Wang, D.; Reese, M. O.; Kopidakis, N.; Gregg, B. A. Chem. Mater. 2008, 20, 6307. (15) Novikov, S. V. Phys. Status Solidi 2003, 236, 119. (16) Frenkel, J. Phys. ReV. 1938, 54, 647. Simmons, J. G. Phys. ReV. 1967, 155, 657.
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