© Copyright 2008 by the American Chemical Society
VOLUME 112, NUMBER 19, MAY 15, 2008
REVIEW ARTICLE The Contribution of Energetic Disorder to Charge Transport in Molecularly Doped Polymers L. B. Schein* Independent Consultant, 7026 Calcaterra Dr., San Jose, California 95120, USA
Andrey Tyutnev£ Moscow State Institute of Electronics and Mathematics, TrekhsVyatitel’skii per. 3/12, Moscow, 109028, Russia ReceiVed: February 4, 2008; In Final Form: March 5, 2008
Energetic disorder plays a central role in many of the theories of charge transport in molecularly doped polymers, MDP. In these theories, energetic disorder determines the energy width of the hopping site manifold. It is generally assumed that the disorder-created energy width arises from interactions between the charge on a dopant molecule and its environment, including the dipole moments of neighboring dopant molecules, the dipole moments of the molecules of the polymer matrix, and van der Waals interactions with neighboring molecules. This assumption can be tested experimentally. Bassler’s formulation of the Gaussian Disorder Model predicts that the energy width can be obtained from the temperature dependence of the mobility: it appears as an activation energy. A review of experimentally characterized MDP systems reveals that there is almost no experimental evidence to support the prediction that the experimentally observed activation energy is due to energetic disorder. Instead, the data appear to suggest that the activation energy is primarily intramolecular in origin.
1. Introduction Molecularly doped polymers are films which consist of a mixture of an electron-donor or -acceptor molecule in a host polymer such as polystyrene PS or polycarbonate PC. Charge transport through these films occurs by charge transfer between neighboring donor or acceptor molecules. This transfer process is assumed to be a hopping mechanism. Attempts to elucidate the hopping mechanism have been the subject of many papers, including this one. There has been considerable interest in the charge-transport properties of these materials because of their use in many technologies such as photoreceptors in electrophotography (the copier and laser printer technology),1 and electroluminescent,2,3 photorefractive,4 photovoltaic,5-7 and transistor8,9 devices. * To whom correspondence should be addressed. Phone and fax: 408997-7946. E-mail:
[email protected]. £ E-mail:
[email protected].
These materials are also scientifically interesting because virtually all of the parameters that can affect the hopping mechanism can be varied. These include the electric field, temperature, the nature of the hopping site (the donor or acceptor dopant molecule), the distance between hopping sites (the concentration of the dopant molecules), and the polymer matrix. With such a rich experimental palette, one might have expected that the physics and chemistry of the hopping process in molecularly doped polymers would have been elucidated by now. However, as will be seen below, several charge-transport models have been suggested, but general agreement among workers has not been achieved. Perhaps the most quoted charge-transport model for these materials is the Gaussian Disorder Model GDM as formulated by Bassler.10 In this model, the electron wave function is assumed to be localized on one dopant molecule, and hopping occurs to neighboring dopant molecules which have their
10.1021/jp801049a CCC: $40.75 © 2008 American Chemical Society Published on Web 04/22/2008
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Lawrence B. Schein received his Ph.D. in experimental solid-state physics from the University of Illinois in 1970. He worked at the Xerox Corporation from 1970 to 1983 and at the IBM Corporation from 1983 to 1994. He is now an independent consultant. He has spent his career working on the physics of electrophotography and contributed to our understanding of toner charging, toner adhesion, and charge-transport mechanisms in organic materials including molecular crystals and molecularly doped polymers. He is the author of “Electrophotography and Development Physics”, a Fellow of the American Physical Society, a Fellow of the Society of Imaging Science and Technology, recipient of the Carlson Memorial Award in 1993, a Senior Member of the IEEE, and a member of the Electrostatics Society of America.
Andrey P. Tyutnev received his Master’s degree in Radiation Chemistry from the Institute of Electrochemistry in 1977 and Ph.D. in Electrophysics and Physical Chemistry from the Institute of Chemical Physics in 1987 (both institutes of the Russian Academy of Sciences, Moscow). He worked at the Moscow Institute of Electromechanics from 1969 until now and at the Moscow Institute of Electronics and Mathematics (part time) from 1997. He spent his career working on radiation effects in polymer and ceramic insulators, studying the radiation-induced conductivity and bulk charging in them. The last 10 years, he studied charge carrier transport in molecularly doped polymers using a novel radiation-induced variety of the time-of-flight method. He is the coauthor of 4 books on the subject (all in Russian).
early to account for MDP charge-transport data energies distributed independently and within a Gaussian density of states (DOS). In this model, it is assumed that the hopping rates are described by an expression of Miller and Abrahams (which assumes that it takes a Boltzmann factor for upward jumps in energy and no activation energy to jump downward), and electron-phonon coupling is sufficiently weak that polaronic effects can be neglected. The density of states of the hopping manifold in the GDM is assumed to be a Gaussian, that is, of the form exp(-2/2σ2), where is the energy relative to the center of the density of states and σ is its Gaussian width. The width σ is a key parameter of the GDM, which can be compared directly with experimental data by studying the temperature dependence of the mobility. By extensive simulations10 with the GDM, it has been predicted that the activation energy of the mobility, -k∂ln(µ(0))/∂(1/T) should be temperature dependent, 2(2σ/3)2/kT, where k is the Boltzmann’s constant, T is absolute temperature, and µ(0) is the mobility extrapolated to zero electric field. The full expression for the mobility,1,10,11 obtained by simulation, is
µ ) µ0 exp[-(2σ/3kT)2] × exp{CE1/2[(σ/kT)2 - Σ2]}
Σ > 1.5 (1)
with Σ2 replaced with 2.5 when Σ < 1.5 for E, the electric field, greater than approximately 4 × 105 V/cm. Σ is a measure of positional disorder (as compared to the energetic disorder, symbolized by σ, which is the focus of this paper), and C is a constant. Note that this expression, in common to all known hopping theories, has the form12
µ ) µ0F2 exp[f1(F)]exp[f2(T,F)]exp[f3(E,T,F)]
(2)
where f3 describes the electric field dependence, f2 describes the temperature dependence, and f1 is the prefactor, which usually includes the overlap integral. The experimental variables, the electric field E, the temperature T, and the distance between hopping sites F, are explicitly included in eq 2. The equation also describes Gill’s13 empirical equation, which was suggested
µ)
[ (
µ0 exp -∆(F)
)] [ (
1 1 1 1 exp βE1/2 kT kT0(F) kT kT0(F)
)]
(3)
where β is a constant, ∆ is the activation energy, and T0, which has been shown to depend on F, is a parameter that was added to account for the fact that at an extrapolated finite temperature, it was observed that the electric field dependence vanished. In analyzing data with the GDM (eq 1), note that plotting ln(µ) versus T-2 at finite electric field convolutes the electric field dependence with the activation energy. Only by plotting ln[µ(0)] versus T-2, where µ(0) is the extrapolated zero electric field mobility, can σ2 be determined because f3 vanishes and the exponential term in eq 1 with the electric field dependence goes to 1. Only such data is considered in this review. In general, with any hopping theory (eq 2), plotting ln[µ(0)] is the only way to obtain the activation energy without convolution. For example, with Gill’s equation, plotting ln[µ(0)] versus T-1 eliminates any convolution of electric field dependence with the activation energy ∆. (In addition, only data in which the mobility is nondispersive, i.e., the transit time is obtained from linear time-current curves, are considered in this review.) The focus of this review is a study of the effect of energetic disorder on the experimentally observed activation energies in published MDP systems. It is argued in the GDM1,10 that three sources of energetic disorder contribute to σ, a dipolar component due to the dipole moments of the dopant molecules in the neighborhood of the charged molecule, σd, a polymer dipole matrix component, σp, and a van der Waals component, σvdw, due to the polarization of neighboring molecules. The argument is made that the electrostatic interaction of the moving charge with a random distribution of dipoles (due to the dopant and the polymer matrix) adds to the local variations of the potentials resulting from van der Waals interactions. It is assumed that these add as
σ2 ) σ2d + σ2p + σ2vdw
(4)
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This formula is valid when the underlying distributions are independent Gaussians or, in keeping with the central limit theorem, when each hopping site energy is determined in a manner that constitutes a sum over a large number of independent random variables. However, it may not be a good approximation for a small number of non-Gaussian distributions, especially for large sigma or low temperatures, when the tails of the distributions are controlling the physics of the process. In this paper, this formula will be used. Both Dieckmann14 and Young15 have related the dipolar disorder σd, the part of σ due to the charge-dipolar interaction, to the concentration of dipole moments of the dopant molecules and their dipole moment. These two formulas are similar and both make intuitively reasonable predictions of the dependence of σd on dipole moment and concentration. In this paper and in many of Borsenberger’s later papers, Young’s expression is used
σd ) 7.04c0.5 p/(δ2)
(5)
where σd is in eV, c is the fraction of sites occupied by randomly oriented dipoles with dipole moment p (in Debye), δ is the minimum intersite distance (in angstroms) between molecules in the material, and is the relative dielectric constant. In this report, we will approximate c by using the fractional mass concentration (in common with Borsenberger). Instead of concentration, experimentalist sometimes calculate F, the average spacing between dipoles, by the lattice gas model, F ) (M/ (AFmc))1/3, where M is the molecular weight, A is Avogadro’s number, and Fm is the mass density. As far as the authors know, no theoretical analysis of σp, the charge-dipole interaction with the molecules of the polymer matrix, exists. We suggest using eq 5 with a simple change: replacing c with (1 - c) to keep the identity of c as the concentration of the dopant molecule
σp ) 7.04(1 - c)0.5 p/(δ2)
(6)
This allows a quantitative estimate of σp. For polystyrene (p ) 0.4 Debye), the maximum value of σp occurs at low concentration of the dopants. Assuming that c ) 0.2, δ ) 8 Å, and ) 3 gives σp equal to 0.013 eV, much smaller than the other components, as will be seen. As far as the authors know, no theoretical analysis of σvdw, the van der Waals component of the energetic disorder due to the neighboring molecules, exists in the MDP literature. This may be because quantum chemical calculations are usually not accurate enough to predict energy differences on the order of 0.1 eV.10 An estimate of the fluctuations due to the intermolecular electronic polarization has been suggested to be between 0.1 and 0.5 eV.16 In the MDP literature, σvdw has been discussed in section 1 of Bassler’s review.10 The σvdw has been ascribed to charge-induced dipole interactions (molecular polarization), which are known to decay much faster (r-4) than the chargedipole interaction (r-2), where r is the distance from the charge. In ref 17, “the occupancy of a hopping site by an excess charge induces a displacement in the electronic orbitals of the nearby molecules. These displacements create a charge-induced dipolar cloud that surrounds the hopping site. The dipolar cloud is determined by the polarizability of the surrounding molecules and the separation distance between these molecules and the occupied hopping site. The fluctuations of the charge-induced dipolar cloud are the physical origin of the van der Waals contribution to the DOS,” that is, the Gaussian density of states. One would intuitively expect the range of this interaction to be much smaller than the range of the charge-dipole interactions
because of the r dependence. As will be seen, the experimentally determined σvdw increases with F. This is rationalized in several experimental papers as follows.18 “The increase in the van der Waals component with increasing intersite distance is not predicted by first-order arguments and possibly reflects an increase in structural randomness upon dilution.” The present authors suggest another possibility: the changing ratio of the dopant and polymer matrix molecules as the calculated F changes. It is now well-known that the GDM, as formulated by Bassler and co-workers, cannot account for the observed electric field dependence of the mobility at low electric fields, which has been shown to be exponential in the square root of the electric field over a very wide range of electric fields.11,19 In order to address this failure, the correlated disorder models (CDM)20-24 have been suggested, which take into account significant longrange correlations in the charge-dipole interactions. It was found that the energy correlation function for charge-dipole interactions decays only as 1/r, where r is the distance from the charge. The consequence is that in disordered dipolar materials, the energies of nearby hopping sites tend to be similar, which is ignored in the GDM. This similarity in hopping energies extends the range over which the electric field affects the hopping rates, leading to stronger electric field effects at lower field, in better agreement with experimental data and just in the region in which the GDM cannot account for the experimentally observed field dependence of the mobility.19 Indeed, the effect of the spatially correlated energies on transport have been determined analytically for one-dimensional transport, with the finding that, for algebraic correlations of the form r-η, it can be shown that the mobility will increase exponentially with the field as exp(Eη/(η+1)). For the case of the charge-dipole (η ) 1) interactions, the mobility is predicted to obey the PooleFrenkel law, in agreement with experiment.21 The physical reason that such behavior arises from the correlations has been discussed at length, and the basic picture which emerges suggests that this is not simply limited to one dimension.22,23 The results of Monte Carlo simulations with charge-dipole disorder in three dimensions are also in good agreement with the Poole-Frenkel law over the field range of interest.24 In keeping with the original analysis of the GDM, these simulation results have also been fit to empirical formulas. The CDM parametrizations will be discussed further in the Discussion section. Polaron theory has been suggested as a possible model for hopping in molecularly doped polymers.25,26 This model assumes that the disorder, which is obviously present in these materials, does not significantly affect the mobility. It assumes, opposite to the GDM, that the electron-phonon coupling is strong and the activation energy is dominated by polaron effects. The predicted mobility is27
µ ) (eF2/kT)P
sinh(eEF/2kT) ω exp[-(Ep/2 - J)/kT] 2π (eEF/2kT)
(7)
which has the same structure as eq 2. Here, e is the electronic charge, J is the overlap integral, P is the probability of charge transfer given an energy coincidence, and ω is the phonon frequency. The activation energy ∆ ) (Ep/2 - J) can be obtained by plotting the ln[µ(0)] versus T-1 (not T-2), as was usually done before the GDM was introduced. Within experimental error, it has been shown that if σ is constant, ∆ is constant; see refs 28 and 29 where both parameters are determined. This theory predicts an electric field dependence inconsistent with experimental data.30 Nevertheless, it is the only theory that can account for the experimental observation, which has now
7298 J. Phys. Chem. C, Vol. 112, No. 19, 2008 observed on four materials, TTA:PC,25 TPD:PC,26 PDA:PCZ,31 and DPH:PCZ,32 that there are some MDP systems in which the prefactor is independent of F. (The initials used to define these materials are given at the end of the article.) However, this result requires J to be larger than can be theoretically justified for a hopping theory (approximately 0.2 eV). There have been some attempts to distinguish the GDM and polaron theory by examining closely the temperature dependence of the log of the zero-field mobility to see whether T-1 or T-2 is a better fit. In many MDP systems, one cannot distinguish the difference due to experimental error; in others, if there is any difference, it is small and near the limits of experimental error. Recently, correlated disorder has been added to the polaron theory.33 Discussion of ref 33 is delayed until the Discussion section. Finally, we mention the dipole trap model, which assumes that the hopping rates are determined by dipole moments associated with the hopping sites.34,35 Similar to the polaron model, this model ignores disorder and the predicted electric field dependence is inconsistent with the data. The idea of a dipole trap was combined with the CDM in refs 36-38. As stated above, the main subject of this paper is an examination of experimental tests of the effect of energetic disorder on the experimentally observed values of σ. We examine three different types of experiments, which will be treated in their own sections in this review: molecularly doped polymers (MDP) in which the concentration of the dopant is changed (section 2A), the polymer matrix is changed (section 2B), and the dipole moment of the dopant molecule is changed (section 2C). Then, we consider MDP, in which the effect of the low concentration of polar additives in MDP is studied (section 3), and finally, we consider data from glasses made from typical dopant molecules (section 4). As will be seen, the conclusion is that there is very little evidence that energetic disorder is the source of the experimentally observed activation energies in MDP. 2. Molecular Doped Polymer Experimental Variations A. Dopant Concentration Varied. In these experiments, the polymer matrix and the dopant molecule are held fixed. The only experimental variable changed is the dopant concentration. If the concentration of dopant molecules with a dipole moment is changed, then the charge-dipole interaction is surely changed without changing other potentially important transport parameters, such as the shape or size of the dopant molecule. This is probably one of the most unambiguous experimental tests of the dependence of σexp on disorder. Consider the MDP system DNTA:PS. DNTA has the highest dipole of any MDP system so far characterized, 5.78 Debye. The equation for the GDM, eq 1, suggests that σ can be obtained from a plot of ln[µ(0)] versus T-2, where µ(0) is the mobility extrapolated to zero electric field. (To clearly distinguish in our discussions between the theoretical parameters such as σ and the experimentally derived parameters, we will add “exp” to the subscript, e.g., σexp.) The experimental observations of σ, σexp, as a function of DNTA concentration or F, calculated as discussed above, are shown in Figure 1. The σexp are observed to be independent of F.39 This data is analyzed as follows in ref 39: the dipolar component of σexp, σd, is calculated from eq 5 (shown in Figure 1). The polymer matrix component is assumed to be zero. The remainder is attributed to σvdw-exp, using eq 4, as shown in the Figure 1. (We again add “exp” to the subscript because this quantity is derived from experimental data.) The reader is immediately faced with a dilemma. Recall from the Introduction that there are no theoretical discussions of σvdw
Schein and Tyutnev
Figure 1. Data and analysis of σexp versus F (labeled a) in DTNA:PS. The data is the constant line labeled σ. The dipolar contribution σd is calculated from eq 5. The polymer component is assumed to be zero. The rest is assigned to σvdw using eq 4 from ref 39(Copyright WileyVCH Verlag GmbH & Co. KGaA. Reproduced with permission.)
in the MDP literature which would allow one to judge whether the values of σvdw-exp and their dependence on F derived from experimental data are reasonable. Indeed, in the authors’ judgment, it is very difficult to understand how such a shortrange interaction such as charge-induced dipole interactions, which is suppose to be the source of σvdw, can vary over such a large range of F, as seen in Figure 1. Quoting the authors, which is repeated in ref 17 and in other publications, as pointed out in the Introduction: “This increase in σvdw with increasing intersite distance, or decreasing dopant concentration, is not predicted from first-order arguments and is believed due to an increase in structural randomness on dilution.” In addition, there are at least two additional surprises (to the authors) in the calculated σvdw-exp curve. First, it is almost zero at high concentrations of dipoles when the dopant molecules are with high probability in contact with each other. The authors would expect σvdw-exp to be large due to interactions with the neighboring dopant molecules. Second, there is a coincidence evident in Figure 1: the van der Waals component and the dopant dipolar component exactly add to give σexp independent of F. Next, consider the MDP system DEASP in PS.29 DEASP has a dipole moment of 4.34 Debye. This system was characterized by Borsenberger and Schein in their respective laboratories, and identical results were obtained (see Figure 2). Within experimental error, σexp is independent of the concentration of DEASP (shown at the top of the graph) or F (shown at the bottom of the graph). A great deal of care was taken to ensure that these results are intrinsic to the material. The measurements were made in both laboratories independently using different sources of DEASP and completely different solvents. At IBM where Schein did his measurements, a nitrogen laser was used to photoexcite the charge carriers; at Kodak where Borsenberger did his work, a Se layer was used to inject holes into the MDP.
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Figure 2. Measurements of σexp for DEASP:PS and DEASP:PC as a function of dopant concentration (top) or the calculated distance between molecules (bottom). Note that σexp is independent of dopant concentration, within the experimental error of about (3% (from ref 29).
The exact same procedures and care were used in characterizing DEH:PS,28 with a dipole moment of 3.16 Debye, by the same authors. And the same results were obtained: σexp is independent of F. In order to understand these results in the context of the GDM, one needs to repeat the exact same arguments as those given for the DNTA:PS system. And once again, one is faced with rationalizing that σvdw-exp is required to be near zero at high dopant concentrations and to increase with F in order to account for this data. In addition, we now have three MDP systems (DNTA:PS, DEASP:PS, DEH:PS) in which the dopant dipolar and van der Waals component of energetic disorder coincidentally add to give σexp independent of F. This coincidental addition is clearly invoked in the GDM literature18 to explain the data in DEH and DEASP; “the dipolar dopant component and the van der Waals component may cancel, thus yielding a DOS that is independent of intersite distance and the polymer host, as observed in DEH and DEASP doped polymers.” A much simpler explanation of the independence of σexp on F is that σexp (or the activation energy) is not determined by the external environment of the dopant molecule, that is, the activation energy is intramolecular in origin.28,29 One wonders how general this result is. We have surveyed the literature for MDP systems in which the dopant molecule’s concentration has been varied and σexp is observed to be independent of F. These systems are shown in Table 1 for both hole and electron transport, along with the dopant molecule’s dipole moment and the observed σexp. The concentration was typically varied from approximately 10 to 70% in these measurements; the exact values can be found in the references (see, for example, Figures 1 and 2). There are 26 such MDP systems; therefore, 26 coincidental additions of the dopant dipolar and van der Waals components are required to understand these data in the context of the GDM. If we consider the large variation of the dopant dipole moment shown in Table 1 and use eq 5, another strange property of σvdw-exp emerges. Note that among the materials listed in Table 1, the dipole moment varies by a factor of 14.5 (5.78/0.4). According to eq 5, σd should vary by this amount. However, note that the experimentally observed σexp hardly varies at all (from 0.15 to 0.086 eV, a factor of 1.7). Using the GDM procedure, eq 4, it must be that σvdw-exp accounts for the rest
J. Phys. Chem. C, Vol. 112, No. 19, 2008 7299 of σexp, that is, σvdw-exp depends strongly on dipole moment, which is internally inconsistent with the definition of σvdw. In some systems, σexp is observed to increase as c decreases (F increases), as shown in Table 2. Again, the range of concentration over which the data is taken is about 10-70%. (The values of σ or ∆ are listed in the Table 2 from high to low concentration.) Shown in Figure 4 are data for TTA:PS (called PS-1 in this figure). The effects of a change in polymer, also shown in the figure, will be discussed in section 2B. In the paper in which these data were published,18 it is argued that the rising σ with F for TTA:PS-1 can be accounted for by assuming that σexp is due entirely to van der Waals component of σ, that is, the dopant dipolar contribution can be ignored. (This is only an approximation since, by using eq 5, it can be shown that the dipolar component is about 30% of the value of the σexp at high concentrations. Inclusion of this component would make the calculated dependence of σvdw-exp on F steeper. A similar assumption is made, and a similar calculation can be made for TAA-A:PS, ENA-B:PS, and TPA:PS.) The identification of σexp in these MDP systems with σvdw is consistent with prior arguments in papers on the GDM applied to MDP systems listed in Table 1 in which σvdw-exp is calculated to be the same order of magnitude as σd and increases with F. However, it does not address the question of whether these properties of σvdw-exp are reasonable. In particular, it does not address whether it is reasonable for σvdw-exp to increase as F increases, given the short range of its interaction. One might argue that a rising σvdw with F is due to the environment around the charged site changing due to “structural randomness”18 or the ratio of the dopant and polymer molecules changing (see Introduction). However, then is it reasonable that σvdw always increases with F? Would one expect that for some reasonable fraction of the MDP systems σvdw-exp would decrease with F? Consider the only other alternative explanation that the authors can conceive of: that the dipolar component is significantly overestimated by eq 5, that is, the van der Waals component dominates the disorder for all of the experiments listed in Tables 1 and 2. One might then interpret Table 1 data as being totally due to van der Waals energetic disorder. This would be strong evidence (26 times) that σvdw-exp is independent of F. However, then is it reasonable that σvdw-exp increases with F in some MDP systems, listed in Table 2? That is, if “structural randomness”18 or the ratio of the dopant and polymer molecules changing is invoked to explain the data in Table 2, why does it not apply to the systems in Table 1, and why is σexp not observed to decrease with F for some fraction of the MDP systems? Note that within the GDM model, all of the σexp data in Tables 1 and 2 are explained in terms of a σvdw which increases with F. In Table 1, the σexp data, which are observed to be independent of F, are accounted for by adding a σvdw that increases with F to a σd that decreases with F. The data σexp in Table 2 always increase; this is explained by noting that σd is small since p is small. The σexp data is assumed to be due primarily to σvdw. Therefore, within the GDM, σexp for lowdipole-moment dopants is expected to be in our Table 2. It must have been quite a surprise in 1995 and 1996 to observe two materials which violated this trend. σexp is constant for TASB: PS.47 TASB has a low dipole moment, 0.54 Debye. In ref 47, it is argued that the dipolar part of the energetic disorder is negligible (based on eq 5); therefore, the observed σexp must be due entirely to the van der Waals energetic disorder: “The absence of a concentration dependence suggest that the van der Waals component is mainly due to an intramolecular interac-
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TABLE 1: A List of MDP Systems in Which It Is Experimentally Observed That σexp or ∆exp Is Independent of Dopant Molecule Concentration within Experimental Errora dipole moment of dopant (D) Hole Transport 1. DNTA:PS (see Figure 1) 2. DEASP:PS (see Figure 2) 3. DEASP:PC (see Figure 2) 4. DEH:PS 5. DEH:PC 6. DEH-A:PC 7. DEH-B:PC 8. DEH-C:PC 9. TPM-E:PS (see Figure 3) 10. TPA-4 ) TAA-4:PS 11. TPA-3 ) TAA-3:PS 12. TPA-2 ) TAA-2:PS 13. TPM-D:PS (see Figure 3) 14. TPM-C:PS (see Figure 3) 15. TPM-B:PS (see Figure 3) 16. PDA:MBDQ 17. TPM ) TPM-A)MPMP:PS (Figure 3) 18. TPM ) TPM-A:PC 19. TPA:PC 20. TASB:PS 21. DOA:PMPS Electron Transport 22. DCAQ:PS 23. DCAQ:PC 24. PTS:PC 25. MBDQ:PC-Z 26. DPQ:PS
σexp (eV)
5.78 4.34 4.34 3.16 3.16 2.67 2.5 2.48 2.1 2.1 2.1 2.0 1.81 1.7 1.51 ≈1.4 1.33 1.33 0.8 0.54 ≈0.5
0.15 0.11 ( 3% 0.11 0.13 ( 4% 0.13 0.13 0.13 0.13 0.110 0.115 0.110 0.110 0.110 0.110 0.108
∆exp (eV)
0.61 0.61 0.60 0.60 0.60 0.60 0.60
0.32 0.106 0.126 ( 0.10 ≈0.095 0.103 0.086
3.3 3.3 2.2 0.5 0.4
0.30
0.132 0.132 0.134 0.49 0.118
reference (location in ref 1) 39 (p 414) 29 (p 473) 29 (p 473) 28, 40 28 (Figure 46) 41 41 41 42 (Figure 10) 43, 44 (Table 7) 43, 44 (Table 7) 43, 44 (Table 7) 42 42 42 31 (Figure 27) 42 (Figure 10) 45 (Figure 1) 46 (p 402) 47 (p 393) 48 (Figure 90) 49 (p 539) 49 (p 539) 50 (p 556) 51 52 (p 543)
a Usually the dopant concentration is varied from 10 to 70%; the exact values can be found in the references (see, for example, Figures 1 and 2). When several different initials have been used in the literature to indicate the same material, the initials are indicated with ) signs. The references include the original reference followed by the page or figure number where the experiment is discussed in Chapter 8 of ref 1. The σexp versus F plots for the various TPM’s are shown in Figure 3 (data taken from ref 25). The value of σexp for TPA:PC (0.095 eV) was estimated using eq 10 in ref 10. The dipole moments with a ≈ sign were estimated by the present authors.
TABLE 2: List of MDP Systems in Which σexp or ∆ Is Observed to Increase as G Increasesa dipole moment of dopant (D) Hole Transport 1. TAA-A:PS 2. DPH:PC-Z 3. TPD:PC 4. TPD:PS 5. ETPD:PC 6. ETPD:PS 7. PDA:PC-Z 8. TAPC:PC 9. TAPC:PS 10. ENA-B:PS 11. ENA-C:PC 12. TTA:PC (Figure 4) 13. TTA ) TPA-1 ) TAA-1:PS (see Figure 4) 14. TTA:PS-2(see Figure 4) 15. TTA:PS-3 (see Figure 4) 16. ENA-A:PC 17. ENA-D:PC
σexp (eV)
2.1 ≈2 1.52 1.52 1.5 1.5 ≈1.4 1.0 1.0 0.86 0.86 0.8 0.8
0.090-0.136 0.067-0.080 0.077-0.097 0.080-0.096 0.110-0.140 0.075-0.116
0.8 0.8 0.66 0.38
0.079-0.108 0.106-0.137 0.079-0.104 0.078-0.100
∆exp (eV)
0.105-.115 0.45-0.52 0.33-0.55 0.22-0.40 0.32-0.45 0.18-0.33 0.35-0.60
0.40-0.60
reference (location in ref 1) 17 (Figure 14) 32 (p 422) 26, 53, 54, (Figure 46, p 395) 54 (p 395) 54 (p 395) 54 (p 395) 31 (Figure 27) 55, 56 (Figure 31) 57 58, 59 (Figure 63) 58 18, 25 (Figure 54, 56) 18, 56, 43, 44 (Figure 56, Table 7) 18 (Figure 56) 18 (Figure 56) 58 58
a The range over which the data is taken is about a 10-70% concentration change. The values of σ or ∆ are listed from high to low concentration in the table. The precise range can be found in the references.
tion...” It is not clear to the present authors what this means. DPQ has an even lower dipole moment, 0.4 Debye. In ref 52, it is argued that the group moment associated with the dopant molecules and not the molecular moments should be considered. As mentioned in the Introduction, polaron theory has also been suggested as a hopping mechanism for MDP. Unfortu-
nately, quantitatively, the required J is larger than can be rationalized within the context of a hopping theory, approximately 0.2 eV. While the magnitude of the J obtained from experimental data may be a challenge within the context of polaron theory, it does suggest a theoretical approach to the data: the activation energy is primarily intramolecular in origin,
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Figure 3. The σexp for various types of TPM-doped PS. Typical experimental error bars of (3% are shown. The curve without data points is the predicted curve for TPM-F using eq 5 and the data point at F ) 11 Å (55% concentration). Data replotted from ref 42.
Figure 4. The σexp versus concentration of TTA in various types of PS and PC polymers; a is the same as F (from ref 18 reprinted with permission from Elsevier).
but in some MDP systems the activation energy is reduced by some interaction energy which decreases as the distance between the dopant molecules increases. The theoretical task would be to identify this interaction energy. Finally, we come to the one system, TPM-F:PS in which σexp is observed to decrease as F increases42 by an amount that is outside of experimental error, as shown in Figure 3. This behavior is unique in MDP systems. As we have seen, in all other characterized MDP systems, σexp is either constant or increases. The observation of a σexp that decreases with F is qualitatively consistent with the GDM if σd is larger than σvdw. If the dopant dipolar disorder dominates σexp in this experiment, then the change from 55 to 10% concentration predicts a change of σd of a factor of 2.34 (square root of the ratio of the change in concentration 55:10). As shown in Figure 3, this predicted change is much more than is experimentally observed, 14%. Therefore, σvdw must contribute to σexp as in all other MDP systems, as argued within the GDM. Therefore, while the data is unique, its explanation within the GDM is the same as that in the other MDP systems: σexp is due to a combination of dopant dipolar and van der Waals energetic disorder. Since this is the only data in which σexp decreases with F among all of the
J. Phys. Chem. C, Vol. 112, No. 19, 2008 7301 MDP systems characterized so far (see Table 1 and 2), it is recommended by the authors that it be repeated to be sure of its validity. As this data has not been verified, the results are ignored in the discussion. B. MDP Polymer Matrix Changed. The change of polymer matrix does affect the mobility. To the authors’ knowledge, in every MDP system, the change from PS to PC lowers the mobility. Two types of dependences of the mobility’s σexp on activation energy are observed, as illustrated in Figures 2 and 4 The first type of dependence of σexp on the polymer matrix is shown in Figure 2, which shows the results for DEASP:PS and DEASP:PC29 as representative of Table 1 data. The σexp is independent of the polymer matrix within experimental error. Quantitative predictions for σ (DEASP:PS) can be made based on the values of σ (DEASP:PC) and the calculated σp’s for PC and PS using eqs 4 and 6
σ2(DEASP:PC) - σ2(DEASP:PS) ) σ2p(PC) - σ2p(PS)
(8)
Assuming σ(DEASP:PC) ) 0.11 eV, c ) 1, p(PC) ) 1 Debye, p(PS) ) 0.4 Debye, and δ ) 8 Å, this is easily solved for σ(DEASP:PS). The prediction is 0.105 eV, close enough to the observed value of 0.11 eV to be within the experimental error of these experiments. This result is also obtained for DEH.28 For Table 1 MDP systems, the change from PC to PS does not affect the electric field dependence.28,29 What the change in polymer affects is the prefactor. Assuming the prefactor has an exponential form, it affects the F0 constant.28,29 Figure 4 (taken from ref 26) shows the second type of the dependence of σexp of the polymer matrix: σexp changes when the polymer matrix is changed (representative of data in Table 2; see also ref 60). In this figure, the MDP system is TTA (0.8 Debye) doped into PS (called PS-1, with a dipole moment of 0.4 Debye) and polystyrene variants PS-2 (0.4 Debye), PS-3 (1.7 Debye), and PC (1.0 Debye). Note that σexp increases with F for all of these systems. Recall that in the analysis of this system by the GDM, σexp for TTA:PS-1 are interpreted as due to σvdw (see section 2A). Accepting for the point of discussion that TTA:PS-1 and TTA:PS-2 are due entirely to the van der Waals component, that implies the polymer dipolar component is approximately independent of F for the TTA:PC and TTA: PS-3 systems (the difference between the two curves). That result is inconsistent with eq 6. Note that PC and PS-3, which have higher dipole moments than PS-1 and PS-2, would be predicted by the GDM to have a higher value of σexp, which they do. However, note that the PC matrix (dipole moment of 1.0) should produce a lower value of σexp than PS-3 (dipole moment 1.7), which is not observed, inconsistent with eq 6. Even more difficulties are encountered if one attempts to compare the data quantitatively with eq 6. Accepting that the TTA:PS-1 curve is due entirely to σvdw, then one can estimate, using eqs 4 and 6, the contribution due to the change to the polymer matrix PC
σ2(TTA:PC) ) σ2p(PC) + σ2vdw(TTA:PS)
(9)
Using Figure 4 and eq 6 at 10% TTA concentration, which is a 90% polymer concentration, gives σvdw(TTA:PS) ) 0.1 eV and σp(PC) ) 0.033 eV. Equation 9 then predicts σ(TTA:PC) ) 0.105 eV, which is much smaller than the experimental observation shown in Figure 4 (0.133 eV). Whether the prefactor is affected by the change of polymer for the MDP systems listed in Table 2 (the second type of dependency) is not clear. From Table 1 of ref 18, the prefactor,
7302 J. Phys. Chem. C, Vol. 112, No. 19, 2008 µ0, is larger by a factor of 2 for PC than PS (PS-1) at 55% concentration. For TPD,53 the prefactor at a high concentration (1.14 mol/cc) is larger by about a factor of 100 for PS than PC (estimated from the data in ref 30). For ENA-B, the prefactor is larger for PS than that for PC at 45% concentration (from Table 8 in ref 1) by about a factor of 2. Independent of any particular hopping theory, we note, as discussed above regarding the MDP systems in Table 1, that the experimental result that σexp is independent of dopant concentration, as in Figure 2, suggests that σexp (or the activation energy, ∆exp) is intramolecular in origin. This suggests a prediction: if σexp is intramolecular in origin, then σexp should be independent of the polymer matrix. That is in fact observed, as can be seen in Figure 2, and this result was repeated in the DEH:PS and DEH:PC systems.28 In the MDP systems listed in Table 2, it is observed that σexp increases with F. This suggests that there is some interaction energy between the charge on a dopant molecule and the other dopant molecules in the system. It is not unreasonable to suggest that this interaction energy depends on the environment between the dopant molecules. One example is the potential energy barrier that a charge would experience as it transfers from one dopant molecule to the other, similar to the behavior of a transfer integral. Even without a detailed model of this interaction energy, one would still be able to predict that if one changes the environment (by, for example, changing from PS to PC), then σ should change, and it does, as can be seen in the Figure 4. The data of Figure 4 suggests that the dipole moment of the polymer matrix may not be the controlling feature of this energy interaction since PS-3 and PC have different dipole moments but basically the same σexp (or activation energy). Therefore, perhaps we have learned a little more about this interaction energy: it not only decreases as the distance between dopant molecules increases, but also, it depends on the polymer matrix. C. Dipole Moment of Dopant Changed. There are several sets of data published in which σexp is plotted versus the dipole moment of the dopant at fixed dopant concentration. This is another view of the data discussed in section 2A; instead of varying the distance between dipolar dopants, the dipole moment is presumably varied at a fixed separation distance. Three sets of data have been identified in the literature in which the dipole moment of the dopant is varied while the concentration of dopant is held fixed: variants of TPM molecules at 35% concentration in PS (ref 61 and page 360, Table 3 and 4 in ref 1), DEH variants at 45% concentration in PS,62 and TPA variants at various concentrations in PS (refs 43, 44, and 63 and Figure 50 and Tables 7 in ref 1). In plotting σexp versus dipole moment, it is implicitly assumed in making such a correlation that the only parameter varied is the dipole moment. However, consider the various molecules used in these studies (they are shown in the references.) It is clear from their structures that the size and shape of the molecules differ. Therefore, the packing in the polymer matrix almost surely differs among these molecules. Since it is the transfer of charge between adjacent molecules that governs charge transport, it appears that some of the potentially critical transport variables are changing as the molecule is changed. That is, trying to draw conclusions from a plot of σexp versus dipole moment ignores the fact that other, possibly important, transport variables are also changing. Equally problematic in making such plots is the implicit assumption that σvdw is unchanged as the molecules are changed. The data are analyzed in GDM by combining eqs 4 and 5 and ignoring the σp contribution (which is a probably a
Schein and Tyutnev
Figure 5. The σ2exp versus p2 for a series of molecules doped into PS at a concentration of 20%. The molecules are TPA ) TAA-1, TPD, TTB, TAA-2, TTA-3, and TPA-4 (from ref 44 reprinted with permission from Elsevier).
reasonable approximation when PS is used, as argued in the Introduction)
σ2 ) [7.04c0.5/(δ2)]2p2 + σ2vdw
(10)
In Figure 5 (Figure 7 from ref 44 and Figure 50 in ref 1) are shown the data for the σ2exp versus p2 for a series of molecules doped at a concentration of 20%. In refs 1 and 44, the molecules are labeled TAA-1 through TAA-4, which are the same as TPA-1 though TPA-4 in ref 43. In addition, the data in Figure 5 include data from TPD and TTB. The order in which the data are plotted from the lowest to highest value of σexp (σexp, the dipole moment, and the molecule) is 0.96 eV, 0.80 D (TAA1); 0.102 eV, 1.52D (TPD); 0.102, 1.56D, (TTB); 0.110 eV, 2.0D (TAA-2); 0.110 eV, 2.1 D (TTA-3); 0.115 eV, 2.60 (TPA4). As can be seen, the range of σexp is quite small, (10%, almost within the range of experimental error of (3%. Note the linearity of line in Figure 5, which is taken as agreement with eq 10. Note also that the p2 ) 0 intercept is quite large, almost equal to the experimental values of σexp. This is taken to represent the σvdw contribution to σexp. However, note that there are no discussions of why σvdw-exp should be constant for all of these diverse molecules; indeed, as pointed out above, with the GDM’s method of determining σvdw-exp, (subtracting quadratically σd from σexp), σvdw-exp actually depends on the dipole moment. In ref 61, it is argued that differences in the van der Waals component can be due to charge delocalization, which should vary among these molecules. In ref 43, this result is extended to various concentrations of TPA-1 through TPA-4 molecules in PS between 5 and 50%. (Note that TTA ) TPA-1 ) TAA-1.) It is found that the data can be accounted for if σvdw increases with F. This is shown in Figure 6 of ref 43 where σvdw goes from 0.064 to 0.116 eV for a 50-5% change in concentration or a 9.5-21 Å change in F. These results are consistent with the GDM analysis of the data in section 1A; to explain the data, σvdw must increase with F. There is another concern about this data analysis: it is easy to choose MDP systems from Tables 1 and 2 and produce plots of σexp versus p that are flat or even decrease as p increases (TASB:PS has p ) 0.54 Debye and σexp) 0.103 eV; TPA:PC has p ) 0.8 Debye and σexp)0.095 eV, TAPC:PS has p ) 1
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J. Phys. Chem. C, Vol. 112, No. 19, 2008 7303
Debye and σexp )0.067 eV at high concentrations). Therefore, even the linear increase of σ2 with p2 seen in Figure 5 is not a fundamental property but is determined in part simply by the choice of which MDP systems to plot. Surely, dipole moment plays some role in determining the mobility of MDPs, but this type of data analysis is not helpful in determining the role of energetic disorder in σexp, in the authors’ opinion. 3. Polar additives in MDP Another type of experiment that probes the effect of dipole disorder on the charge-transport properties of MDP is the addition of polar additives to MDP. This experiment was proposed in 1988 by Vannikov et al.,65 who suggested adding small concentrations (0-6 wt %) of polar molecules which do not act as ordinary hole traps because their ionization energy is higher than that of the dopant molecule. The idea behind these experiments is to change the dipolar disorder which is introduced by the random dipoles of the additive molecules. Many experimental papers were written on this subject.54,64-71 There is general experimental agreement that the addition of polar additives depresses the mobility. We focus our discussion on those papers in which the decreased mobility is experimentally characterized to determine whether the activation energy is changed by plotting ln[µ(0)], the extrapolated zero field mobility, versus T-1 or T-2. Borsenberger and Bassler68 discussed the effect of polar additives in MDP in terms of the GDM formalism. The concept is that the original dipolar disorder should be increased by the introduction of the random dipoles of the additive molecules. They studied the TAPC (1 Debye) doped PC (75 wt %) with DNB additives (up to 2 wt %). At maximum loading effect of p- and m- or o-DNB derivatives, the mobility was depression by 3, 20, and 70 times, respectively, at an electric field of 4 × 104 V/cm. The dipole moments of these molecules are 0.5 Debye (p-DNB), 4 Debye (m-DNB), and 6 Debye (o-DNB). An unexpected result (for the GDM) was that despite the substantial increase of dipolar disorder, σexp proved to be almost independent of the dipole moment of the DNB (see Figure 6). The authors conceded that this result attested to the fact that the dipole moments of the additives (or even surrounding molecules) do not affect σexp. Stasiak and Storch72 used a different approach. Instead of adding dipole additives to a MDP, they created them by UV irradiation in 40% DEH:PC. UV irradiation converts DEH (3.16 Debye) to IND, which has a dipole moment of 2.1 Debye and a higher ionization potential than that of DEH. The IND does not act as a trap; instead, it serves to dilute the DEH (effectively increasing F) and creates an environment with the lower dipole neighbors. To ensure uniformity, the samples were annealed after irradiation. The maximum amount of irradiation corresponded to an effective decrease in concentration of DEH from 41 to 21% measured by UV absorption. This means that the concentration of new dipoles (with p ) 2.1 Debye) is 20%, far higher than that used in ref 47, discussed above (2%). It was observed that the mobility decreased by a factor of 300, much higher than could be accounted for by simple dilution of the DEH molecules (a factor of 15.4, estimated by assuming a prefactor of F2 exp(-2F/F0), using F0 ) 1.7 Å from ref 28 and taking the change in F upon dilution of 10.9 to 13.6 Å from ref 72). Even more striking, the activation energy or σexp was unchanged. This agrees with the Borsenberger and Bassler68 results. The authors72 state, “The observation that the energetic disorder is independent of both the intersite spacing and the
Figure 6. Temperature dependence of hole mobilities in TAPC:PC with various DNB derivative molecules with different dipole moments. The dipole moment of p-DNB is 0.5 Debye, that of m-DNB is 4 Debye, and o-DNB is 6 Debye. Equal slopes of the fitted straight lines indicate equal values of σexp (from ref 68 Reproduced with permission).
electronic background is inconsistent with the disorder model. A better interpretation of the experimental observations is that the hopping process for certain classes of molecules is controlled by intramolecular interactions.” 4. Glasses Some of these donor and acceptor molecules can be made into glasses, that is, 100% concentration. The technical problem that needs to be overcome in order to fabricate these materials at 100% concentration is that they crystallize immediately at room temperature when fabricated normally from solvents. This was overcome by evaporating the molecules in vacuum onto a substrate that was attached to a liquid-N2-cooled stage. It is stated that such samples were amorphous with no indication of crystallization over a period of several weeks. (The test for crystallization was not described.) Note that these materials are prepared very differently than MDP, which are coated from a solvent at room temperature and then the solvent is allowed to evaporate. There is no guarantee that the glass prepared in this manner is the same material as a MDP extrapolated to 100% concentration. In Figure 7 are shown published data on six glasses.73 Plotted is the observed σexp versus the dipole moment. It is fair to say that the correlation is not good. It is puzzling that an attempt to correlate σ with p is attempted with linear axes as shown in Figure 7. According to eq 10, a plot of σ2 versus p2 should be used, as done in Figure 5. Also, there is no discussion of how everything else, including the van der Waals component of σexp-vdw, is held constant. It is pointed out in ref 73 that the prefactor varies by a large amount, 200 among these experiments. Therefore, the effect on the mobility of changing the dipole moment of the donor glasses appears to be dominated by the prefactor (the f1 function of eq 2, which includes the wave function overlap factor associated with all hopping theories) instead of the temperature dependence (which is a
7304 J. Phys. Chem. C, Vol. 112, No. 19, 2008
Figure 7. The σexp versus the dipole moment of various donor molecules made into glasses, that is, 100% concentration (from ref 73).
Schein and Tyutnev Figure 7. This is within the range of typical experimental errors for this type of measurement but does affect the perception of linearity of the top curve in Figure 8. The “other” glasses are labeled “TAA donor glasses” in ref 74 and include TAPC, NDNTA, TPD, TTB, BTNAF, NDNTD, and PPDA. While these are triarylamine compounds, this includes a very wide range of materials which even have separate sections in ref 1 (sections IIIF and IIIH in chapter 8). There is no discussion in ref 74 why σexp-vdw should be constant among this list. In the top curve (the TPM-type molecules), the prefactor of the mobility varies only by a factor of 3, a different result than was obtained in Figure 7. The data are fit to two straight lines on this graph. It is argued that there are two lines because σvdw is different. The σvdw is estimated by extrapolating to p ) 0, which gives 0.085 eV for the top curve and 0.065 eV for the other. (These intercepts appear to have been obtained from plots of σ2 vs p2.) This, of course, assumes that σvdw-exp is different for the two different data sets and constant for each set. The reason for this result is not discussed. For example, if we calculate σexp-vdw for the TPM compounds in Figure 8 by the standard procedure in GDM from the measured values of σexp and eq 5 (using c ) 1, δ ) 8 Å, and ) 3), the results are 0.112 eV (TMP-A), 0.0927 eV (TMP-B), 0.0906 (TMP-C), and 0.0785 eV (TMP-E), which is not constant. Note that values of sigma for all of these glasses are usually different than the observed values for the same molecule in a doped polymer extrapolated to 100% concentration (compare with Tables 1 and 2), as pointed out in the original publications. This suggests that the glass is really not a MDP system extrapolated to 100% concentration in regard to charge-transport properties. The authors suggest that this type of data analysis is not helpful in determining the role of energetic disorder in σexp. 5. Discussion
Figure 8. The σexp for a series of vapor-deposited TPM glasses (TPMA,B,C,E,F in increasing order, open circles) and other glasses (TAPC, NDNTA, TPD, TTB, BTNAF, NDNTD, NPPDA in increasing order, solid circles) (from ref 74 reprinted with permission from the Japanese Journal of Applied Physics). Note that TPM-A ) TPM ) MPMP has a value of 0.98 eV in Figure 7 to 0.93 eV in this figure, which affects the quality of the fit of a straight line to the data of the upper curve.
source of σexp). Attempts73 to analyze the data by estimating the van der Waals part of sigma (by extrapolating to p ) 0) and subtracting the van der Waals part did not improve the correlation. Attempts were made to understand the behavior of MPMP (which is another name for TPM) by internal molecular structure, but then, this type of argument was not extended to the other molecules. Another attempt to find a correlation of σexp with dipole moment in glasses is shown in Figure 8 (taken from ref 74). Again, σ2exp should have been plotted versus p2. Again, there is no discussion of how everything else, including the van der Waals component, is held constant. The σexp for a series of vapor-deposited TPM compounds, TPM-A, B, C, E, F (open circles) and other molecules (solid circles), are plotted. Note that the bottom data point of the top curve is for TPM, which is 0.093 eV in Figure 8 but is 0.098 eV (labeled MPMP) in
From an experimental point of view, the determination of the source of the temperature dependence of the mobility is straightforward and only involves two steps. First, recall that the data for all MDP systems are qualitatively identical; the shape of the current-time curve, the electric field dependence, and the temperature dependence of the mobility are all qualitatively the same. Therefore, it is highly likely that one mechanism determines the hopping physics for all of the MDP systems. Second, one identifies and analyzes the simplest system. Consider DEASP:PC and DEASP:PS. The experimentally observed activation energy (or σexp) is independent of the concentration of dopant or the polymer matrix, as can be seen in Figure 2 and ref 29. This clearly and strongly suggests that the activation energy for hopping does not depend on the neighborhood around the charged molecule, that is, the activation energy is intramolecular in origin. Then, to be more certain, one checks the experiment in another system. This was done on DEH,28 and the same result was obtained. Note that we refer to the change in “neighborhood” in these experiments. Clearly, the distribution and magnitude of dipole moments in the neighborhood of the charged molecule has been changed; this property has been emphasized in the literature and is associated with dipolar disorder. However, in addition, the geometry and identity of the molecules in the neighborhood around the charged molecule has also been changed. In fact, any contributions to disorder that we can envision have been changed during this experiment. One recognizes that in some MDP systems, a dependence of the activation energy (or σexp) on F is observed. As shown in
Review Article Tables 1 and 2, 17 MDP systems out of the total of 43 characterized systems show this effect. One then postulates that there is some “interaction energy” which can lower the activation energy. This interaction energy has been shown to decrease as F increases and to be dependent on the polymer matrix, much as one would expect for a transfer integral. Therefore, in order to understand charge transport in MDP, one needs to understand why the activation energy is primarily intramolecular in origin and the source of this interaction energy. Primarily means that in 60% of the characterized systems, the activation energy is intramolecular in origin; of the other 40%, about 2/3 of the activation energy is intramolecular in origin, and the rest is due to this interaction energy. In the authors’ opinion, this is a reasonable approach and a valid start to a theoretical search for the hopping mechanism. There have been many attempts to understand charge transport in MDP systems within the context of Bassler’s GDM, one MDP system at a time. In this review, we have taken the point of view of trying to understand the trends apparent in the data of all of the MDP-characterized systems. The GDM predicts that the width of the hopping site manifold, σ, can be obtained from the temperature dependence of the mobility. The contribution to the energetic disorder characterized by σ comes from three sources, a dopant molecule dipolar component, a polymer matrix dipole component, and a van der Waals component. We have used an expression of Young’s to characterize the dopant dipolar component. We have suggested that a modified version of eq 5 can be used to characterize the polymer matrix dipolar component. Within GDM, the van der Waals component is obtained by subtracting the experimentally observed σexp from the calculated value of the dipolar components, giving σvdw-exp. We have tested these theoretical results against available data and have found the following. There are 26 characterized MDP systems in which σexp has been observed to be independent of dopant concentration. This is fit within the GDM to a dipolar component due to the dipole moment of the dopant which decreases with F (calculated from eq 5) and a van der Waals component which increases with F, determined by quadratically subtracting from σexp the values calculated from eq 5. This immediately provides the reader with a dilemma. There is no theoretical discussion of σvdw in the MDP literature that would allow one to judge whether the derived σvdw-exp values are reasonable. The properties of the derived σvdw-exp which the authors find unreasonable are that (1) σvdw-exp is predicted to be near zero at high dopant concentrations for high dipole moment dopants, (2) in every case, it has a strong F dependence that increases as F increases, despite its short-range nature, (3) it must coincidentally add quadratically with the dipolar component in all of the 26 characterized systems shown in Table 1 in order to obtain σexp independent of F, and (4) in order to account for the data in Table 1, σvdw-exp must depend on the dipole moment of the dopant molecule, which is internally inconsistent with its definition. A simpler explanation of the independence of σexp on F is that the activation energy is not determined by the neighborhood of the dopant molecule, that is, the activation energy is intramolecular in origin. In 17 characterized MDP systems it is found that σexp increases with F, as shown in Table 2. This increase is on the order of 10-30% as the dopant concentration is changed from about 70 to 10%, depending on the particular dopant molecule and the experiment. In some papers, the increase is attributed to σvdw-exp, and the dipolar component is assumed to be zero because the dipole moment is small. The properties that σvdw-exp
J. Phys. Chem. C, Vol. 112, No. 19, 2008 7305 must have to explain the data, which, in the author’s opinion, do not appear to be reasonable, are (1) σvdw-exp must increase with F, despite its short-range nature and (2) if the increase of σvdw-exp with F is due to “structural randomness” or the changing ratio of molecules in its environment, it should decrease with F in some fraction of the MDP systems, which has never been observed. If one assumes that the dipolar component is significantly overestimated by eq 5, that is, the van der Waals component dominates the disorder for all of these experiments, then one might interpret the Table 1 data as being totally due to van der Waals energetic disorder. That would suggest that σvdw-exp is independent of F (observed 26 times). However, then is it reasonable that σvdw-exp increases with F in some MDP systems, listed in Table 2, or that it does not decrease with F for some fraction of the MDP systems? The effect of the polymer matrix has been evaluated. There are two effects. In those materials in which σexp is constant with F (Table 1 MDP systems), changing the polymer matrix does not change σexp. Using the GDM equations predicts the change from PC to PS should be about at the level of experimental error. Changing the polymer matrix does not affect the electric field dependence of the mobility. It does affect the prefactor. In those materials in which σexp increases with F (Table 2 MDP systems), changing the polymer matrix changes σexp. However, the change does not appear to be related to the dipole moment of the polymer matrix; the change from PS to PC is constant, independent of polymer concentration, inconsistent with eq 6, and is the same for two polymers with very different dipole moments (PC and PS-3). Using the GDM equations to estimate the effect on σexp of a change from PC to PS predicts a much smaller change than what is observed. Again, the electric field dependence does not appear to be affected by the change in polymer matrix. Whether it affects the prefactor is not clear. Consider the logic of these experimental results. It is assumed that the activation energy is due to disorder. The contributions to the disorder energy are listed as (1) dipolar due to the dopant molecule, (2) dipolar due to the polymer matrix, and (3) van der Waals. This seems reasonable. The dipolar contributions due to the dopant molecules and the polymer matrix are calculated using eqs 5 and 6. These formulas have reasonable dependencies on p and c, and even if they are in error by some constant factor, the basic argument would not change. The properties of the van der Waals disorder energy are derived from the data. The properties of the van der Waals disorder energy appear to be unreasonable, in the authors’ opinion. Therefore, the assumption is wrong: the activation energy is not due to disorder. Even if the dipolar disorder is assumed to be zero, the same result is obtained: the properties of the van der Waals disorder energy appear to be unreasonable, implying that the activation energy is not due to disorder. Four of the systems in which σexp is observed to increase with F have been characterized with polaron theory. All four systems have the remarkable (for a hopping theory) property that there is a region in which the prefactor is independent of F. However, this requires within the context of polaron theory an overlap integral of about 0.2 eV, which is larger than can be rationalized within a hopping theory. However, it does lead to a suggested approach. The data can be characterized by an activation energy that is primarily intramolecular in origin, but in some of these MDP systems listed in Table 2, the activation energy is reduced by some interaction energy which decreases as the distance between the dopant molecules increases and
7306 J. Phys. Chem. C, Vol. 112, No. 19, 2008 depends on the polymer matrix, much like a transfer integral. The theoretical task would be to identify this interaction energy. Polar additives have been added to MDP in an attempt to change the dipolar disorder. In the two case in which σexp is obtained by deconvoluting the data (in the two experiments discussed, 2 and 20% additives were used), σexp is observed to not change; instead, the prefactor changes, inconsistent with GDM. There are experiments that have been published in which the dipole moment of the dopant is changed at constant concentration. The problem with these experiments is that it is virtually impossible to plot σexp versus p and hold everything else constant, such as the shape and size of the molecule and the van der Waals component of σ. In addition, the curves obtained depend on the arbitrary choice of molecules used by the particular author. The present authors suggest that this type of data analysis is not helpful in determining the role of energetic disorder in σexp. Glasses made from a 100% concentration of donor molecules have been studied. These studies have problems similar to those of the MDP experiments in which the dipole moment was varied at fixed dopant concentration. It is not possible to guarantee that everything else is held constant, such as the shape and size of the molecule and the van der Waals component of σ. In addition, the curves obtained depend on the arbitrary choice of molecules used by the particular author. Further, it appears that glasses may not be the same, from the point of view of charge transport, as an extrapolated 100% MDP because of the method by which they are fabricated. The authors suggest that this type of data is not helpful in determining the role of energetic disorder in σexp. We now turn our attention to the correlated disorder models (CDM), which have been suggested to better describe the electric field dependence of the mobility, as discussed in the Introduction. The question which we address is whether the same problems that we have identified with the GDM apply to the CDM. Our procedure is to examine the predicted temperature dependence of the zero-field mobility. In ref 24, the mobility is calculated in terms of a parameter σ, which is the width of the dipolar energetic disorder. Just as in the GDM, one would expect σ to increase as the concentration of dipoles in the environment of the charge increases and as the dipole moment of the molecules increases, analogous to eq 5. In eq 5 of ref 24, it is suggested that the temperature dependence for the CDM in three dimensions is virtually identical to the predictions of GDM, with a small change in the numerical factor, 2/3 changes to 3/5. Therefore, all of the objections listed above for the GDM apply to the CDM. There is one paper33 which combines the polaron theory with correlated disorder, which we will call polaron correlated disorder model (PCDM). The significant advance made by this paper is that it solves two problems with the original polaron hopping description of MDP,25,26 the magnitude of J the overlap integral, eq 7, required to fit the data becomes more reasonable and the predicted electric field dependence of the mobility is consistent with the observations. Despite the success of this theory, it does not account for the experimental observation (after proper deconvolution of the data) of a pre-factor that is independent of F, the so-called “adiabatic small polaron hopping” region now observed in four materials TTA:PC,25 TPD: PC,26 PDA:PC-Z,31 and DPH:PC-Z.32 Comparing the fitted three-dimensional version of this theory to the GDM, the GDM temperature-dependent term exp[-(2σ/3kT)2] is replaced with another exponential exp[-0.31(σ/kT)2 - Ep/2kT] (eq 11 of ref
Schein and Tyutnev 33). The first term in the square brackets, which is due to disorder, is about 4 times larger than the second term due to the polaron binding energy Ep using parameters in their Figure 1 (Ep ) 150 meV, σ ) 80 meV). It appears that the contribution to the exponent due to disorder dominates the exponent and, accordingly, the temperature dependence. Therefore, it appears that this theory would have the same problems discussed above for the correlated disorder model and the Gaussian disorder model. Finally, we would like to say that it is clear that MDPs are disordered materials. The disorder may very well affect some aspects of the charge transport. What we have argued here is that disorder does not affect only one aspect of the chargetransport behavior, the temperature dependence of the mobility. Also, it is clear that these molecules have a dipole moment, and a correlation of dipole moment and mobility has been repeatedly demonstrated since the first experiments by Sugiuchi and Nishizawa.75 What we believe we have shown is that dipolar disorder does not contribute to the temperature dependence of the mobility. These results suggest that there are several outstanding problems in charge transport in MDP, which are (1) identification of the molecular property that determines the activation energy, (2) if energetic disorder does not play a role in the activation energy, how does disorder manifest itself, (3) if correlated disorder does not play a role in the activation energy, how does one explain the ubiquitous dependence of the mobility on exp(xE) (see ref 30 and all of the other MDP experimental references) and the direct observation of a temperature at which the electric field dependence vanishes (the so-called T0 parameter in eq 3), which has been observed in other materials44,47,48,59 since its first direct observation76 (see cover), and (4) how does one explain the experimental observation that, in some materials, the prefactor is independent of F (see discussion after eq 7). Clearly, there is much to be discovered about charge transport in MDP. 6. Summary Extensive charge-transport data are now available for MDP systems. We believe our analysis of the data shows that the data are inconsistent with any model of the mobility in which a disorder energy appears exponentially in the temperature dependence. In particular, we believe we have shown that the data are inconsistent with the predictions of the Gaussian disorder model (GDM), the correlated disorder model (CDM), and the polaron correlated disorder model (PCDM). These data suggest that the activation energy is primarily intramolecular in origin. By primarily, we mean that in 40% of the characterized MDP systems, approximately 1/3 of the activation is due to an interaction energy that has properties similar to a transfer integral. We hope these results will encourage new theoretical ideas of the hopping mechanism in molecularly doped polymers and new charge-transport experiments in MDP. Definition of Initials Used in This Paper (“)” indicates a different initial is used for the same material) BTNAF: 9,9-bis{4-[N-(4-tolyl)-N-(2-naphthyl)amino]phenyl}fluorene DCAQ: 2-t-butyl-9,10-N,N′-dicyanoanthragunonediimine DEASP: 1-phenyl-3((diethylamino)styryl)-5-(p-(diethylamino)phenyl)pyrazoline DEH: p-diethylaminobenzaldehyde diphenylhydrazone DEH-A: 9-ethylcarbazole-3-carbaldehyde diphenylhydrazone
Review Article DEH-B: 9-ethylcarbazole-3-carbaldehyde methylphenylhydrazone DEH-C: 1-pyrenecarbaldehyde diphenylhydrazone DNTA: di-p-tolyl-p-nitrophenylamine DOA: plasticizer dioctyl adipate DPH: p-diphenyl-aminobenzaldehyde diphenylhydrazone DPQ: 3,3′-dimethyl-5-5′-di-t-butyldiphenoquinone ENA-A: N,N-bis(2-methyl-2-phenylvinyl)-N,N′-diphenylbenzidine ENA-B: N-(2,2-diphenylvinyl)-4,4′-dimethyldiphenylamine ENA-C: N,N-bis(2,2-diphenylvinyl)-N,N′,diphenylbenzidine ENA-D: N-(2,2-diphenylvinyl)diphenylamine ETPD: N,N′-bis(4-methylphenyl)-N,N′-bis(4-ethylphenyl)(1,1′-3,3′-dimethylbiphenyl)-4,4′-diamine MBDQ: 3,5-dimethyl-3′,5′-di-tert-butyl-4,4′-diphenoquinone NDNTA: N,N′-diphenyl-N,N′-(2-naphthyl)-(1,1′-phenyl)-4,4′diamine NDNTD: N,N′-diphenyl-N,N′-(2-naphthyl)-(p-terphenyl)4,4′’-diamine NPPDA: N,N′-bis(1-naphthalenyl)-N,N′-diphenyl-4,4′-phenyldiamine PC: polycarbonate PC-Z: poly(4,4′-cyclohexylidenediphenyl)carbonate PDA: N,N,N,N′,N′-tetrakis(m-methylphenyl)-1,3-diaminobenzene PMPS: poly(di-n-butylsilylene) PS ) PS-1: polystyrene PS-2: poly(4-t-butylstyrene) PS-3: poly(4-chlorostyrene) PTS: 1,-1-dioxo-2-(4-methylphenyl)-6-phenyl-4-(dicyanomethylidene)thiopyran TAA: triarylamine TAA-A: ethyl ester of 4-[bis(4-methylphenyl)amino]benzenepropanoic acid TAPC: 1,1-bis(di-4-tolylaminophenyl)cyclohexane TASB: bis(ditolylaminostyryl)benzene TMP ) TPM-A ) MPMP: bis(4-N,N-diethylamino-2methylphenyl)-4-methoxyphenylmethane TMP-B: bis(4-N,N-diethylamino-2-methylphenyl)(4-propylphenyl) methane TMP-C: bis(4-N,N-diethylamino-2-methylphenyl)(4-phenylphenyl) methane TMP-D: bis(4-N,N-diethylamino--2-methylphenyl)(4-phenyl) methane TMP-E: bis(4-N,N-diethylamino-2-methylphenyl)(4-methoxyphenyl) methane TMP-F: bis(4-N,N-diethylamino-2-methylphenyl)(4-chlorophenyl) methane TTA18,25,56 ) TPA-143 ) TAA-144: tri-p-tolylamine TPA-243 ) TAA-21,44: tri-p-anisylamine TPA-343 ) TAA-31,44: methyl 3-(p-(di-p-tolylamino)phenylpropionate TPA-443 ) TAA-41,44: 4-bromo-4,4′,4′′-dimethyltriphenylamine TPD: N,N′-diphenyl-N,N-bis(3-methylphenyl)-[1,1′-biphenyl]-4,4′diamine TTA: tri-p-tolylamine TTB: N,N′,N′′,N′′′-tetrakis(4-methylphenyl)-(1,1′-biphenyl)4,4′-diamine Acknowledgment. The authors would like to acknowledge S. V. Novikov, D. H. Dunlap, V. M. Kenkre, P. E. Parris, and A. V. Vannikov for motivating this study when they wrote:24 “We hope that the present analysis will encourage further
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