J. Phys. Chem. C 2009, 113, 1067–1073
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Transient Conductivity Measurements in a Molecularly Doped Polymer Over Wide Dynamic Ranges L. B. Schein* Independent Consultant 7026 Calcaterra DriVe, San Jose, California 95120
Vladimir Saenko, Evgenii D. Pozhidaev, and Andrey Tyutnev Moscow State Institute of Electronics and Mathematics, Bol. TrechsVyatitel. per., 3, Moscow, Russia
D. S. Weiss Graphics Communications Group, Eastman Kodak Company, 2600 Manitou Road, Rochester, New York 14653-4180 ReceiVed: August 15, 2008; ReVised Manuscript ReceiVed: NoVember 17, 2008
Transient conductivity measurements have been carried out in a hydrazone-polycarbonate molecularly doped polymer over wide dynamic ranges of time and current as a function of the applied electric field. These data are plotted using both linear-linear and log-log axes. The plots on linear-linear axes give familiar results: an initial spike followed by a relatively flat current before the transit time; a mobility that equals published values and precisely follows the Poole-Frenkel law, being exponential in the square root of the electric field; and a current after the transit time that falls much more slowly than can be accounted for using Gaussian statistics. The same data plotted using log-log axes reveals the behavior over long times and low currents. Before the transit time, the decrease of the current can be characterized by two power laws, but it only decreases by about a factor of 2 from 10- 2 of the transit time to the transit time and is almost independent of electric field from 5 to 80 V/µm. After the transit time the current decreases algebraically as t-1.75 to a current value of about 10-2 of the value at the transit time and is also almost independent of electric field. Available theoretical models are not consistent with these data. We note a surprising coincidence: the current transients are virtually identical to the current transients obtained in molecular crystals, as if disorder does not play a role in determining the shape of the current transients in molecularly doped polymers. Introduction Molecularly doped polymers are films that consist of a mixture of an electron donor or acceptor molecule in a host polymer such as polystyrene PS or bisphenol A polycarbonate PC. There has been considerable interest in the charge transport properties of these materials because of their use in many technologies.1-5 These materials are also scientifically interesting because virtually all of the parameters that can affect the hopping mechanism can be varied.1,6 Most transient conductivity measurements in molecularly doped polymers have been used to measure the charge carrier mobility. In a transient conductivity experiment, the transit time (t0) is determined by measuring the time at which the current rapidly decreases. It is assumed that at the transit time, charge carriers have begun to exit the sample to the electrode.1 The mobility is then given as L/Et0, where L is the sample thickness and E is the applied electric field. The transit time is only one point on the full current-time curve. One wonders whether other significant information about the charge transport properties of these materials can be revealed by studying the full current-time curve. An experimental study of the full current-time curve is the purpose of this paper. In early transient photoconductivity measurements of chalcoganide glasses7,8 (such as amorphous a-As2Se3 and low * To whom correspondence
[email protected].
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temperature measurements in a-Se) and organic materials such as PVK (poly-N-vinyl carbazole),7,9-13 it was observed that current-time data plotted on linear-linear axes do not show any evidence for a transit time, i.e., a sudden steep drop in current suggesting that charge carriers are beginning to exit the sample. Instead, log-log plots were used, and surprisingly, when plotted in this manner the current-time curves, when normalized, appeared to be independent of electric field. This was termed“universality”.ItwasamajorsuccessoftheScher-Montroll-Lax treatment of stochastic transport in random media to explain these effects in terms of disorder, cast into an algebraic waiting time distribution.7,14-17 The terminology “dispersive transport” was applied to current-time curves which required log-log plots to extract a transit time (although this term has taken on a variety of other meanings). However, as the purity of molecularly doped polymers improved, it became clear that transit times could be observed on linear-linear presentations of the data and that dispersive transport was not a necessary condition of disordered materials. Subsequently, data on molecularly doped polymers were plotted on linear-linear axes and the focus changed, from discussions of dispersive transport to the physical mechanism determining the mobility as obtained from the transit time from linear-linear plots (for recent reviews, see refs 6 and 18). The tacit assumption in the Scher-Montroll (SM) theory is that the disorder arises from positional disorder. Formalisms based on energetic disorder were suggested on the basis of
10.1021/jp807299a CCC: $40.75 2009 American Chemical Society Published on Web 12/30/2008
1068 J. Phys. Chem. C, Vol. 113, No. 3, 2009 multiple trapping (for examples, see refs 19-25) which were then criticized because of the assumptions made, namely, (1) trap-to-trap migration is neglected and (2) transport occurs in an energetically well-defined transport level such as a valence or conduction band or an isoenergetic array of hopping sites. Quoting ref 26, “The inhomogeneous broadening of opticalabsorption spectra of organic glasses and polymers is an unambiguous signature of the splitting of the exciton band into a manifold of localized states.27 By analogy, the same should hold for charge transport states. In that case the above determination of trap release rates would be correct only if the width of the manifold of transport states was much narrower than that of the trap distribution. Previous optical studies argue against this notion.28” With this rationale, an attempt was made to fit the shape of current-time curves with the Gaussian disorder model, GDM, which assumes that hopping occurs within an intrinsic distribution of hopping states within a Gaussian distribution.1,25,29,30 However, the GDM can not fit the electric field dependence of the mobility (as pointed out in ref 31 and addressed in the various correlated disorder models32-37) and has recently been shown to be unable to account for the dependence of the activation energy on experimentally changed disorder.6 (The correlated disorder models32-37 also are unable to account for the dependence of the activation energy on experimentally changed disorder.6) In addition, the GDM is inconsistent with the current-time data that are discussed below, as will be shown. We have carried out transient conductivity experiments in a molecularly doped polymer system over the widest dynamic ranges yet reported, to the authors’ knowledge, and have analyzed the current-time data. As will be discussed, we conclude that the current-time curves and the observed universality (current-time curves independent of electric field) are not consistent with available theories. Experimental Section The molecularly doped polymer MDP used was 30 wt % p-diethylaminobenzaldehyde diphenylhydrazone (DEH) in bisphenyl-A polycarbonate. The materials were dissolved in dichloromethane and blade-coated onto nickelized polyethylene terephthalate (PET, 7 mil). The MDP films were dried by oven curing (80 °C) for 0.5 h. Samples for transient conductivity measurements were prepared by electrode evaporation of an Al electrode (about 26 mm in diameter). The MDP thicknesses, determined by capacitance, were 14-17.5 µm. Charge generation was accomplished with an electron gun38,39 operated in a single-pulse mode with its energy set (7 keV) for surface charge generation. Others have pioneered the use of an electron gun for studying charge transport, including Spear,40 Gross,41 and Hirsch.42 The half-width of the electron pulse is 25 µs which limits the time resolution. Data collection was begun with a 100 µs delay to ensure that no extrinsic circuit effects contribute to the data. Experiments were done in a vacuum of 3 × 10-2 Pa. All measurements were made in the small-signal region, with the ratio of the generated free carriers to charge on the electrode always less than 0.1. In addition, experiments were carried out in which the electron current was reduced by a factor of 10 and the current transient shape was unchanged. The data collection was done digitally, with data rates of 4 × 105 s-1 for up to 10 s. The electric field was varied by sequentially increasing the voltages. Repeatability was verified by returning to a lower voltage and observing that the same current transient was obtained and comparing results on five separate samples.
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Figure 1. Electron beam-generated current transient observed in a 14 µm thick film of 30 wt % DEH:PC plotted on linear-linear axes. The electric field was 5 V/µm. Note the current is approximately flat after an initial current spike and before the transit time. The transit time t0 (0.23 s) and t1/2 (0.47 s) (the time at which the current becomes onehalf of its value at t0) are indicated. w, which equals (t1/2 - t0)/t1/2, is 0.51.
The current transients were typical of prior observations (see almost any prior experimental reference on charge transport in MDP, which are listed in refs 1 and 6) with some having a perfectly flat plateau (see Figure 1). Some samples made from the same MDP coating had currents which were slightly rising and some were slightly falling, all within about 10% of flat. We do not understand why there is a difference in the pulse shape near the transit time, although the obvious suggestion of some surface emission is a possibility. Despite this small variation, the zero field mobility, the Poole-Frenkel effect, and the general features of the extended transients using log-log plots were identical for all samples. By changing the electron energy, the charge generation position can be changed continuously from surface (7 keV) to bulk (50 keV).43,44 The maximum generation rate occurs at a distance approximately equal to 1/3 of the maximum electron range and is a factor of 2.3 larger than at the surface itself. The maximum electron range is 40 µm for 50 keV and 1.5 µm for 7 keV, increasing as (electron energy)1.67.43 The transient currents reported here were obtained with surface generation using 7 keV electrons. Data using high electron energies (50 keV) are not reported here except to note that on log-log current-time plots slopes of t-0.25 were obtained prior to the transit time and t-1.75 were obtained after the transit time, both independent of electric field, within experimental error. Five separate samples were characterized for this experiment. The transit times were obtained as shown in Figure 1, and the electric field dependence is shown in Figure 2. As can be seen, it follows the Poole-Frenkel effect, with the mobility exponential in the square root of electric field with a correlation coefficient of 0.99, in excellent agreement with prior work (refs 45-48 and virtually all other prior experimental references on charge transport in molecularly doped polymers, (see lists in refs 1 and 6). The zero electric field mobility values (at room temperature) obtained for the five samples from the zero field intercept from plots as shown in Figure 2 are 6.2, 3.3, 4.3, 6.1, 5.8 × 10-8 cm2/V · s, also in excellent agreement with prior work.47,48 The question of intrinsic vs extrinsic effects always must be considered in these experiments. We believe the flat plateau is intrinsic to the material used (30% DEH:PC) because (1) flat plateaus have been observed many times before in both this
Transient Conductivity Measurements
Figure 2. Log of the mobility, µ, plotted vs the square root of the electric field (V/µm)1/2. The least-squares linear fit has a correlation coefficient of 0.99. This and all measurements in this paper are at room temperature.
Figure 3. w (see text) vs electric field.
material and in other MDPs and (2) it would require a coincidence for the approximately flat current, observed before the transit time to be made up of a rising and falling current, and this coincidence would have to be repeated many times in all of the transients observed in molecularly doped polymers in which the current before the transit time is approximately independent of time. We believe the mobility observed (obtained from the transit time) is also intrinsic because (1) the zero electric field mobility is agreement with prior published results46-48 and (2) the Poole-Frenkel effect observed is identical within experimental error to prior published results.45,47-49 The most difficult question is whether the falling slope is intrinsic or extrinsic. This falling slope is characterized by its slope and by its half-width w defined as (t1/2 - t0)/t1/245 where t1/2 is time for the current to fall to half the value at t0. This will be dealt with in the Results section. Results Shown in Figure 3 are the half-widths plotted as a function of electric field. The half-width is electric field independent, within experimental error. Field independent half-widths (universality) appear to be a general property of molecularly doped polymers: they have been observed in 50%DEH:PC,46 30%DEH:
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Figure 4. Current transient observed from a 14 µm thick film of 30 wt % DEH in PC plotted on log-log axes. The electric field was 20 V/µm. The transit time (34.1 ms) and the normalization method (see Figure 5) are noted.
PS,46 30%TTA:PC (tri-p-tolyamine doped polycarbonate),46 and 50%TPD:PC (N,N′-diphenyl-N,N-bis(3-methylphenyl)-1,1′-biphenyl]-4,4′-diamine) to first order.50 The only molecularly doped polymer exception appears to be 50%TAPC:PC (1,1bis(dis-4-tolylaminophyenyl) cyclohexane doped polycarbonate) shown in Figure 4 of ref 51. The magnitude of the half-width observed here, 0.5, is somewhat larger than observed previously in this material, 0.31.47 We do not understand the source of this difference, although (1) we believe that the qualitative characteristics that we report here are general properties of MDP, (2) similar magnitudes of discrepancies of w have been reported before, e.g., for 40%TTA:PC w) 0.44 and 0.7 have been reported in two separate papers as reviewed in ref 51 on p 971, and (3) it is possible that differences in sample preparation or the type of polycarbonate, e.g., molecular weight, could affect w. Shown in Figure 4 is a log current-log time transient for 280 V applied across a 14 µm sample (an electric field of 20 V/µm). The transit time, 34.1 ms at the transit time is noted. Note the extrapolation method. The transit time is always slightly larger than the transit time obtained from linear-linear plots, 28.3 ms, for this electric field. Note that the current before the transit time can be characterized by the sum of several power laws but is larger only by about a factor of about 1.16 at 10% of the transit time and 2.2 at 1% of the transit time than at the transit time. After the transit time the current falls rapidly, but not as fast as expected from Gaussian statistics (see below). It appears to fall as ∼t-1.75. Shown in Figure 5 are log-log plots of the current transients for several electric fields, from 5 to 80 V/µm, normalized to the transit time and the current at the transit time (see Figure 4 for the procedure). Note that to first order there is a universal curve, i.e., after rescaling (using the extrapolation method shown in Figure 4) the shape of the current-time curve is independent of electric field. Near the limit of experimental error, we cannot rule out some small electric field dependence to the slopes both after and before the transit time. In none of the five samples did we see a systematic shift of slope with all of the electric fields used, but if some fields are ignored, then some systematic shift can be seen (as in this data set). If the average slope after the transit time is -1.75 the experimental error is about (0.2. Discussion The current before the transit time is observed to decrease, but by only about a factor of 2 from 10-2t0 to t0. To first order
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Figure 5. Normalized log-log plots of current transients from a 14 µm sample as a function of applied electric field. Curves 1-6 are 5, 10, 20, 28.6, 56, and 80 V/µm, respectively. To guide the eye, t-0.06 is plotted before the transit time and t-1.75 is plotted after the transit time.
the current is constant. This could be due to an intrinsic constant mobility or a dynamic equilibrium between an intrinsic transport state and an intrinsic shallow trap (see below). To second order, there is a small decrease of the current with time, which suggests some concentration of deep traps, i.e., traps with trapping times much less than the transit time and release times with times much greater than the transit time. This is consistent with the well-known increase of the slope of this initial current as the temperature is lowered (see Figure 10 of ref 26 for a typical example). The falling current after the transit time can be characterized several ways, by its algebraic slope (-1.75) or its half-width. We will use both to discuss its possible origins. First consider the half-width. As is well known, when the current is time independent before the transit time, it is predicted (Figure 1 in ref 7) that “Gaussian” transport should occur. This is defined as follows.46 During the transit time, t0 ) L/µE, as the charge carriers drift across the film, the charges diffuse relative to their centroid a distance of (2Dt0)1/2 where D is the diffusion constant, which is related to the mobility µ by the Einstein relationship D ) µkT/e where k is Boltzmann’s constant, T is absolute temperature, and e is the electronic charge. Therefore,
w≈
√2Dt0 ) L
2kT ) eLE 2kT eV
(1)
Clearly, our data (Figure 3) are inconsistent with the predicted E dependence (at constant sample thickness). In addition, for typical numbers (L ) 15 µm, T ) 300 K, E ) 20 V/µm) w is predicted by eq 1 to be 0.01, much smaller than is observed (0.5). Remarkably, the same issue arises for molecular crystals. In these materials, charge transport is also characterized by transient photoconductivity and the current transients are usually time independent before the transit time at room temperature and have w’s on the order of 0.4, which is orders of magnitude larger than predicted by Gaussian transport. This type of data can be seen in any publication on charge transport in molecular crystals.52 Figure 6 shows typical transient conductivity data from an anthracene molecular crystal at room temperature.53 Using eq 1, the w predicted for the data in Figure 6 is 0.013, much smaller than is observed, 0.30. For evidence that w is also field independent in molecular crystals, see Figure 8.3 in
Figure 6. Current transient data from a crystal of anthracene in the c′ direction which is 60 µm thick.53 The applied voltage was -300 V, and the transit time, marked by an arrow, is about 350 ns giving a mobility of about 0.4 cm2/V · s. Note the similarity of this transient current-time curve with the data of Figure 1. (The curve goes downward because the electron transient is measured, while in Figure 1 the hole transient is measured.)
ref 54, which shows electron transients in anthracene along the b direction at room temperature for a field variation of factor of 2.6 and ref 55, which shows electron transients in the c′ direction of anthracene at 140 K for a field variation of a factor of 1.5 (in an experiment designed to test whether the field dependence predicted by hopping theories could be observedsit was not). To the authors’ knowledge, Gaussian transport has been observed only once by Karl56 in an anthracene crystal which accounts for about 50% of the observed w. The remainder of w was due to an electric field independent w (which we and all others report). At lower temperatures, the current before the transit time decreases with time in molecular crystals, similar to the behavior observed in MDP. Such behavior has been shown to be extrinsic and due to a deep trap associated with a shallow pretrap level.57 If the width w were due to the width of the carrier generation region G, then w would equal G/L. Even using the full electron range, 1.5 µm, w is predicted to be much smaller than observed (1.5/14 ) 0.107) for this 14 µm thick sample. Furthermore, we have lowered the electron energy from 7 to 3 keV to decrease G. No change in w was observed. In addition, w has been measured as a function of film thickness (using photogeneration), and it was found to be independent of film thickness in this material, 30% DEH:PC, from 3.8 to 42 µm, (ref 46, published in Figure 11 of ref 30), inconsistent with the prediction of this mechanism. The width is not due to field diffusion, an idea suggested by Rudenko and Arkipov,58 because w is independent of electric field. The width is not due to Coulomb repulsion of the charge carriers in the sample because we have ensured in these experiments that the concentration of charge carriers in the sample is very small (see Experimental Section). Consider the effect of intrinsic trapping states. For simplicity of discussion, consider the effect of a single intrinsic trap on the pulse shape, with a trapping time τt and a release time. If the trapping time is much longer than the transit time (at the lowest voltage), there is no effect on the current transient. If the trapping time is much shorter than the transit time and the release time is much longer than the transit time, the current will decrease with time, inconsistent with the data. The only case of interest is if the trapping time and the release time are much shorter than the fastest transit time (highest voltage), so that dynamic equilibrium is attained (giving the experimentally observed approximately time independent current seen in Figure
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1). Then the mobility is said to be trap controlled. (For the mobility to be intrinsic, the shallow trap must be an intrinsic trap). In this case, the width w should be approximately equal to N1/2/N where N is the mean number of traps visited. N is given by the ratio of the trapping time τt to the trap free transit time
w)
1 ) √N
µ0Eτt L
(2)
where µ0 is the microscopic (trap-free) mobility. In this situation, w can be independent of E (as observed, see Figure 3) only if the trapping rate τt-1 is proportional to the drift velocity µ0E. Such might be the case at high electric fields if most hops are in the direction of the field. At low fields, however, carrier motion should be essentially diffusive, τt should be independent of E, µ0 should be field independent (since an exponential becomes constant at low values of the exponent), and w should increase as E1/2. For hopping sites a distance a ) 1 nm apart, the low-field case should apply if eaE e kT, e.g., E e 25 V/µm at T ) 300 K. Our data extend well into this range, but w remains independent of E. The predicted dependence on L is usually not observed. It is noted that only a limited number of measurements have been reported of the dependence of w on L in molecularly doped polymers: w is independent of L in 30% DEH:PS from 3.8 to 42 µm (ref 46, actual data plotted in Figure 11 of ref 30), in 50% DEH:PC from 3.8 to 42 µm and 30% TTA:PC from 10 to 90 µm.31 In TPD:PC,50 it appears that w decreases as L increases. (The data from ref 50 are plotted and discussed with respect to GDM in Figure 11 of ref 30.) Consider next a finite distribution of traps. As shown by Schmidlin,22 Orenstein and Kastner,23 and others if the distribution is exponential, it is equivalent to the Scher-Montroll theory, which is discussed below. If the distribution is finite, that is, it consists of a small number of traps with release times near the transit time, then curves similar to those observed here are predicted at certain electric fields (see, for example, Figures 5-7 of ref 22). However, not only would this be a coincidence, one also expects “a departure from ‘universality’ as the critical trap approached the end of a finite sequence.” (Schmidlin describes this in the caption of Figure 9 of his paper.) The issue of whether the half-width is due to release of carriers from some type of trap remains, especially because the magnitude of w appears to have some variation among reported references, suggesting an extrinsic effect. For w to be electric field independent means that the half-width of the pulse is proportional to the transit time and both must follow the Poole-Frenkel field dependence in exactly the same way. It is difficult to see how a reasonable model of an extrinsic trap can account for these results. If there were a trap which had a trapping time much shorter than 1% of the transit time and a release time which somehow equals the transit time and has the exact same field dependence as the transit time, this might account for our results. However, it is hard to rationalize such a model. Or, if the first moment of distribution of transit times had the exact same field dependence as the transit time and was about 0.5 of the transit time, one might be able to understand these results. But again, this type of model is hard to rationalize. The GDM cannot account for the experimental observations of the effect of the electric field on the mobility and has recently been shown to be unable to account for effect of changing disorder on the activation energy of charge transport.6 Putting these points aside for the moment, the GDM also cannot account
for a current which is independent of time (see Figure 1) for a material such as DEH:PC which has a σ (a measure of energetic disorder obtained by fitting the change in the zero electric field mobility with temperature) of 0.13 eV. This was recognized by Borserberger et al.:26 “these experiments (the observation of a transition from nondispersive to dispersive transport with the word dispersive defined as we have defined it in this paper) typically yield lower transition temperatures, equivalent to a lower degree of energetic disorder, than predicted on the basis of the temperature dependence of the mobility (σ)” (parentheticals added by the present authors). In other words, the GDM predicts for material such as 30% DEH:PC with σ ) 0.13 eV (or σ/kT ) 5 at room temperature) that the current in the regime before the transit time should fall with time much faster than is observed. In Figure 1 of ref 26, the GDM theory prediction is plotted on log-log axes and predicts a slope of -0.33, or a time dependence of t-0.33 from 0.01 to 1.0t0. The data shown here are closer to t-0.06-t°. In addition, the GDM cannot account for the observed field independence of w, as can be see in Figure 12 of ref 30. In order to account for the electric field dependence of the mobility, the GDM was modified to take into account correlated disorder.32-37 These models were more successful in explaining the low field dependence of mobility. It is not clear whether the CDM models would provide a mechanism for understanding the time-of-flight transients reported here. This issue has not been addressed. However, the CDM are, similar to the GDM, unable to account for effects of changing disorder on the activation energy of charge transport.6 Transients have been fit allowing for variable diffusion constant by Hirao and co-workers.59 As expected, the transients could be fit, but the diffusion constant and mobility were not related by the Einstein relation, i.e., the diffusion constants were anomalous, in the words of the authors. The diffusion constants were much bigger than predicted by the Einstein relation and ultimately shifts the discussion to understanding why diffusion constant becomes so large. Scher-Montroll (SM) theory has been applied to some amorphous materials successfully.7 In this theory, the waiting time distribution is much larger than the transit time. Therefore, it predicts that the current should continue to fall right through the transit time, inconsistent with Figure 1. A fit of a transient published elsewhere with SM theory with the highest R that the authors are aware of (0.85) was done by Dunlap.60 This transient clearly has a much stronger decrease of the current with time before the transit time than the data shown here (Figure 1). Attempts to fit the data shown in Figure 1 with SM theory failed because it is not possible to accommodate both the flat plateau before the transit time and the slowly decreasing broad tail after the transit time with any single value of R.61 With the additional assumption that the mean displacement of the charge carrier is proportional to the electric field, SM theory predicts that (eq 54 in ref 7)
t0 ≈
( EL )
1⁄R
(3)
or the mobility
µ≈
( EL )
(1-1⁄R)
(4)
where the time dependence of the current I defines R, that is
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I(t) ≈ t-(1-R) for t < t0, and I(t) ≈ t-(t+R) for t > t0
(5)
Plotting log µ vs log E with the data shown in Figure 2 reveals curved lines, consistent with prior results (note the data points in Figure 9 in ref 45 do not form a straight line), i.e., our data are not consistent with the predicted field dependence of sm theory. Conclusions We have measured electron-beam-generated current transients in a molecularly doped polymer over the widest time and current dynamic ranges yet reported, to the knowledge of the authors, more than 2 orders of magnitude in both axes. We find that the current before the transit time decays by about a factor of 2 from 10-2t0 to t0 and only 1.2 from 10-1t0 to t0. After the transit time, the current decays algebraically by t-1.75. Both before and after the transit time, the shape of current transient is to first order independent of electric field from 5 to 80 V/µm. A small electric field dependence is observed; however, it is not significantly greater than experimental error. The nearly constant current before the transit time suggests that the carriers are hopping with an approximately constant velocity and constant mobility by some intrinsic hopping mechanism or they are in dynamic equilibrium with an intrinsic shallow trap. The small decease in slope could indicate that a small amount of deep trapping exists with a very long release time (with respect to the transit time). This is consistent with the known temperature dependence of this current: it decreases more steeply as the temperature is lowered, suggesting increasing numbers of traps are becoming “deep” traps as the temperature is lowered. Such effects have been observed in molecular crystals, and it was concluded that such behavior is extrinsic and due to a deep trap level associated with shallow pretrap level.57 We find the behavior after the transit time puzzling. We have attempted to explain this behavior with all known theories and have failed to account for it. It is certainly much wider than Gaussian transport would predict, and it is not consistent with the width being determined by the generation region, Rudenko’s field diffusion, Coulomb repulsion, or an intrinsic shallow-trapcontrolled mobility. We have tried to postulate trap properties that would account for these data, but the models do not seem reasonable to us. The data are inconsistent with the predictions of the GDM (putting aside the difficulties that the GDM has explaining the electric field dependence of the mobility and the experimental effects of changing disorder on the activation energy) because the slope of the current before the transit time is observed to be more shallow than predicted and because the field dependence of w is not predicted. As pointed out, field-independent w’s appear to be a general phenomenon of molecularly doped polymers. The data are inconsistent with the SM theory that assumes waiting time distribution functions which are longer than the transit time and its predicted electric field dependence. We note that these data are very similar to data published on molecular crystals in that w is about the same value and is observed to be field independent. The mobility in molecular crystals is about 7 orders of magnitude higher and is electric field independent in the high-temperature region (T > 100 K in naphthalene where the mobility is also almost temperature independent62,63), suggesting a different transport mechanism
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