Two-Layer Mutiple Trapping Model for Universal Current Transients in

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J. Phys. Chem. C 2010, 114, 9076–9088

Two-Layer Mutiple Trapping Model for Universal Current Transients in Molecularly Doped Polymers D. H. Dunlap Department of Physics and Astronomy, UniVersity of New Mexico, 800 Yale BlVd NE, Albuquerque, New Mexico 87131

L. B. Schein* Independent Consultant, 7026 Calcaterra Dr., San Jose, California 95120

Andrey Tyutnev, Vladimir Saenko, and Evgenii D. Pozhidaev Moscow State Institute of Electronics and Mathematics, Bol. TrechsVyatitel. per., 3, Moscow, Russia

P. E. Parris Department of Physics, Missouri UniVersity of Science & Technology, 1315 North Pine Street, Rolla, Missouri 65409-06403

D. S. Weiss Chemical Engineering Department, GaVett Hall, UniVersity of Rochester, Rochester, New York 14627 ReceiVed: February 2, 2010; ReVised Manuscript ReceiVed: March 29, 2010

The mechanism of charge transport in molecularly doped polymers has been the subject of much discussion over the years. In this paper, data obtained from a new experimental variant of the time-of-flight (TOF) technique, called TOF1a, are compared to the predictions of a two-layer multiple trapping model (MTM) with an exponential distribution of traps. In the recently introduced TOF1a experimental variant, the charge generation depth is varied continuously, from surface generation to bulk generation, by varying the energy of the electron-beam excitation source. This produces systematic changes in the shape of the current transient that can be compared to predictions of the two-layer MTM. In the model, one additional assumption is added to the homogeneous MTM, namely: that there exists a surface region, on the order of a micrometer thick, in which the trap distribution is identical to the bulk, but has a higher trap concentration. We find that the characteristic experimental features of an initial spike, a flat plateau, and an anomalously broad tail, as well as the sometimes observed cusp or decreasing current occurring near the transit time, can all be described by such a two-layer model; that is, they can arise as a result of carriers delayed by a trap-rich surface layer. We find that we can semiquantitatively fit current transient data over the whole time range of the experiment, but only by using theoretical parameters that lie in a narrow range, the extent of which we quantify here. I. Introduction We present new experimental results, along with a theoretical analysis, that are intended to explore features of both typical and atypical current-time transients associated with charge generation in molecularly doped polymers (MDPs). At room temperature and above, most current transients in MDPs and many comparable organic semiconductors exhibit similar features in the linear-current, linear-time data presentation: an initial spike followed by a flat plateau that changes at the transit time into a broad tail whose width is comparable to the transit time itself. In approximately 15% of the samples measured (estimates obtained from discussions with several MDP experimentalists) the transients produced are atypical in that, after the initial spike changes into the beginning of a plateau, there is a subsequent increase in the current, which then develops into a maximum before turning over to decay into the characteristic broad tail * To whom correspondence should be addressed. E-mail: schein@ prodigy.net.

seen in more typical current transients. Such local maxima in the current transient are referred to in the literature as “cusps”. Experimentally we use a new variant of the time-the-flight experiment, recently introduced by Tyutnev et al.,1 which uses electrons from a high-energy electron beam as the charge generation source. By varying the energy of the electrons, the depth of the region over which charges are generated in the sample can be continuously varied, from surface to bulk generation.1 In the limit in which the energy of the electron beam is small enough and charge is generated in the immediate vicinity of the contact, the experimental technique reduces to the standard time-of-flight (TOF) experiment studied with photogenerated charge carriers using highly absorbed laser illumination. When the charge generation depth is systematically varied the variant is referred to as a TOF1a experiment. The limit of the experimental technique, in which charge is generated throughout the sample which occurs with a sufficiently energetic electron beam, is referred to as a TOF2 experiment.

10.1021/jp1010132  2010 American Chemical Society Published on Web 04/22/2010

Two-Layer Mutiple Trapping Model Having performed the TOF1a experiment on many different samples, we always find that the observed current transient shapes vary systematically with the depth of the charge generation region, being more cusp-like when the depth is shallow and more plateau-like as it is broadened. Independent of shape, the current transients show universality with electric field. Universality of current-time transients with electric field is a general feature of time-of-flight experiments in amorphous insulators and was first noted in highly dispersive samples.2-4 Universality with field implies that the current and time axes at different field strengths can be scaled so that all of the data for a given sample lies on one master curve.2-4 Usually the data is scaled at the transit time for both time and amplitude. In highly dispersive transients, a transit time in the time-offlight data can only be seen on double-log axes, with the current before the knee decaying in time as t-(1-R), and after the knee as t-(1+R), where the dispersion parameter R is usually5 less than 0.7. These algebraic current-time properties can be understood in the context of the much-studied multiple trapping model (MTM),2-4 which describes transport through a medium containing traps for the case of an exponential energy distribution. While the MTM is only a heuristic model, it has some advantages over numerical simulations of hopping in a spatially and energetically disordered system, in that it can be solved analytically and it provides an intuitive relation between dispersion and trapping. For example, an algebraic decay of the current occurs in the MTM when the assumed exponential distribution, F(ε) ≈ exp(-ε/Ω), of trap energies ε is sufficiently broad that equilibration cannot be achieved. In that case, the width Ω of the distribution determines the value of the dispersion parameter R via the relation R ) kT/Ω, where k is Boltzmann’s constant and T is the absolute temperature.3-5 Calculations with the MTM show that for high enough temperatures, or shallow enough traps, the transients exhibit a plateau and a discernible knee at the transit time when plotted on linear axes.5 This tends to blur the distinction between “dispersive” transients, for which the time for relaxation in the trap manifold is anomalous (infinite), and transients that have a perfectly flat plateau which have often been associated with fully equilibrated transport. As an example of this blurring, it has been observed that there is close agreement between transients calculated from the MTM5,6 and experimental transients observed on 30% DEH:PC (p-diethylaminobenzaldehyde diphenylhydrazone in bisphenol A polycarbonate) at room temperature. This system has been shown to exhibit universality with field7 over a limited dynamic range. If the only thing one knew about the experimental data was that the transients have a flat plateau, it would probably be concluded that they describe equilibrated transport. However, coupled with the results from the MTM, the fact that they also show universality with field suggests that transport in these systems has not in fact equilibrated and raises the question of how these apparently contradictory features can be reconciled. The two-layer MTM model introduced here can reconcile these two features. Many of the transients that we have measured have a perfectly flat plateau and still exhibit universality. On log-log plots the current decays approximately as t-2. On linear-linear plots the half-width of experimental current transients is typically observed to be about twice the value of the transit time, suggesting that charges remain spread throughout the entire sample at the moment that the fastest carriers are leaving it. In addition, about 15% of the transients that we have measured have significant cusps and yet they still exhibit universality. Several checks were made that allow us to rule out the possibility that cusps are

J. Phys. Chem. C, Vol. 114, No. 19, 2010 9077 caused by space charge (see below). Moreover, their universality with field is inconsistent with the idea that cusps are due to delayed injection by interface traps, as originally suggested many years ago by Pfister8 and by Pfister and Scher,9 among others. Instead, we investigate here the idea that the flat plateau, universality, and the approximately t-2 current decay in the tails, as well as the cusps that occasionally occur, may all be caused by a delay of the charge carriers which arises, not just from traps right at the interface but from a trap-rich surface layer that penetrates from the interface into the bulk of the film. The idea that cusps may be caused by the presence of a surface layer in MDPs that has different trapping characteristics than the bulk has been suggested before.10-12 Tyutnev, et al.10,11 have suggested that there is a thin, approximately one micrometer thick, trap-rich surface layer that causes the charge to be delayed before it crosses into the bulk. Indeed, according to this proposal it is the interplay between the depth of the charge generation region and the thickness of the surface region that determines the particular transient shape seen in any given time-of-flight measurement. As we show in the variation of the MTM that we introduce here, to account for increased trapping in the surface layer and at the same time retain strict universality of the transient, the width Ω of the exponential trap energy distribution in the surface layer must be the same as in the bulk; that is, whatever value of dispersion parameter R characterizes the surface must be the same as that which characterizes the bulk. The only feature that distinguishes the two regions, then, is the trap concentration. In this model, in order to obtain a transient which has a cusp, the trap concentration must be higher in the surface layer than in the bulk. This way, charges will move slowly at first in the surface region and then undergo an apparent acceleration as they enter the bulk. In the most general case, the trap distribution can be described as spatially inhomogeneous, with a higher density of traps near the surface. For simplicity, however, we consider in the analysis presented here a model with just two homogeneous layers. This simple two-layer model allows us to examine the interplay between the charge generation depth (determined experimentally by the electron beam’s energy) and the thickness of the surface region where the trap concentration is larger than in the bulk. Our analysis reveals that, with the use of an appropriate set of parameters, current-time transients are predicted that exhibit (i) an initial spike, (ii) either a plateau, a downwardly sloping plateau, or a cusp, depending on the charge generation depth, (iii) a broad tail, and (iv) universality with respect to electric field. The width in parameter space of the theoretical parameters that describe the data is quantitatively determined in this paper for several model systems. The molecularly doped polymers used in these experiments are the charge transport layer in actual organic photoconductors that are part of commercial electrophotographic copiers and printers. A constraint on the properties of the traps is that the trapping and release times must be less than the electrophotographic cycle time, which is about one second. As the transient conductivity experiments reported here are done in the millisecond time frame, this constraint is easily satisfied. In addition it is well-known that a multiple trapping model can describe transport among a set of transport sites with disordered site energies.13-16 The relevant traps in such a system need not be associated with chemical impurities or other defects, but with transport sites lying in the tail of an energetic distribution. Our use of a multiple trapping model to calculate time-of-flight transients allows a more complete investigation of the relevant

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J. Phys. Chem. C, Vol. 114, No. 19, 2010

Dunlap et al.

parameter space than is possible using the more computationally intensive technique of numerical simulations. The paper is organized as follows. In section II we provide details of the experimental procedure, and in section III we present the experimental results. In section IV we present the two-layer multiple trapping model, the results of which are compared in section V to experimental data. The final two sections contain our discussion and summary. II. Experimental Procedure The molecularly doped polymer MDP used in these experiments was 30 wt % p-diethylaminobenzaldehyde diphenylhydrazone (DEH) in bisphenol A polycarbonate PC. The materials were dissolved in dichloromethane and blade coated onto “release paper” at the Eastman Kodak Company in Rochester, New York. “Release paper” is photographic paper which is coated with a thin layer of polyethylene. Since few polymers stick to polyethylene, when a polymer film is coated on this paper, it can be peeled away or “released” from the substrate. We call the surface that was next to the substrate the “release” surface and the opposite surface the “free” surface. The MDP films were dried by oven curing (80 °C) for onehalf hour. The MDP was then peeled off the release paper and cut to appropriate size. Electrode evaporation of Al was done on both sides, with each electrode being about 26 mm in diameter. Al was chosen because of the low dark-current injection compared to other metals. The film thickness L was determined by measuring the capacitance of the film and assuming a dielectric constant of 3. Film thicknesses ranged from 8 to 46 µm. The electric field was varied by sequentially increasing the applied voltage. Repeatability was verified by returning to a lower voltage and verifying that the same current transient shape was obtained. These data have now been repeated many times on similar samples; see for example the data shown in Figures 1-5 of ref 17. Charge generation experiments were conducted using the ELA-65 electron gun facility1,11 at the Moscow State Institute of Electronics and Mathematics. The facility was operated in single-pulse mode with the kinetic energy Ee of the electrons selected in a narrow window, for a set of energies spanning the range from 3-7 keV, for surface charge generation, to 50 keV for bulk generation. In contrast to the exponential attenuation of a laser excitation pulse, for a given electron energy the electron beam absorption has a well-defined maximum electron range lm. The maximum charge generation rate occurs at a distance approximately equal to one-third of the maximum electron range, and is a factor of 2.3 larger than at the surface. The maximum electron range lm is given by the empirical formula1,11

()

lm ) l0

Ee E0

1.67

(1)

where Ee is the energy of the electron beam, and E0 ) 43 keV and l0 ) 31 µm are empirical constants. The charge generation depth l is assumed equal to lm until lm exceeds the sample thickness, at which point l is taken to be equal to the sample thickness L. The maximum electron range is 1.5 µm for 7 keV electrons. Equation 1 accounts for the observed reduction in transit time as Ee is increased, which is caused by the fact that the front edge of the charge generation region has less distance to travel to the counter electrode. Experiments in our MDP

system show good agreement with eq 1, assuming an effective sample length given by L - lm (see below). The irradiated spot was 20 mm in diameter, allowing for large currents while mitigating any space charge effects. All measurements were made in the small signal regime, meaning generated free charge per unit area is less than 0.1CV, where C is the capacitance per unit area of the sample and V the voltage applied across it. To verify that charge generation was in the small signal regime, the electron beam intensity was lowered by a factor of 10. The resulting transient current shape was observed to be unchanged, while its amplitude decreased by about a factor of 10. All experiments were performed at room temperature and in a vacuum of 3 × 10-2 Pa. All data collection was performed digitally, with collection rates of 4 × 105 s -1 for up to 10 s. Whenever holes are generated inside the sample, there is a counter-electron left behind every time an electron-hole pair is made. In addition to these electrons, the beam itself leaves stopped primary electrons that may form a space charge, especially after repeated pulses. The transient is not affected by these electrons with surface generation (TOF experiment), which leaves electrons near the anode, and which occurs with the electron beam set at 7 keV or lower. But the transient can be affected at higher energies when charge is distributed throughout the bulk. In our experiments the e-beam pulse time was fixed at 25 µs. While beam intensities up to 1 mA/cm2 were available with our equipment, the intensities used in the experiment were much smaller, ranging from a minimum of 0.1 µA/cm2 to a maximum of 10 µA/cm2. For a dielectric constant of 3 and an applied field of 105 V/cm, this corresponds to negligible deposited charge due to primary electrons, ranging from 10-4 to 10-2CV per pulse. To calculate the space charge due to the counter electrons when electron-hole pairs are made, one needs to know the number of free electrons that are made for each high-energy electron that enters a sample. This has been measured in ref 18 for our particular MDP and the results are similar to other polymers. At an electric field of 20 V/µm one separated electron-hole pair is made for every 100 eV of energy deposited in the sample. At 7 keV, about 70 such electron-hole pairs are made. In the field ranges used in these experiments the energy required to make an electron-hole pair follows the predictions of the Onsager theory with the initial distance between charges being 6 nm. Except for the holes caught in deep traps, most of them are swept out of the sample during the time between e-beam pulses (about five minutes) possibly leaving behind trapped electrons. An additional precaution was taken to guard against the accumulation of these electrons: after a return to lower voltages indicated a shape change to the current transient, the sample was annealed in air by heating at 65C for 30-40 min. For reference purposes, no annealing is needed for a voltage run, in which the voltages are sequentially increased throughout the experiment and the energy of the electron beam is set at 7 keV. Annealing is typically performed, however, after about 3 data points in a TOF1a experiment. III. Experimental Results Current transients observed on our samples were typical of prior observations (see almost any prior experimental reference on charge transport in MDP, almost all of which are listed in ref 19 and 20), as shown in Figure 1a, which shows the current transient on linear-linear current-time axes, and in Figure 1b, which shows the same current transient on log-log current-time axes. Note in Figure 1a the initial spike at early times, followed by a plateau out to the transit time (defined as the knee in the

Two-Layer Mutiple Trapping Model

Figure 1. a. Current transient from holes which transit an MDP, 30% DEH (p-diethylaminobenzaldehyde diphenylhydrazone) in PC (bisphenol A polycarbonate) presented in linear current-linear time representation. The vertical axis is normalized current and the horizontal axis is time in ms. The sample thickness is 14 µm and the electric field is 43 V/µm. The theoretical predictions of the two-layer model are plotted as circled dots, see text. The transit time is indicated at the knee in the curve. b. log-log plot of the data in Figure 1a. The theoretical predictions of the two-layer model are plotted as circled dots, see text.

curve, as shown in Figure 1a), followed finally by a decaying current that decreases algebraically as t-2.2 (see Figure 1b). Note that in Figure 1b, the spike seen in Figure 1a is actually an algebraic decay, with a time dependence that goes approximately as t-0.5 before it transitions into a plateau at about 1 ms.17,22 (The two-layer model’s quantitative theoretical predictions are plotted on Figure 1a and b as circled dots, as discussed below.) The electric field dependence of the mobility for all samples characterized follows a Poole-Frenkel law: the mobility is exponential in the square root of the electric field, with a correlation coefficient greater than 0.99, in excellent agreement with virtually all other prior experimental references on charge transport in molecularly doped polymers (see lists in refs 19, 20 and 21). In our case, this also includes samples with cusps (discussed below) once the transit time is defined in a consistent manner, such as the intersection of tangents before and after the cusp. Most of our samples had flat plateaus like that which appears after the initial spike in Figure 1a. As noted in the Introduction, however, some (about 15%) of our samples had currents that rise slightly before the transit time, producing a cusp.8 We have studied the behavior of the cusp as a function of electron beam energy, which determines the maximum electron range lm and the charge generation depth l, as given by eq 1. Figure 2a

J. Phys. Chem. C, Vol. 114, No. 19, 2010 9079

Figure 2. a. Current transients from a 16 µm thick film with variable electron beam energy, from 7 to 46 keV corresponding to charge generation depth of 1.5-16 µm (approximately uniform generation). The 16 µm curve’s vertical amplitude was divided by 3 so that it can be seen on the graph. b. log-log plot of the same data as in panel a.

shows the behavior of a cusp in a 16 µm thick film with an electric field of 40 V/µm as a function of electron energy from 7 to 46 keV, or with the charge generation depth l ranging from 1.5 to 16 µm (used to label each curve). With increasing l the cusp becomes less prominent, until at approximately 4.2 µm what remains of the cusp assumes the appearance of a flat plateau. Note that increasing the charge generation depth shortens the “transit time” simply because the front of the charge generation region has less distance to travel to the counter electrode. As is apparent on logarithmic axes (Figure 2b), the power law behavior before the plateau and after the transit time remains unaltered as l is varied (except for the curve corresponding to uniform generation, or l ) 16 µm, in the region before the transit time). The charge generation depth l at which the cusp evolves into a plateau has been observed to be as small as 2.5 µm in a 10 µm sample and as large as 10 µm in a 46 µm sample. Note that, in agreement with the data shown in Figure 1b, the initial spike observed on the linear-linear plots appears as an algebraic decay on log-log plots before it transitions into a plateau (Figure 2b). Cusps are not due to nonuniform electric fields created by space charge. Space charge can be due to the following effects: (1) dark injected charge from the electrodes that could accumulate in deep traps, (2) the generation of hole densities close to the space charge limit, (3) charge accumulation due to the electrons left behind in the bulk when electron-hole pairs are generated, and (4) accumulation of primary electrons from the electron beam as they stop in the sample. Effect (1) was checked

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Figure 3. log-log plot of a 17.5 µm sample for electric fields of 4, 28, and 56 V/µm, showing universality over an electric field range spanning more than one decade. Note that universality also applies to the cusp.

by using aluminum electrodes and noting that the cusp remained even if the measurement was made within one second or minutues after the application of the voltage. Aluminum electrodes have dark currents of 3 × 10-11 A/cm2 at electric fields of 4 × 105 V/cm. Even if all of the electrons generated in the sample in one second were to fall into deep traps, the space charge produced would be much too small to affect the transient current. Effects (2), (3), and (4) were checked, and eliminated as a possibility, as discussed in section II. Cusps are not a bulk effect. A series of experiments were conducted on several samples with 7 keV electron beam radiation going into the sample from both the free and release surfaces. On reversing the orientation of the sample with respect to the electron beam, we observed that in some cases cusps arose in one direction, but not in the other. There was no correlation with the occurrence of a cusp and the side of the sample (free or release) exposed to the electron beam. The fact that cusps were seen in one direction and not in the other shows that cusps are not an effect that arises in the bulk of the sample. Cusps are unpredictable. They appear on some samples and not on others. This observed unpredictability is consistent with observations reported in the literature, starting with Pfister’s paper.8 Only about 15% of our samples exhibited cusps in the manner described above. For the remaining 85% of the samples, the room temperature current evolves into a flat, or rarely, into a slightly decreasing plateau, followed by an algebraic tail as shown in Figure 1. One of the most important experimental results obtained in this work is the experimental observation that the transient current shape is universal with respect to electric field whether it includes a cusp or not.7,17,22 This is demonstrated in Figure 3 over an order of magnitude in electric field for a 17.5 µm thick sample. The electric fields used were 4, 28, and 56 V/µm. It is clear from Figure 3 that this universality extends over a broad window in time, and that it includes not only the cusp region, but also the tail and the initial spike. As we will see from the derivation of the two-layer MTM presented in the next section, this experimentally observed universality constrains the values of the dispersion parameter R to be the same in both regions. IV. Two-Layer Multiple Trapping Model Based upon observations similar to those discussed above, Tyutnev et al. proposed1,10,11 that some of these experimental results could be explained by assuming that additional trapping

Dunlap et al. takes place in a finite region near the anode surface. The basic ideas are as follows. Cusps are formed in the current-time transient when the charge packet accelerates while traversing the sample. A plateau is formed when the charge packet moves at a constant speed while traversing the sample. A current transient that smoothly decays before the transit time results when the speed of the packet decreases while traversing the sample. For a single layer MTM with a sufficiently broad exponential tail in the distribution of trap energies, a decaying transient is predicted. The current is high at first because charges are generated in the free state manifold. The initially decaying spike in the transient describes a decrease in the average velocity, which comes about because as time goes by charges encounter deeper (and rarer) traps which delay them more and more. Adding a surface region to the MTM with an increased trap concentration serves to enhance this delay. Charges that are generated in the surface layer will initially encounter traps more often and slow down more quickly. They will pick up speed, however, when they cross from the surface layer into the bulk, where the concentration of traps and the opportunity for delay is smaller. If sufficient charge is initially generated in the surface layer, this increase in speed can show up in the current transient as a cusp. On the other hand, if the electron beam energy is sufficiently high, more charges will be generated in the bulk than are generated in the surface region. When this occurs, the relatively small amount of charge that is delayed by the surface region will be insufficient to bring about a noticeable cusp and the current transient will resemble the standard algebraic decay characteristic of the homogeneous MTM. As the energy of the electron beam is tuned from low to high, the charge is generated over a region that straddles both the surface and the bulk and the transient changes shape from a rising cusped shape to a downward sloping decay. Between these two extremes, with a fine-tuning of the parameters, the theory will predict plateaus by (1) striking the right balance between the amount of surface-layer and bulk generated charge, (2) striking the right balance in timing so that the characteristic transit time for charge generated in the surface layer is nearly equal to the transit time for charge generated in the bulk, and (3) having a high enough value of R (approximately 0.80) that the underlying transient current in the plateau does not decay too rapidly. An experimentally observed plateau which arises in this way could mistakenly be construed to indicate equilibrated transport. In fact, for such a confluence of characteristics, the only feature of the transient that is not typical of its true character (nonequilibrated transport) is the existence of the plateau. These ideas can be incorporated into a straightforward extension of the MTM by simply taking the trapping and release rates that appear in that model to be spatially inhomogeneous. The starting point, then, is the pair of equations

∂nf ∂ ) - [V(x)nf] ∂t ∂x

∑ Γi(x)nf + ∑ Ri(x)nt(i) i

∂nt(i) ) Γi(x)nf - Ri(x)nt(i) ∂t

(2)

i

(3)

describing trapping and transport along the direction of the applied field, which we take to be the x axis, in the presence of



traps which immobilize the carriers. These equations describe the exchange of charge carriers between the free-state manifold, in which the charge carrier number density is nf(x, t), and the trap manifold in which the density of holes in traps of species i at x is n(i) t (x, t). The capture rate into traps of species i in the neighborhood of the point x is Γi(x), and the corresponding release rate is Ri(x). In the most general case the drift velocity V(x) in the free-state manifold is also a function of position in the sample. The MTM equations are readily integrated in the Laplace domain. For a delta-function initial condition in which N charges are generated in the free state manifold at the point x ) x0, the Laplace transform of the density of charge remaining in the free state manifold is given by

n˜f(s, x) )

N p˜(x, x0, s) V(x)

(4)

where s is the Laplace variable conjugate to time, and

p˜(x, x0, s) ) exp[-sτ(x, x0, s)]

(5)

that can be evaluated by contour integration, giving a branch point singularity in s characterized by a disorder parameter R(x) ) kT/Ω(x), and an activated attempt frequency [ν ) ν0 exp(- βε0)]. In this last equation, an additional factor [πR(x)/ sin πR(x)] on the order of unity has been absorbed into the definition of c(x). Inserting eq 9 in eq 6 we obtain an arrival time distribution of the Scher-Montroll form

[

p˜(x, x0) ) exp -ν0

R(x')

c(x') s ∫xx dx' V(x') (ν) 0

]

(10)

that has been generalized to include spatial inhomogeneity of the trapping parameters. We assume (as in the discussion above) that injected counterions (electrons) are immobile, and that any charge deposited from the electron beam itself is negligible. Under conditions of constant voltage, the current ˜I(s, x0) in the external circuit is given by the integral of the current density

e I˜(s, x0) ) L

∫0L dx V(x)n˜f(s, x) ) Ne ∫L dx p˜(x, x0) L x 0

(11) is the Laplace transform of the distribution of arrival times for a charge starting at position x0 to reach position x. This latter quantity takes the form of a Laplace transform shifting-function, in which the delay time

τ(x, x0, s) )

Γ (x')

x dx' dx' i + ∫x ∫xx V(x') ∑ V(x') s + Ri(x') 0

0

(6)

i

is given by a sum of two terms. The first is the time it takes a charge to drift a distance x - x0 without trapping, and the second accounts for the additional time the charge has spent sitting in traps. Transport is assumed to be trap-limited; that is, the drift time can be neglected compared to the trapping time. Following standard treatments, we assume a thermally activated detrapping rate

Ri ) ν0 exp(-βεi)

(7)

where ν0 is the prefactor frequency, εi is the trap depth, and β ) 1/kT. To satisfy detailed balance, the associated trapping rate

ci(x) Γi ) ν = ci(x)ν0 1 - c(x) 0

(8)

This is equivalent to a spatial average over the distribution of arrival times. For simplicity, we take the proposal of a surface layer literally and model the MDP film as a composite formed from two adjacent uniform layers, the first being a surface layer of thickness d in the vicinity of the anode in which the concentration of traps is high and the second layer describing the bulk film in which the concentration of traps is lower. We denote the former as region 1 and the latter as region 2. Rewriting eq 11 for this case, we have for x0 < d

˜I(s;x0) ) Ne [ L

∫xd dx p˜1(x - x0) + 0

p˜1(d - x0)

∫dL dx p˜2(x - d)]

(12)

which describes the current due to carriers created in the surface layer, while for carriers created at points x0 > d outside the surface layer

˜I(s;x0) ) Ne L

∫xL dx p˜2(x - x0)

(13)

0

In the last two equations, the arrival time distribution must then be proportional to the fraction of hopping sites ci(x) associated with traps of this species, and to the hop frequency ν0, consistent with the mechanism of diffusion-limited trapping. In the right-hand side of the last equation, we have assumed that the total trapping fraction c(x) ) ∑ici(x) is small. At fixed values of x, for a local distribution of trap energies at depth ε0 having an exponential tail of width Ω(x) > kT, the sum in eq 6 can for small s be approximated as an integral

Γ (x)

(

ε-ε

c(x) ∫-∞ dε exp - Ω(x)0 ∑ s +i Ri(x) ∼ Ω(x) i

∞

)

×

ν0 ν0 s ) c(x) s + ν0 exp(-βε) ν ν

()

R(x)-1

(9)

[

p˜i(x - x0) ) exp -

s ( ) (x - x )] ν βea E ci

2

Ri

0

(14)

is transitionally invariant within each region in which Vi, Γi, and Ri are constants. In writing eq 14 in terms of the electric field E, we have assumed that the Einstein relation V ) βeDE is obeyed in the free-state manifold and taken the free-state diffusion constant D ) ν0a2 to be a product of the hopping frequency and the square of the dopant separation a, which has been assumed to be the same in both layers. The bare hopping frequency ν0 subsequently cancels out because of the assumed mechanism of diffusion limited trapping.

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Dunlap et al.

The electron-beam charge-generation method allows for controlled deposition of carriers over a wide region within the sample. To account for this we integrate I˜(s; x0) over the distribution of charge associated with the e-beam deposition pattern. Since the e-beam has a sharply attenuated tail, for simplicity we take the initial distribution of generated charge to be a square pulse of width l, for which the current in the TOF1a experiment can be written as the integral

˜I(s) ) 1 l

∫0l dx0˜I(s;x0)

(15)

of eqs 12 and 13 over the charge generation region. Inserting eqs 12 and 13 into eq 15, this integration may be carried out analytically. The explicit result appears in the Appendix [see, e.g., eqs 27 and 28]. The important thing to note from that result is that for N generated charges the Laplace transform of the current is proportional to a function F of four dimensionless quantities, i.e.

[

I˜(s) ) NeF

c1L

s ( ) ν βea E

R1

2

,

c2L

R2

s ( ) ν βea E 2

d l , , L L

]

(16)

To determine the current time transient I(t) in the time domain it is then necessary to invert the Laplace transform I˜(s) by integrating along the Bromwich contour, i.e.

I(t) )

Ne 2πi

i∞ ds estF(s) ∫-i∞

(17)

To investigate the possibility of universality, we introduce a time scale

t0 )

( )

1 c2L ν βea2E

1/R2

(18)

with which we define a dimensionless time variable τ ) t/t0, and a dimensionless current

i)

t0 I Ne

(19)

Rewriting eq 17 in terms of these variables, and transforming the variable of integration to a dimensionless Laplace variable ξ ) st0, gives the following expression:

i(τ) )

1 2πi

[ ( ) c

i∞ 1 dξ eξτF ξR ∫-i∞ c2

1

c2L 2

βea E

1-R1/R2

]

d l , ξR2, , L L (20)

for the scaled current as a function of the scaled time. From this last expression it is clear that universality of the transient shape with electric field E occurs when the two regions have the same disorder parameter, i.e., when R ) R1 ) R2, for then the function F is independent of E. For this case, the shape of the transient

i(τ) )

1 2πi

[

c

i∞ 1 d l dξ eξτF ξR, ξR, , ∫-i∞ c2 L L

]

(21)

then depends only on the four dimensionless parameters R, d/L, l /L, and c1/c2. In our numerical studies of the transient shape that appear in the following sections, we numerically invert eq 21 for different values of these four parameters. As described below, the sample thickness L and the charge generation depth l are determined experimentally, which leaves us three sample specific theoretical material parameters d, c1/c2, and R, that can be adjusted to best match the shape of the experimental current-time transients. V. Comparison of The Two-Layer MTM with Data In this section we present model calculations in which predictions of the two-layer MTM derived in the last section are compared to the experimental data already presented. In these comparisons, the sample thickness L and the charge generation depth l were obtained experimentally. The dispersion parameter R was taken to be the same in both layers, in keeping with the observed universality of the current transients with electric field. The thickness d of the surface layer, the relative trap concentration c1/c2, and R were varied to obtain current time transients that provide a good fit to the experimental data. A. Plateau Generation in the Two-Layer MTM. As a first example, note the circled dots in Figure 1a and 1b, which represent our best quantitative fit of the two-layer MTM to the experimental data, which possesses a well-developed plateau of the type seen in many experiments on MDPs. To obtain this fit, we set L ) 14 µm and l ) 1.5 µm, consistent with the measured values of the capacitance and the electron beam energy. A least-squares fitting routine was used in an attempt to optimize the fit. Although it is quite easy in the two-layer MTM to produce a transient with a well-defined plateau, we were not able to find a set of parameters that simultaneously fit the tail, the plateau, and the initial spike of the data appearing in that figure. We therefore performed a restricted fit to the region of the experimental data associated with the plateau, the transit time, and the tail region, letting the behavior in the region of the initial spike vary freely. The resulting fit, shown in Figure 1, was obtained with a surface layer d ) 0.65 µm, a trap concentration ratio c1/c2 ) 11.5, and a dispersion parameter R ) 0.77. It clearly does an excellent job reproducing the data in the regions where the fit was made, but it underestimates the strength of the power law associated with the initial spike. This initial calculation does, however, serve to demonstrate one of the main features of the conjecture of Tyutnev, et al., namely, that a current transient with a perfectly flat plateau can be produced in an MTM-type system in which the actual equilibration time is infinite (as it is in this case with R ) 0.77), provided that the MTM parameters are inhomogeneous over the right length scale. Note that in this example the plateau is predicted to appear in a sample in which the charge generation layer l is a little more than twice the thickness d of the surface layer, so that almost equal amounts of charge are generated in the surface layer and in the bulk region immediately beyond it. B. Variation of the Transient with Charge Generation Depth: The TOF1a Experiment. According to the picture of Tyutnev, et al., which provided the motivation for the two-layer model used in this analysis, the plateau appearing in Figure 1 arises from a fine-tuning of the relative widths of the charge generation region and of the surface layer. According to this idea, a shorter charge generation region should alter the transient so as to produce a cusp, which can occur when all the charge is generated in the surface region. Similarly, a wider charge generation region should shorten the plateau, and ultimately lead


Figure 4. a. A fit of the experimental data of Figure 2a using the two-layer MTM with L ) 16 µm. Curves are labeled by their charge generation depths. The vertical axis is normalized current and the horizontal axis is time in ms. The best fit occurs with a surface layer d ) 1.4 µm, a concentration ratio c1/c2 ) 7 and R ) 0.8. Each curve is vertically scaled by the corresponding electron energy, except for the 46 keV curve (labeled by the 16 µm sample thickness) which is also reduced by a factor of 3 to place it on the graph. b. log-log plot of the same calculation presented in panel a. The straight line corresponds to a power law of t-2.

to an algebraically decaying current transient. Both of these general trends are observed in the TOF1a experiments presented in Figure 2a, in which the charge generation depth is systematically varied. To test this behavior on the two-layer MTM derived in the last section, we performed a trial-and-error fit of the model to the TOF1a data appearing in Figure 2, panels a and b, searching for a fixed set of sample parameters (d, c1/c2, R) that qualitatively reproduce the changes that occur with increasing charge generation depth l observed in the TOF1a experiment. For this calculation, we set L ) 16 µm, and ran numerical calculations for the specific charge generation depths l ) 1.5, 3.2, 4.2, 6.5, and 16 µm (obtained from eq 1 and the known electron beam energies) shown on the curves in Figure 2, panels a and b. The results of our fit (d ) 1.4 µm, c1/c2 ) 7, R ) 0.8) presented on double linear and double logarithmic axis, appear in Figure 4, panels a and b, respectively. We emphasize that the different theoretical curves in Figure 4, panels a and b, were all produced with the same assumed surface layer thickness, the same concentration ratio, and the same value of R. The amplitudes of the theoretical curves are arbitrary; in the figure they were scaled by the energy of the electron beam, except for the l ) 16 µm curve, which was also divided by a factor of 3 to allow it to be plotted with the other curves (the same renormalization

J. Phys. Chem. C, Vol. 114, No. 19, 2010 9083 was applied to the l ) 16 µm experimental data in Figure 2). This choice of normalization appears to closely capture the amplitude behavior of the experimental data in Figure 2a. In general, the semiquantative agreement of the entire set of theoretical curves with the corresponding set of experimental curves is rather compelling. There are minor differences, however. Notice in Figure 4a that for charge generation depth l ) 1.5 µm, a cusp is predicted by the theoretical calculation that is a little wider than the cusp that appears in the corresponding experimental curve. For charge generation depth l ) 3.2 µm, a plateau is predicted, while the corresponding experimental data rises very slightly. The theoretical curve for l ) 4.2 µm decays slightly, while experimentally this is where the plateau occurs. At l ) 6.5 µm, the theoretical prediction is of a decaying current, very similar to the experimental data. With bulk generation, at l ) 16 µm, the theory predicts a straight power-law decay that looks very similar to the experimental data. In the log-log plot of Figure 4b, it can be seen that the decay of the tail of the transient, which goes approximately as t-2, is in very good agreement with the experimental data. Before the transit time the predicted slope is t-0.17 which underestimates the experimental result, consistent with the results shown in Figure 1. Most importantly, however, aside from the minor differences noted above, it is clear that the trend of the change in shape with increasing charge generation depth predicted by the two-layer model is exactly as it appears in the TOF1a experiments, supporting the intuition of Tyutnev et al. regarding the consequences of such a surface layer. C. Width in Parameter Space of the Predicted Current Plateau. As we have demonstrated above, it is possible within the two-layer MTM to produce a current transient with a perfectly flat plateau. We have also shown that the two-layer MTM is consistent with experimental observations when the charge generation depth l is varied, as in a TOF1a experiment. In the model, however, the plateau itself can only occur through a fine-tuning of the three theoretical parameters. Since most of the room temperature current transients observed on MDPs do indeed show a plateau, it is important to determine the width in parameter space over which such a plateau is predicted to occur. To address this question we have performed a series of calculations to determine the range in parameter space over which a plateau is produced. For a given sample length L and charge generation depth l, we will refer to the set of parameters (d, c1/c2, R) giving the flattest plateau as a “plateauoptimized set”. To test the width in parameter space for this parameter set, we have systematically varied each of the adjustable parameters above and below its plateau-optimized value until the slope of the plateau varies by 10% (a rough estimate of the experimental uncertainty), or until it develops features that clearly do not match the experiment (such as cusps, or a transient tail that decays too rapidly). We thus present in this section bounds on the width in parameter space that are consistent with (i.e., within 10% of), a plateau-like transient, for several different charge generation depths. 1. Width in Parameter Space for a Sample with L ) 14 µm, l ) 1.5 µm, as in Figure 1. We consider first a system close to that appearing in Figure 1a, which corresponds to a 14 µm sample with a charge generation depth of 1.5 µm. In Figure 1 the theoretical curve resulting from the least-squares fitting routine described above does not actually produce the flattest plateau, but it is close. We find that the plateau-optimized parameter set for this sample thickness and this charge genera-

9084


Dunlap et al. 0.75. Thus, for the data in Figure 1, the width of parameter space that predicts a plateau is

0.3 µm < d < 1.2 µm 10 < c1 /c2 < 30 0.75 < R < 0.85 2. Width in Parameter Space for a Sample with L ) 16 µm, l ) 3.2 µm, as in Figure 4. We now consider a system such as that appearing in the TOF1a data of Figure 2, taken on a sample of length L ) 16 µm. The curve in that figure corresponding to a charge generation depth l ) 4.2 µm also shows a plateau of the type seen in many MDP experiments. The semiquantitative fit of the two-layer MTM for this system, presented in Figure 4a, actually has the plateau occurring at a slightly different charge generation depth, l ) 3.2 µm, as we have already noted. Since the fit is only semiquantitative, and we are interested here in the width in parameter space, we ignore this slight difference in charge generation depth, and consider the behavior of the two-layer model in the neighborhood of the theoretical curve in Figure 4 that exhibits the plateau, i.e., we take L ) 16 µm and l ) 3.2 µm, and plateau-optimized parameters (R ) 0.80, d ) 1.4 µm, c 1/c2 ) 7). In a manner entirely similar to the analysis presented in the last subsection, we determine that a plateau can be produced for this system only with parameters lying in the range

Figure 5. a. The middle curve (d ) 0.7 µm) is the best plateau optimized fit to Figure 1a which has a generation depth of 1.5 µm. The width of the d parameter space is shown in the figure. The plateauoptimized parameters for l ) 1.5 µm and L ) 14 µm are d ) 0.7 µm, R ) 0.8, c1/c2 ) 20. The other curves show what happens when d approaches and exceeds the generation depth l ) 1.5 µm. b. Log-Log plot of the same data as in panel a. Note the double knee after the transit time in the predicted current for d ) 1.2 µm, a feature not seen experimentally.

tion depth actually corresponds to (R ) 0.80, d ) 0.7 µm, c1/c2 ) 20). The plateau predicted by these parameters appears in Figure 5a as the curve with d ) 0.7 µm. Although we do not show a direct comparison, that curve is in reasonable semiquantitative agreement with the experimental data of Figure 1a. The change in the current transient shape that occurs by changing the width d of the surface layer is also shown in Figures 5a and 5b. Note that at d ) 0.3 µm the transient decays before the transit time, while at d ) 1.2 µm a double knee appears in the log-log plot, inconsistent with experimental transients. This double knee arises when the charge that is generated in the surface region is delayed too much, leading to two distinct arrival times that are both visible on a logarithmic scale and appear as an overly broad tail on a linear scale. Through detailed calculations of this sort, we have determined that for this sample length (L ) 14 µm) and charge generation depth (l ) 1.5 µm), a reasonable plateau-like transient (within (10%) can be produced only with a surface layer whose thickness lies in the range 0.3 µm < d < 1.2 µm. Similar calculations can be made in which c1/c2 and R are varied about their plateau-optimized values. We find that when c1/c2 shrinks to 10 a cusp develops, and when the ratio is greater than 30 a double knee appears in the log-log plot. Finally, we find that with d and c1/c2 fixed at their plateau-optimized values, variation of the dispersion parameter R leads to the formation of a cusp when R ) 0.85, and to a tail that decays too rapidly when R )

1.0 µm < d < 1.7 µm 3 < c1 /c2 < 15 0.75 < R < 0.85 3. Width in Parameter Space for a Sample with L ) 14 µm, l ) 0.2 µm. As a final example we consider a theoretical calculation corresponding to a charge generation depth l ) 0.2 µm, which occurs using a nitrogen laser and a highly doped polymer film. (The absorption coefficient for 30% DEH:PC is approximately 0.2 µm. For higher percentage doping the absorption coefficient is correspondingly higher. Since a uniform photogeneration is assumed in the theory, l should probably be taken to be two or three times the absorption coefficient.) For this study the sample thickness was chosen to be L ) 14 µm, and the plateau-optimized parameters were determined through numerical calculation to be (R ) 0.80, d ) 0.08 µm, c1/c2 ) 150). The predicted plateau is shown in Figure 6, along with curves that show the variation in the shape of the current transient produced by a variation in R. Through the process described above, we determine for this system that current transients possessing plateaus similar to those seen in experimental data, require parameters lying in the following ranges:

0.06 µm < d < 0.10 µm 100 < c1 /c2 < 200 0.75 < R < 0.85 D. Changing Sample Thickness. It should be obvious, given the discussions above, that changing the sample thickness will change the plateau-optimized parameters, because the transit time itself changes. As an example, for a sample of thickness L ) 15 µm, and a charge generation depth l ) 1 µm, the plateau-optimized parameters are R ) 0.80, d ) 0.4 µm, and c1/c2 ) 27. For a sample thickness L ) 35 µm, with the same thickness d of surface layer, and the same l ) 1 µm charge


Figure 6. Plateau-optimized transient for a 14 µm thick sample and a generation depth of 0.2 µm. The plateau-optimized parameters are d ) 0.08 µm, c1/c2 ) 150, R ) 0.80. The width of the parameter space associated with alpha is shown in the two curves, R ) 0.85 and 0.75.

generation depth, to obtain a plateau requires the trap concentration ratio to increase to c1/c2 ) 60. If the trap concentration ratio and d were unchanged, the resulting current transient would display a noticeable cusp. If c1/c2 is held constant and d varied, a plateau can not be obtained for any value of d, the surface layer thickness. VI. Discussion Experimentally we have demonstrated that universality extends over a broad window in time, which not only includes the cusp or plateau,17,20 but also includes the tail and the initial spike of the current transient. It is a remarkable experimental observation that universality is observed over the whole curve shape, independent of the charge generation depth and regardless of whether or not the transient exhibits a cusp. We suggest calling this “strong universality” as opposed to “universality” which has been taken to include only the tail and plateau in MDP experiments. We have shown theoretically that charge generation in a twolayer MTM can describe the cusps seen in experiment, as well as the systematic changes in curve shape that occurs as the charge generation depth changes, for a narrow range of parameters. Furthermore, if we assume that the R ’s are the same in the surface and bulk, we are assured to have universality with field as is observed experimentally. Thus the results of our model support the conjecture by Tyutnev, et al.1 We have also been able to theoretically predict current transients displaying plateaus of the type commonly reported in transient conductivity experiments in molecularly doped polymers. Our parameter width studies show, however, that the range of parameters over which a perfectly flat plateau is produced is fairly narrow. It is not clear how such a narrow range of parameters in this two-layer model can account for the 85% of room temperature transients observed experimentally in all MDPs. It is possible that the discontinuity in trapping parameters that occurs at the interface between the surface layer and the bulk reduces the width of the parameter space of the plateau in our model, relative to one in which the trapping parameters vary continuously. Our model demonstrates, however, that the current plateaus commonly reported in the MDP literature cannot be taken as unequivocal proof of equilibrated transport. Indeed, as we have shown, they can arise from a convolution of currents arising from charge carriers created at

J. Phys. Chem. C, Vol. 114, No. 19, 2010 9085 different depths relative to the thickness of a surface layer where enhanced trapping occurs. As can be seen by comparing the values of the plateauoptimized parameters deduced in the last section, a plateau is achieved when the surface layer thickness d is about half the charge generation depth l. As the charge generation region becomes thinner, the value of c1/c2 needed to produce a plateau becomes larger, from a value c1/c2 ) 7 at a charge generation depth of l ) 3.2 µm, to a value of c1/c2 ) 150 at a charge generation depth of l ) 0.2 µm. There are several issues associated with this observation. First, the samples used in these experiments are, as far as we know, identical to those used in earlier studies23 in which photogeneration was accomplished by direct excitation with a nitrogen laser (337 nm). If the theory introduced here is in fact applicable to these samples, one would need to understand why, in basically the same MDP, a charge generation depth l ) 0.2 µm using photogeneration and a charge generation depth l ) 1.5 µm using electron beam excitation both produce plateaus; the two-layer MTM requires very different surface layer thicknesses d and concentration ratios c1/c2 to predict plateaus for these two very different charge generation depths. Second, why is the observed frequency (∼ 15%) of cusps about the same in both the earlier work and that reported here. As we have said and partially demonstrated in Figures 5a,b and 6, the width in parameter space that can produce a plateau is narrow. The region in d space that gives the plateau is about half of the charge generation depth (50%. Outside of this region, transients shapes are predicted which have not been observed (see Figure 5b). For values of d near the charge generation depth, we find a double knee on the logarithmic scale, arising from two distinct arrival times, the first from charges that were generated in the bulk, and the second from charges that were generated in the surface layer. For very small values of d, the transition is dominated by homogeneous MTM behavior, because very few charges are generated in the surface layer. The region in c1/c2 space that gives a plateau appears to get smaller as the charge generation depth shrinks. It is almost (100% at l ) 4.2 µm and (20% at l ) 0.2 µm. The value of R required to fit the plateau, the universality, and the time dependence of the current after the transit time is 0.80 ( 0.05 and must be the same in the surface layer and in bulk regions (see Figure 6). It is well-known that lowering the temperature leads to a disappearance of the plateau, and to a decaying current transient.19 This type of behavior is predicted by the two-layer model as R ) kT/Ω is reduced (see Figure 6). It seems to be a general practice in the MDP literature to mention the experimental observation of cusps, but then to ignore them on the assumption that they are due to an extrinsic effect. The observation of universality, however, even when cusps are observed suggests that cusps are, in some sense, an intrinsic effect. To account for the low 15% fraction of experimental observations with cusps, one could argue that, for material processing reasons that are not well understood, the surface layer thickness is different for different samples, and that only about 15% of samples produced have the correct coincidence of the charge generation depth and the surface layer thickness needed to produce a cusp. This could account, e.g., for our observation of a sample that displays a cusp in one orientation but a plateau in the other. In our experiments, we have found that it is always possible to make the shape of the current transient transition from a cusp to a plateau by simply increasing the electron beam energy, consistent with the

9086


Dunlap et al.

predictions of the two-layer MTM. But we also have data in which a cusp never appears at any electron beam energy, even as low as 2 keV. Within the two-layer model, the inability to form a cusp could be accounted for by postulating a surface layer of negligible thickness. But then it is not clear how a plateau is formed. In our model, we have assumed the existence of a surface layer which has a trap energy profile the same as in the bulk, but with a higher trap concentration. There is evidence in the electret literature that a surface layer with deep traps is formed when a metal is evaporated onto the surface of a polymer.24 However, we have no direct physical evidence for the existence in our materials of the assumed surface layer or its enhanced trapping properties. It is possible that such a region could be caused by surface chemistry between the electrode and the organic material. Alternatively, the hot aluminum evaporation associated with contact formation may create traps in the sample.25 We have shown above that to maintain strict universality it is necessary that the value of R be the same in the surface layer as it is in the bulk. This requires that, if the traps in the surface layer are deeper, or if there are more of them, the logarithmic slope of the tail of the trap distribution must nonetheless remain the same. The simplest explanation for such a coincidence of R ’s is that whatever is causing trapping in the bulk is exactly the same as in the surface layer. The trap distribution has been assumed to be exponential in our two-layer model. Although an equilibrated drift velocity, and hence a true mobility, does not exist in any MTM for the values of R that we have considered, we can operationally define an effective mobility µ ) L/Et0, in terms of the time scale t0 defined in eq 18 (with R2 set equal to the common value of the dispersion parameter R ) kT/Ω). The effective mobility

µ)ν

( )

L c2L E βea2E

-1/R

(22)

where ν ) ν0e-βε0, will then display thermally activated behavior, due to the linear dependence of the dispersion parameter R ) kT/Ω on temperature. In this expression, the activation energy

[(

∆ ) ε0 + Ω ln

) ]

1 c2L -1 βeEa a

(23)

will have a weak logarithmic dependence on temperature and field that could show up as a small deviation from a strict Ahrennius law. It is well established that the mobility derived from the transit time obtained in measurements on MDPs is, in fact, thermally activated. A list of activation energies for all measured MDPs has recently been published.26 If the distribution of trap energies is not exponential, but some other function, then an MTM can still be implemented, but the strong universality with field that arises with an exponential trap distribution will be lost. With a Gaussian distribution, for example, one would probably want to use a displaced Gaussian to represent the trap distribution. The point has been made, however, that as long as that part of the trap distribution that affects the transient in the time window of interest decays sufficiently slowly in energy, for all practical purposes it behaves as an exponential; the transient will then appear to be universal, and a meaningful value of R may be associated with the transient.4

From eq 22 it also follows that the electric field dependence of the apparent mobility arising from the two-layer MTM takes the form

( EL )

µ∝ν

(1-R)/R

(24)

where ν ) ν0e- βε0 is an effective detrapping rate. When R < 1, and the detrapping rate ν is independent of field, the predicted mobility will increase algebraically with the applied field. For the values of R ∼ 0.8 used in our analysis, however, this field dependence is very weak, and cannot account for the much stronger Poole-Frenkel field dependence observed in our samples. Thus, if the two-layer MTM introduced here is be applicable, the detrapping rate ν must be the source of the strong Poole-Frenkel behavior observed. Being essentially heuristic, the MTM does not explicitly identify the nature of the traps upon which it is based. It simply assumes the existence of transport states through which the carriers move across the sample in the presence of an exponential distribution of lower energy traps, with transport between the traps forbidden. In molecularly doped polymers, the transport level is usually assumed to be associated with those states of the dopant molecule through which unhindered transport occurs. In principle, the relevant trap states then could be localized electronic states associated with charged impurities, with neutral impurities having large dipole moments, or they could simply represent the tail of the distribution of site energies associated with localized hole states of the transport molecules themselves, and arise naturally from local environmental fluctuations.13-16 In the case of traps arising from charged impurities, one would expect a Poole-Frenkel field dependence to the detrapping rate arising from the Poole-Frenkel effect, i.e., the bending of the Coulomb barrier by the electric field; but as discussed elsewhere,21 the peak of the Coulomb barrier rp in that case would be identical for all MDPs, and is not the right magnitude to explain the observed mobility and its variation in MDPs. A third, quite different possibility is that the trap states relevant to an MTM description are not actually associated with localized electronic states on specific individual molecules, but that, instead, they are associated with transport sites located in deep and relatively broad energetic valleys embedded in a smooth energy landscape of the type that arises with correlated disorder.27-29 For this situation the effective “detrapping rate” would then correspond to the release rate for carriers escaping over the energy barriers surrounding the deep valleys. For sufficiently long-range correlations, the dynamics of this latter process does indeed obey a Poole-Frenkel law over a wide range of the applied field.27-29 VII. Summary This work has examined the commonly observed plateaus, as well the cusps that are experimentally observed in only about 15% of time-of-flight experiments. Transients displaying cusps are usually ignored by experimentalists, because it is thought that they are due to extrinsic effects. We have discovered experimentally that transients with cusps exhibit universality with field. This is a new result and suggests that cusps are somehow an intrinsic effect. We have systematically ruled out space charge and surface injection as possible sources for cusps. Tyutnev, et al. have suggested that a cusp is caused by delay of the charge packet in a trap-rich, relatively thick, surface layer, that is a generic feature of MDP samples.1 To explore this idea

Two-Layer Mutiple Trapping Model we have implemented a two-layer multiple trapping model with an exponential density of states in the surface and in the bulk. The surface layer has more traps than the bulk but the same trap energy distribution. In the context of this model, cusps can be understood as arising from the increase in current that occurs when charges, initially generated in the surface region where they move slowly due to increased trapping, speed up when they cross over into the bulk. Universality (in the strong sense) is obeyed as long as the disorder parameter R is the same in both regions. For just the right set of parameters, the cusp flattens out, and the transient appears to have a perfectly flat plateau accompanied by a broad tail. Once the thickness of the sample and the charge generation depth are given, the model has only three adjustable parameters, i.e., the thickness d of the surface layer, the concentration ratio c1/c2 of traps in the surface layer relative to traps in the bulk, and the dispersion parameter R. A shortcoming of the model is that it can explain the experimental data only over a narrow range of the three parameters. To predict a plateau-like transient with a tail that resembles those seen in experiment and account for universality, the value of R has to be the same in both regions and in the range R ) 0.80 ( 0.05. With this value of R the plateau and tail of the current transient can be quantitatively fit, but the slope of the initial spike is underestimated. The width d of the surface region must be within about 50% of one-half the width l of the hole generation depth. The value of c1/c2 must increase as the charge generation depth becomes thinner, from a value c1/c2 ) 7 at charge generation depth l ) 3.2 µm, to a value of c1/c2 ) 20 at charge generation depth l ) 1.5 µm, to a value of c1/c2 ) 150 at charge generation depth l ) 0.2 µm. These shortcomings aside, we have shown that the two-layer MTM model is semiquantitatively consistent with experimental observations when the charge generation depth is varied, as in the TOF1a experiment over a wide range of charge generation depths, from surface generation to bulk generation, with only one set of the three theoretical material parameters. Indeed at the moment this is the only theoretical model that we are aware of that has demonstrated its ability to reproduce the behavior of a TOF1a experiment. In this model plateaus are predicted through fine-tuning of the three theoretical material parameters. Nonetheless, we find that the characteristic features of the standard time-of-flight TOF experiment in MDP including the initial spike, a flat plateau, and an anomalously broad tail, as well as the sometimes observed cusp or decreasing current occurring near the transit time, can all be described by this twolayer model. In this model plateaus are not an indication of equilibrated transport but instead are the result of the charge carriers delayed by a trap rich region in a surface layer. Acknowledgment. The authors would like to thank David Capitano of the Eastman Kodak Company for preparing the films used in these studies. Appendix The Green function I˜(s; x0) given in eqs 12 and 13 is the transient for the case that N charges in a sample of cross sectional area A are initially generated at location x0. As expressed by eq 15, the current for a uniform initial charge distribution is the integral of eqs 12 and 13 over a box distribution of initial values x0. The current can be written in terms of the dimensionless Laplace variable ξ, the disorder parameters R1 and R2, the dimensionless ratios c1/c2, d/L, and l/L, and an additional scale factor


λ)

( ) c2L

1-R1/R2

(25)

βea2E

which goes to 1 when R1 ) R2. The overall constant of proportionality is NeA/L. The integration is over exponential functions, and is straightforward, giving

(

c1 d l I˜(s) ) NeF λ ξR1, ξR2, , c2 L L

)

(26)

where the dimensionless function F has two forms, depending on whether d < l or d > l. For d < l, we have

F)

{

(

)

c1 d 1 1 exp -λ ξR1 × 1c1 R1 c1 R1 l c2 L λ ξ λ ξ c2 c2 L c1 l 1 × exp λ ξR1 - 1 + c1 R2 R1 l c2 L λ ξ ξ c2 L c1 R1 l L-d exp λ ξ - 1 1 - exp -ξR2 × c2 L L c1 d exp -λ ξR1 c2 L

[ ( ) ]} {( [ ] )(

[

])

]}

[

(27)

For d > l we have

{

( [

] ) { [ ] [ ])}

c1 d d 1 exp λ ξR1 - 1 × c1 c1 l l c2 L λ ξR1 λ ξR1 c2 c2 L c1 R1 d c1 d 1 exp -λ ξ + exp -λ ξR1 × c1 c2 L c2 L l λ ξR1ξR2 c2 L c1 R1 d L-d exp λ ξ - 1 1 - exp -ξR2 + c2 L L l d d 1 1 1- exp ξR2 - exp ξR2 exp[-ξR2] R2 l l L L ξ ξR2 L (28)

F)

1

]} ( [ ] )( [

{

( [ ]

[ ])

}

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