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PHYSICAL REVIEW B 66, 205208 共2002兲

Nondispersive charge-carrier transport in disordered organic materials containing traps I. I. Fishchuk* Institute for Nuclear Research, National Academy of Sciences of Ukraine, Prospect Nauky 47, 03680 Kyiv, Ukraine

A. K. Kadashchuk Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauky 46, 03028 Kyiv, Ukraine

H. Ba¨ssler Institute of Physical, Nuclear and Macromolecular Chemistry and Material Science Center, Philipps-Universita¨t Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

D. S. Weiss Heidelberg Digital L.L.C., 2600 Manitou Rd., Rochester, New York 14624-1173 共Received 22 April 2002; revised manuscript received 19 August 2002; published 27 November 2002兲 An effective-medium theory has been developed to describe the nondispersive charge-carrier transport in a disordered organic material containing extrinsic traps. The theory can account quantitatively for a variety of basic features of charge-carrier transport, such as the dependence of the charge-carrier mobility on the concentration of traps and the temperature and field dependence of the mobility. The results of calculations are compared to predictions of the Hoesterey-Letson formalism, which has been widely used to describe trapping. We argue that our theory describes more adequately charge transport in the presence of traps, since it accounts for the effects of disorder. The results of the calculation are found to be in good agreement with both experimental data and computer simulations. Our calculations support a notion that the effect of traps can be quantitatively accounted for by the introduction of an effective disorder parameter. Also, it is found that both the relaxation of the ensemble of majority charge carriers within the combined intrinsic and the extrinsic density-of-states distribution and the occurrence of trap-to-trap migration alter the temperature dependence of the charge mobility significantly, notably at lower temperature. DOI: 10.1103/PhysRevB.66.205208

PACS number共s兲: 72.20.Jv, 66.30.⫺h

I. INTRODUCTION

Charge-carrier transport phenomena in disordered organic solids has been the subject of intensive research for many years 共for reviews, see Refs. 1– 4兲, stimulated by the use of these materials in electrophotographic receptors, organic light-emitting devices, organic solar cells, thin-film transistors, etc. Many recent studies of a wide range of molecularly doped polymers 共MDP’s兲 and conjugated main chain- and pendant-group polymers, as well as vapor-deposited molecular glasses, have been described within the framework of the Gaussian disorder model of Ba¨ssler.4,5 The model is premised on the argument that charge-carrier transport occurs by hopping through a Gaussian density of transport states 共DOS兲 of energetic width ␴. Later, Gartstein and Conwell6 suggested that the spatial correlation of the energies of transport sites in disordered media has to be taken into account in order to properly describe the field dependence of mobility over the broad field range. Extensive computer simulations in Refs. 7 and 8 resulted in an amended version of the formalism which provides an adequate basis of experimental analysis. The latter was also supported by an analytic effective-medium theory recently developed by Fishchuk et al.,9 which demonstrated that the disorder formalism can also account for peculiarities of the charge transport in weakly disordered organic systems, provided one uses the correct expression for the jump rates. One of the most important limitations of the disorder for0163-1829/2002/66共20兲/205208共12兲/$20.00

malism is, however, that it does not address carrier transport in the presence of traps. This has led to the recent interest in trapping effects10–20 in disordered organic media. Traps in disordered media are extrinsic localized states that differ from the majority of hopping 共transport兲 states in that they require substantially larger energy input to release the charge carriers back to the intrinsic DOS. For instance, hole or electron transport proceeding via one kind of charge transporting material would be affected by trapping on a molecule of another kind provided the latter has a lower ionization potential or a higher electron affinity, respectively. Traps of structural origin 共such as dimers, aggregates, etc.兲 are also possible. The problem of trap-controlled transport is now recognized as also being very important from the applied point of view in the application of organic materials to electronic devices. For instance, electronic transport in ␲-conjugated polymers is often trap affected, and chargecarrier transport controlled by deep trapping seems to be especially important in photorefractive organic systems where moderately deep traps play an essential role. Thus a reliable theoretical basis for describing trap-controlled transport is necessary for the continued development of materials for organic electronic devices, as well as for better understanding the complexity of the phenomenology of charge transport in disordered molecular solids where accidental impurities may give rise to long release times. A first attempt to extend the disorder model to include the effect of extrinsic trapping was done by Wolf et al.10 and

66 205208-1

©2002 The American Physical Society

¨ SSLER, AND WEISS FISHCHUK, KADASHCHUK, BA


Borsenberger et al.11 for what was referred to as the shallow trapping case. The results of computer simulations and photocurrent transient measurements showed that the presence of a distribution of shallow traps did not change the basic phenomenology of transport, as revealed by the temperature and field dependence of the mobility. It was found that the effect of shallow traps can be quantitatively accounted for by rescaling of ␴ with an effective 共broader兲 width ␴ eff .10,11 To obtain the relationship of the trap-controlled to trap-free mobility the Hoesterey and Letson expression21 has usually been used, although it originally was developed for organic crystals and is based on a discrete trap depth argument 1

␮ 共 c 兲 ⫽ ␮ 共 c⫽0 兲 h ⫺1 ⫽ ␮ 共 c⫽0 兲

1⫹c exp

冉冊 Et k BT

,

共1兲

where h is a trapping factor reflecting the increase of the transit time by the time spent by a carrier in traps, c is the relative trap concentration, ␮ (c) is the trap controlled mobility, ␮ (c⫽0) the trap-free mobility, and k B is Boltzmann’s constant. The employment of the Hoesterey-Letson formalism for the condition c 关 exp(Et /kBT)兴Ⰷ1 yields

冉冊

冉冊冋

␴ eff 2 3k B T ⫽1⫹ ␴ 2␴

2

ln共 c 兲 ⫹

册

Et , k BT

共2兲

where ␴ is the width of the intrinsic DOS in the absence of traps and ␴ eff is the effective disorder parameter as derived from experiment. It was shown that the Hoesterey-Letson formalism provides a reasonable zero-approximation to describe trapping; however, it has certain shortcomings. The key predictions of the formalism are the following: 共i兲 the mobility scales with relative trap concentration as c ⫺1 , and 共ii兲 the concentration at which the mobility is decreased by a factor of 2 is c 1/2 ⫽exp(⫺Et /kBT). However, many recent experimental results12–17 on organic disordered materials are in disagreement with the above predictions. First, the Hoesterey-Letson formalism predicts the onset of the trap-controlled regime at much lower trap concentrations than are actually required, especially for deep traps where the discrepancy is up to a few orders of magnitude.15 Second, experimental results showed that coefficient n in the concentration dependence

␮⬀

1 cn

共3兲

is not a constant but increases with increasing trap depth up to approximately 1.5.15 Another puzzling experimental result, for which no plausible explanation has thus far been offered, is the observation of a coefficient n less than unity.14,15 Thus the above examples demonstrate that the HoestereyLetson formalism, which is probably the simplest approach to describe the effects of trap concentration, eventually fails in the case of amorphous systems. This is not unexpected as the formalism was originally developed for systems devoid of disorder and it is based on a discrete trap depth, an assumption which is probably unrealistic for disordered or-

ganic solids. Therefore, a theoretical approach which can adequately account for the effects of disorder needs to be developed. In the present work an effective, medium approach 共EMA兲 is formulated to describe charge transport in the presence of traps. Here we consider the dependence of the mobility on the trap concentration as well as the temperature and field dependence of the mobility in trap-containing materials. The results of the EMA calculation are found to be in good agreement with both experimental data and computer simulations. II. THEORETICAL FORMULATION

This work extends the recently suggested EMA theory9,22 to describe charge transport in the presence of trapping. The disordered medium is replaced by an effective ordered medium 共a cubic lattice with spacing a which is equal to the average distance between localized states兲. Each lattice site can be either a trap or intrinsic transport 共hopping兲 site with a relative concentration of c or 1⫺c, respectively. We assume a Gaussian DOS distribution of intrinsic transport sites. Trap states are also distributed in energy according to a Gaussian function but they are offset to lower energies with respect to the center of the intrinsic DOS by ⫺E t (E t ⬎0). Thus the cumulative DOS in this case is a superposition of two Gaussians. The case of c⫽0 implies a trap-free disordered system, while c⫽1 means that charge-carrier transport occurs only via the trap manifold. The applied electrical field E is chosen to be directed along one lattice axis, for instance, 0x, i.e., E⫽ 兵 E,0,0 其 . A self-consistent EMA theory based on a two-site cluster approach was recently formulated9,22 to describe nondispersive charge-carrier transport in a trap-free disordered organic system. Fishchuk et al.9 derived a set of three transcendental equations that allow the calculation of three effective param⫺ eters, namely, W ⫹ e , W e , and W e which describe the effective jump rates along, opposite, and normal to the electric field direction, respectively. The effective drift mobility by definition is

␮ e ⫽a

⫺ W⫹ e ⫺W e

E

,

共4兲

where a is a lattice constant. It should be noted that the Einstein diffusion equation was not used in the derivation of Eq. 共4兲 since although it is known to be valid for molecular crystals its validity in disordered systems is still an open question 共for discussions, see, for instance, Ref. 2兲. ⫺ A calculation of the effective parameters W ⫹ e , W e , and W e in the general case when energetic and positional disorder are present in a hopping system, is a very complicated task. In many cases, however, the calculations could be simplified. As shown by theoretical analysis in Refs. 9 and 22 ⫺ the parameters W ⫹ e and W e can be calculated independently for a trap-free energetically disordered system when positional disorder is neglected. For example, with large energetic disorder9 one obtains

205208-2


NONDISPERSIVE CHARGE-CARRIER TRANSPORT IN . . . ⫹ W⫹ e ⫽ 具 W 12典 ,

⫺ W⫺ e ⫽ 具 W 21典 ,

共5兲

where the angular brackets denote configuration averaging. ⫺ In this case the parameters W ⫹ e and W e can be determined using the approximate Miller-Abrahams expression 共for a discussion of the differences between the so-called ‘‘exact’’ and ‘‘approximate’’ Miller-Abrahams expressions, see Ref. 9兲: ⫹ ⫽W 0 W 12

⫺ ⫽W 0 W 21

冉

冊

冉

冊

兩 ␧ 2 ⫺␧ 1 ⫺eaE 兩 ⫹ 共 ␧ 2 ⫺␧ 1 ⫺eaE 兲 exp ⫺ , 2k B T 兩 ␧ 1 ⫺␧ 2 ⫹eaE 兩 ⫹ 共 ␧ 1 ⫺␧ 2 ⫹eaE 兲 exp ⫺ , 2k B T

冋冉冊册冋冉冊册

1 ␧2 exp ⫺ P共 ␧2兲⫽ 2 ␴0 ␴ 0 冑2 ␲ ⫹

c

␴ 1 冑2 ␲

exp ⫺

P共 ␧1兲⫽

⫹

exp ⫺

Ac

␴1 2␲ 1 ␴1 2 k BT

exp

␴0 1 ␧1 ⫹ 2 ␴ 0 k BT

2

⫹

1 ␴0 2 k BT

2

Et 1 ␧1 ␴1 Et ⫺ ⫹ ⫹ k BT 2 ␴ 1 k BT ␴ 1

2

2

,

共9兲

where A⫽

冋冉冊册

1 ␴0 共 1⫺c 兲 exp 2 k BT

2

1

冉冊册

冋

Et 1 ␴1 ⫹ ⫹c exp k BT 2 k BT

2

. 共10兲

In the case of a large degree of energetic disorder ( ␴ 0 /k B T Ⰷ1,␴ 1 /k B TⰇ1), which we are considering in the present work, charge-carrier transport occurs predominantly via effective transport levels within the DOS, which are located slightly below the centers of the transport states 共see Ref. 9兲, and trap state distributions at ␧ ⬘p ⬵⫺ ␴ 0 /2, ␧ ⬙p ⬵⫺(E t ⫹ ␴ 1 /2). In the present work we assume that the transport energy level is located at the maxima of corresponding peaks, i.e., ␧ ⬘p ⬵0, ␧ ⬙p ⬵⫺E t of the DOS distributions. Calculations showed that this approximation has little effect on the results of the calculated temperature and field dependence of the charge mobility. Therefore, instead of Eq. 共8兲 one can use the following approximation for P(␧ 2 ): P 共 ␧ 2 兲 ⫽ 共 1⫺c 兲 ␦ 共 ␧ 2 兲 ⫹c ␦ 共 ␧ 2 ⫹E t 兲 .

共11兲

Substituting Eqs. 共6兲, 共9兲, and 共11兲 into Eq. 共5兲, and perform⫹ ing averaging 具 W 12 典 one obtains an expression for W ⫹ e . When calculating W ⫺ one should take into account that e ⫺ the energies ␧ 2 and ␧ 1 in the expression for 具 W 21 典 correspond to a starting and a target state, respectively. With these calculations for W ⫺ e one obtains a final expression for the nondispersive drift mobility 关Eq. 共4兲兴,

␮ e⫽ ␮ 2

⫺ Y⫹ e ⫺Y e

f

共12兲

,

where ⫾ ⫾ 2 ⫾ Y⫾ e ⫽A 兵共 1⫺c 兲 i 1 ⫹ 共 1⫺c 兲 c 关 i 2 ⫹i 3 exp共 xy 兲兴

2

1 ␧ 2 ⫹E t 2 ␴1

␴ 0 冑2 ␲

共6兲

共7兲

冋冉冊冉冊册冊冋冉冑冉冊册

A 共 1⫺c 兲

⫹

where W 0 ⫽ ␯ 0 exp(⫺2a/b), and b is the localization radius of a charge carrier. An essential point is that, in the derivation of Eq. 共5兲, the value W 0 was taken as being equal for all neighbor sites 共absence of positional disorder兲. Substituting Eqs. 共6兲 and 共7兲 into Eq. 共5兲 and performing the configuration averaging using the concept of an effective transport energy level 共for details see Ref. 9兲 one obtains W ⫹ e and W⫺ e , and consequently ␮ e , by using Eq. 共4兲 over a broad range of temperature and field.9,22 When traps are present in a disordered system, the calcu⫺ lation of W ⫹ e and W e becomes much more complicated. The problem is that the parameter W 0 , which describes a tunneling transition of a carrier between neighbor sites, could in principle be different for the cases of 共i兲 transitions between intrinsic transport sites, 共ii兲 transitions between an intrinsic transport-site and a trap site, and 共iii兲 trap-trap transitions. Therefore the present analysis will be restricted to the case of equal W 0 parameters for all the above-mentioned types of transitions. With this approximation Eq. 共5兲 remains unchanged for the trap-containing systems as well. The presence of traps has to be taken into consideration at the configurational averaging stage of the analysis. First let us calculate parameter W ⫹ e . One has to choose the distribution functions for the target and starting states. As demonstrated 共Refs. 9 and 22兲, one should perform the configurational averaging over the energy distribution of the starting state ␧ 1 and the target state ␧ 2 . In this case the target and starting states are described by the DOS distribution P(␧ 2 ), and the occupational density-of-states 共ODOS兲 distribution P(␧ 1 ), respectively. Let us choose the normalized cumulative DOS distribution function for a trap-containing disordered system as follows: 1⫺c

To obtain an expression for P(␧ 1 ) in the form of an ODOS, one should normalize the product of P(␧ 1 ) 关presented in the form similar to Eq. 共8兲兴 and exp(⫺␧1 /kBT) to unity. One then obtains

⫹c 2 i ⫾ 4 exp共 xy 兲其 ,

2

.

共8兲

Here it is assumed that energy distributions of the density of transport and trap states are described by Gaussian functions of widths ␴ 0 and ␴ 1 respectively. 205208-3

共13兲

冉冊再冋冉冊册冉

x2 1 i⫾ 1 ⫽ exp 2 2

冋冉冊册冎

⫹ 1⫺erf

⫾f

1⫺erf

x⫿ f &

&

,

exp ⫺

x2 ⫾x f 2

冊共14兲


冉冊再冋冉

x2 1 exp i⫾ ⫽ 2 2 2

1⫺erf

冋冉

⫾ f ⫹y &

冊册冉

exp ⫺


x2 ⫹x 共 ⫾ f ⫹y 兲 2

冊册冎冉冊再冋冉冊册冉冊冋冉冊册冎冉冊再冋冉冊册冉冊冋冉冊册冎

⫹ 1⫹erf

x⫿ f ⫺y

x 2␩ 2 1 i⫾ 3 ⫽ exp 2 2

c cr⫽exp ⫺

⫾ f ⫺y

1⫺erf

␩&

␩ x⫿ f ⫹y

x 2␩ 2 1 i⫾ 4 ⫽ exp 2 2 ⫹ 1⫺erf

x⫽

␴0 , k BT

⫾f

1⫺erf

␩ 2 x⫿ f

␮ e⫽ ␮ e共 0 兲 c

␴1 , ␴0

␩&

冉冊

␮ e ⫽ ␮ 2 c ⫺1

共17兲

y⫽

Et , ␴0

f⫽

eaE , ␴0

共18兲

关 1⫹c 2 exp共 xy 兲兴 1 1⫹c exp xy⫹ x 2 共 ␩ 2 ⫺1 兲 2

再冋

册冎

共19兲

.

where ␮ e (0)⫽ ␮ e (c⫽0)⫽ ␮ 2 x exp(⫺x2/2). Let us find a trap concentration c 1/2 at which the charge mobility drops by a factor of 2, ␮ e / ␮ e (0)⫽ 21 , for the condition cⰆ1. It is easy to see that

冋

c 1/2⬵exp ⫺

Then, at cⰇc 1/2 , one has

␮ e⫽ ␮ e共 0 兲

冉冊

Et 1 ␴0 ⫺ k BT 2 k BT

冋

2

册

共 ␩ 2 ⫺1 兲 .

1⫹c 2 exp共 xy 兲 1 exp ⫺xy⫺ x 2 共 ␩ 2 ⫺1 兲 c 2

共20兲

册

c c cr

2

2

冉冊冋

␮ e⫽ ␮ 2c

where erf(z)⫽(2/冑␲ ) 兰 r0 dt exp(⫺t2) is the error function. The result, 关Eq. 共12兲兴 enables a determination of the dependence of the drift charge-carrier mobility ␮ e on temperature (T), electric field (E), relative trap concentration (c), and trap depth (E t ) in disordered systems with large degree of the energetic disorder ( ␴ 0 /k B TⰇ1,␴ 1 /k B TⰇ1) at different ratios of widths of the energetic distributions of hopping and trap states ( ␩ ⫽ ␴ 1 / ␴ 0 ). In the limiting case when f →0 (E→0) and y⬎1, i.e., in the case of moderate and deep traps, from Eqs. 共12兲–共18兲 one obtains

␮ e⫽ ␮ e共 0 兲

c c cr

册

1 exp ⫺ x 2 共 ␩ 2 ⫺1 兲 . 2

共24兲

共21兲

From Eq. 共21兲 one can obtain a critical trap concentration c cr at which the effective charge carrier mobility reaches the minimum value ␮ m e :

冉

␴0 ␴0 Et 1 exp ⫺ ␩ k BT k BT 2 k BT

冊册 2

.

共25兲

Further, for depth (E t ), trap concentrations in the range of c crⰆc⭐1, from Eq. 共24兲 one obtains

␮2

ea 2 ␯ 0 a ⫽ exp ⫺2 , ␴0 b

共23兲

For trap concentrations in the range of c 1/2ⰆcⰆc cr , from Eq. 共24兲 and the expression for ␮ e (0), one obtains

x 2␩ 2 ⫾x f exp ⫺ 2

,

␩␸

␩⫽

␩&

册

冉冊冋冉冊

1⫹

共16兲

,

冋

共22兲

Let us rewrite Eq. 共21兲 in the form

x 2␩ 2 exp ⫺ 2

2

⫹x 共 ⫾ f ⫺y 兲 ⫹ 1⫺erf

冊

1 Et , 2 k BT

1 ␮ me ⫽ ␮ e 共 0 兲 2c cr exp ⫺ x 2 共 ␩ 2 ⫺1 兲 . 2

共15兲

,

&

冉

冊

冉冊冋冉冊册 1 ␴0 ␴0 exp ⫺ ␩ k BT 2 k BT

2

.

共26兲

Thus at c cr one expects a transition from trap-controlled to the trap-to-trap hopping transport, i.e., c cr is the transition point between trap-controlled and trap-to-trap hopping transport, which are described by Eqs. 共25兲 and 共26兲, respectively. III. RESULTS AND DISCUSSION

The EMA theory presented above enables the calculation of charge-carrier mobility in trap-containing disordered organic materials. As shown above, the theory adopts the following material parameters; the average trap depth E t , the width of the intrinsic DOS distribution ␴ 0 , the width of the distribution of traps ␴ 1 , the average intersite distance a, the temperature T, the electric field E, and a prefactor mobility ␮ 2 which can be expressed via the localization radius of a carrier 共wave-function decay parameter兲 in usual way 关cf. Eq. 共18兲兴. These parameters in principle are amenable for estimation from experiment. A. Transition from trap-controlled to the trap-to-trap hopping transport

Results of the EMA calculation of charge carrier mobility versus trap concentration for a hypothetical organic disordered system performed using Eqs. 共12兲–共18兲, with E⫽0, are presented in Fig. 1 共solid curve兲. It should be stressed that all material parameters used for the calculation 共such as E t ⫽0.195 eV, ␴ 0 ⫽0.065 eV, T⫽400 K, etc.兲, were exactly the same as employed earlier in the computer simulation 共Ref. 11兲 of charge transport in this system. Thus, one can directly compare the results of the EMA theory with results of computer simulation11 共symbols in Fig. 1兲 which were obtained for the whole range of the trap concentrations. The latter circumstance was the motivation for a detailed comparison with the results from Ref. 11 where the investigation

205208-4


NONDISPERSIVE CHARGE-CARRIER TRANSPORT IN . . .

FIG. 2. TSL glow curves of a PEPC film doped with DAT: c ⫽0%, 0.5%, 2%, 4%, 7%, and 28% 共curves 1, 2, 3, 4, 5, and 6, respectively兲. The trap depth due to DAT is 0.5 eV.

FIG. 1. The trap concentration dependence of the charge-carrier mobility log(␮/␮2) vs log(c) for an organic disordered system calculated by the EMA theory 共solid curve兲 and obtained by the computer simulation 共Ref. 11兲共symbols兲.

involved hole trapping in polystyrene doped with tri-ptolylamine and varying concentrations of triarylamine traps. It is worth noting that the width of the trap energy distribution was assumed to be equal to the width of the energetic distribution of the intrinsic transport states 共the same assumption was used in the computer simulation兲. In reality, this might not always be the case, especially for conjugated polymers where the origins of the energetic distribution of intrinsic transport states and traps could be different. As one can see from the Fig. 1 共solid curve兲, the present EMA theory gives a nonmonotonic dependence of charge mobility with trap concentration qualitatively similar to what is normally observed in many experiments.23,24 At low trap concentrations the mobility decreases with trap concentration followed by increasing mobility at higher trap concentrations. Thus there is an intermediate trap concentration 共it depends on the ratio E t / ␴ 0 ) at which the mobility has a minimum value. Around this concentration, which according to the calculation is ⬇5% for the chosen parameters, the charge-carrier hopping proceeds via both the intrinsic transport and trap sites. Further increasing the trap concentration leads to trap-to-trap dominated hopping. As the concentration of traps is increased to such levels where hopping among trap sites becomes feasible, the apparent activation energy for mobility begins to decrease. It is interesting that the transition from trap-controlled to trap-to-trap hopping transport can also be seen from thermally stimulated luminescence 共TSL兲 measurements. Phenomenology of the TSL technique is briefly described in the Appendix. It is known that a deep trap is created when di-panisyl-p-tolylamine 共DAT兲 is doped into poly共N-

epoxypropylcarbazole兲共PEPC兲 (E t ⫽0.5 eV) due to the difference in ionization potential of these materials. Figure 2 presents the TSL glow curve of a PEPC film doped with DAT: c⫽0%, 0.5%, 2%, 4%, 7%, and 28%. The TSL glow peak of undoped PEPC 共Fig. 2, curve 1兲 is a broad peak with a maximum at T⬵135 K and an activation energy at the peak maximum of 具 E m 典 ⫽0.33 eV, and is similar to that observed in poly共N-vinylcarbazole兲共PVC兲.25,26 Doping PEPC with small concentrations of DAT results in the appearance of a high-temperature peak at 180–210 K, which depends on the DAT concentration 共Fig. 2兲, while the TSL peak at 135 K remains practically unaffected. The activation energy at the high-temperature maximum is 具 E m 典 ⬇0.5 eV. As one can see, an increase of DAT concentration leads to a relative increase of the new, TSL peak intensity, however at DAT concentrations between 4% and 7% its intensity drastically diminishes 共Fig. 2, curves 4 and 5兲. Moreover, at the highest DAT concentration (c⫽28%, Fig. 2, curve 6兲 the TSL glow curve shifts considerably towards lower temperatures. This means that deep traps create a well-separated manifold of states, while the intrinsic DOS remains nearly unchanged. The drastic reduction of the peak at a DAT concentration between 4% and 7% can be interpreted in terms of percolative-type motion of carriers within the manifold of trapping states, since the distance between DAT molecules becomes sufficiently close for transport. At a very large DAT concentration (c⫽28%) the charge carrier motion proceeds via DAT with no PEPC participation. A shifting of the TSL peak toward lower temperatures indicates that DAT is a more efficient transport material than PEPC, i.e., the effective energetic disorder is less than of PEPC. This agrees well with charge transport measurements in triphenylamine doped PEPC,27 where hole mobility decreases by a few orders of magnitude with an increase of the additive concentration, and reaches the lowest value when the trap concentration is about 3%. However, a further increase of trap concentration results in an increase of the charge mobility. At a triphenylamine concentration of 28% the hole mobility is in excess of that for undoped PEPC by a factor of 3. Thus, at a very low concentration of the additive, the charge transport is trap

205208-5



affected, while at higher concentration 共between 4 and 7%兲 the transition to hopping via the trap sites occurs. As one can see from Fig. 1, the EMA calculation is in good quantitative agreement with results of computer simulation 共symbols兲 at low trap concentrations (0⭐cⰆc cr), although some quantitative discrepancy is found at high trap concentrations when direct trap-to-trap transitions can occur. Let us consider this discrepancy in more detail. Apparently the EMA theory overestimates the mobility at high trap concentrations (c⬎3%) and, consequently, a somewhat lowered value of the trap concentration at the mobility minimum is obtained as compared to the computer simulation. One might conclude from this result that the EMA theory overestimates the number of pathways for charge hopping through the trap manifolds. We think, however, that the deviation is most probably because the formulated EMA theory is valid only for the case of a large degree of the energetic disorder (x Ⰷ1), i.e., when the suggested concept of the transport energy level can be employed.9 In fact, the results presented in Fig. 2 have been obtained at a marginal value of x⫽1.885 and, as shown in Ref. 9 such a value is an intermediate between the weak and strong disorder cases. The problem is that at present there is no appropriate analytical theory describing charge transport in systems with an intermediate degree of the energetic disorder even for trap-free materials 共for a detailed discussion of this issue, see Ref. 9兲. B. Trap concentration dependence of mobility in the trapcontrolled transport regime

It should be noted that for the whole range of c there is a problem in comparing the EMA calculation results with experimental data on trap-containing systems. The point is that the results of EMA calculations 关Eqs. 共12兲–共18兲兴 were obtained assuming equal localization radii of the charge carrier for both transport and trap state, i.e., any possible differences in W 0 , describing the tunneling transition between neighbor states, was ignored. In fact, deep trap states might be, for instance, more localized and hence the parameter W 0 could be smaller for traps in comparison to transport states. Accounting for this fact would considerably complicate the development of a general EMA theory. Therefore we will limit our consideration in this section to the trap-controlled transport regime when c 1/2ⰆcⰆc cr , i.e., when transitions from the starting ODOS states to the target trap states can be ⫾ safely ignored. This corresponds to i ⫾ 2 ⫽i 4 ⫽0 in Eq. 共13兲. From Eq. 共25兲 one obtains

␮ e 共 c 兲 ⫺n ⬀c , ␮ e共 0 兲

共27兲

where n⫽1. Thus, in this case, instead of Eq. 共13兲 one obtains Y⫾ e ⫽

⫾ i⫾ 1 ⫹ci 3 exp共 xy 兲

冉冊

冉

x 2␩ 2 x2 exp ⫹c exp xy⫹ 2 2

冊

.

共28兲

FIG. 3. The concentration dependence of the electron mobility measured in NTDI-doped polystyrene containing the acceptor trap MNQ (E t ⫽0.19 eV) 共symbols兲共Ref. 15兲 and calculated by Eqs. 共12兲, 共28兲, 共14兲, and 共16兲共solid line兲.

This equation is valid for arbitrary values of both electric fields E and trap depths E t , and it is essentially a particular ⫾ case of the Eq. 共13兲 when i ⫾ 2 ⫽i 4 ⫽0. It is noteworthy that in the nondispersive transport regime the apparent coefficient n is predicted not to exceed unity but can be smaller than unity depending on the choice of the certain parameters. This is illustrated by Fig. 3 共solid curve兲, where the concentration dependence of electron mobility was calculated by Eqs. 共12兲, 共28兲, 共14兲, and 共16兲 and fitting with experimental data for N,N ⬘ -bis共1,2-dimethylpropyl兲1,4,5,8-naphthalenetetracarboxylic diimide 共NTDI兲 doped poly共styrene兲共PS兲 containing different concentrations of the acceptor trap, 4-共cyanocarboethoxymethylidene兲-2-methyl1,4-naphthoquinone 共MNQ兲, with a trap depth 0.19 eV.15 This is considered to be a shallow trap. Since the concentration of NDTI with respect to polystyrene was 40 wt % the average intersite distance was about 1 nm. All parameters used in the calculation were taken from the experiment described in Ref. 15, so one can compare the calculation results with experimental data 共given by symbols in Fig. 3兲 for the same system. In the important range of trap concentrations c 1/2ⰆcⰆc cr the charge carrier mobility could be extrapolated by ␮ (c)/ ␮ (0)⬀c ⫺n , where n⫽0.89 from the experimental data and n⫽0.88 from the EMA calculation. The latter is in perfect agreement with recent computer simulations by Novikov,18 which showed that the unusual concentration dependence n⬎1 observed in some experiments is not inherent for non-dispersive transport but rather results from different regimes of charge carrier transport at different trap concentrations. It would be of obvious interest to compare the effect of different trap depths on the trap concentration dependence of charge mobility in the same matrix. Calculations by Eqs.

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FIG. 4. The concentration dependence of the hole mobility in TTA-doped polystyrene containing traps of different depth. Traps are due to DTA (E t ⫽0.08 eV), DAT (E t ⫽0.15 eV), and TTA (E t ⫽0.22 eV) 共Ref. 11兲. Measurements are shown by symbols, and calculations by solid lines. Material parameters and shown in the inset.

FIG. 5. The concentration dependence of the hole mobility measured in DPT-doped polystyrene containing different concentration of traps due to DEH 共symbols兲 and calculated by the present theory 共solid line兲共Ref. 13兲.

共12兲, 共28兲, 共14兲, and 共16兲 of the hole mobility in tri-ptolylamine 共TTA兲-doped polystyrene containing different concentration of traps, namely di-p-tolyl-p-anisylamine 共DTA兲 (E t ⫽0.08 eV), di-p-anisyl-p-tolylamine 共DAT兲 (E t ⫽0.15 eV), and tri-p-anisylamine 共TTA兲 (E t ⫽0.22 eV) are presented in Fig. 4 共solid curves兲. Material parameters were taken from experiment and they are given in the inset of Fig. 4. Note that no adjustable parameters were used in the calculation, the only assumption being that ␴ 1 is equal to ␴ 0 . This seems to be a reasonable assumption since the trap molecules DTA, DAT, and TAA are very similar to the TTA molecules. There is good agreement between theory and experiment.11 The charge transport in these systems was clearly nondispersive, so it is essential that the present theory is able to give an explanation for the apparent coefficient n⬍1. The case of relatively deep trapping is presented in Fig. 5. The dependence of the hole mobility in di-p-tolylphenylamine 共DPT兲-doped polystyrene containing different concentrations of the trap p-diethylaminobenzaldehyde diphenylhydrazone 共DEH兲 (E t ⫽0.32 eV) was calculated by Eqs. 共12兲, 共28兲, 共14兲 and 共16兲共solid line兲 using parameters from the experiment. Experimental results13 are given by symbols. As one can see, the apparent experimental value is n⫽1.26 共i.e., n⬎1) but the calculation yields n⫽0.99 共i.e., n⬵1) for c 1/2Ⰶc Ⰶc cr . This indicates that in the experiment transport was dispersive while the theoretical curve pertains to a nondispersive transport only. As stated above, this conclusion is in agreement with results of analysis of computer simulation results done by Novikov,18 where it was assumed 共cf. Fig. 2 in Ref. 18兲 that the transition from dispersive to

nondispersive transport should be accompanied by a transition from n⬎1 to n⬵1. Such an assumption has been corroborated by our calculations performed in the framework of EMA theory, which predicts n⬍1 for nondispersive transport. When the trap depth increases the apparent n value approaches unity. It should be noted that the minimum in charge carrier mobility experimentally observed at a relatively large trap concentration 共Fig. 5兲 implies that the probability for trap-totrap tunneling transitions is much less than that for the transition between hopping sites. Thus we fail to describe the dependence of charge carrier mobility over the whole trap concentration range due to shortcomings of our theory, which have been mentioned above. Development of the general theory should be done in future. Finally, let us consider more closely the exponent n with increasing trap depth. Considering the trap-controlled transport regime within the trap concentration range (c 1/2Ⰶc Ⰶc cr) one could obtain a line tangent to the log关␮e(c)/␮2兴 vs log(c) curve at a point corresponding to the optimum concentration c opt⫽ 冑c 1/2c cr, employing Eqs. 共12兲 and 共28兲 and taking into account correlation effects. This curve provides better approximation in the case of deeper traps. The exponent n obtained from Eq. 共27兲 depends on parameters of the material x, y, ␩, f. Figure 6 shows dependence of n on E t / ␴ 0 for ␴ 0 /k B T⫽3 and eaE/ ␴ 0 ⫽3 at different values of ␴ 1 / ␴ 0 . The results presented in Fig. 6 suggest that with the arbitrary parameters ␴ 0 , E, and ␴ 1 / ␴ 0 with increasing trap depth the apparent exponent n⬍1 approaches the expected value n⫽1 which follows from the Hoesterey-Letson formalism.

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FIG. 6. Dependence of the exponent n on the trap depth for different ratios ␴ 1 / ␴ 0 . Parameters are given in the inset. C. Temperature dependence of charge mobility in trap-containing disordered systems

A theoretical treatment of the charge-carrier mobility in a disordered material containing traps, over a broad temperature range, reveals a critical temperature T cr at which the transition from trap-controlled to trap-to-trap hopping transport regimes occurs. The expression for T cr at a concentration c can be obtained from Eq. 共22兲 as T cr⫽⫺

Et . 2k B ln共 c 兲

共29兲

Temperature dependence of the charge-carrier mobility calculated by Eqs. 共12兲–共18兲, for different parameters ␩, are presented in Fig. 7. For weak electric fields the asymptotic behavior of the mobility in the temperature range T⬎T cr 共trap-controlled transport regime兲 and in the range T⬍T cr 共trap-to-trap transport regime兲 can be described by Eqs. 共25兲 and 共26兲, respectively. These expressions are quite suitable for the analysis. One can see from Eq. 共25兲 that in the range of T⬎T cr the activation energy of the charge mobility has two contributions. The first contribution is determined by the average trap depth E t , and the second one by the width of the trap distribution in energy, ␴ 1 ⫽ ␩ ␴ 0 . For the parameters used in Fig. 7共a兲, T cr corresponds to ( ␴ 0 /k B T cr) 2 ⬵21. By using the Eq. 共25兲 one can estimate E t and ␴ 1 from the experimental data on the temperature dependence of the charge mobility. On the other hand, it can be seen from Eq. 共26兲 that for T⬍T cr the activation energy of the charge mobility contains only the contribution from the width of the energy trap distribution ␴ 1 ⫽ ␩ ␴ 0 because at such temperatures transport proceeds via the manifold of the traps.

FIG. 7. 共a兲 The calculated temperature dependence of charge mobility in the trap-free 共curve 1兲 and trap-containing disordered systems for different parameters ␩ ⫽ ␴ 1 / ␴ 0 : 0.5 共curve 2兲, 1 共curve 3兲, and 1.25 共curve 4兲 plotted in the ln(␮e /␮2) vs ( ␴ 0 /k B T) 2 representation. The ␮ (T) dependence calculated with the HoestereyLetson formalism is given for comparison 共curve 5兲. 共b兲 The same curves 3 and 5 but replotted plotted in the ln(␮e /␮2) vs ␴ 0 /k B T representation 共curves 3 ⬘ and 5 ⬘ , respectively兲

From Fig. 7共a兲 one can note that the temperature dependence of charge mobility in the trap-containing disordered system depends considerably on the parameter ␩. For instance, at ␩ ⬍1, i.e., when the width of the energy distribution of traps ␴ 1 is smaller than the width of the energy distribution of the intrinsic hopping sites ␴ 0 , the decrease of mobility with decreasing temperature in the range T⬍T cr 关curve 2 in Fig. 7共a兲兴 becomes less pronounced in comparison to that for a trap-free system 共curve 1兲. This can lead to a situation where the charge mobility in trap-containing system at a certain temperature might even exceed that in the trap-free material. For comparison purposes, curve 5 in Fig. 7共a兲 shows the

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measurements29 of a alkoxy-substituted PPV derivative 共DOO-PPV兲. Assuming ␴ 1 ⬍ ␴ 0 , i.e., ␩ ⬍1, with T cr ⫽325 K, a reasonably good agreement between the theoretical calculation and experimental data is obtained. It should be noted that it is commonly assumed that both electron and hole transport properties in PPV-type polymers are governed by extrinsic traps.31 Finally, let us consider the issue of the ‘‘effective disorder parameter’’ for a trap-containing disordered material. As was mentioned in Sec. I, computer simulations and photocurrent transient measurements10,11 suggested that the effect of shallow traps can be quantitatively accounted for by introduction of the effective disorder parameter ␴ eff with no other changes to the disorder formalism. Our calculations support such a notion. Indeed, using Eq. 共25兲 with ␩ ⫽1 one obtains the expression for the effective disorder parameter for a trap containing system ␴ eff which is a function of the trap depth and trap concentration

冉冊

冉冊冋

␴ eff 2 k BT ⫽1⫹2 ␴0 ␴0

FIG. 8. The temperature dependence of the zero-field hole mobility 共symbols兲 measured in PPV-ether 共Ref. 28兲 and that calculated by the above theory 共solid line兲 using parameters from TSL measurements 共Ref. 31兲.

temperature dependence of the charge mobility calculated from the Hoesterey-Letson formalism neglecting the energetic disorder. In this case, for trap concentrations cⰇc 1/2 , one has ln(␮e /␮2)⫽⫺ln(c)⫺(Et /␴0)冑( ␴ 0 /k B T) 2 . Here the activation energy of the mobility E a is equal to the trap depth E a ⫽E t . On the other hand, the present EMA theory, which accounts for the disorder effects, predicts the apparent activation energy of the charge mobility to exceed E a ⫽E t ⫹ ␩ 2 ␴ 20 /k B T over the trap concentration range c 1/2ⰆcⰆc cr (T⬎T cr). For trap concentrations c crⰆc⭐1 (T⬍T cr) one obtains E a ⫽ ␩ 2 ␴ 20 /k B T. This is illustrated by Fig. 7共b兲 where curves 3 and 5 of Fig. 7共a兲 are replotted in Arrhenius coordinates 共curves 3 ⬘ and 5 ⬘ , respectively兲. At a fixed trap concentration the transition from a trapcontrolled to a trap-to-trap hopping regime depends on the temperature and trap depth, so one can also introduce a critical trap depth E cr t determined by the expression E cr t ⫽⫺2k B T ln共 c 兲 ,

ln共 c 兲 ⫹

册

Et . k BT

共31兲

As one can see, Eq. 共31兲 is similar to Eq. 共2兲 obtained earlier from the Hoesterey-Letson formalism.10,11 The parameter ␴ eff can be estimated from the temperature dependence of ␮ e . The Eq. 共31兲 is valid only for the concentration range c 1/2ⰆcⰆc cr , i.e., for the trap-controlled transport regime. In this case the following inequality can be obtained from Eq. 共31兲: 1⬍

冉冊

␴ eff 2 E t k BT ⬍1⫹ . ␴0 ␴0 ␴0

共32兲

From Eq. 共32兲 it follows, that the parameter ␴ eff can be introduced only when it changes weakly with decreasing temperature, say from T 1 down to T 2 . This is possible when E t / ␴ 0 ⬍ ␴ 0 /k B T, i.e., Eq. 共31兲 is valid for the case of relatively shallow traps. Besides, the charge transport can be in a trap-controlled regime only when T 2 ⬎T 1 /2, i.e., the temperature range should be rather narrow. It is remarkable that employing Eq. 共19兲 one can easily obtain an expression for ␴ eff which is also valid for the whole trap concentration range (0⭐c⭐1):

冉冊

冉冊

␴ eff 2 k BT ⫽1⫹2 ␴0 ␴0

共30兲

so that at E t ⬍E cr t charge transport occurs in the trapcontrolled regime described by Eq. 共25兲, while the trap-totrap hopping transport described by Eq. 共26兲 occurs at E t ⬎E cr t . An example comparing experiment and theory is given in Fig. 8. This shows the temperature dependence of the zerofield hole mobility 共symbols兲 measured in polyphenylenevinylene-ether 共PPV-ether兲共Ref. 28兲 and that calculated by the above theory 共solid line兲. To calculate the temperature dependence of the charge mobility we used parameters ␴ 0 , ␴ 1 , E t , and c recently obtained by TSL

2

⫻ln

再

2

1⫹c exp

再

冉冊冋冉冊册冎冉冊

Et 1 ␴0 2 ␴1 ⫹ k BT 2 k BT ␴0 E t 1⫹c 2 exp k BT

2

⫺1

冎

共33兲 Thus, one can use the Eq. 共33兲 for analysis in the general case, i.e., for trap-perturbed (c⬍c 1/2) as well as trapcontrolled 关it transforms to Eq. 共31兲兴 and trap-to-trap (c ⬎c cr) charge-carrier transports. The hypothesis that the introduction of shallow traps in disordered organic photoconducting material leads to an ef-

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concentration and trap depth. Figure 9 summarizes the results of TSL measurements of TTA-doped polystyrene containing various concentrations of the traps DTA, DAT, and TAA 关Figs. 9共a兲, 9共b兲, and 9共c兲, respectively兴. As stated above, these molecules create traps with depths of E t ⫽0.08, 0.15, and 0.22 eV, respectively. One can see that the presence of even a small concentration of traps exerts a rather considerable effect on the TSL. The high-temperature wing of the TSL peak gradually shifts toward higher temperatures with increasing trap concentration, thus demonstrating the appearance of deeper tail states. The effect of traps clearly depends on trap depth; i.e., the TSL shift becomes progressively more pronounced when DAT and TAA traps are used relative to DTA 共Fig. 9兲. The lowest trap concentration level that affects the TSL curve also depends on the trap depth. With DAT and TAA, TSL changes were observed at c⫽0.05%, while a DTA effect was seen only at c⭓1%. The above TSL results are in good agreement with both the charge transport data and calculations presented in Fig. 4. Symbols in Fig. 9共d兲 show effective disorder parameters taken from TSL data of Ref. 32 for different trap depths and trap concentrations presented in Figs. 9共a兲–9共c兲. Solid curves present calculations of ␴ eff by Eq. 共33兲. As one can see, there is a good agreement between experiment and theory. D. Electric-field dependence of charge mobility in the trapcontrolled and trap-to-trap hopping transport regime

FIG. 9. The TSL glow curves of TTA-doped polystyrene containing various concentrations of DTA 共a兲, DAT 共b兲, and TAA 共c兲. The trap depths are 0.08, 0.15, and 0.22, respectively 共Ref. 11兲共d兲 Effective parameters of the energetic disorder estimated from the above TSL data 共symbols兲 for different trap concentrations of DTA, DAT, and TAA traps 共Ref. 32兲. Solid curves present calculations of ␴ eff by Eq. 共33兲共solid curves 1, 2, and 3 where calculated for E t ⫽0.08, 0.15, and 0.24 eV, respectively兲.

fective increase in the energetic disorder is supported by TSL measurements. In our earlier work,19,26 we interpreted the low-temperature TSL of trap-free MDP’s in terms of the thermal release of charge carriers from intrinsic tail states of the DOS. This approach, based on the Gaussian disorder model, provides a reasonable understanding of all observed trends in the TSL. An analysis of the TSL peak of neat MDP’s 共see Refs. 19 and 26 for details兲 yields a width for the density of states profile for localized charge carriers which agrees reasonably well with the MDP’s energetic parameter determined in the usual manner. The evolution of the distribution of localized states in the doubly doped polymers presented in Fig. 4 was studied by TSL with increasing trap

In this section we consider the dependence of charge mobility on the electric field in trap-containing systems. It should be stressed that the present calculations have been done in the framework of the so-called Gaussian disorder model. However, the correlated disorder model 共CDM兲共Refs. 6 – 8兲 recently attracted much attention due to its ability to explain the Poole-Frenkel type of the field dependence of mobility in the range of relatively weak electric fields and now it is commonly accepted that the CDM can provide a correct basis for describing the field dependence of the charge-carrier mobility. It assumes that in an organic medium site energies are correlated smoothly, e.g., by charge-dipole 共charge-quadrupole兲 interactions. Computer simulations of charge transport in three-dimensional hopping systems where energy correlations are taken into account gave results that are well approximated by the typical experimental observation of a Poole-Frenkel type ln ␮e⬀冑E dependence over a broad field range. At the same time, computer simulations of charge transport neglecting energy correlation effects predict the Poole-Frenkel type dependence only for a limited field range of relatively large electric fields. In order to consider the CDM case analytically can use the method suggested in Ref. 9, which enables an accounting of the correlation effects in Eqs. 共12兲–共18兲 obtained in the framework of the Gaussian disorder model. For instance, in order to consider the long-range correlations, which were taken into account under computer simulation,4,5,6,8 one should replace parameters ␴ 0 , x, f, and y in Eqs. 共12兲–共18兲 by ␴ d , x c ⫽ ␴ d /k B T, 冑 f c x c /2 共where f c ⫽eaE/ ␴ d ), and y c ⫽E t / ␴ d respectively. Note that ␴ d is the rescaled width of the DOS and the condition x c Ⰷ1 is retained.

205208-10



tained in computer simulation studies30 of charge transport in three-dimensional disordered systems containing deep traps. IV. CONCLUSION

FIG. 10. 共a兲 The calculated field dependence of charge carrier mobility for a disordered system containing traps with concentrations c⫽10⫺3.20 共curve 2兲, and c⫽c cr⫽10⫺1.95 共curve 3兲 plotted in ln(␮e /␮2) vs the 冑eaE/ ␴ 0 representation. The field dependence of the trap-free system 共curve 1兲 is given for comparison. 共b兲 The same data for the high-field range but replotted in the ln ␮e⬀E representation.

Further, the field dependences of charge mobility in a material containing deep traps could be calculated with the use of Eqs. 共12兲–共18兲 in the framework of the CDM. Figure 10共a兲 shows the dependence of ln(␮e /␮2) versus 冑eaE/ ␴ 0 calculated for different trap concentrations: c⫽0, c ⫽10⫺3.20 (c 1/2ⰆcⰆc cr), and c⫽c cr⫽10⫺1.95 共curves 1, 2, and 3, respectively兲. Parameters used for calculation are given in the inset in Fig. 10共a兲. Curve 3 corresponds to the critical trap calculation c cr at which zero-field charge mobility has a minimal value. As one can see from Fig. 10共a兲, all curves are well approximated by the Poole-Frenkel dependence ln ␮e⬀冑E over a broad field range within k B T/ ␴ 0 ⰆeaE/ ␴ 0 Ⰶ ␴ 0 /k B T. However, at strong electric fields, when curve 1 passes through the maximum, charge mobility in trap-containing systems becomes a linear function of the electric field, ln ␮e⬀E. This is illustrated by Fig. 10共b兲共curve 3兲 where the field dependences of Fig. 10共a兲 within the relevant high-field range are replotted in a ln ␮e⬀E representation. As the electric field further increases the effect of traps diminishes and all curves converge. Similar results were ob-

The effective-medium theory developed in the present work is able to account quantitatively for a variety of basic features of charge-carrier transport in disordered organic materials containing traps, which were revealed earlier in photocurrent transient measurements and computer simulations of charge transport. The important message is that the effect of deep traps in a disordered organic photoconductor cannot be described in terms of the conventional Hoesterey-Letson model which predicts an Arrhenius-type temperature dependence of the charge-carrier mobility where the activation energy is simply the trap depth E t . Our calculations support a notion that effect of traps can be quantitatively accounted for by introduction of the effective disorder parameter, ␴ eff , and an expression for ␴ eff being a function of the trap depth and trap concentration, which is valid for whole concentration range has been obtained. It turns out that both relaxation of the ensemble of majority charge carriers within the combined intrinsic and extrinsic density of state distribution and the occurrence of trap-to-trap migration alters the ␮ (T) dependence significantly, notably at lower temperature when the apparent activation energy can become ⬍E t . Ultimately, ␮ (T) is controlled by the width of the distribution of trap levels. If it is narrower than that of the intrinsic DOS, relaxation of the charge carriers is diminished. As a result ␮ (T) dependence flattens and eventually the mobility in the trap containing system can even exceed that of the undoped system. ACKNOWLEDGMENTS

The authors would like to thank Professor P. W. M. Blom for valuable discussions. This research was supported by NATO Grant No. PST.CLG 978952. H.B. acknowledges financial support from the projects ‘‘Optodynamics’’ at Philipps University. The financial support from Heidelberg Digital L.L.C. 共U.S.兲 is also gratefully acknowledged. APPENDIX Experiment

Films of molecularly doped polymers for TSL measurements with thicknesses of several ␮m were prepared by dissolving the appropriate ratios of charge transporting molecules and polystyrene in dichloromethane 共10% solids兲, and then casting the resulting solutions on a metal substrate. The films were dried for 3 h at 40 °C in air, then at room temperature for 2 h in vacuo. TSL measurements were carried out over a wide temperature range 共4.2–350 K兲. TSL measurements after ultraviolet light excitation were performed with two different methodologies: uniform heating at a rate ␤ ⫽0.15 K/s, and fractional heating. The latter procedure allows a determination of the trap depth when different groups of traps are not well separated in energy or are continuously

205208-11



distributed. The fractional TSL technique,25 being an extension of the initial rise method, is based on cycling the sample with a large number of small temperature oscillations superimposed on a constant heating ramp. The main outcome of

*Corresponding author. Email address: [email protected] 1

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the fractional TSL is the temperature dependence of the mean activation energy, 具 E 典 (T). 25,26 The experimental details of the data processing procedures were described elsewhere.19,20,26

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