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Review Articlephys. stat. sol. (b) 175, 15 (1993)Subject classification: 72.20 and 72.40; 71.35; 71.38; S1 2Fachbereich Physikalische Chemie und Zentrum fur Materialwissenschaftender Philipps- Universitat, Ma rbu rg')Charge Transport in Disordered Organic PhotoconductorsA Monte Carlo Simulation StudyBYH. BASSLER

Contents1 . Introduction2. The model3. Monte Carlo simulation technique4 . Results

4.1 Energetic relaxation4.2 Dependence of the charge carrier mobility on temperature4.3 Field dependence of p and the effect of off-diagonal disorder4.4 Anomalous broadening of time of flight signals4.5 Transition from non-dispersive to dispersive transport4.6 Concentration dependence of ,u

5 . Guideline fo r estimating polaronic effects6 . Concluding remarksReferences

1. IntroductionTo improve the properties of photoconductors requires knowledge concerning the mech-anism by which charge carriers are generated and transported. This recognition hasrecently led to an increased effort, both experimental and theoretical, to clarify thefundamental aspects of charge transport in molecularly doped polymers [1 to 121, mainchain [13 to 181, and pendant group polymers [19, 201. A handicap of earlier work in thisfield was that photocurrent transients, intended to yield the material property called chargecarrier mobility, were notoriously dispersive [21 to 231 indicating that the velocity of a sheet

I ) Hans-Meerwein-Straoe, W-3550 Marburg, FRG.

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16 H. BASSLERof carriers started at time t =0 at the front electrode decreased as the packet traversed thesample. Not only is the mean arrival time of carriers at the exit contact of a sandwichsample a poorly defined quantity in that case, it also reflects a system property, i.e. atransport coefficient, that depends on experimental conditions. Meanwhile it has beenbecome clear that a major contribution to the dispersion was the presence of traps of eitherchemical and physical origin. An example of the latter are incipient dimers in poly-vinylcarbazole [24]. It is now well established that non-dispersive photocurrent transients,documented by well developed plateau regions in a time of flight (TOF) experiment, canbe observed in appropriately selected and prepared samples. The requirements are (i) highchemical purity, (ii) a molecular structure that does not permit the formation of sandwich-type incipient dimers, and (iii) low ionization potential/large electron affinity to ensure thataccidental impurities do not act as traps for holes/electrons. The main intention of thisarticle is to delineate a framework for rationalizing charge transport in polymeric or, moregenerally, disordered organic photoconductors including the transition from non-dispersiveto dispersive transport.There is a recurrent pattern of features recovered when analysing TOF signals observedwith a broad class of molecularly doped polymers like polycarbonate doped with derivativesof triphenylamine [7, 25, 261, hydrazone [5] ,or pyrazolin [27], to mention only a few, ormain chain polymers such as polysilanes [14, 15, 281 or polyphenylenevinylenes [17, 181.They include (i) an activated behavior of the carrier mobility yielding an activation energyof about 0.4 to 0.6 eV independent of chemical constitution and synthesis if analysed interms of Arrhenius equation; (ii) a field dependence of the mobility resembling thePoole-Frenkel law, In p K SE, over an extended range of electric fields [4 to 71; (iii) adeviation of both the magnitude of S and its temperature dependence from the predictionof Poole-Frenkel theory [4, 71; (iv) a reversal of sign of S above a certain temperature [27],(v) the deviation from a In p K E law at low fields [29] and the occasional occurrenceof an increase of p at decreasing field [17,30]; (vi) the sensitivity of p as well as its temperatureand field dependence on composition in binary systems [8]; (vii) the existence of tails ofTOF signals that are anomalously broad if compared with the tail spreading due toconventional (thermal) diffusion [31], and (viii) the transition to dispersive transport atlow temperatures [20, 231.The independence of the temperature coefficient of the mobility on chemical compositionand the ubiquitous occurrence of a In p 3c E1/ behavior is clear evidence against thedominance of impurity effects. One can safely exclude that chemically different systemscontain (i) the same amount of traps having (ii) the same dcpth relatively to the transportlevel and (iii) being charged when empty to account for the In p K El dependence. Instead,one has to conclude that the above features reflect a recurrent intrinsic transport propertyof the various systems. This, in turn, renders multiple trapping models inadequate forrationalizing charge motion except, e.g., n systems like polyvinylcarbazole known to containextrinsic traps.Due to the weak intermolecular coupling valence and conduction bands of molecularcrystals are narrow, typically only some 10 meV wide [32]. As a consequence, the mean freepath a carrier travels between subsequent phonon scattering events is of order of a latticedistance at room temperature. Given the disorder present in non-crystalline organic solidslike molecularly doped polymers which can only shorten the mean free path it isstraightforward to assume that the elementary transport step in such systems is the transferof a charge carrier among adjacent transporting molecules, henceforth called transport sites.

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Charge Transport in Disordered Organic Photoconductors 17In chemical terms this is a redox process involving chemically identical moieties. Thedependence of p on temperature and electric field must reflect the dependence of thatelementary step on T and E . Its activation energy will, in general, be the sum of anintermolecular and an intramolecular contribution. The former arises from the physicalinequivalence of the chemically identical hopping sites due to local disorder that subjectsthe lattice contribution to the energy of a neutral and charged molecule to local, statisticalvariation [33]. The latter is due to the change in molecular conformation upon removal oraddition of an electron from/to the transport molecules (oxidation/reduction). Transfer ofa charge, therefore, requires concomitant activated transfer of the deformation [34 to 371.The essential difference among the currently discussed transport models is related to therelative importance of both contributions. The hopping model assumes that the couplingof the charge to intra(or inter)molecular modes is weak, the activation energy reflectingbasically the static energetic disorder of the hopping sites. The (small) polaron model, onthe other hand, considers the disorder energy to be unimportant relative to the intramoleculardeformation energy.

Unfortunately, there is no direct and easy way to measure the inter- and intramolecularcontributions to the transfer activation energy because the optical spectra of molecularsolids are controlled by excitonic transitions in contrast to valence band +conductionband transitions in inorganic semiconductors. The density of states (DOS) function forcharge transporting states is, thus, not amenable to direct probing by absorption spec-troscopy. Quantum chemical calculations, on the other hand, are usually not accurateenough to predict energy differences of the order 0.1 eV. One has, therefore, to rely on theanalogy of the coupling between an exciton and a charge carrier, on the one hand, and thesurrounding lattice, on the other hand, in order to estimate the width of the disorder-broadened DOS. Consider both a charge carrier and an excited state dipole surroundedby a polarizable medium. To first order approximation their polarization energy P , and

0.15-2db

dX

._0.10

0.05

0

gas to solid shift energy D are determinedby ion-dipole and dipole-dipole coupling,obeying r - 4 and r P 6 dependences on inter-site distances, respectively. Absolute valuesare typically P, z 1.2 eV [38] andD z 0.25 eV, respectively, and exceed theintermolecular exchange energies by almosttwo orders of magnitude [32]. In randommedia intermolecular distances and, con-comitantly, P,and D fluctuate, their relative

Fig. 1. Standard deviations of the widths of theDOS for charge carriers (0,) nd excitations (flex)calculated for P , =1.2 eV and D =0.25 eV . Thewidth of th e DOS for the S , +So0-0 transition inpolyvinylcarbazole is indicated (from [39])

0.0 0.1Ar/r-

2 physica (b ) 17511

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18 H. ASSLERstandard deviations being [39]

ac/P,= 1 Ari j / r i j )4i * j

andacxlD= c @rij /r i jT ,i+j

respectively. The data shown in Fig. 1 were obtained by doing summation and averagingon a computer system consisting of a cubic array of 300 point sites each being allowed tofluctuate about its mean crystallographic position. They demonstrate that the width of theDO8 for charge carriers originating from a given pattern of positional disorder is about1.5 times as big as that for singlet excitation. Measuring the DOS for singlet excitationsthus permits concluding on the width of the DOS for charge carriers as well.

Absorption (and fluorescence) bands of disordered organic solids are usually of Gaussianshape with a standard deviation (Gaussian width) of typically o,,= 500 cm-' (60meV)[40].On the basis of the above argument one expects that the inhomogeneously broadenedDOS for charge carriers are also of Gaussian shape, the Gaussian width a being of order0.1 eV translating into an activation energy of about 0.4 eV (see Section 4.1). Site-selective fluorescence spectroscopy of the pertinent transport materials, on the other hand,yields a fluorescence Stokes shift that is one order of magnitude less [41]. This can beconsidered as a justification for starting with a hopping model as a zero-order approachthat does not include polaronic effects, in particular since an analysis of charge motion ina prototypical molecular crystal (naphthalene) suggested a polaronic binding energy of16 meV only [42].

2. The ModelTo model hopping transport across a disordered single or multiple component organicsolid the charge transporting elements, which can either be the molecules participating intransport or segments of a main chain polymer that are separated by topological defects,are identified as sites whose energies of their hole or electron transporting states are subjectto a Gaussian distribution of energies,

implying that all states are localized. The energy c is measured relative to the center of theDOS. The origin of the energetic (diagonal) disorder is the fluctuation of the latticepolarization energies and/or the distribution of segment length in 71- or o-bonded mainchain polymer. The Gaussian shape of the DOS is suggested by the Gaussian profile ofthe (excitonic) absorption band and by the recognition that the polarization energy isdetermined by a large number of internal coordinates each varying randomly by smallamounts [ 3 3 ] .The tacit assumption contained in (1 ) is that the self-energies of adjacentsites are uncorrelated. This may not strictly be true, yet since structural correlation lengthsin organic glasses do not exceed a few intermolecular distances at most, it appears to be areasonable premise, in particular since a carrier visits typically some lo4 to lo5 sites on itsway across either a simulation or a real world sample.

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Charge Transport in D isordered Organic Photocond uctors 19The jump rate among sites i and j is assumed to be of the Miller-Abrahams type, i.e. isthe product of a prefactor v o , an electronic wave function overlap factor, and a Boltzmannfactor for jumps upward in energy [43],

In (2) a is the average lattice distance. In case of an applied electric field E the site energiesinclude the electrostatic energy term. Equation (2)implies that the electron-phonon couplingis weak enough to render polaronic effects negligible, yet sufficiently strong to guaranteecoupling to the heat bath. Notice that there is no other activation energy except the differencein electronic site energies a carrier has to overcome in order to hop. Hops down in energyare not impeded by an energy matching condition for dissipation of the electronic energydifference. Considering the rich phonon spectra and the low charge exchange rates comparedto typical phonon frequencies this appears to be an uncritical assumption. The above formof the jump rate also implies that downward jumps are not accelerated by an electric field.To consider that (i) in a disordered medium the intersite distance is subject to localvariation and, more importantly, (ii) coupling among the transport molecules, which. areusually non-spherical, depends on mutual orientation [37] and is, therefore, also disorderaffected, the overlap parameter 2ya = r is subjected to distribution as well (off-diagonaldisorder). Operationally this is done by splitting the intersite coupling parameter Tij intotwo site specific contributions T i and T j ,each taken from a Gaussian probability densityof variance oP The variance of Tij is Z = 2G. his is an arguable procedure. It impliessome correlation because all jumps from a given site are affected by the choice of Ti.Choosing a Gaussian for the probability density of Tij appears to be even more criticalsince, as Slowik and Chen [44] ave shown, the overlap parameter is a complicated andstrong function of the mutual molecular angles. In the absence of any explicit knowledgeabout the actual distribution of overlap parameters a Gaussian appears to be a zero-orderchoice, in particular since the relative fluctuation C/2ya required to fit experimental dataturns out to hardly exceed ~ 0 . 3 .n any event, Z should be considered as an operationallydefined measure of the off-diagonal disorder that cannot be directly translated into amicroscopic structural picture in contrast to the parameter characterizing diagonaldisorder.

3. Monte Carlo Simulation TechniqueUnfortunately, the Gaussian form of the DOS in random organic systems prevents closedform analytical solutions of the hopping transport problem. The only analytical theoryavailable to date that retains both the energetic distribution of hopping sites and thedistribution of hopping barriers is the effective medium approach (EMA) developed byMovaghar and coworkers [45]. The higher-order correlation effects appearing when summingover all paths a particle can take on going from site i to j lead in this approximation toan effective reduction of the site density after each jump. It has been shown that for finitetemperatures the EMA provides an excellent description of the hopping process in densesystems [46], its disadvantage being that it does not provide the experimentalist with analyticequations ready to fit experimental data. The EMA is superior to the ultrametric space2*

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20 H. BASSLERconcept [47 to 491 which is mathematically simpler and avoids the averaging techniqueinvolved in the effective medium treatment but does so at the expense of reducing the realenergy structure of the system to an array of isoenergetic hopping sites separated byrandomly distributed barriers.

The alternative approach this article elaborates on is Monte Carlo (MC) simulation [50].It can be considered as an idealized experiment carried out on samples of arbitrarilyadjustable degree of disorder and devoid of any accidental complexity. It allows to determinewhich level of sophistication is required to reproduce the properties of a real world sampleand, by comparison with theory, to check the validity of approximations in analyticaltreatments that are based on the same physical principles.The current M C simulations were performed on a cubic lattice consisting of 70 x 70 x 70sites with site distance a =0.6 nm. Their self-energies and ri-values were chosen accordingto a Gaussian probability density. By applying periodic boundary condition an effectivesample size of up to 8000 x co x 00, equivalent to a sample length of 4.8 pm in x-direction,which is the direction of an applied electric field, was considered. Usually 20 carriers arestarted one at a time on an arbitrary site of a given lattice configuration. Afterwards anew lattice is set up. The total number of carriers considered for a TOF pulse is 100 to 150.This turned out to be a reasonable compromise between computer time and computationalstatistics.The probability that a carrier jumps from a site i to any s i te j within a cube consistingof 7 x 7 x 7 sites is

Pij = Vij/ c v ij .i * j

(3 )A random number xR rom a uniform distribution is chosen and this specifies to which sitethe particle jumps because each site is given a length in random number space accordingto Pi> he time for the jump is determined from

where x C is taken from another exponential distribution of random numbers. The time scaleis defined by the dwell time

t o = (6vo exp ( -2ya))- ' (5)of a carrier on a site in a lattice without disorder. The computational variables are (i) thereduced energetic disorder parameter 6 = c /k T , (ii) C at constant 2ya =10, and (iii) theapplied electric field. The computer kept track of the mean position (x) and thetime-weighted mean energy ( E ) of a carrier as well as the related variances as a functionof time. The carrier mobility is inferred from the mean carrier arrival time at the exit contact.Its determination from the current plateau in a non-dispersive TOF signal yields identicalresults. The apparent diffusivity is calculated from the spatial variance of the carrier packet,

(6)= ((x - (~))')/2t.Keeping track of the evolution of ((x - x))') as the packet moves across the sampleallows concluding on the spreading of the TOF signal in systems of arbitrary length up to8000 lattice planes.

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Charge Transport in Disordered Organic Photoconductors 214. Results4.1 Energetic relaxation

A particle started at arbitrary energy within a Gaussian DOS of an undiluted system ofhopping sites is likely to execute a random walk and relax into tail states. It is a specificfeature of a Gaussian as compared with an exponential DOS that the mean energy, ( E ) ,saturates at long times indicating attainment of dynamic equilibrium. Notice that truethermodynamic equilibrium will never be established since the hopping motion of an excesscarrier, generated externally by photoexcitation or injection, is considered. Contrary to thecase of hopping at the Fermi level of amorphous inorganic semiconductors [51], Fermistatistics is irrelevant since the carrier is considered to move in an otherwise empty DOS.This corresponds to charge carrier densities that are low enough to exclude carrier-carrierinteraction in real experiment. The equilibration energy ( E , ) of a carrier at zero electricfield can be calculated analytically from [46]'r c e ( E ) exp ( - E / k T ) dE-08. (7)2kT-

-a 0( E , ) = lim ( ~ ( t ) ) + c o - - - -s e(4exp ( - c / k T ) dc,

-a2

Note that ( E , ) is independent of the presence of off-diagonal disorder. Fig. 2 portrays thetemporal evolution of a packet of non-interacting carriers relaxing within a Gaussian DOS.As time progresses the shape of the packet becomes slightly asymmetric, the high-energywing of the occupational DOS (DOSO"")equilibrating earlier. In the long-time limit( E , ) = -08 and the shape of the DOS""" becomes Gaussian and acquires a width 0indicating that dynamic equilibrium has been established. The temporal variation of ( ) ,parametric in 8, is shown in Fig. 3. It demonstrates that ( c ( t ) ) relaxes approximately yet

/kT

-1 0 1 2 3 4 5k l ( t / t o l-Fig, 2. Temporal evolution of the distribution of carrier energies in a Gaussian DOS of width 6 = 2.All profiles are broken at the same carrier density illustrating the different relaxation patterns formobile and imm obile carriers. E , denotes the theoretical mean energy in the long-time limit

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H . BASSLER20

IA*- 2

-3

- 43.5

I I I I I I I I I I

Fig. 3. Temporal decayof the meanenergy of an ensemble of carriersmigrating within a Gaussian DOSof variable normalized width 6

not strictly logarithmically with time, in quantitative agreement with analytic EM A results[46]. The relaxation time obeys the empirical relationship [52]t,,,/t, = 10exp (- (1.076)) . (8)

In the T+ 0 limit relaxation is slowed down since intervening thermally activated jumpsthat help a carrier finding additional pathways for relaxation are eliminated [53].There is no direct probe for the relaxation of chargc carriers in organic media. However,triplet excitation can serve as probe particles, albeit with finite lifetime, since they aretransported via exchange interactions as charge carriers are. It was gratifying to note thatthc relaxation of triplet excitation in a benzophenone glass occurs in quantitative agreementwith both simulation and analytic theory [54, 551. This testifies on the adequacy of the basicassumptions underlying the model, in particular, the form of the jump rate (2).

4.2 Dependence of the charge carrier mobility on temperatureFor d =4, i.e. a ~ 0 . 1V at 295 K, which turns out to be a representative value of theenergetic disorder parameter for many experimental systems, ( c , ) =-4a. Since thefractional DOS for c 5 ( c , ) is only, a carrier located at E =(E,)can, on average, continue its motion only after thermal excitations. If all carriers wereexactly located at c = ( c , ) = do and if a transport level existed exactly at the center ofthe DOS (c =0) the transport activation energy were da, i.e. p ( T ) should follow anon-Arrhenius-type temperature dependence, p ( T ) = po exp (- (o/kT)2). oth EMA andMC studies indicate that this characteristic temperature dependence, that has been derivedin other context, too [56 to is actually recovered, albeit with a prefactor less thanunity in the exponent that accounts for the statistics of both the occupational energies and

-40Q ( E ) dc: z 3 x

-

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Charge Transport in Disordered Organic Photoconductors 23103 ( K - ) -

0 1.0 2.0 3.0-1

*.= - 3)c3nL2 - 4?Y0,- -5

-6

-7

. . . . ,\\

\ \ \ A ~ =0.28eV\

\\

\\

I

0 1 2 3 4 0 5 10 1562__c- Fig. 4. Simulation result for th e dependence of the low-ficld charge carrier mobility on the dcgree ofenergetic disorder (6 =a/kT) plotted on In p vs. 6 nd 8 scales, respectively. (In the computationsc was varied.) An Arrhenius fit chosen to match the data in the vicinity of 3 0 0 K is included forcomparisonthe barrier heights,

P(T)= Po exp (- 0 &I2) * (9)The prefactor in the exponent is associated with an incertainty of 4%. Fig. 4 shows p (T )plotted semilogarithmically versus T - l (Fig. 4a) and T -Fig. 4b). The latter representationprovides the better fit although it is also obvious that the distinction becomes the moredifficult the smaller the range of d or temperature is. By differentiation one can infer from(9) the Arrhenius activation energy obtained if one plotted p ( T ) in an Arrhenius fashionand determined the apparent activation energyA, , from the slope at a given temperature 7:

An apparent activation energy of 0.4 eV, inferred from the tangent to an Arrhenius plot at300K would thus be expected for a hopping system characterized by a DOS of widthd x 0.108 eV as expected on the basis of the width of optical absorption bands (Section1). An experimental example for the T-dependence of the hole mobility in an organic glass- a vapor-deposited film of 1,l-bis(di-4-tolylaminophenyl)cyclohexaneTAPC) - ispresented in Fig. 5 (from [60]). Although the deviations are small the In p versusT2it isclearly the better fit. Another example is contained in the early work of Schein et al. onhole transport in p-diethylaminobenzaldehyde-diphenylhydraone DEH) doped into apolycarbonate matrix [5 ] .Although this was the first demonstration of the superiority of the

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24

2 . 0 ~

2Or

H. BASSLER

Fig. 5. Te mp eratu re dependence of the hole mobility in am orp hou s TAPC in In p vs. a ) T - and b)T .- representation, respectively (from [60])

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Charge Transport in Disordered Organic Photoconductors 25In p versus T - 2 epresentation of p ( T )data for organic glasses these authors revised theirinterpretation later on [6, 261 in favor of the conventional Arrhenius analysis in order topromote the polaron concept.An argument in favor of the application of (9) to analyse experimental p ( T )data comesfrom the po da ta obtained by extrapolation to T +00. While the Arrhenius plot notoriou slyoverestimates p o - values are typically x 10cm2 V - ' - ' r larger which is unrealisticalfor a non-crystalline and usually diluted molecular solid - (9) yields values of orderto lo-' cm 2 V - ' s - ' as expected. The compa rison between experimental po-valuesand the mobility in a molecular solid should, however, be considered with some cautionbecause, on the on e hand , the latter is expected to exceed those in an energetically orderedhopping system because of coherence effects and, on the other hand, the former may beenhan ced by off-diagonal disorder (see below).

4.3 Field dependence of p and the effect of off-diagonal disorderThe ho pping mobility must depend o n a n applied electric field E since tilting the DOS byan electrostatic potential reduces the average barrier height for energetic uphill jumps infield direction. As a consequence the equilibration energy (E , ) must increase with E .Heating the carrier gas at high electric fields is a common effect in semiconductors and isusually related to the k-dependence of the electron-phonon coupling. In the present modelit is solely a consequence of the effect the field has on the interplay between upward anddownward jumps of a carrier anticipating that the coupling to the heat bath remainsunaffected. Fig. 6 shows tha t ( E , ) increases with E according to

( E , ) =80 +(E/E,)l.' (1 1)with E , = 1.8 x lo6V cm-' being independent of 8. For 6 = 2 saturation occurs above2 x lo6V cm-'.Thedeviationofthedatapointsfor8 = 3Sandfieldsbelow 1.5 x lo6V cm- 'reflects the fact that at this degree of disorder equilibrium is not fully reached in the

01

ti\A- 1

- 2

-3Fig. 6 . M C result for ( E ) / ( T as a function ofthe electric field, parametric in d-~0 1 2 3E ( MVcrn- ' )

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2.02 53.03 5

H. BASSLF~RFig. 7. Simulated dependence of p onfor variable ci

simulation even with a sample consisting of 8000 lattice planes. The concomitant variationof p with E is portrayed in Fig. 7 for a hopping system subject to energetic disorder only.The S-shaped ( c , ) versus E relationship is retained in the In p versus El l 2 plot [61]. p ( E )saturates at low fields and increases in a Poole-Frenkel fashion at higher fields. The deviationsfrom a In p K E law are small, however. It was for this reason why it has not been recoveredin earlier M C work [62], done on smaller samples. At vanishing disorder, p decreases withE at large fields. This simply reflects the saturation of the drift velocity with field in ahopping system with isoenergetic sites in which the field does not affect intersite jump rates.The data of Fig. 7 demonstrate that the assumption of a hopping system with GaussianDOS is sufficient to reproduce l n p = SE"2 behavior within a limited field interval, theslope parameter S changing sign below a certain value of 6, i.e. above a certain temperature.Meanwhile Movaghar et al. [63] have developed an analytic theory for the p ( E ) behavior.They found that a In p cc En law with 0

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Charge Transport in Disordered Organic Ph otoconductors 27Fig. 8. Logarithm of p against E l l z forPMPS films at various temperatures:0 21 4 K, 0 234 K, 254 K , 27 4 K ,A 294K, n 314 K (from [29])

TOF signals. Fig. 9 illustrates tha t inferring ,ufrom t o extends the field range of the validityof the In p cc El l2 by a factor of 3 towards low er fields. Furthe r, since the essential quantitydetermining the p ( E ) relation is the drop of the electrostatic potential across an intersitedistance, normalized to k'7; the simulation field has to be rescaled by the factor 0.6 nm/aexp,aexp eing the ex perimental mean intersite distance, for com paring experim ental with MCdata. A caution ary note is also in ord er regarding the reliability of low-field mobility data.The larger the transit time is the m ore likely becomes capture by a low conc entration ofdeep traps. This would lower the effective mobility. Delayed injection from an injectionlayer or delayed geminate pair dissociation in the bulk may be another spurious effecttending to bend down the p ( E ) curves a t low fields.Before embarking on a discussion of the T-dependence of the slope parameter S, theeffect of off-diagonal disorder on the p(E)-relation shall be addressed. Superimposing variableoff-diagonal disorder, quantified by the parameter Z, nt o a sam ple characterized by 6 = 3

I I I I

x fromfromw from

Fig. 9. In p vs . El'' plots of the carriermobility based on different definitionsof the transit time (from [64])0

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28 H. BASSLERFig. 10. Mobility p vs . E l l 2 for a hopping systemwith fixed energetic disorder (3 = 3) and variableoff-diagonal disorder (from [61])

10-'00 0.5 2.0 1.5 2.0E 1 /2 ( M V 1 /2 c m - 1 /2 ) -

changes the p ( E ) pattern in a characteristic way. With increasing C the low-field portionof p ( E ) plots tends to bend upward yielding mobilities that increase with decreasing field.Within the lnp cc EL/' regime the values of the slope S decrease and eventually becomenegative (Fig. 10).This behavior can be explained in the following manner. Fluctuation ofthe overlap parameter T i jabout a mean value translates into asymmetric variation of thefactor exp (- T i j ) ontrolling the exchange rate of a charge carrier. The effect is reminiscentof percolation. While certain routes for a carrier will be blocked due to more unfavorableintersite coupling, easy channels are opened that overcompensate for the loss of the former.To first-order approximation one can describe the diffusion enhancement by replacing thedistribution of overlap factors by the ensemble-averaged diffusion coefficient

(12)D)* ~ X P-2g(ril l> 9

U X

Fig. 11. Schematic view of the different routes a charge carrier canfollow to move from D to A. If the electric field acts on the D-Adirection the jump with rate v1 occurs against the field direction

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Charge Transport in Disordered Organic Photoconductors 29where

g(T)=( d- xp [- (r- a ) /P] (13)is the Gaussian weight distribution of the coupling parameter of the individual sites (seeSection 2). Averaging can be done analytically [65] and yields

(14)The microscopic aspect of the effect is illustrated in Fig. 11.While the direct path is impededby poor overlap or larger distance the loop opens a detour that is faster because ofimproved coupling. Fig. 11 also provides a qualitative explanation for the negative effectof an electric field. Suppose the field acts along the direct path. A carrier following the loopwill, therefore, have to execute a jump against the field direction. Consequently, the fieldtends to eliminate the faster detour route. This effect is reminiscent of the phenomenon ofnegative differential resistance in the literature on hopping conduction in amorphoussemiconductors [ 66] .The basic idea proposed by Shklovskii and coworkers [67, 681 for

( D ( Z ) ) =D(Z =0)exp (+Z2) .

c = Of2.12

2.542.833.25

0 2 4 6 8 1 0 1 2 1 4 1 66*-

rationalizing this effect is to invoke field-induced localization by traps or deadends in a random network which fortopological or energetic reasons can onlybe emptied by carrier jumps against thefield direction. At low fields, where theheating effect of the field on the energeticdistribution of hopping carriers is neg-ligible, i.e. the regime of constant p in asystem with 2 =0, it is basically thiseffect that is responsible for the decreaseof p with increasing E. At larger fields,depending on the magnitude of cr, it isovercompensated by the field effect onenergy barriers along the field direction.It leads to an increase of p with E, albeitweaker than in the absence of off-diagonal-disorder, but retaining the In p K Elrelation. At very high fields the effectivedisorder seen by a migrating carrier va-

Fig. 12. The slope S of th e In p vs. El plotsversus bz parametric in C (from [30])

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30 H . BASSLERnishes and p ( E ) must approach the p K E - law expected for a pure hopping system inwhich backward transitions are excluded.

The quantity that is essential for analysing experimental p ( E , T )data is the dependenceof S on the disorder parameters 6 (i.e. temperature) and C. Fig. 12 shows simulation datafor S plotted versus 6 for various values of C. A family of parallel straight lines with slopeC =aS/aB = 2.9 x (cm/Vs) is recovered with systematic deviations occurring for6 2. The parameter (8*) levels off for Z

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Charge Transport in Disordered Organic Photoconductors 31signature of hopping transport in a Gaussian DOS. Provided that the width of the DOSis temperature independent, which appears to be a reasonable approximation at temperaturessufficiently below the glass transition temperature, &* translates into a characteristictemperature T * =a/k6* which turns out to be the equivalent to the Gill temperature (seebelow). Note that (15) predicts a change of sign of the slopes of the In p versus Erelationship even in the absence of off-diagonal disorder. Hence, an earlier statement madein [61] has to be modified in the sense that the presence of off-diagonal disorder suppressesthe temperature at which the sign reversal occurs, yet is not a necessary condition for itsobservation. Neither has a temperature dependent decrease of S to be taken as evidencefor a temperature dependence of C. If C increased with temperature, this would cause anincrease of the slopes aS/a6 and aS/i3TP2, espectively.It is useful to discuss (15) in conjunction with related formulae described in the literature.

10-3c

=329K319309299289279269259249239

229

The first one is that of Gill 1691, whoproposed describing the temperature andfield dependences of the mobility in poly-(N-vinylcarbazole) by the empirical for-mula

xexp _ _(16)

where A , is a zero-field activation energy,p, the pre-factor mobility, and Tff isdefined by l/& =1/T- / T * . Gill se-lected this functional form to account forthe fact that In p ( E , T )versus T- lotsintersect at a finite temperature T * .Equa-tion (16) predicts that the field depend-ence becomes negative for T > T * . Thisequation, however, has no theoreticaljustification.In a study of hole transport in p-diethylaminobenzaldehyde -diphenyl-hydrazone-doped polycarbonate, Scheinet al. [5] suggested factorizing the fieldand temperature dependences of the hole

(E l

Fig. 14. Hole mobility in TAPC/polycarbo-nate against PI2, arametric in temperature(from [30])

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32 H . BASSLERmobility as

p ( ~ , )= p o exp (- (ZY) exp (..( - )).Equation (17) predicts th at the slopes of l n p versus E plots become negative forT >P / y . The constant P was found to agree within a factor of 2 with the Poole-Frenkelcoefficient, while y was an empirical parameter.M ore recent studies of hole transpo rt in tri-p-tolyamine-doped polyca rbona te byBorsenberger [7] confirmed the validity of (17) if modified by replacing 1 /T in the exponentof the field term by (1/T )2. The present sim ulations subs tantia te this reasoning bydemon strating that a mobility obeying (17) is an inheren t signatu re of the random walk ofthe carriers in a random potential with rand om electronic coupling unde r a bias field. Thisprovides a means for determining the degree of built-in d iagona l and off-diagonal disorderfrom p ( E , T ) ata.A textbook example for the observation of the p ( E , T ) behavior outlined above is thework of Borsenberger et al. [30] on hole transport in a TAPC polycarbonate composite(Fig. 14). Low-field (1x lo5V cm -l ) mobility da ta follow (9 )yielding 0 = 0.095 eV for thewidth of the DOS and p o =2 x cm2 V - s- . Knowing o allows plotting the slopevalues of the In p versus El section of p ( E ) plots versus B 2 . F or d2 >14 a straight linewith slope C = 3 . 0 ~ (cm/V)li2 is obtained in excellent agreement with simulationresults (Fig. 15). The intersection of the extrapo lated S(d2)-line with S =0 occurs at( c ? * ) ~=9.6 yielding an off-diagonal disorder p aram eter Z = 3.1.The increase of the slope,

t 6G - 5EVN: 4>xz3

2

1

0

5 10 15 20 2532 ___)

aS/86, noted as S approaches zero indicatesthat for B c 3.75, equivalent to T 2 293 K,Z begins to increase. The most plausibleexplanation is tha t there is a thermally induc-ed contribution of off-diagonal disorder inaddition to the contribution due to thebuilt-in compositional or geometrical dis-order. The deviation of S versus d2 fromlinearity could, in principle, also indicate theonset of a temperature-induced broadeningof the DOS. However, the absence of achange of slope of p ( T ) argues against thispossibility.

Fig. 15. The slope S of In p vs. El/ plots forvarious experimental systems. Data are from[29] (PMPS), [5] (DEH/polycarbonate), [30](TAPCPC), and [96] (FM/PE, (4-n-butoxy-carbonyl-9-fluovenylidene)-malonitrile oped in-to polyester)

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Charge Transport in Disordered Organic Photoconductors 33A stringent test for the applicability of the MC results to analyse those data is theoccurrence of the increase of p at low fields predicted for C x 3. Unfortunately, the dramaticincrease in computer time prevents collecting MC data for both large d and C at low fields.From Fig. 7 and 10 it is, nevertheless, obvious that the general trend borne out by the data

of Fig. 14 is recovered.To document that the above behavioral pattern is indeed a universal feature of transportin molecularly doped polymers and main chain polymers like poly(methylphenylsi1ane)representative data for S(g2) are collected in Fig. 15. Apart from their bearing out the samefunctional pattern, the slope parameters C agree within 20% with the simulation value. Thevarious systems differ with regard to the degree of off-diagonal disorder which is likely tobe related to intermolecular packing constraints affecting both the average electroniccoupling among the sites as well as its fluctuation. Note that the PMPS data suggest C x 0in accord with the absence of a mobility increase at low fields (Fig. 8).4.4 Anomalous broadening of t ime offlight signals

In conventional Gaussian transport the ratio of diffusive spread Ax of an initially &shapedcarrier sheet to its displacement x is- -T (-) ' I 2

(x) eE DtThis implies validity of Einstein's law relating diffusivity to mobility via eD = pkT. For anappliedvoltage U =500 V at 290 K a relative spread of the transit time At& = (2kT/eU)' j2of order is obtained. The experimentally observed spreadings of the tails of whatappears to be non-dispersive TOF signals are much larger [31]. This must be the result ofnon-thermal field-assisted spreading of the drifting carrier packet due to disorder. It isobvious from Fig. 16 showing a series of TOF signals parametric in 6 at vanishingoff-diagonal disorder that this effect is recovered by simulation [64]. The spread of the tail,quantifiable via the dispersion

' Pw = - - = ( g )l I 2 - o ,t 1 / 2

increases with increasing disorder (for definition of to and tl,2. see Section 4.3).The spreading of TOF signals can be accounted for by defining a diffusivity that consistsof the ordinary thermal term Do nd a field- and disorder-dependent contribution Of(&,E ) .The latter can be determined from the spatial variance of the carrier packet in field direction,yielding

The quantityf(8, E ) - 1 thus measures the deviation of a random hopping system underan applied bias field from conventional Einstein behavior. Data forf(6, E ) - 1 bear out aquadratic relationship on the electric field in the field range lo5 to lo6V cm-' as long asd 5 and C =0. At still larger fieldsf(6, E ) - 1 levels-off and begins to decrease (Fig. 17).This is a consequence of the gradual reduction in the effective disorder a random walkerexperiences when migrating in a random potential strongly tilted along the field direction.3 physica (b) 7511

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34

c \\ 3.0r 1 1

H. ASSLER

6

L \

Ir

0 0.5 1.0t/tt,-

Fig. 16. Time of flight (TOF)signals, para-metric in r i (example length 8000 latticeplanes, E =6 x l o 5 V cm- ' )

For E +00 all states become avail-able, implying that the effect ofdisorder vanishes and f ( c ? , E ) ap-proaches its thermal value which itselfis field dependent, however. In ahopping system with no disorderD K cosh e E a ( 2 k T )and the drift velo-cityincreasesasu=p E K sinheEa/(kT)yielding D (d = 0, E ) - 1 =x cotan x% x3/3 with x =eEa/ (2kT) . The si-mulation data fort c? =0 are consis-tent with this analyticresult (Fig. 17).For B =3.5,which turns out to bean experimentally relevant disorderparameter, f(E)- 1 approaches anapproximately linear field dependenceover an appreciable field range. Bymeans of (19) it is immediately seenthat this gives rise to universality ofTOF signals that are non-dispersive,i.e., exhibit a well-developed plateauand yield a thickness-independentcarrier mobility. Recall that in thepast universality has been consideredas a hallmark of dispersive transporttreated in terms of Scher-Montrolltheory [21, 701. The Monte Carlosimulation results demonstrate thatthis is all but so: Universality is asignature of quasi-equilibrium stochas-tic hopping motion within a SUE-ciently wide Gaussian DOS under theaction of a drain field. This is confirmed by experiments which often yield universalyet non-dispersive TOF signals [71, 861 and by simulation [64].Since the quantity directly accessible by experiment is the dispersion w of a T OF pulseit seems appropriate to outline the prediction of Monte Carlo simulation as far as thedependences of w on electric field an sample length L are concerned for systems with

superimposed diagonal and off-diagonal disorder. Fig. 18 illustrates how the functionaldependence of w on E changes from w cc E'/' at small disorder to w const at largerdisorder. At E 2 lo6 V cm- ' decreases because the effective energetic disorder experiencedby a drifting carrier is significantly reduced as eEa becomes comparable to 6.

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Charge Transport in Disordered Organic Photoconductors 35

1

Fig. 17 . J ( E ) - 1 against electric field, parametricin 6 (from [64])

An interesting phenomenon is observedconcerning the evolution of the spatial va-riance of a packet of carriers as a functionof its position within the sample. This allowscalculating w as a function of L . Fig. 19presents data for w at variable 8, ,Z =0, andE =6 x 10' V cm-'. w turns out to be inde-pendent of L at short L and starts decreasingas w K ,! ,-'I2 above a critical length thatincreases with increasing 8. Extrapolating wnormalized to the dispersion wo expected for6= 0 allows predicting the break in w(L)plot for larger disorder parameters and lar-ger samples, respectively. TOF signals arethus expected to be universal for L 2 20 pmprovided that 6 2 4. Remarkably, exper-imental data taken by Young [72] in a 30%DEH/polystyrene sample, known to containonly a small degree of built-in off-diagonaldisorder, yield w =0.4 for 4 pm 5 L 5 40pmindicating that b must be x 4 . 2 at 295 K,equivalent to CJ >0.11 eV. This is consistent with the temperature dependence of the holemobility yielding CT =0.13 eV [73]. On the other hand, the w(L) ata of Yuh and Stolka

[31] on 40% TPD/polycarbonate bear out a L-"' dependence (see Fig. 19) as expectedfor 6 = 4.0, while the &-value nferred from the p(T)-data is 4.2 (0 = 0.108 eV). Consideringthe uncertainty in the extrapolation the agreement is remarkably good. It confirms thatthe scaling behavior of w, otably the change in the functional dependence at large L romw +f(L)o w K L-'I2 is a characteristic feature of hopping motion within a GaussianDOS. The w cc L-"' behavior indicates that the ultimately the long-time limit has beenattained although the spatial variance of the carrier packet maintains being anomalouslylarge.In order to determine whether or not off-diagonal disorder has the same effect on w(L)as energetic disorder a series of computations were carried out for various superpositionsof energetic and geometric disorders. The result shown in Fig. 20 indicates that the latterdoes contribute to the spreading of TOF signals, yet does not yield w +f(L) ith samplesof macroscopic length unless a significant amount of energetic disorder is superimposed.Another way of demonstrating the success of the disorder model to account for thetemporal features of a TOF signals is to compare experimental TOF signals with MonteCarlo simulations based upon the disorder parameters inferred from the dependences ofp on temperature and electric field. This is illustrated in Fig. 21 and 22 for a pure TAPCglass and 75% TAPC/PC. The agreement is striking.3*

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36 H . BASSLER

t3

0.1

3.0 1.5F 2.5 1.52.0; 0

t I I I 1 I I l l 1 I I I I 1 1 1 1 l I 110 106E (Vcrn-1)-Fig. 18. Dispersion w vs. electric field for various com binations of diagonal and off-diagonal disorder.(The sample length was 8000 lattice planes)

For the interpretation of field-induced spreading of TOF tails it is important to notethat it is not due to topological memory effects a random walker may be subject to.Simulations carried out on systems whose site energies were redefined after each jumpyielded virtually identical results [64]. This demonstrates that the effect is related to thedistribution of jump rates as an inherent feature of hopping transport in random media.Recall that ordinary diffusive broadening of a sheet of charge carriers drifting under theaction of a bias field follows Einsteins law relating diffusivity and mobility only if the field

L ( p )-0.1 1 10 100

0.01 0.1 1 10n/8000-Fig. 19. Simulation result for w vs. normalized sample length. Dashed curves are extrapolations.Experimental data are for 30% DEH doped into polystyrene (full squares) (from [72]) and for 40%T PD doped into polycarbonate (open squares) (from [31])

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Charge Transport in Disordered Organic Photoconductors 37L(ym)-0.1 1 10

Fig. 20. w as a function of normalizedsample length for various combinations ofdiagonal and off-diagonal disorder andelectric field

5

0.1

0.01 0.01 0.1 1n/8000-is sm all , i.e. eEal(2kT) 1, an d if the medium is homogeneous. The latter con dition requiresthat, in the case of incoherent hopping transport, the jump rate be a well-defined quantity.It is violated in an energetically or positionally disordered medium. In Section 4.1 it hasbeen described that carriers executing a random walk in a Gaussian DOS tend to settleon average at an energy -a2/(kT) below its center, the variance of the occupational DOS16.0

14.0L

Fig. 21. A comparison of the exper-imentalTOF ignal ( for anamorphousTAPC ayer with the si-mulation result ( 0 ) or u =0.067eV,T = 295 K, and C =2.12 (from [60])

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38 H. BASSLER

t (ys)-Fig. 22 . A comparison of a TOF signal ( ) recorded with a TAPC/PC sample and the simulationresult ( 0 ) or th e paramctcrs 6 =3.5 and C =3.25 that follow from th e analysis of p ( E , T ) ata (from~301)being CT.The jump rate of carriers temporarily localized in the bottom states of the DOS"""will, therefore, be lower than average and vice versa. The concomitant distribution of jumprates must give rise to non-thermal spatial spreading of the carrier packet i f subject to anapplied drain field. Since no field is acting in perpendicular direction, the shape of thecarrier packet becomes cigar-like.A similar effect was first noticed by Rudenko and Arkhipov[74] when analysing transport of charge carriers subject to multiple trapping (MT) by aGaussian distribution of trap levels and later on by Bouchaud and Georges [75] in the courseof their analytic study of hydrodynamic flow in porous media. An elaboration of the M Tcase has recently been given by Haarer et al. [49]. Despite the similarities between M T andhopping in a distribution of intrinsic hopping states derived from the band states viadisorder-induced localization, there are essential differences involved in both approachesas discussed in greater detail in [64].

4.5 Transitionfrom non-dispersive to dispersive transportSince the time a packet of carriers, started randomly in energy, needs to relax to dynamicequilibrium increases faster with time than the transit time across a sample (see Section4.I) , TOF signals must eventually become dispersive once the temperature drops below acertain value that depends on the width of the DOS.To delineate the general phenomenolo-gical pattern, a series of double logarithmic current versus time plots, parametric in 6,covering up to 20 decades in time and up to lodecades in amplitude are presented inFig. 23. The curves for 6 =3, 6, and 8 represent the results of an analytic effective medium

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Charge Transport in Disordered Organic Photoconductors 39ttI I I I I1 5 10 15l o g (t/to)-

Fig. 23. Double logarithmic current vs . time plots. The full and dotted curves are simulation resultsand their extrapolations towards equilibrium. The dashed curves are the result of th e effective med iumapproximation for an infinite sample (from [76])(EMA) calculation of the zero-field diffusivity of a packet of carriers in an infinite sample[45]. Apart from demonstrating the mutual consistency of EMA and simulation results,these plots indicate that in no case a simple power law of the type i ( t ) cc t - ( ' - " ) is obeyedfor times less than the mean arrival time ( t T ) . At short times the time dependence of thecurrent is virtually independent of 8 but tends to saturate earlier with decreasing 8.A more detailed family of current transients, parametric in 6, s shown in Fig. 24 for asample with n, =4000, equivalent to a thickness of 2.4pm.The occurrence of a N D -+Dtransition with increasing 6 s obvious. While the transients reveal a plateau of variabletemporal length for d 4.4.Reducing the samplelength at fixed 8 causes the initial portion of the transients to appear more dispersive. Thisis shown in Fig. 25. The ND -+D transition is more clearly delineated by plotting thereciprocal mean carrier arival time as a function of 8', parametric in sample length, andthe arrival time as a function of n, at variable 8, as shown in Fig. 26 and 27, respectively.Withinthedispersiveregime,t,(n,)deviatesfromlinearity, ( t T ) cc nr,where 1.25

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H . BASSLER0

tc.c01LY00-

ILcLYV0 70

?

d

Log t ime -Fig. 24

Fig. 24. Simulated transients, parametric in 6 fo rE =0.6 M V c m - ' (from [76])

log timeFig. 25

a sample consisting of n, =4000 lattice planes,

Fig. 25. Simulated TOF signals, parametric in n, with 6 = 5.2, E =0.6 M V cm - (from [76])

ND +D transition separates a transport regime in which the mean carrier arrival timeyields a thickness-independent transport coefficient from a regime in which this is no longerthe case. Operationally, the ND +D transition can be defined via the intersection of thelog ( t T ) versus d2 lines representing the average reciprocal carrier arrival times in thenon-dispersive and dispersive regimes, respectively. This yields a correlation betweenthe critical disorder parameter, d,, and sample length. Fig. 28 indicates that the criticaldisorder parameter at which the ND -+D transition occurs increases with the number oflattice planes as 6 = A +B log nzrwhere A = - .6 and B = 6.7 are empirical constants,equivalent to

6 = A' +B log L , (21)

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Charge Transport in Disordered Organic Photoconductors

t I I I 110 15 20 25 30$2 -

41Fig. 26. Simulated reciprocal m eanarrival times vs . dz , parametric inn, (from [76]). The asymptote de-scribing non-dispersive transportwas taken from [30]

Fig. 27 . Simulated mean arrivaltimes vs . n,, parametric in d (from1761)

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42

Fig. 28. The disorder parameter atwhich th e N D +D transition oc-curs vs . sample length

where A' = 44.8 and L is the sample thickness in cm. With a thickness of 20pm and aDOS width of 0.1 eV, transients are then predicted to be dispersive at 232 K. According toa simple criterion used earlier [52] ,a temperature of 286 K would have been estimated.

hI Q P4.88000 (d =4.8 p m )0.6MVcm-'

from logarithmic plots

cime-A transient at 8 = 8, does revealan inflection if plotted on a double-linear scale, yet not a well-developedplateau. This is illustrated in Fig. 29showing a transient for n, = 8000 at

8 =4.8 which is very close to 8,.Obviously, transients that appear mod-erately dispersive on double-linearscales featuring a slope - 1 - xl)z 0.27 in double-logarithmic repre-sentation are still satisfactory for deter-mining mobilities that represent bulkproperties. One reason is that theinflection monitors the arrival of thefastest equilibrating earlier, while re-

Fig. 29. Simulated TOF signal at theND +D transition

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Charge Transport in Disordered Organic Photoconductors 43sidual current relaxation is controlled by slower carriers hopping within the bottom partof the occupational DOS. Fig. 29 also illustrates that the absolute magnitude of the mobilitymay differ up to a factor of 2 depending on how the arrival time is defined. Inferring ( t T )from the intersection of asymptotes in double-linear plots, which is the common experimentalpractice, yields values that are approximately a factor of 2 larger than values derivedfrom ( t T ) which, unfortunately, are not directly accessible in experiment. Another reasonwhy cr, is larger than previously [52] assumed is related to the effect an electric field hason the relaxation behavior of carriers within a Gaussian DOS. A field raises the equilibriumenergy of carriers within the DOS and, concomitantly, shortens the equilibration time.Following Scher and Montroll [21] it has become practice to analyse dispersive TOFsignals in terms of the algebraic decay functions

From Fig. 22 the failure of (22) for rationalizing dispersive transients in a material with aGaussian DOS is obvious. Operationally, one can, nevertheless, determine a dispersionparameter from the current transient in double-logarithmic representation within, say, thelast one and a half decades in time prior to the inflection because the curvature is smallon this time scale. Fig. 30 presents experimental (see below) as well as simulation data forthe related slope -(1 - a l ) of the double-logarithmic curves for t

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44

3.5 I I

-0 0N2.5 - 0 . -b 0 0t 3'0-+F 2.0- -

1.5 - -

H . BASSLER

Fig. 30. Simulated (solid circles)and experimental (opencircles fromFig. 31) values for th e slope parame-ters of Scher-Montroll plots

relaxation of an ensemble of excitations strictly follows a logarithmic time dependence,independent of the absolute time scale. Changing the experimental time scale by varyingthe sample thickness or the field will not, therefore, change the relaxation pattern.Concomitantly, current transients are universal on a double-logarithmic scale. For aGaussian DOS, the energy relaxation function is no longer linear on a logarithmic scalebut tends to settle at a mean energy - 2 / kTbelow the center of the DOS. Changing thetime scale by varying either the sample length or the field will therefore alter the relaxationpattern via the interplay of two opposing effects. Reducing the length will shift the relaxationpattern into a time domain in which i3 AE/a In t is larger, i.e. render the transients moredispersive. On the other hand, increasing the electric field will raise the mean equilibriumenergy of carriers with the consequence that equilibrium is attained faster.A textbook example of the success of the disorder formalism to predict the temperaturedependence of the shape of progressively dispersive TO F signals on the basis of the energeticdisorder parameter determined from high-temperatur p( T ) data is the recent work byBorsenberger et al. on a neat DEH glass [ 76 ] .A series of experimental TOF signals isshown in Fig. 31 on both double-linear and double-logarithmic scales, respectively. The

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Charge Transport in Disordered Organic Photoconductors 45

Lim e-

L

ZJ

I I I I I ILog t ime-Fig. 31 . a) Experimental TOF signals measured with a 5.6 pm thick amorphous DEH film at varioustemperatures ( E =0.6 MV cm-I). b) The same transients in double-logarithmic representation. Thedashed curves are simulations based upon a width of the DOS of D =0.101 eV. The experimentaltransient for 6 =4.45 is coincident with the simulation for 6 =4.4 (from [76])

latter diagram contains simulated TOF signals based upon 0 =0.101 eV as inferredfrom the p ( 7 ) data. Note that there is no adjustable parameter except the trivialscaling factor relating the time frame of the simulation to the experimental time frame. Notonly is the agreement betwen experiment and simulation a crucial consistency test of themodel, it also demonstrates that for this system the entire T-dependence of p must beattributed to disorder. Would it contain a polaronic contribution the onset of dispersionshould have occurred at lower temperatures than observed. The N D .--) D transition isalso evident from a plot of the mobility versus 6 2 or two different sample thicknesses(Fig. 32). The predicted thickness-dependent kink occurs in quantitative accord with modelpredictions.

Superposition of any additional disorder, such as positional disorder or trap-capturinginvolving a distribution of traps, causes further broadening of the tails of dispersive TOFsignals without changing the behavioral pattern in general [77]. This is evident bycomparing the TOF signals Miiller-Horsche et al. [20]reported for PVK with the simulationresults. Translating the activation energy of the mobility Pfister and Griftith [78] measuredfor PVK into a width of the DOS yields 0 =0.11 eV. This allows plotting their 1- tIand 1 +u2 data versus 6 (Fig. 33). While the 1 - 1 values are close to the simulation

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46 H. BASSLER

Fig. 32. Hole mobilities, operatio-nally defined as p = L/E(f,,), forDEH samples of different thick-nesses (from [76])=0.101 ev

10-7 1 4 115 20 25 30 353__c

results, 1 +z2 is smaller by about 0.5. This is readily accounted for in terms oftrapping by carbazole sandwich pairs causing stretching of TOF tails. The results are similarto more recent ones obtained for TAPC/PC [77] in which geometric disorder in additionto energetic disorder is significant.

4.6 Concentration dependence of pFor technological reasons the dependence of the charge carrier mobility on the concentrationof the transport moiety in molecuarly doped polymers has been the subject of early andintense research in this field. It is obvious that p must decrease as the system of transportsites is diluted. On the other hand, dilution prevents aggregation of the photoconductor andimproves the mechanical properties of the film which are largely controlled by the polymericbinder. A compromise has, therefore, to be made concerning the optimum concentrationof the charge transporting species based upon knowledge of p ( c ) .

Experimentally, p is found to decrease by some five orders of magnitude within the relativeconcentration range 1.0 2 c 2 0.1 [l , 4,791 (c is the volume fraction of the transportingmaterial). The simplest way to understand this effect is in terms of the homogeneous latticemodel [22 , 51, 69, 801. One assumes that the only effect of dilution is to increase the mean

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Charge Transport in Disordered Organic Photoconductors 473.5 I I I I I I Fig. 33. Comparison of 1 - 1an d 1 +a2

for PVK measured by Pfister and Grifith[78]. The plot vs. r i is based upon a widthof the DOS of Q =0.11 eV derived fromtheir p ( T )data. T he dashed line (fitting thetriangles) is the simulation result for asystem with energetic disorder only

-3.0--2.5 -U+

7 2.0 -1.5- -1.0 I 1 1 I0.6 I I I I I I 1

l- o.211 0distance among transporting molecules according to

where a is the mean nearest neighbor distance in the undiluted system. Within the frameworkof the concept of incoherent transport this has two consequences: (i) the wave functionoverlap factor decreases with increasing c according to exp (- 2 y a ~ - ~ )nd (ii) the meansquare displacement per jump increases proportional to c - ~ ~ . ence, the prefactor of themobility becomes

where a, = ( e / k T ) u2vo/6) , , being the prefactor in the expression for the jump rate (2) .The full expression for p is

wherefi measures the field dependence of p which depends on T and possibly on c, too,and f i measures the T-dependence of the zero-field mobility which may also depend on c.It is obvious from (25) that in order to determine the functional dependence of ,uo(c)onehas to know the dependences of p on both E and IT; parametric in concentration. I f ~ ( c )s

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48 H . BASSLERmeasured at constant T and E only, data analysis in terms of (25) or any other transportequation can only be a zero-order approach because it tacitly assumesf, and f2 to beindependent of c.The homogeneous lattice model need not, however, be appropriate. Upon dilution afraction of the sites otherwise available for transport are replaced by inert binder molecules.Intuitively, the model of choice for this problem appears to be the site percolation concept[81 to 831. In the absence of disorder it predicts

with c =0.312 for a simple cubic structure and y = 1.5 [82]. The percolation model takesproper account of the discrete molecular structure of the sample, yet is likely to overestimatethe threshold behavior of p(c) near the critical concentration since it ignores the possibilitythat a carrier leaves a dead end of a conducting path by a non-nearest neighbor jumppermitted by virtue of the exponential distance dependence of the intersite jump rate. Thecritical behavior near co is thus likely to be eroded.In the absence of an analytic treatment of site percolation under the premise of exponentialintersite coupling MC simulations were carried out yielding the charge carrier mobility ina diluted cubic system. To save computer time the computations were done on samplesconsisting of 70 lattice planes only and devoid of energetic disorder. The latter conditionimplies that the results be adequate for analysing experimental mobility data extrapolatedto T+ co where the effect of disorder has vanished. Fig. 34 presents MC data for p ( c )parametric in 2ya. They confirm the intuitive notion that erosion of criticality is the morepronounced the smaller 2ya is, i.e. as the spatial extent of the wave function increases.Replotting the data on a In ( p ~ ~ )ersus c- scale reveals, however, that (24) does providea reasonable approximation, at least for c 2 0.1 (Fig. 35). This indicates that the ho-mogeneous lattice model can, in fact, be considered as a zero-order approach to treat theconcentration dependence of carrier motion in molecularly doped polymers. Plottingexperimental ~ ( c )ata in terms of (24)does at least yield the wave function overlap parameter2yu within an accuracy of about 20% [84].

To demonstrate the consistency between simulation and experimental results, the,u0c2-data Borsenberger [7] obtained for the system tritolylamine (TTA)/PC have beenincluded in Fig. 35. They were obtained by plotting the zero-field mobilities on a In p versusT - scale and extrapolating to T -+ co [7]. Data fitting by (24) yields 2ya = 8.5. Had oneplotted the zero-field mobilities - extrapolated according to a In ,u cc E l l 2 relationship -measured at room temperature one would have obtained 2ya = 12.7! The reason forthe difference is that dilution causes the width of the DOS to increase as a result of theincreasing randomness of the internal electric fields due to the randomly oriented dipolesof the polycarbonate binder. This causes the temperature effect on ,u to increase the smallerc becomes.At this stage it appears appropriate to outline and critically examine the small polaronapproach Schein et al. [26] developed to analyse the concentration dependence of the holemobility in the TTA/polycarbonate system. As mentioned in the Introduction the smallpolaron model rests on the ideal that the configuration of a hopping site is altered uponremoval or addition of an extra electron. The charge is, therefore, dressed by a deformationit has to drag while moving. Polaron theory [85] predicts that the associated activationenergy for transport is A =4 E , - where E , is the polaron binding energy and J the

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Charge Transport in Disordered Organic Photoconductors 49

I -3,3A0azvJ - 4I-r\cP -5md

-6

-7

-8

-9

-10

- c, I , I I I I 1

= 3

\ \\0 1.5 2.0 2.5 3.0 3.5(R/a) -Fig. 34 . Monte Carlo simulation results for the concentration dependence of the mobility plotted inaccordance with the ho mogeneo us lattice gas model (24) for various values of the wave function overlapparameter 2ya (from [84]). rand zo are identical with t,, and t o used throughout this paper

charge transfer integral. The zero field mobility is

P denoting the probability that a charge carrier will hop once an energy coincidence betweendonor and acceptor sites has been established via thermal activation. The adiabatic regime4 physica (b) 175/1

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50 H. BASSLERFig. 35. Simulation d at a for p ( c ) ofFig. 34 plotted on a lg ( p O c 2 @ )s. ( R ) / ascale a s suggested by the homogeneouslattice gas model. Dashed lines indicatetheoretical slopes (from [84]). Datapoints are experimcntal and were ob-tained by p lotting the zero field mobili-ties of TTA/PC on a In p vs . T - *scale and extrapolating to T - t a(from [91)

CI)- - 6 --

-8 --

-10 I 1 I I1o 1.5 2.0 2.5 3.0c-113 =(R ) /a -

is defined as P = 1, i.e. charge transfer occurs with unit probability independent of siteseparation. Consequently, the product p ( E = 0) c 2 I 3 should not depend on c as long as Jis large. This condition cannot hold at large intersite distances because J decreasesexponentially with increasing ( R ) . In the non-adiabatic regime

The transition from adiabatic to non-adiabatic small polaron transport should manifestitself in a break in the concentration dependence of po (E =0, T + co) c2/3.At large c , thisquantity should be constant and decrease exponentially with c - i.e. increasing sitedistances, at low concentration.Schein et al. [26] took experimental data for the hole mobility in TTAJexane - he samesystems Borsenberger had used for his study [7] - o verify their model. They plotted thezero-field mobilities, evaluated from In p versus plots, in an Arrhenius diagram yieldingp ( E =0, T+ T ) .A plot of their p ( E =0, T+ co) 213data versus ( R ) / u = c - l i 3 is showni n Fig. 36. A change from distance independent to distance dependent behavior, apparentlyverifying the polaron picture, is obvious. Note, however, that T+ co mobilities saturate at~1 00 cm 2/ Vs hich is a prohibitively large value even for a molecular crystal. Forcomparison, Borsenbergers data, derived from the same raw data set but evaluatedin terms of the hopping model, i.e. from a In p versus T2 lot (see Fig. 28), are neitheranomalously larger nor do they bear out the transition from distance independence todistance dependence at a critical concentration. Note that the difference between thetwo plots is solely a consequence of the fact that extrapolation according to an Arrheniusplot overestimates p(T+ 03). This is a warning example how the choice of modelaffects the final result, and illustrates how hazardous p (T) extrapolations over manydecades are.

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Chargc Transport in Disordcred Organic Photoconductors 51- TA concentration ('10)50 4 0 30 20 10 9 7cccYIIN>$-m.U01,

1.2 1.4 1.6 1.8 2.0 2.2 2.4(R)/a --1/3 =Fig. 36. C omparison of the ,u,,czi3 data for TTA/PC obtained by extrapolating p ( T ) according toArrhenius' law (full symbols, taken from [26]) nd according t o a In p vs . T-' law, respectively (fromFig. 35 using data of Borsenberger et al. [9]). The dashed line is the prediction of the homogeneouslattice model for 2ya =8.5

5. Guideline for Estimating Polaronic EffectsDespite the obvious success of the disorder concept for rationalizing charge transport inmolecularly doped polymers - and, acording to recent work in x- [17, 971 and 0- 29]conjugated main chain polymers as well - t is legitimate to question the reason for thissuccess and to outline problems that need to be addressed in future work. The main criticismthe present approach has to face is the assumption of the Miller-Abrahams ansatz for thejump rate which, on the one hand, ignores polaronic effects but requires moderate electronphonon coupling on the other hand [86]. In a more realistic ansatz the prefactor in thejump rate had to carry a T-dependence. However, the strong T-dependence of p certainlyoverrides any effect of a - presumably - power law dependence of v o on 'Is Far as theconsideration of a polaronic binding energy is concerned a simple estimate may provide aguideline for a crude assessment of its effect on p(T).A first approach to incorporate polaron effects is to invoke intramolecular relaxation ofthe charge carrying species upon oxidation or reduction. Let the average energy gain be Ep- any spread of Ep would be equivalent to increasing disorder - and show up in theexpression for the jump rate by an extra Boltzmann factor carrying an activation energyEJ2. Since all transition rates are affected in the same way the calculation can be done infactorized form and the total, albeit temperature dependent, activation energy of transportwould then be the sum of a disorder and a polaron contribution (see (lo)),

Considering the inherent difficulty to distinguish In p versus T- ' and T - 2 plots coveringa small T-range only, any deviation from a In p versus T - plot would hardly be discernible.4*

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52 H. BASSIXR

0 0.5 1.0APO'/A -

Fig. 37 . Estimate of 8 (left scale) and of the polaronbinding energy E , (right scale) for variable relativepolaron contribution to thc measured total "activa-tion energy" A of the mobility for A =0.5 (- - ),0.4 ( ), 0.3 eV (. . . . . .)

A critical test for a polaronic or any other simplyactivated contribution to T-dependent trans-port such as shallow trapping is, however,provided by the comparison between the tem-perature dependence of the carrier mobility andthe onset of dispersion of TOF signals at lowertemperatures. If both sets of data are consistent,as is the case for DEH (see Section 44, thepolaronic contribution must be negligible ascompared with disorder effects. If, on the otherhand, onset of dispersion starts at lower tem-perature, i.e. smaller 6, this provides a handleon separating disorder from polaron effects.Since only half of E , contributes to A while -because of the offset of the occupational DOSfrom the real DOS by 0 2 / k T (see Section 4.1)- he disorder contribution is a multiple of o,a quite moderate polaronic contribution toA can translate into a quite large ratio of E,/o. This is illustrated in Fig. 37 showing bothr i 2 and E , as a function of the fractional polaron contribution dpo'/dfor d = 0.5, 0.4, and0.3 eV, respectively, while Fig. 38 presents E , / o versus dpo'/A.If, for instance, the o-values

b\nw 43

2

1

0 0.1 0.2 0.3 0.4 0.5A P ~ ' / A- Fig. 38. EJcr vs. Ap"'/A

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Charge Transport in Disordered Organic Photoconductors 53calculated from p(T)-data, on the one hand, and from the temporal evolution of TOFsignals, on the other hand, agreed to within 4% (equivalent to z 10 K uncertainty concerningthe onset temperature of dispersion) any polaronic contribution to A may be 5 8% equivalentto EPIC=0.5. Since CT is typically around 0.1 eV this implies that a 50meV polaronicbinding energy is below the limit an analysis of transport data would allow to separate.The message of this order of magnitude estimate is that small polaronic contributions toA , which must occur because the equilibrium configurations of a neutral transport moleculeand a radical cation/anion are inevitably different and which have been invoked forrationalizing charge transport in crystals [43], are compatible with data analysis in termsof the disorder formalism. In how far larger contributions would modify the expected fielddependence of p remains to be explored in future work. Suffice to mention here that purepolaronic transport should yield a p ( E ) cc E - sinh (eEa/2kT) relation that has never beenobserved in experiment.

6 . Concluding RemarksThe present review summarizes how hopping motion of charge carriers in a Gaussiandensity of localized states with or without superimposed off-diagonal disorder is revealedin experimental observables such as the dependence of the carrier mobility on temperature,electric field, composition in the case of multicomponent systems, and TOF profiles. It hasbeen shown that the recurrent features observed with chemically very different systems canbe understood in terms of a model that is conceptionally simple, yet extremely difficult totackle analytically. Monte Carlo simulations are, however, a convenient tool to determinehow the combination of physical assumptions, for instance, the form of the intersite jumprate, and material parameters such as the degree of diagonal and off-diagonal disordertranslate into macroscopic properties.In terms of the disorder model the similar behavior of a variety of polymeric photo-conductors is a consequence of the similar degree of disorder they contain. This, in turn,is a natural consequence of similar preparation conditions. They are usually solution castby letting the solvent evaporate at room temperature. Therefore, the energetic distributionof metastable conformation trapped within the structure should not be too different. Ittranslates into DOSs typically 0.1 eV wide equivalent to an activation energy of the mobilityof ~ 0 . 4V if inferred from the tangent to an Arrhenius plot at 300 K . It is also manifestin the similar inhomogeneous broadening of optical absorption spectra, their commonorigin being random internal fields in random matrices that act on both a dipole, such asan exciton, or a charge located on a transport molecule [87]. Qualitative support for thisreasoning comes from the matrix dependence of p that translates into a matrix dependenceof the width of the DOS [88] and, notably, a dependence of c on the presence of dipolaradditives [89] and on the dipole moments of either the transport [lo, 901 or host molecules[88] or both [91]. Quantifying such empirical relations by calculating the distribution ofinternal fields and, concomitantly, width and profile of the DOS, is a challenge for futurework. An experimental starting point may be the evaluation of the random internal fieldson the basis of Stark effect measurements employing hole burning spectroscopy ofchromophons embedded in random matrices [92 to 941. The recent work of Novikov andVannikov [95] on the effect of dipolar traps on the field dependence of p offers acomplementary approach.

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54 H . BASSLERAcknowledgements

1 am highly indebted to L. Pautmeier, R. Richert, and B. Ries for their contribution to thiswork. Stimulating discussions and the exchange of ideas and information with M . Abkowitz,P. Borsenberger, B. Movaghar, L. Schein, M. Silver, M. Stolka, and R. Young are gratefullyacknowledged, as is the generous support by the Deutsche Forschungsgemeinschaft andthe Fonds der Chemischen Industrie. The collaboration with M . Silver was supported bya NATO travel grant.

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