Analytic Modeling of Field Dependence of Charge Mobility and

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Analytic Modelling of Field Dependence of Charge Mobility and Applicability of the Effective Transport Level Concept to an Organic Dipole Glass Vladimir R. Nikitenko, Anna Yu. Saunina, Andrey P. Tyutnev, and Oleg V. Prezhdo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01779 • Publication Date (Web): 23 Mar 2017 Downloaded from http://pubs.acs.org on March 30, 2017

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The Journal of Physical Chemistry

Analytic Modelling of Field Dependence of Charge Mobility and Applicability of the Effective Transport Level Concept to an Organic Dipole Glass V.R. Nikitenko a,*, A.Yu. Saunina a, A.P. Tyutnev b, O.V. Prezhdo c,*

(a) National Research Nuclear University «MEPhI» (Moscow Engineering Physics Institute), Moscow, 115409, Russia (b) National Research University Higher School of Economics, Moscow, 101000, Russia (c) Departments of Chemistry, and Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA

Abstract An analytic model of field dependence of charge mobility is developed within the long-range correlated disorder model of a dipole glass. Release of a charge carrier from a deep state is considered as hopping drift and diffusion in a quasi-Coulomb potential well. The analytic results are in good agreement with the fit to the Monte-Carlo simulations, while they contain only one numerical parameter obtained from an independent simulation. The developed approach justifies applicability of the effective transport level concept for modelling of organic materials with large molecular dipoles.

* Corresponding authors: [email protected] (VRN), [email protected] (OVP)

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1. Introduction Modelling of the charge carrier transport characteristics in disordered organic materials is a matter of great interest, because it allows optimization of electronic devices such as OLEDs, OFETs, etc. Description of transport processes in disordered organics is a complicated problem, since, unlike inorganics, organic materials exhibit transport that occurs via tunnel hopping of charge carriers between localized energy states. The mobility of charge carriers, µ, is one of the most important transport characteristics. As a rule, the mobility is studied theoretically by numerical simulation.1-4 At the same time, it is extremely useful to have analytic expressions for mobility dependence on temperature T, energetic disorder scale σ, charge carrier concentration c, and an electric field strength F. The theory of charge mobility is a topic of numerous recent works (see, for example, refs 5-9). The essential role of energetic correlations in the genesis of the so-called Pool-Frenkel-type (PF) field dependence of mobility is widely acknowledged in organic materials,10-16 and is specifically stressed in refs 12, 15. The following expression is commonly used 

µ = µ0 exp  CT + CF 

eaF  . σ 

(1)

Here, F is the electric field strength, e is the elementary charge, and a is the average intermolecular distance. The constants СF and СТ depend on temperature T and the width σ of the Gaussian distribution of energy levels of localized states g(E): g (E) =

 E2  exp  − 2  . 2πσ 2  2σ  1

(2)

Explicit consideration of short-range correlations originating from the interaction of a charge with the field-induced dipoles in the neighboring sites15 leads to a more accurate field dependence in eq 1, over a wider range of field strengths, in comparison with the Gaussian disorder model (GDM) that neglects correlations.17 At the same time, numerical differences in the predictions of these 2 ACS Paragon Plus Environment

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two models are not large as shown in ref 15. The dipole glass model (DGM) that includes long-range correlations is applicable to polar materials with large (> 1D) permanent molecular dipole moments12. This model yields description of PF field dependence including relatively weak fields

eFa / σ < 0.5 . According to the

DGM, 2

9 σ  CT = −   , 25  kT 

 σ 3 2  CF = 0.78  − 2 ,   kT  

(3)

where k is the Boltzmann constant. The value of СF deviates significantly from the result of the Pool-Frenkel model18: CF PF = e 2σ / (πεε 0 a ) , since unoccupied localized states are neutral. The energy of electron (hole) state is calculated in the DGM as the sum of energies of interaction of the electron with randomly oriented intrinsic molecular dipole moments. Fluctuations of this value lead to the Gaussian distribution given by eq 2, except for the very deep tail of distribution which is not affecting the mobility for practically important values of σ/kT < 6.19 Contrary to the GDM,17 energies of neighbor hopping states are strongly correlated in the DGM. Previously, strong correlation of energies of neighbor hopping states led to the conclusion20 that transport level concept21 is not applicable to organic materials with correlated disorder, because the nearest hopping neighbor most probably has an energy close to the initial state, while the transport energy does not depend on the energy of an initial state. Equations 1 and 3 describe both the PF-type field dependence of mobility and its temperature dependence, in good agreement with the experimental results in polar organics12. It should be noted, however, that constants СF and СТ in eq 3 contain numerical factors 9/25, 0.78 and 2, obtained from the Monte-Carlo modelling of mobility for the case of hopping centers forming a cubic lattice. The general applicability of these results can be questioned,22 in particular, since the dependence of a localized state wave function on the inverse localization radius γ was not studied, and only a special case of γa = 5 was considered in the simulations. 3 ACS Paragon Plus Environment


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In this work, an analytic model of the field and temperature dependence of mobility in organic dipole glasses is introduced. The model treats the release of a carrier from a deep state as an escape from a Coulomb-type potential well by means of diffusion-type multiple hopping. Analysis of the results leads to the conclusion that the charge carrier transport in organic materials with correlated disorder can be described by the multiple-trapping formalism based on the effective transport level along with the GDM, although some parameters of the current model vary from those in the GDM.

2. Model for the field-dependent mobility

2.1 Analytic expressions In order to develop an analytic model of charge carrier mobility in disordered organic materials, one should account for the fact that an interaction between carriers and the system of disordered dipoles leads to strong correlations of hopping center energies. It means that the nearest hopping centers are likely to have close energies, at least within the energy spread of σ. Qualitatively, the long-range character of the charge-dipole interaction and a low relative dielectric constant of organic matter result in a significant contribution of numerous dipoles (not only of the nearest neighbors) to the energy of a localized charge carrier. Thus, a shift of a carrier by a typical hopping distance would not cause significant (on the scale of σ) change of its energy. In a typical case of σ / kT >> 1, it follows that deep states (including those with energies about E0 = −σ 2 / kT relative

to the center of the Gaussian distribution) controlling the mobility8,17 are surrounded by a potential well. Taking into account the character of the energy correlation function C (r ) ∝ 1 / r (r>a, where r is the distance from a center under consideration12) the above-mentioned well is of the Coulombtype. Indeed, the conditional probability of the state situated at a distance r from the center with energy Ei to have the energy E is known to be Gaussian 4 ACS Paragon Plus Environment

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with the following “quasi-Coulomb” r-dependence of the average energy U C 23 :

 ( E − U C (r , Ei ) )2  P ( E | Ei ) = exp  − , 2σ d 2 (r ) 2πσ d 2 (r )  

(4)

U C (r , Ei ) = −α Ei a r ,

(5)

σ d ( r ) = σ 1 − (α a r ) ,

(6)

1

2

where α = 0.74. Thus, the carrier in a state at a typical energy, which determines a value of mobility, should overcome an energy barrier formed by the quasi-Coulomb potential given by eq 5 in the presence of an applied external field F: U (r ,θ ) = − α Ei a r − eFr cosθ ,

(7)

where θ is an angle between the direction of the external electric field and the radius-vector of the carrier starting at the initial state. The height of the barrier increases with the absolute value of the initial state energy Ei and decreases with increasing field strength F. Thus, the release of a carrier from deep states is a multistage process involving drift and diffusion over a wide region. In the case of a weak external field, one can define the boundary of a well by the condition

U (r ) ≈ U C (r ) = − kT , which gives

r = rC* = α a Ei kT ,

(8)

in analogy with the Coulomb radius (the Onsager radius) rC = e 2 4πεε 0 kT , where ε is relative permittivity.



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kT

EC

E Ei

1 rC*

a

rF

r Figure 1. Release of a charge carrier from a deep state Ei at a moderate field strength. Dashed line shows the energy dependence given by eq 7, and the number 1 corresponds to a typical first jump from the initial state.

The mobility is calculated according to the ubiquitous (for the lowfield limit) expression based on the Einstein relation 2 1 e ref , µ= 6 kT t

(9)

where t is the field-dependent average time required for a carrier to escape from the potential well, and ref is the typical diffusion length of a carrier during the time t . The field dependence of this length will be discussed below. One can imagine a release process as a quick jump to the nearest neighbor state (the distance is

r0 ≈ a ), see jump 1 in Figure 1, followed by numerous attempts to escape from the quasi-Coulomb well. The time

t

could be expressed as the product of a

reciprocal “attempt to escape frequency” ν = ν 0 exp ( −2γ r0 ) , where ν 0 is a typical phonon frequency (assuming Miller-Abrahams hopping rates1-9,12), and a typical number of attempts to escape, which is the reciprocal probability for a carrier to leave the well after the first jump. It follows from the condition U (r0 ) ≈ U C ( r0 ) = Ei that r0 = α a = 0.74a (at least one nearest neighbor of the 6 ACS Paragon Plus Environment

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initial state is supposed to have an energy close to Ei). The escape probability η(Ei) depends on the initial energy Ei, because the potential in eq 7 depends on Ei. Therefore, the calculation of

t

involves the averaging over Ei with the

distribution function g(Ei):24 ∞

t = ν −1 ∫ dEi g ( Ei )η −1 ( Ei ) .

(10)

−∞

The easiest way of calculating η is according to the Pool-Frenkel model.18,19 Estimation of the integral in eq 10 yields the analytic dependence of the type seen in eq 1. However, calculations show that the field dependence of t

-1

and,

consequently, the mobility is too strong compared to both the experiment and Monte-Carlo simulation within the DGM as eq 3 shows. At the same time, it is known from the theory of separation of photogenerated charge pairs that the Onsager model25 gives weaker dependence for the separation probability in agreement with the experiment26. That is the inherent feature of the Onsager model, since it considers 3D-drift and diffusion of carriers, whereas the PoolFrenkel model is, in fact, one-dimensional. The mobility in the DGM is known to increase significantly faster in 1D than 3D.27 It is to be noted that despite the complexity of the analytical expressions, the Onsager model gives the field dependence close to that of eq 1, over a wide range of field strengths. Applying the exact analytic expression of the Onsager model for the pair separation probability28 with the «quasi-Coulomb» potential given by eq 5 in order to calculate η(Ei) in eq 10 and considering U (r0 ) ≈ U C ( r0 ) = Ei , one obtains l 2

 EC − Ei + eα aF  kT ∞  eα aF   2  η ( Ei ) = exp  − l  eα a Ei F  , (11)  I l  ∑  kT   eα aF l =1  Ei   kT  where Il(x) is the modified Bessel function of the first kind and l-order. Absent in the original Onsager model, the term EC in eq 11 appears due to energetic disorder. It quantifies reduction of the energetic barrier (see Figure 1), and its value is discussed below. The upper limit of integration in eq 10 is assumed to be infinite, since the main energy dependence in the integral is given by the function 7 ACS Paragon Plus Environment


g ( Ei )exp ( − Ei kT ) having maximum at Ei = E0 = − σ 2 kT which is far below the zero point.

2.2 Calculation results The results of calculations of the field dependence of charge carrier mobility, are shown in Figure 2. The data are parametric in disorder parameter σ kT , and

(

)

are obtained assuming ref = rC* −σ 2 kT = α a (σ kT ) in eq 9 and EC = 0 in eq 2

11. 0 -2 -4

T l k e d o m

ln µ

2

1 n o i t a l u m i s C M

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-6

σ/

-8

2.87 3.82 4.17 4.6 5.73

-10 -12 0,0

0,5

1,0

(eaF / σ )

12

Figure 2. Analytic field-dependent mobility in comparison with the results of the Monte-Carlo simulation12 under condition that EC=0 and ref =rc*.

Both the field dependence of ref and the energetic disorder are neglected in this calculation. Therefore, the field dependence of mobility arises only from t . Figure 2 shows that the field dependence of mobility is rather strong in comparison with the results of the MC-calculations based on the DGM12. In order to improve the field dependence, one should be reminded that the mobility ought to be calculated, strictly speaking, as


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µ= x

(F t )

instead of eq 8, if the field is not weak, where x is the mean

displacement of a carrier during time t in the field direction. Operationally, one can introduce the field dependence of the length ref in eq 8 as follows. In a weak field limit, one can estimate ref = rC * (see eq 8) provided that Ei = − σ 2 kT . When the field is stronger, ref should become field-dependent, and it corresponds to the maximum energy in eq 7 (see Figure 1). If θ = 0 and Ei = −σ 2 kT , one obtains rF =

α a Ei eF

≈

α aσ 2 ekTF

.

(12)

In the case of a strong field, rF < rC * and ref should decrease with increasing field strength. In the limit of a strong field, one can expect t ≈ ν −1 ,

x ≈ a and

µ ≈ a t F . Thus, one comes to the following interpolation for ref : −1

ref 2

 * −2 α  σ  2 −2  =  rC +   rF  .  6 kT    

(13)

In order to improve the temperature dependence of mobility, one can estimate EC in eq 11 as an effective transport level in a low-field limit calculated according to the approach given in ref 21 developed in the GDM, because a carrier at r ≈ rC * , being in an energetic proximity to the center of the distribution given by eq 2, has many neighbor centers with close energies both in the DGM and the GDM. Indeed, the Monte-Carlo simulation29 shows a good agreement of the effective transport level values for both correlated and uncorrelated disorder. Calculation results obtained using eq 13 and the effective transport level EC in eq 11 are shown in the Figure 3. The predictions of our model are in good agreement with the numerical results of the DGM. There is a minor disagreement at high values of the field strength, eaF σ ≅ 1 , and at the strongest disorder. One can improve the quantitative accuracy of the model by a more elaborate treatment 9 ACS Paragon Plus Environment


of the field dependence of the length ref , which is defined now by interpolation from eq 13.

0 -2 -4

T l k e d o m

ln µ

2

1 n o i t a l u m i s C M

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-6

σ/

-8

2.87 3.82 4.17 4.6 5.73

-10 -12 0,0

0,5

1,0

(eaF / σ )1 2 Figure 3. Analytic field-dependent mobility in comparison with the results of the Monte-Carlo simulation12 using EC and ref according to eq 14.

3. Discussion In this work, we have established that an analytic expression for charge carrier mobility in organic materials with long-range correlated disorder could be obtained based on the analogy between the release of a carrier from a rather deep state and the separation of a pair of opposite sign carriers bound by the Coulomb interaction. Unlike the well-known formula obtained by fitting the Monte-Carlo results, this expression contains only one numeric constant α = 0.74 obtained from an independent Monte-Carlo simulation12. This constant arises from an energy correlation function that is determined only by features of the charge-dipole interaction, and as a result, it does not depend on the localization radius γ −1 . The latter quantity appears only in the frequency factor ν and implicitly, in the 10 ACS Paragon Plus Environment

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effective transport level EC, as eqs 9 to 11 demonstrate. Thus, the localization radius does not affect the field dependence of mobility, while it has a minor effect on its temperature dependence via EC. This conclusion can be used for experimental verification of the model. Although eqs 9 to 13 are quite complex, their application greatly reduces the calculation time compared not only with the Monte-Carlo modelling but also with other numerical methods (for example, numerical solution of the master equation)1,15. It is important that the developed approach allows justification of applicability of the effective transport level concept to systems with long-range correlated disorder. The applicability has been questioned,20 but recently it has been confirmed by the Monte-Carlo simulations29,30. Indeed, the carrier release from its initial state occurs usually to a state that is close in energy20 (step 1 in Figure 2). Therefore, the carrier has not reached a higher energy transport state. However, the initial carrier release typically does not lead to transport, because this initial step is usually followed by a backward jump. Eventually, the escape of a carrier from the initial state to a transport level does occur, but it happens as a multi-step process requiring numerous attempts to escape. In the course of this process, a charge carrier covers the distance that is significantly longer than the typical hopping distance, and gains а certain energy that is approximately equal to ( EC − Ei ) in a weak electric field; this energy decreases with increasing field. Thus, an average carrier energy after the release from a deep state does not depend on

the

initial

state

energy

Ei .

Indeed,

in

case

of

a

weak

field

η ( Ei ) = exp  − ( EC − Ei ) kT  and eq 10 gives t

−1

2 = ν exp  − (σ kT ) − EC kT  .  

(14)

The above result is the same as in the case of the uncorrelated disorder. The reason for this is as follows. A charge carrier leaves the potential well via sites located at r ≈ rC * . The average energy of hopping is close to zero, while the width of the energy distribution is approximately σ. This means that the energy correlation with the initial state is weak. Moreover, the energy correlation of 11 ACS Paragon Plus Environment


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nearby states with E ≈ EC provides an additional argument in favour of the transport level concept. As the field increases, one can describe the PF dependence of mobility given by eq 1 by formal inclusion of the field-dependent effective transport level in eq 14: EC ( F ) = EC (0) − const F . One can expect an even better applicability of the transport level concept in the cases of short-range correlations in materials with non-polar molecules15. The present calculations use the Miller-Abrahams (MA) model of hopping. The

MA

theory

is

widely

accepted

for

analytic2,5-8,21,24,27,35

and

numerical1,2,12,15,17,20,21,38,40 modelling of hopping transport, in a phenomenological way, because it is mathematically simple and qualitatively consistent with numerous experimental data.1,8,12,17 This model was originally derived for the case of interaction of electrons with acoustic phonons, while the typical energy difference for hopping in disordered organics is too large for this mechanism. The Marcus model, which accounts for the effect of polarization and small-polaron hopping, is more microscopically justified than the MA model, but it is more complicated mathematically. The Marcus theory is also very useful in numerical modelling.31-34,37 The Marcus model has an additional parameter – the reorganization energy, Er , which defines an energy scale. It is important to note that the specific form of the hopping rate manifests itself in our model only in the frequency of horizontal (in energy) jumps, ν , and (implicitly) in the effective transport level, EC . The escape probability from a potential well, together with the maximum energy of the integrand in eq 10, which is equal to −σ 2 kT in the lowfield limit, does not depends on the functional form of the hopping rates. The Marcus model gives ν ≈ ν M 0 exp ( −2γ r0 − Er 4kT ) , neglecting weak power-law temperature dependence of the prefactor ν M 0 . One assumes in eq 10 that the probability of carrier arrival to the state of the energy Ei is proportional to g(Ei). That is true not only for the MA model, in which the rates of downward jumps are independent of the energy difference, but also for the Marcus model, since the carrier arrives most probably from the state that is proximal in space and energy, 12 ACS Paragon Plus Environment

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irrespective to the value of Ei, providing correlated disorder. Thus, transition from the MA model to the Marcus model gives, in the first approximation, only the thermally-activated factor exp ( − Er 4kT ) in mobility. This conclusion is in qualitative agreement with the modelling of the field and temperature dependence of mobility32 (compared with the results of the MA model12) and the temperature dependence of weak-field mobility.2 Provided

Er > σ , the main effect of the

Marcus model is the appearance of the factor exp ( − Er 4kT ) . If Er is too small, the Marcus description is inconsistent with PF-dependence.32 A more precise investigation of the field and temperature dependence of mobility in the DGM, especially in the low-field limit, requires investigation of the transport level EC in the Marcus model. A quantitative analytic description of the field dependence of mobility, subject to the Marcus hopping rate, is outside the scope of this paper. Another important result of this work is an estimation of a spatial scale ref that determines the size of the “quasi-Coulomb trap” surrounding the deep state. In the weak and strong field limits, this value is given by eq 8 with Ei = −σ 2 kT and by eq 12, respectively (again, irrespective to the form of a hopping rate). Note that the description of the hopping transport by the formalism of the multiple trapping model (with an effective transport level EC instead of the mobility edge)35 is possible only if the relevant spatial scale (for example, the thickness of a layer) is much larger than ref. In a weak field, the estimation gives ref ≈ rC* ≈ 12 nm with

σ / kT = 4 and a = 1 nm. Recall that ref is decreasing with increasing field. One should note that the above consideration is valid provided that the quasi-Coulomb potential well is spatially separated from other similar wells, i.e. there is no other state with energy E ≤ Ei inside the well (except for the nearest neighbors of the initial state, see Figure 1 for details). In the low-field limit, the described situation arises if σ kT exceeds 2. This follows from the condition E ref < r* E =−σ 2 kT , where r* ( E ) =  ∫ dE ' g ( E ')   −∞ 

−1 3

is a mean distance between states



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with energies deeper than E. Moreover, deep states form clusters of neighboring states due to energy correlations,8 and a mean distance between them greatly exceeds r* ( E ) . In the variant of the correlated disorder model considered here, energy correlations arise only from charge-dipole interactions12,19,23,32. The charge-charge Coulomb interaction is not considered here. Hence, our consideration is limited to relatively small concentrations of charge carriers. Both experimental8,36 and theoretical1,2,5,8,37 results suggest that the mobility in organics does not depend on concentration if the latter is sufficiently small, which is typically appropriate for applications in light-emitting diodes and photovoltaic cells. For example, measurements produce concentration-independent mobility in poly(p-phenylene vinylene) derivatives, if the relative concentration of charge carriers is smaller than, at least, 10-6. (In general, the threshold value depends on the material parameters, temperature and external field strength). At higher values of charge carrier concentrations, one should consider concentration dependence of mobility, and account for charge-charge interactions as a source for both energy disorder and correlations. Investigations of dynamic charge-charge interactions by Monte-Carlo simulations are scarce and rather contradictory38-40. To our best knowledge, it is usually assumed at present that one can describe the concentration dependence of mobility using the Poisson equation,1,2 accounting for filling of deep states below the quasi-Fermi level and the mean field of space charge. The dynamic Coulomb repulsion of carriers is neglected, except that the occupation of a site by more than one carrier is prohibited.37 However, some MC results show that the effect of Coulomb interactions is no less significant than the usually considered effect of filling of deep states by non-interacting carriers. One can expect long-range correlations induced by charge-charge interactions, due to the long-range character of Coulomb interaction, especially in organic materials with low dielectric constant. Development of an analytic theory of energy disorder and correlations induced by Coulomb interactions is a complicated and challenging problem. It remains a subject of future work. 14 ACS Paragon Plus Environment

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4. Conclusions We have established that the field and temperature dependence of charge carrier mobility in organic materials with long-range correlated disorder (so-called dipole glasses) may be represented analytically in good agreement with the results of the Monte-Carlo simulations12. Carrier release from a deep state is described as a multi-step escape from a long-range quasi-Coulomb potential well, in analogy with the separation of geminate pairs. This approach supports applicability of the effective transport level concept to dipole glasses. However, the approach is legitimate only if the spatial scale exceeds (at the weak field) the quasi-Coulomb radius rC * ≈ 0.74a (σ kT ) . Preliminary Monte-Carlo results show29 that an 2

effective transport level EC is approximately the same as in the uncorrelated Gaussian disorder model21. Improvement of quantitative accuracy for the mobility model requires a more accurate definition of the temperature dependence of EC and the field dependence of the effective hopping distance.

ACKNOWLEDGEMENTS. The authors are indebted to Dr. S.V. Novikov for helpful discussions. V.R.N. and A.Yu.S. acknowledge financial support of the Volkswagen foundation, grant “Understanding the dependence of charge transport on the morphology of organic semiconductor films”, and of the “improving of the competitiveness” program of the National Research Nuclear University “MEPhI. O.V.P. acknowledges financial support of the U.S. National Science Foundation, Grant No. CHE-1565704 and is grateful to the Photochemistry Center of the Russian Science Foundation, project No. 14-43-00052, for hospitality during manuscript preparation.



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Table of Contents Graphic


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Analytic Modeling of Field Dependence of Charge Mobility and

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