Hopping charge transport in amorphous organic and inorganic materials with spatially correlated random energy landscape
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Hopping charge transport in amorphous organic and inorganicmaterials with spatially correlated random energy landscape
S.V. Novikov
A.N. Frumkin Institute of Physical Chemistry and Electrochemistry,Leninsky prosp. 31, Moscow 119071, Russia andNational Research University Higher School of Economics,Myasnitskaya Ulitsa 20, Moscow 101000, Russia
Abstract
General properties of the hopping transport of charge carriers in amorphous organic and inorganicmaterials are discussed. We consider the case where the random energy landscape in the materialsis strongly spatially correlated. This is a very typical situation in the organic materials havingthe Gaussian density of states (DOS) and may be realized in some materials with the exponentialDOS. We demonstrate that the different type of DOS leads to a very different functional form of themobility field dependence even in the case of the identical correlation function of random energy.We provide important arguments in favor of the significant contribution of the local orientationalorder to the total magnitude of the energetic disorder in organic materials. A simple but promisingmodel of charge transport in highly anisotropic composites materials is suggested. eywords: amorphous materials, density of states, spatial correlation, hopping transportof charge carriers. I. INTRODUCTION
Last decades show a sharp surge in the number of papers devoted to various aspects oforganic electronics. Interest to the area is mostly motivated and supported by the devel-opment of various devices: organic light emitting diodes, solar cells, field effect transistors,sensors, and others [1–5]. For the efficient functioning of all such devices a fast transportof charge carriers (electrons and holes) is needed. Essentially, charge transport in amor-phous molecular materials occurs as a series of elementary redox reactions (hops of thecarrier) between two neighbor molecules. A very first hop from the conducting electrode tothe adjacent molecule is a typical example of electrochemical reaction. Interplay betweenelectrochemistry and charge carrier transport is important for the production of conductingorganic materials. Many conducting polymers are obtained by the electrochemical oxidationof various organic compounds [6, 7]. Kinetics of the growth of the conducting polymer filmat the electrode is intimately related with the charge transport properties of the polymer [8].In addition, our consideration of the transport properties of charge carriers in the mediumwith spatially correlated disorder may be applied to the description of the ionic transportin amorphous solid state electrolytes [9–11].Characteristics of a charge carrier transport in amorphous materials are defined by thestatistics of the energetic and positional (or diagonal and non-diagonal) disorder. Energeticdisorder describes random fluctuations of the energies U ( ~r ) of the relevant levels of transportmolecules, while the positional disorder describes fluctuations of the spatial positions ofmolecules. There is a general belief that for the quasi-equilibrium (or nondispersive[ ? ])charge transport the most important is the energetic disorder [13]. The reason is thatthe hopping rate Γ ( ~r ij , ∆ U ij ) depends on distance ~r ij = ~r i − ~r j between any given pairof transport molecules and the energy difference ∆ U ij = U i − U j between correspondingtransport levels in a very different way. Typically, Γ is a symmetric function of ~r ij , Γ( ~r ij ) =Γ( ~r ji ), and in the most cases [14, 15]Γ( ~r ij ) ∝ exp (cid:18) − | ~r ij | r (cid:19) (1)2here r is a localization radius of the wave function of the transport molecule), while forthe dependence Γ(∆ U ij ) the microscopic balance takes place [16], i.e.Γ(∆ U ij ) = Γ(∆ U ji ) exp (cid:18) − ∆ U ij kT (cid:19) (2)and probability for a carrier to jump up in energy is restricted by the exponential factor incomparison to the probability of the downward hop. For the quasi-equilibrium transport theso-called ”bad” sites give the most important contribution. ”Bad” sites are sites which cankeep carrier for the longest time (mean escape time is the longest). For the positional disorderthe ”bad” sites are sites well separated from the nearest neighbors, but the symmetry of Γmeans that the probability to get into such sites is pretty low. Hence, they are not veryimportant. This is not the case for the energetic disorder, where ”bad” sites are sites havingenergy much lower than the energies of the neighbor sites, the probability to get into thosesites is not low, and the contribution of low energy sites to the typical transport time issignificant. This is the reason for the primary importance of the energetic disorder.The simplest and most important characteristic of the energetic disorder is the distribu-tion density of random energies of transport levels, called the density of states. Typically,in amorphous organic materials the DOS has a Gaussian form [17–19] P ( U ) = N (2 πσ ) / exp (cid:18) − U σ (cid:19) (3)with σ ≃ P ( U ) = N U exp ( U/U ) , U < , (4)with U ≃ N is a total concentration of transport sites. Transport prop-erties of the amorphous materials having Gaussian or exponential DOS are very different.The most striking difference is the eventual development of the quasi-equilibrium transportregime with constant average velocity v and diffusivity D for the Gaussian DOS for anytemperature, while for the exponential DOS for low temperature kT < U the process ofthe carrier energetic relaxation is infinite in time and average velocity decreases with timeand, hence, with the thickness of the transport layer L , as v ∝ L − U /kT [21]. Such dramaticdifference in the transport behavior is explained by the very fact that for the GaussianDOS the so-called density of occupied states P occ ( U ) ∝ P ( U ) exp( − U/kT ) does exists forany temperature, while for the exponential DOS this is not so for low temperature. The3ensity P occ ( U ) describes the distribution of the carriers in the quasi-equilibrium regimefor t → ∞ . Moreover, using P occ ( U ) we may immediately provide an estimation for thetemperature dependence of the quasi-equilibrium drift mobility µ = v/E for the GaussianDOS as ln µ ∝ − U a /kT , where U a = − U max = σ /kT is an effective activation energy of thecarrier hops and U max is the position of the maximum of P occ ( U ). The resulting dependenceln µ ∝ − ( σ/kT ) is in a very good agreement with experimental data [17, 22, 23].Some time ago it was realized that the DOS is not the only characteristic of the randomenergy landscape which is relevant for the hopping charge transport. Possible spatial corre-lations in U ( ~r ) are important as well. Correlations are mostly important for the electric fielddependence of the carrier drift mobility µ ( E ). Electric field affects charge transport mostlyby the variation of the ∆ U ij = ∆ U ij − e ~E ( ~r i − ~r j ) and this variation inevitable requiresspatial displacement between sites i and j . Typical magnitude of ∆ U ij strongly depends onthe degree of correlation between U i and U j for a given distance r ij , and that is why spatialcorrelation affects the dependence µ ( E ).Correlation effects for the Gaussian DOS are studied well, especially for the case of 1Dcharge transport. Still, there are some problems that are not clearly understood and weare going to discuss them in the paper. Mostly, we are going to concentrate our attentionon the verification of the important approximate analytical result of Deem and Chandler[24] on the dependence of the transport parameters in the case of zero applied field onthe dimensionality of space and discussion of the effect of the local orientational order inorganic materials on the energetic disorder. For the exponential DOS correlation effects arenot studied at all. Here we are going to provide a first consideration of the correlation effecton the mobility field dependence for the nondispersive quasi-equilibrium regime U /kT < II. SPATIAL CORRELATION OF THE RANDOM ENERGY LANDSCAPE ANDCHARGE TRANSPORT IN ORGANIC MATERIALS: GAUSSIAN DOS
Effect of the spatial correlation in the distribution of random energies U ( ~r ) is best studiedfor the case of amorphous organic materials. In these materials it was found that the contri-4ution from randomly located and oriented permanent dipoles and quadrupoles provides theGaussian DOS with σ ≃ . C ( ~r ) = h U ( ~r ) U (0) i decayingas 1 /r for the dipolar case and 1 /r in the quadrupolar case, correspondingly [25–27] (hereangular brackets mean a statistical averaging). More precisely, contribution of dipoles andquadrupoles provides the Gaussian DOS only for the main body of the distribution and notso far tails, while for the far tails of the DOS its shape is not a Gaussian one [28]. Deviationfrom the Gaussian shape is typically important only for the description of the charge car-rier transport at very low (and, probably, experimentally inaccessible) temperature. Longrange spatial correlation means that the amorphous organic material consists of large clus-ters where every cluster is a set of neighbor transport sites having close values of U (visualrepresentation of such medium and its comparison with the case of spatially noncorrelatedGaussian medium may be found in Ref. [29]).It was found that the correlation effectively governs the field dependence of the driftmobility. 1D transport model suggests that for the Gaussian DOS the power law correlationfunction C ( ~r ) ∝ /r n leads to ln µ ∝ E n/ ( n +1) [30]. This relation for the case of dipolarand quadrupolar disorder has been confirmed by the extensive 3D computer simulation[27, 31–33].For some time it was a puzzle why the so-called Poole-Frenkel mobility field dependenceln µ ∝ E / is ubiquitous in amorphous organic materials. For such dependence the exponentof the correlation function is n = 1 which is perfectly suitable for polar organic materialswith dipoles giving the major contribution to the total energetic disorder. In nonpolarmaterials we should expect that the major contribution is provided by quadrupoles and,correspondingly, n = 3 giving ln µ ∝ E / . A possible solution of the puzzle is that for somereason all experimental studies of the mobility in nonpolar organic materials cover a limitedfield range, no more than one order of magnitude and typically even less [22, 34–37]. In suchnarrow field range it is not possible to make a reliable distinction between ln µ ∝ E / andln µ ∝ E / dependences [26, 27, 38].Majority of the fundamental results for the theory of multidimensional hopping chargecarrier transport in amorphous materials has been obtained using computer simulations[17, 18, 31, 38]. Exact and nontrivial approximate analytical results are essentially limitedto the consideration of 1D charge transport. In fact, we know only one important theoreticalresult which is valid for the case of multidimensional transport and obtained using the5enormalization group (RG) method. Deem and Chandler [24] showed that the leadingasymptotics for the diffusivity in the zero applied field for the Gaussian DOS is D (0) = D exp (cid:20) − d (cid:16) σkT (cid:17) (cid:21) , (5)where D is a diffusivity in the absence of the disorder, and d is the dimensionality of thespace. Due to the validity of the Einstein relation for E = 0 even in the case of strongdisorder, the similar relation is valid for the mobility µ = µ exp (cid:20) − d (cid:16) σkT (cid:17) (cid:21) . (6)Relation (6) provides two nontrivial statements: first, the particular dependence of theexponent of the mobility (or diffusivity) on d and, second, independence of the diffusivity andmobility for E = 0 of the correlation properties of U ( ~r ). We can provide a limited verificationof those statements. First of all, Eq. (6) for d = 1 is identical to the exact solution of 1Dtransport model [30, 39]. For 3D case we can compare Eq. (6) with the simulation data forthree models having different correlation properties: short range correlation C ( ~r ) = 0 for r > C ( ~r ) ∝ /r (dipolar glass(DG) model [31]), and quadrupolar correlation C ( ~r ) ∝ /r (quadrupolar glass (QG) model[40, 41]). For these models ln µ/µ ∝ − C (cid:16) σkT (cid:17) (7)for E → C GDM ≈ . C DG ≈ .
36, and C QG ≈ .
37. We should note that theseparticular values of C are valid only if we use the extrapolation of the mobility to E = 0according to the natural field dependence law, i.e.ln µ/µ ≈ − C (cid:16) σkT (cid:17) + AE m , m = nn + 1 , (8)where n is the exponent in the dependence of the correlation function on distance C ( ~r ) ∝ /r n (for the GDM formally n = ∞ and m = 1) [41]. We see that values of C for differentmodels are indeed pretty close to the RG prediction C = 1 /
3. Moreover, we should expectthat this prediction gives only a leading term of the expansion of C in powers of 1 /d , andthere are corrections to that value proportional to higher powers of 1 /d . Part of the deviationbetween our values of C and 1 /d could be attributed to the very procedure of the calculationof C : it is obtained by the extrapolation of the dependence µ ( E ) to E → µ at E = 0. Hence, small difference6etween 1 / C is not surprising. We may conclude that our simulationdata support the idea of the universality of the transport parameters in the limit E → σ (apart from the direct simulation of the microscopic structure ofa particular amorphous organic material) treats the material as a regular lattice with sitesrandomly occupied (if the fraction of the occupied sites is less than 1) by randomly ori-ented organic molecules (in fact, very simple models of a molecule are used, such as pointdipoles, quadrupoles, etc. [28, 42, 43]) Moreover, usually it is assumed that the molecules areembedded in the continuous dielectric medium described by the only parameter, i.e dielec-tric constant ε . Such models neglect not only the local short range orientational correlationswhich are inevitable for large asymmetric organic molecules, typical for organic charge trans-port materials, but also an inapplicability of the macroscopic ε for the description of theshort range charge–dipole and charge–quadrupole interaction.More reliable consideration of the dielectric properties of polar amorphous organic ma-terials has been carried out by Madigan and Bulovi´c [44]. They considered the lattice DGmodel and introduced a microscopic polarizability α to describe the dielectric properties ofthe medium. Hence, the local dipole moment depends on the local electric field ~p i = ~p i + α ~E i . (9)Then the Claussius-Mossotti equation α = ε − ε + 2 34 π V m (10)has been used to relate α and ε (here V m is a volume per molecule). While for the simple DGmodel σ ∝ /ε , Madigan and Bulovi´c found that for ε → ∞ σ goes to some nonzero constantvalue. The result is quite understandable because in this model the major contribution to σ for large macroscopic ε is provided by the short range interaction with neighbor dipoleseffectively taken at ε = 1.It is interesting to note that the major result of Madigan and Bulovi´c obtained by thetime-consuming computer simulation (Fig. 1, dots) may be reproduced by a very simple7 .40.60.811.2 1 1.5 2 2.5 3 3.5 4 4.5 σ ( ε ) / σ ( ε = ) ε FIG. 1. Dependence of the ratio σ ( ε ) /σ ( ε = 1) on ε for the Madigan-Bulovi´c model [44] (dots).Solid line is calculated using Eq. (12). calculation. For the lattice DG model with a single macroscopic ε [43] σ d = (cid:10) U ( ~r ) (cid:11) = σ ε S, σ = e p c a , S = X ~n | ~n | , (11)where a is the lattice scale, p is the dipole moment, c is the fraction of sites occupied bydipoles, 3D vector ~n with integer components runs over all lattice sites except the origin ~n = 0, and for the simple cubic lattice (SCL) S ≈ .
53. If we assume that ε = 1 for nearestdipoles with | ~n | = 1 and set the macroscopic ε for all other sites, then σ ( ε ) σ ( ε = 1) = (cid:18) − NS (cid:19) ε + NS , (12)where N = 6 is the number of nearest sites for the SCL. This result is shown in Fig. 1(solid line). It is worth to note that for ε = 2 − σ ( ε ) by the factor of 2 in comparison to Eq. (11). Hence, according tothat calculation, typical magnitude of the disorder in amorphous organic materials should be ≃ . C ( ~r ) ≈ . σ d a/r , r ≫ a [25].This means that for the calculation of the correlation function for r ≫ a we have to use σ d ,8stimated by the old Eq. (11), while the total σ tot (magnitude of the total disorder in theDG model) approximately obeys Eq. (12).Computer simulation for the 3D case suggests that for the DG model the mobility de-pendence on T and E has the formln µ/µ ≈ − (cid:18) σ tot kT (cid:19) + C E (cid:20)(cid:16) σ d kT (cid:17) / − Γ (cid:21) p eaE/σ d , (13)where C E ≈ .
78, Γ ≈
2, and the mobility temperature dependence at low fields is governedby the total disorder σ tot , while the mobility field dependence at moderate fields is governedby the correlated component of the disorder, i.e. by σ d [31, 38]. Hence, taking into accountthe Madigan-Bulovi´c correction, we should have the same mobility field dependence butmuch stronger mobility temperature dependence. Experimental data do not support thissuggestion. Typically, the value of σ , estimated from the temperature dependence of µ forlow fields is not significantly greater that σ estimated from the field dependence of µ , and σ tot is still close to 0.1 eV [17, 22, 31, 38]. The most natural explanation of the discrepancyis a contribution from the short range local order in amorphous organic materials whichreduces σ tot but gives no significant correction to σ d . We believe that the results of thepaper [44] clearly indicate an importance of local short range order for the development ofthe random energy landscape in organic materials. III. SPATIAL CORRELATION OF THE RANDOM ENERGY LANDSCAPE ANDCHARGE TRANSPORT IN INORGANIC MATERIALS: EXPONENTIAL DOS
Amorphous inorganic materials are very different from the organic materials. For suchmaterials the exponential DOS described by Eq. (4) is ubiquitous. To the best of ourknowledge an effect of the correlated random energy landscape has not been considered forthe exponential DOS. The obvious difficulty is a problem of introducing the correlation inthe exponential distribution. To overcome this difficulty we use a trick borrowed from theprobability theory, i.e. we a going to produce the correlated exponential distribution usingthe auxiliary Gaussian ones [45]. Indeed, if X and Y are two independent and identicallydistributed random Gaussian variables with zero mean and unit variance, then U = − U (cid:0) X + Y (cid:1) (14)9 l n v / v ln(v a/D ) FIG. 2. Mobility field dependence Eq. (18) for the spatially correlated exponential DOS and variouskinds of the correlation function c ( x ): a / ( x + a ) (curve 1), exp( − x/a ) (curve 2), and θ ( a − x )(curve 3), correspondingly; for all curves 1 − K = 1 × − . Note that v a/D = eaE/kT . has the exponential distribution (4). If the Gaussian variables X and Y are correlated ones,i.e. h X ( x ) X (0) i = c X ( x ) = 0, here x is a spatial variable, then U ( x ) has a correlateddistribution with the correlation function c U ( x ) = h U U i − h U i = U (cid:2) c X ( x ) + c Y ( x ) (cid:3) , (15)and it is obvious that in this way we can model any positive binary correlation function forthe random field U ( x ).A very good approximation for the long time behavior of the hopping transport is a modelof carrier diffusion in the random energy landscape U ( x ). In 1D case the stationary solutionof the diffusion equation gives for the average carrier velocity [46] v = D (cid:0) − e − γL (cid:1) L R dx exp ( − γx ) Z ( x, L ) , Z ( x, L ) = 1 L L Z dy exp (cid:20) U ( y ) − U ( x + y ) kT (cid:21) , γ = v /D . (16)Here v and D are carrier velocity and diffusivity in the absence of the disorder, and L is athickness of the transport layer. We can immediately calculate the average carrier velocity10 l n v / v ln(v a/D ) n exact n fit FIG. 3. Mobility field dependence (solid lines) calculated by Eq. (18) for the power law correlationfunction c ( x ) = a n / ( x + a ) n/ for various n : 0.5, 0.75, 1, 1.5, and 2, from the upmost curve tothe bottom, correspondingly, and for 1 − K = 1 × − . Dotted line shows the limiting dependence v/v = 1 − K , and the broken lines show fits for power law dependence v/v ∝ v n . in the limit L → ∞ , because in that limit Z ( x ) = lim L →∞ Z ( x, L ) = (cid:28) exp (cid:20) U (0) − U ( x ) kT (cid:21)(cid:29) (17)and v = D ∞ R dx exp ( − γx ) Z ( x ) . (18)Using the probability distribution for the correlated Gaussian variables P G ( X , X ) = 12 π √ − c exp (cid:18) − X + X − cX X − c ) (cid:19) , h X X i = c, (19)with the same relation for Y (we assume for simplicity c X = c Y = c ) and taking into accountrelation (14), we obtain Z ( x ) = 11 − K [1 − c ( x )] , K = U /kT. (20)For x → ∞ c ( x ) →
0, while c (0) = 1. Hence, for γ → ∞ v → v , while for γ → v → v (1 − K ). We immediately see that for the infinite medium v could be nonzero11nly for K <
1, and for K = 1 there is a transition to the dispersive non-equilibriumregime. This result also means that the reliable determination of the functional form ofthe dependence v ( v ) can be carried out only in the close vicinity to the transition to thedispersive regime where variation of the ratio v/v is significant. Field dependence of thedimensionless mobility v/v = µ/µ for various kinds of the correlation function c ( x ) isshown in Fig. 2.We can calculate the field dependence of the average velocity (actually, the dependence of v on the bare velocity v which is proportional to E ) for the power-law correlation function c ( x ) = a n / ( x + a ) n/ (this particular form is inspired by the correlation properties ofthe amorphous organic materials). Using a saddle point method we obtain an intermediateasymptotics v ≃ v √ πn (cid:16) γaen (cid:17) n , v (1 − K ) ≪ v ≪ v , (21)which agrees well with the direct calculation using Eq. (18) (see Fig. 3). We see that theasymptotics (21) is developing only for 1 − K ≪ K >
IV. CHARGE TRANSPORT IN COMPOSITE ORGANIC MATERIALS
A very different type of spatial correlation may arise in another class of transport ma-terials, i.e. composite materials. Composite materials are mesoscopically inhomogeneousmaterials, typically having large domains with very different properties. If we assume thatevery domain may be characterized by its own transport properties, then we have a trans-port medium with spatially correlated distribution of transport parameters, such as carrier12 v ( t ) / v ln t/ τ L/a : 10 FIG. 4. Photocurrent transients in the composite material for the bulk regime and various thickness L of the transport layer (shown near the corresponding curve). Other parameters are l a /a = 100, l b /a = 200, τ a /τ = 100, τ b /τ = 1, v a /v = 0 .
01, and v b /v = 1. Here a is a spatial scale (e.g., sizeof a molecule), τ is a time scale, and v = a/τ . For the thick layers one can see formation of thequasi-equilibrium transport regime having a velocity plateau. velocity, diffusivity, etc.Let us consider a simplest model of highly anisotropic composite material having chain-like structure with all chains elongated in the same direction, the hopping charge transportoccurs along the chains and is essentially one-dimensional. Electric field is oriented parallelto the chains and every chain is composed by the clusters of two types of transport materials,material A and material B. We assume that there are distributions of the clusters on length, p a ( l ) and p b ( l ), and the the carrier motion in the clusters is a pure drift, characterized byvelocities v a and v b . In addition, we assume that there are distributions of time p a ( τ ) and p b ( τ ) to cross the interfaces between clusters (index a here means that the carrier goes fromcluster A to cluster B, and index b means the transition B → A).We can write a simple formula for the average carrier velocity for the infinite medium v ∞ = h l a i + h l b i h l a i v a + h l b i v b + h τ a i + h τ b i . (22)The structure of the denominator indicates that there are two very distinct transport13 v ( t ) / v ln t/ τ L/a : 10 FIG. 5. Photocurrent transients in the composite material for the interface regime and variousthickness L of the transport layer (shown near the corresponding curve). Here τ a /τ = 1 × andall other parameters are the same as in Fig. 4. Again, for the thick layers we see formation of thequasi-equilibrium transport regime having a velocity plateau. regimes. In the bulk regime first and second terms in the denominator dominate (mostrelevant is the time for a carrier to drift over clusters), while in the interface regime thirdand forth terms are more important (time to overcome interfaces is dominating). For thediffusivity the corresponding result is D ∞ = v ∞ h ( δt ) i h t i = v ∞ h l a i −h l a i v a + h l b i −h l b i v b + h τ a i − h τ a i + h τ b i − h τ b i (cid:16) h l a i v a + h l b i v b + h τ a i + h τ b i (cid:17) , (23)here t is the time needed to a carrier to travel across the sample having some finite but verylarge thickness L and δt is the fluctuation of that time (more precisely, we have to considerthe limit L → ∞ ). Again, we can differentiate between the bulk and interface regimes, butin addition we have a mixed regime. Indeed, we may quite easily imagine a situation wherethe fluctuation of the clusters’ lengths is negligible (cid:10) l a,b (cid:11) − h l a,b i →
0, while at the sametime the dominant contribution to the denominator in Eq. (23) still comes from the bulkterms. The opposite situation (cid:10) τ a,b (cid:11) − h τ a,b i → τ a,b could be considered originating from the energetic barriers between clusters, in sucha case p ( τ ) ∝ exp( − τ /τ ) with τ ∝ exp( − ∆ /kT ), where ∆ is the height of the barrier. For14he exponential distribution h τ i − h τ i = h τ i and variance of τ can be negligible only if h τ i is negligible. -6-5-4-3-2-1 5 6 7 8 9 10 11 12 l n v L / v ln L/a v ∞ /v = 0.029v ∞ /v = 0.0027 FIG. 6. Dependence of the average carrier velocity v L in the composite material on the thickness L of the transport layer for the bulk ( ▽ ) and interface ( N ) regime. Solid curves show the best fitfor Eq. (24). Some transport characteristics may be very similar for both regimes (e.g., general shapeof current transients), yet there is no necessity to concentrate our attention exclusivelyon the transport properties, so we can expect many possible significant differences. Forexample, suppose that there is some chemical reaction between charge carriers and moleculesof the medium. In the bulk regime carrier spends most time in the interior regions of theclusters, while for the interface regime it dwells mostly near the interfaces. Properties ofthe material could be very different in those domains and we may expect different kineticregimes. Moreover, many purely transport characteristics should be very different in the bulkand interface regimes. Indeed, in the bulk regime, especially for the case of large clusters,the dependence of v on the applied electric field E is expected to have the Poole-Frenkelform ln( v/E ) ∝ E / because velocities v a and v b obey this very law. In the interface regimewe should expect the dependence ln( v/E ) ∝ E because the reduction of the barrier heightis proportional to E .Transport properties for the finite thickness L , especially the shapes of photocurrent15ransients, are impossible to calculate analytically. For this reason we carried out computersimulation and some typical results are shown in Figs. 4 and 5. With the increase of L theaverage carrier velocity v L decreases and its behavior agrees well with the relation v L = v ∞ (cid:20) (cid:18) L L (cid:19) n (cid:21) (24)(see Fig. 6), while the limit velocity v ∞ perfectly agrees with Eq. (22).Evidently, this simple model could demonstrate a wide variety of transport properties.A very tempting task should be to search for the transition between quasi-equilibrium andnon-equilibrium transport regimes, resembling the transition between nondispersive anddispersive regimes. Quite probably, such transition could occur only for a very specifickind of the probability distributions p a ( τ ) and p b ( τ ) (for example, having long tails orexponentially wide relevant domain of τ ). V. CONCLUSION
We considered effects of a spatial correlation of the random energy landscape in amor-phous materials on the transport properties of such materials. Statistical properties of theenergy landscape are of crucial importance for the hopping charge carrier transport. Inorganic materials the DOS usually has the Gaussian shape, while in inorganic materials itusually has the exponential shape. Correlations mostly affect the mobility field dependence.We demonstrated that the resulting mobility field dependences are principally different forthe Gaussian and exponential DOS even for the same type of the binary correlation func-tion of the random energy. We demonstrated that the experimental search of the effects ofcorrelation for the exponential DOS in the nondispersive regime should be carried out ina very close vicinity to the transition to the dispersive regime. We argued that the localorientational order significantly reduces the total energetic disorder in polar organic materi-als. We also suggested the simple model for the description of transport properties of highlyanisotropic composite materials. In spite of its simplicity, the model demonstrate very richbehavior and is promising for further development.16
CKNOWLEDGEMENT
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