Bimolecular Recombination of Charge Carriers in Polar Amorphous Organic Semiconductors: Effect of Spatial Correlation of the Random Energy Landscape
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Bimolecular Recombination of Charge Carriersin Polar Amorphous Organic Semiconductors:Effect of Spatial Correlation of the RandomEnergy Landscape
S.V. Novikov ∗ , † , ‡ † A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Leninsky prosp. 31,119071 Moscow, Russia ‡ National Research University Higher School of Economics, Myasnitskaya Ulitsa 20,Moscow 101000, Russia
E-mail: [email protected]
Phone: +7 495 952 24 28. Fax: +7 495 952 53 081 bstract
We present a simple model of the bimolecular charge carrier recombination in polaramorphous organic semiconductors where the dominant part of the energetic disorderis provided by permanent dipoles and show that the recombination rate constant couldbe much smaller than the corresponding Langevin rate constant. The reason for thestrong decrease of the rate constant is the long range spatial correlation of the ran-dom energy landscape in amorphous dipolar materials, without spatial correlation evenstrong disorder does not modify the Langevin rate constant. Our study shows that thesignificant suppression of the bimolecular recombination could take place in homoge-neous amorphous organic semiconductors and does not need large scale inhomogeneityof the material. Introduction
Charge carrier recombination is one of the most important processes taking place in organicelectronic and optoelectronic devices and to a very large extent determines working param-eters of the devices. In organic light emitting diodes (OLEDs) the recombination givingphotons is a desirable process delivering light, while in organic photovoltaics (OPV) therecombination should be suppressed by all means in order to provide efficient carrier sepa-ration and maximal electric power output. Study of recombination is the area of thrivingexperimental and theoretical endeavor. All recombination processes are divided into twomajor classes: geminate recombination and bimolecular recombination. For the geminaterecombination both carriers are born by the same photon, originate from the same transportsite and initially are located close to each other. In this paper we consider only the bimolec-ular recombination where initial separation and origin of carriers are arbitrary. Assumingspatially homogeneous distribution of carriers, recombination kinetics is governed by theequation d ( n, p ) dt = − γnp (1)where n ( t ) , p ( t ) are the concentration of electrons and holes, correspondingly, and γ is therate constant (we assume that the intrinsic concentration of carriers is negligible).In many papers it is assumed that the bimolecular recombination is in fact the Langevinrecombination with the rate constant γ L = 4 πeε ( µ + + µ − ) (2)where ε is the dielectric constant of the medium and µ + , µ − are mobilities of holes andelectrons, correspondingly. This kind of recombination was considered very long ago by PaulLangevin in his pioneer paper. To a very large extent the use of the Langevin rate constantis explained by the lack of detailed knowledge of the charge recombination in amorphous(and, hence, spatially inhomogeneous) semiconductors. At the same time there is a general3greement that the bimolecular recombination is indeed of the Langevin type in OLEDs, assuggested in many experimental papers.
Nonetheless, in many cases the experimentally measured rate constant is much smallerthan the Langevin constant, the so-called reduction factor ζ = γ/γ L could achieve . , . ,or even × − . Usually the discrepancy is attributed to the specific mesoscopic inho-mogeneous structure of the materials in the device, especially in the case of photovoltaicdevices, where the mesoscopically inhomogeneous structure of the material is specially ar-ranged to achieve a better charge separation.
Typically, such structure is organized bymanufacturing of the working material of the OPV device as a mixture of two different ma-terials, one of which serving as electron donor and another one as electron acceptor, thuscreating separate pathways for electrons and holes.Computer simulations support validity of eq 2 for the case of model disordered materialshaving mesoscopically homogeneous spatial structure and spatially noncorrelated randomenergy landscape with the Gaussian density of states (DOS), thus supporting the idea ofthe necessity of the large scale inhomogeneity for the significant reduction of ζ . In this paperwe are going to demonstrate that the real polar amorphous organic semiconductors whichhave spatially correlated random energy landscape could demonstrate strongly suppressedrecombination and very small ζ factor without mesoscopic inhomogeneity and the decreaseof ζ is directly related to the long range spatial correlation of the random energy landscape. We consider the recombination of carriers in mesoscopically spatially homogeneous polaramorphous semiconductors where the dominant part of the total energetic disorder is pro-vided by randomly located and oriented permanent dipoles. For the high concentration ofdipoles the DOS has the Gaussian shape and slow spatial decay of the electrostatic poten-4ial of the individual dipole leads to the long range spatial correlation of the resulting randomenergy landscape U ( r ) being the sum of the electrostatic contributions of all dipoles. Themodel of the exponential DOS is sometimes considered as a worthy alternative to the modelof the Gaussian DOS, especially for tails of the DOS associated with deep traps.
Nonetheless, various incarnations of the Gaussian DOS model are, probably, the most popu-lar models for description of the transport properties of amorphous organic semiconductors,successfully explaining many general features of hopping charge transport.
For this rea-son we limit our consideration to the case of the Gaussian DOS naturally arising in thedipolar amorphous organic materials.Let us consider the case where mobilities of the opposite kinds of carriers are very different(e.g., because of the large difference of the intermolecular transfer integral): let, for example, µ + ≪ µ − . In addition, we consider the case where concentration of carriers is low, inter-carrier distance is large and typical time before recombination is long enough. In this casewe may assume that carriers have enough time to undergo the full energetic and spatialrelaxation before the recombination event. Hence, the mobile carriers move with mobilitiesdetermined by the energetic disorder of the medium and slow positive carriers mostly dwellin the deep valleys of the energy landscape U ( r ) . If so, then the recombination processcould be considered as a recombination of mobiles electrons with almost static holes. Weassume also that the applied electric field is negligible. Low density of carriers means thatwe consider either the special case where the initial density of carriers is low enough or thelater stage of high density recombination when the majority of carriers already recombined.Let us consider the recombination of electron with the particular hole located at r = 0 in a valley having minimal energy U (0) = − U . Electron is attracted to the trapped holeby the combined effect of the bare Coulomb attraction to that hole and dipolar contributionfrom the potential well localizing the hole. Our crucial approximation is the replacementof the exact fluctuating dipole potential energy U ( r ) around positive charge by its averagevalue (see Figure 1). A similar approach was used by Nikitenko et al. for the analysis of5harge carrier transport in polar amorphous organic materials. We mean here the conditionalaverage taking into account the exact value U (0) = − U . For the random Gaussian landscapethis conditional average is exactly equal to − U C ( r ) /C (0) , where C ( r ) = h U ( r ) U (0) i is thebinary correlation function, angular brackets mean the statistical averaging over all possiblerandom environments, and C (0) = σ is the rms energetic disorder. For r ≫ a , where a isintermolecular distance in the material, C ( r ) ≈ Aσ a/r (practically, this relation is valid withgood accuracy even for r/a ≃ − , see Figure 1b). Typically, A ≃ ; for example, for thelattice model of the disordered polar material where randomly oriented dipoles occupy sitesof the simple cubic lattice (the so-called dipole glass model) A = 0 . .. (we use this veryvalue of A for all further estimations). Hence, for a large distance the complex hole+dipolesprovides the potential ϕ ( r ) = eεr − U e C ( r ) C (0) ≃ eεr − U Aaer (3)and that potential could be considered as a potential generated by the point charge with theeffective charge e ∗ = e − U Aaεe (4)We may assume that the elementary act of the bimolecular recombination in the amorphoussemiconductor could be considered as process somewhat similar to the Langevin recombina-tion in non-disordered medium but taking place between mobile electron with charge − e andstatic hole having effective charge e ∗ . Note that for valleys, where U > and where holesdwell most time, e ∗ < e and could even become negative for especially deep valleys, thusresulting in the effective repulsion between electron and hole. This does not mean that therecombination between electrons and deeply trapped holes becomes impossible, the diffusivemotion of the electrons could still eventually bring charges close to each other leading torecombination, but the corresponding rate constant should be severely diminished. Obvi-ously, for such deeply trapped holes the recombination process becomes very different fromthe usual Langevin recombination. 6 U ( x , y ) / s a) −6−4−202−20 −10 0 10 20 U ( x ) / s x/a b)Figure 1: a) Two dimensional cross-section z = 0 of the 3D sample of the dipole glass inthe vicinity of the deep valley ( U = 5 . σ ) located at x = y = z = 0 , black dots showthe energies of neighbor sites, and a is the lattice scale. Surface created by lines shows theaveraged energy − U C ( x, y, /C (0 , , . b) One dimensional cross-section z = y = 0 of thesame valley, the solid line again shows − U C ( x, , /C (0 , , , the dotted line shows theasymptotics − U Aa/ | x | . 7 Recombination rate constant: general expression
For the calculation of the rate constant for a single hole with the particular value of U we aregoing to use the method of Smoluchowski and Debye (see ref 31), where the rate constant isdefined through the stationary solution for t → ∞ of the equation for the probability density ρ ( r, t ) for the mobile carrier ∂ρ∂t = D − r ∂∂r (cid:20) r (cid:18) ∂ρ∂r + β ∂U∂r ρ (cid:19)(cid:21) (5)where β = 1 /kT and D − is the diffusivity of electrons. Here and later U ( r ) = − ee ∗ /εr and we take into account the spherical symmetry of the problem. In addition we have theboundary conditions: 1) ρ ( R, t ) = 0 , meaning the instant recombination of the pair separatedby the distance R , and 2) ρ ( ∞ , t ) = 1 which is appropriate for the inexhaustible reservoir ofmobile carriers. The stationary density ρ s ( r ) = ρ ( r, t → ∞ ) obeys the equation ddr (cid:20) r (cid:18) dρ s dr + β dUdr ρ s (cid:19)(cid:21) = 0 (6)and the recombination rate constant for that particular hole is defined by the total flux ofmobile carriers through the absorbing sphere of radius Rγ ( U ) = 4 πD − R ∂ρ s ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R (7)Solution of eq 6 is ρ s ( r ) = exp ( − βU ) [1 − S ( r ) /S ( R )] (8) S ( r ) = ∞ Z r dzz exp ( βU ) (9)8n our case U ( r ) = − ee ∗ /εr , so S ( r ) = kT εee ∗ (cid:20) − exp (cid:18) − ee ∗ kT εr (cid:19)(cid:21) (10)and γ ( U ) = 4 πD − S ( R ) (11)This is the rate constant for the particular case where the positive charge is located at thebottom of the valley with depth U . If we assume the quasi-equilibrium distribution of staticcharges after the full relaxation, we obtain the full rate constant by the averaging of the rateconstant γ ( U ) with the density of occupied states P occ ( U ) = 1(2 πσ ) / exp (cid:20) − ( U − U σ ) σ (cid:21) (12)where U σ = σ /kT and after simple transformations we obtain for the full recombinationrate constant γ = h γ ( U ) i = 4 πD − R (2 πδ ) / ∞ Z −∞ dy y exp( y ) − (cid:20) − ( y − y s ) δ (cid:21) (13)where y σ = (cid:0) σkT (cid:1) AaR , y c = R Ons /R , y s = y σ − y c , δ = σkT AaR , R Ons = e /εkT is the Onsagerradius and we should expect R ≃ a . Cut-off for y > provided by the exponent in thedenominator of the fraction in the integral in eq 13 describes the drastic decrease in therecombination rate when the effective charge e ∗ of the static hole becomes negative, thusproviding the repulsion between the hole and approaching electrons.Equation 13 is valid if we assume that the quasi-equilibrium distribution of holes de-scribed by eq 12 is permanently maintained irrespective of recombination for all times (thus,we assume that holes are not totally immobilized in deep valleys of the random energy land-scape). Obviously, it could be possible only for slow recombination when drop of electron9oncentration ∆ n = n ( t ) − n ( t + τ rel ) for the time interval equal to the hole relaxation time τ rel is small compared to n ( t ) (assuming n = p ) ∆ nn ≃ γn τ rel n = γnτ rel ≪ (14)We see that for low concentration of carriers this inequality could be fulfilled for any τ rel , though the actual concentration of carriers where our consideration becomes accuratestrongly depends on τ rel . For low temperature or strong disorder τ rel could be large, butthese very conditions lead to small γ , thus making inequality in eq 14 less restrictive.We may effectively take into account a slow quasi-geminate recombination of carriers at ashort distance described by the rate constant k g by applying a different boundary conditionat r = R , i.e. equating the rate of the slow quasi-geminate recombination to the total fluxthrough the sphere of the radius Rk g ρ ( R, t ) = 4 πD − R (cid:20) ∂ρ∂r + ρkT ∂U∂r (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = R (15)(the so-called radiation boundary condition ). In this case the final expression for theresulting recombination rate constant is γ = 4 πD − R (2 πδ ) / ∞ Z −∞ dy y ( λy + 1) exp( y ) − (cid:20) − ( y − y s ) δ (cid:21) (16)with λ = 4 πD − R/k g and goes to eq 13 for the case of instant quasi-geminate recombination k g → ∞ .Probably, a good estimation for the rate constant for the general case of an arbitraryrelation between D − and D + could be provided by the replacement D − ⇒ D + + D − = D in eqs 13 and 16. This replacement correctly captures both limit cases D + /D − → and D − /D + → and provides a reasonable interpolation for the arbitrary ratio of D − and D + .In addition, for the case of negligible disorder σ/kT → the Gaussian in eq 13 goes to the10elta-function δ ( y − y c ) and the rate constant becomes γ = 4 πD − y c R − exp( − y c ) (17)which in the limit y c ≫ with the suggested replacement D − ⇒ D and assuming the validityof the Einstein relation µ ± = eD ± /kT gives exactly the usual Langevin rate constant γ L .Our approach for incorporation of the slow quasi-geminate recombination is similar tothat developed by Hilczer and Tachiya, though they considered the recombination in non-disordered medium. Naturally, in the limit of vanishing disorder σ → we reproduce theirsresult for the rate constant. According to eq 16, the most general formula of our consideration, the recombination rateconstant has the form γ = γ R F ( y s , δ, λ ) (18)where γ R = 4 πDR and F is a dimensionless function of 3 dimensionless parameters y s , δ ,and λ . Rich structure of eq 16 suggests the possibility of a wide variety of recombinationregimes depending on particular values of y s , δ , and λ . Thorough analysis of eq 16 andcalculation of γ for various cases is provided in the Appendix, where we try to cover thewhole physically meaningful range of parameters. In this Section we consider the behaviorof γ for typical values of parameters relevant for amorphous organic semiconductors.Let us start with the estimation of y s and δ . Numerous experimental transport studiessuggest that σ falls in the range . − . eV, and the typical experimental temperaturevaries from 200K to 350K (most experiments have been carried out around room tempera-ture). In amorphous organic semiconductors typically a ≃ R ≃ − . nm, ε = 3 − ,11o y c ≃ − , δ ≃ − , y s ≃ − − . This estimation means that real organic semi-conductors could possibly demonstrate various recombination regimes, but typically thoseregimes fall in the class "broad Gaussian regime" with δ ≫ . Unfortunately, in the ma-jority of amorphous organic semiconductors realization of the condition ( y s − δ ) /δ ≫ isalmost impossible. This condition could be fulfilled either at very low temperature (whereexperimental measurements are very difficult or even impossible) or for materials with high ε ≃ . − and large a ≃ − nm. Such extreme values of ε and a are very unusual.For this reason the most appropriate way to calculate γ for real semiconductors and typicalexperimental conditions is the use of eq A8 or direct numerical evaluation of the integral ineq 16. Typical behavior of γ/γ L for reasonable values of σ and ε is shown in Figure 2.A possible way to verify our results could be a comparison with the results of computersimulation. Unfortunately, there are no papers considering the simulation of charge carrierrecombination in amorphous dipole medium, but there are papers simulating the recombi-nation in the Gaussian uncorrelated random energy landscape (Bässler’s Gaussian DisorderModel (GDM) ). Comparison of our results with the simulation data for the GDM givesan excellent possibility to verify a very essence of our approach. Indeed, for the GDM thecorrelation function C ( r ) ∝ δ ( r ) and eq 3 immediately tells us that the long range behaviorof the potential of the trapped charge is not modified, e ∗ = e , so γ ≈ γ L . This very behav-ior was indeed observed in simulations. Hence, small ζ in mesoscopically homogeneousamorphous semiconductors is a direct manifestation of the correlated nature of the energylandscape.There is a seeming disagreement between our results and the experimental data forOLEDs, where authors concluded that the bimolecular recombination is the Langevin one. We see several reasons why the significant deviation from the Langevin recombination doesnot occur (i.e., why ζ exp ≃ ). We believe that the most important reason is a very indirectway to extract the bimolecular recombination rate constant γ exp from the experimental data.Direct observation of the decay kinetics p ( t ) or n ( t ) is impossible, and extraction of γ exp usu-12 −6 −4 −2
200 250 300 350 g / g L T, K a) −2 −1
200 250 300 350 g / g L T, K b)Figure 2: Deviation of the bimolecular recombination rate constant from the Langevin con-stant according to eq 13 (hence, we assume the instant quasi-geminate recombination with λ = 0 ). a) Solid lines show the ratio γ/γ L for various values of σ , indicated near the corre-sponding curve. For other relevant parameters the typical values a = R = 1 nm and ε = 3 have been used. b) Plot of the ratio γ/γ L for various values of ε , indicated near the corre-sponding curve, and the same values of a and R . Increase of ε is analogous to the increaseof σ because it again strengthens effect of the disorder (assuming the constant σ = 0 . eV).13lly invoke many assumptions about details of the transport mechanism, trap distribution,etc. Typical examples are provided in refs 2 and 5 where meticulous description of the pro-cedure to estimate γ exp is described. Hence, we have to consider all values of γ exp obtainedin refs 2–8 with great care, they could easily deviate from the true rate constants by one ortwo orders of magnitude. Indeed, detailed description in refs 2 and 5 demonstrates that thebest possible accuracy for γ exp and ζ exp is no better than one order of magnitude.Additional complication is provided by the contribution of trap-assisted recombination(TAR) which is common in some organic semiconductors. Contributions from the trap-assisted and Langevin recombination are not easy to separate, though spectra of the lumi-nescence associated with particular types of recombination are typically different thus givingthe possibility to isolate the individual contributions.
Our recombination scenario to someextent is close to the TAR, with slow carriers being trapped for a long time in deep valleys ofthe random energy landscape. The crucial difference is that for the true TAR energy levelsof traps are separated by some gap from the energy manifold where charge transport occurs,while in our case there is no such gap. Consideration of the true TAR for the spatially cor-related energy landscape could be an interesting and important development of the currentstudy.Taking into account all these complications it should be very useful to measure γ exp inmaterials having very significant disorder with σ ≃ . eV where we should expect verylow ζ . At the moment there are no papers reporting reliable measurements of γ in stronglydisordered organic semiconductors. Nonetheless, in the recent paper it was found thataddition of dopants having large dipole moments and, thus, giving noticeable contribution tothe total dipolar disorder, leads to the suppression of the Langevin recombination. Obviously,further studies are needed for the reliable elucidation of the true mechanism of the influenceof dipole dopants on charge carrier recombination.There is another possible reason for the closeness of γ exp to γ L . We consider here thecase of dipolar materials where the long range behavior of the correlation function is of the14oulomb type C ( r ) ∝ /r . In fact, many materials considered in refs 2–8 have rather lowdipole moments and the dominant part of the random energy landscape is probably gener-ated by randomly located and oriented quadrupoles. For such materials C ( r ) ∝ /r and the effect of disorder on the recombination rate constant could not be described by theeffective charge e ∗ , though the general consideration using Smoluchowski-Debye approachis still possible. Faster decay of C ( r ) leads to the smaller deviation of γ exp from γ L , analo-gously to the case of the GDM. Bimolecular recombination of charge carriers in quadrupolarmaterials will be considered in a separate paper.At last, there is yet another reason why ζ exp is greater than expected; it is associatedwith the effect of the applied electric field E . We consider the recombination for E = 0 only, while in experiments the recombination constant is estimated for nonzero electric field.Computer simulation indicates that ζ grows with E , thus making the difference betweenour results and experimental data less drastic.We have to clarify the difference between our approach and papers of Andriassen andArkhipov, who also considered the deviation of the bimolecular rate constant from γ L indisordered materials. Our results show that the reason for this deviation is not the disorderper se, but the spatial correlation of the disorder. Without correlation ζ ≈ irrespectivelyof the magnitude of the disorder.As we already noted, our approach could be naturally considered as a far extension ofthe Hilczer and Tachiya’s theory to the case of disordered medium. We have to admitthat the independent estimation of λ from first principles is very difficult, in the strikingcontrast to y s and δ , thus greatly complicating estimation of the behavior of γ . Nonetheless,introduction of slow quasi-geminate recombination with k g > gives an inviting possibilityto explain the decrease of ζ with T , observed in some experiments. Indeed, Figure 2demonstrates that for λ = 0 dζdT > . At the same time, the sign of dλdT is arbitrary anddepends on the relation between activation energies of D and k g . Equation A6 hints thatif λ grows with T fast enough, then ζ could decrease with T . Numerical calculation using15 −6 −4 −2
200 250 300 350 g / g L T, K
Figure 3: Effect of the slow quasi-geminate recombination with λ ≫ on the temperaturedependence of the ratio γ/γ L for various values of σ , indicated near the corresponding curve.For other relevant parameters we assume R = 1 nm, ε = 3 , λ = 4 πD R/k = 1 × , and E a = 0 . eV.eq 16 shows that this is indeed so (see Figure 3). For the calculation we used the simplestactivation dependence for k g k g = k exp( − E a /kT ) (19)and the proper relation for D ( T ) D = D exp (cid:20) − (cid:16) σkT (cid:17) (cid:21) (20)which is approximately valid in low field limit, as suggested by the renormalization groupanalysis and computer simulation. In our case the small or moderate disorder is clearly favorable for the emergence of therecombination regime with dζdT < . We can see that even more complicated behavior ispossible, namely the change of the sign of the derivative dζdT . In fact, the very motivationof Hilczer and Tachiya to develop their approach was to explain the decrease of ζ with T in some organic semiconductors. Unfortunately, invocation of the slow quasi-geminate re-16ombination for the explanation of this phenomenon is not fully justified, mostly becausesemiconductors in question are typical mesoscopically inhomogeneous materials specificallydeveloped for the OPV applications. General properties of the charge carrier recombi-nation in such materials are still barely known. For this reason Figure 3 is not provided forthe explanation of the behavior of ζ ( T ) in any particular organic semiconductor but just toillustrate the possibility to obtain dζdT < for reasonable values of relevant parameters evenin mesoscopically spatially homogeneous amorphous semiconductors. We calculate the rate constant of the bimolecular charge carrier recombination in polaramorphous organic semiconductors in the limit of low applied electric field. We show thatthe long range spatial correlation of the random energy landscape typical for such materialsleads to the deviation of the bimolecular recombination from the Langevin-like process withthe resulting rate constant γ being in some cases much smaller than the correspondingLangevin rate constant γ L .The most important qualitative conclusion is that the stronger is the correlation, thelower is the ratio γ/γ L (factor ζ ), and for the total absence of any spatial correlation (theGDM case) our approach gives ζ ≈ in agreement with computer simulations. We mayexpect that in nonpolar amorphous organic materials the deviation of γ from γ L is much lesspronounced due to faster decay of the disorder correlation function.We show that small ζ factor could be achieved in mesoscopically homogeneous amor-phous semiconductors with large σ and ε and could not be unambiguously related to theinhomogeneous structure of the organic material.Suggested approach could be extended to consider the bimolecular recombination innonpolar amorphous organic semiconductors where the dominant part of the random energylandscape is provided by quadrupole molecules or to the case of trap-assisted recombination17n semiconductors with spatially correlated energy landscape. Acknowledgement
Financial support from the FASO State Contract No. 0081-2014-0015 (A.N. Frumkin Insti-tute) and Program of Basic Research of the National Research University Higher School ofEconomics is gratefully acknowledged.
Equation 16 demonstrates rich and complex structure hinting for the possibility of manydifferent recombination regimes. Actual dependence of the recombination rate constant γ onrelevant physical parameters T , σ , ε and others for any particular case is determined by therelation between values of the dimensionless parameters y s , δ , and λ . Typical values of theseparameters (and, hence, typical dependences of γ ) for amorphous organic semiconductors arediscussed in Section 4. In this Appendix we consider the much broader range of possibilities,some of them cannot be realized in today’s semiconducting materials. Nonetheless, we believethat this consideration is not worthless and some recombination regimes, though not feasibletoday, might be observed in semiconducting materials developed in future.Hence, we consider here as many physically meaningful regimes as possible, not limitingour attention to the particular values of y s , δ , and λ , typical for organic semiconductors: wedeal here with the general case of the amorphous material having the spatially correlatedGaussian DOS with the dipolar-like correlation function, and the only necessary conditionsare δ ≥ and λ ≥ .Analytic calculation of the rate constant γ in eq 16 in the general case is not possible.Let us consider various limit cases which could be treated analytically.18 .1 The case of sharp Gaussian δ ≪ The simplest tractable limit is δ ≪ , where the Gaussian in eq 16 goes to the delta functionand γ = γ R y s ( λy s + 1) exp( y s ) − (A1)The most reasonable case of small δ is provided by the negligible disorder σ/kT → , andin this case the resulting rate constant for R Ons /R ≫ goes to the usual Langevin constant γ L (for λ = 0 ).More exotic possibility is the case of strong disorder σ/kT ≫ and huge recombinationradius a/R ≪ , so that δ ≪ but still y s ≫ . In this case the rate constant in eq A1 stillhas an exponentially strong dependence on the effective disorder σ/kT . We have to admitthat at the moment we cannot present any concrete organic semiconductor demonstratingsuch behavior. δ ≫ Let us consider the opposite case δ ≫ . When this inequality is valid, then the mostnatural situation is that for the position of the maximum of the Gaussian we have y s ≫ ,too. Moreover, typically y s ≃ δ , and ratio of the position of the maximum and width of theGaussian obeys the inequality y s /δ ≫ . Hence, we may assume that the relevant region ofthe integration is located far away from y ≃ and we may simplify the integral in eq 16 γ ≃ γ R (2 πδ ) / ∞ Z −∞ dy yλy + 1 exp (cid:20) − y − ( y − y s ) δ (cid:21) (A2)Maximum of the Gaussian in this integral is located at y s − δ , so the more exact conditionfor the validity of the approximate eq A2 is ( y s − δ ) /δ ≫ . We have y s − δ = (cid:16) σkT (cid:17) AaR (cid:18) − AaR (cid:19) − y c (A3)19ertainly, R ≥ a and the most natural choice is R ≈ a , while A < , so the combination y s − δ indeed could be positive, especially at low temperature, even taking into account thenegative contribution from the charge-charge interaction.If λ ( y s − δ ) ≪ , then the first term in the denominator in eq A2 is not relevant and γ ≃ γ R (cid:0) y s − δ (cid:1) exp (cid:18) − y s + 12 δ (cid:19) (A4)while in the opposite case λ ( y s − δ ) ≫ γ ≃ γ R λ exp (cid:18) − y s + 12 δ (cid:19) (A5)A reasonable interpolation between two limits is γ ≃ γ R ( y s − δ )1 + λ ( y s − δ ) exp (cid:18) − y s + 12 δ (cid:19) (A6)and the quality of the interpolation formula could be seen in Figure 4. Even for not so large δ and ( y s − δ ) /δ eq A6 works remarkably well.For λ = 0 we may suggest a better approximation than eq A4 replacing the functionunder the integral in eq 13 by its proper asymptotics y exp( y ) − ⇒ − y, y < y exp( − y ) , y > (A7)the resulting expression takes the form γ ≃ γ R "r π δ exp (cid:18) − y s δ (cid:19) − y s erfc (cid:18) y s δ √ (cid:19) + (A8) +( y s − δ ) exp (cid:18) − y s + 12 δ (cid:19) erfc (cid:18) − y s + δ δ √ (cid:19)(cid:21) .9850.990.99511.005 0 5 10 15 20 g i n t / g l Figure 4: Quality of the interpolation eq A6 is shown. Solid lines show the ratio of γ int ,calculated by eq A6, to the rate constant γ numerically calculated by eq 16 for y s equals to20, 25, 30, and 40 from the lowest curve upward, correspondingly. In all cases δ = 3 .where erfc( x ) is a complimentary error function. For sufficiently large y s eq A8 goes to eqA4. Quality of approximation could be seen in Figure 5. Equation A8 gives a meaningfulresult even for y s = 0 , while eq A4 for δ = 3 gives a negative rate constant for y s < . Yeteq A8 is too cumbersome for the practical use.More exotic case for δ ≫ (strong disorder) is the situation where interaction betweencarriers is so strong that no matter how large is y σ , y s is still negative and | y s | ≫ . In thiscase we may omit the term proportional to exp( y ) in eq 16, so γ ≃ − γ R (2 πδ ) / ∞ Z −∞ dyy exp (cid:20) − ( y − y s ) δ (cid:21) = γ L − πDaA (cid:16) σkT (cid:17) (A9)and the rate constant is essentially equal to the Langevin constant with small correction.This is not surprising due to the dominance of the charge-charge interaction over disorder.In this approximation we assume also that the constant λ is not unusually large (i.e., quasi-geminate recombination unusually slow), namely λ | y s | exp( y s ) ≪ . If the opposite is true,then the rate constant obeys eq A4 but now y s is negative.21 .50.60.70.80.911.1 0 5 10 15 20 25 30 g app / g y s10 −4 −2 g / g R y s Figure 5: Quality of the approximate eq A8 is shown. Solid line shows the ratio of γ app ,calculated by eq A8, to the rate constant γ numerically calculated by eq 13 for δ = 3 . Brokenline shows the corresponding ratio for the rate constant γ app calculated by eq A4. Inset showsthe general behavior of the exact (solid line, eq 13) and approximate (broken line, eq A8)rate constant. References (1) Langevin, P. Recombinaison et Mobilites des Ions dans les Gaz.
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