Vukmirovic, Nenad

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Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleCarrier hopping in disordered semiconducting polymers: How accurate is the Miller-Abrahams model?Permalinkhttps://escholarship.org/uc/item/3951c2d5AuthorVukmirovic, NenadPublication Date2010-07-30 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaCarrier hopping in disordered semiconducting polymers: How accurate is theMiller-Abrahams model?Nenad Vukmirović and Lin-Wang WangComputational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.We performed direct calculations of carrier hopping rates in strongly disordered conjugated poly-mers based on the atomic structure of the system, the corresponding electronic states and theircoupling to all phonon modes. We found that the dependence of hopping rates on distance andthe dependence of the mobility on temperature are significantly different than the ones stemmingfrom the simple Miller-Abrahams model, regardless of the choice of the parameters in the model.A model that satisfactorily describes the hopping rates in the system and avoids the explicit calcu-lation of electron-phonon coupling constants was then proposed and verified. Our results indicatethat, in addition to electronic density of states, the phonon density of states and the spatial overlapof the wavefunctions are the quantities necessary to properly describe carrier hopping in disorderedconjugated polymers.The carrier mobility of conjugated polymer materials isthe most important physical property1–6 for their applica-tion in organic electronic devices. Realistic polymer ma-terials contain both crystalline (ordered) and amorphous(disordered) regions.7 Charge carrier transport is oftenlimited by the presence of amorphous regions. In theseregions the electronic states are localized due to presenceof disorder and carrier transport takes place by phonon-assisted carrier hopping8 between localized states. Suchtransport is traditionally modeled by assuming a certaindensity of electronic states in energy and space and a cer-tain form of hopping probabilities between them.6,9–12 Dif-ferent models are distinguished by the electronic densityof states (DOS) assumed in the model, which is usuallythe tail of the Gaussian9,10,12 or the exponential11 dis-tribution. The transition rates are typically assumed todecay exponentially with the distance between localizedstates,9–12 in the Miller-Abrahams (MA) form.13 Free pa-rameters that appear in the models are fitted to the ex-perimental mobility measurements.From the applications of such models, it is widely un-derstood that the mobility of the material strongly de-pends on the electronic DOS which is therefore believedto be the most important material property when chargetransport is concerned.14 On the other hand, much less ef-fort has been put into understanding how different formsof the transition rates affect the transport and in partic-ular whether the MA expression is suitable at all. Toaddress these questions, in this letter we perform directab-initio calculations of the transition rates between elec-tronic states starting from atomic structure of the system,followed by explicit calculation of electronic state wave-functions and their coupling to phonons. We find strongdeviations of the transition rate dependence on distancefrom the MA form and analyze the consequences of thison electronic transport.Electronic structure calculations were performed usingthe plane wave pseudopotential approach and the chargepatching method for organic systems introduced in Ref. 15which gives the accuracy similar to the density functionaltheory in local density approximation, but with a muchsmaller computational cost. Atomic structure of the sys-0 4 8DOS (1026 eV-1m-3)-0.4-0.200.20.4Energy (eV)0 5 10phonon DOS (eV-1)00.10.20.30.4FIG. 1: (Color online) Wavefunctions of top ten electronicstates in the valence band for the P3HT system of the size58.6×29.3×29.3 Å3 (5020 atoms). The isosurfaces correspondto a 50% probability of finding an electron inside the surface.The hole DOS and the phonon DOS, extracted from our cal-culations are shown in the right.tem was generated from classical molecular dynamics us-ing a simulated annealing procedure, as described in de-tail in Ref. 16. For concreteness, we analyze the poly(3-hexylthiophene) (P3HT) polymer, one of the most widelystudied conjugated polymers for electronics and optoelec-tronics.Electron-phonon (e-ph) coupling constants due to in-teraction with all phonon modes in the system were alsocalculated using the charge patching method, as describedin detail in Ref. 17. The transition rate for downward hopsbetween electronic states i and j was then calculated asWFij = π∑α|Mij,α|2ωα[N(h̄ωα) + 1] δ (εi − εj − h̄ωα) ,(1)where Mij,α = 〈ψi|∂H/∂να|ψj〉 is the e-ph coupling con-stant between electronic states i and j due to phononmode α, ∂H/∂να is the change of the single-particleHamiltonian due to atomic displacements according tophonon mode α, h̄ωα is the energy of that mode,N(h̄ωα) =(exp h̄ωαkBT − 1)−1is the phonon occupationnumber given by the Bose-Einstein distribution at a tem-perature T and εi is the energy of electronic state i.The dependence of transition rates calculated usingEq. (1) on distance for the P3HT system (Fig. 1) is shownas circles in Fig. 2a. Each circle represents one down-2ward transition. Strong deviations from the MA expres-sion (which would yield a straight line in Fig. 2a) are evi-dent. Despite that, one can still argue that some effectivelinear fit of the logWij vs. dij (where dij is the distancebetween the states i and j) dependence could be sufficientfor the description of transport through the system.0 30 60Distance (Å)10-1010-810-610-410-2100S2Full modelModel C - 1.60 20 40 60Model B0 20 40 60Distance (Å)6810121416log 10 (W/s-1)Full modelModel C - 1.6Model A - 1.0, W0fit(b)(a)FIG. 2: (Color online) The dependence of transition rates andwavefunction overlaps on distance for the system presented inFig. 1: (a) The dependence of transition rates for downwardhops on distance in different models. The numbers in legendspecify the value of a in Å for models A and C. (b) The depen-dence of the wavefunction moduli overlap S2ij on the distancebetween the states.To establish whether this is the case, we have calcu-lated the temperature dependence of the hole mobility inP3HT polymer (in the limit of low carrier concentration)using the multiscale methodology introduced in Ref. 17.In Fig. 3a we compare the results of the simulation wheretransition rates were calculated according to Eq. (1) (thatwill be referred to as the full model), with the simula-tion where these were replaced with the MA expression(referred to as model A in what follows) given as:WAij = W0 exp (−dij/a) (2)for downward hops. In all our simulations energies ofelectronic states and their spatial positions were kept thesame. In Eq. (2) the conventional value of W0 = 1014s−1was taken, while simulations were performed for differentvalues of a, specified in legend labels in Fig. 3a. As seenin Fig. 3a, the slope of the temperature dependence of themobility in model A is significantly different than in thefull model and therefore model A with physically reason-able values of W0 cannot reproduce the results of the fullmodel. We find that one has to take the (unrealistic) val-ues of W fit0 = 6.3× 1020 s−1 and a = 1 Å in model A to fitthe temperature dependence from the full model, shownin Fig. 3a.It has been a common practice to fit the experimentallymeasured temperature dependence of the mobility usingthe MA model and extract the electronic DOS from sucha fit.1,11,12 Our results on the other hand imply that theelectronic DOS obtained that way is rather some effectiveDOS, which together with the MA model can reproducethe experimental mobility.It would be desirable to have a model which is stillsufficiently simple and on the other hand accurate enough3 4 51000/T (1/K)10-1510-1210-910-6Mobility (m2 /Vs)Full modelModel A - 1.5Model A - 2.0Model A - 3.0Model B Model C - 1.0Model C - 1.6Model C - 2.0Model C - 3.00 400 800F1/2 (V1/2/cm1/2)11.522.533.5µ(F)/µ(0)Full modelModel A - 1.0, W0fitModel BModel C - 1.6(a)(b)FIG. 3: (Color online) (a) Temperature dependence of the holemobility in P3HT polymer calculated using different models.The numbers in legend labels in the figure specify the value ofa in Å for models A and C. (b) Electric field dependence of themobility calculated using different models. The parameters inthe models have been chosen to yield the same temperaturedependence of mobility.to properly describe the hopping rates and the transport.One can simplify Eq. (1) by introducing an approximationfor the calculation of Mij,α as simply proportional to theoverlap Sij =∫d3r|ψi(r)| · |ψj(r)| of the wavefunctionmoduli. We can then rewrite Eq. (1) as (that we will referto as model B):WBij = β2S2ij [N(εij) + 1]Dph(εij)/εij , (3)where Dph(E) is the phonon DOS normalized such that∫∞0Dph(E)dE = 1 (it is shown in Fig. 1), εij = |εi − εj |and β is the proportionality factor, chosen to be equal to107eVs−1/2 which gives a good fit of model B to the fullmodel, as seen in Fig. 4. The temperature dependenceof the mobility in model B agrees excellently with themobility in the full model, as can be seen from Fig. 3.Model B is simpler than the full model since it requiresonly the evaluation of wavefunctions and not the e-ph cou-pling constants. It is tempting to further simplify thatmodel by assuming that Sij in Eq. 3 decays exponen-tially with distance between the states, which gives theexpression that we will refer to as model C:WCij = β2 exp (−dij/a) [N(εij) + 1]Dph(εij)/εij . (4)However, the actual dependence of Sij on distance, shownin Fig. 2b, is significantly different from the simple expo-nentially decaying function. In a system with spherical36 8 10 12 14 16log10(WF/s-1)6 8 10 12 14 166 8 10 12 14 166810121416log 10(WMODEL /s-1 ) Model B Model C - 1.6 Model A - 1.0, W0fitFIG. 4: (Color online) Comparison of the transition rates infull model (W F ) and models A, B and C for the system shownin Fig. 1. The parameters in the models have been chosen toyield the same temperature dependence of mobility.localized states, it is quite plausible to assume the expo-nentially decaying overlap. On the other hand, the wave-functions in conjugated polymers are significantly differ-ent from spherical wavefunctions, as seen in Fig. 1. Thesestates are localized within the few rings of the main thio-phene chain. Their orientation might be important andtheir localization length can vary. Nevertheless, one canstill wonder whether a model based on the exponential de-cay of the overlap [Eq. (4)] can be effective; e.g. whethera certain fit to the dependence shown in Fig. 2b couldbe satisfactory for the description of transport. Whena = 1.6 Å, the mobility obtained from this model is ingood agreement with the full model (see Fig. 3). How-ever, the corresponding dependence of transition rates ondistance (triangles in Fig. 2a) does not fit well the samedependence in the full model. Therefore, although bothmodels A and C can fit the temperature dependence of themobility, albeit with unrealistic parameters (that do notgive reliable hopping rates, see Fig. 4), it is interesting tosee whether such fitted models would be able to accuratelydescribe other physical properties. To demonstrate this,we have calculated the electric field dependence of the mo-bility using the methodology described in Ref. 18, usingthe parameters that fit the temperature dependence of themobility in the full model. As can be seen from Fig. 3b,both models A and C give different results than the fullmodel.We also note that the results of all the simulationsreported here yield the temperature dependence of themobility of the lnµ ∼ −1/T form, which is the depen-dence that is obtained in the phenomenological modelswith exponential DOS11 and the mobility edge models.7On the other hand, Gaussian disorder model gives thelnµ ∼ −1/T 2 dependence.9,12 It would be very interest-ing to understand how the assumptions inherent to phe-nomenological models (these were outlined in the intro-ductory paragraph) affect the similarities and differencesin the results of our simulations and these models. Thisis however beyond the scope of this letter (except for therole of the form of the transition rates).In conclusion, our results indicate that hopping rates instrongly disordered conjugated polymers are determinedby the spatial overlap of the wavefunctions of the statesinvolved and the phonon DOS, and are not very sensitiveto the details of individual phonon modes. The overlap ofthe wavefunctions, on the other hand, cannot be simplydescribed as a function exponentially decaying with dis-tance. We add to the common wisdom that the electronicDOS determines the charge carrier transport in disorderedpolymers two more important factors: the wavefunctionoverlap and the phonon DOS.This work was supported by the DMS/BES/SC of theU.S. Department of Energy under Contract No. DE-AC02-05CH11231. It used the resources of National En-ergy Research Scientific Computing Center (NERSC) andthe INCITE project allocations within the National Cen-ter for Computational Sciences (NCCS).DISCLAIMERThis document was prepared as an account of worksponsored by the United States Government. Whilethis document is believed to contain correct information,neither the United States Government nor any agencythereof, nor The Regents of the University of California,nor any of their employees, makes any warranty, express orimplied, or assumes any legal responsibility for the accu-racy, completeness, or usefulness of any information, ap-paratus, product, or process disclosed, or represents thatits use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process,or service by its trade name, trademark, manufacturer,or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof, or The Regentsof the University of California. The views and opinionsof authors expressed herein do not necessarily state or re-flect those of the United States Government or any agencythereof or The Regents of the University of California.1 S. M. Tuladhar, M. Sims, J. Kirkpatrick, R. C. Maher, A. J.Chatten, D. D. C. Bradley, J. Nelson, P. G. Etchegoin, C. B.Nielsen, P. Massiot, W. N. George, and J. H. G. Steinke,Phys. Rev. B 79, 035201 (2009).2 V. Coropceanu, J. Cornil, D. A. da Silva Filho, Y. Olivier,R. Silbey, and J.-L. Bredas, Chem. Rev. 107, 926 (2007).3 F. C. Grozema and L. D. A. Siebeless, Int. Rev. Phys.Chem. 27, 87 (2008).4 N. Tessler, Y. Preezant, N. Rappaport, and Y. Roichman,Adv. Mater. 21, 1 (2009).5 D. L. Cheung and A. Troisi, Phys. Chem. Chem. Phys. 10,5941 (2008).6 C. Tanase, E. J. Meijer, P. W. M. Blom, and D. M.de Leeuw, Phys. Rev. Lett. 91, 216601 (2003).47 A. Salleo, T. W. Chen, A. R. Völkel, Y. Wu, P. Liu,B. S. Ong, and R. A. Street, Phys. Rev. B 70, 115311(2004).8 R. Coehoorn, W. F. Pasveer, P. A. Bobbert, and M. A. J.Michels, Phys. Rev. B 72, 155206 (2005).9 S. D. Baranovskii, H. Cordes, F. Hensel, and G. Leising,Phys. Rev. B 62, 7934 (2000).10 V. I. Arkhipov, P. Heremans, E. V. Emelianova, G. J. Adri-aenssens, and H. Bassler, Appl. Phys. Lett. 82, 3245 (2003).11 M. C. J. M. Vissenberg and M. Matters, Phys. Rev. B 57,12964 (1998).12 P. M. Borsenberger, L. Pautmeier, and H. Bassler, J. Chem.Phys. 94, 5447 (1991).13 A. Miller and E. Abrahams, Phys. Rev. 120, 745 (1960).14 D. P. McMahon and A. Troisi, Chem. Phys. Lett. 480, 210(2009).15 N. Vukmirović and L.-W. Wang, J. Chem. Phys. 128,121102 (2008).16 N. Vukmirović and L.-W. Wang, J. Phys. Chem. B 113,409 (2009).17 N. 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Carrier hopping in disordered semiconducting polymers: How accurate is the Miller-Abrahams model?

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Carrier hopping in disordered semiconducting polymers: How accurate is the Miller-Abrahams model?

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