ISSN 10637826, Semiconductors, 2013, Vol. 47, No. 10, pp. 1292–1297. © Pleiades Publishing, Ltd., 2013. Original Russian Text © N.A. Korolev, V.R. Nikitenko, A.P. Tyutnev, 2013, published in Fizika i Tekhnika Poluprovodnikov, 2013, Vol. 47, No. 10, pp. 1304–1309.
ELECTRONIC PROPERTIES OF SEMICONDUCTORS
Numerical Modeling of the Polarization Current of Geminate Pairs in Disordered Polymers with Traps N. A. Koroleva, V. R. Nikitenkoa^, and A. P. Tyutnevb a
National Research Nuclear University “MEPhI”, Moscow, 115409 Russia ^email:
[email protected] b Moscow State Institute of Electronics and Mathematics of the National Research University “Higher School of Economics”, Moscow, 109028 Russia Submitted February 18, 2013; accepted for publication February 21, 2013
Abstract—The time dependences of the polarization current of geminate pairs in disordered media are sim ulated by the Monte Carlo numerical method. A model of hopping transport with exponential energy distri bution of the hopping centers is considered. The dependence of the polarizationcurrent kinetics on the ini tial geminatepair separation, the relative concentration of traps, the temperature, and the strength of the applied electric field is investigated. The limits of the applicability of an analytical theory based on the drift diffusion description of electron and hole motion of the geminate pair in the dispersivetransport mode are discussed. The range of parameters is determined in which the existence of the negative current is possible. DOI: 10.1134/S1063782613100163
1. INTRODUCTION The area of application of organic semiconductors and insulators is constantly being extended; these include datastorage devices, LEDs, solar cells, thin film fieldeffect transistors, etc. Weak intermolecular interaction is inherent to organic materials, in partic ular, polymers, as a consequence of which there occurs the localization of charge carriers in individual mole cules. Transitions between localized states are imple mented by tunnel hopping with the participation of phonons (hopping transport) [1]. Charge carriers in such materials are obtained as a result of injection from electrodes, and photo or radiationinduced generation. Charge generation passes through the stage of the formation of oppositely charged pairs sep arated by the distance r0 (initial pair separation), which can considerably exceed the intermolecular dis tance. Further, two variants are possible: the electron meets a hole, and the pair recombines (geminate recombination), or the carriers drift apart to a distance that considerably exceeds the Onsager radius, rc = 2
e /4πεε 0 kT , where e is the elementary charge, ε is the permittivity, ε0 is the dielectric constant, k is the Bolt zmann constant, and T is the temperature. Since the mobilities are usually markedly different, one of the carriers can be considered motionless. The model of Gaussian disorder [2] is widely used for analyzing hopping transport in organic materials. At the same time, nonsteadystate radiative electrical conduction in many polymers is successfully described by the Rose–Fowler–Weisberg (RFW) model [3], according to which hopping transport proceeds over
the lattice of singleenergy states. The disorder mani fests itself in the fact that a portion of the lattice sites are traps, which are distributed in energy according to exponential law. It is necessary to note that, as in the case of using the model of Gaussian disorder, one also frequently takes into account traps, which are expo nentially distributed in energy (in this case, the “trans port” states are distributed according to Gaussian law) [4]. Although, the typical scale of the Gaussianenergy disorder is 0.1 eV [2], materials with weak disorder are described by Gaussian distribution with a rootmean square variance of 0.05 eV [5] and even lower [6]. In the case of weak Gaussian disorder, the models described above are almost equivalent. It is considered that geminate recombination (GR) proceeds accord ing to the Langevin mechanism so that the Onsager model [7] is applicable. In the case of high generatedradiation energies, the initial energy distribution of charge carriers is far from equilibrium. Therefore, transport of the majority of carriers before their recombination occurs in a highly nonequilibrium (dispersive) mode [3, 8]. In [9– 11], the GR kinetics were investigated on the basis of a solution to the Smoluchowski equation (in the disper sivetransport mode) for the average spacetime distri bution of a more mobile “twin”. It represents the drift diffusion equation in the dispersivetransport mode upon the presence of both an external uniform electric field and the Coulomb field of a motionless “twin”. It is the socalled driftdiffusion approximation, which is applicable, since the difference in the electrostatic energy for the hopping length is negligible in compar ison with the thermal energy kT [12]. However, this
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condition is easily violated if the initial separation is comparable to the hopping length. It is this case that is most interesting when analyzing pair separation in organic photovoltaics [13]. Recently [14], the survival probability of pairs and their recombination rate were modeled by the Monte Carlo (MC) method, and it was shown that the discreteness of the medium leads to significant deviations from the results of the driftdif fusion approximation. However, no analysis of the transient currents induced by the polarization of gem inate pairs in an external field was carried out. Mean while, the results obtained in the driftdiffusion approximation both by analytical methods [9–11, 15] and by numerical solution of the Smoluchowski equa tion [16, 17] show that, under certain conditions, the kinetics of the geminate conductivity are nontrivial: at a certain time interval, the current can be negative, i.e., directed against the external field. In this study, following [14], we investigate the time dependences of the polarization current of geminate pairs by MC numerical modeling. The RFW model is used in this study, the same as previously [14], to allow comparison of the results with those of the driftdiffu sion approximation [9–11]. In contrast to the approx imate analytical approach, these results are not limited to the region of low fields [9, 11] or low temperatures [10]. It is shown that the occurrence of negative cur rents is possible also upon violation of the driftdiffu sion approximation, and the range of parameters, where this effect takes place, is specified. 2. METHOD OF CALCULATIONS An electron appears at the initial moment of time at the distance r0 from a motionless hole as a result of photo or radiationinduced generation. Its initial position is set randomly providing a uniform angular distribution over a sphere with the radius r0 (taking into account the lattice discreteness). Further electron motion occurs as hops over cubiclattice sites. A por tion of the sites are traps, the energy E of which is dis tributed according to exponential law: g(E) = (Nt/E0)exp(–E/E0), where Nt is the trap density, and E0 is the distribution depth. Other sites (“conducting states”) have an identical energy (the zero one disre garding the electrostatic energy). The calculations stop after the carriers either recombine or diverge at a distance exceeding (3–4)rc, when the probability of geminate recombination is negligibly low [14]. The algorithm for modeling carrier transport by the Monte Carlo method is in detail described in [12, 14]. In each test, random trap energies according to the exponen tial distribution are newly generated. To calculate the polarization current in an external electric field applied along the axis x, we first calculated the projec tion of the dipole moment of the geminate pair (GP) onto this axis averaged over all tests. For each short time interval, this projection is determined by multi plication of the number of electrons having (in differ SEMICONDUCTORS
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ent tests) the certain coordinate xi by xi and summa tion over all numbers i. The coordinate x is counted from the “twin”. Electron hops in the plane xi give no contribution to the dipolemoment variation. The polarization current is determined by numerical dif ferentiation (as the ratio of the dipolemoment varia tion for a short time interval to the value of this inter val). The tests are repeated from 5 × 104 to 2 × 105 times depending on the parameters. This number is suffi cient for achieving a comprehensible level of current fluctuation after numerical differentiation. Because of the fluctuations, we failed to obtain smooth current curves. These require further smoothing, which is car ried out by means of the “Origin” program. 3. RESULTS OF CALCULATIONS The calculations are carried out for the following parameter values of the model polymer: ε = 2.0 is the permittivity, ν0 = 1013 s–1 is the frequency factor, a0 = 0.6 nm is the lattice constant, r0 = 3.13 nm is the initial separation of the geminate pair, c = 0.01 and 0.09 are the relative trap concentrations and E0 = 0.07 eV. The polarization current was calculated for various values of the external field F and temperature T on which depend the value of the Onsager radius rc and the value of the dispersion parameter (the disorder parameter) α = kT/E0. The graphs show the values of the current density under the assumption that one pair is generated per 1 m3. The time is normalized to the value of the characteristic hopping time tr = –1
ν 0 exp ( 2γa 0 ) , where γ = 5/a0 is the reciprocal radius of localization of the electron wave function. In Fig. 1, we show the time dependence of the polarization current for various values of r0/rc (the temperature varies). In this case, the intensity of the external field is rather low, eFr c /kT Ⰶ 1, as was also assumed in [9, 11]. In qualitative agreement with the results of the driftdiffusion approximation, the cur rent kinetics prove to change sign at reasonably low temperatures (curves 1, 2). With increasing tempera ture, the time range of the negative current and (to a lesser degree) its absolute value decrease. At larger times, the current again becomes positive decreasing according to the power law tα – 1 (see dash–dotted straight lines). At high temperatures (curves 3 and 4), the current is always positive, and its value increases with temperature. In Fig. 2, we show graphs of the polarization cur rent for two values of relative trap concentrations 0.01 and 0.09. In curves 1 and 2, there are portions of neg ative current, which shift to the right and become broader along the time scale with increasing trap con centration. The absolute values of the current only slightly depend on the concentration, decreasing with
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−9
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Fig. 1. Time dependences of the polarization current for various values of r0/rc: (1, 1') 0.08, (2, 2') 0.14, (3) 0.17, and (4) 0.25. Curves 1', 2 ' are analytical. The value of the relative trap concentration c = 0.09. The ratio of the inten sity of the applied uniform field to that of the Coulomb field at the distance r0, F/F0 = 0.01.
increasing concentration, which is induced by the capture of carriers at traps. These results can be com pared with analytical curves 1' and 2 ' calculated
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according to the method [11] because the weakfield condition is fulfilled. The values of the current obtained by the MC method exceed the analytical results by several orders of magnitude, the difference increasing with decreasing trap concentration. How ever, it is possible to state the good agreement between the times of termination of the negative current (see also curves 1 and 2 in Fig. 1). For curves 3 and 4, the weakfield condition eFr c /kT Ⰶ 1 is not fulfilled, and there is still no negative current. The dependence of the polarization current for c = 0.01 and 0.09 for several values of the external field is shown in Fig. 3. With increasing field, the negative current begins earlier and has a shorter time interval; in this case, the absolute value of the negative current increases, while the field does not become so strong that it excludes this current (curve 3). The presence of the time portion with negative cur rent and its disappearance with increasing tempera ture or strengthening of the external field was qualita tively explained previously in [9–11, 15]. The region of the parameters (temperature and field intensity) at which there is negative current is limited to their rea sonably low values (and also initial pair separation). The parametric region of negativecurrent existence for c = 0.01 and 0.09 is shown in Fig. 4. The curves are plotted by means of averaging the results obtained in different series of tests. The horizontal and vertical bars designate errors. It is possible to note a weak dependence on the trap concentration. In [9] it is shown that negative current exists in the limiting case
2 102
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Fig. 2. Dependences of the polarization current for two values of c: 0.01 and 0.09; (1, 1') c = 0.01, F/F0 = 0.01, r0/rc = 0.14; (2, 2 ') c = 0.09, F/F0 = 0.01, r0/rc = 0.14; (3) c = 0.01, F/F0 = 0.04, r0/rc = 0.12; (4) c = 0.09, F/F0 = 0.04, r0/rc = 0.12; and (1', 2 ') analytical curves.
100
101
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104
105 t/tr
Fig. 3. Dependences of the polarization current for c = 0.01, α = 0.256 (r0/rc = 0.08), and for various values of the intensity of the external uniform field. (1) F/F0 = 0.01, (2) F/F0 = 0.05, and (3) F/F0 = 0.09. SEMICONDUCTORS
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10−11 10−9 10−7 10−5 10−3 10−1 101
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Fig. 4. Region of negative current for c = 0.01 and 0.09; (1) c = 0.01; and (2) c = 0.09.
of a weak external field for r0/rc < 0.16, which was also confirmed by numerical modeling. It is probable that the negativecurrent region disappears completely with increasing external field and decreasing tempera ture as the results of the analytical model predict. 4. DISCUSSION OF RESULTS Curves 1'–4 ' in Fig. 1 are based on analytical solu tion of the Smoluchowski equation [11] in the disper sive mode of transport in a weak uniform field ( eFr c /kT Ⰶ 1). To compare these solutions with the MCmodeling results, it is necessary to ensure corre spondence of the parameters in the RFW model: μ0τ0, where μ0 and τ0 are the mobility and the lifetime of the carrier in the conducting state, respectively, and the hopping lengths a0 in the MC model: μ0τ0 = 2
( e/kT )a 0 〈 n〉 , where 〈n〉 is the average number of hops before capture of the carrier in a trap. The number 〈n〉 is determined by MC modeling [14]. In the case of negative current, its maximum negative values deter mined analytically and by the MC modeling prove to be of the same order of magnitude. The times of termi nation of the negative current are also close. In the rest, the MCmodeling data considerably disagree with the analytical results. So, the highest negative current is achieved later by approximately two orders of magnitude (curves 1 and 1', 2 and 2 ', see also Fig. 2), which is in agreement with the data of [14], according to which the analytical model underesti mates the GRonset time. Thus, the negativecurrent time interval proves to be much narrower than that predicted by the analytical model. This time interval corresponds to the time of SEMICONDUCTORS
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Fig. 5. Time dependences of the survival probability. (1) c = 0.01, F/F0 = 0.01, r0/rc = 0.14; (2) c = 0.09, F/F0 = 0.01, r0/rc = 0.14; (3) c = 0.01, F/F0 = 0.04, r0/rc = 0.12; and (4) c = 0.09, F/F0 = 0.04, r0/rc = 0.12.
the most intense GR. At high temperatures, the intense GR, according to the analytical model [9], gives a fast decrease in the positive current between two asymptotic laws t(α – 1), which correspond to the motion of charges before recombination and after pair separation in the dispersive mode. Since with increas ing temperature, the GR characteristic times sharply decrease, we observe, apparently, only the final asymptotic mode in Fig. 1 (curves 3 and 4). In the case of curve 3, this mode is really well described by the dependence t(α – 1) for α = 0.54, which corresponds to r0/rc = 0.17. For curve 4, α = 0.8; in this case, the transport is poorly described by the approximation of strongly nonequilibrium transport [18, 19]. A more exact description gives a somewhat faster law of decrease, which is in agreement with Fig. 1 [18]. A sharp decrease in the current for long times is related to the release of a major fraction of carriers beyond the bounds of the modeling sphere (of the radius 4rc). For the same parameters, we plot the time depen dences (Fig. 5) of the carrier survival probability in which it can be seen that the survival probability approaches a limiting value (the separation probability or the quantum yield), which is many orders of mag nitude in time later than the time in which the negative current terminates. On the other hand, the MC calcu lations in this study showed surprisingly good coinci dence between the quantum yield and the result for the Onsager model [6–8], Ω∞ = Ωons = exp(–rc/r0), in the limiting case of a weak field. In [9] it was shown that
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Quantum yield of carriers at different initial separations r0 a0, nm
r0, nm
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373 453 666 333 333
0.448 0.544 0.8 0.4 0.4
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1.33 1.27 0.90 1.70 8.48
7.93 × 10–4 2.79 × 10–3 0.01830 0.018 0.13
7.95 × 10–4 2.79 × 10–3 0.01354 0.028 0.17
the analytical results on the survival probability Ω(t) can be used for β < β1: 1
r 1–α β 1 = 1 0 ( 4πξc ) , 3 a0
ξ ≈ 1.
Here the parameter β = [U(r0) – U(r0 – a0)]/kT ≈ 2
r c a 0 /r 0 is the change in the Coulomb energy of the interaction between “twins” in terms of the thermal energy kT. The data in the table show surprisingly good agreement with the Onsager theory for a low value of initial separation; although, the condition β < β1 is almost violated. Apparently, the cause consists in the fact that an increase in the separation probability due to the increased role of diffusion in the discrete medium is compensated by the fact that carriers quickly recombine for the case of a small length of ini tial separation r0 = 3 nm and do not get to the deep traps lowering the separation probability at this moment. Therefore, it can probably be assumed that, in the region of large times, the analytical and MC modeled curves of current in Fig. 2 coincide; although, the different between curves 1 and 1', 2 and 2 ', decreases slowly. An increase in the negativecurrent duration and also the shift of the negativecurrent onset towards larger times with increasing relative trap concentration (Fig. 2, curves 1, 2) is induced by the capture of carri ers at traps including deep ones, the number of which increases, resulting in greater release times and decel eration of the process of recombination. With increas ing external field (curves 3, 4), the time of the drift convergence of carriers decreases, which accelerates their recombination and results in increasing positive current in comparison with that in curves 1 and 2. An increase in the concentration of traps decreases the current for the entire considered time interval (curves 4 and 3). 5. CONCLUSIONS Modeling by the MonteCarlo method has con firmed the possibility of the existence of negative cur rent related to the nonsteadystate polarization of geminate pairs. MC modeling showed that the time of intenseGR onset increases by several orders of mag
nitude in comparison with the results of an analytical model based on the driftdiffusion approximation. It results in later achievement of the highest negative current and smaller duration of its existence as com pared with the analytical result (Figs. 1, 2). This cir cumstance related to the discrete behavior of the medium manifests itself also in the kinetics of a decrease in the survival probability and the recombi nation rate [14]. The result of the analytical model, according to which the negative current exists at values of r0/rc < 0.16 in the limiting case of a weak external field, is confirmed. MC modeling enabled us to estab lish the boundaries of the region of existence of nega tive current in the case of an external field, which can not be considered as weak (Fig. 4). It should be noted that a rather probable spread in the values of the initial pair separation r0 results in the “suppression” of nega tive conductivity [20]. However, even in this case, the GR kinetics considerably affect the photoconductivity because the current can be presented qualitatively as the product of the freecarrier current and the survival probability, which decreases in a wide range from 1 to Ω∞ Ⰶ 1. Investigation of the mechanism of chargecarrier losses, in particular, the ratio of losses for geminate and nongeminate recombination is necessary for increasing the efficiency of photovoltaic cells based on organic materials [21]. For various materials and donor–acceptor systems, conclusions about the dom ination of both the geminate and nongeminate mech anism are made; in this case, nonsteadystate methods of measurements, for example, nonsteadystate con ductivity [22, 23], the electronspin resonance [24], and nonsteadystate photoinduced absorption [25] are widely used. In [25] the GR characteristic times are determined at room temperature in the range of 20– 100 ns. This scale of times fully allows measurements of the nonsteadystate conductivity [22] and corre sponds to the times of the most intense recombination for the parameters used in this study (see Figs. 1–3). At the same time, comparison of the data in Figs. 1 and 5 shows that, even at much larger times in the case of the presence of traps, the GR process is still far from completion and, correspondingly, the GR kinetics still affect the photoconductivity. The insufficiently sub stantiated assumption that the polarization current of SEMICONDUCTORS
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geminate pairs can be neglected led the authors of [22] to the conclusion that the geminate mechanism intro duces no substantial contribution to chargecarrier losses. Meanwhile, the experimental data of this study completely agree with the opposite statement if we assume that, at the moment of time, which is accepted as the initial moment, the intense GR only begins, and the polarization current of pairs introduces the princi pal contribution to the photoconductivity. This exam ple shows the importance of analysis of the polariza tion current of geminate pairs. The authors hope that this study stimulates the interest of researchers to this problem and that it is useful in the analysis of experi mental results. ACKNOWLEDGMENTS The study was supported by the FTsP “Investiga tions and Developments on Priority Lines in the Development of Scientific and Technological Policy of Russia for 2007–2013”, State Contract no. 16.523.11.3004. REFERENCES 1. M. Pope and H. E. Swenberg, Electronic Processes in Organic Crystals and Polymers (Oxford Univ. Press, Oxford, 1999). 2. H. Bassler, Phys. Status Solidi B 107, 9 (1981). 3. A. P. Tyutnev, V. S. Saenko, E. D. Pozhidaev, and N. S. Kostyukov, Dielectric Properties of Polymers in Ionizing Radiation Fields (Nauka, Moscow, 2005) [in Russian]. 4. E. Knapp, R. Häusermann, H. U. Schwarzenbach, and B. Ruhstaller, J. Appl. Phys. 108, 054504 (2010). 5. D. Hertel, H. Bässler, U. Scherf, and H.H. Hörhold, J. Chem. Phys. 110, 9214 (1999). 6. I. I. Fishchuk, D. Hertel, H. Bässler, and A. K. Kada shchuk, Phys. Rev. B 65, 125201 (2002). 7. L. Onsager, Phys. Rev. 54, 554 (1938).
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8. V. I. Arkhipov, A. I. Rudenko, A. M. Andriesh, M. S. Iovu, and S. D. Shutov, Transient Injection Cur rents in Disordered Solids (Shtiintsa, Kishinev, 1983) [in Russian]. 9. V. I. Arkhipov, V. R. Nikitenko, and A. I. Rudenko, Sov. Phys. Semicond. 21, 685 (1987). 10. V. I. Arkhipov, V. R. Nikitenko, and A. I. Rudenko, Sov. Phys. Semicond. 21, 984 (1987). 11. V. R. Nikitenko, A. P. Tyutnev, V. S. Saenko, and E. D. Pozhidaev, Khim. Fiz. 20, 98 (2001). 12. B Ries, G. Schönherr, and H. Bässler, Philos. Mag. B 48, 87 (1983). 13. C. Deibel, T. Ströbel, and V. Dyakonov, Phys. Rev. Lett. 103, 036402 (2009). 14. N. A. Korolev, V. R. Nikitenko, and A. P. Tyutnev, Semiconductors 44, 912 (2010). 15. G. F. Novikov and B. S. Yakovlev, Khim. Vys. Energ. 19, 282 (1985). 16. L. V. Lukin, Chem. Phys. 360, 32 (2009). 17. R. Sh. Ikhsanov, A. P. Tyutnev, V. S. Saenko, and E. D. Pozhidaev, Russ. J. Phys. Chem. B 27, 309 (2008). 18. V. R. Nikitenko, A. P. Tyutnev, V. S. Saenko, and E. D. Pozhidaev, Khim. Fiz. 23, 66 (2004). 19. R. T. Sibatov and V. V. Uchaikin, Phys. Usp. 52, 1019 (2009). 20. A. P. Tyutnev, V. I. Arkhipov, V. R. Nikitenko, and D. N. Sadovnichii, Khim. Vys. Energ. 29, 351 (1995). 21. C. Deibel, T. Ströbel, and V. Dyakonov, Adv. Mater. 22, 4097 (2010). 22. R. A. Street, S. Cowan, and A. J. Heeger, Phys. Rev. B 82, 121301(R) (2010). 23. M. Mingebach, S. Walter, V. Dyakonov, and C. Deibel, Appl. Phys. Lett. 100, 193302 (2012). 24. J. Behrends, A. Sperlich, A. Schnegg, et al., Phys. Rev. B 85, 125206 (2012). 25. S. K. Pal, T. Kesti, M. Maiti, F. Zhang, et al., J. Am. Chem. Soc. 132, 12440 (2010).
Translated by V. Bukhanov
