Ann. Phys. (Berlin) 18, No. 12, 856 – 862 (2009) / DOI 10.1002/andp.200910405 Effect of electric field on diffusion in disordered materials F. Jansson1,2,∗ , A. V. Nenashev3,4 , S. D. Baranovskii5 , F. Gebhard5 , and R. Österbacka2 1 2 3 4 5 Graduate School of Materials Research, Åbo Akademi University, 20500 Turku, Finland Department of Physics and Center for Functional Materials, Åbo Akademi University, 20500 Turku, Finland Institute of Semiconductor Physics, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia Department of Physics and Material Sciences Center, Philipps-University, 35032 Marburg, Germany Received 1 September 2009, accepted 5 September 2009 Published online 11 December 2009 Key words Hopping transport, diffusion, disorder. PACS 72.20.Ht, 72.20.Ee, 72.80.Ng, 72.80.Le The effect of electric field on diffusion of charge carriers in disordered materials is studied by Monte Carlo computer simulations and analytical calculations. It is shown how an electric field enhances the diffusion coefficient in the hopping transport mode. The enhancement essentially depends on the temperature and on the energy scale of the disorder potential. It is shown that in one-dimensional hopping the diffusion coefficient depends linearly on the electric field, while for hopping in three dimensions the dependence is quadratic. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Charge carrier transport in disordered materials - inorganic, organic and biological systems - has been in the focus of intensive experimental and theoretical study for several decades due to various current and potential applications of such materials in modern electronic devices (see, for instance, [1] and references therein). An essential part of the research is dedicated to the study of the mobility of the charge carriers, μ, and their diffusion coefficient, D, as the decisive transport coefficients responsible for performance of most devices. Among other features, the relation between these two transport coefficients is the subject of intensive research, since this relation (called the “Einstein relation”) often provides significant information on the underlying transport mechanism [1]. In numerous experimental studies on organic disordered materials, essential deviations from the conventional form μ= e D, kT (1) of this relation have been claimed [2–8]. In Eq. (1) e is the elementary charge, T is the temperature and k is the Boltzmann constant. According to Einstein, such a relation between μ and D is valid in the case of thermal equilibrium for a non-degenerate system of charge carriers. Deviations from Eq. (1) were predicted theoretically for non-equilibrium transport at low temperatures [9–11] and also for equilibrium transport if the density of states (DOS), which can be used by the charge carriers, strongly depends on energy, for instance, exponentially [12] or according to a Gaussian distribution [13]. The former DOS is usually assumed to apply for inorganic amorphous semiconductors, while the latter one is widely believed to be valid for disordered organic materials, such as molecularly doped and conjugated polymers [14–19]. Since ∗ Corresponding author E-mail: fjansson@abo.fi c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Berlin) 18, No. 12 (2009) 857 most experimental evidence on the violation of Eq. (1) has been reported for organic disordered materials, we focus here on systems with the Gaussian DOS ε2 N √ exp − 2 , g(ε) = (2) 2σ σ 2π where N is the spatial concentration of conducting states and σ is the energy scale of the DOS distribution. All results on the validity or invalidity of Eq. (1) cited above were related to charge transport at low electric fields, when the transport coefficients μ and D do not depend on the strength of the applied electric field F . Remarkably, experiments on disordered organic materials evidence that at relatively low electric fields, at which the carrier mobility μ is field-independent and hence the carrier transport can be treated as Ohmic (low-field regime), the diffusion coefficient D of charge carriers and concomitantly the relation between μ and D become essentially dependent on the magnitude of the applied electric field F [2, 4–6]. Charge transport in disordered organic materials is dominated by incoherent hopping of electrons and holes via localized states with the DOS described by Eq. (2) randomly distributed in space [14–19]. The transition rate between an occupied state i and an empty state j, separated by the distance rij , is described by the Miller-Abrahams expression [20] ⎧ Δε ⎨ e− kTij , Δεij > 0 rij −2 a Γij = Γ0 e , (3) ⎩ 1 , Δεij ≤ 0 where Γ0 is the attempt-to-escape frequency. The energy difference between the sites is Δεij = εj − εi − F e(xj − xi ), (4) where the electric field F is assumed to be directed along the x-direction. The localization length of the charge carriers in the states contributing to hopping transport is a. We assume the latter quantity to be independent of energy and we will neglect correlations between the energies of the localized states, following the so-called Gaussian-disorder-model of Bässler [17–19]. The challenging problem then arises: to describe theoretically field-dependent diffusion of charge carriers in the hopping regime within the Gaussian DOS. This very problem was addressed in the numerical simulations by Richert et al [21]. Using a Monte Carlo algorithm with randomly distributed parameter α (the so-called off-diagonal disorder), it was shown that the diffusion coefficient for hopping transport in the Gaussian DOS depends essentially on the field strength at such low electric fields that the mobility of charge carriers remains field-independent [21]. This result was interpreted in analytical calculations by Bouchaud and Georges [22], who considered a hopping process in a one-dimensional (1D) system of equidistant localized states with transition rates essentially different from those given by Eq. (3). In the calculations of Bouchaud and Georges [22] the transition rates between the neighboring sites were taken as Δi±1,i ± eF d (5) Γi,i±1 = Γ0 exp 2kT with d being the site spacing and Δi,i+1 = Δi+1,i distributed according to g(Δij ) given by Eq. (2). We will call this model the Random-Barrier-Model (RBM) in contrast to the model described by Eqs. (2) and (3), which we call the Random-Energy-Model (REM). Bouchaud and Georges [22] suggested for the fielddependent part of the diffusion coefficient in the RBM the expression D(F )−D(0) ∝ F exp[3σ 2 /8(kT )2 ], which they claimed to be precisely the dependence found in [21]. Later the authors of [21] studied the quantity D(F ) − D(0) by computer simulations in more detail [23] and found a quadratic dependence of D(F ) − D(0) on F at low fields and no turn-over to linear field dependence as suggested by Bouchaud and Georges [22]. The question arises then on whether this discrepancy in the field dependences of the www.ann-phys.org c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 858 F. Jansson et al.: Diffusion in an electric field diffusion constant between computer simulations [23] and analytical calculations [22] is due to different models (RBM [22] against REM [23]) or is it due to different dimensionalities considered in these two approaches (1D in analytical calculations [22] against 3D in computer simulations [23]). The only way to answer this question is to obtain the diffusion coefficient in the REM in 1D and to compare it with the results for the RBM in 1D on one hand and with the results for the REM in 3D on the other hand. In Sect. 2 we examine both of the one-dimensional models. We also present an exact, analytic result for the diffusion coefficient in the RBM, which differs from the one given by Bouchaud and Georges [22]. The results for both RBM and REM give a linear field dependence of the diffusion coefficient at low fields in the 1D case. In Sect. 3 we present the results obtained by computer simulations in 3D systems. The results clearly demonstrate a quadratic field dependence of the diffusion coefficient at low fields, in agreement with previous simulations [23]. One should then conclude that the discrepancy between the linear [22] and the quadratic [23] field dependences of the diffusion constant reported in the literature is due to the different space dimensionalities considered in the two approaches. 2 One-dimensional systems In this section the field dependence of the diffusion coefficient for hopping in a one-dimensional chain will be obtained, both for the RBM and the REM. For one-dimensional, nearest-neighbor hopping in a chain of N sites with periodic boundary conditions the drift velocity v and diffusion coefficient D can be calculated as functions of the hopping rates Γi,i±1 , using equations derived by Derrida [24] (for d = 1): N N N N +2 1 , (6) un irn+i + N Γn,n+1 un rn − v v D = N 2 2 ( n=1 rn ) n=1 n=1 i=1 N Γn+1,n N v = N 1− , (7) Γn,n+1 n=1 n=1 rn ⎤ ⎡ N −1 i Γ 1 n+j,n+j−1 ⎦ ⎣1 + , (8) rn = Γn,n+1 Γ n+j,n+j+1 i=1 j=1 ⎡ un = 1 Γn,n+1 ⎣1 + N −1 i=1 ⎤ i Γn−j+1,n−j ⎦ . Γn−j,n−j+1 j=1 (9) Further, in the case of random and independent hopping rates (Γij and Γji are allowed to be correlated), v and D for infinite chains can be expressed using mean values of the hopping rates: v= 1 Γ→ −1 Γ← 1− , Γ→ 2 1 − Γ← /Γ→ D= 1 − (Γ← /Γ→ )2 1 Γ→ −3 (10) 1 Γ→ Γ← Γ2→ 1 + 2 1 Γ2→ Γ← 1− , Γ→ (11) where Γ← = Γi,i−1 and Γ→ = Γi,i+1 . By evaluating Eqs. (10) and (11) for the RBM defined by Eqs. (2) and (5), we obtain eF d σ2 v = 2dΓ0 exp − sinh , (12) 8(kT )2 2kT σ2 |eF |d σ2 |eF |d 2 − Γ exp D = d2 Γ0 exp − sinh + d . (13) 0 8(kT )2 2kT 8(kT )2 2kT c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 18, No. 12 (2009) 859 Equation (13) differs from the expression given by Bouchaud and Georges, [22] though it is linear in F to first order. Equations (10) and (11) can be applied to the RBM but not to the REM, since in the REM the rates of consecutive hops are correlated. This correlation arises because the rates of two consecutive hops both depend on the energy of the site they have in common. For this reason the REM is studied by computer simulation. A similar simulation is performed for the random barrier case, to verify that the simulation and the analytic result are consistent. The results were obtained by generating several chains with random jump rates (according to the respective model), evaluating Eq. (6) for each chain and averaging the results. Long chains (107 and 108 sites) were needed to obtain a good agreement between different realizations of the chains. For chains of this length, a direct evaluation of Eqs. (6–9) is not practical. Below, the equations are rewritten in a form that can be evaluated in O(N ) steps, using recursion relations. Define gn = Γn,n−1 Γn,n+1 and hn = Γn+1,n , Γn,n+1 and further Gn = 1 + N −1 i gn+j and Hn = 1 + i=1 j=1 N −1 i hn−j , i=1 j=1 so that rn = Gn / Γn,n+1 and un = Hn / Γn,n+1 . All Gn and Hn , and thus rn and un can now be calculated efficiently from Gn−1 = gn Gn − G + 1, Hn+1 = hn Hn − H + 1 where G = H = g1 g2 . . . gN = h1 h2 . . . hN . For the first term in the brackets in Eq. (6), define Sn = N N i=1 irn+i and S = i=1 rn . Now Sn+1 = Sn − S + N rn+1 . These relations are numerically stable if G < 1, which is satisfied if the average drift is to the right (towards larger site indices). The diffusion coefficient is now given by N N Hn S n 1 Gn Hn N +2 D= 2 v . (14) +N −v S Γ Γ 2 n=1 n,n+1 n=1 n,n+1 With this method of evaluation, numerical results for the diffusion coefficient were obtained. The results are shown in Fig. 1 together with the analytical results for the RBM, and in Fig. 2 for the REM. For both models the diffusion coefficient is linear in the electric field (at low fields), see the insets in each figure. 3 Three-dimensional systems To obtain the field dependence of the diffusion coefficient for hopping transport in three dimensions, we perform Monte Carlo simulations. In contrast to the treatment in [17, 21, 23], we simulate transport in an infinite lattice of sites (using periodic boundary conditions) and follow the motion of all charge carriers for a fixed time t, instead of considering a sample of some fixed size. We use a significantly larger lattice than in the earlier simulations (7003 sites), since the finite size effects are severe at low temperatures. Decreasing the system size to 4003 sites resulted in a decrease of the calculated diffusion coefficient by www.ann-phys.org c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 860 F. Jansson et al.: Diffusion in an electric field 12 0.6 10 0.4 8 0.2 12 kT / σ = 0.3 kT / σ = 0.4 kT / σ = 0.5 kT / σ = 1 0.8 0.2 10 0.1 8 0 0.1 0.6 0.5 0.2 D D 14 0 6 0 4 0.7 0.05 0.1 6 4 2 2 0 0 0.5 1 eFd / σ 1.5 2 Fig. 1 (online colour at: www.ann-phys.org) D(F ) in the RBM for different temperatures T . The curves show the analytic solution, Eq. (13), while the symbols show numerical results, Eq. (6) for chains with N = 107 sites. The inset shows the low-field behavior. 0.8 kT / σ = 1.0 0 0 0.5 1 eFd / σ 1.5 2 Fig. 2 (online colour at: www.ann-phys.org) D(F ) in the REM for different temperatures T . Numerical results, Eq. (6), for chains with N = 108 sites are shown. The inset shows the low-field behavior. 33 % for the lowest temperature considered. Further, our simulation does not include off-diagonal disorder. The system is modelled as a cubic lattice of L3 sites with lattice constant d and the site energies i chosen randomly according to Eq. (2). Jumps inside a cube with the side length 7d, centered at the starting site are allowed. The mobility μ is calculated from the average distance that the charge carriers have moved along the direction of the field, while the diffusion constant D is calculated from the width of the carrier packet: x2 − x2 x , D= . (15) Ft 2t Simulation results for the field-dependence of the diffusion coefficient along the field at different temperatures are shown in Fig. 3. The data points are fitted with a square function 2 eF d D(F, T ) = A(T ) + D0 (T ). (16) σ μ= For higher temperatures, it is seen that D(F ) deviates from the simple square function at high fields, and even decreases with increasing field. This behavior can be understood by looking at the hopping rates: when the field is large, almost all jumps will be downwards in energy, which gives the constant factor unity in the hopping rate, Eq. (3). A similar behavior is seen for the one-dimensional REM in Fig. 2. We focus on the field dependence of D at fields lower than the field at which D starts to decrease. The dependence of the fitting parameters A and D0 on temperature is shown in Fig. 4. The diffusion coefficient at zero field, D0 , can be expressed using the Einstein relation (1) and the temperature dependence −bσ2 of the low-field mobility [1]: ln μ ∝ −σ 2 /(kT )2 . Thus D0 (T ) = c kT σ exp( (kT )2 ), for some constants b and c. Based on the simulation results, we conclude that the diffusion constant depends quadratically on the electric field for three-dimensional hopping transport. 4 Conclusions In Sect. 2 it was shown that for hopping in 1D the diffusion coefficient D depends linearly on the electric field for both the RMB and the REM. An analytical expression for D was derived for the RBM case. In Sect. 3 it was shown by Monte Carlo simulation that the field dependence of D in the 3D case is quadratic. We conclude that the different field dependencies reported previously [22, 23] are caused by the different dimensionalities of the systems. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 18, No. 12 (2009) -2 10 -3 10 -4 10 kT = 0.33σ kT = 0.4 σ kT = 0.5 σ kT = 0.7 σ -2 A D0 10-3 10 -4 D A, D0 10 861 10-5 10-5 10 10 -6 0.001 0.01 0.1 eFd / σ 1 -6 10-7 0.3 0.35 0.4 0.45 0.5 0.55 kT / σ 0.6 0.65 0.7 0.75 Fig. 3 (online colour at: www.ann-phys.org) The diffu- Fig. 4 (online colour at: www.ann-phys.org) The temsion coefficient D for hopping in 3D, as a function of the perature dependence of the parameters D0 and A obapplied electric field F . The system is a lattice of 7003 tained from Fig. 3. sites with the localization length a = 0.2d. Packets consisting of 1000 charge carriers were simulated for each data point, and 5 realizations of the disorder were calculated. Acknowledgements Financial support from the Academy of Finland project 116995, from the Deutsche Forschungsgemeinschaft and that of the Fonds der Chemischen Industrie is gratefully acknowledged. Parts of the calculations were done at the facilities of the Finnish IT center for science, CSC. References [1] S. Baranovski (ed.), Charge Transport in Disordered Solids with Applications in Electronics (Wiley & Sons, Chichester, 2006). [2] H.-J. Yuh and M. Stolka, Philos. Mag. B 58(5), 539–549 (1988). [3] P. M. Borsenberger, L. Pautmeier, R. Richert, and H. Bässler, J. Chem. Phys. 94(12), 8276 (1991). [4] P. M. Borsenberger, R. Richert, and H. Bässler, Phys. Rev. B 47(8), 4289–4295 (1993). [5] A. Hirao and H. Nishizawa, Phys. Rev. B 54(7), 4755–4761 (1996). [6] A. Hirao and H. Nishizawa, Phys. Rev. B 56(6), R2904–R2907 (1997). [7] J. M. Lupton and J. Klein, Phys. Rev. B 65(19), 193202 (2002). [8] K. Harada, A. G. Werner, M. Pfeiffer, C. J. Bloom, C. M. Elliott, and K. Leo, Phys. Rev. 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