Finite-Size Effects in Monte Carlo Simulations of the Gaussian Disorder Model

Journal of the Korean Physical Society, Vol. 60, No. 11, June 2012, pp. 1897 1901 Finite-Size Effects in Monte Carlo Simulations of the Gaussian Disorder Model Sunil Kim Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea Joonhyun Yeo Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea Chan Im Department of Chemistry and Konkuk University-Fraunhofer ISE Next Generation Solar Cell Research Center (KFnSC), Konkuk University, Seoul 143-701, Korea Doseok Kim Department of Physics, Sogang University, Seoul 121-742, Korea (Received 9 March 2012, in final form 10 April 2012) We study theoretical issues related to the use of finite size systems in Monte Carlo simulations of the Gaussian disorder model (GDM). The GDM is one of the most widely used models to describe the charge transport in random organic materials. In the GDM, the energy of charge carriers is well known to approach an equilibrium energy in the long-time limit. We find that at low temperatures, the equilibrium energy shows a strong dependence on the system size. Using improved numerical methods, we study system sizes much bigger than those used in earlier works. We show that, below a certain temperature, a system size much larger than those used in conventional studies must be used to correctly describe the temperature dependence of the equilibrium energy. PACS numbers: 72.20.Ee, 72.80.Le, 72.10.Bg Keywords: Charge transport, Random organic solids, Gaussian disorder model, Kinetic Monte Carlo method DOI: 10.3938/jkps.60.1897 I. INTRODUCTION The charge transport in disordered organic materials has been studied both theoretically and experimentally for many years [1]. Among the theoretical models, the so-called Gaussian disorder model (GDM) is probably the most successful and widely used one (see Ref. 2 for a review). In the GDM, a charge carrier hops in an energetically disordered environment, where the energy of localized states is given by a Gaussian distribution with a fixed width. Although there have been some analytical studies [3 6] on this model, many important results, such as a transition from a dispersive to nondispersive transport behavior, have been obtained through numerical Monte Carlo simulations [7 12]. The Monte Carlo simulation on the GDM is relatively easy to implement and has been widely used to understand various problems arising in the charge transport in disordered materials. E-mail: jhyeo@konkuk.ac.kr -1897- These include the effects on the charge transport of the Coulomb interaction among charge carriers [13, 14] and of the morphology of the sample [15]. In order to apply this powerful method to more complicated and realistic situations, it is important to clear any subtle points in the implementation of the method. In this regard, we reexae here a few fundamental aspects of the Monte Carlo simulation of the GDM that have not been received much attention before. One of the key results of the GDM is the relaxation of the energy of charge carriers to an equilibrium energy ɛ, which is responsible for the dispersive-tonondispersive change in the transport behavior [7,8,12]. In this paper, we show that this equilibrium energy crucially depends on the size of the grid points over which the Gaussian energy is distributed. A simple analysis predicts that as a function of temperature T, the equilibrium energy decreases as σ 2 /(k B T ) as the temperature decreases, where σ is the variance of the Gaussian energy distribution and k B is the Boltzmann constant.

-1898- Journal of the Korean Physical Society, Vol. 60, No. 11, June 2012 This trend, however, is not expected to continue all the way down to zero temperature because of the so-called frustration or freezing effect [4, 5, 16]. At very low temperatures, a thermal activation to a higher energy state, which is necessary to achieve an equilibrium, becomes extremely difficult for a charge carrier subject to a shortrange hopping. Therefore, below a certain temperature T c, the equilibrium energy is expected to increase with decreasing temperature, deviating from the σ 2 /(k B T ) behavior [4,5,16]. In this paper, we show that at moderately low temperatures above T c, Monte Carlo simulations of the GDM exhibit another type of deviation of ɛ from the simple 1/T behavior. It is caused by the use of a finite number of grid points. We explicitly show how the temperature dependence of ɛ varies as the size of the system changes in this temperature region. We develop an improved numerical method to perform simulations with systems whose sizes are much bigger than those used in conventional studies [2,9,17]. We find that at low temperatures, one needs a bigger system size than the conventional one to correctly represent the temperature dependence of ɛ in Monte Carlo simulations. In the next section, we briefly review the main ingredients of the GDM and its Monte Carlo simulations. We then discuss the energy relaxation of the charge carriers and its finite size dependence in detail. We conclude in the following section with some discussion. II. GAUSSIAN DISORDER MODEL The Gaussian disorder model (GDM) [2] is a model describing the charge transport in a disordered energetic landscape. In this section, we briefly review key features of the GDM and of the Monte Carlo simulation methods for the model. The model consists of a cubic lattice of N lattice points that represent localized electronic states. Each lattice point is assigned an energy that is extracted from the Gaussian distribution ρ(ɛ) with a standard deviation σ, ρ (ɛ) = 1 σ 2π exp ( ɛ2 2σ 2 ). (1) In many previous works [2,9,17], a lattice with N =70 3 with a periodic boundary condition has been used. One of the main points of this paper is to investigate how physical quantities depend on the number N of the energetic grids over which the Gaussian energies are distributed. A charge carrier can hop from site i to j with the hopping rate ν ij given by the Miller-Abrahams hopping rule [18] [ ( ν ij = ν 0 exp ( 2γ R ij ) 1, exp ɛ )] j ɛ i,(2) k B T where γ is the wave function overlap parameter, which controls the electronic exchange interaction between sites, R ij is the distance between sites i and j, ɛ i is the energy of site i, k B is the Boltzmann constant and T is the temperature. The value of the pre-factor ν 0 depends on the electron interaction mechanism and is assumed to be on the order of the phonon frequency 10 13 s 1. Note that the latter part of Eq. (2) is just the Metropolis algorithm. One can use another algorithm that satisfies the detailed balance condition [19]. Computer simulations on the GDM are based on the so-called kinetic Monte Carlo method [20]. In a kinetic Monte Carlo calculation, one needs to know the normalization factor j i ν ij for the whole configuration space. In usual calculations, however, for computational convenience only a box of finite size M is considered for the calculation of the normalization factor. Conventionally the value of M =7 3 has been frequently used. For the overlap parameter 2γa = 10usedinthe present simulation with the lattice constant a, the hopping rate to sites outside a box of this size can indeed be neglected. The probability of hopping from site i to j is then described by ν ij p ij = M j i ν. (3) ij We randomly select a site j tohopaccordingtothese probabilities. The time taken by this move is evaluated as 1 t ij = M j i ν. (4) ij The simulation is performed by repeating this hopping procedure until the charge carrier reaches the end of the sample. We note that given the overlap parameter 2γa, the system is completely described by the dimensionless inverse temperature βσ, whereβ =1/(k B T ). III. ENERGY RELAXATION In the GDM, the mean energy of the charge carrier ɛ(t) at time t is well known to relax toward the equilibrium energy in the long-time limit, ɛ =lim t ɛ (t). This is responsible for the non-dispersive behavior of the charge transport in this model. In this section, we investigate critically the energy relaxation of the GDM, focusingonthefinite-sizeeffectsinthemontecarlosimulation. The charge carrier starts from some high-energy states and moves downward in energy until it reaches the equilibrium energy. The time at which the energy of the charge carrier reaches ɛ is the relaxation time, which increases very rapidly with decreasing temperature [2]. The equilibrium energy ɛ can easily be calculated as ɛρ (ɛ)exp( βɛ) dɛ ɛ = ρ (ɛ)exp( βɛ) dɛ = βσ2. (5)

Finite-Size Effects in Monte Carlo Simulations of the Gaussian Disorder Model Sunil Kim et al. -1899- In order to achieve an equilibrium state, a charge carrier must be able to be thermally activated to a higher energy state for a subsequent energy relaxation process. At very low temperatures, however, an activated jump to a higher energy state becomes extremely difficult once the charge carrier reaches a sufficiently low-lying state. This frustration effect [16] of the relaxation process results in a freezing of the charge carrier energy ɛ (t) at some value higher than that given in Eq. (5). Thus, the decrease in ɛ with decreasing temperature, as in Eq. (5), does not continue all the way down to zero temperature. Instead, below some temperature T c, ɛ will show an upward curvature even with decreasing temperature [16]. In this paper, we focus on moderately low temperatures with T>T c, where full equilibration of charge carrier energy can be achieved within the simulation times and the expression in Eq. (5) can be applied. We note, however, that this expression is valid only for a system of infinite size. The system studied in a simulation has a finite number of states over which the energies are distributed according to the Gaussian density profile ρ(ɛ). Therefore, there will always be a imum energy ɛ (N) for a given number N of states. The average value of can be estimated from ɛ (N) ɛ (N) ρ(ɛ)dɛ 1 N, (6) or more accurately from averaging over the actual imum energies of the Gaussian energy distributions of size N. Since there is no state available with energy less than ɛ (N) for a system of size N, weexpectthatatverylow temperatures, the equilibrium energy will deviate from the one given by Eq. (5). This is clearly seen in Fig. 1. We can actually evaluate the equilibrium energy for afinitesystembyperforgasimplesamplingintegration of Eq. (5). First, we prepare a set of energy states, {ɛ n }, n =1, 2,...,N, by using a Gaussian random number generator with fixed σ. Then, we evaluate the thermal average as ɛ n = n ɛ np n / n p n,where p n =exp( βɛ n ). By averaging this quantity over a number of different Gaussian distributions, we can obtain ɛ (N), which is the finite-size version of Eq. (5). As can be seen from Fig. 1, the actual simulation results for the equilibrium energy do not follow the infinite-system limit given by Eq. (5), but its finite size version ɛ (N), whichis the solid curve in the main panel of Fig. 1. We can also see that this finite-size equilibrium energy approaches the imum energy ɛ (N) in the zero-temperature limit. In previous Monte Carlo simulations of the GDM, the typical size N of energetic grids over which the Gaussian energies are distributed was N = 70 3 = 343000. As we can see from Fig. 1, the simulated equilibrium energy in this case does not follow Eq. (5) at temperatures below βσ 3. We expect that the finite-size version of the equilibrium energy ɛ (N) will approach the infinite-system result given in Eq. (5) as N. Fig. 1. The filled squares are the equilibrium energy ɛ /σ as a function of the inverse temperature, βσ, obtained from the Monte Carlo simulation of the GDM for energetic grids of N =70 3. The solid curve is the finite-size prediction, ɛ (N), for N =70 3 (see text). The dashed line is the result for an infinite system, as given by Eq. (5), and the horizontal dotted line indicates the average imum energy, ɛ (N), for N =70 3 obtained numerically. The curves in the inset show the finite-size equilibrium energies ɛ (N) for various numbers N. From top to bottom, N =10 3,30 3, and 100 3.Thesymbols are the actual simulation data. The straight line is again the result for the infinite system. Fig. 2. The equilibrium energy of the GDM as a function of the size N of grid points. Boxes, circles and triangles represent three different temperatures, βσ = 3.0, 3.5, and 4.0, respectively. The horizontal lines are infinite-size results. The arrow indicates the size N =70 3 used in previous simulations. This is demonstrated in the inset of Fig. 1. We note that the deviation of the finite-size equilibrium energy ɛ (N) from the infinite-system size limit has also been noticed in Ref. 21. We now present actual Monte Carlo results for the equilibrium energy of the GDM for various values of N. The mean energy ɛ(t) is obtained by averaging the en-

-1900- Journal of the Korean Physical Society, Vol. 60, No. 11, June 2012 ergy of the charge carrier at time t over many trials for a given energetic grid and over different realizations of the Gaussian energy distribution. This quantity decays with time and approaches a constant value ɛ for t greater than the relaxation time. We find that the equilibrium energy ɛ obtained in this way closely follows the finite-size estimate ɛ (N), as shown in Fig. 1. Therefore, in order to evaluate the equilibrium energy correctly at low temperatures βσ 3, one has to use a number N of grid points larger than the conventional 70 3. We perform Monte Carlo simulations on the GDM for a grid size bigger than the one used in conventional simulations. The simulations were performed for size of N up to 640 3. We note that simulating the GDM at such large N is not a trivial task. The fastest way to run the simulation would be to store the hopping rates ν ij for all pairs of sites on the grid. For a box of size 7 3 within which a charge carrier hops at a given time, the required memory size is 7 3 8byte 2.6 Kb. If one consider, for example, 70 3 sites, the required memory is about 70 3 7 3 8byte 0.9 Gb. Therefore, on a contemporary high-end PC with about 8 Gb memory, the biggest grid size one can study in this way will only be about 150 3. On the other hand, if we calculate the hopping rate every time a charge carrier jumps, we find that the computation time is about 10 times longer than in the case where all the hopping rates are stored. In the present work, we use an intermediate method where we store the hopping rates only for the sites two or three steps before the current position of the charge carrier. This costs very little in memory, furthermore, it speeds up the computation time significantly especially in low temperature regions. This is possible because at low temperatures, the charge carrier tends to get trapped in a certain energy configuration so that it spends much time oscillating back and forth between two low energy sites. Figure 2 clearly shows that the finite size effect on the equilibrium energy gradually disappears as the system size is increased. However, the system size N =70 3 conventionally used in the Monte Carlo simulations of the GDM does not give an accurate value of the equilibrium energy at low temperatures below βσ 3. In fact, at low temperatures βσ = 3and3.5, the true equilibrium energy is realized only when the system size is as large as the one used in this paper, i.e., N 640 3. We can also see that a grid size bigger than 10 9 has to be used to correctly describe the equilibrium energy at low temperatures below βσ = 4. IV. DISCUSSION AND CONCLUSION We have investigated in detail some points, which have not been treated in the literature before, in connection with the Monte Carlo simulation of the GDM. Since we are bound to use a finite number of grid points for the Gaussian energy distribution, the temperature dependence of the equilibrium energy will not be that of an infinite-size system, Eq. (5), as commonly assumed. We have calculated a finite-size version of this formula ɛ (N) numerically and have shown that the actual Monte Carlo simulation results follow this finite-size result. We have found that, in order to get rid of any finite-size effects in the equilibrium energy, one needs to consider the size of grid points larger than those used in conventional simulations at low temperatures. In this paper, this has been possible by storing the hopping rates appropriately as the charge carrier hops. As mentioned earlier, all these discussions only apply to moderately low temperatures above T c. At very low temperatures, the relaxation becomes extremely slow [4, 5], and the charge carrier cannot explore the full phase space available to it. Because of this frustration effect, one might say that the charge carrier effectively sees a smaller number of grids. At temperatures below T c, the quasiequilibrium energy will increase with decreasing temperature even for a strictly infinite system. In actual simulations at temperatures below T c, there will also be finite-size effects resulting from the use of a finite number of grids. In this work, the finite-size version of the equilibrium energy ɛ (N) has been calculated by considering only the energetics of localized states, and the actual kinetics of charge carriers is missing, which will be an important factor in the extremely low temperature region. Future numerical works on this quantity at temperatures below T c should show how the frustration effect manifests itself in an actual simulation using a finite number of energetic grids. ACKNOWLEDGMENTS We thank Prof. H. Bässler for useful discussions. This work was supported by the Korean Government (MEST) grant No. 2010-0027660 and by the Seoul R&BD Program (WR090671). S. Kim and J. Yeo were also supported by the WCU program through the KOSEF funded by the MEST (Grant No. R31-2008-000-10057-0). D. Kim acknowledges support by the National Research Foundation (NRF) grant funded by the Korea government (MEST) No. 2011-0017435. REFERENCES [1] See, for example, Charge Transport in Disordered Solids with Applications in Electronics, edited by S. Baranovski (John Wiley & Sons, Ltd., Chichester, 2006). [2] H. Bässler, Phys. Status Solidi B 175, 15 (1993). [3] M. Grünwald, B. Pohlmann, B. Movaghar and D. Wurtz, Philos. Mag. B 49, 341 (1984). [4] B. Movaghar, M. Grünwald, B. Ries and H. 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