Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T07:48:03.220Z Has data issue: false hasContentIssue false

Numerical Formulation of the Effective Medium Approximation: Illustrative Examples and Application to Organic Semiconductors

Published online by Cambridge University Press:  31 January 2011

Peter Graf
Affiliation:
[email protected], National Renewable Energy Laboratory, Golden, Colorado, United States
Muhammet Kose
Affiliation:
[email protected], North Dakota State University, Department of Chemistry and Molecular Biology, Fargo, North Dakota, United States
Kwiseon Kim
Affiliation:
[email protected], National Renewable Energy Laboratory, Golden, Colorado, United States
Get access

Abstract

The effective medium approximation (EMA) approach to calculating the macroscopic properties of disordered materials from an average of microscopic interactions is logically divided into two parts. First, an expression based on the microscale physics of constituents embedded in the effective medium is derived. Second, a configurational average of some sort over is performed, leading to an equation for an effective parameter, which is then solved. Based on this division, we present a numerical approach to the EMA in which the effective parameter ?e satisfies g(ξe) = <f(σ,ξe)> = ∫ f(σ,ξe)ρ(σ)dσ = 0. Here, σ represents possible configurations of the system, i.e. some variable of the physical formulation with respect to which we have knowledge of both i) microscale interactions and ii) the distribution of constituents; f represents our knowledge of the microscale interactions as a function of σ; and ρ represents our knowledge of the distribution of the configurations/constituents with respect to σ. Then g is the expectation of f. The equation says that the average of the microscale interactions embodied in f, which is a function of the effective parameter ξe, over the possible configurations of the system represented by ρ(σ), is zero. This equation allows us to determine the effective parameter ξe, which, for example, we can compare to experiment, or use in a drift/diffusion type of macroscopic simulation. A key point is that our explicit formulation allows specifying ρ and f numerically (e.g. via Monte Carlo simulation).

We present both simple illustrative examples, as well as a realistic application to hopping transport in organic photovoltaics simulation. Preliminary results from this numerical EMA match experimental data remarkably well.

Type
Research Article
Copyright
Copyright © Materials Research Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Köse, M., Graf, P., Kopidakis, N., Shaheen, S., Kim, K., and Rumbles, G., Chem. Phys. Chem, published online (5 Oct 2009), DOI: 10.1002/cphc.200900386.Google Scholar
2 Bruggeman, D.A.G., Ann. Physik (Leipzig) 24, 636 (1935).Google Scholar
3 Landauer, R., J. Appl. Phys. 23, 779 (1952).Google Scholar
4 Jackson, J.D., Classical Electrodynamics, Wiley (1975).Google Scholar
5 Stroud, D., Superlattice. Microst. 23 (3/4), 567 (1998).Google Scholar
6 Movaghar, B. and Schirmacher, W., J. Phys. C 14, 859880 (1981).Google Scholar
7 Parris, P.E. and Bookout, B.D., Phys. Rev. B 48 (13), 9354 (1993).Google Scholar
8 Böttger, H. and Bryksin, V.V., Phys. Stat. Sol. B 96, 219 (1979).Google Scholar
9 Fishchuk, I.I., Philos. Mag. B 81 (6), 561568 (2001).Google Scholar
10 Fishchuk, I.I., Kadashchuk, A., Devi, L.S., Heremans, P., Bässler, H., and Köhler, A., Phys. Rev. B 78, 045211 (2008).Google Scholar
11 Arkhipov, V.I., Fishchuk, I.I., Kadashchuck, A. and Bässler, H., “Charge Transport in Neat and Doped Random Organic Semiconductors,” Semiconducting Polymers, ed. Hadziioannou, G and Malliaras, G.G. (Wiley, 2007) pp. 275383.Google Scholar
12 Miller, A. and Abrahams, E., Phys. Rev. 120, 745 (1960).Google Scholar
13 Marcus, R.A., J. Chem. Phys. 244, 966 (1956).Google Scholar
14 Köse, M.E., Mitchell, W.J., Kopidakis, N., Chang, C.H., Shaheen, S.E., Kim, K. and Rumbles, G., J. Am. Chem. Soc. 129 (46), 1425714270 (2007).Google Scholar