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

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 ξ_e satisfies g(ξ_e) =< f (σ,ξ_e) >= ∫ f (σ,ξ_e) ρ(σ)dσ = 0. (1) 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 ξ_e, over the possible configurations of the system represented by ρ(σ), is zero. This equation allows us to determine the effective parameter ξ_e, 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.