Photovoltaic Effect in Phthalocyanine-Based Organic Solar Cells: 2. Trapping of Molecular Excitons by Impurities

Abstract

Trapping of molecular excitons (MEs) by ionizing impurities is a fast initial step of the photovoltaic effect in organic solar cells. Values for the ME trapping rate constant at a fixed impurity concentration N_I have been found in terms of the model of diffusion-controlled bimolecular reactions. The model takes into account two pathways of quasi-steady-state ME trapping, contact and bulk capture (caused by dipole–dipole interaction with an impurity), which are determined by the ME diffusion length l_S, the minimum radius of approach of partners a₀, and the Förster radius R_S. The trapping quantum yield at R_S > a₀ is Y_S = (1 + b)⁻¹, where b⁻¹ ≈ 4π(2l_DR_S)^3/2N_I. Possible doping conditions when Y_S ~ 1 at N_I ~ 10¹⁸ cm⁻³ are discussed.

APPENDIX

Analysis of Equation (6)

Equation (6) and boundary conditions (11) contain a set of four characteristic lengths a₀, l_D, R_S, and L. To simplify the analysis, one should find dimensionless parameters that are combinations of these lengths, i.e. go to new coordinates. In the low concentration limit for surface trapping (L⁻¹ = 0, R_S = 0), transformation of Eq. (10) leads to radial distribution (12) and trapping current (15) depending on one dimensionless parameter a₀/l_D. For a finite cell size, the solution of Eq. (21) is determined by two parameters: a₀/l_D and L/l_D entering the boundary conditions. For bulk trapping, transformation (16) converts Eq. (6) into inhomogeneous equation (17) with one parameter q. The solution to Eq. (17) in the interval z_L ≤ z ≤ z₀ can be represented as a superposition of fundamental solutions of the homogeneous equation

$${{{{z}^{2}}{{d}^{2}}\Phi } \mathord{\left/ {\vphantom {{{{z}^{2}}{{d}^{2}}\Phi } {d{{z}^{2}}}}} \right. \kern-0em} {d{{z}^{2}}}} + {{zd\Phi } \mathord{\left/ {\vphantom {{zd\Phi } {dz}}} \right. \kern-0em} {dz}} - ({{{{z}^{2}} + 1} \mathord{\left/ {\vphantom {{{{z}^{2}} + 1} {16}}} \right. \kern-0em} {16}})y = 0$$

(A1)

as

$$\Phi (z) = A{{K}_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}}(z) + B{{I}_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}}(z),$$

(A2)

where I_1/4(z) and K_1/4(z) are the modified Bessel functions of the order 1/4, and the particular solution of the homogeneous equation in the neighborhood of z_L, which has the form

$$\begin{gathered} \Phi (z) = {{\Phi }_{0}} + F(z),\,\,\,\,F(z) \\ = {{\left( {z{\text{/}}\alpha } \right)}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 4}} \right. \kern-0em} 4}}}}{{\left( {{{1 + z} \mathord{\left/ {\vphantom {{1 + z} {16q}}} \right. \kern-0em} {16q}} + {{{{z}^{3}}} \mathord{\left/ {\vphantom {{{{z}^{3}}} q}} \right. \kern-0em} q}} \right)}^{{ - 1}}}. \\ \end{gathered} $$

(A3)

The coefficients in Eq. (A2) determine the boundary conditions

$$\begin{gathered} {{K}_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}}({{z}_{0}})A + {{I}_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}}({{z}_{0}})B = - F({{z}_{0}}), \\ K_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}^{'}({{z}_{L}})A + I_{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}^{'}({{z}_{L}})B = - F{\kern 1pt} '({{z}_{L}}), \\ \end{gathered} $$

(A4)

where the prime denotes the derivative with respect to z. The quantum yield of surface and bulk trapping (18) is found numerically from Eqs. (A3) and (A4).