2.6.1 Direct Recombination

The recombination rate of direct recombination rate is given by

\begin{equation} R_{\rm direct}=\gamma (np-n_i^2), \end{equation}
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where \(n_i\) is the intrinsic carrier density, and \(\gamma \) is the direct recombination rate constant.

The rate constant \(\gamma \) can be set via different routes: It can be based on the Langevin expression (\(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} useLangevin}}\)=1), in which case:

\begin{equation} \gamma =\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} preLangevin}}~ \frac{q}{\varepsilon (x)}(\mu _n(x) + \mu _p(x)), \end{equation}
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where the mobilities \(\mu _{n,p}\) and relative dielectric constant \(\varepsilon \) can depend on position. 1 The Langevin prefactor \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} preLangevin}}\) is motivated by the observation that the recombination rate can be (substantially) smaller than the Langevin value in organic solar cells, while the Langevin expression itself serves as an upper limit to the rate in pristine organic semiconductors. Alternatively, the rate constant can be defined directly via \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} k\_ direct}}\), if \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} useLangevin}}\)=0.

If the Onsager-Braun model for geminate recombination is used (see section 2.5), then the direct recombination rate is reduced by a factor of \(1-P(a,T,F)\). [ 13 ]

  1. These parameters can, after all, be different for the different layers in the device. The mobilities, however, can also depend on the position if they depend on the local electric field.