Free-to-free carrier recombination

Free-carrier-to-free-carrier (bi-molecular) recombination is included as an optional loss pathway in the model. This process describes the direct recombination of free electrons and free holes without the involvement of trap states. The basic rate equation is given by:

\[R_{\mathrm{free}} = k_{r} \big(n_{f}p_{f} - n_{0}p_{0}\big) \label{equ:freetofree}\]

Here, \(k_{r}\) is the recombination rate constant, \(n_{f}\) and \(p_{f}\) are the free electron and hole densities, and \(n_{0}\) and \(p_{0}\) are the corresponding equilibrium carrier densities. This formulation captures the net recombination rate after accounting for equilibrium carrier populations.

Empirical λ-power recombination model

In some cases, particularly when fitting empirical rate equations to experimental data, it may be useful to generalize the recombination law by introducing a power-law dependence. This is implemented in the form:

\[R_{\mathrm{free}} = k_{r} \big(n_{f}p_{f} - n_{0}p_{0}\big)^{\tfrac{\lambda+1}{2}} \label{equ:freetofree_lambda}\]

The exponent \(\lambda\) acts as an adjustable parameter that modifies the effective recombination order, allowing the model to reproduce experimental trends such as apparent ideality factors larger than unity. This option can be enabled via the setting Enable \(\lambda\) power in free-to-free recombination in the Configure window of the Electrical parameter editor. This is usually turned off.

Langevin recombination

Langevin recombination is a special case of the general free-to-free (bi-molecular) recombination law, obtained by setting the recombination constant \(k_{r}\) to the Langevin prefactor. In this case, the recombination rate takes the form:

\[R_{\mathrm{Langevin}} = \gamma \big(n p - n_{0} p_{0}\big) \label{equ:langevin}\]

with the Langevin prefactor

\[\gamma = \frac{q}{\varepsilon}\,(\mu_{n} + \mu_{p}) \label{equ:langevin_prefactor}\]

where \(q\) is the elementary charge, \(\varepsilon\) the dielectric permittivity, and \(\mu_{n}, \mu_{p}\) the electron and hole mobilities. This formulation assumes that all carriers remain free and mobile, so recombination occurs as soon as electrons and holes encounter one another under Coulomb attraction.

For real disordered organic semiconductors, the Langevin picture is too simplistic. First, if \(\mu_n\) and \(\mu_p\) are treated as constants and show no explicit carrier-density dependence; in practice, mobilities vary strongly with carrier density in organics (hopping in a disordered DOS), so the recombination rate is intrinsically density-dependent and Langevin misses this unless \(\mu(n,p)\) is modelled. Second, if \(n\) and \(p\) are taken as “free carriers” obtained from Maxwell–Boltzmann statistics, the quasi-Fermi-level–carrier-density relationship is incorrect for Gaussian/exponential DOS with traps, leading to the wrong free/trapped partitioning and an incorrect effective prefactor. Finally, because explicit trap states are absent, trapped charge is not represented and the associated electrostatics (space-charge, screening) and trap-assisted channels are omitted—one reason measured rates are often orders of magnitude below the Langevin limit.

For these reasons, Langevin recombination model should be regarded as a useful benchmark — but not as a realistic description of organic solar cells. It is included here for completeness; accurate modelling requires explicit trap-assisted recombination processes.