Modelling excitons/geminate recombination - organics only
Why you should not model excitons
OghmaNano includes models to handle excited states: modules to simulate excitons and geminate recombination for OPV (e.g. Onsager–Braun for dissociation efficiency), and more detailed excited-state models for OLEDs that track singlets and triplets. However, for OPV device simulations you generally do not need to model excitons explicitly. There have been a few models proposed in the litrature to calculate the number of geminate pairs which get converted to free charge carriers — the Onsager–Braun model, for example, will give you the exciton dissociation efficiency. There are other models which will enable you to calculate the distribution of excitons in a device as a function of position.
However, these models generally require parameters that are rarely known reliably for a given system and are hard to measure (e.g. exciton lifetime, diffusion length, dissociation rate), often needing specialist experiments (TRPL, pump–probe, PL-quenching stacks, bias-dependent EQE/EL) and yielding values that depend on morphology and processing. So although it’s possible (and interesting) to simulate geminate recombination, one is usually better off simply introducing a photon efficiency factor \(\eta_{photon}\). This number ranges between 0.0 and 1.0 and is multiplied by the number of photons absorbed at any point in the device to account for geminate recombination losses.
\[ G = G_{abs}\cdot \eta_{photon} \]
where \(G\) is the charge-carrier generation rate in \(m^{-3}s^{-1}\) in equations [eq:contn] and [eq:contp].
This factor can be obtained to a reasonable degree by comparing the difference between the simulated and experimental \(J_{sc}\). This parameter can be set in the configuration section of the optical simulation window. Therefore, in most cases you should not be modelling excitons explicitly, but rather using the ’photon efficiency factor’. If you still really want to model excitons, read on.
How to model excitons (if you really need to)
If you have read section 13.1 and still want to model excitons, this section explains how to do it in OghmaNano. The exciton solver sits between the optical and electrical models. When the exciton model is off, carrier generation is taken directly from optics via the photon-efficiency shortcut, [eq:contn]: \(G = G_{abs}\,\eta_{\mathrm{photon}}\). When the exciton model is on, optical absorption feeds an exciton diffusion–reaction equation and the electronic generation term comes from exciton dissociation.
\[ \frac{\partial X}{\partial t} = \nabla \!\cdot \!\big(D\,\nabla X\big) + G_{\mathrm{optical}} - k_{\mathrm{dis}}\,X - k_{\mathrm{FRET}}\,X - k_{\mathrm{PL}}\,X - \alpha\,X^{2} \]
Here \(X(\mathbf{r},t)\) is the exciton density (m\(^{-3}\)); \(D\) is the exciton diffusion coefficient (m\(^2\)s\(^{-1}\)); \(G_{\mathrm{optical}}\) is the local exciton generation rate from the optical model (proportional to absorbed photons); \(k_{\mathrm{dis}}\) is the dissociation rate to free charges; \(k_{\mathrm{FRET}}\) is the Förster energy-transfer rate; \(k_{\mathrm{PL}}\) is the radiative decay rate; and \(\alpha\) is the exciton–exciton annihilation coefficient (m\(^3\)s\(^{-1}\)). When the exciton model is enabled, the electronic drift–diffusion generation term is taken as \(G = k_{\mathrm{dis}}\,X\) (volumetric), with optional restrictions to interfacial regions if configured.
The diffusion coefficient is commonly specified via the diffusion length \(L\) and lifetime \(\tau\):
\[ D \;=\; \frac{L^{2}}{\tau} \]
Typical boundary conditions are either quenching (\(X=0\)) at electrodes or quenching layers, and/or no-flux (\(\mathbf{n}\!\cdot\!\nabla X=0\)) at blocking interfaces; interfacial quenching can also be represented with a surface rate constant. Units: \(D\) (m\(^2\)s\(^{-1}\)), \(L\) (m), \(\tau\) (s), rates \(k\) (s\(^{-1}\)).
Practical note: the parameters \(L\), \(\tau\), \(k_{\mathrm{dis}}(E)\), \(k_{\mathrm{FRET}}\), and \(\alpha\) are often hard to determine and can be morphology-dependent. Unless your study targets exciton transport/kinetics explicitly, the simpler \(G = G_{abs}\,\eta_{\mathrm{photon}}\) route is usually more robust for OPV device simulations.