# Quantum Dot Solar Cells: Small Beginnings Have Large Impacts
## 2 Timescales of Excited State Processes
### (Multi)Exciton Dissociation  
Dissociation can be thought of as a special case of charge transport where one half of the exciton (either the electron or hole) has to overcome an additional energy barrier before transferring to another QD, i.e., from the Miller-Abrams rate [131]: \(\Gamma_{dis}=\Gamma_{hop,e+h}\cdot e^{E_{dis}/k_{B}T}\). This additional energy barrier is the exciton binding energy. As discussed by [PERSON] et al. [119] and [PERSON] [132], this energy needed for dissociation, \(E_{dis}\), involves the Coulomb interaction and the cross-polarization energy, expressed as [132]:  
\[E_{dis}=\frac{1.79e^{2}}{4\pi\epsilon_{0}\epsilon_{in}}+\frac{e^{2}}{4\pi\epsilon _{0}R}\frac{\epsilon_{in}-\epsilon_{out}}{\epsilon_{in}\epsilon_{out}} \tag{5}\]  
with \(\epsilon_{0}\) the vacuum permittivity, \(R\) the QD radius, \(e\) the elementary charge and \(\epsilon_{in}\) and \(\epsilon_{out}\) the relative permittivities in the QD and outside the QD, respectively. The physical interpretation of the cross-polarization energy can be understood using the method of image charges. Each charge has a virtual mirror charge outside the QD that can be thought of as a sum of partial induced charges arising from the dielectric nature of the surrounding medium. Here again, the surface chemistry of the QD plays a role in determining the \(\epsilon_{out}\) value, which is a measure of the dielectric polarizability. The surrounding film/media that the exciton dissociates into is effectively a mixture of ligands and other QDs; therefore, \(\epsilon_{out}\) can be calculated given a certain volume fraction, \(f\), via the Bruggeman effective medium approximation taking ligands and QDs into account [119; 133]:  
\[f\frac{\epsilon_{in}-\epsilon_{out}}{\epsilon_{in}+\kappa\epsilon_{out}}=(f-1) \frac{\epsilon_{ligand}-\epsilon_{out}}{\epsilon_{ligand}+\kappa\epsilon_{ out}} \tag{6}\]  
It is an interesting question whether the values for \(\epsilon_{in}\) and \(\epsilon_{ligand}\) should be taken as the static relative permittivity or the high frequency relative permittivity. Intuitively one could consider an exciton a static entity; however, associated with the exciton is an angular frequency as in a classical picture the electron and hole orbit each other. Hence the relevant frequency is determined by the angular frequency of the exciton. Therefore, it is most appropriate to use the higher frequency permittivity if the angular frequency of the exciton is commensurate with that of the optical phonons of the material [114]. For a quantum confined charge carrier, the confinement energy is a kinetic energy and also corresponds to the charge moving around at some frequency in the QD. While the details are beyond the scope of this review, the upshot is that for excitons, and or charges in quantum confined materials in general it is more appropriate to use the high frequency optical dielectric constant. As an example, PbSe QDs with 1,2-ethanediamine ligands \(E_{dis}\) using the low frequency (i.e., static) relative permittivities for the QD and ligands yields around 10 meV, whereas, calculating the \(E_{dis}\) with the relative permittivity at optical frequencies gives around 80 meV [69; 114]. The latter value has been confirmed experimentally [134].