# Quantum Dot Solar Cells: Small Beginnings Have Large Impacts
## 2 Timescales of Excited State Processes
### Absorption and Multiple Exciton Generation
[PERSON] and [PERSON] [63] rewrite (1) and (2) as the absorption coefficient \(\mu\) for a dispersion of QDs with volume fraction \(f\) as:  
\[\mu=\frac{n_{1}}{n_{s}}|f(\omega)|^{2}f\alpha \tag{3}\]  
Figure 2: (a) A plot between the electric field intensity (E\({}^{2}\)) and distance x from the center of the nano-shell (x-axis) shows an increase in the intensity of the electric field only near the nano-shell surface. Inset: a cross-sectional transmission electron microscopic snapshot (TEM—FEI Titan 80–300, Hillsboro, OR, USA) of the incorporation of the nano-shell inside the PbS layer. Adapted with permission from reference [45], copyright 2013 American Chemical Society. (b) A schematic (top) and a scanning electron microscope (SEM—FEI, Hillsboro, Oregon, USA) image (bottom) showing the nanostructured QDSCs with several internal reflections of the incident light. Adapted with permission from reference [62], copyright 2015 American Chemical Society. This clearly shows the high sensitivity of the absorption coefficient to the surrounding environment (\(n_{1}\)/\(n_{\mathrm{s}}\)), which includes the surface ligation of the QDs. For instance, titrating \(p\)-methylbenzenethiolate into a solution of PbS QDs leads to a broadband increase in absorption and a bathochromic shift of the first exciton transition peak [64]. This result is surprising given the fact that \(p\)-methylbenzenethiolate shows negligible absorption in the spectral region of interest and, more interestingly, calculating the change in \(f(\omega)\) defined by the Maxwell-Garnet effective medium theory does not predict the \(\sim\)340% increase in the molar absorption coefficient at 400 nm indicating that the synergistic effect of ligand/QD system is greater than the simple sum of the two. Mixing of the molecular orbitals of the ligand/QD system, calculated using density functional theory, shows an increase in the density of states near the band edge, increasing the number of allowed transitions, thus enhancing the absorption of light. Ligands, such as these, affecting the QD electronic structure have been termed 'non-innocent' ligands and provide a tool for microscopic control of macroscopic properties [65; 66; 67]. While these specific ligand complexes have not yet been applied to devices, they could hold promise for increasing the overall absorption of QDSCs. Further ordering of the QDs into a superlattice has also shown up to 5 fold enhancement of the absorption and represents a potential method to improving device properties as we discuss in the final section [68; 69]. Other strategies include hybrid structures between organic conductors and QDs, as discussed elsewhere [70; 71; 72; 73]. In addition to how strongly a QD absorbs light, the size of the QD determines the photon energy that can be absorbed to create an exciton. If the absorbed photon has energy greater than twice the bandgap (i.e., \(\hbar\omega>2E_{g}\)), then the excitons have multiple paths to decay to their ground state energy levels, each of which occurs over a characteristic timescale. The possibility of an Auger pathway (green arrows, pathway 2a in Figure 3) in QD semiconductors also opens doors to the possibility of an inverse Auger pathway as shown in pathway 1a of Figure 3. Auger processes are three-body events where one carrier interacts with another carrier of opposite charge and transfers energy to a third carrier. The photoexcitation by a high energy photon (\(\hbar\upsilon\) at least \(>2E_{g}\)) can cause the high energy electron-hole pair to split into two or more electron-hole pairs of lesser energy. The first pathway we will discuss is multiple exciton generation (MEG) also referred to as carrier multiplication (CM) in cases where the excitations yield free charges. This phenomenon was observed in QDs at much lower bandgap multiples as compared to bulk semiconductors [74; 75; 76; 77; 78].