# Quantum Dot Solar Cells: Small Beginnings Have Large Impacts
## 2 Timescales of Excited State Processes
### Charge Carrier Transport  
There are several proposed mechanisms for how charges can be transported in nanocrystalline systems; these include: nearest neighbor hopping, variable range hopping (VRH), Efros-Shklovskii VRH (ES-VRH), co-tunneling, and Bloch transport (band-like transport), which are discussed extensively in the literature and reviewed by [PERSON] et al. [30], [PERSON] [33], and [PERSON] et al. [27]. Here we will focus on the nearest neighbor hopping of charge carriers between QDs. We chose to focus on this particular mechanism because it is the dominate transport mechanism for many of the QD solids used in solar cells. Given all the sources of disorder in a film of QDs (energy disorder caused by different sizes of QDs, different spacing and crystal orientation between QDs, variation in impurities in each QD, variation in ligand coverage, etc.) it is unlikely band-like transport or other long-range transport are the dominant energy transfer mechanisms, unless the QDs are extremely strongly coupled by significant necking or, potentially, by semiconducting inorganic ligands [33; 109]. For QD films used in solar cells this strong coupling is usually not the case. Using the Einstein-Smoluchowski relation, we can derive the hopping rate (\(\Gamma_{hop,e+h}=1/\tau_{hop}\)) [110]:  
\[\Gamma_{hop,e+h}=\frac{k_{B}T\cdot\Sigma\mu\cdot NN}{eA^{2}} \tag{4}\]  
where \(k_{B}\) is the Boltzmann constant, \(T\) the temperature, \(\Sigma\mu\) is the sum of the electron and hole mobility, \(NN\) is the number of nearest neighbors (12 in a hexagonal close-packed structure), \(e\) the elementary charge, and \(\Delta\) is the hopping distance. Because the ligands used to grow and disperse the QDs are long, they are exchanged with shorter ligands to improve carrier transport by lowering \(\Delta\)[111]. Using transient absorption techniques, [PERSON] et al. [110] observed fast spectral diffusion of photoexcited carriers as carriers hopped to the largest QDs (i.e., those with the lowest bandgap) that is complete in \(\sim\)2 ps. One design of a QDSC uses this concept as a \"quantum funnel\" where the authors layered different sizes of QDs to tune the energy landscape and drive charges to the contacts [112; 113]. One downside to this is that [PERSON] et al. [114] observed that due to this fast hopping to larger QDs, Auger recombination can be efficient even at low excitation fluence. This is due to multiple carriers congregating in the largest QDs and implies that because energy disorder in QD solids leads to enhanced recombination it can be used for optical gain or fast optical switching [114]. Other groups have since improved the Monte Carlo approach used in these initial studies to analyze the nature of charge transport in QD films [115; 116; 117]. Surface ligands play a strong role as they induce both positional and energetic disorder that lowers the mobility and/or lifetime of photoexcited carriers [71; 118]. As coupling between QDs is increased, the electron and hole wavefunctions delocalize over the QD solid. An Anderson transition occurs as transport goes from thermally activated hopping (due to wavefunction localization caused by electronic disorder) to band-like transport where the mobility increases with decreasing temperature due to reduced phonon scattering [119; 120; 121]. Solar cells exhibiting external quantum efficiencies (EQE) over 100% from CM first employed two different ligand treatments of ethane dithiol (EDT) and hydrazine [79]; without this bilayer structure (discussed more below) CM was not observed in completed devices. In a subsequent study by [PERSON] et al. [80], they also used EDT/hydrazine scheme to show improved extraction with PbSe nanorods (NRs), while PbTe QDs interestingly only required 3-mercaptoethanolpropionic acid (MPA) to exhibit CM with over 120% [81]. [PERSON] et al. [122] showed a strong correlation between the mobility of a QD solid and the CM yield for PbSe QDs by controllably varying the coupling of the QDs with alkyl diamines of different carbon-chain length.