August 13th, 2020
The linear I-V characteristic phenomenologically indicates the source is purely resistive, and maximum power occurs when the load and source impedance are equal. A figure of merit for solar cells that describes how close its I-V characteristic is to the ideal shape is the fill factor FF which is defined as the ratio of P to the maximum power available with the corresponding ideal cell, FF =Pmax / (VOCISC). It ranges from 0 to 1, where 1 indicates an ideal cell. Ideal cells can supply a constant voltage independent on the load resistance up to the maximum current, when the voltage drops quickly to 0. Deviations from the ideal fill factor of 1 are usually due to parasitic resistances, such as shunt and series resistances. Shunt resistances affect behavior in the I-V characteristic close to ISC, while series resistances affect performance close to VOC. For our cell, FF ~ 0.25. We argue that nanotube resistances 1-3 (figure 2) are responsible for this.
FIGURE 4: Optimization strategies for CNSCs. The extracted parameters are presented in table 1. a) I-V characteristics of the cells as indicated. b) Power delivered to the load.
FIGURE 5: Scaling analysis of CNSC performance. The CNSCs characteristics are determined by the metallic and semiconducting carbon nanotube densities, with symbols
corresponding to the cells as in the legend for figure 2. a) VOC « n, Ra. b) Pmax « (nsRq)2
Effectively, it means that the diode in the circuit diagram can be neglected. To estimate the number of nanotubes, we use the measured sheet resistance presented in table 1. Our CNT films are in the percolation limit ,
 . We can therefore use the scaling of sheet resistance with number of CNTs to extract the deposited volume of nanotube dispersion V, via
where VC is the critical volume that determines the onset of conduction . The volume can then be used to extract the surface density of metallic nm and semiconducting nanotubes ns. We assume that the metallic nanotubes dominate the sheet conductance, since their conductance G
is much greater than semiconducting nanotubes Gs. This assumption holds provided the conductance ratio Gm/Gs of metallic to semiconducting nanotubes exceeds the semiconducting to metallic abundance ratio ns/nm. Single-molecule conductance studies of nanotubes indicate a conductance ratio of Gm/Gs ~ 20, ,  which supports our assumption that metallic nanotubes dominate the sheet conductance. We anticipate that for more enriched semiconducting films than studied here, a more detailed analysis will be required that takes into account the nanotube-nanotube contact resistance as well , . The current-generating capacity of our cells is proportional to the number of semiconducting nanotubes ns. Combining both, the open-circuit condition corresponds to an ideal current source (I « ns) connected to CNT1 (RCNT1 « Ro) and the voltage developed across it will be
Voc K nsRo
and our data indeed approximately follows this scaling behavior (figure 5a). The outliers at low VOC are CNT cells where both photoactive and counter electrode are coated with the same composition of carbon nanotubes. Both sides of the cell therefore create a photocurrent in opposite
directions, but the light attenuation in the electrolyte breaks this symmetry and causes a directed current, albeit a smaller one and with a smaller voltage. The enriched cells further tilts the balance in favor of the photoactive side, leading to a VOC that is closer to that expected from the amount of nanotube material deposited on the active side alone.
Our cells have rather large output impedances and cannot maintain constant voltage over a larger range of load impedance. The cell output resistance can be reduced considerably by changing the aspect ratio of the cell, or connecting many cells in parallel. The output voltage can be held constant by a voltage-regulation circuit. However, there are many applications that do not require a low output impedance and would therefore work well with CNSCs, e. g. driving an LCD display or an E-Ink screen.