Semiconducting carbon nanotubes (CNTs) are promising candidates for light-harvesting applications. They exhibit carrier multiplication (CM), and in theory, CNT photodiodes could produce more than one harvested electron per absorbed photon. Such a device could surpass the performance of conventional photovoltaic technology. However, in 2013, the record quantum yield (QY) of a CNT photodiode was only 1 - 5% (the ratio of electrons harvested to photons absorbed). In pursuit of higher QY we fabricated individual suspended carbon nanotube photodiodes. We investigated the unique electronic transport characteristics of these devices by varying the work function of the metal contacts. The resulting electronic transport model gives clues how to improve QY. Using spectrally-resolved scanning photocurrent microscopy we have characterized QY and demonstrated room-temperature QY of up to 70%. Armed with an understanding of the device physics, we outline our recent attempts to achieve QY > 100%.