We investigate single-photon avalanche diodes with a thick absorption zone leading to a high photon detection probability in the near-infrared spectrum, e.g., to 27.9% at 850 nm. Furthermore, modulation doping for tuning the breakdown voltage in single-photon avalanche diodes is used. Modulation doping allows for reduction of the effective doping in the structure during the design phase without process modifications. We compare a modulation doped version with a single-photon avalanche diode not using this technique. We prove that both versions are operational. The modulation doped version shows a reduced dark count rate and afterpulsing probability at the cost of a reduced photon detection probability.