We present a new generation of planar Ge-on-Si single-photon avalanche diode (SPAD) detectors for picosecond resolution timing applications in the wavelength region of 1300-1550 nm. This planar geometry has enabled a significant step-change in performance of this class of SPAD, allowing large excess biases and resulting in a single-photon detection efficiency of 38 % at 125 K at a detection wavelength of 1310 nm. This leads to a record low noise equivalent power for Ge-on-Si SPADs of 2 × 10-16 WHz-1/2, a fifty-fold improvement over the best result for a mesa geometry SPAD. For the first time, InGaAs/InP and Ge-on-Si SPADs have been compared under similar operating conditions, with the Ge-on-Si SPADs exhibiting considerably lower afterpulsing effects, indicating the potential for high count rate operation. These results, utilising the inexpensive Ge-on-Si platform, provide a route towards the development of large arrays of high efficiency, high count rate Ge-on-Si SPADs for use in eye-safe automotive LIDAR and future quantum technology applications.