In this work, we fabricated charge-trapping MemTransistors (CTMTs) on a germanium (Ge) substrate with a single-charge-trapping-layer gate-stack or a double-charge-trapping-layer gate-stack. We first constructed the energy band diagram of two gate stacks using transmission electron microscope (TEM) images and by X-ray photoelectron spectroscopy analysis. We deposited Al2O3 as a tunneling layer and a barrier layer using an atomic layer deposition (ALD) system while depositing HfO2 by ALD as the charge-trapping layer whose conduction band offset with respect to Al2O3 is 1.74 eV. Next, we demonstrated the memory characteristics of the CTMTs. By implementing the double-charge-trapping-layer gate-stack on the CTMT, we were able to enlarge the memory windows by 372 mV, improve the retention by 2.7%, and reduce the read disturbance. Furthermore, we demonstrated the synaptic device characteristics of the CTMTs. With the optimization of pulse schemes, we reduced the nonlinearity of potentiation (alpha(p)) and depression (alpha(d)) from 8.62 and -6.01 to 0.71 and 0.01, respectively, enlarged the ON/OFF ratio from 10.2 to 66.2, and increased the recognition accuracy from 24.5% to 82.1% simultaneously. With the implementation of the double-charge-trapping-layer gate-stack, we could further enlarge the ON/OFF ratio to 75.3 and increase the recognition accuracy to 86.5% simultaneously.