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For the first time, the tradeoffs between higher mobility (smaller bandgap) channel and lower band-to-band tunneling (BTBT) leakage have been investigated. In particular, through detailed experiments and simulations, the transport and leakage in ultrathin (UT) strained germanium (Ge) MOSFETs on bulk and silicon-on-insulator (SOI) have been examined. In the case of strained Ge MOSFETs on bulk Si, the resulting optimal structure obtained was a UT low-defect 2-nm fully strained Ge epi channel on relaxed Si, with a 4-nm Si cap layer. The fabricated device shows very high mobility enhancements >3.5× over bulk Si devices, 2× mobility enhancement and >10× BTBT reduction over 4-nm strained Ge, and surface channel 50% strained SiGe devices. Strained SiGe MOSFETs having UT (TGe<3 nm) very high Ge fraction (∼ 80%) channel and Si cap (TSi cap<3 nm) have also been successfully fabricated on thin relaxed SOI substrates (TSOI=9 nm). The tradeoffs in obtaining a high-mobility (smaller bandgap) channel with low tunneling leakage on UT-SOI have been investigated in detail. The fabricated device shows very high mobility enhancements of >4× over bulk Si devices, >2.5× over strained silicon directly on insulator (SSDOI; strained to 20% relaxed SiGe) devices, and >1.5× over 60% strained SiGe (on relaxed bulk Si) devices.