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An efficient way to reduce the power consumption of electronic devices is to lower the supply voltage, but this voltage is restricted by the thermionic limit of subthreshold swing (SS), 60 millivolts per decade, in field-effect transistors (FETs). We show that a graphene Dirac source (DS) with a much narrower electron density distribution around the Fermi level than that of conventional FETs can lower SS. A DS-FET with a carbon nanotube channel provided an average SS of 40 millivolts per decade over four decades of current at room temperature and high device current I60 of up to 40 microamperes per micrometer at 60 millivolts per decade. When compared with state-of-the-art silicon 14-nanometer node FETs, a similar on-state current Ion is realized but at a much lower supply voltage of 0.5 volts (versus 0.7 volts for silicon) and a much steeper SS below 35 millivolts per decade in the off-state.