In a continuous search for the energy-efficient electronic switches, a great attention is focused on tunnel field-effect transistors (TFETs) demonstrating an abrupt dependence of the source-drain current on the gate voltage. Among all TFETs, those based on one-dimensional (1D) semiconductors exhibit the steepest current switching due to the singular density of states near the band edges, though the current in 1D structures is pretty low. In this paper, we propose a TFET based on 2D graphene bilayer which demonstrates a record steep subthreshold slope enabled by van Hove singularities in the density of states near the edges of conduction and valence bands. Our simulations show the accessibility of 3.5 x 10$^4$ ON/OFF current ratio with 150 mV gate voltage swing, and a maximum subthreshold slope of (20 {mu}V/dec)$^{-1}$ just above the threshold. The high ON-state current of 0.8 mA/{mu}m is enabled by a narrow (~ 0.3 eV) extrinsic band gap, while the smallness of the leakage current is due to an all-electrical doping of the source and drain contacts which suppresses the band-tailing and trap-assisted tunneling.
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