  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.
