  The magnetorotational instability (MRI) drives vigorous turbulence in a
region of protoplanetary disks where the ionization fraction is sufficiently
high. It has recently been shown that the electric field induced by the MRI can
heat up electrons and thereby affect the ionization balance in the gas. In
particular, in a disk where abundant dust grains are present, the electron
heating causes a reduction of the electron abundance, thereby preventing
further growth of the MRI. By using the nonlinear Ohm's law that takes into
account electron heating, we investigate where in protoplanetary disks this
negative feedback between the MRI and ionization chemistry becomes important.
We find that the "e-heating zone," the region where the electron heating limits
the saturation of the MRI, extends out up to 80 AU in the minimum-mass solar
nebula with abundant submicron-sized grains. This region is considerably larger
than the conventional dead zone whose radial extent is $\sim20$ AU in the same
disk model. Scaling arguments show that the MRI turbulence in the e-heating
zone should have a significantly lower saturation level. Submicron-sized grains
in the e-heating zone are so negatively charged that their collisional growth
is unlikely to occur. Our present model neglects ambipolar and Hall diffusion,
but our estimate shows that ambipolar diffusion would also affect the MRI in
the e-heating zone.
