  It is recently proposed that cosmic rays generate a seed magnetic field in
the early universe. In this paper, we propose another generation mechanism of
magnetic fields by cosmic rays, which is the Biermann battery driven by
resistive heating induced by the streaming of cosmic rays. This mechanism is
dominant in small-scale, low-temperature, and strongly-ionized regions,
compared with other previously proposed mechanisms. Because cosmic rays are
expected to be accelerated after the death of the first stars, this mechanism
can work during structure formation in the early universe. We show that it
makes the seed magnetic field with sufficient strength for the subsequent
dynamo to amplify it to the micro Gauss level in the current galaxies.
