  The Biermann battery is a promising generation mechanism of a seed magnetic
field in the universe. We propose a new driving mechanism for the Biermann
battery in an inhomogeneous plasma with a beam component. An inhomogeneous
electron flow induced by the return current can make an electron pressure
gradient which is not parallel to the density gradient. As a result, the
magnetic field is generated by the Biermann battery. We demonstrate the new
generation mechanism of the magnetic field by conducting three-fluid plasma
simulations. By this mechanism, the first cosmic rays can generate the seed
magnetic field with astrophysical scales at redshift around 20.
