Generation of Magnetic Field on the Accretion Disk around a
  Proto-First-Star
  The generation process of magnetic field around a proto-first-star is
studied. Utilizing the recent numerical result of proto-first-star formation
based upon the radiation hydrodynamics simulations, we assess the magnetic
field strength generated by the radiative force and the Biermann battery
effect. We find that magnetic field of \sim 10^{-9} G is generated on the
surface of the accretion disk around the proto-first-star. The field strength
on the accretion disk is smaller by two orders of magnitude than the critical
value, above which the gravitational fragmentation of the disk is suppressed.
Thus, the generated seed magnetic field hardly affect the dynamics of on-site
first star formation directly, unless efficient amplification process is taken
into consideration. We also find that the generated magnetic field is
continuously blown out from the disk on the outflows to the poles, that are
driven by the thermal pressure of photoheated gas. The strength of the diffused
magnetic field in low density regions is \sim 10^{-14}-10^{-13}G at n_H=10^3
cm^{-3} which could play important roles on the next generation star formation,
as well as the seeds of magnetic field exist in present-day universe.
