  Origin of the magnetic field ubiquitous in the Universe is studied based on
the Biermann mechanism, which is expected to work in the non-barotropic region.
We perform a series of two-dimensional MHD simulations of the first generation
supernova remnant (SNR) expanding in the inhomogeneous interstellar matter
(ISM) and study the Biermann mechanism working in the interior of the SNR.
Especially, we pay attention to the relaxation process of electron and ion
temperatures via the Coulomb interaction. In the early SNR in which the
electron temperature is much lower than the ion temperature, the Biermann
mechanism is ineffective, since the gradient of electron pressure is small.
Magnetic fields begin to be generated just behind the shock front when the
electron temperature is sufficiently relaxed. Assuming the explosion energy of
10^52 erg, the total magnetic energy generated reaches about 10^26 erg and does
not depend strongly on the parameters of either SNR or ISM. Analytic expression
to estimate the magnetic total energy is presented and it is shown this agrees
well with our numerical results. Finally we evaluate the expected amplitude of
magnetic fields in protogalaxies as ~10^-19 G, which is sufficient for seed
fields of the observed galactic magnetic fields.
