  We derive the extended Ginzburg-Landau (GL) formalism for a clean s-wave
two-band superconductor by employing a systematic expansion of the free-energy
functional and the corresponding matrix gap equation in powers of the small
deviation from the critical temperature tau = 1-T/T_c. The two lowest orders of
this expansion produce the equation for T_c and the GL theory. It is shown that
in agreement with previous studies, the two-band GL theory maps onto the
single-band GL model and thus fails to describe the difference in the spatial
profiles of the two band condensates. We prove that except for some very
special cases, this difference appears already in the leading correction to the
GL theory, which constitutes the extended GL formalism. We derive linear
differential equations that determine the leading corrections to the band order
parameters and magnetic field, discuss the validity of these equations, and
consider examples of an important interplay between the band condensates.
Finally, we present numerical results for the thermodynamic critical magnetic
field and temperature-dependent band gaps (at zero field), which are in a very
good agreement with those obtained from the full BCS approach in a wide
temperature range. To this end, we emphasize the advantages of our extended GL
theory in comparison with the often used two-component GL-like model based on
an unreconstructed two-band generalization of the Gor'kov derivation.
