We establish the time local well-posedness for large data of a solution to two-dimensional drift-diffusion system in the critical Hardy space H(1) (R(2)). This solution is obtained in the Hardy class, for which the natural free energy is well defined from the initial time. The time global well-posedness result for small data is also obtained in a similar class. The crucial parts of the proof are to employ the end-point type of maximal regularity for the homogeneous heat equation and some new bilinear estimates in the Hardy space. (C) 2008 Elsevier Inc. All rights reserved.