A positron production experiment was performed by irradiating an ultra-intense picosecond laser on solid tantalum targets. Quasi-monoenergetic positron beams were obtained owing to the sheath field on the back of the target. The experiment shows that the peak energy of the positron spectrum has a linear relation with the reciprocal of the target diameter. A simple analytical model of the sheath field was constructed to explain the experimental data, which predicts the positron peak energy in terms of the target diameter and hot electron parameters. Based on the field model, Monte Carlo simulations were conducted to treat the positron production and acceleration self-consistently. The simulated spectra are in good agreement with most experiment results. The disagreement of the 1 mm diameter data reveals that the hot electron propagation along the target flank surface plays an important role in the sheath field set up. Several aspects involved in the positron acceleration are discussed.