  We report a detailed study of the Te vapor annealing effect in
Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ single crystals. Bulk superconductivity can be
gradually induced by annealing with increasing amount of Te, until the molar
ratio of Te to the sample reaches 1 : 10. By further increasing Te molar ratio,
superconducting volume is gradually reduced. Resistivity and Hall effect
measurements manifest that annealing in Te vapor delocalizes the charge
carriers by removing excess Fe. The optimally Te annealed crystal shows
homogeneous critical current densities with a large value of $\sim$ 4.3
$\times$ 10$^5$ A/cm$^2$, which proves that the Te annealing is effective to
induce bulk superconductivity in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$.
