We report a GaAs tunnel junction grown by all-solid-state molecular beam epitaxy (MBE), using tellurium (Te) and magnesium (Mg) as n- and p-type dopants, respectively. The growth conditions, including V/III ratio, and growth rate, growth temperature, were optimized. Through these optimizations, Te- and Mg-doped GaAs with high carrier concentrations as well as good mobilities were obtained. A GaAs tunnel junction with a peak current density of 21 A/cm2 was demonstrated.