At equilibrium, the barrier for electrons to go from metal to semiconductor is higher than the barrier for electrons to go from semiconductor to metal. Shouldn't that mean there are more electrons going from semiconductor to metal and therefore there is a net current flow? I know this can't be true, but I can't seem to reason this out. Where am I thinking wrong?
You are (partially) correct. At the instant the junction is formed, the potentials are as you describe, but then electrons diffuse across the junction, leaving ions behind, which then cause an electric field to build up. This 'bends' the Fermi level to follow the electric field build up. That 'new' electric field acts to prevent further (net) carriers from diffusing across the junction, leading to a equilibrium with a depletion region in the semiconductor, a built-in electric field and a built-In potential across the junction.
