  Using density-functional-theory (DFT) calculations with the HSE06 hybrid
functional, we accurately evaluate the critical thickness of LaAlO3 film for
the intrinsic doping in LaAlO3/SrTiO3 (LAO/STO) heterstructures. The calculated
critical thickness of 6 unit-cell (uc) layers suggests to rule out the
intrinsic doping mechanism. We also calculate the density of oxygen vacancies
on the LAO surface at varying LAO thicknesses, preparation oxygen pressures and
temperatures by using the condition of chemical equilibrium and DFT
calculations. We find that once LAO thickness >=3 uc high-density (~ 10^14
cm^-2 ) oxygen vacancies will inevitably exist on the LAO surface of the
LAO/STO heterstructures even though the samples are grown under high oxygen
pressure. The oxygen vacancies are stabilized by releasing the electrostatic
energy in the LAO film.
