  Lattice anharmonicity is thought to strongly affect vacancy concentrations in
metals at high temperatures. It is however non-trivial to account for this
effect directly using density functional theory (DFT). Here we develop a deep
neural network potential for aluminum that overcomes the limitations inherent
to DFT, and we use it to obtain accurate anharmonic vacancy formation free
energies as a function of temperature. While confirming the important role of
anharmonicity at high temperatures, the calculation unveils a markedly
nonlinear behavior of the vacancy formation entropy and shows that the vacancy
formation free energy only violates Arrhenius law at temperatures above 600 K,
in contrast with previous DFT calculations.
