Macroscopic description of the two-dimensional LaAlO$_3$/SrTiO$_3$ interface

We propose a simple analytical model to explain possible appearance of the metallic conductivity in the two-dimensional (2D) LaAlO$_3$/SrTiO$_3$ interface. Our model considers the interface within a macroscopic approach which is usual to semiconductor heterojunctions and is based on drift-diffusion equations. The solution of these equations allows to obtain the positions of band edges as a function of distances from the interface. We show that for the 2D metallic conductivity to appear at the interface, the constituting substances should have the same type (either electronic or hole) of conductivity; in the opposite case the possible transition to metallic phase has a three-dimensional character. The results of our model calculations are in agreement with available experimental data.