The predictions of the polar catastrophe scenario to explain the occurrence of a metallic interface in heterostructures of the solid solution(LaAlO$_3$)$_{x}$(SrTiO$_3$)$_{1-x}$ (LASTO:x) grown on (001) SrTiO$_3$ were investigated as a function of film thickness and $x$. The films are insulating for the thinnest layers, but above a critical thickness, $t_c$, the interface exhibits a constant finite conductivity which depends in a predictable manner on $x$. It is shown that $t_c$ scales with the strength of the built-in electric field of the polar material, and is immediately understandable in terms of an electronic reconstruction at the nonpolar-polar interface. These results thus conclusively identify the polar-catastrophe model as the intrinsic origin of the doping at this polar oxide interface.
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