Long-range electronic reconstruction to a $d_{xz,yz}$-dominated Fermi surface below the LaAlO$_3$/SrTiO$_3$ interface

Low dimensionality, broken symmetry and easily-modulated carrier concentrations provoke novel electronic phase emergence at oxide interfaces. However, the spatial extent of such reconstructions - i.e. the interfacial "depth" - remains unclear. Examining LaAlO$_3$/SrTiO$_3$ heterostructures at previously unexplored carrier densities $n_{2D}\geq6.9\times10^{14}$ cm$^{-2}$, we observe a Shubnikov-de Haas effect for small in-plane fields, characteristic of an anisotropic 3D Fermi surface with preferential $d_{xz,yz}$ orbital occupancy extending over at least 100~nm perpendicular to the interface. Quantum oscillations from the 3D Fermi surface of bulk doped SrTiO$_3$ emerge simultaneously at higher $n_{2D}$. We distinguish three areas in doped perovskite heterostructures: narrow ($<20$ nm) 2D interfaces housing superconductivity and/or other emergent phases, electronically isotropic regions far ($>120$ nm) from the interface and new intermediate zones where interfacial proximity renormalises the electronic structure relative to the bulk.