Controlled breakdown has recently emerged as a highly accessible technique to fabricate solid-state nanopores. However, in its most common form, controlled breakdown creates a single nanopore at an arbitrary location in the membrane. Here, we introduce a new strategy whereby breakdown is performed by applying the electric field between an on-chip electrode and an electrolyte solution in contact with the opposite side of the membrane. We demonstrate two advantages of this method. First, we can independently fabricate multiple nanopores at given positions in the membrane by localising the applied field to the electrode. Second, we can create nanopores that are self-aligned with complementary nanoelectrodes by applying voltages to the on-chip electrodes to locally heat the membrane during controlled breakdown. This new controlled breakdown method provides a path towards the affordable, rapid, and automatable fabrication of arrays of nanopores self-aligned with complementary on-chip nanostructures.