A nanopore-containing substrate includes a substrate, a membrane on the substrate, and at least one nanoscale electronic element disposed on or embedded in the membrane. The membrane defines at least one nanopore. The nanoscale electronic element is aligned with one of the nanopores such that a shortest distance between an edge of the nanoscale electronic element and the edge of the nanopore is less than 50 nm. The nanopores may be formed by etching through a dielectric layer using a solution while applying a voltage to the nanoscale electronic element relative to the solution. The nanopore-containing substrate can be used to detect or sequence a biopolymer, such as a nucleic acid. The nanopore-containing substrate may be used with a biopolymer detection and/or sequencing system.