  Controlled breakdown has recently emerged as a highly appealing technique to
fabricate solid-state nanopores for a wide range of biosensing applications.
This technique relies on applying an electric field of approximately 0.6-1 V/nm
across the membrane to induce a current, and eventually, breakdown of the
dielectric. However, a detailed description of how electrical conduction
through the dielectric occurs during controlled breakdown has not yet been
reported. Here, we study electrical conduction and nanopore formation in
SiN$_x$ membranes during controlled breakdown. We show that depending on the
membrane stoichiometry, electrical conduction is limited by either oxidation
reactions that must occur at the membrane-electrolyte interface (Si-rich
SiN$_x$), or electron transport across the dielectric (stoichiometric
Si$_3$N$_4$). We provide several important implications resulting from
understanding this process which will aid in further developing controlled
breakdown in the coming years, particularly for extending this technique to
integrate nanopores with on-chip nanostructures.
