1. A method for producing a nonvolatile memory element having the following steps: a) preparation of a carrier material; b) formation of an auxiliary layer; c) formation of a depression in the auxiliary layer; d) filling of the depression with a first electrically conductive material for forming a first electrode; e) formation of at least one field amplification structure at the first electrode; f) formation of a conductivity state changeover material on the first electrode with the field amplifier structure, after which at least two different conductivity states prevailing in the conductivity state changeover material, between which changeover can be repeatedly effected by the application of predetermined programming voltages; and g) formation of a second electrically conductive electrode on the changeover material.