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, wherein the electrically conductive material is deposited in such a way that an adapted depression is produced in a region of the depression; e) formation of at least one field amplification structure at the first electrode, wherein the electrically conductive material is etched back conformally at least as far as a surface of the auxiliary layer by anisotropic etching, and the auxiliary layer is etched back essentially as far as a bottom region of the adapted depression by anisotropic etching; f) formation of a conductivity state changeover material on the first electrode with the at least one 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 conductivity state changeover material.