Annular, linear, and point contact structures are described which exhibit a greatly reduced susceptibility to process deviations caused by lithographic and deposition variations than does a conventional circular contact plug. In one embodiment, a standard conductive material such as carbon or titanium nitride is used to form the contact. In an alternative embodiment, a memory material itself is used to form the contact. These contact structures may be made by various processes, including chemical mechanical planarization and facet etching.