A low heat loss and small contact area electrode structure for a phase change media memory device is disclosed. The memory device includes a composite electrode that includes a dielectric mandrel that is connected with a substrate and having a tapered shape that terminates at a vertex. An electrically conductive material conformally covers the dielectric mandrel and terminates at a tip. A first dielectric layer covers all of the composite electrode except an exposed portion of the composite electrode that is adjacent to the tip. A phase change media is in contact with the exposed portion. The exposed portion is only a small percentage of an overall surface area of the composite electrode so that a contact footprint between the exposed portion and the phase change media is small relative to a surface area of the phase change media and Joule heat transfer from the phase change media into the composite electrode is reduced.