One illustrative memory cell disclosed herein includes at least one layer of insulating material having a first opening and an internal sidewall spacer positioned within the first opening, wherein the internal sidewall spacer includes a spacer opening. The memory cell also includes a bottom electrode positioned within the spacer opening, a memory state material positioned above an upper surface of the bottom electrode and above an upper surface of the internal sidewall spacer, and a top electrode positioned above the memory state material.