The present invention relates to a programmable semiconductor device, preferably a FinFET or tri-gate structure, that contains a first contact element, a second contact element, and at least one fin-shaped fusible link region coupled between the first and second contact elements. The second contact element is laterally spaced apart from the first contact element, and the fin-shaped fusible link region has a vertically notched section. A programming current flowing through the fin-shaped fusible link region causes either significant resistance increase or formation of an electric discontinuity in the vertically notched section. Alternatively, the vertically notched section may contain a dielectric material, and application of a programming voltage between a gate electrode overlaying the vertically notched section and one of the contact elements breaks down the dielectric material and allows current flow between the gate electrode and the fin-shaped fusible link region.