A flip chip interconnect has a tapering interconnect structure, and the area of contact of the interconnect structure with the site on the substrate metallization is less than the area of contact of the interconnect structure with the die pad. Also, a bond-on-lead or bond-on-narrow pad or bond on a small area of a contact pad interconnection includes such tapering flip chip interconnects. Also, methods for making the interconnect structure include providing a die having interconnect pads, providing a substrate having interconnect sites on a patterned conductive layer, providing a bump on a die pad, providing a fusible electrically conductive material either at the interconnect site or on the bump, mating the bump to the interconnect site, and heating to melt the fusible material.