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Efficiency droop, i.e., a decline in brightness of LEDs at high electrical currents, has limited the performance of all commercially available LEDs. Until now, it has limited the output power of sub-micron LEDs and lasers to nanowatt range. Here we present a fin p-n junction LED pixel that eliminates efficiency droop, allowing LEDs brightness to increase linearly with injected current. With record current densities of 1000 KA/cm2 (100 mA), the LEDs transition to lasing within the fin, with high brightness. Despite a light extraction efficiency of only 15%, these devices exceed the output power of any previous electrically-driven sub-micron LED or laser pixel by 100 to 1000 times, while showing comparable external quantum efficiencies. Modeling suggests that spreading of the electron-hole recombination region in fin LEDs at high injection levels suppresses the non-radiative Auger recombination processes. Further refinement of this design is expected to enable development of a new generation of high brightness electrically addressable LED and laser pixels for macro- and micro-scale applications.