Electrically driven light sources are essential in a wide range of applications, from indication and display technologies to high-speed data communication and quantum information processing. Wide-bandgap semiconductors promise to advance solid-state lighting by delivering novel light sources. However, electrical pumping of these devices is still a challenging problem. Many wide-bandgap semiconductor materials, such as SiC, GaN, AlN, ZnS, and Ga2O3, can be easily n-type doped, but their efficient p-type doping is extremely difficult. The lack of holes due to the high activation energy of acceptors greatly limits the performance and practical applicability of wide-bandgap semiconductor devices. Here, we study a novel effect which allows homojunction semiconductor devices, such as p-i-n diodes, to operate well above the limit imposed by doping of the p-type material. Using a rigorous numerical approach, we show that the density of injected holes can exceed the density of holes in the p-type injection layer by up to four orders of magnitude depending on the semiconductor material, dopant, and temperature, which gives the possibility to significantly overcome the doping problem. We present a clear physical explanation of this unexpected feature of wide-bandgap semiconductor p-i-n diodes and closely examine it in 4H-SiC, 3C-SiC, AlN, and ZnS structures. The predicted effect can be exploited to develop bright-light-emitting devices, especially electrically driven nonclassical light sources based on color centers in SiC, AlN, ZnO, and other wide-bandgap semiconductors.