Understanding the physics of defective amorphous semiconductors such as indium gallium zinc oxide ($a$-IGZO) is of paramount importance for engineering thin-film transistors for large-area electronics, like your television. This work introduces a first-principles approach for studying defects in amorphous semiconductors, rationalizing the nature of these defects and offering a coherent description of $a$-IGZO that explains its experimentally observed electrical instabilities. The methodology offers a basis for insightful device optimization.