Invented by Herbert Kroemer at the Central Bureau of Telecommunications Technology of the German Postal Service, in 1953, it continues to influence the design of modern high-speed bipolar junction transistors.
Early drift transistors were made by diffusing the base dopant in a way that caused a higher doping concentration near the emitter reducing towards the collector.
[1]: 307 This graded base happens automatically with the double diffused planar transistor (so they aren't usually called drift transistors).
[2]: 469 Another way to speed the base transit time of this type of transistor is to vary the band gap across the base, e.g. in the SiGe [epitaxial base] BJT the base of Si1−ηGeη can be grown with η approx 0.2 by the collector and reducing to 0 near the emitter (keeping the dopant concentration constant).
[1]: 307 Germanium diffused junction transistors were used by IBM in their Saturated Drift Transistor Resistor Logic (SDTRL), used in the IBM 1620.