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Ordinarily, the two different materials used for a heterojunction must have the same lattice constant (spacing between the atoms). An analogy - imagine pushing together two plastic combs with a slightly different spacing - at regular intervals, you'll see two teeth clump together. In semiconductors, these discontinuities are a kind of "trap", and greatly reduce device performance.
A HEMT where this rule is violated is called a PHEMT or pseudomorphic HEMT. This feat is achieved by using an extremely thin layer of one of the materials - so thin that it simply stretches to fit the other material. This technique allows the construction of transistors with bigger bandgap differences than otherwise possible. This gives them better performance.
To the best of the author's knowledge, PHEMTs and related devices are the fastest transistors available. They can be used to make amplifiers which work at over 200 GHz. Applications are similar to MESFETs - microwave and millimetre wave communications, radar, and radio astronomy.