Propagation of waves in nano-sized GaAs semiconductor induced by electron beam are investigated. A dispersion relation is derived by using quantum hydrodynamics equations including the electrons and holes quantum recoil effects, exchange-correlation potentials, and degenerate pressures. It is found that the propagating modes are instable and strongly depend on the electron beam parameters, as well as the quantum recoil effects and degenerate pressures. The instability region shrinks with the increase of the semiconductor number density. The instability arises because of the energetic electron beam produces electron-hole pairs, which do not keep in phase with the electrostatic potential arising from the pair plasma.

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