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This is the other part of our series of studies concerning the full quantum hydrodynamic (FQHD) model for semiconductors with non-flat doping profile. In this paper, as a sequel to Hu H, Zhang K (Semiconductor full quantum hydrodynamic model with non-flat doping profile: (I) stability. 2021, submitted), we proceed to rigorously justify the semi-classical limit of both steady states and global-in-time solutions as Planck's constant  ε  tends to zero; the strong convergence rates are also obtained. Our main results are proved by using the elaborated energy estimates and compactness analysis.

Semiconductor full quantum hydrodynamic model with non-flat doping profile: (II) semi-classical limit