[MRS90] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser. *Semiconductor equations*. Springer-Verlag, Vienna, 1990.

[MST05] H. Ma, W. Sun, and T. Tang. Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains. *SIAM journal on numerical analysis*, 43(1):58-75, 2005.

[MU93] P. A. Markowich and A. Unterreiter. Vacuum solutions of a stationary drift-diffusion model. *Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)*, 20(3):371-386, 1993.

[MZ07] H. Ma and T. Zhao. A stabilized Hermite spectral method for second-order differential equations in unbounded domains. *Numerical Methods for Partial Differential Equations: An International Journal*, 23(5):968-983, 2007.

[Nis96] K. Nishihara. Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping. *J. Differential Equations*, 131(2):171-188, 1996.

[PCC89] O. W. Purbo, D. T. Cassidy, and S. H. Chisholm. Numerical model for degenerate and heterostructure semiconductor devices. *Journal of Applied Physics*, 66(10):5078-5082, 1989.

[Pou91] F. Poupaud. Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers. *Asymptot. Anal.*, 4(4):293-317, 1991.

[PS00] F. Poupaud and J. Soler. Parabolic limit and stability of the Vlasov-Fokker-Planck system. *Mathematical Models and Methods in Applied Sciences*, 10(07):1027-1045, 2000.

[Rom07] V. Romano. 2D numerical simulation of the MEP energy-transport model with a finite difference scheme. *J. Comput. Phys.*, 221(2):439-468, 2007.

[SG69] D.L. Scharfetter and H.K. Gummel. Large signal analysis of a silicon Read diode. *IEEE Trans. Elec. Dev.*, 16:64-77, 1969.

[Tza05] A. E. Tzavaras. Relative entropy in hyperbolic relaxation. *Communications in Mathematical Sciences*, 3(2):119-132, 2005.

[Vil09] C. Villani. Hypocoercivity. *Mem. Amer. Math. Soc.*, 202:184, 2009.