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It is impossible to cover this topics: as a review, see Quantum Transport in Semiconductors, C. Jacoboni, L. Reggiani, and D. K. Ferry, Eds. New York: Plenum, 1992. The quantum mechanical treatment of transport phenomena are studied in K. K. Thomber and R. P. Feynman, “Velocity acquired by an electron in a finite electric field in a polar crystal,” Phys. Rev. B., vol. 1, pp. 4099–4114, 1970; J. R. Barker and D. K. Ferry, “Self-scattering path-variable formation of high field, time-dependent quantum kinetic equation for semiconductor transport,” Phys. Rev. Lett., vol. 42, pp. 1779–1781, 1979; W. Hansch and G. D. Mahan, “Transport for many-particle system,” Phys. Rev. B., vol. 28, pp. 1902–1922, 1983; Y.-C. Chung, D. Z.-Y. Ting, J. Y. Tang, and K. Hess, “Monte Carlo simulation of impact ionization in GaAs including quantum effects,” Appl. Phys. Lett., vol. 42, pp. 76–78, 1983; G. D. Mahan, “Electron transport in solids,” Phys. Rep., vol. 110, pp. 321–331, 1984; A.-P. Jauho and J. W. Wilkins, “Theory of high-electric - field quantum transport for electron-reasonant impurity systems,” Phys. Rev. B., vol. 29, pp. 1919–1938, 1984; M. V. Fischetti and D. J. DiMaria, “Quantum Monte Carlo simulation of high-field electron transport: an application to silicon dioxide,” Phys. Rev. Lett., 55, pp. 2475–2478, 1985; R. Bertoncini, A. M. Kriman, and D. K. Ferry, “Airy coordinate technique for nonequilibrium Green’s function approach to high-field quantum transport,” Phys. Rev. B., vol.41, pp. 1390–1400, 1990.
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