Master-equation approach to the study of electronic transport in small semiconductor devices

Physical Review B - Condensed Matter and Materials Physics

Volumn 59, Issue 7, 1999, Pages 4901-4917

  Fischetti, M V ^a  

^a IBM T J WATSON RESEARCH CENTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001320219     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.59.4901     Document Type: Article

References (67)

1. L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
2. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962).
3. W. Hänsch and G. D. Mahan, Phys. Rev. B 28, 1902 (1983).
4. A. P. Jauho and J. W. Wilkins, Phys. Rev. B 29, 1919 (1984).
5. S. Sarker, Phys. Rev. B 32, 743 (1985).
6. A. P. Jauho, Phys. Rev. B 32, 2248 (1985).
7. S. Datta, J. Phys.: Condens. Matter 2, 8023 (1990).
8. R. Lake and S. Datta, Phys. Rev. B 45, 6670 (1992).
9. R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, J. Appl. Phys. 81, 7845 (1997).
10. R. Brunetti, C. Jacoboni, and F. Rossi, Phys. Rev. B 39, 10 781 (1989).
11. C. Jacoboni, Semicond. Sci. Technol. 7, B6 (1992);in Quantum Transport in Semiconductors, edited by D. K. Ferry and C. Jacoboni (Plenum, New York, 1992), p. 1.
12. E. Wigner, Phys. Rev. 40, 749 (1932).
13. U. Ravaioli, M. A. Osman, W. Pötz, N. C. Kluksdhal, and D. K. Ferry, Physica B 134, 36 (1985).
14. N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Phys. Rev. B 39, 7720 (1989).
15. W. R. Frensley, Rev. Mod. Phys. 62, 745 (1990).
16. L. Van Hove, Physica (Amsterdam) XXI, 517 (1955).
17. M. V. Fischetti, J. Appl. Phys. 83, 270 (1998).
18. Thanks are due to W. Pötz for having pointed out this possibility based on W. Pötz, J. Math. Phys. 36, 1707 (1995);
19. M. A. Talebian and W. Pötz, Appl. Phys. Lett. 69, 1148 (1996).
20. W. Pauli, Festschrift zum 60. Geburtstage A. Sommerfeld (Hirzel, Leipzig, 1928), p. 30.
21. R. W. Zwanzig, in Quantum Statistical Mechanics, edited by Paul H. E. Meijer (Gordon and Breach, New York, 1966), p. 139.
22. W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 (1957).
23. R. Jancel, Foundations of Classical and Quantum Statistical Mechanics (Pergamon, Braunschweig, 1969).
24. L. S. Schulman, Time’s Arrows and Quantum Measurement (Cambridge University Press, Cambridge, UK, 1997).
25. T. Kuhn and F. Rossi, Phys. Rev. B 46, 7496 (1992).
26. F. Rossi, A. Di Carlo, and P. Lugli, Phys. Rev. Lett. 80, 3348 (1998).
27. H. Johnston and S. Sarkar, J. Phys. A 28, 3071 (1995).
28. J. Schilp, T. Kuhn, and G. Mahler, Phys. Rev. B 50, 5435 (1994).
29. J. Rau and B. Müller, Phys. Rep. 272, 1 (1996).
30. M. Saeki, J. Phys. Soc. Jpn. 55, 1846 (1985).
31. D. Ahn, Phys. Rev. B 50, 8310 (1994).
32. N. Sano and A. Yoshii, J. Appl. Phys. 70, 3127 (1991).
33. P. Bordone, D. Vasileska, and D. K. Ferry, in Hot Carriers in Semiconductors, edited by K. Hess, and U. Ravaioli, J.-P. Leburton (Plenum, New York, 1996), p. 433.
34. R. Brunetti and C. Jacoboni, Phys. Rev. B 57, 1723 (1998).
35. R. Blümel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, Phys. Rev. A 44, 4521 (1991).
36. T. Dittrich, F. Grossman, P. Jung, B. Oelschlägel, and P. Hänggi, Physica A 194, 173 (1993).
37. S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15 932 (1996).
38. F. Rossi (private communication).
39. The appearance of the two-particle density matrix in the equation of motion for the one-particle density matrix is a reflection of the well-known Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, discussed by U. Hohenester and W. Pötz, Phys. Rev. B 56, 13 177 (1997).Also discussed in this reference are the physical effects ignored when invoking the random-phase approximation, which is necessary to truncate the BBGKY hierarchy and obtain a closed equation for the one-particle density matrix.
40. Inserting Eq. (40) into Eq. (39), the second terms on the right-hand side of Eq. (40) cancel under the sum over (Formula presented) in the “diagonal” (Formula presented) case. However, in the general case of an inhomogeneous system considered here, such a cancellation does not occur for (Formula presented) so that these terms give additional contributions to the equation of motion for the off-diagonal elements of the reduced density matrix. See, for example, O. Hess and T. Kuhn, Phys. Rev. A 54, 3347 (1996), for a case in which these contributions matter.
41. G. S. Lent and D. J. Kirkner, J. Appl. Phys. 67, 6353 (1990).
42. W. Pötz, J. Appl. Phys. 66, 2458 (1989).
43. R. K. Mains and G. I. Haddad, University of Michigan report (unpublished);results reported in Ref. 15.
44. Clearly, the case of valence bands degenerate at the expansion point (Formula presented) requires alternative approaches retaining interband effects, such as those by G. Bastard, Wave Mechanics Applied to Semiconductor Microstructures (Halsted Press, New York, 1989);
45. by M. G. Burt, Semicond. Sci. Technol. 2, 460 (1987);
46. or by M. Altarelli, Phys. Rev. B 28, 842 (1983).
47. C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963).
48. J. C. Slater, Phys. Rev. 76, 1592 (1949).
49. J. M. Luttinger, Phys. Rev. 84, 814 (1951).
50. J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).
51. W. Pötz, W. Porod, and D. K. Ferry, Phys. Rev. B 32, 3868 (1985).
52. M. O. Vassell, J. Lee, and H. F. Lockwood, J. Appl. Phys. 54, 5206 (1983).
53. M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 (1988).
54. W. R. Frensley, “Current density operator in systems with nonparabolic, position dependent energy bands,” at the URL: http://www.utdallas.edu/dept/ee/frensley/technical/currden/currden.html
55. Yia-Chung Chang and J. N. Schulman, Phys. Rev. B 25, 3975 (1982).
56. The statement “extrapolated from the closest band” must be intended in the following sense: Two complex dispersions, (Formula presented) and (Formula presented) are computed using the coefficients (Formula presented) for the two bands m and (Formula presented) between which tunneling is considered. For a given energy E, the two values (Formula presented) and (Formula presented) at the given energy are computed and the dispersion corresponding to the smallest value for (Formula presented) is chosen.
57. G. Gilat and L. J. Raubenheimer, Phys. Rev. 144, 390 (1966).
58. M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1966).
59. M. V. Fischetti, N. Sano, S. E. Laux, and K. Natori, IEEE Transaction on Semiconductor Technology Modeling and Simulation (http://www.ieee.org/journal/tcad) at URL http://engine.ieee.org/tcad/journal/accepted/ fischetti-feb97
60. S. Haas, F. Rossi, and T. Kuhn, Phys. Rev. B 53, 12 855 (1996).
61. N. G. Van Kampen, Physica A 194, 543 (1993).
62. A. J. Leggett, in Quantum Implications: Essays in honour of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987).
63. R. Omnès, The Interpretation of Quantum Mechanics (Princeton University Press, Princeton, NJ, 1994).
64. M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993).
65. I. Prigogine, Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962).
66. H. Tsuchiya and T. Miyoshi, J. Appl. Phys. 83, 2574 (1998).
67. L. S. Schulman, A. Ranfagni, and D. Mugnai, Phys. Scr. 49, 536 (1993).