Theory of electron transport in small semiconductor devices using the Pauli master equation

Journal of Applied Physics

Volumn 83, Issue 1, 1998, Pages 270-291

  Fischetti, M V ^a  

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

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; ELECTRON TUNNELING; MONTE CARLO METHODS; SEMICONDUCTOR DEVICES; TUNNEL DIODES;

BOLTZMANN EQUATION; PAULI MASTER EQUATION; SCHRODINGER EQUATION; WAVE FUNCTIONS;

ELECTRON TRANSPORT PROPERTIES;

EID: 0031693941     PISSN: 00218979     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.367149     Document Type: Article

References (100)

1. W. Pauli, Festschrift zum 60, Geburtstage A. Sommerfeld (Hirzel, Leipzig, 1928), p. 30.
2. W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 (1957).
3. L. Van Hove, Physica XXI, 517 (1995); see also the account given by R. W. Zwanzig, in Quantum Statistical Mechanics, edited by P. H. E. Meijer (Gordon and Breach, New York, 1966), p. 139.
4. L. Van Hove, Physica XXI, 517 (1995); see also the account given by R. W. Zwanzig, in Quantum Statistical Mechanics, edited by P. H. E. Meijer (Gordon and Breach, New York, 1966), p. 139.
5. The word "localized" is used here and in the following simply to indicated that the electron is described by a wave packet of finite extent, without any connection whatsoever to the more usually accepted meaning of disorder-induced "localization."
6. W. R. Frenslay, Rev. Mod. Phys. 62, 745 (1990).
7. R. Brunetti, C. Jacoboni, and F. Rossi, Phys. Rev. B 39, 10 781 (1989).
8. C. Jacoboni, Semicond. Sci. Technol. 7, B6 (1992); in Quantum Transport in Semiconductors, edited by D. K. Ferry and C. Jacoboni (Plenum, New York, 1991), p. 1.
9. C. Jacoboni, Semicond. Sci. Technol. 7, B6 (1992); in Quantum Transport in Semiconductors, edited by D. K. Ferry and C. Jacoboni (Plenum, New York, 1991), p. 1.
10. E. Wigner, Phys. Rev. 40, 749 (1932).
11. L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
12. L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
13. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962).
14. See, for instance, M. Charbonneau, K. M. van Vliet, and P. Vasilopoulos, J. Math. Phys. 23, 318 (1982), and references therein.
15. U. Ravaioli, M. A. Osman, W. Pötz, N. C. Kluksdhal, and D. K. Ferry, Physica B+C 134, 36 (1985).
16. W. R. Frensley, Phys. Rev. Lett. 57, 2853 (1986); 60, 1589(E) (1986).
17. W. R. Frensley, Phys. Rev. Lett. 57, 2853 (1986); 60, 1589(E) (1986).
18. N. C. Kluksdahl, A. M. Kriman, and D. K. Ferry, Superlattices Microstruct. 4, 127 (1988).
19. N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Superlatices Microstruct. 5, 397 (1989).
20. N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, Phys. Rev. B 39, 7720 (1989).
21. K. L. Jensen and F. A. Buot, J. Appl. Phys. 65, 5248 (1989).
22. W. Hänsch and G. D. Mahan, Phys. Rev. B 28, 1902 (1983).
23. A. P. Jauho and J. W. Wilkins, Phys. Rev. B 29, 1919 (1984).
24. S. Sarker, Phys. Rev. B 32, 743 (1985).
25. A. P. Jauho, Phys. Rev. B 32, 2248 (1985).
26. L. Reggiani, P. Lugli, and A. P. Jauho, Phys. Rev. B 36, 6602 (1987).
27. S. Datta, J. Phys.: Condens. Matter 2, 8023 (1990).
28. R. Lake and S. Datta, Phys. Rev. B 45, 6670 (1992).
29. R. Lake, G. Klimeck, R. C. Bowen, C. Fernando, D. Jovanovic, D. Blanks, T. S. Moise, Y. C. Kao, M. Leng, and W. R. Frensley, 54th Annual Device Research Conference Digest (IEEE, New York, 1996), p. 174.
30. R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, J. Appl. Phys. 81, 7845 (1997).
31. M. Saeki, J. Phys. Soc. Jpn. 55, 1846 (1986).
32. D. Ahn, Phys. Rev. B 50, 8310 (1994).
33. H. L. Grubin, D. K. Ferry, and R. Akis, Superlattices Microstruct. 20, 531 (1996).
34. W. Pötz, J. Appl. Phys. 66, 2458 (1989).
35. The master equation has been used recently to study transport across quantum dots [B. Kramer, T. Brandes, W. Häusler, K. Jauregui, W. Plaff, and D. Weinmann, Semicond. Sci. Technol. 9, 1871 (1994)], single-electron-tunneling double junctions [W. Krech, A. Hädicke, and F. Seume, Phys. Rev. B 48, 5230 (1993)], and transport via hopping among localized states in disordered AlAs/GaAs superlattices [L.-W. Wang, A. Zunger, and K. A. Mäder, Phys. Rev. B 53, 2010 (1996)]. In the latter reference a Monte Carlo solution is obtained.
36. The master equation has been used recently to study transport across quantum dots [B. Kramer, T. Brandes, W. Häusler, K. Jauregui, W. Plaff, and D. Weinmann, Semicond. Sci. Technol. 9, 1871 (1994)], single-electron-tunneling double junctions [W. Krech, A. Hädicke, and F. Seume, Phys. Rev. B 48, 5230 (1993)], and transport via hopping among localized states in disordered AlAs/GaAs superlattices [L.-W. Wang, A. Zunger, and K. A. Mäder, Phys. Rev. B 53, 2010 (1996)]. In the latter reference a Monte Carlo solution is obtained.
37. The master equation has been used recently to study transport across quantum dots [B. Kramer, T. Brandes, W. Häusler, K. Jauregui, W. Plaff, and D. Weinmann, Semicond. Sci. Technol. 9, 1871 (1994)], single-electron-tunneling double junctions [W. Krech, A. Hädicke, and F. Seume, Phys. Rev. B 48, 5230 (1993)], and transport via hopping among localized states in disordered AlAs/GaAs superlattices [L.-W. Wang, A. Zunger, and K. A. Mäder, Phys. Rev. B 53, 2010 (1996)]. In the latter reference a Monte Carlo solution is obtained.
38. Thanks are due to R. Landauer for making this suggestion, with the strong warning, however, that no formal proof has ever been given that this is indeed true. In a perhaps logically circular argument, such a proof may be obtained only using a formalism which accounts for nondiagonal terms of the density matrix.
39. k,k′, are far from being negligible since the electric field itself creates them. This is, true a fortiori for space-dependent driving forces, as shown by P. J. Price, IBM J. Res. Dev. 10, 395 (1966). The use of the proper basis functions avoids the presence of these terms which cause an unnecessary complication which affects many approaches based on plane waves even under homogeneous conditions. A typical example is given by P. Vasilipoulos, Phys. Rev. B 32, 771 (1985): Using noncurrent-carrying Landau levels as basis functions, conductivity - and so the quantum Hall induced by the (weak) perturbing electric field. This forces the study of linear response only.
40. k,k′, are far from being negligible since the electric field itself creates them. This is, true a fortiori for space-dependent driving forces, as shown by P. J. Price, IBM J. Res. Dev. 10, 395 (1966). The use of the proper basis functions avoids the presence of these terms which cause an unnecessary complication which affects many approaches based on plane waves even under homogeneous conditions. A typical example is given by P. Vasilipoulos, Phys. Rev. B 32, 771 (1985): Using noncurrent-carrying Landau levels as basis functions, conductivity - and so the quantum Hall induced by the (weak) perturbing electric field. This forces the study of linear response only.
41. N. Sano and A. Yoshii, J. Appl. Phys. 70, 3127 (1991).
42. P. Bordone, D. Vasileska, and D. K. Ferry, in Hot Carriers in Semiconductors, edited by K. Hess, J.-P. Leburton, and U. Ravaioli (Plenum, New York, 1996), p. 433.
43. See, for instance, T. Khun and F. Rossi, Phys. Rev. B 46, 7496 (1992), or Ref. 35, Secs. 1, 2, pp 1-230.
44. W. Quade, E. Schöll, F. Rossi, and C. Jacoboni, Phys. Rev. B 50, 7398 (1994).
45. W. Jones and N. H. March, Theoretical Solid State Physics (Wiley-Interscience, Bristol, 1973), Vol. 2, p. 736 ff.; App. A6.1.
46. It is not clear at all that one may write a "reduced" Hamiltonian of this form for the device. Here the device is a "doubly open" system, since it interacts with the usual thermal bath and also with thermal reservoirs of particles. While the Hamiltonian of the thermal bath is just an additive term in the total Hamiltonian, the "Hamiltonian of the contacts" cannot be properly defined separately from the "Hamiltonian of the device," as done in Eq. (1). Perhaps concepts similar to the left and right Hamiltonians introduced in Bardeen's transfer Hamiltonian formalism may be devised, but it not so obvious how. In practice, the crude model used here to describe the device-contact interaction makes the issue moot, since it is treated like a boundary condition rather than a Hamiltonian problem. On the contrary, the interaction with the thermal bath is treated following standard practice: Having traced out the degrees of freedom of all reservoirs, the Markov approximation is essentially invoked for all types of reservoirs, correlation effects in the thermal bath are assumed to decay instantaneously, and, finally, only the "secular terms" are retained. A recent review of these concepts - onset of irreversibility, memory effects, time-scales appering in the system - is given in J. Rau and B. Müller, Phys. Rep. 272, 1, (1996) from a projection-method perspective. The "mysticism" surrounding the quest for the elusive goal of deriving rigorously the master equation and the concept of coarse-graining (tracing-out the irrelevant degrees of freedoms) is beautifully depicted by L. S. Schulman, Time's Arrows and Quantum Measurement (Cambridge University Press, Cambridge, UK, 1997).
47. It is not clear at all that one may write a "reduced" Hamiltonian of this form for the device. Here the device is a "doubly open" system, since it interacts with the usual thermal bath and also with thermal reservoirs of particles. While the Hamiltonian of the thermal bath is just an additive term in the total Hamiltonian, the "Hamiltonian of the contacts" cannot be properly defined separately from the "Hamiltonian of the device," as done in Eq. (1). Perhaps concepts similar to the left and right Hamiltonians introduced in Bardeen's transfer Hamiltonian formalism may be devised, but it not so obvious how. In practice, the crude model used here to describe the device-contact interaction makes the issue moot, since it is treated like a boundary condition rather than a Hamiltonian problem. On the contrary, the interaction with the thermal bath is treated following standard practice: Having traced out the degrees of freedom of all reservoirs, the Markov approximation is essentially invoked for all types of reservoirs, correlation effects in the thermal bath are assumed to decay instantaneously, and, finally, only the "secular terms" are retained. A recent review of these concepts - onset of irreversibility, memory effects, time-scales appering in the system - is given in J. Rau and B. Müller, Phys. Rep. 272, 1, (1996) from a projection-method perspective. The "mysticism" surrounding the quest for the elusive goal of deriving rigorously the master equation and the concept of coarse-graining (tracing-out the irrelevant degrees of freedoms) is beautifully depicted by L. S. Schulman, Time's Arrows and Quantum Measurement (Cambridge University Press, Cambridge, UK, 1997).
48. 2 represent small corrections.
49. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1970), Vol. 1, p. 98 ff. Physically, these properties are implied by unitarity, time-reversal symmetry, and conservation of particle flux.
50. μK± and these off-diagonal terms can be neglected. They may become important in case of highly nonuniformly distributed scatterers, such as the case studied by R. Landauer, IBM J. Res. Dev. 1, 223 (1957), giving rise to spatial fluctuations of the electric field and current density near localized impurities. Whenever impurities are not randomly spatially distributed, the conclusions of both Ref. 2 and Ref. 3 cease to be valid, since the randomizing action of randomly distributed scatterers is required to obtain the phase cancellation which suppresses the off-diagonal elements of ρ. These complications are ignored here, but they may become important when considering statistical fluctuations of dopants in very small devices.
51. Continuity of electric current can be established only by accounting also for the displacement currents associated with the field carried off by the emitted/absorbed phonons and with the rearrangement of the Hartree potential following the charge redistribution after each collision. Presumably, though, it is the continuity of particle probability current as a collision progresses which is under scrunity here. First, the golden rule ignores these transients, as well as the associated current presumably carried by the intermediate virtual states having nonzero amplitude before collapsing into the states selected by the energy-conserving delta function. Second, and more profound, from a positivistic perspective one may deny the meaning of the question "what happens during a collision," since probability-current-continuity is violated in any quantum jump or wave function collapse during scattering or any measurement (see, for example, N. Bohr, Nature (London) 121, 580 (1928). Only a realistic approach different from "conventional" (Copenhagen) interpretation of quantum mechanics, as in D. Bohn and B. J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory, (Rutledge, London, 1993), for example, would accept the question as meaningful. Similarly, the positivistic approach followed here accepts the nonlocality of the collision processes, while alternative views enforce local collision operators (see Ref. 5, Ref. 23). Since the "orthodox" (Copenhagen) interpretation allows, or even demands, nonlocality, one may claim that the introduction of ad hoc local collision operators is a very high price to pay in exchange for "realism."
52. Continuity of electric current can be established only by accounting also for the displacement currents associated with the field carried off by the emitted/absorbed phonons and with the rearrangement of the Hartree potential following the charge redistribution after each collision. Presumably, though, it is the continuity of particle probability current as a collision progresses which is under scrunity here. First, the golden rule ignores these transients, as well as the associated current presumably carried by the intermediate virtual states having nonzero amplitude before collapsing into the states selected by the energy-conserving delta function. Second, and more profound, from a positivistic perspective one may deny the meaning of the question "what happens during a collision," since probability-current-continuity is violated in any quantum jump or wave function collapse during scattering or any measurement (see, for example, N. Bohr, Nature (London) 121, 580 (1928). Only a realistic approach different from "conventional" (Copenhagen) interpretation of quantum mechanics, as in D. Bohn and B. J. Hiley, The Undivided Universe: An Ontological Interpretation of Quantum Theory, (Rutledge, London, 1993), for example, would accept the question as meaningful. Similarly, the positivistic approach followed here accepts the nonlocality of the collision processes, while alternative views enforce local collision operators (see Ref. 5, Ref. 23). Since the "orthodox" (Copenhagen) interpretation allows, or even demands, nonlocality, one may claim that the introduction of ad hoc local collision operators is a very high price to pay in exchange for "realism."
53. J. R. Barker and D. K. Ferry, Phys. Rev. Lett. 42, 1779 (1979).
54. J. Bardeen and W. Shockley, Phys. Rev. 80, 69 (1950).
55. D. C. Herbert and S. J. Till, J. Phys. C 15, 5411 (1982).
56. A. C. Marsh and J. C. Inkson, J. Phys. C 17, 4601 (1984).
57. F. S. Kahn and J. W. Wilkins, Semicond. Sci. Technol. 1, 113, (1986).
58. T. Beleznay, J. Phys. C 19, L447 (1986).
59. M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, and H. Iwai, IEDM Technical Digest (IEEE, New York, 1993), p. 119; IEEE Trans. Electron Devices 42, 1822 (1995); M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, H. S. Momose, and H. Iwai, J. Vac. Sci. Technol. B 13, 1740 (1995).
60. M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, and H. Iwai, IEDM Technical Digest (IEEE, New York, 1993), p. 119; IEEE Trans. Electron Devices 42, 1822 (1995); M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, H. S. Momose, and H. Iwai, J. Vac. Sci. Technol. B 13, 1740 (1995).
61. M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, and H. Iwai, IEDM Technical Digest (IEEE, New York, 1993), p. 119; IEEE Trans. Electron Devices 42, 1822 (1995); M. Ono, M. Saito, T. Yoshitomi, C. Fiegna, T. Ohguro, H. S. Momose, and H. Iwai, J. Vac. Sci. Technol. B 13, 1740 (1995).
62. M. Lundstrom, IEDM Technical Digest (IEEE, New York, 1996), p. 387.
63. G. Y. Wu, Solid State Commun. 90, 397 (1994).
64. See, for example, Ref. 5, App. D, for a general description of how the boundary condition for the Schrödinger equation may be handled in different ways.
65. G. S. Lent and D. J. Kirkner, J. Appl. Phys. 67, 6353 (1990).
66. P. J. Price, Semicond. Semimet. 14, 249 (1979).
67. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).
68. F. Stern, J. Comput. Phys. 6, 56 (1970).
69. A. Spinelli, A. Benvenuti, and A. Pacelli, IEDM Technical Digest (IEEE, New York, 1996), p. 399.
70. S. Ho and K. Yamaguchi, Semicond. Sci. Technol. 7, B430 (1992).
71. R. K. Mains and G. I. Haddad, "Numerical Considerations in the Wigner Function Modeling of Resonant-Tunneling Diodes," University of Michigan preprint; results reported in Ref. 5.
72. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
73. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
74. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
75. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
76. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
77. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
78. 2. Thus, one should account for additional terms involving 〈μ +|ρ|μ-〉, 〈μ+|O|μ-〉, and 〈μ+|μ-〉. However, the off-diagonal terms 〈μ+|ρ|μ-〉 can be ignored, as discussed in Ref. 42. Quite surprisingly, the orthogonality of the terms 〈μ+|μ-〉 constitutes a controversial issue. A. M. Kriman, N. C. Kluksdah, and D. K. Ferry [Phys. Rev. B 36, 5953 (1987)] have argued in favor of their orthogonality in the limit of infinite normalization length, L→∞ [see also E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 499, footnote 6, and W. R. Frensley, in Heterostructures and Quantum Devices, edited by W. R. Frensley and N. G. Einspruch (Academic, San Diego, 1994), Chap. 9)]. However, F. Chevoir and B. Vinter [Phys. Rev. B 47, 7260 (1993)] consider the proof by Kriman et al. suspect and support the conclusion reached by A. D. Stone and A. Szafer [IBM J. Res. Dev. 32, 384 (1988)]. Here, the orthogonality of these states will be considered only a convenient approximation, satisfactory in most cases, but with concerns about its validity in general context. Finally, terms of the type 〈μ+|O|μ-〉 would vanish in the same limit for the operators O considered in the text. Furthermore, in a tunneling context, R. E. Prange [Phys. Rev. 131, 1083 (1963)] has shown that even if nonorthogonal states are chosen as basis functions, the resulting "cross terms" are compensated by yet additional terms which results from the fact that the Hamiltonian describing the left and right reservoirs - written in terms of creation/annihilation operators for these left- and right-traveling states - do not commute.
79. W. Pötz, W. Porod, and D. K. Ferry, Phys. Rev. B 32, 3868 (1985).
80. M. O. Vassell, J. Lee, and H. F. Lockwood, J. Appl. Phys. 54, 5206 (1983).
81. L. V. Keldysh, Sov. Phys. JETP 34, 665 (1958).
82. P. J. Price and J. M. Radcliffe, IBM J. Res. Dev. 3, 364 (1959).
83. E. Mendez, J. Nocera, and W. I. Wang, Phys. Rev. B 45, 3910 (1986).
84. G. Klimech (private communication).
85. P.J. Price, Phys. Rev. B 45, 9042 (1992).
86. F. A. Buot and A. K. Rajagopal, Mater. Sci. Eng. B 35, 303 (1995).
87. R. K. Mains and G. I. Haddad, J. Appl. Phys. 64, 5041 (1988).
88. K. L. Jensen and F. A. Buot, J. Appl. Phys. 67, 7602 (1990).
89. M. L. Leadbeater, E. S. Alves, F. W. Sheard, L. Eaves, M. Henini, O. H. Hughes, and G. A. Toombs, J. Phys.: Condens. Matter 1, 1065 (1989).
90. F. W. Sheard and G. A. Toombs, Semicond. Sci. Technol. 7, B460 (1992).
91. Y. Abe, Semicond. Sci. Technol. 7, B498 (1992).
92. N. C. Kluksdahl, A. M. Kriman, C. Ringhofer, and D. K. Ferry, Solid-State Electron. 31, 742 (1988).
93. R. Lake, G. Klimeck, and S. Dutta, Phys. Rev. B 47, 6427 (1993).
94. R. Lake, G. Klimeck, M. P. Anantram, and S. Datta, Phys. Rev. B 48, 15 132 (1993).
95. J. C. Slater, Phys. Rev. 76, 1592 (1949).
96. J. M. Luttinger, Phys. Rev. 84, 814 (1951).
97. J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).
98. M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 (1993).
99. q is the phonon number in the state labeled by the wave vector q, and ℏωq is the phonon energy.
100. P. J. Price, Ann. Phys. 133, 217 (1981).