Quantum Master Equations in Electronic Transport

Chapter

Abstract

In this chapter we present several quantum master equations (QMEs) that describe the time evolution of the density matrix at various levels of approximations. We emphasize the similarity between the single-particle QME and the Boltzmann transport equation (BTE), starting from truncating the BBGKY chain of equations and ending with similar Monte-Carlo approaches to solve them stochastically and show what kind of boundary conditions are needed to solve the single-particle QME in the light of the open nature of modern electronic devices. The Pauli master equation (PME) and a QME in the perturbation expansion are described and compared both with one another and with the BTE. At the level of the reduced many-particle density matrix, we show several approaches to derive many-particle QMEs starting from the formal Nakajima–Zwanzig equation and ending with the partial-trace-free time-convolutionless equation of motion with memory dressing. Using those results we derive the correct distribution functions of the Landauer-type, for a small, ballistic open system attached to two large reservoirs with ideal black-body absorption characteristics.

Keywords

Quantum transport Master equation Density matrix Distribution function Transient 

References

  1. 1.
    L.E. Reichl, A Modern Course in Statistical Physics (WILEY-VCH, Weinheim, 1980)Google Scholar
  2. 2.
    M. Bonitz, Quantum Kinetic Theory (Teubner, Stuttgart; Leipzig, 1998)Google Scholar
  3. 3.
    W.R. Frensley, Rev. Mod. Phys. 62, 745 (1990)CrossRefGoogle Scholar
  4. 4.
    N.C. Kluksdahl, A.M. Kriman, D.K. Ferry, C. Ringhofer, Phys. Rev. B 39, 7720 (1989)CrossRefGoogle Scholar
  5. 5.
    S.E. Laux, A. Kumar, M.V. Fischetti, IEEE Trans. Nanotechnol. 1, 255 (2002)CrossRefGoogle Scholar
  6. 6.
    W. Pötz, J. Appl. Phys. 66, 2458 (1989)CrossRefGoogle Scholar
  7. 7.
    A. Gehring, S. Selberherr, J. Comput. Electron. 3, 149 (2004)CrossRefGoogle Scholar
  8. 8.
    J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955)MATHGoogle Scholar
  9. 9.
    E. Wigner, Phys. Rev. 40, 749 (1932)MATHCrossRefGoogle Scholar
  10. 10.
    L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962)MATHGoogle Scholar
  11. 11.
    L.V. Keldysh, Sov. Phys. JETP 20, 1018 (1965)MathSciNetGoogle Scholar
  12. 12.
    W. Pauli, Festschrift zum 60. Geburtstage A. Sommerfeld (Hirzel, Leipzig, 1928), p. 30Google Scholar
  13. 13.
    K. Huang, Statistical Mechanics (John Wiley & Sons, New York, 1987)MATHGoogle Scholar
  14. 14.
    H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)MATHGoogle Scholar
  15. 15.
    U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1999)MATHCrossRefGoogle Scholar
  16. 16.
    M.V. Fischetti, J. Appl. Phys. 83, 270 (1998)CrossRefGoogle Scholar
  17. 17.
    M.V. Fischetti, Phys. Rev. B 59, 4901 (1999)CrossRefGoogle Scholar
  18. 18.
    R. Brunetti, C. Jacoboni, F. Rossi, Phys. Rev. B 39, 10781 (1989)CrossRefGoogle Scholar
  19. 19.
    F. Rossi, C. Jacoboni, Europhys. Lett. 18, 169 (1992)CrossRefGoogle Scholar
  20. 20.
    C. Jacoboni, Semicond. Sci. Technol. 7, B6 (1992)CrossRefGoogle Scholar
  21. 21.
    J. von Neumann, Nachr. Ges. Wiss. p. 245 (1927)Google Scholar
  22. 22.
    J.J. Sakurai, Modern Quantum Mechanics (Addison–Wesley, USA, 1994)Google Scholar
  23. 23.
    A.L. Fetter, J.D. Walecka (eds.), Quantum theory of many-particle systems (McGraw-Hill, Inc., 1971)Google Scholar
  24. 24.
    A. Messiah, Quantum Mechanics (Dover, Mineola, 1999)Google Scholar
  25. 25.
    L. van Hove, Rev. Mod. Phys. 21, 517 (1955)MATHGoogle Scholar
  26. 26.
    W. Kohn, J.M. Luttinger, Phys. Rev. 108, 590 (1957)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    J.R. Barker, D.K. Ferry, Phys. Rev. Lett. 42, 1779 (1979)CrossRefGoogle Scholar
  28. 28.
    I.B. Levinson, Y. Yasevichyute, Sov. Phys. JETP 35, 991 (1972)Google Scholar
  29. 29.
    W.V. Houston, Phys. Rev. 57, 184 (1940)MathSciNetCrossRefGoogle Scholar
  30. 30.
    P.J. PriceF, Semicond. Semimet. 14, 249 (1979)CrossRefGoogle Scholar
  31. 31.
    C. Jacoboni, L. Reggiani, Rev. Mod. Phys. 55, 645 (1983)CrossRefGoogle Scholar
  32. 32.
    M.V. Fischetti, S.E. Laux, P.M. Solomon, A. Kumar, J. Comput. Electron. 3, 287 (2004)CrossRefGoogle Scholar
  33. 33.
    D. Vasileska, S.M. Goodnick, Mater. Sci. Eng. R 38, 181 (2002)Google Scholar
  34. 34.
    H. Budd, Phys. Rev. 158, 798 (1967)CrossRefGoogle Scholar
  35. 35.
    R. Chambers, Proc. R. Soc. London 65, 458 (1952)MATHGoogle Scholar
  36. 36.
    M. Lundstrom, Fundamentals of Carrier Transport (Cambridge University Press, Cambridge, 2000)CrossRefGoogle Scholar
  37. 37.
    S. Nakajima, Prog. Theor. Phys. 20, 948 (1958)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    R. Zwanzig, J. Chem. Phys. 33, 1338 (1960)MathSciNetCrossRefGoogle Scholar
  39. 39.
    A.G. Redfield, IBM J. Res. Dev. 1, 19 (1957)CrossRefGoogle Scholar
  40. 40.
    G. Lindblad, Commun. Math. Phys. 48, 199 (1976)MathSciNetCrossRefGoogle Scholar
  41. 41.
    P. Argyres, P. Kelley, Phys. Rev. 134, A98 (1964)CrossRefGoogle Scholar
  42. 42.
    J.R. Barker, D.K. Ferry, Solid-State Electron. 23, 531 (1980)CrossRefGoogle Scholar
  43. 43.
    K. Kassner, Phys. Rev. A 36, 5381 (1987)CrossRefGoogle Scholar
  44. 44.
    M. Sparpaglione, S. Mukamel, J. Chem. Phys. 88, 3263 (1988)CrossRefGoogle Scholar
  45. 45.
    Y. Hu, S. Mukamel, J. Chem. Phys. 91, 6973 (1989)CrossRefGoogle Scholar
  46. 46.
    V. Romero-Rochin, I. Oppenheim, Physica A 155, 52 (1989)CrossRefGoogle Scholar
  47. 47.
    M. Tokuyama, H. Mori, Prog. Theor. Phys. 55, 411 (1975)MathSciNetCrossRefGoogle Scholar
  48. 48.
    F. Shibata, Y. Takahashi, N. Hashitsume, J. Stat. Phys. 17, 171 (1977)MathSciNetCrossRefGoogle Scholar
  49. 49.
    N. Hashitsume, F. Shibata, M. Shingu, J. Stat. Phys. 17, 155 (1977)CrossRefGoogle Scholar
  50. 50.
    M. Saeki, Prog. Theor. Phys. 67, 1313 (1982)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    M. Saeki, Prog. Theor. Phys. 79, 396 (1988)CrossRefGoogle Scholar
  52. 52.
    M. Saeki, Prog. Theor. Phys. 89, 607 (1993)CrossRefGoogle Scholar
  53. 53.
    D. Ahn, Phys. Rev. B 51, 2159 (1995)CrossRefGoogle Scholar
  54. 54.
    D. Ahn, Prog. Quantum Electron. 21, 249 (1997)CrossRefGoogle Scholar
  55. 55.
    D. Ahn, J.H. Oh, K. Kimm, S. Hwang, Phys. Rev. A 61, 052310 (2000)CrossRefGoogle Scholar
  56. 56.
    D. Ahn, J. Lee, M.S. Kim, S.W. Hwang, Phys. Rev. A 66, 012302 (2002)CrossRefGoogle Scholar
  57. 57.
    T.M. Chang, J.L. Skinner, Physica A 193, 483 (1993)CrossRefGoogle Scholar
  58. 58.
    A.A. Golosov, D.R. Reichmann, J. Chem. Phys. 115, 9849 (2001)Google Scholar
  59. 59.
    I. Knezevic, D.K. Ferry, Phys. Rev. E 66, 016131 (2002)CrossRefGoogle Scholar
  60. 60.
    I. Knezevic, D.K. Ferry, Phys. Rev. A 69, 012104 (2004)CrossRefGoogle Scholar
  61. 61.
    I. Knezevic, D.K. Ferry, Phys. Rev. E 67, 066122 (2003)CrossRefGoogle Scholar
  62. 62.
    I. Knezevic, Phys. Rev. B 77, 125301 (2008)CrossRefGoogle Scholar
  63. 63.
    D. Semkat, D. Kremp, M. Bonitz, Phys. Rev. E 59, 1557 (1999)CrossRefGoogle Scholar
  64. 64.
    D. Semkat, D. Kremp, M. Bonitz, J. Math. Phys. 41, 7458 (2000)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    K.M. et al., Phys. Rev. E 63, 020102(R) (2001)Google Scholar
  66. 66.
    W.T. Reid, Riccati Differential Equations (Academic Press, New York, 1972)MATHGoogle Scholar
  67. 67.
    S. Bittanti, A.J. Laub, J.C.Willems (eds.), The Riccati Equation (Springer-Verlag, Berlin, 1991)MATHGoogle Scholar
  68. 68.
    G. Lindblad, J. Phys. A 29, 4197 (1996)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    E.B. Davies, Commun. Math. Phys. 39, 91 (1974)MATHCrossRefGoogle Scholar
  70. 70.
    X.Q. Li, J.Y. Luo, Y.G. Yang, P. Cui, Y.J. Yan, Phys. Rev. B 71, 205304 (2005)CrossRefGoogle Scholar
  71. 71.
    J.N. Pedersen, A. Wacker, Phys. Rev. B 72, 195330 (2005)CrossRefGoogle Scholar
  72. 72.
    D. Bacon, D.A. Lidar, K.B. Whaley, Phys. Rev. A 60, 1944 (1999)CrossRefGoogle Scholar
  73. 73.
    D.A. Lidar, Z. Bihary, K.B. Whaley, Chem. Phys. 268, 35 (2001)CrossRefGoogle Scholar
  74. 74.
    A.M. Kriman, M.J. Kann, D.K. Ferry, R. Joshi, Phys. Rev. Lett. 65, 1619 (1990)CrossRefGoogle Scholar
  75. 75.
    R. Alicki, Phys. Rev. A 40, 4077 (1989)CrossRefGoogle Scholar
  76. 76.
    P. Lugli, D.K. Ferry, IEEE Trans. Electron Devices 32, 2431 (1985)CrossRefGoogle Scholar
  77. 77.
    P. Lugli, D.K. Ferry, Phys. Rev. Lett. 56, 1295 (1986)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.University of Wisconsin-MadisonMadisonUSA

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