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Green’s Functions in Quantum Physics pp 71–96Cite as

Green’s Functions for Tight-Binding Hamiltonians

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Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 7))

Abstract

We introduce the so-called tight-binding Hamiltonians, which have the form

$$ H = \sum\limits_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } } {\left| {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } > {\varepsilon _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } }} < \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } } \right| + \sum\limits_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{m} } {\left| {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } > {V_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\ell } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{m} }} < m} \right|} ,} $$

where each state |ℓz> is an atomic-like orbital centered at the site z; the sites {ℓ} form a lattice. Such Hamiltonians are very important in solid-state physics. Here we calculate the Green’s functions associated with the TBH for various simple lattices. We also review briefly some applications in solid-state physics.

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Reference

  1. C. Kittel: Introduction to Solid State Physics, 5th ed. ( Wiley and Sons, New York 1976 )

    Google Scholar 

  2. C. Kittel: Quantum Theory of Solids ( Wiley and Sons, New York 1973 )

    Google Scholar 

  3. J.M. Ziman: Principles of the Theory of Solids ( Cambridge University Press, London 1964 )

    MATH  Google Scholar 

  4. J.C. Slater, G.F. Koster: Phys. Rev. 94, 1498 (1954)

    Article  ADS  MATH  Google Scholar 

  5. T. Morita: J. Math. Phys. 12, 1744 (1971)

    Article  ADS  Google Scholar 

  6. L. Van Hove: Phys. Rev. 89, 1189 (1953); see also H.P. Rosenstock: Phys. Rev. 97, 290 (1955)

    Google Scholar 

  7. J. Callaway: J. Math. Phys. 5, 783 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  8. S. Katsura, S. Inawashiro: J. Math. Phys. 12, 1622 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. T. Horiguchi: J. Math. Phys. 13, 1411 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  10. T. Horiguchi, C.C. Chen: J. Math. Phys. 15, 659 (1974)

    Article  ADS  Google Scholar 

  11. T. Morita, T. Horiguchi: J. Math. Phys. 12, 981 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. T. Horiguchi: J. Phys. Soc. Jpn. 30, 1261 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. G.S. Joyce: Philos. Trans. Roy. Soc. A273, 583 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. T. Morita: J. Phys. A8, 478 (1975)

    ADS  Google Scholar 

  15. T. Horiguchi, T. Morita: J. Phys. C8, L232 (1975)

    ADS  Google Scholar 

  16. D.J. Austen, P.D. Loly: J. Comput. Phys. 11, 315 (1973)

    Article  ADS  MATH  Google Scholar 

  17. F.T. Hioe: J. Math. Phys. 19, 1065 (1978)

    MathSciNet  ADS  Google Scholar 

  18. M.A. Rashid: J. Math. Phys. 21, 2549 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. S. Inawashiro, S. Katsura, Y. Abe: J. Math. Phys. 14, 560 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  20. S. Katsura, S. Inawashiro, Y. Abe: J. Math. Phys. 12, 895 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. G.S. Joyce: J. Phys. A5, L65 (1972)

    ADS  Google Scholar 

  22. S. Katsura, T. Morita, S. Inawashiro, T. Horiguchi, Y. Abe: J. Math. Phys. 12, 892 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. T. Morita, T. Horiguchi: J. Math. Phys. 12, 986 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. K. Mano: J. Math. Phys. 15, 2175 (1974); 16, 1726 (1975)

    Article  MathSciNet  Google Scholar 

  25. G.S. Joyce: J. Math. Phys. 12, 1390 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. M. Inoue: J. Math. Phys. 15, 704 (1974)

    Article  ADS  Google Scholar 

  27. W.A Harrison: Pseudopotentials in the Theory of Metals ( Benjamin, New York 1966 )

    Google Scholar 

  28. V. Heine, D. Weaire: In Solid State Physics, Vol.24, ed. by H. Ehrenreich, F. Seitz, D. Turnbull ( Academic, New York 1970 ) p. 249

    Google Scholar 

  29. W.A. Harrison: Electronic Structure and the Properties of Solids ( Freeman, San Francisco 1980 )

    Google Scholar 

  30. J.C. Phillips, L. Kleinman: Phys. Rev. 116, 287 (1959)

    Article  ADS  MATH  Google Scholar 

  31. M.H. Cohen, V. Heine: Phys. Rev. 122, 1821 (1961)

    Article  ADS  Google Scholar 

  32. M.L. Cohen, V. Heine: In Solid State Physics, Vol.24, ed. by H. Ehrenreich, F. Seitz, D. Turnbull ( Academic, New York 1970 ) p. 37

    Google Scholar 

  33. J. Hafner, H. Nowotny: Phys. Status Solidi B51, 107 (1972)

    Article  Google Scholar 

  34. J.R. Chelikowsky, M.L. Cohen: Phys. Rev. Lett. 36, 229 (1976); Phys. Rev. B14, 556 (1976)

    ADS  Google Scholar 

  35. D.J. Chadi, M.L. Cohen: Phys. Status Solidi B68, 405 (1975)

    Article  Google Scholar 

  36. J.C. Phillips: Rev. Mod. Phys. 42, 317 (1970)

    Article  ADS  Google Scholar 

  37. S.T. Pantelides, W.A. Harrison: Phys. Rev. B11, 3006 (1975)

    Article  ADS  Google Scholar 

  38. W.A. Harrison: Phys. Rev. B8, 4487 (1973)

    Article  ADS  Google Scholar 

  39. S. Froyen, W.A. Harrison: Phys. Rev. B20, 2420 (1979); W.A. Harrison: Phys. Rev. B24, 5835 (1981)

    Article  Google Scholar 

  40. E.O. Kane, A.B. Kane: Phys. Rev. B17, 2691 (1978)

    Article  ADS  Google Scholar 

  41. J.D. Joannopoulos, M.A. Schlüter, M.L. Cohen: In Proc. 12th Intern. Conf. Physics of Semiconductors (Stuttgart), ed. by M.H. Pilkuhn ( Teubner, Stuttgart 1974 )

    Google Scholar 

  42. R.M. Martin, G. Lucovsky, K. Helliwell: Phys. Rev. B13, 1383 (1976)

    Article  ADS  Google Scholar 

  43. D.J. Chadi: Phys. Rev. B16, 790 and 3572 (1977)

    Google Scholar 

  44. K.C. Pandey, J.C. Phillips: Phys. Rev. B13, 750 (1976)

    Article  ADS  Google Scholar 

  45. D.A. Papaconstantopoulos, E.N. Economou: Phys. Rev. B22, 2903 (1980)

    Article  ADS  Google Scholar 

  46. S.T. Pantelides: Phys. Rev. B11, 5082 (1975)

    Article  ADS  Google Scholar 

  47. J.R. Smith, J.G. Gay: Phys. Rev. B12, 4238 (1975)

    Article  ADS  Google Scholar 

  48. L. Hodges, H. Ehrenreich: Phys. Lett. 16, 203 (1965)

    Article  ADS  Google Scholar 

  49. F.M. Mueller: Phys. Rev. 153, 659 (1967)

    Article  ADS  Google Scholar 

  50. J. Rath, J. Callaway: Phys. Rev. B8, 5398 (1973)

    Article  ADS  Google Scholar 

  51. L.F. Mattheiss: Phys. Rev. B5, 290 and 306 (1972); B6, 4718 (1972) B13, 2433 (1976)

    Google Scholar 

  52. L.F. Mattheiss: Phys. Rev. B12, 2161 (1975)

    Article  ADS  Google Scholar 

  53. B.M. Klein, L.L. Boyer, D.A. Papaconstantopoulos, L.F. Mattheiss: Phys. Rev. B18, 6411 (1978)

    Article  ADS  Google Scholar 

  54. J.D. Joannopoulos, F. Yndurain: Phys. Rev. B1o, 5164 (1974); F. Yndurain, J.D. Joannopoulos: Phys. Rev. B11, 2957 (1975)

    Google Scholar 

  55. D.C. Allan, J.D. Joannopoulos: Phys. Rev. Lett. 44, 43 (1980); J.D. Joannopoulos: J. Non Cryst. Solids 35–36, 781 (1980)

    Google Scholar 

  56. M.F. Thorpe: In Excitations in Disordered Systems, ed. by M.F. Thorpe ( Plenum, New York 1981 ) pp. 85–107

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics, University of Crete, Heraklion, Crete, Greece

    Professor Eleftherios N. Economou PhD

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  1. Professor Eleftherios N. Economou PhD
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© 1983 Springer-Verlag Berlin Heidelberg

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Economou, E.N. (1983). Green’s Functions for Tight-Binding Hamiltonians. In: Green’s Functions in Quantum Physics. Springer Series in Solid-State Sciences, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02369-3_5

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  • DOI: https://doi.org/10.1007/978-3-662-02369-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12266-1

  • Online ISBN: 978-3-662-02369-3

  • eBook Packages: Springer Book Archive

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