Advertisement

Electronic Band Structures

  • Peter Y. Yu
  • Manuel Cardona
Part of the Graduate Texts in Physics book series (GTP)

Summary

A semiconductor sample contains a very large number of atoms. Hence a quantitative quantum mechanical calculation of its physical properties constitutes a rather formidable task. This task can be enormously simplified by bringing into play the symmetry properties of the crystal lattice, i. e., by using group theory. We have shown how wave functions of electrons and vibrational modes (phonons) can be classified according to their behavior under symmetry operations. These classifications involve irreducible representations of the group of symmetry operations. The translational symmetry of crystals led us to Bloch’s theorem and the introduction of Bloch functions for the electrons. We have learnt that their eigenfunctions can be indexed by wave vectors (Bloch vectors) which can be confined to a portion of the reciprocal space called the first Brillouin zone. Similarly, their energy eigenvalues can be represented as functions of wave vectors inside the first Brillouin zone, the so-called electron energy bands. We have reviewed the following main methods for calculating energy bands of semiconductors: the empirical pseudopotential method, the tight-binding or linear combination of atomic orbitals (LCAO) method and the k·p method. We have performed simplified versions of these calculations in order to illustrate the main features of the energy bands in diamond- and zinc-blende-type semiconductors.

Keywords

Wave Function Valence Band Band Structure Irreducible Representation Brillouin Zone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Chapter 2

  1. 2.1
    Quantum Theory of Real Materials (eds. Chelikowsky, J.R., Louie, S.G.) (Kluwer, Dordrecht, 1996)Google Scholar
  2. 2.2
    C. Kittel: Introduction to Solid State Physics, 7th edn. (Wiley, New York 1995) p. 37Google Scholar
  3. 2.3
    L.M. Falicov: Group Theory and its Physical Applications (Univ. Chicago Press, Chicago 1966)Google Scholar
  4. 2.4
    G.F. Koster: Space groups and their representations, in Solid State Physics 5, 173–256 (Academic, New York 1957)Google Scholar
  5. 2.5
    G. Lucovsky, A comparison of the long wavelength optical phonons in trigonal Se and Te, Phys. Stat. Sol. (b) 49, 633 (1972)CrossRefGoogle Scholar
  6. 2.6
    D.M. Greenaway, G. Harbeke: Optical Properties and Band Structure of Semiconductors (Pergamon, New York 1968) p. 44Google Scholar
  7. 2.7
    H. Jones: The Theory of Brillouin Zones and Electronic States in Crystals, 2nd edn. (North-Holland, Amsterdam 1975)Google Scholar
  8. 2.8
    M.L. Cohen, J. Chelikowsky: Electronic Structure and Optical Properties of Semiconductors, 2nd edn., Springer Ser. Solid-State Sci., Vol. 75 (Springer, Berlin, Heidelberg 1989)Google Scholar
  9. 2.9
    J.R. Chelikowsky, D.J. Chadi, M.L. Cohen: Calculated valence band densities of states and photoemission spectra of diamond and zinc-blende semiconductors. Phys. Rev. B 8, 2786–2794 (1973)CrossRefGoogle Scholar
  10. 2.10
    C. Varea de Alvarez, J.P. Walter, R.W. Boyd, M.L. Cohen: Calculated band structures, optical constants and electronic charge densities for InAs and InSb. J. Chem. Phys. Solids 34, 337–345 (1973)CrossRefGoogle Scholar
  11. 2.11
    P. Hohenberg, W. Kohn: Inhomogeneous electron gas. Phys. Rev. B 863, 136 (1964)Google Scholar
  12. 2.12
    W. Kohn, L. Sham: Self-consistent equations including exchange and correlation effects. Phys. Rev. A 113, 140 (1965)Google Scholar
  13. 2.13
    M.S. Hybertsen, S.G. Louie: Electron correlation in semiconductors and insulators. Phys. Rev. B 34, 5390–5413 (1986)CrossRefGoogle Scholar
  14. 2.14
    N. Trouillier, J.L. Martins: Efficient pseudopotentials for plane wave calculations. Phys. Rev. B 43, 1993–2006 (1991)CrossRefGoogle Scholar
  15. 2.15
    E.O. Kane: Band structure of indium antimonide. J. Phys. Chem. Solids 1, 249–261 (1957)CrossRefGoogle Scholar
  16. 2.16
    M. Cardona, F.H. Pollak: Energy-band structure of germanium and silicon. Phys. Rev. 142, 530–543 (1966); see also Vol. 41BCrossRefGoogle Scholar
  17. 2.17
    M. Cardona: Band parameters of semiconductors with zincblende, wurtzite, and germanium structure. J. Phys. Chem. Solids 24, 1543–1555 (1963); erratum: ibid. 26, 1351E (1965)CrossRefGoogle Scholar
  18. 2.18
    O. Madelung, M. Schulz, H. Weiss (eds.): Landolt-Börnstein, Series III, Vol. 17a–h (Semiconductors) (Springer, Berlin, Heidelberg 1987)Google Scholar
  19. 2.19
    E.O. Kane: The k · p method. Semiconductors and Semimetals 1, 75–100 (Academic, New York 1966)Google Scholar
  20. 2.20
    M. Cardona, N.E. Christensen, G. Fasol: Relativistic band structure and spin-orbit splitting of zincblende-type semiconductors. Phys. Rev. B 38, 1806–1827 (1988)CrossRefGoogle Scholar
  21. 2.21
    G. Dresselhaus, A.F. Kip, C. Kittel: Cyclotron resonance of electrons and holes in silicon and germanium crystals. Phys. Rev. 98, 368–384 (1955)CrossRefGoogle Scholar
  22. 2.22
    M. Willatzen, M. Cardona, N.E. Christensen: LMTO and k·p calculation of effective masses and band structure of semiconducting diamond. Phys. Rev. B50, 18054 (1994)CrossRefGoogle Scholar
  23. 2.23
    J.M. Luttinger: Quantum theory of cyclotron resonance in semiconductors: General theory. Phys. Rev. 102, 1030–1041 (1956)CrossRefGoogle Scholar
  24. 2.24
    W.A. Harrison: Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover, New York 1989)Google Scholar
  25. 2.25
    D.J. Chadi, M.L. Cohen: Tight-binding calculations of the valence bands of diamond and zincblende crystals. Phys. Stat. Solidi B 68, 405–419 (1975)CrossRefGoogle Scholar
  26. 2.26
    W.A. Harrison: The physics of solid state chemistry, in Festkörperprobleme 17, 135–155 (Vieweg, Braunschweig, FRG 1977)Google Scholar
  27. 2.27
    F. Herman: Recent progress in energy band theory, in Proc. Int'l Conf. on Physics of Semiconductors (Dunod, Paris 1964) pp. 3–22Google Scholar
  28. 2.28
    T. Dietl, W. Dobrowolski, J. Kosut, B.J. Kowalski, W. Szuskiewicz, Z. Wilamoski, A.M. Witowski: HgSe: Metal or Semiconductor? Phys. Rev. Lett. 81, 1535 (1998); D. Eich, D. Hübner, R. Fink, E. Umbach, K. Ortner, C.R. Becker, G. Landwehr, A. Flezsar: Electronic structure of HgSe investigated by direct and inverse photoemission. Phys. Rev. B61, 12666–12669 (2000)CrossRefGoogle Scholar
  29. 2.29
    T.N. Morgan: Symmetry of electron states in GaP. Phys. Rev. Lett. 21, 819–823 (1968)CrossRefGoogle Scholar
  30. 2.30
    R.M. Wentzcovitch, M. Cardona, M.L. Cohen, N.E. Christensen: X1 and X3 states of electrons and phonons in zincblende-type semiconductors. Solid State Commun. 67, 927–930 (1988)CrossRefGoogle Scholar
  31. 2.31
    S.H. Wei, A. Zunger: Band gaps and spin-orbit splitting of ordered and disordered AlxGa1−xAs and GaAsxSb1−x alloys. Phys. Rev. B 39, 3279–3304 (1989)CrossRefGoogle Scholar

Group Theory and Applications

  1. Burns G.: Introduction to Group Theory and Applications (Academic, New York 1977)Google Scholar
  2. Evarestov R.A., V.P. Smirnov: Site Symmetry in Crystals, Springer Ser. Solid-State Sci., Vol. 108 (Springer, Berlin, Heidelberg 1993)Google Scholar
  3. Falicov L.M.: Group Theory and Its Physical Applications (Univ. Chicago Press, Chicago 1966)Google Scholar
  4. Heine V.: Group Theory in Quantum Mechanics (Pergamon, New York 1960)Google Scholar
  5. Inui T., Y. Tanabe, Y. Onodera: Group Theory and Its Applications in Physics, 2nd edn. Springer Ser. Solid-State Sci., Vol. 78 (Springer, Berlin, Heidelberg 1996)Google Scholar
  6. Jones H.: Groups, Representations, and Physics (Hilger, Bristol 1990)CrossRefGoogle Scholar
  7. Koster G.F.: Space groups and their representations. Solid State Physics 5, 173–256 (Academic, New York 1957)Google Scholar
  8. Lax M.: Symmetry Principles in Solid State and Molecular Physics (Wiley, New York 1974)Google Scholar
  9. Ludwig W., C. Falter: Symmetries in Physics, 2nd edn., Springer Ser. Solid-State Sci., Vol. 64 (Springer, Berlin, Heidelberg 1996)Google Scholar
  10. Tinkham M.: Group Theory and Quantum Mechanics (McGraw-Hill, New York 1964)Google Scholar
  11. Vainshtein B.K.: Fundamentals of Crystals, 2nd edn., Modern Crystallography, Vol. 1 (Springer, Berlin, Heidelberg 1994)Google Scholar

Electronic Band Structures

  1. Cohen M.L., Chelikowsky, J.: Electronic Structure and Optical Properties of Semiconductors, 2nd edn., Springer Ser. Solid-State Sci., Vol. 75 (Springer, Berlin, Heidelberg 1989)Google Scholar
  2. Greenaway D.L., Harbeke, G.: Optical Properties and Band Structure of Semiconductors (Pergamon, New York 1968)Google Scholar
  3. Harrison W.A.: Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover, New York 1989)Google Scholar
  4. Jones H.: The Theory of Brillouin Zones and Electronic States in Crystals (North-Holland, Amsterdam 1962)Google Scholar
  5. Phillips J.C.: Covalent Bonding in Crystals, Molecules, and Polymers (Univ. Chicago Press, Chicago 1969)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Peter Y. Yu
    • 1
  • Manuel Cardona
    • 2
  1. 1.Department of PhysicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Max-Planck-Institut für FestkörperforschungStuttgartGermany

Personalised recommendations