Skip to main content
Springer Nature Link
Log in

Application of the Green’s functions method to the study of the optical properties of semiconductors

  • Published:
Save article
View saved research
La Rivista del Nuovo Cimento (1978-1999) Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. W. Hanke: inFestkörperprobleme XIX, Advanced Solid State Physics (Vieweg, 1979), p. 43.

  2. W. Hanke, H. J. Mattausch, andG. Strinati: inElectron Correlations in Solids, Molecules, and Atoms, edited byJ. T. Devreese andF. Brosens (Plenum Press, New York, N. Y., 1983), p. 289.

    Chapter  Google Scholar 

  3. W. Hanke, N. Meskini, andH. Weiler: inElectronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter, edited byJ. T. Devreese andP. Van Camp (Plenum Press, New York, N. Y., 1985), p. 113.

    Chapter  Google Scholar 

  4. A. A. Abrikosov, L. P. Gorkov, andI. E. Dzyaloshinski:Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliff, N.J., 1963);A. L. Fetter andJ. D. Walecka:Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, N. Y., 1971);G. Rickayzen:Green’s Functions and Condensed Matter (Academic Press, London, 1980);G. D. Mahan:Many-Particle Physics (Plenum Press, New York, N. Y., 1981).

    Google Scholar 

  5. We follow closely the treatment due toL. Hedin andS. Lundqvist: inSolid State Physics, edited byH. Ehrenreich, F. Seitz, andD. Turnbull (Academic Press, New York, N. Y., 1969), Vol.23, p. 1, although our presentation differs from theirs in several technical details.

    Google Scholar 

  6. L. P. Kadanoff andG. Baym:Quantum Statistical Mechanics (Benjamin, Menlo Park, 1962).

    Google Scholar 

  7. A term which ensures the charge neutrality of the system is assumed to be included inV(r).

  8. See,e.g.,C. Csanak,H. S. Taylor, andR. Yaris:Adv. At. Mol. Phys.,7, 287 (1971).

    Article  ADS  Google Scholar 

  9. P. Nozières:Theory of Interacting Fermi Systems (Benjamin, New York, N. Y., 1964).

    Google Scholar 

  10. L. J. Sham andT. M. Rice:Phys. Rev.,144, 708 (1966).

    Article  ADS  Google Scholar 

  11. G. Baym andL. P. Kadanoff:Phys. Rev.,124, 287 (1961).

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Gell-Mann andF. Low:Phys. Rev.,84, 350 (1951).

    Article  MathSciNet  ADS  Google Scholar 

  13. Cf.,e.g., the second of references [4],. sects.13, 32, and52.

    Google Scholar 

  14. G. Baym:Phys. Rev.,127, 1391 (1962).

    Article  MathSciNet  ADS  Google Scholar 

  15. Equation (6.4) can also be interpreted physically as the continuity equation in the presence of the external fieldU whenever the time dependence ofU is slow enough for the adiabatic approximation to hold,i.e. if the external field drives the system back to the ground state after it has exhausted its action.

  16. J. R. Schrieffer:Theory of Superconductivity (Benjamin, New York, N. Y., 1964), Chap. 8.

    Google Scholar 

  17. L. J. Sham andW. Kohn:Phys. Rev.,145, 561 (1966).

    Article  ADS  Google Scholar 

  18. L. J. Sham:Phys. Rev.,150, 720 (1966).

    Article  ADS  Google Scholar 

  19. G. Strinati, H. J. Mattausch, andW. Hanke:Phys. Rev. B,25, 2867 (1982).

    Article  ADS  Google Scholar 

  20. W. Heitler:The Quantum Theory of Radiation (Clarendon, Oxford, 1954), subsect.1.6; see also the last of references [4]A. A. Abrikosov, L. P. Gorkov, andI. E. Dzyaloshinski:Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliff, N.J., 1963), subsect.1.5.

    Google Scholar 

  21. R. Del Sole andE. Fiorino:Phys. Rev. B,29, 4631 (1984).

    Article  ADS  Google Scholar 

  22. Cf.,e.g.,J. D. Jackson:Classical Electrodynamics, (J. Wiley, New York, N.Y., 1962), subject.6.5.

    Google Scholar 

  23. L. Rosenfeld:Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. 2.

    Google Scholar 

  24. H. Ehrenreich: inThe Optical Properties of Solids, Varenna Course XXXIV, edited byJ. Tauc (Academic Press, New York, N.Y., 1966), p. 106.

    Google Scholar 

  25. V. Ambegaokar andW. Kohn:Phys. Rev.,117, 423 (1960).

    Article  MathSciNet  ADS  Google Scholar 

  26. Equations (8.23) generalize to a crystalline material eqs. (6–168) of ref. [9]. which hold for a homogeneous system. The latter, however, hold for any value of the wave vector.

    Google Scholar 

  27. Equation (8.24) generalizes to all orders of perturbation theory the result obtained in ref. [23], within the time-dependent-screened-Hartree-Fock approximation which will be discussed in subsect.8.4.

    Google Scholar 

  28. M. Born andK. Huang:Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1968), Appendix VI.

    Google Scholar 

  29. F. Bassani andM. Altarelli: inHandbook of Synchrotron Radiation, edited byE. E. Koch (North-Holland, Amsterdam, 1983), p. 463.

    Google Scholar 

  30. For a truly isotropic medium an equation similar to eq. (8.42) can be proved forany finite wave vector, since the transverse and the longitudinal components of a tensor that satisfies eq. (8.28) for any rotation ℛ are completely decoupled (cf. ref. [9], subsect.6.6.

    Google Scholar 

  31. W. Hanke andL. J. Sham:Phys. Rev. B,12, 4501 (1975).

    Article  ADS  Google Scholar 

  32. W. Hanke andL. J. Sham:Phys. Rev. B,21, 4656 (1980).

    Article  ADS  Google Scholar 

  33. H. J. Mattausch, W. Hanke, andG. Strinati:Phys. Rev. B,27, 3735 (1983).

    Article  ADS  Google Scholar 

  34. N. Meskini, H. J. Mattausch, andW. Hanke:Solid State Commun.,48, 807 (1983).

    Article  ADS  Google Scholar 

  35. G. Wendin:Phys. Lett. A,51, 291 (1975);M. Ya. Amusia, V. K. Ivanov, andL. V. Chernysheva:Phys. Lett. A,59, 191 (1976);M. Ya. Amusia:Appl. Opt.,19, 4042 (1980);Z. Crljen andG. Wendin:Phys. Scr.,32, 359 (1985).

    Article  ADS  Google Scholar 

  36. L. Hedin:Phys. Rev.,139, A 796 (1965).

    Article  ADS  Google Scholar 

  37. The approximation\(\overline {RPA} \) is equivalent to the independent-electron approximation of the ordinary band theory. Cf.,e.g.,F. Bassani andG. Pastori Parravicini:Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), subsect.5.1.

    Google Scholar 

  38. Cf. ref. [39]. subsect.6.3.

    Article  Google Scholar 

  39. M. del Castillo-Mussot andL. J. Sham:Phys. Rev. B,31, 2092 (1985).

    Article  Google Scholar 

  40. L. J. Sham andM. Schlüter:Phys. Rev. Lett.,51, 1888 (1983);L. J. Sham:Phys. Rev. B,32, 3876 (1985);L. J. Sham andM. Schlüter:Phys. Rev. B,32, 3883 (1985);M. Lannoo, M. Schlüter, andL. J. Sham:Phys. Rev. B,32, 3890 (1985).

    Article  ADS  Google Scholar 

  41. G. Strinati, H. J. Mattausch, andW. Hanke:Phys. Rev. Lett.,45, 290 (1980). See also ref. [19]G. Strinati, H. J. Mattausch, andW. Hanke:Phys. Rev. B,25, 2867 (1982).

    Article  ADS  Google Scholar 

  42. W. Hanke, Th. Gölzen, andH. J. Mattausch:Solid State Commun.,51, 23 (1984).

    Article  ADS  Google Scholar 

  43. C. S. Wang andW. E. Pickett:Phys. Rev. Lett.,51, 597 (1983).

    Article  ADS  Google Scholar 

  44. M. S. Hybertsen andS. G. Louie:Phys. Rev. Lett.,55, 1418 (1985);M. S. Hybertsen andS. G. Louie:Phys. Rev. B,32, 7005 (1985).

    Article  ADS  Google Scholar 

  45. V. M. Galitskii andA. B. Migdal:Sov. Phys. JEPT,7, 96 (1958).

    MathSciNet  Google Scholar 

  46. Physically, the energy dependence of the self-energy is connected to the coupling of multiple electron-hole excitations with the primary one-particle excitation. Cf.G. Strinati:Nuovo Cimento D,4, 397 (1984).

    Article  ADS  Google Scholar 

  47. A. J. Layzer:Phys. Rev.,129, 897 (1963).

    Article  MathSciNet  ADS  Google Scholar 

  48. Cf.,e.g., the second of references [4],, sect.10.

    Google Scholar 

  49. A. Mauger andM. Lannoo:Phys. Rev. B,15, 2324 (1977).

    Article  ADS  Google Scholar 

  50. For numerical convenience, in the rest of this section we shall use the Bohr radius as the unit of length and the rydberg as the unit of energy (ℏ=1,m=1/2,e 2=2 in these units).

  51. For a core hole whose wave function is well localized within a lattice cell, neglecting the energy dependence of the screened interaction leads to a polarization shift which is twice the correct value,i.e. it misses the adiabatic factor of 1/2. Physically, the difference is due to the so-called Coulomb hole term (cf. ref. [5]).

    Google Scholar 

  52. Cf.,e.g., sect.14 of the second of references [4].

    Google Scholar 

  53. For a review on impurity levels see ref. [31], andS. T. Pantelides:Rev. Mod. Phys.,50, 797 (1978).

    Article  ADS  Google Scholar 

  54. H. J. Mattausch, W. Hanke, andG. Strinati:Phys. Rev. B,26, 2302 (1982).

    Article  ADS  Google Scholar 

  55. Cf.,e.g., ref. [39]. Chap. 6.

    Article  Google Scholar 

  56. G. Strinati:Phys. Rev. Lett.,49, 1519 (1982);G. Strinati:Phys. Rev. B,29, 5718 (1984).

    Article  ADS  Google Scholar 

  57. M. Altarelli andF. Bassani:J. Phys. C,4, L328 (1971).

    Article  ADS  Google Scholar 

  58. M. H. Cohen andF. Keffer:Phys. Rev.,99, 1128 (1955).

    Article  ADS  Google Scholar 

  59. R. S. Knox:Theory of Excitons (Academic Press, New York, N.Y., 1963), sect.3 b.

    Google Scholar 

  60. Y. Onodera andY. Toyozawa:J. Phys. Soc. Jpn.,22, 833 (1967).

    Article  ADS  Google Scholar 

  61. H. A. Bethe andE. Salpeter:Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), sect.61.

    Book  Google Scholar 

  62. D. J. Thouless:Nucl. Phys.,22, 78 (1961);M. Ya. Amus’ya, N. A. Cherepkov, andL. V. Chernysheva:Sov. Phys. JETP,33, 90 (1971).

    Article  MathSciNet  Google Scholar 

  63. Cf.,e.g.,N. W. Ashcroft andN. D. Mermin:Solid State Physics (Saunders College, Philadelphia, Penn., 1976), Chapter 27.

    Google Scholar 

  64. U. Fano:Phys. Rev.,118, 451 (1960).

    Article  MathSciNet  ADS  Google Scholar 

  65. This point is discussed in detail byS. K. Sinha, R. P. Gupta, andD. L. Price:Phys. Rev. B,9, 2564 (1974). Within the TDSHF approximation it has been estimated that the screened electron-hole interaction reduces to only about 1/2 of the self-interaction (cf. ref. [34]N. Meskini, H. J. Mattausch, andW. Hanke:Solid State Commun.,48, 807 (1983)).

    Article  ADS  Google Scholar 

  66. Y. Onodera:Prog. Theor. Phys.,49, 37 (1973).

    Article  ADS  Google Scholar 

  67. Quite generally, the condition\(\int\limits_O {dr' \bar \chi } (r, r'; \omega ) = 0\) holds at any finite ω for the full irreducible polarizability including all possible many-body effects. This condition, that follows from the counterpart of the Ward identity (7.11) for the irreducible vertex functions, can be extrapolated at ω=0 for a system which, like a crystalline insulator or semiconductor, has an energy gap in its spectrum. It can thus be taken as a quantum-mechanical characterization of the insulator itself (cf.W. Bardyszewski,R. Del Sole,J. Krupski andG. Strinati:Surf. Sci.,167, 363 (1986)).

Download references

Author information

Author notes

  1. G. Strinati

    Present address: Scuola Normale Superiore, 56100, Pisa, Italy

Authors and Affiliations

  1. Dipartimento di Fisica dell’Università «La Sapienza», 00185, Roma, Italia

    G. Strinati

Authors
  1. G. Strinati
    View author publications

    Search author on:PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Strinati, G. Application of the Green’s functions method to the study of the optical properties of semiconductors. Riv. Nuovo Cim. 11, 1–86 (1988). https://doi.org/10.1007/BF02725962

Download citation

  • Received:

  • Published:

  • Issue date:

  • DOI: https://doi.org/10.1007/BF02725962